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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 1079 ms]
(4) AND
    (5) IRSwT
        (6) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (7) IRSwT
        (8) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (9) IRSwT
        (10) TempFilterProof [SOUND, 7 ms]
        (11) IntTRS
        (12) RankingReductionPairProof [EQUIVALENT, 1 ms]
        (13) YES
    (14) IRSwT
        (15) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (16) IRSwT
        (17) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (18) IRSwT
        (19) TempFilterProof [SOUND, 20 ms]
        (20) IntTRS
        (21) RankingReductionPairProof [EQUIVALENT, 8 ms]
        (22) YES
    (23) IRSwT
        (24) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (25) IRSwT
        (26) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (27) IRSwT
        (28) TempFilterProof [SOUND, 10 ms]
        (29) IntTRS
        (30) PolynomialOrderProcessor [EQUIVALENT, 1 ms]
        (31) YES
    (32) IRSwT
        (33) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (34) IRSwT
        (35) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (36) IRSwT
        (37) TempFilterProof [SOUND, 12 ms]
        (38) IntTRS
        (39) RankingReductionPairProof [EQUIVALENT, 3 ms]
        (40) YES
    (41) IRSwT
        (42) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (43) IRSwT
        (44) TempFilterProof [SOUND, 15 ms]
        (45) IntTRS
        (46) RankingReductionPairProof [EQUIVALENT, 2 ms]
        (47) YES
    (48) IRSwT
        (49) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (50) IRSwT
        (51) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (52) IRSwT
        (53) TempFilterProof [SOUND, 15 ms]
        (54) IntTRS
        (55) RankingReductionPairProof [EQUIVALENT, 6 ms]
        (56) YES


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

(0)
Obligation:
Rules:
f1_0_main_ConstantStackPush(arg1, arg2, arg3, arg4, arg5) -> f500_0_main_GE(arg1P, arg2P, arg3P, arg4P, arg5P) :|: arg2 = arg3P && 0 = arg2P && 0 <= arg1P - 1 && 0 <= arg1 - 1 && -1 <= arg2 - 1 && arg1P <= arg1
f500_0_main_GE(x, x1, x2, x3, x4) -> f500_0_main_GE(x5, x6, x7, x8, x9) :|: x2 = x7 && x1 + 1 = x6 && 0 <= x5 - 1 && 0 <= x - 1 && x5 <= x && 0 <= x2 - 1 && x1 <= x2 - 1
f500_0_main_GE(x10, x11, x12, x13, x14) -> f323_0_rec0_GT(x15, x16, x17, x18, x19) :|: x12 = x16 && 0 = x15 && 0 <= x10 - 1 && 0 <= x12 - 1 && x11 <= x12 - 1
f323_0_rec0_GT(x20, x21, x22, x23, x24) -> f323_0_rec0_GT(x25, x26, x27, x28, x29) :|: x21 = x26 && x20 + 1 = x25 && 0 <= x21 - 1 && -1 <= x20 - 1 && x20 <= x21
f323_0_rec0_GT(x30, x31, x32, x33, x34) -> f376_0_rec1_GT(x35, x36, x37, x38, x39) :|: 2 * x30 = x37 && 0 = x36 && x30 = x35 && x30 <= x31 && 0 <= x31 - 1
f376_0_rec1_GT(x40, x41, x42, x43, x44) -> f376_0_rec1_GT(x45, x46, x47, x48, x49) :|: x42 = x47 && x41 + 1 = x46 && x40 = x45 && -1 <= x41 - 1 && x41 <= x42 && -1 <= x40 - 1
f376_0_rec1_GT(x50, x51, x52, x53, x54) -> f319_0_rec2_LT(x55, x56, x57, x58, x59) :|: x50 + x51 = x57 && x51 = x56 && x50 = x55 && -1 <= x51 - 1 && x51 <= x52 && -1 <= x50 - 1
f319_0_rec2_LT(x60, x61, x62, x63, x64) -> f319_0_rec2_LT(x65, x66, x67, x68, x69) :|: x62 - 1 = x67 && x61 = x66 && x60 = x65 && 0 <= 4 * x62 && 0 <= 2 * x60 + 3 * x61 && 0 <= 2 * x60 && -1 <= x62 - 1 && 0 <= 3 * x61
f319_0_rec2_LT(x70, x71, x72, x73, x74) -> f541_0_rec3_GT(x75, x76, x77, x78, x79) :|: 2 * x70 + 3 * x71 + 4 * x72 = x79 && 0 = x78 && x72 = x77 && x71 = x76 && x70 = x75 && 0 <= 4 * x72 && 0 <= 2 * x70 + 3 * x71 && 0 <= 2 * x70 && -1 <= x72 - 1 && 0 <= 3 * x71
f541_0_rec3_GT(x80, x81, x82, x83, x84) -> f541_0_rec3_GT(x85, x86, x87, x88, x89) :|: x84 = x89 && x83 + 1 = x88 && x82 = x87 && x81 = x86 && x80 = x85 && -1 <= x83 - 1 && 0 <= 1000 * x80 + 100 * x81 + 10 * x82 && -1 <= x82 - 1 && 0 <= 1000 * x80 + 100 * x81 && 0 <= 10 * x82 && 0 <= 1000 * x80 && x83 <= x84 && 0 <= 100 * x81
f541_0_rec3_GT(x90, x91, x92, x93, x94) -> f493_0_rec4_LT(x95, x96, x97, x98, x99) :|: 1000 * x90 + 100 * x91 + 10 * x92 + x93 = x95 && 0 <= 1000 * x90 + 100 * x91 + 10 * x92 && -1 <= x93 - 1 && -1 <= x92 - 1 && 0 <= 1000 * x90 + 100 * x91 && 0 <= 10 * x92 && 0 <= 1000 * x90 && x93 <= x94 && 0 <= 100 * x91
f493_0_rec4_LT(x100, x101, x102, x103, x104) -> f493_0_rec4_LT(x105, x106, x107, x108, x109) :|: x100 - 1 = x105 && -1 <= x100 - 1
__init(x110, x111, x112, x113, x114) -> f1_0_main_ConstantStackPush(x115, x116, x117, x118, x119) :|: 0 <= 0
Start term: __init(arg1, arg2, arg3, arg4, arg5)

