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Proof step 39 from proof btree-mark-dirty (previous step)

# asar0 = m + 1, # asar = m + 1, # zbrar = BRANCH_SIZE, zh[r] = mkznode(rp0, zbrar, false, false, m + 1), union(asar0, 0, m + 1) = alladrs -- adr, union(asar, 0, m + 1) = alladrs -- adr, (( (λ n. ⌜zbrar[n].branch?⌝ ∧ btree(r, zbrar[n].child, zbrar[n].adr, height1, false, key-range(zbrar, n, m + 1, ri), iast, asar[n], asar0[n], ok) )⊗_ (0, m + 1 ) * P) )(zh -- r), m < BRANCH_SIZE, disj(asar0, 0, m + 1), disj(asar, 0, m + 1), nsize(m + 1, nstat), sorted(zbrar, 0, m), unmodified(adr, mkznode(rp0, zbrar, false, false, m + 1), iast), adr ∈ alladrs, adr ∈ iast, r ∈ zh, alladrs ⊆ dom iast, ∀ zh, r, rp0, zbrar, boolvar, height0, rp, adr, ri, iast, alladrs, nstat, P. zh[r] = mkznode(rp0, zbrar, boolvar, false, height0) ∧ ((btree(rp, r, adr, height1, false, ri, iast, alladrs, alladrs, nstat) * P))(zh) ∧ r ∈ zh ∧ adr ∈ alladrs ∧ adr ∈ iast ∧ alladrs ⊆ dom iast ∧ (height1 = 0 → alladrs = ∅ ++ adr) → # alladrs = 0 ∨ alladrs = ∅ ∨ ((btree(rp, r, adr, height1, true, ri, iast, alladrs -- adr, alladrs, nstat) * P))(zh[r, mkznode(rp0, zbrar, boolvar, true, height0)]) ⊦ # alladrs = 0, alladrs = ∅, (( (λ n. ⌜zbrar[n].branch?⌝ ∧ btree(r, zbrar[n].child, zbrar[n].adr, height1, true, key-range(zbrar, n, m + 1, ri), iast, asar[n], asar0[n], ok) )⊗_ (0, m + 1 ) * P) )(zh -- r)

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel> name: sep-forall-reduce seq: ⊦ (∀ n0, P. n0 < m ∧ ((nP(n + n0) * P))(zh) → ((nP2(n + n0) * P))(zh)) ∧ ((nP ⊗_(n, m) * P0))(zh) → ((nP2 ⊗_(n, m) * P0))(zh) subst: [nP, zh, P0, m, n, nP2] → [λ n. ⌜zbrar[n].branch?⌝ ∧ btree (r, zbrar[n].child, zbrar[n].adr, height1, false, key-range(zbrar, n, m + 1, ri), iast, asar[n], asar0[n], ok), zh -- r, P, m + 1, 0, λ n. ⌜zbrar[n].branch?⌝ ∧ btree (r, zbrar[n].child, zbrar[n].adr, height1, true, key-range(zbrar, n, m + 1, ri), iast, asar[n], asar0[n], ok)] flags: <all paths>

Additional information

Used simplifier rules: btree-imp-dirty: ⊦ (( btree(rp, r, adr, height, dirty, fi, iast, as, alladrs, nstat) * P) )(zh) → (( btree(rp, r, adr, height, true, fi, iast, as, alladrs, nstat) * P) )(zh);% ⊦ m < n + 1 ↔ ¬ n < m; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ n + 0 = n; (s)% ⊦ (⌜bv⌝ ∧ P) * P2 = (⌜bv⌝ ∧ P * P2); (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]%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)% ⊦ ⌜true⌝ = true; (s)% ⊦ (true ∧ P0) = P0; (s) Used terms: λ n. ⌜zbrar[n].branch?⌝ ∧ btree (r, zbrar[n].child, zbrar[n].adr, height1, false, key-range(zbrar, n, m + 1, ri), iast, asar[n], asar0[n], ok), zh -- r, P, m + 1, 0, λ n. ⌜zbrar[n].branch?⌝ ∧ btree (r, zbrar[n].child, zbrar[n].adr, height1, true, key-range(zbrar, n, m + 1, ri), iast, asar[n], asar0[n], ok)