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Pygments / examplefiles / output / coq_RelationClasses
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rɆrÊhX(*rˆrÌhj½†rÍhX rΆrÏhj½†rÐhXÚ  Typeclass-based relations, tactics and standard instances

   This is the basic theory needed to formalize morphisms and setoids.

   Author: Matthieu Sozeau
   Institution: LRI, CNRS UMR 8623 - University Paris Sud
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  rI†rJhŒX
irreflexivityrK†rLjsjΆrMhtj=†rNjsjΆrOhŒX	ReflexiverP†rQjsjΆrRhtj6†rShŒX
complementrT†rUjsjΆrVhŒj0†rWhtjD†rXhtjç†rYjsX

rZ†r[jKXHintr\†r]jsjΆr^hŒXExternr_†r`jsjΆrajŸX1rb†rcjsjΆrdhtj6†rehŒX	Reflexiverf†rgjsjΆrhhtj6†rihŒX
complementrj†rkjsjΆrlhtjB†rmhtjD†rnhtjD†rojsjΆrphtX=>rq†rrjsjΆrshŒXclass_applyrt†rujsjΆrvhtX@rw†rxhŒX
irreflexivityry†rzjsjΆr{htj=†r|jsjΆr}hŒXtypeclass_instancesr~†rhtjç†r€jsX

r†r‚jKXClassrƒ†r„jsjΆr…hŒX	Symmetricr††r‡jsjΆrˆhtjW†r‰hŒjY†rŠhtj[†r‹jsjΆrŒhtj6†rhŒj0†rŽjsjΆrhtj=†rjsjΆr‘hŒXrelationr’†r“jsjΆr”hŒjY†r•htjD†r–jsjΆr—htX:=r˜†r™jsX
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r¸†r¹jKXClassrº†r»jsjΆr¼hŒX
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  rцrÒhŒX	asymmetryrÓ†rÔjsjΆrÕhtj=†rÖjsjΆr×jXforallr؆rÙjsjΆrÚhŒjv†rÛjsjΆrÜhŒjy†rÝhtjÀ†rÞjsjΆrßhŒj0†ràjsjΆráhŒjv†râjsjΆrãhŒjy†räjsjΆråhtX->ræ†rçjsjΆrèhŒj0†réjsjΆrêhŒjy†rëjsjΆrìhŒjv†ríjsjΆrîhtX->rï†rðjsjΆrñhŒXFalserò†róhtjç†rôjsX

rõ†röjKXClassr÷†røjsjΆrùhŒX
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irreflexivityrA†rBjsjΆrChtj=†rDjsjΆrEhŒXordrF†rGhtjç†rHjsX

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r:†r;jKXHintr<†r=jsjΆr>hŒXExternr?†r@jsjΆrAjŸX3rB†rCjsjΆrDhtj6†rEhŒX	ReflexiverF†rGjsjΆrHhtj6†rIhŒXfliprJ†rKjsjΆrLhtjB†rMhtjD†rNhtjD†rOjsjΆrPhtX=>rQ†rRjsjΆrSjXapplyrT†rUjsjΆrVhŒXflip_ReflexiverW†rXjsjΆrYhtj=†rZjsjΆr[hŒXtypeclass_instancesr\†r]htjç†r^jsX

r_†r`hŒXProgramra†rbjsjΆrcjKX
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rŽ†rhŒXProgramr†r‘jsjΆr’jKX
Definitionr“†r”jsjΆr•hŒXflip_Symmetricr–†r—jsjΆr˜htj†r™htj6†ršhŒX	Symmetricr›†rœjsjΆrhŒjY†ržjsjΆrŸhŒj0†r htjD†r¡jsjΆr¢htj=†r£jsjΆr¤hŒX	Symmetricr¥†r¦jsjΆr§htj6†r¨hŒXflipr©†rªjsjΆr«hŒj0†r¬htjD†r­jsjΆr®htX:=r¯†r°jsX
  r±†r²jXfunr³†r´jsjΆrµhŒjv†r¶jsjΆr·hŒjy†r¸jsjΆr¹hŒj•†rºjsjΆr»htX=>r¼†r½jsjΆr¾jXsymmetryr¿†rÀjsjΆrÁhtj6†rÂhŒj0†rÃhtX:=rĆrÅe(hŒj0†rÆhtjD†rÇjsjΆrÈhŒj•†rÉhtjç†rÊjsX

rˆrÌhŒXProgramr͆rÎjsjΆrÏjKX
DefinitionrІrÑjsjΆrÒhŒXflip_AsymmetricrÓ†rÔjsjΆrÕhtj†rÖhtj6†r×hŒX
Asymmetricr؆rÙjsjΆrÚhŒjY†rÛjsjΆrÜhŒj0†rÝhtjD†rÞjsjΆrßhtj=†ràjsjΆráhŒX
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r	†r	hŒXProgramr	†r	jsjΆr	jKX
Definitionr	†r	jsjΆr	hŒXflip_Transitiver	†r	jsjΆr	htj†r	htj6†r	hŒX
Transitiver	†r	jsjΆr	hŒjY†r	jsjΆr	hŒj0†r 	htjD†r!	jsjΆr"	htj=†r#	jsjΆr$	hŒX
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r†rhX(*r†rhj½†rhX+ An Equivalence is a PER plus reflexivity. r†rhX*)r†rjsX

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rZ†r[hX(*r\†r]hj½†r^hXO We can now define antisymmetry w.r.t. an equivalence relation on the carrier. r_†r`hX*)ra†rbjsX

rc†rdjKXClassre†rfjsjΆrghŒX
Antisymmetricrh†rijsjΆrjhŒjY†rkjsjΆrlhŒXeqArm†rnjsjΆrohtj†rphtjW†rqhŒXequrr†rsjsjΆrthtj=†rujsjΆrvhŒXEquivalencerw†rxjsjΆryhŒjY†rzjsjΆr{hŒXeqAr|†r}htj[†r~jsjΆrhtj6†r€hŒj0†rjsjΆr‚htj=†rƒjsjΆr„hŒXrelationr…†r†jsjΆr‡hŒjY†rˆhtjD†r‰jsjΆrŠhtX:=r‹†rŒjsX
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r·†r¸hŒXProgramr¹†rºjsjΆr»jKX
Definitionr¼†r½jsjΆr¾hŒXflip_antiSymmetricr¿†rÀjsjΆrÁhtj†rÂhtj6†rÃhŒX
AntisymmetricrĆrÅjsjΆrÆhŒjY†rÇjsjΆrÈhŒXeqArɆrÊjsjΆrËhŒj0†rÌhtjD†rÍjsjΆrÎhtj=†rÏjsX
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firstorderræ†rçhtjç†rèjsjΆréjKXQedrê†rëhtjç†rìjsX

