WORST_CASE(?,O(1))
* Step 1: Desugar WORST_CASE(?,O(1))
    + Considered Problem:
        
        type nat = 0 | S of nat
        ;;
        
        let pred x =
          match x with
          | 0 -> 0
          | S(x') -> x'
        ;;
    + Applied Processor:
        Desugar {analysedFunction = Nothing}
    + Details:
        none
* Step 2: Defunctionalization WORST_CASE(?,O(1))
    + Considered Problem:
        λx : nat.
          (λpred : nat -> nat. pred x) (λx : nat. case x of  | 0 -> 0 | S -> λx' : nat. x') : nat -> nat
        where
          0 :: nat
          S :: nat -> nat
    + Applied Processor:
        Defunctionalization
    + Details:
        none
* Step 3: Inline WORST_CASE(?,O(1))
    + Considered Problem:
        1: main_1(x0) x1 -> x1 x0
        2: cond_pred_x(0()) -> 0()
        3: cond_pred_x(S(x2)) -> x2
        4: pred() x1 -> cond_pred_x(x1)
        5: main(x0) -> main_1(x0) pred()
        where
          0           :: nat
          S           :: nat -> nat
          pred        :: nat -> nat
          main_1      :: nat -> (nat -> nat) -> nat
          cond_pred_x :: nat -> nat
          main        :: nat -> nat
    + Applied Processor:
        Inline {inlineType = InlineRewrite, inlineSelect = <inline non-recursive>}
    + Details:
        none
* Step 4: Inline WORST_CASE(?,O(1))
    + Considered Problem:
        1: main_1(x0) x1 -> x1 x0
        2: cond_pred_x(0()) -> 0()
        3: cond_pred_x(S(x2)) -> x2
        4: pred() x1 -> cond_pred_x(x1)
        5: main(x0) -> pred() x0
        where
          0           :: nat
          S           :: nat -> nat
          pred        :: nat -> nat
          main_1      :: nat -> (nat -> nat) -> nat
          cond_pred_x :: nat -> nat
          main        :: nat -> nat
    + Applied Processor:
        Inline {inlineType = InlineRewrite, inlineSelect = <inline non-recursive>}
    + Details:
        none
* Step 5: Inline WORST_CASE(?,O(1))
    + Considered Problem:
        1: main_1(x0) x1 -> x1 x0
        2: cond_pred_x(0()) -> 0()
        3: cond_pred_x(S(x2)) -> x2
        4: pred() x1 -> cond_pred_x(x1)
        5: main(x2) -> cond_pred_x(x2)
        where
          0           :: nat
          S           :: nat -> nat
          pred        :: nat -> nat
          main_1      :: nat -> (nat -> nat) -> nat
          cond_pred_x :: nat -> nat
          main        :: nat -> nat
    + Applied Processor:
        Inline {inlineType = InlineFull, inlineSelect = <inline case-expression>}
    + Details:
        none
* Step 6: UsableRules WORST_CASE(?,O(1))
    + Considered Problem:
        1: main_1(x0) x1 -> x1 x0
        2: cond_pred_x(0()) -> 0()
        3: cond_pred_x(S(x2)) -> x2
        4: pred() 0() -> 0()
        5: pred() S(x4) -> x4
        6: main(0()) -> 0()
        7: main(S(x4)) -> x4
        where
          0           :: nat
          S           :: nat -> nat
          pred        :: nat -> nat
          main_1      :: nat -> (nat -> nat) -> nat
          cond_pred_x :: nat -> nat
          main        :: nat -> nat
    + Applied Processor:
        UsableRules {urType = Syntactic}
    + Details:
        none
* Step 7: UncurryATRS WORST_CASE(?,O(1))
    + Considered Problem:
        6: main(0()) -> 0()
        7: main(S(x4)) -> x4
        where
          0    :: nat
          S    :: nat -> nat
          main :: nat -> nat
    + Applied Processor:
        UncurryATRS
    + Details:
        none
* Step 8: UsableRules WORST_CASE(?,O(1))
    + Considered Problem:
        1: main(0()) -> 0()
        2: main(S(x4)) -> x4
        where
          0    :: nat
          S    :: nat -> nat
          main :: nat -> nat
    + Applied Processor:
        UsableRules {urType = DFA}
    + Details:
        none
* Step 9: Compression WORST_CASE(?,O(1))
    + Considered Problem:
        1: main(0()) -> 0()
        2: main(S(x4)) -> x4
        where
          0    :: nat
          S    :: nat -> nat
          main :: nat -> nat
    + Applied Processor:
        Compression
    + Details:
        none
* Step 10: ToTctProblem WORST_CASE(?,O(1))
    + Considered Problem:
        1: main(0()) -> 0()
        2: main(S(x4)) -> x4
        where
          0    :: nat
          S    :: nat -> nat
          main :: nat -> nat
    + Applied Processor:
        ToTctProblem
    + Details:
        none
* Step 11: Ara WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict TRS:
            main(0()) -> 0()
            main(S(x4)) -> x4
        - Signature:
            {main/1} / {0/0,S/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {main} and constructors {0,S}
    + Applied Processor:
        Ara {minDegree = 1, maxDegree = 2, araTimeout = 5, araRuleShifting = Nothing}
    + Details:
        Signatures used:
        ----------------
          F (TrsFun "0") :: [] -(0)-> "A"(0)
          F (TrsFun "S") :: ["A"(0)] -(0)-> "A"(0)
          F (TrsFun "main") :: ["A"(0)] -(1)-> "A"(0)
        
        
        Cost-free Signatures used:
        --------------------------
        
        
        
        Base Constructor Signatures used:
        ---------------------------------
        
        

WORST_CASE(?,O(1))