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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 763 ms]
(4) AND
    (5) IRSwT
        (6) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (7) IRSwT
        (8) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (9) IRSwT
        (10) TempFilterProof [SOUND, 15 ms]
        (11) IntTRS
        (12) PolynomialOrderProcessor [EQUIVALENT, 3 ms]
        (13) YES
    (14) IRSwT
        (15) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (16) IRSwT
        (17) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (18) IRSwT
        (19) TempFilterProof [SOUND, 26 ms]
        (20) IntTRS
        (21) PolynomialOrderProcessor [EQUIVALENT, 12 ms]
        (22) YES
    (23) IRSwT
        (24) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (25) IRSwT
        (26) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (27) IRSwT
        (28) TempFilterProof [SOUND, 7 ms]
        (29) IntTRS
        (30) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
        (31) YES


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

(0)
Obligation:
Rules:
l0(i2HAT0, size1010HAT0, size1HAT0, size77HAT0, tmp1111HAT0, tmp88HAT0) -> l1(i2HATpost, size1010HATpost, size1HATpost, size77HATpost, tmp1111HATpost, tmp88HATpost) :|: tmp88HAT0 = tmp88HATpost && tmp1111HAT0 = tmp1111HATpost && size77HAT0 = size77HATpost && size1010HAT0 = size1010HATpost && size1HAT0 = size1HATpost && i2HAT0 = i2HATpost
l2(x, x1, x2, x3, x4, x5) -> l3(x6, x7, x8, x9, x10, x11) :|: x5 = x11 && x4 = x10 && x3 = x9 && x1 = x7 && x2 = x8 && x = x6 && x2 <= x
l2(x12, x13, x14, x15, x16, x17) -> l4(x18, x19, x20, x21, x22, x23) :|: x17 = x23 && x16 = x22 && x15 = x21 && x13 = x19 && x14 = x20 && x18 = 1 + x12 && 1 + x12 <= x14
l5(x24, x25, x26, x27, x28, x29) -> l6(x30, x31, x32, x33, x34, x35) :|: x29 = x35 && x28 = x34 && x27 = x33 && x25 = x31 && x26 = x32 && x24 = x30
l6(x36, x37, x38, x39, x40, x41) -> l4(x42, x43, x44, x45, x46, x47) :|: x41 = x47 && x40 = x46 && x39 = x45 && x37 = x43 && x38 = x44 && x42 = 0 && x38 <= x36
l6(x48, x49, x50, x51, x52, x53) -> l5(x54, x55, x56, x57, x58, x59) :|: x53 = x59 && x52 = x58 && x51 = x57 && x49 = x55 && x50 = x56 && x54 = 1 + x48 && 1 + x48 <= x50
l4(x60, x61, x62, x63, x64, x65) -> l2(x66, x67, x68, x69, x70, x71) :|: x65 = x71 && x64 = x70 && x63 = x69 && x61 = x67 && x62 = x68 && x60 = x66
l1(x72, x73, x74, x75, x76, x77) -> l5(x78, x79, x80, x81, x82, x83) :|: x77 = x83 && x76 = x82 && x75 = x81 && x73 = x79 && x74 = x80 && x78 = 0 && x74 <= x72
l1(x84, x85, x86, x87, x88, x89) -> l0(x90, x91, x92, x93, x94, x95) :|: x89 = x95 && x88 = x94 && x87 = x93 && x85 = x91 && x86 = x92 && x90 = 1 + x84 && 1 + x84 <= x86
l7(x96, x97, x98, x99, x100, x101) -> l0(x102, x103, x104, x105, x106, x107) :|: x102 = 0 && x106 = x106 && x103 = x104 && x107 = x107 && x105 = x104 && x104 = 10
l8(x108, x109, x110, x111, x112, x113) -> l7(x114, x115, x116, x117, x118, x119) :|: x113 = x119 && x112 = x118 && x111 = x117 && x109 = x115 && x110 = x116 && x108 = x114
Start term: l8(i2HAT0, size1010HAT0, size1HAT0, size77HAT0, tmp1111HAT0, tmp88HAT0)

