java.lang.Object
- java.lang.Enum<ProofRule>
- - io.github.cvc5.ProofRule

All Implemented Interfaces:

java.io.Serializable, java.lang.Comparable<ProofRule>
```
public enum ProofRule
extends java.lang.Enum<ProofRule>
```

Enum Constant Summary

Enum Constants
Enum Constant	Description
`ALETHE_RULE`	External – Alethe Place holder for Alethe rules.
`ALPHA_EQUIV`	Quantifiers – Alpha equivalence \[ \inferruleSC{-\mid F, (y_1 \ldots y_n), (z_1,\dots, z_n)} {F = F\{y_1\mapsto z_1,\dots,y_n\mapsto z_n\}} {if $y_1,\dots,y_n, z_1,\dots,z_n$ are unique bound variables} \] Notice that this rule is correct only when $z_1,\dots,z_n$ are not contained in $FV(F) \setminus \{ y_1,\dots, y_n \}$, where $FV(\varphi)$ are the free variables of $\varphi$.
`AND_ELIM`	Boolean – And elimination \[ \inferrule{(F_1 \land \dots \land F_n) \mid i}{F_i} \]
`AND_INTRO`	Boolean – And introduction \[ \inferrule{F_1 \dots F_n \mid -}{(F_1 \land \dots \land F_n)} \]
`ANNOTATION`	Builtin theory – Annotation \[ \inferrule{F \mid a_1 \dots a_n}{F} \] The terms $a_1 \dots a_n$ can be anything used to annotate the proof node, one example is where $a_1$ is a theory.InferenceId.
`ARITH_MULT_NEG`	Arithmetic – Multiplication with negative factor \[ \inferrule{- \mid m, l \diamond r}{(m < 0 \land l \diamond r) \rightarrow m \cdot l \diamond_{inv} m \cdot r} \] where $\diamond$ is a relation symbol and $\diamond_{inv}$ the inverted relation symbol.
`ARITH_MULT_POS`	Arithmetic – Multiplication with positive factor \[ \inferrule{- \mid m, l \diamond r}{(m > 0 \land l \diamond r) \rightarrow m \cdot l \diamond m \cdot r} \] where $\diamond$ is a relation symbol.
`ARITH_MULT_SIGN`	Arithmetic – Sign inference \[ \inferrule{- \mid f_1 \dots f_k, m}{(f_1 \land \dots \land f_k) \rightarrow m \diamond 0} \] where $f_1 \dots f_k$ are variables compared to zero (less, greater or not equal), $m$ is a monomial from these variables and $\diamond$ is the comparison (less or equal) that results from the signs of the variables.
`ARITH_MULT_TANGENT`	Arithmetic – Multiplication tangent plane \[ \inferruleSC{- \mid t, x, y, a, b, \sigma}{(t \leq tplane) \leftrightarrow ((x \leq a \land y \geq b) \lor (x \geq a \land y \leq b))}{if $\sigma = -1$} \inferruleSC{- \mid t, x, y, a, b, \sigma}{(t \geq tplane) \leftrightarrow ((x \leq a \land y \leq b) \lor (x \geq a \land y \geq b))}{if $\sigma = 1$} \] where $x,y$ are real terms (variables or extended terms), $t = x \cdot y$ (possibly under rewriting), $a,b$ 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 $(a,b)$.
`ARITH_NL_COVERING_DIRECT`	Arithmetic – Coverings – Direct conflict We use $\texttt{IRP}_k(poly)$ for an IndexedRootPredicate that is defined as the $k$'th root of the polynomial $poly$.
`ARITH_NL_COVERING_RECURSIVE`	Arithmetic – Coverings – Recursive interval See ARITH_NL_COVERING_DIRECT <cvc5.`ARITH_NL_COVERING_DIRECT`> for the necessary definitions.
`ARITH_OP_ELIM_AXIOM`	Arithmetic – Operator elimination \[ \inferrule{- \mid t}{\texttt{arith::OperatorElim::getAxiomFor(t)}} \]
`ARITH_POLY_NORM`	Arithmetic – Polynomial normalization \[ \inferrule{- \mid t = s}{t = s} \] where $\texttt{arith::PolyNorm::isArithPolyNorm(t, s)} = \top$.
`ARITH_SUM_UB`	Arithmetic – Sum upper bounds \[ \inferrule{P_1 \dots P_n \mid -}{L \diamond R} \] where $P_i$ has the form $L_i \diamond_i R_i$ and $\diamond_i \in \{<, \leq, =\}$.
`ARITH_TRANS_EXP_APPROX_ABOVE_NEG`	Arithmetic – Transcendentals – Exp is approximated from above for negative values \[ \inferrule{- \mid d,t,l,u}{(t \geq l \land t \leq u) \rightarrow exp(t) \leq \texttt{secant}(\exp, l, u, t)} \] where $d$ is an even positive number, $t$ an arithmetic term and $l,u$ are lower and upper bounds on $t$.
`ARITH_TRANS_EXP_APPROX_ABOVE_POS`	Arithmetic – Transcendentals – Exp is approximated from above for positive values \[ \inferrule{- \mid d,t,l,u}{(t \geq l \land t \leq u) \rightarrow exp(t) \leq \texttt{secant-pos}(\exp, l, u, t)} \] where $d$ is an even positive number, $t$ an arithmetic term and $l,u$ are lower and upper bounds on $t$.
`ARITH_TRANS_EXP_APPROX_BELOW`	Arithmetic – Transcendentals – Exp is approximated from below \[ \inferrule{- \mid d,c,t}{t \geq c \rightarrow exp(t) \geq \texttt{maclaurin}(\exp, d, c)} \] where $d$ is an odd positive number, $t$ an arithmetic term and $\texttt{maclaurin}(\exp, d, c)$ is the $d$'th taylor polynomial at zero (also called the Maclaurin series) of the exponential function evaluated at $c$.
`ARITH_TRANS_EXP_NEG`	Arithmetic – Transcendentals – Exp at negative values \[ \inferrule{- \mid t}{(t < 0) \leftrightarrow (\exp(t) < 1)} \]
`ARITH_TRANS_EXP_POSITIVITY`	Arithmetic – Transcendentals – Exp is always positive \[ \inferrule{- \mid t}{\exp(t) > 0} \]
`ARITH_TRANS_EXP_SUPER_LIN`	Arithmetic – Transcendentals – Exp grows super-linearly for positive values \[ \inferrule{- \mid t}{t \leq 0 \lor \exp(t) > t+1} \]
`ARITH_TRANS_EXP_ZERO`	Arithmetic – Transcendentals – Exp at zero \[ \inferrule{- \mid t}{(t=0) \leftrightarrow (\exp(t) = 1)} \]
`ARITH_TRANS_PI`	Arithmetic – Transcendentals – Assert bounds on Pi \[ \inferrule{- \mid l, u}{\texttt{real.pi} \geq l \land \texttt{real.pi} \leq u} \] where $l,u$ are valid lower and upper bounds on $\pi$.
`ARITH_TRANS_SINE_APPROX_ABOVE_NEG`	Arithmetic – Transcendentals – Sine is approximated from above for negative values \[ \inferrule{- \mid d,t,lb,ub,l,u}{(t \geq lb land t \leq ub) \rightarrow \sin(t) \leq \texttt{secant}(\sin, l, u, t)} \] where $d$ is an even positive number, $t$ an arithmetic term, $lb,ub$ are symbolic lower and upper bounds on $t$ (possibly containing $\pi$) and $l,u$ the evaluated lower and upper bounds on $t$.
`ARITH_TRANS_SINE_APPROX_ABOVE_POS`	Arithmetic – Transcendentals – Sine is approximated from above for positive values \[ \inferrule{- \mid d,t,c,lb,ub}{(t \geq lb land t \leq ub) \rightarrow \sin(t) \leq \texttt{upper}(\sin, c)} \] where $d$ is an even positive number, $t$ an arithmetic term, $c$ an arithmetic constant and $lb,ub$ are symbolic lower and upper bounds on $t$ (possibly containing $\pi$).
`ARITH_TRANS_SINE_APPROX_BELOW_NEG`	Arithmetic – Transcendentals – Sine is approximated from below for negative values \[ \inferrule{- \mid d,t,c,lb,ub}{(t \geq lb land t \leq ub) \rightarrow \sin(t) \geq \texttt{lower}(\sin, c)} \] where $d$ is an even positive number, $t$ an arithmetic term, $c$ an arithmetic constant and $lb,ub$ are symbolic lower and upper bounds on $t$ (possibly containing $\pi$).
`ARITH_TRANS_SINE_APPROX_BELOW_POS`	Arithmetic – Transcendentals – Sine is approximated from below for positive values \[ \inferrule{- \mid d,t,lb,ub,l,u}{(t \geq lb land t \leq ub) \rightarrow \sin(t) \geq \texttt{secant}(\sin, l, u, t)} \] where $d$ is an even positive number, $t$ an arithmetic term, $lb,ub$ are symbolic lower and upper bounds on $t$ (possibly containing $\pi$) and $l,u$ the evaluated lower and upper bounds on $t$.
`ARITH_TRANS_SINE_BOUNDS`	Arithmetic – Transcendentals – Sine is always between -1 and 1 \[ \inferrule{- \mid t}{\sin(t) \leq 1 \land \sin(t) \geq -1} \]
`ARITH_TRANS_SINE_SHIFT`	Arithmetic – Transcendentals – Sine is shifted to -pi...pi \[ \inferrule{- \mid x, y, s}{-\pi \leq y \leq \pi \land \sin(y) = \sin(x) \land (\ite{-\pi \leq x \leq \pi}{x = y}{x = y + 2 \pi s})} \] where $x$ is the argument to sine, $y$ is a new real skolem that is $x$ shifted into $-\pi \dots \pi$ and $s$ is a new integer slolem that is the number of phases $y$ is shifted.
`ARITH_TRANS_SINE_SYMMETRY`	Arithmetic – Transcendentals – Sine is symmetric with respect to negation of the argument \[ \inferrule{- \mid t}{\sin(t) - \sin(-t) = 0} \]
`ARITH_TRANS_SINE_TANGENT_PI`	Arithmetic – Transcendentals – Sine is bounded by the tangents at -pi and pi ..
`ARITH_TRANS_SINE_TANGENT_ZERO`	Arithmetic – Transcendentals – Sine is bounded by the tangent at zero ..
`ARITH_TRICHOTOMY`	Arithmetic – Trichotomy of the reals \[ \inferrule{A, B \mid -}{C} \] where $\neg A, \neg B, C$ are $x < c, x = c, x > c$ in some order.
`ARRAYS_EQ_RANGE_EXPAND`	Arrays – Expansion of array range equality \[ \inferrule{-\mid \mathit{eqrange}(a,b,i,j)} {\mathit{eqrange}(a,b,i,j)= \forall x.\> i \leq x \leq j \rightarrow \mathit{select}(a,x)=\mathit{select}(b,x)} \]
`ARRAYS_EXT`	Arrays – Arrays extensionality \[ \inferrule{a \neq b\mid -} {\mathit{select}(a,k)\neq\mathit{select}(b,k)} \] where $k$ is $\texttt{arrays::SkolemCache::getExtIndexSkolem}(a\neq b)$.
`ARRAYS_READ_OVER_WRITE`	Arrays – Read over write \[ \inferrule{i_1 \neq i_2\mid \mathit{select}(\mathit{store}(a,i_1,e),i_2)} {\mathit{select}(\mathit{store}(a,i_1,e),i_2) = \mathit{select}(a,i_2)} \]
`ARRAYS_READ_OVER_WRITE_1`	Arrays – Read over write 1 \[ \inferrule{-\mid \mathit{select}(\mathit{store}(a,i,e),i)} {\mathit{select}(\mathit{store}(a,i,e),i)=e} \]
`ARRAYS_READ_OVER_WRITE_CONTRA`	Arrays – Read over write, contrapositive \[ \inferrule{\mathit{select}(\mathit{store}(a,i_2,e),i_1) \neq \mathit{select}(a,i_1)\mid -}{i_1=i_2} \]
`ASSUME`	Assumption (a leaf) \[ \inferrule{- \mid F}{F} \] This rule has special status, in that an application of assume is an open leaf in a proof that is not (yet) justified.
`BETA_REDUCE`	Equality – Beta reduction \[ \inferrule{\mid \lambda x_1\dots x_n.\> t, t_1,\dots,t_n} {((\lambda x_1\dots x_n.\> t) t_1 \ldots t_n)=t\{x_1\mapsto t_1,\dots,x_n\mapsto t_n\}} \] The right hand side of the equality in the conclusion is computed using standard substitution via Node.substitute.
`BV_BITBLAST_STEP`	Bit-vectors – Bitblast bit-vector constant, variable, and terms For constant and variables: \[ \inferrule{-\mid t}{t = \texttt{bitblast}(t)} \] For terms: \[ \inferrule{-\mid k(\texttt{bitblast}(t_1),\dots,\texttt{bitblast}(t_n))} {k(\texttt{bitblast}(t_1),\dots,\texttt{bitblast}(t_n)) = \texttt{bitblast}(t)} \] where $t$ is $k(t_1,\dots,t_n)$.
