Set Rewrite Rules: Difference between revisions

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imported>Laurent
Improved rule SIMP_COMPSET_EQUAL.
Guillaume (talk | contribs)
51 edits
Rules SIMP_{EMPTY,SINGLE}_PARTITION were implemented
 
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{{RRRow}}|*||{{Rulename|SIMP_SPECIAL_IMP_BFALSE_L}}||<math>  \bfalse  \limp  P \;\;\defi\;\;  \btrue </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_SPECIAL_IMP_BFALSE_L}}||<math>  \bfalse  \limp  P \;\;\defi\;\;  \btrue </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_MULTI_IMP}}||<math>  P \limp  P \;\;\defi\;\;  \btrue </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_MULTI_IMP}}||<math>  P \limp  P \;\;\defi\;\;  \btrue </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_MULTI_IMP_OR}}||<math>  P \land  \ldots  \land  Q \land  \ldots  \land  R \limp  Q \;\;\defi\;\;  \btrue </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_MULTI_IMP_NOT_L}}||<math>\lnot P\limp P\;\;\defi\;\; P</math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_MULTI_IMP_NOT_R}}||<math>P\limp\lnot P\;\;\defi\;\;\lnot P</math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_MULTI_IMP_AND}}||<math>  P \land  \ldots  \land  Q \land  \ldots  \land  R \limp  Q \;\;\defi\;\;  \btrue </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_MULTI_IMP_AND_NOT_R}}||<math>  P \land  \ldots  \land  Q \land  \ldots  \land  R \limp  \lnot\, Q \;\;\defi\;\;  \lnot\,(P \land  \ldots  \land  Q \land  \ldots  \land  R) </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_MULTI_IMP_AND_NOT_R}}||<math>  P \land  \ldots  \land  Q \land  \ldots  \land  R \limp  \lnot\, Q \;\;\defi\;\;  \lnot\,(P \land  \ldots  \land  Q \land  \ldots  \land  R) </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_MULTI_IMP_AND_NOT_L}}||<math>  P \land  \ldots  \land  \lnot\, Q \land  \ldots  \land  R \limp  Q \;\;\defi\;\;  \lnot\,(P \land  \ldots  \land  \lnot\, Q \land  \ldots  \land  R) </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_MULTI_IMP_AND_NOT_L}}||<math>  P \land  \ldots  \land  \lnot\, Q \land  \ldots  \land  R \limp  Q \;\;\defi\;\;  \lnot\,(P \land  \ldots  \land  \lnot\, Q \land  \ldots  \land  R) </math>||  ||  A
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{{RRRow}}|*||{{Rulename|SIMP_SPECIAL_NOT_BTRUE}}||<math>  \lnot\, \btrue  \;\;\defi\;\;  \bfalse </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_SPECIAL_NOT_BTRUE}}||<math>  \lnot\, \btrue  \;\;\defi\;\;  \bfalse </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_SPECIAL_NOT_BFALSE}}||<math>  \lnot\, \bfalse  \;\;\defi\;\;  \btrue </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_SPECIAL_NOT_BFALSE}}||<math>  \lnot\, \bfalse  \;\;\defi\;\;  \btrue </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_NOT_NOT}}||<math>  \lnot\, \lnot\, P \;\;\defi\;\;  P </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_NOT_NOT}}||<math>  \lnot\, \lnot\, P \;\;\defi\;\;  P </math>||  ||  AM
{{RRRow}}|*||{{Rulename|SIMP_NOTEQUAL}}||<math>  E \neq  F \;\;\defi\;\;  \lnot\, E = F </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_NOTEQUAL}}||<math>  E \neq  F \;\;\defi\;\;  \lnot\, E = F </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_NOTIN}}||<math>  E \notin  F \;\;\defi\;\;  \lnot\, E \in  F </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_NOTIN}}||<math>  E \notin  F \;\;\defi\;\;  \lnot\, E \in  F </math>||  ||  A
