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


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


WORST_CASE(Omega(n^1), O(n^2))
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^2).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 159 ms]
(4) CpxRelTRS
(5) NonCtorToCtorProof [UPPER BOUND(ID), 0 ms]
(6) CpxRelTRS
    (7) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxRelTRS
    (9) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
    (10) CdtProblem
    (11) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) CdtProblem
    (13) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) CdtProblem
    (15) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) CdtProblem
    (17) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (18) CdtProblem
    (19) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (20) CdtProblem
    (21) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 59 ms]
    (22) CdtProblem
    (23) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 230 ms]
    (24) CdtProblem
    (25) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 106 ms]
    (26) CdtProblem
    (27) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
    (28) BOUNDS(1, 1)
(29) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(30) CpxRelTRS
    (31) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (32) typed CpxTrs
    (33) OrderProof [LOWER BOUND(ID), 0 ms]
    (34) typed CpxTrs
    (35) RewriteLemmaProof [LOWER BOUND(ID), 217 ms]
    (36) proven lower bound
    (37) LowerBoundPropagationProof [FINISHED, 0 ms]
    (38) BOUNDS(n^1, INF)


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

(0)
Obligation:
The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^2).


The TRS R consists of the following rules:

   app(id, x) -> x
   app(plus, 0) -> id
   app(app(plus, app(s, x)), y) -> app(s, app(app(plus, x), y))

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

(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(id) -> id
   encArg(plus) -> plus
   encArg(0) -> 0
   encArg(s) -> s
   encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2))
   encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2))
   encode_id -> id
   encode_plus -> plus
   encode_0 -> 0
   encode_s -> s


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

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


The TRS R consists of the following rules:

   app(id, x) -> x
   app(plus, 0) -> id
   app(app(plus, app(s, x)), y) -> app(s, app(app(plus, x), y))

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

   encArg(id) -> id
   encArg(plus) -> plus
   encArg(0) -> 0
   encArg(s) -> s
   encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2))
   encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2))
   encode_id -> id
   encode_plus -> plus
   encode_0 -> 0
   encode_s -> s

Rewrite Strategy: FULL
----------------------------------------

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

(4)
Obligation:
The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2).


The TRS R consists of the following rules:

   app(id, x) -> x
   app(plus, 0) -> id
   app(app(plus, app(s, x)), y) -> app(s, app(app(plus, x), y))

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

   encArg(id) -> id
   encArg(plus) -> plus
   encArg(0) -> 0
   encArg(s) -> s
   encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2))
   encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2))
   encode_id -> id
   encode_plus -> plus
   encode_0 -> 0
   encode_s -> s

Rewrite Strategy: FULL
----------------------------------------

(5) NonCtorToCtorProof (UPPER BOUND(ID))
transformed non-ctor to ctor-system
----------------------------------------

(6)
Obligation:
The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

   app(id, x) -> x
   app(plus, 0) -> id
   app(c_app(plus, c_app(s, x)), y) -> app(s, app(app(plus, x), y))

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

   encArg(id) -> id
   encArg(plus) -> plus
   encArg(0) -> 0
   encArg(s) -> s
   encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2))
   encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2))
   encode_id -> id
   encode_plus -> plus
   encode_0 -> 0
   encode_s -> s
   app(x0, x1) -> c_app(x0, x1)

Rewrite Strategy: FULL
----------------------------------------

(7) RcToIrcProof (BOTH BOUNDS(ID, ID))
Converted rc-obligation to irc-obligation.

