NO
proof of prog.inttrs
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


Termination of the given IRSwT could be disproven:

(0) IRSwT
(1) IRSFormatTransformerProof [EQUIVALENT, 0 ms]
(2) IRSwT
(3) IRSwTTerminationDigraphProof [EQUIVALENT, 786 ms]
(4) AND
    (5) IRSwT
        (6) IntTRSCompressionProof [EQUIVALENT, 26 ms]
        (7) IRSwT
        (8) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (9) IRSwT
        (10) TempFilterProof [SOUND, 71 ms]
        (11) IntTRS
        (12) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
        (13) YES
    (14) IRSwT
        (15) IntTRSCompressionProof [EQUIVALENT, 5 ms]
        (16) IRSwT
        (17) TempFilterProof [SOUND, 32 ms]
        (18) IntTRS
        (19) RankingReductionPairProof [EQUIVALENT, 0 ms]
        (20) IntTRS
        (21) RankingReductionPairProof [EQUIVALENT, 0 ms]
        (22) YES
    (23) IRSwT
        (24) IntTRSCompressionProof [EQUIVALENT, 3 ms]
        (25) IRSwT
        (26) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (27) IRSwT
        (28) FilterProof [EQUIVALENT, 0 ms]
        (29) IntTRS
        (30) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (31) IntTRS
        (32) IntTRSPeriodicNontermProof [COMPLETE, 10 ms]
        (33) NO
    (34) IRSwT
        (35) IntTRSCompressionProof [EQUIVALENT, 2 ms]
        (36) IRSwT
        (37) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (38) IRSwT
        (39) FilterProof [EQUIVALENT, 0 ms]
        (40) IntTRS
        (41) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (42) IntTRS
        (43) RankingReductionPairProof [EQUIVALENT, 5 ms]
        (44) YES


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

(0)
Obligation:
Rules:
f93_0_mk_Return(arg1, arg2, arg3) -> f148_0_main_InvokeMethod(arg1P, arg2P, arg3P) :|: arg2 = arg2P && 0 <= arg1P - 1 && 0 <= arg1 - 1 && -1 <= arg2 - 1 && arg1P <= arg1
f1_0_main_ConstantStackPush(x, x1, x2) -> f148_0_main_InvokeMethod(x3, x4, x5) :|: x1 = x4 && 0 <= x3 - 1 && 0 <= x - 1 && -1 <= x1 - 1 && x3 <= x
f148_0_main_InvokeMethod(x6, x7, x8) -> f341_0_main_GE(x9, x10, x11) :|: 0 = x9 && -1 <= x7 - 1 && 0 <= x6 - 1
f341_0_main_GE(x12, x13, x14) -> f475_0_main_GE(x15, x16, x17) :|: 0 = x16 && x12 = x15 && x12 <= 2
f475_0_main_GE(x18, x19, x20) -> f341_0_main_GE(x21, x22, x23) :|: x18 + 1 = x21 && x20 <= x19
f475_0_main_GE(x24, x25, x26) -> f475_0_main_GE(x27, x28, x29) :|: x26 = x29 && x25 + 1 = x28 && x24 = x27 && x25 <= x26 - 1 && x24 <= 2 && 0 <= x26 - 1
f1_0_main_ConstantStackPush(x30, x31, x32) -> f165_0_mk_LE(x33, x34, x35) :|: x31 = x34 && x31 - 1 = x33 && -1 <= x31 - 1 && 0 <= x30 - 1
f148_0_main_InvokeMethod(x36, x37, x38) -> f165_0_mk_LE(x39, x42, x43) :|: x37 = x42 && x37 - 1 = x39 && 0 <= x36 - 1
f148_0_main_InvokeMethod(x44, x45, x48) -> f165_0_mk_LE(x49, x50, x51) :|: x45 = x50 && x45 - 1 = x49 && -1 <= x45 - 1 && 0 <= x44 - 1
f165_0_mk_LE(x52, x53, x54) -> f165_0_mk_LE(x55, x56, x57) :|: x52 = x56 && x52 - 1 = x55 && 0 <= x53 - 1
f341_0_main_GE(x58, x59, x60) -> f291_0_length_NULL(x61, x62, x63) :|: x58 <= 2 && -1 <= x62 - 1 && -1 <= x61 - 1
f291_0_length_NULL(x64, x65, x66) -> f291_0_length_NULL(x67, x68, x69) :|: -1 <= x68 - 1 && -1 <= x67 - 1 && 0 <= x65 - 1 && 0 <= x64 - 1 && x68 + 1 <= x65 && x68 + 1 <= x64 && x67 + 1 <= x65 && x67 + 1 <= x64
f475_0_main_GE(x70, x71, x72) -> f541_0_bubble_NULL(x73, x74, x75) :|: -1 <= x73 - 1 && -1 <= x74 - 1 && x70 <= 2 && x71 <= x72 - 1 && 0 <= x72 - 1
f541_0_bubble_NULL(x76, x77, x78) -> f541_0_bubble_NULL(x79, x80, x81) :|: x79 + 2 <= x76 && x82 <= x83 && x79 + 2 <= x77 && x80 + 2 <= x76 && x80 + 2 <= x77 && 2 <= x76 - 1 && 2 <= x77 - 1 && 0 <= x79 - 1 && 0 <= x80 - 1
f541_0_bubble_NULL(x84, x85, x86) -> f541_0_bubble_NULL(x87, x88, x89) :|: x87 <= x84 && x90 <= x91 - 1 && x87 <= x85 && x88 <= x84 && x88 <= x85 && 2 <= x84 - 1 && 2 <= x85 - 1 && 0 <= x87 - 1 && 0 <= x88 - 1
__init(x92, x93, x94) -> f1_0_main_ConstantStackPush(x95, x96, x97) :|: 0 <= 0
Start term: __init(arg1, arg2, arg3)

