/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^1))
proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 140 ms]
(4) CpxRelTRS
(5) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(6) CdtProblem
    (7) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 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), 214 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:

   .(1, x) -> x
   .(x, 1) -> x
   .(i(x), x) -> 1
   .(x, i(x)) -> 1
   i(1) -> 1
   i(i(x)) -> x
   .(i(y), .(y, z)) -> z
   .(y, .(i(y), z)) -> z

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(cons_.(x_1, x_2)) -> .(encArg(x_1), encArg(x_2))
   encArg(cons_i(x_1)) -> i(encArg(x_1))
   encode_.(x_1, x_2) -> .(encArg(x_1), encArg(x_2))
   encode_1 -> 1
   encode_i(x_1) -> i(encArg(x_1))


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

(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:

   .(1, x) -> x
   .(x, 1) -> x
   .(i(x), x) -> 1
   .(x, i(x)) -> 1
   i(1) -> 1
   i(i(x)) -> x
   .(i(y), .(y, z)) -> z
   .(y, .(i(y), z)) -> z

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

   encArg(1) -> 1
   encArg(cons_.(x_1, x_2)) -> .(encArg(x_1), encArg(x_2))
   encArg(cons_i(x_1)) -> i(encArg(x_1))
   encode_.(x_1, x_2) -> .(encArg(x_1), encArg(x_2))
   encode_1 -> 1
   encode_i(x_1) -> i(encArg(x_1))

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:

   .(1, x) -> x
   .(x, 1) -> x
   .(i(x), x) -> 1
   .(x, i(x)) -> 1
   i(1) -> 1
   i(i(x)) -> x
   .(i(y), .(y, z)) -> z
   .(y, .(i(y), z)) -> z

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

   encArg(1) -> 1
   encArg(cons_.(x_1, x_2)) -> .(encArg(x_1), encArg(x_2))
   encArg(cons_i(x_1)) -> i(encArg(x_1))
   encode_.(x_1, x_2) -> .(encArg(x_1), encArg(x_2))
   encode_1 -> 1
   encode_i(x_1) -> i(encArg(x_1))

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(cons_.(z0, z1)) -> .(encArg(z0), encArg(z1))
   encArg(cons_i(z0)) -> i(encArg(z0))
   encode_.(z0, z1) -> .(encArg(z0), encArg(z1))
   encode_1 -> 1
   encode_i(z0) -> i(encArg(z0))
   .(1, z0) -> z0
   .(z0, 1) -> z0
   .(i(z0), z0) -> 1
   .(z0, i(z0)) -> 1
   .(i(z0), .(z0, z1)) -> z1
   .(z0, .(i(z0), z1)) -> z1
   i(1) -> 1
   i(i(z0)) -> z0
Tuples:
   ENCARG(1) -> c
   ENCARG(cons_.(z0, z1)) -> c1(.'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_i(z0)) -> c2(I(encArg(z0)), ENCARG(z0))
   ENCODE_.(z0, z1) -> c3(.'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_1 -> c4
   ENCODE_I(z0) -> c5(I(encArg(z0)), ENCARG(z0))
   .'(1, z0) -> c6
   .'(z0, 1) -> c7
   .'(i(z0), z0) -> c8
   .'(z0, i(z0)) -> c9
   .'(i(z0), .(z0, z1)) -> c10
   .'(z0, .(i(z0), z1)) -> c11
   I(1) -> c12
   I(i(z0)) -> c13
S tuples:
   .'(1, z0) -> c6
   .'(z0, 1) -> c7
   .'(i(z0), z0) -> c8
   .'(z0, i(z0)) -> c9
   .'(i(z0), .(z0, z1)) -> c10
   .'(z0, .(i(z0), z1)) -> c11
   I(1) -> c12
   I(i(z0)) -> c13
K tuples:none
Defined Rule Symbols:   ._2, i_1, encArg_1, encode_._2, encode_1, encode_i_1

