NO
proof of prog.inttrs
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


Termination of the given IRSwT could be disproven:

(0) IRSwT
(1) IRSFormatTransformerProof [EQUIVALENT, 0 ms]
(2) IRSwT
(3) IRSwTTerminationDigraphProof [EQUIVALENT, 252 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 20 ms]
(6) IRSwT
(7) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
(8) IRSwT
(9) IRSwTChainingProof [EQUIVALENT, 0 ms]
(10) IRSwT
(11) IRSwTTerminationDigraphProof [EQUIVALENT, 34 ms]
(12) IRSwT
(13) IntTRSCompressionProof [EQUIVALENT, 0 ms]
(14) IRSwT
(15) IRSwTChainingProof [EQUIVALENT, 0 ms]
(16) IRSwT
(17) IRSwTTerminationDigraphProof [EQUIVALENT, 30 ms]
(18) IRSwT
(19) IntTRSCompressionProof [EQUIVALENT, 0 ms]
(20) IRSwT
(21) IRSwTChainingProof [EQUIVALENT, 0 ms]
(22) IRSwT
(23) IRSwTTerminationDigraphProof [EQUIVALENT, 31 ms]
(24) IRSwT
(25) IntTRSCompressionProof [EQUIVALENT, 0 ms]
(26) IRSwT
(27) IRSwTChainingProof [EQUIVALENT, 0 ms]
(28) IRSwT
(29) IRSwTTerminationDigraphProof [EQUIVALENT, 75 ms]
(30) AND
    (31) IRSwT
        (32) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (33) IRSwT
        (34) IRSwTChainingProof [EQUIVALENT, 0 ms]
        (35) IRSwT
        (36) IRSwTTerminationDigraphProof [EQUIVALENT, 18 ms]
        (37) IRSwT
        (38) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (39) IRSwT
        (40) IRSwTChainingProof [EQUIVALENT, 0 ms]
        (41) IRSwT
        (42) IRSwTTerminationDigraphProof [EQUIVALENT, 27 ms]
        (43) IRSwT
        (44) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (45) IRSwT
        (46) FilterProof [EQUIVALENT, 0 ms]
        (47) IntTRS
        (48) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (49) IntTRS
        (50) IntTRSPeriodicNontermProof [COMPLETE, 0 ms]
        (51) NO
    (52) IRSwT
        (53) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (54) IRSwT
        (55) IRSwTChainingProof [EQUIVALENT, 0 ms]
        (56) IRSwT
        (57) IRSwTTerminationDigraphProof [EQUIVALENT, 82 ms]
        (58) IRSwT
        (59) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (60) IRSwT


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

(0)
Obligation:
Rules:
l0(xHAT0, yHAT0) -> l1(xHATpost, yHATpost) :|: yHAT0 = yHATpost && xHATpost = 1 + 3 * xHAT0 && xHAT0 <= 1 + 2 * yHAT0 && 1 + 2 * yHAT0 <= xHAT0
l0(x, x1) -> l1(x2, x3) :|: x1 = x3 && x2 = x1 && x <= 2 * x1 && 2 * x1 <= x
l1(x4, x5) -> l2(x6, x7) :|: x5 = x7 && x4 = x6 && 1 + x4 <= 1
l1(x8, x9) -> l2(x10, x11) :|: x9 = x11 && x8 = x10 && 2 <= x8
l2(x12, x13) -> l3(x14, x15) :|: x13 = x15 && x12 = x14 && 1 + x12 <= 2
l2(x16, x17) -> l3(x18, x19) :|: x17 = x19 && x16 = x18 && 3 <= x16
l3(x20, x21) -> l4(x22, x23) :|: x21 = x23 && x20 = x22 && 1 + x20 <= 4
l3(x24, x25) -> l4(x26, x27) :|: x25 = x27 && x24 = x26 && 5 <= x24
l4(x28, x29) -> l0(x30, x31) :|: x28 = x30 && x31 = x31
l5(x32, x33) -> l1(x34, x35) :|: x33 = x35 && 1 <= x34 && x34 = x34
l6(x36, x37) -> l5(x38, x39) :|: x37 = x39 && x36 = x38
Start term: l6(xHAT0, yHAT0)

