ProofRule and ProofRewriteRule 

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

class cvc5. ProofRule ( value ) 

The ProofRule enum

ACI_NORM 

verbatim embed:rst:leading-asterisk 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.++). endverbatim

ALETHE_RULE 

verbatim embed:rst:leading-asterisk 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. endverbatim

ALPHA_EQUIV 

verbatim embed:rst:leading-asterisk 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. endverbatim

AND_ELIM 

verbatim embed:rst:leading-asterisk Boolean – And elimination

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

endverbatim

AND_INTRO 

verbatim embed:rst:leading-asterisk Boolean – And introduction

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

endverbatim

ANNOTATION 

verbatim embed:rst:leading-asterisk 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. endverbatim

ARITH_MULT_NEG 

verbatim embed:rst:leading-asterisk 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. endverbatim

ARITH_MULT_POS 

verbatim embed:rst:leading-asterisk 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. endverbatim

ARITH_MULT_SIGN 

verbatim embed:rst:leading-asterisk 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. endverbatim

ARITH_MULT_TANGENT 

verbatim embed:rst:leading-asterisk 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 \((a,b)\) . endverbatim

ARITH_OP_ELIM_AXIOM 

verbatim embed:rst:leading-asterisk Arithmetic – Operator elimination

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

endverbatim

ARITH_POLY_NORM 

verbatim embed:rst:leading-asterisk 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. endverbatim

ARITH_POLY_NORM_REL 

verbatim embed:rst:leading-asterisk 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 \(1\) . endverbatim

ARITH_SUM_UB 

verbatim embed:rst:leading-asterisk 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 \(R = R_1 + \cdots + R_n\) . endverbatim

ARITH_TRANS_EXP_APPROX_ABOVE_NEG 

verbatim embed:rst:leading-asterisk 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 \(u\) . endverbatim

ARITH_TRANS_EXP_APPROX_ABOVE_POS 

verbatim embed:rst:leading-asterisk 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 \(u\) . endverbatim

ARITH_TRANS_EXP_APPROX_BELOW 

verbatim embed:rst:leading-asterisk 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:

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

endverbatim

ARITH_TRANS_EXP_NEG 

verbatim embed:rst:leading-asterisk Arithmetic – Transcendentals – Exp at negative values

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

endverbatim

ARITH_TRANS_EXP_POSITIVITY 

verbatim embed:rst:leading-asterisk Arithmetic – Transcendentals – Exp is always positive

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

endverbatim

ARITH_TRANS_EXP_SUPER_LIN 

verbatim embed:rst:leading-asterisk Arithmetic – Transcendentals – Exp grows super-linearly for positive values

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

endverbatim

ARITH_TRANS_EXP_ZERO 

verbatim embed:rst:leading-asterisk Arithmetic – Transcendentals – Exp at zero

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

endverbatim

ARITH_TRANS_PI 

verbatim embed:rst:leading-asterisk 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 \(\pi\) . endverbatim

ARITH_TRANS_SINE_APPROX_ABOVE_NEG 

verbatim embed:rst:leading-asterisk 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 \(u\) . endverbatim

ARITH_TRANS_SINE_APPROX_ABOVE_POS 

verbatim embed:rst:leading-asterisk 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 \((lb,ub)\) . endverbatim

ARITH_TRANS_SINE_APPROX_BELOW_NEG 

verbatim embed:rst:leading-asterisk 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 \((lb,ub)\) . endverbatim

ARITH_TRANS_SINE_APPROX_BELOW_POS 

verbatim embed:rst:leading-asterisk 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 \(u\) . endverbatim

ARITH_TRANS_SINE_BOUNDS 

verbatim embed:rst:leading-asterisk Arithmetic – Transcendentals – Sine is always between -1 and 1

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

endverbatim

ARITH_TRANS_SINE_SHIFT 

verbatim embed:rst:leading-asterisk 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 \(x\) . endverbatim

ARITH_TRANS_SINE_SYMMETRY 

verbatim embed:rst:leading-asterisk Arithmetic – Transcendentals – Sine is symmetric with respect to negation of the argument

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

endverbatim

ARITH_TRANS_SINE_TANGENT_PI 

verbatim embed:rst:leading-asterisk Arithmetic – Transcendentals – Sine is bounded by the tangents at -pi and pi

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

ARITH_TRANS_SINE_TANGENT_ZERO 

verbatim embed:rst:leading-asterisk Arithmetic – Transcendentals – Sine is bounded by the tangent at zero

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

ARITH_TRICHOTOMY 

verbatim embed:rst:leading-asterisk 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. endverbatim

ARRAYS_EXT 

verbatim embed:rst:leading-asterisk 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) . endverbatim

ARRAYS_READ_OVER_WRITE 

verbatim embed:rst:leading-asterisk Arrays – Read over write

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

endverbatim

ARRAYS_READ_OVER_WRITE_1 

verbatim embed:rst:leading-asterisk Arrays – Read over write 1

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

endverbatim

ARRAYS_READ_OVER_WRITE_CONTRA 

verbatim embed:rst:leading-asterisk Arrays – Read over write, contrapositive

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

endverbatim

ASSUME 

verbatim embed:rst:leading-asterisk 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). endverbatim

BV_BITBLAST_STEP 

verbatim embed:rst:leading-asterisk 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 \(k(t_1,\dots,t_n)\) . endverbatim

BV_EAGER_ATOM 

verbatim embed:rst:leading-asterisk Bit-vectors – Bit-vector eager atom

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

where \(F\) is of kind BITVECTOR_EAGER_ATOM . endverbatim

CHAIN_RESOLUTION 

verbatim embed:rst:leading-asterisk 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 \(C = C_n'\) endverbatim

