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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 466 ms]
(4) AND
    (5) IRSwT
        (6) IntTRSCompressionProof [EQUIVALENT, 6 ms]
        (7) IRSwT
        (8) TempFilterProof [SOUND, 21 ms]
        (9) IntTRS
        (10) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
        (11) YES
    (12) IRSwT
        (13) IntTRSCompressionProof [EQUIVALENT, 15 ms]
        (14) IRSwT
        (15) IRSwTChainingProof [EQUIVALENT, 0 ms]
        (16) IRSwT
        (17) IRSwTTerminationDigraphProof [EQUIVALENT, 4 ms]
        (18) IRSwT
        (19) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (20) IRSwT
        (21) TempFilterProof [SOUND, 13 ms]
        (22) IntTRS
        (23) RankingReductionPairProof [EQUIVALENT, 0 ms]
        (24) YES


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(0)
Obligation:
Rules:
f1_0_main_Load(arg1, arg2, arg3, arg4, arg5) -> f293_0_loop_LT(arg1P, arg2P, arg3P, arg4P, arg5P) :|: 0 = arg5P && 0 = arg4P && 0 = arg3P && 0 = arg2P && 0 = arg1P && 0 = arg2 && 0 <= arg1 - 1
f1_0_main_Load(x, x1, x2, x3, x4) -> f293_0_loop_LT(x5, x6, x7, x8, x9) :|: 1 = x9 && 1 = x8 && 0 = x7 && 0 = x6 && 1 = x1 && -1 <= x5 - 1 && 0 <= x - 1
f1_0_main_Load(x11, x12, x13, x14, x15) -> f293_0_loop_LT(x16, x17, x18, x20, x21) :|: 2 = x21 && 2 = x20 && 0 = x17 && 2 = x12 && 0 <= x11 - 1 && -1 <= x16 - 1 && -1 <= x18 - 1
f1_0_main_Load(x22, x23, x24, x25, x26) -> f293_0_loop_LT(x27, x28, x29, x30, x31) :|: -1 <= x27 - 1 && 2 <= x23 - 1 && -1 <= x32 - 1 && -1 <= x28 - 1 && 0 <= x22 - 1 && x32 - x28 = x29 && x23 = x30 && 3 = x31
f293_0_loop_LT(x33, x34, x35, x36, x37) -> f293_0_loop_LT(x38, x39, x40, x41, x42) :|: x37 = x42 && x36 = x41 && 10 - (x33 + 1) = x40 && x33 + 1 = x39 && x33 + 1 = x38 && x33 = x34 && -1 <= x36 - 1 && x36 <= x37 && 0 <= x35 - 1 && -1 <= x33 - 1 && x33 <= x33 + 1 - 1
f293_0_loop_LT(x43, x44, x45, x46, x47) -> f293_0_loop_LT(x48, x49, x50, x51, x52) :|: 0 <= x45 - 1 && 1 <= x43 + 1 + x53 && -1 <= x46 - 1 && -1 <= x47 - 1 && x47 <= x46 - 1 && -1 <= x43 - 1 && -1 <= x53 - 1 && x43 = x44 && x43 + 1 + x53 = x48 && x43 + 1 + x53 = x49 && 10 - (x43 + 1 + x53) = x50 && x46 = x51 && x47 + 1 = x52
__init(x54, x55, x56, x57, x58) -> f1_0_main_Load(x59, x60, x61, x62, x63) :|: 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 %).
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(2)
Obligation:
Rules:
f1_0_main_Load(arg1, arg2, arg3, arg4, arg5) -> f293_0_loop_LT(arg1P, arg2P, arg3P, arg4P, arg5P) :|: 0 = arg5P && 0 = arg4P && 0 = arg3P && 0 = arg2P && 0 = arg1P && 0 = arg2 && 0 <= arg1 - 1
