/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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WORST_CASE(Omega(n^1), O(n^2))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Runtime Complexity (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) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CdtProblem
    (7) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 65 ms]
    (8) CdtProblem
    (9) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 21 ms]
    (10) CdtProblem
    (11) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) CdtProblem
    (13) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) CdtProblem
    (15) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) CdtProblem
    (17) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) CdtProblem
    (19) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (20) CdtProblem
    (21) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (22) CdtProblem
    (23) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (24) CdtProblem
    (25) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (26) CdtProblem
    (27) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (28) CdtProblem
    (29) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (30) CdtProblem
    (31) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (32) CdtProblem
    (33) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 1352 ms]
    (34) CdtProblem
    (35) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
    (36) BOUNDS(1, 1)
(37) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(38) CpxTRS
    (39) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (40) typed CpxTrs
    (41) OrderProof [LOWER BOUND(ID), 0 ms]
    (42) typed CpxTrs
    (43) RewriteLemmaProof [LOWER BOUND(ID), 209 ms]
    (44) BEST
        (45) proven lower bound
            (46) LowerBoundPropagationProof [FINISHED, 0 ms]
            (47) BOUNDS(n^1, INF)
        (48) typed CpxTrs


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

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


The TRS R consists of the following rules:

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

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

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

(2)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(0) -> true
   f(1) -> false
   f(s(z0)) -> f(z0)
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   g(s(z0), s(z1)) -> if(f(z0), s(z0), s(z1))
   g(z0, c(z1)) -> g(z0, g(s(c(z1)), z1))
Tuples:
   F(0) -> c1
   F(1) -> c2
   F(s(z0)) -> c3(F(z0))
   IF(true, z0, z1) -> c4
   IF(false, z0, z1) -> c5
   G(s(z0), s(z1)) -> c6(IF(f(z0), s(z0), s(z1)), F(z0))
   G(z0, c(z1)) -> c7(G(z0, g(s(c(z1)), z1)), G(s(c(z1)), z1))
S tuples:
   F(0) -> c1
   F(1) -> c2
   F(s(z0)) -> c3(F(z0))
   IF(true, z0, z1) -> c4
   IF(false, z0, z1) -> c5
   G(s(z0), s(z1)) -> c6(IF(f(z0), s(z0), s(z1)), F(z0))
   G(z0, c(z1)) -> c7(G(z0, g(s(c(z1)), z1)), G(s(c(z1)), z1))
K tuples:none
Defined Rule Symbols:   f_1, if_3, g_2

Defined Pair Symbols:   F_1, IF_3, G_2

Compound Symbols:   c1, c2, c3_1, c4, c5, c6_2, c7_2


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

(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 4 trailing nodes:
   IF(false, z0, z1) -> c5
   F(1) -> c2
   IF(true, z0, z1) -> c4
   F(0) -> c1

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(0) -> true
   f(1) -> false
   f(s(z0)) -> f(z0)
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   g(s(z0), s(z1)) -> if(f(z0), s(z0), s(z1))
   g(z0, c(z1)) -> g(z0, g(s(c(z1)), z1))
Tuples:
   F(s(z0)) -> c3(F(z0))
   G(s(z0), s(z1)) -> c6(IF(f(z0), s(z0), s(z1)), F(z0))
   G(z0, c(z1)) -> c7(G(z0, g(s(c(z1)), z1)), G(s(c(z1)), z1))
S tuples:
   F(s(z0)) -> c3(F(z0))
   G(s(z0), s(z1)) -> c6(IF(f(z0), s(z0), s(z1)), F(z0))
   G(z0, c(z1)) -> c7(G(z0, g(s(c(z1)), z1)), G(s(c(z1)), z1))
K tuples:none
Defined Rule Symbols:   f_1, if_3, g_2

Defined Pair Symbols:   F_1, G_2

Compound Symbols:   c3_1, c6_2, c7_2


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

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(0) -> true
   f(1) -> false
   f(s(z0)) -> f(z0)
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   g(s(z0), s(z1)) -> if(f(z0), s(z0), s(z1))
   g(z0, c(z1)) -> g(z0, g(s(c(z1)), z1))
Tuples:
   F(s(z0)) -> c3(F(z0))
   G(z0, c(z1)) -> c7(G(z0, g(s(c(z1)), z1)), G(s(c(z1)), z1))
   G(s(z0), s(z1)) -> c6(F(z0))
S tuples:
   F(s(z0)) -> c3(F(z0))
   G(z0, c(z1)) -> c7(G(z0, g(s(c(z1)), z1)), G(s(c(z1)), z1))
   G(s(z0), s(z1)) -> c6(F(z0))
K tuples:none
Defined Rule Symbols:   f_1, if_3, g_2

