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Proof step 47 from proof btree-istore-reduce-refed (previous step)

# 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 = ∅, (( ( (λ n. ⌜zbrar[n].branch?⌝ ∧ btree(r, zbrar[n].child, zbrar[n].adr, height0, dirty0, key-range(zbrar, n, m + 1, ri), iast1, asar[n], asar0[n], ok) )⊗_ (0, m + 1 ) * r => mkznode(rp, zbrar, false, dirty0, m + 1)) * P) )(zh0), (∀ n0, P. n0 < m + 1 ∧ (( ( ⌜zbrar[0 + n0].branch?⌝ ∧ btree (r, zbrar[0 + n0].child, zbrar[0 + n0].adr, height0, dirty0, key-range(zbrar, 0 + n0, m + 1, ri), iast0, asar[0 + n0], asar0[0 + n0], ok)) * P) )(zh0) → (( ( ⌜zbrar[0 + n0].branch?⌝ ∧ btree (r, zbrar[0 + n0].child, zbrar[0 + n0].adr, height0, dirty0, key-range(zbrar, 0 + n0, m + 1, ri), iast1, asar[0 + n0], asar0[0 + n0], ok)) * P) )(zh0)) ∧ (( (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 ) * P * r => mkznode(rp, zbrar, false, dirty0, m + 1)) )(zh0)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 48:
# asar0 = m + 1, # asar = m + 1, # zbrar = BRANCH_SIZE, adr ∈ as ↔ ¬ dirty0, union(asar0, 0, m + 1) = alladrs -- adr, union(asar, 0, m + 1) = as -- 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), ((btree(r, zbrar[n].child, zbrar[n].adr, height0, dirty0, key-range(zbrar, n, m + 1, ri), iast0, asar[n], asar0[n], ok) * P0))(zh0), zbrar[n].branch?, 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(adr, mkznode(rp, zbrar, false, dirty0, m + 1), iast0), adr ∈ alladrs, n ≤ m, iast0 | as ⊆ iast1, as ⊆ dom iast0, as ⊆ alladrs, asar[n] ⊆ dom iast0, asar[n] ⊆ asar0[n], 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 = ∅, (( ( (λ n. ⌜zbrar[n].branch?⌝ ∧ btree(r, zbrar[n].child, zbrar[n].adr, height0, dirty0, key-range(zbrar, n, m + 1, ri), iast1, asar[n], asar0[n], ok) )⊗_ (0, m + 1 ) * r => mkznode(rp, zbrar, false, dirty0, m + 1)) * P) )(zh0), ((btree(r, zbrar[n].child, zbrar[n].adr, height0, dirty0, key-range(zbrar, n, m + 1, ri), iast1, asar[n], asar0[n], ok) * P0))(zh0)

Rule argument

Additional information

Used simplifier rules: btree-refed-subseteq-alladrs: ⊦ (( btree(rp, r, adr, height, dirty, fi, iast, as, alladrs, nstat) * P) )(zh) → as ⊆ alladrs;%btree-adrs: ⊦ (( btree (rp, rp0, adr0, height, dirty, fi, iast, as, alladrs, nstat) * P) )(zh) → as ⊆ dom iast;%lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic%fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat% ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ (true ∧ P0) = P0; (s)% ⊦ ⌜true⌝ = true; (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]%and-ass: ⊦ ((P ∧ P3) ∧ P4) = (P ∧ P3 ∧ P4); from specification heap-sep[zheap]% ⊦ (⌜bv⌝ ∧ P) * P2 = (⌜bv⌝ ∧ P * P2); (s)% ⊦ n + 0 = n; (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m < n + 1 ↔ ¬ n < m; (s)