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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 339 ms]
(4) AND
    (5) IRSwT
        (6) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (7) IRSwT
        (8) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (9) IRSwT
        (10) TempFilterProof [SOUND, 91 ms]
        (11) IntTRS
        (12) PolynomialOrderProcessor [EQUIVALENT, 14 ms]
        (13) YES
    (14) IRSwT
        (15) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (16) IRSwT
        (17) TempFilterProof [SOUND, 31 ms]
        (18) IntTRS
        (19) RankingReductionPairProof [EQUIVALENT, 5 ms]
        (20) YES


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(0)
Obligation:
Rules:
f1_0_main_Load(arg1, arg2, arg3, arg4) -> f168_0_main_LE(arg1P, arg2P, arg3P, arg4P) :|: 0 = arg3P && arg2 = arg2P && arg2 - 1 = arg1P && -1 <= arg2 - 1 && 0 <= arg1 - 1
f168_0_main_LE(x, x1, x2, x3) -> f168_0_main_LE(x4, x5, x6, x7) :|: x = x5 && x - 1 = x4 && 0 <= x2 - 1 && x2 <= x6 - 1 && 0 <= x1 - 1
f168_0_main_LE(x8, x9, x10, x11) -> f168_0_main_LE(x12, x13, x14, x15) :|: 1 = x14 && x8 = x13 && x8 - 1 = x12 && 0 <= x9 - 1
f168_0_main_LE(x16, x17, x18, x20) -> f223_0_iterate_EQ(x21, x22, x23, x24) :|: 0 = x24 && x18 = x23 && x18 = x22 && 0 = x21 && 0 <= x18 - 1 && x17 <= 0
f223_0_iterate_EQ(x25, x27, x28, x29) -> f223_0_iterate_EQ(x30, x31, x32, x33) :|: 0 <= x25 - 1 && 0 <= x29 - 1 && 0 <= x28 - 1 && x25 <= x30 - 1 && x25 <= x27 - 1 && x32 <= x28 - 1 && x33 <= x29 - 1 && x25 <= x34 - 1 && x27 = x31
f223_0_iterate_EQ(x35, x36, x37, x38) -> f223_0_iterate_EQ(x39, x40, x41, x42) :|: x41 <= x36 - 1 && 0 <= x36 - 1 && x42 <= x43 - 1 && -1 <= x43 - 1 && x36 = x37 && 1 = x39
__init(x44, x45, x46, x47) -> f1_0_main_Load(x48, x49, x50, x51) :|: 0 <= 0
Start term: __init(arg1, arg2, arg3, arg4)

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(1) IRSFormatTransformerProof (EQUIVALENT)
Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %).
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(2)
Obligation:
Rules:
f1_0_main_Load(arg1, arg2, arg3, arg4) -> f168_0_main_LE(arg1P, arg2P, arg3P, arg4P) :|: 0 = arg3P && arg2 = arg2P && arg2 - 1 = arg1P && -1 <= arg2 - 1 && 0 <= arg1 - 1
f168_0_main_LE(x, x1, x2, x3) -> f168_0_main_LE(x4, x5, x6, x7) :|: x = x5 && x - 1 = x4 && 0 <= x2 - 1 && x2 <= x6 - 1 && 0 <= x1 - 1
f168_0_main_LE(x8, x9, x10, x11) -> f168_0_main_LE(x12, x13, x14, x15) :|: 1 = x14 && x8 = x13 && x8 - 1 = x12 && 0 <= x9 - 1
f168_0_main_LE(x16, x17, x18, x20) -> f223_0_iterate_EQ(x21, x22, x23, x24) :|: 0 = x24 && x18 = x23 && x18 = x22 && 0 = x21 && 0 <= x18 - 1 && x17 <= 0
f223_0_iterate_EQ(x25, x27, x28, x29) -> f223_0_iterate_EQ(x30, x31, x32, x33) :|: 0 <= x25 - 1 && 0 <= x29 - 1 && 0 <= x28 - 1 && x25 <= x30 - 1 && x25 <= x27 - 1 && x32 <= x28 - 1 && x33 <= x29 - 1 && x25 <= x34 - 1 && x27 = x31
f223_0_iterate_EQ(x35, x36, x37, x38) -> f223_0_iterate_EQ(x39, x40, x41, x42) :|: x41 <= x36 - 1 && 0 <= x36 - 1 && x42 <= x43 - 1 && -1 <= x43 - 1 && x36 = x37 && 1 = x39
__init(x44, x45, x46, x47) -> f1_0_main_Load(x48, x49, x50, x51) :|: 0 <= 0
Start term: __init(arg1, arg2, arg3, arg4)

