/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), ?)
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, INF).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 235 ms]
(4) CpxRelTRS
(5) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CpxRelTRS
(7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) typed CpxTrs
(9) OrderProof [LOWER BOUND(ID), 0 ms]
(10) typed CpxTrs
(11) RewriteLemmaProof [LOWER BOUND(ID), 526 ms]
(12) proven lower bound
(13) LowerBoundPropagationProof [FINISHED, 0 ms]
(14) BOUNDS(n^1, INF)


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

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


The TRS R consists of the following rules:

   *(0, x) -> 0
   *(1, x) -> x
   *(2, 2) -> .(1, 0)
   *(3, x) -> .(x, *(min, x))
   *(min, min) -> 1
   *(2, min) -> .(min, 2)
   *(.(x, y), z) -> .(*(x, z), *(y, z))
   *(+(y, z), x) -> +(*(x, y), *(x, z))
   +(0, x) -> x
   +(x, x) -> *(2, x)
   +(1, 2) -> 3
   +(1, min) -> 0
   +(2, min) -> 1
   +(3, x) -> .(1, +(min, x))
   +(.(x, y), z) -> .(x, +(y, z))
   +(*(2, x), x) -> *(3, x)
   +(*(min, x), x) -> 0
   +(*(2, v), *(min, v)) -> v
   .(min, 3) -> min
   .(x, min) -> .(+(min, x), 3)
   .(0, x) -> x
   .(x, .(y, z)) -> .(+(x, y), 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(0) -> 0
   encArg(1) -> 1
   encArg(2) -> 2
   encArg(3) -> 3
   encArg(min) -> min
   encArg(cons_*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2))
   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encArg(cons_.(x_1, x_2)) -> .(encArg(x_1), encArg(x_2))
   encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_1 -> 1
   encode_2 -> 2
   encode_.(x_1, x_2) -> .(encArg(x_1), encArg(x_2))
   encode_3 -> 3
   encode_min -> min
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))


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

(2)
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:

   *(0, x) -> 0
   *(1, x) -> x
   *(2, 2) -> .(1, 0)
   *(3, x) -> .(x, *(min, x))
   *(min, min) -> 1
   *(2, min) -> .(min, 2)
   *(.(x, y), z) -> .(*(x, z), *(y, z))
   *(+(y, z), x) -> +(*(x, y), *(x, z))
   +(0, x) -> x
   +(x, x) -> *(2, x)
   +(1, 2) -> 3
   +(1, min) -> 0
   +(2, min) -> 1
   +(3, x) -> .(1, +(min, x))
   +(.(x, y), z) -> .(x, +(y, z))
   +(*(2, x), x) -> *(3, x)
   +(*(min, x), x) -> 0
   +(*(2, v), *(min, v)) -> v
   .(min, 3) -> min
   .(x, min) -> .(+(min, x), 3)
   .(0, x) -> x
   .(x, .(y, z)) -> .(+(x, y), z)

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

   encArg(0) -> 0
   encArg(1) -> 1
   encArg(2) -> 2
   encArg(3) -> 3
   encArg(min) -> min
   encArg(cons_*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2))
   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encArg(cons_.(x_1, x_2)) -> .(encArg(x_1), encArg(x_2))
   encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_1 -> 1
   encode_2 -> 2
   encode_.(x_1, x_2) -> .(encArg(x_1), encArg(x_2))
   encode_3 -> 3
   encode_min -> min
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))

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, INF).