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

(1) IRSFormatTransformerProof (EQUIVALENT)
Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %).
----------------------------------------

(2)
Obligation:
Rules:
f93_0_mk_Return(arg1, arg2, arg3) -> f148_0_main_InvokeMethod(arg1P, arg2P, arg3P) :|: arg2 = arg2P && 0 <= arg1P - 1 && 0 <= arg1 - 1 && -1 <= arg2 - 1 && arg1P <= arg1
f1_0_main_ConstantStackPush(x, x1, x2) -> f148_0_main_InvokeMethod(x3, x4, x5) :|: x1 = x4 && 0 <= x3 - 1 && 0 <= x - 1 && -1 <= x1 - 1 && x3 <= x
f148_0_main_InvokeMethod(x6, x7, x8) -> f341_0_main_GE(x9, x10, x11) :|: 0 = x9 && -1 <= x7 - 1 && 0 <= x6 - 1
f341_0_main_GE(x12, x13, x14) -> f475_0_main_GE(x15, x16, x17) :|: 0 = x16 && x12 = x15 && x12 <= 2
f475_0_main_GE(x18, x19, x20) -> f341_0_main_GE(x21, x22, x23) :|: x18 + 1 = x21 && x20 <= x19
f475_0_main_GE(x24, x25, x26) -> f475_0_main_GE(x27, x28, x29) :|: x26 = x29 && x25 + 1 = x28 && x24 = x27 && x25 <= x26 - 1 && x24 <= 2 && 0 <= x26 - 1
f1_0_main_ConstantStackPush(x30, x31, x32) -> f165_0_mk_LE(x33, x34, x35) :|: x31 = x34 && x31 - 1 = x33 && -1 <= x31 - 1 && 0 <= x30 - 1
f148_0_main_InvokeMethod(x36, x37, x38) -> f165_0_mk_LE(x39, x42, x43) :|: x37 = x42 && x37 - 1 = x39 && 0 <= x36 - 1
f148_0_main_InvokeMethod(x44, x45, x48) -> f165_0_mk_LE(x49, x50, x51) :|: x45 = x50 && x45 - 1 = x49 && -1 <= x45 - 1 && 0 <= x44 - 1
f165_0_mk_LE(x52, x53, x54) -> f165_0_mk_LE(x55, x56, x57) :|: x52 = x56 && x52 - 1 = x55 && 0 <= x53 - 1
f341_0_main_GE(x58, x59, x60) -> f291_0_length_NULL(x61, x62, x63) :|: x58 <= 2 && -1 <= x62 - 1 && -1 <= x61 - 1
f291_0_length_NULL(x64, x65, x66) -> f291_0_length_NULL(x67, x68, x69) :|: -1 <= x68 - 1 && -1 <= x67 - 1 && 0 <= x65 - 1 && 0 <= x64 - 1 && x68 + 1 <= x65 && x68 + 1 <= x64 && x67 + 1 <= x65 && x67 + 1 <= x64
f475_0_main_GE(x70, x71, x72) -> f541_0_bubble_NULL(x73, x74, x75) :|: -1 <= x73 - 1 && -1 <= x74 - 1 && x70 <= 2 && x71 <= x72 - 1 && 0 <= x72 - 1
f541_0_bubble_NULL(x76, x77, x78) -> f541_0_bubble_NULL(x79, x80, x81) :|: x79 + 2 <= x76 && x82 <= x83 && x79 + 2 <= x77 && x80 + 2 <= x76 && x80 + 2 <= x77 && 2 <= x76 - 1 && 2 <= x77 - 1 && 0 <= x79 - 1 && 0 <= x80 - 1
f541_0_bubble_NULL(x84, x85, x86) -> f541_0_bubble_NULL(x87, x88, x89) :|: x87 <= x84 && x90 <= x91 - 1 && x87 <= x85 && x88 <= x84 && x88 <= x85 && 2 <= x84 - 1 && 2 <= x85 - 1 && 0 <= x87 - 1 && 0 <= x88 - 1
__init(x92, x93, x94) -> f1_0_main_ConstantStackPush(x95, x96, x97) :|: 0 <= 0
Start term: __init(arg1, arg2, arg3)

