aleV0V0F
aleV1V3IaleV2V3IaleV1V2F
aleV5V4OaleV4V5F
a=aminimumaNodeV6V7V8V7F
ais_minimumaminimumV9V9ICfaEmptyais_minimumaminimumV11V11Oais_emptyV11Aa=V11aEmptyqais_emptyV11Aais_minimumaminimumV10V10Oais_emptyV10Aa=V10aEmptyqais_emptyV10aNodeVwVV9ACfaEmptya<c0asizeV13AaheapV13ACfaEmptya=V15V13Oa=V14V13aNodeVwVV9INais_emptyV13Ia=V13aEmptyqais_emptyV13Iais_minimumaminimumV12V12Oais_emptyV12Aa=V12aEmptyqais_emptyV12Aa<c0asizeV12AaheapV12ACfaEmptya=V17V12Oa=V16V12aNodeVwVV9INais_emptyV12Ia=V12aEmptyqais_emptyV12aNodeVwVV9Ia<c0asizeV9AaheapV9F
VC for root_is_min
ais_minimumaminimumV9V9Ia<c0asizeV9AaheapV9F
NamemV11V10FAa=asizeV10c0AaheapV10LaEmpty
VC for empty
NamemV10aemptyFAa=asizeaemptyc0Aaheapaempty
a=V12aminimumV11ICfaEmptya=V12V13aNodewVwV11FACfaEmptytaNodewVwV11Ia<c0asizeV11AaheapV11F
VC for get_min
aheapat1
aheapat2
a=at2aNodeaxax1ax2
a=at1aNodeax3ax4ax5
aleax4ax1
a<a+asizeax5asizeat2a+asizeat1asizeat2Aa<=c0a+asizeat1asizeat2
variant decrease
aheapat2Aaheapax5
precondition
Naleax4ax1
a<a+asizeax2asizeat1a+asizeat1asizeat2Aa<=c0a+asizeat1asizeat2
aheapat1Aaheapax2
a=at2aEmpty
a=at1aEmpty
a=aresultat2
aheaparesult
postcondition
a=at1aNodeawaw1aw2
a=aresultat1
a=aoccV11aresulta+aoccV11ax5aoccV11at2F
a=asizearesulta+asizeax5asizeat2
a=aresult1aNodearesultax4ax3
aheaparesult1
a=aoccV11aresulta+aoccV11ax2aoccV11at1F
a=asizearesulta+asizeax2asizeat1
a=aresult1aNodearesultax1ax
CCa=aresultat2aEmptya=aresultat1wat1aEmptyCa=aresultat2aEmptyia=aresultaNodearesult1V12V11Aa=asizearesult1a+asizeV13asizeat1Aa=aoccV17aresult1a+aoccV17V13aoccV17at1FAaheaparesult1a=aresultaNodearesult2V15V14Aa=asizearesult2a+asizeV16asizeat2Aa=aoccV18aresult2a+aoccV18V16aoccV18at2FAaheaparesult2aleV15V12aNodeVVVat1aNodeVVVat2
a=aoccaearesulta+aoccaeat1aoccaeat2
a=aoccaearesult1a+aoccaeat1aoccaeat2
a=aoccV11aresulta+aoccV11at1aoccV11at2F
a=asizearesulta+asizeat1asizeat2
a=aoccV12aresult1a+aoccV12at1aoccV12at2F
a=asizearesult1a+asizeat1asizeat2
a=aoccV19aresulta+aoccV19at1aoccV19at2F