Defined Pair Symbols:   F_1, G_2

Compound Symbols:   c3_1, c7_2, c6_1


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

(7) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   G(s(z0), s(z1)) -> c6(F(z0))
We considered the (Usable) Rules:
   g(s(z0), s(z1)) -> if(f(z0), s(z0), s(z1))
   if(true, z0, z1) -> z0
   g(z0, c(z1)) -> g(z0, g(s(c(z1)), z1))
   if(false, z0, z1) -> z1
And the Tuples:
   F(s(z0)) -> c3(F(z0))
   G(z0, c(z1)) -> c7(G(z0, g(s(c(z1)), z1)), G(s(c(z1)), z1))
   G(s(z0), s(z1)) -> c6(F(z0))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(1) = [1]
   POL(F(x_1)) = 0
   POL(G(x_1, x_2)) = [1] + x_1 + x_2
   POL(c(x_1)) = [1] + x_1
   POL(c3(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c7(x_1, x_2)) = x_1 + x_2
   POL(f(x_1)) = [1]
   POL(false) = [1]
   POL(g(x_1, x_2)) = 0
   POL(if(x_1, x_2, x_3)) = x_2 + x_3
   POL(s(x_1)) = 0
   POL(true) = [1]

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(0) -> true
   f(1) -> false
   f(s(z0)) -> f(z0)
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   g(s(z0), s(z1)) -> if(f(z0), s(z0), s(z1))
   g(z0, c(z1)) -> g(z0, g(s(c(z1)), z1))
Tuples:
   F(s(z0)) -> c3(F(z0))
   G(z0, c(z1)) -> c7(G(z0, g(s(c(z1)), z1)), G(s(c(z1)), z1))
   G(s(z0), s(z1)) -> c6(F(z0))
S tuples:
   F(s(z0)) -> c3(F(z0))
   G(z0, c(z1)) -> c7(G(z0, g(s(c(z1)), z1)), G(s(c(z1)), z1))
K tuples:
   G(s(z0), s(z1)) -> c6(F(z0))
Defined Rule Symbols:   f_1, if_3, g_2

Defined Pair Symbols:   F_1, G_2

Compound Symbols:   c3_1, c7_2, c6_1


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

(9) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   G(z0, c(z1)) -> c7(G(z0, g(s(c(z1)), z1)), G(s(c(z1)), z1))
We considered the (Usable) Rules:
   g(s(z0), s(z1)) -> if(f(z0), s(z0), s(z1))
   if(true, z0, z1) -> z0
   g(z0, c(z1)) -> g(z0, g(s(c(z1)), z1))
   if(false, z0, z1) -> z1
And the Tuples:
   F(s(z0)) -> c3(F(z0))
   G(z0, c(z1)) -> c7(G(z0, g(s(c(z1)), z1)), G(s(c(z1)), z1))
   G(s(z0), s(z1)) -> c6(F(z0))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(1) = [1]
   POL(F(x_1)) = 0
   POL(G(x_1, x_2)) = x_1 + x_2
   POL(c(x_1)) = [1] + x_1
   POL(c3(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c7(x_1, x_2)) = x_1 + x_2
   POL(f(x_1)) = [1]
   POL(false) = [1]
   POL(g(x_1, x_2)) = 0
   POL(if(x_1, x_2, x_3)) = x_2 + x_3
   POL(s(x_1)) = 0
   POL(true) = [1]

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

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(0) -> true
   f(1) -> false
   f(s(z0)) -> f(z0)
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   g(s(z0), s(z1)) -> if(f(z0), s(z0), s(z1))
   g(z0, c(z1)) -> g(z0, g(s(c(z1)), z1))
Tuples:
   F(s(z0)) -> c3(F(z0))
   G(z0, c(z1)) -> c7(G(z0, g(s(c(z1)), z1)), G(s(c(z1)), z1))
   G(s(z0), s(z1)) -> c6(F(z0))
S tuples:
   F(s(z0)) -> c3(F(z0))
K tuples:
   G(s(z0), s(z1)) -> c6(F(z0))
   G(z0, c(z1)) -> c7(G(z0, g(s(c(z1)), z1)), G(s(c(z1)), z1))
Defined Rule Symbols:   f_1, if_3, g_2

