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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 310 ms]
(4) AND
    (5) IRSwT
        (6) IntTRSCompressionProof [EQUIVALENT, 4 ms]
        (7) IRSwT
        (8) TempFilterProof [SOUND, 6 ms]
        (9) IntTRS
        (10) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
        (11) YES
    (12) IRSwT
        (13) IntTRSCompressionProof [EQUIVALENT, 13 ms]
        (14) IRSwT
        (15) TempFilterProof [SOUND, 56 ms]
        (16) IntTRS
        (17) PolynomialOrderProcessor [EQUIVALENT, 8 ms]
        (18) YES


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

(0)
Obligation:
Rules:
f1_0_main_Load(arg1, arg2, arg3, arg4) -> f532_0_mk_LE(arg1P, arg2P, arg3P, arg4P) :|: 0 = arg4P && 0 = arg3P && 0 = arg2P && -1 = arg1P && 0 = arg2 && 0 <= arg1 - 1
f1_0_main_Load(x, x1, x3, x4) -> f532_0_mk_LE(x7, x8, x9, x10) :|: 0 <= x - 1 && -1 <= x11 - 1 && 1 = x1 && -1 = x7 && 0 = x8 && 1 = x9 && 1 = x10
f1_0_main_Load(x12, x13, x14, x15) -> f532_0_mk_LE(x16, x17, x18, x19) :|: -1 <= x20 - 1 && 1 <= x13 - 1 && -1 <= x21 - 1 && 0 <= x12 - 1 && x21 * x20 - 1 = x16 && x21 * x20 = x17 && x13 = x18 && 2 = x19
f532_0_mk_LE(x22, x23, x24, x25) -> f532_0_mk_LE(x26, x27, x28, x29) :|: x25 = x29 && x24 = x28 && x22 = x27 && x22 - 1 = x26 && -1 <= x24 - 1 && x24 <= x25 && 0 <= x23 - 1
f532_0_mk_LE(x30, x31, x32, x33) -> f532_0_mk_LE(x34, x35, x36, x37) :|: x33 + 1 = x37 && x32 = x36 && x30 = x35 && x30 - 1 = x34 && x33 <= x32 - 1 && -1 <= x33 - 1 && -1 <= x32 - 1 && 0 <= x31 - 1
__init(x38, x39, x40, x41) -> f1_0_main_Load(x42, x43, x44, x45) :|: 0 <= 0
Start term: __init(arg1, arg2, arg3, arg4)

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

(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, arg3, arg4) -> f532_0_mk_LE(arg1P, arg2P, arg3P, arg4P) :|: 0 = arg4P && 0 = arg3P && 0 = arg2P && -1 = arg1P && 0 = arg2 && 0 <= arg1 - 1
f1_0_main_Load(x, x1, x3, x4) -> f532_0_mk_LE(x7, x8, x9, x10) :|: 0 <= x - 1 && -1 <= x11 - 1 && 1 = x1 && -1 = x7 && 0 = x8 && 1 = x9 && 1 = x10
f1_0_main_Load(x12, x13, x14, x15) -> f532_0_mk_LE(x16, x17, x18, x19) :|: -1 <= x20 - 1 && 1 <= x13 - 1 && -1 <= x21 - 1 && 0 <= x12 - 1 && x21 * x20 - 1 = x16 && x21 * x20 = x17 && x13 = x18 && 2 = x19
f532_0_mk_LE(x22, x23, x24, x25) -> f532_0_mk_LE(x26, x27, x28, x29) :|: x25 = x29 && x24 = x28 && x22 = x27 && x22 - 1 = x26 && -1 <= x24 - 1 && x24 <= x25 && 0 <= x23 - 1
f532_0_mk_LE(x30, x31, x32, x33) -> f532_0_mk_LE(x34, x35, x36, x37) :|: x33 + 1 = x37 && x32 = x36 && x30 = x35 && x30 - 1 = x34 && x33 <= x32 - 1 && -1 <= x33 - 1 && -1 <= x32 - 1 && 0 <= x31 - 1
__init(x38, x39, x40, x41) -> f1_0_main_Load(x42, x43, x44, x45) :|: 0 <= 0
Start term: __init(arg1, arg2, arg3, arg4)

