WORST_CASE(Omega(n^3),O(n^3))
proof of input_X7qEONYbs1.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(n^3, n^3).

(0) CpxTRS
(1) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(2) CdtProblem
    (3) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (4) CdtProblem
    (5) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CdtProblem
    (7) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 86 ms]
    (8) CdtProblem
    (9) CdtRuleRemovalProof [UPPER BOUND(ADD(n^3)), 157 ms]
    (10) CdtProblem
    (11) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) BOUNDS(1, 1)
(13) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(14) CdtProblem
    (15) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) CpxRelTRS
    (17) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) CpxRelTRS
    (19) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (20) typed CpxTrs
    (21) OrderProof [LOWER BOUND(ID), 17 ms]
    (22) typed CpxTrs
    (23) RewriteLemmaProof [LOWER BOUND(ID), 304 ms]
    (24) BEST
        (25) proven lower bound
            (26) LowerBoundPropagationProof [FINISHED, 0 ms]
            (27) BOUNDS(n^1, INF)
        (28) typed CpxTrs
            (29) RewriteLemmaProof [LOWER BOUND(ID), 80 ms]
            (30) typed CpxTrs
            (31) RewriteLemmaProof [LOWER BOUND(ID), 58 ms]
            (32) typed CpxTrs
            (33) RewriteLemmaProof [LOWER BOUND(ID), 1922 ms]
            (34) proven lower bound
            (35) LowerBoundPropagationProof [FINISHED, 0 ms]
            (36) BOUNDS(n^3, INF)


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

(0)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(n^3, n^3).


The TRS R consists of the following rules:

   mul0(Cons(x, xs), y) -> add0(mul0(xs, y), y)
   add0(Cons(x, xs), y) -> add0(xs, Cons(S, y))
   mul0(Nil, y) -> Nil
   add0(Nil, y) -> y
   goal(xs, ys) -> mul0(xs, ys)

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

(1) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(2)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   mul0(Cons(z0, z1), z2) -> add0(mul0(z1, z2), z2)
   mul0(Nil, z0) -> Nil
   add0(Cons(z0, z1), z2) -> add0(z1, Cons(S, z2))
   add0(Nil, z0) -> z0
   goal(z0, z1) -> mul0(z0, z1)
Tuples:
   MUL0(Cons(z0, z1), z2) -> c(ADD0(mul0(z1, z2), z2), MUL0(z1, z2))
   MUL0(Nil, z0) -> c1
   ADD0(Cons(z0, z1), z2) -> c2(ADD0(z1, Cons(S, z2)))
   ADD0(Nil, z0) -> c3
   GOAL(z0, z1) -> c4(MUL0(z0, z1))
S tuples:
   MUL0(Cons(z0, z1), z2) -> c(ADD0(mul0(z1, z2), z2), MUL0(z1, z2))
   MUL0(Nil, z0) -> c1
   ADD0(Cons(z0, z1), z2) -> c2(ADD0(z1, Cons(S, z2)))
   ADD0(Nil, z0) -> c3
   GOAL(z0, z1) -> c4(MUL0(z0, z1))
K tuples:none
Defined Rule Symbols:   mul0_2, add0_2, goal_2

Defined Pair Symbols:   MUL0_2, ADD0_2, GOAL_2

Compound Symbols:   c_2, c1, c2_1, c3, c4_1


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

(3) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   GOAL(z0, z1) -> c4(MUL0(z0, z1))
Removed 2 trailing nodes:
   MUL0(Nil, z0) -> c1
   ADD0(Nil, z0) -> c3

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   mul0(Cons(z0, z1), z2) -> add0(mul0(z1, z2), z2)
   mul0(Nil, z0) -> Nil
   add0(Cons(z0, z1), z2) -> add0(z1, Cons(S, z2))
   add0(Nil, z0) -> z0
   goal(z0, z1) -> mul0(z0, z1)
Tuples:
   MUL0(Cons(z0, z1), z2) -> c(ADD0(mul0(z1, z2), z2), MUL0(z1, z2))
   ADD0(Cons(z0, z1), z2) -> c2(ADD0(z1, Cons(S, z2)))
S tuples:
   MUL0(Cons(z0, z1), z2) -> c(ADD0(mul0(z1, z2), z2), MUL0(z1, z2))
   ADD0(Cons(z0, z1), z2) -> c2(ADD0(z1, Cons(S, z2)))
K tuples:none
Defined Rule Symbols:   mul0_2, add0_2, goal_2