`BV_EAGER_ATOM`	Bit-vectors – Bit-vector eager atom \[ \inferrule{-\mid F}{F = F[0]} \] where $F$ is of kind `BITVECTOR_EAGER_ATOM`.
`CHAIN_RESOLUTION`	Boolean – N-ary Resolution \[ \inferrule{C_1 \dots C_n \mid (pol_1 \dots pol_{n-1}), (L_1 \dots L_{n-1})}{C} \] where let $C_1 \dots C_n$ be nodes viewed as clauses, as defined above let $C_1 \diamond_{L,\mathit{pol}} C_2$ represent the resolution of $C_1$ with $C_2$ with pivot $L$ and polarity $pol$, as defined above let $C_1' = C_1$, for each $i > 1$, let $C_i' = C_{i-1} \diamond{L_{i-1}, \mathit{pol}_{i-1}} C_i'$ Note the list of polarities and pivots are provided as s-expressions.
`CNF_AND_NEG`	Boolean – CNF – And Negative \[ \inferrule{- \mid (F_1 \land \dots \land F_n)}{(F_1 \land \dots \land F_n) \lor \neg F_1 \lor \dots \lor \neg F_n} \]
`CNF_AND_POS`	Boolean – CNF – And Positive \[ \inferrule{- \mid (F_1 \land \dots \land F_n), i}{\neg (F_1 \land \dots \land F_n) \lor F_i} \]
`CNF_EQUIV_NEG1`	Boolean – CNF – Equiv Negative 1 \[ \inferrule{- \mid F_1 = F_2}{(F_1 = F_2) \lor F_1 \lor F_2} \]
`CNF_EQUIV_NEG2`	Boolean – CNF – Equiv Negative 2 \[ \inferrule{- \mid F_1 = F_2}{(F_1 = F_2) \lor \neg F_1 \lor \neg F_2} \]
`CNF_EQUIV_POS1`	Boolean – CNF – Equiv Positive 1 \[ \inferrule{- \mid F_1 = F_2}{F_1 \neq F_2 \lor \neg F_1 \lor F_2} \]
`CNF_EQUIV_POS2`	Boolean – CNF – Equiv Positive 2 \[ \inferrule{- \mid F_1 = F_2}{F_1 \neq F_2 \lor F_1 \lor \neg F_2} \]
`CNF_IMPLIES_NEG1`	Boolean – CNF – Implies Negative 1 \[ \inferrule{- \mid F_1 \rightarrow F_2}{(F_1 \rightarrow F_2) \lor F_1} \]
`CNF_IMPLIES_NEG2`	Boolean – CNF – Implies Negative 2 \[ \inferrule{- \mid F_1 \rightarrow F_2}{(F_1 \rightarrow F_2) \lor \neg F_2} \]
`CNF_IMPLIES_POS`	Boolean – CNF – Implies Positive \[ \inferrule{- \mid F_1 \rightarrow F_2}{\neg(F_1 \rightarrow F_2) \lor \neg F_1 \lor F_2} \]
`CNF_ITE_NEG1`	Boolean – CNF – ITE Negative 1 \[ \inferrule{- \mid (\ite{C}{F_1}{F_2})}{(\ite{C}{F_1}{F_2}) \lor \neg C \lor \neg F_1} \]
`CNF_ITE_NEG2`	Boolean – CNF – ITE Negative 2 \[ \inferrule{- \mid (\ite{C}{F_1}{F_2})}{(\ite{C}{F_1}{F_2}) \lor C \lor \neg F_2} \]
`CNF_ITE_NEG3`	Boolean – CNF – ITE Negative 3 \[ \inferrule{- \mid (\ite{C}{F_1}{F_2})}{(\ite{C}{F_1}{F_2}) \lor \neg F_1 \lor \neg F_2} \]
`CNF_ITE_POS1`	Boolean – CNF – ITE Positive 1 \[ \inferrule{- \mid (\ite{C}{F_1}{F_2})}{\neg(\ite{C}{F_1}{F_2}) \lor \neg C \lor F_1} \]
`CNF_ITE_POS2`	Boolean – CNF – ITE Positive 2 \[ \inferrule{- \mid (\ite{C}{F_1}{F_2})}{\neg(\ite{C}{F_1}{F_2}) \lor C \lor F_2} \]
`CNF_ITE_POS3`	Boolean – CNF – ITE Positive 3 \[ \inferrule{- \mid (\ite{C}{F_1}{F_2})}{\neg(\ite{C}{F_1}{F_2}) \lor F_1 \lor F_2} \]
`CNF_OR_NEG`	Boolean – CNF – Or Negative \[ \inferrule{- \mid (F_1 \lor \dots \lor F_n), i}{(F_1 \lor \dots \lor F_n) \lor \neg F_i} \]
`CNF_OR_POS`	Boolean – CNF – Or Positive \[ \inferrule{- \mid (F_1 \lor \dots \lor F_n)}{\neg(F_1 \lor \dots \lor F_n) \lor F_1 \lor \dots \lor F_n} \]
`CNF_XOR_NEG1`	Boolean – CNF – XOR Negative 1 \[ \inferrule{- \mid F_1 \xor F_2}{(F_1 \xor F_2) \lor \neg F_1 \lor F_2} \]
`CNF_XOR_NEG2`	Boolean – CNF – XOR Negative 2 \[ \inferrule{- \mid F_1 \xor F_2}{(F_1 \xor F_2) \lor F_1 \lor \neg F_2} \]
`CNF_XOR_POS1`	Boolean – CNF – XOR Positive 1 \[ \inferrule{- \mid F_1 \xor F_2}{\neg(F_1 \xor F_2) \lor F_1 \lor F_2} \]
`CNF_XOR_POS2`	Boolean – CNF – XOR Positive 2 \[ \inferrule{- \mid F_1 \xor F_2}{\neg(F_1 \xor F_2) \lor \neg F_1 \lor \neg F_2} \]
`CONCAT_CONFLICT`	Strings – Core rules – Concatenation conflict \[ \inferrule{(c_1\cdot t) = (c_2 \cdot s)\mid b}{\bot} \] where $b$ indicates if the direction is reversed, $c_1,\,c_2$ are constants such that $\texttt{Word::splitConstant}(c_1,c_2, \mathit{index},b)$ is null, in other words, neither is a prefix of the other.
`CONCAT_CONFLICT_DEQ`	Strings – Core rules – Concatenation conflict for disequal characters \[ \inferrule{(t_1\cdot t) = (s_1 \cdot s), t_1 \deq s_1 \mid b}{\bot} \] 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.
`CONCAT_CPROP`	Strings – Core rules – Concatenation constant propagation \[ \inferrule{(t_1\cdot w_1\cdot t_2) = (w_2 \cdot s),\, \mathit{len}(t_1) \neq 0\mid \bot}{(t_1 = w_3\cdot r)} \] where $w_1,\,w_2,\,w_3$ are words, $w_3$ is $\mathit{pre}(w_2,p)$, $p$ is $\texttt{Word::overlap}(\mathit{suf}(w_2,1), w_1)$, and $r$ is $\mathit{skolem}(\mathit{suf}(t_1,\mathit{len}(w_3)))$.
`CONCAT_CSPLIT`	Strings – Core rules – Concatenation split for constants \[ \inferrule{(t_1\cdot t_2) = (c \cdot s_2),\, \mathit{len}(t_1) \neq 0\mid \bot}{(t_1 = c\cdot r)} \] where $r$ is $\mathit{skolem}(\mathit{suf}(t_1,1))$.
`CONCAT_EQ`	Strings – Core rules – Concatenation equality \[ \inferrule{(t_1\cdot\ldots \cdot t_n \cdot t) = (t_1 \cdot\ldots \cdot t_n\cdot s)\mid b}{t = s} \] where $\cdot$ stands for string concatenation and $b$ indicates if the direction is reversed.
`CONCAT_LPROP`	Strings – Core rules – Concatenation length propagation \[ \inferrule{(t_1\cdot t_2) = (s_1 \cdot s_2),\, \mathit{len}(t_1) > \mathit{len}(s_1)\mid \bot}{(t_1 = s_1\cdot r_t)} \] where $r_t$ is $\mathit{skolem}(\mathit{suf}(t_1,\mathit{len}(s_1)))$.
`CONCAT_SPLIT`	Strings – Core rules – Concatenation split \[ \inferruleSC{(t_1\cdot t_2) = (s_1 \cdot s_2),\, \mathit{len}(t_1) \neq \mathit{len}(s_1)\mid b}{(t_1 = s_1\cdot r_t) \vee (s_1 = t_1\cdot r_s)}{if $b=\bot$} \] where $r_t$ is $\mathit{skolem}(\mathit{suf}(t_1,\mathit{len}(s_1)))$ and $r_s$ is $\mathit{skolem}(\mathit{suf}(s_1,\mathit{len}(t_1)))$.
`CONCAT_UNIFY`	Strings – Core rules – Concatenation unification \[ \inferrule{(t_1\cdot t_2) = (s_1 \cdot s_2),\, \mathit{len}(t_1) = \mathit{len}(s_1)\mid b}{t_1 = s_1} \] where $b$ indicates if the direction is reversed.
`CONG`	Equality – Congruence \[ \inferrule{t_1=s_1,\dots,t_n=s_n\mid k, f?}{k(f?)(t_1,\dots, t_n) = k(f?)(s_1,\dots, s_n)} \] where $k$ is the application kind.
`CONTRA`	Boolean – Contradiction \[ \inferrule{F, \neg F \mid -}{\bot} \]
`DSL_REWRITE`	Builtin theory – DSL rewrite \[ \inferrule{F_1 \dots F_n \mid id t_1 \dots t_n}{F} \] where the DSL rewrite rule with the given identifier is \(\forall x_1 \dots x_n.
`DT_CLASH`	Datatypes – Clash \[ \inferruleSC{\mathit{is}_{C_i}(t), \mathit{is}_{C_j}(t)\mid -}{\bot} {if $i\neq j$} \]
`DT_COLLAPSE`	Datatypes – Collapse \[ \inferrule{-\mid \mathit{sel}_i(C_j(t_1,\dots,t_n))}{ \mathit{sel}_i(C_j(t_1,\dots,t_n)) = r} \] where $C_j$ is a constructor, $r$ is $t_i$ if $\mathit{sel}_i$ is a correctly applied selector, or `{@link TypeNode#mkGroundTerm()}` of the proper type otherwise.
`DT_INST`	Datatypes – Instantiation \[ \inferrule{-\mid t,n}{\mathit{is}_C(t) = (t = C(\mathit{sel}_1(t),\dots,\mathit{sel}_n(t)))} \] where $C$ is the $n^{\mathit{th}}$ constructor of the type of t, and $\mathit{is}_C$ is the discriminator (tester) for $C$.
`DT_SPLIT`	Datatypes – Split \[ \inferrule{-\mid t}{\mathit{is}_{C_1}(t)\vee\cdots\vee\mathit{is}_{C_n}(t)} \] where $C_1,\dots,C_n$ are all the constructors of the type of $t$.
`DT_UNIF`	Datatypes – Unification \[ \inferrule{C(t_1,\dots,t_n)= C(s_1,\dots,s_n)\mid i}{t_1 = s_i} \] where $C$ is a constructor.
`ENCODE_PRED_TRANSFORM`	Builtin theory – Encode predicate transformation \[ \inferrule{F \mid G}{G} \] where $F$ and $G$ are equivalent up to their encoding in an external proof format.
`EQ_RESOLVE`	Boolean – Equality resolution \[ \inferrule{F_1, (F_1 = F_2) \mid -}{F_2} \] Note this can optionally be seen as a macro for EQUIV_ELIM1 <cvc5.`EQUIV_ELIM1`> + RESOLUTION <cvc5.`RESOLUTION`>.
`EQUIV_ELIM1`	Boolean – Equivalence elimination version 1 \[ \inferrule{F_1 = F_2 \mid -}{\neg F_1 \lor F_2} \]
`EQUIV_ELIM2`	Boolean – Equivalence elimination version 2 \[ \inferrule{F_1 = F_2 \mid -}{F_1 \lor \neg F_2} \]
`EVALUATE`	Builtin theory – Evaluate \[ \inferrule{- \mid t}{t = \texttt{Evaluator::evaluate}(t)} \] Note this is equivalent to: `(REWRITE t {@link MethodId#RW_EVALUATE)}`.
`FACTORING`	Boolean – Factoring \[ \inferrule{C_1 \mid -}{C_2} \] where $C_2$ is the clause $C_1$, but every occurence of a literal after its first occurence is omitted.
`FALSE_ELIM`	Equality – False elim \[ \inferrule{F=\bot\mid -}{\neg F} \]
`FALSE_INTRO`	Equality – False intro \[ \inferrule{\neg F\mid -}{F = \bot} \]
`HO_APP_ENCODE`	Equality – Higher-order application encoding \[ \inferrule{-\mid t}{t= \texttt{TheoryUfRewriter::getHoApplyForApplyUf}(t)} \] For example, this rule concludes $f(x,y) = @(@(f,x),y)$, where $@$ isthe `HO_APPLY` kind.
`HO_CONG`	Equality – Higher-order congruence \[ \inferrule{f=g, t_1=s_1,\dots,t_n=s_n\mid k}{k(f, t_1,\dots, t_n) = k(g, s_1,\dots, s_n)} \] Notice that this rule is only used when the application kind $k$ is either `cvc5.Kind.APPLY_UF` or `cvc5.Kind.HO_APPLY`.
`IMPLIES_ELIM`	Boolean – Implication elimination \[ \inferrule{F_1 \rightarrow F_2 \mid -}{\neg F_1 \lor F_2} \]
`INSTANTIATE`	Quantifiers – Instantiation \[ \inferrule{\forall x_1\dots x_n.\> F\mid (t_1 \dots t_n), (id\, (t)?)?} {F\{x_1\mapsto t_1,\dots,x_n\mapsto t_n\}} \] The list of terms to instantiate $(t_1 \dots t_n)$ is provided as an s-expression as the first argument.
`INT_TIGHT_LB`	Arithmetic – Tighten strict integer lower bounds \[ \inferrule{i > c \mid -}{i \geq \lceil c \rceil} \] where $i$ has integer type.
`INT_TIGHT_UB`	Arithmetic – Tighten strict integer upper bounds \[ \inferrule{i < c \mid -}{i \leq \lfloor c \rfloor} \] where $i$ has integer type.
`ITE_ELIM1`	Boolean – ITE elimination version 1 \[ \inferrule{(\ite{C}{F_1}{F_2}) \mid -}{\neg C \lor F_1} \]
`ITE_ELIM2`	Boolean – ITE elimination version 2 \[ \inferrule{(\ite{C}{F_1}{F_2}) \mid -}{C \lor F_2} \]
