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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 2420 ms]
(4) AND
    (5) IRSwT
        (6) IntTRSCompressionProof [EQUIVALENT, 17 ms]
        (7) IRSwT
        (8) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (9) IRSwT
        (10) TempFilterProof [SOUND, 23 ms]
        (11) IntTRS
        (12) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
        (13) YES
    (14) IRSwT
        (15) IntTRSCompressionProof [EQUIVALENT, 31 ms]
        (16) IRSwT
        (17) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (18) IRSwT
        (19) TempFilterProof [SOUND, 26 ms]
        (20) IntTRS
        (21) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
        (22) YES


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

(0)
Obligation:
Rules:
l0(i2HAT0, iHAT0, rHAT0) -> l1(i2HATpost, iHATpost, rHATpost) :|: rHAT0 = rHATpost && i2HAT0 = i2HATpost && iHATpost = 0 && 10 <= iHAT0
l0(x, x1, x2) -> l2(x3, x4, x5) :|: x2 = x5 && x1 = x4 && x3 = x1 && 1 + x1 <= 10
l3(x6, x7, x8) -> l1(x9, x10, x11) :|: x8 = x11 && x6 = x9 && x10 = 1 + x7
l4(x12, x13, x14) -> l5(x15, x16, x17) :|: x14 = x17 && x12 = x15 && x13 = x16
l5(x18, x19, x20) -> l3(x21, x22, x23) :|: x20 = x23 && x18 = x21 && x19 = x22
l6(x24, x25, x26) -> l0(x27, x28, x29) :|: x26 = x29 && x24 = x27 && x25 = x28
l7(x30, x31, x32) -> l4(x33, x34, x35) :|: x32 = x35 && x30 = x33 && x31 = x34
l7(x36, x37, x38) -> l5(x39, x40, x41) :|: x38 = x41 && x36 = x39 && x37 = x40
l8(x42, x43, x44) -> l7(x45, x46, x47) :|: x42 = x45 && x43 = x46 && x47 = x47
l9(x48, x49, x50) -> l8(x51, x52, x53) :|: x50 = x53 && x48 = x51 && x49 = x52
l9(x54, x55, x56) -> l3(x57, x58, x59) :|: x56 = x59 && x54 = x57 && x55 = x58
l10(x60, x61, x62) -> l11(x63, x64, x65) :|: x62 = x65 && x60 = x63 && x61 = x64 && 10 <= x61
l10(x66, x67, x68) -> l9(x69, x70, x71) :|: x68 = x71 && x66 = x69 && x67 = x70 && 1 + x67 <= 10
l12(x72, x73, x74) -> l6(x75, x76, x77) :|: x74 = x77 && x72 = x75 && x76 = 1 + x73
l1(x78, x79, x80) -> l10(x81, x82, x83) :|: x80 = x83 && x78 = x81 && x79 = x82
l13(x84, x85, x86) -> l14(x87, x88, x89) :|: x86 = x89 && x84 = x87 && x85 = x88
l14(x90, x91, x92) -> l12(x93, x94, x95) :|: x92 = x95 && x90 = x93 && x91 = x94
l15(x96, x97, x98) -> l13(x99, x100, x101) :|: x98 = x101 && x96 = x99 && x97 = x100
l15(x102, x103, x104) -> l14(x105, x106, x107) :|: x104 = x107 && x102 = x105 && x103 = x106
l16(x108, x109, x110) -> l15(x111, x112, x113) :|: x110 = x113 && x108 = x111 && x109 = x112
l2(x114, x115, x116) -> l16(x117, x118, x119) :|: x116 = x119 && x114 = x117 && x115 = x118
l2(x120, x121, x122) -> l12(x123, x124, x125) :|: x122 = x125 && x120 = x123 && x121 = x124
l17(x126, x127, x128) -> l6(x129, x130, x131) :|: x132 = 0 && x130 = 0 && x126 = x129 && x128 = x131
l18(x133, x134, x135) -> l17(x136, x137, x138) :|: x135 = x138 && x133 = x136 && x134 = x137
Start term: l18(i2HAT0, iHAT0, rHAT0)

