Relation Rewrite Rules: Difference between revisions

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imported>Laurent
Removed SIMP_SPECIAL_LAMBDA (subsumed by SIMP_COMPSET_BFALSE) and generalized SIMP_DOM_LAMBDA and SIMP_RAN_LAMBDA.
Guillaume (talk | contribs)
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Rules DEF_EQUAL_FUN_IMAGE and SIMP_SPECIAL_IN_*ID have been implemented in Rodin 3.9
 
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{{RRRow}}|*||{{Rulename|SIMP_DOM_COMPSET}}||<math>  \dom (\{ x \mapsto  a, \ldots , y \mapsto  b\} ) \;\;\defi\;\;  \{ x, \ldots , y\} </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_DOM_SETENUM}}||<math>  \dom (\{ x \mapsto  a, \ldots , y \mapsto  b\} ) \;\;\defi\;\;  \{ x, \ldots , y\} </math>||  ||  A
{{RRRow}}|||{{Rulename|SIMP_DOM_CONVERSE}}||<math>  \dom (r^{-1} ) \;\;\defi\;\;  \ran (r) </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_DOM_CONVERSE}}||<math>  \dom (r^{-1} ) \;\;\defi\;\;  \ran (r) </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_RAN_COMPSET}}||<math>  \ran (\{ x \mapsto  a, \ldots , y \mapsto  b\} ) \;\;\defi\;\;  \{ a, \ldots , b\} </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_RAN_SETENUM}}||<math>  \ran (\{ x \mapsto  a, \ldots , y \mapsto  b\} ) \;\;\defi\;\;  \{ a, \ldots , b\} </math>||  ||  A
{{RRRow}}|||{{Rulename|SIMP_RAN_CONVERSE}}||<math>  \ran (r^{-1} ) \;\;\defi\;\;  \dom (r) </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_RAN_CONVERSE}}||<math>  \ran (r^{-1} ) \;\;\defi\;\;  \dom (r) </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_SPECIAL_OVERL}}||<math>  r \ovl  \ldots  \ovl  \emptyset  \ovl  \ldots  \ovl  s \;\;\defi\;\;  r \ovl  \ldots  \ovl  s </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_SPECIAL_OVERL}}||<math>  r \ovl  \ldots  \ovl  \emptyset  \ovl  \ldots  \ovl  s \;\;\defi\;\;  r \ovl  \ldots  \ovl  s </math>||  ||  A
{{RRRow}}| ||{{Rulename|SIMP_MULTI_OVERL}}||<math>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</math>|| there is a <math>j</math> such that <math>1\leq i < j \leq n</math> and <math>r_i</math> and <math>r_j</math> are syntactically equal. ||  A
{{RRRow}}|*||{{Rulename|SIMP_MULTI_OVERL}}||<math>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</math>|| there is a <math>j</math> such that <math>1\leq i < j \leq n</math> and <math>r_i</math> and <math>r_j</math> are syntactically equal. ||  A
{{RRRow}}| ||{{Rulename|SIMP_TYPE_OVERL_CPROD}}||<math> r \ovl\dots\ovl \mathit{Ty} \;\;\defi\;\; \mathit{Ty} </math>|| where <math>\mathit{Ty}</math> is a type expression ||  A
{{RRRow}}|*||{{Rulename|SIMP_TYPE_OVERL_CPROD}}||<math> r\ovl\cdots\ovl\mathit{Ty}\ovl\cdots\ovl s \;\defi\;\; \mathit{Ty}\ovl\cdots\ovl s </math>|| where <math>\mathit{Ty}</math> is a type expression ||  A
{{RRRow}}| ||{{Rulename|SIMP_SPECIAL_DOMRES_L}}||<math>  \emptyset  \domres  r \;\;\defi\;\;  \emptyset </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_SPECIAL_DOMRES_L}}||<math>  \emptyset  \domres  r \;\;\defi\;\;  \emptyset </math>||  ||  A
{{RRRow}}| ||{{Rulename|SIMP_SPECIAL_DOMRES_R}}||<math>  S \domres  \emptyset  \;\;\defi\;\;  \emptyset </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_SPECIAL_DOMRES_R}}||<math>  S \domres  \emptyset  \;\;\defi\;\;  \emptyset </math>||  ||  A
{{RRRow}}|||{{Rulename|SIMP_TYPE_DOMRES}}||<math> \mathit{Ty} \domres  r \;\;\defi\;\;  r </math>|| where <math>\mathit{Ty}</math> is a type expression ||  A
{{RRRow}}|*||{{Rulename|SIMP_TYPE_DOMRES}}||<math> \mathit{Ty} \domres  r \;\;\defi\;\;  r </math>|| where <math>\mathit{Ty}</math> is a type expression ||  A
{{RRRow}}|||{{Rulename|SIMP_MULTI_DOMRES_DOM}}||<math>  \dom (r) \domres  r \;\;\defi\;\;  r </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_MULTI_DOMRES_DOM}}||<math>  \dom (r) \domres  r \;\;\defi\;\;  r </math>||  ||  A
{{RRRow}}|||{{Rulename|SIMP_MULTI_DOMRES_RAN}}||<math>  \ran (r) \domres  r^{-1}  \;\;\defi\;\;  r^{-1} </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_MULTI_DOMRES_RAN}}||<math>  \ran (r) \domres  r^{-1}  \;\;\defi\;\;  r^{-1} </math>||  ||  A
{{RRRow}}|||{{Rulename|SIMP_DOMRES_ID}}||<math>  S \domres  (T \domres \id) \;\;\defi\;\;  (S \binter  T) \domres \id </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_DOMRES_DOMRES_ID}}||<math>  S \domres  (T \domres \id) \;\;\defi\;\;  (S \binter  T) \domres \id </math>||  ||  A
{{RRRow}}| ||{{Rulename|SIMP_SPECIAL_RANRES_R}}||<math>  r \ranres \emptyset  \;\;\defi\;\;  \emptyset </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_DOMRES_DOMSUB_ID}}||<math>  S \domres (T \domsub \id) \;\;\defi\;\;  (S \setminus  T) \domres \id </math>||  ||  A
{{RRRow}}| ||{{Rulename|SIMP_SPECIAL_RANRES_L}}||<math>  \emptyset \ranres S \;\;\defi\;\;  \emptyset </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_SPECIAL_RANRES_R}}||<math>  r \ranres \emptyset \;\;\defi\;\;  \emptyset </math>||  ||  A
