/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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


WORST_CASE(Omega(n^1), O(n^1))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^1).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 190 ms]
(4) CpxRelTRS
(5) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(6) CdtProblem
    (7) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (8) CdtProblem
    (9) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CdtProblem
    (11) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (12) CdtProblem
    (13) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) CdtProblem
    (15) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 67 ms]
    (16) CdtProblem
    (17) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) BOUNDS(1, 1)
(19) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(20) CpxRelTRS
    (21) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (22) typed CpxTrs
    (23) OrderProof [LOWER BOUND(ID), 0 ms]
    (24) typed CpxTrs
    (25) RewriteLemmaProof [LOWER BOUND(ID), 315 ms]
    (26) proven lower bound
    (27) LowerBoundPropagationProof [FINISHED, 0 ms]
    (28) BOUNDS(n^1, INF)


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

(0)
Obligation:
The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^1).


The TRS R consists of the following rules:

   a(lambda(x), y) -> lambda(a(x, 1))
   a(lambda(x), y) -> lambda(a(x, a(y, t)))
   a(a(x, y), z) -> a(x, a(y, z))
   lambda(x) -> x
   a(x, y) -> x
   a(x, y) -> y

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

(1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID))
The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem:

   encArg(1) -> 1
   encArg(t) -> t
   encArg(cons_a(x_1, x_2)) -> a(encArg(x_1), encArg(x_2))
   encArg(cons_lambda(x_1)) -> lambda(encArg(x_1))
   encode_a(x_1, x_2) -> a(encArg(x_1), encArg(x_2))
   encode_lambda(x_1) -> lambda(encArg(x_1))
   encode_1 -> 1
   encode_t -> t


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

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


The TRS R consists of the following rules:

   a(lambda(x), y) -> lambda(a(x, 1))
   a(lambda(x), y) -> lambda(a(x, a(y, t)))
   a(a(x, y), z) -> a(x, a(y, z))
   lambda(x) -> x
   a(x, y) -> x
   a(x, y) -> y

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

   encArg(1) -> 1
   encArg(t) -> t
   encArg(cons_a(x_1, x_2)) -> a(encArg(x_1), encArg(x_2))
   encArg(cons_lambda(x_1)) -> lambda(encArg(x_1))
   encode_a(x_1, x_2) -> a(encArg(x_1), encArg(x_2))
   encode_lambda(x_1) -> lambda(encArg(x_1))
   encode_1 -> 1
   encode_t -> t

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

(3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
----------------------------------------

(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1).


The TRS R consists of the following rules:

   a(lambda(x), y) -> lambda(a(x, 1))
   a(lambda(x), y) -> lambda(a(x, a(y, t)))
   a(a(x, y), z) -> a(x, a(y, z))
   lambda(x) -> x
   a(x, y) -> x
   a(x, y) -> y

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

   encArg(1) -> 1
   encArg(t) -> t
   encArg(cons_a(x_1, x_2)) -> a(encArg(x_1), encArg(x_2))
   encArg(cons_lambda(x_1)) -> lambda(encArg(x_1))
   encode_a(x_1, x_2) -> a(encArg(x_1), encArg(x_2))
   encode_lambda(x_1) -> lambda(encArg(x_1))
   encode_1 -> 1
   encode_t -> t

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

(5) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS to CDT
----------------------------------------

