WORST_CASE(?,O(n^2)) * Step 1: DependencyPairs WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: 'cklt('EQ()) -> 'false() 'cklt('GT()) -> 'false() 'cklt('LT()) -> 'true() 'compare('0(),'0()) -> 'EQ() 'compare('0(),'neg(y)) -> 'GT() 'compare('0(),'pos(y)) -> 'LT() 'compare('0(),'s(y)) -> 'LT() 'compare('neg(x),'0()) -> 'LT() 'compare('neg(x),'neg(y)) -> 'compare(y,x) 'compare('neg(x),'pos(y)) -> 'LT() 'compare('pos(x),'0()) -> 'GT() 'compare('pos(x),'neg(y)) -> 'GT() 'compare('pos(x),'pos(y)) -> 'compare(x,y) 'compare('s(x),'0()) -> 'GT() 'compare('s(x),'s(y)) -> 'compare(x,y) 'less(x,y) -> 'cklt('compare(x,y)) append(l1,l2) -> append'1(l1,l2) append'1(dd(x,xs),l2) -> dd(x,append(xs,l2)) append'1(nil(),l2) -> l2 flatten(t) -> flatten'1(t) flatten'1(leaf()) -> nil() flatten'1(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2))) flattensort(t) -> insertionsort(flatten(t)) insert(x,l) -> insert'1(l,x) insert'1(dd(y,ys),x) -> insert'2('less(y,x),x,y,ys) insert'1(nil(),x) -> dd(x,nil()) insert'2('false(),x,y,ys) -> dd(x,dd(y,ys)) insert'2('true(),x,y,ys) -> dd(y,insert(x,ys)) insertionsort(l) -> insertionsort'1(l) insertionsort'1(dd(x,xs)) -> insert(x,insertionsort(xs)) insertionsort'1(nil()) -> nil() - Signature: {'cklt/1,'compare/2,'less/2,append/2,append'1/2,flatten/1,flatten'1/1,flattensort/1,insert/2,insert'1/2 ,insert'2/4,insertionsort/1,insertionsort'1/1} / {'0/0,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0 ,dd/2,leaf/0,nil/0,node/3} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt,'compare,'less,append,append'1,flatten,flatten'1 ,flattensort,insert,insert'1,insert'2,insertionsort,insertionsort'1} and constructors {'0,'EQ,'GT,'LT,'false ,'neg,'pos,'s,'true,dd,leaf,nil,node} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs 'cklt#('EQ()) -> c_1() 'cklt#('GT()) -> c_2() 'cklt#('LT()) -> c_3() 'compare#('0(),'0()) -> c_4() 'compare#('0(),'neg(y)) -> c_5() 'compare#('0(),'pos(y)) -> c_6() 'compare#('0(),'s(y)) -> c_7() 'compare#('neg(x),'0()) -> c_8() 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)) 'compare#('neg(x),'pos(y)) -> c_10() 'compare#('pos(x),'0()) -> c_11() 'compare#('pos(x),'neg(y)) -> c_12() 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)) 'compare#('s(x),'0()) -> c_14() 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)) 'less#(x,y) -> c_16('cklt#('compare(x,y)),'compare#(x,y)) append#(l1,l2) -> c_17(append'1#(l1,l2)) append'1#(dd(x,xs),l2) -> c_18(append#(xs,l2)) append'1#(nil(),l2) -> c_19() flatten#(t) -> c_20(flatten'1#(t)) flatten'1#(leaf()) -> c_21() flatten'1#(node(l,t1,t2)) -> c_22(append#(l,append(flatten(t1),flatten(t2))) ,append#(flatten(t1),flatten(t2)) ,flatten#(t1) ,flatten#(t2)) flattensort#(t) -> c_23(insertionsort#(flatten(t)),flatten#(t)) insert#(x,l) -> c_24(insert'1#(l,x)) insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys),'less#(y,x)) insert'1#(nil(),x) -> c_26() insert'2#('false(),x,y,ys) -> c_27() insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)) insertionsort#(l) -> c_29(insertionsort'1#(l)) insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) insertionsort'1#(nil()) -> c_31() Weak DPs and mark the set of starting terms. * Step 2: UsableRules WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: 'cklt#('EQ()) -> c_1() 'cklt#('GT()) -> c_2() 'cklt#('LT()) -> c_3() 'compare#('0(),'0()) -> c_4() 'compare#('0(),'neg(y)) -> c_5() 'compare#('0(),'pos(y)) -> c_6() 'compare#('0(),'s(y)) -> c_7() 'compare#('neg(x),'0()) -> c_8() 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)) 'compare#('neg(x),'pos(y)) -> c_10() 'compare#('pos(x),'0()) -> c_11() 'compare#('pos(x),'neg(y)) -> c_12() 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)) 'compare#('s(x),'0()) -> c_14() 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)) 'less#(x,y) -> c_16('cklt#('compare(x,y)),'compare#(x,y)) append#(l1,l2) -> c_17(append'1#(l1,l2)) append'1#(dd(x,xs),l2) -> c_18(append#(xs,l2)) append'1#(nil(),l2) -> c_19() flatten#(t) -> c_20(flatten'1#(t)) flatten'1#(leaf()) -> c_21() flatten'1#(node(l,t1,t2)) -> c_22(append#(l,append(flatten(t1),flatten(t2))) ,append#(flatten(t1),flatten(t2)) ,flatten#(t1) ,flatten#(t2)) flattensort#(t) -> c_23(insertionsort#(flatten(t)),flatten#(t)) insert#(x,l) -> c_24(insert'1#(l,x)) insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys),'less#(y,x)) insert'1#(nil(),x) -> c_26() insert'2#('false(),x,y,ys) -> c_27() insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)) insertionsort#(l) -> c_29(insertionsort'1#(l)) insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) insertionsort'1#(nil()) -> c_31() - Weak TRS: 'cklt('EQ()) -> 'false() 'cklt('GT()) -> 'false() 'cklt('LT()) -> 'true() 'compare('0(),'0()) -> 'EQ() 'compare('0(),'neg(y)) -> 'GT() 'compare('0(),'pos(y)) -> 'LT() 'compare('0(),'s(y)) -> 'LT() 'compare('neg(x),'0()) -> 'LT() 'compare('neg(x),'neg(y)) -> 'compare(y,x) 'compare('neg(x),'pos(y)) -> 'LT() 'compare('pos(x),'0()) -> 'GT() 'compare('pos(x),'neg(y)) -> 'GT() 'compare('pos(x),'pos(y)) -> 'compare(x,y) 'compare('s(x),'0()) -> 'GT() 'compare('s(x),'s(y)) -> 'compare(x,y) 'less(x,y) -> 'cklt('compare(x,y)) append(l1,l2) -> append'1(l1,l2) append'1(dd(x,xs),l2) -> dd(x,append(xs,l2)) append'1(nil(),l2) -> l2 flatten(t) -> flatten'1(t) flatten'1(leaf()) -> nil() flatten'1(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2))) flattensort(t) -> insertionsort(flatten(t)) insert(x,l) -> insert'1(l,x) insert'1(dd(y,ys),x) -> insert'2('less(y,x),x,y,ys) insert'1(nil(),x) -> dd(x,nil()) insert'2('false(),x,y,ys) -> dd(x,dd(y,ys)) insert'2('true(),x,y,ys) -> dd(y,insert(x,ys)) insertionsort(l) -> insertionsort'1(l) insertionsort'1(dd(x,xs)) -> insert(x,insertionsort(xs)) insertionsort'1(nil()) -> nil() - Signature: {'cklt/1,'compare/2,'less/2,append/2,append'1/2,flatten/1,flatten'1/1,flattensort/1,insert/2,insert'1/2 ,insert'2/4,insertionsort/1,insertionsort'1/1,'cklt#/1,'compare#/2,'less#/2,append#/2,append'1#/2,flatten#/1 ,flatten'1#/1,flattensort#/1,insert#/2,insert'1#/2,insert'2#/4,insertionsort#/1,insertionsort'1#/1} / {'0/0 ,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,leaf/0,nil/0,node/3,c_1/0,c_2/0,c_3/0,c_4/0 ,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/2,c_17/1,c_18/1,c_19/0,c_20/1 ,c_21/0,c_22/4,c_23/2,c_24/1,c_25/2,c_26/0,c_27/0,c_28/1,c_29/1,c_30/2,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,append#,append'1#,flatten# ,flatten'1#,flattensort#,insert#,insert'1#,insert'2#,insertionsort#,insertionsort'1#} and constructors {'0 ,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,leaf,nil,node} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: 'cklt('EQ()) -> 'false() 'cklt('GT()) -> 'false() 'cklt('LT()) -> 'true() 'compare('0(),'0()) -> 'EQ() 'compare('0(),'neg(y)) -> 'GT() 'compare('0(),'pos(y)) -> 'LT() 'compare('0(),'s(y)) -> 'LT() 'compare('neg(x),'0()) -> 'LT() 'compare('neg(x),'neg(y)) -> 'compare(y,x) 'compare('neg(x),'pos(y)) -> 'LT() 'compare('pos(x),'0()) -> 'GT() 'compare('pos(x),'neg(y)) -> 'GT() 'compare('pos(x),'pos(y)) -> 'compare(x,y) 'compare('s(x),'0()) -> 'GT() 'compare('s(x),'s(y)) -> 'compare(x,y) 'less(x,y) -> 'cklt('compare(x,y)) append(l1,l2) -> append'1(l1,l2) append'1(dd(x,xs),l2) -> dd(x,append(xs,l2)) append'1(nil(),l2) -> l2 flatten(t) -> flatten'1(t) flatten'1(leaf()) -> nil() flatten'1(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2))) insert(x,l) -> insert'1(l,x) insert'1(dd(y,ys),x) -> insert'2('less(y,x),x,y,ys) insert'1(nil(),x) -> dd(x,nil()) insert'2('false(),x,y,ys) -> dd(x,dd(y,ys)) insert'2('true(),x,y,ys) -> dd(y,insert(x,ys)) insertionsort(l) -> insertionsort'1(l) insertionsort'1(dd(x,xs)) -> insert(x,insertionsort(xs)) insertionsort'1(nil()) -> nil() 'cklt#('EQ()) -> c_1() 'cklt#('GT()) -> c_2() 'cklt#('LT()) -> c_3() 'compare#('0(),'0()) -> c_4() 'compare#('0(),'neg(y)) -> c_5() 'compare#('0(),'pos(y)) -> c_6() 'compare#('0(),'s(y)) -> c_7() 'compare#('neg(x),'0()) -> c_8() 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)) 'compare#('neg(x),'pos(y)) -> c_10() 'compare#('pos(x),'0()) -> c_11() 'compare#('pos(x),'neg(y)) -> c_12() 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)) 'compare#('s(x),'0()) -> c_14() 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)) 'less#(x,y) -> c_16('cklt#('compare(x,y)),'compare#(x,y)) append#(l1,l2) -> c_17(append'1#(l1,l2)) append'1#(dd(x,xs),l2) -> c_18(append#(xs,l2)) append'1#(nil(),l2) -> c_19() flatten#(t) -> c_20(flatten'1#(t)) flatten'1#(leaf()) -> c_21() flatten'1#(node(l,t1,t2)) -> c_22(append#(l,append(flatten(t1),flatten(t2))) ,append#(flatten(t1),flatten(t2)) ,flatten#(t1) ,flatten#(t2)) flattensort#(t) -> c_23(insertionsort#(flatten(t)),flatten#(t)) insert#(x,l) -> c_24(insert'1#(l,x)) insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys),'less#(y,x)) insert'1#(nil(),x) -> c_26() insert'2#('false(),x,y,ys) -> c_27() insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)) insertionsort#(l) -> c_29(insertionsort'1#(l)) insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) insertionsort'1#(nil()) -> c_31() * Step 3: PredecessorEstimation WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: 'cklt#('EQ()) -> c_1() 'cklt#('GT()) -> c_2() 'cklt#('LT()) -> c_3() 'compare#('0(),'0()) -> c_4() 'compare#('0(),'neg(y)) -> c_5() 'compare#('0(),'pos(y)) -> c_6() 'compare#('0(),'s(y)) -> c_7() 'compare#('neg(x),'0()) -> c_8() 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)) 'compare#('neg(x),'pos(y)) -> c_10() 'compare#('pos(x),'0()) -> c_11() 'compare#('pos(x),'neg(y)) -> c_12() 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)) 'compare#('s(x),'0()) -> c_14() 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)) 'less#(x,y) -> c_16('cklt#('compare(x,y)),'compare#(x,y)) append#(l1,l2) -> c_17(append'1#(l1,l2)) append'1#(dd(x,xs),l2) -> c_18(append#(xs,l2)) append'1#(nil(),l2) -> c_19() flatten#(t) -> c_20(flatten'1#(t)) flatten'1#(leaf()) -> c_21() flatten'1#(node(l,t1,t2)) -> c_22(append#(l,append(flatten(t1),flatten(t2))) ,append#(flatten(t1),flatten(t2)) ,flatten#(t1) ,flatten#(t2)) flattensort#(t) -> c_23(insertionsort#(flatten(t)),flatten#(t)) insert#(x,l) -> c_24(insert'1#(l,x)) insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys),'less#(y,x)) insert'1#(nil(),x) -> c_26() insert'2#('false(),x,y,ys) -> c_27() insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)) insertionsort#(l) -> c_29(insertionsort'1#(l)) insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) insertionsort'1#(nil()) -> c_31() - Weak TRS: 'cklt('EQ()) -> 'false() 'cklt('GT()) -> 'false() 'cklt('LT()) -> 'true() 'compare('0(),'0()) -> 'EQ() 'compare('0(),'neg(y)) -> 'GT() 'compare('0(),'pos(y)) -> 'LT() 'compare('0(),'s(y)) -> 'LT() 'compare('neg(x),'0()) -> 'LT() 'compare('neg(x),'neg(y)) -> 'compare(y,x) 'compare('neg(x),'pos(y)) -> 'LT() 'compare('pos(x),'0()) -> 'GT() 'compare('pos(x),'neg(y)) -> 'GT() 'compare('pos(x),'pos(y)) -> 'compare(x,y) 'compare('s(x),'0()) -> 'GT() 'compare('s(x),'s(y)) -> 'compare(x,y) 'less(x,y) -> 'cklt('compare(x,y)) append(l1,l2) -> append'1(l1,l2) append'1(dd(x,xs),l2) -> dd(x,append(xs,l2)) append'1(nil(),l2) -> l2 flatten(t) -> flatten'1(t) flatten'1(leaf()) -> nil() flatten'1(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2))) insert(x,l) -> insert'1(l,x) insert'1(dd(y,ys),x) -> insert'2('less(y,x),x,y,ys) insert'1(nil(),x) -> dd(x,nil()) insert'2('false(),x,y,ys) -> dd(x,dd(y,ys)) insert'2('true(),x,y,ys) -> dd(y,insert(x,ys)) insertionsort(l) -> insertionsort'1(l) insertionsort'1(dd(x,xs)) -> insert(x,insertionsort(xs)) insertionsort'1(nil()) -> nil() - Signature: {'cklt/1,'compare/2,'less/2,append/2,append'1/2,flatten/1,flatten'1/1,flattensort/1,insert/2,insert'1/2 ,insert'2/4,insertionsort/1,insertionsort'1/1,'cklt#/1,'compare#/2,'less#/2,append#/2,append'1#/2,flatten#/1 ,flatten'1#/1,flattensort#/1,insert#/2,insert'1#/2,insert'2#/4,insertionsort#/1,insertionsort'1#/1} / {'0/0 ,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,leaf/0,nil/0,node/3,c_1/0,c_2/0,c_3/0,c_4/0 ,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/2,c_17/1,c_18/1,c_19/0,c_20/1 ,c_21/0,c_22/4,c_23/2,c_24/1,c_25/2,c_26/0,c_27/0,c_28/1,c_29/1,c_30/2,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,append#,append'1#,flatten# ,flatten'1#,flattensort#,insert#,insert'1#,insert'2#,insertionsort#,insertionsort'1#} and constructors {'0 ,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,leaf,nil,node} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,2,3,4,5,6,7,8,10,11,12,14,19,21,26,27,31} by application of Pre({1,2,3,4,5,6,7,8,10,11,12,14,19,21,26,27,31}) = {9,13,15,16,17,20,24,25,29}. Here rules are labelled as follows: 1: 'cklt#('EQ()) -> c_1() 2: 'cklt#('GT()) -> c_2() 3: 'cklt#('LT()) -> c_3() 4: 'compare#('0(),'0()) -> c_4() 5: 'compare#('0(),'neg(y)) -> c_5() 6: 'compare#('0(),'pos(y)) -> c_6() 7: 'compare#('0(),'s(y)) -> c_7() 8: 'compare#('neg(x),'0()) -> c_8() 9: 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)) 10: 'compare#('neg(x),'pos(y)) -> c_10() 11: 'compare#('pos(x),'0()) -> c_11() 12: 'compare#('pos(x),'neg(y)) -> c_12() 13: 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)) 14: 'compare#('s(x),'0()) -> c_14() 15: 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)) 16: 'less#(x,y) -> c_16('cklt#('compare(x,y)),'compare#(x,y)) 17: append#(l1,l2) -> c_17(append'1#(l1,l2)) 18: append'1#(dd(x,xs),l2) -> c_18(append#(xs,l2)) 19: append'1#(nil(),l2) -> c_19() 20: flatten#(t) -> c_20(flatten'1#(t)) 21: flatten'1#(leaf()) -> c_21() 22: flatten'1#(node(l,t1,t2)) -> c_22(append#(l,append(flatten(t1),flatten(t2))) ,append#(flatten(t1),flatten(t2)) ,flatten#(t1) ,flatten#(t2)) 23: flattensort#(t) -> c_23(insertionsort#(flatten(t)),flatten#(t)) 24: insert#(x,l) -> c_24(insert'1#(l,x)) 25: insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys),'less#(y,x)) 26: insert'1#(nil(),x) -> c_26() 27: insert'2#('false(),x,y,ys) -> c_27() 28: insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)) 29: insertionsort#(l) -> c_29(insertionsort'1#(l)) 30: insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) 31: insertionsort'1#(nil()) -> c_31() * Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)) 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)) 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)) 'less#(x,y) -> c_16('cklt#('compare(x,y)),'compare#(x,y)) append#(l1,l2) -> c_17(append'1#(l1,l2)) append'1#(dd(x,xs),l2) -> c_18(append#(xs,l2)) flatten#(t) -> c_20(flatten'1#(t)) flatten'1#(node(l,t1,t2)) -> c_22(append#(l,append(flatten(t1),flatten(t2))) ,append#(flatten(t1),flatten(t2)) ,flatten#(t1) ,flatten#(t2)) flattensort#(t) -> c_23(insertionsort#(flatten(t)),flatten#(t)) insert#(x,l) -> c_24(insert'1#(l,x)) insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys),'less#(y,x)) insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)) insertionsort#(l) -> c_29(insertionsort'1#(l)) insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) - Weak DPs: 'cklt#('EQ()) -> c_1() 'cklt#('GT()) -> c_2() 'cklt#('LT()) -> c_3() 'compare#('0(),'0()) -> c_4() 'compare#('0(),'neg(y)) -> c_5() 'compare#('0(),'pos(y)) -> c_6() 'compare#('0(),'s(y)) -> c_7() 'compare#('neg(x),'0()) -> c_8() 'compare#('neg(x),'pos(y)) -> c_10() 'compare#('pos(x),'0()) -> c_11() 'compare#('pos(x),'neg(y)) -> c_12() 'compare#('s(x),'0()) -> c_14() append'1#(nil(),l2) -> c_19() flatten'1#(leaf()) -> c_21() insert'1#(nil(),x) -> c_26() insert'2#('false(),x,y,ys) -> c_27() insertionsort'1#(nil()) -> c_31() - Weak TRS: 'cklt('EQ()) -> 'false() 'cklt('GT()) -> 'false() 'cklt('LT()) -> 'true() 'compare('0(),'0()) -> 'EQ() 'compare('0(),'neg(y)) -> 'GT() 'compare('0(),'pos(y)) -> 'LT() 'compare('0(),'s(y)) -> 'LT() 'compare('neg(x),'0()) -> 'LT() 'compare('neg(x),'neg(y)) -> 'compare(y,x) 'compare('neg(x),'pos(y)) -> 'LT() 'compare('pos(x),'0()) -> 'GT() 'compare('pos(x),'neg(y)) -> 'GT() 'compare('pos(x),'pos(y)) -> 'compare(x,y) 'compare('s(x),'0()) -> 'GT() 'compare('s(x),'s(y)) -> 'compare(x,y) 'less(x,y) -> 'cklt('compare(x,y)) append(l1,l2) -> append'1(l1,l2) append'1(dd(x,xs),l2) -> dd(x,append(xs,l2)) append'1(nil(),l2) -> l2 flatten(t) -> flatten'1(t) flatten'1(leaf()) -> nil() flatten'1(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2))) insert(x,l) -> insert'1(l,x) insert'1(dd(y,ys),x) -> insert'2('less(y,x),x,y,ys) insert'1(nil(),x) -> dd(x,nil()) insert'2('false(),x,y,ys) -> dd(x,dd(y,ys)) insert'2('true(),x,y,ys) -> dd(y,insert(x,ys)) insertionsort(l) -> insertionsort'1(l) insertionsort'1(dd(x,xs)) -> insert(x,insertionsort(xs)) insertionsort'1(nil()) -> nil() - Signature: {'cklt/1,'compare/2,'less/2,append/2,append'1/2,flatten/1,flatten'1/1,flattensort/1,insert/2,insert'1/2 ,insert'2/4,insertionsort/1,insertionsort'1/1,'cklt#/1,'compare#/2,'less#/2,append#/2,append'1#/2,flatten#/1 ,flatten'1#/1,flattensort#/1,insert#/2,insert'1#/2,insert'2#/4,insertionsort#/1,insertionsort'1#/1} / {'0/0 ,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,leaf/0,nil/0,node/3,c_1/0,c_2/0,c_3/0,c_4/0 ,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/2,c_17/1,c_18/1,c_19/0,c_20/1 ,c_21/0,c_22/4,c_23/2,c_24/1,c_25/2,c_26/0,c_27/0,c_28/1,c_29/1,c_30/2,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,append#,append'1#,flatten# ,flatten'1#,flattensort#,insert#,insert'1#,insert'2#,insertionsort#,insertionsort'1#} and constructors {'0 ,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,leaf,nil,node} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)) -->_1 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)):3 -->_1 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)):2 -->_1 'compare#('s(x),'0()) -> c_14():26 -->_1 'compare#('pos(x),'neg(y)) -> c_12():25 -->_1 'compare#('pos(x),'0()) -> c_11():24 -->_1 'compare#('neg(x),'pos(y)) -> c_10():23 -->_1 'compare#('neg(x),'0()) -> c_8():22 -->_1 'compare#('0(),'s(y)) -> c_7():21 -->_1 'compare#('0(),'pos(y)) -> c_6():20 -->_1 'compare#('0(),'neg(y)) -> c_5():19 -->_1 'compare#('0(),'0()) -> c_4():18 -->_1 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)):1 2:S:'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)) -->_1 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)):3 -->_1 'compare#('s(x),'0()) -> c_14():26 -->_1 'compare#('pos(x),'neg(y)) -> c_12():25 -->_1 'compare#('pos(x),'0()) -> c_11():24 -->_1 'compare#('neg(x),'pos(y)) -> c_10():23 -->_1 'compare#('neg(x),'0()) -> c_8():22 -->_1 'compare#('0(),'s(y)) -> c_7():21 -->_1 'compare#('0(),'pos(y)) -> c_6():20 -->_1 'compare#('0(),'neg(y)) -> c_5():19 -->_1 'compare#('0(),'0()) -> c_4():18 -->_1 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)):2 -->_1 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)):1 3:S:'compare#('s(x),'s(y)) -> c_15('compare#(x,y)) -->_1 'compare#('s(x),'0()) -> c_14():26 -->_1 'compare#('pos(x),'neg(y)) -> c_12():25 -->_1 'compare#('pos(x),'0()) -> c_11():24 -->_1 'compare#('neg(x),'pos(y)) -> c_10():23 -->_1 'compare#('neg(x),'0()) -> c_8():22 -->_1 'compare#('0(),'s(y)) -> c_7():21 -->_1 'compare#('0(),'pos(y)) -> c_6():20 -->_1 'compare#('0(),'neg(y)) -> c_5():19 -->_1 'compare#('0(),'0()) -> c_4():18 -->_1 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)):3 -->_1 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)):2 -->_1 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)):1 4:S:'less#(x,y) -> c_16('cklt#('compare(x,y)),'compare#(x,y)) -->_2 'compare#('s(x),'0()) -> c_14():26 -->_2 'compare#('pos(x),'neg(y)) -> c_12():25 -->_2 'compare#('pos(x),'0()) -> c_11():24 -->_2 'compare#('neg(x),'pos(y)) -> c_10():23 -->_2 'compare#('neg(x),'0()) -> c_8():22 -->_2 'compare#('0(),'s(y)) -> c_7():21 -->_2 'compare#('0(),'pos(y)) -> c_6():20 -->_2 'compare#('0(),'neg(y)) -> c_5():19 -->_2 'compare#('0(),'0()) -> c_4():18 -->_1 'cklt#('LT()) -> c_3():17 -->_1 'cklt#('GT()) -> c_2():16 -->_1 'cklt#('EQ()) -> c_1():15 -->_2 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)):3 -->_2 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)):2 -->_2 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)):1 5:S:append#(l1,l2) -> c_17(append'1#(l1,l2)) -->_1 append'1#(dd(x,xs),l2) -> c_18(append#(xs,l2)):6 -->_1 append'1#(nil(),l2) -> c_19():27 6:S:append'1#(dd(x,xs),l2) -> c_18(append#(xs,l2)) -->_1 append#(l1,l2) -> c_17(append'1#(l1,l2)):5 7:S:flatten#(t) -> c_20(flatten'1#(t)) -->_1 flatten'1#(node(l,t1,t2)) -> c_22(append#(l,append(flatten(t1),flatten(t2))) ,append#(flatten(t1),flatten(t2)) ,flatten#(t1) ,flatten#(t2)):8 -->_1 flatten'1#(leaf()) -> c_21():28 8:S:flatten'1#(node(l,t1,t2)) -> c_22(append#(l,append(flatten(t1),flatten(t2))) ,append#(flatten(t1),flatten(t2)) ,flatten#(t1) ,flatten#(t2)) -->_4 flatten#(t) -> c_20(flatten'1#(t)):7 -->_3 flatten#(t) -> c_20(flatten'1#(t)):7 -->_2 append#(l1,l2) -> c_17(append'1#(l1,l2)):5 -->_1 append#(l1,l2) -> c_17(append'1#(l1,l2)):5 9:S:flattensort#(t) -> c_23(insertionsort#(flatten(t)),flatten#(t)) -->_1 insertionsort#(l) -> c_29(insertionsort'1#(l)):13 -->_2 flatten#(t) -> c_20(flatten'1#(t)):7 10:S:insert#(x,l) -> c_24(insert'1#(l,x)) -->_1 insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys),'less#(y,x)):11 -->_1 insert'1#(nil(),x) -> c_26():29 11:S:insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys),'less#(y,x)) -->_1 insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)):12 -->_1 insert'2#('false(),x,y,ys) -> c_27():30 -->_2 'less#(x,y) -> c_16('cklt#('compare(x,y)),'compare#(x,y)):4 12:S:insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)) -->_1 insert#(x,l) -> c_24(insert'1#(l,x)):10 13:S:insertionsort#(l) -> c_29(insertionsort'1#(l)) -->_1 insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)):14 -->_1 insertionsort'1#(nil()) -> c_31():31 14:S:insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) -->_2 insertionsort#(l) -> c_29(insertionsort'1#(l)):13 -->_1 insert#(x,l) -> c_24(insert'1#(l,x)):10 15:W:'cklt#('EQ()) -> c_1() 16:W:'cklt#('GT()) -> c_2() 17:W:'cklt#('LT()) -> c_3() 18:W:'compare#('0(),'0()) -> c_4() 19:W:'compare#('0(),'neg(y)) -> c_5() 20:W:'compare#('0(),'pos(y)) -> c_6() 21:W:'compare#('0(),'s(y)) -> c_7() 22:W:'compare#('neg(x),'0()) -> c_8() 23:W:'compare#('neg(x),'pos(y)) -> c_10() 24:W:'compare#('pos(x),'0()) -> c_11() 25:W:'compare#('pos(x),'neg(y)) -> c_12() 26:W:'compare#('s(x),'0()) -> c_14() 27:W:append'1#(nil(),l2) -> c_19() 28:W:flatten'1#(leaf()) -> c_21() 29:W:insert'1#(nil(),x) -> c_26() 30:W:insert'2#('false(),x,y,ys) -> c_27() 31:W:insertionsort'1#(nil()) -> c_31() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 31: insertionsort'1#(nil()) -> c_31() 29: insert'1#(nil(),x) -> c_26() 30: insert'2#('false(),x,y,ys) -> c_27() 28: flatten'1#(leaf()) -> c_21() 27: append'1#(nil(),l2) -> c_19() 15: 'cklt#('EQ()) -> c_1() 16: 'cklt#('GT()) -> c_2() 17: 'cklt#('LT()) -> c_3() 18: 'compare#('0(),'0()) -> c_4() 19: 'compare#('0(),'neg(y)) -> c_5() 20: 'compare#('0(),'pos(y)) -> c_6() 21: 'compare#('0(),'s(y)) -> c_7() 22: 'compare#('neg(x),'0()) -> c_8() 23: 'compare#('neg(x),'pos(y)) -> c_10() 24: 'compare#('pos(x),'0()) -> c_11() 25: 'compare#('pos(x),'neg(y)) -> c_12() 26: 'compare#('s(x),'0()) -> c_14() * Step 5: SimplifyRHS WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)) 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)) 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)) 'less#(x,y) -> c_16('cklt#('compare(x,y)),'compare#(x,y)) append#(l1,l2) -> c_17(append'1#(l1,l2)) append'1#(dd(x,xs),l2) -> c_18(append#(xs,l2)) flatten#(t) -> c_20(flatten'1#(t)) flatten'1#(node(l,t1,t2)) -> c_22(append#(l,append(flatten(t1),flatten(t2))) ,append#(flatten(t1),flatten(t2)) ,flatten#(t1) ,flatten#(t2)) flattensort#(t) -> c_23(insertionsort#(flatten(t)),flatten#(t)) insert#(x,l) -> c_24(insert'1#(l,x)) insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys),'less#(y,x)) insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)) insertionsort#(l) -> c_29(insertionsort'1#(l)) insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) - Weak TRS: 'cklt('EQ()) -> 'false() 'cklt('GT()) -> 'false() 'cklt('LT()) -> 'true() 'compare('0(),'0()) -> 'EQ() 'compare('0(),'neg(y)) -> 'GT() 'compare('0(),'pos(y)) -> 'LT() 'compare('0(),'s(y)) -> 'LT() 'compare('neg(x),'0()) -> 'LT() 'compare('neg(x),'neg(y)) -> 'compare(y,x) 'compare('neg(x),'pos(y)) -> 'LT() 'compare('pos(x),'0()) -> 'GT() 'compare('pos(x),'neg(y)) -> 'GT() 'compare('pos(x),'pos(y)) -> 'compare(x,y) 'compare('s(x),'0()) -> 'GT() 'compare('s(x),'s(y)) -> 'compare(x,y) 'less(x,y) -> 'cklt('compare(x,y)) append(l1,l2) -> append'1(l1,l2) append'1(dd(x,xs),l2) -> dd(x,append(xs,l2)) append'1(nil(),l2) -> l2 flatten(t) -> flatten'1(t) flatten'1(leaf()) -> nil() flatten'1(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2))) insert(x,l) -> insert'1(l,x) insert'1(dd(y,ys),x) -> insert'2('less(y,x),x,y,ys) insert'1(nil(),x) -> dd(x,nil()) insert'2('false(),x,y,ys) -> dd(x,dd(y,ys)) insert'2('true(),x,y,ys) -> dd(y,insert(x,ys)) insertionsort(l) -> insertionsort'1(l) insertionsort'1(dd(x,xs)) -> insert(x,insertionsort(xs)) insertionsort'1(nil()) -> nil() - Signature: {'cklt/1,'compare/2,'less/2,append/2,append'1/2,flatten/1,flatten'1/1,flattensort/1,insert/2,insert'1/2 ,insert'2/4,insertionsort/1,insertionsort'1/1,'cklt#/1,'compare#/2,'less#/2,append#/2,append'1#/2,flatten#/1 ,flatten'1#/1,flattensort#/1,insert#/2,insert'1#/2,insert'2#/4,insertionsort#/1,insertionsort'1#/1} / {'0/0 ,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,leaf/0,nil/0,node/3,c_1/0,c_2/0,c_3/0,c_4/0 ,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/2,c_17/1,c_18/1,c_19/0,c_20/1 ,c_21/0,c_22/4,c_23/2,c_24/1,c_25/2,c_26/0,c_27/0,c_28/1,c_29/1,c_30/2,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,append#,append'1#,flatten# ,flatten'1#,flattensort#,insert#,insert'1#,insert'2#,insertionsort#,insertionsort'1#} and constructors {'0 ,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,leaf,nil,node} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)) -->_1 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)):3 -->_1 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)):2 -->_1 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)):1 2:S:'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)) -->_1 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)):3 -->_1 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)):2 -->_1 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)):1 3:S:'compare#('s(x),'s(y)) -> c_15('compare#(x,y)) -->_1 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)):3 -->_1 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)):2 -->_1 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)):1 4:S:'less#(x,y) -> c_16('cklt#('compare(x,y)),'compare#(x,y)) -->_2 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)):3 -->_2 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)):2 -->_2 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)):1 5:S:append#(l1,l2) -> c_17(append'1#(l1,l2)) -->_1 append'1#(dd(x,xs),l2) -> c_18(append#(xs,l2)):6 6:S:append'1#(dd(x,xs),l2) -> c_18(append#(xs,l2)) -->_1 append#(l1,l2) -> c_17(append'1#(l1,l2)):5 7:S:flatten#(t) -> c_20(flatten'1#(t)) -->_1 