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Proof step 31 from proof btree-mark-dirty (previous step)

# zbrar = BRANCH_SIZE, # asar = # asar0, # asar0 = m + 1, zh[r] = mkznode(rp0, zbrar, false, false, # asar0), union(asar, 0, # asar0) = alladrs -- adr, union(asar0, 0, # asar0) = alladrs -- adr, (( (λ n. ⌜zbrar[n].branch?⌝ ∧ btree(r, zbrar[n].child, zbrar[n].adr, height1, false, key-range(zbrar, n, # asar0, ri), iast, asar[n], asar0[n], ok) )⊗_ (0, # asar0 ) * P) )(zh -- r), disj(asar, 0, # asar0), disj(asar0, 0, # asar0), nsize(# asar0, nstat), sorted(zbrar, 0, m), unmodified(adr, mkznode(rp0, zbrar, false, false, # asar0), iast), r ∈ zh, adr ∈ iast, adr ∈ alladrs, # asar0 ≤ BRANCH_SIZE, alladrs ⊆ dom iast, ∀ zh, r, rp0, zbrar, boolvar, height0, rp, adr, ri, iast, alladrs, nstat, P. zh[r] = mkznode(rp0, zbrar, boolvar, false, height0) ∧ ((btree(rp, r, adr, height1, false, ri, iast, alladrs, alladrs, nstat) * P))(zh) ∧ r ∈ zh ∧ adr ∈ alladrs ∧ adr ∈ iast ∧ alladrs ⊆ dom iast ∧ (height1 = 0 → alladrs = ∅ ++ adr) → # alladrs = 0 ∨ alladrs = ∅ ∨ ((btree(rp, r, adr, height1, true, ri, iast, alladrs -- adr, alladrs, nstat) * P))(zh[r, mkznode(rp0, zbrar, boolvar, true, height0)]) ⊦ # alladrs = 0, # asar0 = 0, alladrs = ∅, ((bnode(r, zbrar, 0, # asar0, # asar0, height1 + 1, true, ri, iast, alladrs -- adr, alladrs -- adr) * P))(zh -- r)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 32:
# zbrar = BRANCH_SIZE, # asar = m + 1, # asar0 = m + 1, zh[r] = mkznode(rp0, zbrar, false, false, m + 1), union(asar, 0, m + 1) = alladrs -- adr, union(asar0, 0, m + 1) = alladrs -- adr, (( (λ n. ⌜zbrar[n].branch?⌝ ∧ btree(r, zbrar[n].child, zbrar[n].adr, height1, false, key-range(zbrar, n, m + 1, ri), iast, asar[n], asar0[n], ok) )⊗_ (0, m + 1 ) * P) )(zh -- r), m < BRANCH_SIZE, disj(asar, 0, m + 1), disj(asar0, 0, m + 1), nsize(m + 1, nstat), sorted(zbrar, 0, m), unmodified(adr, mkznode(rp0, zbrar, false, false, m + 1), iast), r ∈ zh, adr ∈ iast, adr ∈ alladrs, alladrs ⊆ dom iast, ∀ zh, r, rp0, zbrar, boolvar, height0, rp, adr, ri, iast, alladrs, nstat, P. zh[r] = mkznode(rp0, zbrar, boolvar, false, height0) ∧ ((btree(rp, r, adr, height1, false, ri, iast, alladrs, alladrs, nstat) * P))(zh) ∧ r ∈ zh ∧ adr ∈ alladrs ∧ adr ∈ iast ∧ alladrs ⊆ dom iast ∧ (height1 = 0 → alladrs = ∅ ++ adr) → # alladrs = 0 ∨ alladrs = ∅ ∨ ((btree(rp, r, adr, height1, true, ri, iast, alladrs -- adr, alladrs, nstat) * P))(zh[r, mkznode(rp0, zbrar, boolvar, true, height0)]) ⊦ # alladrs = 0, alladrs = ∅, ((bnode(r, zbrar, 0, m + 1, m + 1, height1 + 1, true, ri, iast, alladrs -- adr, alladrs -- adr) * P))(zh -- r)

Rule argument

Additional information

Used simplifier rules: ⊦ BRANCH_SIZE ≠ 0; (s)%lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic%eqle-one: ⊦ m = n + 1 ∧ n < n0 → m ≤ n0; (forward) from specification nat% ⊦ m + 1 ≤ n ↔ m < n; (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m + 1 ≠ 0; (s)