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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 260 ms]
(4) AND
    (5) IRSwT
        (6) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (7) IRSwT
        (8) TempFilterProof [SOUND, 53 ms]
        (9) IntTRS
        (10) RankingReductionPairProof [EQUIVALENT, 17 ms]
        (11) IntTRS
        (12) RankingReductionPairProof [EQUIVALENT, 0 ms]
        (13) YES
    (14) IRSwT
        (15) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (16) IRSwT
        (17) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (18) IRSwT
        (19) TempFilterProof [SOUND, 23 ms]
        (20) IntTRS
        (21) RankingReductionPairProof [EQUIVALENT, 12 ms]
        (22) YES


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(0)
Obligation:
Rules:
f1_0_main_Load(arg1, arg2, arg3) -> f33_0_rec_Cmp(arg1P, arg2P, arg3P) :|: arg2 = arg1P && -1 <= arg2 - 1 && 0 <= arg1 - 1
f33_0_rec_Cmp(x, x1, x2) -> f48_0_rec_GE(x3, x4, x5) :|: x = x5 && x = x4 && x = x3
f48_0_rec_GE(x6, x7, x8) -> f33_0_rec_Cmp(x9, x10, x11) :|: x6 - 1 = x9 && x7 = x8 && 99 <= x7 - 1 && 0 <= x6 - 1 && x6 - 1 <= x6 - 1
f48_0_rec_GE(x12, x13, x14) -> f77_0_descend_LE(x15, x16, x17) :|: x13 = x15 && x13 = x14 && x13 <= 99 && 0 <= x13 - 1
f48_0_rec_GE(x18, x19, x20) -> f77_0_descend_LE(x21, x22, x23) :|: x19 = x21 && x19 = x20 && 0 <= x19 - 1 && x19 <= 99
f48_0_rec_GE(x24, x25, x26) -> f48_0_rec_GE(x27, x28, x29) :|: x25 + 1 = x29 && x25 + 1 = x28 && x24 = x27 && x25 = x26 && 0 <= x25 - 1 && x25 <= 99
f77_0_descend_LE(x30, x31, x32) -> f77_0_descend_LE(x33, x34, x35) :|: x30 - 1 = x33 && x30 - 1 <= x30 - 1 && 0 <= x30 - 1
__init(x36, x37, x38) -> f1_0_main_Load(x39, x40, x41) :|: 0 <= 0
Start term: __init(arg1, arg2, arg3)

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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) -> f33_0_rec_Cmp(arg1P, arg2P, arg3P) :|: arg2 = arg1P && -1 <= arg2 - 1 && 0 <= arg1 - 1
f33_0_rec_Cmp(x, x1, x2) -> f48_0_rec_GE(x3, x4, x5) :|: x = x5 && x = x4 && x = x3
f48_0_rec_GE(x6, x7, x8) -> f33_0_rec_Cmp(x9, x10, x11) :|: x6 - 1 = x9 && x7 = x8 && 99 <= x7 - 1 && 0 <= x6 - 1 && x6 - 1 <= x6 - 1
f48_0_rec_GE(x12, x13, x14) -> f77_0_descend_LE(x15, x16, x17) :|: x13 = x15 && x13 = x14 && x13 <= 99 && 0 <= x13 - 1
f48_0_rec_GE(x18, x19, x20) -> f77_0_descend_LE(x21, x22, x23) :|: x19 = x21 && x19 = x20 && 0 <= x19 - 1 && x19 <= 99
f48_0_rec_GE(x24, x25, x26) -> f48_0_rec_GE(x27, x28, x29) :|: x25 + 1 = x29 && x25 + 1 = x28 && x24 = x27 && x25 = x26 && 0 <= x25 - 1 && x25 <= 99
f77_0_descend_LE(x30, x31, x32) -> f77_0_descend_LE(x33, x34, x35) :|: x30 - 1 = x33 && x30 - 1 <= x30 - 1 && 0 <= x30 - 1
__init(x36, x37, x38) -> f1_0_main_Load(x39, x40, x41) :|: 0 <= 0
Start term: __init(arg1, arg2, arg3)

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

(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) f1_0_main_Load(arg1, arg2, arg3) -> f33_0_rec_Cmp(arg1P, arg2P, arg3P) :|: arg2 = arg1P && -1 <= arg2 - 1 && 0 <= arg1 - 1
(2) f33_0_rec_Cmp(x, x1, x2) -> f48_0_rec_GE(x3, x4, x5) :|: x = x5 && x = x4 && x = x3
(3) f48_0_rec_GE(x6, x7, x8) -> f33_0_rec_Cmp(x9, x10, x11) :|: x6 - 1 = x9 && x7 = x8 && 99 <= x7 - 1 && 0 <= x6 - 1 && x6 - 1 <= x6 - 1
(4) f48_0_rec_GE(x12, x13, x14) -> f77_0_descend_LE(x15, x16, x17) :|: x13 = x15 && x13 = x14 && x13 <= 99 && 0 <= x13 - 1
(5) f48_0_rec_GE(x18, x19, x20) -> f77_0_descend_LE(x21, x22, x23) :|: x19 = x21 && x19 = x20 && 0 <= x19 - 1 && x19 <= 99
(6) f48_0_rec_GE(x24, x25, x26) -> f48_0_rec_GE(x27, x28, x29) :|: x25 + 1 = x29 && x25 + 1 = x28 && x24 = x27 && x25 = x26 && 0 <= x25 - 1 && x25 <= 99
(7) f77_0_descend_LE(x30, x31, x32) -> f77_0_descend_LE(x33, x34, x35) :|: x30 - 1 = x33 && x30 - 1 <= x30 - 1 && 0 <= x30 - 1
(8) __init(x36, x37, x38) -> f1_0_main_Load(x39, x40, x41) :|: 0 <= 0

