# Set Rewrite Rules

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