a=asizeV13a+asizeV11asizeV12Aa=aoccV14V13a+aoccV14V11aoccV14V12FAaheapV13ICCa=V13V12aEmptya=V13V11wV11aEmptyCa=V13V12aEmptyia=V13aNodeV21V16V15Aa=asizeV21a+asizeV17asizeV11Aa=aoccV22V21a+aoccV22V17aoccV22V11FAaheapV21Ea=V13aNodeV23V19V18Aa=asizeV23a+asizeV20asizeV12Aa=aoccV24V23a+aoccV24V20aoccV24V12FAaheapV23EaleV19V16aNodeVVVV11aNodeVVVV12FACCtaEmptytwV11aEmptyCtaEmptyiaheapV11AaheapV27Aa<a+asizeV27asizeV11a+asizeV11asizeV12Aa<=c0a+asizeV11asizeV12aheapV12AaheapV30Aa<a+asizeV30asizeV12a+asizeV11asizeV12Aa<=c0a+asizeV11asizeV12aleV29V26aNodeVVVV11aNodeVVVV12IaheapV12AaheapV11F
VC for merge
a=asizeV14a+asizeV12c1Aa=aoccV15V14aoccV15V12INa=V15V11FAa=aoccV11V14a+aoccV11V12c1AaheapV14Ia=asizeV14a+asizeV13asizeV12Aa=aoccV16V14a+aoccV16V13aoccV16V12FAaheapV14FAaheapV12AaheapV13LaNodeaEmptyV11aEmptyIaheapV12F
VC for add
a=asizeV12a-asizeV11c1Aa=aoccV13V12aoccV13V11INa=V13aminimumV11FAa=aoccaminimumV11V12a-aoccaminimumV11V11c1AaheapV12ICfaEmptya=asizeV12a+asizeV14asizeV15Aa=aoccV16V12a+aoccV16V14aoccV16V15FAaheapV12aNodeVwVV11FACfaEmptyaheapV18AaheapV17aNodeVwVV11Ia<c0asizeV11AaheapV11F
VC for remove_min
a=asizeaemptyc0AaheapaemptyIa<=c0a+V2c1La-V1c1LalengthV0F
loop invariant init
a=a+aoccV3aeltsV0c0V1aocc1V3aemptyaoccV3aeltsV0c0V1FIa=asizeaemptyc0AaheapaemptyIa<=c0a+V2c1La-V1c1LalengthV0F
a<V4alengthV0Aa<=c0V4Ia=a+aoccV5aeltsV0V4V1aocc1V5V3aoccV5aeltsV0c0V1FAa=asizeV3V4AaheapV3Aa<=V4V2Aa<=c0V4FFIa<=c0a+V2c1La-V1c1LalengthV0F
index in array bounds
aheapV3LaV0V4Ia=a+aoccV6aeltsV0V4V1aocc1V6V3aoccV6aeltsV0c0V1FAa=asizeV3V4AaheapV3Aa<=V4V2Aa<=c0V4FFIa<=c0a+V2c1La-V1c1LalengthV0F
a=aoccaV0V4aeltsV0V4V1a+c1aoccaV0V4aeltsV0a+V4c1V1Ia=V7V6FIa=asizeV6a+asizeV3c1Aa=aocc1V8V6aocc1V8V3INa=V8V5FAa=aocc1V5V6a+aocc1V5V3c1AaheapV6FLaV0V4Ia=a+aoccV9aeltsV0V4V1aocc1V9V3aoccV9aeltsV0c0V1FAa=asizeV3V4AaheapV3Aa<=V4V2Aa<=c0V4FFIa<=c0a+V2c1La-V1c1LalengthV0F
assertion
a=asizeV7a+V4c1AaheapV7Ia=aoccaV0V4aeltsV0V4V1a+c1aoccaV0V4aeltsV0a+V4c1V1Ia=V7V6FIa=asizeV6a+asizeV3c1Aa=aocc1V8V6aocc1V8V3INa=V8V5FAa=aocc1V5V6a+aocc1V5V3c1AaheapV6FLaV0V4Ia=a+aoccV9aeltsV0V4V1aocc1V9V3aoccV9aeltsV0c0V1FAa=asizeV3V4AaheapV3Aa<=V4V2Aa<=c0V4FFIa<=c0a+V2c1La-V1c1LalengthV0F
