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


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


KILLED
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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(1, INF).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 226 ms]
(4) CpxRelTRS
(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(6) TRS for Loop Detection
(7) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) CpxRelTRS
    (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) typed CpxTrs
    (11) OrderProof [LOWER BOUND(ID), 0 ms]
    (12) typed CpxTrs
    (13) RewriteLemmaProof [LOWER BOUND(ID), 999 ms]
    (14) BOUNDS(1, INF)
(15) RelTrsToTrsProof [UPPER BOUND(ID), 1 ms]
(16) CpxTRS
(17) NonCtorToCtorProof [UPPER BOUND(ID), 0 ms]
(18) CpxRelTRS
    (19) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (20) CpxWeightedTrs
    (21) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (22) CpxTypedWeightedTrs
    (23) CompletionProof [UPPER BOUND(ID), 0 ms]
    (24) CpxTypedWeightedCompleteTrs
        (25) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (26) CpxRNTS
    (27) CompletionProof [UPPER BOUND(ID), 0 ms]
    (28) CpxTypedWeightedCompleteTrs
        (29) NarrowingProof [BOTH BOUNDS(ID, ID), 137 ms]
        (30) CpxTypedWeightedCompleteTrs
        (31) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (32) CpxRNTS
        (33) InliningProof [UPPER BOUND(ID), 2009 ms]
        (34) CpxRNTS
        (35) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (36) CpxRNTS
        (37) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
        (38) CpxRNTS
        (39) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (40) CpxRNTS
        (41) IntTrsBoundProof [UPPER BOUND(ID), 170 ms]
        (42) CpxRNTS
        (43) IntTrsBoundProof [UPPER BOUND(ID), 55 ms]
        (44) CpxRNTS
        (45) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (46) CpxRNTS
        (47) IntTrsBoundProof [UPPER BOUND(ID), 4 ms]
        (48) CpxRNTS
        (49) IntTrsBoundProof [UPPER BOUND(ID), 1 ms]
        (50) CpxRNTS
        (51) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (52) CpxRNTS
        (53) IntTrsBoundProof [UPPER BOUND(ID), 260 ms]
        (54) CpxRNTS
        (55) IntTrsBoundProof [UPPER BOUND(ID), 3 ms]
        (56) CpxRNTS
        (57) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (58) CpxRNTS
        (59) IntTrsBoundProof [UPPER BOUND(ID), 67 ms]
        (60) CpxRNTS
        (61) IntTrsBoundProof [UPPER BOUND(ID), 1 ms]
        (62) CpxRNTS
        (63) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (64) CpxRNTS
        (65) IntTrsBoundProof [UPPER BOUND(ID), 543 ms]
        (66) CpxRNTS
        (67) IntTrsBoundProof [UPPER BOUND(ID), 165 ms]
        (68) CpxRNTS
        (69) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (70) CpxRNTS
        (71) IntTrsBoundProof [UPPER BOUND(ID), 1960 ms]
        (72) CpxRNTS
        (73) IntTrsBoundProof [UPPER BOUND(ID), 418 ms]
        (74) CpxRNTS
        (75) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (76) CpxRNTS
        (77) IntTrsBoundProof [UPPER BOUND(ID), 2660 ms]
        (78) CpxRNTS
        (79) IntTrsBoundProof [UPPER BOUND(ID), 713 ms]
        (80) CpxRNTS
        (81) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (82) CpxRNTS
        (83) IntTrsBoundProof [UPPER BOUND(ID), 16.3 s]
        (84) CpxRNTS
        (85) IntTrsBoundProof [UPPER BOUND(ID), 328 ms]
        (86) CpxRNTS
(87) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(88) CdtProblem
    (89) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (90) CdtProblem
    (91) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (92) CdtProblem
    (93) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (94) CdtProblem
    (95) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (96) CdtProblem
    (97) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 183 ms]
    (98) CdtProblem
    (99) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 540 ms]
    (100) CdtProblem
    (101) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (102) CdtProblem
    (103) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (104) CdtProblem
    (105) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (106) CdtProblem
    (107) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (108) CdtProblem
    (109) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (110) CdtProblem
    (111) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (112) CdtProblem
    (113) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (114) CdtProblem
    (115) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (116) CdtProblem
    (117) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (118) CdtProblem
    (119) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (120) CdtProblem
    (121) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (122) CdtProblem
    (123) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 1 ms]
    (124) CdtProblem
    (125) CdtRewritingProof [BOTH BOUNDS(ID, ID), 16 ms]
    (126) CdtProblem
    (127) CdtRewritingProof [BOTH BOUNDS(ID, ID), 17 ms]
    (128) CdtProblem


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

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


The TRS R consists of the following rules:

   not(x) -> xor(x, true)
   or(x, y) -> xor(and(x, y), xor(x, y))
   implies(x, y) -> xor(and(x, y), xor(x, true))
   and(x, true) -> x
   and(x, false) -> false
   and(x, x) -> x
   xor(x, false) -> x
   xor(x, x) -> false
   and(xor(x, y), z) -> xor(and(x, z), and(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(true) -> true
   encArg(false) -> false
   encArg(cons_not(x_1)) -> not(encArg(x_1))
   encArg(cons_or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2))
   encArg(cons_implies(x_1, x_2)) -> implies(encArg(x_1), encArg(x_2))
   encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2))
   encArg(cons_xor(x_1, x_2)) -> xor(encArg(x_1), encArg(x_2))
   encode_not(x_1) -> not(encArg(x_1))
   encode_xor(x_1, x_2) -> xor(encArg(x_1), encArg(x_2))
   encode_true -> true
   encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2))
   encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2))
   encode_implies(x_1, x_2) -> implies(encArg(x_1), encArg(x_2))
   encode_false -> false


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

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


The TRS R consists of the following rules:

   not(x) -> xor(x, true)
   or(x, y) -> xor(and(x, y), xor(x, y))
   implies(x, y) -> xor(and(x, y), xor(x, true))
   and(x, true) -> x
   and(x, false) -> false
   and(x, x) -> x
   xor(x, false) -> x
   xor(x, x) -> false
   and(xor(x, y), z) -> xor(and(x, z), and(y, z))

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

   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_not(x_1)) -> not(encArg(x_1))
   encArg(cons_or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2))
   encArg(cons_implies(x_1, x_2)) -> implies(encArg(x_1), encArg(x_2))
   encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2))
   encArg(cons_xor(x_1, x_2)) -> xor(encArg(x_1), encArg(x_2))
   encode_not(x_1) -> not(encArg(x_1))
   encode_xor(x_1, x_2) -> xor(encArg(x_1), encArg(x_2))
   encode_true -> true
   encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2))
   encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2))
   encode_implies(x_1, x_2) -> implies(encArg(x_1), encArg(x_2))
   encode_false -> false

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(1, INF).


The TRS R consists of the following rules:

   not(x) -> xor(x, true)
   or(x, y) -> xor(and(x, y), xor(x, y))
   implies(x, y) -> xor(and(x, y), xor(x, true))
   and(x, true) -> x
   and(x, false) -> false
   and(x, x) -> x
   xor(x, false) -> x
   xor(x, x) -> false
   and(xor(x, y), z) -> xor(and(x, z), and(y, z))

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

   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_not(x_1)) -> not(encArg(x_1))
   encArg(cons_or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2))
   encArg(cons_implies(x_1, x_2)) -> implies(encArg(x_1), encArg(x_2))
   encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2))
   encArg(cons_xor(x_1, x_2)) -> xor(encArg(x_1), encArg(x_2))
   encode_not(x_1) -> not(encArg(x_1))
   encode_xor(x_1, x_2) -> xor(encArg(x_1), encArg(x_2))
   encode_true -> true
   encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2))
   encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2))
   encode_implies(x_1, x_2) -> implies(encArg(x_1), encArg(x_2))
   encode_false -> false

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

(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
----------------------------------------

(6)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   not(x) -> xor(x, true)
   or(x, y) -> xor(and(x, y), xor(x, y))
   implies(x, y) -> xor(and(x, y), xor(x, true))
   and(x, true) -> x
   and(x, false) -> false
   and(x, x) -> x
   xor(x, false) -> x
   xor(x, x) -> false
   and(xor(x, y), z) -> xor(and(x, z), and(y, z))

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

   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_not(x_1)) -> not(encArg(x_1))
   encArg(cons_or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2))
   encArg(cons_implies(x_1, x_2)) -> implies(encArg(x_1), encArg(x_2))
   encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2))
   encArg(cons_xor(x_1, x_2)) -> xor(encArg(x_1), encArg(x_2))
   encode_not(x_1) -> not(encArg(x_1))
   encode_xor(x_1, x_2) -> xor(encArg(x_1), encArg(x_2))
   encode_true -> true
   encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2))
   encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2))
   encode_implies(x_1, x_2) -> implies(encArg(x_1), encArg(x_2))
   encode_false -> false

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

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

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


The TRS R consists of the following rules:

   not(x) -> xor(x, true)
   or(x, y) -> xor(and(x, y), xor(x, y))
   implies(x, y) -> xor(and(x, y), xor(x, true))
   and(x, true) -> x
   and(x, false) -> false
   and(x, x) -> x
   xor(x, false) -> x
   xor(x, x) -> false
   and(xor(x, y), z) -> xor(and(x, z), and(y, z))

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

   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_not(x_1)) -> not(encArg(x_1))
   encArg(cons_or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2))
   encArg(cons_implies(x_1, x_2)) -> implies(encArg(x_1), encArg(x_2))
   encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2))
   encArg(cons_xor(x_1, x_2)) -> xor(encArg(x_1), encArg(x_2))
   encode_not(x_1) -> not(encArg(x_1))
   encode_xor(x_1, x_2) -> xor(encArg(x_1), encArg(x_2))
   encode_true -> true
   encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2))
   encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2))
   encode_implies(x_1, x_2) -> implies(encArg(x_1), encArg(x_2))
   encode_false -> false

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

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

(10)
Obligation:
Innermost TRS:
Rules:
not(x) -> xor(x, true)
or(x, y) -> xor(and(x, y), xor(x, y))
implies(x, y) -> xor(and(x, y), xor(x, true))
and(x, true) -> x
and(x, false) -> false
and(x, x) -> x
xor(x, false) -> x
xor(x, x) -> false
and(xor(x, y), z) -> xor(and(x, z), and(y, z))
encArg(true) -> true
encArg(false) -> false
encArg(cons_not(x_1)) -> not(encArg(x_1))
encArg(cons_or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2))
encArg(cons_implies(x_1, x_2)) -> implies(encArg(x_1), encArg(x_2))
encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2))
encArg(cons_xor(x_1, x_2)) -> xor(encArg(x_1), encArg(x_2))
encode_not(x_1) -> not(encArg(x_1))
encode_xor(x_1, x_2) -> xor(encArg(x_1), encArg(x_2))
encode_true -> true
encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2))
encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2))
encode_implies(x_1, x_2) -> implies(encArg(x_1), encArg(x_2))
encode_false -> false

Types:
not :: true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor
xor :: true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor
true :: true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor
or :: true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor
and :: true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor
implies :: true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor
false :: true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor
encArg :: true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor
cons_not :: true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor
cons_or :: true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor
cons_implies :: true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor
cons_and :: true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor
cons_xor :: true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor
encode_not :: true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor
encode_xor :: true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor
encode_true :: true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor
encode_or :: true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor
encode_and :: true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor
encode_implies :: true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor
encode_false :: true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor
hole_true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor1_0 :: true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor
gen_true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor2_0 :: Nat -> true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor

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

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

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

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

(12)
Obligation:
Innermost TRS:
Rules:
not(x) -> xor(x, true)
or(x, y) -> xor(and(x, y), xor(x, y))
implies(x, y) -> xor(and(x, y), xor(x, true))
and(x, true) -> x
and(x, false) -> false
and(x, x) -> x
xor(x, false) -> x
xor(x, x) -> false
and(xor(x, y), z) -> xor(and(x, z), and(y, z))
encArg(true) -> true
encArg(false) -> false
encArg(cons_not(x_1)) -> not(encArg(x_1))
encArg(cons_or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2))
encArg(cons_implies(x_1, x_2)) -> implies(encArg(x_1), encArg(x_2))
encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2))
encArg(cons_xor(x_1, x_2)) -> xor(encArg(x_1), encArg(x_2))
encode_not(x_1) -> not(encArg(x_1))
encode_xor(x_1, x_2) -> xor(encArg(x_1), encArg(x_2))
encode_true -> true
encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2))
encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2))
encode_implies(x_1, x_2) -> implies(encArg(x_1), encArg(x_2))
encode_false -> false

Types:
not :: true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor
xor :: true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor
true :: true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor
or :: true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor
and :: true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor
implies :: true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor
false :: true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor
encArg :: true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor
cons_not :: true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor
cons_or :: true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor
cons_implies :: true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor
cons_and :: true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor
cons_xor :: true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor
encode_not :: true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor
encode_xor :: true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor
encode_true :: true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor
encode_or :: true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor
encode_and :: true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor
encode_implies :: true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor
encode_false :: true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor
hole_true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor1_0 :: true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor
gen_true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor2_0 :: Nat -> true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor


Generator Equations:
gen_true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor2_0(0) <=> true
gen_true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor2_0(+(x, 1)) <=> cons_not(gen_true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor2_0(x))


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

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

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor2_0(+(1, n26_0))) -> *3_0, rt in Omega(0)

Induction Base:
encArg(gen_true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor2_0(+(1, 0)))

Induction Step:
encArg(gen_true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor2_0(+(1, +(n26_0, 1)))) ->_R^Omega(0)
not(encArg(gen_true:false:cons_not:cons_or:cons_implies:cons_and:cons_xor2_0(+(1, n26_0)))) ->_IH
not(*3_0)

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

(14)
BOUNDS(1, INF)

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

(15) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
----------------------------------------

(16)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   not(x) -> xor(x, true)
   or(x, y) -> xor(and(x, y), xor(x, y))
   implies(x, y) -> xor(and(x, y), xor(x, true))
   and(x, true) -> x
   and(x, false) -> false
   and(x, x) -> x
   xor(x, false) -> x
   xor(x, x) -> false
   and(xor(x, y), z) -> xor(and(x, z), and(y, z))
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_not(x_1)) -> not(encArg(x_1))
   encArg(cons_or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2))
   encArg(cons_implies(x_1, x_2)) -> implies(encArg(x_1), encArg(x_2))
   encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2))
   encArg(cons_xor(x_1, x_2)) -> xor(encArg(x_1), encArg(x_2))
   encode_not(x_1) -> not(encArg(x_1))
   encode_xor(x_1, x_2) -> xor(encArg(x_1), encArg(x_2))
   encode_true -> true
   encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2))
   encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2))
   encode_implies(x_1, x_2) -> implies(encArg(x_1), encArg(x_2))
   encode_false -> false

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

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

(18)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   not(x) -> xor(x, true)
   or(x, y) -> xor(and(x, y), xor(x, y))
   implies(x, y) -> xor(and(x, y), xor(x, true))
   and(x, true) -> x
   and(x, false) -> false
   and(x, x) -> x
   xor(x, false) -> x
   xor(x, x) -> false
   and(c_xor(x, y), z) -> xor(and(x, z), and(y, z))

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

   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_not(x_1)) -> not(encArg(x_1))
   encArg(cons_or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2))
   encArg(cons_implies(x_1, x_2)) -> implies(encArg(x_1), encArg(x_2))
   encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2))
   encArg(cons_xor(x_1, x_2)) -> xor(encArg(x_1), encArg(x_2))
   encode_not(x_1) -> not(encArg(x_1))
   encode_xor(x_1, x_2) -> xor(encArg(x_1), encArg(x_2))
   encode_true -> true
   encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2))
   encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2))
   encode_implies(x_1, x_2) -> implies(encArg(x_1), encArg(x_2))
   encode_false -> false
   xor(x0, x1) -> c_xor(x0, x1)

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

(19) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
Transformed relative TRS to weighted TRS
----------------------------------------

(20)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   not(x) -> xor(x, true) [1]
   or(x, y) -> xor(and(x, y), xor(x, y)) [1]
   implies(x, y) -> xor(and(x, y), xor(x, true)) [1]
   and(x, true) -> x [1]
   and(x, false) -> false [1]
   and(x, x) -> x [1]
   xor(x, false) -> x [1]
   xor(x, x) -> false [1]
   and(c_xor(x, y), z) -> xor(and(x, z), and(y, z)) [1]
   encArg(true) -> true [0]
   encArg(false) -> false [0]
   encArg(cons_not(x_1)) -> not(encArg(x_1)) [0]
   encArg(cons_or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_implies(x_1, x_2)) -> implies(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_xor(x_1, x_2)) -> xor(encArg(x_1), encArg(x_2)) [0]
   encode_not(x_1) -> not(encArg(x_1)) [0]
   encode_xor(x_1, x_2) -> xor(encArg(x_1), encArg(x_2)) [0]
   encode_true -> true [0]
   encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2)) [0]
   encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) [0]
   encode_implies(x_1, x_2) -> implies(encArg(x_1), encArg(x_2)) [0]
   encode_false -> false [0]
   xor(x0, x1) -> c_xor(x0, x1) [0]

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

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

(22)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   not(x) -> xor(x, true) [1]
   or(x, y) -> xor(and(x, y), xor(x, y)) [1]
   implies(x, y) -> xor(and(x, y), xor(x, true)) [1]
   and(x, true) -> x [1]
   and(x, false) -> false [1]
   and(x, x) -> x [1]
   xor(x, false) -> x [1]
   xor(x, x) -> false [1]
   and(c_xor(x, y), z) -> xor(and(x, z), and(y, z)) [1]
   encArg(true) -> true [0]
   encArg(false) -> false [0]
   encArg(cons_not(x_1)) -> not(encArg(x_1)) [0]
   encArg(cons_or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_implies(x_1, x_2)) -> implies(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_xor(x_1, x_2)) -> xor(encArg(x_1), encArg(x_2)) [0]
   encode_not(x_1) -> not(encArg(x_1)) [0]
   encode_xor(x_1, x_2) -> xor(encArg(x_1), encArg(x_2)) [0]
   encode_true -> true [0]
   encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2)) [0]
   encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) [0]
   encode_implies(x_1, x_2) -> implies(encArg(x_1), encArg(x_2)) [0]
   encode_false -> false [0]
   xor(x0, x1) -> c_xor(x0, x1) [0]

The TRS has the following type information:
not :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor
xor :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor
true :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor
or :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor
and :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor
implies :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor
false :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor
c_xor :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor
encArg :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor
cons_not :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor
cons_or :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor
cons_implies :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor
cons_and :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor
cons_xor :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor
encode_not :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor
encode_xor :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor
encode_true :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor
encode_or :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor
encode_and :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor
encode_implies :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor
encode_false :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor

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

(23) CompletionProof (UPPER BOUND(ID))
The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:

   encArg(v0) -> null_encArg [0]
   encode_not(v0) -> null_encode_not [0]
   encode_xor(v0, v1) -> null_encode_xor [0]
   encode_true -> null_encode_true [0]
   encode_or(v0, v1) -> null_encode_or [0]
   encode_and(v0, v1) -> null_encode_and [0]
   encode_implies(v0, v1) -> null_encode_implies [0]
   encode_false -> null_encode_false [0]
   xor(v0, v1) -> null_xor [0]
   and(v0, v1) -> null_and [0]

And the following fresh constants:    null_encArg, null_encode_not, null_encode_xor, null_encode_true, null_encode_or, null_encode_and, null_encode_implies, null_encode_false, null_xor, null_and

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

(24)
Obligation:
Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   not(x) -> xor(x, true) [1]
   or(x, y) -> xor(and(x, y), xor(x, y)) [1]
   implies(x, y) -> xor(and(x, y), xor(x, true)) [1]
   and(x, true) -> x [1]
   and(x, false) -> false [1]
   and(x, x) -> x [1]
   xor(x, false) -> x [1]
   xor(x, x) -> false [1]
   and(c_xor(x, y), z) -> xor(and(x, z), and(y, z)) [1]
   encArg(true) -> true [0]
   encArg(false) -> false [0]
   encArg(cons_not(x_1)) -> not(encArg(x_1)) [0]
   encArg(cons_or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_implies(x_1, x_2)) -> implies(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_xor(x_1, x_2)) -> xor(encArg(x_1), encArg(x_2)) [0]
   encode_not(x_1) -> not(encArg(x_1)) [0]
   encode_xor(x_1, x_2) -> xor(encArg(x_1), encArg(x_2)) [0]
   encode_true -> true [0]
   encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2)) [0]
   encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) [0]
   encode_implies(x_1, x_2) -> implies(encArg(x_1), encArg(x_2)) [0]
   encode_false -> false [0]
   xor(x0, x1) -> c_xor(x0, x1) [0]
   encArg(v0) -> null_encArg [0]
   encode_not(v0) -> null_encode_not [0]
   encode_xor(v0, v1) -> null_encode_xor [0]
   encode_true -> null_encode_true [0]
   encode_or(v0, v1) -> null_encode_or [0]
   encode_and(v0, v1) -> null_encode_and [0]
   encode_implies(v0, v1) -> null_encode_implies [0]
   encode_false -> null_encode_false [0]
   xor(v0, v1) -> null_xor [0]
   and(v0, v1) -> null_and [0]

The TRS has the following type information:
not :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
xor :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
true :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
or :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
and :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
implies :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
false :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
c_xor :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
encArg :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
cons_not :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
cons_or :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
cons_implies :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
cons_and :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
cons_xor :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
encode_not :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
encode_xor :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
encode_true :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
encode_or :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
encode_and :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
encode_implies :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
encode_false :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
null_encArg :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
null_encode_not :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
null_encode_xor :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
null_encode_true :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
null_encode_or :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
null_encode_and :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
null_encode_implies :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
null_encode_false :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
null_xor :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
null_and :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and

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

(25) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

   true => 1
   false => 0
   null_encArg => 0
   null_encode_not => 0
   null_encode_xor => 0
   null_encode_true => 0
   null_encode_or => 0
   null_encode_and => 0
   null_encode_implies => 0
   null_encode_false => 0
   null_xor => 0
   null_and => 0

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

(26)
Obligation:
Complexity RNTS consisting of the following rules:

   and(z', z'') -{ 1 }-> x :|: z' = x, x >= 0, z'' = 1
   and(z', z'') -{ 1 }-> x :|: z' = x, x >= 0, z'' = x
   and(z', z'') -{ 1 }-> xor(and(x, z), and(y, z)) :|: z'' = z, z >= 0, z' = 1 + x + y, x >= 0, y >= 0
   and(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' = x, x >= 0
   and(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   encArg(z') -{ 0 }-> xor(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> not(encArg(x_1)) :|: x_1 >= 0, z' = 1 + x_1
   encArg(z') -{ 0 }-> implies(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> and(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> 1 :|: z' = 1
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0
   encode_and(z', z'') -{ 0 }-> and(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z' = x_1, x_2 >= 0, z'' = x_2
   encode_and(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   encode_false -{ 0 }-> 0 :|: 
   encode_implies(z', z'') -{ 0 }-> implies(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z' = x_1, x_2 >= 0, z'' = x_2
   encode_implies(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   encode_not(z') -{ 0 }-> not(encArg(x_1)) :|: x_1 >= 0, z' = x_1
   encode_not(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0
   encode_or(z', z'') -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z' = x_1, x_2 >= 0, z'' = x_2
   encode_or(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   encode_true -{ 0 }-> 1 :|: 
   encode_true -{ 0 }-> 0 :|: 
   encode_xor(z', z'') -{ 0 }-> xor(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z' = x_1, x_2 >= 0, z'' = x_2
   encode_xor(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   implies(z', z'') -{ 1 }-> xor(and(x, y), xor(x, 1)) :|: z' = x, z'' = y, x >= 0, y >= 0
   not(z') -{ 1 }-> xor(x, 1) :|: z' = x, x >= 0
   or(z', z'') -{ 1 }-> xor(and(x, y), xor(x, y)) :|: z' = x, z'' = y, x >= 0, y >= 0
   xor(z', z'') -{ 1 }-> x :|: z'' = 0, z' = x, x >= 0
   xor(z', z'') -{ 1 }-> 0 :|: z' = x, x >= 0, z'' = x
   xor(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   xor(z', z'') -{ 0 }-> 1 + x0 + x1 :|: z'' = x1, x0 >= 0, x1 >= 0, z' = x0

Only complete derivations are relevant for the runtime complexity.

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

(27) CompletionProof (UPPER BOUND(ID))
The transformation into a RNTS is sound, since:

(a) The obligation is a constructor system where every type has a constant constructor,

(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:
none

(c) The following functions are completely defined:

   and_2
   implies_2
   not_1
   or_2
   encArg_1
   encode_not_1
   encode_xor_2
   encode_true
   encode_or_2
   encode_and_2
   encode_implies_2
   encode_false
   xor_2

Due to the following rules being added:

   encArg(v0) -> null_encArg [0]
   encode_not(v0) -> null_encode_not [0]
   encode_xor(v0, v1) -> null_encode_xor [0]
   encode_true -> null_encode_true [0]
   encode_or(v0, v1) -> null_encode_or [0]
   encode_and(v0, v1) -> null_encode_and [0]
   encode_implies(v0, v1) -> null_encode_implies [0]
   encode_false -> null_encode_false [0]
   xor(v0, v1) -> null_xor [0]
   and(v0, v1) -> null_and [0]

And the following fresh constants:    null_encArg, null_encode_not, null_encode_xor, null_encode_true, null_encode_or, null_encode_and, null_encode_implies, null_encode_false, null_xor, null_and

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

(28)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   not(x) -> xor(x, true) [1]
   or(x, y) -> xor(and(x, y), xor(x, y)) [1]
   implies(x, y) -> xor(and(x, y), xor(x, true)) [1]
   and(x, true) -> x [1]
   and(x, false) -> false [1]
   and(x, x) -> x [1]
   xor(x, false) -> x [1]
   xor(x, x) -> false [1]
   and(c_xor(x, y), z) -> xor(and(x, z), and(y, z)) [1]
   encArg(true) -> true [0]
   encArg(false) -> false [0]
   encArg(cons_not(x_1)) -> not(encArg(x_1)) [0]
   encArg(cons_or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_implies(x_1, x_2)) -> implies(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_xor(x_1, x_2)) -> xor(encArg(x_1), encArg(x_2)) [0]
   encode_not(x_1) -> not(encArg(x_1)) [0]
   encode_xor(x_1, x_2) -> xor(encArg(x_1), encArg(x_2)) [0]
   encode_true -> true [0]
   encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2)) [0]
   encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) [0]
   encode_implies(x_1, x_2) -> implies(encArg(x_1), encArg(x_2)) [0]
   encode_false -> false [0]
   xor(x0, x1) -> c_xor(x0, x1) [0]
   encArg(v0) -> null_encArg [0]
   encode_not(v0) -> null_encode_not [0]
   encode_xor(v0, v1) -> null_encode_xor [0]
   encode_true -> null_encode_true [0]
   encode_or(v0, v1) -> null_encode_or [0]
   encode_and(v0, v1) -> null_encode_and [0]
   encode_implies(v0, v1) -> null_encode_implies [0]
   encode_false -> null_encode_false [0]
   xor(v0, v1) -> null_xor [0]
   and(v0, v1) -> null_and [0]

The TRS has the following type information:
not :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
xor :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
true :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
or :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
and :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
implies :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
false :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
c_xor :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
encArg :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
cons_not :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
cons_or :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
cons_implies :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
cons_and :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
cons_xor :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
encode_not :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
encode_xor :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
encode_true :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
encode_or :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
encode_and :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
encode_implies :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
encode_false :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
null_encArg :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
null_encode_not :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
null_encode_xor :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
null_encode_true :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
null_encode_or :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
null_encode_and :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
null_encode_implies :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
null_encode_false :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
null_xor :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
null_and :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and

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

(29) NarrowingProof (BOTH BOUNDS(ID, ID))
Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
----------------------------------------

(30)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   not(x) -> xor(x, true) [1]
   or(true, true) -> xor(true, false) [3]
   or(x, true) -> xor(x, c_xor(x, true)) [2]
   or(x, true) -> xor(x, null_xor) [2]
   or(x, false) -> xor(false, x) [3]
   or(false, false) -> xor(false, false) [3]
   or(x, false) -> xor(false, c_xor(x, false)) [2]
   or(x, false) -> xor(false, null_xor) [2]
   or(false, false) -> xor(false, false) [3]
   or(x, x) -> xor(x, false) [3]
   or(x, x) -> xor(x, c_xor(x, x)) [2]
   or(x, x) -> xor(x, null_xor) [2]
   or(c_xor(x', y'), false) -> xor(xor(and(x', false), and(y', false)), c_xor(x', y')) [3]
   or(c_xor(x', y'), c_xor(x', y')) -> xor(xor(and(x', c_xor(x', y')), and(y', c_xor(x', y'))), false) [3]
   or(c_xor(x', y'), y) -> xor(xor(and(x', y), and(y', y)), c_xor(c_xor(x', y'), y)) [2]
   or(c_xor(x', y'), y) -> xor(xor(and(x', y), and(y', y)), null_xor) [2]
   or(x, false) -> xor(null_and, x) [2]
   or(x, x) -> xor(null_and, false) [2]
   or(x, y) -> xor(null_and, c_xor(x, y)) [1]
   or(x, y) -> xor(null_and, null_xor) [1]
   implies(true, true) -> xor(true, false) [3]
   implies(x, true) -> xor(x, c_xor(x, true)) [2]
   implies(x, true) -> xor(x, null_xor) [2]
   implies(true, false) -> xor(false, false) [3]
   implies(x, false) -> xor(false, c_xor(x, true)) [2]
   implies(x, false) -> xor(false, null_xor) [2]
   implies(true, true) -> xor(true, false) [3]
   implies(x, x) -> xor(x, c_xor(x, true)) [2]
   implies(x, x) -> xor(x, null_xor) [2]
   implies(c_xor(x'', y''), y) -> xor(xor(and(x'', y), and(y'', y)), c_xor(c_xor(x'', y''), true)) [2]
   implies(c_xor(x'', y''), y) -> xor(xor(and(x'', y), and(y'', y)), null_xor) [2]
   implies(true, y) -> xor(null_and, false) [2]
   implies(x, y) -> xor(null_and, c_xor(x, true)) [1]
   implies(x, y) -> xor(null_and, null_xor) [1]
   and(x, true) -> x [1]
   and(x, false) -> false [1]
   and(x, x) -> x [1]
   xor(x, false) -> x [1]
   xor(x, x) -> false [1]
   and(c_xor(x, y), z) -> xor(and(x, z), and(y, z)) [1]
   encArg(true) -> true [0]
   encArg(false) -> false [0]
   encArg(cons_not(true)) -> not(true) [0]
   encArg(cons_not(false)) -> not(false) [0]
   encArg(cons_not(cons_not(x_1'))) -> not(not(encArg(x_1'))) [0]
   encArg(cons_not(cons_or(x_1'', x_2'))) -> not(or(encArg(x_1''), encArg(x_2'))) [0]
   encArg(cons_not(cons_implies(x_11, x_2''))) -> not(implies(encArg(x_11), encArg(x_2''))) [0]
   encArg(cons_not(cons_and(x_12, x_21))) -> not(and(encArg(x_12), encArg(x_21))) [0]
   encArg(cons_not(cons_xor(x_13, x_22))) -> not(xor(encArg(x_13), encArg(x_22))) [0]
   encArg(cons_not(x_1)) -> not(null_encArg) [0]
   encArg(cons_or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_implies(x_1, x_2)) -> implies(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_xor(x_1, x_2)) -> xor(encArg(x_1), encArg(x_2)) [0]
   encode_not(true) -> not(true) [0]
   encode_not(false) -> not(false) [0]
   encode_not(cons_not(x_1184)) -> not(not(encArg(x_1184))) [0]
   encode_not(cons_or(x_1185, x_2147)) -> not(or(encArg(x_1185), encArg(x_2147))) [0]
   encode_not(cons_implies(x_1186, x_2148)) -> not(implies(encArg(x_1186), encArg(x_2148))) [0]
   encode_not(cons_and(x_1187, x_2149)) -> not(and(encArg(x_1187), encArg(x_2149))) [0]
   encode_not(cons_xor(x_1188, x_2150)) -> not(xor(encArg(x_1188), encArg(x_2150))) [0]
   encode_not(x_1) -> not(null_encArg) [0]
   encode_xor(x_1, x_2) -> xor(encArg(x_1), encArg(x_2)) [0]
   encode_true -> true [0]
   encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2)) [0]
   encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) [0]
   encode_implies(x_1, x_2) -> implies(encArg(x_1), encArg(x_2)) [0]
   encode_false -> false [0]
   xor(x0, x1) -> c_xor(x0, x1) [0]
   encArg(v0) -> null_encArg [0]
   encode_not(v0) -> null_encode_not [0]
   encode_xor(v0, v1) -> null_encode_xor [0]
   encode_true -> null_encode_true [0]
   encode_or(v0, v1) -> null_encode_or [0]
   encode_and(v0, v1) -> null_encode_and [0]
   encode_implies(v0, v1) -> null_encode_implies [0]
   encode_false -> null_encode_false [0]
   xor(v0, v1) -> null_xor [0]
   and(v0, v1) -> null_and [0]

The TRS has the following type information:
not :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
xor :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
true :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
or :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
and :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
implies :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
false :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
c_xor :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
encArg :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
cons_not :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
cons_or :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
cons_implies :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
cons_and :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
cons_xor :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
encode_not :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
encode_xor :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
encode_true :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
encode_or :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
encode_and :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
encode_implies :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and -> true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
encode_false :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
null_encArg :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
null_encode_not :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
null_encode_xor :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
null_encode_true :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
null_encode_or :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
null_encode_and :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
null_encode_implies :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
null_encode_false :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
null_xor :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and
null_and :: true:false:c_xor:cons_not:cons_or:cons_implies:cons_and:cons_xor:null_encArg:null_encode_not:null_encode_xor:null_encode_true:null_encode_or:null_encode_and:null_encode_implies:null_encode_false:null_xor:null_and

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

(31) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

   true => 1
   false => 0
   null_encArg => 0
   null_encode_not => 0
   null_encode_xor => 0
   null_encode_true => 0
   null_encode_or => 0
   null_encode_and => 0
   null_encode_implies => 0
   null_encode_false => 0
   null_xor => 0
   null_and => 0

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

(32)
Obligation:
Complexity RNTS consisting of the following rules:

   and(z', z'') -{ 1 }-> x :|: z' = x, x >= 0, z'' = 1
   and(z', z'') -{ 1 }-> x :|: z' = x, x >= 0, z'' = x
   and(z', z'') -{ 1 }-> xor(and(x, z), and(y, z)) :|: z'' = z, z >= 0, z' = 1 + x + y, x >= 0, y >= 0
   and(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' = x, x >= 0
   and(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   encArg(z') -{ 0 }-> xor(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> not(xor(encArg(x_13), encArg(x_22))) :|: x_13 >= 0, z' = 1 + (1 + x_13 + x_22), x_22 >= 0
   encArg(z') -{ 0 }-> not(or(encArg(x_1''), encArg(x_2'))) :|: x_1'' >= 0, x_2' >= 0, z' = 1 + (1 + x_1'' + x_2')
   encArg(z') -{ 0 }-> not(not(encArg(x_1'))) :|: x_1' >= 0, z' = 1 + (1 + x_1')
   encArg(z') -{ 0 }-> not(implies(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, x_2'' >= 0, z' = 1 + (1 + x_11 + x_2'')
   encArg(z') -{ 0 }-> not(and(encArg(x_12), encArg(x_21))) :|: z' = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0
   encArg(z') -{ 0 }-> not(1) :|: z' = 1 + 1
   encArg(z') -{ 0 }-> not(0) :|: z' = 1 + 0
   encArg(z') -{ 0 }-> not(0) :|: x_1 >= 0, z' = 1 + x_1
   encArg(z') -{ 0 }-> implies(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> and(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> 1 :|: z' = 1
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0
   encode_and(z', z'') -{ 0 }-> and(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z' = x_1, x_2 >= 0, z'' = x_2
   encode_and(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   encode_false -{ 0 }-> 0 :|: 
   encode_implies(z', z'') -{ 0 }-> implies(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z' = x_1, x_2 >= 0, z'' = x_2
   encode_implies(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   encode_not(z') -{ 0 }-> not(xor(encArg(x_1188), encArg(x_2150))) :|: z' = 1 + x_1188 + x_2150, x_1188 >= 0, x_2150 >= 0
   encode_not(z') -{ 0 }-> not(or(encArg(x_1185), encArg(x_2147))) :|: x_1185 >= 0, z' = 1 + x_1185 + x_2147, x_2147 >= 0
   encode_not(z') -{ 0 }-> not(not(encArg(x_1184))) :|: z' = 1 + x_1184, x_1184 >= 0
   encode_not(z') -{ 0 }-> not(implies(encArg(x_1186), encArg(x_2148))) :|: z' = 1 + x_1186 + x_2148, x_1186 >= 0, x_2148 >= 0
   encode_not(z') -{ 0 }-> not(and(encArg(x_1187), encArg(x_2149))) :|: x_2149 >= 0, z' = 1 + x_1187 + x_2149, x_1187 >= 0
   encode_not(z') -{ 0 }-> not(1) :|: z' = 1
   encode_not(z') -{ 0 }-> not(0) :|: z' = 0
   encode_not(z') -{ 0 }-> not(0) :|: x_1 >= 0, z' = x_1
   encode_not(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0
   encode_or(z', z'') -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z' = x_1, x_2 >= 0, z'' = x_2
   encode_or(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   encode_true -{ 0 }-> 1 :|: 
   encode_true -{ 0 }-> 0 :|: 
   encode_xor(z', z'') -{ 0 }-> xor(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z' = x_1, x_2 >= 0, z'' = x_2
   encode_xor(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   implies(z', z'') -{ 2 }-> xor(x, 0) :|: z' = x, x >= 0, z'' = 1
   implies(z', z'') -{ 2 }-> xor(x, 0) :|: z' = x, x >= 0, z'' = x
   implies(z', z'') -{ 2 }-> xor(x, 1 + x + 1) :|: z' = x, x >= 0, z'' = 1
   implies(z', z'') -{ 2 }-> xor(x, 1 + x + 1) :|: z' = x, x >= 0, z'' = x
   implies(z', z'') -{ 2 }-> xor(xor(and(x'', y), and(y'', y)), 0) :|: z'' = y, z' = 1 + x'' + y'', y >= 0, y'' >= 0, x'' >= 0
   implies(z', z'') -{ 2 }-> xor(xor(and(x'', y), and(y'', y)), 1 + (1 + x'' + y'') + 1) :|: z'' = y, z' = 1 + x'' + y'', y >= 0, y'' >= 0, x'' >= 0
   implies(z', z'') -{ 3 }-> xor(1, 0) :|: z' = 1, z'' = 1
   implies(z', z'') -{ 3 }-> xor(0, 0) :|: z'' = 0, z' = 1
   implies(z', z'') -{ 2 }-> xor(0, 0) :|: z'' = 0, z' = x, x >= 0
   implies(z', z'') -{ 2 }-> xor(0, 0) :|: z'' = y, y >= 0, z' = 1
   implies(z', z'') -{ 1 }-> xor(0, 0) :|: z' = x, z'' = y, x >= 0, y >= 0
   implies(z', z'') -{ 2 }-> xor(0, 1 + x + 1) :|: z'' = 0, z' = x, x >= 0
   implies(z', z'') -{ 1 }-> xor(0, 1 + x + 1) :|: z' = x, z'' = y, x >= 0, y >= 0
   not(z') -{ 1 }-> xor(x, 1) :|: z' = x, x >= 0
   or(z', z'') -{ 2 }-> xor(x, 0) :|: z' = x, x >= 0, z'' = 1
   or(z', z'') -{ 3 }-> xor(x, 0) :|: z' = x, x >= 0, z'' = x
   or(z', z'') -{ 2 }-> xor(x, 0) :|: z' = x, x >= 0, z'' = x
   or(z', z'') -{ 2 }-> xor(x, 1 + x + x) :|: z' = x, x >= 0, z'' = x
   or(z', z'') -{ 2 }-> xor(x, 1 + x + 1) :|: z' = x, x >= 0, z'' = 1
   or(z', z'') -{ 2 }-> xor(xor(and(x', y), and(y', y)), 0) :|: z'' = y, z' = 1 + x' + y', x' >= 0, y >= 0, y' >= 0
   or(z', z'') -{ 2 }-> xor(xor(and(x', y), and(y', y)), 1 + (1 + x' + y') + y) :|: z'' = y, z' = 1 + x' + y', x' >= 0, y >= 0, y' >= 0
   or(z', z'') -{ 3 }-> xor(xor(and(x', 0), and(y', 0)), 1 + x' + y') :|: z'' = 0, z' = 1 + x' + y', x' >= 0, y' >= 0
   or(z', z'') -{ 3 }-> xor(xor(and(x', 1 + x' + y'), and(y', 1 + x' + y')), 0) :|: z'' = 1 + x' + y', z' = 1 + x' + y', x' >= 0, y' >= 0
   or(z', z'') -{ 3 }-> xor(1, 0) :|: z' = 1, z'' = 1
   or(z', z'') -{ 3 }-> xor(0, x) :|: z'' = 0, z' = x, x >= 0
   or(z', z'') -{ 2 }-> xor(0, x) :|: z'' = 0, z' = x, x >= 0
   or(z', z'') -{ 3 }-> xor(0, 0) :|: z'' = 0, z' = 0
   or(z', z'') -{ 2 }-> xor(0, 0) :|: z'' = 0, z' = x, x >= 0
   or(z', z'') -{ 2 }-> xor(0, 0) :|: z' = x, x >= 0, z'' = x
   or(z', z'') -{ 1 }-> xor(0, 0) :|: z' = x, z'' = y, x >= 0, y >= 0
   or(z', z'') -{ 1 }-> xor(0, 1 + x + y) :|: z' = x, z'' = y, x >= 0, y >= 0
   or(z', z'') -{ 2 }-> xor(0, 1 + x + 0) :|: z'' = 0, z' = x, x >= 0
   xor(z', z'') -{ 1 }-> x :|: z'' = 0, z' = x, x >= 0
   xor(z', z'') -{ 1 }-> 0 :|: z' = x, x >= 0, z'' = x
   xor(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   xor(z', z'') -{ 0 }-> 1 + x0 + x1 :|: z'' = x1, x0 >= 0, x1 >= 0, z' = x0


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

(33) InliningProof (UPPER BOUND(ID))
Inlined the following terminating rules on right-hand sides where appropriate:

   xor(z', z'') -{ 0 }-> 1 + x0 + x1 :|: z'' = x1, x0 >= 0, x1 >= 0, z' = x0
   xor(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   xor(z', z'') -{ 1 }-> x :|: z'' = 0, z' = x, x >= 0
   xor(z', z'') -{ 1 }-> 0 :|: z' = x, x >= 0, z'' = x
   not(z') -{ 1 }-> xor(x, 1) :|: z' = x, x >= 0

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

(34)
Obligation:
Complexity RNTS consisting of the following rules:

   and(z', z'') -{ 1 }-> x :|: z' = x, x >= 0, z'' = 1
   and(z', z'') -{ 1 }-> x :|: z' = x, x >= 0, z'' = x
   and(z', z'') -{ 1 }-> xor(and(x, z), and(y, z)) :|: z'' = z, z >= 0, z' = 1 + x + y, x >= 0, y >= 0
   and(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' = x, x >= 0
   and(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   encArg(z') -{ 1 }-> xor(x, 1) :|: z' = 1 + 1, 1 = x, x >= 0
   encArg(z') -{ 1 }-> xor(x, 1) :|: z' = 1 + 0, 0 = x, x >= 0
   encArg(z') -{ 1 }-> xor(x, 1) :|: x_1 >= 0, z' = 1 + x_1, 0 = x, x >= 0
   encArg(z') -{ 0 }-> xor(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> not(xor(encArg(x_13), encArg(x_22))) :|: x_13 >= 0, z' = 1 + (1 + x_13 + x_22), x_22 >= 0
   encArg(z') -{ 0 }-> not(or(encArg(x_1''), encArg(x_2'))) :|: x_1'' >= 0, x_2' >= 0, z' = 1 + (1 + x_1'' + x_2')
   encArg(z') -{ 0 }-> not(not(encArg(x_1'))) :|: x_1' >= 0, z' = 1 + (1 + x_1')
   encArg(z') -{ 0 }-> not(implies(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, x_2'' >= 0, z' = 1 + (1 + x_11 + x_2'')
   encArg(z') -{ 0 }-> not(and(encArg(x_12), encArg(x_21))) :|: z' = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0
   encArg(z') -{ 0 }-> implies(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> and(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> 1 :|: z' = 1
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0
   encode_and(z', z'') -{ 0 }-> and(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z' = x_1, x_2 >= 0, z'' = x_2
   encode_and(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   encode_false -{ 0 }-> 0 :|: 
   encode_implies(z', z'') -{ 0 }-> implies(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z' = x_1, x_2 >= 0, z'' = x_2
   encode_implies(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   encode_not(z') -{ 1 }-> xor(x, 1) :|: z' = 1, 1 = x, x >= 0
   encode_not(z') -{ 1 }-> xor(x, 1) :|: z' = 0, 0 = x, x >= 0
   encode_not(z') -{ 1 }-> xor(x, 1) :|: x_1 >= 0, z' = x_1, 0 = x, x >= 0
   encode_not(z') -{ 0 }-> not(xor(encArg(x_1188), encArg(x_2150))) :|: z' = 1 + x_1188 + x_2150, x_1188 >= 0, x_2150 >= 0
   encode_not(z') -{ 0 }-> not(or(encArg(x_1185), encArg(x_2147))) :|: x_1185 >= 0, z' = 1 + x_1185 + x_2147, x_2147 >= 0
   encode_not(z') -{ 0 }-> not(not(encArg(x_1184))) :|: z' = 1 + x_1184, x_1184 >= 0
   encode_not(z') -{ 0 }-> not(implies(encArg(x_1186), encArg(x_2148))) :|: z' = 1 + x_1186 + x_2148, x_1186 >= 0, x_2148 >= 0
   encode_not(z') -{ 0 }-> not(and(encArg(x_1187), encArg(x_2149))) :|: x_2149 >= 0, z' = 1 + x_1187 + x_2149, x_1187 >= 0
   encode_not(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0
   encode_or(z', z'') -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z' = x_1, x_2 >= 0, z'' = x_2
   encode_or(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   encode_true -{ 0 }-> 1 :|: 
   encode_true -{ 0 }-> 0 :|: 
   encode_xor(z', z'') -{ 0 }-> xor(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z' = x_1, x_2 >= 0, z'' = x_2
   encode_xor(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   implies(z', z'') -{ 4 }-> x :|: z' = 1, z'' = 1, 0 = 0, 1 = x, x >= 0
   implies(z', z'') -{ 4 }-> x :|: z'' = 0, z' = 1, 0 = 0, 0 = x, x >= 0
   implies(z', z'') -{ 3 }-> x :|: z'' = y, y >= 0, z' = 1, 0 = 0, 0 = x, x >= 0
   implies(z', z'') -{ 3 }-> x' :|: z' = x, x >= 0, z'' = 1, 0 = 0, x = x', x' >= 0
   implies(z', z'') -{ 3 }-> x' :|: z'' = 0, z' = x, x >= 0, 0 = 0, 0 = x', x' >= 0
   implies(z', z'') -{ 3 }-> x' :|: z' = x, x >= 0, z'' = x, 0 = 0, x = x', x' >= 0
   implies(z', z'') -{ 2 }-> x' :|: z' = x, z'' = y, x >= 0, y >= 0, 0 = 0, 0 = x', x' >= 0
   implies(z', z'') -{ 2 }-> xor(xor(and(x'', y), and(y'', y)), 0) :|: z'' = y, z' = 1 + x'' + y'', y >= 0, y'' >= 0, x'' >= 0
   implies(z', z'') -{ 2 }-> xor(xor(and(x'', y), and(y'', y)), 1 + (1 + x'' + y'') + 1) :|: z'' = y, z' = 1 + x'' + y'', y >= 0, y'' >= 0, x'' >= 0
   implies(z', z'') -{ 3 }-> 0 :|: z' = 1, z'' = 1, v0 >= 0, v1 >= 0, 0 = v1, 1 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z' = x, x >= 0, z'' = 1, v0 >= 0, v1 >= 0, 1 + x + 1 = v1, x = v0
   implies(z', z'') -{ 2 }-> 0 :|: z' = x, x >= 0, z'' = 1, v0 >= 0, v1 >= 0, 0 = v1, x = v0
   implies(z', z'') -{ 3 }-> 0 :|: z' = x, x >= 0, z'' = 1, x = x', x' >= 0, 0 = x'
   implies(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' = 1, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' = 1, 0 = x, x >= 0
   implies(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' = x, x >= 0, v0 >= 0, v1 >= 0, 1 + x + 1 = v1, 0 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' = x, x >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' = x, x >= 0, 0 = x', x' >= 0
   implies(z', z'') -{ 2 }-> 0 :|: z' = x, x >= 0, z'' = x, v0 >= 0, v1 >= 0, 1 + x + 1 = v1, x = v0
   implies(z', z'') -{ 2 }-> 0 :|: z' = x, x >= 0, z'' = x, v0 >= 0, v1 >= 0, 0 = v1, x = v0
   implies(z', z'') -{ 3 }-> 0 :|: z' = x, x >= 0, z'' = x, x = x', x' >= 0, 0 = x'
   implies(z', z'') -{ 2 }-> 0 :|: z'' = y, y >= 0, z' = 1, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 3 }-> 0 :|: z'' = y, y >= 0, z' = 1, 0 = x, x >= 0
   implies(z', z'') -{ 1 }-> 0 :|: z' = x, z'' = y, x >= 0, y >= 0, v0 >= 0, v1 >= 0, 1 + x + 1 = v1, 0 = v0
   implies(z', z'') -{ 1 }-> 0 :|: z' = x, z'' = y, x >= 0, y >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z' = x, z'' = y, x >= 0, y >= 0, 0 = x', x' >= 0
   implies(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z' = 1, z'' = 1, 0 = x1, x0 >= 0, x1 >= 0, 1 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z' = x, x >= 0, z'' = 1, 1 + x + 1 = x1, x0 >= 0, x1 >= 0, x = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z' = x, x >= 0, z'' = 1, 0 = x1, x0 >= 0, x1 >= 0, x = x0
   implies(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z'' = 0, z' = 1, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' = x, x >= 0, 1 + x + 1 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' = x, x >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z' = x, x >= 0, z'' = x, 1 + x + 1 = x1, x0 >= 0, x1 >= 0, x = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z' = x, x >= 0, z'' = x, 0 = x1, x0 >= 0, x1 >= 0, x = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = y, y >= 0, z' = 1, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' = x, z'' = y, x >= 0, y >= 0, 1 + x + 1 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' = x, z'' = y, x >= 0, y >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   not(z') -{ 1 }-> 0 :|: z' = x, x >= 0, v0 >= 0, v1 >= 0, 1 = v1, x = v0
   not(z') -{ 2 }-> 0 :|: z' = x, x >= 0, x = x', x' >= 0, 1 = x'
   not(z') -{ 1 }-> 1 + x0 + x1 :|: z' = x, x >= 0, 1 = x1, x0 >= 0, x1 >= 0, x = x0
   or(z', z'') -{ 4 }-> x :|: z' = 1, z'' = 1, 0 = 0, 1 = x, x >= 0
   or(z', z'') -{ 4 }-> x :|: z'' = 0, z' = 0, 0 = 0, 0 = x, x >= 0
   or(z', z'') -{ 3 }-> x' :|: z' = x, x >= 0, z'' = 1, 0 = 0, x = x', x' >= 0
   or(z', z'') -{ 4 }-> x' :|: z'' = 0, z' = x, x >= 0, x = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z'' = 0, z' = x, x >= 0, 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 4 }-> x' :|: z' = x, x >= 0, z'' = x, 0 = 0, x = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z' = x, x >= 0, z'' = x, 0 = 0, x = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z'' = 0, z' = x, x >= 0, x = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z' = x, x >= 0, z'' = x, 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 2 }-> x' :|: z' = x, z'' = y, x >= 0, y >= 0, 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 2 }-> xor(xor(and(x', y), and(y', y)), 0) :|: z'' = y, z' = 1 + x' + y', x' >= 0, y >= 0, y' >= 0
   or(z', z'') -{ 2 }-> xor(xor(and(x', y), and(y', y)), 1 + (1 + x' + y') + y) :|: z'' = y, z' = 1 + x' + y', x' >= 0, y >= 0, y' >= 0
   or(z', z'') -{ 3 }-> xor(xor(and(x', 0), and(y', 0)), 1 + x' + y') :|: z'' = 0, z' = 1 + x' + y', x' >= 0, y' >= 0
   or(z', z'') -{ 3 }-> xor(xor(and(x', 1 + x' + y'), and(y', 1 + x' + y')), 0) :|: z'' = 1 + x' + y', z' = 1 + x' + y', x' >= 0, y' >= 0
   or(z', z'') -{ 3 }-> 0 :|: z' = 1, z'' = 1, v0 >= 0, v1 >= 0, 0 = v1, 1 = v0
   or(z', z'') -{ 2 }-> 0 :|: z' = x, x >= 0, z'' = 1, v0 >= 0, v1 >= 0, 1 + x + 1 = v1, x = v0
   or(z', z'') -{ 2 }-> 0 :|: z' = x, x >= 0, z'' = 1, v0 >= 0, v1 >= 0, 0 = v1, x = v0
   or(z', z'') -{ 3 }-> 0 :|: z' = x, x >= 0, z'' = 1, x = x', x' >= 0, 0 = x'
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' = x, x >= 0, v0 >= 0, v1 >= 0, x = v1, 0 = v0
   or(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' = x, x >= 0, 0 = x', x' >= 0, x = x'
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' = 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' = 0, 0 = x, x >= 0
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' = x, x >= 0, v0 >= 0, v1 >= 0, 1 + x + 0 = v1, 0 = v0
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' = x, x >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' = x, x >= 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> 0 :|: z' = x, x >= 0, z'' = x, v0 >= 0, v1 >= 0, 0 = v1, x = v0
   or(z', z'') -{ 4 }-> 0 :|: z' = x, x >= 0, z'' = x, x = x', x' >= 0, 0 = x'
   or(z', z'') -{ 2 }-> 0 :|: z' = x, x >= 0, z'' = x, v0 >= 0, v1 >= 0, 1 + x + x = v1, x = v0
   or(z', z'') -{ 2 }-> 0 :|: z' = x, x >= 0, z'' = x, v0 >= 0, v1 >= 0, 0 = v1, x = v0
   or(z', z'') -{ 3 }-> 0 :|: z' = x, x >= 0, z'' = x, x = x', x' >= 0, 0 = x'
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' = x, x >= 0, v0 >= 0, v1 >= 0, x = v1, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' = x, x >= 0, 0 = x', x' >= 0, x = x'
   or(z', z'') -{ 2 }-> 0 :|: z' = x, x >= 0, z'' = x, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z' = x, x >= 0, z'' = x, 0 = x', x' >= 0
   or(z', z'') -{ 1 }-> 0 :|: z' = x, z'' = y, x >= 0, y >= 0, v0 >= 0, v1 >= 0, 1 + x + y = v1, 0 = v0
   or(z', z'') -{ 1 }-> 0 :|: z' = x, z'' = y, x >= 0, y >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 2 }-> 0 :|: z' = x, z'' = y, x >= 0, y >= 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z' = 1, z'' = 1, 0 = x1, x0 >= 0, x1 >= 0, 1 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z' = x, x >= 0, z'' = 1, 1 + x + 1 = x1, x0 >= 0, x1 >= 0, x = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z' = x, x >= 0, z'' = 1, 0 = x1, x0 >= 0, x1 >= 0, x = x0
   or(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z'' = 0, z' = x, x >= 0, x = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z'' = 0, z' = 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' = x, x >= 0, 1 + x + 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' = x, x >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z' = x, x >= 0, z'' = x, 0 = x1, x0 >= 0, x1 >= 0, x = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z' = x, x >= 0, z'' = x, 1 + x + x = x1, x0 >= 0, x1 >= 0, x = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z' = x, x >= 0, z'' = x, 0 = x1, x0 >= 0, x1 >= 0, x = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' = x, x >= 0, x = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z' = x, x >= 0, z'' = x, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' = x, z'' = y, x >= 0, y >= 0, 1 + x + y = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' = x, z'' = y, x >= 0, y >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   xor(z', z'') -{ 1 }-> x :|: z'' = 0, z' = x, x >= 0
   xor(z', z'') -{ 1 }-> 0 :|: z' = x, x >= 0, z'' = x
   xor(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   xor(z', z'') -{ 0 }-> 1 + x0 + x1 :|: z'' = x1, x0 >= 0, x1 >= 0, z' = x0


