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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 267 ms]
(4) AND
    (5) IRSwT
        (6) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (7) IRSwT
        (8) TempFilterProof [SOUND, 30 ms]
        (9) IntTRS
        (10) RankingReductionPairProof [EQUIVALENT, 0 ms]
        (11) YES
    (12) IRSwT
        (13) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (14) IRSwT
        (15) TempFilterProof [SOUND, 23 ms]
        (16) IntTRS
        (17) RankingReductionPairProof [EQUIVALENT, 0 ms]
        (18) YES
    (19) IRSwT
        (20) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (21) IRSwT
        (22) TempFilterProof [SOUND, 20 ms]
        (23) IntTRS
        (24) RankingReductionPairProof [EQUIVALENT, 0 ms]
        (25) YES


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(0)
Obligation:
Rules:
f1_0_main_ConstantStackPush(arg1, arg2, arg3) -> f108_0_add_GT(arg1P, arg2P, arg3P) :|: 0 = arg3P && 0 = arg2P && 0 = arg1P
f108_0_add_GT(x, x1, x2) -> f108_0_add_GT(x3, x4, x5) :|: x1 + 1 = x5 && x1 + 1 = x4 && x + x1 = x3 && x1 = x2 && x1 <= 1000 && -1 <= x - 1 && -1 <= x1 - 1
f108_0_add_GT(x6, x7, x8) -> f208_0_add_GT(x9, x10, x11) :|: 0 = x11 && 0 = x10 && 0 = x9 && x7 = x8 && 1000 <= x7 - 1
f208_0_add_GT(x12, x13, x14) -> f208_0_add_GT(x15, x16, x17) :|: x13 + 2 = x17 && x13 + 2 = x16 && x12 + x13 = x15 && x13 = x14 && x13 <= 1000 && -1 <= x12 - 1 && -1 <= x13 - 1
f208_0_add_GT(x18, x19, x20) -> f311_0_add_GT(x21, x22, x23) :|: 0 = x23 && 0 = x22 && 0 = x21 && x19 = x20 && 1000 <= x19 - 1
f311_0_add_GT(x24, x25, x26) -> f311_0_add_GT(x27, x28, x29) :|: x25 + 3 = x29 && x25 + 3 = x28 && x24 + x25 = x27 && x25 = x26 && x25 <= 1000 && -1 <= x24 - 1 && -1 <= x25 - 1
__init(x30, x31, x32) -> f1_0_main_ConstantStackPush(x33, x34, x35) :|: 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 %).
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(2)
Obligation:
Rules:
f1_0_main_ConstantStackPush(arg1, arg2, arg3) -> f108_0_add_GT(arg1P, arg2P, arg3P) :|: 0 = arg3P && 0 = arg2P && 0 = arg1P
f108_0_add_GT(x, x1, x2) -> f108_0_add_GT(x3, x4, x5) :|: x1 + 1 = x5 && x1 + 1 = x4 && x + x1 = x3 && x1 = x2 && x1 <= 1000 && -1 <= x - 1 && -1 <= x1 - 1
f108_0_add_GT(x6, x7, x8) -> f208_0_add_GT(x9, x10, x11) :|: 0 = x11 && 0 = x10 && 0 = x9 && x7 = x8 && 1000 <= x7 - 1
f208_0_add_GT(x12, x13, x14) -> f208_0_add_GT(x15, x16, x17) :|: x13 + 2 = x17 && x13 + 2 = x16 && x12 + x13 = x15 && x13 = x14 && x13 <= 1000 && -1 <= x12 - 1 && -1 <= x13 - 1
f208_0_add_GT(x18, x19, x20) -> f311_0_add_GT(x21, x22, x23) :|: 0 = x23 && 0 = x22 && 0 = x21 && x19 = x20 && 1000 <= x19 - 1
f311_0_add_GT(x24, x25, x26) -> f311_0_add_GT(x27, x28, x29) :|: x25 + 3 = x29 && x25 + 3 = x28 && x24 + x25 = x27 && x25 = x26 && x25 <= 1000 && -1 <= x24 - 1 && -1 <= x25 - 1
__init(x30, x31, x32) -> f1_0_main_ConstantStackPush(x33, x34, x35) :|: 0 <= 0
Start term: __init(arg1, arg2, arg3)

