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Proof step 17 from proof btree-clean-alladrs (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 18:
# zbrar = BRANCH_SIZE, # asar = m + 1, # asar0 = m + 1, union(asar0, 0, m + 1) = alladrs -- adr, (( ( (λ n. ⌜zbrar[n].branch?⌝ ∧ btree(r0, zbrar[n].child, zbrar[n].adr, height0, false, key-range(zbrar, n, m + 1, ri), iast, asar[n], asar0[n], ok) )⊗_ (0, m + 1 ) * r0 => mkznode(rp, zbrar, false, false, m + 1)) * P) )(zh0), m < BRANCH_SIZE, disj(asar, 0, m + 1), disj(asar0, 0, m + 1), nsize(m + 1, nstat), sorted(zbrar, 0, m), unmodified(adr, mkznode(rp, zbrar, false, false, m + 1), iast), adr ∈ alladrs, adr ∈ iast, (r = null ⊃ P(zh);r0 = r ∧ zh0 = zh), ∀ rp, r, adr, ri, iast, as, alladrs, nstat, P, zh. ((btree(rp, r, adr, height0, false, ri, iast, as, alladrs, nstat) * P))(zh) → as = alladrs, ∀ n. ¬ m < n → asar[n] ⊆ alladrs, ∀ n. ¬ m < n → asar[n] ⊆ dom iast ⊦ # alladrs = 0, union(asar, 0, m + 1) ++ adr = alladrs, alladrs = ∅, adr ∈ union(asar, 0, m + 1)

Rule argument

Additional information

Used simplifier rules: ⊦ ¬ adr ∈ as → (adr ∈ as ++ adr0 ↔ adr = adr0); (s)% ⊦ adr ∈ dom iast ↔ adr ∈ iast; (s)% ⊦ as ++ adr ⊆ as0 ↔ as ⊆ as0 ∧ adr ∈ as0; (ws)% ⊦ m < n + 1 ↔ ¬ n < m; (s)% ⊦ 0 ≤ m; (s)%⊦ union(asar, n, m) ⊆ as ↔ (∀ n0. n ≤ n0 ∧ n0 < n + m → asar[n0] ⊆ as); (s)% ⊦ adr ∈ as → (as0 ++ adr ⊆ as ↔ as0 ⊆ as); (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m + 1 ≠ 0; (s)% ⊦ ¬ adr ∈ as → #(as ++ adr) = # as + 1; (s)% ⊦ ∅ ≠ as ++ adr; (s)