MAYBE
proof of prog.inttrs
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


Termination of the given IRSwT could not be shown:

(0) IRSwT
(1) IRSFormatTransformerProof [EQUIVALENT, 0 ms]
(2) IRSwT
(3) IRSwTTerminationDigraphProof [EQUIVALENT, 6340 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 0 ms]
(6) IRSwT
(7) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
(8) IRSwT
(9) TempFilterProof [SOUND, 425 ms]
(10) IRSwT
(11) IRSwTTerminationDigraphProof [EQUIVALENT, 0 ms]
(12) IRSwT
(13) IntTRSCompressionProof [EQUIVALENT, 0 ms]
(14) IRSwT


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

(0)
Obligation:
Rules:
l0(Result_4HAT0, w_5HAT0, x_6HAT0) -> l2(Result_4HATpost, w_5HATpost, x_6HATpost) :|: Result_4HAT0 = Result_4HATpost && w_5HATpost = 1 + w_5HAT0 && x_6HATpost = 1 + x_6HAT0 && 0 <= 2 - w_5HAT0 && 2 - x_6HAT0 <= 0
l2(x, x1, x2) -> l3(x3, x4, x5) :|: x2 = x5 && x1 = x4 && x = x3 && 1 + x2 <= 2
l2(x6, x7, x8) -> l3(x9, x10, x11) :|: x8 = x11 && x7 = x10 && x6 = x9 && 3 <= x8
l3(x12, x13, x14) -> l1(x15, x16, x17) :|: x14 = x17 && x13 = x16 && x12 = x15 && 1 + x13 <= 2
l3(x18, x19, x20) -> l1(x21, x22, x23) :|: x20 = x23 && x19 = x22 && x18 = x21 && 3 <= x19
l1(x24, x25, x26) -> l0(x27, x28, x29) :|: x26 = x29 && x25 = x28 && x24 = x27
l0(x30, x31, x32) -> l5(x33, x34, x35) :|: x30 = x33 && x34 = 1 + x31 && x35 = 1 + x32 && 0 <= 2 - x31 && 2 - x32 <= 0
l5(x36, x37, x38) -> l6(x39, x40, x41) :|: x38 = x41 && x37 = x40 && x36 = x39 && 1 + x38 <= 2
l5(x42, x43, x44) -> l6(x45, x46, x47) :|: x44 = x47 && x43 = x46 && x42 = x45 && 3 <= x44
l6(x48, x49, x50) -> l4(x51, x52, x53) :|: x50 = x53 && x48 = x51 && x52 = 1 && 2 <= x49 && x49 <= 2
l4(x54, x55, x56) -> l0(x57, x58, x59) :|: x56 = x59 && x55 = x58 && x54 = x57
l0(x60, x61, x62) -> l8(x63, x64, x65) :|: x60 = x63 && x64 = 1 + x61 && x65 = 1 + x62 && 0 <= 1 - x62
l8(x66, x67, x68) -> l9(x69, x70, x71) :|: x68 = x71 && x67 = x70 && x66 = x69 && 1 + x68 <= 2
l8(x72, x73, x74) -> l9(x75, x76, x77) :|: x74 = x77 && x73 = x76 && x72 = x75 && 3 <= x74
l9(x78, x79, x80) -> l7(x81, x82, x83) :|: x80 = x83 && x79 = x82 && x78 = x81 && 1 + x79 <= 2
l9(x84, x85, x86) -> l7(x87, x88, x89) :|: x86 = x89 && x85 = x88 && x84 = x87 && 3 <= x85
l7(x90, x91, x92) -> l0(x93, x94, x95) :|: x92 = x95 && x91 = x94 && x90 = x93
l0(x96, x97, x98) -> l11(x99, x100, x101) :|: x96 = x99 && x100 = 1 + x97 && x101 = 1 + x98 && 0 <= 1 - x98
l11(x102, x103, x104) -> l12(x105, x106, x107) :|: x104 = x107 && x103 = x106 && x102 = x105 && 1 + x104 <= 2
l11(x108, x109, x110) -> l12(x111, x112, x113) :|: x110 = x113 && x109 = x112 && x108 = x111 && 3 <= x110
l12(x114, x115, x116) -> l10(x117, x118, x119) :|: x116 = x119 && x114 = x117 && x118 = 1 && 2 <= x115 && x115 <= 2
l10(x120, x121, x122) -> l0(x123, x124, x125) :|: x122 = x125 && x121 = x124 && x120 = x123
l0(x126, x127, x128) -> l14(x129, x130, x131) :|: x126 = x129 && 2 <= x131 && x131 <= 2 && x130 = 1 + x127 && x131 = 1 + x128 && 0 <= 1 - x128
l14(x132, x133, x134) -> l13(x135, x136, x137) :|: x134 = x137 && x133 = x136 && x132 = x135 && 1 + x133 <= 2
l14(x138, x139, x140) -> l13(x141, x142, x143) :|: x140 = x143 && x139 = x142 && x138 = x141 && 3 <= x139
l13(x144, x145, x146) -> l0(x147, x148, x149) :|: x146 = x149 && x145 = x148 && x144 = x147
l0(x150, x151, x152) -> l15(x153, x154, x155) :|: 0 <= 1 - x152 && x155 = 1 + x152 && x156 = 1 + x151 && x155 <= 2 && 2 <= x155 && x156 <= 2 && 2 <= x156 && x154 = 1 && x150 = x153
l15(x157, x158, x159) -> l0(x160, x161, x162) :|: x159 = x162 && x158 = x161 && x157 = x160
l0(x163, x164, x165) -> l16(x166, x167, x168) :|: x165 = x168 && x164 = x167 && x166 = x166 && 3 - x164 <= 0 && 2 - x165 <= 0
l17(x169, x170, x171) -> l0(x172, x173, x174) :|: x171 = x174 && x170 = x173 && x169 = x172
l18(x175, x176, x177) -> l17(x178, x179, x180) :|: x177 = x180 && x176 = x179 && x175 = x178
Start term: l18(Result_4HAT0, w_5HAT0, x_6HAT0)