rí†rîhX(*rï†rðhj½†rñhX“ Leibinz equality [eq] is an equivalence relation.
   The instance has low priority as it is always applicable
   if only the type is constrained. rò†róhX*)rô†rõjsX

rö†r÷hŒXProgramrø†rùjsjΆrújKXInstancerû†rüjsjΆrýhŒXeq_equivalencerþ†rÿjsjΆrhtj=†rjsjΆrhŒXEquivalencer†rjsjΆrhtj6†rhtjw†rhŒXeqr†r	jsjΆr
hŒjY†rhtjD†rjsjΆr
htjs†rjsjΆrjŸX10r†rhtjç†rjsX

r†rhX(*r†rhj½†rhX7 Logical equivalence [iff] is an equivalence relation. r†rhX*)r†rjsX

r†rhŒXProgramr†rjsjΆr jKXInstancer!†r"jsjΆr#hŒXiff_equivalencer$†r%jsjΆr&htj=†r'jsjΆr(hŒXEquivalencer)†r*jsjΆr+hŒXiffr,†r-htjç†r.jsX

r/†r0hX(*r1†r2hj½†r3hXÅ We now develop a generalization of results on relations for arbitrary predicates.
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r8†r9jKXLocalr:†r;jsjΆr<jKXOpenr=†r>jsjΆr?jKXScoper@†rAjsjΆrBhŒX
list_scoperC†rDhtjç†rEjsX

rF†rGhX(*rH†rIhX' Notation " [ ] " := nil : list_scope. rJ†rKhX*)rL†rMjsjƆrNhX(*rO†rPhX  Notation " [ x ; .. ; y ] " := rQ†rRhj6†rShX
cons x .. rT†rUhj6†rVhX
cons y nilrW†rXhjD†rYhX ..rZ†r[hjD†r\hjΆr]hj6†r^hX
at level 1r_†r`hjD†rahX : list_scope. rb†rchX*)rd†rejsX

rf†rghX(*rh†rihj½†rjhXS A compact representation of non-dependent arities, with the codomain singled-out. rk†rlhX*)rm†rnjsX

ro†rpjKXFixpointrq†rrjsjΆrshŒXarrowsrt†rujsjΆrvhtj6†rwhŒj†rxjsjΆryhtj=†rzjsjΆr{hŒXlistr|†r}jsjΆr~j-XTyper†r€htjD†rjsjΆr‚htj6†rƒhŒXrr„†r…jsjΆr†htj=†r‡jsjΆrˆj-XTyper‰†rŠhtjD†r‹jsjΆrŒhtj=†rjsjΆrŽj-XTyper†rjsjΆr‘htX:=r’†r“jsX
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rȆrÉhX(*rʆrËhj½†rÌhXQ We can define abbreviations for operation and relation types based on [arrows]. r͆rÎhX*)rφrÐjsX

rцrÒjKX
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Definitionrì†ríjsjΆrîhŒXbinary_operationrï†rðjsjΆrñhŒjY†ròjsjΆróhtX:=rô†rõjsjΆröhŒXarrowsr÷†røjsjΆrùhtj6†rúhŒjY†rûhtX::rü†rýhŒjY†rþhtX::rÿ†rhŒXnilr†rhtjD†rjsjΆrhŒjY†rhtjç†rjsjƆrjKX
Definitionr†r	jsjΆr
hŒXternary_operationr†rjsjΆr
hŒjY†rjsjΆrhtX:=r†rjsjΆrhŒXarrowsr†rjsjΆrhtj6†rhŒjY†rhtX::r†rhŒjY†rhtX::r†rhŒjY†rhtX::r†rhŒXnilr †r!htjD†r"jsjΆr#hŒjY†r$htjç†r%jsX

r&†r'hX(*r(†r)hj½†r*hX8 We define n-ary [predicate]s as functions into [Prop]. r+†r,hX*)r-†r.jsX

r/†r0jKXNotationr1†r2jsjΆr3hŒX	predicater4†r5jsjΆr6hŒj†r7jsjΆr8htX:=r9†r:jsjΆr;htj6†r<hŒXarrowsr=†r>jsjΆr?hŒj†r@jsjΆrAj-XProprB†rChtjD†rDhtjç†rEjsX

rF†rGhX(*rH†rIhj½†rJhX Unary predicates, or sets. rK†rLhX*)rM†rNjsX

rO†rPjKX
DefinitionrQ†rRjsjΆrShŒXunary_predicaterT†rUjsjΆrVhŒjY†rWjsjΆrXhtX:=rY†rZjsjΆr[hŒX	predicater\†r]jsjΆr^htj6†r_hŒjY†r`htX::ra†rbhŒXnilrc†rdhtjD†rehtjç†rfjsX

rg†rhhX(*ri†rjhj½†rkhX; Homogeneous binary relations, equivalent to [relation A]. rl†rmhX*)rn†rojsX

rp†rqjKX
Definitionrr†rsjsjΆrthŒXbinary_relationru†rvjsjΆrwhŒjY†rxjsjΆryhtX:=rz†r{jsjΆr|hŒX	predicater}†r~jsjΆrhtj6†r€hŒjY†rhtX::r‚†rƒhŒjY†r„htX::r…†r†hŒXnilr‡†rˆhtjD†r‰htjç†rŠjsX

r‹†rŒhX(*r†rŽhj½†rhXF We can close a predicate by universal or existential quantification. r†r‘hX*)r’†r“jsX

r”†r•jKXFixpointr–†r—jsjΆr˜hŒX
predicate_allr™†ršjsjΆr›htj6†rœhŒj†rjsjΆržhtj=†rŸjsjΆr hŒXlistr¡†r¢jsjΆr£j-XTyper¤†r¥htjD†r¦jsjΆr§htj=†r¨jsjΆr©hŒX	predicaterª†r«jsjΆr¬hŒj†r­jsjΆr®htX->r¯†r°jsjΆr±j-XPropr²†r³jsjΆr´htX:=rµ†r¶jsX
  r·†r¸jXmatchr¹†rºjsjΆr»hŒj†r¼jsjΆr½jXwithr¾†r¿jsX
    rÀ†rÁhtjs†rÂjsjΆrÃhŒXnilrĆrÅjsjΆrÆhtX=>rdžrÈjsjΆrÉjXfunrʆrËjsjΆrÌhŒXfr͆rÎjsjΆrÏhtX=>rІrÑjsjΆrÒhŒj͆rÓjsX
    rÔ†rÕhtjs†rÖjsjΆr×hŒjY†rØjsjΆrÙhtX::rÚ†rÛjsjΆrÜhŒXtlr݆rÞjsjΆrßhtX=>rà†rájsjΆrâjXfunrã†räjsjΆråhŒj͆ræjsjΆrçhtX=>rè†réjsjΆrêjXforallrë†rìjsjΆríhŒjv†rîjsjΆrïhtj=†rðjsjΆrñhŒjY†ròhtjÀ†rójsjΆrôhŒX
predicate_allrõ†röjsjΆr÷hŒXtlrø†rùjsjΆrúhtj6†rûhŒj͆rüjsjΆrýhŒjv†rþhtjD†rÿjsX
  r†rjXendr†rhtjç†rjsX

