WORST_CASE(Omega(n^1),O(n^2))
proof of input_RMnMJy4zPz.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


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

(0) CpxTRS
(1) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(2) CdtProblem
    (3) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CdtProblem
    (5) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 89 ms]
    (6) CdtProblem
    (7) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 82 ms]
    (8) CdtProblem
    (9) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CdtProblem
    (11) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 63 ms]
    (12) CdtProblem
    (13) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) BOUNDS(1, 1)
(15) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(16) CdtProblem
    (17) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) CpxRelTRS
    (19) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (20) CpxRelTRS
    (21) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (22) typed CpxTrs
    (23) OrderProof [LOWER BOUND(ID), 8 ms]
    (24) typed CpxTrs
    (25) RewriteLemmaProof [LOWER BOUND(ID), 558 ms]
    (26) typed CpxTrs
    (27) RewriteLemmaProof [LOWER BOUND(ID), 497 ms]
    (28) typed CpxTrs
    (29) RewriteLemmaProof [LOWER BOUND(ID), 90 ms]
    (30) BEST
        (31) proven lower bound
            (32) LowerBoundPropagationProof [FINISHED, 0 ms]
            (33) BOUNDS(n^1, INF)
        (34) typed CpxTrs


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

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


The TRS R consists of the following rules:

   sort(nil) -> nil
   sort(cons(x, y)) -> insert(x, sort(y))
   insert(x, nil) -> cons(x, nil)
   insert(x, cons(v, w)) -> choose(x, cons(v, w), x, v)
   choose(x, cons(v, w), y, 0) -> cons(x, cons(v, w))
   choose(x, cons(v, w), 0, s(z)) -> cons(v, insert(x, w))
   choose(x, cons(v, w), s(y), s(z)) -> choose(x, cons(v, w), y, z)

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

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

(2)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sort(nil) -> nil
   sort(cons(z0, z1)) -> insert(z0, sort(z1))
   insert(z0, nil) -> cons(z0, nil)
   insert(z0, cons(z1, z2)) -> choose(z0, cons(z1, z2), z0, z1)
   choose(z0, cons(z1, z2), z3, 0) -> cons(z0, cons(z1, z2))
   choose(z0, cons(z1, z2), 0, s(z3)) -> cons(z1, insert(z0, z2))
   choose(z0, cons(z1, z2), s(z3), s(z4)) -> choose(z0, cons(z1, z2), z3, z4)
Tuples:
   SORT(nil) -> c
   SORT(cons(z0, z1)) -> c1(INSERT(z0, sort(z1)), SORT(z1))
   INSERT(z0, nil) -> c2
   INSERT(z0, cons(z1, z2)) -> c3(CHOOSE(z0, cons(z1, z2), z0, z1))
   CHOOSE(z0, cons(z1, z2), z3, 0) -> c4
   CHOOSE(z0, cons(z1, z2), 0, s(z3)) -> c5(INSERT(z0, z2))
   CHOOSE(z0, cons(z1, z2), s(z3), s(z4)) -> c6(CHOOSE(z0, cons(z1, z2), z3, z4))
S tuples:
   SORT(nil) -> c
   SORT(cons(z0, z1)) -> c1(INSERT(z0, sort(z1)), SORT(z1))
   INSERT(z0, nil) -> c2
   INSERT(z0, cons(z1, z2)) -> c3(CHOOSE(z0, cons(z1, z2), z0, z1))
   CHOOSE(z0, cons(z1, z2), z3, 0) -> c4
   CHOOSE(z0, cons(z1, z2), 0, s(z3)) -> c5(INSERT(z0, z2))
   CHOOSE(z0, cons(z1, z2), s(z3), s(z4)) -> c6(CHOOSE(z0, cons(z1, z2), z3, z4))
K tuples:none
Defined Rule Symbols:   sort_1, insert_2, choose_4

Defined Pair Symbols:   SORT_1, INSERT_2, CHOOSE_4

Compound Symbols:   c, c1_2, c2, c3_1, c4, c5_1, c6_1


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

(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 3 trailing nodes:
   CHOOSE(z0, cons(z1, z2), z3, 0) -> c4
   INSERT(z0, nil) -> c2
   SORT(nil) -> c

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sort(nil) -> nil
   sort(cons(z0, z1)) -> insert(z0, sort(z1))
   insert(z0, nil) -> cons(z0, nil)
   insert(z0, cons(z1, z2)) -> choose(z0, cons(z1, z2), z0, z1)
   choose(z0, cons(z1, z2), z3, 0) -> cons(z0, cons(z1, z2))
   choose(z0, cons(z1, z2), 0, s(z3)) -> cons(z1, insert(z0, z2))
   choose(z0, cons(z1, z2), s(z3), s(z4)) -> choose(z0, cons(z1, z2), z3, z4)
Tuples:
   SORT(cons(z0, z1)) -> c1(INSERT(z0, sort(z1)), SORT(z1))
   INSERT(z0, cons(z1, z2)) -> c3(CHOOSE(z0, cons(z1, z2), z0, z1))
   CHOOSE(z0, cons(z1, z2), 0, s(z3)) -> c5(INSERT(z0, z2))
   CHOOSE(z0, cons(z1, z2), s(z3), s(z4)) -> c6(CHOOSE(z0, cons(z1, z2), z3, z4))
S tuples:
   SORT(cons(z0, z1)) -> c1(INSERT(z0, sort(z1)), SORT(z1))
   INSERT(z0, cons(z1, z2)) -> c3(CHOOSE(z0, cons(z1, z2), z0, z1))
   CHOOSE(z0, cons(z1, z2), 0, s(z3)) -> c5(INSERT(z0, z2))
   CHOOSE(z0, cons(z1, z2), s(z3), s(z4)) -> c6(CHOOSE(z0, cons(z1, z2), z3, z4))
K tuples:none
Defined Rule Symbols:   sort_1, insert_2, choose_4