As the TRS is a non-duplicating overlay system, we have rc = irc.
----------------------------------------

(8)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

   app(id, x) -> x
   app(plus, 0) -> id
   app(c_app(plus, c_app(s, x)), y) -> app(s, app(app(plus, x), y))

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

   encArg(id) -> id
   encArg(plus) -> plus
   encArg(0) -> 0
   encArg(s) -> s
   encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2))
   encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2))
   encode_id -> id
   encode_plus -> plus
   encode_0 -> 0
   encode_s -> s
   app(x0, x1) -> c_app(x0, x1)

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

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

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(id) -> id
   encArg(plus) -> plus
   encArg(0) -> 0
   encArg(s) -> s
   encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1))
   encode_app(z0, z1) -> app(encArg(z0), encArg(z1))
   encode_id -> id
   encode_plus -> plus
   encode_0 -> 0
   encode_s -> s
   app(z0, z1) -> c_app(z0, z1)
   app(id, z0) -> z0
   app(plus, 0) -> id
   app(c_app(plus, c_app(s, z0)), z1) -> app(s, app(app(plus, z0), z1))
Tuples:
   ENCARG(id) -> c
   ENCARG(plus) -> c1
   ENCARG(0) -> c2
   ENCARG(s) -> c3
   ENCARG(cons_app(z0, z1)) -> c4(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_APP(z0, z1) -> c5(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_ID -> c6
   ENCODE_PLUS -> c7
   ENCODE_0 -> c8
   ENCODE_S -> c9
   APP(z0, z1) -> c10
   APP(id, z0) -> c11
   APP(plus, 0) -> c12
   APP(c_app(plus, c_app(s, z0)), z1) -> c13(APP(s, app(app(plus, z0), z1)), APP(app(plus, z0), z1), APP(plus, z0))
S tuples:
   APP(id, z0) -> c11
   APP(plus, 0) -> c12
   APP(c_app(plus, c_app(s, z0)), z1) -> c13(APP(s, app(app(plus, z0), z1)), APP(app(plus, z0), z1), APP(plus, z0))
K tuples:none
Defined Rule Symbols:   app_2, encArg_1, encode_app_2, encode_id, encode_plus, encode_0, encode_s

Defined Pair Symbols:   ENCARG_1, ENCODE_APP_2, ENCODE_ID, ENCODE_PLUS, ENCODE_0, ENCODE_S, APP_2

Compound Symbols:   c, c1, c2, c3, c4_3, c5_3, c6, c7, c8, c9, c10, c11, c12, c13_3


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

(11) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 9 trailing nodes:
   ENCARG(s) -> c3
   ENCARG(plus) -> c1
   ENCARG(id) -> c
   ENCARG(0) -> c2
   ENCODE_ID -> c6
   ENCODE_S -> c9
   ENCODE_PLUS -> c7
   APP(z0, z1) -> c10
   ENCODE_0 -> c8

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

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(id) -> id
   encArg(plus) -> plus
   encArg(0) -> 0
   encArg(s) -> s
   encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1))
   encode_app(z0, z1) -> app(encArg(z0), encArg(z1))
   encode_id -> id
   encode_plus -> plus
   encode_0 -> 0
   encode_s -> s
   app(z0, z1) -> c_app(z0, z1)
   app(id, z0) -> z0
   app(plus, 0) -> id
   app(c_app(plus, c_app(s, z0)), z1) -> app(s, app(app(plus, z0), z1))
Tuples:
   ENCARG(cons_app(z0, z1)) -> c4(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_APP(z0, z1) -> c5(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   APP(id, z0) -> c11
   APP(plus, 0) -> c12
   APP(c_app(plus, c_app(s, z0)), z1) -> c13(APP(s, app(app(plus, z0), z1)), APP(app(plus, z0), z1), APP(plus, z0))
S tuples:
   APP(id, z0) -> c11
   APP(plus, 0) -> c12
   APP(c_app(plus, c_app(s, z0)), z1) -> c13(APP(s, app(app(plus, z0), z1)), APP(app(plus, z0), z1), APP(plus, z0))
K tuples:none
Defined Rule Symbols:   app_2, encArg_1, encode_app_2, encode_id, encode_plus, encode_0, encode_s

