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Proof step 15 from proof inv-truncate (previous step)

jvalid0 ↔ jvalid2, as = fi1[datakeys(key, n, fi1)], fi = fi2, fi0 = fi1 \ datakeys(key, n, fi1), gstore = gstore3, gstore0 = gstore1, gstore2 = gstore3, remove-refsize0(gstore1) = gstore1, elems nx ∩ jro = ∅, dom gstore1 ∪ elems nx = dom gstore3 ∪ jro, fi2.inos < pmaxino, fi1.inos < jmaxino, p_os < pmaxino, inv(gstore1), inv(gstore3 \ jro), refsize-cons(gstore1, range fi2), refsize-cons(gstore3, range fi1), ri-inj(fi2), ri-inj(fi1), ri-inj(fi1 \ datakeys(key, n, fi1)), pmaxino ≤ jmaxino, fi2.lebs ⊆ dom gstore1, fi1.lebs ⊆ dom gstore3, jvalid2 ∧ nx ≠ [] → valid-journal-head(nx, gstore3, jro) ⊦ ⟪while as ≠ ∅ do {/* (inv-truncate : invariant (jvalid0 ∧ nx ≠ [] → valid-journal-head(nx, gstore, jro)) ∧ dom gstore = dom gstore` ∧ as ∩ range fi0 = ∅ ∧ as.lebs ⊆ dom gstore ∧ refsize-cons(gstore, range fi0 ∪ as) ∧ inv-journal(nx, gstore, gstore0, jro) ∧ gstore =○refsize gstore` wfbound # as) */; choose n, adr with adr ∈ as in {index_get_gblock_refsize#(adr.lnum, fi0, fi, jro, gstore, gstore0, nx, pmaxino, p_os, jvalid; ; n); index_set_gblock_refsize#(adr.lnum, n - adr.size, fi0, fi, jro, gstore0, nx, pmaxino, p_os, jvalid; gstore); as := as -- adr}}⟫ ( (jvalid2 ∧ nx ≠ [] → valid-journal-head(nx, gstore, jro)) ∧ (fi1 \ datakeys(key, n, fi1)).lebs ⊆ dom gstore ∧ refsize-cons(gstore, range(fi1 \ datakeys(key, n, fi1))) ∧ inv(gstore \ jro) ∧ dom gstore1 ∪ elems nx = dom gstore ∪ jro ∧ gstore =○refsize gstore2), pmaxino = 0, dups(nx)

Heuristic annotation applied rule annotation right

And obtained the following new goals

Proof step 16:
jvalid0 ↔ jvalid2, as = as0, as1 = fi1[datakeys(key, n, fi1)], dom gstore = dom gstore4, fi = fi2, fi0 = fi1 \ datakeys(key, n, fi1), gstore0 = gstore1, gstore2 = gstore3, gstore4 = gstore3, remove-refsize0(gstore1) = gstore1, elems nx ∩ jro = ∅, as ∩ range fi0 = ∅, dom gstore1 ∪ elems nx = dom gstore3 ∪ jro, p_os < pmaxino, fi1.inos < jmaxino, fi2.inos < pmaxino, gstore =○refsize gstore4, inv(gstore3 \ jro), inv(gstore1), inv-journal(nx, gstore, gstore0, jro), refsize-cons(gstore3, range fi1), refsize-cons(gstore1, range fi2), refsize-cons(gstore, range fi0 ∪ as), ri-inj(fi1 \ datakeys(key, n, fi1)), ri-inj(fi1), ri-inj(fi2), pmaxino ≤ jmaxino, fi1.lebs ⊆ dom gstore3, fi2.lebs ⊆ dom gstore1, as.lebs ⊆ dom gstore, jvalid2 ∧ nx ≠ [] → valid-journal-head(nx, gstore3, jro), jvalid0 ∧ nx ≠ [] → valid-journal-head(nx, gstore, jro) ⊦ ⟪choose n, adr with adr ∈ as in {index_get_gblock_refsize#(adr.lnum, fi0, fi, jro, gstore, gstore0, nx, pmaxino, p_os, jvalid; ; n); index_set_gblock_refsize#(adr.lnum, n - adr.size, fi0, fi, jro, gstore0, nx, pmaxino, p_os, jvalid; gstore); as := as -- adr}⟫ ( ( (jvalid0 ∧ nx ≠ [] → valid-journal-head(nx, gstore, jro)) ∧ dom gstore = dom gstore4 ∧ as ∩ range fi0 = ∅ ∧ as.lebs ⊆ dom gstore ∧ refsize-cons(gstore, range fi0 ∪ as) ∧ inv-journal(nx, gstore, gstore0, jro) ∧ gstore =○refsize gstore4) ∧ # as < # as0), as = ∅, pmaxino = 0, dups(nx)

Rule argument

rule: invariant right arg: bound: # as inv: (jvalid0 ∧ nx ≠ [] → valid-journal-head(nx, gstore, jro)) ∧ dom gstore = dom gstore` ∧ as ∩ range fi0 = ∅ ∧ as.lebs ⊆ dom gstore ∧ refsize-cons(gstore, range fi0 ∪ as) ∧ inv-journal(nx, gstore, gstore0, jro) ∧ gstore =○refsize gstore` fmapos: right 1 pos: right 1

Additional information

Used simplifier rules: ⊦ as ∩ ∅ = ∅; (s)%int-a: ⊦ (as ∩ as0) ∩ as1 = as ∩ as0 ∩ as1; from specification set-union[node-store]%int-c: ⊦ as ∩ as0 = as0 ∩ as; from specification set-union[node-store]% ⊦ as ∪ ∅ = as; (s)%una: ⊦ (as ∪ as0) ∪ as1 = as ∪ as0 ∪ as1; from specification set-union[node-store]%unc: ⊦ as ∪ as0 = as0 ∪ as; from specification set-union[node-store]%⊦ inv-journal(nx, gstore, gstore0, ginval) ↔ inv(gstore \ ginval) ∧ inv(gstore0) ∧ elems nx ∩ ginval = ∅ ∧ ¬ dups(nx) ∧ dom gstore0 ∪ elems nx = dom gstore ∪ ginval ∧ remove-refsize0(gstore0) = gstore0; (s)%sub-ntrans-01: ⊦ ns ⊆ ns0 ∧ ¬ ns1 ⊆ ns0 → ¬ ns1 ⊆ ns; (forward) from specification set-basic[nat-set]% ⊦ ri.lebs ⊆ ns → (ri \ ks).lebs ⊆ ns; (ss)% ⊦ ri-inj(ri) → ri[ks] ∩ range(ri \ ks) = ∅; (s)% ⊦ ri.lebs ⊆ ns → ri[ks].lebs ⊆ ns; (s)% ⊦ ri[ks] ∪ range(ri \ ks) = range ri; (s)% ⊦ gstore =○refsize gstore; (s) Used formulas: (jvalid0 ∧ nx ≠ [] → valid-journal-head(nx, gstore, jro)) ∧ dom gstore = dom gstore` ∧ as ∩ range fi0 = ∅ ∧ as.lebs ⊆ dom gstore ∧ refsize-cons(gstore, range fi0 ∪ as) ∧ inv-journal(nx, gstore, gstore0, jro) ∧ gstore =○refsize gstore`, # as = # as