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

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

(2)
Obligation:
Rules:
l0(i2HAT0, size1010HAT0, size1HAT0, size77HAT0, tmp1111HAT0, tmp88HAT0) -> l1(i2HATpost, size1010HATpost, size1HATpost, size77HATpost, tmp1111HATpost, tmp88HATpost) :|: tmp88HAT0 = tmp88HATpost && tmp1111HAT0 = tmp1111HATpost && size77HAT0 = size77HATpost && size1010HAT0 = size1010HATpost && size1HAT0 = size1HATpost && i2HAT0 = i2HATpost
l2(x, x1, x2, x3, x4, x5) -> l3(x6, x7, x8, x9, x10, x11) :|: x5 = x11 && x4 = x10 && x3 = x9 && x1 = x7 && x2 = x8 && x = x6 && x2 <= x
l2(x12, x13, x14, x15, x16, x17) -> l4(x18, x19, x20, x21, x22, x23) :|: x17 = x23 && x16 = x22 && x15 = x21 && x13 = x19 && x14 = x20 && x18 = 1 + x12 && 1 + x12 <= x14
l5(x24, x25, x26, x27, x28, x29) -> l6(x30, x31, x32, x33, x34, x35) :|: x29 = x35 && x28 = x34 && x27 = x33 && x25 = x31 && x26 = x32 && x24 = x30
l6(x36, x37, x38, x39, x40, x41) -> l4(x42, x43, x44, x45, x46, x47) :|: x41 = x47 && x40 = x46 && x39 = x45 && x37 = x43 && x38 = x44 && x42 = 0 && x38 <= x36
l6(x48, x49, x50, x51, x52, x53) -> l5(x54, x55, x56, x57, x58, x59) :|: x53 = x59 && x52 = x58 && x51 = x57 && x49 = x55 && x50 = x56 && x54 = 1 + x48 && 1 + x48 <= x50
l4(x60, x61, x62, x63, x64, x65) -> l2(x66, x67, x68, x69, x70, x71) :|: x65 = x71 && x64 = x70 && x63 = x69 && x61 = x67 && x62 = x68 && x60 = x66
l1(x72, x73, x74, x75, x76, x77) -> l5(x78, x79, x80, x81, x82, x83) :|: x77 = x83 && x76 = x82 && x75 = x81 && x73 = x79 && x74 = x80 && x78 = 0 && x74 <= x72
l1(x84, x85, x86, x87, x88, x89) -> l0(x90, x91, x92, x93, x94, x95) :|: x89 = x95 && x88 = x94 && x87 = x93 && x85 = x91 && x86 = x92 && x90 = 1 + x84 && 1 + x84 <= x86
l7(x96, x97, x98, x99, x100, x101) -> l0(x102, x103, x104, x105, x106, x107) :|: x102 = 0 && x106 = x106 && x103 = x104 && x107 = x107 && x105 = x104 && x104 = 10
l8(x108, x109, x110, x111, x112, x113) -> l7(x114, x115, x116, x117, x118, x119) :|: x113 = x119 && x112 = x118 && x111 = x117 && x109 = x115 && x110 = x116 && x108 = x114
Start term: l8(i2HAT0, size1010HAT0, size1HAT0, size77HAT0, tmp1111HAT0, tmp88HAT0)