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(35) SimplificationProof (BOTH BOUNDS(ID, ID))
Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
----------------------------------------

(36)
Obligation:
Complexity RNTS consisting of the following rules:

   and(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' = 1
   and(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' = z'
   and(z', z'') -{ 1 }-> xor(and(x, z''), and(y, z'')) :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
   and(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
   and(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encArg(z') -{ 1 }-> xor(x, 1) :|: z' = 1 + 1, 1 = x, x >= 0
   encArg(z') -{ 1 }-> xor(x, 1) :|: z' = 1 + 0, 0 = x, x >= 0
   encArg(z') -{ 1 }-> xor(x, 1) :|: z' - 1 >= 0, 0 = x, x >= 0
   encArg(z') -{ 0 }-> xor(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> not(xor(encArg(x_13), encArg(x_22))) :|: x_13 >= 0, z' = 1 + (1 + x_13 + x_22), x_22 >= 0
   encArg(z') -{ 0 }-> not(or(encArg(x_1''), encArg(x_2'))) :|: x_1'' >= 0, x_2' >= 0, z' = 1 + (1 + x_1'' + x_2')
   encArg(z') -{ 0 }-> not(not(encArg(z' - 2))) :|: z' - 2 >= 0
   encArg(z') -{ 0 }-> not(implies(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, x_2'' >= 0, z' = 1 + (1 + x_11 + x_2'')
   encArg(z') -{ 0 }-> not(and(encArg(x_12), encArg(x_21))) :|: z' = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0
   encArg(z') -{ 0 }-> implies(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> and(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> 1 :|: z' = 1
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encode_and(z', z'') -{ 0 }-> and(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_and(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_false -{ 0 }-> 0 :|: 
   encode_implies(z', z'') -{ 0 }-> implies(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_implies(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_not(z') -{ 1 }-> xor(x, 1) :|: z' = 1, 1 = x, x >= 0
   encode_not(z') -{ 1 }-> xor(x, 1) :|: z' = 0, 0 = x, x >= 0
   encode_not(z') -{ 1 }-> xor(x, 1) :|: z' >= 0, 0 = x, x >= 0
   encode_not(z') -{ 0 }-> not(xor(encArg(x_1188), encArg(x_2150))) :|: z' = 1 + x_1188 + x_2150, x_1188 >= 0, x_2150 >= 0
   encode_not(z') -{ 0 }-> not(or(encArg(x_1185), encArg(x_2147))) :|: x_1185 >= 0, z' = 1 + x_1185 + x_2147, x_2147 >= 0
   encode_not(z') -{ 0 }-> not(not(encArg(z' - 1))) :|: z' - 1 >= 0
   encode_not(z') -{ 0 }-> not(implies(encArg(x_1186), encArg(x_2148))) :|: z' = 1 + x_1186 + x_2148, x_1186 >= 0, x_2148 >= 0
   encode_not(z') -{ 0 }-> not(and(encArg(x_1187), encArg(x_2149))) :|: x_2149 >= 0, z' = 1 + x_1187 + x_2149, x_1187 >= 0
   encode_not(z') -{ 0 }-> 0 :|: z' >= 0
   encode_or(z', z'') -{ 0 }-> or(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_or(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_true -{ 0 }-> 1 :|: 
   encode_true -{ 0 }-> 0 :|: 
   encode_xor(z', z'') -{ 0 }-> xor(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_xor(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   implies(z', z'') -{ 4 }-> x :|: z' = 1, z'' = 1, 0 = 0, 1 = x, x >= 0
   implies(z', z'') -{ 4 }-> x :|: z'' = 0, z' = 1, 0 = 0, 0 = x, x >= 0
   implies(z', z'') -{ 3 }-> x :|: z'' >= 0, z' = 1, 0 = 0, 0 = x, x >= 0
   implies(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, 0 = 0, 0 = x', x' >= 0
   implies(z', z'') -{ 2 }-> x' :|: z' >= 0, z'' >= 0, 0 = 0, 0 = x', x' >= 0
   implies(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = 1, 0 = 0
   implies(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   implies(z', z'') -{ 2 }-> xor(xor(and(x'', z''), and(y'', z'')), 0) :|: z' = 1 + x'' + y'', z'' >= 0, y'' >= 0, x'' >= 0
   implies(z', z'') -{ 2 }-> xor(xor(and(x'', z''), and(y'', z'')), 1 + (1 + x'' + y'') + 1) :|: z' = 1 + x'' + y'', z'' >= 0, y'' >= 0, x'' >= 0
   implies(z', z'') -{ 3 }-> 0 :|: z' = 1, z'' = 1, v0 >= 0, v1 >= 0, 0 = v1, 1 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 1 + z' + 1 = v1
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 0 = v1
   implies(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = 1, 0 = z'
   implies(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' = 1, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' = 1, 0 = x, x >= 0
   implies(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 1 = v1, 0 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = x', x' >= 0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 1 + z' + 1 = v1
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   implies(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   implies(z', z'') -{ 2 }-> 0 :|: z'' >= 0, z' = 1, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 3 }-> 0 :|: z'' >= 0, z' = 1, 0 = x, x >= 0
   implies(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 1 = v1, 0 = v0
   implies(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' >= 0, 0 = x', x' >= 0
   implies(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z' = 1, z'' = 1, 0 = x1, x0 >= 0, x1 >= 0, 1 = x0
   implies(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z'' = 0, z' = 1, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 1 + z' + 1 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' >= 0, z' = 1, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 1 + z' + 1 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 1 + z' + 1 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 0 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 1 + z' + 1 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   not(z') -{ 1 }-> 0 :|: z' >= 0, v1 >= 0, 1 = v1
   not(z') -{ 2 }-> 0 :|: z' >= 0, 1 = z'
   not(z') -{ 1 }-> 1 + z' + x1 :|: z' >= 0, 1 = x1, x1 >= 0
   or(z', z'') -{ 4 }-> x :|: z' = 1, z'' = 1, 0 = 0, 1 = x, x >= 0
   or(z', z'') -{ 4 }-> x :|: z'' = 0, z' = 0, 0 = 0, 0 = x, x >= 0
   or(z', z'') -{ 4 }-> x' :|: z'' = 0, z' >= 0, z' = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, z' = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z' >= 0, z'' = z', 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 2 }-> x' :|: z' >= 0, z'' >= 0, 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = 1, 0 = 0
   or(z', z'') -{ 4 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   or(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   or(z', z'') -{ 2 }-> xor(xor(and(x', z''), and(y', z'')), 0) :|: z' = 1 + x' + y', x' >= 0, z'' >= 0, y' >= 0
   or(z', z'') -{ 2 }-> xor(xor(and(x', z''), and(y', z'')), 1 + (1 + x' + y') + z'') :|: z' = 1 + x' + y', x' >= 0, z'' >= 0, y' >= 0
   or(z', z'') -{ 3 }-> xor(xor(and(x', 0), and(y', 0)), 1 + x' + y') :|: z'' = 0, z' = 1 + x' + y', x' >= 0, y' >= 0
   or(z', z'') -{ 3 }-> xor(xor(and(x', 1 + x' + y'), and(y', 1 + x' + y')), 0) :|: z'' = 1 + x' + y', z' = 1 + x' + y', x' >= 0, y' >= 0
   or(z', z'') -{ 3 }-> 0 :|: z' = 1, z'' = 1, v0 >= 0, v1 >= 0, 0 = v1, 1 = v0
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 1 + z' + 1 = v1
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 0 = v1
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = 1, 0 = z'
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, 0 = v0
   or(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' >= 0, 0 = z'
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' = 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' = 0, 0 = x, x >= 0
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 0 = v1, 0 = v0
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   or(z', z'') -{ 4 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 1 + z' + z' = v1
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = x', x' >= 0
   or(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 1 + z' + z'' = v1, 0 = v0
   or(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' >= 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z' = 1, z'' = 1, 0 = x1, x0 >= 0, x1 >= 0, 1 = x0
   or(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z'' = 0, z' = 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 1 + z' + 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z' >= 0, z'' = z', 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 1 + z' + z'' = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 3 }-> 1 + x0 + z' :|: z'' = 0, z' >= 0, x0 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + z' :|: z'' = 0, z' >= 0, x0 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 1 + z' + 1 = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 0 = x1, x1 >= 0
   or(z', z'') -{ 3 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 1 + z' + z' = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   xor(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   xor(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' = z'
   xor(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   xor(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0


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

(37) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID))
Found the following analysis order by SCC decomposition:

   { not }
   { encode_false }
   { xor }
   { encode_true }
   { and }
   { implies }
   { or }
   { encArg }
   { encode_not }
   { encode_xor }
   { encode_implies }
   { encode_and }
   { encode_or }

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

(38)
Obligation:
Complexity RNTS consisting of the following rules:

   and(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' = 1
   and(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' = z'
   and(z', z'') -{ 1 }-> xor(and(x, z''), and(y, z'')) :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
   and(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
   and(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encArg(z') -{ 1 }-> xor(x, 1) :|: z' = 1 + 1, 1 = x, x >= 0
   encArg(z') -{ 1 }-> xor(x, 1) :|: z' = 1 + 0, 0 = x, x >= 0
   encArg(z') -{ 1 }-> xor(x, 1) :|: z' - 1 >= 0, 0 = x, x >= 0
   encArg(z') -{ 0 }-> xor(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> not(xor(encArg(x_13), encArg(x_22))) :|: x_13 >= 0, z' = 1 + (1 + x_13 + x_22), x_22 >= 0
   encArg(z') -{ 0 }-> not(or(encArg(x_1''), encArg(x_2'))) :|: x_1'' >= 0, x_2' >= 0, z' = 1 + (1 + x_1'' + x_2')
   encArg(z') -{ 0 }-> not(not(encArg(z' - 2))) :|: z' - 2 >= 0
   encArg(z') -{ 0 }-> not(implies(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, x_2'' >= 0, z' = 1 + (1 + x_11 + x_2'')
   encArg(z') -{ 0 }-> not(and(encArg(x_12), encArg(x_21))) :|: z' = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0
   encArg(z') -{ 0 }-> implies(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> and(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> 1 :|: z' = 1
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encode_and(z', z'') -{ 0 }-> and(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_and(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_false -{ 0 }-> 0 :|: 
   encode_implies(z', z'') -{ 0 }-> implies(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_implies(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_not(z') -{ 1 }-> xor(x, 1) :|: z' = 1, 1 = x, x >= 0
   encode_not(z') -{ 1 }-> xor(x, 1) :|: z' = 0, 0 = x, x >= 0
   encode_not(z') -{ 1 }-> xor(x, 1) :|: z' >= 0, 0 = x, x >= 0
   encode_not(z') -{ 0 }-> not(xor(encArg(x_1188), encArg(x_2150))) :|: z' = 1 + x_1188 + x_2150, x_1188 >= 0, x_2150 >= 0
   encode_not(z') -{ 0 }-> not(or(encArg(x_1185), encArg(x_2147))) :|: x_1185 >= 0, z' = 1 + x_1185 + x_2147, x_2147 >= 0
   encode_not(z') -{ 0 }-> not(not(encArg(z' - 1))) :|: z' - 1 >= 0
   encode_not(z') -{ 0 }-> not(implies(encArg(x_1186), encArg(x_2148))) :|: z' = 1 + x_1186 + x_2148, x_1186 >= 0, x_2148 >= 0
   encode_not(z') -{ 0 }-> not(and(encArg(x_1187), encArg(x_2149))) :|: x_2149 >= 0, z' = 1 + x_1187 + x_2149, x_1187 >= 0
   encode_not(z') -{ 0 }-> 0 :|: z' >= 0
   encode_or(z', z'') -{ 0 }-> or(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_or(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_true -{ 0 }-> 1 :|: 
   encode_true -{ 0 }-> 0 :|: 
   encode_xor(z', z'') -{ 0 }-> xor(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_xor(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   implies(z', z'') -{ 4 }-> x :|: z' = 1, z'' = 1, 0 = 0, 1 = x, x >= 0
   implies(z', z'') -{ 4 }-> x :|: z'' = 0, z' = 1, 0 = 0, 0 = x, x >= 0
   implies(z', z'') -{ 3 }-> x :|: z'' >= 0, z' = 1, 0 = 0, 0 = x, x >= 0
   implies(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, 0 = 0, 0 = x', x' >= 0
   implies(z', z'') -{ 2 }-> x' :|: z' >= 0, z'' >= 0, 0 = 0, 0 = x', x' >= 0
   implies(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = 1, 0 = 0
   implies(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   implies(z', z'') -{ 2 }-> xor(xor(and(x'', z''), and(y'', z'')), 0) :|: z' = 1 + x'' + y'', z'' >= 0, y'' >= 0, x'' >= 0
   implies(z', z'') -{ 2 }-> xor(xor(and(x'', z''), and(y'', z'')), 1 + (1 + x'' + y'') + 1) :|: z' = 1 + x'' + y'', z'' >= 0, y'' >= 0, x'' >= 0
   implies(z', z'') -{ 3 }-> 0 :|: z' = 1, z'' = 1, v0 >= 0, v1 >= 0, 0 = v1, 1 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 1 + z' + 1 = v1
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 0 = v1
   implies(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = 1, 0 = z'
   implies(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' = 1, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' = 1, 0 = x, x >= 0
   implies(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 1 = v1, 0 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = x', x' >= 0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 1 + z' + 1 = v1
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   implies(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   implies(z', z'') -{ 2 }-> 0 :|: z'' >= 0, z' = 1, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 3 }-> 0 :|: z'' >= 0, z' = 1, 0 = x, x >= 0
   implies(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 1 = v1, 0 = v0
   implies(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' >= 0, 0 = x', x' >= 0
   implies(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z' = 1, z'' = 1, 0 = x1, x0 >= 0, x1 >= 0, 1 = x0
   implies(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z'' = 0, z' = 1, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 1 + z' + 1 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' >= 0, z' = 1, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 1 + z' + 1 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 1 + z' + 1 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 0 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 1 + z' + 1 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   not(z') -{ 1 }-> 0 :|: z' >= 0, v1 >= 0, 1 = v1
   not(z') -{ 2 }-> 0 :|: z' >= 0, 1 = z'
   not(z') -{ 1 }-> 1 + z' + x1 :|: z' >= 0, 1 = x1, x1 >= 0
   or(z', z'') -{ 4 }-> x :|: z' = 1, z'' = 1, 0 = 0, 1 = x, x >= 0
   or(z', z'') -{ 4 }-> x :|: z'' = 0, z' = 0, 0 = 0, 0 = x, x >= 0
   or(z', z'') -{ 4 }-> x' :|: z'' = 0, z' >= 0, z' = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, z' = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z' >= 0, z'' = z', 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 2 }-> x' :|: z' >= 0, z'' >= 0, 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = 1, 0 = 0
   or(z', z'') -{ 4 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   or(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   or(z', z'') -{ 2 }-> xor(xor(and(x', z''), and(y', z'')), 0) :|: z' = 1 + x' + y', x' >= 0, z'' >= 0, y' >= 0
   or(z', z'') -{ 2 }-> xor(xor(and(x', z''), and(y', z'')), 1 + (1 + x' + y') + z'') :|: z' = 1 + x' + y', x' >= 0, z'' >= 0, y' >= 0
   or(z', z'') -{ 3 }-> xor(xor(and(x', 0), and(y', 0)), 1 + x' + y') :|: z'' = 0, z' = 1 + x' + y', x' >= 0, y' >= 0
   or(z', z'') -{ 3 }-> xor(xor(and(x', 1 + x' + y'), and(y', 1 + x' + y')), 0) :|: z'' = 1 + x' + y', z' = 1 + x' + y', x' >= 0, y' >= 0
   or(z', z'') -{ 3 }-> 0 :|: z' = 1, z'' = 1, v0 >= 0, v1 >= 0, 0 = v1, 1 = v0
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 1 + z' + 1 = v1
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 0 = v1
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = 1, 0 = z'
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, 0 = v0
   or(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' >= 0, 0 = z'
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' = 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' = 0, 0 = x, x >= 0
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 0 = v1, 0 = v0
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   or(z', z'') -{ 4 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 1 + z' + z' = v1
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = x', x' >= 0
   or(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 1 + z' + z'' = v1, 0 = v0
   or(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' >= 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z' = 1, z'' = 1, 0 = x1, x0 >= 0, x1 >= 0, 1 = x0
   or(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z'' = 0, z' = 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 1 + z' + 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z' >= 0, z'' = z', 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 1 + z' + z'' = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 3 }-> 1 + x0 + z' :|: z'' = 0, z' >= 0, x0 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + z' :|: z'' = 0, z' >= 0, x0 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 1 + z' + 1 = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 0 = x1, x1 >= 0
   or(z', z'') -{ 3 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 1 + z' + z' = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   xor(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   xor(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' = z'
   xor(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   xor(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {not}, {encode_false}, {xor}, {encode_true}, {and}, {implies}, {or}, {encArg}, {encode_not}, {encode_xor}, {encode_implies}, {encode_and}, {encode_or}

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(39) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
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(40)
Obligation:
Complexity RNTS consisting of the following rules:

   and(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' = 1
   and(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' = z'
   and(z', z'') -{ 1 }-> xor(and(x, z''), and(y, z'')) :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
   and(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
   and(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encArg(z') -{ 1 }-> xor(x, 1) :|: z' = 1 + 1, 1 = x, x >= 0
   encArg(z') -{ 1 }-> xor(x, 1) :|: z' = 1 + 0, 0 = x, x >= 0
   encArg(z') -{ 1 }-> xor(x, 1) :|: z' - 1 >= 0, 0 = x, x >= 0
   encArg(z') -{ 0 }-> xor(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> not(xor(encArg(x_13), encArg(x_22))) :|: x_13 >= 0, z' = 1 + (1 + x_13 + x_22), x_22 >= 0
   encArg(z') -{ 0 }-> not(or(encArg(x_1''), encArg(x_2'))) :|: x_1'' >= 0, x_2' >= 0, z' = 1 + (1 + x_1'' + x_2')
   encArg(z') -{ 0 }-> not(not(encArg(z' - 2))) :|: z' - 2 >= 0
   encArg(z') -{ 0 }-> not(implies(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, x_2'' >= 0, z' = 1 + (1 + x_11 + x_2'')
   encArg(z') -{ 0 }-> not(and(encArg(x_12), encArg(x_21))) :|: z' = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0
   encArg(z') -{ 0 }-> implies(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> and(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> 1 :|: z' = 1
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encode_and(z', z'') -{ 0 }-> and(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_and(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_false -{ 0 }-> 0 :|: 
   encode_implies(z', z'') -{ 0 }-> implies(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_implies(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_not(z') -{ 1 }-> xor(x, 1) :|: z' = 1, 1 = x, x >= 0
   encode_not(z') -{ 1 }-> xor(x, 1) :|: z' = 0, 0 = x, x >= 0
   encode_not(z') -{ 1 }-> xor(x, 1) :|: z' >= 0, 0 = x, x >= 0
   encode_not(z') -{ 0 }-> not(xor(encArg(x_1188), encArg(x_2150))) :|: z' = 1 + x_1188 + x_2150, x_1188 >= 0, x_2150 >= 0
   encode_not(z') -{ 0 }-> not(or(encArg(x_1185), encArg(x_2147))) :|: x_1185 >= 0, z' = 1 + x_1185 + x_2147, x_2147 >= 0
   encode_not(z') -{ 0 }-> not(not(encArg(z' - 1))) :|: z' - 1 >= 0
   encode_not(z') -{ 0 }-> not(implies(encArg(x_1186), encArg(x_2148))) :|: z' = 1 + x_1186 + x_2148, x_1186 >= 0, x_2148 >= 0
   encode_not(z') -{ 0 }-> not(and(encArg(x_1187), encArg(x_2149))) :|: x_2149 >= 0, z' = 1 + x_1187 + x_2149, x_1187 >= 0
   encode_not(z') -{ 0 }-> 0 :|: z' >= 0
   encode_or(z', z'') -{ 0 }-> or(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_or(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_true -{ 0 }-> 1 :|: 
   encode_true -{ 0 }-> 0 :|: 
   encode_xor(z', z'') -{ 0 }-> xor(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_xor(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   implies(z', z'') -{ 4 }-> x :|: z' = 1, z'' = 1, 0 = 0, 1 = x, x >= 0
   implies(z', z'') -{ 4 }-> x :|: z'' = 0, z' = 1, 0 = 0, 0 = x, x >= 0
   implies(z', z'') -{ 3 }-> x :|: z'' >= 0, z' = 1, 0 = 0, 0 = x, x >= 0
   implies(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, 0 = 0, 0 = x', x' >= 0
   implies(z', z'') -{ 2 }-> x' :|: z' >= 0, z'' >= 0, 0 = 0, 0 = x', x' >= 0
   implies(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = 1, 0 = 0
   implies(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   implies(z', z'') -{ 2 }-> xor(xor(and(x'', z''), and(y'', z'')), 0) :|: z' = 1 + x'' + y'', z'' >= 0, y'' >= 0, x'' >= 0
   implies(z', z'') -{ 2 }-> xor(xor(and(x'', z''), and(y'', z'')), 1 + (1 + x'' + y'') + 1) :|: z' = 1 + x'' + y'', z'' >= 0, y'' >= 0, x'' >= 0
   implies(z', z'') -{ 3 }-> 0 :|: z' = 1, z'' = 1, v0 >= 0, v1 >= 0, 0 = v1, 1 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 1 + z' + 1 = v1
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 0 = v1
   implies(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = 1, 0 = z'
   implies(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' = 1, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' = 1, 0 = x, x >= 0
   implies(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 1 = v1, 0 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = x', x' >= 0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 1 + z' + 1 = v1
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   implies(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   implies(z', z'') -{ 2 }-> 0 :|: z'' >= 0, z' = 1, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 3 }-> 0 :|: z'' >= 0, z' = 1, 0 = x, x >= 0
   implies(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 1 = v1, 0 = v0
   implies(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' >= 0, 0 = x', x' >= 0
   implies(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z' = 1, z'' = 1, 0 = x1, x0 >= 0, x1 >= 0, 1 = x0
   implies(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z'' = 0, z' = 1, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 1 + z' + 1 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' >= 0, z' = 1, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 1 + z' + 1 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 1 + z' + 1 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 0 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 1 + z' + 1 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   not(z') -{ 1 }-> 0 :|: z' >= 0, v1 >= 0, 1 = v1
   not(z') -{ 2 }-> 0 :|: z' >= 0, 1 = z'
   not(z') -{ 1 }-> 1 + z' + x1 :|: z' >= 0, 1 = x1, x1 >= 0
   or(z', z'') -{ 4 }-> x :|: z' = 1, z'' = 1, 0 = 0, 1 = x, x >= 0
   or(z', z'') -{ 4 }-> x :|: z'' = 0, z' = 0, 0 = 0, 0 = x, x >= 0
   or(z', z'') -{ 4 }-> x' :|: z'' = 0, z' >= 0, z' = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, z' = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z' >= 0, z'' = z', 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 2 }-> x' :|: z' >= 0, z'' >= 0, 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = 1, 0 = 0
   or(z', z'') -{ 4 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   or(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   or(z', z'') -{ 2 }-> xor(xor(and(x', z''), and(y', z'')), 0) :|: z' = 1 + x' + y', x' >= 0, z'' >= 0, y' >= 0
   or(z', z'') -{ 2 }-> xor(xor(and(x', z''), and(y', z'')), 1 + (1 + x' + y') + z'') :|: z' = 1 + x' + y', x' >= 0, z'' >= 0, y' >= 0
   or(z', z'') -{ 3 }-> xor(xor(and(x', 0), and(y', 0)), 1 + x' + y') :|: z'' = 0, z' = 1 + x' + y', x' >= 0, y' >= 0
   or(z', z'') -{ 3 }-> xor(xor(and(x', 1 + x' + y'), and(y', 1 + x' + y')), 0) :|: z'' = 1 + x' + y', z' = 1 + x' + y', x' >= 0, y' >= 0
   or(z', z'') -{ 3 }-> 0 :|: z' = 1, z'' = 1, v0 >= 0, v1 >= 0, 0 = v1, 1 = v0
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 1 + z' + 1 = v1
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 0 = v1
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = 1, 0 = z'
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, 0 = v0
   or(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' >= 0, 0 = z'
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' = 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' = 0, 0 = x, x >= 0
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 0 = v1, 0 = v0
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   or(z', z'') -{ 4 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 1 + z' + z' = v1
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = x', x' >= 0
   or(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 1 + z' + z'' = v1, 0 = v0
   or(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' >= 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z' = 1, z'' = 1, 0 = x1, x0 >= 0, x1 >= 0, 1 = x0
   or(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z'' = 0, z' = 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 1 + z' + 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z' >= 0, z'' = z', 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 1 + z' + z'' = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 3 }-> 1 + x0 + z' :|: z'' = 0, z' >= 0, x0 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + z' :|: z'' = 0, z' >= 0, x0 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 1 + z' + 1 = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 0 = x1, x1 >= 0
   or(z', z'') -{ 3 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 1 + z' + z' = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   xor(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   xor(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' = z'
   xor(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   xor(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {not}, {encode_false}, {xor}, {encode_true}, {and}, {implies}, {or}, {encArg}, {encode_not}, {encode_xor}, {encode_implies}, {encode_and}, {encode_or}

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

(41) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: not
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 2 + z'

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

(42)
Obligation:
Complexity RNTS consisting of the following rules:

   and(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' = 1
   and(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' = z'
   and(z', z'') -{ 1 }-> xor(and(x, z''), and(y, z'')) :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
   and(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
   and(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encArg(z') -{ 1 }-> xor(x, 1) :|: z' = 1 + 1, 1 = x, x >= 0
   encArg(z') -{ 1 }-> xor(x, 1) :|: z' = 1 + 0, 0 = x, x >= 0
   encArg(z') -{ 1 }-> xor(x, 1) :|: z' - 1 >= 0, 0 = x, x >= 0
   encArg(z') -{ 0 }-> xor(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> not(xor(encArg(x_13), encArg(x_22))) :|: x_13 >= 0, z' = 1 + (1 + x_13 + x_22), x_22 >= 0
   encArg(z') -{ 0 }-> not(or(encArg(x_1''), encArg(x_2'))) :|: x_1'' >= 0, x_2' >= 0, z' = 1 + (1 + x_1'' + x_2')
   encArg(z') -{ 0 }-> not(not(encArg(z' - 2))) :|: z' - 2 >= 0
   encArg(z') -{ 0 }-> not(implies(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, x_2'' >= 0, z' = 1 + (1 + x_11 + x_2'')
   encArg(z') -{ 0 }-> not(and(encArg(x_12), encArg(x_21))) :|: z' = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0
   encArg(z') -{ 0 }-> implies(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> and(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> 1 :|: z' = 1
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encode_and(z', z'') -{ 0 }-> and(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_and(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_false -{ 0 }-> 0 :|: 
   encode_implies(z', z'') -{ 0 }-> implies(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_implies(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_not(z') -{ 1 }-> xor(x, 1) :|: z' = 1, 1 = x, x >= 0
   encode_not(z') -{ 1 }-> xor(x, 1) :|: z' = 0, 0 = x, x >= 0
   encode_not(z') -{ 1 }-> xor(x, 1) :|: z' >= 0, 0 = x, x >= 0
   encode_not(z') -{ 0 }-> not(xor(encArg(x_1188), encArg(x_2150))) :|: z' = 1 + x_1188 + x_2150, x_1188 >= 0, x_2150 >= 0
   encode_not(z') -{ 0 }-> not(or(encArg(x_1185), encArg(x_2147))) :|: x_1185 >= 0, z' = 1 + x_1185 + x_2147, x_2147 >= 0
   encode_not(z') -{ 0 }-> not(not(encArg(z' - 1))) :|: z' - 1 >= 0
   encode_not(z') -{ 0 }-> not(implies(encArg(x_1186), encArg(x_2148))) :|: z' = 1 + x_1186 + x_2148, x_1186 >= 0, x_2148 >= 0
   encode_not(z') -{ 0 }-> not(and(encArg(x_1187), encArg(x_2149))) :|: x_2149 >= 0, z' = 1 + x_1187 + x_2149, x_1187 >= 0
   encode_not(z') -{ 0 }-> 0 :|: z' >= 0
   encode_or(z', z'') -{ 0 }-> or(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_or(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_true -{ 0 }-> 1 :|: 
   encode_true -{ 0 }-> 0 :|: 
   encode_xor(z', z'') -{ 0 }-> xor(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_xor(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   implies(z', z'') -{ 4 }-> x :|: z' = 1, z'' = 1, 0 = 0, 1 = x, x >= 0
   implies(z', z'') -{ 4 }-> x :|: z'' = 0, z' = 1, 0 = 0, 0 = x, x >= 0
   implies(z', z'') -{ 3 }-> x :|: z'' >= 0, z' = 1, 0 = 0, 0 = x, x >= 0
   implies(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, 0 = 0, 0 = x', x' >= 0
   implies(z', z'') -{ 2 }-> x' :|: z' >= 0, z'' >= 0, 0 = 0, 0 = x', x' >= 0
   implies(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = 1, 0 = 0
   implies(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   implies(z', z'') -{ 2 }-> xor(xor(and(x'', z''), and(y'', z'')), 0) :|: z' = 1 + x'' + y'', z'' >= 0, y'' >= 0, x'' >= 0
   implies(z', z'') -{ 2 }-> xor(xor(and(x'', z''), and(y'', z'')), 1 + (1 + x'' + y'') + 1) :|: z' = 1 + x'' + y'', z'' >= 0, y'' >= 0, x'' >= 0
   implies(z', z'') -{ 3 }-> 0 :|: z' = 1, z'' = 1, v0 >= 0, v1 >= 0, 0 = v1, 1 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 1 + z' + 1 = v1
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 0 = v1
   implies(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = 1, 0 = z'
   implies(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' = 1, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' = 1, 0 = x, x >= 0
   implies(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 1 = v1, 0 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = x', x' >= 0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 1 + z' + 1 = v1
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   implies(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   implies(z', z'') -{ 2 }-> 0 :|: z'' >= 0, z' = 1, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 3 }-> 0 :|: z'' >= 0, z' = 1, 0 = x, x >= 0
   implies(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 1 = v1, 0 = v0
   implies(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' >= 0, 0 = x', x' >= 0
   implies(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z' = 1, z'' = 1, 0 = x1, x0 >= 0, x1 >= 0, 1 = x0
   implies(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z'' = 0, z' = 1, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 1 + z' + 1 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' >= 0, z' = 1, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 1 + z' + 1 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 1 + z' + 1 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 0 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 1 + z' + 1 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   not(z') -{ 1 }-> 0 :|: z' >= 0, v1 >= 0, 1 = v1
   not(z') -{ 2 }-> 0 :|: z' >= 0, 1 = z'
   not(z') -{ 1 }-> 1 + z' + x1 :|: z' >= 0, 1 = x1, x1 >= 0
   or(z', z'') -{ 4 }-> x :|: z' = 1, z'' = 1, 0 = 0, 1 = x, x >= 0
   or(z', z'') -{ 4 }-> x :|: z'' = 0, z' = 0, 0 = 0, 0 = x, x >= 0
   or(z', z'') -{ 4 }-> x' :|: z'' = 0, z' >= 0, z' = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, z' = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z' >= 0, z'' = z', 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 2 }-> x' :|: z' >= 0, z'' >= 0, 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = 1, 0 = 0
   or(z', z'') -{ 4 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   or(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   or(z', z'') -{ 2 }-> xor(xor(and(x', z''), and(y', z'')), 0) :|: z' = 1 + x' + y', x' >= 0, z'' >= 0, y' >= 0
   or(z', z'') -{ 2 }-> xor(xor(and(x', z''), and(y', z'')), 1 + (1 + x' + y') + z'') :|: z' = 1 + x' + y', x' >= 0, z'' >= 0, y' >= 0
   or(z', z'') -{ 3 }-> xor(xor(and(x', 0), and(y', 0)), 1 + x' + y') :|: z'' = 0, z' = 1 + x' + y', x' >= 0, y' >= 0
   or(z', z'') -{ 3 }-> xor(xor(and(x', 1 + x' + y'), and(y', 1 + x' + y')), 0) :|: z'' = 1 + x' + y', z' = 1 + x' + y', x' >= 0, y' >= 0
   or(z', z'') -{ 3 }-> 0 :|: z' = 1, z'' = 1, v0 >= 0, v1 >= 0, 0 = v1, 1 = v0
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 1 + z' + 1 = v1
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 0 = v1
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = 1, 0 = z'
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, 0 = v0
   or(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' >= 0, 0 = z'
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' = 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' = 0, 0 = x, x >= 0
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 0 = v1, 0 = v0
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   or(z', z'') -{ 4 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 1 + z' + z' = v1
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = x', x' >= 0
   or(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 1 + z' + z'' = v1, 0 = v0
   or(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' >= 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z' = 1, z'' = 1, 0 = x1, x0 >= 0, x1 >= 0, 1 = x0
   or(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z'' = 0, z' = 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 1 + z' + 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z' >= 0, z'' = z', 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 1 + z' + z'' = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 3 }-> 1 + x0 + z' :|: z'' = 0, z' >= 0, x0 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + z' :|: z'' = 0, z' >= 0, x0 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 1 + z' + 1 = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 0 = x1, x1 >= 0
   or(z', z'') -{ 3 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 1 + z' + z' = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   xor(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   xor(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' = z'
   xor(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   xor(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {not}, {encode_false}, {xor}, {encode_true}, {and}, {implies}, {or}, {encArg}, {encode_not}, {encode_xor}, {encode_implies}, {encode_and}, {encode_or}
Previous analysis results are:
  not: runtime: ?, size: O(n^1) [2 + z']

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

(43) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: not
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 2

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

(44)
Obligation:
Complexity RNTS consisting of the following rules:

   and(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' = 1
   and(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' = z'
   and(z', z'') -{ 1 }-> xor(and(x, z''), and(y, z'')) :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
   and(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
   and(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encArg(z') -{ 1 }-> xor(x, 1) :|: z' = 1 + 1, 1 = x, x >= 0
   encArg(z') -{ 1 }-> xor(x, 1) :|: z' = 1 + 0, 0 = x, x >= 0
   encArg(z') -{ 1 }-> xor(x, 1) :|: z' - 1 >= 0, 0 = x, x >= 0
   encArg(z') -{ 0 }-> xor(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> not(xor(encArg(x_13), encArg(x_22))) :|: x_13 >= 0, z' = 1 + (1 + x_13 + x_22), x_22 >= 0
   encArg(z') -{ 0 }-> not(or(encArg(x_1''), encArg(x_2'))) :|: x_1'' >= 0, x_2' >= 0, z' = 1 + (1 + x_1'' + x_2')
   encArg(z') -{ 0 }-> not(not(encArg(z' - 2))) :|: z' - 2 >= 0
   encArg(z') -{ 0 }-> not(implies(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, x_2'' >= 0, z' = 1 + (1 + x_11 + x_2'')
   encArg(z') -{ 0 }-> not(and(encArg(x_12), encArg(x_21))) :|: z' = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0
   encArg(z') -{ 0 }-> implies(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> and(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> 1 :|: z' = 1
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encode_and(z', z'') -{ 0 }-> and(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_and(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_false -{ 0 }-> 0 :|: 
   encode_implies(z', z'') -{ 0 }-> implies(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_implies(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_not(z') -{ 1 }-> xor(x, 1) :|: z' = 1, 1 = x, x >= 0
   encode_not(z') -{ 1 }-> xor(x, 1) :|: z' = 0, 0 = x, x >= 0
   encode_not(z') -{ 1 }-> xor(x, 1) :|: z' >= 0, 0 = x, x >= 0
   encode_not(z') -{ 0 }-> not(xor(encArg(x_1188), encArg(x_2150))) :|: z' = 1 + x_1188 + x_2150, x_1188 >= 0, x_2150 >= 0
   encode_not(z') -{ 0 }-> not(or(encArg(x_1185), encArg(x_2147))) :|: x_1185 >= 0, z' = 1 + x_1185 + x_2147, x_2147 >= 0
   encode_not(z') -{ 0 }-> not(not(encArg(z' - 1))) :|: z' - 1 >= 0
   encode_not(z') -{ 0 }-> not(implies(encArg(x_1186), encArg(x_2148))) :|: z' = 1 + x_1186 + x_2148, x_1186 >= 0, x_2148 >= 0
   encode_not(z') -{ 0 }-> not(and(encArg(x_1187), encArg(x_2149))) :|: x_2149 >= 0, z' = 1 + x_1187 + x_2149, x_1187 >= 0
   encode_not(z') -{ 0 }-> 0 :|: z' >= 0
   encode_or(z', z'') -{ 0 }-> or(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_or(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_true -{ 0 }-> 1 :|: 
   encode_true -{ 0 }-> 0 :|: 
   encode_xor(z', z'') -{ 0 }-> xor(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_xor(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   implies(z', z'') -{ 4 }-> x :|: z' = 1, z'' = 1, 0 = 0, 1 = x, x >= 0
   implies(z', z'') -{ 4 }-> x :|: z'' = 0, z' = 1, 0 = 0, 0 = x, x >= 0
   implies(z', z'') -{ 3 }-> x :|: z'' >= 0, z' = 1, 0 = 0, 0 = x, x >= 0
   implies(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, 0 = 0, 0 = x', x' >= 0
   implies(z', z'') -{ 2 }-> x' :|: z' >= 0, z'' >= 0, 0 = 0, 0 = x', x' >= 0
   implies(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = 1, 0 = 0
   implies(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   implies(z', z'') -{ 2 }-> xor(xor(and(x'', z''), and(y'', z'')), 0) :|: z' = 1 + x'' + y'', z'' >= 0, y'' >= 0, x'' >= 0
   implies(z', z'') -{ 2 }-> xor(xor(and(x'', z''), and(y'', z'')), 1 + (1 + x'' + y'') + 1) :|: z' = 1 + x'' + y'', z'' >= 0, y'' >= 0, x'' >= 0
   implies(z', z'') -{ 3 }-> 0 :|: z' = 1, z'' = 1, v0 >= 0, v1 >= 0, 0 = v1, 1 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 1 + z' + 1 = v1
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 0 = v1
   implies(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = 1, 0 = z'
   implies(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' = 1, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' = 1, 0 = x, x >= 0
   implies(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 1 = v1, 0 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = x', x' >= 0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 1 + z' + 1 = v1
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   implies(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   implies(z', z'') -{ 2 }-> 0 :|: z'' >= 0, z' = 1, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 3 }-> 0 :|: z'' >= 0, z' = 1, 0 = x, x >= 0
   implies(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 1 = v1, 0 = v0
   implies(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' >= 0, 0 = x', x' >= 0
   implies(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z' = 1, z'' = 1, 0 = x1, x0 >= 0, x1 >= 0, 1 = x0
   implies(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z'' = 0, z' = 1, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 1 + z' + 1 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' >= 0, z' = 1, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 1 + z' + 1 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 1 + z' + 1 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 0 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 1 + z' + 1 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   not(z') -{ 1 }-> 0 :|: z' >= 0, v1 >= 0, 1 = v1
   not(z') -{ 2 }-> 0 :|: z' >= 0, 1 = z'
   not(z') -{ 1 }-> 1 + z' + x1 :|: z' >= 0, 1 = x1, x1 >= 0
   or(z', z'') -{ 4 }-> x :|: z' = 1, z'' = 1, 0 = 0, 1 = x, x >= 0
   or(z', z'') -{ 4 }-> x :|: z'' = 0, z' = 0, 0 = 0, 0 = x, x >= 0
   or(z', z'') -{ 4 }-> x' :|: z'' = 0, z' >= 0, z' = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, z' = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z' >= 0, z'' = z', 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 2 }-> x' :|: z' >= 0, z'' >= 0, 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = 1, 0 = 0
   or(z', z'') -{ 4 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   or(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   or(z', z'') -{ 2 }-> xor(xor(and(x', z''), and(y', z'')), 0) :|: z' = 1 + x' + y', x' >= 0, z'' >= 0, y' >= 0
   or(z', z'') -{ 2 }-> xor(xor(and(x', z''), and(y', z'')), 1 + (1 + x' + y') + z'') :|: z' = 1 + x' + y', x' >= 0, z'' >= 0, y' >= 0
   or(z', z'') -{ 3 }-> xor(xor(and(x', 0), and(y', 0)), 1 + x' + y') :|: z'' = 0, z' = 1 + x' + y', x' >= 0, y' >= 0
   or(z', z'') -{ 3 }-> xor(xor(and(x', 1 + x' + y'), and(y', 1 + x' + y')), 0) :|: z'' = 1 + x' + y', z' = 1 + x' + y', x' >= 0, y' >= 0
   or(z', z'') -{ 3 }-> 0 :|: z' = 1, z'' = 1, v0 >= 0, v1 >= 0, 0 = v1, 1 = v0
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 1 + z' + 1 = v1
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 0 = v1
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = 1, 0 = z'
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, 0 = v0
   or(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' >= 0, 0 = z'
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' = 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' = 0, 0 = x, x >= 0
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 0 = v1, 0 = v0
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   or(z', z'') -{ 4 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 1 + z' + z' = v1
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = x', x' >= 0
   or(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 1 + z' + z'' = v1, 0 = v0
   or(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' >= 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z' = 1, z'' = 1, 0 = x1, x0 >= 0, x1 >= 0, 1 = x0
   or(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z'' = 0, z' = 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 1 + z' + 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z' >= 0, z'' = z', 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 1 + z' + z'' = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 3 }-> 1 + x0 + z' :|: z'' = 0, z' >= 0, x0 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + z' :|: z'' = 0, z' >= 0, x0 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 1 + z' + 1 = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 0 = x1, x1 >= 0
   or(z', z'') -{ 3 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 1 + z' + z' = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   xor(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   xor(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' = z'
   xor(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   xor(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {encode_false}, {xor}, {encode_true}, {and}, {implies}, {or}, {encArg}, {encode_not}, {encode_xor}, {encode_implies}, {encode_and}, {encode_or}
Previous analysis results are:
  not: runtime: O(1) [2], size: O(n^1) [2 + z']

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(45) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(46)
Obligation:
Complexity RNTS consisting of the following rules:

   and(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' = 1
   and(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' = z'
   and(z', z'') -{ 1 }-> xor(and(x, z''), and(y, z'')) :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
   and(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
   and(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encArg(z') -{ 1 }-> xor(x, 1) :|: z' = 1 + 1, 1 = x, x >= 0
   encArg(z') -{ 1 }-> xor(x, 1) :|: z' = 1 + 0, 0 = x, x >= 0
   encArg(z') -{ 1 }-> xor(x, 1) :|: z' - 1 >= 0, 0 = x, x >= 0
   encArg(z') -{ 0 }-> xor(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> not(xor(encArg(x_13), encArg(x_22))) :|: x_13 >= 0, z' = 1 + (1 + x_13 + x_22), x_22 >= 0
   encArg(z') -{ 0 }-> not(or(encArg(x_1''), encArg(x_2'))) :|: x_1'' >= 0, x_2' >= 0, z' = 1 + (1 + x_1'' + x_2')
   encArg(z') -{ 0 }-> not(not(encArg(z' - 2))) :|: z' - 2 >= 0
   encArg(z') -{ 0 }-> not(implies(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, x_2'' >= 0, z' = 1 + (1 + x_11 + x_2'')
   encArg(z') -{ 0 }-> not(and(encArg(x_12), encArg(x_21))) :|: z' = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0
   encArg(z') -{ 0 }-> implies(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> and(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> 1 :|: z' = 1
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encode_and(z', z'') -{ 0 }-> and(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_and(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_false -{ 0 }-> 0 :|: 
   encode_implies(z', z'') -{ 0 }-> implies(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_implies(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_not(z') -{ 1 }-> xor(x, 1) :|: z' = 1, 1 = x, x >= 0
   encode_not(z') -{ 1 }-> xor(x, 1) :|: z' = 0, 0 = x, x >= 0
   encode_not(z') -{ 1 }-> xor(x, 1) :|: z' >= 0, 0 = x, x >= 0
   encode_not(z') -{ 0 }-> not(xor(encArg(x_1188), encArg(x_2150))) :|: z' = 1 + x_1188 + x_2150, x_1188 >= 0, x_2150 >= 0
   encode_not(z') -{ 0 }-> not(or(encArg(x_1185), encArg(x_2147))) :|: x_1185 >= 0, z' = 1 + x_1185 + x_2147, x_2147 >= 0
   encode_not(z') -{ 0 }-> not(not(encArg(z' - 1))) :|: z' - 1 >= 0
   encode_not(z') -{ 0 }-> not(implies(encArg(x_1186), encArg(x_2148))) :|: z' = 1 + x_1186 + x_2148, x_1186 >= 0, x_2148 >= 0
   encode_not(z') -{ 0 }-> not(and(encArg(x_1187), encArg(x_2149))) :|: x_2149 >= 0, z' = 1 + x_1187 + x_2149, x_1187 >= 0
   encode_not(z') -{ 0 }-> 0 :|: z' >= 0
   encode_or(z', z'') -{ 0 }-> or(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_or(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_true -{ 0 }-> 1 :|: 
   encode_true -{ 0 }-> 0 :|: 
   encode_xor(z', z'') -{ 0 }-> xor(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_xor(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   implies(z', z'') -{ 4 }-> x :|: z' = 1, z'' = 1, 0 = 0, 1 = x, x >= 0
   implies(z', z'') -{ 4 }-> x :|: z'' = 0, z' = 1, 0 = 0, 0 = x, x >= 0
   implies(z', z'') -{ 3 }-> x :|: z'' >= 0, z' = 1, 0 = 0, 0 = x, x >= 0
   implies(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, 0 = 0, 0 = x', x' >= 0
   implies(z', z'') -{ 2 }-> x' :|: z' >= 0, z'' >= 0, 0 = 0, 0 = x', x' >= 0
   implies(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = 1, 0 = 0
   implies(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   implies(z', z'') -{ 2 }-> xor(xor(and(x'', z''), and(y'', z'')), 0) :|: z' = 1 + x'' + y'', z'' >= 0, y'' >= 0, x'' >= 0
   implies(z', z'') -{ 2 }-> xor(xor(and(x'', z''), and(y'', z'')), 1 + (1 + x'' + y'') + 1) :|: z' = 1 + x'' + y'', z'' >= 0, y'' >= 0, x'' >= 0
   implies(z', z'') -{ 3 }-> 0 :|: z' = 1, z'' = 1, v0 >= 0, v1 >= 0, 0 = v1, 1 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 1 + z' + 1 = v1
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 0 = v1
   implies(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = 1, 0 = z'
   implies(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' = 1, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' = 1, 0 = x, x >= 0
   implies(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 1 = v1, 0 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = x', x' >= 0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 1 + z' + 1 = v1
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   implies(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   implies(z', z'') -{ 2 }-> 0 :|: z'' >= 0, z' = 1, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 3 }-> 0 :|: z'' >= 0, z' = 1, 0 = x, x >= 0
   implies(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 1 = v1, 0 = v0
   implies(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' >= 0, 0 = x', x' >= 0
   implies(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z' = 1, z'' = 1, 0 = x1, x0 >= 0, x1 >= 0, 1 = x0
   implies(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z'' = 0, z' = 1, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 1 + z' + 1 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' >= 0, z' = 1, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 1 + z' + 1 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 1 + z' + 1 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 0 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 1 + z' + 1 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   not(z') -{ 1 }-> 0 :|: z' >= 0, v1 >= 0, 1 = v1
   not(z') -{ 2 }-> 0 :|: z' >= 0, 1 = z'
   not(z') -{ 1 }-> 1 + z' + x1 :|: z' >= 0, 1 = x1, x1 >= 0
   or(z', z'') -{ 4 }-> x :|: z' = 1, z'' = 1, 0 = 0, 1 = x, x >= 0
   or(z', z'') -{ 4 }-> x :|: z'' = 0, z' = 0, 0 = 0, 0 = x, x >= 0
   or(z', z'') -{ 4 }-> x' :|: z'' = 0, z' >= 0, z' = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, z' = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z' >= 0, z'' = z', 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 2 }-> x' :|: z' >= 0, z'' >= 0, 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = 1, 0 = 0
   or(z', z'') -{ 4 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   or(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   or(z', z'') -{ 2 }-> xor(xor(and(x', z''), and(y', z'')), 0) :|: z' = 1 + x' + y', x' >= 0, z'' >= 0, y' >= 0
   or(z', z'') -{ 2 }-> xor(xor(and(x', z''), and(y', z'')), 1 + (1 + x' + y') + z'') :|: z' = 1 + x' + y', x' >= 0, z'' >= 0, y' >= 0
   or(z', z'') -{ 3 }-> xor(xor(and(x', 0), and(y', 0)), 1 + x' + y') :|: z'' = 0, z' = 1 + x' + y', x' >= 0, y' >= 0
   or(z', z'') -{ 3 }-> xor(xor(and(x', 1 + x' + y'), and(y', 1 + x' + y')), 0) :|: z'' = 1 + x' + y', z' = 1 + x' + y', x' >= 0, y' >= 0
   or(z', z'') -{ 3 }-> 0 :|: z' = 1, z'' = 1, v0 >= 0, v1 >= 0, 0 = v1, 1 = v0
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 1 + z' + 1 = v1
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 0 = v1
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = 1, 0 = z'
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, 0 = v0
   or(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' >= 0, 0 = z'
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' = 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' = 0, 0 = x, x >= 0
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 0 = v1, 0 = v0
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   or(z', z'') -{ 4 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 1 + z' + z' = v1
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = x', x' >= 0
   or(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 1 + z' + z'' = v1, 0 = v0
   or(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' >= 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z' = 1, z'' = 1, 0 = x1, x0 >= 0, x1 >= 0, 1 = x0
   or(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z'' = 0, z' = 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 1 + z' + 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z' >= 0, z'' = z', 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 1 + z' + z'' = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 3 }-> 1 + x0 + z' :|: z'' = 0, z' >= 0, x0 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + z' :|: z'' = 0, z' >= 0, x0 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 1 + z' + 1 = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 0 = x1, x1 >= 0
   or(z', z'') -{ 3 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 1 + z' + z' = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   xor(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   xor(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' = z'
   xor(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   xor(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {encode_false}, {xor}, {encode_true}, {and}, {implies}, {or}, {encArg}, {encode_not}, {encode_xor}, {encode_implies}, {encode_and}, {encode_or}
Previous analysis results are:
  not: runtime: O(1) [2], size: O(n^1) [2 + z']

