# Difference between revisions of "All Rewrite Rules"

This page groups together all the rewrite rules implemented (or planned for implementation) in the Rodin prover. The rules themselves can be found in the following locations (for editing purposes):

Rules that are marked with a * in the first column are implemented in the latest version of Rodin. Rules without a * are planned to be implemented in future versions. Other conventions used in these tables are described in The_Proving_Perspective_(Rodin_User_Manual)#Rewrite_Rules.

Name Rule Side Condition A/M
*
SIMP_SPECIAL_AND_BTRUE
$P \land \ldots \land \btrue \land \ldots \land Q \;\;\defi\;\; P \land \ldots \land Q$ A
*
SIMP_SPECIAL_AND_BFALSE
$P \land \ldots \land \bfalse \land \ldots \land Q \;\;\defi\;\; \bfalse$ A
*
SIMP_MULTI_AND
$P \land \ldots \land Q \land \ldots \land Q \land \ldots \land R \;\;\defi\;\; P \land \ldots \land Q \land \ldots \land R$ A
*
SIMP_MULTI_AND_NOT
$P \land \ldots \land Q \land \ldots \land \lnot\, Q \land \ldots \land R \;\;\defi\;\; \bfalse$ A
*
SIMP_SPECIAL_OR_BTRUE
$P \lor \ldots \lor \btrue \lor \ldots \lor Q \;\;\defi\;\; \btrue$ A
*
SIMP_SPECIAL_OR_BFALSE
$P \lor \ldots \lor \bfalse \lor \ldots \lor Q \;\;\defi\;\; P \lor \ldots \lor Q$ A
*
SIMP_MULTI_OR
$P \lor \ldots \lor Q \lor \ldots \lor Q \lor \ldots \lor R \;\;\defi\;\; P \lor \ldots \lor Q \lor \ldots \lor R$ A
*
SIMP_MULTI_OR_NOT
$P \lor \ldots \lor Q \lor \ldots \lor \lnot\, Q \land \ldots \land R \;\;\defi\;\; \btrue$ A
*
SIMP_SPECIAL_IMP_BTRUE_R
$P \limp \btrue \;\;\defi\;\; \btrue$ A
*
SIMP_SPECIAL_IMP_BTRUE_L
$\btrue \limp P \;\;\defi\;\; P$ A
*
SIMP_SPECIAL_IMP_BFALSE_R
$P \limp \bfalse \;\;\defi\;\; \lnot\, P$ A
*
SIMP_SPECIAL_IMP_BFALSE_L
$\bfalse \limp P \;\;\defi\;\; \btrue$ A
*
SIMP_MULTI_IMP
$P \limp P \;\;\defi\;\; \btrue$ A
*
SIMP_MULTI_IMP_NOT_L
$\lnot P\limp P\;\;\defi\;\; P$ A
*
SIMP_MULTI_IMP_NOT_R
$P\limp\lnot P\;\;\defi\;\;\lnot P$ A
*
SIMP_MULTI_IMP_AND
$P \land \ldots \land Q \land \ldots \land R \limp Q \;\;\defi\;\; \btrue$ A
*
SIMP_MULTI_IMP_AND_NOT_R
$P \land \ldots \land Q \land \ldots \land R \limp \lnot\, Q \;\;\defi\;\; \lnot\,(P \land \ldots \land Q \land \ldots \land R)$ A
*
SIMP_MULTI_IMP_AND_NOT_L
$P \land \ldots \land \lnot\, Q \land \ldots \land R \limp Q \;\;\defi\;\; \lnot\,(P \land \ldots \land \lnot\, Q \land \ldots \land R)$ A
*
SIMP_MULTI_EQV
$P \leqv P \;\;\defi\;\; \btrue$ A
*
SIMP_MULTI_EQV_NOT
$P \leqv \lnot\, P \;\;\defi\;\; \bfalse$ A
*
SIMP_SPECIAL_NOT_BTRUE
$\lnot\, \btrue \;\;\defi\;\; \bfalse$ A
*
SIMP_SPECIAL_NOT_BFALSE
$\lnot\, \bfalse \;\;\defi\;\; \btrue$ A
*
SIMP_NOT_NOT
$\lnot\, \lnot\, P \;\;\defi\;\; P$ AM
*
SIMP_NOTEQUAL
$E \neq F \;\;\defi\;\; \lnot\, E = F$ A
*
SIMP_NOTIN
$E \notin F \;\;\defi\;\; \lnot\, E \in F$ A
*
SIMP_NOTSUBSET
$E \not\subset F \;\;\defi\;\; \lnot\, E \subset F$ A
*
SIMP_NOTSUBSETEQ
$E \not\subseteq F \;\;\defi\;\; \lnot\, E \subseteq F$ A
*
SIMP_NOT_LE
$\lnot\, a \leq b \;\;\defi\;\; a > b$ A
*
SIMP_NOT_GE
$\lnot\, a \geq b \;\;\defi\;\; a < b$ A
*
SIMP_NOT_LT
$\lnot\, a < b \;\;\defi\;\; a \geq b$ A
*
SIMP_NOT_GT
$\lnot\, a > b \;\;\defi\;\; a \leq b$ A
*
SIMP_SPECIAL_NOT_EQUAL_FALSE_R
$\lnot\, (E = \False ) \;\;\defi\;\; (E = \True )$ A
*
SIMP_SPECIAL_NOT_EQUAL_FALSE_L
$\lnot\, (\False = E) \;\;\defi\;\; (\True = E)$ A
*
SIMP_SPECIAL_NOT_EQUAL_TRUE_R
$\lnot\, (E = \True ) \;\;\defi\;\; (E = \False )$ A
*
SIMP_SPECIAL_NOT_EQUAL_TRUE_L
$\lnot\, (\True = E) \;\;\defi\;\; (\False = E)$ A
*
SIMP_FORALL_AND
$\forall x \qdot P \land Q \;\;\defi\;\; (\forall x \qdot P) \land (\forall x \qdot Q)$ A
*
SIMP_EXISTS_OR
$\exists x \qdot P \lor Q \;\;\defi\;\; (\exists x \qdot P) \lor (\exists x \qdot Q)$ A
*
SIMP_EXISTS_IMP
$\exists x\qdot P\limp Q\;\;\defi\;\;(\forall x\qdot P)\limp(\exists x\qdot Q)$ A
*
SIMP_FORALL
$\forall \ldots ,z,\ldots \qdot P(z) \;\;\defi\;\; \forall z \qdot P(z)$ Quantified identifiers other than $z$ do not occur in $P$ A
*
SIMP_EXISTS
$\exists \ldots ,z,\ldots \qdot P(z) \;\;\defi\;\; \exists z \qdot P(z)$ Quantified identifiers other than $z$ do not occur in $P$ A
*
SIMP_MULTI_EQUAL
$E = E \;\;\defi\;\; \btrue$ A
*
SIMP_MULTI_NOTEQUAL
$E \neq E \;\;\defi\;\; \bfalse$ A
*
SIMP_EQUAL_MAPSTO
$E \mapsto F = G \mapsto H \;\;\defi\;\; E = G \land F = H$ A
*
SIMP_EQUAL_SING
$\{ E\} = \{ F\} \;\;\defi\;\; E = F$ A
*
SIMP_SPECIAL_EQUAL_TRUE
$\True = \False \;\;\defi\;\; \bfalse$ A
*
SIMP_TYPE_SUBSETEQ
$S \subseteq \mathit{Ty} \;\;\defi\;\; \btrue$ where $\mathit{Ty}$ is a type expression A
*
SIMP_SUBSETEQ_SING
$\{ E\} \subseteq S \;\;\defi\;\; E \in S$ where $E$ is a single expression A
*
SIMP_SPECIAL_SUBSETEQ
$\emptyset \subseteq S \;\;\defi\;\; \btrue$ A
*
SIMP_MULTI_SUBSETEQ
$S \subseteq S \;\;\defi\;\; \btrue$ A
*
SIMP_SUBSETEQ_BUNION
$S \subseteq A \bunion \ldots \bunion S \bunion \ldots \bunion B \;\;\defi\;\; \btrue$ A
*
SIMP_SUBSETEQ_BINTER
$A \binter \ldots \binter S \binter \ldots \binter B \subseteq S \;\;\defi\;\; \btrue$ A
*
DERIV_SUBSETEQ_BUNION
$A \bunion \ldots \bunion B \subseteq S \;\;\defi\;\; A \subseteq S \land \ldots \land B \subseteq S$ A
*
DERIV_SUBSETEQ_BINTER
$S \subseteq A \binter \ldots \binter B \;\;\defi\;\; S \subseteq A \land \ldots \land S \subseteq B$ A
*
SIMP_SPECIAL_IN
$E \in \emptyset \;\;\defi\;\; \bfalse$ A
*
SIMP_MULTI_IN
$B \in \{ A, \ldots , B, \ldots , C\} \;\;\defi\;\; \btrue$ A
*
SIMP_IN_SING
$E \in \{ F\} \;\;\defi\;\; E = F$ A
*
SIMP_MULTI_SETENUM
$\{ A, \ldots , B, \ldots , B, \ldots , C\} \;\;\defi\;\; \{ A, \ldots , B, \ldots , C\}$ A
*
SIMP_SPECIAL_BINTER
$S \binter \ldots \binter \emptyset \binter \ldots \binter T \;\;\defi\;\; \emptyset$ A
*
SIMP_TYPE_BINTER
$S \binter \ldots \binter \mathit{Ty} \binter \ldots \binter T \;\;\defi\;\; S \binter \ldots \binter T$ where $\mathit{Ty}$ is a type expression A
*
SIMP_MULTI_BINTER
$S \binter \ldots \binter T \binter \ldots \binter T \binter \ldots \binter U \;\;\defi\;\; S \binter \ldots \binter T \binter \ldots \binter U$ A
