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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 158 ms]
(4) CpxRelTRS
(5) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(6) CdtProblem
(7) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) CdtProblem
(9) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
(10) CdtProblem
(11) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
(12) CdtProblem
(13) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
(14) CdtProblem
(15) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
(16) CdtProblem
(17) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 59 ms]
(18) CdtProblem
(19) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 171 ms]
(20) CdtProblem
(21) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
(22) BOUNDS(1, 1)


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

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


The TRS R consists of the following rules:

   +(*(x, y), *(a, y)) -> *(+(x, a), y)
   *(*(x, y), z) -> *(x, *(y, z))

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

(1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID))
The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem:

   encArg(a) -> a
   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encArg(cons_*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2))
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))
   encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2))
   encode_a -> a


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

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


The TRS R consists of the following rules:

   +(*(x, y), *(a, y)) -> *(+(x, a), y)
   *(*(x, y), z) -> *(x, *(y, z))

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

   encArg(a) -> a
   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encArg(cons_*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2))
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))
   encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2))
   encode_a -> a

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

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

(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

   +(*(x, y), *(a, y)) -> *(+(x, a), y)
   *(*(x, y), z) -> *(x, *(y, z))

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

   encArg(a) -> a
   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encArg(cons_*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2))
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))
   encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2))
   encode_a -> a

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

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(a) -> a
   encArg(cons_+(z0, z1)) -> +(encArg(z0), encArg(z1))
   encArg(cons_*(z0, z1)) -> *(encArg(z0), encArg(z1))
   encode_+(z0, z1) -> +(encArg(z0), encArg(z1))
   encode_*(z0, z1) -> *(encArg(z0), encArg(z1))
   encode_a -> a
   +(*(z0, z1), *(a, z1)) -> *(+(z0, a), z1)
   *(*(z0, z1), z2) -> *(z0, *(z1, z2))
Tuples:
   ENCARG(a) -> c
   ENCARG(cons_+(z0, z1)) -> c1(+'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_*(z0, z1)) -> c2(*'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_+(z0, z1) -> c3(+'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_*(z0, z1) -> c4(*'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_A -> c5
   +'(*(z0, z1), *(a, z1)) -> c6(*'(+(z0, a), z1), +'(z0, a))
   *'(*(z0, z1), z2) -> c7(*'(z0, *(z1, z2)), *'(z1, z2))
S tuples:
   +'(*(z0, z1), *(a, z1)) -> c6(*'(+(z0, a), z1), +'(z0, a))
   *'(*(z0, z1), z2) -> c7(*'(z0, *(z1, z2)), *'(z1, z2))
K tuples:none
Defined Rule Symbols:   +_2, *_2, encArg_1, encode_+_2, encode_*_2, encode_a

Defined Pair Symbols:   ENCARG_1, ENCODE_+_2, ENCODE_*_2, ENCODE_A, +'_2, *'_2

Compound Symbols:   c, c1_3, c2_3, c3_3, c4_3, c5, c6_2, c7_2


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

(7) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   ENCARG(a) -> c
   ENCODE_A -> c5

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(a) -> a
   encArg(cons_+(z0, z1)) -> +(encArg(z0), encArg(z1))
   encArg(cons_*(z0, z1)) -> *(encArg(z0), encArg(z1))
   encode_+(z0, z1) -> +(encArg(z0), encArg(z1))
   encode_*(z0, z1) -> *(encArg(z0), encArg(z1))
   encode_a -> a
   +(*(z0, z1), *(a, z1)) -> *(+(z0, a), z1)
   *(*(z0, z1), z2) -> *(z0, *(z1, z2))
Tuples:
   ENCARG(cons_+(z0, z1)) -> c1(+'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_*(z0, z1)) -> c2(*'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_+(z0, z1) -> c3(+'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_*(z0, z1) -> c4(*'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   +'(*(z0, z1), *(a, z1)) -> c6(*'(+(z0, a), z1), +'(z0, a))
   *'(*(z0, z1), z2) -> c7(*'(z0, *(z1, z2)), *'(z1, z2))
S tuples:
   +'(*(z0, z1), *(a, z1)) -> c6(*'(+(z0, a), z1), +'(z0, a))
   *'(*(z0, z1), z2) -> c7(*'(z0, *(z1, z2)), *'(z1, z2))
K tuples:none
Defined Rule Symbols:   +_2, *_2, encArg_1, encode_+_2, encode_*_2, encode_a