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

(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) f93_0_mk_Return(arg1, arg2, arg3) -> f148_0_main_InvokeMethod(arg1P, arg2P, arg3P) :|: arg2 = arg2P && 0 <= arg1P - 1 && 0 <= arg1 - 1 && -1 <= arg2 - 1 && arg1P <= arg1
(2) f1_0_main_ConstantStackPush(x, x1, x2) -> f148_0_main_InvokeMethod(x3, x4, x5) :|: x1 = x4 && 0 <= x3 - 1 && 0 <= x - 1 && -1 <= x1 - 1 && x3 <= x
(3) f148_0_main_InvokeMethod(x6, x7, x8) -> f341_0_main_GE(x9, x10, x11) :|: 0 = x9 && -1 <= x7 - 1 && 0 <= x6 - 1
(4) f341_0_main_GE(x12, x13, x14) -> f475_0_main_GE(x15, x16, x17) :|: 0 = x16 && x12 = x15 && x12 <= 2
(5) f475_0_main_GE(x18, x19, x20) -> f341_0_main_GE(x21, x22, x23) :|: x18 + 1 = x21 && x20 <= x19
(6) f475_0_main_GE(x24, x25, x26) -> f475_0_main_GE(x27, x28, x29) :|: x26 = x29 && x25 + 1 = x28 && x24 = x27 && x25 <= x26 - 1 && x24 <= 2 && 0 <= x26 - 1
(7) f1_0_main_ConstantStackPush(x30, x31, x32) -> f165_0_mk_LE(x33, x34, x35) :|: x31 = x34 && x31 - 1 = x33 && -1 <= x31 - 1 && 0 <= x30 - 1
(8) f148_0_main_InvokeMethod(x36, x37, x38) -> f165_0_mk_LE(x39, x42, x43) :|: x37 = x42 && x37 - 1 = x39 && 0 <= x36 - 1
(9) f148_0_main_InvokeMethod(x44, x45, x48) -> f165_0_mk_LE(x49, x50, x51) :|: x45 = x50 && x45 - 1 = x49 && -1 <= x45 - 1 && 0 <= x44 - 1
(10) f165_0_mk_LE(x52, x53, x54) -> f165_0_mk_LE(x55, x56, x57) :|: x52 = x56 && x52 - 1 = x55 && 0 <= x53 - 1
(11) f341_0_main_GE(x58, x59, x60) -> f291_0_length_NULL(x61, x62, x63) :|: x58 <= 2 && -1 <= x62 - 1 && -1 <= x61 - 1
(12) f291_0_length_NULL(x64, x65, x66) -> f291_0_length_NULL(x67, x68, x69) :|: -1 <= x68 - 1 && -1 <= x67 - 1 && 0 <= x65 - 1 && 0 <= x64 - 1 && x68 + 1 <= x65 && x68 + 1 <= x64 && x67 + 1 <= x65 && x67 + 1 <= x64
(13) f475_0_main_GE(x70, x71, x72) -> f541_0_bubble_NULL(x73, x74, x75) :|: -1 <= x73 - 1 && -1 <= x74 - 1 && x70 <= 2 && x71 <= x72 - 1 && 0 <= x72 - 1
(14) f541_0_bubble_NULL(x76, x77, x78) -> f541_0_bubble_NULL(x79, x80, x81) :|: x79 + 2 <= x76 && x82 <= x83 && x79 + 2 <= x77 && x80 + 2 <= x76 && x80 + 2 <= x77 && 2 <= x76 - 1 && 2 <= x77 - 1 && 0 <= x79 - 1 && 0 <= x80 - 1
(15) f541_0_bubble_NULL(x84, x85, x86) -> f541_0_bubble_NULL(x87, x88, x89) :|: x87 <= x84 && x90 <= x91 - 1 && x87 <= x85 && x88 <= x84 && x88 <= x85 && 2 <= x84 - 1 && 2 <= x85 - 1 && 0 <= x87 - 1 && 0 <= x88 - 1
(16) __init(x92, x93, x94) -> f1_0_main_ConstantStackPush(x95, x96, x97) :|: 0 <= 0

