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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 4346 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 73 ms]
(6) IRSwT
(7) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
(8) IRSwT
(9) TempFilterProof [SOUND, 259 ms]
(10) IntTRS
(11) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
(12) IntTRS
(13) PolynomialOrderProcessor [EQUIVALENT, 8 ms]
(14) AND
    (15) IntTRS
        (16) RankingReductionPairProof [EQUIVALENT, 0 ms]
        (17) IntTRS
        (18) RankingReductionPairProof [EQUIVALENT, 0 ms]
        (19) IntTRS
        (20) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
        (21) IntTRS
        (22) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
        (23) YES
    (24) IntTRS
        (25) RankingReductionPairProof [EQUIVALENT, 0 ms]
        (26) IntTRS
        (27) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
        (28) AND
            (29) IntTRS
                (30) RankingReductionPairProof [EQUIVALENT, 0 ms]
                (31) IntTRS
                (32) RankingReductionPairProof [EQUIVALENT, 0 ms]
                (33) YES
            (34) IntTRS
                (35) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
                (36) IntTRS
                (37) RankingReductionPairProof [EQUIVALENT, 0 ms]
                (38) YES


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

(0)
Obligation:
Rules:
l0(__const_400HAT0, __const_5HAT0, i2HAT0, j3HAT0, k4HAT0, l5HAT0, x1HAT0) -> l1(__const_400HATpost, __const_5HATpost, i2HATpost, j3HATpost, k4HATpost, l5HATpost, x1HATpost) :|: x1HAT0 = x1HATpost && l5HAT0 = l5HATpost && k4HAT0 = k4HATpost && j3HAT0 = j3HATpost && i2HAT0 = i2HATpost && __const_5HAT0 = __const_5HATpost && __const_400HAT0 = __const_400HATpost && __const_5HAT0 <= i2HAT0
l0(x, x1, x2, x3, x4, x5, x6) -> l2(x7, x8, x9, x10, x11, x12, x13) :|: x6 = x13 && x5 = x12 && x4 = x11 && x2 = x9 && x1 = x8 && x = x7 && x10 = 0 && 1 + x2 <= x1
l3(x14, x15, x16, x17, x18, x19, x20) -> l0(x21, x22, x23, x24, x25, x26, x27) :|: x20 = x27 && x19 = x26 && x18 = x25 && x17 = x24 && x16 = x23 && x15 = x22 && x14 = x21
l2(x28, x29, x30, x31, x32, x33, x34) -> l4(x35, x36, x37, x38, x39, x40, x41) :|: x34 = x41 && x33 = x40 && x32 = x39 && x31 = x38 && x30 = x37 && x29 = x36 && x28 = x35
l1(x42, x43, x44, x45, x46, x47, x48) -> l5(x49, x50, x51, x52, x53, x54, x55) :|: x48 = x55 && x47 = x54 && x46 = x53 && x45 = x52 && x44 = x51 && x43 = x50 && x42 = x49
l6(x56, x57, x58, x59, x60, x61, x62) -> l7(x63, x64, x65, x66, x67, x68, x69) :|: x62 = x69 && x61 = x68 && x60 = x67 && x59 = x66 && x58 = x65 && x57 = x64 && x56 = x63
l8(x70, x71, x72, x73, x74, x75, x76) -> l9(x77, x78, x79, x80, x81, x82, x83) :|: x76 = x83 && x74 = x81 && x73 = x80 && x72 = x79 && x71 = x78 && x70 = x77 && x82 = 1 + x75
l9(x84, x85, x86, x87, x88, x89, x90) -> l10(x91, x92, x93, x94, x95, x96, x97) :|: x90 = x97 && x89 = x96 && x88 = x95 && x87 = x94 && x86 = x93 && x85 = x92 && x84 = x91
l11(x98, x99, x100, x101, x102, x103, x104) -> l8(x105, x106, x107, x108, x109, x110, x111) :|: x104 = x111 && x103 = x110 && x102 = x109 && x101 = x108 && x100 = x107 && x99 = x106 && x98 = x105
l11(x112, x113, x114, x115, x116, x117, x118) -> l1(x119, x120, x121, x122, x123, x124, x125) :|: x118 = x125 && x117 = x124 && x116 = x123 && x115 = x122 && x114 = x121 && x113 = x120 && x112 = x119
l10(x126, x127, x128, x129, x130, x131, x132) -> l6(x133, x134, x135, x136, x137, x138, x139) :|: x132 = x139 && x131 = x138 && x129 = x136 && x128 = x135 && x127 = x134 && x126 = x133 && x137 = 1 + x130 && x127 <= x131
l10(x140, x141, x142, x143, x144, x145, x146) -> l11(x147, x148, x149, x150, x151, x152, x153) :|: x146 = x153 && x145 = x152 && x144 = x151 && x143 = x150 && x142 = x149 && x141 = x148 && x140 = x147 && 1 + x145 <= x141
l7(x154, x155, x156, x157, x158, x159, x160) -> l2(x161, x162, x163, x164, x165, x166, x167) :|: x160 = x167 && x159 = x166 && x158 = x165 && x156 = x163 && x155 = x162 && x154 = x161 && x164 = 1 + x157 && x155 <= x158
l7(x168, x169, x170, x171, x172, x173, x174) -> l9(x175, x176, x177, x178, x179, x180, x181) :|: x174 = x181 && x172 = x179 && x171 = x178 && x170 = x177 && x169 = x176 && x168 = x175 && x180 = 0 && 1 + x172 <= x169
l4(x182, x183, x184, x185, x186, x187, x188) -> l3(x189, x190, x191, x192, x193, x194, x195) :|: x188 = x195 && x187 = x194 && x186 = x193 && x185 = x192 && x183 = x190 && x182 = x189 && x191 = 1 + x184 && x183 <= x185
l4(x196, x197, x198, x199, x200, x201, x202) -> l6(x203, x204, x205, x206, x207, x208, x209) :|: x202 = x209 && x201 = x208 && x199 = x206 && x198 = x205 && x197 = x204 && x196 = x203 && x207 = 0 && 1 + x199 <= x197
l12(x210, x211, x212, x213, x214, x215, x216) -> l3(x217, x218, x219, x220, x221, x222, x223) :|: x215 = x222 && x214 = x221 && x213 = x220 && x211 = x218 && x210 = x217 && x219 = 0 && x223 = x210
l13(x224, x225, x226, x227, x228, x229, x230) -> l12(x231, x232, x233, x234, x235, x236, x237) :|: x230 = x237 && x229 = x236 && x228 = x235 && x227 = x234 && x226 = x233 && x225 = x232 && x224 = x231
Start term: l13(__const_400HAT0, __const_5HAT0, i2HAT0, j3HAT0, k4HAT0, l5HAT0, x1HAT0)