Defined Pair Symbols:   F_1, G_2

Compound Symbols:   c3_1, c7_2, c6_1


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

(11) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace G(z0, c(z1)) -> c7(G(z0, g(s(c(z1)), z1)), G(s(c(z1)), z1)) by 
   G(x0, c(s(z1))) -> c7(G(x0, if(f(c(s(z1))), s(c(s(z1))), s(z1))), G(s(c(s(z1))), s(z1)))
   G(x0, c(c(z1))) -> c7(G(x0, g(s(c(c(z1))), g(s(c(z1)), z1))), G(s(c(c(z1))), c(z1)))

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

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(0) -> true
   f(1) -> false
   f(s(z0)) -> f(z0)
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   g(s(z0), s(z1)) -> if(f(z0), s(z0), s(z1))
   g(z0, c(z1)) -> g(z0, g(s(c(z1)), z1))
Tuples:
   F(s(z0)) -> c3(F(z0))
   G(s(z0), s(z1)) -> c6(F(z0))
   G(x0, c(s(z1))) -> c7(G(x0, if(f(c(s(z1))), s(c(s(z1))), s(z1))), G(s(c(s(z1))), s(z1)))
   G(x0, c(c(z1))) -> c7(G(x0, g(s(c(c(z1))), g(s(c(z1)), z1))), G(s(c(c(z1))), c(z1)))
S tuples:
   F(s(z0)) -> c3(F(z0))
K tuples:
   G(s(z0), s(z1)) -> c6(F(z0))
   G(z0, c(z1)) -> c7(G(z0, g(s(c(z1)), z1)), G(s(c(z1)), z1))
Defined Rule Symbols:   f_1, if_3, g_2

Defined Pair Symbols:   F_1, G_2

Compound Symbols:   c3_1, c6_1, c7_2


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

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

(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(0) -> true
   f(1) -> false
   f(s(z0)) -> f(z0)
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   g(s(z0), s(z1)) -> if(f(z0), s(z0), s(z1))
   g(z0, c(z1)) -> g(z0, g(s(c(z1)), z1))
Tuples:
   F(s(z0)) -> c3(F(z0))
   G(s(z0), s(z1)) -> c6(F(z0))
   G(x0, c(c(z1))) -> c7(G(x0, g(s(c(c(z1))), g(s(c(z1)), z1))), G(s(c(c(z1))), c(z1)))
   G(x0, c(s(z1))) -> c7(G(s(c(s(z1))), s(z1)))
S tuples:
   F(s(z0)) -> c3(F(z0))
K tuples:
   G(s(z0), s(z1)) -> c6(F(z0))
   G(z0, c(z1)) -> c7(G(z0, g(s(c(z1)), z1)), G(s(c(z1)), z1))
Defined Rule Symbols:   f_1, if_3, g_2

Defined Pair Symbols:   F_1, G_2

Compound Symbols:   c3_1, c6_1, c7_2, c7_1


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

(15) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace G(x0, c(c(z1))) -> c7(G(x0, g(s(c(c(z1))), g(s(c(z1)), z1))), G(s(c(c(z1))), c(z1))) by 
   G(x0, c(c(s(z1)))) -> c7(G(x0, g(s(c(c(s(z1)))), if(f(c(s(z1))), s(c(s(z1))), s(z1)))), G(s(c(c(s(z1)))), c(s(z1))))
   G(x0, c(c(c(z1)))) -> c7(G(x0, g(s(c(c(c(z1)))), g(s(c(c(z1))), g(s(c(z1)), z1)))), G(s(c(c(c(z1)))), c(c(z1))))
   G(x0, c(c(x1))) -> c7(G(s(c(c(x1))), c(x1)))

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

(16)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(0) -> true
   f(1) -> false
   f(s(z0)) -> f(z0)
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   g(s(z0), s(z1)) -> if(f(z0), s(z0), s(z1))
   g(z0, c(z1)) -> g(z0, g(s(c(z1)), z1))
Tuples:
   F(s(z0)) -> c3(F(z0))
   G(s(z0), s(z1)) -> c6(F(z0))
   G(x0, c(s(z1))) -> c7(G(s(c(s(z1))), s(z1)))
   G(x0, c(c(s(z1)))) -> c7(G(x0, g(s(c(c(s(z1)))), if(f(c(s(z1))), s(c(s(z1))), s(z1)))), G(s(c(c(s(z1)))), c(s(z1))))
   G(x0, c(c(c(z1)))) -> c7(G(x0, g(s(c(c(c(z1)))), g(s(c(c(z1))), g(s(c(z1)), z1)))), G(s(c(c(c(z1)))), c(c(z1))))
   G(x0, c(c(x1))) -> c7(G(s(c(c(x1))), c(x1)))
S tuples:
   F(s(z0)) -> c3(F(z0))
K tuples:
   G(s(z0), s(z1)) -> c6(F(z0))
   G(z0, c(z1)) -> c7(G(z0, g(s(c(z1)), z1)), G(s(c(z1)), z1))
Defined Rule Symbols:   f_1, if_3, g_2

