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doctorondemand / Pygments python

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Pygments / examplefiles / output / example.thy
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  rT†rUjXbyrV†rWjsjˆrXhtj†rYhŒXsubstrZ†r[jsjˆr\hŒXmult_commuter]†r^htj*†r_jsjˆr`htj†rahŒXsimprb†rcjsjˆrdhŒXaddre†rfhtj3†rgjsjˆrhhŒX
power_multri†rjhtj*†rkjsX

rl†rmjKXlemmarn†rojsjˆrphŒXpower_odd_eqrq†rrhtj3†rsjsX
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power_even_eqrŠ†r‹htj*†rŒjsX

r†rŽjKXlemmar†rjsjˆr‘hŒXpower_numeral_evenr’†r“htj3†r”jsX
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r«†r¬jKXlemmar­†r®jsjˆr¯hŒXpower_numeral_oddr°†r±htj3†r²jsX
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add_Suc_rightrĆrÅjsjˆrÆhŒXadd_0_rightrdžrÈjsX
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mult_assocr׆rØjsjˆrÙh‚X..rÚ†rÛjsX

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jsjˆrhtX?r†r
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ultimatelyr…†r†jsjˆr‡jXshowrˆ†r‰jsjˆrŠhtj†r‹jXcaserŒ†rjsjˆrŽjXbyr†rjsjˆr‘htj†r’hŒXsimpr“†r”jsjˆr•hŒXaddr–†r—htj3†r˜jsjˆr™hŒX	power_addrš†r›jsjˆrœhŒX
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mult_assocr£†r¤htj*†r¥jsjÚ†r¦jXqedr§†r¨jsX

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r܆rÝjXendrÞ†rßjsX

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rê†rëjKXlemmarì†ríjsjˆrîhŒXnumeral_sqrrï†rðhtj3†rñjsjˆròjÁj†rójÁX+numeral (Num.sqr k) = numeral k * numeral krô†rõjÁj†röjsX
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r
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r?†r@jKXlemmarA†rBjsjˆrChŒX
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rb†rcjXendrd†rejsX

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r•†r–jKXlemmar—†r˜jsjˆr™hŒXpower_zero_numeralrš†r›jsjˆrœhtje†rhŒXsimprž†rŸhtji†r htj3†r¡jsjˆr¢jÁj†r£jÁX(0::'a) ^ numeral k = 0r¤†r¥jÁj†r¦jsX
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rÙ†rÚjKXlemmarÛ†rÜjsjˆrÝhŒX
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  rJ†rKjRXassumesrL†rMjsjˆrNhŒXgt1rO†rPhtj3†rQjsjˆrRjÁj†rSjÁX1 < arT†rUjÁj†rVjsX
  rW†rXjRXshowsrY†rZjsjˆr[jÁj†r\jÁXa ^ m r]†r^jÝX\<le>r_†r`jÁX a ^ n ra†rbjÝX\<Longrightarrow>rc†rdjÁX m re†rfjÝX\<le>rg†rhjÁX nri†rjjÁj†rkjsjÚ†rljXproofrm†rnjsjˆrohtj†rphŒXinductrq†rrjsjˆrshŒjφrtjsjˆruhŒX	arbitraryrv†rwhtj3†rxjsjˆryhŒjU†rzhtj*†r{jsX
  r|†r}jXcaser~†rjsjˆr€hŒj†rjsX
  r‚†rƒjXshowr„†r…jsjˆr†htj†r‡jXcaserˆ†r‰jsjˆrŠjXbyr‹†rŒjsjˆrhŒXsimprŽ†rjsjÚ†rjXnextr‘†r’jsX
  r“†r”jXcaser•†r–jsjˆr—htj†r˜hŒXSucr™†ršjsjˆr›hŒjφrœhtj*†rjsX
  rž†rŸjXshowr †r¡jsjˆr¢htj†r£jXcaser¤†r¥jsX
  r¦†r§jXproofr¨†r©jsjˆrªhtj†r«hŒXcasesr¬†r­jsjˆr®hŒjU†r¯htj*†r°jsX
    r±†r²jXcaser³†r´jsjˆrµhŒj†r¶jsX
    r·†r¸jXwithr¹†rºjsjˆr»hŒX	Suc.premsr¼†r½jsjˆr¾hŒXSuc.hypsr¿†rÀjsjˆrÁjXhaver†rÃjsjˆrÄjÁj†rÅjÁX
a * a ^ m rƆrÇjÝX\<le>rȆrÉjÁX 1rʆrËjÁj†rÌjsjˆrÍjXbyrΆrÏjsjˆrÐhŒXsimprцrÒjsX
    rÓ†rÔjXwithrÕ†rÖjsjˆr×hŒXgt1r؆rÙjsjˆrÚjXshowrÛ†rÜjsjˆrÝhtj†rÞhŒXthesisr߆ràjsX
      rá†râjXbyrã†räjsjˆråhtj†ræhŒXforcerç†rèjsjˆréhŒXsimprê†rëjsjˆrìhŒXonlyrí†rîhtj3†rïjsjˆrðhŒXpower_gt1_lemmarñ†ròjsX
          ró†rôhŒXnot_lessrõ†röjsjˆr÷htje†røhŒX	symmetricrù†rúhtji†rûhtj*†rüjsX
  rý†rþjXnextrÿ†rjsX
    r†rjXcaser†rjsjˆrhtj†rhŒXSucr†rjsjˆr	hŒjU†r
htj*†rjsX
    r†r
jXwithr†rjsjˆrhŒX	Suc.premsr†rjsjˆrhŒXSuc.hypsr†rjsjˆrjXshowr†rjsjˆrhtj†rhŒXthesisr†rjsX
      r†rjXbyr†r jsjˆr!htj†r"hŒXforcer#†r$jsjˆr%hŒXdestr&†r'htj3†r(jsjˆr)hŒXmult_left_le_imp_ler*†r+jsX
          r,†r-hŒXsimpr.†r/jsjˆr0hŒXaddr1†r2htj3†r3jsjˆr4hŒX
less_transr5†r6jsjˆr7htje†r8hŒXOFr9†r:jsjˆr;hŒX
zero_less_oner<†r=jsjˆr>hŒXgt1r?†r@htji†rAhtj*†rBjsX
  rC†rDjXqedrE†rFjsjÚ†rGjXqedrH†rIjsX

rJ†rKjXtextrL†rMhX{*rN†rOhX:Surely we can strengthen this? It holds for @{text "0<a<1"rP†rQhj:
†rRhX too.rS†rThX*}rU†rVjsjÚ†rWjKXlemmarX†rYjsjˆrZhŒXpower_inject_expr[†r\jsjˆr]htje†r^hŒXsimpr_†r`htji†rahtj3†rbjsX
  rc†rdjÁj†rejÁX1 < a rf†rgjÝX\<Longrightarrow>rh†rijÁX a ^ m = a ^ n rj†rkjÝX\<longleftrightarrow>rl†rmjÁX m = nrn†rojÁj†rpjsX
  rq†rrjXbyrs†rtjsjˆruhtj†rvhŒXforcerw†rxjsjˆryhŒXsimprz†r{jsjˆr|hŒXaddr}†r~htj3†rjsjˆr€hŒX
order_antisymr†r‚jsjˆrƒhŒXpower_le_imp_le_expr„†r…htj*†r†jsX

r‡†rˆjXtextr‰†rŠhX{*r‹†rŒhX+Can relax the first premise to @{term "0<a"r†rŽhj:
†rhX$ in the case of the
natural numbers.r†r‘hX*}r’†r“jsjÚ†r”jKXlemmar•†r–jsjˆr—hŒXpower_less_imp_less_expr˜†r™htj3†ršjsX
  r›†rœjÁj†rjÁX1 < a rž†rŸjÝX\<Longrightarrow>r †r¡jÁX a ^ m < a ^ n r¢†r£jÝX\<Longrightarrow>r¤†r¥jÁX m < nr¦†r§jÁj†r¨jsX
  r©†rªjXbyr«†r¬jsjˆr­htj†r®hŒXsimpr¯†r°jsjˆr±hŒXaddr²†r³htj3†r´jsjˆrµhŒX
order_less_ler¶†r·jsjˆr¸htje†r¹hŒXofrº†r»jsjˆr¼hŒjφr½jsjˆr¾hŒjU†r¿htji†rÀjsjˆrÁhŒXless_ler†rÃjsjˆrÄhtje†rÅhŒXofrƆrÇjsjˆrÈjÁj†rÉjÁXa^mrʆrËjÁj†rÌjsjˆrÍjÁj†rÎjÁXa^nrφrÐjÁj†rÑhtji†rÒjsX
    rÓ†rÔhŒXpower_le_imp_le_exprÕ†rÖhtj*†r×jsX

