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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 361 ms]
(4) AND
    (5) IRSwT
        (6) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (7) IRSwT
        (8) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (9) IRSwT
        (10) TempFilterProof [SOUND, 12 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) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
        (22) YES
    (23) IRSwT
        (24) IntTRSCompressionProof [EQUIVALENT, 6 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, arg4) -> f80_0_create_LE(arg1P, arg2P, arg3P, arg4P) :|: arg2 = arg4P && 0 = arg3P && arg2 - 1 = arg2P && 0 <= arg1P - 1 && 0 <= arg1 - 1 && -1 <= arg2 - 1 && arg1P <= arg1
f80_0_create_LE(x, x1, x2, x3) -> f108_0_main_ArrayAccess(x4, x5, x6, x7) :|: x3 = x6 && 0 <= x4 - 1 && 0 <= x - 1 && x4 <= x && x2 <= x5 - 1 && 0 <= x2 - 1 && x1 <= 0
f80_0_create_LE(x8, x9, x10, x11) -> f108_0_main_ArrayAccess(x12, x13, x14, x15) :|: x11 = x14 && 1 = x13 && 0 <= x12 - 1 && 0 <= x8 - 1 && x9 <= 0 && x12 <= x8
f80_0_create_LE(x16, x17, x18, x19) -> f80_0_create_LE(x20, x21, x22, x24) :|: x19 = x24 && x17 - 1 = x21 && 0 <= x20 - 1 && 0 <= x16 - 1 && 0 <= x17 - 1 && x20 <= x16
f108_0_main_ArrayAccess(x25, x26, x28, x29) -> f147_0_get_LE(x31, x32, x33, x34) :|: 0 <= x28 - 1 && -1 <= x35 - 1 && 0 <= x25 - 1 && x35 - 1 = x31 && x26 = x32
f147_0_get_LE(x36, x37, x38, x39) -> f147_0_get_LE(x40, x41, x42, x43) :|: -1 <= x37 - 1 && 0 <= x44 - 1 && x44 <= x37 - 1 && 0 <= x36 - 1 && x44 <= x41 - 1 && x36 - 1 = x40
f147_0_get_LE(x45, x46, x47, x48) -> f147_0_get_LE(x49, x50, x51, x52) :|: 0 <= x45 - 1 && x53 <= x46 - 1 && -1 <= x46 - 1 && x45 - 1 = x49 && 1 = x50
__init(x54, x55, x56, x57) -> f1_0_main_Load(x58, x59, x60, x61) :|: 0 <= 0
Start term: __init(arg1, arg2, arg3, arg4)

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

(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) -> f80_0_create_LE(arg1P, arg2P, arg3P, arg4P) :|: arg2 = arg4P && 0 = arg3P && arg2 - 1 = arg2P && 0 <= arg1P - 1 && 0 <= arg1 - 1 && -1 <= arg2 - 1 && arg1P <= arg1
f80_0_create_LE(x, x1, x2, x3) -> f108_0_main_ArrayAccess(x4, x5, x6, x7) :|: x3 = x6 && 0 <= x4 - 1 && 0 <= x - 1 && x4 <= x && x2 <= x5 - 1 && 0 <= x2 - 1 && x1 <= 0
f80_0_create_LE(x8, x9, x10, x11) -> f108_0_main_ArrayAccess(x12, x13, x14, x15) :|: x11 = x14 && 1 = x13 && 0 <= x12 - 1 && 0 <= x8 - 1 && x9 <= 0 && x12 <= x8
f80_0_create_LE(x16, x17, x18, x19) -> f80_0_create_LE(x20, x21, x22, x24) :|: x19 = x24 && x17 - 1 = x21 && 0 <= x20 - 1 && 0 <= x16 - 1 && 0 <= x17 - 1 && x20 <= x16
f108_0_main_ArrayAccess(x25, x26, x28, x29) -> f147_0_get_LE(x31, x32, x33, x34) :|: 0 <= x28 - 1 && -1 <= x35 - 1 && 0 <= x25 - 1 && x35 - 1 = x31 && x26 = x32
f147_0_get_LE(x36, x37, x38, x39) -> f147_0_get_LE(x40, x41, x42, x43) :|: -1 <= x37 - 1 && 0 <= x44 - 1 && x44 <= x37 - 1 && 0 <= x36 - 1 && x44 <= x41 - 1 && x36 - 1 = x40
f147_0_get_LE(x45, x46, x47, x48) -> f147_0_get_LE(x49, x50, x51, x52) :|: 0 <= x45 - 1 && x53 <= x46 - 1 && -1 <= x46 - 1 && x45 - 1 = x49 && 1 = x50
__init(x54, x55, x56, x57) -> f1_0_main_Load(x58, x59, x60, x61) :|: 0 <= 0
Start term: __init(arg1, arg2, arg3, arg4)