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

(47) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: encode_false
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

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

(48)
Obligation:
Complexity RNTS consisting of the following rules:

   and(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' = 1
   and(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' = z'
   and(z', z'') -{ 1 }-> xor(and(x, z''), and(y, z'')) :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
   and(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
   and(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encArg(z') -{ 1 }-> xor(x, 1) :|: z' = 1 + 1, 1 = x, x >= 0
   encArg(z') -{ 1 }-> xor(x, 1) :|: z' = 1 + 0, 0 = x, x >= 0
   encArg(z') -{ 1 }-> xor(x, 1) :|: z' - 1 >= 0, 0 = x, x >= 0
   encArg(z') -{ 0 }-> xor(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> not(xor(encArg(x_13), encArg(x_22))) :|: x_13 >= 0, z' = 1 + (1 + x_13 + x_22), x_22 >= 0
   encArg(z') -{ 0 }-> not(or(encArg(x_1''), encArg(x_2'))) :|: x_1'' >= 0, x_2' >= 0, z' = 1 + (1 + x_1'' + x_2')
   encArg(z') -{ 0 }-> not(not(encArg(z' - 2))) :|: z' - 2 >= 0
   encArg(z') -{ 0 }-> not(implies(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, x_2'' >= 0, z' = 1 + (1 + x_11 + x_2'')
   encArg(z') -{ 0 }-> not(and(encArg(x_12), encArg(x_21))) :|: z' = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0
   encArg(z') -{ 0 }-> implies(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> and(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> 1 :|: z' = 1
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encode_and(z', z'') -{ 0 }-> and(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_and(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_false -{ 0 }-> 0 :|: 
   encode_implies(z', z'') -{ 0 }-> implies(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_implies(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_not(z') -{ 1 }-> xor(x, 1) :|: z' = 1, 1 = x, x >= 0
   encode_not(z') -{ 1 }-> xor(x, 1) :|: z' = 0, 0 = x, x >= 0
   encode_not(z') -{ 1 }-> xor(x, 1) :|: z' >= 0, 0 = x, x >= 0
   encode_not(z') -{ 0 }-> not(xor(encArg(x_1188), encArg(x_2150))) :|: z' = 1 + x_1188 + x_2150, x_1188 >= 0, x_2150 >= 0
   encode_not(z') -{ 0 }-> not(or(encArg(x_1185), encArg(x_2147))) :|: x_1185 >= 0, z' = 1 + x_1185 + x_2147, x_2147 >= 0
   encode_not(z') -{ 0 }-> not(not(encArg(z' - 1))) :|: z' - 1 >= 0
   encode_not(z') -{ 0 }-> not(implies(encArg(x_1186), encArg(x_2148))) :|: z' = 1 + x_1186 + x_2148, x_1186 >= 0, x_2148 >= 0
   encode_not(z') -{ 0 }-> not(and(encArg(x_1187), encArg(x_2149))) :|: x_2149 >= 0, z' = 1 + x_1187 + x_2149, x_1187 >= 0
   encode_not(z') -{ 0 }-> 0 :|: z' >= 0
   encode_or(z', z'') -{ 0 }-> or(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_or(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_true -{ 0 }-> 1 :|: 
   encode_true -{ 0 }-> 0 :|: 
   encode_xor(z', z'') -{ 0 }-> xor(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_xor(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   implies(z', z'') -{ 4 }-> x :|: z' = 1, z'' = 1, 0 = 0, 1 = x, x >= 0
   implies(z', z'') -{ 4 }-> x :|: z'' = 0, z' = 1, 0 = 0, 0 = x, x >= 0
   implies(z', z'') -{ 3 }-> x :|: z'' >= 0, z' = 1, 0 = 0, 0 = x, x >= 0
   implies(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, 0 = 0, 0 = x', x' >= 0
   implies(z', z'') -{ 2 }-> x' :|: z' >= 0, z'' >= 0, 0 = 0, 0 = x', x' >= 0
   implies(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = 1, 0 = 0
   implies(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   implies(z', z'') -{ 2 }-> xor(xor(and(x'', z''), and(y'', z'')), 0) :|: z' = 1 + x'' + y'', z'' >= 0, y'' >= 0, x'' >= 0
   implies(z', z'') -{ 2 }-> xor(xor(and(x'', z''), and(y'', z'')), 1 + (1 + x'' + y'') + 1) :|: z' = 1 + x'' + y'', z'' >= 0, y'' >= 0, x'' >= 0
   implies(z', z'') -{ 3 }-> 0 :|: z' = 1, z'' = 1, v0 >= 0, v1 >= 0, 0 = v1, 1 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 1 + z' + 1 = v1
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 0 = v1
   implies(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = 1, 0 = z'
   implies(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' = 1, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' = 1, 0 = x, x >= 0
   implies(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 1 = v1, 0 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = x', x' >= 0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 1 + z' + 1 = v1
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   implies(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   implies(z', z'') -{ 2 }-> 0 :|: z'' >= 0, z' = 1, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 3 }-> 0 :|: z'' >= 0, z' = 1, 0 = x, x >= 0
   implies(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 1 = v1, 0 = v0
   implies(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' >= 0, 0 = x', x' >= 0
   implies(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z' = 1, z'' = 1, 0 = x1, x0 >= 0, x1 >= 0, 1 = x0
   implies(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z'' = 0, z' = 1, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 1 + z' + 1 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' >= 0, z' = 1, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 1 + z' + 1 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 1 + z' + 1 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 0 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 1 + z' + 1 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   not(z') -{ 1 }-> 0 :|: z' >= 0, v1 >= 0, 1 = v1
   not(z') -{ 2 }-> 0 :|: z' >= 0, 1 = z'
   not(z') -{ 1 }-> 1 + z' + x1 :|: z' >= 0, 1 = x1, x1 >= 0
   or(z', z'') -{ 4 }-> x :|: z' = 1, z'' = 1, 0 = 0, 1 = x, x >= 0
   or(z', z'') -{ 4 }-> x :|: z'' = 0, z' = 0, 0 = 0, 0 = x, x >= 0
   or(z', z'') -{ 4 }-> x' :|: z'' = 0, z' >= 0, z' = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, z' = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z' >= 0, z'' = z', 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 2 }-> x' :|: z' >= 0, z'' >= 0, 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = 1, 0 = 0
   or(z', z'') -{ 4 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   or(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   or(z', z'') -{ 2 }-> xor(xor(and(x', z''), and(y', z'')), 0) :|: z' = 1 + x' + y', x' >= 0, z'' >= 0, y' >= 0
   or(z', z'') -{ 2 }-> xor(xor(and(x', z''), and(y', z'')), 1 + (1 + x' + y') + z'') :|: z' = 1 + x' + y', x' >= 0, z'' >= 0, y' >= 0
   or(z', z'') -{ 3 }-> xor(xor(and(x', 0), and(y', 0)), 1 + x' + y') :|: z'' = 0, z' = 1 + x' + y', x' >= 0, y' >= 0
   or(z', z'') -{ 3 }-> xor(xor(and(x', 1 + x' + y'), and(y', 1 + x' + y')), 0) :|: z'' = 1 + x' + y', z' = 1 + x' + y', x' >= 0, y' >= 0
   or(z', z'') -{ 3 }-> 0 :|: z' = 1, z'' = 1, v0 >= 0, v1 >= 0, 0 = v1, 1 = v0
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 1 + z' + 1 = v1
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 0 = v1
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = 1, 0 = z'
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, 0 = v0
   or(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' >= 0, 0 = z'
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' = 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' = 0, 0 = x, x >= 0
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 0 = v1, 0 = v0
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   or(z', z'') -{ 4 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 1 + z' + z' = v1
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = x', x' >= 0
   or(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 1 + z' + z'' = v1, 0 = v0
   or(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' >= 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z' = 1, z'' = 1, 0 = x1, x0 >= 0, x1 >= 0, 1 = x0
   or(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z'' = 0, z' = 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 1 + z' + 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z' >= 0, z'' = z', 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 1 + z' + z'' = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 3 }-> 1 + x0 + z' :|: z'' = 0, z' >= 0, x0 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + z' :|: z'' = 0, z' >= 0, x0 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 1 + z' + 1 = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 0 = x1, x1 >= 0
   or(z', z'') -{ 3 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 1 + z' + z' = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   xor(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   xor(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' = z'
   xor(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   xor(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {encode_false}, {xor}, {encode_true}, {and}, {implies}, {or}, {encArg}, {encode_not}, {encode_xor}, {encode_implies}, {encode_and}, {encode_or}
Previous analysis results are:
  not: runtime: O(1) [2], size: O(n^1) [2 + z']
  encode_false: runtime: ?, size: O(1) [0]

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

(49) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: encode_false
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

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

(50)
Obligation:
Complexity RNTS consisting of the following rules:

   and(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' = 1
   and(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' = z'
   and(z', z'') -{ 1 }-> xor(and(x, z''), and(y, z'')) :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
   and(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
   and(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encArg(z') -{ 1 }-> xor(x, 1) :|: z' = 1 + 1, 1 = x, x >= 0
   encArg(z') -{ 1 }-> xor(x, 1) :|: z' = 1 + 0, 0 = x, x >= 0
   encArg(z') -{ 1 }-> xor(x, 1) :|: z' - 1 >= 0, 0 = x, x >= 0
   encArg(z') -{ 0 }-> xor(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> not(xor(encArg(x_13), encArg(x_22))) :|: x_13 >= 0, z' = 1 + (1 + x_13 + x_22), x_22 >= 0
   encArg(z') -{ 0 }-> not(or(encArg(x_1''), encArg(x_2'))) :|: x_1'' >= 0, x_2' >= 0, z' = 1 + (1 + x_1'' + x_2')
   encArg(z') -{ 0 }-> not(not(encArg(z' - 2))) :|: z' - 2 >= 0
   encArg(z') -{ 0 }-> not(implies(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, x_2'' >= 0, z' = 1 + (1 + x_11 + x_2'')
   encArg(z') -{ 0 }-> not(and(encArg(x_12), encArg(x_21))) :|: z' = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0
   encArg(z') -{ 0 }-> implies(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> and(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> 1 :|: z' = 1
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encode_and(z', z'') -{ 0 }-> and(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_and(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_false -{ 0 }-> 0 :|: 
   encode_implies(z', z'') -{ 0 }-> implies(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_implies(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_not(z') -{ 1 }-> xor(x, 1) :|: z' = 1, 1 = x, x >= 0
   encode_not(z') -{ 1 }-> xor(x, 1) :|: z' = 0, 0 = x, x >= 0
   encode_not(z') -{ 1 }-> xor(x, 1) :|: z' >= 0, 0 = x, x >= 0
   encode_not(z') -{ 0 }-> not(xor(encArg(x_1188), encArg(x_2150))) :|: z' = 1 + x_1188 + x_2150, x_1188 >= 0, x_2150 >= 0
   encode_not(z') -{ 0 }-> not(or(encArg(x_1185), encArg(x_2147))) :|: x_1185 >= 0, z' = 1 + x_1185 + x_2147, x_2147 >= 0
   encode_not(z') -{ 0 }-> not(not(encArg(z' - 1))) :|: z' - 1 >= 0
   encode_not(z') -{ 0 }-> not(implies(encArg(x_1186), encArg(x_2148))) :|: z' = 1 + x_1186 + x_2148, x_1186 >= 0, x_2148 >= 0
   encode_not(z') -{ 0 }-> not(and(encArg(x_1187), encArg(x_2149))) :|: x_2149 >= 0, z' = 1 + x_1187 + x_2149, x_1187 >= 0
   encode_not(z') -{ 0 }-> 0 :|: z' >= 0
   encode_or(z', z'') -{ 0 }-> or(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_or(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_true -{ 0 }-> 1 :|: 
   encode_true -{ 0 }-> 0 :|: 
   encode_xor(z', z'') -{ 0 }-> xor(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_xor(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   implies(z', z'') -{ 4 }-> x :|: z' = 1, z'' = 1, 0 = 0, 1 = x, x >= 0
   implies(z', z'') -{ 4 }-> x :|: z'' = 0, z' = 1, 0 = 0, 0 = x, x >= 0
   implies(z', z'') -{ 3 }-> x :|: z'' >= 0, z' = 1, 0 = 0, 0 = x, x >= 0
   implies(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, 0 = 0, 0 = x', x' >= 0
   implies(z', z'') -{ 2 }-> x' :|: z' >= 0, z'' >= 0, 0 = 0, 0 = x', x' >= 0
   implies(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = 1, 0 = 0
   implies(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   implies(z', z'') -{ 2 }-> xor(xor(and(x'', z''), and(y'', z'')), 0) :|: z' = 1 + x'' + y'', z'' >= 0, y'' >= 0, x'' >= 0
   implies(z', z'') -{ 2 }-> xor(xor(and(x'', z''), and(y'', z'')), 1 + (1 + x'' + y'') + 1) :|: z' = 1 + x'' + y'', z'' >= 0, y'' >= 0, x'' >= 0
   implies(z', z'') -{ 3 }-> 0 :|: z' = 1, z'' = 1, v0 >= 0, v1 >= 0, 0 = v1, 1 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 1 + z' + 1 = v1
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 0 = v1
   implies(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = 1, 0 = z'
   implies(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' = 1, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' = 1, 0 = x, x >= 0
   implies(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 1 = v1, 0 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = x', x' >= 0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 1 + z' + 1 = v1
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   implies(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   implies(z', z'') -{ 2 }-> 0 :|: z'' >= 0, z' = 1, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 3 }-> 0 :|: z'' >= 0, z' = 1, 0 = x, x >= 0
   implies(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 1 = v1, 0 = v0
   implies(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' >= 0, 0 = x', x' >= 0
   implies(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z' = 1, z'' = 1, 0 = x1, x0 >= 0, x1 >= 0, 1 = x0
   implies(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z'' = 0, z' = 1, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 1 + z' + 1 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' >= 0, z' = 1, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 1 + z' + 1 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 1 + z' + 1 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 0 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 1 + z' + 1 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   not(z') -{ 1 }-> 0 :|: z' >= 0, v1 >= 0, 1 = v1
   not(z') -{ 2 }-> 0 :|: z' >= 0, 1 = z'
   not(z') -{ 1 }-> 1 + z' + x1 :|: z' >= 0, 1 = x1, x1 >= 0
   or(z', z'') -{ 4 }-> x :|: z' = 1, z'' = 1, 0 = 0, 1 = x, x >= 0
   or(z', z'') -{ 4 }-> x :|: z'' = 0, z' = 0, 0 = 0, 0 = x, x >= 0
   or(z', z'') -{ 4 }-> x' :|: z'' = 0, z' >= 0, z' = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, z' = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z' >= 0, z'' = z', 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 2 }-> x' :|: z' >= 0, z'' >= 0, 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = 1, 0 = 0
   or(z', z'') -{ 4 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   or(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   or(z', z'') -{ 2 }-> xor(xor(and(x', z''), and(y', z'')), 0) :|: z' = 1 + x' + y', x' >= 0, z'' >= 0, y' >= 0
   or(z', z'') -{ 2 }-> xor(xor(and(x', z''), and(y', z'')), 1 + (1 + x' + y') + z'') :|: z' = 1 + x' + y', x' >= 0, z'' >= 0, y' >= 0
   or(z', z'') -{ 3 }-> xor(xor(and(x', 0), and(y', 0)), 1 + x' + y') :|: z'' = 0, z' = 1 + x' + y', x' >= 0, y' >= 0
   or(z', z'') -{ 3 }-> xor(xor(and(x', 1 + x' + y'), and(y', 1 + x' + y')), 0) :|: z'' = 1 + x' + y', z' = 1 + x' + y', x' >= 0, y' >= 0
   or(z', z'') -{ 3 }-> 0 :|: z' = 1, z'' = 1, v0 >= 0, v1 >= 0, 0 = v1, 1 = v0
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 1 + z' + 1 = v1
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 0 = v1
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = 1, 0 = z'
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, 0 = v0
   or(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' >= 0, 0 = z'
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' = 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' = 0, 0 = x, x >= 0
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 0 = v1, 0 = v0
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   or(z', z'') -{ 4 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 1 + z' + z' = v1
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = x', x' >= 0
   or(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 1 + z' + z'' = v1, 0 = v0
   or(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' >= 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z' = 1, z'' = 1, 0 = x1, x0 >= 0, x1 >= 0, 1 = x0
   or(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z'' = 0, z' = 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 1 + z' + 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z' >= 0, z'' = z', 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 1 + z' + z'' = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 3 }-> 1 + x0 + z' :|: z'' = 0, z' >= 0, x0 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + z' :|: z'' = 0, z' >= 0, x0 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 1 + z' + 1 = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 0 = x1, x1 >= 0
   or(z', z'') -{ 3 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 1 + z' + z' = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   xor(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   xor(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' = z'
   xor(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   xor(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {xor}, {encode_true}, {and}, {implies}, {or}, {encArg}, {encode_not}, {encode_xor}, {encode_implies}, {encode_and}, {encode_or}
Previous analysis results are:
  not: runtime: O(1) [2], size: O(n^1) [2 + z']
  encode_false: runtime: O(1) [0], size: O(1) [0]

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

(51) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(52)
Obligation:
Complexity RNTS consisting of the following rules:

   and(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' = 1
   and(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' = z'
   and(z', z'') -{ 1 }-> xor(and(x, z''), and(y, z'')) :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
   and(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
   and(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encArg(z') -{ 1 }-> xor(x, 1) :|: z' = 1 + 1, 1 = x, x >= 0
   encArg(z') -{ 1 }-> xor(x, 1) :|: z' = 1 + 0, 0 = x, x >= 0
   encArg(z') -{ 1 }-> xor(x, 1) :|: z' - 1 >= 0, 0 = x, x >= 0
   encArg(z') -{ 0 }-> xor(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> not(xor(encArg(x_13), encArg(x_22))) :|: x_13 >= 0, z' = 1 + (1 + x_13 + x_22), x_22 >= 0
   encArg(z') -{ 0 }-> not(or(encArg(x_1''), encArg(x_2'))) :|: x_1'' >= 0, x_2' >= 0, z' = 1 + (1 + x_1'' + x_2')
   encArg(z') -{ 0 }-> not(not(encArg(z' - 2))) :|: z' - 2 >= 0
   encArg(z') -{ 0 }-> not(implies(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, x_2'' >= 0, z' = 1 + (1 + x_11 + x_2'')
   encArg(z') -{ 0 }-> not(and(encArg(x_12), encArg(x_21))) :|: z' = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0
   encArg(z') -{ 0 }-> implies(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> and(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> 1 :|: z' = 1
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encode_and(z', z'') -{ 0 }-> and(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_and(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_false -{ 0 }-> 0 :|: 
   encode_implies(z', z'') -{ 0 }-> implies(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_implies(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_not(z') -{ 1 }-> xor(x, 1) :|: z' = 1, 1 = x, x >= 0
   encode_not(z') -{ 1 }-> xor(x, 1) :|: z' = 0, 0 = x, x >= 0
   encode_not(z') -{ 1 }-> xor(x, 1) :|: z' >= 0, 0 = x, x >= 0
   encode_not(z') -{ 0 }-> not(xor(encArg(x_1188), encArg(x_2150))) :|: z' = 1 + x_1188 + x_2150, x_1188 >= 0, x_2150 >= 0
   encode_not(z') -{ 0 }-> not(or(encArg(x_1185), encArg(x_2147))) :|: x_1185 >= 0, z' = 1 + x_1185 + x_2147, x_2147 >= 0
   encode_not(z') -{ 0 }-> not(not(encArg(z' - 1))) :|: z' - 1 >= 0
   encode_not(z') -{ 0 }-> not(implies(encArg(x_1186), encArg(x_2148))) :|: z' = 1 + x_1186 + x_2148, x_1186 >= 0, x_2148 >= 0
   encode_not(z') -{ 0 }-> not(and(encArg(x_1187), encArg(x_2149))) :|: x_2149 >= 0, z' = 1 + x_1187 + x_2149, x_1187 >= 0
   encode_not(z') -{ 0 }-> 0 :|: z' >= 0
   encode_or(z', z'') -{ 0 }-> or(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_or(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_true -{ 0 }-> 1 :|: 
   encode_true -{ 0 }-> 0 :|: 
   encode_xor(z', z'') -{ 0 }-> xor(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_xor(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   implies(z', z'') -{ 4 }-> x :|: z' = 1, z'' = 1, 0 = 0, 1 = x, x >= 0
   implies(z', z'') -{ 4 }-> x :|: z'' = 0, z' = 1, 0 = 0, 0 = x, x >= 0
   implies(z', z'') -{ 3 }-> x :|: z'' >= 0, z' = 1, 0 = 0, 0 = x, x >= 0
   implies(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, 0 = 0, 0 = x', x' >= 0
   implies(z', z'') -{ 2 }-> x' :|: z' >= 0, z'' >= 0, 0 = 0, 0 = x', x' >= 0
   implies(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = 1, 0 = 0
   implies(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   implies(z', z'') -{ 2 }-> xor(xor(and(x'', z''), and(y'', z'')), 0) :|: z' = 1 + x'' + y'', z'' >= 0, y'' >= 0, x'' >= 0
   implies(z', z'') -{ 2 }-> xor(xor(and(x'', z''), and(y'', z'')), 1 + (1 + x'' + y'') + 1) :|: z' = 1 + x'' + y'', z'' >= 0, y'' >= 0, x'' >= 0
   implies(z', z'') -{ 3 }-> 0 :|: z' = 1, z'' = 1, v0 >= 0, v1 >= 0, 0 = v1, 1 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 1 + z' + 1 = v1
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 0 = v1
   implies(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = 1, 0 = z'
   implies(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' = 1, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' = 1, 0 = x, x >= 0
   implies(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 1 = v1, 0 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = x', x' >= 0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 1 + z' + 1 = v1
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   implies(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   implies(z', z'') -{ 2 }-> 0 :|: z'' >= 0, z' = 1, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 3 }-> 0 :|: z'' >= 0, z' = 1, 0 = x, x >= 0
   implies(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 1 = v1, 0 = v0
   implies(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' >= 0, 0 = x', x' >= 0
   implies(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z' = 1, z'' = 1, 0 = x1, x0 >= 0, x1 >= 0, 1 = x0
   implies(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z'' = 0, z' = 1, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 1 + z' + 1 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' >= 0, z' = 1, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 1 + z' + 1 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 1 + z' + 1 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 0 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 1 + z' + 1 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   not(z') -{ 1 }-> 0 :|: z' >= 0, v1 >= 0, 1 = v1
   not(z') -{ 2 }-> 0 :|: z' >= 0, 1 = z'
   not(z') -{ 1 }-> 1 + z' + x1 :|: z' >= 0, 1 = x1, x1 >= 0
   or(z', z'') -{ 4 }-> x :|: z' = 1, z'' = 1, 0 = 0, 1 = x, x >= 0
   or(z', z'') -{ 4 }-> x :|: z'' = 0, z' = 0, 0 = 0, 0 = x, x >= 0
   or(z', z'') -{ 4 }-> x' :|: z'' = 0, z' >= 0, z' = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, z' = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z' >= 0, z'' = z', 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 2 }-> x' :|: z' >= 0, z'' >= 0, 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = 1, 0 = 0
   or(z', z'') -{ 4 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   or(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   or(z', z'') -{ 2 }-> xor(xor(and(x', z''), and(y', z'')), 0) :|: z' = 1 + x' + y', x' >= 0, z'' >= 0, y' >= 0
   or(z', z'') -{ 2 }-> xor(xor(and(x', z''), and(y', z'')), 1 + (1 + x' + y') + z'') :|: z' = 1 + x' + y', x' >= 0, z'' >= 0, y' >= 0
   or(z', z'') -{ 3 }-> xor(xor(and(x', 0), and(y', 0)), 1 + x' + y') :|: z'' = 0, z' = 1 + x' + y', x' >= 0, y' >= 0
   or(z', z'') -{ 3 }-> xor(xor(and(x', 1 + x' + y'), and(y', 1 + x' + y')), 0) :|: z'' = 1 + x' + y', z' = 1 + x' + y', x' >= 0, y' >= 0
   or(z', z'') -{ 3 }-> 0 :|: z' = 1, z'' = 1, v0 >= 0, v1 >= 0, 0 = v1, 1 = v0
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 1 + z' + 1 = v1
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 0 = v1
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = 1, 0 = z'
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, 0 = v0
   or(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' >= 0, 0 = z'
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' = 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' = 0, 0 = x, x >= 0
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 0 = v1, 0 = v0
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   or(z', z'') -{ 4 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 1 + z' + z' = v1
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = x', x' >= 0
   or(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 1 + z' + z'' = v1, 0 = v0
   or(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' >= 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z' = 1, z'' = 1, 0 = x1, x0 >= 0, x1 >= 0, 1 = x0
   or(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z'' = 0, z' = 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 1 + z' + 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z' >= 0, z'' = z', 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 1 + z' + z'' = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 3 }-> 1 + x0 + z' :|: z'' = 0, z' >= 0, x0 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + z' :|: z'' = 0, z' >= 0, x0 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 1 + z' + 1 = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 0 = x1, x1 >= 0
   or(z', z'') -{ 3 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 1 + z' + z' = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   xor(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   xor(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' = z'
   xor(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   xor(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {xor}, {encode_true}, {and}, {implies}, {or}, {encArg}, {encode_not}, {encode_xor}, {encode_implies}, {encode_and}, {encode_or}
Previous analysis results are:
  not: runtime: O(1) [2], size: O(n^1) [2 + z']
  encode_false: runtime: O(1) [0], size: O(1) [0]

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

(53) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: xor
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 1 + z' + z''

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

(54)
Obligation:
Complexity RNTS consisting of the following rules:

   and(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' = 1
   and(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' = z'
   and(z', z'') -{ 1 }-> xor(and(x, z''), and(y, z'')) :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
   and(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
   and(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encArg(z') -{ 1 }-> xor(x, 1) :|: z' = 1 + 1, 1 = x, x >= 0
   encArg(z') -{ 1 }-> xor(x, 1) :|: z' = 1 + 0, 0 = x, x >= 0
   encArg(z') -{ 1 }-> xor(x, 1) :|: z' - 1 >= 0, 0 = x, x >= 0
   encArg(z') -{ 0 }-> xor(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> not(xor(encArg(x_13), encArg(x_22))) :|: x_13 >= 0, z' = 1 + (1 + x_13 + x_22), x_22 >= 0
   encArg(z') -{ 0 }-> not(or(encArg(x_1''), encArg(x_2'))) :|: x_1'' >= 0, x_2' >= 0, z' = 1 + (1 + x_1'' + x_2')
   encArg(z') -{ 0 }-> not(not(encArg(z' - 2))) :|: z' - 2 >= 0
   encArg(z') -{ 0 }-> not(implies(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, x_2'' >= 0, z' = 1 + (1 + x_11 + x_2'')
   encArg(z') -{ 0 }-> not(and(encArg(x_12), encArg(x_21))) :|: z' = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0
   encArg(z') -{ 0 }-> implies(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> and(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> 1 :|: z' = 1
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encode_and(z', z'') -{ 0 }-> and(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_and(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_false -{ 0 }-> 0 :|: 
   encode_implies(z', z'') -{ 0 }-> implies(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_implies(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_not(z') -{ 1 }-> xor(x, 1) :|: z' = 1, 1 = x, x >= 0
   encode_not(z') -{ 1 }-> xor(x, 1) :|: z' = 0, 0 = x, x >= 0
   encode_not(z') -{ 1 }-> xor(x, 1) :|: z' >= 0, 0 = x, x >= 0
   encode_not(z') -{ 0 }-> not(xor(encArg(x_1188), encArg(x_2150))) :|: z' = 1 + x_1188 + x_2150, x_1188 >= 0, x_2150 >= 0
   encode_not(z') -{ 0 }-> not(or(encArg(x_1185), encArg(x_2147))) :|: x_1185 >= 0, z' = 1 + x_1185 + x_2147, x_2147 >= 0
   encode_not(z') -{ 0 }-> not(not(encArg(z' - 1))) :|: z' - 1 >= 0
   encode_not(z') -{ 0 }-> not(implies(encArg(x_1186), encArg(x_2148))) :|: z' = 1 + x_1186 + x_2148, x_1186 >= 0, x_2148 >= 0
   encode_not(z') -{ 0 }-> not(and(encArg(x_1187), encArg(x_2149))) :|: x_2149 >= 0, z' = 1 + x_1187 + x_2149, x_1187 >= 0
   encode_not(z') -{ 0 }-> 0 :|: z' >= 0
   encode_or(z', z'') -{ 0 }-> or(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_or(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_true -{ 0 }-> 1 :|: 
   encode_true -{ 0 }-> 0 :|: 
   encode_xor(z', z'') -{ 0 }-> xor(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_xor(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   implies(z', z'') -{ 4 }-> x :|: z' = 1, z'' = 1, 0 = 0, 1 = x, x >= 0
   implies(z', z'') -{ 4 }-> x :|: z'' = 0, z' = 1, 0 = 0, 0 = x, x >= 0
   implies(z', z'') -{ 3 }-> x :|: z'' >= 0, z' = 1, 0 = 0, 0 = x, x >= 0
   implies(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, 0 = 0, 0 = x', x' >= 0
   implies(z', z'') -{ 2 }-> x' :|: z' >= 0, z'' >= 0, 0 = 0, 0 = x', x' >= 0
   implies(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = 1, 0 = 0
   implies(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   implies(z', z'') -{ 2 }-> xor(xor(and(x'', z''), and(y'', z'')), 0) :|: z' = 1 + x'' + y'', z'' >= 0, y'' >= 0, x'' >= 0
   implies(z', z'') -{ 2 }-> xor(xor(and(x'', z''), and(y'', z'')), 1 + (1 + x'' + y'') + 1) :|: z' = 1 + x'' + y'', z'' >= 0, y'' >= 0, x'' >= 0
   implies(z', z'') -{ 3 }-> 0 :|: z' = 1, z'' = 1, v0 >= 0, v1 >= 0, 0 = v1, 1 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 1 + z' + 1 = v1
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 0 = v1
   implies(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = 1, 0 = z'
   implies(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' = 1, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' = 1, 0 = x, x >= 0
   implies(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 1 = v1, 0 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = x', x' >= 0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 1 + z' + 1 = v1
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   implies(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   implies(z', z'') -{ 2 }-> 0 :|: z'' >= 0, z' = 1, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 3 }-> 0 :|: z'' >= 0, z' = 1, 0 = x, x >= 0
   implies(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 1 = v1, 0 = v0
   implies(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' >= 0, 0 = x', x' >= 0
   implies(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z' = 1, z'' = 1, 0 = x1, x0 >= 0, x1 >= 0, 1 = x0
   implies(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z'' = 0, z' = 1, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 1 + z' + 1 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' >= 0, z' = 1, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 1 + z' + 1 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 1 + z' + 1 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 0 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 1 + z' + 1 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   not(z') -{ 1 }-> 0 :|: z' >= 0, v1 >= 0, 1 = v1
   not(z') -{ 2 }-> 0 :|: z' >= 0, 1 = z'
   not(z') -{ 1 }-> 1 + z' + x1 :|: z' >= 0, 1 = x1, x1 >= 0
   or(z', z'') -{ 4 }-> x :|: z' = 1, z'' = 1, 0 = 0, 1 = x, x >= 0
   or(z', z'') -{ 4 }-> x :|: z'' = 0, z' = 0, 0 = 0, 0 = x, x >= 0
   or(z', z'') -{ 4 }-> x' :|: z'' = 0, z' >= 0, z' = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, z' = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z' >= 0, z'' = z', 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 2 }-> x' :|: z' >= 0, z'' >= 0, 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = 1, 0 = 0
   or(z', z'') -{ 4 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   or(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   or(z', z'') -{ 2 }-> xor(xor(and(x', z''), and(y', z'')), 0) :|: z' = 1 + x' + y', x' >= 0, z'' >= 0, y' >= 0
   or(z', z'') -{ 2 }-> xor(xor(and(x', z''), and(y', z'')), 1 + (1 + x' + y') + z'') :|: z' = 1 + x' + y', x' >= 0, z'' >= 0, y' >= 0
   or(z', z'') -{ 3 }-> xor(xor(and(x', 0), and(y', 0)), 1 + x' + y') :|: z'' = 0, z' = 1 + x' + y', x' >= 0, y' >= 0
   or(z', z'') -{ 3 }-> xor(xor(and(x', 1 + x' + y'), and(y', 1 + x' + y')), 0) :|: z'' = 1 + x' + y', z' = 1 + x' + y', x' >= 0, y' >= 0
   or(z', z'') -{ 3 }-> 0 :|: z' = 1, z'' = 1, v0 >= 0, v1 >= 0, 0 = v1, 1 = v0
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 1 + z' + 1 = v1
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 0 = v1
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = 1, 0 = z'
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, 0 = v0
   or(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' >= 0, 0 = z'
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' = 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' = 0, 0 = x, x >= 0
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 0 = v1, 0 = v0
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   or(z', z'') -{ 4 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 1 + z' + z' = v1
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = x', x' >= 0
   or(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 1 + z' + z'' = v1, 0 = v0
   or(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' >= 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z' = 1, z'' = 1, 0 = x1, x0 >= 0, x1 >= 0, 1 = x0
   or(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z'' = 0, z' = 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 1 + z' + 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z' >= 0, z'' = z', 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 1 + z' + z'' = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 3 }-> 1 + x0 + z' :|: z'' = 0, z' >= 0, x0 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + z' :|: z'' = 0, z' >= 0, x0 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 1 + z' + 1 = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 0 = x1, x1 >= 0
   or(z', z'') -{ 3 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 1 + z' + z' = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   xor(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   xor(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' = z'
   xor(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   xor(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {xor}, {encode_true}, {and}, {implies}, {or}, {encArg}, {encode_not}, {encode_xor}, {encode_implies}, {encode_and}, {encode_or}
Previous analysis results are:
  not: runtime: O(1) [2], size: O(n^1) [2 + z']
  encode_false: runtime: O(1) [0], size: O(1) [0]
  xor: runtime: ?, size: O(n^1) [1 + z' + z'']

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

(55) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: xor
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

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

(56)
Obligation:
Complexity RNTS consisting of the following rules:

   and(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' = 1
   and(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' = z'
   and(z', z'') -{ 1 }-> xor(and(x, z''), and(y, z'')) :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
   and(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
   and(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encArg(z') -{ 1 }-> xor(x, 1) :|: z' = 1 + 1, 1 = x, x >= 0
   encArg(z') -{ 1 }-> xor(x, 1) :|: z' = 1 + 0, 0 = x, x >= 0
   encArg(z') -{ 1 }-> xor(x, 1) :|: z' - 1 >= 0, 0 = x, x >= 0
   encArg(z') -{ 0 }-> xor(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> not(xor(encArg(x_13), encArg(x_22))) :|: x_13 >= 0, z' = 1 + (1 + x_13 + x_22), x_22 >= 0
   encArg(z') -{ 0 }-> not(or(encArg(x_1''), encArg(x_2'))) :|: x_1'' >= 0, x_2' >= 0, z' = 1 + (1 + x_1'' + x_2')
   encArg(z') -{ 0 }-> not(not(encArg(z' - 2))) :|: z' - 2 >= 0
   encArg(z') -{ 0 }-> not(implies(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, x_2'' >= 0, z' = 1 + (1 + x_11 + x_2'')
   encArg(z') -{ 0 }-> not(and(encArg(x_12), encArg(x_21))) :|: z' = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0
   encArg(z') -{ 0 }-> implies(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> and(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> 1 :|: z' = 1
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encode_and(z', z'') -{ 0 }-> and(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_and(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_false -{ 0 }-> 0 :|: 
   encode_implies(z', z'') -{ 0 }-> implies(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_implies(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_not(z') -{ 1 }-> xor(x, 1) :|: z' = 1, 1 = x, x >= 0
   encode_not(z') -{ 1 }-> xor(x, 1) :|: z' = 0, 0 = x, x >= 0
   encode_not(z') -{ 1 }-> xor(x, 1) :|: z' >= 0, 0 = x, x >= 0
   encode_not(z') -{ 0 }-> not(xor(encArg(x_1188), encArg(x_2150))) :|: z' = 1 + x_1188 + x_2150, x_1188 >= 0, x_2150 >= 0
   encode_not(z') -{ 0 }-> not(or(encArg(x_1185), encArg(x_2147))) :|: x_1185 >= 0, z' = 1 + x_1185 + x_2147, x_2147 >= 0
   encode_not(z') -{ 0 }-> not(not(encArg(z' - 1))) :|: z' - 1 >= 0
   encode_not(z') -{ 0 }-> not(implies(encArg(x_1186), encArg(x_2148))) :|: z' = 1 + x_1186 + x_2148, x_1186 >= 0, x_2148 >= 0
   encode_not(z') -{ 0 }-> not(and(encArg(x_1187), encArg(x_2149))) :|: x_2149 >= 0, z' = 1 + x_1187 + x_2149, x_1187 >= 0
   encode_not(z') -{ 0 }-> 0 :|: z' >= 0
   encode_or(z', z'') -{ 0 }-> or(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_or(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_true -{ 0 }-> 1 :|: 
   encode_true -{ 0 }-> 0 :|: 
   encode_xor(z', z'') -{ 0 }-> xor(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_xor(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   implies(z', z'') -{ 4 }-> x :|: z' = 1, z'' = 1, 0 = 0, 1 = x, x >= 0
   implies(z', z'') -{ 4 }-> x :|: z'' = 0, z' = 1, 0 = 0, 0 = x, x >= 0
   implies(z', z'') -{ 3 }-> x :|: z'' >= 0, z' = 1, 0 = 0, 0 = x, x >= 0
   implies(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, 0 = 0, 0 = x', x' >= 0
   implies(z', z'') -{ 2 }-> x' :|: z' >= 0, z'' >= 0, 0 = 0, 0 = x', x' >= 0
   implies(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = 1, 0 = 0
   implies(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   implies(z', z'') -{ 2 }-> xor(xor(and(x'', z''), and(y'', z'')), 0) :|: z' = 1 + x'' + y'', z'' >= 0, y'' >= 0, x'' >= 0
   implies(z', z'') -{ 2 }-> xor(xor(and(x'', z''), and(y'', z'')), 1 + (1 + x'' + y'') + 1) :|: z' = 1 + x'' + y'', z'' >= 0, y'' >= 0, x'' >= 0
   implies(z', z'') -{ 3 }-> 0 :|: z' = 1, z'' = 1, v0 >= 0, v1 >= 0, 0 = v1, 1 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 1 + z' + 1 = v1
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 0 = v1
   implies(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = 1, 0 = z'
   implies(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' = 1, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' = 1, 0 = x, x >= 0
   implies(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 1 = v1, 0 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = x', x' >= 0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 1 + z' + 1 = v1
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   implies(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   implies(z', z'') -{ 2 }-> 0 :|: z'' >= 0, z' = 1, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 3 }-> 0 :|: z'' >= 0, z' = 1, 0 = x, x >= 0
   implies(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 1 = v1, 0 = v0
   implies(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' >= 0, 0 = x', x' >= 0
   implies(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z' = 1, z'' = 1, 0 = x1, x0 >= 0, x1 >= 0, 1 = x0
   implies(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z'' = 0, z' = 1, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 1 + z' + 1 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' >= 0, z' = 1, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 1 + z' + 1 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 1 + z' + 1 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 0 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 1 + z' + 1 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   not(z') -{ 1 }-> 0 :|: z' >= 0, v1 >= 0, 1 = v1
   not(z') -{ 2 }-> 0 :|: z' >= 0, 1 = z'
   not(z') -{ 1 }-> 1 + z' + x1 :|: z' >= 0, 1 = x1, x1 >= 0
   or(z', z'') -{ 4 }-> x :|: z' = 1, z'' = 1, 0 = 0, 1 = x, x >= 0
   or(z', z'') -{ 4 }-> x :|: z'' = 0, z' = 0, 0 = 0, 0 = x, x >= 0
   or(z', z'') -{ 4 }-> x' :|: z'' = 0, z' >= 0, z' = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, z' = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z' >= 0, z'' = z', 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 2 }-> x' :|: z' >= 0, z'' >= 0, 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = 1, 0 = 0
   or(z', z'') -{ 4 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   or(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   or(z', z'') -{ 2 }-> xor(xor(and(x', z''), and(y', z'')), 0) :|: z' = 1 + x' + y', x' >= 0, z'' >= 0, y' >= 0
   or(z', z'') -{ 2 }-> xor(xor(and(x', z''), and(y', z'')), 1 + (1 + x' + y') + z'') :|: z' = 1 + x' + y', x' >= 0, z'' >= 0, y' >= 0
   or(z', z'') -{ 3 }-> xor(xor(and(x', 0), and(y', 0)), 1 + x' + y') :|: z'' = 0, z' = 1 + x' + y', x' >= 0, y' >= 0
   or(z', z'') -{ 3 }-> xor(xor(and(x', 1 + x' + y'), and(y', 1 + x' + y')), 0) :|: z'' = 1 + x' + y', z' = 1 + x' + y', x' >= 0, y' >= 0
   or(z', z'') -{ 3 }-> 0 :|: z' = 1, z'' = 1, v0 >= 0, v1 >= 0, 0 = v1, 1 = v0
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 1 + z' + 1 = v1
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 0 = v1
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = 1, 0 = z'
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, 0 = v0
   or(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' >= 0, 0 = z'
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' = 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' = 0, 0 = x, x >= 0
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 0 = v1, 0 = v0
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   or(z', z'') -{ 4 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 1 + z' + z' = v1
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = x', x' >= 0
   or(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 1 + z' + z'' = v1, 0 = v0
   or(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' >= 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z' = 1, z'' = 1, 0 = x1, x0 >= 0, x1 >= 0, 1 = x0
   or(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z'' = 0, z' = 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 1 + z' + 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z' >= 0, z'' = z', 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 1 + z' + z'' = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 3 }-> 1 + x0 + z' :|: z'' = 0, z' >= 0, x0 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + z' :|: z'' = 0, z' >= 0, x0 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 1 + z' + 1 = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 0 = x1, x1 >= 0
   or(z', z'') -{ 3 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 1 + z' + z' = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   xor(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   xor(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' = z'
   xor(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   xor(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {encode_true}, {and}, {implies}, {or}, {encArg}, {encode_not}, {encode_xor}, {encode_implies}, {encode_and}, {encode_or}
Previous analysis results are:
  not: runtime: O(1) [2], size: O(n^1) [2 + z']
  encode_false: runtime: O(1) [0], size: O(1) [0]
  xor: runtime: O(1) [1], size: O(n^1) [1 + z' + z'']

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

(57) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(58)
Obligation:
Complexity RNTS consisting of the following rules:

   and(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' = 1
   and(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' = z'
   and(z', z'') -{ 1 }-> xor(and(x, z''), and(y, z'')) :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
   and(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
   and(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encArg(z') -{ 2 }-> s :|: s >= 0, s <= x + 1 + 1, z' = 1 + 1, 1 = x, x >= 0
   encArg(z') -{ 2 }-> s' :|: s' >= 0, s' <= x + 1 + 1, z' = 1 + 0, 0 = x, x >= 0
   encArg(z') -{ 2 }-> s'' :|: s'' >= 0, s'' <= x + 1 + 1, z' - 1 >= 0, 0 = x, x >= 0
   encArg(z') -{ 0 }-> xor(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> not(xor(encArg(x_13), encArg(x_22))) :|: x_13 >= 0, z' = 1 + (1 + x_13 + x_22), x_22 >= 0
   encArg(z') -{ 0 }-> not(or(encArg(x_1''), encArg(x_2'))) :|: x_1'' >= 0, x_2' >= 0, z' = 1 + (1 + x_1'' + x_2')
   encArg(z') -{ 0 }-> not(not(encArg(z' - 2))) :|: z' - 2 >= 0
   encArg(z') -{ 0 }-> not(implies(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, x_2'' >= 0, z' = 1 + (1 + x_11 + x_2'')
   encArg(z') -{ 0 }-> not(and(encArg(x_12), encArg(x_21))) :|: z' = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0
   encArg(z') -{ 0 }-> implies(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> and(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> 1 :|: z' = 1
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encode_and(z', z'') -{ 0 }-> and(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_and(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_false -{ 0 }-> 0 :|: 
   encode_implies(z', z'') -{ 0 }-> implies(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_implies(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_not(z') -{ 2 }-> s1 :|: s1 >= 0, s1 <= x + 1 + 1, z' = 1, 1 = x, x >= 0
   encode_not(z') -{ 2 }-> s2 :|: s2 >= 0, s2 <= x + 1 + 1, z' = 0, 0 = x, x >= 0
   encode_not(z') -{ 2 }-> s3 :|: s3 >= 0, s3 <= x + 1 + 1, z' >= 0, 0 = x, x >= 0
   encode_not(z') -{ 0 }-> not(xor(encArg(x_1188), encArg(x_2150))) :|: z' = 1 + x_1188 + x_2150, x_1188 >= 0, x_2150 >= 0
   encode_not(z') -{ 0 }-> not(or(encArg(x_1185), encArg(x_2147))) :|: x_1185 >= 0, z' = 1 + x_1185 + x_2147, x_2147 >= 0
   encode_not(z') -{ 0 }-> not(not(encArg(z' - 1))) :|: z' - 1 >= 0
   encode_not(z') -{ 0 }-> not(implies(encArg(x_1186), encArg(x_2148))) :|: z' = 1 + x_1186 + x_2148, x_1186 >= 0, x_2148 >= 0
   encode_not(z') -{ 0 }-> not(and(encArg(x_1187), encArg(x_2149))) :|: x_2149 >= 0, z' = 1 + x_1187 + x_2149, x_1187 >= 0
   encode_not(z') -{ 0 }-> 0 :|: z' >= 0
   encode_or(z', z'') -{ 0 }-> or(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_or(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_true -{ 0 }-> 1 :|: 
   encode_true -{ 0 }-> 0 :|: 
   encode_xor(z', z'') -{ 0 }-> xor(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_xor(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   implies(z', z'') -{ 4 }-> x :|: z' = 1, z'' = 1, 0 = 0, 1 = x, x >= 0
   implies(z', z'') -{ 4 }-> x :|: z'' = 0, z' = 1, 0 = 0, 0 = x, x >= 0
   implies(z', z'') -{ 3 }-> x :|: z'' >= 0, z' = 1, 0 = 0, 0 = x, x >= 0
   implies(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, 0 = 0, 0 = x', x' >= 0
   implies(z', z'') -{ 2 }-> x' :|: z' >= 0, z'' >= 0, 0 = 0, 0 = x', x' >= 0
   implies(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = 1, 0 = 0
   implies(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   implies(z', z'') -{ 2 }-> xor(xor(and(x'', z''), and(y'', z'')), 0) :|: z' = 1 + x'' + y'', z'' >= 0, y'' >= 0, x'' >= 0
   implies(z', z'') -{ 2 }-> xor(xor(and(x'', z''), and(y'', z'')), 1 + (1 + x'' + y'') + 1) :|: z' = 1 + x'' + y'', z'' >= 0, y'' >= 0, x'' >= 0
   implies(z', z'') -{ 3 }-> 0 :|: z' = 1, z'' = 1, v0 >= 0, v1 >= 0, 0 = v1, 1 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 1 + z' + 1 = v1
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 0 = v1
   implies(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = 1, 0 = z'
   implies(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' = 1, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' = 1, 0 = x, x >= 0
   implies(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 1 = v1, 0 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = x', x' >= 0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 1 + z' + 1 = v1
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   implies(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   implies(z', z'') -{ 2 }-> 0 :|: z'' >= 0, z' = 1, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 3 }-> 0 :|: z'' >= 0, z' = 1, 0 = x, x >= 0
   implies(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 1 = v1, 0 = v0
   implies(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' >= 0, 0 = x', x' >= 0
   implies(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z' = 1, z'' = 1, 0 = x1, x0 >= 0, x1 >= 0, 1 = x0
   implies(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z'' = 0, z' = 1, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 1 + z' + 1 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' >= 0, z' = 1, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 1 + z' + 1 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 1 + z' + 1 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 0 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 1 + z' + 1 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   not(z') -{ 1 }-> 0 :|: z' >= 0, v1 >= 0, 1 = v1
   not(z') -{ 2 }-> 0 :|: z' >= 0, 1 = z'
   not(z') -{ 1 }-> 1 + z' + x1 :|: z' >= 0, 1 = x1, x1 >= 0
   or(z', z'') -{ 4 }-> x :|: z' = 1, z'' = 1, 0 = 0, 1 = x, x >= 0
   or(z', z'') -{ 4 }-> x :|: z'' = 0, z' = 0, 0 = 0, 0 = x, x >= 0
   or(z', z'') -{ 4 }-> x' :|: z'' = 0, z' >= 0, z' = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, z' = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z' >= 0, z'' = z', 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 2 }-> x' :|: z' >= 0, z'' >= 0, 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = 1, 0 = 0
   or(z', z'') -{ 4 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   or(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   or(z', z'') -{ 2 }-> xor(xor(and(x', z''), and(y', z'')), 0) :|: z' = 1 + x' + y', x' >= 0, z'' >= 0, y' >= 0
   or(z', z'') -{ 2 }-> xor(xor(and(x', z''), and(y', z'')), 1 + (1 + x' + y') + z'') :|: z' = 1 + x' + y', x' >= 0, z'' >= 0, y' >= 0
   or(z', z'') -{ 3 }-> xor(xor(and(x', 0), and(y', 0)), 1 + x' + y') :|: z'' = 0, z' = 1 + x' + y', x' >= 0, y' >= 0
   or(z', z'') -{ 3 }-> xor(xor(and(x', 1 + x' + y'), and(y', 1 + x' + y')), 0) :|: z'' = 1 + x' + y', z' = 1 + x' + y', x' >= 0, y' >= 0
   or(z', z'') -{ 3 }-> 0 :|: z' = 1, z'' = 1, v0 >= 0, v1 >= 0, 0 = v1, 1 = v0
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 1 + z' + 1 = v1
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 0 = v1
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = 1, 0 = z'
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, 0 = v0
   or(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' >= 0, 0 = z'
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' = 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' = 0, 0 = x, x >= 0
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 0 = v1, 0 = v0
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   or(z', z'') -{ 4 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 1 + z' + z' = v1
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = x', x' >= 0
   or(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 1 + z' + z'' = v1, 0 = v0
   or(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' >= 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z' = 1, z'' = 1, 0 = x1, x0 >= 0, x1 >= 0, 1 = x0
   or(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z'' = 0, z' = 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 1 + z' + 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z' >= 0, z'' = z', 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 1 + z' + z'' = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 3 }-> 1 + x0 + z' :|: z'' = 0, z' >= 0, x0 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + z' :|: z'' = 0, z' >= 0, x0 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 1 + z' + 1 = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 0 = x1, x1 >= 0
   or(z', z'') -{ 3 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 1 + z' + z' = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   xor(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   xor(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' = z'
   xor(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   xor(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {encode_true}, {and}, {implies}, {or}, {encArg}, {encode_not}, {encode_xor}, {encode_implies}, {encode_and}, {encode_or}
Previous analysis results are:
  not: runtime: O(1) [2], size: O(n^1) [2 + z']
  encode_false: runtime: O(1) [0], size: O(1) [0]
  xor: runtime: O(1) [1], size: O(n^1) [1 + z' + z'']

