Cvc5ProofRule and Cvc5ProofRewriteRule 

Enum Cvc5ProofRule captures the reasoning steps performed by the SAT solver, the theory solvers and the preprocessor. It represents the inference rules used to derive conclusions within a proof.

Enum Cvc5ProofRewriteRule pertains to rewrites performed on terms. These identifiers are arguments of the proof rules CVC5_PROOF_RULE_THEORY_REWRITE and CVC5_PROOF_RULE_DSL_REWRITE .

enum Cvc5ProofRule 

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 CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_MACRO_REWRITE 

Builtin theory – Rewrite

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

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

enumerator CVC5_PROOF_RULE_EVALUATE 

Builtin theory – Evaluate

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

where \(\texttt{evaluate}\) is implemented by calling the method \(\texttt{Evalutor::evaluate}\) in theory/evaluator.h with an empty substitution. Note this is equivalent to: (REWRITE t MethodId::RW_EVALUATE) .

enumerator CVC5_PROOF_RULE_ACI_NORM 

Builtin theory – associative/commutative/idempotency/identity normalization

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

where \(\texttt{expr::isACNorm(t, s)} = \top\) . For details, see expr/nary_term_util.h . 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 CVC5_PROOF_RULE_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.

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

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 CVC5_PROOF_RULE_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 \(\texttt{rewrite}_{idr}(F \circ \sigma_{ids, ida}(F_n) \circ \cdots \circ \sigma_{ids, ida}(F_1)) = \top\) 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 CVC5_PROOF_RULE_MACRO_SR_PRED_ELIM 

Builtin theory – Substitution + Rewriting predicate elimination

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

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

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

enumerator CVC5_PROOF_RULE_MACRO_SR_PRED_TRANSFORM 

Builtin theory – Substitution + Rewriting predicate elimination

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

where

\[\begin{split}\texttt{rewrite}_{idr}(F \circ \sigma_{ids, ida}(F_n) \circ\cdots \circ \sigma_{ids, ida}(F_1)) =\\ \texttt{rewrite}_{idr}(G \circ \sigma_{ids, ida}(F_n) \circ \cdots \circ \sigma_{ids, ida}(F_1))\end{split}\]

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 CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_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 \(\texttt{expr::narySubstitute}\) (for details, see expr/nary_term_util.h ) which replaces each \(x_i\) with the list \(t_i\) in its place.

enumerator CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_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 \(C_1 \diamond_{L,pol} C_2\) 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 \(C_i' = C_{i-1} \diamond_{L_{i-1}, pol_{i-1}} C_i'\)

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

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

enumerator CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_MACRO_RESOLUTION_TRUST 

Boolean – N-ary Resolution + Factoring + Reordering unchecked

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

enumerator CVC5_PROOF_RULE_SPLIT 

Boolean – Split

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

enumerator CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_NOT_NOT_ELIM 

Boolean – Double negation elimination

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

enumerator CVC5_PROOF_RULE_CONTRA 

Boolean – Contradiction

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

enumerator CVC5_PROOF_RULE_AND_ELIM 

Boolean – And elimination

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

enumerator CVC5_PROOF_RULE_AND_INTRO 

Boolean – And introduction

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

enumerator CVC5_PROOF_RULE_NOT_OR_ELIM 

Boolean – Not Or elimination

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

enumerator CVC5_PROOF_RULE_IMPLIES_ELIM 

Boolean – Implication elimination

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

enumerator CVC5_PROOF_RULE_NOT_IMPLIES_ELIM1 

Boolean – Not Implication elimination version 1

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

enumerator CVC5_PROOF_RULE_NOT_IMPLIES_ELIM2 

Boolean – Not Implication elimination version 2

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

enumerator CVC5_PROOF_RULE_EQUIV_ELIM1 

Boolean – Equivalence elimination version 1

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

enumerator CVC5_PROOF_RULE_EQUIV_ELIM2 

Boolean – Equivalence elimination version 2

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

enumerator CVC5_PROOF_RULE_NOT_EQUIV_ELIM1 

Boolean – Not Equivalence elimination version 1

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

enumerator CVC5_PROOF_RULE_NOT_EQUIV_ELIM2 

Boolean – Not Equivalence elimination version 2

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

enumerator CVC5_PROOF_RULE_XOR_ELIM1 

Boolean – XOR elimination version 1

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

enumerator CVC5_PROOF_RULE_XOR_ELIM2 

Boolean – XOR elimination version 2

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

enumerator CVC5_PROOF_RULE_NOT_XOR_ELIM1 

Boolean – Not XOR elimination version 1

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

enumerator CVC5_PROOF_RULE_NOT_XOR_ELIM2 

Boolean – Not XOR elimination version 2

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

enumerator CVC5_PROOF_RULE_ITE_ELIM1 

Boolean – ITE elimination version 1

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

enumerator CVC5_PROOF_RULE_ITE_ELIM2 

Boolean – ITE elimination version 2

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

enumerator CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_NOT_ITE_ELIM2 

Boolean – Not ITE elimination version 2

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

enumerator CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_CNF_IMPLIES_NEG1 

Boolean – CNF – Implies Negative 1

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

enumerator CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_REFL 

Equality – Reflexivity

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

enumerator CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_TRANS 

Equality – Transitivity

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

enumerator CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_TRUE_INTRO 

Equality – True intro

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

enumerator CVC5_PROOF_RULE_TRUE_ELIM 

Equality – True elim

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

enumerator CVC5_PROOF_RULE_FALSE_INTRO 

Equality – False intro

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

enumerator CVC5_PROOF_RULE_FALSE_ELIM 

Equality – False elim

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

enumerator CVC5_PROOF_RULE_HO_APP_ENCODE 

Equality – Higher-order application encoding

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

where t’ is the higher-order application that is equivalent to t , as implemented by uf::TheoryUfRewriter::getHoApplyForApplyUf . For details see theory/uf/theory_uf_rewriter.h

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

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

enumerator CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_MACRO_BV_BITBLAST 

Bit-vectors – (Macro) Bitblast

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

where \(\texttt{bitblast}\) 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 CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_BV_EAGER_ATOM 

Bit-vectors – Bit-vector eager atom

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

where \(F\) is of kind BITVECTOR_EAGER_ATOM .

enumerator CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_DT_CLASH 

Datatypes – Clash

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

enumerator CVC5_PROOF_RULE_SKOLEM_INTRO 

Quantifiers – Skolem introduction

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

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

enumerator CVC5_PROOF_RULE_SKOLEMIZE 

Quantifiers – Skolemization

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

where \(\sigma\) maps \(x_1,\dots,x_n\) to their representative skolems, which are skolems \(k_1,\dots,k_n\) . For each \(k_i\) , its skolem identifier is QUANTIFIERS_SKOLEMIZE , and its indices are \((\forall x_1\dots x_n.\> F)\) and \(x_i\) .

enumerator CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_SETS_SINGLETON_INJ 

Sets – Singleton injectivity

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

enumerator CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_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 \(z = \epsilon\) .

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 CVC5_PROOF_RULE_CONCAT_UNIFY 

Strings – Core rules – Concatenation unification

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

Alternatively for the reverse:

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

enumerator CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_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 the purification skolem for \(\mathit{ite}( \mathit{len}(t_1) >= \mathit{len}(s_1), \mathit{suf}(t_1,\mathit{len}(s_1)), \mathit{suf}(s_1,\mathit{len}(t_1)))\) 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 CVC5_PROOF_RULE_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 the purification skolem for \(\mathit{suf}(t_1,1)\) .

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 the purification skolem for \(\mathit{pre}(t_2,\mathit{len}(t_2) - 1)\) .

enumerator CVC5_PROOF_RULE_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 \(\mathit{ite}( \mathit{len}(t_1) >= \mathit{len}(s_1), \mathit{suf}(t_1,\mathit{len}(s_1)), \mathit{suf}(s_1,\mathit{len}(t_1)))\) .

