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Proof step 37 from proof btree-clean-alladrs (previous step)

# asar0 = m + 1, # asar = m + 1, # zbrar = BRANCH_SIZE, (( ( (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, n < 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), adr0 ∈ asar0[n], adr ∈ iast, n ≤ m, (r = null ⊃ P(zh);r0 = r ∧ zh0 = zh), ∀ n. ¬ m < n → asar[n] ⊆ dom iast, ∀ n. ¬ m < n → asar[n] ⊆ union(asar0, 0, m + 1) ++ adr, ∀ 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 ⊦ # asar0[n] = 0, # union(asar0, 0, m + 1) = 0, asar0[n] = ∅, union(asar0, 0, m + 1) = ∅, adr ∈ union(asar, 0, m + 1), adr ∈ union(asar0, 0, m + 1), adr0 ∈ union(asar, 0, m + 1)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 38:
⊦ adr ∈ asar[n] ∧ n < m → adr ∈ union(asar, 0, m)
Proof step 39:
# 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), n < BRANCH_SIZE, 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, adr0 ∈ asar0[n], n ≤ m, (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(asar0, 0, m + 1) = 0, # asar0[n] = 0, asar0[n] = ∅, union(asar0, 0, m + 1) = ∅, adr0 ∈ union(asar, 0, m + 1), adr ∈ union(asar0, 0, m + 1), adr ∈ union(asar, 0, m + 1), adr0 ∈ asar[n] ∧ n < m + 1

Rule argument

spec: address-set-array-ops name: in-index-in-union seq: ⊦ adr ∈ asar[n] ∧ n < m → adr ∈ union(asar, 0, m) subst: [n, m, asar, adr] → [n, m + 1, asar, adr0] flags: <all paths>

Additional information

Used terms: n, m + 1, asar, adr0