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

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

(2)
Obligation:
Rules:
l0(xHAT0, yHAT0) -> l1(xHATpost, yHATpost) :|: yHAT0 = yHATpost && xHATpost = 1 + 3 * xHAT0 && xHAT0 <= 1 + 2 * yHAT0 && 1 + 2 * yHAT0 <= xHAT0
l0(x, x1) -> l1(x2, x3) :|: x1 = x3 && x2 = x1 && x <= 2 * x1 && 2 * x1 <= x
l1(x4, x5) -> l2(x6, x7) :|: x5 = x7 && x4 = x6 && 1 + x4 <= 1
l1(x8, x9) -> l2(x10, x11) :|: x9 = x11 && x8 = x10 && 2 <= x8
l2(x12, x13) -> l3(x14, x15) :|: x13 = x15 && x12 = x14 && 1 + x12 <= 2
l2(x16, x17) -> l3(x18, x19) :|: x17 = x19 && x16 = x18 && 3 <= x16
l3(x20, x21) -> l4(x22, x23) :|: x21 = x23 && x20 = x22 && 1 + x20 <= 4
l3(x24, x25) -> l4(x26, x27) :|: x25 = x27 && x24 = x26 && 5 <= x24
l4(x28, x29) -> l0(x30, x31) :|: x28 = x30 && x31 = x31
l5(x32, x33) -> l1(x34, x35) :|: x33 = x35 && 1 <= x34 && x34 = x34
l6(x36, x37) -> l5(x38, x39) :|: x37 = x39 && x36 = x38
Start term: l6(xHAT0, yHAT0)

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

(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(xHAT0, yHAT0) -> l1(xHATpost, yHATpost) :|: yHAT0 = yHATpost && xHATpost = 1 + 3 * xHAT0 && xHAT0 <= 1 + 2 * yHAT0 && 1 + 2 * yHAT0 <= xHAT0
(2) l0(x, x1) -> l1(x2, x3) :|: x1 = x3 && x2 = x1 && x <= 2 * x1 && 2 * x1 <= x
(3) l1(x4, x5) -> l2(x6, x7) :|: x5 = x7 && x4 = x6 && 1 + x4 <= 1
(4) l1(x8, x9) -> l2(x10, x11) :|: x9 = x11 && x8 = x10 && 2 <= x8
(5) l2(x12, x13) -> l3(x14, x15) :|: x13 = x15 && x12 = x14 && 1 + x12 <= 2
(6) l2(x16, x17) -> l3(x18, x19) :|: x17 = x19 && x16 = x18 && 3 <= x16
(7) l3(x20, x21) -> l4(x22, x23) :|: x21 = x23 && x20 = x22 && 1 + x20 <= 4
(8) l3(x24, x25) -> l4(x26, x27) :|: x25 = x27 && x24 = x26 && 5 <= x24
(9) l4(x28, x29) -> l0(x30, x31) :|: x28 = x30 && x31 = x31
(10) l5(x32, x33) -> l1(x34, x35) :|: x33 = x35 && 1 <= x34 && x34 = x34
(11) l6(x36, x37) -> l5(x38, x39) :|: x37 = x39 && x36 = x38

Arcs:
(1) -> (3), (4)
(2) -> (3), (4)
(3) -> (5)
(4) -> (6)
(5) -> (7)
(6) -> (7), (8)
(7) -> (9)
(8) -> (9)
(9) -> (1), (2)
(10) -> (4)
(11) -> (10)

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

(4)
Obligation:

Termination digraph:
Nodes:
(1) l0(xHAT0, yHAT0) -> l1(xHATpost, yHATpost) :|: yHAT0 = yHATpost && xHATpost = 1 + 3 * xHAT0 && xHAT0 <= 1 + 2 * yHAT0 && 1 + 2 * yHAT0 <= xHAT0
(2) l4(x28, x29) -> l0(x30, x31) :|: x28 = x30 && x31 = x31
(3) l3(x24, x25) -> l4(x26, x27) :|: x25 = x27 && x24 = x26 && 5 <= x24
(4) l3(x20, x21) -> l4(x22, x23) :|: x21 = x23 && x20 = x22 && 1 + x20 <= 4
(5) l2(x16, x17) -> l3(x18, x19) :|: x17 = x19 && x16 = x18 && 3 <= x16
(6) l1(x8, x9) -> l2(x10, x11) :|: x9 = x11 && x8 = x10 && 2 <= x8
(7) l2(x12, x13) -> l3(x14, x15) :|: x13 = x15 && x12 = x14 && 1 + x12 <= 2
(8) l1(x4, x5) -> l2(x6, x7) :|: x5 = x7 && x4 = x6 && 1 + x4 <= 1
(9) l0(x, x1) -> l1(x2, x3) :|: x1 = x3 && x2 = x1 && x <= 2 * x1 && 2 * x1 <= x

Arcs:
(1) -> (6), (8)
(2) -> (1), (9)
(3) -> (2)
(4) -> (2)
(5) -> (3), (4)
(6) -> (5)
(7) -> (4)
(8) -> (7)
(9) -> (6), (8)

This digraph is fully evaluated!

