WORST_CASE(?,O(n^4))
* Step 1: Desugar WORST_CASE(?,O(n^4))
+ Considered Problem:
let rec leqNat y x =
match y with
| 0 -> True
| S(y') -> (match x with
| S(x') -> leqNat x' y'
| 0 -> False)
;;
let rec eqNat x y =
match y with
| 0 -> (match x with
| 0 -> True
| S(x') -> False)
| S(y') -> (match x with
| S(x') -> eqNat x' y'
| 0 -> False)
;;
let rec geqNat x y =
match y with
| 0 -> True
| S(y') -> (match x with
| 0 -> False
| S(x') -> geqNat x' y')
;;
let rec ltNat x y =
match y with
| 0 -> False
| S(y') -> (match x with
| 0 -> True
| S(x') -> ltNat x' y')
;;
let rec gtNat x y =
match x with
| 0 -> False
| S(x') -> (match y with
| 0 -> True
| S(y') -> gtNat x' y')
;;
let ifz n th el = match n with
| 0 -> th 0
| S(x) -> el x
;;
let ite b th el = match b with
| True()-> th
| False()-> el
;;
let minus n m =
let rec minus' m n = match m with
| 0 -> 0
| S(x) -> (match n with
| 0 -> m
| S(y) -> minus' x y)
in Pair(minus' n m,m)
;;
let rec plus n m = match m with
| 0 -> n
| S(x) -> S(plus n x)
;;
type ('a,'b,'c) triple = Triple of 'a * 'b * 'c
;;
let rec div_mod n m = match (minus n m) with
| Pair(res,m) -> (match res with
| 0 -> Triple (0,n,m)
| S(x) -> (match (div_mod res m) with
| Triple(a,rest,unusedM) -> Triple(plus S(0) a,rest,m)))
;;
let rec mult n m = match n with
| 0 -> 0
| S(x) -> S(plus (mult x m) m)
;;
type bool = True | False
;;
type 'a list = Nil | Cons of 'a * 'a list
;;
type nat = 0 | S of nat
;;
type Unit = Unit
;;
type ('a,'b) pair = Pair of 'a * 'b
(* * * * * * * * * * *
* Resource Aware ML *
* * * * * * * * * * *
*
* * * Use Cases * *
*
* File:
* example/list_map.raml
*
* Author:
* Jan Hoffmann, Shu-Chun Weng (S(S(0))014)
*
* Description:
* Some variations of list map.
*
*)
;;
(* The usual list map function. *)
let rec map f l =
match l with
| Nil()-> Nil
| Cons(x,xs) ->
let ys = map f xs in
Cons(f x,ys)
;;
(* The usual list rev_map function. *)
let map_rev f l =
let rec rmap l acc =
match l with
| Nil()-> acc
| Cons(x,xs) -> let acc' = Cons(f x,acc) in
rmap xs acc'
in rmap l Nil
;;
(* Iteratively apply two functional arguments. *)
let map_rev2 f1 f2 l =
let rec rmap1 l acc =
match l with
| Nil()-> acc
| Cons(x,xs) -> rmap2 xs (Cons(f1 x,acc))
and rmap2 l acc =
match l with
| Nil()-> acc
| Cons(x,xs) -> rmap1 xs (Cons(f2 x,acc))
in
rmap1 l Nil
;;
let main xs =
let f x = mult x (mult x x) in
let g x = plus x S(S(0)) in
let h x = x in
let map_f_g = map_rev2 f g in
map h (map_rev h (map_f_g xs))
;;
+ Applied Processor:
Desugar {analysedFunction = Nothing}
+ Details:
none
* Step 2: Defunctionalization WORST_CASE(?,O(n^4))
+ Considered Problem:
[1mλxs[0m : nat list.
([1mλplus[0m : nat -> nat -> nat.
([1mλmult[0m : nat -> nat -> nat.
([1mλmap[0m : (nat -> nat) -> nat list -> nat list.
([1mλmap_rev[0m : (nat -> nat) -> nat list -> nat list.
([1mλmap_rev2[0m : (nat -> nat) -> (nat -> nat) -> nat list -> nat list.
([1mλmain[0m : nat list -> nat list. [4mmain[0m [4mxs[0m)
([1mλxs[0m : nat list.
([1mλf[0m : nat -> nat.
([1mλg[0m : nat -> nat.
([1mλh[0m : nat -> nat.
([1mλmap_f_g[0m : nat list -> nat list. [4mmap[0m [4mh[0m ([4mmap_rev[0m [4mh[0m ([4mmap_f_g[0m [4mxs[0m))) ([4mmap_rev2[0m [4mf[0m [4mg[0m))
([1mλx[0m : nat. [4mx[0m))
([1mλx[0m : nat. [4mplus[0m [4mx[0m S(S(0))))
([1mλx[0m : nat. [4mmult[0m [4mx[0m ([4mmult[0m [4mx[0m [4mx[0m))))
([1mλf1[0m : nat -> nat.
[1mλf2[0m : nat -> nat.
[1mλl[0m : nat list.
([1mλrmap1[0m : nat list -> nat list -> nat list.
[1mλrmap2[0m : nat list -> nat list -> nat list. [4mrmap1[0m [4ml[0m Nil)
([1mμ0[0m [[1mλrmap1[0m : nat list -> nat list -> nat list.
[1mλrmap2[0m : nat list -> nat list -> nat list.
[1mλl[0m : nat list.
[1mλacc[0m : nat list.
[1mcase[0m [4ml[0m [1mof[0m
| Nil -> [4macc[0m
| Cons -> [1mλx[0m : nat.
[1mλxs[0m : nat list. [4mrmap2[0m [4mxs[0m Cons([4mf1[0m [4mx[0m,[4macc[0m)
,[1mλrmap1[0m : nat list -> nat list -> nat list.
[1mλrmap2[0m : nat list -> nat list -> nat list.
[1mλl[0m : nat list.
[1mλacc[0m : nat list.
[1mcase[0m [4ml[0m [1mof[0m
| Nil -> [4macc[0m
| Cons -> [1mλx[0m : nat.
[1mλxs[0m : nat list. [4mrmap1[0m [4mxs[0m Cons([4mf2[0m [4mx[0m,[4macc[0m)])
([1mμ1[0m [[1mλrmap1[0m : nat list -> nat list -> nat list.
[1mλrmap2[0m : nat list -> nat list -> nat list.
[1mλl[0m : nat list.
[1mλacc[0m : nat list.
[1mcase[0m [4ml[0m [1mof[0m
| Nil -> [4macc[0m
| Cons -> [1mλx[0m : nat.
[1mλxs[0m : nat list. [4mrmap2[0m [4mxs[0m Cons([4mf1[0m [4mx[0m,[4macc[0m)
,[1mλrmap1[0m : nat list -> nat list -> nat list.
[1mλrmap2[0m : nat list -> nat list -> nat list.
[1mλl[0m : nat list.
[1mλacc[0m : nat list.
[1mcase[0m [4ml[0m [1mof[0m
| Nil -> [4macc[0m
| Cons -> [1mλx[0m : nat.
[1mλxs[0m : nat list. [4mrmap1[0m [4mxs[0m Cons([4mf2[0m [4mx[0m,[4macc[0m)])))
([1mλf[0m : nat -> nat.
[1mλl[0m : nat list.
([1mλrmap[0m : nat list -> nat list -> nat list. [4mrmap[0m [4ml[0m Nil)
([1mμrmap[0m : nat list -> nat list -> nat list.
[1mλl[0m : nat list.
[1mλacc[0m : nat list.
[1mcase[0m [4ml[0m [1mof[0m
| Nil -> [4macc[0m
| Cons -> [1mλx[0m : nat.
[1mλxs[0m : nat list. ([1mλacc'[0m : nat list. [4mrmap[0m [4mxs[0m [4macc'[0m) Cons([4mf[0m [4mx[0m,[4macc[0m))))
([1mμmap[0m : (nat -> nat) -> nat list -> nat list.
[1mλf[0m : nat -> nat.
[1mλl[0m : nat list.
[1mcase[0m [4ml[0m [1mof[0m
| Nil -> Nil
| Cons -> [1mλx[0m : nat.
[1mλxs[0m : nat list. ([1mλys[0m : nat list. Cons([4mf[0m [4mx[0m,[4mys[0m)) ([4mmap[0m [4mf[0m [4mxs[0m)))
([1mμmult[0m : nat -> nat -> nat.
[1mλn[0m : nat. [1mλm[0m : nat. [1mcase[0m [4mn[0m [1mof[0m | 0 -> 0 | S -> [1mλx[0m : nat. S([4mplus[0m ([4mmult[0m [4mx[0m [4mm[0m) [4mm[0m)))
([1mμplus[0m : nat -> nat -> nat. [1mλn[0m : nat. [1mλm[0m : nat. [1mcase[0m [4mm[0m [1mof[0m | 0 -> [4mn[0m | S -> [1mλx[0m : nat. S([4mplus[0m [4mn[0m [4mx[0m)) : nat list
-> nat list
where
0 :: nat
Cons :: 'a -> 'a list -> 'a list
False :: bool
Nil :: 'a list
Pair :: 'a -> 'b -> ('a,'b) pair
S :: nat -> nat
Triple :: 'a -> 'b -> 'c -> ('a,'b,'c) triple
True :: bool
Unit :: Unit
+ Applied Processor:
Defunctionalization
+ Details:
none
* Step 3: Inline WORST_CASE(?,O(n^4))
+ Considered Problem:
1: main_6(x0) x6 -> x6 x0
2: main_xs_3(x3, x4, x6, x9) x10 -> x3 x9 (x4 x9 (x10 x6))
3: main_xs_2(x3, x4, x5, x6, x7, x8) x9 -> main_xs_3(x3, x4, x6, x9) (x5 x7 x8)
4: h() x9 -> x9
5: main_xs_1(x3, x4, x5, x6, x7) x8 -> main_xs_2(x3, x4, x5, x6, x7, x8) h()
6: g(x1) x8 -> x1 x8 S(S(0()))
7: main_xs(x1, x3, x4, x5, x6) x7 -> main_xs_1(x3, x4, x5, x6, x7) g(x1)
8: f(x2) x7 -> x2 x7 (x2 x7 x7)
9: main_7(x1, x2, x3, x4, x5) x6 -> main_xs(x1, x3, x4, x5, x6) f(x2)
10: main_5(x0, x1, x2, x3, x4) x5 -> main_6(x0) main_7(x1, x2, x3, x4, x5)
11: map_rev2_f1_f2_l_1(x7, x8) x9 -> x8 x7 Nil()
12: map_rev2_f1_f2_l(x7) x8 -> map_rev2_f1_f2_l_1(x7, x8)
13: cond_rmap2_l_acc(Nil(), x5, x6, x13) -> x13
14: cond_rmap2_l_acc(Cons(x14, x15), x5, x6, x13) -> rmap1(x5, x6) x15 Cons(x6 x14, x13)
15: rmap2_l(x5, x6, x12) x13 -> cond_rmap2_l_acc(x12, x5, x6, x13)
16: rmap2_2(x5, x6) x12 -> rmap2_l(x5, x6, x12)
17: rmap2(x5, x6) x7 -> rmap2_2(x5, x6) x7
18: cond_rmap1_l_acc(Nil(), x5, x6, x9) -> x9
19: cond_rmap1_l_acc(Cons(x10, x11), x5, x6, x9) -> rmap2(x5, x6) x11 Cons(x5 x10, x9)
20: rmap1_l(x5, x6, x8) x9 -> cond_rmap1_l_acc(x8, x5, x6, x9)
21: rmap1_2(x5, x6) x8 -> rmap1_l(x5, x6, x8)
22: rmap1(x5, x6) x7 -> rmap1_2(x5, x6) x7
23: cond_rmap1_l_acc_1(Nil(), x5, x6, x13) -> x13
24: cond_rmap1_l_acc_1(Cons(x14, x15), x5, x6, x13) -> rmap2_3(x5, x6) x15 Cons(x5 x14, x13)
25: rmap1_l_1(x5, x6, x12) x13 -> cond_rmap1_l_acc_1(x12, x5, x6, x13)
26: rmap1_5(x5, x6) x12 -> rmap1_l_1(x5, x6, x12)
27: rmap1_3(x5, x6) x7 -> rmap1_5(x5, x6) x7
28: cond_rmap2_l_acc_1(Nil(), x5, x6, x9) -> x9
29: cond_rmap2_l_acc_1(Cons(x10, x11), x5, x6, x9) -> rmap1_3(x5, x6) x11 Cons(x6 x10, x9)
30: rmap2_l_1(x5, x6, x8) x9 -> cond_rmap2_l_acc_1(x8, x5, x6, x9)
31: rmap2_5(x5, x6) x8 -> rmap2_l_1(x5, x6, x8)
32: rmap2_3(x5, x6) x7 -> rmap2_5(x5, x6) x7
33: map_rev2_f1_f2(x5, x6) x7 -> map_rev2_f1_f2_l(x7) rmap1(x5, x6) rmap2_3(x5, x6)
34: map_rev2_f1(x5) x6 -> map_rev2_f1_f2(x5, x6)
35: map_rev2() x5 -> map_rev2_f1(x5)
36: main_4(x0, x1, x2, x3) x4 -> main_5(x0, x1, x2, x3, x4) map_rev2()
37: map_rev_f_l(x5) x6 -> x6 x5 Nil()
38: rmap_l_acc_2(x4, x9) x10 -> rmap(x4) x9 x10
39: cond_rmap_l_acc(Nil(), x4, x7) -> x7
40: cond_rmap_l_acc(Cons(x8, x9), x4, x7) -> rmap_l_acc_2(x4, x9) Cons(x4 x8, x7)
41: rmap_l(x4, x6) x7 -> cond_rmap_l_acc(x6, x4, x7)
42: rmap_1(x4) x6 -> rmap_l(x4, x6)
43: rmap(x4) x5 -> rmap_1(x4) x5
44: map_rev_f(x4) x5 -> map_rev_f_l(x5) rmap(x4)
45: map_rev() x4 -> map_rev_f(x4)
46: main_3(x0, x1, x2) x3 -> main_4(x0, x1, x2, x3) map_rev()
47: map_f_l_2(x3, x5) x7 -> Cons(x3 x5, x7)
48: cond_map_f_l(Nil(), x3) -> Nil()
49: cond_map_f_l(Cons(x5, x6), x3) -> map_f_l_2(x3, x5) (map() x3 x6)
50: map_f(x3) x4 -> cond_map_f_l(x4, x3)
51: map_1() x3 -> map_f(x3)
52: map() x0 -> map_1() x0
53: main_2(x0, x1) x2 -> main_3(x0, x1, x2) map()
54: cond_mult_n_m(0(), x1, x3) -> 0()
55: cond_mult_n_m(S(x4), x1, x3) -> S(x1 (mult(x1) x4 x3) x3)
56: mult_n(x1, x2) x3 -> cond_mult_n_m(x2, x1, x3)
57: mult_1(x1) x2 -> mult_n(x1, x2)
58: mult(x1) x2 -> mult_1(x1) x2
59: main_1(x0) x1 -> main_2(x0, x1) mult(x1)
60: cond_plus_n_m(0(), x1) -> x1
61: cond_plus_n_m(S(x3), x1) -> S(plus() x1 x3)
62: plus_n(x1) x2 -> cond_plus_n_m(x2, x1)
63: plus_1() x1 -> plus_n(x1)
64: plus() x0 -> plus_1() x0
65: main(x0) -> main_1(x0) plus()
where
0 :: nat
Cons :: 'a -> 'a list -> 'a list
Nil :: 'a list
S :: nat -> nat
f :: (nat -> nat -> nat) -> nat -> nat
map_f :: (nat -> nat) -> nat list -> nat list
map_rev_f :: (nat -> nat) -> nat list -> nat list
map_rev2_f1 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list
map_rev2_f1_f2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list
g :: (nat -> nat -> nat) -> nat -> nat
h :: nat -> nat
map_rev_f_l :: nat list -> (nat list -> nat list -> nat list) -> nat list
map_rev2_f1_f2_l :: nat list
-> (nat list -> nat list -> nat list)
-> (nat list -> nat list -> nat list)
-> nat list
rmap_l :: (nat -> nat) -> nat list -> nat list -> nat list
rmap1_l :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2_l :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
map_rev :: (nat -> nat) -> nat list -> nat list
map_rev2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list
mult_n :: (nat -> nat -> nat) -> nat -> nat -> nat
plus_n :: nat -> nat -> nat
main_xs :: (nat -> nat -> nat)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> (nat -> nat) -> nat list -> nat list)
-> nat list
-> (nat -> nat)
-> nat list
map_rev2_f1_f2_l_1 :: nat list
-> (nat list -> nat list -> nat list)
-> (nat list -> nat list -> nat list)
-> nat list
rmap1_l_1 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2_l_1 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
main_1 :: nat list -> (nat -> nat -> nat) -> nat list
map_1 :: (nat -> nat) -> nat list -> nat list
mult_1 :: (nat -> nat -> nat) -> nat -> nat -> nat
plus_1 :: nat -> nat -> nat
rmap_1 :: (nat -> nat) -> nat list -> nat list -> nat list
main_xs_1 :: ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> (nat -> nat) -> nat list -> nat list)
-> nat list
-> (nat -> nat)
-> (nat -> nat)
-> nat list
rmap_l_acc_2 :: (nat -> nat) -> nat list -> nat list -> nat list
map_f_l_2 :: (nat -> nat) -> nat -> nat list -> nat list
main_2 :: nat list -> (nat -> nat -> nat) -> (nat -> nat -> nat) -> nat list
rmap1_2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2_2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
main_xs_2 :: ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> (nat -> nat) -> nat list -> nat list)
-> nat list
-> (nat -> nat)
-> (nat -> nat)
-> (nat -> nat)
-> nat list
main_3 :: nat list
-> (nat -> nat -> nat)
-> (nat -> nat -> nat)
-> ((nat -> nat) -> nat list -> nat list)
-> nat list
main_xs_3 :: ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> nat list
-> (nat -> nat)
-> (nat list -> nat list)
-> nat list
main_4 :: nat list
-> (nat -> nat -> nat)
-> (nat -> nat -> nat)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> nat list
main_5 :: nat list
-> (nat -> nat -> nat)
-> (nat -> nat -> nat)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> (nat -> nat) -> nat list -> nat list)
-> nat list
rmap1_5 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2_5 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
main_6 :: nat list -> (nat list -> nat list) -> nat list
main_7 :: (nat -> nat -> nat)
-> (nat -> nat -> nat)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> (nat -> nat) -> nat list -> nat list)
-> nat list
-> nat list
cond_rmap_l_acc :: nat list -> (nat -> nat) -> nat list -> nat list
cond_rmap1_l_acc :: nat list -> (nat -> nat) -> (nat -> nat) -> nat list -> nat list
cond_rmap2_l_acc :: nat list -> (nat -> nat) -> (nat -> nat) -> nat list -> nat list
cond_map_f_l :: nat list -> (nat -> nat) -> nat list
cond_mult_n_m :: nat -> (nat -> nat -> nat) -> nat -> nat
cond_plus_n_m :: nat -> nat -> nat
cond_rmap1_l_acc_1 :: nat list -> (nat -> nat) -> (nat -> nat) -> nat list -> nat list
cond_rmap2_l_acc_1 :: nat list -> (nat -> nat) -> (nat -> nat) -> nat list -> nat list
map :: (nat -> nat) -> nat list -> nat list
mult :: (nat -> nat -> nat) -> nat -> nat -> nat
plus :: nat -> nat -> nat
rmap :: (nat -> nat) -> nat list -> nat list -> nat list
rmap1 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap1_3 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2_3 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
main :: nat list -> nat list
+ Applied Processor:
Inline {inlineType = InlineRewrite, inlineSelect = <inline non-recursive>}
+ Details:
none
* Step 4: Inline WORST_CASE(?,O(n^4))
+ Considered Problem:
1: main_6(x0) x6 -> x6 x0
2: main_xs_3(x3, x4, x6, x9) x10 -> x3 x9 (x4 x9 (x10 x6))
3: main_xs_2(x6, x8, x11, x12, x15, x17) x18 -> x6 x18 (x8 x18 (x11 x15 x17 x12))
4: h() x9 -> x9
5: main_xs_1(x6, x8, x10, x12, x14) x16 -> main_xs_3(x6, x8, x12, h()) (x10 x14 x16)
6: g(x1) x8 -> x1 x8 S(S(0()))
7: main_xs(x3, x6, x8, x10, x12) x14 -> main_xs_2(x6, x8, x10, x12, x14, g(x3)) h()
8: f(x2) x7 -> x2 x7 (x2 x7 x7)
9: main_7(x2, x5, x6, x8, x10) x12 -> main_xs_1(x6, x8, x10, x12, f(x5)) g(x2)
10: main_5(x0, x3, x5, x7, x9) x11 -> main_7(x3, x5, x7, x9, x11) x0
11: map_rev2_f1_f2_l_1(x7, x8) x9 -> x8 x7 Nil()
12: map_rev2_f1_f2_l(x7) x8 -> map_rev2_f1_f2_l_1(x7, x8)
13: cond_rmap2_l_acc(Nil(), x5, x6, x13) -> x13
14: cond_rmap2_l_acc(Cons(x14, x15), x5, x6, x13) -> rmap1(x5, x6) x15 Cons(x6 x14, x13)
15: rmap2_l(x5, x6, x12) x13 -> cond_rmap2_l_acc(x12, x5, x6, x13)
16: rmap2_2(x5, x6) x12 -> rmap2_l(x5, x6, x12)
17: rmap2(x10, x12) x24 -> rmap2_l(x10, x12, x24)
18: cond_rmap1_l_acc(Nil(), x5, x6, x9) -> x9
19: cond_rmap1_l_acc(Cons(x10, x11), x5, x6, x9) -> rmap2(x5, x6) x11 Cons(x5 x10, x9)
20: rmap1_l(x5, x6, x8) x9 -> cond_rmap1_l_acc(x8, x5, x6, x9)
21: rmap1_2(x5, x6) x8 -> rmap1_l(x5, x6, x8)
22: rmap1(x10, x12) x16 -> rmap1_l(x10, x12, x16)
23: cond_rmap1_l_acc_1(Nil(), x5, x6, x13) -> x13
24: cond_rmap1_l_acc_1(Cons(x14, x15), x5, x6, x13) -> rmap2_3(x5, x6) x15 Cons(x5 x14, x13)
25: rmap1_l_1(x5, x6, x12) x13 -> cond_rmap1_l_acc_1(x12, x5, x6, x13)
26: rmap1_5(x5, x6) x12 -> rmap1_l_1(x5, x6, x12)
27: rmap1_3(x10, x12) x24 -> rmap1_l_1(x10, x12, x24)
28: cond_rmap2_l_acc_1(Nil(), x5, x6, x9) -> x9
29: cond_rmap2_l_acc_1(Cons(x10, x11), x5, x6, x9) -> rmap1_3(x5, x6) x11 Cons(x6 x10, x9)
30: rmap2_l_1(x5, x6, x8) x9 -> cond_rmap2_l_acc_1(x8, x5, x6, x9)
31: rmap2_5(x5, x6) x8 -> rmap2_l_1(x5, x6, x8)
32: rmap2_3(x10, x12) x16 -> rmap2_l_1(x10, x12, x16)
33: map_rev2_f1_f2(x11, x13) x14 -> map_rev2_f1_f2_l_1(x14, rmap1(x11, x13)) rmap2_3(x11, x13)
34: map_rev2_f1(x5) x6 -> map_rev2_f1_f2(x5, x6)
35: map_rev2() x5 -> map_rev2_f1(x5)
36: main_4(x0, x2, x4, x6) x8 -> main_6(x0) main_7(x2, x4, x6, x8, map_rev2())
37: map_rev_f_l(x5) x6 -> x6 x5 Nil()
38: rmap_l_acc_2(x4, x9) x10 -> rmap(x4) x9 x10
39: cond_rmap_l_acc(Nil(), x4, x7) -> x7
40: cond_rmap_l_acc(Cons(x17, x18), x8, x15) -> rmap(x8) x18 Cons(x8 x17, x15)
41: rmap_l(x4, x6) x7 -> cond_rmap_l_acc(x6, x4, x7)
42: rmap_1(x4) x6 -> rmap_l(x4, x6)
43: rmap(x8) x12 -> rmap_l(x8, x12)
44: map_rev_f(x9) x10 -> rmap(x9) x10 Nil()
45: map_rev() x4 -> map_rev_f(x4)
46: main_3(x0, x2, x4) x6 -> main_5(x0, x2, x4, x6, map_rev()) map_rev2()
47: map_f_l_2(x3, x5) x7 -> Cons(x3 x5, x7)
48: cond_map_f_l(Nil(), x3) -> Nil()
49: cond_map_f_l(Cons(x10, x13), x6) -> Cons(x6 x10, map() x6 x13)
50: map_f(x3) x4 -> cond_map_f_l(x4, x3)
51: map_1() x3 -> map_f(x3)
52: map() x6 -> map_f(x6)
53: main_2(x0, x2) x4 -> main_4(x0, x2, x4, map()) map_rev()
54: cond_mult_n_m(0(), x1, x3) -> 0()
55: cond_mult_n_m(S(x4), x1, x3) -> S(x1 (mult(x1) x4 x3) x3)
56: mult_n(x1, x2) x3 -> cond_mult_n_m(x2, x1, x3)
57: mult_1(x1) x2 -> mult_n(x1, x2)
58: mult(x2) x4 -> mult_n(x2, x4)
59: main_1(x0) x2 -> main_3(x0, x2, mult(x2)) map()
60: cond_plus_n_m(0(), x1) -> x1
61: cond_plus_n_m(S(x3), x1) -> S(plus() x1 x3)
62: plus_n(x1) x2 -> cond_plus_n_m(x2, x1)
63: plus_1() x1 -> plus_n(x1)
64: plus() x2 -> plus_n(x2)
65: main(x0) -> main_2(x0, plus()) mult(plus())
where
0 :: nat
Cons :: 'a -> 'a list -> 'a list
Nil :: 'a list
S :: nat -> nat
f :: (nat -> nat -> nat) -> nat -> nat
map_f :: (nat -> nat) -> nat list -> nat list
map_rev_f :: (nat -> nat) -> nat list -> nat list
map_rev2_f1 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list
map_rev2_f1_f2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list
g :: (nat -> nat -> nat) -> nat -> nat
h :: nat -> nat
map_rev_f_l :: nat list -> (nat list -> nat list -> nat list) -> nat list
map_rev2_f1_f2_l :: nat list
-> (nat list -> nat list -> nat list)
-> (nat list -> nat list -> nat list)
-> nat list
rmap_l :: (nat -> nat) -> nat list -> nat list -> nat list
rmap1_l :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2_l :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
map_rev :: (nat -> nat) -> nat list -> nat list
map_rev2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list
mult_n :: (nat -> nat -> nat) -> nat -> nat -> nat
plus_n :: nat -> nat -> nat
main_xs :: (nat -> nat -> nat)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> (nat -> nat) -> nat list -> nat list)
-> nat list
-> (nat -> nat)
-> nat list
map_rev2_f1_f2_l_1 :: nat list
-> (nat list -> nat list -> nat list)
-> (nat list -> nat list -> nat list)
-> nat list
rmap1_l_1 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2_l_1 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
main_1 :: nat list -> (nat -> nat -> nat) -> nat list
map_1 :: (nat -> nat) -> nat list -> nat list
mult_1 :: (nat -> nat -> nat) -> nat -> nat -> nat
plus_1 :: nat -> nat -> nat
rmap_1 :: (nat -> nat) -> nat list -> nat list -> nat list
main_xs_1 :: ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> (nat -> nat) -> nat list -> nat list)
-> nat list
-> (nat -> nat)
-> (nat -> nat)
-> nat list
rmap_l_acc_2 :: (nat -> nat) -> nat list -> nat list -> nat list
map_f_l_2 :: (nat -> nat) -> nat -> nat list -> nat list
main_2 :: nat list -> (nat -> nat -> nat) -> (nat -> nat -> nat) -> nat list
rmap1_2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2_2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
main_xs_2 :: ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> (nat -> nat) -> nat list -> nat list)
-> nat list
-> (nat -> nat)
-> (nat -> nat)
-> (nat -> nat)
-> nat list
main_3 :: nat list
-> (nat -> nat -> nat)
-> (nat -> nat -> nat)
-> ((nat -> nat) -> nat list -> nat list)
-> nat list
main_xs_3 :: ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> nat list
-> (nat -> nat)
-> (nat list -> nat list)
-> nat list
main_4 :: nat list
-> (nat -> nat -> nat)
-> (nat -> nat -> nat)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> nat list
main_5 :: nat list
-> (nat -> nat -> nat)
-> (nat -> nat -> nat)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> (nat -> nat) -> nat list -> nat list)
-> nat list
rmap1_5 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2_5 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
main_6 :: nat list -> (nat list -> nat list) -> nat list
main_7 :: (nat -> nat -> nat)
-> (nat -> nat -> nat)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> (nat -> nat) -> nat list -> nat list)
-> nat list
-> nat list
cond_rmap_l_acc :: nat list -> (nat -> nat) -> nat list -> nat list
cond_rmap1_l_acc :: nat list -> (nat -> nat) -> (nat -> nat) -> nat list -> nat list
cond_rmap2_l_acc :: nat list -> (nat -> nat) -> (nat -> nat) -> nat list -> nat list
cond_map_f_l :: nat list -> (nat -> nat) -> nat list
cond_mult_n_m :: nat -> (nat -> nat -> nat) -> nat -> nat
cond_plus_n_m :: nat -> nat -> nat
cond_rmap1_l_acc_1 :: nat list -> (nat -> nat) -> (nat -> nat) -> nat list -> nat list
cond_rmap2_l_acc_1 :: nat list -> (nat -> nat) -> (nat -> nat) -> nat list -> nat list
map :: (nat -> nat) -> nat list -> nat list
mult :: (nat -> nat -> nat) -> nat -> nat -> nat
plus :: nat -> nat -> nat
rmap :: (nat -> nat) -> nat list -> nat list -> nat list
rmap1 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap1_3 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2_3 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
main :: nat list -> nat list
+ Applied Processor:
Inline {inlineType = InlineRewrite, inlineSelect = <inline non-recursive>}
+ Details:
none
* Step 5: Inline WORST_CASE(?,O(n^4))
+ Considered Problem:
1: main_6(x0) x6 -> x6 x0
2: main_xs_3(x3, x4, x6, x9) x10 -> x3 x9 (x4 x9 (x10 x6))
3: main_xs_2(x6, x8, x11, x12, x15, x17) x18 -> x6 x18 (x8 x18 (x11 x15 x17 x12))
4: h() x9 -> x9
5: main_xs_1(x6, x8, x21, x12, x29) x33 -> x6 h() (x8 h() (x21 x29 x33 x12))
6: g(x1) x8 -> x1 x8 S(S(0()))
7: main_xs(x7, x12, x16, x22, x24) x30 -> x12 h() (x16 h() (x22 x30 g(x7) x24))
8: f(x2) x7 -> x2 x7 (x2 x7 x7)
9: main_7(x5, x11, x12, x16, x20) x24 -> main_xs_3(x12, x16, x24, h()) (x20 f(x11) g(x5))
10: main_5(x24, x4, x10, x12, x16) x20 -> main_xs_1(x12, x16, x20, x24, f(x10)) g(x4)
11: map_rev2_f1_f2_l_1(x7, x8) x9 -> x8 x7 Nil()
12: map_rev2_f1_f2_l(x7) x8 -> map_rev2_f1_f2_l_1(x7, x8)
13: cond_rmap2_l_acc(Nil(), x5, x6, x13) -> x13
14: cond_rmap2_l_acc(Cons(x14, x15), x5, x6, x13) -> rmap1(x5, x6) x15 Cons(x6 x14, x13)
15: rmap2_l(x5, x6, x12) x13 -> cond_rmap2_l_acc(x12, x5, x6, x13)
16: rmap2_2(x5, x6) x12 -> rmap2_l(x5, x6, x12)
17: rmap2(x10, x12) x24 -> rmap2_l(x10, x12, x24)
18: cond_rmap1_l_acc(Nil(), x5, x6, x9) -> x9
19: cond_rmap1_l_acc(Cons(x10, x11), x5, x6, x9) -> rmap2(x5, x6) x11 Cons(x5 x10, x9)
20: rmap1_l(x5, x6, x8) x9 -> cond_rmap1_l_acc(x8, x5, x6, x9)
21: rmap1_2(x5, x6) x8 -> rmap1_l(x5, x6, x8)
22: rmap1(x10, x12) x16 -> rmap1_l(x10, x12, x16)
23: cond_rmap1_l_acc_1(Nil(), x5, x6, x13) -> x13
24: cond_rmap1_l_acc_1(Cons(x14, x15), x5, x6, x13) -> rmap2_3(x5, x6) x15 Cons(x5 x14, x13)
25: rmap1_l_1(x5, x6, x12) x13 -> cond_rmap1_l_acc_1(x12, x5, x6, x13)
26: rmap1_5(x5, x6) x12 -> rmap1_l_1(x5, x6, x12)
27: rmap1_3(x10, x12) x24 -> rmap1_l_1(x10, x12, x24)
28: cond_rmap2_l_acc_1(Nil(), x5, x6, x9) -> x9
29: cond_rmap2_l_acc_1(Cons(x10, x11), x5, x6, x9) -> rmap1_3(x5, x6) x11 Cons(x6 x10, x9)
30: rmap2_l_1(x5, x6, x8) x9 -> cond_rmap2_l_acc_1(x8, x5, x6, x9)
31: rmap2_5(x5, x6) x8 -> rmap2_l_1(x5, x6, x8)
32: rmap2_3(x10, x12) x16 -> rmap2_l_1(x10, x12, x16)
33: map_rev2_f1_f2(x23, x27) x14 -> rmap1(x23, x27) x14 Nil()
34: map_rev2_f1(x5) x6 -> map_rev2_f1_f2(x5, x6)
35: map_rev2() x5 -> map_rev2_f1(x5)
36: main_4(x0, x5, x9, x13) x17 -> main_7(x5, x9, x13, x17, map_rev2()) x0
37: map_rev_f_l(x5) x6 -> x6 x5 Nil()
38: rmap_l_acc_2(x4, x9) x10 -> rmap(x4) x9 x10
39: cond_rmap_l_acc(Nil(), x4, x7) -> x7
40: cond_rmap_l_acc(Cons(x17, x18), x8, x15) -> rmap(x8) x18 Cons(x8 x17, x15)
41: rmap_l(x4, x6) x7 -> cond_rmap_l_acc(x6, x4, x7)
42: rmap_1(x4) x6 -> rmap_l(x4, x6)
43: rmap(x8) x12 -> rmap_l(x8, x12)
44: map_rev_f(x9) x10 -> rmap(x9) x10 Nil()
45: map_rev() x4 -> map_rev_f(x4)
46: main_3(x0, x6, x10) x14 -> main_7(x6, x10, x14, map_rev(), map_rev2()) x0
47: map_f_l_2(x3, x5) x7 -> Cons(x3 x5, x7)
48: cond_map_f_l(Nil(), x3) -> Nil()
49: cond_map_f_l(Cons(x10, x13), x6) -> Cons(x6 x10, map() x6 x13)
50: map_f(x3) x4 -> cond_map_f_l(x4, x3)
51: map_1() x3 -> map_f(x3)
52: map() x6 -> map_f(x6)
53: main_2(x0, x4) x8 -> main_6(x0) main_7(x4, x8, map(), map_rev(), map_rev2())
54: cond_mult_n_m(0(), x1, x3) -> 0()
55: cond_mult_n_m(S(x4), x1, x3) -> S(x1 (mult(x1) x4 x3) x3)
56: mult_n(x1, x2) x3 -> cond_mult_n_m(x2, x1, x3)
57: mult_1(x1) x2 -> mult_n(x1, x2)
58: mult(x2) x4 -> mult_n(x2, x4)
59: main_1(x0) x4 -> main_5(x0, x4, mult(x4), map(), map_rev()) map_rev2()
60: cond_plus_n_m(0(), x1) -> x1
61: cond_plus_n_m(S(x3), x1) -> S(plus() x1 x3)
62: plus_n(x1) x2 -> cond_plus_n_m(x2, x1)
63: plus_1() x1 -> plus_n(x1)
64: plus() x2 -> plus_n(x2)
65: main(x0) -> main_4(x0, plus(), mult(plus()), map()) map_rev()
where
0 :: nat
Cons :: 'a -> 'a list -> 'a list
Nil :: 'a list
S :: nat -> nat
f :: (nat -> nat -> nat) -> nat -> nat
map_f :: (nat -> nat) -> nat list -> nat list
map_rev_f :: (nat -> nat) -> nat list -> nat list
map_rev2_f1 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list
map_rev2_f1_f2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list
g :: (nat -> nat -> nat) -> nat -> nat
h :: nat -> nat
map_rev_f_l :: nat list -> (nat list -> nat list -> nat list) -> nat list
map_rev2_f1_f2_l :: nat list
-> (nat list -> nat list -> nat list)
-> (nat list -> nat list -> nat list)
-> nat list
rmap_l :: (nat -> nat) -> nat list -> nat list -> nat list
rmap1_l :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2_l :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
map_rev :: (nat -> nat) -> nat list -> nat list
map_rev2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list
mult_n :: (nat -> nat -> nat) -> nat -> nat -> nat
plus_n :: nat -> nat -> nat
main_xs :: (nat -> nat -> nat)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> (nat -> nat) -> nat list -> nat list)
-> nat list
-> (nat -> nat)
-> nat list
map_rev2_f1_f2_l_1 :: nat list
-> (nat list -> nat list -> nat list)
-> (nat list -> nat list -> nat list)
-> nat list
rmap1_l_1 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2_l_1 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
main_1 :: nat list -> (nat -> nat -> nat) -> nat list
map_1 :: (nat -> nat) -> nat list -> nat list
mult_1 :: (nat -> nat -> nat) -> nat -> nat -> nat
plus_1 :: nat -> nat -> nat
rmap_1 :: (nat -> nat) -> nat list -> nat list -> nat list
main_xs_1 :: ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> (nat -> nat) -> nat list -> nat list)
-> nat list
-> (nat -> nat)
-> (nat -> nat)
-> nat list
rmap_l_acc_2 :: (nat -> nat) -> nat list -> nat list -> nat list
map_f_l_2 :: (nat -> nat) -> nat -> nat list -> nat list
main_2 :: nat list -> (nat -> nat -> nat) -> (nat -> nat -> nat) -> nat list
rmap1_2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2_2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
main_xs_2 :: ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> (nat -> nat) -> nat list -> nat list)
-> nat list
-> (nat -> nat)
-> (nat -> nat)
-> (nat -> nat)
-> nat list
main_3 :: nat list
-> (nat -> nat -> nat)
-> (nat -> nat -> nat)
-> ((nat -> nat) -> nat list -> nat list)
-> nat list
main_xs_3 :: ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> nat list
-> (nat -> nat)
-> (nat list -> nat list)
-> nat list
main_4 :: nat list
-> (nat -> nat -> nat)
-> (nat -> nat -> nat)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> nat list
main_5 :: nat list
-> (nat -> nat -> nat)
-> (nat -> nat -> nat)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> (nat -> nat) -> nat list -> nat list)
-> nat list
rmap1_5 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2_5 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
main_6 :: nat list -> (nat list -> nat list) -> nat list
main_7 :: (nat -> nat -> nat)
-> (nat -> nat -> nat)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> (nat -> nat) -> nat list -> nat list)
-> nat list
-> nat list
cond_rmap_l_acc :: nat list -> (nat -> nat) -> nat list -> nat list
cond_rmap1_l_acc :: nat list -> (nat -> nat) -> (nat -> nat) -> nat list -> nat list