Defined Pair Symbols:   ENCARG_1, ENCODE_._2, ENCODE_1, ENCODE_I_1, .'_2, I_1

Compound Symbols:   c, c1_3, c2_2, c3_3, c4, c5_2, c6, c7, c8, c9, c10, c11, c12, c13


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

(7) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   ENCARG(1) -> c
   ENCODE_1 -> c4

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(1) -> 1
   encArg(cons_.(z0, z1)) -> .(encArg(z0), encArg(z1))
   encArg(cons_i(z0)) -> i(encArg(z0))
   encode_.(z0, z1) -> .(encArg(z0), encArg(z1))
   encode_1 -> 1
   encode_i(z0) -> i(encArg(z0))
   .(1, z0) -> z0
   .(z0, 1) -> z0
   .(i(z0), z0) -> 1
   .(z0, i(z0)) -> 1
   .(i(z0), .(z0, z1)) -> z1
   .(z0, .(i(z0), z1)) -> z1
   i(1) -> 1
   i(i(z0)) -> z0
Tuples:
   ENCARG(cons_.(z0, z1)) -> c1(.'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_i(z0)) -> c2(I(encArg(z0)), ENCARG(z0))
   ENCODE_.(z0, z1) -> c3(.'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_I(z0) -> c5(I(encArg(z0)), ENCARG(z0))
   .'(1, z0) -> c6
   .'(z0, 1) -> c7
   .'(i(z0), z0) -> c8
   .'(z0, i(z0)) -> c9
   .'(i(z0), .(z0, z1)) -> c10
   .'(z0, .(i(z0), z1)) -> c11
   I(1) -> c12
   I(i(z0)) -> c13
S tuples:
   .'(1, z0) -> c6
   .'(z0, 1) -> c7
   .'(i(z0), z0) -> c8
   .'(z0, i(z0)) -> c9
   .'(i(z0), .(z0, z1)) -> c10
   .'(z0, .(i(z0), z1)) -> c11
   I(1) -> c12
   I(i(z0)) -> c13
K tuples:none
Defined Rule Symbols:   ._2, i_1, encArg_1, encode_._2, encode_1, encode_i_1

Defined Pair Symbols:   ENCARG_1, ENCODE_._2, ENCODE_I_1, .'_2, I_1

Compound Symbols:   c1_3, c2_2, c3_3, c5_2, c6, c7, c8, c9, c10, 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(cons_.(z0, z1)) -> .(encArg(z0), encArg(z1))
   encArg(cons_i(z0)) -> i(encArg(z0))
   encode_.(z0, z1) -> .(encArg(z0), encArg(z1))
   encode_1 -> 1
   encode_i(z0) -> i(encArg(z0))
   .(1, z0) -> z0
   .(z0, 1) -> z0
   .(i(z0), z0) -> 1
   .(z0, i(z0)) -> 1
   .(i(z0), .(z0, z1)) -> z1
   .(z0, .(i(z0), z1)) -> z1
   i(1) -> 1
   i(i(z0)) -> z0
Tuples:
   ENCARG(cons_.(z0, z1)) -> c1(.'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_i(z0)) -> c2(I(encArg(z0)), ENCARG(z0))
   .'(1, z0) -> c6
   .'(z0, 1) -> c7
   .'(i(z0), z0) -> c8
   .'(z0, i(z0)) -> c9
   .'(i(z0), .(z0, z1)) -> c10
   .'(z0, .(i(z0), z1)) -> c11
   I(1) -> c12
   I(i(z0)) -> c13
   ENCODE_.(z0, z1) -> c(.'(encArg(z0), encArg(z1)))
   ENCODE_.(z0, z1) -> c(ENCARG(z0))
   ENCODE_.(z0, z1) -> c(ENCARG(z1))
   ENCODE_I(z0) -> c(I(encArg(z0)))
   ENCODE_I(z0) -> c(ENCARG(z0))
S tuples:
   .'(1, z0) -> c6
   .'(z0, 1) -> c7
   .'(i(z0), z0) -> c8
   .'(z0, i(z0)) -> c9
   .'(i(z0), .(z0, z1)) -> c10
   .'(z0, .(i(z0), z1)) -> c11
   I(1) -> c12
   I(i(z0)) -> c13
K tuples:none
Defined Rule Symbols:   ._2, i_1, encArg_1, encode_._2, encode_1, encode_i_1

