YES
proof of prog.inttrs
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


Termination of the given IRSwT could be proven:

(0) IRSwT
(1) IRSFormatTransformerProof [EQUIVALENT, 0 ms]
(2) IRSwT
(3) IRSwTTerminationDigraphProof [EQUIVALENT, 467 ms]
(4) AND
    (5) IRSwT
        (6) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (7) IRSwT
        (8) TempFilterProof [SOUND, 63 ms]
        (9) IntTRS
        (10) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
        (11) YES
    (12) IRSwT
        (13) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (14) IRSwT
        (15) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (16) IRSwT
        (17) FilterProof [EQUIVALENT, 0 ms]
        (18) IntTRS
        (19) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (20) IntTRS
        (21) RankingReductionPairProof [EQUIVALENT, 0 ms]
        (22) YES


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

(0)
Obligation:
Rules:
f1_0_main_New(arg1, arg2, arg3) -> f262_0_main_InvokeMethod(arg1P, arg2P, arg3P) :|: 2 <= arg1P - 1
f1_0_main_New(x, x1, x2) -> f262_0_main_InvokeMethod(x3, x4, x5) :|: 1 <= x3 - 1
f1_0_main_New(x6, x7, x8) -> f76_0__init__LE(x9, x10, x11) :|: 5 = x11 && 5 = x10 && 5 = x9
f76_0__init__LE(x12, x13, x14) -> f76_0__init__LE(x15, x16, x17) :|: x13 - 1 = x17 && x13 - 1 = x16 && x13 - 1 = x15 && x13 = x14 && 1 <= x13 - 1 && 0 <= x12 - 1 && x13 - 1 <= x13 - 1
f76_0__init__LE(x18, x19, x20) -> f288_0__init__InvokeMethod(x21, x22, x23) :|: x19 - 1 = x23 && x18 = x21 && x19 = x20 && 0 <= x18 - 1 && 4 <= x22 - 1 && 1 <= x19 - 1 && x19 - 1 <= x19 - 1
f76_0__init__LE(x24, x25, x26) -> f288_0__init__InvokeMethod(x27, x28, x29) :|: x25 - 1 = x29 && x24 = x27 && x25 = x26 && 0 <= x24 - 1 && 3 <= x28 - 1 && 1 <= x25 - 1 && x25 - 1 <= x25 - 1
f288_0__init__InvokeMethod(x30, x31, x32) -> f76_0__init__LE(x33, x34, x35) :|: x32 = x35 && x32 = x34 && x32 = x33 && 2 <= x31 - 1 && 0 <= x30 - 1 && 0 <= x32 - 1
f262_0_main_InvokeMethod(x36, x37, x38) -> f194_0_height_NONNULL(x39, x40, x41) :|: -1 <= x40 - 1 && 0 <= x39 - 1 && 0 <= x36 - 1 && x40 + 1 <= x36 && x39 <= x36
f194_0_height_NONNULL(x42, x43, x44) -> f194_0_height_NONNULL(x45, x46, x47) :|: -1 <= x46 - 1 && 0 <= x45 - 1 && 0 <= x43 - 1 && 2 <= x42 - 1 && x46 + 1 <= x43 && x46 + 3 <= x42 && x45 <= x43 && x45 + 2 <= x42
f194_0_height_NONNULL(x48, x49, x50) -> f194_0_height_NONNULL(x51, x52, x53) :|: -1 <= x52 - 1 && 0 <= x51 - 1 && -1 <= x49 - 1 && 2 <= x48 - 1 && x52 + 3 <= x48 && x51 + 2 <= x48
__init(x54, x55, x56) -> f1_0_main_New(x57, x58, x59) :|: 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:
f1_0_main_New(arg1, arg2, arg3) -> f262_0_main_InvokeMethod(arg1P, arg2P, arg3P) :|: 2 <= arg1P - 1
f1_0_main_New(x, x1, x2) -> f262_0_main_InvokeMethod(x3, x4, x5) :|: 1 <= x3 - 1
f1_0_main_New(x6, x7, x8) -> f76_0__init__LE(x9, x10, x11) :|: 5 = x11 && 5 = x10 && 5 = x9
f76_0__init__LE(x12, x13, x14) -> f76_0__init__LE(x15, x16, x17) :|: x13 - 1 = x17 && x13 - 1 = x16 && x13 - 1 = x15 && x13 = x14 && 1 <= x13 - 1 && 0 <= x12 - 1 && x13 - 1 <= x13 - 1
f76_0__init__LE(x18, x19, x20) -> f288_0__init__InvokeMethod(x21, x22, x23) :|: x19 - 1 = x23 && x18 = x21 && x19 = x20 && 0 <= x18 - 1 && 4 <= x22 - 1 && 1 <= x19 - 1 && x19 - 1 <= x19 - 1
f76_0__init__LE(x24, x25, x26) -> f288_0__init__InvokeMethod(x27, x28, x29) :|: x25 - 1 = x29 && x24 = x27 && x25 = x26 && 0 <= x24 - 1 && 3 <= x28 - 1 && 1 <= x25 - 1 && x25 - 1 <= x25 - 1
f288_0__init__InvokeMethod(x30, x31, x32) -> f76_0__init__LE(x33, x34, x35) :|: x32 = x35 && x32 = x34 && x32 = x33 && 2 <= x31 - 1 && 0 <= x30 - 1 && 0 <= x32 - 1
f262_0_main_InvokeMethod(x36, x37, x38) -> f194_0_height_NONNULL(x39, x40, x41) :|: -1 <= x40 - 1 && 0 <= x39 - 1 && 0 <= x36 - 1 && x40 + 1 <= x36 && x39 <= x36
f194_0_height_NONNULL(x42, x43, x44) -> f194_0_height_NONNULL(x45, x46, x47) :|: -1 <= x46 - 1 && 0 <= x45 - 1 && 0 <= x43 - 1 && 2 <= x42 - 1 && x46 + 1 <= x43 && x46 + 3 <= x42 && x45 <= x43 && x45 + 2 <= x42
f194_0_height_NONNULL(x48, x49, x50) -> f194_0_height_NONNULL(x51, x52, x53) :|: -1 <= x52 - 1 && 0 <= x51 - 1 && -1 <= x49 - 1 && 2 <= x48 - 1 && x52 + 3 <= x48 && x51 + 2 <= x48
__init(x54, x55, x56) -> f1_0_main_New(x57, x58, x59) :|: 0 <= 0
Start term: __init(arg1, arg2, arg3)

