WORST_CASE(Omega(n^1),O(n^1))
proof of input_O2wsNODBQJ.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).

(0) CpxTRS
(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(2) CpxTRS
    (3) CpxTrsMatchBoundsTAProof [FINISHED, 301 ms]
    (4) BOUNDS(1, n^1)
(5) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CdtProblem
    (7) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxRelTRS
    (9) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CpxRelTRS
    (11) TypeInferenceProof [BOTH BOUNDS(ID, ID), 8 ms]
    (12) typed CpxTrs
    (13) OrderProof [LOWER BOUND(ID), 0 ms]
    (14) typed CpxTrs
    (15) RewriteLemmaProof [LOWER BOUND(ID), 254 ms]
    (16) BEST
        (17) proven lower bound
            (18) LowerBoundPropagationProof [FINISHED, 0 ms]
            (19) BOUNDS(n^1, INF)
        (20) typed CpxTrs
            (21) RewriteLemmaProof [LOWER BOUND(ID), 12 ms]
            (22) typed CpxTrs
            (23) RewriteLemmaProof [LOWER BOUND(ID), 1 ms]
            (24) typed CpxTrs


----------------------------------------

(0)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).


The TRS R consists of the following rules:

   comp_f_g#1(plus_x(x3), comp_f_g(x1, x2), 0) -> plus_x#1(x3, comp_f_g#1(x1, x2, 0))
   comp_f_g#1(plus_x(x3), id, 0) -> plus_x#1(x3, 0)
   map#2(Nil) -> Nil
   map#2(Cons(x16, x6)) -> Cons(plus_x(x16), map#2(x6))
   plus_x#1(0, x6) -> x6
   plus_x#1(S(x8), x10) -> S(plus_x#1(x8, x10))
   foldr_f#3(Nil, 0) -> 0
   foldr_f#3(Cons(x16, x5), x24) -> comp_f_g#1(x16, foldr#3(x5), x24)
   foldr#3(Nil) -> id
   foldr#3(Cons(x32, x6)) -> comp_f_g(x32, foldr#3(x6))
   main(x3) -> foldr_f#3(map#2(x3), 0)

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(1) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
----------------------------------------

(2)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   comp_f_g#1(plus_x(x3), comp_f_g(x1, x2), 0) -> plus_x#1(x3, comp_f_g#1(x1, x2, 0))
   comp_f_g#1(plus_x(x3), id, 0) -> plus_x#1(x3, 0)
   map#2(Nil) -> Nil
   map#2(Cons(x16, x6)) -> Cons(plus_x(x16), map#2(x6))
   plus_x#1(0, x6) -> x6
   plus_x#1(S(x8), x10) -> S(plus_x#1(x8, x10))
   foldr_f#3(Nil, 0) -> 0
   foldr_f#3(Cons(x16, x5), x24) -> comp_f_g#1(x16, foldr#3(x5), x24)
   foldr#3(Nil) -> id
   foldr#3(Cons(x32, x6)) -> comp_f_g(x32, foldr#3(x6))
   main(x3) -> foldr_f#3(map#2(x3), 0)

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(3) CpxTrsMatchBoundsTAProof (FINISHED)
A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 2. 

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
final states : [1, 2, 3, 4, 5, 6]
transitions: 
plus_x0(0) -> 0
comp_f_g0(0, 0) -> 0
00() -> 0
id0() -> 0
Nil0() -> 0
Cons0(0, 0) -> 0
S0(0) -> 0
comp_f_g#10(0, 0, 0) -> 1
map#20(0) -> 2
plus_x#10(0, 0) -> 3
foldr_f#30(0, 0) -> 4
foldr#30(0) -> 5
main0(0) -> 6
01() -> 8
comp_f_g#11(0, 0, 8) -> 7
plus_x#11(0, 7) -> 1
01() -> 9
plus_x#11(0, 9) -> 1
Nil1() -> 2
plus_x1(0) -> 10
map#21(0) -> 11
Cons1(10, 11) -> 2
plus_x#11(0, 0) -> 12
S1(12) -> 3
01() -> 4
foldr#31(0) -> 13
comp_f_g#11(0, 13, 0) -> 4
id1() -> 5
foldr#31(0) -> 14
comp_f_g1(0, 14) -> 5
map#21(0) -> 15
01() -> 16
foldr_f#31(15, 16) -> 6
plus_x#11(0, 7) -> 7
plus_x#11(0, 9) -> 7
Nil1() -> 11
Nil1() -> 15
Cons1(10, 11) -> 11
Cons1(10, 11) -> 15
plus_x#11(0, 7) -> 12
S1(12) -> 1
plus_x#11(0, 9) -> 12
S1(12) -> 12
id1() -> 13
id1() -> 14
comp_f_g1(0, 14) -> 13
comp_f_g1(0, 14) -> 14
comp_f_g#11(0, 14, 8) -> 7
plus_x#11(0, 7) -> 4
plus_x#11(0, 9) -> 4
S1(12) -> 7
02() -> 6
foldr#32(11) -> 17
comp_f_g#12(10, 17, 16) -> 6
id2() -> 17
foldr#32(11) -> 18
comp_f_g2(10, 18) -> 17
id2() -> 18
comp_f_g2(10, 18) -> 18
02() -> 20
comp_f_g#12(10, 18, 20) -> 19
plus_x#12(0, 19) -> 6
02() -> 21
plus_x#12(0, 21) -> 6
plus_x#12(0, 19) -> 19
plus_x#12(0, 21) -> 19
plus_x#11(0, 19) -> 12
S1(12) -> 6
plus_x#11(0, 21) -> 12
S1(12) -> 19
0 -> 3
0 -> 12
7 -> 1
7 -> 12
7 -> 4
9 -> 1
9 -> 7
19 -> 6
19 -> 12
21 -> 6
21 -> 12
21 -> 19

----------------------------------------

(4)
BOUNDS(1, n^1)

