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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


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

(0) CpxTRS
(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
(3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(4) CpxTRS
(5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(6) CpxWeightedTrs
    (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedTrs
    (9) CompletionProof [UPPER BOUND(ID), 0 ms]
    (10) CpxTypedWeightedCompleteTrs
        (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (12) CpxTypedWeightedCompleteTrs
        (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (14) CpxRNTS
        (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (16) CpxRNTS
    (17) CompletionProof [UPPER BOUND(ID), 0 ms]
    (18) CpxTypedWeightedCompleteTrs
        (19) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (20) CpxRNTS
(21) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(22) CdtProblem
    (23) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (24) CdtProblem
    (25) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (26) CdtProblem
    (27) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (28) CdtProblem
    (29) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 21 ms]
    (30) CdtProblem
    (31) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (32) CdtProblem
    (33) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (34) CdtProblem
    (35) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (36) CdtProblem
    (37) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (38) CdtProblem
    (39) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 25 ms]
    (40) CdtProblem
    (41) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (42) CdtProblem
    (43) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (44) CdtProblem
    (45) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (46) CdtProblem
    (47) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (48) CdtProblem


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

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


The TRS R consists of the following rules:

   f(a, g(y)) -> g(g(y))
   f(g(x), a) -> f(x, g(a))
   f(g(x), g(y)) -> h(g(y), x, g(y))
   h(g(x), y, z) -> f(y, h(x, y, z))
   h(a, y, z) -> z

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

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

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


The TRS R consists of the following rules:

   f(a, g(y)) -> g(g(y))
   f(g(x), a) -> f(x, g(a))
   f(g(x), g(y)) -> h(g(y), x, g(y))
   h(g(x), y, z) -> f(y, h(x, y, z))
   h(a, y, z) -> z

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

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

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


The TRS R consists of the following rules:

   f(a, g(y)) -> g(g(y))
   f(g(x), a) -> f(x, g(a))
   f(g(x), g(y)) -> h(g(y), x, g(y))
   h(g(x), y, z) -> f(y, h(x, y, z))
   h(a, y, z) -> z

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

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

(6)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   f(a, g(y)) -> g(g(y)) [1]
   f(g(x), a) -> f(x, g(a)) [1]
   f(g(x), g(y)) -> h(g(y), x, g(y)) [1]
   h(g(x), y, z) -> f(y, h(x, y, z)) [1]
   h(a, y, z) -> z [1]

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

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

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

   f(a, g(y)) -> g(g(y)) [1]
   f(g(x), a) -> f(x, g(a)) [1]
   f(g(x), g(y)) -> h(g(y), x, g(y)) [1]
   h(g(x), y, z) -> f(y, h(x, y, z)) [1]
   h(a, y, z) -> z [1]

The TRS has the following type information:
f :: a:g -> a:g -> a:g
a :: a:g
g :: a:g -> a:g
h :: a:g -> a:g -> a:g -> a:g

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

(9) CompletionProof (UPPER BOUND(ID))
The transformation into a RNTS is sound, since:

(a) The obligation is a constructor system where every type has a constant constructor,

(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:
none

(c) The following functions are completely defined:

   h_3
   f_2

Due to the following rules being added:

   f(v0, v1) -> a [0]

And the following fresh constants: none

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

(10)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   f(a, g(y)) -> g(g(y)) [1]
   f(g(x), a) -> f(x, g(a)) [1]
   f(g(x), g(y)) -> h(g(y), x, g(y)) [1]
   h(g(x), y, z) -> f(y, h(x, y, z)) [1]
   h(a, y, z) -> z [1]
   f(v0, v1) -> a [0]

The TRS has the following type information:
f :: a:g -> a:g -> a:g
a :: a:g
g :: a:g -> a:g
h :: a:g -> a:g -> a:g -> a:g

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

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

(12)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   f(a, g(y)) -> g(g(y)) [1]
   f(g(x), a) -> f(x, g(a)) [1]
   f(g(x), g(y)) -> h(g(y), x, g(y)) [1]
   h(g(g(x')), y, z) -> f(y, f(y, h(x', y, z))) [2]
   h(g(a), y, z) -> f(y, z) [2]
   h(a, y, z) -> z [1]
   f(v0, v1) -> a [0]

The TRS has the following type information:
f :: a:g -> a:g -> a:g
a :: a:g
g :: a:g -> a:g
h :: a:g -> a:g -> a:g -> a:g

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

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

   a => 0

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

(14)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z', z'') -{ 1 }-> h(1 + y, x, 1 + y) :|: z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y
   f(z', z'') -{ 1 }-> f(x, 1 + 0) :|: z'' = 0, z' = 1 + x, x >= 0
   f(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   f(z', z'') -{ 1 }-> 1 + (1 + y) :|: y >= 0, z'' = 1 + y, z' = 0
   h(z', z'', z1) -{ 1 }-> z :|: z1 = z, z >= 0, z'' = y, y >= 0, z' = 0
   h(z', z'', z1) -{ 2 }-> f(y, z) :|: z1 = z, z >= 0, z'' = y, y >= 0, z' = 1 + 0
   h(z', z'', z1) -{ 2 }-> f(y, f(y, h(x', y, z))) :|: z' = 1 + (1 + x'), z1 = z, z >= 0, z'' = y, x' >= 0, y >= 0