Defined Pair Symbols:   ENCARG_1, ENCODE_+_2, ENCODE_*_2, +'_2, *'_2

Compound Symbols:   c1_3, c2_3, c3_3, c4_3, c6_2, c7_2


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

(9) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing tuple parts
----------------------------------------

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(a) -> a
   encArg(cons_+(z0, z1)) -> +(encArg(z0), encArg(z1))
   encArg(cons_*(z0, z1)) -> *(encArg(z0), encArg(z1))
   encode_+(z0, z1) -> +(encArg(z0), encArg(z1))
   encode_*(z0, z1) -> *(encArg(z0), encArg(z1))
   encode_a -> a
   +(*(z0, z1), *(a, z1)) -> *(+(z0, a), z1)
   *(*(z0, z1), z2) -> *(z0, *(z1, z2))
Tuples:
   ENCARG(cons_+(z0, z1)) -> c1(+'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_*(z0, z1)) -> c2(*'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_+(z0, z1) -> c3(+'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_*(z0, z1) -> c4(*'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   *'(*(z0, z1), z2) -> c7(*'(z0, *(z1, z2)), *'(z1, z2))
   +'(*(z0, z1), *(a, z1)) -> c6
S tuples:
   *'(*(z0, z1), z2) -> c7(*'(z0, *(z1, z2)), *'(z1, z2))
   +'(*(z0, z1), *(a, z1)) -> c6
K tuples:none
Defined Rule Symbols:   +_2, *_2, encArg_1, encode_+_2, encode_*_2, encode_a

Defined Pair Symbols:   ENCARG_1, ENCODE_+_2, ENCODE_*_2, *'_2, +'_2

Compound Symbols:   c1_3, c2_3, c3_3, c4_3, c7_2, c6


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

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

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(a) -> a
   encArg(cons_+(z0, z1)) -> +(encArg(z0), encArg(z1))
   encArg(cons_*(z0, z1)) -> *(encArg(z0), encArg(z1))
   encode_+(z0, z1) -> +(encArg(z0), encArg(z1))
   encode_*(z0, z1) -> *(encArg(z0), encArg(z1))
   encode_a -> a
   +(*(z0, z1), *(a, z1)) -> *(+(z0, a), z1)
   *(*(z0, z1), z2) -> *(z0, *(z1, z2))
Tuples:
   ENCARG(cons_+(z0, z1)) -> c1(+'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_*(z0, z1)) -> c2(*'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   *'(*(z0, z1), z2) -> c7(*'(z0, *(z1, z2)), *'(z1, z2))
   +'(*(z0, z1), *(a, z1)) -> c6
   ENCODE_+(z0, z1) -> c(+'(encArg(z0), encArg(z1)))
   ENCODE_+(z0, z1) -> c(ENCARG(z0))
   ENCODE_+(z0, z1) -> c(ENCARG(z1))
   ENCODE_*(z0, z1) -> c(*'(encArg(z0), encArg(z1)))
   ENCODE_*(z0, z1) -> c(ENCARG(z0))
   ENCODE_*(z0, z1) -> c(ENCARG(z1))
S tuples:
   *'(*(z0, z1), z2) -> c7(*'(z0, *(z1, z2)), *'(z1, z2))
   +'(*(z0, z1), *(a, z1)) -> c6
K tuples:none
Defined Rule Symbols:   +_2, *_2, encArg_1, encode_+_2, encode_*_2, encode_a

Defined Pair Symbols:   ENCARG_1, *'_2, +'_2, ENCODE_+_2, ENCODE_*_2

Compound Symbols:   c1_3, c2_3, c7_2, c6, c_1


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

(13) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 4 leading nodes:
   ENCODE_+(z0, z1) -> c(ENCARG(z0))
   ENCODE_+(z0, z1) -> c(ENCARG(z1))
   ENCODE_*(z0, z1) -> c(ENCARG(z0))
   ENCODE_*(z0, z1) -> c(ENCARG(z1))