----------------------------------------

(5) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   comp_f_g#1(plus_x(z0), comp_f_g(z1, z2), 0) -> plus_x#1(z0, comp_f_g#1(z1, z2, 0))
   comp_f_g#1(plus_x(z0), id, 0) -> plus_x#1(z0, 0)
   map#2(Nil) -> Nil
   map#2(Cons(z0, z1)) -> Cons(plus_x(z0), map#2(z1))
   plus_x#1(0, z0) -> z0
   plus_x#1(S(z0), z1) -> S(plus_x#1(z0, z1))
   foldr_f#3(Nil, 0) -> 0
   foldr_f#3(Cons(z0, z1), z2) -> comp_f_g#1(z0, foldr#3(z1), z2)
   foldr#3(Nil) -> id
   foldr#3(Cons(z0, z1)) -> comp_f_g(z0, foldr#3(z1))
   main(z0) -> foldr_f#3(map#2(z0), 0)
Tuples:
   COMP_F_G#1(plus_x(z0), comp_f_g(z1, z2), 0) -> c(PLUS_X#1(z0, comp_f_g#1(z1, z2, 0)), COMP_F_G#1(z1, z2, 0))
   COMP_F_G#1(plus_x(z0), id, 0) -> c1(PLUS_X#1(z0, 0))
   MAP#2(Nil) -> c2
   MAP#2(Cons(z0, z1)) -> c3(MAP#2(z1))
   PLUS_X#1(0, z0) -> c4
   PLUS_X#1(S(z0), z1) -> c5(PLUS_X#1(z0, z1))
   FOLDR_F#3(Nil, 0) -> c6
   FOLDR_F#3(Cons(z0, z1), z2) -> c7(COMP_F_G#1(z0, foldr#3(z1), z2), FOLDR#3(z1))
   FOLDR#3(Nil) -> c8
   FOLDR#3(Cons(z0, z1)) -> c9(FOLDR#3(z1))
   MAIN(z0) -> c10(FOLDR_F#3(map#2(z0), 0), MAP#2(z0))
S tuples:
   COMP_F_G#1(plus_x(z0), comp_f_g(z1, z2), 0) -> c(PLUS_X#1(z0, comp_f_g#1(z1, z2, 0)), COMP_F_G#1(z1, z2, 0))
   COMP_F_G#1(plus_x(z0), id, 0) -> c1(PLUS_X#1(z0, 0))
   MAP#2(Nil) -> c2
   MAP#2(Cons(z0, z1)) -> c3(MAP#2(z1))
   PLUS_X#1(0, z0) -> c4
   PLUS_X#1(S(z0), z1) -> c5(PLUS_X#1(z0, z1))
   FOLDR_F#3(Nil, 0) -> c6
   FOLDR_F#3(Cons(z0, z1), z2) -> c7(COMP_F_G#1(z0, foldr#3(z1), z2), FOLDR#3(z1))
   FOLDR#3(Nil) -> c8
   FOLDR#3(Cons(z0, z1)) -> c9(FOLDR#3(z1))
   MAIN(z0) -> c10(FOLDR_F#3(map#2(z0), 0), MAP#2(z0))
K tuples:none
Defined Rule Symbols:   comp_f_g#1_3, map#2_1, plus_x#1_2, foldr_f#3_2, foldr#3_1, main_1

Defined Pair Symbols:   COMP_F_G#1_3, MAP#2_1, PLUS_X#1_2, FOLDR_F#3_2, FOLDR#3_1, MAIN_1

Compound Symbols:   c_2, c1_1, c2, c3_1, c4, c5_1, c6, c7_2, c8, c9_1, c10_2


----------------------------------------

(7) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID))
Converted S to standard rules, and D \ S as well as R to relative rules.
----------------------------------------

(8)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   COMP_F_G#1(plus_x(z0), comp_f_g(z1, z2), 0) -> c(PLUS_X#1(z0, comp_f_g#1(z1, z2, 0)), COMP_F_G#1(z1, z2, 0))
   COMP_F_G#1(plus_x(z0), id, 0) -> c1(PLUS_X#1(z0, 0))
   MAP#2(Nil) -> c2
   MAP#2(Cons(z0, z1)) -> c3(MAP#2(z1))
   PLUS_X#1(0, z0) -> c4
   PLUS_X#1(S(z0), z1) -> c5(PLUS_X#1(z0, z1))
   FOLDR_F#3(Nil, 0) -> c6
   FOLDR_F#3(Cons(z0, z1), z2) -> c7(COMP_F_G#1(z0, foldr#3(z1), z2), FOLDR#3(z1))
   FOLDR#3(Nil) -> c8
   FOLDR#3(Cons(z0, z1)) -> c9(FOLDR#3(z1))
   MAIN(z0) -> c10(FOLDR_F#3(map#2(z0), 0), MAP#2(z0))

The (relative) TRS S consists of the following rules:

   comp_f_g#1(plus_x(z0), comp_f_g(z1, z2), 0) -> plus_x#1(z0, comp_f_g#1(z1, z2, 0))
   comp_f_g#1(plus_x(z0), id, 0) -> plus_x#1(z0, 0)
   map#2(Nil) -> Nil
   map#2(Cons(z0, z1)) -> Cons(plus_x(z0), map#2(z1))
   plus_x#1(0, z0) -> z0
   plus_x#1(S(z0), z1) -> S(plus_x#1(z0, z1))
   foldr_f#3(Nil, 0) -> 0
   foldr_f#3(Cons(z0, z1), z2) -> comp_f_g#1(z0, foldr#3(z1), z2)
   foldr#3(Nil) -> id
   foldr#3(Cons(z0, z1)) -> comp_f_g(z0, foldr#3(z1))
   main(z0) -> foldr_f#3(map#2(z0), 0)

Rewrite Strategy: INNERMOST
----------------------------------------

(9) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(10)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   COMP_F_G#1(plus_x(z0), comp_f_g(z1, z2), 0') -> c(PLUS_X#1(z0, comp_f_g#1(z1, z2, 0')), COMP_F_G#1(z1, z2, 0'))
   COMP_F_G#1(plus_x(z0), id, 0') -> c1(PLUS_X#1(z0, 0'))
   MAP#2(Nil) -> c2
   MAP#2(Cons(z0, z1)) -> c3(MAP#2(z1))
   PLUS_X#1(0', z0) -> c4
   PLUS_X#1(S(z0), z1) -> c5(PLUS_X#1(z0, z1))
   FOLDR_F#3(Nil, 0') -> c6
   FOLDR_F#3(Cons(z0, z1), z2) -> c7(COMP_F_G#1(z0, foldr#3(z1), z2), FOLDR#3(z1))
   FOLDR#3(Nil) -> c8
   FOLDR#3(Cons(z0, z1)) -> c9(FOLDR#3(z1))
   MAIN(z0) -> c10(FOLDR_F#3(map#2(z0), 0'), MAP#2(z0))