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

(59) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: encode_true
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

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

(60)
Obligation:
Complexity RNTS consisting of the following rules:

   and(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' = 1
   and(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' = z'
   and(z', z'') -{ 1 }-> xor(and(x, z''), and(y, z'')) :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
   and(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
   and(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encArg(z') -{ 2 }-> s :|: s >= 0, s <= x + 1 + 1, z' = 1 + 1, 1 = x, x >= 0
   encArg(z') -{ 2 }-> s' :|: s' >= 0, s' <= x + 1 + 1, z' = 1 + 0, 0 = x, x >= 0
   encArg(z') -{ 2 }-> s'' :|: s'' >= 0, s'' <= x + 1 + 1, z' - 1 >= 0, 0 = x, x >= 0
   encArg(z') -{ 0 }-> xor(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> not(xor(encArg(x_13), encArg(x_22))) :|: x_13 >= 0, z' = 1 + (1 + x_13 + x_22), x_22 >= 0
   encArg(z') -{ 0 }-> not(or(encArg(x_1''), encArg(x_2'))) :|: x_1'' >= 0, x_2' >= 0, z' = 1 + (1 + x_1'' + x_2')
   encArg(z') -{ 0 }-> not(not(encArg(z' - 2))) :|: z' - 2 >= 0
   encArg(z') -{ 0 }-> not(implies(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, x_2'' >= 0, z' = 1 + (1 + x_11 + x_2'')
   encArg(z') -{ 0 }-> not(and(encArg(x_12), encArg(x_21))) :|: z' = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0
   encArg(z') -{ 0 }-> implies(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> and(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> 1 :|: z' = 1
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encode_and(z', z'') -{ 0 }-> and(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_and(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_false -{ 0 }-> 0 :|: 
   encode_implies(z', z'') -{ 0 }-> implies(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_implies(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_not(z') -{ 2 }-> s1 :|: s1 >= 0, s1 <= x + 1 + 1, z' = 1, 1 = x, x >= 0
   encode_not(z') -{ 2 }-> s2 :|: s2 >= 0, s2 <= x + 1 + 1, z' = 0, 0 = x, x >= 0
   encode_not(z') -{ 2 }-> s3 :|: s3 >= 0, s3 <= x + 1 + 1, z' >= 0, 0 = x, x >= 0
   encode_not(z') -{ 0 }-> not(xor(encArg(x_1188), encArg(x_2150))) :|: z' = 1 + x_1188 + x_2150, x_1188 >= 0, x_2150 >= 0
   encode_not(z') -{ 0 }-> not(or(encArg(x_1185), encArg(x_2147))) :|: x_1185 >= 0, z' = 1 + x_1185 + x_2147, x_2147 >= 0
   encode_not(z') -{ 0 }-> not(not(encArg(z' - 1))) :|: z' - 1 >= 0
   encode_not(z') -{ 0 }-> not(implies(encArg(x_1186), encArg(x_2148))) :|: z' = 1 + x_1186 + x_2148, x_1186 >= 0, x_2148 >= 0
   encode_not(z') -{ 0 }-> not(and(encArg(x_1187), encArg(x_2149))) :|: x_2149 >= 0, z' = 1 + x_1187 + x_2149, x_1187 >= 0
   encode_not(z') -{ 0 }-> 0 :|: z' >= 0
   encode_or(z', z'') -{ 0 }-> or(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_or(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_true -{ 0 }-> 1 :|: 
   encode_true -{ 0 }-> 0 :|: 
   encode_xor(z', z'') -{ 0 }-> xor(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_xor(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   implies(z', z'') -{ 4 }-> x :|: z' = 1, z'' = 1, 0 = 0, 1 = x, x >= 0
   implies(z', z'') -{ 4 }-> x :|: z'' = 0, z' = 1, 0 = 0, 0 = x, x >= 0
   implies(z', z'') -{ 3 }-> x :|: z'' >= 0, z' = 1, 0 = 0, 0 = x, x >= 0
   implies(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, 0 = 0, 0 = x', x' >= 0
   implies(z', z'') -{ 2 }-> x' :|: z' >= 0, z'' >= 0, 0 = 0, 0 = x', x' >= 0
   implies(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = 1, 0 = 0
   implies(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   implies(z', z'') -{ 2 }-> xor(xor(and(x'', z''), and(y'', z'')), 0) :|: z' = 1 + x'' + y'', z'' >= 0, y'' >= 0, x'' >= 0
   implies(z', z'') -{ 2 }-> xor(xor(and(x'', z''), and(y'', z'')), 1 + (1 + x'' + y'') + 1) :|: z' = 1 + x'' + y'', z'' >= 0, y'' >= 0, x'' >= 0
   implies(z', z'') -{ 3 }-> 0 :|: z' = 1, z'' = 1, v0 >= 0, v1 >= 0, 0 = v1, 1 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 1 + z' + 1 = v1
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 0 = v1
   implies(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = 1, 0 = z'
   implies(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' = 1, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' = 1, 0 = x, x >= 0
   implies(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 1 = v1, 0 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = x', x' >= 0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 1 + z' + 1 = v1
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   implies(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   implies(z', z'') -{ 2 }-> 0 :|: z'' >= 0, z' = 1, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 3 }-> 0 :|: z'' >= 0, z' = 1, 0 = x, x >= 0
   implies(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 1 = v1, 0 = v0
   implies(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' >= 0, 0 = x', x' >= 0
   implies(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z' = 1, z'' = 1, 0 = x1, x0 >= 0, x1 >= 0, 1 = x0
   implies(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z'' = 0, z' = 1, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 1 + z' + 1 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' >= 0, z' = 1, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 1 + z' + 1 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 1 + z' + 1 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 0 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 1 + z' + 1 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   not(z') -{ 1 }-> 0 :|: z' >= 0, v1 >= 0, 1 = v1
   not(z') -{ 2 }-> 0 :|: z' >= 0, 1 = z'
   not(z') -{ 1 }-> 1 + z' + x1 :|: z' >= 0, 1 = x1, x1 >= 0
   or(z', z'') -{ 4 }-> x :|: z' = 1, z'' = 1, 0 = 0, 1 = x, x >= 0
   or(z', z'') -{ 4 }-> x :|: z'' = 0, z' = 0, 0 = 0, 0 = x, x >= 0
   or(z', z'') -{ 4 }-> x' :|: z'' = 0, z' >= 0, z' = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, z' = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z' >= 0, z'' = z', 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 2 }-> x' :|: z' >= 0, z'' >= 0, 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = 1, 0 = 0
   or(z', z'') -{ 4 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   or(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   or(z', z'') -{ 2 }-> xor(xor(and(x', z''), and(y', z'')), 0) :|: z' = 1 + x' + y', x' >= 0, z'' >= 0, y' >= 0
   or(z', z'') -{ 2 }-> xor(xor(and(x', z''), and(y', z'')), 1 + (1 + x' + y') + z'') :|: z' = 1 + x' + y', x' >= 0, z'' >= 0, y' >= 0
   or(z', z'') -{ 3 }-> xor(xor(and(x', 0), and(y', 0)), 1 + x' + y') :|: z'' = 0, z' = 1 + x' + y', x' >= 0, y' >= 0
   or(z', z'') -{ 3 }-> xor(xor(and(x', 1 + x' + y'), and(y', 1 + x' + y')), 0) :|: z'' = 1 + x' + y', z' = 1 + x' + y', x' >= 0, y' >= 0
   or(z', z'') -{ 3 }-> 0 :|: z' = 1, z'' = 1, v0 >= 0, v1 >= 0, 0 = v1, 1 = v0
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 1 + z' + 1 = v1
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 0 = v1
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = 1, 0 = z'
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, 0 = v0
   or(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' >= 0, 0 = z'
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' = 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' = 0, 0 = x, x >= 0
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 0 = v1, 0 = v0
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   or(z', z'') -{ 4 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 1 + z' + z' = v1
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = x', x' >= 0
   or(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 1 + z' + z'' = v1, 0 = v0
   or(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' >= 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z' = 1, z'' = 1, 0 = x1, x0 >= 0, x1 >= 0, 1 = x0
   or(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z'' = 0, z' = 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 1 + z' + 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z' >= 0, z'' = z', 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 1 + z' + z'' = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 3 }-> 1 + x0 + z' :|: z'' = 0, z' >= 0, x0 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + z' :|: z'' = 0, z' >= 0, x0 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 1 + z' + 1 = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 0 = x1, x1 >= 0
   or(z', z'') -{ 3 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 1 + z' + z' = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   xor(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   xor(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' = z'
   xor(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   xor(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {encode_true}, {and}, {implies}, {or}, {encArg}, {encode_not}, {encode_xor}, {encode_implies}, {encode_and}, {encode_or}
Previous analysis results are:
  not: runtime: O(1) [2], size: O(n^1) [2 + z']
  encode_false: runtime: O(1) [0], size: O(1) [0]
  xor: runtime: O(1) [1], size: O(n^1) [1 + z' + z'']
  encode_true: runtime: ?, size: O(1) [1]

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

(61) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: encode_true
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

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

(62)
Obligation:
Complexity RNTS consisting of the following rules:

   and(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' = 1
   and(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' = z'
   and(z', z'') -{ 1 }-> xor(and(x, z''), and(y, z'')) :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
   and(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
   and(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encArg(z') -{ 2 }-> s :|: s >= 0, s <= x + 1 + 1, z' = 1 + 1, 1 = x, x >= 0
   encArg(z') -{ 2 }-> s' :|: s' >= 0, s' <= x + 1 + 1, z' = 1 + 0, 0 = x, x >= 0
   encArg(z') -{ 2 }-> s'' :|: s'' >= 0, s'' <= x + 1 + 1, z' - 1 >= 0, 0 = x, x >= 0
   encArg(z') -{ 0 }-> xor(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> not(xor(encArg(x_13), encArg(x_22))) :|: x_13 >= 0, z' = 1 + (1 + x_13 + x_22), x_22 >= 0
   encArg(z') -{ 0 }-> not(or(encArg(x_1''), encArg(x_2'))) :|: x_1'' >= 0, x_2' >= 0, z' = 1 + (1 + x_1'' + x_2')
   encArg(z') -{ 0 }-> not(not(encArg(z' - 2))) :|: z' - 2 >= 0
   encArg(z') -{ 0 }-> not(implies(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, x_2'' >= 0, z' = 1 + (1 + x_11 + x_2'')
   encArg(z') -{ 0 }-> not(and(encArg(x_12), encArg(x_21))) :|: z' = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0
   encArg(z') -{ 0 }-> implies(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> and(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> 1 :|: z' = 1
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encode_and(z', z'') -{ 0 }-> and(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_and(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_false -{ 0 }-> 0 :|: 
   encode_implies(z', z'') -{ 0 }-> implies(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_implies(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_not(z') -{ 2 }-> s1 :|: s1 >= 0, s1 <= x + 1 + 1, z' = 1, 1 = x, x >= 0
   encode_not(z') -{ 2 }-> s2 :|: s2 >= 0, s2 <= x + 1 + 1, z' = 0, 0 = x, x >= 0
   encode_not(z') -{ 2 }-> s3 :|: s3 >= 0, s3 <= x + 1 + 1, z' >= 0, 0 = x, x >= 0
   encode_not(z') -{ 0 }-> not(xor(encArg(x_1188), encArg(x_2150))) :|: z' = 1 + x_1188 + x_2150, x_1188 >= 0, x_2150 >= 0
   encode_not(z') -{ 0 }-> not(or(encArg(x_1185), encArg(x_2147))) :|: x_1185 >= 0, z' = 1 + x_1185 + x_2147, x_2147 >= 0
   encode_not(z') -{ 0 }-> not(not(encArg(z' - 1))) :|: z' - 1 >= 0
   encode_not(z') -{ 0 }-> not(implies(encArg(x_1186), encArg(x_2148))) :|: z' = 1 + x_1186 + x_2148, x_1186 >= 0, x_2148 >= 0
   encode_not(z') -{ 0 }-> not(and(encArg(x_1187), encArg(x_2149))) :|: x_2149 >= 0, z' = 1 + x_1187 + x_2149, x_1187 >= 0
   encode_not(z') -{ 0 }-> 0 :|: z' >= 0
   encode_or(z', z'') -{ 0 }-> or(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_or(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_true -{ 0 }-> 1 :|: 
   encode_true -{ 0 }-> 0 :|: 
   encode_xor(z', z'') -{ 0 }-> xor(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_xor(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   implies(z', z'') -{ 4 }-> x :|: z' = 1, z'' = 1, 0 = 0, 1 = x, x >= 0
   implies(z', z'') -{ 4 }-> x :|: z'' = 0, z' = 1, 0 = 0, 0 = x, x >= 0
   implies(z', z'') -{ 3 }-> x :|: z'' >= 0, z' = 1, 0 = 0, 0 = x, x >= 0
   implies(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, 0 = 0, 0 = x', x' >= 0
   implies(z', z'') -{ 2 }-> x' :|: z' >= 0, z'' >= 0, 0 = 0, 0 = x', x' >= 0
   implies(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = 1, 0 = 0
   implies(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   implies(z', z'') -{ 2 }-> xor(xor(and(x'', z''), and(y'', z'')), 0) :|: z' = 1 + x'' + y'', z'' >= 0, y'' >= 0, x'' >= 0
   implies(z', z'') -{ 2 }-> xor(xor(and(x'', z''), and(y'', z'')), 1 + (1 + x'' + y'') + 1) :|: z' = 1 + x'' + y'', z'' >= 0, y'' >= 0, x'' >= 0
   implies(z', z'') -{ 3 }-> 0 :|: z' = 1, z'' = 1, v0 >= 0, v1 >= 0, 0 = v1, 1 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 1 + z' + 1 = v1
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 0 = v1
   implies(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = 1, 0 = z'
   implies(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' = 1, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' = 1, 0 = x, x >= 0
   implies(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 1 = v1, 0 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = x', x' >= 0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 1 + z' + 1 = v1
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   implies(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   implies(z', z'') -{ 2 }-> 0 :|: z'' >= 0, z' = 1, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 3 }-> 0 :|: z'' >= 0, z' = 1, 0 = x, x >= 0
   implies(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 1 = v1, 0 = v0
   implies(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' >= 0, 0 = x', x' >= 0
   implies(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z' = 1, z'' = 1, 0 = x1, x0 >= 0, x1 >= 0, 1 = x0
   implies(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z'' = 0, z' = 1, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 1 + z' + 1 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' >= 0, z' = 1, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 1 + z' + 1 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 1 + z' + 1 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 0 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 1 + z' + 1 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   not(z') -{ 1 }-> 0 :|: z' >= 0, v1 >= 0, 1 = v1
   not(z') -{ 2 }-> 0 :|: z' >= 0, 1 = z'
   not(z') -{ 1 }-> 1 + z' + x1 :|: z' >= 0, 1 = x1, x1 >= 0
   or(z', z'') -{ 4 }-> x :|: z' = 1, z'' = 1, 0 = 0, 1 = x, x >= 0
   or(z', z'') -{ 4 }-> x :|: z'' = 0, z' = 0, 0 = 0, 0 = x, x >= 0
   or(z', z'') -{ 4 }-> x' :|: z'' = 0, z' >= 0, z' = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, z' = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z' >= 0, z'' = z', 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 2 }-> x' :|: z' >= 0, z'' >= 0, 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = 1, 0 = 0
   or(z', z'') -{ 4 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   or(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   or(z', z'') -{ 2 }-> xor(xor(and(x', z''), and(y', z'')), 0) :|: z' = 1 + x' + y', x' >= 0, z'' >= 0, y' >= 0
   or(z', z'') -{ 2 }-> xor(xor(and(x', z''), and(y', z'')), 1 + (1 + x' + y') + z'') :|: z' = 1 + x' + y', x' >= 0, z'' >= 0, y' >= 0
   or(z', z'') -{ 3 }-> xor(xor(and(x', 0), and(y', 0)), 1 + x' + y') :|: z'' = 0, z' = 1 + x' + y', x' >= 0, y' >= 0
   or(z', z'') -{ 3 }-> xor(xor(and(x', 1 + x' + y'), and(y', 1 + x' + y')), 0) :|: z'' = 1 + x' + y', z' = 1 + x' + y', x' >= 0, y' >= 0
   or(z', z'') -{ 3 }-> 0 :|: z' = 1, z'' = 1, v0 >= 0, v1 >= 0, 0 = v1, 1 = v0
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 1 + z' + 1 = v1
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 0 = v1
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = 1, 0 = z'
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, 0 = v0
   or(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' >= 0, 0 = z'
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' = 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' = 0, 0 = x, x >= 0
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 0 = v1, 0 = v0
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   or(z', z'') -{ 4 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 1 + z' + z' = v1
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = x', x' >= 0
   or(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 1 + z' + z'' = v1, 0 = v0
   or(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' >= 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z' = 1, z'' = 1, 0 = x1, x0 >= 0, x1 >= 0, 1 = x0
   or(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z'' = 0, z' = 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 1 + z' + 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z' >= 0, z'' = z', 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 1 + z' + z'' = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 3 }-> 1 + x0 + z' :|: z'' = 0, z' >= 0, x0 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + z' :|: z'' = 0, z' >= 0, x0 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 1 + z' + 1 = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 0 = x1, x1 >= 0
   or(z', z'') -{ 3 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 1 + z' + z' = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   xor(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   xor(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' = z'
   xor(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   xor(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {and}, {implies}, {or}, {encArg}, {encode_not}, {encode_xor}, {encode_implies}, {encode_and}, {encode_or}
Previous analysis results are:
  not: runtime: O(1) [2], size: O(n^1) [2 + z']
  encode_false: runtime: O(1) [0], size: O(1) [0]
  xor: runtime: O(1) [1], size: O(n^1) [1 + z' + z'']
  encode_true: runtime: O(1) [0], size: O(1) [1]

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(63) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(64)
Obligation:
Complexity RNTS consisting of the following rules:

   and(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' = 1
   and(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' = z'
   and(z', z'') -{ 1 }-> xor(and(x, z''), and(y, z'')) :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
   and(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
   and(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encArg(z') -{ 2 }-> s :|: s >= 0, s <= x + 1 + 1, z' = 1 + 1, 1 = x, x >= 0
   encArg(z') -{ 2 }-> s' :|: s' >= 0, s' <= x + 1 + 1, z' = 1 + 0, 0 = x, x >= 0
   encArg(z') -{ 2 }-> s'' :|: s'' >= 0, s'' <= x + 1 + 1, z' - 1 >= 0, 0 = x, x >= 0
   encArg(z') -{ 0 }-> xor(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> not(xor(encArg(x_13), encArg(x_22))) :|: x_13 >= 0, z' = 1 + (1 + x_13 + x_22), x_22 >= 0
   encArg(z') -{ 0 }-> not(or(encArg(x_1''), encArg(x_2'))) :|: x_1'' >= 0, x_2' >= 0, z' = 1 + (1 + x_1'' + x_2')
   encArg(z') -{ 0 }-> not(not(encArg(z' - 2))) :|: z' - 2 >= 0
   encArg(z') -{ 0 }-> not(implies(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, x_2'' >= 0, z' = 1 + (1 + x_11 + x_2'')
   encArg(z') -{ 0 }-> not(and(encArg(x_12), encArg(x_21))) :|: z' = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0
   encArg(z') -{ 0 }-> implies(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> and(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> 1 :|: z' = 1
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encode_and(z', z'') -{ 0 }-> and(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_and(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_false -{ 0 }-> 0 :|: 
   encode_implies(z', z'') -{ 0 }-> implies(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_implies(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_not(z') -{ 2 }-> s1 :|: s1 >= 0, s1 <= x + 1 + 1, z' = 1, 1 = x, x >= 0
   encode_not(z') -{ 2 }-> s2 :|: s2 >= 0, s2 <= x + 1 + 1, z' = 0, 0 = x, x >= 0
   encode_not(z') -{ 2 }-> s3 :|: s3 >= 0, s3 <= x + 1 + 1, z' >= 0, 0 = x, x >= 0
   encode_not(z') -{ 0 }-> not(xor(encArg(x_1188), encArg(x_2150))) :|: z' = 1 + x_1188 + x_2150, x_1188 >= 0, x_2150 >= 0
   encode_not(z') -{ 0 }-> not(or(encArg(x_1185), encArg(x_2147))) :|: x_1185 >= 0, z' = 1 + x_1185 + x_2147, x_2147 >= 0
   encode_not(z') -{ 0 }-> not(not(encArg(z' - 1))) :|: z' - 1 >= 0
   encode_not(z') -{ 0 }-> not(implies(encArg(x_1186), encArg(x_2148))) :|: z' = 1 + x_1186 + x_2148, x_1186 >= 0, x_2148 >= 0
   encode_not(z') -{ 0 }-> not(and(encArg(x_1187), encArg(x_2149))) :|: x_2149 >= 0, z' = 1 + x_1187 + x_2149, x_1187 >= 0
   encode_not(z') -{ 0 }-> 0 :|: z' >= 0
   encode_or(z', z'') -{ 0 }-> or(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_or(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_true -{ 0 }-> 1 :|: 
   encode_true -{ 0 }-> 0 :|: 
   encode_xor(z', z'') -{ 0 }-> xor(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_xor(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   implies(z', z'') -{ 4 }-> x :|: z' = 1, z'' = 1, 0 = 0, 1 = x, x >= 0
   implies(z', z'') -{ 4 }-> x :|: z'' = 0, z' = 1, 0 = 0, 0 = x, x >= 0
   implies(z', z'') -{ 3 }-> x :|: z'' >= 0, z' = 1, 0 = 0, 0 = x, x >= 0
   implies(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, 0 = 0, 0 = x', x' >= 0
   implies(z', z'') -{ 2 }-> x' :|: z' >= 0, z'' >= 0, 0 = 0, 0 = x', x' >= 0
   implies(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = 1, 0 = 0
   implies(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   implies(z', z'') -{ 2 }-> xor(xor(and(x'', z''), and(y'', z'')), 0) :|: z' = 1 + x'' + y'', z'' >= 0, y'' >= 0, x'' >= 0
   implies(z', z'') -{ 2 }-> xor(xor(and(x'', z''), and(y'', z'')), 1 + (1 + x'' + y'') + 1) :|: z' = 1 + x'' + y'', z'' >= 0, y'' >= 0, x'' >= 0
   implies(z', z'') -{ 3 }-> 0 :|: z' = 1, z'' = 1, v0 >= 0, v1 >= 0, 0 = v1, 1 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 1 + z' + 1 = v1
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 0 = v1
   implies(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = 1, 0 = z'
   implies(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' = 1, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' = 1, 0 = x, x >= 0
   implies(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 1 = v1, 0 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = x', x' >= 0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 1 + z' + 1 = v1
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   implies(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   implies(z', z'') -{ 2 }-> 0 :|: z'' >= 0, z' = 1, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 3 }-> 0 :|: z'' >= 0, z' = 1, 0 = x, x >= 0
   implies(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 1 = v1, 0 = v0
   implies(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' >= 0, 0 = x', x' >= 0
   implies(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z' = 1, z'' = 1, 0 = x1, x0 >= 0, x1 >= 0, 1 = x0
   implies(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z'' = 0, z' = 1, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 1 + z' + 1 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' >= 0, z' = 1, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 1 + z' + 1 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 1 + z' + 1 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 0 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 1 + z' + 1 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   not(z') -{ 1 }-> 0 :|: z' >= 0, v1 >= 0, 1 = v1
   not(z') -{ 2 }-> 0 :|: z' >= 0, 1 = z'
   not(z') -{ 1 }-> 1 + z' + x1 :|: z' >= 0, 1 = x1, x1 >= 0
   or(z', z'') -{ 4 }-> x :|: z' = 1, z'' = 1, 0 = 0, 1 = x, x >= 0
   or(z', z'') -{ 4 }-> x :|: z'' = 0, z' = 0, 0 = 0, 0 = x, x >= 0
   or(z', z'') -{ 4 }-> x' :|: z'' = 0, z' >= 0, z' = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, z' = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z' >= 0, z'' = z', 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 2 }-> x' :|: z' >= 0, z'' >= 0, 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = 1, 0 = 0
   or(z', z'') -{ 4 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   or(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   or(z', z'') -{ 2 }-> xor(xor(and(x', z''), and(y', z'')), 0) :|: z' = 1 + x' + y', x' >= 0, z'' >= 0, y' >= 0
   or(z', z'') -{ 2 }-> xor(xor(and(x', z''), and(y', z'')), 1 + (1 + x' + y') + z'') :|: z' = 1 + x' + y', x' >= 0, z'' >= 0, y' >= 0
   or(z', z'') -{ 3 }-> xor(xor(and(x', 0), and(y', 0)), 1 + x' + y') :|: z'' = 0, z' = 1 + x' + y', x' >= 0, y' >= 0
   or(z', z'') -{ 3 }-> xor(xor(and(x', 1 + x' + y'), and(y', 1 + x' + y')), 0) :|: z'' = 1 + x' + y', z' = 1 + x' + y', x' >= 0, y' >= 0
   or(z', z'') -{ 3 }-> 0 :|: z' = 1, z'' = 1, v0 >= 0, v1 >= 0, 0 = v1, 1 = v0
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 1 + z' + 1 = v1
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 0 = v1
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = 1, 0 = z'
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, 0 = v0
   or(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' >= 0, 0 = z'
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' = 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' = 0, 0 = x, x >= 0
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 0 = v1, 0 = v0
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   or(z', z'') -{ 4 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 1 + z' + z' = v1
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = x', x' >= 0
   or(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 1 + z' + z'' = v1, 0 = v0
   or(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' >= 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z' = 1, z'' = 1, 0 = x1, x0 >= 0, x1 >= 0, 1 = x0
   or(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z'' = 0, z' = 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 1 + z' + 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z' >= 0, z'' = z', 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 1 + z' + z'' = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 3 }-> 1 + x0 + z' :|: z'' = 0, z' >= 0, x0 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + z' :|: z'' = 0, z' >= 0, x0 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 1 + z' + 1 = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 0 = x1, x1 >= 0
   or(z', z'') -{ 3 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 1 + z' + z' = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   xor(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   xor(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' = z'
   xor(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   xor(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {and}, {implies}, {or}, {encArg}, {encode_not}, {encode_xor}, {encode_implies}, {encode_and}, {encode_or}
Previous analysis results are:
  not: runtime: O(1) [2], size: O(n^1) [2 + z']
  encode_false: runtime: O(1) [0], size: O(1) [0]
  xor: runtime: O(1) [1], size: O(n^1) [1 + z' + z'']
  encode_true: runtime: O(1) [0], size: O(1) [1]

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

(65) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: and
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 2*z' + z''

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

(66)
Obligation:
Complexity RNTS consisting of the following rules:

   and(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' = 1
   and(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' = z'
   and(z', z'') -{ 1 }-> xor(and(x, z''), and(y, z'')) :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
   and(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
   and(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encArg(z') -{ 2 }-> s :|: s >= 0, s <= x + 1 + 1, z' = 1 + 1, 1 = x, x >= 0
   encArg(z') -{ 2 }-> s' :|: s' >= 0, s' <= x + 1 + 1, z' = 1 + 0, 0 = x, x >= 0
   encArg(z') -{ 2 }-> s'' :|: s'' >= 0, s'' <= x + 1 + 1, z' - 1 >= 0, 0 = x, x >= 0
   encArg(z') -{ 0 }-> xor(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> not(xor(encArg(x_13), encArg(x_22))) :|: x_13 >= 0, z' = 1 + (1 + x_13 + x_22), x_22 >= 0
   encArg(z') -{ 0 }-> not(or(encArg(x_1''), encArg(x_2'))) :|: x_1'' >= 0, x_2' >= 0, z' = 1 + (1 + x_1'' + x_2')
   encArg(z') -{ 0 }-> not(not(encArg(z' - 2))) :|: z' - 2 >= 0
   encArg(z') -{ 0 }-> not(implies(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, x_2'' >= 0, z' = 1 + (1 + x_11 + x_2'')
   encArg(z') -{ 0 }-> not(and(encArg(x_12), encArg(x_21))) :|: z' = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0
   encArg(z') -{ 0 }-> implies(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> and(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> 1 :|: z' = 1
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encode_and(z', z'') -{ 0 }-> and(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_and(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_false -{ 0 }-> 0 :|: 
   encode_implies(z', z'') -{ 0 }-> implies(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_implies(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_not(z') -{ 2 }-> s1 :|: s1 >= 0, s1 <= x + 1 + 1, z' = 1, 1 = x, x >= 0
   encode_not(z') -{ 2 }-> s2 :|: s2 >= 0, s2 <= x + 1 + 1, z' = 0, 0 = x, x >= 0
   encode_not(z') -{ 2 }-> s3 :|: s3 >= 0, s3 <= x + 1 + 1, z' >= 0, 0 = x, x >= 0
   encode_not(z') -{ 0 }-> not(xor(encArg(x_1188), encArg(x_2150))) :|: z' = 1 + x_1188 + x_2150, x_1188 >= 0, x_2150 >= 0
   encode_not(z') -{ 0 }-> not(or(encArg(x_1185), encArg(x_2147))) :|: x_1185 >= 0, z' = 1 + x_1185 + x_2147, x_2147 >= 0
   encode_not(z') -{ 0 }-> not(not(encArg(z' - 1))) :|: z' - 1 >= 0
   encode_not(z') -{ 0 }-> not(implies(encArg(x_1186), encArg(x_2148))) :|: z' = 1 + x_1186 + x_2148, x_1186 >= 0, x_2148 >= 0
   encode_not(z') -{ 0 }-> not(and(encArg(x_1187), encArg(x_2149))) :|: x_2149 >= 0, z' = 1 + x_1187 + x_2149, x_1187 >= 0
   encode_not(z') -{ 0 }-> 0 :|: z' >= 0
   encode_or(z', z'') -{ 0 }-> or(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_or(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_true -{ 0 }-> 1 :|: 
   encode_true -{ 0 }-> 0 :|: 
   encode_xor(z', z'') -{ 0 }-> xor(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_xor(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   implies(z', z'') -{ 4 }-> x :|: z' = 1, z'' = 1, 0 = 0, 1 = x, x >= 0
   implies(z', z'') -{ 4 }-> x :|: z'' = 0, z' = 1, 0 = 0, 0 = x, x >= 0
   implies(z', z'') -{ 3 }-> x :|: z'' >= 0, z' = 1, 0 = 0, 0 = x, x >= 0
   implies(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, 0 = 0, 0 = x', x' >= 0
   implies(z', z'') -{ 2 }-> x' :|: z' >= 0, z'' >= 0, 0 = 0, 0 = x', x' >= 0
   implies(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = 1, 0 = 0
   implies(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   implies(z', z'') -{ 2 }-> xor(xor(and(x'', z''), and(y'', z'')), 0) :|: z' = 1 + x'' + y'', z'' >= 0, y'' >= 0, x'' >= 0
   implies(z', z'') -{ 2 }-> xor(xor(and(x'', z''), and(y'', z'')), 1 + (1 + x'' + y'') + 1) :|: z' = 1 + x'' + y'', z'' >= 0, y'' >= 0, x'' >= 0
   implies(z', z'') -{ 3 }-> 0 :|: z' = 1, z'' = 1, v0 >= 0, v1 >= 0, 0 = v1, 1 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 1 + z' + 1 = v1
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 0 = v1
   implies(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = 1, 0 = z'
   implies(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' = 1, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' = 1, 0 = x, x >= 0
   implies(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 1 = v1, 0 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = x', x' >= 0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 1 + z' + 1 = v1
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   implies(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   implies(z', z'') -{ 2 }-> 0 :|: z'' >= 0, z' = 1, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 3 }-> 0 :|: z'' >= 0, z' = 1, 0 = x, x >= 0
   implies(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 1 = v1, 0 = v0
   implies(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' >= 0, 0 = x', x' >= 0
   implies(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z' = 1, z'' = 1, 0 = x1, x0 >= 0, x1 >= 0, 1 = x0
   implies(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z'' = 0, z' = 1, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 1 + z' + 1 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' >= 0, z' = 1, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 1 + z' + 1 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 1 + z' + 1 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 0 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 1 + z' + 1 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   not(z') -{ 1 }-> 0 :|: z' >= 0, v1 >= 0, 1 = v1
   not(z') -{ 2 }-> 0 :|: z' >= 0, 1 = z'
   not(z') -{ 1 }-> 1 + z' + x1 :|: z' >= 0, 1 = x1, x1 >= 0
   or(z', z'') -{ 4 }-> x :|: z' = 1, z'' = 1, 0 = 0, 1 = x, x >= 0
   or(z', z'') -{ 4 }-> x :|: z'' = 0, z' = 0, 0 = 0, 0 = x, x >= 0
   or(z', z'') -{ 4 }-> x' :|: z'' = 0, z' >= 0, z' = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, z' = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z' >= 0, z'' = z', 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 2 }-> x' :|: z' >= 0, z'' >= 0, 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = 1, 0 = 0
   or(z', z'') -{ 4 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   or(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   or(z', z'') -{ 2 }-> xor(xor(and(x', z''), and(y', z'')), 0) :|: z' = 1 + x' + y', x' >= 0, z'' >= 0, y' >= 0
   or(z', z'') -{ 2 }-> xor(xor(and(x', z''), and(y', z'')), 1 + (1 + x' + y') + z'') :|: z' = 1 + x' + y', x' >= 0, z'' >= 0, y' >= 0
   or(z', z'') -{ 3 }-> xor(xor(and(x', 0), and(y', 0)), 1 + x' + y') :|: z'' = 0, z' = 1 + x' + y', x' >= 0, y' >= 0
   or(z', z'') -{ 3 }-> xor(xor(and(x', 1 + x' + y'), and(y', 1 + x' + y')), 0) :|: z'' = 1 + x' + y', z' = 1 + x' + y', x' >= 0, y' >= 0
   or(z', z'') -{ 3 }-> 0 :|: z' = 1, z'' = 1, v0 >= 0, v1 >= 0, 0 = v1, 1 = v0
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 1 + z' + 1 = v1
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 0 = v1
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = 1, 0 = z'
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, 0 = v0
   or(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' >= 0, 0 = z'
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' = 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' = 0, 0 = x, x >= 0
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 0 = v1, 0 = v0
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   or(z', z'') -{ 4 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 1 + z' + z' = v1
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = x', x' >= 0
   or(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 1 + z' + z'' = v1, 0 = v0
   or(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' >= 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z' = 1, z'' = 1, 0 = x1, x0 >= 0, x1 >= 0, 1 = x0
   or(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z'' = 0, z' = 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 1 + z' + 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z' >= 0, z'' = z', 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 1 + z' + z'' = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 3 }-> 1 + x0 + z' :|: z'' = 0, z' >= 0, x0 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + z' :|: z'' = 0, z' >= 0, x0 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 1 + z' + 1 = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 0 = x1, x1 >= 0
   or(z', z'') -{ 3 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 1 + z' + z' = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   xor(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   xor(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' = z'
   xor(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   xor(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {and}, {implies}, {or}, {encArg}, {encode_not}, {encode_xor}, {encode_implies}, {encode_and}, {encode_or}
Previous analysis results are:
  not: runtime: O(1) [2], size: O(n^1) [2 + z']
  encode_false: runtime: O(1) [0], size: O(1) [0]
  xor: runtime: O(1) [1], size: O(n^1) [1 + z' + z'']
  encode_true: runtime: O(1) [0], size: O(1) [1]
  and: runtime: ?, size: O(n^1) [2*z' + z'']

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

(67) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: and
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 1 + 3*z'

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

(68)
Obligation:
Complexity RNTS consisting of the following rules:

   and(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' = 1
   and(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' = z'
   and(z', z'') -{ 1 }-> xor(and(x, z''), and(y, z'')) :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
   and(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
   and(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encArg(z') -{ 2 }-> s :|: s >= 0, s <= x + 1 + 1, z' = 1 + 1, 1 = x, x >= 0
   encArg(z') -{ 2 }-> s' :|: s' >= 0, s' <= x + 1 + 1, z' = 1 + 0, 0 = x, x >= 0
   encArg(z') -{ 2 }-> s'' :|: s'' >= 0, s'' <= x + 1 + 1, z' - 1 >= 0, 0 = x, x >= 0
   encArg(z') -{ 0 }-> xor(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> not(xor(encArg(x_13), encArg(x_22))) :|: x_13 >= 0, z' = 1 + (1 + x_13 + x_22), x_22 >= 0
   encArg(z') -{ 0 }-> not(or(encArg(x_1''), encArg(x_2'))) :|: x_1'' >= 0, x_2' >= 0, z' = 1 + (1 + x_1'' + x_2')
   encArg(z') -{ 0 }-> not(not(encArg(z' - 2))) :|: z' - 2 >= 0
   encArg(z') -{ 0 }-> not(implies(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, x_2'' >= 0, z' = 1 + (1 + x_11 + x_2'')
   encArg(z') -{ 0 }-> not(and(encArg(x_12), encArg(x_21))) :|: z' = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0
   encArg(z') -{ 0 }-> implies(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> and(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> 1 :|: z' = 1
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encode_and(z', z'') -{ 0 }-> and(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_and(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_false -{ 0 }-> 0 :|: 
   encode_implies(z', z'') -{ 0 }-> implies(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_implies(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_not(z') -{ 2 }-> s1 :|: s1 >= 0, s1 <= x + 1 + 1, z' = 1, 1 = x, x >= 0
   encode_not(z') -{ 2 }-> s2 :|: s2 >= 0, s2 <= x + 1 + 1, z' = 0, 0 = x, x >= 0
   encode_not(z') -{ 2 }-> s3 :|: s3 >= 0, s3 <= x + 1 + 1, z' >= 0, 0 = x, x >= 0
   encode_not(z') -{ 0 }-> not(xor(encArg(x_1188), encArg(x_2150))) :|: z' = 1 + x_1188 + x_2150, x_1188 >= 0, x_2150 >= 0
   encode_not(z') -{ 0 }-> not(or(encArg(x_1185), encArg(x_2147))) :|: x_1185 >= 0, z' = 1 + x_1185 + x_2147, x_2147 >= 0
   encode_not(z') -{ 0 }-> not(not(encArg(z' - 1))) :|: z' - 1 >= 0
   encode_not(z') -{ 0 }-> not(implies(encArg(x_1186), encArg(x_2148))) :|: z' = 1 + x_1186 + x_2148, x_1186 >= 0, x_2148 >= 0
   encode_not(z') -{ 0 }-> not(and(encArg(x_1187), encArg(x_2149))) :|: x_2149 >= 0, z' = 1 + x_1187 + x_2149, x_1187 >= 0
   encode_not(z') -{ 0 }-> 0 :|: z' >= 0
   encode_or(z', z'') -{ 0 }-> or(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_or(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_true -{ 0 }-> 1 :|: 
   encode_true -{ 0 }-> 0 :|: 
   encode_xor(z', z'') -{ 0 }-> xor(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_xor(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   implies(z', z'') -{ 4 }-> x :|: z' = 1, z'' = 1, 0 = 0, 1 = x, x >= 0
   implies(z', z'') -{ 4 }-> x :|: z'' = 0, z' = 1, 0 = 0, 0 = x, x >= 0
   implies(z', z'') -{ 3 }-> x :|: z'' >= 0, z' = 1, 0 = 0, 0 = x, x >= 0
   implies(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, 0 = 0, 0 = x', x' >= 0
   implies(z', z'') -{ 2 }-> x' :|: z' >= 0, z'' >= 0, 0 = 0, 0 = x', x' >= 0
   implies(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = 1, 0 = 0
   implies(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   implies(z', z'') -{ 2 }-> xor(xor(and(x'', z''), and(y'', z'')), 0) :|: z' = 1 + x'' + y'', z'' >= 0, y'' >= 0, x'' >= 0
   implies(z', z'') -{ 2 }-> xor(xor(and(x'', z''), and(y'', z'')), 1 + (1 + x'' + y'') + 1) :|: z' = 1 + x'' + y'', z'' >= 0, y'' >= 0, x'' >= 0
   implies(z', z'') -{ 3 }-> 0 :|: z' = 1, z'' = 1, v0 >= 0, v1 >= 0, 0 = v1, 1 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 1 + z' + 1 = v1
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 0 = v1
   implies(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = 1, 0 = z'
   implies(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' = 1, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' = 1, 0 = x, x >= 0
   implies(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 1 = v1, 0 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = x', x' >= 0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 1 + z' + 1 = v1
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   implies(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   implies(z', z'') -{ 2 }-> 0 :|: z'' >= 0, z' = 1, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 3 }-> 0 :|: z'' >= 0, z' = 1, 0 = x, x >= 0
   implies(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 1 = v1, 0 = v0
   implies(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' >= 0, 0 = x', x' >= 0
   implies(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z' = 1, z'' = 1, 0 = x1, x0 >= 0, x1 >= 0, 1 = x0
   implies(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z'' = 0, z' = 1, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 1 + z' + 1 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' >= 0, z' = 1, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 1 + z' + 1 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 1 + z' + 1 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 0 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 1 + z' + 1 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   not(z') -{ 1 }-> 0 :|: z' >= 0, v1 >= 0, 1 = v1
   not(z') -{ 2 }-> 0 :|: z' >= 0, 1 = z'
   not(z') -{ 1 }-> 1 + z' + x1 :|: z' >= 0, 1 = x1, x1 >= 0
   or(z', z'') -{ 4 }-> x :|: z' = 1, z'' = 1, 0 = 0, 1 = x, x >= 0
   or(z', z'') -{ 4 }-> x :|: z'' = 0, z' = 0, 0 = 0, 0 = x, x >= 0
   or(z', z'') -{ 4 }-> x' :|: z'' = 0, z' >= 0, z' = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, z' = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z' >= 0, z'' = z', 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 2 }-> x' :|: z' >= 0, z'' >= 0, 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = 1, 0 = 0
   or(z', z'') -{ 4 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   or(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   or(z', z'') -{ 2 }-> xor(xor(and(x', z''), and(y', z'')), 0) :|: z' = 1 + x' + y', x' >= 0, z'' >= 0, y' >= 0
   or(z', z'') -{ 2 }-> xor(xor(and(x', z''), and(y', z'')), 1 + (1 + x' + y') + z'') :|: z' = 1 + x' + y', x' >= 0, z'' >= 0, y' >= 0
   or(z', z'') -{ 3 }-> xor(xor(and(x', 0), and(y', 0)), 1 + x' + y') :|: z'' = 0, z' = 1 + x' + y', x' >= 0, y' >= 0
   or(z', z'') -{ 3 }-> xor(xor(and(x', 1 + x' + y'), and(y', 1 + x' + y')), 0) :|: z'' = 1 + x' + y', z' = 1 + x' + y', x' >= 0, y' >= 0
   or(z', z'') -{ 3 }-> 0 :|: z' = 1, z'' = 1, v0 >= 0, v1 >= 0, 0 = v1, 1 = v0
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 1 + z' + 1 = v1
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 0 = v1
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = 1, 0 = z'
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, 0 = v0
   or(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' >= 0, 0 = z'
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' = 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' = 0, 0 = x, x >= 0
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 0 = v1, 0 = v0
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   or(z', z'') -{ 4 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 1 + z' + z' = v1
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = x', x' >= 0
   or(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 1 + z' + z'' = v1, 0 = v0
   or(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' >= 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z' = 1, z'' = 1, 0 = x1, x0 >= 0, x1 >= 0, 1 = x0
   or(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z'' = 0, z' = 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 1 + z' + 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z' >= 0, z'' = z', 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 1 + z' + z'' = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 3 }-> 1 + x0 + z' :|: z'' = 0, z' >= 0, x0 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + z' :|: z'' = 0, z' >= 0, x0 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 1 + z' + 1 = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 0 = x1, x1 >= 0
   or(z', z'') -{ 3 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 1 + z' + z' = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   xor(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   xor(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' = z'
   xor(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   xor(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {implies}, {or}, {encArg}, {encode_not}, {encode_xor}, {encode_implies}, {encode_and}, {encode_or}
Previous analysis results are:
  not: runtime: O(1) [2], size: O(n^1) [2 + z']
  encode_false: runtime: O(1) [0], size: O(1) [0]
  xor: runtime: O(1) [1], size: O(n^1) [1 + z' + z'']
  encode_true: runtime: O(1) [0], size: O(1) [1]
  and: runtime: O(n^1) [1 + 3*z'], size: O(n^1) [2*z' + z'']

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

(69) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(70)
Obligation:
Complexity RNTS consisting of the following rules:

   and(z', z'') -{ 4 + 3*x + 3*y }-> s30 :|: s28 >= 0, s28 <= 2 * x + z'', s29 >= 0, s29 <= 2 * y + z'', s30 >= 0, s30 <= s28 + s29 + 1, z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
   and(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' = 1
   and(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' = z'
   and(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
   and(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encArg(z') -{ 2 }-> s :|: s >= 0, s <= x + 1 + 1, z' = 1 + 1, 1 = x, x >= 0
   encArg(z') -{ 2 }-> s' :|: s' >= 0, s' <= x + 1 + 1, z' = 1 + 0, 0 = x, x >= 0
   encArg(z') -{ 2 }-> s'' :|: s'' >= 0, s'' <= x + 1 + 1, z' - 1 >= 0, 0 = x, x >= 0
   encArg(z') -{ 0 }-> xor(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> not(xor(encArg(x_13), encArg(x_22))) :|: x_13 >= 0, z' = 1 + (1 + x_13 + x_22), x_22 >= 0
   encArg(z') -{ 0 }-> not(or(encArg(x_1''), encArg(x_2'))) :|: x_1'' >= 0, x_2' >= 0, z' = 1 + (1 + x_1'' + x_2')
   encArg(z') -{ 0 }-> not(not(encArg(z' - 2))) :|: z' - 2 >= 0
   encArg(z') -{ 0 }-> not(implies(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, x_2'' >= 0, z' = 1 + (1 + x_11 + x_2'')
   encArg(z') -{ 0 }-> not(and(encArg(x_12), encArg(x_21))) :|: z' = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0
   encArg(z') -{ 0 }-> implies(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> and(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> 1 :|: z' = 1
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encode_and(z', z'') -{ 0 }-> and(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_and(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_false -{ 0 }-> 0 :|: 
   encode_implies(z', z'') -{ 0 }-> implies(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_implies(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_not(z') -{ 2 }-> s1 :|: s1 >= 0, s1 <= x + 1 + 1, z' = 1, 1 = x, x >= 0
   encode_not(z') -{ 2 }-> s2 :|: s2 >= 0, s2 <= x + 1 + 1, z' = 0, 0 = x, x >= 0
   encode_not(z') -{ 2 }-> s3 :|: s3 >= 0, s3 <= x + 1 + 1, z' >= 0, 0 = x, x >= 0
   encode_not(z') -{ 0 }-> not(xor(encArg(x_1188), encArg(x_2150))) :|: z' = 1 + x_1188 + x_2150, x_1188 >= 0, x_2150 >= 0
   encode_not(z') -{ 0 }-> not(or(encArg(x_1185), encArg(x_2147))) :|: x_1185 >= 0, z' = 1 + x_1185 + x_2147, x_2147 >= 0
   encode_not(z') -{ 0 }-> not(not(encArg(z' - 1))) :|: z' - 1 >= 0
   encode_not(z') -{ 0 }-> not(implies(encArg(x_1186), encArg(x_2148))) :|: z' = 1 + x_1186 + x_2148, x_1186 >= 0, x_2148 >= 0
   encode_not(z') -{ 0 }-> not(and(encArg(x_1187), encArg(x_2149))) :|: x_2149 >= 0, z' = 1 + x_1187 + x_2149, x_1187 >= 0
   encode_not(z') -{ 0 }-> 0 :|: z' >= 0
   encode_or(z', z'') -{ 0 }-> or(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_or(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_true -{ 0 }-> 1 :|: 
   encode_true -{ 0 }-> 0 :|: 
   encode_xor(z', z'') -{ 0 }-> xor(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_xor(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   implies(z', z'') -{ 6 + 3*x'' + 3*y'' }-> s23 :|: s20 >= 0, s20 <= 2 * x'' + z'', s21 >= 0, s21 <= 2 * y'' + z'', s22 >= 0, s22 <= s20 + s21 + 1, s23 >= 0, s23 <= s22 + (1 + (1 + x'' + y'') + 1) + 1, z' = 1 + x'' + y'', z'' >= 0, y'' >= 0, x'' >= 0
   implies(z', z'') -{ 6 + 3*x'' + 3*y'' }-> s27 :|: s24 >= 0, s24 <= 2 * x'' + z'', s25 >= 0, s25 <= 2 * y'' + z'', s26 >= 0, s26 <= s24 + s25 + 1, s27 >= 0, s27 <= s26 + 0 + 1, z' = 1 + x'' + y'', z'' >= 0, y'' >= 0, x'' >= 0
   implies(z', z'') -{ 4 }-> x :|: z' = 1, z'' = 1, 0 = 0, 1 = x, x >= 0
   implies(z', z'') -{ 4 }-> x :|: z'' = 0, z' = 1, 0 = 0, 0 = x, x >= 0
   implies(z', z'') -{ 3 }-> x :|: z'' >= 0, z' = 1, 0 = 0, 0 = x, x >= 0
   implies(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, 0 = 0, 0 = x', x' >= 0
   implies(z', z'') -{ 2 }-> x' :|: z' >= 0, z'' >= 0, 0 = 0, 0 = x', x' >= 0
   implies(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = 1, 0 = 0
   implies(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   implies(z', z'') -{ 3 }-> 0 :|: z' = 1, z'' = 1, v0 >= 0, v1 >= 0, 0 = v1, 1 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 1 + z' + 1 = v1
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 0 = v1
   implies(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = 1, 0 = z'
   implies(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' = 1, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' = 1, 0 = x, x >= 0
   implies(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 1 = v1, 0 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = x', x' >= 0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 1 + z' + 1 = v1
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   implies(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   implies(z', z'') -{ 2 }-> 0 :|: z'' >= 0, z' = 1, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 3 }-> 0 :|: z'' >= 0, z' = 1, 0 = x, x >= 0
   implies(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 1 = v1, 0 = v0
   implies(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' >= 0, 0 = x', x' >= 0
   implies(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z' = 1, z'' = 1, 0 = x1, x0 >= 0, x1 >= 0, 1 = x0
   implies(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z'' = 0, z' = 1, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 1 + z' + 1 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' >= 0, z' = 1, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 1 + z' + 1 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 1 + z' + 1 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 0 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 1 + z' + 1 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   not(z') -{ 1 }-> 0 :|: z' >= 0, v1 >= 0, 1 = v1
   not(z') -{ 2 }-> 0 :|: z' >= 0, 1 = z'
   not(z') -{ 1 }-> 1 + z' + x1 :|: z' >= 0, 1 = x1, x1 >= 0
   or(z', z'') -{ 7 + 3*x' + 3*y' }-> s11 :|: s8 >= 0, s8 <= 2 * x' + (1 + x' + y'), s9 >= 0, s9 <= 2 * y' + (1 + x' + y'), s10 >= 0, s10 <= s8 + s9 + 1, s11 >= 0, s11 <= s10 + 0 + 1, z'' = 1 + x' + y', z' = 1 + x' + y', x' >= 0, y' >= 0
   or(z', z'') -{ 6 + 3*x' + 3*y' }-> s15 :|: s12 >= 0, s12 <= 2 * x' + z'', s13 >= 0, s13 <= 2 * y' + z'', s14 >= 0, s14 <= s12 + s13 + 1, s15 >= 0, s15 <= s14 + (1 + (1 + x' + y') + z'') + 1, z' = 1 + x' + y', x' >= 0, z'' >= 0, y' >= 0
   or(z', z'') -{ 6 + 3*x' + 3*y' }-> s19 :|: s16 >= 0, s16 <= 2 * x' + z'', s17 >= 0, s17 <= 2 * y' + z'', s18 >= 0, s18 <= s16 + s17 + 1, s19 >= 0, s19 <= s18 + 0 + 1, z' = 1 + x' + y', x' >= 0, z'' >= 0, y' >= 0
   or(z', z'') -{ 7 + 3*x' + 3*y' }-> s7 :|: s4 >= 0, s4 <= 2 * x' + 0, s5 >= 0, s5 <= 2 * y' + 0, s6 >= 0, s6 <= s4 + s5 + 1, s7 >= 0, s7 <= s6 + (1 + x' + y') + 1, z'' = 0, z' = 1 + x' + y', x' >= 0, y' >= 0
   or(z', z'') -{ 4 }-> x :|: z' = 1, z'' = 1, 0 = 0, 1 = x, x >= 0
   or(z', z'') -{ 4 }-> x :|: z'' = 0, z' = 0, 0 = 0, 0 = x, x >= 0
   or(z', z'') -{ 4 }-> x' :|: z'' = 0, z' >= 0, z' = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, z' = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z' >= 0, z'' = z', 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 2 }-> x' :|: z' >= 0, z'' >= 0, 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = 1, 0 = 0
   or(z', z'') -{ 4 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   or(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   or(z', z'') -{ 3 }-> 0 :|: z' = 1, z'' = 1, v0 >= 0, v1 >= 0, 0 = v1, 1 = v0
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 1 + z' + 1 = v1
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 0 = v1
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = 1, 0 = z'
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, 0 = v0
   or(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' >= 0, 0 = z'
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' = 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' = 0, 0 = x, x >= 0
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 0 = v1, 0 = v0
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   or(z', z'') -{ 4 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 1 + z' + z' = v1
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = x', x' >= 0
   or(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 1 + z' + z'' = v1, 0 = v0
   or(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' >= 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z' = 1, z'' = 1, 0 = x1, x0 >= 0, x1 >= 0, 1 = x0
   or(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z'' = 0, z' = 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 1 + z' + 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z' >= 0, z'' = z', 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 1 + z' + z'' = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 3 }-> 1 + x0 + z' :|: z'' = 0, z' >= 0, x0 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + z' :|: z'' = 0, z' >= 0, x0 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 1 + z' + 1 = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 0 = x1, x1 >= 0
   or(z', z'') -{ 3 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 1 + z' + z' = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   xor(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   xor(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' = z'
   xor(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   xor(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {implies}, {or}, {encArg}, {encode_not}, {encode_xor}, {encode_implies}, {encode_and}, {encode_or}
Previous analysis results are:
  not: runtime: O(1) [2], size: O(n^1) [2 + z']
  encode_false: runtime: O(1) [0], size: O(1) [0]
  xor: runtime: O(1) [1], size: O(n^1) [1 + z' + z'']
  encode_true: runtime: O(1) [0], size: O(1) [1]
  and: runtime: O(n^1) [1 + 3*z'], size: O(n^1) [2*z' + z'']