loop invariant preservation
a=a+aoccV8aeltsV0a+V4c1V1aocc1V8V7aoccV8aeltsV0c0V1FIa=asizeV7a+V4c1AaheapV7Ia=aoccaV0V4aeltsV0V4V1a+c1aoccaV0V4aeltsV0a+V4c1V1Ia=V7V6FIa=asizeV6a+asizeV3c1Aa=aocc1V9V6aocc1V9V3INa=V9V5FAa=aocc1V5V6a+aocc1V5V3c1AaheapV6FLaV0V4Ia=a+aoccV10aeltsV0V4V1aocc1V10V3aoccV10aeltsV0c0V1FAa=asizeV3V4AaheapV3Aa<=V4V2Aa<=c0V4FFIa<=c0a+V2c1La-V1c1LalengthV0F
asorted_subV0c0c0Ia<=c0a+V4c1La-V1c1Ia=a+aoccV5aeltsV0a+V2c1V1aocc1V5V3aoccV5aeltsV0c0V1FAa=asizeV3a+V2c1AaheapV3FIa<=c0a+V2c1La-V1c1LalengthV0F
a=asizeV3a-V1c0AaheapV3Iasorted_subV0c0c0Ia<=c0a+V4c1La-V1c1Ia=a+aoccV5aeltsV0a+V2c1V1aocc1V5V3aoccV5aeltsV0c0V1FAa=asizeV3a+V2c1AaheapV3FIa<=c0a+V2c1La-V1c1LalengthV0F
aleaV0V5V6IamemV6V3FIa<V5c0Aa<=c0V5FIa=asizeV3a-V1c0AaheapV3Iasorted_subV0c0c0Ia<=c0a+V4c1La-V1c1Ia=a+aoccV7aeltsV0a+V2c1V1aocc1V7V3aoccV7aeltsV0c0V1FAa=asizeV3a+V2c1AaheapV3FIa<=c0a+V2c1La-V1c1LalengthV0F
a=a+aoccV5aeltsV0c0c0aocc1V5V3aoccV5aeltsV0c0V1FIaleaV0V6V7IamemV7V3FIa<V6c0Aa<=c0V6FIa=asizeV3a-V1c0AaheapV3Iasorted_subV0c0c0Ia<=c0a+V4c1La-V1c1Ia=a+aoccV8aeltsV0a+V2c1V1aocc1V8V3aoccV8aeltsV0c0V1FAa=asizeV3a+V2c1AaheapV3FIa<=c0a+V2c1La-V1c1LalengthV0F
a<c0asizeV5AaheapV5Ia=a+aoccV8aeltsV6c0V7aocc1V8V5aoccV8aeltsV0c0V1FAaleaV6V9V10IamemV10V5FIa<V9V7Aa<=c0V9FAa=asizeV5a-V1V7AaheapV5Aasorted_subV6c0V7Aa<=V7V4Aa<=c0V7FIa=alengthV6alengthV0FIa<=c0a+V4c1La-V1c1Ia=a+aoccV11aeltsV0a+V2c1V1aocc1V11V3aoccV11aeltsV0c0V1FAa=asizeV3a+V2c1AaheapV3FIa<=c0a+V2c1La-V1c1LalengthV0F
a<V7alengthV6Aa<=c0V7LaminimumV5Ia=a+aoccV9aeltsV6c0V7aocc1V9V5aoccV9aeltsV0c0V1FAaleaV6V10V11IamemV11V5FIa<V10V7Aa<=c0V10FAa=asizeV5a-V1V7AaheapV5Aasorted_subV6c0V7Aa<=V7V4Aa<=c0V7FIa=alengthV6alengthV0FIa<=c0a+V4c1La-V1c1Ia=a+aoccV12aeltsV0a+V2c1V1aocc1V12V3aoccV12aeltsV0c0V1FAa=asizeV3a+V2c1AaheapV3FIa<=c0a+V2c1La-V1c1LalengthV0F