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

(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(i2HAT0, size1010HAT0, size1HAT0, size77HAT0, tmp1111HAT0, tmp88HAT0) -> l1(i2HATpost, size1010HATpost, size1HATpost, size77HATpost, tmp1111HATpost, tmp88HATpost) :|: tmp88HAT0 = tmp88HATpost && tmp1111HAT0 = tmp1111HATpost && size77HAT0 = size77HATpost && size1010HAT0 = size1010HATpost && size1HAT0 = size1HATpost && i2HAT0 = i2HATpost
(2) l2(x, x1, x2, x3, x4, x5) -> l3(x6, x7, x8, x9, x10, x11) :|: x5 = x11 && x4 = x10 && x3 = x9 && x1 = x7 && x2 = x8 && x = x6 && x2 <= x
(3) l2(x12, x13, x14, x15, x16, x17) -> l4(x18, x19, x20, x21, x22, x23) :|: x17 = x23 && x16 = x22 && x15 = x21 && x13 = x19 && x14 = x20 && x18 = 1 + x12 && 1 + x12 <= x14
(4) l5(x24, x25, x26, x27, x28, x29) -> l6(x30, x31, x32, x33, x34, x35) :|: x29 = x35 && x28 = x34 && x27 = x33 && x25 = x31 && x26 = x32 && x24 = x30
(5) l6(x36, x37, x38, x39, x40, x41) -> l4(x42, x43, x44, x45, x46, x47) :|: x41 = x47 && x40 = x46 && x39 = x45 && x37 = x43 && x38 = x44 && x42 = 0 && x38 <= x36
(6) l6(x48, x49, x50, x51, x52, x53) -> l5(x54, x55, x56, x57, x58, x59) :|: x53 = x59 && x52 = x58 && x51 = x57 && x49 = x55 && x50 = x56 && x54 = 1 + x48 && 1 + x48 <= x50
(7) l4(x60, x61, x62, x63, x64, x65) -> l2(x66, x67, x68, x69, x70, x71) :|: x65 = x71 && x64 = x70 && x63 = x69 && x61 = x67 && x62 = x68 && x60 = x66
(8) l1(x72, x73, x74, x75, x76, x77) -> l5(x78, x79, x80, x81, x82, x83) :|: x77 = x83 && x76 = x82 && x75 = x81 && x73 = x79 && x74 = x80 && x78 = 0 && x74 <= x72
(9) l1(x84, x85, x86, x87, x88, x89) -> l0(x90, x91, x92, x93, x94, x95) :|: x89 = x95 && x88 = x94 && x87 = x93 && x85 = x91 && x86 = x92 && x90 = 1 + x84 && 1 + x84 <= x86
(10) l7(x96, x97, x98, x99, x100, x101) -> l0(x102, x103, x104, x105, x106, x107) :|: x102 = 0 && x106 = x106 && x103 = x104 && x107 = x107 && x105 = x104 && x104 = 10
(11) l8(x108, x109, x110, x111, x112, x113) -> l7(x114, x115, x116, x117, x118, x119) :|: x113 = x119 && x112 = x118 && x111 = x117 && x109 = x115 && x110 = x116 && x108 = x114

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

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

(4)
Complex Obligation (AND)

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

(5)
Obligation:

Termination digraph:
Nodes:
(1) l0(i2HAT0, size1010HAT0, size1HAT0, size77HAT0, tmp1111HAT0, tmp88HAT0) -> l1(i2HATpost, size1010HATpost, size1HATpost, size77HATpost, tmp1111HATpost, tmp88HATpost) :|: tmp88HAT0 = tmp88HATpost && tmp1111HAT0 = tmp1111HATpost && size77HAT0 = size77HATpost && size1010HAT0 = size1010HATpost && size1HAT0 = size1HATpost && i2HAT0 = i2HATpost
(2) l1(x84, x85, x86, x87, x88, x89) -> l0(x90, x91, x92, x93, x94, x95) :|: x89 = x95 && x88 = x94 && x87 = x93 && x85 = x91 && x86 = x92 && x90 = 1 + x84 && 1 + x84 <= x86

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

This digraph is fully evaluated!

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

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

(7)
Obligation:
Rules:
l0(i2HAT0:0, size1010HAT0:0, size1HAT0:0, size77HAT0:0, tmp1111HAT0:0, tmp88HAT0:0) -> l0(1 + i2HAT0:0, size1010HAT0:0, size1HAT0:0, size77HAT0:0, tmp1111HAT0:0, tmp88HAT0:0) :|: size1HAT0:0 >= 1 + i2HAT0:0

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

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

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

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

(9)
Obligation:
Rules:
l0(i2HAT0:0, size1HAT0:0) -> l0(1 + i2HAT0:0, size1HAT0:0) :|: size1HAT0:0 >= 1 + i2HAT0:0

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

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

(11)
Obligation:
Rules:
l0(i2HAT0:0, size1HAT0:0) -> l0(c, size1HAT0:0) :|: c = 1 + i2HAT0:0 && size1HAT0:0 >= 1 + i2HAT0:0

