WORST_CASE(?,O(n^4)) * Step 1: DependencyPairs WORST_CASE(?,O(n^4)) + Considered Problem: - Strict TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) -(x,#()) -> x -(#(),x) -> #() -(0(x),0(y)) -> 0(-(x,y)) -(0(x),1(y)) -> 1(-(-(x,y),1(#()))) -(1(x),0(y)) -> 1(-(x,y)) -(1(x),1(y)) -> 0(-(x,y)) 0(#()) -> #() and(x,false()) -> false() and(x,true()) -> x bs(l(x)) -> true() bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z))) ge(x,#()) -> true() ge(#(),0(x)) -> ge(#(),x) ge(#(),1(x)) -> false() ge(0(x),0(y)) -> ge(x,y) ge(0(x),1(y)) -> not(ge(y,x)) ge(1(x),0(y)) -> ge(x,y) ge(1(x),1(y)) -> ge(x,y) if(false(),x,y) -> y if(true(),x,y) -> x max(l(x)) -> x max(n(x,y,z)) -> max(z) min(l(x)) -> x min(n(x,y,z)) -> min(y) not(false()) -> true() not(true()) -> false() size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) val(l(x)) -> x val(n(x,y,z)) -> x wb(l(x)) -> true() wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(1(#()),-(size(y),size(z))),ge(1(#()),-(size(z),size(y)))) ,and(wb(y),wb(z))) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1} / {#/0,1/1,false/0,l/1,n/3,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {+,-,0,and,bs,ge,if,max,min,not,size,val ,wb} and constructors {#,1,false,l,n,true} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs +#(x,#()) -> c_1() +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)) +#(#(),x) -> c_3() +#(0(x),0(y)) -> c_4(0#(+(x,y)),+#(x,y)) +#(0(x),1(y)) -> c_5(+#(x,y)) +#(1(x),0(y)) -> c_6(+#(x,y)) +#(1(x),1(y)) -> c_7(0#(+(+(x,y),1(#()))),+#(+(x,y),1(#())),+#(x,y)) -#(x,#()) -> c_8() -#(#(),x) -> c_9() -#(0(x),0(y)) -> c_10(0#(-(x,y)),-#(x,y)) -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)) -#(1(x),0(y)) -> c_12(-#(x,y)) -#(1(x),1(y)) -> c_13(0#(-(x,y)),-#(x,y)) 0#(#()) -> c_14() and#(x,false()) -> c_15() and#(x,true()) -> c_16() bs#(l(x)) -> c_17() bs#(n(x,y,z)) -> c_18(and#(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z))) ,and#(ge(x,max(y)),ge(min(z),x)) ,ge#(x,max(y)) ,max#(y) ,ge#(min(z),x) ,min#(z) ,and#(bs(y),bs(z)) ,bs#(y) ,bs#(z)) ge#(x,#()) -> c_19() ge#(#(),0(x)) -> c_20(ge#(#(),x)) ge#(#(),1(x)) -> c_21() ge#(0(x),0(y)) -> c_22(ge#(x,y)) ge#(0(x),1(y)) -> c_23(not#(ge(y,x)),ge#(y,x)) ge#(1(x),0(y)) -> c_24(ge#(x,y)) ge#(1(x),1(y)) -> c_25(ge#(x,y)) if#(false(),x,y) -> c_26() if#(true(),x,y) -> c_27() max#(l(x)) -> c_28() max#(n(x,y,z)) -> c_29(max#(z)) min#(l(x)) -> c_30() min#(n(x,y,z)) -> c_31(min#(y)) not#(false()) -> c_32() not#(true()) -> c_33() size#(l(x)) -> c_34() size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)) val#(l(x)) -> c_36() val#(n(x,y,z)) -> c_37() wb#(l(x)) -> c_38() wb#(n(x,y,z)) -> c_39(and#(if(ge(size(y),size(z)) ,ge(1(#()),-(size(y),size(z))) ,ge(1(#()),-(size(z),size(y)))) ,and(wb(y),wb(z))) ,if#(ge(size(y),size(z)),ge(1(#()),-(size(y),size(z))),ge(1(#()),-(size(z),size(y)))) ,ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,-#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,-#(size(z),size(y)) ,size#(z) ,size#(y) ,and#(wb(y),wb(z)) ,wb#(y) ,wb#(z)) Weak DPs and mark the set of starting terms. * Step 2: UsableRules WORST_CASE(?,O(n^4)) + Considered Problem: - Strict DPs: +#(x,#()) -> c_1() +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)) +#(#(),x) -> c_3() +#(0(x),0(y)) -> c_4(0#(+(x,y)),+#(x,y)) +#(0(x),1(y)) -> c_5(+#(x,y)) +#(1(x),0(y)) -> c_6(+#(x,y)) +#(1(x),1(y)) -> c_7(0#(+(+(x,y),1(#()))),+#(+(x,y),1(#())),+#(x,y)) -#(x,#()) -> c_8() -#(#(),x) -> c_9() -#(0(x),0(y)) -> c_10(0#(-(x,y)),-#(x,y)) -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)) -#(1(x),0(y)) -> c_12(-#(x,y)) -#(1(x),1(y)) -> c_13(0#(-(x,y)),-#(x,y)) 0#(#()) -> c_14() and#(x,false()) -> c_15() and#(x,true()) -> c_16() bs#(l(x)) -> c_17() bs#(n(x,y,z)) -> c_18(and#(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z))) ,and#(ge(x,max(y)),ge(min(z),x)) ,ge#(x,max(y)) ,max#(y) ,ge#(min(z),x) ,min#(z) ,and#(bs(y),bs(z)) ,bs#(y) ,bs#(z)) ge#(x,#()) -> c_19() ge#(#(),0(x)) -> c_20(ge#(#(),x)) ge#(#(),1(x)) -> c_21() ge#(0(x),0(y)) -> c_22(ge#(x,y)) ge#(0(x),1(y)) -> c_23(not#(ge(y,x)),ge#(y,x)) ge#(1(x),0(y)) -> c_24(ge#(x,y)) ge#(1(x),1(y)) -> c_25(ge#(x,y)) if#(false(),x,y) -> c_26() if#(true(),x,y) -> c_27() max#(l(x)) -> c_28() max#(n(x,y,z)) -> c_29(max#(z)) min#(l(x)) -> c_30() min#(n(x,y,z)) -> c_31(min#(y)) not#(false()) -> c_32() not#(true()) -> c_33() size#(l(x)) -> c_34() size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)) val#(l(x)) -> c_36() val#(n(x,y,z)) -> c_37() wb#(l(x)) -> c_38() wb#(n(x,y,z)) -> c_39(and#(if(ge(size(y),size(z)) ,ge(1(#()),-(size(y),size(z))) ,ge(1(#()),-(size(z),size(y)))) ,and(wb(y),wb(z))) ,if#(ge(size(y),size(z)),ge(1(#()),-(size(y),size(z))),ge(1(#()),-(size(z),size(y)))) ,ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,-#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,-#(size(z),size(y)) ,size#(z) ,size#(y) ,and#(wb(y),wb(z)) ,wb#(y) ,wb#(z)) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) -(x,#()) -> x -(#(),x) -> #() -(0(x),0(y)) -> 0(-(x,y)) -(0(x),1(y)) -> 1(-(-(x,y),1(#()))) -(1(x),0(y)) -> 1(-(x,y)) -(1(x),1(y)) -> 0(-(x,y)) 0(#()) -> #() and(x,false()) -> false() and(x,true()) -> x bs(l(x)) -> true() bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z))) ge(x,#()) -> true() ge(#(),0(x)) -> ge(#(),x) ge(#(),1(x)) -> false() ge(0(x),0(y)) -> ge(x,y) ge(0(x),1(y)) -> not(ge(y,x)) ge(1(x),0(y)) -> ge(x,y) ge(1(x),1(y)) -> ge(x,y) if(false(),x,y) -> y if(true(),x,y) -> x max(l(x)) -> x max(n(x,y,z)) -> max(z) min(l(x)) -> x min(n(x,y,z)) -> min(y) not(false()) -> true() not(true()) -> false() size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) val(l(x)) -> x val(n(x,y,z)) -> x wb(l(x)) -> true() wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(1(#()),-(size(y),size(z))),ge(1(#()),-(size(z),size(y)))) ,and(wb(y),wb(z))) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/2 ,c_5/1,c_6/1,c_7/3,c_8/0,c_9/0,c_10/2,c_11/2,c_12/1,c_13/2,c_14/0,c_15/0,c_16/0,c_17/0,c_18/9,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/2,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/4 ,c_36/0,c_37/0,c_38/0,c_39/16} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) -(x,#()) -> x -(#(),x) -> #() -(0(x),0(y)) -> 0(-(x,y)) -(0(x),1(y)) -> 1(-(-(x,y),1(#()))) -(1(x),0(y)) -> 1(-(x,y)) -(1(x),1(y)) -> 0(-(x,y)) 0(#()) -> #() and(x,false()) -> false() and(x,true()) -> x bs(l(x)) -> true() bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z))) ge(x,#()) -> true() ge(#(),0(x)) -> ge(#(),x) ge(#(),1(x)) -> false() ge(0(x),0(y)) -> ge(x,y) ge(0(x),1(y)) -> not(ge(y,x)) ge(1(x),0(y)) -> ge(x,y) ge(1(x),1(y)) -> ge(x,y) if(false(),x,y) -> y if(true(),x,y) -> x max(l(x)) -> x max(n(x,y,z)) -> max(z) min(l(x)) -> x min(n(x,y,z)) -> min(y) not(false()) -> true() not(true()) -> false() size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) wb(l(x)) -> true() wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(1(#()),-(size(y),size(z))),ge(1(#()),-(size(z),size(y)))) ,and(wb(y),wb(z))) +#(x,#()) -> c_1() +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)) +#(#(),x) -> c_3() +#(0(x),0(y)) -> c_4(0#(+(x,y)),+#(x,y)) +#(0(x),1(y)) -> c_5(+#(x,y)) +#(1(x),0(y)) -> c_6(+#(x,y)) +#(1(x),1(y)) -> c_7(0#(+(+(x,y),1(#()))),+#(+(x,y),1(#())),+#(x,y)) -#(x,#()) -> c_8() -#(#(),x) -> c_9() -#(0(x),0(y)) -> c_10(0#(-(x,y)),-#(x,y)) -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)) -#(1(x),0(y)) -> c_12(-#(x,y)) -#(1(x),1(y)) -> c_13(0#(-(x,y)),-#(x,y)) 0#(#()) -> c_14() and#(x,false()) -> c_15() and#(x,true()) -> c_16() bs#(l(x)) -> c_17() bs#(n(x,y,z)) -> c_18(and#(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z))) ,and#(ge(x,max(y)),ge(min(z),x)) ,ge#(x,max(y)) ,max#(y) ,ge#(min(z),x) ,min#(z) ,and#(bs(y),bs(z)) ,bs#(y) ,bs#(z)) ge#(x,#()) -> c_19() ge#(#(),0(x)) -> c_20(ge#(#(),x)) ge#(#(),1(x)) -> c_21() ge#(0(x),0(y)) -> c_22(ge#(x,y)) ge#(0(x),1(y)) -> c_23(not#(ge(y,x)),ge#(y,x)) ge#(1(x),0(y)) -> c_24(ge#(x,y)) ge#(1(x),1(y)) -> c_25(ge#(x,y)) if#(false(),x,y) -> c_26() if#(true(),x,y) -> c_27() max#(l(x)) -> c_28() max#(n(x,y,z)) -> c_29(max#(z)) min#(l(x)) -> c_30() min#(n(x,y,z)) -> c_31(min#(y)) not#(false()) -> c_32() not#(true()) -> c_33() size#(l(x)) -> c_34() size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)) val#(l(x)) -> c_36() val#(n(x,y,z)) -> c_37() wb#(l(x)) -> c_38() wb#(n(x,y,z)) -> c_39(and#(if(ge(size(y),size(z)) ,ge(1(#()),-(size(y),size(z))) ,ge(1(#()),-(size(z),size(y)))) ,and(wb(y),wb(z))) ,if#(ge(size(y),size(z)),ge(1(#()),-(size(y),size(z))),ge(1(#()),-(size(z),size(y)))) ,ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,-#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,-#(size(z),size(y)) ,size#(z) ,size#(y) ,and#(wb(y),wb(z)) ,wb#(y) ,wb#(z)) * Step 3: PredecessorEstimation WORST_CASE(?,O(n^4)) + Considered Problem: - Strict DPs: +#(x,#()) -> c_1() +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)) +#(#(),x) -> c_3() +#(0(x),0(y)) -> c_4(0#(+(x,y)),+#(x,y)) +#(0(x),1(y)) -> c_5(+#(x,y)) +#(1(x),0(y)) -> c_6(+#(x,y)) +#(1(x),1(y)) -> c_7(0#(+(+(x,y),1(#()))),+#(+(x,y),1(#())),+#(x,y)) -#(x,#()) -> c_8() -#(#(),x) -> c_9() -#(0(x),0(y)) -> c_10(0#(-(x,y)),-#(x,y)) -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)) -#(1(x),0(y)) -> c_12(-#(x,y)) -#(1(x),1(y)) -> c_13(0#(-(x,y)),-#(x,y)) 0#(#()) -> c_14() and#(x,false()) -> c_15() and#(x,true()) -> c_16() bs#(l(x)) -> c_17() bs#(n(x,y,z)) -> c_18(and#(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z))) ,and#(ge(x,max(y)),ge(min(z),x)) ,ge#(x,max(y)) ,max#(y) ,ge#(min(z),x) ,min#(z) ,and#(bs(y),bs(z)) ,bs#(y) ,bs#(z)) ge#(x,#()) -> c_19() ge#(#(),0(x)) -> c_20(ge#(#(),x)) ge#(#(),1(x)) -> c_21() ge#(0(x),0(y)) -> c_22(ge#(x,y)) ge#(0(x),1(y)) -> c_23(not#(ge(y,x)),ge#(y,x)) ge#(1(x),0(y)) -> c_24(ge#(x,y)) ge#(1(x),1(y)) -> c_25(ge#(x,y)) if#(false(),x,y) -> c_26() if#(true(),x,y) -> c_27() max#(l(x)) -> c_28() max#(n(x,y,z)) -> c_29(max#(z)) min#(l(x)) -> c_30() min#(n(x,y,z)) -> c_31(min#(y)) not#(false()) -> c_32() not#(true()) -> c_33() size#(l(x)) -> c_34() size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)) val#(l(x)) -> c_36() val#(n(x,y,z)) -> c_37() wb#(l(x)) -> c_38() wb#(n(x,y,z)) -> c_39(and#(if(ge(size(y),size(z)) ,ge(1(#()),-(size(y),size(z))) ,ge(1(#()),-(size(z),size(y)))) ,and(wb(y),wb(z))) ,if#(ge(size(y),size(z)),ge(1(#()),-(size(y),size(z))),ge(1(#()),-(size(z),size(y)))) ,ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,-#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,-#(size(z),size(y)) ,size#(z) ,size#(y) ,and#(wb(y),wb(z)) ,wb#(y) ,wb#(z)) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) -(x,#()) -> x -(#(),x) -> #() -(0(x),0(y)) -> 0(-(x,y)) -(0(x),1(y)) -> 1(-(-(x,y),1(#()))) -(1(x),0(y)) -> 1(-(x,y)) -(1(x),1(y)) -> 0(-(x,y)) 0(#()) -> #() and(x,false()) -> false() and(x,true()) -> x bs(l(x)) -> true() bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z))) ge(x,#()) -> true() ge(#(),0(x)) -> ge(#(),x) ge(#(),1(x)) -> false() ge(0(x),0(y)) -> ge(x,y) ge(0(x),1(y)) -> not(ge(y,x)) ge(1(x),0(y)) -> ge(x,y) ge(1(x),1(y)) -> ge(x,y) if(false(),x,y) -> y if(true(),x,y) -> x max(l(x)) -> x max(n(x,y,z)) -> max(z) min(l(x)) -> x min(n(x,y,z)) -> min(y) not(false()) -> true() not(true()) -> false() size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) wb(l(x)) -> true() wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(1(#()),-(size(y),size(z))),ge(1(#()),-(size(z),size(y)))) ,and(wb(y),wb(z))) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/2 ,c_5/1,c_6/1,c_7/3,c_8/0,c_9/0,c_10/2,c_11/2,c_12/1,c_13/2,c_14/0,c_15/0,c_16/0,c_17/0,c_18/9,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/2,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/4 ,c_36/0,c_37/0,c_38/0,c_39/16} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,3,8,9,14,15,16,17,19,21,26,27,28,30,32,33,34,36,37,38} by application of Pre({1,3,8,9,14,15,16,17,19,21,26,27,28,30,32,33,34,36,37,38}) = {2,4,5,6,7,10,11,12,13,18,20,22,23,24,25 ,29,31,35,39}. Here rules are labelled as follows: 1: +#(x,#()) -> c_1() 2: +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)) 3: +#(#(),x) -> c_3() 4: +#(0(x),0(y)) -> c_4(0#(+(x,y)),+#(x,y)) 5: +#(0(x),1(y)) -> c_5(+#(x,y)) 6: +#(1(x),0(y)) -> c_6(+#(x,y)) 7: +#(1(x),1(y)) -> c_7(0#(+(+(x,y),1(#()))),+#(+(x,y),1(#())),+#(x,y)) 8: -#(x,#()) -> c_8() 9: -#(#(),x) -> c_9() 10: -#(0(x),0(y)) -> c_10(0#(-(x,y)),-#(x,y)) 11: -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)) 12: -#(1(x),0(y)) -> c_12(-#(x,y)) 13: -#(1(x),1(y)) -> c_13(0#(-(x,y)),-#(x,y)) 14: 0#(#()) -> c_14() 15: and#(x,false()) -> c_15() 16: and#(x,true()) -> c_16() 17: bs#(l(x)) -> c_17() 18: bs#(n(x,y,z)) -> c_18(and#(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z))) ,and#(ge(x,max(y)),ge(min(z),x)) ,ge#(x,max(y)) ,max#(y) ,ge#(min(z),x) ,min#(z) ,and#(bs(y),bs(z)) ,bs#(y) ,bs#(z)) 19: ge#(x,#()) -> c_19() 20: ge#(#(),0(x)) -> c_20(ge#(#(),x)) 21: ge#(#(),1(x)) -> c_21() 22: ge#(0(x),0(y)) -> c_22(ge#(x,y)) 23: ge#(0(x),1(y)) -> c_23(not#(ge(y,x)),ge#(y,x)) 24: ge#(1(x),0(y)) -> c_24(ge#(x,y)) 25: ge#(1(x),1(y)) -> c_25(ge#(x,y)) 26: if#(false(),x,y) -> c_26() 27: if#(true(),x,y) -> c_27() 28: max#(l(x)) -> c_28() 29: max#(n(x,y,z)) -> c_29(max#(z)) 30: min#(l(x)) -> c_30() 31: min#(n(x,y,z)) -> c_31(min#(y)) 32: not#(false()) -> c_32() 33: not#(true()) -> c_33() 34: size#(l(x)) -> c_34() 35: size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)) 36: val#(l(x)) -> c_36() 37: val#(n(x,y,z)) -> c_37() 38: wb#(l(x)) -> c_38() 39: wb#(n(x,y,z)) -> c_39(and#(if(ge(size(y),size(z)) ,ge(1(#()),-(size(y),size(z))) ,ge(1(#()),-(size(z),size(y)))) ,and(wb(y),wb(z))) ,if#(ge(size(y),size(z)),ge(1(#()),-(size(y),size(z))),ge(1(#()),-(size(z),size(y)))) ,ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,-#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,-#(size(z),size(y)) ,size#(z) ,size#(y) ,and#(wb(y),wb(z)) ,wb#(y) ,wb#(z)) * Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^4)) + Considered Problem: - Strict DPs: +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)) +#(0(x),0(y)) -> c_4(0#(+(x,y)),+#(x,y)) +#(0(x),1(y)) -> c_5(+#(x,y)) +#(1(x),0(y)) -> c_6(+#(x,y)) +#(1(x),1(y)) -> c_7(0#(+(+(x,y),1(#()))),+#(+(x,y),1(#())),+#(x,y)) -#(0(x),0(y)) -> c_10(0#(-(x,y)),-#(x,y)) -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)) -#(1(x),0(y)) -> c_12(-#(x,y)) -#(1(x),1(y)) -> c_13(0#(-(x,y)),-#(x,y)) bs#(n(x,y,z)) -> c_18(and#(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z))) ,and#(ge(x,max(y)),ge(min(z),x)) ,ge#(x,max(y)) ,max#(y) ,ge#(min(z),x) ,min#(z) ,and#(bs(y),bs(z)) ,bs#(y) ,bs#(z)) ge#(#(),0(x)) -> c_20(ge#(#(),x)) ge#(0(x),0(y)) -> c_22(ge#(x,y)) ge#(0(x),1(y)) -> c_23(not#(ge(y,x)),ge#(y,x)) ge#(1(x),0(y)) -> c_24(ge#(x,y)) ge#(1(x),1(y)) -> c_25(ge#(x,y)) max#(n(x,y,z)) -> c_29(max#(z)) min#(n(x,y,z)) -> c_31(min#(y)) size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)) wb#(n(x,y,z)) -> c_39(and#(if(ge(size(y),size(z)) ,ge(1(#()),-(size(y),size(z))) ,ge(1(#()),-(size(z),size(y)))) ,and(wb(y),wb(z))) ,if#(ge(size(y),size(z)),ge(1(#()),-(size(y),size(z))),ge(1(#()),-(size(z),size(y)))) ,ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,-#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,-#(size(z),size(y)) ,size#(z) ,size#(y) ,and#(wb(y),wb(z)) ,wb#(y) ,wb#(z)) - Weak DPs: +#(x,#()) -> c_1() +#(#(),x) -> c_3() -#(x,#()) -> c_8() -#(#(),x) -> c_9() 0#(#()) -> c_14() and#(x,false()) -> c_15() and#(x,true()) -> c_16() bs#(l(x)) -> c_17() ge#(x,#()) -> c_19() ge#(#(),1(x)) -> c_21() if#(false(),x,y) -> c_26() if#(true(),x,y) -> c_27() max#(l(x)) -> c_28() min#(l(x)) -> c_30() not#(false()) -> c_32() not#(true()) -> c_33() size#(l(x)) -> c_34() val#(l(x)) -> c_36() val#(n(x,y,z)) -> c_37() wb#(l(x)) -> c_38() - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) -(x,#()) -> x -(#(),x) -> #() -(0(x),0(y)) -> 0(-(x,y)) -(0(x),1(y)) -> 1(-(-(x,y),1(#()))) -(1(x),0(y)) -> 1(-(x,y)) -(1(x),1(y)) -> 0(-(x,y)) 0(#()) -> #() and(x,false()) -> false() and(x,true()) -> x bs(l(x)) -> true() bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z))) ge(x,#()) -> true() ge(#(),0(x)) -> ge(#(),x) ge(#(),1(x)) -> false() ge(0(x),0(y)) -> ge(x,y) ge(0(x),1(y)) -> not(ge(y,x)) ge(1(x),0(y)) -> ge(x,y) ge(1(x),1(y)) -> ge(x,y) if(false(),x,y) -> y if(true(),x,y) -> x max(l(x)) -> x max(n(x,y,z)) -> max(z) min(l(x)) -> x min(n(x,y,z)) -> min(y) not(false()) -> true() not(true()) -> false() size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) wb(l(x)) -> true() wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(1(#()),-(size(y),size(z))),ge(1(#()),-(size(z),size(y)))) ,and(wb(y),wb(z))) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/2 ,c_5/1,c_6/1,c_7/3,c_8/0,c_9/0,c_10/2,c_11/2,c_12/1,c_13/2,c_14/0,c_15/0,c_16/0,c_17/0,c_18/9,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/2,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/4 ,c_36/0,c_37/0,c_38/0,c_39/16} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:+#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)) -->_2 +#(1(x),1(y)) -> c_7(0#(+(+(x,y),1(#()))),+#(+(x,y),1(#())),+#(x,y)):5 -->_1 +#(1(x),1(y)) -> c_7(0#(+(+(x,y),1(#()))),+#(+(x,y),1(#())),+#(x,y)):5 -->_2 +#(1(x),0(y)) -> c_6(+#(x,y)):4 -->_1 +#(1(x),0(y)) -> c_6(+#(x,y)):4 -->_2 +#(0(x),1(y)) -> c_5(+#(x,y)):3 -->_1 +#(0(x),1(y)) -> c_5(+#(x,y)):3 -->_2 +#(0(x),0(y)) -> c_4(0#(+(x,y)),+#(x,y)):2 -->_1 +#(0(x),0(y)) -> c_4(0#(+(x,y)),+#(x,y)):2 -->_2 +#(#(),x) -> c_3():21 -->_1 +#(#(),x) -> c_3():21 -->_2 +#(x,#()) -> c_1():20 -->_1 +#(x,#()) -> c_1():20 -->_2 +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)):1 -->_1 +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)):1 2:S:+#(0(x),0(y)) -> c_4(0#(+(x,y)),+#(x,y)) -->_2 +#(1(x),1(y)) -> c_7(0#(+(+(x,y),1(#()))),+#(+(x,y),1(#())),+#(x,y)):5 -->_2 +#(1(x),0(y)) -> c_6(+#(x,y)):4 -->_2 +#(0(x),1(y)) -> c_5(+#(x,y)):3 -->_1 0#(#()) -> c_14():24 -->_2 +#(#(),x) -> c_3():21 -->_2 +#(x,#()) -> c_1():20 -->_2 +#(0(x),0(y)) -> c_4(0#(+(x,y)),+#(x,y)):2 -->_2 +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)):1 3:S:+#(0(x),1(y)) -> c_5(+#(x,y)) -->_1 +#(1(x),1(y)) -> c_7(0#(+(+(x,y),1(#()))),+#(+(x,y),1(#())),+#(x,y)):5 -->_1 +#(1(x),0(y)) -> c_6(+#(x,y)):4 -->_1 +#(#(),x) -> c_3():21 -->_1 +#(x,#()) -> c_1():20 -->_1 +#(0(x),1(y)) -> c_5(+#(x,y)):3 -->_1 +#(0(x),0(y)) -> c_4(0#(+(x,y)),+#(x,y)):2 -->_1 +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)):1 4:S:+#(1(x),0(y)) -> c_6(+#(x,y)) -->_1 +#(1(x),1(y)) -> c_7(0#(+(+(x,y),1(#()))),+#(+(x,y),1(#())),+#(x,y)):5 -->_1 +#(#(),x) -> c_3():21 -->_1 +#(x,#()) -> c_1():20 -->_1 +#(1(x),0(y)) -> c_6(+#(x,y)):4 -->_1 +#(0(x),1(y)) -> c_5(+#(x,y)):3 -->_1 +#(0(x),0(y)) -> c_4(0#(+(x,y)),+#(x,y)):2 -->_1 +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)):1 5:S:+#(1(x),1(y)) -> c_7(0#(+(+(x,y),1(#()))),+#(+(x,y),1(#())),+#(x,y)) -->_1 0#(#()) -> c_14():24 -->_3 +#(#(),x) -> c_3():21 -->_2 +#(#(),x) -> c_3():21 -->_3 +#(x,#()) -> c_1():20 -->_3 +#(1(x),1(y)) -> c_7(0#(+(+(x,y),1(#()))),+#(+(x,y),1(#())),+#(x,y)):5 -->_2 +#(1(x),1(y)) -> c_7(0#(+(+(x,y),1(#()))),+#(+(x,y),1(#())),+#(x,y)):5 -->_3 +#(1(x),0(y)) -> c_6(+#(x,y)):4 -->_3 +#(0(x),1(y)) -> c_5(+#(x,y)):3 -->_2 +#(0(x),1(y)) -> c_5(+#(x,y)):3 -->_3 +#(0(x),0(y)) -> c_4(0#(+(x,y)),+#(x,y)):2 -->_3 +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)):1 6:S:-#(0(x),0(y)) -> c_10(0#(-(x,y)),-#(x,y)) -->_2 -#(1(x),1(y)) -> c_13(0#(-(x,y)),-#(x,y)):9 -->_2 -#(1(x),0(y)) -> c_12(-#(x,y)):8 -->_2 -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)):7 -->_1 0#(#()) -> c_14():24 -->_2 -#(#(),x) -> c_9():23 -->_2 -#(x,#()) -> c_8():22 -->_2 -#(0(x),0(y)) -> c_10(0#(-(x,y)),-#(x,y)):6 7:S:-#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)) -->_2 -#(1(x),1(y)) -> c_13(0#(-(x,y)),-#(x,y)):9 -->_1 -#(1(x),1(y)) -> c_13(0#(-(x,y)),-#(x,y)):9 -->_2 -#(1(x),0(y)) -> c_12(-#(x,y)):8 -->_2 -#(#(),x) -> c_9():23 -->_1 -#(#(),x) -> c_9():23 -->_2 -#(x,#()) -> c_8():22 -->_2 -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)):7 -->_1 -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)):7 -->_2 -#(0(x),0(y)) -> c_10(0#(-(x,y)),-#(x,y)):6 8:S:-#(1(x),0(y)) -> c_12(-#(x,y)) -->_1 -#(1(x),1(y)) -> c_13(0#(-(x,y)),-#(x,y)):9 -->_1 -#(#(),x) -> c_9():23 -->_1 -#(x,#()) -> c_8():22 -->_1 -#(1(x),0(y)) -> c_12(-#(x,y)):8 -->_1 -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)):7 -->_1 -#(0(x),0(y)) -> c_10(0#(-(x,y)),-#(x,y)):6 9:S:-#(1(x),1(y)) -> c_13(0#(-(x,y)),-#(x,y)) -->_1 0#(#()) -> c_14():24 -->_2 -#(#(),x) -> c_9():23 -->_2 -#(x,#()) -> c_8():22 -->_2 -#(1(x),1(y)) -> c_13(0#(-(x,y)),-#(x,y)):9 -->_2 -#(1(x),0(y)) -> c_12(-#(x,y)):8 -->_2 -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)):7 -->_2 -#(0(x),0(y)) -> c_10(0#(-(x,y)),-#(x,y)):6 10:S:bs#(n(x,y,z)) -> c_18(and#(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z))) ,and#(ge(x,max(y)),ge(min(z),x)) ,ge#(x,max(y)) ,max#(y) ,ge#(min(z),x) ,min#(z) ,and#(bs(y),bs(z)) ,bs#(y) ,bs#(z)) -->_6 min#(n(x,y,z)) -> c_31(min#(y)):17 -->_4 max#(n(x,y,z)) -> c_29(max#(z)):16 -->_5 ge#(1(x),1(y)) -> c_25(ge#(x,y)):15 -->_3 ge#(1(x),1(y)) -> c_25(ge#(x,y)):15 -->_5 ge#(1(x),0(y)) -> c_24(ge#(x,y)):14 -->_3 ge#(1(x),0(y)) -> c_24(ge#(x,y)):14 -->_5 ge#(0(x),1(y)) -> c_23(not#(ge(y,x)),ge#(y,x)):13 -->_3 ge#(0(x),1(y)) -> c_23(not#(ge(y,x)),ge#(y,x)):13 -->_5 ge#(0(x),0(y)) -> c_22(ge#(x,y)):12 -->_3 ge#(0(x),0(y)) -> c_22(ge#(x,y)):12 -->_5 ge#(#(),0(x)) -> c_20(ge#(#(),x)):11 -->_3 ge#(#(),0(x)) -> c_20(ge#(#(),x)):11 -->_6 min#(l(x)) -> c_30():33 -->_4 max#(l(x)) -> c_28():32 -->_5 ge#(#(),1(x)) -> c_21():29 -->_3 ge#(#(),1(x)) -> c_21():29 -->_5 ge#(x,#()) -> c_19():28 -->_3 ge#(x,#()) -> c_19():28 -->_9 bs#(l(x)) -> c_17():27 -->_8 bs#(l(x)) -> c_17():27 -->_7 and#(x,true()) -> c_16():26 -->_2 and#(x,true()) -> c_16():26 -->_1 and#(x,true()) -> c_16():26 -->_7 and#(x,false()) -> c_15():25 -->_2 and#(x,false()) -> c_15():25 -->_1 and#(x,false()) -> c_15():25 -->_9 bs#(n(x,y,z)) -> c_18(and#(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z))) ,and#(ge(x,max(y)),ge(min(z),x)) ,ge#(x,max(y)) ,max#(y) ,ge#(min(z),x) ,min#(z) ,and#(bs(y),bs(z)) ,bs#(y) ,bs#(z)):10 -->_8 bs#(n(x,y,z)) -> c_18(and#(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z))) ,and#(ge(x,max(y)),ge(min(z),x)) ,ge#(x,max(y)) ,max#(y) ,ge#(min(z),x) ,min#(z) ,and#(bs(y),bs(z)) ,bs#(y) ,bs#(z)):10 11:S:ge#(#(),0(x)) -> c_20(ge#(#(),x)) -->_1 ge#(#(),1(x)) -> c_21():29 -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):11 12:S:ge#(0(x),0(y)) -> c_22(ge#(x,y)) -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):15 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):14 -->_1 ge#(0(x),1(y)) -> c_23(not#(ge(y,x)),ge#(y,x)):13 -->_1 ge#(#(),1(x)) -> c_21():29 -->_1 ge#(x,#()) -> c_19():28 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):12 -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):11 13:S:ge#(0(x),1(y)) -> c_23(not#(ge(y,x)),ge#(y,x)) -->_2 ge#(1(x),1(y)) -> c_25(ge#(x,y)):15 -->_2 ge#(1(x),0(y)) -> c_24(ge#(x,y)):14 -->_1 not#(true()) -> c_33():35 -->_1 not#(false()) -> c_32():34 -->_2 ge#(#(),1(x)) -> c_21():29 -->_2 ge#(x,#()) -> c_19():28 -->_2 ge#(0(x),1(y)) -> c_23(not#(ge(y,x)),ge#(y,x)):13 -->_2 ge#(0(x),0(y)) -> c_22(ge#(x,y)):12 -->_2 ge#(#(),0(x)) -> c_20(ge#(#(),x)):11 14:S:ge#(1(x),0(y)) -> c_24(ge#(x,y)) -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):15 -->_1 ge#(#(),1(x)) -> c_21():29 -->_1 ge#(x,#()) -> c_19():28 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):14 -->_1 ge#(0(x),1(y)) -> c_23(not#(ge(y,x)),ge#(y,x)):13 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):12 -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):11 15:S:ge#(1(x),1(y)) -> c_25(ge#(x,y)) -->_1 ge#(#(),1(x)) -> c_21():29 -->_1 ge#(x,#()) -> c_19():28 -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):15 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):14 -->_1 ge#(0(x),1(y)) -> c_23(not#(ge(y,x)),ge#(y,x)):13 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):12 -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):11 16:S:max#(n(x,y,z)) -> c_29(max#(z)) -->_1 max#(l(x)) -> c_28():32 -->_1 max#(n(x,y,z)) -> c_29(max#(z)):16 17:S:min#(n(x,y,z)) -> c_31(min#(y)) -->_1 min#(l(x)) -> c_30():33 -->_1 min#(n(x,y,z)) -> c_31(min#(y)):17 18:S:size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)) -->_4 size#(l(x)) -> c_34():36 -->_3 size#(l(x)) -> c_34():36 -->_2 +#(#(),x) -> c_3():21 -->_1 +#(#(),x) -> c_3():21 -->_2 +#(x,#()) -> c_1():20 -->_4 size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)):18 -->_3 size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)):18 -->_2 +#(1(x),1(y)) -> c_7(0#(+(+(x,y),1(#()))),+#(+(x,y),1(#())),+#(x,y)):5 -->_1 +#(1(x),1(y)) -> c_7(0#(+(+(x,y),1(#()))),+#(+(x,y),1(#())),+#(x,y)):5 -->_2 +#(1(x),0(y)) -> c_6(+#(x,y)):4 -->_2 +#(0(x),1(y)) -> c_5(+#(x,y)):3 -->_1 +#(0(x),1(y)) -> c_5(+#(x,y)):3 -->_2 +#(0(x),0(y)) -> c_4(0#(+(x,y)),+#(x,y)):2 -->_2 +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)):1 19:S:wb#(n(x,y,z)) -> c_39(and#(if(ge(size(y),size(z)) ,ge(1(#()),-(size(y),size(z))) ,ge(1(#()),-(size(z),size(y)))) ,and(wb(y),wb(z))) ,if#(ge(size(y),size(z)) ,ge(1(#()),-(size(y),size(z))) ,ge(1(#()),-(size(z),size(y)))) ,ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,-#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,-#(size(z),size(y)) ,size#(z) ,size#(y) ,and#(wb(y),wb(z)) ,wb#(y) ,wb#(z)) -->_16 wb#(l(x)) -> c_38():39 -->_15 wb#(l(x)) -> c_38():39 -->_13 size#(l(x)) -> c_34():36 -->_12 size#(l(x)) -> c_34():36 -->_9 size#(l(x)) -> c_34():36 -->_8 size#(l(x)) -> c_34():36 -->_5 size#(l(x)) -> c_34():36 -->_4 size#(l(x)) -> c_34():36 -->_2 if#(true(),x,y) -> c_27():31 -->_2 if#(false(),x,y) -> c_26():30 -->_3 ge#(#(),1(x)) -> c_21():29 -->_10 ge#(x,#()) -> c_19():28 -->_6 ge#(x,#()) -> c_19():28 -->_3 ge#(x,#()) -> c_19():28 -->_14 and#(x,true()) -> c_16():26 -->_1 and#(x,true()) -> c_16():26 -->_14 and#(x,false()) -> c_15():25 -->_1 and#(x,false()) -> c_15():25 -->_11 -#(#(),x) -> c_9():23 -->_7 -#(#(),x) -> c_9():23 -->_11 -#(x,#()) -> c_8():22 -->_7 -#(x,#()) -> c_8():22 -->_16 wb#(n(x,y,z)) -> c_39(and#(if(ge(size(y),size(z)) ,ge(1(#()),-(size(y),size(z))) ,ge(1(#()),-(size(z),size(y)))) ,and(wb(y),wb(z))) ,if#(ge(size(y),size(z)) ,ge(1(#()),-(size(y),size(z))) ,ge(1(#()),-(size(z),size(y)))) ,ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,-#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,-#(size(z),size(y)) ,size#(z) ,size#(y) ,and#(wb(y),wb(z)) ,wb#(y) ,wb#(z)):19 -->_15 wb#(n(x,y,z)) -> c_39(and#(if(ge(size(y),size(z)) ,ge(1(#()),-(size(y),size(z))) ,ge(1(#()),-(size(z),size(y)))) ,and(wb(y),wb(z))) ,if#(ge(size(y),size(z)) ,ge(1(#()),-(size(y),size(z))) ,ge(1(#()),-(size(z),size(y)))) ,ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,-#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,-#(size(z),size(y)) ,size#(z) ,size#(y) ,and#(wb(y),wb(z)) ,wb#(y) ,wb#(z)):19 -->_13 size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)):18 -->_12 size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)):18 -->_9 size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)):18 -->_8 size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)):18 -->_5 size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)):18 -->_4 size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)):18 -->_10 ge#(1(x),1(y)) -> c_25(ge#(x,y)):15 -->_6 ge#(1(x),1(y)) -> c_25(ge#(x,y)):15 -->_3 ge#(1(x),1(y)) -> c_25(ge#(x,y)):15 -->_10 ge#(1(x),0(y)) -> c_24(ge#(x,y)):14 -->_6 ge#(1(x),0(y)) -> c_24(ge#(x,y)):14 -->_3 ge#(1(x),0(y)) -> c_24(ge#(x,y)):14 -->_3 ge#(0(x),1(y)) -> c_23(not#(ge(y,x)),ge#(y,x)):13 -->_3 ge#(0(x),0(y)) -> c_22(ge#(x,y)):12 -->_3 ge#(#(),0(x)) -> c_20(ge#(#(),x)):11 -->_11 -#(1(x),1(y)) -> c_13(0#(-(x,y)),-#(x,y)):9 -->_7 -#(1(x),1(y)) -> c_13(0#(-(x,y)),-#(x,y)):9 -->_11 -#(1(x),0(y)) -> c_12(-#(x,y)):8 -->_7 -#(1(x),0(y)) -> c_12(-#(x,y)):8 -->_11 -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)):7 -->_7 -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)):7 -->_11 -#(0(x),0(y)) -> c_10(0#(-(x,y)),-#(x,y)):6 -->_7 -#(0(x),0(y)) -> c_10(0#(-(x,y)),-#(x,y)):6 20:W:+#(x,#()) -> c_1() 21:W:+#(#(),x) -> c_3() 22:W:-#(x,#()) -> c_8() 23:W:-#(#(),x) -> c_9() 24:W:0#(#()) -> c_14() 25:W:and#(x,false()) -> c_15() 