*
SIMP_MULTI_EQUAL_BINTER
$S \binter \ldots \binter T \binter \ldots \binter U = T \;\;\defi\;\; T \subseteq S \binter \ldots \binter U$ A
*
SIMP_SPECIAL_BUNION
$S \bunion \ldots \bunion \emptyset \bunion \ldots \bunion T \;\;\defi\;\; S \bunion \ldots \bunion T$ A
*
SIMP_TYPE_BUNION
$S \bunion \ldots \bunion \mathit{Ty} \bunion \ldots \bunion T \;\;\defi\;\; \mathit{Ty}$ where $\mathit{Ty}$ is a type expression A
*
SIMP_MULTI_BUNION
$S \bunion \ldots \bunion T \bunion \ldots \bunion T \bunion \ldots \bunion U \;\;\defi\;\; S \bunion \ldots \bunion T \bunion \ldots \bunion U$ A
*
SIMP_MULTI_EQUAL_BUNION
$S \bunion \ldots \bunion T \bunion \ldots \bunion U = T \;\;\defi\;\; S \bunion \ldots \bunion U \subseteq T$ A
*
SIMP_MULTI_SETMINUS
$S \setminus S \;\;\defi\;\; \emptyset$ A
*
SIMP_SPECIAL_SETMINUS_R
$S \setminus \emptyset \;\;\defi\;\; S$ A
*
SIMP_SPECIAL_SETMINUS_L
$\emptyset \setminus S \;\;\defi\;\; \emptyset$ A
*
SIMP_TYPE_SETMINUS
$S \setminus \mathit{Ty} \;\;\defi\;\; \emptyset$ where $\mathit{Ty}$ is a type expression A
*
SIMP_TYPE_SETMINUS_SETMINUS
$\mathit{Ty} \setminus (\mathit{Ty} \setminus S) \;\;\defi\;\; S$ where $\mathit{Ty}$ is a type expression A
*
SIMP_KUNION_POW
$\union (\pow (S)) \;\;\defi\;\; S$ A
*
SIMP_KUNION_POW1
$\union (\pown (S)) \;\;\defi\;\; S$ A
*
SIMP_SPECIAL_KUNION
$\union (\{ \emptyset \} ) \;\;\defi\;\; \emptyset$ A
*
SIMP_SPECIAL_QUNION
$\Union x\qdot \bfalse \mid E \;\;\defi\;\; \emptyset$ A
*
SIMP_SPECIAL_KINTER
$\inter (\{ \emptyset \} ) \;\;\defi\;\; \emptyset$ A
*
SIMP_KINTER_POW
$\inter (\pow (S)) \;\;\defi\;\; \emptyset$ A
*
SIMP_SPECIAL_POW
$\pow (\emptyset ) \;\;\defi\;\; \{ \emptyset \}$ A
*
SIMP_SPECIAL_POW1
$\pown (\emptyset ) \;\;\defi\;\; \emptyset$ A
*
SIMP_SPECIAL_CPROD_R
$S \cprod \emptyset \;\;\defi\;\; \emptyset$ A
*
SIMP_SPECIAL_CPROD_L
$\emptyset \cprod S \;\;\defi\;\; \emptyset$ A
SIMP_COMPSET_EQUAL
$\{ x, y \qdot x = E(y) \land P(y) \mid F(x, y) \} \;\;\defi\;\; \{ y \qdot P(y) \mid F(E(y), y) \}$ where $x$ non free in $E$ and non free in $P$ A
*
SIMP_COMPSET_IN
$\{ x \qdot x \in S \mid x \} \;\;\defi\;\; S$ where $x$ non free in $S$ A
*
SIMP_COMPSET_SUBSETEQ
$\{ x \qdot x \subseteq S \mid x \} \;\;\defi\;\; \pow (S)$ where $x$ non free in $S$ A
*
SIMP_SPECIAL_COMPSET_BFALSE
$\{ x \qdot \bfalse \mid x \} \;\;\defi\;\; \emptyset$ A
*
SIMP_SPECIAL_COMPSET_BTRUE
$\{ x \qdot \btrue \mid E \} \;\;\defi\;\; \mathit{Ty}$ where the type of $E$ is $\mathit{Ty}$ and $E$ is a maplet combination of locally-bound, pairwise-distinct bound identifiers A
*
SIMP_SUBSETEQ_COMPSET_L
$\{ x \qdot P(x) \mid E(x) \} \subseteq S \;\;\defi\;\; \forall y\qdot P(y) \limp E(y) \in S$ where $y$ is fresh A
*
SIMP_IN_COMPSET
$F \in \{ x,y,\ldots \qdot P(x,y,\ldots) \mid E(x,y,\ldots) \} \;\;\defi\;\; \exists x,y,\ldots \qdot P(x,y,\ldots) \land E(x,y,\ldots) = F$ where $x$, $y$, $\ldots$ are not free in $F$ A
*
SIMP_IN_COMPSET_ONEPOINT
$E \in \{ x \qdot P(x) \mid x \} \;\;\defi\;\; P(E)$ Equivalent to general simplification followed by One Point Rule application with the last conjunct predicate A
SIMP_SUBSETEQ_COMPSET_R
$S \subseteq \{ x \qdot P(x) \mid x \} \;\;\defi\;\; \forall y\qdot y \in S \limp P(y)$ where $y$ non free in $S, \{ x \qdot P(x) \mid x \}$ M
*
SIMP_SPECIAL_OVERL
$r \ovl \ldots \ovl \emptyset \ovl \ldots \ovl s \;\;\defi\;\; r \ovl \ldots \ovl s$ A
*
SIMP_SPECIAL_KBOOL_BTRUE
$\bool (\btrue ) \;\;\defi\;\; \True$ A
*
SIMP_SPECIAL_KBOOL_BFALSE
$\bool (\bfalse ) \;\;\defi\;\; \False$ A
DISTRI_SUBSETEQ_BUNION_SING
$S \bunion \{ F\} \subseteq T \;\;\defi\;\; S \subseteq T \land F \in T$ where $F$ is a single expression M
*
DEF_FINITE
$\finite(S) \;\;\defi\;\; \exists n,f\qdot f\in 1\upto n \tbij S$ M
*
SIMP_SPECIAL_FINITE
$\finite (\emptyset ) \;\;\defi\;\; \btrue$ A
*
SIMP_FINITE_SETENUM
$\finite (\{ a, \ldots , b\} ) \;\;\defi\;\; \btrue$ A
*
SIMP_FINITE_BUNION
$\finite (S \bunion T) \;\;\defi\;\; \finite (S) \land \finite (T)$ A
SIMP_FINITE_UNION
$\finite(\union(S)) \;\;\defi\;\; \finite(S)\;\land\;(\forall x\qdot x\in S\limp finite(x))$ M
SIMP_FINITE_QUNION
$\finite(\Union x\qdot P\mid E) \;\;\defi\;\; \finite(\{x\qdot P\mid E\})\;\land\;(\forall x\qdot P\limp finite(E))$ M
*
SIMP_FINITE_POW
$\finite (\pow (S)) \;\;\defi\;\; \finite (S)$ A
*
DERIV_FINITE_CPROD
$\finite (S \cprod T) \;\;\defi\;\; S = \emptyset \lor T = \emptyset \lor (\finite (S) \land \finite (T))$ A
*
SIMP_FINITE_CONVERSE
$\finite (r^{-1} ) \;\;\defi\;\; \finite (r)$ A
*
SIMP_FINITE_UPTO
$\finite (a \upto b) \;\;\defi\;\; \btrue$ A
*
SIMP_FINITE_ID
$\finite (\id) \;\;\defi\;\; \finite (S)$ where $\id$ has type $\pow(S \cprod S)$ A
*
SIMP_FINITE_ID_DOMRES
$\finite (E \domres \id) \;\;\defi\;\; \finite (E)$ A
*
SIMP_FINITE_PRJ1
$\finite (\prjone) \;\;\defi\;\; \finite (S \cprod T)$ where $\prjone$ has type $\pow(S \cprod T \cprod S)$ A
*
SIMP_FINITE_PRJ2
$\finite (\prjtwo) \;\;\defi\;\; \finite (S \cprod T)$ where $\prjtwo$ has type $\pow(S \cprod T \cprod T)$ A
*
SIMP_FINITE_PRJ1_DOMRES
$\finite (E \domres \prjone) \;\;\defi\;\; \finite (E)$ A
*
SIMP_FINITE_PRJ2_DOMRES
$\finite (E \domres \prjtwo) \;\;\defi\;\; \finite (E)$ A
*
SIMP_FINITE_NATURAL
$\finite (\nat ) \;\;\defi\;\; \bfalse$ A
*
SIMP_FINITE_NATURAL1
$\finite (\natn ) \;\;\defi\;\; \bfalse$ A
*
SIMP_FINITE_INTEGER
$\finite (\intg ) \;\;\defi\;\; \bfalse$ A
*
SIMP_FINITE_BOOL
$\finite (\Bool ) \;\;\defi\;\; \btrue$ A
*
SIMP_FINITE_LAMBDA
$\finite(\{x\qdot P\mid E\mapsto F\}) \;\;\defi\;\; \finite(\{x\qdot P\mid E\} )$ where $E$ is a maplet combination of bound identifiers and expressions that are not bound by the comprehension set (i.e., $E$ is syntactically injective) and all identifiers bound by the comprehension set that occur in $F$ also occur in $E$ A
*
SIMP_TYPE_IN
$t \in \mathit{Ty} \;\;\defi\;\; \btrue$ where $\mathit{Ty}$ is a type expression A
*
