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Proof step 6 from proof unmodified-sorted-node (previous step)

BRANCH_SIZE = # zbrar, iast[adr] = indexnode(brar, bv1, nused + 1), indexnode(brar, bv1, nused + 1) = save(mkznode(rp, zbrar, bv1, false, nused + 1)), nused < BRANCH_SIZE, adr ∈ iast ⊦ (∀ m, m0. m < m0 ∧ m0 < nused → load(brar[m]).key < load(brar[m0]).key) ↔ sorted(zbrar, 0, nused)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 7:
n + m ≤ # zbrar ⊦ sorted(zbrar, n, m) ↔ (∀ m0, m1. m0 < m1 ∧ m1 < m → zbrar[n + m0].key < zbrar[n + m1].key)
Proof step 8:
BRANCH_SIZE = # zbrar, iast[adr] = indexnode(brar, bv1, nused + 1), indexnode(brar, bv1, nused + 1) = save(mkznode(rp, zbrar, bv1, false, nused + 1)), nused < BRANCH_SIZE, adr ∈ iast, 0 + nused ≤ # zbrar ⊦ (∀ m, m0. m < m0 ∧ m0 < nused → load(brar[m]).key < load(brar[m0]).key) ↔ (∀ m0, m1. m0 < m1 ∧ m1 < nused → zbrar[0 + m0].key < zbrar[0 + m1].key)

Rule argument

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

Additional information

Used simplifier rules: ⊦ n + 0 = n; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ m ≤ n ↔ ¬ n < m; (ss)%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic%lf: ⊦ m < n ∧ n < n0 → m +1 < n0; (forward) from specification nat-basic%lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic%transitivity: ⊦ n < n0 ∧ n0 < n1 → n < n1; (forward) from specification oelem[nat-set]% ⊦ m < n → n ≠ 0; (s)% ⊦ m +1 = m + 1; (s)% ⊦ ¬ m + n < m; (s) Used terms: nused, 0, zbrar