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

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

(16)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z', z'') -{ 1 }-> h(1 + (z'' - 1), z' - 1, 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0
   f(z', z'') -{ 1 }-> f(z' - 1, 1 + 0) :|: z'' = 0, z' - 1 >= 0
   f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   f(z', z'') -{ 1 }-> 1 + (1 + (z'' - 1)) :|: z'' - 1 >= 0, z' = 0
   h(z', z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z'' >= 0, z' = 0
   h(z', z'', z1) -{ 2 }-> f(z'', z1) :|: z1 >= 0, z'' >= 0, z' = 1 + 0
   h(z', z'', z1) -{ 2 }-> f(z'', f(z'', h(z' - 2, z'', z1))) :|: z1 >= 0, z' - 2 >= 0, z'' >= 0


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

(17) CompletionProof (UPPER BOUND(ID))
The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:

   f(v0, v1) -> null_f [0]
   h(v0, v1, v2) -> null_h [0]

And the following fresh constants:    null_f, null_h

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

(18)
Obligation:
Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   f(a, g(y)) -> g(g(y)) [1]
   f(g(x), a) -> f(x, g(a)) [1]
   f(g(x), g(y)) -> h(g(y), x, g(y)) [1]
   h(g(x), y, z) -> f(y, h(x, y, z)) [1]
   h(a, y, z) -> z [1]
   f(v0, v1) -> null_f [0]
   h(v0, v1, v2) -> null_h [0]

The TRS has the following type information:
f :: a:g:null_f:null_h -> a:g:null_f:null_h -> a:g:null_f:null_h
a :: a:g:null_f:null_h
g :: a:g:null_f:null_h -> a:g:null_f:null_h
h :: a:g:null_f:null_h -> a:g:null_f:null_h -> a:g:null_f:null_h -> a:g:null_f:null_h
null_f :: a:g:null_f:null_h
null_h :: a:g:null_f:null_h

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

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

   a => 0
   null_f => 0
   null_h => 0

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

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

   f(z', z'') -{ 1 }-> h(1 + y, x, 1 + y) :|: z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y
   f(z', z'') -{ 1 }-> f(x, 1 + 0) :|: z'' = 0, z' = 1 + x, x >= 0
   f(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   f(z', z'') -{ 1 }-> 1 + (1 + y) :|: y >= 0, z'' = 1 + y, z' = 0
   h(z', z'', z1) -{ 1 }-> z :|: z1 = z, z >= 0, z'' = y, y >= 0, z' = 0
   h(z', z'', z1) -{ 1 }-> f(y, h(x, y, z)) :|: z' = 1 + x, z1 = z, z >= 0, z'' = y, x >= 0, y >= 0
   h(z', z'', z1) -{ 0 }-> 0 :|: v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, z' = v0

Only complete derivations are relevant for the runtime complexity.

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

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

(22)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(a, g(z0)) -> g(g(z0))
   f(g(z0), a) -> f(z0, g(a))
   f(g(z0), g(z1)) -> h(g(z1), z0, g(z1))
   h(g(z0), z1, z2) -> f(z1, h(z0, z1, z2))
   h(a, z0, z1) -> z1
Tuples:
   F(a, g(z0)) -> c
   F(g(z0), a) -> c1(F(z0, g(a)))
   F(g(z0), g(z1)) -> c2(H(g(z1), z0, g(z1)))
   H(g(z0), z1, z2) -> c3(F(z1, h(z0, z1, z2)), H(z0, z1, z2))
   H(a, z0, z1) -> c4
S tuples:
   F(a, g(z0)) -> c
   F(g(z0), a) -> c1(F(z0, g(a)))
   F(g(z0), g(z1)) -> c2(H(g(z1), z0, g(z1)))
   H(g(z0), z1, z2) -> c3(F(z1, h(z0, z1, z2)), H(z0, z1, z2))
   H(a, z0, z1) -> c4
K tuples:none
Defined Rule Symbols:   f_2, h_3

Defined Pair Symbols:   F_2, H_3

Compound Symbols:   c, c1_1, c2_1, c3_2, c4


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

(23) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   H(a, z0, z1) -> c4
   F(a, g(z0)) -> c

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

(24)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(a, g(z0)) -> g(g(z0))
   f(g(z0), a) -> f(z0, g(a))
   f(g(z0), g(z1)) -> h(g(z1), z0, g(z1))
   h(g(z0), z1, z2) -> f(z1, h(z0, z1, z2))
   h(a, z0, z1) -> z1
Tuples:
   F(g(z0), a) -> c1(F(z0, g(a)))
   F(g(z0), g(z1)) -> c2(H(g(z1), z0, g(z1)))
   H(g(z0), z1, z2) -> c3(F(z1, h(z0, z1, z2)), H(z0, z1, z2))
S tuples:
   F(g(z0), a) -> c1(F(z0, g(a)))
   F(g(z0), g(z1)) -> c2(H(g(z1), z0, g(z1)))
   H(g(z0), z1, z2) -> c3(F(z1, h(z0, z1, z2)), H(z0, z1, z2))
K tuples:none
Defined Rule Symbols:   f_2, h_3