Arcs:
(1) -> (3), (8), (9)
(2) -> (3), (8), (9)
(3) -> (4), (11)
(4) -> (5), (6), (13)
(5) -> (4), (11)
(6) -> (5), (6), (13)
(7) -> (10)
(8) -> (10)
(9) -> (10)
(10) -> (10)
(11) -> (12)
(12) -> (12)
(13) -> (14), (15)
(14) -> (14), (15)
(15) -> (14), (15)
(16) -> (2), (7)

This digraph is fully evaluated!
----------------------------------------

(4)
Complex Obligation (AND)

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

(5)
Obligation:

Termination digraph:
Nodes:
(1) f165_0_mk_LE(x52, x53, x54) -> f165_0_mk_LE(x55, x56, x57) :|: x52 = x56 && x52 - 1 = x55 && 0 <= x53 - 1

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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

(6) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
----------------------------------------

(7)
Obligation:
Rules:
f165_0_mk_LE(x52:0, x53:0, x54:0) -> f165_0_mk_LE(x52:0 - 1, x52:0, x57:0) :|: x53:0 > 0

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

(8) IntTRSUnneededArgumentFilterProof (EQUIVALENT)
Some arguments are removed because they cannot influence termination. We removed arguments according to the following replacements:

   f165_0_mk_LE(x1, x2, x3) -> f165_0_mk_LE(x1, x2)

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

(9)
Obligation:
Rules:
f165_0_mk_LE(x52:0, x53:0) -> f165_0_mk_LE(x52:0 - 1, x52:0) :|: x53:0 > 0

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

(10) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f165_0_mk_LE(VARIABLE, VARIABLE)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(11)
Obligation:
Rules:
f165_0_mk_LE(x52:0, x53:0) -> f165_0_mk_LE(c, x52:0) :|: c = x52:0 - 1 && x53:0 > 0

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

(12) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[f165_0_mk_LE(x, x1)] = x^2 + 2*x1

The following rules are decreasing:
f165_0_mk_LE(x52:0, x53:0) -> f165_0_mk_LE(c, x52:0) :|: c = x52:0 - 1 && x53:0 > 0
The following rules are bounded:
f165_0_mk_LE(x52:0, x53:0) -> f165_0_mk_LE(c, x52:0) :|: c = x52:0 - 1 && x53:0 > 0

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

(13)
YES

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

(14)
Obligation:

Termination digraph:
Nodes:
(1) f341_0_main_GE(x12, x13, x14) -> f475_0_main_GE(x15, x16, x17) :|: 0 = x16 && x12 = x15 && x12 <= 2
(2) f475_0_main_GE(x18, x19, x20) -> f341_0_main_GE(x21, x22, x23) :|: x18 + 1 = x21 && x20 <= x19
(3) f475_0_main_GE(x24, x25, x26) -> f475_0_main_GE(x27, x28, x29) :|: x26 = x29 && x25 + 1 = x28 && x24 = x27 && x25 <= x26 - 1 && x24 <= 2 && 0 <= x26 - 1

Arcs:
(1) -> (2), (3)
(2) -> (1)
(3) -> (2), (3)

This digraph is fully evaluated!

