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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 296 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 19 ms]
(6) IRSwT
(7) IRSwTChainingProof [EQUIVALENT, 0 ms]
(8) IRSwT
(9) IRSwTTerminationDigraphProof [EQUIVALENT, 78 ms]
(10) IRSwT
(11) IntTRSCompressionProof [EQUIVALENT, 0 ms]
(12) IRSwT
(13) IRSwTChainingProof [EQUIVALENT, 0 ms]
(14) IRSwT
(15) IRSwTTerminationDigraphProof [EQUIVALENT, 77 ms]
(16) IRSwT
(17) IntTRSCompressionProof [EQUIVALENT, 0 ms]
(18) IRSwT
(19) IRSwTChainingProof [EQUIVALENT, 0 ms]
(20) IRSwT
(21) IRSwTTerminationDigraphProof [EQUIVALENT, 231 ms]
(22) AND
    (23) IRSwT
        (24) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (25) IRSwT
        (26) TempFilterProof [SOUND, 8 ms]
        (27) IntTRS
        (28) RankingReductionPairProof [EQUIVALENT, 0 ms]
        (29) YES
    (30) IRSwT
        (31) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (32) IRSwT
        (33) TempFilterProof [SOUND, 115 ms]
        (34) IntTRS
        (35) PolynomialOrderProcessor [EQUIVALENT, 9 ms]
        (36) AND
            (37) IntTRS
                (38) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
                (39) IntTRS
                (40) RankingReductionPairProof [EQUIVALENT, 0 ms]
                (41) IntTRS
                (42) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
                (43) IntTRS
                (44) RankingReductionPairProof [EQUIVALENT, 0 ms]
                (45) YES
            (46) IntTRS
                (47) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
                (48) AND
                    (49) IntTRS
                        (50) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
                        (51) IntTRS
                        (52) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
                        (53) YES
                    (54) IntTRS
                        (55) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
                        (56) IntTRS
                        (57) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
                        (58) YES
    (59) IRSwT
        (60) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (61) IRSwT
        (62) TempFilterProof [SOUND, 3 ms]
        (63) IntTRS
        (64) RankingReductionPairProof [EQUIVALENT, 0 ms]
        (65) YES
    (66) IRSwT
        (67) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (68) IRSwT
        (69) TempFilterProof [SOUND, 5 ms]
        (70) IntTRS
        (71) RankingReductionPairProof [EQUIVALENT, 0 ms]
        (72) YES


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

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

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

(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(xHAT0, yHAT0) -> l1(xHATpost, yHATpost) :|: yHAT0 = yHATpost && xHATpost = 1 + xHAT0 && 1 + xHAT0 <= yHAT0 && -1 * xHAT0 <= yHAT0
(2) l0(x, x1) -> l1(x2, x3) :|: x = x2 && x3 = 1 + x1 && 1 + x <= -1 * x1 && x <= x1
(3) l0(x4, x5) -> l1(x6, x7) :|: x5 = x7 && x6 = -1 + x4 && 1 + x5 <= x4 && x5 <= 1 - x4
(4) l0(x8, x9) -> l1(x10, x11) :|: x8 = x10 && x11 = -1 + x9 && 2 - x9 <= x8 && x9 <= x8
(5) l1(x12, x13) -> l2(x14, x15) :|: x13 = x15 && x12 = x14 && 1 + x12 <= 0
(6) l1(x16, x17) -> l2(x18, x19) :|: x17 = x19 && x16 = x18 && 1 <= x16
(7) l2(x20, x21) -> l0(x22, x23) :|: x21 = x23 && x20 = x22 && 1 + x21 <= 0
(8) l2(x24, x25) -> l0(x26, x27) :|: x25 = x27 && x24 = x26 && 1 <= x25
(9) l3(x28, x29) -> l1(x30, x31) :|: x29 = x31 && x28 = x30
(10) l4(x32, x33) -> l3(x34, x35) :|: x33 = x35 && x32 = x34

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

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

(4)
Obligation:

Termination digraph:
Nodes:
(1) l0(xHAT0, yHAT0) -> l1(xHATpost, yHATpost) :|: yHAT0 = yHATpost && xHATpost = 1 + xHAT0 && 1 + xHAT0 <= yHAT0 && -1 * xHAT0 <= yHAT0
(2) l2(x24, x25) -> l0(x26, x27) :|: x25 = x27 && x24 = x26 && 1 <= x25
(3) l1(x12, x13) -> l2(x14, x15) :|: x13 = x15 && x12 = x14 && 1 + x12 <= 0
(4) l0(x, x1) -> l1(x2, x3) :|: x = x2 && x3 = 1 + x1 && 1 + x <= -1 * x1 && x <= x1
(5) l2(x20, x21) -> l0(x22, x23) :|: x21 = x23 && x20 = x22 && 1 + x21 <= 0
(6) l1(x16, x17) -> l2(x18, x19) :|: x17 = x19 && x16 = x18 && 1 <= x16
(7) l0(x8, x9) -> l1(x10, x11) :|: x8 = x10 && x11 = -1 + x9 && 2 - x9 <= x8 && x9 <= x8
(8) l0(x4, x5) -> l1(x6, x7) :|: x5 = x7 && x6 = -1 + x4 && 1 + x5 <= x4 && x5 <= 1 - x4

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

This digraph is fully evaluated!

