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Proof step 30 from proof btree-mark-dirty (previous step)

# asar = # asar0, # zbrar = BRANCH_SIZE, zh[r] = mkznode(rp0, zbrar, false, false, # asar0), union(asar0, 0, # asar0) = alladrs -- adr, union(asar, 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(asar0, 0, # asar0), disj(asar, 0, # asar0), nsize(# asar0, nstat), sorted(zbrar, 0, # asar0 - 1), unmodified(adr, mkznode(rp0, zbrar, false, false, # asar0), iast), adr ∈ alladrs, adr ∈ iast, r ∈ zh, # 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)]) ⊦ # asar0 = 0, # alladrs = 0, alladrs = ∅, ((bnode(r, zbrar, 0, # asar0, # asar0, height1 + 1, true, ri, iast, alladrs -- adr, alladrs -- adr) * P))(zh -- r)

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 31:
# 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)

Rule argument

spec: nat name: elim-pred-c seq: n ≠ 0 ⊦ m = n - 1 ↔ n = m + 1 subst: [n, m] → [# asar0, m]

Additional information

Used simplifier rules: n ≠ 0 ⊦ m = n - 1 ↔ n = m + 1; (elim) Used terms: # asar0, m