r†rjKXFixpointr†rjsjΆr	hŒXpredicate_existsr
†rjsjΆrhtj6†r
hŒj†rjsjΆrhtj=†rjsjΆrhŒXlistr†rjsjΆrj-XTyper†rhtjD†rjsjΆrhtj=†rjsjΆrhŒX	predicater†rjsjΆrhŒj†rjsjΆrhtX->r †r!jsjΆr"j-XPropr#†r$jsjΆr%htX:=r&†r'jsX
  r(†r)jXmatchr*†r+jsjΆr,hŒj†r-jsjΆr.jXwithr/†r0jsX
    r1†r2htjs†r3jsjΆr4hŒXnilr5†r6jsjΆr7htX=>r8†r9jsjΆr:jXfunr;†r<jsjΆr=hŒj͆r>jsjΆr?htX=>r@†rAjsjΆrBhŒj͆rCjsX
    rD†rEhtjs†rFjsjΆrGhŒjY†rHjsjΆrIhtX::rJ†rKjsjΆrLhŒXtlrM†rNjsjΆrOhtX=>rP†rQjsjΆrRjXfunrS†rTjsjΆrUhŒj͆rVjsjΆrWhtX=>rX†rYjsjΆrZjXexistsr[†r\jsjΆr]hŒjv†r^jsjΆr_htj=†r`jsjΆrahŒjY†rbhtjÀ†rcjsjΆrdhŒXpredicate_existsre†rfjsjΆrghŒXtlrh†rijsjΆrjhtj6†rkhŒj͆rljsjΆrmhŒjv†rnhtjD†rojsX
  rp†rqjXendrr†rshtjç†rtjsX

ru†rvhX(*rw†rxhj½†ryhX» Pointwise extension of a binary operation on [T] to a binary operation
   on functions whose codomain is [T].
   For an operator on [Prop] this lifts the operator to a binary operation. rz†r{hX*)r|†r}jsX

r~†rjKXFixpointr€†rjsjΆr‚hŒXpointwise_extensionrƒ†r„jsjΆr…htjW†r†hŒjú†r‡jsjΆrˆhtj=†r‰jsjΆrŠj-XTyper‹†rŒhtj[†rjsjΆrŽhtj6†rhŒXopr†r‘jsjΆr’htj=†r“jsjΆr”hŒXbinary_operationr•†r–jsjΆr—hŒjú†r˜htjD†r™jsX
  rš†r›htj6†rœhŒj†rjsjΆržhtj=†rŸjsjΆr hŒXlistr¡†r¢jsjΆr£j-XTyper¤†r¥htjD†r¦jsjΆr§htj=†r¨jsjΆr©hŒXbinary_operationrª†r«jsjΆr¬htj6†r­hŒXarrowsr®†r¯jsjΆr°hŒj†r±jsjΆr²hŒjú†r³htjD†r´jsjΆrµhtX:=r¶†r·jsX
  r¸†r¹jXmatchrº†r»jsjΆr¼hŒj†r½jsjΆr¾jXwithr¿†rÀjsX
    rÁ†rÂhtjs†rÃjsjΆrÄhŒXnilrņrÆjsjΆrÇhtX=>rȆrÉjsjΆrÊjXfunrˆrÌjsjΆrÍhŒj0†rÎjsjΆrÏhŒXR'rІrÑjsjΆrÒhtX=>rÓ†rÔjsjΆrÕhŒXoprÖ†r×jsjΆrØhŒj0†rÙjsjΆrÚhŒXR'rÛ†rÜjsX
    r݆rÞhtjs†rßjsjΆràhŒjY†rájsjΆrâhtX::rã†räjsjΆråhŒXtlræ†rçjsjΆrèhtX=>ré†rêjsjΆrëjXfunrì†ríjsjΆrîhŒj0†rïjsjΆrðhŒXR'rñ†ròjsjΆróhtX=>rô†rõjsX
      rö†r÷jXfunrø†rùjsjΆrúhŒjv†rûjsjΆrühtX=>rý†rþjsjΆrÿhŒXpointwise_extensionr†rjsjΆrhŒXopr†rjsjΆrhŒXtlr†rjsjΆrhtj6†r	hŒj0†r
jsjΆrhŒjv†rhtjD†r
jsjΆrhtj6†rhŒXR'r†rjsjΆrhŒjv†rhtjD†rjsX
  r†rjXendr†rhtjç†rjsX

r†rhX(*r†rhj½†rhXa Pointwise lifting, equivalent to doing [pointwise_extension] and closing using [predicate_all]. r†r hX*)r!†r"jsX

r#†r$jKXFixpointr%†r&jsjΆr'hŒXpointwise_liftingr(†r)jsjΆr*htj6†r+hŒXopr,†r-jsjΆr.htj=†r/jsjΆr0hŒXbinary_relationr1†r2jsjΆr3j-XPropr4†r5htjD†r6jsX  r7†r8htj6†r9hŒj†r:jsjΆr;htj=†r<jsjΆr=hŒXlistr>†r?jsjΆr@j-XTyperA†rBhtjD†rCjsjΆrDhtj=†rEjsjΆrFhŒXbinary_relationrG†rHjsjΆrIhtj6†rJhŒX	predicaterK†rLjsjΆrMhŒj†rNhtjD†rOjsjΆrPhtX:=rQ†rRjsX
  rS†rTjXmatchrU†rVjsjΆrWhŒj†rXjsjΆrYjXwithrZ†r[jsX
    r\†r]htjs†r^jsjΆr_hŒXnilr`†rajsjΆrbhtX=>rc†rdjsjΆrejXfunrf†rgjsjΆrhhŒj0†rijsjΆrjhŒXR'rk†rljsjΆrmhtX=>rn†rojsjΆrphŒXoprq†rrjsjΆrshŒj0†rtjsjΆruhŒXR'rv†rwjsX
    rx†ryhtjs†rzjsjΆr{hŒjY†r|jsjΆr}htX::r~†rjsjΆr€hŒXtlr†r‚jsjΆrƒhtX=>r„†r…jsjΆr†jXfunr‡†rˆjsjΆr‰hŒj0†rŠjsjΆr‹hŒXR'rŒ†rjsjΆrŽhtX=>r†rjsX
      r‘†r’jXforallr“†r”jsjΆr•hŒjv†r–htjÀ†r—jsjΆr˜hŒXpointwise_liftingr™†ršjsjΆr›hŒXoprœ†rjsjΆržhŒXtlrŸ†r jsjΆr¡htj6†r¢hŒj0†r£jsjΆr¤hŒjv†r¥htjD†r¦jsjΆr§htj6†r¨hŒXR'r©†rªjsjΆr«hŒjv†r¬htjD†r­jsX
  r®†r¯jXendr°†r±htjç†r²jsX