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

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

(6)
Obligation:
Rules:
l1(x10:0, x11:0) -> l2(x10:0, x11:0) :|: x10:0 > 1
l1(x4:0, x5:0) -> l2(x4:0, x5:0) :|: x4:0 < 1
l4(sum~cons_1~times~cons_2~x31:0, x29:0) -> l1(1 + 3 * (1 + 2 * x31:0), x31:0) :|: TRUE && sum~cons_1~times~cons_2~x31:0 = 1 + 2 * x31:0
l3(x20:0, x21:0) -> l4(x20:0, x21:0) :|: x20:0 < 4
l2(x16:0, x17:0) -> l3(x16:0, x17:0) :|: x16:0 > 2
l3(x24:0, x25:0) -> l4(x24:0, x25:0) :|: x24:0 > 4
l4(x, x1) -> l1(x2, x2) :|: TRUE && x = 2 * x2
l2(x12:0, x13:0) -> l3(x12:0, x13:0) :|: x12:0 < 2

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

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

   l1(x1, x2) -> l1(x1)
   l4(x1, x2) -> l4(x1)
   l3(x1, x2) -> l3(x1)
   l2(x1, x2) -> l2(x1)

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

(8)
Obligation:
Rules:
l1(x10:0) -> l2(x10:0) :|: x10:0 > 1
l1(x4:0) -> l2(x4:0) :|: x4:0 < 1
l4(sum~cons_1~times~cons_2~x31:0) -> l1(1 + 3 * (1 + 2 * x31:0)) :|: TRUE && sum~cons_1~times~cons_2~x31:0 = 1 + 2 * x31:0
l3(x20:0) -> l4(x20:0) :|: x20:0 < 4
l2(x16:0) -> l3(x16:0) :|: x16:0 > 2
l3(x24:0) -> l4(x24:0) :|: x24:0 > 4
l4(x) -> l1(x2) :|: TRUE && x = 2 * x2
l2(x12:0) -> l3(x12:0) :|: x12:0 < 2

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

(9) IRSwTChainingProof (EQUIVALENT)
Chaining!
----------------------------------------

(10)
Obligation:
Rules:
l1(x4:0) -> l2(x4:0) :|: x4:0 < 1
l4(sum~cons_1~times~cons_2~x31:0) -> l1(1 + 3 * (1 + 2 * x31:0)) :|: TRUE && sum~cons_1~times~cons_2~x31:0 = 1 + 2 * x31:0
l3(x20:0) -> l4(x20:0) :|: x20:0 < 4
l2(x16:0) -> l3(x16:0) :|: x16:0 > 2
l1(x11) -> l3(x11) :|: TRUE && x11 >= 3
l3(x24:0) -> l4(x24:0) :|: x24:0 > 4
l4(x) -> l1(x2) :|: TRUE && x = 2 * x2
l2(x12:0) -> l3(x12:0) :|: x12:0 < 2

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

(11) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l1(x4:0) -> l2(x4:0) :|: x4:0 < 1
(2) l4(sum~cons_1~times~cons_2~x31:0) -> l1(1 + 3 * (1 + 2 * x31:0)) :|: TRUE && sum~cons_1~times~cons_2~x31:0 = 1 + 2 * x31:0
(3) l3(x20:0) -> l4(x20:0) :|: x20:0 < 4
(4) l2(x16:0) -> l3(x16:0) :|: x16:0 > 2
(5) l1(x11) -> l3(x11) :|: TRUE && x11 >= 3
(6) l3(x24:0) -> l4(x24:0) :|: x24:0 > 4
(7) l4(x) -> l1(x2) :|: TRUE && x = 2 * x2
(8) l2(x12:0) -> l3(x12:0) :|: x12:0 < 2

Arcs:
(1) -> (8)
(2) -> (1), (5)
(3) -> (2), (7)
(4) -> (3), (6)
(5) -> (3), (6)
(6) -> (2), (7)
(7) -> (1), (5)
(8) -> (3)

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

(12)
Obligation:

Termination digraph:
Nodes:
(1) l1(x4:0) -> l2(x4:0) :|: x4:0 < 1
(2) l4(sum~cons_1~times~cons_2~x31:0) -> l1(1 + 3 * (1 + 2 * x31:0)) :|: TRUE && sum~cons_1~times~cons_2~x31:0 = 1 + 2 * x31:0
(3) l3(x20:0) -> l4(x20:0) :|: x20:0 < 4
(4) l2(x12:0) -> l3(x12:0) :|: x12:0 < 2
(5) l1(x11) -> l3(x11) :|: TRUE && x11 >= 3
(6) l4(x) -> l1(x2) :|: TRUE && x = 2 * x2
(7) l3(x24:0) -> l4(x24:0) :|: x24:0 > 4

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

This digraph is fully evaluated!

