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Proof step 35 from proof btree-adrs (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, false, key-range(zbrar, n, m + 1, fi), iast, asar[n], asar0[n], ok) )⊗_ (0, m + 1 ) * P) )(zh1), m < BRANCH_SIZE, ((r => mkznode(rp, zbrar, false, false, m + 1)))(zh2), disj(asar0, 0, m + 1), disj(asar, 0, m + 1), nsize(m + 1, nstat), sorted(zbrar, 0, m), unmodified(adr0, mkznode(rp, zbrar, false, false, m + 1), iast), adr0 ∈ alladrs, zh2 ⊥ zh1, (rp0 = null ⊃ P(zh);r = rp0 ∧ zh2 ∪ zh1 = zh), ∀ n. ¬ m < n → asar[n] ⊆ alladrs, ∀ rp, rp0, adr0, dirty, fi, iast, as, alladrs, nstat, P, zh. ((btree(rp, rp0, adr0, height0, dirty, fi, iast, as, alladrs, nstat) * P))(zh) → as ⊆ dom iast ⊦ # alladrs = 0, alladrs = ∅, adr0 ∈ union(asar, 0, m + 1), union(asar, 0, m + 1) ++ adr0 ⊆ dom iast, alladrs ⊆ dom iast

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 36:
⊦ adr ∈ as → (as0 ++ adr ⊆ as ↔ as0 ⊆ as)
Proof step 37:
# zbrar = BRANCH_SIZE, # asar = m + 1, # asar0 = m + 1, union(asar0, 0, m + 1) = alladrs -- adr0, (( (λ n. ⌜zbrar[n].branch?⌝ ∧ btree(r, zbrar[n].child, zbrar[n].adr, height0, false, key-range(zbrar, n, m + 1, fi), iast, asar[n], asar0[n], ok) )⊗_ (0, m + 1 ) * P) )(zh1), m < BRANCH_SIZE, ((r => mkznode(rp, zbrar, false, false, m + 1)))(zh2), disj(asar, 0, m + 1), disj(asar0, 0, m + 1), nsize(m + 1, nstat), sorted(zbrar, 0, m), unmodified(adr0, mkznode(rp, zbrar, false, false, m + 1), iast), adr0 ∈ alladrs, zh2 ⊥ zh1, (rp0 = null ⊃ P(zh);r = rp0 ∧ zh2 ∪ zh1 = zh), ∀ rp, rp0, adr0, dirty, fi, iast, as, alladrs, nstat, P, zh. ((btree(rp, rp0, adr0, height0, dirty, fi, iast, as, alladrs, nstat) * P))(zh) → as ⊆ dom iast, ∀ n. ¬ m < n → asar[n] ⊆ alladrs ⊦ # alladrs = 0, alladrs = ∅, adr0 ∈ union(asar, 0, m + 1), adr0 ∈ dom iast, alladrs ⊆ dom iast, union(asar, 0, m + 1) ++ adr0 ⊆ dom iast
Proof step 40:
# zbrar = BRANCH_SIZE, # asar = m + 1, # asar0 = m + 1, union(asar0, 0, m + 1) = alladrs -- adr0, (( (λ n. ⌜zbrar[n].branch?⌝ ∧ btree(r, zbrar[n].child, zbrar[n].adr, height0, false, key-range(zbrar, n, m + 1, fi), iast, asar[n], asar0[n], ok) )⊗_ (0, m + 1 ) * P) )(zh1), m < BRANCH_SIZE, ((r => mkznode(rp, zbrar, false, false, m + 1)))(zh2), disj(asar, 0, m + 1), disj(asar0, 0, m + 1), nsize(m + 1, nstat), sorted(zbrar, 0, m), unmodified(adr0, mkznode(rp, zbrar, false, false, m + 1), iast), adr0 ∈ alladrs, adr0 ∈ dom iast, zh2 ⊥ zh1, (rp0 = null ⊃ P(zh);r = rp0 ∧ zh2 ∪ zh1 = zh), ∀ rp, rp0, adr0, dirty, fi, iast, as, alladrs, nstat, P, zh. ((btree(rp, rp0, adr0, height0, dirty, fi, iast, as, alladrs, nstat) * P))(zh) → as ⊆ dom iast, ∀ n. ¬ m < n → asar[n] ⊆ alladrs ⊦ # alladrs = 0, alladrs = ∅, adr0 ∈ union(asar, 0, m + 1), alladrs ⊆ dom iast, union(asar, 0, m + 1) ⊆ dom iast

Rule argument

spec: set-basic, inst: node-store name: sub-add seq: ⊦ adr ∈ as → (as0 ++ adr ⊆ as ↔ as0 ⊆ as) subst: [as, adr, as0] → [dom iast, adr0, union(asar, 0, m + 1)] flags: <all paths>

Additional information

Used terms: dom iast, adr0, union(asar, 0, m + 1)