The (relative) TRS S consists of the following rules:

   comp_f_g#1(plus_x(z0), comp_f_g(z1, z2), 0') -> plus_x#1(z0, comp_f_g#1(z1, z2, 0'))
   comp_f_g#1(plus_x(z0), id, 0') -> plus_x#1(z0, 0')
   map#2(Nil) -> Nil
   map#2(Cons(z0, z1)) -> Cons(plus_x(z0), map#2(z1))
   plus_x#1(0', z0) -> z0
   plus_x#1(S(z0), z1) -> S(plus_x#1(z0, z1))
   foldr_f#3(Nil, 0') -> 0'
   foldr_f#3(Cons(z0, z1), z2) -> comp_f_g#1(z0, foldr#3(z1), z2)
   foldr#3(Nil) -> id
   foldr#3(Cons(z0, z1)) -> comp_f_g(z0, foldr#3(z1))
   main(z0) -> foldr_f#3(map#2(z0), 0')

Rewrite Strategy: INNERMOST
----------------------------------------

(11) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Inferred types.
----------------------------------------

(12)
Obligation:
Innermost TRS:
Rules:
COMP_F_G#1(plus_x(z0), comp_f_g(z1, z2), 0') -> c(PLUS_X#1(z0, comp_f_g#1(z1, z2, 0')), COMP_F_G#1(z1, z2, 0'))
COMP_F_G#1(plus_x(z0), id, 0') -> c1(PLUS_X#1(z0, 0'))
MAP#2(Nil) -> c2
MAP#2(Cons(z0, z1)) -> c3(MAP#2(z1))
PLUS_X#1(0', z0) -> c4
PLUS_X#1(S(z0), z1) -> c5(PLUS_X#1(z0, z1))
FOLDR_F#3(Nil, 0') -> c6
FOLDR_F#3(Cons(z0, z1), z2) -> c7(COMP_F_G#1(z0, foldr#3(z1), z2), FOLDR#3(z1))
FOLDR#3(Nil) -> c8
FOLDR#3(Cons(z0, z1)) -> c9(FOLDR#3(z1))
MAIN(z0) -> c10(FOLDR_F#3(map#2(z0), 0'), MAP#2(z0))
comp_f_g#1(plus_x(z0), comp_f_g(z1, z2), 0') -> plus_x#1(z0, comp_f_g#1(z1, z2, 0'))
comp_f_g#1(plus_x(z0), id, 0') -> plus_x#1(z0, 0')
map#2(Nil) -> Nil
map#2(Cons(z0, z1)) -> Cons(plus_x(z0), map#2(z1))
plus_x#1(0', z0) -> z0
plus_x#1(S(z0), z1) -> S(plus_x#1(z0, z1))
foldr_f#3(Nil, 0') -> 0'
foldr_f#3(Cons(z0, z1), z2) -> comp_f_g#1(z0, foldr#3(z1), z2)
foldr#3(Nil) -> id
foldr#3(Cons(z0, z1)) -> comp_f_g(z0, foldr#3(z1))
main(z0) -> foldr_f#3(map#2(z0), 0')

Types:
COMP_F_G#1 :: plus_x:0':S -> comp_f_g:id -> plus_x:0':S -> c:c1
plus_x :: plus_x:0':S -> plus_x:0':S
comp_f_g :: plus_x:0':S -> comp_f_g:id -> comp_f_g:id
0' :: plus_x:0':S
c :: c4:c5 -> c:c1 -> c:c1
PLUS_X#1 :: plus_x:0':S -> plus_x:0':S -> c4:c5
comp_f_g#1 :: plus_x:0':S -> comp_f_g:id -> plus_x:0':S -> plus_x:0':S
id :: comp_f_g:id
c1 :: c4:c5 -> c:c1
MAP#2 :: Nil:Cons -> c2:c3
Nil :: Nil:Cons
c2 :: c2:c3
Cons :: plus_x:0':S -> Nil:Cons -> Nil:Cons
c3 :: c2:c3 -> c2:c3
c4 :: c4:c5
S :: plus_x:0':S -> plus_x:0':S
c5 :: c4:c5 -> c4:c5
FOLDR_F#3 :: Nil:Cons -> plus_x:0':S -> c6:c7
c6 :: c6:c7
c7 :: c:c1 -> c8:c9 -> c6:c7
foldr#3 :: Nil:Cons -> comp_f_g:id
FOLDR#3 :: Nil:Cons -> c8:c9
c8 :: c8:c9
c9 :: c8:c9 -> c8:c9
MAIN :: Nil:Cons -> c10
c10 :: c6:c7 -> c2:c3 -> c10
map#2 :: Nil:Cons -> Nil:Cons
plus_x#1 :: plus_x:0':S -> plus_x:0':S -> plus_x:0':S
foldr_f#3 :: Nil:Cons -> plus_x:0':S -> plus_x:0':S
main :: Nil:Cons -> plus_x:0':S
hole_c:c11_11 :: c:c1
hole_plus_x:0':S2_11 :: plus_x:0':S
hole_comp_f_g:id3_11 :: comp_f_g:id
hole_c4:c54_11 :: c4:c5
hole_c2:c35_11 :: c2:c3
hole_Nil:Cons6_11 :: Nil:Cons
hole_c6:c77_11 :: c6:c7
hole_c8:c98_11 :: c8:c9
hole_c109_11 :: c10
gen_c:c110_11 :: Nat -> c:c1
gen_plus_x:0':S11_11 :: Nat -> plus_x:0':S
gen_comp_f_g:id12_11 :: Nat -> comp_f_g:id
gen_c4:c513_11 :: Nat -> c4:c5
gen_c2:c314_11 :: Nat -> c2:c3
gen_Nil:Cons15_11 :: Nat -> Nil:Cons
gen_c8:c916_11 :: Nat -> c8:c9

----------------------------------------

(13) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
COMP_F_G#1, PLUS_X#1, comp_f_g#1, MAP#2, foldr#3, FOLDR#3, map#2, plus_x#1

They will be analysed ascendingly in the following order:
PLUS_X#1 < COMP_F_G#1
comp_f_g#1 < COMP_F_G#1
plus_x#1 < comp_f_g#1