CNF_AND_NEG 

verbatim embed:rst:leading-asterisk Boolean – CNF – And Negative

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

endverbatim

CNF_AND_POS 

verbatim embed:rst:leading-asterisk Boolean – CNF – And Positive

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

endverbatim

CNF_EQUIV_NEG1 

verbatim embed:rst:leading-asterisk Boolean – CNF – Equiv Negative 1

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

endverbatim

CNF_EQUIV_NEG2 

verbatim embed:rst:leading-asterisk Boolean – CNF – Equiv Negative 2

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

endverbatim

CNF_EQUIV_POS1 

verbatim embed:rst:leading-asterisk Boolean – CNF – Equiv Positive 1

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

endverbatim

CNF_EQUIV_POS2 

verbatim embed:rst:leading-asterisk Boolean – CNF – Equiv Positive 2

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

endverbatim

CNF_IMPLIES_NEG1 

verbatim embed:rst:leading-asterisk Boolean – CNF – Implies Negative 1

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

endverbatim

CNF_IMPLIES_NEG2 

verbatim embed:rst:leading-asterisk Boolean – CNF – Implies Negative 2

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

endverbatim

CNF_IMPLIES_POS 

verbatim embed:rst:leading-asterisk Boolean – CNF – Implies Positive

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

endverbatim

CNF_ITE_NEG1 

verbatim embed:rst:leading-asterisk Boolean – CNF – ITE Negative 1

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

endverbatim

CNF_ITE_NEG2 

verbatim embed:rst:leading-asterisk Boolean – CNF – ITE Negative 2

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

endverbatim

CNF_ITE_NEG3 

verbatim embed:rst:leading-asterisk Boolean – CNF – ITE Negative 3

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

endverbatim

CNF_ITE_POS1 

verbatim embed:rst:leading-asterisk Boolean – CNF – ITE Positive 1

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

endverbatim

CNF_ITE_POS2 

verbatim embed:rst:leading-asterisk Boolean – CNF – ITE Positive 2

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

endverbatim

CNF_ITE_POS3 

verbatim embed:rst:leading-asterisk Boolean – CNF – ITE Positive 3

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

endverbatim

CNF_OR_NEG 

verbatim embed:rst:leading-asterisk Boolean – CNF – Or Negative

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

endverbatim

CNF_OR_POS 

verbatim embed:rst:leading-asterisk Boolean – CNF – Or Positive

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

endverbatim

CNF_XOR_NEG1 

verbatim embed:rst:leading-asterisk Boolean – CNF – XOR Negative 1

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

endverbatim

CNF_XOR_NEG2 

verbatim embed:rst:leading-asterisk Boolean – CNF – XOR Negative 2

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

endverbatim

CNF_XOR_POS1 

verbatim embed:rst:leading-asterisk Boolean – CNF – XOR Positive 1

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

endverbatim

CNF_XOR_POS2 

verbatim embed:rst:leading-asterisk Boolean – CNF – XOR Positive 2

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

endverbatim

CONCAT_CONFLICT 

verbatim embed:rst:leading-asterisk 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.

endverbatim

CONCAT_CONFLICT_DEQ 

verbatim embed:rst:leading-asterisk 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.

endverbatim

CONCAT_CPROP 

verbatim embed:rst:leading-asterisk 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 \(t_2\) . endverbatim

CONCAT_CSPLIT 

verbatim embed:rst:leading-asterisk 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 \(\mathit{pre}(t_2,\mathit{len}(t_2) - 1)\) . endverbatim

CONCAT_EQ 

verbatim embed:rst:leading-asterisk 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. endverbatim

CONCAT_LPROP 

verbatim embed:rst:leading-asterisk 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 \(\mathit{ite}( \mathit{len}(t_2) >= \mathit{len}(s_2), \mathit{pre}(t_2,\mathit{len}(t_2) - \mathit{len}(s_2)), \mathit{pre}(s_2,\mathit{len}(s_2) - \mathit{len}(t_2)))\) endverbatim

CONCAT_SPLIT 

verbatim embed:rst:leading-asterisk 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 \(\mathit{substr}(x,0,n)\) . endverbatim

CONCAT_UNIFY 

verbatim embed:rst:leading-asterisk 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:

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

endverbatim

CONG 

verbatim embed:rst:leading-asterisk 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. endverbatim

CONTRA 

verbatim embed:rst:leading-asterisk Boolean – Contradiction

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

endverbatim

DRAT_REFUTATION 

verbatim embed:rst:leading-asterisk 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. endverbatim

DSL_REWRITE 

verbatim embed:rst:leading-asterisk 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. endverbatim

DT_CLASH 

verbatim embed:rst:leading-asterisk Datatypes – Clash

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

endverbatim

DT_SPLIT 

verbatim embed:rst:leading-asterisk 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 \(t\) . endverbatim

DT_UNIF 

verbatim embed:rst:leading-asterisk 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. endverbatim

ENCODE_EQ_INTRO 

verbatim embed:rst:leading-asterisk 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. endverbatim

EQUIV_ELIM1 

verbatim embed:rst:leading-asterisk Boolean – Equivalence elimination version 1

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

endverbatim

EQUIV_ELIM2 

verbatim embed:rst:leading-asterisk Boolean – Equivalence elimination version 2

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

endverbatim

EQ_RESOLVE 

verbatim embed:rst:leading-asterisk 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 . endverbatim

EVALUATE 

verbatim embed:rst:leading-asterisk 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) . endverbatim

FACTORING 

verbatim embed:rst:leading-asterisk 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. endverbatim

FALSE_ELIM 

verbatim embed:rst:leading-asterisk Equality – False elim

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

endverbatim

FALSE_INTRO 

verbatim embed:rst:leading-asterisk Equality – False intro

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

endverbatim

HO_APP_ENCODE 

verbatim embed:rst:leading-asterisk 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.

endverbatim

HO_CONG 

verbatim embed:rst:leading-asterisk 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 . endverbatim

IMPLIES_ELIM 

verbatim embed:rst:leading-asterisk Boolean – Implication elimination

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

endverbatim

INSTANTIATE 

verbatim embed:rst:leading-asterisk 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. endverbatim

INT_TIGHT_LB 

verbatim embed:rst:leading-asterisk Arithmetic – Tighten strict integer lower bounds

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

where \(i\) has integer type. endverbatim

INT_TIGHT_UB 

verbatim embed:rst:leading-asterisk Arithmetic – Tighten strict integer upper bounds

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

where \(i\) has integer type. endverbatim

ITE_ELIM1 

verbatim embed:rst:leading-asterisk Boolean – ITE elimination version 1

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

endverbatim

ITE_ELIM2 

verbatim embed:rst:leading-asterisk Boolean – ITE elimination version 2

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

endverbatim

ITE_EQ 

verbatim embed:rst:leading-asterisk Processing rules – If-then-else equivalence

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

endverbatim

LFSC_RULE 

verbatim embed:rst:leading-asterisk 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. endverbatim

MACRO_ARITH_SCALE_SUM_UB 

verbatim embed:rst:leading-asterisk 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:

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

endverbatim

MACRO_BV_BITBLAST 

verbatim embed:rst:leading-asterisk 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 . endverbatim

MACRO_RESOLUTION 

verbatim embed:rst:leading-asterisk 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 \(C_n'\) endverbatim

MACRO_RESOLUTION_TRUST 

verbatim embed:rst:leading-asterisk Boolean – N-ary Resolution + Factoring + Reordering unchecked

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

MACRO_REWRITE 

verbatim embed:rst:leading-asterisk 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. endverbatim

MACRO_RE_ELIM 

verbatim embed:rst:leading-asterisk 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. endverbatim

MACRO_SR_EQ_INTRO 

verbatim embed:rst:leading-asterisk 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 . endverbatim

MACRO_SR_PRED_ELIM 

verbatim embed:rst:leading-asterisk 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 . endverbatim

