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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 903 ms]
(4) AND
    (5) IRSwT
        (6) IntTRSCompressionProof [EQUIVALENT, 8 ms]
        (7) IRSwT
        (8) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (9) IRSwT
        (10) TempFilterProof [SOUND, 6 ms]
        (11) IntTRS
        (12) RankingReductionPairProof [EQUIVALENT, 0 ms]
        (13) YES
    (14) IRSwT
        (15) IntTRSCompressionProof [EQUIVALENT, 3 ms]
        (16) IRSwT
        (17) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (18) IRSwT
        (19) TempFilterProof [SOUND, 5 ms]
        (20) IntTRS
        (21) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
        (22) YES
    (23) IRSwT
        (24) IntTRSCompressionProof [EQUIVALENT, 34 ms]
        (25) IRSwT
        (26) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (27) IRSwT
        (28) TempFilterProof [SOUND, 11 ms]
        (29) IntTRS
        (30) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
        (31) YES
    (32) IRSwT
        (33) IntTRSCompressionProof [EQUIVALENT, 3 ms]
        (34) IRSwT
        (35) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (36) IRSwT
        (37) TempFilterProof [SOUND, 11 ms]
        (38) IntTRS
        (39) RankingReductionPairProof [EQUIVALENT, 0 ms]
        (40) YES


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

(0)
Obligation:
Rules:
l0(i1HAT0, i2HAT0, i3HAT0, i4HAT0, nHAT0) -> l1(i1HATpost, i2HATpost, i3HATpost, i4HATpost, nHATpost) :|: nHAT0 = nHATpost && i3HAT0 = i3HATpost && i2HAT0 = i2HATpost && i1HAT0 = i1HATpost && i4HATpost = -1 + i4HAT0 && 1 <= i4HAT0
l1(x, x1, x2, x3, x4) -> l0(x5, x6, x7, x8, x9) :|: x4 = x9 && x3 = x8 && x2 = x7 && x1 = x6 && x = x5
l2(x10, x11, x12, x13, x14) -> l0(x15, x16, x17, x18, x19) :|: x14 = x19 && x12 = x17 && x11 = x16 && x10 = x15 && x18 = x12 && x14 <= x12
l2(x20, x21, x22, x23, x24) -> l3(x25, x26, x27, x28, x29) :|: x24 = x29 && x23 = x28 && x21 = x26 && x20 = x25 && x27 = 1 + x22 && 1 + x22 <= x24
l3(x30, x31, x32, x33, x34) -> l2(x35, x36, x37, x38, x39) :|: x34 = x39 && x33 = x38 && x32 = x37 && x31 = x36 && x30 = x35
l4(x40, x41, x42, x43, x44) -> l2(x45, x46, x47, x48, x49) :|: x44 = x49 && x43 = x48 && x41 = x46 && x40 = x45 && x47 = x41 && x41 <= 0
l4(x50, x51, x52, x53, x54) -> l5(x55, x56, x57, x58, x59) :|: x54 = x59 && x53 = x58 && x52 = x57 && x50 = x55 && x56 = -1 + x51 && 1 <= x51
l5(x60, x61, x62, x63, x64) -> l4(x65, x66, x67, x68, x69) :|: x64 = x69 && x63 = x68 && x62 = x67 && x61 = x66 && x60 = x65
l6(x70, x71, x72, x73, x74) -> l4(x75, x76, x77, x78, x79) :|: x74 = x79 && x73 = x78 && x72 = x77 && x70 = x75 && x76 = x70 && x74 <= x70
l6(x80, x81, x82, x83, x84) -> l7(x85, x86, x87, x88, x89) :|: x84 = x89 && x83 = x88 && x82 = x87 && x81 = x86 && x85 = 1 + x80 && 1 + x80 <= x84
l7(x90, x91, x92, x93, x94) -> l6(x95, x96, x97, x98, x99) :|: x94 = x99 && x93 = x98 && x92 = x97 && x91 = x96 && x90 = x95
l8(x100, x101, x102, x103, x104) -> l6(x105, x106, x107, x108, x109) :|: x104 = x109 && x103 = x108 && x102 = x107 && x101 = x106 && x100 = x105
l9(x110, x111, x112, x113, x114) -> l8(x115, x116, x117, x118, x119) :|: x114 = x119 && x113 = x118 && x112 = x117 && x111 = x116 && x110 = x115
Start term: l9(i1HAT0, i2HAT0, i3HAT0, i4HAT0, nHAT0)