flatten'1#(node(l,t1,t2)) -> c_22(append#(l,append(flatten(t1),flatten(t2))) ,append#(flatten(t1),flatten(t2)) ,flatten#(t1) ,flatten#(t2)):8 8:S:flatten'1#(node(l,t1,t2)) -> c_22(append#(l,append(flatten(t1),flatten(t2))) ,append#(flatten(t1),flatten(t2)) ,flatten#(t1) ,flatten#(t2)) -->_4 flatten#(t) -> c_20(flatten'1#(t)):7 -->_3 flatten#(t) -> c_20(flatten'1#(t)):7 -->_2 append#(l1,l2) -> c_17(append'1#(l1,l2)):5 -->_1 append#(l1,l2) -> c_17(append'1#(l1,l2)):5 9:S:flattensort#(t) -> c_23(insertionsort#(flatten(t)),flatten#(t)) -->_1 insertionsort#(l) -> c_29(insertionsort'1#(l)):13 -->_2 flatten#(t) -> c_20(flatten'1#(t)):7 10:S:insert#(x,l) -> c_24(insert'1#(l,x)) -->_1 insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys),'less#(y,x)):11 11:S:insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys),'less#(y,x)) -->_1 insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)):12 -->_2 'less#(x,y) -> c_16('cklt#('compare(x,y)),'compare#(x,y)):4 12:S:insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)) -->_1 insert#(x,l) -> c_24(insert'1#(l,x)):10 13:S:insertionsort#(l) -> c_29(insertionsort'1#(l)) -->_1 insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)):14 14:S:insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) -->_2 insertionsort#(l) -> c_29(insertionsort'1#(l)):13 -->_1 insert#(x,l) -> c_24(insert'1#(l,x)):10 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: 'less#(x,y) -> c_16('compare#(x,y)) * Step 6: Decompose WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)) 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)) 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)) 'less#(x,y) -> c_16('compare#(x,y)) append#(l1,l2) -> c_17(append'1#(l1,l2)) append'1#(dd(x,xs),l2) -> c_18(append#(xs,l2)) flatten#(t) -> c_20(flatten'1#(t)) flatten'1#(node(l,t1,t2)) -> c_22(append#(l,append(flatten(t1),flatten(t2))) ,append#(flatten(t1),flatten(t2)) ,flatten#(t1) ,flatten#(t2)) flattensort#(t) -> c_23(insertionsort#(flatten(t)),flatten#(t)) insert#(x,l) -> c_24(insert'1#(l,x)) insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys),'less#(y,x)) insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)) insertionsort#(l) -> c_29(insertionsort'1#(l)) insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) - Weak TRS: 'cklt('EQ()) -> 'false() 'cklt('GT()) -> 'false() 'cklt('LT()) -> 'true() 'compare('0(),'0()) -> 'EQ() 'compare('0(),'neg(y)) -> 'GT() 'compare('0(),'pos(y)) -> 'LT() 'compare('0(),'s(y)) -> 'LT() 'compare('neg(x),'0()) -> 'LT() 'compare('neg(x),'neg(y)) -> 'compare(y,x) 'compare('neg(x),'pos(y)) -> 'LT() 'compare('pos(x),'0()) -> 'GT() 'compare('pos(x),'neg(y)) -> 'GT() 'compare('pos(x),'pos(y)) -> 'compare(x,y) 'compare('s(x),'0()) -> 'GT() 'compare('s(x),'s(y)) -> 'compare(x,y) 'less(x,y) -> 'cklt('compare(x,y)) append(l1,l2) -> append'1(l1,l2) append'1(dd(x,xs),l2) -> dd(x,append(xs,l2)) append'1(nil(),l2) -> l2 flatten(t) -> flatten'1(t) flatten'1(leaf()) -> nil() flatten'1(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2))) insert(x,l) -> insert'1(l,x) insert'1(dd(y,ys),x) -> insert'2('less(y,x),x,y,ys) insert'1(nil(),x) -> dd(x,nil()) insert'2('false(),x,y,ys) -> dd(x,dd(y,ys)) insert'2('true(),x,y,ys) -> dd(y,insert(x,ys)) insertionsort(l) -> insertionsort'1(l) insertionsort'1(dd(x,xs)) -> insert(x,insertionsort(xs)) insertionsort'1(nil()) -> nil() - Signature: {'cklt/1,'compare/2,'less/2,append/2,append'1/2,flatten/1,flatten'1/1,flattensort/1,insert/2,insert'1/2 ,insert'2/4,insertionsort/1,insertionsort'1/1,'cklt#/1,'compare#/2,'less#/2,append#/2,append'1#/2,flatten#/1 ,flatten'1#/1,flattensort#/1,insert#/2,insert'1#/2,insert'2#/4,insertionsort#/1,insertionsort'1#/1} / {'0/0 ,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,leaf/0,nil/0,node/3,c_1/0,c_2/0,c_3/0,c_4/0 ,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/1,c_19/0,c_20/1 ,c_21/0,c_22/4,c_23/2,c_24/1,c_25/2,c_26/0,c_27/0,c_28/1,c_29/1,c_30/2,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,append#,append'1#,flatten# ,flatten'1#,flattensort#,insert#,insert'1#,insert'2#,insertionsort#,insertionsort'1#} and constructors {'0 ,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,leaf,nil,node} + 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: 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)) 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)) 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)) - Weak DPs: 'less#(x,y) -> c_16('compare#(x,y)) append#(l1,l2) -> c_17(append'1#(l1,l2)) append'1#(dd(x,xs),l2) -> c_18(append#(xs,l2)) flatten#(t) -> c_20(flatten'1#(t)) flatten'1#(node(l,t1,t2)) -> c_22(append#(l,append(flatten(t1),flatten(t2))) ,append#(flatten(t1),flatten(t2)) ,flatten#(t1) ,flatten#(t2)) flattensort#(t) -> c_23(insertionsort#(flatten(t)),flatten#(t)) insert#(x,l) -> c_24(insert'1#(l,x)) insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys),'less#(y,x)) insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)) insertionsort#(l) -> c_29(insertionsort'1#(l)) insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) - Weak TRS: 'cklt('EQ()) -> 'false() 'cklt('GT()) -> 'false() 'cklt('LT()) -> 'true() 'compare('0(),'0()) -> 'EQ() 'compare('0(),'neg(y)) -> 'GT() 'compare('0(),'pos(y)) -> 'LT() 'compare('0(),'s(y)) -> 'LT() 'compare('neg(x),'0()) -> 'LT() 'compare('neg(x),'neg(y)) -> 'compare(y,x) 'compare('neg(x),'pos(y)) -> 'LT() 'compare('pos(x),'0()) -> 'GT() 'compare('pos(x),'neg(y)) -> 'GT() 'compare('pos(x),'pos(y)) -> 'compare(x,y) 'compare('s(x),'0()) -> 'GT() 'compare('s(x),'s(y)) -> 'compare(x,y) 'less(x,y) -> 'cklt('compare(x,y)) append(l1,l2) -> append'1(l1,l2) append'1(dd(x,xs),l2) -> dd(x,append(xs,l2)) append'1(nil(),l2) -> l2 flatten(t) -> flatten'1(t) flatten'1(leaf()) -> nil() flatten'1(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2))) insert(x,l) -> insert'1(l,x) insert'1(dd(y,ys),x) -> insert'2('less(y,x),x,y,ys) insert'1(nil(),x) -> dd(x,nil()) insert'2('false(),x,y,ys) -> dd(x,dd(y,ys)) insert'2('true(),x,y,ys) -> dd(y,insert(x,ys)) insertionsort(l) -> insertionsort'1(l) insertionsort'1(dd(x,xs)) -> insert(x,insertionsort(xs)) insertionsort'1(nil()) -> nil() - Signature: {'cklt/1,'compare/2,'less/2,append/2,append'1/2,flatten/1,flatten'1/1,flattensort/1,insert/2,insert'1/2 ,insert'2/4,insertionsort/1,insertionsort'1/1,'cklt#/1,'compare#/2,'less#/2,append#/2,append'1#/2 ,flatten#/1,flatten'1#/1,flattensort#/1,insert#/2,insert'1#/2,insert'2#/4,insertionsort#/1 ,insertionsort'1#/1} / {'0/0,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,leaf/0,nil/0 ,node/3,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1 ,c_16/1,c_17/1,c_18/1,c_19/0,c_20/1,c_21/0,c_22/4,c_23/2,c_24/1,c_25/2,c_26/0,c_27/0,c_28/1,c_29/1,c_30/2 ,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,append#,append'1#,flatten# ,flatten'1#,flattensort#,insert#,insert'1#,insert'2#,insertionsort#,insertionsort'1#} and constructors {'0 ,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,leaf,nil,node} Problem (S) - Strict DPs: 'less#(x,y) -> c_16('compare#(x,y)) append#(l1,l2) -> c_17(append'1#(l1,l2)) append'1#(dd(x,xs),l2) -> c_18(append#(xs,l2)) flatten#(t) -> c_20(flatten'1#(t)) flatten'1#(node(l,t1,t2)) -> c_22(append#(l,append(flatten(t1),flatten(t2))) ,append#(flatten(t1),flatten(t2)) ,flatten#(t1) ,flatten#(t2)) flattensort#(t) -> c_23(insertionsort#(flatten(t)),flatten#(t)) insert#(x,l) -> c_24(insert'1#(l,x)) insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys),'less#(y,x)) insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)) insertionsort#(l) -> c_29(insertionsort'1#(l)) insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) - Weak DPs: 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)) 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)) 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)) - Weak TRS: 'cklt('EQ()) -> 'false() 'cklt('GT()) -> 'false() 'cklt('LT()) -> 'true() 'compare('0(),'0()) -> 'EQ() 'compare('0(),'neg(y)) -> 'GT() 'compare('0(),'pos(y)) -> 'LT() 'compare('0(),'s(y)) -> 'LT() 'compare('neg(x),'0()) -> 'LT() 'compare('neg(x),'neg(y)) -> 'compare(y,x) 'compare('neg(x),'pos(y)) -> 'LT() 'compare('pos(x),'0()) -> 'GT() 'compare('pos(x),'neg(y)) -> 'GT() 'compare('pos(x),'pos(y)) -> 'compare(x,y) 'compare('s(x),'0()) -> 'GT() 'compare('s(x),'s(y)) -> 'compare(x,y) 'less(x,y) -> 'cklt('compare(x,y)) append(l1,l2) -> append'1(l1,l2) append'1(dd(x,xs),l2) -> dd(x,append(xs,l2)) append'1(nil(),l2) -> l2 flatten(t) -> flatten'1(t) flatten'1(leaf()) -> nil() flatten'1(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2))) insert(x,l) -> insert'1(l,x) insert'1(dd(y,ys),x) -> insert'2('less(y,x),x,y,ys) insert'1(nil(),x) -> dd(x,nil()) insert'2('false(),x,y,ys) -> dd(x,dd(y,ys)) insert'2('true(),x,y,ys) -> dd(y,insert(x,ys)) insertionsort(l) -> insertionsort'1(l) insertionsort'1(dd(x,xs)) -> insert(x,insertionsort(xs)) insertionsort'1(nil()) -> nil() - Signature: {'cklt/1,'compare/2,'less/2,append/2,append'1/2,flatten/1,flatten'1/1,flattensort/1,insert/2,insert'1/2 ,insert'2/4,insertionsort/1,insertionsort'1/1,'cklt#/1,'compare#/2,'less#/2,append#/2,append'1#/2 ,flatten#/1,flatten'1#/1,flattensort#/1,insert#/2,insert'1#/2,insert'2#/4,insertionsort#/1 ,insertionsort'1#/1} / {'0/0,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,leaf/0,nil/0 ,node/3,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1 ,c_16/1,c_17/1,c_18/1,c_19/0,c_20/1,c_21/0,c_22/4,c_23/2,c_24/1,c_25/2,c_26/0,c_27/0,c_28/1,c_29/1,c_30/2 ,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,append#,append'1#,flatten# ,flatten'1#,flattensort#,insert#,insert'1#,insert'2#,insertionsort#,insertionsort'1#} and constructors {'0 ,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,leaf,nil,node} ** Step 6.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)) 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)) 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)) - Weak DPs: 'less#(x,y) -> c_16('compare#(x,y)) append#(l1,l2) -> c_17(append'1#(l1,l2)) append'1#(dd(x,xs),l2) -> c_18(append#(xs,l2)) flatten#(t) -> c_20(flatten'1#(t)) flatten'1#(node(l,t1,t2)) -> c_22(append#(l,append(flatten(t1),flatten(t2))) ,append#(flatten(t1),flatten(t2)) ,flatten#(t1) ,flatten#(t2)) flattensort#(t) -> c_23(insertionsort#(flatten(t)),flatten#(t)) insert#(x,l) -> c_24(insert'1#(l,x)) insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys),'less#(y,x)) insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)) insertionsort#(l) -> c_29(insertionsort'1#(l)) insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) - Weak TRS: 'cklt('EQ()) -> 'false() 'cklt('GT()) -> 'false() 'cklt('LT()) -> 'true() 'compare('0(),'0()) -> 'EQ() 'compare('0(),'neg(y)) -> 'GT() 'compare('0(),'pos(y)) -> 'LT() 'compare('0(),'s(y)) -> 'LT() 'compare('neg(x),'0()) -> 'LT() 'compare('neg(x),'neg(y)) -> 'compare(y,x) 'compare('neg(x),'pos(y)) -> 'LT() 'compare('pos(x),'0()) -> 'GT() 'compare('pos(x),'neg(y)) -> 'GT() 'compare('pos(x),'pos(y)) -> 'compare(x,y) 'compare('s(x),'0()) -> 'GT() 'compare('s(x),'s(y)) -> 'compare(x,y) 'less(x,y) -> 'cklt('compare(x,y)) append(l1,l2) -> append'1(l1,l2) append'1(dd(x,xs),l2) -> dd(x,append(xs,l2)) append'1(nil(),l2) -> l2 flatten(t) -> flatten'1(t) flatten'1(leaf()) -> nil() flatten'1(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2))) insert(x,l) -> insert'1(l,x) insert'1(dd(y,ys),x) -> insert'2('less(y,x),x,y,ys) insert'1(nil(),x) -> dd(x,nil()) insert'2('false(),x,y,ys) -> dd(x,dd(y,ys)) insert'2('true(),x,y,ys) -> dd(y,insert(x,ys)) insertionsort(l) -> insertionsort'1(l) insertionsort'1(dd(x,xs)) -> insert(x,insertionsort(xs)) insertionsort'1(nil()) -> nil() - Signature: {'cklt/1,'compare/2,'less/2,append/2,append'1/2,flatten/1,flatten'1/1,flattensort/1,insert/2,insert'1/2 ,insert'2/4,insertionsort/1,insertionsort'1/1,'cklt#/1,'compare#/2,'less#/2,append#/2,append'1#/2,flatten#/1 ,flatten'1#/1,flattensort#/1,insert#/2,insert'1#/2,insert'2#/4,insertionsort#/1,insertionsort'1#/1} / {'0/0 ,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,leaf/0,nil/0,node/3,c_1/0,c_2/0,c_3/0,c_4/0 ,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/1,c_19/0,c_20/1 ,c_21/0,c_22/4,c_23/2,c_24/1,c_25/2,c_26/0,c_27/0,c_28/1,c_29/1,c_30/2,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,append#,append'1#,flatten# ,flatten'1#,flattensort#,insert#,insert'1#,insert'2#,insertionsort#,insertionsort'1#} and constructors {'0 ,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,leaf,nil,node} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)) -->_1 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)):3 -->_1 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)):2 -->_1 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)):1 2:S:'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)) -->_1 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)):1 -->_1 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)):3 -->_1 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)):2 3:S:'compare#('s(x),'s(y)) -> c_15('compare#(x,y)) -->_1 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)):1 -->_1 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)):2 -->_1 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)):3 4:W:'less#(x,y) -> c_16('compare#(x,y)) -->_1 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)):1 -->_1 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)):2 -->_1 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)):3 5:W:append#(l1,l2) -> c_17(append'1#(l1,l2)) -->_1 append'1#(dd(x,xs),l2) -> c_18(append#(xs,l2)):6 6:W:append'1#(dd(x,xs),l2) -> c_18(append#(xs,l2)) -->_1 append#(l1,l2) -> c_17(append'1#(l1,l2)):5 7:W:flatten#(t) -> c_20(flatten'1#(t)) -->_1 flatten'1#(node(l,t1,t2)) -> c_22(append#(l,append(flatten(t1),flatten(t2))) ,append#(flatten(t1),flatten(t2)) ,flatten#(t1) ,flatten#(t2)):8 8:W:flatten'1#(node(l,t1,t2)) -> c_22(append#(l,append(flatten(t1),flatten(t2))) ,append#(flatten(t1),flatten(t2)) ,flatten#(t1) ,flatten#(t2)) -->_2 append#(l1,l2) -> c_17(append'1#(l1,l2)):5 -->_1 append#(l1,l2) -> c_17(append'1#(l1,l2)):5 -->_4 flatten#(t) -> c_20(flatten'1#(t)):7 -->_3 flatten#(t) -> c_20(flatten'1#(t)):7 9:W:flattensort#(t) -> c_23(insertionsort#(flatten(t)),flatten#(t)) -->_2 flatten#(t) -> c_20(flatten'1#(t)):7 -->_1 insertionsort#(l) -> c_29(insertionsort'1#(l)):13 10:W:insert#(x,l) -> c_24(insert'1#(l,x)) -->_1 insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys),'less#(y,x)):11 11:W:insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys),'less#(y,x)) -->_2 'less#(x,y) -> c_16('compare#(x,y)):4 -->_1 insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)):12 12:W:insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)) -->_1 insert#(x,l) -> c_24(insert'1#(l,x)):10 13:W:insertionsort#(l) -> c_29(insertionsort'1#(l)) -->_1 insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)):14 14:W:insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) -->_1 insert#(x,l) -> c_24(insert'1#(l,x)):10 -->_2 insertionsort#(l) -> c_29(insertionsort'1#(l)):13 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 7: flatten#(t) -> c_20(flatten'1#(t)) 8: flatten'1#(node(l,t1,t2)) -> c_22(append#(l,append(flatten(t1),flatten(t2))) ,append#(flatten(t1),flatten(t2)) ,flatten#(t1) ,flatten#(t2)) 5: append#(l1,l2) -> c_17(append'1#(l1,l2)) 6: append'1#(dd(x,xs),l2) -> c_18(append#(xs,l2)) ** Step 6.a:2: SimplifyRHS WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)) 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)) 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)) - Weak DPs: 'less#(x,y) -> c_16('compare#(x,y)) flattensort#(t) -> c_23(insertionsort#(flatten(t)),flatten#(t)) insert#(x,l) -> c_24(insert'1#(l,x)) insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys),'less#(y,x)) insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)) insertionsort#(l) -> c_29(insertionsort'1#(l)) insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) - Weak TRS: 'cklt('EQ()) -> 'false() 'cklt('GT()) -> 'false() 'cklt('LT()) -> 'true() 'compare('0(),'0()) -> 'EQ() 'compare('0(),'neg(y)) -> 'GT() 'compare('0(),'pos(y)) -> 'LT() 'compare('0(),'s(y)) -> 'LT() 'compare('neg(x),'0()) -> 'LT() 'compare('neg(x),'neg(y)) -> 'compare(y,x) 'compare('neg(x),'pos(y)) -> 'LT() 'compare('pos(x),'0()) -> 'GT() 'compare('pos(x),'neg(y)) -> 'GT() 'compare('pos(x),'pos(y)) -> 'compare(x,y) 'compare('s(x),'0()) -> 'GT() 'compare('s(x),'s(y)) -> 'compare(x,y) 'less(x,y) -> 'cklt('compare(x,y)) append(l1,l2) -> append'1(l1,l2) append'1(dd(x,xs),l2) -> dd(x,append(xs,l2)) append'1(nil(),l2) -> l2 flatten(t) -> flatten'1(t) flatten'1(leaf()) -> nil() flatten'1(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2))) insert(x,l) -> insert'1(l,x) insert'1(dd(y,ys),x) -> insert'2('less(y,x),x,y,ys) insert'1(nil(),x) -> dd(x,nil()) insert'2('false(),x,y,ys) -> dd(x,dd(y,ys)) insert'2('true(),x,y,ys) -> dd(y,insert(x,ys)) insertionsort(l) -> insertionsort'1(l) insertionsort'1(dd(x,xs)) -> insert(x,insertionsort(xs)) insertionsort'1(nil()) -> nil() - Signature: {'cklt/1,'compare/2,'less/2,append/2,append'1/2,flatten/1,flatten'1/1,flattensort/1,insert/2,insert'1/2 ,insert'2/4,insertionsort/1,insertionsort'1/1,'cklt#/1,'compare#/2,'less#/2,append#/2,append'1#/2,flatten#/1 ,flatten'1#/1,flattensort#/1,insert#/2,insert'1#/2,insert'2#/4,insertionsort#/1,insertionsort'1#/1} / {'0/0 ,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,leaf/0,nil/0,node/3,c_1/0,c_2/0,c_3/0,c_4/0 ,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/1,c_19/0,c_20/1 ,c_21/0,c_22/4,c_23/2,c_24/1,c_25/2,c_26/0,c_27/0,c_28/1,c_29/1,c_30/2,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,append#,append'1#,flatten# ,flatten'1#,flattensort#,insert#,insert'1#,insert'2#,insertionsort#,insertionsort'1#} and constructors {'0 ,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,leaf,nil,node} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)) -->_1 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)):3 -->_1 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)):2 -->_1 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)):1 2:S:'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)) -->_1 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)):1 -->_1 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)):3 -->_1 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)):2 3:S:'compare#('s(x),'s(y)) -> c_15('compare#(x,y)) -->_1 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)):1 -->_1 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)):2 -->_1 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)):3 4:W:'less#(x,y) -> c_16('compare#(x,y)) -->_1 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)):1 -->_1 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)):2 -->_1 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)):3 9:W:flattensort#(t) -> c_23(insertionsort#(flatten(t)),flatten#(t)) -->_1 insertionsort#(l) -> c_29(insertionsort'1#(l)):13 10:W:insert#(x,l) -> c_24(insert'1#(l,x)) -->_1 insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys),'less#(y,x)):11 11:W:insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys),'less#(y,x)) -->_2 'less#(x,y) -> c_16('compare#(x,y)):4 -->_1 insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)):12 12:W:insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)) -->_1 insert#(x,l) -> c_24(insert'1#(l,x)):10 13:W:insertionsort#(l) -> c_29(insertionsort'1#(l)) -->_1 insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)):14 14:W:insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) -->_1 insert#(x,l) -> c_24(insert'1#(l,x)):10 -->_2 insertionsort#(l) -> c_29(insertionsort'1#(l)):13 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: flattensort#(t) -> c_23(insertionsort#(flatten(t))) ** Step 6.a:3: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)) 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)) 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)) - Weak DPs: 'less#(x,y) -> c_16('compare#(x,y)) flattensort#(t) -> c_23(insertionsort#(flatten(t))) insert#(x,l) -> c_24(insert'1#(l,x)) insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys),'less#(y,x)) insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)) insertionsort#(l) -> c_29(insertionsort'1#(l)) insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) - Weak TRS: 'cklt('EQ()) -> 'false() 'cklt('GT()) -> 'false() 'cklt('LT()) -> 'true() 'compare('0(),'0()) -> 'EQ() 'compare('0(),'neg(y)) -> 'GT() 'compare('0(),'pos(y)) -> 'LT() 'compare('0(),'s(y)) -> 'LT() 'compare('neg(x),'0()) -> 'LT() 'compare('neg(x),'neg(y)) -> 'compare(y,x) 'compare('neg(x),'pos(y)) -> 'LT() 'compare('pos(x),'0()) -> 'GT() 'compare('pos(x),'neg(y)) -> 'GT() 'compare('pos(x),'pos(y)) -> 'compare(x,y) 'compare('s(x),'0()) -> 'GT() 'compare('s(x),'s(y)) -> 'compare(x,y) 'less(x,y) -> 'cklt('compare(x,y)) append(l1,l2) -> append'1(l1,l2) append'1(dd(x,xs),l2) -> dd(x,append(xs,l2)) append'1(nil(),l2) -> l2 flatten(t) -> flatten'1(t) flatten'1(leaf()) -> nil() flatten'1(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2))) insert(x,l) -> insert'1(l,x) insert'1(dd(y,ys),x) -> insert'2('less(y,x),x,y,ys) insert'1(nil(),x) -> dd(x,nil()) insert'2('false(),x,y,ys) -> dd(x,dd(y,ys)) insert'2('true(),x,y,ys) -> dd(y,insert(x,ys)) insertionsort(l) -> insertionsort'1(l) insertionsort'1(dd(x,xs)) -> insert(x,insertionsort(xs)) insertionsort'1(nil()) -> nil() - Signature: {'cklt/1,'compare/2,'less/2,append/2,append'1/2,flatten/1,flatten'1/1,flattensort/1,insert/2,insert'1/2 ,insert'2/4,insertionsort/1,insertionsort'1/1,'cklt#/1,'compare#/2,'less#/2,append#/2,append'1#/2,flatten#/1 ,flatten'1#/1,flattensort#/1,insert#/2,insert'1#/2,insert'2#/4,insertionsort#/1,insertionsort'1#/1} / {'0/0 ,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,leaf/0,nil/0,node/3,c_1/0,c_2/0,c_3/0,c_4/0 ,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/1,c_19/0,c_20/1 ,c_21/0,c_22/4,c_23/1,c_24/1,c_25/2,c_26/0,c_27/0,c_28/1,c_29/1,c_30/2,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,append#,append'1#,flatten# ,flatten'1#,flattensort#,insert#,insert'1#,insert'2#,insertionsort#,insertionsort'1#} and constructors {'0 ,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,leaf,nil,node} + 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: 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)) Consider the set of all dependency pairs 1: 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)) 2: 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)) 3: 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)) 4: 'less#(x,y) -> c_16('compare#(x,y)) 5: flattensort#(t) -> c_23(insertionsort#(flatten(t))) 6: insert#(x,l) -> c_24(insert'1#(l,x)) 7: insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys),'less#(y,x)) 8: insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)) 9: insertionsort#(l) -> c_29(insertionsort'1#(l)) 10: insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) Processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^2)) SPACE(?,?)on application of the dependency pairs {3} These cover all (indirect) predecessors of dependency pairs {3,5} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. *** Step 6.a:3.a:1: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)) 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)) 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)) - Weak DPs: 'less#(x,y) -> c_16('compare#(x,y)) flattensort#(t) -> c_23(insertionsort#(flatten(t))) insert#(x,l) -> c_24(insert'1#(l,x)) insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys),'less#(y,x)) insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)) insertionsort#(l) -> c_29(insertionsort'1#(l)) insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) - Weak TRS: 'cklt('EQ()) -> 'false() 'cklt('GT()) -> 'false() 'cklt('LT()) -> 'true() 'compare('0(),'0()) -> 'EQ() 'compare('0(),'neg(y)) -> 'GT() 'compare('0(),'pos(y)) -> 'LT() 'compare('0(),'s(y)) -> 'LT() 'compare('neg(x),'0()) -> 'LT() 'compare('neg(x),'neg(y)) -> 'compare(y,x) 'compare('neg(x),'pos(y)) -> 'LT() 'compare('pos(x),'0()) -> 'GT() 'compare('pos(x),'neg(y)) -> 'GT() 'compare('pos(x),'pos(y)) -> 'compare(x,y) 'compare('s(x),'0()) -> 'GT() 'compare('s(x),'s(y)) -> 'compare(x,y) 'less(x,y) -> 'cklt('compare(x,y)) append(l1,l2) -> append'1(l1,l2) append'1(dd(x,xs),l2) -> dd(x,append(xs,l2)) append'1(nil(),l2) -> l2 flatten(t) -> flatten'1(t) flatten'1(leaf()) -> nil() flatten'1(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2))) insert(x,l) -> insert'1(l,x) insert'1(dd(y,ys),x) -> insert'2('less(y,x),x,y,ys) insert'1(nil(),x) -> dd(x,nil()) insert'2('false(),x,y,ys) -> dd(x,dd(y,ys)) insert'2('true(),x,y,ys) -> dd(y,insert(x,ys)) insertionsort(l) -> insertionsort'1(l) insertionsort'1(dd(x,xs)) -> insert(x,insertionsort(xs)) insertionsort'1(nil()) -> nil() - Signature: {'cklt/1,'compare/2,'less/2,append/2,append'1/2,flatten/1,flatten'1/1,flattensort/1,insert/2,insert'1/2 ,insert'2/4,insertionsort/1,insertionsort'1/1,'cklt#/1,'compare#/2,'less#/2,append#/2,append'1#/2,flatten#/1 ,flatten'1#/1,flattensort#/1,insert#/2,insert'1#/2,insert'2#/4,insertionsort#/1,insertionsort'1#/1} / {'0/0 ,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,leaf/0,nil/0,node/3,c_1/0,c_2/0,c_3/0,c_4/0 ,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/1,c_19/0,c_20/1 ,c_21/0,c_22/4,c_23/1,c_24/1,c_25/2,c_26/0,c_27/0,c_28/1,c_29/1,c_30/2,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,append#,append'1#,flatten# ,flatten'1#,flattensort#,insert#,insert'1#,insert'2#,insertionsort#,insertionsort'1#} and constructors {'0 ,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,leaf,nil,node} + 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_9) = {1}, uargs(c_13) = {1}, uargs(c_15) = {1}, uargs(c_16) = {1}, uargs(c_23) = {1}, uargs(c_24) = {1}, uargs(c_25) = {1,2}, uargs(c_28) = {1}, uargs(c_29) = {1}, uargs(c_30) = {1,2} Following symbols are considered usable: {append,append'1,flatten,flatten'1,insert,insert'1,insert'2,insertionsort,insertionsort'1,'cklt#,'compare# ,'less#,append#,append'1#,flatten#,flatten'1#,flattensort#,insert#,insert'1#,insert'2#,insertionsort# ,insertionsort'1#} TcT has computed the following interpretation: p('0) = 0 p('EQ) = 0 p('GT) = 0 p('LT) = 0 p('cklt) = 0 p('compare) = 0 p('false) = 1 p('less) = 0 p('neg) = x1 p('pos) = x1 p('s) = 1 + x1 p('true) = 0 p(append) = x1 + x2 p(append'1) = x1 + x2 p(dd) = x1 + x2 p(flatten) = x1 p(flatten'1) = x1 p(flattensort) = 0 p(insert) = x1 + x2 p(insert'1) = x1 + x2 p(insert'2) = x2 + x3 + x4 p(insertionsort) = x1 p(insertionsort'1) = x1 p(leaf) = 0 p(nil) = 0 p(node) = 1 + x1 + x2 + x3 p('cklt#) = 0 p('compare#) = x1*x2 p('less#) = x1*x2 p(append#) = 0 p(append'1#) = 0 p(flatten#) = 0 p(flatten'1#) = 0 p(flattensort#) = 1 + x1 + x1^2 p(insert#) = x1 + x1*x2 p(insert'1#) = x1*x2 + x2 p(insert'2#) = x2 + x2*x4 p(insertionsort#) = x1 + x1^2 p(insertionsort'1#) = 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) = x1 p(c_10) = 0 p(c_11) = 0 p(c_12) = 0 p(c_13) = x1 p(c_14) = 0 p(c_15) = x1 p(c_16) = x1 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) = 1 + x1 p(c_24) = x1 p(c_25) = x1 + x2 p(c_26) = 0 p(c_27) = 0 p(c_28) = x1 p(c_29) = x1 p(c_30) = x1 + x2 p(c_31) = 0 Following rules are strictly oriented: 'compare#('s(x),'s(y)) = 1 + x + x*y + y > x*y = c_15('compare#(x,y)) Following rules are (at-least) weakly oriented: 'compare#('neg(x),'neg(y)) = x*y >= x*y = c_9('compare#(y,x)) 'compare#('pos(x),'pos(y)) = x*y >= x*y = c_13('compare#(x,y)) 'less#(x,y) = x*y >= x*y = c_16('compare#(x,y)) flattensort#(t) = 1 + t + t^2 >= 1 + t + t^2 = c_23(insertionsort#(flatten(t))) insert#(x,l) = l*x + x >= l*x + x = c_24(insert'1#(l,x)) insert'1#(dd(y,ys),x) = x + x*y + x*ys >= x + x*y + x*ys = c_25(insert'2#('less(y,x),x,y,ys),'less#(y,x)) insert'2#('true(),x,y,ys) = x + x*ys >= x + x*ys = c_28(insert#(x,ys)) insertionsort#(l) = l + l^2 >= l + l^2 = c_29(insertionsort'1#(l)) insertionsort'1#(dd(x,xs)) = x + 2*x*xs + x^2 + xs + xs^2 >= x + x*xs + xs + xs^2 = c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) append(l1,l2) = l1 + l2 >= l1 + l2 = append'1(l1,l2) append'1(dd(x,xs),l2) = l2 + x + xs >= l2 + x + xs = dd(x,append(xs,l2)) append'1(nil(),l2) = l2 >= l2 = l2 flatten(t) = t >= t = flatten'1(t) flatten'1(leaf()) = 0 >= 0 = nil() flatten'1(node(l,t1,t2)) = 1 + l + t1 + t2 >= l + t1 + t2 = append(l,append(flatten(t1),flatten(t2))) insert(x,l) = l + x >= l + x = insert'1(l,x) insert'1(dd(y,ys),x) = x + y + ys >= x + y + ys = insert'2('less(y,x),x,y,ys) insert'1(nil(),x) = x >= x = dd(x,nil()) insert'2('false(),x,y,ys) = x + y + ys >= x + y + ys = dd(x,dd(y,ys)) insert'2('true(),x,y,ys) = x + y + ys >= x + y + ys = dd(y,insert(x,ys)) insertionsort(l) = l >= l = insertionsort'1(l) insertionsort'1(dd(x,xs)) = x + xs >= x + xs = insert(x,insertionsort(xs)) insertionsort'1(nil()) = 0 >= 0 = nil() *** Step 6.a:3.