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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 807 ms]
(4) AND
    (5) IRSwT
        (6) IntTRSCompressionProof [EQUIVALENT, 22 ms]
        (7) IRSwT
        (8) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (9) IRSwT
        (10) TempFilterProof [SOUND, 34 ms]
        (11) IntTRS
        (12) RankingReductionPairProof [EQUIVALENT, 0 ms]
        (13) IntTRS
        (14) RankingReductionPairProof [EQUIVALENT, 0 ms]
        (15) YES
    (16) IRSwT
        (17) IntTRSCompressionProof [EQUIVALENT, 11 ms]
        (18) IRSwT
        (19) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (20) IRSwT
        (21) TempFilterProof [SOUND, 99 ms]
        (22) IntTRS
        (23) RankingReductionPairProof [EQUIVALENT, 0 ms]
        (24) IntTRS
        (25) PolynomialOrderProcessor [EQUIVALENT, 6 ms]
        (26) AND
            (27) IntTRS
                (28) RankingReductionPairProof [EQUIVALENT, 0 ms]
                (29) YES
            (30) IntTRS
                (31) RankingReductionPairProof [EQUIVALENT, 0 ms]
                (32) IntTRS
                (33) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
                (34) YES


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

(0)
Obligation:
Rules:
l0(i4HAT0, j5HAT0, k6HAT0) -> l1(i4HATpost, j5HATpost, k6HATpost) :|: k6HAT0 = k6HATpost && j5HAT0 = j5HATpost && i4HAT0 = i4HATpost
l2(x, x1, x2) -> l3(x3, x4, x5) :|: x2 = x5 && x1 = x4 && x = x3
l4(x6, x7, x8) -> l5(x9, x10, x11) :|: x8 = x11 && x7 = x10 && x6 = x9
l6(x12, x13, x14) -> l7(x15, x16, x17) :|: x14 = x17 && x13 = x16 && x12 = x15
l8(x18, x19, x20) -> l6(x21, x22, x23) :|: x20 = x23 && x18 = x21 && x22 = 1 + x19 && 6 <= x20
l8(x24, x25, x26) -> l9(x27, x28, x29) :|: x25 = x28 && x24 = x27 && x29 = 1 + x26 && x26 <= 5
l7(x30, x31, x32) -> l4(x33, x34, x35) :|: x32 = x35 && x31 = x34 && x33 = 1 + x30 && 6 <= x31
l7(x36, x37, x38) -> l9(x39, x40, x41) :|: x37 = x40 && x36 = x39 && x41 = 1 && x37 <= 5
l5(x42, x43, x44) -> l10(x45, x46, x47) :|: x44 = x47 && x43 = x46 && x42 = x45 && 6 <= x42
l5(x48, x49, x50) -> l6(x51, x52, x53) :|: x50 = x53 && x48 = x51 && x52 = 1 && x48 <= 5
l9(x54, x55, x56) -> l8(x57, x58, x59) :|: x56 = x59 && x55 = x58 && x54 = x57
l3(x60, x61, x62) -> l0(x63, x64, x65) :|: x62 = x65 && x61 = x64 && x63 = 1 + x60 && 6 <= x61
l3(x66, x67, x68) -> l2(x69, x70, x71) :|: x68 = x71 && x66 = x69 && x70 = 1 + x67 && x67 <= 5
l1(x72, x73, x74) -> l4(x75, x76, x77) :|: x74 = x77 && x73 = x76 && x75 = 1 && 6 <= x72
l1(x78, x79, x80) -> l2(x81, x82, x83) :|: x80 = x83 && x78 = x81 && x82 = 1 && x78 <= 5
l11(x84, x85, x86) -> l0(x87, x88, x89) :|: x86 = x89 && x85 = x88 && x87 = 1
l12(x90, x91, x92) -> l11(x93, x94, x95) :|: x92 = x95 && x91 = x94 && x90 = x93
Start term: l12(i4HAT0, j5HAT0, k6HAT0)

