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z3_api.h
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z3_api.h
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/*++
Copyright (c) 2015 Microsoft Corporation
--*/
#pragma once
DEFINE_TYPE(Z3_symbol);
DEFINE_TYPE(Z3_literals);
DEFINE_TYPE(Z3_config);
DEFINE_TYPE(Z3_context);
DEFINE_TYPE(Z3_sort);
#define Z3_sort_opt Z3_sort
DEFINE_TYPE(Z3_func_decl);
DEFINE_TYPE(Z3_ast);
#define Z3_ast_opt Z3_ast
DEFINE_TYPE(Z3_app);
DEFINE_TYPE(Z3_pattern);
DEFINE_TYPE(Z3_model);
DEFINE_TYPE(Z3_constructor);
DEFINE_TYPE(Z3_constructor_list);
DEFINE_TYPE(Z3_params);
DEFINE_TYPE(Z3_param_descrs);
DEFINE_TYPE(Z3_parser_context);
DEFINE_TYPE(Z3_goal);
DEFINE_TYPE(Z3_tactic);
DEFINE_TYPE(Z3_probe);
DEFINE_TYPE(Z3_stats);
DEFINE_TYPE(Z3_solver);
DEFINE_TYPE(Z3_solver_callback);
DEFINE_TYPE(Z3_ast_vector);
DEFINE_TYPE(Z3_ast_map);
DEFINE_TYPE(Z3_apply_result);
DEFINE_TYPE(Z3_func_interp);
#define Z3_func_interp_opt Z3_func_interp
DEFINE_TYPE(Z3_func_entry);
DEFINE_TYPE(Z3_fixedpoint);
DEFINE_TYPE(Z3_optimize);
DEFINE_TYPE(Z3_rcf_num);
/** \defgroup capi C API */
/**@{*/
/** @name Types */
/**@{*/
/**
Most of the types in the C API are opaque pointers.
- \c Z3_config: configuration object used to initialize logical contexts.
- \c Z3_context: manager of all other Z3 objects, global configuration options, etc.
- \c Z3_symbol: Lisp-like symbol used to name types, constants, and functions. A symbol can be created using string or integers.
- \c Z3_ast: abstract syntax tree node. That is, the data-structure used in Z3 to represent terms, formulas and types.
- \c Z3_sort: kind of AST used to represent types.
- \c Z3_func_decl: kind of AST used to represent function symbols.
- \c Z3_app: kind of AST used to represent function applications.
- \c Z3_pattern: kind of AST used to represent pattern and multi-patterns used to guide quantifier instantiation.
- \c Z3_constructor: type constructor for a (recursive) datatype.
- \c Z3_constructor_list: list of constructors for a (recursive) datatype.
- \c Z3_params: parameter set used to configure many components such as: simplifiers, tactics, solvers, etc.
- \c Z3_param_descrs: provides a collection of parameter names, their types, default values and documentation strings. Solvers, tactics, and other objects accept different collection of parameters.
- \c Z3_parser_context: context for incrementally parsing strings. Declarations can be added incrementally to the parser state.
- \c Z3_model: model for the constraints asserted into the logical context.
- \c Z3_func_interp: interpretation of a function in a model.
- \c Z3_func_entry: representation of the value of a \c Z3_func_interp at a particular point.
- \c Z3_fixedpoint: context for the recursive predicate solver.
- \c Z3_optimize: context for solving optimization queries.
- \c Z3_ast_vector: vector of \c Z3_ast objects.
- \c Z3_ast_map: mapping from \c Z3_ast to \c Z3_ast objects.
- \c Z3_goal: set of formulas that can be solved and/or transformed using tactics and solvers.
- \c Z3_tactic: basic building block for creating custom solvers for specific problem domains.
- \c Z3_probe: function/predicate used to inspect a goal and collect information that may be used to decide which solver and/or preprocessing step will be used.
- \c Z3_apply_result: collection of subgoals resulting from applying of a tactic to a goal.
- \c Z3_solver: (incremental) solver, possibly specialized by a particular tactic or logic.
- \c Z3_stats: statistical data for a solver.
*/
/**
\brief Z3 string type. It is just an alias for \ccode{const char *}.
*/
typedef const char * Z3_string;
typedef char const* Z3_char_ptr;
typedef Z3_string * Z3_string_ptr;
/**
\brief Lifted Boolean type: \c false, \c undefined, \c true.
