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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 4605 ms]
(4) AND
    (5) IRSwT
        (6) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (7) IRSwT
        (8) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (9) IRSwT
        (10) TempFilterProof [SOUND, 35 ms]
        (11) IntTRS
        (12) PolynomialOrderProcessor [EQUIVALENT, 5 ms]
        (13) YES
    (14) IRSwT
        (15) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (16) IRSwT
        (17) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (18) IRSwT
        (19) TempFilterProof [SOUND, 12 ms]
        (20) IntTRS
        (21) PolynomialOrderProcessor [EQUIVALENT, 1 ms]
        (22) YES


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

(0)
Obligation:
Rules:
l0(__const_10HAT0, ilHAT0, ret_foo5HAT0, ret_foo7HAT0, tmpHAT0, tmp___0HAT0, x4HAT0, x6HAT0) -> l1(__const_10HATpost, ilHATpost, ret_foo5HATpost, ret_foo7HATpost, tmpHATpost, tmp___0HATpost, x4HATpost, x6HATpost) :|: x6HAT0 = x6HATpost && x4HAT0 = x4HATpost && tmp___0HAT0 = tmp___0HATpost && tmpHAT0 = tmpHATpost && ret_foo7HAT0 = ret_foo7HATpost && ret_foo5HAT0 = ret_foo5HATpost && ilHAT0 = ilHATpost && __const_10HAT0 = __const_10HATpost && 0 <= tmpHAT0 && tmpHAT0 <= 0
l0(x, x1, x2, x3, x4, x5, x6, x7) -> l2(x8, x9, x10, x11, x12, x13, x14, x15) :|: x7 = x15 && x6 = x14 && x5 = x13 && x4 = x12 && x3 = x11 && x2 = x10 && x1 = x9 && x = x8 && 1 <= x4
l0(x16, x17, x18, x19, x20, x21, x22, x23) -> l2(x24, x25, x26, x27, x28, x29, x30, x31) :|: x23 = x31 && x22 = x30 && x21 = x29 && x20 = x28 && x19 = x27 && x18 = x26 && x17 = x25 && x16 = x24 && 1 + x20 <= 0
l3(x32, x33, x34, x35, x36, x37, x38, x39) -> l0(x40, x41, x42, x43, x44, x45, x46, x47) :|: x39 = x47 && x37 = x45 && x35 = x43 && x33 = x41 && x32 = x40 && x44 = x42 && x42 = x42 && x46 = x33
l4(x48, x49, x50, x51, x52, x53, x54, x55) -> l5(x56, x57, x58, x59, x60, x61, x62, x63) :|: x55 = x63 && x54 = x62 && x53 = x61 && x52 = x60 && x51 = x59 && x50 = x58 && x49 = x57 && x48 = x56
l6(x64, x65, x66, x67, x68, x69, x70, x71) -> l7(x72, x73, x74, x75, x76, x77, x78, x79) :|: x71 = x79 && x70 = x78 && x69 = x77 && x68 = x76 && x67 = x75 && x66 = x74 && x64 = x72 && x73 = 1 + x65
l8(x80, x81, x82, x83, x84, x85, x86, x87) -> l4(x88, x89, x90, x91, x92, x93, x94, x95) :|: x87 = x95 && x86 = x94 && x85 = x93 && x84 = x92 && x83 = x91 && x82 = x90 && x81 = x89 && x80 = x88 && 0 <= x85 && x85 <= 0
