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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 4297 ms]
(4) AND
    (5) IRSwT
        (6) IntTRSCompressionProof [EQUIVALENT, 72 ms]
        (7) IRSwT
        (8) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (9) IRSwT
        (10) TempFilterProof [SOUND, 28 ms]
        (11) IntTRS
        (12) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
        (13) YES
    (14) IRSwT
        (15) IntTRSCompressionProof [EQUIVALENT, 34 ms]
        (16) IRSwT
        (17) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (18) IRSwT
        (19) TempFilterProof [SOUND, 6 ms]
        (20) IntTRS
        (21) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
        (22) YES


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

(0)
Obligation:
Rules:
l0(__const_100HAT0, i2HAT0, iHAT0, rHAT0) -> l1(__const_100HATpost, i2HATpost, iHATpost, rHATpost) :|: rHAT0 = rHATpost && i2HAT0 = i2HATpost && __const_100HAT0 = __const_100HATpost && iHATpost = 0 && __const_100HAT0 <= iHAT0
l0(x, x1, x2, x3) -> l2(x4, x5, x6, x7) :|: x3 = x7 && x2 = x6 && x = x4 && x5 = x2 && 1 + x2 <= x
l3(x8, x9, x10, x11) -> l1(x12, x13, x14, x15) :|: x11 = x15 && x9 = x13 && x8 = x12 && x14 = 1 + x10
l4(x16, x17, x18, x19) -> l5(x20, x21, x22, x23) :|: x19 = x23 && x17 = x21 && x18 = x22 && x16 = x20
l5(x24, x25, x26, x27) -> l3(x28, x29, x30, x31) :|: x27 = x31 && x25 = x29 && x26 = x30 && x24 = x28
l6(x32, x33, x34, x35) -> l0(x36, x37, x38, x39) :|: x35 = x39 && x33 = x37 && x34 = x38 && x32 = x36
l7(x40, x41, x42, x43) -> l4(x44, x45, x46, x47) :|: x43 = x47 && x41 = x45 && x42 = x46 && x40 = x44
l7(x48, x49, x50, x51) -> l5(x52, x53, x54, x55) :|: x51 = x55 && x49 = x53 && x50 = x54 && x48 = x52
l8(x56, x57, x58, x59) -> l7(x60, x61, x62, x63) :|: x57 = x61 && x58 = x62 && x56 = x60 && x63 = x63
l9(x64, x65, x66, x67) -> l8(x68, x69, x70, x71) :|: x67 = x71 && x65 = x69 && x66 = x70 && x64 = x68
l9(x72, x73, x74, x75) -> l3(x76, x77, x78, x79) :|: x75 = x79 && x73 = x77 && x74 = x78 && x72 = x76
l10(x80, x81, x82, x83) -> l11(x84, x85, x86, x87) :|: x83 = x87 && x81 = x85 && x82 = x86 && x80 = x84 && x80 <= x82
l10(x88, x89, x90, x91) -> l9(x92, x93, x94, x95) :|: x91 = x95 && x89 = x93 && x90 = x94 && x88 = x92 && 1 + x90 <= x88
l12(x96, x97, x98, x99) -> l6(x100, x101, x102, x103) :|: x99 = x103 && x97 = x101 && x96 = x100 && x102 = 1 + x98
l1(x104, x105, x106, x107) -> l10(x108, x109, x110, x111) :|: x107 = x111 && x105 = x109 && x106 = x110 && x104 = x108
l13(x112, x113, x114, x115) -> l14(x116, x117, x118, x119) :|: x115 = x119 && x113 = x117 && x114 = x118 && x112 = x116
l14(x120, x121, x122, x123) -> l12(x124, x125, x126, x127) :|: x123 = x127 && x121 = x125 && x122 = x126 && x120 = x124
l15(x128, x129, x130, x131) -> l13(x132, x133, x134, x135) :|: x131 = x135 && x129 = x133 && x130 = x134 && x128 = x132
l15(x136, x137, x138, x139) -> l14(x140, x141, x142, x143) :|: x139 = x143 && x137 = x141 && x138 = x142 && x136 = x140
l16(x144, x145, x146, x147) -> l15(x148, x149, x150, x151) :|: x147 = x151 && x145 = x149 && x146 = x150 && x144 = x148
l2(x152, x153, x154, x155) -> l16(x156, x157, x158, x159) :|: x155 = x159 && x153 = x157 && x154 = x158 && x152 = x156
l2(x160, x161, x162, x163) -> l12(x164, x165, x166, x167) :|: x163 = x167 && x161 = x165 && x162 = x166 && x160 = x164
l17(x168, x169, x170, x171) -> l6(x172, x173, x174, x175) :|: x176 = 0 && x174 = 0 && x168 = x172 && x169 = x173 && x171 = x175
l18(x177, x178, x179, x180) -> l17(x181, x182, x183, x184) :|: x180 = x184 && x178 = x182 && x179 = x183 && x177 = x181
Start term: l18(__const_100HAT0, i2HAT0, iHAT0, rHAT0)