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

(71) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: implies
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 3 + 3*z' + 2*z''

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

(72)
Obligation:
Complexity RNTS consisting of the following rules:

   and(z', z'') -{ 4 + 3*x + 3*y }-> s30 :|: s28 >= 0, s28 <= 2 * x + z'', s29 >= 0, s29 <= 2 * y + z'', s30 >= 0, s30 <= s28 + s29 + 1, z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
   and(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' = 1
   and(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' = z'
   and(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
   and(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encArg(z') -{ 2 }-> s :|: s >= 0, s <= x + 1 + 1, z' = 1 + 1, 1 = x, x >= 0
   encArg(z') -{ 2 }-> s' :|: s' >= 0, s' <= x + 1 + 1, z' = 1 + 0, 0 = x, x >= 0
   encArg(z') -{ 2 }-> s'' :|: s'' >= 0, s'' <= x + 1 + 1, z' - 1 >= 0, 0 = x, x >= 0
   encArg(z') -{ 0 }-> xor(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> not(xor(encArg(x_13), encArg(x_22))) :|: x_13 >= 0, z' = 1 + (1 + x_13 + x_22), x_22 >= 0
   encArg(z') -{ 0 }-> not(or(encArg(x_1''), encArg(x_2'))) :|: x_1'' >= 0, x_2' >= 0, z' = 1 + (1 + x_1'' + x_2')
   encArg(z') -{ 0 }-> not(not(encArg(z' - 2))) :|: z' - 2 >= 0
   encArg(z') -{ 0 }-> not(implies(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, x_2'' >= 0, z' = 1 + (1 + x_11 + x_2'')
   encArg(z') -{ 0 }-> not(and(encArg(x_12), encArg(x_21))) :|: z' = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0
   encArg(z') -{ 0 }-> implies(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> and(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> 1 :|: z' = 1
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encode_and(z', z'') -{ 0 }-> and(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_and(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_false -{ 0 }-> 0 :|: 
   encode_implies(z', z'') -{ 0 }-> implies(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_implies(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_not(z') -{ 2 }-> s1 :|: s1 >= 0, s1 <= x + 1 + 1, z' = 1, 1 = x, x >= 0
   encode_not(z') -{ 2 }-> s2 :|: s2 >= 0, s2 <= x + 1 + 1, z' = 0, 0 = x, x >= 0
   encode_not(z') -{ 2 }-> s3 :|: s3 >= 0, s3 <= x + 1 + 1, z' >= 0, 0 = x, x >= 0
   encode_not(z') -{ 0 }-> not(xor(encArg(x_1188), encArg(x_2150))) :|: z' = 1 + x_1188 + x_2150, x_1188 >= 0, x_2150 >= 0
   encode_not(z') -{ 0 }-> not(or(encArg(x_1185), encArg(x_2147))) :|: x_1185 >= 0, z' = 1 + x_1185 + x_2147, x_2147 >= 0
   encode_not(z') -{ 0 }-> not(not(encArg(z' - 1))) :|: z' - 1 >= 0
   encode_not(z') -{ 0 }-> not(implies(encArg(x_1186), encArg(x_2148))) :|: z' = 1 + x_1186 + x_2148, x_1186 >= 0, x_2148 >= 0
   encode_not(z') -{ 0 }-> not(and(encArg(x_1187), encArg(x_2149))) :|: x_2149 >= 0, z' = 1 + x_1187 + x_2149, x_1187 >= 0
   encode_not(z') -{ 0 }-> 0 :|: z' >= 0
   encode_or(z', z'') -{ 0 }-> or(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_or(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_true -{ 0 }-> 1 :|: 
   encode_true -{ 0 }-> 0 :|: 
   encode_xor(z', z'') -{ 0 }-> xor(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_xor(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   implies(z', z'') -{ 6 + 3*x'' + 3*y'' }-> s23 :|: s20 >= 0, s20 <= 2 * x'' + z'', s21 >= 0, s21 <= 2 * y'' + z'', s22 >= 0, s22 <= s20 + s21 + 1, s23 >= 0, s23 <= s22 + (1 + (1 + x'' + y'') + 1) + 1, z' = 1 + x'' + y'', z'' >= 0, y'' >= 0, x'' >= 0
   implies(z', z'') -{ 6 + 3*x'' + 3*y'' }-> s27 :|: s24 >= 0, s24 <= 2 * x'' + z'', s25 >= 0, s25 <= 2 * y'' + z'', s26 >= 0, s26 <= s24 + s25 + 1, s27 >= 0, s27 <= s26 + 0 + 1, z' = 1 + x'' + y'', z'' >= 0, y'' >= 0, x'' >= 0
   implies(z', z'') -{ 4 }-> x :|: z' = 1, z'' = 1, 0 = 0, 1 = x, x >= 0
   implies(z', z'') -{ 4 }-> x :|: z'' = 0, z' = 1, 0 = 0, 0 = x, x >= 0
   implies(z', z'') -{ 3 }-> x :|: z'' >= 0, z' = 1, 0 = 0, 0 = x, x >= 0
   implies(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, 0 = 0, 0 = x', x' >= 0
   implies(z', z'') -{ 2 }-> x' :|: z' >= 0, z'' >= 0, 0 = 0, 0 = x', x' >= 0
   implies(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = 1, 0 = 0
   implies(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   implies(z', z'') -{ 3 }-> 0 :|: z' = 1, z'' = 1, v0 >= 0, v1 >= 0, 0 = v1, 1 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 1 + z' + 1 = v1
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 0 = v1
   implies(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = 1, 0 = z'
   implies(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' = 1, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' = 1, 0 = x, x >= 0
   implies(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 1 = v1, 0 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = x', x' >= 0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 1 + z' + 1 = v1
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   implies(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   implies(z', z'') -{ 2 }-> 0 :|: z'' >= 0, z' = 1, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 3 }-> 0 :|: z'' >= 0, z' = 1, 0 = x, x >= 0
   implies(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 1 = v1, 0 = v0
   implies(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' >= 0, 0 = x', x' >= 0
   implies(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z' = 1, z'' = 1, 0 = x1, x0 >= 0, x1 >= 0, 1 = x0
   implies(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z'' = 0, z' = 1, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 1 + z' + 1 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' >= 0, z' = 1, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 1 + z' + 1 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 1 + z' + 1 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 0 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 1 + z' + 1 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   not(z') -{ 1 }-> 0 :|: z' >= 0, v1 >= 0, 1 = v1
   not(z') -{ 2 }-> 0 :|: z' >= 0, 1 = z'
   not(z') -{ 1 }-> 1 + z' + x1 :|: z' >= 0, 1 = x1, x1 >= 0
   or(z', z'') -{ 7 + 3*x' + 3*y' }-> s11 :|: s8 >= 0, s8 <= 2 * x' + (1 + x' + y'), s9 >= 0, s9 <= 2 * y' + (1 + x' + y'), s10 >= 0, s10 <= s8 + s9 + 1, s11 >= 0, s11 <= s10 + 0 + 1, z'' = 1 + x' + y', z' = 1 + x' + y', x' >= 0, y' >= 0
   or(z', z'') -{ 6 + 3*x' + 3*y' }-> s15 :|: s12 >= 0, s12 <= 2 * x' + z'', s13 >= 0, s13 <= 2 * y' + z'', s14 >= 0, s14 <= s12 + s13 + 1, s15 >= 0, s15 <= s14 + (1 + (1 + x' + y') + z'') + 1, z' = 1 + x' + y', x' >= 0, z'' >= 0, y' >= 0
   or(z', z'') -{ 6 + 3*x' + 3*y' }-> s19 :|: s16 >= 0, s16 <= 2 * x' + z'', s17 >= 0, s17 <= 2 * y' + z'', s18 >= 0, s18 <= s16 + s17 + 1, s19 >= 0, s19 <= s18 + 0 + 1, z' = 1 + x' + y', x' >= 0, z'' >= 0, y' >= 0
   or(z', z'') -{ 7 + 3*x' + 3*y' }-> s7 :|: s4 >= 0, s4 <= 2 * x' + 0, s5 >= 0, s5 <= 2 * y' + 0, s6 >= 0, s6 <= s4 + s5 + 1, s7 >= 0, s7 <= s6 + (1 + x' + y') + 1, z'' = 0, z' = 1 + x' + y', x' >= 0, y' >= 0
   or(z', z'') -{ 4 }-> x :|: z' = 1, z'' = 1, 0 = 0, 1 = x, x >= 0
   or(z', z'') -{ 4 }-> x :|: z'' = 0, z' = 0, 0 = 0, 0 = x, x >= 0
   or(z', z'') -{ 4 }-> x' :|: z'' = 0, z' >= 0, z' = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, z' = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z' >= 0, z'' = z', 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 2 }-> x' :|: z' >= 0, z'' >= 0, 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = 1, 0 = 0
   or(z', z'') -{ 4 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   or(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   or(z', z'') -{ 3 }-> 0 :|: z' = 1, z'' = 1, v0 >= 0, v1 >= 0, 0 = v1, 1 = v0
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 1 + z' + 1 = v1
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 0 = v1
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = 1, 0 = z'
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, 0 = v0
   or(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' >= 0, 0 = z'
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' = 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' = 0, 0 = x, x >= 0
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 0 = v1, 0 = v0
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   or(z', z'') -{ 4 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 1 + z' + z' = v1
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = x', x' >= 0
   or(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 1 + z' + z'' = v1, 0 = v0
   or(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' >= 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z' = 1, z'' = 1, 0 = x1, x0 >= 0, x1 >= 0, 1 = x0
   or(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z'' = 0, z' = 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 1 + z' + 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z' >= 0, z'' = z', 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 1 + z' + z'' = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 3 }-> 1 + x0 + z' :|: z'' = 0, z' >= 0, x0 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + z' :|: z'' = 0, z' >= 0, x0 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 1 + z' + 1 = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 0 = x1, x1 >= 0
   or(z', z'') -{ 3 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 1 + z' + z' = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   xor(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   xor(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' = z'
   xor(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   xor(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {implies}, {or}, {encArg}, {encode_not}, {encode_xor}, {encode_implies}, {encode_and}, {encode_or}
Previous analysis results are:
  not: runtime: O(1) [2], size: O(n^1) [2 + z']
  encode_false: runtime: O(1) [0], size: O(1) [0]
  xor: runtime: O(1) [1], size: O(n^1) [1 + z' + z'']
  encode_true: runtime: O(1) [0], size: O(1) [1]
  and: runtime: O(n^1) [1 + 3*z'], size: O(n^1) [2*z' + z'']
  implies: runtime: ?, size: O(n^1) [3 + 3*z' + 2*z'']

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

(73) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: implies
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 4 + 3*z'

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

(74)
Obligation:
Complexity RNTS consisting of the following rules:

   and(z', z'') -{ 4 + 3*x + 3*y }-> s30 :|: s28 >= 0, s28 <= 2 * x + z'', s29 >= 0, s29 <= 2 * y + z'', s30 >= 0, s30 <= s28 + s29 + 1, z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
   and(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' = 1
   and(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' = z'
   and(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
   and(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encArg(z') -{ 2 }-> s :|: s >= 0, s <= x + 1 + 1, z' = 1 + 1, 1 = x, x >= 0
   encArg(z') -{ 2 }-> s' :|: s' >= 0, s' <= x + 1 + 1, z' = 1 + 0, 0 = x, x >= 0
   encArg(z') -{ 2 }-> s'' :|: s'' >= 0, s'' <= x + 1 + 1, z' - 1 >= 0, 0 = x, x >= 0
   encArg(z') -{ 0 }-> xor(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> not(xor(encArg(x_13), encArg(x_22))) :|: x_13 >= 0, z' = 1 + (1 + x_13 + x_22), x_22 >= 0
   encArg(z') -{ 0 }-> not(or(encArg(x_1''), encArg(x_2'))) :|: x_1'' >= 0, x_2' >= 0, z' = 1 + (1 + x_1'' + x_2')
   encArg(z') -{ 0 }-> not(not(encArg(z' - 2))) :|: z' - 2 >= 0
   encArg(z') -{ 0 }-> not(implies(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, x_2'' >= 0, z' = 1 + (1 + x_11 + x_2'')
   encArg(z') -{ 0 }-> not(and(encArg(x_12), encArg(x_21))) :|: z' = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0
   encArg(z') -{ 0 }-> implies(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> and(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> 1 :|: z' = 1
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encode_and(z', z'') -{ 0 }-> and(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_and(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_false -{ 0 }-> 0 :|: 
   encode_implies(z', z'') -{ 0 }-> implies(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_implies(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_not(z') -{ 2 }-> s1 :|: s1 >= 0, s1 <= x + 1 + 1, z' = 1, 1 = x, x >= 0
   encode_not(z') -{ 2 }-> s2 :|: s2 >= 0, s2 <= x + 1 + 1, z' = 0, 0 = x, x >= 0
   encode_not(z') -{ 2 }-> s3 :|: s3 >= 0, s3 <= x + 1 + 1, z' >= 0, 0 = x, x >= 0
   encode_not(z') -{ 0 }-> not(xor(encArg(x_1188), encArg(x_2150))) :|: z' = 1 + x_1188 + x_2150, x_1188 >= 0, x_2150 >= 0
   encode_not(z') -{ 0 }-> not(or(encArg(x_1185), encArg(x_2147))) :|: x_1185 >= 0, z' = 1 + x_1185 + x_2147, x_2147 >= 0
   encode_not(z') -{ 0 }-> not(not(encArg(z' - 1))) :|: z' - 1 >= 0
   encode_not(z') -{ 0 }-> not(implies(encArg(x_1186), encArg(x_2148))) :|: z' = 1 + x_1186 + x_2148, x_1186 >= 0, x_2148 >= 0
   encode_not(z') -{ 0 }-> not(and(encArg(x_1187), encArg(x_2149))) :|: x_2149 >= 0, z' = 1 + x_1187 + x_2149, x_1187 >= 0
   encode_not(z') -{ 0 }-> 0 :|: z' >= 0
   encode_or(z', z'') -{ 0 }-> or(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_or(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_true -{ 0 }-> 1 :|: 
   encode_true -{ 0 }-> 0 :|: 
   encode_xor(z', z'') -{ 0 }-> xor(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_xor(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   implies(z', z'') -{ 6 + 3*x'' + 3*y'' }-> s23 :|: s20 >= 0, s20 <= 2 * x'' + z'', s21 >= 0, s21 <= 2 * y'' + z'', s22 >= 0, s22 <= s20 + s21 + 1, s23 >= 0, s23 <= s22 + (1 + (1 + x'' + y'') + 1) + 1, z' = 1 + x'' + y'', z'' >= 0, y'' >= 0, x'' >= 0
   implies(z', z'') -{ 6 + 3*x'' + 3*y'' }-> s27 :|: s24 >= 0, s24 <= 2 * x'' + z'', s25 >= 0, s25 <= 2 * y'' + z'', s26 >= 0, s26 <= s24 + s25 + 1, s27 >= 0, s27 <= s26 + 0 + 1, z' = 1 + x'' + y'', z'' >= 0, y'' >= 0, x'' >= 0
   implies(z', z'') -{ 4 }-> x :|: z' = 1, z'' = 1, 0 = 0, 1 = x, x >= 0
   implies(z', z'') -{ 4 }-> x :|: z'' = 0, z' = 1, 0 = 0, 0 = x, x >= 0
   implies(z', z'') -{ 3 }-> x :|: z'' >= 0, z' = 1, 0 = 0, 0 = x, x >= 0
   implies(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, 0 = 0, 0 = x', x' >= 0
   implies(z', z'') -{ 2 }-> x' :|: z' >= 0, z'' >= 0, 0 = 0, 0 = x', x' >= 0
   implies(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = 1, 0 = 0
   implies(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   implies(z', z'') -{ 3 }-> 0 :|: z' = 1, z'' = 1, v0 >= 0, v1 >= 0, 0 = v1, 1 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 1 + z' + 1 = v1
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 0 = v1
   implies(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = 1, 0 = z'
   implies(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' = 1, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' = 1, 0 = x, x >= 0
   implies(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 1 = v1, 0 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = x', x' >= 0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 1 + z' + 1 = v1
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   implies(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   implies(z', z'') -{ 2 }-> 0 :|: z'' >= 0, z' = 1, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 3 }-> 0 :|: z'' >= 0, z' = 1, 0 = x, x >= 0
   implies(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 1 = v1, 0 = v0
   implies(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' >= 0, 0 = x', x' >= 0
   implies(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z' = 1, z'' = 1, 0 = x1, x0 >= 0, x1 >= 0, 1 = x0
   implies(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z'' = 0, z' = 1, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 1 + z' + 1 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' >= 0, z' = 1, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 1 + z' + 1 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 1 + z' + 1 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 0 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 1 + z' + 1 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   not(z') -{ 1 }-> 0 :|: z' >= 0, v1 >= 0, 1 = v1
   not(z') -{ 2 }-> 0 :|: z' >= 0, 1 = z'
   not(z') -{ 1 }-> 1 + z' + x1 :|: z' >= 0, 1 = x1, x1 >= 0
   or(z', z'') -{ 7 + 3*x' + 3*y' }-> s11 :|: s8 >= 0, s8 <= 2 * x' + (1 + x' + y'), s9 >= 0, s9 <= 2 * y' + (1 + x' + y'), s10 >= 0, s10 <= s8 + s9 + 1, s11 >= 0, s11 <= s10 + 0 + 1, z'' = 1 + x' + y', z' = 1 + x' + y', x' >= 0, y' >= 0
   or(z', z'') -{ 6 + 3*x' + 3*y' }-> s15 :|: s12 >= 0, s12 <= 2 * x' + z'', s13 >= 0, s13 <= 2 * y' + z'', s14 >= 0, s14 <= s12 + s13 + 1, s15 >= 0, s15 <= s14 + (1 + (1 + x' + y') + z'') + 1, z' = 1 + x' + y', x' >= 0, z'' >= 0, y' >= 0
   or(z', z'') -{ 6 + 3*x' + 3*y' }-> s19 :|: s16 >= 0, s16 <= 2 * x' + z'', s17 >= 0, s17 <= 2 * y' + z'', s18 >= 0, s18 <= s16 + s17 + 1, s19 >= 0, s19 <= s18 + 0 + 1, z' = 1 + x' + y', x' >= 0, z'' >= 0, y' >= 0
   or(z', z'') -{ 7 + 3*x' + 3*y' }-> s7 :|: s4 >= 0, s4 <= 2 * x' + 0, s5 >= 0, s5 <= 2 * y' + 0, s6 >= 0, s6 <= s4 + s5 + 1, s7 >= 0, s7 <= s6 + (1 + x' + y') + 1, z'' = 0, z' = 1 + x' + y', x' >= 0, y' >= 0
   or(z', z'') -{ 4 }-> x :|: z' = 1, z'' = 1, 0 = 0, 1 = x, x >= 0
   or(z', z'') -{ 4 }-> x :|: z'' = 0, z' = 0, 0 = 0, 0 = x, x >= 0
   or(z', z'') -{ 4 }-> x' :|: z'' = 0, z' >= 0, z' = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, z' = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z' >= 0, z'' = z', 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 2 }-> x' :|: z' >= 0, z'' >= 0, 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = 1, 0 = 0
   or(z', z'') -{ 4 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   or(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   or(z', z'') -{ 3 }-> 0 :|: z' = 1, z'' = 1, v0 >= 0, v1 >= 0, 0 = v1, 1 = v0
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 1 + z' + 1 = v1
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 0 = v1
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = 1, 0 = z'
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, 0 = v0
   or(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' >= 0, 0 = z'
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' = 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' = 0, 0 = x, x >= 0
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 0 = v1, 0 = v0
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   or(z', z'') -{ 4 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 1 + z' + z' = v1
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = x', x' >= 0
   or(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 1 + z' + z'' = v1, 0 = v0
   or(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' >= 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z' = 1, z'' = 1, 0 = x1, x0 >= 0, x1 >= 0, 1 = x0
   or(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z'' = 0, z' = 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 1 + z' + 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z' >= 0, z'' = z', 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 1 + z' + z'' = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 3 }-> 1 + x0 + z' :|: z'' = 0, z' >= 0, x0 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + z' :|: z'' = 0, z' >= 0, x0 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 1 + z' + 1 = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 0 = x1, x1 >= 0
   or(z', z'') -{ 3 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 1 + z' + z' = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   xor(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   xor(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' = z'
   xor(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   xor(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {or}, {encArg}, {encode_not}, {encode_xor}, {encode_implies}, {encode_and}, {encode_or}
Previous analysis results are:
  not: runtime: O(1) [2], size: O(n^1) [2 + z']
  encode_false: runtime: O(1) [0], size: O(1) [0]
  xor: runtime: O(1) [1], size: O(n^1) [1 + z' + z'']
  encode_true: runtime: O(1) [0], size: O(1) [1]
  and: runtime: O(n^1) [1 + 3*z'], size: O(n^1) [2*z' + z'']
  implies: runtime: O(n^1) [4 + 3*z'], size: O(n^1) [3 + 3*z' + 2*z'']

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

(75) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(76)
Obligation:
Complexity RNTS consisting of the following rules:

   and(z', z'') -{ 4 + 3*x + 3*y }-> s30 :|: s28 >= 0, s28 <= 2 * x + z'', s29 >= 0, s29 <= 2 * y + z'', s30 >= 0, s30 <= s28 + s29 + 1, z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
   and(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' = 1
   and(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' = z'
   and(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
   and(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encArg(z') -{ 2 }-> s :|: s >= 0, s <= x + 1 + 1, z' = 1 + 1, 1 = x, x >= 0
   encArg(z') -{ 2 }-> s' :|: s' >= 0, s' <= x + 1 + 1, z' = 1 + 0, 0 = x, x >= 0
   encArg(z') -{ 2 }-> s'' :|: s'' >= 0, s'' <= x + 1 + 1, z' - 1 >= 0, 0 = x, x >= 0
   encArg(z') -{ 0 }-> xor(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> not(xor(encArg(x_13), encArg(x_22))) :|: x_13 >= 0, z' = 1 + (1 + x_13 + x_22), x_22 >= 0
   encArg(z') -{ 0 }-> not(or(encArg(x_1''), encArg(x_2'))) :|: x_1'' >= 0, x_2' >= 0, z' = 1 + (1 + x_1'' + x_2')
   encArg(z') -{ 0 }-> not(not(encArg(z' - 2))) :|: z' - 2 >= 0
   encArg(z') -{ 0 }-> not(implies(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, x_2'' >= 0, z' = 1 + (1 + x_11 + x_2'')
   encArg(z') -{ 0 }-> not(and(encArg(x_12), encArg(x_21))) :|: z' = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0
   encArg(z') -{ 0 }-> implies(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> and(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> 1 :|: z' = 1
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encode_and(z', z'') -{ 0 }-> and(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_and(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_false -{ 0 }-> 0 :|: 
   encode_implies(z', z'') -{ 0 }-> implies(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_implies(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_not(z') -{ 2 }-> s1 :|: s1 >= 0, s1 <= x + 1 + 1, z' = 1, 1 = x, x >= 0
   encode_not(z') -{ 2 }-> s2 :|: s2 >= 0, s2 <= x + 1 + 1, z' = 0, 0 = x, x >= 0
   encode_not(z') -{ 2 }-> s3 :|: s3 >= 0, s3 <= x + 1 + 1, z' >= 0, 0 = x, x >= 0
   encode_not(z') -{ 0 }-> not(xor(encArg(x_1188), encArg(x_2150))) :|: z' = 1 + x_1188 + x_2150, x_1188 >= 0, x_2150 >= 0
   encode_not(z') -{ 0 }-> not(or(encArg(x_1185), encArg(x_2147))) :|: x_1185 >= 0, z' = 1 + x_1185 + x_2147, x_2147 >= 0
   encode_not(z') -{ 0 }-> not(not(encArg(z' - 1))) :|: z' - 1 >= 0
   encode_not(z') -{ 0 }-> not(implies(encArg(x_1186), encArg(x_2148))) :|: z' = 1 + x_1186 + x_2148, x_1186 >= 0, x_2148 >= 0
   encode_not(z') -{ 0 }-> not(and(encArg(x_1187), encArg(x_2149))) :|: x_2149 >= 0, z' = 1 + x_1187 + x_2149, x_1187 >= 0
   encode_not(z') -{ 0 }-> 0 :|: z' >= 0
   encode_or(z', z'') -{ 0 }-> or(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_or(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_true -{ 0 }-> 1 :|: 
   encode_true -{ 0 }-> 0 :|: 
   encode_xor(z', z'') -{ 0 }-> xor(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_xor(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   implies(z', z'') -{ 6 + 3*x'' + 3*y'' }-> s23 :|: s20 >= 0, s20 <= 2 * x'' + z'', s21 >= 0, s21 <= 2 * y'' + z'', s22 >= 0, s22 <= s20 + s21 + 1, s23 >= 0, s23 <= s22 + (1 + (1 + x'' + y'') + 1) + 1, z' = 1 + x'' + y'', z'' >= 0, y'' >= 0, x'' >= 0
   implies(z', z'') -{ 6 + 3*x'' + 3*y'' }-> s27 :|: s24 >= 0, s24 <= 2 * x'' + z'', s25 >= 0, s25 <= 2 * y'' + z'', s26 >= 0, s26 <= s24 + s25 + 1, s27 >= 0, s27 <= s26 + 0 + 1, z' = 1 + x'' + y'', z'' >= 0, y'' >= 0, x'' >= 0
   implies(z', z'') -{ 4 }-> x :|: z' = 1, z'' = 1, 0 = 0, 1 = x, x >= 0
   implies(z', z'') -{ 4 }-> x :|: z'' = 0, z' = 1, 0 = 0, 0 = x, x >= 0
   implies(z', z'') -{ 3 }-> x :|: z'' >= 0, z' = 1, 0 = 0, 0 = x, x >= 0
   implies(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, 0 = 0, 0 = x', x' >= 0
   implies(z', z'') -{ 2 }-> x' :|: z' >= 0, z'' >= 0, 0 = 0, 0 = x', x' >= 0
   implies(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = 1, 0 = 0
   implies(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   implies(z', z'') -{ 3 }-> 0 :|: z' = 1, z'' = 1, v0 >= 0, v1 >= 0, 0 = v1, 1 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 1 + z' + 1 = v1
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 0 = v1
   implies(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = 1, 0 = z'
   implies(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' = 1, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' = 1, 0 = x, x >= 0
   implies(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 1 = v1, 0 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = x', x' >= 0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 1 + z' + 1 = v1
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   implies(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   implies(z', z'') -{ 2 }-> 0 :|: z'' >= 0, z' = 1, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 3 }-> 0 :|: z'' >= 0, z' = 1, 0 = x, x >= 0
   implies(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 1 = v1, 0 = v0
   implies(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' >= 0, 0 = x', x' >= 0
   implies(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z' = 1, z'' = 1, 0 = x1, x0 >= 0, x1 >= 0, 1 = x0
   implies(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z'' = 0, z' = 1, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 1 + z' + 1 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' >= 0, z' = 1, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 1 + z' + 1 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 1 + z' + 1 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 0 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 1 + z' + 1 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   not(z') -{ 1 }-> 0 :|: z' >= 0, v1 >= 0, 1 = v1
   not(z') -{ 2 }-> 0 :|: z' >= 0, 1 = z'
   not(z') -{ 1 }-> 1 + z' + x1 :|: z' >= 0, 1 = x1, x1 >= 0
   or(z', z'') -{ 7 + 3*x' + 3*y' }-> s11 :|: s8 >= 0, s8 <= 2 * x' + (1 + x' + y'), s9 >= 0, s9 <= 2 * y' + (1 + x' + y'), s10 >= 0, s10 <= s8 + s9 + 1, s11 >= 0, s11 <= s10 + 0 + 1, z'' = 1 + x' + y', z' = 1 + x' + y', x' >= 0, y' >= 0
   or(z', z'') -{ 6 + 3*x' + 3*y' }-> s15 :|: s12 >= 0, s12 <= 2 * x' + z'', s13 >= 0, s13 <= 2 * y' + z'', s14 >= 0, s14 <= s12 + s13 + 1, s15 >= 0, s15 <= s14 + (1 + (1 + x' + y') + z'') + 1, z' = 1 + x' + y', x' >= 0, z'' >= 0, y' >= 0
   or(z', z'') -{ 6 + 3*x' + 3*y' }-> s19 :|: s16 >= 0, s16 <= 2 * x' + z'', s17 >= 0, s17 <= 2 * y' + z'', s18 >= 0, s18 <= s16 + s17 + 1, s19 >= 0, s19 <= s18 + 0 + 1, z' = 1 + x' + y', x' >= 0, z'' >= 0, y' >= 0
   or(z', z'') -{ 7 + 3*x' + 3*y' }-> s7 :|: s4 >= 0, s4 <= 2 * x' + 0, s5 >= 0, s5 <= 2 * y' + 0, s6 >= 0, s6 <= s4 + s5 + 1, s7 >= 0, s7 <= s6 + (1 + x' + y') + 1, z'' = 0, z' = 1 + x' + y', x' >= 0, y' >= 0
   or(z', z'') -{ 4 }-> x :|: z' = 1, z'' = 1, 0 = 0, 1 = x, x >= 0
   or(z', z'') -{ 4 }-> x :|: z'' = 0, z' = 0, 0 = 0, 0 = x, x >= 0
   or(z', z'') -{ 4 }-> x' :|: z'' = 0, z' >= 0, z' = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, z' = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z' >= 0, z'' = z', 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 2 }-> x' :|: z' >= 0, z'' >= 0, 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = 1, 0 = 0
   or(z', z'') -{ 4 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   or(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   or(z', z'') -{ 3 }-> 0 :|: z' = 1, z'' = 1, v0 >= 0, v1 >= 0, 0 = v1, 1 = v0
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 1 + z' + 1 = v1
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 0 = v1
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = 1, 0 = z'
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, 0 = v0
   or(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' >= 0, 0 = z'
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' = 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' = 0, 0 = x, x >= 0
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 0 = v1, 0 = v0
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   or(z', z'') -{ 4 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 1 + z' + z' = v1
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = x', x' >= 0
   or(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 1 + z' + z'' = v1, 0 = v0
   or(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' >= 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z' = 1, z'' = 1, 0 = x1, x0 >= 0, x1 >= 0, 1 = x0
   or(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z'' = 0, z' = 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 1 + z' + 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z' >= 0, z'' = z', 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 1 + z' + z'' = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 3 }-> 1 + x0 + z' :|: z'' = 0, z' >= 0, x0 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + z' :|: z'' = 0, z' >= 0, x0 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 1 + z' + 1 = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 0 = x1, x1 >= 0
   or(z', z'') -{ 3 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 1 + z' + z' = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   xor(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   xor(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' = z'
   xor(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   xor(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {or}, {encArg}, {encode_not}, {encode_xor}, {encode_implies}, {encode_and}, {encode_or}
Previous analysis results are:
  not: runtime: O(1) [2], size: O(n^1) [2 + z']
  encode_false: runtime: O(1) [0], size: O(1) [0]
  xor: runtime: O(1) [1], size: O(n^1) [1 + z' + z'']
  encode_true: runtime: O(1) [0], size: O(1) [1]
  and: runtime: O(n^1) [1 + 3*z'], size: O(n^1) [2*z' + z'']
  implies: runtime: O(n^1) [4 + 3*z'], size: O(n^1) [3 + 3*z' + 2*z'']

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

(77) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: or
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 2 + 3*z' + 3*z''

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

(78)
Obligation:
Complexity RNTS consisting of the following rules:

   and(z', z'') -{ 4 + 3*x + 3*y }-> s30 :|: s28 >= 0, s28 <= 2 * x + z'', s29 >= 0, s29 <= 2 * y + z'', s30 >= 0, s30 <= s28 + s29 + 1, z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
   and(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' = 1
   and(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' = z'
   and(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
   and(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encArg(z') -{ 2 }-> s :|: s >= 0, s <= x + 1 + 1, z' = 1 + 1, 1 = x, x >= 0
   encArg(z') -{ 2 }-> s' :|: s' >= 0, s' <= x + 1 + 1, z' = 1 + 0, 0 = x, x >= 0
   encArg(z') -{ 2 }-> s'' :|: s'' >= 0, s'' <= x + 1 + 1, z' - 1 >= 0, 0 = x, x >= 0
   encArg(z') -{ 0 }-> xor(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> not(xor(encArg(x_13), encArg(x_22))) :|: x_13 >= 0, z' = 1 + (1 + x_13 + x_22), x_22 >= 0
   encArg(z') -{ 0 }-> not(or(encArg(x_1''), encArg(x_2'))) :|: x_1'' >= 0, x_2' >= 0, z' = 1 + (1 + x_1'' + x_2')
   encArg(z') -{ 0 }-> not(not(encArg(z' - 2))) :|: z' - 2 >= 0
   encArg(z') -{ 0 }-> not(implies(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, x_2'' >= 0, z' = 1 + (1 + x_11 + x_2'')
   encArg(z') -{ 0 }-> not(and(encArg(x_12), encArg(x_21))) :|: z' = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0
   encArg(z') -{ 0 }-> implies(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> and(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> 1 :|: z' = 1
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encode_and(z', z'') -{ 0 }-> and(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_and(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_false -{ 0 }-> 0 :|: 
   encode_implies(z', z'') -{ 0 }-> implies(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_implies(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_not(z') -{ 2 }-> s1 :|: s1 >= 0, s1 <= x + 1 + 1, z' = 1, 1 = x, x >= 0
   encode_not(z') -{ 2 }-> s2 :|: s2 >= 0, s2 <= x + 1 + 1, z' = 0, 0 = x, x >= 0
   encode_not(z') -{ 2 }-> s3 :|: s3 >= 0, s3 <= x + 1 + 1, z' >= 0, 0 = x, x >= 0
   encode_not(z') -{ 0 }-> not(xor(encArg(x_1188), encArg(x_2150))) :|: z' = 1 + x_1188 + x_2150, x_1188 >= 0, x_2150 >= 0
   encode_not(z') -{ 0 }-> not(or(encArg(x_1185), encArg(x_2147))) :|: x_1185 >= 0, z' = 1 + x_1185 + x_2147, x_2147 >= 0
   encode_not(z') -{ 0 }-> not(not(encArg(z' - 1))) :|: z' - 1 >= 0
   encode_not(z') -{ 0 }-> not(implies(encArg(x_1186), encArg(x_2148))) :|: z' = 1 + x_1186 + x_2148, x_1186 >= 0, x_2148 >= 0
   encode_not(z') -{ 0 }-> not(and(encArg(x_1187), encArg(x_2149))) :|: x_2149 >= 0, z' = 1 + x_1187 + x_2149, x_1187 >= 0
   encode_not(z') -{ 0 }-> 0 :|: z' >= 0
   encode_or(z', z'') -{ 0 }-> or(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_or(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_true -{ 0 }-> 1 :|: 
   encode_true -{ 0 }-> 0 :|: 
   encode_xor(z', z'') -{ 0 }-> xor(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_xor(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   implies(z', z'') -{ 6 + 3*x'' + 3*y'' }-> s23 :|: s20 >= 0, s20 <= 2 * x'' + z'', s21 >= 0, s21 <= 2 * y'' + z'', s22 >= 0, s22 <= s20 + s21 + 1, s23 >= 0, s23 <= s22 + (1 + (1 + x'' + y'') + 1) + 1, z' = 1 + x'' + y'', z'' >= 0, y'' >= 0, x'' >= 0
   implies(z', z'') -{ 6 + 3*x'' + 3*y'' }-> s27 :|: s24 >= 0, s24 <= 2 * x'' + z'', s25 >= 0, s25 <= 2 * y'' + z'', s26 >= 0, s26 <= s24 + s25 + 1, s27 >= 0, s27 <= s26 + 0 + 1, z' = 1 + x'' + y'', z'' >= 0, y'' >= 0, x'' >= 0
   implies(z', z'') -{ 4 }-> x :|: z' = 1, z'' = 1, 0 = 0, 1 = x, x >= 0
   implies(z', z'') -{ 4 }-> x :|: z'' = 0, z' = 1, 0 = 0, 0 = x, x >= 0
   implies(z', z'') -{ 3 }-> x :|: z'' >= 0, z' = 1, 0 = 0, 0 = x, x >= 0
   implies(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, 0 = 0, 0 = x', x' >= 0
   implies(z', z'') -{ 2 }-> x' :|: z' >= 0, z'' >= 0, 0 = 0, 0 = x', x' >= 0
   implies(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = 1, 0 = 0
   implies(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   implies(z', z'') -{ 3 }-> 0 :|: z' = 1, z'' = 1, v0 >= 0, v1 >= 0, 0 = v1, 1 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 1 + z' + 1 = v1
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 0 = v1
   implies(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = 1, 0 = z'
   implies(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' = 1, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' = 1, 0 = x, x >= 0
   implies(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 1 = v1, 0 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = x', x' >= 0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 1 + z' + 1 = v1
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   implies(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   implies(z', z'') -{ 2 }-> 0 :|: z'' >= 0, z' = 1, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 3 }-> 0 :|: z'' >= 0, z' = 1, 0 = x, x >= 0
   implies(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 1 = v1, 0 = v0
   implies(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' >= 0, 0 = x', x' >= 0
   implies(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z' = 1, z'' = 1, 0 = x1, x0 >= 0, x1 >= 0, 1 = x0
   implies(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z'' = 0, z' = 1, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 1 + z' + 1 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' >= 0, z' = 1, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 1 + z' + 1 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 1 + z' + 1 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 0 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 1 + z' + 1 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   not(z') -{ 1 }-> 0 :|: z' >= 0, v1 >= 0, 1 = v1
   not(z') -{ 2 }-> 0 :|: z' >= 0, 1 = z'
   not(z') -{ 1 }-> 1 + z' + x1 :|: z' >= 0, 1 = x1, x1 >= 0
   or(z', z'') -{ 7 + 3*x' + 3*y' }-> s11 :|: s8 >= 0, s8 <= 2 * x' + (1 + x' + y'), s9 >= 0, s9 <= 2 * y' + (1 + x' + y'), s10 >= 0, s10 <= s8 + s9 + 1, s11 >= 0, s11 <= s10 + 0 + 1, z'' = 1 + x' + y', z' = 1 + x' + y', x' >= 0, y' >= 0
   or(z', z'') -{ 6 + 3*x' + 3*y' }-> s15 :|: s12 >= 0, s12 <= 2 * x' + z'', s13 >= 0, s13 <= 2 * y' + z'', s14 >= 0, s14 <= s12 + s13 + 1, s15 >= 0, s15 <= s14 + (1 + (1 + x' + y') + z'') + 1, z' = 1 + x' + y', x' >= 0, z'' >= 0, y' >= 0
   or(z', z'') -{ 6 + 3*x' + 3*y' }-> s19 :|: s16 >= 0, s16 <= 2 * x' + z'', s17 >= 0, s17 <= 2 * y' + z'', s18 >= 0, s18 <= s16 + s17 + 1, s19 >= 0, s19 <= s18 + 0 + 1, z' = 1 + x' + y', x' >= 0, z'' >= 0, y' >= 0
   or(z', z'') -{ 7 + 3*x' + 3*y' }-> s7 :|: s4 >= 0, s4 <= 2 * x' + 0, s5 >= 0, s5 <= 2 * y' + 0, s6 >= 0, s6 <= s4 + s5 + 1, s7 >= 0, s7 <= s6 + (1 + x' + y') + 1, z'' = 0, z' = 1 + x' + y', x' >= 0, y' >= 0
   or(z', z'') -{ 4 }-> x :|: z' = 1, z'' = 1, 0 = 0, 1 = x, x >= 0
   or(z', z'') -{ 4 }-> x :|: z'' = 0, z' = 0, 0 = 0, 0 = x, x >= 0
   or(z', z'') -{ 4 }-> x' :|: z'' = 0, z' >= 0, z' = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, z' = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z' >= 0, z'' = z', 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 2 }-> x' :|: z' >= 0, z'' >= 0, 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = 1, 0 = 0
   or(z', z'') -{ 4 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   or(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   or(z', z'') -{ 3 }-> 0 :|: z' = 1, z'' = 1, v0 >= 0, v1 >= 0, 0 = v1, 1 = v0
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 1 + z' + 1 = v1
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 0 = v1
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = 1, 0 = z'
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, 0 = v0
   or(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' >= 0, 0 = z'
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' = 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' = 0, 0 = x, x >= 0
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 0 = v1, 0 = v0
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   or(z', z'') -{ 4 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 1 + z' + z' = v1
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = x', x' >= 0
   or(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 1 + z' + z'' = v1, 0 = v0
   or(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' >= 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z' = 1, z'' = 1, 0 = x1, x0 >= 0, x1 >= 0, 1 = x0
   or(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z'' = 0, z' = 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 1 + z' + 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z' >= 0, z'' = z', 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 1 + z' + z'' = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 3 }-> 1 + x0 + z' :|: z'' = 0, z' >= 0, x0 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + z' :|: z'' = 0, z' >= 0, x0 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 1 + z' + 1 = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 0 = x1, x1 >= 0
   or(z', z'') -{ 3 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 1 + z' + z' = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   xor(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   xor(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' = z'
   xor(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   xor(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {or}, {encArg}, {encode_not}, {encode_xor}, {encode_implies}, {encode_and}, {encode_or}
Previous analysis results are:
  not: runtime: O(1) [2], size: O(n^1) [2 + z']
  encode_false: runtime: O(1) [0], size: O(1) [0]
  xor: runtime: O(1) [1], size: O(n^1) [1 + z' + z'']
  encode_true: runtime: O(1) [0], size: O(1) [1]
  and: runtime: O(n^1) [1 + 3*z'], size: O(n^1) [2*z' + z'']
  implies: runtime: O(n^1) [4 + 3*z'], size: O(n^1) [3 + 3*z' + 2*z'']
  or: runtime: ?, size: O(n^1) [2 + 3*z' + 3*z'']

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

(79) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: or
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 4 + 3*z'

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

(80)
Obligation:
Complexity RNTS consisting of the following rules:

   and(z', z'') -{ 4 + 3*x + 3*y }-> s30 :|: s28 >= 0, s28 <= 2 * x + z'', s29 >= 0, s29 <= 2 * y + z'', s30 >= 0, s30 <= s28 + s29 + 1, z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
   and(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' = 1
   and(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' = z'
   and(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
   and(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encArg(z') -{ 2 }-> s :|: s >= 0, s <= x + 1 + 1, z' = 1 + 1, 1 = x, x >= 0
   encArg(z') -{ 2 }-> s' :|: s' >= 0, s' <= x + 1 + 1, z' = 1 + 0, 0 = x, x >= 0
   encArg(z') -{ 2 }-> s'' :|: s'' >= 0, s'' <= x + 1 + 1, z' - 1 >= 0, 0 = x, x >= 0
   encArg(z') -{ 0 }-> xor(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> not(xor(encArg(x_13), encArg(x_22))) :|: x_13 >= 0, z' = 1 + (1 + x_13 + x_22), x_22 >= 0
   encArg(z') -{ 0 }-> not(or(encArg(x_1''), encArg(x_2'))) :|: x_1'' >= 0, x_2' >= 0, z' = 1 + (1 + x_1'' + x_2')
   encArg(z') -{ 0 }-> not(not(encArg(z' - 2))) :|: z' - 2 >= 0
   encArg(z') -{ 0 }-> not(implies(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, x_2'' >= 0, z' = 1 + (1 + x_11 + x_2'')
   encArg(z') -{ 0 }-> not(and(encArg(x_12), encArg(x_21))) :|: z' = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0
   encArg(z') -{ 0 }-> implies(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> and(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> 1 :|: z' = 1
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encode_and(z', z'') -{ 0 }-> and(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_and(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_false -{ 0 }-> 0 :|: 
   encode_implies(z', z'') -{ 0 }-> implies(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_implies(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_not(z') -{ 2 }-> s1 :|: s1 >= 0, s1 <= x + 1 + 1, z' = 1, 1 = x, x >= 0
   encode_not(z') -{ 2 }-> s2 :|: s2 >= 0, s2 <= x + 1 + 1, z' = 0, 0 = x, x >= 0
   encode_not(z') -{ 2 }-> s3 :|: s3 >= 0, s3 <= x + 1 + 1, z' >= 0, 0 = x, x >= 0
   encode_not(z') -{ 0 }-> not(xor(encArg(x_1188), encArg(x_2150))) :|: z' = 1 + x_1188 + x_2150, x_1188 >= 0, x_2150 >= 0
   encode_not(z') -{ 0 }-> not(or(encArg(x_1185), encArg(x_2147))) :|: x_1185 >= 0, z' = 1 + x_1185 + x_2147, x_2147 >= 0
   encode_not(z') -{ 0 }-> not(not(encArg(z' - 1))) :|: z' - 1 >= 0
   encode_not(z') -{ 0 }-> not(implies(encArg(x_1186), encArg(x_2148))) :|: z' = 1 + x_1186 + x_2148, x_1186 >= 0, x_2148 >= 0
   encode_not(z') -{ 0 }-> not(and(encArg(x_1187), encArg(x_2149))) :|: x_2149 >= 0, z' = 1 + x_1187 + x_2149, x_1187 >= 0
   encode_not(z') -{ 0 }-> 0 :|: z' >= 0
   encode_or(z', z'') -{ 0 }-> or(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_or(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_true -{ 0 }-> 1 :|: 
   encode_true -{ 0 }-> 0 :|: 
   encode_xor(z', z'') -{ 0 }-> xor(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_xor(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   implies(z', z'') -{ 6 + 3*x'' + 3*y'' }-> s23 :|: s20 >= 0, s20 <= 2 * x'' + z'', s21 >= 0, s21 <= 2 * y'' + z'', s22 >= 0, s22 <= s20 + s21 + 1, s23 >= 0, s23 <= s22 + (1 + (1 + x'' + y'') + 1) + 1, z' = 1 + x'' + y'', z'' >= 0, y'' >= 0, x'' >= 0
   implies(z', z'') -{ 6 + 3*x'' + 3*y'' }-> s27 :|: s24 >= 0, s24 <= 2 * x'' + z'', s25 >= 0, s25 <= 2 * y'' + z'', s26 >= 0, s26 <= s24 + s25 + 1, s27 >= 0, s27 <= s26 + 0 + 1, z' = 1 + x'' + y'', z'' >= 0, y'' >= 0, x'' >= 0
   implies(z', z'') -{ 4 }-> x :|: z' = 1, z'' = 1, 0 = 0, 1 = x, x >= 0
   implies(z', z'') -{ 4 }-> x :|: z'' = 0, z' = 1, 0 = 0, 0 = x, x >= 0
   implies(z', z'') -{ 3 }-> x :|: z'' >= 0, z' = 1, 0 = 0, 0 = x, x >= 0
   implies(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, 0 = 0, 0 = x', x' >= 0
   implies(z', z'') -{ 2 }-> x' :|: z' >= 0, z'' >= 0, 0 = 0, 0 = x', x' >= 0
   implies(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = 1, 0 = 0
   implies(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   implies(z', z'') -{ 3 }-> 0 :|: z' = 1, z'' = 1, v0 >= 0, v1 >= 0, 0 = v1, 1 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 1 + z' + 1 = v1
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 0 = v1
   implies(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = 1, 0 = z'
   implies(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' = 1, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' = 1, 0 = x, x >= 0
   implies(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 1 = v1, 0 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = x', x' >= 0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 1 + z' + 1 = v1
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   implies(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   implies(z', z'') -{ 2 }-> 0 :|: z'' >= 0, z' = 1, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 3 }-> 0 :|: z'' >= 0, z' = 1, 0 = x, x >= 0
   implies(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 1 = v1, 0 = v0
   implies(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' >= 0, 0 = x', x' >= 0
   implies(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z' = 1, z'' = 1, 0 = x1, x0 >= 0, x1 >= 0, 1 = x0
   implies(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z'' = 0, z' = 1, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 1 + z' + 1 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' >= 0, z' = 1, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 1 + z' + 1 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 1 + z' + 1 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 0 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 1 + z' + 1 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   not(z') -{ 1 }-> 0 :|: z' >= 0, v1 >= 0, 1 = v1
   not(z') -{ 2 }-> 0 :|: z' >= 0, 1 = z'
   not(z') -{ 1 }-> 1 + z' + x1 :|: z' >= 0, 1 = x1, x1 >= 0
   or(z', z'') -{ 7 + 3*x' + 3*y' }-> s11 :|: s8 >= 0, s8 <= 2 * x' + (1 + x' + y'), s9 >= 0, s9 <= 2 * y' + (1 + x' + y'), s10 >= 0, s10 <= s8 + s9 + 1, s11 >= 0, s11 <= s10 + 0 + 1, z'' = 1 + x' + y', z' = 1 + x' + y', x' >= 0, y' >= 0
   or(z', z'') -{ 6 + 3*x' + 3*y' }-> s15 :|: s12 >= 0, s12 <= 2 * x' + z'', s13 >= 0, s13 <= 2 * y' + z'', s14 >= 0, s14 <= s12 + s13 + 1, s15 >= 0, s15 <= s14 + (1 + (1 + x' + y') + z'') + 1, z' = 1 + x' + y', x' >= 0, z'' >= 0, y' >= 0
   or(z', z'') -{ 6 + 3*x' + 3*y' }-> s19 :|: s16 >= 0, s16 <= 2 * x' + z'', s17 >= 0, s17 <= 2 * y' + z'', s18 >= 0, s18 <= s16 + s17 + 1, s19 >= 0, s19 <= s18 + 0 + 1, z' = 1 + x' + y', x' >= 0, z'' >= 0, y' >= 0
   or(z', z'') -{ 7 + 3*x' + 3*y' }-> s7 :|: s4 >= 0, s4 <= 2 * x' + 0, s5 >= 0, s5 <= 2 * y' + 0, s6 >= 0, s6 <= s4 + s5 + 1, s7 >= 0, s7 <= s6 + (1 + x' + y') + 1, z'' = 0, z' = 1 + x' + y', x' >= 0, y' >= 0
   or(z', z'') -{ 4 }-> x :|: z' = 1, z'' = 1, 0 = 0, 1 = x, x >= 0
   or(z', z'') -{ 4 }-> x :|: z'' = 0, z' = 0, 0 = 0, 0 = x, x >= 0
   or(z', z'') -{ 4 }-> x' :|: z'' = 0, z' >= 0, z' = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, z' = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z' >= 0, z'' = z', 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 2 }-> x' :|: z' >= 0, z'' >= 0, 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = 1, 0 = 0
   or(z', z'') -{ 4 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   or(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   or(z', z'') -{ 3 }-> 0 :|: z' = 1, z'' = 1, v0 >= 0, v1 >= 0, 0 = v1, 1 = v0
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 1 + z' + 1 = v1
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 0 = v1
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = 1, 0 = z'
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, 0 = v0
   or(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' >= 0, 0 = z'
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' = 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' = 0, 0 = x, x >= 0
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 0 = v1, 0 = v0
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   or(z', z'') -{ 4 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 1 + z' + z' = v1
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = x', x' >= 0
   or(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 1 + z' + z'' = v1, 0 = v0
   or(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' >= 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z' = 1, z'' = 1, 0 = x1, x0 >= 0, x1 >= 0, 1 = x0
   or(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z'' = 0, z' = 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 1 + z' + 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z' >= 0, z'' = z', 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 1 + z' + z'' = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 3 }-> 1 + x0 + z' :|: z'' = 0, z' >= 0, x0 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + z' :|: z'' = 0, z' >= 0, x0 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 1 + z' + 1 = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 0 = x1, x1 >= 0
   or(z', z'') -{ 3 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 1 + z' + z' = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   xor(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   xor(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' = z'
   xor(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   xor(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {encArg}, {encode_not}, {encode_xor}, {encode_implies}, {encode_and}, {encode_or}
Previous analysis results are:
  not: runtime: O(1) [2], size: O(n^1) [2 + z']
  encode_false: runtime: O(1) [0], size: O(1) [0]
  xor: runtime: O(1) [1], size: O(n^1) [1 + z' + z'']
  encode_true: runtime: O(1) [0], size: O(1) [1]
  and: runtime: O(n^1) [1 + 3*z'], size: O(n^1) [2*z' + z'']
  implies: runtime: O(n^1) [4 + 3*z'], size: O(n^1) [3 + 3*z' + 2*z'']
  or: runtime: O(n^1) [4 + 3*z'], size: O(n^1) [2 + 3*z' + 3*z'']

