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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 131 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 34 ms]
(6) IRSwT
(7) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
(8) IRSwT
(9) TempFilterProof [SOUND, 13 ms]
(10) IntTRS
(11) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
(12) YES


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

(0)
Obligation:
Rules:
f1_0_main_Load(arg1, arg2) -> f66_0_nest_NE(arg1P, arg2P) :|: 0 <= arg1 - 1 && 0 <= arg2 - 1 && -1 <= arg1P - 1
f66_0_nest_NE(x, x1) -> f66_0_nest_NE(x2, x3) :|: x - 1 = x2 && x - 1 <= x - 1 && 0 <= x - 1
f66_0_nest_NE(x4, x5) -> f115_0_nest_InvokeMethod(x6, x7) :|: 1 = x6 && 1 = x4
f115_0_nest_InvokeMethod(x8, x9) -> f66_0_nest_NE(x10, x11) :|: 0 = x10 && 0 <= x8 - 1
f66_0_nest_NE(x12, x13) -> f115_0_nest_InvokeMethod(x14, x15) :|: x12 = x14 && x12 - 1 <= x12 - 1 && 0 <= x12 - 1
__init(x16, x17) -> f1_0_main_Load(x18, x19) :|: 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) -> f66_0_nest_NE(arg1P, arg2P) :|: 0 <= arg1 - 1 && 0 <= arg2 - 1 && -1 <= arg1P - 1
f66_0_nest_NE(x, x1) -> f66_0_nest_NE(x2, x3) :|: x - 1 = x2 && x - 1 <= x - 1 && 0 <= x - 1
f66_0_nest_NE(x4, x5) -> f115_0_nest_InvokeMethod(x6, x7) :|: 1 = x6 && 1 = x4
f115_0_nest_InvokeMethod(x8, x9) -> f66_0_nest_NE(x10, x11) :|: 0 = x10 && 0 <= x8 - 1
f66_0_nest_NE(x12, x13) -> f115_0_nest_InvokeMethod(x14, x15) :|: x12 = x14 && x12 - 1 <= x12 - 1 && 0 <= x12 - 1
__init(x16, x17) -> f1_0_main_Load(x18, x19) :|: 0 <= 0
Start term: __init(arg1, arg2)

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

(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) f1_0_main_Load(arg1, arg2) -> f66_0_nest_NE(arg1P, arg2P) :|: 0 <= arg1 - 1 && 0 <= arg2 - 1 && -1 <= arg1P - 1
(2) f66_0_nest_NE(x, x1) -> f66_0_nest_NE(x2, x3) :|: x - 1 = x2 && x - 1 <= x - 1 && 0 <= x - 1
(3) f66_0_nest_NE(x4, x5) -> f115_0_nest_InvokeMethod(x6, x7) :|: 1 = x6 && 1 = x4
(4) f115_0_nest_InvokeMethod(x8, x9) -> f66_0_nest_NE(x10, x11) :|: 0 = x10 && 0 <= x8 - 1
(5) f66_0_nest_NE(x12, x13) -> f115_0_nest_InvokeMethod(x14, x15) :|: x12 = x14 && x12 - 1 <= x12 - 1 && 0 <= x12 - 1
(6) __init(x16, x17) -> f1_0_main_Load(x18, x19) :|: 0 <= 0

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

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

(4)
Obligation:

Termination digraph:
Nodes:
(1) f66_0_nest_NE(x, x1) -> f66_0_nest_NE(x2, x3) :|: x - 1 = x2 && x - 1 <= x - 1 && 0 <= x - 1

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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

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

(6)
Obligation:
Rules:
f66_0_nest_NE(x:0, x1:0) -> f66_0_nest_NE(x:0 - 1, x3:0) :|: x:0 > 0

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

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

   f66_0_nest_NE(x1, x2) -> f66_0_nest_NE(x1)

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

(8)
Obligation:
Rules:
f66_0_nest_NE(x:0) -> f66_0_nest_NE(x:0 - 1) :|: x:0 > 0

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

(9) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f66_0_nest_NE(INTEGER)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(10)
Obligation:
Rules:
f66_0_nest_NE(x:0) -> f66_0_nest_NE(c) :|: c = x:0 - 1 && x:0 > 0

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

(11) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[f66_0_nest_NE(x)] = x

The following rules are decreasing:
f66_0_nest_NE(x:0) -> f66_0_nest_NE(c) :|: c = x:0 - 1 && x:0 > 0
The following rules are bounded:
f66_0_nest_NE(x:0) -> f66_0_nest_NE(c) :|: c = x:0 - 1 && x:0 > 0

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

(12)
YES