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

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

(2)
Obligation:
Rules:
l0(i4HAT0, j5HAT0, k6HAT0) -> l1(i4HATpost, j5HATpost, k6HATpost) :|: k6HAT0 = k6HATpost && j5HAT0 = j5HATpost && i4HAT0 = i4HATpost
l2(x, x1, x2) -> l3(x3, x4, x5) :|: x2 = x5 && x1 = x4 && x = x3
l4(x6, x7, x8) -> l5(x9, x10, x11) :|: x8 = x11 && x7 = x10 && x6 = x9
l6(x12, x13, x14) -> l7(x15, x16, x17) :|: x14 = x17 && x13 = x16 && x12 = x15
l8(x18, x19, x20) -> l6(x21, x22, x23) :|: x20 = x23 && x18 = x21 && x22 = 1 + x19 && 6 <= x20
l8(x24, x25, x26) -> l9(x27, x28, x29) :|: x25 = x28 && x24 = x27 && x29 = 1 + x26 && x26 <= 5
l7(x30, x31, x32) -> l4(x33, x34, x35) :|: x32 = x35 && x31 = x34 && x33 = 1 + x30 && 6 <= x31
l7(x36, x37, x38) -> l9(x39, x40, x41) :|: x37 = x40 && x36 = x39 && x41 = 1 && x37 <= 5
l5(x42, x43, x44) -> l10(x45, x46, x47) :|: x44 = x47 && x43 = x46 && x42 = x45 && 6 <= x42
l5(x48, x49, x50) -> l6(x51, x52, x53) :|: x50 = x53 && x48 = x51 && x52 = 1 && x48 <= 5
l9(x54, x55, x56) -> l8(x57, x58, x59) :|: x56 = x59 && x55 = x58 && x54 = x57
l3(x60, x61, x62) -> l0(x63, x64, x65) :|: x62 = x65 && x61 = x64 && x63 = 1 + x60 && 6 <= x61
l3(x66, x67, x68) -> l2(x69, x70, x71) :|: x68 = x71 && x66 = x69 && x70 = 1 + x67 && x67 <= 5
l1(x72, x73, x74) -> l4(x75, x76, x77) :|: x74 = x77 && x73 = x76 && x75 = 1 && 6 <= x72
l1(x78, x79, x80) -> l2(x81, x82, x83) :|: x80 = x83 && x78 = x81 && x82 = 1 && x78 <= 5
l11(x84, x85, x86) -> l0(x87, x88, x89) :|: x86 = x89 && x85 = x88 && x87 = 1
l12(x90, x91, x92) -> l11(x93, x94, x95) :|: x92 = x95 && x91 = x94 && x90 = x93
Start term: l12(i4HAT0, j5HAT0, k6HAT0)