l8(x96, x97, x98, x99, x100, x101, x102, x103) -> l6(x104, x105, x106, x107, x108, x109, x110, x111) :|: x103 = x111 && x102 = x110 && x101 = x109 && x100 = x108 && x99 = x107 && x98 = x106 && x97 = x105 && x96 = x104 && 1 <= x101
l8(x112, x113, x114, x115, x116, x117, x118, x119) -> l6(x120, x121, x122, x123, x124, x125, x126, x127) :|: x119 = x127 && x118 = x126 && x117 = x125 && x116 = x124 && x115 = x123 && x114 = x122 && x113 = x121 && x112 = x120 && 1 + x117 <= 0
l7(x128, x129, x130, x131, x132, x133, x134, x135) -> l9(x136, x137, x138, x139, x140, x141, x142, x143) :|: x135 = x143 && x134 = x142 && x133 = x141 && x132 = x140 && x131 = x139 && x130 = x138 && x129 = x137 && x128 = x136
l9(x144, x145, x146, x147, x148, x149, x150, x151) -> l4(x152, x153, x154, x155, x156, x157, x158, x159) :|: x151 = x159 && x150 = x158 && x149 = x157 && x148 = x156 && x147 = x155 && x146 = x154 && x145 = x153 && x144 = x152 && x144 <= x145
l9(x160, x161, x162, x163, x164, x165, x166, x167) -> l8(x168, x169, x170, x171, x172, x173, x174, x175) :|: x166 = x174 && x164 = x172 && x162 = x170 && x161 = x169 && x160 = x168 && x173 = x171 && x171 = x171 && x175 = x161 && 1 + x161 <= x160
l1(x176, x177, x178, x179, x180, x181, x182, x183) -> l7(x184, x185, x186, x187, x188, x189, x190, x191) :|: x183 = x191 && x182 = x190 && x181 = x189 && x180 = x188 && x179 = x187 && x178 = x186 && x176 = x184 && x185 = 0
l2(x192, x193, x194, x195, x196, x197, x198, x199) -> l1(x200, x201, x202, x203, x204, x205, x206, x207) :|: x199 = x207 && x198 = x206 && x197 = x205 && x196 = x204 && x195 = x203 && x194 = x202 && x193 = x201 && x192 = x200 && x192 <= x193
l2(x208, x209, x210, x211, x212, x213, x214, x215) -> l3(x216, x217, x218, x219, x220, x221, x222, x223) :|: x215 = x223 && x214 = x222 && x213 = x221 && x212 = x220 && x211 = x219 && x210 = x218 && x208 = x216 && x217 = 1 + x209 && 1 + x209 <= x208
l10(x224, x225, x226, x227, x228, x229, x230, x231) -> l3(x232, x233, x234, x235, x236, x237, x238, x239) :|: x231 = x239 && x230 = x238 && x229 = x237 && x228 = x236 && x227 = x235 && x226 = x234 && x224 = x232 && x233 = 0
l11(x240, x241, x242, x243, x244, x245, x246, x247) -> l10(x248, x249, x250, x251, x252, x253, x254, x255) :|: x247 = x255 && x246 = x254 && x245 = x253 && x244 = x252 && x243 = x251 && x242 = x250 && x241 = x249 && x240 = x248
Start term: l11(__const_10HAT0, ilHAT0, ret_foo5HAT0, ret_foo7HAT0, tmpHAT0, tmp___0HAT0, x4HAT0, x6HAT0)