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

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

(6)
Obligation:
Rules:
l2(x24:0, x25:0) -> l0(x24:0, x25:0) :|: x25:0 > 0
l1(x12:0, x13:0) -> l2(x12:0, x13:0) :|: x12:0 < 0
l0(x10:0, x9:0) -> l1(x10:0, -1 + x9:0) :|: x10:0 >= 2 - x9:0
l0(xHAT0:0, yHAT0:0) -> l1(1 + xHAT0:0, yHAT0:0) :|: yHAT0:0 >= -1 * xHAT0:0 && yHAT0:0 >= 1 + xHAT0:0
l0(x4:0, x5:0) -> l1(-1 + x4:0, x5:0) :|: x5:0 <= 1 - x4:0 && x4:0 >= 1 + x5:0
l0(x2:0, x1:0) -> l1(x2:0, 1 + x1:0) :|: x2:0 <= x1:0 && 1 + x2:0 <= -1 * x1:0
l2(x20:0, x21:0) -> l0(x20:0, x21:0) :|: x21:0 < 0
l1(x16:0, x17:0) -> l2(x16:0, x17:0) :|: x16:0 > 0

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

(7) IRSwTChainingProof (EQUIVALENT)
Chaining!
----------------------------------------

(8)
Obligation:
Rules:
l1(x12:0, x13:0) -> l2(x12:0, x13:0) :|: x12:0 < 0
l0(x10:0, x9:0) -> l1(x10:0, -1 + x9:0) :|: x10:0 >= 2 - x9:0
l2(x8, x9) -> l1(x8, -1 + x9) :|: TRUE && x9 >= 1 && x8 + x9 >= 2
l0(xHAT0:0, yHAT0:0) -> l1(1 + xHAT0:0, yHAT0:0) :|: yHAT0:0 >= -1 * xHAT0:0 && yHAT0:0 >= 1 + xHAT0:0
l2(x12, x13) -> l1(1 + x12, x13) :|: TRUE && x13 >= 1 && x13 + x12 >= 0 && x13 + -1 * x12 >= 1
l0(x4:0, x5:0) -> l1(-1 + x4:0, x5:0) :|: x5:0 <= 1 - x4:0 && x4:0 >= 1 + x5:0
l2(x16, x17) -> l1(-1 + x16, x17) :|: TRUE && x17 >= 1 && x17 + x16 <= 1 && x16 + -1 * x17 >= 1
l0(x2:0, x1:0) -> l1(x2:0, 1 + x1:0) :|: x2:0 <= x1:0 && 1 + x2:0 <= -1 * x1:0
l2(x20, x21) -> l1(x20, 1 + x21) :|: TRUE && x21 >= 1 && x20 + -1 * x21 <= 0 && x20 + x21 <= -1
l2(x20:0, x21:0) -> l0(x20:0, x21:0) :|: x21:0 < 0
l1(x16:0, x17:0) -> l2(x16:0, x17:0) :|: x16:0 > 0

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

(9) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l1(x12:0, x13:0) -> l2(x12:0, x13:0) :|: x12:0 < 0
(2) l0(x10:0, x9:0) -> l1(x10:0, -1 + x9:0) :|: x10:0 >= 2 - x9:0
(3) l2(x8, x9) -> l1(x8, -1 + x9) :|: TRUE && x9 >= 1 && x8 + x9 >= 2
(4) l0(xHAT0:0, yHAT0:0) -> l1(1 + xHAT0:0, yHAT0:0) :|: yHAT0:0 >= -1 * xHAT0:0 && yHAT0:0 >= 1 + xHAT0:0
(5) l2(x12, x13) -> l1(1 + x12, x13) :|: TRUE && x13 >= 1 && x13 + x12 >= 0 && x13 + -1 * x12 >= 1
(6) l0(x4:0, x5:0) -> l1(-1 + x4:0, x5:0) :|: x5:0 <= 1 - x4:0 && x4:0 >= 1 + x5:0
(7) l2(x16, x17) -> l1(-1 + x16, x17) :|: TRUE && x17 >= 1 && x17 + x16 <= 1 && x16 + -1 * x17 >= 1
(8) l0(x2:0, x1:0) -> l1(x2:0, 1 + x1:0) :|: x2:0 <= x1:0 && 1 + x2:0 <= -1 * x1:0
(9) l2(x20, x21) -> l1(x20, 1 + x21) :|: TRUE && x21 >= 1 && x20 + -1 * x21 <= 0 && x20 + x21 <= -1
(10) l2(x20:0, x21:0) -> l0(x20:0, x21:0) :|: x21:0 < 0
(11) l1(x16:0, x17:0) -> l2(x16:0, x17:0) :|: x16:0 > 0

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

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

(10)
Obligation:

Termination digraph:
Nodes:
(1) l1(x12:0, x13:0) -> l2(x12:0, x13:0) :|: x12:0 < 0
(2) l2(x20, x21) -> l1(x20, 1 + x21) :|: TRUE && x21 >= 1 && x20 + -1 * x21 <= 0 && x20 + x21 <= -1
(3) l0(x2:0, x1:0) -> l1(x2:0, 1 + x1:0) :|: x2:0 <= x1:0 && 1 + x2:0 <= -1 * x1:0
(4) l0(x10:0, x9:0) -> l1(x10:0, -1 + x9:0) :|: x10:0 >= 2 - x9:0
(5) l2(x20:0, x21:0) -> l0(x20:0, x21:0) :|: x21:0 < 0
(6) l1(x16:0, x17:0) -> l2(x16:0, x17:0) :|: x16:0 > 0
(7) l0(x4:0, x5:0) -> l1(-1 + x4:0, x5:0) :|: x5:0 <= 1 - x4:0 && x4:0 >= 1 + x5:0
(8) l2(x12, x13) -> l1(1 + x12, x13) :|: TRUE && x13 >= 1 && x13 + x12 >= 0 && x13 + -1 * x12 >= 1
(9) l2(x8, x9) -> l1(x8, -1 + x9) :|: TRUE && x9 >= 1 && x8 + x9 >= 2

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

This digraph is fully evaluated!