Defined Pair Symbols:   F_2, H_3

Compound Symbols:   c1_1, c2_1, c3_2


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

(25) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace H(g(z0), z1, z2) -> c3(F(z1, h(z0, z1, z2)), H(z0, z1, z2)) by 
   H(g(g(z0)), z1, z2) -> c3(F(z1, f(z1, h(z0, z1, z2))), H(g(z0), z1, z2))
   H(g(a), z0, z1) -> c3(F(z0, z1), H(a, z0, z1))

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

(26)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(a, g(z0)) -> g(g(z0))
   f(g(z0), a) -> f(z0, g(a))
   f(g(z0), g(z1)) -> h(g(z1), z0, g(z1))
   h(g(z0), z1, z2) -> f(z1, h(z0, z1, z2))
   h(a, z0, z1) -> z1
Tuples:
   F(g(z0), a) -> c1(F(z0, g(a)))
   F(g(z0), g(z1)) -> c2(H(g(z1), z0, g(z1)))
   H(g(g(z0)), z1, z2) -> c3(F(z1, f(z1, h(z0, z1, z2))), H(g(z0), z1, z2))
   H(g(a), z0, z1) -> c3(F(z0, z1), H(a, z0, z1))
S tuples:
   F(g(z0), a) -> c1(F(z0, g(a)))
   F(g(z0), g(z1)) -> c2(H(g(z1), z0, g(z1)))
   H(g(g(z0)), z1, z2) -> c3(F(z1, f(z1, h(z0, z1, z2))), H(g(z0), z1, z2))
   H(g(a), z0, z1) -> c3(F(z0, z1), H(a, z0, z1))
K tuples:none
Defined Rule Symbols:   f_2, h_3

Defined Pair Symbols:   F_2, H_3

Compound Symbols:   c1_1, c2_1, c3_2


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

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

(28)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(a, g(z0)) -> g(g(z0))
   f(g(z0), a) -> f(z0, g(a))
   f(g(z0), g(z1)) -> h(g(z1), z0, g(z1))
   h(g(z0), z1, z2) -> f(z1, h(z0, z1, z2))
   h(a, z0, z1) -> z1
Tuples:
   F(g(z0), a) -> c1(F(z0, g(a)))
   F(g(z0), g(z1)) -> c2(H(g(z1), z0, g(z1)))
   H(g(g(z0)), z1, z2) -> c3(F(z1, f(z1, h(z0, z1, z2))), H(g(z0), z1, z2))
   H(g(a), z0, z1) -> c3(F(z0, z1))
S tuples:
   F(g(z0), a) -> c1(F(z0, g(a)))
   F(g(z0), g(z1)) -> c2(H(g(z1), z0, g(z1)))
   H(g(g(z0)), z1, z2) -> c3(F(z1, f(z1, h(z0, z1, z2))), H(g(z0), z1, z2))
   H(g(a), z0, z1) -> c3(F(z0, z1))
K tuples:none
Defined Rule Symbols:   f_2, h_3

Defined Pair Symbols:   F_2, H_3

Compound Symbols:   c1_1, c2_1, c3_2, c3_1


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

(29) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   F(g(z0), a) -> c1(F(z0, g(a)))
We considered the (Usable) Rules:
   f(g(z0), g(z1)) -> h(g(z1), z0, g(z1))
   f(a, g(z0)) -> g(g(z0))
   h(g(z0), z1, z2) -> f(z1, h(z0, z1, z2))
   f(g(z0), a) -> f(z0, g(a))
And the Tuples:
   F(g(z0), a) -> c1(F(z0, g(a)))
   F(g(z0), g(z1)) -> c2(H(g(z1), z0, g(z1)))
   H(g(g(z0)), z1, z2) -> c3(F(z1, f(z1, h(z0, z1, z2))), H(g(z0), z1, z2))
   H(g(a), z0, z1) -> c3(F(z0, z1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(F(x_1, x_2)) = x_2
   POL(H(x_1, x_2, x_3)) = x_1 + x_3
   POL(a) = [1]
   POL(c1(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(f(x_1, x_2)) = 0
   POL(g(x_1)) = 0
   POL(h(x_1, x_2, x_3)) = x_1 + x_3

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

(30)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(a, g(z0)) -> g(g(z0))
   f(g(z0), a) -> f(z0, g(a))
   f(g(z0), g(z1)) -> h(g(z1), z0, g(z1))
   h(g(z0), z1, z2) -> f(z1, h(z0, z1, z2))
   h(a, z0, z1) -> z1
Tuples:
   F(g(z0), a) -> c1(F(z0, g(a)))
   F(g(z0), g(z1)) -> c2(H(g(z1), z0, g(z1)))
   H(g(g(z0)), z1, z2) -> c3(F(z1, f(z1, h(z0, z1, z2))), H(g(z0), z1, z2))
   H(g(a), z0, z1) -> c3(F(z0, z1))
S tuples:
   F(g(z0), g(z1)) -> c2(H(g(z1), z0, g(z1)))
   H(g(g(z0)), z1, z2) -> c3(F(z1, f(z1, h(z0, z1, z2))), H(g(z0), z1, z2))
   H(g(a), z0, z1) -> c3(F(z0, z1))
K tuples:
   F(g(z0), a) -> c1(F(z0, g(a)))
Defined Rule Symbols:   f_2, h_3