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

(15) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
----------------------------------------

(16)
Obligation:
Rules:
f475_0_main_GE(x24:0, x25:0, x26:0) -> f475_0_main_GE(x24:0, x25:0 + 1, x26:0) :|: x24:0 < 3 && x26:0 - 1 >= x25:0 && x26:0 > 0
f475_0_main_GE(x18:0, x19:0, x20:0) -> f475_0_main_GE(x18:0 + 1, 0, x17:0) :|: x20:0 <= x19:0 && x18:0 < 2

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

(17) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f475_0_main_GE(INTEGER, VARIABLE, VARIABLE)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(18)
Obligation:
Rules:
f475_0_main_GE(x24:0, x25:0, x26:0) -> f475_0_main_GE(x24:0, c, x26:0) :|: c = x25:0 + 1 && (x24:0 < 3 && x26:0 - 1 >= x25:0 && x26:0 > 0)
f475_0_main_GE(x18:0, x19:0, x20:0) -> f475_0_main_GE(c1, c2, x17:0) :|: c2 = 0 && c1 = x18:0 + 1 && (x20:0 <= x19:0 && x18:0 < 2)

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

(19) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ f475_0_main_GE ] = -1*f475_0_main_GE_1

The following rules are decreasing:
f475_0_main_GE(x18:0, x19:0, x20:0) -> f475_0_main_GE(c1, c2, x17:0) :|: c2 = 0 && c1 = x18:0 + 1 && (x20:0 <= x19:0 && x18:0 < 2)

The following rules are bounded:
f475_0_main_GE(x24:0, x25:0, x26:0) -> f475_0_main_GE(x24:0, c, x26:0) :|: c = x25:0 + 1 && (x24:0 < 3 && x26:0 - 1 >= x25:0 && x26:0 > 0)
f475_0_main_GE(x18:0, x19:0, x20:0) -> f475_0_main_GE(c1, c2, x17:0) :|: c2 = 0 && c1 = x18:0 + 1 && (x20:0 <= x19:0 && x18:0 < 2)


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

(20)
Obligation:
Rules:
f475_0_main_GE(x24:0, x25:0, x26:0) -> f475_0_main_GE(x24:0, c, x26:0) :|: c = x25:0 + 1 && (x24:0 < 3 && x26:0 - 1 >= x25:0 && x26:0 > 0)

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

(21) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ f475_0_main_GE ] = f475_0_main_GE_3 + -1*f475_0_main_GE_2

The following rules are decreasing:
f475_0_main_GE(x24:0, x25:0, x26:0) -> f475_0_main_GE(x24:0, c, x26:0) :|: c = x25:0 + 1 && (x24:0 < 3 && x26:0 - 1 >= x25:0 && x26:0 > 0)

The following rules are bounded:
f475_0_main_GE(x24:0, x25:0, x26:0) -> f475_0_main_GE(x24:0, c, x26:0) :|: c = x25:0 + 1 && (x24:0 < 3 && x26:0 - 1 >= x25:0 && x26:0 > 0)


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

(22)
YES

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

(23)
Obligation:

Termination digraph:
Nodes:
(1) f541_0_bubble_NULL(x76, x77, x78) -> f541_0_bubble_NULL(x79, x80, x81) :|: x79 + 2 <= x76 && x82 <= x83 && x79 + 2 <= x77 && x80 + 2 <= x76 && x80 + 2 <= x77 && 2 <= x76 - 1 && 2 <= x77 - 1 && 0 <= x79 - 1 && 0 <= x80 - 1
(2) f541_0_bubble_NULL(x84, x85, x86) -> f541_0_bubble_NULL(x87, x88, x89) :|: x87 <= x84 && x90 <= x91 - 1 && x87 <= x85 && x88 <= x84 && x88 <= x85 && 2 <= x84 - 1 && 2 <= x85 - 1 && 0 <= x87 - 1 && 0 <= x88 - 1

Arcs:
(1) -> (1), (2)
(2) -> (1), (2)

This digraph is fully evaluated!