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

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

(12)
Obligation:
Rules:
l2(x8:0, x9:0) -> l1(x8:0, -1 + x9:0) :|: x8:0 + x9:0 >= 2 && x9:0 > 0
l1(x12:0:0, x13:0:0) -> l2(x12:0:0, x13:0:0) :|: x12:0:0 < 0
l2(x12:0, x13:0) -> l1(1 + x12:0, x13:0) :|: x13:0 + x12:0 >= 0 && x13:0 > 0 && x13:0 + -1 * x12:0 >= 1
l2(x20:0:0, x21:0:0) -> l1(x20:0:0, 1 + x21:0:0) :|: x21:0:0 >= x20:0:0 && 1 + x20:0:0 <= -1 * x21:0:0 && x21:0:0 < 0
l2(x20:0, x21:0) -> l1(x20:0, 1 + x21:0) :|: x20:0 + -1 * x21:0 <= 0 && x21:0 > 0 && x20:0 + x21:0 <= -1
l1(x16:0:0, x17:0:0) -> l2(x16:0:0, x17:0:0) :|: x16:0:0 > 0
l2(x, x1) -> l1(x, -1 + x1) :|: x1 < 0 && x >= 2 - x1
l2(x2, x3) -> l1(-1 + x2, x3) :|: x3 <= 1 - x2 && x2 >= 1 + x3 && x3 < 0

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

(13) IRSwTChainingProof (EQUIVALENT)
Chaining!
----------------------------------------

(14)
Obligation:
Rules:
l1(x12:0:0, x13:0:0) -> l2(x12:0:0, x13:0:0) :|: x12:0:0 < 0
l2(x8, x9) -> l2(x8, -1 + x9) :|: TRUE && x8 + x9 >= 2 && x9 >= 1 && x8 <= -1
l2(x12:0, x13:0) -> l1(1 + x12:0, x13:0) :|: x13:0 + x12:0 >= 0 && x13:0 > 0 && x13:0 + -1 * x12:0 >= 1
l2(x20:0:0, x21:0:0) -> l1(x20:0:0, 1 + x21:0:0) :|: x21:0:0 >= x20:0:0 && 1 + x20:0:0 <= -1 * x21:0:0 && x21:0:0 < 0
l2(x20:0, x21:0) -> l1(x20:0, 1 + x21:0) :|: x20:0 + -1 * x21:0 <= 0 && x21:0 > 0 && x20:0 + x21:0 <= -1
l1(x16:0:0, x17:0:0) -> l2(x16:0:0, x17:0:0) :|: x16:0:0 > 0
l2(x24, x25) -> l2(x24, -1 + x25) :|: TRUE && x24 + x25 >= 2 && x25 >= 1 && x24 >= 1
l2(x, x1) -> l1(x, -1 + x1) :|: x1 < 0 && x >= 2 - x1
l2(x2, x3) -> l1(-1 + x2, x3) :|: x3 <= 1 - x2 && x2 >= 1 + x3 && x3 < 0

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

(15) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l1(x12:0:0, x13:0:0) -> l2(x12:0:0, x13:0:0) :|: x12:0:0 < 0
(2) l2(x8, x9) -> l2(x8, -1 + x9) :|: TRUE && x8 + x9 >= 2 && x9 >= 1 && x8 <= -1
(3) l2(x12:0, x13:0) -> l1(1 + x12:0, x13:0) :|: x13:0 + x12:0 >= 0 && x13:0 > 0 && x13:0 + -1 * x12:0 >= 1
(4) l2(x20:0:0, x21:0:0) -> l1(x20:0:0, 1 + x21:0:0) :|: x21:0:0 >= x20:0:0 && 1 + x20:0:0 <= -1 * x21:0:0 && x21:0:0 < 0
(5) l2(x20:0, x21:0) -> l1(x20:0, 1 + x21:0) :|: x20:0 + -1 * x21:0 <= 0 && x21:0 > 0 && x20:0 + x21:0 <= -1
(6) l1(x16:0:0, x17:0:0) -> l2(x16:0:0, x17:0:0) :|: x16:0:0 > 0
(7) l2(x24, x25) -> l2(x24, -1 + x25) :|: TRUE && x24 + x25 >= 2 && x25 >= 1 && x24 >= 1
(8) l2(x, x1) -> l1(x, -1 + x1) :|: x1 < 0 && x >= 2 - x1
(9) l2(x2, x3) -> l1(-1 + x2, x3) :|: x3 <= 1 - x2 && x2 >= 1 + x3 && x3 < 0

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

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

(16)
Obligation:

Termination digraph:
Nodes:
(1) l1(x12:0:0, x13:0:0) -> l2(x12:0:0, x13:0:0) :|: x12:0:0 < 0
(2) l2(x20:0, x21:0) -> l1(x20:0, 1 + x21:0) :|: x20:0 + -1 * x21:0 <= 0 && x21:0 > 0 && x20:0 + x21:0 <= -1
(3) l2(x20:0:0, x21:0:0) -> l1(x20:0:0, 1 + x21:0:0) :|: x21:0:0 >= x20:0:0 && 1 + x20:0:0 <= -1 * x21:0:0 && x21:0:0 < 0
(4) l2(x12:0, x13:0) -> l1(1 + x12:0, x13:0) :|: x13:0 + x12:0 >= 0 && x13:0 > 0 && x13:0 + -1 * x12:0 >= 1
(5) l2(x24, x25) -> l2(x24, -1 + x25) :|: TRUE && x24 + x25 >= 2 && x25 >= 1 && x24 >= 1
(6) l1(x16:0:0, x17:0:0) -> l2(x16:0:0, x17:0:0) :|: x16:0:0 > 0
(7) l2(x2, x3) -> l1(-1 + x2, x3) :|: x3 <= 1 - x2 && x2 >= 1 + x3 && x3 < 0
(8) l2(x, x1) -> l1(x, -1 + x1) :|: x1 < 0 && x >= 2 - x1
(9) l2(x8, x9) -> l2(x8, -1 + x9) :|: TRUE && x8 + x9 >= 2 && x9 >= 1 && x8 <= -1

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

This digraph is fully evaluated!