{{RRRow}}|||{{Rulename|SIMP_TYPE_RANRES}}||<math>  r \ranres \mathit{Ty} \;\;\defi\;\;  r </math>|| where <math>\mathit{Ty}</math> is a type expression ||  A
{{RRRow}}|*||{{Rulename|SIMP_SPECIAL_RANRES_L}}||<math>  \emptyset \ranres  S \;\;\defi\;\;  \emptyset </math>|| ||  A
{{RRRow}}|||{{Rulename|SIMP_MULTI_RANRES_RAN}}||<math>  r \ranres  \ran (r) \;\;\defi\;\;  r </math>|| ||  A
{{RRRow}}|*||{{Rulename|SIMP_TYPE_RANRES}}||<math>  r \ranres  \mathit{Ty} \;\;\defi\;\;  r </math>|| where <math>\mathit{Ty}</math> is a type expression ||  A
{{RRRow}}|||{{Rulename|SIMP_MULTI_RANRES_DOM}}||<math>  r^{-1}  \ranres  \dom (r) \;\;\defi\;\;  r^{-1} </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_MULTI_RANRES_RAN}}||<math>  r \ranres  \ran (r) \;\;\defi\;\;  r </math>||  ||  A
{{RRRow}}|||{{Rulename|SIMP_RANRES_ID}}||<math>  (S \domres \id) \ranres  T \;\;\defi\;\;  (S \binter  T) \domres \id </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_MULTI_RANRES_DOM}}||<math>  r^{-1}  \ranres  \dom (r) \;\;\defi\;\;  r^{-1} </math>||  ||  A
{{RRRow}}| ||{{Rulename|SIMP_SPECIAL_DOMSUB_L}}||<math>  \emptyset \domsub r \;\;\defi\;\;  r </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_RANRES_ID}}||<math>  \id \ranres S \;\;\defi\;\;  S  \domres \id </math>||  ||  A
{{RRRow}}| ||{{Rulename|SIMP_SPECIAL_DOMSUB_R}}||<math>  S \domsub \emptyset \;\;\defi\;\;  \emptyset </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_RANSUB_ID}}||<math>  \id \ransub S \;\;\defi\;\;  S  \domsub \id </math>||  ||  A
{{RRRow}}|||{{Rulename|SIMP_TYPE_DOMSUB}}||<math> \mathit{Ty} \domsub r \;\;\defi\;\;  \emptyset </math>|| where <math>\mathit{Ty}</math> is a type expression ||  A
{{RRRow}}|*||{{Rulename|SIMP_RANRES_DOMRES_ID}}||<math> (S \domres \id) \ranres T \;\;\defi\;\;  (S \binter  T) \domres \id </math>|| ||  A
{{RRRow}}|||{{Rulename|SIMP_MULTI_DOMSUB_DOM}}||<math>  \dom (r) \domsub r \;\;\defi\;\;  \emptyset </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_RANRES_DOMSUB_ID}}||<math>  (S \domsub \id) \ranres  T \;\;\defi\;\;  (T \setminus  S) \domres \id </math>||  ||  A
{{RRRow}}|||{{Rulename|SIMP_DOMSUB_ID}}||<math>  S \domsub (T \domres \id ) \;\;\defi\;\;  (T \setminus S) \domres \id </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_SPECIAL_DOMSUB_L}}||<math>  \emptyset  \domsub r \;\;\defi\;\;  r </math>||  ||  A
{{RRRow}}| ||{{Rulename|SIMP_SPECIAL_RANSUB_R}}||<math>  r \ransub \emptyset  \;\;\defi\;\;  r </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_SPECIAL_DOMSUB_R}}||<math>  S \domsub \emptyset  \;\;\defi\;\;  \emptyset </math>||  ||  A
{{RRRow}}| ||{{Rulename|SIMP_SPECIAL_RANSUB_L}}||<math> \emptyset  \ransub S \;\;\defi\;\;  \emptyset </math>|| ||  A
{{RRRow}}|*||{{Rulename|SIMP_TYPE_DOMSUB}}||<math> \mathit{Ty} \domsub r \;\;\defi\;\;  \emptyset </math>|| where <math>\mathit{Ty}</math> is a type expression ||  A
{{RRRow}}|||{{Rulename|SIMP_TYPE_RANSUB}}||<math>  r \ransub \mathit{Ty} \;\;\defi\;\;  \emptyset </math>|| where <math>\mathit{Ty}</math> is a type expression ||  A
{{RRRow}}|*||{{Rulename|SIMP_MULTI_DOMSUB_DOM}}||<math>  \dom (r) \domsub r \;\;\defi\;\;  \emptyset </math>|| ||  A
{{RRRow}}|||{{Rulename|SIMP_MULTI_RANSUB_RAN}}||<math> r \ransub \ran (r) \;\;\defi\;\;  \emptyset </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_MULTI_DOMSUB_RAN}}||<math>  \ran (r) \domsub  r^{-1} \;\;\defi\;\;  \emptyset </math>||  ||  A
{{RRRow}}|||{{Rulename|SIMP_RANSUB_ID}}||<math>  (S \domres \id) \ransub  T \;\;\defi\;\;  (S \setminus T) \domres \id </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_DOMSUB_DOMRES_ID}}||<math>  S \domsub (T \domres \id ) \;\;\defi\;\;  (T \setminus S) \domres \id </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_DOMSUB_DOMSUB_ID}}||<math>  S \domsub (T \domsub \id ) \;\;\defi\;\;  (S \bunion T) \domsub \id </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_SPECIAL_RANSUB_R}}||<math>  r \ransub  \emptyset  \;\;\defi\;\;  r </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_SPECIAL_RANSUB_L}}||<math>  \emptyset  \ransub  S \;\;\defi\;\;  \emptyset </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_TYPE_RANSUB}}||<math>  r \ransub  \mathit{Ty} \;\;\defi\;\;  \emptyset </math>|| where <math>\mathit{Ty}</math> is a type expression ||  A
{{RRRow}}|*||{{Rulename|SIMP_MULTI_RANSUB_DOM}}||<math>  r^{-1} \ransub  \dom (r) \;\;\defi\;\;  \emptyset </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_MULTI_RANSUB_RAN}}||<math>  r \ransub  \ran (r) \;\;\defi\;\;  \emptyset </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_RANSUB_DOMRES_ID}}||<math>  (S \domres \id) \ransub  T \;\;\defi\;\;  (S \setminus T) \domres \id </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_RANSUB_DOMSUB_ID}}||<math>  (S \domsub \id) \ransub  T \;\;\defi\;\;  (S \bunion T) \domsub \id </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_SPECIAL_FCOMP}}||<math>  r \fcomp \ldots  \fcomp \emptyset  \fcomp \ldots  \fcomp s \;\;\defi\;\;  \emptyset </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_SPECIAL_FCOMP}}||<math>  r \fcomp \ldots  \fcomp \emptyset  \fcomp \ldots  \fcomp s \;\;\defi\;\;  \emptyset </math>||  ||  A