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)) 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)) - Weak DPs: 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)) 'less#(x,y) -> c_16('compare#(x,y)) flattensort#(t) -> c_23(insertionsort#(flatten(t))) insert#(x,l) -> c_24(insert'1#(l,x)) insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys),'less#(y,x)) insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)) insertionsort#(l) -> c_29(insertionsort'1#(l)) insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) - Weak TRS: 'cklt('EQ()) -> 'false() 'cklt('GT()) -> 'false() 'cklt('LT()) -> 'true() 'compare('0(),'0()) -> 'EQ() 'compare('0(),'neg(y)) -> 'GT() 'compare('0(),'pos(y)) -> 'LT() 'compare('0(),'s(y)) -> 'LT() 'compare('neg(x),'0()) -> 'LT() 'compare('neg(x),'neg(y)) -> 'compare(y,x) 'compare('neg(x),'pos(y)) -> 'LT() 'compare('pos(x),'0()) -> 'GT() 'compare('pos(x),'neg(y)) -> 'GT() 'compare('pos(x),'pos(y)) -> 'compare(x,y) 'compare('s(x),'0()) -> 'GT() 'compare('s(x),'s(y)) -> 'compare(x,y) 'less(x,y) -> 'cklt('compare(x,y)) append(l1,l2) -> append'1(l1,l2) append'1(dd(x,xs),l2) -> dd(x,append(xs,l2)) append'1(nil(),l2) -> l2 flatten(t) -> flatten'1(t) flatten'1(leaf()) -> nil() flatten'1(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2))) insert(x,l) -> insert'1(l,x) insert'1(dd(y,ys),x) -> insert'2('less(y,x),x,y,ys) insert'1(nil(),x) -> dd(x,nil()) insert'2('false(),x,y,ys) -> dd(x,dd(y,ys)) insert'2('true(),x,y,ys) -> dd(y,insert(x,ys)) insertionsort(l) -> insertionsort'1(l) insertionsort'1(dd(x,xs)) -> insert(x,insertionsort(xs)) insertionsort'1(nil()) -> nil() - Signature: {'cklt/1,'compare/2,'less/2,append/2,append'1/2,flatten/1,flatten'1/1,flattensort/1,insert/2,insert'1/2 ,insert'2/4,insertionsort/1,insertionsort'1/1,'cklt#/1,'compare#/2,'less#/2,append#/2,append'1#/2,flatten#/1 ,flatten'1#/1,flattensort#/1,insert#/2,insert'1#/2,insert'2#/4,insertionsort#/1,insertionsort'1#/1} / {'0/0 ,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,leaf/0,nil/0,node/3,c_1/0,c_2/0,c_3/0,c_4/0 ,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/1,c_19/0,c_20/1 ,c_21/0,c_22/4,c_23/1,c_24/1,c_25/2,c_26/0,c_27/0,c_28/1,c_29/1,c_30/2,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,append#,append'1#,flatten# ,flatten'1#,flattensort#,insert#,insert'1#,insert'2#,insertionsort#,insertionsort'1#} and constructors {'0 ,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,leaf,nil,node} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () *** Step 6.a:3.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)) 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)) - Weak DPs: 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)) 'less#(x,y) -> c_16('compare#(x,y)) flattensort#(t) -> c_23(insertionsort#(flatten(t))) insert#(x,l) -> c_24(insert'1#(l,x)) insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys),'less#(y,x)) insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)) insertionsort#(l) -> c_29(insertionsort'1#(l)) insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) - Weak TRS: 'cklt('EQ()) -> 'false() 'cklt('GT()) -> 'false() 'cklt('LT()) -> 'true() 'compare('0(),'0()) -> 'EQ() 'compare('0(),'neg(y)) -> 'GT() 'compare('0(),'pos(y)) -> 'LT() 'compare('0(),'s(y)) -> 'LT() 'compare('neg(x),'0()) -> 'LT() 'compare('neg(x),'neg(y)) -> 'compare(y,x) 'compare('neg(x),'pos(y)) -> 'LT() 'compare('pos(x),'0()) -> 'GT() 'compare('pos(x),'neg(y)) -> 'GT() 'compare('pos(x),'pos(y)) -> 'compare(x,y) 'compare('s(x),'0()) -> 'GT() 'compare('s(x),'s(y)) -> 'compare(x,y) 'less(x,y) -> 'cklt('compare(x,y)) append(l1,l2) -> append'1(l1,l2) append'1(dd(x,xs),l2) -> dd(x,append(xs,l2)) append'1(nil(),l2) -> l2 flatten(t) -> flatten'1(t) flatten'1(leaf()) -> nil() flatten'1(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2))) insert(x,l) -> insert'1(l,x) insert'1(dd(y,ys),x) -> insert'2('less(y,x),x,y,ys) insert'1(nil(),x) -> dd(x,nil()) insert'2('false(),x,y,ys) -> dd(x,dd(y,ys)) insert'2('true(),x,y,ys) -> dd(y,insert(x,ys)) insertionsort(l) -> insertionsort'1(l) insertionsort'1(dd(x,xs)) -> insert(x,insertionsort(xs)) insertionsort'1(nil()) -> nil() - Signature: {'cklt/1,'compare/2,'less/2,append/2,append'1/2,flatten/1,flatten'1/1,flattensort/1,insert/2,insert'1/2 ,insert'2/4,insertionsort/1,insertionsort'1/1,'cklt#/1,'compare#/2,'less#/2,append#/2,append'1#/2,flatten#/1 ,flatten'1#/1,flattensort#/1,insert#/2,insert'1#/2,insert'2#/4,insertionsort#/1,insertionsort'1#/1} / {'0/0 ,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,leaf/0,nil/0,node/3,c_1/0,c_2/0,c_3/0,c_4/0 ,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/1,c_19/0,c_20/1 ,c_21/0,c_22/4,c_23/1,c_24/1,c_25/2,c_26/0,c_27/0,c_28/1,c_29/1,c_30/2,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,append#,append'1#,flatten# ,flatten'1#,flattensort#,insert#,insert'1#,insert'2#,insertionsort#,insertionsort'1#} and constructors {'0 ,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,leaf,nil,node} + 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: 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)) Consider the set of all dependency pairs 1: 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)) 2: 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)) 3: 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)) 4: 'less#(x,y) -> c_16('compare#(x,y)) 5: flattensort#(t) -> c_23(insertionsort#(flatten(t))) 6: insert#(x,l) -> c_24(insert'1#(l,x)) 7: insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys),'less#(y,x)) 8: insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)) 9: insertionsort#(l) -> c_29(insertionsort'1#(l)) 10: insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) Processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^2)) SPACE(?,?)on application of the dependency pairs {2} These cover all (indirect) predecessors of dependency pairs {2,5} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. **** Step 6.a:3.b:1.a:1: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)) 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)) - Weak DPs: 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)) 'less#(x,y) -> c_16('compare#(x,y)) flattensort#(t) -> c_23(insertionsort#(flatten(t))) insert#(x,l) -> c_24(insert'1#(l,x)) insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys),'less#(y,x)) insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)) insertionsort#(l) -> c_29(insertionsort'1#(l)) insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) - Weak TRS: 'cklt('EQ()) -> 'false() 'cklt('GT()) -> 'false() 'cklt('LT()) -> 'true() 'compare('0(),'0()) -> 'EQ() 'compare('0(),'neg(y)) -> 'GT() 'compare('0(),'pos(y)) -> 'LT() 'compare('0(),'s(y)) -> 'LT() 'compare('neg(x),'0()) -> 'LT() 'compare('neg(x),'neg(y)) -> 'compare(y,x) 'compare('neg(x),'pos(y)) -> 'LT() 'compare('pos(x),'0()) -> 'GT() 'compare('pos(x),'neg(y)) -> 'GT() 'compare('pos(x),'pos(y)) -> 'compare(x,y) 'compare('s(x),'0()) -> 'GT() 'compare('s(x),'s(y)) -> 'compare(x,y) 'less(x,y) -> 'cklt('compare(x,y)) append(l1,l2) -> append'1(l1,l2) append'1(dd(x,xs),l2) -> dd(x,append(xs,l2)) append'1(nil(),l2) -> l2 flatten(t) -> flatten'1(t) flatten'1(leaf()) -> nil() flatten'1(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2))) insert(x,l) -> insert'1(l,x) insert'1(dd(y,ys),x) -> insert'2('less(y,x),x,y,ys) insert'1(nil(),x) -> dd(x,nil()) insert'2('false(),x,y,ys) -> dd(x,dd(y,ys)) insert'2('true(),x,y,ys) -> dd(y,insert(x,ys)) insertionsort(l) -> insertionsort'1(l) insertionsort'1(dd(x,xs)) -> insert(x,insertionsort(xs)) insertionsort'1(nil()) -> nil() - Signature: {'cklt/1,'compare/2,'less/2,append/2,append'1/2,flatten/1,flatten'1/1,flattensort/1,insert/2,insert'1/2 ,insert'2/4,insertionsort/1,insertionsort'1/1,'cklt#/1,'compare#/2,'less#/2,append#/2,append'1#/2,flatten#/1 ,flatten'1#/1,flattensort#/1,insert#/2,insert'1#/2,insert'2#/4,insertionsort#/1,insertionsort'1#/1} / {'0/0 ,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,leaf/0,nil/0,node/3,c_1/0,c_2/0,c_3/0,c_4/0 ,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/1,c_19/0,c_20/1 ,c_21/0,c_22/4,c_23/1,c_24/1,c_25/2,c_26/0,c_27/0,c_28/1,c_29/1,c_30/2,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,append#,append'1#,flatten# ,flatten'1#,flattensort#,insert#,insert'1#,insert'2#,insertionsort#,insertionsort'1#} and constructors {'0 ,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,leaf,nil,node} + 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_9) = {1}, uargs(c_13) = {1}, uargs(c_15) = {1}, uargs(c_16) = {1}, uargs(c_23) = {1}, uargs(c_24) = {1}, uargs(c_25) = {1,2}, uargs(c_28) = {1}, uargs(c_29) = {1}, uargs(c_30) = {1,2} Following symbols are considered usable: {append,append'1,flatten,flatten'1,insert,insert'1,insert'2,insertionsort,insertionsort'1,'cklt#,'compare# ,'less#,append#,append'1#,flatten#,flatten'1#,flattensort#,insert#,insert'1#,insert'2#,insertionsort# ,insertionsort'1#} TcT has computed the following interpretation: p('0) = 1 p('EQ) = 0 p('GT) = 1 p('LT) = 0 p('cklt) = 0 p('compare) = x1*x2 + x2 p('false) = 0 p('less) = x1^2 p('neg) = x1 p('pos) = 1 + x1 p('s) = x1 p('true) = 1 p(append) = x1 + x2 p(append'1) = x1 + x2 p(dd) = x1 + x2 p(flatten) = x1 p(flatten'1) = x1 p(flattensort) = 0 p(insert) = x1 + x2 p(insert'1) = x1 + x2 p(insert'2) = x2 + x3 + x4 p(insertionsort) = x1 p(insertionsort'1) = x1 p(leaf) = 0 p(nil) = 0 p(node) = x1 + x2 + x3 p('cklt#) = 0 p('compare#) = x1*x2 p('less#) = x1*x2 p(append#) = 0 p(append'1#) = 0 p(flatten#) = 0 p(flatten'1#) = 0 p(flattensort#) = 1 + x1^2 p(insert#) = x1*x2 p(insert'1#) = x1*x2 p(insert'2#) = x2*x4 p(insertionsort#) = 1 + x1^2 p(insertionsort'1#) = 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) = x1 p(c_10) = 0 p(c_11) = 0 p(c_12) = 0 p(c_13) = x1 p(c_14) = 0 p(c_15) = x1 p(c_16) = x1 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) = x1 p(c_24) = x1 p(c_25) = x1 + x2 p(c_26) = 0 p(c_27) = 0 p(c_28) = x1 p(c_29) = x1 p(c_30) = x1 + x2 p(c_31) = 0 Following rules are strictly oriented: 'compare#('pos(x),'pos(y)) = 1 + x + x*y + y > x*y = c_13('compare#(x,y)) Following rules are (at-least) weakly oriented: 'compare#('neg(x),'neg(y)) = x*y >= x*y = c_9('compare#(y,x)) 'compare#('s(x),'s(y)) = x*y >= x*y = c_15('compare#(x,y)) 'less#(x,y) = x*y >= x*y = c_16('compare#(x,y)) flattensort#(t) = 1 + t^2 >= 1 + t^2 = c_23(insertionsort#(flatten(t))) insert#(x,l) = l*x >= l*x = c_24(insert'1#(l,x)) insert'1#(dd(y,ys),x) = x*y + x*ys >= x*y + x*ys = c_25(insert'2#('less(y,x),x,y,ys),'less#(y,x)) insert'2#('true(),x,y,ys) = x*ys >= x*ys = c_28(insert#(x,ys)) insertionsort#(l) = 1 + l^2 >= 1 + l^2 = c_29(insertionsort'1#(l)) insertionsort'1#(dd(x,xs)) = 1 + 2*x*xs + x^2 + xs^2 >= 1 + x*xs + xs^2 = c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) append(l1,l2) = l1 + l2 >= l1 + l2 = append'1(l1,l2) append'1(dd(x,xs),l2) = l2 + x + xs >= l2 + x + xs = dd(x,append(xs,l2)) append'1(nil(),l2) = l2 >= l2 = l2 flatten(t) = t >= t = flatten'1(t) flatten'1(leaf()) = 0 >= 0 = nil() flatten'1(node(l,t1,t2)) = l + t1 + t2 >= l + t1 + t2 = append(l,append(flatten(t1),flatten(t2))) insert(x,l) = l + x >= l + x = insert'1(l,x) insert'1(dd(y,ys),x) = x + y + ys >= x + y + ys = insert'2('less(y,x),x,y,ys) insert'1(nil(),x) = x >= x = dd(x,nil()) insert'2('false(),x,y,ys) = x + y + ys >= x + y + ys = dd(x,dd(y,ys)) insert'2('true(),x,y,ys) = x + y + ys >= x + y + ys = dd(y,insert(x,ys)) insertionsort(l) = l >= l = insertionsort'1(l) insertionsort'1(dd(x,xs)) = x + xs >= x + xs = insert(x,insertionsort(xs)) insertionsort'1(nil()) = 0 >= 0 = nil() **** Step 6.a:3.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)) - Weak DPs: 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)) 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)) 'less#(x,y) -> c_16('compare#(x,y)) flattensort#(t) -> c_23(insertionsort#(flatten(t))) insert#(x,l) -> c_24(insert'1#(l,x)) insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys),'less#(y,x)) insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)) insertionsort#(l) -> c_29(insertionsort'1#(l)) insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) - Weak TRS: 'cklt('EQ()) -> 'false() 'cklt('GT()) -> 'false() 'cklt('LT()) -> 'true() 'compare('0(),'0()) -> 'EQ() 'compare('0(),'neg(y)) -> 'GT() 'compare('0(),'pos(y)) -> 'LT() 'compare('0(),'s(y)) -> 'LT() 'compare('neg(x),'0()) -> 'LT() 'compare('neg(x),'neg(y)) -> 'compare(y,x) 'compare('neg(x),'pos(y)) -> 'LT() 'compare('pos(x),'0()) -> 'GT() 'compare('pos(x),'neg(y)) -> 'GT() 'compare('pos(x),'pos(y)) -> 'compare(x,y) 'compare('s(x),'0()) -> 'GT() 'compare('s(x),'s(y)) -> 'compare(x,y) 'less(x,y) -> 'cklt('compare(x,y)) append(l1,l2) -> append'1(l1,l2) append'1(dd(x,xs),l2) -> dd(x,append(xs,l2)) append'1(nil(),l2) -> l2 flatten(t) -> flatten'1(t) flatten'1(leaf()) -> nil() flatten'1(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2))) insert(x,l) -> insert'1(l,x) insert'1(dd(y,ys),x) -> insert'2('less(y,x),x,y,ys) insert'1(nil(),x) -> dd(x,nil()) insert'2('false(),x,y,ys) -> dd(x,dd(y,ys)) insert'2('true(),x,y,ys) -> dd(y,insert(x,ys)) insertionsort(l) -> insertionsort'1(l) insertionsort'1(dd(x,xs)) -> insert(x,insertionsort(xs)) insertionsort'1(nil()) -> nil() - Signature: {'cklt/1,'compare/2,'less/2,append/2,append'1/2,flatten/1,flatten'1/1,flattensort/1,insert/2,insert'1/2 ,insert'2/4,insertionsort/1,insertionsort'1/1,'cklt#/1,'compare#/2,'less#/2,append#/2,append'1#/2,flatten#/1 ,flatten'1#/1,flattensort#/1,insert#/2,insert'1#/2,insert'2#/4,insertionsort#/1,insertionsort'1#/1} / {'0/0 ,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,leaf/0,nil/0,node/3,c_1/0,c_2/0,c_3/0,c_4/0 ,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/1,c_19/0,c_20/1 ,c_21/0,c_22/4,c_23/1,c_24/1,c_25/2,c_26/0,c_27/0,c_28/1,c_29/1,c_30/2,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,append#,append'1#,flatten# ,flatten'1#,flattensort#,insert#,insert'1#,insert'2#,insertionsort#,insertionsort'1#} and constructors {'0 ,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,leaf,nil,node} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () **** Step 6.a:3.b:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)) - Weak DPs: 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)) 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)) 'less#(x,y) -> c_16('compare#(x,y)) flattensort#(t) -> c_23(insertionsort#(flatten(t))) insert#(x,l) -> c_24(insert'1#(l,x)) insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys),'less#(y,x)) insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)) insertionsort#(l) -> c_29(insertionsort'1#(l)) insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) - Weak TRS: 'cklt('EQ()) -> 'false() 'cklt('GT()) -> 'false() 'cklt('LT()) -> 'true() 'compare('0(),'0()) -> 'EQ() 'compare('0(),'neg(y)) -> 'GT() 'compare('0(),'pos(y)) -> 'LT() 'compare('0(),'s(y)) -> 'LT() 'compare('neg(x),'0()) -> 'LT() 'compare('neg(x),'neg(y)) -> 'compare(y,x) 'compare('neg(x),'pos(y)) -> 'LT() 'compare('pos(x),'0()) -> 'GT() 'compare('pos(x),'neg(y)) -> 'GT() 'compare('pos(x),'pos(y)) -> 'compare(x,y) 'compare('s(x),'0()) -> 'GT() 'compare('s(x),'s(y)) -> 'compare(x,y) 'less(x,y) -> 'cklt('compare(x,y)) append(l1,l2) -> append'1(l1,l2) append'1(dd(x,xs),l2) -> dd(x,append(xs,l2)) append'1(nil(),l2) -> l2 flatten(t) -> flatten'1(t) flatten'1(leaf()) -> nil() flatten'1(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2))) insert(x,l) -> insert'1(l,x) insert'1(dd(y,ys),x) -> insert'2('less(y,x),x,y,ys) insert'1(nil(),x) -> dd(x,nil()) insert'2('false(),x,y,ys) -> dd(x,dd(y,ys)) insert'2('true(),x,y,ys) -> dd(y,insert(x,ys)) insertionsort(l) -> insertionsort'1(l) insertionsort'1(dd(x,xs)) -> insert(x,insertionsort(xs)) insertionsort'1(nil()) -> nil() - Signature: {'cklt/1,'compare/2,'less/2,append/2,append'1/2,flatten/1,flatten'1/1,flattensort/1,insert/2,insert'1/2 ,insert'2/4,insertionsort/1,insertionsort'1/1,'cklt#/1,'compare#/2,'less#/2,append#/2,append'1#/2,flatten#/1 ,flatten'1#/1,flattensort#/1,insert#/2,insert'1#/2,insert'2#/4,insertionsort#/1,insertionsort'1#/1} / {'0/0 ,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,leaf/0,nil/0,node/3,c_1/0,c_2/0,c_3/0,c_4/0 ,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/1,c_19/0,c_20/1 ,c_21/0,c_22/4,c_23/1,c_24/1,c_25/2,c_26/0,c_27/0,c_28/1,c_29/1,c_30/2,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,append#,append'1#,flatten# ,flatten'1#,flattensort#,insert#,insert'1#,insert'2#,insertionsort#,insertionsort'1#} and constructors {'0 ,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,leaf,nil,node} + 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: 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)) Consider the set of all dependency pairs 1: 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)) 2: 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)) 3: 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)) 4: 'less#(x,y) -> c_16('compare#(x,y)) 5: flattensort#(t) -> c_23(insertionsort#(flatten(t))) 6: insert#(x,l) -> c_24(insert'1#(l,x)) 7: insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys),'less#(y,x)) 8: insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)) 9: insertionsort#(l) -> c_29(insertionsort'1#(l)) 10: insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) Processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^2)) SPACE(?,?)on application of the dependency pairs {1} These cover all (indirect) predecessors of dependency pairs {1,5} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. ***** Step 6.a:3.b:1.b:1.a:1: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)) - Weak DPs: 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)) 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)) 'less#(x,y) -> c_16('compare#(x,y)) flattensort#(t) -> c_23(insertionsort#(flatten(t))) insert#(x,l) -> c_24(insert'1#(l,x)) insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys),'less#(y,x)) insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)) insertionsort#(l) -> c_29(insertionsort'1#(l)) insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) - Weak TRS: 'cklt('EQ()) -> 'false() 'cklt('GT()) -> 'false() 'cklt('LT()) -> 'true() 'compare('0(),'0()) -> 'EQ() 'compare('0(),'neg(y)) -> 'GT() 'compare('0(),'pos(y)) -> 'LT() 'compare('0(),'s(y)) -> 'LT() 'compare('neg(x),'0()) -> 'LT() 'compare('neg(x),'neg(y)) -> 'compare(y,x) 'compare('neg(x),'pos(y)) -> 'LT() 'compare('pos(x),'0()) -> 'GT() 'compare('pos(x),'neg(y)) -> 'GT() 'compare('pos(x),'pos(y)) -> 'compare(x,y) 'compare('s(x),'0()) -> 'GT() 'compare('s(x),'s(y)) -> 'compare(x,y) 'less(x,y) -> 'cklt('compare(x,y)) append(l1,l2) -> append'1(l1,l2) append'1(dd(x,xs),l2) -> dd(x,append(xs,l2)) append'1(nil(),l2) -> l2 flatten(t) -> flatten'1(t) flatten'1(leaf()) -> nil() flatten'1(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2))) insert(x,l) -> insert'1(l,x) insert'1(dd(y,ys),x) -> insert'2('less(y,x),x,y,ys) insert'1(nil(),x) -> dd(x,nil()) insert'2('false(),x,y,ys) -> dd(x,dd(y,ys)) insert'2('true(),x,y,ys) -> dd(y,insert(x,ys)) insertionsort(l) -> insertionsort'1(l) insertionsort'1(dd(x,xs)) -> insert(x,insertionsort(xs)) insertionsort'1(nil()) -> nil() - Signature: {'cklt/1,'compare/2,'less/2,append/2,append'1/2,flatten/1,flatten'1/1,flattensort/1,insert/2,insert'1/2 ,insert'2/4,insertionsort/1,insertionsort'1/1,'cklt#/1,'compare#/2,'less#/2,append#/2,append'1#/2,flatten#/1 ,flatten'1#/1,flattensort#/1,insert#/2,insert'1#/2,insert'2#/4,insertionsort#/1,insertionsort'1#/1} / {'0/0 ,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,leaf/0,nil/0,node/3,c_1/0,c_2/0,c_3/0,c_4/0 ,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/1,c_19/0,c_20/1 ,c_21/0,c_22/4,c_23/1,c_24/1,c_25/2,c_26/0,c_27/0,c_28/1,c_29/1,c_30/2,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,append#,append'1#,flatten# ,flatten'1#,flattensort#,insert#,insert'1#,insert'2#,insertionsort#,insertionsort'1#} and constructors {'0 ,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,leaf,nil,node} + 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_9) = {1}, uargs(c_13) = {1}, uargs(c_15) = {1}, uargs(c_16) = {1}, uargs(c_23) = {1}, uargs(c_24) = {1}, uargs(c_25) = {1,2}, uargs(c_28) = {1}, uargs(c_29) = {1}, uargs(c_30) = {1,2} Following symbols are considered usable: {append,append'1,flatten,flatten'1,insert,insert'1,insert'2,insertionsort,insertionsort'1,'cklt#,'compare# ,'less#,append#,append'1#,flatten#,flatten'1#,flattensort#,insert#,insert'1#,insert'2#,insertionsort# ,insertionsort'1#} TcT has computed the following interpretation: p('0) = 0 p('EQ) = 1 p('GT) = 0 p('LT) = 1 p('cklt) = 0 p('compare) = x1 p('false) = 0 p('less) = 0 p('neg) = 1 + x1 p('pos) = 1 + x1 p('s) = 1 + x1 p('true) = 0 p(append) = x1 + x2 p(append'1) = x1 + x2 p(dd) = x1 + x2 p(flatten) = x1 p(flatten'1) = x1 p(flattensort) = 0 p(insert) = x1 + x2 p(insert'1) = x1 + x2 p(insert'2) = x2 + x3 + x4 p(insertionsort) = x1 p(insertionsort'1) = x1 p(leaf) = 0 p(nil) = 0 p(node) = x1 + x2 + x3 p('cklt#) = 0 p('compare#) = x1*x2 p('less#) = x1*x2 p(append#) = 0 p(append'1#) = 0 p(flatten#) = 0 p(flatten'1#) = 0 p(flattensort#) = x1 + x1^2 p(insert#) = x1 + x1*x2 + x1^2 p(insert'1#) = x1*x2 + x2 + x2^2 p(insert'2#) = x2 + x2*x4 + x2^2 p(insertionsort#) = x1 + x1^2 p(insertionsort'1#) = 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) = x1 p(c_10) = 0 p(c_11) = 0 p(c_12) = 0 p(c_13) = 1 + x1 p(c_14) = 0 p(c_15) = 1 + x1 p(c_16) = x1 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) = x1 p(c_24) = x1 p(c_25) = x1 + x2 p(c_26) = 0 p(c_27) = 0 p(c_28) = x1 p(c_29) = x1 p(c_30) = x1 + x2 p(c_31) = 0 Following rules are strictly oriented: 'compare#('neg(x),'neg(y)) = 1 + x + x*y + y > x*y = c_9('compare#(y,x)) Following rules are (at-least) weakly oriented: 'compare#('pos(x),'pos(y)) = 1 + x + x*y + y >= 1 + x*y = c_13('compare#(x,y)) 'compare#('s(x),'s(y)) = 1 + x + x*y + y >= 1 + x*y = c_15('compare#(x,y)) 'less#(x,y) = x*y >= x*y = c_16('compare#(x,y)) flattensort#(t) = t + t^2 >= t + t^2 = c_23(insertionsort#(flatten(t))) insert#(x,l) = l*x + x + x^2 >= l*x + x + x^2 = c_24(insert'1#(l,x)) insert'1#(dd(y,ys),x) = x + x*y + x*ys + x^2 >= x + x*y + x*ys + x^2 = c_25(insert'2#('less(y,x),x,y,ys),'less#(y,x)) insert'2#('true(),x,y,ys) = x + x*ys + x^2 >= x + x*ys + x^2 = c_28(insert#(x,ys)) insertionsort#(l) = l + l^2 >= l + l^2 = c_29(insertionsort'1#(l)) insertionsort'1#(dd(x,xs)) = x + 2*x*xs + x^2 + xs + xs^2 >= x + x*xs + x^2 + xs + xs^2 = c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) append(l1,l2) = l1 + l2 >= l1 + l2 = append'1(l1,l2) append'1(dd(x,xs),l2) = l2 + x + xs >= l2 + x + xs = dd(x,append(xs,l2)) append'1(nil(),l2) = l2 >= l2 = l2 flatten(t) = t >= t = flatten'1(t) flatten'1(leaf()) = 0 >= 0 = nil() flatten'1(node(l,t1,t2)) = l + t1 + t2 >= l + t1 + t2 = append(l,append(flatten(t1),flatten(t2))) insert(x,l) = l + x >= l + x = insert'1(l,x) insert'1(dd(y,ys),x) = x + y + ys >= x + y + ys = insert'2('less(y,x),x,y,ys) insert'1(nil(),x) = x >= x = dd(x,nil()) insert'2('false(),x,y,ys) = x + y + ys >= x + y + ys = dd(x,dd(y,ys)) insert'2('true(),x,y,ys) = x + y + ys >= x + y + ys = dd(y,insert(x,ys)) insertionsort(l) = l >= l = insertionsort'1(l) insertionsort'1(dd(x,xs)) = x + xs >= x + xs = insert(x,insertionsort(xs)) insertionsort'1(nil()) = 0 >= 0 = nil() ***** Step 6.a:3.b:1.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)) 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)) 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)) 'less#(x,y) -> c_16('compare#(x,y)) flattensort#(t) -> c_23(insertionsort#(flatten(t))) insert#(x,l) -> c_24(insert'1#(l,x)) insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys),'less#(y,x)) insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)) insertionsort#(l) -> c_29(insertionsort'1#(l)) insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) - Weak TRS: 'cklt('EQ()) -> 'false() 'cklt('GT()) -> 'false() 'cklt('LT()) -> 'true() 'compare('0(),'0()) -> 'EQ() 'compare('0(),'neg(y)) -> 'GT() 'compare('0(),'pos(y)) -> 'LT() 'compare('0(),'s(y)) -> 'LT() 'compare('neg(x),'0()) -> 'LT() 'compare('neg(x),'neg(y)) -> 'compare(y,x) 'compare('neg(x),'pos(y)) -> 'LT() 'compare('pos(x),'0()) -> 'GT() 'compare('pos(x),'neg(y)) -> 'GT() 'compare('pos(x),'pos(y)) -> 'compare(x,y) 'compare('s(x),'0()) -> 'GT() 'compare('s(x),'s(y)) -> 'compare(x,y) 'less(x,y) -> 'cklt('compare(x,y)) append(l1,l2) -> append'1(l1,l2) append'1(dd(x,xs),l2) -> dd(x,append(xs,l2)) append'1(nil(),l2) -> l2 flatten(t) -> flatten'1(t) flatten'1(leaf()) -> nil() flatten'1(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2))) insert(x,l) -> insert'1(l,x) insert'1(dd(y,ys),x) -> insert'2('less(y,x),x,y,ys) insert'1(nil(),x) -> dd(x,nil()) insert'2('false(),x,y,ys) -> dd(x,dd(y,ys)) insert'2('true(),x,y,ys) -> dd(y,insert(x,ys)) insertionsort(l) -> insertionsort'1(l) insertionsort'1(dd(x,xs)) -> insert(x,insertionsort(xs)) insertionsort'1(nil()) -> nil() - Signature: {'cklt/1,'compare/2,'less/2,append/2,append'1/2,flatten/1,flatten'1/1,flattensort/1,insert/2,insert'1/2 ,insert'2/4,insertionsort/1,insertionsort'1/1,'cklt#/1,'compare#/2,'less#/2,append#/2,append'1#/2,flatten#/1 ,flatten'1#/1,flattensort#/1,insert#/2,insert'1#/2,insert'2#/4,insertionsort#/1,insertionsort'1#/1} / {'0/0 ,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,leaf/0,nil/0,node/3,c_1/0,c_2/0,c_3/0,c_4/0 ,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/1,c_19/0,c_20/1 ,c_21/0,c_22/4,c_23/1,c_24/1,c_25/2,c_26/0,c_27/0,c_28/1,c_29/1,c_30/2,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,append#,append'1#,flatten# ,flatten'1#,flattensort#,insert#,insert'1#,insert'2#,insertionsort#,insertionsort'1#} and constructors {'0 ,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,leaf,nil,node} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ***** Step 6.a:3.b:1.