Difference between revisions of "Relation Rewrite Rules"

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imported>Nicolas
m (updated rule SIMP_CONVERSE_ID to math V2)
imported>Nicolas
m (updated SIMP_DOM_ID and SIMP_RAN_ID to math language V2)
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{{RRRow}}|*||{{Rulename|SIMP_CONVERSE_COMPSET}}||<math>  \{ x, \ldots , y \qdot  P \mid  x, \ldots , y\} ^{-1}  \;\;\defi\;\;  \{ y, \ldots , x \qdot  P \mid  y, \ldots , x\} </math>||  ||  A
 
{{RRRow}}|*||{{Rulename|SIMP_CONVERSE_COMPSET}}||<math>  \{ x, \ldots , y \qdot  P \mid  x, \ldots , y\} ^{-1}  \;\;\defi\;\;  \{ y, \ldots , x \qdot  P \mid  y, \ldots , x\} </math>||  ||  A
 
{{RRRow}}| ||{{Rulename|SIMP_SPECIAL_ID}}||<math>  \id (\emptyset ) \;\;\defi\;\;  \emptyset </math>||  ||  A
 
{{RRRow}}| ||{{Rulename|SIMP_SPECIAL_ID}}||<math>  \id (\emptyset ) \;\;\defi\;\;  \emptyset </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_DOM_ID}}||<math>  \dom (\id (S)) \;\;\defi\;\;  S </math>|| ||  A
+
{{RRRow}}|*||{{Rulename|SIMP_DOM_ID}}||<math>  \dom (\id) \;\;\defi\;\;  S </math>|| where <math>\id</math> has type <math>\pow(S \cprod S)</math> ||  A
{{RRRow}}|*||{{Rulename|SIMP_RAN_ID}}||<math>  \ran (\id (S)) \;\;\defi\;\;  S </math>|| ||  A
+
{{RRRow}}|*||{{Rulename|SIMP_RAN_ID}}||<math>  \ran (\id) \;\;\defi\;\;  S </math>|| where <math>\id</math> has type <math>\pow(S \cprod S)</math> ||  A
 
{{RRRow}}|*||{{Rulename|SIMP_FCOMP_ID_L}}||<math>  (\id (S) \fcomp r) \;\;\defi\;\;  S \domres  r </math>||  ||  A
 
{{RRRow}}|*||{{Rulename|SIMP_FCOMP_ID_L}}||<math>  (\id (S) \fcomp r) \;\;\defi\;\;  S \domres  r </math>||  ||  A
 
{{RRRow}}|*||{{Rulename|SIMP_FCOMP_ID_R}}||<math>  r \fcomp \id (S) \;\;\defi\;\;  r \ranres  S </math>||  ||  A
 
{{RRRow}}|*||{{Rulename|SIMP_FCOMP_ID_R}}||<math>  r \fcomp \id (S) \;\;\defi\;\;  r \ranres  S </math>||  ||  A

Revision as of 10:34, 23 November 2009

  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 (r)(E \mapsto  F) \;\;\defi\;\;  E A
*
SIMP_FUNIMAGE_PRJ2
  \prjtwo (r)(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
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DISTRI_DOMRES_BUNION_R
  s \domres  (p \bunion  q) \;\;\defi\;\;  (s \domres  p) \bunion  (s \domres  q) M
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DISTRI_DOMRES_BUNION_L
  (s \bunion  t) \domres  r \;\;\defi\;\;  (s \domres  r) \bunion  (t \domres  r) M
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DISTRI_DOMRES_BINTER_R
  s \domres  (p \binter  q) \;\;\defi\;\;  (s \domres  p) \binter  (s \domres  q) M
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DISTRI_DOMRES_BINTER_L
  (s \binter  t) \domres  r \;\;\defi\;\;  (s \domres  r) \binter  (t \domres  r) M
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DISTRI_DOMRES_SETMINUS_R
  s \domres  (p \setminus q) \;\;\defi\;\;  (s \domres  p) \setminus (s \domres  q) M
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DISTRI_DOMRES_SETMINUS_L
  (s \setminus t) \domres  r \;\;\defi\;\;  (s \domres  r) \setminus (t \domres  r) M
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DISTRI_DOMRES_DPROD
  s \domres  (p \dprod  q) \;\;\defi\;\;  (s \domres  p) \dprod  (s \domres  q) M
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DISTRI_DOMRES_OVERL
  s \domres  (r \ovl  q) \;\;\defi\;\;  (s \domres  r) \ovl  (s \domres  q) M
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DISTRI_DOMSUB_BUNION_R
  s \domsub  (p \bunion  q) \;\;\defi\;\;  (s \domsub  p) \bunion  (s \domsub  q) M
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DISTRI_DOMSUB_BUNION_L
  (s \bunion  t) \domsub  r \;\;\defi\;\;  (s \domsub  r) \bunion  (t \domsub  r) M
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DISTRI_DOMSUB_BINTER_R
  s \domsub  (p \binter  q) \;\;\defi\;\;  (s \domsub  p) \binter  (s \domsub  q) M
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DISTRI_DOMSUB_BINTER_L
  (s \binter  t) \domsub  r \;\;\defi\;\;  (s \domsub  r) \binter  (t \domsub  r) M
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DISTRI_DOMSUB_DPROD
  A \domsub  (r \dprod  s) \;\;\defi\;\;  (A \domsub  r) \dprod  (A \domsub  s) M
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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
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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
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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
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DISTRI_RANSUB_BINTER_L
  (p \binter  q) \ransub  s \;\;\defi\;\;  (p \ransub  s) \binter  (q \ransub  s) M
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DISTRI_CONVERSE_BUNION
  (p \bunion  q)^{-1}  \;\;\defi\;\;  p^{-1}  \bunion  q^{-1} M
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DISTRI_CONVERSE_BINTER
  (p \binter  q)^{-1}  \;\;\defi\;\;  p^{-1}  \binter  q^{-1} M
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DISTRI_CONVERSE_SETMINUS
  (r \setminus s)^{-1}  \;\;\defi\;\;  r^{-1}  \setminus s^{-1} M
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DISTRI_CONVERSE_BCOMP
  (r \bcomp  s)^{-1}  \;\;\defi\;\;  (s^{-1}  \bcomp  r^{-1} ) M
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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