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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 4546 ms]
(4) AND
    (5) IRSwT
        (6) IntTRSCompressionProof [EQUIVALENT, 41 ms]
        (7) IRSwT
        (8) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (9) IRSwT
        (10) TempFilterProof [SOUND, 10 ms]
        (11) IntTRS
        (12) PolynomialOrderProcessor [EQUIVALENT, 4 ms]
        (13) YES
    (14) IRSwT
        (15) IntTRSCompressionProof [EQUIVALENT, 17 ms]
        (16) IRSwT
        (17) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (18) IRSwT
        (19) TempFilterProof [SOUND, 28 ms]
        (20) IntTRS
        (21) RankingReductionPairProof [EQUIVALENT, 10 ms]
        (22) IntTRS
        (23) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
        (24) YES
    (25) IRSwT
        (26) IntTRSCompressionProof [EQUIVALENT, 10 ms]
        (27) IRSwT
        (28) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (29) IRSwT
        (30) TempFilterProof [SOUND, 32 ms]
        (31) IntTRS
        (32) PolynomialOrderProcessor [EQUIVALENT, 11 ms]
        (33) YES


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

(0)
Obligation:
Rules:
l0(N6HAT0, NHAT0, i8HAT0, iHAT0, j7HAT0, min9HAT0, t10HAT0, tmpHAT0) -> l1(N6HATpost, NHATpost, i8HATpost, iHATpost, j7HATpost, min9HATpost, t10HATpost, tmpHATpost) :|: tmpHAT0 = tmpHATpost && t10HAT0 = t10HATpost && min9HAT0 = min9HATpost && j7HAT0 = j7HATpost && i8HAT0 = i8HATpost && iHAT0 = iHATpost && N6HAT0 = N6HATpost && NHAT0 = NHATpost
l2(x, x1, x2, x3, x4, x5, x6, x7) -> l3(x8, x9, x10, x11, x12, x13, x14, x15) :|: x7 = x15 && x6 = x14 && x5 = x13 && x4 = x12 && x2 = x10 && x3 = x11 && x = x8 && x1 = x9
l4(x16, x17, x18, x19, x20, x21, x22, x23) -> l5(x24, x25, x26, x27, x28, x29, x30, x31) :|: x23 = x31 && x22 = x30 && x21 = x29 && x20 = x28 && x18 = x26 && x19 = x27 && x16 = x24 && x17 = x25 && -1 + x16 <= x20
l4(x32, x33, x34, x35, x36, x37, x38, x39) -> l6(x40, x41, x42, x43, x44, x45, x46, x47) :|: x39 = x47 && x38 = x46 && x37 = x45 && x34 = x42 && x35 = x43 && x32 = x40 && x33 = x41 && x44 = 1 + x36 && 1 + x36 <= -1 + x32
l7(x48, x49, x50, x51, x52, x53, x54, x55) -> l8(x56, x57, x58, x59, x60, x61, x62, x63) :|: x55 = x63 && x54 = x62 && x53 = x61 && x52 = x60 && x50 = x58 && x51 = x59 && x48 = x56 && x49 = x57
l9(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 && x64 = x72 && x65 = x73 && x74 = 1 + x66
