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Proof step 83 from proof payload-shift-left (previous step)

zbrar[n0] = mkzentry(key, adr), n0 < # zbrar, nused < # zbrar, n < # zbrar, sorted(zbrar, 0, nused + 1), key ∈ payload(shl(zbrar, n, nused + 1), 0, nused), n0 ≤ nused, n ≤ nused ⊦ payload(shl(zbrar, n, nused + 1), 0, nused)[key] = payload(zbrar, 0, nused + 1)[key], zbrar[n].entry?

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 84:
m + nused ≤ # zbrar, n < nused, zbrar[m + n] = mkzentry(key, adr), ∀ n0. n0 < n ∧ zbrar[m + n0].entry? → zbrar[m + n0].key ≠ key ⊦ payload(zbrar, m, nused)[key] = adr
Proof step 85:
zbrar[n0] = mkzentry(key, adr), n < # zbrar, nused < # zbrar, n0 < # zbrar, sorted(zbrar, 0, nused + 1), key ∈ payload(shl(zbrar, n, nused + 1), 0, nused), n ≤ nused, n0 ≤ nused ⊦ payload(shl(zbrar, n, nused + 1), 0, nused)[key] = payload(zbrar, 0, nused + 1)[key], zbrar[n].entry?, 0 + nused + 1 ≤ # zbrar ∧ n0 < nused + 1 ∧ zbrar[0 + n0] = mkzentry(key, adr) ∧ (∀ n1. n1 < n0 ∧ zbrar[0 + n1].entry? → zbrar[0 + n1].key ≠ key)
Proof step 87:
zbrar[n0] = mkzentry(key, adr), zbrar[0 + n0] = mkzentry(key, adr), n < # zbrar, nused < # zbrar, n0 < # zbrar, n0 < nused + 1, sorted(zbrar, 0, nused + 1), key ∈ payload(shl(zbrar, n, nused + 1), 0, nused), n ≤ nused, n0 ≤ nused, 0 + nused + 1 ≤ # zbrar, ∀ n1. n1 < n0 ∧ zbrar[0 + n1].entry? → zbrar[0 + n1].key ≠ key ⊦ payload(shl(zbrar, n, nused + 1), 0, nused)[key] = adr, zbrar[n].entry?

Rule argument

spec: <toplevel> name: at-payload seq: m + nused ≤ # zbrar, n < nused, zbrar[m + n] = mkzentry(key, adr), ∀ n0. n0 < n ∧ zbrar[m + n0].entry? → zbrar[m + n0].key ≠ key ⊦ payload(zbrar, m, nused)[key] = adr subst: [n, adr, key, nused, m, zbrar] → [n0, adr, key, nused + 1, 0, zbrar] flags: <all paths>

Additional information

Used terms: n0, adr, key, nused + 1, 0, zbrar