KILLED
proof of input_9yM2F4QxEd.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


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

(0) CpxTRS
(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
(3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(4) CpxTRS
(5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(6) CpxWeightedTrs
    (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedTrs
    (9) CompletionProof [UPPER BOUND(ID), 0 ms]
    (10) CpxTypedWeightedCompleteTrs
        (11) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (12) CpxRNTS
    (13) CompletionProof [UPPER BOUND(ID), 0 ms]
    (14) CpxTypedWeightedCompleteTrs
        (15) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (16) CpxTypedWeightedCompleteTrs
        (17) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (18) CpxRNTS
        (19) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (20) CpxRNTS
        (21) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
        (22) CpxRNTS
        (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (24) CpxRNTS
        (25) IntTrsBoundProof [UPPER BOUND(ID), 282 ms]
        (26) CpxRNTS
        (27) IntTrsBoundProof [UPPER BOUND(ID), 101 ms]
        (28) CpxRNTS
        (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (30) CpxRNTS
        (31) IntTrsBoundProof [UPPER BOUND(ID), 85 ms]
        (32) CpxRNTS
        (33) IntTrsBoundProof [UPPER BOUND(ID), 74 ms]
        (34) CpxRNTS
        (35) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (36) CpxRNTS
        (37) IntTrsBoundProof [UPPER BOUND(ID), 304 ms]
        (38) CpxRNTS
        (39) IntTrsBoundProof [UPPER BOUND(ID), 73 ms]
        (40) CpxRNTS
        (41) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (42) CpxRNTS
        (43) IntTrsBoundProof [UPPER BOUND(ID), 438 ms]
        (44) CpxRNTS
        (45) IntTrsBoundProof [UPPER BOUND(ID), 6 ms]
        (46) CpxRNTS
        (47) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (48) CpxRNTS
        (49) IntTrsBoundProof [UPPER BOUND(ID), 862 ms]
        (50) CpxRNTS
        (51) IntTrsBoundProof [UPPER BOUND(ID), 134 ms]
        (52) CpxRNTS
(53) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(54) CdtProblem
    (55) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (56) CdtProblem
    (57) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (58) CdtProblem
    (59) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (60) CdtProblem
    (61) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 114 ms]
    (62) CdtProblem
    (63) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 18 ms]
    (64) CdtProblem
    (65) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (66) CdtProblem
    (67) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (68) CdtProblem
    (69) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (70) CdtProblem
    (71) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 14 ms]
    (72) CdtProblem
    (73) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (74) CdtProblem
    (75) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (76) CdtProblem
    (77) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (78) CdtProblem
    (79) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (80) CdtProblem
    (81) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 24 ms]
    (82) CdtProblem
    (83) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (84) CdtProblem
    (85) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (86) CdtProblem
    (87) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 33 ms]
    (88) CdtProblem
    (89) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (90) CdtProblem
    (91) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (92) CdtProblem
    (93) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (94) CdtProblem
    (95) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (96) CdtProblem
    (97) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (98) CdtProblem
    (99) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (100) CdtProblem
    (101) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (102) CdtProblem
    (103) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (104) CdtProblem
    (105) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (106) CdtProblem
    (107) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (108) CdtProblem
    (109) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (110) CdtProblem
    (111) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (112) CdtProblem
    (113) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (114) CdtProblem
    (115) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (116) CdtProblem
    (117) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (118) CdtProblem
    (119) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (120) CdtProblem
    (121) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (122) CdtProblem
    (123) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (124) CdtProblem
    (125) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (126) CdtProblem
    (127) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (128) CdtProblem
    (129) CdtRewritingProof [BOTH BOUNDS(ID, ID), 1 ms]
    (130) CdtProblem
    (131) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (132) CdtProblem
    (133) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (134) CdtProblem
    (135) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (136) CdtProblem
    (137) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (138) CdtProblem
    (139) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (140) CdtProblem
    (141) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (142) CdtProblem
    (143) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (144) CdtProblem
    (145) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (146) CdtProblem
    (147) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (148) CdtProblem
    (149) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (150) CdtProblem
    (151) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (152) CdtProblem
    (153) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (154) CdtProblem
    (155) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (156) CdtProblem
    (157) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (158) CdtProblem
    (159) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (160) CdtProblem
    (161) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (162) CdtProblem
    (163) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (164) CdtProblem
    (165) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (166) CdtProblem
    (167) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (168) CdtProblem
    (169) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (170) CdtProblem
    (171) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (172) CdtProblem
    (173) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (174) CdtProblem


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

(0)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   terms(N) -> cons(recip(sqr(N)), terms(s(N)))
   sqr(0) -> 0
   sqr(s(X)) -> s(add(sqr(X), dbl(X)))
   dbl(0) -> 0
   dbl(s(X)) -> s(s(dbl(X)))
   add(0, X) -> X
   add(s(X), Y) -> s(add(X, Y))
   first(0, X) -> nil
   first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))

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

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

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


The TRS R consists of the following rules:

   terms(N) -> cons(recip(sqr(N)), terms(s(N)))
   sqr(0') -> 0'
   sqr(s(X)) -> s(add(sqr(X), dbl(X)))
   dbl(0') -> 0'
   dbl(s(X)) -> s(s(dbl(X)))
   add(0', X) -> X
   add(s(X), Y) -> s(add(X, Y))
   first(0', X) -> nil
   first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))

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

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

(4)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   terms(N) -> cons(recip(sqr(N)), terms(s(N)))
   sqr(0) -> 0
   sqr(s(X)) -> s(add(sqr(X), dbl(X)))
   dbl(0) -> 0
   dbl(s(X)) -> s(s(dbl(X)))
   add(0, X) -> X
   add(s(X), Y) -> s(add(X, Y))
   first(0, X) -> nil
   first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))

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

(5) RelTrsToWeightedTrsProof (UPPER BOUND(ID))
Transformed relative TRS to weighted TRS
----------------------------------------

(6)
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:

   terms(N) -> cons(recip(sqr(N)), terms(s(N))) [1]
   sqr(0) -> 0 [1]
   sqr(s(X)) -> s(add(sqr(X), dbl(X))) [1]
   dbl(0) -> 0 [1]
   dbl(s(X)) -> s(s(dbl(X))) [1]
   add(0, X) -> X [1]
   add(s(X), Y) -> s(add(X, Y)) [1]
   first(0, X) -> nil [1]
   first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) [1]

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

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

(8)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   terms(N) -> cons(recip(sqr(N)), terms(s(N))) [1]
   sqr(0) -> 0 [1]
   sqr(s(X)) -> s(add(sqr(X), dbl(X))) [1]
   dbl(0) -> 0 [1]
   dbl(s(X)) -> s(s(dbl(X))) [1]
   add(0, X) -> X [1]
   add(s(X), Y) -> s(add(X, Y)) [1]
   first(0, X) -> nil [1]
   first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) [1]

The TRS has the following type information:
terms :: s:0 -> cons:nil
cons :: recip -> cons:nil -> cons:nil
recip :: s:0 -> recip
sqr :: s:0 -> s:0
s :: s:0 -> s:0
0 :: s:0
add :: s:0 -> s:0 -> s:0
dbl :: s:0 -> s:0
first :: s:0 -> cons:nil -> cons:nil
nil :: cons:nil

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

(9) 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:

   first(v0, v1) -> null_first [0]

And the following fresh constants:    null_first, const

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

(10)
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:

   terms(N) -> cons(recip(sqr(N)), terms(s(N))) [1]
   sqr(0) -> 0 [1]
   sqr(s(X)) -> s(add(sqr(X), dbl(X))) [1]
   dbl(0) -> 0 [1]
   dbl(s(X)) -> s(s(dbl(X))) [1]
   add(0, X) -> X [1]
   add(s(X), Y) -> s(add(X, Y)) [1]
   first(0, X) -> nil [1]
   first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) [1]
   first(v0, v1) -> null_first [0]

The TRS has the following type information:
terms :: s:0 -> cons:nil:null_first
cons :: recip -> cons:nil:null_first -> cons:nil:null_first
recip :: s:0 -> recip
sqr :: s:0 -> s:0
s :: s:0 -> s:0
0 :: s:0
add :: s:0 -> s:0 -> s:0
dbl :: s:0 -> s:0
first :: s:0 -> cons:nil:null_first -> cons:nil:null_first
nil :: cons:nil:null_first
null_first :: cons:nil:null_first
const :: recip

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

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

   0 => 0
   nil => 0
   null_first => 0
   const => 0

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

(12)
Obligation:
Complexity RNTS consisting of the following rules:

   add(z, z') -{ 1 }-> X :|: z' = X, X >= 0, z = 0
   add(z, z') -{ 1 }-> 1 + add(X, Y) :|: z = 1 + X, z' = Y, Y >= 0, X >= 0
   dbl(z) -{ 1 }-> 0 :|: z = 0
   dbl(z) -{ 1 }-> 1 + (1 + dbl(X)) :|: z = 1 + X, X >= 0
   first(z, z') -{ 1 }-> 0 :|: z' = X, X >= 0, z = 0
   first(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   first(z, z') -{ 1 }-> 1 + Y + first(X, Z) :|: Z >= 0, z = 1 + X, Y >= 0, X >= 0, z' = 1 + Y + Z
   sqr(z) -{ 1 }-> 0 :|: z = 0
   sqr(z) -{ 1 }-> 1 + add(sqr(X), dbl(X)) :|: z = 1 + X, X >= 0
   terms(z) -{ 1 }-> 1 + (1 + sqr(N)) + terms(1 + N) :|: z = N, N >= 0

Only complete derivations are relevant for the runtime complexity.

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

(13) 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:

   terms_1
   first_2

(c) The following functions are completely defined:

   sqr_1
   dbl_1
   add_2

Due to the following rules being added:
none

And the following fresh constants:    const

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

(14)
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:

   terms(N) -> cons(recip(sqr(N)), terms(s(N))) [1]
   sqr(0) -> 0 [1]
   sqr(s(X)) -> s(add(sqr(X), dbl(X))) [1]
   dbl(0) -> 0 [1]
   dbl(s(X)) -> s(s(dbl(X))) [1]
   add(0, X) -> X [1]
   add(s(X), Y) -> s(add(X, Y)) [1]
   first(0, X) -> nil [1]
   first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) [1]

The TRS has the following type information:
terms :: s:0 -> cons:nil
cons :: recip -> cons:nil -> cons:nil
recip :: s:0 -> recip
sqr :: s:0 -> s:0
s :: s:0 -> s:0
0 :: s:0
add :: s:0 -> s:0 -> s:0
dbl :: s:0 -> s:0
first :: s:0 -> cons:nil -> cons:nil
nil :: cons:nil
const :: recip

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

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

(16)
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:

   terms(N) -> cons(recip(sqr(N)), terms(s(N))) [1]
   sqr(0) -> 0 [1]
   sqr(s(0)) -> s(add(0, 0)) [3]
   sqr(s(s(X'))) -> s(add(s(add(sqr(X'), dbl(X'))), s(s(dbl(X'))))) [3]
   dbl(0) -> 0 [1]
   dbl(s(X)) -> s(s(dbl(X))) [1]
   add(0, X) -> X [1]
   add(s(X), Y) -> s(add(X, Y)) [1]
   first(0, X) -> nil [1]
   first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) [1]

The TRS has the following type information:
terms :: s:0 -> cons:nil
cons :: recip -> cons:nil -> cons:nil
recip :: s:0 -> recip
sqr :: s:0 -> s:0
s :: s:0 -> s:0
0 :: s:0
add :: s:0 -> s:0 -> s:0
dbl :: s:0 -> s:0
first :: s:0 -> cons:nil -> cons:nil
nil :: cons:nil
const :: recip

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

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

   0 => 0
   nil => 0
   const => 0

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

(18)
Obligation:
Complexity RNTS consisting of the following rules:

   add(z, z') -{ 1 }-> X :|: z' = X, X >= 0, z = 0
   add(z, z') -{ 1 }-> 1 + add(X, Y) :|: z = 1 + X, z' = Y, Y >= 0, X >= 0
   dbl(z) -{ 1 }-> 0 :|: z = 0
   dbl(z) -{ 1 }-> 1 + (1 + dbl(X)) :|: z = 1 + X, X >= 0
   first(z, z') -{ 1 }-> 0 :|: z' = X, X >= 0, z = 0
   first(z, z') -{ 1 }-> 1 + Y + first(X, Z) :|: Z >= 0, z = 1 + X, Y >= 0, X >= 0, z' = 1 + Y + Z
   sqr(z) -{ 1 }-> 0 :|: z = 0
   sqr(z) -{ 3 }-> 1 + add(0, 0) :|: z = 1 + 0
   sqr(z) -{ 3 }-> 1 + add(1 + add(sqr(X'), dbl(X')), 1 + (1 + dbl(X'))) :|: X' >= 0, z = 1 + (1 + X')
   terms(z) -{ 1 }-> 1 + (1 + sqr(N)) + terms(1 + N) :|: z = N, N >= 0


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

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

(20)
Obligation:
Complexity RNTS consisting of the following rules:

   add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 1 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0
   dbl(z) -{ 1 }-> 0 :|: z = 0
   dbl(z) -{ 1 }-> 1 + (1 + dbl(z - 1)) :|: z - 1 >= 0
   first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 1 }-> 1 + Y + first(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
   sqr(z) -{ 1 }-> 0 :|: z = 0
   sqr(z) -{ 3 }-> 1 + add(0, 0) :|: z = 1 + 0
   sqr(z) -{ 3 }-> 1 + add(1 + add(sqr(z - 2), dbl(z - 2)), 1 + (1 + dbl(z - 2))) :|: z - 2 >= 0
   terms(z) -{ 1 }-> 1 + (1 + sqr(z)) + terms(1 + z) :|: z >= 0


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

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

   { first }
   { dbl }
   { add }
   { sqr }
   { terms }

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

(22)
Obligation:
Complexity RNTS consisting of the following rules:

   add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 1 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0
   dbl(z) -{ 1 }-> 0 :|: z = 0
   dbl(z) -{ 1 }-> 1 + (1 + dbl(z - 1)) :|: z - 1 >= 0
   first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 1 }-> 1 + Y + first(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
   sqr(z) -{ 1 }-> 0 :|: z = 0
   sqr(z) -{ 3 }-> 1 + add(0, 0) :|: z = 1 + 0
   sqr(z) -{ 3 }-> 1 + add(1 + add(sqr(z - 2), dbl(z - 2)), 1 + (1 + dbl(z - 2))) :|: z - 2 >= 0
   terms(z) -{ 1 }-> 1 + (1 + sqr(z)) + terms(1 + z) :|: z >= 0

Function symbols to be analyzed: {first}, {dbl}, {add}, {sqr}, {terms}

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

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

(24)
Obligation:
Complexity RNTS consisting of the following rules:

   add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 1 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0
   dbl(z) -{ 1 }-> 0 :|: z = 0
   dbl(z) -{ 1 }-> 1 + (1 + dbl(z - 1)) :|: z - 1 >= 0
   first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 1 }-> 1 + Y + first(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
   sqr(z) -{ 1 }-> 0 :|: z = 0
   sqr(z) -{ 3 }-> 1 + add(0, 0) :|: z = 1 + 0
   sqr(z) -{ 3 }-> 1 + add(1 + add(sqr(z - 2), dbl(z - 2)), 1 + (1 + dbl(z - 2))) :|: z - 2 >= 0
   terms(z) -{ 1 }-> 1 + (1 + sqr(z)) + terms(1 + z) :|: z >= 0

Function symbols to be analyzed: {first}, {dbl}, {add}, {sqr}, {terms}

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

(25) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 1 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0
   dbl(z) -{ 1 }-> 0 :|: z = 0
   dbl(z) -{ 1 }-> 1 + (1 + dbl(z - 1)) :|: z - 1 >= 0
   first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 1 }-> 1 + Y + first(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
   sqr(z) -{ 1 }-> 0 :|: z = 0
   sqr(z) -{ 3 }-> 1 + add(0, 0) :|: z = 1 + 0
   sqr(z) -{ 3 }-> 1 + add(1 + add(sqr(z - 2), dbl(z - 2)), 1 + (1 + dbl(z - 2))) :|: z - 2 >= 0
   terms(z) -{ 1 }-> 1 + (1 + sqr(z)) + terms(1 + z) :|: z >= 0

Function symbols to be analyzed: {first}, {dbl}, {add}, {sqr}, {terms}
Previous analysis results are:
  first: runtime: ?, size: O(n^1) [z']