Defined Pair Symbols:   MUL0_2, ADD0_2

Compound Symbols:   c_2, c2_1


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

(5) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   goal(z0, z1) -> mul0(z0, z1)

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   mul0(Cons(z0, z1), z2) -> add0(mul0(z1, z2), z2)
   mul0(Nil, z0) -> Nil
   add0(Cons(z0, z1), z2) -> add0(z1, Cons(S, z2))
   add0(Nil, z0) -> z0
Tuples:
   MUL0(Cons(z0, z1), z2) -> c(ADD0(mul0(z1, z2), z2), MUL0(z1, z2))
   ADD0(Cons(z0, z1), z2) -> c2(ADD0(z1, Cons(S, z2)))
S tuples:
   MUL0(Cons(z0, z1), z2) -> c(ADD0(mul0(z1, z2), z2), MUL0(z1, z2))
   ADD0(Cons(z0, z1), z2) -> c2(ADD0(z1, Cons(S, z2)))
K tuples:none
Defined Rule Symbols:   mul0_2, add0_2

Defined Pair Symbols:   MUL0_2, ADD0_2

Compound Symbols:   c_2, c2_1


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

(7) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   MUL0(Cons(z0, z1), z2) -> c(ADD0(mul0(z1, z2), z2), MUL0(z1, z2))
We considered the (Usable) Rules:none
And the Tuples:
   MUL0(Cons(z0, z1), z2) -> c(ADD0(mul0(z1, z2), z2), MUL0(z1, z2))
   ADD0(Cons(z0, z1), z2) -> c2(ADD0(z1, Cons(S, z2)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(ADD0(x_1, x_2)) = 0
   POL(Cons(x_1, x_2)) = [1] + x_1 + x_2
   POL(MUL0(x_1, x_2)) = x_1
   POL(Nil) = [1]
   POL(S) = [1]
   POL(add0(x_1, x_2)) = [1] + x_1 + x_2
   POL(c(x_1, x_2)) = x_1 + x_2
   POL(c2(x_1)) = x_1
   POL(mul0(x_1, x_2)) = [1] + x_1 + x_2

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   mul0(Cons(z0, z1), z2) -> add0(mul0(z1, z2), z2)
   mul0(Nil, z0) -> Nil
   add0(Cons(z0, z1), z2) -> add0(z1, Cons(S, z2))
   add0(Nil, z0) -> z0
Tuples:
   MUL0(Cons(z0, z1), z2) -> c(ADD0(mul0(z1, z2), z2), MUL0(z1, z2))
   ADD0(Cons(z0, z1), z2) -> c2(ADD0(z1, Cons(S, z2)))
S tuples:
   ADD0(Cons(z0, z1), z2) -> c2(ADD0(z1, Cons(S, z2)))
K tuples:
   MUL0(Cons(z0, z1), z2) -> c(ADD0(mul0(z1, z2), z2), MUL0(z1, z2))
Defined Rule Symbols:   mul0_2, add0_2

Defined Pair Symbols:   MUL0_2, ADD0_2

Compound Symbols:   c_2, c2_1


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

(9) CdtRuleRemovalProof (UPPER BOUND(ADD(n^3)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   ADD0(Cons(z0, z1), z2) -> c2(ADD0(z1, Cons(S, z2)))
We considered the (Usable) Rules:
   mul0(Nil, z0) -> Nil
   add0(Cons(z0, z1), z2) -> add0(z1, Cons(S, z2))
   add0(Nil, z0) -> z0
   mul0(Cons(z0, z1), z2) -> add0(mul0(z1, z2), z2)
And the Tuples:
   MUL0(Cons(z0, z1), z2) -> c(ADD0(mul0(z1, z2), z2), MUL0(z1, z2))
   ADD0(Cons(z0, z1), z2) -> c2(ADD0(z1, Cons(S, z2)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(ADD0(x_1, x_2)) = x_1
   POL(Cons(x_1, x_2)) = [1] + x_2
   POL(MUL0(x_1, x_2)) = x_1^2*x_2 + x_1*x_2^2
   POL(Nil) = 0
   POL(S) = 0
   POL(add0(x_1, x_2)) = x_1 + x_2
   POL(c(x_1, x_2)) = x_1 + x_2
   POL(c2(x_1)) = x_1
   POL(mul0(x_1, x_2)) = x_2 + x_1*x_2