a<c0asizeV5AaheapV5Ia=V9aV6V7V8Aa=aeltsV9asetaeltsV6V7V8Ia=alengthV9alengthV6FLaminimumV5Ia=a+aoccV10aeltsV6c0V7aocc1V10V5aoccV10aeltsV0c0V1FAaleaV6V11V12IamemV12V5FIa<V11V7Aa<=c0V11FAa=asizeV5a-V1V7AaheapV5Aasorted_subV6c0V7Aa<=V7V4Aa<=c0V7FIa=alengthV6alengthV0FIa<=c0a+V4c1La-V1c1Ia=a+aoccV13aeltsV0a+V2c1V1aocc1V13V3aoccV13aeltsV0c0V1FAa=asizeV3a+V2c1AaheapV3FIa<=c0a+V2c1La-V1c1LalengthV0F
asorted_subV9c0a+V7c1Ia=V11V10FIa=asizeV10a-asizeV5c1Aa=aoccV12V10aoccV12V5INa=V12aminimumV5FAa=aoccaminimumV5V10a-aoccaminimumV5V5c1AaheapV10FIa=V9aV6V7V8Aa=aeltsV9asetaeltsV6V7V8Ia=alengthV9alengthV6FLaminimumV5Ia=a+aocc1V13aeltsV6c0V7aoccV13V5aocc1V13aeltsV0c0V1FAaleaV6V14V15IamemV15V5FIa<V14V7Aa<=c0V14FAa=asizeV5a-V1V7AaheapV5Aasorted_subV6c0V7Aa<=V7V4Aa<=c0V7FIa=alengthV6alengthV0FIa<=c0a+V4c1La-V1c1Ia=a+aocc1V16aeltsV0a+V2c1V1aoccV16V3aocc1V16aeltsV0c0V1FAa=asizeV3a+V2c1AaheapV3FIa<=c0a+V2c1La-V1c1LalengthV0F
a=asizeV11a-V1a+V7c1AaheapV11Iasorted_subV9c0a+V7c1Ia=V11V10FIa=asizeV10a-asizeV5c1Aa=aoccV12V10aoccV12V5INa=V12aminimumV5FAa=aoccaminimumV5V10a-aoccaminimumV5V5c1AaheapV10FIa=V9aV6V7V8Aa=aeltsV9asetaeltsV6V7V8Ia=alengthV9alengthV6FLaminimumV5Ia=a+aocc1V13aeltsV6c0V7aoccV13V5aocc1V13aeltsV0c0V1FAaleaV6V14V15IamemV15V5FIa<V14V7Aa<=c0V14FAa=asizeV5a-V1V7AaheapV5Aasorted_subV6c0V7Aa<=V7V4Aa<=c0V7FIa=alengthV6alengthV0FIa<=c0a+V4c1La-V1c1Ia=a+aocc1V16aeltsV0a+V2c1V1aoccV16V3aocc1V16aeltsV0c0V1FAa=asizeV3a+V2c1AaheapV3FIa<=c0a+V2c1La-V1c1LalengthV0F
aleaV9V12V13IamemV13V11FIa<V12a+V7c1Aa<=c0V12FIa=asizeV11a-V1a+V7c1AaheapV11Iasorted_subV9c0a+V7c1Ia=V11V10FIa=asizeV10a-asizeV5c1Aa=aoccV14V10aoccV14V5INa=V14aminimumV5FAa=aoccaminimumV5V10a-aoccaminimumV5V5c1AaheapV10FIa=V9aV6V7V8Aa=aeltsV9asetaeltsV6V7V8Ia=alengthV9alengthV6FLaminimumV5Ia=a+aocc1V15aeltsV6c0V7aoccV15V5aocc1V15aeltsV0c0V1FAaleaV6V16V17IamemV17V5FIa<V16V7Aa<=c0V16FAa=asizeV5a-V1V7AaheapV5Aasorted_subV6c0V7Aa<=V7V4Aa<=c0V7FIa=alengthV6alengthV0FIa<=c0a+V4c1La-V1c1Ia=a+aocc1V18aeltsV0a+V2c1V1aoccV18V3aocc1V18aeltsV0c0V1FAa=asizeV3a+V2c1AaheapV3FIa<=c0a+V2c1La-V1c1LalengthV0F