26:W:and#(x,true()) -> c_16() 27:W:bs#(l(x)) -> c_17() 28:W:ge#(x,#()) -> c_19() 29:W:ge#(#(),1(x)) -> c_21() 30:W:if#(false(),x,y) -> c_26() 31:W:if#(true(),x,y) -> c_27() 32:W:max#(l(x)) -> c_28() 33:W:min#(l(x)) -> c_30() 34:W:not#(false()) -> c_32() 35:W:not#(true()) -> c_33() 36:W:size#(l(x)) -> c_34() 37:W:val#(l(x)) -> c_36() 38:W:val#(n(x,y,z)) -> c_37() 39:W:wb#(l(x)) -> c_38() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 38: val#(n(x,y,z)) -> c_37() 37: val#(l(x)) -> c_36() 30: if#(false(),x,y) -> c_26() 31: if#(true(),x,y) -> c_27() 39: wb#(l(x)) -> c_38() 36: size#(l(x)) -> c_34() 25: and#(x,false()) -> c_15() 26: and#(x,true()) -> c_16() 27: bs#(l(x)) -> c_17() 34: not#(false()) -> c_32() 35: not#(true()) -> c_33() 28: ge#(x,#()) -> c_19() 29: ge#(#(),1(x)) -> c_21() 32: max#(l(x)) -> c_28() 33: min#(l(x)) -> c_30() 22: -#(x,#()) -> c_8() 23: -#(#(),x) -> c_9() 20: +#(x,#()) -> c_1() 21: +#(#(),x) -> c_3() 24: 0#(#()) -> c_14() * Step 5: SimplifyRHS WORST_CASE(?,O(n^4)) + Considered Problem: - Strict DPs: +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)) +#(0(x),0(y)) -> c_4(0#(+(x,y)),+#(x,y)) +#(0(x),1(y)) -> c_5(+#(x,y)) +#(1(x),0(y)) -> c_6(+#(x,y)) +#(1(x),1(y)) -> c_7(0#(+(+(x,y),1(#()))),+#(+(x,y),1(#())),+#(x,y)) -#(0(x),0(y)) -> c_10(0#(-(x,y)),-#(x,y)) -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)) -#(1(x),0(y)) -> c_12(-#(x,y)) -#(1(x),1(y)) -> c_13(0#(-(x,y)),-#(x,y)) bs#(n(x,y,z)) -> c_18(and#(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z))) ,and#(ge(x,max(y)),ge(min(z),x)) ,ge#(x,max(y)) ,max#(y) ,ge#(min(z),x) ,min#(z) ,and#(bs(y),bs(z)) ,bs#(y) ,bs#(z)) ge#(#(),0(x)) -> c_20(ge#(#(),x)) ge#(0(x),0(y)) -> c_22(ge#(x,y)) ge#(0(x),1(y)) -> c_23(not#(ge(y,x)),ge#(y,x)) ge#(1(x),0(y)) -> c_24(ge#(x,y)) ge#(1(x),1(y)) -> c_25(ge#(x,y)) max#(n(x,y,z)) -> c_29(max#(z)) min#(n(x,y,z)) -> c_31(min#(y)) size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)) wb#(n(x,y,z)) -> c_39(and#(if(ge(size(y),size(z)) ,ge(1(#()),-(size(y),size(z))) ,ge(1(#()),-(size(z),size(y)))) ,and(wb(y),wb(z))) ,if#(ge(size(y),size(z)),ge(1(#()),-(size(y),size(z))),ge(1(#()),-(size(z),size(y)))) ,ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,-#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,-#(size(z),size(y)) ,size#(z) ,size#(y) ,and#(wb(y),wb(z)) ,wb#(y) ,wb#(z)) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) -(x,#()) -> x -(#(),x) -> #() -(0(x),0(y)) -> 0(-(x,y)) -(0(x),1(y)) -> 1(-(-(x,y),1(#()))) -(1(x),0(y)) -> 1(-(x,y)) -(1(x),1(y)) -> 0(-(x,y)) 0(#()) -> #() and(x,false()) -> false() and(x,true()) -> x bs(l(x)) -> true() bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z))) ge(x,#()) -> true() ge(#(),0(x)) -> ge(#(),x) ge(#(),1(x)) -> false() ge(0(x),0(y)) -> ge(x,y) ge(0(x),1(y)) -> not(ge(y,x)) ge(1(x),0(y)) -> ge(x,y) ge(1(x),1(y)) -> ge(x,y) if(false(),x,y) -> y if(true(),x,y) -> x max(l(x)) -> x max(n(x,y,z)) -> max(z) min(l(x)) -> x min(n(x,y,z)) -> min(y) not(false()) -> true() not(true()) -> false() size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) wb(l(x)) -> true() wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(1(#()),-(size(y),size(z))),ge(1(#()),-(size(z),size(y)))) ,and(wb(y),wb(z))) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/2 ,c_5/1,c_6/1,c_7/3,c_8/0,c_9/0,c_10/2,c_11/2,c_12/1,c_13/2,c_14/0,c_15/0,c_16/0,c_17/0,c_18/9,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/2,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/4 ,c_36/0,c_37/0,c_38/0,c_39/16} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:+#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)) -->_2 +#(1(x),1(y)) -> c_7(0#(+(+(x,y),1(#()))),+#(+(x,y),1(#())),+#(x,y)):5 -->_1 +#(1(x),1(y)) -> c_7(0#(+(+(x,y),1(#()))),+#(+(x,y),1(#())),+#(x,y)):5 -->_2 +#(1(x),0(y)) -> c_6(+#(x,y)):4 -->_1 +#(1(x),0(y)) -> c_6(+#(x,y)):4 -->_2 +#(0(x),1(y)) -> c_5(+#(x,y)):3 -->_1 +#(0(x),1(y)) -> c_5(+#(x,y)):3 -->_2 +#(0(x),0(y)) -> c_4(0#(+(x,y)),+#(x,y)):2 -->_1 +#(0(x),0(y)) -> c_4(0#(+(x,y)),+#(x,y)):2 -->_2 +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)):1 -->_1 +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)):1 2:S:+#(0(x),0(y)) -> c_4(0#(+(x,y)),+#(x,y)) -->_2 +#(1(x),1(y)) -> c_7(0#(+(+(x,y),1(#()))),+#(+(x,y),1(#())),+#(x,y)):5 -->_2 +#(1(x),0(y)) -> c_6(+#(x,y)):4 -->_2 +#(0(x),1(y)) -> c_5(+#(x,y)):3 -->_2 +#(0(x),0(y)) -> c_4(0#(+(x,y)),+#(x,y)):2 -->_2 +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)):1 3:S:+#(0(x),1(y)) -> c_5(+#(x,y)) -->_1 +#(1(x),1(y)) -> c_7(0#(+(+(x,y),1(#()))),+#(+(x,y),1(#())),+#(x,y)):5 -->_1 +#(1(x),0(y)) -> c_6(+#(x,y)):4 -->_1 +#(0(x),1(y)) -> c_5(+#(x,y)):3 -->_1 +#(0(x),0(y)) -> c_4(0#(+(x,y)),+#(x,y)):2 -->_1 +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)):1 4:S:+#(1(x),0(y)) -> c_6(+#(x,y)) -->_1 +#(1(x),1(y)) -> c_7(0#(+(+(x,y),1(#()))),+#(+(x,y),1(#())),+#(x,y)):5 -->_1 +#(1(x),0(y)) -> c_6(+#(x,y)):4 -->_1 +#(0(x),1(y)) -> c_5(+#(x,y)):3 -->_1 +#(0(x),0(y)) -> c_4(0#(+(x,y)),+#(x,y)):2 -->_1 +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)):1 5:S:+#(1(x),1(y)) -> c_7(0#(+(+(x,y),1(#()))),+#(+(x,y),1(#())),+#(x,y)) -->_3 +#(1(x),1(y)) -> c_7(0#(+(+(x,y),1(#()))),+#(+(x,y),1(#())),+#(x,y)):5 -->_2 +#(1(x),1(y)) -> c_7(0#(+(+(x,y),1(#()))),+#(+(x,y),1(#())),+#(x,y)):5 -->_3 +#(1(x),0(y)) -> c_6(+#(x,y)):4 -->_3 +#(0(x),1(y)) -> c_5(+#(x,y)):3 -->_2 +#(0(x),1(y)) -> c_5(+#(x,y)):3 -->_3 +#(0(x),0(y)) -> c_4(0#(+(x,y)),+#(x,y)):2 -->_3 +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)):1 6:S:-#(0(x),0(y)) -> c_10(0#(-(x,y)),-#(x,y)) -->_2 -#(1(x),1(y)) -> c_13(0#(-(x,y)),-#(x,y)):9 -->_2 -#(1(x),0(y)) -> c_12(-#(x,y)):8 -->_2 -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)):7 -->_2 -#(0(x),0(y)) -> c_10(0#(-(x,y)),-#(x,y)):6 7:S:-#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)) -->_2 -#(1(x),1(y)) -> c_13(0#(-(x,y)),-#(x,y)):9 -->_1 -#(1(x),1(y)) -> c_13(0#(-(x,y)),-#(x,y)):9 -->_2 -#(1(x),0(y)) -> c_12(-#(x,y)):8 -->_2 -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)):7 -->_1 -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)):7 -->_2 -#(0(x),0(y)) -> c_10(0#(-(x,y)),-#(x,y)):6 8:S:-#(1(x),0(y)) -> c_12(-#(x,y)) -->_1 -#(1(x),1(y)) -> c_13(0#(-(x,y)),-#(x,y)):9 -->_1 -#(1(x),0(y)) -> c_12(-#(x,y)):8 -->_1 -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)):7 -->_1 -#(0(x),0(y)) -> c_10(0#(-(x,y)),-#(x,y)):6 9:S:-#(1(x),1(y)) -> c_13(0#(-(x,y)),-#(x,y)) -->_2 -#(1(x),1(y)) -> c_13(0#(-(x,y)),-#(x,y)):9 -->_2 -#(1(x),0(y)) -> c_12(-#(x,y)):8 -->_2 -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)):7 -->_2 -#(0(x),0(y)) -> c_10(0#(-(x,y)),-#(x,y)):6 10:S:bs#(n(x,y,z)) -> c_18(and#(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z))) ,and#(ge(x,max(y)),ge(min(z),x)) ,ge#(x,max(y)) ,max#(y) ,ge#(min(z),x) ,min#(z) ,and#(bs(y),bs(z)) ,bs#(y) ,bs#(z)) -->_6 min#(n(x,y,z)) -> c_31(min#(y)):17 -->_4 max#(n(x,y,z)) -> c_29(max#(z)):16 -->_5 ge#(1(x),1(y)) -> c_25(ge#(x,y)):15 -->_3 ge#(1(x),1(y)) -> c_25(ge#(x,y)):15 -->_5 ge#(1(x),0(y)) -> c_24(ge#(x,y)):14 -->_3 ge#(1(x),0(y)) -> c_24(ge#(x,y)):14 -->_5 ge#(0(x),1(y)) -> c_23(not#(ge(y,x)),ge#(y,x)):13 -->_3 ge#(0(x),1(y)) -> c_23(not#(ge(y,x)),ge#(y,x)):13 -->_5 ge#(0(x),0(y)) -> c_22(ge#(x,y)):12 -->_3 ge#(0(x),0(y)) -> c_22(ge#(x,y)):12 -->_5 ge#(#(),0(x)) -> c_20(ge#(#(),x)):11 -->_3 ge#(#(),0(x)) -> c_20(ge#(#(),x)):11 -->_9 bs#(n(x,y,z)) -> c_18(and#(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z))) ,and#(ge(x,max(y)),ge(min(z),x)) ,ge#(x,max(y)) ,max#(y) ,ge#(min(z),x) ,min#(z) ,and#(bs(y),bs(z)) ,bs#(y) ,bs#(z)):10 -->_8 bs#(n(x,y,z)) -> c_18(and#(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z))) ,and#(ge(x,max(y)),ge(min(z),x)) ,ge#(x,max(y)) ,max#(y) ,ge#(min(z),x) ,min#(z) ,and#(bs(y),bs(z)) ,bs#(y) ,bs#(z)):10 11:S:ge#(#(),0(x)) -> c_20(ge#(#(),x)) -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):11 12:S:ge#(0(x),0(y)) -> c_22(ge#(x,y)) -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):15 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):14 -->_1 ge#(0(x),1(y)) -> c_23(not#(ge(y,x)),ge#(y,x)):13 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):12 -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):11 13:S:ge#(0(x),1(y)) -> c_23(not#(ge(y,x)),ge#(y,x)) -->_2 ge#(1(x),1(y)) -> c_25(ge#(x,y)):15 -->_2 ge#(1(x),0(y)) -> c_24(ge#(x,y)):14 -->_2 ge#(0(x),1(y)) -> c_23(not#(ge(y,x)),ge#(y,x)):13 -->_2 ge#(0(x),0(y)) -> c_22(ge#(x,y)):12 -->_2 ge#(#(),0(x)) -> c_20(ge#(#(),x)):11 14:S:ge#(1(x),0(y)) -> c_24(ge#(x,y)) -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):15 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):14 -->_1 ge#(0(x),1(y)) -> c_23(not#(ge(y,x)),ge#(y,x)):13 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):12 -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):11 15:S:ge#(1(x),1(y)) -> c_25(ge#(x,y)) -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):15 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):14 -->_1 ge#(0(x),1(y)) -> c_23(not#(ge(y,x)),ge#(y,x)):13 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):12 -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):11 16:S:max#(n(x,y,z)) -> c_29(max#(z)) -->_1 max#(n(x,y,z)) -> c_29(max#(z)):16 17:S:min#(n(x,y,z)) -> c_31(min#(y)) -->_1 min#(n(x,y,z)) -> c_31(min#(y)):17 18:S:size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)) -->_4 size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)):18 -->_3 size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)):18 -->_2 +#(1(x),1(y)) -> c_7(0#(+(+(x,y),1(#()))),+#(+(x,y),1(#())),+#(x,y)):5 -->_1 +#(1(x),1(y)) -> c_7(0#(+(+(x,y),1(#()))),+#(+(x,y),1(#())),+#(x,y)):5 -->_2 +#(1(x),0(y)) -> c_6(+#(x,y)):4 -->_2 +#(0(x),1(y)) -> c_5(+#(x,y)):3 -->_1 +#(0(x),1(y)) -> c_5(+#(x,y)):3 -->_2 +#(0(x),0(y)) -> c_4(0#(+(x,y)),+#(x,y)):2 -->_2 +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)):1 19:S:wb#(n(x,y,z)) -> c_39(and#(if(ge(size(y),size(z)) ,ge(1(#()),-(size(y),size(z))) ,ge(1(#()),-(size(z),size(y)))) ,and(wb(y),wb(z))) ,if#(ge(size(y),size(z)) ,ge(1(#()),-(size(y),size(z))) ,ge(1(#()),-(size(z),size(y)))) ,ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,-#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,-#(size(z),size(y)) ,size#(z) ,size#(y) ,and#(wb(y),wb(z)) ,wb#(y) ,wb#(z)) -->_16 wb#(n(x,y,z)) -> c_39(and#(if(ge(size(y),size(z)) ,ge(1(#()),-(size(y),size(z))) ,ge(1(#()),-(size(z),size(y)))) ,and(wb(y),wb(z))) ,if#(ge(size(y),size(z)) ,ge(1(#()),-(size(y),size(z))) ,ge(1(#()),-(size(z),size(y)))) ,ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,-#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,-#(size(z),size(y)) ,size#(z) ,size#(y) ,and#(wb(y),wb(z)) ,wb#(y) ,wb#(z)):19 -->_15 wb#(n(x,y,z)) -> c_39(and#(if(ge(size(y),size(z)) ,ge(1(#()),-(size(y),size(z))) ,ge(1(#()),-(size(z),size(y)))) ,and(wb(y),wb(z))) ,if#(ge(size(y),size(z)) ,ge(1(#()),-(size(y),size(z))) ,ge(1(#()),-(size(z),size(y)))) ,ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,-#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,-#(size(z),size(y)) ,size#(z) ,size#(y) ,and#(wb(y),wb(z)) ,wb#(y) ,wb#(z)):19 -->_13 size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)):18 -->_12 size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)):18 -->_9 size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)):18 -->_8 size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)):18 -->_5 size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)):18 -->_4 size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)):18 -->_10 ge#(1(x),1(y)) -> c_25(ge#(x,y)):15 -->_6 ge#(1(x),1(y)) -> c_25(ge#(x,y)):15 -->_3 ge#(1(x),1(y)) -> c_25(ge#(x,y)):15 -->_10 ge#(1(x),0(y)) -> c_24(ge#(x,y)):14 -->_6 ge#(1(x),0(y)) -> c_24(ge#(x,y)):14 -->_3 ge#(1(x),0(y)) -> c_24(ge#(x,y)):14 -->_3 ge#(0(x),1(y)) -> c_23(not#(ge(y,x)),ge#(y,x)):13 -->_3 ge#(0(x),0(y)) -> c_22(ge#(x,y)):12 -->_3 ge#(#(),0(x)) -> c_20(ge#(#(),x)):11 -->_11 -#(1(x),1(y)) -> c_13(0#(-(x,y)),-#(x,y)):9 -->_7 -#(1(x),1(y)) -> c_13(0#(-(x,y)),-#(x,y)):9 -->_11 -#(1(x),0(y)) -> c_12(-#(x,y)):8 -->_7 -#(1(x),0(y)) -> c_12(-#(x,y)):8 -->_11 -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)):7 -->_7 -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)):7 -->_11 -#(0(x),0(y)) -> c_10(0#(-(x,y)),-#(x,y)):6 -->_7 -#(0(x),0(y)) -> c_10(0#(-(x,y)),-#(x,y)):6 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: +#(0(x),0(y)) -> c_4(+#(x,y)) +#(1(x),1(y)) -> c_7(+#(+(x,y),1(#())),+#(x,y)) -#(0(x),0(y)) -> c_10(-#(x,y)) -#(1(x),1(y)) -> c_13(-#(x,y)) bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),min#(z),bs#(y),bs#(z)) ge#(0(x),1(y)) -> c_23(ge#(y,x)) wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,-#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,-#(size(z),size(y)) ,size#(z) ,size#(y) ,wb#(y) ,wb#(z)) * Step 6: UsableRules WORST_CASE(?,O(n^4)) + Considered Problem: - Strict DPs: +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)) +#(0(x),0(y)) -> c_4(+#(x,y)) +#(0(x),1(y)) -> c_5(+#(x,y)) +#(1(x),0(y)) -> c_6(+#(x,y)) +#(1(x),1(y)) -> c_7(+#(+(x,y),1(#())),+#(x,y)) -#(0(x),0(y)) -> c_10(-#(x,y)) -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)) -#(1(x),0(y)) -> c_12(-#(x,y)) -#(1(x),1(y)) -> c_13(-#(x,y)) bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),min#(z),bs#(y),bs#(z)) ge#(#(),0(x)) -> c_20(ge#(#(),x)) ge#(0(x),0(y)) -> c_22(ge#(x,y)) ge#(0(x),1(y)) -> c_23(ge#(y,x)) ge#(1(x),0(y)) -> c_24(ge#(x,y)) ge#(1(x),1(y)) -> c_25(ge#(x,y)) max#(n(x,y,z)) -> c_29(max#(z)) min#(n(x,y,z)) -> c_31(min#(y)) size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)) wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,-#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,-#(size(z),size(y)) ,size#(z) ,size#(y) ,wb#(y) ,wb#(z)) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) -(x,#()) -> x -(#(),x) -> #() -(0(x),0(y)) -> 0(-(x,y)) -(0(x),1(y)) -> 1(-(-(x,y),1(#()))) -(1(x),0(y)) -> 1(-(x,y)) -(1(x),1(y)) -> 0(-(x,y)) 0(#()) -> #() and(x,false()) -> false() and(x,true()) -> x bs(l(x)) -> true() bs(n(x,y,z)) -> and(and(ge(x,max(y)),ge(min(z),x)),and(bs(y),bs(z))) ge(x,#()) -> true() ge(#(),0(x)) -> ge(#(),x) ge(#(),1(x)) -> false() ge(0(x),0(y)) -> ge(x,y) ge(0(x),1(y)) -> not(ge(y,x)) ge(1(x),0(y)) -> ge(x,y) ge(1(x),1(y)) -> ge(x,y) if(false(),x,y) -> y if(true(),x,y) -> x max(l(x)) -> x max(n(x,y,z)) -> max(z) min(l(x)) -> x min(n(x,y,z)) -> min(y) not(false()) -> true() not(true()) -> false() size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) wb(l(x)) -> true() wb(n(x,y,z)) -> and(if(ge(size(y),size(z)),ge(1(#()),-(size(y),size(z))),ge(1(#()),-(size(z),size(y)))) ,and(wb(y),wb(z))) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/4 ,c_36/0,c_37/0,c_38/0,c_39/13} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) -(x,#()) -> x -(#(),x) -> #() -(0(x),0(y)) -> 0(-(x,y)) -(0(x),1(y)) -> 1(-(-(x,y),1(#()))) -(1(x),0(y)) -> 1(-(x,y)) -(1(x),1(y)) -> 0(-(x,y)) 0(#()) -> #() max(l(x)) -> x max(n(x,y,z)) -> max(z) min(l(x)) -> x min(n(x,y,z)) -> min(y) size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)) +#(0(x),0(y)) -> c_4(+#(x,y)) +#(0(x),1(y)) -> c_5(+#(x,y)) +#(1(x),0(y)) -> c_6(+#(x,y)) +#(1(x),1(y)) -> c_7(+#(+(x,y),1(#())),+#(x,y)) -#(0(x),0(y)) -> c_10(-#(x,y)) -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)) -#(1(x),0(y)) -> c_12(-#(x,y)) -#(1(x),1(y)) -> c_13(-#(x,y)) bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),min#(z),bs#(y),bs#(z)) ge#(#(),0(x)) -> c_20(ge#(#(),x)) ge#(0(x),0(y)) -> c_22(ge#(x,y)) ge#(0(x),1(y)) -> c_23(ge#(y,x)) ge#(1(x),0(y)) -> c_24(ge#(x,y)) ge#(1(x),1(y)) -> c_25(ge#(x,y)) max#(n(x,y,z)) -> c_29(max#(z)) min#(n(x,y,z)) -> c_31(min#(y)) size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)) wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,-#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,-#(size(z),size(y)) ,size#(z) ,size#(y) ,wb#(y) ,wb#(z)) * Step 7: Decompose WORST_CASE(?,O(n^4)) + Considered Problem: - Strict DPs: +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)) +#(0(x),0(y)) -> c_4(+#(x,y)) +#(0(x),1(y)) -> c_5(+#(x,y)) +#(1(x),0(y)) -> c_6(+#(x,y)) +#(1(x),1(y)) -> c_7(+#(+(x,y),1(#())),+#(x,y)) -#(0(x),0(y)) -> c_10(-#(x,y)) -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)) -#(1(x),0(y)) -> c_12(-#(x,y)) -#(1(x),1(y)) -> c_13(-#(x,y)) bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),min#(z),bs#(y),bs#(z)) ge#(#(),0(x)) -> c_20(ge#(#(),x)) ge#(0(x),0(y)) -> c_22(ge#(x,y)) ge#(0(x),1(y)) -> c_23(ge#(y,x)) ge#(1(x),0(y)) -> c_24(ge#(x,y)) ge#(1(x),1(y)) -> c_25(ge#(x,y)) max#(n(x,y,z)) -> c_29(max#(z)) min#(n(x,y,z)) -> c_31(min#(y)) size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)) wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,-#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,-#(size(z),size(y)) ,size#(z) ,size#(y) ,wb#(y) ,wb#(z)) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) -(x,#()) -> x -(#(),x) -> #() -(0(x),0(y)) -> 0(-(x,y)) -(0(x),1(y)) -> 1(-(-(x,y),1(#()))) -(1(x),0(y)) -> 1(-(x,y)) -(1(x),1(y)) -> 0(-(x,y)) 0(#()) -> #() max(l(x)) -> x max(n(x,y,z)) -> max(z) min(l(x)) -> x min(n(x,y,z)) -> min(y) size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/4 ,c_36/0,c_37/0,c_38/0,c_39/13} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd} + Details: We analyse the complexity of following sub-problems (R) and (S). Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component. Problem (R) - Strict DPs: +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)) +#(0(x),0(y)) -> c_4(+#(x,y)) +#(0(x),1(y)) -> c_5(+#(x,y)) +#(1(x),0(y)) -> c_6(+#(x,y)) +#(1(x),1(y)) -> c_7(+#(+(x,y),1(#())),+#(x,y)) - Weak DPs: -#(0(x),0(y)) -> c_10(-#(x,y)) -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)) -#(1(x),0(y)) -> c_12(-#(x,y)) -#(1(x),1(y)) -> c_13(-#(x,y)) bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),min#(z),bs#(y),bs#(z)) ge#(#(),0(x)) -> c_20(ge#(#(),x)) ge#(0(x),0(y)) -> c_22(ge#(x,y)) ge#(0(x),1(y)) -> c_23(ge#(y,x)) ge#(1(x),0(y)) -> c_24(ge#(x,y)) ge#(1(x),1(y)) -> c_25(ge#(x,y)) max#(n(x,y,z)) -> c_29(max#(z)) min#(n(x,y,z)) -> c_31(min#(y)) size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)) wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,-#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,-#(size(z),size(y)) ,size#(z) ,size#(y) ,wb#(y) ,wb#(z)) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) -(x,#()) -> x -(#(),x) -> #() -(0(x),0(y)) -> 0(-(x,y)) -(0(x),1(y)) -> 1(-(-(x,y),1(#()))) -(1(x),0(y)) -> 1(-(x,y)) -(1(x),1(y)) -> 0(-(x,y)) 0(#()) -> #() max(l(x)) -> x max(n(x,y,z)) -> max(z) min(l(x)) -> x min(n(x,y,z)) -> min(y) size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0 ,c_4/1,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0 ,c_20/1,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0 ,c_35/4,c_36/0,c_37/0,c_38/0,c_39/13} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} Problem (S) - Strict DPs: -#(0(x),0(y)) -> c_10(-#(x,y)) -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)) -#(1(x),0(y)) -> c_12(-#(x,y)) -#(1(x),1(y)) -> c_13(-#(x,y)) bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),min#(z),bs#(y),bs#(z)) ge#(#(),0(x)) -> c_20(ge#(#(),x)) ge#(0(x),0(y)) -> c_22(ge#(x,y)) ge#(0(x),1(y)) -> c_23(ge#(y,x)) ge#(1(x),0(y)) -> c_24(ge#(x,y)) ge#(1(x),1(y)) -> c_25(ge#(x,y)) max#(n(x,y,z)) -> c_29(max#(z)) min#(n(x,y,z)) -> c_31(min#(y)) size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)) wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,-#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,-#(size(z),size(y)) ,size#(z) ,size#(y) ,wb#(y) ,wb#(z)) - Weak DPs: +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)) +#(0(x),0(y)) -> c_4(+#(x,y)) +#(0(x),1(y)) -> c_5(+#(x,y)) +#(1(x),0(y)) -> c_6(+#(x,y)) +#(1(x),1(y)) -> c_7(+#(+(x,y),1(#())),+#(x,y)) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) -(x,#()) -> x -(#(),x) -> #() -(0(x),0(y)) -> 0(-(x,y)) -(0(x),1(y)) -> 1(-(-(x,y),1(#()))) -(1(x),0(y)) -> 1(-(x,y)) -(1(x),1(y)) -> 0(-(x,y)) 0(#()) -> #() max(l(x)) -> x max(n(x,y,z)) -> max(z) min(l(x)) -> x min(n(x,y,z)) -> min(y) size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0 ,c_4/1,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0 ,c_20/1,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0 ,c_35/4,c_36/0,c_37/0,c_38/0,c_39/13} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} ** Step 7.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^4)) + Considered Problem: - Strict DPs: +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)) +#(0(x),0(y)) -> c_4(+#(x,y)) +#(0(x),1(y)) -> c_5(+#(x,y)) +#(1(x),0(y)) -> c_6(+#(x,y)) +#(1(x),1(y)) -> c_7(+#(+(x,y),1(#())),+#(x,y)) - Weak DPs: -#(0(x),0(y)) -> c_10(-#(x,y)) -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)) -#(1(x),0(y)) -> c_12(-#(x,y)) -#(1(x),1(y)) -> c_13(-#(x,y)) bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),min#(z),bs#(y),bs#(z)) ge#(#(),0(x)) -> c_20(ge#(#(),x)) ge#(0(x),0(y)) -> c_22(ge#(x,y)) ge#(0(x),1(y)) -> c_23(ge#(y,x)) ge#(1(x),0(y)) -> c_24(ge#(x,y)) ge#(1(x),1(y)) -> c_25(ge#(x,y)) max#(n(x,y,z)) -> c_29(max#(z)) min#(n(x,y,z)) -> c_31(min#(y)) size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)) wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,-#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,-#(size(z),size(y)) ,size#(z) ,size#(y) ,wb#(y) ,wb#(z)) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) -(x,#()) -> x -(#(),x) -> #() -(0(x),0(y)) -> 0(-(x,y)) -(0(x),1(y)) -> 1(-(-(x,y),1(#()))) -(1(x),0(y)) -> 1(-(x,y)) -(1(x),1(y)) -> 0(-(x,y)) 0(#()) -> #() max(l(x)) -> x max(n(x,y,z)) -> max(z) min(l(x)) -> x min(n(x,y,z)) -> min(y) size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/4 ,c_36/0,c_37/0,c_38/0,c_39/13} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:+#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)) -->_2 +#(1(x),1(y)) -> c_7(+#(+(x,y),1(#())),+#(x,y)):5 -->_1 +#(1(x),1(y)) -> c_7(+#(+(x,y),1(#())),+#(x,y)):5 -->_2 +#(1(x),0(y)) -> c_6(+#(x,y)):4 -->_1 +#(1(x),0(y)) -> c_6(+#(x,y)):4 -->_2 +#(0(x),1(y)) -> c_5(+#(x,y)):3 -->_1 +#(0(x),1(y)) -> c_5(+#(x,y)):3 -->_2 +#(0(x),0(y)) -> c_4(+#(x,y)):2 -->_1 +#(0(x),0(y)) -> c_4(+#(x,y)):2 -->_2 +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)):1 -->_1 +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)):1 2:S:+#(0(x),0(y)) -> c_4(+#(x,y)) -->_1 +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)):1 -->_1 +#(1(x),1(y)) -> c_7(+#(+(x,y),1(#())),+#(x,y)):5 -->_1 +#(1(x),0(y)) -> c_6(+#(x,y)):4 -->_1 +#(0(x),1(y)) -> c_5(+#(x,y)):3 -->_1 +#(0(x),0(y)) -> c_4(+#(x,y)):2 3:S:+#(0(x),1(y)) -> c_5(+#(x,y)) -->_1 +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)):1 -->_1 +#(0(x),0(y)) -> c_4(+#(x,y)):2 -->_1 +#(1(x),1(y)) -> c_7(+#(+(x,y),1(#())),+#(x,y)):5 -->_1 +#(1(x),0(y)) -> c_6(+#(x,y)):4 -->_1 +#(0(x),1(y)) -> c_5(+#(x,y)):3 4:S:+#(1(x),0(y)) -> c_6(+#(x,y)) -->_1 +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)):1 -->_1 +#(0(x),0(y)) -> c_4(+#(x,y)):2 -->_1 +#(0(x),1(y)) -> c_5(+#(x,y)):3 -->_1 +#(1(x),1(y)) -> c_7(+#(+(x,y),1(#())),+#(x,y)):5 -->_1 +#(1(x),0(y)) -> c_6(+#(x,y)):4 5:S:+#(1(x),1(y)) -> c_7(+#(+(x,y),1(#())),+#(x,y)) -->_2 +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)):1 -->_2 +#(0(x),0(y)) -> c_4(+#(x,y)):2 -->_2 +#(0(x),1(y)) -> c_5(+#(x,y)):3 -->_1 +#(0(x),1(y)) -> c_5(+#(x,y)):3 -->_2 +#(1(x),0(y)) -> c_6(+#(x,y)):4 -->_2 +#(1(x),1(y)) -> c_7(+#(+(x,y),1(#())),+#(x,y)):5 -->_1 +#(1(x),1(y)) -> c_7(+#(+(x,y),1(#())),+#(x,y)):5 6:W:-#(0(x),0(y)) -> c_10(-#(x,y)) -->_1 -#(1(x),1(y)) -> c_13(-#(x,y)):9 -->_1 -#(1(x),0(y)) -> c_12(-#(x,y)):8 -->_1 -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)):7 -->_1 -#(0(x),0(y)) -> c_10(-#(x,y)):6 7:W:-#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)) -->_2 -#(0(x),0(y)) -> c_10(-#(x,y)):6 -->_2 -#(1(x),1(y)) -> c_13(-#(x,y)):9 -->_1 -#(1(x),1(y)) -> c_13(-#(x,y)):9 -->_2 -#(1(x),0(y)) -> c_12(-#(x,y)):8 -->_2 -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)):7 -->_1 -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)):7 8:W:-#(1(x),0(y)) -> c_12(-#(x,y)) -->_1 -#(0(x),0(y)) -> c_10(-#(x,y)):6 -->_1 -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)):7 -->_1 -#(1(x),1(y)) -> c_13(-#(x,y)):9 -->_1 -#(1(x),0(y)) -> c_12(-#(x,y)):8 9:W:-#(1(x),1(y)) -> c_13(-#(x,y)) -->_1 -#(0(x),0(y)) -> c_10(-#(x,y)):6 -->_1 -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)):7 -->_1 -#(1(x),0(y)) -> c_12(-#(x,y)):8 -->_1 -#(1(x),1(y)) -> c_13(-#(x,y)):9 10:W:bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),min#(z),bs#(y),bs#(z)) -->_4 min#(n(x,y,z)) -> c_31(min#(y)):17 -->_2 max#(n(x,y,z)) -> c_29(max#(z)):16 -->_3 ge#(1(x),1(y)) -> c_25(ge#(x,y)):15 -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):15 -->_3 ge#(1(x),0(y)) -> c_24(ge#(x,y)):14 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):14 -->_3 ge#(0(x),1(y)) -> c_23(ge#(y,x)):13 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):13 -->_3 ge#(0(x),0(y)) -> c_22(ge#(x,y)):12 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):12 -->_3 ge#(#(),0(x)) -> c_20(ge#(#(),x)):11 -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):11 -->_6 bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),min#(z),bs#(y),bs#(z)):10 -->_5 bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),min#(z),bs#(y),bs#(z)):10 11:W:ge#(#(),0(x)) -> c_20(ge#(#(),x)) -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):11 12:W:ge#(0(x),0(y)) -> c_22(ge#(x,y)) -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):11 -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):15 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):14 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):13 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):12 13:W:ge#(0(x),1(y)) -> c_23(ge#(y,x)) -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):11 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):12 -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):15 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):14 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):13 14:W:ge#(1(x),0(y)) -> c_24(ge#(x,y)) -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):11 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):12 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):13 -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):15 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):14 15:W:ge#(1(x),1(y)) -> c_25(ge#(x,y)) -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):11 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):12 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):13 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):14 -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):15 16:W:max#(n(x,y,z)) -> c_29(max#(z)) -->_1 max#(n(x,y,z)) -> c_29(max#(z)):16 17:W:min#(n(x,y,z)) -> c_31(min#(y)) -->_1 min#(n(x,y,z)) -> c_31(min#(y)):17 18:W:size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)) -->_2 +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)):1 -->_2 +#(0(x),0(y)) -> c_4(+#(x,y)):2 -->_2 +#(0(x),1(y)) -> c_5(+#(x,y)):3 -->_1 +#(0(x),1(y)) -> c_5(+#(x,y)):3 -->_2 +#(1(x),0(y)) -> c_6(+#(x,y)):4 -->_2 +#(1(x),1(y)) -> c_7(+#(+(x,y),1(#())),+#(x,y)):5 -->_1 +#(1(x),1(y)) -> c_7(+#(+(x,y),1(#())),+#(x,y)):5 -->_4 size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)):18 -->_3 size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)):18 19:W:wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,-#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,-#(size(z),size(y)) ,size#(z) ,size#(y) ,wb#(y) ,wb#(z)) -->_9 -#(0(x),0(y)) -> c_10(-#(x,y)):6 -->_5 -#(0(x),0(y)) -> c_10(-#(x,y)):6 -->_9 -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)):7 -->_5 -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)):7 -->_9 -#(1(x),0(y)) -> c_12(-#(x,y)):8 -->_5 -#(1(x),0(y)) -> c_12(-#(x,y)):8 -->_9 -#(1(x),1(y)) -> c_13(-#(x,y)):9 -->_5 -#(1(x),1(y)) -> c_13(-#(x,y)):9 -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):11 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):12 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):13 -->_8 ge#(1(x),0(y)) -> c_24(ge#(x,y)):14 -->_4 ge#(1(x),0(y)) -> c_24(ge#(x,y)):14 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):14 -->_8 ge#(1(x),1(y)) -> c_25(ge#(x,y)):15 -->_4 ge#(1(x),1(y)) -> c_25(ge#(x,y)):15 -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):15 -->_11 size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)):18 -->_10 size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)):18 -->_7 size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)):18 -->_6 size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)):18 -->_3 size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)):18 -->_2 size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)):18 -->_13 wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,-#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,-#(size(z),size(y)) ,size#(z) ,size#(y) ,wb#(y) ,wb#(z)):19 -->_12 wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,-#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,-#(size(z),size(y)) ,size#(z) ,size#(y) ,wb#(y) ,wb#(z)):19 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 10: bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),min#(z),bs#(y),bs#(z)) 15: ge#(1(x),1(y)) -> c_25(ge#(x,y)) 14: ge#(1(x),0(y)) -> c_24(ge#(x,y)) 13: ge#(0(x),1(y)) -> c_23(ge#(y,x)) 12: ge#(0(x),0(y)) -> c_22(ge#(x,y)) 11: ge#(#(),0(x)) -> c_20(ge#(#(),x)) 16: max#(n(x,y,z)) -> c_29(max#(z)) 17: min#(n(x,y,z)) -> c_31(min#(y)) 6: -#(0(x),0(y)) -> c_10(-#(x,y)) 9: -#(1(x),1(y)) -> c_13(-#(x,y)) 8: -#(1(x),0(y)) -> c_12(-#(x,y)) 7: -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)) ** Step 7.a:2: SimplifyRHS WORST_CASE(?,O(n^4)) + Considered Problem: - Strict DPs: +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)) +#(0(x),0(y)) -> c_4(+#(x,y)) +#(0(x),1(y)) -> c_5(+#(x,y)) +#(1(x),0(y)) -> c_6(+#(x,y)) +#(1(x),1(y)) -> c_7(+#(+(x,y),1(#())),+#(x,y)) - Weak DPs: size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)) wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,-#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,-#(size(z),size(y)) ,size#(z) ,size#(y) ,wb#(y) ,wb#(z)) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) -(x,#()) -> x -(#(),x) -> #() -(0(x),0(y)) -> 0(-(x,y)) -(0(x),1(y)) -> 1(-(-(x,y),1(#()))) -(1(x),0(y)) -> 1(-(x,y)) -(1(x),1(y)) -> 0(-(x,y)) 0(#()) -> #() max(l(x)) -> x max(n(x,y,z)) -> max(z) min(l(x)) -> x min(n(x,y,z)) -> min(y) size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/4 ,c_36/0,c_37/0,c_38/0,c_39/13} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:+#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)) -->_2 +#(1(x),1(y)) -> c_7(+#(+(x,y),1(#())),+#(x,y)):5 -->_1 +#(1(x),1(y)) -> c_7(+#(+(x,y),1(#())),+#(x,y)):5 -->_2 +#(1(x),0(y)) -> c_6(+#(x,y)):4 -->_1 +#(1(x),0(y)) -> c_6(+#(x,y)):4 -->_2 +#(0(x),1(y)) -> c_5(+#(x,y)):3 -->_1 +#(0(x),1(y)) -> c_5(+#(x,y)):3 -->_2 +#(0(x),0(y)) -> c_4(+#(x,y)):2 -->_1 +#(0(x),0(y)) -> c_4(+#(x,y)):2 -->_2 +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)):1 -->_1 +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)):1 2:S:+#(0(x),0(y)) -> c_4(+#(x,y)) -->_1 +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)):1 -->_1 +#(1(x),1(y)) -> c_7(+#(+(x,y),1(#())),+#(x,y)):5 -->_1 +#(1(x),0(y)) -> c_6(+#(x,y)):4 -->_1 +#(0(x),1(y)) -> c_5(+#(x,y)):3 -->_1 +#(0(x),0(y)) -> c_4(+#(x,y)):2 3:S:+#(0(x),1(y)) -> c_5(+#(x,y)) -->_1 +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)):1 -->_1 +#(0(x),0(y)) -> c_4(+#(x,y)):2 -->_1 +#(1(x),1(y)) -> c_7(+#(+(x,y),1(#())),+#(x,y)):5 -->_1 +#(1(x),0(y)) -> c_6(+#(x,y)):4 -->_1 +#(0(x),1(y)) -> c_5(+#(x,y)):3 4:S:+#(1(x),0(y)) -> c_6(+#(x,y)) -->_1 +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)):1 -->_1 +#(0(x),0(y)) -> c_4(+#(x,y)):2 -->_1 +#(0(x),1(y)) -> c_5(+#(x,y)):3 -->_1 +#(1(x),1(y)) -> c_7(+#(+(x,y),1(#())),+#(x,y)):5 -->_1 +#(1(x),0(y)) -> c_6(+#(x,y)):4 5:S:+#(1(x),1(y)) -> c_7(+#(+(x,y),1(#())),+#(x,y)) -->_2 +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)):1 -->_2 +#(0(x),0(y)) -> c_4(+#(x,y)):2 -->_2 +#(0(x),1(y)) -> c_5(+#(x,y)):3 -->_1 +#(0(x),1(y)) -> c_5(+#(x,y)):3 -->_2 +#(1(x),0(y)) -> c_6(+#(x,y)):4 -->_2 +#(1(x),1(y)) -> c_7(+#(+(x,y),1(#())),+#(x,y)):5 -->_1 +#(1(x),1(y)) -> c_7(+#(+(x,y),1(#())),+#(x,y)):5 18:W:size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)) -->_2 +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)):1 -->_2 +#(0(x),0(y)) -> c_4(+#(x,y)):2 -->_2 +#(0(x),1(y)) -> c_5(+#(x,y)):3 -->_1 +#(0(x),1(y)) -> c_5(+#(x,y)):3 -->_2 +#(1(x),0(y)) -> c_6(+#(x,y)):4 -->_2 +#(1(x),1(y)) -> c_7(+#(+(x,y),1(#())),+#(x,y)):5 -->_1 +#(1(x),1(y)) -> c_7(+#(+(x,y),1(#())),+#(x,y)):5 -->_4 size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)):18 -->_3 size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)):18 19:W:wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,-#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,-#(size(z),size(y)) ,size#(z) ,size#(y) ,wb#(y) ,wb#(z)) -->_11 size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)):18 -->_10 size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)):18 -->_7 size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)):18 -->_6 size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)):18 -->_3 size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)):18 -->_2 size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)):18 -->_13 wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,-#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,-#(size(z),size(y)) ,size#(z) ,size#(y) ,wb#(y) ,wb#(z)):19 -->_12 wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,-#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,-#(size(z),size(y)) ,size#(z) ,size#(y) ,wb#(y) ,wb#(z)):19 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: wb#(n(x,y,z)) -> c_39(size#(y),size#(z),size#(y),size#(z),size#(z),size#(y),wb#(y),wb#(z)) ** Step 7.a:3: UsableRules WORST_CASE(?,O(n^4)) + Considered Problem: - Strict DPs: +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)) +#(0(x),0(y)) -> c_4(+#(x,y)) +#(0(x),1(y)) -> c_5(+#(x,y)) +#(1(x),0(y)) -> c_6(+#(x,y)) +#(1(x),1(y)) -> c_7(+#(+(x,y),1(#())),+#(x,y)) - Weak DPs: size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)) wb#(n(x,y,z)) -> c_39(size#(y),size#(z),size#(y),size#(z),size#(z),size#(y),wb#(y),wb#(z)) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) -(x,#()) -> x -(#(),x) -> #() -(0(x),0(y)) -> 0(-(x,y)) -(0(x),1(y)) -> 1(-(-(x,y),1(#()))) -(1(x),0(y)) -> 1(-(x,y)) -(1(x),1(y)) -> 0(-(x,y)) 0(#()) -> #() max(l(x)) -> x max(n(x,y,z)) -> max(z) min(l(x)) -> x min(n(x,y,z)) -> min(y) size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/4 ,c_36/0,c_37/0,c_38/0,c_39/8} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) 0(#()) -> #() size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)) +#(0(x),0(y)) -> c_4(+#(x,y)) +#(0(x),1(y)) -> c_5(+#(x,y)) +#(1(x),0(y)) -> c_6(+#(x,y)) +#(1(x),1(y)) -> c_7(+#(+(x,y),1(#())),+#(x,y)) size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)) wb#(n(x,y,z)) -> c_39(size#(y),size#(z),size#(y),size#(z),size#(z),size#(y),wb#(y),wb#(z)) ** Step 7.a:4: DecomposeDG WORST_CASE(?,O(n^4)) + Considered Problem: - Strict DPs: +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)) +#(0(x),0(y)) -> c_4(+#(x,y)) +#(0(x),1(y)) -> c_5(+#(x,y)) +#(1(x),0(y)) -> c_6(+#(x,y)) +#(1(x),1(y)) -> c_7(+#(+(x,y),1(#())),+#(x,y)) - Weak DPs: size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)) wb#(n(x,y,z)) -> c_39(size#(y),size#(z),size#(y),size#(z),size#(z),size#(y),wb#(y),wb#(z)) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) 0(#()) -> #() size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/4 ,c_36/0,c_37/0,c_38/0,c_39/8} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing} + Details: We decompose the input problem according to the dependency graph into the upper component wb#(n(x,y,z)) -> c_39(size#(y),size#(z),size#(y),size#(z),size#(z),size#(y),wb#(y),wb#(z)) and a lower component +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)) +#(0(x),0(y)) -> c_4(+#(x,y)) +#(0(x),1(y)) -> c_5(+#(x,y)) +#(1(x),0(y)) -> c_6(+#(x,y)) +#(1(x),1(y)) -> c_7(+#(+(x,y),1(#())),+#(x,y)) size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)) Further, following extension rules are added to the lower component. wb#(n(x,y,z)) -> size#(y) wb#(n(x,y,z)) -> size#(z) wb#(n(x,y,z)) -> wb#(y) wb#(n(x,y,z)) -> wb#(z) *** Step 7.a:4.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: wb#(n(x,y,z)) -> c_39(size#(y),size#(z),size#(y),size#(z),size#(z),size#(y),wb#(y),wb#(z)) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) 0(#()) -> #() size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/4 ,c_36/0,c_37/0,c_38/0,c_39/8} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: wb#(n(x,y,z)) -> c_39(size#(y),size#(z),size#(y),size#(z),size#(z),size#(y),wb#(y),wb#(z)) The strictly oriented rules are moved into the weak component. **** Step 7.a:4.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: wb#(n(x,y,z)) -> c_39(size#(y),size#(z),size#(y),size#(z),size#(z),size#(y),wb#(y),wb#(z)) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) 0(#()) -> #() size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/4 ,c_36/0,c_37/0,c_38/0,c_39/8} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_39) = {7,8} Following symbols are considered usable: {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val#,wb#} TcT has computed the following interpretation: p(#) = [0] p(+) = [1] x2 + [0] p(-) = [0] p(0) = [0] p(1) = [1] x1 + [0] p(and) = [0] p(bs) = [0] p(false) = [0] p(ge) = [0] p(if) = [0] p(l) = [1] x1 + [0] p(max) = [0] p(min) = [0] p(n) = [1] x1 + [1] x2 + [1] x3 + [4] p(not) = [0] p(size) = [0] p(true) = [0] p(val) = [0] p(wb) = [0] p(+#) = [0] p(-#) = [0] p(0#) = [0] p(and#) = [0] p(bs#) = [1] x1 + [0] p(ge#) = [2] p(if#) = [1] p(max#) = [1] x1 + [4] p(min#) = [4] x1 + [2] p(not#) = [1] p(size#) = [1] p(val#) = [1] x1 + [4] p(wb#) = [4] x1 + [0] p(c_1) = [2] p(c_2) = [4] x1 + [1] p(c_3) = [2] p(c_4) = [1] x1 + [2] p(c_5) = [1] p(c_6) = [1] x1 + [0] p(c_7) = [1] x1 + [2] x2 + [2] p(c_8) = [1] p(c_9) = [4] p(c_10) = [1] x1 + [1] p(c_11) = [1] x1 + [1] x2 + [1] p(c_12) = [1] p(c_13) = [4] p(c_14) = [0] p(c_15) = [1] p(c_16) = [0] p(c_17) = [2] p(c_18) = [1] x3 + [1] p(c_19) = [1] p(c_20) = [1] x1 + [0] p(c_21) = [1] p(c_22) = [1] x1 + [8] p(c_23) = [1] p(c_24) = [1] p(c_25) = [0] p(c_26) = [1] p(c_27) = [2] p(c_28) = [0] p(c_29) = [1] x1 + [2] p(c_30) = [0] p(c_31) = [2] p(c_32) = [4] p(c_33) = [0] p(c_34) = [2] p(c_35) = [8] x1 + [1] x4 + [0] p(c_36) = [1] p(c_37) = [1] p(c_38) = [1] p(c_39) = [2] x1 + [7] x2 + [2] x4 + [1] x7 + [1] x8 + [3] Following rules are strictly oriented: wb#(n(x,y,z)) = [4] x + [4] y + [4] z + [16] > [4] y + [4] z + [14] = c_39(size#(y),size#(z),size#(y),size#(z),size#(z),size#(y),wb#(y),wb#(z)) Following rules are (at-least) weakly oriented: **** Step 7.a:4.a:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: wb#(n(x,y,z)) -> c_39(size#(y),size#(z),size#(y),size#(z),size#(z),size#(y),wb#(y),wb#(z)) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) 0(#()) -> #() size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/4 ,c_36/0,c_37/0,c_38/0,c_39/8} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () **** Step 7.a:4.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: wb#(n(x,y,z)) -> c_39(size#(y),size#(z),size#(y),size#(z),size#(z),size#(y),wb#(y),wb#(z)) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) 0(#()) -> #() size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/4 ,c_36/0,c_37/0,c_38/0,c_39/8} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:wb#(n(x,y,z)) -> c_39(size#(y),size#(z),size#(y),size#(z),size#(z),size#(y),wb#(y),wb#(z)) -->_8 wb#(n(x,y,z)) -> c_39(size#(y),size#(z),size#(y),size#(z),size#(z),size#(y),wb#(y),wb#(z)):1 -->_7 wb#(n(x,y,z)) -> c_39(size#(y),size#(z),size#(y),size#(z),size#(z),size#(y),wb#(y),wb#(z)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: wb#(n(x,y,z)) -> c_39(size#(y),size#(z),size#(y),size#(z),size#(z),size#(y),wb#(y),wb#(z)) **** Step 7.a:4.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) 0(#()) -> #() size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/4 ,c_36/0,c_37/0,c_38/0,c_39/8} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). *** Step 7.a:4.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)) +#(0(x),0(y)) -> c_4(+#(x,y)) +#(0(x),1(y)) -> c_5(+#(x,y)) +#(1(x),0(y)) -> c_6(+#(x,y)) +#(1(x),1(y)) -> c_7(+#(+(x,y),1(#())),+#(x,y)) - Weak DPs: size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)) wb#(n(x,y,z)) -> size#(y) wb#(n(x,y,z)) -> size#(z) wb#(n(x,y,z)) -> wb#(y) wb#(n(x,y,z)) -> wb#(z) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) 0(#()) -> #() size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/4 ,c_36/0,c_37/0,c_38/0,c_39/8} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 3: +#(0(x),1(y)) -> c_5(+#(x,y)) The strictly oriented rules are moved into the weak component. **** Step 7.a:4.b:1.a:1: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)) +#(0(x),0(y)) -> c_4(+#(x,y)) +#(0(x),1(y)) -> c_5(+#(x,y)) +#(1(x),0(y)) -> c_6(+#(x,y)) +#(1(x),1(y)) -> c_7(+#(+(x,y),1(#())),+#(x,y)) - Weak DPs: size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)) wb#(n(x,y,z)) -> size#(y) wb#(n(x,y,z)) -> size#(z) wb#(n(x,y,z)) -> wb#(y) wb#(n(x,y,z)) -> wb#(z) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) 0(#()) -> #() size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/4 ,c_36/0,c_37/0,c_38/0,c_39/8} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(mixed(2)): The following argument positions are considered usable: uargs(c_2) = {1,2}, uargs(c_4) = {1}, uargs(c_5) = {1}, uargs(c_6) = {1}, uargs(c_7) = {1,2}, uargs(c_35) = {1,2,3,4} Following symbols are considered usable: {+,0,size,+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val#,wb#} TcT has computed the following interpretation: p(#) = 0 p(+) = x1 + x2 p(-) = 0 p(0) = x1 p(1) = 1 + x1 p(and) = 0 p(bs) = 0 p(false) = 0 p(ge) = 0 p(if) = 0 p(l) = 1 p(max) = 0 p(min) = 0 p(n) = 1 + x1 + x2 + x3 p(not) = 0 p(size) = x1 p(true) = 0 p(val) = 0 p(wb) = 0 p(+#) = x2 p(-#) = 0 p(0#) = 0 p(and#) = 0 p(bs#) = 0 p(ge#) = 0 p(if#) = 0 p(max#) = 0 p(min#) = 0 p(not#) = 0 p(size#) = 1 + x1 + x1^2 p(val#) = 0 p(wb#) = 1 + x1^2 p(c_1) = 0 p(c_2) = x1 + x2 p(c_3) = 0 p(c_4) = x1 p(c_5) = x1 p(c_6) = x1 p(c_7) = x1 + x2 p(c_8) = 0 p(c_9) = 0 p(c_10) = 0 p(c_11) = 0 p(c_12) = 0 p(c_13) = 0 p(c_14) = 0 p(c_15) = 0 p(c_16) = 0 p(c_17) = 0 p(c_18) = 0 p(c_19) = 0 p(c_20) = 0 p(c_21) = 0 p(c_22) = 0 p(c_23) = 0 p(c_24) = 0 p(c_25) = 0 p(c_26) = 0 p(c_27) = 0 p(c_28) = 0 p(c_29) = 0 p(c_30) = 0 p(c_31) = 0 p(c_32) = 0 p(c_33) = 0 p(c_34) = 0 p(c_35) = x1 + x2 + x3 + x4 p(c_36) = 0 p(c_37) = 0 p(c_38) = 0 p(c_39) = 0 Following rules are strictly oriented: +#(0(x),1(y)) = 1 + y > y = c_5(+#(x,y)) Following rules are (at-least) weakly oriented: +#(x,+(y,z)) = y + z >= y + z = c_2(+#(+(x,y),z),+#(x,y)) +#(0(x),0(y)) = y >= y = c_4(+#(x,y)) +#(1(x),0(y)) = y >= y = c_6(+#(x,y)) +#(1(x),1(y)) = 1 + y >= 1 + y = c_7(+#(+(x,y),1(#())),+#(x,y)) size#(n(x,y,z)) = 3 + 3*x + 2*x*y + 2*x*z + x^2 + 3*y + 2*y*z + y^2 + 3*z + z^2 >= 3 + x + x^2 + 2*y + y^2 = c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)) wb#(n(x,y,z)) = 2 + 2*x + 2*x*y + 2*x*z + x^2 + 2*y + 2*y*z + y^2 + 2*z + z^2 >= 1 + y + y^2 = size#(y) wb#(n(x,y,z)) = 2 + 2*x + 2*x*y + 2*x*z + x^2 + 2*y + 2*y*z + y^2 + 2*z + z^2 >= 1 + z + z^2 = size#(z) wb#(n(x,y,z)) = 2 + 2*x + 2*x*y + 2*x*z + x^2 + 2*y + 2*y*z + y^2 + 2*z + z^2 >= 1 + y^2 = wb#(y) wb#(n(x,y,z)) = 2 + 2*x + 2*x*y + 2*x*z + x^2 + 2*y + 2*y*z + y^2 + 2*z + z^2 >= 1 + z^2 = wb#(z) +(x,#()) = x >= x = x +(x,+(y,z)) = x + y + z >= x + y + z = +(+(x,y),z) +(#(),x) = x >= x = x +(0(x),0(y)) = x + y >= x + y = 0(+(x,y)) +(0(x),1(y)) = 1 + x + y >= 1 + x + y = 1(+(x,y)) +(1(x),0(y)) = 1 + x + y >= 1 + x + y = 1(+(x,y)) +(1(x),1(y)) = 2 + x + y >= 1 + x + y = 0(+(+(x,y),1(#()))) 0(#()) = 0 >= 0 = #() size(l(x)) = 1 >= 1 = 1(#()) size(n(x,y,z)) = 1 + x + y + z >= 1 + x + y = +(+(size(x),size(y)),1(#())) **** Step 7.a:4.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)) +#(0(x),0(y)) -> c_4(+#(x,y)) +#(1(x),0(y)) -> c_6(+#(x,y)) +#(1(x),1(y)) -> c_7(+#(+(x,y),1(#())),+#(x,y)) - Weak DPs: +#(0(x),1(y)) -> c_5(+#(x,y)) size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)) wb#(n(x,y,z)) -> size#(y) wb#(n(x,y,z)) -> size#(z) wb#(n(x,y,z)) -> wb#(y) wb#(n(x,y,z)) -> wb#(z) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) 0(#()) -> #() size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/4 ,c_36/0,c_37/0,c_38/0,c_39/8} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () **** Step 7.a:4.b:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)) +#(0(x),0(y)) -> c_4(+#(x,y)) +#(1(x),0(y)) -> c_6(+#(x,y)) +#(1(x),1(y)) -> c_7(+#(+(x,y),1(#())),+#(x,y)) - Weak DPs: +#(0(x),1(y)) -> c_5(+#(x,y)) size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)) wb#(n(x,y,z)) -> size#(y) wb#(n(x,y,z)) -> size#(z) wb#(n(x,y,z)) -> wb#(y) wb#(n(x,y,z)) -> wb#(z) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) 0(#()) -> #() size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/4 ,c_36/0,c_37/0,c_38/0,c_39/8} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 2: +#(0(x),0(y)) -> c_4(+#(x,y)) 3: +#(1(x),0(y)) -> c_6(+#(x,y)) The strictly oriented rules are moved into the weak component. ***** Step 7.a:4.b:1.b:1.a:1: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)) +#(0(x),0(y)) -> c_4(+#(x,y)) +#(1(x),0(y)) -> c_6(+#(x,y)) +#(1(x),1(y)) -> c_7(+#(+(x,y),1(#())),+#(x,y)) - Weak DPs: +#(0(x),1(y)) -> c_5(+#(x,y)) size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)) wb#(n(x,y,z)) -> size#(y) wb#(n(x,y,z)) -> size#(z) wb#(n(x,y,z)) -> wb#(y) wb#(n(x,y,z)) -> wb#(z) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) 0(#()) -> #() size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/4 ,c_36/0,c_37/0,c_38/0,c_39/8} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(mixed(2)): The following argument positions are considered usable: uargs(c_2) = {1,2}, uargs(c_4) = {1}, uargs(c_5) = {1}, uargs(c_6) = {1}, uargs(c_7) = {1,2}, uargs(c_35) = {1,2,3,4} Following symbols are considered usable: {+,0,size,+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val#,wb#} TcT has computed the following interpretation: p(#) = 0 p(+) = x1 + x2 p(-) = 0 p(0) = 1 + x1 p(1) = 1 + x1 p(and) = 0 p(bs) = 0 p(false) = 0 p(ge) = 0 p(if) = 0 p(l) = 1 p(max) = 0 p(min) = 0 p(n) = 1 + x1 + x2 + x3 p(not) = 0 p(size) = x1 p(true) = 0 p(val) = 0 p(wb) = 0 p(+#) = x2 p(-#) = 0 p(0#) = 0 p(and#) = 0 p(bs#) = 0 p(ge#) = 0 p(if#) = 0 p(max#) = 0 p(min#) = 0 p(not#) = 0 p(size#) = x1 + x1^2 p(val#) = 0 p(wb#) = x1^2 p(c_1) = 0 p(c_2) = x1 + x2 p(c_3) = 0 p(c_4) = x1 p(c_5) = 1 + x1 p(c_6) = x1 p(c_7) = x1 + x2 p(c_8) = 0 p(c_9) = 0 p(c_10) = 0 p(c_11) = 0 p(c_12) = 0 p(c_13) = 0 p(c_14) = 0 p(c_15) = 0 p(c_16) = 0 p(c_17) = 0 p(c_18) = 0 p(c_19) = 0 p(c_20) = 0 p(c_21) = 0 p(c_22) = 0 p(c_23) = 0 p(c_24) = 0 p(c_25) = 0 p(c_26) = 0 p(c_27) = 0 p(c_28) = 0 p(c_29) = 0 p(c_30) = 0 p(c_31) = 0 p(c_32) = 0 p(c_33) = 0 p(c_34) = 0 p(c_35) = 1 + x1 + x2 + x3 + x4 p(c_36) = 0 p(c_37) = 0 p(c_38) = 0 p(c_39) = 0 Following rules are strictly oriented: +#(0(x),0(y)) = 1 + y > y = c_4(+#(x,y)) +#(1(x),0(y)) = 1 + y > y = c_6(+#(x,y)) Following rules are (at-least) weakly oriented: +#(x,+(y,z)) = y + z >= y + z = c_2(+#(+(x,y),z),+#(x,y)) +#(0(x),1(y)) = 1 + y >= 1 + y = c_5(+#(x,y)) +#(1(x),1(y)) = 1 + y >= 1 + y = c_7(+#(+(x,y),1(#())),+#(x,y)) size#(n(x,y,z)) = 2 + 3*x + 2*x*y + 2*x*z + x^2 + 3*y + 2*y*z + y^2 + 3*z + z^2 >= 2 + x + x^2 + 2*y + y^2 = c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)) wb#(n(x,y,z)) = 1 + 2*x + 2*x*y + 2*x*z + x^2 + 2*y + 2*y*z + y^2 + 2*z + z^2 >= y + y^2 = size#(y) wb#(n(x,y,z)) = 1 + 2*x + 2*x*y + 2*x*z + x^2 + 2*y + 2*y*z + y^2 + 2*z + z^2 >= z + z^2 = size#(z) wb#(n(x,y,z)) = 1 + 2*x + 2*x*y + 2*x*z + x^2 + 2*y + 2*y*z + y^2 + 2*z + z^2 >= y^2 = wb#(y) wb#(n(x,y,z)) = 1 + 2*x + 2*x*y + 2*x*z + x^2 + 2*y + 2*y*z + y^2 + 2*z + z^2 >= z^2 = wb#(z) +(x,#()) = x >= x = x +(x,+(y,z)) = x + y + z >= x + y + z = +(+(x,y),z) +(#(),x) = x >= x = x +(0(x),0(y)) = 2 + x + y >= 1 + x + y = 0(+(x,y)) +(0(x),1(y)) = 2 + x + y >= 1 + x + y = 1(+(x,y)) +(1(x),0(y)) = 2 + x + y >= 1 + x + y = 1(+(x,y)) +(1(x),1(y)) = 2 + x + y >= 2 + x + y = 0(+(+(x,y),1(#()))) 0(#()) = 1 >= 0 = #() size(l(x)) = 1 >= 1 = 1(#()) size(n(x,y,z)) = 1 + x + y + z >= 1 + x + y = +(+(size(x),size(y)),1(#())) ***** Step 7.a:4.b:1.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)) +#(1(x),1(y)) -> c_7(+#(+(x,y),1(#())),+#(x,y)) - Weak DPs: +#(0(x),0(y)) -> c_4(+#(x,y)) +#(0(x),1(y)) -> c_5(+#(x,y)) +#(1(x),0(y)) -> c_6(+#(x,y)) size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)) wb#(n(x,y,z)) -> size#(y) wb#(n(x,y,z)) -> size#(z) wb#(n(x,y,z)) -> wb#(y) wb#(n(x,y,z)) -> wb#(z) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) 0(#()) -> #() size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/4 ,c_36/0,c_37/0,c_38/0,c_39/8} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ***** Step 7.a:4.b:1.b:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)) +#(1(x),1(y)) -> c_7(+#(+(x,y),1(#())),+#(x,y)) - Weak DPs: +#(0(x),0(y)) -> c_4(+#(x,y)) +#(0(x),1(y)) -> c_5(+#(x,y)) +#(1(x),0(y)) -> c_6(+#(x,y)) size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)) wb#(n(x,y,z)) -> size#(y) wb#(n(x,y,z)) -> size#(z) wb#(n(x,y,z)) -> wb#(y) wb#(n(x,y,z)) -> wb#(z) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) 0(#()) -> #() size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/4 ,c_36/0,c_37/0,c_38/0,c_39/8} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)) The strictly oriented rules are moved into the weak component. ****** Step 7.a:4.b:1.b:1.b:1.a:1: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)) +#(1(x),1(y)) -> c_7(+#(+(x,y),1(#())),+#(x,y)) - Weak DPs: +#(0(x),0(y)) -> c_4(+#(x,y)) +#(0(x),1(y)) -> c_5(+#(x,y)) +#(1(x),0(y)) -> c_6(+#(x,y)) size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)) wb#(n(x,y,z)) -> size#(y) wb#(n(x,y,z)) -> size#(z) wb#(n(x,y,z)) -> wb#(y) wb#(n(x,y,z)) -> wb#(z) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) 0(#()) -> #() size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/4 ,c_36/0,c_37/0,c_38/0,c_39/8} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_2) = {1,2}, uargs(c_4) = {1}, uargs(c_5) = {1}, uargs(c_6) = {1}, uargs(c_7) = {1,2}, uargs(c_35) = {1,2,3,4} Following symbols are considered usable: {+,0,size,+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val#,wb#} TcT has computed the following interpretation: p(#) = [0] [0] p(+) = [1 0] x1 + [2 0] x2 + [0] [0 1] [0 1] [1] p(-) = [2 0] x1 + [0 0] x2 + [0] [0 1] [0 1] [0] p(0) = [0 0] x1 + [0] [0 1] [0] p(1) = [0 0] x1 + [0] [0 1] [1] p(and) = [0 0] x1 + [0] [1 1] [0] p(bs) = [0] [0] p(false) = [1] [0] p(ge) = [0 0] x2 + [0] [1 0] [1] p(if) = [0 0] x3 + [2] [1 1] [0] p(l) = [0] [2] p(max) = [2] [1] p(min) = [0] [0] p(n) = [1 3] x1 + [1 3] x2 + [1 1] x3 + [3] [0 1] [0 1] [0 1] [3] p(not) = [0] [0] p(size) = [0 0] x1 + [0] [0 1] [0] p(true) = [2] [0] p(val) = [2 1] x1 + [0] [2 2] [0] p(wb) = [0 0] x1 + [0] [0 1] [0] p(+#) = [0 2] x2 + [0] [0 0] [1] p(-#) = [0 0] x1 + [0 0] x2 + [0] [0 1] [0 2] [0] p(0#) = [2 0] x1 + [0] [0 0] [0] p(and#) = [0] [0] p(bs#) = [0 0] x1 + [1] [0 1] [0] p(ge#) = [0] [0] p(if#) = [0 0] x2 + [0 0] x3 + [0] [2 0] [0 1] [2] p(max#) = [0 0] x1 + [0] [0 1] [1] p(min#) = [0 2] x1 + [0] [1 1] [2] p(not#) = [0 0] x1 + [0] [1 2] [0] p(size#) = [1 1] x1 + [0] [0 1] [0] p(val#) = [1 0] x1 + [0] [0 0] [1] p(wb#) = [2 0] x1 + [0] [0 2] [0] p(c_1) = [0] [0] p(c_2) = [1 0] x1 + [1 1] x2 + [0] [0 0] [0 1] [0] p(c_3) = [0] [0] p(c_4) = [1 0] x1 + [0] [0 0] [0] p(c_5) = [1 0] x1 + [0] [0 0] [0] p(c_6) = [1 0] x1 + [0] [0 0] [0] p(c_7) = [1 0] x1 + [1 0] x2 + [0] [0 0] [0 1] [0] p(c_8) = [2] [1] p(c_9) = [0] [1] p(c_10) = [0 0] x1 + [1] [0 1] [0] p(c_11) = [2 1] x1 + [2 2] x2 + [1] [0 2] [0 1] [2] p(c_12) = [2 1] x1 + [0] [1 2] [0] p(c_13) = [1 0] x1 + [1] [0 2] [2] p(c_14) = [2] [0] p(c_15) = [1] [0] p(c_16) = [0] [0] p(c_17) = [0] [0] p(c_18) = [0 0] x1 + [0 0] x2 + [0 1] x3 + [0 1] x5 + [0] [2 0] [0 1] [0 0] [0 0] [0] p(c_19) = [0] [0] p(c_20) = [0] [0] p(c_21) = [0] [2] p(c_22) = [1 0] x1 + [1] [0 1] [2] p(c_23) = [0 0] x1 + [0] [1 1] [2] p(c_24) = [0 0] x1 + [0] [1 2] [0] p(c_25) = [0] [0] p(c_26) = [1] [0] p(c_27) = [0] [0] p(c_28) = [0] [2] p(c_29) = [1 0] x1 + [2] [1 0] [0] p(c_30) = [2] [0] p(c_31) = [0] [0] p(c_32) = [0] [2] p(c_33) = [1] [0] p(c_34) = [2] [2] p(c_35) = [1 1] x1 + [1 0] x2 + [1 1] x3 + [1 1] x4 + [3] [0 0] [0 0] [0 0] [0 0] [0] p(c_36) = [0] [0] p(c_37) = [0] [0] p(c_38) = [0] [1] p(c_39) = [0 0] x8 + [0] [0 1] [1] Following rules are strictly oriented: +#(x,+(y,z)) = [0 2] y + [0 2] z + [2] [0 0] [0 0] [1] > [0 2] y + [0 2] z + [1] [0 0] [0 0] [1] = c_2(+#(+(x,y),z),+#(x,y)) Following rules are (at-least) weakly oriented: +#(0(x),0(y)) = [0 2] y + [0] [0 0] [1] >= [0 2] y + [0] [0 0] [0] = c_4(+#(x,y)) +#(0(x),1(y)) = [0 2] y + [2] [0 0] [1] >= [0 2] y + [0] [0 0] [0] = c_5(+#(x,y)) +#(1(x),0(y)) = [0 2] y + [0] [0 0] [1] >= [0 2] y + [0] [0 0] [0] = c_6(+#(x,y)) +#(1(x),1(y)) = [0 2] y + [2] [0 0] [1] >= [0 2] y + [2] [0 0] [1] = c_7(+#(+(x,y),1(#())),+#(x,y)) size#(n(x,y,z)) = [1 4] x + [1 4] y + [1 2] z + [6] [0 1] [0 1] [0 1] [3] >= [1 2] x + [1 4] y + [6] [0 0] [0 0] [0] = c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)) wb#(n(x,y,z)) = [2 6] x + [2 6] y + [2 2] z + [6] [0 2] [0 2] [0 2] [6] >= [1 1] y + [0] [0 1] [0] = size#(y) wb#(n(x,y,z)) = [2 6] x + [2 6] y + [2 2] z + [6] [0 2] [0 2] [0 2] [6] >= [1 1] z + [0] [0 1] [0] = size#(z) wb#(n(x,y,z)) = [2 6] x + [2 6] y + [2 2] z + [6] [0 2] [0 2] [0 2] [6] >= [2 0] y + [0] [0 2] [0] = wb#(y) wb#(n(x,y,z)) = [2 6] x + [2 6] y + [2 2] z + [6] [0 2] [0 2] [0 2] [6] >= [2 0] z + [0] [0 2] [0] = wb#(z) +(x,#()) = [1 0] x + [0] [0 1] [1] >= [1 0] x + [0] [0 1] [0] = x +(x,+(y,z)) = [1 0] x + [2 0] y + [4 0] z + [0] [0 1] [0 1] [0 1] [2] >= [1 0] x + [2 0] y + [2 0] z + [0] [0 1] [0 1] [0 1] [2] = +(+(x,y),z) +(#(),x) = [2 0] x + [0] [0 1] [1] >= [1 0] x + [0] [0 1] [0] = x +(0(x),0(y)) = [0 0] x + [0 0] y + [0] [0 1] [0 1] [1] >= [0 0] x + [0 0] y + [0] [0 1] [0 1] [1] = 0(+(x,y)) +(0(x),1(y)) = [0 0] x + [0 0] y + [0] [0 1] [0 1] [2] >= [0 0] x + [0 0] y + [0] [0 1] [0 1] [2] = 1(+(x,y)) +(1(x),0(y)) = [0 0] x + [0 0] y + [0] [0 1] [0 1] [2] >= [0 0] x + [0 0] y + [0] [0 1] [0 1] [2] = 1(+(x,y)) +(1(x),1(y)) = [0 0] x + [0 0] y + [0] [0 1] [0 1] [3] >= [0 0] x + [0 0] y + [0] [0 1] [0 1] [3] = 0(+(+(x,y),1(#()))) 0(#()) = [0] [0] >= [0] [0] = #() size(l(x)) = [0] [2] >= [0] [1] = 1(#()) size(n(x,y,z)) = [0 0] x + [0 0] y + [0 0] z + [0] [0 1] [0 1] [0 1] [3] >= [0 0] x + [0 0] y + [0] [0 1] [0 1] [3] = +(+(size(x),size(y)),1(#())) ****** Step 7.a:4.b:1.b:1.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: +#(1(x),1(y)) -> c_7(+#(+(x,y),1(#())),+#(x,y)) - Weak DPs: +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)) +#(0(x),0(y)) -> c_4(+#(x,y)) +#(0(x),1(y)) -> c_5(+#(x,y)) +#(1(x),0(y)) -> c_6(+#(x,y)) size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)) wb#(n(x,y,z)) -> size#(y) wb#(n(x,y,z)) -> size#(z) wb#(n(x,y,z)) -> wb#(y) wb#(n(x,y,z)) -> wb#(z) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) 0(#()) -> #() size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/4 ,c_36/0,c_37/0,c_38/0,c_39/8} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ****** Step 7.a:4.b:1.b:1.b:1.b:1: DecomposeDG WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: +#(1(x),1(y)) -> c_7(+#(+(x,y),1(#())),+#(x,y)) - Weak DPs: +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)) +#(0(x),0(y)) -> c_4(+#(x,y)) +#(0(x),1(y)) -> c_5(+#(x,y)) +#(1(x),0(y)) -> c_6(+#(x,y)) size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)) wb#(n(x,y,z)) -> size#(y) wb#(n(x,y,z)) -> size#(z) wb#(n(x,y,z)) -> wb#(y) wb#(n(x,y,z)) -> wb#(z) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) 0(#()) -> #() size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/4 ,c_36/0,c_37/0,c_38/0,c_39/8} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing} + Details: We decompose the input problem according to the dependency graph into the upper component size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)) wb#(n(x,y,z)) -> size#(y) wb#(n(x,y,z)) -> size#(z) wb#(n(x,y,z)) -> wb#(y) wb#(n(x,y,z)) -> wb#(z) and a lower component +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)) +#(0(x),0(y)) -> c_4(+#(x,y)) +#(0(x),1(y)) -> c_5(+#(x,y)) +#(1(x),0(y)) -> c_6(+#(x,y)) +#(1(x),1(y)) -> c_7(+#(+(x,y),1(#())),+#(x,y)) Further, following extension rules are added to the lower component. size#(n(x,y,z)) -> +#(+(size(x),size(y)),1(#())) size#(n(x,y,z)) -> +#(size(x),size(y)) size#(n(x,y,z)) -> size#(x) size#(n(x,y,z)) -> size#(y) wb#(n(x,y,z)) -> size#(y) wb#(n(x,y,z)) -> size#(z) wb#(n(x,y,z)) -> wb#(y) wb#(n(x,y,z)) -> wb#(z) ******* Step 7.a:4.b:1.b:1.b:1.b:1.