l10(x80, x81, x82, x83, x84, x85, x86, x87) -> l9(x88, x89, x90, x91, x92, x93, x94, x95) :|: x87 = x95 && x86 = x94 && x84 = x92 && x82 = x90 && x83 = x91 && x80 = x88 && x81 = x89 && x93 = x82
l10(x96, x97, x98, x99, x100, x101, x102, x103) -> l9(x104, x105, x106, x107, x108, x109, x110, x111) :|: x103 = x111 && x102 = x110 && x101 = x109 && x100 = x108 && x98 = x106 && x99 = x107 && x96 = x104 && x97 = x105
l8(x112, x113, x114, x115, x116, x117, x118, x119) -> l2(x120, x121, x122, x123, x124, x125, x126, x127) :|: x119 = x127 && x117 = x125 && x114 = x122 && x115 = x123 && x112 = x120 && x113 = x121 && x124 = 1 + x116 && x126 = x126 && x112 <= x114
l8(x128, x129, x130, x131, x132, x133, x134, x135) -> l10(x136, x137, x138, x139, x140, x141, x142, x143) :|: x135 = x143 && x134 = x142 && x133 = x141 && x132 = x140 && x130 = x138 && x131 = x139 && x128 = x136 && x129 = x137 && 1 + x130 <= x128
l6(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 && x146 = x154 && x147 = x155 && x144 = x152 && x145 = x153
l3(x160, x161, x162, x163, x164, x165, x166, x167) -> l6(x168, x169, x170, x171, x172, x173, x174, x175) :|: x167 = x175 && x166 = x174 && x165 = x173 && x162 = x170 && x163 = x171 && x160 = x168 && x161 = x169 && x172 = 0 && -1 + x160 <= x164
l3(x176, x177, x178, x179, x180, x181, x182, x183) -> l7(x184, x185, x186, x187, x188, x189, x190, x191) :|: x183 = x191 && x182 = x190 && x180 = x188 && x179 = x187 && x176 = x184 && x177 = x185 && x186 = 1 + x180 && x189 = x180 && 1 + x180 <= -1 + x176
l1(x192, x193, x194, x195, x196, x197, x198, x199) -> l2(x200, x201, x202, x203, x204, x205, x206, x207) :|: x199 = x207 && x198 = x206 && x197 = x205 && x194 = x202 && x195 = x203 && x193 = x201 && x204 = 0 && x200 = x193 && x193 <= x195
l1(x208, x209, x210, x211, x212, x213, x214, x215) -> l0(x216, x217, x218, x219, x220, x221, x222, x223) :|: x215 = x223 && x214 = x222 && x213 = x221 && x212 = x220 && x210 = x218 && x208 = x216 && x209 = x217 && x219 = 1 + x211 && 1 + x211 <= x209
l11(x224, x225, x226, x227, x228, x229, x230, x231) -> l0(x232, x233, x234, x235, x236, x237, x238, x239) :|: x230 = x238 && x229 = x237 && x228 = x236 && x226 = x234 && x224 = x232 && x225 = x233 && x235 = 0 && x239 = x239
l12(x240, x241, x242, x243, x244, x245, x246, x247) -> l11(x248, x249, x250, x251, x252, x253, x254, x255) :|: x247 = x255 && x246 = x254 && x245 = x253 && x244 = x252 && x242 = x250 && x243 = x251 && x240 = x248 && x241 = x249
Start term: l12(N6HAT0, NHAT0, i8HAT0, iHAT0, j7HAT0, min9HAT0, t10HAT0, tmpHAT0)