Defined Pair Symbols:   ENCARG_1, .'_2, I_1, ENCODE_._2, ENCODE_I_1

Compound Symbols:   c1_3, c2_2, c6, c7, c8, c9, c10, c11, c12, c13, c_1


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

(11) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 3 leading nodes:
   ENCODE_.(z0, z1) -> c(ENCARG(z0))
   ENCODE_.(z0, z1) -> c(ENCARG(z1))
   ENCODE_I(z0) -> c(ENCARG(z0))

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

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(1) -> 1
   encArg(cons_.(z0, z1)) -> .(encArg(z0), encArg(z1))
   encArg(cons_i(z0)) -> i(encArg(z0))
   encode_.(z0, z1) -> .(encArg(z0), encArg(z1))
   encode_1 -> 1
   encode_i(z0) -> i(encArg(z0))
   .(1, z0) -> z0
   .(z0, 1) -> z0
   .(i(z0), z0) -> 1
   .(z0, i(z0)) -> 1
   .(i(z0), .(z0, z1)) -> z1
   .(z0, .(i(z0), z1)) -> z1
   i(1) -> 1
   i(i(z0)) -> z0
Tuples:
   ENCARG(cons_.(z0, z1)) -> c1(.'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_i(z0)) -> c2(I(encArg(z0)), ENCARG(z0))
   .'(1, z0) -> c6
   .'(z0, 1) -> c7
   .'(i(z0), z0) -> c8
   .'(z0, i(z0)) -> c9
   .'(i(z0), .(z0, z1)) -> c10
   .'(z0, .(i(z0), z1)) -> c11
   I(1) -> c12
   I(i(z0)) -> c13
   ENCODE_.(z0, z1) -> c(.'(encArg(z0), encArg(z1)))
   ENCODE_I(z0) -> c(I(encArg(z0)))
S tuples:
   .'(1, z0) -> c6
   .'(z0, 1) -> c7
   .'(i(z0), z0) -> c8
   .'(z0, i(z0)) -> c9
   .'(i(z0), .(z0, z1)) -> c10
   .'(z0, .(i(z0), z1)) -> c11
   I(1) -> c12
   I(i(z0)) -> c13
K tuples:none
Defined Rule Symbols:   ._2, i_1, encArg_1, encode_._2, encode_1, encode_i_1

Defined Pair Symbols:   ENCARG_1, .'_2, I_1, ENCODE_._2, ENCODE_I_1

Compound Symbols:   c1_3, c2_2, c6, c7, c8, c9, c10, c11, c12, c13, c_1


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

(13) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   encode_.(z0, z1) -> .(encArg(z0), encArg(z1))
   encode_1 -> 1
   encode_i(z0) -> i(encArg(z0))

