WORST_CASE(Omega(n^1),O(n^2))
proof of /export/starexec/sandbox/benchmark/theBenchmark.trs
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2).

(0) CpxRelTRS
(1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 254 ms]
(2) CpxRelTRS
(3) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(4) CdtProblem
    (5) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CdtProblem
    (7) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CdtProblem
    (9) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CdtProblem
    (11) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 94 ms]
    (12) CdtProblem
    (13) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 57 ms]
    (14) CdtProblem
    (15) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 45 ms]
    (16) CdtProblem
    (17) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 116 ms]
    (18) CdtProblem
    (19) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
    (20) BOUNDS(1, 1)
(21) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(22) CpxRelTRS
    (23) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (24) typed CpxTrs
    (25) OrderProof [LOWER BOUND(ID), 0 ms]
    (26) typed CpxTrs
    (27) RewriteLemmaProof [LOWER BOUND(ID), 2118 ms]
    (28) BEST
        (29) proven lower bound
            (30) LowerBoundPropagationProof [FINISHED, 0 ms]
            (31) BOUNDS(n^1, INF)
        (32) typed CpxTrs
            (33) RewriteLemmaProof [LOWER BOUND(ID), 107 ms]
            (34) typed CpxTrs
            (35) RewriteLemmaProof [LOWER BOUND(ID), 2265 ms]
            (36) typed CpxTrs
            (37) RewriteLemmaProof [LOWER BOUND(ID), 17 ms]
            (38) typed CpxTrs


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

(0)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2).


The TRS R consists of the following rules:

   SQR(0) -> c
   SQR(s(z0)) -> c1(+'(sqr(z0), s(double(z0))), SQR(z0))
   SQR(s(z0)) -> c2(+'(sqr(z0), s(double(z0))), DOUBLE(z0))
   SQR(s(z0)) -> c3(+'(sqr(z0), double(z0)), SQR(z0))
   SQR(s(z0)) -> c4(+'(sqr(z0), double(z0)), DOUBLE(z0))
   DOUBLE(0) -> c5
   DOUBLE(s(z0)) -> c6(DOUBLE(z0))
   +'(z0, 0) -> c7
   +'(z0, s(z1)) -> c8(+'(z0, z1))

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

   sqr(0) -> 0
   sqr(s(z0)) -> +(sqr(z0), s(double(z0)))
   sqr(s(z0)) -> s(+(sqr(z0), double(z0)))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))

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

(1) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
----------------------------------------

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


The TRS R consists of the following rules:

   SQR(0) -> c
   SQR(s(z0)) -> c1(+'(sqr(z0), s(double(z0))), SQR(z0))
   SQR(s(z0)) -> c2(+'(sqr(z0), s(double(z0))), DOUBLE(z0))
   SQR(s(z0)) -> c3(+'(sqr(z0), double(z0)), SQR(z0))
   SQR(s(z0)) -> c4(+'(sqr(z0), double(z0)), DOUBLE(z0))
   DOUBLE(0) -> c5
   DOUBLE(s(z0)) -> c6(DOUBLE(z0))
   +'(z0, 0) -> c7
   +'(z0, s(z1)) -> c8(+'(z0, z1))

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

   sqr(0) -> 0
   sqr(s(z0)) -> +(sqr(z0), s(double(z0)))
   sqr(s(z0)) -> s(+(sqr(z0), double(z0)))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))

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

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sqr(0) -> 0
   sqr(s(z0)) -> +(sqr(z0), s(double(z0)))
   sqr(s(z0)) -> s(+(sqr(z0), double(z0)))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   SQR(0) -> c
   SQR(s(z0)) -> c1(+'(sqr(z0), s(double(z0))), SQR(z0))
   SQR(s(z0)) -> c2(+'(sqr(z0), s(double(z0))), DOUBLE(z0))
   SQR(s(z0)) -> c3(+'(sqr(z0), double(z0)), SQR(z0))
   SQR(s(z0)) -> c4(+'(sqr(z0), double(z0)), DOUBLE(z0))
   DOUBLE(0) -> c5
   DOUBLE(s(z0)) -> c6(DOUBLE(z0))
   +'(z0, 0) -> c7
   +'(z0, s(z1)) -> c8(+'(z0, z1))
Tuples:
   SQR'(0) -> c9
   SQR'(s(z0)) -> c10(+''(sqr(z0), s(double(z0))), SQR'(z0), DOUBLE'(z0))
   SQR'(s(z0)) -> c11(+''(sqr(z0), double(z0)), SQR'(z0), DOUBLE'(z0))
   DOUBLE'(0) -> c12
   DOUBLE'(s(z0)) -> c13(DOUBLE'(z0))
   +''(z0, 0) -> c14
   +''(z0, s(z1)) -> c15(+''(z0, z1))
   SQR''(0) -> c16
   SQR''(s(z0)) -> c17(+'''(sqr(z0), s(double(z0))), SQR'(z0), DOUBLE'(z0), SQR''(z0))
   SQR''(s(z0)) -> c18(+'''(sqr(z0), s(double(z0))), SQR'(z0), DOUBLE'(z0), DOUBLE''(z0))
   SQR''(s(z0)) -> c19(+'''(sqr(z0), double(z0)), SQR'(z0), DOUBLE'(z0), SQR''(z0))
   SQR''(s(z0)) -> c20(+'''(sqr(z0), double(z0)), SQR'(z0), DOUBLE'(z0), DOUBLE''(z0))
   DOUBLE''(0) -> c21
   DOUBLE''(s(z0)) -> c22(DOUBLE''(z0))
   +'''(z0, 0) -> c23
   +'''(z0, s(z1)) -> c24(+'''(z0, z1))
S tuples:
   SQR''(0) -> c16
   SQR''(s(z0)) -> c17(+'''(sqr(z0), s(double(z0))), SQR'(z0), DOUBLE'(z0), SQR''(z0))
   SQR''(s(z0)) -> c18(+'''(sqr(z0), s(double(z0))), SQR'(z0), DOUBLE'(z0), DOUBLE''(z0))
   SQR''(s(z0)) -> c19(+'''(sqr(z0), double(z0)), SQR'(z0), DOUBLE'(z0), SQR''(z0))
   SQR''(s(z0)) -> c20(+'''(sqr(z0), double(z0)), SQR'(z0), DOUBLE'(z0), DOUBLE''(z0))
   DOUBLE''(0) -> c21
   DOUBLE''(s(z0)) -> c22(DOUBLE''(z0))
   +'''(z0, 0) -> c23
   +'''(z0, s(z1)) -> c24(+'''(z0, z1))
K tuples:none
Defined Rule Symbols:   SQR_1, DOUBLE_1, +'_2, sqr_1, double_1, +_2

Defined Pair Symbols:   SQR'_1, DOUBLE'_1, +''_2, SQR''_1, DOUBLE''_1, +'''_2

Compound Symbols:   c9, c10_3, c11_3, c12, c13_1, c14, c15_1, c16, c17_4, c18_4, c19_4, c20_4, c21, c22_1, c23, c24_1


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

(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 6 trailing nodes:
   DOUBLE''(0) -> c21
   SQR'(0) -> c9
   SQR''(0) -> c16
   +''(z0, 0) -> c14
   +'''(z0, 0) -> c23
   DOUBLE'(0) -> c12