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 CVC5_PROOF_RULE_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 the purification skolem for \(\mathit{suf}(t_1,\mathit{len}(w_3))\) . 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 the purification skolem for \(\mathit{pre}(t_2,\mathit{len}(t_2) - \mathit{len}(w_3))\) . 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 CVC5_PROOF_RULE_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 the purification skolem for \(\mathit{pre}(t,n)\) and \(w_2\) is the purification skolem for \(\mathit{suf}(t,n)\) . 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 CVC5_PROOF_RULE_STRING_LENGTH_POS 

Strings – Core rules – Length positive

\[\inferrule{-\mid t}{(\mathit{len}(t) = 0\wedge t= \epsilon)\vee \mathit{len}(t) > 0}\]

enumerator CVC5_PROOF_RULE_STRING_LENGTH_NON_EMPTY 

Strings – Core rules – Length non-empty

\[\inferrule{t\neq \epsilon\mid -}{\mathit{len}(t) \neq 0}\]

enumerator CVC5_PROOF_RULE_STRING_REDUCTION 

Strings – Extended functions – Reduction

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

where \(w\) is \(\texttt{StringsPreprocess::reduce}(t, R, \dots)\) . For details, see theory/strings/theory_strings_preprocess.h . In other words, \(R\) is the reduction predicate for extended term \(t\) , and \(w\) is the purification skolem for \(t\) .

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

enumerator CVC5_PROOF_RULE_STRING_EAGER_REDUCTION 

Strings – Extended functions – Eager reduction

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

where \(R\) is \(\texttt{TermRegistry::eagerReduce}(t)\) . For details, see theory/strings/term_registry.h .

enumerator CVC5_PROOF_RULE_RE_INTER 

Strings – Regular expressions – Intersection

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

enumerator CVC5_PROOF_RULE_RE_UNFOLD_POS 

Strings – Regular expressions – Positive Unfold

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

where \(F\) corresponds to the one-step unfolding of the premise. This is implemented by \(\texttt{RegExpOpr::reduceRegExpPos}(t\in R)\) .

enumerator CVC5_PROOF_RULE_RE_UNFOLD_NEG 

Strings – Regular expressions – Negative Unfold

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

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 CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_MACRO_RE_ELIM 

Strings – Regular expressions – Macro elimination

\[\inferrule{-\mid F,b}{F = F'}\]

where \(F'\) is the result of eliminating regular expressions from \(F\) using the routine \(\texttt{strings::RegExpElimination::eliminate}(F, b)\) . Here, \(b\) is a Boolean indicating whether we are using aggressive eliminations. Notice this rule concludes \(F = F\) if no eliminations are performed for \(F\) .

We do not currently support elaboration of this macro.

enumerator CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_INT_TIGHT_UB 

Arithmetic – Tighten strict integer upper bounds

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

where \(i\) has integer type.

enumerator CVC5_PROOF_RULE_INT_TIGHT_LB 

Arithmetic – Tighten strict integer lower bounds

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

where \(i\) has integer type.

enumerator CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_ARITH_OP_ELIM_AXIOM 

Arithmetic – Operator elimination

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

enumerator CVC5_PROOF_RULE_ARITH_POLY_NORM 

Arithmetic – Polynomial normalization

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

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

enumerator CVC5_PROOF_RULE_ARITH_POLY_NORM_REL 

Arithmetic – Polynomial normalization for relations

\[\inferrule{c_x \cdot (x_1 - x_2) = c_y \cdot (y_1 - y_2) \mid \diamond} {(x_1 \diamond x_2) = (y_1 \diamond y_2)}\]

where \(\diamond \in \{<, \leq, =, \geq, >\}\) for arithmetic and \(\diamond \in \{=\}\) for bitvectors. \(c_x\) and :math:c_y` are scaling factors. For \(<, \leq, \geq, >\) , the scaling factors have the same sign. For bitvectors, they are set to \(1\) .

enumerator CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_ARITH_TRANS_EXP_NEG 

Arithmetic – Transcendentals – Exp at negative values

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

enumerator CVC5_PROOF_RULE_ARITH_TRANS_EXP_POSITIVITY 

Arithmetic – Transcendentals – Exp is always positive

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

enumerator CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_ARITH_TRANS_EXP_ZERO 

Arithmetic – Transcendentals – Exp at zero

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

enumerator CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_ARITH_TRANS_SINE_SHIFT 

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

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

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 skolem that is the number of phases \(y\) is shifted. In particular, \(y\) is the TRANSCENDENTAL_PURIFY_ARG skolem for \(\sin(x)\) and \(s\) is the TRANSCENDENTAL_SINE_PHASE_SHIFT skolem for \(x\) .

enumerator CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_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 CVC5_PROOF_RULE_UNKNOWN 

enumerator CVC5_PROOF_RULE_LAST 

const char * cvc5_proof_rule_to_string ( Cvc5ProofRule rule ) 

Get a string representation of a Cvc5ProofRule.

Parameters :

rule – The proof rule.

Returns :

The string representation.

size_t cvc5_proof_rule_hash ( Cvc5ProofRule rule ) 

Hash function for Cvc5ProofRule.

Parameters :

rule – The proof rule.

Returns :

The hash value.

enum Cvc5ProofRewriteRule 

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 CVC5_PROOF_REWRITE_RULE_NONE 

enumerator CVC5_PROOF_REWRITE_RULE_DISTINCT_ELIM 

Builtin – Distinct elimination

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

enumerator CVC5_PROOF_REWRITE_RULE_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 CVC5_PROOF_REWRITE_RULE_ARITH_DIV_BY_CONST_ELIM 

Arith – Division by constant elimination

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

where \(c\) is a constant.

enumerator CVC5_PROOF_REWRITE_RULE_MACRO_ARITH_STRING_PRED_ENTAIL 

Arithmetic - strings predicate entailment

\[(= s t) = c\]

\[(>= s t) = c\]

where \(c\) is a Boolean constant. This macro is elaborated by applications of EVALUATE , ARITH_POLY_NORM , ARITH_STRING_PRED_ENTAIL , ARITH_STRING_PRED_SAFE_APPROX , as well as other rewrites for normalizing arithmetic predicates.

enumerator CVC5_PROOF_REWRITE_RULE_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 CVC5_PROOF_REWRITE_RULE_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 CVC5_PROOF_REWRITE_RULE_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 CVC5_PROOF_REWRITE_RULE_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 CVC5_PROOF_REWRITE_RULE_EXISTS_ELIM 

Quantifiers – Exists elimination

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

enumerator CVC5_PROOF_REWRITE_RULE_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 CVC5_PROOF_REWRITE_RULE_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 CVC5_PROOF_REWRITE_RULE_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 CVC5_PROOF_REWRITE_RULE_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 CVC5_PROOF_REWRITE_RULE_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 CVC5_PROOF_REWRITE_RULE_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 CVC5_PROOF_REWRITE_RULE_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 CVC5_PROOF_REWRITE_RULE_DT_COLLAPSE_TESTER_SINGLETON 

Datatypes - collapse tester

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

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

enumerator CVC5_PROOF_REWRITE_RULE_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 CVC5_PROOF_REWRITE_RULE_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 CVC5_PROOF_REWRITE_RULE_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 CVC5_PROOF_REWRITE_RULE_BV_ADD_COMBINE_LIKE_TERMS 

Bitvectors - Combine like terms during addition by counting terms

enumerator CVC5_PROOF_REWRITE_RULE_BV_MULT_SIMPLIFY 

Bitvectors - Extract negations from multiplicands

\[bvmul(bvneg(a),\ b,\ c) = bvneg(bvmul(a,\ b,\ c))\]

enumerator CVC5_PROOF_REWRITE_RULE_BV_BITWISE_SLICING 

Bitvectors - Extract continuous substrings of bitvectors

\[bvand(a,\ c) = concat(bvand(a[i_0:j_0],\ c_0) ... bvand(a[i_n:j_n],\ c_n))\]

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

enumerator CVC5_PROOF_REWRITE_RULE_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 CVC5_PROOF_REWRITE_RULE_RE_INTER_UNION_INCLUSION 

Strings - regular expression intersection/union inclusion

\[(re.inter\ R) = \mathit{re.inter}(\mathit{re.none}, R_0)\]

where \(R\) is a list of regular expressions containing r_1 , re.comp(r_2) and the list \(R_0\) where r_2 is a superset of r_1 .