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

(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) f1_0_main_New(arg1, arg2, arg3) -> f262_0_main_InvokeMethod(arg1P, arg2P, arg3P) :|: 2 <= arg1P - 1
(2) f1_0_main_New(x, x1, x2) -> f262_0_main_InvokeMethod(x3, x4, x5) :|: 1 <= x3 - 1
(3) f1_0_main_New(x6, x7, x8) -> f76_0__init__LE(x9, x10, x11) :|: 5 = x11 && 5 = x10 && 5 = x9
(4) f76_0__init__LE(x12, x13, x14) -> f76_0__init__LE(x15, x16, x17) :|: x13 - 1 = x17 && x13 - 1 = x16 && x13 - 1 = x15 && x13 = x14 && 1 <= x13 - 1 && 0 <= x12 - 1 && x13 - 1 <= x13 - 1
(5) f76_0__init__LE(x18, x19, x20) -> f288_0__init__InvokeMethod(x21, x22, x23) :|: x19 - 1 = x23 && x18 = x21 && x19 = x20 && 0 <= x18 - 1 && 4 <= x22 - 1 && 1 <= x19 - 1 && x19 - 1 <= x19 - 1
(6) f76_0__init__LE(x24, x25, x26) -> f288_0__init__InvokeMethod(x27, x28, x29) :|: x25 - 1 = x29 && x24 = x27 && x25 = x26 && 0 <= x24 - 1 && 3 <= x28 - 1 && 1 <= x25 - 1 && x25 - 1 <= x25 - 1
(7) f288_0__init__InvokeMethod(x30, x31, x32) -> f76_0__init__LE(x33, x34, x35) :|: x32 = x35 && x32 = x34 && x32 = x33 && 2 <= x31 - 1 && 0 <= x30 - 1 && 0 <= x32 - 1
(8) f262_0_main_InvokeMethod(x36, x37, x38) -> f194_0_height_NONNULL(x39, x40, x41) :|: -1 <= x40 - 1 && 0 <= x39 - 1 && 0 <= x36 - 1 && x40 + 1 <= x36 && x39 <= x36
(9) f194_0_height_NONNULL(x42, x43, x44) -> f194_0_height_NONNULL(x45, x46, x47) :|: -1 <= x46 - 1 && 0 <= x45 - 1 && 0 <= x43 - 1 && 2 <= x42 - 1 && x46 + 1 <= x43 && x46 + 3 <= x42 && x45 <= x43 && x45 + 2 <= x42
(10) f194_0_height_NONNULL(x48, x49, x50) -> f194_0_height_NONNULL(x51, x52, x53) :|: -1 <= x52 - 1 && 0 <= x51 - 1 && -1 <= x49 - 1 && 2 <= x48 - 1 && x52 + 3 <= x48 && x51 + 2 <= x48
(11) __init(x54, x55, x56) -> f1_0_main_New(x57, x58, x59) :|: 0 <= 0