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

(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(i4HAT0, j5HAT0, k6HAT0) -> l1(i4HATpost, j5HATpost, k6HATpost) :|: k6HAT0 = k6HATpost && j5HAT0 = j5HATpost && i4HAT0 = i4HATpost
(2) l2(x, x1, x2) -> l3(x3, x4, x5) :|: x2 = x5 && x1 = x4 && x = x3
(3) l4(x6, x7, x8) -> l5(x9, x10, x11) :|: x8 = x11 && x7 = x10 && x6 = x9
(4) l6(x12, x13, x14) -> l7(x15, x16, x17) :|: x14 = x17 && x13 = x16 && x12 = x15
(5) l8(x18, x19, x20) -> l6(x21, x22, x23) :|: x20 = x23 && x18 = x21 && x22 = 1 + x19 && 6 <= x20
(6) l8(x24, x25, x26) -> l9(x27, x28, x29) :|: x25 = x28 && x24 = x27 && x29 = 1 + x26 && x26 <= 5
(7) l7(x30, x31, x32) -> l4(x33, x34, x35) :|: x32 = x35 && x31 = x34 && x33 = 1 + x30 && 6 <= x31
(8) l7(x36, x37, x38) -> l9(x39, x40, x41) :|: x37 = x40 && x36 = x39 && x41 = 1 && x37 <= 5
(9) l5(x42, x43, x44) -> l10(x45, x46, x47) :|: x44 = x47 && x43 = x46 && x42 = x45 && 6 <= x42
(10) l5(x48, x49, x50) -> l6(x51, x52, x53) :|: x50 = x53 && x48 = x51 && x52 = 1 && x48 <= 5
(11) l9(x54, x55, x56) -> l8(x57, x58, x59) :|: x56 = x59 && x55 = x58 && x54 = x57
(12) l3(x60, x61, x62) -> l0(x63, x64, x65) :|: x62 = x65 && x61 = x64 && x63 = 1 + x60 && 6 <= x61
(13) l3(x66, x67, x68) -> l2(x69, x70, x71) :|: x68 = x71 && x66 = x69 && x70 = 1 + x67 && x67 <= 5
(14) l1(x72, x73, x74) -> l4(x75, x76, x77) :|: x74 = x77 && x73 = x76 && x75 = 1 && 6 <= x72
(15) l1(x78, x79, x80) -> l2(x81, x82, x83) :|: x80 = x83 && x78 = x81 && x82 = 1 && x78 <= 5
(16) l11(x84, x85, x86) -> l0(x87, x88, x89) :|: x86 = x89 && x85 = x88 && x87 = 1
(17) l12(x90, x91, x92) -> l11(x93, x94, x95) :|: x92 = x95 && x91 = x94 && x90 = x93

Arcs:
(1) -> (14), (15)
(2) -> (12), (13)
(3) -> (9), (10)
(4) -> (7), (8)
(5) -> (4)
(6) -> (11)
(7) -> (3)
(8) -> (11)
(10) -> (4)
(11) -> (5), (6)
(12) -> (1)
(13) -> (2)
(14) -> (3)
(15) -> (2)
(16) -> (1)
(17) -> (16)

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

(4)
Complex Obligation (AND)

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

(5)
Obligation:

Termination digraph:
Nodes:
(1) l0(i4HAT0, j5HAT0, k6HAT0) -> l1(i4HATpost, j5HATpost, k6HATpost) :|: k6HAT0 = k6HATpost && j5HAT0 = j5HATpost && i4HAT0 = i4HATpost
(2) l3(x60, x61, x62) -> l0(x63, x64, x65) :|: x62 = x65 && x61 = x64 && x63 = 1 + x60 && 6 <= x61
(3) l2(x, x1, x2) -> l3(x3, x4, x5) :|: x2 = x5 && x1 = x4 && x = x3
(4) l1(x78, x79, x80) -> l2(x81, x82, x83) :|: x80 = x83 && x78 = x81 && x82 = 1 && x78 <= 5
(5) l3(x66, x67, x68) -> l2(x69, x70, x71) :|: x68 = x71 && x66 = x69 && x70 = 1 + x67 && x67 <= 5

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

This digraph is fully evaluated!

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

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

(7)
Obligation:
Rules:
l2(x3:0, j5HATpost:0, k6HATpost:0) -> l2(1 + x3:0, 1, k6HATpost:0) :|: j5HATpost:0 > 5 && x3:0 < 5
l2(x, x1, x2) -> l2(x, 1 + x1, x2) :|: x1 < 6

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

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

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

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

(9)
Obligation:
Rules:
l2(x3:0, j5HATpost:0) -> l2(1 + x3:0, 1) :|: j5HATpost:0 > 5 && x3:0 < 5
l2(x, x1) -> l2(x, 1 + x1) :|: x1 < 6

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

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

(11)
Obligation:
Rules:
l2(x3:0, j5HATpost:0) -> l2(c, c1) :|: c1 = 1 && c = 1 + x3:0 && (j5HATpost:0 > 5 && x3:0 < 5)
l2(x, x1) -> l2(x, c2) :|: c2 = 1 + x1 && x1 < 6

