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Proof step 14 from proof btree-ri-unique (previous step)

# zbrar = BRANCH_SIZE, as1 -- adr = ∅, adr ∈ as1 ↔ ¬ dirty0, ((btree(rp, rp0, adr, 0, dirty, ri0, iast, as0, ∅ ++ adr, ok) * P0))(zh), ((rp0 => mkznode(rp, zbrar, true, dirty0, nused) * P1))(zh), nused < BRANCH_SIZE, sorted(zbrar, 0, nused), unmodified(adr, mkznode(rp, zbrar, true, dirty0, nused), iast), as0 ⊆ ∅ ++ adr, as1 ⊆ dom iast, as1 ⊆ ∅ ++ adr, as0 ⊆ dom iast, dirty0 → dirty ⊦ ri0 = payload(zbrar, 0, nused)

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, P0, ok, ∅ ++ adr, as0, iast, ri0, dirty, 0, adr, rp0, 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);%maplet-null: ⊦ ((r => znd * P))(zh) → r ≠ null;% ⊦ adr ∈ ∅ ++ adr0 ↔ adr = adr0; (s)% ⊦ nsize(nused, ok) ↔ nused < BRANCH_SIZE; (s)% ⊦ (as ++ adr) -- adr = as -- adr; (s)% ⊦ ∅ -- 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)%lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic% ⊦ m < n → m ≤ n; (s)%eq-objs: ⊦ ((r => znd * P))(zh) ∧ ((r => znd0 * P2))(zh) → znd = znd0; (forward) from specification maplet[zheap]%⊦ mkznode(r, zbrar, boolvar, boolvar0, m) = mkznode(r0, zbrar0, boolvar1, boolvar2, m0) ↔ r = r0 ∧ zbrar = zbrar0 ∧ (boolvar ↔ boolvar1) ∧ (boolvar0 ↔ boolvar2) ∧ m = m0; (ws) Used terms: zh, P0, ok, ∅ ++ adr, as0, iast, ri0, dirty, 0, adr, rp0, rp