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Proof step 144 from proof refine-commit (previous step)

# lpt1 = # avol0, # lpt = # avol0, # avol = # avol0, # lpt2 = # avol0, # lpt0 = # avol0, lpf = LP_INDEX_NODES ↔ n ∈ istore2, lpf = LP_GROUP_NODES ↔ n ∈ gstore2, lpt2[n] = mklp(m1, m2, lpf, m3), remove-refsize0(istore2)[n] = indexblock(indx, lax, m), remove-refsize0(gstore2)[n] = groupblock(gndx, lax0, m0), istore2[n] = indexblock(indx0, lax1, m4), gstore2[n] = groupblock(gndx0, lax2, m5), keyset(gch1) = gc-heap(cleanup-lpt(lpt), []), keyset(gch0) = gc-heap(lpt2, []), keyset(gch) = gc-heap(lpt2, log1), remove-refsize0(gstore1) = gstore1, remove-refsize0(istore1) = istore1, elems log1 ∩ ginval = ∅, dom remove-refsize0(gstore2) ∩ ginval = ∅, dom gstore1 ∪ elems log1 = dom gstore2 ∪ ginval, n < # avol0, ginval < # avol0, avol-cleanup-lpt-unchanged(avol, avol0, lpt), awbuf-inv(avol0, nobuffer), awbuf-inv(avol, nobuffer), awbuf-inv(avol, bleb0), bin-heap-inv(gch1, lpt1), bin-heap-inv(gch0, lpt), bin-heap-inv(gch, lpt2), buf-ro-rw-mode(false, bleb0, lpt2, avol), bufs-log-only(bleb0, log1, lpt2), freelist-cons(cleanup-lpt(lpt), freelist0), freelist-cons(lpt2, freelist), id-idx(cleanup-lpt(lpt), lpt1), id-idx(lpt2, lpt), inv(gstore2 \ ginval), inv(gstore1), inv(istore2), inv(istore1), log-group-block(log1, lpt2), lpt-free-unref(lpt2), lpt-free-unref(lpt0), lpt-leb-size(cleanup-lpt(lpt)), lpt-leb-size(lpt0), lpt-leb-size(lpt2), lpt-ram-flash(lpt1, lpt2, []), lpt-ram-flash(lpt2, lpt0, log1), synced(avol0), synced(avol), istore1 ⊑ istore2, n ∈ istore2 → abs-leb(avol[n], indexblock(indx0, lax1, m4), mklp(m4, m2, lpf, m3)) ∧ m1 = m4, n ∈ gstore2 → m1 = m5, n ∈ gstore2 ∧ ¬ n ∈ ginval → abs-leb(avol[n], groupblock(gndx0, lax2, m5), mklp(m1, m2, lpf, m3)), pstate = unbuffered → bleb0 = nobuffer, pstate.journal? → log1 ≠ [] ∧ pstate.lnum = log1.last ∧ pstate.lnum ∈ gstore2, pstate.index? → pstate.lnum ∈ istore2, pstate ≠ unbuffered → bleb0 = buffered(pstate.lnum), n ∈ log1 → n ∈ gstore2, ∀ n. n < # avol0 → (n ∈ istore1 ↔ lpt0[n].flags = LP_INDEX_NODES ∧ lpt0[n].ref-size ≠ 0) ∧ (n ∈ gstore1 ↔ lpt0[n].flags = LP_GROUP_NODES ∧ lpt0[n].ref-size ≠ 0) ∧ (n ∈ istore1 → lpt0[n].ref-size = istore1[n].ref-size) ∧ (n ∈ gstore1 → lpt0[n].ref-size = gstore1[n].ref-size) ∧ (lpt0[n].flags = LP_FREE → lpt0[n].ref-size = 0) ⊦ pstate = readonly, dups(log1), (n ∈ istore2 → (m4 ≠ 0 ↔ m1 ≠ 0)) ∧ (n ∈ gstore2 → (m5 ≠ 0 ↔ m1 ≠ 0)) ∧ (n ∈ istore2 ∧ m4 ≠ 0 → m1 = m) ∧ (n ∈ gstore2 ∧ m5 ≠ 0 → m1 = m0) ∧ (lpf = LP_FREE → m1 = 0)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 145:
⊦ n ∈ istore ∧ istore[n].ref-size ≠ 0 → remove-refsize0(istore)[n] = istore[n]
Proof step 146:
