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Proof step 37 from proof btree-istore-reduce-refed (previous step)

# zbrar = BRANCH_SIZE, # asar = # asar0, # asar0 = m + 1, adr ∈ as ↔ ¬ dirty0, union(asar, 0, # asar0) = as -- adr, union(asar0, 0, # asar0) = alladrs -- adr, (( ( (λ n. ⌜zbrar[n].branch?⌝ ∧ btree(r, zbrar[n].child, zbrar[n].adr, height0, dirty0, key-range(zbrar, n, # asar0, ri), iast0, asar[n], asar0[n], ok) )⊗_ (0, # asar0 ) * r => mkznode(rp, zbrar, false, dirty0, # asar0)) * P) )(zh0), disj(asar, 0, # asar0), disj(asar0, 0, # asar0), nsize(# asar0, nstat), sorted(zbrar, 0, m), unmodified(adr, mkznode(rp, zbrar, false, dirty0, # asar0), iast0), adr ∈ alladrs, # asar0 ≤ BRANCH_SIZE, as ⊆ alladrs, as ⊆ dom iast0, iast0 | as ⊆ iast1, dirty0 → dirty, (rp0 = null ⊃ adr ∈ as ∧ P(zh);r = rp0 ∧ zh0 = zh), ∀ rp, rp0, adr, dirty, ri, iast0, as, alladrs, nstat, P, zh, iast1. ((btree(rp, rp0, adr, height0, dirty, ri, iast0, as, alladrs, nstat) * P))(zh) ∧ iast0 | as ⊆ iast1 ∧ as ⊆ dom iast0 ∧ as ⊆ alladrs → ((btree(rp, rp0, adr, height0, dirty, ri, iast1, as, alladrs, nstat) * P))(zh) ⊦ # alladrs = 0, # asar0 = 0, alladrs = ∅, (( r => mkznode(rp, zbrar, false, dirty0, # asar0) * bnode(r, zbrar, 0, # asar0, # asar0, height0 + 1, dirty0, ri, iast1, as -- adr, alladrs -- adr) * P) )(zh0)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 38:
# zbrar = BRANCH_SIZE, # asar = m + 1, # asar0 = m + 1, adr ∈ as ↔ ¬ dirty0, union(asar, 0, m + 1) = as -- adr, union(asar0, 0, m + 1) = alladrs -- adr, (( ( (λ n. ⌜zbrar[n].branch?⌝ ∧ btree(r, zbrar[n].child, zbrar[n].adr, height0, dirty0, key-range(zbrar, n, m + 1, ri), iast0, asar[n], asar0[n], ok) )⊗_ (0, m + 1 ) * r => mkznode(rp, zbrar, false, dirty0, m + 1)) * P) )(zh0), m < BRANCH_SIZE, disj(asar, 0, m + 1), disj(asar0, 0, m + 1), nsize(m + 1, nstat), sorted(zbrar, 0, m), unmodified(adr, mkznode(rp, zbrar, false, dirty0, m + 1), iast0), adr ∈ alladrs, as ⊆ alladrs, as ⊆ dom iast0, iast0 | as ⊆ iast1, dirty0 → dirty, (rp0 = null ⊃ adr ∈ as ∧ P(zh);r = rp0 ∧ zh0 = zh), ∀ rp, rp0, adr, dirty, ri, iast0, as, alladrs, nstat, P, zh, iast1. ((btree(rp, rp0, adr, height0, dirty, ri, iast0, as, alladrs, nstat) * P))(zh) ∧ iast0 | as ⊆ iast1 ∧ as ⊆ dom iast0 ∧ as ⊆ alladrs → ((btree(rp, rp0, adr, height0, dirty, ri, iast1, as, alladrs, nstat) * P))(zh) ⊦ # alladrs = 0, alladrs = ∅, ((r => mkznode(rp, zbrar, false, dirty0, m + 1) * bnode(r, zbrar, 0, m + 1, m + 1, height0 + 1, dirty0, ri, iast1, as -- adr, alladrs -- adr) * P) )(zh0)

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)