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

(24) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
----------------------------------------

(25)
Obligation:
Rules:
f541_0_bubble_NULL(x76:0, x77:0, x78:0) -> f541_0_bubble_NULL(x79:0, x80:0, x81:0) :|: x79:0 > 0 && x80:0 > 0 && x77:0 > 2 && x76:0 > 2 && x80:0 + 2 <= x77:0 && x80:0 + 2 <= x76:0 && x79:0 + 2 <= x77:0 && x83:0 >= x82:0 && x79:0 + 2 <= x76:0
f541_0_bubble_NULL(x84:0, x85:0, x86:0) -> f541_0_bubble_NULL(x87:0, x88:0, x89:0) :|: x87:0 > 0 && x88:0 > 0 && x85:0 > 2 && x84:0 > 2 && x88:0 <= x85:0 && x88:0 <= x84:0 && x87:0 <= x85:0 && x91:0 - 1 >= x90:0 && x87:0 <= x84:0

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

(26) IntTRSUnneededArgumentFilterProof (EQUIVALENT)
Some arguments are removed because they cannot influence termination. We removed arguments according to the following replacements:

   f541_0_bubble_NULL(x1, x2, x3) -> f541_0_bubble_NULL(x1, x2)

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

(27)
Obligation:
Rules:
f541_0_bubble_NULL(x76:0, x77:0) -> f541_0_bubble_NULL(x79:0, x80:0) :|: x79:0 > 0 && x80:0 > 0 && x77:0 > 2 && x76:0 > 2 && x80:0 + 2 <= x77:0 && x80:0 + 2 <= x76:0 && x79:0 + 2 <= x77:0 && x83:0 >= x82:0 && x79:0 + 2 <= x76:0
f541_0_bubble_NULL(x84:0, x85:0) -> f541_0_bubble_NULL(x87:0, x88:0) :|: x87:0 > 0 && x88:0 > 0 && x85:0 > 2 && x84:0 > 2 && x88:0 <= x85:0 && x88:0 <= x84:0 && x87:0 <= x85:0 && x91:0 - 1 >= x90:0 && x87:0 <= x84:0

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

(28) FilterProof (EQUIVALENT)
Used the following sort dictionary for filtering: 
f541_0_bubble_NULL(INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(29)
Obligation:
Rules:
f541_0_bubble_NULL(x76:0, x77:0) -> f541_0_bubble_NULL(x79:0, x80:0) :|: x79:0 > 0 && x80:0 > 0 && x77:0 > 2 && x76:0 > 2 && x80:0 + 2 <= x77:0 && x80:0 + 2 <= x76:0 && x79:0 + 2 <= x77:0 && x83:0 >= x82:0 && x79:0 + 2 <= x76:0
f541_0_bubble_NULL(x84:0, x85:0) -> f541_0_bubble_NULL(x87:0, x88:0) :|: x87:0 > 0 && x88:0 > 0 && x85:0 > 2 && x84:0 > 2 && x88:0 <= x85:0 && x88:0 <= x84:0 && x87:0 <= x85:0 && x91:0 - 1 >= x90:0 && x87:0 <= x84:0

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

(30) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
----------------------------------------

(31)
Obligation:
Rules:
f541_0_bubble_NULL(x76:0:0, x77:0:0) -> f541_0_bubble_NULL(x79:0:0, x80:0:0) :|: x83:0:0 >= x82:0:0 && x79:0:0 + 2 <= x76:0:0 && x79:0:0 + 2 <= x77:0:0 && x80:0:0 + 2 <= x76:0:0 && x80:0:0 + 2 <= x77:0:0 && x76:0:0 > 2 && x77:0:0 > 2 && x80:0:0 > 0 && x79:0:0 > 0
f541_0_bubble_NULL(x84:0:0, x85:0:0) -> f541_0_bubble_NULL(x87:0:0, x88:0:0) :|: x91:0:0 - 1 >= x90:0:0 && x87:0:0 <= x84:0:0 && x87:0:0 <= x85:0:0 && x88:0:0 <= x84:0:0 && x88:0:0 <= x85:0:0 && x84:0:0 > 2 && x85:0:0 > 2 && x88:0:0 > 0 && x87:0:0 > 0

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(32) IntTRSPeriodicNontermProof (COMPLETE)
Normalized system to the following form:
f(pc, x76:0:0, x77:0:0) -> f(1, x79:0:0, x80:0:0) :|: pc = 1 && (x83:0:0 >= x82:0:0 && x79:0:0 + 2 <= x76:0:0 && x79:0:0 + 2 <= x77:0:0 && x80:0:0 + 2 <= x76:0:0 && x80:0:0 + 2 <= x77:0:0 && x76:0:0 > 2 && x77:0:0 > 2 && x80:0:0 > 0 && x79:0:0 > 0)
f(pc, x84:0:0, x85:0:0) -> f(1, x87:0:0, x88:0:0) :|: pc = 1 && (x91:0:0 - 1 >= x90:0:0 && x87:0:0 <= x84:0:0 && x87:0:0 <= x85:0:0 && x88:0:0 <= x84:0:0 && x88:0:0 <= x85:0:0 && x84:0:0 > 2 && x85:0:0 > 2 && x88:0:0 > 0 && x87:0:0 > 0)
Witness term starting non-terminating reduction: f(1, 6, 6)
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(33)
NO