f1_0_main_Load(x, x1, x2, x3, x4) -> f293_0_loop_LT(x5, x6, x7, x8, x9) :|: 1 = x9 && 1 = x8 && 0 = x7 && 0 = x6 && 1 = x1 && -1 <= x5 - 1 && 0 <= x - 1
f1_0_main_Load(x11, x12, x13, x14, x15) -> f293_0_loop_LT(x16, x17, x18, x20, x21) :|: 2 = x21 && 2 = x20 && 0 = x17 && 2 = x12 && 0 <= x11 - 1 && -1 <= x16 - 1 && -1 <= x18 - 1
f1_0_main_Load(x22, x23, x24, x25, x26) -> f293_0_loop_LT(x27, x28, x29, x30, x31) :|: -1 <= x27 - 1 && 2 <= x23 - 1 && -1 <= x32 - 1 && -1 <= x28 - 1 && 0 <= x22 - 1 && x32 - x28 = x29 && x23 = x30 && 3 = x31
f293_0_loop_LT(x33, x34, x35, x36, x37) -> f293_0_loop_LT(x38, x39, x40, x41, x42) :|: x37 = x42 && x36 = x41 && 10 - (x33 + 1) = x40 && x33 + 1 = x39 && x33 + 1 = x38 && x33 = x34 && -1 <= x36 - 1 && x36 <= x37 && 0 <= x35 - 1 && -1 <= x33 - 1 && x33 <= x33 + 1 - 1
f293_0_loop_LT(x43, x44, x45, x46, x47) -> f293_0_loop_LT(x48, x49, x50, x51, x52) :|: 0 <= x45 - 1 && 1 <= x43 + 1 + x53 && -1 <= x46 - 1 && -1 <= x47 - 1 && x47 <= x46 - 1 && -1 <= x43 - 1 && -1 <= x53 - 1 && x43 = x44 && x43 + 1 + x53 = x48 && x43 + 1 + x53 = x49 && 10 - (x43 + 1 + x53) = x50 && x46 = x51 && x47 + 1 = x52
__init(x54, x55, x56, x57, x58) -> f1_0_main_Load(x59, x60, x61, x62, x63) :|: 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) -> f293_0_loop_LT(arg1P, arg2P, arg3P, arg4P, arg5P) :|: 0 = arg5P && 0 = arg4P && 0 = arg3P && 0 = arg2P && 0 = arg1P && 0 = arg2 && 0 <= arg1 - 1
(2) f1_0_main_Load(x, x1, x2, x3, x4) -> f293_0_loop_LT(x5, x6, x7, x8, x9) :|: 1 = x9 && 1 = x8 && 0 = x7 && 0 = x6 && 1 = x1 && -1 <= x5 - 1 && 0 <= x - 1
(3) f1_0_main_Load(x11, x12, x13, x14, x15) -> f293_0_loop_LT(x16, x17, x18, x20, x21) :|: 2 = x21 && 2 = x20 && 0 = x17 && 2 = x12 && 0 <= x11 - 1 && -1 <= x16 - 1 && -1 <= x18 - 1
(4) f1_0_main_Load(x22, x23, x24, x25, x26) -> f293_0_loop_LT(x27, x28, x29, x30, x31) :|: -1 <= x27 - 1 && 2 <= x23 - 1 && -1 <= x32 - 1 && -1 <= x28 - 1 && 0 <= x22 - 1 && x32 - x28 = x29 && x23 = x30 && 3 = x31
(5) f293_0_loop_LT(x33, x34, x35, x36, x37) -> f293_0_loop_LT(x38, x39, x40, x41, x42) :|: x37 = x42 && x36 = x41 && 10 - (x33 + 1) = x40 && x33 + 1 = x39 && x33 + 1 = x38 && x33 = x34 && -1 <= x36 - 1 && x36 <= x37 && 0 <= x35 - 1 && -1 <= x33 - 1 && x33 <= x33 + 1 - 1
(6) f293_0_loop_LT(x43, x44, x45, x46, x47) -> f293_0_loop_LT(x48, x49, x50, x51, x52) :|: 0 <= x45 - 1 && 1 <= x43 + 1 + x53 && -1 <= x46 - 1 && -1 <= x47 - 1 && x47 <= x46 - 1 && -1 <= x43 - 1 && -1 <= x53 - 1 && x43 = x44 && x43 + 1 + x53 = x48 && x43 + 1 + x53 = x49 && 10 - (x43 + 1 + x53) = x50 && x46 = x51 && x47 + 1 = x52
(7) __init(x54, x55, x56, x57, x58) -> f1_0_main_Load(x59, x60, x61, x62, x63) :|: 0 <= 0

