Z3DeclKind

The constant true.

The constant false.

The equality predicate.

The n-ary distinct predicate (every argument is mutually distinct).

The ternary if-then-else term.

n-ary conjunction.

n-ary disjunction.

equivalence (binary).

Exclusive or.

Negation.

Implication.

Binary equivalence modulo namings. This binary predicate is used in proof terms. It captures equisatisfiability and equivalence modulo renamings.

Arithmetic numeral.

Arithmetic algebraic numeral. Algebraic numbers are used to represent irrational numbers in Z3.

<=.

=.

<.

.

Addition - Binary.

Binary subtraction.

Unary minus.

Multiplication - Binary.

Division - Binary.

Integer division - Binary.

Remainder - Binary.

Modulus - Binary.

Coercion of integer to real - Unary.

Coercion of real to integer - Unary.

Check if real is also an integer - Unary.

Power operator x^y.

Array store. It satisfies select(store(a,i,v),j) = if i = j then v else select(a,j). Array store takes at least 3 arguments.

Array select.

The constant array. For example, select(const(v),i) = v holds for every v and i. The function is unary.

Array map operator. It satisfies mapf[i] = f(a1[i],…,a_n[i]) for every i.

Default value of arrays. For example default(const(v)) = v. The function is unary.

Set union between two Boolean arrays (two arrays whose range type is Boolean). The function is binary.

Set intersection between two Boolean arrays. The function is binary.

Set difference between two Boolean arrays. The function is binary.

Set complement of a Boolean array. The function is unary.

Subset predicate between two Boolean arrays. The relation is binary.

An array value that behaves as the function graph of the function passed as parameter.

Array extensionality function. It takes two arrays as arguments and produces an index, such that the arrays are different if they are different on the index.

Bit-vector numeral.

One bit bit-vector.

Zero bit bit-vector.

Unary minus.

Binary addition.

Binary subtraction.

Binary multiplication.

Binary signed division.

Binary unsigned division.

Binary signed remainder.

Binary unsigned remainder.

Binary signed modulus.

Unary function. bsdiv(x,0) is congruent to bsdiv0(x).

Unary function. budiv(x,0) is congruent to budiv0(x).

Unary function. bsrem(x,0) is congruent to bsrem0(x).

Unary function. burem(x,0) is congruent to burem0(x).

Unary function. bsmod(x,0) is congruent to bsmod0(x).

Unsigned bit-vector <= - Binary relation.

Signed bit-vector <= - Binary relation.

Unsigned bit-vector >= - Binary relation.

Signed bit-vector >= - Binary relation.

Unsigned bit-vector < - Binary relation.

Signed bit-vector < - Binary relation.

Unsigned bit-vector > - Binary relation.

Signed bit-vector > - Binary relation.

Bit-wise and - Binary.

Bit-wise or - Binary.

Bit-wise not - Unary.

Bit-wise xor - Binary.

Bit-wise nand - Binary.

Bit-wise nor - Binary.

Bit-wise xnor - Binary.

Bit-vector concatenation - Binary.

Bit-vector sign extension.

Bit-vector zero extension.

Bit-vector extraction.

Repeat bit-vector n times.

Bit-vector reduce or - Unary.

Bit-vector reduce and - Unary.

.

Shift left.

Logical shift right.

Arithmetical shift right.

Left rotation.

Right rotation.

(extended) Left rotation. Similar to Z3_OP_ROTATE_LEFT, but it is a binary operator instead of a parametric one.

(extended) Right rotation. Similar to Z3_OP_ROTATE_RIGHT, but it is a binary operator instead of a parametric one.

Coerce integer to bit-vector. NB. This function is not supported by the decision procedures. Only the most rudimentary simplification rules are applied to this function.

Coerce bit-vector to integer. NB. This function is not supported by the decision procedures. Only the most rudimentary simplification rules are applied to this function.