*/
typedef enum
{
Z3_L_FALSE = -1,
Z3_L_UNDEF,
Z3_L_TRUE
} Z3_lbool;
/**
\brief The different kinds of symbol.
In Z3, a symbol can be represented using integers and strings (See #Z3_get_symbol_kind).
\sa Z3_mk_int_symbol
\sa Z3_mk_string_symbol
*/
typedef enum
{
Z3_INT_SYMBOL,
Z3_STRING_SYMBOL
} Z3_symbol_kind;
/**
\brief The different kinds of parameters that can be associated with function symbols.
\sa Z3_get_decl_num_parameters
\sa Z3_get_decl_parameter_kind
- Z3_PARAMETER_INT is used for integer parameters.
- Z3_PARAMETER_DOUBLE is used for double parameters.
- Z3_PARAMETER_RATIONAL is used for parameters that are rational numbers.
- Z3_PARAMETER_SYMBOL is used for parameters that are symbols.
- Z3_PARAMETER_SORT is used for sort parameters.
- Z3_PARAMETER_AST is used for expression parameters.
- Z3_PARAMETER_FUNC_DECL is used for function declaration parameters.
*/
typedef enum
{
Z3_PARAMETER_INT,
Z3_PARAMETER_DOUBLE,
Z3_PARAMETER_RATIONAL,
Z3_PARAMETER_SYMBOL,
Z3_PARAMETER_SORT,
Z3_PARAMETER_AST,
Z3_PARAMETER_FUNC_DECL
} Z3_parameter_kind;
/**
\brief The different kinds of Z3 types (See #Z3_get_sort_kind).
*/
typedef enum
{
Z3_UNINTERPRETED_SORT,
Z3_BOOL_SORT,
Z3_INT_SORT,
Z3_REAL_SORT,
Z3_BV_SORT,
Z3_ARRAY_SORT,
Z3_DATATYPE_SORT,
Z3_RELATION_SORT,
Z3_FINITE_DOMAIN_SORT,
Z3_FLOATING_POINT_SORT,
Z3_ROUNDING_MODE_SORT,
Z3_SEQ_SORT,
Z3_RE_SORT,
Z3_CHAR_SORT,
Z3_UNKNOWN_SORT = 1000
} Z3_sort_kind;
/**
\brief
The different kinds of Z3 AST (abstract syntax trees). That is, terms, formulas and types.
- Z3_APP_AST: constant and applications
- Z3_NUMERAL_AST: numeral constants
- Z3_VAR_AST: bound variables
- Z3_QUANTIFIER_AST: quantifiers
- Z3_SORT_AST: sort
- Z3_FUNC_DECL_AST: function declaration
- Z3_UNKNOWN_AST: internal
*/
typedef enum
{
Z3_NUMERAL_AST,
Z3_APP_AST,
Z3_VAR_AST,
Z3_QUANTIFIER_AST,
Z3_SORT_AST,
Z3_FUNC_DECL_AST,
Z3_UNKNOWN_AST = 1000
} Z3_ast_kind;
/**
\brief The different kinds of interpreted function kinds.
- Z3_OP_TRUE The constant true.
- Z3_OP_FALSE The constant false.
- Z3_OP_EQ The equality predicate.
- Z3_OP_DISTINCT The n-ary distinct predicate (every argument is mutually distinct).
- Z3_OP_ITE The ternary if-then-else term.
- Z3_OP_AND n-ary conjunction.
- Z3_OP_OR n-ary disjunction.
- Z3_OP_IFF equivalence (binary).
- Z3_OP_XOR Exclusive or.
- Z3_OP_NOT Negation.
- Z3_OP_IMPLIES Implication.
- Z3_OP_OEQ Binary equivalence modulo namings. This binary predicate is used in proof terms.
It captures equisatisfiability and equivalence modulo renamings.
- Z3_OP_ANUM Arithmetic numeral.
- Z3_OP_AGNUM Arithmetic algebraic numeral. Algebraic numbers are used to represent irrational numbers in Z3.
- Z3_OP_LE <=.
- Z3_OP_GE >=.
- Z3_OP_LT <.
- Z3_OP_GT >.
- Z3_OP_ADD Addition - Binary.
- Z3_OP_SUB Binary subtraction.
- Z3_OP_UMINUS Unary minus.
- Z3_OP_MUL Multiplication - Binary.
- Z3_OP_DIV Division - Binary.
- Z3_OP_IDIV Integer division - Binary.
- Z3_OP_REM Remainder - Binary.
- Z3_OP_MOD Modulus - Binary.