`LFSC_RULE`	External – LFSC Place holder for LFSC rules.
`MACRO_ARITH_SCALE_SUM_UB`	Arithmetic – Adding inequalities An arithmetic literal is a term of the form $p \diamond c$ where $\diamond \in \{ <, \leq, =, \geq, > \}$, $p$ a polynomial and $c$ a rational constant.
`MACRO_BV_BITBLAST`	Bit-vectors – (Macro) Bitblast \[ \inferrule{-\mid t}{t = \texttt{bitblast}(t)} \] where `bitblast()` represents the result of the bit-blasted term as a bit-vector consisting of the output bits of the bit-blasted circuit representation of the term.
`MACRO_RESOLUTION`	Boolean – N-ary Resolution + Factoring + Reordering \[ \inferrule{C_1 \dots C_n \mid C, pol_1,L_1 \dots pol_{n-1},L_{n-1}}{C} \] where let $C_1 \dots C_n$ be nodes viewed as clauses, as defined in RESOLUTION <cvc5.`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 <cvc5.`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 $C_n'$
`MACRO_RESOLUTION_TRUST`	Boolean – N-ary Resolution + Factoring + Reordering unchecked Same as :cpp:enumerator:`MACRO_RESOLUTION <cvc5.`MACRO_RESOLUTION`>`, but not checked by the internal proof checker.
`MACRO_REWRITE`	Builtin theory – Rewrite \[ \inferrule{- \mid t, idr}{t = \texttt{Rewriter}_{idr}(t)} \] where $idr$ is a MethodId identifier, which determines the kind of rewriter to apply, e.g.
`MACRO_SR_EQ_INTRO`	Builtin theory – Substitution + Rewriting equality introduction In this rule, we provide a term $t$ and conclude that it is equal to its rewritten form under a (proven) substitution.
`MACRO_SR_PRED_ELIM`	Builtin theory – Substitution + Rewriting predicate elimination \[ \inferrule{F, F_1 \dots F_n \mid (ids (ida (idr)?)?)?}{\texttt{Rewriter}_{idr}(F \circ \sigma_{ids, ida}(F_n) \circ \cdots \circ \sigma_{ids, ida}(F_1))} \] where $ids$ and $idr$ are method identifiers.
`MACRO_SR_PRED_INTRO`	Builtin theory – Substitution + Rewriting predicate introduction In this rule, we provide a formula $F$ and conclude it, under the condition that it rewrites to true under a proven substitution.
`MACRO_SR_PRED_TRANSFORM`	Builtin theory – Substitution + Rewriting predicate elimination \[ \inferrule{F, F_1 \dots F_n \mid G, (ids (ida (idr)?)?)?}{G} \] where $\texttt{Rewriter}_{idr}(F \circ \sigma_{ids, ida}(F_n) \circ \cdots \circ \sigma_{ids, ida}(F_1)) = \texttt{Rewriter}_{idr}(G \circ \sigma_{ids, ida}(F_n) \circ \cdots \circ \sigma_{ids, ida}(F_1))$.
`MACRO_STRING_INFERENCE`	Strings – (Macro) String inference \[ \inferrule{?\mid F,\mathit{id},\mathit{isRev},\mathit{exp}}{F} \] used to bookkeep an inference that has not yet been converted via $\texttt{strings::InferProofCons::convert}$.
`MODUS_PONENS`	Boolean – Modus Ponens \[ \inferrule{F_1, (F_1 \rightarrow F_2) \mid -}{F_2} \] Note this can optionally be seen as a macro for IMPLIES_ELIM <cvc5.`IMPLIES_ELIM`> + RESOLUTION <cvc5.`RESOLUTION`>.
`NARY_CONG`	Equality – N-ary Congruence \[ \inferrule{t_1=s_1,\dots,t_n=s_n\mid k}{k(t_1,\dots, t_n) = k(s_1,\dots, s_n)} \] where $k$ is the application kind.
`NOT_AND`	Boolean – De Morgan – Not And \[ \inferrule{\neg(F_1 \land \dots \land F_n) \mid -}{\neg F_1 \lor \dots \lor \neg F_n} \]
`NOT_EQUIV_ELIM1`	Boolean – Not Equivalence elimination version 1 \[ \inferrule{F_1 \neq F_2 \mid -}{F_1 \lor F_2} \]
`NOT_EQUIV_ELIM2`	Boolean – Not Equivalence elimination version 2 \[ \inferrule{F_1 \neq F_2 \mid -}{\neg F_1 \lor \neg F_2} \]
`NOT_IMPLIES_ELIM1`	Boolean – Not Implication elimination version 1 \[ \inferrule{\neg(F_1 \rightarrow F_2) \mid -}{F_1} \]
`NOT_IMPLIES_ELIM2`	Boolean – Not Implication elimination version 2 \[ \inferrule{\neg(F_1 \rightarrow F_2) \mid -}{\neg F_2} \]
`NOT_ITE_ELIM1`	Boolean – Not ITE elimination version 1 \[ \inferrule{\neg(\ite{C}{F_1}{F_2}) \mid -}{\neg C \lor \neg F_1} \]
`NOT_ITE_ELIM2`	Boolean – Not ITE elimination version 2 \[ \inferrule{\neg(\ite{C}{F_1}{F_2}) \mid -}{C \lor \neg F_2} \]
`NOT_NOT_ELIM`	Boolean – Double negation elimination \[ \inferrule{\neg (\neg F) \mid -}{F} \]
`NOT_OR_ELIM`	Boolean – Not Or elimination \[ \inferrule{\neg(F_1 \lor \dots \lor F_n) \mid i}{\neg F_i} \]
`NOT_XOR_ELIM1`	Boolean – Not XOR elimination version 1 \[ \inferrule{\neg(F_1 \xor F_2) \mid -}{F_1 \lor \neg F_2} \]
`NOT_XOR_ELIM2`	Boolean – Not XOR elimination version 2 \[ \inferrule{\neg(F_1 \xor F_2) \mid -}{\neg F_1 \lor F_2} \]
`RE_ELIM`	Strings – Regular expressions – Elimination \[ \inferrule{-\mid F,b}{F = \texttt{strings::RegExpElimination::eliminate}(F, b)} \] where $b$ is a Boolean indicating whether we are using aggressive eliminations.
`RE_INTER`	Strings – Regular expressions – Intersection \[ \inferrule{t\in R_1,\,t\in R_2\mid -}{t\in \mathit{inter}(R_1,R_2)} \]
`RE_UNFOLD_NEG`	Strings – Regular expressions – Negative Unfold \[ \inferrule{t\not\in R\mid -}{\texttt{RegExpOpr::reduceRegExpNeg}(t\not\in R)} \] corresponding to the one-step unfolding of the premise.
`RE_UNFOLD_NEG_CONCAT_FIXED`	Strings – Regular expressions – Unfold negative concatenation, fixed \[ \inferrule{t\not\in R\mid -}{\texttt{RegExpOpr::reduceRegExpNegConcatFixed}(t\not\in R,L,i)} \] where $\texttt{RegExpOpr::getRegExpConcatFixed}(t\not\in R, i) = L$, corresponding to the one-step unfolding of the premise, optimized for fixed length of component $i$ of the regular expression concatenation $R$.
`RE_UNFOLD_POS`	Strings – Regular expressions – Positive Unfold \[ \inferrule{t\in R\mid -}{\texttt{RegExpOpr::reduceRegExpPos}(t\in R)} \] corresponding to the one-step unfolding of the premise.
`REFL`	Equality – Reflexivity \[ \inferrule{-\mid t}{t = t} \]
`REMOVE_TERM_FORMULA_AXIOM`	Processing rules – Remove Term Formulas Axiom \[ \inferrule{- \mid t}{\texttt{RemoveTermFormulas::getAxiomFor}(t)} \]
`REORDERING`	Boolean – Reordering \[ \inferrule{C_1 \mid C_2}{C_2} \] where the set representations of $C_1$ and $C_2$ are the same and the number of literals in $C_2$ is the same of that of $C_1$.
`RESOLUTION`	Boolean – Resolution \[ \inferrule{C_1, C_2 \mid pol, L}{C} \] where $C_1$ and $C_2$ are nodes viewed as clauses, i.e., either an `OR` node with each children viewed as a literal or a node viewed as a literal.
`SAT_REFUTATION`	SAT Refutation for assumption-based unsat cores \[ \inferrule{F_1 \dots F_n \mid -}{\bot} \] where $F_1 \dots F_n$ correspond to the unsat core determined by the SAT solver.
`SCOPE`	Scope (a binder for assumptions) \[ \inferruleSC{F \mid F_1 \dots F_n}{(F_1 \land \dots \land F_n) \Rightarrow F}{if $F\neq\bot$} \textrm{ or } \inferruleSC{F \mid F_1 \dots F_n}{\neg (F_1 \land \dots \land F_n)}{if $F=\bot$} \] This rule has a dual purpose with :cpp:enumerator:`ASSUME <cvc5.`ASSUME`>`.
`SKOLEM_INTRO`	Quantifiers – Skolem introduction \[ \inferrule{-\mid k}{k = t} \] where $t$ is the unpurified form of skolem $k$.
`SKOLEMIZE`	Quantifiers – Skolemization \[ \inferrule{\exists x_1\dots x_n.\> F\mid -}{F\sigma} \] or \[ \inferrule{\neg (\forall x_1\dots x_n.\> F)\mid -}{\neg F\sigma} \] where $\sigma$ maps $x_1,\dots,x_n$ to their representative skolems obtained by `SkolemManager.mkSkolemize`, returned in the skolems argument of that method.
`SPLIT`	Boolean – Split \[ \inferrule{- \mid F}{F \lor \neg F} \]
`STRING_CODE_INJ`	Strings – Code points \[ \inferrule{-\mid t,s}{\mathit{to\_code}(t) = -1 \vee \mathit{to\_code}(t) \neq \mathit{to\_code}(s) \vee t\neq s} \]
`STRING_DECOMPOSE`	Strings – Core rules – String decomposition \[ \inferrule{\mathit{len}(t) \geq n\mid \bot}{t = w_1\cdot w_2 \wedge \mathit{len}(w_1) = n} \] 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 $\mathit{skolem}(\mathit{pre}(t,n)$ and $w_2$ is $\mathit{skolem}(\mathit{suf}(t,n)$.
`STRING_EAGER_REDUCTION`	Strings – Extended functions – Eager reduction \[ \inferrule{-\mid t}{R} \] where $R$ is $\texttt{strings::TermRegistry::eagerReduce}(t)$.
`STRING_LENGTH_NON_EMPTY`	Strings – Core rules – Length non-empty \[ \inferrule{t\neq ''\mid -}{\mathit{len}(t) \neq 0} \]
`STRING_LENGTH_POS`	Strings – Core rules – Length positive \[ \inferrule{-\mid t}{(\mathit{len}(t) = 0\wedge t= '')\vee \mathit{len}(t) > 0} \]
`STRING_REDUCTION`	Strings – Extended functions – Reduction \[ \inferrule{-\mid t}{R\wedge t = w} \] where $w$ is $\texttt{strings::StringsPreprocess::reduce}(t, R, \dots)$.
`STRING_SEQ_UNIT_INJ`	Strings – Sequence unit \[ \inferrule{\mathit{unit}(x) = \mathit{unit}(y)\mid -}{x = y} \] Also applies to the case where $\mathit{unit}(y)$ is a constant sequence of length one.
`SUBS`	Builtin theory – Substitution \[ \inferrule{F_1 \dots F_n \mid t, ids?}{t = t \circ \sigma_{ids}(F_n) \circ \cdots \circ \sigma_{ids}(F_1)} \] where $\sigma_{ids}(F_i)$ are substitutions, which notice are applied in reverse order.
`SYMM`	Equality – Symmetry \[ \inferrule{t_1 = t_2\mid -}{t_2 = t_1} \] or \[ \inferrule{t_1 \neq t_2\mid -}{t_2 \neq t_1} \]
`TRANS`	Equality – Transitivity \[ \inferrule{t_1=t_2,\dots,t_{n-1}=t_n\mid -}{t_1 = t_n} \]
`TRUE_ELIM`	Equality – True elim \[ \inferrule{F=\top\mid -}{F} \]
`TRUE_INTRO`	Equality – True intro \[ \inferrule{F\mid -}{F = \top} \]
`TRUST`	Trusted rule \[ \inferrule{F_1 \dots F_n \mid tid, F, ...}{F} \] where $tid$ is an identifier and $F$ is a formula.
`TRUST_THEORY_REWRITE`	Trusted rules – Theory rewrite \[ \inferrule{- \mid F, tid, rid}{F} \] where $F$ is an equality of the form $t = t'$ where $t'$ is obtained by applying the kind of rewriting given by the method identifier $rid$, which is one of: `RW_REWRITE_THEORY_PRE`, `RW_REWRITE_THEORY_POST`, `RW_REWRITE_EQ_EXT`.
`UNKNOWN`	External – Alethe Place holder for Alethe rules.
`XOR_ELIM1`	Boolean – XOR elimination version 1 \[ \inferrule{F_1 \xor F_2 \mid -}{F_1 \lor F_2} \]
`XOR_ELIM2`	Boolean – XOR elimination version 2 \[ \inferrule{F_1 \xor F_2 \mid -}{\neg F_1 \lor \neg F_2} \]