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)) 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)) 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)) 'less#(x,y) -> c_16('compare#(x,y)) flattensort#(t) -> c_23(insertionsort#(flatten(t))) insert#(x,l) -> c_24(insert'1#(l,x)) insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys),'less#(y,x)) insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)) insertionsort#(l) -> c_29(insertionsort'1#(l)) insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) - Weak TRS: 'cklt('EQ()) -> 'false() 'cklt('GT()) -> 'false() 'cklt('LT()) -> 'true() 'compare('0(),'0()) -> 'EQ() 'compare('0(),'neg(y)) -> 'GT() 'compare('0(),'pos(y)) -> 'LT() 'compare('0(),'s(y)) -> 'LT() 'compare('neg(x),'0()) -> 'LT() 'compare('neg(x),'neg(y)) -> 'compare(y,x) 'compare('neg(x),'pos(y)) -> 'LT() 'compare('pos(x),'0()) -> 'GT() 'compare('pos(x),'neg(y)) -> 'GT() 'compare('pos(x),'pos(y)) -> 'compare(x,y) 'compare('s(x),'0()) -> 'GT() 'compare('s(x),'s(y)) -> 'compare(x,y) 'less(x,y) -> 'cklt('compare(x,y)) append(l1,l2) -> append'1(l1,l2) append'1(dd(x,xs),l2) -> dd(x,append(xs,l2)) append'1(nil(),l2) -> l2 flatten(t) -> flatten'1(t) flatten'1(leaf()) -> nil() flatten'1(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2))) insert(x,l) -> insert'1(l,x) insert'1(dd(y,ys),x) -> insert'2('less(y,x),x,y,ys) insert'1(nil(),x) -> dd(x,nil()) insert'2('false(),x,y,ys) -> dd(x,dd(y,ys)) insert'2('true(),x,y,ys) -> dd(y,insert(x,ys)) insertionsort(l) -> insertionsort'1(l) insertionsort'1(dd(x,xs)) -> insert(x,insertionsort(xs)) insertionsort'1(nil()) -> nil() - Signature: {'cklt/1,'compare/2,'less/2,append/2,append'1/2,flatten/1,flatten'1/1,flattensort/1,insert/2,insert'1/2 ,insert'2/4,insertionsort/1,insertionsort'1/1,'cklt#/1,'compare#/2,'less#/2,append#/2,append'1#/2,flatten#/1 ,flatten'1#/1,flattensort#/1,insert#/2,insert'1#/2,insert'2#/4,insertionsort#/1,insertionsort'1#/1} / {'0/0 ,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,leaf/0,nil/0,node/3,c_1/0,c_2/0,c_3/0,c_4/0 ,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/1,c_19/0,c_20/1 ,c_21/0,c_22/4,c_23/1,c_24/1,c_25/2,c_26/0,c_27/0,c_28/1,c_29/1,c_30/2,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,append#,append'1#,flatten# ,flatten'1#,flattensort#,insert#,insert'1#,insert'2#,insertionsort#,insertionsort'1#} and constructors {'0 ,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,leaf,nil,node} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)) -->_1 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)):3 -->_1 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)):2 -->_1 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)):1 2:W:'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)) -->_1 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)):3 -->_1 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)):2 -->_1 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)):1 3:W:'compare#('s(x),'s(y)) -> c_15('compare#(x,y)) -->_1 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)):3 -->_1 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)):2 -->_1 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)):1 4:W:'less#(x,y) -> c_16('compare#(x,y)) -->_1 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)):3 -->_1 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)):2 -->_1 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)):1 5:W:flattensort#(t) -> c_23(insertionsort#(flatten(t))) -->_1 insertionsort#(l) -> c_29(insertionsort'1#(l)):9 6:W:insert#(x,l) -> c_24(insert'1#(l,x)) -->_1 insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys),'less#(y,x)):7 7:W:insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys),'less#(y,x)) -->_1 insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)):8 -->_2 'less#(x,y) -> c_16('compare#(x,y)):4 8:W:insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)) -->_1 insert#(x,l) -> c_24(insert'1#(l,x)):6 9:W:insertionsort#(l) -> c_29(insertionsort'1#(l)) -->_1 insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)):10 10:W:insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) -->_2 insertionsort#(l) -> c_29(insertionsort'1#(l)):9 -->_1 insert#(x,l) -> c_24(insert'1#(l,x)):6 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 5: flattensort#(t) -> c_23(insertionsort#(flatten(t))) 9: insertionsort#(l) -> c_29(insertionsort'1#(l)) 10: insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) 6: insert#(x,l) -> c_24(insert'1#(l,x)) 8: insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)) 7: insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys),'less#(y,x)) 4: 'less#(x,y) -> c_16('compare#(x,y)) 1: 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)) 3: 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)) 2: 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)) ***** Step 6.a:3.b:1.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: 'cklt('EQ()) -> 'false() 'cklt('GT()) -> 'false() 'cklt('LT()) -> 'true() 'compare('0(),'0()) -> 'EQ() 'compare('0(),'neg(y)) -> 'GT() 'compare('0(),'pos(y)) -> 'LT() 'compare('0(),'s(y)) -> 'LT() 'compare('neg(x),'0()) -> 'LT() 'compare('neg(x),'neg(y)) -> 'compare(y,x) 'compare('neg(x),'pos(y)) -> 'LT() 'compare('pos(x),'0()) -> 'GT() 'compare('pos(x),'neg(y)) -> 'GT() 'compare('pos(x),'pos(y)) -> 'compare(x,y) 'compare('s(x),'0()) -> 'GT() 'compare('s(x),'s(y)) -> 'compare(x,y) 'less(x,y) -> 'cklt('compare(x,y)) append(l1,l2) -> append'1(l1,l2) append'1(dd(x,xs),l2) -> dd(x,append(xs,l2)) append'1(nil(),l2) -> l2 flatten(t) -> flatten'1(t) flatten'1(leaf()) -> nil() flatten'1(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2))) insert(x,l) -> insert'1(l,x) insert'1(dd(y,ys),x) -> insert'2('less(y,x),x,y,ys) insert'1(nil(),x) -> dd(x,nil()) insert'2('false(),x,y,ys) -> dd(x,dd(y,ys)) insert'2('true(),x,y,ys) -> dd(y,insert(x,ys)) insertionsort(l) -> insertionsort'1(l) insertionsort'1(dd(x,xs)) -> insert(x,insertionsort(xs)) insertionsort'1(nil()) -> nil() - Signature: {'cklt/1,'compare/2,'less/2,append/2,append'1/2,flatten/1,flatten'1/1,flattensort/1,insert/2,insert'1/2 ,insert'2/4,insertionsort/1,insertionsort'1/1,'cklt#/1,'compare#/2,'less#/2,append#/2,append'1#/2,flatten#/1 ,flatten'1#/1,flattensort#/1,insert#/2,insert'1#/2,insert'2#/4,insertionsort#/1,insertionsort'1#/1} / {'0/0 ,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,leaf/0,nil/0,node/3,c_1/0,c_2/0,c_3/0,c_4/0 ,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/1,c_19/0,c_20/1 ,c_21/0,c_22/4,c_23/1,c_24/1,c_25/2,c_26/0,c_27/0,c_28/1,c_29/1,c_30/2,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,append#,append'1#,flatten# ,flatten'1#,flattensort#,insert#,insert'1#,insert'2#,insertionsort#,insertionsort'1#} and constructors {'0 ,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,leaf,nil,node} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). ** Step 6.b:1: PredecessorEstimation WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: 'less#(x,y) -> c_16('compare#(x,y)) append#(l1,l2) -> c_17(append'1#(l1,l2)) append'1#(dd(x,xs),l2) -> c_18(append#(xs,l2)) flatten#(t) -> c_20(flatten'1#(t)) flatten'1#(node(l,t1,t2)) -> c_22(append#(l,append(flatten(t1),flatten(t2))) ,append#(flatten(t1),flatten(t2)) ,flatten#(t1) ,flatten#(t2)) flattensort#(t) -> c_23(insertionsort#(flatten(t)),flatten#(t)) insert#(x,l) -> c_24(insert'1#(l,x)) insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys),'less#(y,x)) insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)) insertionsort#(l) -> c_29(insertionsort'1#(l)) insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) - Weak DPs: 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)) 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)) 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)) - Weak TRS: 'cklt('EQ()) -> 'false() 'cklt('GT()) -> 'false() 'cklt('LT()) -> 'true() 'compare('0(),'0()) -> 'EQ() 'compare('0(),'neg(y)) -> 'GT() 'compare('0(),'pos(y)) -> 'LT() 'compare('0(),'s(y)) -> 'LT() 'compare('neg(x),'0()) -> 'LT() 'compare('neg(x),'neg(y)) -> 'compare(y,x) 'compare('neg(x),'pos(y)) -> 'LT() 'compare('pos(x),'0()) -> 'GT() 'compare('pos(x),'neg(y)) -> 'GT() 'compare('pos(x),'pos(y)) -> 'compare(x,y) 'compare('s(x),'0()) -> 'GT() 'compare('s(x),'s(y)) -> 'compare(x,y) 'less(x,y) -> 'cklt('compare(x,y)) append(l1,l2) -> append'1(l1,l2) append'1(dd(x,xs),l2) -> dd(x,append(xs,l2)) append'1(nil(),l2) -> l2 flatten(t) -> flatten'1(t) flatten'1(leaf()) -> nil() flatten'1(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2))) insert(x,l) -> insert'1(l,x) insert'1(dd(y,ys),x) -> insert'2('less(y,x),x,y,ys) insert'1(nil(),x) -> dd(x,nil()) insert'2('false(),x,y,ys) -> dd(x,dd(y,ys)) insert'2('true(),x,y,ys) -> dd(y,insert(x,ys)) insertionsort(l) -> insertionsort'1(l) insertionsort'1(dd(x,xs)) -> insert(x,insertionsort(xs)) insertionsort'1(nil()) -> nil() - Signature: {'cklt/1,'compare/2,'less/2,append/2,append'1/2,flatten/1,flatten'1/1,flattensort/1,insert/2,insert'1/2 ,insert'2/4,insertionsort/1,insertionsort'1/1,'cklt#/1,'compare#/2,'less#/2,append#/2,append'1#/2,flatten#/1 ,flatten'1#/1,flattensort#/1,insert#/2,insert'1#/2,insert'2#/4,insertionsort#/1,insertionsort'1#/1} / {'0/0 ,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,leaf/0,nil/0,node/3,c_1/0,c_2/0,c_3/0,c_4/0 ,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/1,c_19/0,c_20/1 ,c_21/0,c_22/4,c_23/2,c_24/1,c_25/2,c_26/0,c_27/0,c_28/1,c_29/1,c_30/2,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,append#,append'1#,flatten# ,flatten'1#,flattensort#,insert#,insert'1#,insert'2#,insertionsort#,insertionsort'1#} and constructors {'0 ,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,leaf,nil,node} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1} by application of Pre({1}) = {8}. Here rules are labelled as follows: 1: 'less#(x,y) -> c_16('compare#(x,y)) 2: append#(l1,l2) -> c_17(append'1#(l1,l2)) 3: append'1#(dd(x,xs),l2) -> c_18(append#(xs,l2)) 4: flatten#(t) -> c_20(flatten'1#(t)) 5: flatten'1#(node(l,t1,t2)) -> c_22(append#(l,append(flatten(t1),flatten(t2))) ,append#(flatten(t1),flatten(t2)) ,flatten#(t1) ,flatten#(t2)) 6: flattensort#(t) -> c_23(insertionsort#(flatten(t)),flatten#(t)) 7: insert#(x,l) -> c_24(insert'1#(l,x)) 8: insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys),'less#(y,x)) 9: insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)) 10: insertionsort#(l) -> c_29(insertionsort'1#(l)) 11: insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) 12: 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)) 13: 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)) 14: 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)) ** Step 6.b:2: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: append#(l1,l2) -> c_17(append'1#(l1,l2)) append'1#(dd(x,xs),l2) -> c_18(append#(xs,l2)) flatten#(t) -> c_20(flatten'1#(t)) flatten'1#(node(l,t1,t2)) -> c_22(append#(l,append(flatten(t1),flatten(t2))) ,append#(flatten(t1),flatten(t2)) ,flatten#(t1) ,flatten#(t2)) flattensort#(t) -> c_23(insertionsort#(flatten(t)),flatten#(t)) insert#(x,l) -> c_24(insert'1#(l,x)) insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys),'less#(y,x)) insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)) insertionsort#(l) -> c_29(insertionsort'1#(l)) insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) - Weak DPs: 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)) 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)) 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)) 'less#(x,y) -> c_16('compare#(x,y)) - Weak TRS: 'cklt('EQ()) -> 'false() 'cklt('GT()) -> 'false() 'cklt('LT()) -> 'true() 'compare('0(),'0()) -> 'EQ() 'compare('0(),'neg(y)) -> 'GT() 'compare('0(),'pos(y)) -> 'LT() 'compare('0(),'s(y)) -> 'LT() 'compare('neg(x),'0()) -> 'LT() 'compare('neg(x),'neg(y)) -> 'compare(y,x) 'compare('neg(x),'pos(y)) -> 'LT() 'compare('pos(x),'0()) -> 'GT() 'compare('pos(x),'neg(y)) -> 'GT() 'compare('pos(x),'pos(y)) -> 'compare(x,y) 'compare('s(x),'0()) -> 'GT() 'compare('s(x),'s(y)) -> 'compare(x,y) 'less(x,y) -> 'cklt('compare(x,y)) append(l1,l2) -> append'1(l1,l2) append'1(dd(x,xs),l2) -> dd(x,append(xs,l2)) append'1(nil(),l2) -> l2 flatten(t) -> flatten'1(t) flatten'1(leaf()) -> nil() flatten'1(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2))) insert(x,l) -> insert'1(l,x) insert'1(dd(y,ys),x) -> insert'2('less(y,x),x,y,ys) insert'1(nil(),x) -> dd(x,nil()) insert'2('false(),x,y,ys) -> dd(x,dd(y,ys)) insert'2('true(),x,y,ys) -> dd(y,insert(x,ys)) insertionsort(l) -> insertionsort'1(l) insertionsort'1(dd(x,xs)) -> insert(x,insertionsort(xs)) insertionsort'1(nil()) -> nil() - Signature: {'cklt/1,'compare/2,'less/2,append/2,append'1/2,flatten/1,flatten'1/1,flattensort/1,insert/2,insert'1/2 ,insert'2/4,insertionsort/1,insertionsort'1/1,'cklt#/1,'compare#/2,'less#/2,append#/2,append'1#/2,flatten#/1 ,flatten'1#/1,flattensort#/1,insert#/2,insert'1#/2,insert'2#/4,insertionsort#/1,insertionsort'1#/1} / {'0/0 ,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,leaf/0,nil/0,node/3,c_1/0,c_2/0,c_3/0,c_4/0 ,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/1,c_19/0,c_20/1 ,c_21/0,c_22/4,c_23/2,c_24/1,c_25/2,c_26/0,c_27/0,c_28/1,c_29/1,c_30/2,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,append#,append'1#,flatten# ,flatten'1#,flattensort#,insert#,insert'1#,insert'2#,insertionsort#,insertionsort'1#} and constructors {'0 ,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,leaf,nil,node} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:append#(l1,l2) -> c_17(append'1#(l1,l2)) -->_1 append'1#(dd(x,xs),l2) -> c_18(append#(xs,l2)):2 2:S:append'1#(dd(x,xs),l2) -> c_18(append#(xs,l2)) -->_1 append#(l1,l2) -> c_17(append'1#(l1,l2)):1 3:S:flatten#(t) -> c_20(flatten'1#(t)) -->_1 flatten'1#(node(l,t1,t2)) -> c_22(append#(l,append(flatten(t1),flatten(t2))) ,append#(flatten(t1),flatten(t2)) ,flatten#(t1) ,flatten#(t2)):4 4:S:flatten'1#(node(l,t1,t2)) -> c_22(append#(l,append(flatten(t1),flatten(t2))) ,append#(flatten(t1),flatten(t2)) ,flatten#(t1) ,flatten#(t2)) -->_4 flatten#(t) -> c_20(flatten'1#(t)):3 -->_3 flatten#(t) -> c_20(flatten'1#(t)):3 -->_2 append#(l1,l2) -> c_17(append'1#(l1,l2)):1 -->_1 append#(l1,l2) -> c_17(append'1#(l1,l2)):1 5:S:flattensort#(t) -> c_23(insertionsort#(flatten(t)),flatten#(t)) -->_1 insertionsort#(l) -> c_29(insertionsort'1#(l)):9 -->_2 flatten#(t) -> c_20(flatten'1#(t)):3 6:S:insert#(x,l) -> c_24(insert'1#(l,x)) -->_1 insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys),'less#(y,x)):7 7:S:insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys),'less#(y,x)) -->_2 'less#(x,y) -> c_16('compare#(x,y)):14 -->_1 insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)):8 8:S:insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)) -->_1 insert#(x,l) -> c_24(insert'1#(l,x)):6 9:S:insertionsort#(l) -> c_29(insertionsort'1#(l)) -->_1 insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)):10 10:S:insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) -->_2 insertionsort#(l) -> c_29(insertionsort'1#(l)):9 -->_1 insert#(x,l) -> c_24(insert'1#(l,x)):6 11:W:'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)) -->_1 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)):13 -->_1 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)):12 -->_1 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)):11 12:W:'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)) -->_1 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)):13 -->_1 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)):12 -->_1 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)):11 13:W:'compare#('s(x),'s(y)) -> c_15('compare#(x,y)) -->_1 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)):13 -->_1 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)):12 -->_1 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)):11 14:W:'less#(x,y) -> c_16('compare#(x,y)) -->_1 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)):13 -->_1 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)):12 -->_1 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)):11 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 14: 'less#(x,y) -> c_16('compare#(x,y)) 13: 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)) 12: 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)) 11: 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)) ** Step 6.b:3: SimplifyRHS WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: append#(l1,l2) -> c_17(append'1#(l1,l2)) append'1#(dd(x,xs),l2) -> c_18(append#(xs,l2)) flatten#(t) -> c_20(flatten'1#(t)) flatten'1#(node(l,t1,t2)) -> c_22(append#(l,append(flatten(t1),flatten(t2))) ,append#(flatten(t1),flatten(t2)) ,flatten#(t1) ,flatten#(t2)) flattensort#(t) -> c_23(insertionsort#(flatten(t)),flatten#(t)) insert#(x,l) -> c_24(insert'1#(l,x)) insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys),'less#(y,x)) insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)) insertionsort#(l) -> c_29(insertionsort'1#(l)) insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) - Weak TRS: 'cklt('EQ()) -> 'false() 'cklt('GT()) -> 'false() 'cklt('LT()) -> 'true() 'compare('0(),'0()) -> 'EQ() 'compare('0(),'neg(y)) -> 'GT() 'compare('0(),'pos(y)) -> 'LT() 'compare('0(),'s(y)) -> 'LT() 'compare('neg(x),'0()) -> 'LT() 'compare('neg(x),'neg(y)) -> 'compare(y,x) 'compare('neg(x),'pos(y)) -> 'LT() 'compare('pos(x),'0()) -> 'GT() 'compare('pos(x),'neg(y)) -> 'GT() 'compare('pos(x),'pos(y)) -> 'compare(x,y) 'compare('s(x),'0()) -> 'GT() 'compare('s(x),'s(y)) -> 'compare(x,y) 'less(x,y) -> 'cklt('compare(x,y)) append(l1,l2) -> append'1(l1,l2) append'1(dd(x,xs),l2) -> dd(x,append(xs,l2)) append'1(nil(),l2) -> l2 flatten(t) -> flatten'1(t) flatten'1(leaf()) -> nil() flatten'1(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2))) insert(x,l) -> insert'1(l,x) insert'1(dd(y,ys),x) -> insert'2('less(y,x),x,y,ys) insert'1(nil(),x) -> dd(x,nil()) insert'2('false(),x,y,ys) -> dd(x,dd(y,ys)) insert'2('true(),x,y,ys) -> dd(y,insert(x,ys)) insertionsort(l) -> insertionsort'1(l) insertionsort'1(dd(x,xs)) -> insert(x,insertionsort(xs)) insertionsort'1(nil()) -> nil() - Signature: {'cklt/1,'compare/2,'less/2,append/2,append'1/2,flatten/1,flatten'1/1,flattensort/1,insert/2,insert'1/2 ,insert'2/4,insertionsort/1,insertionsort'1/1,'cklt#/1,'compare#/2,'less#/2,append#/2,append'1#/2,flatten#/1 ,flatten'1#/1,flattensort#/1,insert#/2,insert'1#/2,insert'2#/4,insertionsort#/1,insertionsort'1#/1} / {'0/0 ,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,leaf/0,nil/0,node/3,c_1/0,c_2/0,c_3/0,c_4/0 ,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/1,c_19/0,c_20/1 ,c_21/0,c_22/4,c_23/2,c_24/1,c_25/2,c_26/0,c_27/0,c_28/1,c_29/1,c_30/2,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,append#,append'1#,flatten# ,flatten'1#,flattensort#,insert#,insert'1#,insert'2#,insertionsort#,insertionsort'1#} and constructors {'0 ,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,leaf,nil,node} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:append#(l1,l2) -> c_17(append'1#(l1,l2)) -->_1 append'1#(dd(x,xs),l2) -> c_18(append#(xs,l2)):2 2:S:append'1#(dd(x,xs),l2) -> c_18(append#(xs,l2)) -->_1 append#(l1,l2) -> c_17(append'1#(l1,l2)):1 3:S:flatten#(t) -> c_20(flatten'1#(t)) -->_1 flatten'1#(node(l,t1,t2)) -> c_22(append#(l,append(flatten(t1),flatten(t2))) ,append#(flatten(t1),flatten(t2)) ,flatten#(t1) ,flatten#(t2)):4 4:S:flatten'1#(node(l,t1,t2)) -> c_22(append#(l,append(flatten(t1),flatten(t2))) ,append#(flatten(t1),flatten(t2)) ,flatten#(t1) ,flatten#(t2)) -->_4 flatten#(t) -> c_20(flatten'1#(t)):3 -->_3 flatten#(t) -> c_20(flatten'1#(t)):3 -->_2 append#(l1,l2) -> c_17(append'1#(l1,l2)):1 -->_1 append#(l1,l2) -> c_17(append'1#(l1,l2)):1 5:S:flattensort#(t) -> c_23(insertionsort#(flatten(t)),flatten#(t)) -->_1 insertionsort#(l) -> c_29(insertionsort'1#(l)):9 -->_2 flatten#(t) -> c_20(flatten'1#(t)):3 6:S:insert#(x,l) -> c_24(insert'1#(l,x)) -->_1 insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys),'less#(y,x)):7 7:S:insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys),'less#(y,x)) -->_1 insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)):8 8:S:insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)) -->_1 insert#(x,l) -> c_24(insert'1#(l,x)):6 9:S:insertionsort#(l) -> c_29(insertionsort'1#(l)) -->_1 insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)):10 10:S:insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) -->_2 insertionsort#(l) -> c_29(insertionsort'1#(l)):9 -->_1 insert#(x,l) -> c_24(insert'1#(l,x)):6 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys)) ** Step 6.b:4: Decompose WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: append#(l1,l2) -> c_17(append'1#(l1,l2)) append'1#(dd(x,xs),l2) -> c_18(append#(xs,l2)) flatten#(t) -> c_20(flatten'1#(t)) flatten'1#(node(l,t1,t2)) -> c_22(append#(l,append(flatten(t1),flatten(t2))) ,append#(flatten(t1),flatten(t2)) ,flatten#(t1) ,flatten#(t2)) flattensort#(t) -> c_23(insertionsort#(flatten(t)),flatten#(t)) insert#(x,l) -> c_24(insert'1#(l,x)) insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys)) insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)) insertionsort#(l) -> c_29(insertionsort'1#(l)) insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) - Weak TRS: 'cklt('EQ()) -> 'false() 'cklt('GT()) -> 'false() 'cklt('LT()) -> 'true() 'compare('0(),'0()) -> 'EQ() 'compare('0(),'neg(y)) -> 'GT() 'compare('0(),'pos(y)) -> 'LT() 'compare('0(),'s(y)) -> 'LT() 'compare('neg(x),'0()) -> 'LT() 'compare('neg(x),'neg(y)) -> 'compare(y,x) 'compare('neg(x),'pos(y)) -> 'LT() 'compare('pos(x),'0()) -> 'GT() 'compare('pos(x),'neg(y)) -> 'GT() 'compare('pos(x),'pos(y)) -> 'compare(x,y) 'compare('s(x),'0()) -> 'GT() 'compare('s(x),'s(y)) -> 'compare(x,y) 'less(x,y) -> 'cklt('compare(x,y)) append(l1,l2) -> append'1(l1,l2) append'1(dd(x,xs),l2) -> dd(x,append(xs,l2)) append'1(nil(),l2) -> l2 flatten(t) -> flatten'1(t) flatten'1(leaf()) -> nil() flatten'1(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2))) insert(x,l) -> insert'1(l,x) insert'1(dd(y,ys),x) -> insert'2('less(y,x),x,y,ys) insert'1(nil(),x) -> dd(x,nil()) insert'2('false(),x,y,ys) -> dd(x,dd(y,ys)) insert'2('true(),x,y,ys) -> dd(y,insert(x,ys)) insertionsort(l) -> insertionsort'1(l) insertionsort'1(dd(x,xs)) -> insert(x,insertionsort(xs)) insertionsort'1(nil()) -> nil() - Signature: {'cklt/1,'compare/2,'less/2,append/2,append'1/2,flatten/1,flatten'1/1,flattensort/1,insert/2,insert'1/2 ,insert'2/4,insertionsort/1,insertionsort'1/1,'cklt#/1,'compare#/2,'less#/2,append#/2,append'1#/2,flatten#/1 ,flatten'1#/1,flattensort#/1,insert#/2,insert'1#/2,insert'2#/4,insertionsort#/1,insertionsort'1#/1} / {'0/0 ,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,leaf/0,nil/0,node/3,c_1/0,c_2/0,c_3/0,c_4/0 ,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/1,c_19/0,c_20/1 ,c_21/0,c_22/4,c_23/2,c_24/1,c_25/1,c_26/0,c_27/0,c_28/1,c_29/1,c_30/2,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,append#,append'1#,flatten# ,flatten'1#,flattensort#,insert#,insert'1#,insert'2#,insertionsort#,insertionsort'1#} and constructors {'0 ,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,leaf,nil,node} + 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: append#(l1,l2) -> c_17(append'1#(l1,l2)) append'1#(dd(x,xs),l2) -> c_18(append#(xs,l2)) - Weak DPs: flatten#(t) -> c_20(flatten'1#(t)) flatten'1#(node(l,t1,t2)) -> c_22(append#(l,append(flatten(t1),flatten(t2))) ,append#(flatten(t1),flatten(t2)) ,flatten#(t1) ,flatten#(t2)) flattensort#(t) -> c_23(insertionsort#(flatten(t)),flatten#(t)) insert#(x,l) -> c_24(insert'1#(l,x)) insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys)) insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)) insertionsort#(l) -> c_29(insertionsort'1#(l)) insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) - Weak TRS: 'cklt('EQ()) -> 'false() 'cklt('GT()) -> 'false() 'cklt('LT()) -> 'true() 'compare('0(),'0()) -> 'EQ() 'compare('0(),'neg(y)) -> 'GT() 'compare('0(),'pos(y)) -> 'LT() 'compare('0(),'s(y)) -> 'LT() 'compare('neg(x),'0()) -> 'LT() 'compare('neg(x),'neg(y)) -> 'compare(y,x) 'compare('neg(x),'pos(y)) -> 'LT() 'compare('pos(x),'0()) -> 'GT() 'compare('pos(x),'neg(y)) -> 'GT() 'compare('pos(x),'pos(y)) -> 'compare(x,y) 'compare('s(x),'0()) -> 'GT() 'compare('s(x),'s(y)) -> 'compare(x,y) 'less(x,y) -> 'cklt('compare(x,y)) append(l1,l2) -> append'1(l1,l2) append'1(dd(x,xs),l2) -> dd(x,append(xs,l2)) append'1(nil(),l2) -> l2 flatten(t) -> flatten'1(t) flatten'1(leaf()) -> nil() flatten'1(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2))) insert(x,l) -> insert'1(l,x) insert'1(dd(y,ys),x) -> insert'2('less(y,x),x,y,ys) insert'1(nil(),x) -> dd(x,nil()) insert'2('false(),x,y,ys) -> dd(x,dd(y,ys)) insert'2('true(),x,y,ys) -> dd(y,insert(x,ys)) insertionsort(l) -> insertionsort'1(l) insertionsort'1(dd(x,xs)) -> insert(x,insertionsort(xs)) insertionsort'1(nil()) -> nil() - Signature: {'cklt/1,'compare/2,'less/2,append/2,append'1/2,flatten/1,flatten'1/1,flattensort/1,insert/2,insert'1/2 ,insert'2/4,insertionsort/1,insertionsort'1/1,'cklt#/1,'compare#/2,'less#/2,append#/2,append'1#/2 ,flatten#/1,flatten'1#/1,flattensort#/1,insert#/2,insert'1#/2,insert'2#/4,insertionsort#/1 ,insertionsort'1#/1} / {'0/0,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,leaf/0,nil/0 ,node/3,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1 ,c_16/1,c_17/1,c_18/1,c_19/0,c_20/1,c_21/0,c_22/4,c_23/2,c_24/1,c_25/1,c_26/0,c_27/0,c_28/1,c_29/1,c_30/2 ,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,append#,append'1#,flatten# ,flatten'1#,flattensort#,insert#,insert'1#,insert'2#,insertionsort#,insertionsort'1#} and constructors {'0 ,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,leaf,nil,node} Problem (S) - Strict DPs: flatten#(t) -> c_20(flatten'1#(t)) flatten'1#(node(l,t1,t2)) -> c_22(append#(l,append(flatten(t1),flatten(t2))) ,append#(flatten(t1),flatten(t2)) ,flatten#(t1) ,flatten#(t2)) flattensort#(t) -> c_23(insertionsort#(flatten(t)),flatten#(t)) insert#(x,l) -> c_24(insert'1#(l,x)) insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys)) insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)) insertionsort#(l) -> c_29(insertionsort'1#(l)) insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) - Weak DPs: append#(l1,l2) -> c_17(append'1#(l1,l2)) append'1#(dd(x,xs),l2) -> c_18(append#(xs,l2)) - Weak TRS: 'cklt('EQ()) -> 'false() 'cklt('GT()) -> 'false() 'cklt('LT()) -> 'true() 'compare('0(),'0()) -> 'EQ() 'compare('0(),'neg(y)) -> 'GT() 'compare('0(),'pos(y)) -> 'LT() 'compare('0(),'s(y)) -> 'LT() 'compare('neg(x),'0()) -> 'LT() 'compare('neg(x),'neg(y)) -> 'compare(y,x) 'compare('neg(x),'pos(y)) -> 'LT() 'compare('pos(x),'0()) -> 'GT() 'compare('pos(x),'neg(y)) -> 'GT() 'compare('pos(x),'pos(y)) -> 'compare(x,y) 'compare('s(x),'0()) -> 'GT() 'compare('s(x),'s(y)) -> 'compare(x,y) 'less(x,y) -> 'cklt('compare(x,y)) append(l1,l2) -> append'1(l1,l2) append'1(dd(x,xs),l2) -> dd(x,append(xs,l2)) append'1(nil(),l2) -> l2 flatten(t) -> flatten'1(t) flatten'1(leaf()) -> nil() flatten'1(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2))) insert(x,l) -> insert'1(l,x) insert'1(dd(y,ys),x) -> insert'2('less(y,x),x,y,ys) insert'1(nil(),x) -> dd(x,nil()) insert'2('false(),x,y,ys) -> dd(x,dd(y,ys)) insert'2('true(),x,y,ys) -> dd(y,insert(x,ys)) insertionsort(l) -> insertionsort'1(l) insertionsort'1(dd(x,xs)) -> insert(x,insertionsort(xs)) insertionsort'1(nil()) -> nil() - Signature: {'cklt/1,'compare/2,'less/2,append/2,append'1/2,flatten/1,flatten'1/1,flattensort/1,insert/2,insert'1/2 ,insert'2/4,insertionsort/1,insertionsort'1/1,'cklt#/1,'compare#/2,'less#/2,append#/2,append'1#/2 ,flatten#/1,flatten'1#/1,flattensort#/1,insert#/2,insert'1#/2,insert'2#/4,insertionsort#/1 ,insertionsort'1#/1} / {'0/0,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,leaf/0,nil/0 ,node/3,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1 ,c_16/1,c_17/1,c_18/1,c_19/0,c_20/1,c_21/0,c_22/4,c_23/2,c_24/1,c_25/1,c_26/0,c_27/0,c_28/1,c_29/1,c_30/2 ,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,append#,append'1#,flatten# ,flatten'1#,flattensort#,insert#,insert'1#,insert'2#,insertionsort#,insertionsort'1#} and constructors {'0 ,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,leaf,nil,node} *** Step 6.b:4.