- Z3_OP_TO_REAL Coercion of integer to real - Unary.
- Z3_OP_TO_INT Coercion of real to integer - Unary.
- Z3_OP_IS_INT Check if real is also an integer - Unary.
- Z3_OP_POWER Power operator x^y.
- Z3_OP_STORE 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.
- Z3_OP_SELECT Array select.
- Z3_OP_CONST_ARRAY The constant array. For example, select(const(v),i) = v holds for every v and i. The function is unary.
- Z3_OP_ARRAY_DEFAULT Default value of arrays. For example default(const(v)) = v. The function is unary.
- Z3_OP_ARRAY_MAP Array map operator.
It satisfies map[f](a1,..,a_n)[i] = f(a1[i],...,a_n[i]) for every i.
- Z3_OP_SET_UNION Set union between two Boolean arrays (two arrays whose range type is Boolean). The function is binary.
- Z3_OP_SET_INTERSECT Set intersection between two Boolean arrays. The function is binary.
- Z3_OP_SET_DIFFERENCE Set difference between two Boolean arrays. The function is binary.
- Z3_OP_SET_COMPLEMENT Set complement of a Boolean array. The function is unary.
- Z3_OP_SET_SUBSET Subset predicate between two Boolean arrays. The relation is binary.
- Z3_OP_AS_ARRAY An array value that behaves as the function graph of the
function passed as parameter.
- Z3_OP_ARRAY_EXT 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.
- Z3_OP_BNUM Bit-vector numeral.
- Z3_OP_BIT1 One bit bit-vector.
- Z3_OP_BIT0 Zero bit bit-vector.
- Z3_OP_BNEG Unary minus.
- Z3_OP_BADD Binary addition.
- Z3_OP_BSUB Binary subtraction.
- Z3_OP_BMUL Binary multiplication.
- Z3_OP_BSDIV Binary signed division.
- Z3_OP_BUDIV Binary unsigned division.
- Z3_OP_BSREM Binary signed remainder.
- Z3_OP_BUREM Binary unsigned remainder.
- Z3_OP_BSMOD Binary signed modulus.
- Z3_OP_BSDIV0 Unary function. bsdiv(x,0) is congruent to bsdiv0(x).
- Z3_OP_BUDIV0 Unary function. budiv(x,0) is congruent to budiv0(x).
- Z3_OP_BSREM0 Unary function. bsrem(x,0) is congruent to bsrem0(x).
- Z3_OP_BUREM0 Unary function. burem(x,0) is congruent to burem0(x).
- Z3_OP_BSMOD0 Unary function. bsmod(x,0) is congruent to bsmod0(x).
- Z3_OP_ULEQ Unsigned bit-vector <= - Binary relation.
- Z3_OP_SLEQ Signed bit-vector <= - Binary relation.
- Z3_OP_UGEQ Unsigned bit-vector >= - Binary relation.
- Z3_OP_SGEQ Signed bit-vector >= - Binary relation.
- Z3_OP_ULT Unsigned bit-vector < - Binary relation.
- Z3_OP_SLT Signed bit-vector < - Binary relation.
- Z3_OP_UGT Unsigned bit-vector > - Binary relation.
- Z3_OP_SGT Signed bit-vector > - Binary relation.
- Z3_OP_BAND Bit-wise and - Binary.
- Z3_OP_BOR Bit-wise or - Binary.
- Z3_OP_BNOT Bit-wise not - Unary.
- Z3_OP_BXOR Bit-wise xor - Binary.
- Z3_OP_BNAND Bit-wise nand - Binary.
- Z3_OP_BNOR Bit-wise nor - Binary.
- Z3_OP_BXNOR Bit-wise xnor - Binary.
- Z3_OP_CONCAT Bit-vector concatenation - Binary.
- Z3_OP_SIGN_EXT Bit-vector sign extension.
- Z3_OP_ZERO_EXT Bit-vector zero extension.
- Z3_OP_EXTRACT Bit-vector extraction.
- Z3_OP_REPEAT Repeat bit-vector n times.
- Z3_OP_BREDOR Bit-vector reduce or - Unary.
- Z3_OP_BREDAND Bit-vector reduce and - Unary.
- Z3_OP_BCOMP .
- Z3_OP_BSHL Shift left.
- Z3_OP_BLSHR Logical shift right.
- Z3_OP_BASHR Arithmetical shift right.
- Z3_OP_ROTATE_LEFT Left rotation.
