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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 3895 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 0 ms]
(6) IRSwT
(7) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
(8) IRSwT
(9) TempFilterProof [SOUND, 47 ms]
(10) IntTRS
(11) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
(12) IntTRS
(13) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
(14) YES


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

(0)
Obligation:
Rules:
l0(oldX0HAT0, oldX1HAT0, oldX2HAT0, oldX3HAT0, x0HAT0, x1HAT0) -> l1(oldX0HATpost, oldX1HATpost, oldX2HATpost, oldX3HATpost, x0HATpost, x1HATpost) :|: x1HATpost = oldX3HATpost && x0HATpost = oldX2HATpost && oldX3HATpost = oldX3HATpost && oldX2HATpost = oldX2HATpost && oldX1HATpost = x1HAT0 && oldX0HATpost = x0HAT0
l0(x, x1, x2, x3, x4, x5) -> l2(x6, x7, x8, x9, x10, x11) :|: x3 = x9 && x11 = x8 && x10 = -1 + x6 && x8 = x8 && x7 = x5 && x6 = x4
l0(x12, x13, x14, x15, x16, x17) -> l2(x18, x19, x20, x21, x22, x23) :|: x15 = x21 && x14 = x20 && x23 = -1 + x19 && x22 = x18 && x19 = x17 && x18 = x16
l3(x24, x25, x26, x27, x28, x29) -> l1(x30, x31, x32, x33, x34, x35) :|: x35 = x33 && x34 = x32 && x33 = x33 && x32 = x32 && x31 = x29 && x30 = x28
l3(x36, x37, x38, x39, x40, x41) -> l2(x42, x43, x44, x45, x46, x47) :|: x39 = x45 && x38 = x44 && x47 = 1 && x46 = -1 + x42 && x43 = x41 && x42 = x40
l4(x48, x49, x50, x51, x52, x53) -> l0(x54, x55, x56, x57, x58, x59) :|: x51 = x57 && x50 = x56 && x59 = x55 && x58 = x54 && 1 <= x55 && x55 = x53 && x54 = x52
l4(x60, x61, x62, x63, x64, x65) -> l3(x66, x67, x68, x69, x70, x71) :|: x63 = x69 && x62 = x68 && x71 = x67 && x70 = x66 && x67 <= 0 && x67 = x65 && x66 = x64
l5(x72, x73, x74, x75, x76, x77) -> l1(x78, x79, x80, x81, x82, x83) :|: x83 = x81 && x82 = x80 && x81 = x81 && x80 = x80 && x79 = x77 && x78 = x76
