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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 352 ms]
(4) AND
    (5) IRSwT
        (6) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (7) IRSwT
        (8) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (9) IRSwT
        (10) TempFilterProof [SOUND, 23 ms]
        (11) IntTRS
        (12) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
        (13) YES
    (14) IRSwT
        (15) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (16) IRSwT
        (17) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (18) IRSwT
        (19) TempFilterProof [SOUND, 5 ms]
        (20) IntTRS
        (21) RankingReductionPairProof [EQUIVALENT, 0 ms]
        (22) YES
    (23) IRSwT
        (24) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (25) IRSwT
        (26) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (27) IRSwT
        (28) TempFilterProof [SOUND, 4 ms]
        (29) IntTRS
        (30) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
        (31) YES


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(0)
Obligation:
Rules:
f1_0_main_Load(arg1, arg2, arg3) -> f127_0_times_NE(arg1P, arg2P, arg3P) :|: 0 <= arg1 - 1 && -1 <= arg1P - 1 && 1 <= arg2 - 1 && -1 <= arg2P - 1
f127_0_times_NE(x, x1, x2) -> f127_0_times_NE(x3, x4, x5) :|: x1 - 1 = x4 && x = x3 && x1 - 1 <= x1 - 1 && 0 <= x1 - 1
f127_0_times_NE(x6, x7, x8) -> f469_0_times_InvokeMethod(x9, x10, x11) :|: x6 = x11 && 0 = x10 && 1 = x9 && 1 = x7
f127_0_times_NE(x12, x13, x14) -> f469_0_times_InvokeMethod(x15, x16, x17) :|: 0 = x17 && 0 = x16 && x13 = x15 && 0 = x12 && x13 - 1 <= x13 - 1 && 0 <= x13 - 1
f127_0_times_NE(x18, x19, x20) -> f469_0_times_InvokeMethod(x21, x22, x23) :|: x18 = x23 && x19 = x21 && x19 - 1 <= x19 - 1 && 0 <= x19 - 1
f127_0_times_NE(x24, x25, x26) -> f469_0_times_InvokeMethod(x27, x28, x29) :|: 0 = x29 && x25 = x27 && 0 = x24 && x25 - 1 <= x25 - 1 && 0 <= x25 - 1
f218_0_times_Return(x30, x31, x32) -> f469_0_times_InvokeMethod(x33, x34, x35) :|: x31 = x35 && x32 = x34 && x30 = x33
f469_0_times_InvokeMethod(x36, x37, x38) -> f484_0_plus_LE(x39, x40, x41) :|: x38 = x40 && x37 = x39 && 0 <= x36 - 1
f484_0_plus_LE(x42, x43, x44) -> f484_0_plus_LE(x45, x46, x47) :|: x43 - 1 = x46 && x42 = x45 && x43 - 1 <= x43 - 1 && 0 <= x43 - 1
f484_0_plus_LE(x48, x49, x50) -> f484_0_plus_LE(x51, x52, x53) :|: x49 = x52 && x48 - 1 = x51 && 0 <= x48 - 1 && x48 - 1 <= x48 - 1 && x49 <= 0
__init(x54, x55, x56) -> f1_0_main_Load(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_Load(arg1, arg2, arg3) -> f127_0_times_NE(arg1P, arg2P, arg3P) :|: 0 <= arg1 - 1 && -1 <= arg1P - 1 && 1 <= arg2 - 1 && -1 <= arg2P - 1
f127_0_times_NE(x, x1, x2) -> f127_0_times_NE(x3, x4, x5) :|: x1 - 1 = x4 && x = x3 && x1 - 1 <= x1 - 1 && 0 <= x1 - 1
f127_0_times_NE(x6, x7, x8) -> f469_0_times_InvokeMethod(x9, x10, x11) :|: x6 = x11 && 0 = x10 && 1 = x9 && 1 = x7
f127_0_times_NE(x12, x13, x14) -> f469_0_times_InvokeMethod(x15, x16, x17) :|: 0 = x17 && 0 = x16 && x13 = x15 && 0 = x12 && x13 - 1 <= x13 - 1 && 0 <= x13 - 1
f127_0_times_NE(x18, x19, x20) -> f469_0_times_InvokeMethod(x21, x22, x23) :|: x18 = x23 && x19 = x21 && x19 - 1 <= x19 - 1 && 0 <= x19 - 1
f127_0_times_NE(x24, x25, x26) -> f469_0_times_InvokeMethod(x27, x28, x29) :|: 0 = x29 && x25 = x27 && 0 = x24 && x25 - 1 <= x25 - 1 && 0 <= x25 - 1
f218_0_times_Return(x30, x31, x32) -> f469_0_times_InvokeMethod(x33, x34, x35) :|: x31 = x35 && x32 = x34 && x30 = x33
f469_0_times_InvokeMethod(x36, x37, x38) -> f484_0_plus_LE(x39, x40, x41) :|: x38 = x40 && x37 = x39 && 0 <= x36 - 1
f484_0_plus_LE(x42, x43, x44) -> f484_0_plus_LE(x45, x46, x47) :|: x43 - 1 = x46 && x42 = x45 && x43 - 1 <= x43 - 1 && 0 <= x43 - 1
f484_0_plus_LE(x48, x49, x50) -> f484_0_plus_LE(x51, x52, x53) :|: x49 = x52 && x48 - 1 = x51 && 0 <= x48 - 1 && x48 - 1 <= x48 - 1 && x49 <= 0
__init(x54, x55, x56) -> f1_0_main_Load(x57, x58, x59) :|: 0 <= 0
Start term: __init(arg1, arg2, arg3)