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

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

(14)
Obligation:
Rules:
l4(sum~cons_1~times~cons_2~x31:0:0) -> l1(1 + 3 * (1 + 2 * x31:0:0)) :|: TRUE && sum~cons_1~times~cons_2~x31:0:0 = 1 + 2 * x31:0:0
l3(x24:0:0) -> l4(x24:0:0) :|: x24:0:0 > 4
l3(x20:0:0) -> l4(x20:0:0) :|: x20:0:0 < 4
l1(x11:0) -> l3(x11:0) :|: x11:0 > 2
l4(times~cons_2~x2:0) -> l1(x2:0) :|: TRUE && times~cons_2~x2:0 = 2 * x2:0
l1(x4:0:0) -> l3(x4:0:0) :|: x4:0:0 < 1 && x4:0:0 < 2

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

(15) IRSwTChainingProof (EQUIVALENT)
Chaining!
----------------------------------------

(16)
Obligation:
Rules:
l3(x24:0:0) -> l4(x24:0:0) :|: x24:0:0 > 4
l3(x20:0:0) -> l4(x20:0:0) :|: x20:0:0 < 4
l1(x11:0) -> l3(x11:0) :|: x11:0 > 2
l4(x10) -> l3(4 + 6 * x11) :|: TRUE && x10 + -2 * x11 = 1 && x11 >= 0
l4(times~cons_2~x2:0) -> l1(x2:0) :|: TRUE && times~cons_2~x2:0 = 2 * x2:0
l1(x4:0:0) -> l3(x4:0:0) :|: x4:0:0 < 1 && x4:0:0 < 2
l4(x17) -> l3(4 + 6 * x18) :|: TRUE && x17 + -2 * x18 = 1 && x18 <= -1

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

(17) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l3(x24:0:0) -> l4(x24:0:0) :|: x24:0:0 > 4
(2) l3(x20:0:0) -> l4(x20:0:0) :|: x20:0:0 < 4
(3) l1(x11:0) -> l3(x11:0) :|: x11:0 > 2
(4) l4(x10) -> l3(4 + 6 * x11) :|: TRUE && x10 + -2 * x11 = 1 && x11 >= 0
(5) l4(times~cons_2~x2:0) -> l1(x2:0) :|: TRUE && times~cons_2~x2:0 = 2 * x2:0
(6) l1(x4:0:0) -> l3(x4:0:0) :|: x4:0:0 < 1 && x4:0:0 < 2
(7) l4(x17) -> l3(4 + 6 * x18) :|: TRUE && x17 + -2 * x18 = 1 && x18 <= -1

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

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

(18)
Obligation:

Termination digraph:
Nodes:
(1) l3(x24:0:0) -> l4(x24:0:0) :|: x24:0:0 > 4
(2) l4(x10) -> l3(4 + 6 * x11) :|: TRUE && x10 + -2 * x11 = 1 && x11 >= 0
(3) l1(x11:0) -> l3(x11:0) :|: x11:0 > 2
(4) l4(times~cons_2~x2:0) -> l1(x2:0) :|: TRUE && times~cons_2~x2:0 = 2 * x2:0
(5) l3(x20:0:0) -> l4(x20:0:0) :|: x20:0:0 < 4
(6) l4(x17) -> l3(4 + 6 * x18) :|: TRUE && x17 + -2 * x18 = 1 && x18 <= -1
(7) l1(x4:0:0) -> l3(x4:0:0) :|: x4:0:0 < 1 && x4:0:0 < 2

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

This digraph is fully evaluated!

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

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

(20)
Obligation:
Rules:
l4(x17:0) -> l3(4 + 6 * x18:0) :|: x18:0 < 0 && x17:0 + -2 * x18:0 = 1
l3(x24:0:0:0) -> l4(x24:0:0:0) :|: x24:0:0:0 > 4
l4(x10:0) -> l3(4 + 6 * x11:0) :|: x11:0 > -1 && x10:0 + -2 * x11:0 = 1
l4(times~cons_2~x2:0:0) -> l3(x2:0:0) :|: x2:0:0 > 2 && times~cons_2~x2:0:0 = 2 * x2:0:0
l3(x20:0:0:0) -> l4(x20:0:0:0) :|: x20:0:0:0 < 4
l4(x) -> l3(x1) :|: x1 < 1 && x1 < 2 && x = 2 * x1

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

(21) IRSwTChainingProof (EQUIVALENT)
Chaining!
----------------------------------------

(22)
Obligation:
Rules:
l3(x24:0:0:0) -> l4(x24:0:0:0) :|: x24:0:0:0 > 4
l4(x10:0) -> l3(4 + 6 * x11:0) :|: x11:0 > -1 && x10:0 + -2 * x11:0 = 1
l4(times~cons_2~x2:0:0) -> l3(x2:0:0) :|: x2:0:0 > 2 && times~cons_2~x2:0:0 = 2 * x2:0:0
l3(x20:0:0:0) -> l4(x20:0:0:0) :|: x20:0:0:0 < 4
l4(x17) -> l4(4 + 6 * x18) :|: TRUE && x18 <= -1 && x17 + -2 * x18 = 1
l4(x) -> l3(x1) :|: x1 < 1 && x1 < 2 && x = 2 * x1