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

(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(1) -> 1
   encArg(cons_.(z0, z1)) -> .(encArg(z0), encArg(z1))
   encArg(cons_i(z0)) -> i(encArg(z0))
   .(1, z0) -> z0
   .(z0, 1) -> z0
   .(i(z0), z0) -> 1
   .(z0, i(z0)) -> 1
   .(i(z0), .(z0, z1)) -> z1
   .(z0, .(i(z0), z1)) -> z1
   i(1) -> 1
   i(i(z0)) -> z0
Tuples:
   ENCARG(cons_.(z0, z1)) -> c1(.'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_i(z0)) -> c2(I(encArg(z0)), ENCARG(z0))
   .'(1, z0) -> c6
   .'(z0, 1) -> c7
   .'(i(z0), z0) -> c8
   .'(z0, i(z0)) -> c9
   .'(i(z0), .(z0, z1)) -> c10
   .'(z0, .(i(z0), z1)) -> c11
   I(1) -> c12
   I(i(z0)) -> c13
   ENCODE_.(z0, z1) -> c(.'(encArg(z0), encArg(z1)))
   ENCODE_I(z0) -> c(I(encArg(z0)))
S tuples:
   .'(1, z0) -> c6
   .'(z0, 1) -> c7
   .'(i(z0), z0) -> c8
   .'(z0, i(z0)) -> c9
   .'(i(z0), .(z0, z1)) -> c10
   .'(z0, .(i(z0), z1)) -> c11
   I(1) -> c12
   I(i(z0)) -> c13
K tuples:none
Defined Rule Symbols:   encArg_1, ._2, i_1

Defined Pair Symbols:   ENCARG_1, .'_2, I_1, ENCODE_._2, ENCODE_I_1

Compound Symbols:   c1_3, c2_2, c6, c7, c8, c9, c10, 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.
   .'(1, z0) -> c6
   .'(z0, 1) -> c7
   .'(i(z0), z0) -> c8
   .'(z0, i(z0)) -> c9
   .'(i(z0), .(z0, z1)) -> c10
   .'(z0, .(i(z0), z1)) -> c11
   I(1) -> c12
   I(i(z0)) -> c13
We considered the (Usable) Rules:none
And the Tuples:
   ENCARG(cons_.(z0, z1)) -> c1(.'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_i(z0)) -> c2(I(encArg(z0)), ENCARG(z0))
   .'(1, z0) -> c6
   .'(z0, 1) -> c7
   .'(i(z0), z0) -> c8
   .'(z0, i(z0)) -> c9
   .'(i(z0), .(z0, z1)) -> c10
   .'(z0, .(i(z0), z1)) -> c11
   I(1) -> c12
   I(i(z0)) -> c13
   ENCODE_.(z0, z1) -> c(.'(encArg(z0), encArg(z1)))
   ENCODE_I(z0) -> c(I(encArg(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(.(x_1, x_2)) = x_1 + x_2
   POL(.'(x_1, x_2)) = [1]
   POL(1) = [1]
   POL(ENCARG(x_1)) = x_1
   POL(ENCODE_.(x_1, x_2)) = [1]
   POL(ENCODE_I(x_1)) = [1]
   POL(I(x_1)) = [1]
   POL(c(x_1)) = x_1
   POL(c1(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c10) = 0
   POL(c11) = 0
   POL(c12) = 0
   POL(c13) = 0
   POL(c2(x_1, x_2)) = x_1 + x_2
   POL(c6) = 0
   POL(c7) = 0
   POL(c8) = 0
   POL(c9) = 0
   POL(cons_.(x_1, x_2)) = [1] + x_1 + x_2
   POL(cons_i(x_1)) = [1] + x_1
   POL(encArg(x_1)) = [1] + x_1
   POL(i(x_1)) = [1] + x_1

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

(16)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(1) -> 1
   encArg(cons_.(z0, z1)) -> .(encArg(z0), encArg(z1))
   encArg(cons_i(z0)) -> i(encArg(z0))
   .(1, z0) -> z0
   .(z0, 1) -> z0
   .(i(z0), z0) -> 1
   .(z0, i(z0)) -> 1
   .(i(z0), .(z0, z1)) -> z1
   .(z0, .(i(z0), z1)) -> z1
   i(1) -> 1
   i(i(z0)) -> z0
Tuples:
   ENCARG(cons_.(z0, z1)) -> c1(.'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_i(z0)) -> c2(I(encArg(z0)), ENCARG(z0))
   .'(1, z0) -> c6
   .'(z0, 1) -> c7
   .'(i(z0), z0) -> c8
   .'(z0, i(z0)) -> c9
   .'(i(z0), .(z0, z1)) -> c10
   .'(z0, .(i(z0), z1)) -> c11
   I(1) -> c12
   I(i(z0)) -> c13
   ENCODE_.(z0, z1) -> c(.'(encArg(z0), encArg(z1)))
   ENCODE_I(z0) -> c(I(encArg(z0)))
S tuples:none
K tuples:
   .'(1, z0) -> c6
   .'(z0, 1) -> c7
   .'(i(z0), z0) -> c8
   .'(z0, i(z0)) -> c9
   .'(i(z0), .(z0, z1)) -> c10
   .'(z0, .(i(z0), z1)) -> c11
   I(1) -> c12
   I(i(z0)) -> c13
Defined Rule Symbols:   encArg_1, ._2, i_1