r؆rÙjKXlemmarÚ†rÛjsjˆrÜhŒXpower_strict_monor݆rÞjsjˆrßhtje†ràhŒXrule_formatrá†râhtji†rãhtj3†räjsX
  rå†ræjÁj†rçjÁXa < b rè†réjÝX\<Longrightarrow>rê†rëjÁX 0 rì†ríjÝX\<le>rî†rïjÁX a rð†rñjÝX\<Longrightarrow>rò†rójÁX 0 < n rô†rõjÝX\<longrightarrow>rö†r÷jÁX a ^ n < b ^ nrø†rùjÁj†rújsX
  rû†rüjXbyrý†rþjsjˆrÿhtj†rhŒXinductr†rjsjˆrhŒjU†rhtj*†rjsX
   r†rhtj†rhŒXautor	†r
jsjˆrhŒXsimpr†r
jsjˆrhŒXaddr†rhtj3†rjsjˆrhŒXmult_strict_monor†rjsjˆrhŒX
le_less_transr†rjsjˆrhtje†rhŒXofr†rjsjˆrhŒj†rjsjˆrhŒj
†rjsjˆr hŒjƆr!htji†r"htj*†r#jsX

r$†r%jXtextr&†r'hX{*r(†r)hX(Lemma for @{text power_strict_decreasingr*†r+hj:
†r,hX*}r-†r.jsjÚ†r/jKXlemmar0†r1jsjˆr2hŒXpower_Suc_lessr3†r4htj3†r5jsX
  r6†r7jÁj†r8jÁX0 < a r9†r:jÝX\<Longrightarrow>r;†r<jÁX a < 1 r=†r>jÝX\<Longrightarrow>r?†r@jÁX a * a ^ n < a ^ nrA†rBjÁj†rCjsX
  rD†rEjXbyrF†rGjsjˆrHhtj†rIhŒXinductrJ†rKjsjˆrLhŒjU†rMhtj*†rNjsX
    rO†rPhtj†rQhŒXautorR†rSjsjˆrThŒXsimprU†rVjsjˆrWhŒXaddrX†rYhtj3†rZjsjˆr[hŒXmult_strict_left_monor\†r]htj*†r^jsX

r_†r`jKXlemmara†rbjsjˆrchŒXpower_strict_decreasingrd†rejsjˆrfhtje†rghŒXrule_formatrh†rihtji†rjhtj3†rkjsX
  rl†rmjÁj†rnjÁXn < N ro†rpjÝX\<Longrightarrow>rq†rrjÁX 0 < a rs†rtjÝX\<Longrightarrow>ru†rvjÁX a < 1 rw†rxjÝX\<longrightarrow>ry†rzjÁX a ^ N < a ^ nr{†r|jÁj†r}jsjÚ†r~jXproofr†r€jsjˆrhtj†r‚hŒXinductrƒ†r„jsjˆr…hŒXNr††r‡htj*†rˆjsX
  r‰†rŠjXcaser‹†rŒjsjˆrhŒj†rŽjsjˆrjXthenr†r‘jsjˆr’jXshowr“†r”jsjˆr•htj†r–jXcaser—†r˜jsjˆr™jXbyrš†r›jsjˆrœhŒXsimpr†ržjsjÚ†rŸjXnextr †r¡jsX
  r¢†r£jXcaser¤†r¥jsjˆr¦htj†r§hŒXSucr¨†r©jsjˆrªhŒj††r«htj*†r¬jsjˆr­jXthenr®†r¯jsjˆr°jXshowr±†r²jsjˆr³htj†r´jXcaserµ†r¶jsX 
  r·†r¸jRXapplyr¹†rºjsjˆr»htj†r¼hŒXautor½†r¾jsjˆr¿hŒXsimprÀ†rÁjsjˆrÂhŒXaddrÆrÄhtj3†rÅjsjˆrÆhŒXpower_Suc_lessrdžrÈjsjˆrÉhŒXless_Suc_eqrʆrËhtj*†rÌjsX
  r͆rÎjRXapplyrφrÐjsjˆrÑhtj†rÒhŒXsubgoal_tacrÓ†rÔjsjˆrÕjÁj†rÖjÁXa * a^N < 1 * a^nr׆rØjÁj†rÙhtj*†rÚjsX
  rÛ†rÜjRXapplyr݆rÞjsjˆrßhŒXsimprà†rájsX
  râ†rãjRXapplyrä†råjsjˆræhtj†rçhŒXrulerè†réjsjˆrêhŒXmult_strict_monorë†rìhtj*†ríjsjˆrîjRXapplyrï†rðjsjˆrñhŒXautorò†rójsX
  rô†rõjXdonerö†r÷jsjÚ†røjXqedrù†rújsX

rû†rüjXtextrý†rþhX{*rÿ†rhX6Proof resembles that of @{text power_strict_decreasingr†rhj:
†rhX*}r†rjsjÚ†rjKXlemmar†rjsjˆr	hŒXpower_decreasingr
†rjsjˆrhtje†r
hŒXrule_formatr†rhtji†rhtj3†rjsX
  r†rjÁj†rjÁXn r†rjÝX\<le>r†rjÁX N r†rjÝX\<Longrightarrow>r†rjÁX 0 r†rjÝX\<le>r†r jÁX a r!†r"jÝX\<Longrightarrow>r#†r$jÁX a r%†r&jÝX\<le>r'†r(jÁX 1 r)†r*jÝX\<longrightarrow>r+†r,jÁX a ^ N r-†r.jÝX\<le>r/†r0jÁX a ^ nr1†r2jÁj†r3jsjÚ†r4jXproofr5†r6jsjˆr7htj†r8hŒXinductr9†r:jsjˆr;hŒj††r<htj*†r=jsX
  r>†r?jXcaser@†rAjsjˆrBhŒj†rCjsjˆrDjXthenrE†rFjsjˆrGjXshowrH†rIjsjˆrJhtj†rKjXcaserL†rMjsjˆrNjXbyrO†rPjsjˆrQhŒXsimprR†rSjsjÚ†rTjXnextrU†rVjsX
  rW†rXjXcaserY†rZjsjˆr[htj†r\hŒXSucr]†r^jsjˆr_hŒj††r`htj*†rajsjˆrbjXthenrc†rdjsjˆrejXshowrf†rgjsjˆrhhtj†rijXcaserj†rkjsX 
  rl†rmjRXapplyrn†rojsjˆrphtj†rqhŒXautorr†rsjsjˆrthŒXsimpru†rvjsjˆrwhŒXaddrx†ryhtj3†rzjsjˆr{hŒX	le_Suc_eqr|†r}htj*†r~jsX
  r†r€jRXapplyr†r‚jsjˆrƒhtj†r„hŒXsubgoal_tacr…†r†jsjˆr‡jÁj†rˆjÁXa * a^N r‰†rŠjÝX\<le>r‹†rŒjÁX 1 * a^nr†rŽjÁj†rhtj"†rjsjˆr‘hŒXsimpr’†r“htj*†r”jsX
  r•†r–jRXapplyr—†r˜jsjˆr™htj†ršhŒXruler›†rœjsjˆrhŒX	mult_monorž†rŸhtj*†r jsjˆr¡jRXapplyr¢†r£jsjˆr¤hŒXautor¥†r¦jsX
  r§†r¨jXdoner©†rªjsjÚ†r«jXqedr¬†r­jsX

r®†r¯jKXlemmar°†r±jsjˆr²hŒXpower_Suc_less_oner³†r´htj3†rµjsX
  r¶†r·jÁj†r¸jÁX0 < a r¹†rºjÝX\<Longrightarrow>r»†r¼jÁX a < 1 r½†r¾jÝX\<Longrightarrow>r¿†rÀjÁX a ^ Suc n < 1rÁ†rÂjÁj†rÃjsX
  rĆrÅjXusingrƆrÇjsjˆrÈhŒXpower_strict_decreasingrɆrÊjsjˆrËhtje†rÌhŒXofr͆rÎjsjˆrÏhŒj†rÐjsjˆrÑjÁj†rÒjÁXSuc nrÓ†rÔjÁj†rÕjsjˆrÖhŒj
†r×htji†rØjsjˆrÙjXbyrÚ†rÛjsjˆrÜhŒXsimpr݆rÞjsX