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sqr(0) -> 0
   sqr(s(z0)) -> +(sqr(z0), s(double(z0)))
   sqr(s(z0)) -> s(+(sqr(z0), double(z0)))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   SQR(0) -> c
   SQR(s(z0)) -> c1(+'(sqr(z0), s(double(z0))), SQR(z0))
   SQR(s(z0)) -> c2(+'(sqr(z0), s(double(z0))), DOUBLE(z0))
   SQR(s(z0)) -> c3(+'(sqr(z0), double(z0)), SQR(z0))
   SQR(s(z0)) -> c4(+'(sqr(z0), double(z0)), DOUBLE(z0))
   DOUBLE(0) -> c5
   DOUBLE(s(z0)) -> c6(DOUBLE(z0))
   +'(z0, 0) -> c7
   +'(z0, s(z1)) -> c8(+'(z0, z1))
Tuples:
   SQR'(s(z0)) -> c10(+''(sqr(z0), s(double(z0))), SQR'(z0), DOUBLE'(z0))
   SQR'(s(z0)) -> c11(+''(sqr(z0), double(z0)), SQR'(z0), DOUBLE'(z0))
   DOUBLE'(s(z0)) -> c13(DOUBLE'(z0))
   +''(z0, s(z1)) -> c15(+''(z0, z1))
   SQR''(s(z0)) -> c17(+'''(sqr(z0), s(double(z0))), SQR'(z0), DOUBLE'(z0), SQR''(z0))
   SQR''(s(z0)) -> c18(+'''(sqr(z0), s(double(z0))), SQR'(z0), DOUBLE'(z0), DOUBLE''(z0))
   SQR''(s(z0)) -> c19(+'''(sqr(z0), double(z0)), SQR'(z0), DOUBLE'(z0), SQR''(z0))
   SQR''(s(z0)) -> c20(+'''(sqr(z0), double(z0)), SQR'(z0), DOUBLE'(z0), DOUBLE''(z0))
   DOUBLE''(s(z0)) -> c22(DOUBLE''(z0))
   +'''(z0, s(z1)) -> c24(+'''(z0, z1))
S tuples:
   SQR''(s(z0)) -> c17(+'''(sqr(z0), s(double(z0))), SQR'(z0), DOUBLE'(z0), SQR''(z0))
   SQR''(s(z0)) -> c18(+'''(sqr(z0), s(double(z0))), SQR'(z0), DOUBLE'(z0), DOUBLE''(z0))
   SQR''(s(z0)) -> c19(+'''(sqr(z0), double(z0)), SQR'(z0), DOUBLE'(z0), SQR''(z0))
   SQR''(s(z0)) -> c20(+'''(sqr(z0), double(z0)), SQR'(z0), DOUBLE'(z0), DOUBLE''(z0))
   DOUBLE''(s(z0)) -> c22(DOUBLE''(z0))
   +'''(z0, s(z1)) -> c24(+'''(z0, z1))
K tuples:none
Defined Rule Symbols:   SQR_1, DOUBLE_1, +'_2, sqr_1, double_1, +_2

Defined Pair Symbols:   SQR'_1, DOUBLE'_1, +''_2, SQR''_1, DOUBLE''_1, +'''_2

Compound Symbols:   c10_3, c11_3, c13_1, c15_1, c17_4, c18_4, c19_4, c20_4, c22_1, c24_1


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

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sqr(0) -> 0
   sqr(s(z0)) -> +(sqr(z0), s(double(z0)))
   sqr(s(z0)) -> s(+(sqr(z0), double(z0)))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   SQR(0) -> c
   SQR(s(z0)) -> c1(+'(sqr(z0), s(double(z0))), SQR(z0))
   SQR(s(z0)) -> c2(+'(sqr(z0), s(double(z0))), DOUBLE(z0))
   SQR(s(z0)) -> c3(+'(sqr(z0), double(z0)), SQR(z0))
   SQR(s(z0)) -> c4(+'(sqr(z0), double(z0)), DOUBLE(z0))
   DOUBLE(0) -> c5
   DOUBLE(s(z0)) -> c6(DOUBLE(z0))
   +'(z0, 0) -> c7
   +'(z0, s(z1)) -> c8(+'(z0, z1))
Tuples:
   SQR'(s(z0)) -> c10(+''(sqr(z0), s(double(z0))), SQR'(z0), DOUBLE'(z0))
   SQR'(s(z0)) -> c11(+''(sqr(z0), double(z0)), SQR'(z0), DOUBLE'(z0))
   DOUBLE'(s(z0)) -> c13(DOUBLE'(z0))
   +''(z0, s(z1)) -> c15(+''(z0, z1))
   SQR''(s(z0)) -> c17(+'''(sqr(z0), s(double(z0))), SQR'(z0), DOUBLE'(z0), SQR''(z0))
   SQR''(s(z0)) -> c19(+'''(sqr(z0), double(z0)), SQR'(z0), DOUBLE'(z0), SQR''(z0))
   DOUBLE''(s(z0)) -> c22(DOUBLE''(z0))
   +'''(z0, s(z1)) -> c24(+'''(z0, z1))
   SQR''(s(z0)) -> c9(+'''(sqr(z0), s(double(z0))))
   SQR''(s(z0)) -> c9(SQR'(z0))
   SQR''(s(z0)) -> c9(DOUBLE'(z0))
   SQR''(s(z0)) -> c9(DOUBLE''(z0))
   SQR''(s(z0)) -> c9(+'''(sqr(z0), double(z0)))
S tuples:
   SQR''(s(z0)) -> c17(+'''(sqr(z0), s(double(z0))), SQR'(z0), DOUBLE'(z0), SQR''(z0))
   SQR''(s(z0)) -> c19(+'''(sqr(z0), double(z0)), SQR'(z0), DOUBLE'(z0), SQR''(z0))
   DOUBLE''(s(z0)) -> c22(DOUBLE''(z0))
   +'''(z0, s(z1)) -> c24(+'''(z0, z1))
   SQR''(s(z0)) -> c9(+'''(sqr(z0), s(double(z0))))
   SQR''(s(z0)) -> c9(SQR'(z0))
   SQR''(s(z0)) -> c9(DOUBLE'(z0))
   SQR''(s(z0)) -> c9(DOUBLE''(z0))
   SQR''(s(z0)) -> c9(+'''(sqr(z0), double(z0)))
K tuples:none
Defined Rule Symbols:   SQR_1, DOUBLE_1, +'_2, sqr_1, double_1, +_2

Defined Pair Symbols:   SQR'_1, DOUBLE'_1, +''_2, SQR''_1, DOUBLE''_1, +'''_2

Compound Symbols:   c10_3, c11_3, c13_1, c15_1, c17_4, c19_4, c22_1, c24_1, c9_1


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

(9) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   SQR(0) -> c
   SQR(s(z0)) -> c1(+'(sqr(z0), s(double(z0))), SQR(z0))
   SQR(s(z0)) -> c2(+'(sqr(z0), s(double(z0))), DOUBLE(z0))
   SQR(s(z0)) -> c3(+'(sqr(z0), double(z0)), SQR(z0))
   SQR(s(z0)) -> c4(+'(sqr(z0), double(z0)), DOUBLE(z0))
   DOUBLE(0) -> c5
   DOUBLE(s(z0)) -> c6(DOUBLE(z0))
   +'(z0, 0) -> c7
   +'(z0, s(z1)) -> c8(+'(z0, z1))

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

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sqr(0) -> 0
   sqr(s(z0)) -> +(sqr(z0), s(double(z0)))
   sqr(s(z0)) -> s(+(sqr(z0), double(z0)))
   +(z0, s(z1)) -> s(+(z0, z1))
   +(z0, 0) -> z0
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
Tuples:
   SQR'(s(z0)) -> c10(+''(sqr(z0), s(double(z0))), SQR'(z0), DOUBLE'(z0))
   SQR'(s(z0)) -> c11(+''(sqr(z0), double(z0)), SQR'(z0), DOUBLE'(z0))
   DOUBLE'(s(z0)) -> c13(DOUBLE'(z0))
   +''(z0, s(z1)) -> c15(+''(z0, z1))
   SQR''(s(z0)) -> c17(+'''(sqr(z0), s(double(z0))), SQR'(z0), DOUBLE'(z0), SQR''(z0))
   SQR''(s(z0)) -> c19(+'''(sqr(z0), double(z0)), SQR'(z0), DOUBLE'(z0), SQR''(z0))
   DOUBLE''(s(z0)) -> c22(DOUBLE''(z0))
   +'''(z0, s(z1)) -> c24(+'''(z0, z1))
   SQR''(s(z0)) -> c9(+'''(sqr(z0), s(double(z0))))
   SQR''(s(z0)) -> c9(SQR'(z0))
   SQR''(s(z0)) -> c9(DOUBLE'(z0))
   SQR''(s(z0)) -> c9(DOUBLE''(z0))
   SQR''(s(z0)) -> c9(+'''(sqr(z0), double(z0)))
S tuples:
   SQR''(s(z0)) -> c17(+'''(sqr(z0), s(double(z0))), SQR'(z0), DOUBLE'(z0), SQR''(z0))
   SQR''(s(z0)) -> c19(+'''(sqr(z0), double(z0)), SQR'(z0), DOUBLE'(z0), SQR''(z0))
   DOUBLE''(s(z0)) -> c22(DOUBLE''(z0))
   +'''(z0, s(z1)) -> c24(+'''(z0, z1))
   SQR''(s(z0)) -> c9(+'''(sqr(z0), s(double(z0))))
   SQR''(s(z0)) -> c9(SQR'(z0))
   SQR''(s(z0)) -> c9(DOUBLE'(z0))
   SQR''(s(z0)) -> c9(DOUBLE''(z0))
   SQR''(s(z0)) -> c9(+'''(sqr(z0), double(z0)))
K tuples:none
Defined Rule Symbols:   sqr_1, +_2, double_1