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{{RRRow}}|*||{{Rulename|SIMP_FORALL_AND}}||<math>  \forall x \qdot  P \land  Q \;\;\defi\;\;  (\forall x \qdot  P) \land  (\forall x \qdot  Q) </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_FORALL_AND}}||<math>  \forall x \qdot  P \land  Q \;\;\defi\;\;  (\forall x \qdot  P) \land  (\forall x \qdot  Q) </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_EXISTS_OR}}||<math>  \exists x \qdot  P \lor  Q \;\;\defi\;\;  (\exists x \qdot  P) \lor  (\exists x \qdot  Q) </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_EXISTS_OR}}||<math>  \exists x \qdot  P \lor  Q \;\;\defi\;\;  (\exists x \qdot  P) \lor  (\exists x \qdot  Q) </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_EXISTS_IMP}}||<math>\exists x\qdot P\limp Q\;\;\defi\;\;(\forall x\qdot P)\limp(\exists x\qdot Q)</math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_FORALL}}||<math>  \forall \ldots ,z,\ldots  \qdot  P(z) \;\;\defi\;\;  \forall z \qdot  P(z) </math>|| Quantified identifiers other than <math>z</math> do not occur in <math>P</math> ||  A
{{RRRow}}|*||{{Rulename|SIMP_FORALL}}||<math>  \forall \ldots ,z,\ldots  \qdot  P(z) \;\;\defi\;\;  \forall z \qdot  P(z) </math>|| Quantified identifiers other than <math>z</math> do not occur in <math>P</math> ||  A
{{RRRow}}|*||{{Rulename|SIMP_EXISTS}}||<math>  \exists \ldots ,z,\ldots  \qdot  P(z) \;\;\defi\;\;  \exists z \qdot  P(z) </math>|| Quantified identifiers other than <math>z</math> do not occur in <math>P</math> ||  A
{{RRRow}}|*||{{Rulename|SIMP_EXISTS}}||<math>  \exists \ldots ,z,\ldots  \qdot  P(z) \;\;\defi\;\;  \exists z \qdot  P(z) </math>|| Quantified identifiers other than <math>z</math> do not occur in <math>P</math> ||  A
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{{RRRow}}|*||{{Rulename|SIMP_SUBSETEQ_BUNION}}||<math>  S \subseteq  A \bunion  \ldots  \bunion  S \bunion  \ldots  \bunion  B \;\;\defi\;\;  \btrue </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_SUBSETEQ_BUNION}}||<math>  S \subseteq  A \bunion  \ldots  \bunion  S \bunion  \ldots  \bunion  B \;\;\defi\;\;  \btrue </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_SUBSETEQ_BINTER}}||<math>  A \binter  \ldots  \binter  S \binter  \ldots  \binter  B \subseteq  S \;\;\defi\;\;  \btrue </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_SUBSETEQ_BINTER}}||<math>  A \binter  \ldots  \binter  S \binter  \ldots  \binter  B \subseteq  S \;\;\defi\;\;  \btrue </math>||  ||  A
{{RRRow}}|*||{{Rulename|DERIV_SUBSETEQ_BUNION}}||<math>  A \bunion  \ldots  \bunion  B \subseteq  S \;\;\defi\;\;  A \subseteq  S \land  \ldots  \land  B \subseteq  S </math>||  ||  M
{{RRRow}}|*||{{Rulename|DERIV_SUBSETEQ_BUNION}}||<math>  A \bunion  \ldots  \bunion  B \subseteq  S \;\;\defi\;\;  A \subseteq  S \land  \ldots  \land  B \subseteq  S </math>||  ||  A
{{RRRow}}|*||{{Rulename|DERIV_SUBSETEQ_BINTER}}||<math>  S \subseteq  A \binter  \ldots  \binter  B \;\;\defi\;\;  S \subseteq  A \land  \ldots  \land  S \subseteq  B </math>||  ||  M