Defined Pair Symbols:   F_2, H_3

Compound Symbols:   c1_1, c2_1, c3_2, c3_1


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

(31) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace H(g(g(z0)), z1, z2) -> c3(F(z1, f(z1, h(z0, z1, z2))), H(g(z0), z1, z2)) by 
   H(g(g(g(z0))), z1, z2) -> c3(F(z1, f(z1, f(z1, h(z0, z1, z2)))), H(g(g(z0)), z1, z2))
   H(g(g(a)), z0, z1) -> c3(F(z0, f(z0, z1)), H(g(a), z0, z1))

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

(32)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(a, g(z0)) -> g(g(z0))
   f(g(z0), a) -> f(z0, g(a))
   f(g(z0), g(z1)) -> h(g(z1), z0, g(z1))
   h(g(z0), z1, z2) -> f(z1, h(z0, z1, z2))
   h(a, z0, z1) -> z1
Tuples:
   F(g(z0), a) -> c1(F(z0, g(a)))
   F(g(z0), g(z1)) -> c2(H(g(z1), z0, g(z1)))
   H(g(a), z0, z1) -> c3(F(z0, z1))
   H(g(g(g(z0))), z1, z2) -> c3(F(z1, f(z1, f(z1, h(z0, z1, z2)))), H(g(g(z0)), z1, z2))
   H(g(g(a)), z0, z1) -> c3(F(z0, f(z0, z1)), H(g(a), z0, z1))
S tuples:
   F(g(z0), g(z1)) -> c2(H(g(z1), z0, g(z1)))
   H(g(a), z0, z1) -> c3(F(z0, z1))
   H(g(g(g(z0))), z1, z2) -> c3(F(z1, f(z1, f(z1, h(z0, z1, z2)))), H(g(g(z0)), z1, z2))
   H(g(g(a)), z0, z1) -> c3(F(z0, f(z0, z1)), H(g(a), z0, z1))
K tuples:
   F(g(z0), a) -> c1(F(z0, g(a)))
Defined Rule Symbols:   f_2, h_3

Defined Pair Symbols:   F_2, H_3

Compound Symbols:   c1_1, c2_1, c3_1, c3_2


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

(33) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace F(g(z0), a) -> c1(F(z0, g(a))) by 
   F(g(g(y0)), a) -> c1(F(g(y0), g(a)))

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

(34)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(a, g(z0)) -> g(g(z0))
   f(g(z0), a) -> f(z0, g(a))
   f(g(z0), g(z1)) -> h(g(z1), z0, g(z1))
   h(g(z0), z1, z2) -> f(z1, h(z0, z1, z2))
   h(a, z0, z1) -> z1
Tuples:
   F(g(z0), g(z1)) -> c2(H(g(z1), z0, g(z1)))
   H(g(a), z0, z1) -> c3(F(z0, z1))
   H(g(g(g(z0))), z1, z2) -> c3(F(z1, f(z1, f(z1, h(z0, z1, z2)))), H(g(g(z0)), z1, z2))
   H(g(g(a)), z0, z1) -> c3(F(z0, f(z0, z1)), H(g(a), z0, z1))
   F(g(g(y0)), a) -> c1(F(g(y0), g(a)))
S tuples:
   F(g(z0), g(z1)) -> c2(H(g(z1), z0, g(z1)))
   H(g(a), z0, z1) -> c3(F(z0, z1))
   H(g(g(g(z0))), z1, z2) -> c3(F(z1, f(z1, f(z1, h(z0, z1, z2)))), H(g(g(z0)), z1, z2))
   H(g(g(a)), z0, z1) -> c3(F(z0, f(z0, z1)), H(g(a), z0, z1))
K tuples:
   F(g(g(y0)), a) -> c1(F(g(y0), g(a)))
Defined Rule Symbols:   f_2, h_3

Defined Pair Symbols:   F_2, H_3

Compound Symbols:   c2_1, c3_1, c3_2, c1_1


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

(35) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace F(g(z0), g(z1)) -> c2(H(g(z1), z0, g(z1))) by 
   F(g(z0), g(a)) -> c2(H(g(a), z0, g(a)))
   F(g(z0), g(g(g(y0)))) -> c2(H(g(g(g(y0))), z0, g(g(g(y0)))))
   F(g(z0), g(g(a))) -> c2(H(g(g(a)), z0, g(g(a))))

