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Proof step 45 from proof btree-istore-reduce-refed (previous step)

# 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), m < 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, iast0 | as ⊆ iast1, as ⊆ dom iast0, as ⊆ alladrs, 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)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 46:
⊦ (∀ n0, P. n0 < m ∧ ((nP(n + n0) * P))(zh) → ((nP2(n + n0) * P))(zh)) ∧ ((nP ⊗_(n, m) * P0))(zh) → ((nP2 ⊗_(n, m) * P0))(zh)
Proof step 47:
# 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)

Rule argument

spec: <toplevel> name: sep-forall-reduce seq: ⊦ (∀ n0, P. n0 < m ∧ ((nP(n + n0) * P))(zh) → ((nP2(n + n0) * P))(zh)) ∧ ((nP ⊗_(n, m) * P0))(zh) → ((nP2 ⊗_(n, m) * P0))(zh) subst: [nP, zh, P0, m, n, nP2] → [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), zh0, P * r => mkznode(rp, zbrar, false, dirty0, m + 1), m + 1, 0, 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)] flags: <all paths>

Additional information

Used simplifier rules: sep-ass: ⊦ (P * P3) * P4 = P * P3 * P4; from specification heap-sep[zheap]%sep-com: ⊦ P * P2 = P2 * P; from specification heap-sep[zheap] Used terms: 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), zh0, P * r => mkznode(rp, zbrar, false, dirty0, m + 1), m + 1, 0, 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)