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

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

(2)
Obligation:
Rules:
l0(__const_400HAT0, __const_5HAT0, i2HAT0, j3HAT0, k4HAT0, l5HAT0, x1HAT0) -> l1(__const_400HATpost, __const_5HATpost, i2HATpost, j3HATpost, k4HATpost, l5HATpost, x1HATpost) :|: x1HAT0 = x1HATpost && l5HAT0 = l5HATpost && k4HAT0 = k4HATpost && j3HAT0 = j3HATpost && i2HAT0 = i2HATpost && __const_5HAT0 = __const_5HATpost && __const_400HAT0 = __const_400HATpost && __const_5HAT0 <= i2HAT0
l0(x, x1, x2, x3, x4, x5, x6) -> l2(x7, x8, x9, x10, x11, x12, x13) :|: x6 = x13 && x5 = x12 && x4 = x11 && x2 = x9 && x1 = x8 && x = x7 && x10 = 0 && 1 + x2 <= x1
l3(x14, x15, x16, x17, x18, x19, x20) -> l0(x21, x22, x23, x24, x25, x26, x27) :|: x20 = x27 && x19 = x26 && x18 = x25 && x17 = x24 && x16 = x23 && x15 = x22 && x14 = x21
l2(x28, x29, x30, x31, x32, x33, x34) -> l4(x35, x36, x37, x38, x39, x40, x41) :|: x34 = x41 && x33 = x40 && x32 = x39 && x31 = x38 && x30 = x37 && x29 = x36 && x28 = x35
l1(x42, x43, x44, x45, x46, x47, x48) -> l5(x49, x50, x51, x52, x53, x54, x55) :|: x48 = x55 && x47 = x54 && x46 = x53 && x45 = x52 && x44 = x51 && x43 = x50 && x42 = x49
l6(x56, x57, x58, x59, x60, x61, x62) -> l7(x63, x64, x65, x66, x67, x68, x69) :|: x62 = x69 && x61 = x68 && x60 = x67 && x59 = x66 && x58 = x65 && x57 = x64 && x56 = x63
l8(x70, x71, x72, x73, x74, x75, x76) -> l9(x77, x78, x79, x80, x81, x82, x83) :|: x76 = x83 && x74 = x81 && x73 = x80 && x72 = x79 && x71 = x78 && x70 = x77 && x82 = 1 + x75
l9(x84, x85, x86, x87, x88, x89, x90) -> l10(x91, x92, x93, x94, x95, x96, x97) :|: x90 = x97 && x89 = x96 && x88 = x95 && x87 = x94 && x86 = x93 && x85 = x92 && x84 = x91
l11(x98, x99, x100, x101, x102, x103, x104) -> l8(x105, x106, x107, x108, x109, x110, x111) :|: x104 = x111 && x103 = x110 && x102 = x109 && x101 = x108 && x100 = x107 && x99 = x106 && x98 = x105
l11(x112, x113, x114, x115, x116, x117, x118) -> l1(x119, x120, x121, x122, x123, x124, x125) :|: x118 = x125 && x117 = x124 && x116 = x123 && x115 = x122 && x114 = x121 && x113 = x120 && x112 = x119
l10(x126, x127, x128, x129, x130, x131, x132) -> l6(x133, x134, x135, x136, x137, x138, x139) :|: x132 = x139 && x131 = x138 && x129 = x136 && x128 = x135 && x127 = x134 && x126 = x133 && x137 = 1 + x130 && x127 <= x131
l10(x140, x141, x142, x143, x144, x145, x146) -> l11(x147, x148, x149, x150, x151, x152, x153) :|: x146 = x153 && x145 = x152 && x144 = x151 && x143 = x150 && x142 = x149 && x141 = x148 && x140 = x147 && 1 + x145 <= x141
l7(x154, x155, x156, x157, x158, x159, x160) -> l2(x161, x162, x163, x164, x165, x166, x167) :|: x160 = x167 && x159 = x166 && x158 = x165 && x156 = x163 && x155 = x162 && x154 = x161 && x164 = 1 + x157 && x155 <= x158
l7(x168, x169, x170, x171, x172, x173, x174) -> l9(x175, x176, x177, x178, x179, x180, x181) :|: x174 = x181 && x172 = x179 && x171 = x178 && x170 = x177 && x169 = x176 && x168 = x175 && x180 = 0 && 1 + x172 <= x169
l4(x182, x183, x184, x185, x186, x187, x188) -> l3(x189, x190, x191, x192, x193, x194, x195) :|: x188 = x195 && x187 = x194 && x186 = x193 && x185 = x192 && x183 = x190 && x182 = x189 && x191 = 1 + x184 && x183 <= x185
l4(x196, x197, x198, x199, x200, x201, x202) -> l6(x203, x204, x205, x206, x207, x208, x209) :|: x202 = x209 && x201 = x208 && x199 = x206 && x198 = x205 && x197 = x204 && x196 = x203 && x207 = 0 && 1 + x199 <= x197
l12(x210, x211, x212, x213, x214, x215, x216) -> l3(x217, x218, x219, x220, x221, x222, x223) :|: x215 = x222 && x214 = x221 && x213 = x220 && x211 = x218 && x210 = x217 && x219 = 0 && x223 = x210
l13(x224, x225, x226, x227, x228, x229, x230) -> l12(x231, x232, x233, x234, x235, x236, x237) :|: x230 = x237 && x229 = x236 && x228 = x235 && x227 = x234 && x226 = x233 && x225 = x232 && x224 = x231
Start term: l13(__const_400HAT0, __const_5HAT0, i2HAT0, j3HAT0, k4HAT0, l5HAT0, x1HAT0)

