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Proof step 5 from proof btree-adrs (previous step)

∃ r, zh1, zbrar, dirty1, nused. (rp0 = null ⊃ ¬ dirty1 ∧ P(zh);r = rp0 ∧ zh1 = zh) ∧ (dirty1 → dirty) ∧ (adr0 ∈ as ↔ ¬ dirty1) ∧ adr0 ∈ alladrs ∧ nsize(nused, nstat) ∧ unmodified(adr0, mkznode(rp, zbrar, 0 = 0, dirty1, nused), iast) ∧ ((r => mkznode(rp, zbrar, 0 = 0, dirty1, nused) * bnode(r, zbrar, 0, nused, nused, 0, dirty1, fi, iast, as -- adr0, alladrs -- adr0) * P))(zh1) ⊦ as ⊆ dom iast

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 6:
# zbrar = BRANCH_SIZE, as -- adr0 = ∅, adr0 ∈ as ↔ ¬ dirty0, ((r => mkznode(rp, zbrar, true, dirty0, nused) * P))(zh0), nsize(nused, nstat), sorted(zbrar, 0, nused), unmodified(adr0, mkznode(rp, zbrar, true, dirty0, nused), iast), nused ≤ BRANCH_SIZE, dirty0 → dirty, (rp0 = null ⊃ adr0 ∈ as ∧ P(zh);r = rp0 ∧ zh0 = zh) ⊦ as ⊆ dom iast

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);% ⊦ adr ∈ ∅ ++ adr0 ↔ adr = adr0; (s)% ⊦ ∅ ≠ as ++ adr; (s)% ⊦ #(∅ ++ adr) = 1; (s)%size-zero-01: ⊦ as ≠ ∅ → # as ≠ 0; (forward) from specification set-basic[node-store]%in-06: ⊦ adr ∈ as ∧ as = ∅ ++ adr0 → adr = adr0; (forward) from specification set-basic[node-store]%one-size: ⊦ as = ∅ ++ adr → # as = 1; (forward) from specification set-basic[node-store]% ⊦ as -- adr = ∅ ↔ as = ∅ ∨ as = ∅ ++ adr; (ws)%ne-01: ⊦ adr ∈ as → as ≠ ∅; (forward) from specification set-basic[node-store]% ⊦ ⌜bv⌝ (zh) ↔ bv; (s)% ⊦ ((P ∧ P2))(zh) ↔ P(zh) ∧ P2(zh); (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]%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)% ⊦ 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)