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

(36)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(a, g(z0)) -> g(g(z0))
   f(g(z0), a) -> f(z0, g(a))
   f(g(z0), g(z1)) -> h(g(z1), z0, g(z1))
   h(g(z0), z1, z2) -> f(z1, h(z0, z1, z2))
   h(a, z0, z1) -> z1
Tuples:
   H(g(a), z0, z1) -> c3(F(z0, z1))
   H(g(g(g(z0))), z1, z2) -> c3(F(z1, f(z1, f(z1, h(z0, z1, z2)))), H(g(g(z0)), z1, z2))
   H(g(g(a)), z0, z1) -> c3(F(z0, f(z0, z1)), H(g(a), z0, z1))
   F(g(g(y0)), a) -> c1(F(g(y0), g(a)))
   F(g(z0), g(a)) -> c2(H(g(a), z0, g(a)))
   F(g(z0), g(g(g(y0)))) -> c2(H(g(g(g(y0))), z0, g(g(g(y0)))))
   F(g(z0), g(g(a))) -> c2(H(g(g(a)), z0, g(g(a))))
S tuples:
   H(g(a), z0, z1) -> c3(F(z0, z1))
   H(g(g(g(z0))), z1, z2) -> c3(F(z1, f(z1, f(z1, h(z0, z1, z2)))), H(g(g(z0)), z1, z2))
   H(g(g(a)), z0, z1) -> c3(F(z0, f(z0, z1)), H(g(a), z0, z1))
   F(g(z0), g(a)) -> c2(H(g(a), z0, g(a)))
   F(g(z0), g(g(g(y0)))) -> c2(H(g(g(g(y0))), z0, g(g(g(y0)))))
   F(g(z0), g(g(a))) -> c2(H(g(g(a)), z0, g(g(a))))
K tuples:
   F(g(g(y0)), a) -> c1(F(g(y0), g(a)))
Defined Rule Symbols:   f_2, h_3

Defined Pair Symbols:   H_3, F_2

Compound Symbols:   c3_1, c3_2, c1_1, c2_1


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

(37) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace H(g(a), z0, z1) -> c3(F(z0, z1)) by 
   H(g(a), g(g(y0)), a) -> c3(F(g(g(y0)), a))
   H(g(a), g(y0), g(a)) -> c3(F(g(y0), g(a)))
   H(g(a), g(y0), g(g(g(y1)))) -> c3(F(g(y0), g(g(g(y1)))))
   H(g(a), g(y0), g(g(a))) -> c3(F(g(y0), g(g(a))))

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

(38)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(a, g(z0)) -> g(g(z0))
   f(g(z0), a) -> f(z0, g(a))
   f(g(z0), g(z1)) -> h(g(z1), z0, g(z1))
   h(g(z0), z1, z2) -> f(z1, h(z0, z1, z2))
   h(a, z0, z1) -> z1
Tuples:
   H(g(g(g(z0))), z1, z2) -> c3(F(z1, f(z1, f(z1, h(z0, z1, z2)))), H(g(g(z0)), z1, z2))
   H(g(g(a)), z0, z1) -> c3(F(z0, f(z0, z1)), H(g(a), z0, z1))
   F(g(g(y0)), a) -> c1(F(g(y0), g(a)))
   F(g(z0), g(a)) -> c2(H(g(a), z0, g(a)))
   F(g(z0), g(g(g(y0)))) -> c2(H(g(g(g(y0))), z0, g(g(g(y0)))))
   F(g(z0), g(g(a))) -> c2(H(g(g(a)), z0, g(g(a))))
   H(g(a), g(g(y0)), a) -> c3(F(g(g(y0)), a))
   H(g(a), g(y0), g(a)) -> c3(F(g(y0), g(a)))
   H(g(a), g(y0), g(g(g(y1)))) -> c3(F(g(y0), g(g(g(y1)))))
   H(g(a), g(y0), g(g(a))) -> c3(F(g(y0), g(g(a))))
S tuples:
   H(g(g(g(z0))), z1, z2) -> c3(F(z1, f(z1, f(z1, h(z0, z1, z2)))), H(g(g(z0)), z1, z2))
   H(g(g(a)), z0, z1) -> c3(F(z0, f(z0, z1)), H(g(a), z0, z1))
   F(g(z0), g(a)) -> c2(H(g(a), z0, g(a)))
   F(g(z0), g(g(g(y0)))) -> c2(H(g(g(g(y0))), z0, g(g(g(y0)))))
   F(g(z0), g(g(a))) -> c2(H(g(g(a)), z0, g(g(a))))
   H(g(a), g(g(y0)), a) -> c3(F(g(g(y0)), a))
   H(g(a), g(y0), g(a)) -> c3(F(g(y0), g(a)))
   H(g(a), g(y0), g(g(g(y1)))) -> c3(F(g(y0), g(g(g(y1)))))
   H(g(a), g(y0), g(g(a))) -> c3(F(g(y0), g(g(a))))
K tuples:
   F(g(g(y0)), a) -> c1(F(g(y0), g(a)))
Defined Rule Symbols:   f_2, h_3