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(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) f1_0_main_ConstantStackPush(arg1, arg2, arg3) -> f108_0_add_GT(arg1P, arg2P, arg3P) :|: 0 = arg3P && 0 = arg2P && 0 = arg1P
(2) f108_0_add_GT(x, x1, x2) -> f108_0_add_GT(x3, x4, x5) :|: x1 + 1 = x5 && x1 + 1 = x4 && x + x1 = x3 && x1 = x2 && x1 <= 1000 && -1 <= x - 1 && -1 <= x1 - 1
(3) f108_0_add_GT(x6, x7, x8) -> f208_0_add_GT(x9, x10, x11) :|: 0 = x11 && 0 = x10 && 0 = x9 && x7 = x8 && 1000 <= x7 - 1
(4) f208_0_add_GT(x12, x13, x14) -> f208_0_add_GT(x15, x16, x17) :|: x13 + 2 = x17 && x13 + 2 = x16 && x12 + x13 = x15 && x13 = x14 && x13 <= 1000 && -1 <= x12 - 1 && -1 <= x13 - 1
(5) f208_0_add_GT(x18, x19, x20) -> f311_0_add_GT(x21, x22, x23) :|: 0 = x23 && 0 = x22 && 0 = x21 && x19 = x20 && 1000 <= x19 - 1
(6) f311_0_add_GT(x24, x25, x26) -> f311_0_add_GT(x27, x28, x29) :|: x25 + 3 = x29 && x25 + 3 = x28 && x24 + x25 = x27 && x25 = x26 && x25 <= 1000 && -1 <= x24 - 1 && -1 <= x25 - 1
(7) __init(x30, x31, x32) -> f1_0_main_ConstantStackPush(x33, x34, x35) :|: 0 <= 0

Arcs:
(1) -> (2)
(2) -> (2), (3)
(3) -> (4)
(4) -> (4), (5)
(5) -> (6)
(6) -> (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) f108_0_add_GT(x, x1, x2) -> f108_0_add_GT(x3, x4, x5) :|: x1 + 1 = x5 && x1 + 1 = x4 && x + x1 = x3 && x1 = x2 && x1 <= 1000 && -1 <= x - 1 && -1 <= x1 - 1

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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

(7)
Obligation:
Rules:
f108_0_add_GT(x:0, x1:0, x1:0) -> f108_0_add_GT(x:0 + x1:0, x1:0 + 1, x1:0 + 1) :|: x:0 > -1 && x1:0 < 1001 && x1:0 > -1

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(8) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f108_0_add_GT(INTEGER, INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(9)
Obligation:
Rules:
f108_0_add_GT(x:0, x1:0, x1:0) -> f108_0_add_GT(c, c1, c2) :|: c2 = x1:0 + 1 && (c1 = x1:0 + 1 && c = x:0 + x1:0) && (x:0 > -1 && x1:0 < 1001 && x1:0 > -1)

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(10) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ f108_0_add_GT ] = -1*f108_0_add_GT_3

The following rules are decreasing:
f108_0_add_GT(x:0, x1:0, x1:0) -> f108_0_add_GT(c, c1, c2) :|: c2 = x1:0 + 1 && (c1 = x1:0 + 1 && c = x:0 + x1:0) && (x:0 > -1 && x1:0 < 1001 && x1:0 > -1)

The following rules are bounded:
f108_0_add_GT(x:0, x1:0, x1:0) -> f108_0_add_GT(c, c1, c2) :|: c2 = x1:0 + 1 && (c1 = x1:0 + 1 && c = x:0 + x1:0) && (x:0 > -1 && x1:0 < 1001 && x1:0 > -1)


