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Proof step 249 from proof refine-gc (previous step)

# ax0 = # referenced-adrs(lax.to-adr(n1), ri), # lax = # gndx0, gstore2[n1] = groupblock(gndx0, lax, m0), abs-log(nx0, gstore3 ↓) = abs-log(nx1, gstore2 ↓) + ax0, nodes(gstore3 ↓, jro) = nodes(gstore2 ↓, jro0)[ax0, nodes(gstore2 ↓, jro0)[referenced-adrs(lax.to-adr(n1), ri)]], remove-refsize0(gstore1) = gstore1, elems nx0 ∩ jro = ∅, (nx0.toSet \ nx1.toSet) ∩ dom gstore2 = ∅, elems nx1 ∩ jro0 = ∅, dom gstore1 ∪ elems nx0 = dom gstore3 ∪ jro, dom gstore1 ∪ elems nx1 = dom gstore2 ∪ jro0, p_os.keys.inodes?, jro1.keys.inodes?, jro1 < jmaxino0, ri[nodes(gstore2 ↓, jro0)[referenced-adrs(lax.to-adr(n1), ri)].keys, ax0].inos < jmaxino0, jro1 < jmaxino, p_os < pmaxino, ri.inos < jmaxino, fi1.inos < pmaxino, inv(gstore3 \ jro), inv(groupblock(gndx0, lax, m0)), inv(gstore1), inv(gstore2 \ jro0), log-cons(abs-log(nx1, gstore2 ↓), nodes(gstore2 ↓, jro0)), new(ax0, nodes(gstore2 ↓, jro0)), refsize-cons(gstore3, range ri[nodes(gstore2 ↓, jro0)[referenced-adrs(lax.to-adr(n1), ri)].keys, ax0]), refsize-cons(gstore2, range ri), refsize-cons(gstore1, range fi1), ri-cons(ri, nodes(gstore2 ↓, jro0)), ri-inj(ri[nodes(gstore2 ↓, jro0)[referenced-adrs(lax.to-adr(n1), ri)].keys, ax0]), ri-inj(ri), ri-inj(fi1), n1 ∈ gstore1, n1 ∈ gstore2, n1 ∈ gstore3, pmaxino ≤ jmaxino0, jmaxino ≤ jmaxino0, pmaxino ≤ jmaxino, ri.lebs ⊆ dom gstore3, ri[nodes(gstore2 ↓, jro0)[referenced-adrs(lax.to-adr(n1), ri)].keys, ax0].lebs ⊆ dom gstore3, dom gstore2 ⊆ dom gstore3, ri.lebs ⊆ dom gstore2, fi1.lebs ⊆ dom gstore1, range ri ⊆ dom nodes(gstore2 ↓, jro0), ⌊err0⌋ , jvalid5 → jvalid4, jvalid3 ∧ nx1 ≠ [] → valid-journal-head(nx1, gstore2, jro0), jvalid0 ∧ nx0 ≠ [] → valid-journal-head(nx0, gstore3, jro) ⊦ #(ax0.toSet ∩ abs-log(nx1, gstore2 ↓).toSet) = 0, jmaxino = 0, jmaxino0 = 0, m0 = 0, pmaxino = 0, ax0.toSet ∩ abs-log(nx1, gstore2 ↓).toSet = ∅, dups(nodes(gstore2 ↓, jro0)[referenced-adrs(lax.to-adr(n1), ri)].keys), dups(nx1), dups(nx0), dups(ax0), n1 ∈ nx1, n1 ∈ jro0, n1 ∈ nx0, elems nx1 ⊆ dom gstore2 ↓

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: set-basic, inst: nat-set name: Subset seq: ⊦ ns ⊆ ns0 ↔ (∀ n. n ∈ ns → n ∈ ns0) subst: [ns0, ns] → [dom gstore2 ↓, elems nx1] flags: <all paths>

Additional information

Used simplifier rules: ⊦ n ∈ elems nx ↔ n ∈ nx; (s)% ⊦ n ∈ dom gstore ↔ n ∈ gstore; (s)% ⊦ n ∈ gstore ↓ ↔ n ∈ gstore; (s)%int-a: ⊦ (ns ∩ ns0) ∩ ns1 = ns ∩ ns0 ∩ ns1; from specification set-union[nat-set]%int-c: ⊦ ns ∩ ns0 = ns0 ∩ ns; from specification set-union[nat-set]%elems-in-notin-ginval: ⊦ n ∈ nx ∧ elems nx ∩ ns = ∅ → ¬ n ∈ ns; (forward) from specification apersistence-base%una: ⊦ (ns ∪ ns0) ∪ ns1 = ns ∪ ns0 ∪ ns1; from specification set-union[nat-set]%unc: ⊦ ns ∪ ns0 = ns0 ∪ ns; from specification set-union[nat-set]%fwd-in-log-in-gstore: ⊦ n ∈ nx ∧ ginval ∩ elems nx = ∅ ∧ dom gstore0 ∪ elems nx = dom gstore ∪ ginval → n ∈ gstore; (forward) from specification apersistence-base Used terms: dom gstore2 ↓, elems nx1