The TRS R consists of the following rules:

   *(0, x) -> 0
   *(1, x) -> x
   *(2, 2) -> .(1, 0)
   *(3, x) -> .(x, *(min, x))
   *(min, min) -> 1
   *(2, min) -> .(min, 2)
   *(.(x, y), z) -> .(*(x, z), *(y, z))
   *(+(y, z), x) -> +(*(x, y), *(x, z))
   +(0, x) -> x
   +(x, x) -> *(2, x)
   +(1, 2) -> 3
   +(1, min) -> 0
   +(2, min) -> 1
   +(3, x) -> .(1, +(min, x))
   +(.(x, y), z) -> .(x, +(y, z))
   +(*(2, x), x) -> *(3, x)
   +(*(min, x), x) -> 0
   +(*(2, v), *(min, v)) -> v
   .(min, 3) -> min
   .(x, min) -> .(+(min, x), 3)
   .(0, x) -> x
   .(x, .(y, z)) -> .(+(x, y), z)

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

   encArg(0) -> 0
   encArg(1) -> 1
   encArg(2) -> 2
   encArg(3) -> 3
   encArg(min) -> min
   encArg(cons_*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2))
   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encArg(cons_.(x_1, x_2)) -> .(encArg(x_1), encArg(x_2))
   encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_1 -> 1
   encode_2 -> 2
   encode_.(x_1, x_2) -> .(encArg(x_1), encArg(x_2))
   encode_3 -> 3
   encode_min -> min
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))

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

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

(6)
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:

   *'(0', x) -> 0'
   *'(1', x) -> x
   *'(2', 2') -> .(1', 0')
   *'(3', x) -> .(x, *'(min, x))
   *'(min, min) -> 1'
   *'(2', min) -> .(min, 2')
   *'(.(x, y), z) -> .(*'(x, z), *'(y, z))
   *'(+'(y, z), x) -> +'(*'(x, y), *'(x, z))
   +'(0', x) -> x
   +'(x, x) -> *'(2', x)
   +'(1', 2') -> 3'
   +'(1', min) -> 0'
   +'(2', min) -> 1'
   +'(3', x) -> .(1', +'(min, x))
   +'(.(x, y), z) -> .(x, +'(y, z))
   +'(*'(2', x), x) -> *'(3', x)
   +'(*'(min, x), x) -> 0'
   +'(*'(2', v), *'(min, v)) -> v
   .(min, 3') -> min
   .(x, min) -> .(+'(min, x), 3')
   .(0', x) -> x
   .(x, .(y, z)) -> .(+'(x, y), z)

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

   encArg(0') -> 0'
   encArg(1') -> 1'
   encArg(2') -> 2'
   encArg(3') -> 3'
   encArg(min) -> min
   encArg(cons_*(x_1, x_2)) -> *'(encArg(x_1), encArg(x_2))
   encArg(cons_+(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2))
   encArg(cons_.(x_1, x_2)) -> .(encArg(x_1), encArg(x_2))
   encode_*(x_1, x_2) -> *'(encArg(x_1), encArg(x_2))
   encode_0 -> 0'
   encode_1 -> 1'
   encode_2 -> 2'
   encode_.(x_1, x_2) -> .(encArg(x_1), encArg(x_2))
   encode_3 -> 3'
   encode_min -> min
   encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2))

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

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

(8)
Obligation:
Innermost TRS:
Rules:
*'(0', x) -> 0'
*'(1', x) -> x
*'(2', 2') -> .(1', 0')
*'(3', x) -> .(x, *'(min, x))
*'(min, min) -> 1'
*'(2', min) -> .(min, 2')
*'(.(x, y), z) -> .(*'(x, z), *'(y, z))
*'(+'(y, z), x) -> +'(*'(x, y), *'(x, z))
+'(0', x) -> x
+'(x, x) -> *'(2', x)
+'(1', 2') -> 3'
+'(1', min) -> 0'
+'(2', min) -> 1'
+'(3', x) -> .(1', +'(min, x))
+'(.(x, y), z) -> .(x, +'(y, z))
+'(*'(2', x), x) -> *'(3', x)
+'(*'(min, x), x) -> 0'
+'(*'(2', v), *'(min, v)) -> v
.(min, 3') -> min
.(x, min) -> .(+'(min, x), 3')
.(0', x) -> x
.(x, .(y, z)) -> .(+'(x, y), z)
encArg(0') -> 0'
encArg(1') -> 1'
encArg(2') -> 2'
encArg(3') -> 3'
encArg(min) -> min
encArg(cons_*(x_1, x_2)) -> *'(encArg(x_1), encArg(x_2))
encArg(cons_+(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2))
encArg(cons_.(x_1, x_2)) -> .(encArg(x_1), encArg(x_2))
encode_*(x_1, x_2) -> *'(encArg(x_1), encArg(x_2))
encode_0 -> 0'
encode_1 -> 1'
encode_2 -> 2'
encode_.(x_1, x_2) -> .(encArg(x_1), encArg(x_2))
encode_3 -> 3'
encode_min -> min
encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2))