----------------------------------------

(14)
Obligation:
Innermost TRS:
Rules:
COMP_F_G#1(plus_x(z0), comp_f_g(z1, z2), 0') -> c(PLUS_X#1(z0, comp_f_g#1(z1, z2, 0')), COMP_F_G#1(z1, z2, 0'))
COMP_F_G#1(plus_x(z0), id, 0') -> c1(PLUS_X#1(z0, 0'))
MAP#2(Nil) -> c2
MAP#2(Cons(z0, z1)) -> c3(MAP#2(z1))
PLUS_X#1(0', z0) -> c4
PLUS_X#1(S(z0), z1) -> c5(PLUS_X#1(z0, z1))
FOLDR_F#3(Nil, 0') -> c6
FOLDR_F#3(Cons(z0, z1), z2) -> c7(COMP_F_G#1(z0, foldr#3(z1), z2), FOLDR#3(z1))
FOLDR#3(Nil) -> c8
FOLDR#3(Cons(z0, z1)) -> c9(FOLDR#3(z1))
MAIN(z0) -> c10(FOLDR_F#3(map#2(z0), 0'), MAP#2(z0))
comp_f_g#1(plus_x(z0), comp_f_g(z1, z2), 0') -> plus_x#1(z0, comp_f_g#1(z1, z2, 0'))
comp_f_g#1(plus_x(z0), id, 0') -> plus_x#1(z0, 0')
map#2(Nil) -> Nil
map#2(Cons(z0, z1)) -> Cons(plus_x(z0), map#2(z1))
plus_x#1(0', z0) -> z0
plus_x#1(S(z0), z1) -> S(plus_x#1(z0, z1))
foldr_f#3(Nil, 0') -> 0'
foldr_f#3(Cons(z0, z1), z2) -> comp_f_g#1(z0, foldr#3(z1), z2)
foldr#3(Nil) -> id
foldr#3(Cons(z0, z1)) -> comp_f_g(z0, foldr#3(z1))
main(z0) -> foldr_f#3(map#2(z0), 0')

Types:
COMP_F_G#1 :: plus_x:0':S -> comp_f_g:id -> plus_x:0':S -> c:c1
plus_x :: plus_x:0':S -> plus_x:0':S
comp_f_g :: plus_x:0':S -> comp_f_g:id -> comp_f_g:id
0' :: plus_x:0':S
c :: c4:c5 -> c:c1 -> c:c1
PLUS_X#1 :: plus_x:0':S -> plus_x:0':S -> c4:c5
comp_f_g#1 :: plus_x:0':S -> comp_f_g:id -> plus_x:0':S -> plus_x:0':S
id :: comp_f_g:id
c1 :: c4:c5 -> c:c1
MAP#2 :: Nil:Cons -> c2:c3
Nil :: Nil:Cons
c2 :: c2:c3
Cons :: plus_x:0':S -> Nil:Cons -> Nil:Cons
c3 :: c2:c3 -> c2:c3
c4 :: c4:c5
S :: plus_x:0':S -> plus_x:0':S
c5 :: c4:c5 -> c4:c5
FOLDR_F#3 :: Nil:Cons -> plus_x:0':S -> c6:c7
c6 :: c6:c7
c7 :: c:c1 -> c8:c9 -> c6:c7
foldr#3 :: Nil:Cons -> comp_f_g:id
FOLDR#3 :: Nil:Cons -> c8:c9
c8 :: c8:c9
c9 :: c8:c9 -> c8:c9
MAIN :: Nil:Cons -> c10
c10 :: c6:c7 -> c2:c3 -> c10
map#2 :: Nil:Cons -> Nil:Cons
plus_x#1 :: plus_x:0':S -> plus_x:0':S -> plus_x:0':S
foldr_f#3 :: Nil:Cons -> plus_x:0':S -> plus_x:0':S
main :: Nil:Cons -> plus_x:0':S
hole_c:c11_11 :: c:c1
hole_plus_x:0':S2_11 :: plus_x:0':S
hole_comp_f_g:id3_11 :: comp_f_g:id
hole_c4:c54_11 :: c4:c5
hole_c2:c35_11 :: c2:c3
hole_Nil:Cons6_11 :: Nil:Cons
hole_c6:c77_11 :: c6:c7
hole_c8:c98_11 :: c8:c9
hole_c109_11 :: c10
gen_c:c110_11 :: Nat -> c:c1
gen_plus_x:0':S11_11 :: Nat -> plus_x:0':S
gen_comp_f_g:id12_11 :: Nat -> comp_f_g:id
gen_c4:c513_11 :: Nat -> c4:c5
gen_c2:c314_11 :: Nat -> c2:c3
gen_Nil:Cons15_11 :: Nat -> Nil:Cons
gen_c8:c916_11 :: Nat -> c8:c9


Generator Equations:
gen_c:c110_11(0) <=> c1(c4)
gen_c:c110_11(+(x, 1)) <=> c(c4, gen_c:c110_11(x))
gen_plus_x:0':S11_11(0) <=> 0'
gen_plus_x:0':S11_11(+(x, 1)) <=> plus_x(gen_plus_x:0':S11_11(x))
gen_comp_f_g:id12_11(0) <=> id
gen_comp_f_g:id12_11(+(x, 1)) <=> comp_f_g(0', gen_comp_f_g:id12_11(x))
gen_c4:c513_11(0) <=> c4
gen_c4:c513_11(+(x, 1)) <=> c5(gen_c4:c513_11(x))
gen_c2:c314_11(0) <=> c2
gen_c2:c314_11(+(x, 1)) <=> c3(gen_c2:c314_11(x))
gen_Nil:Cons15_11(0) <=> Nil
gen_Nil:Cons15_11(+(x, 1)) <=> Cons(0', gen_Nil:Cons15_11(x))
gen_c8:c916_11(0) <=> c8
gen_c8:c916_11(+(x, 1)) <=> c9(gen_c8:c916_11(x))


The following defined symbols remain to be analysed:
PLUS_X#1, COMP_F_G#1, comp_f_g#1, MAP#2, foldr#3, FOLDR#3, map#2, plus_x#1

They will be analysed ascendingly in the following order:
PLUS_X#1 < COMP_F_G#1
comp_f_g#1 < COMP_F_G#1
plus_x#1 < comp_f_g#1

----------------------------------------

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
MAP#2(gen_Nil:Cons15_11(n27_11)) -> gen_c2:c314_11(n27_11), rt in Omega(1 + n27_11)

Induction Base:
MAP#2(gen_Nil:Cons15_11(0)) ->_R^Omega(1)
c2

Induction Step:
MAP#2(gen_Nil:Cons15_11(+(n27_11, 1))) ->_R^Omega(1)
c3(MAP#2(gen_Nil:Cons15_11(n27_11))) ->_IH
c3(gen_c2:c314_11(c28_11))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(16)
Complex Obligation (BEST)

