Proof Rules

This section introduces two enumerations: ProofRule and ProofRewriteRule.

The ProofRule enumeration captures the reasoning steps performed by the SAT solver, the theory solvers or by the preprocessor. It represents the inference rules used to derive conclusions within a proof.

The ProofRewriteRule enumeration pertains to rewrites performed on terms. These identifiers are arguments of the proof rules THEORY_REWRITE and DSL_REWRITE.



enum class cvc5::ProofRule

An enumeration for proof rules. This enumeration is analogous to Kind for Node objects.

All proof rules are given as inference rules, presented in the following form:

\[\texttt{RULENAME}: \inferruleSC{\varphi_1 \dots \varphi_n \mid t_1 \dots t_m}{\psi}{if $C$}\]

where we call \(\varphi_i\) its premises or children, \(t_i\) its arguments, \(\psi\) its conclusion, and \(C\) its side condition. Alternatively, we can write the application of a proof rule as (RULENAME F1 ... Fn :args t1 ... tm), omitting the conclusion (since it can be uniquely determined from premises and arguments). Note that premises are sometimes given as proofs, i.e., application of proof rules, instead of formulas. This abuses the notation to see proof rule applications and their conclusions interchangeably.

Conceptually, the following proof rules form a calculus whose target user is the Node-level theory solvers. This means that the rules below are designed to reason about, among other things, common operations on Node objects like Rewriter::rewrite or Node::substitute. It is intended to be translated or printed in other formats.

The following ProofRule values include core rules and those categorized by theory, including the theory of equality.

The “core rules” include two distinguished rules which have special status: (1) ASSUME, which represents an open leaf in a proof; and (2) SCOPE, which encloses a scope (a subproof) with a set of scoped assumptions. The core rules additionally correspond to generic operations that are done internally on nodes, e.g., calling Rewriter::rewrite.

Rules with prefix MACRO_ are those that can be defined in terms of other rules. These exist for convenience and can be replaced by their definition in post-processing.

Values:

enumerator ASSUME

Assumption (a leaf)

\[\inferrule{- \mid F}{F}\]

This rule has special status, in that an application of assume is an open leaf in a proof that is not (yet) justified. An assume leaf is analogous to a free variable in a term, where we say “F is a free assumption in proof P” if it contains an application of F that is not bound by SCOPE (see below).

enumerator SCOPE

Scope (a binder for assumptions)

\[\inferruleSC{F \mid F_1 \dots F_n}{(F_1 \land \dots \land F_n) \Rightarrow F}{if $F\neq\bot$} \textrm{ or } \inferruleSC{F \mid F_1 \dots F_n}{\neg (F_1 \land \dots \land F_n)}{if $F=\bot$}\]

This rule has a dual purpose with ASSUME. It is a way to close assumptions in a proof. We require that \(F_1 \dots F_n\) are free assumptions in P and say that \(F_1 \dots F_n\) are not free in (SCOPE P). In other words, they are bound by this application. For example, the proof node: (SCOPE (ASSUME F) :args F) has the conclusion \(F \Rightarrow F\) and has no free assumptions. More generally, a proof with no free assumptions always concludes a valid formula.

enumerator SUBS

Builtin theory – Substitution

\[\inferrule{F_1 \dots F_n \mid t, ids?}{t = t \circ \sigma_{ids}(F_n) \circ \cdots \circ \sigma_{ids}(F_1)}\]

where \(\sigma_{ids}(F_i)\) are substitutions, which notice are applied in reverse order. Notice that \(ids\) is a MethodId identifier, which determines how to convert the formulas \(F_1 \dots F_n\) into substitutions. It is an optional argument, where by default the premises are equalities of the form (= x y) and converted into substitutions \(x\mapsto y\).

enumerator MACRO_REWRITE

Builtin theory – Rewrite

\[\inferrule{- \mid t, idr}{t = \texttt{Rewriter}_{idr}(t)}\]

where \(idr\) is a MethodId identifier, which determines the kind of rewriter to apply, e.g. Rewriter::rewrite.

enumerator EVALUATE

Builtin theory – Evaluate

\[\inferrule{- \mid t}{t = \texttt{Evaluator::evaluate}(t)}\]

Note this is equivalent to: (REWRITE t MethodId::RW_EVALUATE).

enumerator ACI_NORM

Builtin theory – associative/commutative/idempotency/identity normalization

\[\inferrule{- \mid t = s}{t = s}\]

where \(\texttt{expr::isACNorm(t, s)} = \top\). This method normalizes currently based on two kinds of operators: (1) those that are associative, commutative, idempotent, and have an identity element (examples are or, and, bvand), (2) those that are associative and have an identity element (examples are str.++, re.++).

enumerator MACRO_SR_EQ_INTRO

Builtin theory – Substitution + Rewriting equality introduction

In this rule, we provide a term \(t\) and conclude that it is equal to its rewritten form under a (proven) substitution.

\[\inferrule{F_1 \dots F_n \mid t, (ids (ida (idr)?)?)?}{t = \texttt{Rewriter}_{idr}(t \circ \sigma_{ids, ida}(F_n) \circ \cdots \circ \sigma_{ids, ida}(F_1))}\]

In other words, from the point of view of Skolem forms, this rule transforms \(t\) to \(t'\) by standard substitution + rewriting.

The arguments \(ids\), \(ida\) and \(idr\) are optional and specify the identifier of the substitution, the substitution application and rewriter respectively to be used. For details, see theory/builtin/proof_checker.h.

enumerator MACRO_SR_PRED_INTRO

Builtin theory – Substitution + Rewriting predicate introduction

In this rule, we provide a formula \(F\) and conclude it, under the condition that it rewrites to true under a proven substitution.

\[\inferrule{F_1 \dots F_n \mid F, (ids (ida (idr)?)?)?}{F}\]

where \(\texttt{Rewriter}_{idr}(F \circ \sigma_{ids, ida}(F_n) \circ \cdots \circ \sigma_{ids, ida}(F_1)) = \top\) and \(ids\) and \(idr\) are method identifiers.

More generally, this rule also holds when \(\texttt{Rewriter::rewrite}(\texttt{toOriginal}(F')) = \top\) where \(F'\) is the result of the left hand side of the equality above. Here, notice that we apply rewriting on the original form of \(F'\), meaning that this rule may conclude an \(F\) whose Skolem form is justified by the definition of its (fresh) Skolem variables. For example, this rule may justify the conclusion \(k = t\) where \(k\) is the purification Skolem for \(t\), e.g. where the original form of \(k\) is \(t\).

Furthermore, notice that the rewriting and substitution is applied only within the side condition, meaning the rewritten form of the original form of \(F\) does not escape this rule.

enumerator MACRO_SR_PRED_ELIM

Builtin theory – Substitution + Rewriting predicate elimination

\[\inferrule{F, F_1 \dots F_n \mid (ids (ida (idr)?)?)?}{\texttt{Rewriter}_{idr}(F \circ \sigma_{ids, ida}(F_n) \circ \cdots \circ \sigma_{ids, ida}(F_1))}\]

where \(ids\) and \(idr\) are method identifiers.

We rewrite only on the Skolem form of \(F\), similar to MACRO_SR_EQ_INTRO.

enumerator MACRO_SR_PRED_TRANSFORM

Builtin theory – Substitution + Rewriting predicate elimination

\[\inferrule{F, F_1 \dots F_n \mid G, (ids (ida (idr)?)?)?}{G}\]

where \(\texttt{Rewriter}_{idr}(F \circ \sigma_{ids, ida}(F_n) \circ \cdots \circ \sigma_{ids, ida}(F_1)) = \texttt{Rewriter}_{idr}(G \circ \sigma_{ids, ida}(F_n) \circ \cdots \circ \sigma_{ids, ida}(F_1))\).

More generally, this rule also holds when: \(\texttt{Rewriter::rewrite}(\texttt{toOriginal}(F')) = \texttt{Rewriter::rewrite}(\texttt{toOriginal}(G'))\) where \(F'\) and \(G'\) are the result of each side of the equation above. Here, original forms are used in a similar manner to MACRO_SR_PRED_INTRO above.

enumerator ENCODE_EQ_INTRO

Builtin theory – Encode equality introduction

\[\inferrule{- \mid t}{t=t'}\]

where \(t\) and \(t'\) are equivalent up to their encoding in an external proof format.

More specifically, it is the case that \(\texttt{RewriteDbNodeConverter::postConvert}(t) = t;\). This conversion method for instance may drop user patterns from quantified formulas or change the representation of \(t\) in a way that is a no-op in external proof formats.

Note this rule can be treated as a REFL when appropriate in external proof formats.

enumerator DSL_REWRITE

Builtin theory – DSL rewrite

\[\inferrule{F_1 \dots F_n \mid id t_1 \dots t_n}{F}\]

where id is a ProofRewriteRule whose definition in the RARE DSL is \(\forall x_1 \dots x_n. (G_1 \wedge G_n) \Rightarrow G\) where for \(i=1, \dots n\), we have that \(F_i = \sigma(G_i)\) and \(F = \sigma(G)\) where \(\sigma\) is the substitution \(\{x_1\mapsto t_1,\dots,x_n\mapsto t_n\}\).

Notice that the application of the substitution takes into account the possible list semantics of variables \(x_1 \ldots x_n\). If \(x_i\) is a variable with list semantics, then \(t_i\) denotes a list of terms. The substitution implemented by expr::narySubstitute replaces each \(x_i\) with the list \(t_i\) in its place.

enumerator THEORY_REWRITE

Other theory rewrite rules

\[\inferrule{- \mid id, t = t'}{t = t'}\]

where id is the ProofRewriteRule of the theory rewrite rule which transforms \(t\) to \(t'\).

In contrast to DSL_REWRITE, theory rewrite rules used by this proof rule are not necessarily expressible in RARE. Each rule that can be used in this proof rule are documented explicitly in cases within the ProofRewriteRule enum.

enumerator ANNOTATION

Builtin theory – Annotation

\[\inferrule{F \mid a_1 \dots a_n}{F}\]

The terms \(a_1 \dots a_n\) can be anything used to annotate the proof node, one example is where \(a_1\) is a theory::InferenceId.

enumerator ITE_EQ

Processing rules – If-then-else equivalence

\[\inferrule{- \mid \ite{C}{t_1}{t_2}}{\ite{C}{((\ite{C}{t_1}{t_2}) = t_1)}{((\ite{C}{t_1}{t_2}) = t_2)}}\]

enumerator TRUST

Trusted rule

\[\inferrule{F_1 \dots F_n \mid tid, F, ...}{F}\]

where \(tid\) is an identifier and \(F\) is a formula. This rule is used when a formal justification of an inference step cannot be provided. The formulas \(F_1 \dots F_n\) refer to a set of formulas that entail \(F\), which may or may not be provided.

enumerator TRUST_THEORY_REWRITE

Trusted rules – Theory rewrite

\[\inferrule{- \mid F, tid, rid}{F}\]

where \(F\) is an equality of the form \(t = t'\) where \(t'\) is obtained by applying the kind of rewriting given by the method identifier \(rid\), which is one of: RW_REWRITE_THEORY_PRE, RW_REWRITE_THEORY_POST, RW_REWRITE_EQ_EXT. Notice that the checker for this rule does not replay the rewrite to ensure correctness, since theory rewriter methods are not static. For example, the quantifiers rewriter involves constructing new bound variables that are not guaranteed to be consistent on each call.

enumerator SAT_REFUTATION

SAT Refutation for assumption-based unsat cores

\[\inferrule{F_1 \dots F_n \mid -}{\bot}\]

where \(F_1 \dots F_n\) correspond to the unsat core determined by the SAT solver.

enumerator DRAT_REFUTATION

DRAT Refutation

\[\inferrule{F_1 \dots F_n \mid D, P}{\bot}\]

where \(F_1 \dots F_n\) correspond to the clauses in the DIMACS file given by filename D and P is a filename of a file storing a DRAT proof.

enumerator SAT_EXTERNAL_PROVE

SAT external prove Refutation

\[\inferrule{F_1 \dots F_n \mid D}{\bot}\]

where \(F_1 \dots F_n\) correspond to the input clauses in the DIMACS file D.

enumerator RESOLUTION

Boolean – Resolution

\[\inferrule{C_1, C_2 \mid pol, L}{C}\]

where

  • \(C_1\) and \(C_2\) are nodes viewed as clauses, i.e., either an OR node with each children viewed as a literal or a node viewed as a literal. Note that an OR node could also be a literal.

  • \(pol\) is either true or false, representing the polarity of the pivot on the first clause

  • \(L\) is the pivot of the resolution, which occurs as is (resp. under a NOT) in \(C_1\) and negatively (as is) in \(C_2\) if \(pol = \top\) (\(pol = \bot\)).

\(C\) is a clause resulting from collecting all the literals in \(C_1\), minus the first occurrence of the pivot or its negation, and \(C_2\), minus the first occurrence of the pivot or its negation, according to the policy above. If the resulting clause has a single literal, that literal itself is the result; if it has no literals, then the result is false; otherwise it’s an OR node of the resulting literals.

Note that it may be the case that the pivot does not occur in the clauses. In this case the rule is not unsound, but it does not correspond to resolution but rather to a weakening of the clause that did not have a literal eliminated.

enumerator CHAIN_RESOLUTION

Boolean – N-ary Resolution

\[\inferrule{C_1 \dots C_n \mid (pol_1 \dots pol_{n-1}), (L_1 \dots L_{n-1})}{C}\]

where

  • let \(C_1 \dots C_n\) be nodes viewed as clauses, as defined above

  • let \(C_1 \diamond_{L,\mathit{pol}} C_2\) represent the resolution of \(C_1\) with \(C_2\) with pivot \(L\) and polarity \(pol\), as defined above

  • let \(C_1' = C_1\),

  • for each \(i > 1\), let \(C_i' = C_{i-1} \diamond{L_{i-1}, \mathit{pol}_{i-1}} C_i'\)

Note the list of polarities and pivots are provided as s-expressions.

