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Proof step 2 from proof inv-commit-rec-ind (previous step)

adr0 = adr1, istore1 = istore, as ∩ as0 = ∅, ((btree(rp, r0, adr0, height, dirty, ri, nodes(istore) | as, as, alladrs, ok) * P))(zh), zh[r0].dirty, pstate.index?, inv-index(istore, istore0), inv-journal(nx, gstore, gstore0, ginval), inv-state(pstate, nx, gstore, istore, bv), refsize-cons(istore, as0 ∪ as), r0 ∈ zh, as0 ⊆ dom nodes(istore), alladrs ⊆ dom nodes(istore) ⊦ ⟪btree_commit_rec#(r0, adr, istore0, ginval, gstore, gstore0, nx, ino, ns; adr1, err, zh, istore, pstate, bv)⟫ ( istore1 ⊑ istore ∧ inv-state(pstate, nx, gstore, istore, bv) ∧ inv-journal(nx, gstore, gstore0, ginval) ∧ inv-index(istore, istore0) ∧ 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), height, err ≠ ESUCCESS, ri, nodes(istore) | as1, as1, alladrs1, ok) * P))(zh)))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 3:
adr0 = adr1, istore1 = istore, remove-refsize0(istore0) = istore0, remove-refsize0(gstore0) = gstore0, elems nx ∩ ginval = ∅, as ∩ as0 = ∅, dom gstore0 ∪ elems nx = dom gstore ∪ ginval, pstate ≠ unbuffered, ((btree(rp, r0, adr1, height, dirty, ri, nodes(istore) | as, as, alladrs, ok) * P))(zh), zh[r0].dirty, pstate.index?, inv(istore0), inv(istore), inv(gstore0), inv(gstore \ ginval), refsize-cons(istore, as0 ∪ as), pstate.lnum ∈ istore, r0 ∈ zh, as0 ⊆ dom nodes(istore), alladrs ⊆ dom nodes(istore), as ⊆ alladrs, as ⊆ dom nodes(istore), istore0 ⊑ istore, ¬ dups(nx), pstate.journal? → nx ≠ [] ∧ pstate.lnum = nx.last ∧ pstate.lnum ∈ gstore ⊦ ⟪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), height, err ≠ ESUCCESS, ri, nodes(istore) | as1, as1, alladrs1, ok) * P))(zh)))

Rule argument

Additional information

Used simplifier rules: sub-trans: ⊦ as ⊆ as0 ∧ as0 ⊆ as1 → as ⊆ as1; (forward) from specification set-basic[node-store]% ⊦ ¬ unbuffered.index?; (s)%int-c: ⊦ as ∩ as0 = as0 ∩ as; from specification set-union[node-store]%int-a: ⊦ (as ∩ as0) ∩ as1 = as ∩ as0 ∩ as1; from specification set-union[node-store]% ⊦ as ⊆ as0 ∩ as ↔ as ⊆ as0; (s)% ⊦ dom(iast | as) = dom iast ∩ as; (s)%btree-adrs: ⊦ (( btree (rp, rp0, adr0, height, dirty, fi, iast, as, alladrs, nstat) * P) )(zh) → as ⊆ dom iast; (forward) from specification btree-inv%sep-com: ⊦ P * P2 = P2 * P; from specification heap-sep[zheap]%sep-ass: ⊦ (P * P3) * P4 = P * P3 * P4; from specification heap-sep[zheap]%btree-refed-subseteq-alladrs: ⊦ (( btree(rp, r, adr, height, dirty, fi, iast, as, alladrs, nstat) * P) )(zh) → as ⊆ alladrs; (forward) from specification btree-inv%⊦ inv-index(istore, istore0) ↔ inv(istore) ∧ inv(istore0) ∧ istore0 ⊑ istore ∧ remove-refsize0(istore0) = istore0; (s)%⊦ inv-journal(nx, gstore, gstore0, ginval) ↔ inv(gstore \ ginval) ∧ inv(gstore0) ∧ elems nx ∩ ginval = ∅ ∧ ¬ dups(nx) ∧ dom gstore0 ∪ elems nx = dom gstore ∪ ginval ∧ remove-refsize0(gstore0) = gstore0; (s)%⊦ inv-state(pstate, nx, gstore, istore, syn) ↔ ( pstate.journal? → nx ≠ [] ∧ pstate.lnum = nx.last ∧ pstate.lnum ∈ gstore) ∧ (pstate.index? → pstate.lnum ∈ istore) ∧ (pstate = unbuffered → syn); (s)