The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).

0 CpxTRS
    ↳1 CpxTrsMatchBoundsTAProof (⇔, 112 ms)
    ↳2 BOUNDS(1, n^1)
    ↳3 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳4 CdtProblem
        ↳5 CdtLeafRemovalProof (ComplexityIfPolyImplication, 0 ms)
        ↳6 CdtProblem
        ↳7 CdtUsableRulesProof (⇔, 0 ms)
        ↳8 CdtProblem
        ↳9 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 87 ms)
        ↳10 CdtProblem
        ↳11 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 18 ms)
        ↳12 CdtProblem
        ↳13 SIsEmptyProof (BOTH BOUNDS(ID, ID), 0 ms)
        ↳14 BOUNDS(1, 1)

(0) Obligation:

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

@(Cons(x, xs), ys) → Cons(x, @(xs, ys))
@(Nil, ys) → ys
game(p1, Cons(x', xs'), Cons(Capture, xs)) → game(Cons(x', p1), xs', xs)
game(p1, p2, Cons(Swap, xs)) → game(p2, p1, xs)
equal(Capture, Capture) → True
equal(Capture, Swap) → False
equal(Swap, Capture) → False
equal(Swap, Swap) → True
game(p1, p2, Nil) → @(p1, p2)
goal(p1, p2, moves) → game(p1, p2, moves)

Rewrite Strategy: INNERMOST

(1) CpxTrsMatchBoundsTAProof (EQUIVALENT transformation)

A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 2.

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by:
final states : [1, 2, 3, 4]
transitions:
Cons0(0, 0) → 0
Nil0() → 0
Capture0() → 0
Swap0() → 0
True0() → 0
False0() → 0
@0(0, 0) → 1
game0(0, 0, 0) → 2
equal0(0, 0) → 3
goal0(0, 0, 0) → 4
@1(0, 0) → 5
Cons1(0, 5) → 1
Cons1(0, 0) → 6
game1(6, 0, 0) → 2
game1(0, 0, 0) → 2
True1() → 3
False1() → 3
@1(0, 0) → 2
game1(0, 0, 0) → 4
Cons1(0, 5) → 2
Cons1(0, 5) → 5
Cons1(0, 6) → 6
game1(6, 0, 0) → 4
game1(0, 6, 0) → 2
@1(6, 0) → 2
@1(0, 0) → 4
Cons1(0, 5) → 4
@2(0, 0) → 7
Cons2(0, 7) → 2
@2(6, 0) → 7
game1(6, 6, 0) → 2
game1(0, 6, 0) → 4
@1(0, 6) → 2
@1(6, 0) → 4
@1(0, 6) → 5
Cons2(0, 7) → 4
game1(6, 6, 0) → 4
@1(6, 6) → 2
@1(0, 6) → 4
Cons1(0, 5) → 7
Cons2(0, 7) → 7
@2(0, 6) → 7
@2(6, 6) → 7
@1(6, 6) → 4
0 → 1
0 → 2
0 → 5
0 → 4
0 → 7
6 → 2
6 → 4
6 → 5
6 → 7

(2) BOUNDS(1, n^1)

(3) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted Cpx (relative) TRS to CDT

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

@(Cons(z0, z1), z2) → Cons(z0, @(z1, z2))
@(Nil, z0) → z0
game(z0, Cons(z1, z2), Cons(Capture, z3)) → game(Cons(z1, z0), z2, z3)
game(z0, z1, Cons(Swap, z2)) → game(z1, z0, z2)
game(z0, z1, Nil) → @(z0, z1)
equal(Capture, Capture) → True
equal(Capture, Swap) → False
equal(Swap, Capture) → False
equal(Swap, Swap) → True
goal(z0, z1, z2) → game(z0, z1, z2)
Tuples:

@'(Cons(z0, z1), z2) → c(@'(z1, z2))
@'(Nil, z0) → c1
GAME(z0, Cons(z1, z2), Cons(Capture, z3)) → c2(GAME(Cons(z1, z0), z2, z3))
GAME(z0, z1, Cons(Swap, z2)) → c3(GAME(z1, z0, z2))
GAME(z0, z1, Nil) → c4(@'(z0, z1))
EQUAL(Capture, Capture) → c5
EQUAL(Capture, Swap) → c6
EQUAL(Swap, Capture) → c7
EQUAL(Swap, Swap) → c8
GOAL(z0, z1, z2) → c9(GAME(z0, z1, z2))
S tuples:

@'(Cons(z0, z1), z2) → c(@'(z1, z2))
@'(Nil, z0) → c1
GAME(z0, Cons(z1, z2), Cons(Capture, z3)) → c2(GAME(Cons(z1, z0), z2, z3))
GAME(z0, z1, Cons(Swap, z2)) → c3(GAME(z1, z0, z2))
GAME(z0, z1, Nil) → c4(@'(z0, z1))
EQUAL(Capture, Capture) → c5
EQUAL(Capture, Swap) → c6
EQUAL(Swap, Capture) → c7
EQUAL(Swap, Swap) → c8
GOAL(z0, z1, z2) → c9(GAME(z0, z1, z2))
K tuples:none
Defined Rule Symbols:

@, game, equal, goal

Defined Pair Symbols:

@', GAME, EQUAL, GOAL

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9

(5) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 1 leading nodes:

GOAL(z0, z1, z2) → c9(GAME(z0, z1, z2))
Removed 5 trailing nodes:

EQUAL(Swap, Capture) → c7
EQUAL(Capture, Capture) → c5
@'(Nil, z0) → c1
EQUAL(Capture, Swap) → c6
EQUAL(Swap, Swap) → c8

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

@(Cons(z0, z1), z2) → Cons(z0, @(z1, z2))
@(Nil, z0) → z0
game(z0, Cons(z1, z2), Cons(Capture, z3)) → game(Cons(z1, z0), z2, z3)
game(z0, z1, Cons(Swap, z2)) → game(z1, z0, z2)
game(z0, z1, Nil) → @(z0, z1)
equal(Capture, Capture) → True
equal(Capture, Swap) → False
equal(Swap, Capture) → False
equal(Swap, Swap) → True
goal(z0, z1, z2) → game(z0, z1, z2)
Tuples:

@'(Cons(z0, z1), z2) → c(@'(z1, z2))
GAME(z0, Cons(z1, z2), Cons(Capture, z3)) → c2(GAME(Cons(z1, z0), z2, z3))
GAME(z0, z1, Cons(Swap, z2)) → c3(GAME(z1, z0, z2))
GAME(z0, z1, Nil) → c4(@'(z0, z1))
S tuples:

@'(Cons(z0, z1), z2) → c(@'(z1, z2))
GAME(z0, Cons(z1, z2), Cons(Capture, z3)) → c2(GAME(Cons(z1, z0), z2, z3))
GAME(z0, z1, Cons(Swap, z2)) → c3(GAME(z1, z0, z2))
GAME(z0, z1, Nil) → c4(@'(z0, z1))
K tuples:none
Defined Rule Symbols:

@, game, equal, goal

Defined Pair Symbols:

@', GAME

Compound Symbols:

c, c2, c3, c4

(7) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

@(Cons(z0, z1), z2) → Cons(z0, @(z1, z2))
@(Nil, z0) → z0
game(z0, Cons(z1, z2), Cons(Capture, z3)) → game(Cons(z1, z0), z2, z3)
game(z0, z1, Cons(Swap, z2)) → game(z1, z0, z2)
game(z0, z1, Nil) → @(z0, z1)
equal(Capture, Capture) → True
equal(Capture, Swap) → False
equal(Swap, Capture) → False
equal(Swap, Swap) → True
goal(z0, z1, z2) → game(z0, z1, z2)

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

@'(Cons(z0, z1), z2) → c(@'(z1, z2))
GAME(z0, Cons(z1, z2), Cons(Capture, z3)) → c2(GAME(Cons(z1, z0), z2, z3))
GAME(z0, z1, Cons(Swap, z2)) → c3(GAME(z1, z0, z2))
GAME(z0, z1, Nil) → c4(@'(z0, z1))
S tuples:

@'(Cons(z0, z1), z2) → c(@'(z1, z2))
GAME(z0, Cons(z1, z2), Cons(Capture, z3)) → c2(GAME(Cons(z1, z0), z2, z3))
GAME(z0, z1, Cons(Swap, z2)) → c3(GAME(z1, z0, z2))
GAME(z0, z1, Nil) → c4(@'(z0, z1))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