Types:
*' :: 0':1':2':3':min:cons_*:cons_+:cons_. -> 0':1':2':3':min:cons_*:cons_+:cons_. -> 0':1':2':3':min:cons_*:cons_+:cons_.
0' :: 0':1':2':3':min:cons_*:cons_+:cons_.
1' :: 0':1':2':3':min:cons_*:cons_+:cons_.
2' :: 0':1':2':3':min:cons_*:cons_+:cons_.
. :: 0':1':2':3':min:cons_*:cons_+:cons_. -> 0':1':2':3':min:cons_*:cons_+:cons_. -> 0':1':2':3':min:cons_*:cons_+:cons_.
3' :: 0':1':2':3':min:cons_*:cons_+:cons_.
min :: 0':1':2':3':min:cons_*:cons_+:cons_.
+' :: 0':1':2':3':min:cons_*:cons_+:cons_. -> 0':1':2':3':min:cons_*:cons_+:cons_. -> 0':1':2':3':min:cons_*:cons_+:cons_.
encArg :: 0':1':2':3':min:cons_*:cons_+:cons_. -> 0':1':2':3':min:cons_*:cons_+:cons_.
cons_* :: 0':1':2':3':min:cons_*:cons_+:cons_. -> 0':1':2':3':min:cons_*:cons_+:cons_. -> 0':1':2':3':min:cons_*:cons_+:cons_.
cons_+ :: 0':1':2':3':min:cons_*:cons_+:cons_. -> 0':1':2':3':min:cons_*:cons_+:cons_. -> 0':1':2':3':min:cons_*:cons_+:cons_.
cons_. :: 0':1':2':3':min:cons_*:cons_+:cons_. -> 0':1':2':3':min:cons_*:cons_+:cons_. -> 0':1':2':3':min:cons_*:cons_+:cons_.
encode_* :: 0':1':2':3':min:cons_*:cons_+:cons_. -> 0':1':2':3':min:cons_*:cons_+:cons_. -> 0':1':2':3':min:cons_*:cons_+:cons_.
encode_0 :: 0':1':2':3':min:cons_*:cons_+:cons_.
encode_1 :: 0':1':2':3':min:cons_*:cons_+:cons_.
encode_2 :: 0':1':2':3':min:cons_*:cons_+:cons_.
encode_. :: 0':1':2':3':min:cons_*:cons_+:cons_. -> 0':1':2':3':min:cons_*:cons_+:cons_. -> 0':1':2':3':min:cons_*:cons_+:cons_.
encode_3 :: 0':1':2':3':min:cons_*:cons_+:cons_.
encode_min :: 0':1':2':3':min:cons_*:cons_+:cons_.
encode_+ :: 0':1':2':3':min:cons_*:cons_+:cons_. -> 0':1':2':3':min:cons_*:cons_+:cons_. -> 0':1':2':3':min:cons_*:cons_+:cons_.
hole_0':1':2':3':min:cons_*:cons_+:cons_.1_4 :: 0':1':2':3':min:cons_*:cons_+:cons_.
gen_0':1':2':3':min:cons_*:cons_+:cons_.2_4 :: Nat -> 0':1':2':3':min:cons_*:cons_+:cons_.