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: first
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 1 + z

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

(28)
Obligation:
Complexity RNTS consisting of the following rules:

   add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 1 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0
   dbl(z) -{ 1 }-> 0 :|: z = 0
   dbl(z) -{ 1 }-> 1 + (1 + dbl(z - 1)) :|: z - 1 >= 0
   first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 1 }-> 1 + Y + first(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
   sqr(z) -{ 1 }-> 0 :|: z = 0
   sqr(z) -{ 3 }-> 1 + add(0, 0) :|: z = 1 + 0
   sqr(z) -{ 3 }-> 1 + add(1 + add(sqr(z - 2), dbl(z - 2)), 1 + (1 + dbl(z - 2))) :|: z - 2 >= 0
   terms(z) -{ 1 }-> 1 + (1 + sqr(z)) + terms(1 + z) :|: z >= 0

Function symbols to be analyzed: {dbl}, {add}, {sqr}, {terms}
Previous analysis results are:
  first: runtime: O(n^1) [1 + z], size: O(n^1) [z']

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

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

(30)
Obligation:
Complexity RNTS consisting of the following rules:

   add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 1 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0
   dbl(z) -{ 1 }-> 0 :|: z = 0
   dbl(z) -{ 1 }-> 1 + (1 + dbl(z - 1)) :|: z - 1 >= 0
   first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 1 + z }-> 1 + Y + s :|: s >= 0, s <= Z, Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
   sqr(z) -{ 1 }-> 0 :|: z = 0
   sqr(z) -{ 3 }-> 1 + add(0, 0) :|: z = 1 + 0
   sqr(z) -{ 3 }-> 1 + add(1 + add(sqr(z - 2), dbl(z - 2)), 1 + (1 + dbl(z - 2))) :|: z - 2 >= 0
   terms(z) -{ 1 }-> 1 + (1 + sqr(z)) + terms(1 + z) :|: z >= 0

Function symbols to be analyzed: {dbl}, {add}, {sqr}, {terms}
Previous analysis results are:
  first: runtime: O(n^1) [1 + z], size: O(n^1) [z']

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 1 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0
   dbl(z) -{ 1 }-> 0 :|: z = 0
   dbl(z) -{ 1 }-> 1 + (1 + dbl(z - 1)) :|: z - 1 >= 0
   first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 1 + z }-> 1 + Y + s :|: s >= 0, s <= Z, Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
   sqr(z) -{ 1 }-> 0 :|: z = 0
   sqr(z) -{ 3 }-> 1 + add(0, 0) :|: z = 1 + 0
   sqr(z) -{ 3 }-> 1 + add(1 + add(sqr(z - 2), dbl(z - 2)), 1 + (1 + dbl(z - 2))) :|: z - 2 >= 0
   terms(z) -{ 1 }-> 1 + (1 + sqr(z)) + terms(1 + z) :|: z >= 0

Function symbols to be analyzed: {dbl}, {add}, {sqr}, {terms}
Previous analysis results are:
  first: runtime: O(n^1) [1 + z], size: O(n^1) [z']
  dbl: runtime: ?, size: O(n^1) [2*z]

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 1 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0
   dbl(z) -{ 1 }-> 0 :|: z = 0
   dbl(z) -{ 1 }-> 1 + (1 + dbl(z - 1)) :|: z - 1 >= 0
   first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 1 + z }-> 1 + Y + s :|: s >= 0, s <= Z, Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
   sqr(z) -{ 1 }-> 0 :|: z = 0
   sqr(z) -{ 3 }-> 1 + add(0, 0) :|: z = 1 + 0
   sqr(z) -{ 3 }-> 1 + add(1 + add(sqr(z - 2), dbl(z - 2)), 1 + (1 + dbl(z - 2))) :|: z - 2 >= 0
   terms(z) -{ 1 }-> 1 + (1 + sqr(z)) + terms(1 + z) :|: z >= 0

Function symbols to be analyzed: {add}, {sqr}, {terms}
Previous analysis results are:
  first: runtime: O(n^1) [1 + z], size: O(n^1) [z']
  dbl: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]

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

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

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

   add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 1 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0
   dbl(z) -{ 1 }-> 0 :|: z = 0
   dbl(z) -{ 1 + z }-> 1 + (1 + s1) :|: s1 >= 0, s1 <= 2 * (z - 1), z - 1 >= 0
   first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 1 + z }-> 1 + Y + s :|: s >= 0, s <= Z, Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
   sqr(z) -{ 1 }-> 0 :|: z = 0
   sqr(z) -{ 3 }-> 1 + add(0, 0) :|: z = 1 + 0
   sqr(z) -{ 1 + 2*z }-> 1 + add(1 + add(sqr(z - 2), s'), 1 + (1 + s'')) :|: s' >= 0, s' <= 2 * (z - 2), s'' >= 0, s'' <= 2 * (z - 2), z - 2 >= 0
   terms(z) -{ 1 }-> 1 + (1 + sqr(z)) + terms(1 + z) :|: z >= 0

Function symbols to be analyzed: {add}, {sqr}, {terms}
Previous analysis results are:
  first: runtime: O(n^1) [1 + z], size: O(n^1) [z']
  dbl: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]

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

(37) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 1 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0
   dbl(z) -{ 1 }-> 0 :|: z = 0
   dbl(z) -{ 1 + z }-> 1 + (1 + s1) :|: s1 >= 0, s1 <= 2 * (z - 1), z - 1 >= 0
   first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 1 + z }-> 1 + Y + s :|: s >= 0, s <= Z, Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
   sqr(z) -{ 1 }-> 0 :|: z = 0
   sqr(z) -{ 3 }-> 1 + add(0, 0) :|: z = 1 + 0
   sqr(z) -{ 1 + 2*z }-> 1 + add(1 + add(sqr(z - 2), s'), 1 + (1 + s'')) :|: s' >= 0, s' <= 2 * (z - 2), s'' >= 0, s'' <= 2 * (z - 2), z - 2 >= 0
   terms(z) -{ 1 }-> 1 + (1 + sqr(z)) + terms(1 + z) :|: z >= 0

Function symbols to be analyzed: {add}, {sqr}, {terms}
Previous analysis results are:
  first: runtime: O(n^1) [1 + z], size: O(n^1) [z']
  dbl: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  add: runtime: ?, size: O(n^1) [z + z']

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(40)
Obligation:
Complexity RNTS consisting of the following rules:

   add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 1 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0
   dbl(z) -{ 1 }-> 0 :|: z = 0
   dbl(z) -{ 1 + z }-> 1 + (1 + s1) :|: s1 >= 0, s1 <= 2 * (z - 1), z - 1 >= 0
   first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 1 + z }-> 1 + Y + s :|: s >= 0, s <= Z, Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
   sqr(z) -{ 1 }-> 0 :|: z = 0
   sqr(z) -{ 3 }-> 1 + add(0, 0) :|: z = 1 + 0
   sqr(z) -{ 1 + 2*z }-> 1 + add(1 + add(sqr(z - 2), s'), 1 + (1 + s'')) :|: s' >= 0, s' <= 2 * (z - 2), s'' >= 0, s'' <= 2 * (z - 2), z - 2 >= 0
   terms(z) -{ 1 }-> 1 + (1 + sqr(z)) + terms(1 + z) :|: z >= 0

Function symbols to be analyzed: {sqr}, {terms}
Previous analysis results are:
  first: runtime: O(n^1) [1 + z], size: O(n^1) [z']
  dbl: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  add: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']

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

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

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

   add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 1 + z }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + z', z' >= 0, z - 1 >= 0
   dbl(z) -{ 1 }-> 0 :|: z = 0
   dbl(z) -{ 1 + z }-> 1 + (1 + s1) :|: s1 >= 0, s1 <= 2 * (z - 1), z - 1 >= 0
   first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 1 + z }-> 1 + Y + s :|: s >= 0, s <= Z, Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
   sqr(z) -{ 1 }-> 0 :|: z = 0
   sqr(z) -{ 4 }-> 1 + s2 :|: s2 >= 0, s2 <= 0 + 0, z = 1 + 0
   sqr(z) -{ 1 + 2*z }-> 1 + add(1 + add(sqr(z - 2), s'), 1 + (1 + s'')) :|: s' >= 0, s' <= 2 * (z - 2), s'' >= 0, s'' <= 2 * (z - 2), z - 2 >= 0
   terms(z) -{ 1 }-> 1 + (1 + sqr(z)) + terms(1 + z) :|: z >= 0

Function symbols to be analyzed: {sqr}, {terms}
Previous analysis results are:
  first: runtime: O(n^1) [1 + z], size: O(n^1) [z']
  dbl: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  add: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']

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

(43) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: sqr
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 1 + 4*z + 4*z^2

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

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

   add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 1 + z }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + z', z' >= 0, z - 1 >= 0
   dbl(z) -{ 1 }-> 0 :|: z = 0
   dbl(z) -{ 1 + z }-> 1 + (1 + s1) :|: s1 >= 0, s1 <= 2 * (z - 1), z - 1 >= 0
   first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 1 + z }-> 1 + Y + s :|: s >= 0, s <= Z, Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
   sqr(z) -{ 1 }-> 0 :|: z = 0
   sqr(z) -{ 4 }-> 1 + s2 :|: s2 >= 0, s2 <= 0 + 0, z = 1 + 0
   sqr(z) -{ 1 + 2*z }-> 1 + add(1 + add(sqr(z - 2), s'), 1 + (1 + s'')) :|: s' >= 0, s' <= 2 * (z - 2), s'' >= 0, s'' <= 2 * (z - 2), z - 2 >= 0
   terms(z) -{ 1 }-> 1 + (1 + sqr(z)) + terms(1 + z) :|: z >= 0

Function symbols to be analyzed: {sqr}, {terms}
Previous analysis results are:
  first: runtime: O(n^1) [1 + z], size: O(n^1) [z']
  dbl: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  add: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  sqr: runtime: ?, size: O(n^2) [1 + 4*z + 4*z^2]

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

(45) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: sqr
after applying outer abstraction to obtain an ITS,
resulting in: O(n^3) with polynomial bound: 5 + 22*z + 8*z^3

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

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

   add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 1 + z }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + z', z' >= 0, z - 1 >= 0
   dbl(z) -{ 1 }-> 0 :|: z = 0
   dbl(z) -{ 1 + z }-> 1 + (1 + s1) :|: s1 >= 0, s1 <= 2 * (z - 1), z - 1 >= 0
   first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 1 + z }-> 1 + Y + s :|: s >= 0, s <= Z, Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
   sqr(z) -{ 1 }-> 0 :|: z = 0
   sqr(z) -{ 4 }-> 1 + s2 :|: s2 >= 0, s2 <= 0 + 0, z = 1 + 0
   sqr(z) -{ 1 + 2*z }-> 1 + add(1 + add(sqr(z - 2), s'), 1 + (1 + s'')) :|: s' >= 0, s' <= 2 * (z - 2), s'' >= 0, s'' <= 2 * (z - 2), z - 2 >= 0
   terms(z) -{ 1 }-> 1 + (1 + sqr(z)) + terms(1 + z) :|: z >= 0

Function symbols to be analyzed: {terms}
Previous analysis results are:
  first: runtime: O(n^1) [1 + z], size: O(n^1) [z']
  dbl: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  add: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  sqr: runtime: O(n^3) [5 + 22*z + 8*z^3], size: O(n^2) [1 + 4*z + 4*z^2]

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

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

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

   add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 1 + z }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + z', z' >= 0, z - 1 >= 0
   dbl(z) -{ 1 }-> 0 :|: z = 0
   dbl(z) -{ 1 + z }-> 1 + (1 + s1) :|: s1 >= 0, s1 <= 2 * (z - 1), z - 1 >= 0
   first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 1 + z }-> 1 + Y + s :|: s >= 0, s <= Z, Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
   sqr(z) -{ 1 }-> 0 :|: z = 0
   sqr(z) -{ 4 }-> 1 + s2 :|: s2 >= 0, s2 <= 0 + 0, z = 1 + 0
   sqr(z) -{ -99 + s5 + s6 + 120*z + -48*z^2 + 8*z^3 }-> 1 + s7 :|: s5 >= 0, s5 <= 4 * ((z - 2) * (z - 2)) + 4 * (z - 2) + 1, s6 >= 0, s6 <= s5 + s', s7 >= 0, s7 <= 1 + s6 + (1 + (1 + s'')), s' >= 0, s' <= 2 * (z - 2), s'' >= 0, s'' <= 2 * (z - 2), z - 2 >= 0
   terms(z) -{ 6 + 22*z + 8*z^3 }-> 1 + (1 + s4) + terms(1 + z) :|: s4 >= 0, s4 <= 4 * (z * z) + 4 * z + 1, z >= 0

Function symbols to be analyzed: {terms}
Previous analysis results are:
  first: runtime: O(n^1) [1 + z], size: O(n^1) [z']
  dbl: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  add: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  sqr: runtime: O(n^3) [5 + 22*z + 8*z^3], size: O(n^2) [1 + 4*z + 4*z^2]

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

(49) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: terms
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:

   add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 1 + z }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + z', z' >= 0, z - 1 >= 0
   dbl(z) -{ 1 }-> 0 :|: z = 0
   dbl(z) -{ 1 + z }-> 1 + (1 + s1) :|: s1 >= 0, s1 <= 2 * (z - 1), z - 1 >= 0
   first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 1 + z }-> 1 + Y + s :|: s >= 0, s <= Z, Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
   sqr(z) -{ 1 }-> 0 :|: z = 0
   sqr(z) -{ 4 }-> 1 + s2 :|: s2 >= 0, s2 <= 0 + 0, z = 1 + 0
   sqr(z) -{ -99 + s5 + s6 + 120*z + -48*z^2 + 8*z^3 }-> 1 + s7 :|: s5 >= 0, s5 <= 4 * ((z - 2) * (z - 2)) + 4 * (z - 2) + 1, s6 >= 0, s6 <= s5 + s', s7 >= 0, s7 <= 1 + s6 + (1 + (1 + s'')), s' >= 0, s' <= 2 * (z - 2), s'' >= 0, s'' <= 2 * (z - 2), z - 2 >= 0
   terms(z) -{ 6 + 22*z + 8*z^3 }-> 1 + (1 + s4) + terms(1 + z) :|: s4 >= 0, s4 <= 4 * (z * z) + 4 * z + 1, z >= 0

Function symbols to be analyzed: {terms}
Previous analysis results are:
  first: runtime: O(n^1) [1 + z], size: O(n^1) [z']
  dbl: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  add: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  sqr: runtime: O(n^3) [5 + 22*z + 8*z^3], size: O(n^2) [1 + 4*z + 4*z^2]
  terms: runtime: ?, size: O(1) [0]

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

(51) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 1 + z }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + z', z' >= 0, z - 1 >= 0
   dbl(z) -{ 1 }-> 0 :|: z = 0
   dbl(z) -{ 1 + z }-> 1 + (1 + s1) :|: s1 >= 0, s1 <= 2 * (z - 1), z - 1 >= 0
   first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 1 + z }-> 1 + Y + s :|: s >= 0, s <= Z, Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
   sqr(z) -{ 1 }-> 0 :|: z = 0
   sqr(z) -{ 4 }-> 1 + s2 :|: s2 >= 0, s2 <= 0 + 0, z = 1 + 0
   sqr(z) -{ -99 + s5 + s6 + 120*z + -48*z^2 + 8*z^3 }-> 1 + s7 :|: s5 >= 0, s5 <= 4 * ((z - 2) * (z - 2)) + 4 * (z - 2) + 1, s6 >= 0, s6 <= s5 + s', s7 >= 0, s7 <= 1 + s6 + (1 + (1 + s'')), s' >= 0, s' <= 2 * (z - 2), s'' >= 0, s'' <= 2 * (z - 2), z - 2 >= 0
   terms(z) -{ 6 + 22*z + 8*z^3 }-> 1 + (1 + s4) + terms(1 + z) :|: s4 >= 0, s4 <= 4 * (z * z) + 4 * z + 1, z >= 0

Function symbols to be analyzed: {terms}
Previous analysis results are:
  first: runtime: O(n^1) [1 + z], size: O(n^1) [z']
  dbl: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  add: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  sqr: runtime: O(n^3) [5 + 22*z + 8*z^3], size: O(n^2) [1 + 4*z + 4*z^2]
  terms: runtime: INF, size: O(1) [0]

