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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 321 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 49 ms]
(6) IRSwT
(7) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
(8) IRSwT
(9) TempFilterProof [SOUND, 15 ms]
(10) IntTRS
(11) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
(12) YES


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

(0)
Obligation:
Rules:
l0(Result_4HAT0, __disjvr_0HAT0, k_7HAT0, w_8HAT0, x_5HAT0, y_6HAT0) -> l2(Result_4HATpost, __disjvr_0HATpost, k_7HATpost, w_8HATpost, x_5HATpost, y_6HATpost) :|: w_8HAT0 = w_8HATpost && k_7HAT0 = k_7HATpost && __disjvr_0HAT0 = __disjvr_0HATpost && Result_4HAT0 = Result_4HATpost && y_6HATpost = 1 + y_6HAT0 && x_5HATpost = 1 + x_5HAT0 && 0 <= -2 - x_5HAT0
l2(x, x1, x2, x3, x4, x5) -> l1(x6, x7, x8, x9, x10, x11) :|: x5 = x11 && x4 = x10 && x3 = x9 && x2 = x8 && x1 = x7 && x = x6 && x7 = x1
l1(x12, x13, x14, x15, x16, x17) -> l0(x18, x19, x20, x21, x22, x23) :|: x17 = x23 && x16 = x22 && x15 = x21 && x14 = x20 && x13 = x19 && x12 = x18
l0(x24, x25, x26, x27, x28, x29) -> l3(x30, x31, x32, x33, x34, x35) :|: x29 = x35 && x28 = x34 && x27 = x33 && x26 = x32 && x25 = x31 && x30 = x30 && -1 - x28 <= 0
l4(x36, x37, x38, x39, x40, x41) -> l0(x42, x43, x44, x45, x46, x47) :|: x41 = x47 && x40 = x46 && x39 = x45 && x38 = x44 && x37 = x43 && x36 = x42 && 0 <= -2 + x39 && 0 <= -2 + x38
l4(x48, x49, x50, x51, x52, x53) -> l3(x54, x55, x56, x57, x58, x59) :|: x53 = x59 && x52 = x58 && x51 = x57 && x50 = x56 && x49 = x55 && x54 = x54 && -1 + x50 <= 0
l5(x60, x61, x62, x63, x64, x65) -> l4(x66, x67, x68, x69, x70, x71) :|: x65 = x71 && x64 = x70 && x63 = x69 && x62 = x68 && x61 = x67 && x60 = x66
Start term: l5(Result_4HAT0, __disjvr_0HAT0, k_7HAT0, w_8HAT0, 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, __disjvr_0HAT0, k_7HAT0, w_8HAT0, x_5HAT0, y_6HAT0) -> l2(Result_4HATpost, __disjvr_0HATpost, k_7HATpost, w_8HATpost, x_5HATpost, y_6HATpost) :|: w_8HAT0 = w_8HATpost && k_7HAT0 = k_7HATpost && __disjvr_0HAT0 = __disjvr_0HATpost && Result_4HAT0 = Result_4HATpost && y_6HATpost = 1 + y_6HAT0 && x_5HATpost = 1 + x_5HAT0 && 0 <= -2 - x_5HAT0
l2(x, x1, x2, x3, x4, x5) -> l1(x6, x7, x8, x9, x10, x11) :|: x5 = x11 && x4 = x10 && x3 = x9 && x2 = x8 && x1 = x7 && x = x6 && x7 = x1
l1(x12, x13, x14, x15, x16, x17) -> l0(x18, x19, x20, x21, x22, x23) :|: x17 = x23 && x16 = x22 && x15 = x21 && x14 = x20 && x13 = x19 && x12 = x18
l0(x24, x25, x26, x27, x28, x29) -> l3(x30, x31, x32, x33, x34, x35) :|: x29 = x35 && x28 = x34 && x27 = x33 && x26 = x32 && x25 = x31 && x30 = x30 && -1 - x28 <= 0
l4(x36, x37, x38, x39, x40, x41) -> l0(x42, x43, x44, x45, x46, x47) :|: x41 = x47 && x40 = x46 && x39 = x45 && x38 = x44 && x37 = x43 && x36 = x42 && 0 <= -2 + x39 && 0 <= -2 + x38
l4(x48, x49, x50, x51, x52, x53) -> l3(x54, x55, x56, x57, x58, x59) :|: x53 = x59 && x52 = x58 && x51 = x57 && x50 = x56 && x49 = x55 && x54 = x54 && -1 + x50 <= 0
l5(x60, x61, x62, x63, x64, x65) -> l4(x66, x67, x68, x69, x70, x71) :|: x65 = x71 && x64 = x70 && x63 = x69 && x62 = x68 && x61 = x67 && x60 = x66
Start term: l5(Result_4HAT0, __disjvr_0HAT0, k_7HAT0, w_8HAT0, x_5HAT0, y_6HAT0)