{{RRRow}}|||{{Rulename|SIMP_TYPE_FCOMP_ID}}||<math>  r \fcomp \ldots  \fcomp \id \fcomp \ldots  \fcomp s \;\;\defi\;\;  r \fcomp \ldots  \fcomp s </math>|| ||  A
{{RRRow}}|*||{{Rulename|SIMP_TYPE_FCOMP_ID}}||<math>  r \fcomp \ldots  \fcomp \id \fcomp \ldots  \fcomp s \;\;\defi\;\;  r \fcomp \ldots  \fcomp s </math>|| ||  A
{{RRRow}}|||{{Rulename|SIMP_TYPE_FCOMP_R}}||<math>  r \fcomp \mathit{Ty} \;\;\defi\;\;  \dom (r) \cprod  \mathit{Tb} </math>|| where <math>\mathit{Ty}</math> is a type expression equal to <math>\mathit{Ta} \cprod \mathit{Tb}</math> ||  A
{{RRRow}}|*||{{Rulename|SIMP_TYPE_FCOMP_R}}||<math>  r \fcomp \mathit{Ty} \;\;\defi\;\;  \dom (r) \cprod  \mathit{Tb} </math>|| where <math>\mathit{Ty}</math> is a type expression equal to <math>\mathit{Ta} \cprod \mathit{Tb}</math> ||  A
{{RRRow}}|||{{Rulename|SIMP_TYPE_FCOMP_L}}||<math> \mathit{Ty} \fcomp r \;\;\defi\;\;  \mathit{Ta} \cprod  \ran (r) </math>|| where <math>\mathit{Ty}</math> is a type expression equal to <math>\mathit{Ta} \cprod \mathit{Tb}</math> ||  A
{{RRRow}}|*||{{Rulename|SIMP_TYPE_FCOMP_L}}||<math> \mathit{Ty} \fcomp r \;\;\defi\;\;  \mathit{Ta} \cprod  \ran (r) </math>|| where <math>\mathit{Ty}</math> is a type expression equal to <math>\mathit{Ta} \cprod \mathit{Tb}</math> ||  A
{{RRRow}}|||{{Rulename|SIMP_FCOMP_ID}}||<math>  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 </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_FCOMP_ID}}||<math>  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 </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_SPECIAL_BCOMP}}||<math>  r \bcomp  \ldots  \bcomp  \emptyset  \bcomp  \ldots  \bcomp  s \;\;\defi\;\;  \emptyset </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_SPECIAL_BCOMP}}||<math>  r \bcomp  \ldots  \bcomp  \emptyset  \bcomp  \ldots  \bcomp  s \;\;\defi\;\;  \emptyset </math>||  ||  A
{{RRRow}}|||{{Rulename|SIMP_TYPE_BCOMP_ID}}||<math>  r \bcomp  \ldots  \bcomp  \id \bcomp  \ldots  \bcomp  s \;\;\defi\;\;  r \bcomp  \ldots  \bcomp  s </math>|| ||  A
{{RRRow}}|*||{{Rulename|SIMP_TYPE_BCOMP_ID}}||<math>  r \bcomp  \ldots  \bcomp  \id \bcomp  \ldots  \bcomp  s \;\;\defi\;\;  r \bcomp  \ldots  \bcomp  s </math>|| ||  A
{{RRRow}}|||{{Rulename|SIMP_TYPE_BCOMP_L}}||<math> \mathit{Ty} \bcomp  r \;\;\defi\;\;  \dom (r) \cprod  \mathit{Tb} </math>|| where <math>\mathit{Ty}</math> is a type expression equal to <math>\mathit{Ta} \cprod \mathit{Tb}</math> ||  A
{{RRRow}}|*||{{Rulename|SIMP_TYPE_BCOMP_L}}||<math> \mathit{Ty} \bcomp  r \;\;\defi\;\;  \dom (r) \cprod  \mathit{Tb} </math>|| where <math>\mathit{Ty}</math> is a type expression equal to <math>\mathit{Ta} \cprod \mathit{Tb}</math> ||  A
{{RRRow}}|||{{Rulename|SIMP_TYPE_BCOMP_R}}||<math>  r \bcomp  \mathit{Ty} \;\;\defi\;\;  \mathit{Ta} \cprod  \ran (r) </math>|| where <math>\mathit{Ty}</math> is a type expression equal to <math>\mathit{Ta} \cprod \mathit{Tb}</math> ||  A
{{RRRow}}|*||{{Rulename|SIMP_TYPE_BCOMP_R}}||<math>  r \bcomp  \mathit{Ty} \;\;\defi\;\;  \mathit{Ta} \cprod  \ran (r) </math>|| where <math>\mathit{Ty}</math> is a type expression equal to <math>\mathit{Ta} \cprod \mathit{Tb}</math> ||  A
{{RRRow}}|||{{Rulename|SIMP_BCOMP_ID}}||<math>  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 </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_BCOMP_ID}}||<math>  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 </math>||  ||  A
{{RRRow}}| ||{{Rulename|SIMP_SPECIAL_DPROD_R}}||<math>  r \dprod  \emptyset  \;\;\defi\;\;  \emptyset </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_SPECIAL_DPROD_R}}||<math>  r \dprod  \emptyset  \;\;\defi\;\;  \emptyset </math>||  ||  A
{{RRRow}}| ||{{Rulename|SIMP_SPECIAL_DPROD_L}}||<math>  \emptyset  \dprod  r \;\;\defi\;\;  \emptyset </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_SPECIAL_DPROD_L}}||<math>  \emptyset  \dprod  r \;\;\defi\;\;  \emptyset </math>||  ||  A
{{RRRow}}| ||{{Rulename|SIMP_DPROD_CPROD}}||<math>  (\mathit{S} \cprod \mathit{T})  \dprod  (\mathit{U} \cprod \mathit{V})  \;\;\defi\;\;  \mathit{S}  \binter  \mathit{U}  \cprod  (\mathit{T}  \cprod  \mathit{V}) </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_DPROD_CPROD}}||<math>  (\mathit{S} \cprod \mathit{T})  \dprod  (\mathit{U} \cprod \mathit{V})  \;\;\defi\;\;  \mathit{S}  \binter  \mathit{U}  \cprod  (\mathit{T}  \cprod  \mathit{V}) </math>||  ||  A
{{RRRow}}| ||{{Rulename|SIMP_SPECIAL_PPROD_R}}||<math>  r \pprod  \emptyset  \;\;\defi\;\;  \emptyset </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_SPECIAL_PPROD_R}}||<math>  r \pprod  \emptyset  \;\;\defi\;\;  \emptyset </math>||  ||  A