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

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

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(5)
Obligation:

Termination digraph:
Nodes:
(1) f33_0_rec_Cmp(x, x1, x2) -> f48_0_rec_GE(x3, x4, x5) :|: x = x5 && x = x4 && x = x3
(2) f48_0_rec_GE(x6, x7, x8) -> f33_0_rec_Cmp(x9, x10, x11) :|: x6 - 1 = x9 && x7 = x8 && 99 <= x7 - 1 && 0 <= x6 - 1 && x6 - 1 <= x6 - 1
(3) f48_0_rec_GE(x24, x25, x26) -> f48_0_rec_GE(x27, x28, x29) :|: x25 + 1 = x29 && x25 + 1 = x28 && x24 = x27 && x25 = x26 && 0 <= x25 - 1 && x25 <= 99

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

This digraph is fully evaluated!

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

(7)
Obligation:
Rules:
f48_0_rec_GE(x6:0, x7:0, x7:0) -> f48_0_rec_GE(x6:0 - 1, x6:0 - 1, x6:0 - 1) :|: x6:0 > 0 && x7:0 > 99
f48_0_rec_GE(x24:0, x25:0, x25:0) -> f48_0_rec_GE(x24:0, x25:0 + 1, x25:0 + 1) :|: x25:0 < 100 && x25:0 > 0

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(8) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f48_0_rec_GE(VARIABLE, INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(9)
Obligation:
Rules:
f48_0_rec_GE(x6:0, x7:0, x7:0) -> f48_0_rec_GE(c, c1, c2) :|: c2 = x6:0 - 1 && (c1 = x6:0 - 1 && c = x6:0 - 1) && (x6:0 > 0 && x7:0 > 99)
f48_0_rec_GE(x24:0, x25:0, x25:0) -> f48_0_rec_GE(x24:0, c3, c4) :|: c4 = x25:0 + 1 && c3 = x25:0 + 1 && (x25:0 < 100 && x25:0 > 0)

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(10) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ f48_0_rec_GE ] = 2*f48_0_rec_GE_1

The following rules are decreasing:
f48_0_rec_GE(x6:0, x7:0, x7:0) -> f48_0_rec_GE(c, c1, c2) :|: c2 = x6:0 - 1 && (c1 = x6:0 - 1 && c = x6:0 - 1) && (x6:0 > 0 && x7:0 > 99)

The following rules are bounded:
f48_0_rec_GE(x6:0, x7:0, x7:0) -> f48_0_rec_GE(c, c1, c2) :|: c2 = x6:0 - 1 && (c1 = x6:0 - 1 && c = x6:0 - 1) && (x6:0 > 0 && x7:0 > 99)


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(11)
Obligation:
Rules:
f48_0_rec_GE(x24:0, x25:0, x25:0) -> f48_0_rec_GE(x24:0, c3, c4) :|: c4 = x25:0 + 1 && c3 = x25:0 + 1 && (x25:0 < 100 && x25:0 > 0)

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(12) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ f48_0_rec_GE ] = -1*f48_0_rec_GE_3

The following rules are decreasing:
f48_0_rec_GE(x24:0, x25:0, x25:0) -> f48_0_rec_GE(x24:0, c3, c4) :|: c4 = x25:0 + 1 && c3 = x25:0 + 1 && (x25:0 < 100 && x25:0 > 0)

The following rules are bounded:
f48_0_rec_GE(x24:0, x25:0, x25:0) -> f48_0_rec_GE(x24:0, c3, c4) :|: c4 = x25:0 + 1 && c3 = x25:0 + 1 && (x25:0 < 100 && x25:0 > 0)


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(13)
YES

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

(14)
Obligation:

Termination digraph:
Nodes:
(1) f77_0_descend_LE(x30, x31, x32) -> f77_0_descend_LE(x33, x34, x35) :|: x30 - 1 = x33 && x30 - 1 <= x30 - 1 && 0 <= x30 - 1

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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

(16)
Obligation:
Rules:
f77_0_descend_LE(x30:0, x31:0, x32:0) -> f77_0_descend_LE(x30:0 - 1, x34:0, x35:0) :|: x30:0 > 0

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

   f77_0_descend_LE(x1, x2, x3) -> f77_0_descend_LE(x1)

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(18)
Obligation:
Rules:
f77_0_descend_LE(x30:0) -> f77_0_descend_LE(x30:0 - 1) :|: x30:0 > 0

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(19) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f77_0_descend_LE(INTEGER)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(20)
Obligation:
Rules:
f77_0_descend_LE(x30:0) -> f77_0_descend_LE(c) :|: c = x30:0 - 1 && x30:0 > 0

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(21) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ f77_0_descend_LE ] = f77_0_descend_LE_1

The following rules are decreasing:
f77_0_descend_LE(x30:0) -> f77_0_descend_LE(c) :|: c = x30:0 - 1 && x30:0 > 0

The following rules are bounded:
f77_0_descend_LE(x30:0) -> f77_0_descend_LE(c) :|: c = x30:0 - 1 && x30:0 > 0


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(22)
YES