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

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

(2)
Obligation:
Rules:
l0(Result_4HAT0, w_5HAT0, x_6HAT0) -> l2(Result_4HATpost, w_5HATpost, x_6HATpost) :|: Result_4HAT0 = Result_4HATpost && w_5HATpost = 1 + w_5HAT0 && x_6HATpost = 1 + x_6HAT0 && 0 <= 2 - w_5HAT0 && 2 - x_6HAT0 <= 0
l2(x, x1, x2) -> l3(x3, x4, x5) :|: x2 = x5 && x1 = x4 && x = x3 && 1 + x2 <= 2
l2(x6, x7, x8) -> l3(x9, x10, x11) :|: x8 = x11 && x7 = x10 && x6 = x9 && 3 <= x8
l3(x12, x13, x14) -> l1(x15, x16, x17) :|: x14 = x17 && x13 = x16 && x12 = x15 && 1 + x13 <= 2
l3(x18, x19, x20) -> l1(x21, x22, x23) :|: x20 = x23 && x19 = x22 && x18 = x21 && 3 <= x19
l1(x24, x25, x26) -> l0(x27, x28, x29) :|: x26 = x29 && x25 = x28 && x24 = x27
l0(x30, x31, x32) -> l5(x33, x34, x35) :|: x30 = x33 && x34 = 1 + x31 && x35 = 1 + x32 && 0 <= 2 - x31 && 2 - x32 <= 0
l5(x36, x37, x38) -> l6(x39, x40, x41) :|: x38 = x41 && x37 = x40 && x36 = x39 && 1 + x38 <= 2
l5(x42, x43, x44) -> l6(x45, x46, x47) :|: x44 = x47 && x43 = x46 && x42 = x45 && 3 <= x44
l6(x48, x49, x50) -> l4(x51, x52, x53) :|: x50 = x53 && x48 = x51 && x52 = 1 && 2 <= x49 && x49 <= 2
l4(x54, x55, x56) -> l0(x57, x58, x59) :|: x56 = x59 && x55 = x58 && x54 = x57
l0(x60, x61, x62) -> l8(x63, x64, x65) :|: x60 = x63 && x64 = 1 + x61 && x65 = 1 + x62 && 0 <= 1 - x62
l8(x66, x67, x68) -> l9(x69, x70, x71) :|: x68 = x71 && x67 = x70 && x66 = x69 && 1 + x68 <= 2
l8(x72, x73, x74) -> l9(x75, x76, x77) :|: x74 = x77 && x73 = x76 && x72 = x75 && 3 <= x74
l9(x78, x79, x80) -> l7(x81, x82, x83) :|: x80 = x83 && x79 = x82 && x78 = x81 && 1 + x79 <= 2
l9(x84, x85, x86) -> l7(x87, x88, x89) :|: x86 = x89 && x85 = x88 && x84 = x87 && 3 <= x85
l7(x90, x91, x92) -> l0(x93, x94, x95) :|: x92 = x95 && x91 = x94 && x90 = x93
l0(x96, x97, x98) -> l11(x99, x100, x101) :|: x96 = x99 && x100 = 1 + x97 && x101 = 1 + x98 && 0 <= 1 - x98
l11(x102, x103, x104) -> l12(x105, x106, x107) :|: x104 = x107 && x103 = x106 && x102 = x105 && 1 + x104 <= 2
l11(x108, x109, x110) -> l12(x111, x112, x113) :|: x110 = x113 && x109 = x112 && x108 = x111 && 3 <= x110
l12(x114, x115, x116) -> l10(x117, x118, x119) :|: x116 = x119 && x114 = x117 && x118 = 1 && 2 <= x115 && x115 <= 2
l10(x120, x121, x122) -> l0(x123, x124, x125) :|: x122 = x125 && x121 = x124 && x120 = x123
l0(x126, x127, x128) -> l14(x129, x130, x131) :|: x126 = x129 && 2 <= x131 && x131 <= 2 && x130 = 1 + x127 && x131 = 1 + x128 && 0 <= 1 - x128
l14(x132, x133, x134) -> l13(x135, x136, x137) :|: x134 = x137 && x133 = x136 && x132 = x135 && 1 + x133 <= 2
l14(x138, x139, x140) -> l13(x141, x142, x143) :|: x140 = x143 && x139 = x142 && x138 = x141 && 3 <= x139
l13(x144, x145, x146) -> l0(x147, x148, x149) :|: x146 = x149 && x145 = x148 && x144 = x147
l0(x150, x151, x152) -> l15(x153, x154, x155) :|: 0 <= 1 - x152 && x155 = 1 + x152 && x156 = 1 + x151 && x155 <= 2 && 2 <= x155 && x156 <= 2 && 2 <= x156 && x154 = 1 && x150 = x153
l15(x157, x158, x159) -> l0(x160, x161, x162) :|: x159 = x162 && x158 = x161 && x157 = x160
l0(x163, x164, x165) -> l16(x166, x167, x168) :|: x165 = x168 && x164 = x167 && x166 = x166 && 3 - x164 <= 0 && 2 - x165 <= 0
l17(x169, x170, x171) -> l0(x172, x173, x174) :|: x171 = x174 && x170 = x173 && x169 = x172
l18(x175, x176, x177) -> l17(x178, x179, x180) :|: x177 = x180 && x176 = x179 && x175 = x178
Start term: l18(Result_4HAT0, w_5HAT0, x_6HAT0)

