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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 334 ms]
(4) AND
    (5) IRSwT
        (6) IntTRSCompressionProof [EQUIVALENT, 10 ms]
        (7) IRSwT
        (8) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (9) IRSwT
        (10) TempFilterProof [SOUND, 36 ms]
        (11) IntTRS
        (12) PolynomialOrderProcessor [EQUIVALENT, 14 ms]
        (13) YES
    (14) IRSwT
        (15) IntTRSCompressionProof [EQUIVALENT, 6 ms]
        (16) IRSwT
        (17) TempFilterProof [SOUND, 26 ms]
        (18) IntTRS
        (19) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
        (20) IntTRS
        (21) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
        (22) YES


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(0)
Obligation:
Rules:
f1_0_main_Load(arg1, arg2, arg3, arg4) -> f248_0_mod_LT(arg1P, arg2P, arg3P, arg4P) :|: 0 <= arg1 - 1 && -1 <= arg1P - 1 && 1 <= arg2 - 1 && -1 <= arg2P - 1
f248_0_mod_LT(x, x1, x2, x3) -> f319_0_minus_EQ(x4, x5, x6, x7) :|: x1 = x7 && x1 = x6 && x = x5 && x1 = x4 && 0 <= x1 - 1 && x1 <= x
f319_0_minus_EQ(x8, x9, x10, x11) -> f248_0_mod_LT(x12, x13, x14, x15) :|: x8 = x13 && x9 = x12 && 0 = x11 && 0 = x10
f319_0_minus_EQ(x16, x17, x18, x19) -> f319_0_minus_EQ(x20, x21, x22, x23) :|: x18 + 1 = x23 && x18 + 1 = x22 && x17 + 1 = x21 && x16 = x20 && x18 = x19 && x18 <= 0 && x18 <= -1
f319_0_minus_EQ(x24, x25, x26, x27) -> f319_0_minus_EQ(x28, x29, x30, x31) :|: x26 + 1 = x31 && x26 + 1 = x30 && x25 + 1 = x29 && x24 = x28 && x26 = x27 && x26 <= 0 && 0 <= x26 - 1
f319_0_minus_EQ(x32, x33, x34, x35) -> f319_0_minus_EQ(x36, x37, x38, x39) :|: x34 - 1 = x39 && x34 - 1 = x38 && x33 - 1 = x37 && x32 = x36 && x34 = x35 && 0 <= x34 - 1
__init(x40, x41, x42, x43) -> f1_0_main_Load(x44, x45, x46, x47) :|: 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 %).
----------------------------------------