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

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

(18)
Obligation:
Rules:
l1(x12:0:0:0, x13:0:0:0) -> l2(x12:0:0:0, x13:0:0:0) :|: x12:0:0:0 < 0
l2(x20:0:0:0, x21:0:0:0) -> l1(x20:0:0:0, 1 + x21:0:0:0) :|: x21:0:0:0 >= x20:0:0:0 && 1 + x20:0:0:0 <= -1 * x21:0:0:0 && x21:0:0:0 < 0
l2(x24:0, x25:0) -> l2(x24:0, -1 + x25:0) :|: x25:0 > 0 && x24:0 + x25:0 >= 2 && x24:0 > 0
l1(x16:0:0:0, x17:0:0:0) -> l2(x16:0:0:0, x17:0:0:0) :|: x16:0:0:0 > 0
l2(x:0, x1:0) -> l1(x:0, -1 + x1:0) :|: x1:0 < 0 && x:0 >= 2 - x1:0
l2(x8:0, x9:0) -> l2(x8:0, -1 + x9:0) :|: x9:0 > 0 && x8:0 + x9:0 >= 2 && x8:0 < 0
l2(x20:0:0, x21:0:0) -> l1(x20:0:0, 1 + x21:0:0) :|: x20:0:0 + -1 * x21:0:0 <= 0 && x21:0:0 > 0 && x20:0:0 + x21:0:0 <= -1
l2(x2:0, x3:0) -> l1(-1 + x2:0, x3:0) :|: x3:0 <= 1 - x2:0 && x2:0 >= 1 + x3:0 && x3:0 < 0
l2(x12:0:0, x13:0:0) -> l1(1 + x12:0:0, x13:0:0) :|: x13:0:0 + x12:0:0 >= 0 && x13:0:0 > 0 && x13:0:0 + -1 * x12:0:0 >= 1

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

(19) IRSwTChainingProof (EQUIVALENT)
Chaining!
----------------------------------------

(20)
Obligation:
Rules:
l2(x20:0:0:0, x21:0:0:0) -> l1(x20:0:0:0, 1 + x21:0:0:0) :|: x21:0:0:0 >= x20:0:0:0 && 1 + x20:0:0:0 <= -1 * x21:0:0:0 && x21:0:0:0 < 0
l1(x4, x5) -> l1(x4, 1 + x5) :|: TRUE && x4 <= -1 && x5 + -1 * x4 >= 0 && x4 + x5 <= -1 && x5 <= -1
l2(x24:0, x25:0) -> l2(x24:0, -1 + x25:0) :|: x25:0 > 0 && x24:0 + x25:0 >= 2 && x24:0 > 0
l1(x16:0:0:0, x17:0:0:0) -> l2(x16:0:0:0, x17:0:0:0) :|: x16:0:0:0 > 0
l2(x:0, x1:0) -> l1(x:0, -1 + x1:0) :|: x1:0 < 0 && x:0 >= 2 - x1:0
l1(x16, x17) -> l1(x16, -1 + x17) :|: TRUE && x16 <= -1 && x17 <= -1 && x16 + x17 >= 2
l2(x8:0, x9:0) -> l2(x8:0, -1 + x9:0) :|: x9:0 > 0 && x8:0 + x9:0 >= 2 && x8:0 < 0
l1(x20, x21) -> l2(x20, -1 + x21) :|: TRUE && x20 <= -1 && x21 >= 1 && x20 + x21 >= 2
l2(x20:0:0, x21:0:0) -> l1(x20:0:0, 1 + x21:0:0) :|: x20:0:0 + -1 * x21:0:0 <= 0 && x21:0:0 > 0 && x20:0:0 + x21:0:0 <= -1
l1(x24, x25) -> l1(x24, 1 + x25) :|: TRUE && x24 <= -1 && x24 + -1 * x25 <= 0 && x25 >= 1 && x24 + x25 <= -1
l2(x2:0, x3:0) -> l1(-1 + x2:0, x3:0) :|: x3:0 <= 1 - x2:0 && x2:0 >= 1 + x3:0 && x3:0 < 0
l1(x28, x29) -> l1(-1 + x28, x29) :|: TRUE && x28 <= -1 && x29 + x28 <= 1 && x28 + -1 * x29 >= 1 && x29 <= -1
l2(x12:0:0, x13:0:0) -> l1(1 + x12:0:0, x13:0:0) :|: x13:0:0 + x12:0:0 >= 0 && x13:0:0 > 0 && x13:0:0 + -1 * x12:0:0 >= 1
l1(x32, x33) -> l1(1 + x32, x33) :|: TRUE && x32 <= -1 && x33 + x32 >= 0 && x33 >= 1 && x33 + -1 * x32 >= 1

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

(21) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l2(x20:0:0:0, x21:0:0:0) -> l1(x20:0:0:0, 1 + x21:0:0:0) :|: x21:0:0:0 >= x20:0:0:0 && 1 + x20:0:0:0 <= -1 * x21:0:0:0 && x21:0:0:0 < 0
(2) l1(x4, x5) -> l1(x4, 1 + x5) :|: TRUE && x4 <= -1 && x5 + -1 * x4 >= 0 && x4 + x5 <= -1 && x5 <= -1
(3) l2(x24:0, x25:0) -> l2(x24:0, -1 + x25:0) :|: x25:0 > 0 && x24:0 + x25:0 >= 2 && x24:0 > 0
(4) l1(x16:0:0:0, x17:0:0:0) -> l2(x16:0:0:0, x17:0:0:0) :|: x16:0:0:0 > 0
(5) l2(x:0, x1:0) -> l1(x:0, -1 + x1:0) :|: x1:0 < 0 && x:0 >= 2 - x1:0
(6) l1(x16, x17) -> l1(x16, -1 + x17) :|: TRUE && x16 <= -1 && x17 <= -1 && x16 + x17 >= 2
(7) l2(x8:0, x9:0) -> l2(x8:0, -1 + x9:0) :|: x9:0 > 0 && x8:0 + x9:0 >= 2 && x8:0 < 0
(8) l1(x20, x21) -> l2(x20, -1 + x21) :|: TRUE && x20 <= -1 && x21 >= 1 && x20 + x21 >= 2
(9) l2(x20:0:0, x21:0:0) -> l1(x20:0:0, 1 + x21:0:0) :|: x20:0:0 + -1 * x21:0:0 <= 0 && x21:0:0 > 0 && x20:0:0 + x21:0:0 <= -1
(10) l1(x24, x25) -> l1(x24, 1 + x25) :|: TRUE && x24 <= -1 && x24 + -1 * x25 <= 0 && x25 >= 1 && x24 + x25 <= -1
(11) l2(x2:0, x3:0) -> l1(-1 + x2:0, x3:0) :|: x3:0 <= 1 - x2:0 && x2:0 >= 1 + x3:0 && x3:0 < 0
(12) l1(x28, x29) -> l1(-1 + x28, x29) :|: TRUE && x28 <= -1 && x29 + x28 <= 1 && x28 + -1 * x29 >= 1 && x29 <= -1
(13) l2(x12:0:0, x13:0:0) -> l1(1 + x12:0:0, x13:0:0) :|: x13:0:0 + x12:0:0 >= 0 && x13:0:0 > 0 && x13:0:0 + -1 * x12:0:0 >= 1
(14) l1(x32, x33) -> l1(1 + x32, x33) :|: TRUE && x32 <= -1 && x33 + x32 >= 0 && x33 >= 1 && x33 + -1 * x32 >= 1