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

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

(2)
Obligation:
Rules:
f1_0_main_ConstantStackPush(arg1, arg2, arg3, arg4, arg5) -> f500_0_main_GE(arg1P, arg2P, arg3P, arg4P, arg5P) :|: arg2 = arg3P && 0 = arg2P && 0 <= arg1P - 1 && 0 <= arg1 - 1 && -1 <= arg2 - 1 && arg1P <= arg1
f500_0_main_GE(x, x1, x2, x3, x4) -> f500_0_main_GE(x5, x6, x7, x8, x9) :|: x2 = x7 && x1 + 1 = x6 && 0 <= x5 - 1 && 0 <= x - 1 && x5 <= x && 0 <= x2 - 1 && x1 <= x2 - 1
f500_0_main_GE(x10, x11, x12, x13, x14) -> f323_0_rec0_GT(x15, x16, x17, x18, x19) :|: x12 = x16 && 0 = x15 && 0 <= x10 - 1 && 0 <= x12 - 1 && x11 <= x12 - 1
f323_0_rec0_GT(x20, x21, x22, x23, x24) -> f323_0_rec0_GT(x25, x26, x27, x28, x29) :|: x21 = x26 && x20 + 1 = x25 && 0 <= x21 - 1 && -1 <= x20 - 1 && x20 <= x21
f323_0_rec0_GT(x30, x31, x32, x33, x34) -> f376_0_rec1_GT(x35, x36, x37, x38, x39) :|: 2 * x30 = x37 && 0 = x36 && x30 = x35 && x30 <= x31 && 0 <= x31 - 1
f376_0_rec1_GT(x40, x41, x42, x43, x44) -> f376_0_rec1_GT(x45, x46, x47, x48, x49) :|: x42 = x47 && x41 + 1 = x46 && x40 = x45 && -1 <= x41 - 1 && x41 <= x42 && -1 <= x40 - 1
f376_0_rec1_GT(x50, x51, x52, x53, x54) -> f319_0_rec2_LT(x55, x56, x57, x58, x59) :|: x50 + x51 = x57 && x51 = x56 && x50 = x55 && -1 <= x51 - 1 && x51 <= x52 && -1 <= x50 - 1
f319_0_rec2_LT(x60, x61, x62, x63, x64) -> f319_0_rec2_LT(x65, x66, x67, x68, x69) :|: x62 - 1 = x67 && x61 = x66 && x60 = x65 && 0 <= 4 * x62 && 0 <= 2 * x60 + 3 * x61 && 0 <= 2 * x60 && -1 <= x62 - 1 && 0 <= 3 * x61
f319_0_rec2_LT(x70, x71, x72, x73, x74) -> f541_0_rec3_GT(x75, x76, x77, x78, x79) :|: 2 * x70 + 3 * x71 + 4 * x72 = x79 && 0 = x78 && x72 = x77 && x71 = x76 && x70 = x75 && 0 <= 4 * x72 && 0 <= 2 * x70 + 3 * x71 && 0 <= 2 * x70 && -1 <= x72 - 1 && 0 <= 3 * x71
f541_0_rec3_GT(x80, x81, x82, x83, x84) -> f541_0_rec3_GT(x85, x86, x87, x88, x89) :|: x84 = x89 && x83 + 1 = x88 && x82 = x87 && x81 = x86 && x80 = x85 && -1 <= x83 - 1 && 0 <= 1000 * x80 + 100 * x81 + 10 * x82 && -1 <= x82 - 1 && 0 <= 1000 * x80 + 100 * x81 && 0 <= 10 * x82 && 0 <= 1000 * x80 && x83 <= x84 && 0 <= 100 * x81
f541_0_rec3_GT(x90, x91, x92, x93, x94) -> f493_0_rec4_LT(x95, x96, x97, x98, x99) :|: 1000 * x90 + 100 * x91 + 10 * x92 + x93 = x95 && 0 <= 1000 * x90 + 100 * x91 + 10 * x92 && -1 <= x93 - 1 && -1 <= x92 - 1 && 0 <= 1000 * x90 + 100 * x91 && 0 <= 10 * x92 && 0 <= 1000 * x90 && x93 <= x94 && 0 <= 100 * x91
f493_0_rec4_LT(x100, x101, x102, x103, x104) -> f493_0_rec4_LT(x105, x106, x107, x108, x109) :|: x100 - 1 = x105 && -1 <= x100 - 1
__init(x110, x111, x112, x113, x114) -> f1_0_main_ConstantStackPush(x115, x116, x117, x118, x119) :|: 0 <= 0
Start term: __init(arg1, arg2, arg3, arg4, arg5)