or alternatively:

\[\mathit{re.union}(R) = \mathit{re.union}(\mathit{re}.\text{*}(\mathit{re.allchar}),\ R_0)\]

where \(R\) is a list of regular expressions containing r_1 , re.comp(r_2) and the list \(R_0\) , where r_1 is a superset of r_2 .

enumerator CVC5_PROOF_REWRITE_RULE_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 CVC5_PROOF_REWRITE_RULE_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 CVC5_PROOF_REWRITE_RULE_STR_IN_RE_CONCAT_STAR_CHAR 

Strings - string in regular expression concatenation star character

\[\begin{split}\mathit{str.in\_re}(\mathit{str}.\text{++}(s_1, \ldots, s_n), \mathit{re}.\text{*}(R)) =\\ \mathit{str.in\_re}(s_1, \mathit{re}.\text{*}(R)) \wedge \ldots \wedge \mathit{str.in\_re}(s_n, \mathit{re}.\text{*}(R))\end{split}\]

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

enumerator CVC5_PROOF_REWRITE_RULE_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 CVC5_PROOF_REWRITE_RULE_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 CVC5_PROOF_REWRITE_RULE_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 CVC5_PROOF_REWRITE_RULE_SETS_IS_EMPTY_EVAL 

Sets - empty tester evaluation

\[\mathit{sets.is\_empty}(\epsilon) = \top\]

where \(\epsilon\) is the empty set, or alternatively:

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

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

enumerator CVC5_PROOF_REWRITE_RULE_ARITH_PLUS_ZERO 

Auto-generated from RARE rule arith-plus-zero

enumerator CVC5_PROOF_REWRITE_RULE_ARITH_MUL_ONE 

Auto-generated from RARE rule arith-mul-one

enumerator CVC5_PROOF_REWRITE_RULE_ARITH_MUL_ZERO 

Auto-generated from RARE rule arith-mul-zero

enumerator CVC5_PROOF_REWRITE_RULE_ARITH_DIV_TOTAL 

Auto-generated from RARE rule arith-div-total

enumerator CVC5_PROOF_REWRITE_RULE_ARITH_INT_DIV_TOTAL 

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

enumerator CVC5_PROOF_REWRITE_RULE_ARITH_INT_DIV_TOTAL_ONE 

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

enumerator CVC5_PROOF_REWRITE_RULE_ARITH_INT_DIV_TOTAL_ZERO 

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

enumerator CVC5_PROOF_REWRITE_RULE_ARITH_INT_MOD_TOTAL 

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

enumerator CVC5_PROOF_REWRITE_RULE_ARITH_INT_MOD_TOTAL_ONE 

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

enumerator CVC5_PROOF_REWRITE_RULE_ARITH_INT_MOD_TOTAL_ZERO 

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

enumerator CVC5_PROOF_REWRITE_RULE_ARITH_NEG_NEG_ONE 

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

enumerator CVC5_PROOF_REWRITE_RULE_ARITH_ELIM_UMINUS 

Auto-generated from RARE rule arith-elim-uminus

enumerator CVC5_PROOF_REWRITE_RULE_ARITH_ELIM_MINUS 

Auto-generated from RARE rule arith-elim-minus

enumerator CVC5_PROOF_REWRITE_RULE_ARITH_ELIM_GT 

Auto-generated from RARE rule arith-elim-gt

enumerator CVC5_PROOF_REWRITE_RULE_ARITH_ELIM_LT 

Auto-generated from RARE rule arith-elim-lt

enumerator CVC5_PROOF_REWRITE_RULE_ARITH_ELIM_INT_GT 

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

enumerator CVC5_PROOF_REWRITE_RULE_ARITH_ELIM_INT_LT 

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

enumerator CVC5_PROOF_REWRITE_RULE_ARITH_ELIM_LEQ 

Auto-generated from RARE rule arith-elim-leq

enumerator CVC5_PROOF_REWRITE_RULE_ARITH_LEQ_NORM 

Auto-generated from RARE rule arith-leq-norm

enumerator CVC5_PROOF_REWRITE_RULE_ARITH_GEQ_TIGHTEN 

Auto-generated from RARE rule arith-geq-tighten

enumerator CVC5_PROOF_REWRITE_RULE_ARITH_GEQ_NORM1 

Auto-generated from RARE rule arith-geq-norm1

enumerator CVC5_PROOF_REWRITE_RULE_ARITH_GEQ_NORM2 

Auto-generated from RARE rule arith-geq-norm2

enumerator CVC5_PROOF_REWRITE_RULE_ARITH_REFL_LEQ 

Auto-generated from RARE rule arith-refl-leq

enumerator CVC5_PROOF_REWRITE_RULE_ARITH_REFL_LT 

Auto-generated from RARE rule arith-refl-lt

enumerator CVC5_PROOF_REWRITE_RULE_ARITH_REFL_GEQ 

Auto-generated from RARE rule arith-refl-geq

enumerator CVC5_PROOF_REWRITE_RULE_ARITH_REFL_GT 

Auto-generated from RARE rule arith-refl-gt

enumerator CVC5_PROOF_REWRITE_RULE_ARITH_REAL_EQ_ELIM 

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

enumerator CVC5_PROOF_REWRITE_RULE_ARITH_INT_EQ_ELIM 

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

enumerator CVC5_PROOF_REWRITE_RULE_ARITH_PLUS_FLATTEN 

Auto-generated from RARE rule arith-plus-flatten

enumerator CVC5_PROOF_REWRITE_RULE_ARITH_MULT_FLATTEN 

Auto-generated from RARE rule arith-mult-flatten

enumerator CVC5_PROOF_REWRITE_RULE_ARITH_MULT_DIST 

Auto-generated from RARE rule arith-mult-dist

enumerator CVC5_PROOF_REWRITE_RULE_ARITH_PLUS_CANCEL1 

Auto-generated from RARE rule arith-plus-cancel1

enumerator CVC5_PROOF_REWRITE_RULE_ARITH_PLUS_CANCEL2 

Auto-generated from RARE rule arith-plus-cancel2

enumerator CVC5_PROOF_REWRITE_RULE_ARITH_ABS_ELIM 

Auto-generated from RARE rule arith-abs-elim

enumerator CVC5_PROOF_REWRITE_RULE_ARITH_TO_REAL_ELIM 

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

enumerator CVC5_PROOF_REWRITE_RULE_ARITH_TO_INT_ELIM_TO_REAL 

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

enumerator CVC5_PROOF_REWRITE_RULE_ARITH_DIV_ELIM_TO_REAL1 

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

enumerator CVC5_PROOF_REWRITE_RULE_ARITH_DIV_ELIM_TO_REAL2 

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

enumerator CVC5_PROOF_REWRITE_RULE_ARRAY_READ_OVER_WRITE 

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

enumerator CVC5_PROOF_REWRITE_RULE_ARRAY_READ_OVER_WRITE2 

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

enumerator CVC5_PROOF_REWRITE_RULE_ARRAY_STORE_OVERWRITE 

Auto-generated from RARE rule array-store-overwrite

enumerator CVC5_PROOF_REWRITE_RULE_ARRAY_STORE_SELF 

Auto-generated from RARE rule array-store-self

enumerator CVC5_PROOF_REWRITE_RULE_BOOL_DOUBLE_NOT_ELIM 

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

enumerator CVC5_PROOF_REWRITE_RULE_BOOL_NOT_TRUE 

Auto-generated from RARE rule bool-not-true

enumerator CVC5_PROOF_REWRITE_RULE_BOOL_NOT_FALSE 

Auto-generated from RARE rule bool-not-false

enumerator CVC5_PROOF_REWRITE_RULE_BOOL_EQ_TRUE 

Auto-generated from RARE rule bool-eq-true

enumerator CVC5_PROOF_REWRITE_RULE_BOOL_EQ_FALSE 

Auto-generated from RARE rule bool-eq-false

enumerator CVC5_PROOF_REWRITE_RULE_BOOL_EQ_NREFL 

Auto-generated from RARE rule bool-eq-nrefl

enumerator CVC5_PROOF_REWRITE_RULE_BOOL_IMPL_FALSE1 

Auto-generated from RARE rule bool-impl-false1

enumerator CVC5_PROOF_REWRITE_RULE_BOOL_IMPL_FALSE2 

Auto-generated from RARE rule bool-impl-false2

enumerator CVC5_PROOF_REWRITE_RULE_BOOL_IMPL_TRUE1 

Auto-generated from RARE rule bool-impl-true1

enumerator CVC5_PROOF_REWRITE_RULE_BOOL_IMPL_TRUE2 

Auto-generated from RARE rule bool-impl-true2

enumerator CVC5_PROOF_REWRITE_RULE_BOOL_IMPL_ELIM 

Auto-generated from RARE rule bool-impl-elim

enumerator CVC5_PROOF_REWRITE_RULE_BOOL_OR_TRUE 

Auto-generated from RARE rule bool-or-true

enumerator CVC5_PROOF_REWRITE_RULE_BOOL_OR_FALSE 

Auto-generated from RARE rule bool-or-false

enumerator CVC5_PROOF_REWRITE_RULE_BOOL_OR_FLATTEN 

Auto-generated from RARE rule bool-or-flatten

enumerator CVC5_PROOF_REWRITE_RULE_BOOL_OR_DUP 

Auto-generated from RARE rule bool-or-dup

enumerator CVC5_PROOF_REWRITE_RULE_BOOL_AND_TRUE 

Auto-generated from RARE rule bool-and-true

enumerator CVC5_PROOF_REWRITE_RULE_BOOL_AND_FALSE 

Auto-generated from RARE rule bool-and-false

enumerator CVC5_PROOF_REWRITE_RULE_BOOL_AND_FLATTEN 

Auto-generated from RARE rule bool-and-flatten

enumerator CVC5_PROOF_REWRITE_RULE_BOOL_AND_DUP 

Auto-generated from RARE rule bool-and-dup

enumerator CVC5_PROOF_REWRITE_RULE_BOOL_AND_CONF 

Auto-generated from RARE rule bool-and-conf

enumerator CVC5_PROOF_REWRITE_RULE_BOOL_OR_TAUT 

Auto-generated from RARE rule bool-or-taut

enumerator CVC5_PROOF_REWRITE_RULE_BOOL_OR_DE_MORGAN 

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

enumerator CVC5_PROOF_REWRITE_RULE_BOOL_IMPLIES_DE_MORGAN 

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

enumerator CVC5_PROOF_REWRITE_RULE_BOOL_AND_DE_MORGAN 

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

enumerator CVC5_PROOF_REWRITE_RULE_BOOL_XOR_REFL 

Auto-generated from RARE rule bool-xor-refl

enumerator CVC5_PROOF_REWRITE_RULE_BOOL_XOR_NREFL 

Auto-generated from RARE rule bool-xor-nrefl

enumerator CVC5_PROOF_REWRITE_RULE_BOOL_XOR_FALSE 

Auto-generated from RARE rule bool-xor-false

enumerator CVC5_PROOF_REWRITE_RULE_BOOL_XOR_TRUE 

Auto-generated from RARE rule bool-xor-true

enumerator CVC5_PROOF_REWRITE_RULE_BOOL_XOR_COMM 

Auto-generated from RARE rule bool-xor-comm

enumerator CVC5_PROOF_REWRITE_RULE_BOOL_XOR_ELIM 

Auto-generated from RARE rule bool-xor-elim

enumerator CVC5_PROOF_REWRITE_RULE_BOOL_NOT_XOR_ELIM 

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

enumerator CVC5_PROOF_REWRITE_RULE_BOOL_NOT_EQ_ELIM 

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

enumerator CVC5_PROOF_REWRITE_RULE_ITE_NEG_BRANCH 

Auto-generated from RARE rule ite-neg-branch

enumerator CVC5_PROOF_REWRITE_RULE_ITE_THEN_TRUE 

Auto-generated from RARE rule ite-then-true

enumerator CVC5_PROOF_REWRITE_RULE_ITE_ELSE_FALSE 

Auto-generated from RARE rule ite-else-false

enumerator CVC5_PROOF_REWRITE_RULE_ITE_THEN_FALSE 

Auto-generated from RARE rule ite-then-false

enumerator CVC5_PROOF_REWRITE_RULE_ITE_ELSE_TRUE 

Auto-generated from RARE rule ite-else-true

enumerator CVC5_PROOF_REWRITE_RULE_ITE_THEN_LOOKAHEAD_SELF 

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

enumerator CVC5_PROOF_REWRITE_RULE_ITE_ELSE_LOOKAHEAD_SELF 

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

enumerator CVC5_PROOF_REWRITE_RULE_BOOL_NOT_ITE_ELIM 

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

enumerator CVC5_PROOF_REWRITE_RULE_ITE_TRUE_COND 

Auto-generated from RARE rule ite-true-cond

enumerator CVC5_PROOF_REWRITE_RULE_ITE_FALSE_COND 

Auto-generated from RARE rule ite-false-cond

enumerator CVC5_PROOF_REWRITE_RULE_ITE_NOT_COND 

Auto-generated from RARE rule ite-not-cond

enumerator CVC5_PROOF_REWRITE_RULE_ITE_EQ_BRANCH 

Auto-generated from RARE rule ite-eq-branch

enumerator CVC5_PROOF_REWRITE_RULE_ITE_THEN_LOOKAHEAD 

Auto-generated from RARE rule ite-then-lookahead

enumerator CVC5_PROOF_REWRITE_RULE_ITE_ELSE_LOOKAHEAD 

Auto-generated from RARE rule ite-else-lookahead

enumerator CVC5_PROOF_REWRITE_RULE_ITE_THEN_NEG_LOOKAHEAD 

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

enumerator CVC5_PROOF_REWRITE_RULE_ITE_ELSE_NEG_LOOKAHEAD 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_CONCAT_FLATTEN 

Auto-generated from RARE rule bv-concat-flatten

enumerator CVC5_PROOF_REWRITE_RULE_BV_CONCAT_EXTRACT_MERGE 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_EXTRACT_EXTRACT 

Auto-generated from RARE rule bv-extract-extract

enumerator CVC5_PROOF_REWRITE_RULE_BV_EXTRACT_WHOLE 

Auto-generated from RARE rule bv-extract-whole

enumerator CVC5_PROOF_REWRITE_RULE_BV_EXTRACT_CONCAT_1 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_EXTRACT_CONCAT_2 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_EXTRACT_CONCAT_3 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_EXTRACT_CONCAT_4 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_EXTRACT_BITWISE_AND 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_EXTRACT_BITWISE_OR 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_EXTRACT_BITWISE_XOR 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_EXTRACT_NOT 

Auto-generated from RARE rule bv-extract-not

enumerator CVC5_PROOF_REWRITE_RULE_BV_EXTRACT_SIGN_EXTEND_1 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_EXTRACT_SIGN_EXTEND_2 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_EXTRACT_SIGN_EXTEND_3 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_NEG_MULT 

Auto-generated from RARE rule bv-neg-mult

enumerator CVC5_PROOF_REWRITE_RULE_BV_NEG_ADD 

Auto-generated from RARE rule bv-neg-add

enumerator CVC5_PROOF_REWRITE_RULE_BV_MULT_DISTRIB_CONST_NEG 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_MULT_DISTRIB_CONST_ADD 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_MULT_DISTRIB_CONST_SUB 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_MULT_DISTRIB_1 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_MULT_DISTRIB_2 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_NOT_XOR 

Auto-generated from RARE rule bv-not-xor

enumerator CVC5_PROOF_REWRITE_RULE_BV_AND_SIMPLIFY_1 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_AND_SIMPLIFY_2 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_OR_SIMPLIFY_1 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_OR_SIMPLIFY_2 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_XOR_SIMPLIFY_1 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_XOR_SIMPLIFY_2 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_XOR_SIMPLIFY_3 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_ULT_ADD_ONE 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_CONCAT_TO_MULT 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_MULT_SLT_MULT_1 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_MULT_SLT_MULT_2 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_COMMUTATIVE_AND 