Defined Pair Symbols:   F_1, G_2

Compound Symbols:   c3_1, c6_1, c7_1, c7_2


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

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

(18)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(0) -> true
   f(1) -> false
   f(s(z0)) -> f(z0)
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   g(s(z0), s(z1)) -> if(f(z0), s(z0), s(z1))
   g(z0, c(z1)) -> g(z0, g(s(c(z1)), z1))
Tuples:
   F(s(z0)) -> c3(F(z0))
   G(s(z0), s(z1)) -> c6(F(z0))
   G(x0, c(s(z1))) -> c7(G(s(c(s(z1))), s(z1)))
   G(x0, c(c(c(z1)))) -> c7(G(x0, g(s(c(c(c(z1)))), g(s(c(c(z1))), g(s(c(z1)), z1)))), G(s(c(c(c(z1)))), c(c(z1))))
   G(x0, c(c(x1))) -> c7(G(s(c(c(x1))), c(x1)))
   G(x0, c(c(s(z1)))) -> c7(G(s(c(c(s(z1)))), c(s(z1))))
S tuples:
   F(s(z0)) -> c3(F(z0))
K tuples:
   G(s(z0), s(z1)) -> c6(F(z0))
   G(z0, c(z1)) -> c7(G(z0, g(s(c(z1)), z1)), G(s(c(z1)), z1))
Defined Rule Symbols:   f_1, if_3, g_2

Defined Pair Symbols:   F_1, G_2

Compound Symbols:   c3_1, c6_1, c7_1, c7_2


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

(19) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace F(s(z0)) -> c3(F(z0)) by 
   F(s(s(y0))) -> c3(F(s(y0)))

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

(20)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(0) -> true
   f(1) -> false
   f(s(z0)) -> f(z0)
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   g(s(z0), s(z1)) -> if(f(z0), s(z0), s(z1))
   g(z0, c(z1)) -> g(z0, g(s(c(z1)), z1))
Tuples:
   G(s(z0), s(z1)) -> c6(F(z0))
   G(x0, c(s(z1))) -> c7(G(s(c(s(z1))), s(z1)))
   G(x0, c(c(c(z1)))) -> c7(G(x0, g(s(c(c(c(z1)))), g(s(c(c(z1))), g(s(c(z1)), z1)))), G(s(c(c(c(z1)))), c(c(z1))))
   G(x0, c(c(x1))) -> c7(G(s(c(c(x1))), c(x1)))
   G(x0, c(c(s(z1)))) -> c7(G(s(c(c(s(z1)))), c(s(z1))))
   F(s(s(y0))) -> c3(F(s(y0)))
S tuples:
   F(s(s(y0))) -> c3(F(s(y0)))
K tuples:
   G(s(z0), s(z1)) -> c6(F(z0))
   G(z0, c(z1)) -> c7(G(z0, g(s(c(z1)), z1)), G(s(c(z1)), z1))
Defined Rule Symbols:   f_1, if_3, g_2

Defined Pair Symbols:   G_2, F_1

Compound Symbols:   c6_1, c7_1, c7_2, c3_1


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

(21) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace G(s(z0), s(z1)) -> c6(F(z0)) by 
   G(s(s(s(y0))), s(z1)) -> c6(F(s(s(y0))))

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

(22)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(0) -> true
   f(1) -> false
   f(s(z0)) -> f(z0)
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   g(s(z0), s(z1)) -> if(f(z0), s(z0), s(z1))
   g(z0, c(z1)) -> g(z0, g(s(c(z1)), z1))
Tuples:
   G(x0, c(s(z1))) -> c7(G(s(c(s(z1))), s(z1)))
   G(x0, c(c(c(z1)))) -> c7(G(x0, g(s(c(c(c(z1)))), g(s(c(c(z1))), g(s(c(z1)), z1)))), G(s(c(c(c(z1)))), c(c(z1))))
   G(x0, c(c(x1))) -> c7(G(s(c(c(x1))), c(x1)))
   G(x0, c(c(s(z1)))) -> c7(G(s(c(c(s(z1)))), c(s(z1))))
   F(s(s(y0))) -> c3(F(s(y0)))
   G(s(s(s(y0))), s(z1)) -> c6(F(s(s(y0))))
S tuples:
   F(s(s(y0))) -> c3(F(s(y0)))
K tuples:
   G(z0, c(z1)) -> c7(G(z0, g(s(c(z1)), z1)), G(s(c(z1)), z1))
   G(s(s(s(y0))), s(z1)) -> c6(F(s(s(y0))))
Defined Rule Symbols:   f_1, if_3, g_2

