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Proof step 31 from proof btree-write-node (previous step)

# asar0 = m + 1, # asar = m + 1, # zbrar = BRANCH_SIZE, union(asar0, 0, m + 1) = alladrs -- adr0, (( ( (λ n. ⌜zbrar[n].branch?⌝ ∧ btree(r, zbrar[n].child, zbrar[n].adr, height0, true, key-range(zbrar, n, m + 1, ri), iast, asar[n], asar0[n], ok) )⊗_ (0, m + 1 ) * r => mkznode(r0, zbrar, false, true, m + 1)) * P) )(zh), m < BRANCH_SIZE, btree-sub-clean(r, m + 1, zh), disj(asar0, 0, m + 1), disj(asar, 0, m + 1), nsize(m + 1, nstat), sorted(zbrar, 0, m), adr0 ∈ alladrs, ∀ n. ¬ m < n → asar[n] ⊆ dom iast, ∀ n. ¬ m < n → asar[n] ⊆ alladrs ⊦ # alladrs = 0, alladrs = ∅, adr1 ∈ iast, adr1 ∈ alladrs, adr0 ∈ union(asar, 0, m + 1), ¬ adr1 ∈ union(asar, 0, m + 1) ∧ (( ( (λ n. ⌜zbrar[n].branch?⌝ ∧ btree (r, zbrar[n].child, zbrar[n].adr, height0, false, key-range(zbrar, n, m + 1, ri), iast[adr1, save(mkznode(r0, zbrar, false, true, m + 1))], asar[n], asar0[n], ok) )⊗_ (0, m + 1 ) * r =>_) * P) )(zh)

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: set-union, inst: node-store name: delete-shift seq: ⊦ adr ∈ as → (as -- adr = as0 ↔ as = as0 ++ adr ∧ ¬ adr ∈ as0) subst: [adr, as, as0] → [adr0, alladrs, union(asar0, 0, m + 1)] flags: <all paths>

Additional information

Used terms: adr0, alladrs, union(asar0, 0, m + 1)