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(1) -> 1
   encArg(t) -> t
   encArg(cons_a(z0, z1)) -> a(encArg(z0), encArg(z1))
   encArg(cons_lambda(z0)) -> lambda(encArg(z0))
   encode_a(z0, z1) -> a(encArg(z0), encArg(z1))
   encode_lambda(z0) -> lambda(encArg(z0))
   encode_1 -> 1
   encode_t -> t
   a(lambda(z0), z1) -> lambda(a(z0, 1))
   a(lambda(z0), z1) -> lambda(a(z0, a(z1, t)))
   a(a(z0, z1), z2) -> a(z0, a(z1, z2))
   a(z0, z1) -> z0
   a(z0, z1) -> z1
   lambda(z0) -> z0
Tuples:
   ENCARG(1) -> c
   ENCARG(t) -> c1
   ENCARG(cons_a(z0, z1)) -> c2(A(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_lambda(z0)) -> c3(LAMBDA(encArg(z0)), ENCARG(z0))
   ENCODE_A(z0, z1) -> c4(A(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_LAMBDA(z0) -> c5(LAMBDA(encArg(z0)), ENCARG(z0))
   ENCODE_1 -> c6
   ENCODE_T -> c7
   A(lambda(z0), z1) -> c8(LAMBDA(a(z0, 1)), A(z0, 1))
   A(lambda(z0), z1) -> c9(LAMBDA(a(z0, a(z1, t))), A(z0, a(z1, t)), A(z1, t))
   A(a(z0, z1), z2) -> c10(A(z0, a(z1, z2)), A(z1, z2))
   A(z0, z1) -> c11
   A(z0, z1) -> c12
   LAMBDA(z0) -> c13
S tuples:
   A(lambda(z0), z1) -> c8(LAMBDA(a(z0, 1)), A(z0, 1))
   A(lambda(z0), z1) -> c9(LAMBDA(a(z0, a(z1, t))), A(z0, a(z1, t)), A(z1, t))
   A(a(z0, z1), z2) -> c10(A(z0, a(z1, z2)), A(z1, z2))
   A(z0, z1) -> c11
   A(z0, z1) -> c12
   LAMBDA(z0) -> c13
K tuples:none
Defined Rule Symbols:   a_2, lambda_1, encArg_1, encode_a_2, encode_lambda_1, encode_1, encode_t

Defined Pair Symbols:   ENCARG_1, ENCODE_A_2, ENCODE_LAMBDA_1, ENCODE_1, ENCODE_T, A_2, LAMBDA_1

Compound Symbols:   c, c1, c2_3, c3_2, c4_3, c5_2, c6, c7, c8_2, c9_3, c10_2, c11, c12, c13


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

(7) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 3 leading nodes:
   A(lambda(z0), z1) -> c8(LAMBDA(a(z0, 1)), A(z0, 1))
   A(lambda(z0), z1) -> c9(LAMBDA(a(z0, a(z1, t))), A(z0, a(z1, t)), A(z1, t))
   A(a(z0, z1), z2) -> c10(A(z0, a(z1, z2)), A(z1, z2))
Removed 4 trailing nodes:
   ENCODE_1 -> c6
   ENCARG(t) -> c1
   ENCARG(1) -> c
   ENCODE_T -> c7

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(1) -> 1
   encArg(t) -> t
   encArg(cons_a(z0, z1)) -> a(encArg(z0), encArg(z1))
   encArg(cons_lambda(z0)) -> lambda(encArg(z0))
   encode_a(z0, z1) -> a(encArg(z0), encArg(z1))
   encode_lambda(z0) -> lambda(encArg(z0))
   encode_1 -> 1
   encode_t -> t
   a(lambda(z0), z1) -> lambda(a(z0, 1))
   a(lambda(z0), z1) -> lambda(a(z0, a(z1, t)))
   a(a(z0, z1), z2) -> a(z0, a(z1, z2))
   a(z0, z1) -> z0
   a(z0, z1) -> z1
   lambda(z0) -> z0
Tuples:
   ENCARG(cons_a(z0, z1)) -> c2(A(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_lambda(z0)) -> c3(LAMBDA(encArg(z0)), ENCARG(z0))
   ENCODE_A(z0, z1) -> c4(A(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_LAMBDA(z0) -> c5(LAMBDA(encArg(z0)), ENCARG(z0))
   A(z0, z1) -> c11
   A(z0, z1) -> c12
   LAMBDA(z0) -> c13
S tuples:
   A(z0, z1) -> c11
   A(z0, z1) -> c12
   LAMBDA(z0) -> c13
K tuples:none
Defined Rule Symbols:   a_2, lambda_1, encArg_1, encode_a_2, encode_lambda_1, encode_1, encode_t

Defined Pair Symbols:   ENCARG_1, ENCODE_A_2, ENCODE_LAMBDA_1, A_2, LAMBDA_1

Compound Symbols:   c2_3, c3_2, c4_3, c5_2, c11, c12, c13


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

(9) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))
Split RHS of tuples not part of any SCC
----------------------------------------