Defined Pair Symbols:   H_3, F_2

Compound Symbols:   c3_2, c1_1, c2_1, c3_1


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

(39) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   H(g(a), g(g(y0)), a) -> c3(F(g(g(y0)), a))
We considered the (Usable) Rules:none
And the Tuples:
   H(g(g(g(z0))), z1, z2) -> c3(F(z1, f(z1, f(z1, h(z0, z1, z2)))), H(g(g(z0)), z1, z2))
   H(g(g(a)), z0, z1) -> c3(F(z0, f(z0, z1)), H(g(a), z0, z1))
   F(g(g(y0)), a) -> c1(F(g(y0), g(a)))
   F(g(z0), g(a)) -> c2(H(g(a), z0, g(a)))
   F(g(z0), g(g(g(y0)))) -> c2(H(g(g(g(y0))), z0, g(g(g(y0)))))
   F(g(z0), g(g(a))) -> c2(H(g(g(a)), z0, g(g(a))))
   H(g(a), g(g(y0)), a) -> c3(F(g(g(y0)), a))
   H(g(a), g(y0), g(a)) -> c3(F(g(y0), g(a)))
   H(g(a), g(y0), g(g(g(y1)))) -> c3(F(g(y0), g(g(g(y1)))))
   H(g(a), g(y0), g(g(a))) -> c3(F(g(y0), g(g(a))))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(F(x_1, x_2)) = 0
   POL(H(x_1, x_2, x_3)) = x_3
   POL(a) = [1]
   POL(c1(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(f(x_1, x_2)) = [3]
   POL(g(x_1)) = 0
   POL(h(x_1, x_2, x_3)) = [3] + [3]x_2

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

(40)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(a, g(z0)) -> g(g(z0))
   f(g(z0), a) -> f(z0, g(a))
   f(g(z0), g(z1)) -> h(g(z1), z0, g(z1))
   h(g(z0), z1, z2) -> f(z1, h(z0, z1, z2))
   h(a, z0, z1) -> z1
Tuples:
   H(g(g(g(z0))), z1, z2) -> c3(F(z1, f(z1, f(z1, h(z0, z1, z2)))), H(g(g(z0)), z1, z2))
   H(g(g(a)), z0, z1) -> c3(F(z0, f(z0, z1)), H(g(a), z0, z1))
   F(g(g(y0)), a) -> c1(F(g(y0), g(a)))
   F(g(z0), g(a)) -> c2(H(g(a), z0, g(a)))
   F(g(z0), g(g(g(y0)))) -> c2(H(g(g(g(y0))), z0, g(g(g(y0)))))
   F(g(z0), g(g(a))) -> c2(H(g(g(a)), z0, g(g(a))))
   H(g(a), g(g(y0)), a) -> c3(F(g(g(y0)), a))
   H(g(a), g(y0), g(a)) -> c3(F(g(y0), g(a)))
   H(g(a), g(y0), g(g(g(y1)))) -> c3(F(g(y0), g(g(g(y1)))))
   H(g(a), g(y0), g(g(a))) -> c3(F(g(y0), g(g(a))))
S tuples:
   H(g(g(g(z0))), z1, z2) -> c3(F(z1, f(z1, f(z1, h(z0, z1, z2)))), H(g(g(z0)), z1, z2))
   H(g(g(a)), z0, z1) -> c3(F(z0, f(z0, z1)), H(g(a), z0, z1))
   F(g(z0), g(a)) -> c2(H(g(a), z0, g(a)))
   F(g(z0), g(g(g(y0)))) -> c2(H(g(g(g(y0))), z0, g(g(g(y0)))))
   F(g(z0), g(g(a))) -> c2(H(g(g(a)), z0, g(g(a))))
   H(g(a), g(y0), g(a)) -> c3(F(g(y0), g(a)))
   H(g(a), g(y0), g(g(g(y1)))) -> c3(F(g(y0), g(g(g(y1)))))
   H(g(a), g(y0), g(g(a))) -> c3(F(g(y0), g(g(a))))
K tuples:
   F(g(g(y0)), a) -> c1(F(g(y0), g(a)))
   H(g(a), g(g(y0)), a) -> c3(F(g(g(y0)), a))
Defined Rule Symbols:   f_2, h_3

Defined Pair Symbols:   H_3, F_2

Compound Symbols:   c3_2, c1_1, c2_1, c3_1


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

(41) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace F(g(z0), g(a)) -> c2(H(g(a), z0, g(a))) by 
   F(g(g(y0)), g(a)) -> c2(H(g(a), g(y0), g(a)))