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

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

(2)
Obligation:
Rules:
l0(__const_10HAT0, ilHAT0, ret_foo5HAT0, ret_foo7HAT0, tmpHAT0, tmp___0HAT0, x4HAT0, x6HAT0) -> l1(__const_10HATpost, ilHATpost, ret_foo5HATpost, ret_foo7HATpost, tmpHATpost, tmp___0HATpost, x4HATpost, x6HATpost) :|: x6HAT0 = x6HATpost && x4HAT0 = x4HATpost && tmp___0HAT0 = tmp___0HATpost && tmpHAT0 = tmpHATpost && ret_foo7HAT0 = ret_foo7HATpost && ret_foo5HAT0 = ret_foo5HATpost && ilHAT0 = ilHATpost && __const_10HAT0 = __const_10HATpost && 0 <= tmpHAT0 && tmpHAT0 <= 0
l0(x, x1, x2, x3, x4, x5, x6, x7) -> l2(x8, x9, x10, x11, x12, x13, x14, x15) :|: x7 = x15 && x6 = x14 && x5 = x13 && x4 = x12 && x3 = x11 && x2 = x10 && x1 = x9 && x = x8 && 1 <= x4
l0(x16, x17, x18, x19, x20, x21, x22, x23) -> l2(x24, x25, x26, x27, x28, x29, x30, x31) :|: x23 = x31 && x22 = x30 && x21 = x29 && x20 = x28 && x19 = x27 && x18 = x26 && x17 = x25 && x16 = x24 && 1 + x20 <= 0
l3(x32, x33, x34, x35, x36, x37, x38, x39) -> l0(x40, x41, x42, x43, x44, x45, x46, x47) :|: x39 = x47 && x37 = x45 && x35 = x43 && x33 = x41 && x32 = x40 && x44 = x42 && x42 = x42 && x46 = x33
l4(x48, x49, x50, x51, x52, x53, x54, x55) -> l5(x56, x57, x58, x59, x60, x61, x62, x63) :|: x55 = x63 && x54 = x62 && x53 = x61 && x52 = x60 && x51 = x59 && x50 = x58 && x49 = x57 && x48 = x56
l6(x64, x65, x66, x67, x68, x69, x70, x71) -> l7(x72, x73, x74, x75, x76, x77, x78, x79) :|: x71 = x79 && x70 = x78 && x69 = x77 && x68 = x76 && x67 = x75 && x66 = x74 && x64 = x72 && x73 = 1 + x65
l8(x80, x81, x82, x83, x84, x85, x86, x87) -> l4(x88, x89, x90, x91, x92, x93, x94, x95) :|: x87 = x95 && x86 = x94 && x85 = x93 && x84 = x92 && x83 = x91 && x82 = x90 && x81 = x89 && x80 = x88 && 0 <= x85 && x85 <= 0
l8(x96, x97, x98, x99, x100, x101, x102, x103) -> l6(x104, x105, x106, x107, x108, x109, x110, x111) :|: x103 = x111 && x102 = x110 && x101 = x109 && x100 = x108 && x99 = x107 && x98 = x106 && x97 = x105 && x96 = x104 && 1 <= x101