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

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   mul0(Cons(z0, z1), z2) -> add0(mul0(z1, z2), z2)
   mul0(Nil, z0) -> Nil
   add0(Cons(z0, z1), z2) -> add0(z1, Cons(S, z2))
   add0(Nil, z0) -> z0
Tuples:
   MUL0(Cons(z0, z1), z2) -> c(ADD0(mul0(z1, z2), z2), MUL0(z1, z2))
   ADD0(Cons(z0, z1), z2) -> c2(ADD0(z1, Cons(S, z2)))
S tuples:none
K tuples:
   MUL0(Cons(z0, z1), z2) -> c(ADD0(mul0(z1, z2), z2), MUL0(z1, z2))
   ADD0(Cons(z0, z1), z2) -> c2(ADD0(z1, Cons(S, z2)))
Defined Rule Symbols:   mul0_2, add0_2

Defined Pair Symbols:   MUL0_2, ADD0_2

Compound Symbols:   c_2, c2_1


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

(11) SIsEmptyProof (BOTH BOUNDS(ID, ID))
The set S is empty
----------------------------------------

(12)
BOUNDS(1, 1)

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

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

(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   mul0(Cons(z0, z1), z2) -> add0(mul0(z1, z2), z2)
   mul0(Nil, z0) -> Nil
   add0(Cons(z0, z1), z2) -> add0(z1, Cons(S, z2))
   add0(Nil, z0) -> z0
   goal(z0, z1) -> mul0(z0, z1)
Tuples:
   MUL0(Cons(z0, z1), z2) -> c(ADD0(mul0(z1, z2), z2), MUL0(z1, z2))
   MUL0(Nil, z0) -> c1
   ADD0(Cons(z0, z1), z2) -> c2(ADD0(z1, Cons(S, z2)))
   ADD0(Nil, z0) -> c3
   GOAL(z0, z1) -> c4(MUL0(z0, z1))
S tuples:
   MUL0(Cons(z0, z1), z2) -> c(ADD0(mul0(z1, z2), z2), MUL0(z1, z2))
   MUL0(Nil, z0) -> c1
   ADD0(Cons(z0, z1), z2) -> c2(ADD0(z1, Cons(S, z2)))
   ADD0(Nil, z0) -> c3
   GOAL(z0, z1) -> c4(MUL0(z0, z1))
K tuples:none
Defined Rule Symbols:   mul0_2, add0_2, goal_2

Defined Pair Symbols:   MUL0_2, ADD0_2, GOAL_2

Compound Symbols:   c_2, c1, c2_1, c3, c4_1


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

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

(16)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^3, INF).


The TRS R consists of the following rules:

   MUL0(Cons(z0, z1), z2) -> c(ADD0(mul0(z1, z2), z2), MUL0(z1, z2))
   MUL0(Nil, z0) -> c1
   ADD0(Cons(z0, z1), z2) -> c2(ADD0(z1, Cons(S, z2)))
   ADD0(Nil, z0) -> c3
   GOAL(z0, z1) -> c4(MUL0(z0, z1))

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

   mul0(Cons(z0, z1), z2) -> add0(mul0(z1, z2), z2)
   mul0(Nil, z0) -> Nil
   add0(Cons(z0, z1), z2) -> add0(z1, Cons(S, z2))
   add0(Nil, z0) -> z0
   goal(z0, z1) -> mul0(z0, z1)

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

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

(18)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^3, INF).


The TRS R consists of the following rules:

   MUL0(Cons(z0, z1), z2) -> c(ADD0(mul0(z1, z2), z2), MUL0(z1, z2))
   MUL0(Nil, z0) -> c1
   ADD0(Cons(z0, z1), z2) -> c2(ADD0(z1, Cons(S, z2)))
   ADD0(Nil, z0) -> c3
   GOAL(z0, z1) -> c4(MUL0(z0, z1))

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

   mul0(Cons(z0, z1), z2) -> add0(mul0(z1, z2), z2)
   mul0(Nil, z0) -> Nil
   add0(Cons(z0, z1), z2) -> add0(z1, Cons(S, z2))
   add0(Nil, z0) -> z0
   goal(z0, z1) -> mul0(z0, z1)