r߆ràjXtextrá†râhX{*rã†rähX<Proof again resembles that of @{text power_strict_decreasingrå†ræhj:
†rçhX*}rè†réjsjÚ†rêjKXlemmarë†rìjsjˆríhŒXpower_increasingrî†rïjsjˆrðhtje†rñhŒXrule_formatrò†róhtji†rôhtj3†rõjsX
  rö†r÷jÁj†røjÁXn rù†rújÝX\<le>rû†rüjÁX N rý†rþjÝX\<Longrightarrow>rÿ†rjÁX 1 r†rjÝX\<le>r†rjÁX a r†rjÝX\<Longrightarrow>r†rjÁX a ^ n r	†r
jÝX\<le>r†rjÁX a ^ Nr
†rjÁj†rjsjÚ†rjXproofr†rjsjˆrhtj†rhŒXinductr†rjsjˆrhŒj††rhtj*†rjsX
  r†rjXcaser†rjsjˆrhŒj†rjsjˆr jXthenr!†r"jsjˆr#jXshowr$†r%jsjˆr&htj†r'jXcaser(†r)jsjˆr*jXbyr+†r,jsjˆr-hŒXsimpr.†r/jsjÚ†r0jXnextr1†r2jsX
  r3†r4jXcaser5†r6jsjˆr7htj†r8hŒXSucr9†r:jsjˆr;hŒj††r<htj*†r=jsjˆr>jXthenr?†r@jsjˆrAjXshowrB†rCjsjˆrDhtj†rEjXcaserF†rGjsX 
  rH†rIjRXapplyrJ†rKjsjˆrLhtj†rMhŒXautorN†rOjsjˆrPhŒXsimprQ†rRjsjˆrShŒXaddrT†rUhtj3†rVjsjˆrWhŒX	le_Suc_eqrX†rYhtj*†rZjsX
  r[†r\jRXapplyr]†r^jsjˆr_htj†r`hŒXsubgoal_tacra†rbjsjˆrcjÁj†rdjÁX1 * a^n re†rfjÝX\<le>rg†rhjÁX a * a^Nri†rjjÁj†rkhtj"†rljsjˆrmhŒXsimprn†rohtj*†rpjsX
  rq†rrjRXapplyrs†rtjsjˆruhtj†rvhŒXrulerw†rxjsjˆryhŒX	mult_monorz†r{htj*†r|jsjˆr}jRXapplyr~†rjsjˆr€htj†rhŒXautor‚†rƒjsjˆr„hŒXsimpr…†r†jsjˆr‡hŒXaddrˆ†r‰htj3†rŠjsjˆr‹hŒXorder_transrŒ†rjsjˆrŽhtje†rhŒXOFr†r‘jsjˆr’hŒXzero_le_oner“†r”htji†r•htj*†r–jsX
  r—†r˜jXdoner™†ršjsjÚ†r›jXqedrœ†rjsX

rž†rŸjXtextr †r¡hX{*r¢†r£hX(Lemma for @{text power_strict_increasingr¤†r¥hj:
†r¦hX*}r§†r¨jsjÚ†r©jKXlemmarª†r«jsjˆr¬hŒXpower_less_power_Sucr­†r®htj3†r¯jsX
  r°†r±jÁj†r²jÁX1 < a r³†r´jÝX\<Longrightarrow>rµ†r¶jÁX a ^ n < a * a ^ nr·†r¸jÁj†r¹jsX
  rº†r»jXbyr¼†r½jsjˆr¾htj†r¿hŒXinductrÀ†rÁjsjˆrÂhŒjU†rÃhtj*†rÄjsjˆrÅhtj†rÆhŒXautordžrÈjsjˆrÉhŒXsimprʆrËjsjˆrÌhŒXaddr͆rÎhtj3†rÏjsjˆrÐhŒXmult_strict_left_monorцrÒjsjˆrÓhŒX
less_transrÔ†rÕjsjˆrÖhtje†r×hŒXOFr؆rÙjsjˆrÚhŒX
zero_less_onerÛ†rÜhtji†rÝhtj*†rÞjsX

r߆ràjKXlemmará†râjsjˆrãhŒXpower_strict_increasingrä†råjsjˆræhtje†rçhŒXrule_formatrè†réhtji†rêhtj3†rëjsX
  rì†ríjÁj†rîjÁXn < N rï†rðjÝX\<Longrightarrow>rñ†ròjÁX 1 < a ró†rôjÝX\<longrightarrow>rõ†röjÁX a ^ n < a ^ Nr÷†røjÁj†rùjsjÚ†rújXproofrû†rüjsjˆrýhtj†rþhŒXinductrÿ†rjsjˆrhŒj††rhtj*†rjsX
  r†rjXcaser†rjsjˆrhŒj†r	jsjˆr
jXthenr†rjsjˆr
jXshowr†rjsjˆrhtj†rjXcaser†rjsjˆrjXbyr†rjsjˆrhŒXsimpr†rjsjÚ†rjXnextr†rjsX
  r†rjXcaser†r jsjˆr!htj†r"hŒXSucr#†r$jsjˆr%hŒj††r&htj*†r'jsjˆr(jXthenr)†r*jsjˆr+jXshowr,†r-jsjˆr.htj†r/jXcaser0†r1jsX 
  r2†r3jRXapplyr4†r5jsjˆr6htj†r7hŒXautor8†r9jsjˆr:hŒXsimpr;†r<jsjˆr=hŒXaddr>†r?htj3†r@jsjˆrAhŒXpower_less_power_SucrB†rCjsjˆrDhŒXless_Suc_eqrE†rFhtj*†rGjsX
  rH†rIjRXapplyrJ†rKjsjˆrLhtj†rMhŒXsubgoal_tacrN†rOjsjˆrPjÁj†rQjÁX1 * a^n < a * a^NrR†rSjÁj†rThtj"†rUjsjˆrVhŒXsimprW†rXhtj*†rYjsX
  rZ†r[jRXapplyr\†r]jsjˆr^htj†r_hŒXruler`†rajsjˆrbhŒXmult_strict_monorc†rdhtj*†rejsjˆrfe(jRXapplyrg†rhjsjˆrihtj†rjhŒXautork†rljsjˆrmhŒXsimprn†rojsjˆrphŒXaddrq†rrhtj3†rsjsjˆrthŒX
less_transru†rvjsjˆrwhtje†rxhŒXOFry†rzjsjˆr{hŒX
zero_less_oner|†r}htji†r~jsjˆrhŒXless_imp_ler€†rhtj*†r‚jsX
  rƒ†r„jXdoner…†r†jsjÚ†r‡jXqedrˆ†r‰jsX

rŠ†r‹jKXlemmarŒ†rjsjˆrŽhŒXpower_increasing_iffr†rjsjˆr‘htje†r’hŒXsimpr“†r”htji†r•htj3†r–jsX
  r—†r˜jÁj†r™jÁX1 < b rš†r›jÝX\<Longrightarrow>rœ†rjÁX b ^ x rž†rŸjÝX\<le>r †r¡jÁX b ^ y r¢†r£jÝX\<longleftrightarrow>r¤†r¥jÁX x r¦†r§jÝX\<le>r¨†r©jÁX yrª†r«jÁj†r¬jsX
  r­†r®jXbyr¯†r°jsjˆr±htj†r²hŒXblastr³†r´jsjˆrµhŒXintror¶†r·htj3†r¸jsjˆr¹hŒXpower_le_imp_le_exprº†r»jsjˆr¼hŒXpower_increasingr½†r¾jsjˆr¿hŒXless_imp_lerÀ†rÁhtj*†rÂjsX

rÆrÄjKXlemmarņrÆjsjˆrÇhŒXpower_strict_increasing_iffrȆrÉjsjˆrÊhtje†rËhŒXsimpr̆rÍhtji†rÎhtj3†rÏjsX
  rІrÑjÁj†rÒjÁX1 < b rÓ†rÔjÝX\<Longrightarrow>rÕ†rÖjÁX b ^ x < b ^ y r׆rØjÝX\<longleftrightarrow>rÙ†rÚjÁX x < yrÛ†rÜjÁj†rÝjsjÚ†rÞjXbyr߆ràjsjˆráhtj†râhŒXblastrã†räjsjˆråhŒXintroræ†rçhtj3†rèjsjˆréhŒXpower_less_imp_less_exprê†rëjsjˆrìhŒXpower_strict_increasingrí†rîhtj*†rïjsX 