# lpt0 = # avol0, # lpt2 = # avol0, # avol = # avol0, # lpt = # avol0, # lpt1 = # avol0, lpf = LP_GROUP_NODES ↔ n ∈ gstore2, lpf = LP_INDEX_NODES ↔ n ∈ istore2, gstore2[n] = groupblock(gndx0, lax2, m5), istore2[n] = indexblock(indx0, lax1, m4), remove-refsize0(gstore2)[n] = groupblock(gndx, lax0, m0), remove-refsize0(istore2)[n] = indexblock(indx, lax, m), lpt2[n] = mklp(m1, m2, lpf, m3), keyset(gch) = gc-heap(lpt2, log1), keyset(gch0) = gc-heap(lpt2, []), keyset(gch1) = gc-heap(cleanup-lpt(lpt), []), remove-refsize0(istore1) = istore1, remove-refsize0(gstore1) = gstore1, dom remove-refsize0(gstore2) ∩ ginval = ∅, elems log1 ∩ ginval = ∅, dom gstore1 ∪ elems log1 = dom gstore2 ∪ ginval, ginval < # avol0, n < # avol0, avol-cleanup-lpt-unchanged(avol, avol0, lpt), awbuf-inv(avol, bleb0), awbuf-inv(avol, nobuffer), awbuf-inv(avol0, nobuffer), bin-heap-inv(gch, lpt2), bin-heap-inv(gch0, lpt), bin-heap-inv(gch1, lpt1), buf-ro-rw-mode(false, bleb0, lpt2, avol), bufs-log-only(bleb0, log1, lpt2), freelist-cons(lpt2, freelist), freelist-cons(cleanup-lpt(lpt), freelist0), id-idx(lpt2, lpt), id-idx(cleanup-lpt(lpt), lpt1), inv(istore1), inv(istore2), inv(gstore1), inv(gstore2 \ ginval), log-group-block(log1, lpt2), lpt-free-unref(lpt0), lpt-free-unref(lpt2), lpt-leb-size(lpt2), lpt-leb-size(lpt0), lpt-leb-size(cleanup-lpt(lpt)), lpt-ram-flash(lpt2, lpt0, log1), lpt-ram-flash(lpt1, lpt2, []), synced(avol), synced(avol0), istore1 ⊑ istore2, n ∈ log1 → n ∈ gstore2, pstate ≠ unbuffered → bleb0 = buffered(pstate.lnum), pstate.index? → pstate.lnum ∈ istore2, pstate.journal? → log1 ≠ [] ∧ pstate.lnum = log1.last ∧ pstate.lnum ∈ gstore2, pstate = unbuffered → bleb0 = nobuffer, n ∈ gstore2 ∧ ¬ n ∈ ginval → abs-leb(avol[n], groupblock(gndx0, lax2, m5), mklp(m1, m2, lpf, m3)), n ∈ gstore2 → m1 = m5, n ∈ istore2 → abs-leb(avol[n], indexblock(indx0, lax1, m4), mklp(m4, m2, lpf, m3)) ∧ m1 = m4, ∀ n. n < # avol0 → (n ∈ istore1 ↔ lpt0[n].flags = LP_INDEX_NODES ∧ lpt0[n].ref-size ≠ 0) ∧ (n ∈ gstore1 ↔ lpt0[n].flags = LP_GROUP_NODES ∧ lpt0[n].ref-size ≠ 0) ∧ (n ∈ istore1 → lpt0[n].ref-size = istore1[n].ref-size) ∧ (n ∈ gstore1 → lpt0[n].ref-size = gstore1[n].ref-size) ∧ (lpt0[n].flags = LP_FREE → lpt0[n].ref-size = 0) ⊦ pstate = readonly, dups(log1), (n ∈ istore2 → (m4 ≠ 0 ↔ m1 ≠ 0)) ∧ (n ∈ gstore2 → (m5 ≠ 0 ↔ m1 ≠ 0)) ∧ (n ∈ istore2 ∧ m4 ≠ 0 → m1 = m) ∧ (n ∈ gstore2 ∧ m5 ≠ 0 → m1 = m0) ∧ (lpf = LP_FREE → m1 = 0), n ∈ istore2 ∧ istore2[n].ref-size ≠ 0

Rule argument

spec: ablock-store-inv, inst: index-store name: at-remove-refsize0 seq: ⊦ n ∈ istore ∧ istore[n].ref-size ≠ 0 → remove-refsize0(istore)[n] = istore[n] subst: [n, istore] → [n, istore2] flags: <all paths>

Additional information

Used simplifier rules: fwd-inv-iblock: ⊦ inv(istore) ∧ n ∈ istore ∧ istore[n] = iblock → inv(iblock); (forward) from specification apersistence-base%⊦ indexblock(indx, lax, m) = indexblock(indx0, lax0, m0) ↔ indx = indx0 ∧ lax = lax0 ∧ m = m0; (ws)% ⊦ LP_GROUP_NODES ≠ LP_INDEX_NODES; (s)% ⊦ ¬ n ∈ gstore → gstore[n] = (∅)[n]; (s)% ⊦ indexblock(indx, lax, m).ref-size = m; (s)% ⊦ LP_FREE ≠ LP_INDEX_NODES; (s) Used terms: n, istore2