The result of the chain resolution is \(C = C_n'\)

enumerator FACTORING

Boolean – Factoring

\[\inferrule{C_1 \mid -}{C_2}\]

where \(C_2\) is the clause \(C_1\), but every occurrence of a literal after its first occurrence is omitted.

enumerator REORDERING

Boolean – Reordering

\[\inferrule{C_1 \mid C_2}{C_2}\]

where the set representations of \(C_1\) and \(C_2\) are the same and the number of literals in \(C_2\) is the same of that of \(C_1\).

enumerator MACRO_RESOLUTION

Boolean – N-ary Resolution + Factoring + Reordering

\[\inferrule{C_1 \dots C_n \mid C, pol_1,L_1 \dots pol_{n-1},L_{n-1}}{C}\]

where

  • let \(C_1 \dots C_n\) be nodes viewed as clauses, as defined in RESOLUTION

  • let \(C_1 \diamond{L,\mathit{pol}} C_2\) represent the resolution of \(C_1\) with \(C_2\) with pivot \(L\) and polarity \(pol\), as defined in RESOLUTION

  • let \(C_1'\) be equal, in its set representation, to \(C_1\),

  • for each \(i > 1\), let \(C_i'\) be equal, in its set representation, to \(C_{i-1} \diamond{L_{i-1},\mathit{pol}_{i-1}} C_i'\)

The result of the chain resolution is \(C\), which is equal, in its set representation, to \(C_n'\)

enumerator MACRO_RESOLUTION_TRUST

Boolean – N-ary Resolution + Factoring + Reordering unchecked

Same as MACRO_RESOLUTION, but not checked by the internal proof checker.

enumerator SPLIT

Boolean – Split

\[\inferrule{- \mid F}{F \lor \neg F}\]

enumerator EQ_RESOLVE

Boolean – Equality resolution

\[\inferrule{F_1, (F_1 = F_2) \mid -}{F_2}\]

Note this can optionally be seen as a macro for EQUIV_ELIM1 + RESOLUTION.

enumerator MODUS_PONENS

Boolean – Modus Ponens

\[\inferrule{F_1, (F_1 \rightarrow F_2) \mid -}{F_2}\]

Note this can optionally be seen as a macro for IMPLIES_ELIM + RESOLUTION.

enumerator NOT_NOT_ELIM

Boolean – Double negation elimination

\[\inferrule{\neg (\neg F) \mid -}{F}\]

enumerator CONTRA

Boolean – Contradiction

\[\inferrule{F, \neg F \mid -}{\bot}\]

enumerator AND_ELIM

Boolean – And elimination

\[\inferrule{(F_1 \land \dots \land F_n) \mid i}{F_i}\]

enumerator AND_INTRO

Boolean – And introduction

\[\inferrule{F_1 \dots F_n \mid -}{(F_1 \land \dots \land F_n)}\]

enumerator NOT_OR_ELIM

Boolean – Not Or elimination

\[\inferrule{\neg(F_1 \lor \dots \lor F_n) \mid i}{\neg F_i}\]

enumerator IMPLIES_ELIM

Boolean – Implication elimination

\[\inferrule{F_1 \rightarrow F_2 \mid -}{\neg F_1 \lor F_2}\]

enumerator NOT_IMPLIES_ELIM1

Boolean – Not Implication elimination version 1

\[\inferrule{\neg(F_1 \rightarrow F_2) \mid -}{F_1}\]

enumerator NOT_IMPLIES_ELIM2

Boolean – Not Implication elimination version 2

\[\inferrule{\neg(F_1 \rightarrow F_2) \mid -}{\neg F_2}\]

enumerator EQUIV_ELIM1

Boolean – Equivalence elimination version 1

\[\inferrule{F_1 = F_2 \mid -}{\neg F_1 \lor F_2}\]

enumerator EQUIV_ELIM2

Boolean – Equivalence elimination version 2

\[\inferrule{F_1 = F_2 \mid -}{F_1 \lor \neg F_2}\]

enumerator NOT_EQUIV_ELIM1

Boolean – Not Equivalence elimination version 1

\[\inferrule{F_1 \neq F_2 \mid -}{F_1 \lor F_2}\]

enumerator NOT_EQUIV_ELIM2

Boolean – Not Equivalence elimination version 2

\[\inferrule{F_1 \neq F_2 \mid -}{\neg F_1 \lor \neg F_2}\]

enumerator XOR_ELIM1

Boolean – XOR elimination version 1

\[\inferrule{F_1 \xor F_2 \mid -}{F_1 \lor F_2}\]

enumerator XOR_ELIM2

Boolean – XOR elimination version 2

\[\inferrule{F_1 \xor F_2 \mid -}{\neg F_1 \lor \neg F_2}\]

enumerator NOT_XOR_ELIM1

Boolean – Not XOR elimination version 1

\[\inferrule{\neg(F_1 \xor F_2) \mid -}{F_1 \lor \neg F_2}\]

enumerator NOT_XOR_ELIM2

Boolean – Not XOR elimination version 2

\[\inferrule{\neg(F_1 \xor F_2) \mid -}{\neg F_1 \lor F_2}\]

enumerator ITE_ELIM1

Boolean – ITE elimination version 1

\[\inferrule{(\ite{C}{F_1}{F_2}) \mid -}{\neg C \lor F_1}\]

enumerator ITE_ELIM2

Boolean – ITE elimination version 2

\[\inferrule{(\ite{C}{F_1}{F_2}) \mid -}{C \lor F_2}\]

enumerator NOT_ITE_ELIM1

Boolean – Not ITE elimination version 1

\[\inferrule{\neg(\ite{C}{F_1}{F_2}) \mid -}{\neg C \lor \neg F_1}\]

enumerator NOT_ITE_ELIM2

Boolean – Not ITE elimination version 2

\[\inferrule{\neg(\ite{C}{F_1}{F_2}) \mid -}{C \lor \neg F_2}\]

enumerator NOT_AND

Boolean – De Morgan – Not And

\[\inferrule{\neg(F_1 \land \dots \land F_n) \mid -}{\neg F_1 \lor \dots \lor \neg F_n}\]

enumerator CNF_AND_POS

Boolean – CNF – And Positive

\[\inferrule{- \mid (F_1 \land \dots \land F_n), i}{\neg (F_1 \land \dots \land F_n) \lor F_i}\]

enumerator CNF_AND_NEG

Boolean – CNF – And Negative

\[\inferrule{- \mid (F_1 \land \dots \land F_n)}{(F_1 \land \dots \land F_n) \lor \neg F_1 \lor \dots \lor \neg F_n}\]

enumerator CNF_OR_POS

Boolean – CNF – Or Positive

\[\inferrule{- \mid (F_1 \lor \dots \lor F_n)}{\neg(F_1 \lor \dots \lor F_n) \lor F_1 \lor \dots \lor F_n}\]

enumerator CNF_OR_NEG

Boolean – CNF – Or Negative

\[\inferrule{- \mid (F_1 \lor \dots \lor F_n), i}{(F_1 \lor \dots \lor F_n) \lor \neg F_i}\]

enumerator CNF_IMPLIES_POS

Boolean – CNF – Implies Positive

\[\inferrule{- \mid F_1 \rightarrow F_2}{\neg(F_1 \rightarrow F_2) \lor \neg F_1 \lor F_2}\]

enumerator CNF_IMPLIES_NEG1

Boolean – CNF – Implies Negative 1

\[\inferrule{- \mid F_1 \rightarrow F_2}{(F_1 \rightarrow F_2) \lor F_1}\]

enumerator CNF_IMPLIES_NEG2

Boolean – CNF – Implies Negative 2

\[\inferrule{- \mid F_1 \rightarrow F_2}{(F_1 \rightarrow F_2) \lor \neg F_2}\]

enumerator CNF_EQUIV_POS1

Boolean – CNF – Equiv Positive 1

\[\inferrule{- \mid F_1 = F_2}{F_1 \neq F_2 \lor \neg F_1 \lor F_2}\]

enumerator CNF_EQUIV_POS2

Boolean – CNF – Equiv Positive 2

\[\inferrule{- \mid F_1 = F_2}{F_1 \neq F_2 \lor F_1 \lor \neg F_2}\]

enumerator CNF_EQUIV_NEG1

Boolean – CNF – Equiv Negative 1

\[\inferrule{- \mid F_1 = F_2}{(F_1 = F_2) \lor F_1 \lor F_2}\]

enumerator CNF_EQUIV_NEG2

Boolean – CNF – Equiv Negative 2

\[\inferrule{- \mid F_1 = F_2}{(F_1 = F_2) \lor \neg F_1 \lor \neg F_2}\]

enumerator CNF_XOR_POS1

Boolean – CNF – XOR Positive 1

\[\inferrule{- \mid F_1 \xor F_2}{\neg(F_1 \xor F_2) \lor F_1 \lor F_2}\]

enumerator CNF_XOR_POS2

Boolean – CNF – XOR Positive 2

\[\inferrule{- \mid F_1 \xor F_2}{\neg(F_1 \xor F_2) \lor \neg F_1 \lor \neg F_2}\]

enumerator CNF_XOR_NEG1

Boolean – CNF – XOR Negative 1

\[\inferrule{- \mid F_1 \xor F_2}{(F_1 \xor F_2) \lor \neg F_1 \lor F_2}\]

enumerator CNF_XOR_NEG2

Boolean – CNF – XOR Negative 2

\[\inferrule{- \mid F_1 \xor F_2}{(F_1 \xor F_2) \lor F_1 \lor \neg F_2}\]

enumerator CNF_ITE_POS1

Boolean – CNF – ITE Positive 1

\[\inferrule{- \mid (\ite{C}{F_1}{F_2})}{\neg(\ite{C}{F_1}{F_2}) \lor \neg C \lor F_1}\]

enumerator CNF_ITE_POS2

Boolean – CNF – ITE Positive 2

\[\inferrule{- \mid (\ite{C}{F_1}{F_2})}{\neg(\ite{C}{F_1}{F_2}) \lor C \lor F_2}\]

enumerator CNF_ITE_POS3

Boolean – CNF – ITE Positive 3

\[\inferrule{- \mid (\ite{C}{F_1}{F_2})}{\neg(\ite{C}{F_1}{F_2}) \lor F_1 \lor F_2}\]

enumerator CNF_ITE_NEG1

Boolean – CNF – ITE Negative 1

\[\inferrule{- \mid (\ite{C}{F_1}{F_2})}{(\ite{C}{F_1}{F_2}) \lor \neg C \lor \neg F_1}\]

enumerator CNF_ITE_NEG2

Boolean – CNF – ITE Negative 2

\[\inferrule{- \mid (\ite{C}{F_1}{F_2})}{(\ite{C}{F_1}{F_2}) \lor C \lor \neg F_2}\]

enumerator CNF_ITE_NEG3

Boolean – CNF – ITE Negative 3

\[\inferrule{- \mid (\ite{C}{F_1}{F_2})}{(\ite{C}{F_1}{F_2}) \lor \neg F_1 \lor \neg F_2}\]

enumerator REFL

Equality – Reflexivity

\[\inferrule{-\mid t}{t = t}\]

enumerator SYMM

Equality – Symmetry

\[\inferrule{t_1 = t_2\mid -}{t_2 = t_1}\]

or

\[\inferrule{t_1 \neq t_2\mid -}{t_2 \neq t_1}\]

enumerator TRANS

Equality – Transitivity

\[\inferrule{t_1=t_2,\dots,t_{n-1}=t_n\mid -}{t_1 = t_n}\]

enumerator CONG

Equality – Congruence

\[\inferrule{t_1=s_1,\dots,t_n=s_n\mid k, f?}{k(f?)(t_1,\dots, t_n) = k(f?)(s_1,\dots, s_n)}\]

where \(k\) is the application kind. Notice that \(f\) must be provided iff \(k\) is a parameterized kind, e.g. cvc5::Kind::APPLY_UF. The actual node for \(k\) is constructible via ProofRuleChecker::mkKindNode. If \(k\) is a binder kind (e.g. cvc5::Kind::FORALL) then \(f\) is a term of kind cvc5::Kind::VARIABLE_LIST denoting the variables bound by both sides of the conclusion. This rule is used for kinds that have a fixed arity, such as cvc5::Kind::ITE, cvc5::Kind::EQUAL, and so on. It is also used for cvc5::Kind::APPLY_UF where \(f\) must be provided. It is not used for equality between cvc5::Kind::HO_APPLY terms, which should use the HO_CONG proof rule.

enumerator NARY_CONG

Equality – N-ary Congruence

\[\inferrule{t_1=s_1,\dots,t_n=s_n\mid k}{k(t_1,\dots, t_n) = k(s_1,\dots, s_n)}\]

where \(k\) is the application kind. The actual node for \(k\) is constructible via ProofRuleChecker::mkKindNode. This rule is used for kinds that have variadic arity, such as cvc5::Kind::AND, cvc5::Kind::PLUS and so on.

enumerator TRUE_INTRO

Equality – True intro

\[\inferrule{F\mid -}{F = \top}\]

enumerator TRUE_ELIM

Equality – True elim

\[\inferrule{F=\top\mid -}{F}\]

enumerator FALSE_INTRO

Equality – False intro

\[\inferrule{\neg F\mid -}{F = \bot}\]

enumerator FALSE_ELIM

Equality – False elim

\[\inferrule{F=\bot\mid -}{\neg F}\]

enumerator HO_APP_ENCODE

Equality – Higher-order application encoding

\[\inferrule{-\mid t}{t= \texttt{TheoryUfRewriter::getHoApplyForApplyUf}(t)}\]

For example, this rule concludes \(f(x,y) = @(@(f,x),y)\), where \(@\) isthe HO_APPLY kind.