----------------------------------------

(17)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
COMP_F_G#1(plus_x(z0), comp_f_g(z1, z2), 0') -> c(PLUS_X#1(z0, comp_f_g#1(z1, z2, 0')), COMP_F_G#1(z1, z2, 0'))
COMP_F_G#1(plus_x(z0), id, 0') -> c1(PLUS_X#1(z0, 0'))
MAP#2(Nil) -> c2
MAP#2(Cons(z0, z1)) -> c3(MAP#2(z1))
PLUS_X#1(0', z0) -> c4
PLUS_X#1(S(z0), z1) -> c5(PLUS_X#1(z0, z1))
FOLDR_F#3(Nil, 0') -> c6
FOLDR_F#3(Cons(z0, z1), z2) -> c7(COMP_F_G#1(z0, foldr#3(z1), z2), FOLDR#3(z1))
FOLDR#3(Nil) -> c8
FOLDR#3(Cons(z0, z1)) -> c9(FOLDR#3(z1))
MAIN(z0) -> c10(FOLDR_F#3(map#2(z0), 0'), MAP#2(z0))
comp_f_g#1(plus_x(z0), comp_f_g(z1, z2), 0') -> plus_x#1(z0, comp_f_g#1(z1, z2, 0'))
comp_f_g#1(plus_x(z0), id, 0') -> plus_x#1(z0, 0')
map#2(Nil) -> Nil
map#2(Cons(z0, z1)) -> Cons(plus_x(z0), map#2(z1))
plus_x#1(0', z0) -> z0
plus_x#1(S(z0), z1) -> S(plus_x#1(z0, z1))
foldr_f#3(Nil, 0') -> 0'
foldr_f#3(Cons(z0, z1), z2) -> comp_f_g#1(z0, foldr#3(z1), z2)
foldr#3(Nil) -> id
foldr#3(Cons(z0, z1)) -> comp_f_g(z0, foldr#3(z1))
main(z0) -> foldr_f#3(map#2(z0), 0')

Types:
COMP_F_G#1 :: plus_x:0':S -> comp_f_g:id -> plus_x:0':S -> c:c1
plus_x :: plus_x:0':S -> plus_x:0':S
comp_f_g :: plus_x:0':S -> comp_f_g:id -> comp_f_g:id
0' :: plus_x:0':S
c :: c4:c5 -> c:c1 -> c:c1
PLUS_X#1 :: plus_x:0':S -> plus_x:0':S -> c4:c5
comp_f_g#1 :: plus_x:0':S -> comp_f_g:id -> plus_x:0':S -> plus_x:0':S
id :: comp_f_g:id
c1 :: c4:c5 -> c:c1
MAP#2 :: Nil:Cons -> c2:c3
Nil :: Nil:Cons
c2 :: c2:c3
Cons :: plus_x:0':S -> Nil:Cons -> Nil:Cons
c3 :: c2:c3 -> c2:c3
c4 :: c4:c5
S :: plus_x:0':S -> plus_x:0':S
c5 :: c4:c5 -> c4:c5
FOLDR_F#3 :: Nil:Cons -> plus_x:0':S -> c6:c7
c6 :: c6:c7
c7 :: c:c1 -> c8:c9 -> c6:c7
foldr#3 :: Nil:Cons -> comp_f_g:id
FOLDR#3 :: Nil:Cons -> c8:c9
c8 :: c8:c9
c9 :: c8:c9 -> c8:c9
MAIN :: Nil:Cons -> c10
c10 :: c6:c7 -> c2:c3 -> c10
map#2 :: Nil:Cons -> Nil:Cons
plus_x#1 :: plus_x:0':S -> plus_x:0':S -> plus_x:0':S
foldr_f#3 :: Nil:Cons -> plus_x:0':S -> plus_x:0':S
main :: Nil:Cons -> plus_x:0':S
hole_c:c11_11 :: c:c1
hole_plus_x:0':S2_11 :: plus_x:0':S
hole_comp_f_g:id3_11 :: comp_f_g:id
hole_c4:c54_11 :: c4:c5
hole_c2:c35_11 :: c2:c3
hole_Nil:Cons6_11 :: Nil:Cons
hole_c6:c77_11 :: c6:c7
hole_c8:c98_11 :: c8:c9
hole_c109_11 :: c10
gen_c:c110_11 :: Nat -> c:c1
gen_plus_x:0':S11_11 :: Nat -> plus_x:0':S
gen_comp_f_g:id12_11 :: Nat -> comp_f_g:id
gen_c4:c513_11 :: Nat -> c4:c5
gen_c2:c314_11 :: Nat -> c2:c3
gen_Nil:Cons15_11 :: Nat -> Nil:Cons
gen_c8:c916_11 :: Nat -> c8:c9


Generator Equations:
gen_c:c110_11(0) <=> c1(c4)
gen_c:c110_11(+(x, 1)) <=> c(c4, gen_c:c110_11(x))
gen_plus_x:0':S11_11(0) <=> 0'
gen_plus_x:0':S11_11(+(x, 1)) <=> plus_x(gen_plus_x:0':S11_11(x))
gen_comp_f_g:id12_11(0) <=> id
gen_comp_f_g:id12_11(+(x, 1)) <=> comp_f_g(0', gen_comp_f_g:id12_11(x))
gen_c4:c513_11(0) <=> c4
gen_c4:c513_11(+(x, 1)) <=> c5(gen_c4:c513_11(x))
gen_c2:c314_11(0) <=> c2
gen_c2:c314_11(+(x, 1)) <=> c3(gen_c2:c314_11(x))
gen_Nil:Cons15_11(0) <=> Nil
gen_Nil:Cons15_11(+(x, 1)) <=> Cons(0', gen_Nil:Cons15_11(x))
gen_c8:c916_11(0) <=> c8
gen_c8:c916_11(+(x, 1)) <=> c9(gen_c8:c916_11(x))


The following defined symbols remain to be analysed:
MAP#2, COMP_F_G#1, comp_f_g#1, foldr#3, FOLDR#3, map#2, plus_x#1

They will be analysed ascendingly in the following order:
comp_f_g#1 < COMP_F_G#1
plus_x#1 < comp_f_g#1

----------------------------------------

(18) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(19)
BOUNDS(n^1, INF)

