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Proof step 3 from proof inv-commit-rec-ind (previous step)

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)))

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 4:
adr0 = adr1, istore1 = istore, pstate = index(m), remove-refsize0(gstore0) = gstore0, remove-refsize0(istore0) = istore0, as ∩ as0 = ∅, elems nx ∩ ginval = ∅, dom gstore0 ∪ elems nx = dom gstore ∪ ginval, ((btree(rp, r0, adr1, height, dirty, ri, nodes(istore) | as, as, alladrs, ok) * P))(zh), zh[r0].dirty, pstate.index?, inv(gstore \ ginval), inv(gstore0), inv(istore), inv(istore0), refsize-cons(istore, as0 ∪ as), r0 ∈ zh, m ∈ istore, as ⊆ dom nodes(istore), as ⊆ alladrs, alladrs ⊆ dom nodes(istore), as0 ⊆ dom nodes(istore), istore0 ⊑ istore, pstate.journal? → nx ≠ [] ∧ m = nx.last ∧ m ∈ 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))), pstate = unbuffered, dups(nx)

Rule argument

spec: persistence-state name: elim-02 seq: ⊦ pstate.index? → (m = pstate.lnum ↔ pstate = index(m)) subst: [pstate, m] → [pstate, m]

Additional information

Used simplifier rules: ⊦ pstate.index? → (m = pstate.lnum ↔ pstate = index(m)); (elim) Used terms: pstate, m