Arcs:
(1) -> (2)
(2) -> (2)
(3) -> (3), (13)
(4) -> (3), (5), (11), (13)
(5) -> (4)
(7) -> (7), (13)
(8) -> (7), (13)
(9) -> (10), (14)
(10) -> (10), (14)
(11) -> (2), (4), (12)
(12) -> (2), (12)
(13) -> (4), (8), (14)
(14) -> (8), (14)

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

(22)
Complex Obligation (AND)

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

(23)
Obligation:

Termination digraph:
Nodes:
(1) l1(x24, x25) -> l1(x24, 1 + x25) :|: TRUE && x24 <= -1 && x24 + -1 * x25 <= 0 && x25 >= 1 && x24 + x25 <= -1

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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

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

(25)
Obligation:
Rules:
l1(x24:0, x25:0) -> l1(x24:0, 1 + x25:0) :|: x25:0 > 0 && x24:0 + x25:0 <= -1 && x24:0 < 0 && x24:0 + -1 * x25:0 <= 0

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

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

(27)
Obligation:
Rules:
l1(x24:0, x25:0) -> l1(x24:0, c) :|: c = 1 + x25:0 && (x25:0 > 0 && x24:0 + x25:0 <= -1 && x24:0 < 0 && x24:0 + -1 * x25:0 <= 0)

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

(28) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ l1 ] = -1*l1_2 + -1*l1_1

The following rules are decreasing:
l1(x24:0, x25:0) -> l1(x24:0, c) :|: c = 1 + x25:0 && (x25:0 > 0 && x24:0 + x25:0 <= -1 && x24:0 < 0 && x24:0 + -1 * x25:0 <= 0)

The following rules are bounded:
l1(x24:0, x25:0) -> l1(x24:0, c) :|: c = 1 + x25:0 && (x25:0 > 0 && x24:0 + x25:0 <= -1 && x24:0 < 0 && x24:0 + -1 * x25:0 <= 0)


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

(29)
YES

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

(30)
Obligation:

Termination digraph:
Nodes:
(1) l2(x24:0, x25:0) -> l2(x24:0, -1 + x25:0) :|: x25:0 > 0 && x24:0 + x25:0 >= 2 && x24:0 > 0
(2) l1(x16:0:0:0, x17:0:0:0) -> l2(x16:0:0:0, x17:0:0:0) :|: x16:0:0:0 > 0
(3) l2(x12:0:0, x13:0:0) -> l1(1 + x12:0:0, x13:0:0) :|: x13:0:0 + x12:0:0 >= 0 && x13:0:0 > 0 && x13:0:0 + -1 * x12:0:0 >= 1
(4) l2(x8:0, x9:0) -> l2(x8:0, -1 + x9:0) :|: x9:0 > 0 && x8:0 + x9:0 >= 2 && x8:0 < 0
(5) l1(x20, x21) -> l2(x20, -1 + x21) :|: TRUE && x20 <= -1 && x21 >= 1 && x20 + x21 >= 2
(6) l1(x32, x33) -> l1(1 + x32, x33) :|: TRUE && x32 <= -1 && x33 + x32 >= 0 && x33 >= 1 && x33 + -1 * x32 >= 1
(7) l2(x2:0, x3:0) -> l1(-1 + x2:0, x3:0) :|: x3:0 <= 1 - x2:0 && x2:0 >= 1 + x3:0 && x3:0 < 0
(8) l2(x:0, x1:0) -> l1(x:0, -1 + x1:0) :|: x1:0 < 0 && x:0 >= 2 - x1:0

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

This digraph is fully evaluated!

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

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

(32)
Obligation:
Rules:
l2(x8:0:0, x9:0:0) -> l2(x8:0:0, -1 + x9:0:0) :|: x9:0:0 > 0 && x8:0:0 + x9:0:0 >= 2 && x8:0:0 < 0
l2(x24:0:0, x25:0:0) -> l2(x24:0:0, -1 + x25:0:0) :|: x25:0:0 > 0 && x24:0:0 + x25:0:0 >= 2 && x24:0:0 > 0
l1(x16:0:0:0:0, x17:0:0:0:0) -> l2(x16:0:0:0:0, x17:0:0:0:0) :|: x16:0:0:0:0 > 0
l2(x:0:0, x1:0:0) -> l1(x:0:0, -1 + x1:0:0) :|: x1:0:0 < 0 && x:0:0 >= 2 - x1:0:0
l1(x32:0, x33:0) -> l1(1 + x32:0, x33:0) :|: x33:0 > 0 && x33:0 + -1 * x32:0 >= 1 && x32:0 < 0 && x33:0 + x32:0 >= 0
l1(x20:0, x21:0) -> l2(x20:0, -1 + x21:0) :|: x21:0 > 0 && x20:0 < 0 && x20:0 + x21:0 >= 2
l2(x2:0:0, x3:0:0) -> l1(-1 + x2:0:0, x3:0:0) :|: x3:0:0 <= 1 - x2:0:0 && x2:0:0 >= 1 + x3:0:0 && x3:0:0 < 0
l2(x12:0:0:0, x13:0:0:0) -> l1(1 + x12:0:0:0, x13:0:0:0) :|: x13:0:0:0 + x12:0:0:0 >= 0 && x13:0:0:0 > 0 && x13:0:0:0 + -1 * x12:0:0:0 >= 1