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

(53) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(54)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   terms(z0) -> cons(recip(sqr(z0)), terms(s(z0)))
   sqr(0) -> 0
   sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
   dbl(0) -> 0
   dbl(s(z0)) -> s(s(dbl(z0)))
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   first(0, z0) -> nil
   first(s(z0), cons(z1, z2)) -> cons(z1, first(z0, z2))
Tuples:
   TERMS(z0) -> c(SQR(z0))
   TERMS(z0) -> c1(TERMS(s(z0)))
   SQR(0) -> c2
   SQR(s(z0)) -> c3(ADD(sqr(z0), dbl(z0)), SQR(z0))
   SQR(s(z0)) -> c4(ADD(sqr(z0), dbl(z0)), DBL(z0))
   DBL(0) -> c5
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(0, z0) -> c7
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   FIRST(0, z0) -> c9
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
S tuples:
   TERMS(z0) -> c(SQR(z0))
   TERMS(z0) -> c1(TERMS(s(z0)))
   SQR(0) -> c2
   SQR(s(z0)) -> c3(ADD(sqr(z0), dbl(z0)), SQR(z0))
   SQR(s(z0)) -> c4(ADD(sqr(z0), dbl(z0)), DBL(z0))
   DBL(0) -> c5
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(0, z0) -> c7
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   FIRST(0, z0) -> c9
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
K tuples:none
Defined Rule Symbols:   terms_1, sqr_1, dbl_1, add_2, first_2

Defined Pair Symbols:   TERMS_1, SQR_1, DBL_1, ADD_2, FIRST_2

Compound Symbols:   c_1, c1_1, c2, c3_2, c4_2, c5, c6_1, c7, c8_1, c9, c10_1


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

(55) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 4 trailing nodes:
   ADD(0, z0) -> c7
   SQR(0) -> c2
   FIRST(0, z0) -> c9
   DBL(0) -> c5

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

(56)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   terms(z0) -> cons(recip(sqr(z0)), terms(s(z0)))
   sqr(0) -> 0
   sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
   dbl(0) -> 0
   dbl(s(z0)) -> s(s(dbl(z0)))
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   first(0, z0) -> nil
   first(s(z0), cons(z1, z2)) -> cons(z1, first(z0, z2))
Tuples:
   TERMS(z0) -> c(SQR(z0))
   TERMS(z0) -> c1(TERMS(s(z0)))
   SQR(s(z0)) -> c3(ADD(sqr(z0), dbl(z0)), SQR(z0))
   SQR(s(z0)) -> c4(ADD(sqr(z0), dbl(z0)), DBL(z0))
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
S tuples:
   TERMS(z0) -> c(SQR(z0))
   TERMS(z0) -> c1(TERMS(s(z0)))
   SQR(s(z0)) -> c3(ADD(sqr(z0), dbl(z0)), SQR(z0))
   SQR(s(z0)) -> c4(ADD(sqr(z0), dbl(z0)), DBL(z0))
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
K tuples:none
Defined Rule Symbols:   terms_1, sqr_1, dbl_1, add_2, first_2

Defined Pair Symbols:   TERMS_1, SQR_1, DBL_1, ADD_2, FIRST_2

Compound Symbols:   c_1, c1_1, c3_2, c4_2, c6_1, c8_1, c10_1


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

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

(58)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   terms(z0) -> cons(recip(sqr(z0)), terms(s(z0)))
   sqr(0) -> 0
   sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
   dbl(0) -> 0
   dbl(s(z0)) -> s(s(dbl(z0)))
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   first(0, z0) -> nil
   first(s(z0), cons(z1, z2)) -> cons(z1, first(z0, z2))
Tuples:
   TERMS(z0) -> c(SQR(z0))
   TERMS(z0) -> c1(TERMS(s(z0)))
   SQR(s(z0)) -> c3(ADD(sqr(z0), dbl(z0)), SQR(z0))
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(ADD(sqr(z0), dbl(z0)))
   SQR(s(z0)) -> c2(DBL(z0))
S tuples:
   TERMS(z0) -> c(SQR(z0))
   TERMS(z0) -> c1(TERMS(s(z0)))
   SQR(s(z0)) -> c3(ADD(sqr(z0), dbl(z0)), SQR(z0))
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(ADD(sqr(z0), dbl(z0)))
   SQR(s(z0)) -> c2(DBL(z0))
K tuples:none
Defined Rule Symbols:   terms_1, sqr_1, dbl_1, add_2, first_2

Defined Pair Symbols:   TERMS_1, SQR_1, DBL_1, ADD_2, FIRST_2

Compound Symbols:   c_1, c1_1, c3_2, c6_1, c8_1, c10_1, c2_1


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

(59) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   terms(z0) -> cons(recip(sqr(z0)), terms(s(z0)))
   first(0, z0) -> nil
   first(s(z0), cons(z1, z2)) -> cons(z1, first(z0, z2))

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

(60)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sqr(0) -> 0
   sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   dbl(0) -> 0
   dbl(s(z0)) -> s(s(dbl(z0)))
Tuples:
   TERMS(z0) -> c(SQR(z0))
   TERMS(z0) -> c1(TERMS(s(z0)))
   SQR(s(z0)) -> c3(ADD(sqr(z0), dbl(z0)), SQR(z0))
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(ADD(sqr(z0), dbl(z0)))
   SQR(s(z0)) -> c2(DBL(z0))
S tuples:
   TERMS(z0) -> c(SQR(z0))
   TERMS(z0) -> c1(TERMS(s(z0)))
   SQR(s(z0)) -> c3(ADD(sqr(z0), dbl(z0)), SQR(z0))
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(ADD(sqr(z0), dbl(z0)))
   SQR(s(z0)) -> c2(DBL(z0))
K tuples:none
Defined Rule Symbols:   sqr_1, add_2, dbl_1

Defined Pair Symbols:   TERMS_1, SQR_1, DBL_1, ADD_2, FIRST_2

Compound Symbols:   c_1, c1_1, c3_2, c6_1, c8_1, c10_1, c2_1


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

(61) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   TERMS(z0) -> c(SQR(z0))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
We considered the (Usable) Rules:none
And the Tuples:
   TERMS(z0) -> c(SQR(z0))
   TERMS(z0) -> c1(TERMS(s(z0)))
   SQR(s(z0)) -> c3(ADD(sqr(z0), dbl(z0)), SQR(z0))
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(ADD(sqr(z0), dbl(z0)))
   SQR(s(z0)) -> c2(DBL(z0))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(ADD(x_1, x_2)) = 0
   POL(DBL(x_1)) = 0
   POL(FIRST(x_1, x_2)) = x_2
   POL(SQR(x_1)) = 0
   POL(TERMS(x_1)) = [1]
   POL(add(x_1, x_2)) = [1] + [3]x_1 + [2]x_2
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(cons(x_1, x_2)) = [1] + x_2
   POL(dbl(x_1)) = 0
   POL(s(x_1)) = [3] + x_1
   POL(sqr(x_1)) = [3] + [3]x_1

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

(62)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sqr(0) -> 0
   sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   dbl(0) -> 0
   dbl(s(z0)) -> s(s(dbl(z0)))
Tuples:
   TERMS(z0) -> c(SQR(z0))
   TERMS(z0) -> c1(TERMS(s(z0)))
   SQR(s(z0)) -> c3(ADD(sqr(z0), dbl(z0)), SQR(z0))
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(ADD(sqr(z0), dbl(z0)))
   SQR(s(z0)) -> c2(DBL(z0))
S tuples:
   TERMS(z0) -> c1(TERMS(s(z0)))
   SQR(s(z0)) -> c3(ADD(sqr(z0), dbl(z0)), SQR(z0))
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   SQR(s(z0)) -> c2(ADD(sqr(z0), dbl(z0)))
   SQR(s(z0)) -> c2(DBL(z0))
K tuples:
   TERMS(z0) -> c(SQR(z0))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
Defined Rule Symbols:   sqr_1, add_2, dbl_1

Defined Pair Symbols:   TERMS_1, SQR_1, DBL_1, ADD_2, FIRST_2

Compound Symbols:   c_1, c1_1, c3_2, c6_1, c8_1, c10_1, c2_1


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

(63) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   SQR(s(z0)) -> c2(ADD(sqr(z0), dbl(z0)))
   SQR(s(z0)) -> c2(DBL(z0))
We considered the (Usable) Rules:none
And the Tuples:
   TERMS(z0) -> c(SQR(z0))
   TERMS(z0) -> c1(TERMS(s(z0)))
   SQR(s(z0)) -> c3(ADD(sqr(z0), dbl(z0)), SQR(z0))
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(ADD(sqr(z0), dbl(z0)))
   SQR(s(z0)) -> c2(DBL(z0))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(ADD(x_1, x_2)) = 0
   POL(DBL(x_1)) = 0
   POL(FIRST(x_1, x_2)) = x_1 + x_2
   POL(SQR(x_1)) = [1]
   POL(TERMS(x_1)) = [1]
   POL(add(x_1, x_2)) = x_2
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(cons(x_1, x_2)) = x_2
   POL(dbl(x_1)) = [1] + x_1
   POL(s(x_1)) = [1] + x_1
   POL(sqr(x_1)) = [1] + x_1

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

(64)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sqr(0) -> 0
   sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   dbl(0) -> 0
   dbl(s(z0)) -> s(s(dbl(z0)))
Tuples:
   TERMS(z0) -> c(SQR(z0))
   TERMS(z0) -> c1(TERMS(s(z0)))
   SQR(s(z0)) -> c3(ADD(sqr(z0), dbl(z0)), SQR(z0))
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(ADD(sqr(z0), dbl(z0)))
   SQR(s(z0)) -> c2(DBL(z0))
S tuples:
   TERMS(z0) -> c1(TERMS(s(z0)))
   SQR(s(z0)) -> c3(ADD(sqr(z0), dbl(z0)), SQR(z0))
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
K tuples:
   TERMS(z0) -> c(SQR(z0))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(ADD(sqr(z0), dbl(z0)))
   SQR(s(z0)) -> c2(DBL(z0))
Defined Rule Symbols:   sqr_1, add_2, dbl_1

Defined Pair Symbols:   TERMS_1, SQR_1, DBL_1, ADD_2, FIRST_2

Compound Symbols:   c_1, c1_1, c3_2, c6_1, c8_1, c10_1, c2_1


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

(65) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace SQR(s(z0)) -> c3(ADD(sqr(z0), dbl(z0)), SQR(z0)) by 
   SQR(s(0)) -> c3(ADD(sqr(0), 0), SQR(0))
   SQR(s(s(z0))) -> c3(ADD(sqr(s(z0)), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(0)) -> c3(ADD(0, dbl(0)), SQR(0))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0))), SQR(s(z0)))

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

(66)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sqr(0) -> 0
   sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   dbl(0) -> 0
   dbl(s(z0)) -> s(s(dbl(z0)))
Tuples:
   TERMS(z0) -> c(SQR(z0))
   TERMS(z0) -> c1(TERMS(s(z0)))
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(ADD(sqr(z0), dbl(z0)))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(0)) -> c3(ADD(sqr(0), 0), SQR(0))
   SQR(s(s(z0))) -> c3(ADD(sqr(s(z0)), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(0)) -> c3(ADD(0, dbl(0)), SQR(0))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0))), SQR(s(z0)))
S tuples:
   TERMS(z0) -> c1(TERMS(s(z0)))
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   SQR(s(0)) -> c3(ADD(sqr(0), 0), SQR(0))
   SQR(s(s(z0))) -> c3(ADD(sqr(s(z0)), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(0)) -> c3(ADD(0, dbl(0)), SQR(0))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0))), SQR(s(z0)))
K tuples:
   TERMS(z0) -> c(SQR(z0))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(ADD(sqr(z0), dbl(z0)))
   SQR(s(z0)) -> c2(DBL(z0))
Defined Rule Symbols:   sqr_1, add_2, dbl_1

Defined Pair Symbols:   TERMS_1, DBL_1, ADD_2, FIRST_2, SQR_1

Compound Symbols:   c_1, c1_1, c6_1, c8_1, c10_1, c2_1, c3_2


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

(67) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   SQR(s(0)) -> c3(ADD(0, dbl(0)), SQR(0))

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

(68)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sqr(0) -> 0
   sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   dbl(0) -> 0
   dbl(s(z0)) -> s(s(dbl(z0)))
Tuples:
   TERMS(z0) -> c(SQR(z0))
   TERMS(z0) -> c1(TERMS(s(z0)))
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(ADD(sqr(z0), dbl(z0)))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(0)) -> c3(ADD(sqr(0), 0), SQR(0))
   SQR(s(s(z0))) -> c3(ADD(sqr(s(z0)), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0))), SQR(s(z0)))
S tuples:
   TERMS(z0) -> c1(TERMS(s(z0)))
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   SQR(s(0)) -> c3(ADD(sqr(0), 0), SQR(0))
   SQR(s(s(z0))) -> c3(ADD(sqr(s(z0)), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0))), SQR(s(z0)))
K tuples:
   TERMS(z0) -> c(SQR(z0))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(ADD(sqr(z0), dbl(z0)))
   SQR(s(z0)) -> c2(DBL(z0))
Defined Rule Symbols:   sqr_1, add_2, dbl_1

Defined Pair Symbols:   TERMS_1, DBL_1, ADD_2, FIRST_2, SQR_1

Compound Symbols:   c_1, c1_1, c6_1, c8_1, c10_1, c2_1, c3_2


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

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

(70)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sqr(0) -> 0
   sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   dbl(0) -> 0
   dbl(s(z0)) -> s(s(dbl(z0)))
Tuples:
   TERMS(z0) -> c(SQR(z0))
   TERMS(z0) -> c1(TERMS(s(z0)))
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(ADD(sqr(z0), dbl(z0)))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(s(z0))) -> c3(ADD(sqr(s(z0)), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0))), SQR(s(z0)))
   SQR(s(0)) -> c3(ADD(sqr(0), 0))
S tuples:
   TERMS(z0) -> c1(TERMS(s(z0)))
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   SQR(s(s(z0))) -> c3(ADD(sqr(s(z0)), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0))), SQR(s(z0)))
   SQR(s(0)) -> c3(ADD(sqr(0), 0))
K tuples:
   TERMS(z0) -> c(SQR(z0))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(ADD(sqr(z0), dbl(z0)))
   SQR(s(z0)) -> c2(DBL(z0))
Defined Rule Symbols:   sqr_1, add_2, dbl_1

Defined Pair Symbols:   TERMS_1, DBL_1, ADD_2, FIRST_2, SQR_1

Compound Symbols:   c_1, c1_1, c6_1, c8_1, c10_1, c2_1, c3_2, c3_1


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

(71) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   SQR(s(0)) -> c3(ADD(sqr(0), 0))
We considered the (Usable) Rules:
   dbl(0) -> 0
   dbl(s(z0)) -> s(s(dbl(z0)))
And the Tuples:
   TERMS(z0) -> c(SQR(z0))
   TERMS(z0) -> c1(TERMS(s(z0)))
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(ADD(sqr(z0), dbl(z0)))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(s(z0))) -> c3(ADD(sqr(s(z0)), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0))), SQR(s(z0)))
   SQR(s(0)) -> c3(ADD(sqr(0), 0))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(ADD(x_1, x_2)) = x_2
   POL(DBL(x_1)) = [1] + x_1
   POL(FIRST(x_1, x_2)) = x_1
   POL(SQR(x_1)) = [1] + x_1
   POL(TERMS(x_1)) = [1] + x_1
   POL(add(x_1, x_2)) = x_1 + x_2
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(cons(x_1, x_2)) = x_2
   POL(dbl(x_1)) = 0
   POL(s(x_1)) = x_1
   POL(sqr(x_1)) = [1] + x_1

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

(72)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sqr(0) -> 0
   sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   dbl(0) -> 0
   dbl(s(z0)) -> s(s(dbl(z0)))
Tuples:
   TERMS(z0) -> c(SQR(z0))
   TERMS(z0) -> c1(TERMS(s(z0)))
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(ADD(sqr(z0), dbl(z0)))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(s(z0))) -> c3(ADD(sqr(s(z0)), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0))), SQR(s(z0)))
   SQR(s(0)) -> c3(ADD(sqr(0), 0))
S tuples:
   TERMS(z0) -> c1(TERMS(s(z0)))
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   SQR(s(s(z0))) -> c3(ADD(sqr(s(z0)), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0))), SQR(s(z0)))
K tuples:
   TERMS(z0) -> c(SQR(z0))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(ADD(sqr(z0), dbl(z0)))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(0)) -> c3(ADD(sqr(0), 0))
Defined Rule Symbols:   sqr_1, add_2, dbl_1