r³†r´hX(*rµ†r¶hj½†r·hXN The n-ary equivalence relation, defined by lifting the 0-ary [iff] relation. r¸†r¹hX*)rº†r»jsX

r¼†r½jKX
Definitionr¾†r¿jsjΆrÀhŒXpredicate_equivalencerÁ†rÂjsjΆrÃhtjW†rÄhŒj†rÅjsjΆrÆhtj=†rÇjsjΆrÈhŒXlistrɆrÊjsjΆrËj-XTyper̆rÍhtj[†rÎjsjΆrÏhtj=†rÐjsjΆrÑhŒXbinary_relationrÒ†rÓjsjΆrÔhtj6†rÕhŒX	predicaterÖ†r×jsjΆrØhŒj†rÙhtjD†rÚjsjΆrÛhtX:=r܆rÝjsX
  rÞ†rßhŒXpointwise_liftingrà†rájsjΆrâhŒXiffrã†räjsjΆråhŒj†ræhtjç†rçjsX

rè†réhX(*rê†rëhj½†rìhXO The n-ary implication relation, defined by lifting the 0-ary [impl] relation. rí†rîhX*)rï†rðjsX

rñ†ròjKX
Definitionró†rôjsjΆrõhŒXpredicate_implicationrö†r÷jsjΆrøhtjW†rùhŒj†rújsjΆrûhtj=†rüjsjΆrýhŒXlistrþ†rÿjsjΆrj-XTyper†rhtj[†rjsjΆrhtX:=r†rjsX
  r†rhŒXpointwise_liftingr	†r
jsjΆrhŒXimplr†r
jsjΆrhŒj†rhtjç†rjsX

r†rhX(*r†rhj½†rhXD Notations for pointwise equivalence and implication of predicates. r†rhX*)r†rjsX

r†rhŒXInfixr†rjsjΆrjHj>†rjHX<∙>r †r!jHj>†r"jsjΆr#htX:=r$†r%jsjΆr&hŒXpredicate_equivalencer'†r(jsjΆr)htj6†r*hŒXatr+†r,jsjΆr-hŒXlevelr.†r/jsjΆr0jŸX95r1†r2htjÀ†r3jsjΆr4hŒXnor5†r6jsjΆr7hŒX
associativityr8†r9htjD†r:jsjΆr;htj=†r<jsjΆr=hŒXpredicate_scoper>†r?htjç†r@jsjƆrAhŒXInfixrB†rCjsjΆrDjHj>†rEjHX-∙>rF†rGjHj>†rHjsjΆrIhtX:=rJ†rKjsjΆrLe(hŒXpredicate_implicationrM†rNjsjΆrOhtj6†rPhŒXatrQ†rRjsjΆrShŒXlevelrT†rUjsjΆrVjŸX70rW†rXhtjÀ†rYjsjΆrZjXrightr[†r\jsjΆr]hŒX
associativityr^†r_htjD†r`jsjΆrahtj=†rbjsjΆrchŒXpredicate_scoperd†rehtjç†rfjsX

rg†rhjKXOpenri†rjjsjΆrkjKXLocalrl†rmjsjΆrnjKXScopero†rpjsjΆrqhŒXpredicate_scoperr†rshtjç†rtjsX

ru†rvhX(*rw†rxhj½†ryhX The pointwise liftings of conjunction and disjunctions.
   Note that these are [binary_operation]s, building new relations out of old ones. rz†r{hX*)r|†r}jsX

r~†rjKX
Definitionr€†rjsjΆr‚hŒXpredicate_intersectionrƒ†r„jsjΆr…htX:=r††r‡jsjΆrˆhŒXpointwise_extensionr‰†rŠjsjΆr‹hŒXandrŒ†rhtjç†rŽjsjƆrjKX
Definitionr†r‘jsjΆr’hŒXpredicate_unionr“†r”jsjΆr•htX:=r–†r—jsjΆr˜hŒXpointwise_extensionr™†ršjsjΆr›hŒXorrœ†rhtjç†ržjsX

rŸ†r hŒXInfixr¡†r¢jsjΆr£jHj>†r¤jHX/∙\r¥†r¦jHj>†r§jsjΆr¨htX:=r©†rªjsjΆr«hŒXpredicate_intersectionr¬†r­jsjΆr®htj6†r¯hŒXatr°†r±jsjΆr²hŒXlevelr³†r´jsjΆrµjŸX80r¶†r·htjÀ†r¸jsjΆr¹jXrightrº†r»jsjΆr¼hŒX
associativityr½†r¾htjD†r¿jsjΆrÀhtj=†rÁjsjΆrÂhŒXpredicate_scoperÆrÄhtjç†rÅjsjƆrÆhŒXInfixrdžrÈjsjΆrÉjHj>†rÊjHX\∙/rˆrÌjHj>†rÍjsjΆrÎhtX:=rφrÐjsjΆrÑhŒXpredicate_unionrÒ†rÓjsjΆrÔhtj6†rÕhŒXatrÖ†r×jsjΆrØhŒXlevelrÙ†rÚjsjΆrÛjŸX85r܆rÝhtjÀ†rÞjsjΆrßjXrightrà†rájsjΆrâhŒX
associativityrã†rähtjD†råjsjΆræhtj=†rçjsjΆrèhŒXpredicate_scoperé†rêhtjç†rëjsX

rì†ríhX(*rî†rïhj½†rðhX2 The always [True] and always [False] predicates. rñ†ròhX*)ró†rôjsX

rõ†röjKXFixpointr÷†røjsjΆrùjJXtruerú†rûhtjB†rühŒX	predicaterý†rþjsjΆrÿhtjW†rhŒj†rjsjΆrhtj=†rjsjΆrhŒXlistr†rjsjΆrj-XTyper†r	htj[†r
jsjΆrhtj=†rjsjΆr
hŒX	predicater†rjsjΆrhŒj†rjsjΆrhtX:=r†rjsX
  r†rjXmatchr†rjsjΆrhŒj†rjsjΆrjXwithr†rjsX
    r†rhtjs†r jsjΆr!hŒXnilr"†r#jsjΆr$htX=>r%†r&jsjΆr'hŒXTruer(†r)jsX
    r*†r+htjs†r,jsjΆr-hŒjY†r.jsjΆr/htX::r0†r1jsjΆr2hŒXtlr3†r4jsjΆr5htX=>r6†r7jsjΆr8jXfunr9†r:jsjΆr;htjB†r<jsjΆr=htX=>r>†r?jsjΆr@htjw†rAjJXtruerB†rChtjB†rDhŒX	predicaterE†rFjsjΆrGhŒXtlrH†rIjsX
  rJ†rKjXendrL†rMhtjç†rNjsX