Method Summary

All Methods Static Methods Instance Methods Concrete Methods
Modifier and Type	Method	Description
`static ProofRule`	`fromInt(int value)`
`int`	`getValue()`
`static ProofRule`	`valueOf(java.lang.String name)`	Returns the enum constant of this type with the specified name.
`static ProofRule[]`	`values()`	Returns an array containing the constants of this enum type, in the order they are declared.

Methods inherited from class java.lang.Enum
clone, compareTo, equals, finalize, getDeclaringClass, hashCode, name, ordinal, toString, valueOf

Methods inherited from class java.lang.Object
getClass, notify, notifyAll, wait, wait, wait

- Enum Constant Detail
  - ASSUME
```
public static final ProofRule ASSUME
```
    Assumption (a leaf) \[ \inferrule{- \mid F}{F} \] This rule has special status, in that an application of assume is an open leaf in a proof that is not (yet) justified. An assume leaf is analogous to a free variable in a term, where we say "F is a free assumption in proof P" if it contains an application of F that is not bound by SCOPE <cvc5.SCOPE> (see below).
  - SCOPE
```
public static final ProofRule SCOPE
```
    Scope (a binder for assumptions) \[ \inferruleSC{F \mid F_1 \dots F_n}{(F_1 \land \dots \land F_n) \Rightarrow F}{if $F\neq\bot$} \textrm{ or } \inferruleSC{F \mid F_1 \dots F_n}{\neg (F_1 \land \dots \land F_n)}{if $F=\bot$} \] This rule has a dual purpose with :cpp:enumerator:`ASSUME <cvc5.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.
  - SUBS
```
public static final ProofRule SUBS
```
    Builtin theory – Substitution \[ \inferrule{F_1 \dots F_n \mid t, ids?}{t = t \circ \sigma_{ids}(F_n) \circ \cdots \circ \sigma_{ids}(F_1)} \] where $\sigma_{ids}(F_i)$ are substitutions, which notice are applied in reverse order. Notice that $ids$ is a MethodId identifier, which determines how to convert the formulas $F_1 \dots F_n$ into substitutions. It is an optional argument, where by default the premises are equalities of the form `(= x y)` and converted into substitutions $x\mapsto y$.
  - MACRO_REWRITE
```
public static final ProofRule MACRO_REWRITE
```
    Builtin theory – Rewrite \[ \inferrule{- \mid t, idr}{t = \texttt{Rewriter}_{idr}(t)} \] where $idr$ is a MethodId identifier, which determines the kind of rewriter to apply, e.g. Rewriter.rewrite.
  - EVALUATE
```
public static final ProofRule EVALUATE
```
    Builtin theory – Evaluate \[ \inferrule{- \mid t}{t = \texttt{Evaluator::evaluate}(t)} \] Note this is equivalent to: (REWRITE t {@link MethodId#RW_EVALUATE)}.
  - MACRO_SR_EQ_INTRO
```
public static final ProofRule MACRO_SR_EQ_INTRO
```
    Builtin theory – Substitution + Rewriting equality introduction In this rule, we provide a term $t$ and conclude that it is equal to its rewritten form under a (proven) substitution. \[ \inferrule{F_1 \dots F_n \mid t, (ids (ida (idr)?)?)?}{t = \texttt{Rewriter}_{idr}(t \circ \sigma_{ids, ida}(F_n) \circ \cdots \circ \sigma_{ids, ida}(F_1))} \] In other words, from the point of view of Skolem forms, this rule transforms $t$ to $t'$ by standard substitution + rewriting. The arguments $ids$, $ida$ and $idr$ are optional and specify the identifier of the substitution, the substitution application and rewriter respectively to be used. For details, see :cvc5src:`theory/builtin/proof_checker.h`.
  - MACRO_SR_PRED_INTRO
```
public static final ProofRule MACRO_SR_PRED_INTRO
```
    Builtin theory – Substitution + Rewriting predicate introduction In this rule, we provide a formula $F$ and conclude it, under the condition that it rewrites to true under a proven substitution. \[ \inferrule{F_1 \dots F_n \mid F, (ids (ida (idr)?)?)?}{F} \] where $\texttt{Rewriter}_{idr}(F \circ \sigma_{ids, ida}(F_n) \circ \cdots \circ \sigma_{ids, ida}(F_1)) = \top$ and $ids$ and $idr$ are method identifiers. More generally, this rule also holds when $\texttt{Rewriter::rewrite}(\texttt{toOriginal}(F')) = \top$ where $F'$ is the result of the left hand side of the equality above. Here, notice that we apply rewriting on the original form of $F'$, meaning that this rule may conclude an $F$ whose Skolem form is justified by the definition of its (fresh) Skolem variables. For example, this rule may justify the conclusion $k = t$ where $k$ is the purification Skolem for $t$, e.g. where the original form of $k$ is $t$. Furthermore, notice that the rewriting and substitution is applied only within the side condition, meaning the rewritten form of the original form of $F$ does not escape this rule.
  - MACRO_SR_PRED_ELIM
```
public static final ProofRule MACRO_SR_PRED_ELIM
```
    Builtin theory – Substitution + Rewriting predicate elimination \[ \inferrule{F, F_1 \dots F_n \mid (ids (ida (idr)?)?)?}{\texttt{Rewriter}_{idr}(F \circ \sigma_{ids, ida}(F_n) \circ \cdots \circ \sigma_{ids, ida}(F_1))} \] where $ids$ and $idr$ are method identifiers. We rewrite only on the Skolem form of $F$, similar to MACRO_SR_EQ_INTRO <cvc5.MACRO_SR_EQ_INTRO>.
  - MACRO_SR_PRED_TRANSFORM
```
public static final ProofRule MACRO_SR_PRED_TRANSFORM
```
    Builtin theory – Substitution + Rewriting predicate elimination \[ \inferrule{F, F_1 \dots F_n \mid G, (ids (ida (idr)?)?)?}{G} \] where $\texttt{Rewriter}_{idr}(F \circ \sigma_{ids, ida}(F_n) \circ \cdots \circ \sigma_{ids, ida}(F_1)) = \texttt{Rewriter}_{idr}(G \circ \sigma_{ids, ida}(F_n) \circ \cdots \circ \sigma_{ids, ida}(F_1))$. More generally, this rule also holds when: $\texttt{Rewriter::rewrite}(\texttt{toOriginal}(F')) = \texttt{Rewriter::rewrite}(\texttt{toOriginal}(G'))$ where $F'$ and $G'$ are the result of each side of the equation above. Here, original forms are used in a similar manner to MACRO_SR_PRED_INTRO <cvc5.MACRO_SR_PRED_INTRO> above.
  - ENCODE_PRED_TRANSFORM
```
public static final ProofRule ENCODE_PRED_TRANSFORM
```
    Builtin theory – Encode predicate transformation \[ \inferrule{F \mid G}{G} \] where $F$ and $G$ are equivalent up to their encoding in an external proof format. This is currently verified by $\texttt{RewriteDbNodeConverter::convert}(F) = \texttt{RewriteDbNodeConverter::convert}(G)$. This rule can be treated as a no-op when appropriate in external proof formats.
  - DSL_REWRITE
```
public static final ProofRule DSL_REWRITE
```
    Builtin theory – DSL rewrite \[ \inferrule{F_1 \dots F_n \mid id t_1 \dots t_n}{F} \] where the DSL rewrite rule with the given identifier is $\forall x_1 \dots x_n. (G_1 \wedge G_n) \Rightarrow G$ where for $i=1, \dots n$, we have that $F_i = \sigma(G_i)$ and $F = \sigma(G)$ where $\sigma$ is the substitution $\{x_1\mapsto t_1,\dots,x_n\mapsto t_n\}$. Notice that the application of the substitution takes into account the possible list semantics of variables $x_1 \ldots x_n$. If $x_i$ is a variable with list semantics, then $t_i$ denotes a list of terms. The substitution implemented by expr.narySubstitute replaces each $x_i$ with the list $t_i$ in its place.
  - ANNOTATION
```
public static final ProofRule ANNOTATION
```
    Builtin theory – Annotation \[ \inferrule{F \mid a_1 \dots a_n}{F} \] The terms $a_1 \dots a_n$ can be anything used to annotate the proof node, one example is where $a_1$ is a theory.InferenceId.
  - REMOVE_TERM_FORMULA_AXIOM
```
public static final ProofRule REMOVE_TERM_FORMULA_AXIOM
```
    Processing rules – Remove Term Formulas Axiom \[ \inferrule{- \mid t}{\texttt{RemoveTermFormulas::getAxiomFor}(t)} \]
  - TRUST
```
public static final ProofRule TRUST
```
    Trusted rule \[ \inferrule{F_1 \dots F_n \mid tid, F, ...}{F} \] where $tid$ is an identifier and $F$ is a formula. This rule is used when a formal justification of an inference step cannot be provided. The formulas $F_1 \dots F_n$ refer to a set of formulas that entail $F$, which may or may not be provided.
  - TRUST_THEORY_REWRITE
```
public static final ProofRule TRUST_THEORY_REWRITE
```
    Trusted rules – Theory rewrite \[ \inferrule{- \mid F, tid, rid}{F} \] where $F$ is an equality of the form $t = t'$ where $t'$ is obtained by applying the kind of rewriting given by the method identifier $rid$, which is one of: RW_REWRITE_THEORY_PRE, RW_REWRITE_THEORY_POST, RW_REWRITE_EQ_EXT. Notice that the checker for this rule does not replay the rewrite to ensure correctness, since theory rewriter methods are not static. For example, the quantifiers rewriter involves constructing new bound variables that are not guaranteed to be consistent on each call.
  - SAT_REFUTATION
```
public static final ProofRule SAT_REFUTATION
```
    SAT Refutation for assumption-based unsat cores \[ \inferrule{F_1 \dots F_n \mid -}{\bot} \] where $F_1 \dots F_n$ correspond to the unsat core determined by the SAT solver.
  - RESOLUTION
```
public static final ProofRule RESOLUTION
```
    Boolean – Resolution \[ \inferrule{C_1, C_2 \mid pol, L}{C} \] where
    
    $C_1$ and $C_2$ are nodes viewed as clauses, i.e., either an OR node with each children viewed as a literal or a node viewed as a literal. Note that an OR node could also be a literal.
    
    $pol$ is either true or false, representing the polarity of the pivot on the first clause
    
    $L$ is the pivot of the resolution, which occurs as is (resp. under a NOT) in $C_1$ and negatively (as is) in $C_2$ if $pol = \top$ ($pol = \bot$).
    $C$ is a clause resulting from collecting all the literals in $C_1$, minus the first occurrence of the pivot or its negation, and $C_2$, minus the first occurrence of the pivot or its negation, according to the policy above. If the resulting clause has a single literal, that literal itself is the result; if it has no literals, then the result is false; otherwise it's an OR node of the resulting literals. Note that it may be the case that the pivot does not occur in the clauses. In this case the rule is not unsound, but it does not correspond to resolution but rather to a weakening of the clause that did not have a literal eliminated.
  - CHAIN_RESOLUTION
```
public static final ProofRule CHAIN_RESOLUTION
```
    Boolean – N-ary Resolution \[ \inferrule{C_1 \dots C_n \mid (pol_1 \dots pol_{n-1}), (L_1 \dots L_{n-1})}{C} \] where
    
    let $C_1 \dots C_n$ be nodes viewed as clauses, as defined above
    
    let $C_1 \diamond_{L,\mathit{pol}} C_2$ represent the resolution of $C_1$ with $C_2$ with pivot $L$ and polarity $pol$, as defined above
    
    let $C_1' = C_1$,
    
    for each $i > 1$, let $C_i' = C_{i-1} \diamond{L_{i-1}, \mathit{pol}_{i-1}} C_i'$
    Note the list of polarities and pivots are provided as s-expressions. The result of the chain resolution is $C = C_n'$
  - FACTORING
```
public static final ProofRule FACTORING
```
    Boolean – Factoring \[ \inferrule{C_1 \mid -}{C_2} \] where $C_2$ is the clause $C_1$, but every occurence of a literal after its first occurence is omitted.
  - REORDERING
```
public static final ProofRule REORDERING
```
    Boolean – Reordering \[ \inferrule{C_1 \mid C_2}{C_2} \] where the set representations of $C_1$ and $C_2$ are the same and the number of literals in $C_2$ is the same of that of $C_1$.
  - MACRO_RESOLUTION
```
public static final ProofRule MACRO_RESOLUTION
```
    Boolean – N-ary Resolution + Factoring + Reordering \[ \inferrule{C_1 \dots C_n \mid C, pol_1,L_1 \dots pol_{n-1},L_{n-1}}{C} \] where
    