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

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

(2)
Obligation:
Rules:
l0(i1HAT0, i2HAT0, i3HAT0, i4HAT0, nHAT0) -> l1(i1HATpost, i2HATpost, i3HATpost, i4HATpost, nHATpost) :|: nHAT0 = nHATpost && i3HAT0 = i3HATpost && i2HAT0 = i2HATpost && i1HAT0 = i1HATpost && i4HATpost = -1 + i4HAT0 && 1 <= i4HAT0
l1(x, x1, x2, x3, x4) -> l0(x5, x6, x7, x8, x9) :|: x4 = x9 && x3 = x8 && x2 = x7 && x1 = x6 && x = x5
l2(x10, x11, x12, x13, x14) -> l0(x15, x16, x17, x18, x19) :|: x14 = x19 && x12 = x17 && x11 = x16 && x10 = x15 && x18 = x12 && x14 <= x12
l2(x20, x21, x22, x23, x24) -> l3(x25, x26, x27, x28, x29) :|: x24 = x29 && x23 = x28 && x21 = x26 && x20 = x25 && x27 = 1 + x22 && 1 + x22 <= x24
l3(x30, x31, x32, x33, x34) -> l2(x35, x36, x37, x38, x39) :|: x34 = x39 && x33 = x38 && x32 = x37 && x31 = x36 && x30 = x35
l4(x40, x41, x42, x43, x44) -> l2(x45, x46, x47, x48, x49) :|: x44 = x49 && x43 = x48 && x41 = x46 && x40 = x45 && x47 = x41 && x41 <= 0
l4(x50, x51, x52, x53, x54) -> l5(x55, x56, x57, x58, x59) :|: x54 = x59 && x53 = x58 && x52 = x57 && x50 = x55 && x56 = -1 + x51 && 1 <= x51
l5(x60, x61, x62, x63, x64) -> l4(x65, x66, x67, x68, x69) :|: x64 = x69 && x63 = x68 && x62 = x67 && x61 = x66 && x60 = x65
l6(x70, x71, x72, x73, x74) -> l4(x75, x76, x77, x78, x79) :|: x74 = x79 && x73 = x78 && x72 = x77 && x70 = x75 && x76 = x70 && x74 <= x70
l6(x80, x81, x82, x83, x84) -> l7(x85, x86, x87, x88, x89) :|: x84 = x89 && x83 = x88 && x82 = x87 && x81 = x86 && x85 = 1 + x80 && 1 + x80 <= x84
l7(x90, x91, x92, x93, x94) -> l6(x95, x96, x97, x98, x99) :|: x94 = x99 && x93 = x98 && x92 = x97 && x91 = x96 && x90 = x95
l8(x100, x101, x102, x103, x104) -> l6(x105, x106, x107, x108, x109) :|: x104 = x109 && x103 = x108 && x102 = x107 && x101 = x106 && x100 = x105
l9(x110, x111, x112, x113, x114) -> l8(x115, x116, x117, x118, x119) :|: x114 = x119 && x113 = x118 && x112 = x117 && x111 = x116 && x110 = x115
Start term: l9(i1HAT0, i2HAT0, i3HAT0, i4HAT0, nHAT0)