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(1) -> 1
   encArg(t) -> t
   encArg(cons_a(z0, z1)) -> a(encArg(z0), encArg(z1))
   encArg(cons_lambda(z0)) -> lambda(encArg(z0))
   encode_a(z0, z1) -> a(encArg(z0), encArg(z1))
   encode_lambda(z0) -> lambda(encArg(z0))
   encode_1 -> 1
   encode_t -> t
   a(lambda(z0), z1) -> lambda(a(z0, 1))
   a(lambda(z0), z1) -> lambda(a(z0, a(z1, t)))
   a(a(z0, z1), z2) -> a(z0, a(z1, z2))
   a(z0, z1) -> z0
   a(z0, z1) -> z1
   lambda(z0) -> z0
Tuples:
   ENCARG(cons_a(z0, z1)) -> c2(A(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_lambda(z0)) -> c3(LAMBDA(encArg(z0)), ENCARG(z0))
   A(z0, z1) -> c11
   A(z0, z1) -> c12
   LAMBDA(z0) -> c13
   ENCODE_A(z0, z1) -> c(A(encArg(z0), encArg(z1)))
   ENCODE_A(z0, z1) -> c(ENCARG(z0))
   ENCODE_A(z0, z1) -> c(ENCARG(z1))
   ENCODE_LAMBDA(z0) -> c(LAMBDA(encArg(z0)))
   ENCODE_LAMBDA(z0) -> c(ENCARG(z0))
S tuples:
   A(z0, z1) -> c11
   A(z0, z1) -> c12
   LAMBDA(z0) -> c13
K tuples:none
Defined Rule Symbols:   a_2, lambda_1, encArg_1, encode_a_2, encode_lambda_1, encode_1, encode_t

Defined Pair Symbols:   ENCARG_1, A_2, LAMBDA_1, ENCODE_A_2, ENCODE_LAMBDA_1

Compound Symbols:   c2_3, c3_2, c11, c12, c13, c_1


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

(11) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 3 leading nodes:
   ENCODE_A(z0, z1) -> c(ENCARG(z0))
   ENCODE_A(z0, z1) -> c(ENCARG(z1))
   ENCODE_LAMBDA(z0) -> c(ENCARG(z0))

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

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(1) -> 1
   encArg(t) -> t
   encArg(cons_a(z0, z1)) -> a(encArg(z0), encArg(z1))
   encArg(cons_lambda(z0)) -> lambda(encArg(z0))
   encode_a(z0, z1) -> a(encArg(z0), encArg(z1))
   encode_lambda(z0) -> lambda(encArg(z0))
   encode_1 -> 1
   encode_t -> t
   a(lambda(z0), z1) -> lambda(a(z0, 1))
   a(lambda(z0), z1) -> lambda(a(z0, a(z1, t)))
   a(a(z0, z1), z2) -> a(z0, a(z1, z2))
   a(z0, z1) -> z0
   a(z0, z1) -> z1
   lambda(z0) -> z0
Tuples:
   ENCARG(cons_a(z0, z1)) -> c2(A(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_lambda(z0)) -> c3(LAMBDA(encArg(z0)), ENCARG(z0))
   A(z0, z1) -> c11
   A(z0, z1) -> c12
   LAMBDA(z0) -> c13
   ENCODE_A(z0, z1) -> c(A(encArg(z0), encArg(z1)))
   ENCODE_LAMBDA(z0) -> c(LAMBDA(encArg(z0)))
S tuples:
   A(z0, z1) -> c11
   A(z0, z1) -> c12
   LAMBDA(z0) -> c13
K tuples:none
Defined Rule Symbols:   a_2, lambda_1, encArg_1, encode_a_2, encode_lambda_1, encode_1, encode_t

Defined Pair Symbols:   ENCARG_1, A_2, LAMBDA_1, ENCODE_A_2, ENCODE_LAMBDA_1

Compound Symbols:   c2_3, c3_2, c11, c12, c13, c_1


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

(13) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   encode_a(z0, z1) -> a(encArg(z0), encArg(z1))
   encode_lambda(z0) -> lambda(encArg(z0))
   encode_1 -> 1
   encode_t -> t
   a(lambda(z0), z1) -> lambda(a(z0, 1))
   a(lambda(z0), z1) -> lambda(a(z0, a(z1, t)))
   a(a(z0, z1), z2) -> a(z0, a(z1, z2))