- Z3_OP_ROTATE_RIGHT Right rotation.
- Z3_OP_EXT_ROTATE_LEFT (extended) Left rotation. Similar to Z3_OP_ROTATE_LEFT, but it is a binary operator instead of a parametric one.
- Z3_OP_EXT_ROTATE_RIGHT (extended) Right rotation. Similar to Z3_OP_ROTATE_RIGHT, but it is a binary operator instead of a parametric one.
- Z3_OP_INT2BV 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.
- Z3_OP_BV2INT 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.
- Z3_OP_CARRY 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)))
- Z3_OP_XOR3 Compute ternary XOR.
The meaning is given by the equivalence
(xor3 l1 l2 l3) <=> (xor (xor l1 l2) l3)
- Z3_OP_BSMUL_NO_OVFL: 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.
- Z3_OP_BUMUL_NO_OVFL: 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.
- Z3_OP_BSMUL_NO_UDFL: 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.
- Z3_OP_BSDIV_I: 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.
- Z3_OP_BUDIV_I: 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.
- Z3_OP_BSREM_I: 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.
- Z3_OP_BUREM_I: 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.
- Z3_OP_BSMOD_I: 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.
- Z3_OP_PR_UNDEF: Undef/Null proof object.
- Z3_OP_PR_TRUE: Proof for the expression 'true'.
- Z3_OP_PR_ASSERTED: Proof for a fact asserted by the user.
- Z3_OP_PR_GOAL: Proof for a fact (tagged as goal) asserted by the user.
- Z3_OP_PR_MODUS_PONENS: 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).
- Z3_OP_PR_REFLEXIVITY: 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'.
- Z3_OP_PR_SYMMETRY: 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.
- Z3_OP_PR_TRANSITIVITY: 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)
}
- Z3_OP_PR_TRANSITIVITY_STAR: 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.
- Z3_OP_PR_MONOTONICITY: 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.
- Z3_OP_PR_QUANT_INTRO: Given a proof for (~ p q), produces a proof for (~ (forall (x) p) (forall (x) q)).
T1: (~ p q)
[quant-intro T1]: (~ (forall (x) p) (forall (x) q))
- Z3_OP_PR_BIND: 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
- Z3_OP_PR_DISTRIBUTIVITY: 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.
- Z3_OP_PR_AND_ELIM: Given a proof for (and l_1 ... l_n), produces a proof for l_i
T1: (and l_1 ... l_n)
[and-elim T1]: l_i
- Z3_OP_PR_NOT_OR_ELIM: 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)
- Z3_OP_PR_REWRITE: 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)
}
- Z3_OP_PR_REWRITE_STAR: 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)
- Z3_OP_PR_PULL_QUANT: A proof for (iff (f (forall (x) q(x)) r) (forall (x) (f (q x) r))). This proof object has no antecedents.
- Z3_OP_PR_PUSH_QUANT: 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.
- Z3_OP_PR_ELIM_UNUSED_VARS:
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.
- Z3_OP_PR_DER: 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.
- Z3_OP_PR_QUANT_INST: A proof of (or (not (forall (x) (P x))) (P a))
- Z3_OP_PR_HYPOTHESIS: Mark a hypothesis in a natural deduction style proof.
- Z3_OP_PR_LEMMA:
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.
- Z3_OP_PR_UNIT_RESOLUTION:
\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')
}
- Z3_OP_PR_IFF_TRUE:
\nicebox{
T1: p
[iff-true T1]: (iff p true)
}
- Z3_OP_PR_IFF_FALSE:
\nicebox{
T1: (not p)
[iff-false T1]: (iff p false)
}
- Z3_OP_PR_COMMUTATIVITY:
[comm]: (= (f a b) (f b a))
f is a commutative operator.
This proof object has no antecedents.
Remark: if f is bool, then = is iff.
- Z3_OP_PR_DEF_AXIOM: 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).
- Z3_OP_PR_ASSUMPTION_ADD
Clausal proof adding axiom
- Z3_OP_PR_LEMMA_ADD
Clausal proof lemma addition
- Z3_OP_PR_REDUNDANT_DEL
Clausal proof lemma deletion
- Z3_OP_PR_CLAUSE_TRAIL,
Clausal proof trail of additions and deletions
- Z3_OP_PR_DEF_INTRO: 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:
[def-intro]: (and (or n (not e)) (or (not n) e))
or:
[def-intro]: (or (not n) e)
when e only occurs positively.
