Relation Rewrite Rules

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Revision as of 10:41, 23 November 2009 by imported>Nicolas (updated rules SIMP_FUNIMAGE_PRJ1 AND SIMP_FUNIMAGE_PRJ2 to math V2)
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  Name Rule Side Condition A/M
*
SIMP_DOM_COMPSET
 \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_COMPSET
 \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
 \begin{array}{cl} & r \ovl \ldots \ovl s \ovl \ldots \ovl s \ovl \ldots \ovl u \\ \defi & r \ovl \ldots \ovl \ldots \ovl s \ovl \ldots \ovl u \end{array} A
*
SIMP_TYPE_OVERL_CPROD
 r \ovl (\mathit{Ty} \cprod S) \;\;\defi\;\; (\mathit{Ty} \cprod 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_ID
 S \domres \id (T) \;\;\defi\;\; \id (S \binter T) 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 (S) \ranres T \;\;\defi\;\; \id (S \binter T) 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_DOMSUB_ID
 S \domsub \id (T) \;\;\defi\;\; \id (T \setminus S) 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_RAN
 r \ransub \ran (r) \;\;\defi\;\; \emptyset A
*
SIMP_RANSUB_ID
 \id (S) \ransub T \;\;\defi\;\; \id (S \setminus T) 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 \id (S) \fcomp \id (T) \fcomp \ldots s \;\;\defi\;\; r \fcomp \ldots \fcomp \id (S \binter T) \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 \id (S) \bcomp \id (T) \bcomp \ldots \bcomp s \;\;\defi\;\; r \bcomp \ldots \bcomp \id (S \binter T) \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_TYPE_DPROD
 \mathit{Ta} \dprod \mathit{Tb} \;\;\defi\;\; Tc \cprod (Td \cprod Te) where \mathit{Ta} and \mathit{Tb} are type expressions and \mathit{Ta} = \mathit{Tc} \cprod \mathit{Td} and \mathit{Tb} = \mathit{Tc} \cprod \mathit{Te} A
SIMP_SPECIAL_PPROD_R
 r \pprod \emptyset \;\;\defi\;\; \emptyset A
SIMP_SPECIAL_PPROD_L
 \emptyset \pprod r \;\;\defi\;\; \emptyset A
*
SIMP_TYPE_PPROD
 \mathit{Ta} \pprod \mathit{Tb} \;\;\defi\;\; (Tc \cprod Te) \cprod (Td \cprod Tf) where \mathit{Ta} and \mathit{Tb} are type expressions and \mathit{Ta} = \mathit{Tc} \cprod \mathit{Td} and \mathit{Tb} = \mathit{Te} \cprod \mathit{Tf} 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_TYPE_RELIMAGE_ID
 \id[T] \;\;\defi\;\; T A
*
SIMP_RELIMAGE_ID
 \id (S)[T] \;\;\defi\;\; S \binter T 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_TYPE_CONVERSE
 \mathit{Ty}^{-1} \;\;\defi\;\; \mathit{Tb} \cprod \mathit{Ta} where \mathit{Ty} is a type expression equal to \mathit{Ta} \cprod \mathit{Tb} 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, \ldots , y \qdot P \mid x, \ldots , y\} ^{-1} \;\;\defi\;\; \{ y, \ldots , x \qdot P \mid y, \ldots , x\} A
SIMP_SPECIAL_ID
 \id (\emptyset ) \;\;\defi\;\; \emptyset 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
 (\id (S) \fcomp r) \;\;\defi\;\; S \domres r A
*
SIMP_FCOMP_ID_R
 r \fcomp \id (S) \;\;\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_SPECIAL_EQUAL_REL
 A \rel B = \emptyset \;\;\defi\;\; \bfalse idem for operators \pfun \pinj A
SIMP_SPECIAL_EQUAL_RELDOM
 A \trel B = \emptyset \;\;\defi\;\; \lnot\, A = \emptyset \land B = \emptyset idem for operators \tfun \tinj \tsur \tbij A
SIMP_SPECIAL_PRJ1