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

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

(20)
Obligation:
Innermost TRS:
Rules:
MUL0(Cons(z0, z1), z2) -> c(ADD0(mul0(z1, z2), z2), MUL0(z1, z2))
MUL0(Nil, z0) -> c1
ADD0(Cons(z0, z1), z2) -> c2(ADD0(z1, Cons(S, z2)))
ADD0(Nil, z0) -> c3
GOAL(z0, z1) -> c4(MUL0(z0, z1))
mul0(Cons(z0, z1), z2) -> add0(mul0(z1, z2), z2)
mul0(Nil, z0) -> Nil
add0(Cons(z0, z1), z2) -> add0(z1, Cons(S, z2))
add0(Nil, z0) -> z0
goal(z0, z1) -> mul0(z0, z1)

Types:
MUL0 :: Cons:Nil -> Cons:Nil -> c:c1
Cons :: S -> Cons:Nil -> Cons:Nil
c :: c2:c3 -> c:c1 -> c:c1
ADD0 :: Cons:Nil -> Cons:Nil -> c2:c3
mul0 :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
c1 :: c:c1
c2 :: c2:c3 -> c2:c3
S :: S
c3 :: c2:c3
GOAL :: Cons:Nil -> Cons:Nil -> c4
c4 :: c:c1 -> c4
add0 :: Cons:Nil -> Cons:Nil -> Cons:Nil
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil
hole_c:c11_5 :: c:c1
hole_Cons:Nil2_5 :: Cons:Nil
hole_S3_5 :: S
hole_c2:c34_5 :: c2:c3
hole_c45_5 :: c4
gen_c:c16_5 :: Nat -> c:c1
gen_Cons:Nil7_5 :: Nat -> Cons:Nil
gen_c2:c38_5 :: Nat -> c2:c3

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

(21) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
MUL0, ADD0, mul0, add0

They will be analysed ascendingly in the following order:
ADD0 < MUL0
mul0 < MUL0
add0 < mul0

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

(22)
Obligation:
Innermost TRS:
Rules:
MUL0(Cons(z0, z1), z2) -> c(ADD0(mul0(z1, z2), z2), MUL0(z1, z2))
MUL0(Nil, z0) -> c1
ADD0(Cons(z0, z1), z2) -> c2(ADD0(z1, Cons(S, z2)))
ADD0(Nil, z0) -> c3
GOAL(z0, z1) -> c4(MUL0(z0, z1))
mul0(Cons(z0, z1), z2) -> add0(mul0(z1, z2), z2)
mul0(Nil, z0) -> Nil
add0(Cons(z0, z1), z2) -> add0(z1, Cons(S, z2))
add0(Nil, z0) -> z0
goal(z0, z1) -> mul0(z0, z1)

Types:
MUL0 :: Cons:Nil -> Cons:Nil -> c:c1
Cons :: S -> Cons:Nil -> Cons:Nil
c :: c2:c3 -> c:c1 -> c:c1
ADD0 :: Cons:Nil -> Cons:Nil -> c2:c3
mul0 :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
c1 :: c:c1
c2 :: c2:c3 -> c2:c3
S :: S
c3 :: c2:c3
GOAL :: Cons:Nil -> Cons:Nil -> c4
c4 :: c:c1 -> c4
add0 :: Cons:Nil -> Cons:Nil -> Cons:Nil
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil
hole_c:c11_5 :: c:c1
hole_Cons:Nil2_5 :: Cons:Nil
hole_S3_5 :: S
hole_c2:c34_5 :: c2:c3
hole_c45_5 :: c4
gen_c:c16_5 :: Nat -> c:c1
gen_Cons:Nil7_5 :: Nat -> Cons:Nil
gen_c2:c38_5 :: Nat -> c2:c3


Generator Equations:
gen_c:c16_5(0) <=> c1
gen_c:c16_5(+(x, 1)) <=> c(c3, gen_c:c16_5(x))
gen_Cons:Nil7_5(0) <=> Nil
gen_Cons:Nil7_5(+(x, 1)) <=> Cons(S, gen_Cons:Nil7_5(x))
gen_c2:c38_5(0) <=> c3
gen_c2:c38_5(+(x, 1)) <=> c2(gen_c2:c38_5(x))


The following defined symbols remain to be analysed:
ADD0, MUL0, mul0, add0

They will be analysed ascendingly in the following order:
ADD0 < MUL0
mul0 < MUL0
add0 < mul0

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

(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
ADD0(gen_Cons:Nil7_5(n10_5), gen_Cons:Nil7_5(b)) -> gen_c2:c38_5(n10_5), rt in Omega(1 + n10_5)