When e is of the form (ite cond th el):
[def-intro]: (and (or (not cond) (= n th)) (or cond (= n el)))
Otherwise:
[def-intro]: (= n e)
- Z3_OP_PR_APPLY_DEF:
[apply-def T1]: F ~ n
F is 'equivalent' to n, given that T1 is a proof that
n is a name for F.
- Z3_OP_PR_IFF_OEQ:
T1: (iff p q)
[iff~ T1]: (~ p q)
- Z3_OP_PR_NNF_POS: 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'
[nnf-pos T1 T2 T3 T4]: (~ (iff s_1 s_2) (and (or r_1 r_2') (or r_1' 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'.
- Z3_OP_PR_NNF_NEG: Proof for a (negative) NNF step. Examples:
T1: (not s_1) ~ r_1
...
Tn: (not s_n) ~ r_n
[nnf-neg T1 ... Tn]: (not (and s_1 ... s_n)) ~ (or r_1 ... r_n)
and
T1: (not s_1) ~ r_1
...
Tn: (not s_n) ~ r_n
[nnf-neg T1 ... Tn]: (not (or s_1 ... s_n)) ~ (and r_1 ... 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'
[nnf-neg T1 T2 T3 T4]: (~ (not (iff s_1 s_2))
(and (or r_1 r_2) (or r_1' r_2')))
- Z3_OP_PR_SKOLEMIZE: 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.
- Z3_OP_PR_MODUS_PONENS_OEQ: Modus ponens style rule for equi-satisfiability.
T1: p
T2: (~ p q)
[mp~ T1 T2]: q
- Z3_OP_PR_TH_LEMMA: 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.
- Z3_OP_PR_HYPER_RESOLVE: 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.
- Z3_OP_RA_STORE: Insert a record into a relation.
The function takes \c n+1 arguments, where the first argument is the relation and the remaining \c n elements
correspond to the \c n columns of the relation.
- Z3_OP_RA_EMPTY: Creates the empty relation.
- Z3_OP_RA_IS_EMPTY: Tests if the relation is empty.
- Z3_OP_RA_JOIN: Create the relational join.
- Z3_OP_RA_UNION: Create the union or convex hull of two relations.
The function takes two arguments.
- Z3_OP_RA_WIDEN: Widen two relations.
The function takes two arguments.
- Z3_OP_RA_PROJECT: Project the columns (provided as numbers in the parameters).
The function takes one argument.
- Z3_OP_RA_FILTER: 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.
- Z3_OP_RA_NEGATION_FILTER: 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.
- Z3_OP_RA_RENAME: rename columns in the relation.
The function takes one argument.
The parameters contain the renaming as a cycle.
- Z3_OP_RA_COMPLEMENT: Complement the relation.
- Z3_OP_RA_SELECT: Check if a record is an element of the relation.
The function takes \c n+1 arguments, where the first argument is a relation,
and the remaining \c n arguments correspond to a record.
- Z3_OP_RA_CLONE: 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.
- Z3_OP_FD_LT: A less than predicate over the finite domain Z3_FINITE_DOMAIN_SORT.
- Z3_OP_LABEL: 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.
- Z3_OP_LABEL_LIT: A label literal (used by the Boogie Verification condition generator).
A label literal has a set of string parameters. It takes no arguments.
- Z3_OP_DT_CONSTRUCTOR: datatype constructor.
- Z3_OP_DT_RECOGNISER: datatype recognizer.
- Z3_OP_DT_IS: datatype recognizer.
- Z3_OP_DT_ACCESSOR: datatype accessor.
- Z3_OP_DT_UPDATE_FIELD: datatype field update.
- Z3_OP_PB_AT_MOST: Cardinality constraint.
E.g., x + y + z <= 2
- Z3_OP_PB_AT_LEAST: Cardinality constraint.
E.g., x + y + z >= 2
- Z3_OP_PB_LE: Generalized Pseudo-Boolean cardinality constraint.
Example 2*x + 3*y <= 4
- Z3_OP_PB_GE: Generalized Pseudo-Boolean cardinality constraint.
Example 2*x + 3*y + 2*z >= 4
- Z3_OP_PB_EQ: Generalized Pseudo-Boolean equality constraint.