cond_rmap2_l_acc :: nat list -> (nat -> nat) -> (nat -> nat) -> nat list -> nat list
cond_map_f_l :: nat list -> (nat -> nat) -> nat list
cond_mult_n_m :: nat -> (nat -> nat -> nat) -> nat -> nat
cond_plus_n_m :: nat -> nat -> nat
cond_rmap1_l_acc_1 :: nat list -> (nat -> nat) -> (nat -> nat) -> nat list -> nat list
cond_rmap2_l_acc_1 :: nat list -> (nat -> nat) -> (nat -> nat) -> nat list -> nat list
map :: (nat -> nat) -> nat list -> nat list
mult :: (nat -> nat -> nat) -> nat -> nat -> nat
plus :: nat -> nat -> nat
rmap :: (nat -> nat) -> nat list -> nat list -> nat list
rmap1 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap1_3 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2_3 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
main :: nat list -> nat list
+ Applied Processor:
Inline {inlineType = InlineRewrite, inlineSelect = <inline non-recursive>}
+ Details:
none
* Step 6: Inline WORST_CASE(?,O(n^4))
+ Considered Problem:
1: main_6(x0) x6 -> x6 x0
2: main_xs_3(x3, x4, x6, x9) x10 -> x3 x9 (x4 x9 (x10 x6))
3: main_xs_2(x6, x8, x11, x12, x15, x17) x18 -> x6 x18 (x8 x18 (x11 x15 x17 x12))
4: h() x9 -> x9
5: main_xs_1(x6, x8, x21, x12, x29) x33 -> x6 h() (x8 h() (x21 x29 x33 x12))
6: g(x1) x8 -> x1 x8 S(S(0()))
7: main_xs(x7, x12, x16, x22, x24) x30 -> x12 h() (x16 h() (x22 x30 g(x7) x24))
8: f(x2) x7 -> x2 x7 (x2 x7 x7)
9: main_7(x11, x23, x6, x8, x41) x12 -> x6 h() (x8 h() (x41 f(x23) g(x11) x12))
10: main_5(x24, x9, x21, x12, x16) x42 -> x12 h() (x16 h() (x42 f(x21) g(x9) x24))
11: map_rev2_f1_f2_l_1(x7, x8) x9 -> x8 x7 Nil()
12: map_rev2_f1_f2_l(x7) x8 -> map_rev2_f1_f2_l_1(x7, x8)
13: cond_rmap2_l_acc(Nil(), x5, x6, x13) -> x13
14: cond_rmap2_l_acc(Cons(x14, x15), x5, x6, x13) -> rmap1(x5, x6) x15 Cons(x6 x14, x13)
15: rmap2_l(x5, x6, x12) x13 -> cond_rmap2_l_acc(x12, x5, x6, x13)
16: rmap2_2(x5, x6) x12 -> rmap2_l(x5, x6, x12)
17: rmap2(x10, x12) x24 -> rmap2_l(x10, x12, x24)
18: cond_rmap1_l_acc(Nil(), x5, x6, x9) -> x9
19: cond_rmap1_l_acc(Cons(x10, x11), x5, x6, x9) -> rmap2(x5, x6) x11 Cons(x5 x10, x9)
20: rmap1_l(x5, x6, x8) x9 -> cond_rmap1_l_acc(x8, x5, x6, x9)
21: rmap1_2(x5, x6) x8 -> rmap1_l(x5, x6, x8)
22: rmap1(x10, x12) x16 -> rmap1_l(x10, x12, x16)
23: cond_rmap1_l_acc_1(Nil(), x5, x6, x13) -> x13
24: cond_rmap1_l_acc_1(Cons(x14, x15), x5, x6, x13) -> rmap2_3(x5, x6) x15 Cons(x5 x14, x13)
25: rmap1_l_1(x5, x6, x12) x13 -> cond_rmap1_l_acc_1(x12, x5, x6, x13)
26: rmap1_5(x5, x6) x12 -> rmap1_l_1(x5, x6, x12)
27: rmap1_3(x10, x12) x24 -> rmap1_l_1(x10, x12, x24)
28: cond_rmap2_l_acc_1(Nil(), x5, x6, x9) -> x9
29: cond_rmap2_l_acc_1(Cons(x10, x11), x5, x6, x9) -> rmap1_3(x5, x6) x11 Cons(x6 x10, x9)
30: rmap2_l_1(x5, x6, x8) x9 -> cond_rmap2_l_acc_1(x8, x5, x6, x9)
31: rmap2_5(x5, x6) x8 -> rmap2_l_1(x5, x6, x8)
32: rmap2_3(x10, x12) x16 -> rmap2_l_1(x10, x12, x16)
33: map_rev2_f1_f2(x23, x27) x14 -> rmap1(x23, x27) x14 Nil()
34: map_rev2_f1(x5) x6 -> map_rev2_f1_f2(x5, x6)
35: map_rev2() x5 -> map_rev2_f1(x5)
36: main_4(x48, x10, x22, x24) x32 -> main_xs_3(x24, x32, x48, h()) (map_rev2() f(x22) g(x10))
37: map_rev_f_l(x5) x6 -> x6 x5 Nil()
38: rmap_l_acc_2(x4, x9) x10 -> rmap(x4) x9 x10
39: cond_rmap_l_acc(Nil(), x4, x7) -> x7
40: cond_rmap_l_acc(Cons(x17, x18), x8, x15) -> rmap(x8) x18 Cons(x8 x17, x15)
41: rmap_l(x4, x6) x7 -> cond_rmap_l_acc(x6, x4, x7)
42: rmap_1(x4) x6 -> rmap_l(x4, x6)
43: rmap(x8) x12 -> rmap_l(x8, x12)
44: map_rev_f(x9) x10 -> rmap(x9) x10 Nil()
45: map_rev() x4 -> map_rev_f(x4)
46: main_3(x48, x10, x22) x24 -> main_xs_3(x24, map_rev(), x48, h()) (map_rev2() f(x22) g(x10))
47: map_f_l_2(x3, x5) x7 -> Cons(x3 x5, x7)
48: cond_map_f_l(Nil(), x3) -> Nil()
49: cond_map_f_l(Cons(x10, x13), x6) -> Cons(x6 x10, map() x6 x13)
50: map_f(x3) x4 -> cond_map_f_l(x4, x3)
51: map_1() x3 -> map_f(x3)
52: map() x6 -> map_f(x6)
53: main_2(x0, x9) x17 -> main_7(x9, x17, map(), map_rev(), map_rev2()) x0
54: cond_mult_n_m(0(), x1, x3) -> 0()
55: cond_mult_n_m(S(x4), x1, x3) -> S(x1 (mult(x1) x4 x3) x3)
56: mult_n(x1, x2) x3 -> cond_mult_n_m(x2, x1, x3)
57: mult_1(x1) x2 -> mult_n(x1, x2)
58: mult(x2) x4 -> mult_n(x2, x4)
59: main_1(x48) x8 -> main_xs_1(map(), map_rev(), map_rev2(), x48, f(mult(x8))) g(x8)
60: cond_plus_n_m(0(), x1) -> x1
61: cond_plus_n_m(S(x3), x1) -> S(plus() x1 x3)
62: plus_n(x1) x2 -> cond_plus_n_m(x2, x1)
63: plus_1() x1 -> plus_n(x1)
64: plus() x2 -> plus_n(x2)
65: main(x0) -> main_7(plus(), mult(plus()), map(), map_rev(), map_rev2()) x0
where
0 :: nat
Cons :: 'a -> 'a list -> 'a list
Nil :: 'a list
S :: nat -> nat
f :: (nat -> nat -> nat) -> nat -> nat
map_f :: (nat -> nat) -> nat list -> nat list
map_rev_f :: (nat -> nat) -> nat list -> nat list
map_rev2_f1 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list
map_rev2_f1_f2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list
g :: (nat -> nat -> nat) -> nat -> nat
h :: nat -> nat
map_rev_f_l :: nat list -> (nat list -> nat list -> nat list) -> nat list
map_rev2_f1_f2_l :: nat list
-> (nat list -> nat list -> nat list)
-> (nat list -> nat list -> nat list)
-> nat list
rmap_l :: (nat -> nat) -> nat list -> nat list -> nat list
rmap1_l :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2_l :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
map_rev :: (nat -> nat) -> nat list -> nat list
map_rev2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list
mult_n :: (nat -> nat -> nat) -> nat -> nat -> nat
plus_n :: nat -> nat -> nat
main_xs :: (nat -> nat -> nat)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> (nat -> nat) -> nat list -> nat list)
-> nat list
-> (nat -> nat)
-> nat list
map_rev2_f1_f2_l_1 :: nat list
-> (nat list -> nat list -> nat list)
-> (nat list -> nat list -> nat list)
-> nat list
rmap1_l_1 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2_l_1 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
main_1 :: nat list -> (nat -> nat -> nat) -> nat list
map_1 :: (nat -> nat) -> nat list -> nat list
mult_1 :: (nat -> nat -> nat) -> nat -> nat -> nat
plus_1 :: nat -> nat -> nat
rmap_1 :: (nat -> nat) -> nat list -> nat list -> nat list
main_xs_1 :: ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> (nat -> nat) -> nat list -> nat list)
-> nat list
-> (nat -> nat)
-> (nat -> nat)
-> nat list
rmap_l_acc_2 :: (nat -> nat) -> nat list -> nat list -> nat list
map_f_l_2 :: (nat -> nat) -> nat -> nat list -> nat list
main_2 :: nat list -> (nat -> nat -> nat) -> (nat -> nat -> nat) -> nat list
rmap1_2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2_2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
main_xs_2 :: ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> (nat -> nat) -> nat list -> nat list)
-> nat list
-> (nat -> nat)
-> (nat -> nat)
-> (nat -> nat)
-> nat list
main_3 :: nat list
-> (nat -> nat -> nat)
-> (nat -> nat -> nat)
-> ((nat -> nat) -> nat list -> nat list)
-> nat list
main_xs_3 :: ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> nat list
-> (nat -> nat)
-> (nat list -> nat list)
-> nat list
main_4 :: nat list
-> (nat -> nat -> nat)
-> (nat -> nat -> nat)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> nat list
main_5 :: nat list
-> (nat -> nat -> nat)
-> (nat -> nat -> nat)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> (nat -> nat) -> nat list -> nat list)
-> nat list
rmap1_5 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2_5 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
main_6 :: nat list -> (nat list -> nat list) -> nat list
main_7 :: (nat -> nat -> nat)
-> (nat -> nat -> nat)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> (nat -> nat) -> nat list -> nat list)
-> nat list
-> nat list
cond_rmap_l_acc :: nat list -> (nat -> nat) -> nat list -> nat list
cond_rmap1_l_acc :: nat list -> (nat -> nat) -> (nat -> nat) -> nat list -> nat list
cond_rmap2_l_acc :: nat list -> (nat -> nat) -> (nat -> nat) -> nat list -> nat list
cond_map_f_l :: nat list -> (nat -> nat) -> nat list
cond_mult_n_m :: nat -> (nat -> nat -> nat) -> nat -> nat
cond_plus_n_m :: nat -> nat -> nat
cond_rmap1_l_acc_1 :: nat list -> (nat -> nat) -> (nat -> nat) -> nat list -> nat list
cond_rmap2_l_acc_1 :: nat list -> (nat -> nat) -> (nat -> nat) -> nat list -> nat list
map :: (nat -> nat) -> nat list -> nat list
mult :: (nat -> nat -> nat) -> nat -> nat -> nat
plus :: nat -> nat -> nat
rmap :: (nat -> nat) -> nat list -> nat list -> nat list
rmap1 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap1_3 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2_3 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
main :: nat list -> nat list
+ Applied Processor:
Inline {inlineType = InlineRewrite, inlineSelect = <inline non-recursive>}
+ Details:
none
* Step 7: Inline WORST_CASE(?,O(n^4))
+ Considered Problem:
1: main_6(x0) x6 -> x6 x0
2: main_xs_3(x3, x4, x6, x9) x10 -> x3 x9 (x4 x9 (x10 x6))
3: main_xs_2(x6, x8, x11, x12, x15, x17) x18 -> x6 x18 (x8 x18 (x11 x15 x17 x12))
4: h() x9 -> x9
5: main_xs_1(x6, x8, x21, x12, x29) x33 -> x6 h() (x8 h() (x21 x29 x33 x12))
6: g(x1) x8 -> x1 x8 S(S(0()))
7: main_xs(x7, x12, x16, x22, x24) x30 -> x12 h() (x16 h() (x22 x30 g(x7) x24))
8: f(x2) x7 -> x2 x7 (x2 x7 x7)
9: main_7(x11, x23, x6, x8, x41) x12 -> x6 h() (x8 h() (x41 f(x23) g(x11) x12))
10: main_5(x24, x9, x21, x12, x16) x42 -> x12 h() (x16 h() (x42 f(x21) g(x9) x24))
11: map_rev2_f1_f2_l_1(x7, x8) x9 -> x8 x7 Nil()
12: map_rev2_f1_f2_l(x7) x8 -> map_rev2_f1_f2_l_1(x7, x8)
13: cond_rmap2_l_acc(Nil(), x5, x6, x13) -> x13
14: cond_rmap2_l_acc(Cons(x14, x15), x5, x6, x13) -> rmap1(x5, x6) x15 Cons(x6 x14, x13)
15: rmap2_l(x5, x6, x12) x13 -> cond_rmap2_l_acc(x12, x5, x6, x13)
16: rmap2_2(x5, x6) x12 -> rmap2_l(x5, x6, x12)
17: rmap2(x10, x12) x24 -> rmap2_l(x10, x12, x24)
18: cond_rmap1_l_acc(Nil(), x5, x6, x9) -> x9
19: cond_rmap1_l_acc(Cons(x10, x11), x5, x6, x9) -> rmap2(x5, x6) x11 Cons(x5 x10, x9)
20: rmap1_l(x5, x6, x8) x9 -> cond_rmap1_l_acc(x8, x5, x6, x9)
21: rmap1_2(x5, x6) x8 -> rmap1_l(x5, x6, x8)
22: rmap1(x10, x12) x16 -> rmap1_l(x10, x12, x16)
23: cond_rmap1_l_acc_1(Nil(), x5, x6, x13) -> x13
24: cond_rmap1_l_acc_1(Cons(x14, x15), x5, x6, x13) -> rmap2_3(x5, x6) x15 Cons(x5 x14, x13)
25: rmap1_l_1(x5, x6, x12) x13 -> cond_rmap1_l_acc_1(x12, x5, x6, x13)
26: rmap1_5(x5, x6) x12 -> rmap1_l_1(x5, x6, x12)
27: rmap1_3(x10, x12) x24 -> rmap1_l_1(x10, x12, x24)
28: cond_rmap2_l_acc_1(Nil(), x5, x6, x9) -> x9
29: cond_rmap2_l_acc_1(Cons(x10, x11), x5, x6, x9) -> rmap1_3(x5, x6) x11 Cons(x6 x10, x9)
30: rmap2_l_1(x5, x6, x8) x9 -> cond_rmap2_l_acc_1(x8, x5, x6, x9)
31: rmap2_5(x5, x6) x8 -> rmap2_l_1(x5, x6, x8)
32: rmap2_3(x10, x12) x16 -> rmap2_l_1(x10, x12, x16)
33: map_rev2_f1_f2(x23, x27) x14 -> rmap1(x23, x27) x14 Nil()
34: map_rev2_f1(x5) x6 -> map_rev2_f1_f2(x5, x6)
35: map_rev2() x5 -> map_rev2_f1(x5)
36: main_4(x12, x21, x45, x6) x8 -> x6 h() (x8 h() (map_rev2() f(x45) g(x21) x12))
37: map_rev_f_l(x5) x6 -> x6 x5 Nil()
38: rmap_l_acc_2(x4, x9) x10 -> rmap(x4) x9 x10
39: cond_rmap_l_acc(Nil(), x4, x7) -> x7
40: cond_rmap_l_acc(Cons(x17, x18), x8, x15) -> rmap(x8) x18 Cons(x8 x17, x15)
41: rmap_l(x4, x6) x7 -> cond_rmap_l_acc(x6, x4, x7)
42: rmap_1(x4) x6 -> rmap_l(x4, x6)
43: rmap(x8) x12 -> rmap_l(x8, x12)
44: map_rev_f(x9) x10 -> rmap(x9) x10 Nil()
45: map_rev() x4 -> map_rev_f(x4)
46: main_3(x12, x21, x45) x6 -> x6 h() (map_rev() h() (map_rev2() f(x45) g(x21) x12))
47: map_f_l_2(x3, x5) x7 -> Cons(x3 x5, x7)
48: cond_map_f_l(Nil(), x3) -> Nil()
49: cond_map_f_l(Cons(x10, x13), x6) -> Cons(x6 x10, map() x6 x13)
50: map_f(x3) x4 -> cond_map_f_l(x4, x3)
51: map_1() x3 -> map_f(x3)
52: map() x6 -> map_f(x6)
53: main_2(x24, x22) x46 -> map() h() (map_rev() h() (map_rev2() f(x46) g(x22) x24))
54: cond_mult_n_m(0(), x1, x3) -> 0()
55: cond_mult_n_m(S(x4), x1, x3) -> S(x1 (mult(x1) x4 x3) x3)
56: mult_n(x1, x2) x3 -> cond_mult_n_m(x2, x1, x3)
57: mult_1(x1) x2 -> mult_n(x1, x2)
58: mult(x2) x4 -> mult_n(x2, x4)
59: main_1(x24) x17 -> map() h() (map_rev() h() (map_rev2() f(mult(x17)) g(x17) x24))
60: cond_plus_n_m(0(), x1) -> x1
61: cond_plus_n_m(S(x3), x1) -> S(plus() x1 x3)
62: plus_n(x1) x2 -> cond_plus_n_m(x2, x1)
63: plus_1() x1 -> plus_n(x1)
64: plus() x2 -> plus_n(x2)
65: main(x24) -> map() h() (map_rev() h() (map_rev2() f(mult(plus())) g(plus()) x24))
where
0 :: nat
Cons :: 'a -> 'a list -> 'a list
Nil :: 'a list
S :: nat -> nat
f :: (nat -> nat -> nat) -> nat -> nat
map_f :: (nat -> nat) -> nat list -> nat list
map_rev_f :: (nat -> nat) -> nat list -> nat list
map_rev2_f1 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list
map_rev2_f1_f2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list
g :: (nat -> nat -> nat) -> nat -> nat
h :: nat -> nat
map_rev_f_l :: nat list -> (nat list -> nat list -> nat list) -> nat list
map_rev2_f1_f2_l :: nat list
-> (nat list -> nat list -> nat list)
-> (nat list -> nat list -> nat list)
-> nat list
rmap_l :: (nat -> nat) -> nat list -> nat list -> nat list
rmap1_l :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2_l :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
map_rev :: (nat -> nat) -> nat list -> nat list
map_rev2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list
mult_n :: (nat -> nat -> nat) -> nat -> nat -> nat
plus_n :: nat -> nat -> nat
main_xs :: (nat -> nat -> nat)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> (nat -> nat) -> nat list -> nat list)
-> nat list
-> (nat -> nat)
-> nat list
map_rev2_f1_f2_l_1 :: nat list
-> (nat list -> nat list -> nat list)
-> (nat list -> nat list -> nat list)
-> nat list
rmap1_l_1 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2_l_1 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
main_1 :: nat list -> (nat -> nat -> nat) -> nat list
map_1 :: (nat -> nat) -> nat list -> nat list
mult_1 :: (nat -> nat -> nat) -> nat -> nat -> nat
plus_1 :: nat -> nat -> nat
rmap_1 :: (nat -> nat) -> nat list -> nat list -> nat list
main_xs_1 :: ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> (nat -> nat) -> nat list -> nat list)
-> nat list
-> (nat -> nat)
-> (nat -> nat)
-> nat list
rmap_l_acc_2 :: (nat -> nat) -> nat list -> nat list -> nat list
map_f_l_2 :: (nat -> nat) -> nat -> nat list -> nat list
main_2 :: nat list -> (nat -> nat -> nat) -> (nat -> nat -> nat) -> nat list
rmap1_2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2_2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
main_xs_2 :: ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> (nat -> nat) -> nat list -> nat list)
-> nat list
-> (nat -> nat)
-> (nat -> nat)
-> (nat -> nat)
-> nat list
main_3 :: nat list
-> (nat -> nat -> nat)
-> (nat -> nat -> nat)
-> ((nat -> nat) -> nat list -> nat list)
-> nat list
main_xs_3 :: ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> nat list
-> (nat -> nat)
-> (nat list -> nat list)
-> nat list
main_4 :: nat list
-> (nat -> nat -> nat)
-> (nat -> nat -> nat)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> nat list
main_5 :: nat list
-> (nat -> nat -> nat)
-> (nat -> nat -> nat)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> (nat -> nat) -> nat list -> nat list)
-> nat list
rmap1_5 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2_5 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
main_6 :: nat list -> (nat list -> nat list) -> nat list
main_7 :: (nat -> nat -> nat)
-> (nat -> nat -> nat)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> (nat -> nat) -> nat list -> nat list)
-> nat list
-> nat list
cond_rmap_l_acc :: nat list -> (nat -> nat) -> nat list -> nat list
cond_rmap1_l_acc :: nat list -> (nat -> nat) -> (nat -> nat) -> nat list -> nat list
cond_rmap2_l_acc :: nat list -> (nat -> nat) -> (nat -> nat) -> nat list -> nat list
cond_map_f_l :: nat list -> (nat -> nat) -> nat list
cond_mult_n_m :: nat -> (nat -> nat -> nat) -> nat -> nat
cond_plus_n_m :: nat -> nat -> nat
cond_rmap1_l_acc_1 :: nat list -> (nat -> nat) -> (nat -> nat) -> nat list -> nat list
cond_rmap2_l_acc_1 :: nat list -> (nat -> nat) -> (nat -> nat) -> nat list -> nat list
map :: (nat -> nat) -> nat list -> nat list
mult :: (nat -> nat -> nat) -> nat -> nat -> nat
plus :: nat -> nat -> nat
rmap :: (nat -> nat) -> nat list -> nat list -> nat list
rmap1 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap1_3 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2_3 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
main :: nat list -> nat list
+ Applied Processor:
Inline {inlineType = InlineRewrite, inlineSelect = <inline non-recursive>}
+ Details:
none
* Step 8: Inline WORST_CASE(?,O(n^4))
+ Considered Problem:
1: main_6(x0) x6 -> x6 x0
2: main_xs_3(x3, x4, x6, x9) x10 -> x3 x9 (x4 x9 (x10 x6))
3: main_xs_2(x6, x8, x11, x12, x15, x17) x18 -> x6 x18 (x8 x18 (x11 x15 x17 x12))
4: h() x9 -> x9
5: main_xs_1(x6, x8, x21, x12, x29) x33 -> x6 h() (x8 h() (x21 x29 x33 x12))
6: g(x1) x8 -> x1 x8 S(S(0()))
7: main_xs(x7, x12, x16, x22, x24) x30 -> x12 h() (x16 h() (x22 x30 g(x7) x24))
8: f(x2) x7 -> x2 x7 (x2 x7 x7)
9: main_7(x11, x23, x6, x8, x41) x12 -> x6 h() (x8 h() (x41 f(x23) g(x11) x12))
10: main_5(x24, x9, x21, x12, x16) x42 -> x12 h() (x16 h() (x42 f(x21) g(x9) x24))
11: map_rev2_f1_f2_l_1(x7, x8) x9 -> x8 x7 Nil()
12: map_rev2_f1_f2_l(x7) x8 -> map_rev2_f1_f2_l_1(x7, x8)
13: cond_rmap2_l_acc(Nil(), x5, x6, x13) -> x13
14: cond_rmap2_l_acc(Cons(x14, x15), x5, x6, x13) -> rmap1(x5, x6) x15 Cons(x6 x14, x13)
15: rmap2_l(x5, x6, x12) x13 -> cond_rmap2_l_acc(x12, x5, x6, x13)
16: rmap2_2(x5, x6) x12 -> rmap2_l(x5, x6, x12)
17: rmap2(x10, x12) x24 -> rmap2_l(x10, x12, x24)
18: cond_rmap1_l_acc(Nil(), x5, x6, x9) -> x9
19: cond_rmap1_l_acc(Cons(x10, x11), x5, x6, x9) -> rmap2(x5, x6) x11 Cons(x5 x10, x9)
20: rmap1_l(x5, x6, x8) x9 -> cond_rmap1_l_acc(x8, x5, x6, x9)
21: rmap1_2(x5, x6) x8 -> rmap1_l(x5, x6, x8)
22: rmap1(x10, x12) x16 -> rmap1_l(x10, x12, x16)
23: cond_rmap1_l_acc_1(Nil(), x5, x6, x13) -> x13
24: cond_rmap1_l_acc_1(Cons(x14, x15), x5, x6, x13) -> rmap2_3(x5, x6) x15 Cons(x5 x14, x13)
25: rmap1_l_1(x5, x6, x12) x13 -> cond_rmap1_l_acc_1(x12, x5, x6, x13)
26: rmap1_5(x5, x6) x12 -> rmap1_l_1(x5, x6, x12)
27: rmap1_3(x10, x12) x24 -> rmap1_l_1(x10, x12, x24)
28: cond_rmap2_l_acc_1(Nil(), x5, x6, x9) -> x9
29: cond_rmap2_l_acc_1(Cons(x10, x11), x5, x6, x9) -> rmap1_3(x5, x6) x11 Cons(x6 x10, x9)
30: rmap2_l_1(x5, x6, x8) x9 -> cond_rmap2_l_acc_1(x8, x5, x6, x9)
31: rmap2_5(x5, x6) x8 -> rmap2_l_1(x5, x6, x8)
32: rmap2_3(x10, x12) x16 -> rmap2_l_1(x10, x12, x16)
33: map_rev2_f1_f2(x23, x27) x14 -> rmap1(x23, x27) x14 Nil()
34: map_rev2_f1(x5) x6 -> map_rev2_f1_f2(x5, x6)
35: map_rev2() x5 -> map_rev2_f1(x5)
36: main_4(x25, x43, x91, x13) x17 -> x13 h() (x17 h() (map_rev2_f1(f(x91)) g(x43) x25))
37: map_rev_f_l(x5) x6 -> x6 x5 Nil()
38: rmap_l_acc_2(x4, x9) x10 -> rmap(x4) x9 x10
39: cond_rmap_l_acc(Nil(), x4, x7) -> x7
40: cond_rmap_l_acc(Cons(x17, x18), x8, x15) -> rmap(x8) x18 Cons(x8 x17, x15)
41: rmap_l(x4, x6) x7 -> cond_rmap_l_acc(x6, x4, x7)
42: rmap_1(x4) x6 -> rmap_l(x4, x6)
43: rmap(x8) x12 -> rmap_l(x8, x12)
44: map_rev_f(x9) x10 -> rmap(x9) x10 Nil()
45: map_rev() x4 -> map_rev_f(x4)
46: main_3(x25, x43, x91) x13 -> x13 h() (map_rev_f(h()) (map_rev2() f(x91) g(x43) x25))
47: map_f_l_2(x3, x5) x7 -> Cons(x3 x5, x7)
48: cond_map_f_l(Nil(), x3) -> Nil()
49: cond_map_f_l(Cons(x10, x13), x6) -> Cons(x6 x10, map() x6 x13)
50: map_f(x3) x4 -> cond_map_f_l(x4, x3)
51: map_1() x3 -> map_f(x3)
52: map() x6 -> map_f(x6)
53: main_2(x49, x45) x93 -> map() h() (map_rev_f(h()) (map_rev2() f(x93) g(x45) x49))
54: cond_mult_n_m(0(), x1, x3) -> 0()
55: cond_mult_n_m(S(x4), x1, x3) -> S(x1 (mult(x1) x4 x3) x3)
56: mult_n(x1, x2) x3 -> cond_mult_n_m(x2, x1, x3)
57: mult_1(x1) x2 -> mult_n(x1, x2)
58: mult(x2) x4 -> mult_n(x2, x4)
59: main_1(x49) x35 -> map() h() (map_rev_f(h()) (map_rev2() f(mult(x35)) g(x35) x49))
60: cond_plus_n_m(0(), x1) -> x1
61: cond_plus_n_m(S(x3), x1) -> S(plus() x1 x3)
62: plus_n(x1) x2 -> cond_plus_n_m(x2, x1)
63: plus_1() x1 -> plus_n(x1)
64: plus() x2 -> plus_n(x2)
65: main(x49) -> map() h() (map_rev_f(h()) (map_rev2() f(mult(plus())) g(plus()) x49))
where
0 :: nat
Cons :: 'a -> 'a list -> 'a list
Nil :: 'a list
S :: nat -> nat
f :: (nat -> nat -> nat) -> nat -> nat
map_f :: (nat -> nat) -> nat list -> nat list
map_rev_f :: (nat -> nat) -> nat list -> nat list
map_rev2_f1 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list
map_rev2_f1_f2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list
g :: (nat -> nat -> nat) -> nat -> nat
h :: nat -> nat
map_rev_f_l :: nat list -> (nat list -> nat list -> nat list) -> nat list
map_rev2_f1_f2_l :: nat list
-> (nat list -> nat list -> nat list)
-> (nat list -> nat list -> nat list)
-> nat list
rmap_l :: (nat -> nat) -> nat list -> nat list -> nat list
rmap1_l :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2_l :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
map_rev :: (nat -> nat) -> nat list -> nat list
map_rev2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list
mult_n :: (nat -> nat -> nat) -> nat -> nat -> nat
plus_n :: nat -> nat -> nat
main_xs :: (nat -> nat -> nat)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> (nat -> nat) -> nat list -> nat list)
-> nat list
-> (nat -> nat)
-> nat list
map_rev2_f1_f2_l_1 :: nat list
-> (nat list -> nat list -> nat list)
-> (nat list -> nat list -> nat list)
-> nat list
rmap1_l_1 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2_l_1 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
main_1 :: nat list -> (nat -> nat -> nat) -> nat list
map_1 :: (nat -> nat) -> nat list -> nat list
mult_1 :: (nat -> nat -> nat) -> nat -> nat -> nat
plus_1 :: nat -> nat -> nat
rmap_1 :: (nat -> nat) -> nat list -> nat list -> nat list
main_xs_1 :: ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> (nat -> nat) -> nat list -> nat list)
-> nat list
-> (nat -> nat)
-> (nat -> nat)
-> nat list
rmap_l_acc_2 :: (nat -> nat) -> nat list -> nat list -> nat list
map_f_l_2 :: (nat -> nat) -> nat -> nat list -> nat list
main_2 :: nat list -> (nat -> nat -> nat) -> (nat -> nat -> nat) -> nat list
rmap1_2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2_2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
main_xs_2 :: ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> (nat -> nat) -> nat list -> nat list)
-> nat list
-> (nat -> nat)
-> (nat -> nat)
-> (nat -> nat)
-> nat list
main_3 :: nat list
-> (nat -> nat -> nat)
-> (nat -> nat -> nat)
-> ((nat -> nat) -> nat list -> nat list)
-> nat list
main_xs_3 :: ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> nat list
-> (nat -> nat)
-> (nat list -> nat list)
-> nat list
main_4 :: nat list
-> (nat -> nat -> nat)
-> (nat -> nat -> nat)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> nat list
main_5 :: nat list
-> (nat -> nat -> nat)
-> (nat -> nat -> nat)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> (nat -> nat) -> nat list -> nat list)
-> nat list
rmap1_5 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2_5 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
main_6 :: nat list -> (nat list -> nat list) -> nat list
main_7 :: (nat -> nat -> nat)
-> (nat -> nat -> nat)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> (nat -> nat) -> nat list -> nat list)
-> nat list
-> nat list
cond_rmap_l_acc :: nat list -> (nat -> nat) -> nat list -> nat list
cond_rmap1_l_acc :: nat list -> (nat -> nat) -> (nat -> nat) -> nat list -> nat list
cond_rmap2_l_acc :: nat list -> (nat -> nat) -> (nat -> nat) -> nat list -> nat list
cond_map_f_l :: nat list -> (nat -> nat) -> nat list
cond_mult_n_m :: nat -> (nat -> nat -> nat) -> nat -> nat
cond_plus_n_m :: nat -> nat -> nat
cond_rmap1_l_acc_1 :: nat list -> (nat -> nat) -> (nat -> nat) -> nat list -> nat list
cond_rmap2_l_acc_1 :: nat list -> (nat -> nat) -> (nat -> nat) -> nat list -> nat list
map :: (nat -> nat) -> nat list -> nat list
mult :: (nat -> nat -> nat) -> nat -> nat -> nat
plus :: nat -> nat -> nat
rmap :: (nat -> nat) -> nat list -> nat list -> nat list
rmap1 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap1_3 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2_3 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
main :: nat list -> nat list
+ Applied Processor:
Inline {inlineType = InlineRewrite, inlineSelect = <inline non-recursive>}
+ Details:
none
* Step 9: Inline WORST_CASE(?,O(n^4))
+ Considered Problem:
1: main_6(x0) x6 -> x6 x0
2: main_xs_3(x3, x4, x6, x9) x10 -> x3 x9 (x4 x9 (x10 x6))
3: main_xs_2(x6, x8, x11, x12, x15, x17) x18 -> x6 x18 (x8 x18 (x11 x15 x17 x12))
4: h() x9 -> x9
5: main_xs_1(x6, x8, x21, x12, x29) x33 -> x6 h() (x8 h() (x21 x29 x33 x12))
6: g(x1) x8 -> x1 x8 S(S(0()))
7: main_xs(x7, x12, x16, x22, x24) x30 -> x12 h() (x16 h() (x22 x30 g(x7) x24))
8: f(x2) x7 -> x2 x7 (x2 x7 x7)
9: main_7(x11, x23, x6, x8, x41) x12 -> x6 h() (x8 h() (x41 f(x23) g(x11) x12))
10: main_5(x24, x9, x21, x12, x16) x42 -> x12 h() (x16 h() (x42 f(x21) g(x9) x24))
11: map_rev2_f1_f2_l_1(x7, x8) x9 -> x8 x7 Nil()
12: map_rev2_f1_f2_l(x7) x8 -> map_rev2_f1_f2_l_1(x7, x8)
13: cond_rmap2_l_acc(Nil(), x5, x6, x13) -> x13
14: cond_rmap2_l_acc(Cons(x14, x15), x5, x6, x13) -> rmap1(x5, x6) x15 Cons(x6 x14, x13)
15: rmap2_l(x5, x6, x12) x13 -> cond_rmap2_l_acc(x12, x5, x6, x13)
16: rmap2_2(x5, x6) x12 -> rmap2_l(x5, x6, x12)
17: rmap2(x10, x12) x24 -> rmap2_l(x10, x12, x24)
18: cond_rmap1_l_acc(Nil(), x5, x6, x9) -> x9
19: cond_rmap1_l_acc(Cons(x10, x11), x5, x6, x9) -> rmap2(x5, x6) x11 Cons(x5 x10, x9)
20: rmap1_l(x5, x6, x8) x9 -> cond_rmap1_l_acc(x8, x5, x6, x9)
21: rmap1_2(x5, x6) x8 -> rmap1_l(x5, x6, x8)
22: rmap1(x10, x12) x16 -> rmap1_l(x10, x12, x16)
23: cond_rmap1_l_acc_1(Nil(), x5, x6, x13) -> x13
24: cond_rmap1_l_acc_1(Cons(x14, x15), x5, x6, x13) -> rmap2_3(x5, x6) x15 Cons(x5 x14, x13)
25: rmap1_l_1(x5, x6, x12) x13 -> cond_rmap1_l_acc_1(x12, x5, x6, x13)
26: rmap1_5(x5, x6) x12 -> rmap1_l_1(x5, x6, x12)
27: rmap1_3(x10, x12) x24 -> rmap1_l_1(x10, x12, x24)
28: cond_rmap2_l_acc_1(Nil(), x5, x6, x9) -> x9
29: cond_rmap2_l_acc_1(Cons(x10, x11), x5, x6, x9) -> rmap1_3(x5, x6) x11 Cons(x6 x10, x9)
30: rmap2_l_1(x5, x6, x8) x9 -> cond_rmap2_l_acc_1(x8, x5, x6, x9)
31: rmap2_5(x5, x6) x8 -> rmap2_l_1(x5, x6, x8)
32: rmap2_3(x10, x12) x16 -> rmap2_l_1(x10, x12, x16)
33: map_rev2_f1_f2(x23, x27) x14 -> rmap1(x23, x27) x14 Nil()
34: map_rev2_f1(x5) x6 -> map_rev2_f1_f2(x5, x6)
35: map_rev2() x5 -> map_rev2_f1(x5)
36: main_4(x51, x87, x183, x27) x35 -> x27 h() (x35 h() (map_rev2_f1_f2(f(x183), g(x87)) x51))
37: map_rev_f_l(x5) x6 -> x6 x5 Nil()
38: rmap_l_acc_2(x4, x9) x10 -> rmap(x4) x9 x10
39: cond_rmap_l_acc(Nil(), x4, x7) -> x7
40: cond_rmap_l_acc(Cons(x17, x18), x8, x15) -> rmap(x8) x18 Cons(x8 x17, x15)
41: rmap_l(x4, x6) x7 -> cond_rmap_l_acc(x6, x4, x7)
42: rmap_1(x4) x6 -> rmap_l(x4, x6)
43: rmap(x8) x12 -> rmap_l(x8, x12)
44: map_rev_f(x9) x10 -> rmap(x9) x10 Nil()
45: map_rev() x4 -> map_rev_f(x4)
46: main_3(x51, x87, x183) x27 -> x27 h() (rmap(h()) (map_rev2() f(x183) g(x87) x51) Nil())
47: map_f_l_2(x3, x5) x7 -> Cons(x3 x5, x7)
48: cond_map_f_l(Nil(), x3) -> Nil()
49: cond_map_f_l(Cons(x10, x13), x6) -> Cons(x6 x10, map() x6 x13)
50: map_f(x3) x4 -> cond_map_f_l(x4, x3)
51: map_1() x3 -> map_f(x3)
52: map() x6 -> map_f(x6)
53: main_2(x99, x91) x187 -> map() h() (rmap(h()) (map_rev2() f(x187) g(x91) x99) Nil())
54: cond_mult_n_m(0(), x1, x3) -> 0()
55: cond_mult_n_m(S(x4), x1, x3) -> S(x1 (mult(x1) x4 x3) x3)
56: mult_n(x1, x2) x3 -> cond_mult_n_m(x2, x1, x3)
57: mult_1(x1) x2 -> mult_n(x1, x2)
58: mult(x2) x4 -> mult_n(x2, x4)
59: main_1(x99) x71 -> map() h() (rmap(h()) (map_rev2() f(mult(x71)) g(x71) x99) Nil())
60: cond_plus_n_m(0(), x1) -> x1
61: cond_plus_n_m(S(x3), x1) -> S(plus() x1 x3)
62: plus_n(x1) x2 -> cond_plus_n_m(x2, x1)
63: plus_1() x1 -> plus_n(x1)
64: plus() x2 -> plus_n(x2)
65: main(x99) -> map() h() (rmap(h()) (map_rev2() f(mult(plus())) g(plus()) x99) Nil())
where
0 :: nat
Cons :: 'a -> 'a list -> 'a list
Nil :: 'a list
S :: nat -> nat
f :: (nat -> nat -> nat) -> nat -> nat
map_f :: (nat -> nat) -> nat list -> nat list
map_rev_f :: (nat -> nat) -> nat list -> nat list