    let $C_1 \dots C_n$ be nodes viewed as clauses, as defined in RESOLUTION <cvc5.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 <cvc5.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 $C_n'$
  - MACRO_RESOLUTION_TRUST
```
public static final ProofRule MACRO_RESOLUTION_TRUST
```
    Boolean – N-ary Resolution + Factoring + Reordering unchecked Same as :cpp:enumerator:`MACRO_RESOLUTION <cvc5.MACRO_RESOLUTION>`, but not checked by the internal proof checker.
  - SPLIT
```
public static final ProofRule SPLIT
```
    Boolean – Split \[ \inferrule{- \mid F}{F \lor \neg F} \]
  - EQ_RESOLVE
```
public static final ProofRule EQ_RESOLVE
```
    Boolean – Equality resolution \[ \inferrule{F_1, (F_1 = F_2) \mid -}{F_2} \] Note this can optionally be seen as a macro for EQUIV_ELIM1 <cvc5.EQUIV_ELIM1> + RESOLUTION <cvc5.RESOLUTION>.
  - MODUS_PONENS
```
public static final ProofRule MODUS_PONENS
```
    Boolean – Modus Ponens \[ \inferrule{F_1, (F_1 \rightarrow F_2) \mid -}{F_2} \] Note this can optionally be seen as a macro for IMPLIES_ELIM <cvc5.IMPLIES_ELIM> + RESOLUTION <cvc5.RESOLUTION>.
  - NOT_NOT_ELIM
```
public static final ProofRule NOT_NOT_ELIM
```
    Boolean – Double negation elimination \[ \inferrule{\neg (\neg F) \mid -}{F} \]
  - CONTRA
```
public static final ProofRule CONTRA
```
    Boolean – Contradiction \[ \inferrule{F, \neg F \mid -}{\bot} \]
  - AND_ELIM
```
public static final ProofRule AND_ELIM
```
    Boolean – And elimination \[ \inferrule{(F_1 \land \dots \land F_n) \mid i}{F_i} \]
  - AND_INTRO
```
public static final ProofRule AND_INTRO
```
    Boolean – And introduction \[ \inferrule{F_1 \dots F_n \mid -}{(F_1 \land \dots \land F_n)} \]
  - NOT_OR_ELIM
```
public static final ProofRule NOT_OR_ELIM
```
    Boolean – Not Or elimination \[ \inferrule{\neg(F_1 \lor \dots \lor F_n) \mid i}{\neg F_i} \]
  - IMPLIES_ELIM
```
public static final ProofRule IMPLIES_ELIM
```
    Boolean – Implication elimination \[ \inferrule{F_1 \rightarrow F_2 \mid -}{\neg F_1 \lor F_2} \]
  - NOT_IMPLIES_ELIM1
```
public static final ProofRule NOT_IMPLIES_ELIM1
```
    Boolean – Not Implication elimination version 1 \[ \inferrule{\neg(F_1 \rightarrow F_2) \mid -}{F_1} \]
  - NOT_IMPLIES_ELIM2
```
public static final ProofRule NOT_IMPLIES_ELIM2
```
    Boolean – Not Implication elimination version 2 \[ \inferrule{\neg(F_1 \rightarrow F_2) \mid -}{\neg F_2} \]
  - EQUIV_ELIM1
```
public static final ProofRule EQUIV_ELIM1
```
    Boolean – Equivalence elimination version 1 \[ \inferrule{F_1 = F_2 \mid -}{\neg F_1 \lor F_2} \]
  - EQUIV_ELIM2
```
public static final ProofRule EQUIV_ELIM2
```
    Boolean – Equivalence elimination version 2 \[ \inferrule{F_1 = F_2 \mid -}{F_1 \lor \neg F_2} \]
  - NOT_EQUIV_ELIM1
```
public static final ProofRule NOT_EQUIV_ELIM1
```
    Boolean – Not Equivalence elimination version 1 \[ \inferrule{F_1 \neq F_2 \mid -}{F_1 \lor F_2} \]
  - NOT_EQUIV_ELIM2
```
public static final ProofRule NOT_EQUIV_ELIM2
```
    Boolean – Not Equivalence elimination version 2 \[ \inferrule{F_1 \neq F_2 \mid -}{\neg F_1 \lor \neg F_2} \]
  - XOR_ELIM1
```
public static final ProofRule XOR_ELIM1
```
    Boolean – XOR elimination version 1 \[ \inferrule{F_1 \xor F_2 \mid -}{F_1 \lor F_2} \]
  - XOR_ELIM2
```
public static final ProofRule XOR_ELIM2
```
    Boolean – XOR elimination version 2 \[ \inferrule{F_1 \xor F_2 \mid -}{\neg F_1 \lor \neg F_2} \]
  - NOT_XOR_ELIM1
```
public static final ProofRule NOT_XOR_ELIM1
```
    Boolean – Not XOR elimination version 1 \[ \inferrule{\neg(F_1 \xor F_2) \mid -}{F_1 \lor \neg F_2} \]
  - NOT_XOR_ELIM2
```
public static final ProofRule NOT_XOR_ELIM2
```
    Boolean – Not XOR elimination version 2 \[ \inferrule{\neg(F_1 \xor F_2) \mid -}{\neg F_1 \lor F_2} \]
  - ITE_ELIM1
```
public static final ProofRule ITE_ELIM1
```
    Boolean – ITE elimination version 1 \[ \inferrule{(\ite{C}{F_1}{F_2}) \mid -}{\neg C \lor F_1} \]
  - ITE_ELIM2
```
public static final ProofRule ITE_ELIM2
```
    Boolean – ITE elimination version 2 \[ \inferrule{(\ite{C}{F_1}{F_2}) \mid -}{C \lor F_2} \]
  - NOT_ITE_ELIM1
```
public static final ProofRule NOT_ITE_ELIM1
```
    Boolean – Not ITE elimination version 1 \[ \inferrule{\neg(\ite{C}{F_1}{F_2}) \mid -}{\neg C \lor \neg F_1} \]
  - NOT_ITE_ELIM2
```
public static final ProofRule NOT_ITE_ELIM2
```
    Boolean – Not ITE elimination version 2 \[ \inferrule{\neg(\ite{C}{F_1}{F_2}) \mid -}{C \lor \neg F_2} \]
  - NOT_AND
```
public static final ProofRule NOT_AND
```
    Boolean – De Morgan – Not And \[ \inferrule{\neg(F_1 \land \dots \land F_n) \mid -}{\neg F_1 \lor \dots \lor \neg F_n} \]
  - CNF_AND_POS
```
public static final ProofRule CNF_AND_POS
```
    Boolean – CNF – And Positive \[ \inferrule{- \mid (F_1 \land \dots \land F_n), i}{\neg (F_1 \land \dots \land F_n) \lor F_i} \]
  - CNF_AND_NEG
```
public static final ProofRule CNF_AND_NEG
```
    Boolean – CNF – And Negative \[ \inferrule{- \mid (F_1 \land \dots \land F_n)}{(F_1 \land \dots \land F_n) \lor \neg F_1 \lor \dots \lor \neg F_n} \]
  - CNF_OR_POS
```
public static final ProofRule CNF_OR_POS
```
    Boolean – CNF – Or Positive \[ \inferrule{- \mid (F_1 \lor \dots \lor F_n)}{\neg(F_1 \lor \dots \lor F_n) \lor F_1 \lor \dots \lor F_n} \]
  - CNF_OR_NEG
```
public static final ProofRule CNF_OR_NEG
```
    Boolean – CNF – Or Negative \[ \inferrule{- \mid (F_1 \lor \dots \lor F_n), i}{(F_1 \lor \dots \lor F_n) \lor \neg F_i} \]
  - CNF_IMPLIES_POS
```
public static final ProofRule CNF_IMPLIES_POS
```
    Boolean – CNF – Implies Positive \[ \inferrule{- \mid F_1 \rightarrow F_2}{\neg(F_1 \rightarrow F_2) \lor \neg F_1 \lor F_2} \]
  - CNF_IMPLIES_NEG1
```
public static final ProofRule CNF_IMPLIES_NEG1
```
    Boolean – CNF – Implies Negative 1 \[ \inferrule{- \mid F_1 \rightarrow F_2}{(F_1 \rightarrow F_2) \lor F_1} \]
  - CNF_IMPLIES_NEG2
```
public static final ProofRule CNF_IMPLIES_NEG2
```
    Boolean – CNF – Implies Negative 2 \[ \inferrule{- \mid F_1 \rightarrow F_2}{(F_1 \rightarrow F_2) \lor \neg F_2} \]
  - CNF_EQUIV_POS1
```
public static final ProofRule CNF_EQUIV_POS1
```
    Boolean – CNF – Equiv Positive 1 \[ \inferrule{- \mid F_1 = F_2}{F_1 \neq F_2 \lor \neg F_1 \lor F_2} \]
  - CNF_EQUIV_POS2
```
public static final ProofRule CNF_EQUIV_POS2
```
    Boolean – CNF – Equiv Positive 2 \[ \inferrule{- \mid F_1 = F_2}{F_1 \neq F_2 \lor F_1 \lor \neg F_2} \]
  - CNF_EQUIV_NEG1
```
public static final ProofRule CNF_EQUIV_NEG1
```
    Boolean – CNF – Equiv Negative 1 \[ \inferrule{- \mid F_1 = F_2}{(F_1 = F_2) \lor F_1 \lor F_2} \]
  - CNF_EQUIV_NEG2
```
public static final ProofRule CNF_EQUIV_NEG2
```
    Boolean – CNF – Equiv Negative 2 \[ \inferrule{- \mid F_1 = F_2}{(F_1 = F_2) \lor \neg F_1 \lor \neg F_2} \]
  - CNF_XOR_POS1
```
public static final ProofRule CNF_XOR_POS1
```
    Boolean – CNF – XOR Positive 1 \[ \inferrule{- \mid F_1 \xor F_2}{\neg(F_1 \xor F_2) \lor F_1 \lor F_2} \]
  - CNF_XOR_POS2
```
public static final ProofRule CNF_XOR_POS2
```
    Boolean – CNF – XOR Positive 2 \[ \inferrule{- \mid F_1 \xor F_2}{\neg(F_1 \xor F_2) \lor \neg F_1 \lor \neg F_2} \]
  - CNF_XOR_NEG1
```
public static final ProofRule CNF_XOR_NEG1
```
    Boolean – CNF – XOR Negative 1 \[ \inferrule{- \mid F_1 \xor F_2}{(F_1 \xor F_2) \lor \neg F_1 \lor F_2} \]
  - CNF_XOR_NEG2
```
public static final ProofRule CNF_XOR_NEG2
```
    Boolean – CNF – XOR Negative 2 \[ \inferrule{- \mid F_1 \xor F_2}{(F_1 \xor F_2) \lor F_1 \lor \neg F_2} \]
  - CNF_ITE_POS1
```
public static final ProofRule CNF_ITE_POS1
```
    Boolean – CNF – ITE Positive 1 \[ \inferrule{- \mid (\ite{C}{F_1}{F_2})}{\neg(\ite{C}{F_1}{F_2}) \lor \neg C \lor F_1} \]
  - CNF_ITE_POS2
```
public static final ProofRule CNF_ITE_POS2
```
    Boolean – CNF – ITE Positive 2 \[ \inferrule{- \mid (\ite{C}{F_1}{F_2})}{\neg(\ite{C}{F_1}{F_2}) \lor C \lor F_2} \]
  - CNF_ITE_POS3
```
public static final ProofRule CNF_ITE_POS3
```
    Boolean – CNF – ITE Positive 3 \[ \inferrule{- \mid (\ite{C}{F_1}{F_2})}{\neg(\ite{C}{F_1}{F_2}) \lor F_1 \lor F_2} \]
  - CNF_ITE_NEG1
```
public static final ProofRule CNF_ITE_NEG1
```
    Boolean – CNF – ITE Negative 1 \[ \inferrule{- \mid (\ite{C}{F_1}{F_2})}{(\ite{C}{F_1}{F_2}) \lor \neg C \lor \neg F_1} \]
  - CNF_ITE_NEG2
```
public static final ProofRule CNF_ITE_NEG2
```
    Boolean – CNF – ITE Negative 2 \[ \inferrule{- \mid (\ite{C}{F_1}{F_2})}{(\ite{C}{F_1}{F_2}) \lor C \lor \neg F_2} \]
  - CNF_ITE_NEG3
```
public static final ProofRule CNF_ITE_NEG3
```
    Boolean – CNF – ITE Negative 3 \[ \inferrule{- \mid (\ite{C}{F_1}{F_2})}{(\ite{C}{F_1}{F_2}) \lor \neg F_1 \lor \neg F_2} \]
  - REFL
```
public static final ProofRule REFL
```
    Equality – Reflexivity \[ \inferrule{-\mid t}{t = t} \]
  - SYMM
```
public static final ProofRule SYMM
```
    Equality – Symmetry \[ \inferrule{t_1 = t_2\mid -}{t_2 = t_1} \] or \[ \inferrule{t_1 \neq t_2\mid -}{t_2 \neq t_1} \]
  - TRANS
```
public static final ProofRule TRANS
```
    Equality – Transitivity \[ \inferrule{t_1=t_2,\dots,t_{n-1}=t_n\mid -}{t_1 = t_n} \]
  - CONG
```
public static final ProofRule CONG
```
    Equality – Congruence \[ \inferrule{t_1=s_1,\dots,t_n=s_n\mid k, f?}{k(f?)(t_1,\dots, t_n) = k(f?)(s_1,\dots, s_n)} \] where $k$ is the application kind. Notice that $f$ must be provided iff $k$ is a parameterized kind, e.g. `cvc5.Kind.APPLY_UF`. The actual node for $k$ is constructible via ProofRuleChecker.mkKindNode. If $k$ is a binder kind (e.g. cvc5.{@link 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 <cvc5.HO_CONG> proof rule.
  - NARY_CONG
```
public static final ProofRule NARY_CONG
```
    Equality – N-ary Congruence \[ \inferrule{t_1=s_1,\dots,t_n=s_n\mid k}{k(t_1,\dots, t_n) = k(s_1,\dots, s_n)} \] where $k$ is the application kind. The actual node for $k$ is constructible via ProofRuleChecker.mkKindNode. This rule is used for kinds that have variadic arity, such as cvc5.Kind.AND, cvc5.Kind.PLUS and so on.
  - TRUE_INTRO
```
public static final ProofRule TRUE_INTRO
```
    Equality – True intro \[ \inferrule{F\mid -}{F = \top} \]
  - TRUE_ELIM
```
public static final ProofRule TRUE_ELIM
```
    Equality – True elim \[ \inferrule{F=\top\mid -}{F} \]
  - FALSE_INTRO
```
public static final ProofRule FALSE_INTRO
```
    Equality – False intro \[ \inferrule{\neg F\mid -}{F = \bot} \]
  - FALSE_ELIM
```
public static final ProofRule FALSE_ELIM
```
    Equality – False elim \[ \inferrule{F=\bot\mid -}{\neg F} \]
  - HO_APP_ENCODE
```
public static final ProofRule HO_APP_ENCODE
```
    Equality – Higher-order application encoding \[ \inferrule{-\mid t}{t= \texttt{TheoryUfRewriter::getHoApplyForApplyUf}(t)} \] For example, this rule concludes $f(x,y) = @(@(f,x),y)$, where $@$ isthe HO_APPLY kind.
  - HO_CONG
```
public static final ProofRule HO_CONG
```
    Equality – Higher-order congruence \[ \inferrule{f=g, t_1=s_1,\dots,t_n=s_n\mid k}{k(f, t_1,\dots, t_n) = k(g, s_1,\dots, s_n)} \] Notice that this rule is only used when the application kind $k$ is either `cvc5.Kind.APPLY_UF` or `cvc5.Kind.HO_APPLY`.
  - BETA_REDUCE
```
public static final ProofRule BETA_REDUCE
```
    Equality – Beta reduction \[ \inferrule{\mid \lambda x_1\dots x_n.\> t, t_1,\dots,t_n} {((\lambda x_1\dots x_n.\> t) t_1 \ldots t_n)=t\{x_1\mapsto t_1,\dots,x_n\mapsto t_n\}} \] The right hand side of the equality in the conclusion is computed using standard substitution via Node.substitute.
  - ARRAYS_READ_OVER_WRITE
```
public static final ProofRule ARRAYS_READ_OVER_WRITE
```
    Arrays – Read over write \[ \inferrule{i_1 \neq i_2\mid \mathit{select}(\mathit{store}(a,i_1,e),i_2)} {\mathit{select}(\mathit{store}(a,i_1,e),i_2) = \mathit{select}(a,i_2)} \]
  - ARRAYS_READ_OVER_WRITE_CONTRA
```
public static final ProofRule ARRAYS_READ_OVER_WRITE_CONTRA
```
    Arrays – Read over write, contrapositive \[ \inferrule{\mathit{select}(\mathit{store}(a,i_2,e),i_1) \neq \mathit{select}(a,i_1)\mid -}{i_1=i_2} \]
  - ARRAYS_READ_OVER_WRITE_1
```
public static final ProofRule ARRAYS_READ_OVER_WRITE_1
```
    Arrays – Read over write 1 \[ \inferrule{-\mid \mathit{select}(\mathit{store}(a,i,e),i)} {\mathit{select}(\mathit{store}(a,i,e),i)=e} \]
  - ARRAYS_EXT
```
public static final ProofRule ARRAYS_EXT
```
    Arrays – Arrays extensionality \[ \inferrule{a \neq b\mid -} {\mathit{select}(a,k)\neq\mathit{select}(b,k)} \] where $k$ is $\texttt{arrays::SkolemCache::getExtIndexSkolem}(a\neq b)$.
  - ARRAYS_EQ_RANGE_EXPAND
```
public static final ProofRule ARRAYS_EQ_RANGE_EXPAND
```
    Arrays – Expansion of array range equality \[ \inferrule{-\mid \mathit{eqrange}(a,b,i,j)} {\mathit{eqrange}(a,b,i,j)= \forall x.\> i \leq x \leq j \rightarrow \mathit{select}(a,x)=\mathit{select}(b,x)} \]
  - MACRO_BV_BITBLAST
```
public static final ProofRule MACRO_BV_BITBLAST
```
    Bit-vectors – (Macro) Bitblast \[ \inferrule{-\mid t}{t = \texttt{bitblast}(t)} \] where bitblast() represents the result of the bit-blasted term as a bit-vector consisting of the output bits of the bit-blasted circuit representation of the term. Terms are bit-blasted according to the strategies defined in theory/bv/bitblast/bitblast_strategies_template.h.
  - BV_BITBLAST_STEP
```
public static final ProofRule BV_BITBLAST_STEP
```
    Bit-vectors – Bitblast bit-vector constant, variable, and terms For constant and variables: \[ \inferrule{-\mid t}{t = \texttt{bitblast}(t)} \] For terms: \[ \inferrule{-\mid k(\texttt{bitblast}(t_1),\dots,\texttt{bitblast}(t_n))} {k(\texttt{bitblast}(t_1),\dots,\texttt{bitblast}(t_n)) = \texttt{bitblast}(t)} \] where $t$ is $k(t_1,\dots,t_n)$.
  - BV_EAGER_ATOM
```
public static final ProofRule BV_EAGER_ATOM
```
    Bit-vectors – Bit-vector eager atom \[ \inferrule{-\mid F}{F = F[0]} \] where $F$ is of kind BITVECTOR_EAGER_ATOM.