Defined Pair Symbols:   G_2, F_1

Compound Symbols:   c7_1, c7_2, c3_1, c6_1


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

(23) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   G(x0, c(s(z1))) -> c7(G(s(c(s(z1))), s(z1)))
   G(x0, c(c(s(z1)))) -> c7(G(s(c(c(s(z1)))), c(s(z1))))

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

(24)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(0) -> true
   f(1) -> false
   f(s(z0)) -> f(z0)
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   g(s(z0), s(z1)) -> if(f(z0), s(z0), s(z1))
   g(z0, c(z1)) -> g(z0, g(s(c(z1)), z1))
Tuples:
   G(x0, c(c(c(z1)))) -> c7(G(x0, g(s(c(c(c(z1)))), g(s(c(c(z1))), g(s(c(z1)), z1)))), G(s(c(c(c(z1)))), c(c(z1))))
   G(x0, c(c(x1))) -> c7(G(s(c(c(x1))), c(x1)))
   F(s(s(y0))) -> c3(F(s(y0)))
   G(s(s(s(y0))), s(z1)) -> c6(F(s(s(y0))))
S tuples:
   F(s(s(y0))) -> c3(F(s(y0)))
K tuples:
   G(s(s(s(y0))), s(z1)) -> c6(F(s(s(y0))))
Defined Rule Symbols:   f_1, if_3, g_2

Defined Pair Symbols:   G_2, F_1

Compound Symbols:   c7_2, c7_1, c3_1, c6_1


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

(25) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace G(x0, c(c(x1))) -> c7(G(s(c(c(x1))), c(x1))) by 
   G(z0, c(c(c(c(y1))))) -> c7(G(s(c(c(c(c(y1))))), c(c(c(y1)))))
   G(z0, c(c(c(y1)))) -> c7(G(s(c(c(c(y1)))), c(c(y1))))

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

(26)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(0) -> true
   f(1) -> false
   f(s(z0)) -> f(z0)
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   g(s(z0), s(z1)) -> if(f(z0), s(z0), s(z1))
   g(z0, c(z1)) -> g(z0, g(s(c(z1)), z1))
Tuples:
   G(x0, c(c(c(z1)))) -> c7(G(x0, g(s(c(c(c(z1)))), g(s(c(c(z1))), g(s(c(z1)), z1)))), G(s(c(c(c(z1)))), c(c(z1))))
   F(s(s(y0))) -> c3(F(s(y0)))
   G(s(s(s(y0))), s(z1)) -> c6(F(s(s(y0))))
   G(z0, c(c(c(c(y1))))) -> c7(G(s(c(c(c(c(y1))))), c(c(c(y1)))))
   G(z0, c(c(c(y1)))) -> c7(G(s(c(c(c(y1)))), c(c(y1))))
S tuples:
   F(s(s(y0))) -> c3(F(s(y0)))
K tuples:
   G(s(s(s(y0))), s(z1)) -> c6(F(s(s(y0))))
Defined Rule Symbols:   f_1, if_3, g_2

Defined Pair Symbols:   G_2, F_1

Compound Symbols:   c7_2, c3_1, c6_1, c7_1


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

(27) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace F(s(s(y0))) -> c3(F(s(y0))) by 
   F(s(s(s(y0)))) -> c3(F(s(s(y0))))

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

(28)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(0) -> true
   f(1) -> false
   f(s(z0)) -> f(z0)
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   g(s(z0), s(z1)) -> if(f(z0), s(z0), s(z1))
   g(z0, c(z1)) -> g(z0, g(s(c(z1)), z1))
Tuples:
   G(x0, c(c(c(z1)))) -> c7(G(x0, g(s(c(c(c(z1)))), g(s(c(c(z1))), g(s(c(z1)), z1)))), G(s(c(c(c(z1)))), c(c(z1))))
   G(s(s(s(y0))), s(z1)) -> c6(F(s(s(y0))))
   G(z0, c(c(c(c(y1))))) -> c7(G(s(c(c(c(c(y1))))), c(c(c(y1)))))
   G(z0, c(c(c(y1)))) -> c7(G(s(c(c(c(y1)))), c(c(y1))))
   F(s(s(s(y0)))) -> c3(F(s(s(y0))))
S tuples:
   F(s(s(s(y0)))) -> c3(F(s(s(y0))))
K tuples:
   G(s(s(s(y0))), s(z1)) -> c6(F(s(s(y0))))
Defined Rule Symbols:   f_1, if_3, g_2