MACRO_SR_PRED_INTRO 

verbatim embed:rst:leading-asterisk 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. endverbatim

MACRO_SR_PRED_TRANSFORM 

verbatim embed:rst:leading-asterisk 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. endverbatim

MACRO_STRING_INFERENCE 

verbatim embed:rst:leading-asterisk 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 \(\texttt{strings::InferProofCons::convert}\) . endverbatim

MODUS_PONENS 

verbatim embed:rst:leading-asterisk 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 . endverbatim

NARY_CONG 

verbatim embed:rst:leading-asterisk 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. endverbatim

NOT_AND 

verbatim embed:rst:leading-asterisk Boolean – De Morgan – Not And

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

endverbatim

NOT_EQUIV_ELIM1 

verbatim embed:rst:leading-asterisk Boolean – Not Equivalence elimination version 1

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

endverbatim

NOT_EQUIV_ELIM2 

verbatim embed:rst:leading-asterisk Boolean – Not Equivalence elimination version 2

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

endverbatim

NOT_IMPLIES_ELIM1 

verbatim embed:rst:leading-asterisk Boolean – Not Implication elimination version 1

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

endverbatim

NOT_IMPLIES_ELIM2 

verbatim embed:rst:leading-asterisk Boolean – Not Implication elimination version 2

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

endverbatim

NOT_ITE_ELIM1 

verbatim embed:rst:leading-asterisk Boolean – Not ITE elimination version 1

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

endverbatim

NOT_ITE_ELIM2 

verbatim embed:rst:leading-asterisk Boolean – Not ITE elimination version 2

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

endverbatim

NOT_NOT_ELIM 

verbatim embed:rst:leading-asterisk Boolean – Double negation elimination

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

endverbatim

NOT_OR_ELIM 

verbatim embed:rst:leading-asterisk Boolean – Not Or elimination

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

endverbatim

NOT_XOR_ELIM1 

verbatim embed:rst:leading-asterisk Boolean – Not XOR elimination version 1

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

endverbatim

NOT_XOR_ELIM2 

verbatim embed:rst:leading-asterisk Boolean – Not XOR elimination version 2

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

endverbatim

REFL 

verbatim embed:rst:leading-asterisk Equality – Reflexivity

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

endverbatim

REORDERING 

verbatim embed:rst:leading-asterisk 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 \(C_1\) . endverbatim

RESOLUTION 

verbatim embed:rst:leading-asterisk 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. endverbatim

RE_INTER 

verbatim embed:rst:leading-asterisk Strings – Regular expressions – Intersection

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

endverbatim

RE_UNFOLD_NEG 

verbatim embed:rst:leading-asterisk 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” .

endverbatim

RE_UNFOLD_NEG_CONCAT_FIXED 

verbatim embed:rst:leading-asterisk 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 \(L\) .

endverbatim

RE_UNFOLD_POS 

verbatim embed:rst:leading-asterisk 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 \(\texttt{RegExpOpr::reduceRegExpPos}(t\in R)\) . endverbatim

SAT_EXTERNAL_PROVE 

verbatim embed:rst:leading-asterisk 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 . endverbatim

SAT_REFUTATION 

verbatim embed:rst:leading-asterisk 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. endverbatim

SCOPE 

verbatim embed:rst:leading-asterisk 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. endverbatim

SETS_EXT 

verbatim embed:rst:leading-asterisk 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) . endverbatim

SETS_SINGLETON_INJ 

verbatim embed:rst:leading-asterisk Sets – Singleton injectivity

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

endverbatim

SKOLEMIZE 

verbatim embed:rst:leading-asterisk 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 \(x_i\) . endverbatim

SKOLEM_INTRO 

verbatim embed:rst:leading-asterisk Quantifiers – Skolem introduction

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

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

SPLIT 

verbatim embed:rst:leading-asterisk Boolean – Split

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

endverbatim

STRING_CODE_INJ 

verbatim embed:rst:leading-asterisk Strings – Code points

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

endverbatim

STRING_DECOMPOSE 

verbatim embed:rst:leading-asterisk 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 \(\mathit{suf}(t,n)\) . endverbatim

STRING_EAGER_REDUCTION 

verbatim embed:rst:leading-asterisk Strings – Extended functions – Eager reduction

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

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

STRING_LENGTH_NON_EMPTY 

verbatim embed:rst:leading-asterisk Strings – Core rules – Length non-empty

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

endverbatim

STRING_LENGTH_POS 

verbatim embed:rst:leading-asterisk Strings – Core rules – Length positive

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

endverbatim

STRING_REDUCTION 

verbatim embed:rst:leading-asterisk 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 \(t\) . endverbatim

STRING_SEQ_UNIT_INJ 

verbatim embed:rst:leading-asterisk 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. endverbatim

SUBS 

verbatim embed:rst:leading-asterisk 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 \(x\mapsto y\) . endverbatim

SYMM 

verbatim embed:rst:leading-asterisk Equality – Symmetry

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

or

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

endverbatim

THEORY_REWRITE 

verbatim embed:rst:leading-asterisk 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. endverbatim

TRANS 

verbatim embed:rst:leading-asterisk Equality – Transitivity

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

endverbatim

TRUE_ELIM 

verbatim embed:rst:leading-asterisk Equality – True elim

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

endverbatim

TRUE_INTRO 

verbatim embed:rst:leading-asterisk Equality – True intro

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

endverbatim

TRUST 

verbatim embed:rst:leading-asterisk 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. endverbatim

TRUST_THEORY_REWRITE 

verbatim embed:rst:leading-asterisk 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. endverbatim

UNKNOWN 

verbatim embed:rst:leading-asterisk 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. endverbatim

XOR_ELIM1 

verbatim embed:rst:leading-asterisk Boolean – XOR elimination version 1

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

endverbatim

XOR_ELIM2 

verbatim embed:rst:leading-asterisk Boolean – XOR elimination version 2

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

endverbatim

class cvc5. ProofRewriteRule ( value ) 

The ProofRewriteRule enum

ARITH_ABS_ELIM 

Auto-generated from RARE rule arith-abs-elim

ARITH_DIV_BY_CONST_ELIM 

verbatim embed:rst:leading-asterisk Arith – Division by constant elimination

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

where \(c\) is a constant.