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

(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) f1_0_main_ConstantStackPush(arg1, arg2, arg3, arg4, arg5) -> f500_0_main_GE(arg1P, arg2P, arg3P, arg4P, arg5P) :|: arg2 = arg3P && 0 = arg2P && 0 <= arg1P - 1 && 0 <= arg1 - 1 && -1 <= arg2 - 1 && arg1P <= arg1
(2) f500_0_main_GE(x, x1, x2, x3, x4) -> f500_0_main_GE(x5, x6, x7, x8, x9) :|: x2 = x7 && x1 + 1 = x6 && 0 <= x5 - 1 && 0 <= x - 1 && x5 <= x && 0 <= x2 - 1 && x1 <= x2 - 1
(3) f500_0_main_GE(x10, x11, x12, x13, x14) -> f323_0_rec0_GT(x15, x16, x17, x18, x19) :|: x12 = x16 && 0 = x15 && 0 <= x10 - 1 && 0 <= x12 - 1 && x11 <= x12 - 1
(4) f323_0_rec0_GT(x20, x21, x22, x23, x24) -> f323_0_rec0_GT(x25, x26, x27, x28, x29) :|: x21 = x26 && x20 + 1 = x25 && 0 <= x21 - 1 && -1 <= x20 - 1 && x20 <= x21
(5) f323_0_rec0_GT(x30, x31, x32, x33, x34) -> f376_0_rec1_GT(x35, x36, x37, x38, x39) :|: 2 * x30 = x37 && 0 = x36 && x30 = x35 && x30 <= x31 && 0 <= x31 - 1
(6) f376_0_rec1_GT(x40, x41, x42, x43, x44) -> f376_0_rec1_GT(x45, x46, x47, x48, x49) :|: x42 = x47 && x41 + 1 = x46 && x40 = x45 && -1 <= x41 - 1 && x41 <= x42 && -1 <= x40 - 1
(7) f376_0_rec1_GT(x50, x51, x52, x53, x54) -> f319_0_rec2_LT(x55, x56, x57, x58, x59) :|: x50 + x51 = x57 && x51 = x56 && x50 = x55 && -1 <= x51 - 1 && x51 <= x52 && -1 <= x50 - 1
(8) f319_0_rec2_LT(x60, x61, x62, x63, x64) -> f319_0_rec2_LT(x65, x66, x67, x68, x69) :|: x62 - 1 = x67 && x61 = x66 && x60 = x65 && 0 <= 4 * x62 && 0 <= 2 * x60 + 3 * x61 && 0 <= 2 * x60 && -1 <= x62 - 1 && 0 <= 3 * x61
(9) f319_0_rec2_LT(x70, x71, x72, x73, x74) -> f541_0_rec3_GT(x75, x76, x77, x78, x79) :|: 2 * x70 + 3 * x71 + 4 * x72 = x79 && 0 = x78 && x72 = x77 && x71 = x76 && x70 = x75 && 0 <= 4 * x72 && 0 <= 2 * x70 + 3 * x71 && 0 <= 2 * x70 && -1 <= x72 - 1 && 0 <= 3 * x71
(10) f541_0_rec3_GT(x80, x81, x82, x83, x84) -> f541_0_rec3_GT(x85, x86, x87, x88, x89) :|: x84 = x89 && x83 + 1 = x88 && x82 = x87 && x81 = x86 && x80 = x85 && -1 <= x83 - 1 && 0 <= 1000 * x80 + 100 * x81 + 10 * x82 && -1 <= x82 - 1 && 0 <= 1000 * x80 + 100 * x81 && 0 <= 10 * x82 && 0 <= 1000 * x80 && x83 <= x84 && 0 <= 100 * x81