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(81) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(82)
Obligation:
Complexity RNTS consisting of the following rules:

   and(z', z'') -{ 4 + 3*x + 3*y }-> s30 :|: s28 >= 0, s28 <= 2 * x + z'', s29 >= 0, s29 <= 2 * y + z'', s30 >= 0, s30 <= s28 + s29 + 1, z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
   and(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' = 1
   and(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' = z'
   and(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
   and(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encArg(z') -{ 2 }-> s :|: s >= 0, s <= x + 1 + 1, z' = 1 + 1, 1 = x, x >= 0
   encArg(z') -{ 2 }-> s' :|: s' >= 0, s' <= x + 1 + 1, z' = 1 + 0, 0 = x, x >= 0
   encArg(z') -{ 2 }-> s'' :|: s'' >= 0, s'' <= x + 1 + 1, z' - 1 >= 0, 0 = x, x >= 0
   encArg(z') -{ 0 }-> xor(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> not(xor(encArg(x_13), encArg(x_22))) :|: x_13 >= 0, z' = 1 + (1 + x_13 + x_22), x_22 >= 0
   encArg(z') -{ 0 }-> not(or(encArg(x_1''), encArg(x_2'))) :|: x_1'' >= 0, x_2' >= 0, z' = 1 + (1 + x_1'' + x_2')
   encArg(z') -{ 0 }-> not(not(encArg(z' - 2))) :|: z' - 2 >= 0
   encArg(z') -{ 0 }-> not(implies(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, x_2'' >= 0, z' = 1 + (1 + x_11 + x_2'')
   encArg(z') -{ 0 }-> not(and(encArg(x_12), encArg(x_21))) :|: z' = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0
   encArg(z') -{ 0 }-> implies(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> and(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> 1 :|: z' = 1
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encode_and(z', z'') -{ 0 }-> and(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_and(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_false -{ 0 }-> 0 :|: 
   encode_implies(z', z'') -{ 0 }-> implies(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_implies(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_not(z') -{ 2 }-> s1 :|: s1 >= 0, s1 <= x + 1 + 1, z' = 1, 1 = x, x >= 0
   encode_not(z') -{ 2 }-> s2 :|: s2 >= 0, s2 <= x + 1 + 1, z' = 0, 0 = x, x >= 0
   encode_not(z') -{ 2 }-> s3 :|: s3 >= 0, s3 <= x + 1 + 1, z' >= 0, 0 = x, x >= 0
   encode_not(z') -{ 0 }-> not(xor(encArg(x_1188), encArg(x_2150))) :|: z' = 1 + x_1188 + x_2150, x_1188 >= 0, x_2150 >= 0
   encode_not(z') -{ 0 }-> not(or(encArg(x_1185), encArg(x_2147))) :|: x_1185 >= 0, z' = 1 + x_1185 + x_2147, x_2147 >= 0
   encode_not(z') -{ 0 }-> not(not(encArg(z' - 1))) :|: z' - 1 >= 0
   encode_not(z') -{ 0 }-> not(implies(encArg(x_1186), encArg(x_2148))) :|: z' = 1 + x_1186 + x_2148, x_1186 >= 0, x_2148 >= 0
   encode_not(z') -{ 0 }-> not(and(encArg(x_1187), encArg(x_2149))) :|: x_2149 >= 0, z' = 1 + x_1187 + x_2149, x_1187 >= 0
   encode_not(z') -{ 0 }-> 0 :|: z' >= 0
   encode_or(z', z'') -{ 0 }-> or(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_or(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_true -{ 0 }-> 1 :|: 
   encode_true -{ 0 }-> 0 :|: 
   encode_xor(z', z'') -{ 0 }-> xor(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_xor(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   implies(z', z'') -{ 6 + 3*x'' + 3*y'' }-> s23 :|: s20 >= 0, s20 <= 2 * x'' + z'', s21 >= 0, s21 <= 2 * y'' + z'', s22 >= 0, s22 <= s20 + s21 + 1, s23 >= 0, s23 <= s22 + (1 + (1 + x'' + y'') + 1) + 1, z' = 1 + x'' + y'', z'' >= 0, y'' >= 0, x'' >= 0
   implies(z', z'') -{ 6 + 3*x'' + 3*y'' }-> s27 :|: s24 >= 0, s24 <= 2 * x'' + z'', s25 >= 0, s25 <= 2 * y'' + z'', s26 >= 0, s26 <= s24 + s25 + 1, s27 >= 0, s27 <= s26 + 0 + 1, z' = 1 + x'' + y'', z'' >= 0, y'' >= 0, x'' >= 0
   implies(z', z'') -{ 4 }-> x :|: z' = 1, z'' = 1, 0 = 0, 1 = x, x >= 0
   implies(z', z'') -{ 4 }-> x :|: z'' = 0, z' = 1, 0 = 0, 0 = x, x >= 0
   implies(z', z'') -{ 3 }-> x :|: z'' >= 0, z' = 1, 0 = 0, 0 = x, x >= 0
   implies(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, 0 = 0, 0 = x', x' >= 0
   implies(z', z'') -{ 2 }-> x' :|: z' >= 0, z'' >= 0, 0 = 0, 0 = x', x' >= 0
   implies(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = 1, 0 = 0
   implies(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   implies(z', z'') -{ 3 }-> 0 :|: z' = 1, z'' = 1, v0 >= 0, v1 >= 0, 0 = v1, 1 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 1 + z' + 1 = v1
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 0 = v1
   implies(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = 1, 0 = z'
   implies(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' = 1, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' = 1, 0 = x, x >= 0
   implies(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 1 = v1, 0 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = x', x' >= 0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 1 + z' + 1 = v1
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   implies(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   implies(z', z'') -{ 2 }-> 0 :|: z'' >= 0, z' = 1, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 3 }-> 0 :|: z'' >= 0, z' = 1, 0 = x, x >= 0
   implies(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 1 = v1, 0 = v0
   implies(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' >= 0, 0 = x', x' >= 0
   implies(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z' = 1, z'' = 1, 0 = x1, x0 >= 0, x1 >= 0, 1 = x0
   implies(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z'' = 0, z' = 1, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 1 + z' + 1 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' >= 0, z' = 1, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 1 + z' + 1 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 1 + z' + 1 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 0 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 1 + z' + 1 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   not(z') -{ 1 }-> 0 :|: z' >= 0, v1 >= 0, 1 = v1
   not(z') -{ 2 }-> 0 :|: z' >= 0, 1 = z'
   not(z') -{ 1 }-> 1 + z' + x1 :|: z' >= 0, 1 = x1, x1 >= 0
   or(z', z'') -{ 7 + 3*x' + 3*y' }-> s11 :|: s8 >= 0, s8 <= 2 * x' + (1 + x' + y'), s9 >= 0, s9 <= 2 * y' + (1 + x' + y'), s10 >= 0, s10 <= s8 + s9 + 1, s11 >= 0, s11 <= s10 + 0 + 1, z'' = 1 + x' + y', z' = 1 + x' + y', x' >= 0, y' >= 0
   or(z', z'') -{ 6 + 3*x' + 3*y' }-> s15 :|: s12 >= 0, s12 <= 2 * x' + z'', s13 >= 0, s13 <= 2 * y' + z'', s14 >= 0, s14 <= s12 + s13 + 1, s15 >= 0, s15 <= s14 + (1 + (1 + x' + y') + z'') + 1, z' = 1 + x' + y', x' >= 0, z'' >= 0, y' >= 0
   or(z', z'') -{ 6 + 3*x' + 3*y' }-> s19 :|: s16 >= 0, s16 <= 2 * x' + z'', s17 >= 0, s17 <= 2 * y' + z'', s18 >= 0, s18 <= s16 + s17 + 1, s19 >= 0, s19 <= s18 + 0 + 1, z' = 1 + x' + y', x' >= 0, z'' >= 0, y' >= 0
   or(z', z'') -{ 7 + 3*x' + 3*y' }-> s7 :|: s4 >= 0, s4 <= 2 * x' + 0, s5 >= 0, s5 <= 2 * y' + 0, s6 >= 0, s6 <= s4 + s5 + 1, s7 >= 0, s7 <= s6 + (1 + x' + y') + 1, z'' = 0, z' = 1 + x' + y', x' >= 0, y' >= 0
   or(z', z'') -{ 4 }-> x :|: z' = 1, z'' = 1, 0 = 0, 1 = x, x >= 0
   or(z', z'') -{ 4 }-> x :|: z'' = 0, z' = 0, 0 = 0, 0 = x, x >= 0
   or(z', z'') -{ 4 }-> x' :|: z'' = 0, z' >= 0, z' = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, z' = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z' >= 0, z'' = z', 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 2 }-> x' :|: z' >= 0, z'' >= 0, 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = 1, 0 = 0
   or(z', z'') -{ 4 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   or(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   or(z', z'') -{ 3 }-> 0 :|: z' = 1, z'' = 1, v0 >= 0, v1 >= 0, 0 = v1, 1 = v0
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 1 + z' + 1 = v1
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 0 = v1
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = 1, 0 = z'
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, 0 = v0
   or(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' >= 0, 0 = z'
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' = 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' = 0, 0 = x, x >= 0
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 0 = v1, 0 = v0
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   or(z', z'') -{ 4 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 1 + z' + z' = v1
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = x', x' >= 0
   or(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 1 + z' + z'' = v1, 0 = v0
   or(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' >= 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z' = 1, z'' = 1, 0 = x1, x0 >= 0, x1 >= 0, 1 = x0
   or(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z'' = 0, z' = 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 1 + z' + 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z' >= 0, z'' = z', 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 1 + z' + z'' = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 3 }-> 1 + x0 + z' :|: z'' = 0, z' >= 0, x0 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + z' :|: z'' = 0, z' >= 0, x0 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 1 + z' + 1 = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 0 = x1, x1 >= 0
   or(z', z'') -{ 3 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 1 + z' + z' = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   xor(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   xor(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' = z'
   xor(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   xor(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {encArg}, {encode_not}, {encode_xor}, {encode_implies}, {encode_and}, {encode_or}
Previous analysis results are:
  not: runtime: O(1) [2], size: O(n^1) [2 + z']
  encode_false: runtime: O(1) [0], size: O(1) [0]
  xor: runtime: O(1) [1], size: O(n^1) [1 + z' + z'']
  encode_true: runtime: O(1) [0], size: O(1) [1]
  and: runtime: O(n^1) [1 + 3*z'], size: O(n^1) [2*z' + z'']
  implies: runtime: O(n^1) [4 + 3*z'], size: O(n^1) [3 + 3*z' + 2*z'']
  or: runtime: O(n^1) [4 + 3*z'], size: O(n^1) [2 + 3*z' + 3*z'']

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

(83) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: encArg
after applying outer abstraction to obtain an ITS,
resulting in: EXP with polynomial bound: ?

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

(84)
Obligation:
Complexity RNTS consisting of the following rules:

   and(z', z'') -{ 4 + 3*x + 3*y }-> s30 :|: s28 >= 0, s28 <= 2 * x + z'', s29 >= 0, s29 <= 2 * y + z'', s30 >= 0, s30 <= s28 + s29 + 1, z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
   and(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' = 1
   and(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' = z'
   and(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
   and(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encArg(z') -{ 2 }-> s :|: s >= 0, s <= x + 1 + 1, z' = 1 + 1, 1 = x, x >= 0
   encArg(z') -{ 2 }-> s' :|: s' >= 0, s' <= x + 1 + 1, z' = 1 + 0, 0 = x, x >= 0
   encArg(z') -{ 2 }-> s'' :|: s'' >= 0, s'' <= x + 1 + 1, z' - 1 >= 0, 0 = x, x >= 0
   encArg(z') -{ 0 }-> xor(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> not(xor(encArg(x_13), encArg(x_22))) :|: x_13 >= 0, z' = 1 + (1 + x_13 + x_22), x_22 >= 0
   encArg(z') -{ 0 }-> not(or(encArg(x_1''), encArg(x_2'))) :|: x_1'' >= 0, x_2' >= 0, z' = 1 + (1 + x_1'' + x_2')
   encArg(z') -{ 0 }-> not(not(encArg(z' - 2))) :|: z' - 2 >= 0
   encArg(z') -{ 0 }-> not(implies(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, x_2'' >= 0, z' = 1 + (1 + x_11 + x_2'')
   encArg(z') -{ 0 }-> not(and(encArg(x_12), encArg(x_21))) :|: z' = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0
   encArg(z') -{ 0 }-> implies(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> and(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> 1 :|: z' = 1
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encode_and(z', z'') -{ 0 }-> and(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_and(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_false -{ 0 }-> 0 :|: 
   encode_implies(z', z'') -{ 0 }-> implies(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_implies(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_not(z') -{ 2 }-> s1 :|: s1 >= 0, s1 <= x + 1 + 1, z' = 1, 1 = x, x >= 0
   encode_not(z') -{ 2 }-> s2 :|: s2 >= 0, s2 <= x + 1 + 1, z' = 0, 0 = x, x >= 0
   encode_not(z') -{ 2 }-> s3 :|: s3 >= 0, s3 <= x + 1 + 1, z' >= 0, 0 = x, x >= 0
   encode_not(z') -{ 0 }-> not(xor(encArg(x_1188), encArg(x_2150))) :|: z' = 1 + x_1188 + x_2150, x_1188 >= 0, x_2150 >= 0
   encode_not(z') -{ 0 }-> not(or(encArg(x_1185), encArg(x_2147))) :|: x_1185 >= 0, z' = 1 + x_1185 + x_2147, x_2147 >= 0
   encode_not(z') -{ 0 }-> not(not(encArg(z' - 1))) :|: z' - 1 >= 0
   encode_not(z') -{ 0 }-> not(implies(encArg(x_1186), encArg(x_2148))) :|: z' = 1 + x_1186 + x_2148, x_1186 >= 0, x_2148 >= 0
   encode_not(z') -{ 0 }-> not(and(encArg(x_1187), encArg(x_2149))) :|: x_2149 >= 0, z' = 1 + x_1187 + x_2149, x_1187 >= 0
   encode_not(z') -{ 0 }-> 0 :|: z' >= 0
   encode_or(z', z'') -{ 0 }-> or(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_or(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_true -{ 0 }-> 1 :|: 
   encode_true -{ 0 }-> 0 :|: 
   encode_xor(z', z'') -{ 0 }-> xor(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_xor(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   implies(z', z'') -{ 6 + 3*x'' + 3*y'' }-> s23 :|: s20 >= 0, s20 <= 2 * x'' + z'', s21 >= 0, s21 <= 2 * y'' + z'', s22 >= 0, s22 <= s20 + s21 + 1, s23 >= 0, s23 <= s22 + (1 + (1 + x'' + y'') + 1) + 1, z' = 1 + x'' + y'', z'' >= 0, y'' >= 0, x'' >= 0
   implies(z', z'') -{ 6 + 3*x'' + 3*y'' }-> s27 :|: s24 >= 0, s24 <= 2 * x'' + z'', s25 >= 0, s25 <= 2 * y'' + z'', s26 >= 0, s26 <= s24 + s25 + 1, s27 >= 0, s27 <= s26 + 0 + 1, z' = 1 + x'' + y'', z'' >= 0, y'' >= 0, x'' >= 0
   implies(z', z'') -{ 4 }-> x :|: z' = 1, z'' = 1, 0 = 0, 1 = x, x >= 0
   implies(z', z'') -{ 4 }-> x :|: z'' = 0, z' = 1, 0 = 0, 0 = x, x >= 0
   implies(z', z'') -{ 3 }-> x :|: z'' >= 0, z' = 1, 0 = 0, 0 = x, x >= 0
   implies(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, 0 = 0, 0 = x', x' >= 0
   implies(z', z'') -{ 2 }-> x' :|: z' >= 0, z'' >= 0, 0 = 0, 0 = x', x' >= 0
   implies(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = 1, 0 = 0
   implies(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   implies(z', z'') -{ 3 }-> 0 :|: z' = 1, z'' = 1, v0 >= 0, v1 >= 0, 0 = v1, 1 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 1 + z' + 1 = v1
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 0 = v1
   implies(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = 1, 0 = z'
   implies(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' = 1, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' = 1, 0 = x, x >= 0
   implies(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 1 = v1, 0 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = x', x' >= 0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 1 + z' + 1 = v1
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   implies(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   implies(z', z'') -{ 2 }-> 0 :|: z'' >= 0, z' = 1, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 3 }-> 0 :|: z'' >= 0, z' = 1, 0 = x, x >= 0
   implies(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 1 = v1, 0 = v0
   implies(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' >= 0, 0 = x', x' >= 0
   implies(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z' = 1, z'' = 1, 0 = x1, x0 >= 0, x1 >= 0, 1 = x0
   implies(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z'' = 0, z' = 1, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 1 + z' + 1 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' >= 0, z' = 1, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 1 + z' + 1 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 1 + z' + 1 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 0 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 1 + z' + 1 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   not(z') -{ 1 }-> 0 :|: z' >= 0, v1 >= 0, 1 = v1
   not(z') -{ 2 }-> 0 :|: z' >= 0, 1 = z'
   not(z') -{ 1 }-> 1 + z' + x1 :|: z' >= 0, 1 = x1, x1 >= 0
   or(z', z'') -{ 7 + 3*x' + 3*y' }-> s11 :|: s8 >= 0, s8 <= 2 * x' + (1 + x' + y'), s9 >= 0, s9 <= 2 * y' + (1 + x' + y'), s10 >= 0, s10 <= s8 + s9 + 1, s11 >= 0, s11 <= s10 + 0 + 1, z'' = 1 + x' + y', z' = 1 + x' + y', x' >= 0, y' >= 0
   or(z', z'') -{ 6 + 3*x' + 3*y' }-> s15 :|: s12 >= 0, s12 <= 2 * x' + z'', s13 >= 0, s13 <= 2 * y' + z'', s14 >= 0, s14 <= s12 + s13 + 1, s15 >= 0, s15 <= s14 + (1 + (1 + x' + y') + z'') + 1, z' = 1 + x' + y', x' >= 0, z'' >= 0, y' >= 0
   or(z', z'') -{ 6 + 3*x' + 3*y' }-> s19 :|: s16 >= 0, s16 <= 2 * x' + z'', s17 >= 0, s17 <= 2 * y' + z'', s18 >= 0, s18 <= s16 + s17 + 1, s19 >= 0, s19 <= s18 + 0 + 1, z' = 1 + x' + y', x' >= 0, z'' >= 0, y' >= 0
   or(z', z'') -{ 7 + 3*x' + 3*y' }-> s7 :|: s4 >= 0, s4 <= 2 * x' + 0, s5 >= 0, s5 <= 2 * y' + 0, s6 >= 0, s6 <= s4 + s5 + 1, s7 >= 0, s7 <= s6 + (1 + x' + y') + 1, z'' = 0, z' = 1 + x' + y', x' >= 0, y' >= 0
   or(z', z'') -{ 4 }-> x :|: z' = 1, z'' = 1, 0 = 0, 1 = x, x >= 0
   or(z', z'') -{ 4 }-> x :|: z'' = 0, z' = 0, 0 = 0, 0 = x, x >= 0
   or(z', z'') -{ 4 }-> x' :|: z'' = 0, z' >= 0, z' = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, z' = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z' >= 0, z'' = z', 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 2 }-> x' :|: z' >= 0, z'' >= 0, 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = 1, 0 = 0
   or(z', z'') -{ 4 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   or(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   or(z', z'') -{ 3 }-> 0 :|: z' = 1, z'' = 1, v0 >= 0, v1 >= 0, 0 = v1, 1 = v0
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 1 + z' + 1 = v1
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 0 = v1
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = 1, 0 = z'
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, 0 = v0
   or(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' >= 0, 0 = z'
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' = 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' = 0, 0 = x, x >= 0
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 0 = v1, 0 = v0
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   or(z', z'') -{ 4 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 1 + z' + z' = v1
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = x', x' >= 0
   or(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 1 + z' + z'' = v1, 0 = v0
   or(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' >= 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z' = 1, z'' = 1, 0 = x1, x0 >= 0, x1 >= 0, 1 = x0
   or(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z'' = 0, z' = 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 1 + z' + 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z' >= 0, z'' = z', 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 1 + z' + z'' = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 3 }-> 1 + x0 + z' :|: z'' = 0, z' >= 0, x0 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + z' :|: z'' = 0, z' >= 0, x0 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 1 + z' + 1 = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 0 = x1, x1 >= 0
   or(z', z'') -{ 3 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 1 + z' + z' = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   xor(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   xor(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' = z'
   xor(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   xor(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {encArg}, {encode_not}, {encode_xor}, {encode_implies}, {encode_and}, {encode_or}
Previous analysis results are:
  not: runtime: O(1) [2], size: O(n^1) [2 + z']
  encode_false: runtime: O(1) [0], size: O(1) [0]
  xor: runtime: O(1) [1], size: O(n^1) [1 + z' + z'']
  encode_true: runtime: O(1) [0], size: O(1) [1]
  and: runtime: O(n^1) [1 + 3*z'], size: O(n^1) [2*z' + z'']
  implies: runtime: O(n^1) [4 + 3*z'], size: O(n^1) [3 + 3*z' + 2*z'']
  or: runtime: O(n^1) [4 + 3*z'], size: O(n^1) [2 + 3*z' + 3*z'']
  encArg: runtime: ?, size: EXP

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

(85) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: encArg
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

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

(86)
Obligation:
Complexity RNTS consisting of the following rules:

   and(z', z'') -{ 4 + 3*x + 3*y }-> s30 :|: s28 >= 0, s28 <= 2 * x + z'', s29 >= 0, s29 <= 2 * y + z'', s30 >= 0, s30 <= s28 + s29 + 1, z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
   and(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' = 1
   and(z', z'') -{ 1 }-> z' :|: z' >= 0, z'' = z'
   and(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
   and(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encArg(z') -{ 2 }-> s :|: s >= 0, s <= x + 1 + 1, z' = 1 + 1, 1 = x, x >= 0
   encArg(z') -{ 2 }-> s' :|: s' >= 0, s' <= x + 1 + 1, z' = 1 + 0, 0 = x, x >= 0
   encArg(z') -{ 2 }-> s'' :|: s'' >= 0, s'' <= x + 1 + 1, z' - 1 >= 0, 0 = x, x >= 0
   encArg(z') -{ 0 }-> xor(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> or(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> not(xor(encArg(x_13), encArg(x_22))) :|: x_13 >= 0, z' = 1 + (1 + x_13 + x_22), x_22 >= 0
   encArg(z') -{ 0 }-> not(or(encArg(x_1''), encArg(x_2'))) :|: x_1'' >= 0, x_2' >= 0, z' = 1 + (1 + x_1'' + x_2')
   encArg(z') -{ 0 }-> not(not(encArg(z' - 2))) :|: z' - 2 >= 0
   encArg(z') -{ 0 }-> not(implies(encArg(x_11), encArg(x_2''))) :|: x_11 >= 0, x_2'' >= 0, z' = 1 + (1 + x_11 + x_2'')
   encArg(z') -{ 0 }-> not(and(encArg(x_12), encArg(x_21))) :|: z' = 1 + (1 + x_12 + x_21), x_12 >= 0, x_21 >= 0
   encArg(z') -{ 0 }-> implies(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> and(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> 1 :|: z' = 1
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encode_and(z', z'') -{ 0 }-> and(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_and(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_false -{ 0 }-> 0 :|: 
   encode_implies(z', z'') -{ 0 }-> implies(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_implies(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_not(z') -{ 2 }-> s1 :|: s1 >= 0, s1 <= x + 1 + 1, z' = 1, 1 = x, x >= 0
   encode_not(z') -{ 2 }-> s2 :|: s2 >= 0, s2 <= x + 1 + 1, z' = 0, 0 = x, x >= 0
   encode_not(z') -{ 2 }-> s3 :|: s3 >= 0, s3 <= x + 1 + 1, z' >= 0, 0 = x, x >= 0
   encode_not(z') -{ 0 }-> not(xor(encArg(x_1188), encArg(x_2150))) :|: z' = 1 + x_1188 + x_2150, x_1188 >= 0, x_2150 >= 0
   encode_not(z') -{ 0 }-> not(or(encArg(x_1185), encArg(x_2147))) :|: x_1185 >= 0, z' = 1 + x_1185 + x_2147, x_2147 >= 0
   encode_not(z') -{ 0 }-> not(not(encArg(z' - 1))) :|: z' - 1 >= 0
   encode_not(z') -{ 0 }-> not(implies(encArg(x_1186), encArg(x_2148))) :|: z' = 1 + x_1186 + x_2148, x_1186 >= 0, x_2148 >= 0
   encode_not(z') -{ 0 }-> not(and(encArg(x_1187), encArg(x_2149))) :|: x_2149 >= 0, z' = 1 + x_1187 + x_2149, x_1187 >= 0
   encode_not(z') -{ 0 }-> 0 :|: z' >= 0
   encode_or(z', z'') -{ 0 }-> or(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_or(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_true -{ 0 }-> 1 :|: 
   encode_true -{ 0 }-> 0 :|: 
   encode_xor(z', z'') -{ 0 }-> xor(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0
   encode_xor(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   implies(z', z'') -{ 6 + 3*x'' + 3*y'' }-> s23 :|: s20 >= 0, s20 <= 2 * x'' + z'', s21 >= 0, s21 <= 2 * y'' + z'', s22 >= 0, s22 <= s20 + s21 + 1, s23 >= 0, s23 <= s22 + (1 + (1 + x'' + y'') + 1) + 1, z' = 1 + x'' + y'', z'' >= 0, y'' >= 0, x'' >= 0
   implies(z', z'') -{ 6 + 3*x'' + 3*y'' }-> s27 :|: s24 >= 0, s24 <= 2 * x'' + z'', s25 >= 0, s25 <= 2 * y'' + z'', s26 >= 0, s26 <= s24 + s25 + 1, s27 >= 0, s27 <= s26 + 0 + 1, z' = 1 + x'' + y'', z'' >= 0, y'' >= 0, x'' >= 0
   implies(z', z'') -{ 4 }-> x :|: z' = 1, z'' = 1, 0 = 0, 1 = x, x >= 0
   implies(z', z'') -{ 4 }-> x :|: z'' = 0, z' = 1, 0 = 0, 0 = x, x >= 0
   implies(z', z'') -{ 3 }-> x :|: z'' >= 0, z' = 1, 0 = 0, 0 = x, x >= 0
   implies(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, 0 = 0, 0 = x', x' >= 0
   implies(z', z'') -{ 2 }-> x' :|: z' >= 0, z'' >= 0, 0 = 0, 0 = x', x' >= 0
   implies(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = 1, 0 = 0
   implies(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   implies(z', z'') -{ 3 }-> 0 :|: z' = 1, z'' = 1, v0 >= 0, v1 >= 0, 0 = v1, 1 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 1 + z' + 1 = v1
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 0 = v1
   implies(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = 1, 0 = z'
   implies(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' = 1, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' = 1, 0 = x, x >= 0
   implies(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 1 = v1, 0 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = x', x' >= 0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 1 + z' + 1 = v1
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   implies(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   implies(z', z'') -{ 2 }-> 0 :|: z'' >= 0, z' = 1, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 3 }-> 0 :|: z'' >= 0, z' = 1, 0 = x, x >= 0
   implies(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 1 = v1, 0 = v0
   implies(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   implies(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' >= 0, 0 = x', x' >= 0
   implies(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z' = 1, z'' = 1, 0 = x1, x0 >= 0, x1 >= 0, 1 = x0
   implies(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z'' = 0, z' = 1, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 1 + z' + 1 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' >= 0, z' = 1, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 1 + z' + 1 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 1 + z' + 1 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 0 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 1 + z' + 1 = x1, x1 >= 0
   implies(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   not(z') -{ 1 }-> 0 :|: z' >= 0, v1 >= 0, 1 = v1
   not(z') -{ 2 }-> 0 :|: z' >= 0, 1 = z'
   not(z') -{ 1 }-> 1 + z' + x1 :|: z' >= 0, 1 = x1, x1 >= 0
   or(z', z'') -{ 7 + 3*x' + 3*y' }-> s11 :|: s8 >= 0, s8 <= 2 * x' + (1 + x' + y'), s9 >= 0, s9 <= 2 * y' + (1 + x' + y'), s10 >= 0, s10 <= s8 + s9 + 1, s11 >= 0, s11 <= s10 + 0 + 1, z'' = 1 + x' + y', z' = 1 + x' + y', x' >= 0, y' >= 0
   or(z', z'') -{ 6 + 3*x' + 3*y' }-> s15 :|: s12 >= 0, s12 <= 2 * x' + z'', s13 >= 0, s13 <= 2 * y' + z'', s14 >= 0, s14 <= s12 + s13 + 1, s15 >= 0, s15 <= s14 + (1 + (1 + x' + y') + z'') + 1, z' = 1 + x' + y', x' >= 0, z'' >= 0, y' >= 0
   or(z', z'') -{ 6 + 3*x' + 3*y' }-> s19 :|: s16 >= 0, s16 <= 2 * x' + z'', s17 >= 0, s17 <= 2 * y' + z'', s18 >= 0, s18 <= s16 + s17 + 1, s19 >= 0, s19 <= s18 + 0 + 1, z' = 1 + x' + y', x' >= 0, z'' >= 0, y' >= 0
   or(z', z'') -{ 7 + 3*x' + 3*y' }-> s7 :|: s4 >= 0, s4 <= 2 * x' + 0, s5 >= 0, s5 <= 2 * y' + 0, s6 >= 0, s6 <= s4 + s5 + 1, s7 >= 0, s7 <= s6 + (1 + x' + y') + 1, z'' = 0, z' = 1 + x' + y', x' >= 0, y' >= 0
   or(z', z'') -{ 4 }-> x :|: z' = 1, z'' = 1, 0 = 0, 1 = x, x >= 0
   or(z', z'') -{ 4 }-> x :|: z'' = 0, z' = 0, 0 = 0, 0 = x, x >= 0
   or(z', z'') -{ 4 }-> x' :|: z'' = 0, z' >= 0, z' = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z'' = 0, z' >= 0, z' = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> x' :|: z' >= 0, z'' = z', 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 2 }-> x' :|: z' >= 0, z'' >= 0, 0 = 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = 1, 0 = 0
   or(z', z'') -{ 4 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   or(z', z'') -{ 3 }-> z' :|: z' >= 0, z'' = z', 0 = 0
   or(z', z'') -{ 3 }-> 0 :|: z' = 1, z'' = 1, v0 >= 0, v1 >= 0, 0 = v1, 1 = v0
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 1 + z' + 1 = v1
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = 1, v1 >= 0, 0 = v1
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = 1, 0 = z'
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, 0 = v0
   or(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' >= 0, 0 = z'
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' = 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 4 }-> 0 :|: z'' = 0, z' = 0, 0 = x, x >= 0
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 1 + z' + 0 = v1, 0 = v0
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   or(z', z'') -{ 4 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 1 + z' + z' = v1
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v1 >= 0, 0 = v1
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z'' = 0, z' >= 0, v0 >= 0, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z'' = 0, z' >= 0, 0 = z'
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' = z', v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 3 }-> 0 :|: z' >= 0, z'' = z', 0 = x', x' >= 0
   or(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 1 + z' + z'' = v1, 0 = v0
   or(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   or(z', z'') -{ 2 }-> 0 :|: z' >= 0, z'' >= 0, 0 = x', x' >= 0
   or(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z' = 1, z'' = 1, 0 = x1, x0 >= 0, x1 >= 0, 1 = x0
   or(z', z'') -{ 3 }-> 1 + x0 + x1 :|: z'' = 0, z' = 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 1 + z' + 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z'' = 0, z' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + x1 :|: z' >= 0, z'' = z', 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 1 + z' + z'' = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   or(z', z'') -{ 3 }-> 1 + x0 + z' :|: z'' = 0, z' >= 0, x0 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + x0 + z' :|: z'' = 0, z' >= 0, x0 >= 0, 0 = x0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 1 + z' + 1 = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = 1, 0 = x1, x1 >= 0
   or(z', z'') -{ 3 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 1 + z' + z' = x1, x1 >= 0
   or(z', z'') -{ 2 }-> 1 + z' + x1 :|: z' >= 0, z'' = z', 0 = x1, x1 >= 0
   xor(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   xor(z', z'') -{ 1 }-> 0 :|: z' >= 0, z'' = z'
   xor(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   xor(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {encArg}, {encode_not}, {encode_xor}, {encode_implies}, {encode_and}, {encode_or}
Previous analysis results are:
  not: runtime: O(1) [2], size: O(n^1) [2 + z']
  encode_false: runtime: O(1) [0], size: O(1) [0]
  xor: runtime: O(1) [1], size: O(n^1) [1 + z' + z'']
  encode_true: runtime: O(1) [0], size: O(1) [1]
  and: runtime: O(n^1) [1 + 3*z'], size: O(n^1) [2*z' + z'']
  implies: runtime: O(n^1) [4 + 3*z'], size: O(n^1) [3 + 3*z' + 2*z'']
  or: runtime: O(n^1) [4 + 3*z'], size: O(n^1) [2 + 3*z' + 3*z'']
  encArg: runtime: INF, size: EXP

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

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

(88)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_not(z0)) -> not(encArg(z0))
   encArg(cons_or(z0, z1)) -> or(encArg(z0), encArg(z1))
   encArg(cons_implies(z0, z1)) -> implies(encArg(z0), encArg(z1))
   encArg(cons_and(z0, z1)) -> and(encArg(z0), encArg(z1))
   encArg(cons_xor(z0, z1)) -> xor(encArg(z0), encArg(z1))
   encode_not(z0) -> not(encArg(z0))
   encode_xor(z0, z1) -> xor(encArg(z0), encArg(z1))
   encode_true -> true
   encode_or(z0, z1) -> or(encArg(z0), encArg(z1))
   encode_and(z0, z1) -> and(encArg(z0), encArg(z1))
   encode_implies(z0, z1) -> implies(encArg(z0), encArg(z1))
   encode_false -> false
   not(z0) -> xor(z0, true)
   or(z0, z1) -> xor(and(z0, z1), xor(z0, z1))
   implies(z0, z1) -> xor(and(z0, z1), xor(z0, true))
   and(z0, true) -> z0
   and(z0, false) -> false
   and(z0, z0) -> z0
   and(xor(z0, z1), z2) -> xor(and(z0, z2), and(z1, z2))
   xor(z0, false) -> z0
   xor(z0, z0) -> false
Tuples:
   ENCARG(true) -> c
   ENCARG(false) -> c1
   ENCARG(cons_not(z0)) -> c2(NOT(encArg(z0)), ENCARG(z0))
   ENCARG(cons_or(z0, z1)) -> c3(OR(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_implies(z0, z1)) -> c4(IMPLIES(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_and(z0, z1)) -> c5(AND(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_xor(z0, z1)) -> c6(XOR(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_NOT(z0) -> c7(NOT(encArg(z0)), ENCARG(z0))
   ENCODE_XOR(z0, z1) -> c8(XOR(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_TRUE -> c9
   ENCODE_OR(z0, z1) -> c10(OR(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_AND(z0, z1) -> c11(AND(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_IMPLIES(z0, z1) -> c12(IMPLIES(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_FALSE -> c13
   NOT(z0) -> c14(XOR(z0, true))
   OR(z0, z1) -> c15(XOR(and(z0, z1), xor(z0, z1)), AND(z0, z1), XOR(z0, z1))
   IMPLIES(z0, z1) -> c16(XOR(and(z0, z1), xor(z0, true)), AND(z0, z1), XOR(z0, true))
   AND(z0, true) -> c17
   AND(z0, false) -> c18
   AND(z0, z0) -> c19
   AND(xor(z0, z1), z2) -> c20(XOR(and(z0, z2), and(z1, z2)), AND(z0, z2), AND(z1, z2))
   XOR(z0, false) -> c21
   XOR(z0, z0) -> c22
S tuples:
   NOT(z0) -> c14(XOR(z0, true))
   OR(z0, z1) -> c15(XOR(and(z0, z1), xor(z0, z1)), AND(z0, z1), XOR(z0, z1))
   IMPLIES(z0, z1) -> c16(XOR(and(z0, z1), xor(z0, true)), AND(z0, z1), XOR(z0, true))
   AND(z0, true) -> c17
   AND(z0, false) -> c18
   AND(z0, z0) -> c19
   AND(xor(z0, z1), z2) -> c20(XOR(and(z0, z2), and(z1, z2)), AND(z0, z2), AND(z1, z2))
   XOR(z0, false) -> c21
   XOR(z0, z0) -> c22
K tuples:none
Defined Rule Symbols:   not_1, or_2, implies_2, and_2, xor_2, encArg_1, encode_not_1, encode_xor_2, encode_true, encode_or_2, encode_and_2, encode_implies_2, encode_false

Defined Pair Symbols:   ENCARG_1, ENCODE_NOT_1, ENCODE_XOR_2, ENCODE_TRUE, ENCODE_OR_2, ENCODE_AND_2, ENCODE_IMPLIES_2, ENCODE_FALSE, NOT_1, OR_2, IMPLIES_2, AND_2, XOR_2

Compound Symbols:   c, c1, c2_2, c3_3, c4_3, c5_3, c6_3, c7_2, c8_3, c9, c10_3, c11_3, c12_3, c13, c14_1, c15_3, c16_3, c17, c18, c19, c20_3, c21, c22


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

(89) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 4 trailing nodes:
   ENCARG(true) -> c
   ENCODE_TRUE -> c9
   ENCARG(false) -> c1
   ENCODE_FALSE -> c13

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

(90)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_not(z0)) -> not(encArg(z0))
   encArg(cons_or(z0, z1)) -> or(encArg(z0), encArg(z1))
   encArg(cons_implies(z0, z1)) -> implies(encArg(z0), encArg(z1))
   encArg(cons_and(z0, z1)) -> and(encArg(z0), encArg(z1))
   encArg(cons_xor(z0, z1)) -> xor(encArg(z0), encArg(z1))
   encode_not(z0) -> not(encArg(z0))
   encode_xor(z0, z1) -> xor(encArg(z0), encArg(z1))
   encode_true -> true
   encode_or(z0, z1) -> or(encArg(z0), encArg(z1))
   encode_and(z0, z1) -> and(encArg(z0), encArg(z1))
   encode_implies(z0, z1) -> implies(encArg(z0), encArg(z1))
   encode_false -> false
   not(z0) -> xor(z0, true)
   or(z0, z1) -> xor(and(z0, z1), xor(z0, z1))
   implies(z0, z1) -> xor(and(z0, z1), xor(z0, true))
   and(z0, true) -> z0
   and(z0, false) -> false
   and(z0, z0) -> z0
   and(xor(z0, z1), z2) -> xor(and(z0, z2), and(z1, z2))
   xor(z0, false) -> z0
   xor(z0, z0) -> false
Tuples:
   ENCARG(cons_not(z0)) -> c2(NOT(encArg(z0)), ENCARG(z0))
   ENCARG(cons_or(z0, z1)) -> c3(OR(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_implies(z0, z1)) -> c4(IMPLIES(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_and(z0, z1)) -> c5(AND(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_xor(z0, z1)) -> c6(XOR(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_NOT(z0) -> c7(NOT(encArg(z0)), ENCARG(z0))
   ENCODE_XOR(z0, z1) -> c8(XOR(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_OR(z0, z1) -> c10(OR(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_AND(z0, z1) -> c11(AND(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_IMPLIES(z0, z1) -> c12(IMPLIES(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   NOT(z0) -> c14(XOR(z0, true))
   OR(z0, z1) -> c15(XOR(and(z0, z1), xor(z0, z1)), AND(z0, z1), XOR(z0, z1))
   IMPLIES(z0, z1) -> c16(XOR(and(z0, z1), xor(z0, true)), AND(z0, z1), XOR(z0, true))
   AND(z0, true) -> c17
   AND(z0, false) -> c18
   AND(z0, z0) -> c19
   AND(xor(z0, z1), z2) -> c20(XOR(and(z0, z2), and(z1, z2)), AND(z0, z2), AND(z1, z2))
   XOR(z0, false) -> c21
   XOR(z0, z0) -> c22
S tuples:
   NOT(z0) -> c14(XOR(z0, true))
   OR(z0, z1) -> c15(XOR(and(z0, z1), xor(z0, z1)), AND(z0, z1), XOR(z0, z1))
   IMPLIES(z0, z1) -> c16(XOR(and(z0, z1), xor(z0, true)), AND(z0, z1), XOR(z0, true))
   AND(z0, true) -> c17
   AND(z0, false) -> c18
   AND(z0, z0) -> c19
   AND(xor(z0, z1), z2) -> c20(XOR(and(z0, z2), and(z1, z2)), AND(z0, z2), AND(z1, z2))
   XOR(z0, false) -> c21
   XOR(z0, z0) -> c22
K tuples:none
Defined Rule Symbols:   not_1, or_2, implies_2, and_2, xor_2, encArg_1, encode_not_1, encode_xor_2, encode_true, encode_or_2, encode_and_2, encode_implies_2, encode_false

Defined Pair Symbols:   ENCARG_1, ENCODE_NOT_1, ENCODE_XOR_2, ENCODE_OR_2, ENCODE_AND_2, ENCODE_IMPLIES_2, NOT_1, OR_2, IMPLIES_2, AND_2, XOR_2

Compound Symbols:   c2_2, c3_3, c4_3, c5_3, c6_3, c7_2, c8_3, c10_3, c11_3, c12_3, c14_1, c15_3, c16_3, c17, c18, c19, c20_3, c21, c22


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

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

(92)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_not(z0)) -> not(encArg(z0))
   encArg(cons_or(z0, z1)) -> or(encArg(z0), encArg(z1))
   encArg(cons_implies(z0, z1)) -> implies(encArg(z0), encArg(z1))
   encArg(cons_and(z0, z1)) -> and(encArg(z0), encArg(z1))
   encArg(cons_xor(z0, z1)) -> xor(encArg(z0), encArg(z1))
   encode_not(z0) -> not(encArg(z0))
   encode_xor(z0, z1) -> xor(encArg(z0), encArg(z1))
   encode_true -> true
   encode_or(z0, z1) -> or(encArg(z0), encArg(z1))
   encode_and(z0, z1) -> and(encArg(z0), encArg(z1))
   encode_implies(z0, z1) -> implies(encArg(z0), encArg(z1))
   encode_false -> false
   not(z0) -> xor(z0, true)
   or(z0, z1) -> xor(and(z0, z1), xor(z0, z1))
   implies(z0, z1) -> xor(and(z0, z1), xor(z0, true))
   and(z0, true) -> z0
   and(z0, false) -> false
   and(z0, z0) -> z0
   and(xor(z0, z1), z2) -> xor(and(z0, z2), and(z1, z2))
   xor(z0, false) -> z0
   xor(z0, z0) -> false
Tuples:
   ENCARG(cons_not(z0)) -> c2(NOT(encArg(z0)), ENCARG(z0))
   ENCARG(cons_or(z0, z1)) -> c3(OR(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_implies(z0, z1)) -> c4(IMPLIES(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_and(z0, z1)) -> c5(AND(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_xor(z0, z1)) -> c6(XOR(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   NOT(z0) -> c14(XOR(z0, true))
   AND(z0, true) -> c17
   AND(z0, false) -> c18
   AND(z0, z0) -> c19
   AND(xor(z0, z1), z2) -> c20(XOR(and(z0, z2), and(z1, z2)), AND(z0, z2), AND(z1, z2))
   XOR(z0, false) -> c21
   XOR(z0, z0) -> c22
   ENCODE_NOT(z0) -> c(NOT(encArg(z0)))
   ENCODE_NOT(z0) -> c(ENCARG(z0))
   ENCODE_XOR(z0, z1) -> c(XOR(encArg(z0), encArg(z1)))
   ENCODE_XOR(z0, z1) -> c(ENCARG(z0))
   ENCODE_XOR(z0, z1) -> c(ENCARG(z1))
   ENCODE_OR(z0, z1) -> c(OR(encArg(z0), encArg(z1)))
   ENCODE_OR(z0, z1) -> c(ENCARG(z0))
   ENCODE_OR(z0, z1) -> c(ENCARG(z1))
   ENCODE_AND(z0, z1) -> c(AND(encArg(z0), encArg(z1)))
   ENCODE_AND(z0, z1) -> c(ENCARG(z0))
   ENCODE_AND(z0, z1) -> c(ENCARG(z1))
   ENCODE_IMPLIES(z0, z1) -> c(IMPLIES(encArg(z0), encArg(z1)))
   ENCODE_IMPLIES(z0, z1) -> c(ENCARG(z0))
   ENCODE_IMPLIES(z0, z1) -> c(ENCARG(z1))
   OR(z0, z1) -> c(XOR(and(z0, z1), xor(z0, z1)))
   OR(z0, z1) -> c(AND(z0, z1))
   OR(z0, z1) -> c(XOR(z0, z1))
   IMPLIES(z0, z1) -> c(XOR(and(z0, z1), xor(z0, true)))
   IMPLIES(z0, z1) -> c(AND(z0, z1))
   IMPLIES(z0, z1) -> c(XOR(z0, true))
S tuples:
   NOT(z0) -> c14(XOR(z0, true))
   AND(z0, true) -> c17
   AND(z0, false) -> c18
   AND(z0, z0) -> c19
   AND(xor(z0, z1), z2) -> c20(XOR(and(z0, z2), and(z1, z2)), AND(z0, z2), AND(z1, z2))
   XOR(z0, false) -> c21
   XOR(z0, z0) -> c22
   OR(z0, z1) -> c(XOR(and(z0, z1), xor(z0, z1)))
   OR(z0, z1) -> c(AND(z0, z1))
   OR(z0, z1) -> c(XOR(z0, z1))
   IMPLIES(z0, z1) -> c(XOR(and(z0, z1), xor(z0, true)))
   IMPLIES(z0, z1) -> c(AND(z0, z1))
   IMPLIES(z0, z1) -> c(XOR(z0, true))
K tuples:none
Defined Rule Symbols:   not_1, or_2, implies_2, and_2, xor_2, encArg_1, encode_not_1, encode_xor_2, encode_true, encode_or_2, encode_and_2, encode_implies_2, encode_false

Defined Pair Symbols:   ENCARG_1, NOT_1, AND_2, XOR_2, ENCODE_NOT_1, ENCODE_XOR_2, ENCODE_OR_2, ENCODE_AND_2, ENCODE_IMPLIES_2, OR_2, IMPLIES_2

Compound Symbols:   c2_2, c3_3, c4_3, c5_3, c6_3, c14_1, c17, c18, c19, c20_3, c21, c22, c_1


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

(93) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 9 leading nodes:
   ENCODE_NOT(z0) -> c(ENCARG(z0))
   ENCODE_XOR(z0, z1) -> c(ENCARG(z0))
   ENCODE_XOR(z0, z1) -> c(ENCARG(z1))
   ENCODE_OR(z0, z1) -> c(ENCARG(z0))
   ENCODE_OR(z0, z1) -> c(ENCARG(z1))
   ENCODE_AND(z0, z1) -> c(ENCARG(z0))
   ENCODE_AND(z0, z1) -> c(ENCARG(z1))
   ENCODE_IMPLIES(z0, z1) -> c(ENCARG(z0))
   ENCODE_IMPLIES(z0, z1) -> c(ENCARG(z1))

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

(94)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_not(z0)) -> not(encArg(z0))
   encArg(cons_or(z0, z1)) -> or(encArg(z0), encArg(z1))
   encArg(cons_implies(z0, z1)) -> implies(encArg(z0), encArg(z1))
   encArg(cons_and(z0, z1)) -> and(encArg(z0), encArg(z1))
   encArg(cons_xor(z0, z1)) -> xor(encArg(z0), encArg(z1))
   encode_not(z0) -> not(encArg(z0))
   encode_xor(z0, z1) -> xor(encArg(z0), encArg(z1))
   encode_true -> true
   encode_or(z0, z1) -> or(encArg(z0), encArg(z1))
   encode_and(z0, z1) -> and(encArg(z0), encArg(z1))
   encode_implies(z0, z1) -> implies(encArg(z0), encArg(z1))
   encode_false -> false
   not(z0) -> xor(z0, true)
   or(z0, z1) -> xor(and(z0, z1), xor(z0, z1))
   implies(z0, z1) -> xor(and(z0, z1), xor(z0, true))
   and(z0, true) -> z0
   and(z0, false) -> false
   and(z0, z0) -> z0
   and(xor(z0, z1), z2) -> xor(and(z0, z2), and(z1, z2))
   xor(z0, false) -> z0
   xor(z0, z0) -> false
Tuples:
   ENCARG(cons_not(z0)) -> c2(NOT(encArg(z0)), ENCARG(z0))
   ENCARG(cons_or(z0, z1)) -> c3(OR(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_implies(z0, z1)) -> c4(IMPLIES(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_and(z0, z1)) -> c5(AND(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_xor(z0, z1)) -> c6(XOR(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   NOT(z0) -> c14(XOR(z0, true))
   AND(z0, true) -> c17
   AND(z0, false) -> c18
   AND(z0, z0) -> c19
   AND(xor(z0, z1), z2) -> c20(XOR(and(z0, z2), and(z1, z2)), AND(z0, z2), AND(z1, z2))
   XOR(z0, false) -> c21
   XOR(z0, z0) -> c22
   ENCODE_NOT(z0) -> c(NOT(encArg(z0)))
   ENCODE_XOR(z0, z1) -> c(XOR(encArg(z0), encArg(z1)))
   ENCODE_OR(z0, z1) -> c(OR(encArg(z0), encArg(z1)))
   ENCODE_AND(z0, z1) -> c(AND(encArg(z0), encArg(z1)))
   ENCODE_IMPLIES(z0, z1) -> c(IMPLIES(encArg(z0), encArg(z1)))
   OR(z0, z1) -> c(XOR(and(z0, z1), xor(z0, z1)))
   OR(z0, z1) -> c(AND(z0, z1))
   OR(z0, z1) -> c(XOR(z0, z1))
   IMPLIES(z0, z1) -> c(XOR(and(z0, z1), xor(z0, true)))
   IMPLIES(z0, z1) -> c(AND(z0, z1))
   IMPLIES(z0, z1) -> c(XOR(z0, true))
S tuples:
   NOT(z0) -> c14(XOR(z0, true))
   AND(z0, true) -> c17
   AND(z0, false) -> c18
   AND(z0, z0) -> c19
   AND(xor(z0, z1), z2) -> c20(XOR(and(z0, z2), and(z1, z2)), AND(z0, z2), AND(z1, z2))
   XOR(z0, false) -> c21
   XOR(z0, z0) -> c22
   OR(z0, z1) -> c(XOR(and(z0, z1), xor(z0, z1)))
   OR(z0, z1) -> c(AND(z0, z1))
   OR(z0, z1) -> c(XOR(z0, z1))
   IMPLIES(z0, z1) -> c(XOR(and(z0, z1), xor(z0, true)))
   IMPLIES(z0, z1) -> c(AND(z0, z1))
   IMPLIES(z0, z1) -> c(XOR(z0, true))
K tuples:none
Defined Rule Symbols:   not_1, or_2, implies_2, and_2, xor_2, encArg_1, encode_not_1, encode_xor_2, encode_true, encode_or_2, encode_and_2, encode_implies_2, encode_false

