/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 215 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), 194 ms]
(12) BEST
    (13) proven lower bound
        (14) LowerBoundPropagationProof [FINISHED, 0 ms]
        (15) BOUNDS(n^1, INF)
    (16) typed CpxTrs
        (17) RewriteLemmaProof [LOWER BOUND(ID), 216 ms]
        (18) BOUNDS(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:

   f(0) -> true
   f(1) -> false
   f(s(x)) -> f(x)
   if(true, s(x), s(y)) -> s(x)
   if(false, s(x), s(y)) -> s(y)
   g(x, c(y)) -> c(g(x, y))
   g(x, c(y)) -> g(x, if(f(x), c(g(s(x), y)), c(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(0) -> 0
   encArg(true) -> true
   encArg(1) -> 1
   encArg(false) -> false
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(c(x_1)) -> c(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
   encode_f(x_1) -> f(encArg(x_1))
   encode_0 -> 0
   encode_true -> true
   encode_1 -> 1
   encode_false -> false
   encode_s(x_1) -> s(encArg(x_1))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
   encode_c(x_1) -> c(encArg(x_1))


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

(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:

   f(0) -> true
   f(1) -> false
   f(s(x)) -> f(x)
   if(true, s(x), s(y)) -> s(x)
   if(false, s(x), s(y)) -> s(y)
   g(x, c(y)) -> c(g(x, y))
   g(x, c(y)) -> g(x, if(f(x), c(g(s(x), y)), c(y)))

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

   encArg(0) -> 0
   encArg(true) -> true
   encArg(1) -> 1
   encArg(false) -> false
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(c(x_1)) -> c(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
   encode_f(x_1) -> f(encArg(x_1))
   encode_0 -> 0
   encode_true -> true
   encode_1 -> 1
   encode_false -> false
   encode_s(x_1) -> s(encArg(x_1))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
   encode_c(x_1) -> c(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, INF).


The TRS R consists of the following rules:

   f(0) -> true
   f(1) -> false
   f(s(x)) -> f(x)
   if(true, s(x), s(y)) -> s(x)
   if(false, s(x), s(y)) -> s(y)
   g(x, c(y)) -> c(g(x, y))
   g(x, c(y)) -> g(x, if(f(x), c(g(s(x), y)), c(y)))

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

   encArg(0) -> 0
   encArg(true) -> true
   encArg(1) -> 1
   encArg(false) -> false
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(c(x_1)) -> c(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
   encode_f(x_1) -> f(encArg(x_1))
   encode_0 -> 0
   encode_true -> true
   encode_1 -> 1
   encode_false -> false
   encode_s(x_1) -> s(encArg(x_1))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
   encode_c(x_1) -> c(encArg(x_1))

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:

   f(0') -> true
   f(1') -> false
   f(s(x)) -> f(x)
   if(true, s(x), s(y)) -> s(x)
   if(false, s(x), s(y)) -> s(y)
   g(x, c(y)) -> c(g(x, y))
   g(x, c(y)) -> g(x, if(f(x), c(g(s(x), y)), c(y)))

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

   encArg(0') -> 0'
   encArg(true) -> true
   encArg(1') -> 1'
   encArg(false) -> false
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(c(x_1)) -> c(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
   encode_f(x_1) -> f(encArg(x_1))
   encode_0 -> 0'
   encode_true -> true
   encode_1 -> 1'
   encode_false -> false
   encode_s(x_1) -> s(encArg(x_1))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
   encode_c(x_1) -> c(encArg(x_1))