{{RRRow}}|*||{{Rulename|DERIV_SUBSETEQ_BINTER}}||<math>  S \subseteq  A \binter  \ldots  \binter  B \;\;\defi\;\;  S \subseteq  A \land  \ldots  \land  S \subseteq  B </math>||  ||  A
<!--
<!--
{{RRRow}}|||{{Rulename|SIMP_SUBSET_BUNION}}||<math>  A \bunion  \ldots  \bunion  B  \subset  S \;\;\defi\;\;  A  \subset  S \land  \ldots  \land  B  \subset  S </math>||  ||  A
{{RRRow}}|||{{Rulename|SIMP_SUBSET_BUNION}}||<math>  A \bunion  \ldots  \bunion  B  \subset  S \;\;\defi\;\;  A  \subset  S \land  \ldots  \land  B  \subset  S </math>||  ||  A
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{{RRRow}}|*||{{Rulename|SIMP_SPECIAL_CPROD_R}}||<math>  S \cprod  \emptyset  \;\;\defi\;\;  \emptyset </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_SPECIAL_CPROD_R}}||<math>  S \cprod  \emptyset  \;\;\defi\;\;  \emptyset </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_SPECIAL_CPROD_L}}||<math>  \emptyset  \cprod  S \;\;\defi\;\;  \emptyset </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_SPECIAL_CPROD_L}}||<math>  \emptyset  \cprod  S \;\;\defi\;\;  \emptyset </math>||  ||  A
{{RRRow}}|||{{Rulename|SIMP_COMPSET_EQUAL}}||<math>  \{ x\qdot x = E\mid F(x) \}  \;\;\defi\;\;  \{ F(E)\} </math>|| where <math>x</math> non free in <math>E</math> ||  A
{{RRRow}}|||{{Rulename|SIMP_COMPSET_EQUAL}}||<math>  \{ x, y \qdot x = E(y) \land P(y) \mid F(x, y) \}  \;\;\defi\;\;  \{ y \qdot P(y) \mid F(E(y), y) \} </math>|| where <math>x</math> non free in <math>E</math> and non free in <math>P</math> ||  A
{{RRRow}}|*||{{Rulename|SIMP_COMPSET_IN}}||<math>  \{  x \qdot  x \in  S \mid  x \}  \;\;\defi\;\;  S </math>|| where <math>x</math> non free in <math>S</math> ||  A
{{RRRow}}|*||{{Rulename|SIMP_COMPSET_IN}}||<math>  \{  x \qdot  x \in  S \mid  x \}  \;\;\defi\;\;  S </math>|| where <math>x</math> non free in <math>S</math> ||  A
{{RRRow}}|*||{{Rulename|SIMP_COMPSET_SUBSETEQ}}||<math>  \{  x \qdot  x \subseteq  S \mid  x \}  \;\;\defi\;\;  \pow (S)  </math>|| where <math>x</math> non free in <math>S</math> ||  A
{{RRRow}}|*||{{Rulename|SIMP_COMPSET_SUBSETEQ}}||<math>  \{  x \qdot  x \subseteq  S \mid  x \}  \;\;\defi\;\;  \pow (S)  </math>|| where <math>x</math> non free in <math>S</math> ||  A
{{RRRow}}|*||{{Rulename|SIMP_SPECIAL_COMPSET_BFALSE}}||<math>  \{  x \qdot  \bfalse  \mid  x \}  \;\;\defi\;\;  \emptyset </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_SPECIAL_COMPSET_BFALSE}}||<math>  \{  x \qdot  \bfalse  \mid  x \}  \;\;\defi\;\;  \emptyset </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_SPECIAL_COMPSET_BTRUE}}||<math>  \{  x \qdot  \btrue  \mid  x \}  \;\;\defi\;\;  \mathit{Ty} </math>|| where the type of <math>x</math> is <math>\mathit{Ty}</math> ||  A
{{RRRow}}|*||{{Rulename|SIMP_SPECIAL_COMPSET_BTRUE}}||<math>  \{  x \qdot  \btrue  \mid  E \}  \;\;\defi\;\;  \mathit{Ty} </math>|| where the type of <math>E</math> is <math>\mathit{Ty}</math> and <math>E</math> is a maplet combination of locally-bound, pairwise-distinct bound identifiers ||  A
{{RRRow}}|*||{{Rulename|SIMP_SUBSETEQ_COMPSET_L}}||<math>  \{  x \qdot  P(x) \mid  E(x) \}  \subseteq  S \;\;\defi\;\;  \forall y\qdot  P(y) \limp  E(y) \in  S </math>|| where <math>y</math> is fresh ||  A
{{RRRow}}|*||{{Rulename|SIMP_SUBSETEQ_COMPSET_L}}||<math>  \{  x \qdot  P(x) \mid  E(x) \}  \subseteq  S \;\;\defi\;\;  \forall y\qdot  P(y) \limp  E(y) \in  S </math>|| where <math>y</math> is fresh ||  A
