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Proof step 30 from proof btree-refed-subseteq-alladrs (previous step)

# asar0 = # asar, # asar = (n1 + n) + 1, # zbrar = BRANCH_SIZE, zbrar[n] = mkzbranch(key, adr0, r0), (( ( ( ( btree(r, r0, adr0, height0, true, key-range(zbrar, n, # asar, fi), iast, asar[n], asar0[n], ok) * (λ n0. ⌜zbrar[n0].branch?⌝ ∧ btree(r, zbrar[n0].child, zbrar[n0].adr, height0, true, key-range(zbrar, n0, # asar, fi), iast, asar[n0], asar0[n0], ok) )⊗_ (0, n )) * (λ n0. ⌜zbrar[n0].branch?⌝ ∧ btree(r, zbrar[n0].child, zbrar[n0].adr, height0, true, key-range(zbrar, n0, # asar, fi), iast, asar[n0], asar0[n0], ok) )⊗_ (1 + n, n1 )) * r => mkznode(rp, zbrar, false, true, # asar)) * P) )(zh0), n1 + n < BRANCH_SIZE, n < BRANCH_SIZE, n0 < BRANCH_SIZE, disj(asar0, 0, # asar), disj(asar, 0, # asar), nsize(# asar, nstat), sorted(zbrar, 0, n1 + n), n0 ≤ n1 + n, ∀ rp, r, adr, dirty, fi, iast, as, alladrs, nstat, P, zh. ((btree(rp, r, adr, height0, dirty, fi, iast, as, alladrs, nstat) * P))(zh) → as ⊆ alladrs ⊦ adr ∈ union(asar0, 0, # asar), adr ∈ union(asar, 0, # asar), asar[n] ⊆ union(asar0, 0, # asar), asar[n0] ⊆ union(asar0, 0, # asar)

User applied rule all left

And closed the goal!

Rule argument

pos: left 14 terms: r, r0, adr0, true, key-range(zbrar, n, # asar, fi), iast, asar[n], asar0[n], ok, P * r => mkznode(rp, zbrar, false, true, # asar) * (λ n0. ⌜zbrar[n0].branch?⌝ ∧ btree (r, zbrar[n0].child, zbrar[n0].adr, height0, true, key-range(zbrar, n0, # asar, fi), iast, asar[n0], asar0[n0], ok) )⊗_ (1 + n, n1 ) * (λ n0. ⌜zbrar[n0].branch?⌝ ∧ btree (r, zbrar[n0].child, zbrar[n0].adr, height0, true, key-range(zbrar, n0, # asar, fi), iast, asar[n0], asar0[n0], ok) )⊗_ (0, n ), zh0 abort: F discard: T computed: T

Additional information

Used simplifier rules: btree-imp-dirty: ⊦ (( btree(rp, r, adr, height, dirty, fi, iast, as, alladrs, nstat) * P) )(zh) → (( btree(rp, r, adr, height, true, fi, iast, as, alladrs, nstat) * P) )(zh);%nat-le-eq: ⊦ n = n0 + n1 → (n ≤ n0 ↔ n1 = 0 ∧ n = n0);%sep-ass: ⊦ (P * P3) * P4 = P * P3 * P4; from specification heap-sep[zheap]%sep-com: ⊦ P * P2 = P2 * P; from specification heap-sep[zheap]%sub-ntrans: ⊦ as ⊆ as0 ∧ ¬ as ⊆ as1 → ¬ as0 ⊆ as1; (forward) from specification set-basic[node-store]% ⊦ m ≤ m; (s)% ⊦ m < n ↔ ¬ n ≤ m; (ss)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ m + 1 ≠ 0; (s)%n < m, m ≤ # asar ⊦ asar[n] ⊆ union(asar, 0, m); (s) Used terms: r, r0, adr0, true, key-range(zbrar, n, # asar, fi), iast, asar[n], asar0[n], ok, P * r => mkznode(rp, zbrar, false, true, # asar) * (λ n0. ⌜zbrar[n0].branch?⌝ ∧ btree (r, zbrar[n0].child, zbrar[n0].adr, height0, true, key-range(zbrar, n0, # asar, fi), iast, asar[n0], asar0[n0], ok) )⊗_ (1 + n, n1 ) * (λ n0. ⌜zbrar[n0].branch?⌝ ∧ btree (r, zbrar[n0].child, zbrar[n0].adr, height0, true, key-range(zbrar, n0, # asar, fi), iast, asar[n0], asar0[n0], ok) )⊗_ (0, n ), zh0