  - DT_UNIF
```
public static final ProofRule DT_UNIF
```
    Datatypes – Unification \[ \inferrule{C(t_1,\dots,t_n)= C(s_1,\dots,s_n)\mid i}{t_1 = s_i} \] where $C$ is a constructor.
  - DT_INST
```
public static final ProofRule DT_INST
```
    Datatypes – Instantiation \[ \inferrule{-\mid t,n}{\mathit{is}_C(t) = (t = C(\mathit{sel}_1(t),\dots,\mathit{sel}_n(t)))} \] where $C$ is the $n^{\mathit{th}}$ constructor of the type of t, and $\mathit{is}_C$ is the discriminator (tester) for $C$.
  - DT_COLLAPSE
```
public static final ProofRule DT_COLLAPSE
```
    Datatypes – Collapse \[ \inferrule{-\mid \mathit{sel}_i(C_j(t_1,\dots,t_n))}{ \mathit{sel}_i(C_j(t_1,\dots,t_n)) = r} \] where $C_j$ is a constructor, $r$ is $t_i$ if $\mathit{sel}_i$ is a correctly applied selector, or {@link TypeNode#mkGroundTerm()} of the proper type otherwise. Notice that the use of mkGroundTerm differs from the rewriter which uses mkGroundValue in this case.
  - DT_SPLIT
```
public static final ProofRule DT_SPLIT
```
    Datatypes – Split \[ \inferrule{-\mid t}{\mathit{is}_{C_1}(t)\vee\cdots\vee\mathit{is}_{C_n}(t)} \] where $C_1,\dots,C_n$ are all the constructors of the type of $t$.
  - DT_CLASH
```
public static final ProofRule DT_CLASH
```
    Datatypes – Clash \[ \inferruleSC{\mathit{is}_{C_i}(t), \mathit{is}_{C_j}(t)\mid -}{\bot} {if $i\neq j$} \]
  - SKOLEM_INTRO
```
public static final ProofRule SKOLEM_INTRO
```
    Quantifiers – Skolem introduction \[ \inferrule{-\mid k}{k = t} \] where $t$ is the unpurified form of skolem $k$.
  - SKOLEMIZE
```
public static final ProofRule SKOLEMIZE
```
    Quantifiers – Skolemization \[ \inferrule{\exists x_1\dots x_n.\> F\mid -}{F\sigma} \] or \[ \inferrule{\neg (\forall x_1\dots x_n.\> F)\mid -}{\neg F\sigma} \] where $\sigma$ maps $x_1,\dots,x_n$ to their representative skolems obtained by SkolemManager.mkSkolemize, returned in the skolems argument of that method. The witness terms for the returned skolems can be obtained by SkolemManager.getWitnessForm.
  - INSTANTIATE
```
public static final ProofRule INSTANTIATE
```
    Quantifiers – Instantiation \[ \inferrule{\forall x_1\dots x_n.\> F\mid (t_1 \dots t_n), (id\, (t)?)?} {F\{x_1\mapsto t_1,\dots,x_n\mapsto t_n\}} \] The list of terms to instantiate $(t_1 \dots t_n)$ is provided as an s-expression as the first argument. The optional argument $id$ indicates the inference id that caused the instantiation. The term $t$ indicates an additional term (e.g. the trigger) associated with the instantiation, which depends on the id. If the id has prefix QUANTIFIERS_INST_E_MATCHING, then $t$ is the trigger that generated the instantiation.
  - ALPHA_EQUIV
```
public static final ProofRule ALPHA_EQUIV
```
    Quantifiers – Alpha equivalence \[ \inferruleSC{-\mid F, (y_1 \ldots y_n), (z_1,\dots, z_n)} {F = F\{y_1\mapsto z_1,\dots,y_n\mapsto z_n\}} {if $y_1,\dots,y_n, z_1,\dots,z_n$ are unique bound variables} \] Notice that this rule is correct only when $z_1,\dots,z_n$ are not contained in $FV(F) \setminus \{ y_1,\dots, y_n \}$, where $FV(\varphi)$ are the free variables of $\varphi$. The internal quantifiers proof checker does not currently check that this is the case.
  - CONCAT_EQ
```
public static final ProofRule CONCAT_EQ
```
    Strings – Core rules – Concatenation equality \[ \inferrule{(t_1\cdot\ldots \cdot t_n \cdot t) = (t_1 \cdot\ldots \cdot t_n\cdot s)\mid b}{t = s} \] where $\cdot$ stands for string concatenation and $b$ indicates if the direction is reversed. Notice that $t$ or $s$ may be empty, in which case they are implicit in the concatenation above. For example, if the premise is $x\cdot z = x$, then this rule, with argument $\bot$, concludes $z = ''$. Also note that constants are split, such that for $(\mathsf{'abc'} \cdot x) = (\mathsf{'a'} \cdot y)$, this rule, with argument $\bot$, concludes $(\mathsf{'bc'} \cdot x) = y$. This splitting is done only for constants such that Word.splitConstant returns non-null.
  - CONCAT_UNIFY
```
public static final ProofRule CONCAT_UNIFY
```
    Strings – Core rules – Concatenation unification \[ \inferrule{(t_1\cdot t_2) = (s_1 \cdot s_2),\, \mathit{len}(t_1) = \mathit{len}(s_1)\mid b}{t_1 = s_1} \] where $b$ indicates if the direction is reversed.
  - CONCAT_CONFLICT
```
public static final ProofRule CONCAT_CONFLICT
```
    Strings – Core rules – Concatenation conflict \[ \inferrule{(c_1\cdot t) = (c_2 \cdot s)\mid b}{\bot} \] where $b$ indicates if the direction is reversed, $c_1,\,c_2$ are constants such that $\texttt{Word::splitConstant}(c_1,c_2, \mathit{index},b)$ is null, in other words, neither is a prefix of the other. Note it may be the case that one side of the equality denotes the empty string. This rule is used exclusively for strings.
  - CONCAT_CONFLICT_DEQ
```
public static final ProofRule CONCAT_CONFLICT_DEQ
```
    Strings – Core rules – Concatenation conflict for disequal characters \[ \inferrule{(t_1\cdot t) = (s_1 \cdot s), t_1 \deq s_1 \mid b}{\bot} \] 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.
  - CONCAT_SPLIT
```
public static final ProofRule CONCAT_SPLIT
```
    Strings – Core rules – Concatenation split \[ \inferruleSC{(t_1\cdot t_2) = (s_1 \cdot s_2),\, \mathit{len}(t_1) \neq \mathit{len}(s_1)\mid b}{(t_1 = s_1\cdot r_t) \vee (s_1 = t_1\cdot r_s)}{if $b=\bot$} \] where $r_t$ is $\mathit{skolem}(\mathit{suf}(t_1,\mathit{len}(s_1)))$ and $r_s$ is $\mathit{skolem}(\mathit{suf}(s_1,\mathit{len}(t_1)))$. \[ \inferruleSC{(t_1\cdot t_2) = (s_1 \cdot s_2),\, \mathit{len}(t_1) \neq \mathit{len}(s_1)\mid b}{(t_1 = s_1\cdot r_t) \vee (s_1 = t_1\cdot r_s)}{if $b=\top$} \] where $r_t$ is $\mathit{skolem}(\mathit{pre}(t_2,\mathit{len}(t_2) - \mathit{len}(s_2)))$ and $r_s$ is $\mathit{skolem}(\mathit{pre}(s_2,\mathit{len}(s_2) - \mathit{len}(t_2)))$. Above, $\mathit{suf}(x,n)$ is shorthand for $\mathit{substr}(x,n, \mathit{len}(x) - n)$ and $\mathit{pre}(x,n)$ is shorthand for $\mathit{substr}(x,0,n)$.
  - CONCAT_CSPLIT
```
public static final ProofRule CONCAT_CSPLIT
```
    Strings – Core rules – Concatenation split for constants \[ \inferrule{(t_1\cdot t_2) = (c \cdot s_2),\, \mathit{len}(t_1) \neq 0\mid \bot}{(t_1 = c\cdot r)} \] where $r$ is $\mathit{skolem}(\mathit{suf}(t_1,1))$. Alternatively for the reverse: \[ \inferrule{(t_1\cdot t_2) = (s_1 \cdot c),\, \mathit{len}(t_2) \neq 0\mid \top}{(t_2 = r\cdot c)} \] where $r$ is $\mathit{skolem}(\mathit{pre}(t_2,\mathit{len}(t_2) - 1))$.
  - CONCAT_LPROP
```
public static final ProofRule CONCAT_LPROP
```
    Strings – Core rules – Concatenation length propagation \[ \inferrule{(t_1\cdot t_2) = (s_1 \cdot s_2),\, \mathit{len}(t_1) > \mathit{len}(s_1)\mid \bot}{(t_1 = s_1\cdot r_t)} \] where $r_t$ is $\mathit{skolem}(\mathit{suf}(t_1,\mathit{len}(s_1)))$. Alternatively for the reverse: \[ \inferrule{(t_1\cdot t_2) = (s_1 \cdot s_2),\, \mathit{len}(t_2) > \mathit{len}(s_2)\mid \top}{(t_2 = r_t\cdot s_2)} \] where $r_t$ is $\mathit{skolem}(\mathit{pre}(t_2,\mathit{len}(t_2) - \mathit{len}(s_2)))$.
  - CONCAT_CPROP
```
public static final ProofRule CONCAT_CPROP
```
    Strings – Core rules – Concatenation constant propagation \[ \inferrule{(t_1\cdot w_1\cdot t_2) = (w_2 \cdot s),\, \mathit{len}(t_1) \neq 0\mid \bot}{(t_1 = w_3\cdot r)} \] where $w_1,\,w_2,\,w_3$ are words, $w_3$ is $\mathit{pre}(w_2,p)$, $p$ is $\texttt{Word::overlap}(\mathit{suf}(w_2,1), w_1)$, and $r$ is $\mathit{skolem}(\mathit{suf}(t_1,\mathit{len}(w_3)))$. Note that $\mathit{suf}(w_2,p)$ is the largest suffix of $\mathit{suf}(w_2,1)$ that can contain a prefix of $w_1$; since $t_1$ is non-empty, $w_3$ must therefore be contained in $t_1$. Alternatively for the reverse: \[ \inferrule{(t_1\cdot w_1\cdot t_2) = (s \cdot w_2),\, \mathit{len}(t_2) \neq 0\mid \top}{(t_2 = r\cdot w_3)} \] where $w_1,\,w_2,\,w_3$ are words, $w_3$ is $\mathit{suf}(w_2, \mathit{len}(w_2) - p)$, $p$ is $\texttt{Word::roverlap}(\mathit{pre}(w_2, \mathit{len}(w_2) - 1), w_1)$, and $r$ is $\mathit{skolem}(\mathit{pre}(t_2, \mathit{len}(t_2) - \mathit{len}(w_3)))$. Note that $\mathit{pre}(w_2, \mathit{len}(w_2) - p)$ is the largest prefix of $\mathit{pre}(w_2, \mathit{len}(w_2) - 1)$ that can contain a suffix of $w_1$; since $t_2$ is non-empty, $w_3$ must therefore be contained in $t_2$.
  - STRING_DECOMPOSE
```
public static final ProofRule STRING_DECOMPOSE
```
    Strings – Core rules – String decomposition \[ \inferrule{\mathit{len}(t) \geq n\mid \bot}{t = w_1\cdot w_2 \wedge \mathit{len}(w_1) = n} \] 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 $\mathit{skolem}(\mathit{pre}(t,n)$ and $w_2$ is $\mathit{skolem}(\mathit{suf}(t,n)$.
  - STRING_LENGTH_POS
```
public static final ProofRule STRING_LENGTH_POS
```
    Strings – Core rules – Length positive \[ \inferrule{-\mid t}{(\mathit{len}(t) = 0\wedge t= '')\vee \mathit{len}(t) > 0} \]
  - STRING_LENGTH_NON_EMPTY
```
public static final ProofRule STRING_LENGTH_NON_EMPTY
```
    Strings – Core rules – Length non-empty \[ \inferrule{t\neq ''\mid -}{\mathit{len}(t) \neq 0} \]
  - STRING_REDUCTION
```
public static final ProofRule STRING_REDUCTION
```
    Strings – Extended functions – Reduction \[ \inferrule{-\mid t}{R\wedge t = w} \] where $w$ is $\texttt{strings::StringsPreprocess::reduce}(t, R, \dots)$. In other words, $R$ is the reduction predicate for extended term $t$, and $w$ is $skolem(t)$. Notice that the free variables of $R$ are $w$ and the free variables of $t$.
  - STRING_EAGER_REDUCTION
```
public static final ProofRule STRING_EAGER_REDUCTION
```
    Strings – Extended functions – Eager reduction \[ \inferrule{-\mid t}{R} \] where $R$ is $\texttt{strings::TermRegistry::eagerReduce}(t)$.
  - RE_INTER
```
public static final ProofRule RE_INTER
```
    Strings – Regular expressions – Intersection \[ \inferrule{t\in R_1,\,t\in R_2\mid -}{t\in \mathit{inter}(R_1,R_2)} \]
  - RE_UNFOLD_POS
```
public static final ProofRule RE_UNFOLD_POS
```
    Strings – Regular expressions – Positive Unfold \[ \inferrule{t\in R\mid -}{\texttt{RegExpOpr::reduceRegExpPos}(t\in R)} \] corresponding to the one-step unfolding of the premise.
  - RE_UNFOLD_NEG
```
public static final ProofRule RE_UNFOLD_NEG
```
    Strings – Regular expressions – Negative Unfold \[ \inferrule{t\not\in R\mid -}{\texttt{RegExpOpr::reduceRegExpNeg}(t\not\in R)} \] corresponding to the one-step unfolding of the premise.
  - RE_UNFOLD_NEG_CONCAT_FIXED
```
public static final ProofRule RE_UNFOLD_NEG_CONCAT_FIXED
```
    Strings – Regular expressions – Unfold negative concatenation, fixed \[ \inferrule{t\not\in R\mid -}{\texttt{RegExpOpr::reduceRegExpNegConcatFixed}(t\not\in R,L,i)} \] where $\texttt{RegExpOpr::getRegExpConcatFixed}(t\not\in R, i) = L$, corresponding to the one-step unfolding of the premise, optimized for fixed length of component $i$ of the regular expression concatenation $R$.
  - RE_ELIM
```
public static final ProofRule RE_ELIM
```
    Strings – Regular expressions – Elimination \[ \inferrule{-\mid F,b}{F = \texttt{strings::RegExpElimination::eliminate}(F, b)} \] where $b$ is a Boolean indicating whether we are using aggressive eliminations. Notice this rule concludes $F = F$ if no eliminations are performed for $F$.
  - STRING_CODE_INJ
```
public static final ProofRule STRING_CODE_INJ
```
    Strings – Code points \[ \inferrule{-\mid t,s}{\mathit{to\_code}(t) = -1 \vee \mathit{to\_code}(t) \neq \mathit{to\_code}(s) \vee t\neq s} \]
  - STRING_SEQ_UNIT_INJ
```
public static final ProofRule STRING_SEQ_UNIT_INJ
```
    Strings – Sequence unit \[ \inferrule{\mathit{unit}(x) = \mathit{unit}(y)\mid -}{x = y} \] Also applies to the case where $\mathit{unit}(y)$ is a constant sequence of length one.
  - MACRO_STRING_INFERENCE
```
public static final ProofRule MACRO_STRING_INFERENCE
```
    Strings – (Macro) String inference \[ \inferrule{?\mid F,\mathit{id},\mathit{isRev},\mathit{exp}}{F} \] used to bookkeep an inference that has not yet been converted via $\texttt{strings::InferProofCons::convert}$.
  - MACRO_ARITH_SCALE_SUM_UB
```
public static final ProofRule MACRO_ARITH_SCALE_SUM_UB
```
    Arithmetic – Adding inequalities An arithmetic literal is a term of the form $p \diamond c$ where $\diamond \in \{ <, \leq, =, \geq, > \}$, $p$ a polynomial and $c$ a rational constant. \[ \inferrule{l_1 \dots l_n \mid k_1 \dots k_n}{t_1 \diamond t_2} \] where $k_i \in \mathbb{R}, k_i \neq 0$, $\diamond$ is the fusion of the $\diamond_i$ (flipping each if its $k_i$ is negative) such that $\diamond_i \in \{ <, \leq \}$ (this implies that lower bounds have negative $k_i$ and upper bounds have positive $k_i$), $t_1$ is the sum of the scaled polynomials and $t_2$ is the sum of the scaled constants: \[ t_1 \colon= k_1 \cdot p_1 + \cdots + k_n \cdot p_n t_2 \colon= k_1 \cdot c_1 + \cdots + k_n \cdot c_n \]
  - ARITH_SUM_UB
```
public static final ProofRule ARITH_SUM_UB
```
    Arithmetic – Sum upper bounds \[ \inferrule{P_1 \dots P_n \mid -}{L \diamond R} \] where $P_i$ has the form $L_i \diamond_i R_i$ and $\diamond_i \in \{<, \leq, =\}$. Furthermore $\diamond = <$ if $\diamond_i = <$ for any $i$ and $\diamond = \leq$ otherwise, $L = L_1 + \cdots + L_n$ and $R = R_1 + \cdots + R_n$.
  - INT_TIGHT_UB
```
public static final ProofRule INT_TIGHT_UB
```
    Arithmetic – Tighten strict integer upper bounds \[ \inferrule{i < c \mid -}{i \leq \lfloor c \rfloor} \] where $i$ has integer type.
  - INT_TIGHT_LB
```
public static final ProofRule INT_TIGHT_LB
```
    Arithmetic – Tighten strict integer lower bounds \[ \inferrule{i > c \mid -}{i \geq \lceil c \rceil} \] where $i$ has integer type.
  - ARITH_TRICHOTOMY
```
public static final ProofRule ARITH_TRICHOTOMY
```
    Arithmetic – Trichotomy of the reals \[ \inferrule{A, B \mid -}{C} \] where $\neg A, \neg B, C$ are $x < c, x = c, x > c$ in some order. Note that $\neg$ here denotes arithmetic negation, i.e., flipping $\geq$ to $<$ etc.