rð†rñjKXlemmarò†rójsjˆrôhŒXpower_le_imp_le_baserõ†röhtj3†r÷jsX
  rø†rùjRXassumesrú†rûjsjˆrühŒXlerý†rþhtj3†rÿjsjˆrjÁj†rjÁX
a ^ Suc n r†rjÝX\<le>r†rjÁX
 b ^ Suc nr†rjÁj†rjsX
    r	†r
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hŒXynonnegr†rhtj3†rjsjˆrjÁj†rjÁX0 r†rjÝX\<le>r†rjÁX br†rjÁj†rjsX
  r†rjRXshowsr†rjsjˆrjÁj†rjÁXa r †r!jÝX\<le>r"†r#jÁX br$†r%jÁj†r&jsjÚ†r'jXproofr(†r)jsjˆr*htj†r+hŒXruler,†r-jsjˆr.hŒXccontrr/†r0htj*†r1jsX
  r2†r3jXassumer4†r5jsjˆr6jÁj†r7jÁX~ a r8†r9jÝX\<le>r:†r;jÁX br<†r=jÁj†r>jsX
  r?†r@jXthenrA†rBjsjˆrCjXhaverD†rEjsjˆrFjÁj†rGjÁXb < arH†rIjÁj†rJjsjˆrKjXbyrL†rMjsjˆrNhtj†rOhŒXsimprP†rQjsjˆrRhŒXonlyrS†rThtj3†rUjsjˆrVhŒXlinorder_not_lerW†rXhtj*†rYjsX
  rZ†r[jXthenr\†r]jsjˆr^jXhaver_†r`jsjˆrajÁj†rbjÁXb ^ Suc n < a ^ Suc nrc†rdjÁj†rejsX
    rf†rgjXbyrh†rijsjˆrjhtj†rkhŒXsimprl†rmjsjˆrnhŒXonlyro†rphtj3†rqjsjˆrrhŒXassmsrs†rtjsjˆruhŒXpower_strict_monorv†rwhtj*†rxjsX
  ry†rzjXfromr{†r|jsjˆr}hŒXler~†rjsjˆr€jRXandr†r‚jsjˆrƒhŒXthisr„†r…jsjˆr†jXshowr‡†rˆjsjˆr‰hŒXFalserŠ†r‹jsX
    rŒ†rjXbyrŽ†rjsjˆrhtj†r‘hŒXsimpr’†r“jsjˆr”hŒXaddr•†r–htj3†r—jsjˆr˜hŒXlinorder_not_lessr™†ršjsjˆr›htje†rœhŒX	symmetricr†ržhtji†rŸhtj*†r jsjÚ†r¡jXqedr¢†r£jsX

r¤†r¥jKXlemmar¦†r§jsjˆr¨hŒXpower_less_imp_less_baser©†rªhtj3†r«jsX
  r¬†r­jRXassumesr®†r¯jsjˆr°hŒXlessr±†r²htj3†r³jsjˆr´jÁj†rµjÁX
a ^ n < b ^ nr¶†r·jÁj†r¸jsX
  r¹†rºjRXassumesr»†r¼jsjˆr½hŒXnonnegr¾†r¿htj3†rÀjsjˆrÁjÁj†rÂjÁX0 rÆrÄjÝX\<le>rņrÆjÁX brdžrÈjÁj†rÉjsX
  rʆrËjRXshowsr̆rÍjsjˆrÎjÁj†rÏjÁXa < brІrÑjÁj†rÒjsjÚ†rÓjXproofrÔ†rÕjsjˆrÖhtj†r×hŒXruler؆rÙjsjˆrÚhŒXcontrapos_pprÛ†rÜjsjˆrÝhtje†rÞhŒXOFr߆ràjsjˆráhŒXlessrâ†rãhtji†rähtj*†råjsX
  ræ†rçjXassumerè†réjsjˆrêjÁj†rëjÁX~ a < brì†ríjÁj†rîjsX
  rï†rðjXhencerñ†ròjsjˆrójÁj†rôjÁXb rõ†röjÝX\<le>r÷†røjÁX arù†rújÁj†rûjsjˆrüjXbyrý†rþjsjˆrÿhtj†rhŒXsimpr†rjsjˆrhŒXonlyr†rhtj3†rjsjˆrhŒXlinorder_not_lessr†r	htj*†r
jsX
  r†rjXhencer
†rjsjˆrjÁj†rjÁXb ^ n r†rjÝX\<le>r†rjÁX a ^ nr†rjÁj†rjsjˆrjXusingr†rjsjˆrhŒXnonnegr†rjsjˆrjXbyr†r jsjˆr!htj†r"hŒXruler#†r$jsjˆr%hŒX
power_monor&†r'htj*†r(jsX
  r)†r*jXthusr+†r,jsjˆr-jÁj†r.jÝX\<not>r/†r0jÁX a ^ n < b ^ nr1†r2jÁj†r3jsjˆr4jXbyr5†r6jsjˆr7htj†r8hŒXsimpr9†r:jsjˆr;hŒXonlyr<†r=htj3†r>jsjˆr?hŒXlinorder_not_lessr@†rAhtj*†rBjsjÚ†rCjXqedrD†rEjsX

rF†rGjKXlemmarH†rIjsjˆrJhŒXpower_inject_baserK†rLhtj3†rMjsX
  rN†rOjÁj†rPjÁXa ^ Suc n = b ^ Suc n rQ†rRjÝX\<Longrightarrow>rS†rTjÁX 0 rU†rVjÝX\<le>rW†rXjÁX a rY†rZjÝX\<Longrightarrow>r[†r\jÁX 0 r]†r^jÝX\<le>r_†r`jÁX b ra†rbjÝX\<Longrightarrow>rc†rdjÁX a = bre†rfjÁj†rgjsjÚ†rhjXbyri†rjjsjˆrkhtj†rlhŒXblastrm†rnjsjˆrohŒXintrorp†rqhtj3†rrjsjˆrshŒXpower_le_imp_le_basert†rujsjˆrvhŒXantisymrw†rxjsjˆryhŒXeq_reflrz†r{jsjˆr|hŒXsymr}†r~htj*†rjsX

r€†rjKXlemmar‚†rƒjsjˆr„hŒXpower_eq_imp_eq_baser…†r†htj3†r‡jsX
  rˆ†r‰jÁj†rŠjÁXa ^ n = b ^ n r‹†rŒjÝX\<Longrightarrow>r†rŽjÁX 0 r†rjÝX\<le>r‘†r’jÁX a r“†r”jÝX\<Longrightarrow>r•†r–jÁX 0 r—†r˜jÝX\<le>r™†ršjÁX b r›†rœjÝX\<Longrightarrow>r†ržjÁX 0 < n rŸ†r jÝX\<Longrightarrow>r¡†r¢jÁX a = br£†r¤jÁj†r¥jsX
  r¦†r§jXbyr¨†r©jsjˆrªhtj†r«hŒXcasesr¬†r­jsjˆr®hŒjU†r¯htj*†r°jsjˆr±htj†r²hŒXsimp_allr³†r´jsjˆrµhŒXdelr¶†r·htj3†r¸jsjˆr¹hŒX	power_Sucrº†r»htj"†r¼jsjˆr½hŒXruler¾†r¿jsjˆrÀhŒXpower_inject_baserÁ†rÂhtj*†rÃjsX

rĆrÅjKXlemmarƆrÇjsjˆrÈhŒXpower2_le_imp_lerɆrÊhtj3†rËjsX
  r̆rÍjÁj†rÎjÁjцrÏjÁj\†rÐjÁX<^sup>2 rцrÒjÝX\<le>rÓ†rÔjÁX yrÕ†rÖjÁj\†r×jÁX<^sup>2 r؆rÙjÝX\<Longrightarrow>rÚ†rÛjÁX 0 r܆rÝjÝX\<le>rÞ†rßjÁX y rà†rájÝX\<Longrightarrow>râ†rãjÁX x rä†råjÝX\<le>ræ†rçjÁX yrè†réjÁj†rêjsX
  rë†rìjX	unfoldingrí†rîjsjˆrïhŒXnumeral_2_eq_2rð†rñjsjˆròjXbyró†rôjsjˆrõhtj†röhŒXruler÷†røjsjˆrùhŒXpower_le_imp_le_baserú†rûhtj*†rüjsX

rý†rþjKXlemmarÿ†rjsjˆrhŒXpower2_less_imp_lessr†rhtj3†rjsX
  r†rjÁj†rjÁjцrjÁj\†r	jÁX<^sup>2 < yr
†rjÁj\†rjÁX<^sup>2 r
†rjÝX\<Longrightarrow>r†rjÁX 0 r†rjÝX\<le>r†rjÁX y r†rjÝX\<Longrightarrow>r†rjÁX x < yr†rjÁj†rjsX
  r†rjXbyr†rjsjˆr htj†r!hŒXruler"†r#jsjˆr$hŒXpower_less_imp_less_baser%†r&htj*†r'jsX

r(†r)jKXlemmar*†r+jsjˆr,hŒXpower2_eq_imp_eqr-†r.htj3†r/jsX
  r0†r1jÁj†r2jÁjцr3jÁj\†r4jÁX<^sup>2 = yr5†r6jÁj\†r7jÁX<^sup>2 r8†r9jÝX\<Longrightarrow>r:†r;jÁX 0 r<†r=jÝX\<le>r>†r?jÁX x r@†rAjÝX\<Longrightarrow>rB†rCjÁX 0 rD†rEjÝX\<le>rF†rGjÁX y rH†rIjÝX\<Longrightarrow>rJ†rKjÁX x = yrL†rMjÁj†rNjsX
  rO†rPjX	unfoldingrQ†rRjsjˆrShŒXnumeral_2_eq_2rT†rUjsjˆrVjXbyrW†rXjsjˆrYhtj†rZhŒXeruler[†r\jsjˆr]htj†r^hŒX2r_†r`htj*†rajsjˆrbhŒXpower_eq_imp_eq_baserc†rdhtj*†rejsjˆrfhŒXsimprg†rhjsX