l6(x84, x85, x86, x87, x88, x89) -> l4(x90, x91, x92, x93, x94, x95) :|: x87 = x93 && x86 = x92 && x95 = x91 && x94 = x90 && 1 <= x90 && x91 = x89 && x90 = x88
l6(x96, x97, x98, x99, x100, x101) -> l5(x102, x103, x104, x105, x106, x107) :|: x99 = x105 && x98 = x104 && x107 = x103 && x106 = x102 && x102 <= 0 && x103 = x101 && x102 = x100
l2(x108, x109, x110, x111, x112, x113) -> l6(x114, x115, x116, x117, x118, x119) :|: x111 = x117 && x110 = x116 && x119 = x115 && x118 = x114 && x115 = x113 && x114 = x112
l7(x120, x121, x122, x123, x124, x125) -> l0(x126, x127, x128, x129, x130, x131) :|: x125 = x131 && x124 = x130 && x123 = x129 && x122 = x128 && x121 = x127 && x120 = x126
l7(x132, x133, x134, x135, x136, x137) -> l3(x138, x139, x140, x141, x142, x143) :|: x137 = x143 && x136 = x142 && x135 = x141 && x134 = x140 && x133 = x139 && x132 = x138
l7(x144, x145, x146, x147, x148, x149) -> l1(x150, x151, x152, x153, x154, x155) :|: x149 = x155 && x148 = x154 && x147 = x153 && x146 = x152 && x145 = x151 && x144 = x150
l7(x156, x157, x158, x159, x160, x161) -> l4(x162, x163, x164, x165, x166, x167) :|: x161 = x167 && x160 = x166 && x159 = x165 && x158 = x164 && x157 = x163 && x156 = x162
l7(x168, x169, x170, x171, x172, x173) -> l5(x174, x175, x176, x177, x178, x179) :|: x173 = x179 && x172 = x178 && x171 = x177 && x170 = x176 && x169 = x175 && x168 = x174
l7(x180, x181, x182, x183, x184, x185) -> l6(x186, x187, x188, x189, x190, x191) :|: x185 = x191 && x184 = x190 && x183 = x189 && x182 = x188 && x181 = x187 && x180 = x186
l7(x192, x193, x194, x195, x196, x197) -> l2(x198, x199, x200, x201, x202, x203) :|: x197 = x203 && x196 = x202 && x195 = x201 && x194 = x200 && x193 = x199 && x192 = x198
l8(x204, x205, x206, x207, x208, x209) -> l7(x210, x211, x212, x213, x214, x215) :|: x209 = x215 && x208 = x214 && x207 = x213 && x206 = x212 && x205 = x211 && x204 = x210
Start term: l8(oldX0HAT0, oldX1HAT0, oldX2HAT0, oldX3HAT0, x0HAT0, x1HAT0)