Defined Pair Symbols:   ENCARG_1, .'_2, I_1, ENCODE_._2, ENCODE_I_1

Compound Symbols:   c1_3, c2_2, c6, c7, c8, c9, c10, 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:

   .(1', x) -> x
   .(x, 1') -> x
   .(i(x), x) -> 1'
   .(x, i(x)) -> 1'
   i(1') -> 1'
   i(i(x)) -> x
   .(i(y), .(y, z)) -> z
   .(y, .(i(y), z)) -> z

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

   encArg(1') -> 1'
   encArg(cons_.(x_1, x_2)) -> .(encArg(x_1), encArg(x_2))
   encArg(cons_i(x_1)) -> i(encArg(x_1))
   encode_.(x_1, x_2) -> .(encArg(x_1), encArg(x_2))
   encode_1 -> 1'
   encode_i(x_1) -> i(encArg(x_1))

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

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

(22)
Obligation:
Innermost TRS:
Rules:
.(1', x) -> x
.(x, 1') -> x
.(i(x), x) -> 1'
.(x, i(x)) -> 1'
i(1') -> 1'
i(i(x)) -> x
.(i(y), .(y, z)) -> z
.(y, .(i(y), z)) -> z
encArg(1') -> 1'
encArg(cons_.(x_1, x_2)) -> .(encArg(x_1), encArg(x_2))
encArg(cons_i(x_1)) -> i(encArg(x_1))
encode_.(x_1, x_2) -> .(encArg(x_1), encArg(x_2))
encode_1 -> 1'
encode_i(x_1) -> i(encArg(x_1))

Types:
. :: 1':cons_.:cons_i -> 1':cons_.:cons_i -> 1':cons_.:cons_i
1' :: 1':cons_.:cons_i
i :: 1':cons_.:cons_i -> 1':cons_.:cons_i
encArg :: 1':cons_.:cons_i -> 1':cons_.:cons_i
cons_. :: 1':cons_.:cons_i -> 1':cons_.:cons_i -> 1':cons_.:cons_i
cons_i :: 1':cons_.:cons_i -> 1':cons_.:cons_i
encode_. :: 1':cons_.:cons_i -> 1':cons_.:cons_i -> 1':cons_.:cons_i
encode_1 :: 1':cons_.:cons_i
encode_i :: 1':cons_.:cons_i -> 1':cons_.:cons_i
hole_1':cons_.:cons_i1_0 :: 1':cons_.:cons_i
gen_1':cons_.:cons_i2_0 :: Nat -> 1':cons_.:cons_i

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(23) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
encArg
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(24)
Obligation:
Innermost TRS:
Rules:
.(1', x) -> x
.(x, 1') -> x
.(i(x), x) -> 1'
.(x, i(x)) -> 1'
i(1') -> 1'
i(i(x)) -> x
.(i(y), .(y, z)) -> z
.(y, .(i(y), z)) -> z
encArg(1') -> 1'
encArg(cons_.(x_1, x_2)) -> .(encArg(x_1), encArg(x_2))
encArg(cons_i(x_1)) -> i(encArg(x_1))
encode_.(x_1, x_2) -> .(encArg(x_1), encArg(x_2))
encode_1 -> 1'
encode_i(x_1) -> i(encArg(x_1))