a=a+aoccV12aeltsV9c0a+V7c1aocc1V12V11aoccV12aeltsV0c0V1FIaleaV9V13V14IamemV14V11FIa<V13a+V7c1Aa<=c0V13FIa=asizeV11a-V1a+V7c1AaheapV11Iasorted_subV9c0a+V7c1Ia=V11V10FIa=asizeV10a-asizeV5c1Aa=aocc1V15V10aocc1V15V5INa=V15aminimumV5FAa=aocc1aminimumV5V10a-aocc1aminimumV5V5c1AaheapV10FIa=V9aV6V7V8Aa=aeltsV9asetaeltsV6V7V8Ia=alengthV9alengthV6FLaminimumV5Ia=a+aoccV16aeltsV6c0V7aocc1V16V5aoccV16aeltsV0c0V1FAaleaV6V17V18IamemV18V5FIa<V17V7Aa<=c0V17FAa=asizeV5a-V1V7AaheapV5Aasorted_subV6c0V7Aa<=V7V4Aa<=c0V7FIa=alengthV6alengthV0FIa<=c0a+V4c1La-V1c1Ia=a+aoccV19aeltsV0a+V2c1V1aocc1V19V3aoccV19aeltsV0c0V1FAa=asizeV3a+V2c1AaheapV3FIa<=c0a+V2c1La-V1c1LalengthV0F
asortedV6Ia=a+aoccV7aeltsV6c0a+V4c1aocc1V7V5aoccV7aeltsV0c0V1FAaleaV6V8V9IamemV9V5FIa<V8a+V4c1Aa<=c0V8FAa=asizeV5a-V1a+V4c1AaheapV5Aasorted_subV6c0a+V4c1Ia=alengthV6alengthV0FIa<=c0a+V4c1La-V1c1Ia=a+aoccV10aeltsV0a+V2c1V1aocc1V10V3aoccV10aeltsV0c0V1FAa=asizeV3a+V2c1AaheapV3FIa<=c0a+V2c1La-V1c1LalengthV0F
apermut_allV0V6IasortedV6Ia=a+aoccV7aeltsV6c0a+V4c1aocc1V7V5aoccV7aeltsV0c0V1FAaleaV6V8V9IamemV9V5FIa<V8a+V4c1Aa<=c0V8FAa=asizeV5a-V1a+V4c1AaheapV5Aasorted_subV6c0a+V4c1Ia=alengthV6alengthV0FIa<=c0a+V4c1La-V1c1Ia=a+aoccV10aeltsV0a+V2c1V1aocc1V10V3aoccV10aeltsV0c0V1FAa=asizeV3a+V2c1AaheapV3FIa<=c0a+V2c1La-V1c1LalengthV0F
apermut_allV0V0AasortedV0Ia>c0a+V4c1La-V1c1Ia=a+aoccV5aeltsV0a+V2c1V1aocc1V5V3aoccV5aeltsV0c0V1FAa=asizeV3a+V2c1AaheapV3FIa<=c0a+V2c1La-V1c1LalengthV0F
out of loop bounds
apermut_allV0V0AasortedV0Ia>c0a+V3c1Aapermut_allV0V5AasortedV5Ia=a+aoccV6aeltsV5c0a+V3c1aocc1V6V4aoccV6aeltsV0c0V1FAaleaV5V7V8IamemV8V4FIa<V7a+V3c1Aa<=c0V7FAa=asizeV4a-V1a+V3c1AaheapV4Aasorted_subV5c0a+V3c1Aa=a+aoccV14aeltsV11c0a+V9c1aocc1V14V13aoccV14aeltsV0c0V1FAaleaV11V15V16IamemV16V13FIa<V15a+V9c1Aa<=c0V15FAa=asizeV13a-V1a+V9c1AaheapV13Aasorted_subV11c0a+V9c1Ia=V13V12FIa=asizeV12a-asizeV4c1Aa=aocc1V17V12aocc1V17V4INa=V17aminimumV4FAa=aocc1aminimumV4V12a-aocc1aminimumV4V4c1AaheapV12FAa<c0asizeV4AaheapV4Ia=V11aV5V9V10Aa=aeltsV11asetaeltsV5V9V10Ia=alengthV11alengthV5FAa<V9alengthV5Aa<=c0V9LaminimumV4Aa<c0asizeV4AaheapV4Ia=a+aoccV18aeltsV5c0V9aocc1V18V4aoccV18aeltsV0c0V1FAaleaV5V19V20IamemV20V4FIa<V19V9Aa<=c0V19FAa=asizeV4a-V1V9AaheapV4Aasorted_subV5c0V9Aa<=V9V3Aa<=c0V9FIa=alengthV5alengthV0FAa=a+aoccV21aeltsV0c0c0aocc1V21aemptyaoccV21aeltsV0c0V1FAaleaV0V22V23IamemV23aemptyFIa<V22c0Aa<=c0V22FAa=asizeaemptya-V1c0AaheapaemptyAasorted_subV0c0c0Ia<=c0a+V3c1La-V1c1Ia>c0a+V2c1La-V1c1LalengthV0F