Defined Pair Symbols:   ENCARG_1, ENCODE_APP_2, APP_2

Compound Symbols:   c4_3, c5_3, c11, c12, c13_3


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

(13) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(id) -> id
   encArg(plus) -> plus
   encArg(0) -> 0
   encArg(s) -> s
   encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1))
   encode_app(z0, z1) -> app(encArg(z0), encArg(z1))
   encode_id -> id
   encode_plus -> plus
   encode_0 -> 0
   encode_s -> s
   app(z0, z1) -> c_app(z0, z1)
   app(id, z0) -> z0
   app(plus, 0) -> id
   app(c_app(plus, c_app(s, z0)), z1) -> app(s, app(app(plus, z0), z1))
Tuples:
   ENCARG(cons_app(z0, z1)) -> c4(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_APP(z0, z1) -> c5(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   APP(id, z0) -> c11
   APP(plus, 0) -> c12
   APP(c_app(plus, c_app(s, z0)), z1) -> c13(APP(app(plus, z0), z1), APP(plus, z0))
S tuples:
   APP(id, z0) -> c11
   APP(plus, 0) -> c12
   APP(c_app(plus, c_app(s, z0)), z1) -> c13(APP(app(plus, z0), z1), APP(plus, z0))
K tuples:none
Defined Rule Symbols:   app_2, encArg_1, encode_app_2, encode_id, encode_plus, encode_0, encode_s

Defined Pair Symbols:   ENCARG_1, ENCODE_APP_2, APP_2

Compound Symbols:   c4_3, c5_3, c11, c12, c13_2


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

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

(16)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(id) -> id
   encArg(plus) -> plus
   encArg(0) -> 0
   encArg(s) -> s
   encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1))
   encode_app(z0, z1) -> app(encArg(z0), encArg(z1))
   encode_id -> id
   encode_plus -> plus
   encode_0 -> 0
   encode_s -> s
   app(z0, z1) -> c_app(z0, z1)
   app(id, z0) -> z0
   app(plus, 0) -> id
   app(c_app(plus, c_app(s, z0)), z1) -> app(s, app(app(plus, z0), z1))
Tuples:
   ENCARG(cons_app(z0, z1)) -> c4(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   APP(id, z0) -> c11
   APP(plus, 0) -> c12
   APP(c_app(plus, c_app(s, z0)), z1) -> c13(APP(app(plus, z0), z1), APP(plus, z0))
   ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1)))
   ENCODE_APP(z0, z1) -> c(ENCARG(z0))
   ENCODE_APP(z0, z1) -> c(ENCARG(z1))
S tuples:
   APP(id, z0) -> c11
   APP(plus, 0) -> c12
   APP(c_app(plus, c_app(s, z0)), z1) -> c13(APP(app(plus, z0), z1), APP(plus, z0))
K tuples:none
Defined Rule Symbols:   app_2, encArg_1, encode_app_2, encode_id, encode_plus, encode_0, encode_s

Defined Pair Symbols:   ENCARG_1, APP_2, ENCODE_APP_2

Compound Symbols:   c4_3, c11, c12, c13_2, c_1


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

(17) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 2 leading nodes:
   ENCODE_APP(z0, z1) -> c(ENCARG(z0))
   ENCODE_APP(z0, z1) -> c(ENCARG(z1))

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

(18)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(id) -> id
   encArg(plus) -> plus
   encArg(0) -> 0
   encArg(s) -> s
   encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1))
   encode_app(z0, z1) -> app(encArg(z0), encArg(z1))
   encode_id -> id
   encode_plus -> plus
   encode_0 -> 0
   encode_s -> s
   app(z0, z1) -> c_app(z0, z1)
   app(id, z0) -> z0
   app(plus, 0) -> id
   app(c_app(plus, c_app(s, z0)), z1) -> app(s, app(app(plus, z0), z1))
Tuples:
   ENCARG(cons_app(z0, z1)) -> c4(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   APP(id, z0) -> c11
   APP(plus, 0) -> c12
   APP(c_app(plus, c_app(s, z0)), z1) -> c13(APP(app(plus, z0), z1), APP(plus, z0))
   ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1)))
S tuples:
   APP(id, z0) -> c11
   APP(plus, 0) -> c12
   APP(c_app(plus, c_app(s, z0)), z1) -> c13(APP(app(plus, z0), z1), APP(plus, z0))
K tuples:none
Defined Rule Symbols:   app_2, encArg_1, encode_app_2, encode_id, encode_plus, encode_0, encode_s

