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 BV)

(synth-fun inv ((s (BitVec 4)) (t (BitVec 4))) (BitVec 4)
    ((Start (BitVec 4)))
    ((Start (BitVec 4) (s t #x0 #x8 #x7 (bvneg Start) (bvnot Start) (bvadd Start Start) (bvsub Start Start) (bvand Start Start) (bvlshr Start Start) (bvor Start Start) (bvshl Start Start)))))

(declare-var s (BitVec 4))
(declare-var t (BitVec 4))
(define-fun udivtotal ((a (BitVec 4)) (b (BitVec 4))) (BitVec 4)
    (ite (= b #x0) #xF (bvudiv a b)))
(define-fun uremtotal ((a (BitVec 4)) (b (BitVec 4))) (BitVec 4)
    (ite (= b #x0) a (bvurem a b)))
(define-fun min () (BitVec 4)
    (bvnot (bvlshr (bvnot #x0) #x1)))
(define-fun max () (BitVec 4)
    (bvnot min))
(define-fun l ((s (BitVec 4)) (t (BitVec 4))) Bool
    (bvule (bvmul (inv s t) s) t))
(define-fun SC ((s (BitVec 4)) (t (BitVec 4))) Bool
    true)
(constraint (=> (SC s t) (l s t)))

(check-synth)
Solution:
(define-fun inv ((s (BitVec 4)) (t (BitVec 4))) (BitVec 4) #b0000)

Question:
(set-logic BV)

(synth-fun inv ((s (BitVec 4)) (t (BitVec 4))) (BitVec 4)
    ((Start (BitVec 4)))
    ((Start (BitVec 4) (s t #x0 #x8 #x7 (bvneg Start) (bvnot Start) (bvadd Start Start) (bvsub Start Start) (bvand Start Start) (bvlshr Start Start) (bvor Start Start) (bvshl Start Start)))))

(declare-var s (BitVec 4))
(declare-var t (BitVec 4))
(define-fun udivtotal ((a (BitVec 4)) (b (BitVec 4))) (BitVec 4)
    (ite (= b #x0) #xF (bvudiv a b)))
(define-fun uremtotal ((a (BitVec 4)) (b (BitVec 4))) (BitVec 4)
    (ite (= b #x0) a (bvurem a b)))
(define-fun min () (BitVec 4)
    (bvnot (bvlshr (bvnot #x0) #x1)))
(define-fun max () (BitVec 4)
    (bvnot min))
(define-fun l ((s (BitVec 4)) (t (BitVec 4))) Bool
    (distinct (bvmul (inv s t) s) t))
(define-fun SC ((s (BitVec 4)) (t (BitVec 4))) Bool
    (or (distinct t #x0) (distinct s #x0)))
(constraint (=> (SC s t) (l s t)))

(check-synth)
Solution:
(define-fun inv ((s (BitVec 4)) (t (BitVec 4))) (BitVec 4) (bvshl #b0111 t))

Question:
(set-logic BV)

(synth-fun inv ((s (BitVec 4)) (t (BitVec 4))) (BitVec 4)
    ((Start (BitVec 4)))
    ((Start (BitVec 4) (s t #x0 #x8 #x7 (bvneg Start) (bvnot Start) (bvadd Start Start) (bvsub Start Start) (bvand Start Start) (bvlshr Start Start) (bvor Start Start) (bvshl Start Start)))))

(declare-var s (BitVec 4))
(declare-var t (BitVec 4))
(define-fun udivtotal ((a (BitVec 4)) (b (BitVec 4))) (BitVec 4)
    (ite (= b #x0) #xF (bvudiv a b)))
(define-fun uremtotal ((a (BitVec 4)) (b (BitVec 4))) (BitVec 4)
    (ite (= b #x0) a (bvurem a b)))
(define-fun min () (BitVec 4)
    (bvnot (bvlshr (bvnot #x0) #x1)))
(define-fun max () (BitVec 4)
    (bvnot min))
(define-fun l ((s (BitVec 4)) (t (BitVec 4))) Bool
    (bvslt (uremtotal (inv s t) s) t))
(define-fun SC ((s (BitVec 4)) (t (BitVec 4))) Bool
    (bvslt (bvnot t) (bvor (bvneg s) (bvneg t))))
(constraint (=> (SC s t) (l s t)))

(check-synth)
Solution:
(define-fun inv ((s (BitVec 4)) (t (BitVec 4))) (BitVec 4) (bvshl #b1000 (bvshl t t)))

Question:
(set-logic BV)

(synth-fun inv ((s (BitVec 4)) (t (BitVec 4))) (BitVec 4)
    ((Start (BitVec 4)))
    ((Start (BitVec 4) (s t #x0 #x8 #x7 (bvneg Start) (bvnot Start) (bvadd Start Start) (bvsub Start Start) (bvand Start Start) (bvlshr Start Start) (bvor Start Start) (bvshl Start Start)))))

(declare-var s (BitVec 4))
(declare-var t (BitVec 4))
(define-fun udivtotal ((a (BitVec 4)) (b (BitVec 4))) (BitVec 4)
    (ite (= b #x0) #xF (bvudiv a b)))
(define-fun uremtotal ((a (BitVec 4)) (b (BitVec 4))) (BitVec 4)
    (ite (= b #x0) a (bvurem a b)))
(define-fun min () (BitVec 4)
    (bvnot (bvlshr (bvnot #x0) #x1)))
(define-fun max () (BitVec 4)
    (bvnot min))
(define-fun l ((s (BitVec 4)) (t (BitVec 4))) Bool
    (bvugt (uremtotal s (inv s t)) t))
(define-fun SC ((s (BitVec 4)) (t (BitVec 4))) Bool
    (bvult t s))
(constraint (=> (SC s t) (l s t)))

(check-synth)
Solution:


Output grammar:
root ::= "(define-fun inv ((s (BitVec 4)) (t (BitVec 4))) (BitVec 4) " Start ")"
Start ::= "s" | "t" | "#x0" | "#x8" | "#x7" | "(" "bvneg" " " Start ")" | "(" "bvnot" " " Start ")" | "(" "bvadd" " " Start " " Start ")" | "(" "bvsub" " " Start " " Start ")" | "(" "bvand" " " Start " " Start ")" | "(" "bvlshr" " " Start " " Start ")" | "(" "bvor" " " Start " " Start ")" | "(" "bvshl" " " Start " " Start ")"