Defined Pair Symbols:   TERMS_1, DBL_1, ADD_2, FIRST_2, SQR_1

Compound Symbols:   c_1, c1_1, c6_1, c8_1, c10_1, c2_1, c3_2, c3_1


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

(73) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace SQR(s(z0)) -> c2(ADD(sqr(z0), dbl(z0))) by 
   SQR(s(0)) -> c2(ADD(sqr(0), 0))
   SQR(s(s(z0))) -> c2(ADD(sqr(s(z0)), s(s(dbl(z0)))))
   SQR(s(0)) -> c2(ADD(0, dbl(0)))
   SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0))))

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

(74)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sqr(0) -> 0
   sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   dbl(0) -> 0
   dbl(s(z0)) -> s(s(dbl(z0)))
Tuples:
   TERMS(z0) -> c(SQR(z0))
   TERMS(z0) -> c1(TERMS(s(z0)))
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(s(z0))) -> c3(ADD(sqr(s(z0)), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0))), SQR(s(z0)))
   SQR(s(0)) -> c3(ADD(sqr(0), 0))
   SQR(s(0)) -> c2(ADD(sqr(0), 0))
   SQR(s(s(z0))) -> c2(ADD(sqr(s(z0)), s(s(dbl(z0)))))
   SQR(s(0)) -> c2(ADD(0, dbl(0)))
   SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0))))
S tuples:
   TERMS(z0) -> c1(TERMS(s(z0)))
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   SQR(s(s(z0))) -> c3(ADD(sqr(s(z0)), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0))), SQR(s(z0)))
K tuples:
   TERMS(z0) -> c(SQR(z0))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(ADD(sqr(z0), dbl(z0)))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(0)) -> c3(ADD(sqr(0), 0))
Defined Rule Symbols:   sqr_1, add_2, dbl_1

Defined Pair Symbols:   TERMS_1, DBL_1, ADD_2, FIRST_2, SQR_1

Compound Symbols:   c_1, c1_1, c6_1, c8_1, c10_1, c2_1, c3_2, c3_1


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

(75) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   SQR(s(0)) -> c2(ADD(0, dbl(0)))

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

(76)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sqr(0) -> 0
   sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   dbl(0) -> 0
   dbl(s(z0)) -> s(s(dbl(z0)))
Tuples:
   TERMS(z0) -> c(SQR(z0))
   TERMS(z0) -> c1(TERMS(s(z0)))
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(s(z0))) -> c3(ADD(sqr(s(z0)), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0))), SQR(s(z0)))
   SQR(s(0)) -> c3(ADD(sqr(0), 0))
   SQR(s(0)) -> c2(ADD(sqr(0), 0))
   SQR(s(s(z0))) -> c2(ADD(sqr(s(z0)), s(s(dbl(z0)))))
   SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0))))
S tuples:
   TERMS(z0) -> c1(TERMS(s(z0)))
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   SQR(s(s(z0))) -> c3(ADD(sqr(s(z0)), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0))), SQR(s(z0)))
K tuples:
   TERMS(z0) -> c(SQR(z0))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(0)) -> c3(ADD(sqr(0), 0))
Defined Rule Symbols:   sqr_1, add_2, dbl_1

Defined Pair Symbols:   TERMS_1, DBL_1, ADD_2, FIRST_2, SQR_1

Compound Symbols:   c_1, c1_1, c6_1, c8_1, c10_1, c2_1, c3_2, c3_1


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

(77) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace SQR(s(s(z0))) -> c3(ADD(sqr(s(z0)), s(s(dbl(z0)))), SQR(s(z0))) by 
   SQR(s(s(0))) -> c3(ADD(sqr(s(0)), s(s(0))), SQR(s(0)))
   SQR(s(s(s(z0)))) -> c3(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))

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

(78)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sqr(0) -> 0
   sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   dbl(0) -> 0
   dbl(s(z0)) -> s(s(dbl(z0)))
Tuples:
   TERMS(z0) -> c(SQR(z0))
   TERMS(z0) -> c1(TERMS(s(z0)))
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0))), SQR(s(z0)))
   SQR(s(0)) -> c3(ADD(sqr(0), 0))
   SQR(s(0)) -> c2(ADD(sqr(0), 0))
   SQR(s(s(z0))) -> c2(ADD(sqr(s(z0)), s(s(dbl(z0)))))
   SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0))))
   SQR(s(s(0))) -> c3(ADD(sqr(s(0)), s(s(0))), SQR(s(0)))
   SQR(s(s(s(z0)))) -> c3(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
S tuples:
   TERMS(z0) -> c1(TERMS(s(z0)))
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0))), SQR(s(z0)))
   SQR(s(s(0))) -> c3(ADD(sqr(s(0)), s(s(0))), SQR(s(0)))
   SQR(s(s(s(z0)))) -> c3(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
K tuples:
   TERMS(z0) -> c(SQR(z0))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(0)) -> c3(ADD(sqr(0), 0))
Defined Rule Symbols:   sqr_1, add_2, dbl_1

Defined Pair Symbols:   TERMS_1, DBL_1, ADD_2, FIRST_2, SQR_1

Compound Symbols:   c_1, c1_1, c6_1, c8_1, c10_1, c2_1, c3_2, c3_1


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

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

(80)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sqr(0) -> 0
   sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   dbl(0) -> 0
   dbl(s(z0)) -> s(s(dbl(z0)))
Tuples:
   TERMS(z0) -> c(SQR(z0))
   TERMS(z0) -> c1(TERMS(s(z0)))
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0))), SQR(s(z0)))
   SQR(s(0)) -> c3(ADD(sqr(0), 0))
   SQR(s(0)) -> c2(ADD(sqr(0), 0))
   SQR(s(s(z0))) -> c2(ADD(sqr(s(z0)), s(s(dbl(z0)))))
   SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0))))
   SQR(s(s(0))) -> c4(SQR(s(0)))
S tuples:
   TERMS(z0) -> c1(TERMS(s(z0)))
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0))), SQR(s(z0)))
   SQR(s(s(s(z0)))) -> c3(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0))))
   SQR(s(s(0))) -> c4(SQR(s(0)))
K tuples:
   TERMS(z0) -> c(SQR(z0))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(0)) -> c3(ADD(sqr(0), 0))
Defined Rule Symbols:   sqr_1, add_2, dbl_1

Defined Pair Symbols:   TERMS_1, DBL_1, ADD_2, FIRST_2, SQR_1

Compound Symbols:   c_1, c1_1, c6_1, c8_1, c10_1, c2_1, c3_2, c3_1, c4_1


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

(81) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0))))
We considered the (Usable) Rules:none
And the Tuples:
   TERMS(z0) -> c(SQR(z0))
   TERMS(z0) -> c1(TERMS(s(z0)))
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0))), SQR(s(z0)))
   SQR(s(0)) -> c3(ADD(sqr(0), 0))
   SQR(s(0)) -> c2(ADD(sqr(0), 0))
   SQR(s(s(z0))) -> c2(ADD(sqr(s(z0)), s(s(dbl(z0)))))
   SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0))))
   SQR(s(s(0))) -> c4(SQR(s(0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(ADD(x_1, x_2)) = 0
   POL(DBL(x_1)) = [1]
   POL(FIRST(x_1, x_2)) = 0
   POL(SQR(x_1)) = [1]
   POL(TERMS(x_1)) = [1]
   POL(add(x_1, x_2)) = [1] + x_1 + x_2
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(cons(x_1, x_2)) = x_2
   POL(dbl(x_1)) = [1] + x_1
   POL(s(x_1)) = [1]
   POL(sqr(x_1)) = [1] + x_1

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

(82)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sqr(0) -> 0
   sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   dbl(0) -> 0
   dbl(s(z0)) -> s(s(dbl(z0)))
Tuples:
   TERMS(z0) -> c(SQR(z0))
   TERMS(z0) -> c1(TERMS(s(z0)))
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0))), SQR(s(z0)))
   SQR(s(0)) -> c3(ADD(sqr(0), 0))
   SQR(s(0)) -> c2(ADD(sqr(0), 0))
   SQR(s(s(z0))) -> c2(ADD(sqr(s(z0)), s(s(dbl(z0)))))
   SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0))))
   SQR(s(s(0))) -> c4(SQR(s(0)))
S tuples:
   TERMS(z0) -> c1(TERMS(s(z0)))
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0))), SQR(s(z0)))
   SQR(s(s(s(z0)))) -> c3(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(0))) -> c4(SQR(s(0)))
K tuples:
   TERMS(z0) -> c(SQR(z0))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(0)) -> c3(ADD(sqr(0), 0))
   SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0))))
Defined Rule Symbols:   sqr_1, add_2, dbl_1

Defined Pair Symbols:   TERMS_1, DBL_1, ADD_2, FIRST_2, SQR_1

Compound Symbols:   c_1, c1_1, c6_1, c8_1, c10_1, c2_1, c3_2, c3_1, c4_1


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

(83) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0))), SQR(s(z0))) by 
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(0))) -> c3(ADD(s(add(sqr(0), 0)), dbl(s(0))), SQR(s(0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0)))), SQR(s(s(z0))))
   SQR(s(s(0))) -> c3(ADD(s(add(0, dbl(0))), dbl(s(0))), SQR(s(0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0)))), SQR(s(s(z0))))

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

(84)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sqr(0) -> 0
   sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   dbl(0) -> 0
   dbl(s(z0)) -> s(s(dbl(z0)))
Tuples:
   TERMS(z0) -> c(SQR(z0))
   TERMS(z0) -> c1(TERMS(s(z0)))
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(0)) -> c3(ADD(sqr(0), 0))
   SQR(s(0)) -> c2(ADD(sqr(0), 0))
   SQR(s(s(z0))) -> c2(ADD(sqr(s(z0)), s(s(dbl(z0)))))
   SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0))))
   SQR(s(s(0))) -> c4(SQR(s(0)))
   SQR(s(s(0))) -> c3(ADD(s(add(sqr(0), 0)), dbl(s(0))), SQR(s(0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0)))), SQR(s(s(z0))))
   SQR(s(s(0))) -> c3(ADD(s(add(0, dbl(0))), dbl(s(0))), SQR(s(0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0)))), SQR(s(s(z0))))
S tuples:
   TERMS(z0) -> c1(TERMS(s(z0)))
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   SQR(s(s(s(z0)))) -> c3(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(0))) -> c4(SQR(s(0)))
   SQR(s(s(0))) -> c3(ADD(s(add(sqr(0), 0)), dbl(s(0))), SQR(s(0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0)))), SQR(s(s(z0))))
   SQR(s(s(0))) -> c3(ADD(s(add(0, dbl(0))), dbl(s(0))), SQR(s(0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0)))), SQR(s(s(z0))))
K tuples:
   TERMS(z0) -> c(SQR(z0))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(0)) -> c3(ADD(sqr(0), 0))
   SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0))))
Defined Rule Symbols:   sqr_1, add_2, dbl_1

Defined Pair Symbols:   TERMS_1, DBL_1, ADD_2, FIRST_2, SQR_1

Compound Symbols:   c_1, c1_1, c6_1, c8_1, c10_1, c2_1, c3_1, c3_2, c4_1


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

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

(86)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sqr(0) -> 0
   sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   dbl(0) -> 0
   dbl(s(z0)) -> s(s(dbl(z0)))
Tuples:
   TERMS(z0) -> c(SQR(z0))
   TERMS(z0) -> c1(TERMS(s(z0)))
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(0)) -> c3(ADD(sqr(0), 0))
   SQR(s(0)) -> c2(ADD(sqr(0), 0))
   SQR(s(s(z0))) -> c2(ADD(sqr(s(z0)), s(s(dbl(z0)))))
   SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0))))
   SQR(s(s(0))) -> c4(SQR(s(0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0)))), SQR(s(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0)))), SQR(s(s(z0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0))))
   SQR(s(s(0))) -> c5(SQR(s(0)))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0))))
S tuples:
   TERMS(z0) -> c1(TERMS(s(z0)))
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   SQR(s(s(s(z0)))) -> c3(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(0))) -> c4(SQR(s(0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0)))), SQR(s(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0)))), SQR(s(s(z0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0))))
   SQR(s(s(0))) -> c5(SQR(s(0)))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0))))
K tuples:
   TERMS(z0) -> c(SQR(z0))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(0)) -> c3(ADD(sqr(0), 0))
   SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0))))
Defined Rule Symbols:   sqr_1, add_2, dbl_1

Defined Pair Symbols:   TERMS_1, DBL_1, ADD_2, FIRST_2, SQR_1

Compound Symbols:   c_1, c1_1, c6_1, c8_1, c10_1, c2_1, c3_1, c3_2, c4_1, c5_1


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

(87) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0))))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0))))
We considered the (Usable) Rules:none
And the Tuples:
   TERMS(z0) -> c(SQR(z0))
   TERMS(z0) -> c1(TERMS(s(z0)))
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(0)) -> c3(ADD(sqr(0), 0))
   SQR(s(0)) -> c2(ADD(sqr(0), 0))
   SQR(s(s(z0))) -> c2(ADD(sqr(s(z0)), s(s(dbl(z0)))))
   SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0))))
   SQR(s(s(0))) -> c4(SQR(s(0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0)))), SQR(s(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0)))), SQR(s(s(z0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0))))
   SQR(s(s(0))) -> c5(SQR(s(0)))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0))))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(ADD(x_1, x_2)) = 0
   POL(DBL(x_1)) = [1]
   POL(FIRST(x_1, x_2)) = x_2
   POL(SQR(x_1)) = [1]
   POL(TERMS(x_1)) = [1]
   POL(add(x_1, x_2)) = [1] + x_1 + x_2
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(cons(x_1, x_2)) = x_2
   POL(dbl(x_1)) = [1] + x_1
   POL(s(x_1)) = [1]
   POL(sqr(x_1)) = [1] + x_1

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

(88)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sqr(0) -> 0
   sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   dbl(0) -> 0
   dbl(s(z0)) -> s(s(dbl(z0)))
Tuples:
   TERMS(z0) -> c(SQR(z0))
   TERMS(z0) -> c1(TERMS(s(z0)))
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(0)) -> c3(ADD(sqr(0), 0))
   SQR(s(0)) -> c2(ADD(sqr(0), 0))
   SQR(s(s(z0))) -> c2(ADD(sqr(s(z0)), s(s(dbl(z0)))))
   SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0))))
   SQR(s(s(0))) -> c4(SQR(s(0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0)))), SQR(s(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0)))), SQR(s(s(z0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0))))
   SQR(s(s(0))) -> c5(SQR(s(0)))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0))))
S tuples:
   TERMS(z0) -> c1(TERMS(s(z0)))
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   SQR(s(s(s(z0)))) -> c3(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(0))) -> c4(SQR(s(0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0)))), SQR(s(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0)))), SQR(s(s(z0))))
   SQR(s(s(0))) -> c5(SQR(s(0)))
K tuples:
   TERMS(z0) -> c(SQR(z0))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(0)) -> c3(ADD(sqr(0), 0))
   SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0))))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0))))
Defined Rule Symbols:   sqr_1, add_2, dbl_1

Defined Pair Symbols:   TERMS_1, DBL_1, ADD_2, FIRST_2, SQR_1

Compound Symbols:   c_1, c1_1, c6_1, c8_1, c10_1, c2_1, c3_1, c3_2, c4_1, c5_1


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

(89) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace SQR(s(0)) -> c3(ADD(sqr(0), 0)) by 
   SQR(s(0)) -> c3(ADD(0, 0))

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

(90)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sqr(0) -> 0
   sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   dbl(0) -> 0
   dbl(s(z0)) -> s(s(dbl(z0)))
Tuples:
   TERMS(z0) -> c(SQR(z0))
   TERMS(z0) -> c1(TERMS(s(z0)))
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(0)) -> c2(ADD(sqr(0), 0))
   SQR(s(s(z0))) -> c2(ADD(sqr(s(z0)), s(s(dbl(z0)))))
   SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0))))
   SQR(s(s(0))) -> c4(SQR(s(0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0)))), SQR(s(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0)))), SQR(s(s(z0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0))))
   SQR(s(s(0))) -> c5(SQR(s(0)))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0))))
   SQR(s(0)) -> c3(ADD(0, 0))
S tuples:
   TERMS(z0) -> c1(TERMS(s(z0)))
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   SQR(s(s(s(z0)))) -> c3(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(0))) -> c4(SQR(s(0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0)))), SQR(s(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0)))), SQR(s(s(z0))))
   SQR(s(s(0))) -> c5(SQR(s(0)))
K tuples:
   TERMS(z0) -> c(SQR(z0))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(0)) -> c3(ADD(sqr(0), 0))
   SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0))))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0))))
Defined Rule Symbols:   sqr_1, add_2, dbl_1