SIMP_SPECIAL_EQV_BTRUE
$P \leqv \btrue \;\;\defi\;\; P$ A
*
SIMP_SPECIAL_EQV_BFALSE
$P \leqv \bfalse \;\;\defi\;\; \lnot\, P$ A
*
DEF_SUBSET
$A \subset B \;\;\defi\;\; A \subseteq B \land \lnot A = B$ A
*
SIMP_SPECIAL_SUBSET_R
$S \subset \emptyset \;\;\defi\;\; \bfalse$ A
*
SIMP_SPECIAL_SUBSET_L
$\emptyset\subset S \;\;\defi\;\; \lnot\; S = \emptyset$ A
*
SIMP_TYPE_SUBSET_L
$S \subset \mathit{Ty} \;\;\defi\;\; S \neq \mathit{Ty}$ where $\mathit{Ty}$ is a type expression A
*
SIMP_MULTI_SUBSET
$S \subset S \;\;\defi\;\; \bfalse$ A
*
SIMP_EQUAL_CONSTR
$\operatorname{constr} (a_1, \ldots, a_n) = \operatorname{constr} (b_1, \ldots, b_n) \;\;\defi\;\; a_1 = b_1 \land \ldots \land a_n = b_n$ where $\operatorname{constr}$ is a datatype constructor A
*
SIMP_EQUAL_CONSTR_DIFF
$\operatorname{constr_1} (\ldots) = \operatorname{constr_2} (\ldots) \;\;\defi\;\; \bfalse$ where $\operatorname{constr_1}$ and $\operatorname{constr_2}$ are different datatype constructors A
*
SIMP_DESTR_CONSTR
$\operatorname{destr} (\operatorname{constr} (a_1, \ldots, a_n)) \;\;\defi\;\; a_i$ where $\operatorname{destr}$ is the datatype destructor for the i-th argument of datatype constructor $\operatorname{constr}$ A
*
DISTRI_AND_OR
$P \land (Q \lor R) \;\;\defi\;\; (P \land Q) \lor (P \land R)$ M
*
DISTRI_OR_AND
$P \lor (Q \land R) \;\;\defi\;\; (P \lor Q) \land (P \lor R)$ M
*
DEF_OR
$P \lor Q \lor \ldots \lor R \;\;\defi\;\; \lnot\, P \limp (Q \lor \ldots \lor R)$ M
*
DERIV_IMP
$P \limp Q \;\;\defi\;\; \lnot\, Q \limp \lnot\, P$ M
*
DERIV_IMP_IMP
$P \limp (Q \limp R) \;\;\defi\;\; P \land Q \limp R$ M
*
DISTRI_IMP_AND
$P \limp (Q \land R) \;\;\defi\;\; (P \limp Q) \land (P \limp R)$ M
*
DISTRI_IMP_OR
$(P \lor Q) \limp R \;\;\defi\;\; (P \limp R) \land (Q \limp R)$ M
*
DEF_EQV
$P \leqv Q \;\;\defi\;\; (P \limp Q) \land (Q \limp P)$ M
*
DISTRI_NOT_AND
$\lnot\,(P \land Q) \;\;\defi\;\; \lnot\, P \lor \lnot\, Q$ M
*
DISTRI_NOT_OR
$\lnot\,(P \lor Q) \;\;\defi\;\; \lnot\, P \land \lnot\, Q$ M
*
DERIV_NOT_IMP
$\lnot\,(P \limp Q) \;\;\defi\;\; P \land \lnot\, Q$ M
*
DERIV_NOT_FORALL
$\lnot\, \forall x \qdot P \;\;\defi\;\; \exists x \qdot \lnot\, P$ M
*
DERIV_NOT_EXISTS
$\lnot\, \exists x \qdot P \;\;\defi\;\; \forall x \qdot \lnot\, P$ M
*
DEF_IN_MAPSTO
$E \mapsto F \in S \cprod T \;\;\defi\;\; E \in S \land F \in T$ AM
*
DEF_IN_POW
$E \in \pow (S) \;\;\defi\;\; E \subseteq S$ M
*
DEF_IN_POW1
$E \in \pown (S) \;\;\defi\;\; E \in \pow (S) \land S \neq \emptyset$ M
*
DEF_SUBSETEQ
$S \subseteq T \;\;\defi\;\; \forall x \qdot x \in S \limp x \in T$ where $x$ is not free in $S$ or $T$ M
*
DEF_IN_BUNION
$E \in S \bunion T \;\;\defi\;\; E \in S \lor E \in T$ M
*
DEF_IN_BINTER
$E \in S \binter T \;\;\defi\;\; E \in S \land E \in T$ M
*
DEF_IN_SETMINUS
$E \in S \setminus T \;\;\defi\;\; E \in S \land \lnot\,(E \in T)$ M
*
DEF_IN_SETENUM
$E \in \{ A, \ldots , B\} \;\;\defi\;\; E = A \lor \ldots \lor E = B$ M
*
DEF_IN_KUNION
$E \in \union (S) \;\;\defi\;\; \exists s \qdot s \in S \land E \in s$ where $s$ is fresh M
*
DEF_IN_QUNION
$E \in (\Union x \qdot P(x) \mid T(x)) \;\;\defi\;\; \exists s \qdot P(s) \land E \in T(s)$ where $s$ is fresh M
*
DEF_IN_KINTER
$E \in \inter (S) \;\;\defi\;\; \forall s \qdot s \in S \limp E \in s$ where $s$ is fresh M
*
DEF_IN_QINTER
$E \in (\Inter x \qdot P(x) \mid T(x)) \;\;\defi\;\; \forall s \qdot P(s) \limp E \in T(s)$ where $s$ is fresh M
*
DEF_IN_UPTO
$E \in a \upto b \;\;\defi\;\; a \leq E \land E \leq b$ M
*
DISTRI_BUNION_BINTER
$S \bunion (T \binter U) \;\;\defi\;\; (S \bunion T) \binter (S \bunion U)$ M
*
DISTRI_BINTER_BUNION
$S \binter (T \bunion U) \;\;\defi\;\; (S \binter T) \bunion (S \binter U)$ M
DISTRI_BINTER_SETMINUS
$S \binter (T \setminus U) \;\;\defi\;\; (S \binter T) \setminus (S \binter U)$ M
DISTRI_SETMINUS_BUNION
$S \setminus (T \bunion U) \;\;\defi\;\; S \setminus T \setminus U$ M
*
DERIV_TYPE_SETMINUS_BINTER
$\mathit{Ty} \setminus (S \binter T) \;\;\defi\;\; (\mathit{Ty} \setminus S) \bunion (\mathit{Ty} \setminus T)$ where $\mathit{Ty}$ is a type expression M
*
DERIV_TYPE_SETMINUS_BUNION
$\mathit{Ty} \setminus (S \bunion T) \;\;\defi\;\; (\mathit{Ty} \setminus S) \binter (\mathit{Ty} \setminus T)$ where $\mathit{Ty}$ is a type expression M
*
DERIV_TYPE_SETMINUS_SETMINUS
$\mathit{Ty} \setminus (S \setminus T) \;\;\defi\;\; (\mathit{Ty} \setminus S) \bunion T$ where $\mathit{Ty}$ is a type expression M
DISTRI_CPROD_BINTER
$S \cprod (T \binter U) \;\;\defi\;\; (S \cprod T) \binter (S \cprod U)$ M
DISTRI_CPROD_BUNION
$S \cprod (T \bunion U) \;\;\defi\;\; (S \cprod T) \bunion (S \cprod U)$ M
DISTRI_CPROD_SETMINUS
$S \cprod (T \setminus U) \;\;\defi\;\; (S \cprod T) \setminus (S \cprod U)$ M
*
DERIV_SUBSETEQ
$S \subseteq T \;\;\defi\;\; (\mathit{Ty} \setminus T) \subseteq (\mathit{Ty} \setminus S)$ where $\pow (\mathit{Ty})$ is the type of $S$ and $T$ M
*
DERIV_EQUAL
$S = T \;\;\defi\;\; S \subseteq T \land T \subseteq S$ where $\pow (\mathit{Ty})$ is the type of $S$ and $T$ M
*
DERIV_SUBSETEQ_SETMINUS_L
$A \setminus B \subseteq S \;\;\defi\;\; A \subseteq B \bunion S$ M
*
DERIV_SUBSETEQ_SETMINUS_R
$S \subseteq A \setminus B \;\;\defi\;\; S \subseteq A \land S \binter B = \emptyset$ M
*
DEF_PARTITION
$\operatorname{partition} (s, s_1, s_2, \ldots, s_n) \;\;\defi\;\; \begin{array}{ll} & s = s_1\bunion s_2\bunion\cdots\bunion s_n\\ \land& s_1\binter s_2 = \emptyset\\ \vdots\\ \land& s_1\binter s_n = \emptyset\\ \vdots\\ \land& s_{n-1}\binter s_n = \emptyset \end{array}$ AM
SIMP_EMPTY_PARTITION
$\operatorname{partition}(S) \;\;\defi\;\; S = \emptyset$ A
SIMP_SINGLE_PARTITION
$\operatorname{partition}(S, T) \;\;\defi\;\; S = T$ A

Rules that are marked with a * in the first column are implemented in the latest version of Rodin. Rules without a * are planned to be implemented in future versions. Other conventions used in these tables are described in The_Proving_Perspective_(Rodin_User_Manual)#Rewrite_Rules.