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

(42)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(a, g(z0)) -> g(g(z0))
   f(g(z0), a) -> f(z0, g(a))
   f(g(z0), g(z1)) -> h(g(z1), z0, g(z1))
   h(g(z0), z1, z2) -> f(z1, h(z0, z1, z2))
   h(a, z0, z1) -> z1
Tuples:
   H(g(g(g(z0))), z1, z2) -> c3(F(z1, f(z1, f(z1, h(z0, z1, z2)))), H(g(g(z0)), z1, z2))
   H(g(g(a)), z0, z1) -> c3(F(z0, f(z0, z1)), H(g(a), z0, z1))
   F(g(g(y0)), a) -> c1(F(g(y0), g(a)))
   F(g(z0), g(g(g(y0)))) -> c2(H(g(g(g(y0))), z0, g(g(g(y0)))))
   F(g(z0), g(g(a))) -> c2(H(g(g(a)), z0, g(g(a))))
   H(g(a), g(g(y0)), a) -> c3(F(g(g(y0)), a))
   H(g(a), g(y0), g(a)) -> c3(F(g(y0), g(a)))
   H(g(a), g(y0), g(g(g(y1)))) -> c3(F(g(y0), g(g(g(y1)))))
   H(g(a), g(y0), g(g(a))) -> c3(F(g(y0), g(g(a))))
   F(g(g(y0)), g(a)) -> c2(H(g(a), g(y0), g(a)))
S tuples:
   H(g(g(g(z0))), z1, z2) -> c3(F(z1, f(z1, f(z1, h(z0, z1, z2)))), H(g(g(z0)), z1, z2))
   H(g(g(a)), z0, z1) -> c3(F(z0, f(z0, z1)), H(g(a), z0, z1))
   F(g(z0), g(g(g(y0)))) -> c2(H(g(g(g(y0))), z0, g(g(g(y0)))))
   F(g(z0), g(g(a))) -> c2(H(g(g(a)), z0, g(g(a))))
   H(g(a), g(y0), g(a)) -> c3(F(g(y0), g(a)))
   H(g(a), g(y0), g(g(g(y1)))) -> c3(F(g(y0), g(g(g(y1)))))
   H(g(a), g(y0), g(g(a))) -> c3(F(g(y0), g(g(a))))
   F(g(g(y0)), g(a)) -> c2(H(g(a), g(y0), g(a)))
K tuples:
   F(g(g(y0)), a) -> c1(F(g(y0), g(a)))
   H(g(a), g(g(y0)), a) -> c3(F(g(g(y0)), a))
Defined Rule Symbols:   f_2, h_3

Defined Pair Symbols:   H_3, F_2

Compound Symbols:   c3_2, c1_1, c2_1, c3_1


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

(43) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace F(g(g(y0)), a) -> c1(F(g(y0), g(a))) by 
   F(g(g(g(y0))), a) -> c1(F(g(g(y0)), g(a)))

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

(44)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(a, g(z0)) -> g(g(z0))
   f(g(z0), a) -> f(z0, g(a))
   f(g(z0), g(z1)) -> h(g(z1), z0, g(z1))
   h(g(z0), z1, z2) -> f(z1, h(z0, z1, z2))
   h(a, z0, z1) -> z1
Tuples:
   H(g(g(g(z0))), z1, z2) -> c3(F(z1, f(z1, f(z1, h(z0, z1, z2)))), H(g(g(z0)), z1, z2))
   H(g(g(a)), z0, z1) -> c3(F(z0, f(z0, z1)), H(g(a), z0, z1))
   F(g(z0), g(g(g(y0)))) -> c2(H(g(g(g(y0))), z0, g(g(g(y0)))))
   F(g(z0), g(g(a))) -> c2(H(g(g(a)), z0, g(g(a))))
   H(g(a), g(g(y0)), a) -> c3(F(g(g(y0)), a))
   H(g(a), g(y0), g(a)) -> c3(F(g(y0), g(a)))
   H(g(a), g(y0), g(g(g(y1)))) -> c3(F(g(y0), g(g(g(y1)))))
   H(g(a), g(y0), g(g(a))) -> c3(F(g(y0), g(g(a))))
   F(g(g(y0)), g(a)) -> c2(H(g(a), g(y0), g(a)))
   F(g(g(g(y0))), a) -> c1(F(g(g(y0)), g(a)))
S tuples:
   H(g(g(g(z0))), z1, z2) -> c3(F(z1, f(z1, f(z1, h(z0, z1, z2)))), H(g(g(z0)), z1, z2))
   H(g(g(a)), z0, z1) -> c3(F(z0, f(z0, z1)), H(g(a), z0, z1))
   F(g(z0), g(g(g(y0)))) -> c2(H(g(g(g(y0))), z0, g(g(g(y0)))))
   F(g(z0), g(g(a))) -> c2(H(g(g(a)), z0, g(g(a))))
   H(g(a), g(y0), g(a)) -> c3(F(g(y0), g(a)))
   H(g(a), g(y0), g(g(g(y1)))) -> c3(F(g(y0), g(g(g(y1)))))
   H(g(a), g(y0), g(g(a))) -> c3(F(g(y0), g(g(a))))
   F(g(g(y0)), g(a)) -> c2(H(g(a), g(y0), g(a)))
K tuples:
   H(g(a), g(g(y0)), a) -> c3(F(g(g(y0)), a))
   F(g(g(g(y0))), a) -> c1(F(g(g(y0)), g(a)))
Defined Rule Symbols:   f_2, h_3

Defined Pair Symbols:   H_3, F_2

Compound Symbols:   c3_2, c2_1, c3_1, c1_1


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

(45) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace H(g(a), g(g(y0)), a) -> c3(F(g(g(y0)), a)) by 
   H(g(a), g(g(g(y0))), a) -> c3(F(g(g(g(y0))), a))