rO†rPjKXFixpointrQ†rRjsjΆrSjJXfalserT†rUhtjB†rVhŒX	predicaterW†rXjsjΆrYhtjW†rZhŒj†r[jsjΆr\htj=†r]jsjΆr^hŒXlistr_†r`jsjΆraj-XTyperb†rchtj[†rdjsjΆrehtj=†rfjsjΆrghŒX	predicaterh†rijsjΆrjhŒj†rkjsjΆrlhtX:=rm†rnjsX
  ro†rpjXmatchrq†rrjsjΆrshŒj†rtjsjΆrujXwithrv†rwjsX
    rx†ryhtjs†rzjsjΆr{hŒXnilr|†r}jsjΆr~htX=>r†r€jsjΆrhŒXFalser‚†rƒjsX
    r„†r…htjs†r†jsjΆr‡hŒjY†rˆjsjΆr‰htX::rŠ†r‹jsjΆrŒhŒXtlr†rŽjsjΆrhtX=>r†r‘jsjΆr’jXfunr“†r”jsjΆr•htjB†r–jsjΆr—htX=>r˜†r™jsjΆršhtjw†r›jJXfalserœ†rhtjB†ržhŒX	predicaterŸ†r jsjΆr¡hŒXtlr¢†r£jsX
  r¤†r¥jXendr¦†r§htjç†r¨jsX

r©†rªjKXNotationr«†r¬jsjΆr­jHj>†r®jHX	∙⊤∙r¯†r°jHj>†r±jsjΆr²htX:=r³†r´jsjΆrµjJXtruer¶†r·htjB†r¸hŒX	predicater¹†rºjsjΆr»htj=†r¼jsjΆr½hŒXpredicate_scoper¾†r¿htjç†rÀjsjƆrÁjKXNotationr†rÃjsjΆrÄjHj>†rÅjHX	∙⊥∙rƆrÇjHj>†rÈjsjΆrÉhtX:=rʆrËjsjΆrÌjJXfalser͆rÎhtjB†rÏhŒX	predicaterІrÑjsjΆrÒhtj=†rÓjsjΆrÔhŒXpredicate_scoperÕ†rÖhtjç†r×jsX

r؆rÙhX(*rÚ†rÛhj½†rÜhXX Predicate equivalence is an equivalence, and predicate implication defines a preorder. r݆rÞhX*)r߆ràjsX

rá†râhŒXProgramrã†räjsjΆråjKXInstanceræ†rçjsjΆrèhŒX!predicate_equivalence_equivalenceré†rêjsjΆrëhtj=†rìjsjΆríhŒXEquivalencerî†rïjsjΆrðhtj6†rñhtjw†ròhŒXpredicate_equivalenceró†rôjsjΆrõhŒj†röhtjD†r÷htjç†røjsX
  rù†rúhŒXNextrû†rüjsjΆrýhŒX
Obligationrþ†rÿhtjç†rjsX
    r†rjX	inductionr†rjsjΆrhŒj†rjsjΆrhtj·†rjsjΆr	hŒX
firstorderr
†rhtjç†rjsX
  r
†rjKXQedr†rhtjç†rjsX
  r†rhŒXNextr†rjsjΆrhŒX
Obligationr†rhtjç†rjsX
    r†rjX	inductionr†rjsjΆrhŒj†rjsjΆr htj·†r!jsjΆr"hŒX
firstorderr#†r$htjç†r%jsX
  r&†r'jKXQedr(†r)htjç†r*jsX
  r+†r,hŒXNextr-†r.jsjΆr/hŒX
Obligationr0†r1htjç†r2jsX
    r3†r4jXfoldr5†r6jsjΆr7hŒXpointwise_liftingr8†r9htjç†r:jsX
    r;†r<jX	inductionr=†r>jsjΆr?hŒj†r@htjç†rAjsjΆrBhŒX
firstorderrC†rDhtjç†rEjsX
    rF†rGjXintrosrH†rIhtjç†rJjsjΆrKjXsimplrL†rMjsjΆrNjXinrO†rPjsjΆrQhtj½†rRhtjç†rSjsjΆrTjXposerU†rVjsjΆrWhtj6†rXhŒXIHlrY†rZjsjΆr[htj6†r\hŒjv†r]jsjΆr^hŒXx0r_†r`htjD†rajsjΆrbhtj6†rchŒjy†rdjsjΆrehŒXx0rf†rghtjD†rhjsjΆrihtj6†rjhŒj†rkjsjΆrlhŒXx0rm†rnhtjD†rohtjD†rphtjç†rqjsX
    rr†rshŒX
firstorderrt†ruhtjç†rvjsX
  rw†rxjKXQedry†rzhtjç†r{jsX

r|†r}hŒXProgramr~†rjsjΆr€jKXInstancer†r‚jsjΆrƒhŒXpredicate_implication_preorderr„†r…jsjΆr†htj=†r‡jsX
  rˆ†r‰hŒXPreOrderrŠ†r‹jsjΆrŒhtj6†rhtjw†rŽhŒXpredicate_implicationr†rjsjΆr‘hŒj†r’htjD†r“htjç†r”jsX
  r•†r–hŒXNextr—†r˜jsjΆr™hŒX
Obligationrš†r›htjç†rœjsX
    r†ržjX	inductionrŸ†r jsjΆr¡hŒj†r¢jsjΆr£htj·†r¤jsjΆr¥hŒX
firstorderr¦†r§htjç†r¨jsX
  r©†rªjKXQedr«†r¬htjç†r­jsX
  r®†r¯hŒXNextr°†r±jsjΆr²hŒX
Obligationr³†r´htjç†rµjsX
    r¶†r·jX	inductionr¸†r¹jsjΆrºhŒj†r»htjç†r¼jsjΆr½hŒX
firstorderr¾†r¿htjç†rÀjsX
    rÁ†rÂjXunfoldrÆrÄjsjΆrÅhŒXpredicate_implicationrƆrÇjsjΆrÈjXinrɆrÊjsjΆrËhtj½†rÌhtjç†rÍjsjΆrÎjXsimplrφrÐjsjΆrÑjXinrÒ†rÓjsjΆrÔhtj½†rÕhtjç†rÖjsX
    r׆rØjXintrorÙ†rÚhtjç†rÛjsjΆrÜjXposer݆rÞjsjΆrßhtj6†ràhŒXIHlrá†râjsjΆrãhtj6†rähŒjv†råjsjΆræhŒXx0rç†rèhtjD†réjsjΆrêhtj6†rëhŒjy†rìjsjΆríhŒXx0rî†rïhtjD†rðjsjΆrñhtj6†ròhŒj†rójsjΆrôhŒXx0rõ†röhtjD†r÷htjD†røhtjç†rùjsjΆrúhŒX
firstorderrû†rühtjç†rýjsX
  rþ†rÿjKXQedr†rhtjç†rjsX