Defined Pair Symbols:   SORT_1, INSERT_2, CHOOSE_4

Compound Symbols:   c1_2, c3_1, c5_1, c6_1


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

(5) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   SORT(cons(z0, z1)) -> c1(INSERT(z0, sort(z1)), SORT(z1))
We considered the (Usable) Rules:none
And the Tuples:
   SORT(cons(z0, z1)) -> c1(INSERT(z0, sort(z1)), SORT(z1))
   INSERT(z0, cons(z1, z2)) -> c3(CHOOSE(z0, cons(z1, z2), z0, z1))
   CHOOSE(z0, cons(z1, z2), 0, s(z3)) -> c5(INSERT(z0, z2))
   CHOOSE(z0, cons(z1, z2), s(z3), s(z4)) -> c6(CHOOSE(z0, cons(z1, z2), z3, z4))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [3]
   POL(CHOOSE(x_1, x_2, x_3, x_4)) = 0
   POL(INSERT(x_1, x_2)) = 0
   POL(SORT(x_1)) = x_1
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c3(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(choose(x_1, x_2, x_3, x_4)) = [3] + [3]x_1 + [3]x_4
   POL(cons(x_1, x_2)) = [1] + x_2
   POL(insert(x_1, x_2)) = [3] + [3]x_1
   POL(nil) = [3]
   POL(s(x_1)) = x_1
   POL(sort(x_1)) = [3] + [3]x_1

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sort(nil) -> nil
   sort(cons(z0, z1)) -> insert(z0, sort(z1))
   insert(z0, nil) -> cons(z0, nil)
   insert(z0, cons(z1, z2)) -> choose(z0, cons(z1, z2), z0, z1)
   choose(z0, cons(z1, z2), z3, 0) -> cons(z0, cons(z1, z2))
   choose(z0, cons(z1, z2), 0, s(z3)) -> cons(z1, insert(z0, z2))
   choose(z0, cons(z1, z2), s(z3), s(z4)) -> choose(z0, cons(z1, z2), z3, z4)
Tuples:
   SORT(cons(z0, z1)) -> c1(INSERT(z0, sort(z1)), SORT(z1))
   INSERT(z0, cons(z1, z2)) -> c3(CHOOSE(z0, cons(z1, z2), z0, z1))
   CHOOSE(z0, cons(z1, z2), 0, s(z3)) -> c5(INSERT(z0, z2))
   CHOOSE(z0, cons(z1, z2), s(z3), s(z4)) -> c6(CHOOSE(z0, cons(z1, z2), z3, z4))
S tuples:
   INSERT(z0, cons(z1, z2)) -> c3(CHOOSE(z0, cons(z1, z2), z0, z1))
   CHOOSE(z0, cons(z1, z2), 0, s(z3)) -> c5(INSERT(z0, z2))
   CHOOSE(z0, cons(z1, z2), s(z3), s(z4)) -> c6(CHOOSE(z0, cons(z1, z2), z3, z4))
K tuples:
   SORT(cons(z0, z1)) -> c1(INSERT(z0, sort(z1)), SORT(z1))
Defined Rule Symbols:   sort_1, insert_2, choose_4

Defined Pair Symbols:   SORT_1, INSERT_2, CHOOSE_4

Compound Symbols:   c1_2, c3_1, c5_1, c6_1


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

(7) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   CHOOSE(z0, cons(z1, z2), 0, s(z3)) -> c5(INSERT(z0, z2))
We considered the (Usable) Rules:
   choose(z0, cons(z1, z2), 0, s(z3)) -> cons(z1, insert(z0, z2))
   sort(nil) -> nil
   insert(z0, cons(z1, z2)) -> choose(z0, cons(z1, z2), z0, z1)
   sort(cons(z0, z1)) -> insert(z0, sort(z1))
   insert(z0, nil) -> cons(z0, nil)
   choose(z0, cons(z1, z2), z3, 0) -> cons(z0, cons(z1, z2))
   choose(z0, cons(z1, z2), s(z3), s(z4)) -> choose(z0, cons(z1, z2), z3, z4)
And the Tuples:
   SORT(cons(z0, z1)) -> c1(INSERT(z0, sort(z1)), SORT(z1))
   INSERT(z0, cons(z1, z2)) -> c3(CHOOSE(z0, cons(z1, z2), z0, z1))
   CHOOSE(z0, cons(z1, z2), 0, s(z3)) -> c5(INSERT(z0, z2))
   CHOOSE(z0, cons(z1, z2), s(z3), s(z4)) -> c6(CHOOSE(z0, cons(z1, z2), z3, z4))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(CHOOSE(x_1, x_2, x_3, x_4)) = [2]x_2
   POL(INSERT(x_1, x_2)) = [2]x_2
   POL(SORT(x_1)) = [2]x_1^2
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c3(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(choose(x_1, x_2, x_3, x_4)) = [1] + x_2
   POL(cons(x_1, x_2)) = [1] + x_2
   POL(insert(x_1, x_2)) = [1] + x_2
   POL(nil) = 0
   POL(s(x_1)) = 0
   POL(sort(x_1)) = x_1

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sort(nil) -> nil
   sort(cons(z0, z1)) -> insert(z0, sort(z1))
   insert(z0, nil) -> cons(z0, nil)
   insert(z0, cons(z1, z2)) -> choose(z0, cons(z1, z2), z0, z1)
   choose(z0, cons(z1, z2), z3, 0) -> cons(z0, cons(z1, z2))
   choose(z0, cons(z1, z2), 0, s(z3)) -> cons(z1, insert(z0, z2))
   choose(z0, cons(z1, z2), s(z3), s(z4)) -> choose(z0, cons(z1, z2), z3, z4)
Tuples:
   SORT(cons(z0, z1)) -> c1(INSERT(z0, sort(z1)), SORT(z1))
   INSERT(z0, cons(z1, z2)) -> c3(CHOOSE(z0, cons(z1, z2), z0, z1))
   CHOOSE(z0, cons(z1, z2), 0, s(z3)) -> c5(INSERT(z0, z2))
   CHOOSE(z0, cons(z1, z2), s(z3), s(z4)) -> c6(CHOOSE(z0, cons(z1, z2), z3, z4))
S tuples:
   INSERT(z0, cons(z1, z2)) -> c3(CHOOSE(z0, cons(z1, z2), z0, z1))
   CHOOSE(z0, cons(z1, z2), s(z3), s(z4)) -> c6(CHOOSE(z0, cons(z1, z2), z3, z4))
K tuples:
   SORT(cons(z0, z1)) -> c1(INSERT(z0, sort(z1)), SORT(z1))
   CHOOSE(z0, cons(z1, z2), 0, s(z3)) -> c5(INSERT(z0, z2))
Defined Rule Symbols:   sort_1, insert_2, choose_4

