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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 5640 ms]
(4) AND
    (5) IRSwT
        (6) IntTRSCompressionProof [EQUIVALENT, 10 ms]
        (7) IRSwT
        (8) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (9) IRSwT
        (10) TempFilterProof [SOUND, 10 ms]
        (11) IntTRS
        (12) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
        (13) YES
    (14) IRSwT
        (15) IntTRSCompressionProof [EQUIVALENT, 30 ms]
        (16) IRSwT
        (17) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (18) IRSwT
        (19) TempFilterProof [SOUND, 8 ms]
        (20) IntTRS
        (21) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
        (22) YES


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

(0)
Obligation:
Rules:
l0(__const_20HAT0, __const_5HAT0, __const_99HAT0, oldX0HAT0, oldX1HAT0, oldX2HAT0, oldX3HAT0, x0HAT0, x1HAT0) -> l1(__const_20HATpost, __const_5HATpost, __const_99HATpost, oldX0HATpost, oldX1HATpost, oldX2HATpost, oldX3HATpost, x0HATpost, x1HATpost) :|: __const_99HAT0 = __const_99HATpost && __const_5HAT0 = __const_5HATpost && __const_20HAT0 = __const_20HATpost && x1HATpost = oldX3HATpost && x0HATpost = oldX2HATpost && oldX3HATpost = oldX3HATpost && oldX2HATpost = oldX2HATpost && oldX1HATpost = x1HAT0 && oldX0HATpost = x0HAT0
l2(x, x1, x2, x3, x4, x5, x6, x7, x8) -> l3(x9, x10, x11, x12, x13, x14, x15, x16, x17) :|: x6 = x15 && x5 = x14 && x2 = x11 && x1 = x10 && x = x9 && x17 = 3 + x13 && x16 = x12 && x13 = x8 && x12 = x7
l3(x18, x19, x20, x21, x22, x23, x24, x25, x26) -> l0(x27, x28, x29, x30, x31, x32, x33, x34, x35) :|: x24 = x33 && x23 = x32 && x20 = x29 && x19 = x28 && x18 = x27 && x35 = x31 && x34 = x30 && 1 + x18 <= x31 && x31 = x26 && x30 = x25
l3(x36, x37, x38, x39, x40, x41, x42, x43, x44) -> l2(x45, x46, x47, x48, x49, x50, x51, x52, x53) :|: x42 = x51 && x41 = x50 && x38 = x47 && x37 = x46 && x36 = x45 && x53 = x49 && x52 = x48 && x49 <= x36 && x49 = x44 && x48 = x43
l4(x54, x55, x56, x57, x58, x59, x60, x61, x62) -> l5(x63, x64, x65, x66, x67, x68, x69, x70, x71) :|: x60 = x69 && x56 = x65 && x55 = x64 && x54 = x63 && x71 = x68 && x70 = 1 + x66 && x68 = x68 && x67 = x62 && x66 = x61
l5(x72, x73, x74, x75, x76, x77, x78, x79, x80) -> l3(x81, x82, x83, x84, x85, x86, x87, x88, x89) :|: x78 = x87 && x77 = x86 && x74 = x83 && x73 = x82 && x72 = x81 && x89 = x73 && x88 = x84 && 1 + x74 <= x84 && x85 = x80 && x84 = x79
l5(x90, x91, x92, x93, x94, x95, x96, x97, x98) -> l4(x99, x100, x101, x102, x103, x104, x105, x106, x107) :|: x96 = x105 && x92 = x101 && x91 = x100 && x90 = x99 && x107 = x104 && x106 = x102 && x102 <= x92 && x104 = x104 && x103 = x98 && x102 = x97
l6(x108, x109, x110, x111, x112, x113, x114, x115, x116) -> l5(x117, x118, x119, x120, x121, x122, x123, x124, x125) :|: x114 = x123 && x110 = x119 && x109 = x118 && x108 = x117 && x125 = x122 && x124 = 0 && x122 = x122 && x121 = x116 && x120 = x115
l7(x126, x127, x128, x129, x130, x131, x132, x133, x134) -> l6(x135, x136, x137, x138, x139, x140, x141, x142, x143) :|: x128 = x137 && x127 = x136 && x126 = x135 && x143 = x141 && x142 = x140 && x141 = x141 && x140 = x140 && x139 = x134 && x138 = x133