Defined Pair Symbols:   SQR'_1, DOUBLE'_1, +''_2, SQR''_1, DOUBLE''_1, +'''_2

Compound Symbols:   c10_3, c11_3, c13_1, c15_1, c17_4, c19_4, c22_1, c24_1, c9_1


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

(11) 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)) -> c9(+'''(sqr(z0), s(double(z0))))
   SQR''(s(z0)) -> c9(SQR'(z0))
   SQR''(s(z0)) -> c9(DOUBLE'(z0))
   SQR''(s(z0)) -> c9(+'''(sqr(z0), double(z0)))
We considered the (Usable) Rules:none
And the Tuples:
   SQR'(s(z0)) -> c10(+''(sqr(z0), s(double(z0))), SQR'(z0), DOUBLE'(z0))
   SQR'(s(z0)) -> c11(+''(sqr(z0), double(z0)), SQR'(z0), DOUBLE'(z0))
   DOUBLE'(s(z0)) -> c13(DOUBLE'(z0))
   +''(z0, s(z1)) -> c15(+''(z0, z1))
   SQR''(s(z0)) -> c17(+'''(sqr(z0), s(double(z0))), SQR'(z0), DOUBLE'(z0), SQR''(z0))
   SQR''(s(z0)) -> c19(+'''(sqr(z0), double(z0)), SQR'(z0), DOUBLE'(z0), SQR''(z0))
   DOUBLE''(s(z0)) -> c22(DOUBLE''(z0))
   +'''(z0, s(z1)) -> c24(+'''(z0, z1))
   SQR''(s(z0)) -> c9(+'''(sqr(z0), s(double(z0))))
   SQR''(s(z0)) -> c9(SQR'(z0))
   SQR''(s(z0)) -> c9(DOUBLE'(z0))
   SQR''(s(z0)) -> c9(DOUBLE''(z0))
   SQR''(s(z0)) -> c9(+'''(sqr(z0), double(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(+(x_1, x_2)) = [1] + x_1 + x_2
   POL(+''(x_1, x_2)) = 0
   POL(+'''(x_1, x_2)) = 0
   POL(0) = [1]
   POL(DOUBLE'(x_1)) = 0
   POL(DOUBLE''(x_1)) = [1]
   POL(SQR'(x_1)) = 0
   POL(SQR''(x_1)) = [1]
   POL(c10(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c11(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c13(x_1)) = x_1
   POL(c15(x_1)) = x_1
   POL(c17(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
   POL(c19(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
   POL(c22(x_1)) = x_1
   POL(c24(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(double(x_1)) = [1] + x_1
   POL(s(x_1)) = 0
   POL(sqr(x_1)) = [1] + x_1

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

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sqr(0) -> 0
   sqr(s(z0)) -> +(sqr(z0), s(double(z0)))
   sqr(s(z0)) -> s(+(sqr(z0), double(z0)))
   +(z0, s(z1)) -> s(+(z0, z1))
   +(z0, 0) -> z0
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
Tuples:
   SQR'(s(z0)) -> c10(+''(sqr(z0), s(double(z0))), SQR'(z0), DOUBLE'(z0))
   SQR'(s(z0)) -> c11(+''(sqr(z0), double(z0)), SQR'(z0), DOUBLE'(z0))
   DOUBLE'(s(z0)) -> c13(DOUBLE'(z0))
   +''(z0, s(z1)) -> c15(+''(z0, z1))
   SQR''(s(z0)) -> c17(+'''(sqr(z0), s(double(z0))), SQR'(z0), DOUBLE'(z0), SQR''(z0))
   SQR''(s(z0)) -> c19(+'''(sqr(z0), double(z0)), SQR'(z0), DOUBLE'(z0), SQR''(z0))
   DOUBLE''(s(z0)) -> c22(DOUBLE''(z0))
   +'''(z0, s(z1)) -> c24(+'''(z0, z1))
   SQR''(s(z0)) -> c9(+'''(sqr(z0), s(double(z0))))
   SQR''(s(z0)) -> c9(SQR'(z0))
   SQR''(s(z0)) -> c9(DOUBLE'(z0))
   SQR''(s(z0)) -> c9(DOUBLE''(z0))
   SQR''(s(z0)) -> c9(+'''(sqr(z0), double(z0)))
S tuples:
   SQR''(s(z0)) -> c17(+'''(sqr(z0), s(double(z0))), SQR'(z0), DOUBLE'(z0), SQR''(z0))
   SQR''(s(z0)) -> c19(+'''(sqr(z0), double(z0)), SQR'(z0), DOUBLE'(z0), SQR''(z0))
   DOUBLE''(s(z0)) -> c22(DOUBLE''(z0))
   +'''(z0, s(z1)) -> c24(+'''(z0, z1))
   SQR''(s(z0)) -> c9(DOUBLE''(z0))
K tuples:
   SQR''(s(z0)) -> c9(+'''(sqr(z0), s(double(z0))))
   SQR''(s(z0)) -> c9(SQR'(z0))
   SQR''(s(z0)) -> c9(DOUBLE'(z0))
   SQR''(s(z0)) -> c9(+'''(sqr(z0), double(z0)))
Defined Rule Symbols:   sqr_1, +_2, double_1

Defined Pair Symbols:   SQR'_1, DOUBLE'_1, +''_2, SQR''_1, DOUBLE''_1, +'''_2

Compound Symbols:   c10_3, c11_3, c13_1, c15_1, c17_4, c19_4, c22_1, c24_1, c9_1


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

(13) 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)) -> c9(DOUBLE''(z0))
We considered the (Usable) Rules:none
And the Tuples:
   SQR'(s(z0)) -> c10(+''(sqr(z0), s(double(z0))), SQR'(z0), DOUBLE'(z0))
   SQR'(s(z0)) -> c11(+''(sqr(z0), double(z0)), SQR'(z0), DOUBLE'(z0))
   DOUBLE'(s(z0)) -> c13(DOUBLE'(z0))
   +''(z0, s(z1)) -> c15(+''(z0, z1))
   SQR''(s(z0)) -> c17(+'''(sqr(z0), s(double(z0))), SQR'(z0), DOUBLE'(z0), SQR''(z0))
   SQR''(s(z0)) -> c19(+'''(sqr(z0), double(z0)), SQR'(z0), DOUBLE'(z0), SQR''(z0))
   DOUBLE''(s(z0)) -> c22(DOUBLE''(z0))
   +'''(z0, s(z1)) -> c24(+'''(z0, z1))
   SQR''(s(z0)) -> c9(+'''(sqr(z0), s(double(z0))))
   SQR''(s(z0)) -> c9(SQR'(z0))
   SQR''(s(z0)) -> c9(DOUBLE'(z0))
   SQR''(s(z0)) -> c9(DOUBLE''(z0))
   SQR''(s(z0)) -> c9(+'''(sqr(z0), double(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(+(x_1, x_2)) = [3]
   POL(+''(x_1, x_2)) = 0
   POL(+'''(x_1, x_2)) = 0
   POL(0) = 0
   POL(DOUBLE'(x_1)) = 0
   POL(DOUBLE''(x_1)) = [1]
   POL(SQR'(x_1)) = 0
   POL(SQR''(x_1)) = [2]
   POL(c10(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c11(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c13(x_1)) = x_1
   POL(c15(x_1)) = x_1
   POL(c17(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
   POL(c19(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
   POL(c22(x_1)) = x_1
   POL(c24(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(double(x_1)) = 0
   POL(s(x_1)) = 0
   POL(sqr(x_1)) = 0