Defined Pair Symbols:   SORT_1, INSERT_2, CHOOSE_4

Compound Symbols:   c1_2, c3_1, c5_1, c6_1


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

(9) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   INSERT(z0, cons(z1, z2)) -> c3(CHOOSE(z0, cons(z1, z2), z0, z1))
   CHOOSE(z0, cons(z1, z2), 0, s(z3)) -> c5(INSERT(z0, z2))

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

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sort(nil) -> nil
   sort(cons(z0, z1)) -> insert(z0, sort(z1))
   insert(z0, nil) -> cons(z0, nil)
   insert(z0, cons(z1, z2)) -> choose(z0, cons(z1, z2), z0, z1)
   choose(z0, cons(z1, z2), z3, 0) -> cons(z0, cons(z1, z2))
   choose(z0, cons(z1, z2), 0, s(z3)) -> cons(z1, insert(z0, z2))
   choose(z0, cons(z1, z2), s(z3), s(z4)) -> choose(z0, cons(z1, z2), z3, z4)
Tuples:
   SORT(cons(z0, z1)) -> c1(INSERT(z0, sort(z1)), SORT(z1))
   INSERT(z0, cons(z1, z2)) -> c3(CHOOSE(z0, cons(z1, z2), z0, z1))
   CHOOSE(z0, cons(z1, z2), 0, s(z3)) -> c5(INSERT(z0, z2))
   CHOOSE(z0, cons(z1, z2), s(z3), s(z4)) -> c6(CHOOSE(z0, cons(z1, z2), z3, z4))
S tuples:
   CHOOSE(z0, cons(z1, z2), s(z3), s(z4)) -> c6(CHOOSE(z0, cons(z1, z2), z3, z4))
K tuples:
   SORT(cons(z0, z1)) -> c1(INSERT(z0, sort(z1)), SORT(z1))
   CHOOSE(z0, cons(z1, z2), 0, s(z3)) -> c5(INSERT(z0, z2))
   INSERT(z0, cons(z1, z2)) -> c3(CHOOSE(z0, cons(z1, z2), z0, z1))
Defined Rule Symbols:   sort_1, insert_2, choose_4

Defined Pair Symbols:   SORT_1, INSERT_2, CHOOSE_4

Compound Symbols:   c1_2, c3_1, c5_1, c6_1


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

(11) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   CHOOSE(z0, cons(z1, z2), s(z3), s(z4)) -> c6(CHOOSE(z0, cons(z1, z2), z3, z4))
We considered the (Usable) Rules:
   choose(z0, cons(z1, z2), 0, s(z3)) -> cons(z1, insert(z0, z2))
   sort(nil) -> nil
   insert(z0, cons(z1, z2)) -> choose(z0, cons(z1, z2), z0, z1)
   sort(cons(z0, z1)) -> insert(z0, sort(z1))
   insert(z0, nil) -> cons(z0, nil)
   choose(z0, cons(z1, z2), z3, 0) -> cons(z0, cons(z1, z2))
   choose(z0, cons(z1, z2), s(z3), s(z4)) -> choose(z0, cons(z1, z2), z3, z4)
And the Tuples:
   SORT(cons(z0, z1)) -> c1(INSERT(z0, sort(z1)), SORT(z1))
   INSERT(z0, cons(z1, z2)) -> c3(CHOOSE(z0, cons(z1, z2), z0, z1))
   CHOOSE(z0, cons(z1, z2), 0, s(z3)) -> c5(INSERT(z0, z2))
   CHOOSE(z0, cons(z1, z2), s(z3), s(z4)) -> c6(CHOOSE(z0, cons(z1, z2), z3, z4))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(CHOOSE(x_1, x_2, x_3, x_4)) = [2]x_3 + x_1*x_2
   POL(INSERT(x_1, x_2)) = [2]x_1 + x_1*x_2
   POL(SORT(x_1)) = [2]x_1^2
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c3(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(choose(x_1, x_2, x_3, x_4)) = [2] + x_1 + x_2
   POL(cons(x_1, x_2)) = [2] + x_1 + x_2
   POL(insert(x_1, x_2)) = [2] + x_1 + x_2
   POL(nil) = 0
   POL(s(x_1)) = [2] + x_1
   POL(sort(x_1)) = [2] + x_1

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

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sort(nil) -> nil
   sort(cons(z0, z1)) -> insert(z0, sort(z1))
   insert(z0, nil) -> cons(z0, nil)
   insert(z0, cons(z1, z2)) -> choose(z0, cons(z1, z2), z0, z1)
   choose(z0, cons(z1, z2), z3, 0) -> cons(z0, cons(z1, z2))
   choose(z0, cons(z1, z2), 0, s(z3)) -> cons(z1, insert(z0, z2))
   choose(z0, cons(z1, z2), s(z3), s(z4)) -> choose(z0, cons(z1, z2), z3, z4)
Tuples:
   SORT(cons(z0, z1)) -> c1(INSERT(z0, sort(z1)), SORT(z1))
   INSERT(z0, cons(z1, z2)) -> c3(CHOOSE(z0, cons(z1, z2), z0, z1))
   CHOOSE(z0, cons(z1, z2), 0, s(z3)) -> c5(INSERT(z0, z2))
   CHOOSE(z0, cons(z1, z2), s(z3), s(z4)) -> c6(CHOOSE(z0, cons(z1, z2), z3, z4))
S tuples:none
K tuples:
   SORT(cons(z0, z1)) -> c1(INSERT(z0, sort(z1)), SORT(z1))
   CHOOSE(z0, cons(z1, z2), 0, s(z3)) -> c5(INSERT(z0, z2))
   INSERT(z0, cons(z1, z2)) -> c3(CHOOSE(z0, cons(z1, z2), z0, z1))
   CHOOSE(z0, cons(z1, z2), s(z3), s(z4)) -> c6(CHOOSE(z0, cons(z1, z2), z3, z4))
Defined Rule Symbols:   sort_1, insert_2, choose_4

