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Proof step 9 from proof btree-istore-reduce-refed (previous step)

# zbrar = BRANCH_SIZE, as -- adr = ∅, adr ∈ as ↔ ¬ dirty0, ((r => mkznode(rp, zbrar, true, dirty0, nused) * P))(zh0), nsize(nused, nstat), sorted(zbrar, 0, nused), unmodified(adr, mkznode(rp, zbrar, true, dirty0, nused), iast0), nused ≤ BRANCH_SIZE, iast0 | as ⊆ iast1, as ⊆ dom iast0, as ⊆ ∅ ++ adr, dirty0 → dirty, (rp0 = null ⊃ adr ∈ as ∧ P(zh);r = rp0 ∧ zh0 = zh) ⊦ ∃ r, zh1, zbrar0, dirty1, nused0. (rp0 = null ⊃ ¬ dirty1 ∧ P(zh);r = rp0 ∧ zh1 = zh) ∧ (dirty1 → dirty) ∧ (adr ∈ as ↔ ¬ dirty1) ∧ adr ∈ ∅ ++ adr ∧ nsize(nused0, nstat) ∧ unmodified(adr, mkznode(rp, zbrar0, 0 = 0, dirty1, nused0), iast1) ∧ (( r => mkznode(rp, zbrar0, 0 = 0, dirty1, nused0) * bnode(r, zbrar0, 0, nused0, nused0, 0, dirty1, payload(zbrar, 0, nused), iast1, as -- adr, (∅ ++ adr) -- adr) * P) )(zh1)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 10:
# zbrar = BRANCH_SIZE, as -- adr = ∅, adr ∈ as ↔ ¬ dirty0, ((r => mkznode(rp, zbrar, true, dirty0, nused) * P))(zh0), nsize(nused, nstat), sorted(zbrar, 0, nused), unmodified(adr, mkznode(rp, zbrar, true, dirty0, nused), iast0), nused ≤ BRANCH_SIZE, as ⊆ ∅ ++ adr, as ⊆ dom iast0, iast0 | as ⊆ iast1, dirty0 → dirty, (rp0 = null ⊃ adr ∈ as ∧ P(zh);r = rp0 ∧ zh0 = zh) ⊦ ∃ r0, zh1, zbrar0, dirty1, nused0. (rp0 = null ⊃ ¬ dirty1 ∧ P(zh);r0 = rp0 ∧ zh1 = zh) ∧ (dirty1 → dirty) ∧ (adr ∈ as ↔ ¬ dirty1) ∧ nsize(nused0, nstat) ∧ unmodified(adr, mkznode(rp, zbrar0, true, dirty1, nused0), iast1) ∧ nused0 ≤ # zbrar0 ∧ # zbrar0 = BRANCH_SIZE ∧ sorted(zbrar0, 0, nused0) ∧ payload(zbrar, 0, nused) = payload(zbrar0, 0, nused0) ∧ ((P * r0 => mkznode(rp, zbrar0, true, dirty1, nused0)))(zh1)

Rule argument

Additional information

Used simplifier rules: bleaf: ⊦ bnode(rp, zbrar, m, len, nused, 0, dirty, ri, iast, as, alladrs) = ( ⌜m + len ≤ nused ∧ nused ≤ # zbrar ∧ # zbrar = BRANCH_SIZE⌝ ∧ ⌜sorted(zbrar, 0, nused)⌝ ∧ ⌜ri = payload(zbrar, m, len)⌝ ∧ ⌜as = ∅⌝ ∧ ⌜alladrs = ∅⌝ ∧ emp);% ⊦ ⌜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]% ⊦ emp * P0 = P0; (s)% ⊦ (⌜bv⌝ ∧ P) * P2 = (⌜bv⌝ ∧ P * P2); (s)%and-com: ⊦ (P ∧ P2) = (P2 ∧ P); from specification heap-sep[zheap]%and-ass: ⊦ ((P ∧ P3) ∧ P4) = (P ∧ P3 ∧ P4); from specification heap-sep[zheap]% ⊦ (true ∧ P0) = P0; (s)% ⊦ ⌜true⌝ = true; (s)% ⊦ 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)% ⊦ ∅ -- adr = ∅; (s)% ⊦ (as ++ adr) -- adr = as -- adr; (s)% ⊦ adr ∈ ∅ ++ adr0 ↔ adr = adr0; (s)