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

(46)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(a, g(z0)) -> g(g(z0))
   f(g(z0), a) -> f(z0, g(a))
   f(g(z0), g(z1)) -> h(g(z1), z0, g(z1))
   h(g(z0), z1, z2) -> f(z1, h(z0, z1, z2))
   h(a, z0, z1) -> z1
Tuples:
   H(g(g(g(z0))), z1, z2) -> c3(F(z1, f(z1, f(z1, h(z0, z1, z2)))), H(g(g(z0)), z1, z2))
   H(g(g(a)), z0, z1) -> c3(F(z0, f(z0, z1)), H(g(a), z0, z1))
   F(g(z0), g(g(g(y0)))) -> c2(H(g(g(g(y0))), z0, g(g(g(y0)))))
   F(g(z0), g(g(a))) -> c2(H(g(g(a)), z0, g(g(a))))
   H(g(a), g(y0), g(a)) -> c3(F(g(y0), g(a)))
   H(g(a), g(y0), g(g(g(y1)))) -> c3(F(g(y0), g(g(g(y1)))))
   H(g(a), g(y0), g(g(a))) -> c3(F(g(y0), g(g(a))))
   F(g(g(y0)), g(a)) -> c2(H(g(a), g(y0), g(a)))
   F(g(g(g(y0))), a) -> c1(F(g(g(y0)), g(a)))
   H(g(a), g(g(g(y0))), a) -> c3(F(g(g(g(y0))), a))
S tuples:
   H(g(g(g(z0))), z1, z2) -> c3(F(z1, f(z1, f(z1, h(z0, z1, z2)))), H(g(g(z0)), z1, z2))
   H(g(g(a)), z0, z1) -> c3(F(z0, f(z0, z1)), H(g(a), z0, z1))
   F(g(z0), g(g(g(y0)))) -> c2(H(g(g(g(y0))), z0, g(g(g(y0)))))
   F(g(z0), g(g(a))) -> c2(H(g(g(a)), z0, g(g(a))))
   H(g(a), g(y0), g(a)) -> c3(F(g(y0), g(a)))
   H(g(a), g(y0), g(g(g(y1)))) -> c3(F(g(y0), g(g(g(y1)))))
   H(g(a), g(y0), g(g(a))) -> c3(F(g(y0), g(g(a))))
   F(g(g(y0)), g(a)) -> c2(H(g(a), g(y0), g(a)))
K tuples:
   F(g(g(g(y0))), a) -> c1(F(g(g(y0)), g(a)))
   H(g(a), g(g(g(y0))), a) -> c3(F(g(g(g(y0))), a))
Defined Rule Symbols:   f_2, h_3

Defined Pair Symbols:   H_3, F_2

Compound Symbols:   c3_2, c2_1, c3_1, c1_1


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

(47) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace H(g(a), g(y0), g(a)) -> c3(F(g(y0), g(a))) by 
   H(g(a), g(g(y0)), g(a)) -> c3(F(g(g(y0)), g(a)))

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

(48)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(a, g(z0)) -> g(g(z0))
   f(g(z0), a) -> f(z0, g(a))
   f(g(z0), g(z1)) -> h(g(z1), z0, g(z1))
   h(g(z0), z1, z2) -> f(z1, h(z0, z1, z2))
   h(a, z0, z1) -> z1
Tuples:
   H(g(g(g(z0))), z1, z2) -> c3(F(z1, f(z1, f(z1, h(z0, z1, z2)))), H(g(g(z0)), z1, z2))
   H(g(g(a)), z0, z1) -> c3(F(z0, f(z0, z1)), H(g(a), z0, z1))
   F(g(z0), g(g(g(y0)))) -> c2(H(g(g(g(y0))), z0, g(g(g(y0)))))
   F(g(z0), g(g(a))) -> c2(H(g(g(a)), z0, g(g(a))))
   H(g(a), g(y0), g(g(g(y1)))) -> c3(F(g(y0), g(g(g(y1)))))
   H(g(a), g(y0), g(g(a))) -> c3(F(g(y0), g(g(a))))
   F(g(g(y0)), g(a)) -> c2(H(g(a), g(y0), g(a)))
   F(g(g(g(y0))), a) -> c1(F(g(g(y0)), g(a)))
   H(g(a), g(g(g(y0))), a) -> c3(F(g(g(g(y0))), a))
   H(g(a), g(g(y0)), g(a)) -> c3(F(g(g(y0)), g(a)))
S tuples:
   H(g(g(g(z0))), z1, z2) -> c3(F(z1, f(z1, f(z1, h(z0, z1, z2)))), H(g(g(z0)), z1, z2))
   H(g(g(a)), z0, z1) -> c3(F(z0, f(z0, z1)), H(g(a), z0, z1))
   F(g(z0), g(g(g(y0)))) -> c2(H(g(g(g(y0))), z0, g(g(g(y0)))))
   F(g(z0), g(g(a))) -> c2(H(g(g(a)), z0, g(g(a))))
   H(g(a), g(y0), g(g(g(y1)))) -> c3(F(g(y0), g(g(g(y1)))))
   H(g(a), g(y0), g(g(a))) -> c3(F(g(y0), g(g(a))))
   F(g(g(y0)), g(a)) -> c2(H(g(a), g(y0), g(a)))
   H(g(a), g(g(y0)), g(a)) -> c3(F(g(g(y0)), g(a)))
K tuples:
   F(g(g(g(y0))), a) -> c1(F(g(g(y0)), g(a)))
   H(g(a), g(g(g(y0))), a) -> c3(F(g(g(g(y0))), a))
Defined Rule Symbols:   f_2, h_3

Defined Pair Symbols:   H_3, F_2

Compound Symbols:   c3_2, c2_1, c3_1, c1_1