WORST_CASE(Omega(n^1),O(n^1))
proof of input_d7AGwh3hIM.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


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

(0) CpxTRS
(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(2) CpxTRS
    (3) CpxTrsMatchBoundsTAProof [FINISHED, 80 ms]
    (4) BOUNDS(1, n^1)
(5) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CdtProblem
    (7) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxRelTRS
    (9) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CpxRelTRS
    (11) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) typed CpxTrs
    (13) OrderProof [LOWER BOUND(ID), 5 ms]
    (14) typed CpxTrs
    (15) RewriteLemmaProof [LOWER BOUND(ID), 284 ms]
    (16) BEST
        (17) proven lower bound
            (18) LowerBoundPropagationProof [FINISHED, 0 ms]
            (19) BOUNDS(n^1, INF)
        (20) typed CpxTrs


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

(0)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(n^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)

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(1) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
----------------------------------------

(2)
Obligation:
The Runtime Complexity (parallel-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)

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(3) CpxTrsMatchBoundsTAProof (FINISHED)
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

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

(4)
BOUNDS(1, n^1)

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

(5) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(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))
   @'(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:   @_2, game_3, equal_2, goal_3

Defined Pair Symbols:   @'_2, GAME_3, EQUAL_2, GOAL_3

Compound Symbols:   c_1, c1, c2_1, c3_1, c4_1, c5, c6, c7, c8, c9_1


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

(7) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID))
Converted S to standard rules, and D \ S as well as R to relative rules.
----------------------------------------

(8)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   @'(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))

The (relative) TRS S consists of the following 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)

Rewrite Strategy: INNERMOST
----------------------------------------

(9) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(10)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   @'(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))

The (relative) TRS S consists of the following 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)

Rewrite Strategy: INNERMOST
----------------------------------------

(11) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Inferred types.
----------------------------------------

(12)
Obligation:
Innermost TRS:
Rules:
@'(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))
@(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)

Types:
@' :: Cons:Nil -> Cons:Nil -> c:c1
Cons :: Capture:Swap -> Cons:Nil -> Cons:Nil
c :: c:c1 -> c:c1
Nil :: Cons:Nil
c1 :: c:c1
GAME :: Cons:Nil -> Cons:Nil -> Cons:Nil -> c2:c3:c4
Capture :: Capture:Swap
c2 :: c2:c3:c4 -> c2:c3:c4
Swap :: Capture:Swap
c3 :: c2:c3:c4 -> c2:c3:c4
c4 :: c:c1 -> c2:c3:c4
EQUAL :: Capture:Swap -> Capture:Swap -> c5:c6:c7:c8
c5 :: c5:c6:c7:c8
c6 :: c5:c6:c7:c8
c7 :: c5:c6:c7:c8
c8 :: c5:c6:c7:c8
GOAL :: Cons:Nil -> Cons:Nil -> Cons:Nil -> c9
c9 :: c2:c3:c4 -> c9
@ :: Cons:Nil -> Cons:Nil -> Cons:Nil
game :: Cons:Nil -> Cons:Nil -> Cons:Nil -> Cons:Nil
equal :: Capture:Swap -> Capture:Swap -> True:False
True :: True:False
False :: True:False
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil -> Cons:Nil
hole_c:c11_10 :: c:c1
hole_Cons:Nil2_10 :: Cons:Nil
hole_Capture:Swap3_10 :: Capture:Swap
hole_c2:c3:c44_10 :: c2:c3:c4
hole_c5:c6:c7:c85_10 :: c5:c6:c7:c8
hole_c96_10 :: c9
hole_True:False7_10 :: True:False
gen_c:c18_10 :: Nat -> c:c1
gen_Cons:Nil9_10 :: Nat -> Cons:Nil
gen_c2:c3:c410_10 :: Nat -> c2:c3:c4

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

(13) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
@', GAME, @, game

They will be analysed ascendingly in the following order:
@' < GAME
@ < game

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

(14)
Obligation:
Innermost TRS:
Rules:
@'(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))
@(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)