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

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

The following rules are decreasing:
l0(i2HAT0:0, size1HAT0:0) -> l0(c, size1HAT0:0) :|: c = 1 + i2HAT0:0 && size1HAT0:0 >= 1 + i2HAT0:0
The following rules are bounded:
l0(i2HAT0:0, size1HAT0:0) -> l0(c, size1HAT0:0) :|: c = 1 + i2HAT0:0 && size1HAT0:0 >= 1 + i2HAT0:0

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

(13)
YES

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

(14)
Obligation:

Termination digraph:
Nodes:
(1) l5(x24, x25, x26, x27, x28, x29) -> l6(x30, x31, x32, x33, x34, x35) :|: x29 = x35 && x28 = x34 && x27 = x33 && x25 = x31 && x26 = x32 && x24 = x30
(2) l6(x48, x49, x50, x51, x52, x53) -> l5(x54, x55, x56, x57, x58, x59) :|: x53 = x59 && x52 = x58 && x51 = x57 && x49 = x55 && x50 = x56 && x54 = 1 + x48 && 1 + x48 <= x50

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

This digraph is fully evaluated!

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

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

(16)
Obligation:
Rules:
l5(x24:0, x25:0, x26:0, x27:0, x28:0, x29:0) -> l5(1 + x24:0, x25:0, x26:0, x27:0, x28:0, x29:0) :|: x26:0 >= 1 + x24:0

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

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

   l5(x1, x2, x3, x4, x5, x6) -> l5(x1, x3)

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

(18)
Obligation:
Rules:
l5(x24:0, x26:0) -> l5(1 + x24:0, x26:0) :|: x26:0 >= 1 + x24:0

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

(19) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
l5(INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(20)
Obligation:
Rules:
l5(x24:0, x26:0) -> l5(c, x26:0) :|: c = 1 + x24:0 && x26:0 >= 1 + x24:0

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

(21) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l5(x, x1)] = -x + x1

The following rules are decreasing:
l5(x24:0, x26:0) -> l5(c, x26:0) :|: c = 1 + x24:0 && x26:0 >= 1 + x24:0
The following rules are bounded:
l5(x24:0, x26:0) -> l5(c, x26:0) :|: c = 1 + x24:0 && x26:0 >= 1 + x24:0

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

(22)
YES

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

(23)
Obligation:

Termination digraph:
Nodes:
(1) l4(x60, x61, x62, x63, x64, x65) -> l2(x66, x67, x68, x69, x70, x71) :|: x65 = x71 && x64 = x70 && x63 = x69 && x61 = x67 && x62 = x68 && x60 = x66
(2) l2(x12, x13, x14, x15, x16, x17) -> l4(x18, x19, x20, x21, x22, x23) :|: x17 = x23 && x16 = x22 && x15 = x21 && x13 = x19 && x14 = x20 && x18 = 1 + x12 && 1 + x12 <= x14

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

This digraph is fully evaluated!

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

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

(25)
Obligation:
Rules:
l4(x60:0, x19:0, x20:0, x21:0, x22:0, x23:0) -> l4(1 + x60:0, x19:0, x20:0, x21:0, x22:0, x23:0) :|: x20:0 >= 1 + x60:0

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

(26) 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) -> l4(x1, x3)

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

(27)
Obligation:
Rules:
l4(x60:0, x20:0) -> l4(1 + x60:0, x20:0) :|: x20:0 >= 1 + x60:0

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

(28) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
l4(INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(29)
Obligation:
Rules:
l4(x60:0, x20:0) -> l4(c, x20:0) :|: c = 1 + x60:0 && x20:0 >= 1 + x60:0

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

(30) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l4(x, x1)] = -x + x1

The following rules are decreasing:
l4(x60:0, x20:0) -> l4(c, x20:0) :|: c = 1 + x60:0 && x20:0 >= 1 + x60:0
The following rules are bounded:
l4(x60:0, x20:0) -> l4(c, x20:0) :|: c = 1 + x60:0 && x20:0 >= 1 + x60:0

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

(31)
YES