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Proof step 29 from proof btree-mark-dirty (previous step)

zh[r] = mkznode(rp0, zbrar, false, false, height0), (( ( ⌜0 + height0 ≤ height0 ∧ height0 ≤ # zbrar ∧ # zbrar = BRANCH_SIZE⌝ ∧ ⌜height0 ≠ 0 ∧ sorted(zbrar, 0, height0 - 1)⌝ ∧ _∃ (λ asar. _∃ (λ asar0. ⌜ # asar = height0 ∧ disj(asar, 0, height0) ∧ union(asar, 0, height0) = alladrs -- adr ∧ # asar0 = height0 ∧ disj(asar0, 0, height0) ∧ union(asar0, 0, height0) = alladrs -- adr ⌝ ∧ (λ n. ⌜zbrar[n].branch?⌝ ∧ btree(r, zbrar[n].child, zbrar[n].adr, height1, false, key-range(zbrar, n, height0, ri), iast, asar[n], asar0[n], ok) )⊗_ (0, height0 )))) * P) )(zh -- r), nsize(height0, nstat), unmodified(adr, mkznode(rp0, zbrar, false, false, height0), iast), adr ∈ alladrs, adr ∈ iast, r ∈ zh, 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, height0, height0, 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 30:
# 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)

Rule argument

Additional information

Used simplifier rules: ⊦ ((_∃ Pa0 * P0))(zh) ↔ (∃ asar. ((Pa0(asar) * P0))(zh)); (s)% ⊦ ⌜bv⌝ (zh) ↔ bv; (s)% ⊦ ((P ∧ P2))(zh) ↔ P(zh) ∧ P2(zh); (s)%sep-com: ⊦ P * P2 = P2 * P; from specification heap-sep[zheap]%sep-ass: ⊦ (P * P3) * P4 = P * P3 * P4; from specification heap-sep[zheap]%and-com: ⊦ (P ∧ P2) = (P2 ∧ P); from specification heap-sep[zheap]% ⊦ (⌜bv⌝ ∧ P) * P2 = (⌜bv⌝ ∧ P * P2); (s)%and-ass: ⊦ ((P ∧ P3) ∧ P4) = (P ∧ P3 ∧ P4); from specification heap-sep[zheap]% ⊦ m ≤ m; (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ n + 0 = n; (s)