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

(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(1) -> 1
   encArg(t) -> t
   encArg(cons_a(z0, z1)) -> a(encArg(z0), encArg(z1))
   encArg(cons_lambda(z0)) -> lambda(encArg(z0))
   a(z0, z1) -> z0
   a(z0, z1) -> z1
   lambda(z0) -> z0
Tuples:
   ENCARG(cons_a(z0, z1)) -> c2(A(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_lambda(z0)) -> c3(LAMBDA(encArg(z0)), ENCARG(z0))
   A(z0, z1) -> c11
   A(z0, z1) -> c12
   LAMBDA(z0) -> c13
   ENCODE_A(z0, z1) -> c(A(encArg(z0), encArg(z1)))
   ENCODE_LAMBDA(z0) -> c(LAMBDA(encArg(z0)))
S tuples:
   A(z0, z1) -> c11
   A(z0, z1) -> c12
   LAMBDA(z0) -> c13
K tuples:none
Defined Rule Symbols:   encArg_1, a_2, lambda_1

Defined Pair Symbols:   ENCARG_1, A_2, LAMBDA_1, ENCODE_A_2, ENCODE_LAMBDA_1

Compound Symbols:   c2_3, c3_2, c11, c12, c13, c_1


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

(15) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   A(z0, z1) -> c11
   A(z0, z1) -> c12
   LAMBDA(z0) -> c13
We considered the (Usable) Rules:none
And the Tuples:
   ENCARG(cons_a(z0, z1)) -> c2(A(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_lambda(z0)) -> c3(LAMBDA(encArg(z0)), ENCARG(z0))
   A(z0, z1) -> c11
   A(z0, z1) -> c12
   LAMBDA(z0) -> c13
   ENCODE_A(z0, z1) -> c(A(encArg(z0), encArg(z1)))
   ENCODE_LAMBDA(z0) -> c(LAMBDA(encArg(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(1) = [1]
   POL(A(x_1, x_2)) = [1]
   POL(ENCARG(x_1)) = x_1
   POL(ENCODE_A(x_1, x_2)) = [1] + x_2
   POL(ENCODE_LAMBDA(x_1)) = [1]
   POL(LAMBDA(x_1)) = [1]
   POL(a(x_1, x_2)) = x_1 + x_2
   POL(c(x_1)) = x_1
   POL(c11) = 0
   POL(c12) = 0
   POL(c13) = 0
   POL(c2(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(cons_a(x_1, x_2)) = [1] + x_1 + x_2
   POL(cons_lambda(x_1)) = [1] + x_1
   POL(encArg(x_1)) = [1] + x_1
   POL(lambda(x_1)) = x_1
   POL(t) = [1]

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

(16)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(1) -> 1
   encArg(t) -> t
   encArg(cons_a(z0, z1)) -> a(encArg(z0), encArg(z1))
   encArg(cons_lambda(z0)) -> lambda(encArg(z0))
   a(z0, z1) -> z0
   a(z0, z1) -> z1
   lambda(z0) -> z0
Tuples:
   ENCARG(cons_a(z0, z1)) -> c2(A(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_lambda(z0)) -> c3(LAMBDA(encArg(z0)), ENCARG(z0))
   A(z0, z1) -> c11
   A(z0, z1) -> c12
   LAMBDA(z0) -> c13
   ENCODE_A(z0, z1) -> c(A(encArg(z0), encArg(z1)))
   ENCODE_LAMBDA(z0) -> c(LAMBDA(encArg(z0)))
S tuples:none
K tuples:
   A(z0, z1) -> c11
   A(z0, z1) -> c12
   LAMBDA(z0) -> c13
Defined Rule Symbols:   encArg_1, a_2, lambda_1

Defined Pair Symbols:   ENCARG_1, A_2, LAMBDA_1, ENCODE_A_2, ENCODE_LAMBDA_1

Compound Symbols:   c2_3, c3_2, c11, c12, c13, c_1


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

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

(18)
BOUNDS(1, 1)