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

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

(8)
Obligation:
Innermost TRS:
Rules:
f(0') -> true
f(1') -> false
f(s(x)) -> f(x)
if(true, s(x), s(y)) -> s(x)
if(false, s(x), s(y)) -> s(y)
g(x, c(y)) -> c(g(x, y))
g(x, c(y)) -> g(x, if(f(x), c(g(s(x), y)), c(y)))
encArg(0') -> 0'
encArg(true) -> true
encArg(1') -> 1'
encArg(false) -> false
encArg(s(x_1)) -> s(encArg(x_1))
encArg(c(x_1)) -> c(encArg(x_1))
encArg(cons_f(x_1)) -> f(encArg(x_1))
encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
encode_f(x_1) -> f(encArg(x_1))
encode_0 -> 0'
encode_true -> true
encode_1 -> 1'
encode_false -> false
encode_s(x_1) -> s(encArg(x_1))
encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
encode_c(x_1) -> c(encArg(x_1))

Types:
f :: 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g
0' :: 0':true:1':false:s:c:cons_f:cons_if:cons_g
true :: 0':true:1':false:s:c:cons_f:cons_if:cons_g
1' :: 0':true:1':false:s:c:cons_f:cons_if:cons_g
false :: 0':true:1':false:s:c:cons_f:cons_if:cons_g
s :: 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g
if :: 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g
g :: 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g
c :: 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g
encArg :: 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g
cons_f :: 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g
cons_if :: 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g
cons_g :: 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g
encode_f :: 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g
encode_0 :: 0':true:1':false:s:c:cons_f:cons_if:cons_g
encode_true :: 0':true:1':false:s:c:cons_f:cons_if:cons_g
encode_1 :: 0':true:1':false:s:c:cons_f:cons_if:cons_g
encode_false :: 0':true:1':false:s:c:cons_f:cons_if:cons_g
encode_s :: 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g
encode_if :: 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g
encode_g :: 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g
encode_c :: 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g
hole_0':true:1':false:s:c:cons_f:cons_if:cons_g1_4 :: 0':true:1':false:s:c:cons_f:cons_if:cons_g
gen_0':true:1':false:s:c:cons_f:cons_if:cons_g2_4 :: Nat -> 0':true:1':false:s:c:cons_f:cons_if:cons_g

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

(9) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
f, g, encArg

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

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

(10)
Obligation:
Innermost TRS:
Rules:
f(0') -> true
f(1') -> false
f(s(x)) -> f(x)
if(true, s(x), s(y)) -> s(x)
if(false, s(x), s(y)) -> s(y)
g(x, c(y)) -> c(g(x, y))
g(x, c(y)) -> g(x, if(f(x), c(g(s(x), y)), c(y)))
encArg(0') -> 0'
encArg(true) -> true
encArg(1') -> 1'
encArg(false) -> false
encArg(s(x_1)) -> s(encArg(x_1))
encArg(c(x_1)) -> c(encArg(x_1))
encArg(cons_f(x_1)) -> f(encArg(x_1))
encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
encode_f(x_1) -> f(encArg(x_1))
encode_0 -> 0'
encode_true -> true
encode_1 -> 1'
encode_false -> false
encode_s(x_1) -> s(encArg(x_1))
encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
encode_c(x_1) -> c(encArg(x_1))

Types:
f :: 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g
0' :: 0':true:1':false:s:c:cons_f:cons_if:cons_g
true :: 0':true:1':false:s:c:cons_f:cons_if:cons_g
1' :: 0':true:1':false:s:c:cons_f:cons_if:cons_g
false :: 0':true:1':false:s:c:cons_f:cons_if:cons_g
s :: 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g
if :: 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g
g :: 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g
c :: 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g
encArg :: 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g
cons_f :: 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g
cons_if :: 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g
cons_g :: 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g
encode_f :: 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g
encode_0 :: 0':true:1':false:s:c:cons_f:cons_if:cons_g
encode_true :: 0':true:1':false:s:c:cons_f:cons_if:cons_g
encode_1 :: 0':true:1':false:s:c:cons_f:cons_if:cons_g
encode_false :: 0':true:1':false:s:c:cons_f:cons_if:cons_g
encode_s :: 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g
encode_if :: 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g
encode_g :: 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g
encode_c :: 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g
hole_0':true:1':false:s:c:cons_f:cons_if:cons_g1_4 :: 0':true:1':false:s:c:cons_f:cons_if:cons_g
gen_0':true:1':false:s:c:cons_f:cons_if:cons_g2_4 :: Nat -> 0':true:1':false:s:c:cons_f:cons_if:cons_g


