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Proof step 41 from proof btree-refed-subseteq-alladrs (previous step)

# asar0 = # asar, # asar = (n0 + n) + 1, # zbrar = BRANCH_SIZE, zbrar[n] = mkzbranch(key, adr0, r1), (( ( ( ( btree(r0, r1, adr0, height0, false, key-range(zbrar, n, # asar, fi), iast, asar[n], asar0[n], ok) * (λ n1. ⌜zbrar[n1].branch?⌝ ∧ btree(r0, zbrar[n1].child, zbrar[n1].adr, height0, false, key-range(zbrar, n1, # asar, fi), iast, asar[n1], asar0[n1], ok) )⊗_ (0, n )) * (λ n1. ⌜zbrar[n1].branch?⌝ ∧ btree(r0, zbrar[n1].child, zbrar[n1].adr, height0, false, key-range(zbrar, n1, # asar, fi), iast, asar[n1], asar0[n1], ok) )⊗_ (1 + n, n0 )) * r0 => mkznode(rp, zbrar, false, false, # asar)) * P) )(zh0), n < BRANCH_SIZE, n0 + n < BRANCH_SIZE, disj(asar0, 0, # asar), disj(asar, 0, # asar), nsize(# asar, nstat), sorted(zbrar, 0, n0 + n), unmodified(adr, mkznode(rp, zbrar, false, false, # asar), iast), (r = null ⊃ P(zh);r0 = r ∧ zh0 = zh), ∀ 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) ++ adr, union(asar, 0, # asar) ++ adr ⊆ union(asar0, 0, # asar), asar[n] ⊆ union(asar0, 0, # asar)

User applied rule all left

And closed the goal!

Rule argument

pos: left 14 terms: r0, r1, adr0, false, key-range(zbrar, n, # asar, fi), iast, asar[n], asar0[n], ok, P * r0 => mkznode(rp, zbrar, false, false, # asar) * (λ n1. ⌜zbrar[n1].branch?⌝ ∧ btree (r0, zbrar[n1].child, zbrar[n1].adr, height0, false, key-range(zbrar, n1, # asar, fi), iast, asar[n1], asar0[n1], ok) )⊗_ (1 + n, n0 ) * (λ n1. ⌜zbrar[n1].branch?⌝ ∧ btree (r0, zbrar[n1].child, zbrar[n1].adr, height0, false, key-range(zbrar, n1, # asar, fi), iast, asar[n1], asar0[n1], ok) )⊗_ (0, n ), zh0 abort: F discard: T computed: T

Additional information

Used simplifier rules: 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) ++ adr; (s) Used terms: r0, r1, adr0, false, key-range(zbrar, n, # asar, fi), iast, asar[n], asar0[n], ok, P * r0 => mkznode(rp, zbrar, false, false, # asar) * (λ n1. ⌜zbrar[n1].branch?⌝ ∧ btree (r0, zbrar[n1].child, zbrar[n1].adr, height0, false, key-range(zbrar, n1, # asar, fi), iast, asar[n1], asar0[n1], ok) )⊗_ (1 + n, n0 ) * (λ n1. ⌜zbrar[n1].branch?⌝ ∧ btree (r0, zbrar[n1].child, zbrar[n1].adr, height0, false, key-range(zbrar, n1, # asar, fi), iast, asar[n1], asar0[n1], ok) )⊗_ (0, n ), zh0