Defined Pair Symbols:   SORT_1, INSERT_2, CHOOSE_4

Compound Symbols:   c1_2, c3_1, c5_1, c6_1


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

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

(14)
BOUNDS(1, 1)

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

(15) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(16)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sort(nil) -> nil
   sort(cons(z0, z1)) -> insert(z0, sort(z1))
   insert(z0, nil) -> cons(z0, nil)
   insert(z0, cons(z1, z2)) -> choose(z0, cons(z1, z2), z0, z1)
   choose(z0, cons(z1, z2), z3, 0) -> cons(z0, cons(z1, z2))
   choose(z0, cons(z1, z2), 0, s(z3)) -> cons(z1, insert(z0, z2))
   choose(z0, cons(z1, z2), s(z3), s(z4)) -> choose(z0, cons(z1, z2), z3, z4)
Tuples:
   SORT(nil) -> c
   SORT(cons(z0, z1)) -> c1(INSERT(z0, sort(z1)), SORT(z1))
   INSERT(z0, nil) -> c2
   INSERT(z0, cons(z1, z2)) -> c3(CHOOSE(z0, cons(z1, z2), z0, z1))
   CHOOSE(z0, cons(z1, z2), z3, 0) -> c4
   CHOOSE(z0, cons(z1, z2), 0, s(z3)) -> c5(INSERT(z0, z2))
   CHOOSE(z0, cons(z1, z2), s(z3), s(z4)) -> c6(CHOOSE(z0, cons(z1, z2), z3, z4))
S tuples:
   SORT(nil) -> c
   SORT(cons(z0, z1)) -> c1(INSERT(z0, sort(z1)), SORT(z1))
   INSERT(z0, nil) -> c2
   INSERT(z0, cons(z1, z2)) -> c3(CHOOSE(z0, cons(z1, z2), z0, z1))
   CHOOSE(z0, cons(z1, z2), z3, 0) -> c4
   CHOOSE(z0, cons(z1, z2), 0, s(z3)) -> c5(INSERT(z0, z2))
   CHOOSE(z0, cons(z1, z2), s(z3), s(z4)) -> c6(CHOOSE(z0, cons(z1, z2), z3, z4))
K tuples:none
Defined Rule Symbols:   sort_1, insert_2, choose_4

Defined Pair Symbols:   SORT_1, INSERT_2, CHOOSE_4

Compound Symbols:   c, c1_2, c2, c3_1, c4, c5_1, c6_1


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

(17) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID))
Converted S to standard rules, and D \ S as well as R to relative rules.
----------------------------------------

(18)
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:

   SORT(nil) -> c
   SORT(cons(z0, z1)) -> c1(INSERT(z0, sort(z1)), SORT(z1))
   INSERT(z0, nil) -> c2
   INSERT(z0, cons(z1, z2)) -> c3(CHOOSE(z0, cons(z1, z2), z0, z1))
   CHOOSE(z0, cons(z1, z2), z3, 0) -> c4
   CHOOSE(z0, cons(z1, z2), 0, s(z3)) -> c5(INSERT(z0, z2))
   CHOOSE(z0, cons(z1, z2), s(z3), s(z4)) -> c6(CHOOSE(z0, cons(z1, z2), z3, z4))

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

   sort(nil) -> nil
   sort(cons(z0, z1)) -> insert(z0, sort(z1))
   insert(z0, nil) -> cons(z0, nil)
   insert(z0, cons(z1, z2)) -> choose(z0, cons(z1, z2), z0, z1)
   choose(z0, cons(z1, z2), z3, 0) -> cons(z0, cons(z1, z2))
   choose(z0, cons(z1, z2), 0, s(z3)) -> cons(z1, insert(z0, z2))
   choose(z0, cons(z1, z2), s(z3), s(z4)) -> choose(z0, cons(z1, z2), z3, z4)

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

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

(20)
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:

   SORT(nil) -> c
   SORT(cons(z0, z1)) -> c1(INSERT(z0, sort(z1)), SORT(z1))
   INSERT(z0, nil) -> c2
   INSERT(z0, cons(z1, z2)) -> c3(CHOOSE(z0, cons(z1, z2), z0, z1))
   CHOOSE(z0, cons(z1, z2), z3, 0') -> c4
   CHOOSE(z0, cons(z1, z2), 0', s(z3)) -> c5(INSERT(z0, z2))
   CHOOSE(z0, cons(z1, z2), s(z3), s(z4)) -> c6(CHOOSE(z0, cons(z1, z2), z3, z4))

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

   sort(nil) -> nil
   sort(cons(z0, z1)) -> insert(z0, sort(z1))
   insert(z0, nil) -> cons(z0, nil)
   insert(z0, cons(z1, z2)) -> choose(z0, cons(z1, z2), z0, z1)
   choose(z0, cons(z1, z2), z3, 0') -> cons(z0, cons(z1, z2))
   choose(z0, cons(z1, z2), 0', s(z3)) -> cons(z1, insert(z0, z2))
   choose(z0, cons(z1, z2), s(z3), s(z4)) -> choose(z0, cons(z1, z2), z3, z4)