ri†rjjXendrk†rljsX

rm†rnjXcontextro†rpjsjˆrqhŒXlinordered_ring_strictrr†rsjsjÚ†rtjXbeginru†rvjsX

rw†rxjKXlemmary†rzjsjˆr{hŒXsum_squares_eq_zero_iffr|†r}htj3†r~jsX
  r†r€jÁj†rjÁXx * x + y * y = 0 r‚†rƒjÝX\<longleftrightarrow>r„†r…jÁX x = 0 r††r‡jÝX\<and>rˆ†r‰jÁX y = 0rŠ†r‹jÁj†rŒjsX
  r†rŽjXbyr†rjsjˆr‘htj†r’hŒXsimpr“†r”jsjˆr•hŒXaddr–†r—htj3†r˜jsjˆr™hŒXadd_nonneg_eq_0_iffrš†r›htj*†rœjsX

r†ržjKXlemmarŸ†r jsjˆr¡hŒXsum_squares_le_zero_iffr¢†r£htj3†r¤jsX
  r¥†r¦jÁj†r§jÁXx * x + y * y r¨†r©jÝX\<le>rª†r«jÁX 0 r¬†r­jÝX\<longleftrightarrow>r®†r¯jÁX x = 0 r°†r±jÝX\<and>r²†r³jÁX y = 0r´†rµjÁj†r¶jsX
  r·†r¸jXbyr¹†rºjsjˆr»htj†r¼hŒXsimpr½†r¾jsjˆr¿hŒXaddrÀ†rÁhtj3†rÂjsjˆrÃhŒXle_lessrĆrÅjsjˆrÆhŒXnot_sum_squares_lt_zerordžrÈjsjˆrÉhŒXsum_squares_eq_zero_iffrʆrËhtj*†rÌjsX

r͆rÎjKXlemmarφrÐjsjˆrÑhŒXsum_squares_gt_zero_iffrÒ†rÓhtj3†rÔjsX
  rÕ†rÖjÁj†r×jÁX0 < x * x + y * y r؆rÙjÝX\<longleftrightarrow>rÚ†rÛjÁX x r܆rÝjÝX\<noteq>rÞ†rßjÁX 0 rà†rájÝX\<or>râ†rãjÁX y rä†råjÝX\<noteq>ræ†rçjÁX 0rè†réjÁj†rêjsX
  rë†rìjXbyrí†rîjsjˆrïhtj†rðhŒXsimprñ†ròjsjˆróhŒXaddrô†rõhtj3†röjsjˆr÷hŒXnot_lerø†rùjsjˆrúhtje†rûhŒX	symmetricrü†rýhtji†rþjsjˆrÿhŒXsum_squares_le_zero_iffr†rhtj*†rjsX

r†rjXendr†rjsX

r†rjXcontextr	†r
jsjˆrhŒXlinordered_idomr†r
jsjÚ†rjXbeginr†rjsX

r†rjKXlemmar†rjsjˆrhŒX	power_absr†rhtj3†rjsX
  r†rjÁj†rjÁXabs (a ^ n) = abs a ^ nr†rjÁj†rjsX
  r†r jXbyr!†r"jsjˆr#htj†r$hŒXinductr%†r&jsjˆr'hŒjU†r(htj*†r)jsjˆr*htj†r+hŒXautor,†r-jsjˆr.hŒXsimpr/†r0jsjˆr1hŒXaddr2†r3htj3†r4jsjˆr5hŒXabs_multr6†r7htj*†r8jsX

r9†r:jKXlemmar;†r<jsjˆr=hŒXabs_power_minusr>†r?jsjˆr@htje†rAhŒXsimprB†rChtji†rDhtj3†rEjsX
  rF†rGjÁj†rHjÁXabs ((-a) ^ n) = abs (a ^ n)rI†rJjÁj†rKjsX
  rL†rMjXbyrN†rOjsjˆrPhtj†rQhŒXsimprR†rSjsjˆrThŒXaddrU†rVhtj3†rWjsjˆrXhŒX	power_absrY†rZhtj*†r[jsX

r\†r]jKXlemmar^†r_jsjˆr`hŒXzero_less_power_abs_iffra†rbjsjˆrchtje†rdhŒXsimpre†rfhtj"†rgjsjˆrhhŒXno_atpri†rjhtji†rkhtj3†rljsX
  rm†rnjÁj†rojÁX0 < abs a ^ n rp†rqjÝX\<longleftrightarrow>rr†rsjÁX a rt†rujÝX\<noteq>rv†rwjÁX 0 rx†ryjÝX\<or>rz†r{jÁX n = 0r|†r}jÁj†r~jsjÚ†rjXproofr€†rjsjˆr‚htj†rƒhŒXinductr„†r…jsjˆr†hŒjU†r‡htj*†rˆjsX
  r‰†rŠjXcaser‹†rŒjsjˆrhŒj†rŽjsjˆrjXshowr†r‘jsjˆr’htj†r“jXcaser”†r•jsjˆr–jXbyr—†r˜jsjˆr™hŒXsimprš†r›jsjÚ†rœjXnextr†ržjsX
  rŸ†r jXcaser¡†r¢jsjˆr£htj†r¤hŒXSucr¥†r¦jsjˆr§hŒjU†r¨htj*†r©jsjˆrªjXshowr«†r¬jsjˆr­htj†r®jXcaser¯†r°jsjˆr±jXbyr²†r³jsjˆr´htj†rµhŒXautor¶†r·jsjˆr¸hŒXsimpr¹†rºjsjˆr»hŒXaddr¼†r½htj3†r¾jsjˆr¿hŒXSucrÀ†rÁjsjˆrÂhŒXzero_less_mult_iffrÆrÄhtj*†rÅjsjÚ†rÆjXqedrdžrÈjsX

rɆrÊjKXlemmarˆrÌjsjˆrÍhŒXzero_le_power_absrΆrÏjsjˆrÐhtje†rÑhŒXsimprÒ†rÓhtji†rÔhtj3†rÕjsX
  rÖ†r×jÁj†rØjÁX0 rÙ†rÚjÝX\<le>rÛ†rÜjÁX
 abs a ^ nr݆rÞjÁj†rßjsX
  rà†rájXbyrâ†rãjsjˆrähtj†råhŒXruleræ†rçjsjˆrèhŒX
zero_le_powerré†rêjsjˆrëhtje†rìhŒXOFrí†rîjsjˆrïhŒXabs_ge_zerorð†rñhtji†ròhtj*†rójsX

rô†rõjKXlemmarö†r÷jsjˆrøhŒXzero_le_power2rù†rújsjˆrûhtje†rühŒXsimprý†rþhtji†rÿhtj3†rjsX
  r†rjÁj†rjÁX0 r†rjÝX\<le>r†rjÁX ar†r	jÁj\†r
jÁX<^sup>2r†rjÁj†r
jsX
  r†rjXbyr†rjsjˆrhtj†rhŒXsimpr†rjsjˆrhŒXaddr†rhtj3†rjsjˆrhŒXpower2_eq_squarer†rhtj*†rjsX

r†rjKXlemmar †r!jsjˆr"hŒXzero_less_power2r#†r$jsjˆr%htje†r&hŒXsimpr'†r(htji†r)htj3†r*jsX
  r+†r,jÁj†r-jÁX0 < ar.†r/jÁj\†r0jÁX<^sup>2 r1†r2jÝX\<longleftrightarrow>r3†r4jÁX a r5†r6jÝX\<noteq>r7†r8jÁX 0r9†r:jÁj†r;jsX
  r<†r=jXbyr>†r?jsjˆr@htj†rAhŒXforcerB†rCjsjˆrDhŒXsimprE†rFjsjˆrGhŒXaddrH†rIhtj3†rJjsjˆrKhŒXpower2_eq_squarerL†rMjsjˆrNhŒXzero_less_mult_iffrO†rPjsjˆrQhŒXlinorder_neq_iffrR†rShtj*†rTjsX

rU†rVjKXlemmarW†rXjsjˆrYhŒX
power2_less_0rZ†r[jsjˆr\htje†r]hŒXsimpr^†r_htji†r`htj3†rajsX
  rb†rcjÁj†rdjÝX\<not>re†rfjÁX arg†rhjÁj\†rijÁX<^sup>2 < 0rj†rkjÁj†rljsX
  rm†rnjXbyro†rpjsjˆrqhtj†rrhŒXforcers†rtjsjˆruhŒXsimprv†rwjsjˆrxhŒXaddry†rzhtj3†r{jsjˆr|hŒXpower2_eq_squarer}†r~jsjˆrhŒXmult_less_0_iffr€†rhtj*†r‚jsX