Auto-generated from RARE rule bv-commutative-and

enumerator CVC5_PROOF_REWRITE_RULE_BV_COMMUTATIVE_OR 

Auto-generated from RARE rule bv-commutative-or

enumerator CVC5_PROOF_REWRITE_RULE_BV_COMMUTATIVE_XOR 

Auto-generated from RARE rule bv-commutative-xor

enumerator CVC5_PROOF_REWRITE_RULE_BV_COMMUTATIVE_MUL 

Auto-generated from RARE rule bv-commutative-mul

enumerator CVC5_PROOF_REWRITE_RULE_BV_OR_ZERO 

Auto-generated from RARE rule bv-or-zero

enumerator CVC5_PROOF_REWRITE_RULE_BV_MUL_ONE 

Auto-generated from RARE rule bv-mul-one

enumerator CVC5_PROOF_REWRITE_RULE_BV_MUL_ZERO 

Auto-generated from RARE rule bv-mul-zero

enumerator CVC5_PROOF_REWRITE_RULE_BV_ADD_ZERO 

Auto-generated from RARE rule bv-add-zero

enumerator CVC5_PROOF_REWRITE_RULE_BV_ADD_TWO 

Auto-generated from RARE rule bv-add-two

enumerator CVC5_PROOF_REWRITE_RULE_BV_ZERO_EXTEND_ELIMINATE_0 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_SIGN_EXTEND_ELIMINATE_0 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_NOT_NEQ 

Auto-generated from RARE rule bv-not-neq

enumerator CVC5_PROOF_REWRITE_RULE_BV_ULT_ONES 

Auto-generated from RARE rule bv-ult-ones

enumerator CVC5_PROOF_REWRITE_RULE_BV_OR_FLATTEN 

Auto-generated from RARE rule bv-or-flatten

enumerator CVC5_PROOF_REWRITE_RULE_BV_XOR_FLATTEN 

Auto-generated from RARE rule bv-xor-flatten

enumerator CVC5_PROOF_REWRITE_RULE_BV_AND_FLATTEN 

Auto-generated from RARE rule bv-and-flatten

enumerator CVC5_PROOF_REWRITE_RULE_BV_MUL_FLATTEN 

Auto-generated from RARE rule bv-mul-flatten

enumerator CVC5_PROOF_REWRITE_RULE_BV_CONCAT_MERGE_CONST 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_COMMUTATIVE_ADD 

Auto-generated from RARE rule bv-commutative-add

enumerator CVC5_PROOF_REWRITE_RULE_BV_NEG_SUB 

Auto-generated from RARE rule bv-neg-sub

enumerator CVC5_PROOF_REWRITE_RULE_BV_NEG_IDEMP 

Auto-generated from RARE rule bv-neg-idemp

enumerator CVC5_PROOF_REWRITE_RULE_BV_SUB_ELIMINATE 

Auto-generated from RARE rule bv-sub-eliminate

enumerator CVC5_PROOF_REWRITE_RULE_BV_UGT_ELIMINATE 

Auto-generated from RARE rule bv-ugt-eliminate

enumerator CVC5_PROOF_REWRITE_RULE_BV_UGE_ELIMINATE 

Auto-generated from RARE rule bv-uge-eliminate

enumerator CVC5_PROOF_REWRITE_RULE_BV_SGT_ELIMINATE 

Auto-generated from RARE rule bv-sgt-eliminate

enumerator CVC5_PROOF_REWRITE_RULE_BV_SGE_ELIMINATE 

Auto-generated from RARE rule bv-sge-eliminate

enumerator CVC5_PROOF_REWRITE_RULE_BV_SLT_ELIMINATE 

Auto-generated from RARE rule bv-slt-eliminate

enumerator CVC5_PROOF_REWRITE_RULE_BV_SLE_ELIMINATE 

Auto-generated from RARE rule bv-sle-eliminate

enumerator CVC5_PROOF_REWRITE_RULE_BV_REDOR_ELIMINATE 

Auto-generated from RARE rule bv-redor-eliminate

enumerator CVC5_PROOF_REWRITE_RULE_BV_REDAND_ELIMINATE 

Auto-generated from RARE rule bv-redand-eliminate

enumerator CVC5_PROOF_REWRITE_RULE_BV_ULE_ELIMINATE 

Auto-generated from RARE rule bv-ule-eliminate

enumerator CVC5_PROOF_REWRITE_RULE_BV_COMP_ELIMINATE 

Auto-generated from RARE rule bv-comp-eliminate

enumerator CVC5_PROOF_REWRITE_RULE_BV_REPEAT_ELIMINATE_1 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_REPEAT_ELIMINATE_2 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_ROTATE_LEFT_ELIMINATE_1 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_ROTATE_LEFT_ELIMINATE_2 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_ROTATE_RIGHT_ELIMINATE_1 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_ROTATE_RIGHT_ELIMINATE_2 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_NAND_ELIMINATE 

Auto-generated from RARE rule bv-nand-eliminate

enumerator CVC5_PROOF_REWRITE_RULE_BV_NOR_ELIMINATE 

Auto-generated from RARE rule bv-nor-eliminate

enumerator CVC5_PROOF_REWRITE_RULE_BV_XNOR_ELIMINATE 

Auto-generated from RARE rule bv-xnor-eliminate

enumerator CVC5_PROOF_REWRITE_RULE_BV_SDIV_ELIMINATE 

Auto-generated from RARE rule bv-sdiv-eliminate

enumerator CVC5_PROOF_REWRITE_RULE_BV_SDIV_ELIMINATE_FEWER_BITWISE_OPS 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_ZERO_EXTEND_ELIMINATE 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_SIGN_EXTEND_ELIMINATE 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_UADDO_ELIMINATE 

Auto-generated from RARE rule bv-uaddo-eliminate

enumerator CVC5_PROOF_REWRITE_RULE_BV_SADDO_ELIMINATE 

Auto-generated from RARE rule bv-saddo-eliminate

enumerator CVC5_PROOF_REWRITE_RULE_BV_SDIVO_ELIMINATE 

Auto-generated from RARE rule bv-sdivo-eliminate

enumerator CVC5_PROOF_REWRITE_RULE_BV_SMOD_ELIMINATE 

Auto-generated from RARE rule bv-smod-eliminate

enumerator CVC5_PROOF_REWRITE_RULE_BV_SMOD_ELIMINATE_FEWER_BITWISE_OPS 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_SREM_ELIMINATE 

Auto-generated from RARE rule bv-srem-eliminate

enumerator CVC5_PROOF_REWRITE_RULE_BV_SREM_ELIMINATE_FEWER_BITWISE_OPS 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_USUBO_ELIMINATE 

Auto-generated from RARE rule bv-usubo-eliminate

enumerator CVC5_PROOF_REWRITE_RULE_BV_SSUBO_ELIMINATE 

Auto-generated from RARE rule bv-ssubo-eliminate

enumerator CVC5_PROOF_REWRITE_RULE_BV_ITE_EQUAL_CHILDREN 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_ITE_CONST_CHILDREN_1 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_ITE_CONST_CHILDREN_2 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_ITE_EQUAL_COND_1 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_ITE_EQUAL_COND_2 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_ITE_EQUAL_COND_3 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_ITE_MERGE_THEN_IF 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_ITE_MERGE_ELSE_IF 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_ITE_MERGE_THEN_ELSE 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_ITE_MERGE_ELSE_ELSE 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_SHL_BY_CONST_0 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_SHL_BY_CONST_1 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_SHL_BY_CONST_2 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_LSHR_BY_CONST_0 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_LSHR_BY_CONST_1 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_LSHR_BY_CONST_2 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_ASHR_BY_CONST_0 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_ASHR_BY_CONST_1 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_ASHR_BY_CONST_2 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_AND_CONCAT_PULLUP 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_OR_CONCAT_PULLUP 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_XOR_CONCAT_PULLUP 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_BITWISE_IDEMP_1 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_BITWISE_IDEMP_2 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_AND_ZERO 

Auto-generated from RARE rule bv-and-zero

enumerator CVC5_PROOF_REWRITE_RULE_BV_AND_ONE 

Auto-generated from RARE rule bv-and-one

enumerator CVC5_PROOF_REWRITE_RULE_BV_OR_ONE 

Auto-generated from RARE rule bv-or-one

enumerator CVC5_PROOF_REWRITE_RULE_BV_XOR_DUPLICATE 

Auto-generated from RARE rule bv-xor-duplicate

enumerator CVC5_PROOF_REWRITE_RULE_BV_XOR_ONES 

Auto-generated from RARE rule bv-xor-ones

enumerator CVC5_PROOF_REWRITE_RULE_BV_XOR_ZERO 

Auto-generated from RARE rule bv-xor-zero

enumerator CVC5_PROOF_REWRITE_RULE_BV_BITWISE_NOT_AND 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_BITWISE_NOT_OR 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_XOR_NOT 