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: append#(l1,l2) -> c_17(append'1#(l1,l2)) append'1#(dd(x,xs),l2) -> c_18(append#(xs,l2)) - Weak DPs: flatten#(t) -> c_20(flatten'1#(t)) flatten'1#(node(l,t1,t2)) -> c_22(append#(l,append(flatten(t1),flatten(t2))) ,append#(flatten(t1),flatten(t2)) ,flatten#(t1) ,flatten#(t2)) flattensort#(t) -> c_23(insertionsort#(flatten(t)),flatten#(t)) insert#(x,l) -> c_24(insert'1#(l,x)) insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys)) insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)) insertionsort#(l) -> c_29(insertionsort'1#(l)) insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) - Weak TRS: 'cklt('EQ()) -> 'false() 'cklt('GT()) -> 'false() 'cklt('LT()) -> 'true() 'compare('0(),'0()) -> 'EQ() 'compare('0(),'neg(y)) -> 'GT() 'compare('0(),'pos(y)) -> 'LT() 'compare('0(),'s(y)) -> 'LT() 'compare('neg(x),'0()) -> 'LT() 'compare('neg(x),'neg(y)) -> 'compare(y,x) 'compare('neg(x),'pos(y)) -> 'LT() 'compare('pos(x),'0()) -> 'GT() 'compare('pos(x),'neg(y)) -> 'GT() 'compare('pos(x),'pos(y)) -> 'compare(x,y) 'compare('s(x),'0()) -> 'GT() 'compare('s(x),'s(y)) -> 'compare(x,y) 'less(x,y) -> 'cklt('compare(x,y)) append(l1,l2) -> append'1(l1,l2) append'1(dd(x,xs),l2) -> dd(x,append(xs,l2)) append'1(nil(),l2) -> l2 flatten(t) -> flatten'1(t) flatten'1(leaf()) -> nil() flatten'1(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2))) insert(x,l) -> insert'1(l,x) insert'1(dd(y,ys),x) -> insert'2('less(y,x),x,y,ys) insert'1(nil(),x) -> dd(x,nil()) insert'2('false(),x,y,ys) -> dd(x,dd(y,ys)) insert'2('true(),x,y,ys) -> dd(y,insert(x,ys)) insertionsort(l) -> insertionsort'1(l) insertionsort'1(dd(x,xs)) -> insert(x,insertionsort(xs)) insertionsort'1(nil()) -> nil() - Signature: {'cklt/1,'compare/2,'less/2,append/2,append'1/2,flatten/1,flatten'1/1,flattensort/1,insert/2,insert'1/2 ,insert'2/4,insertionsort/1,insertionsort'1/1,'cklt#/1,'compare#/2,'less#/2,append#/2,append'1#/2,flatten#/1 ,flatten'1#/1,flattensort#/1,insert#/2,insert'1#/2,insert'2#/4,insertionsort#/1,insertionsort'1#/1} / {'0/0 ,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,leaf/0,nil/0,node/3,c_1/0,c_2/0,c_3/0,c_4/0 ,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/1,c_19/0,c_20/1 ,c_21/0,c_22/4,c_23/2,c_24/1,c_25/1,c_26/0,c_27/0,c_28/1,c_29/1,c_30/2,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,append#,append'1#,flatten# ,flatten'1#,flattensort#,insert#,insert'1#,insert'2#,insertionsort#,insertionsort'1#} and constructors {'0 ,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,leaf,nil,node} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:append#(l1,l2) -> c_17(append'1#(l1,l2)) -->_1 append'1#(dd(x,xs),l2) -> c_18(append#(xs,l2)):2 2:S:append'1#(dd(x,xs),l2) -> c_18(append#(xs,l2)) -->_1 append#(l1,l2) -> c_17(append'1#(l1,l2)):1 3:W:flatten#(t) -> c_20(flatten'1#(t)) -->_1 flatten'1#(node(l,t1,t2)) -> c_22(append#(l,append(flatten(t1),flatten(t2))) ,append#(flatten(t1),flatten(t2)) ,flatten#(t1) ,flatten#(t2)):4 4:W:flatten'1#(node(l,t1,t2)) -> c_22(append#(l,append(flatten(t1),flatten(t2))) ,append#(flatten(t1),flatten(t2)) ,flatten#(t1) ,flatten#(t2)) -->_2 append#(l1,l2) -> c_17(append'1#(l1,l2)):1 -->_1 append#(l1,l2) -> c_17(append'1#(l1,l2)):1 -->_4 flatten#(t) -> c_20(flatten'1#(t)):3 -->_3 flatten#(t) -> c_20(flatten'1#(t)):3 5:W:flattensort#(t) -> c_23(insertionsort#(flatten(t)),flatten#(t)) -->_2 flatten#(t) -> c_20(flatten'1#(t)):3 -->_1 insertionsort#(l) -> c_29(insertionsort'1#(l)):9 6:W:insert#(x,l) -> c_24(insert'1#(l,x)) -->_1 insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys)):7 7:W:insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys)) -->_1 insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)):8 8:W:insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)) -->_1 insert#(x,l) -> c_24(insert'1#(l,x)):6 9:W:insertionsort#(l) -> c_29(insertionsort'1#(l)) -->_1 insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)):10 10:W:insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) -->_1 insert#(x,l) -> c_24(insert'1#(l,x)):6 -->_2 insertionsort#(l) -> c_29(insertionsort'1#(l)):9 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 9: insertionsort#(l) -> c_29(insertionsort'1#(l)) 10: insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) 6: insert#(x,l) -> c_24(insert'1#(l,x)) 8: insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)) 7: insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys)) *** Step 6.b:4.a:2: SimplifyRHS WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: append#(l1,l2) -> c_17(append'1#(l1,l2)) append'1#(dd(x,xs),l2) -> c_18(append#(xs,l2)) - Weak DPs: flatten#(t) -> c_20(flatten'1#(t)) flatten'1#(node(l,t1,t2)) -> c_22(append#(l,append(flatten(t1),flatten(t2))) ,append#(flatten(t1),flatten(t2)) ,flatten#(t1) ,flatten#(t2)) flattensort#(t) -> c_23(insertionsort#(flatten(t)),flatten#(t)) - Weak TRS: 'cklt('EQ()) -> 'false() 'cklt('GT()) -> 'false() 'cklt('LT()) -> 'true() 'compare('0(),'0()) -> 'EQ() 'compare('0(),'neg(y)) -> 'GT() 'compare('0(),'pos(y)) -> 'LT() 'compare('0(),'s(y)) -> 'LT() 'compare('neg(x),'0()) -> 'LT() 'compare('neg(x),'neg(y)) -> 'compare(y,x) 'compare('neg(x),'pos(y)) -> 'LT() 'compare('pos(x),'0()) -> 'GT() 'compare('pos(x),'neg(y)) -> 'GT() 'compare('pos(x),'pos(y)) -> 'compare(x,y) 'compare('s(x),'0()) -> 'GT() 'compare('s(x),'s(y)) -> 'compare(x,y) 'less(x,y) -> 'cklt('compare(x,y)) append(l1,l2) -> append'1(l1,l2) append'1(dd(x,xs),l2) -> dd(x,append(xs,l2)) append'1(nil(),l2) -> l2 flatten(t) -> flatten'1(t) flatten'1(leaf()) -> nil() flatten'1(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2))) insert(x,l) -> insert'1(l,x) insert'1(dd(y,ys),x) -> insert'2('less(y,x),x,y,ys) insert'1(nil(),x) -> dd(x,nil()) insert'2('false(),x,y,ys) -> dd(x,dd(y,ys)) insert'2('true(),x,y,ys) -> dd(y,insert(x,ys)) insertionsort(l) -> insertionsort'1(l) insertionsort'1(dd(x,xs)) -> insert(x,insertionsort(xs)) insertionsort'1(nil()) -> nil() - Signature: {'cklt/1,'compare/2,'less/2,append/2,append'1/2,flatten/1,flatten'1/1,flattensort/1,insert/2,insert'1/2 ,insert'2/4,insertionsort/1,insertionsort'1/1,'cklt#/1,'compare#/2,'less#/2,append#/2,append'1#/2,flatten#/1 ,flatten'1#/1,flattensort#/1,insert#/2,insert'1#/2,insert'2#/4,insertionsort#/1,insertionsort'1#/1} / {'0/0 ,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,leaf/0,nil/0,node/3,c_1/0,c_2/0,c_3/0,c_4/0 ,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/1,c_19/0,c_20/1 ,c_21/0,c_22/4,c_23/2,c_24/1,c_25/1,c_26/0,c_27/0,c_28/1,c_29/1,c_30/2,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,append#,append'1#,flatten# ,flatten'1#,flattensort#,insert#,insert'1#,insert'2#,insertionsort#,insertionsort'1#} and constructors {'0 ,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,leaf,nil,node} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:append#(l1,l2) -> c_17(append'1#(l1,l2)) -->_1 append'1#(dd(x,xs),l2) -> c_18(append#(xs,l2)):2 2:S:append'1#(dd(x,xs),l2) -> c_18(append#(xs,l2)) -->_1 append#(l1,l2) -> c_17(append'1#(l1,l2)):1 3:W:flatten#(t) -> c_20(flatten'1#(t)) -->_1 flatten'1#(node(l,t1,t2)) -> c_22(append#(l,append(flatten(t1),flatten(t2))) ,append#(flatten(t1),flatten(t2)) ,flatten#(t1) ,flatten#(t2)):4 4:W:flatten'1#(node(l,t1,t2)) -> c_22(append#(l,append(flatten(t1),flatten(t2))) ,append#(flatten(t1),flatten(t2)) ,flatten#(t1) ,flatten#(t2)) -->_2 append#(l1,l2) -> c_17(append'1#(l1,l2)):1 -->_1 append#(l1,l2) -> c_17(append'1#(l1,l2)):1 -->_4 flatten#(t) -> c_20(flatten'1#(t)):3 -->_3 flatten#(t) -> c_20(flatten'1#(t)):3 5:W:flattensort#(t) -> c_23(insertionsort#(flatten(t)),flatten#(t)) -->_2 flatten#(t) -> c_20(flatten'1#(t)):3 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: flattensort#(t) -> c_23(flatten#(t)) *** Step 6.b:4.a:3: UsableRules WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: append#(l1,l2) -> c_17(append'1#(l1,l2)) append'1#(dd(x,xs),l2) -> c_18(append#(xs,l2)) - Weak DPs: flatten#(t) -> c_20(flatten'1#(t)) flatten'1#(node(l,t1,t2)) -> c_22(append#(l,append(flatten(t1),flatten(t2))) ,append#(flatten(t1),flatten(t2)) ,flatten#(t1) ,flatten#(t2)) flattensort#(t) -> c_23(flatten#(t)) - Weak TRS: 'cklt('EQ()) -> 'false() 'cklt('GT()) -> 'false() 'cklt('LT()) -> 'true() 'compare('0(),'0()) -> 'EQ() 'compare('0(),'neg(y)) -> 'GT() 'compare('0(),'pos(y)) -> 'LT() 'compare('0(),'s(y)) -> 'LT() 'compare('neg(x),'0()) -> 'LT() 'compare('neg(x),'neg(y)) -> 'compare(y,x) 'compare('neg(x),'pos(y)) -> 'LT() 'compare('pos(x),'0()) -> 'GT() 'compare('pos(x),'neg(y)) -> 'GT() 'compare('pos(x),'pos(y)) -> 'compare(x,y) 'compare('s(x),'0()) -> 'GT() 'compare('s(x),'s(y)) -> 'compare(x,y) 'less(x,y) -> 'cklt('compare(x,y)) append(l1,l2) -> append'1(l1,l2) append'1(dd(x,xs),l2) -> dd(x,append(xs,l2)) append'1(nil(),l2) -> l2 flatten(t) -> flatten'1(t) flatten'1(leaf()) -> nil() flatten'1(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2))) insert(x,l) -> insert'1(l,x) insert'1(dd(y,ys),x) -> insert'2('less(y,x),x,y,ys) insert'1(nil(),x) -> dd(x,nil()) insert'2('false(),x,y,ys) -> dd(x,dd(y,ys)) insert'2('true(),x,y,ys) -> dd(y,insert(x,ys)) insertionsort(l) -> insertionsort'1(l) insertionsort'1(dd(x,xs)) -> insert(x,insertionsort(xs)) insertionsort'1(nil()) -> nil() - Signature: {'cklt/1,'compare/2,'less/2,append/2,append'1/2,flatten/1,flatten'1/1,flattensort/1,insert/2,insert'1/2 ,insert'2/4,insertionsort/1,insertionsort'1/1,'cklt#/1,'compare#/2,'less#/2,append#/2,append'1#/2,flatten#/1 ,flatten'1#/1,flattensort#/1,insert#/2,insert'1#/2,insert'2#/4,insertionsort#/1,insertionsort'1#/1} / {'0/0 ,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,leaf/0,nil/0,node/3,c_1/0,c_2/0,c_3/0,c_4/0 ,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/1,c_19/0,c_20/1 ,c_21/0,c_22/4,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/1,c_29/1,c_30/2,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,append#,append'1#,flatten# ,flatten'1#,flattensort#,insert#,insert'1#,insert'2#,insertionsort#,insertionsort'1#} and constructors {'0 ,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,leaf,nil,node} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: append(l1,l2) -> append'1(l1,l2) append'1(dd(x,xs),l2) -> dd(x,append(xs,l2)) append'1(nil(),l2) -> l2 flatten(t) -> flatten'1(t) flatten'1(leaf()) -> nil() flatten'1(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2))) append#(l1,l2) -> c_17(append'1#(l1,l2)) append'1#(dd(x,xs),l2) -> c_18(append#(xs,l2)) flatten#(t) -> c_20(flatten'1#(t)) flatten'1#(node(l,t1,t2)) -> c_22(append#(l,append(flatten(t1),flatten(t2))) ,append#(flatten(t1),flatten(t2)) ,flatten#(t1) ,flatten#(t2)) flattensort#(t) -> c_23(flatten#(t)) *** Step 6.b:4.a:4: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: append#(l1,l2) -> c_17(append'1#(l1,l2)) append'1#(dd(x,xs),l2) -> c_18(append#(xs,l2)) - Weak DPs: flatten#(t) -> c_20(flatten'1#(t)) flatten'1#(node(l,t1,t2)) -> c_22(append#(l,append(flatten(t1),flatten(t2))) ,append#(flatten(t1),flatten(t2)) ,flatten#(t1) ,flatten#(t2)) flattensort#(t) -> c_23(flatten#(t)) - Weak TRS: append(l1,l2) -> append'1(l1,l2) append'1(dd(x,xs),l2) -> dd(x,append(xs,l2)) append'1(nil(),l2) -> l2 flatten(t) -> flatten'1(t) flatten'1(leaf()) -> nil() flatten'1(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2))) - Signature: {'cklt/1,'compare/2,'less/2,append/2,append'1/2,flatten/1,flatten'1/1,flattensort/1,insert/2,insert'1/2 ,insert'2/4,insertionsort/1,insertionsort'1/1,'cklt#/1,'compare#/2,'less#/2,append#/2,append'1#/2,flatten#/1 ,flatten'1#/1,flattensort#/1,insert#/2,insert'1#/2,insert'2#/4,insertionsort#/1,insertionsort'1#/1} / {'0/0 ,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,leaf/0,nil/0,node/3,c_1/0,c_2/0,c_3/0,c_4/0 ,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/1,c_19/0,c_20/1 ,c_21/0,c_22/4,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/1,c_29/1,c_30/2,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,append#,append'1#,flatten# ,flatten'1#,flattensort#,insert#,insert'1#,insert'2#,insertionsort#,insertionsort'1#} and constructors {'0 ,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,leaf,nil,node} + 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: append'1#(dd(x,xs),l2) -> c_18(append#(xs,l2)) Consider the set of all dependency pairs 1: append#(l1,l2) -> c_17(append'1#(l1,l2)) 2: append'1#(dd(x,xs),l2) -> c_18(append#(xs,l2)) 3: flatten#(t) -> c_20(flatten'1#(t)) 4: flatten'1#(node(l,t1,t2)) -> c_22(append#(l,append(flatten(t1),flatten(t2))) ,append#(flatten(t1),flatten(t2)) ,flatten#(t1) ,flatten#(t2)) 5: flattensort#(t) -> c_23(flatten#(t)) Processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^2)) SPACE(?,?)on application of the dependency pairs {2} These cover all (indirect) predecessors of dependency pairs {2,5} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. **** Step 6.b:4.a:4.a:1: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: append#(l1,l2) -> c_17(append'1#(l1,l2)) append'1#(dd(x,xs),l2) -> c_18(append#(xs,l2)) - Weak DPs: flatten#(t) -> c_20(flatten'1#(t)) flatten'1#(node(l,t1,t2)) -> c_22(append#(l,append(flatten(t1),flatten(t2))) ,append#(flatten(t1),flatten(t2)) ,flatten#(t1) ,flatten#(t2)) flattensort#(t) -> c_23(flatten#(t)) - Weak TRS: append(l1,l2) -> append'1(l1,l2) append'1(dd(x,xs),l2) -> dd(x,append(xs,l2)) append'1(nil(),l2) -> l2 flatten(t) -> flatten'1(t) flatten'1(leaf()) -> nil() flatten'1(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2))) - Signature: {'cklt/1,'compare/2,'less/2,append/2,append'1/2,flatten/1,flatten'1/1,flattensort/1,insert/2,insert'1/2 ,insert'2/4,insertionsort/1,insertionsort'1/1,'cklt#/1,'compare#/2,'less#/2,append#/2,append'1#/2,flatten#/1 ,flatten'1#/1,flattensort#/1,insert#/2,insert'1#/2,insert'2#/4,insertionsort#/1,insertionsort'1#/1} / {'0/0 ,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,leaf/0,nil/0,node/3,c_1/0,c_2/0,c_3/0,c_4/0 ,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/1,c_19/0,c_20/1 ,c_21/0,c_22/4,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/1,c_29/1,c_30/2,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,append#,append'1#,flatten# ,flatten'1#,flattensort#,insert#,insert'1#,insert'2#,insertionsort#,insertionsort'1#} and constructors {'0 ,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,leaf,nil,node} + 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_17) = {1}, uargs(c_18) = {1}, uargs(c_20) = {1}, uargs(c_22) = {1,2,3,4}, uargs(c_23) = {1} Following symbols are considered usable: {append,append'1,flatten,flatten'1,'cklt#,'compare#,'less#,append#,append'1#,flatten#,flatten'1# ,flattensort#,insert#,insert'1#,insert'2#,insertionsort#,insertionsort'1#} TcT has computed the following interpretation: p('0) = 0 p('EQ) = 0 p('GT) = 0 p('LT) = 0 p('cklt) = 0 p('compare) = 0 p('false) = 0 p('less) = 0 p('neg) = 0 p('pos) = 0 p('s) = 0 p('true) = 0 p(append) = x1 + x2 p(append'1) = x1 + x2 p(dd) = 1 + x2 p(flatten) = x1 p(flatten'1) = x1 p(flattensort) = 0 p(insert) = 0 p(insert'1) = 0 p(insert'2) = 0 p(insertionsort) = 0 p(insertionsort'1) = 0 p(leaf) = 0 p(nil) = 0 p(node) = 1 + x1 + x2 + x3 p('cklt#) = 0 p('compare#) = 0 p('less#) = 0 p(append#) = x1 p(append'1#) = x1 p(flatten#) = x1^2 p(flatten'1#) = x1^2 p(flattensort#) = 1 + x1 + x1^2 p(insert#) = 0 p(insert'1#) = 0 p(insert'2#) = 0 p(insertionsort#) = 0 p(insertionsort'1#) = 0 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) = x1 p(c_18) = x1 p(c_19) = 0 p(c_20) = x1 p(c_21) = 0 p(c_22) = x1 + x2 + x3 + x4 p(c_23) = 1 + x1 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 Following rules are strictly oriented: append'1#(dd(x,xs),l2) = 1 + xs > xs = c_18(append#(xs,l2)) Following rules are (at-least) weakly oriented: append#(l1,l2) = l1 >= l1 = c_17(append'1#(l1,l2)) flatten#(t) = t^2 >= t^2 = c_20(flatten'1#(t)) flatten'1#(node(l,t1,t2)) = 1 + 2*l + 2*l*t1 + 2*l*t2 + l^2 + 2*t1 + 2*t1*t2 + t1^2 + 2*t2 + t2^2 >= l + t1 + t1^2 + t2^2 = c_22(append#(l,append(flatten(t1),flatten(t2))),append#(flatten(t1),flatten(t2)),flatten#(t1),flatten#(t2)) flattensort#(t) = 1 + t + t^2 >= 1 + t^2 = c_23(flatten#(t)) append(l1,l2) = l1 + l2 >= l1 + l2 = append'1(l1,l2) append'1(dd(x,xs),l2) = 1 + l2 + xs >= 1 + l2 + xs = dd(x,append(xs,l2)) append'1(nil(),l2) = l2 >= l2 = l2 flatten(t) = t >= t = flatten'1(t) flatten'1(leaf()) = 0 >= 0 = nil() flatten'1(node(l,t1,t2)) = 1 + l + t1 + t2 >= l + t1 + t2 = append(l,append(flatten(t1),flatten(t2))) **** Step 6.b:4.a:4.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: append#(l1,l2) -> c_17(append'1#(l1,l2)) - Weak DPs: append'1#(dd(x,xs),l2) -> c_18(append#(xs,l2)) flatten#(t) -> c_20(flatten'1#(t)) flatten'1#(node(l,t1,t2)) -> c_22(append#(l,append(flatten(t1),flatten(t2))) ,append#(flatten(t1),flatten(t2)) ,flatten#(t1) ,flatten#(t2)) flattensort#(t) -> c_23(flatten#(t)) - Weak TRS: append(l1,l2) -> append'1(l1,l2) append'1(dd(x,xs),l2) -> dd(x,append(xs,l2)) append'1(nil(),l2) -> l2 flatten(t) -> flatten'1(t) flatten'1(leaf()) -> nil() flatten'1(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2))) - Signature: {'cklt/1,'compare/2,'less/2,append/2,append'1/2,flatten/1,flatten'1/1,flattensort/1,insert/2,insert'1/2 ,insert'2/4,insertionsort/1,insertionsort'1/1,'cklt#/1,'compare#/2,'less#/2,append#/2,append'1#/2,flatten#/1 ,flatten'1#/1,flattensort#/1,insert#/2,insert'1#/2,insert'2#/4,insertionsort#/1,insertionsort'1#/1} / {'0/0 ,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,leaf/0,nil/0,node/3,c_1/0,c_2/0,c_3/0,c_4/0 ,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/1,c_19/0,c_20/1 ,c_21/0,c_22/4,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/1,c_29/1,c_30/2,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,append#,append'1#,flatten# ,flatten'1#,flattensort#,insert#,insert'1#,insert'2#,insertionsort#,insertionsort'1#} and constructors {'0 ,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,leaf,nil,node} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () **** Step 6.b:4.a:4.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: append#(l1,l2) -> c_17(append'1#(l1,l2)) - Weak DPs: append'1#(dd(x,xs),l2) -> c_18(append#(xs,l2)) flatten#(t) -> c_20(flatten'1#(t)) flatten'1#(node(l,t1,t2)) -> c_22(append#(l,append(flatten(t1),flatten(t2))) ,append#(flatten(t1),flatten(t2)) ,flatten#(t1) ,flatten#(t2)) flattensort#(t) -> c_23(flatten#(t)) - Weak TRS: append(l1,l2) -> append'1(l1,l2) append'1(dd(x,xs),l2) -> dd(x,append(xs,l2)) append'1(nil(),l2) -> l2 flatten(t) -> flatten'1(t) flatten'1(leaf()) -> nil() flatten'1(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2))) - Signature: {'cklt/1,'compare/2,'less/2,append/2,append'1/2,flatten/1,flatten'1/1,flattensort/1,insert/2,insert'1/2 ,insert'2/4,insertionsort/1,insertionsort'1/1,'cklt#/1,'compare#/2,'less#/2,append#/2,append'1#/2,flatten#/1 ,flatten'1#/1,flattensort#/1,insert#/2,insert'1#/2,insert'2#/4,insertionsort#/1,insertionsort'1#/1} / {'0/0 ,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,leaf/0,nil/0,node/3,c_1/0,c_2/0,c_3/0,c_4/0 ,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/1,c_19/0,c_20/1 ,c_21/0,c_22/4,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/1,c_29/1,c_30/2,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,append#,append'1#,flatten# ,flatten'1#,flattensort#,insert#,insert'1#,insert'2#,insertionsort#,insertionsort'1#} and constructors {'0 ,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,leaf,nil,node} + 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: append#(l1,l2) -> c_17(append'1#(l1,l2)) Consider the set of all dependency pairs 1: append#(l1,l2) -> c_17(append'1#(l1,l2)) 2: append'1#(dd(x,xs),l2) -> c_18(append#(xs,l2)) 3: flatten#(t) -> c_20(flatten'1#(t)) 4: flatten'1#(node(l,t1,t2)) -> c_22(append#(l,append(flatten(t1),flatten(t2))) ,append#(flatten(t1),flatten(t2)) ,flatten#(t1) ,flatten#(t2)) 5: flattensort#(t) -> c_23(flatten#(t)) Processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^2)) SPACE(?,?)on application of the dependency pairs {1} These cover all (indirect) predecessors of dependency pairs {1,2,5} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. ***** Step 6.b:4.a:4.b:1.a:1: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: append#(l1,l2) -> c_17(append'1#(l1,l2)) - Weak DPs: append'1#(dd(x,xs),l2) -> c_18(append#(xs,l2)) flatten#(t) -> c_20(flatten'1#(t)) flatten'1#(node(l,t1,t2)) -> c_22(append#(l,append(flatten(t1),flatten(t2))) ,append#(flatten(t1),flatten(t2)) ,flatten#(t1) ,flatten#(t2)) flattensort#(t) -> c_23(flatten#(t)) - Weak TRS: append(l1,l2) -> append'1(l1,l2) append'1(dd(x,xs),l2) -> dd(x,append(xs,l2)) append'1(nil(),l2) -> l2 flatten(t) -> flatten'1(t) flatten'1(leaf()) -> nil() flatten'1(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2))) - Signature: {'cklt/1,'compare/2,'less/2,append/2,append'1/2,flatten/1,flatten'1/1,flattensort/1,insert/2,insert'1/2 ,insert'2/4,insertionsort/1,insertionsort'1/1,'cklt#/1,'compare#/2,'less#/2,append#/2,append'1#/2,flatten#/1 ,flatten'1#/1,flattensort#/1,insert#/2,insert'1#/2,insert'2#/4,insertionsort#/1,insertionsort'1#/1} / {'0/0 ,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,leaf/0,nil/0,node/3,c_1/0,c_2/0,c_3/0,c_4/0 ,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/1,c_19/0,c_20/1 ,c_21/0,c_22/4,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/1,c_29/1,c_30/2,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,append#,append'1#,flatten# ,flatten'1#,flattensort#,insert#,insert'1#,insert'2#,insertionsort#,insertionsort'1#} and constructors {'0 ,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,leaf,nil,node} + 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_17) = {1}, uargs(c_18) = {1}, uargs(c_20) = {1}, uargs(c_22) = {1,2,3,4}, uargs(c_23) = {1} Following symbols are considered usable: {append,append'1,flatten,flatten'1,'cklt#,'compare#,'less#,append#,append'1#,flatten#,flatten'1# ,flattensort#,insert#,insert'1#,insert'2#,insertionsort#,insertionsort'1#} TcT has computed the following interpretation: p('0) = 0 p('EQ) = 0 p('GT) = 0 p('LT) = 0 p('cklt) = 0 p('compare) = 0 p('false) = 0 p('less) = 0 p('neg) = 0 p('pos) = 0 p('s) = 0 p('true) = 0 p(append) = x1 + x2 p(append'1) = x1 + x2 p(dd) = 1 + x2 p(flatten) = x1 p(flatten'1) = x1 p(flattensort) = 0 p(insert) = 0 p(insert'1) = 0 p(insert'2) = 0 p(insertionsort) = 0 p(insertionsort'1) = 0 p(leaf) = 0 p(nil) = 0 p(node) = 1 + x1 + x2 + x3 p('cklt#) = 0 p('compare#) = 0 p('less#) = 0 p(append#) = 1 + x1 p(append'1#) = x1 p(flatten#) = x1 + x1^2 p(flatten'1#) = x1 + x1^2 p(flattensort#) = x1 + x1^2 p(insert#) = 0 p(insert'1#) = 0 p(insert'2#) = 0 p(insertionsort#) = 0 p(insertionsort'1#) = 0 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) = x1 p(c_18) = x1 p(c_19) = 0 p(c_20) = x1 p(c_21) = 0 p(c_22) = x1 + x2 + x3 + x4 p(c_23) = x1 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 Following rules are strictly oriented: append#(l1,l2) = 1 + l1 > l1 = c_17(append'1#(l1,l2)) Following rules are (at-least) weakly oriented: append'1#(dd(x,xs),l2) = 1 + xs >= 1 + xs = c_18(append#(xs,l2)) flatten#(t) = t + t^2 >= t + t^2 = c_20(flatten'1#(t)) flatten'1#(node(l,t1,t2)) = 2 + 3*l + 2*l*t1 + 2*l*t2 + l^2 + 3*t1 + 2*t1*t2 + t1^2 + 3*t2 + t2^2 >= 2 + l + 2*t1 + t1^2 + t2 + t2^2 = c_22(append#(l,append(flatten(t1),flatten(t2))),append#(flatten(t1),flatten(t2)),flatten#(t1),flatten#(t2)) flattensort#(t) = t + t^2 >= t + t^2 = c_23(flatten#(t)) append(l1,l2) = l1 + l2 >= l1 + l2 = append'1(l1,l2) append'1(dd(x,xs),l2) = 1 + l2 + xs >= 1 + l2 + xs = dd(x,append(xs,l2)) append'1(nil(),l2) = l2 >= l2 = l2 flatten(t) = t >= t = flatten'1(t) flatten'1(leaf()) = 0 >= 0 = nil() flatten'1(node(l,t1,t2)) = 1 + l + t1 + t2 >= l + t1 + t2 = append(l,append(flatten(t1),flatten(t2))) ***** Step 6.b:4.a:4.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: append#(l1,l2) -> c_17(append'1#(l1,l2)) append'1#(dd(x,xs),l2) -> c_18(append#(xs,l2)) flatten#(t) -> c_20(flatten'1#(t)) flatten'1#(node(l,t1,t2)) -> c_22(append#(l,append(flatten(t1),flatten(t2))) ,append#(flatten(t1),flatten(t2)) ,flatten#(t1) ,flatten#(t2)) flattensort#(t) -> c_23(flatten#(t)) - Weak TRS: append(l1,l2) -> append'1(l1,l2) append'1(dd(x,xs),l2) -> dd(x,append(xs,l2)) append'1(nil(),l2) -> l2 flatten(t) -> flatten'1(t) flatten'1(leaf()) -> nil() flatten'1(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2))) - Signature: {'cklt/1,'compare/2,'less/2,append/2,append'1/2,flatten/1,flatten'1/1,flattensort/1,insert/2,insert'1/2 ,insert'2/4,insertionsort/1,insertionsort'1/1,'cklt#/1,'compare#/2,'less#/2,append#/2,append'1#/2,flatten#/1 ,flatten'1#/1,flattensort#/1,insert#/2,insert'1#/2,insert'2#/4,insertionsort#/1,insertionsort'1#/1} / {'0/0 ,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,leaf/0,nil/0,node/3,c_1/0,c_2/0,c_3/0,c_4/0 ,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/1,c_19/0,c_20/1 ,c_21/0,c_22/4,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/1,c_29/1,c_30/2,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,append#,append'1#,flatten# ,flatten'1#,flattensort#,insert#,insert'1#,insert'2#,insertionsort#,insertionsort'1#} and constructors {'0 ,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,leaf,nil,node} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ***** Step 6.b:4.a:4.