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Proof step 20 from proof payload-mod (previous step)

payload(zbrar[(m + 1) + n0, mkzentry(zbrar[(m + 1) + n0].key, adr)], m + 1, nused0) = payload(zbrar, m + 1, nused0)[zbrar[(m + 1) + n0].key, adr], n0 < nused0, m + nused0 < # zbrar, m < # zbrar, sorted(zbrar, m, nused0 + 1) ⊦ (zbrar[m].entry? ⊃ payload(zbrar, m + 1, nused0)[zbrar[(m + 1) + n0].key, adr][zbrar[m].key, zbrar[m].adr] ;payload(zbrar, m + 1, nused0)[zbrar[(m + 1) + n0].key, adr] ) = (zbrar[m].entry? ⊃ payload(zbrar, m + 1, nused0)[zbrar[m].key, zbrar[m].adr];payload(zbrar, m + 1, nused0))[zbrar[m + n0 + 1].key, adr]

User applied rule case distinction

And obtained the following new goals

Proof step 21:
payload(zbrar[(m + 1) + n0, mkzentry(zbrar[(m + 1) + n0].key, adr)], m + 1, nused0) = payload(zbrar, m + 1, nused0)[zbrar[(m + 1) + n0].key, adr], zbrar[m].entry?, m < # zbrar, m + nused0 < # zbrar, n0 < nused0, sorted(zbrar, m, nused0 + 1) ⊦ payload(zbrar, m + 1, nused0)[zbrar[(m + 1) + n0].key, adr][zbrar[m].key, zbrar[m].adr] = (zbrar[m].entry? ⊃ payload(zbrar, m + 1, nused0)[zbrar[m].key, zbrar[m].adr];payload(zbrar, m + 1, nused0))[zbrar[m + n0 + 1].key, adr]

Rule argument

pos: right 1

Additional information

Used simplifier rules: a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic