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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 247 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 31 ms]
(6) IRSwT
(7) TempFilterProof [SOUND, 48 ms]
(8) IntTRS
(9) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
(10) IntTRS
(11) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
(12) YES


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(0)
Obligation:
Rules:
f1_0_main_Load(arg1, arg2) -> f180_0_ack_GT(arg1P, arg2P) :|: 0 = arg2P && 0 = arg1P && 0 = arg2 && 0 <= arg1 - 1
f1_0_main_Load(x, x1) -> f180_0_ack_GT(x2, x3) :|: 0 = x2 && 1 = x1 && -1 <= x3 - 1 && 0 <= x - 1
f1_0_main_Load(x4, x5) -> f180_0_ack_GT(x6, x7) :|: 0 <= x4 - 1 && -1 <= x7 - 1 && 1 <= x5 - 1 && -1 <= x6 - 1
f180_0_ack_GT(x8, x9) -> f180_0_ack_GT(x10, x11) :|: x9 - 1 = x11 && 1 = x10 && 0 = x8 && x9 - 1 <= x9 - 1 && 0 <= x9 - 1
f180_0_ack_GT(x12, x13) -> f180_0_ack_GT(x14, x15) :|: x13 = x15 && x12 - 1 = x14 && 0 <= x13 - 1 && x13 - 1 <= x13 - 1 && 0 <= x12 - 1
f180_0_ack_GT(x16, x17) -> f180_0_ack_GT(x18, x19) :|: x17 - 1 = x19 && x17 - 1 <= x17 - 1 && 0 <= x18 - 1 && 0 <= x17 - 1 && 0 <= x16 - 1
__init(x20, x21) -> f1_0_main_Load(x22, x23) :|: 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_Load(arg1, arg2) -> f180_0_ack_GT(arg1P, arg2P) :|: 0 = arg2P && 0 = arg1P && 0 = arg2 && 0 <= arg1 - 1
f1_0_main_Load(x, x1) -> f180_0_ack_GT(x2, x3) :|: 0 = x2 && 1 = x1 && -1 <= x3 - 1 && 0 <= x - 1
f1_0_main_Load(x4, x5) -> f180_0_ack_GT(x6, x7) :|: 0 <= x4 - 1 && -1 <= x7 - 1 && 1 <= x5 - 1 && -1 <= x6 - 1
f180_0_ack_GT(x8, x9) -> f180_0_ack_GT(x10, x11) :|: x9 - 1 = x11 && 1 = x10 && 0 = x8 && x9 - 1 <= x9 - 1 && 0 <= x9 - 1
f180_0_ack_GT(x12, x13) -> f180_0_ack_GT(x14, x15) :|: x13 = x15 && x12 - 1 = x14 && 0 <= x13 - 1 && x13 - 1 <= x13 - 1 && 0 <= x12 - 1
f180_0_ack_GT(x16, x17) -> f180_0_ack_GT(x18, x19) :|: x17 - 1 = x19 && x17 - 1 <= x17 - 1 && 0 <= x18 - 1 && 0 <= x17 - 1 && 0 <= x16 - 1
__init(x20, x21) -> f1_0_main_Load(x22, x23) :|: 0 <= 0
Start term: __init(arg1, arg2)

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(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) f1_0_main_Load(arg1, arg2) -> f180_0_ack_GT(arg1P, arg2P) :|: 0 = arg2P && 0 = arg1P && 0 = arg2 && 0 <= arg1 - 1
(2) f1_0_main_Load(x, x1) -> f180_0_ack_GT(x2, x3) :|: 0 = x2 && 1 = x1 && -1 <= x3 - 1 && 0 <= x - 1
(3) f1_0_main_Load(x4, x5) -> f180_0_ack_GT(x6, x7) :|: 0 <= x4 - 1 && -1 <= x7 - 1 && 1 <= x5 - 1 && -1 <= x6 - 1
(4) f180_0_ack_GT(x8, x9) -> f180_0_ack_GT(x10, x11) :|: x9 - 1 = x11 && 1 = x10 && 0 = x8 && x9 - 1 <= x9 - 1 && 0 <= x9 - 1
(5) f180_0_ack_GT(x12, x13) -> f180_0_ack_GT(x14, x15) :|: x13 = x15 && x12 - 1 = x14 && 0 <= x13 - 1 && x13 - 1 <= x13 - 1 && 0 <= x12 - 1
(6) f180_0_ack_GT(x16, x17) -> f180_0_ack_GT(x18, x19) :|: x17 - 1 = x19 && x17 - 1 <= x17 - 1 && 0 <= x18 - 1 && 0 <= x17 - 1 && 0 <= x16 - 1
(7) __init(x20, x21) -> f1_0_main_Load(x22, x23) :|: 0 <= 0