----------------------------------------

(20)
Obligation:
Innermost TRS:
Rules:
COMP_F_G#1(plus_x(z0), comp_f_g(z1, z2), 0') -> c(PLUS_X#1(z0, comp_f_g#1(z1, z2, 0')), COMP_F_G#1(z1, z2, 0'))
COMP_F_G#1(plus_x(z0), id, 0') -> c1(PLUS_X#1(z0, 0'))
MAP#2(Nil) -> c2
MAP#2(Cons(z0, z1)) -> c3(MAP#2(z1))
PLUS_X#1(0', z0) -> c4
PLUS_X#1(S(z0), z1) -> c5(PLUS_X#1(z0, z1))
FOLDR_F#3(Nil, 0') -> c6
FOLDR_F#3(Cons(z0, z1), z2) -> c7(COMP_F_G#1(z0, foldr#3(z1), z2), FOLDR#3(z1))
FOLDR#3(Nil) -> c8
FOLDR#3(Cons(z0, z1)) -> c9(FOLDR#3(z1))
MAIN(z0) -> c10(FOLDR_F#3(map#2(z0), 0'), MAP#2(z0))
comp_f_g#1(plus_x(z0), comp_f_g(z1, z2), 0') -> plus_x#1(z0, comp_f_g#1(z1, z2, 0'))
comp_f_g#1(plus_x(z0), id, 0') -> plus_x#1(z0, 0')
map#2(Nil) -> Nil
map#2(Cons(z0, z1)) -> Cons(plus_x(z0), map#2(z1))
plus_x#1(0', z0) -> z0
plus_x#1(S(z0), z1) -> S(plus_x#1(z0, z1))
foldr_f#3(Nil, 0') -> 0'
foldr_f#3(Cons(z0, z1), z2) -> comp_f_g#1(z0, foldr#3(z1), z2)
foldr#3(Nil) -> id
foldr#3(Cons(z0, z1)) -> comp_f_g(z0, foldr#3(z1))
main(z0) -> foldr_f#3(map#2(z0), 0')

Types:
COMP_F_G#1 :: plus_x:0':S -> comp_f_g:id -> plus_x:0':S -> c:c1
plus_x :: plus_x:0':S -> plus_x:0':S
comp_f_g :: plus_x:0':S -> comp_f_g:id -> comp_f_g:id
0' :: plus_x:0':S
c :: c4:c5 -> c:c1 -> c:c1
PLUS_X#1 :: plus_x:0':S -> plus_x:0':S -> c4:c5
comp_f_g#1 :: plus_x:0':S -> comp_f_g:id -> plus_x:0':S -> plus_x:0':S
id :: comp_f_g:id
c1 :: c4:c5 -> c:c1
MAP#2 :: Nil:Cons -> c2:c3
Nil :: Nil:Cons
c2 :: c2:c3
Cons :: plus_x:0':S -> Nil:Cons -> Nil:Cons
c3 :: c2:c3 -> c2:c3
c4 :: c4:c5
S :: plus_x:0':S -> plus_x:0':S
c5 :: c4:c5 -> c4:c5
FOLDR_F#3 :: Nil:Cons -> plus_x:0':S -> c6:c7
c6 :: c6:c7
c7 :: c:c1 -> c8:c9 -> c6:c7
foldr#3 :: Nil:Cons -> comp_f_g:id
FOLDR#3 :: Nil:Cons -> c8:c9
c8 :: c8:c9
c9 :: c8:c9 -> c8:c9
MAIN :: Nil:Cons -> c10
c10 :: c6:c7 -> c2:c3 -> c10
map#2 :: Nil:Cons -> Nil:Cons
plus_x#1 :: plus_x:0':S -> plus_x:0':S -> plus_x:0':S
foldr_f#3 :: Nil:Cons -> plus_x:0':S -> plus_x:0':S
main :: Nil:Cons -> plus_x:0':S
hole_c:c11_11 :: c:c1
hole_plus_x:0':S2_11 :: plus_x:0':S
hole_comp_f_g:id3_11 :: comp_f_g:id
hole_c4:c54_11 :: c4:c5
hole_c2:c35_11 :: c2:c3
hole_Nil:Cons6_11 :: Nil:Cons
hole_c6:c77_11 :: c6:c7
hole_c8:c98_11 :: c8:c9
hole_c109_11 :: c10
gen_c:c110_11 :: Nat -> c:c1
gen_plus_x:0':S11_11 :: Nat -> plus_x:0':S
gen_comp_f_g:id12_11 :: Nat -> comp_f_g:id
gen_c4:c513_11 :: Nat -> c4:c5
gen_c2:c314_11 :: Nat -> c2:c3
gen_Nil:Cons15_11 :: Nat -> Nil:Cons
gen_c8:c916_11 :: Nat -> c8:c9


Lemmas:
MAP#2(gen_Nil:Cons15_11(n27_11)) -> gen_c2:c314_11(n27_11), rt in Omega(1 + n27_11)


Generator Equations:
gen_c:c110_11(0) <=> c1(c4)
gen_c:c110_11(+(x, 1)) <=> c(c4, gen_c:c110_11(x))
gen_plus_x:0':S11_11(0) <=> 0'
gen_plus_x:0':S11_11(+(x, 1)) <=> plus_x(gen_plus_x:0':S11_11(x))
gen_comp_f_g:id12_11(0) <=> id
gen_comp_f_g:id12_11(+(x, 1)) <=> comp_f_g(0', gen_comp_f_g:id12_11(x))
gen_c4:c513_11(0) <=> c4
gen_c4:c513_11(+(x, 1)) <=> c5(gen_c4:c513_11(x))
gen_c2:c314_11(0) <=> c2
gen_c2:c314_11(+(x, 1)) <=> c3(gen_c2:c314_11(x))
gen_Nil:Cons15_11(0) <=> Nil
gen_Nil:Cons15_11(+(x, 1)) <=> Cons(0', gen_Nil:Cons15_11(x))
gen_c8:c916_11(0) <=> c8
gen_c8:c916_11(+(x, 1)) <=> c9(gen_c8:c916_11(x))


The following defined symbols remain to be analysed:
foldr#3, COMP_F_G#1, comp_f_g#1, FOLDR#3, map#2, plus_x#1

They will be analysed ascendingly in the following order:
comp_f_g#1 < COMP_F_G#1
plus_x#1 < comp_f_g#1

----------------------------------------

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
foldr#3(gen_Nil:Cons15_11(n471_11)) -> gen_comp_f_g:id12_11(n471_11), rt in Omega(0)

Induction Base:
foldr#3(gen_Nil:Cons15_11(0)) ->_R^Omega(0)
id

Induction Step:
foldr#3(gen_Nil:Cons15_11(+(n471_11, 1))) ->_R^Omega(0)
comp_f_g(0', foldr#3(gen_Nil:Cons15_11(n471_11))) ->_IH
comp_f_g(0', gen_comp_f_g:id12_11(c472_11))