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

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

(22)
Obligation:
Innermost TRS:
Rules:
SORT(nil) -> c
SORT(cons(z0, z1)) -> c1(INSERT(z0, sort(z1)), SORT(z1))
INSERT(z0, nil) -> c2
INSERT(z0, cons(z1, z2)) -> c3(CHOOSE(z0, cons(z1, z2), z0, z1))
CHOOSE(z0, cons(z1, z2), z3, 0') -> c4
CHOOSE(z0, cons(z1, z2), 0', s(z3)) -> c5(INSERT(z0, z2))
CHOOSE(z0, cons(z1, z2), s(z3), s(z4)) -> c6(CHOOSE(z0, cons(z1, z2), z3, z4))
sort(nil) -> nil
sort(cons(z0, z1)) -> insert(z0, sort(z1))
insert(z0, nil) -> cons(z0, nil)
insert(z0, cons(z1, z2)) -> choose(z0, cons(z1, z2), z0, z1)
choose(z0, cons(z1, z2), z3, 0') -> cons(z0, cons(z1, z2))
choose(z0, cons(z1, z2), 0', s(z3)) -> cons(z1, insert(z0, z2))
choose(z0, cons(z1, z2), s(z3), s(z4)) -> choose(z0, cons(z1, z2), z3, z4)

Types:
SORT :: nil:cons -> c:c1
nil :: nil:cons
c :: c:c1
cons :: 0':s -> nil:cons -> nil:cons
c1 :: c2:c3 -> c:c1 -> c:c1
INSERT :: 0':s -> nil:cons -> c2:c3
sort :: nil:cons -> nil:cons
c2 :: c2:c3
c3 :: c4:c5:c6 -> c2:c3
CHOOSE :: 0':s -> nil:cons -> 0':s -> 0':s -> c4:c5:c6
0' :: 0':s
c4 :: c4:c5:c6
s :: 0':s -> 0':s
c5 :: c2:c3 -> c4:c5:c6
c6 :: c4:c5:c6 -> c4:c5:c6
insert :: 0':s -> nil:cons -> nil:cons
choose :: 0':s -> nil:cons -> 0':s -> 0':s -> nil:cons
hole_c:c11_7 :: c:c1
hole_nil:cons2_7 :: nil:cons
hole_0':s3_7 :: 0':s
hole_c2:c34_7 :: c2:c3
hole_c4:c5:c65_7 :: c4:c5:c6
gen_c:c16_7 :: Nat -> c:c1
gen_nil:cons7_7 :: Nat -> nil:cons
gen_0':s8_7 :: Nat -> 0':s
gen_c4:c5:c69_7 :: Nat -> c4:c5:c6

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

(23) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
SORT, INSERT, sort, CHOOSE, insert, choose

They will be analysed ascendingly in the following order:
INSERT < SORT
sort < SORT
INSERT = CHOOSE
insert < sort
insert = choose

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

(24)
Obligation:
Innermost TRS:
Rules:
SORT(nil) -> c
SORT(cons(z0, z1)) -> c1(INSERT(z0, sort(z1)), SORT(z1))
INSERT(z0, nil) -> c2
INSERT(z0, cons(z1, z2)) -> c3(CHOOSE(z0, cons(z1, z2), z0, z1))
CHOOSE(z0, cons(z1, z2), z3, 0') -> c4
CHOOSE(z0, cons(z1, z2), 0', s(z3)) -> c5(INSERT(z0, z2))
CHOOSE(z0, cons(z1, z2), s(z3), s(z4)) -> c6(CHOOSE(z0, cons(z1, z2), z3, z4))
sort(nil) -> nil
sort(cons(z0, z1)) -> insert(z0, sort(z1))
insert(z0, nil) -> cons(z0, nil)
insert(z0, cons(z1, z2)) -> choose(z0, cons(z1, z2), z0, z1)
choose(z0, cons(z1, z2), z3, 0') -> cons(z0, cons(z1, z2))
choose(z0, cons(z1, z2), 0', s(z3)) -> cons(z1, insert(z0, z2))
choose(z0, cons(z1, z2), s(z3), s(z4)) -> choose(z0, cons(z1, z2), z3, z4)

Types:
SORT :: nil:cons -> c:c1
nil :: nil:cons
c :: c:c1
cons :: 0':s -> nil:cons -> nil:cons
c1 :: c2:c3 -> c:c1 -> c:c1
INSERT :: 0':s -> nil:cons -> c2:c3
sort :: nil:cons -> nil:cons
c2 :: c2:c3
c3 :: c4:c5:c6 -> c2:c3
CHOOSE :: 0':s -> nil:cons -> 0':s -> 0':s -> c4:c5:c6
0' :: 0':s
c4 :: c4:c5:c6
s :: 0':s -> 0':s
c5 :: c2:c3 -> c4:c5:c6
c6 :: c4:c5:c6 -> c4:c5:c6
insert :: 0':s -> nil:cons -> nil:cons
choose :: 0':s -> nil:cons -> 0':s -> 0':s -> nil:cons
hole_c:c11_7 :: c:c1
hole_nil:cons2_7 :: nil:cons
hole_0':s3_7 :: 0':s
hole_c2:c34_7 :: c2:c3
hole_c4:c5:c65_7 :: c4:c5:c6
gen_c:c16_7 :: Nat -> c:c1
gen_nil:cons7_7 :: Nat -> nil:cons
gen_0':s8_7 :: Nat -> 0':s
gen_c4:c5:c69_7 :: Nat -> c4:c5:c6


Generator Equations:
gen_c:c16_7(0) <=> c
gen_c:c16_7(+(x, 1)) <=> c1(c2, gen_c:c16_7(x))
gen_nil:cons7_7(0) <=> nil
gen_nil:cons7_7(+(x, 1)) <=> cons(0', gen_nil:cons7_7(x))
gen_0':s8_7(0) <=> 0'
gen_0':s8_7(+(x, 1)) <=> s(gen_0':s8_7(x))
gen_c4:c5:c69_7(0) <=> c4
gen_c4:c5:c69_7(+(x, 1)) <=> c6(gen_c4:c5:c69_7(x))


The following defined symbols remain to be analysed:
choose, SORT, INSERT, sort, CHOOSE, insert