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

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

(2)
Obligation:
Rules:
l0(__const_100HAT0, i2HAT0, iHAT0, rHAT0) -> l1(__const_100HATpost, i2HATpost, iHATpost, rHATpost) :|: rHAT0 = rHATpost && i2HAT0 = i2HATpost && __const_100HAT0 = __const_100HATpost && iHATpost = 0 && __const_100HAT0 <= iHAT0
l0(x, x1, x2, x3) -> l2(x4, x5, x6, x7) :|: x3 = x7 && x2 = x6 && x = x4 && x5 = x2 && 1 + x2 <= x
l3(x8, x9, x10, x11) -> l1(x12, x13, x14, x15) :|: x11 = x15 && x9 = x13 && x8 = x12 && x14 = 1 + x10
l4(x16, x17, x18, x19) -> l5(x20, x21, x22, x23) :|: x19 = x23 && x17 = x21 && x18 = x22 && x16 = x20
l5(x24, x25, x26, x27) -> l3(x28, x29, x30, x31) :|: x27 = x31 && x25 = x29 && x26 = x30 && x24 = x28
l6(x32, x33, x34, x35) -> l0(x36, x37, x38, x39) :|: x35 = x39 && x33 = x37 && x34 = x38 && x32 = x36
l7(x40, x41, x42, x43) -> l4(x44, x45, x46, x47) :|: x43 = x47 && x41 = x45 && x42 = x46 && x40 = x44
l7(x48, x49, x50, x51) -> l5(x52, x53, x54, x55) :|: x51 = x55 && x49 = x53 && x50 = x54 && x48 = x52
l8(x56, x57, x58, x59) -> l7(x60, x61, x62, x63) :|: x57 = x61 && x58 = x62 && x56 = x60 && x63 = x63
l9(x64, x65, x66, x67) -> l8(x68, x69, x70, x71) :|: x67 = x71 && x65 = x69 && x66 = x70 && x64 = x68
l9(x72, x73, x74, x75) -> l3(x76, x77, x78, x79) :|: x75 = x79 && x73 = x77 && x74 = x78 && x72 = x76
l10(x80, x81, x82, x83) -> l11(x84, x85, x86, x87) :|: x83 = x87 && x81 = x85 && x82 = x86 && x80 = x84 && x80 <= x82
l10(x88, x89, x90, x91) -> l9(x92, x93, x94, x95) :|: x91 = x95 && x89 = x93 && x90 = x94 && x88 = x92 && 1 + x90 <= x88
l12(x96, x97, x98, x99) -> l6(x100, x101, x102, x103) :|: x99 = x103 && x97 = x101 && x96 = x100 && x102 = 1 + x98
l1(x104, x105, x106, x107) -> l10(x108, x109, x110, x111) :|: x107 = x111 && x105 = x109 && x106 = x110 && x104 = x108
l13(x112, x113, x114, x115) -> l14(x116, x117, x118, x119) :|: x115 = x119 && x113 = x117 && x114 = x118 && x112 = x116
l14(x120, x121, x122, x123) -> l12(x124, x125, x126, x127) :|: x123 = x127 && x121 = x125 && x122 = x126 && x120 = x124
l15(x128, x129, x130, x131) -> l13(x132, x133, x134, x135) :|: x131 = x135 && x129 = x133 && x130 = x134 && x128 = x132
l15(x136, x137, x138, x139) -> l14(x140, x141, x142, x143) :|: x139 = x143 && x137 = x141 && x138 = x142 && x136 = x140
l16(x144, x145, x146, x147) -> l15(x148, x149, x150, x151) :|: x147 = x151 && x145 = x149 && x146 = x150 && x144 = x148
l2(x152, x153, x154, x155) -> l16(x156, x157, x158, x159) :|: x155 = x159 && x153 = x157 && x154 = x158 && x152 = x156
l2(x160, x161, x162, x163) -> l12(x164, x165, x166, x167) :|: x163 = x167 && x161 = x165 && x162 = x166 && x160 = x164
l17(x168, x169, x170, x171) -> l6(x172, x173, x174, x175) :|: x176 = 0 && x174 = 0 && x168 = x172 && x169 = x173 && x171 = x175
l18(x177, x178, x179, x180) -> l17(x181, x182, x183, x184) :|: x180 = x184 && x178 = x182 && x179 = x183 && x177 = x181
Start term: l18(__const_100HAT0, i2HAT0, iHAT0, rHAT0)