map_rev2_f1 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list
map_rev2_f1_f2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list
g :: (nat -> nat -> nat) -> nat -> nat
h :: nat -> nat
map_rev_f_l :: nat list -> (nat list -> nat list -> nat list) -> nat list
map_rev2_f1_f2_l :: nat list
-> (nat list -> nat list -> nat list)
-> (nat list -> nat list -> nat list)
-> nat list
rmap_l :: (nat -> nat) -> nat list -> nat list -> nat list
rmap1_l :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2_l :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
map_rev :: (nat -> nat) -> nat list -> nat list
map_rev2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list
mult_n :: (nat -> nat -> nat) -> nat -> nat -> nat
plus_n :: nat -> nat -> nat
main_xs :: (nat -> nat -> nat)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> (nat -> nat) -> nat list -> nat list)
-> nat list
-> (nat -> nat)
-> nat list
map_rev2_f1_f2_l_1 :: nat list
-> (nat list -> nat list -> nat list)
-> (nat list -> nat list -> nat list)
-> nat list
rmap1_l_1 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2_l_1 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
main_1 :: nat list -> (nat -> nat -> nat) -> nat list
map_1 :: (nat -> nat) -> nat list -> nat list
mult_1 :: (nat -> nat -> nat) -> nat -> nat -> nat
plus_1 :: nat -> nat -> nat
rmap_1 :: (nat -> nat) -> nat list -> nat list -> nat list
main_xs_1 :: ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> (nat -> nat) -> nat list -> nat list)
-> nat list
-> (nat -> nat)
-> (nat -> nat)
-> nat list
rmap_l_acc_2 :: (nat -> nat) -> nat list -> nat list -> nat list
map_f_l_2 :: (nat -> nat) -> nat -> nat list -> nat list
main_2 :: nat list -> (nat -> nat -> nat) -> (nat -> nat -> nat) -> nat list
rmap1_2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2_2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
main_xs_2 :: ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> (nat -> nat) -> nat list -> nat list)
-> nat list
-> (nat -> nat)
-> (nat -> nat)
-> (nat -> nat)
-> nat list
main_3 :: nat list
-> (nat -> nat -> nat)
-> (nat -> nat -> nat)
-> ((nat -> nat) -> nat list -> nat list)
-> nat list
main_xs_3 :: ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> nat list
-> (nat -> nat)
-> (nat list -> nat list)
-> nat list
main_4 :: nat list
-> (nat -> nat -> nat)
-> (nat -> nat -> nat)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> nat list
main_5 :: nat list
-> (nat -> nat -> nat)
-> (nat -> nat -> nat)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> (nat -> nat) -> nat list -> nat list)
-> nat list
rmap1_5 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2_5 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
main_6 :: nat list -> (nat list -> nat list) -> nat list
main_7 :: (nat -> nat -> nat)
-> (nat -> nat -> nat)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> (nat -> nat) -> nat list -> nat list)
-> nat list
-> nat list
cond_rmap_l_acc :: nat list -> (nat -> nat) -> nat list -> nat list
cond_rmap1_l_acc :: nat list -> (nat -> nat) -> (nat -> nat) -> nat list -> nat list
cond_rmap2_l_acc :: nat list -> (nat -> nat) -> (nat -> nat) -> nat list -> nat list
cond_map_f_l :: nat list -> (nat -> nat) -> nat list
cond_mult_n_m :: nat -> (nat -> nat -> nat) -> nat -> nat
cond_plus_n_m :: nat -> nat -> nat
cond_rmap1_l_acc_1 :: nat list -> (nat -> nat) -> (nat -> nat) -> nat list -> nat list
cond_rmap2_l_acc_1 :: nat list -> (nat -> nat) -> (nat -> nat) -> nat list -> nat list
map :: (nat -> nat) -> nat list -> nat list
mult :: (nat -> nat -> nat) -> nat -> nat -> nat
plus :: nat -> nat -> nat
rmap :: (nat -> nat) -> nat list -> nat list -> nat list
rmap1 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap1_3 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2_3 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
main :: nat list -> nat list
+ Applied Processor:
Inline {inlineType = InlineRewrite, inlineSelect = <inline non-recursive>}
+ Details:
none
* Step 10: Inline WORST_CASE(?,O(n^4))
+ Considered Problem:
1: main_6(x0) x6 -> x6 x0
2: main_xs_3(x3, x4, x6, x9) x10 -> x3 x9 (x4 x9 (x10 x6))
3: main_xs_2(x6, x8, x11, x12, x15, x17) x18 -> x6 x18 (x8 x18 (x11 x15 x17 x12))
4: h() x9 -> x9
5: main_xs_1(x6, x8, x21, x12, x29) x33 -> x6 h() (x8 h() (x21 x29 x33 x12))
6: g(x1) x8 -> x1 x8 S(S(0()))
7: main_xs(x7, x12, x16, x22, x24) x30 -> x12 h() (x16 h() (x22 x30 g(x7) x24))
8: f(x2) x7 -> x2 x7 (x2 x7 x7)
9: main_7(x11, x23, x6, x8, x41) x12 -> x6 h() (x8 h() (x41 f(x23) g(x11) x12))
10: main_5(x24, x9, x21, x12, x16) x42 -> x12 h() (x16 h() (x42 f(x21) g(x9) x24))
11: map_rev2_f1_f2_l_1(x7, x8) x9 -> x8 x7 Nil()
12: map_rev2_f1_f2_l(x7) x8 -> map_rev2_f1_f2_l_1(x7, x8)
13: cond_rmap2_l_acc(Nil(), x5, x6, x13) -> x13
14: cond_rmap2_l_acc(Cons(x14, x15), x5, x6, x13) -> rmap1(x5, x6) x15 Cons(x6 x14, x13)
15: rmap2_l(x5, x6, x12) x13 -> cond_rmap2_l_acc(x12, x5, x6, x13)
16: rmap2_2(x5, x6) x12 -> rmap2_l(x5, x6, x12)
17: rmap2(x10, x12) x24 -> rmap2_l(x10, x12, x24)
18: cond_rmap1_l_acc(Nil(), x5, x6, x9) -> x9
19: cond_rmap1_l_acc(Cons(x10, x11), x5, x6, x9) -> rmap2(x5, x6) x11 Cons(x5 x10, x9)
20: rmap1_l(x5, x6, x8) x9 -> cond_rmap1_l_acc(x8, x5, x6, x9)
21: rmap1_2(x5, x6) x8 -> rmap1_l(x5, x6, x8)
22: rmap1(x10, x12) x16 -> rmap1_l(x10, x12, x16)
23: cond_rmap1_l_acc_1(Nil(), x5, x6, x13) -> x13
24: cond_rmap1_l_acc_1(Cons(x14, x15), x5, x6, x13) -> rmap2_3(x5, x6) x15 Cons(x5 x14, x13)
25: rmap1_l_1(x5, x6, x12) x13 -> cond_rmap1_l_acc_1(x12, x5, x6, x13)
26: rmap1_5(x5, x6) x12 -> rmap1_l_1(x5, x6, x12)
27: rmap1_3(x10, x12) x24 -> rmap1_l_1(x10, x12, x24)
28: cond_rmap2_l_acc_1(Nil(), x5, x6, x9) -> x9
29: cond_rmap2_l_acc_1(Cons(x10, x11), x5, x6, x9) -> rmap1_3(x5, x6) x11 Cons(x6 x10, x9)
30: rmap2_l_1(x5, x6, x8) x9 -> cond_rmap2_l_acc_1(x8, x5, x6, x9)
31: rmap2_5(x5, x6) x8 -> rmap2_l_1(x5, x6, x8)
32: rmap2_3(x10, x12) x16 -> rmap2_l_1(x10, x12, x16)
33: map_rev2_f1_f2(x23, x27) x14 -> rmap1(x23, x27) x14 Nil()
34: map_rev2_f1(x5) x6 -> map_rev2_f1_f2(x5, x6)
35: map_rev2() x5 -> map_rev2_f1(x5)
36: main_4(x28, x175, x367, x55) x71 -> x55 h() (x71 h() (rmap1(f(x367), g(x175)) x28 Nil()))
37: map_rev_f_l(x5) x6 -> x6 x5 Nil()
38: rmap_l_acc_2(x4, x9) x10 -> rmap(x4) x9 x10
39: cond_rmap_l_acc(Nil(), x4, x7) -> x7
40: cond_rmap_l_acc(Cons(x17, x18), x8, x15) -> rmap(x8) x18 Cons(x8 x17, x15)
41: rmap_l(x4, x6) x7 -> cond_rmap_l_acc(x6, x4, x7)
42: rmap_1(x4) x6 -> rmap_l(x4, x6)
43: rmap(x8) x12 -> rmap_l(x8, x12)
44: map_rev_f(x9) x10 -> rmap(x9) x10 Nil()
45: map_rev() x4 -> map_rev_f(x4)
46: main_3(x103, x175, x367) x55 -> x55 h() (rmap(h()) (map_rev2_f1(f(x367)) g(x175) x103) Nil())
47: map_f_l_2(x3, x5) x7 -> Cons(x3 x5, x7)
48: cond_map_f_l(Nil(), x3) -> Nil()
49: cond_map_f_l(Cons(x10, x13), x6) -> Cons(x6 x10, map() x6 x13)
50: map_f(x3) x4 -> cond_map_f_l(x4, x3)
51: map_1() x3 -> map_f(x3)
52: map() x6 -> map_f(x6)
53: main_2(x199, x183) x375 -> map() h() (rmap(h()) (map_rev2_f1(f(x375)) g(x183) x199) Nil())
54: cond_mult_n_m(0(), x1, x3) -> 0()
55: cond_mult_n_m(S(x4), x1, x3) -> S(x1 (mult(x1) x4 x3) x3)
56: mult_n(x1, x2) x3 -> cond_mult_n_m(x2, x1, x3)
57: mult_1(x1) x2 -> mult_n(x1, x2)
58: mult(x2) x4 -> mult_n(x2, x4)
59: main_1(x199) x143 -> map() h() (rmap(h()) (map_rev2_f1(f(mult(x143))) g(x143) x199) Nil())
60: cond_plus_n_m(0(), x1) -> x1
61: cond_plus_n_m(S(x3), x1) -> S(plus() x1 x3)
62: plus_n(x1) x2 -> cond_plus_n_m(x2, x1)
63: plus_1() x1 -> plus_n(x1)
64: plus() x2 -> plus_n(x2)
65: main(x199) -> map() h() (rmap(h()) (map_rev2_f1(f(mult(plus()))) g(plus()) x199) Nil())
where
0 :: nat
Cons :: 'a -> 'a list -> 'a list
Nil :: 'a list
S :: nat -> nat
f :: (nat -> nat -> nat) -> nat -> nat
map_f :: (nat -> nat) -> nat list -> nat list
map_rev_f :: (nat -> nat) -> nat list -> nat list
map_rev2_f1 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list
map_rev2_f1_f2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list
g :: (nat -> nat -> nat) -> nat -> nat
h :: nat -> nat
map_rev_f_l :: nat list -> (nat list -> nat list -> nat list) -> nat list
map_rev2_f1_f2_l :: nat list
-> (nat list -> nat list -> nat list)
-> (nat list -> nat list -> nat list)
-> nat list
rmap_l :: (nat -> nat) -> nat list -> nat list -> nat list
rmap1_l :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2_l :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
map_rev :: (nat -> nat) -> nat list -> nat list
map_rev2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list
mult_n :: (nat -> nat -> nat) -> nat -> nat -> nat
plus_n :: nat -> nat -> nat
main_xs :: (nat -> nat -> nat)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> (nat -> nat) -> nat list -> nat list)
-> nat list
-> (nat -> nat)
-> nat list
map_rev2_f1_f2_l_1 :: nat list
-> (nat list -> nat list -> nat list)
-> (nat list -> nat list -> nat list)
-> nat list
rmap1_l_1 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2_l_1 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
main_1 :: nat list -> (nat -> nat -> nat) -> nat list
map_1 :: (nat -> nat) -> nat list -> nat list
mult_1 :: (nat -> nat -> nat) -> nat -> nat -> nat
plus_1 :: nat -> nat -> nat
rmap_1 :: (nat -> nat) -> nat list -> nat list -> nat list
main_xs_1 :: ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> (nat -> nat) -> nat list -> nat list)
-> nat list
-> (nat -> nat)
-> (nat -> nat)
-> nat list
rmap_l_acc_2 :: (nat -> nat) -> nat list -> nat list -> nat list
map_f_l_2 :: (nat -> nat) -> nat -> nat list -> nat list
main_2 :: nat list -> (nat -> nat -> nat) -> (nat -> nat -> nat) -> nat list
rmap1_2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2_2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
main_xs_2 :: ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> (nat -> nat) -> nat list -> nat list)
-> nat list
-> (nat -> nat)
-> (nat -> nat)
-> (nat -> nat)
-> nat list
main_3 :: nat list
-> (nat -> nat -> nat)
-> (nat -> nat -> nat)
-> ((nat -> nat) -> nat list -> nat list)
-> nat list
main_xs_3 :: ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> nat list
-> (nat -> nat)
-> (nat list -> nat list)
-> nat list
main_4 :: nat list
-> (nat -> nat -> nat)
-> (nat -> nat -> nat)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> nat list
main_5 :: nat list
-> (nat -> nat -> nat)
-> (nat -> nat -> nat)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> (nat -> nat) -> nat list -> nat list)
-> nat list
rmap1_5 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2_5 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
main_6 :: nat list -> (nat list -> nat list) -> nat list
main_7 :: (nat -> nat -> nat)
-> (nat -> nat -> nat)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> (nat -> nat) -> nat list -> nat list)
-> nat list
-> nat list
cond_rmap_l_acc :: nat list -> (nat -> nat) -> nat list -> nat list
cond_rmap1_l_acc :: nat list -> (nat -> nat) -> (nat -> nat) -> nat list -> nat list
cond_rmap2_l_acc :: nat list -> (nat -> nat) -> (nat -> nat) -> nat list -> nat list
cond_map_f_l :: nat list -> (nat -> nat) -> nat list
cond_mult_n_m :: nat -> (nat -> nat -> nat) -> nat -> nat
cond_plus_n_m :: nat -> nat -> nat
cond_rmap1_l_acc_1 :: nat list -> (nat -> nat) -> (nat -> nat) -> nat list -> nat list
cond_rmap2_l_acc_1 :: nat list -> (nat -> nat) -> (nat -> nat) -> nat list -> nat list
map :: (nat -> nat) -> nat list -> nat list
mult :: (nat -> nat -> nat) -> nat -> nat -> nat
plus :: nat -> nat -> nat
rmap :: (nat -> nat) -> nat list -> nat list -> nat list
rmap1 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap1_3 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2_3 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
main :: nat list -> nat list
+ Applied Processor:
Inline {inlineType = InlineRewrite, inlineSelect = <inline non-recursive>}
+ Details:
none
* Step 11: Inline WORST_CASE(?,O(n^4))
+ Considered Problem:
1: main_6(x0) x6 -> x6 x0
2: main_xs_3(x3, x4, x6, x9) x10 -> x3 x9 (x4 x9 (x10 x6))
3: main_xs_2(x6, x8, x11, x12, x15, x17) x18 -> x6 x18 (x8 x18 (x11 x15 x17 x12))
4: h() x9 -> x9
5: main_xs_1(x6, x8, x21, x12, x29) x33 -> x6 h() (x8 h() (x21 x29 x33 x12))
6: g(x1) x8 -> x1 x8 S(S(0()))
7: main_xs(x7, x12, x16, x22, x24) x30 -> x12 h() (x16 h() (x22 x30 g(x7) x24))
8: f(x2) x7 -> x2 x7 (x2 x7 x7)
9: main_7(x11, x23, x6, x8, x41) x12 -> x6 h() (x8 h() (x41 f(x23) g(x11) x12))
10: main_5(x24, x9, x21, x12, x16) x42 -> x12 h() (x16 h() (x42 f(x21) g(x9) x24))
11: map_rev2_f1_f2_l_1(x7, x8) x9 -> x8 x7 Nil()
12: map_rev2_f1_f2_l(x7) x8 -> map_rev2_f1_f2_l_1(x7, x8)
13: cond_rmap2_l_acc(Nil(), x5, x6, x13) -> x13
14: cond_rmap2_l_acc(Cons(x14, x15), x5, x6, x13) -> rmap1(x5, x6) x15 Cons(x6 x14, x13)
15: rmap2_l(x5, x6, x12) x13 -> cond_rmap2_l_acc(x12, x5, x6, x13)
16: rmap2_2(x5, x6) x12 -> rmap2_l(x5, x6, x12)
17: rmap2(x10, x12) x24 -> rmap2_l(x10, x12, x24)
18: cond_rmap1_l_acc(Nil(), x5, x6, x9) -> x9
19: cond_rmap1_l_acc(Cons(x10, x11), x5, x6, x9) -> rmap2(x5, x6) x11 Cons(x5 x10, x9)
20: rmap1_l(x5, x6, x8) x9 -> cond_rmap1_l_acc(x8, x5, x6, x9)
21: rmap1_2(x5, x6) x8 -> rmap1_l(x5, x6, x8)
22: rmap1(x10, x12) x16 -> rmap1_l(x10, x12, x16)
23: cond_rmap1_l_acc_1(Nil(), x5, x6, x13) -> x13
24: cond_rmap1_l_acc_1(Cons(x14, x15), x5, x6, x13) -> rmap2_3(x5, x6) x15 Cons(x5 x14, x13)
25: rmap1_l_1(x5, x6, x12) x13 -> cond_rmap1_l_acc_1(x12, x5, x6, x13)
26: rmap1_5(x5, x6) x12 -> rmap1_l_1(x5, x6, x12)
27: rmap1_3(x10, x12) x24 -> rmap1_l_1(x10, x12, x24)
28: cond_rmap2_l_acc_1(Nil(), x5, x6, x9) -> x9
29: cond_rmap2_l_acc_1(Cons(x10, x11), x5, x6, x9) -> rmap1_3(x5, x6) x11 Cons(x6 x10, x9)
30: rmap2_l_1(x5, x6, x8) x9 -> cond_rmap2_l_acc_1(x8, x5, x6, x9)
31: rmap2_5(x5, x6) x8 -> rmap2_l_1(x5, x6, x8)
32: rmap2_3(x10, x12) x16 -> rmap2_l_1(x10, x12, x16)
33: map_rev2_f1_f2(x23, x27) x14 -> rmap1(x23, x27) x14 Nil()
34: map_rev2_f1(x5) x6 -> map_rev2_f1_f2(x5, x6)
35: map_rev2() x5 -> map_rev2_f1(x5)
36: main_4(x28, x175, x367, x55) x71 -> x55 h() (x71 h() (rmap1(f(x367), g(x175)) x28 Nil()))
37: map_rev_f_l(x5) x6 -> x6 x5 Nil()
38: rmap_l_acc_2(x4, x9) x10 -> rmap(x4) x9 x10
39: cond_rmap_l_acc(Nil(), x4, x7) -> x7
40: cond_rmap_l_acc(Cons(x17, x18), x8, x15) -> rmap(x8) x18 Cons(x8 x17, x15)
41: rmap_l(x4, x6) x7 -> cond_rmap_l_acc(x6, x4, x7)
42: rmap_1(x4) x6 -> rmap_l(x4, x6)
43: rmap(x8) x12 -> rmap_l(x8, x12)
44: map_rev_f(x9) x10 -> rmap(x9) x10 Nil()
45: map_rev() x4 -> map_rev_f(x4)
46: main_3(x207, x351, x735) x111 -> x111 h() (rmap(h()) (map_rev2_f1_f2(f(x735), g(x351)) x207) Nil())
47: map_f_l_2(x3, x5) x7 -> Cons(x3 x5, x7)
48: cond_map_f_l(Nil(), x3) -> Nil()
49: cond_map_f_l(Cons(x10, x13), x6) -> Cons(x6 x10, map() x6 x13)
50: map_f(x3) x4 -> cond_map_f_l(x4, x3)
51: map_1() x3 -> map_f(x3)
52: map() x6 -> map_f(x6)
53: main_2(x399, x367) x751 -> map() h() (rmap(h()) (map_rev2_f1_f2(f(x751), g(x367)) x399) Nil())
54: cond_mult_n_m(0(), x1, x3) -> 0()
55: cond_mult_n_m(S(x4), x1, x3) -> S(x1 (mult(x1) x4 x3) x3)
56: mult_n(x1, x2) x3 -> cond_mult_n_m(x2, x1, x3)
57: mult_1(x1) x2 -> mult_n(x1, x2)
58: mult(x2) x4 -> mult_n(x2, x4)
59: main_1(x399) x287 -> map() h() (rmap(h()) (map_rev2_f1_f2(f(mult(x287)), g(x287)) x399) Nil())
60: cond_plus_n_m(0(), x1) -> x1
61: cond_plus_n_m(S(x3), x1) -> S(plus() x1 x3)
62: plus_n(x1) x2 -> cond_plus_n_m(x2, x1)
63: plus_1() x1 -> plus_n(x1)
64: plus() x2 -> plus_n(x2)
65: main(x399) -> map() h() (rmap(h()) (map_rev2_f1_f2(f(mult(plus())), g(plus())) x399) Nil())
where
0 :: nat
Cons :: 'a -> 'a list -> 'a list
Nil :: 'a list
S :: nat -> nat
f :: (nat -> nat -> nat) -> nat -> nat
map_f :: (nat -> nat) -> nat list -> nat list
map_rev_f :: (nat -> nat) -> nat list -> nat list
map_rev2_f1 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list
map_rev2_f1_f2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list
g :: (nat -> nat -> nat) -> nat -> nat
h :: nat -> nat
map_rev_f_l :: nat list -> (nat list -> nat list -> nat list) -> nat list
map_rev2_f1_f2_l :: nat list
-> (nat list -> nat list -> nat list)
-> (nat list -> nat list -> nat list)
-> nat list
rmap_l :: (nat -> nat) -> nat list -> nat list -> nat list
rmap1_l :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2_l :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
map_rev :: (nat -> nat) -> nat list -> nat list
map_rev2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list
mult_n :: (nat -> nat -> nat) -> nat -> nat -> nat
plus_n :: nat -> nat -> nat
main_xs :: (nat -> nat -> nat)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> (nat -> nat) -> nat list -> nat list)
-> nat list
-> (nat -> nat)
-> nat list
map_rev2_f1_f2_l_1 :: nat list
-> (nat list -> nat list -> nat list)
-> (nat list -> nat list -> nat list)
-> nat list
rmap1_l_1 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2_l_1 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
main_1 :: nat list -> (nat -> nat -> nat) -> nat list
map_1 :: (nat -> nat) -> nat list -> nat list
mult_1 :: (nat -> nat -> nat) -> nat -> nat -> nat
plus_1 :: nat -> nat -> nat
rmap_1 :: (nat -> nat) -> nat list -> nat list -> nat list
main_xs_1 :: ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> (nat -> nat) -> nat list -> nat list)
-> nat list
-> (nat -> nat)
-> (nat -> nat)
-> nat list
rmap_l_acc_2 :: (nat -> nat) -> nat list -> nat list -> nat list
map_f_l_2 :: (nat -> nat) -> nat -> nat list -> nat list
main_2 :: nat list -> (nat -> nat -> nat) -> (nat -> nat -> nat) -> nat list
rmap1_2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2_2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
main_xs_2 :: ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> (nat -> nat) -> nat list -> nat list)
-> nat list
-> (nat -> nat)
-> (nat -> nat)
-> (nat -> nat)
-> nat list
main_3 :: nat list
-> (nat -> nat -> nat)
-> (nat -> nat -> nat)
-> ((nat -> nat) -> nat list -> nat list)
-> nat list
main_xs_3 :: ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> nat list
-> (nat -> nat)
-> (nat list -> nat list)
-> nat list
main_4 :: nat list
-> (nat -> nat -> nat)
-> (nat -> nat -> nat)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> nat list
main_5 :: nat list
-> (nat -> nat -> nat)
-> (nat -> nat -> nat)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> (nat -> nat) -> nat list -> nat list)
-> nat list
rmap1_5 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2_5 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
main_6 :: nat list -> (nat list -> nat list) -> nat list
main_7 :: (nat -> nat -> nat)
-> (nat -> nat -> nat)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> (nat -> nat) -> nat list -> nat list)
-> nat list
-> nat list
cond_rmap_l_acc :: nat list -> (nat -> nat) -> nat list -> nat list
cond_rmap1_l_acc :: nat list -> (nat -> nat) -> (nat -> nat) -> nat list -> nat list
cond_rmap2_l_acc :: nat list -> (nat -> nat) -> (nat -> nat) -> nat list -> nat list
cond_map_f_l :: nat list -> (nat -> nat) -> nat list
cond_mult_n_m :: nat -> (nat -> nat -> nat) -> nat -> nat
cond_plus_n_m :: nat -> nat -> nat
cond_rmap1_l_acc_1 :: nat list -> (nat -> nat) -> (nat -> nat) -> nat list -> nat list
cond_rmap2_l_acc_1 :: nat list -> (nat -> nat) -> (nat -> nat) -> nat list -> nat list
map :: (nat -> nat) -> nat list -> nat list
mult :: (nat -> nat -> nat) -> nat -> nat -> nat
plus :: nat -> nat -> nat
rmap :: (nat -> nat) -> nat list -> nat list -> nat list
rmap1 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap1_3 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2_3 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
main :: nat list -> nat list
+ Applied Processor:
Inline {inlineType = InlineRewrite, inlineSelect = <inline non-recursive>}
+ Details:
none
* Step 12: Inline WORST_CASE(?,O(n^4))
+ Considered Problem:
1: main_6(x0) x6 -> x6 x0
2: main_xs_3(x3, x4, x6, x9) x10 -> x3 x9 (x4 x9 (x10 x6))
3: main_xs_2(x6, x8, x11, x12, x15, x17) x18 -> x6 x18 (x8 x18 (x11 x15 x17 x12))
4: h() x9 -> x9
5: main_xs_1(x6, x8, x21, x12, x29) x33 -> x6 h() (x8 h() (x21 x29 x33 x12))
6: g(x1) x8 -> x1 x8 S(S(0()))
7: main_xs(x7, x12, x16, x22, x24) x30 -> x12 h() (x16 h() (x22 x30 g(x7) x24))
8: f(x2) x7 -> x2 x7 (x2 x7 x7)
9: main_7(x11, x23, x6, x8, x41) x12 -> x6 h() (x8 h() (x41 f(x23) g(x11) x12))
10: main_5(x24, x9, x21, x12, x16) x42 -> x12 h() (x16 h() (x42 f(x21) g(x9) x24))
11: map_rev2_f1_f2_l_1(x7, x8) x9 -> x8 x7 Nil()
12: map_rev2_f1_f2_l(x7) x8 -> map_rev2_f1_f2_l_1(x7, x8)
13: cond_rmap2_l_acc(Nil(), x5, x6, x13) -> x13
14: cond_rmap2_l_acc(Cons(x14, x15), x5, x6, x13) -> rmap1(x5, x6) x15 Cons(x6 x14, x13)
15: rmap2_l(x5, x6, x12) x13 -> cond_rmap2_l_acc(x12, x5, x6, x13)
16: rmap2_2(x5, x6) x12 -> rmap2_l(x5, x6, x12)
17: rmap2(x10, x12) x24 -> rmap2_l(x10, x12, x24)
18: cond_rmap1_l_acc(Nil(), x5, x6, x9) -> x9
19: cond_rmap1_l_acc(Cons(x10, x11), x5, x6, x9) -> rmap2(x5, x6) x11 Cons(x5 x10, x9)
20: rmap1_l(x5, x6, x8) x9 -> cond_rmap1_l_acc(x8, x5, x6, x9)
21: rmap1_2(x5, x6) x8 -> rmap1_l(x5, x6, x8)
22: rmap1(x10, x12) x16 -> rmap1_l(x10, x12, x16)
23: cond_rmap1_l_acc_1(Nil(), x5, x6, x13) -> x13
24: cond_rmap1_l_acc_1(Cons(x14, x15), x5, x6, x13) -> rmap2_3(x5, x6) x15 Cons(x5 x14, x13)
25: rmap1_l_1(x5, x6, x12) x13 -> cond_rmap1_l_acc_1(x12, x5, x6, x13)
26: rmap1_5(x5, x6) x12 -> rmap1_l_1(x5, x6, x12)
27: rmap1_3(x10, x12) x24 -> rmap1_l_1(x10, x12, x24)
28: cond_rmap2_l_acc_1(Nil(), x5, x6, x9) -> x9
29: cond_rmap2_l_acc_1(Cons(x10, x11), x5, x6, x9) -> rmap1_3(x5, x6) x11 Cons(x6 x10, x9)
30: rmap2_l_1(x5, x6, x8) x9 -> cond_rmap2_l_acc_1(x8, x5, x6, x9)
31: rmap2_5(x5, x6) x8 -> rmap2_l_1(x5, x6, x8)
32: rmap2_3(x10, x12) x16 -> rmap2_l_1(x10, x12, x16)
33: map_rev2_f1_f2(x23, x27) x14 -> rmap1(x23, x27) x14 Nil()
34: map_rev2_f1(x5) x6 -> map_rev2_f1_f2(x5, x6)
35: map_rev2() x5 -> map_rev2_f1(x5)
36: main_4(x28, x175, x367, x55) x71 -> x55 h() (x71 h() (rmap1(f(x367), g(x175)) x28 Nil()))
37: map_rev_f_l(x5) x6 -> x6 x5 Nil()
38: rmap_l_acc_2(x4, x9) x10 -> rmap(x4) x9 x10
39: cond_rmap_l_acc(Nil(), x4, x7) -> x7
40: cond_rmap_l_acc(Cons(x17, x18), x8, x15) -> rmap(x8) x18 Cons(x8 x17, x15)
41: rmap_l(x4, x6) x7 -> cond_rmap_l_acc(x6, x4, x7)
42: rmap_1(x4) x6 -> rmap_l(x4, x6)
43: rmap(x8) x12 -> rmap_l(x8, x12)
44: map_rev_f(x9) x10 -> rmap(x9) x10 Nil()
45: map_rev() x4 -> map_rev_f(x4)
46: main_3(x28, x703, x1471) x223 -> x223 h() (rmap(h()) (rmap1(f(x1471), g(x703)) x28 Nil()) Nil())
47: map_f_l_2(x3, x5) x7 -> Cons(x3 x5, x7)
48: cond_map_f_l(Nil(), x3) -> Nil()
49: cond_map_f_l(Cons(x10, x13), x6) -> Cons(x6 x10, map() x6 x13)
50: map_f(x3) x4 -> cond_map_f_l(x4, x3)
51: map_1() x3 -> map_f(x3)
52: map() x6 -> map_f(x6)
53: main_2(x28, x735) x1503 -> map() h() (rmap(h()) (rmap1(f(x1503), g(x735)) x28 Nil()) Nil())
54: cond_mult_n_m(0(), x1, x3) -> 0()
55: cond_mult_n_m(S(x4), x1, x3) -> S(x1 (mult(x1) x4 x3) x3)
56: mult_n(x1, x2) x3 -> cond_mult_n_m(x2, x1, x3)
57: mult_1(x1) x2 -> mult_n(x1, x2)
58: mult(x2) x4 -> mult_n(x2, x4)
59: main_1(x28) x575 -> map() h() (rmap(h()) (rmap1(f(mult(x575)), g(x575)) x28 Nil()) Nil())
60: cond_plus_n_m(0(), x1) -> x1
61: cond_plus_n_m(S(x3), x1) -> S(plus() x1 x3)
62: plus_n(x1) x2 -> cond_plus_n_m(x2, x1)
63: plus_1() x1 -> plus_n(x1)
64: plus() x2 -> plus_n(x2)
65: main(x28) -> map() h() (rmap(h()) (rmap1(f(mult(plus())), g(plus())) x28 Nil()) Nil())
where
0 :: nat
Cons :: 'a -> 'a list -> 'a list
Nil :: 'a list
S :: nat -> nat
f :: (nat -> nat -> nat) -> nat -> nat
map_f :: (nat -> nat) -> nat list -> nat list
map_rev_f :: (nat -> nat) -> nat list -> nat list
map_rev2_f1 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list
map_rev2_f1_f2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list
g :: (nat -> nat -> nat) -> nat -> nat
h :: nat -> nat
map_rev_f_l :: nat list -> (nat list -> nat list -> nat list) -> nat list
map_rev2_f1_f2_l :: nat list
-> (nat list -> nat list -> nat list)
-> (nat list -> nat list -> nat list)
-> nat list
rmap_l :: (nat -> nat) -> nat list -> nat list -> nat list
rmap1_l :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2_l :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
map_rev :: (nat -> nat) -> nat list -> nat list
map_rev2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list
mult_n :: (nat -> nat -> nat) -> nat -> nat -> nat
plus_n :: nat -> nat -> nat
main_xs :: (nat -> nat -> nat)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> (nat -> nat) -> nat list -> nat list)
-> nat list
-> (nat -> nat)
-> nat list
map_rev2_f1_f2_l_1 :: nat list
-> (nat list -> nat list -> nat list)
-> (nat list -> nat list -> nat list)
-> nat list
rmap1_l_1 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2_l_1 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
main_1 :: nat list -> (nat -> nat -> nat) -> nat list
map_1 :: (nat -> nat) -> nat list -> nat list
mult_1 :: (nat -> nat -> nat) -> nat -> nat -> nat
plus_1 :: nat -> nat -> nat
rmap_1 :: (nat -> nat) -> nat list -> nat list -> nat list
main_xs_1 :: ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> (nat -> nat) -> nat list -> nat list)
-> nat list
-> (nat -> nat)
-> (nat -> nat)
-> nat list
rmap_l_acc_2 :: (nat -> nat) -> nat list -> nat list -> nat list
map_f_l_2 :: (nat -> nat) -> nat -> nat list -> nat list
main_2 :: nat list -> (nat -> nat -> nat) -> (nat -> nat -> nat) -> nat list
rmap1_2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2_2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
main_xs_2 :: ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> (nat -> nat) -> nat list -> nat list)
-> nat list
-> (nat -> nat)
-> (nat -> nat)
-> (nat -> nat)
-> nat list
main_3 :: nat list
-> (nat -> nat -> nat)
-> (nat -> nat -> nat)
-> ((nat -> nat) -> nat list -> nat list)
-> nat list
main_xs_3 :: ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> nat list
-> (nat -> nat)
-> (nat list -> nat list)
-> nat list
main_4 :: nat list
-> (nat -> nat -> nat)
-> (nat -> nat -> nat)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> nat list
main_5 :: nat list
-> (nat -> nat -> nat)
-> (nat -> nat -> nat)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> (nat -> nat) -> nat list -> nat list)
-> nat list
rmap1_5 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2_5 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
main_6 :: nat list -> (nat list -> nat list) -> nat list
main_7 :: (nat -> nat -> nat)
-> (nat -> nat -> nat)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> (nat -> nat) -> nat list -> nat list)
-> nat list
-> nat list
cond_rmap_l_acc :: nat list -> (nat -> nat) -> nat list -> nat list
cond_rmap1_l_acc :: nat list -> (nat -> nat) -> (nat -> nat) -> nat list -> nat list
cond_rmap2_l_acc :: nat list -> (nat -> nat) -> (nat -> nat) -> nat list -> nat list
cond_map_f_l :: nat list -> (nat -> nat) -> nat list
cond_mult_n_m :: nat -> (nat -> nat -> nat) -> nat -> nat
cond_plus_n_m :: nat -> nat -> nat
cond_rmap1_l_acc_1 :: nat list -> (nat -> nat) -> (nat -> nat) -> nat list -> nat list
cond_rmap2_l_acc_1 :: nat list -> (nat -> nat) -> (nat -> nat) -> nat list -> nat list
map :: (nat -> nat) -> nat list -> nat list
mult :: (nat -> nat -> nat) -> nat -> nat -> nat
plus :: nat -> nat -> nat
rmap :: (nat -> nat) -> nat list -> nat list -> nat list
rmap1 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap1_3 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2_3 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
main :: nat list -> nat list
+ Applied Processor:
Inline {inlineType = InlineFull, inlineSelect = <inline case-expression>}
+ Details:
none
* Step 13: UsableRules WORST_CASE(?,O(n^4))
+ Considered Problem:
1: main_6(x0) x6 -> x6 x0
2: main_xs_3(x3, x4, x6, x9) x10 -> x3 x9 (x4 x9 (x10 x6))
3: main_xs_2(x6, x8, x11, x12, x15, x17) x18 -> x6 x18 (x8 x18 (x11 x15 x17 x12))
4: h() x9 -> x9
5: main_xs_1(x6, x8, x21, x12, x29) x33 -> x6 h() (x8 h() (x21 x29 x33 x12))
6: g(x1) x8 -> x1 x8 S(S(0()))
7: main_xs(x7, x12, x16, x22, x24) x30 -> x12 h() (x16 h() (x22 x30 g(x7) x24))
8: f(x2) x7 -> x2 x7 (x2 x7 x7)
9: main_7(x11, x23, x6, x8, x41) x12 -> x6 h() (x8 h() (x41 f(x23) g(x11) x12))