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

(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(Result_4HAT0, w_5HAT0, x_6HAT0) -> l2(Result_4HATpost, w_5HATpost, x_6HATpost) :|: Result_4HAT0 = Result_4HATpost && w_5HATpost = 1 + w_5HAT0 && x_6HATpost = 1 + x_6HAT0 && 0 <= 2 - w_5HAT0 && 2 - x_6HAT0 <= 0
(2) l2(x, x1, x2) -> l3(x3, x4, x5) :|: x2 = x5 && x1 = x4 && x = x3 && 1 + x2 <= 2
(3) l2(x6, x7, x8) -> l3(x9, x10, x11) :|: x8 = x11 && x7 = x10 && x6 = x9 && 3 <= x8
(4) l3(x12, x13, x14) -> l1(x15, x16, x17) :|: x14 = x17 && x13 = x16 && x12 = x15 && 1 + x13 <= 2
(5) l3(x18, x19, x20) -> l1(x21, x22, x23) :|: x20 = x23 && x19 = x22 && x18 = x21 && 3 <= x19
(6) l1(x24, x25, x26) -> l0(x27, x28, x29) :|: x26 = x29 && x25 = x28 && x24 = x27
(7) l0(x30, x31, x32) -> l5(x33, x34, x35) :|: x30 = x33 && x34 = 1 + x31 && x35 = 1 + x32 && 0 <= 2 - x31 && 2 - x32 <= 0
(8) l5(x36, x37, x38) -> l6(x39, x40, x41) :|: x38 = x41 && x37 = x40 && x36 = x39 && 1 + x38 <= 2
(9) l5(x42, x43, x44) -> l6(x45, x46, x47) :|: x44 = x47 && x43 = x46 && x42 = x45 && 3 <= x44
(10) l6(x48, x49, x50) -> l4(x51, x52, x53) :|: x50 = x53 && x48 = x51 && x52 = 1 && 2 <= x49 && x49 <= 2
(11) l4(x54, x55, x56) -> l0(x57, x58, x59) :|: x56 = x59 && x55 = x58 && x54 = x57
(12) l0(x60, x61, x62) -> l8(x63, x64, x65) :|: x60 = x63 && x64 = 1 + x61 && x65 = 1 + x62 && 0 <= 1 - x62
(13) l8(x66, x67, x68) -> l9(x69, x70, x71) :|: x68 = x71 && x67 = x70 && x66 = x69 && 1 + x68 <= 2
(14) l8(x72, x73, x74) -> l9(x75, x76, x77) :|: x74 = x77 && x73 = x76 && x72 = x75 && 3 <= x74
(15) l9(x78, x79, x80) -> l7(x81, x82, x83) :|: x80 = x83 && x79 = x82 && x78 = x81 && 1 + x79 <= 2
(16) l9(x84, x85, x86) -> l7(x87, x88, x89) :|: x86 = x89 && x85 = x88 && x84 = x87 && 3 <= x85
(17) l7(x90, x91, x92) -> l0(x93, x94, x95) :|: x92 = x95 && x91 = x94 && x90 = x93
(18) l0(x96, x97, x98) -> l11(x99, x100, x101) :|: x96 = x99 && x100 = 1 + x97 && x101 = 1 + x98 && 0 <= 1 - x98
(19) l11(x102, x103, x104) -> l12(x105, x106, x107) :|: x104 = x107 && x103 = x106 && x102 = x105 && 1 + x104 <= 2
(20) l11(x108, x109, x110) -> l12(x111, x112, x113) :|: x110 = x113 && x109 = x112 && x108 = x111 && 3 <= x110
(21) l12(x114, x115, x116) -> l10(x117, x118, x119) :|: x116 = x119 && x114 = x117 && x118 = 1 && 2 <= x115 && x115 <= 2
(22) l10(x120, x121, x122) -> l0(x123, x124, x125) :|: x122 = x125 && x121 = x124 && x120 = x123
(23) l0(x126, x127, x128) -> l14(x129, x130, x131) :|: x126 = x129 && 2 <= x131 && x131 <= 2 && x130 = 1 + x127 && x131 = 1 + x128 && 0 <= 1 - x128
(24) l14(x132, x133, x134) -> l13(x135, x136, x137) :|: x134 = x137 && x133 = x136 && x132 = x135 && 1 + x133 <= 2
(25) l14(x138, x139, x140) -> l13(x141, x142, x143) :|: x140 = x143 && x139 = x142 && x138 = x141 && 3 <= x139
(26) l13(x144, x145, x146) -> l0(x147, x148, x149) :|: x146 = x149 && x145 = x148 && x144 = x147
(27) l0(x150, x151, x152) -> l15(x153, x154, x155) :|: 0 <= 1 - x152 && x155 = 1 + x152 && x156 = 1 + x151 && x155 <= 2 && 2 <= x155 && x156 <= 2 && 2 <= x156 && x154 = 1 && x150 = x153
(28) l15(x157, x158, x159) -> l0(x160, x161, x162) :|: x159 = x162 && x158 = x161 && x157 = x160
(29) l0(x163, x164, x165) -> l16(x166, x167, x168) :|: x165 = x168 && x164 = x167 && x166 = x166 && 3 - x164 <= 0 && 2 - x165 <= 0
(30) l17(x169, x170, x171) -> l0(x172, x173, x174) :|: x171 = x174 && x170 = x173 && x169 = x172
(31) l18(x175, x176, x177) -> l17(x178, x179, x180) :|: x177 = x180 && x176 = x179 && x175 = x178