Arcs:
(1) -> (8)
(2) -> (8)
(3) -> (4), (5), (6)
(4) -> (4), (5), (6)
(5) -> (7)
(6) -> (7)
(7) -> (4), (5), (6)
(8) -> (9), (10)
(9) -> (9), (10)
(10) -> (9), (10)
(11) -> (1), (2), (3)

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

(4)
Complex Obligation (AND)

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

(5)
Obligation:

Termination digraph:
Nodes:
(1) f76_0__init__LE(x12, x13, x14) -> f76_0__init__LE(x15, x16, x17) :|: x13 - 1 = x17 && x13 - 1 = x16 && x13 - 1 = x15 && x13 = x14 && 1 <= x13 - 1 && 0 <= x12 - 1 && x13 - 1 <= x13 - 1
(2) f288_0__init__InvokeMethod(x30, x31, x32) -> f76_0__init__LE(x33, x34, x35) :|: x32 = x35 && x32 = x34 && x32 = x33 && 2 <= x31 - 1 && 0 <= x30 - 1 && 0 <= x32 - 1
(3) f76_0__init__LE(x24, x25, x26) -> f288_0__init__InvokeMethod(x27, x28, x29) :|: x25 - 1 = x29 && x24 = x27 && x25 = x26 && 0 <= x24 - 1 && 3 <= x28 - 1 && 1 <= x25 - 1 && x25 - 1 <= x25 - 1
(4) f76_0__init__LE(x18, x19, x20) -> f288_0__init__InvokeMethod(x21, x22, x23) :|: x19 - 1 = x23 && x18 = x21 && x19 = x20 && 0 <= x18 - 1 && 4 <= x22 - 1 && 1 <= x19 - 1 && x19 - 1 <= x19 - 1

Arcs:
(1) -> (1), (3), (4)
(2) -> (1), (3), (4)
(3) -> (2)
(4) -> (2)

This digraph is fully evaluated!

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

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

(7)
Obligation:
Rules:
f76_0__init__LE(x12:0, x13:0, x13:0) -> f76_0__init__LE(x13:0 - 1, x13:0 - 1, x13:0 - 1) :|: x12:0 > 0 && x13:0 > 1
f76_0__init__LE(x18:0, x19:0, x19:0) -> f76_0__init__LE(x19:0 - 1, x19:0 - 1, x19:0 - 1) :|: x19:0 > 1 && x22:0 > 4 && x18:0 > 0
f76_0__init__LE(x24:0, x25:0, x25:0) -> f76_0__init__LE(x25:0 - 1, x25:0 - 1, x25:0 - 1) :|: x25:0 > 1 && x28:0 > 3 && x24:0 > 0

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

(8) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f76_0__init__LE(INTEGER, INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(9)
Obligation:
Rules:
f76_0__init__LE(x12:0, x13:0, x13:0) -> f76_0__init__LE(c, c1, c2) :|: c2 = x13:0 - 1 && (c1 = x13:0 - 1 && c = x13:0 - 1) && (x12:0 > 0 && x13:0 > 1)
f76_0__init__LE(x18:0, x19:0, x19:0) -> f76_0__init__LE(c3, c4, c5) :|: c5 = x19:0 - 1 && (c4 = x19:0 - 1 && c3 = x19:0 - 1) && (x19:0 > 1 && x22:0 > 4 && x18:0 > 0)
f76_0__init__LE(x24:0, x25:0, x25:0) -> f76_0__init__LE(c6, c7, c8) :|: c8 = x25:0 - 1 && (c7 = x25:0 - 1 && c6 = x25:0 - 1) && (x25:0 > 1 && x28:0 > 3 && x24:0 > 0)