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(11)
YES

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

(12)
Obligation:

Termination digraph:
Nodes:
(1) f208_0_add_GT(x12, x13, x14) -> f208_0_add_GT(x15, x16, x17) :|: x13 + 2 = x17 && x13 + 2 = x16 && x12 + x13 = x15 && x13 = x14 && x13 <= 1000 && -1 <= x12 - 1 && -1 <= x13 - 1

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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

(14)
Obligation:
Rules:
f208_0_add_GT(x12:0, x13:0, x13:0) -> f208_0_add_GT(x12:0 + x13:0, x13:0 + 2, x13:0 + 2) :|: x12:0 > -1 && x13:0 < 1001 && x13:0 > -1

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(15) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f208_0_add_GT(INTEGER, INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(16)
Obligation:
Rules:
f208_0_add_GT(x12:0, x13:0, x13:0) -> f208_0_add_GT(c, c1, c2) :|: c2 = x13:0 + 2 && (c1 = x13:0 + 2 && c = x12:0 + x13:0) && (x12:0 > -1 && x13:0 < 1001 && x13:0 > -1)

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(17) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ f208_0_add_GT ] = -1/2*f208_0_add_GT_3

The following rules are decreasing:
f208_0_add_GT(x12:0, x13:0, x13:0) -> f208_0_add_GT(c, c1, c2) :|: c2 = x13:0 + 2 && (c1 = x13:0 + 2 && c = x12:0 + x13:0) && (x12:0 > -1 && x13:0 < 1001 && x13:0 > -1)

The following rules are bounded:
f208_0_add_GT(x12:0, x13:0, x13:0) -> f208_0_add_GT(c, c1, c2) :|: c2 = x13:0 + 2 && (c1 = x13:0 + 2 && c = x12:0 + x13:0) && (x12:0 > -1 && x13:0 < 1001 && x13:0 > -1)


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(18)
YES

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

(19)
Obligation:

Termination digraph:
Nodes:
(1) f311_0_add_GT(x24, x25, x26) -> f311_0_add_GT(x27, x28, x29) :|: x25 + 3 = x29 && x25 + 3 = x28 && x24 + x25 = x27 && x25 = x26 && x25 <= 1000 && -1 <= x24 - 1 && -1 <= x25 - 1

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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

(21)
Obligation:
Rules:
f311_0_add_GT(x24:0, x25:0, x25:0) -> f311_0_add_GT(x24:0 + x25:0, x25:0 + 3, x25:0 + 3) :|: x24:0 > -1 && x25:0 < 1001 && x25:0 > -1

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(22) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f311_0_add_GT(INTEGER, INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(23)
Obligation:
Rules:
f311_0_add_GT(x24:0, x25:0, x25:0) -> f311_0_add_GT(c, c1, c2) :|: c2 = x25:0 + 3 && (c1 = x25:0 + 3 && c = x24:0 + x25:0) && (x24:0 > -1 && x25:0 < 1001 && x25:0 > -1)

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(24) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ f311_0_add_GT ] = -1/3*f311_0_add_GT_3

The following rules are decreasing:
f311_0_add_GT(x24:0, x25:0, x25:0) -> f311_0_add_GT(c, c1, c2) :|: c2 = x25:0 + 3 && (c1 = x25:0 + 3 && c = x24:0 + x25:0) && (x24:0 > -1 && x25:0 < 1001 && x25:0 > -1)

The following rules are bounded:
f311_0_add_GT(x24:0, x25:0, x25:0) -> f311_0_add_GT(c, c1, c2) :|: c2 = x25:0 + 3 && (c1 = x25:0 + 3 && c = x24:0 + x25:0) && (x24:0 > -1 && x25:0 < 1001 && x25:0 > -1)


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(25)
YES