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

(9) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
*', ., +', encArg

They will be analysed ascendingly in the following order:
*' = .
*' = +'
*' < encArg
. = +'
. < encArg
+' < encArg

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

(10)
Obligation:
Innermost TRS:
Rules:
*'(0', x) -> 0'
*'(1', x) -> x
*'(2', 2') -> .(1', 0')
*'(3', x) -> .(x, *'(min, x))
*'(min, min) -> 1'
*'(2', min) -> .(min, 2')
*'(.(x, y), z) -> .(*'(x, z), *'(y, z))
*'(+'(y, z), x) -> +'(*'(x, y), *'(x, z))
+'(0', x) -> x
+'(x, x) -> *'(2', x)
+'(1', 2') -> 3'
+'(1', min) -> 0'
+'(2', min) -> 1'
+'(3', x) -> .(1', +'(min, x))
+'(.(x, y), z) -> .(x, +'(y, z))
+'(*'(2', x), x) -> *'(3', x)
+'(*'(min, x), x) -> 0'
+'(*'(2', v), *'(min, v)) -> v
.(min, 3') -> min
.(x, min) -> .(+'(min, x), 3')
.(0', x) -> x
.(x, .(y, z)) -> .(+'(x, y), z)
encArg(0') -> 0'
encArg(1') -> 1'
encArg(2') -> 2'
encArg(3') -> 3'
encArg(min) -> min
encArg(cons_*(x_1, x_2)) -> *'(encArg(x_1), encArg(x_2))
encArg(cons_+(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2))
encArg(cons_.(x_1, x_2)) -> .(encArg(x_1), encArg(x_2))
encode_*(x_1, x_2) -> *'(encArg(x_1), encArg(x_2))
encode_0 -> 0'
encode_1 -> 1'
encode_2 -> 2'
encode_.(x_1, x_2) -> .(encArg(x_1), encArg(x_2))
encode_3 -> 3'
encode_min -> min
encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2))

Types:
*' :: 0':1':2':3':min:cons_*:cons_+:cons_. -> 0':1':2':3':min:cons_*:cons_+:cons_. -> 0':1':2':3':min:cons_*:cons_+:cons_.
0' :: 0':1':2':3':min:cons_*:cons_+:cons_.
1' :: 0':1':2':3':min:cons_*:cons_+:cons_.
2' :: 0':1':2':3':min:cons_*:cons_+:cons_.
. :: 0':1':2':3':min:cons_*:cons_+:cons_. -> 0':1':2':3':min:cons_*:cons_+:cons_. -> 0':1':2':3':min:cons_*:cons_+:cons_.
3' :: 0':1':2':3':min:cons_*:cons_+:cons_.
min :: 0':1':2':3':min:cons_*:cons_+:cons_.
+' :: 0':1':2':3':min:cons_*:cons_+:cons_. -> 0':1':2':3':min:cons_*:cons_+:cons_. -> 0':1':2':3':min:cons_*:cons_+:cons_.
encArg :: 0':1':2':3':min:cons_*:cons_+:cons_. -> 0':1':2':3':min:cons_*:cons_+:cons_.
cons_* :: 0':1':2':3':min:cons_*:cons_+:cons_. -> 0':1':2':3':min:cons_*:cons_+:cons_. -> 0':1':2':3':min:cons_*:cons_+:cons_.
cons_+ :: 0':1':2':3':min:cons_*:cons_+:cons_. -> 0':1':2':3':min:cons_*:cons_+:cons_. -> 0':1':2':3':min:cons_*:cons_+:cons_.
cons_. :: 0':1':2':3':min:cons_*:cons_+:cons_. -> 0':1':2':3':min:cons_*:cons_+:cons_. -> 0':1':2':3':min:cons_*:cons_+:cons_.
encode_* :: 0':1':2':3':min:cons_*:cons_+:cons_. -> 0':1':2':3':min:cons_*:cons_+:cons_. -> 0':1':2':3':min:cons_*:cons_+:cons_.
encode_0 :: 0':1':2':3':min:cons_*:cons_+:cons_.
encode_1 :: 0':1':2':3':min:cons_*:cons_+:cons_.
encode_2 :: 0':1':2':3':min:cons_*:cons_+:cons_.
encode_. :: 0':1':2':3':min:cons_*:cons_+:cons_. -> 0':1':2':3':min:cons_*:cons_+:cons_. -> 0':1':2':3':min:cons_*:cons_+:cons_.
encode_3 :: 0':1':2':3':min:cons_*:cons_+:cons_.
encode_min :: 0':1':2':3':min:cons_*:cons_+:cons_.
encode_+ :: 0':1':2':3':min:cons_*:cons_+:cons_. -> 0':1':2':3':min:cons_*:cons_+:cons_. -> 0':1':2':3':min:cons_*:cons_+:cons_.
hole_0':1':2':3':min:cons_*:cons_+:cons_.1_4 :: 0':1':2':3':min:cons_*:cons_+:cons_.
gen_0':1':2':3':min:cons_*:cons_+:cons_.2_4 :: Nat -> 0':1':2':3':min:cons_*:cons_+:cons_.