r†rhX(*r†rhj½†rhXj We define the various operations which define the algebra on binary relations,
   from the general ones. r†r	hX*)r
†rjsX

r†r
jKX
Definitionr†rjsjΆrhŒXrelation_equivalencer†rjsjΆrhtjW†rhŒjY†rjsjΆrhtj=†rjsjΆrj-XTyper†rhtj[†rjsjΆrhtj=†rjsjΆrhŒXrelationr†r jsjΆr!htj6†r"hŒXrelationr#†r$jsjΆr%hŒjY†r&htjD†r'jsjΆr(htX:=r)†r*jsX
  r+†r,htjw†r-hŒXpredicate_equivalencer.†r/jsjΆr0htj6†r1htjB†r2htX::r3†r4htjB†r5htX::r6†r7hŒXnilr8†r9htjD†r:htjç†r;jsX

r<†r=jKXClassr>†r?jsjΆr@hŒXsubrelationrA†rBjsjΆrChtjW†rDhŒjY†rEhtj=†rFj-XTyperG†rHhtj[†rIjsjΆrJhtj6†rKhŒj0†rLjsjΆrMhŒXR'rN†rOjsjΆrPhtj=†rQjsjΆrRhŒXrelationrS†rTjsjΆrUhŒjY†rVhtjD†rWjsjΆrXhtj=†rYjsjΆrZj-XPropr[†r\jsjΆr]htX:=r^†r_jsX
  r`†rahŒXis_subrelationrb†rcjsjΆrdhtj=†rejsjΆrfhtjw†rghŒXpredicate_implicationrh†rijsjΆrjhtj6†rkhŒjY†rlhtX::rm†rnhŒjY†rohtX::rp†rqhŒXnilrr†rshtjD†rtjsjΆruhŒj0†rvjsjΆrwhŒXR'rx†ryhtjç†rzjsX

r{†r|jKXImplicitr}†r~jsjΆrjKX	Argumentsr€†rjsjΆr‚hŒXsubrelationrƒ†r„jsjΆr…htjv†r†htjv†r‡hŒjY†rˆhtj‡†r‰htj‡†rŠhtjç†r‹jsX

rŒ†rjKX
DefinitionrŽ†rjsjΆrhŒXrelation_conjunctionr‘†r’jsjΆr“htjW†r”hŒjY†r•htj[†r–jsjΆr—htj6†r˜hŒj0†r™jsjΆršhtj=†r›jsjΆrœhŒXrelationr†ržjsjΆrŸhŒjY†r htjD†r¡jsjΆr¢htj6†r£hŒXR'r¤†r¥jsjΆr¦htj=†r§jsjΆr¨hŒXrelationr©†rªjsjΆr«hŒjY†r¬htjD†r­jsjΆr®htj=†r¯jsjΆr°hŒXrelationr±†r²jsjΆr³hŒjY†r´jsjΆrµhtX:=r¶†r·jsX
  r¸†r¹htjw†rºhŒXpredicate_intersectionr»†r¼jsjΆr½htj6†r¾hŒjY†r¿htX::rÀ†rÁhŒjY†rÂhtX::rÆrÄhŒXnilrņrÆhtjD†rÇjsjΆrÈhŒj0†rÉjsjΆrÊhŒXR'rˆrÌhtjç†rÍjsX

rΆrÏjKX
DefinitionrІrÑjsjΆrÒhŒXrelation_disjunctionrÓ†rÔjsjΆrÕhtjW†rÖhŒjY†r×htj[†rØjsjΆrÙhtj6†rÚhŒj0†rÛjsjΆrÜhtj=†rÝjsjΆrÞhŒXrelationr߆ràjsjΆráhŒjY†râhtjD†rãjsjΆrähtj6†råhŒXR'ræ†rçjsjΆrèhtj=†réjsjΆrêhŒXrelationrë†rìjsjΆríhŒjY†rîhtjD†rïjsjΆrðhtj=†rñjsjΆròhŒXrelationró†rôjsjΆrõhŒjY†röjsjΆr÷htX:=rø†rùjsX
  rú†rûhtjw†rühŒXpredicate_unionrý†rþjsjΆrÿhtj6†rhŒjY†rhtX::r†rhŒjY†rhtX::r†rhŒXnilr†rhtjD†r	jsjΆr
hŒj0†rjsjΆrhŒXR'r
†rhtjç†rjsX

r†rhX(*r†rhj½†rhXR Relation equivalence is an equivalence, and subrelation defines a partial order. r†rhX*)r†rjsX

r†rjKXSetr†rjsjΆrhŒX	Automaticr†rjsjΆr hŒXIntroductionr!†r"htjç†r#jsX

r$†r%jKXInstancer&†r'jsjΆr(hŒX relation_equivalence_equivalencer)†r*jsjΆr+htj6†r,hŒjY†r-jsjΆr.htj=†r/jsjΆr0j-XTyper1†r2htjD†r3jsjΆr4htj=†r5jsX
  r6†r7hŒXEquivalencer8†r9jsjΆr:htj6†r;htjw†r<hŒXrelation_equivalencer=†r>jsjΆr?hŒjY†r@htjD†rAhtjç†rBjsjƆrCjKXProofrD†rEhtjç†rFjsjΆrGjRXexactrH†rIjsjΆrJhtj6†rKhtjw†rLhŒX!predicate_equivalence_equivalencerM†rNjsjΆrOhtj6†rPhŒjY†rQhtX::rR†rShŒjY†rThtX::rU†rVhŒXnilrW†rXhtjD†rYhtjD†rZhtjç†r[jsjΆr\jKXQedr]†r^htjç†r_jsX

r`†rajKXInstancerb†rcjsjΆrdhŒXrelation_implication_preorderre†rfjsjΆrghŒjY†rhjsjΆrihtj=†rjjsjΆrkhŒXPreOrderrl†rmjsjΆrnhtj6†rohtjw†rphŒXsubrelationrq†rrjsjΆrshŒjY†rthtjD†ruhtjç†rvjsjƆrwjKXProofrx†ryhtjç†rzjsjΆr{jRXexactr|†r}jsjΆr~htj6†rhtjw†r€hŒXpredicate_implication_preorderr†r‚jsjΆrƒhtj6†r„hŒjY†r…htX::r††r‡hŒjY†rˆhtX::r‰†rŠhŒXnilr‹†rŒhtjD†rhtjD†rŽhtjç†rjsjΆrjKXQedr‘†r’htjç†r“jsX