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

(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sqr(0) -> 0
   sqr(s(z0)) -> +(sqr(z0), s(double(z0)))
   sqr(s(z0)) -> s(+(sqr(z0), double(z0)))
   +(z0, s(z1)) -> s(+(z0, z1))
   +(z0, 0) -> z0
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
Tuples:
   SQR'(s(z0)) -> c10(+''(sqr(z0), s(double(z0))), SQR'(z0), DOUBLE'(z0))
   SQR'(s(z0)) -> c11(+''(sqr(z0), double(z0)), SQR'(z0), DOUBLE'(z0))
   DOUBLE'(s(z0)) -> c13(DOUBLE'(z0))
   +''(z0, s(z1)) -> c15(+''(z0, z1))
   SQR''(s(z0)) -> c17(+'''(sqr(z0), s(double(z0))), SQR'(z0), DOUBLE'(z0), SQR''(z0))
   SQR''(s(z0)) -> c19(+'''(sqr(z0), double(z0)), SQR'(z0), DOUBLE'(z0), SQR''(z0))
   DOUBLE''(s(z0)) -> c22(DOUBLE''(z0))
   +'''(z0, s(z1)) -> c24(+'''(z0, z1))
   SQR''(s(z0)) -> c9(+'''(sqr(z0), s(double(z0))))
   SQR''(s(z0)) -> c9(SQR'(z0))
   SQR''(s(z0)) -> c9(DOUBLE'(z0))
   SQR''(s(z0)) -> c9(DOUBLE''(z0))
   SQR''(s(z0)) -> c9(+'''(sqr(z0), double(z0)))
S tuples:
   SQR''(s(z0)) -> c17(+'''(sqr(z0), s(double(z0))), SQR'(z0), DOUBLE'(z0), SQR''(z0))
   SQR''(s(z0)) -> c19(+'''(sqr(z0), double(z0)), SQR'(z0), DOUBLE'(z0), SQR''(z0))
   DOUBLE''(s(z0)) -> c22(DOUBLE''(z0))
   +'''(z0, s(z1)) -> c24(+'''(z0, z1))
K tuples:
   SQR''(s(z0)) -> c9(+'''(sqr(z0), s(double(z0))))
   SQR''(s(z0)) -> c9(SQR'(z0))
   SQR''(s(z0)) -> c9(DOUBLE'(z0))
   SQR''(s(z0)) -> c9(+'''(sqr(z0), double(z0)))
   SQR''(s(z0)) -> c9(DOUBLE''(z0))
Defined Rule Symbols:   sqr_1, +_2, double_1

Defined Pair Symbols:   SQR'_1, DOUBLE'_1, +''_2, SQR''_1, DOUBLE''_1, +'''_2

Compound Symbols:   c10_3, c11_3, c13_1, c15_1, c17_4, c19_4, c22_1, c24_1, c9_1


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

(15) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   SQR''(s(z0)) -> c17(+'''(sqr(z0), s(double(z0))), SQR'(z0), DOUBLE'(z0), SQR''(z0))
   SQR''(s(z0)) -> c19(+'''(sqr(z0), double(z0)), SQR'(z0), DOUBLE'(z0), SQR''(z0))
   DOUBLE''(s(z0)) -> c22(DOUBLE''(z0))
We considered the (Usable) Rules:none
And the Tuples:
   SQR'(s(z0)) -> c10(+''(sqr(z0), s(double(z0))), SQR'(z0), DOUBLE'(z0))
   SQR'(s(z0)) -> c11(+''(sqr(z0), double(z0)), SQR'(z0), DOUBLE'(z0))
   DOUBLE'(s(z0)) -> c13(DOUBLE'(z0))
   +''(z0, s(z1)) -> c15(+''(z0, z1))
   SQR''(s(z0)) -> c17(+'''(sqr(z0), s(double(z0))), SQR'(z0), DOUBLE'(z0), SQR''(z0))
   SQR''(s(z0)) -> c19(+'''(sqr(z0), double(z0)), SQR'(z0), DOUBLE'(z0), SQR''(z0))
   DOUBLE''(s(z0)) -> c22(DOUBLE''(z0))
   +'''(z0, s(z1)) -> c24(+'''(z0, z1))
   SQR''(s(z0)) -> c9(+'''(sqr(z0), s(double(z0))))
   SQR''(s(z0)) -> c9(SQR'(z0))
   SQR''(s(z0)) -> c9(DOUBLE'(z0))
   SQR''(s(z0)) -> c9(DOUBLE''(z0))
   SQR''(s(z0)) -> c9(+'''(sqr(z0), double(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(+(x_1, x_2)) = x_2
   POL(+''(x_1, x_2)) = 0
   POL(+'''(x_1, x_2)) = 0
   POL(0) = [1]
   POL(DOUBLE'(x_1)) = 0
   POL(DOUBLE''(x_1)) = [2] + [3]x_1
   POL(SQR'(x_1)) = 0
   POL(SQR''(x_1)) = [1] + [3]x_1
   POL(c10(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c11(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c13(x_1)) = x_1
   POL(c15(x_1)) = x_1
   POL(c17(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
   POL(c19(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
   POL(c22(x_1)) = x_1
   POL(c24(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(double(x_1)) = [2] + [3]x_1
   POL(s(x_1)) = [3] + x_1
   POL(sqr(x_1)) = [2] + [3]x_1

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

(16)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sqr(0) -> 0
   sqr(s(z0)) -> +(sqr(z0), s(double(z0)))
   sqr(s(z0)) -> s(+(sqr(z0), double(z0)))
   +(z0, s(z1)) -> s(+(z0, z1))
   +(z0, 0) -> z0
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
Tuples:
   SQR'(s(z0)) -> c10(+''(sqr(z0), s(double(z0))), SQR'(z0), DOUBLE'(z0))
   SQR'(s(z0)) -> c11(+''(sqr(z0), double(z0)), SQR'(z0), DOUBLE'(z0))
   DOUBLE'(s(z0)) -> c13(DOUBLE'(z0))
   +''(z0, s(z1)) -> c15(+''(z0, z1))
   SQR''(s(z0)) -> c17(+'''(sqr(z0), s(double(z0))), SQR'(z0), DOUBLE'(z0), SQR''(z0))
   SQR''(s(z0)) -> c19(+'''(sqr(z0), double(z0)), SQR'(z0), DOUBLE'(z0), SQR''(z0))
   DOUBLE''(s(z0)) -> c22(DOUBLE''(z0))
   +'''(z0, s(z1)) -> c24(+'''(z0, z1))
   SQR''(s(z0)) -> c9(+'''(sqr(z0), s(double(z0))))
   SQR''(s(z0)) -> c9(SQR'(z0))
   SQR''(s(z0)) -> c9(DOUBLE'(z0))
   SQR''(s(z0)) -> c9(DOUBLE''(z0))
   SQR''(s(z0)) -> c9(+'''(sqr(z0), double(z0)))
S tuples:
   +'''(z0, s(z1)) -> c24(+'''(z0, z1))
K tuples:
   SQR''(s(z0)) -> c9(+'''(sqr(z0), s(double(z0))))
   SQR''(s(z0)) -> c9(SQR'(z0))
   SQR''(s(z0)) -> c9(DOUBLE'(z0))
   SQR''(s(z0)) -> c9(+'''(sqr(z0), double(z0)))
   SQR''(s(z0)) -> c9(DOUBLE''(z0))
   SQR''(s(z0)) -> c17(+'''(sqr(z0), s(double(z0))), SQR'(z0), DOUBLE'(z0), SQR''(z0))
   SQR''(s(z0)) -> c19(+'''(sqr(z0), double(z0)), SQR'(z0), DOUBLE'(z0), SQR''(z0))
   DOUBLE''(s(z0)) -> c22(DOUBLE''(z0))
Defined Rule Symbols:   sqr_1, +_2, double_1