Defined Pair Symbols:   G_2, F_1

Compound Symbols:   c7_2, c6_1, c7_1, c3_1


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

(29) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace G(s(s(s(y0))), s(z1)) -> c6(F(s(s(y0)))) by 
   G(s(s(s(s(y0)))), s(z1)) -> c6(F(s(s(s(y0)))))

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

(30)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(0) -> true
   f(1) -> false
   f(s(z0)) -> f(z0)
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   g(s(z0), s(z1)) -> if(f(z0), s(z0), s(z1))
   g(z0, c(z1)) -> g(z0, g(s(c(z1)), z1))
Tuples:
   G(x0, c(c(c(z1)))) -> c7(G(x0, g(s(c(c(c(z1)))), g(s(c(c(z1))), g(s(c(z1)), z1)))), G(s(c(c(c(z1)))), c(c(z1))))
   G(z0, c(c(c(c(y1))))) -> c7(G(s(c(c(c(c(y1))))), c(c(c(y1)))))
   G(z0, c(c(c(y1)))) -> c7(G(s(c(c(c(y1)))), c(c(y1))))
   F(s(s(s(y0)))) -> c3(F(s(s(y0))))
   G(s(s(s(s(y0)))), s(z1)) -> c6(F(s(s(s(y0)))))
S tuples:
   F(s(s(s(y0)))) -> c3(F(s(s(y0))))
K tuples:
   G(s(s(s(s(y0)))), s(z1)) -> c6(F(s(s(s(y0)))))
Defined Rule Symbols:   f_1, if_3, g_2

Defined Pair Symbols:   G_2, F_1

Compound Symbols:   c7_2, c7_1, c3_1, c6_1


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

(31) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace G(z0, c(c(c(y1)))) -> c7(G(s(c(c(c(y1)))), c(c(y1)))) by 
   G(z0, c(c(c(c(y1))))) -> c7(G(s(c(c(c(c(y1))))), c(c(c(y1)))))
   G(z0, c(c(c(c(c(y1)))))) -> c7(G(s(c(c(c(c(c(y1)))))), c(c(c(c(y1))))))

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

(32)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(0) -> true
   f(1) -> false
   f(s(z0)) -> f(z0)
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   g(s(z0), s(z1)) -> if(f(z0), s(z0), s(z1))
   g(z0, c(z1)) -> g(z0, g(s(c(z1)), z1))
Tuples:
   G(x0, c(c(c(z1)))) -> c7(G(x0, g(s(c(c(c(z1)))), g(s(c(c(z1))), g(s(c(z1)), z1)))), G(s(c(c(c(z1)))), c(c(z1))))
   G(z0, c(c(c(c(y1))))) -> c7(G(s(c(c(c(c(y1))))), c(c(c(y1)))))
   F(s(s(s(y0)))) -> c3(F(s(s(y0))))
   G(s(s(s(s(y0)))), s(z1)) -> c6(F(s(s(s(y0)))))
   G(z0, c(c(c(c(c(y1)))))) -> c7(G(s(c(c(c(c(c(y1)))))), c(c(c(c(y1))))))
S tuples:
   F(s(s(s(y0)))) -> c3(F(s(s(y0))))
K tuples:
   G(s(s(s(s(y0)))), s(z1)) -> c6(F(s(s(s(y0)))))
Defined Rule Symbols:   f_1, if_3, g_2

Defined Pair Symbols:   G_2, F_1

Compound Symbols:   c7_2, c7_1, c3_1, c6_1


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

(33) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   F(s(s(s(y0)))) -> c3(F(s(s(y0))))
We considered the (Usable) Rules:none
And the Tuples:
   G(x0, c(c(c(z1)))) -> c7(G(x0, g(s(c(c(c(z1)))), g(s(c(c(z1))), g(s(c(z1)), z1)))), G(s(c(c(c(z1)))), c(c(z1))))
   G(z0, c(c(c(c(y1))))) -> c7(G(s(c(c(c(c(y1))))), c(c(c(y1)))))
   F(s(s(s(y0)))) -> c3(F(s(s(y0))))
   G(s(s(s(s(y0)))), s(z1)) -> c6(F(s(s(s(y0)))))
   G(z0, c(c(c(c(c(y1)))))) -> c7(G(s(c(c(c(c(c(y1)))))), c(c(c(c(y1))))))
The order we found is given by the following interpretation:

