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Proof step 63 from proof recovery (previous step)

# avol2 = # avol0, # lpt2 = # avol0, # avol = # avol0, # lpt4 = # avol0, # lpt = # avol0, keyset(gch1) = gc-heap(lpt2, log3), keyset(gch) = gc-heap(replay-lpt(replay-log(lpt4, log3), log3, # avol0), log3), remove-refsize0(istore2) = istore2, remove-refsize0(gstore2) = gstore2, elems log3 ∩ ns3 = ∅, dom gstore2 ∪ elems log3 = dom gstore1 ∪ ns3, ns3 < # avol0, abs-gstore(avol0, lpt2, gstore1, ns3), abs-istore(avol0, lpt2, istore1), avol-replay-lpt-unchanged(avol, avol2, replay-log(lpt4, log3), log3), awbuf-inv(avol0, bleb1), awbuf-inv(avol2, nobuffer), bin-heap-inv(gch1, lpt2), bin-heap-inv(gch, lpt), buf-ro-rw-mode(pstate0 = readonly, bleb1, lpt2, avol0), bufs-log-only(bleb1, log3, lpt2), flash-lpt-cons(lpt4, istore2, gstore2), freelist-cons(lpt2, log4), freelist-cons(replay-lpt(replay-log(lpt4, log3), log3, # avol0), freelist), id-idx(replay-lpt(replay-log(lpt4, log3), log3, # avol0), lpt), inv(istore2), inv(istore1), inv(gstore2), inv(gstore1 \ ns3), log-group-block(log3, lpt2), log-group-block(log3, lpt), lpt-free-unref(lpt2), lpt-free-unref(lpt4), lpt-leb-size(lpt2), lpt-leb-size(lpt4), lpt-leb-size(replay-lpt(replay-log(lpt4, log3), log3, # avol0)), lpt-ram-flash(lpt2, lpt4, log3), lpt-ram-flash(lpt, lpt4, log3), synced(avol), synced(avol0), synced(avol2), valid-indices(lpt4, log3), avol ↓ avol0, istore2 ⊑ istore1, pstate0 = unbuffered → isbuf0, pstate0.index? → pstate0.lnum ∈ istore1, pstate0.journal? → log3 ≠ [] ∧ pstate0.lnum = log3.last ∧ pstate0.lnum ∈ gstore1, pstate0 ≠ unbuffered ∧ pstate0 ≠ readonly → bleb1 = buffered(pstate0.lnum), pstate0 = unbuffered → bleb1 = nobuffer ⊦ dups(log3), abs-istore(avol, replay-lpt(replay-log(lpt4, log3), log3, # avol0), remove-refsize0(crash(istore1, istore2))) ∧ abs-gstore(avol, replay-lpt(replay-log(lpt4, log3), log3, # avol0), crash(gstore1, gstore2, ns3, log3), ns3)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 64:
n = # lpt0, # avol0 = # lpt, # lpt0 = # lpt ⊦ abs-istore(avol0, lpt, istore) ∧ avol1 ↓ avol0 ∧ flash-lpt-cons(lpt0, istore0, gstore0) ∧ istore0 ⊑ istore ∧ synced(avol0) ∧ dom istore ∩ elems nx = ∅ → abs-istore(avol1, replay-lpt(replay-log(lpt0, nx), nx, n), remove-refsize0(crash(istore, istore0)))
Proof step 65:
# lpt = # avol0, # avol = # avol0, # avol2 = # avol0, # avol0 = # lpt4, # avol0 = # lpt2, # lpt4 = # lpt2, keyset(gch) = gc-heap(replay-lpt(replay-log(lpt4, log3), log3, # avol0), log3), keyset(gch1) = gc-heap(lpt2, log3), remove-refsize0(gstore2) = gstore2, remove-refsize0(istore2) = istore2, elems log3 ∩ ns3 = ∅, dom istore1 ∩ elems log3 = ∅, dom gstore2 ∪ elems log3 = dom gstore1 ∪ ns3, ns3 < # avol0, abs-gstore(avol0, lpt2, gstore1, ns3), abs-istore(avol0, lpt2, istore1), avol-replay-lpt-unchanged(avol, avol2, replay-log(lpt4, log3), log3), awbuf-inv(avol2, nobuffer), awbuf-inv(avol0, bleb1), bin-heap-inv(gch, lpt), bin-heap-inv(gch1, lpt2), buf-ro-rw-mode(pstate0 = readonly, bleb1, lpt2, avol0), bufs-log-only(bleb1, log3, lpt2), flash-lpt-cons(lpt4, istore2, gstore2), freelist-cons(replay-lpt(replay-log(lpt4, log3), log3, # avol0), freelist), freelist-cons(lpt2, log4), id-idx(replay-lpt(replay-log(lpt4, log3), log3, # avol0), lpt), inv(gstore1 \ ns3), inv(gstore2), inv(istore1), inv(istore2), log-group-block(log3, lpt), log-group-block(log3, lpt2), lpt-free-unref(lpt4), lpt-free-unref(lpt2), lpt-leb-size(replay-lpt(replay-log(lpt4, log3), log3, # avol0)), lpt-leb-size(lpt4), lpt-leb-size(lpt2), lpt-ram-flash(lpt, lpt4, log3), lpt-ram-flash(lpt2, lpt4, log3), synced(avol2), synced(avol), synced(avol0), valid-indices(lpt4, log3), avol ↓ avol0, istore2 ⊑ istore1, pstate0 = unbuffered → bleb1 = nobuffer, pstate0 ≠ unbuffered ∧ pstate0 ≠ readonly → bleb1 = buffered(pstate0.lnum), pstate0.journal? → log3 ≠ [] ∧ pstate0.lnum = log3.last ∧ pstate0.lnum ∈ gstore1, pstate0.index? → pstate0.lnum ∈ istore1, pstate0 = unbuffered → isbuf0 ⊦ dups(log3), true ∧ abs-gstore(avol, replay-lpt(replay-log(lpt4, log3), log3, # avol0), crash(gstore1, gstore2, ns3, log3), ns3)