{{RRRow}}|*||{{Rulename|SIMP_SPECIAL_EQUAL_COMPSET}}||<math>  \{  x \qdot  P(x) \mid  E \}  = \emptyset  \;\;\defi\;\;  \forall x\qdot  \lnot\, P(x) </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_IN_COMPSET}}||<math>  F \in  \{  x,y,\ldots \qdot  P(x,y,\ldots) \mid  E(x,y,\ldots) \}  \;\;\defi\;\;  \exists x,y,\ldots \qdot P(x,y,\ldots) \land E(x,y,\ldots) = F </math>|| where <math>x</math>, <math>y</math>, <math>\ldots</math> are not free in <math>F</math> ||  A
{{RRRow}}|*||{{Rulename|SIMP_IN_COMPSET}}||<math>  F \in  \{  x,y,\ldots \qdot  P(x,y,\ldots) \mid  E(x,y,\ldots) \}  \;\;\defi\;\;  \exists x,y,\ldots \qdot P(x,y,\ldots) \land E(x,y,\ldots) = F </math>|| where <math>x</math>, <math>y</math>, <math>\ldots</math> are not free in <math>F</math> ||  A
{{RRRow}}|*||{{Rulename|SIMP_IN_COMPSET_ONEPOINT}}||<math>  E \in  \{  x \qdot  P(x) \mid  x \}  \;\;\defi\;\;  P(E) </math>|| Equivalent to general simplification followed by One Point Rule application with the last conjunct predicate ||  A
{{RRRow}}|*||{{Rulename|SIMP_IN_COMPSET_ONEPOINT}}||<math>  E \in  \{  x \qdot  P(x) \mid  x \}  \;\;\defi\;\;  P(E) </math>|| Equivalent to general simplification followed by One Point Rule application with the last conjunct predicate ||  A
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{{RRRow}}|*||{{Rulename|SIMP_SPECIAL_KBOOL_BFALSE}}||<math>  \bool (\bfalse ) \;\;\defi\;\;  \False </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_SPECIAL_KBOOL_BFALSE}}||<math>  \bool (\bfalse ) \;\;\defi\;\;  \False </math>||  ||  A
{{RRRow}}|||{{Rulename|DISTRI_SUBSETEQ_BUNION_SING}}||<math>  S \bunion  \{ F\}  \subseteq  T \;\;\defi\;\;  S \subseteq  T \land  F \in  T </math>|| where <math>F</math> is a single expression ||  M
{{RRRow}}|||{{Rulename|DISTRI_SUBSETEQ_BUNION_SING}}||<math>  S \bunion  \{ F\}  \subseteq  T \;\;\defi\;\;  S \subseteq  T \land  F \in  T </math>|| where <math>F</math> is a single expression ||  M
{{RRRow}}| ||{{Rulename|DEF_FINITE}}||<math>  \finite(S) \;\;\defi\;\;  \exists n,f\qdot n\in\nat\land f\in 1\upto n \tbij S </math>||  ||  M
{{RRRow}}|*||{{Rulename|DEF_FINITE}}||<math>  \finite(S) \;\;\defi\;\;  \exists n,f\qdot f\in 1\upto n \tbij S </math>||  ||  M
{{RRRow}}|*||{{Rulename|SIMP_SPECIAL_FINITE}}||<math>  \finite (\emptyset ) \;\;\defi\;\;  \btrue </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_SPECIAL_FINITE}}||<math>  \finite (\emptyset ) \;\;\defi\;\;  \btrue </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_FINITE_SETENUM}}||<math>  \finite (\{ a, \ldots , b\} ) \;\;\defi\;\;  \btrue </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_FINITE_SETENUM}}||<math>  \finite (\{ a, \ldots , b\} ) \;\;\defi\;\;  \btrue </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_FINITE_BUNION}}||<math>  \finite (S \bunion  T) \;\;\defi\;\;  \finite (S) \land  \finite (T) </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_FINITE_BUNION}}||<math>  \finite (S \bunion  T) \;\;\defi\;\;  \finite (S) \land  \finite (T) </math>||  ||  A
{{RRRow}}| ||{{Rulename|SIMP_FINITE_UNION}}||<math>\finite(\union(S)) \;\;\defi\;\; \finite(S)\;\land\;(\forall x\qdot x\in S\limp finite(x))</math>||  ||  M