{{RRRow}}| ||{{Rulename|SIMP_SPECIAL_PPROD_L}}||<math>  \emptyset  \pprod  r \;\;\defi\;\;  \emptyset </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_SPECIAL_PPROD_L}}||<math>  \emptyset  \pprod  r \;\;\defi\;\;  \emptyset </math>||  ||  A
{{RRRow}}| ||{{Rulename|SIMP_PPROD_CPROD}}||<math>  (\mathit{S} \cprod \mathit{T})  \pprod  (\mathit{U} \cprod \mathit{V}) \;\;\defi\;\;  (\mathit{S} \cprod \mathit{U}) \cprod (\mathit{T} \cprod \mathit{V}) </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_PPROD_CPROD}}||<math>  (\mathit{S} \cprod \mathit{T})  \pprod  (\mathit{U} \cprod \mathit{V}) \;\;\defi\;\;  (\mathit{S} \cprod \mathit{U}) \cprod (\mathit{T} \cprod \mathit{V}) </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_SPECIAL_RELIMAGE_R}}||<math>  r[\emptyset ] \;\;\defi\;\;  \emptyset </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_SPECIAL_RELIMAGE_R}}||<math>  r[\emptyset ] \;\;\defi\;\;  \emptyset </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_SPECIAL_RELIMAGE_L}}||<math>  \emptyset [S] \;\;\defi\;\;  \emptyset </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_SPECIAL_RELIMAGE_L}}||<math>  \emptyset [S] \;\;\defi\;\;  \emptyset </math>||  ||  A
{{RRRow}}|||{{Rulename|SIMP_TYPE_RELIMAGE}}||<math>  r[Ty] \;\;\defi\;\;  \ran (r) </math>|| where <math>\mathit{Ty}</math> is a type expression ||  A
{{RRRow}}|*||{{Rulename|SIMP_TYPE_RELIMAGE}}||<math>  r[Ty] \;\;\defi\;\;  \ran (r) </math>|| where <math>\mathit{Ty}</math> is a type expression ||  A
{{RRRow}}|||{{Rulename|SIMP_MULTI_RELIMAGE_DOM}}||<math>  r[\dom (r)] \;\;\defi\;\;  \ran (r) </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_MULTI_RELIMAGE_DOM}}||<math>  r[\dom (r)] \;\;\defi\;\;  \ran (r) </math>||  ||  A
{{RRRow}}|||{{Rulename|SIMP_TYPE_RELIMAGE_ID}}||<math>  \id[T] \;\;\defi\;\;  T </math>|| ||  A
{{RRRow}}|*||{{Rulename|SIMP_RELIMAGE_ID}}||<math>  \id[T] \;\;\defi\;\;  T </math>|| ||  A
{{RRRow}}|||{{Rulename|SIMP_RELIMAGE_ID}}||<math>  (S \domres \id)[T] \;\;\defi\;\;  S \binter  T </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_RELIMAGE_DOMRES_ID}}||<math>  (S \domres \id)[T] \;\;\defi\;\;  S \binter  T </math>||  ||  A
{{RRRow}}|||{{Rulename|SIMP_MULTI_RELIMAGE_CPROD_SING}}||<math>  (\{ E\}  \cprod  S)[\{ E\} ] \;\;\defi\;\;  S </math>|| where <math>E</math> is a single expression ||  A
{{RRRow}}|*||{{Rulename|SIMP_RELIMAGE_DOMSUB_ID}}||<math>  (S \domsub \id)[T] \;\;\defi\;\;  T \setminus S </math>||  ||  A
{{RRRow}}|||{{Rulename|SIMP_MULTI_RELIMAGE_SING_MAPSTO}}||<math>  \{ E \mapsto  F\} [\{ E\} ] \;\;\defi\;\;  \{ F\} </math>|| where <math>E</math> is a single expression ||  A
{{RRRow}}|*||{{Rulename|SIMP_MULTI_RELIMAGE_CPROD_SING}}||<math>  (\{ E\}  \cprod  S)[\{ E\} ] \;\;\defi\;\;  S </math>|| where <math>E</math> is a single expression ||  A
{{RRRow}}|||{{Rulename|SIMP_MULTI_RELIMAGE_CONVERSE_RANSUB}}||<math>  (r \ransub  S)^{-1} [S] \;\;\defi\;\;  \emptyset </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_MULTI_RELIMAGE_SING_MAPSTO}}||<math>  \{ E \mapsto  F\} [\{ E\} ] \;\;\defi\;\;  \{ F\} </math>|| where <math>E</math> is a single expression ||  A
{{RRRow}}|||{{Rulename|SIMP_MULTI_RELIMAGE_CONVERSE_RANRES}}||<math>  (r \ranres  S)^{-1} [S] \;\;\defi\;\;  r^{-1} [S] </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_MULTI_RELIMAGE_CONVERSE_RANSUB}}||<math>  (r \ransub  S)^{-1} [S] \;\;\defi\;\;  \emptyset </math>||  ||  A
{{RRRow}}|||{{Rulename|SIMP_RELIMAGE_CONVERSE_DOMSUB}}||<math>  (S \domsub  r)^{-1} [T] \;\;\defi\;\;  r^{-1} [T] \setminus S </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_MULTI_RELIMAGE_CONVERSE_RANRES}}||<math>  (r \ranres  S)^{-1} [S] \;\;\defi\;\;  r^{-1} [S] </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_RELIMAGE_CONVERSE_DOMSUB}}||<math>  (S \domsub  r)^{-1} [T] \;\;\defi\;\;  r^{-1} [T] \setminus S </math>||  ||  A
{{RRRow}}|||{{Rulename|DERIV_RELIMAGE_RANSUB}}||<math>  (r \ransub  S)[T] \;\;\defi\;\;  r[T] \setminus S </math>||  ||  M
{{RRRow}}|||{{Rulename|DERIV_RELIMAGE_RANSUB}}||<math>  (r \ransub  S)[T] \;\;\defi\;\;  r[T] \setminus S </math>||  ||  M
{{RRRow}}|||{{Rulename|DERIV_RELIMAGE_RANRES}}||<math>  (r \ranres  S)[T] \;\;\defi\;\;  r[T] \binter  S </math>||  ||  M
{{RRRow}}|||{{Rulename|DERIV_RELIMAGE_RANRES}}||<math>  (r \ranres  S)[T] \;\;\defi\;\;  r[T] \binter  S </math>||  ||  M
{{RRRow}}|||{{Rulename|SIMP_MULTI_RELIMAGE_DOMSUB}}||<math>  (S \domsub  r)[S] \;\;\defi\;\;  \emptyset </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_MULTI_RELIMAGE_DOMSUB}}||<math>  (S \domsub  r)[S] \;\;\defi\;\;  \emptyset </math>||  ||  A
{{RRRow}}|||{{Rulename|DERIV_RELIMAGE_DOMSUB}}||<math>  (S \domsub  r)[T] \;\;\defi\;\;  r[T \setminus S] </math>||  ||  M
{{RRRow}}|||{{Rulename|DERIV_RELIMAGE_DOMSUB}}||<math>  (S \domsub  r)[T] \;\;\defi\;\;  r[T \setminus S] </math>||  ||  M