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

(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(__const_100HAT0, i2HAT0, iHAT0, rHAT0) -> l1(__const_100HATpost, i2HATpost, iHATpost, rHATpost) :|: rHAT0 = rHATpost && i2HAT0 = i2HATpost && __const_100HAT0 = __const_100HATpost && iHATpost = 0 && __const_100HAT0 <= iHAT0
(2) l0(x, x1, x2, x3) -> l2(x4, x5, x6, x7) :|: x3 = x7 && x2 = x6 && x = x4 && x5 = x2 && 1 + x2 <= x
(3) l3(x8, x9, x10, x11) -> l1(x12, x13, x14, x15) :|: x11 = x15 && x9 = x13 && x8 = x12 && x14 = 1 + x10
(4) l4(x16, x17, x18, x19) -> l5(x20, x21, x22, x23) :|: x19 = x23 && x17 = x21 && x18 = x22 && x16 = x20
(5) l5(x24, x25, x26, x27) -> l3(x28, x29, x30, x31) :|: x27 = x31 && x25 = x29 && x26 = x30 && x24 = x28
(6) l6(x32, x33, x34, x35) -> l0(x36, x37, x38, x39) :|: x35 = x39 && x33 = x37 && x34 = x38 && x32 = x36
(7) l7(x40, x41, x42, x43) -> l4(x44, x45, x46, x47) :|: x43 = x47 && x41 = x45 && x42 = x46 && x40 = x44
(8) l7(x48, x49, x50, x51) -> l5(x52, x53, x54, x55) :|: x51 = x55 && x49 = x53 && x50 = x54 && x48 = x52
(9) l8(x56, x57, x58, x59) -> l7(x60, x61, x62, x63) :|: x57 = x61 && x58 = x62 && x56 = x60 && x63 = x63
(10) l9(x64, x65, x66, x67) -> l8(x68, x69, x70, x71) :|: x67 = x71 && x65 = x69 && x66 = x70 && x64 = x68
(11) l9(x72, x73, x74, x75) -> l3(x76, x77, x78, x79) :|: x75 = x79 && x73 = x77 && x74 = x78 && x72 = x76
(12) l10(x80, x81, x82, x83) -> l11(x84, x85, x86, x87) :|: x83 = x87 && x81 = x85 && x82 = x86 && x80 = x84 && x80 <= x82
(13) l10(x88, x89, x90, x91) -> l9(x92, x93, x94, x95) :|: x91 = x95 && x89 = x93 && x90 = x94 && x88 = x92 && 1 + x90 <= x88
(14) l12(x96, x97, x98, x99) -> l6(x100, x101, x102, x103) :|: x99 = x103 && x97 = x101 && x96 = x100 && x102 = 1 + x98
(15) l1(x104, x105, x106, x107) -> l10(x108, x109, x110, x111) :|: x107 = x111 && x105 = x109 && x106 = x110 && x104 = x108
(16) l13(x112, x113, x114, x115) -> l14(x116, x117, x118, x119) :|: x115 = x119 && x113 = x117 && x114 = x118 && x112 = x116
(17) l14(x120, x121, x122, x123) -> l12(x124, x125, x126, x127) :|: x123 = x127 && x121 = x125 && x122 = x126 && x120 = x124
(18) l15(x128, x129, x130, x131) -> l13(x132, x133, x134, x135) :|: x131 = x135 && x129 = x133 && x130 = x134 && x128 = x132
(19) l15(x136, x137, x138, x139) -> l14(x140, x141, x142, x143) :|: x139 = x143 && x137 = x141 && x138 = x142 && x136 = x140
(20) l16(x144, x145, x146, x147) -> l15(x148, x149, x150, x151) :|: x147 = x151 && x145 = x149 && x146 = x150 && x144 = x148
(21) l2(x152, x153, x154, x155) -> l16(x156, x157, x158, x159) :|: x155 = x159 && x153 = x157 && x154 = x158 && x152 = x156
(22) l2(x160, x161, x162, x163) -> l12(x164, x165, x166, x167) :|: x163 = x167 && x161 = x165 && x162 = x166 && x160 = x164
(23) l17(x168, x169, x170, x171) -> l6(x172, x173, x174, x175) :|: x176 = 0 && x174 = 0 && x168 = x172 && x169 = x173 && x171 = x175
(24) l18(x177, x178, x179, x180) -> l17(x181, x182, x183, x184) :|: x180 = x184 && x178 = x182 && x179 = x183 && x177 = x181

