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Proof step 39 from proof btree-refed-subseteq-alladrs (previous step)

# zbrar = BRANCH_SIZE, # asar = (n0 + n) + 1, # asar0 = # asar, (( ( ( ( btree(r0, zbrar[n].child, zbrar[n].adr, 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), zbrar[n].branch?, n0 + n < BRANCH_SIZE, n < BRANCH_SIZE, disj(asar, 0, # asar), disj(asar0, 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(asar, 0, # asar), adr ∈ union(asar0, 0, # asar), asar[n] ⊆ union(asar0, 0, # asar), union(asar, 0, # asar) ++ adr ⊆ union(asar0, 0, # asar), asar[n] ⊆ union(asar0, 0, # asar) ++ adr

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 40:
# 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), zbrar[n].branch?, 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)

Rule argument

spec: zbranch name: elim-01 seq: ⊦ zbr.branch? → ( (key = zbr.key ∧ adr = zbr.adr) ∧ r = zbr.child ↔ zbr = mkzbranch(key, adr, r)) subst: [zbr, key, adr, r] → [zbrar[n], key, adr0, r1]

Additional information

Used simplifier rules: ⊦ zbr.branch? → ( (key = zbr.key ∧ adr = zbr.adr) ∧ r = zbr.child ↔ zbr = mkzbranch(key, adr, r)); (elim) Used terms: zbrar[n], key, adr0, r1