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

(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(__const_400HAT0, __const_5HAT0, i2HAT0, j3HAT0, k4HAT0, l5HAT0, x1HAT0) -> l1(__const_400HATpost, __const_5HATpost, i2HATpost, j3HATpost, k4HATpost, l5HATpost, x1HATpost) :|: x1HAT0 = x1HATpost && l5HAT0 = l5HATpost && k4HAT0 = k4HATpost && j3HAT0 = j3HATpost && i2HAT0 = i2HATpost && __const_5HAT0 = __const_5HATpost && __const_400HAT0 = __const_400HATpost && __const_5HAT0 <= i2HAT0
(2) l0(x, x1, x2, x3, x4, x5, x6) -> l2(x7, x8, x9, x10, x11, x12, x13) :|: x6 = x13 && x5 = x12 && x4 = x11 && x2 = x9 && x1 = x8 && x = x7 && x10 = 0 && 1 + x2 <= x1
(3) l3(x14, x15, x16, x17, x18, x19, x20) -> l0(x21, x22, x23, x24, x25, x26, x27) :|: x20 = x27 && x19 = x26 && x18 = x25 && x17 = x24 && x16 = x23 && x15 = x22 && x14 = x21
(4) l2(x28, x29, x30, x31, x32, x33, x34) -> l4(x35, x36, x37, x38, x39, x40, x41) :|: x34 = x41 && x33 = x40 && x32 = x39 && x31 = x38 && x30 = x37 && x29 = x36 && x28 = x35
(5) l1(x42, x43, x44, x45, x46, x47, x48) -> l5(x49, x50, x51, x52, x53, x54, x55) :|: x48 = x55 && x47 = x54 && x46 = x53 && x45 = x52 && x44 = x51 && x43 = x50 && x42 = x49
(6) l6(x56, x57, x58, x59, x60, x61, x62) -> l7(x63, x64, x65, x66, x67, x68, x69) :|: x62 = x69 && x61 = x68 && x60 = x67 && x59 = x66 && x58 = x65 && x57 = x64 && x56 = x63
(7) l8(x70, x71, x72, x73, x74, x75, x76) -> l9(x77, x78, x79, x80, x81, x82, x83) :|: x76 = x83 && x74 = x81 && x73 = x80 && x72 = x79 && x71 = x78 && x70 = x77 && x82 = 1 + x75
(8) l9(x84, x85, x86, x87, x88, x89, x90) -> l10(x91, x92, x93, x94, x95, x96, x97) :|: x90 = x97 && x89 = x96 && x88 = x95 && x87 = x94 && x86 = x93 && x85 = x92 && x84 = x91
(9) l11(x98, x99, x100, x101, x102, x103, x104) -> l8(x105, x106, x107, x108, x109, x110, x111) :|: x104 = x111 && x103 = x110 && x102 = x109 && x101 = x108 && x100 = x107 && x99 = x106 && x98 = x105
(10) l11(x112, x113, x114, x115, x116, x117, x118) -> l1(x119, x120, x121, x122, x123, x124, x125) :|: x118 = x125 && x117 = x124 && x116 = x123 && x115 = x122 && x114 = x121 && x113 = x120 && x112 = x119
(11) l10(x126, x127, x128, x129, x130, x131, x132) -> l6(x133, x134, x135, x136, x137, x138, x139) :|: x132 = x139 && x131 = x138 && x129 = x136 && x128 = x135 && x127 = x134 && x126 = x133 && x137 = 1 + x130 && x127 <= x131
(12) l10(x140, x141, x142, x143, x144, x145, x146) -> l11(x147, x148, x149, x150, x151, x152, x153) :|: x146 = x153 && x145 = x152 && x144 = x151 && x143 = x150 && x142 = x149 && x141 = x148 && x140 = x147 && 1 + x145 <= x141
(13) l7(x154, x155, x156, x157, x158, x159, x160) -> l2(x161, x162, x163, x164, x165, x166, x167) :|: x160 = x167 && x159 = x166 && x158 = x165 && x156 = x163 && x155 = x162 && x154 = x161 && x164 = 1 + x157 && x155 <= x158
(14) l7(x168, x169, x170, x171, x172, x173, x174) -> l9(x175, x176, x177, x178, x179, x180, x181) :|: x174 = x181 && x172 = x179 && x171 = x178 && x170 = x177 && x169 = x176 && x168 = x175 && x180 = 0 && 1 + x172 <= x169
(15) l4(x182, x183, x184, x185, x186, x187, x188) -> l3(x189, x190, x191, x192, x193, x194, x195) :|: x188 = x195 && x187 = x194 && x186 = x193 && x185 = x192 && x183 = x190 && x182 = x189 && x191 = 1 + x184 && x183 <= x185
(16) l4(x196, x197, x198, x199, x200, x201, x202) -> l6(x203, x204, x205, x206, x207, x208, x209) :|: x202 = x209 && x201 = x208 && x199 = x206 && x198 = x205 && x197 = x204 && x196 = x203 && x207 = 0 && 1 + x199 <= x197
(17) l12(x210, x211, x212, x213, x214, x215, x216) -> l3(x217, x218, x219, x220, x221, x222, x223) :|: x215 = x222 && x214 = x221 && x213 = x220 && x211 = x218 && x210 = x217 && x219 = 0 && x223 = x210
(18) l13(x224, x225, x226, x227, x228, x229, x230) -> l12(x231, x232, x233, x234, x235, x236, x237) :|: x230 = x237 && x229 = x236 && x228 = x235 && x227 = x234 && x226 = x233 && x225 = x232 && x224 = x231

