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Proof step 3 from proof inv-io-save (previous step)

adr0 = adr1, istore1 = istore, remove-refsize0(istore0) = istore0, remove-refsize0(gstore0) = gstore0, elems nx ∩ ginval = ∅, dom gstore0 ∪ elems nx = dom gstore ∪ ginval, pstate ≠ unbuffered, ((btree(rp, r, adr1, height, dirty, ri, nodes(istore), as, alladrs, ok) * P))(zh), zh[r].dirty, pstate.index?, btree-sub-clean(r, zh[r].usedsize, zh), inv(istore0), inv(istore), inv(gstore0), inv(gstore \ ginval), refsize-cons(istore, as0 ∪ as), pstate.lnum ∈ istore, r ∈ zh, as ⊆ alladrs, as ⊆ dom nodes(istore), alladrs ⊆ dom nodes(istore), as0 ⊆ dom nodes(istore), istore0 ⊑ istore, ¬ dups(nx), pstate.journal? → nx ≠ [] ∧ pstate.lnum = nx.last ∧ pstate.lnum ∈ gstore ⊦ ⟪btree_io_save#(r, adr, istore0, ginval, gstore, gstore0, nx, maxino, ginval0; adr1, zh, istore, pstate, syn; err)⟫ ( refsize-cons(istore, as0 ∪ (err = ESUCCESS ⊃ as ++ adr1;as)) ∧ inv(istore) ∧ istore0 ⊑ istore ∧ (pstate.journal? → nx ≠ [] ∧ pstate.lnum = nx.last ∧ pstate.lnum ∈ gstore) ∧ (pstate.index? → pstate.lnum ∈ istore) ∧ (pstate = unbuffered → syn) ∧ istore1 ⊑ istore ∧ r ∈ zh ∧ (err = ESUCCESS → ¬ adr1 ∈ as) ∧ dom nodes(istore) = dom nodes(istore1) ∪ (err = ESUCCESS ⊃ ∅ ++ adr1;∅) ∧ (err = ESUCCESS → ¬ adr1 ∈ istore1) ∧ (( btree (rp, r, (err = ESUCCESS ⊃ adr1;adr0), height, err ≠ ESUCCESS, ri, nodes(istore), (err = ESUCCESS ⊃ as ++ adr1;as), (err = ESUCCESS ⊃ (alladrs -- adr0) ++ adr1;alladrs), ok) * P) )(zh) ∧ (err = ESUCCESS ∨ ⌊err⌋ ) ∧ (err = ESUCCESS → pstate.index? ∧ ¬ zh[r].dirty))

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, elems nx ∩ ginval = ∅, dom gstore0 ∪ elems nx = dom gstore ∪ ginval, ((btree(rp, r, adr1, height, dirty, ri, nodes(istore), as, alladrs, ok) * P))(zh), zh[r].dirty, pstate.index?, btree-sub-clean(r, zh[r].usedsize, zh), inv(gstore \ ginval), inv(gstore0), inv(istore), inv(istore0), refsize-cons(istore, as0 ∪ as), r ∈ zh, m ∈ istore, as0 ⊆ dom nodes(istore), alladrs ⊆ dom nodes(istore), as ⊆ dom nodes(istore), as ⊆ alladrs, istore0 ⊑ istore, pstate.journal? → nx ≠ [] ∧ m = nx.last ∧ m ∈ gstore ⊦ ⟪btree_io_save#(r, adr, istore0, ginval, gstore, gstore0, nx, maxino, ginval0; adr1, zh, istore, pstate, syn; err)⟫ ( refsize-cons(istore, as0 ∪ (err = ESUCCESS ⊃ as ++ adr1;as)) ∧ inv(istore) ∧ istore0 ⊑ istore ∧ (pstate.journal? → nx ≠ [] ∧ pstate.lnum = nx.last ∧ pstate.lnum ∈ gstore) ∧ (pstate.index? → pstate.lnum ∈ istore) ∧ (pstate = unbuffered → syn) ∧ istore1 ⊑ istore ∧ r ∈ zh ∧ (err = ESUCCESS → ¬ adr1 ∈ as) ∧ dom nodes(istore) = dom nodes(istore1) ∪ (err = ESUCCESS ⊃ ∅ ++ adr1;∅) ∧ (err = ESUCCESS → ¬ adr1 ∈ istore1) ∧ (( btree (rp, r, (err = ESUCCESS ⊃ adr1;adr0), height, err ≠ ESUCCESS, ri, nodes(istore), (err = ESUCCESS ⊃ as ++ adr1;as), (err = ESUCCESS ⊃ (alladrs -- adr0) ++ adr1;alladrs), ok) * P) )(zh) ∧ (err = ESUCCESS ∨ ⌊err⌋ ) ∧ (err = ESUCCESS → pstate.index? ∧ ¬ zh[r].dirty)), 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