Types:
@' :: Cons:Nil -> Cons:Nil -> c:c1
Cons :: Capture:Swap -> Cons:Nil -> Cons:Nil
c :: c:c1 -> c:c1
Nil :: Cons:Nil
c1 :: c:c1
GAME :: Cons:Nil -> Cons:Nil -> Cons:Nil -> c2:c3:c4
Capture :: Capture:Swap
c2 :: c2:c3:c4 -> c2:c3:c4
Swap :: Capture:Swap
c3 :: c2:c3:c4 -> c2:c3:c4
c4 :: c:c1 -> c2:c3:c4
EQUAL :: Capture:Swap -> Capture:Swap -> c5:c6:c7:c8
c5 :: c5:c6:c7:c8
c6 :: c5:c6:c7:c8
c7 :: c5:c6:c7:c8
c8 :: c5:c6:c7:c8
GOAL :: Cons:Nil -> Cons:Nil -> Cons:Nil -> c9
c9 :: c2:c3:c4 -> c9
@ :: Cons:Nil -> Cons:Nil -> Cons:Nil
game :: Cons:Nil -> Cons:Nil -> Cons:Nil -> Cons:Nil
equal :: Capture:Swap -> Capture:Swap -> True:False
True :: True:False
False :: True:False
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil -> Cons:Nil
hole_c:c11_10 :: c:c1
hole_Cons:Nil2_10 :: Cons:Nil
hole_Capture:Swap3_10 :: Capture:Swap
hole_c2:c3:c44_10 :: c2:c3:c4
hole_c5:c6:c7:c85_10 :: c5:c6:c7:c8
hole_c96_10 :: c9
hole_True:False7_10 :: True:False
gen_c:c18_10 :: Nat -> c:c1
gen_Cons:Nil9_10 :: Nat -> Cons:Nil
gen_c2:c3:c410_10 :: Nat -> c2:c3:c4


Generator Equations:
gen_c:c18_10(0) <=> c1
gen_c:c18_10(+(x, 1)) <=> c(gen_c:c18_10(x))
gen_Cons:Nil9_10(0) <=> Nil
gen_Cons:Nil9_10(+(x, 1)) <=> Cons(Capture, gen_Cons:Nil9_10(x))
gen_c2:c3:c410_10(0) <=> c4(c1)
gen_c2:c3:c410_10(+(x, 1)) <=> c2(gen_c2:c3:c410_10(x))


The following defined symbols remain to be analysed:
@', GAME, @, game

They will be analysed ascendingly in the following order:
@' < GAME
@ < game

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

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
@'(gen_Cons:Nil9_10(n12_10), gen_Cons:Nil9_10(b)) -> gen_c:c18_10(n12_10), rt in Omega(1 + n12_10)

Induction Base:
@'(gen_Cons:Nil9_10(0), gen_Cons:Nil9_10(b)) ->_R^Omega(1)
c1

Induction Step:
@'(gen_Cons:Nil9_10(+(n12_10, 1)), gen_Cons:Nil9_10(b)) ->_R^Omega(1)
c(@'(gen_Cons:Nil9_10(n12_10), gen_Cons:Nil9_10(b))) ->_IH
c(gen_c:c18_10(c13_10))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(16)
Complex Obligation (BEST)

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

(17)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
@'(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))
@(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)

Types:
@' :: Cons:Nil -> Cons:Nil -> c:c1
Cons :: Capture:Swap -> Cons:Nil -> Cons:Nil
c :: c:c1 -> c:c1
Nil :: Cons:Nil
c1 :: c:c1
GAME :: Cons:Nil -> Cons:Nil -> Cons:Nil -> c2:c3:c4
Capture :: Capture:Swap
c2 :: c2:c3:c4 -> c2:c3:c4
Swap :: Capture:Swap
c3 :: c2:c3:c4 -> c2:c3:c4
c4 :: c:c1 -> c2:c3:c4
EQUAL :: Capture:Swap -> Capture:Swap -> c5:c6:c7:c8
c5 :: c5:c6:c7:c8
c6 :: c5:c6:c7:c8
c7 :: c5:c6:c7:c8
c8 :: c5:c6:c7:c8
GOAL :: Cons:Nil -> Cons:Nil -> Cons:Nil -> c9
c9 :: c2:c3:c4 -> c9
@ :: Cons:Nil -> Cons:Nil -> Cons:Nil
game :: Cons:Nil -> Cons:Nil -> Cons:Nil -> Cons:Nil
equal :: Capture:Swap -> Capture:Swap -> True:False
True :: True:False
False :: True:False
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil -> Cons:Nil
hole_c:c11_10 :: c:c1
hole_Cons:Nil2_10 :: Cons:Nil
hole_Capture:Swap3_10 :: Capture:Swap
hole_c2:c3:c44_10 :: c2:c3:c4
hole_c5:c6:c7:c85_10 :: c5:c6:c7:c8
hole_c96_10 :: c9
hole_True:False7_10 :: True:False
gen_c:c18_10 :: Nat -> c:c1
gen_Cons:Nil9_10 :: Nat -> Cons:Nil
gen_c2:c3:c410_10 :: Nat -> c2:c3:c4