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

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

(4)
Obligation:

Termination digraph:
Nodes:
(1) l0(x, x1, x2, x3, x4, x5, x6) -> l2(x7, x8, x9, x10, x11, x12, x13) :|: x6 = x13 && x5 = x12 && x4 = x11 && x2 = x9 && x1 = x8 && x = x7 && x10 = 0 && 1 + x2 <= x1
(2) l3(x14, x15, x16, x17, x18, x19, x20) -> l0(x21, x22, x23, x24, x25, x26, x27) :|: x20 = x27 && x19 = x26 && x18 = x25 && x17 = x24 && x16 = x23 && x15 = x22 && x14 = x21
(3) l4(x182, x183, x184, x185, x186, x187, x188) -> l3(x189, x190, x191, x192, x193, x194, x195) :|: x188 = x195 && x187 = x194 && x186 = x193 && x185 = x192 && x183 = x190 && x182 = x189 && x191 = 1 + x184 && x183 <= x185
(4) l2(x28, x29, x30, x31, x32, x33, x34) -> l4(x35, x36, x37, x38, x39, x40, x41) :|: x34 = x41 && x33 = x40 && x32 = x39 && x31 = x38 && x30 = x37 && x29 = x36 && x28 = x35
(5) l7(x154, x155, x156, x157, x158, x159, x160) -> l2(x161, x162, x163, x164, x165, x166, x167) :|: x160 = x167 && x159 = x166 && x158 = x165 && x156 = x163 && x155 = x162 && x154 = x161 && x164 = 1 + x157 && x155 <= x158
(6) l6(x56, x57, x58, x59, x60, x61, x62) -> l7(x63, x64, x65, x66, x67, x68, x69) :|: x62 = x69 && x61 = x68 && x60 = x67 && x59 = x66 && x58 = x65 && x57 = x64 && x56 = x63
(7) l4(x196, x197, x198, x199, x200, x201, x202) -> l6(x203, x204, x205, x206, x207, x208, x209) :|: x202 = x209 && x201 = x208 && x199 = x206 && x198 = x205 && x197 = x204 && x196 = x203 && x207 = 0 && 1 + x199 <= x197
(8) l10(x126, x127, x128, x129, x130, x131, x132) -> l6(x133, x134, x135, x136, x137, x138, x139) :|: x132 = x139 && x131 = x138 && x129 = x136 && x128 = x135 && x127 = x134 && x126 = x133 && x137 = 1 + x130 && x127 <= x131
(9) l9(x84, x85, x86, x87, x88, x89, x90) -> l10(x91, x92, x93, x94, x95, x96, x97) :|: x90 = x97 && x89 = x96 && x88 = x95 && x87 = x94 && x86 = x93 && x85 = x92 && x84 = x91
(10) l7(x168, x169, x170, x171, x172, x173, x174) -> l9(x175, x176, x177, x178, x179, x180, x181) :|: x174 = x181 && x172 = x179 && x171 = x178 && x170 = x177 && x169 = x176 && x168 = x175 && x180 = 0 && 1 + x172 <= x169
(11) l8(x70, x71, x72, x73, x74, x75, x76) -> l9(x77, x78, x79, x80, x81, x82, x83) :|: x76 = x83 && x74 = x81 && x73 = x80 && x72 = x79 && x71 = x78 && x70 = x77 && x82 = 1 + x75
(12) l11(x98, x99, x100, x101, x102, x103, x104) -> l8(x105, x106, x107, x108, x109, x110, x111) :|: x104 = x111 && x103 = x110 && x102 = x109 && x101 = x108 && x100 = x107 && x99 = x106 && x98 = x105
(13) l10(x140, x141, x142, x143, x144, x145, x146) -> l11(x147, x148, x149, x150, x151, x152, x153) :|: x146 = x153 && x145 = x152 && x144 = x151 && x143 = x150 && x142 = x149 && x141 = x148 && x140 = x147 && 1 + x145 <= x141