r”†r•hX(*r–†r—hj½†r˜hjΆr™hj½†ršhj½†r›hj½†rœhX¬ Partial Order.
   A partial order is a preorder which is additionally antisymmetric.
   We give an equivalent definition, up-to an equivalence relation
   on the carrier. r†ržhX*)rŸ†r jsX

r¡†r¢jKXClassr£†r¤jsjΆr¥hŒXPartialOrderr¦†r§jsjΆr¨htjW†r©hŒjY†rªhtj[†r«jsjΆr¬hŒXeqAr­†r®jsjΆr¯htj†r°htjW†r±hŒXequr²†r³jsjΆr´htj=†rµjsjΆr¶hŒXEquivalencer·†r¸jsjΆr¹hŒjY†rºjsjΆr»hŒXeqAr¼†r½htj[†r¾jsjΆr¿hŒj0†rÀjsjΆrÁhtj†rÂhtjW†rÃhŒXpreorĆrÅjsjΆrÆhtj=†rÇjsjΆrÈhŒXPreOrderrɆrÊjsjΆrËhŒjY†rÌjsjΆrÍhŒj0†rÎhtj[†rÏjsjΆrÐhtX:=rцrÒjsX
  rÓ†rÔhŒXpartial_order_equivalencerÕ†rÖjsjΆr×htj=†rØjsjΆrÙhŒXrelation_equivalencerÚ†rÛjsjΆrÜhŒXeqAr݆rÞjsjΆrßhtj6†ràhŒXrelation_conjunctionrá†râjsjΆrãhŒj0†räjsjΆråhtj6†ræhŒXinverserç†rèjsjΆréhŒj0†rêhtjD†rëhtjD†rìhtjç†ríjsX

rî†rïhX(*rð†rñhj½†ròhX` The equivalence proof is sufficient for proving that [R] must be a morphism
   for equivalence ró†rôhj6†rõhX
see Morphismsrö†r÷hjD†røhXJ.
   It is also sufficient to show that [R] is antisymmetric w.r.t. [eqA] rù†rúhX*)rû†rüjsX

rý†rþjKXInstancerÿ†rjsjΆrhŒXpartial_order_antisymr†rjsjΆrhtj†rhtj6†rhŒXPartialOrderr†rjsjΆr	hŒjY†r
jsjΆrhŒXeqAr†r
jsjΆrhŒj0†rhtjD†rjsjΆrhtj=†rjsjΆrhtX!r†rjsjΆrhŒX
Antisymmetricr†rjsjΆrhŒjY†rjsjΆrhŒXeqAr†rjsjΆrhŒj0†rhtjç†r jsjƆr!jKXProofr"†r#jsjΆr$jXwithr%†r&jsjΆr'jXautor(†r)htjç†r*jsX
  r+†r,hŒXreduce_goalr-†r.htjç†r/jsX
  r0†r1jXposer2†r3jsjΆr4hŒXproofr5†r6jsjΆr7hŒXpartial_order_equivalencer8†r9jsjΆr:jXasr;†r<jsjΆr=hŒXpoer>†r?htjç†r@jsjΆrAjbXdorB†rCjsjΆrDjŸjB†rEjsjΆrFjXredrG†rHjsjΆrIjXinrJ†rKjsjΆrLhŒXpoerM†rNhtjç†rOjsX
  rP†rQjXapplyrR†rSjsjΆrThtX<-rU†rVjsjΆrWhŒXpoerX†rYhtjç†rZjsjΆr[hŒX
firstorderr\†r]htjç†r^jsjƆr_jKXQedr`†rahtjç†rbjsX

rc†rdhX(*re†rfhj½†rghXD The partial order defined by subrelation and relation equivalence. rh†rihX*)rj†rkjsX

rl†rmhŒXProgramrn†rojsjΆrpjKXInstancerq†rrjsjΆrshŒXsubrelation_partial_orderrt†rujsjΆrvhtj=†rwjsX
  rx†ryhtj†rzjsjΆr{hŒXPartialOrderr|†r}jsjΆr~htj6†rhŒXrelationr€†rjsjΆr‚hŒjY†rƒhtjD†r„jsjΆr…hŒXrelation_equivalencer††r‡jsjΆrˆhŒXsubrelationr‰†rŠhtjç†r‹jsX

  rŒ†rhŒXNextrŽ†rjsjΆrhŒX
Obligationr‘†r’htjç†r“jsX
  r”†r•jKXProofr–†r—htjç†r˜jsX
    r™†ršjXunfoldr›†rœjsjΆrhŒXrelation_equivalencerž†rŸjsjΆr jXinr¡†r¢jsjΆr£htj½†r¤htjç†r¥jsjΆr¦hŒX
firstorderr§†r¨htjç†r©jsX
  rª†r«jKXQedr¬†r­htjç†r®jsX

r¯†r°e(hŒXTypeclassesr±†r²jsjΆr³hŒXOpaquer´†rµjsjΆr¶hŒXarrowsr·†r¸jsjΆr¹hŒXpredicate_implicationrº†r»jsjΆr¼hŒXpredicate_equivalencer½†r¾jsX
  r¿†rÀhŒXrelation_equivalencerÁ†rÂjsjΆrÃhŒXpointwise_liftingrĆrÅhtjç†rÆjsX

rdžrÈhX(*rɆrÊhj½†rËhX§ Rewrite relation on a given support: declares a relation as a rewrite
   relation for use by the generalized rewriting tactic.
   It helps choosing if a rewrite should be handled
   by the generalized or the regular rewriting tactic using leibniz equality.
   Users can declare an [RewriteRelation A RA] anywhere to declare default
   relations. This is also done automatically by the [Declare Relation A RA]
   commands. r̆rÍhX*)rΆrÏjsX

rІrÑjKXClassrÒ†rÓjsjΆrÔhŒXRewriteRelationrÕ†rÖjsjΆr×htjW†rØhŒjY†rÙjsjΆrÚhtj=†rÛjsjΆrÜj-XTyper݆rÞhtj[†rßjsjΆràhtj6†ráhŒXRArâ†rãjsjΆrähtj=†råjsjΆræhŒXrelationrç†rèjsjΆréhŒjY†rêhtjD†rëhtjç†rìjsX

rí†rîjKXInstancerï†rðhtj=†rñjsjΆròhŒXRewriteRelationró†rôjsjΆrõhŒXimplrö†r÷htjç†røjsjƆrùjKXInstancerú†rûhtj=†rüjsjΆrýhŒXRewriteRelationrþ†rÿjsjΆrhŒXiffr†rhtjç†rjsjƆrjKXInstancer†rhtj=†rjsjΆrhŒXRewriteRelationr	†r
jsjΆrhtj6†rhtjw†r
hŒXrelation_equivalencer†rjsjΆrhŒjY†rhtjD†rhtjç†rjsX