They will be analysed ascendingly in the following order:
INSERT < SORT
sort < SORT
INSERT = CHOOSE
insert < sort
insert = choose

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

(25) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
choose(gen_0':s8_7(a), gen_nil:cons7_7(1), gen_0':s8_7(n11_7), gen_0':s8_7(n11_7)) -> cons(gen_0':s8_7(a), gen_nil:cons7_7(1)), rt in Omega(0)

Induction Base:
choose(gen_0':s8_7(a), gen_nil:cons7_7(1), gen_0':s8_7(0), gen_0':s8_7(0)) ->_R^Omega(0)
cons(gen_0':s8_7(a), cons(0', gen_nil:cons7_7(0)))

Induction Step:
choose(gen_0':s8_7(a), gen_nil:cons7_7(1), gen_0':s8_7(+(n11_7, 1)), gen_0':s8_7(+(n11_7, 1))) ->_R^Omega(0)
choose(gen_0':s8_7(a), cons(0', gen_nil:cons7_7(0)), gen_0':s8_7(n11_7), gen_0':s8_7(n11_7)) ->_IH
cons(gen_0':s8_7(a), gen_nil:cons7_7(1))

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

(26)
Obligation:
Innermost TRS:
Rules:
SORT(nil) -> c
SORT(cons(z0, z1)) -> c1(INSERT(z0, sort(z1)), SORT(z1))
INSERT(z0, nil) -> c2
INSERT(z0, cons(z1, z2)) -> c3(CHOOSE(z0, cons(z1, z2), z0, z1))
CHOOSE(z0, cons(z1, z2), z3, 0') -> c4
CHOOSE(z0, cons(z1, z2), 0', s(z3)) -> c5(INSERT(z0, z2))
CHOOSE(z0, cons(z1, z2), s(z3), s(z4)) -> c6(CHOOSE(z0, cons(z1, z2), z3, z4))
sort(nil) -> nil
sort(cons(z0, z1)) -> insert(z0, sort(z1))
insert(z0, nil) -> cons(z0, nil)
insert(z0, cons(z1, z2)) -> choose(z0, cons(z1, z2), z0, z1)
choose(z0, cons(z1, z2), z3, 0') -> cons(z0, cons(z1, z2))
choose(z0, cons(z1, z2), 0', s(z3)) -> cons(z1, insert(z0, z2))
choose(z0, cons(z1, z2), s(z3), s(z4)) -> choose(z0, cons(z1, z2), z3, z4)

Types:
SORT :: nil:cons -> c:c1
nil :: nil:cons
c :: c:c1
cons :: 0':s -> nil:cons -> nil:cons
c1 :: c2:c3 -> c:c1 -> c:c1
INSERT :: 0':s -> nil:cons -> c2:c3
sort :: nil:cons -> nil:cons
c2 :: c2:c3
c3 :: c4:c5:c6 -> c2:c3
CHOOSE :: 0':s -> nil:cons -> 0':s -> 0':s -> c4:c5:c6
0' :: 0':s
c4 :: c4:c5:c6
s :: 0':s -> 0':s
c5 :: c2:c3 -> c4:c5:c6
c6 :: c4:c5:c6 -> c4:c5:c6
insert :: 0':s -> nil:cons -> nil:cons
choose :: 0':s -> nil:cons -> 0':s -> 0':s -> nil:cons
hole_c:c11_7 :: c:c1
hole_nil:cons2_7 :: nil:cons
hole_0':s3_7 :: 0':s
hole_c2:c34_7 :: c2:c3
hole_c4:c5:c65_7 :: c4:c5:c6
gen_c:c16_7 :: Nat -> c:c1
gen_nil:cons7_7 :: Nat -> nil:cons
gen_0':s8_7 :: Nat -> 0':s
gen_c4:c5:c69_7 :: Nat -> c4:c5:c6


Lemmas:
choose(gen_0':s8_7(a), gen_nil:cons7_7(1), gen_0':s8_7(n11_7), gen_0':s8_7(n11_7)) -> cons(gen_0':s8_7(a), gen_nil:cons7_7(1)), rt in Omega(0)


Generator Equations:
gen_c:c16_7(0) <=> c
gen_c:c16_7(+(x, 1)) <=> c1(c2, gen_c:c16_7(x))
gen_nil:cons7_7(0) <=> nil
gen_nil:cons7_7(+(x, 1)) <=> cons(0', gen_nil:cons7_7(x))
gen_0':s8_7(0) <=> 0'
gen_0':s8_7(+(x, 1)) <=> s(gen_0':s8_7(x))
gen_c4:c5:c69_7(0) <=> c4
gen_c4:c5:c69_7(+(x, 1)) <=> c6(gen_c4:c5:c69_7(x))


The following defined symbols remain to be analysed:
insert, SORT, INSERT, sort, CHOOSE

They will be analysed ascendingly in the following order:
INSERT < SORT
sort < SORT
INSERT = CHOOSE
insert < sort
insert = choose

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

(27) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
sort(gen_nil:cons7_7(n5274_7)) -> *10_7, rt in Omega(0)

Induction Base:
sort(gen_nil:cons7_7(0))

Induction Step:
sort(gen_nil:cons7_7(+(n5274_7, 1))) ->_R^Omega(0)
insert(0', sort(gen_nil:cons7_7(n5274_7))) ->_IH
insert(0', *10_7)

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

(28)
Obligation:
Innermost TRS:
Rules:
SORT(nil) -> c
SORT(cons(z0, z1)) -> c1(INSERT(z0, sort(z1)), SORT(z1))
INSERT(z0, nil) -> c2
INSERT(z0, cons(z1, z2)) -> c3(CHOOSE(z0, cons(z1, z2), z0, z1))
CHOOSE(z0, cons(z1, z2), z3, 0') -> c4
CHOOSE(z0, cons(z1, z2), 0', s(z3)) -> c5(INSERT(z0, z2))
CHOOSE(z0, cons(z1, z2), s(z3), s(z4)) -> c6(CHOOSE(z0, cons(z1, z2), z3, z4))
sort(nil) -> nil
sort(cons(z0, z1)) -> insert(z0, sort(z1))
insert(z0, nil) -> cons(z0, nil)
insert(z0, cons(z1, z2)) -> choose(z0, cons(z1, z2), z0, z1)
choose(z0, cons(z1, z2), z3, 0') -> cons(z0, cons(z1, z2))
choose(z0, cons(z1, z2), 0', s(z3)) -> cons(z1, insert(z0, z2))
choose(z0, cons(z1, z2), s(z3), s(z4)) -> choose(z0, cons(z1, z2), z3, z4)