endverbatim

ARITH_DIV_ELIM_TO_REAL1 

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

ARITH_DIV_ELIM_TO_REAL2 

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

ARITH_DIV_TOTAL 

Auto-generated from RARE rule arith-div-total

ARITH_ELIM_GT 

Auto-generated from RARE rule arith-elim-gt

ARITH_ELIM_INT_GT 

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

ARITH_ELIM_INT_LT 

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

ARITH_ELIM_LEQ 

Auto-generated from RARE rule arith-elim-leq

ARITH_ELIM_LT 

Auto-generated from RARE rule arith-elim-lt

ARITH_ELIM_MINUS 

Auto-generated from RARE rule arith-elim-minus

ARITH_ELIM_UMINUS 

Auto-generated from RARE rule arith-elim-uminus

ARITH_GEQ_NORM1 

Auto-generated from RARE rule arith-geq-norm1

ARITH_GEQ_NORM2 

Auto-generated from RARE rule arith-geq-norm2

ARITH_GEQ_TIGHTEN 

Auto-generated from RARE rule arith-geq-tighten

ARITH_INT_DIV_TOTAL 

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

ARITH_INT_DIV_TOTAL_ONE 

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

ARITH_INT_DIV_TOTAL_ZERO 

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

ARITH_INT_EQ_ELIM 

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

ARITH_INT_MOD_TOTAL 

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

ARITH_INT_MOD_TOTAL_ONE 

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

ARITH_INT_MOD_TOTAL_ZERO 

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

ARITH_LEQ_NORM 

Auto-generated from RARE rule arith-leq-norm

ARITH_MULT_DIST 

Auto-generated from RARE rule arith-mult-dist

ARITH_MULT_FLATTEN 

Auto-generated from RARE rule arith-mult-flatten

ARITH_MUL_ONE 

Auto-generated from RARE rule arith-mul-one

ARITH_MUL_ZERO 

Auto-generated from RARE rule arith-mul-zero

ARITH_NEG_NEG_ONE 

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

ARITH_PLUS_CANCEL1 

Auto-generated from RARE rule arith-plus-cancel1

ARITH_PLUS_CANCEL2 

Auto-generated from RARE rule arith-plus-cancel2

ARITH_PLUS_FLATTEN 

Auto-generated from RARE rule arith-plus-flatten

ARITH_PLUS_ZERO 

Auto-generated from RARE rule arith-plus-zero

ARITH_REAL_EQ_ELIM 

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

ARITH_REFL_GEQ 

Auto-generated from RARE rule arith-refl-geq

ARITH_REFL_GT 

Auto-generated from RARE rule arith-refl-gt

ARITH_REFL_LEQ 

Auto-generated from RARE rule arith-refl-leq

ARITH_REFL_LT 

Auto-generated from RARE rule arith-refl-lt

ARITH_STRING_PRED_ENTAIL 

verbatim embed:rst:leading-asterisk 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.

endverbatim

ARITH_STRING_PRED_SAFE_APPROX 

verbatim embed:rst:leading-asterisk 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”.

endverbatim

ARITH_TO_INT_ELIM_TO_REAL 

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

ARITH_TO_REAL_ELIM 

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

ARRAYS_EQ_RANGE_EXPAND 

verbatim embed:rst:leading-asterisk Arrays – Expansion of array range equality

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

endverbatim

ARRAY_READ_OVER_WRITE 

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

ARRAY_READ_OVER_WRITE2 

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

ARRAY_STORE_OVERWRITE 

Auto-generated from RARE rule array-store-overwrite

ARRAY_STORE_SELF 

Auto-generated from RARE rule array-store-self

BETA_REDUCE 

verbatim embed:rst:leading-asterisk 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 . endverbatim

BOOL_AND_CONF 

Auto-generated from RARE rule bool-and-conf

BOOL_AND_DE_MORGAN 

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

BOOL_AND_DUP 

Auto-generated from RARE rule bool-and-dup

BOOL_AND_FALSE 

Auto-generated from RARE rule bool-and-false

BOOL_AND_FLATTEN 

Auto-generated from RARE rule bool-and-flatten

BOOL_AND_TRUE 

Auto-generated from RARE rule bool-and-true

BOOL_DOUBLE_NOT_ELIM 

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

BOOL_EQ_FALSE 

Auto-generated from RARE rule bool-eq-false

BOOL_EQ_NREFL 

Auto-generated from RARE rule bool-eq-nrefl

BOOL_EQ_TRUE 

Auto-generated from RARE rule bool-eq-true

BOOL_IMPLIES_DE_MORGAN 

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

BOOL_IMPL_ELIM 

Auto-generated from RARE rule bool-impl-elim

BOOL_IMPL_FALSE1 

Auto-generated from RARE rule bool-impl-false1

BOOL_IMPL_FALSE2 

Auto-generated from RARE rule bool-impl-false2

BOOL_IMPL_TRUE1 

Auto-generated from RARE rule bool-impl-true1

BOOL_IMPL_TRUE2 

Auto-generated from RARE rule bool-impl-true2

BOOL_NOT_EQ_ELIM 

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

BOOL_NOT_FALSE 

Auto-generated from RARE rule bool-not-false

BOOL_NOT_ITE_ELIM 

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

BOOL_NOT_TRUE 

Auto-generated from RARE rule bool-not-true

BOOL_NOT_XOR_ELIM 

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

BOOL_OR_DE_MORGAN 

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

BOOL_OR_DUP 

Auto-generated from RARE rule bool-or-dup

BOOL_OR_FALSE 

Auto-generated from RARE rule bool-or-false

BOOL_OR_FLATTEN 

Auto-generated from RARE rule bool-or-flatten

BOOL_OR_TAUT 

Auto-generated from RARE rule bool-or-taut

BOOL_OR_TRUE 

Auto-generated from RARE rule bool-or-true

BOOL_XOR_COMM 

Auto-generated from RARE rule bool-xor-comm

BOOL_XOR_ELIM 

Auto-generated from RARE rule bool-xor-elim

BOOL_XOR_FALSE 

Auto-generated from RARE rule bool-xor-false

BOOL_XOR_NREFL 

Auto-generated from RARE rule bool-xor-nrefl

BOOL_XOR_REFL 

Auto-generated from RARE rule bool-xor-refl

BOOL_XOR_TRUE 

Auto-generated from RARE rule bool-xor-true

BV_ADD_COMBINE_LIKE_TERMS 

verbatim embed:rst:leading-asterisk Bitvectors - Combine like terms during addition by counting terms endverbatim

BV_ADD_TWO 

Auto-generated from RARE rule bv-add-two

BV_ADD_ZERO 

Auto-generated from RARE rule bv-add-zero

BV_AND_CONCAT_PULLUP 

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

BV_AND_FLATTEN 

Auto-generated from RARE rule bv-and-flatten

BV_AND_ONE 

Auto-generated from RARE rule bv-and-one

BV_AND_SIMPLIFY_1 

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

BV_AND_SIMPLIFY_2 

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

BV_AND_ZERO 

Auto-generated from RARE rule bv-and-zero

BV_ASHR_BY_CONST_0 

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

BV_ASHR_BY_CONST_1 

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

BV_ASHR_BY_CONST_2 

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

BV_ASHR_ZERO 

Auto-generated from RARE rule bv-ashr-zero

BV_BITWISE_IDEMP_1 

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

BV_BITWISE_IDEMP_2 

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

BV_BITWISE_NOT_AND 

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

BV_BITWISE_NOT_OR 

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

BV_BITWISE_SLICING 

verbatim embed:rst:leading-asterisk 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 endverbatim

