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Proof step 70 from proof refine-lookup (previous step)

remove-refsize0(gstore1) = gstore1, elems nx ∩ jro0 = ∅, dom gstore1 ∪ elems nx = dom gstore2 ∪ jro0, p_os.keys.inodes?, jro1.keys.inodes?, fi1.inos < pmaxino, ri.inos < jmaxino, p_os < pmaxino, jro1 < jmaxino, inv(gstore1), inv(gstore2 \ jro0), log-cons(abs-log(nx, gstore2 ↓), nodes(gstore2 ↓, jro0)), refsize-cons(gstore1, range fi1), refsize-cons(gstore2, range ri), ri-cons(ri, nodes(gstore2 ↓, jro0)), ri-inj(fi1), ri-inj(ri), key ∈ ri, pmaxino ≤ jmaxino, fi1.lebs ⊆ dom gstore1, ri.lebs ⊆ dom gstore2, jvalid1 ∧ nx ≠ [] → valid-journal-head(nx, gstore2, jro0), jvalid4 → jvalid0 ⊦ jmaxino = 0, pmaxino = 0, dups(nx), ri[key] ∈ gstore2 ∧ ¬ ri[key].lnum ∈ jro0

Heuristic elimination applied rule insert elim lemma

And closed the goal!

Rule argument

spec: address-data name: elim seq: ⊦ (m = adr.lnum ∧ m0 = adr.pos) ∧ m1 = adr.size ↔ adr = @(m, m0, m1) subst: [adr, m, m0, m1] → [ri[key], m, m0, m1]

Additional information

Used simplifier rules: ⊦ key ∈ ri ∧ ri[key] = @(m0, m1, m2) ∧ ri-cons(ri, nodes(gstore ↓, ginval)) → (m0 ∈ ginval ↔ false); (as)%lebs-in-ri-in-gstore-explicit: ⊦ key ∈ ri ∧ ri[key] = adr ∧ ri.lebs ⊆ dom gstore → adr.lnum ∈ gstore; (forward) from specification gjournal-base% ⊦ @(m, m0, m1).lnum = m; (s)%⊦ key ∈ ri ∧ ri[key] = adr ∧ ri-cons(ri, nodes(gstore ↓, ginval)) → adr ∈ gstore; (ss)% ⊦ (m = adr.lnum ∧ m0 = adr.pos) ∧ m1 = adr.size ↔ adr = @(m, m0, m1); (elim) Used terms: ri[key], m, m0, m1