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

(10) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[f76_0__init__LE(x, x1, x2)] = x2

The following rules are decreasing:
f76_0__init__LE(x12:0, x13:0, x13:0) -> f76_0__init__LE(c, c1, c2) :|: c2 = x13:0 - 1 && (c1 = x13:0 - 1 && c = x13:0 - 1) && (x12:0 > 0 && x13:0 > 1)
f76_0__init__LE(x18:0, x19:0, x19:0) -> f76_0__init__LE(c3, c4, c5) :|: c5 = x19:0 - 1 && (c4 = x19:0 - 1 && c3 = x19:0 - 1) && (x19:0 > 1 && x22:0 > 4 && x18:0 > 0)
f76_0__init__LE(x24:0, x25:0, x25:0) -> f76_0__init__LE(c6, c7, c8) :|: c8 = x25:0 - 1 && (c7 = x25:0 - 1 && c6 = x25:0 - 1) && (x25:0 > 1 && x28:0 > 3 && x24:0 > 0)
The following rules are bounded:
f76_0__init__LE(x12:0, x13:0, x13:0) -> f76_0__init__LE(c, c1, c2) :|: c2 = x13:0 - 1 && (c1 = x13:0 - 1 && c = x13:0 - 1) && (x12:0 > 0 && x13:0 > 1)
f76_0__init__LE(x18:0, x19:0, x19:0) -> f76_0__init__LE(c3, c4, c5) :|: c5 = x19:0 - 1 && (c4 = x19:0 - 1 && c3 = x19:0 - 1) && (x19:0 > 1 && x22:0 > 4 && x18:0 > 0)
f76_0__init__LE(x24:0, x25:0, x25:0) -> f76_0__init__LE(c6, c7, c8) :|: c8 = x25:0 - 1 && (c7 = x25:0 - 1 && c6 = x25:0 - 1) && (x25:0 > 1 && x28:0 > 3 && x24:0 > 0)

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

(11)
YES

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

(12)
Obligation:

Termination digraph:
Nodes:
(1) f194_0_height_NONNULL(x42, x43, x44) -> f194_0_height_NONNULL(x45, x46, x47) :|: -1 <= x46 - 1 && 0 <= x45 - 1 && 0 <= x43 - 1 && 2 <= x42 - 1 && x46 + 1 <= x43 && x46 + 3 <= x42 && x45 <= x43 && x45 + 2 <= x42
(2) f194_0_height_NONNULL(x48, x49, x50) -> f194_0_height_NONNULL(x51, x52, x53) :|: -1 <= x52 - 1 && 0 <= x51 - 1 && -1 <= x49 - 1 && 2 <= x48 - 1 && x52 + 3 <= x48 && x51 + 2 <= x48

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

This digraph is fully evaluated!

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

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

(14)
Obligation:
Rules:
f194_0_height_NONNULL(x48:0, x49:0, x50:0) -> f194_0_height_NONNULL(x51:0, x52:0, x53:0) :|: x52:0 + 3 <= x48:0 && x51:0 + 2 <= x48:0 && x48:0 > 2 && x49:0 > -1 && x51:0 > 0 && x52:0 > -1
f194_0_height_NONNULL(x42:0, x43:0, x44:0) -> f194_0_height_NONNULL(x45:0, x46:0, x47:0) :|: x45:0 <= x43:0 && x45:0 + 2 <= x42:0 && x46:0 + 3 <= x42:0 && x46:0 + 1 <= x43:0 && x42:0 > 2 && x43:0 > 0 && x45:0 > 0 && x46:0 > -1

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

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

   f194_0_height_NONNULL(x1, x2, x3) -> f194_0_height_NONNULL(x1, x2)