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(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) f1_0_main_Load(arg1, arg2, arg3, arg4) -> f168_0_main_LE(arg1P, arg2P, arg3P, arg4P) :|: 0 = arg3P && arg2 = arg2P && arg2 - 1 = arg1P && -1 <= arg2 - 1 && 0 <= arg1 - 1
(2) f168_0_main_LE(x, x1, x2, x3) -> f168_0_main_LE(x4, x5, x6, x7) :|: x = x5 && x - 1 = x4 && 0 <= x2 - 1 && x2 <= x6 - 1 && 0 <= x1 - 1
(3) f168_0_main_LE(x8, x9, x10, x11) -> f168_0_main_LE(x12, x13, x14, x15) :|: 1 = x14 && x8 = x13 && x8 - 1 = x12 && 0 <= x9 - 1
(4) f168_0_main_LE(x16, x17, x18, x20) -> f223_0_iterate_EQ(x21, x22, x23, x24) :|: 0 = x24 && x18 = x23 && x18 = x22 && 0 = x21 && 0 <= x18 - 1 && x17 <= 0
(5) f223_0_iterate_EQ(x25, x27, x28, x29) -> f223_0_iterate_EQ(x30, x31, x32, x33) :|: 0 <= x25 - 1 && 0 <= x29 - 1 && 0 <= x28 - 1 && x25 <= x30 - 1 && x25 <= x27 - 1 && x32 <= x28 - 1 && x33 <= x29 - 1 && x25 <= x34 - 1 && x27 = x31
(6) f223_0_iterate_EQ(x35, x36, x37, x38) -> f223_0_iterate_EQ(x39, x40, x41, x42) :|: x41 <= x36 - 1 && 0 <= x36 - 1 && x42 <= x43 - 1 && -1 <= x43 - 1 && x36 = x37 && 1 = x39
(7) __init(x44, x45, x46, x47) -> f1_0_main_Load(x48, x49, x50, x51) :|: 0 <= 0

Arcs:
(1) -> (3)
(2) -> (2), (3), (4)
(3) -> (2), (3), (4)
(4) -> (6)
(5) -> (5), (6)
(6) -> (5), (6)
(7) -> (1)

This digraph is fully evaluated!
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(4)
Complex Obligation (AND)

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(5)
Obligation:

Termination digraph:
Nodes:
(1) f168_0_main_LE(x8, x9, x10, x11) -> f168_0_main_LE(x12, x13, x14, x15) :|: 1 = x14 && x8 = x13 && x8 - 1 = x12 && 0 <= x9 - 1
(2) f168_0_main_LE(x, x1, x2, x3) -> f168_0_main_LE(x4, x5, x6, x7) :|: x = x5 && x - 1 = x4 && 0 <= x2 - 1 && x2 <= x6 - 1 && 0 <= x1 - 1

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

This digraph is fully evaluated!

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(6) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(7)
Obligation:
Rules:
f168_0_main_LE(x5:0, x1:0, x2:0, x3:0) -> f168_0_main_LE(x5:0 - 1, x5:0, x6:0, x7:0) :|: x6:0 - 1 >= x2:0 && x2:0 > 0 && x1:0 > 0
f168_0_main_LE(x13:0, x9:0, x10:0, x11:0) -> f168_0_main_LE(x13:0 - 1, x13:0, 1, x15:0) :|: x9:0 > 0

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

   f168_0_main_LE(x1, x2, x3, x4) -> f168_0_main_LE(x1, x2, x3)

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(9)
Obligation:
Rules:
f168_0_main_LE(x5:0, x1:0, x2:0) -> f168_0_main_LE(x5:0 - 1, x5:0, x6:0) :|: x6:0 - 1 >= x2:0 && x2:0 > 0 && x1:0 > 0
f168_0_main_LE(x13:0, x9:0, x10:0) -> f168_0_main_LE(x13:0 - 1, x13:0, 1) :|: x9:0 > 0

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(10) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f168_0_main_LE(VARIABLE, VARIABLE, VARIABLE)
Replaced non-predefined constructor symbols by 0.
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(11)
Obligation:
Rules:
f168_0_main_LE(x5:0, x1:0, x2:0) -> f168_0_main_LE(c, x5:0, x6:0) :|: c = x5:0 - 1 && (x6:0 - 1 >= x2:0 && x2:0 > 0 && x1:0 > 0)
f168_0_main_LE(x13:0, x9:0, x10:0) -> f168_0_main_LE(c1, x13:0, c2) :|: c2 = 1 && c1 = x13:0 - 1 && x9:0 > 0

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(12) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[f168_0_main_LE(x, x1, x2)] = -2 + x^2 + 2*x1