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

(12) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ l2 ] = -1*l2_1

The following rules are decreasing:
l2(x3:0, j5HATpost:0) -> l2(c, c1) :|: c1 = 1 && c = 1 + x3:0 && (j5HATpost:0 > 5 && x3:0 < 5)

The following rules are bounded:
l2(x3:0, j5HATpost:0) -> l2(c, c1) :|: c1 = 1 && c = 1 + x3:0 && (j5HATpost:0 > 5 && x3:0 < 5)


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

(13)
Obligation:
Rules:
l2(x, x1) -> l2(x, c2) :|: c2 = 1 + x1 && x1 < 6

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

(14) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ l2 ] = -1*l2_2

The following rules are decreasing:
l2(x, x1) -> l2(x, c2) :|: c2 = 1 + x1 && x1 < 6

The following rules are bounded:
l2(x, x1) -> l2(x, c2) :|: c2 = 1 + x1 && x1 < 6


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

(15)
YES

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

(16)
Obligation:

Termination digraph:
Nodes:
(1) l4(x6, x7, x8) -> l5(x9, x10, x11) :|: x8 = x11 && x7 = x10 && x6 = x9
(2) l7(x30, x31, x32) -> l4(x33, x34, x35) :|: x32 = x35 && x31 = x34 && x33 = 1 + x30 && 6 <= x31
(3) l6(x12, x13, x14) -> l7(x15, x16, x17) :|: x14 = x17 && x13 = x16 && x12 = x15
(4) l5(x48, x49, x50) -> l6(x51, x52, x53) :|: x50 = x53 && x48 = x51 && x52 = 1 && x48 <= 5
(5) l8(x18, x19, x20) -> l6(x21, x22, x23) :|: x20 = x23 && x18 = x21 && x22 = 1 + x19 && 6 <= x20
(6) l9(x54, x55, x56) -> l8(x57, x58, x59) :|: x56 = x59 && x55 = x58 && x54 = x57
(7) l7(x36, x37, x38) -> l9(x39, x40, x41) :|: x37 = x40 && x36 = x39 && x41 = 1 && x37 <= 5
(8) l8(x24, x25, x26) -> l9(x27, x28, x29) :|: x25 = x28 && x24 = x27 && x29 = 1 + x26 && x26 <= 5

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

This digraph is fully evaluated!

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

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

(18)
Obligation:
Rules:
l6(x12:0, x13:0, x14:0) -> l9(x12:0, x13:0, 1) :|: x13:0 < 6
l9(x27:0, x28:0, x56:0) -> l9(x27:0, x28:0, 1 + x56:0) :|: x56:0 < 6
l9(x21:0, x55:0, x23:0) -> l6(x21:0, 1 + x55:0, x23:0) :|: x23:0 > 5
l6(x, x1, x2) -> l6(1 + x, 1, x2) :|: x1 > 5 && x < 5

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

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

   l6(x1, x2, x3) -> l6(x1, x2)

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

(20)
Obligation:
Rules:
l6(x12:0, x13:0) -> l9(x12:0, x13:0, 1) :|: x13:0 < 6
l9(x27:0, x28:0, x56:0) -> l9(x27:0, x28:0, 1 + x56:0) :|: x56:0 < 6
l9(x21:0, x55:0, x23:0) -> l6(x21:0, 1 + x55:0) :|: x23:0 > 5
l6(x, x1) -> l6(1 + x, 1) :|: x1 > 5 && x < 5

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

(21) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
l6(VARIABLE, VARIABLE)
l9(VARIABLE, VARIABLE, VARIABLE)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(22)
Obligation:
Rules:
l6(x12:0, x13:0) -> l9(x12:0, x13:0, c) :|: c = 1 && x13:0 < 6
l9(x27:0, x28:0, x56:0) -> l9(x27:0, x28:0, c1) :|: c1 = 1 + x56:0 && x56:0 < 6
l9(x21:0, x55:0, x23:0) -> l6(x21:0, c2) :|: c2 = 1 + x55:0 && x23:0 > 5
l6(x, x1) -> l6(c3, c4) :|: c4 = 1 && c3 = 1 + x && (x1 > 5 && x < 5)

