You are an expert in program synthesis. You are tasked with solving a Syntax-Guided Synthesis (SyGuS) problem. Your goal is to output a function that should produce outputs that satisfy a series of constraints when given specific inputs.

Question:
(set-logic SLIA)

(synth-fun f ((name String)) String
    ((Start String) (ntString String) (ntInt Int) (ntBool Bool))
    ((Start String (ntString))
    (ntString String (name " " (str.++ ntString ntString) (str.replace ntString ntString ntString) (str.at ntString ntInt) (str.from_int ntInt) (str.substr ntString ntInt ntInt)))
    (ntInt Int (0 1 (+ ntInt ntInt) (- ntInt ntInt) (str.len ntString) (str.to_int ntString) (str.indexof ntString ntString ntInt)))
    (ntBool Bool (true false (str.prefixof ntString ntString) (str.suffixof ntString ntString) (str.contains ntString ntString)))))

(declare-var name String)
(constraint (= (f "Nancy FreeHafer") "FreeHafer"))
(constraint (= (f "Andrew Cencici") "Cencici"))
(constraint (= (f "Jan Kotas") "Kotas"))
(constraint (= (f "Mariya Sergienko") "Sergienko"))

(check-synth)
Solution:
(define-fun f ((name String)) String (str.substr name (+ 1 (str.indexof name " " 0)) (str.len name)))

Question:
(set-logic SLIA)

(synth-fun f ((name String)) String
    ((Start String) (ntString String) (ntInt Int) (ntBool Bool))
    ((Start String (ntString))
    (ntString String (name " " "." (str.++ ntString ntString) (str.replace ntString ntString ntString) (str.at ntString ntInt) (str.from_int ntInt) (str.substr ntString ntInt ntInt)))
    (ntInt Int (0 1 2 (+ ntInt ntInt) (- ntInt ntInt) (str.len ntString) (str.to_int ntString) (str.indexof ntString ntString ntInt)))
    (ntBool Bool (true false (str.prefixof ntString ntString) (str.suffixof ntString ntString) (str.contains ntString ntString)))))

(declare-var name String)
(constraint (= (f "Nancy FreeHafer") "N.F."))
(constraint (= (f "Andrew Cencici") "A.C."))
(constraint (= (f "Jan Kotas") "J.K."))
(constraint (= (f "Mariya Sergienko") "M.S."))

(check-synth)
Solution:
(define-fun f ((name String)) String (str.++ (str.at name 0) (str.++ "." (str.++ (str.at name (+ 1 (str.indexof name " " 0))) "."))))

Question:
(set-logic SLIA)

(synth-fun f ((name String)) String
    ((Start String) (ntString String) (ntInt Int) (ntBool Bool))
    ((Start String (ntString))
    (ntString String (name " " (str.++ ntString ntString) (str.replace ntString ntString ntString) (str.at ntString ntInt) (str.substr ntString ntInt ntInt)))
    (ntInt Int (0 1 (+ ntInt ntInt) (- ntInt ntInt) (str.len ntString) (str.indexof ntString ntString ntInt)))
    (ntBool Bool (true false (str.prefixof ntString ntString) (str.suffixof ntString ntString)))))

(declare-var name String)
(constraint (= (f "Nancy FreeHafer") "FreeHafer"))
(constraint (= (f "Andrew Cencici") "Cencici"))
(constraint (= (f "Jan Kotas") "Kotas"))
(constraint (= (f "Mariya Sergienko") "Sergienko"))

(check-synth)
Solution:
(define-fun f ((name String)) String (str.substr name (+ 1 (str.indexof name " " 0)) (str.len name)))

Question:
(set-logic SLIA)

(synth-fun f ((name String)) String
    ((Start String) (ntString String) (ntInt Int) (ntBool Bool))
    ((Start String (ntString))
    (ntString String (name " " (str.++ ntString ntString) (str.replace ntString ntString ntString) (str.at ntString ntInt) (str.from_int ntInt) (str.substr ntString ntInt ntInt)))
    (ntInt Int (0 1 (+ ntInt ntInt) (- ntInt ntInt) (str.len ntString) (str.to_int ntString) (str.indexof ntString ntString ntInt)))
    (ntBool Bool (true false (str.prefixof ntString ntString) (str.suffixof ntString ntString) (str.contains ntString ntString)))))

(declare-var name String)
(constraint (= (f "Nancy FreeHafer") "FreeHafer"))
(constraint (= (f "Andrew Cencici") "Cencici"))
(constraint (= (f "Jan Kotas") "Kotas"))
(constraint (= (f "Mariya Sergienko") "Sergienko"))

(check-synth)
Solution:


Output grammar:
root ::= "(define-fun f ((name String)) String " Start ")"
Start ::= ntString
ntString ::= "name" | "\" \"" | "(" "str.++" " " ntString " " ntString ")" | "(" "str.replace" " " ntString " " ntString " " ntString ")" | "(" "str.at" " " ntString " " ntInt ")" | "(" "str.from_int" " " ntInt ")" | "(" "str.substr" " " ntString " " ntInt " " ntInt ")"
ntInt ::= "0" | "1" | "(" "+" " " ntInt " " ntInt ")" | "(" "-" " " ntInt " " ntInt ")" | "(" "str.len" " " ntString ")" | "(" "str.to_int" " " ntString ")" | "(" "str.indexof" " " ntString " " ntString " " ntInt ")"
ntBool ::= "true" | "false" | "(" "str.prefixof" " " ntString " " ntString ")" | "(" "str.suffixof" " " ntString " " ntString ")" | "(" "str.contains" " " ntString " " ntString ")"