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

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

(20)
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:

   a(lambda(x), y) -> lambda(a(x, 1'))
   a(lambda(x), y) -> lambda(a(x, a(y, t)))
   a(a(x, y), z) -> a(x, a(y, z))
   lambda(x) -> x
   a(x, y) -> x
   a(x, y) -> y

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

   encArg(1') -> 1'
   encArg(t) -> t
   encArg(cons_a(x_1, x_2)) -> a(encArg(x_1), encArg(x_2))
   encArg(cons_lambda(x_1)) -> lambda(encArg(x_1))
   encode_a(x_1, x_2) -> a(encArg(x_1), encArg(x_2))
   encode_lambda(x_1) -> lambda(encArg(x_1))
   encode_1 -> 1'
   encode_t -> t

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

(21) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(22)
Obligation:
Innermost TRS:
Rules:
a(lambda(x), y) -> lambda(a(x, 1'))
a(lambda(x), y) -> lambda(a(x, a(y, t)))
a(a(x, y), z) -> a(x, a(y, z))
lambda(x) -> x
a(x, y) -> x
a(x, y) -> y
encArg(1') -> 1'
encArg(t) -> t
encArg(cons_a(x_1, x_2)) -> a(encArg(x_1), encArg(x_2))
encArg(cons_lambda(x_1)) -> lambda(encArg(x_1))
encode_a(x_1, x_2) -> a(encArg(x_1), encArg(x_2))
encode_lambda(x_1) -> lambda(encArg(x_1))
encode_1 -> 1'
encode_t -> t

Types:
a :: 1':t:cons_a:cons_lambda -> 1':t:cons_a:cons_lambda -> 1':t:cons_a:cons_lambda
lambda :: 1':t:cons_a:cons_lambda -> 1':t:cons_a:cons_lambda
1' :: 1':t:cons_a:cons_lambda
t :: 1':t:cons_a:cons_lambda
encArg :: 1':t:cons_a:cons_lambda -> 1':t:cons_a:cons_lambda
cons_a :: 1':t:cons_a:cons_lambda -> 1':t:cons_a:cons_lambda -> 1':t:cons_a:cons_lambda
cons_lambda :: 1':t:cons_a:cons_lambda -> 1':t:cons_a:cons_lambda
encode_a :: 1':t:cons_a:cons_lambda -> 1':t:cons_a:cons_lambda -> 1':t:cons_a:cons_lambda
encode_lambda :: 1':t:cons_a:cons_lambda -> 1':t:cons_a:cons_lambda
encode_1 :: 1':t:cons_a:cons_lambda
encode_t :: 1':t:cons_a:cons_lambda
hole_1':t:cons_a:cons_lambda1_0 :: 1':t:cons_a:cons_lambda
gen_1':t:cons_a:cons_lambda2_0 :: Nat -> 1':t:cons_a:cons_lambda

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

(23) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
a, encArg

They will be analysed ascendingly in the following order:
a < encArg

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

(24)
Obligation:
Innermost TRS:
Rules:
a(lambda(x), y) -> lambda(a(x, 1'))
a(lambda(x), y) -> lambda(a(x, a(y, t)))
a(a(x, y), z) -> a(x, a(y, z))
lambda(x) -> x
a(x, y) -> x
a(x, y) -> y
encArg(1') -> 1'
encArg(t) -> t
encArg(cons_a(x_1, x_2)) -> a(encArg(x_1), encArg(x_2))
encArg(cons_lambda(x_1)) -> lambda(encArg(x_1))
encode_a(x_1, x_2) -> a(encArg(x_1), encArg(x_2))
encode_lambda(x_1) -> lambda(encArg(x_1))
encode_1 -> 1'
encode_t -> t