Note this rule can be treated as a REFL when appropriate in external proof formats.

enumerator HO_CONG

Equality – Higher-order congruence

\[\inferrule{f=g, t_1=s_1,\dots,t_n=s_n\mid k}{k(f, t_1,\dots, t_n) = k(g, s_1,\dots, s_n)}\]

Notice that this rule is only used when the application kind \(k\) is either cvc5::Kind::APPLY_UF or cvc5::Kind::HO_APPLY.

enumerator ARRAYS_READ_OVER_WRITE

Arrays – Read over write

\[\inferrule{i_1 \neq i_2\mid \mathit{select}(\mathit{store}(a,i_1,e),i_2)} {\mathit{select}(\mathit{store}(a,i_1,e),i_2) = \mathit{select}(a,i_2)}\]

enumerator ARRAYS_READ_OVER_WRITE_CONTRA

Arrays – Read over write, contrapositive

\[\inferrule{\mathit{select}(\mathit{store}(a,i_2,e),i_1) \neq \mathit{select}(a,i_1)\mid -}{i_1=i_2}\]

enumerator ARRAYS_READ_OVER_WRITE_1

Arrays – Read over write 1

\[\inferrule{-\mid \mathit{select}(\mathit{store}(a,i,e),i)} {\mathit{select}(\mathit{store}(a,i,e),i)=e}\]

enumerator ARRAYS_EXT

Arrays – Arrays extensionality

\[\inferrule{a \neq b\mid -} {\mathit{select}(a,k)\neq\mathit{select}(b,k)}\]

where \(k\) is the \(\texttt{ARRAY_DEQ_DIFF}\) skolem for (a, b).

enumerator MACRO_BV_BITBLAST

Bit-vectors – (Macro) Bitblast

\[\inferrule{-\mid t}{t = \texttt{bitblast}(t)}\]

where bitblast() represents the result of the bit-blasted term as a bit-vector consisting of the output bits of the bit-blasted circuit representation of the term. Terms are bit-blasted according to the strategies defined in theory/bv/bitblast/bitblast_strategies_template.h.

enumerator BV_BITBLAST_STEP

Bit-vectors – Bitblast bit-vector constant, variable, and terms

For constant and variables:

\[\inferrule{-\mid t}{t = \texttt{bitblast}(t)}\]

For terms:

\[\inferrule{-\mid k(\texttt{bitblast}(t_1),\dots,\texttt{bitblast}(t_n))} {k(\texttt{bitblast}(t_1),\dots,\texttt{bitblast}(t_n)) = \texttt{bitblast}(t)}\]

where \(t\) is \(k(t_1,\dots,t_n)\).

enumerator BV_EAGER_ATOM

Bit-vectors – Bit-vector eager atom

\[\inferrule{-\mid F}{F = F[0]}\]

where \(F\) is of kind BITVECTOR_EAGER_ATOM.

enumerator DT_UNIF

Datatypes – Unification

\[\inferrule{C(t_1,\dots,t_n)= C(s_1,\dots,s_n)\mid i}{t_1 = s_i}\]

where \(C\) is a constructor.

enumerator DT_SPLIT

Datatypes – Split

\[\inferrule{-\mid t}{\mathit{is}_{C_1}(t)\vee\cdots\vee\mathit{is}_{C_n}(t)}\]

where \(C_1,\dots,C_n\) are all the constructors of the type of \(t\).

enumerator DT_CLASH

Datatypes – Clash

\[\inferruleSC{\mathit{is}_{C_i}(t), \mathit{is}_{C_j}(t)\mid -}{\bot} {if $i\neq j$}\]

enumerator SKOLEM_INTRO

Quantifiers – Skolem introduction

\[\inferrule{-\mid k}{k = t}\]

where \(t\) is the unpurified form of skolem \(k\).

enumerator SKOLEMIZE

Quantifiers – Skolemization

\[\inferrule{\exists x_1\dots x_n.\> F\mid -}{F\sigma}\]

or

\[\inferrule{\neg (\forall x_1\dots x_n.\> F)\mid -}{\neg F\sigma}\]

where \(\sigma\) maps \(x_1,\dots,x_n\) to their representative skolems obtained by SkolemManager::mkSkolemize, returned in the skolems argument of that method. The witness terms for the returned skolems can be obtained by SkolemManager::getWitnessForm.

enumerator INSTANTIATE

Quantifiers – Instantiation

\[\inferrule{\forall x_1\dots x_n.\> F\mid (t_1 \dots t_n), (id\, (t)?)?} {F\{x_1\mapsto t_1,\dots,x_n\mapsto t_n\}}\]

The list of terms to instantiate \((t_1 \dots t_n)\) is provided as an s-expression as the first argument. The optional argument \(id\) indicates the inference id that caused the instantiation. The term \(t\) indicates an additional term (e.g. the trigger) associated with the instantiation, which depends on the id. If the id has prefix QUANTIFIERS_INST_E_MATCHING, then \(t\) is the trigger that generated the instantiation.

enumerator ALPHA_EQUIV

Quantifiers – Alpha equivalence

\[\inferruleSC{-\mid F, (y_1 \ldots y_n), (z_1,\dots, z_n)} {F = F\{y_1\mapsto z_1,\dots,y_n\mapsto z_n\}} {if $y_1,\dots,y_n, z_1,\dots,z_n$ are unique bound variables}\]

Notice that this rule is correct only when \(z_1,\dots,z_n\) are not contained in \(FV(F) \setminus \{ y_1,\dots, y_n \}\), where \(FV(\varphi)\) are the free variables of \(\varphi\). The internal quantifiers proof checker does not currently check that this is the case.

enumerator SETS_SINGLETON_INJ

Sets – Singleton injectivity

\[\inferrule{\mathit{set.singleton}(t) = \mathit{set.singleton}(s)\mid -}{t=s}\]

enumerator SETS_EXT

Sets – Sets extensionality

\[\inferrule{a \neq b\mid -} {\mathit{set.member}(k,a)\neq\mathit{set.member}(k,b)}\]

where \(k\) is the \(\texttt{SETS_DEQ_DIFF}\) skolem for (a, b).

enumerator CONCAT_EQ

Strings – Core rules – Concatenation equality

\[\inferrule{(t_1\cdot\ldots \cdot t_n \cdot t) = (t_1 \cdot\ldots \cdot t_n\cdot s)\mid b}{t = s}\]

where \(\cdot\) stands for string concatenation and \(b\) indicates if the direction is reversed.

Notice that \(t\) or \(s\) may be empty, in which case they are implicit in the concatenation above. For example, if the premise is \(x\cdot z = x\), then this rule, with argument \(\bot\), concludes \(z = ''\).

Also note that constants are split, such that for \((\mathsf{'abc'} \cdot x) = (\mathsf{'a'} \cdot y)\), this rule, with argument \(\bot\), concludes \((\mathsf{'bc'} \cdot x) = y\). This splitting is done only for constants such that Word::splitConstant returns non-null.

enumerator CONCAT_UNIFY

Strings – Core rules – Concatenation unification

\[\inferrule{(t_1\cdot t_2) = (s_1 \cdot s_2),\, \mathit{len}(t_1) = \mathit{len}(s_1)\mid b}{t_1 = s_1}\]

where \(b\) indicates if the direction is reversed.

enumerator CONCAT_CONFLICT

Strings – Core rules – Concatenation conflict

\[\inferrule{(c_1\cdot t) = (c_2 \cdot s)\mid b}{\bot}\]

where \(b\) indicates if the direction is reversed, \(c_1,\,c_2\) are constants such that \(\texttt{Word::splitConstant}(c_1,c_2, \mathit{index},b)\) is null, in other words, neither is a prefix of the other. Note it may be the case that one side of the equality denotes the empty string.

This rule is used exclusively for strings.

enumerator CONCAT_CONFLICT_DEQ

Strings – Core rules – Concatenation conflict for disequal characters

\[\inferrule{(t_1\cdot t) = (s_1 \cdot s), t_1 \neq s_1 \mid b}{\bot}\]

where $t_1$ and $s_1$ are constants of length one, or otherwise one side of the equality is the empty sequence and $t_1$ or $s_1$ corresponding to that side is the empty sequence.

This rule is used exclusively for sequences.

enumerator CONCAT_SPLIT

Strings – Core rules – Concatenation split

\[\inferruleSC{(t_1\cdot t_2) = (s_1 \cdot s_2),\, \mathit{len}(t_1) \neq \mathit{len}(s_1)\mid b}{((t_1 = s_1\cdot r) \vee (s_1 = t_1\cdot r)) \wedge r \neq \epsilon \wedge \mathit{len}(r)>0}{if $b=\bot$}\]

where \(r\) is \(\mathit{skolem}(\mathit{ite}( \mathit{len}(t_1) >= \mathit{len}(s_1), \mathit{suf}(t_1,\mathit{len}(s_1)), \mathit{suf}(s_1,\mathit{len}(t_1))))\) and epsilon is the empty string (or sequence).

\[\inferruleSC{(t_1\cdot t_2) = (s_1 \cdot s_2),\, \mathit{len}(t_2) \neq \mathit{len}(s_2)\mid b}{((t_2 = r \cdot s_2) \vee (s_2 = r \cdot t_2)) \wedge r \neq \epsilon \wedge \mathit{len}(r)>0}{if $b=\top$}\]

where \(r\) is the purification Skolem for \(\mathit{ite}( \mathit{len}(t_2) >= \mathit{len}(s_2), \mathit{pre}(t_2,\mathit{len}(t_2) - \mathit{len}(s_2)), \mathit{pre}(s_2,\mathit{len}(s_2) - \mathit{len}(t_2)))\) and epsilon is the empty string (or sequence).

Above, \(\mathit{suf}(x,n)\) is shorthand for \(\mathit{substr}(x,n, \mathit{len}(x) - n)\) and \(\mathit{pre}(x,n)\) is shorthand for \(\mathit{substr}(x,0,n)\).

enumerator CONCAT_CSPLIT

Strings – Core rules – Concatenation split for constants

\[\inferrule{(t_1\cdot t_2) = (c \cdot s_2),\, \mathit{len}(t_1) \neq 0\mid \bot}{(t_1 = c\cdot r)}\]

where \(r\) is \(\mathit{skolem}(\mathit{suf}(t_1,1))\).

Alternatively for the reverse:

\[\inferrule{(t_1\cdot t_2) = (s_1 \cdot c),\, \mathit{len}(t_2) \neq 0\mid \top}{(t_2 = r\cdot c)}\]

where \(r\) is \(\mathit{skolem}(\mathit{pre}(t_2,\mathit{len}(t_2) - 1))\).

enumerator CONCAT_LPROP

Strings – Core rules – Concatenation length propagation

\[\inferrule{(t_1\cdot t_2) = (s_1 \cdot s_2),\, \mathit{len}(t_1) > \mathit{len}(s_1)\mid \bot}{(t_1 = s_1\cdot r)}\]

where \(r\) is the purification Skolem for \(\mathit{skolem}(\mathit{ite}( \mathit{len}(t_1) >= \mathit{len}(s_1), \mathit{suf}(t_1,\mathit{len}(s_1)), \mathit{suf}(s_1,\mathit{len}(t_1))))\).

Alternatively for the reverse:

\[\inferrule{(t_1\cdot t_2) = (s_1 \cdot s_2),\, \mathit{len}(t_2) > \mathit{len}(s_2)\mid \top}{(t_2 = r \cdot s_2)}\]

where \(r\) is the purification Skolem for \(\mathit{ite}( \mathit{len}(t_2) >= \mathit{len}(s_2), \mathit{pre}(t_2,\mathit{len}(t_2) - \mathit{len}(s_2)), \mathit{pre}(s_2,\mathit{len}(s_2) - \mathit{len}(t_2)))\)

enumerator CONCAT_CPROP

Strings – Core rules – Concatenation constant propagation

\[\inferrule{(t_1\cdot w_1\cdot t_2) = (w_2 \cdot s),\, \mathit{len}(t_1) \neq 0\mid \bot}{(t_1 = t_3\cdot r)}\]

where \(w_1,\,w_2\) are words, \(t_3\) is \(\mathit{pre}(w_2,p)\), \(p\) is \(\texttt{Word::overlap}(\mathit{suf}(w_2,1), w_1)\), and \(r\) is \(\mathit{skolem}(\mathit{suf}(t_1,\mathit{len}(w_3)))\). Note that \(\mathit{suf}(w_2,p)\) is the largest suffix of \(\mathit{suf}(w_2,1)\) that can contain a prefix of \(w_1\); since \(t_1\) is non-empty, \(w_3\) must therefore be contained in \(t_1\).

Alternatively for the reverse:

\[\inferrule{(t_1\cdot w_1\cdot t_2) = (s \cdot w_2),\, \mathit{len}(t_2) \neq 0\mid \top}{(t_2 = r\cdot t_3)}\]

where \(w_1,\,w_2\) are words, \(t_3\) is \(\mathit{substr}(w_2, \mathit{len}(w_2) - p, p)\), \(p\) is \(\texttt{Word::roverlap}(\mathit{pre}(w_2, \mathit{len}(w_2) - 1), w_1)\), and \(r\) is \(\mathit{skolem}(\mathit{pre}(t_2, \mathit{len}(t_2) - \mathit{len}(w_3)))\). Note that \(\mathit{pre}(w_2, \mathit{len}(w_2) - p)\) is the largest prefix of \(\mathit{pre}(w_2, \mathit{len}(w_2) - 1)\) that can contain a suffix of \(w_1\); since \(t_2\) is non-empty, \(w_3\) must therefore be contained in \(t_2\).

enumerator STRING_DECOMPOSE

Strings – Core rules – String decomposition

\[\inferrule{\mathit{len}(t) \geq n\mid \bot}{t = w_1\cdot w_2 \wedge \mathit{len}(w_1) = n}\]

where \(w_1\) is \(\mathit{skolem}(\mathit{pre}(t,n)\) and \(w_2\) is \(\mathit{skolem}(\mathit{suf}(t,n)\). Or alternatively for the reverse:

\[\inferrule{\mathit{len}(t) \geq n\mid \top}{t = w_1\cdot w_2 \wedge \mathit{len}(w_2) = n}\]

where \(w_1\) is the purification skolem for \(\mathit{pre}(t,n)\) and \(w_2\) is the purification skolem for \(\mathit{suf}(t,n)\).

enumerator STRING_LENGTH_POS

Strings – Core rules – Length positive

\[\inferrule{-\mid t}{(\mathit{len}(t) = 0\wedge t= '')\vee \mathit{len}(t) > 0}\]

enumerator STRING_LENGTH_NON_EMPTY

Strings – Core rules – Length non-empty

\[\inferrule{t\neq ''\mid -}{\mathit{len}(t) \neq 0}\]

enumerator STRING_REDUCTION

Strings – Extended functions – Reduction

\[\inferrule{-\mid t}{R\wedge t = w}\]

where \(w\) is \(\texttt{strings::StringsPreprocess::reduce}(t, R, \dots)\). In other words, \(R\) is the reduction predicate for extended term \(t\), and \(w\) is \(skolem(t)\).

