plus0yyplussxysplusxytimes0y0timessxyplusytimesxyeq_nat0syfalseeq_nat00trueeq_natsxsyeq_natxyeq_natsx0falseeq_ratpairabpaircdeq_nattimesadtimesbcdiv_ratpairabpaircdpairtimesadtimesbctimes_ratpairabpaircdpairtimesactimesbdplus_ratpairabpaircdpairplustimesadtimescbtimesbdnon_zero_ratpairabnon_zero_nattimesabnon_zero_nats0truenon_zero_nat0falsenon_zero_natssxnon_zero_natsx2pairss0s0minus_natsxsyminus_natxysqrt2rif_approxnon_zero_ratreq_rat2times_ratrrrif_approxtruetruerrif_approxtruefalsersqrt2div_ratplus_ratrdiv_rat2r2initsqrt22run_againxrun_againinit2.1plussxyplusxytimessxyplusytimesxytimessxytimesxyeq_natsxsyeq_natxyeq_ratpairabpaircdeq_nattimesadtimesbceq_ratpairabpaircdtimesadeq_ratpairabpaircdtimesbcdiv_ratpairabpaircdtimesaddiv_ratpairabpaircdtimesbctimes_ratpairabpaircdtimesactimes_ratpairabpaircdtimesbdplus_ratpairabpaircdplustimesadtimescbplus_ratpairabpaircdtimesadplus_ratpairabpaircdtimescbplus_ratpairabpaircdtimesbdnon_zero_ratpairabnon_zero_nattimesabnon_zero_ratpairabtimesabnon_zero_natssxnon_zero_natsxminus_natsxsyminus_natxysqrt2rif_approxnon_zero_ratreq_rat2times_ratrrrsqrt2rnon_zero_ratrsqrt2req_rat2times_ratrrsqrt2r2sqrt2rtimes_ratrrif_approxtruefalsersqrt2div_ratplus_ratrdiv_rat2r2if_approxtruefalserdiv_ratplus_ratrdiv_rat2r2if_approxtruefalserplus_ratrdiv_rat2rif_approxtruefalserdiv_rat2rif_approxtruefalser2initsqrt22init2run_againxrun_againinitrun_againxinittruerun_againxrun_againinitplus0yyplussxysplusxytimes0y0timessxyplusytimesxyeq_nat0syfalseeq_nat00trueeq_natsxsyeq_natxyeq_natsx0falseeq_ratpairabpaircdeq_nattimesadtimesbcdiv_ratpairabpaircdpairtimesadtimesbctimes_ratpairabpaircdpairtimesactimesbdplus_ratpairabpaircdpairplustimesadtimescbtimesbdnon_zero_ratpairabnon_zero_nattimesabnon_zero_nats0truenon_zero_nat0falsenon_zero_natssxnon_zero_natsx2pairss0s0minus_natsxsyminus_natxysqrt2rif_approxnon_zero_ratreq_rat2times_ratrrrif_approxtruetruerrif_approxtruefalsersqrt2div_ratplus_ratrdiv_rat2r2initsqrt22run_againxrun_againinitrun_againxrun_againinitinitsqrt222pairss0s0if_approxtruefalsersqrt2div_ratplus_ratrdiv_rat2r2sqrt2rif_approxnon_zero_ratreq_rat2times_ratrrrdiv_ratpairabpaircdpairtimesadtimesbcplus_ratpairabpaircdpairplustimesadtimescbtimesbdnon_zero_ratpairabnon_zero_nattimesabtimes_ratpairabpaircdpairtimesactimesbdeq_ratpairabpaircdeq_nattimesadtimesbcif_approxtruetruerrtimes0y0timessxyplusytimesxyeq_nat0syfalseeq_nat00trueeq_natsxsyeq_natxyeq_natsx0falseplus0yyplussxysplusxynon_zero_nats0truenon_zero_nat0falsenon_zero_natssxnon_zero_natsxplus0x0plussx0x1times0x0timessx0x1eq_nat0sx0eq_nat00eq_natsx0sx1eq_natsx00eq_ratpairx0x1pairx2x3div_ratpairx0x1pairx2x3times_ratpairx0x1pairx2x3plus_ratpairx0x1pairx2x3non_zero_ratpairx0x1non_zero_nats0non_zero_nat0non_zero_natssx02sqrt2x0if_approxtruetruex0if_approxtruefalsex0initrun_againxrun_againinit1initsqrt22run_againsqrt22run_againxrun_againsqrt22initsqrt222pairss0s0sqrt2rif_approxnon_zero_ratreq_rat2times_ratrrrif_approxtruefalsersqrt2div_ratplus_ratrdiv_rat2r2non_zero_ratpairabnon_zero_nattimesabtimes_ratpairabpaircdpairtimesactimesbdeq_ratpairabpaircdeq_nattimesadtimesbcif_approxtruetruerrdiv_ratpairabpaircdpairtimesadtimesbcplus_ratpairabpaircdpairplustimesadtimescbtimesbdtimes0y0timessxyplusytimesxyplus0yyplussxysplusxyeq_nat0syfalseeq_nat00trueeq_natsxsyeq_natxyeq_natsx0falsenon_zero_nats0truenon_zero_nat0falsenon_zero_natssxnon_zero_natsxrun_againxrun_againsqrt222pairss0s0sqrt2rif_approxnon_zero_ratreq_rat2times_ratrrrif_approxtruefalsersqrt2div_ratplus_ratrdiv_rat2r2non_zero_ratpairabnon_zero_nattimesabtimes_ratpairabpaircdpairtimesactimesbdeq_ratpairabpaircdeq_nattimesadtimesbcif_approxtruetruerrdiv_ratpairabpaircdpairtimesadtimesbcplus_ratpairabpaircdpairplustimesadtimescbtimesbdtimes0y0timessxyplusytimesxyplus0yyplussxysplusxyeq_nat0syfalseeq_nat00trueeq_natsxsyeq_natxyeq_natsx0falsenon_zero_nats0truenon_zero_nat0falsenon_zero_natssxnon_zero_natsxplus0x0plussx0x1times0x0timessx0x1eq_nat0sx0eq_nat00eq_natsx0sx1eq_natsx00eq_ratpairx0x1pairx2x3div_ratpairx0x1pairx2x3times_ratpairx0x1pairx2x3plus_ratpairx0x1pairx2x3non_zero_ratpairx0x1non_zero_nats0non_zero_nat0non_zero_natssx02sqrt2x0if_approxtruetruex0if_approxtruefalsex0run_againxrun_againsqrt221sqrt2rif_approxnon_zero_ratreq_rat2times_ratrrrrun_againif_approxnon_zero_rat2eq_rat2times_rat222run_againxrun_againif_approxnon_zero_rat2eq_rat2times_rat2222pairss0s0if_approxtruefalsersqrt2div_ratplus_ratrdiv_rat2r2sqrt2rif_approxnon_zero_ratreq_rat2times_ratrrrdiv_ratpairabpaircdpairtimesadtimesbcplus_ratpairabpaircdpairplustimesadtimescbtimesbdnon_zero_ratpairabnon_zero_nattimesabtimes_ratpairabpaircdpairtimesactimesbdeq_ratpairabpaircdeq_nattimesadtimesbcif_approxtruetruerrtimes0y0timessxyplusytimesxyeq_nat0syfalseeq_nat00trueeq_natsxsyeq_natxyeq_natsx0falseplus0yyplussxysplusxynon_zero_nats0truenon_zero_nat0falsenon_zero_natssxnon_zero_natsxrun_againxrun_againif_approxnon_zero_rat2eq_rat2times_rat2221112pairss0s0run_againif_approxnon_zero_ratpairss0s0eq_rat2times_rat222run_againx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for single proof: 100,00 % (84 real / 0 unknown / 0 assumptions / 84 total proof steps)http://aprove.informatik.rwth-aachen.deJohnDoe