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

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

(2)
Obligation:
Rules:
l0(oldX0HAT0, oldX1HAT0, oldX2HAT0, oldX3HAT0, x0HAT0, x1HAT0) -> l1(oldX0HATpost, oldX1HATpost, oldX2HATpost, oldX3HATpost, x0HATpost, x1HATpost) :|: x1HATpost = oldX3HATpost && x0HATpost = oldX2HATpost && oldX3HATpost = oldX3HATpost && oldX2HATpost = oldX2HATpost && oldX1HATpost = x1HAT0 && oldX0HATpost = x0HAT0
l0(x, x1, x2, x3, x4, x5) -> l2(x6, x7, x8, x9, x10, x11) :|: x3 = x9 && x11 = x8 && x10 = -1 + x6 && x8 = x8 && x7 = x5 && x6 = x4
l0(x12, x13, x14, x15, x16, x17) -> l2(x18, x19, x20, x21, x22, x23) :|: x15 = x21 && x14 = x20 && x23 = -1 + x19 && x22 = x18 && x19 = x17 && x18 = x16
l3(x24, x25, x26, x27, x28, x29) -> l1(x30, x31, x32, x33, x34, x35) :|: x35 = x33 && x34 = x32 && x33 = x33 && x32 = x32 && x31 = x29 && x30 = x28
l3(x36, x37, x38, x39, x40, x41) -> l2(x42, x43, x44, x45, x46, x47) :|: x39 = x45 && x38 = x44 && x47 = 1 && x46 = -1 + x42 && x43 = x41 && x42 = x40
l4(x48, x49, x50, x51, x52, x53) -> l0(x54, x55, x56, x57, x58, x59) :|: x51 = x57 && x50 = x56 && x59 = x55 && x58 = x54 && 1 <= x55 && x55 = x53 && x54 = x52
l4(x60, x61, x62, x63, x64, x65) -> l3(x66, x67, x68, x69, x70, x71) :|: x63 = x69 && x62 = x68 && x71 = x67 && x70 = x66 && x67 <= 0 && x67 = x65 && x66 = x64
l5(x72, x73, x74, x75, x76, x77) -> l1(x78, x79, x80, x81, x82, x83) :|: x83 = x81 && x82 = x80 && x81 = x81 && x80 = x80 && x79 = x77 && x78 = x76
l6(x84, x85, x86, x87, x88, x89) -> l4(x90, x91, x92, x93, x94, x95) :|: x87 = x93 && x86 = x92 && x95 = x91 && x94 = x90 && 1 <= x90 && x91 = x89 && x90 = x88
l6(x96, x97, x98, x99, x100, x101) -> l5(x102, x103, x104, x105, x106, x107) :|: x99 = x105 && x98 = x104 && x107 = x103 && x106 = x102 && x102 <= 0 && x103 = x101 && x102 = x100
l2(x108, x109, x110, x111, x112, x113) -> l6(x114, x115, x116, x117, x118, x119) :|: x111 = x117 && x110 = x116 && x119 = x115 && x118 = x114 && x115 = x113 && x114 = x112
l7(x120, x121, x122, x123, x124, x125) -> l0(x126, x127, x128, x129, x130, x131) :|: x125 = x131 && x124 = x130 && x123 = x129 && x122 = x128 && x121 = x127 && x120 = x126
l7(x132, x133, x134, x135, x136, x137) -> l3(x138, x139, x140, x141, x142, x143) :|: x137 = x143 && x136 = x142 && x135 = x141 && x134 = x140 && x133 = x139 && x132 = x138
l7(x144, x145, x146, x147, x148, x149) -> l1(x150, x151, x152, x153, x154, x155) :|: x149 = x155 && x148 = x154 && x147 = x153 && x146 = x152 && x145 = x151 && x144 = x150
l7(x156, x157, x158, x159, x160, x161) -> l4(x162, x163, x164, x165, x166, x167) :|: x161 = x167 && x160 = x166 && x159 = x165 && x158 = x164 && x157 = x163 && x156 = x162
l7(x168, x169, x170, x171, x172, x173) -> l5(x174, x175, x176, x177, x178, x179) :|: x173 = x179 && x172 = x178 && x171 = x177 && x170 = x176 && x169 = x175 && x168 = x174
l7(x180, x181, x182, x183, x184, x185) -> l6(x186, x187, x188, x189, x190, x191) :|: x185 = x191 && x184 = x190 && x183 = x189 && x182 = x188 && x181 = x187 && x180 = x186
l7(x192, x193, x194, x195, x196, x197) -> l2(x198, x199, x200, x201, x202, x203) :|: x197 = x203 && x196 = x202 && x195 = x201 && x194 = x200 && x193 = x199 && x192 = x198
l8(x204, x205, x206, x207, x208, x209) -> l7(x210, x211, x212, x213, x214, x215) :|: x209 = x215 && x208 = x214 && x207 = x213 && x206 = x212 && x205 = x211 && x204 = x210
Start term: l8(oldX0HAT0, oldX1HAT0, oldX2HAT0, oldX3HAT0, x0HAT0, x1HAT0)