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

(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) f1_0_main_Load(arg1, arg2, arg3) -> f127_0_times_NE(arg1P, arg2P, arg3P) :|: 0 <= arg1 - 1 && -1 <= arg1P - 1 && 1 <= arg2 - 1 && -1 <= arg2P - 1
(2) f127_0_times_NE(x, x1, x2) -> f127_0_times_NE(x3, x4, x5) :|: x1 - 1 = x4 && x = x3 && x1 - 1 <= x1 - 1 && 0 <= x1 - 1
(3) f127_0_times_NE(x6, x7, x8) -> f469_0_times_InvokeMethod(x9, x10, x11) :|: x6 = x11 && 0 = x10 && 1 = x9 && 1 = x7
(4) f127_0_times_NE(x12, x13, x14) -> f469_0_times_InvokeMethod(x15, x16, x17) :|: 0 = x17 && 0 = x16 && x13 = x15 && 0 = x12 && x13 - 1 <= x13 - 1 && 0 <= x13 - 1
(5) f127_0_times_NE(x18, x19, x20) -> f469_0_times_InvokeMethod(x21, x22, x23) :|: x18 = x23 && x19 = x21 && x19 - 1 <= x19 - 1 && 0 <= x19 - 1
(6) f127_0_times_NE(x24, x25, x26) -> f469_0_times_InvokeMethod(x27, x28, x29) :|: 0 = x29 && x25 = x27 && 0 = x24 && x25 - 1 <= x25 - 1 && 0 <= x25 - 1
(7) f218_0_times_Return(x30, x31, x32) -> f469_0_times_InvokeMethod(x33, x34, x35) :|: x31 = x35 && x32 = x34 && x30 = x33
(8) f469_0_times_InvokeMethod(x36, x37, x38) -> f484_0_plus_LE(x39, x40, x41) :|: x38 = x40 && x37 = x39 && 0 <= x36 - 1
(9) f484_0_plus_LE(x42, x43, x44) -> f484_0_plus_LE(x45, x46, x47) :|: x43 - 1 = x46 && x42 = x45 && x43 - 1 <= x43 - 1 && 0 <= x43 - 1
(10) f484_0_plus_LE(x48, x49, x50) -> f484_0_plus_LE(x51, x52, x53) :|: x49 = x52 && x48 - 1 = x51 && 0 <= x48 - 1 && x48 - 1 <= x48 - 1 && x49 <= 0
(11) __init(x54, x55, x56) -> f1_0_main_Load(x57, x58, x59) :|: 0 <= 0

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

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

(4)
Complex Obligation (AND)

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

(5)
Obligation:

Termination digraph:
Nodes:
(1) f127_0_times_NE(x, x1, x2) -> f127_0_times_NE(x3, x4, x5) :|: x1 - 1 = x4 && x = x3 && x1 - 1 <= x1 - 1 && 0 <= x1 - 1

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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