{{RRRow}}|||{{Rulename|DERIV_RELIMAGE_DOMRES}}||<math>  (S \domres  r)[T] \;\;\defi\;\;  r[S \binter  T] </math>||  ||  M
{{RRRow}}|||{{Rulename|DERIV_RELIMAGE_DOMRES}}||<math>  (S \domres  r)[T] \;\;\defi\;\;  r[S \binter  T] </math>||  ||  M
{{RRRow}}| ||{{Rulename|SIMP_SPECIAL_CONVERSE}}||<math>  \emptyset ^{-1}  \;\;\defi\;\;  \emptyset </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_SPECIAL_CONVERSE}}||<math>  \emptyset ^{-1}  \;\;\defi\;\;  \emptyset </math>||  ||  A
{{RRRow}}|||{{Rulename|SIMP_CONVERSE_ID}}||<math>  \id^{-1}  \;\;\defi\;\;  \id </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_CONVERSE_ID}}||<math>  \id^{-1}  \;\;\defi\;\;  \id </math>||  ||  A
{{RRRow}}|||{{Rulename|SIMP_CONVERSE_CPROD}}||<math>  (\mathit{S} \cprod \mathit{T})^{-1}  \;\;\defi\;\;  \mathit{T} \cprod \mathit{S} </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_CONVERSE_CPROD}}||<math>  (\mathit{S} \cprod \mathit{T})^{-1}  \;\;\defi\;\;  \mathit{T} \cprod \mathit{S} </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_CONVERSE_SETENUM}}||<math>  \{ x \mapsto  a, \ldots , y \mapsto  b\} ^{-1}  \;\;\defi\;\;  \{ a \mapsto  x, \ldots , b \mapsto  y\} </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_CONVERSE_SETENUM}}||<math>  \{ x \mapsto  a, \ldots , y \mapsto  b\} ^{-1}  \;\;\defi\;\;  \{ a \mapsto  x, \ldots , b \mapsto  y\} </math>||  ||  A
{{RRRow}}| ||{{Rulename|SIMP_CONVERSE_COMPSET}}||<math>  \{ X \qdot  P \mid  x\mapsto y\} ^{-1}  \;\;\defi\;\;  \{ X \qdot  P \mid  y\mapsto x\} </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_CONVERSE_COMPSET}}||<math>  \{ X \qdot  P \mid  x\mapsto y\} ^{-1}  \;\;\defi\;\;  \{ X \qdot  P \mid  y\mapsto x\} </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_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) \;\;\defi\;\;  S </math>|| where <math>\id</math> has type <math>\pow(S \cprod 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>  (S \domres \id) \fcomp r \;\;\defi\;\;  S \domres  r </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_FCOMP_ID_L}}||<math>  (S \domres \id) \fcomp r \;\;\defi\;\;  S \domres  r </math>||  ||  A
{{RRRow}}|||{{Rulename|SIMP_FCOMP_ID_R}}||<math>  r \fcomp (S \domres \id) \;\;\defi\;\;  r \ranres  S </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_FCOMP_ID_R}}||<math>  r \fcomp (S \domres \id) \;\;\defi\;\;  r \ranres  S </math>||  ||  A
{{RRRow}}| ||{{Rulename|SIMP_SPECIAL_REL_R}}||<math>  S \rel  \emptyset  \;\;\defi\;\;  \{ \emptyset \} </math>|| idem for operators <math>\srel  \pfun  \pinj  \psur</math> ||  A
{{RRRow}}|*||{{Rulename|SIMP_SPECIAL_REL_R}}||<math>  S \rel  \emptyset  \;\;\defi\;\;  \{ \emptyset \} </math>|| idem for operators <math>\srel  \pfun  \pinj  \psur</math> ||  A
{{RRRow}}| ||{{Rulename|SIMP_SPECIAL_REL_L}}||<math>  \emptyset  \rel  S \;\;\defi\;\;  \{ \emptyset \} </math>|| idem for operators <math>\trel  \pfun  \tfun  \pinj  \tinj</math> ||  A
{{RRRow}}|*||{{Rulename|SIMP_SPECIAL_REL_L}}||<math>  \emptyset  \rel  S \;\;\defi\;\;  \{ \emptyset \} </math>|| idem for operators <math>\trel  \pfun  \tfun  \pinj  \tinj</math> ||  A
{{RRRow}}| ||{{Rulename|SIMP_SPECIAL_EQUAL_REL}}||<math>  A \rel  B = \emptyset  \;\;\defi\;\;  \bfalse </math>|| idem for operators <math>\pfun  \pinj</math> ||  A
{{RRRow}}|*||{{Rulename|SIMP_FUNIMAGE_PRJ1}}||<math>  \prjone (E \mapsto  F) \;\;\defi\;\;  E </math>||  ||  A
{{RRRow}}| ||{{Rulename|SIMP_SPECIAL_EQUAL_RELDOM}}||<math>  A \trel  B = \emptyset  \;\;\defi\;\;  \lnot\, A = \emptyset  \land  B = \emptyset </math>|| idem for operators <math>\tfun  \tinj  \tsur  \tbij</math> ||  A
{{RRRow}}|*||{{Rulename|SIMP_FUNIMAGE_PRJ2}}||<math>  \prjtwo (E \mapsto  F) \;\;\defi\;\;  F </math>||  ||  A
{{RRRow}}|||{{Rulename|SIMP_FUNIMAGE_PRJ1}}||<math>  \prjone (E \mapsto  F) \;\;\defi\;\;  E </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_DOM_PRJ1}}||<math>  \dom (\prjone) \;\;\defi\;\;  S \cprod T </math>|| where <math>\prjone</math> has type <math>\pow(S \cprod T \cprod S)</math> ||  A
{{RRRow}}|||{{Rulename|SIMP_FUNIMAGE_PRJ2}}||<math>  \prjtwo (E \mapsto  F) \;\;\defi\;\;  F </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_DOM_PRJ2}}||<math>  \dom (\prjtwo) \;\;\defi\;\; S \cprod T </math>|| where <math>\prjtwo</math> has type <math>\pow(S \cprod T \cprod T)</math> ||  A
{{RRRow}}|||{{Rulename|SIMP_DOM_PRJ1}}||<math>  \dom (\prjone) \;\;\defi\;\;  S \cprod T </math>|| where <math>\prjone</math> has type <math>\pow(S \cprod T \cprod S)</math> ||  A
{{RRRow}}|*||{{Rulename|SIMP_RAN_PRJ1}}||<math>  \ran (\prjone) \;\;\defi\;\;  S </math>|| where <math>\prjone</math> has type <math>\pow(S \cprod T \cprod S)</math> ||  A
{{RRRow}}|||{{Rulename|SIMP_DOM_PRJ2}}||<math>  \dom (\prjtwo) \;\;\defi\;\; S \cprod T </math>|| where <math>\prjtwo</math> has type <math>\pow(S \cprod T \cprod T)</math> ||  A