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)) - Weak DPs: wb#(n(x,y,z)) -> size#(y) wb#(n(x,y,z)) -> size#(z) wb#(n(x,y,z)) -> wb#(y) wb#(n(x,y,z)) -> wb#(z) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) 0(#()) -> #() size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/4 ,c_36/0,c_37/0,c_38/0,c_39/8} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)) The strictly oriented rules are moved into the weak component. ******** Step 7.a:4.b:1.b:1.b:1.b:1.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)) - Weak DPs: wb#(n(x,y,z)) -> size#(y) wb#(n(x,y,z)) -> size#(z) wb#(n(x,y,z)) -> wb#(y) wb#(n(x,y,z)) -> wb#(z) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) 0(#()) -> #() size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/4 ,c_36/0,c_37/0,c_38/0,c_39/8} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_35) = {1,2,3,4} Following symbols are considered usable: {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val#,wb#} TcT has computed the following interpretation: p(#) = [0] p(+) = [4] x1 + [0] p(-) = [0] p(0) = [1] x1 + [5] p(1) = [1] x1 + [1] p(and) = [0] p(bs) = [0] p(false) = [0] p(ge) = [0] p(if) = [0] p(l) = [0] p(max) = [0] p(min) = [0] p(n) = [1] x1 + [1] x2 + [1] x3 + [7] p(not) = [0] p(size) = [1] p(true) = [0] p(val) = [0] p(wb) = [0] p(+#) = [2] p(-#) = [0] p(0#) = [0] p(and#) = [0] p(bs#) = [0] p(ge#) = [0] p(if#) = [1] x3 + [0] p(max#) = [1] x1 + [0] p(min#) = [1] x1 + [0] p(not#) = [0] p(size#) = [4] x1 + [0] p(val#) = [8] x1 + [0] p(wb#) = [4] x1 + [1] p(c_1) = [0] p(c_2) = [1] x2 + [0] p(c_3) = [0] p(c_4) = [0] p(c_5) = [4] x1 + [0] p(c_6) = [2] x1 + [0] p(c_7) = [1] x2 + [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [0] p(c_11) = [1] x1 + [0] p(c_12) = [0] p(c_13) = [1] x1 + [0] p(c_14) = [0] p(c_15) = [0] p(c_16) = [0] p(c_17) = [0] p(c_18) = [1] x2 + [1] x3 + [1] x5 + [4] x6 + [0] p(c_19) = [0] p(c_20) = [2] x1 + [2] p(c_21) = [1] p(c_22) = [1] x1 + [8] p(c_23) = [1] x1 + [1] p(c_24) = [0] p(c_25) = [8] x1 + [1] p(c_26) = [1] p(c_27) = [0] p(c_28) = [1] p(c_29) = [0] p(c_30) = [2] p(c_31) = [2] x1 + [0] p(c_32) = [2] p(c_33) = [2] p(c_34) = [1] p(c_35) = [7] x1 + [4] x2 + [1] x3 + [1] x4 + [4] p(c_36) = [4] p(c_37) = [0] p(c_38) = [2] p(c_39) = [1] x2 + [1] x5 + [1] x8 + [0] Following rules are strictly oriented: size#(n(x,y,z)) = [4] x + [4] y + [4] z + [28] > [4] x + [4] y + [26] = c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)) Following rules are (at-least) weakly oriented: wb#(n(x,y,z)) = [4] x + [4] y + [4] z + [29] >= [4] y + [0] = size#(y) wb#(n(x,y,z)) = [4] x + [4] y + [4] z + [29] >= [4] z + [0] = size#(z) wb#(n(x,y,z)) = [4] x + [4] y + [4] z + [29] >= [4] y + [1] = wb#(y) wb#(n(x,y,z)) = [4] x + [4] y + [4] z + [29] >= [4] z + [1] = wb#(z) ******** Step 7.a:4.b:1.b:1.b:1.b:1.a:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)) wb#(n(x,y,z)) -> size#(y) wb#(n(x,y,z)) -> size#(z) wb#(n(x,y,z)) -> wb#(y) wb#(n(x,y,z)) -> wb#(z) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) 0(#()) -> #() size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/4 ,c_36/0,c_37/0,c_38/0,c_39/8} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ******** Step 7.a:4.b:1.b:1.b:1.b:1.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)) wb#(n(x,y,z)) -> size#(y) wb#(n(x,y,z)) -> size#(z) wb#(n(x,y,z)) -> wb#(y) wb#(n(x,y,z)) -> wb#(z) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) 0(#()) -> #() size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/4 ,c_36/0,c_37/0,c_38/0,c_39/8} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)) -->_4 size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)):1 -->_3 size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)):1 2:W:wb#(n(x,y,z)) -> size#(y) -->_1 size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)):1 3:W:wb#(n(x,y,z)) -> size#(z) -->_1 size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)):1 4:W:wb#(n(x,y,z)) -> wb#(y) -->_1 wb#(n(x,y,z)) -> wb#(z):5 -->_1 wb#(n(x,y,z)) -> wb#(y):4 -->_1 wb#(n(x,y,z)) -> size#(z):3 -->_1 wb#(n(x,y,z)) -> size#(y):2 5:W:wb#(n(x,y,z)) -> wb#(z) -->_1 wb#(n(x,y,z)) -> wb#(z):5 -->_1 wb#(n(x,y,z)) -> wb#(y):4 -->_1 wb#(n(x,y,z)) -> size#(z):3 -->_1 wb#(n(x,y,z)) -> size#(y):2 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 4: wb#(n(x,y,z)) -> wb#(y) 5: wb#(n(x,y,z)) -> wb#(z) 3: wb#(n(x,y,z)) -> size#(z) 2: wb#(n(x,y,z)) -> size#(y) 1: size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)) ******** Step 7.a:4.b:1.b:1.b:1.b:1.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) 0(#()) -> #() size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/4 ,c_36/0,c_37/0,c_38/0,c_39/8} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). ******* Step 7.a:4.b:1.b:1.b:1.b:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: +#(1(x),1(y)) -> c_7(+#(+(x,y),1(#())),+#(x,y)) - Weak DPs: +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)) +#(0(x),0(y)) -> c_4(+#(x,y)) +#(0(x),1(y)) -> c_5(+#(x,y)) +#(1(x),0(y)) -> c_6(+#(x,y)) size#(n(x,y,z)) -> +#(+(size(x),size(y)),1(#())) size#(n(x,y,z)) -> +#(size(x),size(y)) size#(n(x,y,z)) -> size#(x) size#(n(x,y,z)) -> size#(y) wb#(n(x,y,z)) -> size#(y) wb#(n(x,y,z)) -> size#(z) wb#(n(x,y,z)) -> wb#(y) wb#(n(x,y,z)) -> wb#(z) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) 0(#()) -> #() size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/4 ,c_36/0,c_37/0,c_38/0,c_39/8} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: +#(1(x),1(y)) -> c_7(+#(+(x,y),1(#())),+#(x,y)) The strictly oriented rules are moved into the weak component. ******** Step 7.a:4.b:1.b:1.b:1.b:1.b:1.a:1: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: +#(1(x),1(y)) -> c_7(+#(+(x,y),1(#())),+#(x,y)) - Weak DPs: +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)) +#(0(x),0(y)) -> c_4(+#(x,y)) +#(0(x),1(y)) -> c_5(+#(x,y)) +#(1(x),0(y)) -> c_6(+#(x,y)) size#(n(x,y,z)) -> +#(+(size(x),size(y)),1(#())) size#(n(x,y,z)) -> +#(size(x),size(y)) size#(n(x,y,z)) -> size#(x) size#(n(x,y,z)) -> size#(y) wb#(n(x,y,z)) -> size#(y) wb#(n(x,y,z)) -> size#(z) wb#(n(x,y,z)) -> wb#(y) wb#(n(x,y,z)) -> wb#(z) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) 0(#()) -> #() size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/4 ,c_36/0,c_37/0,c_38/0,c_39/8} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(mixed(2)): The following argument positions are considered usable: uargs(c_2) = {1,2}, uargs(c_4) = {1}, uargs(c_5) = {1}, uargs(c_6) = {1}, uargs(c_7) = {1,2} Following symbols are considered usable: {+,0,size,+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val#,wb#} TcT has computed the following interpretation: p(#) = 0 p(+) = x1 + x2 p(-) = 0 p(0) = x1 p(1) = 1 + x1 p(and) = 0 p(bs) = 0 p(false) = 0 p(ge) = 0 p(if) = 0 p(l) = 1 + x1 p(max) = 0 p(min) = 0 p(n) = 1 + x1 + x2 + x3 p(not) = 0 p(size) = x1 p(true) = 0 p(val) = 0 p(wb) = 0 p(+#) = x1*x2 + x2^2 p(-#) = 0 p(0#) = 0 p(and#) = 0 p(bs#) = 0 p(ge#) = 0 p(if#) = 0 p(max#) = 0 p(min#) = 0 p(not#) = 0 p(size#) = x1^2 p(val#) = 0 p(wb#) = x1^2 p(c_1) = 0 p(c_2) = x1 + x2 p(c_3) = 0 p(c_4) = x1 p(c_5) = 1 + x1 p(c_6) = x1 p(c_7) = x1 + x2 p(c_8) = 0 p(c_9) = 0 p(c_10) = 0 p(c_11) = 0 p(c_12) = 0 p(c_13) = 0 p(c_14) = 0 p(c_15) = 0 p(c_16) = 0 p(c_17) = 0 p(c_18) = 0 p(c_19) = 0 p(c_20) = 0 p(c_21) = 0 p(c_22) = 0 p(c_23) = 0 p(c_24) = 0 p(c_25) = 0 p(c_26) = 0 p(c_27) = 0 p(c_28) = 0 p(c_29) = 0 p(c_30) = 0 p(c_31) = 0 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) = 0 Following rules are strictly oriented: +#(1(x),1(y)) = 2 + x + x*y + 3*y + y^2 > 1 + x + x*y + y + y^2 = c_7(+#(+(x,y),1(#())),+#(x,y)) Following rules are (at-least) weakly oriented: +#(x,+(y,z)) = x*y + x*z + 2*y*z + y^2 + z^2 >= x*y + x*z + y*z + y^2 + z^2 = c_2(+#(+(x,y),z),+#(x,y)) +#(0(x),0(y)) = x*y + y^2 >= x*y + y^2 = c_4(+#(x,y)) +#(0(x),1(y)) = 1 + x + x*y + 2*y + y^2 >= 1 + x*y + y^2 = c_5(+#(x,y)) +#(1(x),0(y)) = x*y + y + y^2 >= x*y + y^2 = c_6(+#(x,y)) size#(n(x,y,z)) = 1 + 2*x + 2*x*y + 2*x*z + x^2 + 2*y + 2*y*z + y^2 + 2*z + z^2 >= 1 + x + y = +#(+(size(x),size(y)),1(#())) size#(n(x,y,z)) = 1 + 2*x + 2*x*y + 2*x*z + x^2 + 2*y + 2*y*z + y^2 + 2*z + z^2 >= x*y + y^2 = +#(size(x),size(y)) size#(n(x,y,z)) = 1 + 2*x + 2*x*y + 2*x*z + x^2 + 2*y + 2*y*z + y^2 + 2*z + z^2 >= x^2 = size#(x) size#(n(x,y,z)) = 1 + 2*x + 2*x*y + 2*x*z + x^2 + 2*y + 2*y*z + y^2 + 2*z + z^2 >= y^2 = size#(y) wb#(n(x,y,z)) = 1 + 2*x + 2*x*y + 2*x*z + x^2 + 2*y + 2*y*z + y^2 + 2*z + z^2 >= y^2 = size#(y) wb#(n(x,y,z)) = 1 + 2*x + 2*x*y + 2*x*z + x^2 + 2*y + 2*y*z + y^2 + 2*z + z^2 >= z^2 = size#(z) wb#(n(x,y,z)) = 1 + 2*x + 2*x*y + 2*x*z + x^2 + 2*y + 2*y*z + y^2 + 2*z + z^2 >= y^2 = wb#(y) wb#(n(x,y,z)) = 1 + 2*x + 2*x*y + 2*x*z + x^2 + 2*y + 2*y*z + y^2 + 2*z + z^2 >= z^2 = wb#(z) +(x,#()) = x >= x = x +(x,+(y,z)) = x + y + z >= x + y + z = +(+(x,y),z) +(#(),x) = x >= x = x +(0(x),0(y)) = x + y >= x + y = 0(+(x,y)) +(0(x),1(y)) = 1 + x + y >= 1 + x + y = 1(+(x,y)) +(1(x),0(y)) = 1 + x + y >= 1 + x + y = 1(+(x,y)) +(1(x),1(y)) = 2 + x + y >= 1 + x + y = 0(+(+(x,y),1(#()))) 0(#()) = 0 >= 0 = #() size(l(x)) = 1 + x >= 1 = 1(#()) size(n(x,y,z)) = 1 + x + y + z >= 1 + x + y = +(+(size(x),size(y)),1(#())) ******** Step 7.a:4.b:1.b:1.b:1.b:1.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)) +#(0(x),0(y)) -> c_4(+#(x,y)) +#(0(x),1(y)) -> c_5(+#(x,y)) +#(1(x),0(y)) -> c_6(+#(x,y)) +#(1(x),1(y)) -> c_7(+#(+(x,y),1(#())),+#(x,y)) size#(n(x,y,z)) -> +#(+(size(x),size(y)),1(#())) size#(n(x,y,z)) -> +#(size(x),size(y)) size#(n(x,y,z)) -> size#(x) size#(n(x,y,z)) -> size#(y) wb#(n(x,y,z)) -> size#(y) wb#(n(x,y,z)) -> size#(z) wb#(n(x,y,z)) -> wb#(y) wb#(n(x,y,z)) -> wb#(z) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) 0(#()) -> #() size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/4 ,c_36/0,c_37/0,c_38/0,c_39/8} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ******** Step 7.a:4.b:1.b:1.b:1.b:1.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)) +#(0(x),0(y)) -> c_4(+#(x,y)) +#(0(x),1(y)) -> c_5(+#(x,y)) +#(1(x),0(y)) -> c_6(+#(x,y)) +#(1(x),1(y)) -> c_7(+#(+(x,y),1(#())),+#(x,y)) size#(n(x,y,z)) -> +#(+(size(x),size(y)),1(#())) size#(n(x,y,z)) -> +#(size(x),size(y)) size#(n(x,y,z)) -> size#(x) size#(n(x,y,z)) -> size#(y) wb#(n(x,y,z)) -> size#(y) wb#(n(x,y,z)) -> size#(z) wb#(n(x,y,z)) -> wb#(y) wb#(n(x,y,z)) -> wb#(z) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) 0(#()) -> #() size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/4 ,c_36/0,c_37/0,c_38/0,c_39/8} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:+#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)) -->_2 +#(1(x),1(y)) -> c_7(+#(+(x,y),1(#())),+#(x,y)):5 -->_1 +#(1(x),1(y)) -> c_7(+#(+(x,y),1(#())),+#(x,y)):5 -->_2 +#(1(x),0(y)) -> c_6(+#(x,y)):4 -->_1 +#(1(x),0(y)) -> c_6(+#(x,y)):4 -->_2 +#(0(x),1(y)) -> c_5(+#(x,y)):3 -->_1 +#(0(x),1(y)) -> c_5(+#(x,y)):3 -->_2 +#(0(x),0(y)) -> c_4(+#(x,y)):2 -->_1 +#(0(x),0(y)) -> c_4(+#(x,y)):2 -->_2 +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)):1 -->_1 +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)):1 2:W:+#(0(x),0(y)) -> c_4(+#(x,y)) -->_1 +#(1(x),1(y)) -> c_7(+#(+(x,y),1(#())),+#(x,y)):5 -->_1 +#(1(x),0(y)) -> c_6(+#(x,y)):4 -->_1 +#(0(x),1(y)) -> c_5(+#(x,y)):3 -->_1 +#(0(x),0(y)) -> c_4(+#(x,y)):2 -->_1 +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)):1 3:W:+#(0(x),1(y)) -> c_5(+#(x,y)) -->_1 +#(1(x),1(y)) -> c_7(+#(+(x,y),1(#())),+#(x,y)):5 -->_1 +#(1(x),0(y)) -> c_6(+#(x,y)):4 -->_1 +#(0(x),1(y)) -> c_5(+#(x,y)):3 -->_1 +#(0(x),0(y)) -> c_4(+#(x,y)):2 -->_1 +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)):1 4:W:+#(1(x),0(y)) -> c_6(+#(x,y)) -->_1 +#(1(x),1(y)) -> c_7(+#(+(x,y),1(#())),+#(x,y)):5 -->_1 +#(1(x),0(y)) -> c_6(+#(x,y)):4 -->_1 +#(0(x),1(y)) -> c_5(+#(x,y)):3 -->_1 +#(0(x),0(y)) -> c_4(+#(x,y)):2 -->_1 +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)):1 5:W:+#(1(x),1(y)) -> c_7(+#(+(x,y),1(#())),+#(x,y)) -->_2 +#(1(x),1(y)) -> c_7(+#(+(x,y),1(#())),+#(x,y)):5 -->_1 +#(1(x),1(y)) -> c_7(+#(+(x,y),1(#())),+#(x,y)):5 -->_2 +#(1(x),0(y)) -> c_6(+#(x,y)):4 -->_2 +#(0(x),1(y)) -> c_5(+#(x,y)):3 -->_1 +#(0(x),1(y)) -> c_5(+#(x,y)):3 -->_2 +#(0(x),0(y)) -> c_4(+#(x,y)):2 -->_2 +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)):1 6:W:size#(n(x,y,z)) -> +#(+(size(x),size(y)),1(#())) -->_1 +#(1(x),1(y)) -> c_7(+#(+(x,y),1(#())),+#(x,y)):5 -->_1 +#(0(x),1(y)) -> c_5(+#(x,y)):3 7:W:size#(n(x,y,z)) -> +#(size(x),size(y)) -->_1 +#(1(x),1(y)) -> c_7(+#(+(x,y),1(#())),+#(x,y)):5 -->_1 +#(1(x),0(y)) -> c_6(+#(x,y)):4 -->_1 +#(0(x),1(y)) -> c_5(+#(x,y)):3 -->_1 +#(0(x),0(y)) -> c_4(+#(x,y)):2 -->_1 +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)):1 8:W:size#(n(x,y,z)) -> size#(x) -->_1 size#(n(x,y,z)) -> size#(y):9 -->_1 size#(n(x,y,z)) -> size#(x):8 -->_1 size#(n(x,y,z)) -> +#(size(x),size(y)):7 -->_1 size#(n(x,y,z)) -> +#(+(size(x),size(y)),1(#())):6 9:W:size#(n(x,y,z)) -> size#(y) -->_1 size#(n(x,y,z)) -> size#(y):9 -->_1 size#(n(x,y,z)) -> size#(x):8 -->_1 size#(n(x,y,z)) -> +#(size(x),size(y)):7 -->_1 size#(n(x,y,z)) -> +#(+(size(x),size(y)),1(#())):6 10:W:wb#(n(x,y,z)) -> size#(y) -->_1 size#(n(x,y,z)) -> size#(y):9 -->_1 size#(n(x,y,z)) -> size#(x):8 -->_1 size#(n(x,y,z)) -> +#(size(x),size(y)):7 -->_1 size#(n(x,y,z)) -> +#(+(size(x),size(y)),1(#())):6 11:W:wb#(n(x,y,z)) -> size#(z) -->_1 size#(n(x,y,z)) -> size#(y):9 -->_1 size#(n(x,y,z)) -> size#(x):8 -->_1 size#(n(x,y,z)) -> +#(size(x),size(y)):7 -->_1 size#(n(x,y,z)) -> +#(+(size(x),size(y)),1(#())):6 12:W:wb#(n(x,y,z)) -> wb#(y) -->_1 wb#(n(x,y,z)) -> wb#(z):13 -->_1 wb#(n(x,y,z)) -> wb#(y):12 -->_1 wb#(n(x,y,z)) -> size#(z):11 -->_1 wb#(n(x,y,z)) -> size#(y):10 13:W:wb#(n(x,y,z)) -> wb#(z) -->_1 wb#(n(x,y,z)) -> wb#(z):13 -->_1 wb#(n(x,y,z)) -> wb#(y):12 -->_1 wb#(n(x,y,z)) -> size#(z):11 -->_1 wb#(n(x,y,z)) -> size#(y):10 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 12: wb#(n(x,y,z)) -> wb#(y) 13: wb#(n(x,y,z)) -> wb#(z) 11: wb#(n(x,y,z)) -> size#(z) 10: wb#(n(x,y,z)) -> size#(y) 8: size#(n(x,y,z)) -> size#(x) 9: size#(n(x,y,z)) -> size#(y) 7: size#(n(x,y,z)) -> +#(size(x),size(y)) 6: size#(n(x,y,z)) -> +#(+(size(x),size(y)),1(#())) 1: +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)) 5: +#(1(x),1(y)) -> c_7(+#(+(x,y),1(#())),+#(x,y)) 4: +#(1(x),0(y)) -> c_6(+#(x,y)) 3: +#(0(x),1(y)) -> c_5(+#(x,y)) 2: +#(0(x),0(y)) -> c_4(+#(x,y)) ******** Step 7.a:4.b:1.b:1.b:1.b:1.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) 0(#()) -> #() size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/4 ,c_36/0,c_37/0,c_38/0,c_39/8} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). ** Step 7.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: -#(0(x),0(y)) -> c_10(-#(x,y)) -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)) -#(1(x),0(y)) -> c_12(-#(x,y)) -#(1(x),1(y)) -> c_13(-#(x,y)) bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),min#(z),bs#(y),bs#(z)) ge#(#(),0(x)) -> c_20(ge#(#(),x)) ge#(0(x),0(y)) -> c_22(ge#(x,y)) ge#(0(x),1(y)) -> c_23(ge#(y,x)) ge#(1(x),0(y)) -> c_24(ge#(x,y)) ge#(1(x),1(y)) -> c_25(ge#(x,y)) max#(n(x,y,z)) -> c_29(max#(z)) min#(n(x,y,z)) -> c_31(min#(y)) size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)) wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,-#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,-#(size(z),size(y)) ,size#(z) ,size#(y) ,wb#(y) ,wb#(z)) - Weak DPs: +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)) +#(0(x),0(y)) -> c_4(+#(x,y)) +#(0(x),1(y)) -> c_5(+#(x,y)) +#(1(x),0(y)) -> c_6(+#(x,y)) +#(1(x),1(y)) -> c_7(+#(+(x,y),1(#())),+#(x,y)) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) -(x,#()) -> x -(#(),x) -> #() -(0(x),0(y)) -> 0(-(x,y)) -(0(x),1(y)) -> 1(-(-(x,y),1(#()))) -(1(x),0(y)) -> 1(-(x,y)) -(1(x),1(y)) -> 0(-(x,y)) 0(#()) -> #() max(l(x)) -> x max(n(x,y,z)) -> max(z) min(l(x)) -> x min(n(x,y,z)) -> min(y) size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/4 ,c_36/0,c_37/0,c_38/0,c_39/13} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:-#(0(x),0(y)) -> c_10(-#(x,y)) -->_1 -#(1(x),1(y)) -> c_13(-#(x,y)):4 -->_1 -#(1(x),0(y)) -> c_12(-#(x,y)):3 -->_1 -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)):2 -->_1 -#(0(x),0(y)) -> c_10(-#(x,y)):1 2:S:-#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)) -->_2 -#(1(x),1(y)) -> c_13(-#(x,y)):4 -->_1 -#(1(x),1(y)) -> c_13(-#(x,y)):4 -->_2 -#(1(x),0(y)) -> c_12(-#(x,y)):3 -->_2 -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)):2 -->_1 -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)):2 -->_2 -#(0(x),0(y)) -> c_10(-#(x,y)):1 3:S:-#(1(x),0(y)) -> c_12(-#(x,y)) -->_1 -#(1(x),1(y)) -> c_13(-#(x,y)):4 -->_1 -#(1(x),0(y)) -> c_12(-#(x,y)):3 -->_1 -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)):2 -->_1 -#(0(x),0(y)) -> c_10(-#(x,y)):1 4:S:-#(1(x),1(y)) -> c_13(-#(x,y)) -->_1 -#(1(x),1(y)) -> c_13(-#(x,y)):4 -->_1 -#(1(x),0(y)) -> c_12(-#(x,y)):3 -->_1 -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)):2 -->_1 -#(0(x),0(y)) -> c_10(-#(x,y)):1 5:S:bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),min#(z),bs#(y),bs#(z)) -->_4 min#(n(x,y,z)) -> c_31(min#(y)):12 -->_2 max#(n(x,y,z)) -> c_29(max#(z)):11 -->_3 ge#(1(x),1(y)) -> c_25(ge#(x,y)):10 -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):10 -->_3 ge#(1(x),0(y)) -> c_24(ge#(x,y)):9 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):9 -->_3 ge#(0(x),1(y)) -> c_23(ge#(y,x)):8 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):8 -->_3 ge#(0(x),0(y)) -> c_22(ge#(x,y)):7 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):7 -->_3 ge#(#(),0(x)) -> c_20(ge#(#(),x)):6 -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):6 -->_6 bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),min#(z),bs#(y),bs#(z)):5 -->_5 bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),min#(z),bs#(y),bs#(z)):5 6:S:ge#(#(),0(x)) -> c_20(ge#(#(),x)) -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):6 7:S:ge#(0(x),0(y)) -> c_22(ge#(x,y)) -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):10 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):9 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):8 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):7 -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):6 8:S:ge#(0(x),1(y)) -> c_23(ge#(y,x)) -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):10 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):9 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):8 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):7 -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):6 9:S:ge#(1(x),0(y)) -> c_24(ge#(x,y)) -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):10 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):9 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):8 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):7 -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):6 10:S:ge#(1(x),1(y)) -> c_25(ge#(x,y)) -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):10 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):9 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):8 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):7 -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):6 11:S:max#(n(x,y,z)) -> c_29(max#(z)) -->_1 max#(n(x,y,z)) -> c_29(max#(z)):11 12:S:min#(n(x,y,z)) -> c_31(min#(y)) -->_1 min#(n(x,y,z)) -> c_31(min#(y)):12 13:S:size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)) -->_2 +#(1(x),1(y)) -> c_7(+#(+(x,y),1(#())),+#(x,y)):19 -->_1 +#(1(x),1(y)) -> c_7(+#(+(x,y),1(#())),+#(x,y)):19 -->_2 +#(1(x),0(y)) -> c_6(+#(x,y)):18 -->_2 +#(0(x),1(y)) -> c_5(+#(x,y)):17 -->_1 +#(0(x),1(y)) -> c_5(+#(x,y)):17 -->_2 +#(0(x),0(y)) -> c_4(+#(x,y)):16 -->_2 +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)):15 -->_4 size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)):13 -->_3 size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)):13 14:S:wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,-#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,-#(size(z),size(y)) ,size#(z) ,size#(y) ,wb#(y) ,wb#(z)) -->_13 wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,-#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,-#(size(z),size(y)) ,size#(z) ,size#(y) ,wb#(y) ,wb#(z)):14 -->_12 wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,-#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,-#(size(z),size(y)) ,size#(z) ,size#(y) ,wb#(y) ,wb#(z)):14 -->_11 size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)):13 -->_10 size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)):13 -->_7 size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)):13 -->_6 size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)):13 -->_3 size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)):13 -->_2 size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)):13 -->_8 ge#(1(x),1(y)) -> c_25(ge#(x,y)):10 -->_4 ge#(1(x),1(y)) -> c_25(ge#(x,y)):10 -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):10 -->_8 ge#(1(x),0(y)) -> c_24(ge#(x,y)):9 -->_4 ge#(1(x),0(y)) -> c_24(ge#(x,y)):9 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):9 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):8 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):7 -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):6 -->_9 -#(1(x),1(y)) -> c_13(-#(x,y)):4 -->_5 -#(1(x),1(y)) -> c_13(-#(x,y)):4 -->_9 -#(1(x),0(y)) -> c_12(-#(x,y)):3 -->_5 -#(1(x),0(y)) -> c_12(-#(x,y)):3 -->_9 -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)):2 -->_5 -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)):2 -->_9 -#(0(x),0(y)) -> c_10(-#(x,y)):1 -->_5 -#(0(x),0(y)) -> c_10(-#(x,y)):1 15:W:+#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)) -->_2 +#(1(x),1(y)) -> c_7(+#(+(x,y),1(#())),+#(x,y)):19 -->_1 +#(1(x),1(y)) -> c_7(+#(+(x,y),1(#())),+#(x,y)):19 -->_2 +#(1(x),0(y)) -> c_6(+#(x,y)):18 -->_1 +#(1(x),0(y)) -> c_6(+#(x,y)):18 -->_2 +#(0(x),1(y)) -> c_5(+#(x,y)):17 -->_1 +#(0(x),1(y)) -> c_5(+#(x,y)):17 -->_2 +#(0(x),0(y)) -> c_4(+#(x,y)):16 -->_1 +#(0(x),0(y)) -> c_4(+#(x,y)):16 -->_2 +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)):15 -->_1 +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)):15 16:W:+#(0(x),0(y)) -> c_4(+#(x,y)) -->_1 +#(1(x),1(y)) -> c_7(+#(+(x,y),1(#())),+#(x,y)):19 -->_1 +#(1(x),0(y)) -> c_6(+#(x,y)):18 -->_1 +#(0(x),1(y)) -> c_5(+#(x,y)):17 -->_1 +#(0(x),0(y)) -> c_4(+#(x,y)):16 -->_1 +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)):15 17:W:+#(0(x),1(y)) -> c_5(+#(x,y)) -->_1 +#(1(x),1(y)) -> c_7(+#(+(x,y),1(#())),+#(x,y)):19 -->_1 +#(1(x),0(y)) -> c_6(+#(x,y)):18 -->_1 +#(0(x),1(y)) -> c_5(+#(x,y)):17 -->_1 +#(0(x),0(y)) -> c_4(+#(x,y)):16 -->_1 +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)):15 18:W:+#(1(x),0(y)) -> c_6(+#(x,y)) -->_1 +#(1(x),1(y)) -> c_7(+#(+(x,y),1(#())),+#(x,y)):19 -->_1 +#(1(x),0(y)) -> c_6(+#(x,y)):18 -->_1 +#(0(x),1(y)) -> c_5(+#(x,y)):17 -->_1 +#(0(x),0(y)) -> c_4(+#(x,y)):16 -->_1 +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)):15 19:W:+#(1(x),1(y)) -> c_7(+#(+(x,y),1(#())),+#(x,y)) -->_2 +#(1(x),1(y)) -> c_7(+#(+(x,y),1(#())),+#(x,y)):19 -->_1 +#(1(x),1(y)) -> c_7(+#(+(x,y),1(#())),+#(x,y)):19 -->_2 +#(1(x),0(y)) -> c_6(+#(x,y)):18 -->_2 +#(0(x),1(y)) -> c_5(+#(x,y)):17 -->_1 +#(0(x),1(y)) -> c_5(+#(x,y)):17 -->_2 +#(0(x),0(y)) -> c_4(+#(x,y)):16 -->_2 +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)):15 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 19: +#(1(x),1(y)) -> c_7(+#(+(x,y),1(#())),+#(x,y)) 18: +#(1(x),0(y)) -> c_6(+#(x,y)) 17: +#(0(x),1(y)) -> c_5(+#(x,y)) 16: +#(0(x),0(y)) -> c_4(+#(x,y)) 15: +#(x,+(y,z)) -> c_2(+#(+(x,y),z),+#(x,y)) ** Step 7.b:2: SimplifyRHS WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: -#(0(x),0(y)) -> c_10(-#(x,y)) -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)) -#(1(x),0(y)) -> c_12(-#(x,y)) -#(1(x),1(y)) -> c_13(-#(x,y)) bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),min#(z),bs#(y),bs#(z)) ge#(#(),0(x)) -> c_20(ge#(#(),x)) ge#(0(x),0(y)) -> c_22(ge#(x,y)) ge#(0(x),1(y)) -> c_23(ge#(y,x)) ge#(1(x),0(y)) -> c_24(ge#(x,y)) ge#(1(x),1(y)) -> c_25(ge#(x,y)) max#(n(x,y,z)) -> c_29(max#(z)) min#(n(x,y,z)) -> c_31(min#(y)) size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)) wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,-#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,-#(size(z),size(y)) ,size#(z) ,size#(y) ,wb#(y) ,wb#(z)) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) -(x,#()) -> x -(#(),x) -> #() -(0(x),0(y)) -> 0(-(x,y)) -(0(x),1(y)) -> 1(-(-(x,y),1(#()))) -(1(x),0(y)) -> 1(-(x,y)) -(1(x),1(y)) -> 0(-(x,y)) 0(#()) -> #() max(l(x)) -> x max(n(x,y,z)) -> max(z) min(l(x)) -> x min(n(x,y,z)) -> min(y) size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/4 ,c_36/0,c_37/0,c_38/0,c_39/13} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:-#(0(x),0(y)) -> c_10(-#(x,y)) -->_1 -#(1(x),1(y)) -> c_13(-#(x,y)):4 -->_1 -#(1(x),0(y)) -> c_12(-#(x,y)):3 -->_1 -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)):2 -->_1 -#(0(x),0(y)) -> c_10(-#(x,y)):1 2:S:-#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)) -->_2 -#(1(x),1(y)) -> c_13(-#(x,y)):4 -->_1 -#(1(x),1(y)) -> c_13(-#(x,y)):4 -->_2 -#(1(x),0(y)) -> c_12(-#(x,y)):3 -->_2 -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)):2 -->_1 -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)):2 -->_2 -#(0(x),0(y)) -> c_10(-#(x,y)):1 3:S:-#(1(x),0(y)) -> c_12(-#(x,y)) -->_1 -#(1(x),1(y)) -> c_13(-#(x,y)):4 -->_1 -#(1(x),0(y)) -> c_12(-#(x,y)):3 -->_1 -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)):2 -->_1 -#(0(x),0(y)) -> c_10(-#(x,y)):1 4:S:-#(1(x),1(y)) -> c_13(-#(x,y)) -->_1 -#(1(x),1(y)) -> c_13(-#(x,y)):4 -->_1 -#(1(x),0(y)) -> c_12(-#(x,y)):3 -->_1 -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)):2 -->_1 -#(0(x),0(y)) -> c_10(-#(x,y)):1 5:S:bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),min#(z),bs#(y),bs#(z)) -->_4 min#(n(x,y,z)) -> c_31(min#(y)):12 -->_2 max#(n(x,y,z)) -> c_29(max#(z)):11 -->_3 ge#(1(x),1(y)) -> c_25(ge#(x,y)):10 -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):10 -->_3 ge#(1(x),0(y)) -> c_24(ge#(x,y)):9 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):9 -->_3 ge#(0(x),1(y)) -> c_23(ge#(y,x)):8 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):8 -->_3 ge#(0(x),0(y)) -> c_22(ge#(x,y)):7 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):7 -->_3 ge#(#(),0(x)) -> c_20(ge#(#(),x)):6 -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):6 -->_6 bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),min#(z),bs#(y),bs#(z)):5 -->_5 bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),min#(z),bs#(y),bs#(z)):5 6:S:ge#(#(),0(x)) -> c_20(ge#(#(),x)) -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):6 7:S:ge#(0(x),0(y)) -> c_22(ge#(x,y)) -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):10 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):9 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):8 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):7 -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):6 8:S:ge#(0(x),1(y)) -> c_23(ge#(y,x)) -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):10 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):9 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):8 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):7 -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):6 9:S:ge#(1(x),0(y)) -> c_24(ge#(x,y)) -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):10 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):9 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):8 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):7 -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):6 10:S:ge#(1(x),1(y)) -> c_25(ge#(x,y)) -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):10 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):9 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):8 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):7 -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):6 11:S:max#(n(x,y,z)) -> c_29(max#(z)) -->_1 max#(n(x,y,z)) -> c_29(max#(z)):11 12:S:min#(n(x,y,z)) -> c_31(min#(y)) -->_1 min#(n(x,y,z)) -> c_31(min#(y)):12 13:S:size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)) -->_4 size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)):13 -->_3 size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)):13 14:S:wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,-#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,-#(size(z),size(y)) ,size#(z) ,size#(y) ,wb#(y) ,wb#(z)) -->_13 wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,-#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,-#(size(z),size(y)) ,size#(z) ,size#(y) ,wb#(y) ,wb#(z)):14 -->_12 wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,-#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,-#(size(z),size(y)) ,size#(z) ,size#(y) ,wb#(y) ,wb#(z)):14 -->_11 size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)):13 -->_10 size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)):13 -->_7 size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)):13 -->_6 size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)):13 -->_3 size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)):13 -->_2 size#(n(x,y,z)) -> c_35(+#(+(size(x),size(y)),1(#())),+#(size(x),size(y)),size#(x),size#(y)):13 -->_8 ge#(1(x),1(y)) -> c_25(ge#(x,y)):10 -->_4 ge#(1(x),1(y)) -> c_25(ge#(x,y)):10 -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):10 -->_8 ge#(1(x),0(y)) -> c_24(ge#(x,y)):9 -->_4 ge#(1(x),0(y)) -> c_24(ge#(x,y)):9 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):9 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):8 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):7 -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):6 -->_9 -#(1(x),1(y)) -> c_13(-#(x,y)):4 -->_5 -#(1(x),1(y)) -> c_13(-#(x,y)):4 -->_9 -#(1(x),0(y)) -> c_12(-#(x,y)):3 -->_5 -#(1(x),0(y)) -> c_12(-#(x,y)):3 -->_9 -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)):2 -->_5 -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)):2 -->_9 -#(0(x),0(y)) -> c_10(-#(x,y)):1 -->_5 -#(0(x),0(y)) -> c_10(-#(x,y)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: size#(n(x,y,z)) -> c_35(size#(x),size#(y)) ** Step 7.b:3: Decompose WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: -#(0(x),0(y)) -> c_10(-#(x,y)) -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)) -#(1(x),0(y)) -> c_12(-#(x,y)) -#(1(x),1(y)) -> c_13(-#(x,y)) bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),min#(z),bs#(y),bs#(z)) ge#(#(),0(x)) -> c_20(ge#(#(),x)) ge#(0(x),0(y)) -> c_22(ge#(x,y)) ge#(0(x),1(y)) -> c_23(ge#(y,x)) ge#(1(x),0(y)) -> c_24(ge#(x,y)) ge#(1(x),1(y)) -> c_25(ge#(x,y)) max#(n(x,y,z)) -> c_29(max#(z)) min#(n(x,y,z)) -> c_31(min#(y)) size#(n(x,y,z)) -> c_35(size#(x),size#(y)) wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,-#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,-#(size(z),size(y)) ,size#(z) ,size#(y) ,wb#(y) ,wb#(z)) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) -(x,#()) -> x -(#(),x) -> #() -(0(x),0(y)) -> 0(-(x,y)) -(0(x),1(y)) -> 1(-(-(x,y),1(#()))) -(1(x),0(y)) -> 1(-(x,y)) -(1(x),1(y)) -> 0(-(x,y)) 0(#()) -> #() max(l(x)) -> x max(n(x,y,z)) -> max(z) min(l(x)) -> x min(n(x,y,z)) -> min(y) size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/2 ,c_36/0,c_37/0,c_38/0,c_39/13} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd} + Details: We analyse the complexity of following sub-problems (R) and (S). Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component. Problem (R) - Strict DPs: -#(0(x),0(y)) -> c_10(-#(x,y)) -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)) -#(1(x),0(y)) -> c_12(-#(x,y)) -#(1(x),1(y)) -> c_13(-#(x,y)) - Weak DPs: bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),min#(z),bs#(y),bs#(z)) ge#(#(),0(x)) -> c_20(ge#(#(),x)) ge#(0(x),0(y)) -> c_22(ge#(x,y)) ge#(0(x),1(y)) -> c_23(ge#(y,x)) ge#(1(x),0(y)) -> c_24(ge#(x,y)) ge#(1(x),1(y)) -> c_25(ge#(x,y)) max#(n(x,y,z)) -> c_29(max#(z)) min#(n(x,y,z)) -> c_31(min#(y)) size#(n(x,y,z)) -> c_35(size#(x),size#(y)) wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,-#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,-#(size(z),size(y)) ,size#(z) ,size#(y) ,wb#(y) ,wb#(z)) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) -(x,#()) -> x -(#(),x) -> #() -(0(x),0(y)) -> 0(-(x,y)) -(0(x),1(y)) -> 1(-(-(x,y),1(#()))) -(1(x),0(y)) -> 1(-(x,y)) -(1(x),1(y)) -> 0(-(x,y)) 0(#()) -> #() max(l(x)) -> x max(n(x,y,z)) -> max(z) min(l(x)) -> x min(n(x,y,z)) -> min(y) size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0 ,c_4/1,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0 ,c_20/1,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0 ,c_35/2,c_36/0,c_37/0,c_38/0,c_39/13} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} Problem (S) - Strict DPs: bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),min#(z),bs#(y),bs#(z)) ge#(#(),0(x)) -> c_20(ge#(#(),x)) ge#(0(x),0(y)) -> c_22(ge#(x,y)) ge#(0(x),1(y)) -> c_23(ge#(y,x)) ge#(1(x),0(y)) -> c_24(ge#(x,y)) ge#(1(x),1(y)) -> c_25(ge#(x,y)) max#(n(x,y,z)) -> c_29(max#(z)) min#(n(x,y,z)) -> c_31(min#(y)) size#(n(x,y,z)) -> c_35(size#(x),size#(y)) wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,-#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,-#(size(z),size(y)) ,size#(z) ,size#(y) ,wb#(y) ,wb#(z)) - Weak DPs: -#(0(x),0(y)) -> c_10(-#(x,y)) -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)) -#(1(x),0(y)) -> c_12(-#(x,y)) -#(1(x),1(y)) -> c_13(-#(x,y)) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) -(x,#()) -> x -(#(),x) -> #() -(0(x),0(y)) -> 0(-(x,y)) -(0(x),1(y)) -> 1(-(-(x,y),1(#()))) -(1(x),0(y)) -> 1(-(x,y)) -(1(x),1(y)) -> 0(-(x,y)) 0(#()) -> #() max(l(x)) -> x max(n(x,y,z)) -> max(z) min(l(x)) -> x min(n(x,y,z)) -> min(y) size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0 ,c_4/1,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0 ,c_20/1,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0 ,c_35/2,c_36/0,c_37/0,c_38/0,c_39/13} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} *** Step 7.b:3.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: -#(0(x),0(y)) -> c_10(-#(x,y)) -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)) -#(1(x),0(y)) -> c_12(-#(x,y)) -#(1(x),1(y)) -> c_13(-#(x,y)) - Weak DPs: bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),min#(z),bs#(y),bs#(z)) ge#(#(),0(x)) -> c_20(ge#(#(),x)) ge#(0(x),0(y)) -> c_22(ge#(x,y)) ge#(0(x),1(y)) -> c_23(ge#(y,x)) ge#(1(x),0(y)) -> c_24(ge#(x,y)) ge#(1(x),1(y)) -> c_25(ge#(x,y)) max#(n(x,y,z)) -> c_29(max#(z)) min#(n(x,y,z)) -> c_31(min#(y)) size#(n(x,y,z)) -> c_35(size#(x),size#(y)) wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,-#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,-#(size(z),size(y)) ,size#(z) ,size#(y) ,wb#(y) ,wb#(z)) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) -(x,#()) -> x -(#(),x) -> #() -(0(x),0(y)) -> 0(-(x,y)) -(0(x),1(y)) -> 1(-(-(x,y),1(#()))) -(1(x),0(y)) -> 1(-(x,y)) -(1(x),1(y)) -> 0(-(x,y)) 0(#()) -> #() max(l(x)) -> x max(n(x,y,z)) -> max(z) min(l(x)) -> x min(n(x,y,z)) -> min(y) size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/2 ,c_36/0,c_37/0,c_38/0,c_39/13} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:-#(0(x),0(y)) -> c_10(-#(x,y)) -->_1 -#(1(x),1(y)) -> c_13(-#(x,y)):4 -->_1 -#(1(x),0(y)) -> c_12(-#(x,y)):3 -->_1 -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)):2 -->_1 -#(0(x),0(y)) -> c_10(-#(x,y)):1 2:S:-#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)) -->_2 -#(0(x),0(y)) -> c_10(-#(x,y)):1 -->_2 -#(1(x),1(y)) -> c_13(-#(x,y)):4 -->_1 -#(1(x),1(y)) -> c_13(-#(x,y)):4 -->_2 -#(1(x),0(y)) -> c_12(-#(x,y)):3 -->_2 -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)):2 -->_1 -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)):2 3:S:-#(1(x),0(y)) -> c_12(-#(x,y)) -->_1 -#(0(x),0(y)) -> c_10(-#(x,y)):1 -->_1 -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)):2 -->_1 -#(1(x),1(y)) -> c_13(-#(x,y)):4 -->_1 -#(1(x),0(y)) -> c_12(-#(x,y)):3 4:S:-#(1(x),1(y)) -> c_13(-#(x,y)) -->_1 -#(0(x),0(y)) -> c_10(-#(x,y)):1 -->_1 -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)):2 -->_1 -#(1(x),0(y)) -> c_12(-#(x,y)):3 -->_1 -#(1(x),1(y)) -> c_13(-#(x,y)):4 5:W:bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),min#(z),bs#(y),bs#(z)) -->_4 min#(n(x,y,z)) -> c_31(min#(y)):12 -->_2 max#(n(x,y,z)) -> c_29(max#(z)):11 -->_3 ge#(1(x),1(y)) -> c_25(ge#(x,y)):10 -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):10 -->_3 ge#(1(x),0(y)) -> c_24(ge#(x,y)):9 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):9 -->_3 ge#(0(x),1(y)) -> c_23(ge#(y,x)):8 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):8 -->_3 ge#(0(x),0(y)) -> c_22(ge#(x,y)):7 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):7 -->_3 ge#(#(),0(x)) -> c_20(ge#(#(),x)):6 -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):6 -->_6 bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),min#(z),bs#(y),bs#(z)):5 -->_5 bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),min#(z),bs#(y),bs#(z)):5 6:W:ge#(#(),0(x)) -> c_20(ge#(#(),x)) -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):6 7:W:ge#(0(x),0(y)) -> c_22(ge#(x,y)) -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):6 -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):10 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):9 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):8 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):7 8:W:ge#(0(x),1(y)) -> c_23(ge#(y,x)) -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):6 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):7 -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):10 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):9 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):8 9:W:ge#(1(x),0(y)) -> c_24(ge#(x,y)) -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):6 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):7 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):8 -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):10 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):9 10:W:ge#(1(x),1(y)) -> c_25(ge#(x,y)) -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):6 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):7 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):8 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):9 -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):10 11:W:max#(n(x,y,z)) -> c_29(max#(z)) -->_1 max#(n(x,y,z)) -> c_29(max#(z)):11 12:W:min#(n(x,y,z)) -> c_31(min#(y)) -->_1 min#(n(x,y,z)) -> c_31(min#(y)):12 13:W:size#(n(x,y,z)) -> c_35(size#(x),size#(y)) -->_2 size#(n(x,y,z)) -> c_35(size#(x),size#(y)):13 -->_1 size#(n(x,y,z)) -> c_35(size#(x),size#(y)):13 14:W:wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,-#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,-#(size(z),size(y)) ,size#(z) ,size#(y) ,wb#(y) ,wb#(z)) -->_9 -#(0(x),0(y)) -> c_10(-#(x,y)):1 -->_5 -#(0(x),0(y)) -> c_10(-#(x,y)):1 -->_9 -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)):2 -->_5 -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)):2 -->_9 -#(1(x),0(y)) -> c_12(-#(x,y)):3 -->_5 -#(1(x),0(y)) -> c_12(-#(x,y)):3 -->_9 -#(1(x),1(y)) -> c_13(-#(x,y)):4 -->_5 -#(1(x),1(y)) -> c_13(-#(x,y)):4 -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):6 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):7 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):8 -->_8 ge#(1(x),0(y)) -> c_24(ge#(x,y)):9 -->_4 ge#(1(x),0(y)) -> c_24(ge#(x,y)):9 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):9 -->_8 ge#(1(x),1(y)) -> c_25(ge#(x,y)):10 -->_4 ge#(1(x),1(y)) -> c_25(ge#(x,y)):10 -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):10 -->_11 size#(n(x,y,z)) -> c_35(size#(x),size#(y)):13 -->_10 size#(n(x,y,z)) -> c_35(size#(x),size#(y)):13 -->_7 size#(n(x,y,z)) -> c_35(size#(x),size#(y)):13 -->_6 size#(n(x,y,z)) -> c_35(size#(x),size#(y)):13 -->_3 size#(n(x,y,z)) -> c_35(size#(x),size#(y)):13 -->_2 size#(n(x,y,z)) -> c_35(size#(x),size#(y)):13 -->_13 wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,-#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,-#(size(z),size(y)) ,size#(z) ,size#(y) ,wb#(y) ,wb#(z)):14 -->_12 wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,-#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,-#(size(z),size(y)) ,size#(z) ,size#(y) ,wb#(y) ,wb#(z)):14 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 13: size#(n(x,y,z)) -> c_35(size#(x),size#(y)) 5: bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),min#(z),bs#(y),bs#(z)) 10: ge#(1(x),1(y)) -> c_25(ge#(x,y)) 9: ge#(1(x),0(y)) -> c_24(ge#(x,y)) 8: ge#(0(x),1(y)) -> c_23(ge#(y,x)) 7: ge#(0(x),0(y)) -> c_22(ge#(x,y)) 6: ge#(#(),0(x)) -> c_20(ge#(#(),x)) 11: max#(n(x,y,z)) -> c_29(max#(z)) 12: min#(n(x,y,z)) -> c_31(min#(y)) *** Step 7.b:3.a:2: SimplifyRHS WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: -#(0(x),0(y)) -> c_10(-#(x,y)) -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)) -#(1(x),0(y)) -> c_12(-#(x,y)) -#(1(x),1(y)) -> c_13(-#(x,y)) - Weak DPs: wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,-#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,-#(size(z),size(y)) ,size#(z) ,size#(y) ,wb#(y) ,wb#(z)) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) -(x,#()) -> x -(#(),x) -> #() -(0(x),0(y)) -> 0(-(x,y)) -(0(x),1(y)) -> 1(-(-(x,y),1(#()))) -(1(x),0(y)) -> 1(-(x,y)) -(1(x),1(y)) -> 0(-(x,y)) 0(#()) -> #() max(l(x)) -> x max(n(x,y,z)) -> max(z) min(l(x)) -> x min(n(x,y,z)) -> min(y) size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/2 ,c_36/0,c_37/0,c_38/0,c_39/13} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:-#(0(x),0(y)) -> c_10(-#(x,y)) -->_1 -#(1(x),1(y)) -> c_13(-#(x,y)):4 -->_1 -#(1(x),0(y)) -> c_12(-#(x,y)):3 -->_1 -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)):2 -->_1 -#(0(x),0(y)) -> c_10(-#(x,y)):1 2:S:-#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)) -->_2 -#(0(x),0(y)) -> c_10(-#(x,y)):1 -->_2 -#(1(x),1(y)) -> c_13(-#(x,y)):4 -->_1 -#(1(x),1(y)) -> c_13(-#(x,y)):4 -->_2 -#(1(x),0(y)) -> c_12(-#(x,y)):3 -->_2 -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)):2 -->_1 -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)):2 3:S:-#(1(x),0(y)) -> c_12(-#(x,y)) -->_1 -#(0(x),0(y)) -> c_10(-#(x,y)):1 -->_1 -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)):2 -->_1 -#(1(x),1(y)) -> c_13(-#(x,y)):4 -->_1 -#(1(x),0(y)) -> c_12(-#(x,y)):3 4:S:-#(1(x),1(y)) -> c_13(-#(x,y)) -->_1 -#(0(x),0(y)) -> c_10(-#(x,y)):1 -->_1 -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)):2 -->_1 -#(1(x),0(y)) -> c_12(-#(x,y)):3 -->_1 -#(1(x),1(y)) -> c_13(-#(x,y)):4 14:W:wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,-#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,-#(size(z),size(y)) ,size#(z) ,size#(y) ,wb#(y) ,wb#(z)) -->_9 -#(0(x),0(y)) -> c_10(-#(x,y)):1 -->_5 -#(0(x),0(y)) -> c_10(-#(x,y)):1 -->_9 -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)):2 -->_5 -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)):2 -->_9 -#(1(x),0(y)) -> c_12(-#(x,y)):3 -->_5 -#(1(x),0(y)) -> c_12(-#(x,y)):3 -->_9 -#(1(x),1(y)) -> c_13(-#(x,y)):4 -->_5 -#(1(x),1(y)) -> c_13(-#(x,y)):4 -->_13 wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,-#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,-#(size(z),size(y)) ,size#(z) ,size#(y) ,wb#(y) ,wb#(z)):14 -->_12 wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,-#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,-#(size(z),size(y)) ,size#(z) ,size#(y) ,wb#(y) ,wb#(z)):14 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: wb#(n(x,y,z)) -> c_39(-#(size(y),size(z)),-#(size(z),size(y)),wb#(y),wb#(z)) *** Step 7.b:3.a:3: UsableRules WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: -#(0(x),0(y)) -> c_10(-#(x,y)) -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)) -#(1(x),0(y)) -> c_12(-#(x,y)) -#(1(x),1(y)) -> c_13(-#(x,y)) - Weak DPs: wb#(n(x,y,z)) -> c_39(-#(size(y),size(z)),-#(size(z),size(y)),wb#(y),wb#(z)) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) -(x,#()) -> x -(#(),x) -> #() -(0(x),0(y)) -> 0(-(x,y)) -(0(x),1(y)) -> 1(-(-(x,y),1(#()))) -(1(x),0(y)) -> 1(-(x,y)) -(1(x),1(y)) -> 0(-(x,y)) 0(#()) -> #() max(l(x)) -> x max(n(x,y,z)) -> max(z) min(l(x)) -> x min(n(x,y,z)) -> min(y) size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/2 ,c_36/0,c_37/0,c_38/0,c_39/4} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) -(x,#()) -> x -(#(),x) -> #() -(0(x),0(y)) -> 0(-(x,y)) -(0(x),1(y)) -> 1(-(-(x,y),1(#()))) -(1(x),0(y)) -> 1(-(x,y)) -(1(x),1(y)) -> 0(-(x,y)) 0(#()) -> #() size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) -#(0(x),0(y)) -> c_10(-#(x,y)) -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)) -#(1(x),0(y)) -> c_12(-#(x,y)) -#(1(x),1(y)) -> c_13(-#(x,y)) wb#(n(x,y,z)) -> c_39(-#(size(y),size(z)),-#(size(z),size(y)),wb#(y),wb#(z)) *** Step 7.b:3.a:4: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: -#(0(x),0(y)) -> c_10(-#(x,y)) -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)) -#(1(x),0(y)) -> c_12(-#(x,y)) -#(1(x),1(y)) -> c_13(-#(x,y)) - Weak DPs: wb#(n(x,y,z)) -> c_39(-#(size(y),size(z)),-#(size(z),size(y)),wb#(y),wb#(z)) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) -(x,#()) -> x -(#(),x) -> #() -(0(x),0(y)) -> 0(-(x,y)) -(0(x),1(y)) -> 1(-(-(x,y),1(#()))) -(1(x),0(y)) -> 1(-(x,y)) -(1(x),1(y)) -> 0(-(x,y)) 0(#()) -> #() size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/2 ,c_36/0,c_37/0,c_38/0,c_39/4} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: -#(0(x),0(y)) -> c_10(-#(x,y)) 4: -#(1(x),1(y)) -> c_13(-#(x,y)) The strictly oriented rules are moved into the weak component. **** Step 7.b:3.a:4.a:1: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: -#(0(x),0(y)) -> c_10(-#(x,y)) -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)) -#(1(x),0(y)) -> c_12(-#(x,y)) -#(1(x),1(y)) -> c_13(-#(x,y)) - Weak DPs: wb#(n(x,y,z)) -> c_39(-#(size(y),size(z)),-#(size(z),size(y)),wb#(y),wb#(z)) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) -(x,#()) -> x -(#(),x) -> #() -(0(x),0(y)) -> 0(-(x,y)) -(0(x),1(y)) -> 1(-(-(x,y),1(#()))) -(1(x),0(y)) -> 1(-(x,y)) -(1(x),1(y)) -> 0(-(x,y)) 0(#()) -> #() size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/2 ,c_36/0,c_37/0,c_38/0,c_39/4} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(mixed(2)): The following argument positions are considered usable: uargs(c_10) = {1}, uargs(c_11) = {1,2}, uargs(c_12) = {1}, uargs(c_13) = {1}, uargs(c_39) = {1,2,3,4} Following symbols are considered usable: {+,0,size,+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val#,wb#} TcT has computed the following interpretation: p(#) = 0 p(+) = x1 + x2 p(-) = 0 p(0) = 1 + x1 p(1) = 1 + x1 p(and) = 0 p(bs) = 0 p(false) = 0 p(ge) = 0 p(if) = 0 p(l) = 1 p(max) = 0 p(min) = 0 p(n) = 1 + x1 + x2 + x3 p(not) = 0 p(size) = x1 p(true) = 0 p(val) = 0 p(wb) = 0 p(+#) = 0 p(-#) = x2 p(0#) = 0 p(and#) = 0 p(bs#) = 0 p(ge#) = 0 p(if#) = 0 p(max#) = 0 p(min#) = 0 p(not#) = 0 p(size#) = 0 p(val#) = 0 p(wb#) = 1 + x1^2 p(c_1) = 0 p(c_2) = 0 p(c_3) = 0 p(c_4) = 0 p(c_5) = 0 p(c_6) = 0 p(c_7) = 0 p(c_8) = 0 p(c_9) = 0 p(c_10) = x1 p(c_11) = x1 + x2 p(c_12) = 1 + x1 p(c_13) = x1 p(c_14) = 0 p(c_15) = 0 p(c_16) = 0 p(c_17) = 0 p(c_18) = 0 p(c_19) = 0 p(c_20) = 0 p(c_21) = 0 p(c_22) = 0 p(c_23) = 0 p(c_24) = 0 p(c_25) = 0 p(c_26) = 0 p(c_27) = 0 p(c_28) = 0 p(c_29) = 0 p(c_30) = 0 p(c_31) = 0 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) = x1 + x2 + x3 + x4 Following rules are strictly oriented: -#(0(x),0(y)) = 1 + y > y = c_10(-#(x,y)) -#(1(x),1(y)) = 1 + y > y = c_13(-#(x,y)) Following rules are (at-least) weakly oriented: -#(0(x),1(y)) = 1 + y >= 1 + y = c_11(-#(-(x,y),1(#())),-#(x,y)) -#(1(x),0(y)) = 1 + y >= 1 + y = c_12(-#(x,y)) wb#(n(x,y,z)) = 2 + 2*x + 2*x*y + 2*x*z + x^2 + 2*y + 2*y*z + y^2 + 2*z + z^2 >= 2 + y + y^2 + z + z^2 = c_39(-#(size(y),size(z)),-#(size(z),size(y)),wb#(y),wb#(z)) +(x,#()) = x >= x = x +(x,+(y,z)) = x + y + z >= x + y + z = +(+(x,y),z) +(#(),x) = x >= x = x +(0(x),0(y)) = 2 + x + y >= 1 + x + y = 0(+(x,y)) +(0(x),1(y)) = 2 + x + y >= 1 + x + y = 1(+(x,y)) +(1(x),0(y)) = 2 + x + y >= 1 + x + y = 1(+(x,y)) +(1(x),1(y)) = 2 + x + y >= 2 + x + y = 0(+(+(x,y),1(#()))) 0(#()) = 1 >= 0 = #() size(l(x)) = 1 >= 1 = 1(#()) size(n(x,y,z)) = 1 + x + y + z >= 1 + x + y = +(+(size(x),size(y)),1(#())) **** Step 7.b:3.a:4.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)) -#(1(x),0(y)) -> c_12(-#(x,y)) - Weak DPs: -#(0(x),0(y)) -> c_10(-#(x,y)) -#(1(x),1(y)) -> c_13(-#(x,y)) wb#(n(x,y,z)) -> c_39(-#(size(y),size(z)),-#(size(z),size(y)),wb#(y),wb#(z)) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) -(x,#()) -> x -(#(),x) -> #() -(0(x),0(y)) -> 0(-(x,y)) -(0(x),1(y)) -> 1(-(-(x,y),1(#()))) -(1(x),0(y)) -> 1(-(x,y)) -(1(x),1(y)) -> 0(-(x,y)) 0(#()) -> #() size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/2 ,c_36/0,c_37/0,c_38/0,c_39/4} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () **** Step 7.b:3.a:4.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)) -#(1(x),0(y)) -> c_12(-#(x,y)) - Weak DPs: -#(0(x),0(y)) -> c_10(-#(x,y)) -#(1(x),1(y)) -> c_13(-#(x,y)) wb#(n(x,y,z)) -> c_39(-#(size(y),size(z)),-#(size(z),size(y)),wb#(y),wb#(z)) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) -(x,#()) -> x -(#(),x) -> #() -(0(x),0(y)) -> 0(-(x,y)) -(0(x),1(y)) -> 1(-(-(x,y),1(#()))) -(1(x),0(y)) -> 1(-(x,y)) -(1(x),1(y)) -> 0(-(x,y)) 0(#()) -> #() size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/2 ,c_36/0,c_37/0,c_38/0,c_39/4} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 2: -#(1(x),0(y)) -> c_12(-#(x,y)) The strictly oriented rules are moved into the weak component. ***** Step 7.b:3.a:4.b:1.a:1: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)) -#(1(x),0(y)) -> c_12(-#(x,y)) - Weak DPs: -#(0(x),0(y)) -> c_10(-#(x,y)) -#(1(x),1(y)) -> c_13(-#(x,y)) wb#(n(x,y,z)) -> c_39(-#(size(y),size(z)),-#(size(z),size(y)),wb#(y),wb#(z)) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) -(x,#()) -> x -(#(),x) -> #() -(0(x),0(y)) -> 0(-(x,y)) -(0(x),1(y)) -> 1(-(-(x,y),1(#()))) -(1(x),0(y)) -> 1(-(x,y)) -(1(x),1(y)) -> 0(-(x,y)) 0(#()) -> #() size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/2 ,c_36/0,c_37/0,c_38/0,c_39/4} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_10) = {1}, uargs(c_11) = {1,2}, uargs(c_12) = {1}, uargs(c_13) = {1}, uargs(c_39) = {1,2,3,4} Following symbols are considered usable: {+,-,0,size,+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val#,wb#} TcT has computed the following interpretation: p(#) = [0] [0] p(+) = [1 0] x1 + [1 0] x2 + [1] [0 1] [0 1] [0] p(-) = [1 2] x1 + [3] [0 1] [0] p(0) = [0 0] x1 + [0] [0 1] [1] p(1) = [0 0] x1 + [0] [0 1] [1] p(and) = [1 0] x1 + [2 1] x2 + [2] [2 1] [2 1] [0] p(bs) = [0] [2] p(false) = [2] [0] p(ge) = [0 0] x2 + [0] [0 2] [0] p(if) = [0 0] x1 + [2] [2 0] [0] p(l) = [1 1] x1 + [1] [0 0] [1] p(max) = [0] [1] p(min) = [1] [1] p(n) = [1 0] x1 + [1 2] x2 + [1 3] x3 + [2] [0 1] [0 1] [0 1] [1] p(not) = [0] [1] p(size) = [2 0] x1 + [1] [0 1] [0] p(true) = [0] [0] p(val) = [2 0] x1 + [0] [0 2] [1] p(wb) = [0 0] x1 + [1] [1 0] [1] p(+#) = [0 1] x2 + [2] [0 1] [0] p(-#) = [0 0] x1 + [0 2] x2 + [0] [1 3] [0 1] [0] p(0#) = [0 2] x1 + [2] [0 1] [0] p(and#) = [0 0] x1 + [0] [0 2] [2] p(bs#) = [2] [0] p(ge#) = [2] [2] p(if#) = [0 0] x3 + [0] [2 1] [1] p(max#) = [0 0] x1 + [2] [2 0] [1] p(min#) = [0 0] x1 + [0] [0 2] [2] p(not#) = [0 0] x1 + [2] [2 0] [0] p(size#) = [1 0] x1 + [2] [2 2] [0] p(val#) = [0] [0] p(wb#) = [1 2] x1 + [1] [2 0] [0] p(c_1) = [2] [1] p(c_2) = [1 0] x2 + [0] [1 0] [0] p(c_3) = [0] [0] p(c_4) = [0] [0] p(c_5) = [1 2] x1 + [2] [1 1] [0] p(c_6) = [1 0] x1 + [2] [0 0] [0] p(c_7) = [1 2] x1 + [2] [0 0] [2] p(c_8) = [1] [0] p(c_9) = [1] [0] p(c_10) = [1 0] x1 + [0] [0 0] [0] p(c_11) = [1 0] x1 + [1 0] x2 + [0] [2 0] [0 0] [0] p(c_12) = [1 0] x1 + [0] [0 0] [0] p(c_13) = [1 0] x1 + [2] [0 0] [3] p(c_14) = [2] [0] p(c_15) = [1] [1] p(c_16) = [0] [1] p(c_17) = [2] [2] p(c_18) = [0 1] x3 + [1 0] x4 + [0 0] x6 + [2] [0 0] [2 0] [1 2] [1] p(c_19) = [0] [0] p(c_20) = [2] [0] p(c_21) = [0] [1] p(c_22) = [0] [0] p(c_23) = [2] [0] p(c_24) = [2 0] x1 + [0] [2 0] [2] p(c_25) = [2] [1] p(c_26) = [0] [1] p(c_27) = [1] [1] p(c_28) = [0] [0] p(c_29) = [0] [0] p(c_30) = [1] [2] p(c_31) = [0 2] x1 + [0] [0 0] [0] p(c_32) = [0] [0] p(c_33) = [0] [2] p(c_34) = [0] [1] p(c_35) = [2 0] x2 + [0] [1 0] [0] p(c_36) = [0] [0] p(c_37) = [0] [0] p(c_38) = [0] [2] p(c_39) = [1 0] x1 + [1 0] x2 + [1 0] x3 + [1 0] x4 + [2] [2 1] [0 0] [0 0] [0 1] [3] Following rules are strictly oriented: -#(1(x),0(y)) = [0 0] x + [0 2] y + [2] [0 3] [0 1] [4] > [0 2] y + [0] [0 0] [0] = c_12(-#(x,y)) Following rules are (at-least) weakly oriented: -#(0(x),0(y)) = [0 0] x + [0 2] y + [2] [0 3] [0 1] [4] >= [0 2] y + [0] [0 0] [0] = c_10(-#(x,y)) -#(0(x),1(y)) = [0 0] x + [0 2] y + [2] [0 3] [0 1] [4] >= [0 2] y + [2] [0 0] [4] = c_11(-#(-(x,y),1(#())),-#(x,y)) -#(1(x),1(y)) = [0 0] x + [0 2] y + [2] [0 3] [0 1] [4] >= [0 2] y + [2] [0 0] [3] = c_13(-#(x,y)) wb#(n(x,y,z)) = [1 2] x + [1 4] y + [1 5] z + [5] [2 0] [2 4] [2 6] [4] >= [1 4] y + [1 4] z + [4] [2 3] [2 5] [4] = c_39(-#(size(y),size(z)),-#(size(z),size(y)),wb#(y),wb#(z)) +(x,#()) = [1 0] x + [1] [0 1] [0] >= [1 0] x + [0] [0 1] [0] = x +(x,+(y,z)) = [1 0] x + [1 0] y + [1 0] z + [2] [0 1] [0 1] [0 1] [0] >= [1 0] x + [1 0] y + [1 0] z + [2] [0 1] [0 1] [0 1] [0] = +(+(x,y),z) +(#(),x) = [1 0] x + [1] [0 1] [0] >= [1 0] x + [0] [0 1] [0] = x +(0(x),0(y)) = [0 0] x + [0 0] y + [1] [0 1] [0 1] [2] >= [0 0] x + [0 0] y + [0] [0 1] [0 1] [1] = 0(+(x,y)) +(0(x),1(y)) = [0 0] x + [0 0] y + [1] [0 1] [0 1] [2] >= [0 0] x + [0 0] y + [0] [0 1] [0 1] [1] = 1(+(x,y)) +(1(x),0(y)) = [0 0] x + [0 0] y + [1] [0 1] [0 1] [2] >= [0 0] x + [0 0] y + [0] [0 1] [0 1] [1] = 1(+(x,y)) +(1(x),1(y)) = [0 0] x + [0 0] y + [1] [0 1] [0 1] [2] >= [0 0] x + [0 0] y + [0] [0 1] [0 1] [2] = 0(+(+(x,y),1(#()))) -(x,#()) = [1 2] x + [3] [0 1] [0] >= [1 0] x + [0] [0 1] [0] = x -(#(),x) = [3] [0] >= [0] [0] = #() -(0(x),0(y)) = [0 2] x + [5] [0 1] [1] >= [0 0] x + [0] [0 1] [1] = 0(-(x,y)) -(0(x),1(y)) = [0 2] x + [5] [0 1] [1] >= [0 0] x + [0] [0 1] [1] = 1(-(-(x,y),1(#()))) -(1(x),0(y)) = [0 2] x + [5] [0 1] [1] >= [0 0] x + [0] [0 1] [1] = 1(-(x,y)) -(1(x),1(y)) = [0 2] x + [5] [0 1] [1] >= [0 0] x + [0] [0 1] [1] = 0(-(x,y)) 0(#()) = [0] [1] >= [0] [0] = #() size(l(x)) = [2 2] x + [3] [0 0] [1] >= [0] [1] = 1(#()) size(n(x,y,z)) = [2 0] x + [2 4] y + [2 6] z + [5] [0 1] [0 1] [0 1] [1] >= [2 0] x + [2 0] y + [4] [0 1] [0 1] [1] = +(+(size(x),size(y)),1(#())) ***** Step 7.b:3.a:4.