Generator Equations:
gen_0':true:1':false:s:c:cons_f:cons_if:cons_g2_4(0) <=> 0'
gen_0':true:1':false:s:c:cons_f:cons_if:cons_g2_4(+(x, 1)) <=> s(gen_0':true:1':false:s:c:cons_f:cons_if:cons_g2_4(x))


The following defined symbols remain to be analysed:
f, g, encArg

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

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

(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
f(gen_0':true:1':false:s:c:cons_f:cons_if:cons_g2_4(n4_4)) -> true, rt in Omega(1 + n4_4)

Induction Base:
f(gen_0':true:1':false:s:c:cons_f:cons_if:cons_g2_4(0)) ->_R^Omega(1)
true

Induction Step:
f(gen_0':true:1':false:s:c:cons_f:cons_if:cons_g2_4(+(n4_4, 1))) ->_R^Omega(1)
f(gen_0':true:1':false:s:c:cons_f:cons_if:cons_g2_4(n4_4)) ->_IH
true

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

(12)
Complex Obligation (BEST)

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

(13)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
f(0') -> true
f(1') -> false
f(s(x)) -> f(x)
if(true, s(x), s(y)) -> s(x)
if(false, s(x), s(y)) -> s(y)
g(x, c(y)) -> c(g(x, y))
g(x, c(y)) -> g(x, if(f(x), c(g(s(x), y)), c(y)))
encArg(0') -> 0'
encArg(true) -> true
encArg(1') -> 1'
encArg(false) -> false
encArg(s(x_1)) -> s(encArg(x_1))
encArg(c(x_1)) -> c(encArg(x_1))
encArg(cons_f(x_1)) -> f(encArg(x_1))
encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
encode_f(x_1) -> f(encArg(x_1))
encode_0 -> 0'
encode_true -> true
encode_1 -> 1'
encode_false -> false
encode_s(x_1) -> s(encArg(x_1))
encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
encode_c(x_1) -> c(encArg(x_1))

Types:
f :: 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g
0' :: 0':true:1':false:s:c:cons_f:cons_if:cons_g
true :: 0':true:1':false:s:c:cons_f:cons_if:cons_g
1' :: 0':true:1':false:s:c:cons_f:cons_if:cons_g
false :: 0':true:1':false:s:c:cons_f:cons_if:cons_g
s :: 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g
if :: 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g
g :: 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g
c :: 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g
encArg :: 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g
cons_f :: 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g
cons_if :: 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g
cons_g :: 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g
encode_f :: 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g
encode_0 :: 0':true:1':false:s:c:cons_f:cons_if:cons_g
encode_true :: 0':true:1':false:s:c:cons_f:cons_if:cons_g
encode_1 :: 0':true:1':false:s:c:cons_f:cons_if:cons_g
encode_false :: 0':true:1':false:s:c:cons_f:cons_if:cons_g
encode_s :: 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g
encode_if :: 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g
encode_g :: 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g
encode_c :: 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g
hole_0':true:1':false:s:c:cons_f:cons_if:cons_g1_4 :: 0':true:1':false:s:c:cons_f:cons_if:cons_g
gen_0':true:1':false:s:c:cons_f:cons_if:cons_g2_4 :: Nat -> 0':true:1':false:s:c:cons_f:cons_if:cons_g


Generator Equations:
gen_0':true:1':false:s:c:cons_f:cons_if:cons_g2_4(0) <=> 0'
gen_0':true:1':false:s:c:cons_f:cons_if:cons_g2_4(+(x, 1)) <=> s(gen_0':true:1':false:s:c:cons_f:cons_if:cons_g2_4(x))


