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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 427 ms]
(4) AND
    (5) IRSwT
        (6) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (7) IRSwT
        (8) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (9) IRSwT
        (10) TempFilterProof [SOUND, 28 ms]
        (11) IntTRS
        (12) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
        (13) YES
    (14) IRSwT
        (15) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (16) IRSwT
        (17) TempFilterProof [SOUND, 43 ms]
        (18) IntTRS
        (19) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
        (20) IntTRS
        (21) PolynomialOrderProcessor [EQUIVALENT, 4 ms]
        (22) YES


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(0)
Obligation:
Rules:
f1_0_main_Load(arg1, arg2, arg3, arg4, arg5) -> f370_0_div_LT(arg1P, arg2P, arg3P, arg4P, arg5P) :|: 0 = arg1P && 0 <= arg1 - 1 && -1 <= arg2P - 1 && -1 <= arg2 - 1 && -1 <= arg3P - 1
f370_0_div_LT(x, x1, x2, x3, x4) -> f398_0_minus_EQ(x5, x6, x7, x8, x9) :|: x2 = x9 && x2 = x8 && x1 = x7 && x = x6 && x2 = x5 && 0 <= x2 - 1 && x2 <= x1
f398_0_minus_EQ(x10, x11, x12, x13, x14) -> f398_0_minus_EQ(x15, x16, x17, x18, x19) :|: x13 + 1 = x19 && x13 + 1 = x18 && x12 + 1 = x17 && x11 = x16 && x10 = x15 && x13 = x14 && x13 <= 0 && x13 <= -1
f398_0_minus_EQ(x20, x21, x22, x23, x24) -> f398_0_minus_EQ(x25, x26, x27, x28, x29) :|: x23 + 1 = x29 && x23 + 1 = x28 && x22 + 1 = x27 && x21 = x26 && x20 = x25 && x23 = x24 && x23 <= 0 && 0 <= x23 - 1
f398_0_minus_EQ(x30, x31, x32, x33, x34) -> f398_0_minus_EQ(x35, x36, x37, x38, x39) :|: x33 - 1 = x39 && x33 - 1 = x38 && x32 - 1 = x37 && x31 = x36 && x30 = x35 && x33 = x34 && 0 <= x33 - 1
f398_0_minus_EQ(x40, x41, x42, x43, x44) -> f370_0_div_LT(x45, x46, x47, x48, x49) :|: x40 = x47 && x42 = x46 && x41 + 1 = x45 && 0 = x44 && 0 = x43 && -1 <= x41 - 1
__init(x50, x51, x52, x53, x54) -> f1_0_main_Load(x55, x56, x57, x58, x59) :|: 0 <= 0
Start term: __init(arg1, arg2, arg3, arg4, arg5)

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(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, arg4, arg5) -> f370_0_div_LT(arg1P, arg2P, arg3P, arg4P, arg5P) :|: 0 = arg1P && 0 <= arg1 - 1 && -1 <= arg2P - 1 && -1 <= arg2 - 1 && -1 <= arg3P - 1
f370_0_div_LT(x, x1, x2, x3, x4) -> f398_0_minus_EQ(x5, x6, x7, x8, x9) :|: x2 = x9 && x2 = x8 && x1 = x7 && x = x6 && x2 = x5 && 0 <= x2 - 1 && x2 <= x1
f398_0_minus_EQ(x10, x11, x12, x13, x14) -> f398_0_minus_EQ(x15, x16, x17, x18, x19) :|: x13 + 1 = x19 && x13 + 1 = x18 && x12 + 1 = x17 && x11 = x16 && x10 = x15 && x13 = x14 && x13 <= 0 && x13 <= -1
f398_0_minus_EQ(x20, x21, x22, x23, x24) -> f398_0_minus_EQ(x25, x26, x27, x28, x29) :|: x23 + 1 = x29 && x23 + 1 = x28 && x22 + 1 = x27 && x21 = x26 && x20 = x25 && x23 = x24 && x23 <= 0 && 0 <= x23 - 1
f398_0_minus_EQ(x30, x31, x32, x33, x34) -> f398_0_minus_EQ(x35, x36, x37, x38, x39) :|: x33 - 1 = x39 && x33 - 1 = x38 && x32 - 1 = x37 && x31 = x36 && x30 = x35 && x33 = x34 && 0 <= x33 - 1
f398_0_minus_EQ(x40, x41, x42, x43, x44) -> f370_0_div_LT(x45, x46, x47, x48, x49) :|: x40 = x47 && x42 = x46 && x41 + 1 = x45 && 0 = x44 && 0 = x43 && -1 <= x41 - 1
__init(x50, x51, x52, x53, x54) -> f1_0_main_Load(x55, x56, x57, x58, x59) :|: 0 <= 0
Start term: __init(arg1, arg2, arg3, arg4, arg5)