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: append#(l1,l2) -> c_17(append'1#(l1,l2)) append'1#(dd(x,xs),l2) -> c_18(append#(xs,l2)) flatten#(t) -> c_20(flatten'1#(t)) flatten'1#(node(l,t1,t2)) -> c_22(append#(l,append(flatten(t1),flatten(t2))) ,append#(flatten(t1),flatten(t2)) ,flatten#(t1) ,flatten#(t2)) flattensort#(t) -> c_23(flatten#(t)) - Weak TRS: append(l1,l2) -> append'1(l1,l2) append'1(dd(x,xs),l2) -> dd(x,append(xs,l2)) append'1(nil(),l2) -> l2 flatten(t) -> flatten'1(t) flatten'1(leaf()) -> nil() flatten'1(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2))) - Signature: {'cklt/1,'compare/2,'less/2,append/2,append'1/2,flatten/1,flatten'1/1,flattensort/1,insert/2,insert'1/2 ,insert'2/4,insertionsort/1,insertionsort'1/1,'cklt#/1,'compare#/2,'less#/2,append#/2,append'1#/2,flatten#/1 ,flatten'1#/1,flattensort#/1,insert#/2,insert'1#/2,insert'2#/4,insertionsort#/1,insertionsort'1#/1} / {'0/0 ,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,leaf/0,nil/0,node/3,c_1/0,c_2/0,c_3/0,c_4/0 ,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/1,c_19/0,c_20/1 ,c_21/0,c_22/4,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/1,c_29/1,c_30/2,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,append#,append'1#,flatten# ,flatten'1#,flattensort#,insert#,insert'1#,insert'2#,insertionsort#,insertionsort'1#} and constructors {'0 ,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,leaf,nil,node} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:append#(l1,l2) -> c_17(append'1#(l1,l2)) -->_1 append'1#(dd(x,xs),l2) -> c_18(append#(xs,l2)):2 2:W:append'1#(dd(x,xs),l2) -> c_18(append#(xs,l2)) -->_1 append#(l1,l2) -> c_17(append'1#(l1,l2)):1 3:W:flatten#(t) -> c_20(flatten'1#(t)) -->_1 flatten'1#(node(l,t1,t2)) -> c_22(append#(l,append(flatten(t1),flatten(t2))) ,append#(flatten(t1),flatten(t2)) ,flatten#(t1) ,flatten#(t2)):4 4:W:flatten'1#(node(l,t1,t2)) -> c_22(append#(l,append(flatten(t1),flatten(t2))) ,append#(flatten(t1),flatten(t2)) ,flatten#(t1) ,flatten#(t2)) -->_4 flatten#(t) -> c_20(flatten'1#(t)):3 -->_3 flatten#(t) -> c_20(flatten'1#(t)):3 -->_2 append#(l1,l2) -> c_17(append'1#(l1,l2)):1 -->_1 append#(l1,l2) -> c_17(append'1#(l1,l2)):1 5:W:flattensort#(t) -> c_23(flatten#(t)) -->_1 flatten#(t) -> c_20(flatten'1#(t)):3 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 5: flattensort#(t) -> c_23(flatten#(t)) 3: flatten#(t) -> c_20(flatten'1#(t)) 4: flatten'1#(node(l,t1,t2)) -> c_22(append#(l,append(flatten(t1),flatten(t2))) ,append#(flatten(t1),flatten(t2)) ,flatten#(t1) ,flatten#(t2)) 1: append#(l1,l2) -> c_17(append'1#(l1,l2)) 2: append'1#(dd(x,xs),l2) -> c_18(append#(xs,l2)) ***** Step 6.b:4.a:4.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: append(l1,l2) -> append'1(l1,l2) append'1(dd(x,xs),l2) -> dd(x,append(xs,l2)) append'1(nil(),l2) -> l2 flatten(t) -> flatten'1(t) flatten'1(leaf()) -> nil() flatten'1(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2))) - Signature: {'cklt/1,'compare/2,'less/2,append/2,append'1/2,flatten/1,flatten'1/1,flattensort/1,insert/2,insert'1/2 ,insert'2/4,insertionsort/1,insertionsort'1/1,'cklt#/1,'compare#/2,'less#/2,append#/2,append'1#/2,flatten#/1 ,flatten'1#/1,flattensort#/1,insert#/2,insert'1#/2,insert'2#/4,insertionsort#/1,insertionsort'1#/1} / {'0/0 ,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,leaf/0,nil/0,node/3,c_1/0,c_2/0,c_3/0,c_4/0 ,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/1,c_19/0,c_20/1 ,c_21/0,c_22/4,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/1,c_29/1,c_30/2,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,append#,append'1#,flatten# ,flatten'1#,flattensort#,insert#,insert'1#,insert'2#,insertionsort#,insertionsort'1#} and constructors {'0 ,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,leaf,nil,node} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). *** Step 6.b:4.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: flatten#(t) -> c_20(flatten'1#(t)) flatten'1#(node(l,t1,t2)) -> c_22(append#(l,append(flatten(t1),flatten(t2))) ,append#(flatten(t1),flatten(t2)) ,flatten#(t1) ,flatten#(t2)) flattensort#(t) -> c_23(insertionsort#(flatten(t)),flatten#(t)) insert#(x,l) -> c_24(insert'1#(l,x)) insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys)) insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)) insertionsort#(l) -> c_29(insertionsort'1#(l)) insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) - Weak DPs: append#(l1,l2) -> c_17(append'1#(l1,l2)) append'1#(dd(x,xs),l2) -> c_18(append#(xs,l2)) - Weak TRS: 'cklt('EQ()) -> 'false() 'cklt('GT()) -> 'false() 'cklt('LT()) -> 'true() 'compare('0(),'0()) -> 'EQ() 'compare('0(),'neg(y)) -> 'GT() 'compare('0(),'pos(y)) -> 'LT() 'compare('0(),'s(y)) -> 'LT() 'compare('neg(x),'0()) -> 'LT() 'compare('neg(x),'neg(y)) -> 'compare(y,x) 'compare('neg(x),'pos(y)) -> 'LT() 'compare('pos(x),'0()) -> 'GT() 'compare('pos(x),'neg(y)) -> 'GT() 'compare('pos(x),'pos(y)) -> 'compare(x,y) 'compare('s(x),'0()) -> 'GT() 'compare('s(x),'s(y)) -> 'compare(x,y) 'less(x,y) -> 'cklt('compare(x,y)) append(l1,l2) -> append'1(l1,l2) append'1(dd(x,xs),l2) -> dd(x,append(xs,l2)) append'1(nil(),l2) -> l2 flatten(t) -> flatten'1(t) flatten'1(leaf()) -> nil() flatten'1(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2))) insert(x,l) -> insert'1(l,x) insert'1(dd(y,ys),x) -> insert'2('less(y,x),x,y,ys) insert'1(nil(),x) -> dd(x,nil()) insert'2('false(),x,y,ys) -> dd(x,dd(y,ys)) insert'2('true(),x,y,ys) -> dd(y,insert(x,ys)) insertionsort(l) -> insertionsort'1(l) insertionsort'1(dd(x,xs)) -> insert(x,insertionsort(xs)) insertionsort'1(nil()) -> nil() - Signature: {'cklt/1,'compare/2,'less/2,append/2,append'1/2,flatten/1,flatten'1/1,flattensort/1,insert/2,insert'1/2 ,insert'2/4,insertionsort/1,insertionsort'1/1,'cklt#/1,'compare#/2,'less#/2,append#/2,append'1#/2,flatten#/1 ,flatten'1#/1,flattensort#/1,insert#/2,insert'1#/2,insert'2#/4,insertionsort#/1,insertionsort'1#/1} / {'0/0 ,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,leaf/0,nil/0,node/3,c_1/0,c_2/0,c_3/0,c_4/0 ,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/1,c_19/0,c_20/1 ,c_21/0,c_22/4,c_23/2,c_24/1,c_25/1,c_26/0,c_27/0,c_28/1,c_29/1,c_30/2,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,append#,append'1#,flatten# ,flatten'1#,flattensort#,insert#,insert'1#,insert'2#,insertionsort#,insertionsort'1#} and constructors {'0 ,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,leaf,nil,node} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:flatten#(t) -> c_20(flatten'1#(t)) -->_1 flatten'1#(node(l,t1,t2)) -> c_22(append#(l,append(flatten(t1),flatten(t2))) ,append#(flatten(t1),flatten(t2)) ,flatten#(t1) ,flatten#(t2)):2 2:S:flatten'1#(node(l,t1,t2)) -> c_22(append#(l,append(flatten(t1),flatten(t2))) ,append#(flatten(t1),flatten(t2)) ,flatten#(t1) ,flatten#(t2)) -->_2 append#(l1,l2) -> c_17(append'1#(l1,l2)):9 -->_1 append#(l1,l2) -> c_17(append'1#(l1,l2)):9 -->_4 flatten#(t) -> c_20(flatten'1#(t)):1 -->_3 flatten#(t) -> c_20(flatten'1#(t)):1 3:S:flattensort#(t) -> c_23(insertionsort#(flatten(t)),flatten#(t)) -->_1 insertionsort#(l) -> c_29(insertionsort'1#(l)):7 -->_2 flatten#(t) -> c_20(flatten'1#(t)):1 4:S:insert#(x,l) -> c_24(insert'1#(l,x)) -->_1 insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys)):5 5:S:insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys)) -->_1 insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)):6 6:S:insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)) -->_1 insert#(x,l) -> c_24(insert'1#(l,x)):4 7:S:insertionsort#(l) -> c_29(insertionsort'1#(l)) -->_1 insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)):8 8:S:insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) -->_2 insertionsort#(l) -> c_29(insertionsort'1#(l)):7 -->_1 insert#(x,l) -> c_24(insert'1#(l,x)):4 9:W:append#(l1,l2) -> c_17(append'1#(l1,l2)) -->_1 append'1#(dd(x,xs),l2) -> c_18(append#(xs,l2)):10 10:W:append'1#(dd(x,xs),l2) -> c_18(append#(xs,l2)) -->_1 append#(l1,l2) -> c_17(append'1#(l1,l2)):9 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 9: append#(l1,l2) -> c_17(append'1#(l1,l2)) 10: append'1#(dd(x,xs),l2) -> c_18(append#(xs,l2)) *** Step 6.b:4.b:2: SimplifyRHS WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: flatten#(t) -> c_20(flatten'1#(t)) flatten'1#(node(l,t1,t2)) -> c_22(append#(l,append(flatten(t1),flatten(t2))) ,append#(flatten(t1),flatten(t2)) ,flatten#(t1) ,flatten#(t2)) flattensort#(t) -> c_23(insertionsort#(flatten(t)),flatten#(t)) insert#(x,l) -> c_24(insert'1#(l,x)) insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys)) insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)) insertionsort#(l) -> c_29(insertionsort'1#(l)) insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) - Weak TRS: 'cklt('EQ()) -> 'false() 'cklt('GT()) -> 'false() 'cklt('LT()) -> 'true() 'compare('0(),'0()) -> 'EQ() 'compare('0(),'neg(y)) -> 'GT() 'compare('0(),'pos(y)) -> 'LT() 'compare('0(),'s(y)) -> 'LT() 'compare('neg(x),'0()) -> 'LT() 'compare('neg(x),'neg(y)) -> 'compare(y,x) 'compare('neg(x),'pos(y)) -> 'LT() 'compare('pos(x),'0()) -> 'GT() 'compare('pos(x),'neg(y)) -> 'GT() 'compare('pos(x),'pos(y)) -> 'compare(x,y) 'compare('s(x),'0()) -> 'GT() 'compare('s(x),'s(y)) -> 'compare(x,y) 'less(x,y) -> 'cklt('compare(x,y)) append(l1,l2) -> append'1(l1,l2) append'1(dd(x,xs),l2) -> dd(x,append(xs,l2)) append'1(nil(),l2) -> l2 flatten(t) -> flatten'1(t) flatten'1(leaf()) -> nil() flatten'1(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2))) insert(x,l) -> insert'1(l,x) insert'1(dd(y,ys),x) -> insert'2('less(y,x),x,y,ys) insert'1(nil(),x) -> dd(x,nil()) insert'2('false(),x,y,ys) -> dd(x,dd(y,ys)) insert'2('true(),x,y,ys) -> dd(y,insert(x,ys)) insertionsort(l) -> insertionsort'1(l) insertionsort'1(dd(x,xs)) -> insert(x,insertionsort(xs)) insertionsort'1(nil()) -> nil() - Signature: {'cklt/1,'compare/2,'less/2,append/2,append'1/2,flatten/1,flatten'1/1,flattensort/1,insert/2,insert'1/2 ,insert'2/4,insertionsort/1,insertionsort'1/1,'cklt#/1,'compare#/2,'less#/2,append#/2,append'1#/2,flatten#/1 ,flatten'1#/1,flattensort#/1,insert#/2,insert'1#/2,insert'2#/4,insertionsort#/1,insertionsort'1#/1} / {'0/0 ,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,leaf/0,nil/0,node/3,c_1/0,c_2/0,c_3/0,c_4/0 ,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/1,c_19/0,c_20/1 ,c_21/0,c_22/4,c_23/2,c_24/1,c_25/1,c_26/0,c_27/0,c_28/1,c_29/1,c_30/2,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,append#,append'1#,flatten# ,flatten'1#,flattensort#,insert#,insert'1#,insert'2#,insertionsort#,insertionsort'1#} and constructors {'0 ,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,leaf,nil,node} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:flatten#(t) -> c_20(flatten'1#(t)) -->_1 flatten'1#(node(l,t1,t2)) -> c_22(append#(l,append(flatten(t1),flatten(t2))) ,append#(flatten(t1),flatten(t2)) ,flatten#(t1) ,flatten#(t2)):2 2:S:flatten'1#(node(l,t1,t2)) -> c_22(append#(l,append(flatten(t1),flatten(t2))) ,append#(flatten(t1),flatten(t2)) ,flatten#(t1) ,flatten#(t2)) -->_4 flatten#(t) -> c_20(flatten'1#(t)):1 -->_3 flatten#(t) -> c_20(flatten'1#(t)):1 3:S:flattensort#(t) -> c_23(insertionsort#(flatten(t)),flatten#(t)) -->_1 insertionsort#(l) -> c_29(insertionsort'1#(l)):7 -->_2 flatten#(t) -> c_20(flatten'1#(t)):1 4:S:insert#(x,l) -> c_24(insert'1#(l,x)) -->_1 insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys)):5 5:S:insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys)) -->_1 insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)):6 6:S:insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)) -->_1 insert#(x,l) -> c_24(insert'1#(l,x)):4 7:S:insertionsort#(l) -> c_29(insertionsort'1#(l)) -->_1 insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)):8 8:S:insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) -->_2 insertionsort#(l) -> c_29(insertionsort'1#(l)):7 -->_1 insert#(x,l) -> c_24(insert'1#(l,x)):4 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: flatten'1#(node(l,t1,t2)) -> c_22(flatten#(t1),flatten#(t2)) *** Step 6.b:4.b:3: Decompose WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: flatten#(t) -> c_20(flatten'1#(t)) flatten'1#(node(l,t1,t2)) -> c_22(flatten#(t1),flatten#(t2)) flattensort#(t) -> c_23(insertionsort#(flatten(t)),flatten#(t)) insert#(x,l) -> c_24(insert'1#(l,x)) insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys)) insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)) insertionsort#(l) -> c_29(insertionsort'1#(l)) insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) - Weak TRS: 'cklt('EQ()) -> 'false() 'cklt('GT()) -> 'false() 'cklt('LT()) -> 'true() 'compare('0(),'0()) -> 'EQ() 'compare('0(),'neg(y)) -> 'GT() 'compare('0(),'pos(y)) -> 'LT() 'compare('0(),'s(y)) -> 'LT() 'compare('neg(x),'0()) -> 'LT() 'compare('neg(x),'neg(y)) -> 'compare(y,x) 'compare('neg(x),'pos(y)) -> 'LT() 'compare('pos(x),'0()) -> 'GT() 'compare('pos(x),'neg(y)) -> 'GT() 'compare('pos(x),'pos(y)) -> 'compare(x,y) 'compare('s(x),'0()) -> 'GT() 'compare('s(x),'s(y)) -> 'compare(x,y) 'less(x,y) -> 'cklt('compare(x,y)) append(l1,l2) -> append'1(l1,l2) append'1(dd(x,xs),l2) -> dd(x,append(xs,l2)) append'1(nil(),l2) -> l2 flatten(t) -> flatten'1(t) flatten'1(leaf()) -> nil() flatten'1(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2))) insert(x,l) -> insert'1(l,x) insert'1(dd(y,ys),x) -> insert'2('less(y,x),x,y,ys) insert'1(nil(),x) -> dd(x,nil()) insert'2('false(),x,y,ys) -> dd(x,dd(y,ys)) insert'2('true(),x,y,ys) -> dd(y,insert(x,ys)) insertionsort(l) -> insertionsort'1(l) insertionsort'1(dd(x,xs)) -> insert(x,insertionsort(xs)) insertionsort'1(nil()) -> nil() - Signature: {'cklt/1,'compare/2,'less/2,append/2,append'1/2,flatten/1,flatten'1/1,flattensort/1,insert/2,insert'1/2 ,insert'2/4,insertionsort/1,insertionsort'1/1,'cklt#/1,'compare#/2,'less#/2,append#/2,append'1#/2,flatten#/1 ,flatten'1#/1,flattensort#/1,insert#/2,insert'1#/2,insert'2#/4,insertionsort#/1,insertionsort'1#/1} / {'0/0 ,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,leaf/0,nil/0,node/3,c_1/0,c_2/0,c_3/0,c_4/0 ,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/1,c_19/0,c_20/1 ,c_21/0,c_22/2,c_23/2,c_24/1,c_25/1,c_26/0,c_27/0,c_28/1,c_29/1,c_30/2,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,append#,append'1#,flatten# ,flatten'1#,flattensort#,insert#,insert'1#,insert'2#,insertionsort#,insertionsort'1#} and constructors {'0 ,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,leaf,nil,node} + 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: flatten#(t) -> c_20(flatten'1#(t)) flatten'1#(node(l,t1,t2)) -> c_22(flatten#(t1),flatten#(t2)) - Weak DPs: flattensort#(t) -> c_23(insertionsort#(flatten(t)),flatten#(t)) insert#(x,l) -> c_24(insert'1#(l,x)) insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys)) insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)) insertionsort#(l) -> c_29(insertionsort'1#(l)) insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) - Weak TRS: 'cklt('EQ()) -> 'false() 'cklt('GT()) -> 'false() 'cklt('LT()) -> 'true() 'compare('0(),'0()) -> 'EQ() 'compare('0(),'neg(y)) -> 'GT() 'compare('0(),'pos(y)) -> 'LT() 'compare('0(),'s(y)) -> 'LT() 'compare('neg(x),'0()) -> 'LT() 'compare('neg(x),'neg(y)) -> 'compare(y,x) 'compare('neg(x),'pos(y)) -> 'LT() 'compare('pos(x),'0()) -> 'GT() 'compare('pos(x),'neg(y)) -> 'GT() 'compare('pos(x),'pos(y)) -> 'compare(x,y) 'compare('s(x),'0()) -> 'GT() 'compare('s(x),'s(y)) -> 'compare(x,y) 'less(x,y) -> 'cklt('compare(x,y)) append(l1,l2) -> append'1(l1,l2) append'1(dd(x,xs),l2) -> dd(x,append(xs,l2)) append'1(nil(),l2) -> l2 flatten(t) -> flatten'1(t) flatten'1(leaf()) -> nil() flatten'1(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2))) insert(x,l) -> insert'1(l,x) insert'1(dd(y,ys),x) -> insert'2('less(y,x),x,y,ys) insert'1(nil(),x) -> dd(x,nil()) insert'2('false(),x,y,ys) -> dd(x,dd(y,ys)) insert'2('true(),x,y,ys) -> dd(y,insert(x,ys)) insertionsort(l) -> insertionsort'1(l) insertionsort'1(dd(x,xs)) -> insert(x,insertionsort(xs)) insertionsort'1(nil()) -> nil() - Signature: {'cklt/1,'compare/2,'less/2,append/2,append'1/2,flatten/1,flatten'1/1,flattensort/1,insert/2,insert'1/2 ,insert'2/4,insertionsort/1,insertionsort'1/1,'cklt#/1,'compare#/2,'less#/2,append#/2,append'1#/2 ,flatten#/1,flatten'1#/1,flattensort#/1,insert#/2,insert'1#/2,insert'2#/4,insertionsort#/1 ,insertionsort'1#/1} / {'0/0,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,leaf/0,nil/0 ,node/3,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1 ,c_16/1,c_17/1,c_18/1,c_19/0,c_20/1,c_21/0,c_22/2,c_23/2,c_24/1,c_25/1,c_26/0,c_27/0,c_28/1,c_29/1,c_30/2 ,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,append#,append'1#,flatten# ,flatten'1#,flattensort#,insert#,insert'1#,insert'2#,insertionsort#,insertionsort'1#} and constructors {'0 ,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,leaf,nil,node} Problem (S) - Strict DPs: flattensort#(t) -> c_23(insertionsort#(flatten(t)),flatten#(t)) insert#(x,l) -> c_24(insert'1#(l,x)) insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys)) insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)) insertionsort#(l) -> c_29(insertionsort'1#(l)) insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) - Weak DPs: flatten#(t) -> c_20(flatten'1#(t)) flatten'1#(node(l,t1,t2)) -> c_22(flatten#(t1),flatten#(t2)) - Weak TRS: 'cklt('EQ()) -> 'false() 'cklt('GT()) -> 'false() 'cklt('LT()) -> 'true() 'compare('0(),'0()) -> 'EQ() 'compare('0(),'neg(y)) -> 'GT() 'compare('0(),'pos(y)) -> 'LT() 'compare('0(),'s(y)) -> 'LT() 'compare('neg(x),'0()) -> 'LT() 'compare('neg(x),'neg(y)) -> 'compare(y,x) 'compare('neg(x),'pos(y)) -> 'LT() 'compare('pos(x),'0()) -> 'GT() 'compare('pos(x),'neg(y)) -> 'GT() 'compare('pos(x),'pos(y)) -> 'compare(x,y) 'compare('s(x),'0()) -> 'GT() 'compare('s(x),'s(y)) -> 'compare(x,y) 'less(x,y) -> 'cklt('compare(x,y)) append(l1,l2) -> append'1(l1,l2) append'1(dd(x,xs),l2) -> dd(x,append(xs,l2)) append'1(nil(),l2) -> l2 flatten(t) -> flatten'1(t) flatten'1(leaf()) -> nil() flatten'1(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2))) insert(x,l) -> insert'1(l,x) insert'1(dd(y,ys),x) -> insert'2('less(y,x),x,y,ys) insert'1(nil(),x) -> dd(x,nil()) insert'2('false(),x,y,ys) -> dd(x,dd(y,ys)) insert'2('true(),x,y,ys) -> dd(y,insert(x,ys)) insertionsort(l) -> insertionsort'1(l) insertionsort'1(dd(x,xs)) -> insert(x,insertionsort(xs)) insertionsort'1(nil()) -> nil() - Signature: {'cklt/1,'compare/2,'less/2,append/2,append'1/2,flatten/1,flatten'1/1,flattensort/1,insert/2,insert'1/2 ,insert'2/4,insertionsort/1,insertionsort'1/1,'cklt#/1,'compare#/2,'less#/2,append#/2,append'1#/2 ,flatten#/1,flatten'1#/1,flattensort#/1,insert#/2,insert'1#/2,insert'2#/4,insertionsort#/1 ,insertionsort'1#/1} / {'0/0,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,leaf/0,nil/0 ,node/3,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1 ,c_16/1,c_17/1,c_18/1,c_19/0,c_20/1,c_21/0,c_22/2,c_23/2,c_24/1,c_25/1,c_26/0,c_27/0,c_28/1,c_29/1,c_30/2 ,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,append#,append'1#,flatten# ,flatten'1#,flattensort#,insert#,insert'1#,insert'2#,insertionsort#,insertionsort'1#} and constructors {'0 ,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,leaf,nil,node} **** Step 6.b:4.b:3.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: flatten#(t) -> c_20(flatten'1#(t)) flatten'1#(node(l,t1,t2)) -> c_22(flatten#(t1),flatten#(t2)) - Weak DPs: flattensort#(t) -> c_23(insertionsort#(flatten(t)),flatten#(t)) insert#(x,l) -> c_24(insert'1#(l,x)) insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys)) insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)) insertionsort#(l) -> c_29(insertionsort'1#(l)) insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) - Weak TRS: 'cklt('EQ()) -> 'false() 'cklt('GT()) -> 'false() 'cklt('LT()) -> 'true() 'compare('0(),'0()) -> 'EQ() 'compare('0(),'neg(y)) -> 'GT() 'compare('0(),'pos(y)) -> 'LT() 'compare('0(),'s(y)) -> 'LT() 'compare('neg(x),'0()) -> 'LT() 'compare('neg(x),'neg(y)) -> 'compare(y,x) 'compare('neg(x),'pos(y)) -> 'LT() 'compare('pos(x),'0()) -> 'GT() 'compare('pos(x),'neg(y)) -> 'GT() 'compare('pos(x),'pos(y)) -> 'compare(x,y) 'compare('s(x),'0()) -> 'GT() 'compare('s(x),'s(y)) -> 'compare(x,y) 'less(x,y) -> 'cklt('compare(x,y)) append(l1,l2) -> append'1(l1,l2) append'1(dd(x,xs),l2) -> dd(x,append(xs,l2)) append'1(nil(),l2) -> l2 flatten(t) -> flatten'1(t) flatten'1(leaf()) -> nil() flatten'1(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2))) insert(x,l) -> insert'1(l,x) insert'1(dd(y,ys),x) -> insert'2('less(y,x),x,y,ys) insert'1(nil(),x) -> dd(x,nil()) insert'2('false(),x,y,ys) -> dd(x,dd(y,ys)) insert'2('true(),x,y,ys) -> dd(y,insert(x,ys)) insertionsort(l) -> insertionsort'1(l) insertionsort'1(dd(x,xs)) -> insert(x,insertionsort(xs)) insertionsort'1(nil()) -> nil() - Signature: {'cklt/1,'compare/2,'less/2,append/2,append'1/2,flatten/1,flatten'1/1,flattensort/1,insert/2,insert'1/2 ,insert'2/4,insertionsort/1,insertionsort'1/1,'cklt#/1,'compare#/2,'less#/2,append#/2,append'1#/2,flatten#/1 ,flatten'1#/1,flattensort#/1,insert#/2,insert'1#/2,insert'2#/4,insertionsort#/1,insertionsort'1#/1} / {'0/0 ,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,leaf/0,nil/0,node/3,c_1/0,c_2/0,c_3/0,c_4/0 ,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/1,c_19/0,c_20/1 ,c_21/0,c_22/2,c_23/2,c_24/1,c_25/1,c_26/0,c_27/0,c_28/1,c_29/1,c_30/2,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,append#,append'1#,flatten# ,flatten'1#,flattensort#,insert#,insert'1#,insert'2#,insertionsort#,insertionsort'1#} and constructors {'0 ,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,leaf,nil,node} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:flatten#(t) -> c_20(flatten'1#(t)) -->_1 flatten'1#(node(l,t1,t2)) -> c_22(flatten#(t1),flatten#(t2)):2 2:S:flatten'1#(node(l,t1,t2)) -> c_22(flatten#(t1),flatten#(t2)) -->_2 flatten#(t) -> c_20(flatten'1#(t)):1 -->_1 flatten#(t) -> c_20(flatten'1#(t)):1 3:W:flattensort#(t) -> c_23(insertionsort#(flatten(t)),flatten#(t)) -->_2 flatten#(t) -> c_20(flatten'1#(t)):1 -->_1 insertionsort#(l) -> c_29(insertionsort'1#(l)):7 4:W:insert#(x,l) -> c_24(insert'1#(l,x)) -->_1 insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys)):5 5:W:insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys)) -->_1 insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)):6 6:W:insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)) -->_1 insert#(x,l) -> c_24(insert'1#(l,x)):4 7:W:insertionsort#(l) -> c_29(insertionsort'1#(l)) -->_1 insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)):8 8:W:insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) -->_1 insert#(x,l) -> c_24(insert'1#(l,x)):4 -->_2 insertionsort#(l) -> c_29(insertionsort'1#(l)):7 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 7: insertionsort#(l) -> c_29(insertionsort'1#(l)) 8: insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) 4: insert#(x,l) -> c_24(insert'1#(l,x)) 6: insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)) 5: insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys)) **** Step 6.b:4.b:3.a:2: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: flatten#(t) -> c_20(flatten'1#(t)) flatten'1#(node(l,t1,t2)) -> c_22(flatten#(t1),flatten#(t2)) - Weak DPs: flattensort#(t) -> c_23(insertionsort#(flatten(t)),flatten#(t)) - Weak TRS: 'cklt('EQ()) -> 'false() 'cklt('GT()) -> 'false() 'cklt('LT()) -> 'true() 'compare('0(),'0()) -> 'EQ() 'compare('0(),'neg(y)) -> 'GT() 'compare('0(),'pos(y)) -> 'LT() 'compare('0(),'s(y)) -> 'LT() 'compare('neg(x),'0()) -> 'LT() 'compare('neg(x),'neg(y)) -> 'compare(y,x) 'compare('neg(x),'pos(y)) -> 'LT() 'compare('pos(x),'0()) -> 'GT() 'compare('pos(x),'neg(y)) -> 'GT() 'compare('pos(x),'pos(y)) -> 'compare(x,y) 'compare('s(x),'0()) -> 'GT() 'compare('s(x),'s(y)) -> 'compare(x,y) 'less(x,y) -> 'cklt('compare(x,y)) append(l1,l2) -> append'1(l1,l2) append'1(dd(x,xs),l2) -> dd(x,append(xs,l2)) append'1(nil(),l2) -> l2 flatten(t) -> flatten'1(t) flatten'1(leaf()) -> nil() flatten'1(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2))) insert(x,l) -> insert'1(l,x) insert'1(dd(y,ys),x) -> insert'2('less(y,x),x,y,ys) insert'1(nil(),x) -> dd(x,nil()) insert'2('false(),x,y,ys) -> dd(x,dd(y,ys)) insert'2('true(),x,y,ys) -> dd(y,insert(x,ys)) insertionsort(l) -> insertionsort'1(l) insertionsort'1(dd(x,xs)) -> insert(x,insertionsort(xs)) insertionsort'1(nil()) -> nil() - Signature: {'cklt/1,'compare/2,'less/2,append/2,append'1/2,flatten/1,flatten'1/1,flattensort/1,insert/2,insert'1/2 ,insert'2/4,insertionsort/1,insertionsort'1/1,'cklt#/1,'compare#/2,'less#/2,append#/2,append'1#/2,flatten#/1 ,flatten'1#/1,flattensort#/1,insert#/2,insert'1#/2,insert'2#/4,insertionsort#/1,insertionsort'1#/1} / {'0/0 ,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,leaf/0,nil/0,node/3,c_1/0,c_2/0,c_3/0,c_4/0 ,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/1,c_19/0,c_20/1 ,c_21/0,c_22/2,c_23/2,c_24/1,c_25/1,c_26/0,c_27/0,c_28/1,c_29/1,c_30/2,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,append#,append'1#,flatten# ,flatten'1#,flattensort#,insert#,insert'1#,insert'2#,insertionsort#,insertionsort'1#} and constructors {'0 ,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,leaf,nil,node} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:flatten#(t) -> c_20(flatten'1#(t)) -->_1 flatten'1#(node(l,t1,t2)) -> c_22(flatten#(t1),flatten#(t2)):2 2:S:flatten'1#(node(l,t1,t2)) -> c_22(flatten#(t1),flatten#(t2)) -->_2 flatten#(t) -> c_20(flatten'1#(t)):1 -->_1 flatten#(t) -> c_20(flatten'1#(t)):1 3:W:flattensort#(t) -> c_23(insertionsort#(flatten(t)),flatten#(t)) -->_2 flatten#(t) -> c_20(flatten'1#(t)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: flattensort#(t) -> c_23(flatten#(t)) **** Step 6.b:4.b:3.a:3: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: flatten#(t) -> c_20(flatten'1#(t)) flatten'1#(node(l,t1,t2)) -> c_22(flatten#(t1),flatten#(t2)) - Weak DPs: flattensort#(t) -> c_23(flatten#(t)) - Weak TRS: 'cklt('EQ()) -> 'false() 'cklt('GT()) -> 'false() 'cklt('LT()) -> 'true() 'compare('0(),'0()) -> 'EQ() 'compare('0(),'neg(y)) -> 'GT() 'compare('0(),'pos(y)) -> 'LT() 'compare('0(),'s(y)) -> 'LT() 'compare('neg(x),'0()) -> 'LT() 'compare('neg(x),'neg(y)) -> 'compare(y,x) 'compare('neg(x),'pos(y)) -> 'LT() 'compare('pos(x),'0()) -> 'GT() 'compare('pos(x),'neg(y)) -> 'GT() 'compare('pos(x),'pos(y)) -> 'compare(x,y) 'compare('s(x),'0()) -> 'GT() 'compare('s(x),'s(y)) -> 'compare(x,y) 'less(x,y) -> 'cklt('compare(x,y)) append(l1,l2) -> append'1(l1,l2) append'1(dd(x,xs),l2) -> dd(x,append(xs,l2)) append'1(nil(),l2) -> l2 flatten(t) -> flatten'1(t) flatten'1(leaf()) -> nil() flatten'1(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2))) insert(x,l) -> insert'1(l,x) insert'1(dd(y,ys),x) -> insert'2('less(y,x),x,y,ys) insert'1(nil(),x) -> dd(x,nil()) insert'2('false(),x,y,ys) -> dd(x,dd(y,ys)) insert'2('true(),x,y,ys) -> dd(y,insert(x,ys)) insertionsort(l) -> insertionsort'1(l) insertionsort'1(dd(x,xs)) -> insert(x,insertionsort(xs)) insertionsort'1(nil()) -> nil() - Signature: {'cklt/1,'compare/2,'less/2,append/2,append'1/2,flatten/1,flatten'1/1,flattensort/1,insert/2,insert'1/2 ,insert'2/4,insertionsort/1,insertionsort'1/1,'cklt#/1,'compare#/2,'less#/2,append#/2,append'1#/2,flatten#/1 ,flatten'1#/1,flattensort#/1,insert#/2,insert'1#/2,insert'2#/4,insertionsort#/1,insertionsort'1#/1} / {'0/0 ,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,leaf/0,nil/0,node/3,c_1/0,c_2/0,c_3/0,c_4/0 ,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/1,c_19/0,c_20/1 ,c_21/0,c_22/2,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/1,c_29/1,c_30/2,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,append#,append'1#,flatten# ,flatten'1#,flattensort#,insert#,insert'1#,insert'2#,insertionsort#,insertionsort'1#} and constructors {'0 ,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,leaf,nil,node} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: flatten#(t) -> c_20(flatten'1#(t)) flatten'1#(node(l,t1,t2)) -> c_22(flatten#(t1),flatten#(t2)) flattensort#(t) -> c_23(flatten#(t)) **** Step 6.b:4.b:3.a:4: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: flatten#(t) -> c_20(flatten'1#(t)) flatten'1#(node(l,t1,t2)) -> c_22(flatten#(t1),flatten#(t2)) - Weak DPs: flattensort#(t) -> c_23(flatten#(t)) - Signature: {'cklt/1,'compare/2,'less/2,append/2,append'1/2,flatten/1,flatten'1/1,flattensort/1,insert/2,insert'1/2 ,insert'2/4,insertionsort/1,insertionsort'1/1,'cklt#/1,'compare#/2,'less#/2,append#/2,append'1#/2,flatten#/1 ,flatten'1#/1,flattensort#/1,insert#/2,insert'1#/2,insert'2#/4,insertionsort#/1,insertionsort'1#/1} / {'0/0 ,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,leaf/0,nil/0,node/3,c_1/0,c_2/0,c_3/0,c_4/0 ,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/1,c_19/0,c_20/1 ,c_21/0,c_22/2,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/1,c_29/1,c_30/2,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,append#,append'1#,flatten# ,flatten'1#,flattensort#,insert#,insert'1#,insert'2#,insertionsort#,insertionsort'1#} and constructors {'0 ,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,leaf,nil,node} + 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: 2: flatten'1#(node(l,t1,t2)) -> c_22(flatten#(t1),flatten#(t2)) Consider the set of all dependency pairs 1: flatten#(t) -> c_20(flatten'1#(t)) 2: flatten'1#(node(l,t1,t2)) -> c_22(flatten#(t1),flatten#(t2)) 3: flattensort#(t) -> c_23(flatten#(t)) Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1)) SPACE(?,?)on application of the dependency pairs {2} These cover all (indirect) predecessors of dependency pairs {1,2,3} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. ***** Step 6.b:4.b:3.a:4.