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

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

(2)
Obligation:
Rules:
l0(N6HAT0, NHAT0, i8HAT0, iHAT0, j7HAT0, min9HAT0, t10HAT0, tmpHAT0) -> l1(N6HATpost, NHATpost, i8HATpost, iHATpost, j7HATpost, min9HATpost, t10HATpost, tmpHATpost) :|: tmpHAT0 = tmpHATpost && t10HAT0 = t10HATpost && min9HAT0 = min9HATpost && j7HAT0 = j7HATpost && i8HAT0 = i8HATpost && iHAT0 = iHATpost && N6HAT0 = N6HATpost && NHAT0 = NHATpost
l2(x, x1, x2, x3, x4, x5, x6, x7) -> l3(x8, x9, x10, x11, x12, x13, x14, x15) :|: x7 = x15 && x6 = x14 && x5 = x13 && x4 = x12 && x2 = x10 && x3 = x11 && x = x8 && x1 = x9
l4(x16, x17, x18, x19, x20, x21, x22, x23) -> l5(x24, x25, x26, x27, x28, x29, x30, x31) :|: x23 = x31 && x22 = x30 && x21 = x29 && x20 = x28 && x18 = x26 && x19 = x27 && x16 = x24 && x17 = x25 && -1 + x16 <= x20
l4(x32, x33, x34, x35, x36, x37, x38, x39) -> l6(x40, x41, x42, x43, x44, x45, x46, x47) :|: x39 = x47 && x38 = x46 && x37 = x45 && x34 = x42 && x35 = x43 && x32 = x40 && x33 = x41 && x44 = 1 + x36 && 1 + x36 <= -1 + x32
l7(x48, x49, x50, x51, x52, x53, x54, x55) -> l8(x56, x57, x58, x59, x60, x61, x62, x63) :|: x55 = x63 && x54 = x62 && x53 = x61 && x52 = x60 && x50 = x58 && x51 = x59 && x48 = x56 && x49 = x57
l9(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 && x64 = x72 && x65 = x73 && x74 = 1 + x66
l10(x80, x81, x82, x83, x84, x85, x86, x87) -> l9(x88, x89, x90, x91, x92, x93, x94, x95) :|: x87 = x95 && x86 = x94 && x84 = x92 && x82 = x90 && x83 = x91 && x80 = x88 && x81 = x89 && x93 = x82
l10(x96, x97, x98, x99, x100, x101, x102, x103) -> l9(x104, x105, x106, x107, x108, x109, x110, x111) :|: x103 = x111 && x102 = x110 && x101 = x109 && x100 = x108 && x98 = x106 && x99 = x107 && x96 = x104 && x97 = x105
l8(x112, x113, x114, x115, x116, x117, x118, x119) -> l2(x120, x121, x122, x123, x124, x125, x126, x127) :|: x119 = x127 && x117 = x125 && x114 = x122 && x115 = x123 && x112 = x120 && x113 = x121 && x124 = 1 + x116 && x126 = x126 && x112 <= x114
l8(x128, x129, x130, x131, x132, x133, x134, x135) -> l10(x136, x137, x138, x139, x140, x141, x142, x143) :|: x135 = x143 && x134 = x142 && x133 = x141 && x132 = x140 && x130 = x138 && x131 = x139 && x128 = x136 && x129 = x137 && 1 + x130 <= x128
l6(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 && x146 = x154 && x147 = x155 && x144 = x152 && x145 = x153
l3(x160, x161, x162, x163, x164, x165, x166, x167) -> l6(x168, x169, x170, x171, x172, x173, x174, x175) :|: x167 = x175 && x166 = x174 && x165 = x173 && x162 = x170 && x163 = x171 && x160 = x168 && x161 = x169 && x172 = 0 && -1 + x160 <= x164
l3(x176, x177, x178, x179, x180, x181, x182, x183) -> l7(x184, x185, x186, x187, x188, x189, x190, x191) :|: x183 = x191 && x182 = x190 && x180 = x188 && x179 = x187 && x176 = x184 && x177 = x185 && x186 = 1 + x180 && x189 = x180 && 1 + x180 <= -1 + x176
l1(x192, x193, x194, x195, x196, x197, x198, x199) -> l2(x200, x201, x202, x203, x204, x205, x206, x207) :|: x199 = x207 && x198 = x206 && x197 = x205 && x194 = x202 && x195 = x203 && x193 = x201 && x204 = 0 && x200 = x193 && x193 <= x195
l1(x208, x209, x210, x211, x212, x213, x214, x215) -> l0(x216, x217, x218, x219, x220, x221, x222, x223) :|: x215 = x223 && x214 = x222 && x213 = x221 && x212 = x220 && x210 = x218 && x208 = x216 && x209 = x217 && x219 = 1 + x211 && 1 + x211 <= x209
l11(x224, x225, x226, x227, x228, x229, x230, x231) -> l0(x232, x233, x234, x235, x236, x237, x238, x239) :|: x230 = x238 && x229 = x237 && x228 = x236 && x226 = x234 && x224 = x232 && x225 = x233 && x235 = 0 && x239 = x239
l12(x240, x241, x242, x243, x244, x245, x246, x247) -> l11(x248, x249, x250, x251, x252, x253, x254, x255) :|: x247 = x255 && x246 = x254 && x245 = x253 && x244 = x252 && x242 = x250 && x243 = x251 && x240 = x248 && x241 = x249
Start term: l12(N6HAT0, NHAT0, i8HAT0, iHAT0, j7HAT0, min9HAT0, t10HAT0, tmpHAT0)

