/export/starexec/sandbox2/solver/bin/starexec_run_tct_rci /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1),O(n^3)) * Step 1: Sum. WORST_CASE(Omega(n^1),O(n^3)) + Considered Problem: - Strict TRS: app(add(n,x),y) -> add(n,app(x,y)) app(nil(),y) -> y concat(cons(u,v),y) -> cons(u,concat(v,y)) concat(leaf(),y) -> y less_leaves(x,leaf()) -> false() less_leaves(cons(u,v),cons(w,z)) -> less_leaves(concat(u,v),concat(w,z)) less_leaves(leaf(),cons(w,z)) -> true() minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) reverse(add(n,x)) -> app(reverse(x),add(n,nil())) reverse(nil()) -> nil() shuffle(add(n,x)) -> add(n,shuffle(reverse(x))) shuffle(nil()) -> nil() - Signature: {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0 ,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {app,concat,less_leaves,minus,quot,reverse ,shuffle} and constructors {0,add,cons,false,leaf,nil,s,true} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: Sum. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: app(add(n,x),y) -> add(n,app(x,y)) app(nil(),y) -> y concat(cons(u,v),y) -> cons(u,concat(v,y)) concat(leaf(),y) -> y less_leaves(x,leaf()) -> false() less_leaves(cons(u,v),cons(w,z)) -> less_leaves(concat(u,v),concat(w,z)) less_leaves(leaf(),cons(w,z)) -> true() minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) reverse(add(n,x)) -> app(reverse(x),add(n,nil())) reverse(nil()) -> nil() shuffle(add(n,x)) -> add(n,shuffle(reverse(x))) shuffle(nil()) -> nil() - Signature: {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0 ,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {app,concat,less_leaves,minus,quot,reverse ,shuffle} and constructors {0,add,cons,false,leaf,nil,s,true} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:2: DecreasingLoops. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: app(add(n,x),y) -> add(n,app(x,y)) app(nil(),y) -> y concat(cons(u,v),y) -> cons(u,concat(v,y)) concat(leaf(),y) -> y less_leaves(x,leaf()) -> false() less_leaves(cons(u,v),cons(w,z)) -> less_leaves(concat(u,v),concat(w,z)) less_leaves(leaf(),cons(w,z)) -> true() minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) reverse(add(n,x)) -> app(reverse(x),add(n,nil())) reverse(nil()) -> nil() shuffle(add(n,x)) -> add(n,shuffle(reverse(x))) shuffle(nil()) -> nil() - Signature: {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0 ,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {app,concat,less_leaves,minus,quot,reverse ,shuffle} and constructors {0,add,cons,false,leaf,nil,s,true} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: app(y,z){y -> add(x,y)} = app(add(x,y),z) ->^+ add(x,app(y,z)) = C[app(y,z) = app(y,z){}] ** Step 1.b:1: DependencyPairs. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: app(add(n,x),y) -> add(n,app(x,y)) app(nil(),y) -> y concat(cons(u,v),y) -> cons(u,concat(v,y)) concat(leaf(),y) -> y less_leaves(x,leaf()) -> false() less_leaves(cons(u,v),cons(w,z)) -> less_leaves(concat(u,v),concat(w,z)) less_leaves(leaf(),cons(w,z)) -> true() minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) reverse(add(n,x)) -> app(reverse(x),add(n,nil())) reverse(nil()) -> nil() shuffle(add(n,x)) -> add(n,shuffle(reverse(x))) shuffle(nil()) -> nil() - Signature: {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0 ,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {app,concat,less_leaves,minus,quot,reverse ,shuffle} and constructors {0,add,cons,false,leaf,nil,s,true} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs app#(add(n,x),y) -> c_1(app#(x,y)) app#(nil(),y) -> c_2() concat#(cons(u,v),y) -> c_3(concat#(v,y)) concat#(leaf(),y) -> c_4() less_leaves#(x,leaf()) -> c_5() less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)),concat#(u,v),concat#(w,z)) less_leaves#(leaf(),cons(w,z)) -> c_7() minus#(x,0()) -> c_8() minus#(s(x),s(y)) -> c_9(minus#(x,y)) quot#(0(),s(y)) -> c_10() quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y)) reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x)) reverse#(nil()) -> c_13() shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x)) shuffle#(nil()) -> c_15() Weak DPs and mark the set of starting terms. ** Step 1.b:2: PredecessorEstimation. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: app#(add(n,x),y) -> c_1(app#(x,y)) app#(nil(),y) -> c_2() concat#(cons(u,v),y) -> c_3(concat#(v,y)) concat#(leaf(),y) -> c_4() less_leaves#(x,leaf()) -> c_5() less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)),concat#(u,v),concat#(w,z)) less_leaves#(leaf(),cons(w,z)) -> c_7() minus#(x,0()) -> c_8() minus#(s(x),s(y)) -> c_9(minus#(x,y)) quot#(0(),s(y)) -> c_10() quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y)) reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x)) reverse#(nil()) -> c_13() shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x)) shuffle#(nil()) -> c_15() - Weak TRS: app(add(n,x),y) -> add(n,app(x,y)) app(nil(),y) -> y concat(cons(u,v),y) -> cons(u,concat(v,y)) concat(leaf(),y) -> y less_leaves(x,leaf()) -> false() less_leaves(cons(u,v),cons(w,z)) -> less_leaves(concat(u,v),concat(w,z)) less_leaves(leaf(),cons(w,z)) -> true() minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) reverse(add(n,x)) -> app(reverse(x),add(n,nil())) reverse(nil()) -> nil() shuffle(add(n,x)) -> add(n,shuffle(reverse(x))) shuffle(nil()) -> nil() - Signature: {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2 ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0 ,c_5/0,c_6/3,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/2,c_13/0,c_14/2,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse# ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {2,4,5,7,8,10,13,15} by application of Pre({2,4,5,7,8,10,13,15}) = {1,3,6,9,11,12,14}. Here rules are labelled as follows: 1: app#(add(n,x),y) -> c_1(app#(x,y)) 2: app#(nil(),y) -> c_2() 3: concat#(cons(u,v),y) -> c_3(concat#(v,y)) 4: concat#(leaf(),y) -> c_4() 5: less_leaves#(x,leaf()) -> c_5() 6: less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)),concat#(u,v),concat#(w,z)) 7: less_leaves#(leaf(),cons(w,z)) -> c_7() 8: minus#(x,0()) -> c_8() 9: minus#(s(x),s(y)) -> c_9(minus#(x,y)) 10: quot#(0(),s(y)) -> c_10() 11: quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y)) 12: reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x)) 13: reverse#(nil()) -> c_13() 14: shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x)) 15: shuffle#(nil()) -> c_15() ** Step 1.b:3: RemoveWeakSuffixes. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: app#(add(n,x),y) -> c_1(app#(x,y)) concat#(cons(u,v),y) -> c_3(concat#(v,y)) less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)),concat#(u,v),concat#(w,z)) minus#(s(x),s(y)) -> c_9(minus#(x,y)) quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y)) reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x)) shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x)) - Weak DPs: app#(nil(),y) -> c_2() concat#(leaf(),y) -> c_4() less_leaves#(x,leaf()) -> c_5() less_leaves#(leaf(),cons(w,z)) -> c_7() minus#(x,0()) -> c_8() quot#(0(),s(y)) -> c_10() reverse#(nil()) -> c_13() shuffle#(nil()) -> c_15() - Weak TRS: app(add(n,x),y) -> add(n,app(x,y)) app(nil(),y) -> y concat(cons(u,v),y) -> cons(u,concat(v,y)) concat(leaf(),y) -> y less_leaves(x,leaf()) -> false() less_leaves(cons(u,v),cons(w,z)) -> less_leaves(concat(u,v),concat(w,z)) less_leaves(leaf(),cons(w,z)) -> true() minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) reverse(add(n,x)) -> app(reverse(x),add(n,nil())) reverse(nil()) -> nil() shuffle(add(n,x)) -> add(n,shuffle(reverse(x))) shuffle(nil()) -> nil() - Signature: {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2 ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0 ,c_5/0,c_6/3,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/2,c_13/0,c_14/2,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse# ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:app#(add(n,x),y) -> c_1(app#(x,y)) -->_1 app#(nil(),y) -> c_2():8 -->_1 app#(add(n,x),y) -> c_1(app#(x,y)):1 2:S:concat#(cons(u,v),y) -> c_3(concat#(v,y)) -->_1 concat#(leaf(),y) -> c_4():9 -->_1 concat#(cons(u,v),y) -> c_3(concat#(v,y)):2 3:S:less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)) ,concat#(u,v) ,concat#(w,z)) -->_1 less_leaves#(leaf(),cons(w,z)) -> c_7():11 -->_1 less_leaves#(x,leaf()) -> c_5():10 -->_3 concat#(leaf(),y) -> c_4():9 -->_2 concat#(leaf(),y) -> c_4():9 -->_1 less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)) ,concat#(u,v) ,concat#(w,z)):3 -->_3 concat#(cons(u,v),y) -> c_3(concat#(v,y)):2 -->_2 concat#(cons(u,v),y) -> c_3(concat#(v,y)):2 4:S:minus#(s(x),s(y)) -> c_9(minus#(x,y)) -->_1 minus#(x,0()) -> c_8():12 -->_1 minus#(s(x),s(y)) -> c_9(minus#(x,y)):4 5:S:quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y)) -->_1 quot#(0(),s(y)) -> c_10():13 -->_2 minus#(x,0()) -> c_8():12 -->_1 quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y)):5 -->_2 minus#(s(x),s(y)) -> c_9(minus#(x,y)):4 6:S:reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x)) -->_2 reverse#(nil()) -> c_13():14 -->_1 app#(nil(),y) -> c_2():8 -->_2 reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x)):6 -->_1 app#(add(n,x),y) -> c_1(app#(x,y)):1 7:S:shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x)) -->_1 shuffle#(nil()) -> c_15():15 -->_2 reverse#(nil()) -> c_13():14 -->_1 shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x)):7 -->_2 reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x)):6 8:W:app#(nil(),y) -> c_2() 9:W:concat#(leaf(),y) -> c_4() 10:W:less_leaves#(x,leaf()) -> c_5() 11:W:less_leaves#(leaf(),cons(w,z)) -> c_7() 12:W:minus#(x,0()) -> c_8() 13:W:quot#(0(),s(y)) -> c_10() 14:W:reverse#(nil()) -> c_13() 15:W:shuffle#(nil()) -> c_15() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 15: shuffle#(nil()) -> c_15() 14: reverse#(nil()) -> c_13() 13: quot#(0(),s(y)) -> c_10() 12: minus#(x,0()) -> c_8() 10: less_leaves#(x,leaf()) -> c_5() 11: less_leaves#(leaf(),cons(w,z)) -> c_7() 9: concat#(leaf(),y) -> c_4() 8: app#(nil(),y) -> c_2() ** Step 1.b:4: UsableRules. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: app#(add(n,x),y) -> c_1(app#(x,y)) concat#(cons(u,v),y) -> c_3(concat#(v,y)) less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)),concat#(u,v),concat#(w,z)) minus#(s(x),s(y)) -> c_9(minus#(x,y)) quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y)) reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x)) shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x)) - Weak TRS: app(add(n,x),y) -> add(n,app(x,y)) app(nil(),y) -> y concat(cons(u,v),y) -> cons(u,concat(v,y)) concat(leaf(),y) -> y less_leaves(x,leaf()) -> false() less_leaves(cons(u,v),cons(w,z)) -> less_leaves(concat(u,v),concat(w,z)) less_leaves(leaf(),cons(w,z)) -> true() minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) reverse(add(n,x)) -> app(reverse(x),add(n,nil())) reverse(nil()) -> nil() shuffle(add(n,x)) -> add(n,shuffle(reverse(x))) shuffle(nil()) -> nil() - Signature: {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2 ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0 ,c_5/0,c_6/3,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/2,c_13/0,c_14/2,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse# ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: app(add(n,x),y) -> add(n,app(x,y)) app(nil(),y) -> y concat(cons(u,v),y) -> cons(u,concat(v,y)) concat(leaf(),y) -> y minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) reverse(add(n,x)) -> app(reverse(x),add(n,nil())) reverse(nil()) -> nil() app#(add(n,x),y) -> c_1(app#(x,y)) concat#(cons(u,v),y) -> c_3(concat#(v,y)) less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)),concat#(u,v),concat#(w,z)) minus#(s(x),s(y)) -> c_9(minus#(x,y)) quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y)) reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x)) shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x)) ** Step 1.b:5: DecomposeDG. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: app#(add(n,x),y) -> c_1(app#(x,y)) concat#(cons(u,v),y) -> c_3(concat#(v,y)) less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)),concat#(u,v),concat#(w,z)) minus#(s(x),s(y)) -> c_9(minus#(x,y)) quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y)) reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x)) shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x)) - Weak TRS: app(add(n,x),y) -> add(n,app(x,y)) app(nil(),y) -> y concat(cons(u,v),y) -> cons(u,concat(v,y)) concat(leaf(),y) -> y minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) reverse(add(n,x)) -> app(reverse(x),add(n,nil())) reverse(nil()) -> nil() - Signature: {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2 ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0 ,c_5/0,c_6/3,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/2,c_13/0,c_14/2,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse# ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true} + Applied Processor: DecomposeDG {onSelection = all below first cut in WDG, onUpper = Nothing, onLower = Nothing} + Details: We decompose the input problem according to the dependency graph into the upper component less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)),concat#(u,v),concat#(w,z)) quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y)) shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x)) and a lower component app#(add(n,x),y) -> c_1(app#(x,y)) concat#(cons(u,v),y) -> c_3(concat#(v,y)) minus#(s(x),s(y)) -> c_9(minus#(x,y)) reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x)) Further, following extension rules are added to the lower component. less_leaves#(cons(u,v),cons(w,z)) -> concat#(u,v) less_leaves#(cons(u,v),cons(w,z)) -> concat#(w,z) less_leaves#(cons(u,v),cons(w,z)) -> less_leaves#(concat(u,v),concat(w,z)) quot#(s(x),s(y)) -> minus#(x,y) quot#(s(x),s(y)) -> quot#(minus(x,y),s(y)) shuffle#(add(n,x)) -> reverse#(x) shuffle#(add(n,x)) -> shuffle#(reverse(x)) *** Step 1.b:5.a:1: SimplifyRHS. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)),concat#(u,v),concat#(w,z)) quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y)) shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x)) - Weak TRS: app(add(n,x),y) -> add(n,app(x,y)) app(nil(),y) -> y concat(cons(u,v),y) -> cons(u,concat(v,y)) concat(leaf(),y) -> y minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) reverse(add(n,x)) -> app(reverse(x),add(n,nil())) reverse(nil()) -> nil() - Signature: {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2 ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0 ,c_5/0,c_6/3,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/2,c_13/0,c_14/2,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse# ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)) ,concat#(u,v) ,concat#(w,z)) -->_1 less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)) ,concat#(u,v) ,concat#(w,z)):1 2:S:quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y)) -->_1 quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y)):2 3:S:shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x)) -->_1 shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x)):3 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z))) quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y))) shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x))) *** Step 1.b:5.a:2: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z))) quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y))) shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x))) - Weak TRS: app(add(n,x),y) -> add(n,app(x,y)) app(nil(),y) -> y concat(cons(u,v),y) -> cons(u,concat(v,y)) concat(leaf(),y) -> y minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) reverse(add(n,x)) -> app(reverse(x),add(n,nil())) reverse(nil()) -> nil() - Signature: {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2 ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0 ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/2,c_13/0,c_14/1,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse# ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_6) = {1}, uargs(c_11) = {1}, uargs(c_14) = {1} Following symbols are considered usable: {concat,minus,app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#} TcT has computed the following interpretation: p(0) = [1] p(add) = [1] x1 + [2] p(app) = [4] x1 + [6] x2 + [1] p(concat) = [1] x1 + [1] x2 + [1] p(cons) = [1] x1 + [1] x2 + [1] p(false) = [2] p(leaf) = [6] p(less_leaves) = [4] x2 + [1] p(minus) = [1] x1 + [0] p(nil) = [0] p(quot) = [1] x1 + [8] p(reverse) = [0] p(s) = [1] x1 + [2] p(shuffle) = [2] x1 + [1] p(true) = [0] p(app#) = [1] x2 + [1] p(concat#) = [1] x1 + [1] x2 + [4] p(less_leaves#) = [4] x1 + [14] x2 + [12] p(minus#) = [1] p(quot#) = [4] x1 + [1] x2 + [1] p(reverse#) = [1] x1 + [1] p(shuffle#) = [0] p(c_1) = [2] x1 + [1] p(c_2) = [0] p(c_3) = [1] x1 + [1] p(c_4) = [0] p(c_5) = [2] p(c_6) = [1] x1 + [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [2] p(c_10) = [0] p(c_11) = [1] x1 + [4] p(c_12) = [8] x2 + [0] p(c_13) = [1] p(c_14) = [2] x1 + [0] p(c_15) = [1] Following rules are strictly oriented: quot#(s(x),s(y)) = [4] x + [1] y + [11] > [4] x + [1] y + [7] = c_11(quot#(minus(x,y),s(y))) Following rules are (at-least) weakly oriented: less_leaves#(cons(u,v),cons(w,z)) = [4] u + [4] v + [14] w + [14] z + [30] >= [4] u + [4] v + [14] w + [14] z + [30] = c_6(less_leaves#(concat(u,v),concat(w,z))) shuffle#(add(n,x)) = [0] >= [0] = c_14(shuffle#(reverse(x))) concat(cons(u,v),y) = [1] u + [1] v + [1] y + [2] >= [1] u + [1] v + [1] y + [2] = cons(u,concat(v,y)) concat(leaf(),y) = [1] y + [7] >= [1] y + [0] = y minus(x,0()) = [1] x + [0] >= [1] x + [0] = x minus(s(x),s(y)) = [1] x + [2] >= [1] x + [0] = minus(x,y) *** Step 1.b:5.a:3: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z))) shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x))) - Weak DPs: quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y))) - Weak TRS: app(add(n,x),y) -> add(n,app(x,y)) app(nil(),y) -> y concat(cons(u,v),y) -> cons(u,concat(v,y)) concat(leaf(),y) -> y minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) reverse(add(n,x)) -> app(reverse(x),add(n,nil())) reverse(nil()) -> nil() - Signature: {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2 ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0 ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/2,c_13/0,c_14/1,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse# ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_6) = {1}, uargs(c_11) = {1}, uargs(c_14) = {1} Following symbols are considered usable: {concat,app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#} TcT has computed the following interpretation: p(0) = [1] p(add) = [1] x1 + [2] p(app) = [14] x2 + [1] p(concat) = [1] x1 + [1] x2 + [0] p(cons) = [1] x1 + [1] x2 + [1] p(false) = [1] p(leaf) = [10] p(less_leaves) = [0] p(minus) = [1] x2 + [8] p(nil) = [0] p(quot) = [2] x2 + [2] p(reverse) = [0] p(s) = [1] p(shuffle) = [1] x1 + [0] p(true) = [0] p(app#) = [1] x2 + [2] p(concat#) = [2] x1 + [1] x2 + [0] p(less_leaves#) = [8] x2 + [0] p(minus#) = [1] x1 + [2] x2 + [1] p(quot#) = [12] x2 + [8] p(reverse#) = [1] x1 + [1] p(shuffle#) = [0] p(c_1) = [0] p(c_2) = [1] p(c_3) = [1] p(c_4) = [1] p(c_5) = [0] p(c_6) = [1] x1 + [7] p(c_7) = [0] p(c_8) = [0] p(c_9) = [1] p(c_10) = [8] p(c_11) = [1] x1 + [0] p(c_12) = [1] p(c_13) = [1] p(c_14) = [2] x1 + [0] p(c_15) = [1] Following rules are strictly oriented: less_leaves#(cons(u,v),cons(w,z)) = [8] w + [8] z + [8] > [8] w + [8] z + [7] = c_6(less_leaves#(concat(u,v),concat(w,z))) Following rules are (at-least) weakly oriented: quot#(s(x),s(y)) = [20] >= [20] = c_11(quot#(minus(x,y),s(y))) shuffle#(add(n,x)) = [0] >= [0] = c_14(shuffle#(reverse(x))) concat(cons(u,v),y) = [1] u + [1] v + [1] y + [1] >= [1] u + [1] v + [1] y + [1] = cons(u,concat(v,y)) concat(leaf(),y) = [1] y + [10] >= [1] y + [0] = y *** Step 1.b:5.a:4: WeightGap. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x))) - Weak DPs: less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z))) quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y))) - Weak TRS: app(add(n,x),y) -> add(n,app(x,y)) app(nil(),y) -> y concat(cons(u,v),y) -> cons(u,concat(v,y)) concat(leaf(),y) -> y minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) reverse(add(n,x)) -> app(reverse(x),add(n,nil())) reverse(nil()) -> nil() - Signature: {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2 ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0 ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/2,c_13/0,c_14/1,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse# ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(add) = {2}, uargs(app) = {1}, uargs(cons) = {2}, uargs(less_leaves#) = {1,2}, uargs(quot#) = {1}, uargs(shuffle#) = {1}, uargs(c_6) = {1}, uargs(c_11) = {1}, uargs(c_14) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(add) = [1] x2 + [2] p(app) = [1] x1 + [1] x2 + [0] p(concat) = [1] x2 + [0] p(cons) = [1] x2 + [0] p(false) = [0] p(leaf) = [0] p(less_leaves) = [4] x1 + [0] p(minus) = [1] x1 + [1] p(nil) = [0] p(quot) = [0] p(reverse) = [1] x1 + [1] p(s) = [1] x1 + [5] p(shuffle) = [0] p(true) = [0] p(app#) = [0] p(concat#) = [0] p(less_leaves#) = [1] x1 + [1] x2 + [0] p(minus#) = [2] x2 + [0] p(quot#) = [1] x1 + [4] p(reverse#) = [1] p(shuffle#) = [1] x1 + [7] p(c_1) = [2] x1 + [0] p(c_2) = [1] p(c_3) = [1] x1 + [1] p(c_4) = [2] p(c_5) = [0] p(c_6) = [1] x1 + [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [4] x1 + [1] p(c_10) = [4] p(c_11) = [1] x1 + [4] p(c_12) = [2] x1 + [1] x2 + [4] p(c_13) = [0] p(c_14) = [1] x1 + [0] p(c_15) = [0] Following rules are strictly oriented: shuffle#(add(n,x)) = [1] x + [9] > [1] x + [8] = c_14(shuffle#(reverse(x))) Following rules are (at-least) weakly oriented: less_leaves#(cons(u,v),cons(w,z)) = [1] v + [1] z + [0] >= [1] v + [1] z + [0] = c_6(less_leaves#(concat(u,v),concat(w,z))) quot#(s(x),s(y)) = [1] x + [9] >= [1] x + [9] = c_11(quot#(minus(x,y),s(y))) app(add(n,x),y) = [1] x + [1] y + [2] >= [1] x + [1] y + [2] = add(n,app(x,y)) app(nil(),y) = [1] y + [0] >= [1] y + [0] = y concat(cons(u,v),y) = [1] y + [0] >= [1] y + [0] = cons(u,concat(v,y)) concat(leaf(),y) = [1] y + [0] >= [1] y + [0] = y minus(x,0()) = [1] x + [1] >= [1] x + [0] = x minus(s(x),s(y)) = [1] x + [6] >= [1] x + [1] = minus(x,y) reverse(add(n,x)) = [1] x + [3] >= [1] x + [3] = app(reverse(x),add(n,nil())) reverse(nil()) = [1] >= [0] = nil() Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** Step 1.b:5.a:5: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z))) quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y))) shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x))) - Weak TRS: app(add(n,x),y) -> add(n,app(x,y)) app(nil(),y) -> y concat(cons(u,v),y) -> cons(u,concat(v,y)) concat(leaf(),y) -> y minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) reverse(add(n,x)) -> app(reverse(x),add(n,nil())) reverse(nil()) -> nil() - Signature: {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2 ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0 ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/2,c_13/0,c_14/1,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse# ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). *** Step 1.b:5.b:1: DecomposeDG. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: app#(add(n,x),y) -> c_1(app#(x,y)) concat#(cons(u,v),y) -> c_3(concat#(v,y)) minus#(s(x),s(y)) -> c_9(minus#(x,y)) reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x)) - Weak DPs: less_leaves#(cons(u,v),cons(w,z)) -> concat#(u,v) less_leaves#(cons(u,v),cons(w,z)) -> concat#(w,z) less_leaves#(cons(u,v),cons(w,z)) -> less_leaves#(concat(u,v),concat(w,z)) quot#(s(x),s(y)) -> minus#(x,y) quot#(s(x),s(y)) -> quot#(minus(x,y),s(y)) shuffle#(add(n,x)) -> reverse#(x) shuffle#(add(n,x)) -> shuffle#(reverse(x)) - Weak TRS: app(add(n,x),y) -> add(n,app(x,y)) app(nil(),y) -> y concat(cons(u,v),y) -> cons(u,concat(v,y)) concat(leaf(),y) -> y minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) reverse(add(n,x)) -> app(reverse(x),add(n,nil())) reverse(nil()) -> nil() - Signature: {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2 ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0 ,c_5/0,c_6/3,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/2,c_13/0,c_14/2,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse# ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true} + Applied Processor: DecomposeDG {onSelection = all below first cut in WDG, onUpper = Nothing, onLower = Nothing} + Details: We decompose the input problem according to the dependency graph into the upper component concat#(cons(u,v),y) -> c_3(concat#(v,y)) less_leaves#(cons(u,v),cons(w,z)) -> concat#(u,v) less_leaves#(cons(u,v),cons(w,z)) -> concat#(w,z) less_leaves#(cons(u,v),cons(w,z)) -> less_leaves#(concat(u,v),concat(w,z)) minus#(s(x),s(y)) -> c_9(minus#(x,y)) quot#(s(x),s(y)) -> minus#(x,y) quot#(s(x),s(y)) -> quot#(minus(x,y),s(y)) reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x)) shuffle#(add(n,x)) -> reverse#(x) shuffle#(add(n,x)) -> shuffle#(reverse(x)) and a lower component app#(add(n,x),y) -> c_1(app#(x,y)) Further, following extension rules are added to the lower component. concat#(cons(u,v),y) -> concat#(v,y) less_leaves#(cons(u,v),cons(w,z)) -> concat#(u,v) less_leaves#(cons(u,v),cons(w,z)) -> concat#(w,z) less_leaves#(cons(u,v),cons(w,z)) -> less_leaves#(concat(u,v),concat(w,z)) minus#(s(x),s(y)) -> minus#(x,y) quot#(s(x),s(y)) -> minus#(x,y) quot#(s(x),s(y)) -> quot#(minus(x,y),s(y)) reverse#(add(n,x)) -> app#(reverse(x),add(n,nil())) reverse#(add(n,x)) -> reverse#(x) shuffle#(add(n,x)) -> reverse#(x) shuffle#(add(n,x)) -> shuffle#(reverse(x)) **** Step 1.b:5.b:1.a:1: SimplifyRHS. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: concat#(cons(u,v),y) -> c_3(concat#(v,y)) minus#(s(x),s(y)) -> c_9(minus#(x,y)) reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x)) - Weak DPs: less_leaves#(cons(u,v),cons(w,z)) -> concat#(u,v) less_leaves#(cons(u,v),cons(w,z)) -> concat#(w,z) less_leaves#(cons(u,v),cons(w,z)) -> less_leaves#(concat(u,v),concat(w,z)) quot#(s(x),s(y)) -> minus#(x,y) quot#(s(x),s(y)) -> quot#(minus(x,y),s(y)) shuffle#(add(n,x)) -> reverse#(x) shuffle#(add(n,x)) -> shuffle#(reverse(x)) - Weak TRS: app(add(n,x),y) -> add(n,app(x,y)) app(nil(),y) -> y concat(cons(u,v),y) -> cons(u,concat(v,y)) concat(leaf(),y) -> y minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) reverse(add(n,x)) -> app(reverse(x),add(n,nil())) reverse(nil()) -> nil() - Signature: {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2 ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0 ,c_5/0,c_6/3,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/2,c_13/0,c_14/2,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse# ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:concat#(cons(u,v),y) -> c_3(concat#(v,y)) -->_1 concat#(cons(u,v),y) -> c_3(concat#(v,y)):1 2:S:minus#(s(x),s(y)) -> c_9(minus#(x,y)) -->_1 minus#(s(x),s(y)) -> c_9(minus#(x,y)):2 3:S:reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x)) -->_2 reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x)):3 4:W:less_leaves#(cons(u,v),cons(w,z)) -> concat#(u,v) -->_1 concat#(cons(u,v),y) -> c_3(concat#(v,y)):1 5:W:less_leaves#(cons(u,v),cons(w,z)) -> concat#(w,z) -->_1 concat#(cons(u,v),y) -> c_3(concat#(v,y)):1 6:W:less_leaves#(cons(u,v),cons(w,z)) -> less_leaves#(concat(u,v),concat(w,z)) -->_1 less_leaves#(cons(u,v),cons(w,z)) -> less_leaves#(concat(u,v),concat(w,z)):6 -->_1 less_leaves#(cons(u,v),cons(w,z)) -> concat#(w,z):5 -->_1 less_leaves#(cons(u,v),cons(w,z)) -> concat#(u,v):4 7:W:quot#(s(x),s(y)) -> minus#(x,y) -->_1 minus#(s(x),s(y)) -> c_9(minus#(x,y)):2 8:W:quot#(s(x),s(y)) -> quot#(minus(x,y),s(y)) -->_1 quot#(s(x),s(y)) -> quot#(minus(x,y),s(y)):8 -->_1 quot#(s(x),s(y)) -> minus#(x,y):7 9:W:shuffle#(add(n,x)) -> reverse#(x) -->_1 reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x)):3 10:W:shuffle#(add(n,x)) -> shuffle#(reverse(x)) -->_1 shuffle#(add(n,x)) -> shuffle#(reverse(x)):10 -->_1 shuffle#(add(n,x)) -> reverse#(x):9 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: reverse#(add(n,x)) -> c_12(reverse#(x)) **** Step 