Defined Pair Symbols:   TERMS_1, DBL_1, ADD_2, FIRST_2, SQR_1

Compound Symbols:   c_1, c1_1, c6_1, c8_1, c10_1, c2_1, c3_2, c4_1, c5_1, c3_1


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

(91) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   SQR(s(0)) -> c3(ADD(0, 0))

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

(92)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sqr(0) -> 0
   sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   dbl(0) -> 0
   dbl(s(z0)) -> s(s(dbl(z0)))
Tuples:
   TERMS(z0) -> c(SQR(z0))
   TERMS(z0) -> c1(TERMS(s(z0)))
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(0)) -> c2(ADD(sqr(0), 0))
   SQR(s(s(z0))) -> c2(ADD(sqr(s(z0)), s(s(dbl(z0)))))
   SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0))))
   SQR(s(s(0))) -> c4(SQR(s(0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0)))), SQR(s(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0)))), SQR(s(s(z0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0))))
   SQR(s(s(0))) -> c5(SQR(s(0)))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0))))
S tuples:
   TERMS(z0) -> c1(TERMS(s(z0)))
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   SQR(s(s(s(z0)))) -> c3(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(0))) -> c4(SQR(s(0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0)))), SQR(s(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0)))), SQR(s(s(z0))))
   SQR(s(s(0))) -> c5(SQR(s(0)))
K tuples:
   TERMS(z0) -> c(SQR(z0))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0))))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0))))
Defined Rule Symbols:   sqr_1, add_2, dbl_1

Defined Pair Symbols:   TERMS_1, DBL_1, ADD_2, FIRST_2, SQR_1

Compound Symbols:   c_1, c1_1, c6_1, c8_1, c10_1, c2_1, c3_2, c4_1, c5_1


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

(93) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace SQR(s(0)) -> c2(ADD(sqr(0), 0)) by 
   SQR(s(0)) -> c2(ADD(0, 0))

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

(94)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sqr(0) -> 0
   sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   dbl(0) -> 0
   dbl(s(z0)) -> s(s(dbl(z0)))
Tuples:
   TERMS(z0) -> c(SQR(z0))
   TERMS(z0) -> c1(TERMS(s(z0)))
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(s(z0))) -> c2(ADD(sqr(s(z0)), s(s(dbl(z0)))))
   SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0))))
   SQR(s(s(0))) -> c4(SQR(s(0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0)))), SQR(s(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0)))), SQR(s(s(z0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0))))
   SQR(s(s(0))) -> c5(SQR(s(0)))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0))))
   SQR(s(0)) -> c2(ADD(0, 0))
S tuples:
   TERMS(z0) -> c1(TERMS(s(z0)))
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   SQR(s(s(s(z0)))) -> c3(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(0))) -> c4(SQR(s(0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0)))), SQR(s(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0)))), SQR(s(s(z0))))
   SQR(s(s(0))) -> c5(SQR(s(0)))
K tuples:
   TERMS(z0) -> c(SQR(z0))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0))))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0))))
Defined Rule Symbols:   sqr_1, add_2, dbl_1

Defined Pair Symbols:   TERMS_1, DBL_1, ADD_2, FIRST_2, SQR_1

Compound Symbols:   c_1, c1_1, c6_1, c8_1, c10_1, c2_1, c3_2, c4_1, c5_1


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

(95) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   SQR(s(0)) -> c2(ADD(0, 0))

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

(96)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sqr(0) -> 0
   sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   dbl(0) -> 0
   dbl(s(z0)) -> s(s(dbl(z0)))
Tuples:
   TERMS(z0) -> c(SQR(z0))
   TERMS(z0) -> c1(TERMS(s(z0)))
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(s(z0))) -> c2(ADD(sqr(s(z0)), s(s(dbl(z0)))))
   SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0))))
   SQR(s(s(0))) -> c4(SQR(s(0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0)))), SQR(s(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0)))), SQR(s(s(z0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0))))
   SQR(s(s(0))) -> c5(SQR(s(0)))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0))))
S tuples:
   TERMS(z0) -> c1(TERMS(s(z0)))
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   SQR(s(s(s(z0)))) -> c3(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(0))) -> c4(SQR(s(0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0)))), SQR(s(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0)))), SQR(s(s(z0))))
   SQR(s(s(0))) -> c5(SQR(s(0)))
K tuples:
   TERMS(z0) -> c(SQR(z0))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0))))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0))))
Defined Rule Symbols:   sqr_1, add_2, dbl_1

Defined Pair Symbols:   TERMS_1, DBL_1, ADD_2, FIRST_2, SQR_1

Compound Symbols:   c_1, c1_1, c6_1, c8_1, c10_1, c2_1, c3_2, c4_1, c5_1


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

(97) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace SQR(s(s(z0))) -> c2(ADD(sqr(s(z0)), s(s(dbl(z0))))) by 
   SQR(s(s(0))) -> c2(ADD(sqr(s(0)), s(s(0))))
   SQR(s(s(s(z0)))) -> c2(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0)))))))
   SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))))

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

(98)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sqr(0) -> 0
   sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   dbl(0) -> 0
   dbl(s(z0)) -> s(s(dbl(z0)))
Tuples:
   TERMS(z0) -> c(SQR(z0))
   TERMS(z0) -> c1(TERMS(s(z0)))
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0))))
   SQR(s(s(0))) -> c4(SQR(s(0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0)))), SQR(s(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0)))), SQR(s(s(z0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0))))
   SQR(s(s(0))) -> c5(SQR(s(0)))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0))))
   SQR(s(s(0))) -> c2(ADD(sqr(s(0)), s(s(0))))
   SQR(s(s(s(z0)))) -> c2(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0)))))))
   SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))))
S tuples:
   TERMS(z0) -> c1(TERMS(s(z0)))
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   SQR(s(s(s(z0)))) -> c3(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(0))) -> c4(SQR(s(0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0)))), SQR(s(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0)))), SQR(s(s(z0))))
   SQR(s(s(0))) -> c5(SQR(s(0)))
K tuples:
   TERMS(z0) -> c(SQR(z0))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0))))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0))))
Defined Rule Symbols:   sqr_1, add_2, dbl_1

Defined Pair Symbols:   TERMS_1, DBL_1, ADD_2, FIRST_2, SQR_1

Compound Symbols:   c_1, c1_1, c6_1, c8_1, c10_1, c2_1, c3_2, c4_1, c5_1


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

(99) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))) by 
   SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))))
   SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), dbl(s(0))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0)))))
   SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), dbl(s(0))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0)))))

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

(100)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sqr(0) -> 0
   sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   dbl(0) -> 0
   dbl(s(z0)) -> s(s(dbl(z0)))
Tuples:
   TERMS(z0) -> c(SQR(z0))
   TERMS(z0) -> c1(TERMS(s(z0)))
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(s(s(z0)))) -> c3(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0))))
   SQR(s(s(0))) -> c4(SQR(s(0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0)))), SQR(s(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0)))), SQR(s(s(z0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0))))
   SQR(s(s(0))) -> c5(SQR(s(0)))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0))))
   SQR(s(s(0))) -> c2(ADD(sqr(s(0)), s(s(0))))
   SQR(s(s(s(z0)))) -> c2(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0)))))))
   SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))))
   SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), dbl(s(0))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0)))))
   SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), dbl(s(0))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0)))))
S tuples:
   TERMS(z0) -> c1(TERMS(s(z0)))
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   SQR(s(s(s(z0)))) -> c3(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(0))) -> c4(SQR(s(0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0)))), SQR(s(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0)))), SQR(s(s(z0))))
   SQR(s(s(0))) -> c5(SQR(s(0)))
K tuples:
   TERMS(z0) -> c(SQR(z0))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0))))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0))))
Defined Rule Symbols:   sqr_1, add_2, dbl_1

Defined Pair Symbols:   TERMS_1, DBL_1, ADD_2, FIRST_2, SQR_1

Compound Symbols:   c_1, c1_1, c6_1, c8_1, c10_1, c2_1, c3_2, c4_1, c5_1


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

(101) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace SQR(s(s(s(z0)))) -> c3(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) by SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
----------------------------------------

(102)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sqr(0) -> 0
   sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   dbl(0) -> 0
   dbl(s(z0)) -> s(s(dbl(z0)))
Tuples:
   TERMS(z0) -> c(SQR(z0))
   TERMS(z0) -> c1(TERMS(s(z0)))
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0))))
   SQR(s(s(0))) -> c4(SQR(s(0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0)))), SQR(s(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0)))), SQR(s(s(z0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0))))
   SQR(s(s(0))) -> c5(SQR(s(0)))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0))))
   SQR(s(s(0))) -> c2(ADD(sqr(s(0)), s(s(0))))
   SQR(s(s(s(z0)))) -> c2(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0)))))))
   SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))))
   SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), dbl(s(0))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0)))))
   SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), dbl(s(0))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0)))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
S tuples:
   TERMS(z0) -> c1(TERMS(s(z0)))
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(0))) -> c4(SQR(s(0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0)))), SQR(s(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0)))), SQR(s(s(z0))))
   SQR(s(s(0))) -> c5(SQR(s(0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
K tuples:
   TERMS(z0) -> c(SQR(z0))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0))))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0))))
Defined Rule Symbols:   sqr_1, add_2, dbl_1

Defined Pair Symbols:   TERMS_1, DBL_1, ADD_2, FIRST_2, SQR_1

Compound Symbols:   c_1, c1_1, c6_1, c8_1, c10_1, c2_1, c3_2, c4_1, c5_1


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

(103) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0)))) by SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), dbl(0))), s(s(0))))
----------------------------------------

(104)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sqr(0) -> 0
   sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   dbl(0) -> 0
   dbl(s(z0)) -> s(s(dbl(z0)))
Tuples:
   TERMS(z0) -> c(SQR(z0))
   TERMS(z0) -> c1(TERMS(s(z0)))
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(0))) -> c4(SQR(s(0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0)))), SQR(s(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0)))), SQR(s(s(z0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0))))
   SQR(s(s(0))) -> c5(SQR(s(0)))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0))))
   SQR(s(s(0))) -> c2(ADD(sqr(s(0)), s(s(0))))
   SQR(s(s(s(z0)))) -> c2(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0)))))))
   SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))))
   SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), dbl(s(0))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0)))))
   SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), dbl(s(0))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0)))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), dbl(0))), s(s(0))))
S tuples:
   TERMS(z0) -> c1(TERMS(s(z0)))
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(0))) -> c4(SQR(s(0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0)))), SQR(s(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0)))), SQR(s(s(z0))))
   SQR(s(s(0))) -> c5(SQR(s(0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
K tuples:
   TERMS(z0) -> c(SQR(z0))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0))))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0))))
Defined Rule Symbols:   sqr_1, add_2, dbl_1

Defined Pair Symbols:   TERMS_1, DBL_1, ADD_2, FIRST_2, SQR_1

Compound Symbols:   c_1, c1_1, c6_1, c8_1, c10_1, c2_1, c3_2, c4_1, c5_1


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

(105) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0)))), SQR(s(s(z0)))) by SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
----------------------------------------

(106)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sqr(0) -> 0
   sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   dbl(0) -> 0
   dbl(s(z0)) -> s(s(dbl(z0)))
Tuples:
   TERMS(z0) -> c(SQR(z0))
   TERMS(z0) -> c1(TERMS(s(z0)))
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(0))) -> c4(SQR(s(0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0)))), SQR(s(s(z0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0))))
   SQR(s(s(0))) -> c5(SQR(s(0)))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0))))
   SQR(s(s(0))) -> c2(ADD(sqr(s(0)), s(s(0))))
   SQR(s(s(s(z0)))) -> c2(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0)))))))
   SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))))
   SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), dbl(s(0))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0)))))
   SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), dbl(s(0))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0)))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), dbl(0))), s(s(0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
S tuples:
   TERMS(z0) -> c1(TERMS(s(z0)))
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(0))) -> c4(SQR(s(0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0)))), SQR(s(s(z0))))
   SQR(s(s(0))) -> c5(SQR(s(0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
K tuples:
   TERMS(z0) -> c(SQR(z0))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0))))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0))))
Defined Rule Symbols:   sqr_1, add_2, dbl_1

Defined Pair Symbols:   TERMS_1, DBL_1, ADD_2, FIRST_2, SQR_1

Compound Symbols:   c_1, c1_1, c6_1, c8_1, c10_1, c2_1, c3_2, c4_1, c5_1


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

(107) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0)))), SQR(s(s(z0)))) by SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
----------------------------------------

(108)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sqr(0) -> 0
   sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   dbl(0) -> 0
   dbl(s(z0)) -> s(s(dbl(z0)))
Tuples:
   TERMS(z0) -> c(SQR(z0))
   TERMS(z0) -> c1(TERMS(s(z0)))
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(0))) -> c4(SQR(s(0)))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0))))
   SQR(s(s(0))) -> c5(SQR(s(0)))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0))))
   SQR(s(s(0))) -> c2(ADD(sqr(s(0)), s(s(0))))
   SQR(s(s(s(z0)))) -> c2(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0)))))))
   SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))))
   SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), dbl(s(0))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0)))))
   SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), dbl(s(0))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0)))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), dbl(0))), s(s(0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
S tuples:
   TERMS(z0) -> c1(TERMS(s(z0)))
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(0))) -> c4(SQR(s(0)))
   SQR(s(s(0))) -> c5(SQR(s(0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
K tuples:
   TERMS(z0) -> c(SQR(z0))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0))))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0))))
Defined Rule Symbols:   sqr_1, add_2, dbl_1

Defined Pair Symbols:   TERMS_1, DBL_1, ADD_2, FIRST_2, SQR_1

Compound Symbols:   c_1, c1_1, c6_1, c8_1, c10_1, c2_1, c3_2, c4_1, c5_1


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

(109) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0)))) by SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), s(s(dbl(0)))))
----------------------------------------

(110)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sqr(0) -> 0
   sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   dbl(0) -> 0
   dbl(s(z0)) -> s(s(dbl(z0)))
Tuples:
   TERMS(z0) -> c(SQR(z0))
   TERMS(z0) -> c1(TERMS(s(z0)))
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(0))) -> c4(SQR(s(0)))
   SQR(s(s(0))) -> c5(SQR(s(0)))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0))))
   SQR(s(s(0))) -> c2(ADD(sqr(s(0)), s(s(0))))
   SQR(s(s(s(z0)))) -> c2(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0)))))))
   SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))))
   SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), dbl(s(0))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0)))))
   SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), dbl(s(0))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0)))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), dbl(0))), s(s(0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), s(s(dbl(0)))))
S tuples:
   TERMS(z0) -> c1(TERMS(s(z0)))
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(0))) -> c4(SQR(s(0)))
   SQR(s(s(0))) -> c5(SQR(s(0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
K tuples:
   TERMS(z0) -> c(SQR(z0))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0))))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0))))
Defined Rule Symbols:   sqr_1, add_2, dbl_1

Defined Pair Symbols:   TERMS_1, DBL_1, ADD_2, FIRST_2, SQR_1

Compound Symbols:   c_1, c1_1, c6_1, c8_1, c10_1, c2_1, c3_2, c4_1, c5_1


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

(111) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0)))) by SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), s(s(dbl(0)))))
----------------------------------------

(112)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sqr(0) -> 0
   sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   dbl(0) -> 0
   dbl(s(z0)) -> s(s(dbl(z0)))
Tuples:
   TERMS(z0) -> c(SQR(z0))
   TERMS(z0) -> c1(TERMS(s(z0)))
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(0))) -> c4(SQR(s(0)))
   SQR(s(s(0))) -> c5(SQR(s(0)))
   SQR(s(s(0))) -> c2(ADD(sqr(s(0)), s(s(0))))
   SQR(s(s(s(z0)))) -> c2(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0)))))))
   SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))))
   SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), dbl(s(0))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0)))))
   SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), dbl(s(0))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0)))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), dbl(0))), s(s(0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), s(s(dbl(0)))))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), s(s(dbl(0)))))
S tuples:
   TERMS(z0) -> c1(TERMS(s(z0)))
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(0))) -> c4(SQR(s(0)))
   SQR(s(s(0))) -> c5(SQR(s(0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
K tuples:
   TERMS(z0) -> c(SQR(z0))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0))))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0))))
Defined Rule Symbols:   sqr_1, add_2, dbl_1