Defined Pair Symbols:   ENCARG_1, APP_2, ENCODE_APP_2

Compound Symbols:   c4_3, c11, c12, c13_2, c_1


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

(19) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   encode_app(z0, z1) -> app(encArg(z0), encArg(z1))
   encode_id -> id
   encode_plus -> plus
   encode_0 -> 0
   encode_s -> s

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

(20)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(id) -> id
   encArg(plus) -> plus
   encArg(0) -> 0
   encArg(s) -> s
   encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1))
   app(z0, z1) -> c_app(z0, z1)
   app(id, z0) -> z0
   app(plus, 0) -> id
   app(c_app(plus, c_app(s, z0)), z1) -> app(s, app(app(plus, z0), z1))
Tuples:
   ENCARG(cons_app(z0, z1)) -> c4(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   APP(id, z0) -> c11
   APP(plus, 0) -> c12
   APP(c_app(plus, c_app(s, z0)), z1) -> c13(APP(app(plus, z0), z1), APP(plus, z0))
   ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1)))
S tuples:
   APP(id, z0) -> c11
   APP(plus, 0) -> c12
   APP(c_app(plus, c_app(s, z0)), z1) -> c13(APP(app(plus, z0), z1), APP(plus, z0))
K tuples:none
Defined Rule Symbols:   encArg_1, app_2

Defined Pair Symbols:   ENCARG_1, APP_2, ENCODE_APP_2

Compound Symbols:   c4_3, c11, c12, c13_2, c_1


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

(21) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   APP(id, z0) -> c11
We considered the (Usable) Rules:
   app(z0, z1) -> c_app(z0, z1)
   encArg(plus) -> plus
   encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1))
   encArg(id) -> id
   app(c_app(plus, c_app(s, z0)), z1) -> app(s, app(app(plus, z0), z1))
   encArg(s) -> s
   encArg(0) -> 0
   app(id, z0) -> z0
   app(plus, 0) -> id
And the Tuples:
   ENCARG(cons_app(z0, z1)) -> c4(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   APP(id, z0) -> c11
   APP(plus, 0) -> c12
   APP(c_app(plus, c_app(s, z0)), z1) -> c13(APP(app(plus, z0), z1), APP(plus, z0))
   ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(APP(x_1, x_2)) = x_1
   POL(ENCARG(x_1)) = x_1
   POL(ENCODE_APP(x_1, x_2)) = [1]
   POL(app(x_1, x_2)) = x_2
   POL(c(x_1)) = x_1
   POL(c11) = 0
   POL(c12) = 0
   POL(c13(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c_app(x_1, x_2)) = x_2
   POL(cons_app(x_1, x_2)) = [1] + x_1 + x_2
   POL(encArg(x_1)) = [1]
   POL(id) = [1]
   POL(plus) = 0
   POL(s) = 0

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

(22)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(id) -> id
   encArg(plus) -> plus
   encArg(0) -> 0
   encArg(s) -> s
   encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1))
   app(z0, z1) -> c_app(z0, z1)
   app(id, z0) -> z0
   app(plus, 0) -> id
   app(c_app(plus, c_app(s, z0)), z1) -> app(s, app(app(plus, z0), z1))
Tuples:
   ENCARG(cons_app(z0, z1)) -> c4(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   APP(id, z0) -> c11
   APP(plus, 0) -> c12
   APP(c_app(plus, c_app(s, z0)), z1) -> c13(APP(app(plus, z0), z1), APP(plus, z0))
   ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1)))
S tuples:
   APP(plus, 0) -> c12
   APP(c_app(plus, c_app(s, z0)), z1) -> c13(APP(app(plus, z0), z1), APP(plus, z0))
K tuples:
   APP(id, z0) -> c11
Defined Rule Symbols:   encArg_1, app_2