Types:
. :: 1':cons_.:cons_i -> 1':cons_.:cons_i -> 1':cons_.:cons_i
1' :: 1':cons_.:cons_i
i :: 1':cons_.:cons_i -> 1':cons_.:cons_i
encArg :: 1':cons_.:cons_i -> 1':cons_.:cons_i
cons_. :: 1':cons_.:cons_i -> 1':cons_.:cons_i -> 1':cons_.:cons_i
cons_i :: 1':cons_.:cons_i -> 1':cons_.:cons_i
encode_. :: 1':cons_.:cons_i -> 1':cons_.:cons_i -> 1':cons_.:cons_i
encode_1 :: 1':cons_.:cons_i
encode_i :: 1':cons_.:cons_i -> 1':cons_.:cons_i
hole_1':cons_.:cons_i1_0 :: 1':cons_.:cons_i
gen_1':cons_.:cons_i2_0 :: Nat -> 1':cons_.:cons_i


Generator Equations:
gen_1':cons_.:cons_i2_0(0) <=> 1'
gen_1':cons_.:cons_i2_0(+(x, 1)) <=> cons_.(1', gen_1':cons_.:cons_i2_0(x))


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

(25) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_1':cons_.:cons_i2_0(n4_0)) -> gen_1':cons_.:cons_i2_0(0), rt in Omega(n4_0)

Induction Base:
encArg(gen_1':cons_.:cons_i2_0(0)) ->_R^Omega(0)
1'

Induction Step:
encArg(gen_1':cons_.:cons_i2_0(+(n4_0, 1))) ->_R^Omega(0)
.(encArg(1'), encArg(gen_1':cons_.:cons_i2_0(n4_0))) ->_R^Omega(0)
.(1', encArg(gen_1':cons_.:cons_i2_0(n4_0))) ->_IH
.(1', gen_1':cons_.:cons_i2_0(0)) ->_R^Omega(1)
gen_1':cons_.:cons_i2_0(0)

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

(26)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
.(1', x) -> x
.(x, 1') -> x
.(i(x), x) -> 1'
.(x, i(x)) -> 1'
i(1') -> 1'
i(i(x)) -> x
.(i(y), .(y, z)) -> z
.(y, .(i(y), z)) -> z
encArg(1') -> 1'
encArg(cons_.(x_1, x_2)) -> .(encArg(x_1), encArg(x_2))
encArg(cons_i(x_1)) -> i(encArg(x_1))
encode_.(x_1, x_2) -> .(encArg(x_1), encArg(x_2))
encode_1 -> 1'
encode_i(x_1) -> i(encArg(x_1))

Types:
. :: 1':cons_.:cons_i -> 1':cons_.:cons_i -> 1':cons_.:cons_i
1' :: 1':cons_.:cons_i
i :: 1':cons_.:cons_i -> 1':cons_.:cons_i
encArg :: 1':cons_.:cons_i -> 1':cons_.:cons_i
cons_. :: 1':cons_.:cons_i -> 1':cons_.:cons_i -> 1':cons_.:cons_i
cons_i :: 1':cons_.:cons_i -> 1':cons_.:cons_i
encode_. :: 1':cons_.:cons_i -> 1':cons_.:cons_i -> 1':cons_.:cons_i
encode_1 :: 1':cons_.:cons_i
encode_i :: 1':cons_.:cons_i -> 1':cons_.:cons_i
hole_1':cons_.:cons_i1_0 :: 1':cons_.:cons_i
gen_1':cons_.:cons_i2_0 :: Nat -> 1':cons_.:cons_i


Generator Equations:
gen_1':cons_.:cons_i2_0(0) <=> 1'
gen_1':cons_.:cons_i2_0(+(x, 1)) <=> cons_.(1', gen_1':cons_.:cons_i2_0(x))


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

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

(28)
BOUNDS(n^1, INF)