rƒ†r„jKXlemmar…†r†jsjˆr‡hŒX
abs_power2rˆ†r‰jsjˆrŠhtje†r‹hŒXsimprŒ†rhtji†rŽhtj3†rjsX
  r†r‘jÁj†r’jÁXabs (ar“†r”jÁj\†r•jÁX<^sup>2) = ar–†r—jÁj\†r˜jÁX<^sup>2r™†ršjÁj†r›jsX
  rœ†rjXbyrž†rŸjsjˆr htj†r¡hŒXsimpr¢†r£jsjˆr¤hŒXaddr¥†r¦htj3†r§jsjˆr¨hŒXpower2_eq_squarer©†rªjsjˆr«hŒXabs_multr¬†r­jsjˆr®hŒX
abs_mult_selfr¯†r°htj*†r±jsX

r²†r³jKXlemmar´†rµjsjˆr¶hŒX
power2_absr·†r¸jsjˆr¹htje†rºhŒXsimpr»†r¼htji†r½htj3†r¾jsX
  r¿†rÀjÁj†rÁjÁX(abs a)r†rÃjÁj\†rÄjÁX<^sup>2 = arņrÆjÁj\†rÇjÁX<^sup>2rȆrÉjÁj†rÊjsX
  rˆrÌjXbyr͆rÎjsjˆrÏhtj†rÐhŒXsimprцrÒjsjˆrÓhŒXaddrÔ†rÕhtj3†rÖjsjˆr×hŒXpower2_eq_squarer؆rÙjsjˆrÚhŒX
abs_mult_selfrÛ†rÜhtj*†rÝjsX

rÞ†rßjKXlemmarà†rájsjˆrâhŒXodd_power_less_zerorã†rähtj3†råjsX
  ræ†rçjÁj†rèjÁXa < 0 ré†rêjÝX\<Longrightarrow>rë†rìjÁX a ^ Suc (2*n) < 0rí†rîjÁj†rïjsjÚ†rðjXproofrñ†ròjsjˆróhtj†rôhŒXinductrõ†röjsjˆr÷hŒjU†røhtj*†rùjsX
  rú†rûjXcaserü†rýjsjˆrþhŒj†rÿjsX
  r†rjXthenr†rjsjˆrjXshowr†rjsjˆrhtj†rjXcaser	†r
jsjˆrjXbyr†r
jsjˆrhŒXsimpr†rjsjÚ†rjXnextr†rjsX
  r†rjXcaser†rjsjˆrhtj†rhŒXSucr†rjsjˆrhŒjU†rhtj*†rjsX
  r†r jXhaver!†r"jsjˆr#jÁj†r$jÁX*a ^ Suc (2 * Suc n) = (a*a) * a ^ Suc(2*n)r%†r&jÁj†r'jsX
    r(†r)jXbyr*†r+jsjˆr,htj†r-hŒXsimpr.†r/jsjˆr0hŒXaddr1†r2htj3†r3jsjˆr4hŒXmult_acr5†r6jsjˆr7hŒX	power_addr8†r9jsjˆr:hŒXpower2_eq_squarer;†r<htj*†r=jsX
  r>†r?jXthusr@†rAjsjˆrBhtj†rCjXcaserD†rEjsX
    rF†rGjXbyrH†rIjsjˆrJhtj†rKhŒXsimprL†rMjsjˆrNhŒXdelrO†rPhtj3†rQjsjˆrRhŒX	power_SucrS†rTjsjˆrUe(hŒXaddrV†rWhtj3†rXjsjˆrYhŒXSucrZ†r[jsjˆr\hŒXmult_less_0_iffr]†r^jsjˆr_hŒXmult_neg_negr`†rahtj*†rbjsjÚ†rcjXqedrd†rejsX

rf†rgjKXlemmarh†rijsjˆrjhŒXodd_0_le_power_imp_0_lerk†rlhtj3†rmjsX
  rn†rojÁj†rpjÁX0 rq†rrjÝX\<le>rs†rtjÁX a ^ Suc (2*n) ru†rvjÝX\<Longrightarrow>rw†rxjÁX 0 ry†rzjÝX\<le>r{†r|jÁX ar}†r~jÁj†rjsX
  r€†rjXusingr‚†rƒjsjˆr„hŒXodd_power_less_zeror…†r†jsjˆr‡htje†rˆhŒXofr‰†rŠjsjˆr‹hŒj
†rŒjsjˆrhŒjU†rŽhtji†rjsX
    r†r‘jXbyr’†r“jsjˆr”htj†r•hŒXforcer–†r—jsjˆr˜hŒXsimpr™†ršjsjˆr›hŒXaddrœ†rhtj3†ržjsjˆrŸhŒXlinorder_not_lessr †r¡jsjˆr¢htje†r£hŒX	symmetricr¤†r¥htji†r¦htj*†r§jsX 

r¨†r©jKXlemmarª†r«jsjˆr¬hŒXzero_le_even_power'r­†r®htje†r¯hŒXsimpr°†r±htji†r²htj3†r³jsX
  r´†rµjÁj†r¶jÁX0 r·†r¸jÝX\<le>r¹†rºjÁX
 a ^ (2*n)r»†r¼jÁj†r½jsjÚ†r¾jXproofr¿†rÀjsjˆrÁhtj†rÂhŒXinductrÆrÄjsjˆrÅhŒjU†rÆhtj*†rÇjsX
  rȆrÉjXcaserʆrËjsjˆrÌhŒj†rÍjsX
    rΆrÏjXshowrІrÑjsjˆrÒhtj†rÓjXcaserÔ†rÕjsjˆrÖjXbyr׆rØjsjˆrÙhŒXsimprÚ†rÛjsjÚ†rÜjXnextr݆rÞjsX
  r߆ràjXcaserá†râjsjˆrãhtj†rähŒXSucrå†ræjsjˆrçhŒjU†rèhtj*†réjsX
    rê†rëjXhaverì†ríjsjˆrîjÁj†rïjÁX#a ^ (2 * Suc n) = (a*a) * a ^ (2*n)rð†rñjÁj†ròjsX 
      ró†rôjXbyrõ†röjsjˆr÷htj†røhŒXsimprù†rújsjˆrûhŒXaddrü†rýhtj3†rþjsjˆrÿhŒXmult_acr†rjsjˆrhŒX	power_addr†rjsjˆrhŒXpower2_eq_squarer†rhtj*†rjsX
    r	†r
jXthusr†rjsjˆr
htj†rjXcaser†rjsX
      r†rjXbyr†rjsjˆrhtj†rhŒXsimpr†rjsjˆrhŒXaddr†rhtj3†rjsjˆrhŒXSucr†rjsjˆr hŒXzero_le_mult_iffr!†r"htj*†r#jsjÚ†r$jXqedr%†r&jsX

r'†r(jKXlemmar)†r*jsjˆr+hŒXsum_power2_ge_zeror,†r-htj3†r.jsX
  r/†r0jÁj†r1jÁX0 r2†r3jÝX\<le>r4†r5jÁX xr6†r7jÁj\†r8jÁX<^sup>2 + yr9†r:jÁj\†r;jÁX<^sup>2r<†r=jÁj†r>jsX
  r?†r@jXbyrA†rBjsjˆrChtj†rDhŒXintrorE†rFjsjˆrGhŒXadd_nonneg_nonnegrH†rIjsjˆrJhŒXzero_le_power2rK†rLhtj*†rMjsX

rN†rOjKXlemmarP†rQjsjˆrRhŒXnot_sum_power2_lt_zerorS†rThtj3†rUjsX
  rV†rWjÁj†rXjÝX\<not>rY†rZjÁX xr[†r\jÁj\†r]jÁX<^sup>2 + yr^†r_jÁj\†r`jÁX<^sup>2 < 0ra†rbjÁj†rcjsX
  rd†rejX	unfoldingrf†rgjsjˆrhhŒXnot_lessri†rjjsjˆrkjXbyrl†rmjsjˆrnhtj†rohŒXrulerp†rqjsjˆrrhŒXsum_power2_ge_zerors†rthtj*†rujsX

rv†rwjKXlemmarx†ryjsjˆrzhŒXsum_power2_eq_zero_iffr{†r|htj3†r}jsX
  r~†rjÁj†r€jÁjцrjÁj\†r‚jÁX<^sup>2 + yrƒ†r„jÁj\†r…jÁX<^sup>2 = 0 r††r‡jÝX\<longleftrightarrow>rˆ†r‰jÁX x = 0 rŠ†r‹jÝX\<and>rŒ†rjÁX y = 0rŽ†rjÁj†rjsX
  r‘†r’jX	unfoldingr“†r”jsjˆr•hŒXpower2_eq_squarer–†r—jsjˆr˜jXbyr™†ršjsjˆr›htj†rœhŒXsimpr†ržjsjˆrŸhŒXaddr †r¡htj3†r¢jsjˆr£hŒXadd_nonneg_eq_0_iffr¤†r¥htj*†r¦jsX