Compute the carry bit in a full-adder. The meaning is given by the equivalence (carry l1 l2 l3) <=> (or (and l1 l2) (and l1 l3) (and l2 l3)))

Compute ternary XOR. The meaning is given by the equivalence (xor3 l1 l2 l3) <=> (xor (xor l1 l2) l3)

a predicate to check that bit-wise signed multiplication does not overflow. Signed multiplication overflows if the operands have the same sign and the result of multiplication does not fit within the available bits. \sa Z3_mk_bvmul_no_overflow.

check that bit-wise unsigned multiplication does not overflow. Unsigned multiplication overflows if the result does not fit within the available bits. \sa Z3_mk_bvmul_no_overflow.

check that bit-wise signed multiplication does not underflow. Signed multiplication underflows if the operands have opposite signs and the result of multiplication does not fit within the available bits. Z3_mk_bvmul_no_underflow.

Binary signed division. It has the same semantics as Z3_OP_BSDIV, but created in a context where the second operand can be assumed to be non-zero.

Binary unsigned division. It has the same semantics as Z3_OP_BUDIV, but created in a context where the second operand can be assumed to be non-zero.

Binary signed remainder. It has the same semantics as Z3_OP_BSREM, but created in a context where the second operand can be assumed to be non-zero.

Binary unsigned remainder. It has the same semantics as Z3_OP_BUREM, but created in a context where the second operand can be assumed to be non-zero.

Binary signed modulus. It has the same semantics as Z3_OP_BSMOD, but created in a context where the second operand can be assumed to be non-zero.

Undef/Null proof object.

Proof for the expression ‘true’.

Proof for a fact asserted by the user.

Proof for a fact (tagged as goal) asserted by the user.

Given a proof for p and a proof for (implies p q), produces a proof for q.

 T1: p
 T2: (implies p q)
 [mp T1 T2]: q

The second antecedents may also be a proof for (iff p q).

A proof for (R t t), where R is a reflexive relation. This proof object has no antecedents. The only reflexive relations that are used are equivalence modulo namings, equality and equivalence. That is, R is either ‘~’, ‘=’ or ‘iff’.

Given an symmetric relation R and a proof for (R t s), produces a proof for (R s t). \nicebox{ T1: (R t s) [symmetry T1]: (R s t) } T1 is the antecedent of this proof object.

Given a transitive relation R, and proofs for (R t s) and (R s u), produces a proof for (R t u). \nicebox{ T1: (R t s) T2: (R s u) [trans T1 T2]: (R t u) }

Condensed transitivity proof. It combines several symmetry and transitivity proofs. Example: \nicebox{ T1: (R a b) T2: (R c b) T3: (R c d) [trans* T1 T2 T3]: (R a d) } R must be a symmetric and transitive relation.

 Assuming that this proof object is a proof for (R s t), then
 a proof checker must check if it is possible to prove (R s t)
 using the antecedents, symmetry and transitivity.  That is,
 if there is a path from s to t, if we view every
 antecedent (R a b) as an edge between a and b.

Monotonicity proof object.

 T1: (R t_1 s_1)
 ...
 Tn: (R t_n s_n)
 [monotonicity T1 ... Tn]: (R (f t_1 ... t_n) (f s_1 ... s_n))

Remark: if t_i == s_i, then the antecedent Ti is suppressed. That is, reflexivity proofs are suppressed to save space.

Given a proof for (~ p q), produces a proof for (~ (forall (x) p) (forall (x) q)).

T1: (~ p q)

Given a proof p, produces a proof of lambda x . p, where x are free variables in p.

T1: f [proof-bind T1] forall (x) f

Distributivity proof object. Given that f (= or) distributes over g (= and), produces a proof for \nicebox{ (= (f a (g c d)) (g (f a c) (f a d))) } If f and g are associative, this proof also justifies the following equality: \nicebox{ (= (f (g a b) (g c d)) (g (f a c) (f a d) (f b c) (f b d))) } where each f and g can have arbitrary number of arguments.

This proof object has no antecedents. Remark. This rule is used by the CNF conversion pass and instantiated by f = or, and g = and.