Arcs:
(1) -> (3)
(2) -> (4), (5)
(3) -> (4), (5)
(4) -> (6)
(5) -> (6)
(6) -> (1), (7), (12), (18), (23), (27), (29)
(7) -> (9)
(8) -> (10)
(9) -> (10)
(10) -> (11)
(11) -> (1), (7), (12), (18), (23), (27), (29)
(12) -> (13)
(13) -> (15), (16)
(14) -> (15), (16)
(15) -> (17)
(16) -> (17)
(17) -> (1), (7), (12), (18), (23), (27), (29)
(18) -> (19)
(19) -> (21)
(20) -> (21)
(21) -> (22)
(22) -> (1), (7), (12), (18), (23), (27), (29)
(23) -> (24), (25)
(24) -> (26)
(25) -> (26)
(26) -> (1), (7), (12), (18), (23), (27), (29)
(27) -> (28)
(28) -> (1), (7), (12), (18), (23), (27), (29)
(30) -> (1), (7), (12), (18), (23), (27), (29)
(31) -> (30)

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

(4)
Obligation:

Termination digraph:
Nodes:
(1) l0(Result_4HAT0, w_5HAT0, x_6HAT0) -> l2(Result_4HATpost, w_5HATpost, x_6HATpost) :|: Result_4HAT0 = Result_4HATpost && w_5HATpost = 1 + w_5HAT0 && x_6HATpost = 1 + x_6HAT0 && 0 <= 2 - w_5HAT0 && 2 - x_6HAT0 <= 0
(2) l4(x54, x55, x56) -> l0(x57, x58, x59) :|: x56 = x59 && x55 = x58 && x54 = x57
(3) l6(x48, x49, x50) -> l4(x51, x52, x53) :|: x50 = x53 && x48 = x51 && x52 = 1 && 2 <= x49 && x49 <= 2
(4) l5(x42, x43, x44) -> l6(x45, x46, x47) :|: x44 = x47 && x43 = x46 && x42 = x45 && 3 <= x44
(5) l0(x30, x31, x32) -> l5(x33, x34, x35) :|: x30 = x33 && x34 = 1 + x31 && x35 = 1 + x32 && 0 <= 2 - x31 && 2 - x32 <= 0
(6) l7(x90, x91, x92) -> l0(x93, x94, x95) :|: x92 = x95 && x91 = x94 && x90 = x93
(7) l9(x84, x85, x86) -> l7(x87, x88, x89) :|: x86 = x89 && x85 = x88 && x84 = x87 && 3 <= x85
(8) l9(x78, x79, x80) -> l7(x81, x82, x83) :|: x80 = x83 && x79 = x82 && x78 = x81 && 1 + x79 <= 2
(9) l8(x66, x67, x68) -> l9(x69, x70, x71) :|: x68 = x71 && x67 = x70 && x66 = x69 && 1 + x68 <= 2
(10) l0(x60, x61, x62) -> l8(x63, x64, x65) :|: x60 = x63 && x64 = 1 + x61 && x65 = 1 + x62 && 0 <= 1 - x62
(11) l10(x120, x121, x122) -> l0(x123, x124, x125) :|: x122 = x125 && x121 = x124 && x120 = x123
(12) l12(x114, x115, x116) -> l10(x117, x118, x119) :|: x116 = x119 && x114 = x117 && x118 = 1 && 2 <= x115 && x115 <= 2
(13) l11(x102, x103, x104) -> l12(x105, x106, x107) :|: x104 = x107 && x103 = x106 && x102 = x105 && 1 + x104 <= 2
(14) l0(x96, x97, x98) -> l11(x99, x100, x101) :|: x96 = x99 && x100 = 1 + x97 && x101 = 1 + x98 && 0 <= 1 - x98
(15) l13(x144, x145, x146) -> l0(x147, x148, x149) :|: x146 = x149 && x145 = x148 && x144 = x147
(16) l14(x138, x139, x140) -> l13(x141, x142, x143) :|: x140 = x143 && x139 = x142 && x138 = x141 && 3 <= x139
(17) l14(x132, x133, x134) -> l13(x135, x136, x137) :|: x134 = x137 && x133 = x136 && x132 = x135 && 1 + x133 <= 2
(18) l0(x126, x127, x128) -> l14(x129, x130, x131) :|: x126 = x129 && 2 <= x131 && x131 <= 2 && x130 = 1 + x127 && x131 = 1 + x128 && 0 <= 1 - x128
(19) l15(x157, x158, x159) -> l0(x160, x161, x162) :|: x159 = x162 && x158 = x161 && x157 = x160
(20) l0(x150, x151, x152) -> l15(x153, x154, x155) :|: 0 <= 1 - x152 && x155 = 1 + x152 && x156 = 1 + x151 && x155 <= 2 && 2 <= x155 && x156 <= 2 && 2 <= x156 && x154 = 1 && x150 = x153
(21) l1(x24, x25, x26) -> l0(x27, x28, x29) :|: x26 = x29 && x25 = x28 && x24 = x27
(22) l3(x18, x19, x20) -> l1(x21, x22, x23) :|: x20 = x23 && x19 = x22 && x18 = x21 && 3 <= x19
(23) l3(x12, x13, x14) -> l1(x15, x16, x17) :|: x14 = x17 && x13 = x16 && x12 = x15 && 1 + x13 <= 2
(24) l2(x6, x7, x8) -> l3(x9, x10, x11) :|: x8 = x11 && x7 = x10 && x6 = x9 && 3 <= x8

