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Proof step 10 from proof btree-mark-dirty (previous step)

# zbrar = BRANCH_SIZE, ((r => mkznode(rp0, zbrar, true, false, height0) * P))(zh), nsize(height0, nstat), sorted(zbrar, 0, height0), unmodified(adr, mkznode(rp0, zbrar, true, false, height0), iast), adr ∈ iast, height0 ≤ BRANCH_SIZE ⊦ ∃ r1, zh1, zbrar0, dirty, nused. (r = null ⊃ ¬ dirty ∧ P(zh[r, mkznode(rp0, zbrar, true, true, height0)]);r1 = r ∧ zh1 = zh[r, mkznode(rp0, zbrar, true, true, height0)]) ∧ (dirty → true) ∧ (adr ∈ ∅ ↔ ¬ dirty) ∧ adr ∈ ∅ ++ adr ∧ nsize(nused, nstat) ∧ unmodified(adr, mkznode(rp0, zbrar0, 0 = 0, dirty, nused), iast) ∧ (( r1 => mkznode(rp0, zbrar0, 0 = 0, dirty, nused) * bnode(r1, zbrar0, 0, nused, nused, 0, dirty, payload(zbrar, 0, height0), iast, ∅ -- adr, (∅ ++ adr) -- adr) * P) )(zh1)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 11:
# zbrar = BRANCH_SIZE, ((r => mkznode(rp0, zbrar, true, false, height0) * P))(zh), nsize(height0, nstat), sorted(zbrar, 0, height0), unmodified(adr, mkznode(rp0, zbrar, true, false, height0), iast), adr ∈ iast, height0 ≤ BRANCH_SIZE ⊦ ∃ zbrar0, nused. nsize(nused, nstat) ∧ nused ≤ # zbrar0 ∧ # zbrar0 = BRANCH_SIZE ∧ sorted(zbrar0, 0, nused) ∧ payload(zbrar, 0, height0) = payload(zbrar0, 0, nused) ∧ ((P * r => mkznode(rp0, zbrar0, true, true, nused)))(zh[r, mkznode(rp0, zbrar, true, true, height0)])

Rule argument

Additional information

Used simplifier rules: maplet-null: ⊦ ((r => znd * P))(zh) → r ≠ null;%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)% ⊦ 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)% ⊦ (as ++ adr) -- adr = as -- adr; (s)% ⊦ ∅ -- adr = ∅; (s)% ⊦ unmodified(adr, mkznode(rp, zbrar, bv, true, nused), iast); (s)% ⊦ adr ∈ ∅ ++ adr0 ↔ adr = adr0; (s)% ⊦ ¬ adr ∈ ∅; (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]