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

(33) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
l2(INTEGER, VARIABLE)
l1(INTEGER, VARIABLE)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(34)
Obligation:
Rules:
l2(x8:0:0, x9:0:0) -> l2(x8:0:0, c) :|: c = -1 + x9:0:0 && (x9:0:0 > 0 && x8:0:0 + x9:0:0 >= 2 && x8:0:0 < 0)
l2(x24:0:0, x25:0:0) -> l2(x24:0:0, c1) :|: c1 = -1 + x25:0:0 && (x25:0:0 > 0 && x24:0:0 + x25:0:0 >= 2 && x24:0:0 > 0)
l1(x16:0:0:0:0, x17:0:0:0:0) -> l2(x16:0:0:0:0, x17:0:0:0:0) :|: x16:0:0:0:0 > 0
l2(x:0:0, x1:0:0) -> l1(x:0:0, c2) :|: c2 = -1 + x1:0:0 && (x1:0:0 < 0 && x:0:0 >= 2 - x1:0:0)
l1(x32:0, x33:0) -> l1(c3, x33:0) :|: c3 = 1 + x32:0 && (x33:0 > 0 && x33:0 + -1 * x32:0 >= 1 && x32:0 < 0 && x33:0 + x32:0 >= 0)
l1(x20:0, x21:0) -> l2(x20:0, c4) :|: c4 = -1 + x21:0 && (x21:0 > 0 && x20:0 < 0 && x20:0 + x21:0 >= 2)
l2(x2:0:0, x3:0:0) -> l1(c5, x3:0:0) :|: c5 = -1 + x2:0:0 && (x3:0:0 <= 1 - x2:0:0 && x2:0:0 >= 1 + x3:0:0 && x3:0:0 < 0)
l2(x12:0:0:0, x13:0:0:0) -> l1(c6, x13:0:0:0) :|: c6 = 1 + x12:0:0:0 && (x13:0:0:0 + x12:0:0:0 >= 0 && x13:0:0:0 > 0 && x13:0:0:0 + -1 * x12:0:0:0 >= 1)

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

(35) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l2(x, x1)] = -1 + x1
[l1(x2, x3)] = -1 + x3

The following rules are decreasing:
l2(x8:0:0, x9:0:0) -> l2(x8:0:0, c) :|: c = -1 + x9:0:0 && (x9:0:0 > 0 && x8:0:0 + x9:0:0 >= 2 && x8:0:0 < 0)
l2(x24:0:0, x25:0:0) -> l2(x24:0:0, c1) :|: c1 = -1 + x25:0:0 && (x25:0:0 > 0 && x24:0:0 + x25:0:0 >= 2 && x24:0:0 > 0)
l2(x:0:0, x1:0:0) -> l1(x:0:0, c2) :|: c2 = -1 + x1:0:0 && (x1:0:0 < 0 && x:0:0 >= 2 - x1:0:0)
l1(x20:0, x21:0) -> l2(x20:0, c4) :|: c4 = -1 + x21:0 && (x21:0 > 0 && x20:0 < 0 && x20:0 + x21:0 >= 2)
The following rules are bounded:
l2(x8:0:0, x9:0:0) -> l2(x8:0:0, c) :|: c = -1 + x9:0:0 && (x9:0:0 > 0 && x8:0:0 + x9:0:0 >= 2 && x8:0:0 < 0)
l2(x24:0:0, x25:0:0) -> l2(x24:0:0, c1) :|: c1 = -1 + x25:0:0 && (x25:0:0 > 0 && x24:0:0 + x25:0:0 >= 2 && x24:0:0 > 0)
l1(x32:0, x33:0) -> l1(c3, x33:0) :|: c3 = 1 + x32:0 && (x33:0 > 0 && x33:0 + -1 * x32:0 >= 1 && x32:0 < 0 && x33:0 + x32:0 >= 0)
l1(x20:0, x21:0) -> l2(x20:0, c4) :|: c4 = -1 + x21:0 && (x21:0 > 0 && x20:0 < 0 && x20:0 + x21:0 >= 2)
l2(x12:0:0:0, x13:0:0:0) -> l1(c6, x13:0:0:0) :|: c6 = 1 + x12:0:0:0 && (x13:0:0:0 + x12:0:0:0 >= 0 && x13:0:0:0 > 0 && x13:0:0:0 + -1 * x12:0:0:0 >= 1)

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

(36)
Complex Obligation (AND)

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

(37)
Obligation:
Rules:
l1(x16:0:0:0:0, x17:0:0:0:0) -> l2(x16:0:0:0:0, x17:0:0:0:0) :|: x16:0:0:0:0 > 0
l1(x32:0, x33:0) -> l1(c3, x33:0) :|: c3 = 1 + x32:0 && (x33:0 > 0 && x33:0 + -1 * x32:0 >= 1 && x32:0 < 0 && x33:0 + x32:0 >= 0)
l2(x2:0:0, x3:0:0) -> l1(c5, x3:0:0) :|: c5 = -1 + x2:0:0 && (x3:0:0 <= 1 - x2:0:0 && x2:0:0 >= 1 + x3:0:0 && x3:0:0 < 0)
l2(x12:0:0:0, x13:0:0:0) -> l1(c6, x13:0:0:0) :|: c6 = 1 + x12:0:0:0 && (x13:0:0:0 + x12:0:0:0 >= 0 && x13:0:0:0 > 0 && x13:0:0:0 + -1 * x12:0:0:0 >= 1)

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

(38) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l1(x, x1)] = -2 + x - x*x1 - 6*x1 + x1^2
[l2(x2, x3)] = -2 + x2 - x2*x3 - 6*x3 + x3^2

The following rules are decreasing:
l2(x2:0:0, x3:0:0) -> l1(c5, x3:0:0) :|: c5 = -1 + x2:0:0 && (x3:0:0 <= 1 - x2:0:0 && x2:0:0 >= 1 + x3:0:0 && x3:0:0 < 0)
The following rules are bounded:
l2(x2:0:0, x3:0:0) -> l1(c5, x3:0:0) :|: c5 = -1 + x2:0:0 && (x3:0:0 <= 1 - x2:0:0 && x2:0:0 >= 1 + x3:0:0 && x3:0:0 < 0)