Notice that the free variables of \(R\) are \(w\) and the free variables of \(t\).

enumerator STRING_EAGER_REDUCTION

Strings – Extended functions – Eager reduction

\[\inferrule{-\mid t}{R}\]

where \(R\) is \(\texttt{strings::TermRegistry::eagerReduce}(t)\).

enumerator RE_INTER

Strings – Regular expressions – Intersection

\[\inferrule{t\in R_1,\,t\in R_2\mid -}{t\in \mathit{inter}(R_1,R_2)}\]

enumerator RE_UNFOLD_POS

Strings – Regular expressions – Positive Unfold

\[\inferrule{t\in R\mid -}{\texttt{RegExpOpr::reduceRegExpPos}(t\in R)}\]

corresponding to the one-step unfolding of the premise.

enumerator RE_UNFOLD_NEG

Strings – Regular expressions – Negative Unfold

\[\inferrule{t \not \in \mathit{re}.\text{*}(R) \mid -}{t \neq \ '' \ \wedge \forall L. L \leq 0 \vee \mathit{str.len}(t) < L \vee \mathit{pre}(t, L) \not \in R \vee \mathit{suf}(t, L) \not \in \mathit{re}.\text{*}(R)}\]

Or alternatively for regular expression concatenation:

\[\inferrule{t \not \in \mathit{re}.\text{++}(R_1, \ldots, R_n)\mid -}{\forall L. L < 0 \vee \mathit{str.len}(t) < L \vee \mathit{pre}(t, L) \not \in R_1 \vee \mathit{suf}(t, L) \not \in \mathit{re}.\text{++}(R_2, \ldots, R_n)}\]

Note that in either case the varaible \(L\) has type \(Int\) and name “@var.str_index”.

enumerator RE_UNFOLD_NEG_CONCAT_FIXED

Strings – Regular expressions – Unfold negative concatenation, fixed

\[ \inferrule{t\not\in \mathit{re}.\text{re.++}(r_1, \ldots, r_n) \mid \bot}{ \mathit{pre}(t, L) \not \in r_1 \vee \mathit{suf}(t, L) \not \in \mathit{re}.\text{re.++}(r_2, \ldots, r_n)}\]

where \(r_1\) has fixed length \(L\).

or alternatively for the reverse:

\[\inferrule{t \not \in \mathit{re}.\text{re.++}(r_1, \ldots, r_n) \mid \top}{ \mathit{suf}(t, str.len(t) - L) \not \in r_n \vee \mathit{pre}(t, str.len(t) - L) \not \in \mathit{re}.\text{re.++}(r_1, \ldots, r_{n-1})}\]

where \(r_n\) has fixed length \(L\).

enumerator STRING_CODE_INJ

Strings – Code points

\[\inferrule{-\mid t,s}{\mathit{to\_code}(t) = -1 \vee \mathit{to\_code}(t) \neq \mathit{to\_code}(s) \vee t = s}\]

enumerator STRING_SEQ_UNIT_INJ

Strings – Sequence unit

\[\inferrule{\mathit{unit}(x) = \mathit{unit}(y)\mid -}{x = y}\]

Also applies to the case where \(\mathit{unit}(y)\) is a constant sequence of length one.

enumerator MACRO_STRING_INFERENCE

Strings – (Macro) String inference

\[\inferrule{?\mid F,\mathit{id},\mathit{isRev},\mathit{exp}}{F}\]

used to bookkeep an inference that has not yet been converted via \(\texttt{strings::InferProofCons::convert}\).

enumerator MACRO_RE_ELIM

Strings – Regular expressions – Macro elimination

\[\inferrule{-\mid F,b}{F = \texttt{strings::RegExpElimination::eliminate}(F, b)}\]

where \(b\) is a Boolean indicating whether we are using aggressive eliminations. Notice this rule concludes \(F = F\) if no eliminations are performed for \(F\).

rst .. note:: We do not currently support elaboration of this macro.

enumerator MACRO_ARITH_SCALE_SUM_UB

Arithmetic – Adding inequalities

An arithmetic literal is a term of the form \(p \diamond c\) where \(\diamond \in \{ <, \leq, =, \geq, > \}\), \(p\) a polynomial and \(c\) a rational constant.

\[\inferrule{l_1 \dots l_n \mid k_1 \dots k_n}{t_1 \diamond t_2}\]

where \(k_i \in \mathbb{R}, k_i \neq 0\), \(\diamond\) is the fusion of the \(\diamond_i\) (flipping each if its \(k_i\) is negative) such that \(\diamond_i \in \{ <, \leq \}\) (this implies that lower bounds have negative \(k_i\) and upper bounds have positive \(k_i\)), \(t_1\) is the sum of the scaled polynomials and \(t_2\) is the sum of the scaled constants:

\[ \begin{align}\begin{aligned}t_1 \colon= k_1 \cdot p_1 + \cdots + k_n \cdot p_n\\t_2 \colon= k_1 \cdot c_1 + \cdots + k_n \cdot c_n\end{aligned}\end{align} \]

enumerator ARITH_SUM_UB

Arithmetic – Sum upper bounds

\[\inferrule{P_1 \dots P_n \mid -}{L \diamond R}\]

where \(P_i\) has the form \(L_i \diamond_i R_i\) and \(\diamond_i \in \{<, \leq, =\}\). Furthermore \(\diamond = <\) if \(\diamond_i = <\) for any \(i\) and \(\diamond = \leq\) otherwise, \(L = L_1 + \cdots + L_n\) and \(R = R_1 + \cdots + R_n\).

enumerator INT_TIGHT_UB

Arithmetic – Tighten strict integer upper bounds

\[\inferrule{i < c \mid -}{i \leq \lfloor c \rfloor}\]

where \(i\) has integer type.

enumerator INT_TIGHT_LB

Arithmetic – Tighten strict integer lower bounds

\[\inferrule{i > c \mid -}{i \geq \lceil c \rceil}\]

where \(i\) has integer type.

enumerator ARITH_TRICHOTOMY

Arithmetic – Trichotomy of the reals

\[\inferrule{A, B \mid -}{C}\]

where \(\neg A, \neg B, C\) are \(x < c, x = c, x > c\) in some order. Note that \(\neg\) here denotes arithmetic negation, i.e., flipping \(\geq\) to \(<\) etc.

enumerator ARITH_OP_ELIM_AXIOM

Arithmetic – Operator elimination

\[\inferrule{- \mid t}{\texttt{arith::OperatorElim::getAxiomFor(t)}}\]

enumerator ARITH_POLY_NORM

Arithmetic – Polynomial normalization

\[\inferrule{- \mid t = s}{t = s}\]

where \(\texttt{arith::PolyNorm::isArithPolyNorm(t, s)} = \top\). This method normalizes polynomials over arithmetic or bitvectors.

enumerator ARITH_MULT_SIGN

Arithmetic – Sign inference

\[\inferrule{- \mid f_1 \dots f_k, m}{(f_1 \land \dots \land f_k) \rightarrow m \diamond 0}\]

where \(f_1 \dots f_k\) are variables compared to zero (less, greater or not equal), \(m\) is a monomial from these variables and \(\diamond\) is the comparison (less or equal) that results from the signs of the variables. All variables with even exponent in \(m\) should be given as not equal to zero while all variables with odd exponent in \(m\) should be given as less or greater than zero.

enumerator ARITH_MULT_POS

Arithmetic – Multiplication with positive factor

\[\inferrule{- \mid m, l \diamond r}{(m > 0 \land l \diamond r) \rightarrow m \cdot l \diamond m \cdot r}\]

where \(\diamond\) is a relation symbol.

enumerator ARITH_MULT_NEG

Arithmetic – Multiplication with negative factor

\[\inferrule{- \mid m, l \diamond r}{(m < 0 \land l \diamond r) \rightarrow m \cdot l \diamond_{inv} m \cdot r}\]

where \(\diamond\) is a relation symbol and \(\diamond_{inv}\) the inverted relation symbol.

enumerator ARITH_MULT_TANGENT

Arithmetic – Multiplication tangent plane

\[ \begin{align}\begin{aligned}\inferruleSC{- \mid x, y, a, b, \sigma}{(t \leq tplane) \leftrightarrow ((x \leq a \land y \geq b) \lor (x \geq a \land y \leq b))}{if $\sigma = -1$}\\\inferruleSC{- \mid x, y, a, b, \sigma}{(t \geq tplane) \leftrightarrow ((x \leq a \land y \leq b) \lor (x \geq a \land y \geq b))}{if $\sigma = 1$}\end{aligned}\end{align} \]

where \(x,y\) are real terms (variables or extended terms), \(t = x \cdot y\), \(a,b\) are real constants, \(\sigma \in \{ 1, -1\}\) and \(tplane := b \cdot x + a \cdot y - a \cdot b\) is the tangent plane of \(x \cdot y\) at \((a,b)\).

enumerator ARITH_TRANS_PI

Arithmetic – Transcendentals – Assert bounds on Pi

\[\inferrule{- \mid l, u}{\texttt{real.pi} \geq l \land \texttt{real.pi} \leq u}\]

where \(l,u\) are valid lower and upper bounds on \(\pi\).

enumerator ARITH_TRANS_EXP_NEG

Arithmetic – Transcendentals – Exp at negative values

\[\inferrule{- \mid t}{(t < 0) \leftrightarrow (\exp(t) < 1)}\]

enumerator ARITH_TRANS_EXP_POSITIVITY

Arithmetic – Transcendentals – Exp is always positive

\[\inferrule{- \mid t}{\exp(t) > 0}\]

enumerator ARITH_TRANS_EXP_SUPER_LIN

Arithmetic – Transcendentals – Exp grows super-linearly for positive values

\[\inferrule{- \mid t}{t \leq 0 \lor \exp(t) > t+1}\]

enumerator ARITH_TRANS_EXP_ZERO

Arithmetic – Transcendentals – Exp at zero

\[\inferrule{- \mid t}{(t=0) \leftrightarrow (\exp(t) = 1)}\]

enumerator ARITH_TRANS_EXP_APPROX_ABOVE_NEG

Arithmetic – Transcendentals – Exp is approximated from above for negative values

\[\inferrule{- \mid d,t,l,u}{(t \geq l \land t \leq u) \rightarrow exp(t) \leq \texttt{secant}(\exp, l, u, t)}\]

where \(d\) is an even positive number, \(t\) an arithmetic term and \(l,u\) are lower and upper bounds on \(t\). Let \(p\) be the \(d\)’th taylor polynomial at zero (also called the Maclaurin series) of the exponential function. \(\texttt{secant}(\exp, l, u, t)\) denotes the secant of \(p\) from \((l, \exp(l))\) to \((u, \exp(u))\) evaluated at \(t\), calculated as follows:

\[\frac{p(l) - p(u)}{l - u} \cdot (t - l) + p(l)\]

The lemma states that if \(t\) is between \(l\) and \(u\), then \(\exp(t\) is below the secant of \(p\) from \(l\) to \(u\).

enumerator ARITH_TRANS_EXP_APPROX_ABOVE_POS

Arithmetic – Transcendentals – Exp is approximated from above for positive values

\[\inferrule{- \mid d,t,l,u}{(t \geq l \land t \leq u) \rightarrow exp(t) \leq \texttt{secant-pos}(\exp, l, u, t)}\]

where \(d\) is an even positive number, \(t\) an arithmetic term and \(l,u\) are lower and upper bounds on \(t\). Let \(p^*\) be a modification of the \(d\)’th taylor polynomial at zero (also called the Maclaurin series) of the exponential function as follows where \(p(d-1)\) is the regular Maclaurin series of degree \(d-1\):

\[p^* := p(d-1) \cdot \frac{1 + t^n}{n!}\]

\(\texttt{secant-pos}(\exp, l, u, t)\) denotes the secant of \(p\) from \((l, \exp(l))\) to \((u, \exp(u))\) evaluated at \(t\), calculated as follows:

\[\frac{p(l) - p(u)}{l - u} \cdot (t - l) + p(l)\]

The lemma states that if \(t\) is between \(l\) and \(u\), then \(\exp(t\) is below the secant of \(p\) from \(l\) to \(u\).

enumerator ARITH_TRANS_EXP_APPROX_BELOW

Arithmetic – Transcendentals – Exp is approximated from below

\[\inferrule{- \mid d,c,t}{t \geq c \rightarrow exp(t) \geq \texttt{maclaurin}(\exp, d, c)}\]

where \(d\) is an odd positive number, \(t\) an arithmetic term and \(\texttt{maclaurin}(\exp, d, c)\) is the \(d\)’th taylor polynomial at zero (also called the Maclaurin series) of the exponential function evaluated at \(c\). The Maclaurin series for the exponential function is the following:

\[\exp(x) = \sum_{n=0}^{\infty} \frac{x^n}{n!}\]

enumerator ARITH_TRANS_SINE_BOUNDS

Arithmetic – Transcendentals – Sine is always between -1 and 1

\[\inferrule{- \mid t}{\sin(t) \leq 1 \land \sin(t) \geq -1}\]

enumerator ARITH_TRANS_SINE_SHIFT

Arithmetic – Transcendentals – Sine is shifted to -pi…pi

\[\inferrule{- \mid x, y, s}{-\pi \leq y \leq \pi \land \sin(y) = \sin(x) \land (\ite{-\pi \leq x \leq \pi}{x = y}{x = y + 2 \pi s})}\]

where \(x\) is the argument to sine, \(y\) is a new real skolem that is \(x\) shifted into \(-\pi \dots \pi\) and \(s\) is a new integer slolem that is the number of phases \(y\) is shifted.

enumerator ARITH_TRANS_SINE_SYMMETRY

Arithmetic – Transcendentals – Sine is symmetric with respect to negation of the argument

\[\inferrule{- \mid t}{\sin(t) - \sin(-t) = 0}\]

enumerator ARITH_TRANS_SINE_TANGENT_ZERO

Arithmetic – Transcendentals – Sine is bounded by the tangent at zero

\[\inferrule{- \mid t}{(t > 0 \rightarrow \sin(t) < t) \land (t < 0 \rightarrow \sin(t) > t)} \]

enumerator ARITH_TRANS_SINE_TANGENT_PI

Arithmetic – Transcendentals – Sine is bounded by the tangents at -pi and pi

\[\inferrule{- \mid t}{(t > -\pi \rightarrow \sin(t) > -\pi - t) \land (t < \pi \rightarrow \sin(t) < \pi - t)} \]

enumerator ARITH_TRANS_SINE_APPROX_ABOVE_NEG

Arithmetic – Transcendentals – Sine is approximated from above for negative values

\[\inferrule{- \mid d,t,lb,ub,l,u}{(t \geq lb land t \leq ub) \rightarrow \sin(t) \leq \texttt{secant}(\sin, l, u, t)}\]

where \(d\) is an even positive number, \(t\) an arithmetic term, \(lb,ub\) are symbolic lower and upper bounds on \(t\) (possibly containing \(\pi\)) and \(l,u\) the evaluated lower and upper bounds on \(t\). Let \(p\) be the \(d\)’th taylor polynomial at zero (also called the Maclaurin series) of the sine function. \(\texttt{secant}(\sin, l, u, t)\) denotes the secant of \(p\) from \((l, \sin(l))\) to \((u, \sin(u))\) evaluated at \(t\), calculated as follows:

\[\frac{p(l) - p(u)}{l - u} \cdot (t - l) + p(l)\]

The lemma states that if \(t\) is between \(l\) and \(u\), then \(\sin(t)\) is below the secant of \(p\) from \(l\) to \(u\).