(2)
Obligation:
Rules:
f1_0_main_Load(arg1, arg2, arg3, arg4) -> f248_0_mod_LT(arg1P, arg2P, arg3P, arg4P) :|: 0 <= arg1 - 1 && -1 <= arg1P - 1 && 1 <= arg2 - 1 && -1 <= arg2P - 1
f248_0_mod_LT(x, x1, x2, x3) -> f319_0_minus_EQ(x4, x5, x6, x7) :|: x1 = x7 && x1 = x6 && x = x5 && x1 = x4 && 0 <= x1 - 1 && x1 <= x
f319_0_minus_EQ(x8, x9, x10, x11) -> f248_0_mod_LT(x12, x13, x14, x15) :|: x8 = x13 && x9 = x12 && 0 = x11 && 0 = x10
f319_0_minus_EQ(x16, x17, x18, x19) -> f319_0_minus_EQ(x20, x21, x22, x23) :|: x18 + 1 = x23 && x18 + 1 = x22 && x17 + 1 = x21 && x16 = x20 && x18 = x19 && x18 <= 0 && x18 <= -1
f319_0_minus_EQ(x24, x25, x26, x27) -> f319_0_minus_EQ(x28, x29, x30, x31) :|: x26 + 1 = x31 && x26 + 1 = x30 && x25 + 1 = x29 && x24 = x28 && x26 = x27 && x26 <= 0 && 0 <= x26 - 1
f319_0_minus_EQ(x32, x33, x34, x35) -> f319_0_minus_EQ(x36, x37, x38, x39) :|: x34 - 1 = x39 && x34 - 1 = x38 && x33 - 1 = x37 && x32 = x36 && x34 = x35 && 0 <= x34 - 1
__init(x40, x41, x42, x43) -> f1_0_main_Load(x44, x45, x46, x47) :|: 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) -> f248_0_mod_LT(arg1P, arg2P, arg3P, arg4P) :|: 0 <= arg1 - 1 && -1 <= arg1P - 1 && 1 <= arg2 - 1 && -1 <= arg2P - 1
(2) f248_0_mod_LT(x, x1, x2, x3) -> f319_0_minus_EQ(x4, x5, x6, x7) :|: x1 = x7 && x1 = x6 && x = x5 && x1 = x4 && 0 <= x1 - 1 && x1 <= x
(3) f319_0_minus_EQ(x8, x9, x10, x11) -> f248_0_mod_LT(x12, x13, x14, x15) :|: x8 = x13 && x9 = x12 && 0 = x11 && 0 = x10
(4) f319_0_minus_EQ(x16, x17, x18, x19) -> f319_0_minus_EQ(x20, x21, x22, x23) :|: x18 + 1 = x23 && x18 + 1 = x22 && x17 + 1 = x21 && x16 = x20 && x18 = x19 && x18 <= 0 && x18 <= -1
(5) f319_0_minus_EQ(x24, x25, x26, x27) -> f319_0_minus_EQ(x28, x29, x30, x31) :|: x26 + 1 = x31 && x26 + 1 = x30 && x25 + 1 = x29 && x24 = x28 && x26 = x27 && x26 <= 0 && 0 <= x26 - 1
(6) f319_0_minus_EQ(x32, x33, x34, x35) -> f319_0_minus_EQ(x36, x37, x38, x39) :|: x34 - 1 = x39 && x34 - 1 = x38 && x33 - 1 = x37 && x32 = x36 && x34 = x35 && 0 <= x34 - 1
(7) __init(x40, x41, x42, x43) -> f1_0_main_Load(x44, x45, x46, x47) :|: 0 <= 0

Arcs:
(1) -> (2)
(2) -> (6)
(3) -> (2)
(4) -> (3), (4)
(6) -> (3), (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) f319_0_minus_EQ(x16, x17, x18, x19) -> f319_0_minus_EQ(x20, x21, x22, x23) :|: x18 + 1 = x23 && x18 + 1 = x22 && x17 + 1 = x21 && x16 = x20 && x18 = x19 && x18 <= 0 && x18 <= -1

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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

(7)
Obligation:
Rules:
f319_0_minus_EQ(x16:0, x17:0, x18:0, x18:0) -> f319_0_minus_EQ(x16:0, x17:0 + 1, x18:0 + 1, x18:0 + 1) :|: x18:0 < 0 && x18:0 < 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:

   f319_0_minus_EQ(x1, x2, x3, x4) -> f319_0_minus_EQ(x3, x4)

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(9)
Obligation:
Rules:
f319_0_minus_EQ(x18:0, x18:0) -> f319_0_minus_EQ(x18:0 + 1, x18:0 + 1) :|: x18:0 < 0 && x18:0 < 1

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(10) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f319_0_minus_EQ(INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
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(11)
Obligation:
Rules:
f319_0_minus_EQ(x18:0, x18:0) -> f319_0_minus_EQ(c, c1) :|: c1 = x18:0 + 1 && c = x18:0 + 1 && (x18:0 < 0 && x18:0 < 1)

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(12) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[f319_0_minus_EQ(x, x1)] = -x1

The following rules are decreasing:
f319_0_minus_EQ(x18:0, x18:0) -> f319_0_minus_EQ(c, c1) :|: c1 = x18:0 + 1 && c = x18:0 + 1 && (x18:0 < 0 && x18:0 < 1)
The following rules are bounded:
f319_0_minus_EQ(x18:0, x18:0) -> f319_0_minus_EQ(c, c1) :|: c1 = x18:0 + 1 && c = x18:0 + 1 && (x18:0 < 0 && x18:0 < 1)