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

This digraph is fully evaluated!

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(5) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(6)
Obligation:
Rules:
l4(x182:0, x183:0, x184:0, x185:0, x11:0, x12:0, x13:0) -> l4(x182:0, x183:0, 1 + x184:0, 0, x11:0, x12:0, x13:0) :|: x185:0 >= x183:0 && x183:0 >= 1 + (1 + x184:0)
l9(x105:0, x106:0, x107:0, x108:0, x109:0, x110:0, x111:0) -> l9(x105:0, x106:0, x107:0, x108:0, x109:0, 1 + x110:0, x111:0) :|: x106:0 >= 1 + x110:0
l9(x133:0, x134:0, x135:0, x136:0, x88:0, x138:0, x139:0) -> l6(x133:0, x134:0, x135:0, x136:0, 1 + x88:0, x138:0, x139:0) :|: x138:0 >= x134:0
l4(x196:0, x197:0, x198:0, x199:0, x200:0, x201:0, x202:0) -> l6(x196:0, x197:0, x198:0, x199:0, 0, x201:0, x202:0) :|: x197:0 >= 1 + x199:0
l6(x161:0, x162:0, x163:0, x59:0, x165:0, x166:0, x167:0) -> l4(x161:0, x162:0, x163:0, 1 + x59:0, x165:0, x166:0, x167:0) :|: x165:0 >= x162:0
l6(x175:0, x176:0, x177:0, x178:0, x179:0, x61:0, x181:0) -> l9(x175:0, x176:0, x177:0, x178:0, x179:0, 0, x181:0) :|: x176:0 >= 1 + x179:0

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(7) 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) -> l4(x2, x3, x4)
   l9(x1, x2, x3, x4, x5, x6, x7) -> l9(x2, x3, x4, x5, x6)
   l6(x1, x2, x3, x4, x5, x6, x7) -> l6(x2, x3, x4, x5)

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(8)
Obligation:
Rules:
l4(x183:0, x184:0, x185:0) -> l4(x183:0, 1 + x184:0, 0) :|: x185:0 >= x183:0 && x183:0 >= 1 + (1 + x184:0)
l9(x106:0, x107:0, x108:0, x109:0, x110:0) -> l9(x106:0, x107:0, x108:0, x109:0, 1 + x110:0) :|: x106:0 >= 1 + x110:0
l9(x134:0, x135:0, x136:0, x88:0, x138:0) -> l6(x134:0, x135:0, x136:0, 1 + x88:0) :|: x138:0 >= x134:0
l4(x197:0, x198:0, x199:0) -> l6(x197:0, x198:0, x199:0, 0) :|: x197:0 >= 1 + x199:0
l6(x162:0, x163:0, x59:0, x165:0) -> l4(x162:0, x163:0, 1 + x59:0) :|: x165:0 >= x162:0
l6(x176:0, x177:0, x178:0, x179:0) -> l9(x176:0, x177:0, x178:0, x179:0, 0) :|: x176:0 >= 1 + x179:0

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(9) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
l4(INTEGER, VARIABLE, VARIABLE)
l9(INTEGER, VARIABLE, VARIABLE, VARIABLE, VARIABLE)
l6(INTEGER, VARIABLE, VARIABLE, VARIABLE)
Replaced non-predefined constructor symbols by 0.
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(10)
Obligation:
Rules:
l4(x183:0, x184:0, x185:0) -> l4(x183:0, c, c1) :|: c1 = 0 && c = 1 + x184:0 && (x185:0 >= x183:0 && x183:0 >= 1 + (1 + x184:0))
l9(x106:0, x107:0, x108:0, x109:0, x110:0) -> l9(x106:0, x107:0, x108:0, x109:0, c2) :|: c2 = 1 + x110:0 && x106:0 >= 1 + x110:0
l9(x134:0, x135:0, x136:0, x88:0, x138:0) -> l6(x134:0, x135:0, x136:0, c3) :|: c3 = 1 + x88:0 && x138:0 >= x134:0
l4(x197:0, x198:0, x199:0) -> l6(x197:0, x198:0, x199:0, c4) :|: c4 = 0 && x197:0 >= 1 + x199:0
l6(x162:0, x163:0, x59:0, x165:0) -> l4(x162:0, x163:0, c5) :|: c5 = 1 + x59:0 && x165:0 >= x162:0
l6(x176:0, x177:0, x178:0, x179:0) -> l9(x176:0, x177:0, x178:0, x179:0, c6) :|: c6 = 0 && x176:0 >= 1 + x179:0