Defined Pair Symbols:   ENCARG_1, APP_2, ENCODE_APP_2

Compound Symbols:   c4_3, c11, c12, c13_2, c_1


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

(23) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   APP(c_app(plus, c_app(s, z0)), z1) -> c13(APP(app(plus, z0), z1), APP(plus, z0))
We considered the (Usable) Rules:
   app(z0, z1) -> c_app(z0, z1)
   encArg(plus) -> plus
   encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1))
   encArg(id) -> id
   app(c_app(plus, c_app(s, z0)), z1) -> app(s, app(app(plus, z0), z1))
   encArg(s) -> s
   encArg(0) -> 0
   app(id, z0) -> z0
   app(plus, 0) -> id
And the Tuples:
   ENCARG(cons_app(z0, z1)) -> c4(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   APP(id, z0) -> c11
   APP(plus, 0) -> c12
   APP(c_app(plus, c_app(s, z0)), z1) -> c13(APP(app(plus, z0), z1), APP(plus, z0))
   ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(APP(x_1, x_2)) = [2]x_1
   POL(ENCARG(x_1)) = [2]x_1 + x_1^2
   POL(ENCODE_APP(x_1, x_2)) = [2] + [2]x_1 + [2]x_2 + [2]x_2^2 + [2]x_1*x_2 + [2]x_1^2
   POL(app(x_1, x_2)) = x_1 + x_2
   POL(c(x_1)) = x_1
   POL(c11) = 0
   POL(c12) = 0
   POL(c13(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c_app(x_1, x_2)) = x_1 + x_2
   POL(cons_app(x_1, x_2)) = [2] + x_1 + x_2
   POL(encArg(x_1)) = x_1
   POL(id) = 0
   POL(plus) = 0
   POL(s) = [1]

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

(24)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(id) -> id
   encArg(plus) -> plus
   encArg(0) -> 0
   encArg(s) -> s
   encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1))
   app(z0, z1) -> c_app(z0, z1)
   app(id, z0) -> z0
   app(plus, 0) -> id
   app(c_app(plus, c_app(s, z0)), z1) -> app(s, app(app(plus, z0), z1))
Tuples:
   ENCARG(cons_app(z0, z1)) -> c4(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   APP(id, z0) -> c11
   APP(plus, 0) -> c12
   APP(c_app(plus, c_app(s, z0)), z1) -> c13(APP(app(plus, z0), z1), APP(plus, z0))
   ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1)))
S tuples:
   APP(plus, 0) -> c12
K tuples:
   APP(id, z0) -> c11
   APP(c_app(plus, c_app(s, z0)), z1) -> c13(APP(app(plus, z0), z1), APP(plus, z0))
Defined Rule Symbols:   encArg_1, app_2

Defined Pair Symbols:   ENCARG_1, APP_2, ENCODE_APP_2

Compound Symbols:   c4_3, c11, c12, c13_2, c_1


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

(25) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   APP(plus, 0) -> c12
We considered the (Usable) Rules:
   app(z0, z1) -> c_app(z0, z1)
   encArg(plus) -> plus
   encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1))
   encArg(id) -> id
   app(c_app(plus, c_app(s, z0)), z1) -> app(s, app(app(plus, z0), z1))
   encArg(s) -> s
   encArg(0) -> 0
   app(id, z0) -> z0
   app(plus, 0) -> id
And the Tuples:
   ENCARG(cons_app(z0, z1)) -> c4(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   APP(id, z0) -> c11
   APP(plus, 0) -> c12
   APP(c_app(plus, c_app(s, z0)), z1) -> c13(APP(app(plus, z0), z1), APP(plus, z0))
   ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(APP(x_1, x_2)) = [2]x_1
   POL(ENCARG(x_1)) = x_1^2
   POL(ENCODE_APP(x_1, x_2)) = [2] + [2]x_1 + [2]x_2^2 + [2]x_1*x_2 + [2]x_1^2
   POL(app(x_1, x_2)) = x_1 + x_2
   POL(c(x_1)) = x_1
   POL(c11) = 0
   POL(c12) = 0
   POL(c13(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c_app(x_1, x_2)) = x_1 + x_2
   POL(cons_app(x_1, x_2)) = [1] + x_1 + x_2
   POL(encArg(x_1)) = x_1
   POL(id) = 0
   POL(plus) = [1]
   POL(s) = [1]