Matrix interpretation [MATRO]:

Non-tuple symbols: 
<<<
 M( s_1(x_1) ) =  	[[0], [2]] 	+ 	[[1, 4], [0, 1]] 	* 	x_1
>>>

<<<
 M( true ) =  	[[0], [0]]
>>>

<<<
 M( c_1(x_1) ) =  	[[0], [0]] 	+ 	[[0, 0], [0, 0]] 	* 	x_1
>>>

<<<
 M( f_1(x_1) ) =  	[[4], [0]] 	+ 	[[0, 0], [0, 4]] 	* 	x_1
>>>

<<<
 M( if_3(x_1, ..., x_3) ) =  	[[0], [0]] 	+ 	[[0, 0], [0, 0]] 	* 	x_1 	+ 	[[0, 0], [0, 0]] 	* 	x_2 	+ 	[[0, 0], [0, 0]] 	* 	x_3
>>>

<<<
 M( 0 ) =  	[[0], [2]]
>>>

<<<
 M( 1 ) =  	[[0], [4]]
>>>

<<<
 M( g_2(x_1, x_2) ) =  	[[0], [0]] 	+ 	[[0, 0], [0, 0]] 	* 	x_1 	+ 	[[0, 0], [0, 0]] 	* 	x_2
>>>

<<<
 M( false ) =  	[[1], [4]]
>>>

Tuple symbols: 
<<<
 M( G_2(x_1, x_2) ) =  	[[0], [0]] 	+ 	[[1, 0], [0, 0]] 	* 	x_1 	+ 	[[0, 0], [0, 0]] 	* 	x_2
>>>

<<<
 M( c6_1(x_1) ) =  	[[0], [0]] 	+ 	[[1, 0], [0, 0]] 	* 	x_1
>>>

<<<
 M( c3_1(x_1) ) =  	[[0], [2]] 	+ 	[[1, 2], [0, 1]] 	* 	x_1
>>>

<<<
 M( F_1(x_1) ) =  	[[0], [0]] 	+ 	[[1, 0], [0, 1]] 	* 	x_1
>>>

<<<
 M( c7_2(x_1, x_2) ) =  	[[0], [0]] 	+ 	[[1, 0], [0, 0]] 	* 	x_1 	+ 	[[1, 0], [0, 0]] 	* 	x_2
>>>

<<<
 M( c7_1(x_1) ) =  	[[0], [0]] 	+ 	[[1, 0], [0, 0]] 	* 	x_1
>>>



Matrix type: 

We used a basic matrix type which is not further parametrizeable.





As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order.
----------------------------------------

(34)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(0) -> true
   f(1) -> false
   f(s(z0)) -> f(z0)
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   g(s(z0), s(z1)) -> if(f(z0), s(z0), s(z1))
   g(z0, c(z1)) -> g(z0, g(s(c(z1)), z1))
Tuples:
   G(x0, c(c(c(z1)))) -> c7(G(x0, g(s(c(c(c(z1)))), g(s(c(c(z1))), g(s(c(z1)), z1)))), G(s(c(c(c(z1)))), c(c(z1))))
   G(z0, c(c(c(c(y1))))) -> c7(G(s(c(c(c(c(y1))))), c(c(c(y1)))))
   F(s(s(s(y0)))) -> c3(F(s(s(y0))))
   G(s(s(s(s(y0)))), s(z1)) -> c6(F(s(s(s(y0)))))
   G(z0, c(c(c(c(c(y1)))))) -> c7(G(s(c(c(c(c(c(y1)))))), c(c(c(c(y1))))))
S tuples:none
K tuples:
   G(s(s(s(s(y0)))), s(z1)) -> c6(F(s(s(s(y0)))))
   F(s(s(s(y0)))) -> c3(F(s(s(y0))))
Defined Rule Symbols:   f_1, if_3, g_2

Defined Pair Symbols:   G_2, F_1

Compound Symbols:   c7_2, c7_1, c3_1, c6_1


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

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

(36)
BOUNDS(1, 1)

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

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

(38)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

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

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

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

(40)
Obligation:
Innermost TRS:
Rules:
f(0') -> true
f(1') -> false
f(s(x)) -> f(x)
if(true, x, y) -> x
if(false, x, y) -> y
g(s(x), s(y)) -> if(f(x), s(x), s(y))
g(x, c(y)) -> g(x, g(s(c(y)), y))