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(22)
Obligation:
Innermost TRS:
Rules:
COMP_F_G#1(plus_x(z0), comp_f_g(z1, z2), 0') -> c(PLUS_X#1(z0, comp_f_g#1(z1, z2, 0')), COMP_F_G#1(z1, z2, 0'))
COMP_F_G#1(plus_x(z0), id, 0') -> c1(PLUS_X#1(z0, 0'))
MAP#2(Nil) -> c2
MAP#2(Cons(z0, z1)) -> c3(MAP#2(z1))
PLUS_X#1(0', z0) -> c4
PLUS_X#1(S(z0), z1) -> c5(PLUS_X#1(z0, z1))
FOLDR_F#3(Nil, 0') -> c6
FOLDR_F#3(Cons(z0, z1), z2) -> c7(COMP_F_G#1(z0, foldr#3(z1), z2), FOLDR#3(z1))
FOLDR#3(Nil) -> c8
FOLDR#3(Cons(z0, z1)) -> c9(FOLDR#3(z1))
MAIN(z0) -> c10(FOLDR_F#3(map#2(z0), 0'), MAP#2(z0))
comp_f_g#1(plus_x(z0), comp_f_g(z1, z2), 0') -> plus_x#1(z0, comp_f_g#1(z1, z2, 0'))
comp_f_g#1(plus_x(z0), id, 0') -> plus_x#1(z0, 0')
map#2(Nil) -> Nil
map#2(Cons(z0, z1)) -> Cons(plus_x(z0), map#2(z1))
plus_x#1(0', z0) -> z0
plus_x#1(S(z0), z1) -> S(plus_x#1(z0, z1))
foldr_f#3(Nil, 0') -> 0'
foldr_f#3(Cons(z0, z1), z2) -> comp_f_g#1(z0, foldr#3(z1), z2)
foldr#3(Nil) -> id
foldr#3(Cons(z0, z1)) -> comp_f_g(z0, foldr#3(z1))
main(z0) -> foldr_f#3(map#2(z0), 0')

Types:
COMP_F_G#1 :: plus_x:0':S -> comp_f_g:id -> plus_x:0':S -> c:c1
plus_x :: plus_x:0':S -> plus_x:0':S
comp_f_g :: plus_x:0':S -> comp_f_g:id -> comp_f_g:id
0' :: plus_x:0':S
c :: c4:c5 -> c:c1 -> c:c1
PLUS_X#1 :: plus_x:0':S -> plus_x:0':S -> c4:c5
comp_f_g#1 :: plus_x:0':S -> comp_f_g:id -> plus_x:0':S -> plus_x:0':S
id :: comp_f_g:id
c1 :: c4:c5 -> c:c1
MAP#2 :: Nil:Cons -> c2:c3
Nil :: Nil:Cons
c2 :: c2:c3
Cons :: plus_x:0':S -> Nil:Cons -> Nil:Cons
c3 :: c2:c3 -> c2:c3
c4 :: c4:c5
S :: plus_x:0':S -> plus_x:0':S
c5 :: c4:c5 -> c4:c5
FOLDR_F#3 :: Nil:Cons -> plus_x:0':S -> c6:c7
c6 :: c6:c7
c7 :: c:c1 -> c8:c9 -> c6:c7
foldr#3 :: Nil:Cons -> comp_f_g:id
FOLDR#3 :: Nil:Cons -> c8:c9
c8 :: c8:c9
c9 :: c8:c9 -> c8:c9
MAIN :: Nil:Cons -> c10
c10 :: c6:c7 -> c2:c3 -> c10
map#2 :: Nil:Cons -> Nil:Cons
plus_x#1 :: plus_x:0':S -> plus_x:0':S -> plus_x:0':S
foldr_f#3 :: Nil:Cons -> plus_x:0':S -> plus_x:0':S
main :: Nil:Cons -> plus_x:0':S
hole_c:c11_11 :: c:c1
hole_plus_x:0':S2_11 :: plus_x:0':S
hole_comp_f_g:id3_11 :: comp_f_g:id
hole_c4:c54_11 :: c4:c5
hole_c2:c35_11 :: c2:c3
hole_Nil:Cons6_11 :: Nil:Cons
hole_c6:c77_11 :: c6:c7
hole_c8:c98_11 :: c8:c9
hole_c109_11 :: c10
gen_c:c110_11 :: Nat -> c:c1
gen_plus_x:0':S11_11 :: Nat -> plus_x:0':S
gen_comp_f_g:id12_11 :: Nat -> comp_f_g:id
gen_c4:c513_11 :: Nat -> c4:c5
gen_c2:c314_11 :: Nat -> c2:c3
gen_Nil:Cons15_11 :: Nat -> Nil:Cons
gen_c8:c916_11 :: Nat -> c8:c9


Lemmas:
MAP#2(gen_Nil:Cons15_11(n27_11)) -> gen_c2:c314_11(n27_11), rt in Omega(1 + n27_11)
foldr#3(gen_Nil:Cons15_11(n471_11)) -> gen_comp_f_g:id12_11(n471_11), rt in Omega(0)


Generator Equations:
gen_c:c110_11(0) <=> c1(c4)
gen_c:c110_11(+(x, 1)) <=> c(c4, gen_c:c110_11(x))
gen_plus_x:0':S11_11(0) <=> 0'
gen_plus_x:0':S11_11(+(x, 1)) <=> plus_x(gen_plus_x:0':S11_11(x))
gen_comp_f_g:id12_11(0) <=> id
gen_comp_f_g:id12_11(+(x, 1)) <=> comp_f_g(0', gen_comp_f_g:id12_11(x))
gen_c4:c513_11(0) <=> c4
gen_c4:c513_11(+(x, 1)) <=> c5(gen_c4:c513_11(x))
gen_c2:c314_11(0) <=> c2
gen_c2:c314_11(+(x, 1)) <=> c3(gen_c2:c314_11(x))
gen_Nil:Cons15_11(0) <=> Nil
gen_Nil:Cons15_11(+(x, 1)) <=> Cons(0', gen_Nil:Cons15_11(x))
gen_c8:c916_11(0) <=> c8
gen_c8:c916_11(+(x, 1)) <=> c9(gen_c8:c916_11(x))


The following defined symbols remain to be analysed:
FOLDR#3, COMP_F_G#1, comp_f_g#1, map#2, plus_x#1

They will be analysed ascendingly in the following order:
comp_f_g#1 < COMP_F_G#1
plus_x#1 < comp_f_g#1

----------------------------------------

(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
FOLDR#3(gen_Nil:Cons15_11(n851_11)) -> gen_c8:c916_11(n851_11), rt in Omega(1 + n851_11)