10: main_5(x24, x9, x21, x12, x16) x42 -> x12 h() (x16 h() (x42 f(x21) g(x9) x24))
11: map_rev2_f1_f2_l_1(x7, x8) x9 -> x8 x7 Nil()
12: map_rev2_f1_f2_l(x7) x8 -> map_rev2_f1_f2_l_1(x7, x8)
13: cond_rmap2_l_acc(Nil(), x5, x6, x13) -> x13
14: cond_rmap2_l_acc(Cons(x14, x15), x5, x6, x13) -> rmap1(x5, x6) x15 Cons(x6 x14, x13)
15: rmap2_l(x10, x12, Nil()) x26 -> x26
16: rmap2_l(x10, x12, Cons(x28, x30)) x26 -> rmap1(x10, x12) x30 Cons(x12 x28, x26)
17: rmap2_2(x5, x6) x12 -> rmap2_l(x5, x6, x12)
18: rmap2(x10, x12) x24 -> rmap2_l(x10, x12, x24)
19: cond_rmap1_l_acc(Nil(), x5, x6, x9) -> x9
20: cond_rmap1_l_acc(Cons(x10, x11), x5, x6, x9) -> rmap2(x5, x6) x11 Cons(x5 x10, x9)
21: rmap1_l(x10, x12, Nil()) x18 -> x18
22: rmap1_l(x10, x12, Cons(x20, x22)) x18 -> rmap2(x10, x12) x22 Cons(x10 x20, x18)
23: rmap1_2(x5, x6) x8 -> rmap1_l(x5, x6, x8)
24: rmap1(x10, x12) x16 -> rmap1_l(x10, x12, x16)
25: cond_rmap1_l_acc_1(Nil(), x5, x6, x13) -> x13
26: cond_rmap1_l_acc_1(Cons(x14, x15), x5, x6, x13) -> rmap2_3(x5, x6) x15 Cons(x5 x14, x13)
27: rmap1_l_1(x10, x12, Nil()) x26 -> x26
28: rmap1_l_1(x10, x12, Cons(x28, x30)) x26 -> rmap2_3(x10, x12) x30 Cons(x10 x28, x26)
29: rmap1_5(x5, x6) x12 -> rmap1_l_1(x5, x6, x12)
30: rmap1_3(x10, x12) x24 -> rmap1_l_1(x10, x12, x24)
31: cond_rmap2_l_acc_1(Nil(), x5, x6, x9) -> x9
32: cond_rmap2_l_acc_1(Cons(x10, x11), x5, x6, x9) -> rmap1_3(x5, x6) x11 Cons(x6 x10, x9)
33: rmap2_l_1(x10, x12, Nil()) x18 -> x18
34: rmap2_l_1(x10, x12, Cons(x20, x22)) x18 -> rmap1_3(x10, x12) x22 Cons(x12 x20, x18)
35: rmap2_5(x5, x6) x8 -> rmap2_l_1(x5, x6, x8)
36: rmap2_3(x10, x12) x16 -> rmap2_l_1(x10, x12, x16)
37: map_rev2_f1_f2(x23, x27) x14 -> rmap1(x23, x27) x14 Nil()
38: map_rev2_f1(x5) x6 -> map_rev2_f1_f2(x5, x6)
39: map_rev2() x5 -> map_rev2_f1(x5)
40: main_4(x28, x175, x367, x55) x71 -> x55 h() (x71 h() (rmap1(f(x367), g(x175)) x28 Nil()))
41: map_rev_f_l(x5) x6 -> x6 x5 Nil()
42: rmap_l_acc_2(x4, x9) x10 -> rmap(x4) x9 x10
43: cond_rmap_l_acc(Nil(), x4, x7) -> x7
44: cond_rmap_l_acc(Cons(x17, x18), x8, x15) -> rmap(x8) x18 Cons(x8 x17, x15)
45: rmap_l(x8, Nil()) x14 -> x14
46: rmap_l(x16, Cons(x34, x36)) x30 -> rmap(x16) x36 Cons(x16 x34, x30)
47: rmap_1(x4) x6 -> rmap_l(x4, x6)
48: rmap(x8) x12 -> rmap_l(x8, x12)
49: map_rev_f(x9) x10 -> rmap(x9) x10 Nil()
50: map_rev() x4 -> map_rev_f(x4)
51: main_3(x28, x703, x1471) x223 -> x223 h() (rmap(h()) (rmap1(f(x1471), g(x703)) x28 Nil()) Nil())
52: map_f_l_2(x3, x5) x7 -> Cons(x3 x5, x7)
53: cond_map_f_l(Nil(), x3) -> Nil()
54: cond_map_f_l(Cons(x10, x13), x6) -> Cons(x6 x10, map() x6 x13)
55: map_f(x6) Nil() -> Nil()
56: map_f(x12) Cons(x20, x26) -> Cons(x12 x20, map() x12 x26)
57: map_1() x3 -> map_f(x3)
58: map() x6 -> map_f(x6)
59: main_2(x28, x735) x1503 -> map() h() (rmap(h()) (rmap1(f(x1503), g(x735)) x28 Nil()) Nil())
60: cond_mult_n_m(0(), x1, x3) -> 0()
61: cond_mult_n_m(S(x4), x1, x3) -> S(x1 (mult(x1) x4 x3) x3)
62: mult_n(x2, 0()) x6 -> 0()
63: mult_n(x2, S(x8)) x6 -> S(x2 (mult(x2) x8 x6) x6)
64: mult_1(x1) x2 -> mult_n(x1, x2)
65: mult(x2) x4 -> mult_n(x2, x4)
66: main_1(x28) x575 -> map() h() (rmap(h()) (rmap1(f(mult(x575)), g(x575)) x28 Nil()) Nil())
67: cond_plus_n_m(0(), x1) -> x1
68: cond_plus_n_m(S(x3), x1) -> S(plus() x1 x3)
69: plus_n(x2) 0() -> x2
70: plus_n(x2) S(x6) -> S(plus() x2 x6)
71: plus_1() x1 -> plus_n(x1)
72: plus() x2 -> plus_n(x2)
73: main(x28) -> map() h() (rmap(h()) (rmap1(f(mult(plus())), g(plus())) x28 Nil()) Nil())
where
0 :: nat
Cons :: 'a -> 'a list -> 'a list
Nil :: 'a list
S :: nat -> nat
f :: (nat -> nat -> nat) -> nat -> nat
map_f :: (nat -> nat) -> nat list -> nat list
map_rev_f :: (nat -> nat) -> nat list -> nat list
map_rev2_f1 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list
map_rev2_f1_f2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list
g :: (nat -> nat -> nat) -> nat -> nat
h :: nat -> nat
map_rev_f_l :: nat list -> (nat list -> nat list -> nat list) -> nat list
map_rev2_f1_f2_l :: nat list
-> (nat list -> nat list -> nat list)
-> (nat list -> nat list -> nat list)
-> nat list
rmap_l :: (nat -> nat) -> nat list -> nat list -> nat list
rmap1_l :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2_l :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
map_rev :: (nat -> nat) -> nat list -> nat list
map_rev2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list
mult_n :: (nat -> nat -> nat) -> nat -> nat -> nat
plus_n :: nat -> nat -> nat
main_xs :: (nat -> nat -> nat)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> (nat -> nat) -> nat list -> nat list)
-> nat list
-> (nat -> nat)
-> nat list
map_rev2_f1_f2_l_1 :: nat list
-> (nat list -> nat list -> nat list)
-> (nat list -> nat list -> nat list)
-> nat list
rmap1_l_1 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2_l_1 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
main_1 :: nat list -> (nat -> nat -> nat) -> nat list
map_1 :: (nat -> nat) -> nat list -> nat list
mult_1 :: (nat -> nat -> nat) -> nat -> nat -> nat
plus_1 :: nat -> nat -> nat
rmap_1 :: (nat -> nat) -> nat list -> nat list -> nat list
main_xs_1 :: ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> (nat -> nat) -> nat list -> nat list)
-> nat list
-> (nat -> nat)
-> (nat -> nat)
-> nat list
rmap_l_acc_2 :: (nat -> nat) -> nat list -> nat list -> nat list
map_f_l_2 :: (nat -> nat) -> nat -> nat list -> nat list
main_2 :: nat list -> (nat -> nat -> nat) -> (nat -> nat -> nat) -> nat list
rmap1_2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2_2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
main_xs_2 :: ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> (nat -> nat) -> nat list -> nat list)
-> nat list
-> (nat -> nat)
-> (nat -> nat)
-> (nat -> nat)
-> nat list
main_3 :: nat list
-> (nat -> nat -> nat)
-> (nat -> nat -> nat)
-> ((nat -> nat) -> nat list -> nat list)
-> nat list
main_xs_3 :: ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> nat list
-> (nat -> nat)
-> (nat list -> nat list)
-> nat list
main_4 :: nat list
-> (nat -> nat -> nat)
-> (nat -> nat -> nat)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> nat list
main_5 :: nat list
-> (nat -> nat -> nat)
-> (nat -> nat -> nat)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> (nat -> nat) -> nat list -> nat list)
-> nat list
rmap1_5 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2_5 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
main_6 :: nat list -> (nat list -> nat list) -> nat list
main_7 :: (nat -> nat -> nat)
-> (nat -> nat -> nat)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> (nat -> nat) -> nat list -> nat list)
-> nat list
-> nat list
cond_rmap_l_acc :: nat list -> (nat -> nat) -> nat list -> nat list
cond_rmap1_l_acc :: nat list -> (nat -> nat) -> (nat -> nat) -> nat list -> nat list
cond_rmap2_l_acc :: nat list -> (nat -> nat) -> (nat -> nat) -> nat list -> nat list
cond_map_f_l :: nat list -> (nat -> nat) -> nat list
cond_mult_n_m :: nat -> (nat -> nat -> nat) -> nat -> nat
cond_plus_n_m :: nat -> nat -> nat
cond_rmap1_l_acc_1 :: nat list -> (nat -> nat) -> (nat -> nat) -> nat list -> nat list
cond_rmap2_l_acc_1 :: nat list -> (nat -> nat) -> (nat -> nat) -> nat list -> nat list
map :: (nat -> nat) -> nat list -> nat list
mult :: (nat -> nat -> nat) -> nat -> nat -> nat
plus :: nat -> nat -> nat
rmap :: (nat -> nat) -> nat list -> nat list -> nat list
rmap1 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap1_3 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2_3 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
main :: nat list -> nat list
+ Applied Processor:
UsableRules {urType = Syntactic}
+ Details:
none
* Step 14: NeededRules WORST_CASE(?,O(n^4))
+ Considered Problem:
1: main_6(x0) x6 -> x6 x0
2: main_xs_3(x3, x4, x6, x9) x10 -> x3 x9 (x4 x9 (x10 x6))
3: main_xs_2(x6, x8, x11, x12, x15, x17) x18 -> x6 x18 (x8 x18 (x11 x15 x17 x12))
4: h() x9 -> x9
5: main_xs_1(x6, x8, x21, x12, x29) x33 -> x6 h() (x8 h() (x21 x29 x33 x12))
6: g(x1) x8 -> x1 x8 S(S(0()))
7: main_xs(x7, x12, x16, x22, x24) x30 -> x12 h() (x16 h() (x22 x30 g(x7) x24))
8: f(x2) x7 -> x2 x7 (x2 x7 x7)
9: main_7(x11, x23, x6, x8, x41) x12 -> x6 h() (x8 h() (x41 f(x23) g(x11) x12))
10: main_5(x24, x9, x21, x12, x16) x42 -> x12 h() (x16 h() (x42 f(x21) g(x9) x24))
11: map_rev2_f1_f2_l_1(x7, x8) x9 -> x8 x7 Nil()
12: map_rev2_f1_f2_l(x7) x8 -> map_rev2_f1_f2_l_1(x7, x8)
15: rmap2_l(x10, x12, Nil()) x26 -> x26
16: rmap2_l(x10, x12, Cons(x28, x30)) x26 -> rmap1(x10, x12) x30 Cons(x12 x28, x26)
17: rmap2_2(x5, x6) x12 -> rmap2_l(x5, x6, x12)
18: rmap2(x10, x12) x24 -> rmap2_l(x10, x12, x24)
21: rmap1_l(x10, x12, Nil()) x18 -> x18
22: rmap1_l(x10, x12, Cons(x20, x22)) x18 -> rmap2(x10, x12) x22 Cons(x10 x20, x18)
23: rmap1_2(x5, x6) x8 -> rmap1_l(x5, x6, x8)
24: rmap1(x10, x12) x16 -> rmap1_l(x10, x12, x16)
27: rmap1_l_1(x10, x12, Nil()) x26 -> x26
28: rmap1_l_1(x10, x12, Cons(x28, x30)) x26 -> rmap2_3(x10, x12) x30 Cons(x10 x28, x26)
29: rmap1_5(x5, x6) x12 -> rmap1_l_1(x5, x6, x12)
30: rmap1_3(x10, x12) x24 -> rmap1_l_1(x10, x12, x24)
33: rmap2_l_1(x10, x12, Nil()) x18 -> x18
34: rmap2_l_1(x10, x12, Cons(x20, x22)) x18 -> rmap1_3(x10, x12) x22 Cons(x12 x20, x18)
35: rmap2_5(x5, x6) x8 -> rmap2_l_1(x5, x6, x8)
36: rmap2_3(x10, x12) x16 -> rmap2_l_1(x10, x12, x16)
37: map_rev2_f1_f2(x23, x27) x14 -> rmap1(x23, x27) x14 Nil()
38: map_rev2_f1(x5) x6 -> map_rev2_f1_f2(x5, x6)
39: map_rev2() x5 -> map_rev2_f1(x5)
40: main_4(x28, x175, x367, x55) x71 -> x55 h() (x71 h() (rmap1(f(x367), g(x175)) x28 Nil()))
41: map_rev_f_l(x5) x6 -> x6 x5 Nil()
42: rmap_l_acc_2(x4, x9) x10 -> rmap(x4) x9 x10
45: rmap_l(x8, Nil()) x14 -> x14
46: rmap_l(x16, Cons(x34, x36)) x30 -> rmap(x16) x36 Cons(x16 x34, x30)
47: rmap_1(x4) x6 -> rmap_l(x4, x6)
48: rmap(x8) x12 -> rmap_l(x8, x12)
49: map_rev_f(x9) x10 -> rmap(x9) x10 Nil()
50: map_rev() x4 -> map_rev_f(x4)
51: main_3(x28, x703, x1471) x223 -> x223 h() (rmap(h()) (rmap1(f(x1471), g(x703)) x28 Nil()) Nil())
52: map_f_l_2(x3, x5) x7 -> Cons(x3 x5, x7)
55: map_f(x6) Nil() -> Nil()
56: map_f(x12) Cons(x20, x26) -> Cons(x12 x20, map() x12 x26)
57: map_1() x3 -> map_f(x3)
58: map() x6 -> map_f(x6)
59: main_2(x28, x735) x1503 -> map() h() (rmap(h()) (rmap1(f(x1503), g(x735)) x28 Nil()) Nil())
62: mult_n(x2, 0()) x6 -> 0()
63: mult_n(x2, S(x8)) x6 -> S(x2 (mult(x2) x8 x6) x6)
64: mult_1(x1) x2 -> mult_n(x1, x2)
65: mult(x2) x4 -> mult_n(x2, x4)
66: main_1(x28) x575 -> map() h() (rmap(h()) (rmap1(f(mult(x575)), g(x575)) x28 Nil()) Nil())
69: plus_n(x2) 0() -> x2
70: plus_n(x2) S(x6) -> S(plus() x2 x6)
71: plus_1() x1 -> plus_n(x1)
72: plus() x2 -> plus_n(x2)
73: main(x28) -> map() h() (rmap(h()) (rmap1(f(mult(plus())), g(plus())) x28 Nil()) Nil())
where
0 :: nat
Cons :: 'a -> 'a list -> 'a list
Nil :: 'a list
S :: nat -> nat
f :: (nat -> nat -> nat) -> nat -> nat
map_f :: (nat -> nat) -> nat list -> nat list
map_rev_f :: (nat -> nat) -> nat list -> nat list
map_rev2_f1 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list
map_rev2_f1_f2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list
g :: (nat -> nat -> nat) -> nat -> nat
h :: nat -> nat
map_rev_f_l :: nat list -> (nat list -> nat list -> nat list) -> nat list
map_rev2_f1_f2_l :: nat list
-> (nat list -> nat list -> nat list)
-> (nat list -> nat list -> nat list)
-> nat list
rmap_l :: (nat -> nat) -> nat list -> nat list -> nat list
rmap1_l :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2_l :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
map_rev :: (nat -> nat) -> nat list -> nat list
map_rev2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list
mult_n :: (nat -> nat -> nat) -> nat -> nat -> nat
plus_n :: nat -> nat -> nat
main_xs :: (nat -> nat -> nat)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> (nat -> nat) -> nat list -> nat list)
-> nat list
-> (nat -> nat)
-> nat list
map_rev2_f1_f2_l_1 :: nat list
-> (nat list -> nat list -> nat list)
-> (nat list -> nat list -> nat list)
-> nat list
rmap1_l_1 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2_l_1 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
main_1 :: nat list -> (nat -> nat -> nat) -> nat list
map_1 :: (nat -> nat) -> nat list -> nat list
mult_1 :: (nat -> nat -> nat) -> nat -> nat -> nat
plus_1 :: nat -> nat -> nat
rmap_1 :: (nat -> nat) -> nat list -> nat list -> nat list
main_xs_1 :: ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> (nat -> nat) -> nat list -> nat list)
-> nat list
-> (nat -> nat)
-> (nat -> nat)
-> nat list
rmap_l_acc_2 :: (nat -> nat) -> nat list -> nat list -> nat list
map_f_l_2 :: (nat -> nat) -> nat -> nat list -> nat list
main_2 :: nat list -> (nat -> nat -> nat) -> (nat -> nat -> nat) -> nat list
rmap1_2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2_2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
main_xs_2 :: ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> (nat -> nat) -> nat list -> nat list)
-> nat list
-> (nat -> nat)
-> (nat -> nat)
-> (nat -> nat)
-> nat list
main_3 :: nat list
-> (nat -> nat -> nat)
-> (nat -> nat -> nat)
-> ((nat -> nat) -> nat list -> nat list)
-> nat list
main_xs_3 :: ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> nat list
-> (nat -> nat)
-> (nat list -> nat list)
-> nat list
main_4 :: nat list
-> (nat -> nat -> nat)
-> (nat -> nat -> nat)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> nat list
main_5 :: nat list
-> (nat -> nat -> nat)
-> (nat -> nat -> nat)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> (nat -> nat) -> nat list -> nat list)
-> nat list
rmap1_5 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2_5 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
main_6 :: nat list -> (nat list -> nat list) -> nat list
main_7 :: (nat -> nat -> nat)
-> (nat -> nat -> nat)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> (nat -> nat) -> nat list -> nat list)
-> nat list
-> nat list
map :: (nat -> nat) -> nat list -> nat list
mult :: (nat -> nat -> nat) -> nat -> nat -> nat
plus :: nat -> nat -> nat
rmap :: (nat -> nat) -> nat list -> nat list -> nat list
rmap1 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap1_3 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2_3 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
main :: nat list -> nat list
+ Applied Processor:
NeededRules
+ Details:
none
* Step 15: CFA WORST_CASE(?,O(n^4))
+ Considered Problem:
1: main_6(x0) x6 -> x6 x0
2: main_xs_3(x3, x4, x6, x9) x10 -> x3 x9 (x4 x9 (x10 x6))
3: main_xs_2(x6, x8, x11, x12, x15, x17) x18 -> x6 x18 (x8 x18 (x11 x15 x17 x12))
4: h() x9 -> x9
5: main_xs_1(x6, x8, x21, x12, x29) x33 -> x6 h() (x8 h() (x21 x29 x33 x12))
6: g(x1) x8 -> x1 x8 S(S(0()))
7: main_xs(x7, x12, x16, x22, x24) x30 -> x12 h() (x16 h() (x22 x30 g(x7) x24))
8: f(x2) x7 -> x2 x7 (x2 x7 x7)
9: main_7(x11, x23, x6, x8, x41) x12 -> x6 h() (x8 h() (x41 f(x23) g(x11) x12))
10: main_5(x24, x9, x21, x12, x16) x42 -> x12 h() (x16 h() (x42 f(x21) g(x9) x24))
11: map_rev2_f1_f2_l_1(x7, x8) x9 -> x8 x7 Nil()
12: map_rev2_f1_f2_l(x7) x8 -> map_rev2_f1_f2_l_1(x7, x8)
13: rmap2_l(x10, x12, Nil()) x26 -> x26
14: rmap2_l(x10, x12, Cons(x28, x30)) x26 -> rmap1(x10, x12) x30 Cons(x12 x28, x26)
15: rmap2_2(x5, x6) x12 -> rmap2_l(x5, x6, x12)
16: rmap2(x10, x12) x24 -> rmap2_l(x10, x12, x24)
17: rmap1_l(x10, x12, Nil()) x18 -> x18
18: rmap1_l(x10, x12, Cons(x20, x22)) x18 -> rmap2(x10, x12) x22 Cons(x10 x20, x18)
19: rmap1_2(x5, x6) x8 -> rmap1_l(x5, x6, x8)
20: rmap1(x10, x12) x16 -> rmap1_l(x10, x12, x16)
21: rmap1_l_1(x10, x12, Nil()) x26 -> x26
22: rmap1_l_1(x10, x12, Cons(x28, x30)) x26 -> rmap2_3(x10, x12) x30 Cons(x10 x28, x26)
23: rmap1_5(x5, x6) x12 -> rmap1_l_1(x5, x6, x12)
24: rmap1_3(x10, x12) x24 -> rmap1_l_1(x10, x12, x24)
25: rmap2_l_1(x10, x12, Nil()) x18 -> x18
26: rmap2_l_1(x10, x12, Cons(x20, x22)) x18 -> rmap1_3(x10, x12) x22 Cons(x12 x20, x18)
27: rmap2_5(x5, x6) x8 -> rmap2_l_1(x5, x6, x8)
28: rmap2_3(x10, x12) x16 -> rmap2_l_1(x10, x12, x16)
29: map_rev2_f1_f2(x23, x27) x14 -> rmap1(x23, x27) x14 Nil()
30: map_rev2_f1(x5) x6 -> map_rev2_f1_f2(x5, x6)
31: map_rev2() x5 -> map_rev2_f1(x5)
32: main_4(x28, x175, x367, x55) x71 -> x55 h() (x71 h() (rmap1(f(x367), g(x175)) x28 Nil()))
33: map_rev_f_l(x5) x6 -> x6 x5 Nil()
34: rmap_l_acc_2(x4, x9) x10 -> rmap(x4) x9 x10
35: rmap_l(x8, Nil()) x14 -> x14
36: rmap_l(x16, Cons(x34, x36)) x30 -> rmap(x16) x36 Cons(x16 x34, x30)
37: rmap_1(x4) x6 -> rmap_l(x4, x6)
38: rmap(x8) x12 -> rmap_l(x8, x12)
39: map_rev_f(x9) x10 -> rmap(x9) x10 Nil()
40: map_rev() x4 -> map_rev_f(x4)
41: main_3(x28, x703, x1471) x223 -> x223 h() (rmap(h()) (rmap1(f(x1471), g(x703)) x28 Nil()) Nil())
42: map_f_l_2(x3, x5) x7 -> Cons(x3 x5, x7)
43: map_f(x6) Nil() -> Nil()
44: map_f(x12) Cons(x20, x26) -> Cons(x12 x20, map() x12 x26)
45: map_1() x3 -> map_f(x3)
46: map() x6 -> map_f(x6)
47: main_2(x28, x735) x1503 -> map() h() (rmap(h()) (rmap1(f(x1503), g(x735)) x28 Nil()) Nil())
48: mult_n(x2, 0()) x6 -> 0()
49: mult_n(x2, S(x8)) x6 -> S(x2 (mult(x2) x8 x6) x6)
50: mult_1(x1) x2 -> mult_n(x1, x2)
51: mult(x2) x4 -> mult_n(x2, x4)
52: main_1(x28) x575 -> map() h() (rmap(h()) (rmap1(f(mult(x575)), g(x575)) x28 Nil()) Nil())
53: plus_n(x2) 0() -> x2
54: plus_n(x2) S(x6) -> S(plus() x2 x6)
55: plus_1() x1 -> plus_n(x1)
56: plus() x2 -> plus_n(x2)
57: main(x28) -> map() h() (rmap(h()) (rmap1(f(mult(plus())), g(plus())) x28 Nil()) Nil())
where
0 :: nat
Cons :: 'a -> 'a list -> 'a list
Nil :: 'a list
S :: nat -> nat
f :: (nat -> nat -> nat) -> nat -> nat
map_f :: (nat -> nat) -> nat list -> nat list
map_rev_f :: (nat -> nat) -> nat list -> nat list
map_rev2_f1 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list
map_rev2_f1_f2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list
g :: (nat -> nat -> nat) -> nat -> nat
h :: nat -> nat
map_rev_f_l :: nat list -> (nat list -> nat list -> nat list) -> nat list
map_rev2_f1_f2_l :: nat list
-> (nat list -> nat list -> nat list)
-> (nat list -> nat list -> nat list)
-> nat list
rmap_l :: (nat -> nat) -> nat list -> nat list -> nat list
rmap1_l :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2_l :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
map_rev :: (nat -> nat) -> nat list -> nat list
map_rev2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list
mult_n :: (nat -> nat -> nat) -> nat -> nat -> nat
plus_n :: nat -> nat -> nat
main_xs :: (nat -> nat -> nat)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> (nat -> nat) -> nat list -> nat list)
-> nat list
-> (nat -> nat)
-> nat list
map_rev2_f1_f2_l_1 :: nat list
-> (nat list -> nat list -> nat list)
-> (nat list -> nat list -> nat list)
-> nat list
rmap1_l_1 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2_l_1 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
main_1 :: nat list -> (nat -> nat -> nat) -> nat list
map_1 :: (nat -> nat) -> nat list -> nat list
mult_1 :: (nat -> nat -> nat) -> nat -> nat -> nat
plus_1 :: nat -> nat -> nat
rmap_1 :: (nat -> nat) -> nat list -> nat list -> nat list
main_xs_1 :: ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> (nat -> nat) -> nat list -> nat list)
-> nat list
-> (nat -> nat)
-> (nat -> nat)
-> nat list
rmap_l_acc_2 :: (nat -> nat) -> nat list -> nat list -> nat list
map_f_l_2 :: (nat -> nat) -> nat -> nat list -> nat list
main_2 :: nat list -> (nat -> nat -> nat) -> (nat -> nat -> nat) -> nat list
rmap1_2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2_2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
main_xs_2 :: ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> (nat -> nat) -> nat list -> nat list)
-> nat list
-> (nat -> nat)
-> (nat -> nat)
-> (nat -> nat)
-> nat list
main_3 :: nat list
-> (nat -> nat -> nat)
-> (nat -> nat -> nat)
-> ((nat -> nat) -> nat list -> nat list)
-> nat list
main_xs_3 :: ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> nat list
-> (nat -> nat)
-> (nat list -> nat list)
-> nat list
main_4 :: nat list
-> (nat -> nat -> nat)
-> (nat -> nat -> nat)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> nat list
main_5 :: nat list
-> (nat -> nat -> nat)
-> (nat -> nat -> nat)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> (nat -> nat) -> nat list -> nat list)
-> nat list
rmap1_5 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2_5 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
main_6 :: nat list -> (nat list -> nat list) -> nat list
main_7 :: (nat -> nat -> nat)
-> (nat -> nat -> nat)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> nat list -> nat list)
-> ((nat -> nat) -> (nat -> nat) -> nat list -> nat list)
-> nat list
-> nat list
map :: (nat -> nat) -> nat list -> nat list
mult :: (nat -> nat -> nat) -> nat -> nat -> nat
plus :: nat -> nat -> nat
rmap :: (nat -> nat) -> nat list -> nat list -> nat list
rmap1 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap1_3 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2_3 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
main :: nat list -> nat list
+ Applied Processor:
CFA {cfaRefinement = <control flow analysis>}
+ Details:
{X{*} -> Nil()
| Nil()
| S(X{*})
| S(X{*})
| 0()
| Nil()
| Nil()
| S(X{*})
| S(X{*})
| 0()
| 0()
| Nil()
| Nil()
| Cons(X{*},X{*})
| Cons(X{*},X{*})
| Nil()
| Nil()
| Cons(X{*},X{*})
| Nil()
| Nil()
| Nil()
| Cons(X{*},X{*})
| Cons(X{*},X{*})
| Nil()
| Nil()
| Nil()
| Nil()
| Cons(X{*},X{*})
| Cons(X{*},X{*})
| Nil()
| Cons(X{*},X{*})
| Cons(X{*},X{*})
| Nil()
| Cons(X{*},X{*})
| Cons(X{*},X{*})
| Nil()
| Cons(X{*},X{*})
| Cons(X{*},X{*})
| Nil()
| Nil()
| 0()
| S(X{*})
| S(X{*})
,V{x1_6} -> plus()
,V{x2_8} -> mult(plus())
,V{x2_48} -> V{x2_51}
,V{x2_49} -> V{x2_51}
,V{x2_51} -> V{x2_49}
| plus()
,V{x2_53} -> V{x2_56}
,V{x2_54} -> V{x2_56}
,V{x2_56} -> V{x2_54}
| R{49}
| R{48}
| V{x8_6}
,V{x4_51} -> V{x8_49}
| V{x7_8}
,V{x6_43} -> V{x6_46}
,V{x6_46} -> V{x12_44}
| h()
,V{x6_48} -> R{49}
| V{x6_49}
| R{48}
| V{x7_8}
,V{x6_49} -> R{49}
| V{x6_49}
| R{48}
| V{x7_8}
,V{x6_54} -> @(@(plus(),V{x2_54}),V{x6_54})
| @(R{56},V{x6_54})
| 0()
| @(@(V{x2_49},@(@(mult(V{x2_49}),V{x8_49}),V{x6_49})),V{x6_49})
| @(@(V{x2_49},@(R{51},V{x6_49})),V{x6_49})
| @(@(V{x2_49},R{48}),V{x6_49})
| @(@(V{x2_49},R{49}),V{x6_49})
| @(R{56},V{x6_49})
| R{53}
| R{54}
| X{*}
| S(0())
,V{x7_8} -> V{x20_18}
,V{x8_6} -> V{x28_14}
,V{x8_35} -> V{x8_38}
,V{x8_38} -> V{x16_36}
| h()
,V{x8_49} -> X{*}
,V{x9_4} -> V{x20_44}
| V{x34_36}
,V{x10_13} -> V{x10_16}
,V{x10_14} -> V{x10_16}
,V{x10_16} -> V{x10_18}
,V{x10_17} -> V{x10_20}
,V{x10_18} -> V{x10_20}
,V{x10_20} -> V{x10_14}
| f(mult(plus()))
,V{x12_13} -> V{x12_16}
,V{x12_14} -> V{x12_16}
,V{x12_16} -> V{x12_18}
,V{x12_17} -> V{x12_20}
,V{x12_18} -> V{x12_20}
,V{x12_20} -> V{x12_14}
| g(plus())
,V{x12_38} -> V{x36_36}
| R{18}
| R{17}
,V{x12_44} -> V{x6_46}
,V{x14_35} -> Cons(R{4},V{x30_36})
| Nil()
,V{x16_20} -> V{x30_14}
| V{x28_57}
,V{x16_36} -> V{x8_38}
,V{x18_17} -> Cons(R{6},V{x26_14})
| Nil()
,V{x18_18} -> Cons(R{6},V{x26_14})
| Nil()
,V{x20_18} -> X{*}
,V{x20_44} -> R{4}
,V{x22_18} -> X{*}
,V{x24_16} -> V{x22_18}
,V{x26_13} -> Cons(R{8},V{x18_18})
,V{x26_14} -> Cons(R{8},V{x18_18})
,V{x26_44} -> V{x30_36}
,V{x28_14} -> X{*}
,V{x28_57} -> X{*}
,V{x30_14} -> X{*}
,V{x30_36} -> Cons(R{4},V{x30_36})
| Nil()
,V{x34_36} -> R{6}
| R{8}
,V{x36_36} -> V{x26_14}
| V{x18_18}
,R{0} -> R{57}
| main(X{*})
,R{4} -> V{x9_4}
,R{6} -> R{54}
| @(R{56},S(S(0())))
| @(@(V{x1_6},V{x8_6}),S(S(0())))
,R{8} -> R{49}
| R{48}
| @(R{51},R{48})
| @(R{51},R{49})
| @(@(V{x2_8},V{x7_8}),R{49})
| @(@(V{x2_8},V{x7_8}),R{48})
| @(R{51},@(R{51},V{x7_8}))
| @(@(V{x2_8},V{x7_8}),@(R{51},V{x7_8}))
| @(R{51},@(@(V{x2_8},V{x7_8}),V{x7_8}))
| @(@(V{x2_8},V{x7_8}),@(@(V{x2_8},V{x7_8}),V{x7_8}))
,R{13} -> V{x26_13}
,R{14} -> R{18}
| R{17}
| @(R{20},Cons(R{6},V{x26_14}))
| @(@(rmap1(V{x10_14},V{x12_14}),V{x30_14}),Cons(R{6},V{x26_14}))
| @(R{20},Cons(@(V{x12_14},V{x28_14}),V{x26_14}))
| @(@(rmap1(V{x10_14},V{x12_14}),V{x30_14}),Cons(@(V{x12_14},V{x28_14}),V{x26_14}))
,R{16} -> rmap2_l(V{x10_16},V{x12_16},V{x24_16})
,R{17} -> V{x18_17}
,R{18} -> R{14}
| R{13}
| @(R{16},Cons(R{8},V{x18_18}))
| @(@(rmap2(V{x10_18},V{x12_18}),V{x22_18}),Cons(R{8},V{x18_18}))
| @(R{16},Cons(@(V{x10_18},V{x20_18}),V{x18_18}))
| @(@(rmap2(V{x10_18},V{x12_18}),V{x22_18}),Cons(@(V{x10_18},V{x20_18}),V{x18_18}))
,R{20} -> rmap1_l(V{x10_20},V{x12_20},V{x16_20})
,R{35} -> V{x14_35}
,R{36} -> R{36}
| R{35}
| @(R{38},Cons(R{4},V{x30_36}))
| @(@(rmap(V{x16_36}),V{x36_36}),Cons(R{4},V{x30_36}))
| @(R{38},Cons(@(V{x16_36},V{x34_36}),V{x30_36}))
| @(@(rmap(V{x16_36}),V{x36_36}),Cons(@(V{x16_36},V{x34_36}),V{x30_36}))
,R{38} -> rmap_l(V{x8_38},V{x12_38})
,R{43} -> Nil()
,R{44} -> Cons(R{4},R{43})
| Cons(R{4},R{44})
| Cons(@(V{x12_44},V{x20_44}),R{44})
| Cons(@(V{x12_44},V{x20_44}),R{43})
| Cons(R{4},@(R{46},V{x26_44}))
| Cons(@(V{x12_44},V{x20_44}),@(R{46},V{x26_44}))
| Cons(R{4},@(@(map(),V{x12_44}),V{x26_44}))
| Cons(@(V{x12_44},V{x20_44}),@(@(map(),V{x12_44}),V{x26_44}))
,R{46} -> map_f(V{x6_46})
,R{48} -> 0()
,R{49} -> S(R{54})
| S(R{53})
| S(@(R{56},V{x6_49}))
| S(@(@(V{x2_49},R{49}),V{x6_49}))
| S(@(@(V{x2_49},R{48}),V{x6_49}))
| S(@(@(V{x2_49},@(R{51},V{x6_49})),V{x6_49}))
| S(@(@(V{x2_49},@(@(mult(V{x2_49}),V{x8_49}),V{x6_49})),V{x6_49}))
,R{51} -> mult_n(V{x2_51},V{x4_51})
,R{53} -> V{x2_53}
,R{54} -> S(R{54})
| S(R{53})
| S(@(R{56},V{x6_54}))
| S(@(@(plus(),V{x2_54}),V{x6_54}))
,R{56} -> plus_n(V{x2_56})
,R{57} -> R{44}
| @(@(map(),h()),R{36})
| @(R{46},R{36})
| R{43}
| @(R{46},R{35})
| @(@(map(),h()),R{35})
| @(R{46},@(R{38},Nil()))
| @(@(map(),h()),@(R{38},Nil()))
| @(R{46},@(@(rmap(h()),R{17}),Nil()))
| @(R{46},@(@(rmap(h()),R{18}),Nil()))
| @(@(map(),h()),@(@(rmap(h()),R{18}),Nil()))
| @(@(map(),h()),@(@(rmap(h()),R{17}),Nil()))
| @(R{46},@(@(rmap(h()),@(R{20},Nil())),Nil()))
| @(@(map(),h()),@(@(rmap(h()),@(R{20},Nil())),Nil()))
| @(R{46},@(@(rmap(h()),@(@(rmap1(f(mult(plus())),g(plus())),V{x28_57}),Nil())),Nil()))
| @(@(map(),h()),@(@(rmap(h()),@(@(rmap1(f(mult(plus())),g(plus())),V{x28_57}),Nil())),Nil()))}
* Step 16: UncurryATRS WORST_CASE(?,O(n^4))
+ Considered Problem:
4: h() x9 -> x9
6: g(plus()) x1 -> plus() x1 S(S(0()))
8: f(mult(plus())) x1 -> mult(plus()) x1 (mult(plus()) x1 x1)
13: rmap2_l(f(mult(plus())), g(plus()), Nil()) Cons(x1, x2) -> Cons(x1, x2)
14: rmap2_l(f(mult(plus())), g(plus()), Cons(x4, x3)) Cons(x1, x2) -> rmap1(f(mult(plus())), g(plus())) x3
Cons(g(plus()) x4, Cons(x1, x2))
16: rmap2(f(mult(plus())), g(plus())) x1 -> rmap2_l(f(mult(plus())), g(plus()), x1)
17: rmap1_l(f(mult(plus())), g(plus()), Nil()) x1 -> x1
18: rmap1_l(f(mult(plus())), g(plus()), Cons(x3, x2)) x1 -> rmap2(f(mult(plus())), g(plus())) x2
Cons(f(mult(plus())) x3, x1)
20: rmap1(f(mult(plus())), g(plus())) x1 -> rmap1_l(f(mult(plus())), g(plus()), x1)
35: rmap_l(h(), Nil()) x1 -> x1
36: rmap_l(h(), Cons(x3, x2)) x1 -> rmap(h()) x2 Cons(h() x3, x1)
38: rmap(h()) x1 -> rmap_l(h(), x1)
43: map_f(h()) Nil() -> Nil()
44: map_f(h()) Cons(x2, x1) -> Cons(h() x2, map() h() x1)
46: map() h() -> map_f(h())
48: mult_n(plus(), 0()) x1 -> 0()
49: mult_n(plus(), S(x2)) x1 -> S(plus() (mult(plus()) x2 x1) x1)
51: mult(plus()) x1 -> mult_n(plus(), x1)
53: plus_n(x2) 0() -> x2
54: plus_n(x2) S(x1) -> S(plus() x2 x1)
56: plus() x2 -> plus_n(x2)
57: main(x1) -> map() h() (rmap(h()) (rmap1(f(mult(plus())), g(plus())) x1 Nil()) Nil())
where
0 :: nat
Cons :: 'a -> 'a list -> 'a list
Nil :: 'a list
S :: nat -> nat
f :: (nat -> nat -> nat) -> nat -> nat
map_f :: (nat -> nat) -> nat list -> nat list
g :: (nat -> nat -> nat) -> nat -> nat
h :: nat -> nat
rmap_l :: (nat -> nat) -> nat list -> nat list -> nat list
rmap1_l :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2_l :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
mult_n :: (nat -> nat -> nat) -> nat -> nat -> nat
plus_n :: nat -> nat -> nat
map :: (nat -> nat) -> nat list -> nat list
mult :: (nat -> nat -> nat) -> nat -> nat -> nat
plus :: nat -> nat -> nat
rmap :: (nat -> nat) -> nat list -> nat list -> nat list
rmap1 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
main :: nat list -> nat list
+ Applied Processor:
UncurryATRS
+ Details:
none
* Step 17: Inline WORST_CASE(?,O(n^4))