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

(7)
Obligation:
Rules:
f127_0_times_NE(x3:0, x1:0, x2:0) -> f127_0_times_NE(x3:0, x1:0 - 1, x5:0) :|: x1:0 > 0

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

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

   f127_0_times_NE(x1, x2, x3) -> f127_0_times_NE(x2)

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

(9)
Obligation:
Rules:
f127_0_times_NE(x1:0) -> f127_0_times_NE(x1:0 - 1) :|: x1:0 > 0

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(10) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f127_0_times_NE(INTEGER)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(11)
Obligation:
Rules:
f127_0_times_NE(x1:0) -> f127_0_times_NE(c) :|: c = x1:0 - 1 && x1:0 > 0

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

(12) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[f127_0_times_NE(x)] = x

The following rules are decreasing:
f127_0_times_NE(x1:0) -> f127_0_times_NE(c) :|: c = x1:0 - 1 && x1:0 > 0
The following rules are bounded:
f127_0_times_NE(x1:0) -> f127_0_times_NE(c) :|: c = x1:0 - 1 && x1:0 > 0

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

(13)
YES

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

(14)
Obligation:

Termination digraph:
Nodes:
(1) f484_0_plus_LE(x42, x43, x44) -> f484_0_plus_LE(x45, x46, x47) :|: x43 - 1 = x46 && x42 = x45 && x43 - 1 <= x43 - 1 && 0 <= x43 - 1

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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

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

(16)
Obligation:
Rules:
f484_0_plus_LE(x42:0, x43:0, x44:0) -> f484_0_plus_LE(x42:0, x43:0 - 1, x47:0) :|: x43:0 > 0

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

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

   f484_0_plus_LE(x1, x2, x3) -> f484_0_plus_LE(x2)

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

(18)
Obligation:
Rules:
f484_0_plus_LE(x43:0) -> f484_0_plus_LE(x43:0 - 1) :|: x43:0 > 0

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

(19) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f484_0_plus_LE(INTEGER)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(20)
Obligation:
Rules:
f484_0_plus_LE(x43:0) -> f484_0_plus_LE(c) :|: c = x43:0 - 1 && x43:0 > 0

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

(21) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ f484_0_plus_LE ] = f484_0_plus_LE_1

The following rules are decreasing:
f484_0_plus_LE(x43:0) -> f484_0_plus_LE(c) :|: c = x43:0 - 1 && x43:0 > 0

The following rules are bounded:
f484_0_plus_LE(x43:0) -> f484_0_plus_LE(c) :|: c = x43:0 - 1 && x43:0 > 0


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

(22)
YES

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

(23)
Obligation:

Termination digraph:
Nodes:
(1) f484_0_plus_LE(x48, x49, x50) -> f484_0_plus_LE(x51, x52, x53) :|: x49 = x52 && x48 - 1 = x51 && 0 <= x48 - 1 && x48 - 1 <= x48 - 1 && x49 <= 0

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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

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

(25)
Obligation:
Rules:
f484_0_plus_LE(x48:0, x49:0, x50:0) -> f484_0_plus_LE(x48:0 - 1, x49:0, x53:0) :|: x49:0 < 1 && x48:0 > 0

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

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

   f484_0_plus_LE(x1, x2, x3) -> f484_0_plus_LE(x1, x2)

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

(27)
Obligation:
Rules:
f484_0_plus_LE(x48:0, x49:0) -> f484_0_plus_LE(x48:0 - 1, x49:0) :|: x49:0 < 1 && x48:0 > 0

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

(28) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f484_0_plus_LE(INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(29)
Obligation:
Rules:
f484_0_plus_LE(x48:0, x49:0) -> f484_0_plus_LE(c, x49:0) :|: c = x48:0 - 1 && (x49:0 < 1 && x48:0 > 0)

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

(30) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[f484_0_plus_LE(x, x1)] = x

The following rules are decreasing:
f484_0_plus_LE(x48:0, x49:0) -> f484_0_plus_LE(c, x49:0) :|: c = x48:0 - 1 && (x49:0 < 1 && x48:0 > 0)
The following rules are bounded:
f484_0_plus_LE(x48:0, x49:0) -> f484_0_plus_LE(c, x49:0) :|: c = x48:0 - 1 && (x49:0 < 1 && x48:0 > 0)

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(31)
YES