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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 355 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 42 ms]
(6) IRSwT
(7) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
(8) IRSwT
(9) TempFilterProof [SOUND, 14 ms]
(10) IntTRS
(11) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
(12) YES


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

(0)
Obligation:
Rules:
l0(oldX0HAT0, oldX1HAT0, x0HAT0) -> l1(oldX0HATpost, oldX1HATpost, x0HATpost) :|: x0HATpost = oldX1HATpost && oldX1HATpost = oldX1HATpost && oldX0HATpost = x0HAT0
l2(x, x1, x2) -> l3(x3, x4, x5) :|: x1 = x4 && x5 = 1 + x3 && x3 = x2
l3(x6, x7, x8) -> l0(x9, x10, x11) :|: x7 = x10 && x11 = x9 && 20 <= x9 && x9 = x8
l3(x12, x13, x14) -> l2(x15, x16, x17) :|: x13 = x16 && x17 = x15 && x15 <= 19 && x15 = x14
l4(x18, x19, x20) -> l3(x21, x22, x23) :|: x19 = x22 && x23 = 0 && x21 = x20
l5(x24, x25, x26) -> l4(x27, x28, x29) :|: x29 = x28 && x28 = x28 && x27 = x26
l5(x30, x31, x32) -> l1(x33, x34, x35) :|: x32 = x35 && x31 = x34 && x30 = x33
l5(x36, x37, x38) -> l0(x39, x40, x41) :|: x38 = x41 && x37 = x40 && x36 = x39
l5(x42, x43, x44) -> l2(x45, x46, x47) :|: x44 = x47 && x43 = x46 && x42 = x45
l5(x48, x49, x50) -> l3(x51, x52, x53) :|: x50 = x53 && x49 = x52 && x48 = x51
l5(x54, x55, x56) -> l4(x57, x58, x59) :|: x56 = x59 && x55 = x58 && x54 = x57
l6(x60, x61, x62) -> l5(x63, x64, x65) :|: x62 = x65 && x61 = x64 && x60 = x63
Start term: l6(oldX0HAT0, oldX1HAT0, x0HAT0)

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

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

(2)
Obligation:
Rules:
l0(oldX0HAT0, oldX1HAT0, x0HAT0) -> l1(oldX0HATpost, oldX1HATpost, x0HATpost) :|: x0HATpost = oldX1HATpost && oldX1HATpost = oldX1HATpost && oldX0HATpost = x0HAT0
l2(x, x1, x2) -> l3(x3, x4, x5) :|: x1 = x4 && x5 = 1 + x3 && x3 = x2
l3(x6, x7, x8) -> l0(x9, x10, x11) :|: x7 = x10 && x11 = x9 && 20 <= x9 && x9 = x8
l3(x12, x13, x14) -> l2(x15, x16, x17) :|: x13 = x16 && x17 = x15 && x15 <= 19 && x15 = x14
l4(x18, x19, x20) -> l3(x21, x22, x23) :|: x19 = x22 && x23 = 0 && x21 = x20
l5(x24, x25, x26) -> l4(x27, x28, x29) :|: x29 = x28 && x28 = x28 && x27 = x26
l5(x30, x31, x32) -> l1(x33, x34, x35) :|: x32 = x35 && x31 = x34 && x30 = x33
l5(x36, x37, x38) -> l0(x39, x40, x41) :|: x38 = x41 && x37 = x40 && x36 = x39
l5(x42, x43, x44) -> l2(x45, x46, x47) :|: x44 = x47 && x43 = x46 && x42 = x45
l5(x48, x49, x50) -> l3(x51, x52, x53) :|: x50 = x53 && x49 = x52 && x48 = x51
l5(x54, x55, x56) -> l4(x57, x58, x59) :|: x56 = x59 && x55 = x58 && x54 = x57
l6(x60, x61, x62) -> l5(x63, x64, x65) :|: x62 = x65 && x61 = x64 && x60 = x63
Start term: l6(oldX0HAT0, oldX1HAT0, x0HAT0)

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

(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(oldX0HAT0, oldX1HAT0, x0HAT0) -> l1(oldX0HATpost, oldX1HATpost, x0HATpost) :|: x0HATpost = oldX1HATpost && oldX1HATpost = oldX1HATpost && oldX0HATpost = x0HAT0
(2) l2(x, x1, x2) -> l3(x3, x4, x5) :|: x1 = x4 && x5 = 1 + x3 && x3 = x2
(3) l3(x6, x7, x8) -> l0(x9, x10, x11) :|: x7 = x10 && x11 = x9 && 20 <= x9 && x9 = x8
(4) l3(x12, x13, x14) -> l2(x15, x16, x17) :|: x13 = x16 && x17 = x15 && x15 <= 19 && x15 = x14
(5) l4(x18, x19, x20) -> l3(x21, x22, x23) :|: x19 = x22 && x23 = 0 && x21 = x20
(6) l5(x24, x25, x26) -> l4(x27, x28, x29) :|: x29 = x28 && x28 = x28 && x27 = x26
(7) l5(x30, x31, x32) -> l1(x33, x34, x35) :|: x32 = x35 && x31 = x34 && x30 = x33
(8) l5(x36, x37, x38) -> l0(x39, x40, x41) :|: x38 = x41 && x37 = x40 && x36 = x39
(9) l5(x42, x43, x44) -> l2(x45, x46, x47) :|: x44 = x47 && x43 = x46 && x42 = x45
(10) l5(x48, x49, x50) -> l3(x51, x52, x53) :|: x50 = x53 && x49 = x52 && x48 = x51
(11) l5(x54, x55, x56) -> l4(x57, x58, x59) :|: x56 = x59 && x55 = x58 && x54 = x57
(12) l6(x60, x61, x62) -> l5(x63, x64, x65) :|: x62 = x65 && x61 = x64 && x60 = x63

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

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

(4)
Obligation:

Termination digraph:
Nodes:
(1) l2(x, x1, x2) -> l3(x3, x4, x5) :|: x1 = x4 && x5 = 1 + x3 && x3 = x2
(2) l3(x12, x13, x14) -> l2(x15, x16, x17) :|: x13 = x16 && x17 = x15 && x15 <= 19 && x15 = x14

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

This digraph is fully evaluated!

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

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

(6)
Obligation:
Rules:
l2(x:0, x16:0, x2:0) -> l2(1 + x2:0, x16:0, 1 + x2:0) :|: x2:0 < 19

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

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

   l2(x1, x2, x3) -> l2(x3)

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

(8)
Obligation:
Rules:
l2(x2:0) -> l2(1 + x2:0) :|: x2:0 < 19

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

(9) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
l2(INTEGER)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(10)
Obligation:
Rules:
l2(x2:0) -> l2(c) :|: c = 1 + x2:0 && x2:0 < 19

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

(11) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l2(x)] = 18 - x

The following rules are decreasing:
l2(x2:0) -> l2(c) :|: c = 1 + x2:0 && x2:0 < 19
The following rules are bounded:
l2(x2:0) -> l2(c) :|: c = 1 + x2:0 && x2:0 < 19

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

(12)
YES