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

(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(a) -> a
   encArg(cons_+(z0, z1)) -> +(encArg(z0), encArg(z1))
   encArg(cons_*(z0, z1)) -> *(encArg(z0), encArg(z1))
   encode_+(z0, z1) -> +(encArg(z0), encArg(z1))
   encode_*(z0, z1) -> *(encArg(z0), encArg(z1))
   encode_a -> a
   +(*(z0, z1), *(a, z1)) -> *(+(z0, a), z1)
   *(*(z0, z1), z2) -> *(z0, *(z1, z2))
Tuples:
   ENCARG(cons_+(z0, z1)) -> c1(+'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_*(z0, z1)) -> c2(*'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   *'(*(z0, z1), z2) -> c7(*'(z0, *(z1, z2)), *'(z1, z2))
   +'(*(z0, z1), *(a, z1)) -> c6
   ENCODE_+(z0, z1) -> c(+'(encArg(z0), encArg(z1)))
   ENCODE_*(z0, z1) -> c(*'(encArg(z0), encArg(z1)))
S tuples:
   *'(*(z0, z1), z2) -> c7(*'(z0, *(z1, z2)), *'(z1, z2))
   +'(*(z0, z1), *(a, z1)) -> c6
K tuples:none
Defined Rule Symbols:   +_2, *_2, encArg_1, encode_+_2, encode_*_2, encode_a

Defined Pair Symbols:   ENCARG_1, *'_2, +'_2, ENCODE_+_2, ENCODE_*_2

Compound Symbols:   c1_3, c2_3, c7_2, c6, c_1


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

(15) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   encode_+(z0, z1) -> +(encArg(z0), encArg(z1))
   encode_*(z0, z1) -> *(encArg(z0), encArg(z1))
   encode_a -> a

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

(16)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(a) -> a
   encArg(cons_+(z0, z1)) -> +(encArg(z0), encArg(z1))
   encArg(cons_*(z0, z1)) -> *(encArg(z0), encArg(z1))
   +(*(z0, z1), *(a, z1)) -> *(+(z0, a), z1)
   *(*(z0, z1), z2) -> *(z0, *(z1, z2))
Tuples:
   ENCARG(cons_+(z0, z1)) -> c1(+'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_*(z0, z1)) -> c2(*'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   *'(*(z0, z1), z2) -> c7(*'(z0, *(z1, z2)), *'(z1, z2))
   +'(*(z0, z1), *(a, z1)) -> c6
   ENCODE_+(z0, z1) -> c(+'(encArg(z0), encArg(z1)))
   ENCODE_*(z0, z1) -> c(*'(encArg(z0), encArg(z1)))
S tuples:
   *'(*(z0, z1), z2) -> c7(*'(z0, *(z1, z2)), *'(z1, z2))
   +'(*(z0, z1), *(a, z1)) -> c6
K tuples:none
Defined Rule Symbols:   encArg_1, +_2, *_2

Defined Pair Symbols:   ENCARG_1, *'_2, +'_2, ENCODE_+_2, ENCODE_*_2

Compound Symbols:   c1_3, c2_3, c7_2, c6, c_1


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

(17) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   +'(*(z0, z1), *(a, z1)) -> c6
We considered the (Usable) Rules:none
And the Tuples:
   ENCARG(cons_+(z0, z1)) -> c1(+'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_*(z0, z1)) -> c2(*'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   *'(*(z0, z1), z2) -> c7(*'(z0, *(z1, z2)), *'(z1, z2))
   +'(*(z0, z1), *(a, z1)) -> c6
   ENCODE_+(z0, z1) -> c(+'(encArg(z0), encArg(z1)))
   ENCODE_*(z0, z1) -> c(*'(encArg(z0), encArg(z1)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(*(x_1, x_2)) = [1] + x_2
   POL(*'(x_1, x_2)) = 0
   POL(+(x_1, x_2)) = [1] + x_2
   POL(+'(x_1, x_2)) = [1]
   POL(ENCARG(x_1)) = x_1
   POL(ENCODE_*(x_1, x_2)) = x_1 + x_2
   POL(ENCODE_+(x_1, x_2)) = [1] + x_2
   POL(a) = [1]
   POL(c(x_1)) = x_1
   POL(c1(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c2(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c6) = 0
   POL(c7(x_1, x_2)) = x_1 + x_2
   POL(cons_*(x_1, x_2)) = [1] + x_1 + x_2
   POL(cons_+(x_1, x_2)) = [1] + x_1 + x_2
   POL(encArg(x_1)) = [1] + x_1