Defined Pair Symbols:   SQR'_1, DOUBLE'_1, +''_2, SQR''_1, DOUBLE''_1, +'''_2

Compound Symbols:   c10_3, c11_3, c13_1, c15_1, c17_4, c19_4, c22_1, c24_1, c9_1


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

(17) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   +'''(z0, s(z1)) -> c24(+'''(z0, z1))
We considered the (Usable) Rules:
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
And the Tuples:
   SQR'(s(z0)) -> c10(+''(sqr(z0), s(double(z0))), SQR'(z0), DOUBLE'(z0))
   SQR'(s(z0)) -> c11(+''(sqr(z0), double(z0)), SQR'(z0), DOUBLE'(z0))
   DOUBLE'(s(z0)) -> c13(DOUBLE'(z0))
   +''(z0, s(z1)) -> c15(+''(z0, z1))
   SQR''(s(z0)) -> c17(+'''(sqr(z0), s(double(z0))), SQR'(z0), DOUBLE'(z0), SQR''(z0))
   SQR''(s(z0)) -> c19(+'''(sqr(z0), double(z0)), SQR'(z0), DOUBLE'(z0), SQR''(z0))
   DOUBLE''(s(z0)) -> c22(DOUBLE''(z0))
   +'''(z0, s(z1)) -> c24(+'''(z0, z1))
   SQR''(s(z0)) -> c9(+'''(sqr(z0), s(double(z0))))
   SQR''(s(z0)) -> c9(SQR'(z0))
   SQR''(s(z0)) -> c9(DOUBLE'(z0))
   SQR''(s(z0)) -> c9(DOUBLE''(z0))
   SQR''(s(z0)) -> c9(+'''(sqr(z0), double(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(+(x_1, x_2)) = [1]
   POL(+''(x_1, x_2)) = 0
   POL(+'''(x_1, x_2)) = x_2
   POL(0) = 0
   POL(DOUBLE'(x_1)) = 0
   POL(DOUBLE''(x_1)) = [2] + [2]x_1 + x_1^2
   POL(SQR'(x_1)) = [2]
   POL(SQR''(x_1)) = x_1^2
   POL(c10(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c11(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c13(x_1)) = x_1
   POL(c15(x_1)) = x_1
   POL(c17(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
   POL(c19(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
   POL(c22(x_1)) = x_1
   POL(c24(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(double(x_1)) = [2]x_1
   POL(s(x_1)) = [2] + x_1
   POL(sqr(x_1)) = 0

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

(18)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sqr(0) -> 0
   sqr(s(z0)) -> +(sqr(z0), s(double(z0)))
   sqr(s(z0)) -> s(+(sqr(z0), double(z0)))
   +(z0, s(z1)) -> s(+(z0, z1))
   +(z0, 0) -> z0
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
Tuples:
   SQR'(s(z0)) -> c10(+''(sqr(z0), s(double(z0))), SQR'(z0), DOUBLE'(z0))
   SQR'(s(z0)) -> c11(+''(sqr(z0), double(z0)), SQR'(z0), DOUBLE'(z0))
   DOUBLE'(s(z0)) -> c13(DOUBLE'(z0))
   +''(z0, s(z1)) -> c15(+''(z0, z1))
   SQR''(s(z0)) -> c17(+'''(sqr(z0), s(double(z0))), SQR'(z0), DOUBLE'(z0), SQR''(z0))
   SQR''(s(z0)) -> c19(+'''(sqr(z0), double(z0)), SQR'(z0), DOUBLE'(z0), SQR''(z0))
   DOUBLE''(s(z0)) -> c22(DOUBLE''(z0))
   +'''(z0, s(z1)) -> c24(+'''(z0, z1))
   SQR''(s(z0)) -> c9(+'''(sqr(z0), s(double(z0))))
   SQR''(s(z0)) -> c9(SQR'(z0))
   SQR''(s(z0)) -> c9(DOUBLE'(z0))
   SQR''(s(z0)) -> c9(DOUBLE''(z0))
   SQR''(s(z0)) -> c9(+'''(sqr(z0), double(z0)))
S tuples:none
K tuples:
   SQR''(s(z0)) -> c9(+'''(sqr(z0), s(double(z0))))
   SQR''(s(z0)) -> c9(SQR'(z0))
   SQR''(s(z0)) -> c9(DOUBLE'(z0))
   SQR''(s(z0)) -> c9(+'''(sqr(z0), double(z0)))
   SQR''(s(z0)) -> c9(DOUBLE''(z0))
   SQR''(s(z0)) -> c17(+'''(sqr(z0), s(double(z0))), SQR'(z0), DOUBLE'(z0), SQR''(z0))
   SQR''(s(z0)) -> c19(+'''(sqr(z0), double(z0)), SQR'(z0), DOUBLE'(z0), SQR''(z0))
   DOUBLE''(s(z0)) -> c22(DOUBLE''(z0))
   +'''(z0, s(z1)) -> c24(+'''(z0, z1))
Defined Rule Symbols:   sqr_1, +_2, double_1

Defined Pair Symbols:   SQR'_1, DOUBLE'_1, +''_2, SQR''_1, DOUBLE''_1, +'''_2

Compound Symbols:   c10_3, c11_3, c13_1, c15_1, c17_4, c19_4, c22_1, c24_1, c9_1


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

(19) SIsEmptyProof (BOTH BOUNDS(ID, ID))
The set S is empty
----------------------------------------

(20)
BOUNDS(1, 1)

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

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

(22)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   SQR(0') -> c
   SQR(s(z0)) -> c1(+'(sqr(z0), s(double(z0))), SQR(z0))
   SQR(s(z0)) -> c2(+'(sqr(z0), s(double(z0))), DOUBLE(z0))
   SQR(s(z0)) -> c3(+'(sqr(z0), double(z0)), SQR(z0))
   SQR(s(z0)) -> c4(+'(sqr(z0), double(z0)), DOUBLE(z0))
   DOUBLE(0') -> c5
   DOUBLE(s(z0)) -> c6(DOUBLE(z0))
   +'(z0, 0') -> c7
   +'(z0, s(z1)) -> c8(+'(z0, z1))

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

   sqr(0') -> 0'
   sqr(s(z0)) -> +'(sqr(z0), s(double(z0)))
   sqr(s(z0)) -> s(+'(sqr(z0), double(z0)))
   double(0') -> 0'
   double(s(z0)) -> s(s(double(z0)))
   +'(z0, 0') -> z0
   +'(z0, s(z1)) -> s(+'(z0, z1))

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

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

(24)
Obligation:
Innermost TRS:
Rules:
SQR(0') -> c
SQR(s(z0)) -> c1(+'(sqr(z0), s(double(z0))), SQR(z0))
SQR(s(z0)) -> c2(+'(sqr(z0), s(double(z0))), DOUBLE(z0))
SQR(s(z0)) -> c3(+'(sqr(z0), double(z0)), SQR(z0))
SQR(s(z0)) -> c4(+'(sqr(z0), double(z0)), DOUBLE(z0))
DOUBLE(0') -> c5
DOUBLE(s(z0)) -> c6(DOUBLE(z0))
+'(z0, 0') -> c7
+'(z0, s(z1)) -> c8(+'(z0, z1))
sqr(0') -> 0'
sqr(s(z0)) -> +'(sqr(z0), s(double(z0)))
sqr(s(z0)) -> s(+'(sqr(z0), double(z0)))
double(0') -> 0'
double(s(z0)) -> s(s(double(z0)))
+'(z0, 0') -> z0
+'(z0, s(z1)) -> s(+'(z0, z1))

Types:
SQR :: 0':s:c7:c8 -> c:c1:c2:c3:c4
0' :: 0':s:c7:c8
c :: c:c1:c2:c3:c4
s :: 0':s:c7:c8 -> 0':s:c7:c8
c1 :: 0':s:c7:c8 -> c:c1:c2:c3:c4 -> c:c1:c2:c3:c4
+' :: 0':s:c7:c8 -> 0':s:c7:c8 -> 0':s:c7:c8
sqr :: 0':s:c7:c8 -> 0':s:c7:c8
double :: 0':s:c7:c8 -> 0':s:c7:c8
c2 :: 0':s:c7:c8 -> c5:c6 -> c:c1:c2:c3:c4
DOUBLE :: 0':s:c7:c8 -> c5:c6
c3 :: 0':s:c7:c8 -> c:c1:c2:c3:c4 -> c:c1:c2:c3:c4
c4 :: 0':s:c7:c8 -> c5:c6 -> c:c1:c2:c3:c4
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
c7 :: 0':s:c7:c8
c8 :: 0':s:c7:c8 -> 0':s:c7:c8
hole_c:c1:c2:c3:c41_9 :: c:c1:c2:c3:c4
hole_0':s:c7:c82_9 :: 0':s:c7:c8
hole_c5:c63_9 :: c5:c6
gen_c:c1:c2:c3:c44_9 :: Nat -> c:c1:c2:c3:c4
gen_0':s:c7:c85_9 :: Nat -> 0':s:c7:c8
gen_c5:c66_9 :: Nat -> c5:c6

