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Proof step 4 from proof btree-clean-alladrs (previous step)

((btree(rp, r, adr, 0, false, ri, iast, ∅, ∅ ++ adr, nstat) * P))(zh) ⊦

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel> name: btree seq: ⊦ (( btree (rp, r0, adr, height, dirty0, ri, iast, as, alladrs, nstat) * P) )(zh0) ↔ (∃ r, zh, zbrar, dirty, nused. (r0 = null ⊃ ¬ dirty ∧ P(zh0);r = r0 ∧ zh = zh0) ∧ (dirty → dirty0) ∧ (adr ∈ as ↔ ¬ dirty) ∧ adr ∈ alladrs ∧ nsize(nused, nstat) ∧ unmodified (adr, mkznode(rp, zbrar, height = 0, dirty, nused), iast) ∧ (( r => mkznode(rp, zbrar, height = 0, dirty, nused) * bnode (r, zbrar, 0, nused, nused, height, dirty, ri, iast, as -- adr, alladrs -- adr) * P) )(zh)) subst: [zh0, P, nstat, alladrs, as, iast, ri, dirty0, height, adr, r0, rp] → [zh, P, nstat, ∅ ++ adr, ∅, iast, ri, false, 0, adr, r, rp] flags: <all paths>

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 ∈ ∅; (s)% ⊦ adr ∈ ∅ ++ adr0 ↔ adr = adr0; (s)% ⊦ ∅ -- adr = ∅; (s)% ⊦ (as ++ adr) -- adr = as -- adr; (s)% ⊦ n + 0 = n; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ m ≤ m; (s)% ⊦ ⌜true⌝ = true; (s)% ⊦ (true ∧ P0) = P0; (s)%and-ass: ⊦ ((P ∧ P3) ∧ P4) = (P ∧ P3 ∧ P4); from specification heap-sep[zheap]%and-com: ⊦ (P ∧ P2) = (P2 ∧ P); from specification heap-sep[zheap]% ⊦ (⌜bv⌝ ∧ P) * P2 = (⌜bv⌝ ∧ P * P2); (s)% ⊦ emp * P0 = P0; (s)%sep-ass: ⊦ (P * P3) * P4 = P * P3 * P4; from specification heap-sep[zheap]%sep-com: ⊦ P * P2 = P2 * P; from specification heap-sep[zheap]% ⊦ ((P ∧ P2))(zh) ↔ P(zh) ∧ P2(zh); (s)% ⊦ ⌜bv⌝ (zh) ↔ bv; (s) Used terms: zh, P, nstat, ∅ ++ adr, ∅, iast, ri, false, 0, adr, r, rp