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Proof step 30 from proof payload-shift-left (previous step)

zbrar[n] = mkzentry(key, adr), zbrar[n2 + 1] = mkzentry(key0, adr0), n < # zbrar, nused < # zbrar, n2 < nused, n2 + 1 < # zbrar, n2 < # zbrar, sorted(zbrar, 0, nused + 1), key0 ∈ payload(shl(zbrar, n, nused + 1), 0, nused), n ≤ n2 + 1, n ≤ nused, ∀ n0. ¬ n2 < n0 ∧ zbrar[n0].entry? → zbrar[n0].key ≠ key0 ⊦ payload(shl(zbrar, n, nused + 1), 0, nused)[key0] = adr0, key0 = key

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 31:
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 32:
zbrar[n2 + 1] = mkzentry(key0, adr0), zbrar[n] = mkzentry(key, adr), n2 < # zbrar, n2 + 1 < # zbrar, n2 < nused, nused < # zbrar, n < # zbrar, sorted(zbrar, 0, nused + 1), key0 ∈ payload(shl(zbrar, n, nused + 1), 0, nused), n ≤ nused, n ≤ n2 + 1, ∀ n0. ¬ n2 < n0 ∧ zbrar[n0].entry? → zbrar[n0].key ≠ key0 ⊦ payload(shl(zbrar, n, nused + 1), 0, nused)[key0] = adr0, key0 = key, 0 + nused ≤ # shl(zbrar, n, nused + 1) ∧ n2 < nused ∧ shl(zbrar, n, nused + 1)[0 + n2] = mkzentry(key0, adr0) ∧ (∀ n0. n0 < n2 ∧ shl(zbrar, n, nused + 1)[0 + n0].entry? → shl(zbrar, n, nused + 1)[0 + n0].key ≠ key0)

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] → [n2, adr0, key0, nused, 0, shl(zbrar, n, nused + 1)] flags: <all paths>

Additional information

Used terms: n2, adr0, key0, nused, 0, shl(zbrar, n, nused + 1)