Types:
a :: 1':t:cons_a:cons_lambda -> 1':t:cons_a:cons_lambda -> 1':t:cons_a:cons_lambda
lambda :: 1':t:cons_a:cons_lambda -> 1':t:cons_a:cons_lambda
1' :: 1':t:cons_a:cons_lambda
t :: 1':t:cons_a:cons_lambda
encArg :: 1':t:cons_a:cons_lambda -> 1':t:cons_a:cons_lambda
cons_a :: 1':t:cons_a:cons_lambda -> 1':t:cons_a:cons_lambda -> 1':t:cons_a:cons_lambda
cons_lambda :: 1':t:cons_a:cons_lambda -> 1':t:cons_a:cons_lambda
encode_a :: 1':t:cons_a:cons_lambda -> 1':t:cons_a:cons_lambda -> 1':t:cons_a:cons_lambda
encode_lambda :: 1':t:cons_a:cons_lambda -> 1':t:cons_a:cons_lambda
encode_1 :: 1':t:cons_a:cons_lambda
encode_t :: 1':t:cons_a:cons_lambda
hole_1':t:cons_a:cons_lambda1_0 :: 1':t:cons_a:cons_lambda
gen_1':t:cons_a:cons_lambda2_0 :: Nat -> 1':t:cons_a:cons_lambda


Generator Equations:
gen_1':t:cons_a:cons_lambda2_0(0) <=> 1'
gen_1':t:cons_a:cons_lambda2_0(+(x, 1)) <=> cons_a(1', gen_1':t:cons_a:cons_lambda2_0(x))


The following defined symbols remain to be analysed:
a, encArg

They will be analysed ascendingly in the following order:
a < encArg

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(25) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_1':t:cons_a:cons_lambda2_0(n50_0)) -> gen_1':t:cons_a:cons_lambda2_0(0), rt in Omega(n50_0)

Induction Base:
encArg(gen_1':t:cons_a:cons_lambda2_0(0)) ->_R^Omega(0)
1'

Induction Step:
encArg(gen_1':t:cons_a:cons_lambda2_0(+(n50_0, 1))) ->_R^Omega(0)
a(encArg(1'), encArg(gen_1':t:cons_a:cons_lambda2_0(n50_0))) ->_R^Omega(0)
a(1', encArg(gen_1':t:cons_a:cons_lambda2_0(n50_0))) ->_IH
a(1', gen_1':t:cons_a:cons_lambda2_0(0)) ->_R^Omega(1)
1'

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(26)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
a(lambda(x), y) -> lambda(a(x, 1'))
a(lambda(x), y) -> lambda(a(x, a(y, t)))
a(a(x, y), z) -> a(x, a(y, z))
lambda(x) -> x
a(x, y) -> x
a(x, y) -> y
encArg(1') -> 1'
encArg(t) -> t
encArg(cons_a(x_1, x_2)) -> a(encArg(x_1), encArg(x_2))
encArg(cons_lambda(x_1)) -> lambda(encArg(x_1))
encode_a(x_1, x_2) -> a(encArg(x_1), encArg(x_2))
encode_lambda(x_1) -> lambda(encArg(x_1))
encode_1 -> 1'
encode_t -> t

Types:
a :: 1':t:cons_a:cons_lambda -> 1':t:cons_a:cons_lambda -> 1':t:cons_a:cons_lambda
lambda :: 1':t:cons_a:cons_lambda -> 1':t:cons_a:cons_lambda
1' :: 1':t:cons_a:cons_lambda
t :: 1':t:cons_a:cons_lambda
encArg :: 1':t:cons_a:cons_lambda -> 1':t:cons_a:cons_lambda
cons_a :: 1':t:cons_a:cons_lambda -> 1':t:cons_a:cons_lambda -> 1':t:cons_a:cons_lambda
cons_lambda :: 1':t:cons_a:cons_lambda -> 1':t:cons_a:cons_lambda
encode_a :: 1':t:cons_a:cons_lambda -> 1':t:cons_a:cons_lambda -> 1':t:cons_a:cons_lambda
encode_lambda :: 1':t:cons_a:cons_lambda -> 1':t:cons_a:cons_lambda
encode_1 :: 1':t:cons_a:cons_lambda
encode_t :: 1':t:cons_a:cons_lambda
hole_1':t:cons_a:cons_lambda1_0 :: 1':t:cons_a:cons_lambda
gen_1':t:cons_a:cons_lambda2_0 :: Nat -> 1':t:cons_a:cons_lambda


Generator Equations:
gen_1':t:cons_a:cons_lambda2_0(0) <=> 1'
gen_1':t:cons_a:cons_lambda2_0(+(x, 1)) <=> cons_a(1', gen_1':t:cons_a:cons_lambda2_0(x))


The following defined symbols remain to be analysed:
encArg
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(27) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(28)
BOUNDS(n^1, INF)