l7(x144, x145, x146, x147, x148, x149, x150, x151, x152) -> l1(x153, x154, x155, x156, x157, x158, x159, x160, x161) :|: x152 = x161 && x151 = x160 && x150 = x159 && x149 = x158 && x148 = x157 && x147 = x156 && x146 = x155 && x145 = x154 && x144 = x153
l7(x162, x163, x164, x165, x166, x167, x168, x169, x170) -> l0(x171, x172, x173, x174, x175, x176, x177, x178, x179) :|: x170 = x179 && x169 = x178 && x168 = x177 && x167 = x176 && x166 = x175 && x165 = x174 && x164 = x173 && x163 = x172 && x162 = x171
l7(x180, x181, x182, x183, x184, x185, x186, x187, x188) -> l2(x189, x190, x191, x192, x193, x194, x195, x196, x197) :|: x188 = x197 && x187 = x196 && x186 = x195 && x185 = x194 && x184 = x193 && x183 = x192 && x182 = x191 && x181 = x190 && x180 = x189
l7(x198, x199, x200, x201, x202, x203, x204, x205, x206) -> l3(x207, x208, x209, x210, x211, x212, x213, x214, x215) :|: x206 = x215 && x205 = x214 && x204 = x213 && x203 = x212 && x202 = x211 && x201 = x210 && x200 = x209 && x199 = x208 && x198 = x207
l7(x216, x217, x218, x219, x220, x221, x222, x223, x224) -> l4(x225, x226, x227, x228, x229, x230, x231, x232, x233) :|: x224 = x233 && x223 = x232 && x222 = x231 && x221 = x230 && x220 = x229 && x219 = x228 && x218 = x227 && x217 = x226 && x216 = x225
l7(x234, x235, x236, x237, x238, x239, x240, x241, x242) -> l5(x243, x244, x245, x246, x247, x248, x249, x250, x251) :|: x242 = x251 && x241 = x250 && x240 = x249 && x239 = x248 && x238 = x247 && x237 = x246 && x236 = x245 && x235 = x244 && x234 = x243
l7(x252, x253, x254, x255, x256, x257, x258, x259, x260) -> l6(x261, x262, x263, x264, x265, x266, x267, x268, x269) :|: x260 = x269 && x259 = x268 && x258 = x267 && x257 = x266 && x256 = x265 && x255 = x264 && x254 = x263 && x253 = x262 && x252 = x261
l8(x270, x271, x272, x273, x274, x275, x276, x277, x278) -> l7(x279, x280, x281, x282, x283, x284, x285, x286, x287) :|: x278 = x287 && x277 = x286 && x276 = x285 && x275 = x284 && x274 = x283 && x273 = x282 && x272 = x281 && x271 = x280 && x270 = x279
Start term: l8(__const_20HAT0, __const_5HAT0, __const_99HAT0, 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(__const_20HAT0, __const_5HAT0, __const_99HAT0, oldX0HAT0, oldX1HAT0, oldX2HAT0, oldX3HAT0, x0HAT0, x1HAT0) -> l1(__const_20HATpost, __const_5HATpost, __const_99HATpost, oldX0HATpost, oldX1HATpost, oldX2HATpost, oldX3HATpost, x0HATpost, x1HATpost) :|: __const_99HAT0 = __const_99HATpost && __const_5HAT0 = __const_5HATpost && __const_20HAT0 = __const_20HATpost && x1HATpost = oldX3HATpost && x0HATpost = oldX2HATpost && oldX3HATpost = oldX3HATpost && oldX2HATpost = oldX2HATpost && oldX1HATpost = x1HAT0 && oldX0HATpost = x0HAT0
l2(x, x1, x2, x3, x4, x5, x6, x7, x8) -> l3(x9, x10, x11, x12, x13, x14, x15, x16, x17) :|: x6 = x15 && x5 = x14 && x2 = x11 && x1 = x10 && x = x9 && x17 = 3 + x13 && x16 = x12 && x13 = x8 && x12 = x7
l3(x18, x19, x20, x21, x22, x23, x24, x25, x26) -> l0(x27, x28, x29, x30, x31, x32, x33, x34, x35) :|: x24 = x33 && x23 = x32 && x20 = x29 && x19 = x28 && x18 = x27 && x35 = x31 && x34 = x30 && 1 + x18 <= x31 && x31 = x26 && x30 = x25
l3(x36, x37, x38, x39, x40, x41, x42, x43, x44) -> l2(x45, x46, x47, x48, x49, x50, x51, x52, x53) :|: x42 = x51 && x41 = x50 && x38 = x47 && x37 = x46 && x36 = x45 && x53 = x49 && x52 = x48 && x49 <= x36 && x49 = x44 && x48 = x43
