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Proof step 55 from proof btree-istore-reduce-refed (previous step)

# asar0 = m + 1, # asar = m + 1, # zbrar = BRANCH_SIZE, adr ∈ as ↔ ¬ dirty0, zbrar[n] = mkzbranch(key, adr0, r0), union(asar0, 0, m + 1) = alladrs -- adr, union(asar, 0, m + 1) = as -- adr, (( ( (λ n. ⌜zbrar[n].branch?⌝ ∧ btree(r, zbrar[n].child, zbrar[n].adr, height0, dirty0, key-range(zbrar, n, m + 1, ri), iast0, asar[n], asar0[n], ok) )⊗_ (0, m + 1 ) * r => mkznode(rp, zbrar, false, dirty0, m + 1)) * P) )(zh0), ((btree(r, r0, adr0, height0, dirty0, key-range(zbrar, n, m + 1, ri), iast0, asar[n], asar0[n], ok) * P0))(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, dirty0, m + 1), iast0), adr ∈ alladrs, n ≤ m, iast0 | as ⊆ iast1, as ⊆ dom iast0, as ⊆ alladrs, asar[n] ⊆ dom iast0, asar[n] ⊆ asar0[n], dirty0 → dirty, (rp0 = null ⊃ adr ∈ as ∧ P(zh);r = rp0 ∧ zh0 = zh) ⊦ # alladrs = 0, alladrs = ∅, (( ( (λ n. ⌜zbrar[n].branch?⌝ ∧ btree(r, zbrar[n].child, zbrar[n].adr, height0, dirty0, key-range(zbrar, n, m + 1, ri), iast1, asar[n], asar0[n], ok) )⊗_ (0, m + 1 ) * r => mkznode(rp, zbrar, false, dirty0, m + 1)) * P) )(zh0), ((btree(r, r0, adr0, height0, dirty0, key-range(zbrar, n, m + 1, ri), iast1, asar[n], asar0[n], ok) * P0))(zh0), asar[n] ⊆ as, iast0 | asar[n] ⊆ iast1, asar0[n] ⊆ as

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] → [adr, as, union(asar, 0, m + 1)] flags: <all paths>

Additional information

Used simplifier rules: maplet-null: ⊦ ((r => znd * P))(zh) → r ≠ null;% ⊦ ¬ adr ∈ as → (adr ∈ as ++ adr0 ↔ adr = adr0); (s)% ⊦ as ++ adr ⊆ as0 ↔ as ⊆ as0 ∧ adr ∈ as0; (ws)%⊦ union(asar, n, m) ⊆ as ↔ (∀ n0. n ≤ n0 ∧ n0 < n + m → asar[n0] ⊆ as); (s)% ⊦ 0 ≤ m; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ m < n + 1 ↔ ¬ n < m; (s)% ⊦ adr ∈ dom iast ↔ adr ∈ iast; (s)% ⊦ adr ∈ as → (as0 ++ adr ⊆ as ↔ as0 ⊆ as); (s)% ⊦ m ≤ n → (n < m ↔ false); (s)% ⊦ m + n ≤ m + n0 ↔ n ≤ n0; (s)%n < m, m ≤ # asar ⊦ asar[n] ⊆ union(asar, 0, m) ++ adr; (s)% ⊦ ¬ adr ∈ as → as -- adr = as; (s)%sep-ass: ⊦ (P * P3) * P4 = P * P3 * P4; from specification heap-sep[zheap]%sep-com: ⊦ P * P2 = P2 * P; from specification heap-sep[zheap]% ⊦ unmodified(adr, mkznode(rp, zbrar, bv, true, nused), iast); (s)%n < m, m ≤ # asar ⊦ asar[n] ⊆ union(asar, 0, m); (s) Used terms: adr, as, union(asar, 0, m + 1)