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

(39)
Obligation:
Rules:
l1(x16:0:0:0:0, x17:0:0:0:0) -> l2(x16:0:0:0:0, x17:0:0:0:0) :|: x16:0:0:0:0 > 0
l1(x32:0, x33:0) -> l1(c3, x33:0) :|: c3 = 1 + x32:0 && (x33:0 > 0 && x33:0 + -1 * x32:0 >= 1 && x32:0 < 0 && x33:0 + x32:0 >= 0)
l2(x12:0:0:0, x13:0:0:0) -> l1(c6, x13:0:0:0) :|: c6 = 1 + x12:0:0:0 && (x13:0:0:0 + x12:0:0:0 >= 0 && x13:0:0:0 > 0 && x13:0:0:0 + -1 * x12:0:0:0 >= 1)

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

(40) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ l1 ] = -2*l1_1 + 2*l1_2 + 1
[ l2 ] = -2*l2_1 + 2*l2_2

The following rules are decreasing:
l1(x16:0:0:0:0, x17:0:0:0:0) -> l2(x16:0:0:0:0, x17:0:0:0:0) :|: x16:0:0:0:0 > 0
l1(x32:0, x33:0) -> l1(c3, x33:0) :|: c3 = 1 + x32:0 && (x33:0 > 0 && x33:0 + -1 * x32:0 >= 1 && x32:0 < 0 && x33:0 + x32:0 >= 0)
l2(x12:0:0:0, x13:0:0:0) -> l1(c6, x13:0:0:0) :|: c6 = 1 + x12:0:0:0 && (x13:0:0:0 + x12:0:0:0 >= 0 && x13:0:0:0 > 0 && x13:0:0:0 + -1 * x12:0:0:0 >= 1)

The following rules are bounded:
l2(x12:0:0:0, x13:0:0:0) -> l1(c6, x13:0:0:0) :|: c6 = 1 + x12:0:0:0 && (x13:0:0:0 + x12:0:0:0 >= 0 && x13:0:0:0 > 0 && x13:0:0:0 + -1 * x12:0:0:0 >= 1)


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

(41)
Obligation:
Rules:
l1(x16:0:0:0:0, x17:0:0:0:0) -> l2(x16:0:0:0:0, x17:0:0:0:0) :|: x16:0:0:0:0 > 0
l1(x32:0, x33:0) -> l1(c3, x33:0) :|: c3 = 1 + x32:0 && (x33:0 > 0 && x33:0 + -1 * x32:0 >= 1 && x32:0 < 0 && x33:0 + x32:0 >= 0)

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

(42) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l1(x, x1)] = 1
[l2(x2, x3)] = 0

The following rules are decreasing:
l1(x16:0:0:0:0, x17:0:0:0:0) -> l2(x16:0:0:0:0, x17:0:0:0:0) :|: x16:0:0:0:0 > 0
The following rules are bounded:
l1(x16:0:0:0:0, x17:0:0:0:0) -> l2(x16:0:0:0:0, x17:0:0:0:0) :|: x16:0:0:0:0 > 0
l1(x32:0, x33:0) -> l1(c3, x33:0) :|: c3 = 1 + x32:0 && (x33:0 > 0 && x33:0 + -1 * x32:0 >= 1 && x32:0 < 0 && x33:0 + x32:0 >= 0)

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

(43)
Obligation:
Rules:
l1(x32:0, x33:0) -> l1(c3, x33:0) :|: c3 = 1 + x32:0 && (x33:0 > 0 && x33:0 + -1 * x32:0 >= 1 && x32:0 < 0 && x33:0 + x32:0 >= 0)

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

(44) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ l1 ] = -1*l1_1

The following rules are decreasing:
l1(x32:0, x33:0) -> l1(c3, x33:0) :|: c3 = 1 + x32:0 && (x33:0 > 0 && x33:0 + -1 * x32:0 >= 1 && x32:0 < 0 && x33:0 + x32:0 >= 0)

The following rules are bounded:
l1(x32:0, x33:0) -> l1(c3, x33:0) :|: c3 = 1 + x32:0 && (x33:0 > 0 && x33:0 + -1 * x32:0 >= 1 && x32:0 < 0 && x33:0 + x32:0 >= 0)


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(45)
YES

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(46)
Obligation:
Rules:
l1(x16:0:0:0:0, x17:0:0:0:0) -> l2(x16:0:0:0:0, x17:0:0:0:0) :|: x16:0:0:0:0 > 0
l2(x:0:0, x1:0:0) -> l1(x:0:0, c2) :|: c2 = -1 + x1:0:0 && (x1:0:0 < 0 && x:0:0 >= 2 - x1:0:0)
l2(x2:0:0, x3:0:0) -> l1(c5, x3:0:0) :|: c5 = -1 + x2:0:0 && (x3:0:0 <= 1 - x2:0:0 && x2:0:0 >= 1 + x3:0:0 && x3:0:0 < 0)

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(47) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l1(x, x1)] = -2 + x
[l2(x2, x3)] = -2 + x2

The following rules are decreasing:
l2(x2:0:0, x3:0:0) -> l1(c5, x3:0:0) :|: c5 = -1 + x2:0:0 && (x3:0:0 <= 1 - x2:0:0 && x2:0:0 >= 1 + x3:0:0 && x3:0:0 < 0)
The following rules are bounded:
l2(x:0:0, x1:0:0) -> l1(x:0:0, c2) :|: c2 = -1 + x1:0:0 && (x1:0:0 < 0 && x:0:0 >= 2 - x1:0:0)

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(48)
Complex Obligation (AND)

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(49)
Obligation:
Rules:
l1(x16:0:0:0:0, x17:0:0:0:0) -> l2(x16:0:0:0:0, x17:0:0:0:0) :|: x16:0:0:0:0 > 0
l2(x:0:0, x1:0:0) -> l1(x:0:0, c2) :|: c2 = -1 + x1:0:0 && (x1:0:0 < 0 && x:0:0 >= 2 - x1:0:0)

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(50) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l1(x, x1)] = -2 + x + x1
[l2(x2, x3)] = -2 + x2 + x3

