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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 3669 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 10 ms]
(6) IRSwT
(7) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
(8) IRSwT
(9) FilterProof [EQUIVALENT, 0 ms]
(10) IntTRS
(11) IntTRSCompressionProof [EQUIVALENT, 0 ms]
(12) IntTRS
(13) RankingReductionPairProof [EQUIVALENT, 20 ms]
(14) YES


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

(0)
Obligation:
Rules:
l0(oldX0HAT0, oldX1HAT0, oldX2HAT0, oldX3HAT0, oldX4HAT0, oldX5HAT0, oldX6HAT0, x0HAT0, x1HAT0, x2HAT0) -> l1(oldX0HATpost, oldX1HATpost, oldX2HATpost, oldX3HATpost, oldX4HATpost, oldX5HATpost, oldX6HATpost, x0HATpost, x1HATpost, x2HATpost) :|: oldX6HAT0 = oldX6HATpost && x2HATpost = oldX5HATpost && x1HATpost = oldX4HATpost && x0HATpost = oldX3HATpost && oldX5HATpost = oldX5HATpost && oldX4HATpost = oldX4HATpost && oldX3HATpost = oldX3HATpost && oldX2HATpost = x2HAT0 && oldX1HATpost = x1HAT0 && oldX0HATpost = x0HAT0
l2(x, x1, x2, x3, x4, x5, x6, x7, x8, x9) -> l1(x10, x11, x12, x13, x14, x15, x16, x17, x18, x19) :|: x19 = x15 && x18 = x14 && x17 = x13 && 2 <= x12 - 2 * x16 && x16 = x16 && x15 = x15 && x14 = x14 && x13 = x13 && x12 = x9 && x11 = x8 && x10 = x7
l2(x20, x21, x22, x23, x24, x25, x26, x27, x28, x29) -> l1(x30, x31, x32, x33, x34, x35, x36, x37, x38, x39) :|: x39 = x35 && x38 = x34 && x37 = x33 && 1 + x32 - 2 * x36 <= 0 && x36 = x36 && x35 = x35 && x34 = x34 && x33 = x33 && x32 = x29 && x31 = x28 && x30 = x27
l2(x40, x41, x42, x43, x44, x45, x46, x47, x48, x49) -> l3(x50, x51, x52, x53, x54, x55, x56, x57, x58, x59) :|: x46 = x56 && x45 = x55 && x44 = x54 && x59 = x53 && x58 = 1 + x51 && x57 = x50 && 1 + x52 - 2 * x53 <= 2 && 0 <= x52 - 2 * x53 && x53 = x53 && x52 = x49 && x51 = x48 && x50 = x47
l3(x60, x61, x62, x63, x64, x65, x66, x67, x68, x69) -> l0(x70, x71, x72, x73, x74, x75, x76, x77, x78, x79) :|: x66 = x76 && x65 = x75 && x64 = x74 && x63 = x73 && x79 = x72 && x78 = x71 && x77 = x70 && x72 <= 1 && x72 = x69 && x71 = x68 && x70 = x67
l3(x80, x81, x82, x83, x84, x85, x86, x87, x88, x89) -> l2(x90, x91, x92, x93, x94, x95, x96, x97, x98, x99) :|: x86 = x96 && x85 = x95 && x84 = x94 && x83 = x93 && x99 = x92 && x98 = x91 && x97 = x90 && 2 <= x92 && x92 = x89 && x91 = x88 && x90 = x87