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: flatten#(t) -> c_20(flatten'1#(t)) flatten'1#(node(l,t1,t2)) -> c_22(flatten#(t1),flatten#(t2)) - Weak DPs: flattensort#(t) -> c_23(flatten#(t)) - Signature: {'cklt/1,'compare/2,'less/2,append/2,append'1/2,flatten/1,flatten'1/1,flattensort/1,insert/2,insert'1/2 ,insert'2/4,insertionsort/1,insertionsort'1/1,'cklt#/1,'compare#/2,'less#/2,append#/2,append'1#/2,flatten#/1 ,flatten'1#/1,flattensort#/1,insert#/2,insert'1#/2,insert'2#/4,insertionsort#/1,insertionsort'1#/1} / {'0/0 ,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,leaf/0,nil/0,node/3,c_1/0,c_2/0,c_3/0,c_4/0 ,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/1,c_19/0,c_20/1 ,c_21/0,c_22/2,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/1,c_29/1,c_30/2,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,append#,append'1#,flatten# ,flatten'1#,flattensort#,insert#,insert'1#,insert'2#,insertionsort#,insertionsort'1#} and constructors {'0 ,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,leaf,nil,node} + 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_20) = {1}, uargs(c_22) = {1,2}, uargs(c_23) = {1} Following symbols are considered usable: {'cklt#,'compare#,'less#,append#,append'1#,flatten#,flatten'1#,flattensort#,insert#,insert'1#,insert'2# ,insertionsort#,insertionsort'1#} TcT has computed the following interpretation: p('0) = [0] p('EQ) = [0] p('GT) = [0] p('LT) = [0] p('cklt) = [0] p('compare) = [0] p('false) = [0] p('less) = [0] p('neg) = [1] x1 + [0] p('pos) = [1] x1 + [0] p('s) = [1] x1 + [0] p('true) = [1] p(append) = [1] p(append'1) = [2] x1 + [4] p(dd) = [1] x2 + [0] p(flatten) = [1] p(flatten'1) = [1] x1 + [1] p(flattensort) = [8] x1 + [1] p(insert) = [1] x1 + [1] p(insert'1) = [2] x2 + [0] p(insert'2) = [1] x1 + [1] x2 + [2] x3 + [2] p(insertionsort) = [0] p(insertionsort'1) = [2] x1 + [2] p(leaf) = [2] p(nil) = [1] p(node) = [1] x2 + [1] x3 + [2] p('cklt#) = [0] p('compare#) = [1] x2 + [0] p('less#) = [2] x1 + [2] x2 + [4] p(append#) = [0] p(append'1#) = [8] x1 + [0] p(flatten#) = [8] x1 + [2] p(flatten'1#) = [8] x1 + [2] p(flattensort#) = [8] x1 + [3] p(insert#) = [4] x1 + [1] x2 + [0] p(insert'1#) = [0] p(insert'2#) = [1] x1 + [1] x2 + [4] x4 + [0] p(insertionsort#) = [1] p(insertionsort'1#) = [2] x1 + [1] p(c_1) = [0] p(c_2) = [1] p(c_3) = [8] p(c_4) = [1] p(c_5) = [2] p(c_6) = [0] p(c_7) = [2] p(c_8) = [1] p(c_9) = [2] p(c_10) = [1] p(c_11) = [0] p(c_12) = [0] p(c_13) = [1] x1 + [2] p(c_14) = [8] p(c_15) = [2] p(c_16) = [1] p(c_17) = [1] x1 + [1] p(c_18) = [1] p(c_19) = [0] p(c_20) = [1] x1 + [0] p(c_21) = [0] p(c_22) = [1] x1 + [1] x2 + [6] p(c_23) = [1] x1 + [1] p(c_24) = [1] p(c_25) = [8] p(c_26) = [1] p(c_27) = [2] p(c_28) = [2] p(c_29) = [8] x1 + [1] p(c_30) = [2] x1 + [1] x2 + [1] p(c_31) = [1] Following rules are strictly oriented: flatten'1#(node(l,t1,t2)) = [8] t1 + [8] t2 + [18] > [8] t1 + [8] t2 + [10] = c_22(flatten#(t1),flatten#(t2)) Following rules are (at-least) weakly oriented: flatten#(t) = [8] t + [2] >= [8] t + [2] = c_20(flatten'1#(t)) flattensort#(t) = [8] t + [3] >= [8] t + [3] = c_23(flatten#(t)) ***** Step 6.b:4.b:3.a:4.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: flatten#(t) -> c_20(flatten'1#(t)) - Weak DPs: flatten'1#(node(l,t1,t2)) -> c_22(flatten#(t1),flatten#(t2)) flattensort#(t) -> c_23(flatten#(t)) - Signature: {'cklt/1,'compare/2,'less/2,append/2,append'1/2,flatten/1,flatten'1/1,flattensort/1,insert/2,insert'1/2 ,insert'2/4,insertionsort/1,insertionsort'1/1,'cklt#/1,'compare#/2,'less#/2,append#/2,append'1#/2,flatten#/1 ,flatten'1#/1,flattensort#/1,insert#/2,insert'1#/2,insert'2#/4,insertionsort#/1,insertionsort'1#/1} / {'0/0 ,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,leaf/0,nil/0,node/3,c_1/0,c_2/0,c_3/0,c_4/0 ,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/1,c_19/0,c_20/1 ,c_21/0,c_22/2,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/1,c_29/1,c_30/2,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,append#,append'1#,flatten# ,flatten'1#,flattensort#,insert#,insert'1#,insert'2#,insertionsort#,insertionsort'1#} and constructors {'0 ,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,leaf,nil,node} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ***** Step 6.b:4.b:3.a:4.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: flatten#(t) -> c_20(flatten'1#(t)) flatten'1#(node(l,t1,t2)) -> c_22(flatten#(t1),flatten#(t2)) flattensort#(t) -> c_23(flatten#(t)) - Signature: {'cklt/1,'compare/2,'less/2,append/2,append'1/2,flatten/1,flatten'1/1,flattensort/1,insert/2,insert'1/2 ,insert'2/4,insertionsort/1,insertionsort'1/1,'cklt#/1,'compare#/2,'less#/2,append#/2,append'1#/2,flatten#/1 ,flatten'1#/1,flattensort#/1,insert#/2,insert'1#/2,insert'2#/4,insertionsort#/1,insertionsort'1#/1} / {'0/0 ,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,leaf/0,nil/0,node/3,c_1/0,c_2/0,c_3/0,c_4/0 ,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/1,c_19/0,c_20/1 ,c_21/0,c_22/2,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/1,c_29/1,c_30/2,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,append#,append'1#,flatten# ,flatten'1#,flattensort#,insert#,insert'1#,insert'2#,insertionsort#,insertionsort'1#} and constructors {'0 ,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,leaf,nil,node} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:flatten#(t) -> c_20(flatten'1#(t)) -->_1 flatten'1#(node(l,t1,t2)) -> c_22(flatten#(t1),flatten#(t2)):2 2:W:flatten'1#(node(l,t1,t2)) -> c_22(flatten#(t1),flatten#(t2)) -->_2 flatten#(t) -> c_20(flatten'1#(t)):1 -->_1 flatten#(t) -> c_20(flatten'1#(t)):1 3:W:flattensort#(t) -> c_23(flatten#(t)) -->_1 flatten#(t) -> c_20(flatten'1#(t)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 3: flattensort#(t) -> c_23(flatten#(t)) 1: flatten#(t) -> c_20(flatten'1#(t)) 2: flatten'1#(node(l,t1,t2)) -> c_22(flatten#(t1),flatten#(t2)) ***** Step 6.b:4.b:3.a:4.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Signature: {'cklt/1,'compare/2,'less/2,append/2,append'1/2,flatten/1,flatten'1/1,flattensort/1,insert/2,insert'1/2 ,insert'2/4,insertionsort/1,insertionsort'1/1,'cklt#/1,'compare#/2,'less#/2,append#/2,append'1#/2,flatten#/1 ,flatten'1#/1,flattensort#/1,insert#/2,insert'1#/2,insert'2#/4,insertionsort#/1,insertionsort'1#/1} / {'0/0 ,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,leaf/0,nil/0,node/3,c_1/0,c_2/0,c_3/0,c_4/0 ,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/1,c_19/0,c_20/1 ,c_21/0,c_22/2,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/1,c_29/1,c_30/2,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,append#,append'1#,flatten# ,flatten'1#,flattensort#,insert#,insert'1#,insert'2#,insertionsort#,insertionsort'1#} and constructors {'0 ,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,leaf,nil,node} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). **** Step 6.b:4.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: flattensort#(t) -> c_23(insertionsort#(flatten(t)),flatten#(t)) insert#(x,l) -> c_24(insert'1#(l,x)) insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys)) insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)) insertionsort#(l) -> c_29(insertionsort'1#(l)) insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) - Weak DPs: flatten#(t) -> c_20(flatten'1#(t)) flatten'1#(node(l,t1,t2)) -> c_22(flatten#(t1),flatten#(t2)) - Weak TRS: 'cklt('EQ()) -> 'false() 'cklt('GT()) -> 'false() 'cklt('LT()) -> 'true() 'compare('0(),'0()) -> 'EQ() 'compare('0(),'neg(y)) -> 'GT() 'compare('0(),'pos(y)) -> 'LT() 'compare('0(),'s(y)) -> 'LT() 'compare('neg(x),'0()) -> 'LT() 'compare('neg(x),'neg(y)) -> 'compare(y,x) 'compare('neg(x),'pos(y)) -> 'LT() 'compare('pos(x),'0()) -> 'GT() 'compare('pos(x),'neg(y)) -> 'GT() 'compare('pos(x),'pos(y)) -> 'compare(x,y) 'compare('s(x),'0()) -> 'GT() 'compare('s(x),'s(y)) -> 'compare(x,y) 'less(x,y) -> 'cklt('compare(x,y)) append(l1,l2) -> append'1(l1,l2) append'1(dd(x,xs),l2) -> dd(x,append(xs,l2)) append'1(nil(),l2) -> l2 flatten(t) -> flatten'1(t) flatten'1(leaf()) -> nil() flatten'1(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2))) insert(x,l) -> insert'1(l,x) insert'1(dd(y,ys),x) -> insert'2('less(y,x),x,y,ys) insert'1(nil(),x) -> dd(x,nil()) insert'2('false(),x,y,ys) -> dd(x,dd(y,ys)) insert'2('true(),x,y,ys) -> dd(y,insert(x,ys)) insertionsort(l) -> insertionsort'1(l) insertionsort'1(dd(x,xs)) -> insert(x,insertionsort(xs)) insertionsort'1(nil()) -> nil() - Signature: {'cklt/1,'compare/2,'less/2,append/2,append'1/2,flatten/1,flatten'1/1,flattensort/1,insert/2,insert'1/2 ,insert'2/4,insertionsort/1,insertionsort'1/1,'cklt#/1,'compare#/2,'less#/2,append#/2,append'1#/2,flatten#/1 ,flatten'1#/1,flattensort#/1,insert#/2,insert'1#/2,insert'2#/4,insertionsort#/1,insertionsort'1#/1} / {'0/0 ,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,leaf/0,nil/0,node/3,c_1/0,c_2/0,c_3/0,c_4/0 ,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/1,c_19/0,c_20/1 ,c_21/0,c_22/2,c_23/2,c_24/1,c_25/1,c_26/0,c_27/0,c_28/1,c_29/1,c_30/2,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,append#,append'1#,flatten# ,flatten'1#,flattensort#,insert#,insert'1#,insert'2#,insertionsort#,insertionsort'1#} and constructors {'0 ,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,leaf,nil,node} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:flattensort#(t) -> c_23(insertionsort#(flatten(t)),flatten#(t)) -->_2 flatten#(t) -> c_20(flatten'1#(t)):7 -->_1 insertionsort#(l) -> c_29(insertionsort'1#(l)):5 2:S:insert#(x,l) -> c_24(insert'1#(l,x)) -->_1 insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys)):3 3:S:insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys)) -->_1 insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)):4 4:S:insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)) -->_1 insert#(x,l) -> c_24(insert'1#(l,x)):2 5:S:insertionsort#(l) -> c_29(insertionsort'1#(l)) -->_1 insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)):6 6:S:insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) -->_2 insertionsort#(l) -> c_29(insertionsort'1#(l)):5 -->_1 insert#(x,l) -> c_24(insert'1#(l,x)):2 7:W:flatten#(t) -> c_20(flatten'1#(t)) -->_1 flatten'1#(node(l,t1,t2)) -> c_22(flatten#(t1),flatten#(t2)):8 8:W:flatten'1#(node(l,t1,t2)) -> c_22(flatten#(t1),flatten#(t2)) -->_2 flatten#(t) -> c_20(flatten'1#(t)):7 -->_1 flatten#(t) -> c_20(flatten'1#(t)):7 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 7: flatten#(t) -> c_20(flatten'1#(t)) 8: flatten'1#(node(l,t1,t2)) -> c_22(flatten#(t1),flatten#(t2)) **** Step 6.b:4.b:3.b:2: SimplifyRHS WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: flattensort#(t) -> c_23(insertionsort#(flatten(t)),flatten#(t)) insert#(x,l) -> c_24(insert'1#(l,x)) insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys)) insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)) insertionsort#(l) -> c_29(insertionsort'1#(l)) insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) - Weak TRS: 'cklt('EQ()) -> 'false() 'cklt('GT()) -> 'false() 'cklt('LT()) -> 'true() 'compare('0(),'0()) -> 'EQ() 'compare('0(),'neg(y)) -> 'GT() 'compare('0(),'pos(y)) -> 'LT() 'compare('0(),'s(y)) -> 'LT() 'compare('neg(x),'0()) -> 'LT() 'compare('neg(x),'neg(y)) -> 'compare(y,x) 'compare('neg(x),'pos(y)) -> 'LT() 'compare('pos(x),'0()) -> 'GT() 'compare('pos(x),'neg(y)) -> 'GT() 'compare('pos(x),'pos(y)) -> 'compare(x,y) 'compare('s(x),'0()) -> 'GT() 'compare('s(x),'s(y)) -> 'compare(x,y) 'less(x,y) -> 'cklt('compare(x,y)) append(l1,l2) -> append'1(l1,l2) append'1(dd(x,xs),l2) -> dd(x,append(xs,l2)) append'1(nil(),l2) -> l2 flatten(t) -> flatten'1(t) flatten'1(leaf()) -> nil() flatten'1(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2))) insert(x,l) -> insert'1(l,x) insert'1(dd(y,ys),x) -> insert'2('less(y,x),x,y,ys) insert'1(nil(),x) -> dd(x,nil()) insert'2('false(),x,y,ys) -> dd(x,dd(y,ys)) insert'2('true(),x,y,ys) -> dd(y,insert(x,ys)) insertionsort(l) -> insertionsort'1(l) insertionsort'1(dd(x,xs)) -> insert(x,insertionsort(xs)) insertionsort'1(nil()) -> nil() - Signature: {'cklt/1,'compare/2,'less/2,append/2,append'1/2,flatten/1,flatten'1/1,flattensort/1,insert/2,insert'1/2 ,insert'2/4,insertionsort/1,insertionsort'1/1,'cklt#/1,'compare#/2,'less#/2,append#/2,append'1#/2,flatten#/1 ,flatten'1#/1,flattensort#/1,insert#/2,insert'1#/2,insert'2#/4,insertionsort#/1,insertionsort'1#/1} / {'0/0 ,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,leaf/0,nil/0,node/3,c_1/0,c_2/0,c_3/0,c_4/0 ,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/1,c_19/0,c_20/1 ,c_21/0,c_22/2,c_23/2,c_24/1,c_25/1,c_26/0,c_27/0,c_28/1,c_29/1,c_30/2,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,append#,append'1#,flatten# ,flatten'1#,flattensort#,insert#,insert'1#,insert'2#,insertionsort#,insertionsort'1#} and constructors {'0 ,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,leaf,nil,node} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:flattensort#(t) -> c_23(insertionsort#(flatten(t)),flatten#(t)) -->_1 insertionsort#(l) -> c_29(insertionsort'1#(l)):5 2:S:insert#(x,l) -> c_24(insert'1#(l,x)) -->_1 insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys)):3 3:S:insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys)) -->_1 insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)):4 4:S:insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)) -->_1 insert#(x,l) -> c_24(insert'1#(l,x)):2 5:S:insertionsort#(l) -> c_29(insertionsort'1#(l)) -->_1 insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)):6 6:S:insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) -->_2 insertionsort#(l) -> c_29(insertionsort'1#(l)):5 -->_1 insert#(x,l) -> c_24(insert'1#(l,x)):2 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: flattensort#(t) -> c_23(insertionsort#(flatten(t))) **** Step 6.b:4.b:3.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: flattensort#(t) -> c_23(insertionsort#(flatten(t))) insert#(x,l) -> c_24(insert'1#(l,x)) insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys)) insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)) insertionsort#(l) -> c_29(insertionsort'1#(l)) insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) - Weak TRS: 'cklt('EQ()) -> 'false() 'cklt('GT()) -> 'false() 'cklt('LT()) -> 'true() 'compare('0(),'0()) -> 'EQ() 'compare('0(),'neg(y)) -> 'GT() 'compare('0(),'pos(y)) -> 'LT() 'compare('0(),'s(y)) -> 'LT() 'compare('neg(x),'0()) -> 'LT() 'compare('neg(x),'neg(y)) -> 'compare(y,x) 'compare('neg(x),'pos(y)) -> 'LT() 'compare('pos(x),'0()) -> 'GT() 'compare('pos(x),'neg(y)) -> 'GT() 'compare('pos(x),'pos(y)) -> 'compare(x,y) 'compare('s(x),'0()) -> 'GT() 'compare('s(x),'s(y)) -> 'compare(x,y) 'less(x,y) -> 'cklt('compare(x,y)) append(l1,l2) -> append'1(l1,l2) append'1(dd(x,xs),l2) -> dd(x,append(xs,l2)) append'1(nil(),l2) -> l2 flatten(t) -> flatten'1(t) flatten'1(leaf()) -> nil() flatten'1(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2))) insert(x,l) -> insert'1(l,x) insert'1(dd(y,ys),x) -> insert'2('less(y,x),x,y,ys) insert'1(nil(),x) -> dd(x,nil()) insert'2('false(),x,y,ys) -> dd(x,dd(y,ys)) insert'2('true(),x,y,ys) -> dd(y,insert(x,ys)) insertionsort(l) -> insertionsort'1(l) insertionsort'1(dd(x,xs)) -> insert(x,insertionsort(xs)) insertionsort'1(nil()) -> nil() - Signature: {'cklt/1,'compare/2,'less/2,append/2,append'1/2,flatten/1,flatten'1/1,flattensort/1,insert/2,insert'1/2 ,insert'2/4,insertionsort/1,insertionsort'1/1,'cklt#/1,'compare#/2,'less#/2,append#/2,append'1#/2,flatten#/1 ,flatten'1#/1,flattensort#/1,insert#/2,insert'1#/2,insert'2#/4,insertionsort#/1,insertionsort'1#/1} / {'0/0 ,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,leaf/0,nil/0,node/3,c_1/0,c_2/0,c_3/0,c_4/0 ,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/1,c_19/0,c_20/1 ,c_21/0,c_22/2,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/1,c_29/1,c_30/2,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,append#,append'1#,flatten# ,flatten'1#,flattensort#,insert#,insert'1#,insert'2#,insertionsort#,insertionsort'1#} and constructors {'0 ,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,leaf,nil,node} + 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: insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys)) Consider the set of all dependency pairs 1: flattensort#(t) -> c_23(insertionsort#(flatten(t))) 2: insert#(x,l) -> c_24(insert'1#(l,x)) 3: insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys)) 4: insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)) 5: insertionsort#(l) -> c_29(insertionsort'1#(l)) 6: insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) Processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^2)) SPACE(?,?)on application of the dependency pairs {3} These cover all (indirect) predecessors of dependency pairs {1,3,4} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. ***** Step 6.b:4.b:3.b:3.a:1: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: flattensort#(t) -> c_23(insertionsort#(flatten(t))) insert#(x,l) -> c_24(insert'1#(l,x)) insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys)) insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)) insertionsort#(l) -> c_29(insertionsort'1#(l)) insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) - Weak TRS: 'cklt('EQ()) -> 'false() 'cklt('GT()) -> 'false() 'cklt('LT()) -> 'true() 'compare('0(),'0()) -> 'EQ() 'compare('0(),'neg(y)) -> 'GT() 'compare('0(),'pos(y)) -> 'LT() 'compare('0(),'s(y)) -> 'LT() 'compare('neg(x),'0()) -> 'LT() 'compare('neg(x),'neg(y)) -> 'compare(y,x) 'compare('neg(x),'pos(y)) -> 'LT() 'compare('pos(x),'0()) -> 'GT() 'compare('pos(x),'neg(y)) -> 'GT() 'compare('pos(x),'pos(y)) -> 'compare(x,y) 'compare('s(x),'0()) -> 'GT() 'compare('s(x),'s(y)) -> 'compare(x,y) 'less(x,y) -> 'cklt('compare(x,y)) append(l1,l2) -> append'1(l1,l2) append'1(dd(x,xs),l2) -> dd(x,append(xs,l2)) append'1(nil(),l2) -> l2 flatten(t) -> flatten'1(t) flatten'1(leaf()) -> nil() flatten'1(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2))) insert(x,l) -> insert'1(l,x) insert'1(dd(y,ys),x) -> insert'2('less(y,x),x,y,ys) insert'1(nil(),x) -> dd(x,nil()) insert'2('false(),x,y,ys) -> dd(x,dd(y,ys)) insert'2('true(),x,y,ys) -> dd(y,insert(x,ys)) insertionsort(l) -> insertionsort'1(l) insertionsort'1(dd(x,xs)) -> insert(x,insertionsort(xs)) insertionsort'1(nil()) -> nil() - Signature: {'cklt/1,'compare/2,'less/2,append/2,append'1/2,flatten/1,flatten'1/1,flattensort/1,insert/2,insert'1/2 ,insert'2/4,insertionsort/1,insertionsort'1/1,'cklt#/1,'compare#/2,'less#/2,append#/2,append'1#/2,flatten#/1 ,flatten'1#/1,flattensort#/1,insert#/2,insert'1#/2,insert'2#/4,insertionsort#/1,insertionsort'1#/1} / {'0/0 ,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,leaf/0,nil/0,node/3,c_1/0,c_2/0,c_3/0,c_4/0 ,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/1,c_19/0,c_20/1 ,c_21/0,c_22/2,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/1,c_29/1,c_30/2,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,append#,append'1#,flatten# ,flatten'1#,flattensort#,insert#,insert'1#,insert'2#,insertionsort#,insertionsort'1#} and constructors {'0 ,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,leaf,nil,node} + 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_23) = {1}, uargs(c_24) = {1}, uargs(c_25) = {1}, uargs(c_28) = {1}, uargs(c_29) = {1}, uargs(c_30) = {1,2} Following symbols are considered usable: {'cklt,'compare,'less,append,append'1,flatten,flatten'1,insert,insert'1,insert'2,insertionsort ,insertionsort'1,'cklt#,'compare#,'less#,append#,append'1#,flatten#,flatten'1#,flattensort#,insert# ,insert'1#,insert'2#,insertionsort#,insertionsort'1#} TcT has computed the following interpretation: p('0) = 0 p('EQ) = 1 p('GT) = 1 p('LT) = 1 p('cklt) = x1^2 p('compare) = 1 p('false) = 1 p('less) = 1 p('neg) = 0 p('pos) = 0 p('s) = 0 p('true) = 1 p(append) = x1 + x2 p(append'1) = x1 + x2 p(dd) = 1 + x2 p(flatten) = x1 p(flatten'1) = x1 p(flattensort) = 0 p(insert) = 1 + x2 p(insert'1) = 1 + x1 p(insert'2) = 1 + x1*x4 + x1^2 p(insertionsort) = x1 p(insertionsort'1) = x1 p(leaf) = 1 p(nil) = 0 p(node) = x1 + x2 + x3 p('cklt#) = 0 p('compare#) = 0 p('less#) = 0 p(append#) = 0 p(append'1#) = 0 p(flatten#) = 0 p(flatten'1#) = 0 p(flattensort#) = 1 + x1^2 p(insert#) = x2 p(insert'1#) = x1 p(insert'2#) = x4 p(insertionsort#) = 1 + x1^2 p(insertionsort'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) = 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) = x1 p(c_24) = x1 p(c_25) = x1 p(c_26) = 0 p(c_27) = 0 p(c_28) = x1 p(c_29) = 1 + x1 p(c_30) = x1 + x2 p(c_31) = 0 Following rules are strictly oriented: insert'1#(dd(y,ys),x) = 1 + ys > ys = c_25(insert'2#('less(y,x),x,y,ys)) Following rules are (at-least) weakly oriented: flattensort#(t) = 1 + t^2 >= 1 + t^2 = c_23(insertionsort#(flatten(t))) insert#(x,l) = l >= l = c_24(insert'1#(l,x)) insert'2#('true(),x,y,ys) = ys >= ys = c_28(insert#(x,ys)) insertionsort#(l) = 1 + l^2 >= 1 + l^2 = c_29(insertionsort'1#(l)) insertionsort'1#(dd(x,xs)) = 1 + 2*xs + xs^2 >= 1 + xs + xs^2 = c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) 'cklt('EQ()) = 1 >= 1 = 'false() 'cklt('GT()) = 1 >= 1 = 'false() 'cklt('LT()) = 1 >= 1 = 'true() 'compare('0(),'0()) = 1 >= 1 = 'EQ() 'compare('0(),'neg(y)) = 1 >= 1 = 'GT() 'compare('0(),'pos(y)) = 1 >= 1 = 'LT() 'compare('0(),'s(y)) = 1 >= 1 = 'LT() 'compare('neg(x),'0()) = 1 >= 1 = 'LT() 'compare('neg(x),'neg(y)) = 1 >= 1 = 'compare(y,x) 'compare('neg(x),'pos(y)) = 1 >= 1 = 'LT() 'compare('pos(x),'0()) = 1 >= 1 = 'GT() 'compare('pos(x),'neg(y)) = 1 >= 1 = 'GT() 'compare('pos(x),'pos(y)) = 1 >= 1 = 'compare(x,y) 'compare('s(x),'0()) = 1 >= 1 = 'GT() 'compare('s(x),'s(y)) = 1 >= 1 = 'compare(x,y) 'less(x,y) = 1 >= 1 = 'cklt('compare(x,y)) append(l1,l2) = l1 + l2 >= l1 + l2 = append'1(l1,l2) append'1(dd(x,xs),l2) = 1 + l2 + xs >= 1 + l2 + xs = dd(x,append(xs,l2)) append'1(nil(),l2) = l2 >= l2 = l2 flatten(t) = t >= t = flatten'1(t) flatten'1(leaf()) = 1 >= 0 = nil() flatten'1(node(l,t1,t2)) = l + t1 + t2 >= l + t1 + t2 = append(l,append(flatten(t1),flatten(t2))) insert(x,l) = 1 + l >= 1 + l = insert'1(l,x) insert'1(dd(y,ys),x) = 2 + ys >= 2 + ys = insert'2('less(y,x),x,y,ys) insert'1(nil(),x) = 1 >= 1 = dd(x,nil()) insert'2('false(),x,y,ys) = 2 + ys >= 2 + ys = dd(x,dd(y,ys)) insert'2('true(),x,y,ys) = 2 + ys >= 2 + ys = dd(y,insert(x,ys)) insertionsort(l) = l >= l = insertionsort'1(l) insertionsort'1(dd(x,xs)) = 1 + xs >= 1 + xs = insert(x,insertionsort(xs)) insertionsort'1(nil()) = 0 >= 0 = nil() ***** Step 6.b:4.b:3.b:3.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: flattensort#(t) -> c_23(insertionsort#(flatten(t))) insert#(x,l) -> c_24(insert'1#(l,x)) insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)) insertionsort#(l) -> c_29(insertionsort'1#(l)) insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) - Weak DPs: insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys)) - Weak TRS: 'cklt('EQ()) -> 'false() 'cklt('GT()) -> 'false() 'cklt('LT()) -> 'true() 'compare('0(),'0()) -> 'EQ() 'compare('0(),'neg(y)) -> 'GT() 'compare('0(),'pos(y)) -> 'LT() 'compare('0(),'s(y)) -> 'LT() 'compare('neg(x),'0()) -> 'LT() 'compare('neg(x),'neg(y)) -> 'compare(y,x) 'compare('neg(x),'pos(y)) -> 'LT() 'compare('pos(x),'0()) -> 'GT() 'compare('pos(x),'neg(y)) -> 'GT() 'compare('pos(x),'pos(y)) -> 'compare(x,y) 'compare('s(x),'0()) -> 'GT() 'compare('s(x),'s(y)) -> 'compare(x,y) 'less(x,y) -> 'cklt('compare(x,y)) append(l1,l2) -> append'1(l1,l2) append'1(dd(x,xs),l2) -> dd(x,append(xs,l2)) append'1(nil(),l2) -> l2 flatten(t) -> flatten'1(t) flatten'1(leaf()) -> nil() flatten'1(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2))) insert(x,l) -> insert'1(l,x) insert'1(dd(y,ys),x) -> insert'2('less(y,x),x,y,ys) insert'1(nil(),x) -> dd(x,nil()) insert'2('false(),x,y,ys) -> dd(x,dd(y,ys)) insert'2('true(),x,y,ys) -> dd(y,insert(x,ys)) insertionsort(l) -> insertionsort'1(l) insertionsort'1(dd(x,xs)) -> insert(x,insertionsort(xs)) insertionsort'1(nil()) -> nil() - Signature: {'cklt/1,'compare/2,'less/2,append/2,append'1/2,flatten/1,flatten'1/1,flattensort/1,insert/2,insert'1/2 ,insert'2/4,insertionsort/1,insertionsort'1/1,'cklt#/1,'compare#/2,'less#/2,append#/2,append'1#/2,flatten#/1 ,flatten'1#/1,flattensort#/1,insert#/2,insert'1#/2,insert'2#/4,insertionsort#/1,insertionsort'1#/1} / {'0/0 ,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,leaf/0,nil/0,node/3,c_1/0,c_2/0,c_3/0,c_4/0 ,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/1,c_19/0,c_20/1 ,c_21/0,c_22/2,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/1,c_29/1,c_30/2,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,append#,append'1#,flatten# ,flatten'1#,flattensort#,insert#,insert'1#,insert'2#,insertionsort#,insertionsort'1#} and constructors {'0 ,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,leaf,nil,node} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ***** Step 6.b:4.b:3.b:3.