Generator Equations:
gen_0':1':2':3':min:cons_*:cons_+:cons_.2_4(0) <=> 0'
gen_0':1':2':3':min:cons_*:cons_+:cons_.2_4(+(x, 1)) <=> cons_*(0', gen_0':1':2':3':min:cons_*:cons_+:cons_.2_4(x))


The following defined symbols remain to be analysed:
., *', +', encArg

They will be analysed ascendingly in the following order:
*' = .
*' = +'
*' < encArg
. = +'
. < encArg
+' < encArg

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

(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_0':1':2':3':min:cons_*:cons_+:cons_.2_4(n4161_4)) -> gen_0':1':2':3':min:cons_*:cons_+:cons_.2_4(0), rt in Omega(n4161_4)

Induction Base:
encArg(gen_0':1':2':3':min:cons_*:cons_+:cons_.2_4(0)) ->_R^Omega(0)
0'

Induction Step:
encArg(gen_0':1':2':3':min:cons_*:cons_+:cons_.2_4(+(n4161_4, 1))) ->_R^Omega(0)
*'(encArg(0'), encArg(gen_0':1':2':3':min:cons_*:cons_+:cons_.2_4(n4161_4))) ->_R^Omega(0)
*'(0', encArg(gen_0':1':2':3':min:cons_*:cons_+:cons_.2_4(n4161_4))) ->_IH
*'(0', gen_0':1':2':3':min:cons_*:cons_+:cons_.2_4(0)) ->_R^Omega(1)
0'

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

(12)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
*'(0', x) -> 0'
*'(1', x) -> x
*'(2', 2') -> .(1', 0')
*'(3', x) -> .(x, *'(min, x))
*'(min, min) -> 1'
*'(2', min) -> .(min, 2')
*'(.(x, y), z) -> .(*'(x, z), *'(y, z))
*'(+'(y, z), x) -> +'(*'(x, y), *'(x, z))
+'(0', x) -> x
+'(x, x) -> *'(2', x)
+'(1', 2') -> 3'
+'(1', min) -> 0'
+'(2', min) -> 1'
+'(3', x) -> .(1', +'(min, x))
+'(.(x, y), z) -> .(x, +'(y, z))
+'(*'(2', x), x) -> *'(3', x)
+'(*'(min, x), x) -> 0'
+'(*'(2', v), *'(min, v)) -> v
.(min, 3') -> min
.(x, min) -> .(+'(min, x), 3')
.(0', x) -> x
.(x, .(y, z)) -> .(+'(x, y), z)
encArg(0') -> 0'
encArg(1') -> 1'
encArg(2') -> 2'
encArg(3') -> 3'
encArg(min) -> min
encArg(cons_*(x_1, x_2)) -> *'(encArg(x_1), encArg(x_2))
encArg(cons_+(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2))
encArg(cons_.(x_1, x_2)) -> .(encArg(x_1), encArg(x_2))
encode_*(x_1, x_2) -> *'(encArg(x_1), encArg(x_2))
encode_0 -> 0'
encode_1 -> 1'
encode_2 -> 2'
encode_.(x_1, x_2) -> .(encArg(x_1), encArg(x_2))
encode_3 -> 3'
encode_min -> min
encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2))