Arcs:
(1) -> (15)
(2) -> (21), (22)
(3) -> (15)
(4) -> (5)
(5) -> (3)
(6) -> (1), (2)
(7) -> (4)
(8) -> (5)
(9) -> (7), (8)
(10) -> (9)
(11) -> (3)
(13) -> (10), (11)
(14) -> (6)
(15) -> (12), (13)
(16) -> (17)
(17) -> (14)
(18) -> (16)
(19) -> (17)
(20) -> (18), (19)
(21) -> (20)
(22) -> (14)
(23) -> (6)
(24) -> (23)

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

(4)
Complex Obligation (AND)

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

(5)
Obligation:

Termination digraph:
Nodes:
(1) l0(x, x1, x2, x3) -> l2(x4, x5, x6, x7) :|: x3 = x7 && x2 = x6 && x = x4 && x5 = x2 && 1 + x2 <= x
(2) l6(x32, x33, x34, x35) -> l0(x36, x37, x38, x39) :|: x35 = x39 && x33 = x37 && x34 = x38 && x32 = x36
(3) l12(x96, x97, x98, x99) -> l6(x100, x101, x102, x103) :|: x99 = x103 && x97 = x101 && x96 = x100 && x102 = 1 + x98
(4) l2(x160, x161, x162, x163) -> l12(x164, x165, x166, x167) :|: x163 = x167 && x161 = x165 && x162 = x166 && x160 = x164
(5) l14(x120, x121, x122, x123) -> l12(x124, x125, x126, x127) :|: x123 = x127 && x121 = x125 && x122 = x126 && x120 = x124
(6) l15(x136, x137, x138, x139) -> l14(x140, x141, x142, x143) :|: x139 = x143 && x137 = x141 && x138 = x142 && x136 = x140
(7) l13(x112, x113, x114, x115) -> l14(x116, x117, x118, x119) :|: x115 = x119 && x113 = x117 && x114 = x118 && x112 = x116
(8) l15(x128, x129, x130, x131) -> l13(x132, x133, x134, x135) :|: x131 = x135 && x129 = x133 && x130 = x134 && x128 = x132
(9) l16(x144, x145, x146, x147) -> l15(x148, x149, x150, x151) :|: x147 = x151 && x145 = x149 && x146 = x150 && x144 = x148
(10) l2(x152, x153, x154, x155) -> l16(x156, x157, x158, x159) :|: x155 = x159 && x153 = x157 && x154 = x158 && x152 = x156

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

This digraph is fully evaluated!