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

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

(2)
Obligation:
Rules:
l0(i2HAT0, iHAT0, rHAT0) -> l1(i2HATpost, iHATpost, rHATpost) :|: rHAT0 = rHATpost && i2HAT0 = i2HATpost && iHATpost = 0 && 10 <= iHAT0
l0(x, x1, x2) -> l2(x3, x4, x5) :|: x2 = x5 && x1 = x4 && x3 = x1 && 1 + x1 <= 10
l3(x6, x7, x8) -> l1(x9, x10, x11) :|: x8 = x11 && x6 = x9 && x10 = 1 + x7
l4(x12, x13, x14) -> l5(x15, x16, x17) :|: x14 = x17 && x12 = x15 && x13 = x16
l5(x18, x19, x20) -> l3(x21, x22, x23) :|: x20 = x23 && x18 = x21 && x19 = x22
l6(x24, x25, x26) -> l0(x27, x28, x29) :|: x26 = x29 && x24 = x27 && x25 = x28
l7(x30, x31, x32) -> l4(x33, x34, x35) :|: x32 = x35 && x30 = x33 && x31 = x34
l7(x36, x37, x38) -> l5(x39, x40, x41) :|: x38 = x41 && x36 = x39 && x37 = x40
l8(x42, x43, x44) -> l7(x45, x46, x47) :|: x42 = x45 && x43 = x46 && x47 = x47
l9(x48, x49, x50) -> l8(x51, x52, x53) :|: x50 = x53 && x48 = x51 && x49 = x52
l9(x54, x55, x56) -> l3(x57, x58, x59) :|: x56 = x59 && x54 = x57 && x55 = x58
l10(x60, x61, x62) -> l11(x63, x64, x65) :|: x62 = x65 && x60 = x63 && x61 = x64 && 10 <= x61
l10(x66, x67, x68) -> l9(x69, x70, x71) :|: x68 = x71 && x66 = x69 && x67 = x70 && 1 + x67 <= 10
l12(x72, x73, x74) -> l6(x75, x76, x77) :|: x74 = x77 && x72 = x75 && x76 = 1 + x73
l1(x78, x79, x80) -> l10(x81, x82, x83) :|: x80 = x83 && x78 = x81 && x79 = x82
l13(x84, x85, x86) -> l14(x87, x88, x89) :|: x86 = x89 && x84 = x87 && x85 = x88
l14(x90, x91, x92) -> l12(x93, x94, x95) :|: x92 = x95 && x90 = x93 && x91 = x94
l15(x96, x97, x98) -> l13(x99, x100, x101) :|: x98 = x101 && x96 = x99 && x97 = x100
l15(x102, x103, x104) -> l14(x105, x106, x107) :|: x104 = x107 && x102 = x105 && x103 = x106
l16(x108, x109, x110) -> l15(x111, x112, x113) :|: x110 = x113 && x108 = x111 && x109 = x112
l2(x114, x115, x116) -> l16(x117, x118, x119) :|: x116 = x119 && x114 = x117 && x115 = x118
l2(x120, x121, x122) -> l12(x123, x124, x125) :|: x122 = x125 && x120 = x123 && x121 = x124
l17(x126, x127, x128) -> l6(x129, x130, x131) :|: x132 = 0 && x130 = 0 && x126 = x129 && x128 = x131
l18(x133, x134, x135) -> l17(x136, x137, x138) :|: x135 = x138 && x133 = x136 && x134 = x137
Start term: l18(i2HAT0, iHAT0, rHAT0)