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

(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) f1_0_main_Load(arg1, arg2, arg3, arg4) -> f80_0_create_LE(arg1P, arg2P, arg3P, arg4P) :|: arg2 = arg4P && 0 = arg3P && arg2 - 1 = arg2P && 0 <= arg1P - 1 && 0 <= arg1 - 1 && -1 <= arg2 - 1 && arg1P <= arg1
(2) f80_0_create_LE(x, x1, x2, x3) -> f108_0_main_ArrayAccess(x4, x5, x6, x7) :|: x3 = x6 && 0 <= x4 - 1 && 0 <= x - 1 && x4 <= x && x2 <= x5 - 1 && 0 <= x2 - 1 && x1 <= 0
(3) f80_0_create_LE(x8, x9, x10, x11) -> f108_0_main_ArrayAccess(x12, x13, x14, x15) :|: x11 = x14 && 1 = x13 && 0 <= x12 - 1 && 0 <= x8 - 1 && x9 <= 0 && x12 <= x8
(4) f80_0_create_LE(x16, x17, x18, x19) -> f80_0_create_LE(x20, x21, x22, x24) :|: x19 = x24 && x17 - 1 = x21 && 0 <= x20 - 1 && 0 <= x16 - 1 && 0 <= x17 - 1 && x20 <= x16
(5) f108_0_main_ArrayAccess(x25, x26, x28, x29) -> f147_0_get_LE(x31, x32, x33, x34) :|: 0 <= x28 - 1 && -1 <= x35 - 1 && 0 <= x25 - 1 && x35 - 1 = x31 && x26 = x32
(6) f147_0_get_LE(x36, x37, x38, x39) -> f147_0_get_LE(x40, x41, x42, x43) :|: -1 <= x37 - 1 && 0 <= x44 - 1 && x44 <= x37 - 1 && 0 <= x36 - 1 && x44 <= x41 - 1 && x36 - 1 = x40
(7) f147_0_get_LE(x45, x46, x47, x48) -> f147_0_get_LE(x49, x50, x51, x52) :|: 0 <= x45 - 1 && x53 <= x46 - 1 && -1 <= x46 - 1 && x45 - 1 = x49 && 1 = x50
(8) __init(x54, x55, x56, x57) -> f1_0_main_Load(x58, x59, x60, x61) :|: 0 <= 0

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

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

(4)
Complex Obligation (AND)

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

(5)
Obligation:

Termination digraph:
Nodes:
(1) f80_0_create_LE(x16, x17, x18, x19) -> f80_0_create_LE(x20, x21, x22, x24) :|: x19 = x24 && x17 - 1 = x21 && 0 <= x20 - 1 && 0 <= x16 - 1 && 0 <= x17 - 1 && x20 <= x16

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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

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

(7)
Obligation:
Rules:
f80_0_create_LE(x16:0, x17:0, x18:0, x19:0) -> f80_0_create_LE(x20:0, x17:0 - 1, x22:0, x19:0) :|: x17:0 > 0 && x20:0 <= x16:0 && x20:0 > 0 && x16:0 > 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:

   f80_0_create_LE(x1, x2, x3, x4) -> f80_0_create_LE(x1, x2)

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

(9)
Obligation:
Rules:
f80_0_create_LE(x16:0, x17:0) -> f80_0_create_LE(x20:0, x17:0 - 1) :|: x17:0 > 0 && x20:0 <= x16:0 && x20:0 > 0 && x16:0 > 0

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(10) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f80_0_create_LE(INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(11)
Obligation:
Rules:
f80_0_create_LE(x16:0, x17:0) -> f80_0_create_LE(x20:0, c) :|: c = x17:0 - 1 && (x17:0 > 0 && x20:0 <= x16:0 && x20:0 > 0 && x16:0 > 0)

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(12) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[f80_0_create_LE(x, x1)] = x1

The following rules are decreasing:
f80_0_create_LE(x16:0, x17:0) -> f80_0_create_LE(x20:0, c) :|: c = x17:0 - 1 && (x17:0 > 0 && x20:0 <= x16:0 && x20:0 > 0 && x16:0 > 0)
The following rules are bounded:
f80_0_create_LE(x16:0, x17:0) -> f80_0_create_LE(x20:0, c) :|: c = x17:0 - 1 && (x17:0 > 0 && x20:0 <= x16:0 && x20:0 > 0 && x16:0 > 0)