BV_COMMUTATIVE_ADD 

Auto-generated from RARE rule bv-commutative-add

BV_COMMUTATIVE_AND 

Auto-generated from RARE rule bv-commutative-and

BV_COMMUTATIVE_MUL 

Auto-generated from RARE rule bv-commutative-mul

BV_COMMUTATIVE_OR 

Auto-generated from RARE rule bv-commutative-or

BV_COMMUTATIVE_XOR 

Auto-generated from RARE rule bv-commutative-xor

BV_COMP_ELIMINATE 

Auto-generated from RARE rule bv-comp-eliminate

BV_CONCAT_EXTRACT_MERGE 

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

BV_CONCAT_FLATTEN 

Auto-generated from RARE rule bv-concat-flatten

BV_CONCAT_MERGE_CONST 

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

BV_CONCAT_TO_MULT 

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

BV_EXTRACT_BITWISE_AND 

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

BV_EXTRACT_BITWISE_OR 

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

BV_EXTRACT_BITWISE_XOR 

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

BV_EXTRACT_CONCAT_1 

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

BV_EXTRACT_CONCAT_2 

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

BV_EXTRACT_CONCAT_3 

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

BV_EXTRACT_CONCAT_4 

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

BV_EXTRACT_EXTRACT 

Auto-generated from RARE rule bv-extract-extract

BV_EXTRACT_MULT_LEADING_BIT 

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

BV_EXTRACT_NOT 

Auto-generated from RARE rule bv-extract-not

BV_EXTRACT_SIGN_EXTEND_1 

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

BV_EXTRACT_SIGN_EXTEND_2 

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

BV_EXTRACT_SIGN_EXTEND_3 

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

BV_EXTRACT_WHOLE 

Auto-generated from RARE rule bv-extract-whole

BV_ITE_CONST_CHILDREN_1 

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

BV_ITE_CONST_CHILDREN_2 

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

BV_ITE_EQUAL_CHILDREN 

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

BV_ITE_EQUAL_COND_1 

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

BV_ITE_EQUAL_COND_2 

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

BV_ITE_EQUAL_COND_3 

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

BV_ITE_MERGE_ELSE_ELSE 

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

BV_ITE_MERGE_ELSE_IF 

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

BV_ITE_MERGE_THEN_ELSE 

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

BV_ITE_MERGE_THEN_IF 

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

BV_LSHR_BY_CONST_0 

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

BV_LSHR_BY_CONST_1 

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

BV_LSHR_BY_CONST_2 

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

BV_LSHR_ZERO 

Auto-generated from RARE rule bv-lshr-zero

BV_LT_SELF 

Auto-generated from RARE rule bv-lt-self

BV_MERGE_SIGN_EXTEND_1 

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

BV_MERGE_SIGN_EXTEND_2 

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

BV_MERGE_SIGN_EXTEND_3 

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

BV_MULT_DISTRIB_1 

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

BV_MULT_DISTRIB_2 

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

BV_MULT_DISTRIB_CONST_ADD 

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

BV_MULT_DISTRIB_CONST_NEG 

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

BV_MULT_DISTRIB_CONST_SUB 

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

BV_MULT_POW2_1 

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

BV_MULT_POW2_2 

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

BV_MULT_POW2_2B 

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

BV_MULT_SIMPLIFY 

verbatim embed:rst:leading-asterisk Bitvectors - Extract negations from multiplicands