{{RRRow}}| ||{{Rulename|SIMP_FINITE_QUNION}}||<math>\finite(\Union x\qdot P\mid E) \;\;\defi\;\; \finite(\{x\qdot P\mid E\})\;\land\;(\forall x\qdot P\limp finite(E))</math>||  ||  M
{{RRRow}}|*||{{Rulename|SIMP_FINITE_POW}}||<math>  \finite (\pow (S)) \;\;\defi\;\;  \finite (S) </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_FINITE_POW}}||<math>  \finite (\pow (S)) \;\;\defi\;\;  \finite (S) </math>||  ||  A
{{RRRow}}|*||{{Rulename|DERIV_FINITE_CPROD}}||<math>  \finite (S \cprod  T) \;\;\defi\;\;  S = \emptyset  \lor  T = \emptyset  \lor  (\finite (S) \land  \finite (T)) </math>||  ||  A
{{RRRow}}|*||{{Rulename|DERIV_FINITE_CPROD}}||<math>  \finite (S \cprod  T) \;\;\defi\;\;  S = \emptyset  \lor  T = \emptyset  \lor  (\finite (S) \land  \finite (T)) </math>||  ||  A
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{{RRRow}}|*||{{Rulename|SIMP_FINITE_UPTO}}||<math>  \finite (a \upto  b) \;\;\defi\;\;  \btrue </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_FINITE_UPTO}}||<math>  \finite (a \upto  b) \;\;\defi\;\;  \btrue </math>||  ||  A
<!-- Disabled rules (some are false, need more thinking to check which)
<!-- Disabled rules (some are false, need more thinking to check which)
{{RRRow}}|||{{Rulename|SIMP_FINITE_UNION}}||<math>\finite(\union(S)) \;\;\defi\;\; \finite(S)\;\land\;(\forall x\qdot x\in S\limp \finite(x))</math>||  ||  M
{{RRRow}}|||{{Rulename|SIMP_FINITE_QUNION}}||<math>\finite(\Union x\qdot P\mid E) \;\;\defi\;\; \finite(\{x\qdot P\mid E\})\;\land\;(\forall x\qdot P\limp \finite(E))</math>|| this equivalence is false: for example, the union of an infinite set of empty sets is finite; the right-to-left implication is true ||  M
{{RRRow}}|||{{Rulename|SIMP_FINITE_BINTER_L}}||<math>  \finite (S \binter  T) \;\;\defi\;\;  \finite (S) </math>||  ||  M
{{RRRow}}|||{{Rulename|SIMP_FINITE_BINTER_L}}||<math>  \finite (S \binter  T) \;\;\defi\;\;  \finite (S) </math>||  ||  M
{{RRRow}}|||{{Rulename|SIMP_FINITE_BINTER_R}}||<math>  \finite (S \binter  T) \;\;\defi\;\;  \finite (T) </math>||  ||  M
{{RRRow}}|||{{Rulename|SIMP_FINITE_BINTER_R}}||<math>  \finite (S \binter  T) \;\;\defi\;\;  \finite (T) </math>||  ||  M
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{{RRRow}}|*||{{Rulename|SIMP_FINITE_BOOL}}||<math>  \finite (\Bool ) \;\;\defi\;\;  \btrue </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_FINITE_BOOL}}||<math>  \finite (\Bool ) \;\;\defi\;\;  \btrue </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_FINITE_LAMBDA}}||<math>  \finite(\{x\qdot P\mid E\mapsto F\}) \;\;\defi\;\;  \finite(\{x\qdot P\mid E\} ) </math>|| where <math>E</math> is a maplet combination of bound identifiers and expressions that are not bound by the comprehension set (i.e., <math>E</math> is syntactically injective) and all identifiers bound by the comprehension set that occur in <math>F</math> also occur in <math>E</math> ||  A
{{RRRow}}|*||{{Rulename|SIMP_FINITE_LAMBDA}}||<math>  \finite(\{x\qdot P\mid E\mapsto F\}) \;\;\defi\;\;  \finite(\{x\qdot P\mid E\} ) </math>|| where <math>E</math> is a maplet combination of bound identifiers and expressions that are not bound by the comprehension set (i.e., <math>E</math> is syntactically injective) and all identifiers bound by the comprehension set that occur in <math>F</math> also occur in <math>E</math> ||  A