Defined Pair Symbols:   TERMS_1, DBL_1, ADD_2, FIRST_2, SQR_1

Compound Symbols:   c_1, c1_1, c6_1, c8_1, c10_1, c2_1, c3_2, c4_1, c5_1


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

(113) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace TERMS(z0) -> c(SQR(z0)) by 
   TERMS(s(x0)) -> c(SQR(s(x0)))

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

(114)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sqr(0) -> 0
   sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   dbl(0) -> 0
   dbl(s(z0)) -> s(s(dbl(z0)))
Tuples:
   TERMS(z0) -> c1(TERMS(s(z0)))
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(0))) -> c4(SQR(s(0)))
   SQR(s(s(0))) -> c5(SQR(s(0)))
   SQR(s(s(0))) -> c2(ADD(sqr(s(0)), s(s(0))))
   SQR(s(s(s(z0)))) -> c2(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0)))))))
   SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))))
   SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), dbl(s(0))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0)))))
   SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), dbl(s(0))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0)))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), dbl(0))), s(s(0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), s(s(dbl(0)))))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), s(s(dbl(0)))))
   TERMS(s(x0)) -> c(SQR(s(x0)))
S tuples:
   TERMS(z0) -> c1(TERMS(s(z0)))
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(0))) -> c4(SQR(s(0)))
   SQR(s(s(0))) -> c5(SQR(s(0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
K tuples:
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0))))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0))))
   TERMS(s(x0)) -> c(SQR(s(x0)))
Defined Rule Symbols:   sqr_1, add_2, dbl_1

Defined Pair Symbols:   TERMS_1, DBL_1, ADD_2, FIRST_2, SQR_1

Compound Symbols:   c1_1, c6_1, c8_1, c10_1, c2_1, c3_2, c4_1, c5_1, c_1


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

(115) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace SQR(s(s(0))) -> c2(ADD(sqr(s(0)), s(s(0)))) by SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), dbl(0))), s(s(0))))
----------------------------------------

(116)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sqr(0) -> 0
   sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   dbl(0) -> 0
   dbl(s(z0)) -> s(s(dbl(z0)))
Tuples:
   TERMS(z0) -> c1(TERMS(s(z0)))
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(0))) -> c4(SQR(s(0)))
   SQR(s(s(0))) -> c5(SQR(s(0)))
   SQR(s(s(s(z0)))) -> c2(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0)))))))
   SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))))
   SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), dbl(s(0))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0)))))
   SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), dbl(s(0))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0)))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), dbl(0))), s(s(0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), s(s(dbl(0)))))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), s(s(dbl(0)))))
   TERMS(s(x0)) -> c(SQR(s(x0)))
   SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), dbl(0))), s(s(0))))
S tuples:
   TERMS(z0) -> c1(TERMS(s(z0)))
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(0))) -> c4(SQR(s(0)))
   SQR(s(s(0))) -> c5(SQR(s(0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
K tuples:
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0))))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0))))
   TERMS(s(x0)) -> c(SQR(s(x0)))
Defined Rule Symbols:   sqr_1, add_2, dbl_1

Defined Pair Symbols:   TERMS_1, DBL_1, ADD_2, FIRST_2, SQR_1

Compound Symbols:   c1_1, c6_1, c8_1, c10_1, c2_1, c3_2, c4_1, c5_1, c_1


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

(117) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace SQR(s(s(s(z0)))) -> c2(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0))))))) by SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))))
----------------------------------------

(118)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sqr(0) -> 0
   sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   dbl(0) -> 0
   dbl(s(z0)) -> s(s(dbl(z0)))
Tuples:
   TERMS(z0) -> c1(TERMS(s(z0)))
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(0))) -> c4(SQR(s(0)))
   SQR(s(s(0))) -> c5(SQR(s(0)))
   SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))))
   SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), dbl(s(0))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0)))))
   SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), dbl(s(0))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0)))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), dbl(0))), s(s(0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), s(s(dbl(0)))))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), s(s(dbl(0)))))
   TERMS(s(x0)) -> c(SQR(s(x0)))
   SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), dbl(0))), s(s(0))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))))
S tuples:
   TERMS(z0) -> c1(TERMS(s(z0)))
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(0))) -> c4(SQR(s(0)))
   SQR(s(s(0))) -> c5(SQR(s(0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
K tuples:
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0))))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0))))
   TERMS(s(x0)) -> c(SQR(s(x0)))
Defined Rule Symbols:   sqr_1, add_2, dbl_1

Defined Pair Symbols:   TERMS_1, DBL_1, ADD_2, FIRST_2, SQR_1

Compound Symbols:   c1_1, c6_1, c8_1, c10_1, c2_1, c3_2, c4_1, c5_1, c_1


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

(119) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace TERMS(z0) -> c1(TERMS(s(z0))) by 
   TERMS(s(x0)) -> c1(TERMS(s(s(x0))))

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

(120)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sqr(0) -> 0
   sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   dbl(0) -> 0
   dbl(s(z0)) -> s(s(dbl(z0)))
Tuples:
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(0))) -> c4(SQR(s(0)))
   SQR(s(s(0))) -> c5(SQR(s(0)))
   SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))))
   SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), dbl(s(0))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0)))))
   SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), dbl(s(0))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0)))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), dbl(0))), s(s(0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), s(s(dbl(0)))))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), s(s(dbl(0)))))
   TERMS(s(x0)) -> c(SQR(s(x0)))
   SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), dbl(0))), s(s(0))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))))
   TERMS(s(x0)) -> c1(TERMS(s(s(x0))))
S tuples:
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(0))) -> c4(SQR(s(0)))
   SQR(s(s(0))) -> c5(SQR(s(0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
   TERMS(s(x0)) -> c1(TERMS(s(s(x0))))
K tuples:
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0))))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0))))
   TERMS(s(x0)) -> c(SQR(s(x0)))
Defined Rule Symbols:   sqr_1, add_2, dbl_1

Defined Pair Symbols:   DBL_1, ADD_2, FIRST_2, SQR_1, TERMS_1

Compound Symbols:   c6_1, c8_1, c10_1, c2_1, c3_2, c4_1, c5_1, c_1, c1_1


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

(121) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), dbl(s(0)))) by SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), s(s(dbl(0)))))
----------------------------------------

(122)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sqr(0) -> 0
   sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   dbl(0) -> 0
   dbl(s(z0)) -> s(s(dbl(z0)))
Tuples:
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(0))) -> c4(SQR(s(0)))
   SQR(s(s(0))) -> c5(SQR(s(0)))
   SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0)))))
   SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), dbl(s(0))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0)))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), dbl(0))), s(s(0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), s(s(dbl(0)))))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), s(s(dbl(0)))))
   TERMS(s(x0)) -> c(SQR(s(x0)))
   SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), dbl(0))), s(s(0))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))))
   TERMS(s(x0)) -> c1(TERMS(s(s(x0))))
   SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), s(s(dbl(0)))))
S tuples:
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(0))) -> c4(SQR(s(0)))
   SQR(s(s(0))) -> c5(SQR(s(0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
   TERMS(s(x0)) -> c1(TERMS(s(s(x0))))
K tuples:
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0))))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0))))
   TERMS(s(x0)) -> c(SQR(s(x0)))
Defined Rule Symbols:   sqr_1, add_2, dbl_1

Defined Pair Symbols:   DBL_1, ADD_2, FIRST_2, SQR_1, TERMS_1

Compound Symbols:   c6_1, c8_1, c10_1, c2_1, c3_2, c4_1, c5_1, c_1, c1_1


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

(123) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0))))) by SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))))
----------------------------------------

(124)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sqr(0) -> 0
   sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   dbl(0) -> 0
   dbl(s(z0)) -> s(s(dbl(z0)))
Tuples:
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(0))) -> c4(SQR(s(0)))
   SQR(s(s(0))) -> c5(SQR(s(0)))
   SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))))
   SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), dbl(s(0))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0)))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), dbl(0))), s(s(0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), s(s(dbl(0)))))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), s(s(dbl(0)))))
   TERMS(s(x0)) -> c(SQR(s(x0)))
   SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), dbl(0))), s(s(0))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))))
   TERMS(s(x0)) -> c1(TERMS(s(s(x0))))
   SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), s(s(dbl(0)))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))))
S tuples:
   DBL(s(z0)) -> c6(DBL(z0))
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(0))) -> c4(SQR(s(0)))
   SQR(s(s(0))) -> c5(SQR(s(0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
   TERMS(s(x0)) -> c1(TERMS(s(s(x0))))
K tuples:
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0))))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0))))
   TERMS(s(x0)) -> c(SQR(s(x0)))
Defined Rule Symbols:   sqr_1, add_2, dbl_1

Defined Pair Symbols:   DBL_1, ADD_2, FIRST_2, SQR_1, TERMS_1

Compound Symbols:   c6_1, c8_1, c10_1, c2_1, c3_2, c4_1, c5_1, c_1, c1_1


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

(125) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace DBL(s(z0)) -> c6(DBL(z0)) by 
   DBL(s(s(y0))) -> c6(DBL(s(y0)))

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

(126)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sqr(0) -> 0
   sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   dbl(0) -> 0
   dbl(s(z0)) -> s(s(dbl(z0)))
Tuples:
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(0))) -> c4(SQR(s(0)))
   SQR(s(s(0))) -> c5(SQR(s(0)))
   SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))))
   SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), dbl(s(0))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0)))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), dbl(0))), s(s(0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), s(s(dbl(0)))))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), s(s(dbl(0)))))
   TERMS(s(x0)) -> c(SQR(s(x0)))
   SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), dbl(0))), s(s(0))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))))
   TERMS(s(x0)) -> c1(TERMS(s(s(x0))))
   SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), s(s(dbl(0)))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))))
   DBL(s(s(y0))) -> c6(DBL(s(y0)))
S tuples:
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(0))) -> c4(SQR(s(0)))
   SQR(s(s(0))) -> c5(SQR(s(0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
   TERMS(s(x0)) -> c1(TERMS(s(s(x0))))
   DBL(s(s(y0))) -> c6(DBL(s(y0)))
K tuples:
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0))))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0))))
   TERMS(s(x0)) -> c(SQR(s(x0)))
Defined Rule Symbols:   sqr_1, add_2, dbl_1

Defined Pair Symbols:   ADD_2, FIRST_2, SQR_1, TERMS_1, DBL_1

Compound Symbols:   c8_1, c10_1, c2_1, c3_2, c4_1, c5_1, c_1, c1_1, c6_1


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

(127) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), dbl(s(0)))) by SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), s(s(dbl(0)))))
----------------------------------------

(128)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sqr(0) -> 0
   sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   dbl(0) -> 0
   dbl(s(z0)) -> s(s(dbl(z0)))
Tuples:
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(0))) -> c4(SQR(s(0)))
   SQR(s(s(0))) -> c5(SQR(s(0)))
   SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0)))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), dbl(0))), s(s(0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), s(s(dbl(0)))))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), s(s(dbl(0)))))
   TERMS(s(x0)) -> c(SQR(s(x0)))
   SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), dbl(0))), s(s(0))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))))
   TERMS(s(x0)) -> c1(TERMS(s(s(x0))))
   SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), s(s(dbl(0)))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))))
   DBL(s(s(y0))) -> c6(DBL(s(y0)))
   SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), s(s(dbl(0)))))
S tuples:
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(0))) -> c4(SQR(s(0)))
   SQR(s(s(0))) -> c5(SQR(s(0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
   TERMS(s(x0)) -> c1(TERMS(s(s(x0))))
   DBL(s(s(y0))) -> c6(DBL(s(y0)))
K tuples:
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0))))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0))))
   TERMS(s(x0)) -> c(SQR(s(x0)))
Defined Rule Symbols:   sqr_1, add_2, dbl_1

Defined Pair Symbols:   ADD_2, FIRST_2, SQR_1, TERMS_1, DBL_1

Compound Symbols:   c8_1, c10_1, c2_1, c3_2, c4_1, c5_1, c_1, c1_1, c6_1


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

(129) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0))))) by SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))))
----------------------------------------

(130)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sqr(0) -> 0
   sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   dbl(0) -> 0
   dbl(s(z0)) -> s(s(dbl(z0)))
Tuples:
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(0))) -> c4(SQR(s(0)))
   SQR(s(s(0))) -> c5(SQR(s(0)))
   SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), dbl(0))), s(s(0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), s(s(dbl(0)))))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), s(s(dbl(0)))))
   TERMS(s(x0)) -> c(SQR(s(x0)))
   SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), dbl(0))), s(s(0))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))))
   TERMS(s(x0)) -> c1(TERMS(s(s(x0))))
   SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), s(s(dbl(0)))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))))
   DBL(s(s(y0))) -> c6(DBL(s(y0)))
   SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), s(s(dbl(0)))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))))
S tuples:
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(0))) -> c4(SQR(s(0)))
   SQR(s(s(0))) -> c5(SQR(s(0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
   TERMS(s(x0)) -> c1(TERMS(s(s(x0))))
   DBL(s(s(y0))) -> c6(DBL(s(y0)))
K tuples:
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0))))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0))))
   TERMS(s(x0)) -> c(SQR(s(x0)))
Defined Rule Symbols:   sqr_1, add_2, dbl_1

Defined Pair Symbols:   ADD_2, FIRST_2, SQR_1, TERMS_1, DBL_1

Compound Symbols:   c8_1, c10_1, c2_1, c3_2, c4_1, c5_1, c_1, c1_1, c6_1


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

(131) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) by SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
----------------------------------------

(132)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sqr(0) -> 0
   sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   dbl(0) -> 0
   dbl(s(z0)) -> s(s(dbl(z0)))
Tuples:
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(0))) -> c4(SQR(s(0)))
   SQR(s(s(0))) -> c5(SQR(s(0)))
   SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))))
   SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), dbl(0))), s(s(0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), s(s(dbl(0)))))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), s(s(dbl(0)))))
   TERMS(s(x0)) -> c(SQR(s(x0)))
   SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), dbl(0))), s(s(0))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))))
   TERMS(s(x0)) -> c1(TERMS(s(s(x0))))
   SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), s(s(dbl(0)))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))))
   DBL(s(s(y0))) -> c6(DBL(s(y0)))
   SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), s(s(dbl(0)))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
S tuples:
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(0))) -> c4(SQR(s(0)))
   SQR(s(s(0))) -> c5(SQR(s(0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
   TERMS(s(x0)) -> c1(TERMS(s(s(x0))))
   DBL(s(s(y0))) -> c6(DBL(s(y0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
K tuples:
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0))))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0))))
   TERMS(s(x0)) -> c(SQR(s(x0)))
Defined Rule Symbols:   sqr_1, add_2, dbl_1

Defined Pair Symbols:   ADD_2, FIRST_2, SQR_1, TERMS_1, DBL_1

Compound Symbols:   c8_1, c10_1, c2_1, c3_2, c4_1, c5_1, c_1, c1_1, c6_1


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

(133) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace TERMS(s(x0)) -> c(SQR(s(x0))) by 
   TERMS(s(s(x0))) -> c(SQR(s(s(x0))))

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

(134)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sqr(0) -> 0
   sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   dbl(0) -> 0
   dbl(s(z0)) -> s(s(dbl(z0)))
Tuples:
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(0))) -> c4(SQR(s(0)))
   SQR(s(s(0))) -> c5(SQR(s(0)))
   SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))))
   SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), dbl(0))), s(s(0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), s(s(dbl(0)))))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), s(s(dbl(0)))))
   SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), dbl(0))), s(s(0))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))))
   TERMS(s(x0)) -> c1(TERMS(s(s(x0))))
   SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), s(s(dbl(0)))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))))
   DBL(s(s(y0))) -> c6(DBL(s(y0)))
   SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), s(s(dbl(0)))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   TERMS(s(s(x0))) -> c(SQR(s(s(x0))))
S tuples:
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(0))) -> c4(SQR(s(0)))
   SQR(s(s(0))) -> c5(SQR(s(0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
   TERMS(s(x0)) -> c1(TERMS(s(s(x0))))
   DBL(s(s(y0))) -> c6(DBL(s(y0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
K tuples:
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0))))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0))))
   TERMS(s(s(x0))) -> c(SQR(s(s(x0))))
Defined Rule Symbols:   sqr_1, add_2, dbl_1