Defined Pair Symbols:   ENCARG_1, NOT_1, AND_2, XOR_2, ENCODE_NOT_1, ENCODE_XOR_2, ENCODE_OR_2, ENCODE_AND_2, ENCODE_IMPLIES_2, OR_2, IMPLIES_2

Compound Symbols:   c2_2, c3_3, c4_3, c5_3, c6_3, c14_1, c17, c18, c19, c20_3, c21, c22, c_1


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

(95) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   encode_not(z0) -> not(encArg(z0))
   encode_xor(z0, z1) -> xor(encArg(z0), encArg(z1))
   encode_true -> true
   encode_or(z0, z1) -> or(encArg(z0), encArg(z1))
   encode_and(z0, z1) -> and(encArg(z0), encArg(z1))
   encode_implies(z0, z1) -> implies(encArg(z0), encArg(z1))
   encode_false -> false

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

(96)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_not(z0)) -> not(encArg(z0))
   encArg(cons_or(z0, z1)) -> or(encArg(z0), encArg(z1))
   encArg(cons_implies(z0, z1)) -> implies(encArg(z0), encArg(z1))
   encArg(cons_and(z0, z1)) -> and(encArg(z0), encArg(z1))
   encArg(cons_xor(z0, z1)) -> xor(encArg(z0), encArg(z1))
   not(z0) -> xor(z0, true)
   or(z0, z1) -> xor(and(z0, z1), xor(z0, z1))
   implies(z0, z1) -> xor(and(z0, z1), xor(z0, true))
   and(z0, true) -> z0
   and(z0, false) -> false
   and(z0, z0) -> z0
   and(xor(z0, z1), z2) -> xor(and(z0, z2), and(z1, z2))
   xor(z0, false) -> z0
   xor(z0, z0) -> false
Tuples:
   ENCARG(cons_not(z0)) -> c2(NOT(encArg(z0)), ENCARG(z0))
   ENCARG(cons_or(z0, z1)) -> c3(OR(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_implies(z0, z1)) -> c4(IMPLIES(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_and(z0, z1)) -> c5(AND(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_xor(z0, z1)) -> c6(XOR(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   NOT(z0) -> c14(XOR(z0, true))
   AND(z0, true) -> c17
   AND(z0, false) -> c18
   AND(z0, z0) -> c19
   AND(xor(z0, z1), z2) -> c20(XOR(and(z0, z2), and(z1, z2)), AND(z0, z2), AND(z1, z2))
   XOR(z0, false) -> c21
   XOR(z0, z0) -> c22
   ENCODE_NOT(z0) -> c(NOT(encArg(z0)))
   ENCODE_XOR(z0, z1) -> c(XOR(encArg(z0), encArg(z1)))
   ENCODE_OR(z0, z1) -> c(OR(encArg(z0), encArg(z1)))
   ENCODE_AND(z0, z1) -> c(AND(encArg(z0), encArg(z1)))
   ENCODE_IMPLIES(z0, z1) -> c(IMPLIES(encArg(z0), encArg(z1)))
   OR(z0, z1) -> c(XOR(and(z0, z1), xor(z0, z1)))
   OR(z0, z1) -> c(AND(z0, z1))
   OR(z0, z1) -> c(XOR(z0, z1))
   IMPLIES(z0, z1) -> c(XOR(and(z0, z1), xor(z0, true)))
   IMPLIES(z0, z1) -> c(AND(z0, z1))
   IMPLIES(z0, z1) -> c(XOR(z0, true))
S tuples:
   NOT(z0) -> c14(XOR(z0, true))
   AND(z0, true) -> c17
   AND(z0, false) -> c18
   AND(z0, z0) -> c19
   AND(xor(z0, z1), z2) -> c20(XOR(and(z0, z2), and(z1, z2)), AND(z0, z2), AND(z1, z2))
   XOR(z0, false) -> c21
   XOR(z0, z0) -> c22
   OR(z0, z1) -> c(XOR(and(z0, z1), xor(z0, z1)))
   OR(z0, z1) -> c(AND(z0, z1))
   OR(z0, z1) -> c(XOR(z0, z1))
   IMPLIES(z0, z1) -> c(XOR(and(z0, z1), xor(z0, true)))
   IMPLIES(z0, z1) -> c(AND(z0, z1))
   IMPLIES(z0, z1) -> c(XOR(z0, true))
K tuples:none
Defined Rule Symbols:   encArg_1, not_1, or_2, implies_2, and_2, xor_2

Defined Pair Symbols:   ENCARG_1, NOT_1, AND_2, XOR_2, ENCODE_NOT_1, ENCODE_XOR_2, ENCODE_OR_2, ENCODE_AND_2, ENCODE_IMPLIES_2, OR_2, IMPLIES_2

Compound Symbols:   c2_2, c3_3, c4_3, c5_3, c6_3, c14_1, c17, c18, c19, c20_3, c21, c22, c_1


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

(97) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   NOT(z0) -> c14(XOR(z0, true))
   IMPLIES(z0, z1) -> c(XOR(and(z0, z1), xor(z0, true)))
   IMPLIES(z0, z1) -> c(AND(z0, z1))
   IMPLIES(z0, z1) -> c(XOR(z0, true))
We considered the (Usable) Rules:none
And the Tuples:
   ENCARG(cons_not(z0)) -> c2(NOT(encArg(z0)), ENCARG(z0))
   ENCARG(cons_or(z0, z1)) -> c3(OR(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_implies(z0, z1)) -> c4(IMPLIES(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_and(z0, z1)) -> c5(AND(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_xor(z0, z1)) -> c6(XOR(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   NOT(z0) -> c14(XOR(z0, true))
   AND(z0, true) -> c17
   AND(z0, false) -> c18
   AND(z0, z0) -> c19
   AND(xor(z0, z1), z2) -> c20(XOR(and(z0, z2), and(z1, z2)), AND(z0, z2), AND(z1, z2))
   XOR(z0, false) -> c21
   XOR(z0, z0) -> c22
   ENCODE_NOT(z0) -> c(NOT(encArg(z0)))
   ENCODE_XOR(z0, z1) -> c(XOR(encArg(z0), encArg(z1)))
   ENCODE_OR(z0, z1) -> c(OR(encArg(z0), encArg(z1)))
   ENCODE_AND(z0, z1) -> c(AND(encArg(z0), encArg(z1)))
   ENCODE_IMPLIES(z0, z1) -> c(IMPLIES(encArg(z0), encArg(z1)))
   OR(z0, z1) -> c(XOR(and(z0, z1), xor(z0, z1)))
   OR(z0, z1) -> c(AND(z0, z1))
   OR(z0, z1) -> c(XOR(z0, z1))
   IMPLIES(z0, z1) -> c(XOR(and(z0, z1), xor(z0, true)))
   IMPLIES(z0, z1) -> c(AND(z0, z1))
   IMPLIES(z0, z1) -> c(XOR(z0, true))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(AND(x_1, x_2)) = 0
   POL(ENCARG(x_1)) = x_1
   POL(ENCODE_AND(x_1, x_2)) = x_1
   POL(ENCODE_IMPLIES(x_1, x_2)) = [1] + x_1 + x_2
   POL(ENCODE_NOT(x_1)) = [1]
   POL(ENCODE_OR(x_1, x_2)) = 0
   POL(ENCODE_XOR(x_1, x_2)) = x_1 + x_2
   POL(IMPLIES(x_1, x_2)) = [1]
   POL(NOT(x_1)) = [1]
   POL(OR(x_1, x_2)) = 0
   POL(XOR(x_1, x_2)) = 0
   POL(and(x_1, x_2)) = 0
   POL(c(x_1)) = x_1
   POL(c14(x_1)) = x_1
   POL(c17) = 0
   POL(c18) = 0
   POL(c19) = 0
   POL(c2(x_1, x_2)) = x_1 + x_2
   POL(c20(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c21) = 0
   POL(c22) = 0
   POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c4(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c5(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c6(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(cons_and(x_1, x_2)) = [1] + x_1 + x_2
   POL(cons_implies(x_1, x_2)) = [1] + x_1 + x_2
   POL(cons_not(x_1)) = [1] + x_1
   POL(cons_or(x_1, x_2)) = [1] + x_1 + x_2
   POL(cons_xor(x_1, x_2)) = [1] + x_1 + x_2
   POL(encArg(x_1)) = [1] + x_1
   POL(false) = [1]
   POL(implies(x_1, x_2)) = [1] + x_1 + x_2
   POL(not(x_1)) = [1] + x_1
   POL(or(x_1, x_2)) = [1] + x_1 + x_2
   POL(true) = [1]
   POL(xor(x_1, x_2)) = 0

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

(98)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_not(z0)) -> not(encArg(z0))
   encArg(cons_or(z0, z1)) -> or(encArg(z0), encArg(z1))
   encArg(cons_implies(z0, z1)) -> implies(encArg(z0), encArg(z1))
   encArg(cons_and(z0, z1)) -> and(encArg(z0), encArg(z1))
   encArg(cons_xor(z0, z1)) -> xor(encArg(z0), encArg(z1))
   not(z0) -> xor(z0, true)
   or(z0, z1) -> xor(and(z0, z1), xor(z0, z1))
   implies(z0, z1) -> xor(and(z0, z1), xor(z0, true))
   and(z0, true) -> z0
   and(z0, false) -> false
   and(z0, z0) -> z0
   and(xor(z0, z1), z2) -> xor(and(z0, z2), and(z1, z2))
   xor(z0, false) -> z0
   xor(z0, z0) -> false
Tuples:
   ENCARG(cons_not(z0)) -> c2(NOT(encArg(z0)), ENCARG(z0))
   ENCARG(cons_or(z0, z1)) -> c3(OR(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_implies(z0, z1)) -> c4(IMPLIES(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_and(z0, z1)) -> c5(AND(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_xor(z0, z1)) -> c6(XOR(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   NOT(z0) -> c14(XOR(z0, true))
   AND(z0, true) -> c17
   AND(z0, false) -> c18
   AND(z0, z0) -> c19
   AND(xor(z0, z1), z2) -> c20(XOR(and(z0, z2), and(z1, z2)), AND(z0, z2), AND(z1, z2))
   XOR(z0, false) -> c21
   XOR(z0, z0) -> c22
   ENCODE_NOT(z0) -> c(NOT(encArg(z0)))
   ENCODE_XOR(z0, z1) -> c(XOR(encArg(z0), encArg(z1)))
   ENCODE_OR(z0, z1) -> c(OR(encArg(z0), encArg(z1)))
   ENCODE_AND(z0, z1) -> c(AND(encArg(z0), encArg(z1)))
   ENCODE_IMPLIES(z0, z1) -> c(IMPLIES(encArg(z0), encArg(z1)))
   OR(z0, z1) -> c(XOR(and(z0, z1), xor(z0, z1)))
   OR(z0, z1) -> c(AND(z0, z1))
   OR(z0, z1) -> c(XOR(z0, z1))
   IMPLIES(z0, z1) -> c(XOR(and(z0, z1), xor(z0, true)))
   IMPLIES(z0, z1) -> c(AND(z0, z1))
   IMPLIES(z0, z1) -> c(XOR(z0, true))
S tuples:
   AND(z0, true) -> c17
   AND(z0, false) -> c18
   AND(z0, z0) -> c19
   AND(xor(z0, z1), z2) -> c20(XOR(and(z0, z2), and(z1, z2)), AND(z0, z2), AND(z1, z2))
   XOR(z0, false) -> c21
   XOR(z0, z0) -> c22
   OR(z0, z1) -> c(XOR(and(z0, z1), xor(z0, z1)))
   OR(z0, z1) -> c(AND(z0, z1))
   OR(z0, z1) -> c(XOR(z0, z1))
K tuples:
   NOT(z0) -> c14(XOR(z0, true))
   IMPLIES(z0, z1) -> c(XOR(and(z0, z1), xor(z0, true)))
   IMPLIES(z0, z1) -> c(AND(z0, z1))
   IMPLIES(z0, z1) -> c(XOR(z0, true))
Defined Rule Symbols:   encArg_1, not_1, or_2, implies_2, and_2, xor_2

Defined Pair Symbols:   ENCARG_1, NOT_1, AND_2, XOR_2, ENCODE_NOT_1, ENCODE_XOR_2, ENCODE_OR_2, ENCODE_AND_2, ENCODE_IMPLIES_2, OR_2, IMPLIES_2

Compound Symbols:   c2_2, c3_3, c4_3, c5_3, c6_3, c14_1, c17, c18, c19, c20_3, c21, c22, c_1


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

(99) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   OR(z0, z1) -> c(XOR(and(z0, z1), xor(z0, z1)))
   OR(z0, z1) -> c(AND(z0, z1))
   OR(z0, z1) -> c(XOR(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:
   ENCARG(cons_not(z0)) -> c2(NOT(encArg(z0)), ENCARG(z0))
   ENCARG(cons_or(z0, z1)) -> c3(OR(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_implies(z0, z1)) -> c4(IMPLIES(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_and(z0, z1)) -> c5(AND(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_xor(z0, z1)) -> c6(XOR(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   NOT(z0) -> c14(XOR(z0, true))
   AND(z0, true) -> c17
   AND(z0, false) -> c18
   AND(z0, z0) -> c19
   AND(xor(z0, z1), z2) -> c20(XOR(and(z0, z2), and(z1, z2)), AND(z0, z2), AND(z1, z2))
   XOR(z0, false) -> c21
   XOR(z0, z0) -> c22
   ENCODE_NOT(z0) -> c(NOT(encArg(z0)))
   ENCODE_XOR(z0, z1) -> c(XOR(encArg(z0), encArg(z1)))
   ENCODE_OR(z0, z1) -> c(OR(encArg(z0), encArg(z1)))
   ENCODE_AND(z0, z1) -> c(AND(encArg(z0), encArg(z1)))
   ENCODE_IMPLIES(z0, z1) -> c(IMPLIES(encArg(z0), encArg(z1)))
   OR(z0, z1) -> c(XOR(and(z0, z1), xor(z0, z1)))
   OR(z0, z1) -> c(AND(z0, z1))
   OR(z0, z1) -> c(XOR(z0, z1))
   IMPLIES(z0, z1) -> c(XOR(and(z0, z1), xor(z0, true)))
   IMPLIES(z0, z1) -> c(AND(z0, z1))
   IMPLIES(z0, z1) -> c(XOR(z0, true))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(AND(x_1, x_2)) = 0
   POL(ENCARG(x_1)) = x_1^2
   POL(ENCODE_AND(x_1, x_2)) = [2] + [2]x_1 + x_2 + x_2^2 + [2]x_1*x_2 + x_1^2
   POL(ENCODE_IMPLIES(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(ENCODE_NOT(x_1)) = [2] + [2]x_1 + [2]x_1^2
   POL(ENCODE_OR(x_1, x_2)) = [2] + [2]x_1 + [2]x_2 + [2]x_2^2 + x_1*x_2 + [2]x_1^2
   POL(ENCODE_XOR(x_1, x_2)) = [1] + [2]x_2^2 + [2]x_1*x_2 + [2]x_1^2
   POL(IMPLIES(x_1, x_2)) = 0
   POL(NOT(x_1)) = 0
   POL(OR(x_1, x_2)) = [2]
   POL(XOR(x_1, x_2)) = 0
   POL(and(x_1, x_2)) = 0
   POL(c(x_1)) = x_1
   POL(c14(x_1)) = x_1
   POL(c17) = 0
   POL(c18) = 0
   POL(c19) = 0
   POL(c2(x_1, x_2)) = x_1 + x_2
   POL(c20(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c21) = 0
   POL(c22) = 0
   POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c4(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c5(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c6(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(cons_and(x_1, x_2)) = [1] + x_1 + x_2
   POL(cons_implies(x_1, x_2)) = x_1 + x_2
   POL(cons_not(x_1)) = x_1
   POL(cons_or(x_1, x_2)) = [2] + x_1 + x_2
   POL(cons_xor(x_1, x_2)) = x_1 + x_2
   POL(encArg(x_1)) = 0
   POL(false) = 0
   POL(implies(x_1, x_2)) = 0
   POL(not(x_1)) = 0
   POL(or(x_1, x_2)) = [2]x_1^2
   POL(true) = 0
   POL(xor(x_1, x_2)) = 0

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

(100)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_not(z0)) -> not(encArg(z0))
   encArg(cons_or(z0, z1)) -> or(encArg(z0), encArg(z1))
   encArg(cons_implies(z0, z1)) -> implies(encArg(z0), encArg(z1))
   encArg(cons_and(z0, z1)) -> and(encArg(z0), encArg(z1))
   encArg(cons_xor(z0, z1)) -> xor(encArg(z0), encArg(z1))
   not(z0) -> xor(z0, true)
   or(z0, z1) -> xor(and(z0, z1), xor(z0, z1))
   implies(z0, z1) -> xor(and(z0, z1), xor(z0, true))
   and(z0, true) -> z0
   and(z0, false) -> false
   and(z0, z0) -> z0
   and(xor(z0, z1), z2) -> xor(and(z0, z2), and(z1, z2))
   xor(z0, false) -> z0
   xor(z0, z0) -> false
Tuples:
   ENCARG(cons_not(z0)) -> c2(NOT(encArg(z0)), ENCARG(z0))
   ENCARG(cons_or(z0, z1)) -> c3(OR(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_implies(z0, z1)) -> c4(IMPLIES(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_and(z0, z1)) -> c5(AND(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_xor(z0, z1)) -> c6(XOR(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   NOT(z0) -> c14(XOR(z0, true))
   AND(z0, true) -> c17
   AND(z0, false) -> c18
   AND(z0, z0) -> c19
   AND(xor(z0, z1), z2) -> c20(XOR(and(z0, z2), and(z1, z2)), AND(z0, z2), AND(z1, z2))
   XOR(z0, false) -> c21
   XOR(z0, z0) -> c22
   ENCODE_NOT(z0) -> c(NOT(encArg(z0)))
   ENCODE_XOR(z0, z1) -> c(XOR(encArg(z0), encArg(z1)))
   ENCODE_OR(z0, z1) -> c(OR(encArg(z0), encArg(z1)))
   ENCODE_AND(z0, z1) -> c(AND(encArg(z0), encArg(z1)))
   ENCODE_IMPLIES(z0, z1) -> c(IMPLIES(encArg(z0), encArg(z1)))
   OR(z0, z1) -> c(XOR(and(z0, z1), xor(z0, z1)))
   OR(z0, z1) -> c(AND(z0, z1))
   OR(z0, z1) -> c(XOR(z0, z1))
   IMPLIES(z0, z1) -> c(XOR(and(z0, z1), xor(z0, true)))
   IMPLIES(z0, z1) -> c(AND(z0, z1))
   IMPLIES(z0, z1) -> c(XOR(z0, true))
S tuples:
   AND(z0, true) -> c17
   AND(z0, false) -> c18
   AND(z0, z0) -> c19
   AND(xor(z0, z1), z2) -> c20(XOR(and(z0, z2), and(z1, z2)), AND(z0, z2), AND(z1, z2))
   XOR(z0, false) -> c21
   XOR(z0, z0) -> c22
K tuples:
   NOT(z0) -> c14(XOR(z0, true))
   IMPLIES(z0, z1) -> c(XOR(and(z0, z1), xor(z0, true)))
   IMPLIES(z0, z1) -> c(AND(z0, z1))
   IMPLIES(z0, z1) -> c(XOR(z0, true))
   OR(z0, z1) -> c(XOR(and(z0, z1), xor(z0, z1)))
   OR(z0, z1) -> c(AND(z0, z1))
   OR(z0, z1) -> c(XOR(z0, z1))
Defined Rule Symbols:   encArg_1, not_1, or_2, implies_2, and_2, xor_2

Defined Pair Symbols:   ENCARG_1, NOT_1, AND_2, XOR_2, ENCODE_NOT_1, ENCODE_XOR_2, ENCODE_OR_2, ENCODE_AND_2, ENCODE_IMPLIES_2, OR_2, IMPLIES_2

Compound Symbols:   c2_2, c3_3, c4_3, c5_3, c6_3, c14_1, c17, c18, c19, c20_3, c21, c22, c_1


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

(101) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace OR(z0, z1) -> c(XOR(and(z0, z1), xor(z0, z1))) by 
   OR(z0, false) -> c(XOR(and(z0, false), z0))
   OR(z0, z0) -> c(XOR(and(z0, z0), false))
   OR(z0, true) -> c(XOR(z0, xor(z0, true)))
   OR(z0, false) -> c(XOR(false, xor(z0, false)))
   OR(z0, z0) -> c(XOR(z0, xor(z0, z0)))
   OR(xor(z0, z1), z2) -> c(XOR(xor(and(z0, z2), and(z1, z2)), xor(xor(z0, z1), z2)))

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

(102)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_not(z0)) -> not(encArg(z0))
   encArg(cons_or(z0, z1)) -> or(encArg(z0), encArg(z1))
   encArg(cons_implies(z0, z1)) -> implies(encArg(z0), encArg(z1))
   encArg(cons_and(z0, z1)) -> and(encArg(z0), encArg(z1))
   encArg(cons_xor(z0, z1)) -> xor(encArg(z0), encArg(z1))
   not(z0) -> xor(z0, true)
   or(z0, z1) -> xor(and(z0, z1), xor(z0, z1))
   implies(z0, z1) -> xor(and(z0, z1), xor(z0, true))
   and(z0, true) -> z0
   and(z0, false) -> false
   and(z0, z0) -> z0
   and(xor(z0, z1), z2) -> xor(and(z0, z2), and(z1, z2))
   xor(z0, false) -> z0
   xor(z0, z0) -> false
Tuples:
   ENCARG(cons_not(z0)) -> c2(NOT(encArg(z0)), ENCARG(z0))
   ENCARG(cons_or(z0, z1)) -> c3(OR(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_implies(z0, z1)) -> c4(IMPLIES(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_and(z0, z1)) -> c5(AND(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_xor(z0, z1)) -> c6(XOR(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   NOT(z0) -> c14(XOR(z0, true))
   AND(z0, true) -> c17
   AND(z0, false) -> c18
   AND(z0, z0) -> c19
   AND(xor(z0, z1), z2) -> c20(XOR(and(z0, z2), and(z1, z2)), AND(z0, z2), AND(z1, z2))
   XOR(z0, false) -> c21
   XOR(z0, z0) -> c22
   ENCODE_NOT(z0) -> c(NOT(encArg(z0)))
   ENCODE_XOR(z0, z1) -> c(XOR(encArg(z0), encArg(z1)))
   ENCODE_OR(z0, z1) -> c(OR(encArg(z0), encArg(z1)))
   ENCODE_AND(z0, z1) -> c(AND(encArg(z0), encArg(z1)))
   ENCODE_IMPLIES(z0, z1) -> c(IMPLIES(encArg(z0), encArg(z1)))
   OR(z0, z1) -> c(AND(z0, z1))
   OR(z0, z1) -> c(XOR(z0, z1))
   IMPLIES(z0, z1) -> c(XOR(and(z0, z1), xor(z0, true)))
   IMPLIES(z0, z1) -> c(AND(z0, z1))
   IMPLIES(z0, z1) -> c(XOR(z0, true))
   OR(z0, false) -> c(XOR(and(z0, false), z0))
   OR(z0, z0) -> c(XOR(and(z0, z0), false))
   OR(z0, true) -> c(XOR(z0, xor(z0, true)))
   OR(z0, false) -> c(XOR(false, xor(z0, false)))
   OR(z0, z0) -> c(XOR(z0, xor(z0, z0)))
   OR(xor(z0, z1), z2) -> c(XOR(xor(and(z0, z2), and(z1, z2)), xor(xor(z0, z1), z2)))
S tuples:
   AND(z0, true) -> c17
   AND(z0, false) -> c18
   AND(z0, z0) -> c19
   AND(xor(z0, z1), z2) -> c20(XOR(and(z0, z2), and(z1, z2)), AND(z0, z2), AND(z1, z2))
   XOR(z0, false) -> c21
   XOR(z0, z0) -> c22
K tuples:
   NOT(z0) -> c14(XOR(z0, true))
   IMPLIES(z0, z1) -> c(XOR(and(z0, z1), xor(z0, true)))
   IMPLIES(z0, z1) -> c(AND(z0, z1))
   IMPLIES(z0, z1) -> c(XOR(z0, true))
   OR(z0, z1) -> c(XOR(and(z0, z1), xor(z0, z1)))
   OR(z0, z1) -> c(AND(z0, z1))
   OR(z0, z1) -> c(XOR(z0, z1))
Defined Rule Symbols:   encArg_1, not_1, or_2, implies_2, and_2, xor_2

Defined Pair Symbols:   ENCARG_1, NOT_1, AND_2, XOR_2, ENCODE_NOT_1, ENCODE_XOR_2, ENCODE_OR_2, ENCODE_AND_2, ENCODE_IMPLIES_2, OR_2, IMPLIES_2

Compound Symbols:   c2_2, c3_3, c4_3, c5_3, c6_3, c14_1, c17, c18, c19, c20_3, c21, c22, c_1


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

(103) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IMPLIES(z0, z1) -> c(XOR(and(z0, z1), xor(z0, true))) by 
   IMPLIES(true, x1) -> c(XOR(and(true, x1), false))
   IMPLIES(z0, true) -> c(XOR(z0, xor(z0, true)))
   IMPLIES(z0, false) -> c(XOR(false, xor(z0, true)))
   IMPLIES(z0, z0) -> c(XOR(z0, xor(z0, true)))
   IMPLIES(xor(z0, z1), z2) -> c(XOR(xor(and(z0, z2), and(z1, z2)), xor(xor(z0, z1), true)))

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

(104)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_not(z0)) -> not(encArg(z0))
   encArg(cons_or(z0, z1)) -> or(encArg(z0), encArg(z1))
   encArg(cons_implies(z0, z1)) -> implies(encArg(z0), encArg(z1))
   encArg(cons_and(z0, z1)) -> and(encArg(z0), encArg(z1))
   encArg(cons_xor(z0, z1)) -> xor(encArg(z0), encArg(z1))
   not(z0) -> xor(z0, true)
   or(z0, z1) -> xor(and(z0, z1), xor(z0, z1))
   implies(z0, z1) -> xor(and(z0, z1), xor(z0, true))
   and(z0, true) -> z0
   and(z0, false) -> false
   and(z0, z0) -> z0
   and(xor(z0, z1), z2) -> xor(and(z0, z2), and(z1, z2))
   xor(z0, false) -> z0
   xor(z0, z0) -> false
Tuples:
   ENCARG(cons_not(z0)) -> c2(NOT(encArg(z0)), ENCARG(z0))
   ENCARG(cons_or(z0, z1)) -> c3(OR(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_implies(z0, z1)) -> c4(IMPLIES(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_and(z0, z1)) -> c5(AND(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_xor(z0, z1)) -> c6(XOR(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   NOT(z0) -> c14(XOR(z0, true))
   AND(z0, true) -> c17
   AND(z0, false) -> c18
   AND(z0, z0) -> c19
   AND(xor(z0, z1), z2) -> c20(XOR(and(z0, z2), and(z1, z2)), AND(z0, z2), AND(z1, z2))
   XOR(z0, false) -> c21
   XOR(z0, z0) -> c22
   ENCODE_NOT(z0) -> c(NOT(encArg(z0)))
   ENCODE_XOR(z0, z1) -> c(XOR(encArg(z0), encArg(z1)))
   ENCODE_OR(z0, z1) -> c(OR(encArg(z0), encArg(z1)))
   ENCODE_AND(z0, z1) -> c(AND(encArg(z0), encArg(z1)))
   ENCODE_IMPLIES(z0, z1) -> c(IMPLIES(encArg(z0), encArg(z1)))
   OR(z0, z1) -> c(AND(z0, z1))
   OR(z0, z1) -> c(XOR(z0, z1))
   IMPLIES(z0, z1) -> c(AND(z0, z1))
   IMPLIES(z0, z1) -> c(XOR(z0, true))
   OR(z0, false) -> c(XOR(and(z0, false), z0))
   OR(z0, z0) -> c(XOR(and(z0, z0), false))
   OR(z0, true) -> c(XOR(z0, xor(z0, true)))
   OR(z0, false) -> c(XOR(false, xor(z0, false)))
   OR(z0, z0) -> c(XOR(z0, xor(z0, z0)))
   OR(xor(z0, z1), z2) -> c(XOR(xor(and(z0, z2), and(z1, z2)), xor(xor(z0, z1), z2)))
   IMPLIES(true, x1) -> c(XOR(and(true, x1), false))
   IMPLIES(z0, true) -> c(XOR(z0, xor(z0, true)))
   IMPLIES(z0, false) -> c(XOR(false, xor(z0, true)))
   IMPLIES(z0, z0) -> c(XOR(z0, xor(z0, true)))
   IMPLIES(xor(z0, z1), z2) -> c(XOR(xor(and(z0, z2), and(z1, z2)), xor(xor(z0, z1), true)))
S tuples:
   AND(z0, true) -> c17
   AND(z0, false) -> c18
   AND(z0, z0) -> c19
   AND(xor(z0, z1), z2) -> c20(XOR(and(z0, z2), and(z1, z2)), AND(z0, z2), AND(z1, z2))
   XOR(z0, false) -> c21
   XOR(z0, z0) -> c22
K tuples:
   NOT(z0) -> c14(XOR(z0, true))
   IMPLIES(z0, z1) -> c(XOR(and(z0, z1), xor(z0, true)))
   IMPLIES(z0, z1) -> c(AND(z0, z1))
   IMPLIES(z0, z1) -> c(XOR(z0, true))
   OR(z0, z1) -> c(XOR(and(z0, z1), xor(z0, z1)))
   OR(z0, z1) -> c(AND(z0, z1))
   OR(z0, z1) -> c(XOR(z0, z1))
Defined Rule Symbols:   encArg_1, not_1, or_2, implies_2, and_2, xor_2

Defined Pair Symbols:   ENCARG_1, NOT_1, AND_2, XOR_2, ENCODE_NOT_1, ENCODE_XOR_2, ENCODE_OR_2, ENCODE_AND_2, ENCODE_IMPLIES_2, OR_2, IMPLIES_2

Compound Symbols:   c2_2, c3_3, c4_3, c5_3, c6_3, c14_1, c17, c18, c19, c20_3, c21, c22, c_1


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

(105) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace OR(z0, false) -> c(XOR(and(z0, false), z0)) by 
   OR(z0, false) -> c(XOR(false, z0))
   OR(false, false) -> c(XOR(false, false))
   OR(xor(z0, z1), false) -> c(XOR(xor(and(z0, false), and(z1, false)), xor(z0, z1)))

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

(106)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_not(z0)) -> not(encArg(z0))
   encArg(cons_or(z0, z1)) -> or(encArg(z0), encArg(z1))
   encArg(cons_implies(z0, z1)) -> implies(encArg(z0), encArg(z1))
   encArg(cons_and(z0, z1)) -> and(encArg(z0), encArg(z1))
   encArg(cons_xor(z0, z1)) -> xor(encArg(z0), encArg(z1))
   not(z0) -> xor(z0, true)
   or(z0, z1) -> xor(and(z0, z1), xor(z0, z1))
   implies(z0, z1) -> xor(and(z0, z1), xor(z0, true))
   and(z0, true) -> z0
   and(z0, false) -> false
   and(z0, z0) -> z0
   and(xor(z0, z1), z2) -> xor(and(z0, z2), and(z1, z2))
   xor(z0, false) -> z0
   xor(z0, z0) -> false
Tuples:
   ENCARG(cons_not(z0)) -> c2(NOT(encArg(z0)), ENCARG(z0))
   ENCARG(cons_or(z0, z1)) -> c3(OR(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_implies(z0, z1)) -> c4(IMPLIES(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_and(z0, z1)) -> c5(AND(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_xor(z0, z1)) -> c6(XOR(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   NOT(z0) -> c14(XOR(z0, true))
   AND(z0, true) -> c17
   AND(z0, false) -> c18
   AND(z0, z0) -> c19
   AND(xor(z0, z1), z2) -> c20(XOR(and(z0, z2), and(z1, z2)), AND(z0, z2), AND(z1, z2))
   XOR(z0, false) -> c21
   XOR(z0, z0) -> c22
   ENCODE_NOT(z0) -> c(NOT(encArg(z0)))
   ENCODE_XOR(z0, z1) -> c(XOR(encArg(z0), encArg(z1)))
   ENCODE_OR(z0, z1) -> c(OR(encArg(z0), encArg(z1)))
   ENCODE_AND(z0, z1) -> c(AND(encArg(z0), encArg(z1)))
   ENCODE_IMPLIES(z0, z1) -> c(IMPLIES(encArg(z0), encArg(z1)))
   OR(z0, z1) -> c(AND(z0, z1))
   OR(z0, z1) -> c(XOR(z0, z1))
   IMPLIES(z0, z1) -> c(AND(z0, z1))
   IMPLIES(z0, z1) -> c(XOR(z0, true))
   OR(z0, z0) -> c(XOR(and(z0, z0), false))
   OR(z0, true) -> c(XOR(z0, xor(z0, true)))
   OR(z0, false) -> c(XOR(false, xor(z0, false)))
   OR(z0, z0) -> c(XOR(z0, xor(z0, z0)))
   OR(xor(z0, z1), z2) -> c(XOR(xor(and(z0, z2), and(z1, z2)), xor(xor(z0, z1), z2)))
   IMPLIES(true, x1) -> c(XOR(and(true, x1), false))
   IMPLIES(z0, true) -> c(XOR(z0, xor(z0, true)))
   IMPLIES(z0, false) -> c(XOR(false, xor(z0, true)))
   IMPLIES(z0, z0) -> c(XOR(z0, xor(z0, true)))
   IMPLIES(xor(z0, z1), z2) -> c(XOR(xor(and(z0, z2), and(z1, z2)), xor(xor(z0, z1), true)))
   OR(z0, false) -> c(XOR(false, z0))
   OR(false, false) -> c(XOR(false, false))
   OR(xor(z0, z1), false) -> c(XOR(xor(and(z0, false), and(z1, false)), xor(z0, z1)))
S tuples:
   AND(z0, true) -> c17
   AND(z0, false) -> c18
   AND(z0, z0) -> c19
   AND(xor(z0, z1), z2) -> c20(XOR(and(z0, z2), and(z1, z2)), AND(z0, z2), AND(z1, z2))
   XOR(z0, false) -> c21
   XOR(z0, z0) -> c22
K tuples:
   NOT(z0) -> c14(XOR(z0, true))
   IMPLIES(z0, z1) -> c(XOR(and(z0, z1), xor(z0, true)))
   IMPLIES(z0, z1) -> c(AND(z0, z1))
   IMPLIES(z0, z1) -> c(XOR(z0, true))
   OR(z0, z1) -> c(XOR(and(z0, z1), xor(z0, z1)))
   OR(z0, z1) -> c(AND(z0, z1))
   OR(z0, z1) -> c(XOR(z0, z1))
Defined Rule Symbols:   encArg_1, not_1, or_2, implies_2, and_2, xor_2

Defined Pair Symbols:   ENCARG_1, NOT_1, AND_2, XOR_2, ENCODE_NOT_1, ENCODE_XOR_2, ENCODE_OR_2, ENCODE_AND_2, ENCODE_IMPLIES_2, OR_2, IMPLIES_2

Compound Symbols:   c2_2, c3_3, c4_3, c5_3, c6_3, c14_1, c17, c18, c19, c20_3, c21, c22, c_1


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

(107) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace OR(z0, z0) -> c(XOR(and(z0, z0), false)) by 
   OR(true, true) -> c(XOR(true, false))
   OR(false, false) -> c(XOR(false, false))
   OR(z0, z0) -> c(XOR(z0, false))
   OR(xor(z0, z1), xor(z0, z1)) -> c(XOR(xor(and(z0, xor(z0, z1)), and(z1, xor(z0, z1))), false))

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

(108)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_not(z0)) -> not(encArg(z0))
   encArg(cons_or(z0, z1)) -> or(encArg(z0), encArg(z1))
   encArg(cons_implies(z0, z1)) -> implies(encArg(z0), encArg(z1))
   encArg(cons_and(z0, z1)) -> and(encArg(z0), encArg(z1))
   encArg(cons_xor(z0, z1)) -> xor(encArg(z0), encArg(z1))
   not(z0) -> xor(z0, true)
   or(z0, z1) -> xor(and(z0, z1), xor(z0, z1))
   implies(z0, z1) -> xor(and(z0, z1), xor(z0, true))
   and(z0, true) -> z0
   and(z0, false) -> false
   and(z0, z0) -> z0
   and(xor(z0, z1), z2) -> xor(and(z0, z2), and(z1, z2))
   xor(z0, false) -> z0
   xor(z0, z0) -> false
Tuples:
   ENCARG(cons_not(z0)) -> c2(NOT(encArg(z0)), ENCARG(z0))
   ENCARG(cons_or(z0, z1)) -> c3(OR(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_implies(z0, z1)) -> c4(IMPLIES(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_and(z0, z1)) -> c5(AND(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_xor(z0, z1)) -> c6(XOR(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   NOT(z0) -> c14(XOR(z0, true))
   AND(z0, true) -> c17
   AND(z0, false) -> c18
   AND(z0, z0) -> c19
   AND(xor(z0, z1), z2) -> c20(XOR(and(z0, z2), and(z1, z2)), AND(z0, z2), AND(z1, z2))
   XOR(z0, false) -> c21
   XOR(z0, z0) -> c22
   ENCODE_NOT(z0) -> c(NOT(encArg(z0)))
   ENCODE_XOR(z0, z1) -> c(XOR(encArg(z0), encArg(z1)))
   ENCODE_OR(z0, z1) -> c(OR(encArg(z0), encArg(z1)))
   ENCODE_AND(z0, z1) -> c(AND(encArg(z0), encArg(z1)))
   ENCODE_IMPLIES(z0, z1) -> c(IMPLIES(encArg(z0), encArg(z1)))
   OR(z0, z1) -> c(AND(z0, z1))
   OR(z0, z1) -> c(XOR(z0, z1))
   IMPLIES(z0, z1) -> c(AND(z0, z1))
   IMPLIES(z0, z1) -> c(XOR(z0, true))
   OR(z0, true) -> c(XOR(z0, xor(z0, true)))
   OR(z0, false) -> c(XOR(false, xor(z0, false)))
   OR(z0, z0) -> c(XOR(z0, xor(z0, z0)))
   OR(xor(z0, z1), z2) -> c(XOR(xor(and(z0, z2), and(z1, z2)), xor(xor(z0, z1), z2)))
   IMPLIES(true, x1) -> c(XOR(and(true, x1), false))
   IMPLIES(z0, true) -> c(XOR(z0, xor(z0, true)))
   IMPLIES(z0, false) -> c(XOR(false, xor(z0, true)))
   IMPLIES(z0, z0) -> c(XOR(z0, xor(z0, true)))
   IMPLIES(xor(z0, z1), z2) -> c(XOR(xor(and(z0, z2), and(z1, z2)), xor(xor(z0, z1), true)))
   OR(z0, false) -> c(XOR(false, z0))
   OR(false, false) -> c(XOR(false, false))
   OR(xor(z0, z1), false) -> c(XOR(xor(and(z0, false), and(z1, false)), xor(z0, z1)))
   OR(true, true) -> c(XOR(true, false))
   OR(z0, z0) -> c(XOR(z0, false))
   OR(xor(z0, z1), xor(z0, z1)) -> c(XOR(xor(and(z0, xor(z0, z1)), and(z1, xor(z0, z1))), false))
S tuples:
   AND(z0, true) -> c17
   AND(z0, false) -> c18
   AND(z0, z0) -> c19
   AND(xor(z0, z1), z2) -> c20(XOR(and(z0, z2), and(z1, z2)), AND(z0, z2), AND(z1, z2))
   XOR(z0, false) -> c21
   XOR(z0, z0) -> c22
K tuples:
   NOT(z0) -> c14(XOR(z0, true))
   IMPLIES(z0, z1) -> c(XOR(and(z0, z1), xor(z0, true)))
   IMPLIES(z0, z1) -> c(AND(z0, z1))
   IMPLIES(z0, z1) -> c(XOR(z0, true))
   OR(z0, z1) -> c(XOR(and(z0, z1), xor(z0, z1)))
   OR(z0, z1) -> c(AND(z0, z1))
   OR(z0, z1) -> c(XOR(z0, z1))
Defined Rule Symbols:   encArg_1, not_1, or_2, implies_2, and_2, xor_2

Defined Pair Symbols:   ENCARG_1, NOT_1, AND_2, XOR_2, ENCODE_NOT_1, ENCODE_XOR_2, ENCODE_OR_2, ENCODE_AND_2, ENCODE_IMPLIES_2, OR_2, IMPLIES_2

Compound Symbols:   c2_2, c3_3, c4_3, c5_3, c6_3, c14_1, c17, c18, c19, c20_3, c21, c22, c_1


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

(109) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace OR(z0, true) -> c(XOR(z0, xor(z0, true))) by 
   OR(true, true) -> c(XOR(true, false))

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

(110)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_not(z0)) -> not(encArg(z0))
   encArg(cons_or(z0, z1)) -> or(encArg(z0), encArg(z1))
   encArg(cons_implies(z0, z1)) -> implies(encArg(z0), encArg(z1))
   encArg(cons_and(z0, z1)) -> and(encArg(z0), encArg(z1))
   encArg(cons_xor(z0, z1)) -> xor(encArg(z0), encArg(z1))
   not(z0) -> xor(z0, true)
   or(z0, z1) -> xor(and(z0, z1), xor(z0, z1))
   implies(z0, z1) -> xor(and(z0, z1), xor(z0, true))
   and(z0, true) -> z0
   and(z0, false) -> false
   and(z0, z0) -> z0
   and(xor(z0, z1), z2) -> xor(and(z0, z2), and(z1, z2))
   xor(z0, false) -> z0
   xor(z0, z0) -> false
Tuples:
   ENCARG(cons_not(z0)) -> c2(NOT(encArg(z0)), ENCARG(z0))
   ENCARG(cons_or(z0, z1)) -> c3(OR(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_implies(z0, z1)) -> c4(IMPLIES(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_and(z0, z1)) -> c5(AND(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_xor(z0, z1)) -> c6(XOR(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   NOT(z0) -> c14(XOR(z0, true))
   AND(z0, true) -> c17
   AND(z0, false) -> c18
   AND(z0, z0) -> c19
   AND(xor(z0, z1), z2) -> c20(XOR(and(z0, z2), and(z1, z2)), AND(z0, z2), AND(z1, z2))
   XOR(z0, false) -> c21
   XOR(z0, z0) -> c22
   ENCODE_NOT(z0) -> c(NOT(encArg(z0)))
   ENCODE_XOR(z0, z1) -> c(XOR(encArg(z0), encArg(z1)))
   ENCODE_OR(z0, z1) -> c(OR(encArg(z0), encArg(z1)))
   ENCODE_AND(z0, z1) -> c(AND(encArg(z0), encArg(z1)))
   ENCODE_IMPLIES(z0, z1) -> c(IMPLIES(encArg(z0), encArg(z1)))
   OR(z0, z1) -> c(AND(z0, z1))
   OR(z0, z1) -> c(XOR(z0, z1))
   IMPLIES(z0, z1) -> c(AND(z0, z1))
   IMPLIES(z0, z1) -> c(XOR(z0, true))
   OR(z0, false) -> c(XOR(false, xor(z0, false)))
   OR(z0, z0) -> c(XOR(z0, xor(z0, z0)))
   OR(xor(z0, z1), z2) -> c(XOR(xor(and(z0, z2), and(z1, z2)), xor(xor(z0, z1), z2)))
   IMPLIES(true, x1) -> c(XOR(and(true, x1), false))
   IMPLIES(z0, true) -> c(XOR(z0, xor(z0, true)))
   IMPLIES(z0, false) -> c(XOR(false, xor(z0, true)))
   IMPLIES(z0, z0) -> c(XOR(z0, xor(z0, true)))
   IMPLIES(xor(z0, z1), z2) -> c(XOR(xor(and(z0, z2), and(z1, z2)), xor(xor(z0, z1), true)))
   OR(z0, false) -> c(XOR(false, z0))
   OR(false, false) -> c(XOR(false, false))
   OR(xor(z0, z1), false) -> c(XOR(xor(and(z0, false), and(z1, false)), xor(z0, z1)))
   OR(true, true) -> c(XOR(true, false))
   OR(z0, z0) -> c(XOR(z0, false))
   OR(xor(z0, z1), xor(z0, z1)) -> c(XOR(xor(and(z0, xor(z0, z1)), and(z1, xor(z0, z1))), false))
S tuples:
   AND(z0, true) -> c17
   AND(z0, false) -> c18
   AND(z0, z0) -> c19
   AND(xor(z0, z1), z2) -> c20(XOR(and(z0, z2), and(z1, z2)), AND(z0, z2), AND(z1, z2))
   XOR(z0, false) -> c21
   XOR(z0, z0) -> c22
K tuples:
   NOT(z0) -> c14(XOR(z0, true))
   IMPLIES(z0, z1) -> c(XOR(and(z0, z1), xor(z0, true)))
   IMPLIES(z0, z1) -> c(AND(z0, z1))
   IMPLIES(z0, z1) -> c(XOR(z0, true))
   OR(z0, z1) -> c(XOR(and(z0, z1), xor(z0, z1)))
   OR(z0, z1) -> c(AND(z0, z1))
   OR(z0, z1) -> c(XOR(z0, z1))
Defined Rule Symbols:   encArg_1, not_1, or_2, implies_2, and_2, xor_2

Defined Pair Symbols:   ENCARG_1, NOT_1, AND_2, XOR_2, ENCODE_NOT_1, ENCODE_XOR_2, ENCODE_OR_2, ENCODE_AND_2, ENCODE_IMPLIES_2, OR_2, IMPLIES_2

Compound Symbols:   c2_2, c3_3, c4_3, c5_3, c6_3, c14_1, c17, c18, c19, c20_3, c21, c22, c_1


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

(111) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace OR(z0, false) -> c(XOR(false, xor(z0, false))) by 
   OR(z0, false) -> c(XOR(false, z0))
   OR(false, false) -> c(XOR(false, false))

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

(112)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_not(z0)) -> not(encArg(z0))
   encArg(cons_or(z0, z1)) -> or(encArg(z0), encArg(z1))
   encArg(cons_implies(z0, z1)) -> implies(encArg(z0), encArg(z1))
   encArg(cons_and(z0, z1)) -> and(encArg(z0), encArg(z1))
   encArg(cons_xor(z0, z1)) -> xor(encArg(z0), encArg(z1))
   not(z0) -> xor(z0, true)
   or(z0, z1) -> xor(and(z0, z1), xor(z0, z1))
   implies(z0, z1) -> xor(and(z0, z1), xor(z0, true))
   and(z0, true) -> z0
   and(z0, false) -> false
   and(z0, z0) -> z0
   and(xor(z0, z1), z2) -> xor(and(z0, z2), and(z1, z2))
   xor(z0, false) -> z0
   xor(z0, z0) -> false
Tuples:
   ENCARG(cons_not(z0)) -> c2(NOT(encArg(z0)), ENCARG(z0))
   ENCARG(cons_or(z0, z1)) -> c3(OR(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_implies(z0, z1)) -> c4(IMPLIES(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_and(z0, z1)) -> c5(AND(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_xor(z0, z1)) -> c6(XOR(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   NOT(z0) -> c14(XOR(z0, true))
   AND(z0, true) -> c17
   AND(z0, false) -> c18
   AND(z0, z0) -> c19
   AND(xor(z0, z1), z2) -> c20(XOR(and(z0, z2), and(z1, z2)), AND(z0, z2), AND(z1, z2))
   XOR(z0, false) -> c21
   XOR(z0, z0) -> c22
   ENCODE_NOT(z0) -> c(NOT(encArg(z0)))
   ENCODE_XOR(z0, z1) -> c(XOR(encArg(z0), encArg(z1)))
   ENCODE_OR(z0, z1) -> c(OR(encArg(z0), encArg(z1)))
   ENCODE_AND(z0, z1) -> c(AND(encArg(z0), encArg(z1)))
   ENCODE_IMPLIES(z0, z1) -> c(IMPLIES(encArg(z0), encArg(z1)))
   OR(z0, z1) -> c(AND(z0, z1))
   OR(z0, z1) -> c(XOR(z0, z1))
   IMPLIES(z0, z1) -> c(AND(z0, z1))
   IMPLIES(z0, z1) -> c(XOR(z0, true))
   OR(z0, z0) -> c(XOR(z0, xor(z0, z0)))
   OR(xor(z0, z1), z2) -> c(XOR(xor(and(z0, z2), and(z1, z2)), xor(xor(z0, z1), z2)))
   IMPLIES(true, x1) -> c(XOR(and(true, x1), false))
   IMPLIES(z0, true) -> c(XOR(z0, xor(z0, true)))
   IMPLIES(z0, false) -> c(XOR(false, xor(z0, true)))
   IMPLIES(z0, z0) -> c(XOR(z0, xor(z0, true)))
   IMPLIES(xor(z0, z1), z2) -> c(XOR(xor(and(z0, z2), and(z1, z2)), xor(xor(z0, z1), true)))
   OR(z0, false) -> c(XOR(false, z0))
   OR(false, false) -> c(XOR(false, false))
   OR(xor(z0, z1), false) -> c(XOR(xor(and(z0, false), and(z1, false)), xor(z0, z1)))
   OR(true, true) -> c(XOR(true, false))
   OR(z0, z0) -> c(XOR(z0, false))
   OR(xor(z0, z1), xor(z0, z1)) -> c(XOR(xor(and(z0, xor(z0, z1)), and(z1, xor(z0, z1))), false))
S tuples:
   AND(z0, true) -> c17
   AND(z0, false) -> c18
   AND(z0, z0) -> c19
   AND(xor(z0, z1), z2) -> c20(XOR(and(z0, z2), and(z1, z2)), AND(z0, z2), AND(z1, z2))
   XOR(z0, false) -> c21
   XOR(z0, z0) -> c22
K tuples:
   NOT(z0) -> c14(XOR(z0, true))
   IMPLIES(z0, z1) -> c(XOR(and(z0, z1), xor(z0, true)))
   IMPLIES(z0, z1) -> c(AND(z0, z1))
   IMPLIES(z0, z1) -> c(XOR(z0, true))
   OR(z0, z1) -> c(XOR(and(z0, z1), xor(z0, z1)))
   OR(z0, z1) -> c(AND(z0, z1))
   OR(z0, z1) -> c(XOR(z0, z1))
Defined Rule Symbols:   encArg_1, not_1, or_2, implies_2, and_2, xor_2

Defined Pair Symbols:   ENCARG_1, NOT_1, AND_2, XOR_2, ENCODE_NOT_1, ENCODE_XOR_2, ENCODE_OR_2, ENCODE_AND_2, ENCODE_IMPLIES_2, OR_2, IMPLIES_2

Compound Symbols:   c2_2, c3_3, c4_3, c5_3, c6_3, c14_1, c17, c18, c19, c20_3, c21, c22, c_1


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

(113) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace OR(z0, z0) -> c(XOR(z0, xor(z0, z0))) by 
   OR(false, false) -> c(XOR(false, false))
   OR(z0, z0) -> c(XOR(z0, false))

