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Proof step 2 from proof refine-deallocate-gnd (previous step)

syn ↔ syn0, fi1 = fi0, ginval = ginval0, ginval1 = ginval2, gstore = gstore0, gstore1 = gstore2, maxino = maxino0, nx = nx0, ri = fi, syn, btree-inv(rp, adr, istore, istore0, zh, ri, fi1), btree-inv(rp, adr, istore, istore0, zh, fi, fi0), inv-index(istore, istore0), inv-journal(nx0, gstore0, gstore2, ginval0), inv-journal(nx, gstore, gstore1, ginval), inv-state(pstate, nx0, gstore0, istore, syn0), ri-inj(fi0), ri-inj(fi), n ∈ gstore ⊦ ⟪btree_deallocate_gnd#(n, adr, istore, istore0, gstore2, nx0, maxino0, ginval2, pstate, syn0; ginval0, gstore0; err)⟫ ⟨index_deallocate_gnd#(n, fi, fi0, gstore1, nx, maxino, ginval1, syn; ginval, gstore; err0)⟩ ( ( ginval = ginval0 ∧ gstore = gstore0 ∧ gstore1 = gstore2 ∧ nx = nx0 ∧ maxino = maxino0 ∧ ginval1 = ginval2 ∧ btree-inv(rp, adr, istore, istore0, zh, fi, fi0) ∧ ri = fi ∧ fi1 = fi0 ∧ (syn ↔ syn0)) ∧ err = err0), n ∈ nx

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 3:
ginval = ginval0, gstore = gstore0, remove-refsize0(gstore2) = gstore2, remove-refsize0(istore0) = istore0, elems nx0 ∩ ginval0 = ∅, dom gstore2 ∪ elems nx0 = dom gstore0 ∪ ginval0, btree-inv(rp, adr, istore, istore0, zh, fi, fi0), inv(gstore0 \ ginval0), inv(gstore2), inv(istore), inv(istore0), ri-inj(fi), ri-inj(fi0), n ∈ gstore0, n ∈ gstore2, istore0 ⊑ istore, ¬ dups(nx0), ¬ n ∈ nx0, pstate.journal? → nx0 ≠ [] ∧ pstate.lnum = nx0.last ∧ pstate.lnum ∈ gstore0, pstate.index? → pstate.lnum ∈ istore ⊦ ⟪btree_deallocate_gnd#(n, adr, istore, istore0, gstore2, nx0, maxino0, ginval2, pstate, true; ginval0, gstore0; err)⟫ ⟨index_deallocate_gnd#(n, fi, fi0, gstore2, nx0, maxino0, ginval2, true; ginval, gstore; err0)⟩ (ginval = ginval0 ∧ gstore = gstore0 ∧ err = err0)

Rule argument

Additional information

Used simplifier rules: fwd-in-gstore0: ⊦ n ∈ gstore ∧ ¬ n ∈ nx ∧ dom gstore0 ∪ elems nx = dom gstore ∪ ginval → n ∈ gstore0; (forward) from specification apersistence-base%unc: ⊦ ns ∪ ns0 = ns0 ∪ ns; from specification set-union[nat-set]%una: ⊦ (ns ∪ ns0) ∪ ns1 = ns ∪ ns0 ∪ ns1; from specification set-union[nat-set]%⊦ inv-index(istore, istore0) ↔ inv(istore) ∧ inv(istore0) ∧ istore0 ⊑ istore ∧ remove-refsize0(istore0) = istore0; (s)%⊦ 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)%⊦ inv-state(pstate, nx, gstore, istore, syn) ↔ ( pstate.journal? → nx ≠ [] ∧ pstate.lnum = nx.last ∧ pstate.lnum ∈ gstore) ∧ (pstate.index? → pstate.lnum ∈ istore) ∧ (pstate = unbuffered → syn); (s)