Name Rule Side Condition A/M
*
SIMP_DOM_SETENUM
$\dom (\{ x \mapsto a, \ldots , y \mapsto b\} ) \;\;\defi\;\; \{ x, \ldots , y\}$ A
*
SIMP_DOM_CONVERSE
$\dom (r^{-1} ) \;\;\defi\;\; \ran (r)$ A
*
SIMP_RAN_SETENUM
$\ran (\{ x \mapsto a, \ldots , y \mapsto b\} ) \;\;\defi\;\; \{ a, \ldots , b\}$ A
*
SIMP_RAN_CONVERSE
$\ran (r^{-1} ) \;\;\defi\;\; \dom (r)$ A
*
SIMP_SPECIAL_OVERL
$r \ovl \ldots \ovl \emptyset \ovl \ldots \ovl s \;\;\defi\;\; r \ovl \ldots \ovl s$ A
*
SIMP_MULTI_OVERL
$r_1 \ovl \cdots \ovl r_n \defi r_1 \ovl \cdots \ovl r_{i-1} \ovl r_{i+1} \ovl \cdots \ovl r_n$ there is a $j$ such that $1\leq i < j \leq n$ and $r_i$ and $r_j$ are syntactically equal. A
*
SIMP_TYPE_OVERL_CPROD
$r\ovl\cdots\ovl\mathit{Ty}\ovl\cdots\ovl s \;\defi\;\; \mathit{Ty}\ovl\cdots\ovl s$ where $\mathit{Ty}$ is a type expression A
*
SIMP_SPECIAL_DOMRES_L
$\emptyset \domres r \;\;\defi\;\; \emptyset$ A
*
SIMP_SPECIAL_DOMRES_R
$S \domres \emptyset \;\;\defi\;\; \emptyset$ A
*
SIMP_TYPE_DOMRES
$\mathit{Ty} \domres r \;\;\defi\;\; r$ where $\mathit{Ty}$ is a type expression A
*
SIMP_MULTI_DOMRES_DOM
$\dom (r) \domres r \;\;\defi\;\; r$ A
*
SIMP_MULTI_DOMRES_RAN
$\ran (r) \domres r^{-1} \;\;\defi\;\; r^{-1}$ A
*
SIMP_DOMRES_DOMRES_ID
$S \domres (T \domres \id) \;\;\defi\;\; (S \binter T) \domres \id$ A
*
SIMP_DOMRES_DOMSUB_ID
$S \domres (T \domsub \id) \;\;\defi\;\; (S \setminus T) \domres \id$ A
*
SIMP_SPECIAL_RANRES_R
$r \ranres \emptyset \;\;\defi\;\; \emptyset$ A
*
SIMP_SPECIAL_RANRES_L
$\emptyset \ranres S \;\;\defi\;\; \emptyset$ A
*
SIMP_TYPE_RANRES
$r \ranres \mathit{Ty} \;\;\defi\;\; r$ where $\mathit{Ty}$ is a type expression A
*
SIMP_MULTI_RANRES_RAN
$r \ranres \ran (r) \;\;\defi\;\; r$ A
*
SIMP_MULTI_RANRES_DOM
$r^{-1} \ranres \dom (r) \;\;\defi\;\; r^{-1}$ A
*
SIMP_RANRES_ID
$\id \ranres S \;\;\defi\;\; S \domres \id$ A
*
SIMP_RANSUB_ID
$\id \ransub S \;\;\defi\;\; S \domsub \id$ A
*
SIMP_RANRES_DOMRES_ID
$(S \domres \id) \ranres T \;\;\defi\;\; (S \binter T) \domres \id$ A
*
SIMP_RANRES_DOMSUB_ID
$(S \domsub \id) \ranres T \;\;\defi\;\; (T \setminus S) \domres \id$ A
*
SIMP_SPECIAL_DOMSUB_L
$\emptyset \domsub r \;\;\defi\;\; r$ A
*
SIMP_SPECIAL_DOMSUB_R
$S \domsub \emptyset \;\;\defi\;\; \emptyset$ A
*
SIMP_TYPE_DOMSUB
$\mathit{Ty} \domsub r \;\;\defi\;\; \emptyset$ where $\mathit{Ty}$ is a type expression A
*
SIMP_MULTI_DOMSUB_DOM
$\dom (r) \domsub r \;\;\defi\;\; \emptyset$ A
*
SIMP_MULTI_DOMSUB_RAN
$\ran (r) \domsub r^{-1} \;\;\defi\;\; \emptyset$ A
*
SIMP_DOMSUB_DOMRES_ID
$S \domsub (T \domres \id ) \;\;\defi\;\; (T \setminus S) \domres \id$ A
*
SIMP_DOMSUB_DOMSUB_ID
$S \domsub (T \domsub \id ) \;\;\defi\;\; (S \bunion T) \domsub \id$ A
*
SIMP_SPECIAL_RANSUB_R
$r \ransub \emptyset \;\;\defi\;\; r$ A
*
SIMP_SPECIAL_RANSUB_L
$\emptyset \ransub S \;\;\defi\;\; \emptyset$ A
*
SIMP_TYPE_RANSUB
$r \ransub \mathit{Ty} \;\;\defi\;\; \emptyset$ where $\mathit{Ty}$ is a type expression A
*
SIMP_MULTI_RANSUB_DOM
$r^{-1} \ransub \dom (r) \;\;\defi\;\; \emptyset$ A
*
SIMP_MULTI_RANSUB_RAN
$r \ransub \ran (r) \;\;\defi\;\; \emptyset$ A
*
SIMP_RANSUB_DOMRES_ID
$(S \domres \id) \ransub T \;\;\defi\;\; (S \setminus T) \domres \id$ A
*
SIMP_RANSUB_DOMSUB_ID
$(S \domsub \id) \ransub T \;\;\defi\;\; (S \bunion T) \domsub \id$ A
*
SIMP_SPECIAL_FCOMP
$r \fcomp \ldots \fcomp \emptyset \fcomp \ldots \fcomp s \;\;\defi\;\; \emptyset$ A
*
SIMP_TYPE_FCOMP_ID
$r \fcomp \ldots \fcomp \id \fcomp \ldots \fcomp s \;\;\defi\;\; r \fcomp \ldots \fcomp s$ A
*
SIMP_TYPE_FCOMP_R
$r \fcomp \mathit{Ty} \;\;\defi\;\; \dom (r) \cprod \mathit{Tb}$ where $\mathit{Ty}$ is a type expression equal to $\mathit{Ta} \cprod \mathit{Tb}$ A
*
SIMP_TYPE_FCOMP_L
$\mathit{Ty} \fcomp r \;\;\defi\;\; \mathit{Ta} \cprod \ran (r)$ where $\mathit{Ty}$ is a type expression equal to $\mathit{Ta} \cprod \mathit{Tb}$ A
*
SIMP_FCOMP_ID
$r \fcomp \ldots \fcomp S \domres \id \fcomp T \domres \id \fcomp \ldots s \;\;\defi\;\; r \fcomp \ldots \fcomp (S \binter T) \domres \id \fcomp \ldots \fcomp s$ A
*
SIMP_SPECIAL_BCOMP
$r \bcomp \ldots \bcomp \emptyset \bcomp \ldots \bcomp s \;\;\defi\;\; \emptyset$ A
*
SIMP_TYPE_BCOMP_ID
$r \bcomp \ldots \bcomp \id \bcomp \ldots \bcomp s \;\;\defi\;\; r \bcomp \ldots \bcomp s$ A
*
SIMP_TYPE_BCOMP_L
$\mathit{Ty} \bcomp r \;\;\defi\;\; \dom (r) \cprod \mathit{Tb}$ where $\mathit{Ty}$ is a type expression equal to $\mathit{Ta} \cprod \mathit{Tb}$ A
*
SIMP_TYPE_BCOMP_R
$r \bcomp \mathit{Ty} \;\;\defi\;\; \mathit{Ta} \cprod \ran (r)$ where $\mathit{Ty}$ is a type expression equal to $\mathit{Ta} \cprod \mathit{Tb}$ A
*
SIMP_BCOMP_ID
$r \bcomp \ldots \bcomp S \domres \id \bcomp T \domres \id \bcomp \ldots \bcomp s \;\;\defi\;\; r \bcomp \ldots \bcomp (S \binter T) \domres \id \bcomp \ldots \bcomp s$ A
*
SIMP_SPECIAL_DPROD_R
$r \dprod \emptyset \;\;\defi\;\; \emptyset$ A
*
SIMP_SPECIAL_DPROD_L
$\emptyset \dprod r \;\;\defi\;\; \emptyset$ A
*
SIMP_DPROD_CPROD
$(\mathit{S} \cprod \mathit{T}) \dprod (\mathit{U} \cprod \mathit{V}) \;\;\defi\;\; \mathit{S} \binter \mathit{U} \cprod (\mathit{T} \cprod \mathit{V})$ A
*
SIMP_SPECIAL_PPROD_R
$r \pprod \emptyset \;\;\defi\;\; \emptyset$ A
*
SIMP_SPECIAL_PPROD_L
$\emptyset \pprod r \;\;\defi\;\; \emptyset$ A
*
SIMP_PPROD_CPROD
$(\mathit{S} \cprod \mathit{T}) \pprod (\mathit{U} \cprod \mathit{V}) \;\;\defi\;\; (\mathit{S} \cprod \mathit{U}) \cprod (\mathit{T} \cprod \mathit{V})$ A
*
SIMP_SPECIAL_RELIMAGE_R
$r[\emptyset ] \;\;\defi\;\; \emptyset$ A
*
SIMP_SPECIAL_RELIMAGE_L
$\emptyset [S] \;\;\defi\;\; \emptyset$ A
*
SIMP_TYPE_RELIMAGE
$r[Ty] \;\;\defi\;\; \ran (r)$ where $\mathit{Ty}$ is a type expression A
*
SIMP_MULTI_RELIMAGE_DOM
$r[\dom (r)] \;\;\defi\;\; \ran (r)$ A
*
SIMP_RELIMAGE_ID
$\id[T] \;\;\defi\;\; T$ A
*
SIMP_RELIMAGE_DOMRES_ID
$(S \domres \id)[T] \;\;\defi\;\; S \binter T$ A
*
SIMP_RELIMAGE_DOMSUB_ID
$(S \domsub \id)[T] \;\;\defi\;\; T \setminus S$ A
*
SIMP_MULTI_RELIMAGE_CPROD_SING