The following rules are decreasing:
l2(x:0:0, x1:0:0) -> l1(x:0:0, c2) :|: c2 = -1 + x1:0:0 && (x1:0:0 < 0 && x:0:0 >= 2 - x1:0:0)
The following rules are bounded:
l2(x:0:0, x1:0:0) -> l1(x:0:0, c2) :|: c2 = -1 + x1:0:0 && (x1:0:0 < 0 && x:0:0 >= 2 - x1:0:0)

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(51)
Obligation:
Rules:
l1(x16:0:0:0:0, x17:0:0:0:0) -> l2(x16:0:0:0:0, x17:0:0:0:0) :|: x16:0:0:0:0 > 0

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(52) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l1(x, x1)] = 1
[l2(x2, x3)] = 0

The following rules are decreasing:
l1(x16:0:0:0:0, x17:0:0:0:0) -> l2(x16:0:0:0:0, x17:0:0:0:0) :|: x16:0:0:0:0 > 0
The following rules are bounded:
l1(x16:0:0:0:0, x17:0:0:0:0) -> l2(x16:0:0:0:0, x17:0:0:0:0) :|: x16:0:0:0:0 > 0

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(53)
YES

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(54)
Obligation:
Rules:
l1(x16:0:0:0:0, x17:0:0:0:0) -> l2(x16:0:0:0:0, x17:0:0:0:0) :|: x16:0:0:0:0 > 0
l2(x2:0:0, x3:0:0) -> l1(c5, x3:0:0) :|: c5 = -1 + x2:0:0 && (x3:0:0 <= 1 - x2:0:0 && x2:0:0 >= 1 + x3:0:0 && x3:0:0 < 0)

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(55) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l1(x, x1)] = -2 + x - 2*x1
[l2(x2, x3)] = -2 + x2 - 2*x3

The following rules are decreasing:
l2(x2:0:0, x3:0:0) -> l1(c5, x3:0:0) :|: c5 = -1 + x2:0:0 && (x3:0:0 <= 1 - x2:0:0 && x2:0:0 >= 1 + x3:0:0 && x3:0:0 < 0)
The following rules are bounded:
l2(x2:0:0, x3:0:0) -> l1(c5, x3:0:0) :|: c5 = -1 + x2:0:0 && (x3:0:0 <= 1 - x2:0:0 && x2:0:0 >= 1 + x3:0:0 && x3:0:0 < 0)

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(56)
Obligation:
Rules:
l1(x16:0:0:0:0, x17:0:0:0:0) -> l2(x16:0:0:0:0, x17:0:0:0:0) :|: x16:0:0:0:0 > 0

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(57) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l1(x, x1)] = 1
[l2(x2, x3)] = 0

The following rules are decreasing:
l1(x16:0:0:0:0, x17:0:0:0:0) -> l2(x16:0:0:0:0, x17:0:0:0:0) :|: x16:0:0:0:0 > 0
The following rules are bounded:
l1(x16:0:0:0:0, x17:0:0:0:0) -> l2(x16:0:0:0:0, x17:0:0:0:0) :|: x16:0:0:0:0 > 0

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(58)
YES

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(59)
Obligation:

Termination digraph:
Nodes:
(1) l1(x28, x29) -> l1(-1 + x28, x29) :|: TRUE && x28 <= -1 && x29 + x28 <= 1 && x28 + -1 * x29 >= 1 && x29 <= -1

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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(60) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(61)
Obligation:
Rules:
l1(x28:0, x29:0) -> l1(-1 + x28:0, x29:0) :|: x28:0 + -1 * x29:0 >= 1 && x29:0 < 0 && x28:0 < 0 && x29:0 + x28:0 <= 1

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(62) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
l1(INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
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(63)
Obligation:
Rules:
l1(x28:0, x29:0) -> l1(c, x29:0) :|: c = -1 + x28:0 && (x28:0 + -1 * x29:0 >= 1 && x29:0 < 0 && x28:0 < 0 && x29:0 + x28:0 <= 1)

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(64) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ l1 ] = l1_1 + -1*l1_2

The following rules are decreasing:
l1(x28:0, x29:0) -> l1(c, x29:0) :|: c = -1 + x28:0 && (x28:0 + -1 * x29:0 >= 1 && x29:0 < 0 && x28:0 < 0 && x29:0 + x28:0 <= 1)

The following rules are bounded:
l1(x28:0, x29:0) -> l1(c, x29:0) :|: c = -1 + x28:0 && (x28:0 + -1 * x29:0 >= 1 && x29:0 < 0 && x28:0 < 0 && x29:0 + x28:0 <= 1)


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(65)
YES

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(66)
Obligation:

Termination digraph:
Nodes:
(1) l1(x4, x5) -> l1(x4, 1 + x5) :|: TRUE && x4 <= -1 && x5 + -1 * x4 >= 0 && x4 + x5 <= -1 && x5 <= -1

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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(67) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(68)
Obligation:
Rules:
l1(x4:0, x5:0) -> l1(x4:0, 1 + x5:0) :|: x4:0 + x5:0 <= -1 && x5:0 < 0 && x4:0 < 0 && x5:0 + -1 * x4:0 >= 0

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(69) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
l1(INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
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(70)
Obligation:
Rules:
l1(x4:0, x5:0) -> l1(x4:0, c) :|: c = 1 + x5:0 && (x4:0 + x5:0 <= -1 && x5:0 < 0 && x4:0 < 0 && x5:0 + -1 * x4:0 >= 0)

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(71) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ l1 ] = -1*l1_2

The following rules are decreasing:
l1(x4:0, x5:0) -> l1(x4:0, c) :|: c = 1 + x5:0 && (x4:0 + x5:0 <= -1 && x5:0 < 0 && x4:0 < 0 && x5:0 + -1 * x4:0 >= 0)

The following rules are bounded:
l1(x4:0, x5:0) -> l1(x4:0, c) :|: c = 1 + x5:0 && (x4:0 + x5:0 <= -1 && x5:0 < 0 && x4:0 < 0 && x5:0 + -1 * x4:0 >= 0)


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(72)
YES