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

(25) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
SQR, +', sqr, double, DOUBLE

They will be analysed ascendingly in the following order:
+' < SQR
sqr < SQR
double < SQR
DOUBLE < SQR
+' < sqr
double < sqr

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

(26)
Obligation:
Innermost TRS:
Rules:
SQR(0') -> c
SQR(s(z0)) -> c1(+'(sqr(z0), s(double(z0))), SQR(z0))
SQR(s(z0)) -> c2(+'(sqr(z0), s(double(z0))), DOUBLE(z0))
SQR(s(z0)) -> c3(+'(sqr(z0), double(z0)), SQR(z0))
SQR(s(z0)) -> c4(+'(sqr(z0), double(z0)), DOUBLE(z0))
DOUBLE(0') -> c5
DOUBLE(s(z0)) -> c6(DOUBLE(z0))
+'(z0, 0') -> c7
+'(z0, s(z1)) -> c8(+'(z0, z1))
sqr(0') -> 0'
sqr(s(z0)) -> +'(sqr(z0), s(double(z0)))
sqr(s(z0)) -> s(+'(sqr(z0), double(z0)))
double(0') -> 0'
double(s(z0)) -> s(s(double(z0)))
+'(z0, 0') -> z0
+'(z0, s(z1)) -> s(+'(z0, z1))

Types:
SQR :: 0':s:c7:c8 -> c:c1:c2:c3:c4
0' :: 0':s:c7:c8
c :: c:c1:c2:c3:c4
s :: 0':s:c7:c8 -> 0':s:c7:c8
c1 :: 0':s:c7:c8 -> c:c1:c2:c3:c4 -> c:c1:c2:c3:c4
+' :: 0':s:c7:c8 -> 0':s:c7:c8 -> 0':s:c7:c8
sqr :: 0':s:c7:c8 -> 0':s:c7:c8
double :: 0':s:c7:c8 -> 0':s:c7:c8
c2 :: 0':s:c7:c8 -> c5:c6 -> c:c1:c2:c3:c4
DOUBLE :: 0':s:c7:c8 -> c5:c6
c3 :: 0':s:c7:c8 -> c:c1:c2:c3:c4 -> c:c1:c2:c3:c4
c4 :: 0':s:c7:c8 -> c5:c6 -> c:c1:c2:c3:c4
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
c7 :: 0':s:c7:c8
c8 :: 0':s:c7:c8 -> 0':s:c7:c8
hole_c:c1:c2:c3:c41_9 :: c:c1:c2:c3:c4
hole_0':s:c7:c82_9 :: 0':s:c7:c8
hole_c5:c63_9 :: c5:c6
gen_c:c1:c2:c3:c44_9 :: Nat -> c:c1:c2:c3:c4
gen_0':s:c7:c85_9 :: Nat -> 0':s:c7:c8
gen_c5:c66_9 :: Nat -> c5:c6


Generator Equations:
gen_c:c1:c2:c3:c44_9(0) <=> c
gen_c:c1:c2:c3:c44_9(+(x, 1)) <=> c1(0', gen_c:c1:c2:c3:c44_9(x))
gen_0':s:c7:c85_9(0) <=> 0'
gen_0':s:c7:c85_9(+(x, 1)) <=> s(gen_0':s:c7:c85_9(x))
gen_c5:c66_9(0) <=> c5
gen_c5:c66_9(+(x, 1)) <=> c6(gen_c5:c66_9(x))


The following defined symbols remain to be analysed:
+', SQR, sqr, double, DOUBLE

They will be analysed ascendingly in the following order:
+' < SQR
sqr < SQR
double < SQR
DOUBLE < SQR
+' < sqr
double < sqr

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

(27) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
+'(gen_0':s:c7:c85_9(a), gen_0':s:c7:c85_9(+(1, n8_9))) -> *7_9, rt in Omega(n8_9)

Induction Base:
+'(gen_0':s:c7:c85_9(a), gen_0':s:c7:c85_9(+(1, 0)))

Induction Step:
+'(gen_0':s:c7:c85_9(a), gen_0':s:c7:c85_9(+(1, +(n8_9, 1)))) ->_R^Omega(1)
c8(+'(gen_0':s:c7:c85_9(a), gen_0':s:c7:c85_9(+(1, n8_9)))) ->_IH
c8(*7_9)

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

(28)
Complex Obligation (BEST)

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

(29)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
SQR(0') -> c
SQR(s(z0)) -> c1(+'(sqr(z0), s(double(z0))), SQR(z0))
SQR(s(z0)) -> c2(+'(sqr(z0), s(double(z0))), DOUBLE(z0))
SQR(s(z0)) -> c3(+'(sqr(z0), double(z0)), SQR(z0))
SQR(s(z0)) -> c4(+'(sqr(z0), double(z0)), DOUBLE(z0))
DOUBLE(0') -> c5
DOUBLE(s(z0)) -> c6(DOUBLE(z0))
+'(z0, 0') -> c7
+'(z0, s(z1)) -> c8(+'(z0, z1))
sqr(0') -> 0'
sqr(s(z0)) -> +'(sqr(z0), s(double(z0)))
sqr(s(z0)) -> s(+'(sqr(z0), double(z0)))
double(0') -> 0'
double(s(z0)) -> s(s(double(z0)))
+'(z0, 0') -> z0
+'(z0, s(z1)) -> s(+'(z0, z1))

Types:
SQR :: 0':s:c7:c8 -> c:c1:c2:c3:c4
0' :: 0':s:c7:c8
c :: c:c1:c2:c3:c4
s :: 0':s:c7:c8 -> 0':s:c7:c8
c1 :: 0':s:c7:c8 -> c:c1:c2:c3:c4 -> c:c1:c2:c3:c4
+' :: 0':s:c7:c8 -> 0':s:c7:c8 -> 0':s:c7:c8
sqr :: 0':s:c7:c8 -> 0':s:c7:c8
double :: 0':s:c7:c8 -> 0':s:c7:c8
c2 :: 0':s:c7:c8 -> c5:c6 -> c:c1:c2:c3:c4
DOUBLE :: 0':s:c7:c8 -> c5:c6
c3 :: 0':s:c7:c8 -> c:c1:c2:c3:c4 -> c:c1:c2:c3:c4
c4 :: 0':s:c7:c8 -> c5:c6 -> c:c1:c2:c3:c4
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
c7 :: 0':s:c7:c8
c8 :: 0':s:c7:c8 -> 0':s:c7:c8
hole_c:c1:c2:c3:c41_9 :: c:c1:c2:c3:c4
hole_0':s:c7:c82_9 :: 0':s:c7:c8
hole_c5:c63_9 :: c5:c6
gen_c:c1:c2:c3:c44_9 :: Nat -> c:c1:c2:c3:c4
gen_0':s:c7:c85_9 :: Nat -> 0':s:c7:c8
gen_c5:c66_9 :: Nat -> c5:c6


Generator Equations:
gen_c:c1:c2:c3:c44_9(0) <=> c
gen_c:c1:c2:c3:c44_9(+(x, 1)) <=> c1(0', gen_c:c1:c2:c3:c44_9(x))
gen_0':s:c7:c85_9(0) <=> 0'
gen_0':s:c7:c85_9(+(x, 1)) <=> s(gen_0':s:c7:c85_9(x))
gen_c5:c66_9(0) <=> c5
gen_c5:c66_9(+(x, 1)) <=> c6(gen_c5:c66_9(x))


The following defined symbols remain to be analysed:
+', SQR, sqr, double, DOUBLE

They will be analysed ascendingly in the following order:
+' < SQR
sqr < SQR
double < SQR
DOUBLE < SQR
+' < sqr
double < sqr

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

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

(31)
BOUNDS(n^1, INF)