The following defined symbols remain to be analysed:
f, g, encArg

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

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

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

(15)
BOUNDS(n^1, INF)

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

(16)
Obligation:
Innermost TRS:
Rules:
f(0') -> true
f(1') -> false
f(s(x)) -> f(x)
if(true, s(x), s(y)) -> s(x)
if(false, s(x), s(y)) -> s(y)
g(x, c(y)) -> c(g(x, y))
g(x, c(y)) -> g(x, if(f(x), c(g(s(x), y)), c(y)))
encArg(0') -> 0'
encArg(true) -> true
encArg(1') -> 1'
encArg(false) -> false
encArg(s(x_1)) -> s(encArg(x_1))
encArg(c(x_1)) -> c(encArg(x_1))
encArg(cons_f(x_1)) -> f(encArg(x_1))
encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
encode_f(x_1) -> f(encArg(x_1))
encode_0 -> 0'
encode_true -> true
encode_1 -> 1'
encode_false -> false
encode_s(x_1) -> s(encArg(x_1))
encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
encode_c(x_1) -> c(encArg(x_1))

Types:
f :: 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g
0' :: 0':true:1':false:s:c:cons_f:cons_if:cons_g
true :: 0':true:1':false:s:c:cons_f:cons_if:cons_g
1' :: 0':true:1':false:s:c:cons_f:cons_if:cons_g
false :: 0':true:1':false:s:c:cons_f:cons_if:cons_g
s :: 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g
if :: 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g
g :: 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g
c :: 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g
encArg :: 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g
cons_f :: 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g
cons_if :: 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g
cons_g :: 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g
encode_f :: 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g
encode_0 :: 0':true:1':false:s:c:cons_f:cons_if:cons_g
encode_true :: 0':true:1':false:s:c:cons_f:cons_if:cons_g
encode_1 :: 0':true:1':false:s:c:cons_f:cons_if:cons_g
encode_false :: 0':true:1':false:s:c:cons_f:cons_if:cons_g
encode_s :: 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g
encode_if :: 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g
encode_g :: 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g
encode_c :: 0':true:1':false:s:c:cons_f:cons_if:cons_g -> 0':true:1':false:s:c:cons_f:cons_if:cons_g
hole_0':true:1':false:s:c:cons_f:cons_if:cons_g1_4 :: 0':true:1':false:s:c:cons_f:cons_if:cons_g
gen_0':true:1':false:s:c:cons_f:cons_if:cons_g2_4 :: Nat -> 0':true:1':false:s:c:cons_f:cons_if:cons_g


Lemmas:
f(gen_0':true:1':false:s:c:cons_f:cons_if:cons_g2_4(n4_4)) -> true, rt in Omega(1 + n4_4)


Generator Equations:
gen_0':true:1':false:s:c:cons_f:cons_if:cons_g2_4(0) <=> 0'
gen_0':true:1':false:s:c:cons_f:cons_if:cons_g2_4(+(x, 1)) <=> s(gen_0':true:1':false:s:c:cons_f:cons_if:cons_g2_4(x))


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

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

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_0':true:1':false:s:c:cons_f:cons_if:cons_g2_4(n251_4)) -> gen_0':true:1':false:s:c:cons_f:cons_if:cons_g2_4(n251_4), rt in Omega(0)

Induction Base:
encArg(gen_0':true:1':false:s:c:cons_f:cons_if:cons_g2_4(0)) ->_R^Omega(0)
0'

Induction Step:
encArg(gen_0':true:1':false:s:c:cons_f:cons_if:cons_g2_4(+(n251_4, 1))) ->_R^Omega(0)
s(encArg(gen_0':true:1':false:s:c:cons_f:cons_if:cons_g2_4(n251_4))) ->_IH
s(gen_0':true:1':false:s:c:cons_f:cons_if:cons_g2_4(c252_4))

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

(18)
BOUNDS(1, INF)