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Proof step 12 from proof btree-mark-dirty (previous step)

zh[r] = mkznode(rp0, zbrar, boolvar, false, height0), ((btree(rp, r, adr, height1 +1, false, ri, iast, alladrs, alladrs, nstat) * P))(zh), adr ∈ iast, adr ∈ alladrs, r ∈ zh, alladrs ⊆ dom iast, height1 +1 = 0 → alladrs = ∅ ++ adr, ∀ 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 = ∅, ((btree(rp, r, adr, height1 +1, true, ri, iast, alladrs -- adr, alladrs, nstat) * P))(zh[r, mkznode(rp0, zbrar, boolvar, true, height0)])

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 13:
zh[r] = mkznode(rp0, zbrar, boolvar, false, height0), ((btree(rp, r, adr, height1 + 1, false, ri, iast, alladrs, alladrs, nstat) * P))(zh), adr ∈ iast, adr ∈ alladrs, 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 = ∅, ((btree(rp, r, adr, height1 + 1, true, ri, iast, alladrs -- adr, alladrs, nstat) * P))(zh[r, mkznode(rp0, zbrar, boolvar, true, height0)])

Rule argument

Additional information

Used simplifier rules: btree-clean-adr-in: ⊦ r ∈ zh ∧ zh[r] = mkznode(rp0, zbrar, boolvar, false, height0) ∧ (( btree (rp, r, adr, height, dirty, ri, iast, as, alladrs, nstat) * P) )(zh) → adr ∈ as ∧ (height = 0 → as = ∅ ++ adr);%sep-com: ⊦ P * P2 = P2 * P; from specification heap-sep[zheap]%sep-ass: ⊦ (P * P3) * P4 = P * P3 * P4; from specification heap-sep[zheap]%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m + 1 ≠ 0; (s)% ⊦ m +1 = m + 1; (s)