Given a proof for (and l_1 … l_n), produces a proof for l_i

T1: (and l_1 … l_n)

Given a proof for (not (or l_1 … l_n)), produces a proof for (not l_i).

T1: (not (or l_1 … l_n)) [not-or-elim T1]: (not l_i)

A proof for a local rewriting step (= t s). The head function symbol of t is interpreted.

This proof object has no antecedents. The conclusion of a rewrite rule is either an equality (= t s), an equivalence (iff t s), or equi-satisfiability (~ t s). Remark: if f is bool, then = is iff. Examples: \nicebox{ (= (+ x 0) x) (= (+ x 1 2) (+ 3 x)) (iff (or x false) x) }

A proof for rewriting an expression t into an expression s. This proof object can have n antecedents. The antecedents are proofs for equalities used as substitution rules. The proof rule is used in a few cases. The cases are:

• When applying contextual simplification (CONTEXT_SIMPLIFIER=true)
• When converting bit-vectors to Booleans (BIT2BOOL=true)

A proof for (iff (f (forall (x) q(x)) r) (forall (x) (f (q x) r))). This proof object has no antecedents.

A proof for: \nicebox{ (iff (forall (x_1 … x_m) (and p_1[x_1 … x_m] … p_n[x_1 … x_m])) (and (forall (x_1 … x_m) p_1[x_1 … x_m]) … (forall (x_1 … x_m) p_n[x_1 … x_m]))) } This proof object has no antecedents.

A proof for (iff (forall (x_1 … x_n y_1 … y_m) p[x_1 … x_n]) (forall (x_1 … x_n) p[x_1 … x_n]))

It is used to justify the elimination of unused variables. This proof object has no antecedents.

A proof for destructive equality resolution: (iff (forall (x) (or (not (= x t)) P[x])) P[t]) if x does not occur in t.

This proof object has no antecedents.

Several variables can be eliminated simultaneously.

A proof of (or (not (forall (x) (P x))) (P a))

Mark a hypothesis in a natural deduction style proof.

T1: false
[lemma T1]: (or (not l_1) ... (not l_n))

This proof object has one antecedent: a hypothetical proof for false. It converts the proof in a proof for (or (not l_1) … (not l_n)), when T1 contains the open hypotheses: l_1, …, l_n. The hypotheses are closed after an application of a lemma. Furthermore, there are no other open hypotheses in the subtree covered by the lemma.

\nicebox{ T1: (or l_1 … l_n l_1’ … l_m’) T2: (not l_1) … T(n+1): (not l_n) [unit-resolution T1 … T(n+1)]: (or l_1’ … l_m’) }

\nicebox{ T1: p [iff-true T1]: (iff p true) }

\nicebox{ T1: (not p) [iff-false T1]: (iff p false) }

f is a commutative operator.

This proof object has no antecedents. Remark: if f is bool, then = is iff.

Proof object used to justify Tseitin’s like axioms: \nicebox{ (or (not (and p q)) p) (or (not (and p q)) q) (or (not (and p q r)) p) (or (not (and p q r)) q) (or (not (and p q r)) r) … (or (and p q) (not p) (not q)) (or (not (or p q)) p q) (or (or p q) (not p)) (or (or p q) (not q)) (or (not (iff p q)) (not p) q) (or (not (iff p q)) p (not q)) (or (iff p q) (not p) (not q)) (or (iff p q) p q) (or (not (ite a b c)) (not a) b) (or (not (ite a b c)) a c) (or (ite a b c) (not a) (not b)) (or (ite a b c) a (not c)) (or (not (not a)) (not a)) (or (not a) a) } This proof object has no antecedents. Note: all axioms are propositional tautologies. Note also that ‘and’ and ‘or’ can take multiple arguments. You can recover the propositional tautologies by unfolding the Boolean connectives in the axioms a small bounded number of steps (=3).

Clausal proof adding axiom

Clausal proof lemma addition

Clausal proof lemma deletion

Introduces a name for a formula/term. Suppose e is an expression with free variables x, and def-intro introduces the name n(x). The possible cases are:

When e is of Boolean type:

or:

when e only occurs positively.