enumerator ARITH_TRANS_SINE_APPROX_ABOVE_POS

Arithmetic – Transcendentals – Sine is approximated from above for positive values

\[\inferrule{- \mid d,t,c,lb,ub}{(t \geq lb land t \leq ub) \rightarrow \sin(t) \leq \texttt{upper}(\sin, c)}\]

where \(d\) is an even positive number, \(t\) an arithmetic term, \(c\) an arithmetic constant and \(lb,ub\) are symbolic lower and upper bounds on \(t\) (possibly containing \(\pi\)). Let \(p\) be the \(d\)’th taylor polynomial at zero (also called the Maclaurin series) of the sine function. \(\texttt{upper}(\sin, c)\) denotes the upper bound on \(\sin(c)\) given by \(p\) and \(lb,up\) such that \(\sin(t)\) is the maximum of the sine function on \((lb,ub)\).

enumerator ARITH_TRANS_SINE_APPROX_BELOW_NEG

Arithmetic – Transcendentals – Sine is approximated from below for negative values

\[\inferrule{- \mid d,t,c,lb,ub}{(t \geq lb land t \leq ub) \rightarrow \sin(t) \geq \texttt{lower}(\sin, c)}\]

where \(d\) is an even positive number, \(t\) an arithmetic term, \(c\) an arithmetic constant and \(lb,ub\) are symbolic lower and upper bounds on \(t\) (possibly containing \(\pi\)). Let \(p\) be the \(d\)’th taylor polynomial at zero (also called the Maclaurin series) of the sine function. \(\texttt{lower}(\sin, c)\) denotes the lower bound on \(\sin(c)\) given by \(p\) and \(lb,up\) such that \(\sin(t)\) is the minimum of the sine function on \((lb,ub)\).

enumerator ARITH_TRANS_SINE_APPROX_BELOW_POS

Arithmetic – Transcendentals – Sine is approximated from below for positive values

\[\inferrule{- \mid d,t,lb,ub,l,u}{(t \geq lb land t \leq ub) \rightarrow \sin(t) \geq \texttt{secant}(\sin, l, u, t)}\]

where \(d\) is an even positive number, \(t\) an arithmetic term, \(lb,ub\) are symbolic lower and upper bounds on \(t\) (possibly containing \(\pi\)) and \(l,u\) the evaluated lower and upper bounds on \(t\). Let \(p\) be the \(d\)’th taylor polynomial at zero (also called the Maclaurin series) of the sine function. \(\texttt{secant}(\sin, l, u, t)\) denotes the secant of \(p\) from \((l, \sin(l))\) to \((u, \sin(u))\) evaluated at \(t\), calculated as follows:

\[\frac{p(l) - p(u)}{l - u} \cdot (t - l) + p(l)\]

The lemma states that if \(t\) is between \(l\) and \(u\), then \(\sin(t)\) is above the secant of \(p\) from \(l\) to \(u\).

enumerator LFSC_RULE

External – LFSC

Place holder for LFSC rules.

\[\inferrule{P_1, \dots, P_n\mid \texttt{id}, Q, A_1,\dots, A_m}{Q}\]

Note that the premises and arguments are arbitrary. It’s expected that \(\texttt{id}\) refer to a proof rule in the external LFSC calculus.

enumerator ALETHE_RULE

External – Alethe

Place holder for Alethe rules.

\[\inferrule{P_1, \dots, P_n\mid \texttt{id}, Q, Q', A_1,\dots, A_m}{Q}\]

Note that the premises and arguments are arbitrary. It’s expected that \(\texttt{id}\) refer to a proof rule in the external Alethe calculus, and that \(Q'\) be the representation of Q to be printed by the Alethe printer.

enumerator UNKNOWN
std::ostream &cvc5::operator<<(std::ostream &out, ProofRule rule)

Writes a proof rule name to a stream.

Parameters:
  • out – The stream to write to

  • rule – The proof rule to write to the stream

Returns:

The stream

std::string std::to_string(cvc5::ProofRule rule)

Converts a proof rule to a string.

Parameters:

rule – The proof rule

Returns:

The name of the proof rule

template<>
struct hash<cvc5::ProofRule>

Hash function for ProofRules.

Public Functions

size_t operator()(cvc5::ProofRule rule) const

Hashes a ProofRule to a size_t.

Parameters:

rule – The proof rule.

Returns:

The hash value.


enum class cvc5::ProofRewriteRule

This enumeration represents the rewrite rules used in a rewrite proof. Some of the rules are internal ad-hoc rewrites, while others are rewrites specified by the RARE DSL. This enumeration is used as the first argument to the DSL_REWRITE proof rule and the THEORY_REWRITE proof rule.

Values:

enumerator NONE
enumerator DISTINCT_ELIM

Builtin – Distinct elimination

\[\texttt{distinct}(t_1, \ldots, tn) = \bigwedge_{i \neq j} t_i \neq t_j\]

enumerator MACRO_BOOL_NNF_NORM

Booleans – Negation Normal Form with normalization

\[F = G\]

where \(G\) is the result of applying negation normal form to \(F\) with additional normalizations, see TheoryBoolRewriter::computeNnfNorm.

enumerator ARITH_DIV_BY_CONST_ELIM

Arith – Division by constant elimination

\[t / c = t * 1/c\]

where \(c\) is a constant.

enumerator ARITH_STRING_PRED_ENTAIL

Arithmetic - strings predicate entailment

\[(>= n 0) = true\]

Where \(n\) can be shown to be greater than or equal to \(0\) by reasoning about string length being positive and basic properties of addition and multiplication.

enumerator ARITH_STRING_PRED_SAFE_APPROX

Arithmetic - strings predicate entailment

\[(>= n 0) = (>= m 0)\]

Where \(m\) is a safe under-approximation of \(n\), namely we have that \((>= n m)\) and \((>= m 0)\).

In detail, subterms of \(n\) may be replaced with other terms to obtain \(m\) based on the reasoning described in the paper Reynolds et al, CAV 2019, “High-Level Abstractions for Simplifying Extended String Constraints in SMT”.

enumerator BETA_REDUCE

Equality – Beta reduction

\[((\lambda x_1 \ldots x_n.\> t) \ t_1 \ldots t_n) = t\{x_1 \mapsto t_1, \ldots, x_n \mapsto t_n\}\]

The right hand side of the equality in the conclusion is computed using standard substitution via Node::substitute.

enumerator ARRAYS_EQ_RANGE_EXPAND

Arrays – Expansion of array range equality

\[\mathit{eqrange}(a,b,i,j)= \forall x.\> i \leq x \leq j \rightarrow \mathit{select}(a,x)=\mathit{select}(b,x)\]

enumerator EXISTS_ELIM

Quantifiers – Exists elimination

\[\exists x_1\dots x_n.\> F = \neg \forall x_1\dots x_n.\> \neg F\]

enumerator QUANT_UNUSED_VARS

Quantifiers – Unused variables

\[\forall X.\> F = \forall X_1.\> F\]

where \(X_1\) is the subset of \(X\) that appear free in \(F\).

enumerator QUANT_MERGE_PRENEX

Quantifiers – Merge prenex

\[\forall X_1.\> \ldots \forall X_n.\> F = \forall X_1 \ldots X_n.\> F\]

where \(X_1 \ldots X_n\) are lists of variables.

enumerator QUANT_MINISCOPE

Quantifiers – Miniscoping

\[\forall X.\> F_1 \wedge \ldots \wedge F_n = (\forall X.\> F_1) \wedge \ldots \wedge (\forall X.\> F_n)\]

enumerator MACRO_QUANT_PARTITION_CONNECTED_FV

Quantifiers – Macro connected free variable partitioning

\[\forall X.\> F_1 \vee \ldots \vee F_n = (\forall X_1.\> F_{1,1} \vee \ldots \vee F_{1,k_1}) \vee \ldots \vee (\forall X_m.\> F_{m,1} \vee \ldots \vee F_{m,k_m})\]

where \(X_1, \ldots, X_m\) is a partition of \(X\). This is determined by computing the connected components when considering two variables in \(X\) to be connected if they occur in the same \(F_i\).

enumerator DT_INST

Datatypes – Instantiation

\[\mathit{is}_C(t) = (t = C(\mathit{sel}_1(t),\dots,\mathit{sel}_n(t)))\]

where \(C\) is the \(n^{\mathit{th}}\) constructor of the type of \(t\), and \(\mathit{is}_C\) is the discriminator (tester) for \(C\).

enumerator DT_COLLAPSE_SELECTOR

Datatypes - collapse selector

\[s_i(c(t_1, \ldots, t_n)) = t_i\]

where \(s_i\) is the \(i^{th}\) selector for constructor \(c\).

enumerator DT_COLLAPSE_TESTER

Datatypes - collapse tester

\[\mathit{is}_c(c(t_1, \ldots, t_n)) = true\]

or alternatively

\[\mathit{is}_c(d(t_1, \ldots, t_n)) = false\]

where \(c\) and \(d\) are distinct constructors.

enumerator DT_COLLAPSE_TESTER_SINGLETON

Datatypes - collapse tester

\[\mathit{is}_c(t) = true\]

where \(c\) is the only constructor of its associated datatype.

enumerator DT_CONS_EQ

Datatypes - constructor equality

\[(c(t_1, \ldots, t_n) = c(s_1, \ldots, s_n)) = (t_1 = s_1 \wedge \ldots \wedge t_n = s_n)\]

or alternatively

\[(c(t_1, \ldots, t_n) = d(s_1, \ldots, s_m)) = false\]

where \(c\) and \(d\) are distinct constructors.

enumerator BV_UMULO_ELIMINATE

Bitvectors - Unsigned multiplication overflow detection elimination

See M.Gok, M.J. Schulte, P.I. Balzola, “Efficient integer multiplication overflow detection circuits”, 2001. http://ieeexplore.ieee.org/document/987767

enumerator BV_SMULO_ELIMINATE

Bitvectors - Unsigned multiplication overflow detection elimination

See M.Gok, M.J. Schulte, P.I. Balzola, “Efficient integer multiplication overflow detection circuits”, 2001. http://ieeexplore.ieee.org/document/987767

enumerator BV_ADD_COMBINE_LIKE_TERMS

Bitvectors - Combine like terms during addition by counting terms

enumerator BV_MULT_SIMPLIFY

Bitvectors - Extract negations from multiplicands

\[(-a bvmul b bvmul c) \to -(a bvmul b c)\]

enumerator BV_BITWISE_SLICING

Bitvectors - Extract continuous substrings of bitvectors

\[(a bvand c) \to (concat (bvand a[i0:j0] c0) ... (bvand a[in:jn] cn))\]

where c0,…, cn are maximally continuous substrings of 0 or 1 in the constant c

enumerator RE_LOOP_ELIM

Strings - regular expression loop elimination

\[((\_\ re.loop\ l\ u)\ R) = (re.union\ R^l \ldots R^u)\]

where \(u \geq l\).

enumerator STR_IN_RE_EVAL

Strings - regular expression membership evaluation

\[\mathit{str.in\_re}(s, R) = c\]

where \(s\) is a constant string, \(R\) is a constant regular expression and \(c\) is true or false.

enumerator STR_IN_RE_CONSUME

Strings - regular expression membership consume

\[\mathit{str.in_re}(s, R) = b\]

where \(b\) is either \(false\) or the result of stripping entailed prefixes and suffixes off of \(s\) and \(R\).

enumerator STR_IN_RE_CONCAT_STAR_CHAR

Strings - string in regular expression concatenation star character

\[\mathit{str.in\_re}(\mathit{str}.\text{++}(s_1, \ldots, s_n),\]

mathit{re}.text{*}(R)) = mathit{str.in_re}(s_1, mathit{re}.text{*}(R)) wedge ldots wedge mathit{str.in_re}(s_n, mathit{re}.text{*}(R))

where all strings in \(R\) have length one.

enumerator STR_IN_RE_SIGMA

Strings - string in regular expression sigma

\[\mathit{str.in\_re}(s, \mathit{re}.\text{++}(\mathit{re.allchar}, \ldots,\]
mathit{re.allchar})) =

(mathit{str.len}(s) = n)

or alternatively:

\[\mathit{str.in\_re}(s, \mathit{re}.\text{++}(\mathit{re.allchar}, \ldots,\]
mathit{re.allchar}, mathit{re}.text{*}(mathit{re.allchar}))) =

(mathit{str.len}(s) ge n)

enumerator STR_IN_RE_SIGMA_STAR

Strings - string in regular expression sigma star

\[\mathit{str.in\_re}(s,\]

mathit{re}.text{*}(mathit{re}.text{++}(mathit{re.allchar}, ldots, mathit{re.allchar}))) =

(mathit{str.len}(s) % n = 0)

where \(n\) is the number of \(\mathit{re.allchar}\) arguments to \(\mathit{re}.\text{++}\).

enumerator MACRO_SUBSTR_STRIP_SYM_LENGTH

Strings - strings substring strip symbolic length

\[(str.substr s n m) = t\]

where \(t\) is obtained by fully or partially stripping components of \(s\) based on \(n\) and \(m\).

enumerator SETS_IS_EMPTY_EVAL

Sets - empty tester evaluation

\[\mathit{sets.is\_empty}(as \ \mathit{set.empty} \ (\mathit{Set} \ T)) =\]

top

or alternatively:

\[\mathit{sets.is\_empty}(c) = \bot\]

where \(c\) is a constant set that is not the empty set.