{{RRRow}}|*||{{Rulename|SIMP_RAN_PRJ2}}||<math>  \ran (\prjtwo) \;\;\defi\;\;  T </math>|| where <math>\prjtwo</math> has type <math>\pow(S \cprod T \cprod T)</math> ||  A
{{RRRow}}|||{{Rulename|SIMP_RAN_PRJ1}}||<math>  \ran (\prjone) \;\;\defi\;\;  S </math>|| where <math>\prjone</math> has type <math>\pow(S \cprod T \cprod S)</math> ||  A
{{RRRow}}|||{{Rulename|SIMP_RAN_PRJ2}}||<math>  \ran (\prjtwo) \;\;\defi\;\;  T </math>|| where <math>\prjtwo</math> has type <math>\pow(S \cprod T \cprod T)</math> ||  A
{{RRRow}}|*||{{Rulename|SIMP_FUNIMAGE_LAMBDA}}||<math>  (\lambda x \qdot  P(x) \mid  E(x))(y) \;\;\defi\;\;  E(y) </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_FUNIMAGE_LAMBDA}}||<math>  (\lambda x \qdot  P(x) \mid  E(x))(y) \;\;\defi\;\;  E(y) </math>||  ||  A
{{RRRow}}|||{{Rulename|SIMP_DOM_LAMBDA}}||<math>\dom(\{x\qdot P\mid E\mapsto F) \;\;\defi\;\; \{x\qdot P\mid E\}</math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_DOM_LAMBDA}}||<math>\dom(\{x\qdot P\mid E\mapsto F) \;\;\defi\;\; \{x\qdot P\mid E\}</math>||  ||  A
{{RRRow}}|||{{Rulename|SIMP_RAN_LAMBDA}}||<math>\ran(\{x\qdot P\mid E\mapsto F) \;\;\defi\;\; \{x\qdot P\mid F\} </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_RAN_LAMBDA}}||<math>\ran(\{x\qdot P\mid E\mapsto F) \;\;\defi\;\; \{x\qdot P\mid F\} </math>||  ||  A
{{RRRow}}|||{{Rulename|SIMP_IN_FUNIMAGE}}||<math>E\mapsto F(E)\in F \;\;\defi\;\; \btrue</math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_IN_FUNIMAGE}}||<math>E\mapsto F(E)\in F \;\;\defi\;\; \btrue</math>||  ||  A
{{RRRow}}|||{{Rulename|SIMP_IN_FUNIMAGE_CONVERSE_L}}||<math>F^{-1}(E)\mapsto E\in F \;\;\defi\;\; \btrue</math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_IN_FUNIMAGE_CONVERSE_L}}||<math>F^{-1}(E)\mapsto E\in F \;\;\defi\;\; \btrue</math>||  ||  A
{{RRRow}}|||{{Rulename|SIMP_IN_FUNIMAGE_CONVERSE_R}}||<math>F(E)\mapsto E\in F^{-1} \;\;\defi\;\; \btrue</math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_IN_FUNIMAGE_CONVERSE_R}}||<math>F(E)\mapsto E\in F^{-1} \;\;\defi\;\; \btrue</math>||  ||  A
{{RRRow}}|||{{Rulename|SIMP_MULTI_FUNIMAGE_SETENUM_LL}}||<math>  \{ A \mapsto  E, \ldots  , B \mapsto  E\} (x) \;\;\defi\;\;  E </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_MULTI_FUNIMAGE_SETENUM_LL}}||<math>  \{ A \mapsto  E, \ldots  , B \mapsto  E\} (x) \;\;\defi\;\;  E </math>||  ||  A
{{RRRow}}|||{{Rulename|SIMP_MULTI_FUNIMAGE_SETENUM_LR}}||<math>  \{ A \mapsto  E, \ldots  , x \mapsto  y, \ldots  , B \mapsto  F\} (x) \;\;\defi\;\;  y </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_MULTI_FUNIMAGE_SETENUM_LR}}||<math>  \{ E, \ldots  , x \mapsto  y, \ldots  , F\} (x) \;\;\defi\;\;  y </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_MULTI_FUNIMAGE_OVERL_SETENUM}}||<math>  (r \ovl  \ldots  \ovl  \{ A \mapsto  E, \ldots  , x \mapsto  y, \ldots  , B \mapsto  F\} )(x) \;\;\defi\;\;  y </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_MULTI_FUNIMAGE_OVERL_SETENUM}}||<math>  (r \ovl  \ldots  \ovl  \{ E, \ldots  , x \mapsto  y, \ldots  , F\} )(x) \;\;\defi\;\;  y </math>||  ||  A
{{RRRow}}|||{{Rulename|SIMP_MULTI_FUNIMAGE_BUNION_SETENUM}}||<math>  (r \bunion  \ldots  \bunion  \{ A \mapsto  E, \ldots  , x \mapsto  y, \ldots  , B \mapsto  F\} )(x) \;\;\defi\;\;  y </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_MULTI_FUNIMAGE_BUNION_SETENUM}}||<math>  (r \bunion  \ldots  \bunion  \{ E, \ldots  , x \mapsto  y, \ldots  , F\} )(x) \;\;\defi\;\;  y </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_FUNIMAGE_CPROD}}||<math>  (S \cprod  \{ F\} )(x) \;\;\defi\;\;  F </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_FUNIMAGE_CPROD}}||<math>  (S \cprod  \{ F\} )(x) \;\;\defi\;\;  F </math>||  ||  A
{{RRRow}}|||{{Rulename|SIMP_FUNIMAGE_ID}}||<math>  \id (x) \;\;\defi\;\;  x </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_FUNIMAGE_ID}}||<math>  \id (x) \;\;\defi\;\;  x </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_FUNIMAGE_FUNIMAGE_CONVERSE}}||<math>  f(f^{-1} (E)) \;\;\defi\;\;  E </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_FUNIMAGE_FUNIMAGE_CONVERSE}}||<math>  f(f^{-1} (E)) \;\;\defi\;\;  E </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_FUNIMAGE_CONVERSE_FUNIMAGE}}||<math>  f^{-1}(f(E)) \;\;\defi\;\;  E </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_FUNIMAGE_CONVERSE_FUNIMAGE}}||<math>  f^{-1}(f(E)) \;\;\defi\;\;  E </math>||  ||  A
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{{RRRow}}|*||{{Rulename|SIMP_FUNIMAGE_RANSUB}}||<math>(F \ransub E)(G)\;\;\defi\;\;F(G)</math> ||  with hypothesis<math> F \in \mathit{A} \ \mathit{op}\  \mathit{B}</math> where <math>\mathit{op}</math> is one of <math>\pfun</math>, <math>\tfun</math>, <math>\pinj</math>, <math>\tinj</math>, <math>\psur</math>, <math>\tsur</math>, <math>\tbij</math>.  || AM