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

(23) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l3(x24:0:0:0) -> l4(x24:0:0:0) :|: x24:0:0:0 > 4
(2) l4(x10:0) -> l3(4 + 6 * x11:0) :|: x11:0 > -1 && x10:0 + -2 * x11:0 = 1
(3) l4(times~cons_2~x2:0:0) -> l3(x2:0:0) :|: x2:0:0 > 2 && times~cons_2~x2:0:0 = 2 * x2:0:0
(4) l3(x20:0:0:0) -> l4(x20:0:0:0) :|: x20:0:0:0 < 4
(5) l4(x17) -> l4(4 + 6 * x18) :|: TRUE && x18 <= -1 && x17 + -2 * x18 = 1
(6) l4(x) -> l3(x1) :|: x1 < 1 && x1 < 2 && x = 2 * x1

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

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

(24)
Obligation:

Termination digraph:
Nodes:
(1) l3(x24:0:0:0) -> l4(x24:0:0:0) :|: x24:0:0:0 > 4
(2) l4(x10:0) -> l3(4 + 6 * x11:0) :|: x11:0 > -1 && x10:0 + -2 * x11:0 = 1
(3) l3(x20:0:0:0) -> l4(x20:0:0:0) :|: x20:0:0:0 < 4
(4) l4(x) -> l3(x1) :|: x1 < 1 && x1 < 2 && x = 2 * x1
(5) l4(x17) -> l4(4 + 6 * x18) :|: TRUE && x18 <= -1 && x17 + -2 * x18 = 1
(6) l4(times~cons_2~x2:0:0) -> l3(x2:0:0) :|: x2:0:0 > 2 && times~cons_2~x2:0:0 = 2 * x2:0:0

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

This digraph is fully evaluated!

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

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

(26)
Obligation:
Rules:
l4(times~cons_2~x2:0:0:0) -> l3(x2:0:0:0) :|: x2:0:0:0 > 2 && times~cons_2~x2:0:0:0 = 2 * x2:0:0:0
l3(x24:0:0:0:0) -> l4(x24:0:0:0:0) :|: x24:0:0:0:0 > 4
l4(x10:0:0) -> l3(4 + 6 * x11:0:0) :|: x11:0:0 > -1 && x10:0:0 + -2 * x11:0:0 = 1
l4(times~cons_2~x1:0) -> l3(x1:0) :|: x1:0 < 1 && x1:0 < 2 && times~cons_2~x1:0 = 2 * x1:0
l3(x20:0:0:0:0) -> l4(x20:0:0:0:0) :|: x20:0:0:0:0 < 4
l4(x17:0) -> l4(4 + 6 * x18:0) :|: x17:0 + -2 * x18:0 = 1 && x18:0 < 0

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

(27) IRSwTChainingProof (EQUIVALENT)
Chaining!
----------------------------------------

(28)
Obligation:
Rules:
l3(x24:0:0:0:0) -> l4(x24:0:0:0:0) :|: x24:0:0:0:0 > 4
l4(x4) -> l4(x5) :|: TRUE && x4 + -2 * x5 = 0 && x5 >= 5
l4(x10:0:0) -> l3(4 + 6 * x11:0:0) :|: x11:0:0 > -1 && x10:0:0 + -2 * x11:0:0 = 1
l4(times~cons_2~x1:0) -> l3(x1:0) :|: x1:0 < 1 && x1:0 < 2 && times~cons_2~x1:0 = 2 * x1:0
l3(x20:0:0:0:0) -> l4(x20:0:0:0:0) :|: x20:0:0:0:0 < 4
l4(x15) -> l4(x16) :|: TRUE && x16 >= 3 && x15 + -2 * x16 = 0 && x16 <= 3
l4(x17:0) -> l4(4 + 6 * x18:0) :|: x17:0 + -2 * x18:0 = 1 && x18:0 < 0

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

(29) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l3(x24:0:0:0:0) -> l4(x24:0:0:0:0) :|: x24:0:0:0:0 > 4
(2) l4(x4) -> l4(x5) :|: TRUE && x4 + -2 * x5 = 0 && x5 >= 5
(3) l4(x10:0:0) -> l3(4 + 6 * x11:0:0) :|: x11:0:0 > -1 && x10:0:0 + -2 * x11:0:0 = 1
(4) l4(times~cons_2~x1:0) -> l3(x1:0) :|: x1:0 < 1 && x1:0 < 2 && times~cons_2~x1:0 = 2 * x1:0
(5) l3(x20:0:0:0:0) -> l4(x20:0:0:0:0) :|: x20:0:0:0:0 < 4
(6) l4(x15) -> l4(x16) :|: TRUE && x16 >= 3 && x15 + -2 * x16 = 0 && x16 <= 3
(7) l4(x17:0) -> l4(4 + 6 * x18:0) :|: x17:0 + -2 * x18:0 = 1 && x18:0 < 0

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

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

(30)
Complex Obligation (AND)

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

(31)
Obligation:

Termination digraph:
Nodes:
(1) l4(times~cons_2~x1:0) -> l3(x1:0) :|: x1:0 < 1 && x1:0 < 2 && times~cons_2~x1:0 = 2 * x1:0
(2) l4(x17:0) -> l4(4 + 6 * x18:0) :|: x17:0 + -2 * x18:0 = 1 && x18:0 < 0
(3) l3(x20:0:0:0:0) -> l4(x20:0:0:0:0) :|: x20:0:0:0:0 < 4

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

This digraph is fully evaluated!