l4(x54, x55, x56, x57, x58, x59, x60, x61, x62) -> l5(x63, x64, x65, x66, x67, x68, x69, x70, x71) :|: x60 = x69 && x56 = x65 && x55 = x64 && x54 = x63 && x71 = x68 && x70 = 1 + x66 && x68 = x68 && x67 = x62 && x66 = x61
l5(x72, x73, x74, x75, x76, x77, x78, x79, x80) -> l3(x81, x82, x83, x84, x85, x86, x87, x88, x89) :|: x78 = x87 && x77 = x86 && x74 = x83 && x73 = x82 && x72 = x81 && x89 = x73 && x88 = x84 && 1 + x74 <= x84 && x85 = x80 && x84 = x79
l5(x90, x91, x92, x93, x94, x95, x96, x97, x98) -> l4(x99, x100, x101, x102, x103, x104, x105, x106, x107) :|: x96 = x105 && x92 = x101 && x91 = x100 && x90 = x99 && x107 = x104 && x106 = x102 && x102 <= x92 && x104 = x104 && x103 = x98 && x102 = x97
l6(x108, x109, x110, x111, x112, x113, x114, x115, x116) -> l5(x117, x118, x119, x120, x121, x122, x123, x124, x125) :|: x114 = x123 && x110 = x119 && x109 = x118 && x108 = x117 && x125 = x122 && x124 = 0 && x122 = x122 && x121 = x116 && x120 = x115
l7(x126, x127, x128, x129, x130, x131, x132, x133, x134) -> l6(x135, x136, x137, x138, x139, x140, x141, x142, x143) :|: x128 = x137 && x127 = x136 && x126 = x135 && x143 = x141 && x142 = x140 && x141 = x141 && x140 = x140 && x139 = x134 && x138 = x133
l7(x144, x145, x146, x147, x148, x149, x150, x151, x152) -> l1(x153, x154, x155, x156, x157, x158, x159, x160, x161) :|: x152 = x161 && x151 = x160 && x150 = x159 && x149 = x158 && x148 = x157 && x147 = x156 && x146 = x155 && x145 = x154 && x144 = x153
l7(x162, x163, x164, x165, x166, x167, x168, x169, x170) -> l0(x171, x172, x173, x174, x175, x176, x177, x178, x179) :|: x170 = x179 && x169 = x178 && x168 = x177 && x167 = x176 && x166 = x175 && x165 = x174 && x164 = x173 && x163 = x172 && x162 = x171
l7(x180, x181, x182, x183, x184, x185, x186, x187, x188) -> l2(x189, x190, x191, x192, x193, x194, x195, x196, x197) :|: x188 = x197 && x187 = x196 && x186 = x195 && x185 = x194 && x184 = x193 && x183 = x192 && x182 = x191 && x181 = x190 && x180 = x189
l7(x198, x199, x200, x201, x202, x203, x204, x205, x206) -> l3(x207, x208, x209, x210, x211, x212, x213, x214, x215) :|: x206 = x215 && x205 = x214 && x204 = x213 && x203 = x212 && x202 = x211 && x201 = x210 && x200 = x209 && x199 = x208 && x198 = x207
l7(x216, x217, x218, x219, x220, x221, x222, x223, x224) -> l4(x225, x226, x227, x228, x229, x230, x231, x232, x233) :|: x224 = x233 && x223 = x232 && x222 = x231 && x221 = x230 && x220 = x229 && x219 = x228 && x218 = x227 && x217 = x226 && x216 = x225
l7(x234, x235, x236, x237, x238, x239, x240, x241, x242) -> l5(x243, x244, x245, x246, x247, x248, x249, x250, x251) :|: x242 = x251 && x241 = x250 && x240 = x249 && x239 = x248 && x238 = x247 && x237 = x246 && x236 = x245 && x235 = x244 && x234 = x243
l7(x252, x253, x254, x255, x256, x257, x258, x259, x260) -> l6(x261, x262, x263, x264, x265, x266, x267, x268, x269) :|: x260 = x269 && x259 = x268 && x258 = x267 && x257 = x266 && x256 = x265 && x255 = x264 && x254 = x263 && x253 = x262 && x252 = x261
l8(x270, x271, x272, x273, x274, x275, x276, x277, x278) -> l7(x279, x280, x281, x282, x283, x284, x285, x286, x287) :|: x278 = x287 && x277 = x286 && x276 = x285 && x275 = x284 && x274 = x283 && x273 = x282 && x272 = x281 && x271 = x280 && x270 = x279
Start term: l8(__const_20HAT0, __const_5HAT0, __const_99HAT0, oldX0HAT0, oldX1HAT0, oldX2HAT0, oldX3HAT0, x0HAT0, x1HAT0)