l4(x100, x101, x102, x103, x104, x105, x106, x107, x108, x109) -> l3(x110, x111, x112, x113, x114, x115, x116, x117, x118, x119) :|: x106 = x116 && x105 = x115 && x104 = x114 && x103 = x113 && x119 = x110 && x118 = 0 && x117 = x110 && x112 = x109 && x111 = x108 && x110 = x107
l5(x120, x121, x122, x123, x124, x125, x126, x127, x128, x129) -> l4(x130, x131, x132, x133, x134, x135, x136, x137, x138, x139) :|: x126 = x136 && x125 = x135 && x139 = x134 && x138 = x133 && x137 = x130 && x134 = x134 && x133 = x133 && x132 = x129 && x131 = x128 && x130 = x127
l5(x140, x141, x142, x143, x144, x145, x146, x147, x148, x149) -> l1(x150, x151, x152, x153, x154, x155, x156, x157, x158, x159) :|: x149 = x159 && x148 = x158 && x147 = x157 && x146 = x156 && x145 = x155 && x144 = x154 && x143 = x153 && x142 = x152 && x141 = x151 && x140 = x150
l5(x160, x161, x162, x163, x164, x165, x166, x167, x168, x169) -> l0(x170, x171, x172, x173, x174, x175, x176, x177, x178, x179) :|: x169 = x179 && x168 = x178 && x167 = x177 && x166 = x176 && x165 = x175 && x164 = x174 && x163 = x173 && x162 = x172 && x161 = x171 && x160 = x170
l5(x180, x181, x182, x183, x184, x185, x186, x187, x188, x189) -> l2(x190, x191, x192, x193, x194, x195, x196, x197, x198, x199) :|: x189 = x199 && x188 = x198 && x187 = x197 && x186 = x196 && x185 = x195 && x184 = x194 && x183 = x193 && x182 = x192 && x181 = x191 && x180 = x190
l5(x200, x201, x202, x203, x204, x205, x206, x207, x208, x209) -> l3(x210, x211, x212, x213, x214, x215, x216, x217, x218, x219) :|: x209 = x219 && x208 = x218 && x207 = x217 && x206 = x216 && x205 = x215 && x204 = x214 && x203 = x213 && x202 = x212 && x201 = x211 && x200 = x210
l5(x220, x221, x222, x223, x224, x225, x226, x227, x228, x229) -> l4(x230, x231, x232, x233, x234, x235, x236, x237, x238, x239) :|: x229 = x239 && x228 = x238 && x227 = x237 && x226 = x236 && x225 = x235 && x224 = x234 && x223 = x233 && x222 = x232 && x221 = x231 && x220 = x230
l6(x240, x241, x242, x243, x244, x245, x246, x247, x248, x249) -> l5(x250, x251, x252, x253, x254, x255, x256, x257, x258, x259) :|: x249 = x259 && x248 = x258 && x247 = x257 && x246 = x256 && x245 = x255 && x244 = x254 && x243 = x253 && x242 = x252 && x241 = x251 && x240 = x250
Start term: l6(oldX0HAT0, oldX1HAT0, oldX2HAT0, oldX3HAT0, oldX4HAT0, oldX5HAT0, oldX6HAT0, x0HAT0, x1HAT0, x2HAT0)

