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Proof step 191 from proof inv-commit-rec-ind (previous step)

as1 ++ adr = (alladrs -- adr3) ++ adr, zh1[r0] = mkznode(r1, zbrar, false, true, m0), zh[r0] = mkznode(r2, zbrar0, false, true, n), zh0[r0] = mkznode(r3, zbrar1, boolvar0, false, m2), dom nodes(istore1) = dom nodes(istore) ++ adr, remove-refsize0(istore0) = istore0, remove-refsize0(gstore1) = gstore1, elems nx ∩ ginval1 = ∅, as ∩ as0 = ∅, as1 ∩ as0 = ∅, dom gstore1 ∪ elems nx = dom gstore2 ∪ ginval1, ((btree(rp, r0, adr, height, false, ri, nodes(istore1), as1 ++ adr, as1 ++ adr, ok) * P))(zh0), ((btree(rp, r0, adr3, height, dirty, ri, nodes(istore) | as1, as1, alladrs, ok) * P))(zh), ((btree(rp, r0, adr3, height, dirty, ri, nodes(istore2) | as, as, alladrs, ok) * P))(zh1), ((btree(rp, r0, adr3, height, dirty, ri, nodes(istore), as1, alladrs, ok) * P))(zh), btree-sub-clean(r0, n, zh), inv(istore1), inv(istore0), inv(istore2), inv(gstore1), inv(gstore2 \ ginval1), inv(istore), refsize-cons(istore1, (as1 ∪ as0) ++ adr), refsize-cons(istore2, as0 ∪ as), refsize-cons(istore, as0 ∪ as1), r0 ∈ zh0, m ∈ istore, m1 ∈ istore, m ∈ istore2, r0 ∈ zh1, r0 ∈ zh, m ∈ istore1, m1 ∈ istore1, adr ∈ istore1, m3 ∈ istore1, as1 ⊆ alladrs, as1 ∩ dom nodes(istore2) ⊆ alladrs, as0 ⊆ dom nodes(istore2), alladrs ⊆ dom nodes(istore2), as ⊆ alladrs, as ⊆ dom nodes(istore2), as1 ∩ dom nodes(istore2) ⊆ as, as1 ⊆ dom nodes(istore2), as0 ⊆ dom nodes(istore), alladrs ⊆ dom nodes(istore), as1 ∩ dom nodes(istore2) ⊆ dom nodes(istore), as1 ⊆ dom nodes(istore), istore0 ⊑ istore1, istore ⊑ istore1, istore0 ⊑ istore, istore0 ⊑ istore2, istore2 ⊑ istore, istore2 ⊑ istore1, ∀ height0, zh, r0, r, zbrar, boolvar, m0, adr0, adr1, istore1, istore, pstate, m, istore0, gstore0, nx, ginval, as, as0, gstore, rp, dirty, ri, alladrs, P, adr, ino, ns, err, bv. height0 < height → zh[r0] = mkznode(r, zbrar, boolvar, true, m0) ∧ adr0 = adr1 ∧ istore1 = istore ∧ pstate = index(m) ∧ remove-refsize0(istore0) = istore0 ∧ remove-refsize0(gstore0) = gstore0 ∧ elems nx ∩ ginval = ∅ ∧ as ∩ as0 = ∅ ∧ dom gstore0 ∪ elems nx = dom gstore ∪ ginval ∧ ((btree(rp, r0, adr1, height0, dirty, ri, nodes(istore) | as, as, alladrs, ok) * P))(zh) ∧ inv(istore0) ∧ inv(istore) ∧ inv(gstore0) ∧ inv(gstore \ ginval) ∧ refsize-cons(istore, as0 ∪ as) ∧ m ∈ istore ∧ r0 ∈ zh ∧ as0 ⊆ dom nodes(istore) ∧ alladrs ⊆ dom nodes(istore) ∧ as ⊆ alladrs ∧ as ⊆ dom nodes(istore) ∧ istore0 ⊑ istore ∧ ¬ dups(nx) ∧ ¬ adr1 ∈ as ∧ (height0 = 0 → as = ∅) → ⟪btree_commit_rec#(r0, adr, istore0, ginval, gstore, gstore0, nx, ino, ns; adr1, err, zh, istore, pstate, bv)⟫ ( istore1 ⊑ istore ∧ (pstate.journal? → nx ≠ [] ∧ pstate.lnum = nx.last ∧ pstate.lnum ∈ gstore) ∧ (pstate.index? → pstate.lnum ∈ istore) ∧ (pstate = unbuffered → bv) ∧ inv(istore) ∧ istore0 ⊑ istore ∧ r0 ∈ zh ∧ (err = ESUCCESS → ¬ zh[r0].dirty ∧ pstate.index?) ∧ (∃ as1, alladrs1. refsize-cons(istore, as0 ∪ as1) ∧ as1 ∩ dom nodes(istore1) ⊆ as ∧ as1 ∩ as0 = ∅ ∧ alladrs1 ⊆ alladrs ∪ as1 ∧ ((btree(rp, r0, (err = ESUCCESS ⊃ adr1;adr0), height0, err ≠ ESUCCESS, ri, nodes(istore) | as1, as1, alladrs1, ok) * P))(zh))) ⊦ height = 0, dups(nx), adr ∈ as1, adr3 ∈ as, adr3 ∈ as1, adr ∈ istore, adr ∈ istore2, adr ∈ istore0, alladrs -- adr3 ⊆ (as1 ∪ alladrs) ++ adr

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: set-union, inst: node-store name: sub-del-03 seq: ⊦ as ⊆ as0 → as -- adr ⊆ as0 subst: [as0, adr, as] → [(as1 ∪ alladrs) ++ adr, adr3, alladrs] flags: <all paths>

Additional information

Used simplifier rules: ⊦ as ⊆ (as ∪ as0) ++ adr; (s)%una: ⊦ (as ∪ as0) ∪ as1 = as ∪ as0 ∪ as1; from specification set-union[node-store]%unc: ⊦ as ∪ as0 = as0 ∪ as; from specification set-union[node-store] Used terms: (as1 ∪ alladrs) ++ adr, adr3, alladrs