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

(32)
Obligation:
Innermost TRS:
Rules:
SQR(0') -> c
SQR(s(z0)) -> c1(+'(sqr(z0), s(double(z0))), SQR(z0))
SQR(s(z0)) -> c2(+'(sqr(z0), s(double(z0))), DOUBLE(z0))
SQR(s(z0)) -> c3(+'(sqr(z0), double(z0)), SQR(z0))
SQR(s(z0)) -> c4(+'(sqr(z0), double(z0)), DOUBLE(z0))
DOUBLE(0') -> c5
DOUBLE(s(z0)) -> c6(DOUBLE(z0))
+'(z0, 0') -> c7
+'(z0, s(z1)) -> c8(+'(z0, z1))
sqr(0') -> 0'
sqr(s(z0)) -> +'(sqr(z0), s(double(z0)))
sqr(s(z0)) -> s(+'(sqr(z0), double(z0)))
double(0') -> 0'
double(s(z0)) -> s(s(double(z0)))
+'(z0, 0') -> z0
+'(z0, s(z1)) -> s(+'(z0, z1))

Types:
SQR :: 0':s:c7:c8 -> c:c1:c2:c3:c4
0' :: 0':s:c7:c8
c :: c:c1:c2:c3:c4
s :: 0':s:c7:c8 -> 0':s:c7:c8
c1 :: 0':s:c7:c8 -> c:c1:c2:c3:c4 -> c:c1:c2:c3:c4
+' :: 0':s:c7:c8 -> 0':s:c7:c8 -> 0':s:c7:c8
sqr :: 0':s:c7:c8 -> 0':s:c7:c8
double :: 0':s:c7:c8 -> 0':s:c7:c8
c2 :: 0':s:c7:c8 -> c5:c6 -> c:c1:c2:c3:c4
DOUBLE :: 0':s:c7:c8 -> c5:c6
c3 :: 0':s:c7:c8 -> c:c1:c2:c3:c4 -> c:c1:c2:c3:c4
c4 :: 0':s:c7:c8 -> c5:c6 -> c:c1:c2:c3:c4
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
c7 :: 0':s:c7:c8
c8 :: 0':s:c7:c8 -> 0':s:c7:c8
hole_c:c1:c2:c3:c41_9 :: c:c1:c2:c3:c4
hole_0':s:c7:c82_9 :: 0':s:c7:c8
hole_c5:c63_9 :: c5:c6
gen_c:c1:c2:c3:c44_9 :: Nat -> c:c1:c2:c3:c4
gen_0':s:c7:c85_9 :: Nat -> 0':s:c7:c8
gen_c5:c66_9 :: Nat -> c5:c6


Lemmas:
+'(gen_0':s:c7:c85_9(a), gen_0':s:c7:c85_9(+(1, n8_9))) -> *7_9, rt in Omega(n8_9)


Generator Equations:
gen_c:c1:c2:c3:c44_9(0) <=> c
gen_c:c1:c2:c3:c44_9(+(x, 1)) <=> c1(0', gen_c:c1:c2:c3:c44_9(x))
gen_0':s:c7:c85_9(0) <=> 0'
gen_0':s:c7:c85_9(+(x, 1)) <=> s(gen_0':s:c7:c85_9(x))
gen_c5:c66_9(0) <=> c5
gen_c5:c66_9(+(x, 1)) <=> c6(gen_c5:c66_9(x))


The following defined symbols remain to be analysed:
double, SQR, sqr, DOUBLE

They will be analysed ascendingly in the following order:
sqr < SQR
double < SQR
DOUBLE < SQR
double < sqr

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

(33) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
double(gen_0':s:c7:c85_9(n1930_9)) -> gen_0':s:c7:c85_9(*(2, n1930_9)), rt in Omega(0)

Induction Base:
double(gen_0':s:c7:c85_9(0)) ->_R^Omega(0)
0'

Induction Step:
double(gen_0':s:c7:c85_9(+(n1930_9, 1))) ->_R^Omega(0)
s(s(double(gen_0':s:c7:c85_9(n1930_9)))) ->_IH
s(s(gen_0':s:c7:c85_9(*(2, c1931_9))))

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

(34)
Obligation:
Innermost TRS:
Rules:
SQR(0') -> c
SQR(s(z0)) -> c1(+'(sqr(z0), s(double(z0))), SQR(z0))
SQR(s(z0)) -> c2(+'(sqr(z0), s(double(z0))), DOUBLE(z0))
SQR(s(z0)) -> c3(+'(sqr(z0), double(z0)), SQR(z0))
SQR(s(z0)) -> c4(+'(sqr(z0), double(z0)), DOUBLE(z0))
DOUBLE(0') -> c5
DOUBLE(s(z0)) -> c6(DOUBLE(z0))
+'(z0, 0') -> c7
+'(z0, s(z1)) -> c8(+'(z0, z1))
sqr(0') -> 0'
sqr(s(z0)) -> +'(sqr(z0), s(double(z0)))
sqr(s(z0)) -> s(+'(sqr(z0), double(z0)))
double(0') -> 0'
double(s(z0)) -> s(s(double(z0)))
+'(z0, 0') -> z0
+'(z0, s(z1)) -> s(+'(z0, z1))

Types:
SQR :: 0':s:c7:c8 -> c:c1:c2:c3:c4
0' :: 0':s:c7:c8
c :: c:c1:c2:c3:c4
s :: 0':s:c7:c8 -> 0':s:c7:c8
c1 :: 0':s:c7:c8 -> c:c1:c2:c3:c4 -> c:c1:c2:c3:c4
+' :: 0':s:c7:c8 -> 0':s:c7:c8 -> 0':s:c7:c8
sqr :: 0':s:c7:c8 -> 0':s:c7:c8
double :: 0':s:c7:c8 -> 0':s:c7:c8
c2 :: 0':s:c7:c8 -> c5:c6 -> c:c1:c2:c3:c4
DOUBLE :: 0':s:c7:c8 -> c5:c6
c3 :: 0':s:c7:c8 -> c:c1:c2:c3:c4 -> c:c1:c2:c3:c4
c4 :: 0':s:c7:c8 -> c5:c6 -> c:c1:c2:c3:c4
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
c7 :: 0':s:c7:c8
c8 :: 0':s:c7:c8 -> 0':s:c7:c8
hole_c:c1:c2:c3:c41_9 :: c:c1:c2:c3:c4
hole_0':s:c7:c82_9 :: 0':s:c7:c8
hole_c5:c63_9 :: c5:c6
gen_c:c1:c2:c3:c44_9 :: Nat -> c:c1:c2:c3:c4
gen_0':s:c7:c85_9 :: Nat -> 0':s:c7:c8
gen_c5:c66_9 :: Nat -> c5:c6


Lemmas:
+'(gen_0':s:c7:c85_9(a), gen_0':s:c7:c85_9(+(1, n8_9))) -> *7_9, rt in Omega(n8_9)
double(gen_0':s:c7:c85_9(n1930_9)) -> gen_0':s:c7:c85_9(*(2, n1930_9)), rt in Omega(0)


Generator Equations:
gen_c:c1:c2:c3:c44_9(0) <=> c
gen_c:c1:c2:c3:c44_9(+(x, 1)) <=> c1(0', gen_c:c1:c2:c3:c44_9(x))
gen_0':s:c7:c85_9(0) <=> 0'
gen_0':s:c7:c85_9(+(x, 1)) <=> s(gen_0':s:c7:c85_9(x))
gen_c5:c66_9(0) <=> c5
gen_c5:c66_9(+(x, 1)) <=> c6(gen_c5:c66_9(x))


The following defined symbols remain to be analysed:
sqr, SQR, DOUBLE

They will be analysed ascendingly in the following order:
sqr < SQR
DOUBLE < SQR

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

(35) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
sqr(gen_0':s:c7:c85_9(+(1, n2443_9))) -> *7_9, rt in Omega(0)

Induction Base:
sqr(gen_0':s:c7:c85_9(+(1, 0)))

Induction Step:
sqr(gen_0':s:c7:c85_9(+(1, +(n2443_9, 1)))) ->_R^Omega(0)
+'(sqr(gen_0':s:c7:c85_9(+(1, n2443_9))), s(double(gen_0':s:c7:c85_9(+(1, n2443_9))))) ->_IH
+'(*7_9, s(double(gen_0':s:c7:c85_9(+(1, n2443_9))))) ->_L^Omega(0)
+'(*7_9, s(gen_0':s:c7:c85_9(*(2, +(1, n2443_9)))))