enumerator ARITH_PLUS_ZERO

Auto-generated from RARE rule arith-plus-zero

enumerator ARITH_MUL_ONE

Auto-generated from RARE rule arith-mul-one

enumerator ARITH_MUL_ZERO

Auto-generated from RARE rule arith-mul-zero

enumerator ARITH_DIV_TOTAL

Auto-generated from RARE rule arith-div-total

enumerator ARITH_INT_DIV_TOTAL

Auto-generated from RARE rule arith-int-div-total

enumerator ARITH_INT_DIV_TOTAL_ONE

Auto-generated from RARE rule arith-int-div-total-one

enumerator ARITH_INT_DIV_TOTAL_ZERO

Auto-generated from RARE rule arith-int-div-total-zero

enumerator ARITH_INT_MOD_TOTAL

Auto-generated from RARE rule arith-int-mod-total

enumerator ARITH_INT_MOD_TOTAL_ONE

Auto-generated from RARE rule arith-int-mod-total-one

enumerator ARITH_INT_MOD_TOTAL_ZERO

Auto-generated from RARE rule arith-int-mod-total-zero

enumerator ARITH_NEG_NEG_ONE

Auto-generated from RARE rule arith-neg-neg-one

enumerator ARITH_ELIM_UMINUS

Auto-generated from RARE rule arith-elim-uminus

enumerator ARITH_ELIM_MINUS

Auto-generated from RARE rule arith-elim-minus

enumerator ARITH_ELIM_GT

Auto-generated from RARE rule arith-elim-gt

enumerator ARITH_ELIM_LT

Auto-generated from RARE rule arith-elim-lt

enumerator ARITH_ELIM_INT_GT

Auto-generated from RARE rule arith-elim-int-gt

enumerator ARITH_ELIM_INT_LT

Auto-generated from RARE rule arith-elim-int-lt

enumerator ARITH_ELIM_LEQ

Auto-generated from RARE rule arith-elim-leq

enumerator ARITH_LEQ_NORM

Auto-generated from RARE rule arith-leq-norm

enumerator ARITH_GEQ_TIGHTEN

Auto-generated from RARE rule arith-geq-tighten

enumerator ARITH_GEQ_NORM1

Auto-generated from RARE rule arith-geq-norm1

enumerator ARITH_GEQ_NORM2

Auto-generated from RARE rule arith-geq-norm2

enumerator ARITH_REFL_LEQ

Auto-generated from RARE rule arith-refl-leq

enumerator ARITH_REFL_LT

Auto-generated from RARE rule arith-refl-lt

enumerator ARITH_REFL_GEQ

Auto-generated from RARE rule arith-refl-geq

enumerator ARITH_REFL_GT

Auto-generated from RARE rule arith-refl-gt

enumerator ARITH_REAL_EQ_ELIM

Auto-generated from RARE rule arith-real-eq-elim

enumerator ARITH_INT_EQ_ELIM

Auto-generated from RARE rule arith-int-eq-elim

enumerator ARITH_PLUS_FLATTEN

Auto-generated from RARE rule arith-plus-flatten

enumerator ARITH_MULT_FLATTEN

Auto-generated from RARE rule arith-mult-flatten

enumerator ARITH_MULT_DIST

Auto-generated from RARE rule arith-mult-dist

enumerator ARITH_PLUS_CANCEL1

Auto-generated from RARE rule arith-plus-cancel1

enumerator ARITH_PLUS_CANCEL2

Auto-generated from RARE rule arith-plus-cancel2

enumerator ARITH_ABS_ELIM

Auto-generated from RARE rule arith-abs-elim

enumerator ARITH_TO_REAL_ELIM

Auto-generated from RARE rule arith-to-real-elim

enumerator ARITH_TO_INT_ELIM_TO_REAL

Auto-generated from RARE rule arith-to-int-elim-to-real

enumerator ARITH_DIV_ELIM_TO_REAL1

Auto-generated from RARE rule arith-div-elim-to-real1

enumerator ARITH_DIV_ELIM_TO_REAL2

Auto-generated from RARE rule arith-div-elim-to-real2

enumerator ARRAY_READ_OVER_WRITE

Auto-generated from RARE rule array-read-over-write

enumerator ARRAY_READ_OVER_WRITE2

Auto-generated from RARE rule array-read-over-write2

enumerator ARRAY_STORE_OVERWRITE

Auto-generated from RARE rule array-store-overwrite

enumerator ARRAY_STORE_SELF

Auto-generated from RARE rule array-store-self

enumerator BOOL_DOUBLE_NOT_ELIM

Auto-generated from RARE rule bool-double-not-elim

enumerator BOOL_NOT_TRUE

Auto-generated from RARE rule bool-not-true

enumerator BOOL_NOT_FALSE

Auto-generated from RARE rule bool-not-false

enumerator BOOL_EQ_TRUE

Auto-generated from RARE rule bool-eq-true

enumerator BOOL_EQ_FALSE

Auto-generated from RARE rule bool-eq-false

enumerator BOOL_EQ_NREFL

Auto-generated from RARE rule bool-eq-nrefl

enumerator BOOL_IMPL_FALSE1

Auto-generated from RARE rule bool-impl-false1

enumerator BOOL_IMPL_FALSE2

Auto-generated from RARE rule bool-impl-false2

enumerator BOOL_IMPL_TRUE1

Auto-generated from RARE rule bool-impl-true1

enumerator BOOL_IMPL_TRUE2

Auto-generated from RARE rule bool-impl-true2

enumerator BOOL_IMPL_ELIM

Auto-generated from RARE rule bool-impl-elim

enumerator BOOL_OR_TRUE

Auto-generated from RARE rule bool-or-true

enumerator BOOL_OR_FALSE

Auto-generated from RARE rule bool-or-false

enumerator BOOL_OR_FLATTEN

Auto-generated from RARE rule bool-or-flatten

enumerator BOOL_OR_DUP

Auto-generated from RARE rule bool-or-dup

enumerator BOOL_AND_TRUE

Auto-generated from RARE rule bool-and-true

enumerator BOOL_AND_FALSE

Auto-generated from RARE rule bool-and-false

enumerator BOOL_AND_FLATTEN

Auto-generated from RARE rule bool-and-flatten

enumerator BOOL_AND_DUP

Auto-generated from RARE rule bool-and-dup

enumerator BOOL_AND_CONF

Auto-generated from RARE rule bool-and-conf

enumerator BOOL_OR_TAUT

Auto-generated from RARE rule bool-or-taut

enumerator BOOL_OR_DE_MORGAN

Auto-generated from RARE rule bool-or-de-morgan

enumerator BOOL_IMPLIES_DE_MORGAN

Auto-generated from RARE rule bool-implies-de-morgan

enumerator BOOL_AND_DE_MORGAN

Auto-generated from RARE rule bool-and-de-morgan

enumerator BOOL_XOR_REFL

Auto-generated from RARE rule bool-xor-refl

enumerator BOOL_XOR_NREFL

Auto-generated from RARE rule bool-xor-nrefl

enumerator BOOL_XOR_FALSE

Auto-generated from RARE rule bool-xor-false

enumerator BOOL_XOR_TRUE

Auto-generated from RARE rule bool-xor-true

enumerator BOOL_XOR_COMM

Auto-generated from RARE rule bool-xor-comm

enumerator BOOL_XOR_ELIM

Auto-generated from RARE rule bool-xor-elim

enumerator BOOL_NOT_XOR_ELIM

Auto-generated from RARE rule bool-not-xor-elim

enumerator BOOL_NOT_EQ_ELIM

Auto-generated from RARE rule bool-not-eq-elim

enumerator ITE_NEG_BRANCH

Auto-generated from RARE rule ite-neg-branch

enumerator ITE_THEN_TRUE

Auto-generated from RARE rule ite-then-true

enumerator ITE_ELSE_FALSE

Auto-generated from RARE rule ite-else-false

enumerator ITE_THEN_FALSE

Auto-generated from RARE rule ite-then-false

enumerator ITE_ELSE_TRUE

Auto-generated from RARE rule ite-else-true

enumerator ITE_THEN_LOOKAHEAD_SELF

Auto-generated from RARE rule ite-then-lookahead-self

enumerator ITE_ELSE_LOOKAHEAD_SELF

Auto-generated from RARE rule ite-else-lookahead-self

enumerator BOOL_NOT_ITE_ELIM

Auto-generated from RARE rule bool-not-ite-elim

enumerator ITE_TRUE_COND

Auto-generated from RARE rule ite-true-cond

enumerator ITE_FALSE_COND

Auto-generated from RARE rule ite-false-cond

enumerator ITE_NOT_COND

Auto-generated from RARE rule ite-not-cond

enumerator ITE_EQ_BRANCH

Auto-generated from RARE rule ite-eq-branch

enumerator ITE_THEN_LOOKAHEAD

Auto-generated from RARE rule ite-then-lookahead

enumerator ITE_ELSE_LOOKAHEAD

Auto-generated from RARE rule ite-else-lookahead

enumerator ITE_THEN_NEG_LOOKAHEAD

Auto-generated from RARE rule ite-then-neg-lookahead

enumerator ITE_ELSE_NEG_LOOKAHEAD

Auto-generated from RARE rule ite-else-neg-lookahead

enumerator BV_CONCAT_FLATTEN

Auto-generated from RARE rule bv-concat-flatten

enumerator BV_CONCAT_EXTRACT_MERGE

Auto-generated from RARE rule bv-concat-extract-merge

enumerator BV_EXTRACT_EXTRACT

Auto-generated from RARE rule bv-extract-extract

enumerator BV_EXTRACT_WHOLE

Auto-generated from RARE rule bv-extract-whole

enumerator BV_EXTRACT_CONCAT_1

Auto-generated from RARE rule bv-extract-concat-1

enumerator BV_EXTRACT_CONCAT_2

Auto-generated from RARE rule bv-extract-concat-2

enumerator BV_EXTRACT_CONCAT_3

Auto-generated from RARE rule bv-extract-concat-3

enumerator BV_EXTRACT_CONCAT_4

Auto-generated from RARE rule bv-extract-concat-4

enumerator BV_EXTRACT_BITWISE_AND

Auto-generated from RARE rule bv-extract-bitwise-and

enumerator BV_EXTRACT_BITWISE_OR

Auto-generated from RARE rule bv-extract-bitwise-or

enumerator BV_EXTRACT_BITWISE_XOR

Auto-generated from RARE rule bv-extract-bitwise-xor

enumerator BV_EXTRACT_NOT

Auto-generated from RARE rule bv-extract-not

enumerator BV_EXTRACT_SIGN_EXTEND_1

Auto-generated from RARE rule bv-extract-sign-extend-1

enumerator BV_EXTRACT_SIGN_EXTEND_2

Auto-generated from RARE rule bv-extract-sign-extend-2

enumerator BV_EXTRACT_SIGN_EXTEND_3

Auto-generated from RARE rule bv-extract-sign-extend-3

enumerator BV_NEG_MULT

Auto-generated from RARE rule bv-neg-mult

enumerator BV_NEG_ADD

Auto-generated from RARE rule bv-neg-add

enumerator BV_MULT_DISTRIB_CONST_NEG

Auto-generated from RARE rule bv-mult-distrib-const-neg

enumerator BV_MULT_DISTRIB_CONST_ADD

Auto-generated from RARE rule bv-mult-distrib-const-add

enumerator BV_MULT_DISTRIB_CONST_SUB

Auto-generated from RARE rule bv-mult-distrib-const-sub

enumerator BV_MULT_DISTRIB_1

Auto-generated from RARE rule bv-mult-distrib-1

enumerator BV_MULT_DISTRIB_2

Auto-generated from RARE rule bv-mult-distrib-2

enumerator BV_NOT_XOR

Auto-generated from RARE rule bv-not-xor

enumerator BV_AND_SIMPLIFY_1

Auto-generated from RARE rule bv-and-simplify-1

enumerator BV_AND_SIMPLIFY_2

Auto-generated from RARE rule bv-and-simplify-2

enumerator BV_OR_SIMPLIFY_1

Auto-generated from RARE rule bv-or-simplify-1

enumerator BV_OR_SIMPLIFY_2

Auto-generated from RARE rule bv-or-simplify-2

enumerator BV_XOR_SIMPLIFY_1

Auto-generated from RARE rule bv-xor-simplify-1

enumerator BV_XOR_SIMPLIFY_2

Auto-generated from RARE rule bv-xor-simplify-2

enumerator BV_XOR_SIMPLIFY_3

Auto-generated from RARE rule bv-xor-simplify-3

enumerator BV_ULT_ADD_ONE

Auto-generated from RARE rule bv-ult-add-one

enumerator BV_CONCAT_TO_MULT

Auto-generated from RARE rule bv-concat-to-mult

enumerator BV_MULT_SLT_MULT_1

Auto-generated from RARE rule bv-mult-slt-mult-1

enumerator BV_MULT_SLT_MULT_2

Auto-generated from RARE rule bv-mult-slt-mult-2

enumerator BV_COMMUTATIVE_AND

Auto-generated from RARE rule bv-commutative-and

enumerator BV_COMMUTATIVE_OR

Auto-generated from RARE rule bv-commutative-or

enumerator BV_COMMUTATIVE_XOR

Auto-generated from RARE rule bv-commutative-xor

enumerator BV_COMMUTATIVE_MUL

Auto-generated from RARE rule bv-commutative-mul

enumerator BV_OR_ZERO

Auto-generated from RARE rule bv-or-zero

enumerator BV_MUL_ONE

Auto-generated from RARE rule bv-mul-one

enumerator BV_MUL_ZERO

Auto-generated from RARE rule bv-mul-zero

enumerator BV_ADD_ZERO

Auto-generated from RARE rule bv-add-zero

enumerator BV_ADD_TWO

Auto-generated from RARE rule bv-add-two

enumerator BV_ZERO_EXTEND_ELIMINATE_0

Auto-generated from RARE rule bv-zero-extend-eliminate-0

enumerator BV_SIGN_EXTEND_ELIMINATE_0

Auto-generated from RARE rule bv-sign-extend-eliminate-0

enumerator BV_NOT_NEQ

Auto-generated from RARE rule bv-not-neq

enumerator BV_ULT_ONES

Auto-generated from RARE rule bv-ult-ones

enumerator BV_OR_FLATTEN

Auto-generated from RARE rule bv-or-flatten

enumerator BV_XOR_FLATTEN

Auto-generated from RARE rule bv-xor-flatten

enumerator BV_AND_FLATTEN

Auto-generated from RARE rule bv-and-flatten

enumerator BV_MUL_FLATTEN

Auto-generated from RARE rule bv-mul-flatten

enumerator BV_CONCAT_MERGE_CONST

Auto-generated from RARE rule bv-concat-merge-const

enumerator BV_COMMUTATIVE_ADD

Auto-generated from RARE rule bv-commutative-add

enumerator BV_NEG_SUB

Auto-generated from RARE rule bv-neg-sub

enumerator BV_NEG_IDEMP

