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Proof step 210 from proof inv-commit-rec-ind (previous step)

(err = ESUCCESS ⊃ ∅ ++ adr;∅ ++ adr3) = ∅ ++ (err = ESUCCESS ⊃ adr;adr3), zh[r0] = mkznode(r1, zbrar, true, true, m0), istore[adr3] = istore2[adr3], zh0[r0] = mkznode(r2, zbrar0, boolvar0, boolvar1, m1), dom nodes(istore2) = dom nodes(istore) ∪ (err = ESUCCESS ⊃ ∅ ++ adr;∅), remove-refsize0(gstore1) = gstore1, remove-refsize0(istore0) = istore0, elems nx ∩ ginval1 = ∅, dom gstore1 ∪ elems nx = dom gstore2 ∪ ginval1, (( btree (rp, r0, (err = ESUCCESS ⊃ adr;adr3), 0, err ≠ ESUCCESS, ri, nodes(istore2), (err = ESUCCESS ⊃ ∅ ++ adr;∅), ∅ ++ (err = ESUCCESS ⊃ adr;adr3), ok) * P) )(zh0), ((btree(rp, r0, adr3, 0, dirty, ri, ∅, ∅, ∅ ++ adr3, ok) * P))(zh), ((btree(rp, r0, adr3, 0, dirty, ri, nodes(istore), ∅, ∅ ++ adr3, ok) * P))(zh), btree-sub-clean(r0, m0, zh), inv(istore2), inv(gstore2 \ ginval1), inv(gstore1), inv(istore), inv(istore0), refsize-cons(istore2, as0 ∪ (err = ESUCCESS ⊃ ∅ ++ adr;∅)), refsize-cons(istore, as0), r0 ∈ zh0, m ∈ istore, r0 ∈ zh, adr3 ∈ istore, m ∈ istore2, adr3 ∈ istore2, (err = ESUCCESS ⊃ ∅ ++ adr;∅) ⊆ ∅ ++ (err = ESUCCESS ⊃ adr;adr3), as0 ⊆ dom nodes(istore), (err = ESUCCESS ⊃ ∅ ++ adr;∅) ⊆ dom nodes(istore2), istore0 ⊑ istore2, istore ⊑ istore2, istore0 ⊑ istore, pstate0.journal? → nx ≠ [] ∧ pstate0.lnum = nx.last ∧ pstate0.lnum ∈ gstore2, pstate0.index? → pstate0.lnum ∈ istore2, pstate0 = unbuffered → syn, err = ESUCCESS → ¬ adr ∈ istore, err = ESUCCESS → pstate0.index? ∧ ¬ boolvar1, err = ESUCCESS ∨ ⌊err⌋ , ∀ 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 < 0 → 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))) ⊦ dups(nx), (err = ESUCCESS → ¬ boolvar1 ∧ pstate0.index?) ∧ (∃ as, alladrs. refsize-cons(istore2, as0 ∪ as) ∧ as ∩ dom nodes(istore) = ∅ ∧ alladrs ⊆ as ++ adr3 ∧ ((btree(rp, r0, (err = ESUCCESS ⊃ adr;adr3), 0, err ≠ ESUCCESS, ri, nodes(istore2) | as, as, alladrs, ok) * P))(zh0))

User applied rule case distinction

And obtained the following new goals

Proof step 211:
(err = ESUCCESS ⊃ ∅ ++ adr;∅ ++ adr3) = ∅ ++ (err = ESUCCESS ⊃ adr;adr3), zh0[r0] = mkznode(r2, zbrar0, boolvar0, boolvar1, m1), istore[adr3] = istore2[adr3], zh[r0] = mkznode(r1, zbrar, true, true, m0), dom nodes(istore2) = dom nodes(istore) ∪ (err = ESUCCESS ⊃ ∅ ++ adr;∅), remove-refsize0(istore0) = istore0, remove-refsize0(gstore1) = gstore1, elems nx ∩ ginval1 = ∅, dom gstore1 ∪ elems nx = dom gstore2 ∪ ginval1, ((btree(rp, r0, adr3, 0, dirty, ri, nodes(istore), ∅, ∅ ++ adr3, ok) * P))(zh), ((btree(rp, r0, adr3, 0, dirty, ri, ∅, ∅, ∅ ++ adr3, ok) * P))(zh), (( btree (rp, r0, (err = ESUCCESS ⊃ adr;adr3), 0, err ≠ ESUCCESS, ri, nodes(istore2), (err = ESUCCESS ⊃ ∅ ++ adr;∅), ∅ ++ (err = ESUCCESS ⊃ adr;adr3), ok) * P) )(zh0), btree-sub-clean(r0, m0, zh), inv(istore0), inv(istore), inv(gstore1), inv(gstore2 \ ginval1), inv(istore2), refsize-cons(istore, as0), refsize-cons(istore2, as0 ∪ (err = ESUCCESS ⊃ ∅ ++ adr;∅)), adr3 ∈ istore2, m ∈ istore2, adr3 ∈ istore, r0 ∈ zh, m ∈ istore, r0 ∈ zh0, (err = ESUCCESS ⊃ ∅ ++ adr;∅) ⊆ dom nodes(istore2), as0 ⊆ dom nodes(istore), (err = ESUCCESS ⊃ ∅ ++ adr;∅) ⊆ ∅ ++ (err = ESUCCESS ⊃ adr;adr3), istore0 ⊑ istore, istore ⊑ istore2, istore0 ⊑ istore2, err = ESUCCESS → pstate0.index? ∧ ¬ boolvar1, err = ESUCCESS → ¬ adr ∈ istore, pstate0 = unbuffered → syn, pstate0.index? → pstate0.lnum ∈ istore2, pstate0.journal? → nx ≠ [] ∧ pstate0.lnum = nx.last ∧ pstate0.lnum ∈ gstore2, err = ESUCCESS ∨ ⌊err⌋ , ∀ 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 < 0 → 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))) ⊦ dups(nx), ∃ as, alladrs. refsize-cons(istore2, as0 ∪ as) ∧ as ∩ dom nodes(istore) = ∅ ∧ alladrs ⊆ as ++ adr3 ∧ ((btree(rp, r0, (err = ESUCCESS ⊃ adr;adr3), 0, err ≠ ESUCCESS, ri, nodes(istore2) | as, as, alladrs, ok) * P))(zh0)

Rule argument

pos: right 2

Additional information

Used simplifier rules: ⊦ (as ++ adr) ∪ as0 = (as ∪ as0) ++ adr; (s)% ⊦ as ∪ ∅ = as; (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]% ⊦ ∅ ++ adr ⊆ as ↔ adr ∈ as; (s)% ⊦ adr ∈ ∅ ++ adr0 ↔ adr = adr0; (s)% ⊦ adr ∈ dom iast ↔ adr ∈ iast; (s)% ⊦ adr ∈ nodes(istore) ↔ adr ∈ istore; (s)