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

(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, oldX4HAT0, oldX5HAT0, oldX6HAT0, x0HAT0, x1HAT0, x2HAT0) -> l1(oldX0HATpost, oldX1HATpost, oldX2HATpost, oldX3HATpost, oldX4HATpost, oldX5HATpost, oldX6HATpost, x0HATpost, x1HATpost, x2HATpost) :|: oldX6HAT0 = oldX6HATpost && x2HATpost = oldX5HATpost && x1HATpost = oldX4HATpost && x0HATpost = oldX3HATpost && oldX5HATpost = oldX5HATpost && oldX4HATpost = oldX4HATpost && oldX3HATpost = oldX3HATpost && oldX2HATpost = x2HAT0 && oldX1HATpost = x1HAT0 && oldX0HATpost = x0HAT0
l2(x, x1, x2, x3, x4, x5, x6, x7, x8, x9) -> l1(x10, x11, x12, x13, x14, x15, x16, x17, x18, x19) :|: x19 = x15 && x18 = x14 && x17 = x13 && 2 <= x12 - 2 * x16 && x16 = x16 && x15 = x15 && x14 = x14 && x13 = x13 && x12 = x9 && x11 = x8 && x10 = x7
l2(x20, x21, x22, x23, x24, x25, x26, x27, x28, x29) -> l1(x30, x31, x32, x33, x34, x35, x36, x37, x38, x39) :|: x39 = x35 && x38 = x34 && x37 = x33 && 1 + x32 - 2 * x36 <= 0 && x36 = x36 && x35 = x35 && x34 = x34 && x33 = x33 && x32 = x29 && x31 = x28 && x30 = x27
l2(x40, x41, x42, x43, x44, x45, x46, x47, x48, x49) -> l3(x50, x51, x52, x53, x54, x55, x56, x57, x58, x59) :|: x46 = x56 && x45 = x55 && x44 = x54 && x59 = x53 && x58 = 1 + x51 && x57 = x50 && 1 + x52 - 2 * x53 <= 2 && 0 <= x52 - 2 * x53 && x53 = x53 && x52 = x49 && x51 = x48 && x50 = x47
l3(x60, x61, x62, x63, x64, x65, x66, x67, x68, x69) -> l0(x70, x71, x72, x73, x74, x75, x76, x77, x78, x79) :|: x66 = x76 && x65 = x75 && x64 = x74 && x63 = x73 && x79 = x72 && x78 = x71 && x77 = x70 && x72 <= 1 && x72 = x69 && x71 = x68 && x70 = x67
l3(x80, x81, x82, x83, x84, x85, x86, x87, x88, x89) -> l2(x90, x91, x92, x93, x94, x95, x96, x97, x98, x99) :|: x86 = x96 && x85 = x95 && x84 = x94 && x83 = x93 && x99 = x92 && x98 = x91 && x97 = x90 && 2 <= x92 && x92 = x89 && x91 = x88 && x90 = x87
l4(x100, x101, x102, x103, x104, x105, x106, x107, x108, x109) -> l3(x110, x111, x112, x113, x114, x115, x116, x117, x118, x119) :|: x106 = x116 && x105 = x115 && x104 = x114 && x103 = x113 && x119 = x110 && x118 = 0 && x117 = x110 && x112 = x109 && x111 = x108 && x110 = x107
l5(x120, x121, x122, x123, x124, x125, x126, x127, x128, x129) -> l4(x130, x131, x132, x133, x134, x135, x136, x137, x138, x139) :|: x126 = x136 && x125 = x135 && x139 = x134 && x138 = x133 && x137 = x130 && x134 = x134 && x133 = x133 && x132 = x129 && x131 = x128 && x130 = x127
l5(x140, x141, x142, x143, x144, x145, x146, x147, x148, x149) -> l1(x150, x151, x152, x153, x154, x155, x156, x157, x158, x159) :|: x149 = x159 && x148 = x158 && x147 = x157 && x146 = x156 && x145 = x155 && x144 = x154 && x143 = x153 && x142 = x152 && x141 = x151 && x140 = x150
l5(x160, x161, x162, x163, x164, x165, x166, x167, x168, x169) -> l0(x170, x171, x172, x173, x174, x175, x176, x177, x178, x179) :|: x169 = x179 && x168 = x178 && x167 = x177 && x166 = x176 && x165 = x175 && x164 = x174 && x163 = x173 && x162 = x172 && x161 = x171 && x160 = x170
l5(x180, x181, x182, x183, x184, x185, x186, x187, x188, x189) -> l2(x190, x191, x192, x193, x194, x195, x196, x197, x198, x199) :|: x189 = x199 && x188 = x198 && x187 = x197 && x186 = x196 && x185 = x195 && x184 = x194 && x183 = x193 && x182 = x192 && x181 = x191 && x180 = x190
l5(x200, x201, x202, x203, x204, x205, x206, x207, x208, x209) -> l3(x210, x211, x212, x213, x214, x215, x216, x217, x218, x219) :|: x209 = x219 && x208 = x218 && x207 = x217 && x206 = x216 && x205 = x215 && x204 = x214 && x203 = x213 && x202 = x212 && x201 = x211 && x200 = x210
l5(x220, x221, x222, x223, x224, x225, x226, x227, x228, x229) -> l4(x230, x231, x232, x233, x234, x235, x236, x237, x238, x239) :|: x229 = x239 && x228 = x238 && x227 = x237 && x226 = x236 && x225 = x235 && x224 = x234 && x223 = x233 && x222 = x232 && x221 = x231 && x220 = x230
l6(x240, x241, x242, x243, x244, x245, x246, x247, x248, x249) -> l5(x250, x251, x252, x253, x254, x255, x256, x257, x258, x259) :|: x249 = x259 && x248 = x258 && x247 = x257 && x246 = x256 && x245 = x255 && x244 = x254 && x243 = x253 && x242 = x252 && x241 = x251 && x240 = x250
Start term: l6(oldX0HAT0, oldX1HAT0, oldX2HAT0, oldX3HAT0, oldX4HAT0, oldX5HAT0, oldX6HAT0, x0HAT0, x1HAT0, x2HAT0)