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

(36)
Obligation:
Innermost TRS:
Rules:
SQR(0') -> c
SQR(s(z0)) -> c1(+'(sqr(z0), s(double(z0))), SQR(z0))
SQR(s(z0)) -> c2(+'(sqr(z0), s(double(z0))), DOUBLE(z0))
SQR(s(z0)) -> c3(+'(sqr(z0), double(z0)), SQR(z0))
SQR(s(z0)) -> c4(+'(sqr(z0), double(z0)), DOUBLE(z0))
DOUBLE(0') -> c5
DOUBLE(s(z0)) -> c6(DOUBLE(z0))
+'(z0, 0') -> c7
+'(z0, s(z1)) -> c8(+'(z0, z1))
sqr(0') -> 0'
sqr(s(z0)) -> +'(sqr(z0), s(double(z0)))
sqr(s(z0)) -> s(+'(sqr(z0), double(z0)))
double(0') -> 0'
double(s(z0)) -> s(s(double(z0)))
+'(z0, 0') -> z0
+'(z0, s(z1)) -> s(+'(z0, z1))

Types:
SQR :: 0':s:c7:c8 -> c:c1:c2:c3:c4
0' :: 0':s:c7:c8
c :: c:c1:c2:c3:c4
s :: 0':s:c7:c8 -> 0':s:c7:c8
c1 :: 0':s:c7:c8 -> c:c1:c2:c3:c4 -> c:c1:c2:c3:c4
+' :: 0':s:c7:c8 -> 0':s:c7:c8 -> 0':s:c7:c8
sqr :: 0':s:c7:c8 -> 0':s:c7:c8
double :: 0':s:c7:c8 -> 0':s:c7:c8
c2 :: 0':s:c7:c8 -> c5:c6 -> c:c1:c2:c3:c4
DOUBLE :: 0':s:c7:c8 -> c5:c6
c3 :: 0':s:c7:c8 -> c:c1:c2:c3:c4 -> c:c1:c2:c3:c4
c4 :: 0':s:c7:c8 -> c5:c6 -> c:c1:c2:c3:c4
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
c7 :: 0':s:c7:c8
c8 :: 0':s:c7:c8 -> 0':s:c7:c8
hole_c:c1:c2:c3:c41_9 :: c:c1:c2:c3:c4
hole_0':s:c7:c82_9 :: 0':s:c7:c8
hole_c5:c63_9 :: c5:c6
gen_c:c1:c2:c3:c44_9 :: Nat -> c:c1:c2:c3:c4
gen_0':s:c7:c85_9 :: Nat -> 0':s:c7:c8
gen_c5:c66_9 :: Nat -> c5:c6


Lemmas:
+'(gen_0':s:c7:c85_9(a), gen_0':s:c7:c85_9(+(1, n8_9))) -> *7_9, rt in Omega(n8_9)
double(gen_0':s:c7:c85_9(n1930_9)) -> gen_0':s:c7:c85_9(*(2, n1930_9)), rt in Omega(0)
sqr(gen_0':s:c7:c85_9(+(1, n2443_9))) -> *7_9, rt in Omega(0)


Generator Equations:
gen_c:c1:c2:c3:c44_9(0) <=> c
gen_c:c1:c2:c3:c44_9(+(x, 1)) <=> c1(0', gen_c:c1:c2:c3:c44_9(x))
gen_0':s:c7:c85_9(0) <=> 0'
gen_0':s:c7:c85_9(+(x, 1)) <=> s(gen_0':s:c7:c85_9(x))
gen_c5:c66_9(0) <=> c5
gen_c5:c66_9(+(x, 1)) <=> c6(gen_c5:c66_9(x))


The following defined symbols remain to be analysed:
DOUBLE, SQR

They will be analysed ascendingly in the following order:
DOUBLE < SQR

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

(37) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
DOUBLE(gen_0':s:c7:c85_9(n44676_9)) -> gen_c5:c66_9(n44676_9), rt in Omega(1 + n44676_9)

Induction Base:
DOUBLE(gen_0':s:c7:c85_9(0)) ->_R^Omega(1)
c5

Induction Step:
DOUBLE(gen_0':s:c7:c85_9(+(n44676_9, 1))) ->_R^Omega(1)
c6(DOUBLE(gen_0':s:c7:c85_9(n44676_9))) ->_IH
c6(gen_c5:c66_9(c44677_9))

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

(38)
Obligation:
Innermost TRS:
Rules:
SQR(0') -> c
SQR(s(z0)) -> c1(+'(sqr(z0), s(double(z0))), SQR(z0))
SQR(s(z0)) -> c2(+'(sqr(z0), s(double(z0))), DOUBLE(z0))
SQR(s(z0)) -> c3(+'(sqr(z0), double(z0)), SQR(z0))
SQR(s(z0)) -> c4(+'(sqr(z0), double(z0)), DOUBLE(z0))
DOUBLE(0') -> c5
DOUBLE(s(z0)) -> c6(DOUBLE(z0))
+'(z0, 0') -> c7
+'(z0, s(z1)) -> c8(+'(z0, z1))
sqr(0') -> 0'
sqr(s(z0)) -> +'(sqr(z0), s(double(z0)))
sqr(s(z0)) -> s(+'(sqr(z0), double(z0)))
double(0') -> 0'
double(s(z0)) -> s(s(double(z0)))
+'(z0, 0') -> z0
+'(z0, s(z1)) -> s(+'(z0, z1))

Types:
SQR :: 0':s:c7:c8 -> c:c1:c2:c3:c4
0' :: 0':s:c7:c8
c :: c:c1:c2:c3:c4
s :: 0':s:c7:c8 -> 0':s:c7:c8
c1 :: 0':s:c7:c8 -> c:c1:c2:c3:c4 -> c:c1:c2:c3:c4
+' :: 0':s:c7:c8 -> 0':s:c7:c8 -> 0':s:c7:c8
sqr :: 0':s:c7:c8 -> 0':s:c7:c8
double :: 0':s:c7:c8 -> 0':s:c7:c8
c2 :: 0':s:c7:c8 -> c5:c6 -> c:c1:c2:c3:c4
DOUBLE :: 0':s:c7:c8 -> c5:c6
c3 :: 0':s:c7:c8 -> c:c1:c2:c3:c4 -> c:c1:c2:c3:c4
c4 :: 0':s:c7:c8 -> c5:c6 -> c:c1:c2:c3:c4
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
c7 :: 0':s:c7:c8
c8 :: 0':s:c7:c8 -> 0':s:c7:c8
hole_c:c1:c2:c3:c41_9 :: c:c1:c2:c3:c4
hole_0':s:c7:c82_9 :: 0':s:c7:c8
hole_c5:c63_9 :: c5:c6
gen_c:c1:c2:c3:c44_9 :: Nat -> c:c1:c2:c3:c4
gen_0':s:c7:c85_9 :: Nat -> 0':s:c7:c8
gen_c5:c66_9 :: Nat -> c5:c6


Lemmas:
+'(gen_0':s:c7:c85_9(a), gen_0':s:c7:c85_9(+(1, n8_9))) -> *7_9, rt in Omega(n8_9)
double(gen_0':s:c7:c85_9(n1930_9)) -> gen_0':s:c7:c85_9(*(2, n1930_9)), rt in Omega(0)
sqr(gen_0':s:c7:c85_9(+(1, n2443_9))) -> *7_9, rt in Omega(0)
DOUBLE(gen_0':s:c7:c85_9(n44676_9)) -> gen_c5:c66_9(n44676_9), rt in Omega(1 + n44676_9)


Generator Equations:
gen_c:c1:c2:c3:c44_9(0) <=> c
gen_c:c1:c2:c3:c44_9(+(x, 1)) <=> c1(0', gen_c:c1:c2:c3:c44_9(x))
gen_0':s:c7:c85_9(0) <=> 0'
gen_0':s:c7:c85_9(+(x, 1)) <=> s(gen_0':s:c7:c85_9(x))
gen_c5:c66_9(0) <=> c5
gen_c5:c66_9(+(x, 1)) <=> c6(gen_c5:c66_9(x))


The following defined symbols remain to be analysed:
SQR