(11) f541_0_rec3_GT(x90, x91, x92, x93, x94) -> f493_0_rec4_LT(x95, x96, x97, x98, x99) :|: 1000 * x90 + 100 * x91 + 10 * x92 + x93 = x95 && 0 <= 1000 * x90 + 100 * x91 + 10 * x92 && -1 <= x93 - 1 && -1 <= x92 - 1 && 0 <= 1000 * x90 + 100 * x91 && 0 <= 10 * x92 && 0 <= 1000 * x90 && x93 <= x94 && 0 <= 100 * x91
(12) f493_0_rec4_LT(x100, x101, x102, x103, x104) -> f493_0_rec4_LT(x105, x106, x107, x108, x109) :|: x100 - 1 = x105 && -1 <= x100 - 1
(13) __init(x110, x111, x112, x113, x114) -> f1_0_main_ConstantStackPush(x115, x116, x117, x118, x119) :|: 0 <= 0

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

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

(4)
Complex Obligation (AND)

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

(5)
Obligation:

Termination digraph:
Nodes:
(1) f500_0_main_GE(x, x1, x2, x3, x4) -> f500_0_main_GE(x5, x6, x7, x8, x9) :|: x2 = x7 && x1 + 1 = x6 && 0 <= x5 - 1 && 0 <= x - 1 && x5 <= x && 0 <= x2 - 1 && x1 <= x2 - 1

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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

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

(7)
Obligation:
Rules:
f500_0_main_GE(x:0, x1:0, x2:0, x3:0, x4:0) -> f500_0_main_GE(x5:0, x1:0 + 1, x2:0, x8:0, x9:0) :|: x2:0 > 0 && x2:0 - 1 >= x1:0 && x:0 >= x5:0 && x5:0 > 0 && x:0 > 0

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

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

   f500_0_main_GE(x1, x2, x3, x4, x5) -> f500_0_main_GE(x1, x2, x3)

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

(9)
Obligation:
Rules:
f500_0_main_GE(x:0, x1:0, x2:0) -> f500_0_main_GE(x5:0, x1:0 + 1, x2:0) :|: x2:0 > 0 && x2:0 - 1 >= x1:0 && x:0 >= x5:0 && x5:0 > 0 && x:0 > 0

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

(10) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f500_0_main_GE(INTEGER, INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(11)
Obligation:
Rules:
f500_0_main_GE(x:0, x1:0, x2:0) -> f500_0_main_GE(x5:0, c, x2:0) :|: c = x1:0 + 1 && (x2:0 > 0 && x2:0 - 1 >= x1:0 && x:0 >= x5:0 && x5:0 > 0 && x:0 > 0)