+ Considered Problem:
1: h#1(x9) -> x9
2: g#1(plus(), x1) -> plus#2(x1, S(S(0())))
3: f#1(mult(plus()), x1) -> mult#2(plus(), x1, mult#2(plus(), x1, x1))
4: rmap2_l#1(f(mult(plus())), g(plus()), Nil(), Cons(x1, x2)) -> Cons(x1, x2)
5: rmap2_l#1(f(mult(plus())), g(plus()), Cons(x4, x3), Cons(x1, x2)) -> rmap1#2(f(mult(plus())), g(plus())
, x3
, Cons(g#1(plus(), x4), Cons(x1
, x2)))
6: rmap2#1(f(mult(plus())), g(plus()), x1) -> rmap2_l(f(mult(plus())), g(plus()), x1)
7: rmap2#2(f(mult(plus())), g(plus()), x1, x2) -> rmap2_l#1(f(mult(plus())), g(plus()), x1, x2)
8: rmap1_l#1(f(mult(plus())), g(plus()), Nil(), x1) -> x1
9: rmap1_l#1(f(mult(plus())), g(plus()), Cons(x3, x2), x1) -> rmap2#2(f(mult(plus())), g(plus())
, x2
, Cons(f#1(mult(plus()), x3), x1))
10: rmap1#1(f(mult(plus())), g(plus()), x1) -> rmap1_l(f(mult(plus())), g(plus()), x1)
11: rmap1#2(f(mult(plus())), g(plus()), x1, x2) -> rmap1_l#1(f(mult(plus())), g(plus()), x1, x2)
12: rmap_l#1(h(), Nil(), x1) -> x1
13: rmap_l#1(h(), Cons(x3, x2), x1) -> rmap#2(h(), x2, Cons(h#1(x3), x1))
14: rmap#1(h(), x1) -> rmap_l(h(), x1)
15: rmap#2(h(), x1, x2) -> rmap_l#1(h(), x1, x2)
16: map_f#1(h(), Nil()) -> Nil()
17: map_f#1(h(), Cons(x2, x1)) -> Cons(h#1(x2), map#2(h(), x1))
18: map#1(h()) -> map_f(h())
19: map#2(h(), x0) -> map_f#1(h(), x0)
20: mult_n#1(plus(), 0(), x1) -> 0()
21: mult_n#1(plus(), S(x2), x1) -> S(plus#2(mult#2(plus(), x2, x1), x1))
22: mult#1(plus(), x1) -> mult_n(plus(), x1)
23: mult#2(plus(), x1, x2) -> mult_n#1(plus(), x1, x2)
24: plus_n#1(x2, 0()) -> x2
25: plus_n#1(x2, S(x1)) -> S(plus#2(x2, x1))
26: plus#1(x2) -> plus_n(x2)
27: plus#2(x2, x3) -> plus_n#1(x2, x3)
28: main(x1) -> map#2(h(), rmap#2(h(), rmap1#2(f(mult(plus())), g(plus()), x1, Nil()), Nil()))
where
0 :: nat
Cons :: 'a -> 'a list -> 'a list
Nil :: 'a list
S :: nat -> nat
f :: (nat -> nat -> nat) -> nat -> nat
f#1 :: (nat -> nat -> nat) -> nat -> nat
g :: (nat -> nat -> nat) -> nat -> nat
g#1 :: (nat -> nat -> nat) -> nat -> nat
h :: nat -> nat
h#1 :: nat -> nat
main :: nat list -> nat list
map#1 :: (nat -> nat) -> nat list -> nat list
map#2 :: (nat -> nat) -> nat list -> nat list
map_f :: (nat -> nat) -> nat list -> nat list
map_f#1 :: (nat -> nat) -> nat list -> nat list
mult :: (nat -> nat -> nat) -> nat -> nat -> nat
mult#1 :: (nat -> nat -> nat) -> nat -> nat -> nat
mult#2 :: (nat -> nat -> nat) -> nat -> nat -> nat
mult_n :: (nat -> nat -> nat) -> nat -> nat -> nat
mult_n#1 :: (nat -> nat -> nat) -> nat -> nat -> nat
plus :: nat -> nat -> nat
plus#1 :: nat -> nat -> nat
plus#2 :: nat -> nat -> nat
plus_n :: nat -> nat -> nat
plus_n#1 :: nat -> nat -> nat
rmap#1 :: (nat -> nat) -> nat list -> nat list -> nat list
rmap#2 :: (nat -> nat) -> nat list -> nat list -> nat list
rmap1#1 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap1#2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap1_l :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap1_l#1 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2#1 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2#2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2_l :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2_l#1 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap_l :: (nat -> nat) -> nat list -> nat list -> nat list
rmap_l#1 :: (nat -> nat) -> nat list -> nat list -> nat list
+ Applied Processor:
Inline {inlineType = InlineFull, inlineSelect = <inline decreasing>}
+ Details:
none
* Step 18: UsableRules WORST_CASE(?,O(n^4))
+ Considered Problem:
1: h#1(x9) -> x9
2: g#1(plus(), x1) -> plus#2(x1, S(S(0())))
3: f#1(mult(plus()), x1) -> mult#2(plus(), x1, mult#2(plus(), x1, x1))
4: rmap2_l#1(f(mult(plus())), g(plus()), Nil(), Cons(x1, x2)) -> Cons(x1, x2)
5: rmap2_l#1(f(mult(plus())), g(plus()), Cons(x4, x3), Cons(x1, x2)) -> rmap1#2(f(mult(plus())), g(plus())
, x3
, Cons(g#1(plus(), x4), Cons(x1
, x2)))
6: rmap2#1(f(mult(plus())), g(plus()), x1) -> rmap2_l(f(mult(plus())), g(plus()), x1)
7: rmap2#2(f(mult(plus())), g(plus()), x1, x2) -> rmap2_l#1(f(mult(plus())), g(plus()), x1, x2)
8: rmap1_l#1(f(mult(plus())), g(plus()), Nil(), x1) -> x1
9: rmap1_l#1(f(mult(plus())), g(plus()), Cons(x3, x2), x1) -> rmap2#2(f(mult(plus())), g(plus())
, x2
, Cons(f#1(mult(plus()), x3), x1))
10: rmap1#1(f(mult(plus())), g(plus()), x1) -> rmap1_l(f(mult(plus())), g(plus()), x1)
11: rmap1#2(f(mult(plus())), g(plus()), x1, x2) -> rmap1_l#1(f(mult(plus())), g(plus()), x1, x2)
12: rmap_l#1(h(), Nil(), x1) -> x1
13: rmap_l#1(h(), Cons(x18, x5), x3) -> rmap#2(h(), x5, Cons(x18, x3))
14: rmap#1(h(), x1) -> rmap_l(h(), x1)
15: rmap#2(h(), x1, x2) -> rmap_l#1(h(), x1, x2)
16: map_f#1(h(), Nil()) -> Nil()
17: map_f#1(h(), Cons(x18, x3)) -> Cons(x18, map#2(h(), x3))
18: map#1(h()) -> map_f(h())
19: map#2(h(), x0) -> map_f#1(h(), x0)
20: mult_n#1(plus(), 0(), x1) -> 0()
21: mult_n#1(plus(), S(x2), x1) -> S(plus#2(mult#2(plus(), x2, x1), x1))
22: mult#1(plus(), x1) -> mult_n(plus(), x1)
23: mult#2(plus(), x1, x2) -> mult_n#1(plus(), x1, x2)
24: plus_n#1(x2, 0()) -> x2
25: plus_n#1(x2, S(x1)) -> S(plus#2(x2, x1))
26: plus#1(x2) -> plus_n(x2)
27: plus#2(x2, x3) -> plus_n#1(x2, x3)
28: main(x1) -> map#2(h(), rmap#2(h(), rmap1#2(f(mult(plus())), g(plus()), x1, Nil()), Nil()))
where
0 :: nat
Cons :: 'a -> 'a list -> 'a list
Nil :: 'a list
S :: nat -> nat
f :: (nat -> nat -> nat) -> nat -> nat
f#1 :: (nat -> nat -> nat) -> nat -> nat
g :: (nat -> nat -> nat) -> nat -> nat
g#1 :: (nat -> nat -> nat) -> nat -> nat
h :: nat -> nat
h#1 :: nat -> nat
main :: nat list -> nat list
map#1 :: (nat -> nat) -> nat list -> nat list
map#2 :: (nat -> nat) -> nat list -> nat list
map_f :: (nat -> nat) -> nat list -> nat list
map_f#1 :: (nat -> nat) -> nat list -> nat list
mult :: (nat -> nat -> nat) -> nat -> nat -> nat
mult#1 :: (nat -> nat -> nat) -> nat -> nat -> nat
mult#2 :: (nat -> nat -> nat) -> nat -> nat -> nat
mult_n :: (nat -> nat -> nat) -> nat -> nat -> nat
mult_n#1 :: (nat -> nat -> nat) -> nat -> nat -> nat
plus :: nat -> nat -> nat
plus#1 :: nat -> nat -> nat
plus#2 :: nat -> nat -> nat
plus_n :: nat -> nat -> nat
plus_n#1 :: nat -> nat -> nat
rmap#1 :: (nat -> nat) -> nat list -> nat list -> nat list
rmap#2 :: (nat -> nat) -> nat list -> nat list -> nat list
rmap1#1 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap1#2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap1_l :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap1_l#1 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2#1 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2#2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2_l :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2_l#1 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap_l :: (nat -> nat) -> nat list -> nat list -> nat list
rmap_l#1 :: (nat -> nat) -> nat list -> nat list -> nat list
+ Applied Processor:
UsableRules {urType = Syntactic}
+ Details:
none
* Step 19: Inline WORST_CASE(?,O(n^4))
+ Considered Problem:
2: g#1(plus(), x1) -> plus#2(x1, S(S(0())))
3: f#1(mult(plus()), x1) -> mult#2(plus(), x1, mult#2(plus(), x1, x1))
4: rmap2_l#1(f(mult(plus())), g(plus()), Nil(), Cons(x1, x2)) -> Cons(x1, x2)
5: rmap2_l#1(f(mult(plus())), g(plus()), Cons(x4, x3), Cons(x1, x2)) -> rmap1#2(f(mult(plus())), g(plus())
, x3
, Cons(g#1(plus(), x4), Cons(x1
, x2)))
7: rmap2#2(f(mult(plus())), g(plus()), x1, x2) -> rmap2_l#1(f(mult(plus())), g(plus()), x1, x2)
8: rmap1_l#1(f(mult(plus())), g(plus()), Nil(), x1) -> x1
9: rmap1_l#1(f(mult(plus())), g(plus()), Cons(x3, x2), x1) -> rmap2#2(f(mult(plus())), g(plus())
, x2
, Cons(f#1(mult(plus()), x3), x1))
11: rmap1#2(f(mult(plus())), g(plus()), x1, x2) -> rmap1_l#1(f(mult(plus())), g(plus()), x1, x2)
12: rmap_l#1(h(), Nil(), x1) -> x1
13: rmap_l#1(h(), Cons(x18, x5), x3) -> rmap#2(h(), x5, Cons(x18, x3))
15: rmap#2(h(), x1, x2) -> rmap_l#1(h(), x1, x2)
16: map_f#1(h(), Nil()) -> Nil()
17: map_f#1(h(), Cons(x18, x3)) -> Cons(x18, map#2(h(), x3))
19: map#2(h(), x0) -> map_f#1(h(), x0)
20: mult_n#1(plus(), 0(), x1) -> 0()
21: mult_n#1(plus(), S(x2), x1) -> S(plus#2(mult#2(plus(), x2, x1), x1))
23: mult#2(plus(), x1, x2) -> mult_n#1(plus(), x1, x2)
24: plus_n#1(x2, 0()) -> x2
25: plus_n#1(x2, S(x1)) -> S(plus#2(x2, x1))
27: plus#2(x2, x3) -> plus_n#1(x2, x3)
28: main(x1) -> map#2(h(), rmap#2(h(), rmap1#2(f(mult(plus())), g(plus()), x1, Nil()), Nil()))
where
0 :: nat
Cons :: 'a -> 'a list -> 'a list
Nil :: 'a list
S :: nat -> nat
f :: (nat -> nat -> nat) -> nat -> nat
f#1 :: (nat -> nat -> nat) -> nat -> nat
g :: (nat -> nat -> nat) -> nat -> nat
g#1 :: (nat -> nat -> nat) -> nat -> nat
h :: nat -> nat
main :: nat list -> nat list
map#2 :: (nat -> nat) -> nat list -> nat list
map_f#1 :: (nat -> nat) -> nat list -> nat list
mult :: (nat -> nat -> nat) -> nat -> nat -> nat
mult#2 :: (nat -> nat -> nat) -> nat -> nat -> nat
mult_n#1 :: (nat -> nat -> nat) -> nat -> nat -> nat
plus :: nat -> nat -> nat
plus#2 :: nat -> nat -> nat
plus_n#1 :: nat -> nat -> nat
rmap#2 :: (nat -> nat) -> nat list -> nat list -> nat list
rmap1#2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap1_l#1 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2#2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2_l#1 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap_l#1 :: (nat -> nat) -> nat list -> nat list -> nat list
+ Applied Processor:
Inline {inlineType = InlineFull, inlineSelect = <inline decreasing>}
+ Details:
none
* Step 20: UsableRules WORST_CASE(?,O(n^4))
+ Considered Problem:
1: g#1(plus(), x1) -> plus#2(x1, S(S(0())))
2: f#1(mult(plus()), x1) -> mult#2(plus(), x1, mult#2(plus(), x1, x1))
3: rmap2_l#1(f(mult(plus())), g(plus()), Nil(), Cons(x1, x2)) -> Cons(x1, x2)
4: rmap2_l#1(f(mult(plus())), g(plus()), Cons(x2, x7), Cons(x3, x5)) -> rmap1#2(f(mult(plus())), g(plus())
, x7
, Cons(plus#2(x2, S(S(0())))
, Cons(x3, x5)))
5: rmap2#2(f(mult(plus())), g(plus()), Nil(), Cons(x2, x4)) -> Cons(x2, x4)
6: rmap2#2(f(mult(plus())), g(plus()), Cons(x8, x6), Cons(x2, x4)) -> rmap1#2(f(mult(plus())), g(plus())
, x6
, Cons(g#1(plus(), x8), Cons(x2
, x4)))
7: rmap1_l#1(f(mult(plus())), g(plus()), Nil(), x1) -> x1
8: rmap1_l#1(f(mult(plus())), g(plus()), Cons(x7, x2), x3) -> rmap2_l#1(f(mult(plus())), g(plus())
, x2
, Cons(f#1(mult(plus()), x7), x3))
9: rmap1#2(f(mult(plus())), g(plus()), Nil(), x2) -> x2
10: rmap1#2(f(mult(plus())), g(plus()), Cons(x6, x4), x2) -> rmap2#2(f(mult(plus())), g(plus())
, x4
, Cons(f#1(mult(plus()), x6), x2))
11: rmap_l#1(h(), Nil(), x1) -> x1
12: rmap_l#1(h(), Cons(x18, x5), x3) -> rmap#2(h(), x5, Cons(x18, x3))
13: rmap#2(h(), Nil(), x2) -> x2
14: rmap#2(h(), Cons(x36, x10), x6) -> rmap#2(h(), x10, Cons(x36, x6))
15: map_f#1(h(), Nil()) -> Nil()
16: map_f#1(h(), Cons(x18, x3)) -> Cons(x18, map#2(h(), x3))
17: map#2(h(), Nil()) -> Nil()
18: map#2(h(), Cons(x36, x6)) -> Cons(x36, map#2(h(), x6))
19: mult_n#1(plus(), 0(), x1) -> 0()
20: mult_n#1(plus(), S(x2), x1) -> S(plus#2(mult#2(plus(), x2, x1), x1))
21: mult#2(plus(), 0(), x2) -> 0()
22: mult#2(plus(), S(x4), x2) -> S(plus#2(mult#2(plus(), x4, x2), x2))
23: plus_n#1(x2, 0()) -> x2
24: plus_n#1(x2, S(x1)) -> S(plus#2(x2, x1))
25: plus#2(x4, 0()) -> x4
26: plus#2(x4, S(x2)) -> S(plus#2(x4, x2))
27: main(x1) -> map#2(h(), rmap#2(h(), rmap1#2(f(mult(plus())), g(plus()), x1, Nil()), Nil()))
where
0 :: nat
Cons :: 'a -> 'a list -> 'a list
Nil :: 'a list
S :: nat -> nat
f :: (nat -> nat -> nat) -> nat -> nat
f#1 :: (nat -> nat -> nat) -> nat -> nat
g :: (nat -> nat -> nat) -> nat -> nat
g#1 :: (nat -> nat -> nat) -> nat -> nat
h :: nat -> nat
main :: nat list -> nat list
map#2 :: (nat -> nat) -> nat list -> nat list
map_f#1 :: (nat -> nat) -> nat list -> nat list
mult :: (nat -> nat -> nat) -> nat -> nat -> nat
mult#2 :: (nat -> nat -> nat) -> nat -> nat -> nat
mult_n#1 :: (nat -> nat -> nat) -> nat -> nat -> nat
plus :: nat -> nat -> nat
plus#2 :: nat -> nat -> nat
plus_n#1 :: nat -> nat -> nat
rmap#2 :: (nat -> nat) -> nat list -> nat list -> nat list
rmap1#2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap1_l#1 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2#2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2_l#1 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap_l#1 :: (nat -> nat) -> nat list -> nat list -> nat list
+ Applied Processor:
UsableRules {urType = Syntactic}
+ Details:
none
* Step 21: Inline WORST_CASE(?,O(n^4))
+ Considered Problem:
1: g#1(plus(), x1) -> plus#2(x1, S(S(0())))
2: f#1(mult(plus()), x1) -> mult#2(plus(), x1, mult#2(plus(), x1, x1))
5: rmap2#2(f(mult(plus())), g(plus()), Nil(), Cons(x2, x4)) -> Cons(x2, x4)
6: rmap2#2(f(mult(plus())), g(plus()), Cons(x8, x6), Cons(x2, x4)) -> rmap1#2(f(mult(plus())), g(plus())
, x6
, Cons(g#1(plus(), x8), Cons(x2
, x4)))
9: rmap1#2(f(mult(plus())), g(plus()), Nil(), x2) -> x2
10: rmap1#2(f(mult(plus())), g(plus()), Cons(x6, x4), x2) -> rmap2#2(f(mult(plus())), g(plus())
, x4
, Cons(f#1(mult(plus()), x6), x2))
13: rmap#2(h(), Nil(), x2) -> x2
14: rmap#2(h(), Cons(x36, x10), x6) -> rmap#2(h(), x10, Cons(x36, x6))
17: map#2(h(), Nil()) -> Nil()
18: map#2(h(), Cons(x36, x6)) -> Cons(x36, map#2(h(), x6))
21: mult#2(plus(), 0(), x2) -> 0()
22: mult#2(plus(), S(x4), x2) -> S(plus#2(mult#2(plus(), x4, x2), x2))
25: plus#2(x4, 0()) -> x4
26: plus#2(x4, S(x2)) -> S(plus#2(x4, x2))
27: main(x1) -> map#2(h(), rmap#2(h(), rmap1#2(f(mult(plus())), g(plus()), x1, Nil()), Nil()))
where
0 :: nat
Cons :: 'a -> 'a list -> 'a list
Nil :: 'a list
S :: nat -> nat
f :: (nat -> nat -> nat) -> nat -> nat
f#1 :: (nat -> nat -> nat) -> nat -> nat
g :: (nat -> nat -> nat) -> nat -> nat
g#1 :: (nat -> nat -> nat) -> nat -> nat
h :: nat -> nat
main :: nat list -> nat list
map#2 :: (nat -> nat) -> nat list -> nat list
mult :: (nat -> nat -> nat) -> nat -> nat -> nat
mult#2 :: (nat -> nat -> nat) -> nat -> nat -> nat
plus :: nat -> nat -> nat
plus#2 :: nat -> nat -> nat
rmap#2 :: (nat -> nat) -> nat list -> nat list -> nat list
rmap1#2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2#2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
+ Applied Processor:
Inline {inlineType = InlineFull, inlineSelect = <inline decreasing>}
+ Details:
none
* Step 22: UsableRules WORST_CASE(?,O(n^4))
+ Considered Problem:
1: g#1(plus(), x1) -> plus#2(x1, S(S(0())))
2: f#1(mult(plus()), x1) -> mult#2(plus(), x1, mult#2(plus(), x1, x1))
3: rmap2#2(f(mult(plus())), g(plus()), Nil(), Cons(x2, x4)) -> Cons(x2, x4)
4: rmap2#2(f(mult(plus())), g(plus()), Cons(x2, x13), Cons(x5, x9)) -> rmap1#2(f(mult(plus())), g(plus())
, x13
, Cons(plus#2(x2, S(S(0())))
, Cons(x5, x9)))
5: rmap1#2(f(mult(plus())), g(plus()), Nil(), x2) -> x2
6: rmap1#2(f(mult(plus())), g(plus()), Cons(x13, Nil()), x8) -> Cons(f#1(mult(plus()), x13), x8)
7: rmap1#2(f(mult(plus())), g(plus()), Cons(x13, Cons(x16, x12)), x8) -> rmap1#2(f(mult(plus())), g(plus())
, x12
, Cons(g#1(plus(), x16)
, Cons(f#1(mult(plus())
, x13), x8)))
8: rmap#2(h(), Nil(), x2) -> x2
9: rmap#2(h(), Cons(x36, x10), x6) -> rmap#2(h(), x10, Cons(x36, x6))
10: map#2(h(), Nil()) -> Nil()
11: map#2(h(), Cons(x36, x6)) -> Cons(x36, map#2(h(), x6))
12: mult#2(plus(), 0(), x2) -> 0()
13: mult#2(plus(), S(x4), x2) -> S(plus#2(mult#2(plus(), x4, x2), x2))
14: plus#2(x4, 0()) -> x4
15: plus#2(x4, S(x2)) -> S(plus#2(x4, x2))
16: main(x1) -> map#2(h(), rmap#2(h(), rmap1#2(f(mult(plus())), g(plus()), x1, Nil()), Nil()))
where
0 :: nat
Cons :: 'a -> 'a list -> 'a list
Nil :: 'a list
S :: nat -> nat
f :: (nat -> nat -> nat) -> nat -> nat
f#1 :: (nat -> nat -> nat) -> nat -> nat
g :: (nat -> nat -> nat) -> nat -> nat
g#1 :: (nat -> nat -> nat) -> nat -> nat
h :: nat -> nat
main :: nat list -> nat list
map#2 :: (nat -> nat) -> nat list -> nat list
mult :: (nat -> nat -> nat) -> nat -> nat -> nat
mult#2 :: (nat -> nat -> nat) -> nat -> nat -> nat
plus :: nat -> nat -> nat
plus#2 :: nat -> nat -> nat
rmap#2 :: (nat -> nat) -> nat list -> nat list -> nat list
rmap1#2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
rmap2#2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
+ Applied Processor:
UsableRules {urType = Syntactic}
+ Details:
none
* Step 23: Inline WORST_CASE(?,O(n^4))
+ Considered Problem:
1: g#1(plus(), x1) -> plus#2(x1, S(S(0())))
2: f#1(mult(plus()), x1) -> mult#2(plus(), x1, mult#2(plus(), x1, x1))
5: rmap1#2(f(mult(plus())), g(plus()), Nil(), x2) -> x2
6: rmap1#2(f(mult(plus())), g(plus()), Cons(x13, Nil()), x8) -> Cons(f#1(mult(plus()), x13), x8)
7: rmap1#2(f(mult(plus())), g(plus()), Cons(x13, Cons(x16, x12)), x8) -> rmap1#2(f(mult(plus())), g(plus())
, x12
, Cons(g#1(plus(), x16)
, Cons(f#1(mult(plus())
, x13), x8)))
8: rmap#2(h(), Nil(), x2) -> x2
9: rmap#2(h(), Cons(x36, x10), x6) -> rmap#2(h(), x10, Cons(x36, x6))
10: map#2(h(), Nil()) -> Nil()
11: map#2(h(), Cons(x36, x6)) -> Cons(x36, map#2(h(), x6))
12: mult#2(plus(), 0(), x2) -> 0()
13: mult#2(plus(), S(x4), x2) -> S(plus#2(mult#2(plus(), x4, x2), x2))
14: plus#2(x4, 0()) -> x4
15: plus#2(x4, S(x2)) -> S(plus#2(x4, x2))
16: main(x1) -> map#2(h(), rmap#2(h(), rmap1#2(f(mult(plus())), g(plus()), x1, Nil()), Nil()))
where
0 :: nat
Cons :: 'a -> 'a list -> 'a list
Nil :: 'a list
S :: nat -> nat
f :: (nat -> nat -> nat) -> nat -> nat
f#1 :: (nat -> nat -> nat) -> nat -> nat
g :: (nat -> nat -> nat) -> nat -> nat
g#1 :: (nat -> nat -> nat) -> nat -> nat
h :: nat -> nat
main :: nat list -> nat list
map#2 :: (nat -> nat) -> nat list -> nat list
mult :: (nat -> nat -> nat) -> nat -> nat -> nat
mult#2 :: (nat -> nat -> nat) -> nat -> nat -> nat
plus :: nat -> nat -> nat
plus#2 :: nat -> nat -> nat
rmap#2 :: (nat -> nat) -> nat list -> nat list -> nat list
rmap1#2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
+ Applied Processor:
Inline {inlineType = InlineFull, inlineSelect = <inline decreasing>}
+ Details:
none
* Step 24: UsableRules WORST_CASE(?,O(n^4))
+ Considered Problem:
1: g#1(plus(), x1) -> plus#2(x1, S(S(0())))
2: f#1(mult(plus()), x1) -> mult#2(plus(), x1, mult#2(plus(), x1, x1))
3: rmap1#2(f(mult(plus())), g(plus()), Nil(), x2) -> x2
4: rmap1#2(f(mult(plus())), g(plus()), Cons(x13, Nil()), x8) -> Cons(f#1(mult(plus()), x13), x8)
5: rmap1#2(f(mult(plus())), g(plus()), Cons(x27, Cons(x2, x25)), x17) -> rmap1#2(f(mult(plus())), g(plus())
, x25
, Cons(plus#2(x2, S(S(0())))
, Cons(f#1(mult(plus())
, x27)
, x17)))
6: rmap#2(h(), Nil(), x2) -> x2
7: rmap#2(h(), Cons(x36, x10), x6) -> rmap#2(h(), x10, Cons(x36, x6))
8: map#2(h(), Nil()) -> Nil()
9: map#2(h(), Cons(x36, x6)) -> Cons(x36, map#2(h(), x6))
10: mult#2(plus(), 0(), x2) -> 0()
11: mult#2(plus(), S(x4), x2) -> S(plus#2(mult#2(plus(), x4, x2), x2))
12: plus#2(x4, 0()) -> x4
13: plus#2(x4, S(x2)) -> S(plus#2(x4, x2))
14: main(x1) -> map#2(h(), rmap#2(h(), rmap1#2(f(mult(plus())), g(plus()), x1, Nil()), Nil()))
where
0 :: nat
Cons :: 'a -> 'a list -> 'a list
Nil :: 'a list
S :: nat -> nat
f :: (nat -> nat -> nat) -> nat -> nat
f#1 :: (nat -> nat -> nat) -> nat -> nat
g :: (nat -> nat -> nat) -> nat -> nat
g#1 :: (nat -> nat -> nat) -> nat -> nat
h :: nat -> nat
main :: nat list -> nat list
map#2 :: (nat -> nat) -> nat list -> nat list
mult :: (nat -> nat -> nat) -> nat -> nat -> nat
mult#2 :: (nat -> nat -> nat) -> nat -> nat -> nat
plus :: nat -> nat -> nat
plus#2 :: nat -> nat -> nat
rmap#2 :: (nat -> nat) -> nat list -> nat list -> nat list
rmap1#2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
+ Applied Processor:
UsableRules {urType = Syntactic}
+ Details:
none
* Step 25: Compression WORST_CASE(?,O(n^4))
+ Considered Problem:
2: f#1(mult(plus()), x1) -> mult#2(plus(), x1, mult#2(plus(), x1, x1))
3: rmap1#2(f(mult(plus())), g(plus()), Nil(), x2) -> x2
4: rmap1#2(f(mult(plus())), g(plus()), Cons(x13, Nil()), x8) -> Cons(f#1(mult(plus()), x13), x8)
5: rmap1#2(f(mult(plus())), g(plus()), Cons(x27, Cons(x2, x25)), x17) -> rmap1#2(f(mult(plus())), g(plus())
, x25
, Cons(plus#2(x2, S(S(0())))
, Cons(f#1(mult(plus())
, x27)
, x17)))
6: rmap#2(h(), Nil(), x2) -> x2
7: rmap#2(h(), Cons(x36, x10), x6) -> rmap#2(h(), x10, Cons(x36, x6))
8: map#2(h(), Nil()) -> Nil()
9: map#2(h(), Cons(x36, x6)) -> Cons(x36, map#2(h(), x6))
10: mult#2(plus(), 0(), x2) -> 0()
11: mult#2(plus(), S(x4), x2) -> S(plus#2(mult#2(plus(), x4, x2), x2))
12: plus#2(x4, 0()) -> x4
13: plus#2(x4, S(x2)) -> S(plus#2(x4, x2))
14: main(x1) -> map#2(h(), rmap#2(h(), rmap1#2(f(mult(plus())), g(plus()), x1, Nil()), Nil()))
where
0 :: nat
Cons :: 'a -> 'a list -> 'a list
Nil :: 'a list
S :: nat -> nat
f :: (nat -> nat -> nat) -> nat -> nat
f#1 :: (nat -> nat -> nat) -> nat -> nat
g :: (nat -> nat -> nat) -> nat -> nat
h :: nat -> nat
main :: nat list -> nat list
map#2 :: (nat -> nat) -> nat list -> nat list
mult :: (nat -> nat -> nat) -> nat -> nat -> nat
mult#2 :: (nat -> nat -> nat) -> nat -> nat -> nat
plus :: nat -> nat -> nat
plus#2 :: nat -> nat -> nat
rmap#2 :: (nat -> nat) -> nat list -> nat list -> nat list
rmap1#2 :: (nat -> nat) -> (nat -> nat) -> nat list -> nat list -> nat list
+ Applied Processor:
Compression
+ Details:
none
* Step 26: ToTctProblem WORST_CASE(?,O(n^4))
+ Considered Problem:
1: f#1(x1) -> mult#2(x1, mult#2(x1, x1))
2: rmap1#2(Nil(), x2) -> x2
3: rmap1#2(Cons(x13, Nil()), x8) -> Cons(f#1(x13), x8)
4: rmap1#2(Cons(x27, Cons(x2, x25)), x17) -> rmap1#2(x25, Cons(plus#2(x2, S(S(0()))), Cons(f#1(x27), x17)))
5: rmap#2(Nil(), x2) -> x2
6: rmap#2(Cons(x36, x10), x6) -> rmap#2(x10, Cons(x36, x6))
7: map#2(Nil()) -> Nil()
8: map#2(Cons(x36, x6)) -> Cons(x36, map#2(x6))
9: mult#2(0(), x2) -> 0()
10: mult#2(S(x4), x2) -> S(plus#2(mult#2(x4, x2), x2))
11: plus#2(x4, 0()) -> x4
12: plus#2(x4, S(x2)) -> S(plus#2(x4, x2))
13: main(x1) -> map#2(rmap#2(rmap1#2(x1, Nil()), Nil()))
where
0 :: nat
Cons :: 'a -> 'a list -> 'a list
Nil :: 'a list
S :: nat -> nat
f#1 :: nat -> nat
main :: nat list -> nat list
map#2 :: nat list -> nat list
mult#2 :: nat -> nat -> nat
plus#2 :: nat -> nat -> nat
rmap#2 :: nat list -> nat list -> nat list
rmap1#2 :: nat list -> nat list -> nat list
+ Applied Processor:
ToTctProblem
+ Details:
none
* Step 27: DependencyPairs WORST_CASE(?,O(n^4))
+ Considered Problem:
- Strict TRS:
f#1(x1) -> mult#2(x1,mult#2(x1,x1))
main(x1) -> map#2(rmap#2(rmap1#2(x1,Nil()),Nil()))
map#2(Cons(x36,x6)) -> Cons(x36,map#2(x6))
map#2(Nil()) -> Nil()
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> S(plus#2(mult#2(x4,x2),x2))
plus#2(x4,0()) -> x4
plus#2(x4,S(x2)) -> S(plus#2(x4,x2))
rmap#2(Cons(x36,x10),x6) -> rmap#2(x10,Cons(x36,x6))
rmap#2(Nil(),x2) -> x2
rmap1#2(Cons(x13,Nil()),x8) -> Cons(f#1(x13),x8)
rmap1#2(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
rmap1#2(Nil(),x2) -> x2
- Signature:
{f#1/1,main/1,map#2/1,mult#2/2,plus#2/2,rmap#2/2,rmap1#2/2} / {0/0,Cons/2,Nil/0,S/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {main} and constructors {0,Cons,Nil,S}
+ Applied Processor:
DependencyPairs {dpKind_ = DT}
+ Details:
We add the following dependency tuples:
Strict DPs
f#1#(x1) -> c_1(mult#2#(x1,mult#2(x1,x1)),mult#2#(x1,x1))
main#(x1) -> c_2(map#2#(rmap#2(rmap1#2(x1,Nil()),Nil()))
,rmap#2#(rmap1#2(x1,Nil()),Nil())
,rmap1#2#(x1,Nil()))
map#2#(Cons(x36,x6)) -> c_3(map#2#(x6))
map#2#(Nil()) -> c_4()
mult#2#(0(),x2) -> c_5()
mult#2#(S(x4),x2) -> c_6(plus#2#(mult#2(x4,x2),x2),mult#2#(x4,x2))
plus#2#(x4,0()) -> c_7()
plus#2#(x4,S(x2)) -> c_8(plus#2#(x4,x2))
rmap#2#(Cons(x36,x10),x6) -> c_9(rmap#2#(x10,Cons(x36,x6)))
rmap#2#(Nil(),x2) -> c_10()
rmap1#2#(Cons(x13,Nil()),x8) -> c_11(f#1#(x13))
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> c_12(rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
,plus#2#(x2,S(S(0())))
,f#1#(x27))
rmap1#2#(Nil(),x2) -> c_13()
Weak DPs
and mark the set of starting terms.
* Step 28: UsableRules WORST_CASE(?,O(n^4))
+ Considered Problem:
- Strict DPs:
f#1#(x1) -> c_1(mult#2#(x1,mult#2(x1,x1)),mult#2#(x1,x1))
main#(x1) -> c_2(map#2#(rmap#2(rmap1#2(x1,Nil()),Nil()))
,rmap#2#(rmap1#2(x1,Nil()),Nil())
,rmap1#2#(x1,Nil()))
map#2#(Cons(x36,x6)) -> c_3(map#2#(x6))
map#2#(Nil()) -> c_4()
mult#2#(0(),x2) -> c_5()
mult#2#(S(x4),x2) -> c_6(plus#2#(mult#2(x4,x2),x2),mult#2#(x4,x2))
plus#2#(x4,0()) -> c_7()
plus#2#(x4,S(x2)) -> c_8(plus#2#(x4,x2))
rmap#2#(Cons(x36,x10),x6) -> c_9(rmap#2#(x10,Cons(x36,x6)))
rmap#2#(Nil(),x2) -> c_10()
rmap1#2#(Cons(x13,Nil()),x8) -> c_11(f#1#(x13))
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> c_12(rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
,plus#2#(x2,S(S(0())))
,f#1#(x27))
rmap1#2#(Nil(),x2) -> c_13()
- Weak TRS:
f#1(x1) -> mult#2(x1,mult#2(x1,x1))
main(x1) -> map#2(rmap#2(rmap1#2(x1,Nil()),Nil()))
map#2(Cons(x36,x6)) -> Cons(x36,map#2(x6))
map#2(Nil()) -> Nil()
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> S(plus#2(mult#2(x4,x2),x2))
plus#2(x4,0()) -> x4
plus#2(x4,S(x2)) -> S(plus#2(x4,x2))
rmap#2(Cons(x36,x10),x6) -> rmap#2(x10,Cons(x36,x6))
rmap#2(Nil(),x2) -> x2
rmap1#2(Cons(x13,Nil()),x8) -> Cons(f#1(x13),x8)
rmap1#2(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
rmap1#2(Nil(),x2) -> x2
- Signature:
{f#1/1,main/1,map#2/1,mult#2/2,plus#2/2,rmap#2/2,rmap1#2/2,f#1#/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2
,rmap#2#/2,rmap1#2#/2} / {0/0,Cons/2,Nil/0,S/1,c_1/2,c_2/3,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/1,c_9/1,c_10/0
,c_11/1,c_12/3,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {main#} and constructors {0,Cons,Nil,S}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
f#1(x1) -> mult#2(x1,mult#2(x1,x1))
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> S(plus#2(mult#2(x4,x2),x2))
plus#2(x4,0()) -> x4
plus#2(x4,S(x2)) -> S(plus#2(x4,x2))
rmap#2(Cons(x36,x10),x6) -> rmap#2(x10,Cons(x36,x6))
rmap#2(Nil(),x2) -> x2
rmap1#2(Cons(x13,Nil()),x8) -> Cons(f#1(x13),x8)
rmap1#2(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
rmap1#2(Nil(),x2) -> x2
f#1#(x1) -> c_1(mult#2#(x1,mult#2(x1,x1)),mult#2#(x1,x1))
main#(x1) -> c_2(map#2#(rmap#2(rmap1#2(x1,Nil()),Nil()))
,rmap#2#(rmap1#2(x1,Nil()),Nil())
,rmap1#2#(x1,Nil()))
map#2#(Cons(x36,x6)) -> c_3(map#2#(x6))
map#2#(Nil()) -> c_4()
mult#2#(0(),x2) -> c_5()
mult#2#(S(x4),x2) -> c_6(plus#2#(mult#2(x4,x2),x2),mult#2#(x4,x2))
plus#2#(x4,0()) -> c_7()
plus#2#(x4,S(x2)) -> c_8(plus#2#(x4,x2))
rmap#2#(Cons(x36,x10),x6) -> c_9(rmap#2#(x10,Cons(x36,x6)))
rmap#2#(Nil(),x2) -> c_10()
rmap1#2#(Cons(x13,Nil()),x8) -> c_11(f#1#(x13))
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> c_12(rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
,plus#2#(x2,S(S(0())))
,f#1#(x27))
rmap1#2#(Nil(),x2) -> c_13()
* Step 29: PredecessorEstimation WORST_CASE(?,O(n^4))
+ Considered Problem:
- Strict DPs:
f#1#(x1) -> c_1(mult#2#(x1,mult#2(x1,x1)),mult#2#(x1,x1))
main#(x1) -> c_2(map#2#(rmap#2(rmap1#2(x1,Nil()),Nil()))
,rmap#2#(rmap1#2(x1,Nil()),Nil())
,rmap1#2#(x1,Nil()))
map#2#(Cons(x36,x6)) -> c_3(map#2#(x6))
map#2#(Nil()) -> c_4()
mult#2#(0(),x2) -> c_5()
mult#2#(S(x4),x2) -> c_6(plus#2#(mult#2(x4,x2),x2),mult#2#(x4,x2))
plus#2#(x4,0()) -> c_7()
plus#2#(x4,S(x2)) -> c_8(plus#2#(x4,x2))
rmap#2#(Cons(x36,x10),x6) -> c_9(rmap#2#(x10,Cons(x36,x6)))
rmap#2#(Nil(),x2) -> c_10()
rmap1#2#(Cons(x13,Nil()),x8) -> c_11(f#1#(x13))
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> c_12(rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
,plus#2#(x2,S(S(0())))
,f#1#(x27))
rmap1#2#(Nil(),x2) -> c_13()
- Weak TRS:
f#1(x1) -> mult#2(x1,mult#2(x1,x1))
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> S(plus#2(mult#2(x4,x2),x2))
plus#2(x4,0()) -> x4
plus#2(x4,S(x2)) -> S(plus#2(x4,x2))
rmap#2(Cons(x36,x10),x6) -> rmap#2(x10,Cons(x36,x6))
rmap#2(Nil(),x2) -> x2
rmap1#2(Cons(x13,Nil()),x8) -> Cons(f#1(x13),x8)
rmap1#2(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
rmap1#2(Nil(),x2) -> x2
- Signature:
{f#1/1,main/1,map#2/1,mult#2/2,plus#2/2,rmap#2/2,rmap1#2/2,f#1#/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2
,rmap#2#/2,rmap1#2#/2} / {0/0,Cons/2,Nil/0,S/1,c_1/2,c_2/3,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/1,c_9/1,c_10/0
,c_11/1,c_12/3,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {main#} and constructors {0,Cons,Nil,S}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{4,5,7,10,13}
by application of
Pre({4,5,7,10,13}) = {1,2,3,6,8,9,12}.