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

(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(i1HAT0, i2HAT0, i3HAT0, i4HAT0, nHAT0) -> l1(i1HATpost, i2HATpost, i3HATpost, i4HATpost, nHATpost) :|: nHAT0 = nHATpost && i3HAT0 = i3HATpost && i2HAT0 = i2HATpost && i1HAT0 = i1HATpost && i4HATpost = -1 + i4HAT0 && 1 <= i4HAT0
(2) l1(x, x1, x2, x3, x4) -> l0(x5, x6, x7, x8, x9) :|: x4 = x9 && x3 = x8 && x2 = x7 && x1 = x6 && x = x5
(3) l2(x10, x11, x12, x13, x14) -> l0(x15, x16, x17, x18, x19) :|: x14 = x19 && x12 = x17 && x11 = x16 && x10 = x15 && x18 = x12 && x14 <= x12
(4) l2(x20, x21, x22, x23, x24) -> l3(x25, x26, x27, x28, x29) :|: x24 = x29 && x23 = x28 && x21 = x26 && x20 = x25 && x27 = 1 + x22 && 1 + x22 <= x24
(5) l3(x30, x31, x32, x33, x34) -> l2(x35, x36, x37, x38, x39) :|: x34 = x39 && x33 = x38 && x32 = x37 && x31 = x36 && x30 = x35
(6) l4(x40, x41, x42, x43, x44) -> l2(x45, x46, x47, x48, x49) :|: x44 = x49 && x43 = x48 && x41 = x46 && x40 = x45 && x47 = x41 && x41 <= 0
(7) l4(x50, x51, x52, x53, x54) -> l5(x55, x56, x57, x58, x59) :|: x54 = x59 && x53 = x58 && x52 = x57 && x50 = x55 && x56 = -1 + x51 && 1 <= x51
(8) l5(x60, x61, x62, x63, x64) -> l4(x65, x66, x67, x68, x69) :|: x64 = x69 && x63 = x68 && x62 = x67 && x61 = x66 && x60 = x65
(9) l6(x70, x71, x72, x73, x74) -> l4(x75, x76, x77, x78, x79) :|: x74 = x79 && x73 = x78 && x72 = x77 && x70 = x75 && x76 = x70 && x74 <= x70
(10) l6(x80, x81, x82, x83, x84) -> l7(x85, x86, x87, x88, x89) :|: x84 = x89 && x83 = x88 && x82 = x87 && x81 = x86 && x85 = 1 + x80 && 1 + x80 <= x84
(11) l7(x90, x91, x92, x93, x94) -> l6(x95, x96, x97, x98, x99) :|: x94 = x99 && x93 = x98 && x92 = x97 && x91 = x96 && x90 = x95
(12) l8(x100, x101, x102, x103, x104) -> l6(x105, x106, x107, x108, x109) :|: x104 = x109 && x103 = x108 && x102 = x107 && x101 = x106 && x100 = x105
(13) l9(x110, x111, x112, x113, x114) -> l8(x115, x116, x117, x118, x119) :|: x114 = x119 && x113 = x118 && x112 = x117 && x111 = x116 && x110 = x115

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

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

(4)
Complex Obligation (AND)

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

(5)
Obligation:

Termination digraph:
Nodes:
(1) l6(x80, x81, x82, x83, x84) -> l7(x85, x86, x87, x88, x89) :|: x84 = x89 && x83 = x88 && x82 = x87 && x81 = x86 && x85 = 1 + x80 && 1 + x80 <= x84
(2) l7(x90, x91, x92, x93, x94) -> l6(x95, x96, x97, x98, x99) :|: x94 = x99 && x93 = x98 && x92 = x97 && x91 = x96 && x90 = x95

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

This digraph is fully evaluated!

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

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

(7)
Obligation:
Rules:
l6(x80:0, x81:0, x82:0, x83:0, x84:0) -> l6(1 + x80:0, x81:0, x82:0, x83:0, x84:0) :|: x84:0 >= 1 + x80:0

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

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

   l6(x1, x2, x3, x4, x5) -> l6(x1, x5)

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

(9)
Obligation:
Rules:
l6(x80:0, x84:0) -> l6(1 + x80:0, x84:0) :|: x84:0 >= 1 + x80:0

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

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

(11)
Obligation:
Rules:
l6(x80:0, x84:0) -> l6(c, x84:0) :|: c = 1 + x80:0 && x84:0 >= 1 + x80:0

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

(12) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ l6 ] = l6_2 + -1*l6_1

The following rules are decreasing:
l6(x80:0, x84:0) -> l6(c, x84:0) :|: c = 1 + x80:0 && x84:0 >= 1 + x80:0

The following rules are bounded:
l6(x80:0, x84:0) -> l6(c, x84:0) :|: c = 1 + x80:0 && x84:0 >= 1 + x80:0


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

(13)
YES

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

(14)
Obligation:

Termination digraph:
Nodes:
(1) l4(x50, x51, x52, x53, x54) -> l5(x55, x56, x57, x58, x59) :|: x54 = x59 && x53 = x58 && x52 = x57 && x50 = x55 && x56 = -1 + x51 && 1 <= x51
(2) l5(x60, x61, x62, x63, x64) -> l4(x65, x66, x67, x68, x69) :|: x64 = x69 && x63 = x68 && x62 = x67 && x61 = x66 && x60 = x65

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

This digraph is fully evaluated!