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

(16)
Obligation:
Rules:
f194_0_height_NONNULL(x48:0, x49:0) -> f194_0_height_NONNULL(x51:0, x52:0) :|: x52:0 + 3 <= x48:0 && x51:0 + 2 <= x48:0 && x48:0 > 2 && x49:0 > -1 && x51:0 > 0 && x52:0 > -1
f194_0_height_NONNULL(x42:0, x43:0) -> f194_0_height_NONNULL(x45:0, x46:0) :|: x45:0 <= x43:0 && x45:0 + 2 <= x42:0 && x46:0 + 3 <= x42:0 && x46:0 + 1 <= x43:0 && x42:0 > 2 && x43:0 > 0 && x45:0 > 0 && x46:0 > -1

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

(17) FilterProof (EQUIVALENT)
Used the following sort dictionary for filtering: 
f194_0_height_NONNULL(INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(18)
Obligation:
Rules:
f194_0_height_NONNULL(x48:0, x49:0) -> f194_0_height_NONNULL(x51:0, x52:0) :|: x52:0 + 3 <= x48:0 && x51:0 + 2 <= x48:0 && x48:0 > 2 && x49:0 > -1 && x51:0 > 0 && x52:0 > -1
f194_0_height_NONNULL(x42:0, x43:0) -> f194_0_height_NONNULL(x45:0, x46:0) :|: x45:0 <= x43:0 && x45:0 + 2 <= x42:0 && x46:0 + 3 <= x42:0 && x46:0 + 1 <= x43:0 && x42:0 > 2 && x43:0 > 0 && x45:0 > 0 && x46:0 > -1

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

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

(20)
Obligation:
Rules:
f194_0_height_NONNULL(x42:0:0, x43:0:0) -> f194_0_height_NONNULL(x45:0:0, x46:0:0) :|: x45:0:0 > 0 && x46:0:0 > -1 && x43:0:0 > 0 && x42:0:0 > 2 && x46:0:0 + 1 <= x43:0:0 && x46:0:0 + 3 <= x42:0:0 && x45:0:0 + 2 <= x42:0:0 && x45:0:0 <= x43:0:0
f194_0_height_NONNULL(x48:0:0, x49:0:0) -> f194_0_height_NONNULL(x51:0:0, x52:0:0) :|: x51:0:0 > 0 && x52:0:0 > -1 && x49:0:0 > -1 && x48:0:0 > 2 && x51:0:0 + 2 <= x48:0:0 && x52:0:0 + 3 <= x48:0:0

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

(21) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ f194_0_height_NONNULL ] = 1/2*f194_0_height_NONNULL_1

The following rules are decreasing:
f194_0_height_NONNULL(x42:0:0, x43:0:0) -> f194_0_height_NONNULL(x45:0:0, x46:0:0) :|: x45:0:0 > 0 && x46:0:0 > -1 && x43:0:0 > 0 && x42:0:0 > 2 && x46:0:0 + 1 <= x43:0:0 && x46:0:0 + 3 <= x42:0:0 && x45:0:0 + 2 <= x42:0:0 && x45:0:0 <= x43:0:0
f194_0_height_NONNULL(x48:0:0, x49:0:0) -> f194_0_height_NONNULL(x51:0:0, x52:0:0) :|: x51:0:0 > 0 && x52:0:0 > -1 && x49:0:0 > -1 && x48:0:0 > 2 && x51:0:0 + 2 <= x48:0:0 && x52:0:0 + 3 <= x48:0:0

The following rules are bounded:
f194_0_height_NONNULL(x42:0:0, x43:0:0) -> f194_0_height_NONNULL(x45:0:0, x46:0:0) :|: x45:0:0 > 0 && x46:0:0 > -1 && x43:0:0 > 0 && x42:0:0 > 2 && x46:0:0 + 1 <= x43:0:0 && x46:0:0 + 3 <= x42:0:0 && x45:0:0 + 2 <= x42:0:0 && x45:0:0 <= x43:0:0
f194_0_height_NONNULL(x48:0:0, x49:0:0) -> f194_0_height_NONNULL(x51:0:0, x52:0:0) :|: x51:0:0 > 0 && x52:0:0 > -1 && x49:0:0 > -1 && x48:0:0 > 2 && x51:0:0 + 2 <= x48:0:0 && x52:0:0 + 3 <= x48:0:0


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

(22)
YES