When e is of the form (ite cond th el):

Otherwise: def-intro: (= n e)

[apply-def T1]: F ~ n

F is ‘equivalent’ to n, given that T1 is a proof that n is a name for F.

T1: (iff p q) [iff~ T1]: (~ p q)

Proof for a (positive) NNF step. Example:

T1: (not s_1) ~ r_1 T2: (not s_2) ~ r_2 T3: s_1 ~ r_1’ T4: s_2 ~ r_2’

The negation normal form steps NNF_POS and NNF_NEG are used in the following cases: (a) When creating the NNF of a positive force quantifier. The quantifier is retained (unless the bound variables are eliminated). Example

 T1: q ~ q_new
 [nnf-pos T1]: (~ (forall (x T) q) (forall (x T) q_new))

(b) When recursively creating NNF over Boolean formulas, where the top-level connective is changed during NNF conversion. The relevant Boolean connectives for NNF_POS are ‘implies’, ‘iff’, ‘xor’, ‘ite’. NNF_NEG furthermore handles the case where negation is pushed over Boolean connectives ‘and’ and ‘or’.

Proof for a (negative) NNF step. Examples:

T1: (not s_1) ~ r_1 … Tn: (not s_n) ~ r_n

and

T1: (not s_1) ~ r_1 … Tn: (not s_n) ~ r_n

and

T1: (not s_1) ~ r_1 T2: (not s_2) ~ r_2 T3: s_1 ~ r_1’ T4: s_2 ~ r_2’

                     (and (or r_1 r_2) (or r_1' r_2')))

Proof for:

 [sk]: (~ (not (forall x (p x y))) (not (p (sk y) y)))
 [sk]: (~ (exists x (p x y)) (p (sk y) y))

This proof object has no antecedents.

Modus ponens style rule for equi-satisfiability.

T1: p T2: (~ p q)

Generic proof for theory lemmas. The theory lemma function comes with one or more parameters. The first parameter indicates the name of the theory. For the theory of arithmetic, additional parameters provide hints for checking the theory lemma. The hints for arithmetic are:

- farkas - followed by rational coefficients. Multiply the coefficients to the
  inequalities in the lemma, add the (negated) inequalities and obtain a contradiction.

- triangle-eq - Indicates a lemma related to the equivalence:

   (iff (= t1 t2) (and (<= t1 t2) (<= t2 t1)))

- gcd-test - Indicates an integer linear arithmetic lemma that uses a gcd test.

Hyper-resolution rule.

The premises of the rules is a sequence of clauses. The first clause argument is the main clause of the rule. with a literal from the first (main) clause.

Premises of the rules are of the form \nicebox{ (or l0 l1 l2 .. ln) } or \nicebox{ (=> (and l1 l2 .. ln) l0) } or in the most general (ground) form: \nicebox{ (=> (and ln+1 ln+2 .. ln+m) (or l0 l1 .. ln)) } In other words we use the following (Prolog style) convention for Horn implications: The head of a Horn implication is position 0, the first conjunct in the body of an implication is position 1 the second conjunct in the body of an implication is position 2

For general implications where the head is a disjunction, the first n positions correspond to the n disjuncts in the head. The next m positions correspond to the m conjuncts in the body.

The premises can be universally quantified so that the most general non-ground form is:

\nicebox{ (forall (vars) (=> (and ln+1 ln+2 .. ln+m) (or l0 l1 .. ln))) }

The hyper-resolution rule takes a sequence of parameters. The parameters are substitutions of bound variables separated by pairs of literal positions from the main clause and side clause.

Insert a record into a relation. The function takes n+1 arguments, where the first argument is the relation and the remaining n elements correspond to the n columns of the relation.

Creates the empty relation.

Tests if the relation is empty.

Create the relational join.

Create the union or convex hull of two relations. The function takes two arguments.

Widen two relations. The function takes two arguments.

Project the columns (provided as numbers in the parameters). The function takes one argument.