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

(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(__const_20HAT0, __const_5HAT0, __const_99HAT0, oldX0HAT0, oldX1HAT0, oldX2HAT0, oldX3HAT0, x0HAT0, x1HAT0) -> l1(__const_20HATpost, __const_5HATpost, __const_99HATpost, oldX0HATpost, oldX1HATpost, oldX2HATpost, oldX3HATpost, x0HATpost, x1HATpost) :|: __const_99HAT0 = __const_99HATpost && __const_5HAT0 = __const_5HATpost && __const_20HAT0 = __const_20HATpost && x1HATpost = oldX3HATpost && x0HATpost = oldX2HATpost && oldX3HATpost = oldX3HATpost && oldX2HATpost = oldX2HATpost && oldX1HATpost = x1HAT0 && oldX0HATpost = x0HAT0
(2) l2(x, x1, x2, x3, x4, x5, x6, x7, x8) -> l3(x9, x10, x11, x12, x13, x14, x15, x16, x17) :|: x6 = x15 && x5 = x14 && x2 = x11 && x1 = x10 && x = x9 && x17 = 3 + x13 && x16 = x12 && x13 = x8 && x12 = x7
(3) l3(x18, x19, x20, x21, x22, x23, x24, x25, x26) -> l0(x27, x28, x29, x30, x31, x32, x33, x34, x35) :|: x24 = x33 && x23 = x32 && x20 = x29 && x19 = x28 && x18 = x27 && x35 = x31 && x34 = x30 && 1 + x18 <= x31 && x31 = x26 && x30 = x25
(4) l3(x36, x37, x38, x39, x40, x41, x42, x43, x44) -> l2(x45, x46, x47, x48, x49, x50, x51, x52, x53) :|: x42 = x51 && x41 = x50 && x38 = x47 && x37 = x46 && x36 = x45 && x53 = x49 && x52 = x48 && x49 <= x36 && x49 = x44 && x48 = x43
(5) l4(x54, x55, x56, x57, x58, x59, x60, x61, x62) -> l5(x63, x64, x65, x66, x67, x68, x69, x70, x71) :|: x60 = x69 && x56 = x65 && x55 = x64 && x54 = x63 && x71 = x68 && x70 = 1 + x66 && x68 = x68 && x67 = x62 && x66 = x61
(6) l5(x72, x73, x74, x75, x76, x77, x78, x79, x80) -> l3(x81, x82, x83, x84, x85, x86, x87, x88, x89) :|: x78 = x87 && x77 = x86 && x74 = x83 && x73 = x82 && x72 = x81 && x89 = x73 && x88 = x84 && 1 + x74 <= x84 && x85 = x80 && x84 = x79
(7) l5(x90, x91, x92, x93, x94, x95, x96, x97, x98) -> l4(x99, x100, x101, x102, x103, x104, x105, x106, x107) :|: x96 = x105 && x92 = x101 && x91 = x100 && x90 = x99 && x107 = x104 && x106 = x102 && x102 <= x92 && x104 = x104 && x103 = x98 && x102 = x97
(8) l6(x108, x109, x110, x111, x112, x113, x114, x115, x116) -> l5(x117, x118, x119, x120, x121, x122, x123, x124, x125) :|: x114 = x123 && x110 = x119 && x109 = x118 && x108 = x117 && x125 = x122 && x124 = 0 && x122 = x122 && x121 = x116 && x120 = x115