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

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

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(5)
Obligation:

Termination digraph:
Nodes:
(1) f293_0_loop_LT(x43, x44, x45, x46, x47) -> f293_0_loop_LT(x48, x49, x50, x51, x52) :|: 0 <= x45 - 1 && 1 <= x43 + 1 + x53 && -1 <= x46 - 1 && -1 <= x47 - 1 && x47 <= x46 - 1 && -1 <= x43 - 1 && -1 <= x53 - 1 && x43 = x44 && x43 + 1 + x53 = x48 && x43 + 1 + x53 = x49 && 10 - (x43 + 1 + x53) = x50 && x46 = x51 && x47 + 1 = x52

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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(6) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(7)
Obligation:
Rules:
f293_0_loop_LT(x43:0, x43:0, x45:0, x46:0, x47:0) -> f293_0_loop_LT(x43:0 + 1 + x53:0, x43:0 + 1 + x53:0, 10 - (x43:0 + 1 + x53:0), x46:0, x47:0 + 1) :|: x43:0 > -1 && x53:0 > -1 && x47:0 <= x46:0 - 1 && x47:0 > -1 && x46:0 > -1 && x43:0 + 1 + x53:0 >= 1 && x45:0 > 0

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(8) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f293_0_loop_LT(INTEGER, INTEGER, INTEGER, INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(9)
Obligation:
Rules:
f293_0_loop_LT(x43:0, x43:0, x45:0, x46:0, x47:0) -> f293_0_loop_LT(c, c1, c2, x46:0, c3) :|: c3 = x47:0 + 1 && (c2 = 10 - (x43:0 + 1 + x53:0) && (c1 = x43:0 + 1 + x53:0 && c = x43:0 + 1 + x53:0)) && (x43:0 > -1 && x53:0 > -1 && x47:0 <= x46:0 - 1 && x47:0 > -1 && x46:0 > -1 && x43:0 + 1 + x53:0 >= 1 && x45:0 > 0)

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(10) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[f293_0_loop_LT(x, x1, x2, x3, x4)] = x3 - x4

The following rules are decreasing:
f293_0_loop_LT(x43:0, x43:0, x45:0, x46:0, x47:0) -> f293_0_loop_LT(c, c1, c2, x46:0, c3) :|: c3 = x47:0 + 1 && (c2 = 10 - (x43:0 + 1 + x53:0) && (c1 = x43:0 + 1 + x53:0 && c = x43:0 + 1 + x53:0)) && (x43:0 > -1 && x53:0 > -1 && x47:0 <= x46:0 - 1 && x47:0 > -1 && x46:0 > -1 && x43:0 + 1 + x53:0 >= 1 && x45:0 > 0)
The following rules are bounded:
f293_0_loop_LT(x43:0, x43:0, x45:0, x46:0, x47:0) -> f293_0_loop_LT(c, c1, c2, x46:0, c3) :|: c3 = x47:0 + 1 && (c2 = 10 - (x43:0 + 1 + x53:0) && (c1 = x43:0 + 1 + x53:0 && c = x43:0 + 1 + x53:0)) && (x43:0 > -1 && x53:0 > -1 && x47:0 <= x46:0 - 1 && x47:0 > -1 && x46:0 > -1 && x43:0 + 1 + x53:0 >= 1 && x45:0 > 0)

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(11)
YES

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(12)
Obligation:

Termination digraph:
Nodes:
(1) f293_0_loop_LT(x33, x34, x35, x36, x37) -> f293_0_loop_LT(x38, x39, x40, x41, x42) :|: x37 = x42 && x36 = x41 && 10 - (x33 + 1) = x40 && x33 + 1 = x39 && x33 + 1 = x38 && x33 = x34 && -1 <= x36 - 1 && x36 <= x37 && 0 <= x35 - 1 && -1 <= x33 - 1 && x33 <= x33 + 1 - 1

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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

(14)
Obligation:
Rules:
f293_0_loop_LT(x33:0, x33:0, x35:0, x36:0, x37:0) -> f293_0_loop_LT(x33:0 + 1, x33:0 + 1, 10 - (x33:0 + 1), x36:0, x37:0) :|: x35:0 > 0 && x33:0 > -1 && x36:0 > -1 && x37:0 >= x36:0

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(15) IRSwTChainingProof (EQUIVALENT)
Chaining!
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(16)
Obligation:
Rules:
f293_0_loop_LT(x, x, x1, x2, x3) -> f293_0_loop_LT(x + 2, x + 2, 8 + -1 * x, x2, x3) :|: TRUE && x1 >= 1 && x >= 0 && x2 >= 0 && x3 + -1 * x2 >= 0 && -1 * x >= -8

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(17) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) f293_0_loop_LT(x, x, x1, x2, x3) -> f293_0_loop_LT(x + 2, x + 2, 8 + -1 * x, x2, x3) :|: TRUE && x1 >= 1 && x >= 0 && x2 >= 0 && x3 + -1 * x2 >= 0 && -1 * x >= -8

Arcs:
(1) -> (1)

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

(18)
Obligation:

Termination digraph:
Nodes:
(1) f293_0_loop_LT(x, x, x1, x2, x3) -> f293_0_loop_LT(x + 2, x + 2, 8 + -1 * x, x2, x3) :|: TRUE && x1 >= 1 && x >= 0 && x2 >= 0 && x3 + -1 * x2 >= 0 && -1 * x >= -8

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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

(20)
Obligation:
Rules:
f293_0_loop_LT(x:0, x:0, x1:0, x2:0, x3:0) -> f293_0_loop_LT(x:0 + 2, x:0 + 2, 8 + -1 * x:0, x2:0, x3:0) :|: x3:0 + -1 * x2:0 >= 0 && -8 <= -1 * x:0 && x2:0 > -1 && x1:0 > 0 && x:0 > -1

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(21) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f293_0_loop_LT(INTEGER, INTEGER, INTEGER, INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(22)
Obligation:
Rules:
f293_0_loop_LT(x:0, x:0, x1:0, x2:0, x3:0) -> f293_0_loop_LT(c, c1, c2, x2:0, x3:0) :|: c2 = 8 + -1 * x:0 && (c1 = x:0 + 2 && c = x:0 + 2) && (x3:0 + -1 * x2:0 >= 0 && -8 <= -1 * x:0 && x2:0 > -1 && x1:0 > 0 && x:0 > -1)

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(23) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ f293_0_loop_LT ] = -1/2*f293_0_loop_LT_2

The following rules are decreasing:
f293_0_loop_LT(x:0, x:0, x1:0, x2:0, x3:0) -> f293_0_loop_LT(c, c1, c2, x2:0, x3:0) :|: c2 = 8 + -1 * x:0 && (c1 = x:0 + 2 && c = x:0 + 2) && (x3:0 + -1 * x2:0 >= 0 && -8 <= -1 * x:0 && x2:0 > -1 && x1:0 > 0 && x:0 > -1)

The following rules are bounded:
f293_0_loop_LT(x:0, x:0, x1:0, x2:0, x3:0) -> f293_0_loop_LT(c, c1, c2, x2:0, x3:0) :|: c2 = 8 + -1 * x:0 && (c1 = x:0 + 2 && c = x:0 + 2) && (x3:0 + -1 * x2:0 >= 0 && -8 <= -1 * x:0 && x2:0 > -1 && x1:0 > 0 && x:0 > -1)


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(24)
YES