$(\{ E\} \cprod S)[\{ E\} ] \;\;\defi\;\; S$ where $E$ is a single expression A
*
SIMP_MULTI_RELIMAGE_SING_MAPSTO
$\{ E \mapsto F\} [\{ E\} ] \;\;\defi\;\; \{ F\}$ where $E$ is a single expression A
*
SIMP_MULTI_RELIMAGE_CONVERSE_RANSUB
$(r \ransub S)^{-1} [S] \;\;\defi\;\; \emptyset$ A
*
SIMP_MULTI_RELIMAGE_CONVERSE_RANRES
$(r \ranres S)^{-1} [S] \;\;\defi\;\; r^{-1} [S]$ A
*
SIMP_RELIMAGE_CONVERSE_DOMSUB
$(S \domsub r)^{-1} [T] \;\;\defi\;\; r^{-1} [T] \setminus S$ A
DERIV_RELIMAGE_RANSUB
$(r \ransub S)[T] \;\;\defi\;\; r[T] \setminus S$ M
DERIV_RELIMAGE_RANRES
$(r \ranres S)[T] \;\;\defi\;\; r[T] \binter S$ M
*
SIMP_MULTI_RELIMAGE_DOMSUB
$(S \domsub r)[S] \;\;\defi\;\; \emptyset$ A
DERIV_RELIMAGE_DOMSUB
$(S \domsub r)[T] \;\;\defi\;\; r[T \setminus S]$ M
DERIV_RELIMAGE_DOMRES
$(S \domres r)[T] \;\;\defi\;\; r[S \binter T]$ M
*
SIMP_SPECIAL_CONVERSE
$\emptyset ^{-1} \;\;\defi\;\; \emptyset$ A
*
SIMP_CONVERSE_ID
$\id^{-1} \;\;\defi\;\; \id$ A
*
SIMP_CONVERSE_CPROD
$(\mathit{S} \cprod \mathit{T})^{-1} \;\;\defi\;\; \mathit{T} \cprod \mathit{S}$ A
*
SIMP_CONVERSE_SETENUM
$\{ x \mapsto a, \ldots , y \mapsto b\} ^{-1} \;\;\defi\;\; \{ a \mapsto x, \ldots , b \mapsto y\}$ A
*
SIMP_CONVERSE_COMPSET
$\{ X \qdot P \mid x\mapsto y\} ^{-1} \;\;\defi\;\; \{ X \qdot P \mid y\mapsto x\}$ A
*
SIMP_DOM_ID
$\dom (\id) \;\;\defi\;\; S$ where $\id$ has type $\pow(S \cprod S)$ A
*
SIMP_RAN_ID
$\ran (\id) \;\;\defi\;\; S$ where $\id$ has type $\pow(S \cprod S)$ A
*
SIMP_FCOMP_ID_L
$(S \domres \id) \fcomp r \;\;\defi\;\; S \domres r$ A
*
SIMP_FCOMP_ID_R
$r \fcomp (S \domres \id) \;\;\defi\;\; r \ranres S$ A
*
SIMP_SPECIAL_REL_R
$S \rel \emptyset \;\;\defi\;\; \{ \emptyset \}$ idem for operators $\srel \pfun \pinj \psur$ A
*
SIMP_SPECIAL_REL_L
$\emptyset \rel S \;\;\defi\;\; \{ \emptyset \}$ idem for operators $\trel \pfun \tfun \pinj \tinj$ A
*
SIMP_FUNIMAGE_PRJ1
$\prjone (E \mapsto F) \;\;\defi\;\; E$ A
*
SIMP_FUNIMAGE_PRJ2
$\prjtwo (E \mapsto F) \;\;\defi\;\; F$ A
*
SIMP_DOM_PRJ1
$\dom (\prjone) \;\;\defi\;\; S \cprod T$ where $\prjone$ has type $\pow(S \cprod T \cprod S)$ A
*
SIMP_DOM_PRJ2
$\dom (\prjtwo) \;\;\defi\;\; S \cprod T$ where $\prjtwo$ has type $\pow(S \cprod T \cprod T)$ A
*
SIMP_RAN_PRJ1
$\ran (\prjone) \;\;\defi\;\; S$ where $\prjone$ has type $\pow(S \cprod T \cprod S)$ A
*
SIMP_RAN_PRJ2
$\ran (\prjtwo) \;\;\defi\;\; T$ where $\prjtwo$ has type $\pow(S \cprod T \cprod T)$ A
*
SIMP_FUNIMAGE_LAMBDA
$(\lambda x \qdot P(x) \mid E(x))(y) \;\;\defi\;\; E(y)$ A
*
SIMP_DOM_LAMBDA
$\dom(\{x\qdot P\mid E\mapsto F) \;\;\defi\;\; \{x\qdot P\mid E\}$ A
*
SIMP_RAN_LAMBDA
$\ran(\{x\qdot P\mid E\mapsto F) \;\;\defi\;\; \{x\qdot P\mid F\}$ A
*
SIMP_IN_FUNIMAGE
$E\mapsto F(E)\in F \;\;\defi\;\; \btrue$ A
*
SIMP_IN_FUNIMAGE_CONVERSE_L
$F^{-1}(E)\mapsto E\in F \;\;\defi\;\; \btrue$ A
*
SIMP_IN_FUNIMAGE_CONVERSE_R
$F(E)\mapsto E\in F^{-1} \;\;\defi\;\; \btrue$ A
*
SIMP_MULTI_FUNIMAGE_SETENUM_LL
$\{ A \mapsto E, \ldots , B \mapsto E\} (x) \;\;\defi\;\; E$ A
*
SIMP_MULTI_FUNIMAGE_SETENUM_LR
$\{ E, \ldots , x \mapsto y, \ldots , F\} (x) \;\;\defi\;\; y$ A
*
SIMP_MULTI_FUNIMAGE_OVERL_SETENUM
$(r \ovl \ldots \ovl \{ E, \ldots , x \mapsto y, \ldots , F\} )(x) \;\;\defi\;\; y$ A
*
SIMP_MULTI_FUNIMAGE_BUNION_SETENUM
$(r \bunion \ldots \bunion \{ E, \ldots , x \mapsto y, \ldots , F\} )(x) \;\;\defi\;\; y$ A
*
SIMP_FUNIMAGE_CPROD
$(S \cprod \{ F\} )(x) \;\;\defi\;\; F$ A
*
SIMP_FUNIMAGE_ID
$\id (x) \;\;\defi\;\; x$ A
*
SIMP_FUNIMAGE_FUNIMAGE_CONVERSE
$f(f^{-1} (E)) \;\;\defi\;\; E$ A
*
SIMP_FUNIMAGE_CONVERSE_FUNIMAGE
$f^{-1}(f(E)) \;\;\defi\;\; E$ A
*
SIMP_FUNIMAGE_FUNIMAGE_CONVERSE_SETENUM
$\{x \mapsto a, \ldots, y \mapsto b\}(\{a \mapsto x, \ldots, b \mapsto y\}(E)) \;\;\defi\;\; E$ A
*
SIMP_FUNIMAGE_DOMRES
$(E \domres F)(G)\;\;\defi\;\;F(G)$ with hypothesis$F \in \mathit{A} \ \mathit{op}\ \mathit{B}$ where $\mathit{op}$ is one of $\pfun$, $\tfun$, $\pinj$, $\tinj$, $\psur$, $\tsur$, $\tbij$. AM
*
SIMP_FUNIMAGE_DOMSUB
$(E \domsub F)(G)\;\;\defi\;\;F(G)$ with hypothesis$F \in \mathit{A} \ \mathit{op}\ \mathit{B}$ where $\mathit{op}$ is one of $\pfun$, $\tfun$, $\pinj$, $\tinj$, $\psur$, $\tsur$, $\tbij$. AM
*
SIMP_FUNIMAGE_RANRES
$(F\ranres E)(G)\;\;\defi\;\;F(G)$ with hypothesis$F \in \mathit{A} \ \mathit{op}\ \mathit{B}$ where $\mathit{op}$ is one of $\pfun$, $\tfun$, $\pinj$, $\tinj$, $\psur$, $\tsur$, $\tbij$. AM
*
SIMP_FUNIMAGE_RANSUB
$(F \ransub E)(G)\;\;\defi\;\;F(G)$ with hypothesis$F \in \mathit{A} \ \mathit{op}\ \mathit{B}$ where $\mathit{op}$ is one of $\pfun$, $\tfun$, $\pinj$, $\tinj$, $\psur$, $\tsur$, $\tbij$. AM
*
SIMP_FUNIMAGE_SETMINUS
$(F \setminus E)(G)\;\;\defi\;\;F(G)$ with hypothesis$F \in \mathit{A} \ \mathit{op}\ \mathit{B}$ where $\mathit{op}$ is one of $\pfun$, $\tfun$, $\pinj$, $\tinj$, $\psur$, $\tsur$, $\tbij$. AM
DEF_BCOMP
$r \bcomp \ldots \bcomp s \;\;\defi\;\; s \fcomp \ldots \fcomp r$ M
DERIV_ID_SING
$\{ E\} \domres \id \;\;\defi\;\; \{ E \mapsto E\}$ where $E$ is a single expression M
*
SIMP_SPECIAL_DOM
$\dom (\emptyset ) \;\;\defi\;\; \emptyset$ A
*
SIMP_SPECIAL_RAN
$\ran (\emptyset ) \;\;\defi\;\; \emptyset$ A
*
SIMP_CONVERSE_CONVERSE
$r^{-1-1} \;\;\defi\;\; r$ A
*
DERIV_RELIMAGE_FCOMP
$(p \fcomp q)[s] \;\;\defi\;\; q[p[s]]$ M
*
DERIV_FCOMP_DOMRES
$(s \domres p) \fcomp q \;\;\defi\;\; s \domres (p \fcomp q)$ M
*
DERIV_FCOMP_DOMSUB
$(s \domsub p) \fcomp q \;\;\defi\;\; s \domsub (p \fcomp q)$ M
*
DERIV_FCOMP_RANRES
$p \fcomp (q \ranres s) \;\;\defi\;\; (p \fcomp q) \ranres s$ M
*
DERIV_FCOMP_RANSUB
$p \fcomp (q \ransub s) \;\;\defi\;\; (p \fcomp q) \ransub s$ M
DERIV_FCOMP_SING
$\{E\mapsto F\}\fcomp\{F\mapsto G\} \;\;\defi\;\; \{E\mapsto G\}$ A
*
SIMP_SPECIAL_EQUAL_RELDOMRAN
$\emptyset \strel \emptyset \;\;\defi\;\; \{ \emptyset \}$ idem for operators $\tsur \tbij$ A
*
SIMP_TYPE_DOM
$\dom (\mathit{Ty}) \;\;\defi\;\; \mathit{Ta}$ where $\mathit{Ty}$ is a type expression equal to $\mathit{Ta} \cprod \mathit{Tb}$ A
*
SIMP_TYPE_RAN
