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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 127 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 11 ms]
(6) IRSwT
(7) TempFilterProof [SOUND, 14 ms]
(8) IntTRS
(9) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
(10) YES


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

(0)
Obligation:
Rules:
l0(iHAT0) -> l1(iHATpost) :|: iHAT0 = iHATpost && 3 <= iHAT0
l0(x) -> l2(x1) :|: x1 = 1 + x && 1 + x <= 3
l3(x2) -> l0(x3) :|: x2 = x3
l3(x4) -> l1(x5) :|: x4 = x5
l2(x6) -> l3(x7) :|: x6 = x7
l4(x8) -> l5(x9) :|: x8 = x9
l1(x10) -> l6(x11) :|: x10 = x11
l6(x12) -> l4(x13) :|: x12 = x13 && 1 + x12 <= 2
l6(x14) -> l4(x15) :|: x14 = x15 && 2 <= x14
l7(x16) -> l2(x17) :|: x17 = 0
l8(x18) -> l7(x19) :|: x18 = x19
Start term: l8(iHAT0)

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

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

(2)
Obligation:
Rules:
l0(iHAT0) -> l1(iHATpost) :|: iHAT0 = iHATpost && 3 <= iHAT0
l0(x) -> l2(x1) :|: x1 = 1 + x && 1 + x <= 3
l3(x2) -> l0(x3) :|: x2 = x3
l3(x4) -> l1(x5) :|: x4 = x5
l2(x6) -> l3(x7) :|: x6 = x7
l4(x8) -> l5(x9) :|: x8 = x9
l1(x10) -> l6(x11) :|: x10 = x11
l6(x12) -> l4(x13) :|: x12 = x13 && 1 + x12 <= 2
l6(x14) -> l4(x15) :|: x14 = x15 && 2 <= x14
l7(x16) -> l2(x17) :|: x17 = 0
l8(x18) -> l7(x19) :|: x18 = x19
Start term: l8(iHAT0)

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

(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(iHAT0) -> l1(iHATpost) :|: iHAT0 = iHATpost && 3 <= iHAT0
(2) l0(x) -> l2(x1) :|: x1 = 1 + x && 1 + x <= 3
(3) l3(x2) -> l0(x3) :|: x2 = x3
(4) l3(x4) -> l1(x5) :|: x4 = x5
(5) l2(x6) -> l3(x7) :|: x6 = x7
(6) l4(x8) -> l5(x9) :|: x8 = x9
(7) l1(x10) -> l6(x11) :|: x10 = x11
(8) l6(x12) -> l4(x13) :|: x12 = x13 && 1 + x12 <= 2
(9) l6(x14) -> l4(x15) :|: x14 = x15 && 2 <= x14
(10) l7(x16) -> l2(x17) :|: x17 = 0
(11) l8(x18) -> l7(x19) :|: x18 = x19

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

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

(4)
Obligation:

Termination digraph:
Nodes:
(1) l0(x) -> l2(x1) :|: x1 = 1 + x && 1 + x <= 3
(2) l3(x2) -> l0(x3) :|: x2 = x3
(3) l2(x6) -> l3(x7) :|: x6 = x7

Arcs:
(1) -> (3)
(2) -> (1)
(3) -> (2)

This digraph is fully evaluated!

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

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

(6)
Obligation:
Rules:
l3(x2:0) -> l3(1 + x2:0) :|: x2:0 < 3

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

(7) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
l3(INTEGER)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(8)
Obligation:
Rules:
l3(x2:0) -> l3(c) :|: c = 1 + x2:0 && x2:0 < 3

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

(9) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[l3(x)] = 2 - x

The following rules are decreasing:
l3(x2:0) -> l3(c) :|: c = 1 + x2:0 && x2:0 < 3
The following rules are bounded:
l3(x2:0) -> l3(c) :|: c = 1 + x2:0 && x2:0 < 3

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

(10)
YES