Auto-generated from RARE rule bv-xor-not

enumerator CVC5_PROOF_REWRITE_RULE_BV_NOT_IDEMP 

Auto-generated from RARE rule bv-not-idemp

enumerator CVC5_PROOF_REWRITE_RULE_BV_ULT_ZERO_1 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_ULT_ZERO_2 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_ULT_SELF 

Auto-generated from RARE rule bv-ult-self

enumerator CVC5_PROOF_REWRITE_RULE_BV_LT_SELF 

Auto-generated from RARE rule bv-lt-self

enumerator CVC5_PROOF_REWRITE_RULE_BV_ULE_SELF 

Auto-generated from RARE rule bv-ule-self

enumerator CVC5_PROOF_REWRITE_RULE_BV_ULE_ZERO 

Auto-generated from RARE rule bv-ule-zero

enumerator CVC5_PROOF_REWRITE_RULE_BV_ZERO_ULE 

Auto-generated from RARE rule bv-zero-ule

enumerator CVC5_PROOF_REWRITE_RULE_BV_SLE_SELF 

Auto-generated from RARE rule bv-sle-self

enumerator CVC5_PROOF_REWRITE_RULE_BV_ULE_MAX 

Auto-generated from RARE rule bv-ule-max

enumerator CVC5_PROOF_REWRITE_RULE_BV_NOT_ULT 

Auto-generated from RARE rule bv-not-ult

enumerator CVC5_PROOF_REWRITE_RULE_BV_NOT_ULE 

Auto-generated from RARE rule bv-not-ule

enumerator CVC5_PROOF_REWRITE_RULE_BV_NOT_SLE 

Auto-generated from RARE rule bv-not-sle

enumerator CVC5_PROOF_REWRITE_RULE_BV_MULT_POW2_1 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_MULT_POW2_2 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_MULT_POW2_2B 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_EXTRACT_MULT_LEADING_BIT 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_UDIV_POW2_NOT_ONE 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_UDIV_ZERO 

Auto-generated from RARE rule bv-udiv-zero

enumerator CVC5_PROOF_REWRITE_RULE_BV_UDIV_ONE 

Auto-generated from RARE rule bv-udiv-one

enumerator CVC5_PROOF_REWRITE_RULE_BV_UREM_POW2_NOT_ONE 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_UREM_ONE 

Auto-generated from RARE rule bv-urem-one

enumerator CVC5_PROOF_REWRITE_RULE_BV_UREM_SELF 

Auto-generated from RARE rule bv-urem-self

enumerator CVC5_PROOF_REWRITE_RULE_BV_SHL_ZERO 

Auto-generated from RARE rule bv-shl-zero

enumerator CVC5_PROOF_REWRITE_RULE_BV_LSHR_ZERO 

Auto-generated from RARE rule bv-lshr-zero

enumerator CVC5_PROOF_REWRITE_RULE_BV_ASHR_ZERO 

Auto-generated from RARE rule bv-ashr-zero

enumerator CVC5_PROOF_REWRITE_RULE_BV_UGT_UREM 

Auto-generated from RARE rule bv-ugt-urem

enumerator CVC5_PROOF_REWRITE_RULE_BV_ULT_ONE 

Auto-generated from RARE rule bv-ult-one

enumerator CVC5_PROOF_REWRITE_RULE_BV_SLT_ZERO 

Auto-generated from RARE rule bv-slt-zero

enumerator CVC5_PROOF_REWRITE_RULE_BV_MERGE_SIGN_EXTEND_1 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_MERGE_SIGN_EXTEND_2 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_MERGE_SIGN_EXTEND_3 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_SIGN_EXTEND_EQ_CONST_1 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_SIGN_EXTEND_EQ_CONST_2 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_ZERO_EXTEND_EQ_CONST_1 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_ZERO_EXTEND_EQ_CONST_2 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_ZERO_EXTEND_ULT_CONST_1 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_ZERO_EXTEND_ULT_CONST_2 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_SIGN_EXTEND_ULT_CONST_1 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_SIGN_EXTEND_ULT_CONST_2 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_SIGN_EXTEND_ULT_CONST_3 