  - ARITH_OP_ELIM_AXIOM
```
public static final ProofRule ARITH_OP_ELIM_AXIOM
```
    Arithmetic – Operator elimination \[ \inferrule{- \mid t}{\texttt{arith::OperatorElim::getAxiomFor(t)}} \]
  - ARITH_POLY_NORM
```
public static final ProofRule ARITH_POLY_NORM
```
    Arithmetic – Polynomial normalization \[ \inferrule{- \mid t = s}{t = s} \] where $\texttt{arith::PolyNorm::isArithPolyNorm(t, s)} = \top$. This method normalizes polynomials over arithmetic or bitvectors.
  - ARITH_MULT_SIGN
```
public static final ProofRule ARITH_MULT_SIGN
```
    Arithmetic – Sign inference \[ \inferrule{- \mid f_1 \dots f_k, m}{(f_1 \land \dots \land f_k) \rightarrow m \diamond 0} \] where $f_1 \dots f_k$ are variables compared to zero (less, greater or not equal), $m$ is a monomial from these variables and $\diamond$ is the comparison (less or equal) that results from the signs of the variables. All variables with even exponent in $m$ should be given as not equal to zero while all variables with odd exponent in $m$ should be given as less or greater than zero.
  - ARITH_MULT_POS
```
public static final ProofRule ARITH_MULT_POS
```
    Arithmetic – Multiplication with positive factor \[ \inferrule{- \mid m, l \diamond r}{(m > 0 \land l \diamond r) \rightarrow m \cdot l \diamond m \cdot r} \] where $\diamond$ is a relation symbol.
  - ARITH_MULT_NEG
```
public static final ProofRule ARITH_MULT_NEG
```
    Arithmetic – Multiplication with negative factor \[ \inferrule{- \mid m, l \diamond r}{(m < 0 \land l \diamond r) \rightarrow m \cdot l \diamond_{inv} m \cdot r} \] where $\diamond$ is a relation symbol and $\diamond_{inv}$ the inverted relation symbol.
  - ARITH_MULT_TANGENT
```
public static final ProofRule ARITH_MULT_TANGENT
```
    Arithmetic – Multiplication tangent plane \[ \inferruleSC{- \mid t, x, y, a, b, \sigma}{(t \leq tplane) \leftrightarrow ((x \leq a \land y \geq b) \lor (x \geq a \land y \leq b))}{if $\sigma = -1$} \inferruleSC{- \mid t, x, y, a, b, \sigma}{(t \geq tplane) \leftrightarrow ((x \leq a \land y \leq b) \lor (x \geq a \land y \geq b))}{if $\sigma = 1$} \] where $x,y$ are real terms (variables or extended terms), $t = x \cdot y$ (possibly under rewriting), $a,b$ 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 $(a,b)$.
  - ARITH_TRANS_PI
```
public static final ProofRule ARITH_TRANS_PI
```
    Arithmetic – Transcendentals – Assert bounds on Pi \[ \inferrule{- \mid l, u}{\texttt{real.pi} \geq l \land \texttt{real.pi} \leq u} \] where $l,u$ are valid lower and upper bounds on $\pi$.
  - ARITH_TRANS_EXP_NEG
```
public static final ProofRule ARITH_TRANS_EXP_NEG
```
    Arithmetic – Transcendentals – Exp at negative values \[ \inferrule{- \mid t}{(t < 0) \leftrightarrow (\exp(t) < 1)} \]
  - ARITH_TRANS_EXP_POSITIVITY
```
public static final ProofRule ARITH_TRANS_EXP_POSITIVITY
```
    Arithmetic – Transcendentals – Exp is always positive \[ \inferrule{- \mid t}{\exp(t) > 0} \]
  - ARITH_TRANS_EXP_SUPER_LIN
```
public static final ProofRule ARITH_TRANS_EXP_SUPER_LIN
```
    Arithmetic – Transcendentals – Exp grows super-linearly for positive values \[ \inferrule{- \mid t}{t \leq 0 \lor \exp(t) > t+1} \]
  - ARITH_TRANS_EXP_ZERO
```
public static final ProofRule ARITH_TRANS_EXP_ZERO
```
    Arithmetic – Transcendentals – Exp at zero \[ \inferrule{- \mid t}{(t=0) \leftrightarrow (\exp(t) = 1)} \]
  - ARITH_TRANS_EXP_APPROX_ABOVE_NEG
```
public static final ProofRule ARITH_TRANS_EXP_APPROX_ABOVE_NEG
```
    Arithmetic – Transcendentals – Exp is approximated from above for negative values \[ \inferrule{- \mid d,t,l,u}{(t \geq l \land t \leq u) \rightarrow exp(t) \leq \texttt{secant}(\exp, l, u, t)} \] where $d$ is an even positive number, $t$ an arithmetic term and $l,u$ are lower and upper bounds on $t$. Let $p$ be the $d$'th taylor polynomial at zero (also called the Maclaurin series) of the exponential function. $\texttt{secant}(\exp, l, u, t)$ denotes the secant of $p$ from $(l, \exp(l))$ to $(u, \exp(u))$ evaluated at $t$, calculated as follows: \[ \frac{p(l) - p(u)}{l - u} \cdot (t - l) + p(l) \] The lemma states that if $t$ is between $l$ and $u$, then $\exp(t$ is below the secant of $p$ from $l$ to $u$.
  - ARITH_TRANS_EXP_APPROX_ABOVE_POS
```
public static final ProofRule ARITH_TRANS_EXP_APPROX_ABOVE_POS
```
    Arithmetic – Transcendentals – Exp is approximated from above for positive values \[ \inferrule{- \mid d,t,l,u}{(t \geq l \land t \leq u) \rightarrow exp(t) \leq \texttt{secant-pos}(\exp, l, u, t)} \] where $d$ is an even positive number, $t$ an arithmetic term and $l,u$ are lower and upper bounds on $t$. Let $p^*$ be a modification of the $d$'th taylor polynomial at zero (also called the Maclaurin series) of the exponential function as follows where $p(d-1)$ is the regular Maclaurin series of degree $d-1$: \[ p^* := p(d-1) \cdot \frac{1 + t^n}{n!} \] $\texttt{secant-pos}(\exp, l, u, t)$ denotes the secant of $p$ from $(l, \exp(l))$ to $(u, \exp(u))$ evaluated at $t$, calculated as follows: \[ \frac{p(l) - p(u)}{l - u} \cdot (t - l) + p(l) \] The lemma states that if $t$ is between $l$ and $u$, then $\exp(t$ is below the secant of $p$ from $l$ to $u$.
  - ARITH_TRANS_EXP_APPROX_BELOW
```
public static final ProofRule ARITH_TRANS_EXP_APPROX_BELOW
```
    Arithmetic – Transcendentals – Exp is approximated from below \[ \inferrule{- \mid d,c,t}{t \geq c \rightarrow exp(t) \geq \texttt{maclaurin}(\exp, d, c)} \] where $d$ is an odd positive number, $t$ an arithmetic term and $\texttt{maclaurin}(\exp, d, c)$ is the $d$'th taylor polynomial at zero (also called the Maclaurin series) of the exponential function evaluated at $c$. The Maclaurin series for the exponential function is the following: \[ \exp(x) = \sum_{n=0}^{\infty} \frac{x^n}{n!} \]
  - ARITH_TRANS_SINE_BOUNDS
```
public static final ProofRule ARITH_TRANS_SINE_BOUNDS
```
    Arithmetic – Transcendentals – Sine is always between -1 and 1 \[ \inferrule{- \mid t}{\sin(t) \leq 1 \land \sin(t) \geq -1} \]
  - ARITH_TRANS_SINE_SHIFT
```
public static final ProofRule ARITH_TRANS_SINE_SHIFT
```
    Arithmetic – Transcendentals – Sine is shifted to -pi...pi \[ \inferrule{- \mid x, y, s}{-\pi \leq y \leq \pi \land \sin(y) = \sin(x) \land (\ite{-\pi \leq x \leq \pi}{x = y}{x = y + 2 \pi s})} \] where $x$ is the argument to sine, $y$ is a new real skolem that is $x$ shifted into $-\pi \dots \pi$ and $s$ is a new integer slolem that is the number of phases $y$ is shifted.
  - ARITH_TRANS_SINE_SYMMETRY
```
public static final ProofRule ARITH_TRANS_SINE_SYMMETRY
```
    Arithmetic – Transcendentals – Sine is symmetric with respect to negation of the argument \[ \inferrule{- \mid t}{\sin(t) - \sin(-t) = 0} \]
  - ARITH_TRANS_SINE_TANGENT_ZERO
```
public static final ProofRule ARITH_TRANS_SINE_TANGENT_ZERO
```
    Arithmetic – Transcendentals – Sine is bounded by the tangent at zero .. math. \inferrule{- \mid t}{(t > 0 \rightarrow \sin(t) < t) \land (t < 0 \rightarrow \sin(t) > t)}
  - ARITH_TRANS_SINE_TANGENT_PI
```
public static final ProofRule ARITH_TRANS_SINE_TANGENT_PI
```
    Arithmetic – Transcendentals – Sine is bounded by the tangents at -pi and pi .. math. \inferrule{- \mid t}{(t > -\pi \rightarrow \sin(t) > -\pi - t) \land (t < \pi \rightarrow \sin(t) < \pi - t)}
  - ARITH_TRANS_SINE_APPROX_ABOVE_NEG
```
public static final ProofRule ARITH_TRANS_SINE_APPROX_ABOVE_NEG
```
    Arithmetic – Transcendentals – Sine is approximated from above for negative values \[ \inferrule{- \mid d,t,lb,ub,l,u}{(t \geq lb land t \leq ub) \rightarrow \sin(t) \leq \texttt{secant}(\sin, l, u, t)} \] where $d$ is an even positive number, $t$ an arithmetic term, $lb,ub$ are symbolic lower and upper bounds on $t$ (possibly containing $\pi$) and $l,u$ the evaluated lower and upper bounds on $t$. Let $p$ be the $d$'th taylor polynomial at zero (also called the Maclaurin series) of the sine function. $\texttt{secant}(\sin, l, u, t)$ denotes the secant of $p$ from $(l, \sin(l))$ to $(u, \sin(u))$ evaluated at $t$, calculated as follows: \[ \frac{p(l) - p(u)}{l - u} \cdot (t - l) + p(l) \] The lemma states that if $t$ is between $l$ and $u$, then $\sin(t)$ is below the secant of $p$ from $l$ to $u$.
  - ARITH_TRANS_SINE_APPROX_ABOVE_POS
```
public static final ProofRule ARITH_TRANS_SINE_APPROX_ABOVE_POS
```
    Arithmetic – Transcendentals – Sine is approximated from above for positive values \[ \inferrule{- \mid d,t,c,lb,ub}{(t \geq lb land t \leq ub) \rightarrow \sin(t) \leq \texttt{upper}(\sin, c)} \] where $d$ is an even positive number, $t$ an arithmetic term, $c$ an arithmetic constant and $lb,ub$ are symbolic lower and upper bounds on $t$ (possibly containing $\pi$). Let $p$ be the $d$'th taylor polynomial at zero (also called the Maclaurin series) of the sine function. $\texttt{upper}(\sin, c)$ denotes the upper bound on $\sin(c)$ given by $p$ and $lb,up$ such that $\sin(t)$ is the maximum of the sine function on $(lb,ub)$.
  - ARITH_TRANS_SINE_APPROX_BELOW_NEG
```
public static final ProofRule ARITH_TRANS_SINE_APPROX_BELOW_NEG
```
    Arithmetic – Transcendentals – Sine is approximated from below for negative values \[ \inferrule{- \mid d,t,c,lb,ub}{(t \geq lb land t \leq ub) \rightarrow \sin(t) \geq \texttt{lower}(\sin, c)} \] where $d$ is an even positive number, $t$ an arithmetic term, $c$ an arithmetic constant and $lb,ub$ are symbolic lower and upper bounds on $t$ (possibly containing $\pi$). Let $p$ be the $d$'th taylor polynomial at zero (also called the Maclaurin series) of the sine function. $\texttt{lower}(\sin, c)$ denotes the lower bound on $\sin(c)$ given by $p$ and $lb,up$ such that $\sin(t)$ is the minimum of the sine function on $(lb,ub)$.
  - ARITH_TRANS_SINE_APPROX_BELOW_POS
```
public static final ProofRule ARITH_TRANS_SINE_APPROX_BELOW_POS
```
    Arithmetic – Transcendentals – Sine is approximated from below for positive values \[ \inferrule{- \mid d,t,lb,ub,l,u}{(t \geq lb land t \leq ub) \rightarrow \sin(t) \geq \texttt{secant}(\sin, l, u, t)} \] where $d$ is an even positive number, $t$ an arithmetic term, $lb,ub$ are symbolic lower and upper bounds on $t$ (possibly containing $\pi$) and $l,u$ the evaluated lower and upper bounds on $t$. Let $p$ be the $d$'th taylor polynomial at zero (also called the Maclaurin series) of the sine function. $\texttt{secant}(\sin, l, u, t)$ denotes the secant of $p$ from $(l, \sin(l))$ to $(u, \sin(u))$ evaluated at $t$, calculated as follows: \[ \frac{p(l) - p(u)}{l - u} \cdot (t - l) + p(l) \] The lemma states that if $t$ is between $l$ and $u$, then $\sin(t)$ is above the secant of $p$ from $l$ to $u$.
  - ARITH_NL_COVERING_DIRECT
```
public static final ProofRule ARITH_NL_COVERING_DIRECT
```
    Arithmetic – Coverings – Direct conflict We use $\texttt{IRP}_k(poly)$ for an IndexedRootPredicate that is defined as the $k$'th root of the polynomial $poly$. Note that $poly$ may not be univariate; in this case, the value of $\texttt{IRP}_k(poly)$ can only be calculated with respect to a (partial) model for all but one variable of $poly$. A formula $\texttt{Interval}(x_i)$ describes that a variable $x_i$ is within a particular interval whose bounds are given as IRPs. It is either an open interval or a point interval: \[ \texttt{IRP}_k(poly) < x_i < \texttt{IRP}_k(poly) x_i = \texttt{IRP}_k(poly) \] A formula $\texttt{Cell}(x_1 \dots x_i)$ describes a portion of the real space in the following form: \[ \texttt{Interval}(x_1) \land \dots \land \texttt{Interval}(x_i) \] A cell can also be empty (for $i = 0$). A formula $\texttt{Covering}(x_i)$ is a set of intervals, implying that $x_i$ can be in neither of these intervals. To be a covering (of the real line), the union of these intervals should be the real numbers. \[ \inferrule{\texttt{Cell}, A \mid -}{\bot} \] A direct interval is generated from an assumption $A$ (in variables $x_1 \dots x_i$) over a $\texttt{Cell}(x_1 \dots x_i)$. It derives that $A$ evaluates to false over the cell. In the actual algorithm, it means that $x_i$ can not be in the topmost interval of the cell.
  - ARITH_NL_COVERING_RECURSIVE
```
public static final ProofRule ARITH_NL_COVERING_RECURSIVE
```
    Arithmetic – Coverings – Recursive interval See ARITH_NL_COVERING_DIRECT <cvc5.ARITH_NL_COVERING_DIRECT> for the necessary definitions. \[ \inferrule{\texttt{Cell}, \texttt{Covering} \mid -}{\bot} \] A recursive interval is generated from $\texttt{Covering}(x_i)$ over $\texttt{Cell}(x_1 \dots x_{i-1})$. It generates the conclusion that no $x_i$ exists that extends the cell and satisfies all assumptions.
  - LFSC_RULE
```
public static final ProofRule LFSC_RULE
```
    External – LFSC Place holder for LFSC rules. \[ \inferrule{P_1, \dots, P_n\mid \texttt{id}, Q, A_1,\dots, A_m}{Q} \] Note that the premises and arguments are arbitrary. It's expected that $\texttt{id}$ refer to a proof rule in the external LFSC calculus.
  - ALETHE_RULE
```
public static final ProofRule ALETHE_RULE
```
    External – Alethe Place holder for Alethe rules. \[ \inferrule{P_1, \dots, P_n\mid \texttt{id}, Q, Q', A_1,\dots, A_m}{Q} \] Note that the premises and arguments are arbitrary. It's expected that $\texttt{id}$ refer to a proof rule in the external Alethe calculus, and that $Q'$ be the representation of Q to be printed by the Alethe printer.
  - UNKNOWN
```
public static final ProofRule UNKNOWN
```
    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.
- Method Detail
  - values
```
public static ProofRule[] values()
```
    Returns an array containing the constants of this enum type, in the order they are declared. This method may be used to iterate over the constants as follows:
    for (ProofRule c : ProofRule.values()) System.out.println(c);
    Returns:
    