r§†r¨jKXlemmar©†rªjsjˆr«hŒXsum_power2_le_zero_iffr¬†r­htj3†r®jsX
  r¯†r°jÁj†r±jÁjцr²jÁj\†r³jÁX<^sup>2 + yr´†rµjÁj\†r¶jÁX<^sup>2 r·†r¸jÝX\<le>r¹†rºjÁX 0 r»†r¼jÝX\<longleftrightarrow>r½†r¾jÁX x = 0 r¿†rÀjÝX\<and>rÁ†rÂjÁX y = 0rÆrÄjÁj†rÅjsX
  rƆrÇjXbyrȆrÉjsjˆrÊhtj†rËhŒXsimpr̆rÍjsjˆrÎhŒXaddrφrÐhtj3†rÑjsjˆrÒhŒXle_lessrÓ†rÔjsjˆrÕhŒXsum_power2_eq_zero_iffrÖ†r×jsjˆrØhŒXnot_sum_power2_lt_zerorÙ†rÚhtj*†rÛjsX

r܆rÝjKXlemmarÞ†rßjsjˆràhŒXsum_power2_gt_zero_iffrá†râhtj3†rãjsX
  rä†råjÁj†ræjÁX0 < xrç†rèjÁj\†réjÁX<^sup>2 + yrê†rëjÁj\†rìjÁX<^sup>2 rí†rîjÝX\<longleftrightarrow>rï†rðjÁX x rñ†ròjÝX\<noteq>ró†rôjÁX 0 rõ†röjÝX\<or>r÷†røjÁX y rù†rújÝX\<noteq>rû†rüjÁX 0rý†rþjÁj†rÿjsX
  r†rjX	unfoldingr†rjsjˆrhŒXnot_ler†rjsjˆrhtje†rhŒX	symmetricr	†r
htji†rjsjˆrjXbyr
†rjsjˆrhtj†rhŒXsimpr†rjsjˆrhŒXaddr†rhtj3†rjsjˆrhŒXsum_power2_le_zero_iffr†rhtj*†rjsX

r†rjXendr†rjsX


r†r jÉX
subsectionr!†r"jsjˆr#hX{*r$†r%hX Miscellaneous rules r&†r'hX*}r(†r)jsX

r*†r+jKXlemmar,†r-jsjˆr.hŒXpower_eq_ifr/†r0htj3†r1jsjˆr2jÁj†r3jÁX.p ^ m = (if m=0 then 1 else p * (p ^ (m - 1)))r4†r5jÁj†r6jsX
  r7†r8jX	unfoldingr9†r:jsjˆr;hŒXOne_nat_defr<†r=jsjˆr>jXbyr?†r@jsjˆrAhtj†rBhŒXcasesrC†rDjsjˆrEhŒjφrFhtj*†rGjsjˆrHhŒXsimp_allrI†rJjsX

rK†rLjKXlemmarM†rNjsjˆrOhŒX
power2_sumrP†rQhtj3†rRjsX
  rS†rTjRXfixesrU†rVjsjˆrWhŒjцrXjsjˆrYhŒXyrZ†r[jsjˆr\htX::r]†r^jsjˆr_jÁj†r`jÁX'a::comm_semiring_1ra†rbjÁj†rcjsX
  rd†rejRXshowsrf†rgjsjˆrhjÁj†rijÁX(x + y)rj†rkjÁj\†rljÁX<^sup>2 = xrm†rnjÁj\†rojÁX<^sup>2 + yrp†rqjÁj\†rrjÁX<^sup>2 + 2 * x * yrs†rtjÁj†rujsX
  rv†rwjXbyrx†ryjsjˆrzhtj†r{hŒXsimpr|†r}jsjˆr~hŒXaddr†r€htj3†rjsjˆr‚hŒX
algebra_simpsrƒ†r„jsjˆr…hŒXpower2_eq_squarer††r‡jsjˆrˆhŒXmult_2_rightr‰†rŠhtj*†r‹jsX

rŒ†rjKXlemmarŽ†rjsjˆrhŒXpower2_diffr‘†r’htj3†r“jsX
  r”†r•jRXfixesr–†r—jsjˆr˜hŒjцr™jsjˆršhŒjZ†r›jsjˆrœhtX::r†ržjsjˆrŸjÁj†r jÁX'a::comm_ring_1r¡†r¢jÁj†r£jsX
  r¤†r¥jRXshowsr¦†r§jsjˆr¨jÁj†r©jÁX(x - y)rª†r«jÁj\†r¬jÁX<^sup>2 = xr­†r®jÁj\†r¯jÁX<^sup>2 + yr°†r±jÁj\†r²jÁX<^sup>2 - 2 * x * yr³†r´jÁj†rµjsX
  r¶†r·jXbyr¸†r¹jsjˆrºhtj†r»hŒXsimpr¼†r½jsjˆr¾hŒXaddr¿†rÀhtj3†rÁjsjˆrÂhŒX
ring_distribsrÆrÄjsjˆrÅhŒXpower2_eq_squarerƆrÇjsjˆrÈhŒXmult_2rɆrÊhtj*†rËjsjˆrÌhtj†rÍhŒXrulerΆrÏjsjˆrÐhŒXmult_commuterцrÒhtj*†rÓjsX

rÔ†rÕjKXlemmarÖ†r×jsjˆrØhŒXpower_0_SucrÙ†rÚjsjˆrÛhtje†rÜhŒXsimpr݆rÞhtji†rßhtj3†ràjsX
  rá†râjÁj†rãjÁX((0::'a::{power, semiring_0}) ^ Suc n = 0rä†råjÁj†ræjsX
  rç†rèjXbyré†rêjsjˆrëhŒXsimprì†ríjsX

rî†rïjXtextrð†rñhX{*rò†róhX@It looks plausible as a simprule, but its effect can be strange.rô†rõhX*}rö†r÷jsjÚ†røjKXlemmarù†rújsjˆrûhŒXpower_0_leftrü†rýhtj3†rþjsX
  rÿ†rjÁj†rjÁX;0 ^ n = (if n = 0 then 1 else (0::'a::{power, semiring_0}))r†rjÁj†rjsX
  r†rjXbyr†rjsjˆr	htj†r
hŒXinductr†rjsjˆr
hŒjU†rhtj*†rjsjˆrhŒXsimp_allr†rjsX

r†rjKXlemmar†rjsjˆrhŒXpower_eq_0_iffr†rjsjˆrhtje†rhŒXsimpr†rhtji†rhtj3†rjsX
  r †r!jÁj†r"jÁX
a ^ n = 0 r#†r$jÝX\<longleftrightarrow>r%†r&jÁXC
     a = (0::'a::{mult_zero,zero_neq_one,no_zero_divisors,power}) r'†r(jÝX\<and>r)†r*jÁX n r+†r,jÝX\<noteq>r-†r.jÁX 0r/†r0jÁj†r1jsX
  r2†r3jXbyr4†r5jsjˆr6htj†r7hŒXinductr8†r9jsjˆr:hŒjU†r;htj*†r<jsX
    r=†r>htj†r?hŒXautor@†rAjsjˆrBhŒXsimprC†rDjsjˆrEhŒXaddrF†rGhtj3†rHjsjˆrIhŒXno_zero_divisorsrJ†rKjsjˆrLhŒXelimrM†rNhtj3†rOjsjˆrPhŒXcontrapos_pprQ†rRhtj*†rSjsX

rT†rUjKXlemmarV†rWjsjˆrXhtj†rYjRXinrZ†r[jsjˆr\hŒXfieldr]†r^htj*†r_jsjˆr`hŒX
power_diffra†rbhtj3†rcjsX
  rd†rejRXassumesrf†rgjsjˆrhhŒXnzri†rjhtj3†rkjsjˆrljÁj†rmjÁXa rn†rojÝX\<noteq>rp†rqjÁX 0rr†rsjÁj†rtjsX
  ru†rvjRXshowsrw†rxjsjˆryjÁj†rzjÁXn r{†r|jÝX\<le>r}†r~jÁX m r†r€jÝX\<Longrightarrow>r†r‚jÁX a ^ (m - n) = a ^ m / a ^ nrƒ†r„jÁj†r…jsX
  r††r‡jXbyrˆ†r‰jsjˆrŠhtj†r‹hŒXinductrŒ†rjsjˆrŽhŒjφrjsjˆrhŒjU†r‘jsjˆr’hŒXruler“†r”htj3†r•jsjˆr–hŒXdiff_inductr—†r˜htj*†r™jsjˆršhtj†r›hŒXsimp_allrœ†rjsjˆržhŒXaddrŸ†r htj3†r¡jsjˆr¢hŒXnzr£†r¤jsjˆr¥hŒXfield_power_not_zeror¦†r§htj*†r¨jsX