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)) - Weak DPs: -#(0(x),0(y)) -> c_10(-#(x,y)) -#(1(x),0(y)) -> c_12(-#(x,y)) -#(1(x),1(y)) -> c_13(-#(x,y)) wb#(n(x,y,z)) -> c_39(-#(size(y),size(z)),-#(size(z),size(y)),wb#(y),wb#(z)) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) -(x,#()) -> x -(#(),x) -> #() -(0(x),0(y)) -> 0(-(x,y)) -(0(x),1(y)) -> 1(-(-(x,y),1(#()))) -(1(x),0(y)) -> 1(-(x,y)) -(1(x),1(y)) -> 0(-(x,y)) 0(#()) -> #() size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/2 ,c_36/0,c_37/0,c_38/0,c_39/4} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ***** Step 7.b:3.a:4.b:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)) - Weak DPs: -#(0(x),0(y)) -> c_10(-#(x,y)) -#(1(x),0(y)) -> c_12(-#(x,y)) -#(1(x),1(y)) -> c_13(-#(x,y)) wb#(n(x,y,z)) -> c_39(-#(size(y),size(z)),-#(size(z),size(y)),wb#(y),wb#(z)) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) -(x,#()) -> x -(#(),x) -> #() -(0(x),0(y)) -> 0(-(x,y)) -(0(x),1(y)) -> 1(-(-(x,y),1(#()))) -(1(x),0(y)) -> 1(-(x,y)) -(1(x),1(y)) -> 0(-(x,y)) 0(#()) -> #() size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/2 ,c_36/0,c_37/0,c_38/0,c_39/4} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)) The strictly oriented rules are moved into the weak component. ****** Step 7.b:3.a:4.b:1.b:1.a:1: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)) - Weak DPs: -#(0(x),0(y)) -> c_10(-#(x,y)) -#(1(x),0(y)) -> c_12(-#(x,y)) -#(1(x),1(y)) -> c_13(-#(x,y)) wb#(n(x,y,z)) -> c_39(-#(size(y),size(z)),-#(size(z),size(y)),wb#(y),wb#(z)) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) -(x,#()) -> x -(#(),x) -> #() -(0(x),0(y)) -> 0(-(x,y)) -(0(x),1(y)) -> 1(-(-(x,y),1(#()))) -(1(x),0(y)) -> 1(-(x,y)) -(1(x),1(y)) -> 0(-(x,y)) 0(#()) -> #() size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/2 ,c_36/0,c_37/0,c_38/0,c_39/4} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(mixed(2)): The following argument positions are considered usable: uargs(c_10) = {1}, uargs(c_11) = {1,2}, uargs(c_12) = {1}, uargs(c_13) = {1}, uargs(c_39) = {1,2,3,4} Following symbols are considered usable: {+,-,0,size,+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val#,wb#} TcT has computed the following interpretation: p(#) = 0 p(+) = x1 + x2 p(-) = x1 p(0) = 1 + x1 p(1) = 1 + x1 p(and) = 0 p(bs) = 0 p(false) = 0 p(ge) = 0 p(if) = 0 p(l) = 1 p(max) = 0 p(min) = 0 p(n) = 1 + x1 + x2 + x3 p(not) = 0 p(size) = x1 p(true) = 0 p(val) = 0 p(wb) = 0 p(+#) = 0 p(-#) = x1*x2 p(0#) = 0 p(and#) = 0 p(bs#) = 0 p(ge#) = 0 p(if#) = 0 p(max#) = 0 p(min#) = 0 p(not#) = 0 p(size#) = 0 p(val#) = 0 p(wb#) = x1^2 p(c_1) = 0 p(c_2) = 0 p(c_3) = 0 p(c_4) = 0 p(c_5) = 0 p(c_6) = 0 p(c_7) = 0 p(c_8) = 0 p(c_9) = 0 p(c_10) = 1 + x1 p(c_11) = x1 + x2 p(c_12) = x1 p(c_13) = x1 p(c_14) = 0 p(c_15) = 0 p(c_16) = 0 p(c_17) = 0 p(c_18) = 0 p(c_19) = 0 p(c_20) = 0 p(c_21) = 0 p(c_22) = 0 p(c_23) = 0 p(c_24) = 0 p(c_25) = 0 p(c_26) = 0 p(c_27) = 0 p(c_28) = 0 p(c_29) = 0 p(c_30) = 0 p(c_31) = 0 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) = 1 + x1 + x2 + x3 + x4 Following rules are strictly oriented: -#(0(x),1(y)) = 1 + x + x*y + y > x + x*y = c_11(-#(-(x,y),1(#())),-#(x,y)) Following rules are (at-least) weakly oriented: -#(0(x),0(y)) = 1 + x + x*y + y >= 1 + x*y = c_10(-#(x,y)) -#(1(x),0(y)) = 1 + x + x*y + y >= x*y = c_12(-#(x,y)) -#(1(x),1(y)) = 1 + x + x*y + y >= x*y = c_13(-#(x,y)) wb#(n(x,y,z)) = 1 + 2*x + 2*x*y + 2*x*z + x^2 + 2*y + 2*y*z + y^2 + 2*z + z^2 >= 1 + 2*y*z + y^2 + z^2 = c_39(-#(size(y),size(z)),-#(size(z),size(y)),wb#(y),wb#(z)) +(x,#()) = x >= x = x +(x,+(y,z)) = x + y + z >= x + y + z = +(+(x,y),z) +(#(),x) = x >= x = x +(0(x),0(y)) = 2 + x + y >= 1 + x + y = 0(+(x,y)) +(0(x),1(y)) = 2 + x + y >= 1 + x + y = 1(+(x,y)) +(1(x),0(y)) = 2 + x + y >= 1 + x + y = 1(+(x,y)) +(1(x),1(y)) = 2 + x + y >= 2 + x + y = 0(+(+(x,y),1(#()))) -(x,#()) = x >= x = x -(#(),x) = 0 >= 0 = #() -(0(x),0(y)) = 1 + x >= 1 + x = 0(-(x,y)) -(0(x),1(y)) = 1 + x >= 1 + x = 1(-(-(x,y),1(#()))) -(1(x),0(y)) = 1 + x >= 1 + x = 1(-(x,y)) -(1(x),1(y)) = 1 + x >= 1 + x = 0(-(x,y)) 0(#()) = 1 >= 0 = #() size(l(x)) = 1 >= 1 = 1(#()) size(n(x,y,z)) = 1 + x + y + z >= 1 + x + y = +(+(size(x),size(y)),1(#())) ****** Step 7.b:3.a:4.b:1.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: -#(0(x),0(y)) -> c_10(-#(x,y)) -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)) -#(1(x),0(y)) -> c_12(-#(x,y)) -#(1(x),1(y)) -> c_13(-#(x,y)) wb#(n(x,y,z)) -> c_39(-#(size(y),size(z)),-#(size(z),size(y)),wb#(y),wb#(z)) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) -(x,#()) -> x -(#(),x) -> #() -(0(x),0(y)) -> 0(-(x,y)) -(0(x),1(y)) -> 1(-(-(x,y),1(#()))) -(1(x),0(y)) -> 1(-(x,y)) -(1(x),1(y)) -> 0(-(x,y)) 0(#()) -> #() size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/2 ,c_36/0,c_37/0,c_38/0,c_39/4} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ****** Step 7.b:3.a:4.b:1.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: -#(0(x),0(y)) -> c_10(-#(x,y)) -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)) -#(1(x),0(y)) -> c_12(-#(x,y)) -#(1(x),1(y)) -> c_13(-#(x,y)) wb#(n(x,y,z)) -> c_39(-#(size(y),size(z)),-#(size(z),size(y)),wb#(y),wb#(z)) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) -(x,#()) -> x -(#(),x) -> #() -(0(x),0(y)) -> 0(-(x,y)) -(0(x),1(y)) -> 1(-(-(x,y),1(#()))) -(1(x),0(y)) -> 1(-(x,y)) -(1(x),1(y)) -> 0(-(x,y)) 0(#()) -> #() size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/2 ,c_36/0,c_37/0,c_38/0,c_39/4} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:-#(0(x),0(y)) -> c_10(-#(x,y)) -->_1 -#(1(x),1(y)) -> c_13(-#(x,y)):4 -->_1 -#(1(x),0(y)) -> c_12(-#(x,y)):3 -->_1 -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)):2 -->_1 -#(0(x),0(y)) -> c_10(-#(x,y)):1 2:W:-#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)) -->_2 -#(1(x),1(y)) -> c_13(-#(x,y)):4 -->_1 -#(1(x),1(y)) -> c_13(-#(x,y)):4 -->_2 -#(1(x),0(y)) -> c_12(-#(x,y)):3 -->_2 -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)):2 -->_1 -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)):2 -->_2 -#(0(x),0(y)) -> c_10(-#(x,y)):1 3:W:-#(1(x),0(y)) -> c_12(-#(x,y)) -->_1 -#(1(x),1(y)) -> c_13(-#(x,y)):4 -->_1 -#(1(x),0(y)) -> c_12(-#(x,y)):3 -->_1 -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)):2 -->_1 -#(0(x),0(y)) -> c_10(-#(x,y)):1 4:W:-#(1(x),1(y)) -> c_13(-#(x,y)) -->_1 -#(1(x),1(y)) -> c_13(-#(x,y)):4 -->_1 -#(1(x),0(y)) -> c_12(-#(x,y)):3 -->_1 -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)):2 -->_1 -#(0(x),0(y)) -> c_10(-#(x,y)):1 5:W:wb#(n(x,y,z)) -> c_39(-#(size(y),size(z)),-#(size(z),size(y)),wb#(y),wb#(z)) -->_4 wb#(n(x,y,z)) -> c_39(-#(size(y),size(z)),-#(size(z),size(y)),wb#(y),wb#(z)):5 -->_3 wb#(n(x,y,z)) -> c_39(-#(size(y),size(z)),-#(size(z),size(y)),wb#(y),wb#(z)):5 -->_2 -#(1(x),1(y)) -> c_13(-#(x,y)):4 -->_1 -#(1(x),1(y)) -> c_13(-#(x,y)):4 -->_2 -#(1(x),0(y)) -> c_12(-#(x,y)):3 -->_1 -#(1(x),0(y)) -> c_12(-#(x,y)):3 -->_2 -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)):2 -->_1 -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)):2 -->_2 -#(0(x),0(y)) -> c_10(-#(x,y)):1 -->_1 -#(0(x),0(y)) -> c_10(-#(x,y)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 5: wb#(n(x,y,z)) -> c_39(-#(size(y),size(z)),-#(size(z),size(y)),wb#(y),wb#(z)) 1: -#(0(x),0(y)) -> c_10(-#(x,y)) 4: -#(1(x),1(y)) -> c_13(-#(x,y)) 3: -#(1(x),0(y)) -> c_12(-#(x,y)) 2: -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)) ****** Step 7.b:3.a:4.b:1.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) -(x,#()) -> x -(#(),x) -> #() -(0(x),0(y)) -> 0(-(x,y)) -(0(x),1(y)) -> 1(-(-(x,y),1(#()))) -(1(x),0(y)) -> 1(-(x,y)) -(1(x),1(y)) -> 0(-(x,y)) 0(#()) -> #() size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/2 ,c_36/0,c_37/0,c_38/0,c_39/4} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). *** Step 7.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),min#(z),bs#(y),bs#(z)) ge#(#(),0(x)) -> c_20(ge#(#(),x)) ge#(0(x),0(y)) -> c_22(ge#(x,y)) ge#(0(x),1(y)) -> c_23(ge#(y,x)) ge#(1(x),0(y)) -> c_24(ge#(x,y)) ge#(1(x),1(y)) -> c_25(ge#(x,y)) max#(n(x,y,z)) -> c_29(max#(z)) min#(n(x,y,z)) -> c_31(min#(y)) size#(n(x,y,z)) -> c_35(size#(x),size#(y)) wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,-#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,-#(size(z),size(y)) ,size#(z) ,size#(y) ,wb#(y) ,wb#(z)) - Weak DPs: -#(0(x),0(y)) -> c_10(-#(x,y)) -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)) -#(1(x),0(y)) -> c_12(-#(x,y)) -#(1(x),1(y)) -> c_13(-#(x,y)) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) -(x,#()) -> x -(#(),x) -> #() -(0(x),0(y)) -> 0(-(x,y)) -(0(x),1(y)) -> 1(-(-(x,y),1(#()))) -(1(x),0(y)) -> 1(-(x,y)) -(1(x),1(y)) -> 0(-(x,y)) 0(#()) -> #() max(l(x)) -> x max(n(x,y,z)) -> max(z) min(l(x)) -> x min(n(x,y,z)) -> min(y) size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/2 ,c_36/0,c_37/0,c_38/0,c_39/13} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),min#(z),bs#(y),bs#(z)) -->_4 min#(n(x,y,z)) -> c_31(min#(y)):8 -->_2 max#(n(x,y,z)) -> c_29(max#(z)):7 -->_3 ge#(1(x),1(y)) -> c_25(ge#(x,y)):6 -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):6 -->_3 ge#(1(x),0(y)) -> c_24(ge#(x,y)):5 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):5 -->_3 ge#(0(x),1(y)) -> c_23(ge#(y,x)):4 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):4 -->_3 ge#(0(x),0(y)) -> c_22(ge#(x,y)):3 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):3 -->_3 ge#(#(),0(x)) -> c_20(ge#(#(),x)):2 -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):2 -->_6 bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),min#(z),bs#(y),bs#(z)):1 -->_5 bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),min#(z),bs#(y),bs#(z)):1 2:S:ge#(#(),0(x)) -> c_20(ge#(#(),x)) -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):2 3:S:ge#(0(x),0(y)) -> c_22(ge#(x,y)) -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):6 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):5 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):4 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):3 -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):2 4:S:ge#(0(x),1(y)) -> c_23(ge#(y,x)) -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):6 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):5 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):4 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):3 -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):2 5:S:ge#(1(x),0(y)) -> c_24(ge#(x,y)) -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):6 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):5 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):4 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):3 -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):2 6:S:ge#(1(x),1(y)) -> c_25(ge#(x,y)) -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):6 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):5 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):4 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):3 -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):2 7:S:max#(n(x,y,z)) -> c_29(max#(z)) -->_1 max#(n(x,y,z)) -> c_29(max#(z)):7 8:S:min#(n(x,y,z)) -> c_31(min#(y)) -->_1 min#(n(x,y,z)) -> c_31(min#(y)):8 9:S:size#(n(x,y,z)) -> c_35(size#(x),size#(y)) -->_2 size#(n(x,y,z)) -> c_35(size#(x),size#(y)):9 -->_1 size#(n(x,y,z)) -> c_35(size#(x),size#(y)):9 10:S:wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,-#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,-#(size(z),size(y)) ,size#(z) ,size#(y) ,wb#(y) ,wb#(z)) -->_9 -#(1(x),1(y)) -> c_13(-#(x,y)):14 -->_5 -#(1(x),1(y)) -> c_13(-#(x,y)):14 -->_9 -#(1(x),0(y)) -> c_12(-#(x,y)):13 -->_5 -#(1(x),0(y)) -> c_12(-#(x,y)):13 -->_9 -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)):12 -->_5 -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)):12 -->_9 -#(0(x),0(y)) -> c_10(-#(x,y)):11 -->_5 -#(0(x),0(y)) -> c_10(-#(x,y)):11 -->_13 wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,-#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,-#(size(z),size(y)) ,size#(z) ,size#(y) ,wb#(y) ,wb#(z)):10 -->_12 wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,-#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,-#(size(z),size(y)) ,size#(z) ,size#(y) ,wb#(y) ,wb#(z)):10 -->_11 size#(n(x,y,z)) -> c_35(size#(x),size#(y)):9 -->_10 size#(n(x,y,z)) -> c_35(size#(x),size#(y)):9 -->_7 size#(n(x,y,z)) -> c_35(size#(x),size#(y)):9 -->_6 size#(n(x,y,z)) -> c_35(size#(x),size#(y)):9 -->_3 size#(n(x,y,z)) -> c_35(size#(x),size#(y)):9 -->_2 size#(n(x,y,z)) -> c_35(size#(x),size#(y)):9 -->_8 ge#(1(x),1(y)) -> c_25(ge#(x,y)):6 -->_4 ge#(1(x),1(y)) -> c_25(ge#(x,y)):6 -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):6 -->_8 ge#(1(x),0(y)) -> c_24(ge#(x,y)):5 -->_4 ge#(1(x),0(y)) -> c_24(ge#(x,y)):5 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):5 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):4 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):3 -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):2 11:W:-#(0(x),0(y)) -> c_10(-#(x,y)) -->_1 -#(1(x),1(y)) -> c_13(-#(x,y)):14 -->_1 -#(1(x),0(y)) -> c_12(-#(x,y)):13 -->_1 -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)):12 -->_1 -#(0(x),0(y)) -> c_10(-#(x,y)):11 12:W:-#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)) -->_2 -#(1(x),1(y)) -> c_13(-#(x,y)):14 -->_1 -#(1(x),1(y)) -> c_13(-#(x,y)):14 -->_2 -#(1(x),0(y)) -> c_12(-#(x,y)):13 -->_2 -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)):12 -->_1 -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)):12 -->_2 -#(0(x),0(y)) -> c_10(-#(x,y)):11 13:W:-#(1(x),0(y)) -> c_12(-#(x,y)) -->_1 -#(1(x),1(y)) -> c_13(-#(x,y)):14 -->_1 -#(1(x),0(y)) -> c_12(-#(x,y)):13 -->_1 -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)):12 -->_1 -#(0(x),0(y)) -> c_10(-#(x,y)):11 14:W:-#(1(x),1(y)) -> c_13(-#(x,y)) -->_1 -#(1(x),1(y)) -> c_13(-#(x,y)):14 -->_1 -#(1(x),0(y)) -> c_12(-#(x,y)):13 -->_1 -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)):12 -->_1 -#(0(x),0(y)) -> c_10(-#(x,y)):11 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 14: -#(1(x),1(y)) -> c_13(-#(x,y)) 13: -#(1(x),0(y)) -> c_12(-#(x,y)) 12: -#(0(x),1(y)) -> c_11(-#(-(x,y),1(#())),-#(x,y)) 11: -#(0(x),0(y)) -> c_10(-#(x,y)) *** Step 7.b:3.b:2: SimplifyRHS WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),min#(z),bs#(y),bs#(z)) ge#(#(),0(x)) -> c_20(ge#(#(),x)) ge#(0(x),0(y)) -> c_22(ge#(x,y)) ge#(0(x),1(y)) -> c_23(ge#(y,x)) ge#(1(x),0(y)) -> c_24(ge#(x,y)) ge#(1(x),1(y)) -> c_25(ge#(x,y)) max#(n(x,y,z)) -> c_29(max#(z)) min#(n(x,y,z)) -> c_31(min#(y)) size#(n(x,y,z)) -> c_35(size#(x),size#(y)) wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,-#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,-#(size(z),size(y)) ,size#(z) ,size#(y) ,wb#(y) ,wb#(z)) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) -(x,#()) -> x -(#(),x) -> #() -(0(x),0(y)) -> 0(-(x,y)) -(0(x),1(y)) -> 1(-(-(x,y),1(#()))) -(1(x),0(y)) -> 1(-(x,y)) -(1(x),1(y)) -> 0(-(x,y)) 0(#()) -> #() max(l(x)) -> x max(n(x,y,z)) -> max(z) min(l(x)) -> x min(n(x,y,z)) -> min(y) size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/2 ,c_36/0,c_37/0,c_38/0,c_39/13} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),min#(z),bs#(y),bs#(z)) -->_4 min#(n(x,y,z)) -> c_31(min#(y)):8 -->_2 max#(n(x,y,z)) -> c_29(max#(z)):7 -->_3 ge#(1(x),1(y)) -> c_25(ge#(x,y)):6 -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):6 -->_3 ge#(1(x),0(y)) -> c_24(ge#(x,y)):5 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):5 -->_3 ge#(0(x),1(y)) -> c_23(ge#(y,x)):4 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):4 -->_3 ge#(0(x),0(y)) -> c_22(ge#(x,y)):3 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):3 -->_3 ge#(#(),0(x)) -> c_20(ge#(#(),x)):2 -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):2 -->_6 bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),min#(z),bs#(y),bs#(z)):1 -->_5 bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),min#(z),bs#(y),bs#(z)):1 2:S:ge#(#(),0(x)) -> c_20(ge#(#(),x)) -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):2 3:S:ge#(0(x),0(y)) -> c_22(ge#(x,y)) -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):6 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):5 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):4 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):3 -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):2 4:S:ge#(0(x),1(y)) -> c_23(ge#(y,x)) -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):6 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):5 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):4 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):3 -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):2 5:S:ge#(1(x),0(y)) -> c_24(ge#(x,y)) -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):6 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):5 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):4 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):3 -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):2 6:S:ge#(1(x),1(y)) -> c_25(ge#(x,y)) -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):6 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):5 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):4 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):3 -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):2 7:S:max#(n(x,y,z)) -> c_29(max#(z)) -->_1 max#(n(x,y,z)) -> c_29(max#(z)):7 8:S:min#(n(x,y,z)) -> c_31(min#(y)) -->_1 min#(n(x,y,z)) -> c_31(min#(y)):8 9:S:size#(n(x,y,z)) -> c_35(size#(x),size#(y)) -->_2 size#(n(x,y,z)) -> c_35(size#(x),size#(y)):9 -->_1 size#(n(x,y,z)) -> c_35(size#(x),size#(y)):9 10:S:wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,-#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,-#(size(z),size(y)) ,size#(z) ,size#(y) ,wb#(y) ,wb#(z)) -->_13 wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,-#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,-#(size(z),size(y)) ,size#(z) ,size#(y) ,wb#(y) ,wb#(z)):10 -->_12 wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,-#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,-#(size(z),size(y)) ,size#(z) ,size#(y) ,wb#(y) ,wb#(z)):10 -->_11 size#(n(x,y,z)) -> c_35(size#(x),size#(y)):9 -->_10 size#(n(x,y,z)) -> c_35(size#(x),size#(y)):9 -->_7 size#(n(x,y,z)) -> c_35(size#(x),size#(y)):9 -->_6 size#(n(x,y,z)) -> c_35(size#(x),size#(y)):9 -->_3 size#(n(x,y,z)) -> c_35(size#(x),size#(y)):9 -->_2 size#(n(x,y,z)) -> c_35(size#(x),size#(y)):9 -->_8 ge#(1(x),1(y)) -> c_25(ge#(x,y)):6 -->_4 ge#(1(x),1(y)) -> c_25(ge#(x,y)):6 -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):6 -->_8 ge#(1(x),0(y)) -> c_24(ge#(x,y)):5 -->_4 ge#(1(x),0(y)) -> c_24(ge#(x,y)):5 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):5 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):4 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):3 -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):2 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,size#(z) ,size#(y) ,wb#(y) ,wb#(z)) *** Step 7.b:3.b:3: Decompose WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),min#(z),bs#(y),bs#(z)) ge#(#(),0(x)) -> c_20(ge#(#(),x)) ge#(0(x),0(y)) -> c_22(ge#(x,y)) ge#(0(x),1(y)) -> c_23(ge#(y,x)) ge#(1(x),0(y)) -> c_24(ge#(x,y)) ge#(1(x),1(y)) -> c_25(ge#(x,y)) max#(n(x,y,z)) -> c_29(max#(z)) min#(n(x,y,z)) -> c_31(min#(y)) size#(n(x,y,z)) -> c_35(size#(x),size#(y)) wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,size#(z) ,size#(y) ,wb#(y) ,wb#(z)) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) -(x,#()) -> x -(#(),x) -> #() -(0(x),0(y)) -> 0(-(x,y)) -(0(x),1(y)) -> 1(-(-(x,y),1(#()))) -(1(x),0(y)) -> 1(-(x,y)) -(1(x),1(y)) -> 0(-(x,y)) 0(#()) -> #() max(l(x)) -> x max(n(x,y,z)) -> max(z) min(l(x)) -> x min(n(x,y,z)) -> min(y) size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/2 ,c_36/0,c_37/0,c_38/0,c_39/11} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd} + Details: We analyse the complexity of following sub-problems (R) and (S). Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component. Problem (R) - Strict DPs: bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),min#(z),bs#(y),bs#(z)) ge#(#(),0(x)) -> c_20(ge#(#(),x)) ge#(0(x),0(y)) -> c_22(ge#(x,y)) ge#(0(x),1(y)) -> c_23(ge#(y,x)) ge#(1(x),0(y)) -> c_24(ge#(x,y)) ge#(1(x),1(y)) -> c_25(ge#(x,y)) max#(n(x,y,z)) -> c_29(max#(z)) min#(n(x,y,z)) -> c_31(min#(y)) - Weak DPs: size#(n(x,y,z)) -> c_35(size#(x),size#(y)) wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,size#(z) ,size#(y) ,wb#(y) ,wb#(z)) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) -(x,#()) -> x -(#(),x) -> #() -(0(x),0(y)) -> 0(-(x,y)) -(0(x),1(y)) -> 1(-(-(x,y),1(#()))) -(1(x),0(y)) -> 1(-(x,y)) -(1(x),1(y)) -> 0(-(x,y)) 0(#()) -> #() max(l(x)) -> x max(n(x,y,z)) -> max(z) min(l(x)) -> x min(n(x,y,z)) -> min(y) size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0 ,c_4/1,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0 ,c_20/1,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0 ,c_35/2,c_36/0,c_37/0,c_38/0,c_39/11} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} Problem (S) - Strict DPs: size#(n(x,y,z)) -> c_35(size#(x),size#(y)) wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,size#(z) ,size#(y) ,wb#(y) ,wb#(z)) - Weak DPs: bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),min#(z),bs#(y),bs#(z)) ge#(#(),0(x)) -> c_20(ge#(#(),x)) ge#(0(x),0(y)) -> c_22(ge#(x,y)) ge#(0(x),1(y)) -> c_23(ge#(y,x)) ge#(1(x),0(y)) -> c_24(ge#(x,y)) ge#(1(x),1(y)) -> c_25(ge#(x,y)) max#(n(x,y,z)) -> c_29(max#(z)) min#(n(x,y,z)) -> c_31(min#(y)) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) -(x,#()) -> x -(#(),x) -> #() -(0(x),0(y)) -> 0(-(x,y)) -(0(x),1(y)) -> 1(-(-(x,y),1(#()))) -(1(x),0(y)) -> 1(-(x,y)) -(1(x),1(y)) -> 0(-(x,y)) 0(#()) -> #() max(l(x)) -> x max(n(x,y,z)) -> max(z) min(l(x)) -> x min(n(x,y,z)) -> min(y) size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0 ,c_4/1,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0 ,c_20/1,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0 ,c_35/2,c_36/0,c_37/0,c_38/0,c_39/11} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} **** Step 7.b:3.b:3.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),min#(z),bs#(y),bs#(z)) ge#(#(),0(x)) -> c_20(ge#(#(),x)) ge#(0(x),0(y)) -> c_22(ge#(x,y)) ge#(0(x),1(y)) -> c_23(ge#(y,x)) ge#(1(x),0(y)) -> c_24(ge#(x,y)) ge#(1(x),1(y)) -> c_25(ge#(x,y)) max#(n(x,y,z)) -> c_29(max#(z)) min#(n(x,y,z)) -> c_31(min#(y)) - Weak DPs: size#(n(x,y,z)) -> c_35(size#(x),size#(y)) wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,size#(z) ,size#(y) ,wb#(y) ,wb#(z)) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) -(x,#()) -> x -(#(),x) -> #() -(0(x),0(y)) -> 0(-(x,y)) -(0(x),1(y)) -> 1(-(-(x,y),1(#()))) -(1(x),0(y)) -> 1(-(x,y)) -(1(x),1(y)) -> 0(-(x,y)) 0(#()) -> #() max(l(x)) -> x max(n(x,y,z)) -> max(z) min(l(x)) -> x min(n(x,y,z)) -> min(y) size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/2 ,c_36/0,c_37/0,c_38/0,c_39/11} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),min#(z),bs#(y),bs#(z)) -->_4 min#(n(x,y,z)) -> c_31(min#(y)):8 -->_2 max#(n(x,y,z)) -> c_29(max#(z)):7 -->_3 ge#(1(x),1(y)) -> c_25(ge#(x,y)):6 -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):6 -->_3 ge#(1(x),0(y)) -> c_24(ge#(x,y)):5 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):5 -->_3 ge#(0(x),1(y)) -> c_23(ge#(y,x)):4 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):4 -->_3 ge#(0(x),0(y)) -> c_22(ge#(x,y)):3 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):3 -->_3 ge#(#(),0(x)) -> c_20(ge#(#(),x)):2 -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):2 -->_6 bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),min#(z),bs#(y),bs#(z)):1 -->_5 bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),min#(z),bs#(y),bs#(z)):1 2:S:ge#(#(),0(x)) -> c_20(ge#(#(),x)) -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):2 3:S:ge#(0(x),0(y)) -> c_22(ge#(x,y)) -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):2 -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):6 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):5 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):4 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):3 4:S:ge#(0(x),1(y)) -> c_23(ge#(y,x)) -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):2 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):3 -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):6 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):5 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):4 5:S:ge#(1(x),0(y)) -> c_24(ge#(x,y)) -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):2 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):3 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):4 -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):6 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):5 6:S:ge#(1(x),1(y)) -> c_25(ge#(x,y)) -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):2 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):3 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):4 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):5 -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):6 7:S:max#(n(x,y,z)) -> c_29(max#(z)) -->_1 max#(n(x,y,z)) -> c_29(max#(z)):7 8:S:min#(n(x,y,z)) -> c_31(min#(y)) -->_1 min#(n(x,y,z)) -> c_31(min#(y)):8 9:W:size#(n(x,y,z)) -> c_35(size#(x),size#(y)) -->_2 size#(n(x,y,z)) -> c_35(size#(x),size#(y)):9 -->_1 size#(n(x,y,z)) -> c_35(size#(x),size#(y)):9 10:W:wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,size#(z) ,size#(y) ,wb#(y) ,wb#(z)) -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):2 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):3 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):4 -->_7 ge#(1(x),0(y)) -> c_24(ge#(x,y)):5 -->_4 ge#(1(x),0(y)) -> c_24(ge#(x,y)):5 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):5 -->_7 ge#(1(x),1(y)) -> c_25(ge#(x,y)):6 -->_4 ge#(1(x),1(y)) -> c_25(ge#(x,y)):6 -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):6 -->_9 size#(n(x,y,z)) -> c_35(size#(x),size#(y)):9 -->_8 size#(n(x,y,z)) -> c_35(size#(x),size#(y)):9 -->_6 size#(n(x,y,z)) -> c_35(size#(x),size#(y)):9 -->_5 size#(n(x,y,z)) -> c_35(size#(x),size#(y)):9 -->_3 size#(n(x,y,z)) -> c_35(size#(x),size#(y)):9 -->_2 size#(n(x,y,z)) -> c_35(size#(x),size#(y)):9 -->_11 wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,size#(z) ,size#(y) ,wb#(y) ,wb#(z)):10 -->_10 wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,size#(z) ,size#(y) ,wb#(y) ,wb#(z)):10 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 9: size#(n(x,y,z)) -> c_35(size#(x),size#(y)) **** Step 7.b:3.b:3.a:2: SimplifyRHS WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),min#(z),bs#(y),bs#(z)) ge#(#(),0(x)) -> c_20(ge#(#(),x)) ge#(0(x),0(y)) -> c_22(ge#(x,y)) ge#(0(x),1(y)) -> c_23(ge#(y,x)) ge#(1(x),0(y)) -> c_24(ge#(x,y)) ge#(1(x),1(y)) -> c_25(ge#(x,y)) max#(n(x,y,z)) -> c_29(max#(z)) min#(n(x,y,z)) -> c_31(min#(y)) - Weak DPs: wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,size#(z) ,size#(y) ,wb#(y) ,wb#(z)) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) -(x,#()) -> x -(#(),x) -> #() -(0(x),0(y)) -> 0(-(x,y)) -(0(x),1(y)) -> 1(-(-(x,y),1(#()))) -(1(x),0(y)) -> 1(-(x,y)) -(1(x),1(y)) -> 0(-(x,y)) 0(#()) -> #() max(l(x)) -> x max(n(x,y,z)) -> max(z) min(l(x)) -> x min(n(x,y,z)) -> min(y) size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/2 ,c_36/0,c_37/0,c_38/0,c_39/11} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),min#(z),bs#(y),bs#(z)) -->_4 min#(n(x,y,z)) -> c_31(min#(y)):8 -->_2 max#(n(x,y,z)) -> c_29(max#(z)):7 -->_3 ge#(1(x),1(y)) -> c_25(ge#(x,y)):6 -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):6 -->_3 ge#(1(x),0(y)) -> c_24(ge#(x,y)):5 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):5 -->_3 ge#(0(x),1(y)) -> c_23(ge#(y,x)):4 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):4 -->_3 ge#(0(x),0(y)) -> c_22(ge#(x,y)):3 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):3 -->_3 ge#(#(),0(x)) -> c_20(ge#(#(),x)):2 -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):2 -->_6 bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),min#(z),bs#(y),bs#(z)):1 -->_5 bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),min#(z),bs#(y),bs#(z)):1 2:S:ge#(#(),0(x)) -> c_20(ge#(#(),x)) -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):2 3:S:ge#(0(x),0(y)) -> c_22(ge#(x,y)) -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):2 -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):6 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):5 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):4 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):3 4:S:ge#(0(x),1(y)) -> c_23(ge#(y,x)) -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):2 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):3 -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):6 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):5 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):4 5:S:ge#(1(x),0(y)) -> c_24(ge#(x,y)) -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):2 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):3 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):4 -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):6 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):5 6:S:ge#(1(x),1(y)) -> c_25(ge#(x,y)) -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):2 