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(34)
Obligation:

Termination digraph:
Nodes:
(1) f291_0_length_NULL(x64, x65, x66) -> f291_0_length_NULL(x67, x68, x69) :|: -1 <= x68 - 1 && -1 <= x67 - 1 && 0 <= x65 - 1 && 0 <= x64 - 1 && x68 + 1 <= x65 && x68 + 1 <= x64 && x67 + 1 <= x65 && x67 + 1 <= x64

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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(35) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(36)
Obligation:
Rules:
f291_0_length_NULL(x64:0, x65:0, x66:0) -> f291_0_length_NULL(x67:0, x68:0, x69:0) :|: x67:0 + 1 <= x65:0 && x67:0 + 1 <= x64:0 && x68:0 + 1 <= x64:0 && x68:0 + 1 <= x65:0 && x64:0 > 0 && x65:0 > 0 && x67:0 > -1 && x68:0 > -1

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(37) IntTRSUnneededArgumentFilterProof (EQUIVALENT)
Some arguments are removed because they cannot influence termination. We removed arguments according to the following replacements:

   f291_0_length_NULL(x1, x2, x3) -> f291_0_length_NULL(x1, x2)

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(38)
Obligation:
Rules:
f291_0_length_NULL(x64:0, x65:0) -> f291_0_length_NULL(x67:0, x68:0) :|: x67:0 + 1 <= x65:0 && x67:0 + 1 <= x64:0 && x68:0 + 1 <= x64:0 && x68:0 + 1 <= x65:0 && x64:0 > 0 && x65:0 > 0 && x67:0 > -1 && x68:0 > -1

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(39) FilterProof (EQUIVALENT)
Used the following sort dictionary for filtering: 
f291_0_length_NULL(INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
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(40)
Obligation:
Rules:
f291_0_length_NULL(x64:0, x65:0) -> f291_0_length_NULL(x67:0, x68:0) :|: x67:0 + 1 <= x65:0 && x67:0 + 1 <= x64:0 && x68:0 + 1 <= x64:0 && x68:0 + 1 <= x65:0 && x64:0 > 0 && x65:0 > 0 && x67:0 > -1 && x68:0 > -1

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(41) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
----------------------------------------

(42)
Obligation:
Rules:
f291_0_length_NULL(x64:0:0, x65:0:0) -> f291_0_length_NULL(x67:0:0, x68:0:0) :|: x67:0:0 > -1 && x68:0:0 > -1 && x65:0:0 > 0 && x64:0:0 > 0 && x68:0:0 + 1 <= x65:0:0 && x68:0:0 + 1 <= x64:0:0 && x67:0:0 + 1 <= x64:0:0 && x67:0:0 + 1 <= x65:0:0

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(43) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ f291_0_length_NULL ] = f291_0_length_NULL_1

The following rules are decreasing:
f291_0_length_NULL(x64:0:0, x65:0:0) -> f291_0_length_NULL(x67:0:0, x68:0:0) :|: x67:0:0 > -1 && x68:0:0 > -1 && x65:0:0 > 0 && x64:0:0 > 0 && x68:0:0 + 1 <= x65:0:0 && x68:0:0 + 1 <= x64:0:0 && x67:0:0 + 1 <= x64:0:0 && x67:0:0 + 1 <= x65:0:0

The following rules are bounded:
f291_0_length_NULL(x64:0:0, x65:0:0) -> f291_0_length_NULL(x67:0:0, x68:0:0) :|: x67:0:0 > -1 && x68:0:0 > -1 && x65:0:0 > 0 && x64:0:0 > 0 && x68:0:0 + 1 <= x65:0:0 && x68:0:0 + 1 <= x64:0:0 && x67:0:0 + 1 <= x64:0:0 && x67:0:0 + 1 <= x65:0:0


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(44)
YES