Rule argument

spec: persistence-base name: abs-istore-crash seq: n = # lpt0, # avol0 = # lpt, # lpt0 = # lpt ⊦ abs-istore(avol0, lpt, istore) ∧ avol1 ↓ avol0 ∧ flash-lpt-cons(lpt0, istore0, gstore0) ∧ istore0 ⊑ istore ∧ synced(avol0) ∧ dom istore ∩ elems nx = ∅ → abs-istore (avol1, replay-lpt(replay-log(lpt0, nx), nx, n), remove-refsize0(crash(istore, istore0))) subst: [avol0, lpt, gstore0, istore0, istore, n, nx, lpt0, avol1] → [avol0, lpt2, gstore2, istore2, istore1, # avol0, log3, lpt4, avol] flags: <all paths>

Additional information

Used simplifier rules: una: ⊦ (ns ∪ ns0) ∪ ns1 = ns ∪ ns0 ∪ ns1; from specification set-union[nat-set]%unc: ⊦ ns ∪ ns0 = ns0 ∪ ns; from specification set-union[nat-set]%int-a: ⊦ (ns ∩ ns0) ∩ ns1 = ns ∩ ns0 ∩ ns1; from specification set-union[nat-set]%int-c: ⊦ ns ∩ ns0 = ns0 ∩ ns; from specification set-union[nat-set]%⊦ dom gstore0 ∪ elems nx = dom gstore ∪ ginval ∧ elems nx ∩ ginval = ∅ ∧ abs-istore(avol, lpt, istore) ∧ abs-gstore(avol, lpt, gstore, ginval) → dom istore ∩ elems nx = ∅; (s) Used terms: avol0, lpt2, gstore2, istore2, istore1, # avol0, log3, lpt4, avol