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: insert#(x,l) -> c_24(insert'1#(l,x)) insertionsort#(l) -> c_29(insertionsort'1#(l)) insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) - Weak DPs: flattensort#(t) -> c_23(insertionsort#(flatten(t))) insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys)) insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)) - Weak TRS: 'cklt('EQ()) -> 'false() 'cklt('GT()) -> 'false() 'cklt('LT()) -> 'true() 'compare('0(),'0()) -> 'EQ() 'compare('0(),'neg(y)) -> 'GT() 'compare('0(),'pos(y)) -> 'LT() 'compare('0(),'s(y)) -> 'LT() 'compare('neg(x),'0()) -> 'LT() 'compare('neg(x),'neg(y)) -> 'compare(y,x) 'compare('neg(x),'pos(y)) -> 'LT() 'compare('pos(x),'0()) -> 'GT() 'compare('pos(x),'neg(y)) -> 'GT() 'compare('pos(x),'pos(y)) -> 'compare(x,y) 'compare('s(x),'0()) -> 'GT() 'compare('s(x),'s(y)) -> 'compare(x,y) 'less(x,y) -> 'cklt('compare(x,y)) append(l1,l2) -> append'1(l1,l2) append'1(dd(x,xs),l2) -> dd(x,append(xs,l2)) append'1(nil(),l2) -> l2 flatten(t) -> flatten'1(t) flatten'1(leaf()) -> nil() flatten'1(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2))) insert(x,l) -> insert'1(l,x) insert'1(dd(y,ys),x) -> insert'2('less(y,x),x,y,ys) insert'1(nil(),x) -> dd(x,nil()) insert'2('false(),x,y,ys) -> dd(x,dd(y,ys)) insert'2('true(),x,y,ys) -> dd(y,insert(x,ys)) insertionsort(l) -> insertionsort'1(l) insertionsort'1(dd(x,xs)) -> insert(x,insertionsort(xs)) insertionsort'1(nil()) -> nil() - Signature: {'cklt/1,'compare/2,'less/2,append/2,append'1/2,flatten/1,flatten'1/1,flattensort/1,insert/2,insert'1/2 ,insert'2/4,insertionsort/1,insertionsort'1/1,'cklt#/1,'compare#/2,'less#/2,append#/2,append'1#/2,flatten#/1 ,flatten'1#/1,flattensort#/1,insert#/2,insert'1#/2,insert'2#/4,insertionsort#/1,insertionsort'1#/1} / {'0/0 ,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,leaf/0,nil/0,node/3,c_1/0,c_2/0,c_3/0,c_4/0 ,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/1,c_19/0,c_20/1 ,c_21/0,c_22/2,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/1,c_29/1,c_30/2,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,append#,append'1#,flatten# ,flatten'1#,flattensort#,insert#,insert'1#,insert'2#,insertionsort#,insertionsort'1#} and constructors {'0 ,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,leaf,nil,node} + 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: 2: insertionsort#(l) -> c_29(insertionsort'1#(l)) Consider the set of all dependency pairs 1: insert#(x,l) -> c_24(insert'1#(l,x)) 2: insertionsort#(l) -> c_29(insertionsort'1#(l)) 3: insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) 4: flattensort#(t) -> c_23(insertionsort#(flatten(t))) 5: insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys)) 6: insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)) Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1)) SPACE(?,?)on application of the dependency pairs {2} These cover all (indirect) predecessors of dependency pairs {2,3,4} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. ****** Step 6.b:4.b:3.b:3.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: insert#(x,l) -> c_24(insert'1#(l,x)) insertionsort#(l) -> c_29(insertionsort'1#(l)) insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) - Weak DPs: flattensort#(t) -> c_23(insertionsort#(flatten(t))) insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys)) insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)) - Weak TRS: 'cklt('EQ()) -> 'false() 'cklt('GT()) -> 'false() 'cklt('LT()) -> 'true() 'compare('0(),'0()) -> 'EQ() 'compare('0(),'neg(y)) -> 'GT() 'compare('0(),'pos(y)) -> 'LT() 'compare('0(),'s(y)) -> 'LT() 'compare('neg(x),'0()) -> 'LT() 'compare('neg(x),'neg(y)) -> 'compare(y,x) 'compare('neg(x),'pos(y)) -> 'LT() 'compare('pos(x),'0()) -> 'GT() 'compare('pos(x),'neg(y)) -> 'GT() 'compare('pos(x),'pos(y)) -> 'compare(x,y) 'compare('s(x),'0()) -> 'GT() 'compare('s(x),'s(y)) -> 'compare(x,y) 'less(x,y) -> 'cklt('compare(x,y)) append(l1,l2) -> append'1(l1,l2) append'1(dd(x,xs),l2) -> dd(x,append(xs,l2)) append'1(nil(),l2) -> l2 flatten(t) -> flatten'1(t) flatten'1(leaf()) -> nil() flatten'1(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2))) insert(x,l) -> insert'1(l,x) insert'1(dd(y,ys),x) -> insert'2('less(y,x),x,y,ys) insert'1(nil(),x) -> dd(x,nil()) insert'2('false(),x,y,ys) -> dd(x,dd(y,ys)) insert'2('true(),x,y,ys) -> dd(y,insert(x,ys)) insertionsort(l) -> insertionsort'1(l) insertionsort'1(dd(x,xs)) -> insert(x,insertionsort(xs)) insertionsort'1(nil()) -> nil() - Signature: {'cklt/1,'compare/2,'less/2,append/2,append'1/2,flatten/1,flatten'1/1,flattensort/1,insert/2,insert'1/2 ,insert'2/4,insertionsort/1,insertionsort'1/1,'cklt#/1,'compare#/2,'less#/2,append#/2,append'1#/2,flatten#/1 ,flatten'1#/1,flattensort#/1,insert#/2,insert'1#/2,insert'2#/4,insertionsort#/1,insertionsort'1#/1} / {'0/0 ,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,leaf/0,nil/0,node/3,c_1/0,c_2/0,c_3/0,c_4/0 ,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/1,c_19/0,c_20/1 ,c_21/0,c_22/2,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/1,c_29/1,c_30/2,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,append#,append'1#,flatten# ,flatten'1#,flattensort#,insert#,insert'1#,insert'2#,insertionsort#,insertionsort'1#} and constructors {'0 ,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,leaf,nil,node} + 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_23) = {1}, uargs(c_24) = {1}, uargs(c_25) = {1}, uargs(c_28) = {1}, uargs(c_29) = {1}, uargs(c_30) = {1,2} Following symbols are considered usable: {append,append'1,flatten,flatten'1,'cklt#,'compare#,'less#,append#,append'1#,flatten#,flatten'1# ,flattensort#,insert#,insert'1#,insert'2#,insertionsort#,insertionsort'1#} TcT has computed the following interpretation: p('0) = [2] p('EQ) = [0] p('GT) = [4] p('LT) = [1] p('cklt) = [0] p('compare) = [4] x1 + [1] p('false) = [0] p('less) = [3] x1 + [4] x2 + [0] p('neg) = [0] p('pos) = [3] p('s) = [3] p('true) = [0] p(append) = [1] x1 + [1] x2 + [0] p(append'1) = [1] x1 + [1] x2 + [0] p(dd) = [1] x2 + [2] p(flatten) = [4] x1 + [0] p(flatten'1) = [4] x1 + [0] p(flattensort) = [0] p(insert) = [2] p(insert'1) = [0] p(insert'2) = [2] x1 + [1] x2 + [2] x3 + [4] p(insertionsort) = [0] p(insertionsort'1) = [0] p(leaf) = [2] p(nil) = [0] p(node) = [1] x1 + [1] x2 + [1] x3 + [0] p('cklt#) = [1] x1 + [1] p('compare#) = [0] p('less#) = [0] p(append#) = [1] x1 + [4] x2 + [4] p(append'1#) = [4] x1 + [2] x2 + [2] p(flatten#) = [1] x1 + [0] p(flatten'1#) = [4] p(flattensort#) = [5] x1 + [4] p(insert#) = [0] p(insert'1#) = [0] p(insert'2#) = [0] p(insertionsort#) = [1] x1 + [1] p(insertionsort'1#) = [1] x1 + [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [1] p(c_4) = [0] p(c_5) = [0] p(c_6) = [0] p(c_7) = [4] p(c_8) = [1] p(c_9) = [0] p(c_10) = [1] p(c_11) = [1] p(c_12) = [4] p(c_13) = [2] x1 + [1] p(c_14) = [4] p(c_15) = [2] x1 + [0] p(c_16) = [0] p(c_17) = [1] x1 + [0] p(c_18) = [2] x1 + [4] p(c_19) = [1] p(c_20) = [1] p(c_21) = [1] p(c_22) = [4] p(c_23) = [1] x1 + [3] p(c_24) = [2] x1 + [0] p(c_25) = [1] x1 + [0] p(c_26) = [1] p(c_27) = [1] p(c_28) = [1] x1 + [0] p(c_29) = [1] x1 + [0] p(c_30) = [4] x1 + [1] x2 + [1] p(c_31) = [4] Following rules are strictly oriented: insertionsort#(l) = [1] l + [1] > [1] l + [0] = c_29(insertionsort'1#(l)) Following rules are (at-least) weakly oriented: flattensort#(t) = [5] t + [4] >= [4] t + [4] = c_23(insertionsort#(flatten(t))) insert#(x,l) = [0] >= [0] = c_24(insert'1#(l,x)) insert'1#(dd(y,ys),x) = [0] >= [0] = c_25(insert'2#('less(y,x),x,y,ys)) insert'2#('true(),x,y,ys) = [0] >= [0] = c_28(insert#(x,ys)) insertionsort'1#(dd(x,xs)) = [1] xs + [2] >= [1] xs + [2] = c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) append(l1,l2) = [1] l1 + [1] l2 + [0] >= [1] l1 + [1] l2 + [0] = append'1(l1,l2) append'1(dd(x,xs),l2) = [1] l2 + [1] xs + [2] >= [1] l2 + [1] xs + [2] = dd(x,append(xs,l2)) append'1(nil(),l2) = [1] l2 + [0] >= [1] l2 + [0] = l2 flatten(t) = [4] t + [0] >= [4] t + [0] = flatten'1(t) flatten'1(leaf()) = [8] >= [0] = nil() flatten'1(node(l,t1,t2)) = [4] l + [4] t1 + [4] t2 + [0] >= [1] l + [4] t1 + [4] t2 + [0] = append(l,append(flatten(t1),flatten(t2))) ****** Step 6.b:4.b:3.b:3.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: insert#(x,l) -> c_24(insert'1#(l,x)) insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) - Weak DPs: flattensort#(t) -> c_23(insertionsort#(flatten(t))) insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys)) insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)) insertionsort#(l) -> c_29(insertionsort'1#(l)) - Weak TRS: 'cklt('EQ()) -> 'false() 'cklt('GT()) -> 'false() 'cklt('LT()) -> 'true() 'compare('0(),'0()) -> 'EQ() 'compare('0(),'neg(y)) -> 'GT() 'compare('0(),'pos(y)) -> 'LT() 'compare('0(),'s(y)) -> 'LT() 'compare('neg(x),'0()) -> 'LT() 'compare('neg(x),'neg(y)) -> 'compare(y,x) 'compare('neg(x),'pos(y)) -> 'LT() 'compare('pos(x),'0()) -> 'GT() 'compare('pos(x),'neg(y)) -> 'GT() 'compare('pos(x),'pos(y)) -> 'compare(x,y) 'compare('s(x),'0()) -> 'GT() 'compare('s(x),'s(y)) -> 'compare(x,y) 'less(x,y) -> 'cklt('compare(x,y)) append(l1,l2) -> append'1(l1,l2) append'1(dd(x,xs),l2) -> dd(x,append(xs,l2)) append'1(nil(),l2) -> l2 flatten(t) -> flatten'1(t) flatten'1(leaf()) -> nil() flatten'1(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2))) insert(x,l) -> insert'1(l,x) insert'1(dd(y,ys),x) -> insert'2('less(y,x),x,y,ys) insert'1(nil(),x) -> dd(x,nil()) insert'2('false(),x,y,ys) -> dd(x,dd(y,ys)) insert'2('true(),x,y,ys) -> dd(y,insert(x,ys)) insertionsort(l) -> insertionsort'1(l) insertionsort'1(dd(x,xs)) -> insert(x,insertionsort(xs)) insertionsort'1(nil()) -> nil() - Signature: {'cklt/1,'compare/2,'less/2,append/2,append'1/2,flatten/1,flatten'1/1,flattensort/1,insert/2,insert'1/2 ,insert'2/4,insertionsort/1,insertionsort'1/1,'cklt#/1,'compare#/2,'less#/2,append#/2,append'1#/2,flatten#/1 ,flatten'1#/1,flattensort#/1,insert#/2,insert'1#/2,insert'2#/4,insertionsort#/1,insertionsort'1#/1} / {'0/0 ,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,leaf/0,nil/0,node/3,c_1/0,c_2/0,c_3/0,c_4/0 ,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/1,c_19/0,c_20/1 ,c_21/0,c_22/2,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/1,c_29/1,c_30/2,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,append#,append'1#,flatten# ,flatten'1#,flattensort#,insert#,insert'1#,insert'2#,insertionsort#,insertionsort'1#} and constructors {'0 ,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,leaf,nil,node} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ****** Step 6.b:4.b:3.b:3.b:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: insert#(x,l) -> c_24(insert'1#(l,x)) - Weak DPs: flattensort#(t) -> c_23(insertionsort#(flatten(t))) insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys)) insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)) insertionsort#(l) -> c_29(insertionsort'1#(l)) insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) - Weak TRS: 'cklt('EQ()) -> 'false() 'cklt('GT()) -> 'false() 'cklt('LT()) -> 'true() 'compare('0(),'0()) -> 'EQ() 'compare('0(),'neg(y)) -> 'GT() 'compare('0(),'pos(y)) -> 'LT() 'compare('0(),'s(y)) -> 'LT() 'compare('neg(x),'0()) -> 'LT() 'compare('neg(x),'neg(y)) -> 'compare(y,x) 'compare('neg(x),'pos(y)) -> 'LT() 'compare('pos(x),'0()) -> 'GT() 'compare('pos(x),'neg(y)) -> 'GT() 'compare('pos(x),'pos(y)) -> 'compare(x,y) 'compare('s(x),'0()) -> 'GT() 'compare('s(x),'s(y)) -> 'compare(x,y) 'less(x,y) -> 'cklt('compare(x,y)) append(l1,l2) -> append'1(l1,l2) append'1(dd(x,xs),l2) -> dd(x,append(xs,l2)) append'1(nil(),l2) -> l2 flatten(t) -> flatten'1(t) flatten'1(leaf()) -> nil() flatten'1(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2))) insert(x,l) -> insert'1(l,x) insert'1(dd(y,ys),x) -> insert'2('less(y,x),x,y,ys) insert'1(nil(),x) -> dd(x,nil()) insert'2('false(),x,y,ys) -> dd(x,dd(y,ys)) insert'2('true(),x,y,ys) -> dd(y,insert(x,ys)) insertionsort(l) -> insertionsort'1(l) insertionsort'1(dd(x,xs)) -> insert(x,insertionsort(xs)) insertionsort'1(nil()) -> nil() - Signature: {'cklt/1,'compare/2,'less/2,append/2,append'1/2,flatten/1,flatten'1/1,flattensort/1,insert/2,insert'1/2 ,insert'2/4,insertionsort/1,insertionsort'1/1,'cklt#/1,'compare#/2,'less#/2,append#/2,append'1#/2,flatten#/1 ,flatten'1#/1,flattensort#/1,insert#/2,insert'1#/2,insert'2#/4,insertionsort#/1,insertionsort'1#/1} / {'0/0 ,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,leaf/0,nil/0,node/3,c_1/0,c_2/0,c_3/0,c_4/0 ,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/1,c_19/0,c_20/1 ,c_21/0,c_22/2,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/1,c_29/1,c_30/2,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,append#,append'1#,flatten# ,flatten'1#,flattensort#,insert#,insert'1#,insert'2#,insertionsort#,insertionsort'1#} and constructors {'0 ,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,leaf,nil,node} + 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: insert#(x,l) -> c_24(insert'1#(l,x)) Consider the set of all dependency pairs 1: insert#(x,l) -> c_24(insert'1#(l,x)) 2: flattensort#(t) -> c_23(insertionsort#(flatten(t))) 3: insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys)) 4: insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)) 5: insertionsort#(l) -> c_29(insertionsort'1#(l)) 6: insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) Processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^2)) SPACE(?,?)on application of the dependency pairs {1} These cover all (indirect) predecessors of dependency pairs {1,2,3,4} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. ******* Step 6.b:4.b:3.b:3.b:1.b:1.a:1: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: insert#(x,l) -> c_24(insert'1#(l,x)) - Weak DPs: flattensort#(t) -> c_23(insertionsort#(flatten(t))) insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys)) insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)) insertionsort#(l) -> c_29(insertionsort'1#(l)) insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) - Weak TRS: 'cklt('EQ()) -> 'false() 'cklt('GT()) -> 'false() 'cklt('LT()) -> 'true() 'compare('0(),'0()) -> 'EQ() 'compare('0(),'neg(y)) -> 'GT() 'compare('0(),'pos(y)) -> 'LT() 'compare('0(),'s(y)) -> 'LT() 'compare('neg(x),'0()) -> 'LT() 'compare('neg(x),'neg(y)) -> 'compare(y,x) 'compare('neg(x),'pos(y)) -> 'LT() 'compare('pos(x),'0()) -> 'GT() 'compare('pos(x),'neg(y)) -> 'GT() 'compare('pos(x),'pos(y)) -> 'compare(x,y) 'compare('s(x),'0()) -> 'GT() 'compare('s(x),'s(y)) -> 'compare(x,y) 'less(x,y) -> 'cklt('compare(x,y)) append(l1,l2) -> append'1(l1,l2) append'1(dd(x,xs),l2) -> dd(x,append(xs,l2)) append'1(nil(),l2) -> l2 flatten(t) -> flatten'1(t) flatten'1(leaf()) -> nil() flatten'1(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2))) insert(x,l) -> insert'1(l,x) insert'1(dd(y,ys),x) -> insert'2('less(y,x),x,y,ys) insert'1(nil(),x) -> dd(x,nil()) insert'2('false(),x,y,ys) -> dd(x,dd(y,ys)) insert'2('true(),x,y,ys) -> dd(y,insert(x,ys)) insertionsort(l) -> insertionsort'1(l) insertionsort'1(dd(x,xs)) -> insert(x,insertionsort(xs)) insertionsort'1(nil()) -> nil() - Signature: {'cklt/1,'compare/2,'less/2,append/2,append'1/2,flatten/1,flatten'1/1,flattensort/1,insert/2,insert'1/2 ,insert'2/4,insertionsort/1,insertionsort'1/1,'cklt#/1,'compare#/2,'less#/2,append#/2,append'1#/2,flatten#/1 ,flatten'1#/1,flattensort#/1,insert#/2,insert'1#/2,insert'2#/4,insertionsort#/1,insertionsort'1#/1} / {'0/0 ,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,leaf/0,nil/0,node/3,c_1/0,c_2/0,c_3/0,c_4/0 ,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/1,c_19/0,c_20/1 ,c_21/0,c_22/2,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/1,c_29/1,c_30/2,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,append#,append'1#,flatten# ,flatten'1#,flattensort#,insert#,insert'1#,insert'2#,insertionsort#,insertionsort'1#} and constructors {'0 ,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,leaf,nil,node} + 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_23) = {1}, uargs(c_24) = {1}, uargs(c_25) = {1}, uargs(c_28) = {1}, uargs(c_29) = {1}, uargs(c_30) = {1,2} Following symbols are considered usable: {'cklt,'compare,'less,append,append'1,flatten,flatten'1,insert,insert'1,insert'2,insertionsort ,insertionsort'1,'cklt#,'compare#,'less#,append#,append'1#,flatten#,flatten'1#,flattensort#,insert# ,insert'1#,insert'2#,insertionsort#,insertionsort'1#} TcT has computed the following interpretation: p('0) = 0 p('EQ) = 1 p('GT) = 1 p('LT) = 1 p('cklt) = x1^2 p('compare) = 1 p('false) = 1 p('less) = 1 p('neg) = 0 p('pos) = 0 p('s) = 0 p('true) = 1 p(append) = x1 + x2 p(append'1) = x1 + x2 p(dd) = 1 + x2 p(flatten) = x1 p(flatten'1) = x1 p(flattensort) = 0 p(insert) = 1 + x2 p(insert'1) = 1 + x1 p(insert'2) = 1 + x1^2 + x4 p(insertionsort) = x1 p(insertionsort'1) = x1 p(leaf) = 0 p(nil) = 0 p(node) = x1 + x2 + x3 p('cklt#) = 0 p('compare#) = 0 p('less#) = 0 p(append#) = 0 p(append'1#) = 0 p(flatten#) = 0 p(flatten'1#) = 0 p(flattensort#) = x1 + x1^2 p(insert#) = 1 + x2 p(insert'1#) = x1 p(insert'2#) = x1*x4 + x1^2 p(insertionsort#) = x1 + x1^2 p(insertionsort'1#) = 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) = 0 p(c_19) = 0 p(c_20) = 0 p(c_21) = 0 p(c_22) = 0 p(c_23) = x1 p(c_24) = x1 p(c_25) = x1 p(c_26) = 0 p(c_27) = 0 p(c_28) = x1 p(c_29) = x1 p(c_30) = x1 + x2 p(c_31) = 0 Following rules are strictly oriented: insert#(x,l) = 1 + l > l = c_24(insert'1#(l,x)) Following rules are (at-least) weakly oriented: flattensort#(t) = t + t^2 >= t + t^2 = c_23(insertionsort#(flatten(t))) insert'1#(dd(y,ys),x) = 1 + ys >= 1 + ys = c_25(insert'2#('less(y,x),x,y,ys)) insert'2#('true(),x,y,ys) = 1 + ys >= 1 + ys = c_28(insert#(x,ys)) insertionsort#(l) = l + l^2 >= l + l^2 = c_29(insertionsort'1#(l)) insertionsort'1#(dd(x,xs)) = 2 + 3*xs + xs^2 >= 1 + 2*xs + xs^2 = c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) 'cklt('EQ()) = 1 >= 1 = 'false() 'cklt('GT()) = 1 >= 1 = 'false() 'cklt('LT()) = 1 >= 1 = 'true() 'compare('0(),'0()) = 1 >= 1 = 'EQ() 'compare('0(),'neg(y)) = 1 >= 1 = 'GT() 'compare('0(),'pos(y)) = 1 >= 1 = 'LT() 'compare('0(),'s(y)) = 1 >= 1 = 'LT() 'compare('neg(x),'0()) = 1 >= 1 = 'LT() 'compare('neg(x),'neg(y)) = 1 >= 1 = 'compare(y,x) 'compare('neg(x),'pos(y)) = 1 >= 1 = 'LT() 'compare('pos(x),'0()) = 1 >= 1 = 'GT() 'compare('pos(x),'neg(y)) = 1 >= 1 = 'GT() 'compare('pos(x),'pos(y)) = 1 >= 1 = 'compare(x,y) 'compare('s(x),'0()) = 1 >= 1 = 'GT() 'compare('s(x),'s(y)) = 1 >= 1 = 'compare(x,y) 'less(x,y) = 1 >= 1 = 'cklt('compare(x,y)) append(l1,l2) = l1 + l2 >= l1 + l2 = append'1(l1,l2) append'1(dd(x,xs),l2) = 1 + l2 + xs >= 1 + l2 + xs = dd(x,append(xs,l2)) append'1(nil(),l2) = l2 >= l2 = l2 flatten(t) = t >= t = flatten'1(t) flatten'1(leaf()) = 0 >= 0 = nil() flatten'1(node(l,t1,t2)) = l + t1 + t2 >= l + t1 + t2 = append(l,append(flatten(t1),flatten(t2))) insert(x,l) = 1 + l >= 1 + l = insert'1(l,x) insert'1(dd(y,ys),x) = 2 + ys >= 2 + ys = insert'2('less(y,x),x,y,ys) insert'1(nil(),x) = 1 >= 1 = dd(x,nil()) insert'2('false(),x,y,ys) = 2 + ys >= 2 + ys = dd(x,dd(y,ys)) insert'2('true(),x,y,ys) = 2 + ys >= 2 + ys = dd(y,insert(x,ys)) insertionsort(l) = l >= l = insertionsort'1(l) insertionsort'1(dd(x,xs)) = 1 + xs >= 1 + xs = insert(x,insertionsort(xs)) insertionsort'1(nil()) = 0 >= 0 = nil() ******* Step 6.b:4.b:3.b:3.b:1.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: flattensort#(t) -> c_23(insertionsort#(flatten(t))) insert#(x,l) -> c_24(insert'1#(l,x)) insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys)) insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)) insertionsort#(l) -> c_29(insertionsort'1#(l)) insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) - Weak TRS: 'cklt('EQ()) -> 'false() 'cklt('GT()) -> 'false() 'cklt('LT()) -> 'true() 'compare('0(),'0()) -> 'EQ() 'compare('0(),'neg(y)) -> 'GT() 'compare('0(),'pos(y)) -> 'LT() 'compare('0(),'s(y)) -> 'LT() 'compare('neg(x),'0()) -> 'LT() 'compare('neg(x),'neg(y)) -> 'compare(y,x) 'compare('neg(x),'pos(y)) -> 'LT() 'compare('pos(x),'0()) -> 'GT() 'compare('pos(x),'neg(y)) -> 'GT() 'compare('pos(x),'pos(y)) -> 'compare(x,y) 'compare('s(x),'0()) -> 'GT() 'compare('s(x),'s(y)) -> 'compare(x,y) 'less(x,y) -> 'cklt('compare(x,y)) append(l1,l2) -> append'1(l1,l2) append'1(dd(x,xs),l2) -> dd(x,append(xs,l2)) append'1(nil(),l2) -> l2 flatten(t) -> flatten'1(t) flatten'1(leaf()) -> nil() flatten'1(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2))) insert(x,l) -> insert'1(l,x) insert'1(dd(y,ys),x) -> insert'2('less(y,x),x,y,ys) insert'1(nil(),x) -> dd(x,nil()) insert'2('false(),x,y,ys) -> dd(x,dd(y,ys)) insert'2('true(),x,y,ys) -> dd(y,insert(x,ys)) insertionsort(l) -> insertionsort'1(l) insertionsort'1(dd(x,xs)) -> insert(x,insertionsort(xs)) insertionsort'1(nil()) -> nil() - Signature: {'cklt/1,'compare/2,'less/2,append/2,append'1/2,flatten/1,flatten'1/1,flattensort/1,insert/2,insert'1/2 ,insert'2/4,insertionsort/1,insertionsort'1/1,'cklt#/1,'compare#/2,'less#/2,append#/2,append'1#/2,flatten#/1 ,flatten'1#/1,flattensort#/1,insert#/2,insert'1#/2,insert'2#/4,insertionsort#/1,insertionsort'1#/1} / {'0/0 ,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,leaf/0,nil/0,node/3,c_1/0,c_2/0,c_3/0,c_4/0 ,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/1,c_19/0,c_20/1 ,c_21/0,c_22/2,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/1,c_29/1,c_30/2,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,append#,append'1#,flatten# ,flatten'1#,flattensort#,insert#,insert'1#,insert'2#,insertionsort#,insertionsort'1#} and constructors {'0 ,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,leaf,nil,node} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ******* Step 6.b:4.b:3.b:3.b:1.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: flattensort#(t) -> c_23(insertionsort#(flatten(t))) insert#(x,l) -> c_24(insert'1#(l,x)) insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys)) insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)) insertionsort#(l) -> c_29(insertionsort'1#(l)) insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) - Weak TRS: 'cklt('EQ()) -> 'false() 'cklt('GT()) -> 'false() 'cklt('LT()) -> 'true() 'compare('0(),'0()) -> 'EQ() 'compare('0(),'neg(y)) -> 'GT() 'compare('0(),'pos(y)) -> 'LT() 'compare('0(),'s(y)) -> 'LT() 'compare('neg(x),'0()) -> 'LT() 'compare('neg(x),'neg(y)) -> 'compare(y,x) 'compare('neg(x),'pos(y)) -> 'LT() 'compare('pos(x),'0()) -> 'GT() 'compare('pos(x),'neg(y)) -> 'GT() 'compare('pos(x),'pos(y)) -> 'compare(x,y) 'compare('s(x),'0()) -> 'GT() 'compare('s(x),'s(y)) -> 'compare(x,y) 'less(x,y) -> 'cklt('compare(x,y)) append(l1,l2) -> append'1(l1,l2) append'1(dd(x,xs),l2) -> dd(x,append(xs,l2)) append'1(nil(),l2) -> l2 flatten(t) -> flatten'1(t) flatten'1(leaf()) -> nil() flatten'1(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2))) insert(x,l) -> insert'1(l,x) insert'1(dd(y,ys),x) -> insert'2('less(y,x),x,y,ys) insert'1(nil(),x) -> dd(x,nil()) insert'2('false(),x,y,ys) -> dd(x,dd(y,ys)) insert'2('true(),x,y,ys) -> dd(y,insert(x,ys)) insertionsort(l) -> insertionsort'1(l) insertionsort'1(dd(x,xs)) -> insert(x,insertionsort(xs)) insertionsort'1(nil()) -> nil() - Signature: {'cklt/1,'compare/2,'less/2,append/2,append'1/2,flatten/1,flatten'1/1,flattensort/1,insert/2,insert'1/2 ,insert'2/4,insertionsort/1,insertionsort'1/1,'cklt#/1,'compare#/2,'less#/2,append#/2,append'1#/2,flatten#/1 ,flatten'1#/1,flattensort#/1,insert#/2,insert'1#/2,insert'2#/4,insertionsort#/1,insertionsort'1#/1} / {'0/0 ,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,leaf/0,nil/0,node/3,c_1/0,c_2/0,c_3/0,c_4/0 ,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/1,c_19/0,c_20/1 ,c_21/0,c_22/2,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/1,c_29/1,c_30/2,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,append#,append'1#,flatten# ,flatten'1#,flattensort#,insert#,insert'1#,insert'2#,insertionsort#,insertionsort'1#} and constructors {'0 ,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,leaf,nil,node} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:flattensort#(t) -> c_23(insertionsort#(flatten(t))) -->_1 insertionsort#(l) -> c_29(insertionsort'1#(l)):5 2:W:insert#(x,l) -> c_24(insert'1#(l,x)) -->_1 insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys)):3 3:W:insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys)) -->_1 insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)):4 4:W:insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)) -->_1 insert#(x,l) -> c_24(insert'1#(l,x)):2 5:W:insertionsort#(l) -> c_29(insertionsort'1#(l)) -->_1 insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)):6 6:W:insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) -->_2 insertionsort#(l) -> c_29(insertionsort'1#(l)):5 -->_1 insert#(x,l) -> c_24(insert'1#(l,x)):2 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: flattensort#(t) -> c_23(insertionsort#(flatten(t))) 5: insertionsort#(l) -> c_29(insertionsort'1#(l)) 6: insertionsort'1#(dd(x,xs)) -> c_30(insert#(x,insertionsort(xs)),insertionsort#(xs)) 2: insert#(x,l) -> c_24(insert'1#(l,x)) 4: insert'2#('true(),x,y,ys) -> c_28(insert#(x,ys)) 3: insert'1#(dd(y,ys),x) -> c_25(insert'2#('less(y,x),x,y,ys)) ******* Step 6.b:4.b:3.b:3.b:1.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: 'cklt('EQ()) -> 'false() 'cklt('GT()) -> 'false() 'cklt('LT()) -> 'true() 'compare('0(),'0()) -> 'EQ() 'compare('0(),'neg(y)) -> 'GT() 'compare('0(),'pos(y)) -> 'LT() 'compare('0(),'s(y)) -> 'LT() 'compare('neg(x),'0()) -> 'LT() 'compare('neg(x),'neg(y)) -> 'compare(y,x) 'compare('neg(x),'pos(y)) -> 'LT() 'compare('pos(x),'0()) -> 'GT() 'compare('pos(x),'neg(y)) -> 'GT() 'compare('pos(x),'pos(y)) -> 'compare(x,y) 'compare('s(x),'0()) -> 'GT() 'compare('s(x),'s(y)) -> 'compare(x,y) 'less(x,y) -> 'cklt('compare(x,y)) append(l1,l2) -> append'1(l1,l2) append'1(dd(x,xs),l2) -> dd(x,append(xs,l2)) append'1(nil(),l2) -> l2 flatten(t) -> flatten'1(t) flatten'1(leaf()) -> nil() flatten'1(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2))) insert(x,l) -> insert'1(l,x) insert'1(dd(y,ys),x) -> insert'2('less(y,x),x,y,ys) insert'1(nil(),x) -> dd(x,nil()) insert'2('false(),x,y,ys) -> dd(x,dd(y,ys)) insert'2('true(),x,y,ys) -> dd(y,insert(x,ys)) insertionsort(l) -> insertionsort'1(l) insertionsort'1(dd(x,xs)) -> insert(x,insertionsort(xs)) insertionsort'1(nil()) -> nil() - Signature: {'cklt/1,'compare/2,'less/2,append/2,append'1/2,flatten/1,flatten'1/1,flattensort/1,insert/2,insert'1/2 ,insert'2/4,insertionsort/1,insertionsort'1/1,'cklt#/1,'compare#/2,'less#/2,append#/2,append'1#/2,flatten#/1 ,flatten'1#/1,flattensort#/1,insert#/2,insert'1#/2,insert'2#/4,insertionsort#/1,insertionsort'1#/1} / {'0/0 ,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,leaf/0,nil/0,node/3,c_1/0,c_2/0,c_3/0,c_4/0 ,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/1,c_19/0,c_20/1 ,c_21/0,c_22/2,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/1,c_29/1,c_30/2,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,append#,append'1#,flatten# ,flatten'1#,flattensort#,insert#,insert'1#,insert'2#,insertionsort#,insertionsort'1#} and constructors {'0 ,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,leaf,nil,node} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))