Defined Pair Symbols:   ADD_2, FIRST_2, SQR_1, TERMS_1, DBL_1

Compound Symbols:   c8_1, c10_1, c2_1, c3_2, c4_1, c5_1, c1_1, c6_1, c_1


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

(135) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace TERMS(s(x0)) -> c1(TERMS(s(s(x0)))) by 
   TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0)))))

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

(136)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sqr(0) -> 0
   sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   dbl(0) -> 0
   dbl(s(z0)) -> s(s(dbl(z0)))
Tuples:
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(0))) -> c4(SQR(s(0)))
   SQR(s(s(0))) -> c5(SQR(s(0)))
   SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))))
   SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), dbl(0))), s(s(0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), s(s(dbl(0)))))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), s(s(dbl(0)))))
   SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), dbl(0))), s(s(0))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))))
   SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), s(s(dbl(0)))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))))
   DBL(s(s(y0))) -> c6(DBL(s(y0)))
   SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), s(s(dbl(0)))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   TERMS(s(s(x0))) -> c(SQR(s(s(x0))))
   TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0)))))
S tuples:
   ADD(s(z0), z1) -> c8(ADD(z0, z1))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(0))) -> c4(SQR(s(0)))
   SQR(s(s(0))) -> c5(SQR(s(0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
   DBL(s(s(y0))) -> c6(DBL(s(y0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0)))))
K tuples:
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0))))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0))))
   TERMS(s(s(x0))) -> c(SQR(s(s(x0))))
Defined Rule Symbols:   sqr_1, add_2, dbl_1

Defined Pair Symbols:   ADD_2, FIRST_2, SQR_1, DBL_1, TERMS_1

Compound Symbols:   c8_1, c10_1, c2_1, c3_2, c4_1, c5_1, c6_1, c_1, c1_1


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

(137) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace ADD(s(z0), z1) -> c8(ADD(z0, z1)) by 
   ADD(s(s(y0)), z1) -> c8(ADD(s(y0), z1))

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

(138)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sqr(0) -> 0
   sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   dbl(0) -> 0
   dbl(s(z0)) -> s(s(dbl(z0)))
Tuples:
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(0))) -> c4(SQR(s(0)))
   SQR(s(s(0))) -> c5(SQR(s(0)))
   SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))))
   SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), dbl(0))), s(s(0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), s(s(dbl(0)))))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), s(s(dbl(0)))))
   SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), dbl(0))), s(s(0))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))))
   SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), s(s(dbl(0)))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))))
   DBL(s(s(y0))) -> c6(DBL(s(y0)))
   SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), s(s(dbl(0)))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   TERMS(s(s(x0))) -> c(SQR(s(s(x0))))
   TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0)))))
   ADD(s(s(y0)), z1) -> c8(ADD(s(y0), z1))
S tuples:
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(0))) -> c4(SQR(s(0)))
   SQR(s(s(0))) -> c5(SQR(s(0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
   DBL(s(s(y0))) -> c6(DBL(s(y0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0)))))
   ADD(s(s(y0)), z1) -> c8(ADD(s(y0), z1))
K tuples:
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0))))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0))))
   TERMS(s(s(x0))) -> c(SQR(s(s(x0))))
Defined Rule Symbols:   sqr_1, add_2, dbl_1

Defined Pair Symbols:   FIRST_2, SQR_1, DBL_1, TERMS_1, ADD_2

Compound Symbols:   c10_1, c2_1, c3_2, c4_1, c5_1, c6_1, c_1, c1_1, c8_1


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

(139) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), dbl(0))), s(s(0)))) by SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), 0)), s(s(0))))
----------------------------------------

(140)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sqr(0) -> 0
   sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   dbl(0) -> 0
   dbl(s(z0)) -> s(s(dbl(z0)))
Tuples:
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(0))) -> c4(SQR(s(0)))
   SQR(s(s(0))) -> c5(SQR(s(0)))
   SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), s(s(dbl(0)))))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), s(s(dbl(0)))))
   SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), dbl(0))), s(s(0))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))))
   SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), s(s(dbl(0)))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))))
   DBL(s(s(y0))) -> c6(DBL(s(y0)))
   SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), s(s(dbl(0)))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   TERMS(s(s(x0))) -> c(SQR(s(s(x0))))
   TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0)))))
   ADD(s(s(y0)), z1) -> c8(ADD(s(y0), z1))
   SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), 0)), s(s(0))))
S tuples:
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(0))) -> c4(SQR(s(0)))
   SQR(s(s(0))) -> c5(SQR(s(0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
   DBL(s(s(y0))) -> c6(DBL(s(y0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0)))))
   ADD(s(s(y0)), z1) -> c8(ADD(s(y0), z1))
K tuples:
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0))))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0))))
   TERMS(s(s(x0))) -> c(SQR(s(s(x0))))
Defined Rule Symbols:   sqr_1, add_2, dbl_1

Defined Pair Symbols:   FIRST_2, SQR_1, DBL_1, TERMS_1, ADD_2

Compound Symbols:   c10_1, c2_1, c3_2, c4_1, c5_1, c6_1, c_1, c1_1, c8_1


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

(141) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) by SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
----------------------------------------

(142)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sqr(0) -> 0
   sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   dbl(0) -> 0
   dbl(s(z0)) -> s(s(dbl(z0)))
Tuples:
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(0))) -> c4(SQR(s(0)))
   SQR(s(s(0))) -> c5(SQR(s(0)))
   SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), s(s(dbl(0)))))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), s(s(dbl(0)))))
   SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), dbl(0))), s(s(0))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))))
   SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), s(s(dbl(0)))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))))
   DBL(s(s(y0))) -> c6(DBL(s(y0)))
   SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), s(s(dbl(0)))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   TERMS(s(s(x0))) -> c(SQR(s(s(x0))))
   TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0)))))
   ADD(s(s(y0)), z1) -> c8(ADD(s(y0), z1))
   SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), 0)), s(s(0))))
S tuples:
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(0))) -> c4(SQR(s(0)))
   SQR(s(s(0))) -> c5(SQR(s(0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
   DBL(s(s(y0))) -> c6(DBL(s(y0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0)))))
   ADD(s(s(y0)), z1) -> c8(ADD(s(y0), z1))
K tuples:
   FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2))
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0))))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0))))
   TERMS(s(s(x0))) -> c(SQR(s(s(x0))))
Defined Rule Symbols:   sqr_1, add_2, dbl_1

Defined Pair Symbols:   FIRST_2, SQR_1, DBL_1, TERMS_1, ADD_2

Compound Symbols:   c10_1, c2_1, c3_2, c4_1, c5_1, c6_1, c_1, c1_1, c8_1


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

(143) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) by 
   FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c10(FIRST(s(y0), cons(y1, y2)))

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

(144)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sqr(0) -> 0
   sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   dbl(0) -> 0
   dbl(s(z0)) -> s(s(dbl(z0)))
Tuples:
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(0))) -> c4(SQR(s(0)))
   SQR(s(s(0))) -> c5(SQR(s(0)))
   SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), s(s(dbl(0)))))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), s(s(dbl(0)))))
   SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), dbl(0))), s(s(0))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))))
   SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), s(s(dbl(0)))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))))
   DBL(s(s(y0))) -> c6(DBL(s(y0)))
   SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), s(s(dbl(0)))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   TERMS(s(s(x0))) -> c(SQR(s(s(x0))))
   TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0)))))
   ADD(s(s(y0)), z1) -> c8(ADD(s(y0), z1))
   SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), 0)), s(s(0))))
   FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c10(FIRST(s(y0), cons(y1, y2)))
S tuples:
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(0))) -> c4(SQR(s(0)))
   SQR(s(s(0))) -> c5(SQR(s(0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
   DBL(s(s(y0))) -> c6(DBL(s(y0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0)))))
   ADD(s(s(y0)), z1) -> c8(ADD(s(y0), z1))
K tuples:
   SQR(s(z0)) -> c2(DBL(z0))
   SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0))))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0))))
   TERMS(s(s(x0))) -> c(SQR(s(s(x0))))
   FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c10(FIRST(s(y0), cons(y1, y2)))
Defined Rule Symbols:   sqr_1, add_2, dbl_1

Defined Pair Symbols:   SQR_1, DBL_1, TERMS_1, ADD_2, FIRST_2

Compound Symbols:   c2_1, c3_2, c4_1, c5_1, c6_1, c_1, c1_1, c8_1, c10_1


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

(145) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace SQR(s(z0)) -> c2(DBL(z0)) by 
   SQR(s(s(s(y0)))) -> c2(DBL(s(s(y0))))

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

(146)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sqr(0) -> 0
   sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   dbl(0) -> 0
   dbl(s(z0)) -> s(s(dbl(z0)))
Tuples:
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(0))) -> c4(SQR(s(0)))
   SQR(s(s(0))) -> c5(SQR(s(0)))
   SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), s(s(dbl(0)))))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), s(s(dbl(0)))))
   SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), dbl(0))), s(s(0))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))))
   SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), s(s(dbl(0)))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))))
   DBL(s(s(y0))) -> c6(DBL(s(y0)))
   SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), s(s(dbl(0)))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   TERMS(s(s(x0))) -> c(SQR(s(s(x0))))
   TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0)))))
   ADD(s(s(y0)), z1) -> c8(ADD(s(y0), z1))
   SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), 0)), s(s(0))))
   FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c10(FIRST(s(y0), cons(y1, y2)))
   SQR(s(s(s(y0)))) -> c2(DBL(s(s(y0))))
S tuples:
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(0))) -> c4(SQR(s(0)))
   SQR(s(s(0))) -> c5(SQR(s(0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
   DBL(s(s(y0))) -> c6(DBL(s(y0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0)))))
   ADD(s(s(y0)), z1) -> c8(ADD(s(y0), z1))
K tuples:
   SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0))))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0))))
   TERMS(s(s(x0))) -> c(SQR(s(s(x0))))
   FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c10(FIRST(s(y0), cons(y1, y2)))
   SQR(s(s(s(y0)))) -> c2(DBL(s(s(y0))))
Defined Rule Symbols:   sqr_1, add_2, dbl_1

Defined Pair Symbols:   SQR_1, DBL_1, TERMS_1, ADD_2, FIRST_2

Compound Symbols:   c3_2, c4_1, c5_1, c2_1, c6_1, c_1, c1_1, c8_1, c10_1


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

(147) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   SQR(s(s(0))) -> c5(SQR(s(0)))
   SQR(s(s(0))) -> c4(SQR(s(0)))

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

(148)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sqr(0) -> 0
   sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   dbl(0) -> 0
   dbl(s(z0)) -> s(s(dbl(z0)))
Tuples:
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), s(s(dbl(0)))))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), s(s(dbl(0)))))
   SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), dbl(0))), s(s(0))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))))
   SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), s(s(dbl(0)))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))))
   DBL(s(s(y0))) -> c6(DBL(s(y0)))
   SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), s(s(dbl(0)))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   TERMS(s(s(x0))) -> c(SQR(s(s(x0))))
   TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0)))))
   ADD(s(s(y0)), z1) -> c8(ADD(s(y0), z1))
   SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), 0)), s(s(0))))
   FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c10(FIRST(s(y0), cons(y1, y2)))
   SQR(s(s(s(y0)))) -> c2(DBL(s(s(y0))))
S tuples:
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0))))
   DBL(s(s(y0))) -> c6(DBL(s(y0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0)))))
   ADD(s(s(y0)), z1) -> c8(ADD(s(y0), z1))
K tuples:
   TERMS(s(s(x0))) -> c(SQR(s(s(x0))))
   FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c10(FIRST(s(y0), cons(y1, y2)))
   SQR(s(s(s(y0)))) -> c2(DBL(s(s(y0))))
Defined Rule Symbols:   sqr_1, add_2, dbl_1

Defined Pair Symbols:   SQR_1, DBL_1, TERMS_1, ADD_2, FIRST_2

Compound Symbols:   c3_2, c2_1, c5_1, c6_1, c_1, c1_1, c8_1, c4_1, c10_1


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

(149) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) by SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
----------------------------------------

(150)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sqr(0) -> 0
   sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   dbl(0) -> 0
   dbl(s(z0)) -> s(s(dbl(z0)))
Tuples:
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), s(s(dbl(0)))))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), s(s(dbl(0)))))
   SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), dbl(0))), s(s(0))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))))
   SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), s(s(dbl(0)))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))))
   DBL(s(s(y0))) -> c6(DBL(s(y0)))
   SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), s(s(dbl(0)))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   TERMS(s(s(x0))) -> c(SQR(s(s(x0))))
   TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0)))))
   ADD(s(s(y0)), z1) -> c8(ADD(s(y0), z1))
   SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), 0)), s(s(0))))
   FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c10(FIRST(s(y0), cons(y1, y2)))
   SQR(s(s(s(y0)))) -> c2(DBL(s(s(y0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
S tuples:
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   DBL(s(s(y0))) -> c6(DBL(s(y0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0)))))
   ADD(s(s(y0)), z1) -> c8(ADD(s(y0), z1))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
K tuples:
   TERMS(s(s(x0))) -> c(SQR(s(s(x0))))
   FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c10(FIRST(s(y0), cons(y1, y2)))
   SQR(s(s(s(y0)))) -> c2(DBL(s(s(y0))))
Defined Rule Symbols:   sqr_1, add_2, dbl_1

Defined Pair Symbols:   SQR_1, DBL_1, TERMS_1, ADD_2, FIRST_2

Compound Symbols:   c3_2, c2_1, c5_1, c6_1, c_1, c1_1, c8_1, c4_1, c10_1


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

(151) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), s(s(dbl(0))))) by SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), s(s(0))))
----------------------------------------

(152)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sqr(0) -> 0
   sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   dbl(0) -> 0
   dbl(s(z0)) -> s(s(dbl(z0)))
Tuples:
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), s(s(dbl(0)))))
   SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), dbl(0))), s(s(0))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))))
   SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), s(s(dbl(0)))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))))
   DBL(s(s(y0))) -> c6(DBL(s(y0)))
   SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), s(s(dbl(0)))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   TERMS(s(s(x0))) -> c(SQR(s(s(x0))))
   TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0)))))
   ADD(s(s(y0)), z1) -> c8(ADD(s(y0), z1))
   SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), 0)), s(s(0))))
   FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c10(FIRST(s(y0), cons(y1, y2)))
   SQR(s(s(s(y0)))) -> c2(DBL(s(s(y0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), s(s(0))))
S tuples:
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   DBL(s(s(y0))) -> c6(DBL(s(y0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0)))))
   ADD(s(s(y0)), z1) -> c8(ADD(s(y0), z1))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
K tuples:
   TERMS(s(s(x0))) -> c(SQR(s(s(x0))))
   FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c10(FIRST(s(y0), cons(y1, y2)))
   SQR(s(s(s(y0)))) -> c2(DBL(s(s(y0))))
Defined Rule Symbols:   sqr_1, add_2, dbl_1

Defined Pair Symbols:   SQR_1, DBL_1, TERMS_1, ADD_2, FIRST_2

Compound Symbols:   c3_2, c2_1, c5_1, c6_1, c_1, c1_1, c8_1, c4_1, c10_1


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

(153) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), s(s(dbl(0))))) by SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), s(s(0))))
----------------------------------------

(154)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sqr(0) -> 0
   sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   dbl(0) -> 0
   dbl(s(z0)) -> s(s(dbl(z0)))
Tuples:
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))))
   SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), dbl(0))), s(s(0))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))))
   SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), s(s(dbl(0)))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))))
   DBL(s(s(y0))) -> c6(DBL(s(y0)))
   SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), s(s(dbl(0)))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   TERMS(s(s(x0))) -> c(SQR(s(s(x0))))
   TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0)))))
   ADD(s(s(y0)), z1) -> c8(ADD(s(y0), z1))
   SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), 0)), s(s(0))))
   FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c10(FIRST(s(y0), cons(y1, y2)))
   SQR(s(s(s(y0)))) -> c2(DBL(s(s(y0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), s(s(0))))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), s(s(0))))
S tuples:
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   DBL(s(s(y0))) -> c6(DBL(s(y0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0)))))
   ADD(s(s(y0)), z1) -> c8(ADD(s(y0), z1))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
K tuples:
   TERMS(s(s(x0))) -> c(SQR(s(s(x0))))
   FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c10(FIRST(s(y0), cons(y1, y2)))
   SQR(s(s(s(y0)))) -> c2(DBL(s(s(y0))))
Defined Rule Symbols:   sqr_1, add_2, dbl_1