Arcs:
(1) -> (24)
(2) -> (1), (5), (10), (14), (18), (20)
(3) -> (2)
(4) -> (3)
(5) -> (4)
(6) -> (1), (5), (10), (14), (18), (20)
(7) -> (6)
(8) -> (6)
(9) -> (7), (8)
(10) -> (9)
(11) -> (1), (5), (10), (14), (18), (20)
(12) -> (11)
(13) -> (12)
(14) -> (13)
(15) -> (1), (5), (10), (14), (18), (20)
(16) -> (15)
(17) -> (15)
(18) -> (16), (17)
(19) -> (1), (5), (10), (14), (18), (20)
(20) -> (19)
(21) -> (1), (5), (10), (14), (18), (20)
(22) -> (21)
(23) -> (21)
(24) -> (22), (23)

This digraph is fully evaluated!

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

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

(6)
Obligation:
Rules:
l0(x126:0, x127:0, x128:0) -> l0(x126:0, 1 + x127:0, 1 + x128:0) :|: x127:0 < 1 && x128:0 > 0 && x128:0 < 2
l0(x105:0, x97:0, x98:0) -> l0(x105:0, 1, 1 + x98:0) :|: x98:0 < 1 && x98:0 < 2 && x97:0 > 0 && x97:0 < 2
l0(Result_4HAT0:0, w_5HAT0:0, x_6HAT0:0) -> l0(Result_4HAT0:0, 1 + w_5HAT0:0, 1 + x_6HAT0:0) :|: x_6HAT0:0 > 1 && w_5HAT0:0 < 3 && w_5HAT0:0 > 1
l0(x60:0, x61:0, x62:0) -> l0(x60:0, 1 + x61:0, 1 + x62:0) :|: x62:0 < 1 && x61:0 > 1 && x62:0 < 2
l0(x30:0, x31:0, x32:0) -> l0(x30:0, 1, 1 + x32:0) :|: x31:0 < 3 && x32:0 > 1 && x31:0 > 0 && x31:0 < 2
l0(x, x1, x2) -> l0(x, 1 + x1, 1 + x2) :|: x1 > 1 && x2 > 0 && x2 < 2
l0(x3, x4, x5) -> l0(x3, 1 + x4, 1 + x5) :|: x5 < 1 && x4 < 1 && x5 < 2
l0(x6, x7, x8) -> l0(x6, 1 + x7, 1 + x8) :|: x8 > 1 && x7 < 3 && x7 < 1
l0(x150:0, x151:0, x152:0) -> l0(x150:0, 1, 1 + x152:0) :|: x151:0 < 2 && x151:0 > 0 && x152:0 < 2 && x152:0 > 0

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

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

   l0(x1, x2, x3) -> l0(x2, x3)

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

(8)
Obligation:
Rules:
l0(x127:0, x128:0) -> l0(1 + x127:0, 1 + x128:0) :|: x127:0 < 1 && x128:0 > 0 && x128:0 < 2
l0(x97:0, x98:0) -> l0(1, 1 + x98:0) :|: x98:0 < 1 && x98:0 < 2 && x97:0 > 0 && x97:0 < 2
l0(w_5HAT0:0, x_6HAT0:0) -> l0(1 + w_5HAT0:0, 1 + x_6HAT0:0) :|: x_6HAT0:0 > 1 && w_5HAT0:0 < 3 && w_5HAT0:0 > 1
l0(x61:0, x62:0) -> l0(1 + x61:0, 1 + x62:0) :|: x62:0 < 1 && x61:0 > 1 && x62:0 < 2
l0(x31:0, x32:0) -> l0(1, 1 + x32:0) :|: x31:0 < 3 && x32:0 > 1 && x31:0 > 0 && x31:0 < 2
l0(x1, x2) -> l0(1 + x1, 1 + x2) :|: x1 > 1 && x2 > 0 && x2 < 2
l0(x4, x5) -> l0(1 + x4, 1 + x5) :|: x5 < 1 && x4 < 1 && x5 < 2
l0(x7, x8) -> l0(1 + x7, 1 + x8) :|: x8 > 1 && x7 < 3 && x7 < 1
l0(x151:0, x152:0) -> l0(1, 1 + x152:0) :|: x151:0 < 2 && x151:0 > 0 && x152:0 < 2 && x152:0 > 0

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

(9) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
l0(VARIABLE, INTEGER)
Replaced non-predefined constructor symbols by 0.The following proof was generated: 
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


Termination of the given IntTRS could not be shown:



- IntTRS
  - PolynomialOrderProcessor

Rules:
l0(x127:0, x128:0) -> l0(c, c1) :|: c1 = 1 + x128:0 && c = 1 + x127:0 && (x127:0 < 1 && x128:0 > 0 && x128:0 < 2)
l0(x97:0, x98:0) -> l0(c2, c3) :|: c3 = 1 + x98:0 && c2 = 1 && (x98:0 < 1 && x98:0 < 2 && x97:0 > 0 && x97:0 < 2)
l0(w_5HAT0:0, x_6HAT0:0) -> l0(c4, c5) :|: c5 = 1 + x_6HAT0:0 && c4 = 1 + w_5HAT0:0 && (x_6HAT0:0 > 1 && w_5HAT0:0 < 3 && w_5HAT0:0 > 1)
l0(x61:0, x62:0) -> l0(c6, c7) :|: c7 = 1 + x62:0 && c6 = 1 + x61:0 && (x62:0 < 1 && x61:0 > 1 && x62:0 < 2)
l0(x31:0, x32:0) -> l0(c8, c9) :|: c9 = 1 + x32:0 && c8 = 1 && (x31:0 < 3 && x32:0 > 1 && x31:0 > 0 && x31:0 < 2)
l0(x1, x2) -> l0(c10, c11) :|: c11 = 1 + x2 && c10 = 1 + x1 && (x1 > 1 && x2 > 0 && x2 < 2)
l0(x4, x5) -> l0(c12, c13) :|: c13 = 1 + x5 && c12 = 1 + x4 && (x5 < 1 && x4 < 1 && x5 < 2)
l0(x7, x8) -> l0(c14, c15) :|: c15 = 1 + x8 && c14 = 1 + x7 && (x8 > 1 && x7 < 3 && x7 < 1)
l0(x151:0, x152:0) -> l0(c16, c17) :|: c17 = 1 + x152:0 && c16 = 1 && (x151:0 < 2 && x151:0 > 0 && x152:0 < 2 && x152:0 > 0)

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

The following rules are decreasing:
l0(x127:0, x128:0) -> l0(c, c1) :|: c1 = 1 + x128:0 && c = 1 + x127:0 && (x127:0 < 1 && x128:0 > 0 && x128:0 < 2)
l0(w_5HAT0:0, x_6HAT0:0) -> l0(c4, c5) :|: c5 = 1 + x_6HAT0:0 && c4 = 1 + w_5HAT0:0 && (x_6HAT0:0 > 1 && w_5HAT0:0 < 3 && w_5HAT0:0 > 1)
l0(x61:0, x62:0) -> l0(c6, c7) :|: c7 = 1 + x62:0 && c6 = 1 + x61:0 && (x62:0 < 1 && x61:0 > 1 && x62:0 < 2)
l0(x1, x2) -> l0(c10, c11) :|: c11 = 1 + x2 && c10 = 1 + x1 && (x1 > 1 && x2 > 0 && x2 < 2)
l0(x4, x5) -> l0(c12, c13) :|: c13 = 1 + x5 && c12 = 1 + x4 && (x5 < 1 && x4 < 1 && x5 < 2)
l0(x7, x8) -> l0(c14, c15) :|: c15 = 1 + x8 && c14 = 1 + x7 && (x8 > 1 && x7 < 3 && x7 < 1)
The following rules are bounded:
l0(x127:0, x128:0) -> l0(c, c1) :|: c1 = 1 + x128:0 && c = 1 + x127:0 && (x127:0 < 1 && x128:0 > 0 && x128:0 < 2)
l0(x4, x5) -> l0(c12, c13) :|: c13 = 1 + x5 && c12 = 1 + x4 && (x5 < 1 && x4 < 1 && x5 < 2)
l0(x7, x8) -> l0(c14, c15) :|: c15 = 1 + x8 && c14 = 1 + x7 && (x8 > 1 && x7 < 3 && x7 < 1)


- IntTRS
  - PolynomialOrderProcessor
    - IntTRS
      - PolynomialOrderProcessor

Rules:
l0(x97:0, x98:0) -> l0(c2, c3) :|: c3 = 1 + x98:0 && c2 = 1 && (x98:0 < 1 && x98:0 < 2 && x97:0 > 0 && x97:0 < 2)
l0(w_5HAT0:0, x_6HAT0:0) -> l0(c4, c5) :|: c5 = 1 + x_6HAT0:0 && c4 = 1 + w_5HAT0:0 && (x_6HAT0:0 > 1 && w_5HAT0:0 < 3 && w_5HAT0:0 > 1)
l0(x61:0, x62:0) -> l0(c6, c7) :|: c7 = 1 + x62:0 && c6 = 1 + x61:0 && (x62:0 < 1 && x61:0 > 1 && x62:0 < 2)
l0(x31:0, x32:0) -> l0(c8, c9) :|: c9 = 1 + x32:0 && c8 = 1 && (x31:0 < 3 && x32:0 > 1 && x31:0 > 0 && x31:0 < 2)
l0(x1, x2) -> l0(c10, c11) :|: c11 = 1 + x2 && c10 = 1 + x1 && (x1 > 1 && x2 > 0 && x2 < 2)
l0(x151:0, x152:0) -> l0(c16, c17) :|: c17 = 1 + x152:0 && c16 = 1 && (x151:0 < 2 && x151:0 > 0 && x152:0 < 2 && x152:0 > 0)

Found the following polynomial interpretation:
[l0(x, x1)] = -1 + x - x1

The following rules are decreasing:
l0(x97:0, x98:0) -> l0(c2, c3) :|: c3 = 1 + x98:0 && c2 = 1 && (x98:0 < 1 && x98:0 < 2 && x97:0 > 0 && x97:0 < 2)
l0(x31:0, x32:0) -> l0(c8, c9) :|: c9 = 1 + x32:0 && c8 = 1 && (x31:0 < 3 && x32:0 > 1 && x31:0 > 0 && x31:0 < 2)
l0(x151:0, x152:0) -> l0(c16, c17) :|: c17 = 1 + x152:0 && c16 = 1 && (x151:0 < 2 && x151:0 > 0 && x152:0 < 2 && x152:0 > 0)
The following rules are bounded:
l0(x97:0, x98:0) -> l0(c2, c3) :|: c3 = 1 + x98:0 && c2 = 1 && (x98:0 < 1 && x98:0 < 2 && x97:0 > 0 && x97:0 < 2)
l0(x61:0, x62:0) -> l0(c6, c7) :|: c7 = 1 + x62:0 && c6 = 1 + x61:0 && (x62:0 < 1 && x61:0 > 1 && x62:0 < 2)
l0(x1, x2) -> l0(c10, c11) :|: c11 = 1 + x2 && c10 = 1 + x1 && (x1 > 1 && x2 > 0 && x2 < 2)