{{RRRow}}|*||{{Rulename|SIMP_FUNIMAGE_RANSUB}}||<math>(F \ransub E)(G)\;\;\defi\;\;F(G)</math> ||  with hypothesis<math> F \in \mathit{A} \ \mathit{op}\  \mathit{B}</math> where <math>\mathit{op}</math> is one of <math>\pfun</math>, <math>\tfun</math>, <math>\pinj</math>, <math>\tinj</math>, <math>\psur</math>, <math>\tsur</math>, <math>\tbij</math>.  || AM
{{RRRow}}|*||{{Rulename|SIMP_FUNIMAGE_SETMINUS}}||<math>(F \setminus E)(G)\;\;\defi\;\;F(G)</math> ||  with hypothesis<math> F \in \mathit{A} \ \mathit{op}\  \mathit{B}</math> where <math>\mathit{op}</math> is one of <math>\pfun</math>, <math>\tfun</math>, <math>\pinj</math>, <math>\tinj</math>, <math>\psur</math>, <math>\tsur</math>, <math>\tbij</math>.  || AM
{{RRRow}}|*||{{Rulename|SIMP_FUNIMAGE_SETMINUS}}||<math>(F \setminus E)(G)\;\;\defi\;\;F(G)</math> ||  with hypothesis<math> F \in \mathit{A} \ \mathit{op}\  \mathit{B}</math> where <math>\mathit{op}</math> is one of <math>\pfun</math>, <math>\tfun</math>, <math>\pinj</math>, <math>\tinj</math>, <math>\psur</math>, <math>\tsur</math>, <math>\tbij</math>.  || AM
{{RRRow}}|||{{Rulename|DEF_BCOMP}}||<math>  r \bcomp  \ldots  \bcomp  s \;\;\defi\;\;  s \fcomp \ldots  \fcomp r </math>||  ||  M
{{RRRow}}|*||{{Rulename|DEF_EQUAL_FUNIMAGE}}||<math> f(x) = y \;\;\defi\;\; x \mapsto y \in f </math>||  ||  M
{{RRRow}}|*||{{Rulename|DEF_BCOMP}}||<math>  r \bcomp  \ldots  \bcomp  s \;\;\defi\;\;  s \fcomp \ldots  \fcomp r </math>||  ||  M
{{RRRow}}|||{{Rulename|DERIV_ID_SING}}||<math>  \{ E\} \domres \id \;\;\defi\;\;  \{ E \mapsto  E\} </math>|| where <math>E</math> is a single expression ||  M
{{RRRow}}|||{{Rulename|DERIV_ID_SING}}||<math>  \{ E\} \domres \id \;\;\defi\;\;  \{ E \mapsto  E\} </math>|| where <math>E</math> is a single expression ||  M
{{RRRow}}|*||{{Rulename|SIMP_SPECIAL_DOM}}||<math>  \dom (\emptyset ) \;\;\defi\;\;  \emptyset </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_SPECIAL_DOM}}||<math>  \dom (\emptyset ) \;\;\defi\;\;  \emptyset </math>||  ||  A
Line 114: Line 122:
{{RRRow}}|*||{{Rulename|DERIV_FCOMP_RANRES}}||<math>  p \fcomp (q \ranres  s) \;\;\defi\;\;  (p \fcomp q) \ranres  s </math>||  ||  M
{{RRRow}}|*||{{Rulename|DERIV_FCOMP_RANRES}}||<math>  p \fcomp (q \ranres  s) \;\;\defi\;\;  (p \fcomp q) \ranres  s </math>||  ||  M
{{RRRow}}|*||{{Rulename|DERIV_FCOMP_RANSUB}}||<math>  p \fcomp (q \ransub  s) \;\;\defi\;\;  (p \fcomp q) \ransub  s </math>||  ||  M
{{RRRow}}|*||{{Rulename|DERIV_FCOMP_RANSUB}}||<math>  p \fcomp (q \ransub  s) \;\;\defi\;\;  (p \fcomp q) \ransub  s </math>||  ||  M
{{RRRow}}| ||{{Rulename|SIMP_SPECIAL_EQUAL_RELDOMRAN}}||<math>  \emptyset  \strel  \emptyset  \;\;\defi\;\;  \{ \emptyset \} </math>|| idem for operators <math>\tsur  \tbij</math> ||  A
{{RRRow}}| ||{{Rulename|DERIV_FCOMP_SING}}||<math>  \{E\mapsto F\}\fcomp\{F\mapsto G\} \;\;\defi\;\;  \{E\mapsto G\} </math>||  ||  A
{{RRRow}}|||{{Rulename|SIMP_TYPE_DOM}}||<math>  \dom (\mathit{Ty}) \;\;\defi\;\;  \mathit{Ta} </math>|| where <math>\mathit{Ty}</math> is a type expression equal to <math>\mathit{Ta} \cprod \mathit{Tb}</math> ||  A
{{RRRow}}|*||{{Rulename|SIMP_SPECIAL_EQUAL_RELDOMRAN}}||<math>  \emptyset  \strel  \emptyset  \;\;\defi\;\;  \{ \emptyset \} </math>|| idem for operators <math>\tsur  \tbij</math> ||  A
{{RRRow}}|||{{Rulename|SIMP_TYPE_RAN}}||<math>  \ran (\mathit{Ty}) \;\;\defi\;\;  \mathit{Tb} </math>|| where <math>\mathit{Ty}</math> is a type expression equal to <math>\mathit{Ta} \cprod \mathit{Tb}</math> ||  A
{{RRRow}}|*||{{Rulename|SIMP_TYPE_DOM}}||<math>  \dom (\mathit{Ty}) \;\;\defi\;\;  \mathit{Ta} </math>|| where <math>\mathit{Ty}</math> is a type expression equal to <math>\mathit{Ta} \cprod \mathit{Tb}</math> ||  A
{{RRRow}}|||{{Rulename|SIMP_MULTI_DOM_CPROD}}||<math>  \dom (E \cprod  E) \;\;\defi\;\;  E </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_TYPE_RAN}}||<math>  \ran (\mathit{Ty}) \;\;\defi\;\;  \mathit{Tb} </math>|| where <math>\mathit{Ty}</math> is a type expression equal to <math>\mathit{Ta} \cprod \mathit{Tb}</math> ||  A
{{RRRow}}|||{{Rulename|SIMP_MULTI_RAN_CPROD}}||<math>  \ran (E \cprod  E) \;\;\defi\;\;  E </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_MULTI_DOM_CPROD}}||<math>  \dom (E \cprod  E) \;\;\defi\;\;  E </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_MULTI_RAN_CPROD}}||<math>  \ran (E \cprod  E) \;\;\defi\;\;  E </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_MULTI_DOM_DOMRES}}||<math>\dom(A\domres f) \;\;\defi\;\; \dom(f)\binter A</math>|| || A
{{RRRow}}|*||{{Rulename|SIMP_MULTI_DOM_DOMSUB}}||<math>\dom(A\domsub f) \;\;\defi\;\; \dom(f)\setminus A</math>|| || A
{{RRRow}}|*||{{Rulename|SIMP_MULTI_RAN_RANRES}}||<math>\ran(f\ranres A) \;\;\defi\;\; \ran(f)\binter A</math>|| || A
{{RRRow}}|*||{{Rulename|SIMP_MULTI_RAN_RANSUB}}||<math>\ran(f\ransub A) \;\;\defi\;\; \ran(f)\setminus A</math>|| || A
