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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 71 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 0 ms]
(6) IRSwT
(7) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
(8) IRSwT
(9) FilterProof [EQUIVALENT, 0 ms]
(10) IntTRS
(11) IntTRSCompressionProof [EQUIVALENT, 0 ms]
(12) IntTRS
(13) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
(14) YES


----------------------------------------

(0)
Obligation:
Rules:
f1_0_main_Load(arg1, arg2) -> f153_0_run_NULL(arg1P, arg2P) :|: 3 <= arg1P - 1 && 0 <= arg1 - 1 && -1 <= arg2 - 1 && arg1P - 3 <= arg1
f153_0_run_NULL(x, x1) -> f153_0_run_NULL(x2, x3) :|: -1 <= x2 - 1 && 0 <= x - 1 && x2 + 1 <= x
__init(x4, x5) -> f1_0_main_Load(x6, x7) :|: 0 <= 0
Start term: __init(arg1, arg2)

----------------------------------------

(1) IRSFormatTransformerProof (EQUIVALENT)
Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %).
----------------------------------------

(2)
Obligation:
Rules:
f1_0_main_Load(arg1, arg2) -> f153_0_run_NULL(arg1P, arg2P) :|: 3 <= arg1P - 1 && 0 <= arg1 - 1 && -1 <= arg2 - 1 && arg1P - 3 <= arg1
f153_0_run_NULL(x, x1) -> f153_0_run_NULL(x2, x3) :|: -1 <= x2 - 1 && 0 <= x - 1 && x2 + 1 <= x
__init(x4, x5) -> f1_0_main_Load(x6, x7) :|: 0 <= 0
Start term: __init(arg1, arg2)

----------------------------------------

(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) f1_0_main_Load(arg1, arg2) -> f153_0_run_NULL(arg1P, arg2P) :|: 3 <= arg1P - 1 && 0 <= arg1 - 1 && -1 <= arg2 - 1 && arg1P - 3 <= arg1
(2) f153_0_run_NULL(x, x1) -> f153_0_run_NULL(x2, x3) :|: -1 <= x2 - 1 && 0 <= x - 1 && x2 + 1 <= x
(3) __init(x4, x5) -> f1_0_main_Load(x6, x7) :|: 0 <= 0

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

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

(4)
Obligation:

Termination digraph:
Nodes:
(1) f153_0_run_NULL(x, x1) -> f153_0_run_NULL(x2, x3) :|: -1 <= x2 - 1 && 0 <= x - 1 && x2 + 1 <= x

Arcs:
(1) -> (1)

This digraph is fully evaluated!

----------------------------------------

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

(6)
Obligation:
Rules:
f153_0_run_NULL(x:0, x1:0) -> f153_0_run_NULL(x2:0, x3:0) :|: x2:0 > -1 && x:0 > 0 && x:0 >= x2:0 + 1

----------------------------------------

(7) IntTRSUnneededArgumentFilterProof (EQUIVALENT)
Some arguments are removed because they cannot influence termination. We removed arguments according to the following replacements:

   f153_0_run_NULL(x1, x2) -> f153_0_run_NULL(x1)

----------------------------------------

(8)
Obligation:
Rules:
f153_0_run_NULL(x:0) -> f153_0_run_NULL(x2:0) :|: x2:0 > -1 && x:0 > 0 && x:0 >= x2:0 + 1

----------------------------------------

(9) FilterProof (EQUIVALENT)
Used the following sort dictionary for filtering: 
f153_0_run_NULL(INTEGER)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(10)
Obligation:
Rules:
f153_0_run_NULL(x:0) -> f153_0_run_NULL(x2:0) :|: x2:0 > -1 && x:0 > 0 && x:0 >= x2:0 + 1

----------------------------------------

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

(12)
Obligation:
Rules:
f153_0_run_NULL(x:0:0) -> f153_0_run_NULL(x2:0:0) :|: x2:0:0 > -1 && x:0:0 > 0 && x:0:0 >= x2:0:0 + 1

----------------------------------------

(13) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[f153_0_run_NULL(x)] = x

The following rules are decreasing:
f153_0_run_NULL(x:0:0) -> f153_0_run_NULL(x2:0:0) :|: x2:0:0 > -1 && x:0:0 > 0 && x:0:0 >= x2:0:0 + 1
The following rules are bounded:
f153_0_run_NULL(x:0:0) -> f153_0_run_NULL(x2:0:0) :|: x2:0:0 > -1 && x:0:0 > 0 && x:0:0 >= x2:0:0 + 1

----------------------------------------

(14)
YES