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

(18)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(a) -> a
   encArg(cons_+(z0, z1)) -> +(encArg(z0), encArg(z1))
   encArg(cons_*(z0, z1)) -> *(encArg(z0), encArg(z1))
   +(*(z0, z1), *(a, z1)) -> *(+(z0, a), z1)
   *(*(z0, z1), z2) -> *(z0, *(z1, z2))
Tuples:
   ENCARG(cons_+(z0, z1)) -> c1(+'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_*(z0, z1)) -> c2(*'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   *'(*(z0, z1), z2) -> c7(*'(z0, *(z1, z2)), *'(z1, z2))
   +'(*(z0, z1), *(a, z1)) -> c6
   ENCODE_+(z0, z1) -> c(+'(encArg(z0), encArg(z1)))
   ENCODE_*(z0, z1) -> c(*'(encArg(z0), encArg(z1)))
S tuples:
   *'(*(z0, z1), z2) -> c7(*'(z0, *(z1, z2)), *'(z1, z2))
K tuples:
   +'(*(z0, z1), *(a, z1)) -> c6
Defined Rule Symbols:   encArg_1, +_2, *_2

Defined Pair Symbols:   ENCARG_1, *'_2, +'_2, ENCODE_+_2, ENCODE_*_2

Compound Symbols:   c1_3, c2_3, c7_2, c6, c_1


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

(19) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   *'(*(z0, z1), z2) -> c7(*'(z0, *(z1, z2)), *'(z1, z2))
We considered the (Usable) Rules:
   encArg(cons_+(z0, z1)) -> +(encArg(z0), encArg(z1))
   encArg(cons_*(z0, z1)) -> *(encArg(z0), encArg(z1))
   +(*(z0, z1), *(a, z1)) -> *(+(z0, a), z1)
   *(*(z0, z1), z2) -> *(z0, *(z1, z2))
   encArg(a) -> a
And the Tuples:
   ENCARG(cons_+(z0, z1)) -> c1(+'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_*(z0, z1)) -> c2(*'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   *'(*(z0, z1), z2) -> c7(*'(z0, *(z1, z2)), *'(z1, z2))
   +'(*(z0, z1), *(a, z1)) -> c6
   ENCODE_+(z0, z1) -> c(+'(encArg(z0), encArg(z1)))
   ENCODE_*(z0, z1) -> c(*'(encArg(z0), encArg(z1)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(*(x_1, x_2)) = [2] + x_1 + x_2
   POL(*'(x_1, x_2)) = [2]x_1
   POL(+(x_1, x_2)) = x_1
   POL(+'(x_1, x_2)) = 0
   POL(ENCARG(x_1)) = [2]x_1 + x_1^2
   POL(ENCODE_*(x_1, x_2)) = [1] + [2]x_1 + x_2 + [2]x_2^2 + [2]x_1*x_2 + [2]x_1^2
   POL(ENCODE_+(x_1, x_2)) = [1] + x_1 + [2]x_2^2 + [2]x_1*x_2 + [2]x_1^2
   POL(a) = 0
   POL(c(x_1)) = x_1
   POL(c1(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c2(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c6) = 0
   POL(c7(x_1, x_2)) = x_1 + x_2
   POL(cons_*(x_1, x_2)) = [2] + x_1 + x_2
   POL(cons_+(x_1, x_2)) = x_1 + x_2
   POL(encArg(x_1)) = x_1

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

(20)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(a) -> a
   encArg(cons_+(z0, z1)) -> +(encArg(z0), encArg(z1))
   encArg(cons_*(z0, z1)) -> *(encArg(z0), encArg(z1))
   +(*(z0, z1), *(a, z1)) -> *(+(z0, a), z1)
   *(*(z0, z1), z2) -> *(z0, *(z1, z2))
Tuples:
   ENCARG(cons_+(z0, z1)) -> c1(+'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCARG(cons_*(z0, z1)) -> c2(*'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   *'(*(z0, z1), z2) -> c7(*'(z0, *(z1, z2)), *'(z1, z2))
   +'(*(z0, z1), *(a, z1)) -> c6
   ENCODE_+(z0, z1) -> c(+'(encArg(z0), encArg(z1)))
   ENCODE_*(z0, z1) -> c(*'(encArg(z0), encArg(z1)))
S tuples:none
K tuples:
   +'(*(z0, z1), *(a, z1)) -> c6
   *'(*(z0, z1), z2) -> c7(*'(z0, *(z1, z2)), *'(z1, z2))
Defined Rule Symbols:   encArg_1, +_2, *_2

Defined Pair Symbols:   ENCARG_1, *'_2, +'_2, ENCODE_+_2, ENCODE_*_2

Compound Symbols:   c1_3, c2_3, c7_2, c6, c_1


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

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

(22)
BOUNDS(1, 1)