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

endverbatim

BV_MULT_SLT_MULT_1 

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

BV_MULT_SLT_MULT_2 

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

BV_MUL_FLATTEN 

Auto-generated from RARE rule bv-mul-flatten

BV_MUL_ONE 

Auto-generated from RARE rule bv-mul-one

BV_MUL_ZERO 

Auto-generated from RARE rule bv-mul-zero

BV_NAND_ELIMINATE 

Auto-generated from RARE rule bv-nand-eliminate

BV_NEG_ADD 

Auto-generated from RARE rule bv-neg-add

BV_NEG_IDEMP 

Auto-generated from RARE rule bv-neg-idemp

BV_NEG_MULT 

Auto-generated from RARE rule bv-neg-mult

BV_NEG_SUB 

Auto-generated from RARE rule bv-neg-sub

BV_NOR_ELIMINATE 

Auto-generated from RARE rule bv-nor-eliminate

BV_NOT_IDEMP 

Auto-generated from RARE rule bv-not-idemp

BV_NOT_NEQ 

Auto-generated from RARE rule bv-not-neq

BV_NOT_SLE 

Auto-generated from RARE rule bv-not-sle

BV_NOT_ULE 

Auto-generated from RARE rule bv-not-ule

BV_NOT_ULT 

Auto-generated from RARE rule bv-not-ult

BV_NOT_XOR 

Auto-generated from RARE rule bv-not-xor

BV_OR_CONCAT_PULLUP 

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

BV_OR_FLATTEN 

Auto-generated from RARE rule bv-or-flatten

BV_OR_ONE 

Auto-generated from RARE rule bv-or-one

BV_OR_SIMPLIFY_1 

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

BV_OR_SIMPLIFY_2 

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

BV_OR_ZERO 

Auto-generated from RARE rule bv-or-zero

BV_REDAND_ELIMINATE 

Auto-generated from RARE rule bv-redand-eliminate

BV_REDOR_ELIMINATE 

Auto-generated from RARE rule bv-redor-eliminate

BV_REPEAT_ELIMINATE_1 

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

BV_REPEAT_ELIMINATE_2 

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

BV_ROTATE_LEFT_ELIMINATE_1 

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

BV_ROTATE_LEFT_ELIMINATE_2 

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

BV_ROTATE_RIGHT_ELIMINATE_1 

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

BV_ROTATE_RIGHT_ELIMINATE_2 

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

BV_SADDO_ELIMINATE 

Auto-generated from RARE rule bv-saddo-eliminate

BV_SDIVO_ELIMINATE 

Auto-generated from RARE rule bv-sdivo-eliminate

BV_SDIV_ELIMINATE 

Auto-generated from RARE rule bv-sdiv-eliminate

BV_SDIV_ELIMINATE_FEWER_BITWISE_OPS 

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

BV_SGE_ELIMINATE 

Auto-generated from RARE rule bv-sge-eliminate

BV_SGT_ELIMINATE 

Auto-generated from RARE rule bv-sgt-eliminate

BV_SHL_BY_CONST_0 

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

BV_SHL_BY_CONST_1 

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

BV_SHL_BY_CONST_2 

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

BV_SHL_ZERO 

Auto-generated from RARE rule bv-shl-zero

BV_SIGN_EXTEND_ELIMINATE 

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

BV_SIGN_EXTEND_ELIMINATE_0 

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

BV_SIGN_EXTEND_EQ_CONST_1 

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

BV_SIGN_EXTEND_EQ_CONST_2 

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

BV_SIGN_EXTEND_ULT_CONST_1 

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

BV_SIGN_EXTEND_ULT_CONST_2 

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

BV_SIGN_EXTEND_ULT_CONST_3 

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

BV_SIGN_EXTEND_ULT_CONST_4 

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

BV_SLE_ELIMINATE 

Auto-generated from RARE rule bv-sle-eliminate

BV_SLE_SELF 

Auto-generated from RARE rule bv-sle-self

BV_SLT_ELIMINATE 

Auto-generated from RARE rule bv-slt-eliminate

BV_SLT_ZERO 

Auto-generated from RARE rule bv-slt-zero

BV_SMOD_ELIMINATE 

Auto-generated from RARE rule bv-smod-eliminate

BV_SMOD_ELIMINATE_FEWER_BITWISE_OPS 

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

BV_SMULO_ELIMINATE 

verbatim embed:rst:leading-asterisk Bitvectors - Unsigned multiplication overflow detection elimination

See M.Gok, M.J. Schulte, P.I. Balzola, “Efficient integer multiplication overflow detection circuits”, 2001. http: endverbatim

BV_SREM_ELIMINATE 

Auto-generated from RARE rule bv-srem-eliminate

BV_SREM_ELIMINATE_FEWER_BITWISE_OPS 

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

BV_SSUBO_ELIMINATE 

Auto-generated from RARE rule bv-ssubo-eliminate

BV_SUB_ELIMINATE 

Auto-generated from RARE rule bv-sub-eliminate

BV_UADDO_ELIMINATE 

Auto-generated from RARE rule bv-uaddo-eliminate

BV_UDIV_ONE 

Auto-generated from RARE rule bv-udiv-one

BV_UDIV_POW2_NOT_ONE 

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

BV_UDIV_ZERO 

Auto-generated from RARE rule bv-udiv-zero

BV_UGE_ELIMINATE 

Auto-generated from RARE rule bv-uge-eliminate

BV_UGT_ELIMINATE 

Auto-generated from RARE rule bv-ugt-eliminate

BV_UGT_UREM 

Auto-generated from RARE rule bv-ugt-urem

BV_ULE_ELIMINATE 

Auto-generated from RARE rule bv-ule-eliminate

BV_ULE_MAX 

Auto-generated from RARE rule bv-ule-max

BV_ULE_SELF 

Auto-generated from RARE rule bv-ule-self

BV_ULE_ZERO 

Auto-generated from RARE rule bv-ule-zero

BV_ULT_ADD_ONE 

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

BV_ULT_ONE 

Auto-generated from RARE rule bv-ult-one

BV_ULT_ONES 

Auto-generated from RARE rule bv-ult-ones

BV_ULT_SELF 

Auto-generated from RARE rule bv-ult-self

BV_ULT_ZERO_1 

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

BV_ULT_ZERO_2 

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

BV_UMULO_ELIMINATE 

verbatim embed:rst:leading-asterisk Bitvectors - Unsigned multiplication overflow detection elimination

See M.Gok, M.J. Schulte, P.I. Balzola, “Efficient integer multiplication overflow detection circuits”, 2001. http: endverbatim

BV_UREM_ONE 

Auto-generated from RARE rule bv-urem-one

BV_UREM_POW2_NOT_ONE 

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

BV_UREM_SELF 

Auto-generated from RARE rule bv-urem-self

BV_USUBO_ELIMINATE 

Auto-generated from RARE rule bv-usubo-eliminate

BV_XNOR_ELIMINATE 

Auto-generated from RARE rule bv-xnor-eliminate

BV_XOR_CONCAT_PULLUP 

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

BV_XOR_DUPLICATE 

Auto-generated from RARE rule bv-xor-duplicate

BV_XOR_FLATTEN 

Auto-generated from RARE rule bv-xor-flatten

BV_XOR_NOT 

Auto-generated from RARE rule bv-xor-not

BV_XOR_ONES 

Auto-generated from RARE rule bv-xor-ones

BV_XOR_SIMPLIFY_1 

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

BV_XOR_SIMPLIFY_2 

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

BV_XOR_SIMPLIFY_3 

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

BV_XOR_ZERO 

Auto-generated from RARE rule bv-xor-zero

BV_ZERO_EXTEND_ELIMINATE 

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

BV_ZERO_EXTEND_ELIMINATE_0 

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

BV_ZERO_EXTEND_EQ_CONST_1 

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

BV_ZERO_EXTEND_EQ_CONST_2 

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

BV_ZERO_EXTEND_ULT_CONST_1 

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

BV_ZERO_EXTEND_ULT_CONST_2 

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

BV_ZERO_ULE 

Auto-generated from RARE rule bv-zero-ule

DISTINCT_BINARY_ELIM 

Auto-generated from RARE rule distinct-binary-elim

DISTINCT_ELIM 

verbatim embed:rst:leading-asterisk Builtin – Distinct elimination

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

endverbatim

DT_COLLAPSE_SELECTOR 

verbatim embed:rst:leading-asterisk Datatypes - collapse selector

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

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

endverbatim

DT_COLLAPSE_TESTER 

verbatim embed:rst:leading-asterisk 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.

endverbatim

DT_COLLAPSE_TESTER_SINGLETON 

verbatim embed:rst:leading-asterisk Datatypes - collapse tester

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

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

endverbatim

DT_CONS_EQ 

verbatim embed:rst:leading-asterisk 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.

endverbatim

DT_INST 

verbatim embed:rst:leading-asterisk 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 \(C\) . endverbatim

EQ_REFL 

Auto-generated from RARE rule eq-refl

EQ_SYMM 

Auto-generated from RARE rule eq-symm

EXISTS_ELIM 

verbatim embed:rst:leading-asterisk Quantifiers – Exists elimination

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

endverbatim

ITE_ELSE_FALSE 

Auto-generated from RARE rule ite-else-false

ITE_ELSE_LOOKAHEAD 

Auto-generated from RARE rule ite-else-lookahead

ITE_ELSE_LOOKAHEAD_SELF 

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

ITE_ELSE_NEG_LOOKAHEAD 

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

ITE_ELSE_TRUE 

Auto-generated from RARE rule ite-else-true

ITE_EQ_BRANCH 

Auto-generated from RARE rule ite-eq-branch

ITE_FALSE_COND 

Auto-generated from RARE rule ite-false-cond

ITE_NEG_BRANCH 

Auto-generated from RARE rule ite-neg-branch

ITE_NOT_COND 

Auto-generated from RARE rule ite-not-cond

ITE_THEN_FALSE 

Auto-generated from RARE rule ite-then-false

ITE_THEN_LOOKAHEAD 

Auto-generated from RARE rule ite-then-lookahead

ITE_THEN_LOOKAHEAD_SELF 

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

ITE_THEN_NEG_LOOKAHEAD 

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

ITE_THEN_TRUE 

Auto-generated from RARE rule ite-then-true

ITE_TRUE_COND 

Auto-generated from RARE rule ite-true-cond

MACRO_ARITH_STRING_PRED_ENTAIL 

verbatim embed:rst:leading-asterisk 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.