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

(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(N6HAT0, NHAT0, i8HAT0, iHAT0, j7HAT0, min9HAT0, t10HAT0, tmpHAT0) -> l1(N6HATpost, NHATpost, i8HATpost, iHATpost, j7HATpost, min9HATpost, t10HATpost, tmpHATpost) :|: tmpHAT0 = tmpHATpost && t10HAT0 = t10HATpost && min9HAT0 = min9HATpost && j7HAT0 = j7HATpost && i8HAT0 = i8HATpost && iHAT0 = iHATpost && N6HAT0 = N6HATpost && NHAT0 = NHATpost
(2) l2(x, x1, x2, x3, x4, x5, x6, x7) -> l3(x8, x9, x10, x11, x12, x13, x14, x15) :|: x7 = x15 && x6 = x14 && x5 = x13 && x4 = x12 && x2 = x10 && x3 = x11 && x = x8 && x1 = x9
(3) l4(x16, x17, x18, x19, x20, x21, x22, x23) -> l5(x24, x25, x26, x27, x28, x29, x30, x31) :|: x23 = x31 && x22 = x30 && x21 = x29 && x20 = x28 && x18 = x26 && x19 = x27 && x16 = x24 && x17 = x25 && -1 + x16 <= x20
(4) l4(x32, x33, x34, x35, x36, x37, x38, x39) -> l6(x40, x41, x42, x43, x44, x45, x46, x47) :|: x39 = x47 && x38 = x46 && x37 = x45 && x34 = x42 && x35 = x43 && x32 = x40 && x33 = x41 && x44 = 1 + x36 && 1 + x36 <= -1 + x32
(5) l7(x48, x49, x50, x51, x52, x53, x54, x55) -> l8(x56, x57, x58, x59, x60, x61, x62, x63) :|: x55 = x63 && x54 = x62 && x53 = x61 && x52 = x60 && x50 = x58 && x51 = x59 && x48 = x56 && x49 = x57
(6) l9(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 && x64 = x72 && x65 = x73 && x74 = 1 + x66
(7) l10(x80, x81, x82, x83, x84, x85, x86, x87) -> l9(x88, x89, x90, x91, x92, x93, x94, x95) :|: x87 = x95 && x86 = x94 && x84 = x92 && x82 = x90 && x83 = x91 && x80 = x88 && x81 = x89 && x93 = x82
(8) l10(x96, x97, x98, x99, x100, x101, x102, x103) -> l9(x104, x105, x106, x107, x108, x109, x110, x111) :|: x103 = x111 && x102 = x110 && x101 = x109 && x100 = x108 && x98 = x106 && x99 = x107 && x96 = x104 && x97 = x105
(9) l8(x112, x113, x114, x115, x116, x117, x118, x119) -> l2(x120, x121, x122, x123, x124, x125, x126, x127) :|: x119 = x127 && x117 = x125 && x114 = x122 && x115 = x123 && x112 = x120 && x113 = x121 && x124 = 1 + x116 && x126 = x126 && x112 <= x114
(10) l8(x128, x129, x130, x131, x132, x133, x134, x135) -> l10(x136, x137, x138, x139, x140, x141, x142, x143) :|: x135 = x143 && x134 = x142 && x133 = x141 && x132 = x140 && x130 = x138 && x131 = x139 && x128 = x136 && x129 = x137 && 1 + x130 <= x128
(11) l6(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 && x146 = x154 && x147 = x155 && x144 = x152 && x145 = x153
(12) l3(x160, x161, x162, x163, x164, x165, x166, x167) -> l6(x168, x169, x170, x171, x172, x173, x174, x175) :|: x167 = x175 && x166 = x174 && x165 = x173 && x162 = x170 && x163 = x171 && x160 = x168 && x161 = x169 && x172 = 0 && -1 + x160 <= x164
(13) l3(x176, x177, x178, x179, x180, x181, x182, x183) -> l7(x184, x185, x186, x187, x188, x189, x190, x191) :|: x183 = x191 && x182 = x190 && x180 = x188 && x179 = x187 && x176 = x184 && x177 = x185 && x186 = 1 + x180 && x189 = x180 && 1 + x180 <= -1 + x176
(14) l1(x192, x193, x194, x195, x196, x197, x198, x199) -> l2(x200, x201, x202, x203, x204, x205, x206, x207) :|: x199 = x207 && x198 = x206 && x197 = x205 && x194 = x202 && x195 = x203 && x193 = x201 && x204 = 0 && x200 = x193 && x193 <= x195
(15) l1(x208, x209, x210, x211, x212, x213, x214, x215) -> l0(x216, x217, x218, x219, x220, x221, x222, x223) :|: x215 = x223 && x214 = x222 && x213 = x221 && x212 = x220 && x210 = x218 && x208 = x216 && x209 = x217 && x219 = 1 + x211 && 1 + x211 <= x209
(16) l11(x224, x225, x226, x227, x228, x229, x230, x231) -> l0(x232, x233, x234, x235, x236, x237, x238, x239) :|: x230 = x238 && x229 = x237 && x228 = x236 && x226 = x234 && x224 = x232 && x225 = x233 && x235 = 0 && x239 = x239
(17) l12(x240, x241, x242, x243, x244, x245, x246, x247) -> l11(x248, x249, x250, x251, x252, x253, x254, x255) :|: x247 = x255 && x246 = x254 && x245 = x253 && x244 = x252 && x242 = x250 && x243 = x251 && x240 = x248 && x241 = x249