$\ran (\mathit{Ty}) \;\;\defi\;\; \mathit{Tb}$ where $\mathit{Ty}$ is a type expression equal to $\mathit{Ta} \cprod \mathit{Tb}$ A
*
SIMP_MULTI_DOM_CPROD
$\dom (E \cprod E) \;\;\defi\;\; E$ A
*
SIMP_MULTI_RAN_CPROD
$\ran (E \cprod E) \;\;\defi\;\; E$ A
*
SIMP_MULTI_DOM_DOMRES
$\dom(A\domres f) \;\;\defi\;\; \dom(f)\binter A$ A
*
SIMP_MULTI_DOM_DOMSUB
$\dom(A\domsub f) \;\;\defi\;\; \dom(f)\setminus A$ A
*
SIMP_MULTI_RAN_RANRES
$\ran(f\ranres A) \;\;\defi\;\; \ran(f)\binter A$ A
*
SIMP_MULTI_RAN_RANSUB
$\ran(f\ransub A) \;\;\defi\;\; \ran(f)\setminus A$ A
*
DEF_IN_DOM
$E \in \dom (r) \;\;\defi\;\; \exists y \qdot E \mapsto y \in r$ M
*
DEF_IN_RAN
$F \in \ran (r) \;\;\defi\;\; \exists x \qdot x \mapsto F \in r$ M
*
DEF_IN_CONVERSE
$E \mapsto F \in r^{-1} \;\;\defi\;\; F \mapsto E \in r$ M
*
DEF_IN_DOMRES
$E \mapsto F \in S \domres r \;\;\defi\;\; E \in S \land E \mapsto F \in r$ M
*
DEF_IN_RANRES
$E \mapsto F \in r \ranres T \;\;\defi\;\; E \mapsto F \in r \land F \in T$ M
*
DEF_IN_DOMSUB
$E \mapsto F \in S \domsub r \;\;\defi\;\; E \notin S \land E \mapsto F \in r$ M
*
DEF_IN_RANSUB
$E \mapsto F \in r \ranres T \;\;\defi\;\; E \mapsto F \in r \land F \notin T$ M
*
DEF_IN_RELIMAGE
$F \in r[w] \;\;\defi\;\; \exists x \qdot x \in w \land x \mapsto F \in r$ M
*
DEF_IN_FCOMP
$E \mapsto F \in (p \fcomp q) \;\;\defi\;\; \exists x \qdot E \mapsto x \in p \land x \mapsto F \in q$ M
*
DEF_OVERL
$p \ovl q \;\;\defi\;\; (\dom (q) \domsub p) \bunion q$ M
*
DEF_IN_ID
$E \mapsto F \in \id \;\;\defi\;\; E = F$ M
*
DEF_IN_DPROD
$E \mapsto (F \mapsto G) \in p \dprod q \;\;\defi\;\; E \mapsto F \in p \land E \mapsto G \in q$ M
*
DEF_IN_PPROD
$(E \mapsto G) \mapsto (F \mapsto H) \in p \pprod q \;\;\defi\;\; E \mapsto F \in p \land G \mapsto H \in q$ M
*
DEF_IN_REL
$r \in S \rel T \;\;\defi\;\; r\subseteq S\cprod T$ M
*
DEF_IN_RELDOM
$r \in S \trel T \;\;\defi\;\; r \in S \rel T \land \dom (r) = S$ M
*
DEF_IN_RELRAN
$r \in S \srel T \;\;\defi\;\; r \in S \rel T \land \ran (r) = T$ M
*
DEF_IN_RELDOMRAN
$r \in S \strel T \;\;\defi\;\; r \in S \rel T \land \dom (r) = S \land \ran (r) = T$ M
*
DEF_IN_FCT
$\begin{array}{rcl} f \in S \pfun T & \defi & f \in S \rel T \\ & \land & (\forall x,y,z \qdot x \mapsto y \in f \land x \mapsto z \in f \limp y = z) \\ \end{array}$ M
*
DEF_IN_TFCT
$f \in S \tfun T \;\;\defi\;\; f \in S \pfun T \land \dom (f) = S$ M
*
DEF_IN_INJ
$f \in S \pinj T \;\;\defi\;\; f \in S \pfun T \land f^{-1} \in T \pfun S$ M
*
DEF_IN_TINJ
$f \in S \tinj T \;\;\defi\;\; f \in S \pinj T \land \dom (f) = S$ M
*
DEF_IN_SURJ
$f \in S \psur T \;\;\defi\;\; f \in S \pfun T \land \ran (f) = T$ M
*
DEF_IN_TSURJ
$f \in S \tsur T \;\;\defi\;\; f \in S \psur T \land \dom (f) = S$ M
*
DEF_IN_BIJ
$f \in S \tbij T \;\;\defi\;\; f \in S \tinj T \land \ran (f) = T$ M
DISTRI_BCOMP_BUNION
$r \bcomp (s \bunion t) \;\;\defi\;\; (r \bcomp s) \bunion (r \bcomp t)$ M
*
DISTRI_FCOMP_BUNION_R
$p \fcomp (q \bunion r) \;\;\defi\;\; (p \fcomp q) \bunion (p \fcomp r)$ M
*
DISTRI_FCOMP_BUNION_L
$(q \bunion r) \fcomp p \;\;\defi\;\; (q \fcomp p) \bunion (r \fcomp p)$ M
DISTRI_DPROD_BUNION
$r \dprod (s \bunion t) \;\;\defi\;\; (r \dprod s) \bunion (r \dprod t)$ M
DISTRI_DPROD_BINTER
$r \dprod (s \binter t) \;\;\defi\;\; (r \dprod s) \binter (r \dprod t)$ M
DISTRI_DPROD_SETMINUS
$r \dprod (s \setminus t) \;\;\defi\;\; (r \dprod s) \setminus (r \dprod t)$ M
DISTRI_DPROD_OVERL
$r \dprod (s \ovl t) \;\;\defi\;\; (r \dprod s) \ovl (r \dprod t)$ M
DISTRI_PPROD_BUNION
$r \pprod (s \bunion t) \;\;\defi\;\; (r \pprod s) \bunion (r \pprod t)$ M
DISTRI_PPROD_BINTER
$r \pprod (s \binter t) \;\;\defi\;\; (r \pprod s) \binter (r \pprod t)$ M
DISTRI_PPROD_SETMINUS
$r \pprod (s \setminus t) \;\;\defi\;\; (r \pprod s) \setminus (r \pprod t)$ M
DISTRI_PPROD_OVERL
$r \pprod (s \ovl t) \;\;\defi\;\; (r \pprod s) \ovl (r \pprod t)$ M
DISTRI_OVERL_BUNION_L
$(p \bunion q) \ovl r \;\;\defi\;\; (p \ovl r) \bunion (q \ovl r)$ M
DISTRI_OVERL_BINTER_L
$(p \binter q) \ovl r \;\;\defi\;\; (p \ovl r) \binter (q \ovl r)$ M
*
DISTRI_DOMRES_BUNION_R
$s \domres (p \bunion q) \;\;\defi\;\; (s \domres p) \bunion (s \domres q)$ M
*
DISTRI_DOMRES_BUNION_L
$(s \bunion t) \domres r \;\;\defi\;\; (s \domres r) \bunion (t \domres r)$ M
*
DISTRI_DOMRES_BINTER_R
$s \domres (p \binter q) \;\;\defi\;\; (s \domres p) \binter (s \domres q)$ M
*
DISTRI_DOMRES_BINTER_L
$(s \binter t) \domres r \;\;\defi\;\; (s \domres r) \binter (t \domres r)$ M
DISTRI_DOMRES_SETMINUS_R
$s \domres (p \setminus q) \;\;\defi\;\; (s \domres p) \setminus (s \domres q)$ M
DISTRI_DOMRES_SETMINUS_L
$(s \setminus t) \domres r \;\;\defi\;\; (s \domres r) \setminus (t \domres r)$ M
DISTRI_DOMRES_DPROD
$s \domres (p \dprod q) \;\;\defi\;\; (s \domres p) \dprod (s \domres q)$ M
DISTRI_DOMRES_OVERL
$s \domres (r \ovl q) \;\;\defi\;\; (s \domres r) \ovl (s \domres q)$ M
*
DISTRI_DOMSUB_BUNION_R
$s \domsub (p \bunion q) \;\;\defi\;\; (s \domsub p) \bunion (s \domsub q)$ M
*
DISTRI_DOMSUB_BUNION_L
$(s \bunion t) \domsub r \;\;\defi\;\; (s \domsub r) \binter (t \domsub r)$ M
*
DISTRI_DOMSUB_BINTER_R
$s \domsub (p \binter q) \;\;\defi\;\; (s \domsub p) \binter (s \domsub q)$ M
*
DISTRI_DOMSUB_BINTER_L
$(s \binter t) \domsub r \;\;\defi\;\; (s \domsub r) \bunion (t \domsub r)$ M
DISTRI_DOMSUB_DPROD
$A \domsub (r \dprod s) \;\;\defi\;\; (A \domsub r) \dprod (A \domsub s)$ M
DISTRI_DOMSUB_OVERL
$A \domsub (r \ovl s) \;\;\defi\;\; (A \domsub r) \ovl (A \domsub s)$ M
*
DISTRI_RANRES_BUNION_R
$r \ranres (s \bunion t) \;\;\defi\;\; (r \ranres s) \bunion (r \ranres t)$ M
*
DISTRI_RANRES_BUNION_L
$(p \bunion q) \ranres s \;\;\defi\;\; (p \ranres s) \bunion (q \ranres s)$ M
*
DISTRI_RANRES_BINTER_R
$r \ranres (s \binter t) \;\;\defi\;\; (r \ranres s) \binter (r \ranres t)$ M
*
DISTRI_RANRES_BINTER_L
$(p \binter q) \ranres s \;\;\defi\;\; (p \ranres s) \binter (q \ranres s)$ M
DISTRI_RANRES_SETMINUS_R
$r \ranres (s \setminus t) \;\;\defi\;\; (r \ranres s) \setminus (r \ranres t)$ M