Filter (restrict) a relation with respect to a predicate. The first argument is a relation. The second argument is a predicate with free de-Bruijn indices corresponding to the columns of the relation. So the first column in the relation has index 0.

Intersect the first relation with respect to negation of the second relation (the function takes two arguments). Logically, the specification can be described by a function

target = filter_by_negation(pos, neg, columns)

where columns are pairs c1, d1, .., cN, dN of columns from pos and neg, such that target are elements in x in pos, such that there is no y in neg that agrees with x on the columns c1, d1, .., cN, dN.

rename columns in the relation. The function takes one argument. The parameters contain the renaming as a cycle.

Complement the relation.

Check if a record is an element of the relation. The function takes n+1 arguments, where the first argument is a relation, and the remaining n arguments correspond to a record.

Create a fresh copy (clone) of a relation. The function is logically the identity, but in the context of a register machine allows for #Z3_OP_RA_UNION to perform destructive updates to the first argument.

A less than predicate over the finite domain Z3_FINITE_DOMAIN_SORT.

A label (used by the Boogie Verification condition generator). The label has two parameters, a string and a Boolean polarity. It takes one argument, a formula.

A label literal (used by the Boogie Verification condition generator). A label literal has a set of string parameters. It takes no arguments.

datatype constructor.

datatype recognizer.

datatype recognizer.

datatype accessor.

datatype field update.

Cardinality constraint. E.g., x + y + z <= 2

Cardinality constraint. E.g., x + y + z >= 2

Generalized Pseudo-Boolean cardinality constraint. Example 2*x + 3*y <= 4

Generalized Pseudo-Boolean cardinality constraint. Example 2*x + 3*y + 2*z >= 4

Generalized Pseudo-Boolean equality constraint. Example 2*x + 1*y + 2*z + 1*u = 4

A relation that is a total linear order

A relation that is a partial order

A relation that is a piecewise linear order

A relation that is a tree order

Transitive closure of a relation

Transitive reflexive closure of a relation

Floating-point rounding mode RNE

Floating-point rounding mode RNA

Floating-point rounding mode RTP

Floating-point rounding mode RTN

Floating-point rounding mode RTZ

Floating-point value

Floating-point +oo

Floating-point -oo

Floating-point NaN

Floating-point +zero

Floating-point -zero

Floating-point addition

Floating-point subtraction

Floating-point negation

Floating-point multiplication

Floating-point division

Floating-point remainder

Floating-point absolute value

Floating-point minimum

Floating-point maximum

Floating-point fused multiply-add

Floating-point square root

Floating-point round to integral

Floating-point equality

Floating-point less than

Floating-point greater than

Floating-point less than or equal

Floating-point greater than or equal

Floating-point isNaN

Floating-point isInfinite

Floating-point isZero

Floating-point isNormal

Floating-point isSubnormal

Floating-point isNegative

Floating-point isPositive

Floating-point constructor from 3 bit-vectors

Floating-point conversion (various)

Floating-point conversion from unsigned bit-vector

Floating-point conversion to unsigned bit-vector

Floating-point conversion to signed bit-vector

Floating-point conversion to real number

Floating-point conversion to IEEE-754 bit-vector

(Implicitly) represents the internal bitvector- representation of a floating-point term (used for the lazy encoding of non-relevant terms in theory_fpa)

Conversion of a 3-bit bit-vector term to a floating-point rounding-mode term

The conversion uses the following values: 0 = 000 = Z3_OP_FPA_RM_NEAREST_TIES_TO_EVEN, 1 = 001 = Z3_OP_FPA_RM_NEAREST_TIES_TO_AWAY, 2 = 010 = Z3_OP_FPA_RM_TOWARD_POSITIVE, 3 = 011 = Z3_OP_FPA_RM_TOWARD_NEGATIVE, 4 = 100 = Z3_OP_FPA_RM_TOWARD_ZERO.

internal (often interpreted) symbol, but no additional information is exposed. Tools may use the string representation of the function declaration to obtain more information.

function declared as recursive

kind used for uninterpreted symbols.