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

This digraph is fully evaluated!
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(4)
Obligation:

Termination digraph:
Nodes:
(1) f180_0_ack_GT(x8, x9) -> f180_0_ack_GT(x10, x11) :|: x9 - 1 = x11 && 1 = x10 && 0 = x8 && x9 - 1 <= x9 - 1 && 0 <= x9 - 1
(2) f180_0_ack_GT(x12, x13) -> f180_0_ack_GT(x14, x15) :|: x13 = x15 && x12 - 1 = x14 && 0 <= x13 - 1 && x13 - 1 <= x13 - 1 && 0 <= x12 - 1
(3) f180_0_ack_GT(x16, x17) -> f180_0_ack_GT(x18, x19) :|: x17 - 1 = x19 && x17 - 1 <= x17 - 1 && 0 <= x18 - 1 && 0 <= x17 - 1 && 0 <= x16 - 1

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

This digraph is fully evaluated!

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(5) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(6)
Obligation:
Rules:
f180_0_ack_GT(x12:0, x13:0) -> f180_0_ack_GT(x12:0 - 1, x13:0) :|: x12:0 > 0 && x13:0 > 0
f180_0_ack_GT(x16:0, x17:0) -> f180_0_ack_GT(x18:0, x17:0 - 1) :|: x17:0 > 0 && x18:0 > 0 && x16:0 > 0
f180_0_ack_GT(cons_0, x9:0) -> f180_0_ack_GT(1, x9:0 - 1) :|: x9:0 > 0 && cons_0 = 0

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(7) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f180_0_ack_GT(VARIABLE, INTEGER)
Replaced non-predefined constructor symbols by 0.
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(8)
Obligation:
Rules:
f180_0_ack_GT(x12:0, x13:0) -> f180_0_ack_GT(c, x13:0) :|: c = x12:0 - 1 && (x12:0 > 0 && x13:0 > 0)
f180_0_ack_GT(x16:0, x17:0) -> f180_0_ack_GT(x18:0, c1) :|: c1 = x17:0 - 1 && (x17:0 > 0 && x18:0 > 0 && x16:0 > 0)
f180_0_ack_GT(c2, x9:0) -> f180_0_ack_GT(c3, c4) :|: c4 = x9:0 - 1 && (c3 = 1 && c2 = 0) && (x9:0 > 0 && cons_0 = 0)

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(9) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[f180_0_ack_GT(x, x1)] = -1 + x1

The following rules are decreasing:
f180_0_ack_GT(x16:0, x17:0) -> f180_0_ack_GT(x18:0, c1) :|: c1 = x17:0 - 1 && (x17:0 > 0 && x18:0 > 0 && x16:0 > 0)
f180_0_ack_GT(c2, x9:0) -> f180_0_ack_GT(c3, c4) :|: c4 = x9:0 - 1 && (c3 = 1 && c2 = 0) && (x9:0 > 0 && cons_0 = 0)
The following rules are bounded:
f180_0_ack_GT(x12:0, x13:0) -> f180_0_ack_GT(c, x13:0) :|: c = x12:0 - 1 && (x12:0 > 0 && x13:0 > 0)
f180_0_ack_GT(x16:0, x17:0) -> f180_0_ack_GT(x18:0, c1) :|: c1 = x17:0 - 1 && (x17:0 > 0 && x18:0 > 0 && x16:0 > 0)
f180_0_ack_GT(c2, x9:0) -> f180_0_ack_GT(c3, c4) :|: c4 = x9:0 - 1 && (c3 = 1 && c2 = 0) && (x9:0 > 0 && cons_0 = 0)

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(10)
Obligation:
Rules:
f180_0_ack_GT(x12:0, x13:0) -> f180_0_ack_GT(c, x13:0) :|: c = x12:0 - 1 && (x12:0 > 0 && x13:0 > 0)

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(11) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[f180_0_ack_GT(x, x1)] = x

The following rules are decreasing:
f180_0_ack_GT(x12:0, x13:0) -> f180_0_ack_GT(c, x13:0) :|: c = x12:0 - 1 && (x12:0 > 0 && x13:0 > 0)
The following rules are bounded:
f180_0_ack_GT(x12:0, x13:0) -> f180_0_ack_GT(c, x13:0) :|: c = x12:0 - 1 && (x12:0 > 0 && x13:0 > 0)

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(12)
YES