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Proof step 113 from proof payload-shift-left (previous step)

nused < # zbrar, n < # zbrar, sorted(zbrar, 0, nused + 1), n ≤ nused, ∃ n0. n0 < nused ∧ (n0 < n ⊃ zbrar[n0];zbrar[n0 + 1]).entry? ∧ (n0 < n ⊃ zbrar[n0];zbrar[n0 + 1]).key = key ⊦ zbrar[n].entry?, ∃ n0. ¬ nused < n0 ∧ zbrar[n0].entry? ∧ zbrar[n0].key = key

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 114:
(n0 < n ⊃ zbrar[n0];zbrar[n0 + 1]).entry?, n0 < # zbrar, nused < # zbrar, n < # zbrar, n0 < nused, n0 + 1 < # zbrar, sorted(zbrar, 0, nused + 1), n ≤ nused ⊦ zbrar[n].entry?, ∃ n1. ¬ nused < n1 ∧ zbrar[n1].entry? ∧ zbrar[n1].key = (n0 < n ⊃ zbrar[n0];zbrar[n0 + 1]).key

Rule argument

Additional information

Used simplifier rules: ⊦ m +1 = m + 1; (s)% ⊦ m < n → n ≠ 0; (s)%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