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(11) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l4(x, x1, x2)] = -2 + x - x1
[l9(x3, x4, x5, x6, x7)] = -2 + x3 - x4
[l6(x8, x9, x10, x11)] = -2 + x8 - x9

The following rules are decreasing:
l4(x183:0, x184:0, x185:0) -> l4(x183:0, c, c1) :|: c1 = 0 && c = 1 + x184:0 && (x185:0 >= x183:0 && x183:0 >= 1 + (1 + x184:0))
The following rules are bounded:
l4(x183:0, x184:0, x185:0) -> l4(x183:0, c, c1) :|: c1 = 0 && c = 1 + x184:0 && (x185:0 >= x183:0 && x183:0 >= 1 + (1 + x184:0))

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(12)
Obligation:
Rules:
l9(x106:0, x107:0, x108:0, x109:0, x110:0) -> l9(x106:0, x107:0, x108:0, x109:0, c2) :|: c2 = 1 + x110:0 && x106:0 >= 1 + x110:0
l9(x134:0, x135:0, x136:0, x88:0, x138:0) -> l6(x134:0, x135:0, x136:0, c3) :|: c3 = 1 + x88:0 && x138:0 >= x134:0
l4(x197:0, x198:0, x199:0) -> l6(x197:0, x198:0, x199:0, c4) :|: c4 = 0 && x197:0 >= 1 + x199:0
l6(x162:0, x163:0, x59:0, x165:0) -> l4(x162:0, x163:0, c5) :|: c5 = 1 + x59:0 && x165:0 >= x162:0
l6(x176:0, x177:0, x178:0, x179:0) -> l9(x176:0, x177:0, x178:0, x179:0, c6) :|: c6 = 0 && x176:0 >= 1 + x179:0

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(13) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l9(x, x1, x2, x3, x4)] = -1 + x - x2
[l6(x5, x6, x7, x8)] = -1 + x5 - x7
[l4(x9, x10, x11)] = -1 - x11 + x9

The following rules are decreasing:
l6(x162:0, x163:0, x59:0, x165:0) -> l4(x162:0, x163:0, c5) :|: c5 = 1 + x59:0 && x165:0 >= x162:0
The following rules are bounded:
l4(x197:0, x198:0, x199:0) -> l6(x197:0, x198:0, x199:0, c4) :|: c4 = 0 && x197:0 >= 1 + x199:0

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(14)
Complex Obligation (AND)

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(15)
Obligation:
Rules:
l9(x106:0, x107:0, x108:0, x109:0, x110:0) -> l9(x106:0, x107:0, x108:0, x109:0, c2) :|: c2 = 1 + x110:0 && x106:0 >= 1 + x110:0
l9(x134:0, x135:0, x136:0, x88:0, x138:0) -> l6(x134:0, x135:0, x136:0, c3) :|: c3 = 1 + x88:0 && x138:0 >= x134:0
l4(x197:0, x198:0, x199:0) -> l6(x197:0, x198:0, x199:0, c4) :|: c4 = 0 && x197:0 >= 1 + x199:0
l6(x176:0, x177:0, x178:0, x179:0) -> l9(x176:0, x177:0, x178:0, x179:0, c6) :|: c6 = 0 && x176:0 >= 1 + x179:0

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(16) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ l9 ] = -1
[ l6 ] = -1
[ l4 ] = 0

The following rules are decreasing:
l4(x197:0, x198:0, x199:0) -> l6(x197:0, x198:0, x199:0, c4) :|: c4 = 0 && x197:0 >= 1 + x199:0

The following rules are bounded:
l9(x106:0, x107:0, x108:0, x109:0, x110:0) -> l9(x106:0, x107:0, x108:0, x109:0, c2) :|: c2 = 1 + x110:0 && x106:0 >= 1 + x110:0
l9(x134:0, x135:0, x136:0, x88:0, x138:0) -> l6(x134:0, x135:0, x136:0, c3) :|: c3 = 1 + x88:0 && x138:0 >= x134:0
l4(x197:0, x198:0, x199:0) -> l6(x197:0, x198:0, x199:0, c4) :|: c4 = 0 && x197:0 >= 1 + x199:0
l6(x176:0, x177:0, x178:0, x179:0) -> l9(x176:0, x177:0, x178:0, x179:0, c6) :|: c6 = 0 && x176:0 >= 1 + x179:0


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(17)
Obligation:
Rules:
l9(x106:0, x107:0, x108:0, x109:0, x110:0) -> l9(x106:0, x107:0, x108:0, x109:0, c2) :|: c2 = 1 + x110:0 && x106:0 >= 1 + x110:0
l9(x134:0, x135:0, x136:0, x88:0, x138:0) -> l6(x134:0, x135:0, x136:0, c3) :|: c3 = 1 + x88:0 && x138:0 >= x134:0
l6(x176:0, x177:0, x178:0, x179:0) -> l9(x176:0, x177:0, x178:0, x179:0, c6) :|: c6 = 0 && x176:0 >= 1 + x179:0