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

enumerator CVC5_PROOF_REWRITE_RULE_BV_SIGN_EXTEND_ULT_CONST_4 

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

enumerator CVC5_PROOF_REWRITE_RULE_SETS_EQ_SINGLETON_EMP 

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

enumerator CVC5_PROOF_REWRITE_RULE_SETS_MEMBER_SINGLETON 

Auto-generated from RARE rule sets-member-singleton

enumerator CVC5_PROOF_REWRITE_RULE_SETS_MEMBER_EMP 

Auto-generated from RARE rule sets-member-emp

enumerator CVC5_PROOF_REWRITE_RULE_SETS_SUBSET_ELIM 

Auto-generated from RARE rule sets-subset-elim

enumerator CVC5_PROOF_REWRITE_RULE_SETS_UNION_COMM 

Auto-generated from RARE rule sets-union-comm

enumerator CVC5_PROOF_REWRITE_RULE_SETS_INTER_COMM 

Auto-generated from RARE rule sets-inter-comm

enumerator CVC5_PROOF_REWRITE_RULE_SETS_INTER_EMP1 

Auto-generated from RARE rule sets-inter-emp1

enumerator CVC5_PROOF_REWRITE_RULE_SETS_INTER_EMP2 

Auto-generated from RARE rule sets-inter-emp2

enumerator CVC5_PROOF_REWRITE_RULE_SETS_MINUS_EMP1 

Auto-generated from RARE rule sets-minus-emp1

enumerator CVC5_PROOF_REWRITE_RULE_SETS_MINUS_EMP2 

Auto-generated from RARE rule sets-minus-emp2

enumerator CVC5_PROOF_REWRITE_RULE_SETS_UNION_EMP1 

Auto-generated from RARE rule sets-union-emp1

enumerator CVC5_PROOF_REWRITE_RULE_SETS_UNION_EMP2 

Auto-generated from RARE rule sets-union-emp2

enumerator CVC5_PROOF_REWRITE_RULE_SETS_INTER_MEMBER 

Auto-generated from RARE rule sets-inter-member

enumerator CVC5_PROOF_REWRITE_RULE_SETS_MINUS_MEMBER 

Auto-generated from RARE rule sets-minus-member

enumerator CVC5_PROOF_REWRITE_RULE_SETS_UNION_MEMBER 

Auto-generated from RARE rule sets-union-member

enumerator CVC5_PROOF_REWRITE_RULE_SETS_CHOOSE_SINGLETON 

Auto-generated from RARE rule sets-choose-singleton

enumerator CVC5_PROOF_REWRITE_RULE_SETS_CARD_SINGLETON 

Auto-generated from RARE rule sets-card-singleton

enumerator CVC5_PROOF_REWRITE_RULE_SETS_CARD_UNION 

Auto-generated from RARE rule sets-card-union

enumerator CVC5_PROOF_REWRITE_RULE_SETS_CARD_MINUS 

Auto-generated from RARE rule sets-card-minus

enumerator CVC5_PROOF_REWRITE_RULE_SETS_CARD_EMP 

Auto-generated from RARE rule sets-card-emp

enumerator CVC5_PROOF_REWRITE_RULE_STR_EQ_CTN_FALSE 

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

enumerator CVC5_PROOF_REWRITE_RULE_STR_EQ_CTN_FULL_FALSE1 

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

enumerator CVC5_PROOF_REWRITE_RULE_STR_EQ_CTN_FULL_FALSE2 

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

enumerator CVC5_PROOF_REWRITE_RULE_STR_CONCAT_FLATTEN 

Auto-generated from RARE rule str-concat-flatten

enumerator CVC5_PROOF_REWRITE_RULE_STR_CONCAT_FLATTEN_EQ 

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

enumerator CVC5_PROOF_REWRITE_RULE_STR_CONCAT_FLATTEN_EQ_REV 

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

enumerator CVC5_PROOF_REWRITE_RULE_STR_SUBSTR_EMPTY_STR 

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

enumerator CVC5_PROOF_REWRITE_RULE_STR_SUBSTR_EMPTY_RANGE 

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

enumerator CVC5_PROOF_REWRITE_RULE_STR_SUBSTR_EMPTY_START 

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

enumerator CVC5_PROOF_REWRITE_RULE_STR_SUBSTR_EMPTY_START_NEG 

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

enumerator CVC5_PROOF_REWRITE_RULE_STR_SUBSTR_EQ_EMPTY 

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

enumerator CVC5_PROOF_REWRITE_RULE_STR_LEN_REPLACE_INV 

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

enumerator CVC5_PROOF_REWRITE_RULE_STR_LEN_UPDATE_INV 

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

enumerator CVC5_PROOF_REWRITE_RULE_STR_LEN_SUBSTR_IN_RANGE 

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

enumerator CVC5_PROOF_REWRITE_RULE_STR_LEN_SUBSTR_UB1 

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

enumerator CVC5_PROOF_REWRITE_RULE_STR_LEN_SUBSTR_UB2 

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

enumerator CVC5_PROOF_REWRITE_RULE_STR_CONCAT_CLASH 

Auto-generated from RARE rule str-concat-clash

enumerator CVC5_PROOF_REWRITE_RULE_STR_CONCAT_CLASH_REV 

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

enumerator CVC5_PROOF_REWRITE_RULE_STR_CONCAT_CLASH2 

Auto-generated from RARE rule str-concat-clash2

enumerator CVC5_PROOF_REWRITE_RULE_STR_CONCAT_CLASH2_REV 

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

enumerator CVC5_PROOF_REWRITE_RULE_STR_CONCAT_UNIFY 

Auto-generated from RARE rule str-concat-unify

enumerator CVC5_PROOF_REWRITE_RULE_STR_CONCAT_UNIFY_REV 

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

enumerator CVC5_PROOF_REWRITE_RULE_STR_CONCAT_UNIFY_BASE 

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

enumerator CVC5_PROOF_REWRITE_RULE_STR_CONCAT_UNIFY_BASE_REV 

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

enumerator CVC5_PROOF_REWRITE_RULE_STR_CONCAT_CLASH_CHAR 

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

enumerator CVC5_PROOF_REWRITE_RULE_STR_CONCAT_CLASH_CHAR_REV 

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

enumerator CVC5_PROOF_REWRITE_RULE_STR_PREFIXOF_ELIM 

Auto-generated from RARE rule str-prefixof-elim

enumerator CVC5_PROOF_REWRITE_RULE_STR_SUFFIXOF_ELIM 

Auto-generated from RARE rule str-suffixof-elim

enumerator CVC5_PROOF_REWRITE_RULE_STR_PREFIXOF_ONE 

Auto-generated from RARE rule str-prefixof-one

enumerator CVC5_PROOF_REWRITE_RULE_STR_SUFFIXOF_ONE 

Auto-generated from RARE rule str-suffixof-one

enumerator CVC5_PROOF_REWRITE_RULE_STR_SUBSTR_COMBINE1 

Auto-generated from RARE rule str-substr-combine1

enumerator CVC5_PROOF_REWRITE_RULE_STR_SUBSTR_COMBINE2 

Auto-generated from RARE rule str-substr-combine2

enumerator CVC5_PROOF_REWRITE_RULE_STR_SUBSTR_COMBINE3 

Auto-generated from RARE rule str-substr-combine3

enumerator CVC5_PROOF_REWRITE_RULE_STR_SUBSTR_COMBINE4 

Auto-generated from RARE rule str-substr-combine4

enumerator CVC5_PROOF_REWRITE_RULE_STR_SUBSTR_CONCAT1 

Auto-generated from RARE rule str-substr-concat1

enumerator CVC5_PROOF_REWRITE_RULE_STR_SUBSTR_CONCAT2 

Auto-generated from RARE rule str-substr-concat2

enumerator CVC5_PROOF_REWRITE_RULE_STR_SUBSTR_FULL 

Auto-generated from RARE rule str-substr-full

enumerator CVC5_PROOF_REWRITE_RULE_STR_SUBSTR_FULL_EQ 

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

enumerator CVC5_PROOF_REWRITE_RULE_STR_CONTAINS_REFL 

Auto-generated from RARE rule str-contains-refl

enumerator CVC5_PROOF_REWRITE_RULE_STR_CONTAINS_CONCAT_FIND 

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

enumerator CVC5_PROOF_REWRITE_RULE_STR_CONTAINS_SPLIT_CHAR 

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

enumerator CVC5_PROOF_REWRITE_RULE_STR_CONTAINS_LT_LEN 

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

enumerator CVC5_PROOF_REWRITE_RULE_STR_CONTAINS_LEQ_LEN_EQ 

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

enumerator CVC5_PROOF_REWRITE_RULE_STR_CONTAINS_EMP 

Auto-generated from RARE rule str-contains-emp

enumerator CVC5_PROOF_REWRITE_RULE_STR_CONTAINS_IS_EMP 

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

enumerator CVC5_PROOF_REWRITE_RULE_STR_CONCAT_EMP 

Auto-generated from RARE rule str-concat-emp

enumerator CVC5_PROOF_REWRITE_RULE_STR_AT_ELIM 

Auto-generated from RARE rule str-at-elim

enumerator CVC5_PROOF_REWRITE_RULE_STR_REPLACE_SELF 

Auto-generated from RARE rule str-replace-self

enumerator CVC5_PROOF_REWRITE_RULE_STR_REPLACE_NO_CONTAINS 

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

enumerator CVC5_PROOF_REWRITE_RULE_STR_REPLACE_EMPTY 

Auto-generated from RARE rule str-replace-empty

enumerator CVC5_PROOF_REWRITE_RULE_STR_LEN_CONCAT_REC 

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

enumerator CVC5_PROOF_REWRITE_RULE_STR_INDEXOF_SELF 

Auto-generated from RARE rule str-indexof-self

enumerator CVC5_PROOF_REWRITE_RULE_STR_INDEXOF_NO_CONTAINS 

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

enumerator CVC5_PROOF_REWRITE_RULE_STR_TO_LOWER_CONCAT 

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

enumerator CVC5_PROOF_REWRITE_RULE_STR_TO_UPPER_CONCAT 

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

enumerator CVC5_PROOF_REWRITE_RULE_STR_TO_LOWER_UPPER 

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