@', GAME

Compound Symbols:

c, c2, c3, c4

(9) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

GAME(z0, z1, Nil) → c4(@'(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:

@'(Cons(z0, z1), z2) → c(@'(z1, z2))
GAME(z0, Cons(z1, z2), Cons(Capture, z3)) → c2(GAME(Cons(z1, z0), z2, z3))
GAME(z0, z1, Cons(Swap, z2)) → c3(GAME(z1, z0, z2))
GAME(z0, z1, Nil) → c4(@'(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(@'(x1, x2)) = 0   
POL(Capture) = 0   
POL(Cons(x1, x2)) = 0   
POL(GAME(x1, x2, x3)) = [2]   
POL(Nil) = 0   
POL(Swap) = 0   
POL(c(x1)) = x1   
POL(c2(x1)) = x1   
POL(c3(x1)) = x1   
POL(c4(x1)) = x1   

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

@'(Cons(z0, z1), z2) → c(@'(z1, z2))
GAME(z0, Cons(z1, z2), Cons(Capture, z3)) → c2(GAME(Cons(z1, z0), z2, z3))
GAME(z0, z1, Cons(Swap, z2)) → c3(GAME(z1, z0, z2))
GAME(z0, z1, Nil) → c4(@'(z0, z1))
S tuples:

@'(Cons(z0, z1), z2) → c(@'(z1, z2))
GAME(z0, Cons(z1, z2), Cons(Capture, z3)) → c2(GAME(Cons(z1, z0), z2, z3))
GAME(z0, z1, Cons(Swap, z2)) → c3(GAME(z1, z0, z2))
K tuples:

GAME(z0, z1, Nil) → c4(@'(z0, z1))
Defined Rule Symbols:none

Defined Pair Symbols:

@', GAME

Compound Symbols:

c, c2, c3, c4

(11) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

@'(Cons(z0, z1), z2) → c(@'(z1, z2))
GAME(z0, Cons(z1, z2), Cons(Capture, z3)) → c2(GAME(Cons(z1, z0), z2, z3))
GAME(z0, z1, Cons(Swap, z2)) → c3(GAME(z1, z0, z2))
We considered the (Usable) Rules:none
And the Tuples:

@'(Cons(z0, z1), z2) → c(@'(z1, z2))
GAME(z0, Cons(z1, z2), Cons(Capture, z3)) → c2(GAME(Cons(z1, z0), z2, z3))
GAME(z0, z1, Cons(Swap, z2)) → c3(GAME(z1, z0, z2))
GAME(z0, z1, Nil) → c4(@'(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(@'(x1, x2)) = [2]x1 + [3]x2   
POL(Capture) = [2]   
POL(Cons(x1, x2)) = [2] + x1 + x2   
POL(GAME(x1, x2, x3)) = [1] + [3]x1 + [3]x2 + [2]x3   
POL(Nil) = 0   
POL(Swap) = 0   
POL(c(x1)) = x1   
POL(c2(x1)) = x1   
POL(c3(x1)) = x1   
POL(c4(x1)) = x1   

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

@'(Cons(z0, z1), z2) → c(@'(z1, z2))
GAME(z0, Cons(z1, z2), Cons(Capture, z3)) → c2(GAME(Cons(z1, z0), z2, z3))
GAME(z0, z1, Cons(Swap, z2)) → c3(GAME(z1, z0, z2))
GAME(z0, z1, Nil) → c4(@'(z0, z1))
S tuples:none
K tuples:

GAME(z0, z1, Nil) → c4(@'(z0, z1))
@'(Cons(z0, z1), z2) → c(@'(z1, z2))
GAME(z0, Cons(z1, z2), Cons(Capture, z3)) → c2(GAME(Cons(z1, z0), z2, z3))
GAME(z0, z1, Cons(Swap, z2)) → c3(GAME(z1, z0, z2))
Defined Rule Symbols:none

Defined Pair Symbols:

@', GAME

Compound Symbols:

c, c2, c3, c4

(13) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)

The set S is empty

(14) BOUNDS(1, 1)