{{RRRow}}|*||{{Rulename|DEF_IN_DOM}}||<math>  E \in  \dom (r) \;\;\defi\;\;  \exists y \qdot  E \mapsto  y \in  r </math>||  ||  M
{{RRRow}}|*||{{Rulename|DEF_IN_DOM}}||<math>  E \in  \dom (r) \;\;\defi\;\;  \exists y \qdot  E \mapsto  y \in  r </math>||  ||  M
{{RRRow}}|*||{{Rulename|DEF_IN_RAN}}||<math>  F \in  \ran (r) \;\;\defi\;\;  \exists x \qdot  x \mapsto  F  \in  r </math>||  ||  M
{{RRRow}}|*||{{Rulename|DEF_IN_RAN}}||<math>  F \in  \ran (r) \;\;\defi\;\;  \exists x \qdot  x \mapsto  F  \in  r </math>||  ||  M
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{{RRRow}}|*||{{Rulename|DERIV_DOM_TOTALREL}}||<math>  \dom (r) \;\;\defi\;\;  E </math> || with hypothesis <math>r \in E \ \mathit{op}\  F</math>, where <math>\mathit{op}</math> is one of <math>\trel, \strel, \tfun, \tinj, \tsur, \tbij</math> ||  M
{{RRRow}}|*||{{Rulename|DERIV_DOM_TOTALREL}}||<math>  \dom (r) \;\;\defi\;\;  E </math> || with hypothesis <math>r \in E \ \mathit{op}\  F</math>, where <math>\mathit{op}</math> is one of <math>\trel, \strel, \tfun, \tinj, \tsur, \tbij</math> ||  M
{{RRRow}}| ||{{Rulename|DERIV_RAN_SURJREL}}||<math>  \ran (r) \;\;\defi\;\;  F </math> || with hypothesis <math>r \in E \ \mathit{op}\  F</math>, where <math>\mathit{op}</math> is one of <math>\srel,\strel, \psur, \tsur, \tbij</math> ||  M
{{RRRow}}| ||{{Rulename|DERIV_RAN_SURJREL}}||<math>  \ran (r) \;\;\defi\;\;  F </math> || with hypothesis <math>r \in E \ \mathit{op}\  F</math>, where <math>\mathit{op}</math> is one of <math>\srel,\strel, \psur, \tsur, \tbij</math> ||  M
{{RRRow}}|b||{{Rulename|prjone-functional}}||<math> \prjone \in E\ \mathit{op}\ F \;\;\defi\;\; \btrue </math> || where <math>\mathit{op}</math> is one of <math> \pfun, \tfun, \trel </math> ||  A
{{RRRow}}|*||{{Rulename|DERIV_PRJ1_SURJ}}||<math>\prjone \in\mathit{Ty}_1\ \mathit{op}\ \mathit{Ty}_2\;\;\defi\;\; \btrue </math> || where <math>\mathit{Ty}_1</math> and <math>\mathit{Ty}_2</math> are types and <math>\mathit{op}</math> is one of <math>\rel, \trel, \srel, \strel, \pfun, \tfun, \psur, \tsur </math> ||  A
{{RRRow}}|b||{{Rulename|prjtwo-functional}}||<math> \prjtwo \in E\ \mathit{op}\ F \;\;\defi\;\; \btrue </math> || where <math>\mathit{op}</math> is one of <math> \pfun, \tfun, \trel </math> ||  A
{{RRRow}}|*||{{Rulename|DERIV_PRJ2_SURJ}}||<math>\prjtwo \in\mathit{Ty}_1\ \mathit{op}\ \mathit{Ty}_2\;\;\defi\;\; \btrue </math> || where <math>\mathit{Ty}_1</math> and <math>\mathit{Ty}_2</math> are types and <math>\mathit{op}</math> is one of <math>\rel, \trel, \srel, \strel, \pfun, \tfun, \psur, \tsur </math> ||  A
{{RRRow}}| ||{{Rulename|prj-expand}}||<math> x \;\;\defi\;\; \prjone(x) \mapsto \prjtwo(x) </math> || ||  M
{{RRRow}}|*||{{Rulename|DERIV_ID_BIJ}}||<math>\id \in\mathit{Ty}\ \mathit{op}\ \mathit{Ty}\;\;\defi\;\; \btrue </math> || where <math>\mathit{Ty}</math> is a type and <math>\mathit{op}</math> is any arrow ||  A
{{RRRow}}|*||{{Rulename|SIMP_MAPSTO_PRJ1_PRJ2}}||<math>\prjone(E)\mapsto\prjtwo(E)\;\;\defi\;\; E </math> ||  ||  A
{{RRRow}}| ||{{Rulename|DERIV_EXPAND_PRJS}}||<math> E \;\;\defi\;\; \prjone(E) \mapsto \prjtwo(E) </math> || ||  M
{{RRRow}}|*||{{Rulename|SIMP_DOM_SUCC}}||<math>\dom(\usucc) \;\;\defi\;\; \intg</math> || ||  A
{{RRRow}}|*||{{Rulename|SIMP_RAN_SUCC}}||<math>\ran(\usucc) \;\;\defi\;\; \intg</math> || ||  A
{{RRRow}}|*||{{Rulename|DERIV_MULTI_IN_BUNION}}||<math> E\in A\bunion\cdots\bunion\left\{\cdots, E,\cdots\right\}\bunion\cdots\bunion B\;\;\defi\;\; \btrue</math> || ||  A
{{RRRow}}|*||{{Rulename|DERIV_MULTI_IN_SETMINUS}}||<math> E\in S\setminus\left\{\cdots, E,\cdots\right\} \;\;\defi\;\; \bfalse</math> || ||  A
{{RRRow}}|*||{{Rulename|DEF_PRED}}||<math> \upred\;\;\defi\;\; \usucc^{-1}</math> || ||  A
{{RRRow}}|*||{{Rulename|SIMP_SPECIAL_IN_ID}}||<math> E \mapsto E \in \id \;\;\defi\;\; \btrue</math> || ||  A
{{RRRow}}|*||{{Rulename|SIMP_SPECIAL_IN_SETMINUS_ID}}||<math>E \mapsto E \in r \setminus \id \;\;\defi\;\; \bfalse</math> || ||  A
{{RRRow}}|*||{{Rulename|SIMP_SPECIAL_IN_DOMRES_ID}}||<math>E \mapsto E \in S \domres \id \;\;\defi\;\; E \in S</math> || ||  A
{{RRRow}}|*||{{Rulename|SIMP_SPECIAL_IN_SETMINUS_DOMRES_ID}}||<math>E \mapsto E \in r \setminus (S \domres \id) \;\;\defi\;\; E \mapsto E \in S \domsub r</math> || ||  A


|}
|}

Latest revision as of 15:13, 3 June 2024

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_EQUAL_FUNIMAGE
 f(x) = y \;\;\defi\;\; x \mapsto y \in f M
*
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
*
SIMP_SPECIAL_IN_ID
 E \mapsto E \in \id \;\;\defi\;\; \btrue A
*
SIMP_SPECIAL_IN_SETMINUS_ID
 E \mapsto E \in r \setminus \id \;\;\defi\;\; \bfalse A
*
SIMP_SPECIAL_IN_DOMRES_ID
 E \mapsto E \in S \domres \id \;\;\defi\;\; E \in S A
*
SIMP_SPECIAL_IN_SETMINUS_DOMRES_ID
 E \mapsto E \in r \setminus (S \domres \id) \;\;\defi\;\; E \mapsto E \in S \domsub r A
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