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

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

(16)
Obligation:
Rules:
l4(x50:0, x51:0, x52:0, x53:0, x54:0) -> l4(x50:0, -1 + x51:0, x52:0, x53:0, x54:0) :|: x51:0 > 0

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

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

   l4(x1, x2, x3, x4, x5) -> l4(x2)

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

(18)
Obligation:
Rules:
l4(x51:0) -> l4(-1 + x51:0) :|: x51:0 > 0

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

(19) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
l4(INTEGER)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(20)
Obligation:
Rules:
l4(x51:0) -> l4(c) :|: c = -1 + x51:0 && x51:0 > 0

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

(21) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l4(x)] = x

The following rules are decreasing:
l4(x51:0) -> l4(c) :|: c = -1 + x51:0 && x51:0 > 0
The following rules are bounded:
l4(x51:0) -> l4(c) :|: c = -1 + x51:0 && x51:0 > 0

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

(22)
YES

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

(23)
Obligation:

Termination digraph:
Nodes:
(1) l2(x20, x21, x22, x23, x24) -> l3(x25, x26, x27, x28, x29) :|: x24 = x29 && x23 = x28 && x21 = x26 && x20 = x25 && x27 = 1 + x22 && 1 + x22 <= x24
(2) l3(x30, x31, x32, x33, x34) -> l2(x35, x36, x37, x38, x39) :|: x34 = x39 && x33 = x38 && x32 = x37 && x31 = x36 && x30 = x35

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

This digraph is fully evaluated!

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

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

(25)
Obligation:
Rules:
l2(x20:0, x21:0, x22:0, x23:0, x24:0) -> l2(x20:0, x21:0, 1 + x22:0, x23:0, x24:0) :|: x24:0 >= 1 + x22:0

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

(26) 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(x3, x5)

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

(27)
Obligation:
Rules:
l2(x22:0, x24:0) -> l2(1 + x22:0, x24:0) :|: x24:0 >= 1 + x22:0

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

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

(29)
Obligation:
Rules:
l2(x22:0, x24:0) -> l2(c, x24:0) :|: c = 1 + x22:0 && x24:0 >= 1 + x22:0

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

(30) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l2(x, x1)] = -x + x1

The following rules are decreasing:
l2(x22:0, x24:0) -> l2(c, x24:0) :|: c = 1 + x22:0 && x24:0 >= 1 + x22:0
The following rules are bounded:
l2(x22:0, x24:0) -> l2(c, x24:0) :|: c = 1 + x22:0 && x24:0 >= 1 + x22:0

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

(31)
YES

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

(32)
Obligation:

Termination digraph:
Nodes:
(1) l0(i1HAT0, i2HAT0, i3HAT0, i4HAT0, nHAT0) -> l1(i1HATpost, i2HATpost, i3HATpost, i4HATpost, nHATpost) :|: nHAT0 = nHATpost && i3HAT0 = i3HATpost && i2HAT0 = i2HATpost && i1HAT0 = i1HATpost && i4HATpost = -1 + i4HAT0 && 1 <= i4HAT0
(2) l1(x, x1, x2, x3, x4) -> l0(x5, x6, x7, x8, x9) :|: x4 = x9 && x3 = x8 && x2 = x7 && x1 = x6 && x = x5

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

This digraph is fully evaluated!

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

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

(34)
Obligation:
Rules:
l0(i1HAT0:0, i2HAT0:0, i3HAT0:0, i4HAT0:0, nHAT0:0) -> l0(i1HAT0:0, i2HAT0:0, i3HAT0:0, -1 + i4HAT0:0, nHAT0:0) :|: i4HAT0:0 > 0

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

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

   l0(x1, x2, x3, x4, x5) -> l0(x4)

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

(36)
Obligation:
Rules:
l0(i4HAT0:0) -> l0(-1 + i4HAT0:0) :|: i4HAT0:0 > 0

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

(37) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
l0(INTEGER)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(38)
Obligation:
Rules:
l0(i4HAT0:0) -> l0(c) :|: c = -1 + i4HAT0:0 && i4HAT0:0 > 0

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

(39) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ l0 ] = l0_1

The following rules are decreasing:
l0(i4HAT0:0) -> l0(c) :|: c = -1 + i4HAT0:0 && i4HAT0:0 > 0

The following rules are bounded:
l0(i4HAT0:0) -> l0(c) :|: c = -1 + i4HAT0:0 && i4HAT0:0 > 0


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

(40)
YES