Induction Base:
ADD0(gen_Cons:Nil7_5(0), gen_Cons:Nil7_5(b)) ->_R^Omega(1)
c3

Induction Step:
ADD0(gen_Cons:Nil7_5(+(n10_5, 1)), gen_Cons:Nil7_5(b)) ->_R^Omega(1)
c2(ADD0(gen_Cons:Nil7_5(n10_5), Cons(S, gen_Cons:Nil7_5(b)))) ->_IH
c2(gen_c2:c38_5(c11_5))

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

(24)
Complex Obligation (BEST)

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

(25)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
MUL0(Cons(z0, z1), z2) -> c(ADD0(mul0(z1, z2), z2), MUL0(z1, z2))
MUL0(Nil, z0) -> c1
ADD0(Cons(z0, z1), z2) -> c2(ADD0(z1, Cons(S, z2)))
ADD0(Nil, z0) -> c3
GOAL(z0, z1) -> c4(MUL0(z0, z1))
mul0(Cons(z0, z1), z2) -> add0(mul0(z1, z2), z2)
mul0(Nil, z0) -> Nil
add0(Cons(z0, z1), z2) -> add0(z1, Cons(S, z2))
add0(Nil, z0) -> z0
goal(z0, z1) -> mul0(z0, z1)

Types:
MUL0 :: Cons:Nil -> Cons:Nil -> c:c1
Cons :: S -> Cons:Nil -> Cons:Nil
c :: c2:c3 -> c:c1 -> c:c1
ADD0 :: Cons:Nil -> Cons:Nil -> c2:c3
mul0 :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
c1 :: c:c1
c2 :: c2:c3 -> c2:c3
S :: S
c3 :: c2:c3
GOAL :: Cons:Nil -> Cons:Nil -> c4
c4 :: c:c1 -> c4
add0 :: Cons:Nil -> Cons:Nil -> Cons:Nil
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil
hole_c:c11_5 :: c:c1
hole_Cons:Nil2_5 :: Cons:Nil
hole_S3_5 :: S
hole_c2:c34_5 :: c2:c3
hole_c45_5 :: c4
gen_c:c16_5 :: Nat -> c:c1
gen_Cons:Nil7_5 :: Nat -> Cons:Nil
gen_c2:c38_5 :: Nat -> c2:c3


Generator Equations:
gen_c:c16_5(0) <=> c1
gen_c:c16_5(+(x, 1)) <=> c(c3, gen_c:c16_5(x))
gen_Cons:Nil7_5(0) <=> Nil
gen_Cons:Nil7_5(+(x, 1)) <=> Cons(S, gen_Cons:Nil7_5(x))
gen_c2:c38_5(0) <=> c3
gen_c2:c38_5(+(x, 1)) <=> c2(gen_c2:c38_5(x))


The following defined symbols remain to be analysed:
ADD0, MUL0, mul0, add0

They will be analysed ascendingly in the following order:
ADD0 < MUL0
mul0 < MUL0
add0 < mul0

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

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

(27)
BOUNDS(n^1, INF)

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

(28)
Obligation:
Innermost TRS:
Rules:
MUL0(Cons(z0, z1), z2) -> c(ADD0(mul0(z1, z2), z2), MUL0(z1, z2))
MUL0(Nil, z0) -> c1
ADD0(Cons(z0, z1), z2) -> c2(ADD0(z1, Cons(S, z2)))
ADD0(Nil, z0) -> c3
GOAL(z0, z1) -> c4(MUL0(z0, z1))
mul0(Cons(z0, z1), z2) -> add0(mul0(z1, z2), z2)
mul0(Nil, z0) -> Nil
add0(Cons(z0, z1), z2) -> add0(z1, Cons(S, z2))
add0(Nil, z0) -> z0
goal(z0, z1) -> mul0(z0, z1)

Types:
MUL0 :: Cons:Nil -> Cons:Nil -> c:c1
Cons :: S -> Cons:Nil -> Cons:Nil
c :: c2:c3 -> c:c1 -> c:c1
ADD0 :: Cons:Nil -> Cons:Nil -> c2:c3
mul0 :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
c1 :: c:c1
c2 :: c2:c3 -> c2:c3
S :: S
c3 :: c2:c3
GOAL :: Cons:Nil -> Cons:Nil -> c4
c4 :: c:c1 -> c4
add0 :: Cons:Nil -> Cons:Nil -> Cons:Nil
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil
hole_c:c11_5 :: c:c1
hole_Cons:Nil2_5 :: Cons:Nil
hole_S3_5 :: S
hole_c2:c34_5 :: c2:c3
hole_c45_5 :: c4
gen_c:c16_5 :: Nat -> c:c1
gen_Cons:Nil7_5 :: Nat -> Cons:Nil
gen_c2:c38_5 :: Nat -> c2:c3