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

(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) f1_0_main_Load(arg1, arg2, arg3, arg4) -> f532_0_mk_LE(arg1P, arg2P, arg3P, arg4P) :|: 0 = arg4P && 0 = arg3P && 0 = arg2P && -1 = arg1P && 0 = arg2 && 0 <= arg1 - 1
(2) f1_0_main_Load(x, x1, x3, x4) -> f532_0_mk_LE(x7, x8, x9, x10) :|: 0 <= x - 1 && -1 <= x11 - 1 && 1 = x1 && -1 = x7 && 0 = x8 && 1 = x9 && 1 = x10
(3) f1_0_main_Load(x12, x13, x14, x15) -> f532_0_mk_LE(x16, x17, x18, x19) :|: -1 <= x20 - 1 && 1 <= x13 - 1 && -1 <= x21 - 1 && 0 <= x12 - 1 && x21 * x20 - 1 = x16 && x21 * x20 = x17 && x13 = x18 && 2 = x19
(4) f532_0_mk_LE(x22, x23, x24, x25) -> f532_0_mk_LE(x26, x27, x28, x29) :|: x25 = x29 && x24 = x28 && x22 = x27 && x22 - 1 = x26 && -1 <= x24 - 1 && x24 <= x25 && 0 <= x23 - 1
(5) f532_0_mk_LE(x30, x31, x32, x33) -> f532_0_mk_LE(x34, x35, x36, x37) :|: x33 + 1 = x37 && x32 = x36 && x30 = x35 && x30 - 1 = x34 && x33 <= x32 - 1 && -1 <= x33 - 1 && -1 <= x32 - 1 && 0 <= x31 - 1
(6) __init(x38, x39, x40, x41) -> f1_0_main_Load(x42, x43, x44, x45) :|: 0 <= 0

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

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

(4)
Complex Obligation (AND)

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

(5)
Obligation:

Termination digraph:
Nodes:
(1) f532_0_mk_LE(x30, x31, x32, x33) -> f532_0_mk_LE(x34, x35, x36, x37) :|: x33 + 1 = x37 && x32 = x36 && x30 = x35 && x30 - 1 = x34 && x33 <= x32 - 1 && -1 <= x33 - 1 && -1 <= x32 - 1 && 0 <= x31 - 1

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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

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

(7)
Obligation:
Rules:
f532_0_mk_LE(x30:0, x31:0, x32:0, x33:0) -> f532_0_mk_LE(x30:0 - 1, x30:0, x32:0, x33:0 + 1) :|: x32:0 > -1 && x31:0 > 0 && x33:0 <= x32:0 - 1 && x33:0 > -1

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

(8) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f532_0_mk_LE(VARIABLE, VARIABLE, INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(9)
Obligation:
Rules:
f532_0_mk_LE(x30:0, x31:0, x32:0, x33:0) -> f532_0_mk_LE(c, x30:0, x32:0, c1) :|: c1 = x33:0 + 1 && c = x30:0 - 1 && (x32:0 > -1 && x31:0 > 0 && x33:0 <= x32:0 - 1 && x33:0 > -1)

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

(10) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[f532_0_mk_LE(x, x1, x2, x3)] = x2 - x3

The following rules are decreasing:
f532_0_mk_LE(x30:0, x31:0, x32:0, x33:0) -> f532_0_mk_LE(c, x30:0, x32:0, c1) :|: c1 = x33:0 + 1 && c = x30:0 - 1 && (x32:0 > -1 && x31:0 > 0 && x33:0 <= x32:0 - 1 && x33:0 > -1)
The following rules are bounded:
f532_0_mk_LE(x30:0, x31:0, x32:0, x33:0) -> f532_0_mk_LE(c, x30:0, x32:0, c1) :|: c1 = x33:0 + 1 && c = x30:0 - 1 && (x32:0 > -1 && x31:0 > 0 && x33:0 <= x32:0 - 1 && x33:0 > -1)

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

(11)
YES

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

(12)
Obligation:

Termination digraph:
Nodes:
(1) f532_0_mk_LE(x22, x23, x24, x25) -> f532_0_mk_LE(x26, x27, x28, x29) :|: x25 = x29 && x24 = x28 && x22 = x27 && x22 - 1 = x26 && -1 <= x24 - 1 && x24 <= x25 && 0 <= x23 - 1

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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

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

(14)
Obligation:
Rules:
f532_0_mk_LE(x22:0, x23:0, x24:0, x25:0) -> f532_0_mk_LE(x22:0 - 1, x22:0, x24:0, x25:0) :|: x25:0 >= x24:0 && x24:0 > -1 && x23:0 > 0

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

(15) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f532_0_mk_LE(VARIABLE, VARIABLE, INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(16)
Obligation:
Rules:
f532_0_mk_LE(x22:0, x23:0, x24:0, x25:0) -> f532_0_mk_LE(c, x22:0, x24:0, x25:0) :|: c = x22:0 - 1 && (x25:0 >= x24:0 && x24:0 > -1 && x23:0 > 0)

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

(17) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[f532_0_mk_LE(x, x1, x2, x3)] = x^2 + 2*x1

The following rules are decreasing:
f532_0_mk_LE(x22:0, x23:0, x24:0, x25:0) -> f532_0_mk_LE(c, x22:0, x24:0, x25:0) :|: c = x22:0 - 1 && (x25:0 >= x24:0 && x24:0 > -1 && x23:0 > 0)
The following rules are bounded:
f532_0_mk_LE(x22:0, x23:0, x24:0, x25:0) -> f532_0_mk_LE(c, x22:0, x24:0, x25:0) :|: c = x22:0 - 1 && (x25:0 >= x24:0 && x24:0 > -1 && x23:0 > 0)

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

(18)
YES