r†rhX(*r†rhj½†rhX^ Any [Equivalence] declared in the context is automatically considered
   a rewrite relation. r†rhX*)r†rjsX

r†rjKXInstancer†r jsjΆr!hŒXequivalence_rewrite_relationr"†r#jsjΆr$htj†r%htj6†r&hŒXEquivalencer'†r(jsjΆr)hŒjY†r*jsjΆr+hŒXeqAr,†r-htjD†r.jsjΆr/htj=†r0jsjΆr1hŒXRewriteRelationr2†r3jsjΆr4hŒXeqAr5†r6htjç†r7jsX

r8†r9hX(*r:†r;hj½†r<hX Strict Order r=†r>hX*)r?†r@jsX

rA†rBjKXClassrC†rDjsjΆrEhŒXStrictOrderrF†rGjsjΆrHhtjW†rIhŒjY†rJjsjΆrKhtj=†rLjsjΆrMj-XTyperN†rOhtj[†rPjsjΆrQhtj6†rRhŒj0†rSjsjΆrThtj=†rUjsjΆrVhŒXrelationrW†rXjsjΆrYhŒjY†rZhtjD†r[jsjΆr\htX:=r]†r^jsjΆr_htjW†r`jsX
  ra†rbhŒXStrictOrder_Irreflexiverc†rdjsjΆrehtX:>rf†rgjsjΆrhhŒXIrreflexiveri†rjjsjΆrkhŒj0†rljsjΆrmhtj·†rnjsX
  ro†rphŒXStrictOrder_Transitiverq†rrjsjΆrshtX:>rt†rujsjΆrvhŒX
Transitiverw†rxjsjΆryhŒj0†rzjsjƆr{htj[†r|htjç†r}jsX

r~†rjKXInstancer€†rjsjΆr‚hŒXStrictOrder_Asymmetricrƒ†r„jsjΆr…htj†r†htj6†r‡hŒXStrictOrderrˆ†r‰jsjΆrŠhŒjY†r‹jsjΆrŒhŒj0†rhtjD†rŽjsjΆrhtj=†rjsjΆr‘hŒX
Asymmetricr’†r“jsjΆr”hŒj0†r•htjç†r–jsjƆr—jKXProofr˜†r™htjç†ršjsjΆr›hŒX
firstorderrœ†rhtjç†ržjsjΆrŸjKXQedr †r¡htjç†r¢jsX

r£†r¤hX(*r¥†r¦hj½†r§hX7 Inversing a [StrictOrder] gives another [StrictOrder] r¨†r©hX*)rª†r«jsX

r¬†r­jKXLemmar®†r¯jsjΆr°hŒXStrictOrder_inverser±†r²jsjΆr³htj†r´htj6†rµhŒXStrictOrderr¶†r·jsjΆr¸hŒjY†r¹jsjΆrºhŒj0†r»htjD†r¼jsjΆr½htj=†r¾jsjΆr¿hŒXStrictOrderrÀ†rÁjsjΆrÂhtj6†rÃhŒXinverserĆrÅjsjΆrÆhŒj0†rÇhtjD†rÈhtjç†rÉjsjƆrÊjKXProofrˆrÌhtjç†rÍjsjΆrÎhŒX
firstorderrφrÐhtjç†rÑjsjΆrÒjKXQedrÓ†rÔhtjç†rÕjsX

rÖ†r×hX(*r؆rÙhj½†rÚhX Same for [PartialOrder]. rÛ†rÜhX*)r݆rÞjsX

r߆ràjKXLemmará†râjsjΆrãhŒXPreOrder_inverserä†råjsjΆræhtj†rçhtj6†rèhŒXPreOrderré†rêjsjΆrëhŒjY†rìjsjΆríhŒj0†rîhtjD†rïjsjΆrðhtj=†rñjsjΆròhŒXPreOrderró†rôjsjΆrõhtj6†röhŒXinverser÷†røjsjΆrùhŒj0†rúhtjD†rûhtjç†rüjsjƆrýjKXProofrþ†rÿhtjç†rjsjΆrhŒX
firstorderr†rhtjç†rjsjΆrjKXQedr†rhtjç†rjsX

r	†r
jKXHintr†rjsjΆr
hŒXExternr†rjsjΆrjŸjB†rjsjΆrhtj6†rhŒXStrictOrderr†rjsjΆrhtj6†rhŒXinverser†rjsjΆrhtjB†rhtjD†rhtjD†rjsjΆrhtX=>r†r jsjΆr!hŒXclass_applyr"†r#jsjΆr$hŒXStrictOrder_inverser%†r&jsjΆr'htj=†r(jsjΆr)hŒXtypeclass_instancesr*†r+htjç†r,jsjƆr-jKXHintr.†r/jsjΆr0hŒXExternr1†r2jsjΆr3jŸjB†r4jsjΆr5htj6†r6hŒXPreOrderr7†r8jsjΆr9htj6†r:hŒXinverser;†r<jsjΆr=htjB†r>htjD†r?htjD†r@jsjΆrAhtX=>rB†rCjsjΆrDhŒXclass_applyrE†rFjsjΆrGhŒXPreOrder_inverserH†rIjsjΆrJhtj=†rKjsjΆrLhŒXtypeclass_instancesrM†rNhtjç†rOjsX

rP†rQjKXLemmarR†rSjsjΆrThŒXPartialOrder_inverserU†rVjsjΆrWhtj†rXhtj6†rYhŒXPartialOrderrZ†r[jsjΆr\hŒjY†r]jsjΆr^hŒXeqAr_†r`jsjΆrahŒj0†rbhtjD†rcjsjΆrdhtj=†rejsjΆrfhŒXPartialOrderrg†rhjsjΆrihŒXeqArj†rkjsjΆrlhtj6†rmhŒXinversern†rojsjΆrphŒj0†rqhtjD†rrhtjç†rsjsjƆrtjKXProofru†rvhtjç†rwjsjΆrxhŒX
firstorderry†rzhtjç†r{jsjΆr|jKXQedr}†r~htjç†rjsX

r€†rjKXHintr‚†rƒjsjΆr„hŒXExternr…†r†jsjΆr‡jŸjB†rˆjsjΆr‰htj6†rŠhŒXPartialOrderr‹†rŒjsjΆrhtj6†rŽhŒXinverser†rjsjΆr‘htjB†r’htjD†r“htjD†r”jsjΆr•htX=>r–†r—jsjΆr˜hŒXclass_applyr™†ršjsjΆr›hŒXPartialOrder_inverserœ†rjsjΆržhtj=†rŸjsjΆr hŒXtypeclass_instancesr¡†r¢htjç†r£jsjƆr¤e.

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