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
    (bvsge (uremtotal s (inv s t)) t))
(define-fun SC ((s (BitVec 4)) (t (BitVec 4))) Bool
    (and (=> (bvsge s #x0) (bvsge s t)) (=> (and (bvslt s #x0) (bvsge t #x0)) (bvugt (bvsub s t) t))))
(constraint (=> (SC s t) (l s t)))

(check-synth)
Solution:
(define-fun inv ((s (BitVec 4)) (t (BitVec 4))) (BitVec 4) (bvsub (bvor s #b1000) (bvand t #b0111)))

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
    (bvuge (bvand (inv s t) s) t))
(define-fun SC ((s (BitVec 4)) (t (BitVec 4))) Bool
    (not (bvult s t)))
(constraint (=> (SC s t) (l s t)))

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

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
    (bvsle (bvshl (inv s t) s) t))
(define-fun SC ((s (BitVec 4)) (t (BitVec 4))) Bool
    (bvult (bvlshr t (bvlshr t s)) min))
(constraint (=> (SC s t) (l s t)))

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

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 (bvneg (inv s t)) t))
(define-fun SC ((s (BitVec 4)) (t (BitVec 4))) Bool
    true)
(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 ")"