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(18) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ l9 ] = 3*l9_1 + -3*l9_4 + -1
[ l6 ] = 3*l6_1 + -3*l6_4

The following rules are decreasing:
l9(x134:0, x135:0, x136:0, x88:0, x138:0) -> l6(x134:0, x135:0, x136:0, c3) :|: c3 = 1 + x88:0 && x138:0 >= x134:0
l6(x176:0, x177:0, x178:0, x179:0) -> l9(x176:0, x177:0, x178:0, x179:0, c6) :|: c6 = 0 && x176:0 >= 1 + x179:0

The following rules are bounded:
l6(x176:0, x177:0, x178:0, x179:0) -> l9(x176:0, x177:0, x178:0, x179:0, c6) :|: c6 = 0 && x176:0 >= 1 + x179:0


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(19)
Obligation:
Rules:
l9(x106:0, x107:0, x108:0, x109:0, x110:0) -> l9(x106:0, x107:0, x108:0, x109:0, c2) :|: c2 = 1 + x110:0 && x106:0 >= 1 + x110:0
l9(x134:0, x135:0, x136:0, x88:0, x138:0) -> l6(x134:0, x135:0, x136:0, c3) :|: c3 = 1 + x88:0 && x138:0 >= x134:0

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(20) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l9(x, x1, x2, x3, x4)] = 1
[l6(x5, x6, x7, x8)] = 0

The following rules are decreasing:
l9(x134:0, x135:0, x136:0, x88:0, x138:0) -> l6(x134:0, x135:0, x136:0, c3) :|: c3 = 1 + x88:0 && x138:0 >= x134:0
The following rules are bounded:
l9(x106:0, x107:0, x108:0, x109:0, x110:0) -> l9(x106:0, x107:0, x108:0, x109:0, c2) :|: c2 = 1 + x110:0 && x106:0 >= 1 + x110:0
l9(x134:0, x135:0, x136:0, x88:0, x138:0) -> l6(x134:0, x135:0, x136:0, c3) :|: c3 = 1 + x88:0 && x138:0 >= x134:0

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(21)
Obligation:
Rules:
l9(x106:0, x107:0, x108:0, x109:0, x110:0) -> l9(x106:0, x107:0, x108:0, x109:0, c2) :|: c2 = 1 + x110:0 && x106:0 >= 1 + x110:0

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(22) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l9(x, x1, x2, x3, x4)] = x - x4

The following rules are decreasing:
l9(x106:0, x107:0, x108:0, x109:0, x110:0) -> l9(x106:0, x107:0, x108:0, x109:0, c2) :|: c2 = 1 + x110:0 && x106:0 >= 1 + x110:0
The following rules are bounded:
l9(x106:0, x107:0, x108:0, x109:0, x110:0) -> l9(x106:0, x107:0, x108:0, x109:0, c2) :|: c2 = 1 + x110:0 && x106:0 >= 1 + x110:0

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(23)
YES

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(24)
Obligation:
Rules:
l9(x106:0, x107:0, x108:0, x109:0, x110:0) -> l9(x106:0, x107:0, x108:0, x109:0, c2) :|: c2 = 1 + x110:0 && x106:0 >= 1 + x110:0
l9(x134:0, x135:0, x136:0, x88:0, x138:0) -> l6(x134:0, x135:0, x136:0, c3) :|: c3 = 1 + x88:0 && x138:0 >= x134:0
l6(x162:0, x163:0, x59:0, x165:0) -> l4(x162:0, x163:0, c5) :|: c5 = 1 + x59:0 && x165:0 >= x162:0
l6(x176:0, x177:0, x178:0, x179:0) -> l9(x176:0, x177:0, x178:0, x179:0, c6) :|: c6 = 0 && x176:0 >= 1 + x179:0

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(25) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ l9 ] = 0
[ l6 ] = 0
[ l4 ] = -1

The following rules are decreasing:
l6(x162:0, x163:0, x59:0, x165:0) -> l4(x162:0, x163:0, c5) :|: c5 = 1 + x59:0 && x165:0 >= x162:0

The following rules are bounded:
l9(x106:0, x107:0, x108:0, x109:0, x110:0) -> l9(x106:0, x107:0, x108:0, x109:0, c2) :|: c2 = 1 + x110:0 && x106:0 >= 1 + x110:0
l9(x134:0, x135:0, x136:0, x88:0, x138:0) -> l6(x134:0, x135:0, x136:0, c3) :|: c3 = 1 + x88:0 && x138:0 >= x134:0
l6(x162:0, x163:0, x59:0, x165:0) -> l4(x162:0, x163:0, c5) :|: c5 = 1 + x59:0 && x165:0 >= x162:0
l6(x176:0, x177:0, x178:0, x179:0) -> l9(x176:0, x177:0, x178:0, x179:0, c6) :|: c6 = 0 && x176:0 >= 1 + x179:0