Here rules are labelled as follows:
1: f#1#(x1) -> c_1(mult#2#(x1,mult#2(x1,x1)),mult#2#(x1,x1))
2: main#(x1) -> c_2(map#2#(rmap#2(rmap1#2(x1,Nil()),Nil()))
,rmap#2#(rmap1#2(x1,Nil()),Nil())
,rmap1#2#(x1,Nil()))
3: map#2#(Cons(x36,x6)) -> c_3(map#2#(x6))
4: map#2#(Nil()) -> c_4()
5: mult#2#(0(),x2) -> c_5()
6: mult#2#(S(x4),x2) -> c_6(plus#2#(mult#2(x4,x2),x2),mult#2#(x4,x2))
7: plus#2#(x4,0()) -> c_7()
8: plus#2#(x4,S(x2)) -> c_8(plus#2#(x4,x2))
9: rmap#2#(Cons(x36,x10),x6) -> c_9(rmap#2#(x10,Cons(x36,x6)))
10: rmap#2#(Nil(),x2) -> c_10()
11: rmap1#2#(Cons(x13,Nil()),x8) -> c_11(f#1#(x13))
12: rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> c_12(rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
,plus#2#(x2,S(S(0())))
,f#1#(x27))
13: rmap1#2#(Nil(),x2) -> c_13()
* Step 30: RemoveWeakSuffixes WORST_CASE(?,O(n^4))
+ Considered Problem:
- Strict DPs:
f#1#(x1) -> c_1(mult#2#(x1,mult#2(x1,x1)),mult#2#(x1,x1))
main#(x1) -> c_2(map#2#(rmap#2(rmap1#2(x1,Nil()),Nil()))
,rmap#2#(rmap1#2(x1,Nil()),Nil())
,rmap1#2#(x1,Nil()))
map#2#(Cons(x36,x6)) -> c_3(map#2#(x6))
mult#2#(S(x4),x2) -> c_6(plus#2#(mult#2(x4,x2),x2),mult#2#(x4,x2))
plus#2#(x4,S(x2)) -> c_8(plus#2#(x4,x2))
rmap#2#(Cons(x36,x10),x6) -> c_9(rmap#2#(x10,Cons(x36,x6)))
rmap1#2#(Cons(x13,Nil()),x8) -> c_11(f#1#(x13))
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> c_12(rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
,plus#2#(x2,S(S(0())))
,f#1#(x27))
- Weak DPs:
map#2#(Nil()) -> c_4()
mult#2#(0(),x2) -> c_5()
plus#2#(x4,0()) -> c_7()
rmap#2#(Nil(),x2) -> c_10()
rmap1#2#(Nil(),x2) -> c_13()
- Weak TRS:
f#1(x1) -> mult#2(x1,mult#2(x1,x1))
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> S(plus#2(mult#2(x4,x2),x2))
plus#2(x4,0()) -> x4
plus#2(x4,S(x2)) -> S(plus#2(x4,x2))
rmap#2(Cons(x36,x10),x6) -> rmap#2(x10,Cons(x36,x6))
rmap#2(Nil(),x2) -> x2
rmap1#2(Cons(x13,Nil()),x8) -> Cons(f#1(x13),x8)
rmap1#2(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
rmap1#2(Nil(),x2) -> x2
- Signature:
{f#1/1,main/1,map#2/1,mult#2/2,plus#2/2,rmap#2/2,rmap1#2/2,f#1#/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2
,rmap#2#/2,rmap1#2#/2} / {0/0,Cons/2,Nil/0,S/1,c_1/2,c_2/3,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/1,c_9/1,c_10/0
,c_11/1,c_12/3,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {main#} and constructors {0,Cons,Nil,S}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:f#1#(x1) -> c_1(mult#2#(x1,mult#2(x1,x1)),mult#2#(x1,x1))
-->_2 mult#2#(S(x4),x2) -> c_6(plus#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)):4
-->_1 mult#2#(S(x4),x2) -> c_6(plus#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)):4
-->_2 mult#2#(0(),x2) -> c_5():10
-->_1 mult#2#(0(),x2) -> c_5():10
2:S:main#(x1) -> c_2(map#2#(rmap#2(rmap1#2(x1,Nil()),Nil()))
,rmap#2#(rmap1#2(x1,Nil()),Nil())
,rmap1#2#(x1,Nil()))
-->_3 rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> c_12(rmap1#2#(x25
,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
,plus#2#(x2,S(S(0())))
,f#1#(x27)):8
-->_3 rmap1#2#(Cons(x13,Nil()),x8) -> c_11(f#1#(x13)):7
-->_2 rmap#2#(Cons(x36,x10),x6) -> c_9(rmap#2#(x10,Cons(x36,x6))):6
-->_1 map#2#(Cons(x36,x6)) -> c_3(map#2#(x6)):3
-->_3 rmap1#2#(Nil(),x2) -> c_13():13
-->_2 rmap#2#(Nil(),x2) -> c_10():12
-->_1 map#2#(Nil()) -> c_4():9
3:S:map#2#(Cons(x36,x6)) -> c_3(map#2#(x6))
-->_1 map#2#(Nil()) -> c_4():9
-->_1 map#2#(Cons(x36,x6)) -> c_3(map#2#(x6)):3
4:S:mult#2#(S(x4),x2) -> c_6(plus#2#(mult#2(x4,x2),x2),mult#2#(x4,x2))
-->_1 plus#2#(x4,S(x2)) -> c_8(plus#2#(x4,x2)):5
-->_1 plus#2#(x4,0()) -> c_7():11
-->_2 mult#2#(0(),x2) -> c_5():10
-->_2 mult#2#(S(x4),x2) -> c_6(plus#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)):4
5:S:plus#2#(x4,S(x2)) -> c_8(plus#2#(x4,x2))
-->_1 plus#2#(x4,0()) -> c_7():11
-->_1 plus#2#(x4,S(x2)) -> c_8(plus#2#(x4,x2)):5
6:S:rmap#2#(Cons(x36,x10),x6) -> c_9(rmap#2#(x10,Cons(x36,x6)))
-->_1 rmap#2#(Nil(),x2) -> c_10():12
-->_1 rmap#2#(Cons(x36,x10),x6) -> c_9(rmap#2#(x10,Cons(x36,x6))):6
7:S:rmap1#2#(Cons(x13,Nil()),x8) -> c_11(f#1#(x13))
-->_1 f#1#(x1) -> c_1(mult#2#(x1,mult#2(x1,x1)),mult#2#(x1,x1)):1
8:S:rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> c_12(rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
,plus#2#(x2,S(S(0())))
,f#1#(x27))
-->_1 rmap1#2#(Nil(),x2) -> c_13():13
-->_1 rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> c_12(rmap1#2#(x25
,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
,plus#2#(x2,S(S(0())))
,f#1#(x27)):8
-->_1 rmap1#2#(Cons(x13,Nil()),x8) -> c_11(f#1#(x13)):7
-->_2 plus#2#(x4,S(x2)) -> c_8(plus#2#(x4,x2)):5
-->_3 f#1#(x1) -> c_1(mult#2#(x1,mult#2(x1,x1)),mult#2#(x1,x1)):1
9:W:map#2#(Nil()) -> c_4()
10:W:mult#2#(0(),x2) -> c_5()
11:W:plus#2#(x4,0()) -> c_7()
12:W:rmap#2#(Nil(),x2) -> c_10()
13:W:rmap1#2#(Nil(),x2) -> c_13()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
9: map#2#(Nil()) -> c_4()
12: rmap#2#(Nil(),x2) -> c_10()
13: rmap1#2#(Nil(),x2) -> c_13()
10: mult#2#(0(),x2) -> c_5()
11: plus#2#(x4,0()) -> c_7()
* Step 31: Decompose WORST_CASE(?,O(n^4))
+ Considered Problem:
- Strict DPs:
f#1#(x1) -> c_1(mult#2#(x1,mult#2(x1,x1)),mult#2#(x1,x1))
main#(x1) -> c_2(map#2#(rmap#2(rmap1#2(x1,Nil()),Nil()))
,rmap#2#(rmap1#2(x1,Nil()),Nil())
,rmap1#2#(x1,Nil()))
map#2#(Cons(x36,x6)) -> c_3(map#2#(x6))
mult#2#(S(x4),x2) -> c_6(plus#2#(mult#2(x4,x2),x2),mult#2#(x4,x2))
plus#2#(x4,S(x2)) -> c_8(plus#2#(x4,x2))
rmap#2#(Cons(x36,x10),x6) -> c_9(rmap#2#(x10,Cons(x36,x6)))
rmap1#2#(Cons(x13,Nil()),x8) -> c_11(f#1#(x13))
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> c_12(rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
,plus#2#(x2,S(S(0())))
,f#1#(x27))
- Weak TRS:
f#1(x1) -> mult#2(x1,mult#2(x1,x1))
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> S(plus#2(mult#2(x4,x2),x2))
plus#2(x4,0()) -> x4
plus#2(x4,S(x2)) -> S(plus#2(x4,x2))
rmap#2(Cons(x36,x10),x6) -> rmap#2(x10,Cons(x36,x6))
rmap#2(Nil(),x2) -> x2
rmap1#2(Cons(x13,Nil()),x8) -> Cons(f#1(x13),x8)
rmap1#2(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
rmap1#2(Nil(),x2) -> x2
- Signature:
{f#1/1,main/1,map#2/1,mult#2/2,plus#2/2,rmap#2/2,rmap1#2/2,f#1#/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2
,rmap#2#/2,rmap1#2#/2} / {0/0,Cons/2,Nil/0,S/1,c_1/2,c_2/3,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/1,c_9/1,c_10/0
,c_11/1,c_12/3,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {main#} and constructors {0,Cons,Nil,S}
+ 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:
f#1#(x1) -> c_1(mult#2#(x1,mult#2(x1,x1)),mult#2#(x1,x1))
mult#2#(S(x4),x2) -> c_6(plus#2#(mult#2(x4,x2),x2),mult#2#(x4,x2))
plus#2#(x4,S(x2)) -> c_8(plus#2#(x4,x2))
- Weak DPs:
main#(x1) -> c_2(map#2#(rmap#2(rmap1#2(x1,Nil()),Nil()))
,rmap#2#(rmap1#2(x1,Nil()),Nil())
,rmap1#2#(x1,Nil()))
map#2#(Cons(x36,x6)) -> c_3(map#2#(x6))
rmap#2#(Cons(x36,x10),x6) -> c_9(rmap#2#(x10,Cons(x36,x6)))
rmap1#2#(Cons(x13,Nil()),x8) -> c_11(f#1#(x13))
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> c_12(rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
,plus#2#(x2,S(S(0())))
,f#1#(x27))
- Weak TRS:
f#1(x1) -> mult#2(x1,mult#2(x1,x1))
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> S(plus#2(mult#2(x4,x2),x2))
plus#2(x4,0()) -> x4
plus#2(x4,S(x2)) -> S(plus#2(x4,x2))
rmap#2(Cons(x36,x10),x6) -> rmap#2(x10,Cons(x36,x6))
rmap#2(Nil(),x2) -> x2
rmap1#2(Cons(x13,Nil()),x8) -> Cons(f#1(x13),x8)
rmap1#2(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
rmap1#2(Nil(),x2) -> x2
- Signature:
{f#1/1,main/1,map#2/1,mult#2/2,plus#2/2,rmap#2/2,rmap1#2/2,f#1#/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2
,rmap#2#/2,rmap1#2#/2} / {0/0,Cons/2,Nil/0,S/1,c_1/2,c_2/3,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/1,c_9/1
,c_10/0,c_11/1,c_12/3,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {main#} and constructors {0,Cons,Nil,S}
Problem (S)
- Strict DPs:
main#(x1) -> c_2(map#2#(rmap#2(rmap1#2(x1,Nil()),Nil()))
,rmap#2#(rmap1#2(x1,Nil()),Nil())
,rmap1#2#(x1,Nil()))
map#2#(Cons(x36,x6)) -> c_3(map#2#(x6))
rmap#2#(Cons(x36,x10),x6) -> c_9(rmap#2#(x10,Cons(x36,x6)))
rmap1#2#(Cons(x13,Nil()),x8) -> c_11(f#1#(x13))
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> c_12(rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
,plus#2#(x2,S(S(0())))
,f#1#(x27))
- Weak DPs:
f#1#(x1) -> c_1(mult#2#(x1,mult#2(x1,x1)),mult#2#(x1,x1))
mult#2#(S(x4),x2) -> c_6(plus#2#(mult#2(x4,x2),x2),mult#2#(x4,x2))
plus#2#(x4,S(x2)) -> c_8(plus#2#(x4,x2))
- Weak TRS:
f#1(x1) -> mult#2(x1,mult#2(x1,x1))
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> S(plus#2(mult#2(x4,x2),x2))
plus#2(x4,0()) -> x4
plus#2(x4,S(x2)) -> S(plus#2(x4,x2))
rmap#2(Cons(x36,x10),x6) -> rmap#2(x10,Cons(x36,x6))
rmap#2(Nil(),x2) -> x2
rmap1#2(Cons(x13,Nil()),x8) -> Cons(f#1(x13),x8)
rmap1#2(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
rmap1#2(Nil(),x2) -> x2
- Signature:
{f#1/1,main/1,map#2/1,mult#2/2,plus#2/2,rmap#2/2,rmap1#2/2,f#1#/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2
,rmap#2#/2,rmap1#2#/2} / {0/0,Cons/2,Nil/0,S/1,c_1/2,c_2/3,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/1,c_9/1
,c_10/0,c_11/1,c_12/3,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {main#} and constructors {0,Cons,Nil,S}
** Step 31.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^4))
+ Considered Problem:
- Strict DPs:
f#1#(x1) -> c_1(mult#2#(x1,mult#2(x1,x1)),mult#2#(x1,x1))
mult#2#(S(x4),x2) -> c_6(plus#2#(mult#2(x4,x2),x2),mult#2#(x4,x2))
plus#2#(x4,S(x2)) -> c_8(plus#2#(x4,x2))
- Weak DPs:
main#(x1) -> c_2(map#2#(rmap#2(rmap1#2(x1,Nil()),Nil()))
,rmap#2#(rmap1#2(x1,Nil()),Nil())
,rmap1#2#(x1,Nil()))
map#2#(Cons(x36,x6)) -> c_3(map#2#(x6))
rmap#2#(Cons(x36,x10),x6) -> c_9(rmap#2#(x10,Cons(x36,x6)))
rmap1#2#(Cons(x13,Nil()),x8) -> c_11(f#1#(x13))
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> c_12(rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
,plus#2#(x2,S(S(0())))
,f#1#(x27))
- Weak TRS:
f#1(x1) -> mult#2(x1,mult#2(x1,x1))
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> S(plus#2(mult#2(x4,x2),x2))
plus#2(x4,0()) -> x4
plus#2(x4,S(x2)) -> S(plus#2(x4,x2))
rmap#2(Cons(x36,x10),x6) -> rmap#2(x10,Cons(x36,x6))
rmap#2(Nil(),x2) -> x2
rmap1#2(Cons(x13,Nil()),x8) -> Cons(f#1(x13),x8)
rmap1#2(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
rmap1#2(Nil(),x2) -> x2
- Signature:
{f#1/1,main/1,map#2/1,mult#2/2,plus#2/2,rmap#2/2,rmap1#2/2,f#1#/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2
,rmap#2#/2,rmap1#2#/2} / {0/0,Cons/2,Nil/0,S/1,c_1/2,c_2/3,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/1,c_9/1,c_10/0
,c_11/1,c_12/3,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {main#} and constructors {0,Cons,Nil,S}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:f#1#(x1) -> c_1(mult#2#(x1,mult#2(x1,x1)),mult#2#(x1,x1))
-->_2 mult#2#(S(x4),x2) -> c_6(plus#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)):4
-->_1 mult#2#(S(x4),x2) -> c_6(plus#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)):4
2:W:main#(x1) -> c_2(map#2#(rmap#2(rmap1#2(x1,Nil()),Nil()))
,rmap#2#(rmap1#2(x1,Nil()),Nil())
,rmap1#2#(x1,Nil()))
-->_3 rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> c_12(rmap1#2#(x25
,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
,plus#2#(x2,S(S(0())))
,f#1#(x27)):8
-->_3 rmap1#2#(Cons(x13,Nil()),x8) -> c_11(f#1#(x13)):7
-->_2 rmap#2#(Cons(x36,x10),x6) -> c_9(rmap#2#(x10,Cons(x36,x6))):6
-->_1 map#2#(Cons(x36,x6)) -> c_3(map#2#(x6)):3
3:W:map#2#(Cons(x36,x6)) -> c_3(map#2#(x6))
-->_1 map#2#(Cons(x36,x6)) -> c_3(map#2#(x6)):3
4:S:mult#2#(S(x4),x2) -> c_6(plus#2#(mult#2(x4,x2),x2),mult#2#(x4,x2))
-->_1 plus#2#(x4,S(x2)) -> c_8(plus#2#(x4,x2)):5
-->_2 mult#2#(S(x4),x2) -> c_6(plus#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)):4
5:S:plus#2#(x4,S(x2)) -> c_8(plus#2#(x4,x2))
-->_1 plus#2#(x4,S(x2)) -> c_8(plus#2#(x4,x2)):5
6:W:rmap#2#(Cons(x36,x10),x6) -> c_9(rmap#2#(x10,Cons(x36,x6)))
-->_1 rmap#2#(Cons(x36,x10),x6) -> c_9(rmap#2#(x10,Cons(x36,x6))):6
7:W:rmap1#2#(Cons(x13,Nil()),x8) -> c_11(f#1#(x13))
-->_1 f#1#(x1) -> c_1(mult#2#(x1,mult#2(x1,x1)),mult#2#(x1,x1)):1
8:W:rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> c_12(rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
,plus#2#(x2,S(S(0())))
,f#1#(x27))
-->_3 f#1#(x1) -> c_1(mult#2#(x1,mult#2(x1,x1)),mult#2#(x1,x1)):1
-->_2 plus#2#(x4,S(x2)) -> c_8(plus#2#(x4,x2)):5
-->_1 rmap1#2#(Cons(x13,Nil()),x8) -> c_11(f#1#(x13)):7
-->_1 rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> c_12(rmap1#2#(x25
,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
,plus#2#(x2,S(S(0())))
,f#1#(x27)):8
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
3: map#2#(Cons(x36,x6)) -> c_3(map#2#(x6))
6: rmap#2#(Cons(x36,x10),x6) -> c_9(rmap#2#(x10,Cons(x36,x6)))
** Step 31.a:2: SimplifyRHS WORST_CASE(?,O(n^4))
+ Considered Problem:
- Strict DPs:
f#1#(x1) -> c_1(mult#2#(x1,mult#2(x1,x1)),mult#2#(x1,x1))
mult#2#(S(x4),x2) -> c_6(plus#2#(mult#2(x4,x2),x2),mult#2#(x4,x2))
plus#2#(x4,S(x2)) -> c_8(plus#2#(x4,x2))
- Weak DPs:
main#(x1) -> c_2(map#2#(rmap#2(rmap1#2(x1,Nil()),Nil()))
,rmap#2#(rmap1#2(x1,Nil()),Nil())
,rmap1#2#(x1,Nil()))
rmap1#2#(Cons(x13,Nil()),x8) -> c_11(f#1#(x13))
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> c_12(rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
,plus#2#(x2,S(S(0())))
,f#1#(x27))
- Weak TRS:
f#1(x1) -> mult#2(x1,mult#2(x1,x1))
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> S(plus#2(mult#2(x4,x2),x2))
plus#2(x4,0()) -> x4
plus#2(x4,S(x2)) -> S(plus#2(x4,x2))
rmap#2(Cons(x36,x10),x6) -> rmap#2(x10,Cons(x36,x6))
rmap#2(Nil(),x2) -> x2
rmap1#2(Cons(x13,Nil()),x8) -> Cons(f#1(x13),x8)
rmap1#2(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
rmap1#2(Nil(),x2) -> x2
- Signature:
{f#1/1,main/1,map#2/1,mult#2/2,plus#2/2,rmap#2/2,rmap1#2/2,f#1#/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2
,rmap#2#/2,rmap1#2#/2} / {0/0,Cons/2,Nil/0,S/1,c_1/2,c_2/3,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/1,c_9/1,c_10/0
,c_11/1,c_12/3,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {main#} and constructors {0,Cons,Nil,S}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:f#1#(x1) -> c_1(mult#2#(x1,mult#2(x1,x1)),mult#2#(x1,x1))
-->_2 mult#2#(S(x4),x2) -> c_6(plus#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)):4
-->_1 mult#2#(S(x4),x2) -> c_6(plus#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)):4
2:W:main#(x1) -> c_2(map#2#(rmap#2(rmap1#2(x1,Nil()),Nil()))
,rmap#2#(rmap1#2(x1,Nil()),Nil())
,rmap1#2#(x1,Nil()))
-->_3 rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> c_12(rmap1#2#(x25
,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
,plus#2#(x2,S(S(0())))
,f#1#(x27)):8
-->_3 rmap1#2#(Cons(x13,Nil()),x8) -> c_11(f#1#(x13)):7
4:S:mult#2#(S(x4),x2) -> c_6(plus#2#(mult#2(x4,x2),x2),mult#2#(x4,x2))
-->_1 plus#2#(x4,S(x2)) -> c_8(plus#2#(x4,x2)):5
-->_2 mult#2#(S(x4),x2) -> c_6(plus#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)):4
5:S:plus#2#(x4,S(x2)) -> c_8(plus#2#(x4,x2))
-->_1 plus#2#(x4,S(x2)) -> c_8(plus#2#(x4,x2)):5
7:W:rmap1#2#(Cons(x13,Nil()),x8) -> c_11(f#1#(x13))
-->_1 f#1#(x1) -> c_1(mult#2#(x1,mult#2(x1,x1)),mult#2#(x1,x1)):1
8:W:rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> c_12(rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
,plus#2#(x2,S(S(0())))
,f#1#(x27))
-->_3 f#1#(x1) -> c_1(mult#2#(x1,mult#2(x1,x1)),mult#2#(x1,x1)):1
-->_2 plus#2#(x4,S(x2)) -> c_8(plus#2#(x4,x2)):5
-->_1 rmap1#2#(Cons(x13,Nil()),x8) -> c_11(f#1#(x13)):7
-->_1 rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> c_12(rmap1#2#(x25
,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
,plus#2#(x2,S(S(0())))
,f#1#(x27)):8
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
main#(x1) -> c_2(rmap1#2#(x1,Nil()))
** Step 31.a:3: UsableRules WORST_CASE(?,O(n^4))
+ Considered Problem:
- Strict DPs:
f#1#(x1) -> c_1(mult#2#(x1,mult#2(x1,x1)),mult#2#(x1,x1))
mult#2#(S(x4),x2) -> c_6(plus#2#(mult#2(x4,x2),x2),mult#2#(x4,x2))
plus#2#(x4,S(x2)) -> c_8(plus#2#(x4,x2))
- Weak DPs:
main#(x1) -> c_2(rmap1#2#(x1,Nil()))
rmap1#2#(Cons(x13,Nil()),x8) -> c_11(f#1#(x13))
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> c_12(rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
,plus#2#(x2,S(S(0())))
,f#1#(x27))
- Weak TRS:
f#1(x1) -> mult#2(x1,mult#2(x1,x1))
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> S(plus#2(mult#2(x4,x2),x2))
plus#2(x4,0()) -> x4
plus#2(x4,S(x2)) -> S(plus#2(x4,x2))
rmap#2(Cons(x36,x10),x6) -> rmap#2(x10,Cons(x36,x6))
rmap#2(Nil(),x2) -> x2
rmap1#2(Cons(x13,Nil()),x8) -> Cons(f#1(x13),x8)
rmap1#2(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
rmap1#2(Nil(),x2) -> x2
- Signature:
{f#1/1,main/1,map#2/1,mult#2/2,plus#2/2,rmap#2/2,rmap1#2/2,f#1#/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2
,rmap#2#/2,rmap1#2#/2} / {0/0,Cons/2,Nil/0,S/1,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/1,c_9/1,c_10/0
,c_11/1,c_12/3,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {main#} and constructors {0,Cons,Nil,S}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
f#1(x1) -> mult#2(x1,mult#2(x1,x1))
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> S(plus#2(mult#2(x4,x2),x2))
plus#2(x4,0()) -> x4
plus#2(x4,S(x2)) -> S(plus#2(x4,x2))
f#1#(x1) -> c_1(mult#2#(x1,mult#2(x1,x1)),mult#2#(x1,x1))
main#(x1) -> c_2(rmap1#2#(x1,Nil()))
mult#2#(S(x4),x2) -> c_6(plus#2#(mult#2(x4,x2),x2),mult#2#(x4,x2))
plus#2#(x4,S(x2)) -> c_8(plus#2#(x4,x2))
rmap1#2#(Cons(x13,Nil()),x8) -> c_11(f#1#(x13))
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> c_12(rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
,plus#2#(x2,S(S(0())))
,f#1#(x27))
** Step 31.a:4: DecomposeDG WORST_CASE(?,O(n^4))
+ Considered Problem:
- Strict DPs:
f#1#(x1) -> c_1(mult#2#(x1,mult#2(x1,x1)),mult#2#(x1,x1))
mult#2#(S(x4),x2) -> c_6(plus#2#(mult#2(x4,x2),x2),mult#2#(x4,x2))
plus#2#(x4,S(x2)) -> c_8(plus#2#(x4,x2))
- Weak DPs:
main#(x1) -> c_2(rmap1#2#(x1,Nil()))
rmap1#2#(Cons(x13,Nil()),x8) -> c_11(f#1#(x13))
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> c_12(rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
,plus#2#(x2,S(S(0())))
,f#1#(x27))
- Weak TRS:
f#1(x1) -> mult#2(x1,mult#2(x1,x1))
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> S(plus#2(mult#2(x4,x2),x2))
plus#2(x4,0()) -> x4
plus#2(x4,S(x2)) -> S(plus#2(x4,x2))
- Signature:
{f#1/1,main/1,map#2/1,mult#2/2,plus#2/2,rmap#2/2,rmap1#2/2,f#1#/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2
,rmap#2#/2,rmap1#2#/2} / {0/0,Cons/2,Nil/0,S/1,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/1,c_9/1,c_10/0
,c_11/1,c_12/3,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {main#} and constructors {0,Cons,Nil,S}
+ Applied Processor:
DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing}
+ Details:
We decompose the input problem according to the dependency graph into the upper component
main#(x1) -> c_2(rmap1#2#(x1,Nil()))
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> c_12(rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
,plus#2#(x2,S(S(0())))
,f#1#(x27))
and a lower component
f#1#(x1) -> c_1(mult#2#(x1,mult#2(x1,x1)),mult#2#(x1,x1))
mult#2#(S(x4),x2) -> c_6(plus#2#(mult#2(x4,x2),x2),mult#2#(x4,x2))
plus#2#(x4,S(x2)) -> c_8(plus#2#(x4,x2))
rmap1#2#(Cons(x13,Nil()),x8) -> c_11(f#1#(x13))
Further, following extension rules are added to the lower component.
main#(x1) -> rmap1#2#(x1,Nil())
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> f#1#(x27)
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> plus#2#(x2,S(S(0())))
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
*** Step 31.a:4.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
main#(x1) -> c_2(rmap1#2#(x1,Nil()))
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> c_12(rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
,plus#2#(x2,S(S(0())))
,f#1#(x27))
- Weak TRS:
f#1(x1) -> mult#2(x1,mult#2(x1,x1))
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> S(plus#2(mult#2(x4,x2),x2))
plus#2(x4,0()) -> x4
plus#2(x4,S(x2)) -> S(plus#2(x4,x2))
- Signature:
{f#1/1,main/1,map#2/1,mult#2/2,plus#2/2,rmap#2/2,rmap1#2/2,f#1#/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2
,rmap#2#/2,rmap1#2#/2} / {0/0,Cons/2,Nil/0,S/1,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/1,c_9/1,c_10/0
,c_11/1,c_12/3,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {main#} and constructors {0,Cons,Nil,S}
+ Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
+ Details:
We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
1: main#(x1) -> c_2(rmap1#2#(x1,Nil()))
2: rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> c_12(rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
,plus#2#(x2,S(S(0())))
,f#1#(x27))
The strictly oriented rules are moved into the weak component.
**** Step 31.a:4.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
main#(x1) -> c_2(rmap1#2#(x1,Nil()))
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> c_12(rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
,plus#2#(x2,S(S(0())))
,f#1#(x27))
- Weak TRS:
f#1(x1) -> mult#2(x1,mult#2(x1,x1))
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> S(plus#2(mult#2(x4,x2),x2))
plus#2(x4,0()) -> x4
plus#2(x4,S(x2)) -> S(plus#2(x4,x2))
- Signature:
{f#1/1,main/1,map#2/1,mult#2/2,plus#2/2,rmap#2/2,rmap1#2/2,f#1#/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2
,rmap#2#/2,rmap1#2#/2} / {0/0,Cons/2,Nil/0,S/1,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/1,c_9/1,c_10/0
,c_11/1,c_12/3,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {main#} and constructors {0,Cons,Nil,S}
+ 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_2) = {1},
uargs(c_12) = {1}
Following symbols are considered usable:
{main#,rmap1#2#}
TcT has computed the following interpretation:
p(0) = [0]
p(Cons) = [1] x1 + [1] x2 + [2]
p(Nil) = [0]
p(S) = [0]
p(f#1) = [8]
p(main) = [0]
p(map#2) = [0]
p(mult#2) = [1] x2 + [2]
p(plus#2) = [7] x1 + [10]
p(rmap#2) = [0]
p(rmap1#2) = [2] x2 + [0]
p(f#1#) = [2] x1 + [0]
p(main#) = [8] x1 + [9]
p(map#2#) = [2] x1 + [1]
p(mult#2#) = [1] x2 + [0]
p(plus#2#) = [1] x1 + [1] x2 + [4]
p(rmap#2#) = [4] x1 + [1] x2 + [0]
p(rmap1#2#) = [4] x1 + [0]
p(c_1) = [1] x1 + [2] x2 + [4]
p(c_2) = [2] x1 + [0]
p(c_3) = [1] x1 + [1]
p(c_4) = [0]
p(c_5) = [1]
p(c_6) = [2] x1 + [1] x2 + [1]
p(c_7) = [1]
p(c_8) = [1] x1 + [0]
p(c_9) = [1]
p(c_10) = [2]
p(c_11) = [2] x1 + [0]
p(c_12) = [1] x1 + [2] x2 + [2] x3 + [7]
p(c_13) = [2]
Following rules are strictly oriented:
main#(x1) = [8] x1 + [9]
> [8] x1 + [0]
= c_2(rmap1#2#(x1,Nil()))
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) = [4] x2 + [4] x25 + [4] x27 + [16]
> [2] x2 + [4] x25 + [4] x27 + [15]
= c_12(rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17))),plus#2#(x2,S(S(0()))),f#1#(x27))
Following rules are (at-least) weakly oriented:
**** Step 31.a:4.a:1.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
main#(x1) -> c_2(rmap1#2#(x1,Nil()))
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> c_12(rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
,plus#2#(x2,S(S(0())))
,f#1#(x27))
- Weak TRS:
f#1(x1) -> mult#2(x1,mult#2(x1,x1))
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> S(plus#2(mult#2(x4,x2),x2))
plus#2(x4,0()) -> x4
plus#2(x4,S(x2)) -> S(plus#2(x4,x2))
- Signature:
{f#1/1,main/1,map#2/1,mult#2/2,plus#2/2,rmap#2/2,rmap1#2/2,f#1#/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2
,rmap#2#/2,rmap1#2#/2} / {0/0,Cons/2,Nil/0,S/1,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/1,c_9/1,c_10/0
,c_11/1,c_12/3,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {main#} and constructors {0,Cons,Nil,S}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
**** Step 31.a:4.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
main#(x1) -> c_2(rmap1#2#(x1,Nil()))
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> c_12(rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
,plus#2#(x2,S(S(0())))
,f#1#(x27))
- Weak TRS:
f#1(x1) -> mult#2(x1,mult#2(x1,x1))
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> S(plus#2(mult#2(x4,x2),x2))
plus#2(x4,0()) -> x4
plus#2(x4,S(x2)) -> S(plus#2(x4,x2))
- Signature:
{f#1/1,main/1,map#2/1,mult#2/2,plus#2/2,rmap#2/2,rmap1#2/2,f#1#/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2
,rmap#2#/2,rmap1#2#/2} / {0/0,Cons/2,Nil/0,S/1,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/1,c_9/1,c_10/0
,c_11/1,c_12/3,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {main#} and constructors {0,Cons,Nil,S}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:main#(x1) -> c_2(rmap1#2#(x1,Nil()))
-->_1 rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> c_12(rmap1#2#(x25
,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
,plus#2#(x2,S(S(0())))
,f#1#(x27)):2
2:W:rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> c_12(rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
,plus#2#(x2,S(S(0())))
,f#1#(x27))
-->_1 rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> c_12(rmap1#2#(x25
,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
,plus#2#(x2,S(S(0())))
,f#1#(x27)):2
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: main#(x1) -> c_2(rmap1#2#(x1,Nil()))
2: rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> c_12(rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
,plus#2#(x2,S(S(0())))
,f#1#(x27))
**** Step 31.a:4.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
f#1(x1) -> mult#2(x1,mult#2(x1,x1))
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> S(plus#2(mult#2(x4,x2),x2))
plus#2(x4,0()) -> x4
plus#2(x4,S(x2)) -> S(plus#2(x4,x2))
- Signature:
{f#1/1,main/1,map#2/1,mult#2/2,plus#2/2,rmap#2/2,rmap1#2/2,f#1#/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2
,rmap#2#/2,rmap1#2#/2} / {0/0,Cons/2,Nil/0,S/1,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/1,c_9/1,c_10/0
,c_11/1,c_12/3,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {main#} and constructors {0,Cons,Nil,S}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
*** Step 31.a:4.b:1: DecomposeDG WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
f#1#(x1) -> c_1(mult#2#(x1,mult#2(x1,x1)),mult#2#(x1,x1))
mult#2#(S(x4),x2) -> c_6(plus#2#(mult#2(x4,x2),x2),mult#2#(x4,x2))
plus#2#(x4,S(x2)) -> c_8(plus#2#(x4,x2))
- Weak DPs:
main#(x1) -> rmap1#2#(x1,Nil())
rmap1#2#(Cons(x13,Nil()),x8) -> c_11(f#1#(x13))
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> f#1#(x27)
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> plus#2#(x2,S(S(0())))
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
- Weak TRS:
f#1(x1) -> mult#2(x1,mult#2(x1,x1))
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> S(plus#2(mult#2(x4,x2),x2))
plus#2(x4,0()) -> x4
plus#2(x4,S(x2)) -> S(plus#2(x4,x2))
- Signature:
{f#1/1,main/1,map#2/1,mult#2/2,plus#2/2,rmap#2/2,rmap1#2/2,f#1#/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2
,rmap#2#/2,rmap1#2#/2} / {0/0,Cons/2,Nil/0,S/1,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/1,c_9/1,c_10/0
,c_11/1,c_12/3,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {main#} and constructors {0,Cons,Nil,S}
+ Applied Processor:
DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing}
+ Details:
We decompose the input problem according to the dependency graph into the upper component
f#1#(x1) -> c_1(mult#2#(x1,mult#2(x1,x1)),mult#2#(x1,x1))
main#(x1) -> rmap1#2#(x1,Nil())
mult#2#(S(x4),x2) -> c_6(plus#2#(mult#2(x4,x2),x2),mult#2#(x4,x2))
rmap1#2#(Cons(x13,Nil()),x8) -> c_11(f#1#(x13))
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> f#1#(x27)
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> plus#2#(x2,S(S(0())))
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
and a lower component
plus#2#(x4,S(x2)) -> c_8(plus#2#(x4,x2))
Further, following extension rules are added to the lower component.
f#1#(x1) -> mult#2#(x1,x1)
f#1#(x1) -> mult#2#(x1,mult#2(x1,x1))
main#(x1) -> rmap1#2#(x1,Nil())
mult#2#(S(x4),x2) -> mult#2#(x4,x2)
mult#2#(S(x4),x2) -> plus#2#(mult#2(x4,x2),x2)
rmap1#2#(Cons(x13,Nil()),x8) -> f#1#(x13)
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> f#1#(x27)
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> plus#2#(x2,S(S(0())))
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
**** Step 31.a:4.b:1.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f#1#(x1) -> c_1(mult#2#(x1,mult#2(x1,x1)),mult#2#(x1,x1))
mult#2#(S(x4),x2) -> c_6(plus#2#(mult#2(x4,x2),x2),mult#2#(x4,x2))
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> plus#2#(x2,S(S(0())))
- Weak DPs:
main#(x1) -> rmap1#2#(x1,Nil())
rmap1#2#(Cons(x13,Nil()),x8) -> c_11(f#1#(x13))
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> f#1#(x27)
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
- Weak TRS:
f#1(x1) -> mult#2(x1,mult#2(x1,x1))
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> S(plus#2(mult#2(x4,x2),x2))
plus#2(x4,0()) -> x4
plus#2(x4,S(x2)) -> S(plus#2(x4,x2))
- Signature:
{f#1/1,main/1,map#2/1,mult#2/2,plus#2/2,rmap#2/2,rmap1#2/2,f#1#/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2
,rmap#2#/2,rmap1#2#/2} / {0/0,Cons/2,Nil/0,S/1,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/1,c_9/1,c_10/0
,c_11/1,c_12/3,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {main#} and constructors {0,Cons,Nil,S}
+ Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
+ Details:
We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
1: f#1#(x1) -> c_1(mult#2#(x1,mult#2(x1,x1)),mult#2#(x1,x1))
3: rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> plus#2#(x2,S(S(0())))
Consider the set of all dependency pairs
1: f#1#(x1) -> c_1(mult#2#(x1,mult#2(x1,x1)),mult#2#(x1,x1))
2: mult#2#(S(x4),x2) -> c_6(plus#2#(mult#2(x4,x2),x2),mult#2#(x4,x2))
3: rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> plus#2#(x2,S(S(0())))
4: main#(x1) -> rmap1#2#(x1,Nil())
5: rmap1#2#(Cons(x13,Nil()),x8) -> c_11(f#1#(x13))
6: rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> f#1#(x27)
7: rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
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
{1,3}
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 31.a:4.b:1.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f#1#(x1) -> c_1(mult#2#(x1,mult#2(x1,x1)),mult#2#(x1,x1))
mult#2#(S(x4),x2) -> c_6(plus#2#(mult#2(x4,x2),x2),mult#2#(x4,x2))
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> plus#2#(x2,S(S(0())))
- Weak DPs:
main#(x1) -> rmap1#2#(x1,Nil())
rmap1#2#(Cons(x13,Nil()),x8) -> c_11(f#1#(x13))
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> f#1#(x27)
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
- Weak TRS:
f#1(x1) -> mult#2(x1,mult#2(x1,x1))
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> S(plus#2(mult#2(x4,x2),x2))
plus#2(x4,0()) -> x4
plus#2(x4,S(x2)) -> S(plus#2(x4,x2))
- Signature:
{f#1/1,main/1,map#2/1,mult#2/2,plus#2/2,rmap#2/2,rmap1#2/2,f#1#/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2
,rmap#2#/2,rmap1#2#/2} / {0/0,Cons/2,Nil/0,S/1,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/1,c_9/1,c_10/0
,c_11/1,c_12/3,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {main#} and constructors {0,Cons,Nil,S}
+ 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_1) = {1,2},
uargs(c_6) = {1,2},
uargs(c_11) = {1}
Following symbols are considered usable:
{f#1#,main#,mult#2#,rmap1#2#}
TcT has computed the following interpretation:
p(0) = [3]
p(Cons) = [1] x1 + [4]
p(Nil) = [0]
p(S) = [1] x1 + [2]
p(f#1) = [8] x1 + [8]
p(main) = [2]
p(map#2) = [1]
p(mult#2) = [10]
p(plus#2) = [1] x1 + [8]
p(rmap#2) = [2] x1 + [1] x2 + [2]
p(rmap1#2) = [1] x1 + [1] x2 + [0]
p(f#1#) = [12]
p(main#) = [12]
p(map#2#) = [8]
p(mult#2#) = [0]
p(plus#2#) = [0]
p(rmap#2#) = [2] x1 + [8] x2 + [1]
p(rmap1#2#) = [12]
p(c_1) = [8] x1 + [2] x2 + [1]
p(c_2) = [2]
p(c_3) = [2] x1 + [0]
p(c_4) = [1]
p(c_5) = [0]
p(c_6) = [4] x1 + [2] x2 + [0]
p(c_7) = [1]
p(c_8) = [1]
p(c_9) = [1]
p(c_10) = [1]
p(c_11) = [1] x1 + [0]
p(c_12) = [1] x1 + [4] x2 + [1] x3 + [2]
p(c_13) = [1]
Following rules are strictly oriented:
f#1#(x1) = [12]
> [1]
= c_1(mult#2#(x1,mult#2(x1,x1)),mult#2#(x1,x1))
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) = [12]
> [0]
= plus#2#(x2,S(S(0())))
Following rules are (at-least) weakly oriented:
main#(x1) = [12]
>= [12]
= rmap1#2#(x1,Nil())
mult#2#(S(x4),x2) = [0]
>= [0]
= c_6(plus#2#(mult#2(x4,x2),x2),mult#2#(x4,x2))
rmap1#2#(Cons(x13,Nil()),x8) = [12]
>= [12]
= c_11(f#1#(x13))
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) = [12]
>= [12]
= f#1#(x27)
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) = [12]
>= [12]
= rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
***** Step 31.a:4.b:1.a:1.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
mult#2#(S(x4),x2) -> c_6(plus#2#(mult#2(x4,x2),x2),mult#2#(x4,x2))
- Weak DPs:
f#1#(x1) -> c_1(mult#2#(x1,mult#2(x1,x1)),mult#2#(x1,x1))
main#(x1) -> rmap1#2#(x1,Nil())
rmap1#2#(Cons(x13,Nil()),x8) -> c_11(f#1#(x13))
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> f#1#(x27)
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> plus#2#(x2,S(S(0())))
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
- Weak TRS:
f#1(x1) -> mult#2(x1,mult#2(x1,x1))
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> S(plus#2(mult#2(x4,x2),x2))
plus#2(x4,0()) -> x4
plus#2(x4,S(x2)) -> S(plus#2(x4,x2))
- Signature:
{f#1/1,main/1,map#2/1,mult#2/2,plus#2/2,rmap#2/2,rmap1#2/2,f#1#/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2
,rmap#2#/2,rmap1#2#/2} / {0/0,Cons/2,Nil/0,S/1,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/1,c_9/1,c_10/0
,c_11/1,c_12/3,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {main#} and constructors {0,Cons,Nil,S}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
***** Step 31.a:4.b:1.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
mult#2#(S(x4),x2) -> c_6(plus#2#(mult#2(x4,x2),x2),mult#2#(x4,x2))
- Weak DPs:
f#1#(x1) -> c_1(mult#2#(x1,mult#2(x1,x1)),mult#2#(x1,x1))
main#(x1) -> rmap1#2#(x1,Nil())
rmap1#2#(Cons(x13,Nil()),x8) -> c_11(f#1#(x13))
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> f#1#(x27)
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> plus#2#(x2,S(S(0())))
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
- Weak TRS:
f#1(x1) -> mult#2(x1,mult#2(x1,x1))
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> S(plus#2(mult#2(x4,x2),x2))
plus#2(x4,0()) -> x4
plus#2(x4,S(x2)) -> S(plus#2(x4,x2))
- Signature:
{f#1/1,main/1,map#2/1,mult#2/2,plus#2/2,rmap#2/2,rmap1#2/2,f#1#/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2
,rmap#2#/2,rmap1#2#/2} / {0/0,Cons/2,Nil/0,S/1,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/1,c_9/1,c_10/0
,c_11/1,c_12/3,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {main#} and constructors {0,Cons,Nil,S}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:mult#2#(S(x4),x2) -> c_6(plus#2#(mult#2(x4,x2),x2),mult#2#(x4,x2))
-->_2 mult#2#(S(x4),x2) -> c_6(plus#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)):1
2:W:f#1#(x1) -> c_1(mult#2#(x1,mult#2(x1,x1)),mult#2#(x1,x1))
-->_2 mult#2#(S(x4),x2) -> c_6(plus#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)):1
-->_1 mult#2#(S(x4),x2) -> c_6(plus#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)):1
3:W:main#(x1) -> rmap1#2#(x1,Nil())
-->_1 rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2#(x25
,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17))):7
-->_1 rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> f#1#(x27):5
-->_1 rmap1#2#(Cons(x13,Nil()),x8) -> c_11(f#1#(x13)):4
-->_1 rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> plus#2#(x2,S(S(0()))):6
4:W:rmap1#2#(Cons(x13,Nil()),x8) -> c_11(f#1#(x13))
-->_1 f#1#(x1) -> c_1(mult#2#(x1,mult#2(x1,x1)),mult#2#(x1,x1)):2
5:W:rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> f#1#(x27)
-->_1 f#1#(x1) -> c_1(mult#2#(x1,mult#2(x1,x1)),mult#2#(x1,x1)):2
6:W:rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> plus#2#(x2,S(S(0())))
7:W:rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
-->_1 rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2#(x25
,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17))):7
-->_1 rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> plus#2#(x2,S(S(0()))):6
-->_1 rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> f#1#(x27):5
-->_1 rmap1#2#(Cons(x13,Nil()),x8) -> c_11(f#1#(x13)):4
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
6: rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> plus#2#(x2,S(S(0())))
***** Step 31.a:4.b:1.a:1.b:2: SimplifyRHS WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
mult#2#(S(x4),x2) -> c_6(plus#2#(mult#2(x4,x2),x2),mult#2#(x4,x2))
- Weak DPs:
f#1#(x1) -> c_1(mult#2#(x1,mult#2(x1,x1)),mult#2#(x1,x1))
main#(x1) -> rmap1#2#(x1,Nil())
rmap1#2#(Cons(x13,Nil()),x8) -> c_11(f#1#(x13))
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> f#1#(x27)
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
- Weak TRS:
f#1(x1) -> mult#2(x1,mult#2(x1,x1))
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> S(plus#2(mult#2(x4,x2),x2))
plus#2(x4,0()) -> x4
plus#2(x4,S(x2)) -> S(plus#2(x4,x2))
- Signature:
{f#1/1,main/1,map#2/1,mult#2/2,plus#2/2,rmap#2/2,rmap1#2/2,f#1#/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2
,rmap#2#/2,rmap1#2#/2} / {0/0,Cons/2,Nil/0,S/1,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/1,c_9/1,c_10/0
,c_11/1,c_12/3,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {main#} and constructors {0,Cons,Nil,S}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:mult#2#(S(x4),x2) -> c_6(plus#2#(mult#2(x4,x2),x2),mult#2#(x4,x2))
-->_2 mult#2#(S(x4),x2) -> c_6(plus#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)):1
2:W:f#1#(x1) -> c_1(mult#2#(x1,mult#2(x1,x1)),mult#2#(x1,x1))
-->_2 mult#2#(S(x4),x2) -> c_6(plus#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)):1
-->_1 mult#2#(S(x4),x2) -> c_6(plus#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)):1
3:W:main#(x1) -> rmap1#2#(x1,Nil())
-->_1 rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2#(x25
,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17))):7
-->_1 rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> f#1#(x27):5
-->_1 rmap1#2#(Cons(x13,Nil()),x8) -> c_11(f#1#(x13)):4
4:W:rmap1#2#(Cons(x13,Nil()),x8) -> c_11(f#1#(x13))
-->_1 f#1#(x1) -> c_1(mult#2#(x1,mult#2(x1,x1)),mult#2#(x1,x1)):2
5:W:rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> f#1#(x27)
-->_1 f#1#(x1) -> c_1(mult#2#(x1,mult#2(x1,x1)),mult#2#(x1,x1)):2
7:W:rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
-->_1 rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2#(x25
,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17))):7
-->_1 rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> f#1#(x27):5
-->_1 rmap1#2#(Cons(x13,Nil()),x8) -> c_11(f#1#(x13)):4
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
mult#2#(S(x4),x2) -> c_6(mult#2#(x4,x2))
***** Step 31.a:4.b:1.a:1.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
mult#2#(S(x4),x2) -> c_6(mult#2#(x4,x2))
- Weak DPs:
f#1#(x1) -> c_1(mult#2#(x1,mult#2(x1,x1)),mult#2#(x1,x1))
main#(x1) -> rmap1#2#(x1,Nil())
rmap1#2#(Cons(x13,Nil()),x8) -> c_11(f#1#(x13))
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> f#1#(x27)
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
- Weak TRS:
f#1(x1) -> mult#2(x1,mult#2(x1,x1))
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> S(plus#2(mult#2(x4,x2),x2))
plus#2(x4,0()) -> x4
plus#2(x4,S(x2)) -> S(plus#2(x4,x2))
- Signature:
{f#1/1,main/1,map#2/1,mult#2/2,plus#2/2,rmap#2/2,rmap1#2/2,f#1#/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2
,rmap#2#/2,rmap1#2#/2} / {0/0,Cons/2,Nil/0,S/1,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/1,c_7/0,c_8/1,c_9/1,c_10/0
,c_11/1,c_12/3,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {main#} and constructors {0,Cons,Nil,S}
+ Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
+ Details:
We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
1: mult#2#(S(x4),x2) -> c_6(mult#2#(x4,x2))
Consider the set of all dependency pairs
1: mult#2#(S(x4),x2) -> c_6(mult#2#(x4,x2))
2: f#1#(x1) -> c_1(mult#2#(x1,mult#2(x1,x1)),mult#2#(x1,x1))
3: main#(x1) -> rmap1#2#(x1,Nil())
4: rmap1#2#(Cons(x13,Nil()),x8) -> c_11(f#1#(x13))
5: rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> f#1#(x27)
6: rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
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
{1}
These cover all (indirect) predecessors of dependency pairs
{1,3}
their number of applications is equally bounded.
The dependency pairs are shifted into the weak component.