Induction Base:
FOLDR#3(gen_Nil:Cons15_11(0)) ->_R^Omega(1)
c8

Induction Step:
FOLDR#3(gen_Nil:Cons15_11(+(n851_11, 1))) ->_R^Omega(1)
c9(FOLDR#3(gen_Nil:Cons15_11(n851_11))) ->_IH
c9(gen_c8:c916_11(c852_11))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(24)
Obligation:
Innermost TRS:
Rules:
COMP_F_G#1(plus_x(z0), comp_f_g(z1, z2), 0') -> c(PLUS_X#1(z0, comp_f_g#1(z1, z2, 0')), COMP_F_G#1(z1, z2, 0'))
COMP_F_G#1(plus_x(z0), id, 0') -> c1(PLUS_X#1(z0, 0'))
MAP#2(Nil) -> c2
MAP#2(Cons(z0, z1)) -> c3(MAP#2(z1))
PLUS_X#1(0', z0) -> c4
PLUS_X#1(S(z0), z1) -> c5(PLUS_X#1(z0, z1))
FOLDR_F#3(Nil, 0') -> c6
FOLDR_F#3(Cons(z0, z1), z2) -> c7(COMP_F_G#1(z0, foldr#3(z1), z2), FOLDR#3(z1))
FOLDR#3(Nil) -> c8
FOLDR#3(Cons(z0, z1)) -> c9(FOLDR#3(z1))
MAIN(z0) -> c10(FOLDR_F#3(map#2(z0), 0'), MAP#2(z0))
comp_f_g#1(plus_x(z0), comp_f_g(z1, z2), 0') -> plus_x#1(z0, comp_f_g#1(z1, z2, 0'))
comp_f_g#1(plus_x(z0), id, 0') -> plus_x#1(z0, 0')
map#2(Nil) -> Nil
map#2(Cons(z0, z1)) -> Cons(plus_x(z0), map#2(z1))
plus_x#1(0', z0) -> z0
plus_x#1(S(z0), z1) -> S(plus_x#1(z0, z1))
foldr_f#3(Nil, 0') -> 0'
foldr_f#3(Cons(z0, z1), z2) -> comp_f_g#1(z0, foldr#3(z1), z2)
foldr#3(Nil) -> id
foldr#3(Cons(z0, z1)) -> comp_f_g(z0, foldr#3(z1))
main(z0) -> foldr_f#3(map#2(z0), 0')

Types:
COMP_F_G#1 :: plus_x:0':S -> comp_f_g:id -> plus_x:0':S -> c:c1
plus_x :: plus_x:0':S -> plus_x:0':S
comp_f_g :: plus_x:0':S -> comp_f_g:id -> comp_f_g:id
0' :: plus_x:0':S
c :: c4:c5 -> c:c1 -> c:c1
PLUS_X#1 :: plus_x:0':S -> plus_x:0':S -> c4:c5
comp_f_g#1 :: plus_x:0':S -> comp_f_g:id -> plus_x:0':S -> plus_x:0':S
id :: comp_f_g:id
c1 :: c4:c5 -> c:c1
MAP#2 :: Nil:Cons -> c2:c3
Nil :: Nil:Cons
c2 :: c2:c3
Cons :: plus_x:0':S -> Nil:Cons -> Nil:Cons
c3 :: c2:c3 -> c2:c3
c4 :: c4:c5
S :: plus_x:0':S -> plus_x:0':S
c5 :: c4:c5 -> c4:c5
FOLDR_F#3 :: Nil:Cons -> plus_x:0':S -> c6:c7
c6 :: c6:c7
c7 :: c:c1 -> c8:c9 -> c6:c7
foldr#3 :: Nil:Cons -> comp_f_g:id
FOLDR#3 :: Nil:Cons -> c8:c9
c8 :: c8:c9
c9 :: c8:c9 -> c8:c9
MAIN :: Nil:Cons -> c10
c10 :: c6:c7 -> c2:c3 -> c10
map#2 :: Nil:Cons -> Nil:Cons
plus_x#1 :: plus_x:0':S -> plus_x:0':S -> plus_x:0':S
foldr_f#3 :: Nil:Cons -> plus_x:0':S -> plus_x:0':S
main :: Nil:Cons -> plus_x:0':S
hole_c:c11_11 :: c:c1
hole_plus_x:0':S2_11 :: plus_x:0':S
hole_comp_f_g:id3_11 :: comp_f_g:id
hole_c4:c54_11 :: c4:c5
hole_c2:c35_11 :: c2:c3
hole_Nil:Cons6_11 :: Nil:Cons
hole_c6:c77_11 :: c6:c7
hole_c8:c98_11 :: c8:c9
hole_c109_11 :: c10
gen_c:c110_11 :: Nat -> c:c1
gen_plus_x:0':S11_11 :: Nat -> plus_x:0':S
gen_comp_f_g:id12_11 :: Nat -> comp_f_g:id
gen_c4:c513_11 :: Nat -> c4:c5
gen_c2:c314_11 :: Nat -> c2:c3
gen_Nil:Cons15_11 :: Nat -> Nil:Cons
gen_c8:c916_11 :: Nat -> c8:c9


Lemmas:
MAP#2(gen_Nil:Cons15_11(n27_11)) -> gen_c2:c314_11(n27_11), rt in Omega(1 + n27_11)
foldr#3(gen_Nil:Cons15_11(n471_11)) -> gen_comp_f_g:id12_11(n471_11), rt in Omega(0)
FOLDR#3(gen_Nil:Cons15_11(n851_11)) -> gen_c8:c916_11(n851_11), rt in Omega(1 + n851_11)


Generator Equations:
gen_c:c110_11(0) <=> c1(c4)
gen_c:c110_11(+(x, 1)) <=> c(c4, gen_c:c110_11(x))
gen_plus_x:0':S11_11(0) <=> 0'
gen_plus_x:0':S11_11(+(x, 1)) <=> plus_x(gen_plus_x:0':S11_11(x))
gen_comp_f_g:id12_11(0) <=> id
gen_comp_f_g:id12_11(+(x, 1)) <=> comp_f_g(0', gen_comp_f_g:id12_11(x))
gen_c4:c513_11(0) <=> c4
gen_c4:c513_11(+(x, 1)) <=> c5(gen_c4:c513_11(x))
gen_c2:c314_11(0) <=> c2
gen_c2:c314_11(+(x, 1)) <=> c3(gen_c2:c314_11(x))
gen_Nil:Cons15_11(0) <=> Nil
gen_Nil:Cons15_11(+(x, 1)) <=> Cons(0', gen_Nil:Cons15_11(x))
gen_c8:c916_11(0) <=> c8
gen_c8:c916_11(+(x, 1)) <=> c9(gen_c8:c916_11(x))


The following defined symbols remain to be analysed:
map#2, COMP_F_G#1, comp_f_g#1, plus_x#1

They will be analysed ascendingly in the following order:
comp_f_g#1 < COMP_F_G#1
plus_x#1 < comp_f_g#1