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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 292 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 0 ms]
(6) IRSwT
(7) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
(8) IRSwT
(9) TempFilterProof [SOUND, 63 ms]
(10) IntTRS
(11) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
(12) IntTRS
(13) RankingReductionPairProof [EQUIVALENT, 0 ms]
(14) YES


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

(0)
Obligation:
Rules:
l0(Result_4HAT0, __const_21HAT0, __const_31HAT0, x_5HAT0, y_6HAT0) -> l1(Result_4HATpost, __const_21HATpost, __const_31HATpost, x_5HATpost, y_6HATpost) :|: y_6HAT0 = y_6HATpost && x_5HAT0 = x_5HATpost && __const_31HAT0 = __const_31HATpost && __const_21HAT0 = __const_21HATpost && Result_4HATpost = Result_4HATpost && 1 - __const_21HAT0 + x_5HAT0 <= 0
l0(x, x1, x2, x3, x4) -> l2(x5, x6, x7, x8, x9) :|: x4 = x9 && x3 = x8 && x2 = x7 && x1 = x6 && x = x5 && 0 <= -1 * x1 + x3
l3(x10, x11, x12, x13, x14) -> l0(x15, x16, x17, x18, x19) :|: x14 = x19 && x13 = x18 && x12 = x17 && x11 = x16 && x10 = x15
l2(x20, x21, x22, x23, x24) -> l0(x25, x26, x27, x28, x29) :|: x24 = x29 && x22 = x27 && x21 = x26 && x20 = x25 && x28 = -1 + x23 && 1 - x22 + x24 <= 0
l2(x30, x31, x32, x33, x34) -> l4(x35, x36, x37, x38, x39) :|: x33 = x38 && x32 = x37 && x31 = x36 && x30 = x35 && x39 = -1 + x34 && 0 <= -1 * x32 + x34
l4(x40, x41, x42, x43, x44) -> l2(x45, x46, x47, x48, x49) :|: x44 = x49 && x43 = x48 && x42 = x47 && x41 = x46 && x40 = x45
l5(x50, x51, x52, x53, x54) -> l3(x55, x56, x57, x58, x59) :|: x54 = x59 && x53 = x58 && x52 = x57 && x51 = x56 && x50 = x55
Start term: l5(Result_4HAT0, __const_21HAT0, __const_31HAT0, x_5HAT0, y_6HAT0)

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

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

(2)
Obligation:
Rules:
l0(Result_4HAT0, __const_21HAT0, __const_31HAT0, x_5HAT0, y_6HAT0) -> l1(Result_4HATpost, __const_21HATpost, __const_31HATpost, x_5HATpost, y_6HATpost) :|: y_6HAT0 = y_6HATpost && x_5HAT0 = x_5HATpost && __const_31HAT0 = __const_31HATpost && __const_21HAT0 = __const_21HATpost && Result_4HATpost = Result_4HATpost && 1 - __const_21HAT0 + x_5HAT0 <= 0
l0(x, x1, x2, x3, x4) -> l2(x5, x6, x7, x8, x9) :|: x4 = x9 && x3 = x8 && x2 = x7 && x1 = x6 && x = x5 && 0 <= -1 * x1 + x3
l3(x10, x11, x12, x13, x14) -> l0(x15, x16, x17, x18, x19) :|: x14 = x19 && x13 = x18 && x12 = x17 && x11 = x16 && x10 = x15
l2(x20, x21, x22, x23, x24) -> l0(x25, x26, x27, x28, x29) :|: x24 = x29 && x22 = x27 && x21 = x26 && x20 = x25 && x28 = -1 + x23 && 1 - x22 + x24 <= 0
l2(x30, x31, x32, x33, x34) -> l4(x35, x36, x37, x38, x39) :|: x33 = x38 && x32 = x37 && x31 = x36 && x30 = x35 && x39 = -1 + x34 && 0 <= -1 * x32 + x34
l4(x40, x41, x42, x43, x44) -> l2(x45, x46, x47, x48, x49) :|: x44 = x49 && x43 = x48 && x42 = x47 && x41 = x46 && x40 = x45
l5(x50, x51, x52, x53, x54) -> l3(x55, x56, x57, x58, x59) :|: x54 = x59 && x53 = x58 && x52 = x57 && x51 = x56 && x50 = x55
Start term: l5(Result_4HAT0, __const_21HAT0, __const_31HAT0, x_5HAT0, y_6HAT0)

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

(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(Result_4HAT0, __const_21HAT0, __const_31HAT0, x_5HAT0, y_6HAT0) -> l1(Result_4HATpost, __const_21HATpost, __const_31HATpost, x_5HATpost, y_6HATpost) :|: y_6HAT0 = y_6HATpost && x_5HAT0 = x_5HATpost && __const_31HAT0 = __const_31HATpost && __const_21HAT0 = __const_21HATpost && Result_4HATpost = Result_4HATpost && 1 - __const_21HAT0 + x_5HAT0 <= 0
(2) l0(x, x1, x2, x3, x4) -> l2(x5, x6, x7, x8, x9) :|: x4 = x9 && x3 = x8 && x2 = x7 && x1 = x6 && x = x5 && 0 <= -1 * x1 + x3
(3) l3(x10, x11, x12, x13, x14) -> l0(x15, x16, x17, x18, x19) :|: x14 = x19 && x13 = x18 && x12 = x17 && x11 = x16 && x10 = x15
(4) l2(x20, x21, x22, x23, x24) -> l0(x25, x26, x27, x28, x29) :|: x24 = x29 && x22 = x27 && x21 = x26 && x20 = x25 && x28 = -1 + x23 && 1 - x22 + x24 <= 0
(5) l2(x30, x31, x32, x33, x34) -> l4(x35, x36, x37, x38, x39) :|: x33 = x38 && x32 = x37 && x31 = x36 && x30 = x35 && x39 = -1 + x34 && 0 <= -1 * x32 + x34
(6) l4(x40, x41, x42, x43, x44) -> l2(x45, x46, x47, x48, x49) :|: x44 = x49 && x43 = x48 && x42 = x47 && x41 = x46 && x40 = x45
(7) l5(x50, x51, x52, x53, x54) -> l3(x55, x56, x57, x58, x59) :|: x54 = x59 && x53 = x58 && x52 = x57 && x51 = x56 && x50 = x55

