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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 151 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 12 ms]
(6) IRSwT
(7) IRSwTChainingProof [EQUIVALENT, 0 ms]
(8) IRSwT
(9) IRSwTTerminationDigraphProof [EQUIVALENT, 3 ms]
(10) IRSwT
(11) IntTRSCompressionProof [EQUIVALENT, 0 ms]
(12) IRSwT
(13) TempFilterProof [SOUND, 5 ms]
(14) IntTRS
(15) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
(16) YES


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(0)
Obligation:
Rules:
f1_0_main_Load(arg1, arg2, arg3) -> f252_0_min_GE(arg1P, arg2P, arg3P) :|: arg1P - 1 = arg2P && 0 <= arg1 - 1 && -1 <= arg3P - 1 && -1 <= arg2 - 1 && -1 <= arg1P - 1
f252_0_min_GE(x, x1, x2) -> f261_0_main_NE(x3, x4, x5) :|: x2 = x5 && x1 = x4 && x = x3 && x1 <= x2 - 1
f252_0_min_GE(x6, x7, x8) -> f261_0_main_NE(x9, x10, x11) :|: x8 = x11 && x8 = x10 && x6 = x9 && x8 <= x7
f261_0_main_NE(x12, x13, x14) -> f252_0_min_GE(x15, x16, x17) :|: x13 + 1 = x17 && x12 - 1 = x16 && x12 = x15 && x13 = x14 && -1 <= x12 - 1
__init(x18, x19, x20) -> f1_0_main_Load(x21, x22, x23) :|: 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_Load(arg1, arg2, arg3) -> f252_0_min_GE(arg1P, arg2P, arg3P) :|: arg1P - 1 = arg2P && 0 <= arg1 - 1 && -1 <= arg3P - 1 && -1 <= arg2 - 1 && -1 <= arg1P - 1
f252_0_min_GE(x, x1, x2) -> f261_0_main_NE(x3, x4, x5) :|: x2 = x5 && x1 = x4 && x = x3 && x1 <= x2 - 1
f252_0_min_GE(x6, x7, x8) -> f261_0_main_NE(x9, x10, x11) :|: x8 = x11 && x8 = x10 && x6 = x9 && x8 <= x7
f261_0_main_NE(x12, x13, x14) -> f252_0_min_GE(x15, x16, x17) :|: x13 + 1 = x17 && x12 - 1 = x16 && x12 = x15 && x13 = x14 && -1 <= x12 - 1
__init(x18, x19, x20) -> f1_0_main_Load(x21, x22, x23) :|: 0 <= 0
Start term: __init(arg1, arg2, arg3)

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(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) f1_0_main_Load(arg1, arg2, arg3) -> f252_0_min_GE(arg1P, arg2P, arg3P) :|: arg1P - 1 = arg2P && 0 <= arg1 - 1 && -1 <= arg3P - 1 && -1 <= arg2 - 1 && -1 <= arg1P - 1
(2) f252_0_min_GE(x, x1, x2) -> f261_0_main_NE(x3, x4, x5) :|: x2 = x5 && x1 = x4 && x = x3 && x1 <= x2 - 1
(3) f252_0_min_GE(x6, x7, x8) -> f261_0_main_NE(x9, x10, x11) :|: x8 = x11 && x8 = x10 && x6 = x9 && x8 <= x7
(4) f261_0_main_NE(x12, x13, x14) -> f252_0_min_GE(x15, x16, x17) :|: x13 + 1 = x17 && x12 - 1 = x16 && x12 = x15 && x13 = x14 && -1 <= x12 - 1
(5) __init(x18, x19, x20) -> f1_0_main_Load(x21, x22, x23) :|: 0 <= 0

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

This digraph is fully evaluated!
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(4)
Obligation:

Termination digraph:
Nodes:
(1) f252_0_min_GE(x6, x7, x8) -> f261_0_main_NE(x9, x10, x11) :|: x8 = x11 && x8 = x10 && x6 = x9 && x8 <= x7
(2) f261_0_main_NE(x12, x13, x14) -> f252_0_min_GE(x15, x16, x17) :|: x13 + 1 = x17 && x12 - 1 = x16 && x12 = x15 && x13 = x14 && -1 <= x12 - 1

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

This digraph is fully evaluated!

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(5) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(6)
Obligation:
Rules:
f252_0_min_GE(x15:0, x7:0, x10:0) -> f252_0_min_GE(x15:0, x15:0 - 1, x10:0 + 1) :|: x7:0 >= x10:0 && x15:0 > -1

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(7) IRSwTChainingProof (EQUIVALENT)
Chaining!
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(8)
Obligation:
Rules:
f252_0_min_GE(x, x1, x2) -> f252_0_min_GE(x, x + -1, x2 + 2) :|: TRUE && x1 + -1 * x2 >= 0 && x >= 0 && x + -1 * x2 >= 2

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(9) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) f252_0_min_GE(x, x1, x2) -> f252_0_min_GE(x, x + -1, x2 + 2) :|: TRUE && x1 + -1 * x2 >= 0 && x >= 0 && x + -1 * x2 >= 2

Arcs:
(1) -> (1)

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

(10)
Obligation:

Termination digraph:
Nodes:
(1) f252_0_min_GE(x, x1, x2) -> f252_0_min_GE(x, x + -1, x2 + 2) :|: TRUE && x1 + -1 * x2 >= 0 && x >= 0 && x + -1 * x2 >= 2

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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

(12)
Obligation:
Rules:
f252_0_min_GE(x:0, x1:0, x2:0) -> f252_0_min_GE(x:0, x:0 - 1, x2:0 + 2) :|: x:0 > -1 && x1:0 + -1 * x2:0 >= 0 && x:0 + -1 * x2:0 >= 2

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(13) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f252_0_min_GE(INTEGER, INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
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(14)
Obligation:
Rules:
f252_0_min_GE(x:0, x1:0, x2:0) -> f252_0_min_GE(x:0, c, c1) :|: c1 = x2:0 + 2 && c = x:0 - 1 && (x:0 > -1 && x1:0 + -1 * x2:0 >= 0 && x:0 + -1 * x2:0 >= 2)

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(15) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[f252_0_min_GE(x, x1, x2)] = -1 + x - x2

The following rules are decreasing:
f252_0_min_GE(x:0, x1:0, x2:0) -> f252_0_min_GE(x:0, c, c1) :|: c1 = x2:0 + 2 && c = x:0 - 1 && (x:0 > -1 && x1:0 + -1 * x2:0 >= 0 && x:0 + -1 * x2:0 >= 2)
The following rules are bounded:
f252_0_min_GE(x:0, x1:0, x2:0) -> f252_0_min_GE(x:0, c, c1) :|: c1 = x2:0 + 2 && c = x:0 - 1 && (x:0 > -1 && x1:0 + -1 * x2:0 >= 0 && x:0 + -1 * x2:0 >= 2)

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(16)
YES