r©†rªjXtextr«†r¬hX{*r­†r®hX"Perhaps these should be simprules.r¯†r°hX*}r±†r²jsjÚ†r³jKXlemmar´†rµjsjˆr¶hŒX
power_inverser·†r¸htj3†r¹jsX
  rº†r»jRXfixesr¼†r½jsjˆr¾hŒj
†r¿jsjˆrÀhtX::rÁ†rÂjsjˆrÃjÁj†rÄjÁX'a::division_ring_inverse_zerorņrÆjÁj†rÇjsX
  rȆrÉjRXshowsrʆrËjsjˆrÌjÁj†rÍjÁXinverse (a ^ n) = inverse a ^ nrΆrÏjÁj†rÐjsjÚ†rÑjRXapplyrÒ†rÓjsjˆrÔhtj†rÕhŒXcasesrÖ†r×jsjˆrØjÁj†rÙjÁXa = 0rÚ†rÛjÁj†rÜhtj*†rÝjsjÚ†rÞjRXapplyr߆ràjsjˆráhtj†râhŒXsimprã†räjsjˆråhŒXaddræ†rçhtj3†rèjsjˆréhŒXpower_0_leftrê†rëhtj*†rìjsjÚ†ríjRXapplyrî†rïjsjˆrðhtj†rñhŒXsimprò†rójsjˆrôhŒXaddrõ†röhtj3†r÷jsjˆrøhŒXnonzero_power_inverserù†rúhtj*†rûjsjÚ†rüjXdonerý†rþjsjˆrÿhX(*r†rhX+ TODO: reorient or rename to inverse_power r†rhX*)r†rjsX

r†rjKXlemmar†r	jsjˆr
hŒXpower_one_overr†rhtj3†r
jsX
  r†rjÁj†rjÁX;1 / (a::'a::{field_inverse_zero, power}) ^ n =  (1 / a) ^ nr†rjÁj†rjsX
  r†rjXbyr†rjsjˆrhtj†rhŒXsimpr†rjsjˆrhŒXaddr†rhtj3†rjsjˆr hŒXdivide_inverser!†r"htj*†r#jsjˆr$htj†r%hŒXruler&†r'jsjˆr(hŒX
power_inverser)†r*htj*†r+jsX

r,†r-jKXlemmar.†r/jsjˆr0hŒXpower_divider1†r2htj3†r3jsX
  r4†r5jÁj†r6jÁX5(a / b) ^ n = (a::'a::field_inverse_zero) ^ n / b ^ nr7†r8jÁj†r9jsjÚ†r:jRXapplyr;†r<jsjˆr=htj†r>hŒXcasesr?†r@jsjˆrAjÁj†rBjÁXb = 0rC†rDjÁj†rEhtj*†rFjsjÚ†rGjRXapplyrH†rIjsjˆrJhtj†rKhŒXsimprL†rMjsjˆrNhŒXaddrO†rPhtj3†rQjsjˆrRhŒXpower_0_leftrS†rThtj*†rUjsjÚ†rVjRXapplyrW†rXjsjˆrYhtj†rZhŒXruler[†r\jsjˆr]hŒXnonzero_power_divider^†r_htj*†r`jsjÚ†rajRXapplyrb†rcjsjˆrdhŒX
assumptionre†rfjsjÚ†rgjXdonerh†rijsX

rj†rkjXtextrl†rmjsjˆrnhX{*ro†rphXB Simprules for comparisons where common factors can be cancelled. rq†rrhX*}rs†rtjsX

ru†rvjXlemmasrw†rxjsjˆryhŒXzero_compare_simpsrz†r{jsjˆr|htj÷†r}jsX
    r~†rhŒXadd_strict_increasingr€†rjsjˆr‚hŒXadd_strict_increasing2rƒ†r„jsjˆr…hŒXadd_increasingr††r‡jsX
    rˆ†r‰hŒXzero_le_mult_iffrŠ†r‹jsjˆrŒhŒXzero_le_divide_iffr†rŽjsX 
    r†rhŒXzero_less_mult_iffr‘†r’jsjˆr“hŒXzero_less_divide_iffr”†r•jsX 
    r–†r—hŒX
mult_le_0_iffr˜†r™jsjˆršhŒXdivide_le_0_iffr›†rœjsX 
    r†ržhŒXmult_less_0_iffrŸ†r jsjˆr¡hŒXdivide_less_0_iffr¢†r£jsX 
    r¤†r¥hŒXzero_le_power2r¦†r§jsjˆr¨hŒX
power2_less_0r©†rªjsX


r«†r¬jÉX
subsectionr­†r®jsjˆr¯hX{*r°†r±hX( Exponentiation for the Natural Numbers r²†r³hX*}r´†rµjsX

r¶†r·jKXlemmar¸†r¹jsjˆrºhŒXnat_one_le_powerr»†r¼jsjˆr½htje†r¾hŒXsimpr¿†rÀhtji†rÁhtj3†rÂjsX
  rÆrÄjÁj†rÅjÁXSuc 0 rƆrÇjÝX\<le>rȆrÉjÁX i rʆrËjÝX\<Longrightarrow>r̆rÍjÁX Suc 0 rΆrÏjÝX\<le>rІrÑjÁX i ^ nrÒ†rÓjÁj†rÔjsX
  rÕ†rÖjXbyr׆rØjsjˆrÙhtj†rÚhŒXrulerÛ†rÜjsjˆrÝhŒXone_le_powerrÞ†rßjsjˆràhtje†ráhŒXofrâ†rãjsjˆrähŒXirå†ræjsjˆrçhŒjU†rèhtj"†réjsjˆrêhŒXunfoldedrë†rìjsjˆríhŒXOne_nat_defrî†rïhtji†rðhtj*†rñjsX

rò†rójKXlemmarô†rõjsjˆröhŒXnat_zero_less_power_iffr÷†røjsjˆrùhtje†rúhŒXsimprû†rühtji†rýhtj3†rþjsX
  rÿ†r jÁj†r jÁX
x ^ n > 0 r †r jÝX\<longleftrightarrow>r †r jÁX x > (0::nat) r †r jÝX\<or>r †r	 jÁX n = 0r
 †r jÁj†r jsX
  r
 †r jXbyr †r jsjˆr htj†r hŒXinductr †r jsjˆr hŒjU†r htj*†r jsjˆr hŒXautor †r jsX

r †r jKXlemmar †r jsjˆr hŒXnat_power_eq_Suc_0_iffr  †r! jsjˆr" htje†r# hŒXsimpr$ †r% htji†r& htj3†r' jsX 
  r( †r) jÁj†r* jÁXx ^ m = Suc 0 r+ †r, jÝX\<longleftrightarrow>r- †r. jÁX m = 0 r/ †r0 jÝX\<or>r1 †r2 jÁX
 x = Suc 0r3 †r4 jÁj†r5 jsX
  r6 †r7 jXbyr8 †r9 jsjˆr: htj†r; hŒXinductr< †r= jsjˆr> hŒjφr? htj*†r@ jsjˆrA hŒXautorB †rC jsX

rD †rE jKXlemmarF †rG jsjˆrH hŒXpower_Suc_0rI †rJ jsjˆrK htje†rL hŒXsimprM †rN htji†rO htj3†rP jsX
  rQ †rR jÁj†rS jÁXSuc 0 ^ n = Suc 0rT †rU jÁj†rV jsX
  rW †rX jXbyrY †rZ jsjˆr[ hŒXsimpr\ †r] jsX

r^ †r_ jXtextr` †ra hX{*rb †rc hX1Valid for the naturals, but what if @{text"0<i<1"rd †re hj:
†rf hXC?
Premises cannot be weakened: consider the case where @{term "i=0"rg †rh hj:
†ri hX,
@{term "m=1"rj †rk hj:
†rl hX and @{term "n=0"rm †rn hj:
†ro hj7†rp hX*}rq †rr jsjÚ†rs jKXlemmart †ru jsjˆrv hŒXnat_power_less_imp_lessrw †rx htj3†ry jsX
  rz †r{ jRXassumesr| †r} jsjˆr~ hŒXnonnegr †r€ htj3†r jsjˆr‚ jÁj†rƒ jÁX0 < (ir„ †r… jÝX\<Colon>r† †r‡ jÁXnat)rˆ †r‰ jÁj†rŠ jsX
  r‹ †rŒ jRXassumesr †rŽ jsjˆr hŒXlessr †r‘ htj3†r’ jsjˆr“ jÁj†r” jÁX
i ^ m < i ^ nr• †r– jÁj†r— jsX
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