Arcs:
(2) -> (4), (5)
(3) -> (1), (2)
(4) -> (1), (2)
(5) -> (6)
(6) -> (4), (5)
(7) -> (3)

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

(4)
Obligation:

Termination digraph:
Nodes:
(1) l0(x, x1, x2, x3, x4) -> l2(x5, x6, x7, x8, x9) :|: x4 = x9 && x3 = x8 && x2 = x7 && x1 = x6 && x = x5 && 0 <= -1 * x1 + x3
(2) l2(x20, x21, x22, x23, x24) -> l0(x25, x26, x27, x28, x29) :|: x24 = x29 && x22 = x27 && x21 = x26 && x20 = x25 && x28 = -1 + x23 && 1 - x22 + x24 <= 0
(3) l4(x40, x41, x42, x43, x44) -> l2(x45, x46, x47, x48, x49) :|: x44 = x49 && x43 = x48 && x42 = x47 && x41 = x46 && x40 = x45
(4) l2(x30, x31, x32, x33, x34) -> l4(x35, x36, x37, x38, x39) :|: x33 = x38 && x32 = x37 && x31 = x36 && x30 = x35 && x39 = -1 + x34 && 0 <= -1 * x32 + x34

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

This digraph is fully evaluated!

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

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

(6)
Obligation:
Rules:
l2(x30:0, x31:0, x32:0, x33:0, x34:0) -> l2(x30:0, x31:0, x32:0, x33:0, -1 + x34:0) :|: 0 <= -1 * x32:0 + x34:0
l2(x20:0, x21:0, x22:0, x23:0, x24:0) -> l2(x20:0, x21:0, x22:0, -1 + x23:0, x24:0) :|: 1 - x22:0 + x24:0 <= 0 && 0 <= -1 * x21:0 + (-1 + x23:0)

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

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

   l2(x1, x2, x3, x4, x5) -> l2(x2, x3, x4, x5)

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

(8)
Obligation:
Rules:
l2(x31:0, x32:0, x33:0, x34:0) -> l2(x31:0, x32:0, x33:0, -1 + x34:0) :|: 0 <= -1 * x32:0 + x34:0
l2(x21:0, x22:0, x23:0, x24:0) -> l2(x21:0, x22:0, -1 + x23:0, x24:0) :|: 1 - x22:0 + x24:0 <= 0 && 0 <= -1 * x21:0 + (-1 + x23:0)

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

(9) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
l2(VARIABLE, INTEGER, VARIABLE, INTEGER)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(10)
Obligation:
Rules:
l2(x31:0, x32:0, x33:0, x34:0) -> l2(x31:0, x32:0, x33:0, c) :|: c = -1 + x34:0 && 0 <= -1 * x32:0 + x34:0
l2(x21:0, x22:0, x23:0, x24:0) -> l2(x21:0, x22:0, c1, x24:0) :|: c1 = -1 + x23:0 && (1 - x22:0 + x24:0 <= 0 && 0 <= -1 * x21:0 + (-1 + x23:0))

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

(11) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l2(x, x1, x2, x3)] = -1 - x + x2

The following rules are decreasing:
l2(x21:0, x22:0, x23:0, x24:0) -> l2(x21:0, x22:0, c1, x24:0) :|: c1 = -1 + x23:0 && (1 - x22:0 + x24:0 <= 0 && 0 <= -1 * x21:0 + (-1 + x23:0))
The following rules are bounded:
l2(x21:0, x22:0, x23:0, x24:0) -> l2(x21:0, x22:0, c1, x24:0) :|: c1 = -1 + x23:0 && (1 - x22:0 + x24:0 <= 0 && 0 <= -1 * x21:0 + (-1 + x23:0))

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

(12)
Obligation:
Rules:
l2(x31:0, x32:0, x33:0, x34:0) -> l2(x31:0, x32:0, x33:0, c) :|: c = -1 + x34:0 && 0 <= -1 * x32:0 + x34:0

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

(13) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ l2 ] = -1*l2_2 + l2_4

The following rules are decreasing:
l2(x31:0, x32:0, x33:0, x34:0) -> l2(x31:0, x32:0, x33:0, c) :|: c = -1 + x34:0 && 0 <= -1 * x32:0 + x34:0

The following rules are bounded:
l2(x31:0, x32:0, x33:0, x34:0) -> l2(x31:0, x32:0, x33:0, c) :|: c = -1 + x34:0 && 0 <= -1 * x32:0 + x34:0


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

(14)
YES