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

(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(oldX0HAT0, oldX1HAT0, oldX2HAT0, oldX3HAT0, x0HAT0, x1HAT0) -> l1(oldX0HATpost, oldX1HATpost, oldX2HATpost, oldX3HATpost, x0HATpost, x1HATpost) :|: x1HATpost = oldX3HATpost && x0HATpost = oldX2HATpost && oldX3HATpost = oldX3HATpost && oldX2HATpost = oldX2HATpost && oldX1HATpost = x1HAT0 && oldX0HATpost = x0HAT0
(2) l0(x, x1, x2, x3, x4, x5) -> l2(x6, x7, x8, x9, x10, x11) :|: x3 = x9 && x11 = x8 && x10 = -1 + x6 && x8 = x8 && x7 = x5 && x6 = x4
(3) l0(x12, x13, x14, x15, x16, x17) -> l2(x18, x19, x20, x21, x22, x23) :|: x15 = x21 && x14 = x20 && x23 = -1 + x19 && x22 = x18 && x19 = x17 && x18 = x16
(4) l3(x24, x25, x26, x27, x28, x29) -> l1(x30, x31, x32, x33, x34, x35) :|: x35 = x33 && x34 = x32 && x33 = x33 && x32 = x32 && x31 = x29 && x30 = x28
(5) l3(x36, x37, x38, x39, x40, x41) -> l2(x42, x43, x44, x45, x46, x47) :|: x39 = x45 && x38 = x44 && x47 = 1 && x46 = -1 + x42 && x43 = x41 && x42 = x40
(6) l4(x48, x49, x50, x51, x52, x53) -> l0(x54, x55, x56, x57, x58, x59) :|: x51 = x57 && x50 = x56 && x59 = x55 && x58 = x54 && 1 <= x55 && x55 = x53 && x54 = x52
(7) l4(x60, x61, x62, x63, x64, x65) -> l3(x66, x67, x68, x69, x70, x71) :|: x63 = x69 && x62 = x68 && x71 = x67 && x70 = x66 && x67 <= 0 && x67 = x65 && x66 = x64
(8) l5(x72, x73, x74, x75, x76, x77) -> l1(x78, x79, x80, x81, x82, x83) :|: x83 = x81 && x82 = x80 && x81 = x81 && x80 = x80 && x79 = x77 && x78 = x76
(9) l6(x84, x85, x86, x87, x88, x89) -> l4(x90, x91, x92, x93, x94, x95) :|: x87 = x93 && x86 = x92 && x95 = x91 && x94 = x90 && 1 <= x90 && x91 = x89 && x90 = x88
(10) l6(x96, x97, x98, x99, x100, x101) -> l5(x102, x103, x104, x105, x106, x107) :|: x99 = x105 && x98 = x104 && x107 = x103 && x106 = x102 && x102 <= 0 && x103 = x101 && x102 = x100
(11) l2(x108, x109, x110, x111, x112, x113) -> l6(x114, x115, x116, x117, x118, x119) :|: x111 = x117 && x110 = x116 && x119 = x115 && x118 = x114 && x115 = x113 && x114 = x112
(12) l7(x120, x121, x122, x123, x124, x125) -> l0(x126, x127, x128, x129, x130, x131) :|: x125 = x131 && x124 = x130 && x123 = x129 && x122 = x128 && x121 = x127 && x120 = x126
(13) l7(x132, x133, x134, x135, x136, x137) -> l3(x138, x139, x140, x141, x142, x143) :|: x137 = x143 && x136 = x142 && x135 = x141 && x134 = x140 && x133 = x139 && x132 = x138
(14) l7(x144, x145, x146, x147, x148, x149) -> l1(x150, x151, x152, x153, x154, x155) :|: x149 = x155 && x148 = x154 && x147 = x153 && x146 = x152 && x145 = x151 && x144 = x150
(15) l7(x156, x157, x158, x159, x160, x161) -> l4(x162, x163, x164, x165, x166, x167) :|: x161 = x167 && x160 = x166 && x159 = x165 && x158 = x164 && x157 = x163 && x156 = x162
(16) l7(x168, x169, x170, x171, x172, x173) -> l5(x174, x175, x176, x177, x178, x179) :|: x173 = x179 && x172 = x178 && x171 = x177 && x170 = x176 && x169 = x175 && x168 = x174
(17) l7(x180, x181, x182, x183, x184, x185) -> l6(x186, x187, x188, x189, x190, x191) :|: x185 = x191 && x184 = x190 && x183 = x189 && x182 = x188 && x181 = x187 && x180 = x186
(18) l7(x192, x193, x194, x195, x196, x197) -> l2(x198, x199, x200, x201, x202, x203) :|: x197 = x203 && x196 = x202 && x195 = x201 && x194 = x200 && x193 = x199 && x192 = x198
(19) l8(x204, x205, x206, x207, x208, x209) -> l7(x210, x211, x212, x213, x214, x215) :|: x209 = x215 && x208 = x214 && x207 = x213 && x206 = x212 && x205 = x211 && x204 = x210

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

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

(4)
Obligation:

Termination digraph:
Nodes:
(1) l0(x, x1, x2, x3, x4, x5) -> l2(x6, x7, x8, x9, x10, x11) :|: x3 = x9 && x11 = x8 && x10 = -1 + x6 && x8 = x8 && x7 = x5 && x6 = x4
(2) l4(x48, x49, x50, x51, x52, x53) -> l0(x54, x55, x56, x57, x58, x59) :|: x51 = x57 && x50 = x56 && x59 = x55 && x58 = x54 && 1 <= x55 && x55 = x53 && x54 = x52
(3) l6(x84, x85, x86, x87, x88, x89) -> l4(x90, x91, x92, x93, x94, x95) :|: x87 = x93 && x86 = x92 && x95 = x91 && x94 = x90 && 1 <= x90 && x91 = x89 && x90 = x88
(4) l2(x108, x109, x110, x111, x112, x113) -> l6(x114, x115, x116, x117, x118, x119) :|: x111 = x117 && x110 = x116 && x119 = x115 && x118 = x114 && x115 = x113 && x114 = x112
(5) l3(x36, x37, x38, x39, x40, x41) -> l2(x42, x43, x44, x45, x46, x47) :|: x39 = x45 && x38 = x44 && x47 = 1 && x46 = -1 + x42 && x43 = x41 && x42 = x40
(6) l4(x60, x61, x62, x63, x64, x65) -> l3(x66, x67, x68, x69, x70, x71) :|: x63 = x69 && x62 = x68 && x71 = x67 && x70 = x66 && x67 <= 0 && x67 = x65 && x66 = x64
(7) l0(x12, x13, x14, x15, x16, x17) -> l2(x18, x19, x20, x21, x22, x23) :|: x15 = x21 && x14 = x20 && x23 = -1 + x19 && x22 = x18 && x19 = x17 && x18 = x16

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

This digraph is fully evaluated!

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

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

(6)
Obligation:
Rules:
l6(x84:0, x85:0, x116:0, x117:0, x42:0, x43:0) -> l6(-1 + x42:0, 1, x116:0, x117:0, -1 + x42:0, 1) :|: x42:0 > 0 && x43:0 < 1
l6(x, x1, x2, x3, x4, x5) -> l6(-1 + x4, x6, x6, x3, -1 + x4, x6) :|: x4 > 0 && x5 > 0
l6(x7, x8, x9, x10, x11, x12) -> l6(x11, -1 + x12, x9, x10, x11, -1 + x12) :|: x11 > 0 && x12 > 0

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

(7) 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, x6) -> l6(x5, x6)

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

(8)
Obligation:
Rules:
l6(x42:0, x43:0) -> l6(-1 + x42:0, 1) :|: x42:0 > 0 && x43:0 < 1
l6(x4, x5) -> l6(-1 + x4, x6) :|: x4 > 0 && x5 > 0
l6(x11, x12) -> l6(x11, -1 + x12) :|: x11 > 0 && x12 > 0

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

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

(10)
Obligation:
Rules:
l6(x42:0, x43:0) -> l6(c, c1) :|: c1 = 1 && c = -1 + x42:0 && (x42:0 > 0 && x43:0 < 1)
l6(x4, x5) -> l6(c2, x6) :|: c2 = -1 + x4 && (x4 > 0 && x5 > 0)
l6(x11, x12) -> l6(x11, c3) :|: c3 = -1 + x12 && (x11 > 0 && x12 > 0)

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

(11) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l6(x, x1)] = x

The following rules are decreasing:
l6(x42:0, x43:0) -> l6(c, c1) :|: c1 = 1 && c = -1 + x42:0 && (x42:0 > 0 && x43:0 < 1)
l6(x4, x5) -> l6(c2, x6) :|: c2 = -1 + x4 && (x4 > 0 && x5 > 0)
The following rules are bounded:
l6(x42:0, x43:0) -> l6(c, c1) :|: c1 = 1 && c = -1 + x42:0 && (x42:0 > 0 && x43:0 < 1)
l6(x4, x5) -> l6(c2, x6) :|: c2 = -1 + x4 && (x4 > 0 && x5 > 0)
l6(x11, x12) -> l6(x11, c3) :|: c3 = -1 + x12 && (x11 > 0 && x12 > 0)

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

(12)
Obligation:
Rules:
l6(x11, x12) -> l6(x11, c3) :|: c3 = -1 + x12 && (x11 > 0 && x12 > 0)

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

(13) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l6(x, x1)] = x1

The following rules are decreasing:
l6(x11, x12) -> l6(x11, c3) :|: c3 = -1 + x12 && (x11 > 0 && x12 > 0)
The following rules are bounded:
l6(x11, x12) -> l6(x11, c3) :|: c3 = -1 + x12 && (x11 > 0 && x12 > 0)

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

(14)
YES