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Proof step 12 from proof btree-repl-dirty-address (previous step)

((r => mkznode(rp, zbrar0, height = 0, true, nused) * bnode(r, zbrar0, 0, nused, nused, height, true, ri, iast, as, alladrs -- adr0) * P))(zh), nsize(nused, nstat), adr0 ∈ alladrs, as ⊆ alladrs, as ⊆ dom iast, height = 0 → as = ∅ ⊦ # alladrs = 0, alladrs = ∅, adr0 ∈ as, adr1 ∈ alladrs, (adr1 ∈ as ↔ ¬ true) ∧ nsize(nused, nstat) ∧ unmodified(adr1, mkznode(rp, zbrar0, height = 0, true, nused), iast) ∧ (( r => mkznode(rp, zbrar0, height = 0, true, nused) * bnode(r, zbrar0, 0, nused, nused, height, true, ri, iast, as -- adr1, alladrs -- adr0 -- adr1) * P) )(zh)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 13:
((r => mkznode(rp, zbrar0, false, true, nused) * bnode(r, zbrar0, 0, nused, nused, height, true, ri, iast, as, alladrs -- adr0) * P))(zh), nsize(nused, nstat), adr0 ∈ alladrs, as ⊆ alladrs, as ⊆ dom iast ⊦ # alladrs = 0, alladrs = ∅, height = 0, ((r => mkznode(rp, zbrar0, false, true, nused) * bnode(r, zbrar0, 0, nused, nused, height, true, ri, iast, as, alladrs -- adr0 -- adr1) * P))(zh), adr1 ∈ as, adr0 ∈ as, adr1 ∈ alladrs

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);% ⊦ true(zh); (s)% ⊦ (true ∧ P0) = P0; (s)% ⊦ ⌜true⌝ = true; (s)% ⊦ ∅ -- adr = ∅; (s)% ⊦ as = ∅ ++ adr → as -- adr = ∅; (s)% ⊦ ⌜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)% ⊦ as -- adr = ∅ ↔ as = ∅ ∨ as = ∅ ++ adr; (ws)% ⊦ 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 ∈ dom iast ↔ adr ∈ iast; (s)%ne-01: ⊦ adr ∈ as → as ≠ ∅; (forward) from specification set-basic[node-store]%sub-in: ⊦ adr ∈ as ∧ as ⊆ as0 → adr ∈ as0; (forward) from specification set-basic[node-store]% ⊦ ¬ adr ∈ as → as -- adr = as; (s)% ⊦ unmodified(adr, mkznode(rp, zbrar, bv, true, nused), iast); (s)