{{RRRow}}|*||{{Rulename|SIMP_TYPE_EQUAL_EMPTY}}||<math> \mathit{Ty} = \emptyset  \;\;\defi\;\;  \bfalse </math>|| where <math>\mathit{Ty}</math> is a type expression ||  A
{{RRRow}}|*||{{Rulename|SIMP_TYPE_IN}}||<math>  t \in  \mathit{Ty} \;\;\defi\;\;  \btrue </math>|| where <math>\mathit{Ty}</math> is a type expression ||  A
{{RRRow}}|*||{{Rulename|SIMP_TYPE_IN}}||<math>  t \in  \mathit{Ty} \;\;\defi\;\;  \btrue </math>|| where <math>\mathit{Ty}</math> is a type expression ||  A
{{RRRow}}|*||{{Rulename|SIMP_SPECIAL_EQV_BTRUE}}||<math>  P \leqv  \btrue  \;\;\defi\;\;  P </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_SPECIAL_EQV_BTRUE}}||<math>  P \leqv  \btrue  \;\;\defi\;\;  P </math>||  ||  A
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{{RRRow}}|*||{{Rulename|DERIV_NOT_FORALL}}||<math>  \lnot\, \forall x \qdot  P \;\;\defi\;\;  \exists x \qdot  \lnot\, P </math>||  ||  M
{{RRRow}}|*||{{Rulename|DERIV_NOT_FORALL}}||<math>  \lnot\, \forall x \qdot  P \;\;\defi\;\;  \exists x \qdot  \lnot\, P </math>||  ||  M
{{RRRow}}|*||{{Rulename|DERIV_NOT_EXISTS}}||<math>  \lnot\, \exists x \qdot  P \;\;\defi\;\;  \forall x \qdot  \lnot\, P </math>||  ||  M
{{RRRow}}|*||{{Rulename|DERIV_NOT_EXISTS}}||<math>  \lnot\, \exists x \qdot  P \;\;\defi\;\;  \forall x \qdot  \lnot\, P </math>||  ||  M
{{RRRow}}|*||{{Rulename|DEF_SPECIAL_NOT_EQUAL}}||<math>  \lnot\, S = \emptyset  \;\;\defi\;\;  \exists x \qdot  x \in  S </math>|| where <math>x</math> is not free in <math>S</math> ||  M
{{RRRow}}|*||{{Rulename|DEF_IN_MAPSTO}}||<math>  E \mapsto  F \in  S \cprod  T \;\;\defi\;\;  E \in  S \land  F \in  T </math>||  ||  AM
{{RRRow}}|*||{{Rulename|DEF_IN_MAPSTO}}||<math>  E \mapsto  F \in  S \cprod  T \;\;\defi\;\;  E \in  S \land  F \in  T </math>||  ||  M
{{RRRow}}|*||{{Rulename|DEF_IN_POW}}||<math>  E \in  \pow (S) \;\;\defi\;\;  E \subseteq  S </math>||  ||  M
{{RRRow}}|*||{{Rulename|DEF_IN_POW}}||<math>  E \in  \pow (S) \;\;\defi\;\;  E \subseteq  S </math>||  ||  M
{{RRRow}}|*||{{Rulename|DEF_IN_POW1}}||<math>  E \in  \pown (S) \;\;\defi\;\; E \in  \pow (S) \land S \neq \emptyset </math>||  ||  M
{{RRRow}}|*||{{Rulename|DEF_IN_POW1}}||<math>  E \in  \pown (S) \;\;\defi\;\; E \in  \pow (S) \land S \neq \emptyset </math>||  ||  M
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\land& s_{n-1}\binter s_n = \emptyset
\land& s_{n-1}\binter s_n = \emptyset
\end{array}</math>||  ||  AM
\end{array}</math>||  ||  AM
{{RRRow}}|*||{{Rulename|SIMP_EMPTY_PARTITION}}||<math>\operatorname{partition}(S)  \;\;\defi\;\;  S = \emptyset </math>||  ||  A
{{RRRow}}|*||{{Rulename|SIMP_SINGLE_PARTITION}}||<math>\operatorname{partition}(S, T)  \;\;\defi\;\;  S = T </math>||  ||  A
{{RRRow}}|*||{{Rulename|DEF_EQUAL_CARD}}||<math>\operatorname{card}(S) = k \;\;\defi\;\; \exists f \qdot f \in 1..k \tbij S</math>|| also works for <math>k = \operatorname{card}(S)</math> || M
{{RRRow}}|*||{{Rulename|SIMP_EQUAL_CARD}}||<math>\operatorname{card}(S) = \operatorname{card}(T) \;\;\defi\;\; \exists f \qdot f \in S \tbij T</math>|| || M
|}
|}


Latest revision as of 14:33, 13 April 2023

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


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