Types:
f :: 0':1':s:c -> true:false
0' :: 0':1':s:c
true :: true:false
1' :: 0':1':s:c
false :: true:false
s :: 0':1':s:c -> 0':1':s:c
if :: true:false -> 0':1':s:c -> 0':1':s:c -> 0':1':s:c
g :: 0':1':s:c -> 0':1':s:c -> 0':1':s:c
c :: 0':1':s:c -> 0':1':s:c
hole_true:false1_0 :: true:false
hole_0':1':s:c2_0 :: 0':1':s:c
gen_0':1':s:c3_0 :: Nat -> 0':1':s:c

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

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

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

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

(42)
Obligation:
Innermost TRS:
Rules:
f(0') -> true
f(1') -> false
f(s(x)) -> f(x)
if(true, x, y) -> x
if(false, x, y) -> y
g(s(x), s(y)) -> if(f(x), s(x), s(y))
g(x, c(y)) -> g(x, g(s(c(y)), y))

Types:
f :: 0':1':s:c -> true:false
0' :: 0':1':s:c
true :: true:false
1' :: 0':1':s:c
false :: true:false
s :: 0':1':s:c -> 0':1':s:c
if :: true:false -> 0':1':s:c -> 0':1':s:c -> 0':1':s:c
g :: 0':1':s:c -> 0':1':s:c -> 0':1':s:c
c :: 0':1':s:c -> 0':1':s:c
hole_true:false1_0 :: true:false
hole_0':1':s:c2_0 :: 0':1':s:c
gen_0':1':s:c3_0 :: Nat -> 0':1':s:c


Generator Equations:
gen_0':1':s:c3_0(0) <=> 0'
gen_0':1':s:c3_0(+(x, 1)) <=> s(gen_0':1':s:c3_0(x))


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

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

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

(43) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
f(gen_0':1':s:c3_0(n5_0)) -> true, rt in Omega(1 + n5_0)

Induction Base:
f(gen_0':1':s:c3_0(0)) ->_R^Omega(1)
true

Induction Step:
f(gen_0':1':s:c3_0(+(n5_0, 1))) ->_R^Omega(1)
f(gen_0':1':s:c3_0(n5_0)) ->_IH
true

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

(44)
Complex Obligation (BEST)

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

(45)
Obligation:
Proved the lower bound n^1 for the following obligation:

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

Types:
f :: 0':1':s:c -> true:false
0' :: 0':1':s:c
true :: true:false
1' :: 0':1':s:c
false :: true:false
s :: 0':1':s:c -> 0':1':s:c
if :: true:false -> 0':1':s:c -> 0':1':s:c -> 0':1':s:c
g :: 0':1':s:c -> 0':1':s:c -> 0':1':s:c
c :: 0':1':s:c -> 0':1':s:c
hole_true:false1_0 :: true:false
hole_0':1':s:c2_0 :: 0':1':s:c
gen_0':1':s:c3_0 :: Nat -> 0':1':s:c


Generator Equations:
gen_0':1':s:c3_0(0) <=> 0'
gen_0':1':s:c3_0(+(x, 1)) <=> s(gen_0':1':s:c3_0(x))


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

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

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

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

(47)
BOUNDS(n^1, INF)

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

(48)
Obligation:
Innermost TRS:
Rules:
f(0') -> true
f(1') -> false
f(s(x)) -> f(x)
if(true, x, y) -> x
if(false, x, y) -> y
g(s(x), s(y)) -> if(f(x), s(x), s(y))
g(x, c(y)) -> g(x, g(s(c(y)), y))

Types:
f :: 0':1':s:c -> true:false
0' :: 0':1':s:c
true :: true:false
1' :: 0':1':s:c
false :: true:false
s :: 0':1':s:c -> 0':1':s:c
if :: true:false -> 0':1':s:c -> 0':1':s:c -> 0':1':s:c
g :: 0':1':s:c -> 0':1':s:c -> 0':1':s:c
c :: 0':1':s:c -> 0':1':s:c
hole_true:false1_0 :: true:false
hole_0':1':s:c2_0 :: 0':1':s:c
gen_0':1':s:c3_0 :: Nat -> 0':1':s:c


Lemmas:
f(gen_0':1':s:c3_0(n5_0)) -> true, rt in Omega(1 + n5_0)


Generator Equations:
gen_0':1':s:c3_0(0) <=> 0'
gen_0':1':s:c3_0(+(x, 1)) <=> s(gen_0':1':s:c3_0(x))


The following defined symbols remain to be analysed:
g