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

(114)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_not(z0)) -> not(encArg(z0))
   encArg(cons_or(z0, z1)) -> or(encArg(z0), encArg(z1))
   encArg(cons_implies(z0, z1)) -> implies(encArg(z0), encArg(z1))
   encArg(cons_and(z0, z1)) -> and(encArg(z0), encArg(z1))
   encArg(cons_xor(z0, z1)) -> xor(encArg(z0), encArg(z1))
   not(z0) -> xor(z0, true)
   or(z0, z1) -> xor(and(z0, z1), xor(z0, z1))
   implies(z0, z1) -> xor(and(z0, z1), xor(z0, true))
   and(z0, true) -> z0
   and(z0, false) -> false
   and(z0, z0) -> z0
   and(xor(z0, z1), z2) -> xor(and(z0, z2), and(z1, z2))
   xor(z0, false) -> z0
   xor(z0, z0) -> false
Tuples:
   ENCARG(cons_not(z0)) -> c2(NOT(encArg(z0)), ENCARG(z0))
   ENCARG(cons_or(z0, z1)) -> c3(OR(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_implies(z0, z1)) -> c4(IMPLIES(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_and(z0, z1)) -> c5(AND(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_xor(z0, z1)) -> c6(XOR(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   NOT(z0) -> c14(XOR(z0, true))
   AND(z0, true) -> c17
   AND(z0, false) -> c18
   AND(z0, z0) -> c19
   AND(xor(z0, z1), z2) -> c20(XOR(and(z0, z2), and(z1, z2)), AND(z0, z2), AND(z1, z2))
   XOR(z0, false) -> c21
   XOR(z0, z0) -> c22
   ENCODE_NOT(z0) -> c(NOT(encArg(z0)))
   ENCODE_XOR(z0, z1) -> c(XOR(encArg(z0), encArg(z1)))
   ENCODE_OR(z0, z1) -> c(OR(encArg(z0), encArg(z1)))
   ENCODE_AND(z0, z1) -> c(AND(encArg(z0), encArg(z1)))
   ENCODE_IMPLIES(z0, z1) -> c(IMPLIES(encArg(z0), encArg(z1)))
   OR(z0, z1) -> c(AND(z0, z1))
   OR(z0, z1) -> c(XOR(z0, z1))
   IMPLIES(z0, z1) -> c(AND(z0, z1))
   IMPLIES(z0, z1) -> c(XOR(z0, true))
   OR(xor(z0, z1), z2) -> c(XOR(xor(and(z0, z2), and(z1, z2)), xor(xor(z0, z1), z2)))
   IMPLIES(true, x1) -> c(XOR(and(true, x1), false))
   IMPLIES(z0, true) -> c(XOR(z0, xor(z0, true)))
   IMPLIES(z0, false) -> c(XOR(false, xor(z0, true)))
   IMPLIES(z0, z0) -> c(XOR(z0, xor(z0, true)))
   IMPLIES(xor(z0, z1), z2) -> c(XOR(xor(and(z0, z2), and(z1, z2)), xor(xor(z0, z1), true)))
   OR(z0, false) -> c(XOR(false, z0))
   OR(false, false) -> c(XOR(false, false))
   OR(xor(z0, z1), false) -> c(XOR(xor(and(z0, false), and(z1, false)), xor(z0, z1)))
   OR(true, true) -> c(XOR(true, false))
   OR(z0, z0) -> c(XOR(z0, false))
   OR(xor(z0, z1), xor(z0, z1)) -> c(XOR(xor(and(z0, xor(z0, z1)), and(z1, xor(z0, z1))), false))
S tuples:
   AND(z0, true) -> c17
   AND(z0, false) -> c18
   AND(z0, z0) -> c19
   AND(xor(z0, z1), z2) -> c20(XOR(and(z0, z2), and(z1, z2)), AND(z0, z2), AND(z1, z2))
   XOR(z0, false) -> c21
   XOR(z0, z0) -> c22
K tuples:
   NOT(z0) -> c14(XOR(z0, true))
   IMPLIES(z0, z1) -> c(XOR(and(z0, z1), xor(z0, true)))
   IMPLIES(z0, z1) -> c(AND(z0, z1))
   IMPLIES(z0, z1) -> c(XOR(z0, true))
   OR(z0, z1) -> c(XOR(and(z0, z1), xor(z0, z1)))
   OR(z0, z1) -> c(AND(z0, z1))
   OR(z0, z1) -> c(XOR(z0, z1))
Defined Rule Symbols:   encArg_1, not_1, or_2, implies_2, and_2, xor_2

Defined Pair Symbols:   ENCARG_1, NOT_1, AND_2, XOR_2, ENCODE_NOT_1, ENCODE_XOR_2, ENCODE_OR_2, ENCODE_AND_2, ENCODE_IMPLIES_2, OR_2, IMPLIES_2

Compound Symbols:   c2_2, c3_3, c4_3, c5_3, c6_3, c14_1, c17, c18, c19, c20_3, c21, c22, c_1


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

(115) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace OR(xor(z0, z1), z2) -> c(XOR(xor(and(z0, z2), and(z1, z2)), xor(xor(z0, z1), z2))) by 
   OR(xor(x0, x1), false) -> c(XOR(xor(and(x0, false), and(x1, false)), xor(x0, x1)))
   OR(xor(x0, x1), xor(x0, x1)) -> c(XOR(xor(and(x0, xor(x0, x1)), and(x1, xor(x0, x1))), false))
   OR(xor(x0, z0), true) -> c(XOR(xor(and(x0, true), z0), xor(xor(x0, z0), true)))
   OR(xor(x0, z0), false) -> c(XOR(xor(and(x0, false), false), xor(xor(x0, z0), false)))
   OR(xor(x0, z0), z0) -> c(XOR(xor(and(x0, z0), z0), xor(xor(x0, z0), z0)))
   OR(xor(x0, xor(z0, z1)), z2) -> c(XOR(xor(and(x0, z2), xor(and(z0, z2), and(z1, z2))), xor(xor(x0, xor(z0, z1)), z2)))
   OR(xor(z0, x1), true) -> c(XOR(xor(z0, and(x1, true)), xor(xor(z0, x1), true)))
   OR(xor(z0, x1), false) -> c(XOR(xor(false, and(x1, false)), xor(xor(z0, x1), false)))
   OR(xor(z0, x1), z0) -> c(XOR(xor(z0, and(x1, z0)), xor(xor(z0, x1), z0)))
   OR(xor(xor(z0, z1), x1), z2) -> c(XOR(xor(xor(and(z0, z2), and(z1, z2)), and(x1, z2)), xor(xor(xor(z0, z1), x1), z2)))

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

(116)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_not(z0)) -> not(encArg(z0))
   encArg(cons_or(z0, z1)) -> or(encArg(z0), encArg(z1))
   encArg(cons_implies(z0, z1)) -> implies(encArg(z0), encArg(z1))
   encArg(cons_and(z0, z1)) -> and(encArg(z0), encArg(z1))
   encArg(cons_xor(z0, z1)) -> xor(encArg(z0), encArg(z1))
   not(z0) -> xor(z0, true)
   or(z0, z1) -> xor(and(z0, z1), xor(z0, z1))
   implies(z0, z1) -> xor(and(z0, z1), xor(z0, true))
   and(z0, true) -> z0
   and(z0, false) -> false
   and(z0, z0) -> z0
   and(xor(z0, z1), z2) -> xor(and(z0, z2), and(z1, z2))
   xor(z0, false) -> z0
   xor(z0, z0) -> false
Tuples:
   ENCARG(cons_not(z0)) -> c2(NOT(encArg(z0)), ENCARG(z0))
   ENCARG(cons_or(z0, z1)) -> c3(OR(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_implies(z0, z1)) -> c4(IMPLIES(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_and(z0, z1)) -> c5(AND(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_xor(z0, z1)) -> c6(XOR(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   NOT(z0) -> c14(XOR(z0, true))
   AND(z0, true) -> c17
   AND(z0, false) -> c18
   AND(z0, z0) -> c19
   AND(xor(z0, z1), z2) -> c20(XOR(and(z0, z2), and(z1, z2)), AND(z0, z2), AND(z1, z2))
   XOR(z0, false) -> c21
   XOR(z0, z0) -> c22
   ENCODE_NOT(z0) -> c(NOT(encArg(z0)))
   ENCODE_XOR(z0, z1) -> c(XOR(encArg(z0), encArg(z1)))
   ENCODE_OR(z0, z1) -> c(OR(encArg(z0), encArg(z1)))
   ENCODE_AND(z0, z1) -> c(AND(encArg(z0), encArg(z1)))
   ENCODE_IMPLIES(z0, z1) -> c(IMPLIES(encArg(z0), encArg(z1)))
   OR(z0, z1) -> c(AND(z0, z1))
   OR(z0, z1) -> c(XOR(z0, z1))
   IMPLIES(z0, z1) -> c(AND(z0, z1))
   IMPLIES(z0, z1) -> c(XOR(z0, true))
   IMPLIES(true, x1) -> c(XOR(and(true, x1), false))
   IMPLIES(z0, true) -> c(XOR(z0, xor(z0, true)))
   IMPLIES(z0, false) -> c(XOR(false, xor(z0, true)))
   IMPLIES(z0, z0) -> c(XOR(z0, xor(z0, true)))
   IMPLIES(xor(z0, z1), z2) -> c(XOR(xor(and(z0, z2), and(z1, z2)), xor(xor(z0, z1), true)))
   OR(z0, false) -> c(XOR(false, z0))
   OR(false, false) -> c(XOR(false, false))
   OR(xor(z0, z1), false) -> c(XOR(xor(and(z0, false), and(z1, false)), xor(z0, z1)))
   OR(true, true) -> c(XOR(true, false))
   OR(z0, z0) -> c(XOR(z0, false))
   OR(xor(z0, z1), xor(z0, z1)) -> c(XOR(xor(and(z0, xor(z0, z1)), and(z1, xor(z0, z1))), false))
   OR(xor(x0, z0), true) -> c(XOR(xor(and(x0, true), z0), xor(xor(x0, z0), true)))
   OR(xor(x0, z0), false) -> c(XOR(xor(and(x0, false), false), xor(xor(x0, z0), false)))
   OR(xor(x0, z0), z0) -> c(XOR(xor(and(x0, z0), z0), xor(xor(x0, z0), z0)))
   OR(xor(x0, xor(z0, z1)), z2) -> c(XOR(xor(and(x0, z2), xor(and(z0, z2), and(z1, z2))), xor(xor(x0, xor(z0, z1)), z2)))
   OR(xor(z0, x1), true) -> c(XOR(xor(z0, and(x1, true)), xor(xor(z0, x1), true)))
   OR(xor(z0, x1), false) -> c(XOR(xor(false, and(x1, false)), xor(xor(z0, x1), false)))
   OR(xor(z0, x1), z0) -> c(XOR(xor(z0, and(x1, z0)), xor(xor(z0, x1), z0)))
   OR(xor(xor(z0, z1), x1), z2) -> c(XOR(xor(xor(and(z0, z2), and(z1, z2)), and(x1, z2)), xor(xor(xor(z0, z1), x1), z2)))
S tuples:
   AND(z0, true) -> c17
   AND(z0, false) -> c18
   AND(z0, z0) -> c19
   AND(xor(z0, z1), z2) -> c20(XOR(and(z0, z2), and(z1, z2)), AND(z0, z2), AND(z1, z2))
   XOR(z0, false) -> c21
   XOR(z0, z0) -> c22
K tuples:
   NOT(z0) -> c14(XOR(z0, true))
   IMPLIES(z0, z1) -> c(XOR(and(z0, z1), xor(z0, true)))
   IMPLIES(z0, z1) -> c(AND(z0, z1))
   IMPLIES(z0, z1) -> c(XOR(z0, true))
   OR(z0, z1) -> c(XOR(and(z0, z1), xor(z0, z1)))
   OR(z0, z1) -> c(AND(z0, z1))
   OR(z0, z1) -> c(XOR(z0, z1))
Defined Rule Symbols:   encArg_1, not_1, or_2, implies_2, and_2, xor_2

Defined Pair Symbols:   ENCARG_1, NOT_1, AND_2, XOR_2, ENCODE_NOT_1, ENCODE_XOR_2, ENCODE_OR_2, ENCODE_AND_2, ENCODE_IMPLIES_2, OR_2, IMPLIES_2

Compound Symbols:   c2_2, c3_3, c4_3, c5_3, c6_3, c14_1, c17, c18, c19, c20_3, c21, c22, c_1


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

(117) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IMPLIES(z0, true) -> c(XOR(z0, xor(z0, true))) by 
   IMPLIES(true, true) -> c(XOR(true, false))

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

(118)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_not(z0)) -> not(encArg(z0))
   encArg(cons_or(z0, z1)) -> or(encArg(z0), encArg(z1))
   encArg(cons_implies(z0, z1)) -> implies(encArg(z0), encArg(z1))
   encArg(cons_and(z0, z1)) -> and(encArg(z0), encArg(z1))
   encArg(cons_xor(z0, z1)) -> xor(encArg(z0), encArg(z1))
   not(z0) -> xor(z0, true)
   or(z0, z1) -> xor(and(z0, z1), xor(z0, z1))
   implies(z0, z1) -> xor(and(z0, z1), xor(z0, true))
   and(z0, true) -> z0
   and(z0, false) -> false
   and(z0, z0) -> z0
   and(xor(z0, z1), z2) -> xor(and(z0, z2), and(z1, z2))
   xor(z0, false) -> z0
   xor(z0, z0) -> false
Tuples:
   ENCARG(cons_not(z0)) -> c2(NOT(encArg(z0)), ENCARG(z0))
   ENCARG(cons_or(z0, z1)) -> c3(OR(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_implies(z0, z1)) -> c4(IMPLIES(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_and(z0, z1)) -> c5(AND(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_xor(z0, z1)) -> c6(XOR(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   NOT(z0) -> c14(XOR(z0, true))
   AND(z0, true) -> c17
   AND(z0, false) -> c18
   AND(z0, z0) -> c19
   AND(xor(z0, z1), z2) -> c20(XOR(and(z0, z2), and(z1, z2)), AND(z0, z2), AND(z1, z2))
   XOR(z0, false) -> c21
   XOR(z0, z0) -> c22
   ENCODE_NOT(z0) -> c(NOT(encArg(z0)))
   ENCODE_XOR(z0, z1) -> c(XOR(encArg(z0), encArg(z1)))
   ENCODE_OR(z0, z1) -> c(OR(encArg(z0), encArg(z1)))
   ENCODE_AND(z0, z1) -> c(AND(encArg(z0), encArg(z1)))
   ENCODE_IMPLIES(z0, z1) -> c(IMPLIES(encArg(z0), encArg(z1)))
   OR(z0, z1) -> c(AND(z0, z1))
   OR(z0, z1) -> c(XOR(z0, z1))
   IMPLIES(z0, z1) -> c(AND(z0, z1))
   IMPLIES(z0, z1) -> c(XOR(z0, true))
   IMPLIES(true, x1) -> c(XOR(and(true, x1), false))
   IMPLIES(z0, false) -> c(XOR(false, xor(z0, true)))
   IMPLIES(z0, z0) -> c(XOR(z0, xor(z0, true)))
   IMPLIES(xor(z0, z1), z2) -> c(XOR(xor(and(z0, z2), and(z1, z2)), xor(xor(z0, z1), true)))
   OR(z0, false) -> c(XOR(false, z0))
   OR(false, false) -> c(XOR(false, false))
   OR(xor(z0, z1), false) -> c(XOR(xor(and(z0, false), and(z1, false)), xor(z0, z1)))
   OR(true, true) -> c(XOR(true, false))
   OR(z0, z0) -> c(XOR(z0, false))
   OR(xor(z0, z1), xor(z0, z1)) -> c(XOR(xor(and(z0, xor(z0, z1)), and(z1, xor(z0, z1))), false))
   OR(xor(x0, z0), true) -> c(XOR(xor(and(x0, true), z0), xor(xor(x0, z0), true)))
   OR(xor(x0, z0), false) -> c(XOR(xor(and(x0, false), false), xor(xor(x0, z0), false)))
   OR(xor(x0, z0), z0) -> c(XOR(xor(and(x0, z0), z0), xor(xor(x0, z0), z0)))
   OR(xor(x0, xor(z0, z1)), z2) -> c(XOR(xor(and(x0, z2), xor(and(z0, z2), and(z1, z2))), xor(xor(x0, xor(z0, z1)), z2)))
   OR(xor(z0, x1), true) -> c(XOR(xor(z0, and(x1, true)), xor(xor(z0, x1), true)))
   OR(xor(z0, x1), false) -> c(XOR(xor(false, and(x1, false)), xor(xor(z0, x1), false)))
   OR(xor(z0, x1), z0) -> c(XOR(xor(z0, and(x1, z0)), xor(xor(z0, x1), z0)))
   OR(xor(xor(z0, z1), x1), z2) -> c(XOR(xor(xor(and(z0, z2), and(z1, z2)), and(x1, z2)), xor(xor(xor(z0, z1), x1), z2)))
   IMPLIES(true, true) -> c(XOR(true, false))
S tuples:
   AND(z0, true) -> c17
   AND(z0, false) -> c18
   AND(z0, z0) -> c19
   AND(xor(z0, z1), z2) -> c20(XOR(and(z0, z2), and(z1, z2)), AND(z0, z2), AND(z1, z2))
   XOR(z0, false) -> c21
   XOR(z0, z0) -> c22
K tuples:
   NOT(z0) -> c14(XOR(z0, true))
   IMPLIES(z0, z1) -> c(XOR(and(z0, z1), xor(z0, true)))
   IMPLIES(z0, z1) -> c(AND(z0, z1))
   IMPLIES(z0, z1) -> c(XOR(z0, true))
   OR(z0, z1) -> c(XOR(and(z0, z1), xor(z0, z1)))
   OR(z0, z1) -> c(AND(z0, z1))
   OR(z0, z1) -> c(XOR(z0, z1))
Defined Rule Symbols:   encArg_1, not_1, or_2, implies_2, and_2, xor_2

Defined Pair Symbols:   ENCARG_1, NOT_1, AND_2, XOR_2, ENCODE_NOT_1, ENCODE_XOR_2, ENCODE_OR_2, ENCODE_AND_2, ENCODE_IMPLIES_2, OR_2, IMPLIES_2

Compound Symbols:   c2_2, c3_3, c4_3, c5_3, c6_3, c14_1, c17, c18, c19, c20_3, c21, c22, c_1


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

(119) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IMPLIES(z0, false) -> c(XOR(false, xor(z0, true))) by 
   IMPLIES(true, false) -> c(XOR(false, false))

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

(120)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_not(z0)) -> not(encArg(z0))
   encArg(cons_or(z0, z1)) -> or(encArg(z0), encArg(z1))
   encArg(cons_implies(z0, z1)) -> implies(encArg(z0), encArg(z1))
   encArg(cons_and(z0, z1)) -> and(encArg(z0), encArg(z1))
   encArg(cons_xor(z0, z1)) -> xor(encArg(z0), encArg(z1))
   not(z0) -> xor(z0, true)
   or(z0, z1) -> xor(and(z0, z1), xor(z0, z1))
   implies(z0, z1) -> xor(and(z0, z1), xor(z0, true))
   and(z0, true) -> z0
   and(z0, false) -> false
   and(z0, z0) -> z0
   and(xor(z0, z1), z2) -> xor(and(z0, z2), and(z1, z2))
   xor(z0, false) -> z0
   xor(z0, z0) -> false
Tuples:
   ENCARG(cons_not(z0)) -> c2(NOT(encArg(z0)), ENCARG(z0))
   ENCARG(cons_or(z0, z1)) -> c3(OR(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_implies(z0, z1)) -> c4(IMPLIES(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_and(z0, z1)) -> c5(AND(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_xor(z0, z1)) -> c6(XOR(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   NOT(z0) -> c14(XOR(z0, true))
   AND(z0, true) -> c17
   AND(z0, false) -> c18
   AND(z0, z0) -> c19
   AND(xor(z0, z1), z2) -> c20(XOR(and(z0, z2), and(z1, z2)), AND(z0, z2), AND(z1, z2))
   XOR(z0, false) -> c21
   XOR(z0, z0) -> c22
   ENCODE_NOT(z0) -> c(NOT(encArg(z0)))
   ENCODE_XOR(z0, z1) -> c(XOR(encArg(z0), encArg(z1)))
   ENCODE_OR(z0, z1) -> c(OR(encArg(z0), encArg(z1)))
   ENCODE_AND(z0, z1) -> c(AND(encArg(z0), encArg(z1)))
   ENCODE_IMPLIES(z0, z1) -> c(IMPLIES(encArg(z0), encArg(z1)))
   OR(z0, z1) -> c(AND(z0, z1))
   OR(z0, z1) -> c(XOR(z0, z1))
   IMPLIES(z0, z1) -> c(AND(z0, z1))
   IMPLIES(z0, z1) -> c(XOR(z0, true))
   IMPLIES(true, x1) -> c(XOR(and(true, x1), false))
   IMPLIES(z0, z0) -> c(XOR(z0, xor(z0, true)))
   IMPLIES(xor(z0, z1), z2) -> c(XOR(xor(and(z0, z2), and(z1, z2)), xor(xor(z0, z1), true)))
   OR(z0, false) -> c(XOR(false, z0))
   OR(false, false) -> c(XOR(false, false))
   OR(xor(z0, z1), false) -> c(XOR(xor(and(z0, false), and(z1, false)), xor(z0, z1)))
   OR(true, true) -> c(XOR(true, false))
   OR(z0, z0) -> c(XOR(z0, false))
   OR(xor(z0, z1), xor(z0, z1)) -> c(XOR(xor(and(z0, xor(z0, z1)), and(z1, xor(z0, z1))), false))
   OR(xor(x0, z0), true) -> c(XOR(xor(and(x0, true), z0), xor(xor(x0, z0), true)))
   OR(xor(x0, z0), false) -> c(XOR(xor(and(x0, false), false), xor(xor(x0, z0), false)))
   OR(xor(x0, z0), z0) -> c(XOR(xor(and(x0, z0), z0), xor(xor(x0, z0), z0)))
   OR(xor(x0, xor(z0, z1)), z2) -> c(XOR(xor(and(x0, z2), xor(and(z0, z2), and(z1, z2))), xor(xor(x0, xor(z0, z1)), z2)))
   OR(xor(z0, x1), true) -> c(XOR(xor(z0, and(x1, true)), xor(xor(z0, x1), true)))
   OR(xor(z0, x1), false) -> c(XOR(xor(false, and(x1, false)), xor(xor(z0, x1), false)))
   OR(xor(z0, x1), z0) -> c(XOR(xor(z0, and(x1, z0)), xor(xor(z0, x1), z0)))
   OR(xor(xor(z0, z1), x1), z2) -> c(XOR(xor(xor(and(z0, z2), and(z1, z2)), and(x1, z2)), xor(xor(xor(z0, z1), x1), z2)))
   IMPLIES(true, true) -> c(XOR(true, false))
   IMPLIES(true, false) -> c(XOR(false, false))
S tuples:
   AND(z0, true) -> c17
   AND(z0, false) -> c18
   AND(z0, z0) -> c19
   AND(xor(z0, z1), z2) -> c20(XOR(and(z0, z2), and(z1, z2)), AND(z0, z2), AND(z1, z2))
   XOR(z0, false) -> c21
   XOR(z0, z0) -> c22
K tuples:
   NOT(z0) -> c14(XOR(z0, true))
   IMPLIES(z0, z1) -> c(XOR(and(z0, z1), xor(z0, true)))
   IMPLIES(z0, z1) -> c(AND(z0, z1))
   IMPLIES(z0, z1) -> c(XOR(z0, true))
   OR(z0, z1) -> c(XOR(and(z0, z1), xor(z0, z1)))
   OR(z0, z1) -> c(AND(z0, z1))
   OR(z0, z1) -> c(XOR(z0, z1))
Defined Rule Symbols:   encArg_1, not_1, or_2, implies_2, and_2, xor_2

Defined Pair Symbols:   ENCARG_1, NOT_1, AND_2, XOR_2, ENCODE_NOT_1, ENCODE_XOR_2, ENCODE_OR_2, ENCODE_AND_2, ENCODE_IMPLIES_2, OR_2, IMPLIES_2

Compound Symbols:   c2_2, c3_3, c4_3, c5_3, c6_3, c14_1, c17, c18, c19, c20_3, c21, c22, c_1


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

(121) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IMPLIES(z0, z0) -> c(XOR(z0, xor(z0, true))) by 
   IMPLIES(true, true) -> c(XOR(true, false))

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

(122)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_not(z0)) -> not(encArg(z0))
   encArg(cons_or(z0, z1)) -> or(encArg(z0), encArg(z1))
   encArg(cons_implies(z0, z1)) -> implies(encArg(z0), encArg(z1))
   encArg(cons_and(z0, z1)) -> and(encArg(z0), encArg(z1))
   encArg(cons_xor(z0, z1)) -> xor(encArg(z0), encArg(z1))
   not(z0) -> xor(z0, true)
   or(z0, z1) -> xor(and(z0, z1), xor(z0, z1))
   implies(z0, z1) -> xor(and(z0, z1), xor(z0, true))
   and(z0, true) -> z0
   and(z0, false) -> false
   and(z0, z0) -> z0
   and(xor(z0, z1), z2) -> xor(and(z0, z2), and(z1, z2))
   xor(z0, false) -> z0
   xor(z0, z0) -> false
Tuples:
   ENCARG(cons_not(z0)) -> c2(NOT(encArg(z0)), ENCARG(z0))
   ENCARG(cons_or(z0, z1)) -> c3(OR(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_implies(z0, z1)) -> c4(IMPLIES(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_and(z0, z1)) -> c5(AND(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_xor(z0, z1)) -> c6(XOR(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   NOT(z0) -> c14(XOR(z0, true))
   AND(z0, true) -> c17
   AND(z0, false) -> c18
   AND(z0, z0) -> c19
   AND(xor(z0, z1), z2) -> c20(XOR(and(z0, z2), and(z1, z2)), AND(z0, z2), AND(z1, z2))
   XOR(z0, false) -> c21
   XOR(z0, z0) -> c22
   ENCODE_NOT(z0) -> c(NOT(encArg(z0)))
   ENCODE_XOR(z0, z1) -> c(XOR(encArg(z0), encArg(z1)))
   ENCODE_OR(z0, z1) -> c(OR(encArg(z0), encArg(z1)))
   ENCODE_AND(z0, z1) -> c(AND(encArg(z0), encArg(z1)))
   ENCODE_IMPLIES(z0, z1) -> c(IMPLIES(encArg(z0), encArg(z1)))
   OR(z0, z1) -> c(AND(z0, z1))
   OR(z0, z1) -> c(XOR(z0, z1))
   IMPLIES(z0, z1) -> c(AND(z0, z1))
   IMPLIES(z0, z1) -> c(XOR(z0, true))
   IMPLIES(true, x1) -> c(XOR(and(true, x1), false))
   IMPLIES(xor(z0, z1), z2) -> c(XOR(xor(and(z0, z2), and(z1, z2)), xor(xor(z0, z1), true)))
   OR(z0, false) -> c(XOR(false, z0))
   OR(false, false) -> c(XOR(false, false))
   OR(xor(z0, z1), false) -> c(XOR(xor(and(z0, false), and(z1, false)), xor(z0, z1)))
   OR(true, true) -> c(XOR(true, false))
   OR(z0, z0) -> c(XOR(z0, false))
   OR(xor(z0, z1), xor(z0, z1)) -> c(XOR(xor(and(z0, xor(z0, z1)), and(z1, xor(z0, z1))), false))
   OR(xor(x0, z0), true) -> c(XOR(xor(and(x0, true), z0), xor(xor(x0, z0), true)))
   OR(xor(x0, z0), false) -> c(XOR(xor(and(x0, false), false), xor(xor(x0, z0), false)))
   OR(xor(x0, z0), z0) -> c(XOR(xor(and(x0, z0), z0), xor(xor(x0, z0), z0)))
   OR(xor(x0, xor(z0, z1)), z2) -> c(XOR(xor(and(x0, z2), xor(and(z0, z2), and(z1, z2))), xor(xor(x0, xor(z0, z1)), z2)))
   OR(xor(z0, x1), true) -> c(XOR(xor(z0, and(x1, true)), xor(xor(z0, x1), true)))
   OR(xor(z0, x1), false) -> c(XOR(xor(false, and(x1, false)), xor(xor(z0, x1), false)))
   OR(xor(z0, x1), z0) -> c(XOR(xor(z0, and(x1, z0)), xor(xor(z0, x1), z0)))
   OR(xor(xor(z0, z1), x1), z2) -> c(XOR(xor(xor(and(z0, z2), and(z1, z2)), and(x1, z2)), xor(xor(xor(z0, z1), x1), z2)))
   IMPLIES(true, true) -> c(XOR(true, false))
   IMPLIES(true, false) -> c(XOR(false, false))
S tuples:
   AND(z0, true) -> c17
   AND(z0, false) -> c18
   AND(z0, z0) -> c19
   AND(xor(z0, z1), z2) -> c20(XOR(and(z0, z2), and(z1, z2)), AND(z0, z2), AND(z1, z2))
   XOR(z0, false) -> c21
   XOR(z0, z0) -> c22
K tuples:
   NOT(z0) -> c14(XOR(z0, true))
   IMPLIES(z0, z1) -> c(XOR(and(z0, z1), xor(z0, true)))
   IMPLIES(z0, z1) -> c(AND(z0, z1))
   IMPLIES(z0, z1) -> c(XOR(z0, true))
   OR(z0, z1) -> c(XOR(and(z0, z1), xor(z0, z1)))
   OR(z0, z1) -> c(AND(z0, z1))
   OR(z0, z1) -> c(XOR(z0, z1))
Defined Rule Symbols:   encArg_1, not_1, or_2, implies_2, and_2, xor_2

Defined Pair Symbols:   ENCARG_1, NOT_1, AND_2, XOR_2, ENCODE_NOT_1, ENCODE_XOR_2, ENCODE_OR_2, ENCODE_AND_2, ENCODE_IMPLIES_2, OR_2, IMPLIES_2

Compound Symbols:   c2_2, c3_3, c4_3, c5_3, c6_3, c14_1, c17, c18, c19, c20_3, c21, c22, c_1


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

(123) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IMPLIES(xor(z0, z1), z2) -> c(XOR(xor(and(z0, z2), and(z1, z2)), xor(xor(z0, z1), true))) by 
   IMPLIES(xor(x0, z0), true) -> c(XOR(xor(and(x0, true), z0), xor(xor(x0, z0), true)))
   IMPLIES(xor(x0, z0), false) -> c(XOR(xor(and(x0, false), false), xor(xor(x0, z0), true)))
   IMPLIES(xor(x0, z0), z0) -> c(XOR(xor(and(x0, z0), z0), xor(xor(x0, z0), true)))
   IMPLIES(xor(x0, xor(z0, z1)), z2) -> c(XOR(xor(and(x0, z2), xor(and(z0, z2), and(z1, z2))), xor(xor(x0, xor(z0, z1)), true)))
   IMPLIES(xor(z0, x1), true) -> c(XOR(xor(z0, and(x1, true)), xor(xor(z0, x1), true)))
   IMPLIES(xor(z0, x1), false) -> c(XOR(xor(false, and(x1, false)), xor(xor(z0, x1), true)))
   IMPLIES(xor(z0, x1), z0) -> c(XOR(xor(z0, and(x1, z0)), xor(xor(z0, x1), true)))
   IMPLIES(xor(xor(z0, z1), x1), z2) -> c(XOR(xor(xor(and(z0, z2), and(z1, z2)), and(x1, z2)), xor(xor(xor(z0, z1), x1), true)))

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

(124)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_not(z0)) -> not(encArg(z0))
   encArg(cons_or(z0, z1)) -> or(encArg(z0), encArg(z1))
   encArg(cons_implies(z0, z1)) -> implies(encArg(z0), encArg(z1))
   encArg(cons_and(z0, z1)) -> and(encArg(z0), encArg(z1))
   encArg(cons_xor(z0, z1)) -> xor(encArg(z0), encArg(z1))
   not(z0) -> xor(z0, true)
   or(z0, z1) -> xor(and(z0, z1), xor(z0, z1))
   implies(z0, z1) -> xor(and(z0, z1), xor(z0, true))
   and(z0, true) -> z0
   and(z0, false) -> false
   and(z0, z0) -> z0
   and(xor(z0, z1), z2) -> xor(and(z0, z2), and(z1, z2))
   xor(z0, false) -> z0
   xor(z0, z0) -> false
Tuples:
   ENCARG(cons_not(z0)) -> c2(NOT(encArg(z0)), ENCARG(z0))
   ENCARG(cons_or(z0, z1)) -> c3(OR(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_implies(z0, z1)) -> c4(IMPLIES(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_and(z0, z1)) -> c5(AND(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_xor(z0, z1)) -> c6(XOR(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   NOT(z0) -> c14(XOR(z0, true))
   AND(z0, true) -> c17
   AND(z0, false) -> c18
   AND(z0, z0) -> c19
   AND(xor(z0, z1), z2) -> c20(XOR(and(z0, z2), and(z1, z2)), AND(z0, z2), AND(z1, z2))
   XOR(z0, false) -> c21
   XOR(z0, z0) -> c22
   ENCODE_NOT(z0) -> c(NOT(encArg(z0)))
   ENCODE_XOR(z0, z1) -> c(XOR(encArg(z0), encArg(z1)))
   ENCODE_OR(z0, z1) -> c(OR(encArg(z0), encArg(z1)))
   ENCODE_AND(z0, z1) -> c(AND(encArg(z0), encArg(z1)))
   ENCODE_IMPLIES(z0, z1) -> c(IMPLIES(encArg(z0), encArg(z1)))
   OR(z0, z1) -> c(AND(z0, z1))
   OR(z0, z1) -> c(XOR(z0, z1))
   IMPLIES(z0, z1) -> c(AND(z0, z1))
   IMPLIES(z0, z1) -> c(XOR(z0, true))
   IMPLIES(true, x1) -> c(XOR(and(true, x1), false))
   OR(z0, false) -> c(XOR(false, z0))
   OR(false, false) -> c(XOR(false, false))
   OR(xor(z0, z1), false) -> c(XOR(xor(and(z0, false), and(z1, false)), xor(z0, z1)))
   OR(true, true) -> c(XOR(true, false))
   OR(z0, z0) -> c(XOR(z0, false))
   OR(xor(z0, z1), xor(z0, z1)) -> c(XOR(xor(and(z0, xor(z0, z1)), and(z1, xor(z0, z1))), false))
   OR(xor(x0, z0), true) -> c(XOR(xor(and(x0, true), z0), xor(xor(x0, z0), true)))
   OR(xor(x0, z0), false) -> c(XOR(xor(and(x0, false), false), xor(xor(x0, z0), false)))
   OR(xor(x0, z0), z0) -> c(XOR(xor(and(x0, z0), z0), xor(xor(x0, z0), z0)))
   OR(xor(x0, xor(z0, z1)), z2) -> c(XOR(xor(and(x0, z2), xor(and(z0, z2), and(z1, z2))), xor(xor(x0, xor(z0, z1)), z2)))
   OR(xor(z0, x1), true) -> c(XOR(xor(z0, and(x1, true)), xor(xor(z0, x1), true)))
   OR(xor(z0, x1), false) -> c(XOR(xor(false, and(x1, false)), xor(xor(z0, x1), false)))
   OR(xor(z0, x1), z0) -> c(XOR(xor(z0, and(x1, z0)), xor(xor(z0, x1), z0)))
   OR(xor(xor(z0, z1), x1), z2) -> c(XOR(xor(xor(and(z0, z2), and(z1, z2)), and(x1, z2)), xor(xor(xor(z0, z1), x1), z2)))
   IMPLIES(true, true) -> c(XOR(true, false))
   IMPLIES(true, false) -> c(XOR(false, false))
   IMPLIES(xor(x0, z0), true) -> c(XOR(xor(and(x0, true), z0), xor(xor(x0, z0), true)))
   IMPLIES(xor(x0, z0), false) -> c(XOR(xor(and(x0, false), false), xor(xor(x0, z0), true)))
   IMPLIES(xor(x0, z0), z0) -> c(XOR(xor(and(x0, z0), z0), xor(xor(x0, z0), true)))
   IMPLIES(xor(x0, xor(z0, z1)), z2) -> c(XOR(xor(and(x0, z2), xor(and(z0, z2), and(z1, z2))), xor(xor(x0, xor(z0, z1)), true)))
   IMPLIES(xor(z0, x1), true) -> c(XOR(xor(z0, and(x1, true)), xor(xor(z0, x1), true)))
   IMPLIES(xor(z0, x1), false) -> c(XOR(xor(false, and(x1, false)), xor(xor(z0, x1), true)))
   IMPLIES(xor(z0, x1), z0) -> c(XOR(xor(z0, and(x1, z0)), xor(xor(z0, x1), true)))
   IMPLIES(xor(xor(z0, z1), x1), z2) -> c(XOR(xor(xor(and(z0, z2), and(z1, z2)), and(x1, z2)), xor(xor(xor(z0, z1), x1), true)))
S tuples:
   AND(z0, true) -> c17
   AND(z0, false) -> c18
   AND(z0, z0) -> c19
   AND(xor(z0, z1), z2) -> c20(XOR(and(z0, z2), and(z1, z2)), AND(z0, z2), AND(z1, z2))
   XOR(z0, false) -> c21
   XOR(z0, z0) -> c22
K tuples:
   NOT(z0) -> c14(XOR(z0, true))
   IMPLIES(z0, z1) -> c(XOR(and(z0, z1), xor(z0, true)))
   IMPLIES(z0, z1) -> c(AND(z0, z1))
   IMPLIES(z0, z1) -> c(XOR(z0, true))
   OR(z0, z1) -> c(XOR(and(z0, z1), xor(z0, z1)))
   OR(z0, z1) -> c(AND(z0, z1))
   OR(z0, z1) -> c(XOR(z0, z1))
Defined Rule Symbols:   encArg_1, not_1, or_2, implies_2, and_2, xor_2

Defined Pair Symbols:   ENCARG_1, NOT_1, AND_2, XOR_2, ENCODE_NOT_1, ENCODE_XOR_2, ENCODE_OR_2, ENCODE_AND_2, ENCODE_IMPLIES_2, OR_2, IMPLIES_2

Compound Symbols:   c2_2, c3_3, c4_3, c5_3, c6_3, c14_1, c17, c18, c19, c20_3, c21, c22, c_1


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

(125) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace OR(xor(x0, z0), false) -> c(XOR(xor(and(x0, false), false), xor(xor(x0, z0), false))) by OR(xor(z0, z1), false) -> c(XOR(xor(and(z0, false), false), xor(z0, z1)))
----------------------------------------

(126)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_not(z0)) -> not(encArg(z0))
   encArg(cons_or(z0, z1)) -> or(encArg(z0), encArg(z1))
   encArg(cons_implies(z0, z1)) -> implies(encArg(z0), encArg(z1))
   encArg(cons_and(z0, z1)) -> and(encArg(z0), encArg(z1))
   encArg(cons_xor(z0, z1)) -> xor(encArg(z0), encArg(z1))
   not(z0) -> xor(z0, true)
   or(z0, z1) -> xor(and(z0, z1), xor(z0, z1))
   implies(z0, z1) -> xor(and(z0, z1), xor(z0, true))
   and(z0, true) -> z0
   and(z0, false) -> false
   and(z0, z0) -> z0
   and(xor(z0, z1), z2) -> xor(and(z0, z2), and(z1, z2))
   xor(z0, false) -> z0
   xor(z0, z0) -> false
Tuples:
   ENCARG(cons_not(z0)) -> c2(NOT(encArg(z0)), ENCARG(z0))
   ENCARG(cons_or(z0, z1)) -> c3(OR(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_implies(z0, z1)) -> c4(IMPLIES(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_and(z0, z1)) -> c5(AND(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_xor(z0, z1)) -> c6(XOR(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   NOT(z0) -> c14(XOR(z0, true))
   AND(z0, true) -> c17
   AND(z0, false) -> c18
   AND(z0, z0) -> c19
   AND(xor(z0, z1), z2) -> c20(XOR(and(z0, z2), and(z1, z2)), AND(z0, z2), AND(z1, z2))
   XOR(z0, false) -> c21
   XOR(z0, z0) -> c22
   ENCODE_NOT(z0) -> c(NOT(encArg(z0)))
   ENCODE_XOR(z0, z1) -> c(XOR(encArg(z0), encArg(z1)))
   ENCODE_OR(z0, z1) -> c(OR(encArg(z0), encArg(z1)))
   ENCODE_AND(z0, z1) -> c(AND(encArg(z0), encArg(z1)))
   ENCODE_IMPLIES(z0, z1) -> c(IMPLIES(encArg(z0), encArg(z1)))
   OR(z0, z1) -> c(AND(z0, z1))
   OR(z0, z1) -> c(XOR(z0, z1))
   IMPLIES(z0, z1) -> c(AND(z0, z1))
   IMPLIES(z0, z1) -> c(XOR(z0, true))
   IMPLIES(true, x1) -> c(XOR(and(true, x1), false))
   OR(z0, false) -> c(XOR(false, z0))
   OR(false, false) -> c(XOR(false, false))
   OR(xor(z0, z1), false) -> c(XOR(xor(and(z0, false), and(z1, false)), xor(z0, z1)))
   OR(true, true) -> c(XOR(true, false))
   OR(z0, z0) -> c(XOR(z0, false))
   OR(xor(z0, z1), xor(z0, z1)) -> c(XOR(xor(and(z0, xor(z0, z1)), and(z1, xor(z0, z1))), false))
   OR(xor(x0, z0), true) -> c(XOR(xor(and(x0, true), z0), xor(xor(x0, z0), true)))
   OR(xor(x0, z0), z0) -> c(XOR(xor(and(x0, z0), z0), xor(xor(x0, z0), z0)))
   OR(xor(x0, xor(z0, z1)), z2) -> c(XOR(xor(and(x0, z2), xor(and(z0, z2), and(z1, z2))), xor(xor(x0, xor(z0, z1)), z2)))
   OR(xor(z0, x1), true) -> c(XOR(xor(z0, and(x1, true)), xor(xor(z0, x1), true)))
   OR(xor(z0, x1), false) -> c(XOR(xor(false, and(x1, false)), xor(xor(z0, x1), false)))
   OR(xor(z0, x1), z0) -> c(XOR(xor(z0, and(x1, z0)), xor(xor(z0, x1), z0)))
   OR(xor(xor(z0, z1), x1), z2) -> c(XOR(xor(xor(and(z0, z2), and(z1, z2)), and(x1, z2)), xor(xor(xor(z0, z1), x1), z2)))
   IMPLIES(true, true) -> c(XOR(true, false))
   IMPLIES(true, false) -> c(XOR(false, false))
   IMPLIES(xor(x0, z0), true) -> c(XOR(xor(and(x0, true), z0), xor(xor(x0, z0), true)))
   IMPLIES(xor(x0, z0), false) -> c(XOR(xor(and(x0, false), false), xor(xor(x0, z0), true)))
   IMPLIES(xor(x0, z0), z0) -> c(XOR(xor(and(x0, z0), z0), xor(xor(x0, z0), true)))
   IMPLIES(xor(x0, xor(z0, z1)), z2) -> c(XOR(xor(and(x0, z2), xor(and(z0, z2), and(z1, z2))), xor(xor(x0, xor(z0, z1)), true)))
   IMPLIES(xor(z0, x1), true) -> c(XOR(xor(z0, and(x1, true)), xor(xor(z0, x1), true)))
   IMPLIES(xor(z0, x1), false) -> c(XOR(xor(false, and(x1, false)), xor(xor(z0, x1), true)))
   IMPLIES(xor(z0, x1), z0) -> c(XOR(xor(z0, and(x1, z0)), xor(xor(z0, x1), true)))
   IMPLIES(xor(xor(z0, z1), x1), z2) -> c(XOR(xor(xor(and(z0, z2), and(z1, z2)), and(x1, z2)), xor(xor(xor(z0, z1), x1), true)))
   OR(xor(z0, z1), false) -> c(XOR(xor(and(z0, false), false), xor(z0, z1)))
S tuples:
   AND(z0, true) -> c17
   AND(z0, false) -> c18
   AND(z0, z0) -> c19
   AND(xor(z0, z1), z2) -> c20(XOR(and(z0, z2), and(z1, z2)), AND(z0, z2), AND(z1, z2))
   XOR(z0, false) -> c21
   XOR(z0, z0) -> c22
K tuples:
   NOT(z0) -> c14(XOR(z0, true))
   IMPLIES(z0, z1) -> c(XOR(and(z0, z1), xor(z0, true)))
   IMPLIES(z0, z1) -> c(AND(z0, z1))
   IMPLIES(z0, z1) -> c(XOR(z0, true))
   OR(z0, z1) -> c(XOR(and(z0, z1), xor(z0, z1)))
   OR(z0, z1) -> c(AND(z0, z1))
   OR(z0, z1) -> c(XOR(z0, z1))
Defined Rule Symbols:   encArg_1, not_1, or_2, implies_2, and_2, xor_2

Defined Pair Symbols:   ENCARG_1, NOT_1, AND_2, XOR_2, ENCODE_NOT_1, ENCODE_XOR_2, ENCODE_OR_2, ENCODE_AND_2, ENCODE_IMPLIES_2, OR_2, IMPLIES_2

Compound Symbols:   c2_2, c3_3, c4_3, c5_3, c6_3, c14_1, c17, c18, c19, c20_3, c21, c22, c_1


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

(127) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace OR(xor(z0, x1), false) -> c(XOR(xor(false, and(x1, false)), xor(xor(z0, x1), false))) by OR(xor(z0, z1), false) -> c(XOR(xor(false, and(z1, false)), xor(z0, z1)))
----------------------------------------

(128)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_not(z0)) -> not(encArg(z0))
   encArg(cons_or(z0, z1)) -> or(encArg(z0), encArg(z1))
   encArg(cons_implies(z0, z1)) -> implies(encArg(z0), encArg(z1))
   encArg(cons_and(z0, z1)) -> and(encArg(z0), encArg(z1))
   encArg(cons_xor(z0, z1)) -> xor(encArg(z0), encArg(z1))
   not(z0) -> xor(z0, true)
   or(z0, z1) -> xor(and(z0, z1), xor(z0, z1))
   implies(z0, z1) -> xor(and(z0, z1), xor(z0, true))
   and(z0, true) -> z0
   and(z0, false) -> false
   and(z0, z0) -> z0
   and(xor(z0, z1), z2) -> xor(and(z0, z2), and(z1, z2))
   xor(z0, false) -> z0
   xor(z0, z0) -> false
Tuples:
   ENCARG(cons_not(z0)) -> c2(NOT(encArg(z0)), ENCARG(z0))
   ENCARG(cons_or(z0, z1)) -> c3(OR(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_implies(z0, z1)) -> c4(IMPLIES(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_and(z0, z1)) -> c5(AND(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_xor(z0, z1)) -> c6(XOR(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   NOT(z0) -> c14(XOR(z0, true))
   AND(z0, true) -> c17
   AND(z0, false) -> c18
   AND(z0, z0) -> c19
   AND(xor(z0, z1), z2) -> c20(XOR(and(z0, z2), and(z1, z2)), AND(z0, z2), AND(z1, z2))
   XOR(z0, false) -> c21
   XOR(z0, z0) -> c22
   ENCODE_NOT(z0) -> c(NOT(encArg(z0)))
   ENCODE_XOR(z0, z1) -> c(XOR(encArg(z0), encArg(z1)))
   ENCODE_OR(z0, z1) -> c(OR(encArg(z0), encArg(z1)))
   ENCODE_AND(z0, z1) -> c(AND(encArg(z0), encArg(z1)))
   ENCODE_IMPLIES(z0, z1) -> c(IMPLIES(encArg(z0), encArg(z1)))
   OR(z0, z1) -> c(AND(z0, z1))
   OR(z0, z1) -> c(XOR(z0, z1))
   IMPLIES(z0, z1) -> c(AND(z0, z1))
   IMPLIES(z0, z1) -> c(XOR(z0, true))
   IMPLIES(true, x1) -> c(XOR(and(true, x1), false))
   OR(z0, false) -> c(XOR(false, z0))
   OR(false, false) -> c(XOR(false, false))
   OR(xor(z0, z1), false) -> c(XOR(xor(and(z0, false), and(z1, false)), xor(z0, z1)))
   OR(true, true) -> c(XOR(true, false))
   OR(z0, z0) -> c(XOR(z0, false))
   OR(xor(z0, z1), xor(z0, z1)) -> c(XOR(xor(and(z0, xor(z0, z1)), and(z1, xor(z0, z1))), false))
   OR(xor(x0, z0), true) -> c(XOR(xor(and(x0, true), z0), xor(xor(x0, z0), true)))
   OR(xor(x0, z0), z0) -> c(XOR(xor(and(x0, z0), z0), xor(xor(x0, z0), z0)))
   OR(xor(x0, xor(z0, z1)), z2) -> c(XOR(xor(and(x0, z2), xor(and(z0, z2), and(z1, z2))), xor(xor(x0, xor(z0, z1)), z2)))
   OR(xor(z0, x1), true) -> c(XOR(xor(z0, and(x1, true)), xor(xor(z0, x1), true)))
   OR(xor(z0, x1), z0) -> c(XOR(xor(z0, and(x1, z0)), xor(xor(z0, x1), z0)))
   OR(xor(xor(z0, z1), x1), z2) -> c(XOR(xor(xor(and(z0, z2), and(z1, z2)), and(x1, z2)), xor(xor(xor(z0, z1), x1), z2)))
   IMPLIES(true, true) -> c(XOR(true, false))
   IMPLIES(true, false) -> c(XOR(false, false))
   IMPLIES(xor(x0, z0), true) -> c(XOR(xor(and(x0, true), z0), xor(xor(x0, z0), true)))
   IMPLIES(xor(x0, z0), false) -> c(XOR(xor(and(x0, false), false), xor(xor(x0, z0), true)))
   IMPLIES(xor(x0, z0), z0) -> c(XOR(xor(and(x0, z0), z0), xor(xor(x0, z0), true)))
   IMPLIES(xor(x0, xor(z0, z1)), z2) -> c(XOR(xor(and(x0, z2), xor(and(z0, z2), and(z1, z2))), xor(xor(x0, xor(z0, z1)), true)))
   IMPLIES(xor(z0, x1), true) -> c(XOR(xor(z0, and(x1, true)), xor(xor(z0, x1), true)))
   IMPLIES(xor(z0, x1), false) -> c(XOR(xor(false, and(x1, false)), xor(xor(z0, x1), true)))
   IMPLIES(xor(z0, x1), z0) -> c(XOR(xor(z0, and(x1, z0)), xor(xor(z0, x1), true)))
   IMPLIES(xor(xor(z0, z1), x1), z2) -> c(XOR(xor(xor(and(z0, z2), and(z1, z2)), and(x1, z2)), xor(xor(xor(z0, z1), x1), true)))
   OR(xor(z0, z1), false) -> c(XOR(xor(and(z0, false), false), xor(z0, z1)))
   OR(xor(z0, z1), false) -> c(XOR(xor(false, and(z1, false)), xor(z0, z1)))
S tuples:
   AND(z0, true) -> c17
   AND(z0, false) -> c18
   AND(z0, z0) -> c19
   AND(xor(z0, z1), z2) -> c20(XOR(and(z0, z2), and(z1, z2)), AND(z0, z2), AND(z1, z2))
   XOR(z0, false) -> c21
   XOR(z0, z0) -> c22
K tuples:
   NOT(z0) -> c14(XOR(z0, true))
   IMPLIES(z0, z1) -> c(XOR(and(z0, z1), xor(z0, true)))
   IMPLIES(z0, z1) -> c(AND(z0, z1))
   IMPLIES(z0, z1) -> c(XOR(z0, true))
   OR(z0, z1) -> c(XOR(and(z0, z1), xor(z0, z1)))
   OR(z0, z1) -> c(AND(z0, z1))
   OR(z0, z1) -> c(XOR(z0, z1))
Defined Rule Symbols:   encArg_1, not_1, or_2, implies_2, and_2, xor_2

Defined Pair Symbols:   ENCARG_1, NOT_1, AND_2, XOR_2, ENCODE_NOT_1, ENCODE_XOR_2, ENCODE_OR_2, ENCODE_AND_2, ENCODE_IMPLIES_2, OR_2, IMPLIES_2

Compound Symbols:   c2_2, c3_3, c4_3, c5_3, c6_3, c14_1, c17, c18, c19, c20_3, c21, c22, c_1