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(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(oldX0HAT0, oldX1HAT0, oldX2HAT0, oldX3HAT0, oldX4HAT0, oldX5HAT0, oldX6HAT0, x0HAT0, x1HAT0, x2HAT0) -> l1(oldX0HATpost, oldX1HATpost, oldX2HATpost, oldX3HATpost, oldX4HATpost, oldX5HATpost, oldX6HATpost, x0HATpost, x1HATpost, x2HATpost) :|: oldX6HAT0 = oldX6HATpost && x2HATpost = oldX5HATpost && x1HATpost = oldX4HATpost && x0HATpost = oldX3HATpost && oldX5HATpost = oldX5HATpost && oldX4HATpost = oldX4HATpost && oldX3HATpost = oldX3HATpost && oldX2HATpost = x2HAT0 && oldX1HATpost = x1HAT0 && oldX0HATpost = x0HAT0
(2) l2(x, x1, x2, x3, x4, x5, x6, x7, x8, x9) -> l1(x10, x11, x12, x13, x14, x15, x16, x17, x18, x19) :|: x19 = x15 && x18 = x14 && x17 = x13 && 2 <= x12 - 2 * x16 && x16 = x16 && x15 = x15 && x14 = x14 && x13 = x13 && x12 = x9 && x11 = x8 && x10 = x7
(3) l2(x20, x21, x22, x23, x24, x25, x26, x27, x28, x29) -> l1(x30, x31, x32, x33, x34, x35, x36, x37, x38, x39) :|: x39 = x35 && x38 = x34 && x37 = x33 && 1 + x32 - 2 * x36 <= 0 && x36 = x36 && x35 = x35 && x34 = x34 && x33 = x33 && x32 = x29 && x31 = x28 && x30 = x27
(4) l2(x40, x41, x42, x43, x44, x45, x46, x47, x48, x49) -> l3(x50, x51, x52, x53, x54, x55, x56, x57, x58, x59) :|: x46 = x56 && x45 = x55 && x44 = x54 && x59 = x53 && x58 = 1 + x51 && x57 = x50 && 1 + x52 - 2 * x53 <= 2 && 0 <= x52 - 2 * x53 && x53 = x53 && x52 = x49 && x51 = x48 && x50 = x47
(5) l3(x60, x61, x62, x63, x64, x65, x66, x67, x68, x69) -> l0(x70, x71, x72, x73, x74, x75, x76, x77, x78, x79) :|: x66 = x76 && x65 = x75 && x64 = x74 && x63 = x73 && x79 = x72 && x78 = x71 && x77 = x70 && x72 <= 1 && x72 = x69 && x71 = x68 && x70 = x67
(6) l3(x80, x81, x82, x83, x84, x85, x86, x87, x88, x89) -> l2(x90, x91, x92, x93, x94, x95, x96, x97, x98, x99) :|: x86 = x96 && x85 = x95 && x84 = x94 && x83 = x93 && x99 = x92 && x98 = x91 && x97 = x90 && 2 <= x92 && x92 = x89 && x91 = x88 && x90 = x87
(7) l4(x100, x101, x102, x103, x104, x105, x106, x107, x108, x109) -> l3(x110, x111, x112, x113, x114, x115, x116, x117, x118, x119) :|: x106 = x116 && x105 = x115 && x104 = x114 && x103 = x113 && x119 = x110 && x118 = 0 && x117 = x110 && x112 = x109 && x111 = x108 && x110 = x107
(8) l5(x120, x121, x122, x123, x124, x125, x126, x127, x128, x129) -> l4(x130, x131, x132, x133, x134, x135, x136, x137, x138, x139) :|: x126 = x136 && x125 = x135 && x139 = x134 && x138 = x133 && x137 = x130 && x134 = x134 && x133 = x133 && x132 = x129 && x131 = x128 && x130 = x127
(9) l5(x140, x141, x142, x143, x144, x145, x146, x147, x148, x149) -> l1(x150, x151, x152, x153, x154, x155, x156, x157, x158, x159) :|: x149 = x159 && x148 = x158 && x147 = x157 && x146 = x156 && x145 = x155 && x144 = x154 && x143 = x153 && x142 = x152 && x141 = x151 && x140 = x150
(10) l5(x160, x161, x162, x163, x164, x165, x166, x167, x168, x169) -> l0(x170, x171, x172, x173, x174, x175, x176, x177, x178, x179) :|: x169 = x179 && x168 = x178 && x167 = x177 && x166 = x176 && x165 = x175 && x164 = x174 && x163 = x173 && x162 = x172 && x161 = x171 && x160 = x170
(11) l5(x180, x181, x182, x183, x184, x185, x186, x187, x188, x189) -> l2(x190, x191, x192, x193, x194, x195, x196, x197, x198, x199) :|: x189 = x199 && x188 = x198 && x187 = x197 && x186 = x196 && x185 = x195 && x184 = x194 && x183 = x193 && x182 = x192 && x181 = x191 && x180 = x190
(12) l5(x200, x201, x202, x203, x204, x205, x206, x207, x208, x209) -> l3(x210, x211, x212, x213, x214, x215, x216, x217, x218, x219) :|: x209 = x219 && x208 = x218 && x207 = x217 && x206 = x216 && x205 = x215 && x204 = x214 && x203 = x213 && x202 = x212 && x201 = x211 && x200 = x210
(13) l5(x220, x221, x222, x223, x224, x225, x226, x227, x228, x229) -> l4(x230, x231, x232, x233, x234, x235, x236, x237, x238, x239) :|: x229 = x239 && x228 = x238 && x227 = x237 && x226 = x236 && x225 = x235 && x224 = x234 && x223 = x233 && x222 = x232 && x221 = x231 && x220 = x230
(14) l6(x240, x241, x242, x243, x244, x245, x246, x247, x248, x249) -> l5(x250, x251, x252, x253, x254, x255, x256, x257, x258, x259) :|: x249 = x259 && x248 = x258 && x247 = x257 && x246 = x256 && x245 = x255 && x244 = x254 && x243 = x253 && x242 = x252 && x241 = x251 && x240 = x250