Example 2*x + 1*y + 2*z + 1*u = 4
- Z3_OP_SPECIAL_RELATION_LO: A relation that is a total linear order
- Z3_OP_SPECIAL_RELATION_PO: A relation that is a partial order
- Z3_OP_SPECIAL_RELATION_PLO: A relation that is a piecewise linear order
- Z3_OP_SPECIAL_RELATION_TO: A relation that is a tree order
- Z3_OP_SPECIAL_RELATION_TC: Transitive closure of a relation
- Z3_OP_SPECIAL_RELATION_TRC: Transitive reflexive closure of a relation
- Z3_OP_FPA_RM_NEAREST_TIES_TO_EVEN: Floating-point rounding mode RNE
- Z3_OP_FPA_RM_NEAREST_TIES_TO_AWAY: Floating-point rounding mode RNA
- Z3_OP_FPA_RM_TOWARD_POSITIVE: Floating-point rounding mode RTP
- Z3_OP_FPA_RM_TOWARD_NEGATIVE: Floating-point rounding mode RTN
- Z3_OP_FPA_RM_TOWARD_ZERO: Floating-point rounding mode RTZ
- Z3_OP_FPA_NUM: Floating-point value
- Z3_OP_FPA_PLUS_INF: Floating-point +oo
- Z3_OP_FPA_MINUS_INF: Floating-point -oo
- Z3_OP_FPA_NAN: Floating-point NaN
- Z3_OP_FPA_PLUS_ZERO: Floating-point +zero
- Z3_OP_FPA_MINUS_ZERO: Floating-point -zero
- Z3_OP_FPA_ADD: Floating-point addition
- Z3_OP_FPA_SUB: Floating-point subtraction
- Z3_OP_FPA_NEG: Floating-point negation
- Z3_OP_FPA_MUL: Floating-point multiplication
- Z3_OP_FPA_DIV: Floating-point division
- Z3_OP_FPA_REM: Floating-point remainder
- Z3_OP_FPA_ABS: Floating-point absolute value
- Z3_OP_FPA_MIN: Floating-point minimum
- Z3_OP_FPA_MAX: Floating-point maximum
- Z3_OP_FPA_FMA: Floating-point fused multiply-add
- Z3_OP_FPA_SQRT: Floating-point square root
- Z3_OP_FPA_ROUND_TO_INTEGRAL: Floating-point round to integral
- Z3_OP_FPA_EQ: Floating-point equality
- Z3_OP_FPA_LT: Floating-point less than
- Z3_OP_FPA_GT: Floating-point greater than
- Z3_OP_FPA_LE: Floating-point less than or equal
- Z3_OP_FPA_GE: Floating-point greater than or equal
- Z3_OP_FPA_IS_NAN: Floating-point isNaN
- Z3_OP_FPA_IS_INF: Floating-point isInfinite
- Z3_OP_FPA_IS_ZERO: Floating-point isZero
- Z3_OP_FPA_IS_NORMAL: Floating-point isNormal
- Z3_OP_FPA_IS_SUBNORMAL: Floating-point isSubnormal
- Z3_OP_FPA_IS_NEGATIVE: Floating-point isNegative
- Z3_OP_FPA_IS_POSITIVE: Floating-point isPositive
- Z3_OP_FPA_FP: Floating-point constructor from 3 bit-vectors
- Z3_OP_FPA_TO_FP: Floating-point conversion (various)
- Z3_OP_FPA_TO_FP_UNSIGNED: Floating-point conversion from unsigned bit-vector
- Z3_OP_FPA_TO_UBV: Floating-point conversion to unsigned bit-vector
- Z3_OP_FPA_TO_SBV: Floating-point conversion to signed bit-vector
- Z3_OP_FPA_TO_REAL: Floating-point conversion to real number
- Z3_OP_FPA_TO_IEEE_BV: Floating-point conversion to IEEE-754 bit-vector
- Z3_OP_FPA_BVWRAP: (Implicitly) represents the internal bitvector-
representation of a floating-point term (used for the lazy encoding
of non-relevant terms in theory_fpa)
- Z3_OP_FPA_BV2RM: 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.
- Z3_OP_INTERNAL: internal (often interpreted) symbol, but no additional
information is exposed. Tools may use the string representation of the
function declaration to obtain more information.
- Z3_OP_RECURSIVE: function declared as recursive
- Z3_OP_UNINTERPRETED: kind used for uninterpreted symbols.
*/
typedef enum {
// Basic
Z3_OP_TRUE = 0x100,
Z3_OP_FALSE,
Z3_OP_EQ,
Z3_OP_DISTINCT,
Z3_OP_ITE,
Z3_OP_AND,
Z3_OP_OR,
Z3_OP_IFF,
Z3_OP_XOR,