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Proof step 101 from proof payload-shift-left (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 102:
zbrar[n2 + 1] = mkzentry((n0 < n ⊃ zbrar[n0];zbrar[n0 + 1]).key, adr), (n0 < n ⊃ zbrar[n0];zbrar[n0 + 1]).entry?, n0 + 1 < # zbrar, n < # zbrar, nused < # zbrar, n2 < nused, n2 + 1 < # zbrar, n2 < # zbrar, n0 < n2, n < nused, n0 < nused, n0 < # zbrar, n0 + 1 < nused, n + 1 < # zbrar, 2 + n0 < # zbrar, sorted(zbrar, 0, nused + 1), (n0 < n ⊃ zbrar[n0];zbrar[n0 + 1]).key ∈ payload(shl(zbrar, n, nused + 1), 0, nused), n ≤ n2, n ≤ n2 + 1, ∀ n1. ¬ n2 < n1 ∧ zbrar[n1].entry? → zbrar[n1].key ≠ (n0 < n ⊃ zbrar[n0];zbrar[n0 + 1]).key ⊦ payload(shl(zbrar, n, nused + 1), 0, nused)[(n0 < n ⊃ zbrar[n0];zbrar[n0 + 1]).key] = adr, zbrar[n].entry?

Rule argument

Additional information

Used simplifier rules: c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m < n → m ≤ n; (s)% ⊦ m +1 = m + 1; (s)% ⊦ m < n → n ≠ 0; (s)% ⊦ m < n → (n ≤ m ↔ false); (s)%fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat%transitivity: ⊦ n < n0 ∧ n0 < n1 → n < n1; (forward) from specification oelem[nat-set]%lf: ⊦ m < n ∧ n < n0 → m +1 < n0; (forward) from specification nat-basic%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic% ⊦ m ≤ n → (n < m ↔ false); (s)% ⊦ m < n ↔ ¬ n ≤ m; (ss)