(9) l7(x126, x127, x128, x129, x130, x131, x132, x133, x134) -> l6(x135, x136, x137, x138, x139, x140, x141, x142, x143) :|: x128 = x137 && x127 = x136 && x126 = x135 && x143 = x141 && x142 = x140 && x141 = x141 && x140 = x140 && x139 = x134 && x138 = x133
(10) l7(x144, x145, x146, x147, x148, x149, x150, x151, x152) -> l1(x153, x154, x155, x156, x157, x158, x159, x160, x161) :|: x152 = x161 && x151 = x160 && x150 = x159 && x149 = x158 && x148 = x157 && x147 = x156 && x146 = x155 && x145 = x154 && x144 = x153
(11) l7(x162, x163, x164, x165, x166, x167, x168, x169, x170) -> l0(x171, x172, x173, x174, x175, x176, x177, x178, x179) :|: x170 = x179 && x169 = x178 && x168 = x177 && x167 = x176 && x166 = x175 && x165 = x174 && x164 = x173 && x163 = x172 && x162 = x171
(12) l7(x180, x181, x182, x183, x184, x185, x186, x187, x188) -> l2(x189, x190, x191, x192, x193, x194, x195, x196, x197) :|: x188 = x197 && x187 = x196 && x186 = x195 && x185 = x194 && x184 = x193 && x183 = x192 && x182 = x191 && x181 = x190 && x180 = x189
(13) l7(x198, x199, x200, x201, x202, x203, x204, x205, x206) -> l3(x207, x208, x209, x210, x211, x212, x213, x214, x215) :|: x206 = x215 && x205 = x214 && x204 = x213 && x203 = x212 && x202 = x211 && x201 = x210 && x200 = x209 && x199 = x208 && x198 = x207
(14) l7(x216, x217, x218, x219, x220, x221, x222, x223, x224) -> l4(x225, x226, x227, x228, x229, x230, x231, x232, x233) :|: x224 = x233 && x223 = x232 && x222 = x231 && x221 = x230 && x220 = x229 && x219 = x228 && x218 = x227 && x217 = x226 && x216 = x225
(15) l7(x234, x235, x236, x237, x238, x239, x240, x241, x242) -> l5(x243, x244, x245, x246, x247, x248, x249, x250, x251) :|: x242 = x251 && x241 = x250 && x240 = x249 && x239 = x248 && x238 = x247 && x237 = x246 && x236 = x245 && x235 = x244 && x234 = x243
(16) l7(x252, x253, x254, x255, x256, x257, x258, x259, x260) -> l6(x261, x262, x263, x264, x265, x266, x267, x268, x269) :|: x260 = x269 && x259 = x268 && x258 = x267 && x257 = x266 && x256 = x265 && x255 = x264 && x254 = x263 && x253 = x262 && x252 = x261
(17) l8(x270, x271, x272, x273, x274, x275, x276, x277, x278) -> l7(x279, x280, x281, x282, x283, x284, x285, x286, x287) :|: x278 = x287 && x277 = x286 && x276 = x285 && x275 = x284 && x274 = x283 && x273 = x282 && x272 = x281 && x271 = x280 && x270 = x279

