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Proof step 18 from proof btree-mark-dirty (previous step)

(( r => mkznode(rp0, zbrar, false, false, height0) * bnode(r, zbrar, 0, height0, height0, height1 + 1, false, ri, iast, alladrs -- adr, alladrs -- adr) * P) )(zh), nsize(height0, nstat), unmodified(adr, mkznode(rp0, zbrar, false, false, height0), iast), adr ∈ alladrs, adr ∈ iast, 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 = ∅, ∃ r1, zh1, zbrar0, dirty, nused. (r = null ⊃ ¬ dirty ∧ P(zh[r, mkznode(rp0, zbrar, false, true, height0)]);r1 = r ∧ zh1 = zh[r, mkznode(rp0, zbrar, false, true, height0)]) ∧ (dirty → true) ∧ (adr ∈ alladrs -- adr ↔ ¬ dirty) ∧ adr ∈ alladrs ∧ nsize(nused, nstat) ∧ unmodified(adr, mkznode(rp0, zbrar0, height1 + 1 = 0, dirty, nused), iast) ∧ (( r1 => mkznode(rp0, zbrar0, height1 + 1 = 0, dirty, nused) * bnode(r1, zbrar0, 0, nused, nused, height1 + 1, dirty, ri, iast, alladrs -- adr -- adr, alladrs -- adr) * P) )(zh1)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 19:
(( r => mkznode(rp0, zbrar, false, false, height0) * bnode(r, zbrar, 0, height0, height0, height1 + 1, false, ri, iast, alladrs -- adr, alladrs -- adr) * P) )(zh), nsize(height0, nstat), unmodified(adr, mkznode(rp0, zbrar, false, false, height0), iast), adr ∈ iast, adr ∈ alladrs, 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 = ∅, ∃ zbrar0, nused. nsize(nused, nstat) ∧ (( r => mkznode(rp0, zbrar0, false, true, nused) * bnode(r, zbrar0, 0, nused, nused, height1 + 1, true, ri, iast, alladrs -- adr, alladrs -- adr) * P) )(zh[r, mkznode(rp0, zbrar, false, true, height0)])

Rule argument

Additional information

Used simplifier rules: maplet-null: ⊦ ((r => znd * P))(zh) → r ≠ null;% ⊦ as -- adr -- adr = as -- adr; (s)% ⊦ unmodified(adr, mkznode(rp, zbrar, bv, true, nused), iast); (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m + 1 ≠ 0; (s)% ⊦ adr ∈ as -- adr0 ↔ adr ≠ adr0 ∧ adr ∈ as; (s)%sep-com: ⊦ P * P2 = P2 * P; from specification heap-sep[zheap]%sep-ass: ⊦ (P * P3) * P4 = P * P3 * P4; from specification heap-sep[zheap]