****** Step 31.a:4.b:1.a:1.b:3.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
mult#2#(S(x4),x2) -> c_6(mult#2#(x4,x2))
- Weak DPs:
f#1#(x1) -> c_1(mult#2#(x1,mult#2(x1,x1)),mult#2#(x1,x1))
main#(x1) -> rmap1#2#(x1,Nil())
rmap1#2#(Cons(x13,Nil()),x8) -> c_11(f#1#(x13))
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> f#1#(x27)
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
- Weak TRS:
f#1(x1) -> mult#2(x1,mult#2(x1,x1))
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> S(plus#2(mult#2(x4,x2),x2))
plus#2(x4,0()) -> x4
plus#2(x4,S(x2)) -> S(plus#2(x4,x2))
- Signature:
{f#1/1,main/1,map#2/1,mult#2/2,plus#2/2,rmap#2/2,rmap1#2/2,f#1#/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2
,rmap#2#/2,rmap1#2#/2} / {0/0,Cons/2,Nil/0,S/1,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/1,c_7/0,c_8/1,c_9/1,c_10/0
,c_11/1,c_12/3,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {main#} and constructors {0,Cons,Nil,S}
+ 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_1) = {1,2},
uargs(c_6) = {1},
uargs(c_11) = {1}
Following symbols are considered usable:
{f#1#,main#,mult#2#,rmap1#2#}
TcT has computed the following interpretation:
p(0) = [8]
p(Cons) = [1] x1 + [1] x2 + [0]
p(Nil) = [0]
p(S) = [1] x1 + [8]
p(f#1) = [1]
p(main) = [2]
p(map#2) = [1] x1 + [0]
p(mult#2) = [4] x2 + [2]
p(plus#2) = [15]
p(rmap#2) = [2]
p(rmap1#2) = [1] x2 + [0]
p(f#1#) = [4] x1 + [0]
p(main#) = [9] x1 + [5]
p(map#2#) = [1]
p(mult#2#) = [1] x1 + [0]
p(plus#2#) = [1] x1 + [8]
p(rmap#2#) = [0]
p(rmap1#2#) = [8] x1 + [4]
p(c_1) = [1] x1 + [2] x2 + [0]
p(c_2) = [0]
p(c_3) = [0]
p(c_4) = [2]
p(c_5) = [1]
p(c_6) = [1] x1 + [4]
p(c_7) = [1]
p(c_8) = [1] x1 + [0]
p(c_9) = [0]
p(c_10) = [1]
p(c_11) = [1] x1 + [4]
p(c_12) = [0]
p(c_13) = [1]
Following rules are strictly oriented:
mult#2#(S(x4),x2) = [1] x4 + [8]
> [1] x4 + [4]
= c_6(mult#2#(x4,x2))
Following rules are (at-least) weakly oriented:
f#1#(x1) = [4] x1 + [0]
>= [3] x1 + [0]
= c_1(mult#2#(x1,mult#2(x1,x1)),mult#2#(x1,x1))
main#(x1) = [9] x1 + [5]
>= [8] x1 + [4]
= rmap1#2#(x1,Nil())
rmap1#2#(Cons(x13,Nil()),x8) = [8] x13 + [4]
>= [4] x13 + [4]
= c_11(f#1#(x13))
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) = [8] x2 + [8] x25 + [8] x27 + [4]
>= [4] x27 + [0]
= f#1#(x27)
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) = [8] x2 + [8] x25 + [8] x27 + [4]
>= [8] x25 + [4]
= rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
****** Step 31.a:4.b:1.a:1.b:3.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
f#1#(x1) -> c_1(mult#2#(x1,mult#2(x1,x1)),mult#2#(x1,x1))
main#(x1) -> rmap1#2#(x1,Nil())
mult#2#(S(x4),x2) -> c_6(mult#2#(x4,x2))
rmap1#2#(Cons(x13,Nil()),x8) -> c_11(f#1#(x13))
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> f#1#(x27)
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
- Weak TRS:
f#1(x1) -> mult#2(x1,mult#2(x1,x1))
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> S(plus#2(mult#2(x4,x2),x2))
plus#2(x4,0()) -> x4
plus#2(x4,S(x2)) -> S(plus#2(x4,x2))
- Signature:
{f#1/1,main/1,map#2/1,mult#2/2,plus#2/2,rmap#2/2,rmap1#2/2,f#1#/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2
,rmap#2#/2,rmap1#2#/2} / {0/0,Cons/2,Nil/0,S/1,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/1,c_7/0,c_8/1,c_9/1,c_10/0
,c_11/1,c_12/3,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {main#} and constructors {0,Cons,Nil,S}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
****** Step 31.a:4.b:1.a:1.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
f#1#(x1) -> c_1(mult#2#(x1,mult#2(x1,x1)),mult#2#(x1,x1))
main#(x1) -> rmap1#2#(x1,Nil())
mult#2#(S(x4),x2) -> c_6(mult#2#(x4,x2))
rmap1#2#(Cons(x13,Nil()),x8) -> c_11(f#1#(x13))
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> f#1#(x27)
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
- Weak TRS:
f#1(x1) -> mult#2(x1,mult#2(x1,x1))
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> S(plus#2(mult#2(x4,x2),x2))
plus#2(x4,0()) -> x4
plus#2(x4,S(x2)) -> S(plus#2(x4,x2))
- Signature:
{f#1/1,main/1,map#2/1,mult#2/2,plus#2/2,rmap#2/2,rmap1#2/2,f#1#/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2
,rmap#2#/2,rmap1#2#/2} / {0/0,Cons/2,Nil/0,S/1,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/1,c_7/0,c_8/1,c_9/1,c_10/0
,c_11/1,c_12/3,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {main#} and constructors {0,Cons,Nil,S}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:f#1#(x1) -> c_1(mult#2#(x1,mult#2(x1,x1)),mult#2#(x1,x1))
-->_2 mult#2#(S(x4),x2) -> c_6(mult#2#(x4,x2)):3
-->_1 mult#2#(S(x4),x2) -> c_6(mult#2#(x4,x2)):3
2:W:main#(x1) -> rmap1#2#(x1,Nil())
-->_1 rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2#(x25
,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17))):6
-->_1 rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> f#1#(x27):5
-->_1 rmap1#2#(Cons(x13,Nil()),x8) -> c_11(f#1#(x13)):4
3:W:mult#2#(S(x4),x2) -> c_6(mult#2#(x4,x2))
-->_1 mult#2#(S(x4),x2) -> c_6(mult#2#(x4,x2)):3
4:W:rmap1#2#(Cons(x13,Nil()),x8) -> c_11(f#1#(x13))
-->_1 f#1#(x1) -> c_1(mult#2#(x1,mult#2(x1,x1)),mult#2#(x1,x1)):1
5:W:rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> f#1#(x27)
-->_1 f#1#(x1) -> c_1(mult#2#(x1,mult#2(x1,x1)),mult#2#(x1,x1)):1
6:W:rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
-->_1 rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2#(x25
,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17))):6
-->_1 rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> f#1#(x27):5
-->_1 rmap1#2#(Cons(x13,Nil()),x8) -> c_11(f#1#(x13)):4
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
2: main#(x1) -> rmap1#2#(x1,Nil())
6: rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
4: rmap1#2#(Cons(x13,Nil()),x8) -> c_11(f#1#(x13))
5: rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> f#1#(x27)
1: f#1#(x1) -> c_1(mult#2#(x1,mult#2(x1,x1)),mult#2#(x1,x1))
3: mult#2#(S(x4),x2) -> c_6(mult#2#(x4,x2))
****** Step 31.a:4.b:1.a:1.b:3.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
f#1(x1) -> mult#2(x1,mult#2(x1,x1))
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> S(plus#2(mult#2(x4,x2),x2))
plus#2(x4,0()) -> x4
plus#2(x4,S(x2)) -> S(plus#2(x4,x2))
- Signature:
{f#1/1,main/1,map#2/1,mult#2/2,plus#2/2,rmap#2/2,rmap1#2/2,f#1#/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2
,rmap#2#/2,rmap1#2#/2} / {0/0,Cons/2,Nil/0,S/1,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/1,c_7/0,c_8/1,c_9/1,c_10/0
,c_11/1,c_12/3,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {main#} and constructors {0,Cons,Nil,S}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
**** Step 31.a:4.b:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
plus#2#(x4,S(x2)) -> c_8(plus#2#(x4,x2))
- Weak DPs:
f#1#(x1) -> mult#2#(x1,x1)
f#1#(x1) -> mult#2#(x1,mult#2(x1,x1))
main#(x1) -> rmap1#2#(x1,Nil())
mult#2#(S(x4),x2) -> mult#2#(x4,x2)
mult#2#(S(x4),x2) -> plus#2#(mult#2(x4,x2),x2)
rmap1#2#(Cons(x13,Nil()),x8) -> f#1#(x13)
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> f#1#(x27)
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> plus#2#(x2,S(S(0())))
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
- Weak TRS:
f#1(x1) -> mult#2(x1,mult#2(x1,x1))
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> S(plus#2(mult#2(x4,x2),x2))
plus#2(x4,0()) -> x4
plus#2(x4,S(x2)) -> S(plus#2(x4,x2))
- Signature:
{f#1/1,main/1,map#2/1,mult#2/2,plus#2/2,rmap#2/2,rmap1#2/2,f#1#/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2
,rmap#2#/2,rmap1#2#/2} / {0/0,Cons/2,Nil/0,S/1,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/1,c_9/1,c_10/0
,c_11/1,c_12/3,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {main#} and constructors {0,Cons,Nil,S}
+ 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: plus#2#(x4,S(x2)) -> c_8(plus#2#(x4,x2))
Consider the set of all dependency pairs
1: plus#2#(x4,S(x2)) -> c_8(plus#2#(x4,x2))
2: f#1#(x1) -> mult#2#(x1,x1)
3: f#1#(x1) -> mult#2#(x1,mult#2(x1,x1))
4: main#(x1) -> rmap1#2#(x1,Nil())
5: mult#2#(S(x4),x2) -> mult#2#(x4,x2)
6: mult#2#(S(x4),x2) -> plus#2#(mult#2(x4,x2),x2)
7: rmap1#2#(Cons(x13,Nil()),x8) -> f#1#(x13)
8: rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> f#1#(x27)
9: rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> plus#2#(x2,S(S(0())))
10: rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
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
{1,4}
their number of applications is equally bounded.
The dependency pairs are shifted into the weak component.
***** Step 31.a:4.b:1.b:1.a:1: NaturalPI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
plus#2#(x4,S(x2)) -> c_8(plus#2#(x4,x2))
- Weak DPs:
f#1#(x1) -> mult#2#(x1,x1)
f#1#(x1) -> mult#2#(x1,mult#2(x1,x1))
main#(x1) -> rmap1#2#(x1,Nil())
mult#2#(S(x4),x2) -> mult#2#(x4,x2)
mult#2#(S(x4),x2) -> plus#2#(mult#2(x4,x2),x2)
rmap1#2#(Cons(x13,Nil()),x8) -> f#1#(x13)
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> f#1#(x27)
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> plus#2#(x2,S(S(0())))
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
- Weak TRS:
f#1(x1) -> mult#2(x1,mult#2(x1,x1))
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> S(plus#2(mult#2(x4,x2),x2))
plus#2(x4,0()) -> x4
plus#2(x4,S(x2)) -> S(plus#2(x4,x2))
- Signature:
{f#1/1,main/1,map#2/1,mult#2/2,plus#2/2,rmap#2/2,rmap1#2/2,f#1#/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2
,rmap#2#/2,rmap1#2#/2} / {0/0,Cons/2,Nil/0,S/1,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/1,c_9/1,c_10/0
,c_11/1,c_12/3,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {main#} and constructors {0,Cons,Nil,S}
+ 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_8) = {1}
Following symbols are considered usable:
{mult#2,plus#2,f#1#,main#,mult#2#,plus#2#,rmap1#2#}
TcT has computed the following interpretation:
p(0) = 0
p(Cons) = x1 + x2
p(Nil) = 1
p(S) = 1 + x1
p(f#1) = 0
p(main) = 0
p(map#2) = 2 + x1
p(mult#2) = x1 + 2*x1*x2
p(plus#2) = x1 + x2
p(rmap#2) = 0
p(rmap1#2) = 2*x1^2
p(f#1#) = 2 + x1 + 2*x1^2
p(main#) = 3 + 2*x1 + 2*x1^2
p(map#2#) = 2
p(mult#2#) = 2 + x2
p(plus#2#) = x2
p(rmap#2#) = 2 + 2*x1*x2 + 2*x2
p(rmap1#2#) = 3 + x1 + 2*x1^2
p(c_1) = 0
p(c_2) = 0
p(c_3) = 0
p(c_4) = 0
p(c_5) = 1
p(c_6) = 1
p(c_7) = 2
p(c_8) = x1
p(c_9) = 2
p(c_10) = 0
p(c_11) = 1 + x1
p(c_12) = x3
p(c_13) = 1
Following rules are strictly oriented:
plus#2#(x4,S(x2)) = 1 + x2
> x2
= c_8(plus#2#(x4,x2))
Following rules are (at-least) weakly oriented:
f#1#(x1) = 2 + x1 + 2*x1^2
>= 2 + x1
= mult#2#(x1,x1)
f#1#(x1) = 2 + x1 + 2*x1^2
>= 2 + x1 + 2*x1^2
= mult#2#(x1,mult#2(x1,x1))
main#(x1) = 3 + 2*x1 + 2*x1^2
>= 3 + x1 + 2*x1^2
= rmap1#2#(x1,Nil())
mult#2#(S(x4),x2) = 2 + x2
>= 2 + x2
= mult#2#(x4,x2)
mult#2#(S(x4),x2) = 2 + x2
>= x2
= plus#2#(mult#2(x4,x2),x2)
rmap1#2#(Cons(x13,Nil()),x8) = 6 + 5*x13 + 2*x13^2
>= 2 + x13 + 2*x13^2
= f#1#(x13)
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) = 3 + x2 + 4*x2*x25 + 4*x2*x27 + 2*x2^2 + x25 + 4*x25*x27 + 2*x25^2 + x27 + 2*x27^2
>= 2 + x27 + 2*x27^2
= f#1#(x27)
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) = 3 + x2 + 4*x2*x25 + 4*x2*x27 + 2*x2^2 + x25 + 4*x25*x27 + 2*x25^2 + x27 + 2*x27^2
>= 2
= plus#2#(x2,S(S(0())))
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) = 3 + x2 + 4*x2*x25 + 4*x2*x27 + 2*x2^2 + x25 + 4*x25*x27 + 2*x25^2 + x27 + 2*x27^2
>= 3 + x25 + 2*x25^2
= rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
mult#2(0(),x2) = 0
>= 0
= 0()
mult#2(S(x4),x2) = 1 + 2*x2 + 2*x2*x4 + x4
>= 1 + x2 + 2*x2*x4 + x4
= S(plus#2(mult#2(x4,x2),x2))
plus#2(x4,0()) = x4
>= x4
= x4
plus#2(x4,S(x2)) = 1 + x2 + x4
>= 1 + x2 + x4
= S(plus#2(x4,x2))
***** Step 31.a:4.b:1.b:1.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
f#1#(x1) -> mult#2#(x1,x1)
f#1#(x1) -> mult#2#(x1,mult#2(x1,x1))
main#(x1) -> rmap1#2#(x1,Nil())
mult#2#(S(x4),x2) -> mult#2#(x4,x2)
mult#2#(S(x4),x2) -> plus#2#(mult#2(x4,x2),x2)
plus#2#(x4,S(x2)) -> c_8(plus#2#(x4,x2))
rmap1#2#(Cons(x13,Nil()),x8) -> f#1#(x13)
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> f#1#(x27)
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> plus#2#(x2,S(S(0())))
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
- Weak TRS:
f#1(x1) -> mult#2(x1,mult#2(x1,x1))
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> S(plus#2(mult#2(x4,x2),x2))
plus#2(x4,0()) -> x4
plus#2(x4,S(x2)) -> S(plus#2(x4,x2))
- Signature:
{f#1/1,main/1,map#2/1,mult#2/2,plus#2/2,rmap#2/2,rmap1#2/2,f#1#/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2
,rmap#2#/2,rmap1#2#/2} / {0/0,Cons/2,Nil/0,S/1,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/1,c_9/1,c_10/0
,c_11/1,c_12/3,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {main#} and constructors {0,Cons,Nil,S}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
***** Step 31.a:4.b:1.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
f#1#(x1) -> mult#2#(x1,x1)
f#1#(x1) -> mult#2#(x1,mult#2(x1,x1))
main#(x1) -> rmap1#2#(x1,Nil())
mult#2#(S(x4),x2) -> mult#2#(x4,x2)
mult#2#(S(x4),x2) -> plus#2#(mult#2(x4,x2),x2)
plus#2#(x4,S(x2)) -> c_8(plus#2#(x4,x2))
rmap1#2#(Cons(x13,Nil()),x8) -> f#1#(x13)
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> f#1#(x27)
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> plus#2#(x2,S(S(0())))
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
- Weak TRS:
f#1(x1) -> mult#2(x1,mult#2(x1,x1))
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> S(plus#2(mult#2(x4,x2),x2))
plus#2(x4,0()) -> x4
plus#2(x4,S(x2)) -> S(plus#2(x4,x2))
- Signature:
{f#1/1,main/1,map#2/1,mult#2/2,plus#2/2,rmap#2/2,rmap1#2/2,f#1#/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2
,rmap#2#/2,rmap1#2#/2} / {0/0,Cons/2,Nil/0,S/1,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/1,c_9/1,c_10/0
,c_11/1,c_12/3,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {main#} and constructors {0,Cons,Nil,S}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:f#1#(x1) -> mult#2#(x1,x1)
-->_1 mult#2#(S(x4),x2) -> plus#2#(mult#2(x4,x2),x2):5
-->_1 mult#2#(S(x4),x2) -> mult#2#(x4,x2):4
2:W:f#1#(x1) -> mult#2#(x1,mult#2(x1,x1))
-->_1 mult#2#(S(x4),x2) -> plus#2#(mult#2(x4,x2),x2):5
-->_1 mult#2#(S(x4),x2) -> mult#2#(x4,x2):4
3:W:main#(x1) -> rmap1#2#(x1,Nil())
-->_1 rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2#(x25
,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17))):10
-->_1 rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> plus#2#(x2,S(S(0()))):9
-->_1 rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> f#1#(x27):8
-->_1 rmap1#2#(Cons(x13,Nil()),x8) -> f#1#(x13):7
4:W:mult#2#(S(x4),x2) -> mult#2#(x4,x2)
-->_1 mult#2#(S(x4),x2) -> plus#2#(mult#2(x4,x2),x2):5
-->_1 mult#2#(S(x4),x2) -> mult#2#(x4,x2):4
5:W:mult#2#(S(x4),x2) -> plus#2#(mult#2(x4,x2),x2)
-->_1 plus#2#(x4,S(x2)) -> c_8(plus#2#(x4,x2)):6
6:W:plus#2#(x4,S(x2)) -> c_8(plus#2#(x4,x2))
-->_1 plus#2#(x4,S(x2)) -> c_8(plus#2#(x4,x2)):6
7:W:rmap1#2#(Cons(x13,Nil()),x8) -> f#1#(x13)
-->_1 f#1#(x1) -> mult#2#(x1,mult#2(x1,x1)):2
-->_1 f#1#(x1) -> mult#2#(x1,x1):1
8:W:rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> f#1#(x27)
-->_1 f#1#(x1) -> mult#2#(x1,mult#2(x1,x1)):2
-->_1 f#1#(x1) -> mult#2#(x1,x1):1
9:W:rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> plus#2#(x2,S(S(0())))
-->_1 plus#2#(x4,S(x2)) -> c_8(plus#2#(x4,x2)):6
10:W:rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
-->_1 rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2#(x25
,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17))):10
-->_1 rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> plus#2#(x2,S(S(0()))):9
-->_1 rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> f#1#(x27):8
-->_1 rmap1#2#(Cons(x13,Nil()),x8) -> f#1#(x13):7
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
3: main#(x1) -> rmap1#2#(x1,Nil())
10: rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
7: rmap1#2#(Cons(x13,Nil()),x8) -> f#1#(x13)
8: rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> f#1#(x27)
9: rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> plus#2#(x2,S(S(0())))
2: f#1#(x1) -> mult#2#(x1,mult#2(x1,x1))
1: f#1#(x1) -> mult#2#(x1,x1)
4: mult#2#(S(x4),x2) -> mult#2#(x4,x2)
5: mult#2#(S(x4),x2) -> plus#2#(mult#2(x4,x2),x2)
6: plus#2#(x4,S(x2)) -> c_8(plus#2#(x4,x2))
***** Step 31.a:4.b:1.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
f#1(x1) -> mult#2(x1,mult#2(x1,x1))
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> S(plus#2(mult#2(x4,x2),x2))
plus#2(x4,0()) -> x4
plus#2(x4,S(x2)) -> S(plus#2(x4,x2))
- Signature:
{f#1/1,main/1,map#2/1,mult#2/2,plus#2/2,rmap#2/2,rmap1#2/2,f#1#/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2
,rmap#2#/2,rmap1#2#/2} / {0/0,Cons/2,Nil/0,S/1,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/1,c_9/1,c_10/0
,c_11/1,c_12/3,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {main#} and constructors {0,Cons,Nil,S}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
** Step 31.b:1: PredecessorEstimation WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
main#(x1) -> c_2(map#2#(rmap#2(rmap1#2(x1,Nil()),Nil()))
,rmap#2#(rmap1#2(x1,Nil()),Nil())
,rmap1#2#(x1,Nil()))
map#2#(Cons(x36,x6)) -> c_3(map#2#(x6))
rmap#2#(Cons(x36,x10),x6) -> c_9(rmap#2#(x10,Cons(x36,x6)))
rmap1#2#(Cons(x13,Nil()),x8) -> c_11(f#1#(x13))
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> c_12(rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
,plus#2#(x2,S(S(0())))
,f#1#(x27))
- Weak DPs:
f#1#(x1) -> c_1(mult#2#(x1,mult#2(x1,x1)),mult#2#(x1,x1))
mult#2#(S(x4),x2) -> c_6(plus#2#(mult#2(x4,x2),x2),mult#2#(x4,x2))
plus#2#(x4,S(x2)) -> c_8(plus#2#(x4,x2))
- Weak TRS:
f#1(x1) -> mult#2(x1,mult#2(x1,x1))
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> S(plus#2(mult#2(x4,x2),x2))
plus#2(x4,0()) -> x4
plus#2(x4,S(x2)) -> S(plus#2(x4,x2))
rmap#2(Cons(x36,x10),x6) -> rmap#2(x10,Cons(x36,x6))
rmap#2(Nil(),x2) -> x2
rmap1#2(Cons(x13,Nil()),x8) -> Cons(f#1(x13),x8)
rmap1#2(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
rmap1#2(Nil(),x2) -> x2
- Signature:
{f#1/1,main/1,map#2/1,mult#2/2,plus#2/2,rmap#2/2,rmap1#2/2,f#1#/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2
,rmap#2#/2,rmap1#2#/2} / {0/0,Cons/2,Nil/0,S/1,c_1/2,c_2/3,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/1,c_9/1,c_10/0
,c_11/1,c_12/3,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {main#} and constructors {0,Cons,Nil,S}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{4}
by application of
Pre({4}) = {1,5}.
Here rules are labelled as follows:
1: main#(x1) -> c_2(map#2#(rmap#2(rmap1#2(x1,Nil()),Nil()))
,rmap#2#(rmap1#2(x1,Nil()),Nil())
,rmap1#2#(x1,Nil()))
2: map#2#(Cons(x36,x6)) -> c_3(map#2#(x6))
3: rmap#2#(Cons(x36,x10),x6) -> c_9(rmap#2#(x10,Cons(x36,x6)))
4: rmap1#2#(Cons(x13,Nil()),x8) -> c_11(f#1#(x13))
5: rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> c_12(rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
,plus#2#(x2,S(S(0())))
,f#1#(x27))
6: f#1#(x1) -> c_1(mult#2#(x1,mult#2(x1,x1)),mult#2#(x1,x1))
7: mult#2#(S(x4),x2) -> c_6(plus#2#(mult#2(x4,x2),x2),mult#2#(x4,x2))
8: plus#2#(x4,S(x2)) -> c_8(plus#2#(x4,x2))
** Step 31.b:2: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
main#(x1) -> c_2(map#2#(rmap#2(rmap1#2(x1,Nil()),Nil()))
,rmap#2#(rmap1#2(x1,Nil()),Nil())
,rmap1#2#(x1,Nil()))
map#2#(Cons(x36,x6)) -> c_3(map#2#(x6))
rmap#2#(Cons(x36,x10),x6) -> c_9(rmap#2#(x10,Cons(x36,x6)))
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> c_12(rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
,plus#2#(x2,S(S(0())))
,f#1#(x27))
- Weak DPs:
f#1#(x1) -> c_1(mult#2#(x1,mult#2(x1,x1)),mult#2#(x1,x1))
mult#2#(S(x4),x2) -> c_6(plus#2#(mult#2(x4,x2),x2),mult#2#(x4,x2))
plus#2#(x4,S(x2)) -> c_8(plus#2#(x4,x2))
rmap1#2#(Cons(x13,Nil()),x8) -> c_11(f#1#(x13))
- Weak TRS:
f#1(x1) -> mult#2(x1,mult#2(x1,x1))
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> S(plus#2(mult#2(x4,x2),x2))
plus#2(x4,0()) -> x4
plus#2(x4,S(x2)) -> S(plus#2(x4,x2))
rmap#2(Cons(x36,x10),x6) -> rmap#2(x10,Cons(x36,x6))
rmap#2(Nil(),x2) -> x2
rmap1#2(Cons(x13,Nil()),x8) -> Cons(f#1(x13),x8)
rmap1#2(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
rmap1#2(Nil(),x2) -> x2
- Signature:
{f#1/1,main/1,map#2/1,mult#2/2,plus#2/2,rmap#2/2,rmap1#2/2,f#1#/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2
,rmap#2#/2,rmap1#2#/2} / {0/0,Cons/2,Nil/0,S/1,c_1/2,c_2/3,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/1,c_9/1,c_10/0
,c_11/1,c_12/3,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {main#} and constructors {0,Cons,Nil,S}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:main#(x1) -> c_2(map#2#(rmap#2(rmap1#2(x1,Nil()),Nil()))
,rmap#2#(rmap1#2(x1,Nil()),Nil())
,rmap1#2#(x1,Nil()))
-->_3 rmap1#2#(Cons(x13,Nil()),x8) -> c_11(f#1#(x13)):8
-->_3 rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> c_12(rmap1#2#(x25
,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
,plus#2#(x2,S(S(0())))
,f#1#(x27)):4
-->_2 rmap#2#(Cons(x36,x10),x6) -> c_9(rmap#2#(x10,Cons(x36,x6))):3
-->_1 map#2#(Cons(x36,x6)) -> c_3(map#2#(x6)):2
2:S:map#2#(Cons(x36,x6)) -> c_3(map#2#(x6))
-->_1 map#2#(Cons(x36,x6)) -> c_3(map#2#(x6)):2
3:S:rmap#2#(Cons(x36,x10),x6) -> c_9(rmap#2#(x10,Cons(x36,x6)))
-->_1 rmap#2#(Cons(x36,x10),x6) -> c_9(rmap#2#(x10,Cons(x36,x6))):3
4:S:rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> c_12(rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
,plus#2#(x2,S(S(0())))
,f#1#(x27))
-->_1 rmap1#2#(Cons(x13,Nil()),x8) -> c_11(f#1#(x13)):8
-->_2 plus#2#(x4,S(x2)) -> c_8(plus#2#(x4,x2)):7
-->_3 f#1#(x1) -> c_1(mult#2#(x1,mult#2(x1,x1)),mult#2#(x1,x1)):5
-->_1 rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> c_12(rmap1#2#(x25
,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
,plus#2#(x2,S(S(0())))
,f#1#(x27)):4
5:W:f#1#(x1) -> c_1(mult#2#(x1,mult#2(x1,x1)),mult#2#(x1,x1))
-->_2 mult#2#(S(x4),x2) -> c_6(plus#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)):6
-->_1 mult#2#(S(x4),x2) -> c_6(plus#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)):6
6:W:mult#2#(S(x4),x2) -> c_6(plus#2#(mult#2(x4,x2),x2),mult#2#(x4,x2))
-->_1 plus#2#(x4,S(x2)) -> c_8(plus#2#(x4,x2)):7
-->_2 mult#2#(S(x4),x2) -> c_6(plus#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)):6
7:W:plus#2#(x4,S(x2)) -> c_8(plus#2#(x4,x2))
-->_1 plus#2#(x4,S(x2)) -> c_8(plus#2#(x4,x2)):7
8:W:rmap1#2#(Cons(x13,Nil()),x8) -> c_11(f#1#(x13))
-->_1 f#1#(x1) -> c_1(mult#2#(x1,mult#2(x1,x1)),mult#2#(x1,x1)):5
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
8: rmap1#2#(Cons(x13,Nil()),x8) -> c_11(f#1#(x13))
5: f#1#(x1) -> c_1(mult#2#(x1,mult#2(x1,x1)),mult#2#(x1,x1))
6: mult#2#(S(x4),x2) -> c_6(plus#2#(mult#2(x4,x2),x2),mult#2#(x4,x2))
7: plus#2#(x4,S(x2)) -> c_8(plus#2#(x4,x2))
** Step 31.b:3: SimplifyRHS WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
main#(x1) -> c_2(map#2#(rmap#2(rmap1#2(x1,Nil()),Nil()))
,rmap#2#(rmap1#2(x1,Nil()),Nil())
,rmap1#2#(x1,Nil()))
map#2#(Cons(x36,x6)) -> c_3(map#2#(x6))
rmap#2#(Cons(x36,x10),x6) -> c_9(rmap#2#(x10,Cons(x36,x6)))
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> c_12(rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
,plus#2#(x2,S(S(0())))
,f#1#(x27))
- Weak TRS:
f#1(x1) -> mult#2(x1,mult#2(x1,x1))
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> S(plus#2(mult#2(x4,x2),x2))
plus#2(x4,0()) -> x4
plus#2(x4,S(x2)) -> S(plus#2(x4,x2))
rmap#2(Cons(x36,x10),x6) -> rmap#2(x10,Cons(x36,x6))
rmap#2(Nil(),x2) -> x2
rmap1#2(Cons(x13,Nil()),x8) -> Cons(f#1(x13),x8)
rmap1#2(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
rmap1#2(Nil(),x2) -> x2
- Signature:
{f#1/1,main/1,map#2/1,mult#2/2,plus#2/2,rmap#2/2,rmap1#2/2,f#1#/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2
,rmap#2#/2,rmap1#2#/2} / {0/0,Cons/2,Nil/0,S/1,c_1/2,c_2/3,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/1,c_9/1,c_10/0
,c_11/1,c_12/3,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {main#} and constructors {0,Cons,Nil,S}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:main#(x1) -> c_2(map#2#(rmap#2(rmap1#2(x1,Nil()),Nil()))
,rmap#2#(rmap1#2(x1,Nil()),Nil())
,rmap1#2#(x1,Nil()))
-->_3 rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> c_12(rmap1#2#(x25
,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
,plus#2#(x2,S(S(0())))
,f#1#(x27)):4
-->_2 rmap#2#(Cons(x36,x10),x6) -> c_9(rmap#2#(x10,Cons(x36,x6))):3
-->_1 map#2#(Cons(x36,x6)) -> c_3(map#2#(x6)):2
2:S:map#2#(Cons(x36,x6)) -> c_3(map#2#(x6))
-->_1 map#2#(Cons(x36,x6)) -> c_3(map#2#(x6)):2
3:S:rmap#2#(Cons(x36,x10),x6) -> c_9(rmap#2#(x10,Cons(x36,x6)))
-->_1 rmap#2#(Cons(x36,x10),x6) -> c_9(rmap#2#(x10,Cons(x36,x6))):3
4:S:rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> c_12(rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
,plus#2#(x2,S(S(0())))
,f#1#(x27))
-->_1 rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> c_12(rmap1#2#(x25
,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
,plus#2#(x2,S(S(0())))
,f#1#(x27)):4
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> c_12(rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17))))
** Step 31.b:4: Decompose WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
main#(x1) -> c_2(map#2#(rmap#2(rmap1#2(x1,Nil()),Nil()))
,rmap#2#(rmap1#2(x1,Nil()),Nil())
,rmap1#2#(x1,Nil()))
map#2#(Cons(x36,x6)) -> c_3(map#2#(x6))
rmap#2#(Cons(x36,x10),x6) -> c_9(rmap#2#(x10,Cons(x36,x6)))
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> c_12(rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17))))
- Weak TRS:
f#1(x1) -> mult#2(x1,mult#2(x1,x1))
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> S(plus#2(mult#2(x4,x2),x2))
plus#2(x4,0()) -> x4
plus#2(x4,S(x2)) -> S(plus#2(x4,x2))
rmap#2(Cons(x36,x10),x6) -> rmap#2(x10,Cons(x36,x6))
rmap#2(Nil(),x2) -> x2
rmap1#2(Cons(x13,Nil()),x8) -> Cons(f#1(x13),x8)
rmap1#2(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
rmap1#2(Nil(),x2) -> x2
- Signature:
{f#1/1,main/1,map#2/1,mult#2/2,plus#2/2,rmap#2/2,rmap1#2/2,f#1#/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2
,rmap#2#/2,rmap1#2#/2} / {0/0,Cons/2,Nil/0,S/1,c_1/2,c_2/3,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/1,c_9/1,c_10/0
,c_11/1,c_12/1,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {main#} and constructors {0,Cons,Nil,S}
+ 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:
main#(x1) -> c_2(map#2#(rmap#2(rmap1#2(x1,Nil()),Nil()))
,rmap#2#(rmap1#2(x1,Nil()),Nil())
,rmap1#2#(x1,Nil()))
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> c_12(rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17))))
- Weak DPs:
map#2#(Cons(x36,x6)) -> c_3(map#2#(x6))
rmap#2#(Cons(x36,x10),x6) -> c_9(rmap#2#(x10,Cons(x36,x6)))
- Weak TRS:
f#1(x1) -> mult#2(x1,mult#2(x1,x1))
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> S(plus#2(mult#2(x4,x2),x2))
plus#2(x4,0()) -> x4
plus#2(x4,S(x2)) -> S(plus#2(x4,x2))
rmap#2(Cons(x36,x10),x6) -> rmap#2(x10,Cons(x36,x6))
rmap#2(Nil(),x2) -> x2
rmap1#2(Cons(x13,Nil()),x8) -> Cons(f#1(x13),x8)
rmap1#2(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
rmap1#2(Nil(),x2) -> x2
- Signature:
{f#1/1,main/1,map#2/1,mult#2/2,plus#2/2,rmap#2/2,rmap1#2/2,f#1#/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2
,rmap#2#/2,rmap1#2#/2} / {0/0,Cons/2,Nil/0,S/1,c_1/2,c_2/3,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/1,c_9/1
,c_10/0,c_11/1,c_12/1,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {main#} and constructors {0,Cons,Nil,S}
Problem (S)
- Strict DPs:
map#2#(Cons(x36,x6)) -> c_3(map#2#(x6))
rmap#2#(Cons(x36,x10),x6) -> c_9(rmap#2#(x10,Cons(x36,x6)))
- Weak DPs:
main#(x1) -> c_2(map#2#(rmap#2(rmap1#2(x1,Nil()),Nil()))
,rmap#2#(rmap1#2(x1,Nil()),Nil())
,rmap1#2#(x1,Nil()))
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> c_12(rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17))))
- Weak TRS:
f#1(x1) -> mult#2(x1,mult#2(x1,x1))
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> S(plus#2(mult#2(x4,x2),x2))
plus#2(x4,0()) -> x4
plus#2(x4,S(x2)) -> S(plus#2(x4,x2))
rmap#2(Cons(x36,x10),x6) -> rmap#2(x10,Cons(x36,x6))
rmap#2(Nil(),x2) -> x2
rmap1#2(Cons(x13,Nil()),x8) -> Cons(f#1(x13),x8)
rmap1#2(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
rmap1#2(Nil(),x2) -> x2
- Signature:
{f#1/1,main/1,map#2/1,mult#2/2,plus#2/2,rmap#2/2,rmap1#2/2,f#1#/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2
,rmap#2#/2,rmap1#2#/2} / {0/0,Cons/2,Nil/0,S/1,c_1/2,c_2/3,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/1,c_9/1
,c_10/0,c_11/1,c_12/1,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {main#} and constructors {0,Cons,Nil,S}
*** Step 31.b:4.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
main#(x1) -> c_2(map#2#(rmap#2(rmap1#2(x1,Nil()),Nil()))
,rmap#2#(rmap1#2(x1,Nil()),Nil())
,rmap1#2#(x1,Nil()))
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> c_12(rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17))))
- Weak DPs:
map#2#(Cons(x36,x6)) -> c_3(map#2#(x6))
rmap#2#(Cons(x36,x10),x6) -> c_9(rmap#2#(x10,Cons(x36,x6)))
- Weak TRS:
f#1(x1) -> mult#2(x1,mult#2(x1,x1))
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> S(plus#2(mult#2(x4,x2),x2))
plus#2(x4,0()) -> x4
plus#2(x4,S(x2)) -> S(plus#2(x4,x2))
rmap#2(Cons(x36,x10),x6) -> rmap#2(x10,Cons(x36,x6))
rmap#2(Nil(),x2) -> x2
rmap1#2(Cons(x13,Nil()),x8) -> Cons(f#1(x13),x8)
rmap1#2(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
rmap1#2(Nil(),x2) -> x2
- Signature:
{f#1/1,main/1,map#2/1,mult#2/2,plus#2/2,rmap#2/2,rmap1#2/2,f#1#/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2
,rmap#2#/2,rmap1#2#/2} / {0/0,Cons/2,Nil/0,S/1,c_1/2,c_2/3,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/1,c_9/1,c_10/0
,c_11/1,c_12/1,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {main#} and constructors {0,Cons,Nil,S}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:main#(x1) -> c_2(map#2#(rmap#2(rmap1#2(x1,Nil()),Nil()))
,rmap#2#(rmap1#2(x1,Nil()),Nil())
,rmap1#2#(x1,Nil()))
-->_3 rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> c_12(rmap1#2#(x25
,Cons(plus#2(x2,S(S(0())))
,Cons(f#1(x27),x17)))):4
-->_2 rmap#2#(Cons(x36,x10),x6) -> c_9(rmap#2#(x10,Cons(x36,x6))):3
-->_1 map#2#(Cons(x36,x6)) -> c_3(map#2#(x6)):2
2:W:map#2#(Cons(x36,x6)) -> c_3(map#2#(x6))
-->_1 map#2#(Cons(x36,x6)) -> c_3(map#2#(x6)):2
3:W:rmap#2#(Cons(x36,x10),x6) -> c_9(rmap#2#(x10,Cons(x36,x6)))
-->_1 rmap#2#(Cons(x36,x10),x6) -> c_9(rmap#2#(x10,Cons(x36,x6))):3
4:S:rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> c_12(rmap1#2#(x25
,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17))))
-->_1 rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> c_12(rmap1#2#(x25
,Cons(plus#2(x2,S(S(0())))
,Cons(f#1(x27),x17)))):4
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
2: map#2#(Cons(x36,x6)) -> c_3(map#2#(x6))
3: rmap#2#(Cons(x36,x10),x6) -> c_9(rmap#2#(x10,Cons(x36,x6)))
*** Step 31.b:4.a:2: SimplifyRHS WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
main#(x1) -> c_2(map#2#(rmap#2(rmap1#2(x1,Nil()),Nil()))
,rmap#2#(rmap1#2(x1,Nil()),Nil())
,rmap1#2#(x1,Nil()))
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> c_12(rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17))))
- Weak TRS:
f#1(x1) -> mult#2(x1,mult#2(x1,x1))
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> S(plus#2(mult#2(x4,x2),x2))
plus#2(x4,0()) -> x4
plus#2(x4,S(x2)) -> S(plus#2(x4,x2))
rmap#2(Cons(x36,x10),x6) -> rmap#2(x10,Cons(x36,x6))
rmap#2(Nil(),x2) -> x2
rmap1#2(Cons(x13,Nil()),x8) -> Cons(f#1(x13),x8)
rmap1#2(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
rmap1#2(Nil(),x2) -> x2
- Signature:
{f#1/1,main/1,map#2/1,mult#2/2,plus#2/2,rmap#2/2,rmap1#2/2,f#1#/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2
,rmap#2#/2,rmap1#2#/2} / {0/0,Cons/2,Nil/0,S/1,c_1/2,c_2/3,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/1,c_9/1,c_10/0
,c_11/1,c_12/1,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {main#} and constructors {0,Cons,Nil,S}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:main#(x1) -> c_2(map#2#(rmap#2(rmap1#2(x1,Nil()),Nil()))
,rmap#2#(rmap1#2(x1,Nil()),Nil())
,rmap1#2#(x1,Nil()))
-->_3 rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> c_12(rmap1#2#(x25
,Cons(plus#2(x2,S(S(0())))
,Cons(f#1(x27),x17)))):4
4:S:rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> c_12(rmap1#2#(x25
,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17))))
-->_1 rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> c_12(rmap1#2#(x25
,Cons(plus#2(x2,S(S(0())))
,Cons(f#1(x27),x17)))):4
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
main#(x1) -> c_2(rmap1#2#(x1,Nil()))
*** Step 31.b:4.a:3: UsableRules WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
main#(x1) -> c_2(rmap1#2#(x1,Nil()))
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> c_12(rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17))))
- Weak TRS:
f#1(x1) -> mult#2(x1,mult#2(x1,x1))
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> S(plus#2(mult#2(x4,x2),x2))
plus#2(x4,0()) -> x4
plus#2(x4,S(x2)) -> S(plus#2(x4,x2))
rmap#2(Cons(x36,x10),x6) -> rmap#2(x10,Cons(x36,x6))
rmap#2(Nil(),x2) -> x2
rmap1#2(Cons(x13,Nil()),x8) -> Cons(f#1(x13),x8)
rmap1#2(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
rmap1#2(Nil(),x2) -> x2
- Signature:
{f#1/1,main/1,map#2/1,mult#2/2,plus#2/2,rmap#2/2,rmap1#2/2,f#1#/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2
,rmap#2#/2,rmap1#2#/2} / {0/0,Cons/2,Nil/0,S/1,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/1,c_9/1,c_10/0
,c_11/1,c_12/1,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {main#} and constructors {0,Cons,Nil,S}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
f#1(x1) -> mult#2(x1,mult#2(x1,x1))
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> S(plus#2(mult#2(x4,x2),x2))
plus#2(x4,0()) -> x4
plus#2(x4,S(x2)) -> S(plus#2(x4,x2))
main#(x1) -> c_2(rmap1#2#(x1,Nil()))
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> c_12(rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17))))
*** Step 31.b:4.a:4: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
main#(x1) -> c_2(rmap1#2#(x1,Nil()))
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> c_12(rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17))))
- Weak TRS:
f#1(x1) -> mult#2(x1,mult#2(x1,x1))
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> S(plus#2(mult#2(x4,x2),x2))
plus#2(x4,0()) -> x4
plus#2(x4,S(x2)) -> S(plus#2(x4,x2))
- Signature:
{f#1/1,main/1,map#2/1,mult#2/2,plus#2/2,rmap#2/2,rmap1#2/2,f#1#/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2
,rmap#2#/2,rmap1#2#/2} / {0/0,Cons/2,Nil/0,S/1,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/1,c_9/1,c_10/0
,c_11/1,c_12/1,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {main#} and constructors {0,Cons,Nil,S}
+ Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
+ Details:
We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
1: main#(x1) -> c_2(rmap1#2#(x1,Nil()))
2: rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> c_12(rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17))))
The strictly oriented rules are moved into the weak component.