Defined Pair Symbols:   SQR_1, DBL_1, TERMS_1, ADD_2, FIRST_2

Compound Symbols:   c3_2, c2_1, c6_1, c_1, c1_1, c8_1, c4_1, c10_1, c5_1


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

(155) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), dbl(0))), s(s(0)))) by SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), s(s(0))))
----------------------------------------

(156)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sqr(0) -> 0
   sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   dbl(0) -> 0
   dbl(s(z0)) -> s(s(dbl(z0)))
Tuples:
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))))
   SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), s(s(dbl(0)))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))))
   DBL(s(s(y0))) -> c6(DBL(s(y0)))
   SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), s(s(dbl(0)))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   TERMS(s(s(x0))) -> c(SQR(s(s(x0))))
   TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0)))))
   ADD(s(s(y0)), z1) -> c8(ADD(s(y0), z1))
   SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), 0)), s(s(0))))
   FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c10(FIRST(s(y0), cons(y1, y2)))
   SQR(s(s(s(y0)))) -> c2(DBL(s(s(y0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), s(s(0))))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), s(s(0))))
   SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), s(s(0))))
S tuples:
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   DBL(s(s(y0))) -> c6(DBL(s(y0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0)))))
   ADD(s(s(y0)), z1) -> c8(ADD(s(y0), z1))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
K tuples:
   TERMS(s(s(x0))) -> c(SQR(s(s(x0))))
   FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c10(FIRST(s(y0), cons(y1, y2)))
   SQR(s(s(s(y0)))) -> c2(DBL(s(s(y0))))
Defined Rule Symbols:   sqr_1, add_2, dbl_1

Defined Pair Symbols:   SQR_1, DBL_1, TERMS_1, ADD_2, FIRST_2

Compound Symbols:   c3_2, c2_1, c6_1, c_1, c1_1, c8_1, c4_1, c10_1, c5_1


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

(157) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0))))))) by SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))))
----------------------------------------

(158)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sqr(0) -> 0
   sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   dbl(0) -> 0
   dbl(s(z0)) -> s(s(dbl(z0)))
Tuples:
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))))
   SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), s(s(dbl(0)))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))))
   DBL(s(s(y0))) -> c6(DBL(s(y0)))
   SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), s(s(dbl(0)))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   TERMS(s(s(x0))) -> c(SQR(s(s(x0))))
   TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0)))))
   ADD(s(s(y0)), z1) -> c8(ADD(s(y0), z1))
   SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), 0)), s(s(0))))
   FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c10(FIRST(s(y0), cons(y1, y2)))
   SQR(s(s(s(y0)))) -> c2(DBL(s(s(y0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), s(s(0))))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), s(s(0))))
   SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), s(s(0))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))))
S tuples:
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   DBL(s(s(y0))) -> c6(DBL(s(y0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0)))))
   ADD(s(s(y0)), z1) -> c8(ADD(s(y0), z1))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
K tuples:
   TERMS(s(s(x0))) -> c(SQR(s(s(x0))))
   FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c10(FIRST(s(y0), cons(y1, y2)))
   SQR(s(s(s(y0)))) -> c2(DBL(s(s(y0))))
Defined Rule Symbols:   sqr_1, add_2, dbl_1

Defined Pair Symbols:   SQR_1, DBL_1, TERMS_1, ADD_2, FIRST_2

Compound Symbols:   c3_2, c2_1, c6_1, c_1, c1_1, c8_1, c4_1, c10_1, c5_1


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

(159) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), s(s(dbl(0))))) by SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), s(s(0))))
----------------------------------------

(160)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sqr(0) -> 0
   sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   dbl(0) -> 0
   dbl(s(z0)) -> s(s(dbl(z0)))
Tuples:
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))))
   DBL(s(s(y0))) -> c6(DBL(s(y0)))
   SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), s(s(dbl(0)))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   TERMS(s(s(x0))) -> c(SQR(s(s(x0))))
   TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0)))))
   ADD(s(s(y0)), z1) -> c8(ADD(s(y0), z1))
   SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), 0)), s(s(0))))
   FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c10(FIRST(s(y0), cons(y1, y2)))
   SQR(s(s(s(y0)))) -> c2(DBL(s(s(y0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), s(s(0))))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), s(s(0))))
   SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), s(s(0))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))))
S tuples:
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   DBL(s(s(y0))) -> c6(DBL(s(y0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0)))))
   ADD(s(s(y0)), z1) -> c8(ADD(s(y0), z1))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
K tuples:
   TERMS(s(s(x0))) -> c(SQR(s(s(x0))))
   FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c10(FIRST(s(y0), cons(y1, y2)))
   SQR(s(s(s(y0)))) -> c2(DBL(s(s(y0))))
Defined Rule Symbols:   sqr_1, add_2, dbl_1

Defined Pair Symbols:   SQR_1, DBL_1, TERMS_1, ADD_2, FIRST_2

Compound Symbols:   c3_2, c2_1, c6_1, c_1, c1_1, c8_1, c4_1, c10_1, c5_1


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

(161) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0)))))) by SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))))
----------------------------------------

(162)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sqr(0) -> 0
   sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   dbl(0) -> 0
   dbl(s(z0)) -> s(s(dbl(z0)))
Tuples:
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))))
   DBL(s(s(y0))) -> c6(DBL(s(y0)))
   SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), s(s(dbl(0)))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   TERMS(s(s(x0))) -> c(SQR(s(s(x0))))
   TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0)))))
   ADD(s(s(y0)), z1) -> c8(ADD(s(y0), z1))
   SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), 0)), s(s(0))))
   FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c10(FIRST(s(y0), cons(y1, y2)))
   SQR(s(s(s(y0)))) -> c2(DBL(s(s(y0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), s(s(0))))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), s(s(0))))
   SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), s(s(0))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))))
S tuples:
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   DBL(s(s(y0))) -> c6(DBL(s(y0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0)))))
   ADD(s(s(y0)), z1) -> c8(ADD(s(y0), z1))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
K tuples:
   TERMS(s(s(x0))) -> c(SQR(s(s(x0))))
   FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c10(FIRST(s(y0), cons(y1, y2)))
   SQR(s(s(s(y0)))) -> c2(DBL(s(s(y0))))
Defined Rule Symbols:   sqr_1, add_2, dbl_1

Defined Pair Symbols:   SQR_1, DBL_1, TERMS_1, ADD_2, FIRST_2

Compound Symbols:   c3_2, c2_1, c6_1, c_1, c1_1, c8_1, c4_1, c10_1, c5_1


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

(163) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), s(s(dbl(0))))) by SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), s(s(0))))
----------------------------------------

(164)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sqr(0) -> 0
   sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   dbl(0) -> 0
   dbl(s(z0)) -> s(s(dbl(z0)))
Tuples:
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))))
   DBL(s(s(y0))) -> c6(DBL(s(y0)))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   TERMS(s(s(x0))) -> c(SQR(s(s(x0))))
   TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0)))))
   ADD(s(s(y0)), z1) -> c8(ADD(s(y0), z1))
   SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), 0)), s(s(0))))
   FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c10(FIRST(s(y0), cons(y1, y2)))
   SQR(s(s(s(y0)))) -> c2(DBL(s(s(y0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), s(s(0))))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), s(s(0))))
   SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), s(s(0))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))))
   SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), s(s(0))))
S tuples:
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   DBL(s(s(y0))) -> c6(DBL(s(y0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0)))))
   ADD(s(s(y0)), z1) -> c8(ADD(s(y0), z1))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
K tuples:
   TERMS(s(s(x0))) -> c(SQR(s(s(x0))))
   FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c10(FIRST(s(y0), cons(y1, y2)))
   SQR(s(s(s(y0)))) -> c2(DBL(s(s(y0))))
Defined Rule Symbols:   sqr_1, add_2, dbl_1

Defined Pair Symbols:   SQR_1, DBL_1, TERMS_1, ADD_2, FIRST_2

Compound Symbols:   c3_2, c2_1, c6_1, c_1, c1_1, c8_1, c4_1, c10_1, c5_1


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

(165) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0)))))) by SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(s(s(dbl(z0)))))))
----------------------------------------

(166)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sqr(0) -> 0
   sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   dbl(0) -> 0
   dbl(s(z0)) -> s(s(dbl(z0)))
Tuples:
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))))
   DBL(s(s(y0))) -> c6(DBL(s(y0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   TERMS(s(s(x0))) -> c(SQR(s(s(x0))))
   TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0)))))
   ADD(s(s(y0)), z1) -> c8(ADD(s(y0), z1))
   SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), 0)), s(s(0))))
   FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c10(FIRST(s(y0), cons(y1, y2)))
   SQR(s(s(s(y0)))) -> c2(DBL(s(s(y0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), s(s(0))))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), s(s(0))))
   SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), s(s(0))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))))
   SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), s(s(0))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(s(s(dbl(z0)))))))
S tuples:
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   DBL(s(s(y0))) -> c6(DBL(s(y0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0)))))
   ADD(s(s(y0)), z1) -> c8(ADD(s(y0), z1))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
K tuples:
   TERMS(s(s(x0))) -> c(SQR(s(s(x0))))
   FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c10(FIRST(s(y0), cons(y1, y2)))
   SQR(s(s(s(y0)))) -> c2(DBL(s(s(y0))))
Defined Rule Symbols:   sqr_1, add_2, dbl_1

Defined Pair Symbols:   SQR_1, DBL_1, TERMS_1, ADD_2, FIRST_2

Compound Symbols:   c3_2, c2_1, c6_1, c_1, c1_1, c8_1, c4_1, c10_1, c5_1


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

(167) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace DBL(s(s(y0))) -> c6(DBL(s(y0))) by 
   DBL(s(s(s(y0)))) -> c6(DBL(s(s(y0))))

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

(168)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sqr(0) -> 0
   sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   dbl(0) -> 0
   dbl(s(z0)) -> s(s(dbl(z0)))
Tuples:
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   TERMS(s(s(x0))) -> c(SQR(s(s(x0))))
   TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0)))))
   ADD(s(s(y0)), z1) -> c8(ADD(s(y0), z1))
   SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), 0)), s(s(0))))
   FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c10(FIRST(s(y0), cons(y1, y2)))
   SQR(s(s(s(y0)))) -> c2(DBL(s(s(y0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), s(s(0))))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), s(s(0))))
   SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), s(s(0))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))))
   SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), s(s(0))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(s(s(dbl(z0)))))))
   DBL(s(s(s(y0)))) -> c6(DBL(s(s(y0))))
S tuples:
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0)))))
   ADD(s(s(y0)), z1) -> c8(ADD(s(y0), z1))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   DBL(s(s(s(y0)))) -> c6(DBL(s(s(y0))))
K tuples:
   TERMS(s(s(x0))) -> c(SQR(s(s(x0))))
   FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c10(FIRST(s(y0), cons(y1, y2)))
   SQR(s(s(s(y0)))) -> c2(DBL(s(s(y0))))
Defined Rule Symbols:   sqr_1, add_2, dbl_1

Defined Pair Symbols:   SQR_1, TERMS_1, ADD_2, FIRST_2, DBL_1

Compound Symbols:   c3_2, c2_1, c_1, c1_1, c8_1, c4_1, c10_1, c5_1, c6_1


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

(169) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace ADD(s(s(y0)), z1) -> c8(ADD(s(y0), z1)) by 
   ADD(s(s(s(y0))), z1) -> c8(ADD(s(s(y0)), z1))

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

(170)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sqr(0) -> 0
   sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   dbl(0) -> 0
   dbl(s(z0)) -> s(s(dbl(z0)))
Tuples:
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   TERMS(s(s(x0))) -> c(SQR(s(s(x0))))
   TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0)))))
   SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), 0)), s(s(0))))
   FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c10(FIRST(s(y0), cons(y1, y2)))
   SQR(s(s(s(y0)))) -> c2(DBL(s(s(y0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), s(s(0))))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), s(s(0))))
   SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), s(s(0))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))))
   SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), s(s(0))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(s(s(dbl(z0)))))))
   DBL(s(s(s(y0)))) -> c6(DBL(s(s(y0))))
   ADD(s(s(s(y0))), z1) -> c8(ADD(s(s(y0)), z1))
S tuples:
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0)))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   DBL(s(s(s(y0)))) -> c6(DBL(s(s(y0))))
   ADD(s(s(s(y0))), z1) -> c8(ADD(s(s(y0)), z1))
K tuples:
   TERMS(s(s(x0))) -> c(SQR(s(s(x0))))
   FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c10(FIRST(s(y0), cons(y1, y2)))
   SQR(s(s(s(y0)))) -> c2(DBL(s(s(y0))))
Defined Rule Symbols:   sqr_1, add_2, dbl_1

Defined Pair Symbols:   SQR_1, TERMS_1, FIRST_2, DBL_1, ADD_2

Compound Symbols:   c3_2, c2_1, c_1, c1_1, c4_1, c10_1, c5_1, c6_1, c8_1


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

(171) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c10(FIRST(s(y0), cons(y1, y2))) by 
   FIRST(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c10(FIRST(s(s(y0)), cons(z2, cons(y2, y3))))

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

(172)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sqr(0) -> 0
   sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   dbl(0) -> 0
   dbl(s(z0)) -> s(s(dbl(z0)))
Tuples:
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   TERMS(s(s(x0))) -> c(SQR(s(s(x0))))
   TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0)))))
   SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), 0)), s(s(0))))
   SQR(s(s(s(y0)))) -> c2(DBL(s(s(y0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), s(s(0))))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), s(s(0))))
   SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), s(s(0))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))))
   SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), s(s(0))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(s(s(dbl(z0)))))))
   DBL(s(s(s(y0)))) -> c6(DBL(s(s(y0))))
   ADD(s(s(s(y0))), z1) -> c8(ADD(s(s(y0)), z1))
   FIRST(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c10(FIRST(s(s(y0)), cons(z2, cons(y2, y3))))
S tuples:
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0)))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   DBL(s(s(s(y0)))) -> c6(DBL(s(s(y0))))
   ADD(s(s(s(y0))), z1) -> c8(ADD(s(s(y0)), z1))
K tuples:
   TERMS(s(s(x0))) -> c(SQR(s(s(x0))))
   SQR(s(s(s(y0)))) -> c2(DBL(s(s(y0))))
   FIRST(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c10(FIRST(s(s(y0)), cons(z2, cons(y2, y3))))
Defined Rule Symbols:   sqr_1, add_2, dbl_1

Defined Pair Symbols:   SQR_1, TERMS_1, DBL_1, ADD_2, FIRST_2

Compound Symbols:   c3_2, c2_1, c_1, c1_1, c4_1, c5_1, c6_1, c8_1, c10_1


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

(173) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace SQR(s(s(s(y0)))) -> c2(DBL(s(s(y0)))) by 
   SQR(s(s(s(s(y0))))) -> c2(DBL(s(s(s(y0)))))

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

(174)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sqr(0) -> 0
   sqr(s(z0)) -> s(add(sqr(z0), dbl(z0)))
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   dbl(0) -> 0
   dbl(s(z0)) -> s(s(dbl(z0)))
Tuples:
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   TERMS(s(s(x0))) -> c(SQR(s(s(x0))))
   TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0)))))
   SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), 0)), s(s(0))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), s(s(0))))
   SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), s(s(0))))
   SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), s(s(0))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))))
   SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), s(s(0))))
   SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(s(s(dbl(z0)))))))
   DBL(s(s(s(y0)))) -> c6(DBL(s(s(y0))))
   ADD(s(s(s(y0))), z1) -> c8(ADD(s(s(y0)), z1))
   FIRST(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c10(FIRST(s(s(y0)), cons(z2, cons(y2, y3))))
   SQR(s(s(s(s(y0))))) -> c2(DBL(s(s(s(y0)))))
S tuples:
   SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0)))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0)))))
   SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0))))
   DBL(s(s(s(y0)))) -> c6(DBL(s(s(y0))))
   ADD(s(s(s(y0))), z1) -> c8(ADD(s(s(y0)), z1))
K tuples:
   TERMS(s(s(x0))) -> c(SQR(s(s(x0))))
   FIRST(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c10(FIRST(s(s(y0)), cons(z2, cons(y2, y3))))
   SQR(s(s(s(s(y0))))) -> c2(DBL(s(s(s(y0)))))
Defined Rule Symbols:   sqr_1, add_2, dbl_1

Defined Pair Symbols:   SQR_1, TERMS_1, DBL_1, ADD_2, FIRST_2

Compound Symbols:   c3_2, c2_1, c_1, c1_1, c4_1, c5_1, c6_1, c8_1, c10_1