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

(12) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ f500_0_main_GE ] = f500_0_main_GE_3 + -1*f500_0_main_GE_2

The following rules are decreasing:
f500_0_main_GE(x:0, x1:0, x2:0) -> f500_0_main_GE(x5:0, c, x2:0) :|: c = x1:0 + 1 && (x2:0 > 0 && x2:0 - 1 >= x1:0 && x:0 >= x5:0 && x5:0 > 0 && x:0 > 0)

The following rules are bounded:
f500_0_main_GE(x:0, x1:0, x2:0) -> f500_0_main_GE(x5:0, c, x2:0) :|: c = x1:0 + 1 && (x2:0 > 0 && x2:0 - 1 >= x1:0 && x:0 >= x5:0 && x5:0 > 0 && x:0 > 0)


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

(13)
YES

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

(14)
Obligation:

Termination digraph:
Nodes:
(1) f323_0_rec0_GT(x20, x21, x22, x23, x24) -> f323_0_rec0_GT(x25, x26, x27, x28, x29) :|: x21 = x26 && x20 + 1 = x25 && 0 <= x21 - 1 && -1 <= x20 - 1 && x20 <= x21

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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

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

(16)
Obligation:
Rules:
f323_0_rec0_GT(x20:0, x21:0, x22:0, x23:0, x24:0) -> f323_0_rec0_GT(x20:0 + 1, x21:0, x27:0, x28:0, x29:0) :|: x20:0 > -1 && x21:0 > 0 && x21:0 >= x20:0

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

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

   f323_0_rec0_GT(x1, x2, x3, x4, x5) -> f323_0_rec0_GT(x1, x2)

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

(18)
Obligation:
Rules:
f323_0_rec0_GT(x20:0, x21:0) -> f323_0_rec0_GT(x20:0 + 1, x21:0) :|: x20:0 > -1 && x21:0 > 0 && x21:0 >= x20:0

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

(19) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f323_0_rec0_GT(INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(20)
Obligation:
Rules:
f323_0_rec0_GT(x20:0, x21:0) -> f323_0_rec0_GT(c, x21:0) :|: c = x20:0 + 1 && (x20:0 > -1 && x21:0 > 0 && x21:0 >= x20:0)

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

(21) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ f323_0_rec0_GT ] = -1*f323_0_rec0_GT_1 + f323_0_rec0_GT_2

The following rules are decreasing:
f323_0_rec0_GT(x20:0, x21:0) -> f323_0_rec0_GT(c, x21:0) :|: c = x20:0 + 1 && (x20:0 > -1 && x21:0 > 0 && x21:0 >= x20:0)

The following rules are bounded:
f323_0_rec0_GT(x20:0, x21:0) -> f323_0_rec0_GT(c, x21:0) :|: c = x20:0 + 1 && (x20:0 > -1 && x21:0 > 0 && x21:0 >= x20:0)


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

(22)
YES

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

(23)
Obligation:

Termination digraph:
Nodes:
(1) f376_0_rec1_GT(x40, x41, x42, x43, x44) -> f376_0_rec1_GT(x45, x46, x47, x48, x49) :|: x42 = x47 && x41 + 1 = x46 && x40 = x45 && -1 <= x41 - 1 && x41 <= x42 && -1 <= x40 - 1

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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

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

(25)
Obligation:
Rules:
f376_0_rec1_GT(x40:0, x41:0, x42:0, x43:0, x44:0) -> f376_0_rec1_GT(x40:0, x41:0 + 1, x42:0, x48:0, x49:0) :|: x42:0 >= x41:0 && x41:0 > -1 && x40:0 > -1

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

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

   f376_0_rec1_GT(x1, x2, x3, x4, x5) -> f376_0_rec1_GT(x1, x2, x3)

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

(27)
Obligation:
Rules:
f376_0_rec1_GT(x40:0, x41:0, x42:0) -> f376_0_rec1_GT(x40:0, x41:0 + 1, x42:0) :|: x42:0 >= x41:0 && x41:0 > -1 && x40:0 > -1