**** Step 31.b:4.a:4.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
main#(x1) -> c_2(rmap1#2#(x1,Nil()))
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> c_12(rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17))))
- Weak TRS:
f#1(x1) -> mult#2(x1,mult#2(x1,x1))
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> S(plus#2(mult#2(x4,x2),x2))
plus#2(x4,0()) -> x4
plus#2(x4,S(x2)) -> S(plus#2(x4,x2))
- Signature:
{f#1/1,main/1,map#2/1,mult#2/2,plus#2/2,rmap#2/2,rmap1#2/2,f#1#/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2
,rmap#2#/2,rmap1#2#/2} / {0/0,Cons/2,Nil/0,S/1,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/1,c_9/1,c_10/0
,c_11/1,c_12/1,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {main#} and constructors {0,Cons,Nil,S}
+ 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_2) = {1},
uargs(c_12) = {1}
Following symbols are considered usable:
{main#,rmap1#2#}
TcT has computed the following interpretation:
p(0) = [0]
p(Cons) = [1] x2 + [4]
p(Nil) = [1]
p(S) = [1] x1 + [4]
p(f#1) = [0]
p(main) = [1]
p(map#2) = [1] x1 + [1]
p(mult#2) = [5] x1 + [3] x2 + [4]
p(plus#2) = [0]
p(rmap#2) = [8] x1 + [1] x2 + [1]
p(rmap1#2) = [1] x2 + [0]
p(f#1#) = [1] x1 + [1]
p(main#) = [8] x1 + [13]
p(map#2#) = [1] x1 + [1]
p(mult#2#) = [4] x1 + [1] x2 + [0]
p(plus#2#) = [1] x1 + [8]
p(rmap#2#) = [1] x2 + [1]
p(rmap1#2#) = [2] x1 + [0]
p(c_1) = [1] x1 + [2] x2 + [4]
p(c_2) = [4] x1 + [2]
p(c_3) = [1]
p(c_4) = [1]
p(c_5) = [0]
p(c_6) = [1]
p(c_7) = [0]
p(c_8) = [2] x1 + [8]
p(c_9) = [1] x1 + [0]
p(c_10) = [1]
p(c_11) = [2] x1 + [2]
p(c_12) = [1] x1 + [15]
p(c_13) = [2]
Following rules are strictly oriented:
main#(x1) = [8] x1 + [13]
> [8] x1 + [2]
= c_2(rmap1#2#(x1,Nil()))
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) = [2] x25 + [16]
> [2] x25 + [15]
= c_12(rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17))))
Following rules are (at-least) weakly oriented:
**** Step 31.b:4.a:4.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
main#(x1) -> c_2(rmap1#2#(x1,Nil()))
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> c_12(rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17))))
- Weak TRS:
f#1(x1) -> mult#2(x1,mult#2(x1,x1))
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> S(plus#2(mult#2(x4,x2),x2))
plus#2(x4,0()) -> x4
plus#2(x4,S(x2)) -> S(plus#2(x4,x2))
- Signature:
{f#1/1,main/1,map#2/1,mult#2/2,plus#2/2,rmap#2/2,rmap1#2/2,f#1#/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2
,rmap#2#/2,rmap1#2#/2} / {0/0,Cons/2,Nil/0,S/1,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/1,c_9/1,c_10/0
,c_11/1,c_12/1,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {main#} and constructors {0,Cons,Nil,S}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
**** Step 31.b:4.a:4.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
main#(x1) -> c_2(rmap1#2#(x1,Nil()))
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> c_12(rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17))))
- Weak TRS:
f#1(x1) -> mult#2(x1,mult#2(x1,x1))
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> S(plus#2(mult#2(x4,x2),x2))
plus#2(x4,0()) -> x4
plus#2(x4,S(x2)) -> S(plus#2(x4,x2))
- Signature:
{f#1/1,main/1,map#2/1,mult#2/2,plus#2/2,rmap#2/2,rmap1#2/2,f#1#/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2
,rmap#2#/2,rmap1#2#/2} / {0/0,Cons/2,Nil/0,S/1,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/1,c_9/1,c_10/0
,c_11/1,c_12/1,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {main#} and constructors {0,Cons,Nil,S}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:main#(x1) -> c_2(rmap1#2#(x1,Nil()))
-->_1 rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> c_12(rmap1#2#(x25
,Cons(plus#2(x2,S(S(0())))
,Cons(f#1(x27),x17)))):2
2:W:rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> c_12(rmap1#2#(x25
,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17))))
-->_1 rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> c_12(rmap1#2#(x25
,Cons(plus#2(x2,S(S(0())))
,Cons(f#1(x27),x17)))):2
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: main#(x1) -> c_2(rmap1#2#(x1,Nil()))
2: rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> c_12(rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17))))
**** Step 31.b:4.a:4.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
f#1(x1) -> mult#2(x1,mult#2(x1,x1))
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> S(plus#2(mult#2(x4,x2),x2))
plus#2(x4,0()) -> x4
plus#2(x4,S(x2)) -> S(plus#2(x4,x2))
- Signature:
{f#1/1,main/1,map#2/1,mult#2/2,plus#2/2,rmap#2/2,rmap1#2/2,f#1#/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2
,rmap#2#/2,rmap1#2#/2} / {0/0,Cons/2,Nil/0,S/1,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/1,c_9/1,c_10/0
,c_11/1,c_12/1,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {main#} and constructors {0,Cons,Nil,S}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
*** Step 31.b:4.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
map#2#(Cons(x36,x6)) -> c_3(map#2#(x6))
rmap#2#(Cons(x36,x10),x6) -> c_9(rmap#2#(x10,Cons(x36,x6)))
- Weak DPs:
main#(x1) -> c_2(map#2#(rmap#2(rmap1#2(x1,Nil()),Nil()))
,rmap#2#(rmap1#2(x1,Nil()),Nil())
,rmap1#2#(x1,Nil()))
rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> c_12(rmap1#2#(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17))))
- Weak TRS:
f#1(x1) -> mult#2(x1,mult#2(x1,x1))
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> S(plus#2(mult#2(x4,x2),x2))
plus#2(x4,0()) -> x4
plus#2(x4,S(x2)) -> S(plus#2(x4,x2))
rmap#2(Cons(x36,x10),x6) -> rmap#2(x10,Cons(x36,x6))
rmap#2(Nil(),x2) -> x2
rmap1#2(Cons(x13,Nil()),x8) -> Cons(f#1(x13),x8)
rmap1#2(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
rmap1#2(Nil(),x2) -> x2
- Signature:
{f#1/1,main/1,map#2/1,mult#2/2,plus#2/2,rmap#2/2,rmap1#2/2,f#1#/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2
,rmap#2#/2,rmap1#2#/2} / {0/0,Cons/2,Nil/0,S/1,c_1/2,c_2/3,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/1,c_9/1,c_10/0
,c_11/1,c_12/1,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {main#} and constructors {0,Cons,Nil,S}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:map#2#(Cons(x36,x6)) -> c_3(map#2#(x6))
-->_1 map#2#(Cons(x36,x6)) -> c_3(map#2#(x6)):1
2:S:rmap#2#(Cons(x36,x10),x6) -> c_9(rmap#2#(x10,Cons(x36,x6)))
-->_1 rmap#2#(Cons(x36,x10),x6) -> c_9(rmap#2#(x10,Cons(x36,x6))):2
3:W:main#(x1) -> c_2(map#2#(rmap#2(rmap1#2(x1,Nil()),Nil()))
,rmap#2#(rmap1#2(x1,Nil()),Nil())
,rmap1#2#(x1,Nil()))
-->_3 rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> c_12(rmap1#2#(x25
,Cons(plus#2(x2,S(S(0())))
,Cons(f#1(x27),x17)))):4
-->_2 rmap#2#(Cons(x36,x10),x6) -> c_9(rmap#2#(x10,Cons(x36,x6))):2
-->_1 map#2#(Cons(x36,x6)) -> c_3(map#2#(x6)):1
4:W:rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> c_12(rmap1#2#(x25
,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17))))
-->_1 rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> c_12(rmap1#2#(x25
,Cons(plus#2(x2,S(S(0())))
,Cons(f#1(x27),x17)))):4
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
4: rmap1#2#(Cons(x27,Cons(x2,x25)),x17) -> c_12(rmap1#2#(x25
,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17))))
*** Step 31.b:4.b:2: SimplifyRHS WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
map#2#(Cons(x36,x6)) -> c_3(map#2#(x6))
rmap#2#(Cons(x36,x10),x6) -> c_9(rmap#2#(x10,Cons(x36,x6)))
- Weak DPs:
main#(x1) -> c_2(map#2#(rmap#2(rmap1#2(x1,Nil()),Nil()))
,rmap#2#(rmap1#2(x1,Nil()),Nil())
,rmap1#2#(x1,Nil()))
- Weak TRS:
f#1(x1) -> mult#2(x1,mult#2(x1,x1))
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> S(plus#2(mult#2(x4,x2),x2))
plus#2(x4,0()) -> x4
plus#2(x4,S(x2)) -> S(plus#2(x4,x2))
rmap#2(Cons(x36,x10),x6) -> rmap#2(x10,Cons(x36,x6))
rmap#2(Nil(),x2) -> x2
rmap1#2(Cons(x13,Nil()),x8) -> Cons(f#1(x13),x8)
rmap1#2(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
rmap1#2(Nil(),x2) -> x2
- Signature:
{f#1/1,main/1,map#2/1,mult#2/2,plus#2/2,rmap#2/2,rmap1#2/2,f#1#/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2
,rmap#2#/2,rmap1#2#/2} / {0/0,Cons/2,Nil/0,S/1,c_1/2,c_2/3,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/1,c_9/1,c_10/0
,c_11/1,c_12/1,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {main#} and constructors {0,Cons,Nil,S}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:map#2#(Cons(x36,x6)) -> c_3(map#2#(x6))
-->_1 map#2#(Cons(x36,x6)) -> c_3(map#2#(x6)):1
2:S:rmap#2#(Cons(x36,x10),x6) -> c_9(rmap#2#(x10,Cons(x36,x6)))
-->_1 rmap#2#(Cons(x36,x10),x6) -> c_9(rmap#2#(x10,Cons(x36,x6))):2
3:W:main#(x1) -> c_2(map#2#(rmap#2(rmap1#2(x1,Nil()),Nil()))
,rmap#2#(rmap1#2(x1,Nil()),Nil())
,rmap1#2#(x1,Nil()))
-->_2 rmap#2#(Cons(x36,x10),x6) -> c_9(rmap#2#(x10,Cons(x36,x6))):2
-->_1 map#2#(Cons(x36,x6)) -> c_3(map#2#(x6)):1
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
main#(x1) -> c_2(map#2#(rmap#2(rmap1#2(x1,Nil()),Nil())),rmap#2#(rmap1#2(x1,Nil()),Nil()))
*** Step 31.b:4.b:3: Decompose WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
map#2#(Cons(x36,x6)) -> c_3(map#2#(x6))
rmap#2#(Cons(x36,x10),x6) -> c_9(rmap#2#(x10,Cons(x36,x6)))
- Weak DPs:
main#(x1) -> c_2(map#2#(rmap#2(rmap1#2(x1,Nil()),Nil())),rmap#2#(rmap1#2(x1,Nil()),Nil()))
- Weak TRS:
f#1(x1) -> mult#2(x1,mult#2(x1,x1))
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> S(plus#2(mult#2(x4,x2),x2))
plus#2(x4,0()) -> x4
plus#2(x4,S(x2)) -> S(plus#2(x4,x2))
rmap#2(Cons(x36,x10),x6) -> rmap#2(x10,Cons(x36,x6))
rmap#2(Nil(),x2) -> x2
rmap1#2(Cons(x13,Nil()),x8) -> Cons(f#1(x13),x8)
rmap1#2(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
rmap1#2(Nil(),x2) -> x2
- Signature:
{f#1/1,main/1,map#2/1,mult#2/2,plus#2/2,rmap#2/2,rmap1#2/2,f#1#/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2
,rmap#2#/2,rmap1#2#/2} / {0/0,Cons/2,Nil/0,S/1,c_1/2,c_2/2,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/1,c_9/1,c_10/0
,c_11/1,c_12/1,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {main#} and constructors {0,Cons,Nil,S}
+ 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:
map#2#(Cons(x36,x6)) -> c_3(map#2#(x6))
- Weak DPs:
main#(x1) -> c_2(map#2#(rmap#2(rmap1#2(x1,Nil()),Nil())),rmap#2#(rmap1#2(x1,Nil()),Nil()))
rmap#2#(Cons(x36,x10),x6) -> c_9(rmap#2#(x10,Cons(x36,x6)))
- Weak TRS:
f#1(x1) -> mult#2(x1,mult#2(x1,x1))
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> S(plus#2(mult#2(x4,x2),x2))
plus#2(x4,0()) -> x4
plus#2(x4,S(x2)) -> S(plus#2(x4,x2))
rmap#2(Cons(x36,x10),x6) -> rmap#2(x10,Cons(x36,x6))
rmap#2(Nil(),x2) -> x2
rmap1#2(Cons(x13,Nil()),x8) -> Cons(f#1(x13),x8)
rmap1#2(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
rmap1#2(Nil(),x2) -> x2
- Signature:
{f#1/1,main/1,map#2/1,mult#2/2,plus#2/2,rmap#2/2,rmap1#2/2,f#1#/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2
,rmap#2#/2,rmap1#2#/2} / {0/0,Cons/2,Nil/0,S/1,c_1/2,c_2/2,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/1,c_9/1
,c_10/0,c_11/1,c_12/1,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {main#} and constructors {0,Cons,Nil,S}
Problem (S)
- Strict DPs:
rmap#2#(Cons(x36,x10),x6) -> c_9(rmap#2#(x10,Cons(x36,x6)))
- Weak DPs:
main#(x1) -> c_2(map#2#(rmap#2(rmap1#2(x1,Nil()),Nil())),rmap#2#(rmap1#2(x1,Nil()),Nil()))
map#2#(Cons(x36,x6)) -> c_3(map#2#(x6))
- Weak TRS:
f#1(x1) -> mult#2(x1,mult#2(x1,x1))
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> S(plus#2(mult#2(x4,x2),x2))
plus#2(x4,0()) -> x4
plus#2(x4,S(x2)) -> S(plus#2(x4,x2))
rmap#2(Cons(x36,x10),x6) -> rmap#2(x10,Cons(x36,x6))
rmap#2(Nil(),x2) -> x2
rmap1#2(Cons(x13,Nil()),x8) -> Cons(f#1(x13),x8)
rmap1#2(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
rmap1#2(Nil(),x2) -> x2
- Signature:
{f#1/1,main/1,map#2/1,mult#2/2,plus#2/2,rmap#2/2,rmap1#2/2,f#1#/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2
,rmap#2#/2,rmap1#2#/2} / {0/0,Cons/2,Nil/0,S/1,c_1/2,c_2/2,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/1,c_9/1
,c_10/0,c_11/1,c_12/1,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {main#} and constructors {0,Cons,Nil,S}
**** Step 31.b:4.b:3.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
map#2#(Cons(x36,x6)) -> c_3(map#2#(x6))
- Weak DPs:
main#(x1) -> c_2(map#2#(rmap#2(rmap1#2(x1,Nil()),Nil())),rmap#2#(rmap1#2(x1,Nil()),Nil()))
rmap#2#(Cons(x36,x10),x6) -> c_9(rmap#2#(x10,Cons(x36,x6)))
- Weak TRS:
f#1(x1) -> mult#2(x1,mult#2(x1,x1))
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> S(plus#2(mult#2(x4,x2),x2))
plus#2(x4,0()) -> x4
plus#2(x4,S(x2)) -> S(plus#2(x4,x2))
rmap#2(Cons(x36,x10),x6) -> rmap#2(x10,Cons(x36,x6))
rmap#2(Nil(),x2) -> x2
rmap1#2(Cons(x13,Nil()),x8) -> Cons(f#1(x13),x8)
rmap1#2(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
rmap1#2(Nil(),x2) -> x2
- Signature:
{f#1/1,main/1,map#2/1,mult#2/2,plus#2/2,rmap#2/2,rmap1#2/2,f#1#/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2
,rmap#2#/2,rmap1#2#/2} / {0/0,Cons/2,Nil/0,S/1,c_1/2,c_2/2,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/1,c_9/1,c_10/0
,c_11/1,c_12/1,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {main#} and constructors {0,Cons,Nil,S}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:map#2#(Cons(x36,x6)) -> c_3(map#2#(x6))
-->_1 map#2#(Cons(x36,x6)) -> c_3(map#2#(x6)):1
2:W:rmap#2#(Cons(x36,x10),x6) -> c_9(rmap#2#(x10,Cons(x36,x6)))
-->_1 rmap#2#(Cons(x36,x10),x6) -> c_9(rmap#2#(x10,Cons(x36,x6))):2
3:W:main#(x1) -> c_2(map#2#(rmap#2(rmap1#2(x1,Nil()),Nil())),rmap#2#(rmap1#2(x1,Nil()),Nil()))
-->_1 map#2#(Cons(x36,x6)) -> c_3(map#2#(x6)):1
-->_2 rmap#2#(Cons(x36,x10),x6) -> c_9(rmap#2#(x10,Cons(x36,x6))):2
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
2: rmap#2#(Cons(x36,x10),x6) -> c_9(rmap#2#(x10,Cons(x36,x6)))
**** Step 31.b:4.b:3.a:2: SimplifyRHS WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
map#2#(Cons(x36,x6)) -> c_3(map#2#(x6))
- Weak DPs:
main#(x1) -> c_2(map#2#(rmap#2(rmap1#2(x1,Nil()),Nil())),rmap#2#(rmap1#2(x1,Nil()),Nil()))
- Weak TRS:
f#1(x1) -> mult#2(x1,mult#2(x1,x1))
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> S(plus#2(mult#2(x4,x2),x2))
plus#2(x4,0()) -> x4
plus#2(x4,S(x2)) -> S(plus#2(x4,x2))
rmap#2(Cons(x36,x10),x6) -> rmap#2(x10,Cons(x36,x6))
rmap#2(Nil(),x2) -> x2
rmap1#2(Cons(x13,Nil()),x8) -> Cons(f#1(x13),x8)
rmap1#2(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
rmap1#2(Nil(),x2) -> x2
- Signature:
{f#1/1,main/1,map#2/1,mult#2/2,plus#2/2,rmap#2/2,rmap1#2/2,f#1#/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2
,rmap#2#/2,rmap1#2#/2} / {0/0,Cons/2,Nil/0,S/1,c_1/2,c_2/2,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/1,c_9/1,c_10/0
,c_11/1,c_12/1,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {main#} and constructors {0,Cons,Nil,S}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:map#2#(Cons(x36,x6)) -> c_3(map#2#(x6))
-->_1 map#2#(Cons(x36,x6)) -> c_3(map#2#(x6)):1
3:W:main#(x1) -> c_2(map#2#(rmap#2(rmap1#2(x1,Nil()),Nil())),rmap#2#(rmap1#2(x1,Nil()),Nil()))
-->_1 map#2#(Cons(x36,x6)) -> c_3(map#2#(x6)):1
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
main#(x1) -> c_2(map#2#(rmap#2(rmap1#2(x1,Nil()),Nil())))
**** Step 31.b:4.b:3.a:3: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
map#2#(Cons(x36,x6)) -> c_3(map#2#(x6))
- Weak DPs:
main#(x1) -> c_2(map#2#(rmap#2(rmap1#2(x1,Nil()),Nil())))
- Weak TRS:
f#1(x1) -> mult#2(x1,mult#2(x1,x1))
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> S(plus#2(mult#2(x4,x2),x2))
plus#2(x4,0()) -> x4
plus#2(x4,S(x2)) -> S(plus#2(x4,x2))
rmap#2(Cons(x36,x10),x6) -> rmap#2(x10,Cons(x36,x6))
rmap#2(Nil(),x2) -> x2
rmap1#2(Cons(x13,Nil()),x8) -> Cons(f#1(x13),x8)
rmap1#2(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
rmap1#2(Nil(),x2) -> x2
- Signature:
{f#1/1,main/1,map#2/1,mult#2/2,plus#2/2,rmap#2/2,rmap1#2/2,f#1#/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2
,rmap#2#/2,rmap1#2#/2} / {0/0,Cons/2,Nil/0,S/1,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/1,c_9/1,c_10/0
,c_11/1,c_12/1,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {main#} and constructors {0,Cons,Nil,S}
+ Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
+ Details:
We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
1: map#2#(Cons(x36,x6)) -> c_3(map#2#(x6))
Consider the set of all dependency pairs
1: map#2#(Cons(x36,x6)) -> c_3(map#2#(x6))
2: main#(x1) -> c_2(map#2#(rmap#2(rmap1#2(x1,Nil()),Nil())))
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
{1}
These cover all (indirect) predecessors of dependency pairs
{1,2}
their number of applications is equally bounded.
The dependency pairs are shifted into the weak component.
***** Step 31.b:4.b:3.a:3.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
map#2#(Cons(x36,x6)) -> c_3(map#2#(x6))
- Weak DPs:
main#(x1) -> c_2(map#2#(rmap#2(rmap1#2(x1,Nil()),Nil())))
- Weak TRS:
f#1(x1) -> mult#2(x1,mult#2(x1,x1))
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> S(plus#2(mult#2(x4,x2),x2))
plus#2(x4,0()) -> x4
plus#2(x4,S(x2)) -> S(plus#2(x4,x2))
rmap#2(Cons(x36,x10),x6) -> rmap#2(x10,Cons(x36,x6))
rmap#2(Nil(),x2) -> x2
rmap1#2(Cons(x13,Nil()),x8) -> Cons(f#1(x13),x8)
rmap1#2(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
rmap1#2(Nil(),x2) -> x2
- Signature:
{f#1/1,main/1,map#2/1,mult#2/2,plus#2/2,rmap#2/2,rmap1#2/2,f#1#/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2
,rmap#2#/2,rmap1#2#/2} / {0/0,Cons/2,Nil/0,S/1,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/1,c_9/1,c_10/0
,c_11/1,c_12/1,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {main#} and constructors {0,Cons,Nil,S}
+ 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_2) = {1},
uargs(c_3) = {1}
Following symbols are considered usable:
{rmap#2,rmap1#2,main#,map#2#}
TcT has computed the following interpretation:
p(0) = [4]
p(Cons) = [1] x2 + [4]
p(Nil) = [0]
p(S) = [4]
p(f#1) = [0]
p(main) = [1] x1 + [4]
p(map#2) = [1]
p(mult#2) = [5]
p(plus#2) = [4] x2 + [2]
p(rmap#2) = [1] x1 + [1] x2 + [0]
p(rmap1#2) = [2] x1 + [2] x2 + [0]
p(f#1#) = [1] x1 + [0]
p(main#) = [9] x1 + [8]
p(map#2#) = [4] x1 + [0]
p(mult#2#) = [1]
p(plus#2#) = [1]
p(rmap#2#) = [1] x2 + [1]
p(rmap1#2#) = [1] x1 + [8] x2 + [1]
p(c_1) = [4] x1 + [0]
p(c_2) = [1] x1 + [1]
p(c_3) = [1] x1 + [6]
p(c_4) = [1]
p(c_5) = [4]
p(c_6) = [2]
p(c_7) = [1]
p(c_8) = [1] x1 + [0]
p(c_9) = [2]
p(c_10) = [1]
p(c_11) = [1]
p(c_12) = [0]
p(c_13) = [0]
Following rules are strictly oriented:
map#2#(Cons(x36,x6)) = [4] x6 + [16]
> [4] x6 + [6]
= c_3(map#2#(x6))
Following rules are (at-least) weakly oriented:
main#(x1) = [9] x1 + [8]
>= [8] x1 + [1]
= c_2(map#2#(rmap#2(rmap1#2(x1,Nil()),Nil())))
rmap#2(Cons(x36,x10),x6) = [1] x10 + [1] x6 + [4]
>= [1] x10 + [1] x6 + [4]
= rmap#2(x10,Cons(x36,x6))
rmap#2(Nil(),x2) = [1] x2 + [0]
>= [1] x2 + [0]
= x2
rmap1#2(Cons(x13,Nil()),x8) = [2] x8 + [8]
>= [1] x8 + [4]
= Cons(f#1(x13),x8)
rmap1#2(Cons(x27,Cons(x2,x25)),x17) = [2] x17 + [2] x25 + [16]
>= [2] x17 + [2] x25 + [16]
= rmap1#2(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
rmap1#2(Nil(),x2) = [2] x2 + [0]
>= [1] x2 + [0]
= x2
***** Step 31.b:4.b:3.a:3.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
main#(x1) -> c_2(map#2#(rmap#2(rmap1#2(x1,Nil()),Nil())))
map#2#(Cons(x36,x6)) -> c_3(map#2#(x6))
- Weak TRS:
f#1(x1) -> mult#2(x1,mult#2(x1,x1))
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> S(plus#2(mult#2(x4,x2),x2))
plus#2(x4,0()) -> x4
plus#2(x4,S(x2)) -> S(plus#2(x4,x2))
rmap#2(Cons(x36,x10),x6) -> rmap#2(x10,Cons(x36,x6))
rmap#2(Nil(),x2) -> x2
rmap1#2(Cons(x13,Nil()),x8) -> Cons(f#1(x13),x8)
rmap1#2(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
rmap1#2(Nil(),x2) -> x2
- Signature:
{f#1/1,main/1,map#2/1,mult#2/2,plus#2/2,rmap#2/2,rmap1#2/2,f#1#/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2
,rmap#2#/2,rmap1#2#/2} / {0/0,Cons/2,Nil/0,S/1,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/1,c_9/1,c_10/0
,c_11/1,c_12/1,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {main#} and constructors {0,Cons,Nil,S}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
***** Step 31.b:4.b:3.a:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
main#(x1) -> c_2(map#2#(rmap#2(rmap1#2(x1,Nil()),Nil())))
map#2#(Cons(x36,x6)) -> c_3(map#2#(x6))
- Weak TRS:
f#1(x1) -> mult#2(x1,mult#2(x1,x1))
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> S(plus#2(mult#2(x4,x2),x2))
plus#2(x4,0()) -> x4
plus#2(x4,S(x2)) -> S(plus#2(x4,x2))
rmap#2(Cons(x36,x10),x6) -> rmap#2(x10,Cons(x36,x6))
rmap#2(Nil(),x2) -> x2
rmap1#2(Cons(x13,Nil()),x8) -> Cons(f#1(x13),x8)
rmap1#2(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
rmap1#2(Nil(),x2) -> x2
- Signature:
{f#1/1,main/1,map#2/1,mult#2/2,plus#2/2,rmap#2/2,rmap1#2/2,f#1#/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2
,rmap#2#/2,rmap1#2#/2} / {0/0,Cons/2,Nil/0,S/1,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/1,c_9/1,c_10/0
,c_11/1,c_12/1,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {main#} and constructors {0,Cons,Nil,S}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:main#(x1) -> c_2(map#2#(rmap#2(rmap1#2(x1,Nil()),Nil())))
-->_1 map#2#(Cons(x36,x6)) -> c_3(map#2#(x6)):2
2:W:map#2#(Cons(x36,x6)) -> c_3(map#2#(x6))
-->_1 map#2#(Cons(x36,x6)) -> c_3(map#2#(x6)):2
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: main#(x1) -> c_2(map#2#(rmap#2(rmap1#2(x1,Nil()),Nil())))
2: map#2#(Cons(x36,x6)) -> c_3(map#2#(x6))
***** Step 31.b:4.b:3.a:3.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
f#1(x1) -> mult#2(x1,mult#2(x1,x1))
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> S(plus#2(mult#2(x4,x2),x2))
plus#2(x4,0()) -> x4
plus#2(x4,S(x2)) -> S(plus#2(x4,x2))
rmap#2(Cons(x36,x10),x6) -> rmap#2(x10,Cons(x36,x6))
rmap#2(Nil(),x2) -> x2
rmap1#2(Cons(x13,Nil()),x8) -> Cons(f#1(x13),x8)
rmap1#2(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
rmap1#2(Nil(),x2) -> x2
- Signature:
{f#1/1,main/1,map#2/1,mult#2/2,plus#2/2,rmap#2/2,rmap1#2/2,f#1#/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2
,rmap#2#/2,rmap1#2#/2} / {0/0,Cons/2,Nil/0,S/1,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/1,c_9/1,c_10/0
,c_11/1,c_12/1,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {main#} and constructors {0,Cons,Nil,S}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
**** Step 31.b:4.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
rmap#2#(Cons(x36,x10),x6) -> c_9(rmap#2#(x10,Cons(x36,x6)))
- Weak DPs:
main#(x1) -> c_2(map#2#(rmap#2(rmap1#2(x1,Nil()),Nil())),rmap#2#(rmap1#2(x1,Nil()),Nil()))
map#2#(Cons(x36,x6)) -> c_3(map#2#(x6))
- Weak TRS:
f#1(x1) -> mult#2(x1,mult#2(x1,x1))
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> S(plus#2(mult#2(x4,x2),x2))
plus#2(x4,0()) -> x4
plus#2(x4,S(x2)) -> S(plus#2(x4,x2))
rmap#2(Cons(x36,x10),x6) -> rmap#2(x10,Cons(x36,x6))
rmap#2(Nil(),x2) -> x2
rmap1#2(Cons(x13,Nil()),x8) -> Cons(f#1(x13),x8)
rmap1#2(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
rmap1#2(Nil(),x2) -> x2
- Signature:
{f#1/1,main/1,map#2/1,mult#2/2,plus#2/2,rmap#2/2,rmap1#2/2,f#1#/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2
,rmap#2#/2,rmap1#2#/2} / {0/0,Cons/2,Nil/0,S/1,c_1/2,c_2/2,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/1,c_9/1,c_10/0
,c_11/1,c_12/1,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {main#} and constructors {0,Cons,Nil,S}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:rmap#2#(Cons(x36,x10),x6) -> c_9(rmap#2#(x10,Cons(x36,x6)))
-->_1 rmap#2#(Cons(x36,x10),x6) -> c_9(rmap#2#(x10,Cons(x36,x6))):1
2:W:main#(x1) -> c_2(map#2#(rmap#2(rmap1#2(x1,Nil()),Nil())),rmap#2#(rmap1#2(x1,Nil()),Nil()))
-->_1 map#2#(Cons(x36,x6)) -> c_3(map#2#(x6)):3
-->_2 rmap#2#(Cons(x36,x10),x6) -> c_9(rmap#2#(x10,Cons(x36,x6))):1
3:W:map#2#(Cons(x36,x6)) -> c_3(map#2#(x6))
-->_1 map#2#(Cons(x36,x6)) -> c_3(map#2#(x6)):3
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
3: map#2#(Cons(x36,x6)) -> c_3(map#2#(x6))
**** Step 31.b:4.b:3.b:2: SimplifyRHS WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
rmap#2#(Cons(x36,x10),x6) -> c_9(rmap#2#(x10,Cons(x36,x6)))
- Weak DPs:
main#(x1) -> c_2(map#2#(rmap#2(rmap1#2(x1,Nil()),Nil())),rmap#2#(rmap1#2(x1,Nil()),Nil()))
- Weak TRS:
f#1(x1) -> mult#2(x1,mult#2(x1,x1))
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> S(plus#2(mult#2(x4,x2),x2))
plus#2(x4,0()) -> x4
plus#2(x4,S(x2)) -> S(plus#2(x4,x2))
rmap#2(Cons(x36,x10),x6) -> rmap#2(x10,Cons(x36,x6))
rmap#2(Nil(),x2) -> x2
rmap1#2(Cons(x13,Nil()),x8) -> Cons(f#1(x13),x8)
rmap1#2(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
rmap1#2(Nil(),x2) -> x2
- Signature:
{f#1/1,main/1,map#2/1,mult#2/2,plus#2/2,rmap#2/2,rmap1#2/2,f#1#/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2
,rmap#2#/2,rmap1#2#/2} / {0/0,Cons/2,Nil/0,S/1,c_1/2,c_2/2,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/1,c_9/1,c_10/0
,c_11/1,c_12/1,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {main#} and constructors {0,Cons,Nil,S}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:rmap#2#(Cons(x36,x10),x6) -> c_9(rmap#2#(x10,Cons(x36,x6)))
-->_1 rmap#2#(Cons(x36,x10),x6) -> c_9(rmap#2#(x10,Cons(x36,x6))):1
2:W:main#(x1) -> c_2(map#2#(rmap#2(rmap1#2(x1,Nil()),Nil())),rmap#2#(rmap1#2(x1,Nil()),Nil()))
-->_2 rmap#2#(Cons(x36,x10),x6) -> c_9(rmap#2#(x10,Cons(x36,x6))):1
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
main#(x1) -> c_2(rmap#2#(rmap1#2(x1,Nil()),Nil()))
**** Step 31.b:4.b:3.b:3: UsableRules WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
rmap#2#(Cons(x36,x10),x6) -> c_9(rmap#2#(x10,Cons(x36,x6)))
- Weak DPs:
main#(x1) -> c_2(rmap#2#(rmap1#2(x1,Nil()),Nil()))
- Weak TRS:
f#1(x1) -> mult#2(x1,mult#2(x1,x1))
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> S(plus#2(mult#2(x4,x2),x2))
plus#2(x4,0()) -> x4
plus#2(x4,S(x2)) -> S(plus#2(x4,x2))
rmap#2(Cons(x36,x10),x6) -> rmap#2(x10,Cons(x36,x6))
rmap#2(Nil(),x2) -> x2
rmap1#2(Cons(x13,Nil()),x8) -> Cons(f#1(x13),x8)
rmap1#2(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
rmap1#2(Nil(),x2) -> x2
- Signature:
{f#1/1,main/1,map#2/1,mult#2/2,plus#2/2,rmap#2/2,rmap1#2/2,f#1#/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2
,rmap#2#/2,rmap1#2#/2} / {0/0,Cons/2,Nil/0,S/1,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/1,c_9/1,c_10/0
,c_11/1,c_12/1,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {main#} and constructors {0,Cons,Nil,S}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
f#1(x1) -> mult#2(x1,mult#2(x1,x1))
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> S(plus#2(mult#2(x4,x2),x2))
plus#2(x4,0()) -> x4
plus#2(x4,S(x2)) -> S(plus#2(x4,x2))
rmap1#2(Cons(x13,Nil()),x8) -> Cons(f#1(x13),x8)
rmap1#2(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
rmap1#2(Nil(),x2) -> x2
main#(x1) -> c_2(rmap#2#(rmap1#2(x1,Nil()),Nil()))
rmap#2#(Cons(x36,x10),x6) -> c_9(rmap#2#(x10,Cons(x36,x6)))
**** Step 31.b:4.b:3.b:4: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
rmap#2#(Cons(x36,x10),x6) -> c_9(rmap#2#(x10,Cons(x36,x6)))
- Weak DPs:
main#(x1) -> c_2(rmap#2#(rmap1#2(x1,Nil()),Nil()))
- Weak TRS:
f#1(x1) -> mult#2(x1,mult#2(x1,x1))
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> S(plus#2(mult#2(x4,x2),x2))
plus#2(x4,0()) -> x4
plus#2(x4,S(x2)) -> S(plus#2(x4,x2))
rmap1#2(Cons(x13,Nil()),x8) -> Cons(f#1(x13),x8)
rmap1#2(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
rmap1#2(Nil(),x2) -> x2
- Signature:
{f#1/1,main/1,map#2/1,mult#2/2,plus#2/2,rmap#2/2,rmap1#2/2,f#1#/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2
,rmap#2#/2,rmap1#2#/2} / {0/0,Cons/2,Nil/0,S/1,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/1,c_9/1,c_10/0
,c_11/1,c_12/1,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {main#} and constructors {0,Cons,Nil,S}
+ Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
+ Details:
We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
1: rmap#2#(Cons(x36,x10),x6) -> c_9(rmap#2#(x10,Cons(x36,x6)))
Consider the set of all dependency pairs
1: rmap#2#(Cons(x36,x10),x6) -> c_9(rmap#2#(x10,Cons(x36,x6)))
2: main#(x1) -> c_2(rmap#2#(rmap1#2(x1,Nil()),Nil()))
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
{1}
These cover all (indirect) predecessors of dependency pairs
{1,2}
their number of applications is equally bounded.
The dependency pairs are shifted into the weak component.
***** Step 31.b:4.b:3.b:4.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
rmap#2#(Cons(x36,x10),x6) -> c_9(rmap#2#(x10,Cons(x36,x6)))
- Weak DPs:
main#(x1) -> c_2(rmap#2#(rmap1#2(x1,Nil()),Nil()))
- Weak TRS:
f#1(x1) -> mult#2(x1,mult#2(x1,x1))
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> S(plus#2(mult#2(x4,x2),x2))
plus#2(x4,0()) -> x4
plus#2(x4,S(x2)) -> S(plus#2(x4,x2))
rmap1#2(Cons(x13,Nil()),x8) -> Cons(f#1(x13),x8)
rmap1#2(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
rmap1#2(Nil(),x2) -> x2
- Signature:
{f#1/1,main/1,map#2/1,mult#2/2,plus#2/2,rmap#2/2,rmap1#2/2,f#1#/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2
,rmap#2#/2,rmap1#2#/2} / {0/0,Cons/2,Nil/0,S/1,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/1,c_9/1,c_10/0
,c_11/1,c_12/1,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {main#} and constructors {0,Cons,Nil,S}
+ 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_2) = {1},
uargs(c_9) = {1}
Following symbols are considered usable:
{rmap1#2,main#,rmap#2#}
TcT has computed the following interpretation:
p(0) = [3]
p(Cons) = [1] x2 + [4]
p(Nil) = [0]
p(S) = [3]
p(f#1) = [4] x1 + [1]
p(main) = [2] x1 + [0]
p(map#2) = [1] x1 + [1]
p(mult#2) = [7] x1 + [9]
p(plus#2) = [10] x2 + [1]
p(rmap#2) = [1] x1 + [1] x2 + [1]
p(rmap1#2) = [2] x1 + [2] x2 + [4]
p(f#1#) = [1] x1 + [2]
p(main#) = [8] x1 + [10]
p(map#2#) = [1] x1 + [1]
p(mult#2#) = [1] x1 + [2] x2 + [1]
p(plus#2#) = [1] x2 + [4]
p(rmap#2#) = [1] x1 + [0]
p(rmap1#2#) = [1] x1 + [1] x2 + [2]
p(c_1) = [1] x1 + [2] x2 + [4]
p(c_2) = [2] x1 + [2]
p(c_3) = [1] x1 + [1]
p(c_4) = [1]
p(c_5) = [0]
p(c_6) = [1] x1 + [1]
p(c_7) = [1]
p(c_8) = [0]
p(c_9) = [1] x1 + [2]
p(c_10) = [1]
p(c_11) = [1]
p(c_12) = [2] x1 + [1]
p(c_13) = [0]
Following rules are strictly oriented:
rmap#2#(Cons(x36,x10),x6) = [1] x10 + [4]
> [1] x10 + [2]
= c_9(rmap#2#(x10,Cons(x36,x6)))
Following rules are (at-least) weakly oriented:
main#(x1) = [8] x1 + [10]
>= [4] x1 + [10]
= c_2(rmap#2#(rmap1#2(x1,Nil()),Nil()))
rmap1#2(Cons(x13,Nil()),x8) = [2] x8 + [12]
>= [1] x8 + [4]
= Cons(f#1(x13),x8)
rmap1#2(Cons(x27,Cons(x2,x25)),x17) = [2] x17 + [2] x25 + [20]
>= [2] x17 + [2] x25 + [20]
= rmap1#2(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
rmap1#2(Nil(),x2) = [2] x2 + [4]
>= [1] x2 + [0]
= x2
***** Step 31.b:4.b:3.b:4.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
main#(x1) -> c_2(rmap#2#(rmap1#2(x1,Nil()),Nil()))
rmap#2#(Cons(x36,x10),x6) -> c_9(rmap#2#(x10,Cons(x36,x6)))
- Weak TRS:
f#1(x1) -> mult#2(x1,mult#2(x1,x1))
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> S(plus#2(mult#2(x4,x2),x2))
plus#2(x4,0()) -> x4
plus#2(x4,S(x2)) -> S(plus#2(x4,x2))
rmap1#2(Cons(x13,Nil()),x8) -> Cons(f#1(x13),x8)
rmap1#2(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
rmap1#2(Nil(),x2) -> x2
- Signature:
{f#1/1,main/1,map#2/1,mult#2/2,plus#2/2,rmap#2/2,rmap1#2/2,f#1#/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2
,rmap#2#/2,rmap1#2#/2} / {0/0,Cons/2,Nil/0,S/1,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/1,c_9/1,c_10/0
,c_11/1,c_12/1,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {main#} and constructors {0,Cons,Nil,S}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
***** Step 31.b:4.b:3.b:4.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
main#(x1) -> c_2(rmap#2#(rmap1#2(x1,Nil()),Nil()))
rmap#2#(Cons(x36,x10),x6) -> c_9(rmap#2#(x10,Cons(x36,x6)))
- Weak TRS:
f#1(x1) -> mult#2(x1,mult#2(x1,x1))
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> S(plus#2(mult#2(x4,x2),x2))
plus#2(x4,0()) -> x4
plus#2(x4,S(x2)) -> S(plus#2(x4,x2))
rmap1#2(Cons(x13,Nil()),x8) -> Cons(f#1(x13),x8)
rmap1#2(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
rmap1#2(Nil(),x2) -> x2
- Signature:
{f#1/1,main/1,map#2/1,mult#2/2,plus#2/2,rmap#2/2,rmap1#2/2,f#1#/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2
,rmap#2#/2,rmap1#2#/2} / {0/0,Cons/2,Nil/0,S/1,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/1,c_9/1,c_10/0
,c_11/1,c_12/1,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {main#} and constructors {0,Cons,Nil,S}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:main#(x1) -> c_2(rmap#2#(rmap1#2(x1,Nil()),Nil()))
-->_1 rmap#2#(Cons(x36,x10),x6) -> c_9(rmap#2#(x10,Cons(x36,x6))):2
2:W:rmap#2#(Cons(x36,x10),x6) -> c_9(rmap#2#(x10,Cons(x36,x6)))
-->_1 rmap#2#(Cons(x36,x10),x6) -> c_9(rmap#2#(x10,Cons(x36,x6))):2
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: main#(x1) -> c_2(rmap#2#(rmap1#2(x1,Nil()),Nil()))
2: rmap#2#(Cons(x36,x10),x6) -> c_9(rmap#2#(x10,Cons(x36,x6)))
***** Step 31.b:4.b:3.b:4.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
f#1(x1) -> mult#2(x1,mult#2(x1,x1))
mult#2(0(),x2) -> 0()
mult#2(S(x4),x2) -> S(plus#2(mult#2(x4,x2),x2))
plus#2(x4,0()) -> x4
plus#2(x4,S(x2)) -> S(plus#2(x4,x2))
rmap1#2(Cons(x13,Nil()),x8) -> Cons(f#1(x13),x8)
rmap1#2(Cons(x27,Cons(x2,x25)),x17) -> rmap1#2(x25,Cons(plus#2(x2,S(S(0()))),Cons(f#1(x27),x17)))
rmap1#2(Nil(),x2) -> x2
- Signature:
{f#1/1,main/1,map#2/1,mult#2/2,plus#2/2,rmap#2/2,rmap1#2/2,f#1#/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2
,rmap#2#/2,rmap1#2#/2} / {0/0,Cons/2,Nil/0,S/1,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/1,c_9/1,c_10/0
,c_11/1,c_12/1,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {main#} and constructors {0,Cons,Nil,S}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(?,O(n^4))