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):3 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):4 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):5 -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):6 7:S:max#(n(x,y,z)) -> c_29(max#(z)) -->_1 max#(n(x,y,z)) -> c_29(max#(z)):7 8:S:min#(n(x,y,z)) -> c_31(min#(y)) -->_1 min#(n(x,y,z)) -> c_31(min#(y)):8 10:W:wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,size#(z) ,size#(y) ,wb#(y) ,wb#(z)) -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):2 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):3 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):4 -->_7 ge#(1(x),0(y)) -> c_24(ge#(x,y)):5 -->_4 ge#(1(x),0(y)) -> c_24(ge#(x,y)):5 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):5 -->_7 ge#(1(x),1(y)) -> c_25(ge#(x,y)):6 -->_4 ge#(1(x),1(y)) -> c_25(ge#(x,y)):6 -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):6 -->_11 wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,size#(z) ,size#(y) ,wb#(y) ,wb#(z)):10 -->_10 wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,size#(z) ,size#(y) ,wb#(y) ,wb#(z)):10 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,ge#(1(#()),-(size(y),size(z))) ,ge#(1(#()),-(size(z),size(y))) ,wb#(y) ,wb#(z)) **** Step 7.b:3.b:3.a:3: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),min#(z),bs#(y),bs#(z)) ge#(#(),0(x)) -> c_20(ge#(#(),x)) ge#(0(x),0(y)) -> c_22(ge#(x,y)) ge#(0(x),1(y)) -> c_23(ge#(y,x)) ge#(1(x),0(y)) -> c_24(ge#(x,y)) ge#(1(x),1(y)) -> c_25(ge#(x,y)) max#(n(x,y,z)) -> c_29(max#(z)) min#(n(x,y,z)) -> c_31(min#(y)) - Weak DPs: wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,ge#(1(#()),-(size(y),size(z))) ,ge#(1(#()),-(size(z),size(y))) ,wb#(y) ,wb#(z)) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) -(x,#()) -> x -(#(),x) -> #() -(0(x),0(y)) -> 0(-(x,y)) -(0(x),1(y)) -> 1(-(-(x,y),1(#()))) -(1(x),0(y)) -> 1(-(x,y)) -(1(x),1(y)) -> 0(-(x,y)) 0(#()) -> #() max(l(x)) -> x max(n(x,y,z)) -> max(z) min(l(x)) -> x min(n(x,y,z)) -> min(y) size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/2 ,c_36/0,c_37/0,c_38/0,c_39/5} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 8: min#(n(x,y,z)) -> c_31(min#(y)) The strictly oriented rules are moved into the weak component. ***** Step 7.b:3.b:3.a:3.a:1: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),min#(z),bs#(y),bs#(z)) ge#(#(),0(x)) -> c_20(ge#(#(),x)) ge#(0(x),0(y)) -> c_22(ge#(x,y)) ge#(0(x),1(y)) -> c_23(ge#(y,x)) ge#(1(x),0(y)) -> c_24(ge#(x,y)) ge#(1(x),1(y)) -> c_25(ge#(x,y)) max#(n(x,y,z)) -> c_29(max#(z)) min#(n(x,y,z)) -> c_31(min#(y)) - Weak DPs: wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,ge#(1(#()),-(size(y),size(z))) ,ge#(1(#()),-(size(z),size(y))) ,wb#(y) ,wb#(z)) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) -(x,#()) -> x -(#(),x) -> #() -(0(x),0(y)) -> 0(-(x,y)) -(0(x),1(y)) -> 1(-(-(x,y),1(#()))) -(1(x),0(y)) -> 1(-(x,y)) -(1(x),1(y)) -> 0(-(x,y)) 0(#()) -> #() max(l(x)) -> x max(n(x,y,z)) -> max(z) min(l(x)) -> x min(n(x,y,z)) -> min(y) size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/2 ,c_36/0,c_37/0,c_38/0,c_39/5} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(mixed(2)): The following argument positions are considered usable: uargs(c_18) = {1,2,3,4,5,6}, uargs(c_20) = {1}, uargs(c_22) = {1}, uargs(c_23) = {1}, uargs(c_24) = {1}, uargs(c_25) = {1}, uargs(c_29) = {1}, uargs(c_31) = {1}, uargs(c_39) = {1,2,3,4,5} Following symbols are considered usable: {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val#,wb#} TcT has computed the following interpretation: p(#) = 1 p(+) = 0 p(-) = 0 p(0) = 1 p(1) = 0 p(and) = 0 p(bs) = 0 p(false) = 0 p(ge) = 0 p(if) = 0 p(l) = 0 p(max) = 0 p(min) = x1 p(n) = 1 + x2 + x3 p(not) = 0 p(size) = 0 p(true) = 0 p(val) = 0 p(wb) = 0 p(+#) = 0 p(-#) = 0 p(0#) = 0 p(and#) = 0 p(bs#) = 1 + x1 + x1^2 p(ge#) = 0 p(if#) = 0 p(max#) = 1 p(min#) = x1 p(not#) = 0 p(size#) = 0 p(val#) = 0 p(wb#) = x1 + x1^2 p(c_1) = 0 p(c_2) = 0 p(c_3) = 0 p(c_4) = 0 p(c_5) = 0 p(c_6) = 0 p(c_7) = 0 p(c_8) = 0 p(c_9) = 0 p(c_10) = 0 p(c_11) = 0 p(c_12) = 0 p(c_13) = 0 p(c_14) = 0 p(c_15) = 0 p(c_16) = 0 p(c_17) = 0 p(c_18) = x1 + x2 + x3 + x4 + x5 + x6 p(c_19) = 0 p(c_20) = x1 p(c_21) = 0 p(c_22) = x1 p(c_23) = x1 p(c_24) = x1 p(c_25) = x1 p(c_26) = 0 p(c_27) = 0 p(c_28) = 0 p(c_29) = x1 p(c_30) = 0 p(c_31) = x1 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) = 1 + x1 + x2 + x3 + x4 + x5 Following rules are strictly oriented: min#(n(x,y,z)) = 1 + y + z > y = c_31(min#(y)) Following rules are (at-least) weakly oriented: bs#(n(x,y,z)) = 3 + 3*y + 2*y*z + y^2 + 3*z + z^2 >= 3 + y + y^2 + 2*z + z^2 = c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),min#(z),bs#(y),bs#(z)) ge#(#(),0(x)) = 0 >= 0 = c_20(ge#(#(),x)) ge#(0(x),0(y)) = 0 >= 0 = c_22(ge#(x,y)) ge#(0(x),1(y)) = 0 >= 0 = c_23(ge#(y,x)) ge#(1(x),0(y)) = 0 >= 0 = c_24(ge#(x,y)) ge#(1(x),1(y)) = 0 >= 0 = c_25(ge#(x,y)) max#(n(x,y,z)) = 1 >= 1 = c_29(max#(z)) wb#(n(x,y,z)) = 2 + 3*y + 2*y*z + y^2 + 3*z + z^2 >= 1 + y + y^2 + z + z^2 = c_39(ge#(size(y),size(z)),ge#(1(#()),-(size(y),size(z))),ge#(1(#()),-(size(z),size(y))),wb#(y),wb#(z)) ***** Step 7.b:3.b:3.a:3.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),min#(z),bs#(y),bs#(z)) ge#(#(),0(x)) -> c_20(ge#(#(),x)) ge#(0(x),0(y)) -> c_22(ge#(x,y)) ge#(0(x),1(y)) -> c_23(ge#(y,x)) ge#(1(x),0(y)) -> c_24(ge#(x,y)) ge#(1(x),1(y)) -> c_25(ge#(x,y)) max#(n(x,y,z)) -> c_29(max#(z)) - Weak DPs: min#(n(x,y,z)) -> c_31(min#(y)) wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,ge#(1(#()),-(size(y),size(z))) ,ge#(1(#()),-(size(z),size(y))) ,wb#(y) ,wb#(z)) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) -(x,#()) -> x -(#(),x) -> #() -(0(x),0(y)) -> 0(-(x,y)) -(0(x),1(y)) -> 1(-(-(x,y),1(#()))) -(1(x),0(y)) -> 1(-(x,y)) -(1(x),1(y)) -> 0(-(x,y)) 0(#()) -> #() max(l(x)) -> x max(n(x,y,z)) -> max(z) min(l(x)) -> x min(n(x,y,z)) -> min(y) size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/2 ,c_36/0,c_37/0,c_38/0,c_39/5} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ***** Step 7.b:3.b:3.a:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),min#(z),bs#(y),bs#(z)) ge#(#(),0(x)) -> c_20(ge#(#(),x)) ge#(0(x),0(y)) -> c_22(ge#(x,y)) ge#(0(x),1(y)) -> c_23(ge#(y,x)) ge#(1(x),0(y)) -> c_24(ge#(x,y)) ge#(1(x),1(y)) -> c_25(ge#(x,y)) max#(n(x,y,z)) -> c_29(max#(z)) - Weak DPs: min#(n(x,y,z)) -> c_31(min#(y)) wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,ge#(1(#()),-(size(y),size(z))) ,ge#(1(#()),-(size(z),size(y))) ,wb#(y) ,wb#(z)) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) -(x,#()) -> x -(#(),x) -> #() -(0(x),0(y)) -> 0(-(x,y)) -(0(x),1(y)) -> 1(-(-(x,y),1(#()))) -(1(x),0(y)) -> 1(-(x,y)) -(1(x),1(y)) -> 0(-(x,y)) 0(#()) -> #() max(l(x)) -> x max(n(x,y,z)) -> max(z) min(l(x)) -> x min(n(x,y,z)) -> min(y) size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/2 ,c_36/0,c_37/0,c_38/0,c_39/5} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),min#(z),bs#(y),bs#(z)) -->_4 min#(n(x,y,z)) -> c_31(min#(y)):8 -->_2 max#(n(x,y,z)) -> c_29(max#(z)):7 -->_3 ge#(1(x),1(y)) -> c_25(ge#(x,y)):6 -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):6 -->_3 ge#(1(x),0(y)) -> c_24(ge#(x,y)):5 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):5 -->_3 ge#(0(x),1(y)) -> c_23(ge#(y,x)):4 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):4 -->_3 ge#(0(x),0(y)) -> c_22(ge#(x,y)):3 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):3 -->_3 ge#(#(),0(x)) -> c_20(ge#(#(),x)):2 -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):2 -->_6 bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),min#(z),bs#(y),bs#(z)):1 -->_5 bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),min#(z),bs#(y),bs#(z)):1 2:S:ge#(#(),0(x)) -> c_20(ge#(#(),x)) -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):2 3:S:ge#(0(x),0(y)) -> c_22(ge#(x,y)) -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):6 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):5 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):4 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):3 -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):2 4:S:ge#(0(x),1(y)) -> c_23(ge#(y,x)) -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):6 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):5 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):4 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):3 -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):2 5:S:ge#(1(x),0(y)) -> c_24(ge#(x,y)) -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):6 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):5 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):4 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):3 -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):2 6:S:ge#(1(x),1(y)) -> c_25(ge#(x,y)) -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):6 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):5 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):4 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):3 -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):2 7:S:max#(n(x,y,z)) -> c_29(max#(z)) -->_1 max#(n(x,y,z)) -> c_29(max#(z)):7 8:W:min#(n(x,y,z)) -> c_31(min#(y)) -->_1 min#(n(x,y,z)) -> c_31(min#(y)):8 9:W:wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,ge#(1(#()),-(size(y),size(z))) ,ge#(1(#()),-(size(z),size(y))) ,wb#(y) ,wb#(z)) -->_5 wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,ge#(1(#()),-(size(y),size(z))) ,ge#(1(#()),-(size(z),size(y))) ,wb#(y) ,wb#(z)):9 -->_4 wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,ge#(1(#()),-(size(y),size(z))) ,ge#(1(#()),-(size(z),size(y))) ,wb#(y) ,wb#(z)):9 -->_3 ge#(1(x),1(y)) -> c_25(ge#(x,y)):6 -->_2 ge#(1(x),1(y)) -> c_25(ge#(x,y)):6 -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):6 -->_3 ge#(1(x),0(y)) -> c_24(ge#(x,y)):5 -->_2 ge#(1(x),0(y)) -> c_24(ge#(x,y)):5 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):5 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):4 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):3 -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):2 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 8: min#(n(x,y,z)) -> c_31(min#(y)) ***** Step 7.b:3.b:3.a:3.b:2: SimplifyRHS WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),min#(z),bs#(y),bs#(z)) ge#(#(),0(x)) -> c_20(ge#(#(),x)) ge#(0(x),0(y)) -> c_22(ge#(x,y)) ge#(0(x),1(y)) -> c_23(ge#(y,x)) ge#(1(x),0(y)) -> c_24(ge#(x,y)) ge#(1(x),1(y)) -> c_25(ge#(x,y)) max#(n(x,y,z)) -> c_29(max#(z)) - Weak DPs: wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,ge#(1(#()),-(size(y),size(z))) ,ge#(1(#()),-(size(z),size(y))) ,wb#(y) ,wb#(z)) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) -(x,#()) -> x -(#(),x) -> #() -(0(x),0(y)) -> 0(-(x,y)) -(0(x),1(y)) -> 1(-(-(x,y),1(#()))) -(1(x),0(y)) -> 1(-(x,y)) -(1(x),1(y)) -> 0(-(x,y)) 0(#()) -> #() max(l(x)) -> x max(n(x,y,z)) -> max(z) min(l(x)) -> x min(n(x,y,z)) -> min(y) size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/2 ,c_36/0,c_37/0,c_38/0,c_39/5} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),min#(z),bs#(y),bs#(z)) -->_2 max#(n(x,y,z)) -> c_29(max#(z)):7 -->_3 ge#(1(x),1(y)) -> c_25(ge#(x,y)):6 -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):6 -->_3 ge#(1(x),0(y)) -> c_24(ge#(x,y)):5 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):5 -->_3 ge#(0(x),1(y)) -> c_23(ge#(y,x)):4 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):4 -->_3 ge#(0(x),0(y)) -> c_22(ge#(x,y)):3 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):3 -->_3 ge#(#(),0(x)) -> c_20(ge#(#(),x)):2 -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):2 -->_6 bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),min#(z),bs#(y),bs#(z)):1 -->_5 bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),min#(z),bs#(y),bs#(z)):1 2:S:ge#(#(),0(x)) -> c_20(ge#(#(),x)) -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):2 3:S:ge#(0(x),0(y)) -> c_22(ge#(x,y)) -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):6 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):5 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):4 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):3 -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):2 4:S:ge#(0(x),1(y)) -> c_23(ge#(y,x)) -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):6 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):5 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):4 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):3 -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):2 5:S:ge#(1(x),0(y)) -> c_24(ge#(x,y)) -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):6 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):5 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):4 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):3 -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):2 6:S:ge#(1(x),1(y)) -> c_25(ge#(x,y)) -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):6 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):5 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):4 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):3 -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):2 7:S:max#(n(x,y,z)) -> c_29(max#(z)) -->_1 max#(n(x,y,z)) -> c_29(max#(z)):7 9:W:wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,ge#(1(#()),-(size(y),size(z))) ,ge#(1(#()),-(size(z),size(y))) ,wb#(y) ,wb#(z)) -->_5 wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,ge#(1(#()),-(size(y),size(z))) ,ge#(1(#()),-(size(z),size(y))) ,wb#(y) ,wb#(z)):9 -->_4 wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,ge#(1(#()),-(size(y),size(z))) ,ge#(1(#()),-(size(z),size(y))) ,wb#(y) ,wb#(z)):9 -->_3 ge#(1(x),1(y)) -> c_25(ge#(x,y)):6 -->_2 ge#(1(x),1(y)) -> c_25(ge#(x,y)):6 -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):6 -->_3 ge#(1(x),0(y)) -> c_24(ge#(x,y)):5 -->_2 ge#(1(x),0(y)) -> c_24(ge#(x,y)):5 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):5 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):4 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):3 -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):2 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),bs#(y),bs#(z)) ***** Step 7.b:3.b:3.a:3.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),bs#(y),bs#(z)) ge#(#(),0(x)) -> c_20(ge#(#(),x)) ge#(0(x),0(y)) -> c_22(ge#(x,y)) ge#(0(x),1(y)) -> c_23(ge#(y,x)) ge#(1(x),0(y)) -> c_24(ge#(x,y)) ge#(1(x),1(y)) -> c_25(ge#(x,y)) max#(n(x,y,z)) -> c_29(max#(z)) - Weak DPs: wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,ge#(1(#()),-(size(y),size(z))) ,ge#(1(#()),-(size(z),size(y))) ,wb#(y) ,wb#(z)) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) -(x,#()) -> x -(#(),x) -> #() -(0(x),0(y)) -> 0(-(x,y)) -(0(x),1(y)) -> 1(-(-(x,y),1(#()))) -(1(x),0(y)) -> 1(-(x,y)) -(1(x),1(y)) -> 0(-(x,y)) 0(#()) -> #() max(l(x)) -> x max(n(x,y,z)) -> max(z) min(l(x)) -> x min(n(x,y,z)) -> min(y) size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/5,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/2 ,c_36/0,c_37/0,c_38/0,c_39/5} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),bs#(y),bs#(z)) 7: max#(n(x,y,z)) -> c_29(max#(z)) The strictly oriented rules are moved into the weak component. ****** Step 7.b:3.b:3.a:3.b:3.a:1: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),bs#(y),bs#(z)) ge#(#(),0(x)) -> c_20(ge#(#(),x)) ge#(0(x),0(y)) -> c_22(ge#(x,y)) ge#(0(x),1(y)) -> c_23(ge#(y,x)) ge#(1(x),0(y)) -> c_24(ge#(x,y)) ge#(1(x),1(y)) -> c_25(ge#(x,y)) max#(n(x,y,z)) -> c_29(max#(z)) - Weak DPs: wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,ge#(1(#()),-(size(y),size(z))) ,ge#(1(#()),-(size(z),size(y))) ,wb#(y) ,wb#(z)) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) -(x,#()) -> x -(#(),x) -> #() -(0(x),0(y)) -> 0(-(x,y)) -(0(x),1(y)) -> 1(-(-(x,y),1(#()))) -(1(x),0(y)) -> 1(-(x,y)) -(1(x),1(y)) -> 0(-(x,y)) 0(#()) -> #() max(l(x)) -> x max(n(x,y,z)) -> max(z) min(l(x)) -> x min(n(x,y,z)) -> min(y) size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/5,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/2 ,c_36/0,c_37/0,c_38/0,c_39/5} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(mixed(2)): The following argument positions are considered usable: uargs(c_18) = {1,2,3,4,5}, uargs(c_20) = {1}, uargs(c_22) = {1}, uargs(c_23) = {1}, uargs(c_24) = {1}, uargs(c_25) = {1}, uargs(c_29) = {1}, uargs(c_39) = {1,2,3,4,5} Following symbols are considered usable: {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val#,wb#} TcT has computed the following interpretation: p(#) = 0 p(+) = x2 p(-) = x1 p(0) = 1 + x1 p(1) = x1 p(and) = 0 p(bs) = 0 p(false) = 0 p(ge) = 0 p(if) = 0 p(l) = 0 p(max) = 1 p(min) = 0 p(n) = 1 + x2 + x3 p(not) = 0 p(size) = 0 p(true) = 0 p(val) = 0 p(wb) = 0 p(+#) = 0 p(-#) = 0 p(0#) = 0 p(and#) = 0 p(bs#) = x1^2 p(ge#) = 0 p(if#) = 0 p(max#) = x1 p(min#) = 0 p(not#) = 0 p(size#) = 0 p(val#) = 0 p(wb#) = x1^2 p(c_1) = 0 p(c_2) = 0 p(c_3) = 0 p(c_4) = 0 p(c_5) = 0 p(c_6) = 0 p(c_7) = 0 p(c_8) = 0 p(c_9) = 0 p(c_10) = 0 p(c_11) = 0 p(c_12) = 0 p(c_13) = 0 p(c_14) = 0 p(c_15) = 0 p(c_16) = 0 p(c_17) = 0 p(c_18) = x1 + x2 + x3 + x4 + x5 p(c_19) = 0 p(c_20) = x1 p(c_21) = 0 p(c_22) = x1 p(c_23) = x1 p(c_24) = x1 p(c_25) = x1 p(c_26) = 0 p(c_27) = 0 p(c_28) = 0 p(c_29) = x1 p(c_30) = 0 p(c_31) = 0 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) = 1 + x1 + x2 + x3 + x4 + x5 Following rules are strictly oriented: bs#(n(x,y,z)) = 1 + 2*y + 2*y*z + y^2 + 2*z + z^2 > y + y^2 + z^2 = c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),bs#(y),bs#(z)) max#(n(x,y,z)) = 1 + y + z > z = c_29(max#(z)) Following rules are (at-least) weakly oriented: ge#(#(),0(x)) = 0 >= 0 = c_20(ge#(#(),x)) ge#(0(x),0(y)) = 0 >= 0 = c_22(ge#(x,y)) ge#(0(x),1(y)) = 0 >= 0 = c_23(ge#(y,x)) ge#(1(x),0(y)) = 0 >= 0 = c_24(ge#(x,y)) ge#(1(x),1(y)) = 0 >= 0 = c_25(ge#(x,y)) wb#(n(x,y,z)) = 1 + 2*y + 2*y*z + y^2 + 2*z + z^2 >= 1 + y^2 + z^2 = c_39(ge#(size(y),size(z)),ge#(1(#()),-(size(y),size(z))),ge#(1(#()),-(size(z),size(y))),wb#(y),wb#(z)) ****** Step 7.b:3.b:3.a:3.b:3.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: ge#(#(),0(x)) -> c_20(ge#(#(),x)) ge#(0(x),0(y)) -> c_22(ge#(x,y)) ge#(0(x),1(y)) -> c_23(ge#(y,x)) ge#(1(x),0(y)) -> c_24(ge#(x,y)) ge#(1(x),1(y)) -> c_25(ge#(x,y)) - Weak DPs: bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),bs#(y),bs#(z)) max#(n(x,y,z)) -> c_29(max#(z)) wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,ge#(1(#()),-(size(y),size(z))) ,ge#(1(#()),-(size(z),size(y))) ,wb#(y) ,wb#(z)) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) -(x,#()) -> x -(#(),x) -> #() -(0(x),0(y)) -> 0(-(x,y)) -(0(x),1(y)) -> 1(-(-(x,y),1(#()))) -(1(x),0(y)) -> 1(-(x,y)) -(1(x),1(y)) -> 0(-(x,y)) 0(#()) -> #() max(l(x)) -> x max(n(x,y,z)) -> max(z) min(l(x)) -> x min(n(x,y,z)) -> min(y) size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/5,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/2 ,c_36/0,c_37/0,c_38/0,c_39/5} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ****** Step 7.b:3.b:3.a:3.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: ge#(#(),0(x)) -> c_20(ge#(#(),x)) ge#(0(x),0(y)) -> c_22(ge#(x,y)) ge#(0(x),1(y)) -> c_23(ge#(y,x)) ge#(1(x),0(y)) -> c_24(ge#(x,y)) ge#(1(x),1(y)) -> c_25(ge#(x,y)) - Weak DPs: bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),bs#(y),bs#(z)) max#(n(x,y,z)) -> c_29(max#(z)) wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,ge#(1(#()),-(size(y),size(z))) ,ge#(1(#()),-(size(z),size(y))) ,wb#(y) ,wb#(z)) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) -(x,#()) -> x -(#(),x) -> #() -(0(x),0(y)) -> 0(-(x,y)) -(0(x),1(y)) -> 1(-(-(x,y),1(#()))) -(1(x),0(y)) -> 1(-(x,y)) -(1(x),1(y)) -> 0(-(x,y)) 0(#()) -> #() max(l(x)) -> x max(n(x,y,z)) -> max(z) min(l(x)) -> x min(n(x,y,z)) -> min(y) size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/5,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/2 ,c_36/0,c_37/0,c_38/0,c_39/5} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:ge#(#(),0(x)) -> c_20(ge#(#(),x)) -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):1 2:S:ge#(0(x),0(y)) -> c_22(ge#(x,y)) -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):5 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):4 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):3 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):2 -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):1 3:S:ge#(0(x),1(y)) -> c_23(ge#(y,x)) -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):5 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):4 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):3 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):2 -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):1 4:S:ge#(1(x),0(y)) -> c_24(ge#(x,y)) -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):5 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):4 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):3 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):2 -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):1 5:S:ge#(1(x),1(y)) -> c_25(ge#(x,y)) -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):5 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):4 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):3 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):2 -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):1 6:W:bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),bs#(y),bs#(z)) -->_2 max#(n(x,y,z)) -> c_29(max#(z)):7 -->_5 bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),bs#(y),bs#(z)):6 -->_4 bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),bs#(y),bs#(z)):6 -->_3 ge#(1(x),1(y)) -> c_25(ge#(x,y)):5 -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):5 -->_3 ge#(1(x),0(y)) -> c_24(ge#(x,y)):4 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):4 -->_3 ge#(0(x),1(y)) -> c_23(ge#(y,x)):3 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):3 -->_3 ge#(0(x),0(y)) -> c_22(ge#(x,y)):2 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):2 -->_3 ge#(#(),0(x)) -> c_20(ge#(#(),x)):1 -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):1 7:W:max#(n(x,y,z)) -> c_29(max#(z)) -->_1 max#(n(x,y,z)) -> c_29(max#(z)):7 8:W:wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,ge#(1(#()),-(size(y),size(z))) ,ge#(1(#()),-(size(z),size(y))) ,wb#(y) ,wb#(z)) -->_5 wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,ge#(1(#()),-(size(y),size(z))) ,ge#(1(#()),-(size(z),size(y))) ,wb#(y) ,wb#(z)):8 -->_4 wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,ge#(1(#()),-(size(y),size(z))) ,ge#(1(#()),-(size(z),size(y))) ,wb#(y) ,wb#(z)):8 -->_3 ge#(1(x),1(y)) -> c_25(ge#(x,y)):5 -->_2 ge#(1(x),1(y)) -> c_25(ge#(x,y)):5 -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):5 -->_3 ge#(1(x),0(y)) -> c_24(ge#(x,y)):4 -->_2 ge#(1(x),0(y)) -> c_24(ge#(x,y)):4 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):4 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):3 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):2 -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 7: max#(n(x,y,z)) -> c_29(max#(z)) ****** Step 7.b:3.b:3.a:3.b:3.b:2: SimplifyRHS WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: ge#(#(),0(x)) -> c_20(ge#(#(),x)) ge#(0(x),0(y)) -> c_22(ge#(x,y)) ge#(0(x),1(y)) -> c_23(ge#(y,x)) ge#(1(x),0(y)) -> c_24(ge#(x,y)) ge#(1(x),1(y)) -> c_25(ge#(x,y)) - Weak DPs: bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),bs#(y),bs#(z)) wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,ge#(1(#()),-(size(y),size(z))) ,ge#(1(#()),-(size(z),size(y))) ,wb#(y) ,wb#(z)) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) -(x,#()) -> x -(#(),x) -> #() -(0(x),0(y)) -> 0(-(x,y)) -(0(x),1(y)) -> 1(-(-(x,y),1(#()))) -(1(x),0(y)) -> 1(-(x,y)) -(1(x),1(y)) -> 0(-(x,y)) 0(#()) -> #() max(l(x)) -> x max(n(x,y,z)) -> max(z) min(l(x)) -> x min(n(x,y,z)) -> min(y) size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/5,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/2 ,c_36/0,c_37/0,c_38/0,c_39/5} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:ge#(#(),0(x)) -> c_20(ge#(#(),x)) -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):1 2:S:ge#(0(x),0(y)) -> c_22(ge#(x,y)) -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):5 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):4 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):3 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):2 -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):1 3:S:ge#(0(x),1(y)) -> c_23(ge#(y,x)) -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):5 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):4 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):3 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):2 -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):1 4:S:ge#(1(x),0(y)) -> c_24(ge#(x,y)) -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):5 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):4 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):3 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):2 -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):1 5:S:ge#(1(x),1(y)) -> c_25(ge#(x,y)) -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):5 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):4 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):3 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):2 -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):1 6:W:bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),bs#(y),bs#(z)) -->_5 bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),bs#(y),bs#(z)):6 -->_4 bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),bs#(y),bs#(z)):6 -->_3 ge#(1(x),1(y)) -> c_25(ge#(x,y)):5 -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):5 -->_3 ge#(1(x),0(y)) -> c_24(ge#(x,y)):4 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):4 -->_3 ge#(0(x),1(y)) -> c_23(ge#(y,x)):3 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):3 -->_3 ge#(0(x),0(y)) -> c_22(ge#(x,y)):2 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):2 -->_3 ge#(#(),0(x)) -> c_20(ge#(#(),x)):1 -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):1 8:W:wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,ge#(1(#()),-(size(y),size(z))) ,ge#(1(#()),-(size(z),size(y))) ,wb#(y) ,wb#(z)) -->_5 wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,ge#(1(#()),-(size(y),size(z))) ,ge#(1(#()),-(size(z),size(y))) ,wb#(y) ,wb#(z)):8 -->_4 wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,ge#(1(#()),-(size(y),size(z))) ,ge#(1(#()),-(size(z),size(y))) ,wb#(y) ,wb#(z)):8 -->_3 ge#(1(x),1(y)) -> c_25(ge#(x,y)):5 -->_2 ge#(1(x),1(y)) -> c_25(ge#(x,y)):5 -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):5 -->_3 ge#(1(x),0(y)) -> c_24(ge#(x,y)):4 -->_2 ge#(1(x),0(y)) -> c_24(ge#(x,y)):4 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):4 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):3 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):2 -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),ge#(min(z),x),bs#(y),bs#(z)) ****** Step 7.b:3.b:3.a:3.b:3.