- IntTRS
  - PolynomialOrderProcessor
    - IntTRS
      - PolynomialOrderProcessor
        - AND
          - IntTRS
          - IntTRS
            - RankingReductionPairProof
          - IntTRS

Rules:
l0(w_5HAT0:0, x_6HAT0:0) -> l0(c4, c5) :|: c5 = 1 + x_6HAT0:0 && c4 = 1 + w_5HAT0:0 && (x_6HAT0:0 > 1 && w_5HAT0:0 < 3 && w_5HAT0:0 > 1)
l0(x61:0, x62:0) -> l0(c6, c7) :|: c7 = 1 + x62:0 && c6 = 1 + x61:0 && (x62:0 < 1 && x61:0 > 1 && x62:0 < 2)
l0(x1, x2) -> l0(c10, c11) :|: c11 = 1 + x2 && c10 = 1 + x1 && (x1 > 1 && x2 > 0 && x2 < 2)

Interpretation:
[ l0 ] = -3*l0_2

The following rules are decreasing:
l0(w_5HAT0:0, x_6HAT0:0) -> l0(c4, c5) :|: c5 = 1 + x_6HAT0:0 && c4 = 1 + w_5HAT0:0 && (x_6HAT0:0 > 1 && w_5HAT0:0 < 3 && w_5HAT0:0 > 1)
l0(x61:0, x62:0) -> l0(c6, c7) :|: c7 = 1 + x62:0 && c6 = 1 + x61:0 && (x62:0 < 1 && x61:0 > 1 && x62:0 < 2)
l0(x1, x2) -> l0(c10, c11) :|: c11 = 1 + x2 && c10 = 1 + x1 && (x1 > 1 && x2 > 0 && x2 < 2)

The following rules are bounded:
l0(x61:0, x62:0) -> l0(c6, c7) :|: c7 = 1 + x62:0 && c6 = 1 + x61:0 && (x62:0 < 1 && x61:0 > 1 && x62:0 < 2)



- IntTRS
  - PolynomialOrderProcessor
    - IntTRS
      - PolynomialOrderProcessor
        - AND
          - IntTRS
          - IntTRS
            - RankingReductionPairProof
              - IntTRS
                - RankingReductionPairProof
          - IntTRS

Rules:
l0(w_5HAT0:0, x_6HAT0:0) -> l0(c4, c5) :|: c5 = 1 + x_6HAT0:0 && c4 = 1 + w_5HAT0:0 && (x_6HAT0:0 > 1 && w_5HAT0:0 < 3 && w_5HAT0:0 > 1)
l0(x1, x2) -> l0(c10, c11) :|: c11 = 1 + x2 && c10 = 1 + x1 && (x1 > 1 && x2 > 0 && x2 < 2)

Interpretation:
[ l0 ] = -3*l0_1

The following rules are decreasing:
l0(w_5HAT0:0, x_6HAT0:0) -> l0(c4, c5) :|: c5 = 1 + x_6HAT0:0 && c4 = 1 + w_5HAT0:0 && (x_6HAT0:0 > 1 && w_5HAT0:0 < 3 && w_5HAT0:0 > 1)
l0(x1, x2) -> l0(c10, c11) :|: c11 = 1 + x2 && c10 = 1 + x1 && (x1 > 1 && x2 > 0 && x2 < 2)

The following rules are bounded:
l0(w_5HAT0:0, x_6HAT0:0) -> l0(c4, c5) :|: c5 = 1 + x_6HAT0:0 && c4 = 1 + w_5HAT0:0 && (x_6HAT0:0 > 1 && w_5HAT0:0 < 3 && w_5HAT0:0 > 1)



- IntTRS
  - PolynomialOrderProcessor
    - IntTRS
      - PolynomialOrderProcessor
        - AND
          - IntTRS
          - IntTRS
            - RankingReductionPairProof
              - IntTRS
                - RankingReductionPairProof
                  - IntTRS
                    - PolynomialOrderProcessor
          - IntTRS

Rules:
l0(x1, x2) -> l0(c10, c11) :|: c11 = 1 + x2 && c10 = 1 + x1 && (x1 > 1 && x2 > 0 && x2 < 2)

Found the following polynomial interpretation:
[l0(x, x1)] = 1 - x1

The following rules are decreasing:
l0(x1, x2) -> l0(c10, c11) :|: c11 = 1 + x2 && c10 = 1 + x1 && (x1 > 1 && x2 > 0 && x2 < 2)
The following rules are bounded:
l0(x1, x2) -> l0(c10, c11) :|: c11 = 1 + x2 && c10 = 1 + x1 && (x1 > 1 && x2 > 0 && x2 < 2)



- IntTRS
  - PolynomialOrderProcessor
    - IntTRS
      - PolynomialOrderProcessor
        - AND
          - IntTRS
          - IntTRS
          - IntTRS
            - PolynomialOrderProcessor

Rules:
l0(w_5HAT0:0, x_6HAT0:0) -> l0(c4, c5) :|: c5 = 1 + x_6HAT0:0 && c4 = 1 + w_5HAT0:0 && (x_6HAT0:0 > 1 && w_5HAT0:0 < 3 && w_5HAT0:0 > 1)
l0(x31:0, x32:0) -> l0(c8, c9) :|: c9 = 1 + x32:0 && c8 = 1 && (x31:0 < 3 && x32:0 > 1 && x31:0 > 0 && x31:0 < 2)
l0(x151:0, x152:0) -> l0(c16, c17) :|: c17 = 1 + x152:0 && c16 = 1 && (x151:0 < 2 && x151:0 > 0 && x152:0 < 2 && x152:0 > 0)