apermut_allV0V0AasortedV0Ia>c0a+V3c1Aapermut_allV0V5AasortedV5Ia=a+aoccV6aeltsV5c0a+V3c1aocc1V6V4aoccV6aeltsV0c0V1FAaleaV5V7V8IamemV8V4FIa<V7a+V3c1Aa<=c0V7FAa=asizeV4a-V1a+V3c1AaheapV4Aasorted_subV5c0a+V3c1Aa=a+aoccV14aeltsV11c0a+V9c1aocc1V14V13aoccV14aeltsV0c0V1FAaleaV11V15V16IamemV16V13FIa<V15a+V9c1Aa<=c0V15FAa=asizeV13a-V1a+V9c1AaheapV13Aasorted_subV11c0a+V9c1Ia=V13V12FIa=asizeV12a-asizeV4c1Aa=aocc1V17V12aocc1V17V4INa=V17aminimumV4FAa=aocc1aminimumV4V12a-aocc1aminimumV4V4c1AaheapV12FAa<c0asizeV4AaheapV4Ia=V11aV5V9V10Aa=aeltsV11asetaeltsV5V9V10Ia=alengthV11alengthV5FAa<V9alengthV5Aa<=c0V9LaminimumV4Aa<c0asizeV4AaheapV4Ia=a+aoccV18aeltsV5c0V9aocc1V18V4aoccV18aeltsV0c0V1FAaleaV5V19V20IamemV20V4FIa<V19V9Aa<=c0V19FAa=asizeV4a-V1V9AaheapV4Aasorted_subV5c0V9Aa<=V9V3Aa<=c0V9FIa=alengthV5alengthV0FAa=a+aoccV21aeltsV0c0c0aocc1V21aemptyaoccV21aeltsV0c0V1FAaleaV0V22V23IamemV23aemptyFIa<V22c0Aa<=c0V22FAa=asizeaemptya-V1c0AaheapaemptyAasorted_subV0c0c0Ia<=c0a+V3c1La-V1c1Ia>c0a+V2c1Aapermut_allV0V0AasortedV0Ia>c0a+V25c1A
VC for heapsort

1f3a4c40a73a033af7c3f6386b3bf376 5H4H3H2H1H0
f8b09903a1c9dc93cddab540a7474478 8H7H6H3H2H1H0
e730f6548ce59e532fed33a25ba4b962 11H10H9H6H3H2H1H0
747c816396479366c794ce0b17d9bbf9 47H46H9H6H3H2H1H0
c6916d25270e083744f3cd8b701165ce 18H17H16H15H14H13H12H9H6H3H2H1H0
831c4d5c95be96da7766c6e874b352c7 20H19H16H15H14H13H12H9H6H3H2H1H0
672e6d16bc9371fe99f263157e3b93a1 18H22H21H15H14H13H12H9H6H3H2H1H0
cf03cae037d58b1e7d38f9d17c6faad4 20H23H21H15H14H13H12H9H6H3H2H1H0
fc7c3789d74d4e4ea22600977cb20eaa 28H27H38H13H12H9H6H3H2H1H0
f0f537c3b73e2de0e18d7ceb53c6762f 28H27H26H25H24H13H12H9H6H3H2H1H0
a28b8bb9378f92ae24389307ca2a75e4 28H27H30H29H24H13H12H9H6H3H2H1H0
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