    an array containing the constants of this enum type, in the order they are declared
  - valueOf
```
public static ProofRule valueOf(java.lang.String name)
```
    Returns the enum constant of this type with the specified name. The string must match exactly an identifier used to declare an enum constant in this type. (Extraneous whitespace characters are not permitted.)
    
    Parameters:
    
    name - the name of the enum constant to be returned.
    
    Returns:
    
    the enum constant with the specified name
    
    Throws:
    
    java.lang.IllegalArgumentException - if this enum type has no constant with the specified name
    
    java.lang.NullPointerException - if the argument is null
  - fromInt
```
public static ProofRule fromInt(int value)
                         throws CVC5ApiException
```
    Throws:
    
    CVC5ApiException
  - getValue
```
public int getValue()
```

Enum ProofRule

Enum Constant Summary

Method Summary

Methods inherited from class java.lang.Enum

Methods inherited from class java.lang.Object

Enum Constant Detail

ASSUME

SCOPE

SUBS

MACRO_REWRITE

EVALUATE

MACRO_SR_EQ_INTRO

MACRO_SR_PRED_INTRO

MACRO_SR_PRED_ELIM

MACRO_SR_PRED_TRANSFORM

ENCODE_PRED_TRANSFORM

DSL_REWRITE

ANNOTATION

REMOVE_TERM_FORMULA_AXIOM

TRUST

TRUST_THEORY_REWRITE

SAT_REFUTATION

RESOLUTION

CHAIN_RESOLUTION

FACTORING

REORDERING

MACRO_RESOLUTION

MACRO_RESOLUTION_TRUST

SPLIT

EQ_RESOLVE

MODUS_PONENS

NOT_NOT_ELIM

CONTRA

AND_ELIM

AND_INTRO

NOT_OR_ELIM

IMPLIES_ELIM

NOT_IMPLIES_ELIM1

NOT_IMPLIES_ELIM2

EQUIV_ELIM1

EQUIV_ELIM2

NOT_EQUIV_ELIM1

NOT_EQUIV_ELIM2

XOR_ELIM1

XOR_ELIM2

NOT_XOR_ELIM1

NOT_XOR_ELIM2

ITE_ELIM1

ITE_ELIM2

NOT_ITE_ELIM1

NOT_ITE_ELIM2

NOT_AND

CNF_AND_POS

CNF_AND_NEG

CNF_OR_POS

CNF_OR_NEG

CNF_IMPLIES_POS

CNF_IMPLIES_NEG1

CNF_IMPLIES_NEG2

CNF_EQUIV_POS1

CNF_EQUIV_POS2

CNF_EQUIV_NEG1

CNF_EQUIV_NEG2

CNF_XOR_POS1

CNF_XOR_POS2

CNF_XOR_NEG1

CNF_XOR_NEG2

CNF_ITE_POS1

CNF_ITE_POS2

CNF_ITE_POS3

CNF_ITE_NEG1

CNF_ITE_NEG2

CNF_ITE_NEG3

REFL

SYMM

TRANS

CONG

NARY_CONG

TRUE_INTRO

TRUE_ELIM