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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 148 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 0 ms]
(6) IRSwT
(7) IRSwTChainingProof [EQUIVALENT, 0 ms]
(8) IRSwT
(9) IRSwTTerminationDigraphProof [EQUIVALENT, 5 ms]
(10) IRSwT
(11) IntTRSCompressionProof [EQUIVALENT, 0 ms]
(12) IRSwT
(13) TempFilterProof [SOUND, 16 ms]
(14) IntTRS
(15) RankingReductionPairProof [EQUIVALENT, 0 ms]
(16) YES


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(0)
Obligation:
Rules:
f1_0_main_Load(arg1, arg2, arg3, arg4) -> f218_0_main_LE(arg1P, arg2P, arg3P, arg4P) :|: arg1P + arg2P = arg4P && 0 <= arg1 - 1 && -1 <= arg2P - 1 && -1 <= arg1P - 1 && -1 <= arg2 - 1 && -1 <= arg3P - 1
f218_0_main_LE(x, x1, x2, x3) -> f218_0_main_LE(x4, x5, x6, x7) :|: x + 1 + x1 + 1 = x7 && x2 = x6 && x1 + 1 = x5 && x + 1 = x4 && -2 <= x1 - 1 && -2 <= x - 1 && x3 <= x2 - 1
__init(x8, x9, x10, x11) -> f1_0_main_Load(x12, x13, x14, x15) :|: 0 <= 0
Start term: __init(arg1, arg2, arg3, arg4)

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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, arg4) -> f218_0_main_LE(arg1P, arg2P, arg3P, arg4P) :|: arg1P + arg2P = arg4P && 0 <= arg1 - 1 && -1 <= arg2P - 1 && -1 <= arg1P - 1 && -1 <= arg2 - 1 && -1 <= arg3P - 1
f218_0_main_LE(x, x1, x2, x3) -> f218_0_main_LE(x4, x5, x6, x7) :|: x + 1 + x1 + 1 = x7 && x2 = x6 && x1 + 1 = x5 && x + 1 = x4 && -2 <= x1 - 1 && -2 <= x - 1 && x3 <= x2 - 1
__init(x8, x9, x10, x11) -> f1_0_main_Load(x12, x13, x14, x15) :|: 0 <= 0
Start term: __init(arg1, arg2, arg3, arg4)

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(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) f1_0_main_Load(arg1, arg2, arg3, arg4) -> f218_0_main_LE(arg1P, arg2P, arg3P, arg4P) :|: arg1P + arg2P = arg4P && 0 <= arg1 - 1 && -1 <= arg2P - 1 && -1 <= arg1P - 1 && -1 <= arg2 - 1 && -1 <= arg3P - 1
(2) f218_0_main_LE(x, x1, x2, x3) -> f218_0_main_LE(x4, x5, x6, x7) :|: x + 1 + x1 + 1 = x7 && x2 = x6 && x1 + 1 = x5 && x + 1 = x4 && -2 <= x1 - 1 && -2 <= x - 1 && x3 <= x2 - 1
(3) __init(x8, x9, x10, x11) -> f1_0_main_Load(x12, x13, x14, x15) :|: 0 <= 0

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

This digraph is fully evaluated!
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(4)
Obligation:

Termination digraph:
Nodes:
(1) f218_0_main_LE(x, x1, x2, x3) -> f218_0_main_LE(x4, x5, x6, x7) :|: x + 1 + x1 + 1 = x7 && x2 = x6 && x1 + 1 = x5 && x + 1 = x4 && -2 <= x1 - 1 && -2 <= x - 1 && x3 <= x2 - 1

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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(5) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(6)
Obligation:
Rules:
f218_0_main_LE(x:0, x1:0, x2:0, x3:0) -> f218_0_main_LE(x:0 + 1, x1:0 + 1, x2:0, x:0 + 1 + x1:0 + 1) :|: x:0 > -2 && x1:0 > -2 && x3:0 <= x2:0 - 1

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(7) IRSwTChainingProof (EQUIVALENT)
Chaining!
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(8)
Obligation:
Rules:
f218_0_main_LE(x, x1, x2, x3) -> f218_0_main_LE(x + 2, x1 + 2, x2, x + 4 + x1) :|: TRUE && x >= -1 && x1 >= -1 && x3 + -1 * x2 <= -1 && x + x1 + -1 * x2 <= -3

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(9) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) f218_0_main_LE(x, x1, x2, x3) -> f218_0_main_LE(x + 2, x1 + 2, x2, x + 4 + x1) :|: TRUE && x >= -1 && x1 >= -1 && x3 + -1 * x2 <= -1 && x + x1 + -1 * x2 <= -3

Arcs:
(1) -> (1)

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

(10)
Obligation:

Termination digraph:
Nodes:
(1) f218_0_main_LE(x, x1, x2, x3) -> f218_0_main_LE(x + 2, x1 + 2, x2, x + 4 + x1) :|: TRUE && x >= -1 && x1 >= -1 && x3 + -1 * x2 <= -1 && x + x1 + -1 * x2 <= -3

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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

(12)
Obligation:
Rules:
f218_0_main_LE(x:0, x1:0, x2:0, x3:0) -> f218_0_main_LE(x:0 + 2, x1:0 + 2, x2:0, x:0 + 4 + x1:0) :|: x3:0 + -1 * x2:0 <= -1 && x:0 + x1:0 + -1 * x2:0 <= -3 && x:0 > -2 && x1:0 > -2

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(13) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f218_0_main_LE(INTEGER, INTEGER, INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(14)
Obligation:
Rules:
f218_0_main_LE(x:0, x1:0, x2:0, x3:0) -> f218_0_main_LE(c, c1, x2:0, c2) :|: c2 = x:0 + 4 + x1:0 && (c1 = x1:0 + 2 && c = x:0 + 2) && (x3:0 + -1 * x2:0 <= -1 && x:0 + x1:0 + -1 * x2:0 <= -3 && x:0 > -2 && x1:0 > -2)

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(15) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ f218_0_main_LE ] = 1/2*f218_0_main_LE_3 + -1/2*f218_0_main_LE_1

The following rules are decreasing:
f218_0_main_LE(x:0, x1:0, x2:0, x3:0) -> f218_0_main_LE(c, c1, x2:0, c2) :|: c2 = x:0 + 4 + x1:0 && (c1 = x1:0 + 2 && c = x:0 + 2) && (x3:0 + -1 * x2:0 <= -1 && x:0 + x1:0 + -1 * x2:0 <= -3 && x:0 > -2 && x1:0 > -2)

The following rules are bounded:
f218_0_main_LE(x:0, x1:0, x2:0, x3:0) -> f218_0_main_LE(c, c1, x2:0, c2) :|: c2 = x:0 + 4 + x1:0 && (c1 = x1:0 + 2 && c = x:0 + 2) && (x3:0 + -1 * x2:0 <= -1 && x:0 + x1:0 + -1 * x2:0 <= -3 && x:0 > -2 && x1:0 > -2)


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(16)
YES