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

(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(Result_4HAT0, __disjvr_0HAT0, k_7HAT0, w_8HAT0, x_5HAT0, y_6HAT0) -> l2(Result_4HATpost, __disjvr_0HATpost, k_7HATpost, w_8HATpost, x_5HATpost, y_6HATpost) :|: w_8HAT0 = w_8HATpost && k_7HAT0 = k_7HATpost && __disjvr_0HAT0 = __disjvr_0HATpost && Result_4HAT0 = Result_4HATpost && y_6HATpost = 1 + y_6HAT0 && x_5HATpost = 1 + x_5HAT0 && 0 <= -2 - x_5HAT0
(2) l2(x, x1, x2, x3, x4, x5) -> l1(x6, x7, x8, x9, x10, x11) :|: x5 = x11 && x4 = x10 && x3 = x9 && x2 = x8 && x1 = x7 && x = x6 && x7 = x1
(3) l1(x12, x13, x14, x15, x16, x17) -> l0(x18, x19, x20, x21, x22, x23) :|: x17 = x23 && x16 = x22 && x15 = x21 && x14 = x20 && x13 = x19 && x12 = x18
(4) l0(x24, x25, x26, x27, x28, x29) -> l3(x30, x31, x32, x33, x34, x35) :|: x29 = x35 && x28 = x34 && x27 = x33 && x26 = x32 && x25 = x31 && x30 = x30 && -1 - x28 <= 0
(5) l4(x36, x37, x38, x39, x40, x41) -> l0(x42, x43, x44, x45, x46, x47) :|: x41 = x47 && x40 = x46 && x39 = x45 && x38 = x44 && x37 = x43 && x36 = x42 && 0 <= -2 + x39 && 0 <= -2 + x38
(6) l4(x48, x49, x50, x51, x52, x53) -> l3(x54, x55, x56, x57, x58, x59) :|: x53 = x59 && x52 = x58 && x51 = x57 && x50 = x56 && x49 = x55 && x54 = x54 && -1 + x50 <= 0
(7) l5(x60, x61, x62, x63, x64, x65) -> l4(x66, x67, x68, x69, x70, x71) :|: x65 = x71 && x64 = x70 && x63 = x69 && x62 = x68 && x61 = x67 && x60 = x66

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

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

(4)
Obligation:

Termination digraph:
Nodes:
(1) l0(Result_4HAT0, __disjvr_0HAT0, k_7HAT0, w_8HAT0, x_5HAT0, y_6HAT0) -> l2(Result_4HATpost, __disjvr_0HATpost, k_7HATpost, w_8HATpost, x_5HATpost, y_6HATpost) :|: w_8HAT0 = w_8HATpost && k_7HAT0 = k_7HATpost && __disjvr_0HAT0 = __disjvr_0HATpost && Result_4HAT0 = Result_4HATpost && y_6HATpost = 1 + y_6HAT0 && x_5HATpost = 1 + x_5HAT0 && 0 <= -2 - x_5HAT0
(2) l1(x12, x13, x14, x15, x16, x17) -> l0(x18, x19, x20, x21, x22, x23) :|: x17 = x23 && x16 = x22 && x15 = x21 && x14 = x20 && x13 = x19 && x12 = x18
(3) l2(x, x1, x2, x3, x4, x5) -> l1(x6, x7, x8, x9, x10, x11) :|: x5 = x11 && x4 = x10 && x3 = x9 && x2 = x8 && x1 = x7 && x = x6 && x7 = x1

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

This digraph is fully evaluated!

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

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

(6)
Obligation:
Rules:
l1(Result_4HATpost:0, __disjvr_0HATpost:0, k_7HATpost:0, w_8HATpost:0, x16:0, x17:0) -> l1(Result_4HATpost:0, __disjvr_0HATpost:0, k_7HATpost:0, w_8HATpost:0, 1 + x16:0, 1 + x17:0) :|: x16:0 < -1

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

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

   l1(x1, x2, x3, x4, x5, x6) -> l1(x5)

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

(8)
Obligation:
Rules:
l1(x16:0) -> l1(1 + x16:0) :|: x16:0 < -1

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

(9) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
l1(INTEGER)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(10)
Obligation:
Rules:
l1(x16:0) -> l1(c) :|: c = 1 + x16:0 && x16:0 < -1

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

(11) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l1(x)] = -x

The following rules are decreasing:
l1(x16:0) -> l1(c) :|: c = 1 + x16:0 && x16:0 < -1
The following rules are bounded:
l1(x16:0) -> l1(c) :|: c = 1 + x16:0 && x16:0 < -1

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

(12)
YES