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: DependencyPairs 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:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          main#(0()) -> c_1()
          main#(S(x4)) -> c_2()
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 12: PredecessorEstimation WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            main#(0()) -> c_1()
            main#(S(x4)) -> c_2()
        - Weak TRS:
            main(0()) -> 0()
            main(S(x4)) -> x4
        - Signature:
            {main/1,main#/1} / {0/0,S/1,c_1/0,c_2/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {main#} and constructors {0,S}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,2}
        by application of
          Pre({1,2}) = {}.
        Here rules are labelled as follows:
          1: main#(0()) -> c_1()
          2: main#(S(x4)) -> c_2()
* Step 13: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            main#(0()) -> c_1()
            main#(S(x4)) -> c_2()
        - Weak TRS:
            main(0()) -> 0()
            main(S(x4)) -> x4
        - Signature:
            {main/1,main#/1} / {0/0,S/1,c_1/0,c_2/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {main#} and constructors {0,S}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:main#(0()) -> c_1()
             
          
          2:W:main#(S(x4)) -> c_2()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: main#(S(x4)) -> c_2()
          1: main#(0()) -> c_1()
* Step 14: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            main(0()) -> 0()
            main(S(x4)) -> x4
        - Signature:
            {main/1,main#/1} / {0/0,S/1,c_1/0,c_2/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {main#} and constructors {0,S}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(1))