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

(26)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(id) -> id
   encArg(plus) -> plus
   encArg(0) -> 0
   encArg(s) -> s
   encArg(cons_app(z0, z1)) -> app(encArg(z0), encArg(z1))
   app(z0, z1) -> c_app(z0, z1)
   app(id, z0) -> z0
   app(plus, 0) -> id
   app(c_app(plus, c_app(s, z0)), z1) -> app(s, app(app(plus, z0), z1))
Tuples:
   ENCARG(cons_app(z0, z1)) -> c4(APP(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   APP(id, z0) -> c11
   APP(plus, 0) -> c12
   APP(c_app(plus, c_app(s, z0)), z1) -> c13(APP(app(plus, z0), z1), APP(plus, z0))
   ENCODE_APP(z0, z1) -> c(APP(encArg(z0), encArg(z1)))
S tuples:none
K tuples:
   APP(id, z0) -> c11
   APP(c_app(plus, c_app(s, z0)), z1) -> c13(APP(app(plus, z0), z1), APP(plus, z0))
   APP(plus, 0) -> c12
Defined Rule Symbols:   encArg_1, app_2

Defined Pair Symbols:   ENCARG_1, APP_2, ENCODE_APP_2

Compound Symbols:   c4_3, c11, c12, c13_2, c_1


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

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

(28)
BOUNDS(1, 1)

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

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

(30)
Obligation:
The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   app(id, x) -> x
   app(plus, 0') -> id
   app(app(plus, app(s, x)), y) -> app(s, app(app(plus, x), y))

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

   encArg(id) -> id
   encArg(plus) -> plus
   encArg(0') -> 0'
   encArg(s) -> s
   encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2))
   encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2))
   encode_id -> id
   encode_plus -> plus
   encode_0 -> 0'
   encode_s -> s

Rewrite Strategy: FULL
----------------------------------------

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

(32)
Obligation:
TRS:
Rules:
app(id, x) -> x
app(plus, 0') -> id
app(app(plus, app(s, x)), y) -> app(s, app(app(plus, x), y))
encArg(id) -> id
encArg(plus) -> plus
encArg(0') -> 0'
encArg(s) -> s
encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2))
encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2))
encode_id -> id
encode_plus -> plus
encode_0 -> 0'
encode_s -> s

Types:
app :: id:plus:0':s:cons_app -> id:plus:0':s:cons_app -> id:plus:0':s:cons_app
id :: id:plus:0':s:cons_app
plus :: id:plus:0':s:cons_app
0' :: id:plus:0':s:cons_app
s :: id:plus:0':s:cons_app
encArg :: id:plus:0':s:cons_app -> id:plus:0':s:cons_app
cons_app :: id:plus:0':s:cons_app -> id:plus:0':s:cons_app -> id:plus:0':s:cons_app
encode_app :: id:plus:0':s:cons_app -> id:plus:0':s:cons_app -> id:plus:0':s:cons_app
encode_id :: id:plus:0':s:cons_app
encode_plus :: id:plus:0':s:cons_app
encode_0 :: id:plus:0':s:cons_app
encode_s :: id:plus:0':s:cons_app
hole_id:plus:0':s:cons_app1_3 :: id:plus:0':s:cons_app
gen_id:plus:0':s:cons_app2_3 :: Nat -> id:plus:0':s:cons_app

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

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

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

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

(34)
Obligation:
TRS:
Rules:
app(id, x) -> x
app(plus, 0') -> id
app(app(plus, app(s, x)), y) -> app(s, app(app(plus, x), y))
encArg(id) -> id
encArg(plus) -> plus
encArg(0') -> 0'
encArg(s) -> s
encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2))
encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2))
encode_id -> id
encode_plus -> plus
encode_0 -> 0'
encode_s -> s