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

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

(4)
Complex Obligation (AND)

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

(5)
Obligation:

Termination digraph:
Nodes:
(1) l4(x54, x55, x56, x57, x58, x59, x60, x61, x62) -> l5(x63, x64, x65, x66, x67, x68, x69, x70, x71) :|: x60 = x69 && x56 = x65 && x55 = x64 && x54 = x63 && x71 = x68 && x70 = 1 + x66 && x68 = x68 && x67 = x62 && x66 = x61
(2) l5(x90, x91, x92, x93, x94, x95, x96, x97, x98) -> l4(x99, x100, x101, x102, x103, x104, x105, x106, x107) :|: x96 = x105 && x92 = x101 && x91 = x100 && x90 = x99 && x107 = x104 && x106 = x102 && x102 <= x92 && x104 = x104 && x103 = x98 && x102 = x97

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

This digraph is fully evaluated!

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

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

(7)
Obligation:
Rules:
l4(x54:0, x100:0, x101:0, x57:0, x58:0, x59:0, x105:0, x61:0, x62:0) -> l4(x54:0, x100:0, x101:0, 1 + x61:0, x103:0, x104:0, x105:0, 1 + x61:0, x104:0) :|: x101:0 >= 1 + x61:0

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

(8) 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, x6, x7, x8, x9) -> l4(x3, x8)

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

(9)
Obligation:
Rules:
l4(x101:0, x61:0) -> l4(x101:0, 1 + x61:0) :|: x101:0 >= 1 + x61:0

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

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

(11)
Obligation:
Rules:
l4(x101:0, x61:0) -> l4(x101:0, c) :|: c = 1 + x61:0 && x101:0 >= 1 + x61:0

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

(12) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l4(x, x1)] = x - x1

The following rules are decreasing:
l4(x101:0, x61:0) -> l4(x101:0, c) :|: c = 1 + x61:0 && x101:0 >= 1 + x61:0
The following rules are bounded:
l4(x101:0, x61:0) -> l4(x101:0, c) :|: c = 1 + x61:0 && x101:0 >= 1 + x61:0

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

(13)
YES

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

(14)
Obligation:

Termination digraph:
Nodes:
(1) l2(x, x1, x2, x3, x4, x5, x6, x7, x8) -> l3(x9, x10, x11, x12, x13, x14, x15, x16, x17) :|: x6 = x15 && x5 = x14 && x2 = x11 && x1 = x10 && x = x9 && x17 = 3 + x13 && x16 = x12 && x13 = x8 && x12 = x7
(2) l3(x36, x37, x38, x39, x40, x41, x42, x43, x44) -> l2(x45, x46, x47, x48, x49, x50, x51, x52, x53) :|: x42 = x51 && x41 = x50 && x38 = x47 && x37 = x46 && x36 = x45 && x53 = x49 && x52 = x48 && x49 <= x36 && x49 = x44 && x48 = x43

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

This digraph is fully evaluated!

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

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

(16)
Obligation:
Rules:
l2(x45:0, x10:0, x11:0, x3:0, x4:0, x14:0, x15:0, x12:0, x13:0) -> l2(x45:0, x10:0, x11:0, x12:0, 3 + x13:0, x14:0, x15:0, x12:0, 3 + x13:0) :|: x45:0 >= 3 + x13:0

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

(17) 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) -> l2(x1, x9)

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

(18)
Obligation:
Rules:
l2(x45:0, x13:0) -> l2(x45:0, 3 + x13:0) :|: x45:0 >= 3 + x13:0

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

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

(20)
Obligation:
Rules:
l2(x45:0, x13:0) -> l2(x45:0, c) :|: c = 3 + x13:0 && x45:0 >= 3 + x13:0

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

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

The following rules are decreasing:
l2(x45:0, x13:0) -> l2(x45:0, c) :|: c = 3 + x13:0 && x45:0 >= 3 + x13:0
The following rules are bounded:
l2(x45:0, x13:0) -> l2(x45:0, c) :|: c = 3 + x13:0 && x45:0 >= 3 + x13:0

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

(22)
YES