Auto-generated from RARE rule bv-neg-idemp

enumerator BV_SUB_ELIMINATE

Auto-generated from RARE rule bv-sub-eliminate

enumerator BV_UGT_ELIMINATE

Auto-generated from RARE rule bv-ugt-eliminate

enumerator BV_UGE_ELIMINATE

Auto-generated from RARE rule bv-uge-eliminate

enumerator BV_SGT_ELIMINATE

Auto-generated from RARE rule bv-sgt-eliminate

enumerator BV_SGE_ELIMINATE

Auto-generated from RARE rule bv-sge-eliminate

enumerator BV_SLT_ELIMINATE

Auto-generated from RARE rule bv-slt-eliminate

enumerator BV_SLE_ELIMINATE

Auto-generated from RARE rule bv-sle-eliminate

enumerator BV_REDOR_ELIMINATE

Auto-generated from RARE rule bv-redor-eliminate

enumerator BV_REDAND_ELIMINATE

Auto-generated from RARE rule bv-redand-eliminate

enumerator BV_ULE_ELIMINATE

Auto-generated from RARE rule bv-ule-eliminate

enumerator BV_COMP_ELIMINATE

Auto-generated from RARE rule bv-comp-eliminate

enumerator BV_REPEAT_ELIMINATE_1

Auto-generated from RARE rule bv-repeat-eliminate-1

enumerator BV_REPEAT_ELIMINATE_2

Auto-generated from RARE rule bv-repeat-eliminate-2

enumerator BV_ROTATE_LEFT_ELIMINATE_1

Auto-generated from RARE rule bv-rotate-left-eliminate-1

enumerator BV_ROTATE_LEFT_ELIMINATE_2

Auto-generated from RARE rule bv-rotate-left-eliminate-2

enumerator BV_ROTATE_RIGHT_ELIMINATE_1

Auto-generated from RARE rule bv-rotate-right-eliminate-1

enumerator BV_ROTATE_RIGHT_ELIMINATE_2

Auto-generated from RARE rule bv-rotate-right-eliminate-2

enumerator BV_NAND_ELIMINATE

Auto-generated from RARE rule bv-nand-eliminate

enumerator BV_NOR_ELIMINATE

Auto-generated from RARE rule bv-nor-eliminate

enumerator BV_XNOR_ELIMINATE

Auto-generated from RARE rule bv-xnor-eliminate

enumerator BV_SDIV_ELIMINATE

Auto-generated from RARE rule bv-sdiv-eliminate

enumerator BV_SDIV_ELIMINATE_FEWER_BITWISE_OPS

Auto-generated from RARE rule bv-sdiv-eliminate-fewer-bitwise-ops

enumerator BV_ZERO_EXTEND_ELIMINATE

Auto-generated from RARE rule bv-zero-extend-eliminate

enumerator BV_SIGN_EXTEND_ELIMINATE

Auto-generated from RARE rule bv-sign-extend-eliminate

enumerator BV_UADDO_ELIMINATE

Auto-generated from RARE rule bv-uaddo-eliminate

enumerator BV_SADDO_ELIMINATE

Auto-generated from RARE rule bv-saddo-eliminate

enumerator BV_SDIVO_ELIMINATE

Auto-generated from RARE rule bv-sdivo-eliminate

enumerator BV_SMOD_ELIMINATE

Auto-generated from RARE rule bv-smod-eliminate

enumerator BV_SMOD_ELIMINATE_FEWER_BITWISE_OPS

Auto-generated from RARE rule bv-smod-eliminate-fewer-bitwise-ops

enumerator BV_SREM_ELIMINATE

Auto-generated from RARE rule bv-srem-eliminate

enumerator BV_SREM_ELIMINATE_FEWER_BITWISE_OPS

Auto-generated from RARE rule bv-srem-eliminate-fewer-bitwise-ops

enumerator BV_USUBO_ELIMINATE

Auto-generated from RARE rule bv-usubo-eliminate

enumerator BV_SSUBO_ELIMINATE

Auto-generated from RARE rule bv-ssubo-eliminate

enumerator BV_ITE_EQUAL_CHILDREN

Auto-generated from RARE rule bv-ite-equal-children

enumerator BV_ITE_CONST_CHILDREN_1

Auto-generated from RARE rule bv-ite-const-children-1

enumerator BV_ITE_CONST_CHILDREN_2

Auto-generated from RARE rule bv-ite-const-children-2

enumerator BV_ITE_EQUAL_COND_1

Auto-generated from RARE rule bv-ite-equal-cond-1

enumerator BV_ITE_EQUAL_COND_2

Auto-generated from RARE rule bv-ite-equal-cond-2

enumerator BV_ITE_EQUAL_COND_3

Auto-generated from RARE rule bv-ite-equal-cond-3

enumerator BV_ITE_MERGE_THEN_IF

Auto-generated from RARE rule bv-ite-merge-then-if

enumerator BV_ITE_MERGE_ELSE_IF

Auto-generated from RARE rule bv-ite-merge-else-if

enumerator BV_ITE_MERGE_THEN_ELSE

Auto-generated from RARE rule bv-ite-merge-then-else

enumerator BV_ITE_MERGE_ELSE_ELSE

Auto-generated from RARE rule bv-ite-merge-else-else

enumerator BV_SHL_BY_CONST_0

Auto-generated from RARE rule bv-shl-by-const-0

enumerator BV_SHL_BY_CONST_1

Auto-generated from RARE rule bv-shl-by-const-1

enumerator BV_SHL_BY_CONST_2

Auto-generated from RARE rule bv-shl-by-const-2

enumerator BV_LSHR_BY_CONST_0

Auto-generated from RARE rule bv-lshr-by-const-0

enumerator BV_LSHR_BY_CONST_1

Auto-generated from RARE rule bv-lshr-by-const-1

enumerator BV_LSHR_BY_CONST_2

Auto-generated from RARE rule bv-lshr-by-const-2

enumerator BV_ASHR_BY_CONST_0

Auto-generated from RARE rule bv-ashr-by-const-0

enumerator BV_ASHR_BY_CONST_1

Auto-generated from RARE rule bv-ashr-by-const-1

enumerator BV_ASHR_BY_CONST_2

Auto-generated from RARE rule bv-ashr-by-const-2

enumerator BV_AND_CONCAT_PULLUP

Auto-generated from RARE rule bv-and-concat-pullup

enumerator BV_OR_CONCAT_PULLUP

Auto-generated from RARE rule bv-or-concat-pullup

enumerator BV_XOR_CONCAT_PULLUP

Auto-generated from RARE rule bv-xor-concat-pullup

enumerator BV_BITWISE_IDEMP_1

Auto-generated from RARE rule bv-bitwise-idemp-1

enumerator BV_BITWISE_IDEMP_2

Auto-generated from RARE rule bv-bitwise-idemp-2

enumerator BV_AND_ZERO

Auto-generated from RARE rule bv-and-zero

enumerator BV_AND_ONE

Auto-generated from RARE rule bv-and-one

enumerator BV_OR_ONE

Auto-generated from RARE rule bv-or-one

enumerator BV_XOR_DUPLICATE

Auto-generated from RARE rule bv-xor-duplicate

enumerator BV_XOR_ONES

Auto-generated from RARE rule bv-xor-ones

enumerator BV_XOR_ZERO

Auto-generated from RARE rule bv-xor-zero

enumerator BV_BITWISE_NOT_AND

Auto-generated from RARE rule bv-bitwise-not-and

enumerator BV_BITWISE_NOT_OR

Auto-generated from RARE rule bv-bitwise-not-or

enumerator BV_XOR_NOT

Auto-generated from RARE rule bv-xor-not

enumerator BV_NOT_IDEMP

Auto-generated from RARE rule bv-not-idemp

enumerator BV_ULT_ZERO_1

Auto-generated from RARE rule bv-ult-zero-1

enumerator BV_ULT_ZERO_2

Auto-generated from RARE rule bv-ult-zero-2

enumerator BV_ULT_SELF

Auto-generated from RARE rule bv-ult-self

enumerator BV_LT_SELF

Auto-generated from RARE rule bv-lt-self

enumerator BV_ULE_SELF

Auto-generated from RARE rule bv-ule-self

enumerator BV_ULE_ZERO

Auto-generated from RARE rule bv-ule-zero

enumerator BV_ZERO_ULE

Auto-generated from RARE rule bv-zero-ule

enumerator BV_SLE_SELF

Auto-generated from RARE rule bv-sle-self

enumerator BV_ULE_MAX

Auto-generated from RARE rule bv-ule-max

enumerator BV_NOT_ULT

Auto-generated from RARE rule bv-not-ult

enumerator BV_NOT_ULE

Auto-generated from RARE rule bv-not-ule

enumerator BV_NOT_SLE

Auto-generated from RARE rule bv-not-sle

enumerator BV_MULT_POW2_1

Auto-generated from RARE rule bv-mult-pow2-1

enumerator BV_MULT_POW2_2

Auto-generated from RARE rule bv-mult-pow2-2

enumerator BV_MULT_POW2_2B

Auto-generated from RARE rule bv-mult-pow2-2b

enumerator BV_EXTRACT_MULT_LEADING_BIT

Auto-generated from RARE rule bv-extract-mult-leading-bit

enumerator BV_UDIV_POW2_NOT_ONE

Auto-generated from RARE rule bv-udiv-pow2-not-one

enumerator BV_UDIV_ZERO

Auto-generated from RARE rule bv-udiv-zero

enumerator BV_UDIV_ONE

Auto-generated from RARE rule bv-udiv-one

enumerator BV_UREM_POW2_NOT_ONE

Auto-generated from RARE rule bv-urem-pow2-not-one

enumerator BV_UREM_ONE

Auto-generated from RARE rule bv-urem-one

enumerator BV_UREM_SELF

Auto-generated from RARE rule bv-urem-self

enumerator BV_SHL_ZERO

Auto-generated from RARE rule bv-shl-zero

enumerator BV_LSHR_ZERO

Auto-generated from RARE rule bv-lshr-zero

enumerator BV_ASHR_ZERO

Auto-generated from RARE rule bv-ashr-zero

enumerator BV_UGT_UREM

Auto-generated from RARE rule bv-ugt-urem

enumerator BV_ULT_ONE

Auto-generated from RARE rule bv-ult-one

enumerator BV_SLT_ZERO

Auto-generated from RARE rule bv-slt-zero

enumerator BV_MERGE_SIGN_EXTEND_1

Auto-generated from RARE rule bv-merge-sign-extend-1

enumerator BV_MERGE_SIGN_EXTEND_2

Auto-generated from RARE rule bv-merge-sign-extend-2

enumerator BV_MERGE_SIGN_EXTEND_3

Auto-generated from RARE rule bv-merge-sign-extend-3

enumerator BV_SIGN_EXTEND_EQ_CONST_1

Auto-generated from RARE rule bv-sign-extend-eq-const-1

enumerator BV_SIGN_EXTEND_EQ_CONST_2

Auto-generated from RARE rule bv-sign-extend-eq-const-2

enumerator BV_ZERO_EXTEND_EQ_CONST_1

Auto-generated from RARE rule bv-zero-extend-eq-const-1

enumerator BV_ZERO_EXTEND_EQ_CONST_2

Auto-generated from RARE rule bv-zero-extend-eq-const-2

enumerator BV_ZERO_EXTEND_ULT_CONST_1

Auto-generated from RARE rule bv-zero-extend-ult-const-1

enumerator BV_ZERO_EXTEND_ULT_CONST_2

Auto-generated from RARE rule bv-zero-extend-ult-const-2

enumerator BV_SIGN_EXTEND_ULT_CONST_1

Auto-generated from RARE rule bv-sign-extend-ult-const-1

enumerator BV_SIGN_EXTEND_ULT_CONST_2

Auto-generated from RARE rule bv-sign-extend-ult-const-2

enumerator BV_SIGN_EXTEND_ULT_CONST_3

Auto-generated from RARE rule bv-sign-extend-ult-const-3

enumerator BV_SIGN_EXTEND_ULT_CONST_4

Auto-generated from RARE rule bv-sign-extend-ult-const-4

enumerator SETS_EQ_SINGLETON_EMP

Auto-generated from RARE rule sets-eq-singleton-emp

enumerator SETS_MEMBER_SINGLETON

Auto-generated from RARE rule sets-member-singleton

enumerator SETS_MEMBER_EMP

Auto-generated from RARE rule sets-member-emp

enumerator SETS_SUBSET_ELIM

Auto-generated from RARE rule sets-subset-elim

enumerator SETS_UNION_COMM

Auto-generated from RARE rule sets-union-comm

enumerator SETS_INTER_COMM

Auto-generated from RARE rule sets-inter-comm

enumerator SETS_INTER_EMP1

Auto-generated from RARE rule sets-inter-emp1

enumerator SETS_INTER_EMP2

Auto-generated from RARE rule sets-inter-emp2

enumerator SETS_MINUS_EMP1

Auto-generated from RARE rule sets-minus-emp1

enumerator SETS_MINUS_EMP2

Auto-generated from RARE rule sets-minus-emp2

enumerator SETS_UNION_EMP1

Auto-generated from RARE rule sets-union-emp1

enumerator SETS_UNION_EMP2

Auto-generated from RARE rule sets-union-emp2

enumerator SETS_INTER_MEMBER

Auto-generated from RARE rule sets-inter-member

enumerator SETS_MINUS_MEMBER

Auto-generated from RARE rule sets-minus-member

enumerator SETS_UNION_MEMBER

Auto-generated from RARE rule sets-union-member

enumerator SETS_CHOOSE_SINGLETON

Auto-generated from RARE rule sets-choose-singleton

enumerator SETS_CARD_SINGLETON

Auto-generated from RARE rule sets-card-singleton

enumerator SETS_CARD_UNION

Auto-generated from RARE rule sets-card-union

enumerator SETS_CARD_MINUS

Auto-generated from RARE rule sets-card-minus

enumerator SETS_CARD_EMP

Auto-generated from RARE rule sets-card-emp

enumerator STR_EQ_CTN_FALSE

Auto-generated from RARE rule str-eq-ctn-false

enumerator STR_EQ_CTN_FULL_FALSE1

Auto-generated from RARE rule str-eq-ctn-full-false1

enumerator STR_EQ_CTN_FULL_FALSE2

Auto-generated from RARE rule str-eq-ctn-full-false2

enumerator STR_CONCAT_FLATTEN

Auto-generated from RARE rule str-concat-flatten

enumerator STR_CONCAT_FLATTEN_EQ

Auto-generated from RARE rule str-concat-flatten-eq

enumerator STR_CONCAT_FLATTEN_EQ_REV

Auto-generated from RARE rule str-concat-flatten-eq-rev

enumerator STR_SUBSTR_EMPTY_STR

Auto-generated from RARE rule str-substr-empty-str

enumerator STR_SUBSTR_EMPTY_RANGE

Auto-generated from RARE rule str-substr-empty-range

enumerator STR_SUBSTR_EMPTY_START

Auto-generated from RARE rule str-substr-empty-start

enumerator STR_SUBSTR_EMPTY_START_NEG

Auto-generated from RARE rule str-substr-empty-start-neg

enumerator STR_SUBSTR_EQ_EMPTY

Auto-generated from RARE rule str-substr-eq-empty

enumerator STR_LEN_REPLACE_INV

