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Proof step 22 from proof btree-refed-subseteq-alladrs (previous step)

# asar0 = m + 1, # asar = m + 1, # zbrar = BRANCH_SIZE, adr ∈ as ↔ ¬ dirty0, union(asar, 0, m + 1) = as -- adr, (( ( (λ n. ⌜zbrar[n].branch?⌝ ∧ btree(r0, zbrar[n].child, zbrar[n].adr, height0, dirty0, key-range(zbrar, n, m + 1, fi), iast, asar[n], asar0[n], ok) )⊗_ (0, m + 1 ) * r0 => mkznode(rp, zbrar, false, dirty0, m + 1)) * P) )(zh0), m < BRANCH_SIZE, disj(asar0, 0, m + 1), disj(asar, 0, m + 1), nsize(m + 1, nstat), sorted(zbrar, 0, m), unmodified(adr, mkznode(rp, zbrar, false, dirty0, m + 1), iast), dirty0 → dirty, (r = null ⊃ adr ∈ as ∧ P(zh);r0 = r ∧ zh0 = zh), ∀ rp, r, adr, dirty, fi, iast, as, alladrs, nstat, P, zh. ((btree(rp, r, adr, height0, dirty, fi, iast, as, alladrs, nstat) * P))(zh) → as ⊆ alladrs ⊦ adr ∈ union(asar0, 0, m + 1), adr ∈ as, as ⊆ union(asar0, 0, m + 1) ++ adr, as ⊆ union(asar0, 0, m + 1)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 23:
# asar0 = m + 1, # asar = m + 1, # zbrar = BRANCH_SIZE, (( ( (λ n. ⌜zbrar[n].branch?⌝ ∧ btree(r, zbrar[n].child, zbrar[n].adr, height0, true, key-range(zbrar, n, m + 1, fi), iast, asar[n], asar0[n], ok) )⊗_ (0, m + 1 ) * r => mkznode(rp, zbrar, false, true, m + 1)) * P) )(zh0), m < BRANCH_SIZE, n < BRANCH_SIZE, n0 < BRANCH_SIZE, disj(asar0, 0, m + 1), disj(asar, 0, m + 1), nsize(m + 1, nstat), sorted(zbrar, 0, m), n ≤ m, n0 ≤ m, ∀ rp, r, adr, dirty, fi, iast, as, alladrs, nstat, P, zh. ((btree(rp, r, adr, height0, dirty, fi, iast, as, alladrs, nstat) * P))(zh) → as ⊆ alladrs ⊦ adr ∈ union(asar0, 0, m + 1), adr ∈ union(asar, 0, m + 1), asar[n] ⊆ union(asar0, 0, m + 1), asar[n0] ⊆ union(asar0, 0, m + 1)

Rule argument

Additional information

Used simplifier rules: maplet-null: ⊦ ((r => znd * P))(zh) → r ≠ null;%lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic%fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat% ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ m < n + 1 ↔ ¬ n < m; (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ 0 ≤ m; (s)%⊦ union(asar, n, m) ⊆ as ↔ (∀ n0. n ≤ n0 ∧ n0 < n + m → asar[n0] ⊆ as); (s)% ⊦ unmodified(adr, mkznode(rp, zbrar, bv, true, nused), iast); (s)%sep-com: ⊦ P * P2 = P2 * P; from specification heap-sep[zheap]%sep-ass: ⊦ (P * P3) * P4 = P * P3 * P4; from specification heap-sep[zheap]% ⊦ ¬ adr ∈ as → (as ⊆ as0 ++ adr ↔ as ⊆ as0); (s)% ⊦ ¬ adr ∈ as → as -- adr = as; (s)