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(13)
YES

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

(14)
Obligation:

Termination digraph:
Nodes:
(1) f147_0_get_LE(x36, x37, x38, x39) -> f147_0_get_LE(x40, x41, x42, x43) :|: -1 <= x37 - 1 && 0 <= x44 - 1 && x44 <= x37 - 1 && 0 <= x36 - 1 && x44 <= x41 - 1 && x36 - 1 = x40

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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

(16)
Obligation:
Rules:
f147_0_get_LE(x36:0, x37:0, x38:0, x39:0) -> f147_0_get_LE(x36:0 - 1, x41:0, x42:0, x43:0) :|: x36:0 > 0 && x44:0 <= x41:0 - 1 && x44:0 <= x37:0 - 1 && x44:0 > 0 && x37:0 > -1

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

   f147_0_get_LE(x1, x2, x3, x4) -> f147_0_get_LE(x1, x2)

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

(18)
Obligation:
Rules:
f147_0_get_LE(x36:0, x37:0) -> f147_0_get_LE(x36:0 - 1, x41:0) :|: x36:0 > 0 && x44:0 <= x41:0 - 1 && x44:0 <= x37:0 - 1 && x44:0 > 0 && x37:0 > -1

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

(19) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f147_0_get_LE(INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(20)
Obligation:
Rules:
f147_0_get_LE(x36:0, x37:0) -> f147_0_get_LE(c, x41:0) :|: c = x36:0 - 1 && (x36:0 > 0 && x44:0 <= x41:0 - 1 && x44:0 <= x37:0 - 1 && x44:0 > 0 && x37:0 > -1)

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(21) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[f147_0_get_LE(x, x1)] = x

The following rules are decreasing:
f147_0_get_LE(x36:0, x37:0) -> f147_0_get_LE(c, x41:0) :|: c = x36:0 - 1 && (x36:0 > 0 && x44:0 <= x41:0 - 1 && x44:0 <= x37:0 - 1 && x44:0 > 0 && x37:0 > -1)
The following rules are bounded:
f147_0_get_LE(x36:0, x37:0) -> f147_0_get_LE(c, x41:0) :|: c = x36:0 - 1 && (x36:0 > 0 && x44:0 <= x41:0 - 1 && x44:0 <= x37:0 - 1 && x44:0 > 0 && x37:0 > -1)

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

(22)
YES

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

(23)
Obligation:

Termination digraph:
Nodes:
(1) f147_0_get_LE(x45, x46, x47, x48) -> f147_0_get_LE(x49, x50, x51, x52) :|: 0 <= x45 - 1 && x53 <= x46 - 1 && -1 <= x46 - 1 && x45 - 1 = x49 && 1 = x50

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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

(25)
Obligation:
Rules:
f147_0_get_LE(x45:0, x46:0, x47:0, x48:0) -> f147_0_get_LE(x45:0 - 1, 1, x51:0, x52:0) :|: x45:0 > 0 && x53:0 <= x46:0 - 1 && x46:0 > -1

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

   f147_0_get_LE(x1, x2, x3, x4) -> f147_0_get_LE(x1, x2)

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

(27)
Obligation:
Rules:
f147_0_get_LE(x45:0, x46:0) -> f147_0_get_LE(x45:0 - 1, 1) :|: x45:0 > 0 && x53:0 <= x46:0 - 1 && x46:0 > -1

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

(28) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f147_0_get_LE(INTEGER, VARIABLE)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(29)
Obligation:
Rules:
f147_0_get_LE(x45:0, x46:0) -> f147_0_get_LE(c, c1) :|: c1 = 1 && c = x45:0 - 1 && (x45:0 > 0 && x53:0 <= x46:0 - 1 && x46:0 > -1)

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

(30) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[f147_0_get_LE(x, x1)] = x

The following rules are decreasing:
f147_0_get_LE(x45:0, x46:0) -> f147_0_get_LE(c, c1) :|: c1 = 1 && c = x45:0 - 1 && (x45:0 > 0 && x53:0 <= x46:0 - 1 && x46:0 > -1)
The following rules are bounded:
f147_0_get_LE(x45:0, x46:0) -> f147_0_get_LE(c, c1) :|: c1 = 1 && c = x45:0 - 1 && (x45:0 > 0 && x53:0 <= x46:0 - 1 && x46:0 > -1)

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(31)
YES