l8(x112, x113, x114, x115, x116, x117, x118, x119) -> l6(x120, x121, x122, x123, x124, x125, x126, x127) :|: x119 = x127 && x118 = x126 && x117 = x125 && x116 = x124 && x115 = x123 && x114 = x122 && x113 = x121 && x112 = x120 && 1 + x117 <= 0
l7(x128, x129, x130, x131, x132, x133, x134, x135) -> l9(x136, x137, x138, x139, x140, x141, x142, x143) :|: x135 = x143 && x134 = x142 && x133 = x141 && x132 = x140 && x131 = x139 && x130 = x138 && x129 = x137 && x128 = x136
l9(x144, x145, x146, x147, x148, x149, x150, x151) -> l4(x152, x153, x154, x155, x156, x157, x158, x159) :|: x151 = x159 && x150 = x158 && x149 = x157 && x148 = x156 && x147 = x155 && x146 = x154 && x145 = x153 && x144 = x152 && x144 <= x145
l9(x160, x161, x162, x163, x164, x165, x166, x167) -> l8(x168, x169, x170, x171, x172, x173, x174, x175) :|: x166 = x174 && x164 = x172 && x162 = x170 && x161 = x169 && x160 = x168 && x173 = x171 && x171 = x171 && x175 = x161 && 1 + x161 <= x160
l1(x176, x177, x178, x179, x180, x181, x182, x183) -> l7(x184, x185, x186, x187, x188, x189, x190, x191) :|: x183 = x191 && x182 = x190 && x181 = x189 && x180 = x188 && x179 = x187 && x178 = x186 && x176 = x184 && x185 = 0
l2(x192, x193, x194, x195, x196, x197, x198, x199) -> l1(x200, x201, x202, x203, x204, x205, x206, x207) :|: x199 = x207 && x198 = x206 && x197 = x205 && x196 = x204 && x195 = x203 && x194 = x202 && x193 = x201 && x192 = x200 && x192 <= x193
l2(x208, x209, x210, x211, x212, x213, x214, x215) -> l3(x216, x217, x218, x219, x220, x221, x222, x223) :|: x215 = x223 && x214 = x222 && x213 = x221 && x212 = x220 && x211 = x219 && x210 = x218 && x208 = x216 && x217 = 1 + x209 && 1 + x209 <= x208
l10(x224, x225, x226, x227, x228, x229, x230, x231) -> l3(x232, x233, x234, x235, x236, x237, x238, x239) :|: x231 = x239 && x230 = x238 && x229 = x237 && x228 = x236 && x227 = x235 && x226 = x234 && x224 = x232 && x233 = 0
l11(x240, x241, x242, x243, x244, x245, x246, x247) -> l10(x248, x249, x250, x251, x252, x253, x254, x255) :|: x247 = x255 && x246 = x254 && x245 = x253 && x244 = x252 && x243 = x251 && x242 = x250 && x241 = x249 && x240 = x248
Start term: l11(__const_10HAT0, ilHAT0, ret_foo5HAT0, ret_foo7HAT0, tmpHAT0, tmp___0HAT0, x4HAT0, x6HAT0)

