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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 135 ms]
(4) AND
    (5) IRSwT
        (6) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (7) IRSwT
        (8) TempFilterProof [SOUND, 22 ms]
        (9) IntTRS
        (10) RankingReductionPairProof [EQUIVALENT, 7 ms]
        (11) YES
    (12) IRSwT
        (13) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (14) IRSwT
        (15) TempFilterProof [SOUND, 18 ms]
        (16) IntTRS
        (17) PolynomialOrderProcessor [EQUIVALENT, 4 ms]
        (18) YES


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(0)
Obligation:
Rules:
f1_0_main_Load(arg1, arg2) -> f145_0_main_EQ(arg1P, arg2P) :|: 0 <= arg1 - 1 && -1 <= arg1P - 1 && -1 <= arg2 - 1 && -1 <= arg2P - 1
f145_0_main_EQ(x, x1) -> f145_0_main_EQ(x2, x3) :|: x1 = x3 && x + 1 = x2 && x <= x1 - 1
f145_0_main_EQ(x4, x5) -> f145_0_main_EQ(x6, x7) :|: x5 + 1 = x7 && x4 = x6 && x5 <= x4 - 1
__init(x8, x9) -> f1_0_main_Load(x10, x11) :|: 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) -> f145_0_main_EQ(arg1P, arg2P) :|: 0 <= arg1 - 1 && -1 <= arg1P - 1 && -1 <= arg2 - 1 && -1 <= arg2P - 1
f145_0_main_EQ(x, x1) -> f145_0_main_EQ(x2, x3) :|: x1 = x3 && x + 1 = x2 && x <= x1 - 1
f145_0_main_EQ(x4, x5) -> f145_0_main_EQ(x6, x7) :|: x5 + 1 = x7 && x4 = x6 && x5 <= x4 - 1
__init(x8, x9) -> f1_0_main_Load(x10, x11) :|: 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) -> f145_0_main_EQ(arg1P, arg2P) :|: 0 <= arg1 - 1 && -1 <= arg1P - 1 && -1 <= arg2 - 1 && -1 <= arg2P - 1
(2) f145_0_main_EQ(x, x1) -> f145_0_main_EQ(x2, x3) :|: x1 = x3 && x + 1 = x2 && x <= x1 - 1
(3) f145_0_main_EQ(x4, x5) -> f145_0_main_EQ(x6, x7) :|: x5 + 1 = x7 && x4 = x6 && x5 <= x4 - 1
(4) __init(x8, x9) -> f1_0_main_Load(x10, x11) :|: 0 <= 0

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

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

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(5)
Obligation:

Termination digraph:
Nodes:
(1) f145_0_main_EQ(x4, x5) -> f145_0_main_EQ(x6, x7) :|: x5 + 1 = x7 && x4 = x6 && x5 <= x4 - 1

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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(6) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(7)
Obligation:
Rules:
f145_0_main_EQ(x4:0, x5:0) -> f145_0_main_EQ(x4:0, x5:0 + 1) :|: x5:0 <= x4:0 - 1

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(8) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f145_0_main_EQ(INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
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(9)
Obligation:
Rules:
f145_0_main_EQ(x4:0, x5:0) -> f145_0_main_EQ(x4:0, c) :|: c = x5:0 + 1 && x5:0 <= x4:0 - 1

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(10) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ f145_0_main_EQ ] = -1*f145_0_main_EQ_2 + f145_0_main_EQ_1

The following rules are decreasing:
f145_0_main_EQ(x4:0, x5:0) -> f145_0_main_EQ(x4:0, c) :|: c = x5:0 + 1 && x5:0 <= x4:0 - 1

The following rules are bounded:
f145_0_main_EQ(x4:0, x5:0) -> f145_0_main_EQ(x4:0, c) :|: c = x5:0 + 1 && x5:0 <= x4:0 - 1


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(11)
YES

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(12)
Obligation:

Termination digraph:
Nodes:
(1) f145_0_main_EQ(x, x1) -> f145_0_main_EQ(x2, x3) :|: x1 = x3 && x + 1 = x2 && x <= x1 - 1

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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(13) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(14)
Obligation:
Rules:
f145_0_main_EQ(x:0, x1:0) -> f145_0_main_EQ(x:0 + 1, x1:0) :|: x:0 <= x1:0 - 1

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(15) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f145_0_main_EQ(INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
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(16)
Obligation:
Rules:
f145_0_main_EQ(x:0, x1:0) -> f145_0_main_EQ(c, x1:0) :|: c = x:0 + 1 && x:0 <= x1:0 - 1

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(17) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[f145_0_main_EQ(x, x1)] = -x + x1

The following rules are decreasing:
f145_0_main_EQ(x:0, x1:0) -> f145_0_main_EQ(c, x1:0) :|: c = x:0 + 1 && x:0 <= x1:0 - 1
The following rules are bounded:
f145_0_main_EQ(x:0, x1:0) -> f145_0_main_EQ(c, x1:0) :|: c = x:0 + 1 && x:0 <= x1:0 - 1

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(18)
YES