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(13)
YES

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(14)
Obligation:

Termination digraph:
Nodes:
(1) f248_0_mod_LT(x, x1, x2, x3) -> f319_0_minus_EQ(x4, x5, x6, x7) :|: x1 = x7 && x1 = x6 && x = x5 && x1 = x4 && 0 <= x1 - 1 && x1 <= x
(2) f319_0_minus_EQ(x8, x9, x10, x11) -> f248_0_mod_LT(x12, x13, x14, x15) :|: x8 = x13 && x9 = x12 && 0 = x11 && 0 = x10
(3) f319_0_minus_EQ(x32, x33, x34, x35) -> f319_0_minus_EQ(x36, x37, x38, x39) :|: x34 - 1 = x39 && x34 - 1 = x38 && x33 - 1 = x37 && x32 = x36 && x34 = x35 && 0 <= x34 - 1

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

This digraph is fully evaluated!

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(15) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(16)
Obligation:
Rules:
f319_0_minus_EQ(x32:0, x33:0, x34:0, x34:0) -> f319_0_minus_EQ(x32:0, x33:0 - 1, x34:0 - 1, x34:0 - 1) :|: x34:0 > 0
f319_0_minus_EQ(x13:0, x12:0, cons_0, cons_01) -> f319_0_minus_EQ(x13:0, x12:0, x13:0, x13:0) :|: x13:0 <= x12:0 && x13:0 > 0 && cons_0 = 0 && cons_01 = 0

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(17) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f319_0_minus_EQ(VARIABLE, VARIABLE, INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(18)
Obligation:
Rules:
f319_0_minus_EQ(x32:0, x33:0, x34:0, x34:0) -> f319_0_minus_EQ(x32:0, c, c1, c2) :|: c2 = x34:0 - 1 && (c1 = x34:0 - 1 && c = x33:0 - 1) && x34:0 > 0
f319_0_minus_EQ(x13:0, x12:0, c3, c4) -> f319_0_minus_EQ(x13:0, x12:0, x13:0, x13:0) :|: c4 = 0 && c3 = 0 && (x13:0 <= x12:0 && x13:0 > 0 && cons_0 = 0 && cons_01 = 0)

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(19) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[f319_0_minus_EQ(x, x1, x2, x3)] = x + x1 - x2

The following rules are decreasing:
f319_0_minus_EQ(x13:0, x12:0, c3, c4) -> f319_0_minus_EQ(x13:0, x12:0, x13:0, x13:0) :|: c4 = 0 && c3 = 0 && (x13:0 <= x12:0 && x13:0 > 0 && cons_0 = 0 && cons_01 = 0)
The following rules are bounded:
f319_0_minus_EQ(x13:0, x12:0, c3, c4) -> f319_0_minus_EQ(x13:0, x12:0, x13:0, x13:0) :|: c4 = 0 && c3 = 0 && (x13:0 <= x12:0 && x13:0 > 0 && cons_0 = 0 && cons_01 = 0)

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(20)
Obligation:
Rules:
f319_0_minus_EQ(x32:0, x33:0, x34:0, x34:0) -> f319_0_minus_EQ(x32:0, c, c1, c2) :|: c2 = x34:0 - 1 && (c1 = x34:0 - 1 && c = x33:0 - 1) && x34:0 > 0

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(21) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[f319_0_minus_EQ(x, x1, x2, x3)] = x2

The following rules are decreasing:
f319_0_minus_EQ(x32:0, x33:0, x34:0, x34:0) -> f319_0_minus_EQ(x32:0, c, c1, c2) :|: c2 = x34:0 - 1 && (c1 = x34:0 - 1 && c = x33:0 - 1) && x34:0 > 0
The following rules are bounded:
f319_0_minus_EQ(x32:0, x33:0, x34:0, x34:0) -> f319_0_minus_EQ(x32:0, c, c1, c2) :|: c2 = x34:0 - 1 && (c1 = x34:0 - 1 && c = x33:0 - 1) && x34:0 > 0

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(22)
YES