Types:
app :: id:plus:0':s:cons_app -> id:plus:0':s:cons_app -> id:plus:0':s:cons_app
id :: id:plus:0':s:cons_app
plus :: id:plus:0':s:cons_app
0' :: id:plus:0':s:cons_app
s :: id:plus:0':s:cons_app
encArg :: id:plus:0':s:cons_app -> id:plus:0':s:cons_app
cons_app :: id:plus:0':s:cons_app -> id:plus:0':s:cons_app -> id:plus:0':s:cons_app
encode_app :: id:plus:0':s:cons_app -> id:plus:0':s:cons_app -> id:plus:0':s:cons_app
encode_id :: id:plus:0':s:cons_app
encode_plus :: id:plus:0':s:cons_app
encode_0 :: id:plus:0':s:cons_app
encode_s :: id:plus:0':s:cons_app
hole_id:plus:0':s:cons_app1_3 :: id:plus:0':s:cons_app
gen_id:plus:0':s:cons_app2_3 :: Nat -> id:plus:0':s:cons_app


Generator Equations:
gen_id:plus:0':s:cons_app2_3(0) <=> id
gen_id:plus:0':s:cons_app2_3(+(x, 1)) <=> cons_app(id, gen_id:plus:0':s:cons_app2_3(x))


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

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

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

(35) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_id:plus:0':s:cons_app2_3(n20_3)) -> gen_id:plus:0':s:cons_app2_3(0), rt in Omega(n20_3)

Induction Base:
encArg(gen_id:plus:0':s:cons_app2_3(0)) ->_R^Omega(0)
id

Induction Step:
encArg(gen_id:plus:0':s:cons_app2_3(+(n20_3, 1))) ->_R^Omega(0)
app(encArg(id), encArg(gen_id:plus:0':s:cons_app2_3(n20_3))) ->_R^Omega(0)
app(id, encArg(gen_id:plus:0':s:cons_app2_3(n20_3))) ->_IH
app(id, gen_id:plus:0':s:cons_app2_3(0)) ->_R^Omega(1)
gen_id:plus:0':s:cons_app2_3(0)

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

(36)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
app(id, x) -> x
app(plus, 0') -> id
app(app(plus, app(s, x)), y) -> app(s, app(app(plus, x), y))
encArg(id) -> id
encArg(plus) -> plus
encArg(0') -> 0'
encArg(s) -> s
encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2))
encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2))
encode_id -> id
encode_plus -> plus
encode_0 -> 0'
encode_s -> s

Types:
app :: id:plus:0':s:cons_app -> id:plus:0':s:cons_app -> id:plus:0':s:cons_app
id :: id:plus:0':s:cons_app
plus :: id:plus:0':s:cons_app
0' :: id:plus:0':s:cons_app
s :: id:plus:0':s:cons_app
encArg :: id:plus:0':s:cons_app -> id:plus:0':s:cons_app
cons_app :: id:plus:0':s:cons_app -> id:plus:0':s:cons_app -> id:plus:0':s:cons_app
encode_app :: id:plus:0':s:cons_app -> id:plus:0':s:cons_app -> id:plus:0':s:cons_app
encode_id :: id:plus:0':s:cons_app
encode_plus :: id:plus:0':s:cons_app
encode_0 :: id:plus:0':s:cons_app
encode_s :: id:plus:0':s:cons_app
hole_id:plus:0':s:cons_app1_3 :: id:plus:0':s:cons_app
gen_id:plus:0':s:cons_app2_3 :: Nat -> id:plus:0':s:cons_app


Generator Equations:
gen_id:plus:0':s:cons_app2_3(0) <=> id
gen_id:plus:0':s:cons_app2_3(+(x, 1)) <=> cons_app(id, gen_id:plus:0':s:cons_app2_3(x))


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

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

(38)
BOUNDS(n^1, INF)