Types:
SORT :: nil:cons -> c:c1
nil :: nil:cons
c :: c:c1
cons :: 0':s -> nil:cons -> nil:cons
c1 :: c2:c3 -> c:c1 -> c:c1
INSERT :: 0':s -> nil:cons -> c2:c3
sort :: nil:cons -> nil:cons
c2 :: c2:c3
c3 :: c4:c5:c6 -> c2:c3
CHOOSE :: 0':s -> nil:cons -> 0':s -> 0':s -> c4:c5:c6
0' :: 0':s
c4 :: c4:c5:c6
s :: 0':s -> 0':s
c5 :: c2:c3 -> c4:c5:c6
c6 :: c4:c5:c6 -> c4:c5:c6
insert :: 0':s -> nil:cons -> nil:cons
choose :: 0':s -> nil:cons -> 0':s -> 0':s -> nil:cons
hole_c:c11_7 :: c:c1
hole_nil:cons2_7 :: nil:cons
hole_0':s3_7 :: 0':s
hole_c2:c34_7 :: c2:c3
hole_c4:c5:c65_7 :: c4:c5:c6
gen_c:c16_7 :: Nat -> c:c1
gen_nil:cons7_7 :: Nat -> nil:cons
gen_0':s8_7 :: Nat -> 0':s
gen_c4:c5:c69_7 :: Nat -> c4:c5:c6


Lemmas:
choose(gen_0':s8_7(a), gen_nil:cons7_7(1), gen_0':s8_7(n11_7), gen_0':s8_7(n11_7)) -> cons(gen_0':s8_7(a), gen_nil:cons7_7(1)), rt in Omega(0)
sort(gen_nil:cons7_7(n5274_7)) -> *10_7, rt in Omega(0)


Generator Equations:
gen_c:c16_7(0) <=> c
gen_c:c16_7(+(x, 1)) <=> c1(c2, gen_c:c16_7(x))
gen_nil:cons7_7(0) <=> nil
gen_nil:cons7_7(+(x, 1)) <=> cons(0', gen_nil:cons7_7(x))
gen_0':s8_7(0) <=> 0'
gen_0':s8_7(+(x, 1)) <=> s(gen_0':s8_7(x))
gen_c4:c5:c69_7(0) <=> c4
gen_c4:c5:c69_7(+(x, 1)) <=> c6(gen_c4:c5:c69_7(x))


The following defined symbols remain to be analysed:
CHOOSE, SORT, INSERT

They will be analysed ascendingly in the following order:
INSERT < SORT
INSERT = CHOOSE

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

(29) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
CHOOSE(gen_0':s8_7(a), gen_nil:cons7_7(1), gen_0':s8_7(n9351_7), gen_0':s8_7(n9351_7)) -> gen_c4:c5:c69_7(n9351_7), rt in Omega(1 + n9351_7)

Induction Base:
CHOOSE(gen_0':s8_7(a), gen_nil:cons7_7(1), gen_0':s8_7(0), gen_0':s8_7(0)) ->_R^Omega(1)
c4

Induction Step:
CHOOSE(gen_0':s8_7(a), gen_nil:cons7_7(1), gen_0':s8_7(+(n9351_7, 1)), gen_0':s8_7(+(n9351_7, 1))) ->_R^Omega(1)
c6(CHOOSE(gen_0':s8_7(a), cons(0', gen_nil:cons7_7(0)), gen_0':s8_7(n9351_7), gen_0':s8_7(n9351_7))) ->_IH
c6(gen_c4:c5:c69_7(c9352_7))

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

(30)
Complex Obligation (BEST)

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

(31)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
SORT(nil) -> c
SORT(cons(z0, z1)) -> c1(INSERT(z0, sort(z1)), SORT(z1))
INSERT(z0, nil) -> c2
INSERT(z0, cons(z1, z2)) -> c3(CHOOSE(z0, cons(z1, z2), z0, z1))
CHOOSE(z0, cons(z1, z2), z3, 0') -> c4
CHOOSE(z0, cons(z1, z2), 0', s(z3)) -> c5(INSERT(z0, z2))
CHOOSE(z0, cons(z1, z2), s(z3), s(z4)) -> c6(CHOOSE(z0, cons(z1, z2), z3, z4))
sort(nil) -> nil
sort(cons(z0, z1)) -> insert(z0, sort(z1))
insert(z0, nil) -> cons(z0, nil)
insert(z0, cons(z1, z2)) -> choose(z0, cons(z1, z2), z0, z1)
choose(z0, cons(z1, z2), z3, 0') -> cons(z0, cons(z1, z2))
choose(z0, cons(z1, z2), 0', s(z3)) -> cons(z1, insert(z0, z2))
choose(z0, cons(z1, z2), s(z3), s(z4)) -> choose(z0, cons(z1, z2), z3, z4)