enumerator CVC5_PROOF_REWRITE_RULE_STR_TO_UPPER_LOWER 

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

enumerator CVC5_PROOF_REWRITE_RULE_RE_ALL_ELIM 

Auto-generated from RARE rule re-all-elim

enumerator CVC5_PROOF_REWRITE_RULE_RE_OPT_ELIM 

Auto-generated from RARE rule re-opt-elim

enumerator CVC5_PROOF_REWRITE_RULE_RE_DIFF_ELIM 

Auto-generated from RARE rule re-diff-elim

enumerator CVC5_PROOF_REWRITE_RULE_RE_CONCAT_EMP 

Auto-generated from RARE rule re-concat-emp

enumerator CVC5_PROOF_REWRITE_RULE_RE_CONCAT_NONE 

Auto-generated from RARE rule re-concat-none

enumerator CVC5_PROOF_REWRITE_RULE_RE_CONCAT_FLATTEN 

Auto-generated from RARE rule re-concat-flatten

enumerator CVC5_PROOF_REWRITE_RULE_RE_CONCAT_STAR_SWAP 

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

enumerator CVC5_PROOF_REWRITE_RULE_RE_CONCAT_MERGE 

Auto-generated from RARE rule re-concat-merge

enumerator CVC5_PROOF_REWRITE_RULE_RE_UNION_ALL 

Auto-generated from RARE rule re-union-all

enumerator CVC5_PROOF_REWRITE_RULE_RE_UNION_NONE 

Auto-generated from RARE rule re-union-none

enumerator CVC5_PROOF_REWRITE_RULE_RE_UNION_FLATTEN 

Auto-generated from RARE rule re-union-flatten

enumerator CVC5_PROOF_REWRITE_RULE_RE_UNION_DUP 

Auto-generated from RARE rule re-union-dup

enumerator CVC5_PROOF_REWRITE_RULE_RE_INTER_ALL 

Auto-generated from RARE rule re-inter-all

enumerator CVC5_PROOF_REWRITE_RULE_RE_INTER_NONE 

Auto-generated from RARE rule re-inter-none

enumerator CVC5_PROOF_REWRITE_RULE_RE_INTER_FLATTEN 

Auto-generated from RARE rule re-inter-flatten

enumerator CVC5_PROOF_REWRITE_RULE_RE_INTER_DUP 

Auto-generated from RARE rule re-inter-dup

enumerator CVC5_PROOF_REWRITE_RULE_RE_LOOP_NEG 

Auto-generated from RARE rule re-loop-neg

enumerator CVC5_PROOF_REWRITE_RULE_RE_INTER_CSTRING 

Auto-generated from RARE rule re-inter-cstring

enumerator CVC5_PROOF_REWRITE_RULE_RE_INTER_CSTRING_NEG 

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

enumerator CVC5_PROOF_REWRITE_RULE_STR_NTH_ELIM_CODE 

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

enumerator CVC5_PROOF_REWRITE_RULE_STR_SUBSTR_LEN_INCLUDE 

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

enumerator CVC5_PROOF_REWRITE_RULE_STR_SUBSTR_LEN_INCLUDE_PRE 

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

enumerator CVC5_PROOF_REWRITE_RULE_STR_SUBSTR_LEN_SKIP 

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

enumerator CVC5_PROOF_REWRITE_RULE_SEQ_REV_CONCAT 

Auto-generated from RARE rule seq-rev-concat

enumerator CVC5_PROOF_REWRITE_RULE_SEQ_LEN_UNIT 

Auto-generated from RARE rule seq-len-unit

enumerator CVC5_PROOF_REWRITE_RULE_SEQ_NTH_UNIT 

Auto-generated from RARE rule seq-nth-unit

enumerator CVC5_PROOF_REWRITE_RULE_SEQ_REV_UNIT 

Auto-generated from RARE rule seq-rev-unit

enumerator CVC5_PROOF_REWRITE_RULE_RE_IN_EMPTY 

Auto-generated from RARE rule re-in-empty

enumerator CVC5_PROOF_REWRITE_RULE_RE_IN_SIGMA 

Auto-generated from RARE rule re-in-sigma

enumerator CVC5_PROOF_REWRITE_RULE_RE_IN_SIGMA_STAR 

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

enumerator CVC5_PROOF_REWRITE_RULE_RE_IN_CSTRING 

Auto-generated from RARE rule re-in-cstring

enumerator CVC5_PROOF_REWRITE_RULE_RE_IN_COMP 

Auto-generated from RARE rule re-in-comp

enumerator CVC5_PROOF_REWRITE_RULE_STR_IN_RE_UNION_ELIM 

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

enumerator CVC5_PROOF_REWRITE_RULE_STR_IN_RE_INTER_ELIM 

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

enumerator CVC5_PROOF_REWRITE_RULE_STR_IN_RE_RANGE_ELIM 

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

enumerator CVC5_PROOF_REWRITE_RULE_STR_IN_RE_CONTAINS 

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

enumerator CVC5_PROOF_REWRITE_RULE_STR_IN_RE_STRIP_PREFIX 

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

enumerator CVC5_PROOF_REWRITE_RULE_STR_IN_RE_STRIP_PREFIX_NEG 

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

enumerator CVC5_PROOF_REWRITE_RULE_STR_IN_RE_STRIP_PREFIX_SR_SINGLE 

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

enumerator CVC5_PROOF_REWRITE_RULE_STR_IN_RE_STRIP_PREFIX_SR_SINGLE_NEG 

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

enumerator CVC5_PROOF_REWRITE_RULE_STR_IN_RE_STRIP_PREFIX_SRS_SINGLE 

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

enumerator CVC5_PROOF_REWRITE_RULE_STR_IN_RE_STRIP_PREFIX_SRS_SINGLE_NEG 

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

enumerator CVC5_PROOF_REWRITE_RULE_STR_IN_RE_STRIP_PREFIX_S_SINGLE 

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

enumerator CVC5_PROOF_REWRITE_RULE_STR_IN_RE_STRIP_PREFIX_S_SINGLE_NEG 

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

enumerator CVC5_PROOF_REWRITE_RULE_STR_IN_RE_STRIP_PREFIX_BASE 

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

enumerator CVC5_PROOF_REWRITE_RULE_STR_IN_RE_STRIP_PREFIX_BASE_NEG 

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

enumerator CVC5_PROOF_REWRITE_RULE_STR_IN_RE_STRIP_PREFIX_BASE_S_SINGLE 

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

enumerator CVC5_PROOF_REWRITE_RULE_STR_IN_RE_STRIP_PREFIX_BASE_S_SINGLE_NEG 

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

enumerator CVC5_PROOF_REWRITE_RULE_STR_IN_RE_STRIP_CHAR 

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

enumerator CVC5_PROOF_REWRITE_RULE_STR_IN_RE_STRIP_CHAR_S_SINGLE 

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

enumerator CVC5_PROOF_REWRITE_RULE_STR_IN_RE_STRIP_PREFIX_REV 

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

enumerator CVC5_PROOF_REWRITE_RULE_STR_IN_RE_STRIP_PREFIX_NEG_REV 

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

enumerator CVC5_PROOF_REWRITE_RULE_STR_IN_RE_STRIP_PREFIX_SR_SINGLE_REV 

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

enumerator CVC5_PROOF_REWRITE_RULE_STR_IN_RE_STRIP_PREFIX_SR_SINGLE_NEG_REV 

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

enumerator CVC5_PROOF_REWRITE_RULE_STR_IN_RE_STRIP_PREFIX_SRS_SINGLE_REV 

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

enumerator CVC5_PROOF_REWRITE_RULE_STR_IN_RE_STRIP_PREFIX_SRS_SINGLE_NEG_REV 

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

enumerator CVC5_PROOF_REWRITE_RULE_STR_IN_RE_STRIP_PREFIX_S_SINGLE_REV 

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

enumerator CVC5_PROOF_REWRITE_RULE_STR_IN_RE_STRIP_PREFIX_S_SINGLE_NEG_REV 

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

enumerator CVC5_PROOF_REWRITE_RULE_STR_IN_RE_STRIP_PREFIX_BASE_REV 

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

enumerator CVC5_PROOF_REWRITE_RULE_STR_IN_RE_STRIP_PREFIX_BASE_NEG_REV 

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

enumerator CVC5_PROOF_REWRITE_RULE_STR_IN_RE_STRIP_PREFIX_BASE_S_SINGLE_REV 

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

enumerator CVC5_PROOF_REWRITE_RULE_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 CVC5_PROOF_REWRITE_RULE_STR_IN_RE_STRIP_CHAR_REV 

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

enumerator CVC5_PROOF_REWRITE_RULE_STR_IN_RE_STRIP_CHAR_S_SINGLE_REV 

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

enumerator CVC5_PROOF_REWRITE_RULE_STR_IN_RE_REQ_UNFOLD 

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

enumerator CVC5_PROOF_REWRITE_RULE_STR_IN_RE_REQ_UNFOLD_REV 

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

enumerator CVC5_PROOF_REWRITE_RULE_STR_IN_RE_SKIP_UNFOLD 

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

enumerator CVC5_PROOF_REWRITE_RULE_STR_IN_RE_SKIP_UNFOLD_REV 

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

enumerator CVC5_PROOF_REWRITE_RULE_STR_IN_RE_TEST_UNFOLD 

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

enumerator CVC5_PROOF_REWRITE_RULE_STR_IN_RE_TEST_UNFOLD_REV 

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

enumerator CVC5_PROOF_REWRITE_RULE_EQ_REFL 

Auto-generated from RARE rule eq-refl

enumerator CVC5_PROOF_REWRITE_RULE_EQ_SYMM 

Auto-generated from RARE rule eq-symm

enumerator CVC5_PROOF_REWRITE_RULE_DISTINCT_BINARY_ELIM 

Auto-generated from RARE rule distinct-binary-elim

enumerator CVC5_PROOF_REWRITE_RULE_UF_BV2NAT_GEQ_ELIM 

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

enumerator CVC5_PROOF_REWRITE_RULE_LAST 