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

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

(33)
Obligation:
Rules:
l4(x17:0:0) -> l4(4 + 6 * x18:0:0) :|: x17:0:0 + -2 * x18:0:0 = 1 && x18:0:0 < 0
l4(times~cons_2~x1:0:0) -> l4(x1:0:0) :|: x1:0:0 < 1 && x1:0:0 < 2 && x1:0:0 < 4 && times~cons_2~x1:0:0 = 2 * x1:0:0

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

(34) IRSwTChainingProof (EQUIVALENT)
Chaining!
----------------------------------------

(35)
Obligation:
Rules:
l4(x) -> l4(4 + 6 * x3) :|: TRUE && x + -2 * x1 = 1 && x1 <= -1 && 6 * x1 + -2 * x3 = -3 && x3 <= -1
l4(times~cons_2~x1:0:0) -> l4(x1:0:0) :|: x1:0:0 < 1 && x1:0:0 < 2 && x1:0:0 < 4 && times~cons_2~x1:0:0 = 2 * x1:0:0
l4(x4) -> l4(x7) :|: TRUE && x4 + -2 * x5 = 1 && x5 <= -1 && x7 <= 0 && 3 * x5 + -1 * x7 = -2

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

(36) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l4(x) -> l4(4 + 6 * x3) :|: TRUE && x + -2 * x1 = 1 && x1 <= -1 && 6 * x1 + -2 * x3 = -3 && x3 <= -1
(2) l4(times~cons_2~x1:0:0) -> l4(x1:0:0) :|: x1:0:0 < 1 && x1:0:0 < 2 && x1:0:0 < 4 && times~cons_2~x1:0:0 = 2 * x1:0:0
(3) l4(x4) -> l4(x7) :|: TRUE && x4 + -2 * x5 = 1 && x5 <= -1 && x7 <= 0 && 3 * x5 + -1 * x7 = -2

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

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

(37)
Obligation:

Termination digraph:
Nodes:
(1) l4(times~cons_2~x1:0:0) -> l4(x1:0:0) :|: x1:0:0 < 1 && x1:0:0 < 2 && x1:0:0 < 4 && times~cons_2~x1:0:0 = 2 * x1:0:0
(2) l4(x4) -> l4(x7) :|: TRUE && x4 + -2 * x5 = 1 && x5 <= -1 && x7 <= 0 && 3 * x5 + -1 * x7 = -2

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

This digraph is fully evaluated!

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

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

(39)
Obligation:
Rules:
l4(times~cons_2~x1:0:0:0) -> l4(x1:0:0:0) :|: x1:0:0:0 < 1 && x1:0:0:0 < 2 && x1:0:0:0 < 4 && times~cons_2~x1:0:0:0 = 2 * x1:0:0:0
l4(x4:0) -> l4(x7:0) :|: x7:0 < 1 && 3 * x5:0 + -1 * x7:0 = -2 && x4:0 + -2 * x5:0 = 1 && x5:0 < 0

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

(40) IRSwTChainingProof (EQUIVALENT)
Chaining!
----------------------------------------

(41)
Obligation:
Rules:
l4(x) -> l4(x3) :|: TRUE && x1 <= 0 && x + -2 * x1 = 0 && x3 <= 0 && x1 + -2 * x3 = 0
l4(x4:0) -> l4(x7:0) :|: x7:0 < 1 && 3 * x5:0 + -1 * x7:0 = -2 && x4:0 + -2 * x5:0 = 1 && x5:0 < 0
l4(x4) -> l4(x7) :|: TRUE && x5 <= 0 && x4 + -2 * x5 = 0 && x7 <= 0 && 3 * x8 + -1 * x7 = -2 && x5 + -2 * x8 = 1 && x8 <= -1

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

(42) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l4(x) -> l4(x3) :|: TRUE && x1 <= 0 && x + -2 * x1 = 0 && x3 <= 0 && x1 + -2 * x3 = 0
(2) l4(x4:0) -> l4(x7:0) :|: x7:0 < 1 && 3 * x5:0 + -1 * x7:0 = -2 && x4:0 + -2 * x5:0 = 1 && x5:0 < 0
(3) l4(x4) -> l4(x7) :|: TRUE && x5 <= 0 && x4 + -2 * x5 = 0 && x7 <= 0 && 3 * x8 + -1 * x7 = -2 && x5 + -2 * x8 = 1 && x8 <= -1

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

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

(43)
Obligation:

Termination digraph:
Nodes:
(1) l4(x) -> l4(x3) :|: TRUE && x1 <= 0 && x + -2 * x1 = 0 && x3 <= 0 && x1 + -2 * x3 = 0
(2) l4(x4:0) -> l4(x7:0) :|: x7:0 < 1 && 3 * x5:0 + -1 * x7:0 = -2 && x4:0 + -2 * x5:0 = 1 && x5:0 < 0
(3) l4(x4) -> l4(x7) :|: TRUE && x5 <= 0 && x4 + -2 * x5 = 0 && x7 <= 0 && 3 * x8 + -1 * x7 = -2 && x5 + -2 * x8 = 1 && x8 <= -1

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

This digraph is fully evaluated!

