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Proof step 32 from proof payload-shift-left (previous step)

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)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 33:
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, (n2 < n ⊃ zbrar[n2];mkzentry(key0, adr0)) = mkzentry(key0, adr0) ∧ (n0 < n2 ∧ (n0 < n ∨ nused ≤ n0 ⊃ zbrar[n0];zbrar[n0 + 1]).entry? → (n0 < n ∨ nused ≤ n0 ⊃ zbrar[n0];zbrar[n0 + 1]).key ≠ key0)

Rule argument

Additional information

Used simplifier rules: fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat% ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ m < n → (n ≤ m ↔ false); (s)% ⊦ m + n ≤ m + n0 ↔ n ≤ n0; (s)%n ≤ m ⊦ (m - n) + m0 ≤ n0 ↔ m + m0 ≤ n0 + n; (s)% ⊦ m ≤ n → (n < m ↔ false); (s)% ⊦ n ≤ m → m + n0 - n = (m - n) + n0; (s)%m0 < # zbrar, n ≤ m ⊦ shl(zbrar, n, m)[m0] = (m0 < n ∨ n + (m - n - 1) ≤ m0 ⊃ zbrar[m0];zbrar[m0 + 1]); (s)% ⊦ m < n → ¬ n < m; (s)% ⊦ m + n < m + n0 ↔ n < n0; (s)% ⊦ m +1 = m + 1; (s)% ⊦ m < n → n ≠ 0; (s)%transitivity: ⊦ n < n0 ∧ n0 < n1 → n < n1; (forward) from specification oelem[nat-set]%fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat%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; (ss)% ⊦ m < n → m ≤ n; (s)% ⊦ # shl(zbrar, n, m) = # zbrar; (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ n + 0 = n; (s)