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

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

(4)
Complex Obligation (AND)

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

(5)
Obligation:

Termination digraph:
Nodes:
(1) l0(N6HAT0, NHAT0, i8HAT0, iHAT0, j7HAT0, min9HAT0, t10HAT0, tmpHAT0) -> l1(N6HATpost, NHATpost, i8HATpost, iHATpost, j7HATpost, min9HATpost, t10HATpost, tmpHATpost) :|: tmpHAT0 = tmpHATpost && t10HAT0 = t10HATpost && min9HAT0 = min9HATpost && j7HAT0 = j7HATpost && i8HAT0 = i8HATpost && iHAT0 = iHATpost && N6HAT0 = N6HATpost && NHAT0 = NHATpost
(2) l1(x208, x209, x210, x211, x212, x213, x214, x215) -> l0(x216, x217, x218, x219, x220, x221, x222, x223) :|: x215 = x223 && x214 = x222 && x213 = x221 && x212 = x220 && x210 = x218 && x208 = x216 && x209 = x217 && x219 = 1 + x211 && 1 + x211 <= x209

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

This digraph is fully evaluated!

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

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

(7)
Obligation:
Rules:
l0(N6HAT0:0, NHAT0:0, i8HAT0:0, iHAT0:0, j7HAT0:0, min9HAT0:0, t10HAT0:0, tmpHAT0:0) -> l0(N6HAT0:0, NHAT0:0, i8HAT0:0, 1 + iHAT0:0, j7HAT0:0, min9HAT0:0, t10HAT0:0, tmpHAT0:0) :|: NHAT0:0 >= 1 + iHAT0:0

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

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

   l0(x1, x2, x3, x4, x5, x6, x7, x8) -> l0(x2, x4)

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

(9)
Obligation:
Rules:
l0(NHAT0:0, iHAT0:0) -> l0(NHAT0:0, 1 + iHAT0:0) :|: NHAT0:0 >= 1 + iHAT0:0

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

(10) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
l0(INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(11)
Obligation:
Rules:
l0(NHAT0:0, iHAT0:0) -> l0(NHAT0:0, c) :|: c = 1 + iHAT0:0 && NHAT0:0 >= 1 + iHAT0:0