endverbatim

MACRO_BOOL_NNF_NORM 

verbatim embed:rst:leading-asterisk 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.

endverbatim

MACRO_QUANT_PARTITION_CONNECTED_FV 

verbatim embed:rst:leading-asterisk 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 \(F_i\) . endverbatim

MACRO_SUBSTR_STRIP_SYM_LENGTH 

verbatim embed:rst:leading-asterisk 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 \(m\) .

endverbatim

NONE 

verbatim embed:rst:leading-asterisk 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. endverbatim

QUANT_MERGE_PRENEX 

verbatim embed:rst:leading-asterisk 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.

endverbatim

QUANT_MINISCOPE 

verbatim embed:rst:leading-asterisk Quantifiers – Miniscoping

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

endverbatim

QUANT_UNUSED_VARS 

verbatim embed:rst:leading-asterisk Quantifiers – Unused variables

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

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

endverbatim

RE_ALL_ELIM 

Auto-generated from RARE rule re-all-elim

RE_CONCAT_EMP 

Auto-generated from RARE rule re-concat-emp

RE_CONCAT_FLATTEN 

Auto-generated from RARE rule re-concat-flatten

RE_CONCAT_MERGE 

Auto-generated from RARE rule re-concat-merge

RE_CONCAT_NONE 

Auto-generated from RARE rule re-concat-none

RE_CONCAT_STAR_SWAP 

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

RE_DIFF_ELIM 

Auto-generated from RARE rule re-diff-elim

RE_INTER_ALL 

Auto-generated from RARE rule re-inter-all

RE_INTER_CSTRING 

Auto-generated from RARE rule re-inter-cstring

RE_INTER_CSTRING_NEG 

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

RE_INTER_DUP 

Auto-generated from RARE rule re-inter-dup

RE_INTER_FLATTEN 

Auto-generated from RARE rule re-inter-flatten

RE_INTER_NONE 

Auto-generated from RARE rule re-inter-none

RE_INTER_UNION_INCLUSION 

verbatim embed:rst:leading-asterisk 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 .

endverbatim

RE_IN_COMP 

Auto-generated from RARE rule re-in-comp

RE_IN_CSTRING 

Auto-generated from RARE rule re-in-cstring

RE_IN_EMPTY 

Auto-generated from RARE rule re-in-empty

RE_IN_SIGMA 

Auto-generated from RARE rule re-in-sigma

RE_IN_SIGMA_STAR 

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

RE_LOOP_ELIM 

verbatim embed:rst:leading-asterisk Strings - regular expression loop elimination

\[re.loop_{l,u}(R) = re.union(R^l, \ldots, R^u)\]

where \(u \geq l\) .

endverbatim

RE_LOOP_NEG 

Auto-generated from RARE rule re-loop-neg

RE_OPT_ELIM 

Auto-generated from RARE rule re-opt-elim

RE_UNION_ALL 

Auto-generated from RARE rule re-union-all

RE_UNION_DUP 

Auto-generated from RARE rule re-union-dup

RE_UNION_FLATTEN 

Auto-generated from RARE rule re-union-flatten

RE_UNION_NONE 

Auto-generated from RARE rule re-union-none

SEQ_LEN_UNIT 

Auto-generated from RARE rule seq-len-unit

SEQ_NTH_UNIT 

Auto-generated from RARE rule seq-nth-unit

SEQ_REV_CONCAT 

Auto-generated from RARE rule seq-rev-concat

SEQ_REV_UNIT 

Auto-generated from RARE rule seq-rev-unit

SETS_CARD_EMP 

Auto-generated from RARE rule sets-card-emp

SETS_CARD_MINUS 

Auto-generated from RARE rule sets-card-minus

SETS_CARD_SINGLETON 

Auto-generated from RARE rule sets-card-singleton

SETS_CARD_UNION 

Auto-generated from RARE rule sets-card-union

SETS_CHOOSE_SINGLETON 

Auto-generated from RARE rule sets-choose-singleton

SETS_EQ_SINGLETON_EMP 

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

SETS_INTER_COMM 

Auto-generated from RARE rule sets-inter-comm

SETS_INTER_EMP1 

Auto-generated from RARE rule sets-inter-emp1

SETS_INTER_EMP2 

Auto-generated from RARE rule sets-inter-emp2

SETS_INTER_MEMBER 

Auto-generated from RARE rule sets-inter-member

SETS_IS_EMPTY_EVAL 

verbatim embed:rst:leading-asterisk 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.