Generator Equations:
gen_c:c18_10(0) <=> c1
gen_c:c18_10(+(x, 1)) <=> c(gen_c:c18_10(x))
gen_Cons:Nil9_10(0) <=> Nil
gen_Cons:Nil9_10(+(x, 1)) <=> Cons(Capture, gen_Cons:Nil9_10(x))
gen_c2:c3:c410_10(0) <=> c4(c1)
gen_c2:c3:c410_10(+(x, 1)) <=> c2(gen_c2:c3:c410_10(x))


The following defined symbols remain to be analysed:
@', GAME, @, game

They will be analysed ascendingly in the following order:
@' < GAME
@ < game

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

(18) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(19)
BOUNDS(n^1, INF)

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

(20)
Obligation:
Innermost TRS:
Rules:
@'(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))
@(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)

Types:
@' :: Cons:Nil -> Cons:Nil -> c:c1
Cons :: Capture:Swap -> Cons:Nil -> Cons:Nil
c :: c:c1 -> c:c1
Nil :: Cons:Nil
c1 :: c:c1
GAME :: Cons:Nil -> Cons:Nil -> Cons:Nil -> c2:c3:c4
Capture :: Capture:Swap
c2 :: c2:c3:c4 -> c2:c3:c4
Swap :: Capture:Swap
c3 :: c2:c3:c4 -> c2:c3:c4
c4 :: c:c1 -> c2:c3:c4
EQUAL :: Capture:Swap -> Capture:Swap -> c5:c6:c7:c8
c5 :: c5:c6:c7:c8
c6 :: c5:c6:c7:c8
c7 :: c5:c6:c7:c8
c8 :: c5:c6:c7:c8
GOAL :: Cons:Nil -> Cons:Nil -> Cons:Nil -> c9
c9 :: c2:c3:c4 -> c9
@ :: Cons:Nil -> Cons:Nil -> Cons:Nil
game :: Cons:Nil -> Cons:Nil -> Cons:Nil -> Cons:Nil
equal :: Capture:Swap -> Capture:Swap -> True:False
True :: True:False
False :: True:False
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil -> Cons:Nil
hole_c:c11_10 :: c:c1
hole_Cons:Nil2_10 :: Cons:Nil
hole_Capture:Swap3_10 :: Capture:Swap
hole_c2:c3:c44_10 :: c2:c3:c4
hole_c5:c6:c7:c85_10 :: c5:c6:c7:c8
hole_c96_10 :: c9
hole_True:False7_10 :: True:False
gen_c:c18_10 :: Nat -> c:c1
gen_Cons:Nil9_10 :: Nat -> Cons:Nil
gen_c2:c3:c410_10 :: Nat -> c2:c3:c4


Lemmas:
@'(gen_Cons:Nil9_10(n12_10), gen_Cons:Nil9_10(b)) -> gen_c:c18_10(n12_10), rt in Omega(1 + n12_10)


Generator Equations:
gen_c:c18_10(0) <=> c1
gen_c:c18_10(+(x, 1)) <=> c(gen_c:c18_10(x))
gen_Cons:Nil9_10(0) <=> Nil
gen_Cons:Nil9_10(+(x, 1)) <=> Cons(Capture, gen_Cons:Nil9_10(x))
gen_c2:c3:c410_10(0) <=> c4(c1)
gen_c2:c3:c410_10(+(x, 1)) <=> c2(gen_c2:c3:c410_10(x))


The following defined symbols remain to be analysed:
GAME, @, game

They will be analysed ascendingly in the following order:
@ < game