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

(12) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l0(x, x1)] = x - x1

The following rules are decreasing:
l0(NHAT0:0, iHAT0:0) -> l0(NHAT0:0, c) :|: c = 1 + iHAT0:0 && NHAT0:0 >= 1 + iHAT0:0
The following rules are bounded:
l0(NHAT0:0, iHAT0:0) -> l0(NHAT0:0, c) :|: c = 1 + iHAT0:0 && NHAT0:0 >= 1 + iHAT0:0

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

(13)
YES

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

(14)
Obligation:

Termination digraph:
Nodes:
(1) l2(x, x1, x2, x3, x4, x5, x6, x7) -> l3(x8, x9, x10, x11, x12, x13, x14, x15) :|: x7 = x15 && x6 = x14 && x5 = x13 && x4 = x12 && x2 = x10 && x3 = x11 && x = x8 && x1 = x9
(2) l8(x112, x113, x114, x115, x116, x117, x118, x119) -> l2(x120, x121, x122, x123, x124, x125, x126, x127) :|: x119 = x127 && x117 = x125 && x114 = x122 && x115 = x123 && x112 = x120 && x113 = x121 && x124 = 1 + x116 && x126 = x126 && x112 <= x114
(3) l7(x48, x49, x50, x51, x52, x53, x54, x55) -> l8(x56, x57, x58, x59, x60, x61, x62, x63) :|: x55 = x63 && x54 = x62 && x53 = x61 && x52 = x60 && x50 = x58 && x51 = x59 && x48 = x56 && x49 = x57
(4) l3(x176, x177, x178, x179, x180, x181, x182, x183) -> l7(x184, x185, x186, x187, x188, x189, x190, x191) :|: x183 = x191 && x182 = x190 && x180 = x188 && x179 = x187 && x176 = x184 && x177 = x185 && x186 = 1 + x180 && x189 = x180 && 1 + x180 <= -1 + x176
(5) l9(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 && x64 = x72 && x65 = x73 && x74 = 1 + x66
(6) l10(x96, x97, x98, x99, x100, x101, x102, x103) -> l9(x104, x105, x106, x107, x108, x109, x110, x111) :|: x103 = x111 && x102 = x110 && x101 = x109 && x100 = x108 && x98 = x106 && x99 = x107 && x96 = x104 && x97 = x105
(7) l10(x80, x81, x82, x83, x84, x85, x86, x87) -> l9(x88, x89, x90, x91, x92, x93, x94, x95) :|: x87 = x95 && x86 = x94 && x84 = x92 && x82 = x90 && x83 = x91 && x80 = x88 && x81 = x89 && x93 = x82
(8) l8(x128, x129, x130, x131, x132, x133, x134, x135) -> l10(x136, x137, x138, x139, x140, x141, x142, x143) :|: x135 = x143 && x134 = x142 && x133 = x141 && x132 = x140 && x130 = x138 && x131 = x139 && x128 = x136 && x129 = x137 && 1 + x130 <= x128

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

This digraph is fully evaluated!

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(15) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(16)
Obligation:
Rules:
l7(x104:0, x105:0, x106:0, x107:0, x108:0, x109:0, x110:0, x111:0) -> l7(x104:0, x105:0, 1 + x106:0, x107:0, x108:0, x109:0, x110:0, x111:0) :|: x104:0 >= 1 + x106:0
l7(x136:0, x137:0, x138:0, x139:0, x140:0, x141:0, x142:0, x143:0) -> l7(x136:0, x137:0, 1 + x138:0, x139:0, x140:0, x138:0, x142:0, x143:0) :|: x136:0 >= 1 + x138:0
l7(x120:0, x121:0, x10:0, x11:0, x52:0, x125:0, x54:0, x127:0) -> l7(x120:0, x121:0, 1 + (1 + x52:0), x11:0, 1 + x52:0, 1 + x52:0, x126:0, x127:0) :|: x120:0 <= x10:0 && 1 + (1 + x52:0) <= -1 + x120:0

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

   l7(x1, x2, x3, x4, x5, x6, x7, x8) -> l7(x1, x3, x5)

