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Proof step 70 from proof btree-write-other (previous step)

# asar0 = m + 1, # asar = m + 1, # zbrar = BRANCH_SIZE, adr0 ∈ as ↔ ¬ dirty0, zbrar[n] = mkzbranch(key, adr1, r1), as = union(asar, 0, m + 1) ++ adr0, union(asar0, 0, m + 1) = alladrs -- adr0, (( ( (λ n. ⌜zbrar[n].branch?⌝ ∧ btree(r0, zbrar[n].child, zbrar[n].adr, height0, dirty0, key-range(zbrar, n, m + 1, ri), iast, asar[n], asar0[n], ok) )⊗_ (0, m + 1 ) * r0 => mkznode(rp, zbrar, false, dirty0, m + 1)) * P) )(zh0), ((btree(r0, r1, adr1, height0, dirty0, key-range(zbrar, n, m + 1, ri), iast, asar[n], asar0[n], ok) * P0))(zh0), m < BRANCH_SIZE, n < BRANCH_SIZE, disj(asar0, 0, m + 1), disj(asar, 0, m + 1), nsize(m + 1, nstat), sorted(zbrar, 0, m), unmodified(adr0, mkznode(rp, zbrar, false, dirty0, m + 1), iast), adr0 ∈ alladrs, adr0 ∈ as, n ≤ m, asar[n] ⊆ dom iast, asar[n] ⊆ asar0[n], dirty0 → dirty, (r = null ⊃ adr0 ∈ as ∧ P(zh);r0 = r ∧ zh0 = zh), ∀ rp, r, adr0, dirty, ri, iast, adr, ind, as, alladrs, nstat, P, zh. ( ((btree(rp, r, adr0, height0, dirty, ri, iast[adr, ind], as, alladrs, nstat) * P))(zh) ↔ ((btree(rp, r, adr0, height0, dirty, ri, iast, as, alladrs, nstat) * P))(zh)) ∨ adr ∈ as ⊦ # alladrs = 0, alladrs = ∅, (( ( (λ n. ⌜zbrar[n].branch?⌝ ∧ btree(r0, zbrar[n].child, zbrar[n].adr, height0, dirty0, key-range(zbrar, n, m + 1, ri), iast[adr, ind], asar[n], asar0[n], ok) )⊗_ (0, m + 1 ) * r0 => mkznode(rp, zbrar, false, dirty0, m + 1)) * P) )(zh0), ((btree(r0, r1, adr1, height0, dirty0, key-range(zbrar, n, m + 1, ri), iast[adr, ind], asar[n], asar0[n], ok) * P0))(zh0), adr ∈ as, adr0 ∈ union(asar, 0, m + 1)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 71:
# zbrar = BRANCH_SIZE, # asar = m + 1, # asar0 = m + 1, zbrar[n] = mkzbranch(key, adr1, r1), union(asar0, 0, m + 1) = alladrs -- adr0, ((btree(r0, r1, adr1, height0, false, key-range(zbrar, n, m + 1, ri), iast, asar[n], asar0[n], ok) * P0))(zh0), (( ( (λ n. ⌜zbrar[n].branch?⌝ ∧ btree(r0, zbrar[n].child, zbrar[n].adr, height0, false, key-range(zbrar, n, m + 1, ri), iast, asar[n], asar0[n], ok) )⊗_ (0, m + 1 ) * r0 => mkznode(rp, zbrar, false, false, m + 1)) * P) )(zh0), n < BRANCH_SIZE, m < BRANCH_SIZE, disj(asar, 0, m + 1), disj(asar0, 0, m + 1), nsize(m + 1, nstat), sorted(zbrar, 0, m), unmodified(adr0, mkznode(rp, zbrar, false, false, m + 1), iast), adr0 ∈ alladrs, n ≤ m, asar[n] ⊆ asar0[n], asar[n] ⊆ dom iast, (r = null ⊃ P(zh);r0 = r ∧ zh0 = zh), ∀ rp, r, adr0, dirty, ri, iast, adr, ind, as, alladrs, nstat, P, zh. ( ((btree(rp, r, adr0, height0, dirty, ri, iast[adr, ind], as, alladrs, nstat) * P))(zh) ↔ ((btree(rp, r, adr0, height0, dirty, ri, iast, as, alladrs, nstat) * P))(zh)) ∨ adr ∈ as ⊦ # alladrs = 0, adr = adr0, alladrs = ∅, ((btree(r0, r1, adr1, height0, false, key-range(zbrar, n, m + 1, ri), iast[adr, ind], asar[n], asar0[n], ok) * P0))(zh0), (( ( (λ n. ⌜zbrar[n].branch?⌝ ∧ btree(r0, zbrar[n].child, zbrar[n].adr, height0, false, key-range(zbrar, n, m + 1, ri), iast[adr, ind], asar[n], asar0[n], ok) )⊗_ (0, m + 1 ) * r0 => mkznode(rp, zbrar, false, false, m + 1)) * P) )(zh0), adr0 ∈ union(asar, 0, m + 1), adr ∈ union(asar, 0, m + 1)

Rule argument

Additional information

Used simplifier rules: ⊦ mkzbranch(key, adr, r) = mkzbranch(key0, adr0, r0) ↔ key = key0 ∧ adr = adr0 ∧ r = r0; (ws)%unmodified-clean-fwd-zbrar: n < nused, nused ≤ BRANCH_SIZE, # zbrar0 = # zbrar1, # zbrar0 = BRANCH_SIZE ⊦ zbrar0[n] = mkzbranch(key, adr0, r2) ∧ unmodified(adr, mkznode(rp0, zbrar0, bv, false, nused), iast) ∧ unmodified(adr, mkznode(rp1, zbrar1, bv, false, nused), iast) → (∃ r0. zbrar1[n] = mkzbranch(key, adr0, r0)); (forward) from specification zbranch-array-ops%succ-01: ⊦ m + 1 ≤ n ↔ m < n; from specification nat%c: ⊦ m + n = n + m; from specification nat-basic2%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic2%le-ls-simp: ⊦ m ≤ n → (n < m ↔ false); from specification nat%s: ⊦ m < n + 1 ↔ ¬ n < m; from specification nat-basic2%add: ⊦ ¬ adr ∈ as ++ adr0 ↔ adr ≠ adr0 ∧ ¬ adr ∈ as; from specification set-basic[node-store]%in-03: ⊦ ¬ adr ∈ as → (adr ∈ as ++ adr0 ↔ adr = adr0); from specification set-basic[node-store]