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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 192 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 0 ms]
(6) IRSwT
(7) TempFilterProof [SOUND, 54 ms]
(8) IntTRS
(9) PolynomialOrderProcessor [EQUIVALENT, 16 ms]
(10) IntTRS
(11) RankingReductionPairProof [EQUIVALENT, 0 ms]
(12) YES


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(0)
Obligation:
Rules:
f1_0_main_ConstantStackPush(arg1, arg2) -> f138_0_ack_GT(arg1P, arg2P) :|: 10 = arg2P && 12 = arg1P
f138_0_ack_GT(x, x1) -> f138_0_ack_GT(x2, x3) :|: x1 - 1 = x3 && 1 = x2 && x <= 0 && 0 <= x1 - 1
f138_0_ack_GT(x4, x5) -> f138_0_ack_GT(x6, x7) :|: x5 = x7 && x4 - 1 = x6 && 0 <= x5 - 1 && x5 - 1 <= x5 - 1 && 0 <= x4 - 1
f138_0_ack_GT(x8, x9) -> f138_0_ack_GT(x10, x11) :|: x9 - 1 = x11 && 0 <= x9 - 1 && x9 - 1 <= x9 - 1 && 0 <= x8 - 1
__init(x12, x13) -> f1_0_main_ConstantStackPush(x14, x15) :|: 0 <= 0
Start term: __init(arg1, arg2)

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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) -> f138_0_ack_GT(arg1P, arg2P) :|: 10 = arg2P && 12 = arg1P
f138_0_ack_GT(x, x1) -> f138_0_ack_GT(x2, x3) :|: x1 - 1 = x3 && 1 = x2 && x <= 0 && 0 <= x1 - 1
f138_0_ack_GT(x4, x5) -> f138_0_ack_GT(x6, x7) :|: x5 = x7 && x4 - 1 = x6 && 0 <= x5 - 1 && x5 - 1 <= x5 - 1 && 0 <= x4 - 1
f138_0_ack_GT(x8, x9) -> f138_0_ack_GT(x10, x11) :|: x9 - 1 = x11 && 0 <= x9 - 1 && x9 - 1 <= x9 - 1 && 0 <= x8 - 1
__init(x12, x13) -> f1_0_main_ConstantStackPush(x14, x15) :|: 0 <= 0
Start term: __init(arg1, arg2)

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(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) f1_0_main_ConstantStackPush(arg1, arg2) -> f138_0_ack_GT(arg1P, arg2P) :|: 10 = arg2P && 12 = arg1P
(2) f138_0_ack_GT(x, x1) -> f138_0_ack_GT(x2, x3) :|: x1 - 1 = x3 && 1 = x2 && x <= 0 && 0 <= x1 - 1
(3) f138_0_ack_GT(x4, x5) -> f138_0_ack_GT(x6, x7) :|: x5 = x7 && x4 - 1 = x6 && 0 <= x5 - 1 && x5 - 1 <= x5 - 1 && 0 <= x4 - 1
(4) f138_0_ack_GT(x8, x9) -> f138_0_ack_GT(x10, x11) :|: x9 - 1 = x11 && 0 <= x9 - 1 && x9 - 1 <= x9 - 1 && 0 <= x8 - 1
(5) __init(x12, x13) -> f1_0_main_ConstantStackPush(x14, x15) :|: 0 <= 0

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

This digraph is fully evaluated!
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(4)
Obligation:

Termination digraph:
Nodes:
(1) f138_0_ack_GT(x4, x5) -> f138_0_ack_GT(x6, x7) :|: x5 = x7 && x4 - 1 = x6 && 0 <= x5 - 1 && x5 - 1 <= x5 - 1 && 0 <= x4 - 1
(2) f138_0_ack_GT(x, x1) -> f138_0_ack_GT(x2, x3) :|: x1 - 1 = x3 && 1 = x2 && x <= 0 && 0 <= x1 - 1
(3) f138_0_ack_GT(x8, x9) -> f138_0_ack_GT(x10, x11) :|: x9 - 1 = x11 && 0 <= x9 - 1 && x9 - 1 <= x9 - 1 && 0 <= x8 - 1

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

This digraph is fully evaluated!

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(5) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(6)
Obligation:
Rules:
f138_0_ack_GT(x8:0, x9:0) -> f138_0_ack_GT(x10:0, x9:0 - 1) :|: x8:0 > 0 && x9:0 > 0
f138_0_ack_GT(x:0, x1:0) -> f138_0_ack_GT(1, x1:0 - 1) :|: x1:0 > 0 && x:0 < 1
f138_0_ack_GT(x4:0, x5:0) -> f138_0_ack_GT(x4:0 - 1, x5:0) :|: x4:0 > 0 && x5:0 > 0

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(7) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f138_0_ack_GT(VARIABLE, INTEGER)
Replaced non-predefined constructor symbols by 0.
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(8)
Obligation:
Rules:
f138_0_ack_GT(x8:0, x9:0) -> f138_0_ack_GT(x10:0, c) :|: c = x9:0 - 1 && (x8:0 > 0 && x9:0 > 0)
f138_0_ack_GT(x:0, x1:0) -> f138_0_ack_GT(c1, c2) :|: c2 = x1:0 - 1 && c1 = 1 && (x1:0 > 0 && x:0 < 1)
f138_0_ack_GT(x4:0, x5:0) -> f138_0_ack_GT(c3, x5:0) :|: c3 = x4:0 - 1 && (x4:0 > 0 && x5:0 > 0)

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(9) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[f138_0_ack_GT(x, x1)] = x1

The following rules are decreasing:
f138_0_ack_GT(x8:0, x9:0) -> f138_0_ack_GT(x10:0, c) :|: c = x9:0 - 1 && (x8:0 > 0 && x9:0 > 0)
f138_0_ack_GT(x:0, x1:0) -> f138_0_ack_GT(c1, c2) :|: c2 = x1:0 - 1 && c1 = 1 && (x1:0 > 0 && x:0 < 1)
The following rules are bounded:
f138_0_ack_GT(x8:0, x9:0) -> f138_0_ack_GT(x10:0, c) :|: c = x9:0 - 1 && (x8:0 > 0 && x9:0 > 0)
f138_0_ack_GT(x:0, x1:0) -> f138_0_ack_GT(c1, c2) :|: c2 = x1:0 - 1 && c1 = 1 && (x1:0 > 0 && x:0 < 1)
f138_0_ack_GT(x4:0, x5:0) -> f138_0_ack_GT(c3, x5:0) :|: c3 = x4:0 - 1 && (x4:0 > 0 && x5:0 > 0)

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(10)
Obligation:
Rules:
f138_0_ack_GT(x4:0, x5:0) -> f138_0_ack_GT(c3, x5:0) :|: c3 = x4:0 - 1 && (x4:0 > 0 && x5:0 > 0)

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(11) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ f138_0_ack_GT ] = f138_0_ack_GT_1

The following rules are decreasing:
f138_0_ack_GT(x4:0, x5:0) -> f138_0_ack_GT(c3, x5:0) :|: c3 = x4:0 - 1 && (x4:0 > 0 && x5:0 > 0)

The following rules are bounded:
f138_0_ack_GT(x4:0, x5:0) -> f138_0_ack_GT(c3, x5:0) :|: c3 = x4:0 - 1 && (x4:0 > 0 && x5:0 > 0)


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(12)
YES