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: ge#(#(),0(x)) -> c_20(ge#(#(),x)) ge#(0(x),0(y)) -> c_22(ge#(x,y)) ge#(0(x),1(y)) -> c_23(ge#(y,x)) ge#(1(x),0(y)) -> c_24(ge#(x,y)) ge#(1(x),1(y)) -> c_25(ge#(x,y)) - Weak DPs: bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),ge#(min(z),x),bs#(y),bs#(z)) wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,ge#(1(#()),-(size(y),size(z))) ,ge#(1(#()),-(size(z),size(y))) ,wb#(y) ,wb#(z)) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) -(x,#()) -> x -(#(),x) -> #() -(0(x),0(y)) -> 0(-(x,y)) -(0(x),1(y)) -> 1(-(-(x,y),1(#()))) -(1(x),0(y)) -> 1(-(x,y)) -(1(x),1(y)) -> 0(-(x,y)) 0(#()) -> #() max(l(x)) -> x max(n(x,y,z)) -> max(z) min(l(x)) -> x min(n(x,y,z)) -> min(y) size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/4,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/2 ,c_36/0,c_37/0,c_38/0,c_39/5} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: ge#(#(),0(x)) -> c_20(ge#(#(),x)) 2: ge#(0(x),0(y)) -> c_22(ge#(x,y)) 3: ge#(0(x),1(y)) -> c_23(ge#(y,x)) 4: ge#(1(x),0(y)) -> c_24(ge#(x,y)) 5: ge#(1(x),1(y)) -> c_25(ge#(x,y)) The strictly oriented rules are moved into the weak component. ******* Step 7.b:3.b:3.a:3.b:3.b:3.a:1: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: ge#(#(),0(x)) -> c_20(ge#(#(),x)) ge#(0(x),0(y)) -> c_22(ge#(x,y)) ge#(0(x),1(y)) -> c_23(ge#(y,x)) ge#(1(x),0(y)) -> c_24(ge#(x,y)) ge#(1(x),1(y)) -> c_25(ge#(x,y)) - Weak DPs: bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),ge#(min(z),x),bs#(y),bs#(z)) wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,ge#(1(#()),-(size(y),size(z))) ,ge#(1(#()),-(size(z),size(y))) ,wb#(y) ,wb#(z)) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) -(x,#()) -> x -(#(),x) -> #() -(0(x),0(y)) -> 0(-(x,y)) -(0(x),1(y)) -> 1(-(-(x,y),1(#()))) -(1(x),0(y)) -> 1(-(x,y)) -(1(x),1(y)) -> 0(-(x,y)) 0(#()) -> #() max(l(x)) -> x max(n(x,y,z)) -> max(z) min(l(x)) -> x min(n(x,y,z)) -> min(y) size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/4,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/2 ,c_36/0,c_37/0,c_38/0,c_39/5} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(mixed(2)): The following argument positions are considered usable: uargs(c_18) = {1,2,3,4}, uargs(c_20) = {1}, uargs(c_22) = {1}, uargs(c_23) = {1}, uargs(c_24) = {1}, uargs(c_25) = {1}, uargs(c_39) = {1,2,3,4,5} Following symbols are considered usable: {+,-,0,max,min,size,+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val#,wb#} TcT has computed the following interpretation: p(#) = 0 p(+) = x1 + x2 p(-) = x1 p(0) = 1 + x1 p(1) = 1 + x1 p(and) = 0 p(bs) = 0 p(false) = 0 p(ge) = 0 p(if) = 0 p(l) = 1 + x1 p(max) = x1 p(min) = x1 p(n) = 1 + x1 + x2 + x3 p(not) = 0 p(size) = x1 p(true) = 0 p(val) = 0 p(wb) = 0 p(+#) = 0 p(-#) = 0 p(0#) = 0 p(and#) = 0 p(bs#) = x1^2 p(ge#) = x1 + x2 p(if#) = 0 p(max#) = 0 p(min#) = 0 p(not#) = 0 p(size#) = 0 p(val#) = 0 p(wb#) = x1 + x1^2 p(c_1) = 0 p(c_2) = 0 p(c_3) = 0 p(c_4) = 0 p(c_5) = 0 p(c_6) = 0 p(c_7) = 0 p(c_8) = 0 p(c_9) = 0 p(c_10) = 0 p(c_11) = 0 p(c_12) = 0 p(c_13) = 0 p(c_14) = 0 p(c_15) = 0 p(c_16) = 0 p(c_17) = 0 p(c_18) = 1 + x1 + x2 + x3 + x4 p(c_19) = 0 p(c_20) = x1 p(c_21) = 0 p(c_22) = x1 p(c_23) = x1 p(c_24) = x1 p(c_25) = x1 p(c_26) = 0 p(c_27) = 0 p(c_28) = 0 p(c_29) = 0 p(c_30) = 0 p(c_31) = 0 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) = x1 + x2 + x3 + x4 + x5 Following rules are strictly oriented: ge#(#(),0(x)) = 1 + x > x = c_20(ge#(#(),x)) ge#(0(x),0(y)) = 2 + x + y > x + y = c_22(ge#(x,y)) ge#(0(x),1(y)) = 2 + x + y > x + y = c_23(ge#(y,x)) ge#(1(x),0(y)) = 2 + x + y > x + y = c_24(ge#(x,y)) ge#(1(x),1(y)) = 2 + x + y > x + y = c_25(ge#(x,y)) Following rules are (at-least) weakly oriented: bs#(n(x,y,z)) = 1 + 2*x + 2*x*y + 2*x*z + x^2 + 2*y + 2*y*z + y^2 + 2*z + z^2 >= 1 + 2*x + y + y^2 + z + z^2 = c_18(ge#(x,max(y)),ge#(min(z),x),bs#(y),bs#(z)) wb#(n(x,y,z)) = 2 + 3*x + 2*x*y + 2*x*z + x^2 + 3*y + 2*y*z + y^2 + 3*z + z^2 >= 2 + 3*y + y^2 + 3*z + z^2 = c_39(ge#(size(y),size(z)),ge#(1(#()),-(size(y),size(z))),ge#(1(#()),-(size(z),size(y))),wb#(y),wb#(z)) +(x,#()) = x >= x = x +(x,+(y,z)) = x + y + z >= x + y + z = +(+(x,y),z) +(#(),x) = x >= x = x +(0(x),0(y)) = 2 + x + y >= 1 + x + y = 0(+(x,y)) +(0(x),1(y)) = 2 + x + y >= 1 + x + y = 1(+(x,y)) +(1(x),0(y)) = 2 + x + y >= 1 + x + y = 1(+(x,y)) +(1(x),1(y)) = 2 + x + y >= 2 + x + y = 0(+(+(x,y),1(#()))) -(x,#()) = x >= x = x -(#(),x) = 0 >= 0 = #() -(0(x),0(y)) = 1 + x >= 1 + x = 0(-(x,y)) -(0(x),1(y)) = 1 + x >= 1 + x = 1(-(-(x,y),1(#()))) -(1(x),0(y)) = 1 + x >= 1 + x = 1(-(x,y)) -(1(x),1(y)) = 1 + x >= 1 + x = 0(-(x,y)) 0(#()) = 1 >= 0 = #() max(l(x)) = 1 + x >= x = x max(n(x,y,z)) = 1 + x + y + z >= z = max(z) min(l(x)) = 1 + x >= x = x min(n(x,y,z)) = 1 + x + y + z >= y = min(y) size(l(x)) = 1 + x >= 1 = 1(#()) size(n(x,y,z)) = 1 + x + y + z >= 1 + x + y = +(+(size(x),size(y)),1(#())) ******* Step 7.b:3.b:3.a:3.b:3.b:3.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),ge#(min(z),x),bs#(y),bs#(z)) ge#(#(),0(x)) -> c_20(ge#(#(),x)) ge#(0(x),0(y)) -> c_22(ge#(x,y)) ge#(0(x),1(y)) -> c_23(ge#(y,x)) ge#(1(x),0(y)) -> c_24(ge#(x,y)) ge#(1(x),1(y)) -> c_25(ge#(x,y)) wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,ge#(1(#()),-(size(y),size(z))) ,ge#(1(#()),-(size(z),size(y))) ,wb#(y) ,wb#(z)) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) -(x,#()) -> x -(#(),x) -> #() -(0(x),0(y)) -> 0(-(x,y)) -(0(x),1(y)) -> 1(-(-(x,y),1(#()))) -(1(x),0(y)) -> 1(-(x,y)) -(1(x),1(y)) -> 0(-(x,y)) 0(#()) -> #() max(l(x)) -> x max(n(x,y,z)) -> max(z) min(l(x)) -> x min(n(x,y,z)) -> min(y) size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/4,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/2 ,c_36/0,c_37/0,c_38/0,c_39/5} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ******* Step 7.b:3.b:3.a:3.b:3.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),ge#(min(z),x),bs#(y),bs#(z)) ge#(#(),0(x)) -> c_20(ge#(#(),x)) ge#(0(x),0(y)) -> c_22(ge#(x,y)) ge#(0(x),1(y)) -> c_23(ge#(y,x)) ge#(1(x),0(y)) -> c_24(ge#(x,y)) ge#(1(x),1(y)) -> c_25(ge#(x,y)) wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,ge#(1(#()),-(size(y),size(z))) ,ge#(1(#()),-(size(z),size(y))) ,wb#(y) ,wb#(z)) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) -(x,#()) -> x -(#(),x) -> #() -(0(x),0(y)) -> 0(-(x,y)) -(0(x),1(y)) -> 1(-(-(x,y),1(#()))) -(1(x),0(y)) -> 1(-(x,y)) -(1(x),1(y)) -> 0(-(x,y)) 0(#()) -> #() max(l(x)) -> x max(n(x,y,z)) -> max(z) min(l(x)) -> x min(n(x,y,z)) -> min(y) size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/4,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/2 ,c_36/0,c_37/0,c_38/0,c_39/5} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),ge#(min(z),x),bs#(y),bs#(z)) -->_2 ge#(1(x),1(y)) -> c_25(ge#(x,y)):6 -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):6 -->_2 ge#(1(x),0(y)) -> c_24(ge#(x,y)):5 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):5 -->_2 ge#(0(x),1(y)) -> c_23(ge#(y,x)):4 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):4 -->_2 ge#(0(x),0(y)) -> c_22(ge#(x,y)):3 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):3 -->_2 ge#(#(),0(x)) -> c_20(ge#(#(),x)):2 -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):2 -->_4 bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),ge#(min(z),x),bs#(y),bs#(z)):1 -->_3 bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),ge#(min(z),x),bs#(y),bs#(z)):1 2:W:ge#(#(),0(x)) -> c_20(ge#(#(),x)) -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):2 3:W:ge#(0(x),0(y)) -> c_22(ge#(x,y)) -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):6 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):5 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):4 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):3 -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):2 4:W:ge#(0(x),1(y)) -> c_23(ge#(y,x)) -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):6 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):5 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):4 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):3 -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):2 5:W:ge#(1(x),0(y)) -> c_24(ge#(x,y)) -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):6 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):5 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):4 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):3 -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):2 6:W:ge#(1(x),1(y)) -> c_25(ge#(x,y)) -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):6 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):5 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):4 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):3 -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):2 7:W:wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,ge#(1(#()),-(size(y),size(z))) ,ge#(1(#()),-(size(z),size(y))) ,wb#(y) ,wb#(z)) -->_5 wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,ge#(1(#()),-(size(y),size(z))) ,ge#(1(#()),-(size(z),size(y))) ,wb#(y) ,wb#(z)):7 -->_4 wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,ge#(1(#()),-(size(y),size(z))) ,ge#(1(#()),-(size(z),size(y))) ,wb#(y) ,wb#(z)):7 -->_3 ge#(1(x),1(y)) -> c_25(ge#(x,y)):6 -->_2 ge#(1(x),1(y)) -> c_25(ge#(x,y)):6 -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):6 -->_3 ge#(1(x),0(y)) -> c_24(ge#(x,y)):5 -->_2 ge#(1(x),0(y)) -> c_24(ge#(x,y)):5 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):5 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):4 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):3 -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):2 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 7: wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,ge#(1(#()),-(size(y),size(z))) ,ge#(1(#()),-(size(z),size(y))) ,wb#(y) ,wb#(z)) 1: bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),ge#(min(z),x),bs#(y),bs#(z)) 6: ge#(1(x),1(y)) -> c_25(ge#(x,y)) 5: ge#(1(x),0(y)) -> c_24(ge#(x,y)) 4: ge#(0(x),1(y)) -> c_23(ge#(y,x)) 3: ge#(0(x),0(y)) -> c_22(ge#(x,y)) 2: ge#(#(),0(x)) -> c_20(ge#(#(),x)) ******* Step 7.b:3.b:3.a:3.b:3.b:3.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) -(x,#()) -> x -(#(),x) -> #() -(0(x),0(y)) -> 0(-(x,y)) -(0(x),1(y)) -> 1(-(-(x,y),1(#()))) -(1(x),0(y)) -> 1(-(x,y)) -(1(x),1(y)) -> 0(-(x,y)) 0(#()) -> #() max(l(x)) -> x max(n(x,y,z)) -> max(z) min(l(x)) -> x min(n(x,y,z)) -> min(y) size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/4,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/2 ,c_36/0,c_37/0,c_38/0,c_39/5} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). **** Step 7.b:3.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: size#(n(x,y,z)) -> c_35(size#(x),size#(y)) wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,size#(z) ,size#(y) ,wb#(y) ,wb#(z)) - Weak DPs: bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),min#(z),bs#(y),bs#(z)) ge#(#(),0(x)) -> c_20(ge#(#(),x)) ge#(0(x),0(y)) -> c_22(ge#(x,y)) ge#(0(x),1(y)) -> c_23(ge#(y,x)) ge#(1(x),0(y)) -> c_24(ge#(x,y)) ge#(1(x),1(y)) -> c_25(ge#(x,y)) max#(n(x,y,z)) -> c_29(max#(z)) min#(n(x,y,z)) -> c_31(min#(y)) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) -(x,#()) -> x -(#(),x) -> #() -(0(x),0(y)) -> 0(-(x,y)) -(0(x),1(y)) -> 1(-(-(x,y),1(#()))) -(1(x),0(y)) -> 1(-(x,y)) -(1(x),1(y)) -> 0(-(x,y)) 0(#()) -> #() max(l(x)) -> x max(n(x,y,z)) -> max(z) min(l(x)) -> x min(n(x,y,z)) -> min(y) size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/2 ,c_36/0,c_37/0,c_38/0,c_39/11} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:size#(n(x,y,z)) -> c_35(size#(x),size#(y)) -->_2 size#(n(x,y,z)) -> c_35(size#(x),size#(y)):1 -->_1 size#(n(x,y,z)) -> c_35(size#(x),size#(y)):1 2:S:wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,size#(z) ,size#(y) ,wb#(y) ,wb#(z)) -->_7 ge#(1(x),1(y)) -> c_25(ge#(x,y)):8 -->_4 ge#(1(x),1(y)) -> c_25(ge#(x,y)):8 -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):8 -->_7 ge#(1(x),0(y)) -> c_24(ge#(x,y)):7 -->_4 ge#(1(x),0(y)) -> c_24(ge#(x,y)):7 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):7 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):6 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):5 -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):4 -->_11 wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,size#(z) ,size#(y) ,wb#(y) ,wb#(z)):2 -->_10 wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,size#(z) ,size#(y) ,wb#(y) ,wb#(z)):2 -->_9 size#(n(x,y,z)) -> c_35(size#(x),size#(y)):1 -->_8 size#(n(x,y,z)) -> c_35(size#(x),size#(y)):1 -->_6 size#(n(x,y,z)) -> c_35(size#(x),size#(y)):1 -->_5 size#(n(x,y,z)) -> c_35(size#(x),size#(y)):1 -->_3 size#(n(x,y,z)) -> c_35(size#(x),size#(y)):1 -->_2 size#(n(x,y,z)) -> c_35(size#(x),size#(y)):1 3:W:bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),min#(z),bs#(y),bs#(z)) -->_4 min#(n(x,y,z)) -> c_31(min#(y)):10 -->_2 max#(n(x,y,z)) -> c_29(max#(z)):9 -->_3 ge#(1(x),1(y)) -> c_25(ge#(x,y)):8 -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):8 -->_3 ge#(1(x),0(y)) -> c_24(ge#(x,y)):7 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):7 -->_3 ge#(0(x),1(y)) -> c_23(ge#(y,x)):6 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):6 -->_3 ge#(0(x),0(y)) -> c_22(ge#(x,y)):5 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):5 -->_3 ge#(#(),0(x)) -> c_20(ge#(#(),x)):4 -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):4 -->_6 bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),min#(z),bs#(y),bs#(z)):3 -->_5 bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),min#(z),bs#(y),bs#(z)):3 4:W:ge#(#(),0(x)) -> c_20(ge#(#(),x)) -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):4 5:W:ge#(0(x),0(y)) -> c_22(ge#(x,y)) -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):8 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):7 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):6 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):5 -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):4 6:W:ge#(0(x),1(y)) -> c_23(ge#(y,x)) -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):8 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):7 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):6 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):5 -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):4 7:W:ge#(1(x),0(y)) -> c_24(ge#(x,y)) -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):8 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):7 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):6 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):5 -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):4 8:W:ge#(1(x),1(y)) -> c_25(ge#(x,y)) -->_1 ge#(1(x),1(y)) -> c_25(ge#(x,y)):8 -->_1 ge#(1(x),0(y)) -> c_24(ge#(x,y)):7 -->_1 ge#(0(x),1(y)) -> c_23(ge#(y,x)):6 -->_1 ge#(0(x),0(y)) -> c_22(ge#(x,y)):5 -->_1 ge#(#(),0(x)) -> c_20(ge#(#(),x)):4 9:W:max#(n(x,y,z)) -> c_29(max#(z)) -->_1 max#(n(x,y,z)) -> c_29(max#(z)):9 10:W:min#(n(x,y,z)) -> c_31(min#(y)) -->_1 min#(n(x,y,z)) -> c_31(min#(y)):10 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 3: bs#(n(x,y,z)) -> c_18(ge#(x,max(y)),max#(y),ge#(min(z),x),min#(z),bs#(y),bs#(z)) 9: max#(n(x,y,z)) -> c_29(max#(z)) 10: min#(n(x,y,z)) -> c_31(min#(y)) 8: ge#(1(x),1(y)) -> c_25(ge#(x,y)) 7: ge#(1(x),0(y)) -> c_24(ge#(x,y)) 6: ge#(0(x),1(y)) -> c_23(ge#(y,x)) 5: ge#(0(x),0(y)) -> c_22(ge#(x,y)) 4: ge#(#(),0(x)) -> c_20(ge#(#(),x)) **** Step 7.b:3.b:3.b:2: SimplifyRHS WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: size#(n(x,y,z)) -> c_35(size#(x),size#(y)) wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,size#(z) ,size#(y) ,wb#(y) ,wb#(z)) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) -(x,#()) -> x -(#(),x) -> #() -(0(x),0(y)) -> 0(-(x,y)) -(0(x),1(y)) -> 1(-(-(x,y),1(#()))) -(1(x),0(y)) -> 1(-(x,y)) -(1(x),1(y)) -> 0(-(x,y)) 0(#()) -> #() max(l(x)) -> x max(n(x,y,z)) -> max(z) min(l(x)) -> x min(n(x,y,z)) -> min(y) size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/2 ,c_36/0,c_37/0,c_38/0,c_39/11} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:size#(n(x,y,z)) -> c_35(size#(x),size#(y)) -->_2 size#(n(x,y,z)) -> c_35(size#(x),size#(y)):1 -->_1 size#(n(x,y,z)) -> c_35(size#(x),size#(y)):1 2:S:wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,size#(z) ,size#(y) ,wb#(y) ,wb#(z)) -->_11 wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,size#(z) ,size#(y) ,wb#(y) ,wb#(z)):2 -->_10 wb#(n(x,y,z)) -> c_39(ge#(size(y),size(z)) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(y),size(z))) ,size#(y) ,size#(z) ,ge#(1(#()),-(size(z),size(y))) ,size#(z) ,size#(y) ,wb#(y) ,wb#(z)):2 -->_9 size#(n(x,y,z)) -> c_35(size#(x),size#(y)):1 -->_8 size#(n(x,y,z)) -> c_35(size#(x),size#(y)):1 -->_6 size#(n(x,y,z)) -> c_35(size#(x),size#(y)):1 -->_5 size#(n(x,y,z)) -> c_35(size#(x),size#(y)):1 -->_3 size#(n(x,y,z)) -> c_35(size#(x),size#(y)):1 -->_2 size#(n(x,y,z)) -> c_35(size#(x),size#(y)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: wb#(n(x,y,z)) -> c_39(size#(y),size#(z),size#(y),size#(z),size#(z),size#(y),wb#(y),wb#(z)) **** Step 7.b:3.b:3.b:3: UsableRules WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: size#(n(x,y,z)) -> c_35(size#(x),size#(y)) wb#(n(x,y,z)) -> c_39(size#(y),size#(z),size#(y),size#(z),size#(z),size#(y),wb#(y),wb#(z)) - Weak TRS: +(x,#()) -> x +(x,+(y,z)) -> +(+(x,y),z) +(#(),x) -> x +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) -(x,#()) -> x -(#(),x) -> #() -(0(x),0(y)) -> 0(-(x,y)) -(0(x),1(y)) -> 1(-(-(x,y),1(#()))) -(1(x),0(y)) -> 1(-(x,y)) -(1(x),1(y)) -> 0(-(x,y)) 0(#()) -> #() max(l(x)) -> x max(n(x,y,z)) -> max(z) min(l(x)) -> x min(n(x,y,z)) -> min(y) size(l(x)) -> 1(#()) size(n(x,y,z)) -> +(+(size(x),size(y)),1(#())) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/2 ,c_36/0,c_37/0,c_38/0,c_39/8} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: size#(n(x,y,z)) -> c_35(size#(x),size#(y)) wb#(n(x,y,z)) -> c_39(size#(y),size#(z),size#(y),size#(z),size#(z),size#(y),wb#(y),wb#(z)) **** Step 7.b:3.b:3.b:4: Decompose WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: size#(n(x,y,z)) -> c_35(size#(x),size#(y)) wb#(n(x,y,z)) -> c_39(size#(y),size#(z),size#(y),size#(z),size#(z),size#(y),wb#(y),wb#(z)) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/2 ,c_36/0,c_37/0,c_38/0,c_39/8} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd} + Details: We analyse the complexity of following sub-problems (R) and (S). Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component. Problem (R) - Strict DPs: size#(n(x,y,z)) -> c_35(size#(x),size#(y)) - Weak DPs: wb#(n(x,y,z)) -> c_39(size#(y),size#(z),size#(y),size#(z),size#(z),size#(y),wb#(y),wb#(z)) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0 ,c_4/1,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0 ,c_20/1,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0 ,c_35/2,c_36/0,c_37/0,c_38/0,c_39/8} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} Problem (S) - Strict DPs: wb#(n(x,y,z)) -> c_39(size#(y),size#(z),size#(y),size#(z),size#(z),size#(y),wb#(y),wb#(z)) - Weak DPs: size#(n(x,y,z)) -> c_35(size#(x),size#(y)) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0 ,c_4/1,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0 ,c_20/1,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0 ,c_35/2,c_36/0,c_37/0,c_38/0,c_39/8} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} ***** Step 7.b:3.b:3.b:4.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: size#(n(x,y,z)) -> c_35(size#(x),size#(y)) - Weak DPs: wb#(n(x,y,z)) -> c_39(size#(y),size#(z),size#(y),size#(z),size#(z),size#(y),wb#(y),wb#(z)) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/2 ,c_36/0,c_37/0,c_38/0,c_39/8} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: size#(n(x,y,z)) -> c_35(size#(x),size#(y)) The strictly oriented rules are moved into the weak component. ****** Step 7.b:3.b:3.b:4.a:1.a:1: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: size#(n(x,y,z)) -> c_35(size#(x),size#(y)) - Weak DPs: wb#(n(x,y,z)) -> c_39(size#(y),size#(z),size#(y),size#(z),size#(z),size#(y),wb#(y),wb#(z)) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/2 ,c_36/0,c_37/0,c_38/0,c_39/8} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(mixed(2)): The following argument positions are considered usable: uargs(c_35) = {1,2}, uargs(c_39) = {1,2,3,4,5,6,7,8} Following symbols are considered usable: {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val#,wb#} TcT has computed the following interpretation: p(#) = 0 p(+) = 1 + x2 + x2^2 p(-) = 4 + x1*x2 + 2*x1^2 + x2^2 p(0) = 1 + x1 + x1^2 p(1) = 4 p(and) = x1 + x2 + x2^2 p(bs) = 1 + x1 + x1^2 p(false) = 0 p(ge) = 2*x1 + x1^2 p(if) = x2 + x2^2 p(l) = 4 + x1 p(max) = 1 + x1 p(min) = 1 + 2*x1 + x1^2 p(n) = 1 + x1 + x2 + x3 p(not) = 0 p(size) = 4*x1 + 4*x1^2 p(true) = 0 p(val) = 1 p(wb) = 4 + 2*x1^2 p(+#) = 1 + x1 + x1*x2 + x2 p(-#) = x1^2 + 4*x2 + 4*x2^2 p(0#) = 0 p(and#) = x1*x2 + x1^2 + x2 p(bs#) = 1 p(ge#) = x1^2 p(if#) = x1*x2 + 2*x1^2 + 4*x2 + 4*x2*x3 + 2*x3 + 4*x3^2 p(max#) = 1 + x1 + x1^2 p(min#) = x1^2 p(not#) = 1 p(size#) = 2*x1 p(val#) = 4 + 4*x1^2 p(wb#) = 6*x1 + 4*x1^2 p(c_1) = 2 p(c_2) = 1 p(c_3) = 1 p(c_4) = 1 p(c_5) = 0 p(c_6) = 1 p(c_7) = 4 + x2 p(c_8) = 1 p(c_9) = 0 p(c_10) = 0 p(c_11) = 1 p(c_12) = 4 + x1 p(c_13) = 1 p(c_14) = 0 p(c_15) = 0 p(c_16) = 0 p(c_17) = 4 p(c_18) = 1 + x1 + x2 + x6 p(c_19) = 4 p(c_20) = 1 + x1 p(c_21) = 1 p(c_22) = 0 p(c_23) = 2 + x1 p(c_24) = 1 p(c_25) = 4 p(c_26) = 1 p(c_27) = 0 p(c_28) = 4 p(c_29) = 1 p(c_30) = 0 p(c_31) = 0 p(c_32) = 2 p(c_33) = 0 p(c_34) = 1 p(c_35) = x1 + x2 p(c_36) = 1 p(c_37) = 0 p(c_38) = 1 p(c_39) = x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 Following rules are strictly oriented: size#(n(x,y,z)) = 2 + 2*x + 2*y + 2*z > 2*x + 2*y = c_35(size#(x),size#(y)) Following rules are (at-least) weakly oriented: wb#(n(x,y,z)) = 10 + 14*x + 8*x*y + 8*x*z + 4*x^2 + 14*y + 8*y*z + 4*y^2 + 14*z + 4*z^2 >= 12*y + 4*y^2 + 12*z + 4*z^2 = c_39(size#(y),size#(z),size#(y),size#(z),size#(z),size#(y),wb#(y),wb#(z)) ****** Step 7.b:3.b:3.b:4.a:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: size#(n(x,y,z)) -> c_35(size#(x),size#(y)) wb#(n(x,y,z)) -> c_39(size#(y),size#(z),size#(y),size#(z),size#(z),size#(y),wb#(y),wb#(z)) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/2 ,c_36/0,c_37/0,c_38/0,c_39/8} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ****** Step 7.b:3.b:3.b:4.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: size#(n(x,y,z)) -> c_35(size#(x),size#(y)) wb#(n(x,y,z)) -> c_39(size#(y),size#(z),size#(y),size#(z),size#(z),size#(y),wb#(y),wb#(z)) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/2 ,c_36/0,c_37/0,c_38/0,c_39/8} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:size#(n(x,y,z)) -> c_35(size#(x),size#(y)) -->_2 size#(n(x,y,z)) -> c_35(size#(x),size#(y)):1 -->_1 size#(n(x,y,z)) -> c_35(size#(x),size#(y)):1 2:W:wb#(n(x,y,z)) -> c_39(size#(y),size#(z),size#(y),size#(z),size#(z),size#(y),wb#(y),wb#(z)) -->_8 wb#(n(x,y,z)) -> c_39(size#(y),size#(z),size#(y),size#(z),size#(z),size#(y),wb#(y),wb#(z)):2 -->_7 wb#(n(x,y,z)) -> c_39(size#(y),size#(z),size#(y),size#(z),size#(z),size#(y),wb#(y),wb#(z)):2 -->_6 size#(n(x,y,z)) -> c_35(size#(x),size#(y)):1 -->_5 size#(n(x,y,z)) -> c_35(size#(x),size#(y)):1 -->_4 size#(n(x,y,z)) -> c_35(size#(x),size#(y)):1 -->_3 size#(n(x,y,z)) -> c_35(size#(x),size#(y)):1 -->_2 size#(n(x,y,z)) -> c_35(size#(x),size#(y)):1 -->_1 size#(n(x,y,z)) -> c_35(size#(x),size#(y)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 2: wb#(n(x,y,z)) -> c_39(size#(y),size#(z),size#(y),size#(z),size#(z),size#(y),wb#(y),wb#(z)) 1: size#(n(x,y,z)) -> c_35(size#(x),size#(y)) ****** Step 7.b:3.b:3.b:4.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/2 ,c_36/0,c_37/0,c_38/0,c_39/8} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). ***** Step 7.b:3.b:3.b:4.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: wb#(n(x,y,z)) -> c_39(size#(y),size#(z),size#(y),size#(z),size#(z),size#(y),wb#(y),wb#(z)) - Weak DPs: size#(n(x,y,z)) -> c_35(size#(x),size#(y)) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/2 ,c_36/0,c_37/0,c_38/0,c_39/8} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:wb#(n(x,y,z)) -> c_39(size#(y),size#(z),size#(y),size#(z),size#(z),size#(y),wb#(y),wb#(z)) -->_6 size#(n(x,y,z)) -> c_35(size#(x),size#(y)):2 -->_5 size#(n(x,y,z)) -> c_35(size#(x),size#(y)):2 -->_4 size#(n(x,y,z)) -> c_35(size#(x),size#(y)):2 -->_3 size#(n(x,y,z)) -> c_35(size#(x),size#(y)):2 -->_2 size#(n(x,y,z)) -> c_35(size#(x),size#(y)):2 -->_1 size#(n(x,y,z)) -> c_35(size#(x),size#(y)):2 -->_8 wb#(n(x,y,z)) -> c_39(size#(y),size#(z),size#(y),size#(z),size#(z),size#(y),wb#(y),wb#(z)):1 -->_7 wb#(n(x,y,z)) -> c_39(size#(y),size#(z),size#(y),size#(z),size#(z),size#(y),wb#(y),wb#(z)):1 2:W:size#(n(x,y,z)) -> c_35(size#(x),size#(y)) -->_2 size#(n(x,y,z)) -> c_35(size#(x),size#(y)):2 -->_1 size#(n(x,y,z)) -> c_35(size#(x),size#(y)):2 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 2: size#(n(x,y,z)) -> c_35(size#(x),size#(y)) ***** Step 7.b:3.b:3.b:4.b:2: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: wb#(n(x,y,z)) -> c_39(size#(y),size#(z),size#(y),size#(z),size#(z),size#(y),wb#(y),wb#(z)) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/2 ,c_36/0,c_37/0,c_38/0,c_39/8} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:wb#(n(x,y,z)) -> c_39(size#(y),size#(z),size#(y),size#(z),size#(z),size#(y),wb#(y),wb#(z)) -->_8 wb#(n(x,y,z)) -> c_39(size#(y),size#(z),size#(y),size#(z),size#(z),size#(y),wb#(y),wb#(z)):1 -->_7 wb#(n(x,y,z)) -> c_39(size#(y),size#(z),size#(y),size#(z),size#(z),size#(y),wb#(y),wb#(z)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: wb#(n(x,y,z)) -> c_39(wb#(y),wb#(z)) ***** Step 7.b:3.b:3.b:4.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: wb#(n(x,y,z)) -> c_39(wb#(y),wb#(z)) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/2 ,c_36/0,c_37/0,c_38/0,c_39/2} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: wb#(n(x,y,z)) -> c_39(wb#(y),wb#(z)) The strictly oriented rules are moved into the weak component. ****** Step 7.b:3.b:3.b:4.b:3.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: wb#(n(x,y,z)) -> c_39(wb#(y),wb#(z)) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/2 ,c_36/0,c_37/0,c_38/0,c_39/2} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_39) = {1,2} Following symbols are considered usable: {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val#,wb#} TcT has computed the following interpretation: p(#) = [1] p(+) = [4] p(-) = [0] p(0) = [0] p(1) = [1] x1 + [0] p(and) = [1] p(bs) = [0] p(false) = [0] p(ge) = [1] x1 + [1] x2 + [0] p(if) = [1] x2 + [2] x3 + [1] p(l) = [1] p(max) = [1] p(min) = [0] p(n) = [1] x1 + [1] x2 + [1] x3 + [1] p(not) = [0] p(size) = [0] p(true) = [0] p(val) = [0] p(wb) = [0] p(+#) = [0] p(-#) = [0] p(0#) = [0] p(and#) = [0] p(bs#) = [0] p(ge#) = [0] p(if#) = [0] p(max#) = [0] p(min#) = [0] p(not#) = [0] p(size#) = [0] p(val#) = [0] p(wb#) = [9] x1 + [7] p(c_1) = [0] p(c_2) = [2] x1 + [1] x2 + [0] p(c_3) = [0] p(c_4) = [1] x1 + [0] p(c_5) = [8] x1 + [0] p(c_6) = [1] x1 + [0] p(c_7) = [1] x2 + [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [1] x1 + [0] p(c_11) = [8] x2 + [0] p(c_12) = [1] x1 + [0] p(c_13) = [1] x1 + [0] p(c_14) = [0] p(c_15) = [0] p(c_16) = [0] p(c_17) = [0] p(c_18) = [1] x1 + [2] x2 + [1] x3 + [1] x4 + [8] x5 + [0] p(c_19) = [0] p(c_20) = [0] p(c_21) = [0] p(c_22) = [1] x1 + [0] p(c_23) = [0] p(c_24) = [1] x1 + [0] p(c_25) = [1] x1 + [0] p(c_26) = [0] p(c_27) = [0] p(c_28) = [0] p(c_29) = [0] p(c_30) = [0] p(c_31) = [2] x1 + [0] p(c_32) = [2] p(c_33) = [4] p(c_34) = [0] p(c_35) = [1] p(c_36) = [2] p(c_37) = [1] p(c_38) = [8] p(c_39) = [1] x1 + [1] x2 + [1] Following rules are strictly oriented: wb#(n(x,y,z)) = [9] x + [9] y + [9] z + [16] > [9] y + [9] z + [15] = c_39(wb#(y),wb#(z)) Following rules are (at-least) weakly oriented: ****** Step 7.b:3.b:3.b:4.b:3.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: wb#(n(x,y,z)) -> c_39(wb#(y),wb#(z)) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/2 ,c_36/0,c_37/0,c_38/0,c_39/2} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ****** Step 7.b:3.b:3.b:4.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: wb#(n(x,y,z)) -> c_39(wb#(y),wb#(z)) - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/2 ,c_36/0,c_37/0,c_38/0,c_39/2} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:wb#(n(x,y,z)) -> c_39(wb#(y),wb#(z)) -->_2 wb#(n(x,y,z)) -> c_39(wb#(y),wb#(z)):1 -->_1 wb#(n(x,y,z)) -> c_39(wb#(y),wb#(z)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: wb#(n(x,y,z)) -> c_39(wb#(y),wb#(z)) ****** Step 7.b:3.b:3.b:4.b:3.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Signature: {+/2,-/2,0/1,and/2,bs/1,ge/2,if/3,max/1,min/1,not/1,size/1,val/1,wb/1,+#/2,-#/2,0#/1,and#/2,bs#/1,ge#/2 ,if#/3,max#/1,min#/1,not#/1,size#/1,val#/1,wb#/1} / {#/0,1/1,false/0,l/1,n/3,true/0,c_1/0,c_2/2,c_3/0,c_4/1 ,c_5/1,c_6/1,c_7/2,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/1,c_14/0,c_15/0,c_16/0,c_17/0,c_18/6,c_19/0,c_20/1 ,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/1,c_32/0,c_33/0,c_34/0,c_35/2 ,c_36/0,c_37/0,c_38/0,c_39/2} - Obligation: innermost runtime complexity wrt. defined symbols {+#,-#,0#,and#,bs#,ge#,if#,max#,min#,not#,size#,val# ,wb#} and constructors {#,1,false,l,n,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^4))