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Proof step 26 from proof btree-write-node (previous step)

(( r => mkznode(r0, zbrar, false, true, m0) * ( ⌜0 + m0 ≤ m0 ∧ m0 ≤ # zbrar ∧ # zbrar = BRANCH_SIZE⌝ ∧ ⌜m0 ≠ 0 ∧ sorted(zbrar, 0, m0 - 1)⌝ ∧ _∃ (λ asar. _∃ (λ asar0. ⌜ # asar = m0 ∧ disj(asar, 0, m0) ∧ union(asar, 0, m0) = as ∧ # asar0 = m0 ∧ disj(asar0, 0, m0) ∧ union(asar0, 0, m0) = alladrs -- adr0 ⌝ ∧ (λ n. ⌜zbrar[n].branch?⌝ ∧ btree(r, zbrar[n].child, zbrar[n].adr, height0, true, key-range(zbrar, n, m0, ri), iast, asar[n], asar0[n], ok) )⊗_ (0, m0 )))) * P) )(zh), btree-sub-clean(r, m0, zh), nsize(m0, nstat), adr0 ∈ alladrs, as ⊆ dom iast, as ⊆ alladrs ⊦ # alladrs = 0, alladrs = ∅, adr1 ∈ iast, adr1 ∈ alladrs, adr0 ∈ as, m0 ≤ # zbrar ∧ # zbrar = BRANCH_SIZE ∧ m0 ≠ 0 ∧ sorted(zbrar, 0, m0 - 1) ∧ (∃ asar, asar0. # asar = # asar0 ∧ disj(asar, 0, # asar0) ∧ union(asar, 0, # asar0) = as -- adr1 ∧ # asar0 = m0 ∧ disj(asar0, 0, # asar0) ∧ union(asar0, 0, # asar0) = alladrs -- adr0 -- adr1 ∧ (( ( (λ n. ⌜zbrar[n].branch?⌝ ∧ btree (r, zbrar[n].child, zbrar[n].adr, height0, false, key-range(zbrar, n, # asar0, ri), iast[adr1, save(mkznode(r0, zbrar, false, true, # asar0))], asar[n], asar0[n], ok) )⊗_ (0, # asar0 ) * r =>_) * P) )(zh))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 27:
# asar = # asar0, # zbrar = BRANCH_SIZE, union(asar0, 0, # asar0) = alladrs -- adr0, (( ( (λ n. ⌜zbrar[n].branch?⌝ ∧ btree(r, zbrar[n].child, zbrar[n].adr, height0, true, key-range(zbrar, n, # asar0, ri), iast, asar[n], asar0[n], ok) )⊗_ (0, # asar0 ) * r => mkznode(r0, zbrar, false, true, # asar0)) * P) )(zh), btree-sub-clean(r, # asar0, zh), disj(asar0, 0, # asar0), disj(asar, 0, # asar0), nsize(# asar0, nstat), sorted(zbrar, 0, # asar0 - 1), adr0 ∈ alladrs, # asar0 ≤ BRANCH_SIZE, ∀ n. n < # asar0 → asar[n] ⊆ dom iast, ∀ n. n < # asar0 → asar[n] ⊆ alladrs ⊦ # asar0 = 0, # alladrs = 0, alladrs = ∅, adr1 ∈ iast, adr1 ∈ alladrs, adr0 ∈ union(asar, 0, # asar0), ∃ asar2, asar1. # asar2 = # asar1 ∧ disj(asar2, 0, # asar1) ∧ union(asar2, 0, # asar1) = union(asar, 0, # asar0) -- adr1 ∧ # asar1 = # asar0 ∧ disj(asar1, 0, # asar1) ∧ union(asar1, 0, # asar1) = alladrs -- adr0 -- adr1 ∧ (( ( (λ n. ⌜zbrar[n].branch?⌝ ∧ btree (r, zbrar[n].child, zbrar[n].adr, height0, false, key-range(zbrar, n, # asar1, ri), iast[adr1, save(mkznode(r0, zbrar, false, true, # asar1))], asar2[n], asar1[n], ok) )⊗_ (0, # asar1 ) * r =>_) * P) )(zh)

Rule argument

Additional information

Used simplifier rules: ⊦ 0 ≤ m; (s)%⊦ union(asar, n, m) ⊆ as ↔ (∀ n0. n ≤ n0 ∧ n0 < n + m → asar[n0] ⊆ as); (s)% ⊦ ((_∃ 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)