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Proof step 16 from proof sorted-shift-left (previous step)

n + 1 < # zbrar, n + 1 < nused, n < nused, n < m, m0 + 1 < # zbrar, 2 + m0 < # zbrar, m0 < nused, nused < # zbrar, n < # zbrar, m < nused, m0 < m, m + 1 < # zbrar, m < # zbrar, m0 + 1 < nused, m0 < # zbrar, 2 + n < # zbrar, sorted(zbrar, 0, nused + 1), n ≤ m0 ⊦ zbrar[m0 + 1].key < zbrar[m + 1].key

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 17:
⊦ m0 < m ∧ m1 < m ∧ sorted(zbrar, 0, m) → (zbrar[m0].key < zbrar[m1].key ↔ m0 < m1)

Rule argument

spec: zbranch-array-sorted name: sorted-lt seq: ⊦ m0 < m ∧ m1 < m ∧ sorted(zbrar, 0, m) → (zbrar[m0].key < zbrar[m1].key ↔ m0 < m1) subst: [m, m1, m0, zbrar] → [nused + 1, m + 1, m0 + 1, zbrar] flags: <all paths>

Additional information

Used simplifier rules: ⊦ m + n < m + n0 ↔ n < n0; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ m + n < m0 → m + n < m0 + n0; (s)% ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ m < n → (n ≤ m ↔ false); (s) Used terms: nused + 1, m + 1, m0 + 1, zbrar