Lemmas:
ADD0(gen_Cons:Nil7_5(n10_5), gen_Cons:Nil7_5(b)) -> gen_c2:c38_5(n10_5), rt in Omega(1 + n10_5)


Generator Equations:
gen_c:c16_5(0) <=> c1
gen_c:c16_5(+(x, 1)) <=> c(c3, gen_c:c16_5(x))
gen_Cons:Nil7_5(0) <=> Nil
gen_Cons:Nil7_5(+(x, 1)) <=> Cons(S, gen_Cons:Nil7_5(x))
gen_c2:c38_5(0) <=> c3
gen_c2:c38_5(+(x, 1)) <=> c2(gen_c2:c38_5(x))


The following defined symbols remain to be analysed:
add0, MUL0, mul0

They will be analysed ascendingly in the following order:
mul0 < MUL0
add0 < mul0

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

(29) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
add0(gen_Cons:Nil7_5(n471_5), gen_Cons:Nil7_5(b)) -> gen_Cons:Nil7_5(+(n471_5, b)), rt in Omega(0)

Induction Base:
add0(gen_Cons:Nil7_5(0), gen_Cons:Nil7_5(b)) ->_R^Omega(0)
gen_Cons:Nil7_5(b)

Induction Step:
add0(gen_Cons:Nil7_5(+(n471_5, 1)), gen_Cons:Nil7_5(b)) ->_R^Omega(0)
add0(gen_Cons:Nil7_5(n471_5), Cons(S, gen_Cons:Nil7_5(b))) ->_IH
gen_Cons:Nil7_5(+(+(b, 1), c472_5))

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

(30)
Obligation:
Innermost TRS:
Rules:
MUL0(Cons(z0, z1), z2) -> c(ADD0(mul0(z1, z2), z2), MUL0(z1, z2))
MUL0(Nil, z0) -> c1
ADD0(Cons(z0, z1), z2) -> c2(ADD0(z1, Cons(S, z2)))
ADD0(Nil, z0) -> c3
GOAL(z0, z1) -> c4(MUL0(z0, z1))
mul0(Cons(z0, z1), z2) -> add0(mul0(z1, z2), z2)
mul0(Nil, z0) -> Nil
add0(Cons(z0, z1), z2) -> add0(z1, Cons(S, z2))
add0(Nil, z0) -> z0
goal(z0, z1) -> mul0(z0, z1)

Types:
MUL0 :: Cons:Nil -> Cons:Nil -> c:c1
Cons :: S -> Cons:Nil -> Cons:Nil
c :: c2:c3 -> c:c1 -> c:c1
ADD0 :: Cons:Nil -> Cons:Nil -> c2:c3
mul0 :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
c1 :: c:c1
c2 :: c2:c3 -> c2:c3
S :: S
c3 :: c2:c3
GOAL :: Cons:Nil -> Cons:Nil -> c4
c4 :: c:c1 -> c4
add0 :: Cons:Nil -> Cons:Nil -> Cons:Nil
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil
hole_c:c11_5 :: c:c1
hole_Cons:Nil2_5 :: Cons:Nil
hole_S3_5 :: S
hole_c2:c34_5 :: c2:c3
hole_c45_5 :: c4
gen_c:c16_5 :: Nat -> c:c1
gen_Cons:Nil7_5 :: Nat -> Cons:Nil
gen_c2:c38_5 :: Nat -> c2:c3


Lemmas:
ADD0(gen_Cons:Nil7_5(n10_5), gen_Cons:Nil7_5(b)) -> gen_c2:c38_5(n10_5), rt in Omega(1 + n10_5)
add0(gen_Cons:Nil7_5(n471_5), gen_Cons:Nil7_5(b)) -> gen_Cons:Nil7_5(+(n471_5, b)), rt in Omega(0)


Generator Equations:
gen_c:c16_5(0) <=> c1
gen_c:c16_5(+(x, 1)) <=> c(c3, gen_c:c16_5(x))
gen_Cons:Nil7_5(0) <=> Nil
gen_Cons:Nil7_5(+(x, 1)) <=> Cons(S, gen_Cons:Nil7_5(x))
gen_c2:c38_5(0) <=> c3
gen_c2:c38_5(+(x, 1)) <=> c2(gen_c2:c38_5(x))