The following rules are decreasing:
f168_0_main_LE(x5:0, x1:0, x2:0) -> f168_0_main_LE(c, x5:0, x6:0) :|: c = x5:0 - 1 && (x6:0 - 1 >= x2:0 && x2:0 > 0 && x1:0 > 0)
f168_0_main_LE(x13:0, x9:0, x10:0) -> f168_0_main_LE(c1, x13:0, c2) :|: c2 = 1 && c1 = x13:0 - 1 && x9:0 > 0
The following rules are bounded:
f168_0_main_LE(x5:0, x1:0, x2:0) -> f168_0_main_LE(c, x5:0, x6:0) :|: c = x5:0 - 1 && (x6:0 - 1 >= x2:0 && x2:0 > 0 && x1:0 > 0)
f168_0_main_LE(x13:0, x9:0, x10:0) -> f168_0_main_LE(c1, x13:0, c2) :|: c2 = 1 && c1 = x13:0 - 1 && x9:0 > 0

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(13)
YES

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(14)
Obligation:

Termination digraph:
Nodes:
(1) f223_0_iterate_EQ(x35, x36, x37, x38) -> f223_0_iterate_EQ(x39, x40, x41, x42) :|: x41 <= x36 - 1 && 0 <= x36 - 1 && x42 <= x43 - 1 && -1 <= x43 - 1 && x36 = x37 && 1 = x39
(2) f223_0_iterate_EQ(x25, x27, x28, x29) -> f223_0_iterate_EQ(x30, x31, x32, x33) :|: 0 <= x25 - 1 && 0 <= x29 - 1 && 0 <= x28 - 1 && x25 <= x30 - 1 && x25 <= x27 - 1 && x32 <= x28 - 1 && x33 <= x29 - 1 && x25 <= x34 - 1 && x27 = x31

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

This digraph is fully evaluated!

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

(16)
Obligation:
Rules:
f223_0_iterate_EQ(x35:0, x36:0, x36:0, x38:0) -> f223_0_iterate_EQ(1, x40:0, x41:0, x42:0) :|: x43:0 - 1 >= x42:0 && x43:0 > -1 && x36:0 > 0 && x41:0 <= x36:0 - 1
f223_0_iterate_EQ(x25:0, x27:0, x28:0, x29:0) -> f223_0_iterate_EQ(x30:0, x27:0, x32:0, x33:0) :|: x33:0 <= x29:0 - 1 && x34:0 - 1 >= x25:0 && x32:0 <= x28:0 - 1 && x27:0 - 1 >= x25:0 && x30:0 - 1 >= x25:0 && x28:0 > 0 && x29:0 > 0 && x25:0 > 0

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(17) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f223_0_iterate_EQ(VARIABLE, VARIABLE, INTEGER, VARIABLE)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(18)
Obligation:
Rules:
f223_0_iterate_EQ(x35:0, x36:0, x36:0, x38:0) -> f223_0_iterate_EQ(c, x40:0, x41:0, x42:0) :|: c = 1 && (x43:0 - 1 >= x42:0 && x43:0 > -1 && x36:0 > 0 && x41:0 <= x36:0 - 1)
f223_0_iterate_EQ(x25:0, x27:0, x28:0, x29:0) -> f223_0_iterate_EQ(x30:0, x27:0, x32:0, x33:0) :|: x33:0 <= x29:0 - 1 && x34:0 - 1 >= x25:0 && x32:0 <= x28:0 - 1 && x27:0 - 1 >= x25:0 && x30:0 - 1 >= x25:0 && x28:0 > 0 && x29:0 > 0 && x25:0 > 0

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(19) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ f223_0_iterate_EQ ] = f223_0_iterate_EQ_3

The following rules are decreasing:
f223_0_iterate_EQ(x35:0, x36:0, x36:0, x38:0) -> f223_0_iterate_EQ(c, x40:0, x41:0, x42:0) :|: c = 1 && (x43:0 - 1 >= x42:0 && x43:0 > -1 && x36:0 > 0 && x41:0 <= x36:0 - 1)
f223_0_iterate_EQ(x25:0, x27:0, x28:0, x29:0) -> f223_0_iterate_EQ(x30:0, x27:0, x32:0, x33:0) :|: x33:0 <= x29:0 - 1 && x34:0 - 1 >= x25:0 && x32:0 <= x28:0 - 1 && x27:0 - 1 >= x25:0 && x30:0 - 1 >= x25:0 && x28:0 > 0 && x29:0 > 0 && x25:0 > 0

The following rules are bounded:
f223_0_iterate_EQ(x35:0, x36:0, x36:0, x38:0) -> f223_0_iterate_EQ(c, x40:0, x41:0, x42:0) :|: c = 1 && (x43:0 - 1 >= x42:0 && x43:0 > -1 && x36:0 > 0 && x41:0 <= x36:0 - 1)
f223_0_iterate_EQ(x25:0, x27:0, x28:0, x29:0) -> f223_0_iterate_EQ(x30:0, x27:0, x32:0, x33:0) :|: x33:0 <= x29:0 - 1 && x34:0 - 1 >= x25:0 && x32:0 <= x28:0 - 1 && x27:0 - 1 >= x25:0 && x30:0 - 1 >= x25:0 && x28:0 > 0 && x29:0 > 0 && x25:0 > 0


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(20)
YES