1.b:5.b:1.a:2: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: concat#(cons(u,v),y) -> c_3(concat#(v,y)) minus#(s(x),s(y)) -> c_9(minus#(x,y)) reverse#(add(n,x)) -> c_12(reverse#(x)) - Weak DPs: less_leaves#(cons(u,v),cons(w,z)) -> concat#(u,v) less_leaves#(cons(u,v),cons(w,z)) -> concat#(w,z) less_leaves#(cons(u,v),cons(w,z)) -> less_leaves#(concat(u,v),concat(w,z)) quot#(s(x),s(y)) -> minus#(x,y) quot#(s(x),s(y)) -> quot#(minus(x,y),s(y)) shuffle#(add(n,x)) -> reverse#(x) shuffle#(add(n,x)) -> shuffle#(reverse(x)) - Weak TRS: app(add(n,x),y) -> add(n,app(x,y)) app(nil(),y) -> y concat(cons(u,v),y) -> cons(u,concat(v,y)) concat(leaf(),y) -> y minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) reverse(add(n,x)) -> app(reverse(x),add(n,nil())) reverse(nil()) -> nil() - Signature: {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2 ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0 ,c_5/0,c_6/3,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse# ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_3) = {1}, uargs(c_9) = {1}, uargs(c_12) = {1} Following symbols are considered usable: {app,reverse,app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#} TcT has computed the following interpretation: p(0) = [2] p(add) = [1] x2 + [4] p(app) = [1] x1 + [1] x2 + [0] p(concat) = [0] p(cons) = [0] p(false) = [1] p(leaf) = [4] p(less_leaves) = [4] p(minus) = [5] x2 + [2] p(nil) = [0] p(quot) = [2] x1 + [0] p(reverse) = [1] x1 + [0] p(s) = [0] p(shuffle) = [1] p(true) = [4] p(app#) = [4] x1 + [1] p(concat#) = [0] p(less_leaves#) = [6] p(minus#) = [0] p(quot#) = [0] p(reverse#) = [1] x1 + [4] p(shuffle#) = [1] x1 + [1] p(c_1) = [0] p(c_2) = [0] p(c_3) = [4] x1 + [0] p(c_4) = [1] p(c_5) = [1] p(c_6) = [0] p(c_7) = [1] p(c_8) = [0] p(c_9) = [1] x1 + [0] p(c_10) = [2] p(c_11) = [1] x1 + [4] x2 + [1] p(c_12) = [1] x1 + [0] p(c_13) = [1] p(c_14) = [1] x1 + [2] x2 + [0] p(c_15) = [1] Following rules are strictly oriented: reverse#(add(n,x)) = [1] x + [8] > [1] x + [4] = c_12(reverse#(x)) Following rules are (at-least) weakly oriented: concat#(cons(u,v),y) = [0] >= [0] = c_3(concat#(v,y)) less_leaves#(cons(u,v),cons(w,z)) = [6] >= [0] = concat#(u,v) less_leaves#(cons(u,v),cons(w,z)) = [6] >= [0] = concat#(w,z) less_leaves#(cons(u,v),cons(w,z)) = [6] >= [6] = less_leaves#(concat(u,v),concat(w,z)) minus#(s(x),s(y)) = [0] >= [0] = c_9(minus#(x,y)) quot#(s(x),s(y)) = [0] >= [0] = minus#(x,y) quot#(s(x),s(y)) = [0] >= [0] = quot#(minus(x,y),s(y)) shuffle#(add(n,x)) = [1] x + [5] >= [1] x + [4] = reverse#(x) shuffle#(add(n,x)) = [1] x + [5] >= [1] x + [1] = shuffle#(reverse(x)) app(add(n,x),y) = [1] x + [1] y + [4] >= [1] x + [1] y + [4] = add(n,app(x,y)) app(nil(),y) = [1] y + [0] >= [1] y + [0] = y reverse(add(n,x)) = [1] x + [4] >= [1] x + [4] = app(reverse(x),add(n,nil())) reverse(nil()) = [0] >= [0] = nil() **** Step 1.b:5.b:1.a:3: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: concat#(cons(u,v),y) -> c_3(concat#(v,y)) minus#(s(x),s(y)) -> c_9(minus#(x,y)) - Weak DPs: less_leaves#(cons(u,v),cons(w,z)) -> concat#(u,v) less_leaves#(cons(u,v),cons(w,z)) -> concat#(w,z) less_leaves#(cons(u,v),cons(w,z)) -> less_leaves#(concat(u,v),concat(w,z)) quot#(s(x),s(y)) -> minus#(x,y) quot#(s(x),s(y)) -> quot#(minus(x,y),s(y)) reverse#(add(n,x)) -> c_12(reverse#(x)) shuffle#(add(n,x)) -> reverse#(x) shuffle#(add(n,x)) -> shuffle#(reverse(x)) - Weak TRS: app(add(n,x),y) -> add(n,app(x,y)) app(nil(),y) -> y concat(cons(u,v),y) -> cons(u,concat(v,y)) concat(leaf(),y) -> y minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) reverse(add(n,x)) -> app(reverse(x),add(n,nil())) reverse(nil()) -> nil() - Signature: {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2 ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0 ,c_5/0,c_6/3,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse# ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_3) = {1}, uargs(c_9) = {1}, uargs(c_12) = {1} Following symbols are considered usable: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#} TcT has computed the following interpretation: p(0) = [0] p(add) = [1] x1 + [0] p(app) = [1] x1 + [0] p(concat) = [0] p(cons) = [1] x2 + [0] p(false) = [0] p(leaf) = [0] p(less_leaves) = [0] p(minus) = [4] x1 + [0] p(nil) = [0] p(quot) = [0] p(reverse) = [1] p(s) = [1] x1 + [3] p(shuffle) = [0] p(true) = [0] p(app#) = [0] p(concat#) = [0] p(less_leaves#) = [0] p(minus#) = [1] x2 + [0] p(quot#) = [1] x2 + [6] p(reverse#) = [0] p(shuffle#) = [0] p(c_1) = [0] p(c_2) = [2] p(c_3) = [4] x1 + [0] p(c_4) = [0] p(c_5) = [0] p(c_6) = [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [1] x1 + [0] p(c_10) = [0] p(c_11) = [1] x1 + [0] p(c_12) = [4] x1 + [0] p(c_13) = [0] p(c_14) = [0] p(c_15) = [0] Following rules are strictly oriented: minus#(s(x),s(y)) = [1] y + [3] > [1] y + [0] = c_9(minus#(x,y)) Following rules are (at-least) weakly oriented: concat#(cons(u,v),y) = [0] >= [0] = c_3(concat#(v,y)) less_leaves#(cons(u,v),cons(w,z)) = [0] >= [0] = concat#(u,v) less_leaves#(cons(u,v),cons(w,z)) = [0] >= [0] = concat#(w,z) less_leaves#(cons(u,v),cons(w,z)) = [0] >= [0] = less_leaves#(concat(u,v),concat(w,z)) quot#(s(x),s(y)) = [1] y + [9] >= [1] y + [0] = minus#(x,y) quot#(s(x),s(y)) = [1] y + [9] >= [1] y + [9] = quot#(minus(x,y),s(y)) reverse#(add(n,x)) = [0] >= [0] = c_12(reverse#(x)) shuffle#(add(n,x)) = [0] >= [0] = reverse#(x) shuffle#(add(n,x)) = [0] >= [0] = shuffle#(reverse(x)) **** Step 1.b:5.b:1.a:4: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: concat#(cons(u,v),y) -> c_3(concat#(v,y)) - Weak DPs: less_leaves#(cons(u,v),cons(w,z)) -> concat#(u,v) less_leaves#(cons(u,v),cons(w,z)) -> concat#(w,z) less_leaves#(cons(u,v),cons(w,z)) -> less_leaves#(concat(u,v),concat(w,z)) minus#(s(x),s(y)) -> c_9(minus#(x,y)) quot#(s(x),s(y)) -> minus#(x,y) quot#(s(x),s(y)) -> quot#(minus(x,y),s(y)) reverse#(add(n,x)) -> c_12(reverse#(x)) shuffle#(add(n,x)) -> reverse#(x) shuffle#(add(n,x)) -> shuffle#(reverse(x)) - Weak TRS: app(add(n,x),y) -> add(n,app(x,y)) app(nil(),y) -> y concat(cons(u,v),y) -> cons(u,concat(v,y)) concat(leaf(),y) -> y minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) reverse(add(n,x)) -> app(reverse(x),add(n,nil())) reverse(nil()) -> nil() - Signature: {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2 ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0 ,c_5/0,c_6/3,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse# ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_3) = {1}, uargs(c_9) = {1}, uargs(c_12) = {1} Following symbols are considered usable: {app,concat,reverse,app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#} TcT has computed the following interpretation: p(0) = [0] p(add) = [0] p(app) = [1] x2 + [0] p(concat) = [1] x1 + [1] x2 + [1] p(cons) = [1] x1 + [1] x2 + [1] p(false) = [0] p(leaf) = [0] p(less_leaves) = [0] p(minus) = [1] p(nil) = [0] p(quot) = [0] p(reverse) = [0] p(s) = [4] p(shuffle) = [0] p(true) = [0] p(app#) = [0] p(concat#) = [4] x1 + [0] p(less_leaves#) = [6] x1 + [4] x2 + [2] p(minus#) = [0] p(quot#) = [5] p(reverse#) = [0] p(shuffle#) = [4] x1 + [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [1] x1 + [0] p(c_4) = [0] p(c_5) = [1] p(c_6) = [1] x2 + [2] p(c_7) = [1] p(c_8) = [0] p(c_9) = [4] x1 + [0] p(c_10) = [1] p(c_11) = [4] x1 + [1] x2 + [0] p(c_12) = [1] x1 + [0] p(c_13) = [1] p(c_14) = [1] x1 + [1] x2 + [0] p(c_15) = [0] Following rules are strictly oriented: concat#(cons(u,v),y) = [4] u + [4] v + [4] > [4] v + [0] = c_3(concat#(v,y)) Following rules are (at-least) weakly oriented: less_leaves#(cons(u,v),cons(w,z)) = [6] u + [6] v + [4] w + [4] z + [12] >= [4] u + [0] = concat#(u,v) less_leaves#(cons(u,v),cons(w,z)) = [6] u + [6] v + [4] w + [4] z + [12] >= [4] w + [0] = concat#(w,z) less_leaves#(cons(u,v),cons(w,z)) = [6] u + [6] v + [4] w + [4] z + [12] >= [6] u + [6] v + [4] w + [4] z + [12] = less_leaves#(concat(u,v),concat(w,z)) minus#(s(x),s(y)) = [0] >= [0] = c_9(minus#(x,y)) quot#(s(x),s(y)) = [5] >= [0] = minus#(x,y) quot#(s(x),s(y)) = [5] >= [5] = quot#(minus(x,y),s(y)) reverse#(add(n,x)) = [0] >= [0] = c_12(reverse#(x)) shuffle#(add(n,x)) = [0] >= [0] = reverse#(x) shuffle#(add(n,x)) = [0] >= [0] = shuffle#(reverse(x)) app(add(n,x),y) = [1] y + [0] >= [0] = add(n,app(x,y)) app(nil(),y) = [1] y + [0] >= [1] y + [0] = y concat(cons(u,v),y) = [1] u + [1] v + [1] y + [2] >= [1] u + [1] v + [1] y + [2] = cons(u,concat(v,y)) concat(leaf(),y) = [1] y + [1] >= [1] y + [0] = y reverse(add(n,x)) = [0] >= [0] = app(reverse(x),add(n,nil())) reverse(nil()) = [0] >= [0] = nil() **** Step 1.b:5.b:1.a:5: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: concat#(cons(u,v),y) -> c_3(concat#(v,y)) less_leaves#(cons(u,v),cons(w,z)) -> concat#(u,v) less_leaves#(cons(u,v),cons(w,z)) -> concat#(w,z) less_leaves#(cons(u,v),cons(w,z)) -> less_leaves#(concat(u,v),concat(w,z)) minus#(s(x),s(y)) -> c_9(minus#(x,y)) quot#(s(x),s(y)) -> minus#(x,y) quot#(s(x),s(y)) -> quot#(minus(x,y),s(y)) reverse#(add(n,x)) -> c_12(reverse#(x)) shuffle#(add(n,x)) -> reverse#(x) shuffle#(add(n,x)) -> shuffle#(reverse(x)) - Weak TRS: app(add(n,x),y) -> add(n,app(x,y)) app(nil(),y) -> y concat(cons(u,v),y) -> cons(u,concat(v,y)) concat(leaf(),y) -> y minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) reverse(add(n,x)) -> app(reverse(x),add(n,nil())) reverse(nil()) -> nil() - Signature: {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2 ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0 ,c_5/0,c_6/3,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse# ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). **** Step 1.b:5.b:1.b:1: RemoveWeakSuffixes. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: app#(add(n,x),y) -> c_1(app#(x,y)) - Weak DPs: concat#(cons(u,v),y) -> concat#(v,y) less_leaves#(cons(u,v),cons(w,z)) -> concat#(u,v) less_leaves#(cons(u,v),cons(w,z)) -> concat#(w,z) less_leaves#(cons(u,v),cons(w,z)) -> less_leaves#(concat(u,v),concat(w,z)) minus#(s(x),s(y)) -> minus#(x,y) quot#(s(x),s(y)) -> minus#(x,y) quot#(s(x),s(y)) -> quot#(minus(x,y),s(y)) reverse#(add(n,x)) -> app#(reverse(x),add(n,nil())) reverse#(add(n,x)) -> reverse#(x) shuffle#(add(n,x)) -> reverse#(x) shuffle#(add(n,x)) -> shuffle#(reverse(x)) - Weak TRS: app(add(n,x),y) -> add(n,app(x,y)) app(nil(),y) -> y concat(cons(u,v),y) -> cons(u,concat(v,y)) concat(leaf(),y) -> y minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) reverse(add(n,x)) -> app(reverse(x),add(n,nil())) reverse(nil()) -> nil() - Signature: {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2 ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0 ,c_5/0,c_6/3,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/2,c_13/0,c_14/2,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse# ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:app#(add(n,x),y) -> c_1(app#(x,y)) -->_1 app#(add(n,x),y) -> c_1(app#(x,y)):1 2:W:concat#(cons(u,v),y) -> concat#(v,y) -->_1 concat#(cons(u,v),y) -> concat#(v,y):2 3:W:less_leaves#(cons(u,v),cons(w,z)) -> concat#(u,v) -->_1 concat#(cons(u,v),y) -> concat#(v,y):2 4:W:less_leaves#(cons(u,v),cons(w,z)) -> concat#(w,z) -->_1 concat#(cons(u,v),y) -> concat#(v,y):2 5:W:less_leaves#(cons(u,v),cons(w,z)) -> less_leaves#(concat(u,v),concat(w,z)) -->_1 less_leaves#(cons(u,v),cons(w,z)) -> less_leaves#(concat(u,v),concat(w,z)):5 -->_1 less_leaves#(cons(u,v),cons(w,z)) -> concat#(w,z):4 -->_1 less_leaves#(cons(u,v),cons(w,z)) -> concat#(u,v):3 6:W:minus#(s(x),s(y)) -> minus#(x,y) -->_1 minus#(s(x),s(y)) -> minus#(x,y):6 7:W:quot#(s(x),s(y)) -> minus#(x,y) -->_1 minus#(s(x),s(y)) -> minus#(x,y):6 8:W:quot#(s(x),s(y)) -> quot#(minus(x,y),s(y)) -->_1 quot#(s(x),s(y)) -> quot#(minus(x,y),s(y)):8 -->_1 quot#(s(x),s(y)) -> minus#(x,y):7 9:W:reverse#(add(n,x)) -> app#(reverse(x),add(n,nil())) -->_1 app#(add(n,x),y) -> c_1(app#(x,y)):1 10:W:reverse#(add(n,x)) -> reverse#(x) -->_1 reverse#(add(n,x)) -> reverse#(x):10 -->_1 reverse#(add(n,x)) -> app#(reverse(x),add(n,nil())):9 11:W:shuffle#(add(n,x)) -> reverse#(x) -->_1 reverse#(add(n,x)) -> reverse#(x):10 -->_1 reverse#(add(n,x)) -> app#(reverse(x),add(n,nil())):9 12:W:shuffle#(add(n,x)) -> shuffle#(reverse(x)) -->_1 shuffle#(add(n,x)) -> shuffle#(reverse(x)):12 -->_1 shuffle#(add(n,x)) -> reverse#(x):11 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 8: quot#(s(x),s(y)) -> quot#(minus(x,y),s(y)) 7: quot#(s(x),s(y)) -> minus#(x,y) 6: minus#(s(x),s(y)) -> minus#(x,y) 5: less_leaves#(cons(u,v),cons(w,z)) -> less_leaves#(concat(u,v),concat(w,z)) 4: less_leaves#(cons(u,v),cons(w,z)) -> concat#(w,z) 3: less_leaves#(cons(u,v),cons(w,z)) -> concat#(u,v) 2: concat#(cons(u,v),y) -> concat#(v,y) **** Step 1.b:5.b:1.b:2: UsableRules. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: app#(add(n,x),y) -> c_1(app#(x,y)) - Weak DPs: reverse#(add(n,x)) -> app#(reverse(x),add(n,nil())) reverse#(add(n,x)) -> reverse#(x) shuffle#(add(n,x)) -> reverse#(x) shuffle#(add(n,x)) -> shuffle#(reverse(x)) - Weak TRS: app(add(n,x),y) -> add(n,app(x,y)) app(nil(),y) -> y concat(cons(u,v),y) -> cons(u,concat(v,y)) concat(leaf(),y) -> y minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) reverse(add(n,x)) -> app(reverse(x),add(n,nil())) reverse(nil()) -> nil() - Signature: {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2 ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0 ,c_5/0,c_6/3,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/2,c_13/0,c_14/2,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse# ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: app(add(n,x),y) -> add(n,app(x,y)) app(nil(),y) -> y reverse(add(n,x)) -> app(reverse(x),add(n,nil())) reverse(nil()) -> nil() app#(add(n,x),y) -> c_1(app#(x,y)) reverse#(add(n,x)) -> app#(reverse(x),add(n,nil())) reverse#(add(n,x)) -> reverse#(x) shuffle#(add(n,x)) -> reverse#(x) shuffle#(add(n,x)) -> shuffle#(reverse(x)) **** Step 1.b:5.b:1.b:3: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: app#(add(n,x),y) -> c_1(app#(x,y)) - Weak DPs: reverse#(add(n,x)) -> app#(reverse(x),add(n,nil())) reverse#(add(n,x)) -> reverse#(x) shuffle#(add(n,x)) -> reverse#(x) shuffle#(add(n,x)) -> shuffle#(reverse(x)) - Weak TRS: app(add(n,x),y) -> add(n,app(x,y)) app(nil(),y) -> y reverse(add(n,x)) -> app(reverse(x),add(n,nil())) reverse(nil()) -> nil() - Signature: {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2 ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0 ,c_5/0,c_6/3,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/2,c_13/0,c_14/2,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse# ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_1) = {1} Following symbols are considered usable: {app,reverse,app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#} TcT has computed the following interpretation: p(0) = [1] p(add) = [1] x2 + [2] p(app) = [1] x1 + [1] x2 + [0] p(concat) = [1] x1 + [1] x2 + [2] p(cons) = [2] p(false) = [0] p(leaf) = [1] p(less_leaves) = [1] x2 + [1] p(minus) = [1] x1 + [1] x2 + [0] p(nil) = [0] p(quot) = [2] x2 + [2] p(reverse) = [1] x1 + [0] p(s) = [1] x1 + [0] p(shuffle) = [1] x1 + [0] p(true) = [1] p(app#) = [8] x1 + [0] p(concat#) = [0] p(less_leaves#) = [1] p(minus#) = [1] x2 + [8] p(quot#) = [2] x1 + [0] p(reverse#) = [8] x1 + [6] p(shuffle#) = [8] x1 + [1] p(c_1) = [1] x1 + [12] p(c_2) = [0] p(c_3) = [1] p(c_4) = [0] p(c_5) = [1] p(c_6) = [2] x1 + [1] x2 + [1] p(c_7) = [0] p(c_8) = [0] p(c_9) = [4] x1 + [0] p(c_10) = [1] p(c_11) = [1] x1 + [1] p(c_12) = [1] x1 + [1] x2 + [0] p(c_13) = [1] p(c_14) = [1] x1 + [4] x2 + [0] p(c_15) = [2] Following rules are strictly oriented: app#(add(n,x),y) = [8] x + [16] > [8] x + [12] = c_1(app#(x,y)) Following rules are (at-least) weakly oriented: reverse#(add(n,x)) = [8] x + [22] >= [8] x + [0] = app#(reverse(x),add(n,nil())) reverse#(add(n,x)) = [8] x + [22] >= [8] x + [6] = reverse#(x) shuffle#(add(n,x)) = [8] x + [17] >= [8] x + [6] = reverse#(x) shuffle#(add(n,x)) = [8] x + [17] >= [8] x + [1] = shuffle#(reverse(x)) app(add(n,x),y) = [1] x + [1] y + [2] >= [1] x + [1] y + [2] = add(n,app(x,y)) app(nil(),y) = [1] y + [0] >= [1] y + [0] = y reverse(add(n,x)) = [1] x + [2] >= [1] x + [2] = app(reverse(x),add(n,nil())) reverse(nil()) = [0] >= [0] = nil() **** Step 1.b:5.b:1.b:4: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: app#(add(n,x),y) -> c_1(app#(x,y)) reverse#(add(n,x)) -> app#(reverse(x),add(n,nil())) reverse#(add(n,x)) -> reverse#(x) shuffle#(add(n,x)) -> reverse#(x) shuffle#(add(n,x)) -> shuffle#(reverse(x)) - Weak TRS: app(add(n,x),y) -> add(n,app(x,y)) app(nil(),y) -> y reverse(add(n,x)) -> app(reverse(x),add(n,nil())) reverse(nil()) -> nil() - Signature: {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2 ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0 ,c_5/0,c_6/3,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/2,c_13/0,c_14/2,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse# ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),O(n^3))