Found the following polynomial interpretation:
[l0(x, x1)] = 1 - x

The following rules are decreasing:
l0(w_5HAT0:0, x_6HAT0:0) -> l0(c4, c5) :|: c5 = 1 + x_6HAT0:0 && c4 = 1 + w_5HAT0:0 && (x_6HAT0:0 > 1 && w_5HAT0:0 < 3 && w_5HAT0:0 > 1)
The following rules are bounded:
l0(x31:0, x32:0) -> l0(c8, c9) :|: c9 = 1 + x32:0 && c8 = 1 && (x31:0 < 3 && x32:0 > 1 && x31:0 > 0 && x31:0 < 2)
l0(x151:0, x152:0) -> l0(c16, c17) :|: c17 = 1 + x152:0 && c16 = 1 && (x151:0 < 2 && x151:0 > 0 && x152:0 < 2 && x152:0 > 0)



- IntTRS
  - PolynomialOrderProcessor
    - IntTRS
      - PolynomialOrderProcessor
        - AND
          - IntTRS
          - IntTRS
          - IntTRS
            - PolynomialOrderProcessor
              - AND
                - IntTRS
                - IntTRS
                  - PolynomialOrderProcessor
                - IntTRS

Rules:
l0(x31:0, x32:0) -> l0(c8, c9) :|: c9 = 1 + x32:0 && c8 = 1 && (x31:0 < 3 && x32:0 > 1 && x31:0 > 0 && x31:0 < 2)
l0(x151:0, x152:0) -> l0(c16, c17) :|: c17 = 1 + x152:0 && c16 = 1 && (x151:0 < 2 && x151:0 > 0 && x152:0 < 2 && x152:0 > 0)

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

The following rules are decreasing:
l0(x31:0, x32:0) -> l0(c8, c9) :|: c9 = 1 + x32:0 && c8 = 1 && (x31:0 < 3 && x32:0 > 1 && x31:0 > 0 && x31:0 < 2)
l0(x151:0, x152:0) -> l0(c16, c17) :|: c17 = 1 + x152:0 && c16 = 1 && (x151:0 < 2 && x151:0 > 0 && x152:0 < 2 && x152:0 > 0)
The following rules are bounded:
l0(x151:0, x152:0) -> l0(c16, c17) :|: c17 = 1 + x152:0 && c16 = 1 && (x151:0 < 2 && x151:0 > 0 && x152:0 < 2 && x152:0 > 0)


- IntTRS
  - PolynomialOrderProcessor
    - IntTRS
      - PolynomialOrderProcessor
        - AND
          - IntTRS
          - IntTRS
          - IntTRS
            - PolynomialOrderProcessor
              - AND
                - IntTRS
                - IntTRS
                  - PolynomialOrderProcessor
                    - IntTRS
                - IntTRS

Rules:
l0(x31:0, x32:0) -> l0(c8, c9) :|: c9 = 1 + x32:0 && c8 = 1 && (x31:0 < 3 && x32:0 > 1 && x31:0 > 0 && x31:0 < 2)



- IntTRS
  - PolynomialOrderProcessor
    - IntTRS
      - PolynomialOrderProcessor
        - AND
          - IntTRS
          - IntTRS
          - IntTRS
            - PolynomialOrderProcessor
              - AND
                - IntTRS
                - IntTRS
                - IntTRS
                  - RankingReductionPairProof

Rules:
l0(w_5HAT0:0, x_6HAT0:0) -> l0(c4, c5) :|: c5 = 1 + x_6HAT0:0 && c4 = 1 + w_5HAT0:0 && (x_6HAT0:0 > 1 && w_5HAT0:0 < 3 && w_5HAT0:0 > 1)

Interpretation:
[ l0 ] = -1*l0_1

The following rules are decreasing:
l0(w_5HAT0:0, x_6HAT0:0) -> l0(c4, c5) :|: c5 = 1 + x_6HAT0:0 && c4 = 1 + w_5HAT0:0 && (x_6HAT0:0 > 1 && w_5HAT0:0 < 3 && w_5HAT0:0 > 1)

The following rules are bounded:
l0(w_5HAT0:0, x_6HAT0:0) -> l0(c4, c5) :|: c5 = 1 + x_6HAT0:0 && c4 = 1 + w_5HAT0:0 && (x_6HAT0:0 > 1 && w_5HAT0:0 < 3 && w_5HAT0:0 > 1)




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

(10)
Obligation:
Rules:
l0(x31:0, x32:0) -> l0(1, 1 + x32:0) :|: x31:0 < 3 && x32:0 > 1 && x31:0 > 0 && x31:0 < 2

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

(11) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(x31:0, x32:0) -> l0(1, 1 + x32:0) :|: x31:0 < 3 && x32:0 > 1 && x31:0 > 0 && x31:0 < 2

Arcs:
(1) -> (1)

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

(12)
Obligation:

Termination digraph:
Nodes:
(1) l0(x31:0, x32:0) -> l0(1, 1 + x32:0) :|: x31:0 < 3 && x32:0 > 1 && x31:0 > 0 && x31:0 < 2

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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

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

(14)
Obligation:
Rules:
l0(x31:0:0, x32:0:0) -> l0(1, 1 + x32:0:0) :|: x31:0:0 > 0 && x31:0:0 < 2 && x32:0:0 > 1 && x31:0:0 < 3