Types:
*' :: 0':1':2':3':min:cons_*:cons_+:cons_. -> 0':1':2':3':min:cons_*:cons_+:cons_. -> 0':1':2':3':min:cons_*:cons_+:cons_.
0' :: 0':1':2':3':min:cons_*:cons_+:cons_.
1' :: 0':1':2':3':min:cons_*:cons_+:cons_.
2' :: 0':1':2':3':min:cons_*:cons_+:cons_.
. :: 0':1':2':3':min:cons_*:cons_+:cons_. -> 0':1':2':3':min:cons_*:cons_+:cons_. -> 0':1':2':3':min:cons_*:cons_+:cons_.
3' :: 0':1':2':3':min:cons_*:cons_+:cons_.
min :: 0':1':2':3':min:cons_*:cons_+:cons_.
+' :: 0':1':2':3':min:cons_*:cons_+:cons_. -> 0':1':2':3':min:cons_*:cons_+:cons_. -> 0':1':2':3':min:cons_*:cons_+:cons_.
encArg :: 0':1':2':3':min:cons_*:cons_+:cons_. -> 0':1':2':3':min:cons_*:cons_+:cons_.
cons_* :: 0':1':2':3':min:cons_*:cons_+:cons_. -> 0':1':2':3':min:cons_*:cons_+:cons_. -> 0':1':2':3':min:cons_*:cons_+:cons_.
cons_+ :: 0':1':2':3':min:cons_*:cons_+:cons_. -> 0':1':2':3':min:cons_*:cons_+:cons_. -> 0':1':2':3':min:cons_*:cons_+:cons_.
cons_. :: 0':1':2':3':min:cons_*:cons_+:cons_. -> 0':1':2':3':min:cons_*:cons_+:cons_. -> 0':1':2':3':min:cons_*:cons_+:cons_.
encode_* :: 0':1':2':3':min:cons_*:cons_+:cons_. -> 0':1':2':3':min:cons_*:cons_+:cons_. -> 0':1':2':3':min:cons_*:cons_+:cons_.
encode_0 :: 0':1':2':3':min:cons_*:cons_+:cons_.
encode_1 :: 0':1':2':3':min:cons_*:cons_+:cons_.
encode_2 :: 0':1':2':3':min:cons_*:cons_+:cons_.
encode_. :: 0':1':2':3':min:cons_*:cons_+:cons_. -> 0':1':2':3':min:cons_*:cons_+:cons_. -> 0':1':2':3':min:cons_*:cons_+:cons_.
encode_3 :: 0':1':2':3':min:cons_*:cons_+:cons_.
encode_min :: 0':1':2':3':min:cons_*:cons_+:cons_.
encode_+ :: 0':1':2':3':min:cons_*:cons_+:cons_. -> 0':1':2':3':min:cons_*:cons_+:cons_. -> 0':1':2':3':min:cons_*:cons_+:cons_.
hole_0':1':2':3':min:cons_*:cons_+:cons_.1_4 :: 0':1':2':3':min:cons_*:cons_+:cons_.
gen_0':1':2':3':min:cons_*:cons_+:cons_.2_4 :: Nat -> 0':1':2':3':min:cons_*:cons_+:cons_.


Generator Equations:
gen_0':1':2':3':min:cons_*:cons_+:cons_.2_4(0) <=> 0'
gen_0':1':2':3':min:cons_*:cons_+:cons_.2_4(+(x, 1)) <=> cons_*(0', gen_0':1':2':3':min:cons_*:cons_+:cons_.2_4(x))


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

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

(14)
BOUNDS(n^1, INF)