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(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) f1_0_main_Load(arg1, arg2, arg3, arg4, arg5) -> f370_0_div_LT(arg1P, arg2P, arg3P, arg4P, arg5P) :|: 0 = arg1P && 0 <= arg1 - 1 && -1 <= arg2P - 1 && -1 <= arg2 - 1 && -1 <= arg3P - 1
(2) f370_0_div_LT(x, x1, x2, x3, x4) -> f398_0_minus_EQ(x5, x6, x7, x8, x9) :|: x2 = x9 && x2 = x8 && x1 = x7 && x = x6 && x2 = x5 && 0 <= x2 - 1 && x2 <= x1
(3) f398_0_minus_EQ(x10, x11, x12, x13, x14) -> f398_0_minus_EQ(x15, x16, x17, x18, x19) :|: x13 + 1 = x19 && x13 + 1 = x18 && x12 + 1 = x17 && x11 = x16 && x10 = x15 && x13 = x14 && x13 <= 0 && x13 <= -1
(4) f398_0_minus_EQ(x20, x21, x22, x23, x24) -> f398_0_minus_EQ(x25, x26, x27, x28, x29) :|: x23 + 1 = x29 && x23 + 1 = x28 && x22 + 1 = x27 && x21 = x26 && x20 = x25 && x23 = x24 && x23 <= 0 && 0 <= x23 - 1
(5) f398_0_minus_EQ(x30, x31, x32, x33, x34) -> f398_0_minus_EQ(x35, x36, x37, x38, x39) :|: x33 - 1 = x39 && x33 - 1 = x38 && x32 - 1 = x37 && x31 = x36 && x30 = x35 && x33 = x34 && 0 <= x33 - 1
(6) f398_0_minus_EQ(x40, x41, x42, x43, x44) -> f370_0_div_LT(x45, x46, x47, x48, x49) :|: x40 = x47 && x42 = x46 && x41 + 1 = x45 && 0 = x44 && 0 = x43 && -1 <= x41 - 1
(7) __init(x50, x51, x52, x53, x54) -> f1_0_main_Load(x55, x56, x57, x58, x59) :|: 0 <= 0

Arcs:
(1) -> (2)
(2) -> (5)
(3) -> (3), (6)
(5) -> (5), (6)
(6) -> (2)
(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) f398_0_minus_EQ(x10, x11, x12, x13, x14) -> f398_0_minus_EQ(x15, x16, x17, x18, x19) :|: x13 + 1 = x19 && x13 + 1 = x18 && x12 + 1 = x17 && x11 = x16 && x10 = x15 && x13 = x14 && x13 <= 0 && x13 <= -1

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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

(7)
Obligation:
Rules:
f398_0_minus_EQ(x10:0, x11:0, x12:0, x13:0, x13:0) -> f398_0_minus_EQ(x10:0, x11:0, x12:0 + 1, x13:0 + 1, x13:0 + 1) :|: x13:0 < 0 && x13:0 < 1

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

   f398_0_minus_EQ(x1, x2, x3, x4, x5) -> f398_0_minus_EQ(x4, x5)

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(9)
Obligation:
Rules:
f398_0_minus_EQ(x13:0, x13:0) -> f398_0_minus_EQ(x13:0 + 1, x13:0 + 1) :|: x13:0 < 0 && x13:0 < 1

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(10) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f398_0_minus_EQ(INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
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(11)
Obligation:
Rules:
f398_0_minus_EQ(x13:0, x13:0) -> f398_0_minus_EQ(c, c1) :|: c1 = x13:0 + 1 && c = x13:0 + 1 && (x13:0 < 0 && x13:0 < 1)

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(12) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[f398_0_minus_EQ(x, x1)] = -x1

The following rules are decreasing:
f398_0_minus_EQ(x13:0, x13:0) -> f398_0_minus_EQ(c, c1) :|: c1 = x13:0 + 1 && c = x13:0 + 1 && (x13:0 < 0 && x13:0 < 1)
The following rules are bounded:
f398_0_minus_EQ(x13:0, x13:0) -> f398_0_minus_EQ(c, c1) :|: c1 = x13:0 + 1 && c = x13:0 + 1 && (x13:0 < 0 && x13:0 < 1)

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(13)
YES

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(14)
Obligation:

Termination digraph:
Nodes:
(1) f370_0_div_LT(x, x1, x2, x3, x4) -> f398_0_minus_EQ(x5, x6, x7, x8, x9) :|: x2 = x9 && x2 = x8 && x1 = x7 && x = x6 && x2 = x5 && 0 <= x2 - 1 && x2 <= x1
(2) f398_0_minus_EQ(x40, x41, x42, x43, x44) -> f370_0_div_LT(x45, x46, x47, x48, x49) :|: x40 = x47 && x42 = x46 && x41 + 1 = x45 && 0 = x44 && 0 = x43 && -1 <= x41 - 1
(3) f398_0_minus_EQ(x30, x31, x32, x33, x34) -> f398_0_minus_EQ(x35, x36, x37, x38, x39) :|: x33 - 1 = x39 && x33 - 1 = x38 && x32 - 1 = x37 && x31 = x36 && x30 = x35 && x33 = x34 && 0 <= x33 - 1

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

This digraph is fully evaluated!

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

(16)
Obligation:
Rules:
f398_0_minus_EQ(x40:0, x41:0, x42:0, cons_0, cons_01) -> f398_0_minus_EQ(x40:0, x41:0 + 1, x42:0, x40:0, x40:0) :|: x42:0 >= x40:0 && x40:0 > 0 && x41:0 > -1 && cons_0 = 0 && cons_01 = 0
f398_0_minus_EQ(x30:0, x31:0, x32:0, x33:0, x33:0) -> f398_0_minus_EQ(x30:0, x31:0, x32:0 - 1, x33:0 - 1, x33:0 - 1) :|: x33:0 > 0

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(17) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f398_0_minus_EQ(VARIABLE, VARIABLE, VARIABLE, INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(18)
Obligation:
Rules:
f398_0_minus_EQ(x40:0, x41:0, x42:0, c, c1) -> f398_0_minus_EQ(x40:0, c2, x42:0, x40:0, x40:0) :|: c2 = x41:0 + 1 && (c1 = 0 && c = 0) && (x42:0 >= x40:0 && x40:0 > 0 && x41:0 > -1 && cons_0 = 0 && cons_01 = 0)
f398_0_minus_EQ(x30:0, x31:0, x32:0, x33:0, x33:0) -> f398_0_minus_EQ(x30:0, x31:0, c3, c4, c5) :|: c5 = x33:0 - 1 && (c4 = x33:0 - 1 && c3 = x32:0 - 1) && x33:0 > 0

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(19) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[f398_0_minus_EQ(x, x1, x2, x3, x4)] = x + x1 + 2*x2 - 2*x3

The following rules are decreasing:
f398_0_minus_EQ(x40:0, x41:0, x42:0, c, c1) -> f398_0_minus_EQ(x40:0, c2, x42:0, x40:0, x40:0) :|: c2 = x41:0 + 1 && (c1 = 0 && c = 0) && (x42:0 >= x40:0 && x40:0 > 0 && x41:0 > -1 && cons_0 = 0 && cons_01 = 0)
The following rules are bounded:
f398_0_minus_EQ(x40:0, x41:0, x42:0, c, c1) -> f398_0_minus_EQ(x40:0, c2, x42:0, x40:0, x40:0) :|: c2 = x41:0 + 1 && (c1 = 0 && c = 0) && (x42:0 >= x40:0 && x40:0 > 0 && x41:0 > -1 && cons_0 = 0 && cons_01 = 0)

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(20)
Obligation:
Rules:
f398_0_minus_EQ(x30:0, x31:0, x32:0, x33:0, x33:0) -> f398_0_minus_EQ(x30:0, x31:0, c3, c4, c5) :|: c5 = x33:0 - 1 && (c4 = x33:0 - 1 && c3 = x32:0 - 1) && x33:0 > 0

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(21) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[f398_0_minus_EQ(x, x1, x2, x3, x4)] = x3

The following rules are decreasing:
f398_0_minus_EQ(x30:0, x31:0, x32:0, x33:0, x33:0) -> f398_0_minus_EQ(x30:0, x31:0, c3, c4, c5) :|: c5 = x33:0 - 1 && (c4 = x33:0 - 1 && c3 = x32:0 - 1) && x33:0 > 0
The following rules are bounded:
f398_0_minus_EQ(x30:0, x31:0, x32:0, x33:0, x33:0) -> f398_0_minus_EQ(x30:0, x31:0, c3, c4, c5) :|: c5 = x33:0 - 1 && (c4 = x33:0 - 1 && c3 = x32:0 - 1) && x33:0 > 0

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(22)
YES