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

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

(4)
Obligation:

Termination digraph:
Nodes:
(1) l2(x40, x41, x42, x43, x44, x45, x46, x47, x48, x49) -> l3(x50, x51, x52, x53, x54, x55, x56, x57, x58, x59) :|: x46 = x56 && x45 = x55 && x44 = x54 && x59 = x53 && x58 = 1 + x51 && x57 = x50 && 1 + x52 - 2 * x53 <= 2 && 0 <= x52 - 2 * x53 && x53 = x53 && x52 = x49 && x51 = x48 && x50 = x47
(2) l3(x80, x81, x82, x83, x84, x85, x86, x87, x88, x89) -> l2(x90, x91, x92, x93, x94, x95, x96, x97, x98, x99) :|: x86 = x96 && x85 = x95 && x84 = x94 && x83 = x93 && x99 = x92 && x98 = x91 && x97 = x90 && 2 <= x92 && x92 = x89 && x91 = x88 && x90 = x87

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

This digraph is fully evaluated!

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

(6)
Obligation:
Rules:
l2(x40:0, x41:0, x42:0, x43:0, x44:0, x45:0, x46:0, x47:0, x48:0, x49:0) -> l2(x47:0, 1 + x48:0, x53:0, x53:0, x44:0, x45:0, x46:0, x47:0, 1 + x48:0, x53:0) :|: x49:0 - 2 * x53:0 >= 0 && 2 >= 1 + x49:0 - 2 * x53:0 && x53:0 > 1

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(7) 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, x6, x7, x8, x9, x10) -> l2(x10)

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(8)
Obligation:
Rules:
l2(x49:0) -> l2(x53:0) :|: x49:0 - 2 * x53:0 >= 0 && 2 >= 1 + x49:0 - 2 * x53:0 && x53:0 > 1

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(9) FilterProof (EQUIVALENT)
Used the following sort dictionary for filtering: 
l2(INTEGER)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(10)
Obligation:
Rules:
l2(x49:0) -> l2(x53:0) :|: x49:0 - 2 * x53:0 >= 0 && 2 >= 1 + x49:0 - 2 * x53:0 && x53:0 > 1

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

(12)
Obligation:
Rules:
l2(x49:0:0) -> l2(x53:0:0) :|: x49:0:0 - 2 * x53:0:0 >= 0 && 2 >= 1 + x49:0:0 - 2 * x53:0:0 && x53:0:0 > 1

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(13) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ l2 ] = 1/2*l2_1

The following rules are decreasing:
l2(x49:0:0) -> l2(x53:0:0) :|: x49:0:0 - 2 * x53:0:0 >= 0 && 2 >= 1 + x49:0:0 - 2 * x53:0:0 && x53:0:0 > 1

The following rules are bounded:
l2(x49:0:0) -> l2(x53:0:0) :|: x49:0:0 - 2 * x53:0:0 >= 0 && 2 >= 1 + x49:0:0 - 2 * x53:0:0 && x53:0:0 > 1


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(14)
YES