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Proof step 20 from proof btree-clean-alladrs (previous step)

# asar0 = m + 1, # asar = m + 1, # zbrar = BRANCH_SIZE, alladrs = union(asar0, 0, m + 1) ++ 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, (r = null ⊃ P(zh);r0 = r ∧ zh0 = zh), ∀ n. ¬ m < n → asar[n] ⊆ dom iast, ∀ n. ¬ m < n → asar[n] ⊆ alladrs, ∀ 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, union(asar, 0, m + 1) ++ adr = alladrs, alladrs = ∅, adr ∈ union(asar, 0, m + 1), adr ∈ union(asar0, 0, m + 1)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 21:
# zbrar = BRANCH_SIZE, # asar = m + 1, # asar0 = m + 1, (( ( (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 ∈ 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] ⊆ union(asar0, 0, m + 1) ++ adr, ∀ n. ¬ m < n → asar[n] ⊆ dom iast ⊦ union(asar, 0, m + 1) = union(asar0, 0, m + 1), adr ∈ union(asar0, 0, m + 1), adr ∈ union(asar, 0, m + 1)

Rule argument

Additional information

Used simplifier rules: ⊦ ¬ adr ∈ as → (adr ∈ as ++ adr0 ↔ adr = adr0); (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)% ⊦ ¬ adr ∈ as ∧ ¬ adr ∈ as0 → (as ++ adr = as0 ++ adr ↔ as = as0); (s)% ⊦ ∅ ≠ as ++ adr; (s)