DISTRI_RANRES_SETMINUS_L
$(p \setminus q) \ranres s \;\;\defi\;\; (p \ranres s) \setminus (q \ranres s)$ M
*
DISTRI_RANSUB_BUNION_R
$r \ransub (s\bunion t) \;\;\defi\;\; (r \ransub s) \binter (r \ransub t)$ M
*
DISTRI_RANSUB_BUNION_L
$(p \bunion q) \ransub s \;\;\defi\;\; (p \ransub s) \bunion (q \ransub s)$ M
*
DISTRI_RANSUB_BINTER_R
$r \ransub (s \binter t) \;\;\defi\;\; (r \ransub s) \bunion (r \ransub t)$ M
*
DISTRI_RANSUB_BINTER_L
$(p \binter q) \ransub s \;\;\defi\;\; (p \ransub s) \binter (q \ransub s)$ M
*
DISTRI_CONVERSE_BUNION
$(p \bunion q)^{-1} \;\;\defi\;\; p^{-1} \bunion q^{-1}$ M
DISTRI_CONVERSE_BINTER
$(p \binter q)^{-1} \;\;\defi\;\; p^{-1} \binter q^{-1}$ M
DISTRI_CONVERSE_SETMINUS
$(r \setminus s)^{-1} \;\;\defi\;\; r^{-1} \setminus s^{-1}$ M
DISTRI_CONVERSE_BCOMP
$(r \bcomp s)^{-1} \;\;\defi\;\; (s^{-1} \bcomp r^{-1} )$ M
DISTRI_CONVERSE_FCOMP
$(p \fcomp q)^{-1} \;\;\defi\;\; (q^{-1} \fcomp p^{-1} )$ M
DISTRI_CONVERSE_PPROD
$(r \pprod s)^{-1} \;\;\defi\;\; r^{-1} \pprod s^{-1}$ M
DISTRI_CONVERSE_DOMRES
$(s \domres r)^{-1} \;\;\defi\;\; r^{-1} \ranres s$ M
DISTRI_CONVERSE_DOMSUB
$(s \domsub r)^{-1} \;\;\defi\;\; r^{-1} \ransub s$ M
DISTRI_CONVERSE_RANRES
$(r \ranres s)^{-1} \;\;\defi\;\; s \domres r^{-1}$ M
DISTRI_CONVERSE_RANSUB
$(r \ransub s)^{-1} \;\;\defi\;\; s \domsub r^{-1}$ M
*
DISTRI_DOM_BUNION
$\dom (r \bunion s) \;\;\defi\;\; \dom (r) \bunion \dom (s)$ M
*
DISTRI_RAN_BUNION
$\ran (r \bunion s) \;\;\defi\;\; \ran (r) \bunion \ran (s)$ M
*
DISTRI_RELIMAGE_BUNION_R
$r[S \bunion T] \;\;\defi\;\; r[S] \bunion r[T]$ M
*
DISTRI_RELIMAGE_BUNION_L
$(p \bunion q)[S] \;\;\defi\;\; p[S] \bunion q[S]$ M
*
DERIV_DOM_TOTALREL
$\dom (r) \;\;\defi\;\; E$ with hypothesis $r \in E \ \mathit{op}\ F$, where $\mathit{op}$ is one of $\trel, \strel, \tfun, \tinj, \tsur, \tbij$ M
DERIV_RAN_SURJREL
$\ran (r) \;\;\defi\;\; F$ with hypothesis $r \in E \ \mathit{op}\ F$, where $\mathit{op}$ is one of $\srel,\strel, \psur, \tsur, \tbij$ M
*
DERIV_PRJ1_SURJ
$\prjone \in\mathit{Ty}_1\ \mathit{op}\ \mathit{Ty}_2\;\;\defi\;\; \btrue$ where $\mathit{Ty}_1$ and $\mathit{Ty}_2$ are types and $\mathit{op}$ is one of $\rel, \trel, \srel, \strel, \pfun, \tfun, \psur, \tsur$ A
*
DERIV_PRJ2_SURJ
$\prjtwo \in\mathit{Ty}_1\ \mathit{op}\ \mathit{Ty}_2\;\;\defi\;\; \btrue$ where $\mathit{Ty}_1$ and $\mathit{Ty}_2$ are types and $\mathit{op}$ is one of $\rel, \trel, \srel, \strel, \pfun, \tfun, \psur, \tsur$ A
*
DERIV_ID_BIJ
$\id \in\mathit{Ty}\ \mathit{op}\ \mathit{Ty}\;\;\defi\;\; \btrue$ where $\mathit{Ty}$ is a type and $\mathit{op}$ is any arrow A
*
SIMP_MAPSTO_PRJ1_PRJ2
$\prjone(E)\mapsto\prjtwo(E)\;\;\defi\;\; E$ A
DERIV_EXPAND_PRJS
$E \;\;\defi\;\; \prjone(E) \mapsto \prjtwo(E)$ M
*
SIMP_DOM_SUCC
$\dom(\usucc) \;\;\defi\;\; \intg$ A
*
SIMP_RAN_SUCC
$\ran(\usucc) \;\;\defi\;\; \intg$ A
*
DERIV_MULTI_IN_BUNION
$E\in A\bunion\cdots\bunion\left\{\cdots, E,\cdots\right\}\bunion\cdots\bunion B\;\;\defi\;\; \btrue$ A
*
DERIV_MULTI_IN_SETMINUS
$E\in S\setminus\left\{\cdots, E,\cdots\right\} \;\;\defi\;\; \bfalse$ A
*
DEF_PRED
$\upred\;\;\defi\;\; \usucc^{-1}$ A

Rules that are marked with a * in the first column are implemented in the latest version of Rodin. Rules without a * are planned to be implemented in future versions. Other conventions used in these tables are described in The_Proving_Perspective_(Rodin_User_Manual)#Rewrite_Rules.

Name Rule Side Condition A/M
*
SIMP_SPECIAL_MOD_0
$0 \,\bmod\, E \;\;\defi\;\; 0$ A
*
SIMP_SPECIAL_MOD_1
$E \,\bmod\, 1 \;\;\defi\;\; 0$ A
*
SIMP_MIN_SING
$\min (\{ E\} ) \;\;\defi\;\; E$ where $E$ is a single expression A
*
SIMP_MAX_SING
$\max (\{ E\} ) \;\;\defi\;\; E$ where $E$ is a single expression A
*
SIMP_MIN_NATURAL
$\min (\nat ) \;\;\defi\;\; 0$ A
*
SIMP_MIN_NATURAL1
$\min (\natn ) \;\;\defi\;\; 1$ A
*
SIMP_MIN_BUNION_SING
$\begin{array}{cl} & \min (S \bunion \ldots \bunion \{ \min (T)\} \bunion \ldots \bunion U) \\ \defi & \min (S \bunion \ldots \bunion T \bunion \ldots \bunion U) \\ \end{array}$ A
*
SIMP_MAX_BUNION_SING
$\begin{array}{cl} & \max (S \bunion \ldots \bunion \{ \max (T)\} \bunion \ldots \bunion U) \\ \defi & \max (S \bunion \ldots \bunion T \bunion \ldots \bunion U) \\ \end{array}$ A
*
SIMP_MIN_UPTO
$\min (E \upto F) \;\;\defi\;\; E$ A
*
SIMP_MAX_UPTO
$\max (E \upto F) \;\;\defi\;\; F$ A
*
SIMP_LIT_MIN
$\min (\{ E, \ldots , i, \ldots , j, \ldots , H\} ) \;\;\defi\;\; \min (\{ E, \ldots , i, \ldots , H\} )$ where $i$ and $j$ are literals and $i \leq j$ A
*
SIMP_LIT_MAX
$\max (\{ E, \ldots , i, \ldots , j, \ldots , H\} ) \;\;\defi\;\; \max (\{ E, \ldots , i, \ldots , H\} )$ where $i$ and $j$ are literals and $i \geq j$ A
*
SIMP_SPECIAL_CARD
$\card (\emptyset ) \;\;\defi\;\; 0$ A
*
SIMP_CARD_SING
$\card (\{ E\} ) \;\;\defi\;\; 1$ where $E$ is a single expression A
*
SIMP_SPECIAL_EQUAL_CARD
$\card (S) = 0 \;\;\defi\;\; S = \emptyset$ A
*
SIMP_CARD_POW
$\card (\pow (S)) \;\;\defi\;\; 2\expn{\card(S)}$ A
*
SIMP_CARD_BUNION
$\card (S \bunion T) \;\;\defi\;\; \card (S) + \card (T) - \card (S \binter T)$ A
SIMP_CARD_SETMINUS
$\card(S\setminus T)\;\;\defi\;\;\card(S) - \card(T)$ with hypotheses $T\subseteq S$ and either $\finite(S)$ or $\finite(T)$ A
SIMP_CARD_SETMINUS_SETENUM
$\card(S\setminus\{E_1,\ldots,E_n\})\;\;\defi\;\;\card(S) - \card(\{E_1,\ldots,E_n\})$ with hypotheses $E_i\in S$ for all $i\in 1\upto n$ A
*
SIMP_CARD_CONVERSE
$\card (r^{-1} ) \;\;\defi\;\; \card (r)$ A
*
SIMP_CARD_ID
$\card (\id) \;\;\defi\;\; \card (S)$ where $\id$ has type $\pow (S \cprod S)$ A
*
SIMP_CARD_ID_DOMRES
$\card (S\domres\id) \;\;\defi\;\; \card (S)$ A
*
SIMP_CARD_PRJ1
$\card (\prjone) \;\;\defi\;\; \card (S \cprod T)$ where $\prjone$ has type $\pow(S \cprod T \cprod S)$ A
*
SIMP_CARD_PRJ2
$\card (\prjtwo) \;\;\defi\;\; \card (S \cprod T)$ where $\prjtwo$ has type $\pow(S \cprod T \cprod T)$ A
*
SIMP_CARD_PRJ1_DOMRES
$\card (E \domres \prjone) \;\;\defi\;\; \card (E)$ A
*
SIMP_CARD_PRJ2_DOMRES