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

(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(__const_10HAT0, ilHAT0, ret_foo5HAT0, ret_foo7HAT0, tmpHAT0, tmp___0HAT0, x4HAT0, x6HAT0) -> l1(__const_10HATpost, ilHATpost, ret_foo5HATpost, ret_foo7HATpost, tmpHATpost, tmp___0HATpost, x4HATpost, x6HATpost) :|: x6HAT0 = x6HATpost && x4HAT0 = x4HATpost && tmp___0HAT0 = tmp___0HATpost && tmpHAT0 = tmpHATpost && ret_foo7HAT0 = ret_foo7HATpost && ret_foo5HAT0 = ret_foo5HATpost && ilHAT0 = ilHATpost && __const_10HAT0 = __const_10HATpost && 0 <= tmpHAT0 && tmpHAT0 <= 0
(2) l0(x, x1, x2, x3, x4, x5, x6, x7) -> l2(x8, x9, x10, x11, x12, x13, x14, x15) :|: x7 = x15 && x6 = x14 && x5 = x13 && x4 = x12 && x3 = x11 && x2 = x10 && x1 = x9 && x = x8 && 1 <= x4
(3) l0(x16, x17, x18, x19, x20, x21, x22, x23) -> l2(x24, x25, x26, x27, x28, x29, x30, x31) :|: x23 = x31 && x22 = x30 && x21 = x29 && x20 = x28 && x19 = x27 && x18 = x26 && x17 = x25 && x16 = x24 && 1 + x20 <= 0
(4) l3(x32, x33, x34, x35, x36, x37, x38, x39) -> l0(x40, x41, x42, x43, x44, x45, x46, x47) :|: x39 = x47 && x37 = x45 && x35 = x43 && x33 = x41 && x32 = x40 && x44 = x42 && x42 = x42 && x46 = x33
(5) l4(x48, x49, x50, x51, x52, x53, x54, x55) -> l5(x56, x57, x58, x59, x60, x61, x62, x63) :|: x55 = x63 && x54 = x62 && x53 = x61 && x52 = x60 && x51 = x59 && x50 = x58 && x49 = x57 && x48 = x56
(6) l6(x64, x65, x66, x67, x68, x69, x70, x71) -> l7(x72, x73, x74, x75, x76, x77, x78, x79) :|: x71 = x79 && x70 = x78 && x69 = x77 && x68 = x76 && x67 = x75 && x66 = x74 && x64 = x72 && x73 = 1 + x65
(7) l8(x80, x81, x82, x83, x84, x85, x86, x87) -> l4(x88, x89, x90, x91, x92, x93, x94, x95) :|: x87 = x95 && x86 = x94 && x85 = x93 && x84 = x92 && x83 = x91 && x82 = x90 && x81 = x89 && x80 = x88 && 0 <= x85 && x85 <= 0
(8) l8(x96, x97, x98, x99, x100, x101, x102, x103) -> l6(x104, x105, x106, x107, x108, x109, x110, x111) :|: x103 = x111 && x102 = x110 && x101 = x109 && x100 = x108 && x99 = x107 && x98 = x106 && x97 = x105 && x96 = x104 && 1 <= x101
(9) l8(x112, x113, x114, x115, x116, x117, x118, x119) -> l6(x120, x121, x122, x123, x124, x125, x126, x127) :|: x119 = x127 && x118 = x126 && x117 = x125 && x116 = x124 && x115 = x123 && x114 = x122 && x113 = x121 && x112 = x120 && 1 + x117 <= 0
(10) l7(x128, x129, x130, x131, x132, x133, x134, x135) -> l9(x136, x137, x138, x139, x140, x141, x142, x143) :|: x135 = x143 && x134 = x142 && x133 = x141 && x132 = x140 && x131 = x139 && x130 = x138 && x129 = x137 && x128 = x136
(11) l9(x144, x145, x146, x147, x148, x149, x150, x151) -> l4(x152, x153, x154, x155, x156, x157, x158, x159) :|: x151 = x159 && x150 = x158 && x149 = x157 && x148 = x156 && x147 = x155 && x146 = x154 && x145 = x153 && x144 = x152 && x144 <= x145
(12) l9(x160, x161, x162, x163, x164, x165, x166, x167) -> l8(x168, x169, x170, x171, x172, x173, x174, x175) :|: x166 = x174 && x164 = x172 && x162 = x170 && x161 = x169 && x160 = x168 && x173 = x171 && x171 = x171 && x175 = x161 && 1 + x161 <= x160
(13) l1(x176, x177, x178, x179, x180, x181, x182, x183) -> l7(x184, x185, x186, x187, x188, x189, x190, x191) :|: x183 = x191 && x182 = x190 && x181 = x189 && x180 = x188 && x179 = x187 && x178 = x186 && x176 = x184 && x185 = 0
(14) l2(x192, x193, x194, x195, x196, x197, x198, x199) -> l1(x200, x201, x202, x203, x204, x205, x206, x207) :|: x199 = x207 && x198 = x206 && x197 = x205 && x196 = x204 && x195 = x203 && x194 = x202 && x193 = x201 && x192 = x200 && x192 <= x193
(15) l2(x208, x209, x210, x211, x212, x213, x214, x215) -> l3(x216, x217, x218, x219, x220, x221, x222, x223) :|: x215 = x223 && x214 = x222 && x213 = x221 && x212 = x220 && x211 = x219 && x210 = x218 && x208 = x216 && x217 = 1 + x209 && 1 + x209 <= x208
(16) l10(x224, x225, x226, x227, x228, x229, x230, x231) -> l3(x232, x233, x234, x235, x236, x237, x238, x239) :|: x231 = x239 && x230 = x238 && x229 = x237 && x228 = x236 && x227 = x235 && x226 = x234 && x224 = x232 && x233 = 0
(17) l11(x240, x241, x242, x243, x244, x245, x246, x247) -> l10(x248, x249, x250, x251, x252, x253, x254, x255) :|: x247 = x255 && x246 = x254 && x245 = x253 && x244 = x252 && x243 = x251 && x242 = x250 && x241 = x249 && x240 = x248

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

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

(4)
Complex Obligation (AND)

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

(5)
Obligation:

Termination digraph:
Nodes:
(1) l0(x, x1, x2, x3, x4, x5, x6, x7) -> l2(x8, x9, x10, x11, x12, x13, x14, x15) :|: x7 = x15 && x6 = x14 && x5 = x13 && x4 = x12 && x3 = x11 && x2 = x10 && x1 = x9 && x = x8 && 1 <= x4
(2) l3(x32, x33, x34, x35, x36, x37, x38, x39) -> l0(x40, x41, x42, x43, x44, x45, x46, x47) :|: x39 = x47 && x37 = x45 && x35 = x43 && x33 = x41 && x32 = x40 && x44 = x42 && x42 = x42 && x46 = x33
(3) l2(x208, x209, x210, x211, x212, x213, x214, x215) -> l3(x216, x217, x218, x219, x220, x221, x222, x223) :|: x215 = x223 && x214 = x222 && x213 = x221 && x212 = x220 && x211 = x219 && x210 = x218 && x208 = x216 && x217 = 1 + x209 && 1 + x209 <= x208
(4) l0(x16, x17, x18, x19, x20, x21, x22, x23) -> l2(x24, x25, x26, x27, x28, x29, x30, x31) :|: x23 = x31 && x22 = x30 && x21 = x29 && x20 = x28 && x19 = x27 && x18 = x26 && x17 = x25 && x16 = x24 && 1 + x20 <= 0

Arcs:
(1) -> (3)
(2) -> (1), (4)
(3) -> (2)
(4) -> (3)

This digraph is fully evaluated!

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

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

(7)
Obligation:
Rules:
l3(x216:0, x14:0, x34:0, x11:0, x36:0, x13:0, x38:0, x15:0) -> l3(x216:0, 1 + x14:0, x10:0, x11:0, x10:0, x13:0, x14:0, x15:0) :|: x10:0 > 0 && x216:0 >= 1 + x14:0
l3(x, x1, x2, x3, x4, x5, x6, x7) -> l3(x, 1 + x1, x8, x3, x8, x5, x1, x7) :|: x8 < 0 && x >= 1 + x1

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

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

   l3(x1, x2, x3, x4, x5, x6, x7, x8) -> l3(x1, x2)

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

(9)
Obligation:
Rules:
l3(x216:0, x14:0) -> l3(x216:0, 1 + x14:0) :|: x10:0 > 0 && x216:0 >= 1 + x14:0
l3(x, x1) -> l3(x, 1 + x1) :|: x8 < 0 && x >= 1 + x1

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

(10) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
l3(INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(11)
Obligation:
Rules:
l3(x216:0, x14:0) -> l3(x216:0, c) :|: c = 1 + x14:0 && (x10:0 > 0 && x216:0 >= 1 + x14:0)
l3(x, x1) -> l3(x, c1) :|: c1 = 1 + x1 && (x8 < 0 && x >= 1 + x1)

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

(12) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l3(x, x1)] = x - x1

The following rules are decreasing:
l3(x216:0, x14:0) -> l3(x216:0, c) :|: c = 1 + x14:0 && (x10:0 > 0 && x216:0 >= 1 + x14:0)
l3(x, x1) -> l3(x, c1) :|: c1 = 1 + x1 && (x8 < 0 && x >= 1 + x1)
The following rules are bounded:
l3(x216:0, x14:0) -> l3(x216:0, c) :|: c = 1 + x14:0 && (x10:0 > 0 && x216:0 >= 1 + x14:0)
l3(x, x1) -> l3(x, c1) :|: c1 = 1 + x1 && (x8 < 0 && x >= 1 + x1)

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(13)
YES

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(14)
Obligation:

Termination digraph:
Nodes:
(1) l7(x128, x129, x130, x131, x132, x133, x134, x135) -> l9(x136, x137, x138, x139, x140, x141, x142, x143) :|: x135 = x143 && x134 = x142 && x133 = x141 && x132 = x140 && x131 = x139 && x130 = x138 && x129 = x137 && x128 = x136
(2) l6(x64, x65, x66, x67, x68, x69, x70, x71) -> l7(x72, x73, x74, x75, x76, x77, x78, x79) :|: x71 = x79 && x70 = x78 && x69 = x77 && x68 = x76 && x67 = x75 && x66 = x74 && x64 = x72 && x73 = 1 + x65
(3) l8(x112, x113, x114, x115, x116, x117, x118, x119) -> l6(x120, x121, x122, x123, x124, x125, x126, x127) :|: x119 = x127 && x118 = x126 && x117 = x125 && x116 = x124 && x115 = x123 && x114 = x122 && x113 = x121 && x112 = x120 && 1 + x117 <= 0
(4) l8(x96, x97, x98, x99, x100, x101, x102, x103) -> l6(x104, x105, x106, x107, x108, x109, x110, x111) :|: x103 = x111 && x102 = x110 && x101 = x109 && x100 = x108 && x99 = x107 && x98 = x106 && x97 = x105 && x96 = x104 && 1 <= x101
(5) l9(x160, x161, x162, x163, x164, x165, x166, x167) -> l8(x168, x169, x170, x171, x172, x173, x174, x175) :|: x166 = x174 && x164 = x172 && x162 = x170 && x161 = x169 && x160 = x168 && x173 = x171 && x171 = x171 && x175 = x161 && 1 + x161 <= x160

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

This digraph is fully evaluated!

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(15) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(16)
Obligation:
Rules:
l8(x112:0, x113:0, x114:0, x115:0, x116:0, x117:0, x118:0, x119:0) -> l8(x112:0, 1 + x113:0, x114:0, x171:0, x116:0, x171:0, x118:0, 1 + x113:0) :|: x117:0 < 0 && x112:0 >= 1 + (1 + x113:0)
l8(x, x1, x2, x3, x4, x5, x6, x7) -> l8(x, 1 + x1, x2, x8, x4, x8, x6, 1 + x1) :|: x5 > 0 && x >= 1 + (1 + x1)

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

   l8(x1, x2, x3, x4, x5, x6, x7, x8) -> l8(x1, x2, x6)

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(18)
Obligation:
Rules:
l8(x112:0, x113:0, x117:0) -> l8(x112:0, 1 + x113:0, x171:0) :|: x117:0 < 0 && x112:0 >= 1 + (1 + x113:0)
l8(x, x1, x5) -> l8(x, 1 + x1, x8) :|: x5 > 0 && x >= 1 + (1 + x1)

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(19) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
l8(INTEGER, INTEGER, VARIABLE)
Replaced non-predefined constructor symbols by 0.
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(20)
Obligation:
Rules:
l8(x112:0, x113:0, x117:0) -> l8(x112:0, c, x171:0) :|: c = 1 + x113:0 && (x117:0 < 0 && x112:0 >= 1 + (1 + x113:0))
l8(x, x1, x5) -> l8(x, c1, x8) :|: c1 = 1 + x1 && (x5 > 0 && x >= 1 + (1 + x1))

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(21) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l8(x, x1, x2)] = -2 + x - x1

The following rules are decreasing:
l8(x112:0, x113:0, x117:0) -> l8(x112:0, c, x171:0) :|: c = 1 + x113:0 && (x117:0 < 0 && x112:0 >= 1 + (1 + x113:0))
l8(x, x1, x5) -> l8(x, c1, x8) :|: c1 = 1 + x1 && (x5 > 0 && x >= 1 + (1 + x1))
The following rules are bounded:
l8(x112:0, x113:0, x117:0) -> l8(x112:0, c, x171:0) :|: c = 1 + x113:0 && (x117:0 < 0 && x112:0 >= 1 + (1 + x113:0))
l8(x, x1, x5) -> l8(x, c1, x8) :|: c1 = 1 + x1 && (x5 > 0 && x >= 1 + (1 + x1))

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(22)
YES