Auto-generated from RARE rule str-len-replace-inv

enumerator STR_LEN_UPDATE_INV

Auto-generated from RARE rule str-len-update-inv

enumerator STR_LEN_SUBSTR_IN_RANGE

Auto-generated from RARE rule str-len-substr-in-range

enumerator STR_LEN_SUBSTR_UB1

Auto-generated from RARE rule str-len-substr-ub1

enumerator STR_LEN_SUBSTR_UB2

Auto-generated from RARE rule str-len-substr-ub2

enumerator STR_CONCAT_CLASH

Auto-generated from RARE rule str-concat-clash

enumerator STR_CONCAT_CLASH_REV

Auto-generated from RARE rule str-concat-clash-rev

enumerator STR_CONCAT_CLASH2

Auto-generated from RARE rule str-concat-clash2

enumerator STR_CONCAT_CLASH2_REV

Auto-generated from RARE rule str-concat-clash2-rev

enumerator STR_CONCAT_UNIFY

Auto-generated from RARE rule str-concat-unify

enumerator STR_CONCAT_UNIFY_REV

Auto-generated from RARE rule str-concat-unify-rev

enumerator STR_CONCAT_UNIFY_BASE

Auto-generated from RARE rule str-concat-unify-base

enumerator STR_CONCAT_UNIFY_BASE_REV

Auto-generated from RARE rule str-concat-unify-base-rev

enumerator STR_CONCAT_CLASH_CHAR

Auto-generated from RARE rule str-concat-clash-char

enumerator STR_CONCAT_CLASH_CHAR_REV

Auto-generated from RARE rule str-concat-clash-char-rev

enumerator STR_PREFIXOF_ELIM

Auto-generated from RARE rule str-prefixof-elim

enumerator STR_SUFFIXOF_ELIM

Auto-generated from RARE rule str-suffixof-elim

enumerator STR_PREFIXOF_ONE

Auto-generated from RARE rule str-prefixof-one

enumerator STR_SUFFIXOF_ONE

Auto-generated from RARE rule str-suffixof-one

enumerator STR_SUBSTR_COMBINE1

Auto-generated from RARE rule str-substr-combine1

enumerator STR_SUBSTR_COMBINE2

Auto-generated from RARE rule str-substr-combine2

enumerator STR_SUBSTR_COMBINE3

Auto-generated from RARE rule str-substr-combine3

enumerator STR_SUBSTR_COMBINE4

Auto-generated from RARE rule str-substr-combine4

enumerator STR_SUBSTR_CONCAT1

Auto-generated from RARE rule str-substr-concat1

enumerator STR_SUBSTR_CONCAT2

Auto-generated from RARE rule str-substr-concat2

enumerator STR_SUBSTR_FULL

Auto-generated from RARE rule str-substr-full

enumerator STR_SUBSTR_FULL_EQ

Auto-generated from RARE rule str-substr-full-eq

enumerator STR_CONTAINS_REFL

Auto-generated from RARE rule str-contains-refl

enumerator STR_CONTAINS_CONCAT_FIND

Auto-generated from RARE rule str-contains-concat-find

enumerator STR_CONTAINS_SPLIT_CHAR

Auto-generated from RARE rule str-contains-split-char

enumerator STR_CONTAINS_LT_LEN

Auto-generated from RARE rule str-contains-lt-len

enumerator STR_CONTAINS_LEQ_LEN_EQ

Auto-generated from RARE rule str-contains-leq-len-eq

enumerator STR_CONTAINS_EMP

Auto-generated from RARE rule str-contains-emp

enumerator STR_CONTAINS_IS_EMP

Auto-generated from RARE rule str-contains-is-emp

enumerator STR_CONCAT_EMP

Auto-generated from RARE rule str-concat-emp

enumerator STR_AT_ELIM

Auto-generated from RARE rule str-at-elim

enumerator STR_REPLACE_SELF

Auto-generated from RARE rule str-replace-self

enumerator STR_REPLACE_NO_CONTAINS

Auto-generated from RARE rule str-replace-no-contains

enumerator STR_REPLACE_EMPTY

Auto-generated from RARE rule str-replace-empty

enumerator STR_LEN_CONCAT_REC

Auto-generated from RARE rule str-len-concat-rec

enumerator STR_INDEXOF_SELF

Auto-generated from RARE rule str-indexof-self

enumerator STR_INDEXOF_NO_CONTAINS

Auto-generated from RARE rule str-indexof-no-contains

enumerator STR_TO_LOWER_CONCAT

Auto-generated from RARE rule str-to-lower-concat

enumerator STR_TO_UPPER_CONCAT

Auto-generated from RARE rule str-to-upper-concat

enumerator STR_TO_LOWER_UPPER

Auto-generated from RARE rule str-to-lower-upper

enumerator STR_TO_UPPER_LOWER

Auto-generated from RARE rule str-to-upper-lower

enumerator RE_ALL_ELIM

Auto-generated from RARE rule re-all-elim

enumerator RE_OPT_ELIM

Auto-generated from RARE rule re-opt-elim

enumerator RE_DIFF_ELIM

Auto-generated from RARE rule re-diff-elim

enumerator RE_CONCAT_EMP

Auto-generated from RARE rule re-concat-emp

enumerator RE_CONCAT_NONE

Auto-generated from RARE rule re-concat-none

enumerator RE_CONCAT_FLATTEN

Auto-generated from RARE rule re-concat-flatten

enumerator RE_CONCAT_STAR_SWAP

Auto-generated from RARE rule re-concat-star-swap

enumerator RE_CONCAT_MERGE

Auto-generated from RARE rule re-concat-merge

enumerator RE_UNION_ALL

Auto-generated from RARE rule re-union-all

enumerator RE_UNION_NONE

Auto-generated from RARE rule re-union-none

enumerator RE_UNION_FLATTEN

Auto-generated from RARE rule re-union-flatten

enumerator RE_UNION_DUP

Auto-generated from RARE rule re-union-dup

enumerator RE_INTER_ALL

Auto-generated from RARE rule re-inter-all

enumerator RE_INTER_NONE

Auto-generated from RARE rule re-inter-none

enumerator RE_INTER_FLATTEN

Auto-generated from RARE rule re-inter-flatten

enumerator RE_INTER_DUP

Auto-generated from RARE rule re-inter-dup

enumerator RE_LOOP_NEG

Auto-generated from RARE rule re-loop-neg

enumerator RE_INTER_CSTRING

Auto-generated from RARE rule re-inter-cstring

enumerator RE_INTER_CSTRING_NEG

Auto-generated from RARE rule re-inter-cstring-neg

enumerator STR_NTH_ELIM_CODE

Auto-generated from RARE rule str-nth-elim-code

enumerator STR_SUBSTR_LEN_INCLUDE

Auto-generated from RARE rule str-substr-len-include

enumerator STR_SUBSTR_LEN_INCLUDE_PRE

Auto-generated from RARE rule str-substr-len-include-pre

enumerator STR_SUBSTR_LEN_SKIP

Auto-generated from RARE rule str-substr-len-skip

enumerator SEQ_REV_CONCAT

Auto-generated from RARE rule seq-rev-concat

enumerator SEQ_LEN_UNIT

Auto-generated from RARE rule seq-len-unit

enumerator SEQ_NTH_UNIT

Auto-generated from RARE rule seq-nth-unit

enumerator SEQ_REV_UNIT

Auto-generated from RARE rule seq-rev-unit

enumerator RE_IN_EMPTY

Auto-generated from RARE rule re-in-empty

enumerator RE_IN_SIGMA

Auto-generated from RARE rule re-in-sigma

enumerator RE_IN_SIGMA_STAR

Auto-generated from RARE rule re-in-sigma-star

enumerator RE_IN_CSTRING

Auto-generated from RARE rule re-in-cstring

enumerator RE_IN_COMP

Auto-generated from RARE rule re-in-comp

enumerator STR_IN_RE_UNION_ELIM

Auto-generated from RARE rule str-in-re-union-elim

enumerator STR_IN_RE_INTER_ELIM

Auto-generated from RARE rule str-in-re-inter-elim

enumerator STR_IN_RE_RANGE_ELIM

Auto-generated from RARE rule str-in-re-range-elim

enumerator STR_IN_RE_CONTAINS

Auto-generated from RARE rule str-in-re-contains

enumerator STR_IN_RE_STRIP_PREFIX

Auto-generated from RARE rule str-in-re-strip-prefix

enumerator STR_IN_RE_STRIP_PREFIX_NEG

Auto-generated from RARE rule str-in-re-strip-prefix-neg

enumerator STR_IN_RE_STRIP_PREFIX_SR_SINGLE

Auto-generated from RARE rule str-in-re-strip-prefix-sr-single

enumerator STR_IN_RE_STRIP_PREFIX_SR_SINGLE_NEG

Auto-generated from RARE rule str-in-re-strip-prefix-sr-single-neg

enumerator STR_IN_RE_STRIP_PREFIX_SRS_SINGLE

Auto-generated from RARE rule str-in-re-strip-prefix-srs-single

enumerator STR_IN_RE_STRIP_PREFIX_SRS_SINGLE_NEG

Auto-generated from RARE rule str-in-re-strip-prefix-srs-single-neg

enumerator STR_IN_RE_STRIP_PREFIX_S_SINGLE

Auto-generated from RARE rule str-in-re-strip-prefix-s-single

enumerator STR_IN_RE_STRIP_PREFIX_S_SINGLE_NEG

Auto-generated from RARE rule str-in-re-strip-prefix-s-single-neg

enumerator STR_IN_RE_STRIP_PREFIX_BASE

Auto-generated from RARE rule str-in-re-strip-prefix-base

enumerator STR_IN_RE_STRIP_PREFIX_BASE_NEG

Auto-generated from RARE rule str-in-re-strip-prefix-base-neg

enumerator STR_IN_RE_STRIP_PREFIX_BASE_S_SINGLE

Auto-generated from RARE rule str-in-re-strip-prefix-base-s-single

enumerator STR_IN_RE_STRIP_PREFIX_BASE_S_SINGLE_NEG

Auto-generated from RARE rule str-in-re-strip-prefix-base-s-single-neg

enumerator STR_IN_RE_STRIP_CHAR

Auto-generated from RARE rule str-in-re-strip-char

enumerator STR_IN_RE_STRIP_CHAR_S_SINGLE

Auto-generated from RARE rule str-in-re-strip-char-s-single

enumerator STR_IN_RE_STRIP_PREFIX_REV

Auto-generated from RARE rule str-in-re-strip-prefix-rev

enumerator STR_IN_RE_STRIP_PREFIX_NEG_REV

Auto-generated from RARE rule str-in-re-strip-prefix-neg-rev

enumerator STR_IN_RE_STRIP_PREFIX_SR_SINGLE_REV

Auto-generated from RARE rule str-in-re-strip-prefix-sr-single-rev

enumerator STR_IN_RE_STRIP_PREFIX_SR_SINGLE_NEG_REV

Auto-generated from RARE rule str-in-re-strip-prefix-sr-single-neg-rev

enumerator STR_IN_RE_STRIP_PREFIX_SRS_SINGLE_REV

Auto-generated from RARE rule str-in-re-strip-prefix-srs-single-rev

enumerator STR_IN_RE_STRIP_PREFIX_SRS_SINGLE_NEG_REV

Auto-generated from RARE rule str-in-re-strip-prefix-srs-single-neg-rev

enumerator STR_IN_RE_STRIP_PREFIX_S_SINGLE_REV

Auto-generated from RARE rule str-in-re-strip-prefix-s-single-rev

enumerator STR_IN_RE_STRIP_PREFIX_S_SINGLE_NEG_REV

Auto-generated from RARE rule str-in-re-strip-prefix-s-single-neg-rev

enumerator STR_IN_RE_STRIP_PREFIX_BASE_REV

Auto-generated from RARE rule str-in-re-strip-prefix-base-rev

enumerator STR_IN_RE_STRIP_PREFIX_BASE_NEG_REV

Auto-generated from RARE rule str-in-re-strip-prefix-base-neg-rev

enumerator STR_IN_RE_STRIP_PREFIX_BASE_S_SINGLE_REV

Auto-generated from RARE rule str-in-re-strip-prefix-base-s-single-rev

enumerator STR_IN_RE_STRIP_PREFIX_BASE_S_SINGLE_NEG_REV

Auto-generated from RARE rule str-in-re-strip-prefix-base-s-single-neg-rev

enumerator STR_IN_RE_STRIP_CHAR_REV

Auto-generated from RARE rule str-in-re-strip-char-rev

enumerator STR_IN_RE_STRIP_CHAR_S_SINGLE_REV

Auto-generated from RARE rule str-in-re-strip-char-s-single-rev

enumerator STR_IN_RE_NO_PREFIX

Auto-generated from RARE rule str-in-re-no-prefix

enumerator STR_IN_RE_NO_PREFIX_REV

Auto-generated from RARE rule str-in-re-no-prefix-rev

enumerator STR_IN_RE_REQ_UNFOLD

Auto-generated from RARE rule str-in-re-req-unfold

enumerator STR_IN_RE_REQ_UNFOLD_REV

Auto-generated from RARE rule str-in-re-req-unfold-rev

enumerator STR_IN_RE_SKIP_UNFOLD

Auto-generated from RARE rule str-in-re-skip-unfold

enumerator STR_IN_RE_SKIP_UNFOLD_REV

Auto-generated from RARE rule str-in-re-skip-unfold-rev

enumerator STR_IN_RE_TEST_UNFOLD

Auto-generated from RARE rule str-in-re-test-unfold

enumerator STR_IN_RE_TEST_UNFOLD_REV

Auto-generated from RARE rule str-in-re-test-unfold-rev

enumerator EQ_REFL

Auto-generated from RARE rule eq-refl

enumerator EQ_SYMM

Auto-generated from RARE rule eq-symm

enumerator DISTINCT_BINARY_ELIM

Auto-generated from RARE rule distinct-binary-elim

enumerator UF_BV2NAT_GEQ_ELIM

Auto-generated from RARE rule uf-bv2nat-geq-elim

std::ostream &cvc5::operator<<(std::ostream &out, ProofRewriteRule rule)

Writes a proof rewrite rule name to a stream.

Parameters:
  • out – The stream to write to

  • rule – The proof rewrite rule to write to the stream

Returns:

The stream

std::string std::to_string(cvc5::ProofRewriteRule rule)

Converts a proof rewrite rule to a string.

Parameters:

rule – The proof rewrite rule

Returns:

The name of the proof rewrite rule

template<>
struct hash<cvc5::ProofRewriteRule>

Hash function for ProofRewriteRules.

Public Functions

size_t operator()(cvc5::ProofRewriteRule rule) const

Hashes a ProofRewriteRule to a size_t.

Parameters:

rule – The proof rewrite rule.

Returns:

The hash value.