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Proof step 9 from proof payload-shift-left (previous step)

zbrar[n] = mkzentry(key, adr), n < # zbrar, nused < # zbrar, sorted(zbrar, 0, nused + 1), n ≤ nused ⊦ (key0 ∈ payload(shl(zbrar, n, nused + 1), 0, nused) ↔ key0 ≠ key ∧ key0 ∈ payload(zbrar, 0, nused + 1)) ∧ (key0 ∈ payload(shl(zbrar, n, nused + 1), 0, nused) → payload(shl(zbrar, n, nused + 1), 0, nused)[key0] = payload(zbrar, 0, nused + 1)[key0])

User applied rule cut formula

And obtained the following new goals

Proof step 10:
key0 ∈ payload(shl(zbrar, n, nused + 1), 0, nused) ↔ key0 ∈ payload(zbrar, 0, nused + 1) -- key, zbrar[n] = mkzentry(key, adr), nused < # zbrar, n < # zbrar, sorted(zbrar, 0, nused + 1), n ≤ nused ⊦ (key0 ∈ payload(shl(zbrar, n, nused + 1), 0, nused) ↔ key0 ≠ key ∧ key0 ∈ payload(zbrar, 0, nused + 1)) ∧ (key0 ∈ payload(shl(zbrar, n, nused + 1), 0, nused) → payload(shl(zbrar, n, nused + 1), 0, nused)[key0] = payload(zbrar, 0, nused + 1)[key0])
Proof step 38:
zbrar[n] = mkzentry(key, adr), nused < # zbrar, n < # zbrar, sorted(zbrar, 0, nused + 1), n ≤ nused ⊦ key0 ∈ payload(shl(zbrar, n, nused + 1), 0, nused) ↔ key0 ∈ payload(zbrar, 0, nused + 1) -- key, (key0 ∈ payload(shl(zbrar, n, nused + 1), 0, nused) ↔ key0 ≠ key ∧ key0 ∈ payload(zbrar, 0, nused + 1)) ∧ (key0 ∈ payload(shl(zbrar, n, nused + 1), 0, nused) → payload(shl(zbrar, n, nused + 1), 0, nused)[key0] = payload(zbrar, 0, nused + 1)[key0])

Rule argument

fma: key0 ∈ payload(shl(zbrar, n, nused + 1), 0, nused) ↔ key0 ∈ payload(zbrar, 0, nused + 1) -- key

Additional information

Used formulas: key0 ∈ payload(shl(zbrar, n, nused + 1), 0, nused) ↔ key0 ∈ payload(zbrar, 0, nused + 1) -- key