$\card (E \domres \prjtwo) \;\;\defi\;\; \card (E)$ A
*
SIMP_CARD_LAMBDA
$\card(\{x\qdot P\mid E\mapsto F\}) \;\;\defi\;\; \card(\{x\qdot P\mid E\} )$ where $E$ is a maplet combination of bound identifiers and expressions that are not bound by the comprehension set (i.e., $E$ is syntactically injective) and all identifiers bound by the comprehension set that occur in $F$ also occur in $E$ A
*
SIMP_LIT_CARD_UPTO
$\card (i \upto j) \;\;\defi\;\; j-i+1$ where $i$ and $j$ are literals and $i \leq j$ A
SIMP_TYPE_CARD
$\card (\mathit{Tenum}) \;\;\defi\;\; N$ where $\mathit{Tenum}$ is a carrier set containing $N$ elements A
*
SIMP_LIT_GE_CARD_1
$\card (S) \geq 1 \;\;\defi\;\; \lnot\, S = \emptyset$ A
*
SIMP_LIT_LE_CARD_1
$1 \leq \card (S) \;\;\defi\;\; \lnot\, S = \emptyset$ A
*
SIMP_LIT_LE_CARD_0
$0 \leq \card (S) \;\;\defi\;\; \btrue$ A
*
SIMP_LIT_GE_CARD_0
$\card (S) \geq 0 \;\;\defi\;\; \btrue$ A
*
SIMP_LIT_GT_CARD_0
$\card (S) > 0 \;\;\defi\;\; \lnot\, S = \emptyset$ A
*
SIMP_LIT_LT_CARD_0
$0 < \card (S) \;\;\defi\;\; \lnot\, S = \emptyset$ A
*
SIMP_LIT_EQUAL_CARD_1
$\card (S) = 1 \;\;\defi\;\; \exists x \qdot S = \{ x\}$ A
*
SIMP_CARD_NATURAL
$\card (S) \in \nat \;\;\defi\;\; \btrue$ A
*
SIMP_CARD_NATURAL1
$\card (S) \in \natn \;\;\defi\;\; \lnot\, S = \emptyset$ A
*
SIMP_LIT_IN_NATURAL
$i \in \nat \;\;\defi\;\; \btrue$ where $i$ is a non-negative literal A
*
SIMP_SPECIAL_IN_NATURAL1
$0 \in \natn \;\;\defi\;\; \bfalse$ A
*
SIMP_LIT_IN_NATURAL1
$i \in \natn \;\;\defi\;\; \btrue$ where $i$ is a positive literal A
*
SIMP_LIT_UPTO
$i \upto j \;\;\defi\;\; \emptyset$ where $i$ and $j$ are literals and $j < i$ A
*
SIMP_LIT_IN_MINUS_NATURAL
$-i \in \nat \;\;\defi\;\; \bfalse$ where $i$ is a positive literal A
*
SIMP_LIT_IN_MINUS_NATURAL1
$-i \in \natn \;\;\defi\;\; \bfalse$ where $i$ is a non-negative literal A
*
DEF_IN_NATURAL
$x \in \nat \;\;\defi\;\; 0 \leq x$ M
*
DEF_IN_NATURAL1
$x \in \natn \;\;\defi\;\; 1 \leq x$ M
*
SIMP_LIT_EQUAL_KBOOL_TRUE
$\bool (P) = \True \;\;\defi\;\; P$ A
*
SIMP_LIT_EQUAL_KBOOL_FALSE
$\bool (P) = \False \;\;\defi\;\; \lnot\, P$ A
DEF_EQUAL_MIN
$E = \min (S) \;\;\defi\;\; E \in S \land (\forall x \qdot x \in S \limp E \leq x)$ where $x$ non free in $S, E$ M
DEF_EQUAL_MAX
$E = \max (S) \;\;\defi\;\; E \in S \land (\forall x \qdot x \in S \limp E \geq x)$ where $x$ non free in $S, E$ M
*
SIMP_SPECIAL_PLUS
$E + \ldots + 0 + \ldots + F \;\;\defi\;\; E + \ldots + F$ A
*
SIMP_SPECIAL_MINUS_R
$E - 0 \;\;\defi\;\; E$ A
*
SIMP_SPECIAL_MINUS_L
$0 - E \;\;\defi\;\; -E$ A
*
SIMP_MINUS_MINUS
$- (- E) \;\;\defi\;\; E$ A
*
SIMP_MINUS_UNMINUS
$E - (- F) \;\;\defi\;\; E + F$ where $(-F)$ is a unary minus expression or a negative literal M
*
SIMP_MULTI_MINUS
$E - E \;\;\defi\;\; 0$ A
*
SIMP_MULTI_MINUS_PLUS_L
$(A + \ldots + C + \ldots + B) - C \;\;\defi\;\; A + \ldots + B$ M
*
SIMP_MULTI_MINUS_PLUS_R
$C - (A + \ldots + C + \ldots + B) \;\;\defi\;\; -(A + \ldots + B)$ M
*
SIMP_MULTI_MINUS_PLUS_PLUS
$(A + \ldots + E + \ldots + B) - (C + \ldots + E + \ldots + D) \;\;\defi\;\; (A + \ldots + B) - (C + \ldots + D)$ M
*
SIMP_MULTI_PLUS_MINUS
$(A + \ldots + D + \ldots + (C - D) + \ldots + B) \;\;\defi\;\; A + \ldots + C + \ldots + B$ M
*
SIMP_MULTI_ARITHREL_PLUS_PLUS
$A + \ldots + E + \ldots + B < C + \ldots + E + \ldots + D \;\;\defi\;\; A + \ldots + B < C + \ldots + D$ where the root relation ($<$ here) is one of $\{=, <, \leq, >, \geq\}$ M
*
SIMP_MULTI_ARITHREL_PLUS_R
$C < A + \ldots + C \ldots + B \;\;\defi\;\; 0 < A + \ldots + B$ where the root relation ($<$ here) is one of $\{=, <, \leq, >, \geq\}$ M
*
SIMP_MULTI_ARITHREL_PLUS_L
$A + \ldots + C \ldots + B < C \;\;\defi\;\; A + \ldots + B < 0$ where the root relation ($<$ here) is one of $\{=, <, \leq, >, \geq\}$ M
*
SIMP_MULTI_ARITHREL_MINUS_MINUS_R
$A - C < B - C \;\;\defi\;\; A < B$ where the root relation ($<$ here) is one of $\{=, <, \leq, >, \geq\}$ M
*
SIMP_MULTI_ARITHREL_MINUS_MINUS_L
$C - A < C - B \;\;\defi\;\; B < A$ where the root relation ($<$ here) is one of $\{=, <, \leq, >, \geq\}$ M
*
SIMP_SPECIAL_PROD_0
$E * \ldots * 0 * \ldots * F \;\;\defi\;\; 0$ A
*
SIMP_SPECIAL_PROD_1
$E * \ldots * 1 * \ldots * F \;\;\defi\;\; E * \ldots * F$ A
*
SIMP_SPECIAL_PROD_MINUS_EVEN
$(-E) * \ldots * (-F) \;\;\defi\;\; E * \ldots * F$ if an even number of $-$ A
*
SIMP_SPECIAL_PROD_MINUS_ODD
$(-E) * \ldots * (-F) \;\;\defi\;\; -(E * \ldots * F)$ if an odd number of $-$ A
*
SIMP_LIT_MINUS
$- (i) \;\;\defi\;\; (-i)$ where $i$ is a literal A
*
SIMP_LIT_EQUAL
$i = j \;\;\defi\;\; \btrue \;or\; \bfalse \;\;(computation)$ where $i$ and $j$ are literals A
*
SIMP_LIT_LE
$i \leq j \;\;\defi\;\; \btrue \;or\; \bfalse \;\;(computation)$ where $i$ and $j$ are literals A
*
SIMP_LIT_LT
$i < j \;\;\defi\;\; \btrue \;or\; \bfalse \;\;(computation)$ where $i$ and $j$ are literals A
*
SIMP_LIT_GE
$i \geq j \;\;\defi\;\; \btrue \;or\; \bfalse \;\;(computation)$ where $i$ and $j$ are literals A
*
SIMP_LIT_GT
$i > j \;\;\defi\;\; \btrue \;or\; \bfalse \;\;(computation)$ where $i$ and $j$ are literals A
*
SIMP_DIV_MINUS
$(- E) \div (-F) \;\;\defi\;\; E \div F$ A
*
SIMP_SPECIAL_DIV_1
$E \div 1 \;\;\defi\;\; E$ A
*
SIMP_SPECIAL_DIV_0
$0 \div E \;\;\defi\;\; 0$ A
*
SIMP_SPECIAL_EXPN_1_R
$E ^ 1 \;\;\defi\;\; E$ A
*
SIMP_SPECIAL_EXPN_1_L
$1 ^ E \;\;\defi\;\; 1$ A
*
SIMP_SPECIAL_EXPN_0
$E ^ 0 \;\;\defi\;\; 1$ A
*
SIMP_MULTI_LE
$E \leq E \;\;\defi\;\; \btrue$ A
*
SIMP_MULTI_LT
$E < E \;\;\defi\;\; \bfalse$ A
*
SIMP_MULTI_GE
$E \geq E \;\;\defi\;\; \btrue$ A
*
SIMP_MULTI_GT
$E > E \;\;\defi\;\; \bfalse$ A
*
SIMP_MULTI_DIV
$E \div E \;\;\defi\;\; 1$ A
*
SIMP_MULTI_DIV_PROD
$(X * \ldots * E * \ldots * Y) \div E \;\;\defi\;\; X * \ldots * Y$ A
*
SIMP_MULTI_MOD
$E \,\bmod\, E \;\;\defi\;\; 0$ A
DISTRI_PROD_PLUS
$a * (b + c) \;\;\defi\;\; (a * b) + (a * c)$ M
DISTRI_PROD_MINUS
$a * (b - c) \;\;\defi\;\; (a * b) - (a * c)$ M
DERIV_NOT_EQUAL
$\lnot E = F \;\;\defi\;\; E < F \lor E > F$ $E$ and $F$ must be of Integer type M