endverbatim

SETS_MEMBER_EMP 

Auto-generated from RARE rule sets-member-emp

SETS_MEMBER_SINGLETON 

Auto-generated from RARE rule sets-member-singleton

SETS_MINUS_EMP1 

Auto-generated from RARE rule sets-minus-emp1

SETS_MINUS_EMP2 

Auto-generated from RARE rule sets-minus-emp2

SETS_MINUS_MEMBER 

Auto-generated from RARE rule sets-minus-member

SETS_SUBSET_ELIM 

Auto-generated from RARE rule sets-subset-elim

SETS_UNION_COMM 

Auto-generated from RARE rule sets-union-comm

SETS_UNION_EMP1 

Auto-generated from RARE rule sets-union-emp1

SETS_UNION_EMP2 

Auto-generated from RARE rule sets-union-emp2

SETS_UNION_MEMBER 

Auto-generated from RARE rule sets-union-member

STR_AT_ELIM 

Auto-generated from RARE rule str-at-elim

STR_CONCAT_CLASH 

Auto-generated from RARE rule str-concat-clash

STR_CONCAT_CLASH2 

Auto-generated from RARE rule str-concat-clash2

STR_CONCAT_CLASH2_REV 

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

STR_CONCAT_CLASH_CHAR 

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

STR_CONCAT_CLASH_CHAR_REV 

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

STR_CONCAT_CLASH_REV 

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

STR_CONCAT_EMP 

Auto-generated from RARE rule str-concat-emp

STR_CONCAT_FLATTEN 

Auto-generated from RARE rule str-concat-flatten

STR_CONCAT_FLATTEN_EQ 

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

STR_CONCAT_FLATTEN_EQ_REV 

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

STR_CONCAT_UNIFY 

Auto-generated from RARE rule str-concat-unify

STR_CONCAT_UNIFY_BASE 

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

STR_CONCAT_UNIFY_BASE_REV 

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

STR_CONCAT_UNIFY_REV 

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

STR_CONTAINS_CONCAT_FIND 

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

STR_CONTAINS_EMP 

Auto-generated from RARE rule str-contains-emp

STR_CONTAINS_IS_EMP 

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

STR_CONTAINS_LEQ_LEN_EQ 

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

STR_CONTAINS_LT_LEN 

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

STR_CONTAINS_REFL 

Auto-generated from RARE rule str-contains-refl

STR_CONTAINS_SPLIT_CHAR 

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

STR_EQ_CTN_FALSE 

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

STR_EQ_CTN_FULL_FALSE1 

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

STR_EQ_CTN_FULL_FALSE2 

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

STR_INDEXOF_NO_CONTAINS 

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

STR_INDEXOF_SELF 

Auto-generated from RARE rule str-indexof-self

STR_IN_RE_CONCAT_STAR_CHAR 

verbatim embed:rst:leading-asterisk 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.

endverbatim

STR_IN_RE_CONSUME 

verbatim embed:rst:leading-asterisk 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 \(R\) .

endverbatim

STR_IN_RE_CONTAINS 

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

STR_IN_RE_EVAL 

verbatim embed:rst:leading-asterisk 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.

endverbatim

STR_IN_RE_INTER_ELIM 

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

STR_IN_RE_RANGE_ELIM 

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

STR_IN_RE_REQ_UNFOLD 

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

STR_IN_RE_REQ_UNFOLD_REV 

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

STR_IN_RE_SIGMA 

verbatim embed:rst:leading-asterisk 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:

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

endverbatim

STR_IN_RE_SIGMA_STAR 

verbatim embed:rst:leading-asterisk 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 \(\mathit{re}.\text{++}\) .

endverbatim

STR_IN_RE_SKIP_UNFOLD 

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

STR_IN_RE_SKIP_UNFOLD_REV 

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

STR_IN_RE_STRIP_CHAR 

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

STR_IN_RE_STRIP_CHAR_REV 

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

STR_IN_RE_STRIP_CHAR_S_SINGLE 

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

STR_IN_RE_STRIP_CHAR_S_SINGLE_REV 

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

STR_IN_RE_STRIP_PREFIX 

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

STR_IN_RE_STRIP_PREFIX_BASE 

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

STR_IN_RE_STRIP_PREFIX_BASE_NEG 

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

STR_IN_RE_STRIP_PREFIX_BASE_NEG_REV 

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

STR_IN_RE_STRIP_PREFIX_BASE_REV 

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

STR_IN_RE_STRIP_PREFIX_BASE_S_SINGLE 

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

STR_IN_RE_STRIP_PREFIX_BASE_S_SINGLE_NEG 

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

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

STR_IN_RE_STRIP_PREFIX_BASE_S_SINGLE_REV 

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

STR_IN_RE_STRIP_PREFIX_NEG 

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

STR_IN_RE_STRIP_PREFIX_NEG_REV 

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

STR_IN_RE_STRIP_PREFIX_REV 

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

STR_IN_RE_STRIP_PREFIX_SRS_SINGLE 

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

STR_IN_RE_STRIP_PREFIX_SRS_SINGLE_NEG 

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

STR_IN_RE_STRIP_PREFIX_SRS_SINGLE_NEG_REV 

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

STR_IN_RE_STRIP_PREFIX_SRS_SINGLE_REV 

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

STR_IN_RE_STRIP_PREFIX_SR_SINGLE 

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

STR_IN_RE_STRIP_PREFIX_SR_SINGLE_NEG 

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

STR_IN_RE_STRIP_PREFIX_SR_SINGLE_NEG_REV 

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

STR_IN_RE_STRIP_PREFIX_SR_SINGLE_REV 

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

STR_IN_RE_STRIP_PREFIX_S_SINGLE 

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

STR_IN_RE_STRIP_PREFIX_S_SINGLE_NEG 

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

STR_IN_RE_STRIP_PREFIX_S_SINGLE_NEG_REV 

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

STR_IN_RE_STRIP_PREFIX_S_SINGLE_REV 

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

STR_IN_RE_TEST_UNFOLD 

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

STR_IN_RE_TEST_UNFOLD_REV 

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

STR_IN_RE_UNION_ELIM 

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

STR_LEN_CONCAT_REC 

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

STR_LEN_REPLACE_INV 

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

STR_LEN_SUBSTR_IN_RANGE 

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

STR_LEN_SUBSTR_UB1 

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

STR_LEN_SUBSTR_UB2 

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

STR_LEN_UPDATE_INV 

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

STR_NTH_ELIM_CODE 

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

STR_PREFIXOF_ELIM 

Auto-generated from RARE rule str-prefixof-elim

STR_PREFIXOF_ONE 

Auto-generated from RARE rule str-prefixof-one

STR_REPLACE_EMPTY 

Auto-generated from RARE rule str-replace-empty

STR_REPLACE_NO_CONTAINS 

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

STR_REPLACE_SELF 

Auto-generated from RARE rule str-replace-self

STR_SUBSTR_COMBINE1 

Auto-generated from RARE rule str-substr-combine1

STR_SUBSTR_COMBINE2 

Auto-generated from RARE rule str-substr-combine2

STR_SUBSTR_COMBINE3 

Auto-generated from RARE rule str-substr-combine3

STR_SUBSTR_COMBINE4 

Auto-generated from RARE rule str-substr-combine4

STR_SUBSTR_CONCAT1 

Auto-generated from RARE rule str-substr-concat1

STR_SUBSTR_CONCAT2 

Auto-generated from RARE rule str-substr-concat2

STR_SUBSTR_EMPTY_RANGE 

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

STR_SUBSTR_EMPTY_START 

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

STR_SUBSTR_EMPTY_START_NEG 

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

STR_SUBSTR_EMPTY_STR 

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

STR_SUBSTR_EQ_EMPTY 

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

STR_SUBSTR_FULL 

Auto-generated from RARE rule str-substr-full

STR_SUBSTR_FULL_EQ 

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

STR_SUBSTR_LEN_INCLUDE 

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

STR_SUBSTR_LEN_INCLUDE_PRE 

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

STR_SUBSTR_LEN_SKIP 

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

STR_SUFFIXOF_ELIM 

Auto-generated from RARE rule str-suffixof-elim

STR_SUFFIXOF_ONE 

Auto-generated from RARE rule str-suffixof-one

STR_TO_LOWER_CONCAT 

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

STR_TO_LOWER_UPPER 

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

STR_TO_UPPER_CONCAT 

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

STR_TO_UPPER_LOWER 

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

UF_BV2NAT_GEQ_ELIM 

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