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Proof step 69 from proof refine-check-commit (previous step)

remove-refsize0(gstore1) = gstore1, elems nx ∩ jro2 = ∅, dom gstore1 ∪ elems nx = dom gstore2 ∪ jro2, p_os.keys.inodes?, jro1.keys.inodes?, fi1.inos < pmaxino, ri.inos < jmaxino, p_os < pmaxino, jro1 < jmaxino, inv(gstore1), inv(gstore2 \ jro2), log-cons(abs-log(nx, gstore2 ↓), nodes(gstore2 ↓, jro2)), refsize-cons(gstore1, range fi1), refsize-cons(gstore2, range ri), ri-cons(ri, nodes(gstore2 ↓, jro2)), ri-inj(fi1), ri-inj(ri), pmaxino ≤ jmaxino, fi1.lebs ⊆ dom gstore1, ri.lebs ⊆ dom gstore2, jvalid1 ∧ nx ≠ [] → valid-journal-head(nx, gstore2, jro2), jvalid → jvalid0 ⊦ #(dom remove-refsize0(gstore2) ∩ jro2) = 0, jmaxino = 0, pmaxino = 0, dom remove-refsize0(gstore2) ∩ jro2 = ∅, dups(nx)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 70:
⊦ ns = ns0 ↔ (∀ n. n ∈ ns ↔ n ∈ ns0)
Proof step 71:
remove-refsize0(gstore1) = gstore1, elems nx ∩ jro2 = ∅, dom gstore1 ∪ elems nx = dom gstore2 ∪ jro2, jro1.keys.inodes?, p_os.keys.inodes?, jro1 < jmaxino, p_os < pmaxino, ri.inos < jmaxino, fi1.inos < pmaxino, inv(gstore2 \ jro2), inv(gstore1), log-cons(abs-log(nx, gstore2 ↓), nodes(gstore2 ↓, jro2)), refsize-cons(gstore2, range ri), refsize-cons(gstore1, range fi1), ri-cons(ri, nodes(gstore2 ↓, jro2)), ri-inj(ri), ri-inj(fi1), pmaxino ≤ jmaxino, ri.lebs ⊆ dom gstore2, fi1.lebs ⊆ dom gstore1, jvalid → jvalid0, jvalid1 ∧ nx ≠ [] → valid-journal-head(nx, gstore2, jro2) ⊦ #(dom remove-refsize0(gstore2) ∩ jro2) = 0, jmaxino = 0, pmaxino = 0, dups(nx), ∀ n. n ∈ dom remove-refsize0(gstore2) ∩ jro2 ↔ n ∈ ∅

Rule argument

spec: set-basic, inst: nat-set name: Extension seq: ⊦ ns = ns0 ↔ (∀ n. n ∈ ns ↔ n ∈ ns0) subst: [ns0, ns] → [∅, dom remove-refsize0(gstore2) ∩ jro2] flags: <all paths>

Additional information

Used terms: ∅, dom remove-refsize0(gstore2) ∩ jro2