The following defined symbols remain to be analysed:
mul0, MUL0

They will be analysed ascendingly in the following order:
mul0 < MUL0

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

(31) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
mul0(gen_Cons:Nil7_5(n1349_5), gen_Cons:Nil7_5(b)) -> gen_Cons:Nil7_5(*(n1349_5, b)), rt in Omega(0)

Induction Base:
mul0(gen_Cons:Nil7_5(0), gen_Cons:Nil7_5(b)) ->_R^Omega(0)
Nil

Induction Step:
mul0(gen_Cons:Nil7_5(+(n1349_5, 1)), gen_Cons:Nil7_5(b)) ->_R^Omega(0)
add0(mul0(gen_Cons:Nil7_5(n1349_5), gen_Cons:Nil7_5(b)), gen_Cons:Nil7_5(b)) ->_IH
add0(gen_Cons:Nil7_5(*(c1350_5, b)), gen_Cons:Nil7_5(b)) ->_L^Omega(0)
gen_Cons:Nil7_5(+(*(n1349_5, b), b))

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

(32)
Obligation:
Innermost TRS:
Rules:
MUL0(Cons(z0, z1), z2) -> c(ADD0(mul0(z1, z2), z2), MUL0(z1, z2))
MUL0(Nil, z0) -> c1
ADD0(Cons(z0, z1), z2) -> c2(ADD0(z1, Cons(S, z2)))
ADD0(Nil, z0) -> c3
GOAL(z0, z1) -> c4(MUL0(z0, z1))
mul0(Cons(z0, z1), z2) -> add0(mul0(z1, z2), z2)
mul0(Nil, z0) -> Nil
add0(Cons(z0, z1), z2) -> add0(z1, Cons(S, z2))
add0(Nil, z0) -> z0
goal(z0, z1) -> mul0(z0, z1)

Types:
MUL0 :: Cons:Nil -> Cons:Nil -> c:c1
Cons :: S -> Cons:Nil -> Cons:Nil
c :: c2:c3 -> c:c1 -> c:c1
ADD0 :: Cons:Nil -> Cons:Nil -> c2:c3
mul0 :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
c1 :: c:c1
c2 :: c2:c3 -> c2:c3
S :: S
c3 :: c2:c3
GOAL :: Cons:Nil -> Cons:Nil -> c4
c4 :: c:c1 -> c4
add0 :: Cons:Nil -> Cons:Nil -> Cons:Nil
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil
hole_c:c11_5 :: c:c1
hole_Cons:Nil2_5 :: Cons:Nil
hole_S3_5 :: S
hole_c2:c34_5 :: c2:c3
hole_c45_5 :: c4
gen_c:c16_5 :: Nat -> c:c1
gen_Cons:Nil7_5 :: Nat -> Cons:Nil
gen_c2:c38_5 :: Nat -> c2:c3


Lemmas:
ADD0(gen_Cons:Nil7_5(n10_5), gen_Cons:Nil7_5(b)) -> gen_c2:c38_5(n10_5), rt in Omega(1 + n10_5)
add0(gen_Cons:Nil7_5(n471_5), gen_Cons:Nil7_5(b)) -> gen_Cons:Nil7_5(+(n471_5, b)), rt in Omega(0)
mul0(gen_Cons:Nil7_5(n1349_5), gen_Cons:Nil7_5(b)) -> gen_Cons:Nil7_5(*(n1349_5, b)), rt in Omega(0)


Generator Equations:
gen_c:c16_5(0) <=> c1
gen_c:c16_5(+(x, 1)) <=> c(c3, gen_c:c16_5(x))
gen_Cons:Nil7_5(0) <=> Nil
gen_Cons:Nil7_5(+(x, 1)) <=> Cons(S, gen_Cons:Nil7_5(x))
gen_c2:c38_5(0) <=> c3
gen_c2:c38_5(+(x, 1)) <=> c2(gen_c2:c38_5(x))


The following defined symbols remain to be analysed:
MUL0
----------------------------------------

(33) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
MUL0(gen_Cons:Nil7_5(+(1, n2336_5)), gen_Cons:Nil7_5(b)) -> *9_5, rt in Omega(b*n2336_5 + b*n2336_5^2 + n2336_5)

