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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 129 ms]
(4) AND
    (5) IRSwT
        (6) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (7) IRSwT
        (8) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (9) IRSwT
        (10) TempFilterProof [SOUND, 20 ms]
        (11) IntTRS
        (12) RankingReductionPairProof [EQUIVALENT, 0 ms]
        (13) YES
    (14) IRSwT
        (15) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (16) IRSwT
        (17) TempFilterProof [SOUND, 10 ms]
        (18) IntTRS
        (19) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
        (20) YES


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

(0)
Obligation:
Rules:
f1_0_main_Load(arg1, arg2) -> f142_0_main_GT(arg1P, arg2P) :|: 0 <= arg1 - 1 && -1 <= arg2P - 1 && -1 <= arg2 - 1 && -1 <= arg1P - 1
f142_0_main_GT(x, x1) -> f142_0_main_GT(x2, x3) :|: x1 - 1 = x3 && x = x2 && 0 <= x1 - 1
f142_0_main_GT(x4, x5) -> f142_0_main_GT(x6, x7) :|: 0 = x7 && x4 - 1 = x6 && 0 = x5 && 0 <= x4 - 1
__init(x8, x9) -> f1_0_main_Load(x10, x11) :|: 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) -> f142_0_main_GT(arg1P, arg2P) :|: 0 <= arg1 - 1 && -1 <= arg2P - 1 && -1 <= arg2 - 1 && -1 <= arg1P - 1
f142_0_main_GT(x, x1) -> f142_0_main_GT(x2, x3) :|: x1 - 1 = x3 && x = x2 && 0 <= x1 - 1
f142_0_main_GT(x4, x5) -> f142_0_main_GT(x6, x7) :|: 0 = x7 && x4 - 1 = x6 && 0 = x5 && 0 <= x4 - 1
__init(x8, x9) -> f1_0_main_Load(x10, x11) :|: 0 <= 0
Start term: __init(arg1, arg2)

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

(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) f1_0_main_Load(arg1, arg2) -> f142_0_main_GT(arg1P, arg2P) :|: 0 <= arg1 - 1 && -1 <= arg2P - 1 && -1 <= arg2 - 1 && -1 <= arg1P - 1
(2) f142_0_main_GT(x, x1) -> f142_0_main_GT(x2, x3) :|: x1 - 1 = x3 && x = x2 && 0 <= x1 - 1
(3) f142_0_main_GT(x4, x5) -> f142_0_main_GT(x6, x7) :|: 0 = x7 && x4 - 1 = x6 && 0 = x5 && 0 <= x4 - 1
(4) __init(x8, x9) -> f1_0_main_Load(x10, x11) :|: 0 <= 0

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

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

(4)
Complex Obligation (AND)

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

(5)
Obligation:

Termination digraph:
Nodes:
(1) f142_0_main_GT(x, x1) -> f142_0_main_GT(x2, x3) :|: x1 - 1 = x3 && x = x2 && 0 <= x1 - 1

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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

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

(7)
Obligation:
Rules:
f142_0_main_GT(x2:0, x1:0) -> f142_0_main_GT(x2:0, x1:0 - 1) :|: x1:0 > 0

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

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

   f142_0_main_GT(x1, x2) -> f142_0_main_GT(x2)

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

(9)
Obligation:
Rules:
f142_0_main_GT(x1:0) -> f142_0_main_GT(x1:0 - 1) :|: x1:0 > 0

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

(10) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f142_0_main_GT(INTEGER)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(11)
Obligation:
Rules:
f142_0_main_GT(x1:0) -> f142_0_main_GT(c) :|: c = x1:0 - 1 && x1:0 > 0

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

(12) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ f142_0_main_GT ] = f142_0_main_GT_1

The following rules are decreasing:
f142_0_main_GT(x1:0) -> f142_0_main_GT(c) :|: c = x1:0 - 1 && x1:0 > 0

The following rules are bounded:
f142_0_main_GT(x1:0) -> f142_0_main_GT(c) :|: c = x1:0 - 1 && x1:0 > 0


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

(13)
YES

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

(14)
Obligation:

Termination digraph:
Nodes:
(1) f142_0_main_GT(x4, x5) -> f142_0_main_GT(x6, x7) :|: 0 = x7 && x4 - 1 = x6 && 0 = x5 && 0 <= x4 - 1

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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

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

(16)
Obligation:
Rules:
f142_0_main_GT(x4:0, cons_0) -> f142_0_main_GT(x4:0 - 1, 0) :|: x4:0 > 0 && cons_0 = 0

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

(17) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f142_0_main_GT(INTEGER, VARIABLE)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(18)
Obligation:
Rules:
f142_0_main_GT(x4:0, c) -> f142_0_main_GT(c1, c2) :|: c2 = 0 && (c1 = x4:0 - 1 && c = 0) && (x4:0 > 0 && cons_0 = 0)

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

(19) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[f142_0_main_GT(x, x1)] = x + c1*x1

The following rules are decreasing:
f142_0_main_GT(x4:0, c) -> f142_0_main_GT(c1, c2) :|: c2 = 0 && (c1 = x4:0 - 1 && c = 0) && (x4:0 > 0 && cons_0 = 0)
The following rules are bounded:
f142_0_main_GT(x4:0, c) -> f142_0_main_GT(c1, c2) :|: c2 = 0 && (c1 = x4:0 - 1 && c = 0) && (x4:0 > 0 && cons_0 = 0)

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

(20)
YES