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(18)
Obligation:
Rules:
l7(x104:0, x106:0, x108:0) -> l7(x104:0, 1 + x106:0, x108:0) :|: x104:0 >= 1 + x106:0
l7(x120:0, x10:0, x52:0) -> l7(x120:0, 1 + (1 + x52:0), 1 + x52:0) :|: x120:0 <= x10:0 && 1 + (1 + x52:0) <= -1 + x120:0

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(19) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
l7(INTEGER, INTEGER, VARIABLE)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(20)
Obligation:
Rules:
l7(x104:0, x106:0, x108:0) -> l7(x104:0, c, x108:0) :|: c = 1 + x106:0 && x104:0 >= 1 + x106:0
l7(x120:0, x10:0, x52:0) -> l7(x120:0, c1, c2) :|: c2 = 1 + x52:0 && c1 = 1 + (1 + x52:0) && (x120:0 <= x10:0 && 1 + (1 + x52:0) <= -1 + x120:0)

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(21) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ l7 ] = 2*l7_1 + -2*l7_3

The following rules are decreasing:
l7(x120:0, x10:0, x52:0) -> l7(x120:0, c1, c2) :|: c2 = 1 + x52:0 && c1 = 1 + (1 + x52:0) && (x120:0 <= x10:0 && 1 + (1 + x52:0) <= -1 + x120:0)

The following rules are bounded:
l7(x120:0, x10:0, x52:0) -> l7(x120:0, c1, c2) :|: c2 = 1 + x52:0 && c1 = 1 + (1 + x52:0) && (x120:0 <= x10:0 && 1 + (1 + x52:0) <= -1 + x120:0)


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(22)
Obligation:
Rules:
l7(x104:0, x106:0, x108:0) -> l7(x104:0, c, x108:0) :|: c = 1 + x106:0 && x104:0 >= 1 + x106:0

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(23) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l7(x, x1, x2)] = x - x1

The following rules are decreasing:
l7(x104:0, x106:0, x108:0) -> l7(x104:0, c, x108:0) :|: c = 1 + x106:0 && x104:0 >= 1 + x106:0
The following rules are bounded:
l7(x104:0, x106:0, x108:0) -> l7(x104:0, c, x108:0) :|: c = 1 + x106:0 && x104:0 >= 1 + x106:0

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(24)
YES

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(25)
Obligation:

Termination digraph:
Nodes:
(1) l6(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 && x146 = x154 && x147 = x155 && x144 = x152 && x145 = x153
(2) l4(x32, x33, x34, x35, x36, x37, x38, x39) -> l6(x40, x41, x42, x43, x44, x45, x46, x47) :|: x39 = x47 && x38 = x46 && x37 = x45 && x34 = x42 && x35 = x43 && x32 = x40 && x33 = x41 && x44 = 1 + x36 && 1 + x36 <= -1 + x32

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

This digraph is fully evaluated!

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

(27)
Obligation:
Rules:
l6(x144:0, x145:0, x146:0, x147:0, x148:0, x149:0, x150:0, x151:0) -> l6(x144:0, x145:0, x146:0, x147:0, 1 + x148:0, x149:0, x150:0, x151:0) :|: 1 + x148:0 <= -1 + x144:0

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(28) 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, x7, x8) -> l6(x1, x5)

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

(29)
Obligation:
Rules:
l6(x144:0, x148:0) -> l6(x144:0, 1 + x148:0) :|: 1 + x148:0 <= -1 + x144:0

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

(31)
Obligation:
Rules:
l6(x144:0, x148:0) -> l6(x144:0, c) :|: c = 1 + x148:0 && 1 + x148:0 <= -1 + x144:0

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

The following rules are decreasing:
l6(x144:0, x148:0) -> l6(x144:0, c) :|: c = 1 + x148:0 && 1 + x148:0 <= -1 + x144:0
The following rules are bounded:
l6(x144:0, x148:0) -> l6(x144:0, c) :|: c = 1 + x148:0 && 1 + x148:0 <= -1 + x144:0

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(33)
YES