Induction Base:
MUL0(gen_Cons:Nil7_5(+(1, 0)), gen_Cons:Nil7_5(b))

Induction Step:
MUL0(gen_Cons:Nil7_5(+(1, +(n2336_5, 1))), gen_Cons:Nil7_5(b)) ->_R^Omega(1)
c(ADD0(mul0(gen_Cons:Nil7_5(+(1, n2336_5)), gen_Cons:Nil7_5(b)), gen_Cons:Nil7_5(b)), MUL0(gen_Cons:Nil7_5(+(1, n2336_5)), gen_Cons:Nil7_5(b))) ->_L^Omega(0)
c(ADD0(gen_Cons:Nil7_5(*(+(1, n2336_5), b)), gen_Cons:Nil7_5(b)), MUL0(gen_Cons:Nil7_5(+(1, n2336_5)), gen_Cons:Nil7_5(b))) ->_L^Omega(1 + b + b*n2336_5)
c(gen_c2:c38_5(+(b, *(n2336_5, b))), MUL0(gen_Cons:Nil7_5(+(1, n2336_5)), gen_Cons:Nil7_5(b))) ->_IH
c(gen_c2:c38_5(+(b, *(n2336_5, b))), *9_5)

We have rt in Omega(n^3) and sz in O(n). Thus, we have irc_R in Omega(n^3).
----------------------------------------

(34)
Obligation:
Proved the lower bound n^3 for the following obligation:

Innermost TRS:
Rules:
MUL0(Cons(z0, z1), z2) -> c(ADD0(mul0(z1, z2), z2), MUL0(z1, z2))
MUL0(Nil, z0) -> c1
ADD0(Cons(z0, z1), z2) -> c2(ADD0(z1, Cons(S, z2)))
ADD0(Nil, z0) -> c3
GOAL(z0, z1) -> c4(MUL0(z0, z1))
mul0(Cons(z0, z1), z2) -> add0(mul0(z1, z2), z2)
mul0(Nil, z0) -> Nil
add0(Cons(z0, z1), z2) -> add0(z1, Cons(S, z2))
add0(Nil, z0) -> z0
goal(z0, z1) -> mul0(z0, z1)

Types:
MUL0 :: Cons:Nil -> Cons:Nil -> c:c1
Cons :: S -> Cons:Nil -> Cons:Nil
c :: c2:c3 -> c:c1 -> c:c1
ADD0 :: Cons:Nil -> Cons:Nil -> c2:c3
mul0 :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
c1 :: c:c1
c2 :: c2:c3 -> c2:c3
S :: S
c3 :: c2:c3
GOAL :: Cons:Nil -> Cons:Nil -> c4
c4 :: c:c1 -> c4
add0 :: Cons:Nil -> Cons:Nil -> Cons:Nil
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil
hole_c:c11_5 :: c:c1
hole_Cons:Nil2_5 :: Cons:Nil
hole_S3_5 :: S
hole_c2:c34_5 :: c2:c3
hole_c45_5 :: c4
gen_c:c16_5 :: Nat -> c:c1
gen_Cons:Nil7_5 :: Nat -> Cons:Nil
gen_c2:c38_5 :: Nat -> c2:c3


Lemmas:
ADD0(gen_Cons:Nil7_5(n10_5), gen_Cons:Nil7_5(b)) -> gen_c2:c38_5(n10_5), rt in Omega(1 + n10_5)
add0(gen_Cons:Nil7_5(n471_5), gen_Cons:Nil7_5(b)) -> gen_Cons:Nil7_5(+(n471_5, b)), rt in Omega(0)
mul0(gen_Cons:Nil7_5(n1349_5), gen_Cons:Nil7_5(b)) -> gen_Cons:Nil7_5(*(n1349_5, b)), rt in Omega(0)


Generator Equations:
gen_c:c16_5(0) <=> c1
gen_c:c16_5(+(x, 1)) <=> c(c3, gen_c:c16_5(x))
gen_Cons:Nil7_5(0) <=> Nil
gen_Cons:Nil7_5(+(x, 1)) <=> Cons(S, gen_Cons:Nil7_5(x))
gen_c2:c38_5(0) <=> c3
gen_c2:c38_5(+(x, 1)) <=> c2(gen_c2:c38_5(x))


The following defined symbols remain to be analysed:
MUL0
----------------------------------------

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

(36)
BOUNDS(n^3, INF)