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

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

(45)
Obligation:
Rules:
l4(x4:0) -> l4(x7:0) :|: x5:0 + -2 * x8:0 = 1 && x8:0 < 0 && 3 * x8:0 + -1 * x7:0 = -2 && x7:0 < 1 && x5:0 < 1 && x4:0 + -2 * x5:0 = 0
l4(x:0) -> l4(x3:0) :|: x3:0 < 1 && x1:0 + -2 * x3:0 = 0 && x1:0 < 1 && x:0 + -2 * x1:0 = 0
l4(x4:0:0) -> l4(x7:0:0) :|: x4:0:0 + -2 * x5:0:0 = 1 && x5:0:0 < 0 && 3 * x5:0:0 + -1 * x7:0:0 = -2 && x7:0:0 < 1

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

(46) FilterProof (EQUIVALENT)
Used the following sort dictionary for filtering: 
l4(INTEGER)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(47)
Obligation:
Rules:
l4(x4:0) -> l4(x7:0) :|: x5:0 + -2 * x8:0 = 1 && x8:0 < 0 && 3 * x8:0 + -1 * x7:0 = -2 && x7:0 < 1 && x5:0 < 1 && x4:0 + -2 * x5:0 = 0
l4(x:0) -> l4(x3:0) :|: x3:0 < 1 && x1:0 + -2 * x3:0 = 0 && x1:0 < 1 && x:0 + -2 * x1:0 = 0
l4(x4:0:0) -> l4(x7:0:0) :|: x4:0:0 + -2 * x5:0:0 = 1 && x5:0:0 < 0 && 3 * x5:0:0 + -1 * x7:0:0 = -2 && x7:0:0 < 1

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

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

(49)
Obligation:
Rules:
l4(x4:0:0) -> l4(x7:0:0) :|: x5:0:0 < 1 && x4:0:0 + -2 * x5:0:0 = 0 && x7:0:0 < 1 && 3 * x8:0:0 + -1 * x7:0:0 = -2 && x8:0:0 < 0 && x5:0:0 + -2 * x8:0:0 = 1
l4(x4:0:0:0) -> l4(x7:0:0:0) :|: 3 * x5:0:0:0 + -1 * x7:0:0:0 = -2 && x7:0:0:0 < 1 && x5:0:0:0 < 0 && x4:0:0:0 + -2 * x5:0:0:0 = 1
l4(x:0:0) -> l4(x3:0:0) :|: x1:0:0 < 1 && x:0:0 + -2 * x1:0:0 = 0 && x1:0:0 + -2 * x3:0:0 = 0 && x3:0:0 < 1

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

(50) IntTRSPeriodicNontermProof (COMPLETE)
Normalized system to the following form:
f(pc, x4:0:0) -> f(1, x7:0:0) :|: pc = 1 && (x5:0:0 < 1 && x4:0:0 + -2 * x5:0:0 = 0 && x7:0:0 < 1 && 3 * x8:0:0 + -1 * x7:0:0 = -2 && x8:0:0 < 0 && x5:0:0 + -2 * x8:0:0 = 1)
f(pc, x4:0:0:0) -> f(1, x7:0:0:0) :|: pc = 1 && (3 * x5:0:0:0 + -1 * x7:0:0:0 = -2 && x7:0:0:0 < 1 && x5:0:0:0 < 0 && x4:0:0:0 + -2 * x5:0:0:0 = 1)
f(pc, x:0:0) -> f(1, x3:0:0) :|: pc = 1 && (x1:0:0 < 1 && x:0:0 + -2 * x1:0:0 = 0 && x1:0:0 + -2 * x3:0:0 = 0 && x3:0:0 < 1)
Witness term starting non-terminating reduction: f(1, 0)
----------------------------------------

(51)
NO

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

(52)
Obligation:

Termination digraph:
Nodes:
(1) l3(x24:0:0:0:0) -> l4(x24:0:0:0:0) :|: x24:0:0:0:0 > 4
(2) l4(x10:0:0) -> l3(4 + 6 * x11:0:0) :|: x11:0:0 > -1 && x10:0:0 + -2 * x11:0:0 = 1
(3) l4(x15) -> l4(x16) :|: TRUE && x16 >= 3 && x15 + -2 * x16 = 0 && x16 <= 3
(4) l4(x4) -> l4(x5) :|: TRUE && x4 + -2 * x5 = 0 && x5 >= 5

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

This digraph is fully evaluated!