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

(28) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f376_0_rec1_GT(INTEGER, INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(29)
Obligation:
Rules:
f376_0_rec1_GT(x40:0, x41:0, x42:0) -> f376_0_rec1_GT(x40:0, c, x42:0) :|: c = x41:0 + 1 && (x42:0 >= x41:0 && x41:0 > -1 && x40:0 > -1)

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

(30) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[f376_0_rec1_GT(x, x1, x2)] = -x1 + x2

The following rules are decreasing:
f376_0_rec1_GT(x40:0, x41:0, x42:0) -> f376_0_rec1_GT(x40:0, c, x42:0) :|: c = x41:0 + 1 && (x42:0 >= x41:0 && x41:0 > -1 && x40:0 > -1)
The following rules are bounded:
f376_0_rec1_GT(x40:0, x41:0, x42:0) -> f376_0_rec1_GT(x40:0, c, x42:0) :|: c = x41:0 + 1 && (x42:0 >= x41:0 && x41:0 > -1 && x40:0 > -1)

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

(31)
YES

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

(32)
Obligation:

Termination digraph:
Nodes:
(1) f319_0_rec2_LT(x60, x61, x62, x63, x64) -> f319_0_rec2_LT(x65, x66, x67, x68, x69) :|: x62 - 1 = x67 && x61 = x66 && x60 = x65 && 0 <= 4 * x62 && 0 <= 2 * x60 + 3 * x61 && 0 <= 2 * x60 && -1 <= x62 - 1 && 0 <= 3 * x61

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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

(34)
Obligation:
Rules:
f319_0_rec2_LT(x60:0, x61:0, x62:0, x63:0, x64:0) -> f319_0_rec2_LT(x60:0, x61:0, x62:0 - 1, x68:0, x69:0) :|: x62:0 > -1 && 3 * x61:0 >= 0 && 2 * x60:0 >= 0 && 4 * x62:0 >= 0 && 2 * x60:0 + 3 * x61:0 >= 0

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

   f319_0_rec2_LT(x1, x2, x3, x4, x5) -> f319_0_rec2_LT(x1, x2, x3)

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(36)
Obligation:
Rules:
f319_0_rec2_LT(x60:0, x61:0, x62:0) -> f319_0_rec2_LT(x60:0, x61:0, x62:0 - 1) :|: x62:0 > -1 && 3 * x61:0 >= 0 && 2 * x60:0 >= 0 && 4 * x62:0 >= 0 && 2 * x60:0 + 3 * x61:0 >= 0

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(37) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f319_0_rec2_LT(INTEGER, INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(38)
Obligation:
Rules:
f319_0_rec2_LT(x60:0, x61:0, x62:0) -> f319_0_rec2_LT(x60:0, x61:0, c) :|: c = x62:0 - 1 && (x62:0 > -1 && 3 * x61:0 >= 0 && 2 * x60:0 >= 0 && 4 * x62:0 >= 0 && 2 * x60:0 + 3 * x61:0 >= 0)

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(39) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ f319_0_rec2_LT ] = f319_0_rec2_LT_3

The following rules are decreasing:
f319_0_rec2_LT(x60:0, x61:0, x62:0) -> f319_0_rec2_LT(x60:0, x61:0, c) :|: c = x62:0 - 1 && (x62:0 > -1 && 3 * x61:0 >= 0 && 2 * x60:0 >= 0 && 4 * x62:0 >= 0 && 2 * x60:0 + 3 * x61:0 >= 0)

The following rules are bounded:
f319_0_rec2_LT(x60:0, x61:0, x62:0) -> f319_0_rec2_LT(x60:0, x61:0, c) :|: c = x62:0 - 1 && (x62:0 > -1 && 3 * x61:0 >= 0 && 2 * x60:0 >= 0 && 4 * x62:0 >= 0 && 2 * x60:0 + 3 * x61:0 >= 0)


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(40)
YES

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(41)
Obligation:

Termination digraph:
Nodes:
(1) f541_0_rec3_GT(x80, x81, x82, x83, x84) -> f541_0_rec3_GT(x85, x86, x87, x88, x89) :|: x84 = x89 && x83 + 1 = x88 && x82 = x87 && x81 = x86 && x80 = x85 && -1 <= x83 - 1 && 0 <= 1000 * x80 + 100 * x81 + 10 * x82 && -1 <= x82 - 1 && 0 <= 1000 * x80 + 100 * x81 && 0 <= 10 * x82 && 0 <= 1000 * x80 && x83 <= x84 && 0 <= 100 * x81