Types:
SORT :: nil:cons -> c:c1
nil :: nil:cons
c :: c:c1
cons :: 0':s -> nil:cons -> nil:cons
c1 :: c2:c3 -> c:c1 -> c:c1
INSERT :: 0':s -> nil:cons -> c2:c3
sort :: nil:cons -> nil:cons
c2 :: c2:c3
c3 :: c4:c5:c6 -> c2:c3
CHOOSE :: 0':s -> nil:cons -> 0':s -> 0':s -> c4:c5:c6
0' :: 0':s
c4 :: c4:c5:c6
s :: 0':s -> 0':s
c5 :: c2:c3 -> c4:c5:c6
c6 :: c4:c5:c6 -> c4:c5:c6
insert :: 0':s -> nil:cons -> nil:cons
choose :: 0':s -> nil:cons -> 0':s -> 0':s -> nil:cons
hole_c:c11_7 :: c:c1
hole_nil:cons2_7 :: nil:cons
hole_0':s3_7 :: 0':s
hole_c2:c34_7 :: c2:c3
hole_c4:c5:c65_7 :: c4:c5:c6
gen_c:c16_7 :: Nat -> c:c1
gen_nil:cons7_7 :: Nat -> nil:cons
gen_0':s8_7 :: Nat -> 0':s
gen_c4:c5:c69_7 :: Nat -> c4:c5:c6


Lemmas:
choose(gen_0':s8_7(a), gen_nil:cons7_7(1), gen_0':s8_7(n11_7), gen_0':s8_7(n11_7)) -> cons(gen_0':s8_7(a), gen_nil:cons7_7(1)), rt in Omega(0)
sort(gen_nil:cons7_7(n5274_7)) -> *10_7, rt in Omega(0)


Generator Equations:
gen_c:c16_7(0) <=> c
gen_c:c16_7(+(x, 1)) <=> c1(c2, gen_c:c16_7(x))
gen_nil:cons7_7(0) <=> nil
gen_nil:cons7_7(+(x, 1)) <=> cons(0', gen_nil:cons7_7(x))
gen_0':s8_7(0) <=> 0'
gen_0':s8_7(+(x, 1)) <=> s(gen_0':s8_7(x))
gen_c4:c5:c69_7(0) <=> c4
gen_c4:c5:c69_7(+(x, 1)) <=> c6(gen_c4:c5:c69_7(x))


The following defined symbols remain to be analysed:
CHOOSE, SORT, INSERT

They will be analysed ascendingly in the following order:
INSERT < SORT
INSERT = CHOOSE

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

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

(33)
BOUNDS(n^1, INF)

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

(34)
Obligation:
Innermost TRS:
Rules:
SORT(nil) -> c
SORT(cons(z0, z1)) -> c1(INSERT(z0, sort(z1)), SORT(z1))
INSERT(z0, nil) -> c2
INSERT(z0, cons(z1, z2)) -> c3(CHOOSE(z0, cons(z1, z2), z0, z1))
CHOOSE(z0, cons(z1, z2), z3, 0') -> c4
CHOOSE(z0, cons(z1, z2), 0', s(z3)) -> c5(INSERT(z0, z2))
CHOOSE(z0, cons(z1, z2), s(z3), s(z4)) -> c6(CHOOSE(z0, cons(z1, z2), z3, z4))
sort(nil) -> nil
sort(cons(z0, z1)) -> insert(z0, sort(z1))
insert(z0, nil) -> cons(z0, nil)
insert(z0, cons(z1, z2)) -> choose(z0, cons(z1, z2), z0, z1)
choose(z0, cons(z1, z2), z3, 0') -> cons(z0, cons(z1, z2))
choose(z0, cons(z1, z2), 0', s(z3)) -> cons(z1, insert(z0, z2))
choose(z0, cons(z1, z2), s(z3), s(z4)) -> choose(z0, cons(z1, z2), z3, z4)

Types:
SORT :: nil:cons -> c:c1
nil :: nil:cons
c :: c:c1
cons :: 0':s -> nil:cons -> nil:cons
c1 :: c2:c3 -> c:c1 -> c:c1
INSERT :: 0':s -> nil:cons -> c2:c3
sort :: nil:cons -> nil:cons
c2 :: c2:c3
c3 :: c4:c5:c6 -> c2:c3
CHOOSE :: 0':s -> nil:cons -> 0':s -> 0':s -> c4:c5:c6
0' :: 0':s
c4 :: c4:c5:c6
s :: 0':s -> 0':s
c5 :: c2:c3 -> c4:c5:c6
c6 :: c4:c5:c6 -> c4:c5:c6
insert :: 0':s -> nil:cons -> nil:cons
choose :: 0':s -> nil:cons -> 0':s -> 0':s -> nil:cons
hole_c:c11_7 :: c:c1
hole_nil:cons2_7 :: nil:cons
hole_0':s3_7 :: 0':s
hole_c2:c34_7 :: c2:c3
hole_c4:c5:c65_7 :: c4:c5:c6
gen_c:c16_7 :: Nat -> c:c1
gen_nil:cons7_7 :: Nat -> nil:cons
gen_0':s8_7 :: Nat -> 0':s
gen_c4:c5:c69_7 :: Nat -> c4:c5:c6


Lemmas:
choose(gen_0':s8_7(a), gen_nil:cons7_7(1), gen_0':s8_7(n11_7), gen_0':s8_7(n11_7)) -> cons(gen_0':s8_7(a), gen_nil:cons7_7(1)), rt in Omega(0)
sort(gen_nil:cons7_7(n5274_7)) -> *10_7, rt in Omega(0)
CHOOSE(gen_0':s8_7(a), gen_nil:cons7_7(1), gen_0':s8_7(n9351_7), gen_0':s8_7(n9351_7)) -> gen_c4:c5:c69_7(n9351_7), rt in Omega(1 + n9351_7)


Generator Equations:
gen_c:c16_7(0) <=> c
gen_c:c16_7(+(x, 1)) <=> c1(c2, gen_c:c16_7(x))
gen_nil:cons7_7(0) <=> nil
gen_nil:cons7_7(+(x, 1)) <=> cons(0', gen_nil:cons7_7(x))
gen_0':s8_7(0) <=> 0'
gen_0':s8_7(+(x, 1)) <=> s(gen_0':s8_7(x))
gen_c4:c5:c69_7(0) <=> c4
gen_c4:c5:c69_7(+(x, 1)) <=> c6(gen_c4:c5:c69_7(x))


The following defined symbols remain to be analysed:
INSERT, SORT

They will be analysed ascendingly in the following order:
INSERT < SORT
INSERT = CHOOSE