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Proof step 60 from proof inv-truncate (previous step)

m2 + m1 = ((range(fi1 \ datakeys(key, n0, fi1)) | m) ∪ (as0 | m)).size, gstore2[m] = groupblock(gndx1, lax1, m2 + m1), dom gstore2 = dom gstore3, remove-refsize0(gstore1) = gstore1, elems nx ∩ jro = ∅, as0 ∩ range(fi1 \ datakeys(key, n0, fi1)) = ∅, dom gstore1 ∪ elems nx = dom gstore3 ∪ jro, p_os < pmaxino, fi1.inos < jmaxino, fi2.inos < pmaxino, gstore2 =○refsize gstore3, inv(gstore3 \ jro), inv(gstore1), inv(gstore2 \ jro), inv(gstore2[m, groupblock(gndx1, lax1, m2)] \ jro), refsize-cons(gstore3, range fi1), refsize-cons(gstore1, range fi2), ri-inj(fi1), ri-inj(fi1 \ datakeys(key, n0, fi1)), ri-inj(fi2), @(m, m0, m1) ∈ as0, m ∈ gstore2, pmaxino ≤ jmaxino, fi1.lebs ⊆ dom gstore3, fi2.lebs ⊆ dom gstore1, as0.lebs ⊆ dom gstore3, jvalid2 ∧ nx ≠ [] → valid-journal-head(nx, gstore3, jro), jvalid2 ∧ nx ≠ [] → valid-journal-head(nx, gstore2, jro) ⊦ # as0 = 0, as0 = ∅, m2 = ((range(fi1 \ datakeys(key, n0, fi1)) | m) ∪ ((as0 | m) -- @(m, m0, m1))).size, pmaxino = 0, dups(nx)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 61:
⊦ ¬ adr ∈ as → as ∪ (as0 -- adr) = (as ∪ as0) -- adr
Proof step 62:
m2 + m1 = ((range(fi1 \ datakeys(key, n0, fi1)) | m) ∪ (as0 | m)).size, gstore2[m] = groupblock(gndx1, lax1, m2 + m1), dom gstore2 = dom gstore3, remove-refsize0(gstore1) = gstore1, as0 ∩ range(fi1 \ datakeys(key, n0, fi1)) = ∅, elems nx ∩ jro = ∅, dom gstore1 ∪ elems nx = dom gstore3 ∪ jro, fi2.inos < pmaxino, fi1.inos < jmaxino, p_os < pmaxino, gstore2 =○refsize gstore3, inv(gstore2[m, groupblock(gndx1, lax1, m2)] \ jro), inv(gstore2 \ jro), inv(gstore1), inv(gstore3 \ jro), refsize-cons(gstore1, range fi2), refsize-cons(gstore3, range fi1), ri-inj(fi2), ri-inj(fi1 \ datakeys(key, n0, fi1)), ri-inj(fi1), @(m, m0, m1) ∈ range(fi1 \ datakeys(key, n0, fi1)) | m, m ∈ gstore2, @(m, m0, m1) ∈ as0, pmaxino ≤ jmaxino, as0.lebs ⊆ dom gstore3, fi2.lebs ⊆ dom gstore1, fi1.lebs ⊆ dom gstore3, jvalid2 ∧ nx ≠ [] → valid-journal-head(nx, gstore2, jro), jvalid2 ∧ nx ≠ [] → valid-journal-head(nx, gstore3, jro) ⊦ # as0 = 0, as0 = ∅, m2 = ((range(fi1 \ datakeys(key, n0, fi1)) | m) ∪ ((as0 | m) -- @(m, m0, m1))).size, pmaxino = 0, dups(nx)
Proof step 67:
m2 + m1 = ((range(fi1 \ datakeys(key, n0, fi1)) | m) ∪ (as0 | m)).size, gstore2[m] = groupblock(gndx1, lax1, m2 + m1), dom gstore2 = dom gstore3, remove-refsize0(gstore1) = gstore1, as0 ∩ range(fi1 \ datakeys(key, n0, fi1)) = ∅, elems nx ∩ jro = ∅, dom gstore1 ∪ elems nx = dom gstore3 ∪ jro, fi2.inos < pmaxino, fi1.inos < jmaxino, p_os < pmaxino, gstore2 =○refsize gstore3, inv(gstore2[m, groupblock(gndx1, lax1, m2)] \ jro), inv(gstore2 \ jro), inv(gstore1), inv(gstore3 \ jro), refsize-cons(gstore1, range fi2), refsize-cons(gstore3, range fi1), ri-inj(fi2), ri-inj(fi1 \ datakeys(key, n0, fi1)), ri-inj(fi1), m ∈ gstore2, @(m, m0, m1) ∈ as0, pmaxino ≤ jmaxino, as0.lebs ⊆ dom gstore3, fi2.lebs ⊆ dom gstore1, fi1.lebs ⊆ dom gstore3, jvalid2 ∧ nx ≠ [] → valid-journal-head(nx, gstore2, jro), jvalid2 ∧ nx ≠ [] → valid-journal-head(nx, gstore3, jro) ⊦ # as0 = 0, as0 = ∅, m2 = (((range(fi1 \ datakeys(key, n0, fi1)) | m) ∪ (as0 | m)) -- @(m, m0, m1)).size, pmaxino = 0, dups(nx), @(m, m0, m1) ∈ range(fi1 \ datakeys(key, n0, fi1)) | m

Rule argument

spec: set-union, inst: node-store name: undel-03 seq: ⊦ ¬ adr ∈ as → as ∪ (as0 -- adr) = (as ∪ as0) -- adr subst: [adr, as0, as] → [@(m, m0, m1), as0 | m, range(fi1 \ datakeys(key, n0, fi1)) | m] flags: <all paths>

Additional information

Used terms: @(m, m0, m1), as0 | m, range(fi1 \ datakeys(key, n0, fi1)) | m