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

(23) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ l6 ] = -2*l6_1
[ l9 ] = -2*l9_1

The following rules are decreasing:
l6(x, x1) -> l6(c3, c4) :|: c4 = 1 && c3 = 1 + x && (x1 > 5 && x < 5)

The following rules are bounded:
l6(x, x1) -> l6(c3, c4) :|: c4 = 1 && c3 = 1 + x && (x1 > 5 && x < 5)


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

(24)
Obligation:
Rules:
l6(x12:0, x13:0) -> l9(x12:0, x13:0, c) :|: c = 1 && x13:0 < 6
l9(x27:0, x28:0, x56:0) -> l9(x27:0, x28:0, c1) :|: c1 = 1 + x56:0 && x56:0 < 6
l9(x21:0, x55:0, x23:0) -> l6(x21:0, c2) :|: c2 = 1 + x55:0 && x23:0 > 5

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

(25) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l6(x, x1)] = 5 - x1
[l9(x2, x3, x4)] = 5 - x3

The following rules are decreasing:
l9(x21:0, x55:0, x23:0) -> l6(x21:0, c2) :|: c2 = 1 + x55:0 && x23:0 > 5
The following rules are bounded:
l6(x12:0, x13:0) -> l9(x12:0, x13:0, c) :|: c = 1 && x13:0 < 6

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

(26)
Complex Obligation (AND)

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

(27)
Obligation:
Rules:
l6(x12:0, x13:0) -> l9(x12:0, x13:0, c) :|: c = 1 && x13:0 < 6
l9(x27:0, x28:0, x56:0) -> l9(x27:0, x28:0, c1) :|: c1 = 1 + x56:0 && x56:0 < 6

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

(28) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ l6 ] = 0
[ l9 ] = -1*l9_3

The following rules are decreasing:
l6(x12:0, x13:0) -> l9(x12:0, x13:0, c) :|: c = 1 && x13:0 < 6
l9(x27:0, x28:0, x56:0) -> l9(x27:0, x28:0, c1) :|: c1 = 1 + x56:0 && x56:0 < 6

The following rules are bounded:
l6(x12:0, x13:0) -> l9(x12:0, x13:0, c) :|: c = 1 && x13:0 < 6
l9(x27:0, x28:0, x56:0) -> l9(x27:0, x28:0, c1) :|: c1 = 1 + x56:0 && x56:0 < 6


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

(29)
YES

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

(30)
Obligation:
Rules:
l9(x27:0, x28:0, x56:0) -> l9(x27:0, x28:0, c1) :|: c1 = 1 + x56:0 && x56:0 < 6
l9(x21:0, x55:0, x23:0) -> l6(x21:0, c2) :|: c2 = 1 + x55:0 && x23:0 > 5

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

(31) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ l9 ] = 0
[ l6 ] = -1

The following rules are decreasing:
l9(x21:0, x55:0, x23:0) -> l6(x21:0, c2) :|: c2 = 1 + x55:0 && x23:0 > 5

The following rules are bounded:
l9(x27:0, x28:0, x56:0) -> l9(x27:0, x28:0, c1) :|: c1 = 1 + x56:0 && x56:0 < 6
l9(x21:0, x55:0, x23:0) -> l6(x21:0, c2) :|: c2 = 1 + x55:0 && x23:0 > 5


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

(32)
Obligation:
Rules:
l9(x27:0, x28:0, x56:0) -> l9(x27:0, x28:0, c1) :|: c1 = 1 + x56:0 && x56:0 < 6

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

(33) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l9(x, x1, x2)] = 5 - x2

The following rules are decreasing:
l9(x27:0, x28:0, x56:0) -> l9(x27:0, x28:0, c1) :|: c1 = 1 + x56:0 && x56:0 < 6
The following rules are bounded:
l9(x27:0, x28:0, x56:0) -> l9(x27:0, x28:0, c1) :|: c1 = 1 + x56:0 && x56:0 < 6

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

(34)
YES