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

(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(i2HAT0, iHAT0, rHAT0) -> l1(i2HATpost, iHATpost, rHATpost) :|: rHAT0 = rHATpost && i2HAT0 = i2HATpost && iHATpost = 0 && 10 <= iHAT0
(2) l0(x, x1, x2) -> l2(x3, x4, x5) :|: x2 = x5 && x1 = x4 && x3 = x1 && 1 + x1 <= 10
(3) l3(x6, x7, x8) -> l1(x9, x10, x11) :|: x8 = x11 && x6 = x9 && x10 = 1 + x7
(4) l4(x12, x13, x14) -> l5(x15, x16, x17) :|: x14 = x17 && x12 = x15 && x13 = x16
(5) l5(x18, x19, x20) -> l3(x21, x22, x23) :|: x20 = x23 && x18 = x21 && x19 = x22
(6) l6(x24, x25, x26) -> l0(x27, x28, x29) :|: x26 = x29 && x24 = x27 && x25 = x28
(7) l7(x30, x31, x32) -> l4(x33, x34, x35) :|: x32 = x35 && x30 = x33 && x31 = x34
(8) l7(x36, x37, x38) -> l5(x39, x40, x41) :|: x38 = x41 && x36 = x39 && x37 = x40
(9) l8(x42, x43, x44) -> l7(x45, x46, x47) :|: x42 = x45 && x43 = x46 && x47 = x47
(10) l9(x48, x49, x50) -> l8(x51, x52, x53) :|: x50 = x53 && x48 = x51 && x49 = x52
(11) l9(x54, x55, x56) -> l3(x57, x58, x59) :|: x56 = x59 && x54 = x57 && x55 = x58
(12) l10(x60, x61, x62) -> l11(x63, x64, x65) :|: x62 = x65 && x60 = x63 && x61 = x64 && 10 <= x61
(13) l10(x66, x67, x68) -> l9(x69, x70, x71) :|: x68 = x71 && x66 = x69 && x67 = x70 && 1 + x67 <= 10
(14) l12(x72, x73, x74) -> l6(x75, x76, x77) :|: x74 = x77 && x72 = x75 && x76 = 1 + x73
(15) l1(x78, x79, x80) -> l10(x81, x82, x83) :|: x80 = x83 && x78 = x81 && x79 = x82
(16) l13(x84, x85, x86) -> l14(x87, x88, x89) :|: x86 = x89 && x84 = x87 && x85 = x88
(17) l14(x90, x91, x92) -> l12(x93, x94, x95) :|: x92 = x95 && x90 = x93 && x91 = x94
(18) l15(x96, x97, x98) -> l13(x99, x100, x101) :|: x98 = x101 && x96 = x99 && x97 = x100
(19) l15(x102, x103, x104) -> l14(x105, x106, x107) :|: x104 = x107 && x102 = x105 && x103 = x106
(20) l16(x108, x109, x110) -> l15(x111, x112, x113) :|: x110 = x113 && x108 = x111 && x109 = x112
(21) l2(x114, x115, x116) -> l16(x117, x118, x119) :|: x116 = x119 && x114 = x117 && x115 = x118
(22) l2(x120, x121, x122) -> l12(x123, x124, x125) :|: x122 = x125 && x120 = x123 && x121 = x124
(23) l17(x126, x127, x128) -> l6(x129, x130, x131) :|: x132 = 0 && x130 = 0 && x126 = x129 && x128 = x131
(24) l18(x133, x134, x135) -> l17(x136, x137, x138) :|: x135 = x138 && x133 = x136 && x134 = x137

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

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

(4)
Complex Obligation (AND)

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

(5)
Obligation:

Termination digraph:
Nodes:
(1) l0(x, x1, x2) -> l2(x3, x4, x5) :|: x2 = x5 && x1 = x4 && x3 = x1 && 1 + x1 <= 10
(2) l6(x24, x25, x26) -> l0(x27, x28, x29) :|: x26 = x29 && x24 = x27 && x25 = x28
(3) l12(x72, x73, x74) -> l6(x75, x76, x77) :|: x74 = x77 && x72 = x75 && x76 = 1 + x73
(4) l2(x120, x121, x122) -> l12(x123, x124, x125) :|: x122 = x125 && x120 = x123 && x121 = x124
(5) l14(x90, x91, x92) -> l12(x93, x94, x95) :|: x92 = x95 && x90 = x93 && x91 = x94
(6) l15(x102, x103, x104) -> l14(x105, x106, x107) :|: x104 = x107 && x102 = x105 && x103 = x106
(7) l13(x84, x85, x86) -> l14(x87, x88, x89) :|: x86 = x89 && x84 = x87 && x85 = x88
(8) l15(x96, x97, x98) -> l13(x99, x100, x101) :|: x98 = x101 && x96 = x99 && x97 = x100
(9) l16(x108, x109, x110) -> l15(x111, x112, x113) :|: x110 = x113 && x108 = x111 && x109 = x112
(10) l2(x114, x115, x116) -> l16(x117, x118, x119) :|: x116 = x119 && x114 = x117 && x115 = x118