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

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

(54)
Obligation:
Rules:
l4(x4:0) -> l4(x5:0) :|: x5:0 > 4 && x4:0 + -2 * x5:0 = 0
l4(x15:0) -> l4(x16:0) :|: x15:0 + -2 * x16:0 = 0 && x16:0 > 2 && x16:0 < 4
l4(x10:0:0:0) -> l4(4 + 6 * x11:0:0:0) :|: x11:0:0:0 > -1 && x10:0:0:0 + -2 * x11:0:0:0 = 1 && 6 * x11:0:0:0 > 0

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

(55) IRSwTChainingProof (EQUIVALENT)
Chaining!
----------------------------------------

(56)
Obligation:
Rules:
l4(x) -> l4(x3) :|: TRUE && x1 >= 5 && x + -2 * x1 = 0 && x3 >= 5 && x1 + -2 * x3 = 0
l4(x15:0) -> l4(x16:0) :|: x15:0 + -2 * x16:0 = 0 && x16:0 > 2 && x16:0 < 4
l4(x4) -> l4(x7) :|: TRUE && x5 >= 5 && x4 + -2 * x5 = 0 && x5 + -2 * x7 = 0 && x7 >= 3 && x7 <= 3
l4(x10:0:0:0) -> l4(4 + 6 * x11:0:0:0) :|: x11:0:0:0 > -1 && x10:0:0:0 + -2 * x11:0:0:0 = 1 && 6 * x11:0:0:0 > 0
l4(x8) -> l4(4 + 6 * x11) :|: TRUE && x9 >= 5 && x8 + -2 * x9 = 0 && x9 + -2 * x11 = 1 && x11 >= 1

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

(57) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l4(x) -> l4(x3) :|: TRUE && x1 >= 5 && x + -2 * x1 = 0 && x3 >= 5 && x1 + -2 * x3 = 0
(2) l4(x15:0) -> l4(x16:0) :|: x15:0 + -2 * x16:0 = 0 && x16:0 > 2 && x16:0 < 4
(3) l4(x4) -> l4(x7) :|: TRUE && x5 >= 5 && x4 + -2 * x5 = 0 && x5 + -2 * x7 = 0 && x7 >= 3 && x7 <= 3
(4) l4(x10:0:0:0) -> l4(4 + 6 * x11:0:0:0) :|: x11:0:0:0 > -1 && x10:0:0:0 + -2 * x11:0:0:0 = 1 && 6 * x11:0:0:0 > 0
(5) l4(x8) -> l4(4 + 6 * x11) :|: TRUE && x9 >= 5 && x8 + -2 * x9 = 0 && x9 + -2 * x11 = 1 && x11 >= 1

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

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

(58)
Obligation:

Termination digraph:
Nodes:
(1) l4(x) -> l4(x3) :|: TRUE && x1 >= 5 && x + -2 * x1 = 0 && x3 >= 5 && x1 + -2 * x3 = 0
(2) l4(x8) -> l4(4 + 6 * x11) :|: TRUE && x9 >= 5 && x8 + -2 * x9 = 0 && x9 + -2 * x11 = 1 && x11 >= 1
(3) l4(x10:0:0:0) -> l4(4 + 6 * x11:0:0:0) :|: x11:0:0:0 > -1 && x10:0:0:0 + -2 * x11:0:0:0 = 1 && 6 * x11:0:0:0 > 0
(4) l4(x4) -> l4(x7) :|: TRUE && x5 >= 5 && x4 + -2 * x5 = 0 && x5 + -2 * x7 = 0 && x7 >= 3 && x7 <= 3
(5) l4(x15:0) -> l4(x16:0) :|: x15:0 + -2 * x16:0 = 0 && x16:0 > 2 && x16:0 < 4

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

This digraph is fully evaluated!

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

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

(60)
Obligation:
Rules:
l4(x:0) -> l4(x3:0) :|: x3:0 > 4 && x1:0 + -2 * x3:0 = 0 && x1:0 > 4 && x:0 + -2 * x1:0 = 0
l4(x8:0) -> l4(4 + 6 * x11:0) :|: x9:0 + -2 * x11:0 = 1 && x11:0 > 0 && x9:0 > 4 && x8:0 + -2 * x9:0 = 0
l4(x4:0) -> l4(x7:0) :|: x7:0 > 2 && x7:0 < 4 && x5:0 + -2 * x7:0 = 0 && x5:0 > 4 && x4:0 + -2 * x5:0 = 0
l4(x15:0:0) -> l4(x16:0:0) :|: x15:0:0 + -2 * x16:0:0 = 0 && x16:0:0 > 2 && x16:0:0 < 4
l4(x10:0:0:0:0) -> l4(4 + 6 * x11:0:0:0:0) :|: x11:0:0:0:0 > -1 && x10:0:0:0:0 + -2 * x11:0:0:0:0 = 1 && 6 * x11:0:0:0:0 > 0