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(26)
Obligation:
Rules:
l9(x106:0, x107:0, x108:0, x109:0, x110:0) -> l9(x106:0, x107:0, x108:0, x109:0, c2) :|: c2 = 1 + x110:0 && x106:0 >= 1 + x110:0
l9(x134:0, x135:0, x136:0, x88:0, x138:0) -> l6(x134:0, x135:0, x136:0, c3) :|: c3 = 1 + x88:0 && x138:0 >= x134:0
l6(x176:0, x177:0, x178:0, x179:0) -> l9(x176:0, x177:0, x178:0, x179:0, c6) :|: c6 = 0 && x176:0 >= 1 + x179:0

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(27) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l9(x, x1, x2, x3, x4)] = -1 + x - x3
[l6(x5, x6, x7, x8)] = -1 + x5 - x8

The following rules are decreasing:
l9(x134:0, x135:0, x136:0, x88:0, x138:0) -> l6(x134:0, x135:0, x136:0, c3) :|: c3 = 1 + x88:0 && x138:0 >= x134:0
The following rules are bounded:
l6(x176:0, x177:0, x178:0, x179:0) -> l9(x176:0, x177:0, x178:0, x179:0, c6) :|: c6 = 0 && x176:0 >= 1 + x179:0

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(28)
Complex Obligation (AND)

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(29)
Obligation:
Rules:
l9(x106:0, x107:0, x108:0, x109:0, x110:0) -> l9(x106:0, x107:0, x108:0, x109:0, c2) :|: c2 = 1 + x110:0 && x106:0 >= 1 + x110:0
l6(x176:0, x177:0, x178:0, x179:0) -> l9(x176:0, x177:0, x178:0, x179:0, c6) :|: c6 = 0 && x176:0 >= 1 + x179:0

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(30) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ l9 ] = 0
[ l6 ] = 1

The following rules are decreasing:
l6(x176:0, x177:0, x178:0, x179:0) -> l9(x176:0, x177:0, x178:0, x179:0, c6) :|: c6 = 0 && x176:0 >= 1 + x179:0

The following rules are bounded:
l9(x106:0, x107:0, x108:0, x109:0, x110:0) -> l9(x106:0, x107:0, x108:0, x109:0, c2) :|: c2 = 1 + x110:0 && x106:0 >= 1 + x110:0
l6(x176:0, x177:0, x178:0, x179:0) -> l9(x176:0, x177:0, x178:0, x179:0, c6) :|: c6 = 0 && x176:0 >= 1 + x179:0


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(31)
Obligation:
Rules:
l9(x106:0, x107:0, x108:0, x109:0, x110:0) -> l9(x106:0, x107:0, x108:0, x109:0, c2) :|: c2 = 1 + x110:0 && x106:0 >= 1 + x110:0

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(32) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ l9 ] = l9_1 + -1*l9_5

The following rules are decreasing:
l9(x106:0, x107:0, x108:0, x109:0, x110:0) -> l9(x106:0, x107:0, x108:0, x109:0, c2) :|: c2 = 1 + x110:0 && x106:0 >= 1 + x110:0

The following rules are bounded:
l9(x106:0, x107:0, x108:0, x109:0, x110:0) -> l9(x106:0, x107:0, x108:0, x109:0, c2) :|: c2 = 1 + x110:0 && x106:0 >= 1 + x110:0


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(33)
YES

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(34)
Obligation:
Rules:
l9(x106:0, x107:0, x108:0, x109:0, x110:0) -> l9(x106:0, x107:0, x108:0, x109:0, c2) :|: c2 = 1 + x110:0 && x106:0 >= 1 + x110:0
l9(x134:0, x135:0, x136:0, x88:0, x138:0) -> l6(x134:0, x135:0, x136:0, c3) :|: c3 = 1 + x88:0 && x138:0 >= x134:0

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(35) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l9(x, x1, x2, x3, x4)] = 1
[l6(x5, x6, x7, x8)] = 0

The following rules are decreasing:
l9(x134:0, x135:0, x136:0, x88:0, x138:0) -> l6(x134:0, x135:0, x136:0, c3) :|: c3 = 1 + x88:0 && x138:0 >= x134:0
The following rules are bounded:
l9(x106:0, x107:0, x108:0, x109:0, x110:0) -> l9(x106:0, x107:0, x108:0, x109:0, c2) :|: c2 = 1 + x110:0 && x106:0 >= 1 + x110:0
l9(x134:0, x135:0, x136:0, x88:0, x138:0) -> l6(x134:0, x135:0, x136:0, c3) :|: c3 = 1 + x88:0 && x138:0 >= x134:0

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(36)
Obligation:
Rules:
l9(x106:0, x107:0, x108:0, x109:0, x110:0) -> l9(x106:0, x107:0, x108:0, x109:0, c2) :|: c2 = 1 + x110:0 && x106:0 >= 1 + x110:0

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(37) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ l9 ] = l9_1 + -1*l9_5

The following rules are decreasing:
l9(x106:0, x107:0, x108:0, x109:0, x110:0) -> l9(x106:0, x107:0, x108:0, x109:0, c2) :|: c2 = 1 + x110:0 && x106:0 >= 1 + x110:0

The following rules are bounded:
l9(x106:0, x107:0, x108:0, x109:0, x110:0) -> l9(x106:0, x107:0, x108:0, x109:0, c2) :|: c2 = 1 + x110:0 && x106:0 >= 1 + x110:0


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(38)
YES