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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

(43)
Obligation:
Rules:
f541_0_rec3_GT(x80:0, x81:0, x82:0, x83:0, x84:0) -> f541_0_rec3_GT(x80:0, x81:0, x82:0, x83:0 + 1, x84:0) :|: x84:0 >= x83:0 && 100 * x81:0 >= 0 && 1000 * x80:0 >= 0 && 10 * x82:0 >= 0 && 1000 * x80:0 + 100 * x81:0 >= 0 && x82:0 > -1 && x83:0 > -1 && 1000 * x80:0 + 100 * x81:0 + 10 * x82:0 >= 0

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

(44) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f541_0_rec3_GT(INTEGER, INTEGER, INTEGER, INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(45)
Obligation:
Rules:
f541_0_rec3_GT(x80:0, x81:0, x82:0, x83:0, x84:0) -> f541_0_rec3_GT(x80:0, x81:0, x82:0, c, x84:0) :|: c = x83:0 + 1 && (x84:0 >= x83:0 && 100 * x81:0 >= 0 && 1000 * x80:0 >= 0 && 10 * x82:0 >= 0 && 1000 * x80:0 + 100 * x81:0 >= 0 && x82:0 > -1 && x83:0 > -1 && 1000 * x80:0 + 100 * x81:0 + 10 * x82:0 >= 0)

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(46) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ f541_0_rec3_GT ] = f541_0_rec3_GT_5 + -1*f541_0_rec3_GT_4

The following rules are decreasing:
f541_0_rec3_GT(x80:0, x81:0, x82:0, x83:0, x84:0) -> f541_0_rec3_GT(x80:0, x81:0, x82:0, c, x84:0) :|: c = x83:0 + 1 && (x84:0 >= x83:0 && 100 * x81:0 >= 0 && 1000 * x80:0 >= 0 && 10 * x82:0 >= 0 && 1000 * x80:0 + 100 * x81:0 >= 0 && x82:0 > -1 && x83:0 > -1 && 1000 * x80:0 + 100 * x81:0 + 10 * x82:0 >= 0)

The following rules are bounded:
f541_0_rec3_GT(x80:0, x81:0, x82:0, x83:0, x84:0) -> f541_0_rec3_GT(x80:0, x81:0, x82:0, c, x84:0) :|: c = x83:0 + 1 && (x84:0 >= x83:0 && 100 * x81:0 >= 0 && 1000 * x80:0 >= 0 && 10 * x82:0 >= 0 && 1000 * x80:0 + 100 * x81:0 >= 0 && x82:0 > -1 && x83:0 > -1 && 1000 * x80:0 + 100 * x81:0 + 10 * x82:0 >= 0)


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(47)
YES

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(48)
Obligation:

Termination digraph:
Nodes:
(1) f493_0_rec4_LT(x100, x101, x102, x103, x104) -> f493_0_rec4_LT(x105, x106, x107, x108, x109) :|: x100 - 1 = x105 && -1 <= x100 - 1

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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

(50)
Obligation:
Rules:
f493_0_rec4_LT(x100:0, x101:0, x102:0, x103:0, x104:0) -> f493_0_rec4_LT(x100:0 - 1, x106:0, x107:0, x108:0, x109:0) :|: x100:0 > -1

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

   f493_0_rec4_LT(x1, x2, x3, x4, x5) -> f493_0_rec4_LT(x1)

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

(52)
Obligation:
Rules:
f493_0_rec4_LT(x100:0) -> f493_0_rec4_LT(x100:0 - 1) :|: x100:0 > -1

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

(53) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f493_0_rec4_LT(INTEGER)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(54)
Obligation:
Rules:
f493_0_rec4_LT(x100:0) -> f493_0_rec4_LT(c) :|: c = x100:0 - 1 && x100:0 > -1

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(55) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ f493_0_rec4_LT ] = f493_0_rec4_LT_1

The following rules are decreasing:
f493_0_rec4_LT(x100:0) -> f493_0_rec4_LT(c) :|: c = x100:0 - 1 && x100:0 > -1

The following rules are bounded:
f493_0_rec4_LT(x100:0) -> f493_0_rec4_LT(c) :|: c = x100:0 - 1 && x100:0 > -1


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(56)
YES