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

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

(7)
Obligation:
Rules:
l12(x100:0, x101:0, x98:0, x103:0) -> l12(x100:0, 1 + x98:0, 1 + x98:0, x103:0) :|: x100:0 >= 1 + (1 + x98:0)

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

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

   l12(x1, x2, x3, x4) -> l12(x1, x3)

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

(9)
Obligation:
Rules:
l12(x100:0, x98:0) -> l12(x100:0, 1 + x98:0) :|: x100:0 >= 1 + (1 + x98:0)

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

(10) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
l12(INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(11)
Obligation:
Rules:
l12(x100:0, x98:0) -> l12(x100:0, c) :|: c = 1 + x98:0 && x100:0 >= 1 + (1 + x98:0)

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

(12) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l12(x, x1)] = x - x1

The following rules are decreasing:
l12(x100:0, x98:0) -> l12(x100:0, c) :|: c = 1 + x98:0 && x100:0 >= 1 + (1 + x98:0)
The following rules are bounded:
l12(x100:0, x98:0) -> l12(x100:0, c) :|: c = 1 + x98:0 && x100:0 >= 1 + (1 + x98:0)

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

(13)
YES

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

(14)
Obligation:

Termination digraph:
Nodes:
(1) l1(x104, x105, x106, x107) -> l10(x108, x109, x110, x111) :|: x107 = x111 && x105 = x109 && x106 = x110 && x104 = x108
(2) l3(x8, x9, x10, x11) -> l1(x12, x13, x14, x15) :|: x11 = x15 && x9 = x13 && x8 = x12 && x14 = 1 + x10
(3) l9(x72, x73, x74, x75) -> l3(x76, x77, x78, x79) :|: x75 = x79 && x73 = x77 && x74 = x78 && x72 = x76
(4) l5(x24, x25, x26, x27) -> l3(x28, x29, x30, x31) :|: x27 = x31 && x25 = x29 && x26 = x30 && x24 = x28
(5) l7(x48, x49, x50, x51) -> l5(x52, x53, x54, x55) :|: x51 = x55 && x49 = x53 && x50 = x54 && x48 = x52
(6) l4(x16, x17, x18, x19) -> l5(x20, x21, x22, x23) :|: x19 = x23 && x17 = x21 && x18 = x22 && x16 = x20
(7) l7(x40, x41, x42, x43) -> l4(x44, x45, x46, x47) :|: x43 = x47 && x41 = x45 && x42 = x46 && x40 = x44
(8) l8(x56, x57, x58, x59) -> l7(x60, x61, x62, x63) :|: x57 = x61 && x58 = x62 && x56 = x60 && x63 = x63
(9) l9(x64, x65, x66, x67) -> l8(x68, x69, x70, x71) :|: x67 = x71 && x65 = x69 && x66 = x70 && x64 = x68
(10) l10(x88, x89, x90, x91) -> l9(x92, x93, x94, x95) :|: x91 = x95 && x89 = x93 && x90 = x94 && x88 = x92 && 1 + x90 <= x88

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

This digraph is fully evaluated!

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

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

(16)
Obligation:
Rules:
l9(x108:0, x109:0, x22:0, x67:0) -> l9(x108:0, x109:0, 1 + x22:0, x111:0) :|: x108:0 >= 1 + (1 + x22:0)
l9(x, x1, x2, x3) -> l9(x, x1, 1 + x2, x3) :|: x >= 1 + (1 + x2)

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

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

   l9(x1, x2, x3, x4) -> l9(x1, x3)

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

(18)
Obligation:
Rules:
l9(x108:0, x22:0) -> l9(x108:0, 1 + x22:0) :|: x108:0 >= 1 + (1 + x22:0)

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

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

(20)
Obligation:
Rules:
l9(x108:0, x22:0) -> l9(x108:0, c) :|: c = 1 + x22:0 && x108:0 >= 1 + (1 + x22:0)

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

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

The following rules are decreasing:
l9(x108:0, x22:0) -> l9(x108:0, c) :|: c = 1 + x22:0 && x108:0 >= 1 + (1 + x22:0)
The following rules are bounded:
l9(x108:0, x22:0) -> l9(x108:0, c) :|: c = 1 + x22:0 && x108:0 >= 1 + (1 + x22:0)

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

(22)
YES