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

This digraph is fully evaluated!

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

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

(7)
Obligation:
Rules:
l12(x27:0, x73:0, x107:0) -> l12(1 + x73:0, 1 + x73:0, x107:0) :|: x73:0 < 9

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

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

   l12(x1, x2, x3) -> l12(x2)

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

(9)
Obligation:
Rules:
l12(x73:0) -> l12(1 + x73:0) :|: x73:0 < 9

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

(10) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
l12(INTEGER)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(11)
Obligation:
Rules:
l12(x73:0) -> l12(c) :|: c = 1 + x73:0 && x73:0 < 9

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

(12) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l12(x)] = 8 - x

The following rules are decreasing:
l12(x73:0) -> l12(c) :|: c = 1 + x73:0 && x73:0 < 9
The following rules are bounded:
l12(x73:0) -> l12(c) :|: c = 1 + x73:0 && x73:0 < 9

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

(13)
YES

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

(14)
Obligation:

Termination digraph:
Nodes:
(1) l1(x78, x79, x80) -> l10(x81, x82, x83) :|: x80 = x83 && x78 = x81 && x79 = x82
(2) l3(x6, x7, x8) -> l1(x9, x10, x11) :|: x8 = x11 && x6 = x9 && x10 = 1 + x7
(3) l9(x54, x55, x56) -> l3(x57, x58, x59) :|: x56 = x59 && x54 = x57 && x55 = x58
(4) l5(x18, x19, x20) -> l3(x21, x22, x23) :|: x20 = x23 && x18 = x21 && x19 = x22
(5) l7(x36, x37, x38) -> l5(x39, x40, x41) :|: x38 = x41 && x36 = x39 && x37 = x40
(6) l4(x12, x13, x14) -> l5(x15, x16, x17) :|: x14 = x17 && x12 = x15 && x13 = x16
(7) l7(x30, x31, x32) -> l4(x33, x34, x35) :|: x32 = x35 && x30 = x33 && x31 = x34
(8) l8(x42, x43, x44) -> l7(x45, x46, x47) :|: x42 = x45 && x43 = x46 && x47 = x47
(9) l9(x48, x49, x50) -> l8(x51, x52, x53) :|: x50 = x53 && x48 = x51 && x49 = x52
(10) l10(x66, x67, x68) -> l9(x69, x70, x71) :|: x68 = x71 && x66 = x69 && x67 = x70 && 1 + x67 <= 10

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

This digraph is fully evaluated!

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

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

(16)
Obligation:
Rules:
l9(x54:0, x55:0, x11:0) -> l9(x54:0, 1 + x55:0, x11:0) :|: x55:0 < 9
l9(x, x1, x2) -> l9(x, 1 + x1, x3) :|: x1 < 9

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

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

   l9(x1, x2, x3) -> l9(x2)

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

(18)
Obligation:
Rules:
l9(x55:0) -> l9(1 + x55:0) :|: x55:0 < 9

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

(19) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
l9(INTEGER)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(20)
Obligation:
Rules:
l9(x55:0) -> l9(c) :|: c = 1 + x55:0 && x55:0 < 9

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

(21) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l9(x)] = 8 - x

The following rules are decreasing:
l9(x55:0) -> l9(c) :|: c = 1 + x55:0 && x55:0 < 9
The following rules are bounded:
l9(x55:0) -> l9(c) :|: c = 1 + x55:0 && x55:0 < 9

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

(22)
YES