 \prjone (\emptyset ) \;\;\defi\;\; \emptyset A
SIMP_SPECIAL_PRJ2
 \prjtwo (\emptyset ) \;\;\defi\;\; \emptyset 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 (r)) \;\;\defi\;\; r A
*
SIMP_DOM_PRJ2
 \dom (\prjtwo (r)) \;\;\defi\;\; r A
*
SIMP_RAN_PRJ1
 \ran (\prjone (r)) \;\;\defi\;\; \dom (r) A
*
SIMP_RAN_PRJ2
 \ran (\prjtwo (r)) \;\;\defi\;\; \ran (r) A
SIMP_SPECIAL_LAMBDA
 (\lambda x \qdot \bfalse \mid E) \;\;\defi\;\; \emptyset A
*
SIMP_FUNIMAGE_LAMBDA
 (\lambda x \qdot P(x) \mid E(x))(y) \;\;\defi\;\; E(y) A
*
SIMP_DOM_LAMBDA
 \dom (\lambda x \qdot P \mid E) \;\;\defi\;\; \{ x\qdot P \mid x\} A
*
SIMP_RAN_LAMBDA
 \ran (\lambda x \qdot P \mid E) \;\;\defi\;\; \{ x\qdot P \mid E\} A
*
SIMP_MULTI_FUNIMAGE_SETENUM_LL
 \{ A \mapsto E, \ldots , B \mapsto E\} (x) \;\;\defi\;\; E A
*
SIMP_MULTI_FUNIMAGE_SETENUM_LR
 \{ A \mapsto E, \ldots , x \mapsto y, \ldots , B \mapsto F\} (x) \;\;\defi\;\; y A
*
SIMP_MULTI_FUNIMAGE_OVERL_SETENUM
 (r \ovl \ldots \ovl \{ A \mapsto E, \ldots , x \mapsto y, \ldots , B \mapsto F\} )(x) \;\;\defi\;\; y A
*
SIMP_MULTI_FUNIMAGE_BUNION_SETENUM
 (r \bunion \ldots \bunion \{ A \mapsto E, \ldots , x \mapsto y, \ldots , B \mapsto F\} )(x) \;\;\defi\;\; y A
*
SIMP_FUNIMAGE_CPROD
 (S \cprod \{ F\} )(x) \;\;\defi\;\; F A
*
SIMP_FUNIMAGE_ID
 \id (S)(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
*
DEF_BCOMP
 r \bcomp \ldots \bcomp s \;\;\defi\;\; s \fcomp \ldots \fcomp r M
*
DERIV_ID_SING
 \id (\{ E\} ) \;\;\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
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} \rel \mathit{Tb} A
*
SIMP_TYPE_RAN
 \ran (\mathit{Ty}) \;\;\defi\;\; \mathit{Tb} where \mathit{Ty} is a type expression equal to \mathit{Ta} \rel \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
*
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 (S) \;\;\defi\;\; E \in S \land F = E 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_RELDOM
 r \in S \trel T \;\;\defi\;\; r \in S \rel T \land \dom (r) = S M
*
DEF_IN_RELRAN
 r \in S \trel 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 \pfun 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 s) 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) \bunion (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) \binter (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 \bunion s) \ransub t \;\;\defi\;\; (r \ransub t) \bunion (s \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 \binter s) \ransub t \;\;\defi\;\; (r \ransub t) \binter (s \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
*
DISTRI_DOM_BUNION
 \dom (p \bunion q) \;\;\defi\;\; \dom (p) \bunion \dom (q) M
*
DISTRI_RAN_BUNION
 \ran (p \bunion q) \;\;\defi\;\; \ran (p) \bunion \ran (q) 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
prjone-total
 z \in \dom (\prjone) \;\;\defi\;\; \btrue A
prjtwo-total
 z \in \dom (\prjtwo) \;\;\defi\;\; \btrue A
prjone-functional
 \prjone \in E\ \mathit{op}\ F \;\;\defi\;\; \btrue where \mathit{op} is one of \pfun, \tfun, \trel A
prjtwo-functional
 \prjtwo \in E\ \mathit{op}\ F \;\;\defi\;\; \btrue where \mathit{op} is one of \pfun, \tfun, \trel A
prj-expand
 x \;\;\defi\;\; \prjone(x) \mapsto \prjtwo(x) M
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