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Proof step 24 from proof refine-create-volume (previous step)

# apebs = # wlar, # apebs0 = # wlar0, apebs0[vols[v, fill(mkebatbl(n), unmapped)][v0][m].pnum] = mkpeb(aehdr, avhdr, buf, m0, boolvar), abs(avols, apebs, vols, q), bflipset-cons(bflipset, wlar), bflipset-cons(bflipset0, wlar0), ebm-cons(avols), id(apebs0, apebs, mapped-pebs(vols)), ubi-io-inv(apebs), ubi-io-inv(apebs0), unique-sqns(apebs), unique-sqns(apebs0), valid-ebatbl(apebs, wlar, vols, vtblpnum), valid-ebatbl(apebs0, wlar0, vols[v, fill(mkebatbl(n), unmapped)], vtblpnum0), valid-eraseq(apebs, q), valid-eraseq(wlar, q), valid-eraseq(q), valid-eraseq(apebs0, q0), valid-eraseq(wlar0, q0), valid-eraseq(q0), valid-global-sqn(apebs, sqnum), valid-global-sqn(apebs0, sqnum0), valid-vtbl(apebs, vols, wlar, vtblpnum), valid-vtbl(apebs0, vols[v, fill(mkebatbl(n), unmapped)], wlar0, vtblpnum0), valid-wl(apebs, wlar), valid-wl(apebs0, wlar0), q0.lebrefs ⊆ ∅ ++ (VTBL_VOLID × VTBL_LNUM), ∀ v. (v ∈ vols ↔ v ∈ avols) ∧ ( v ∈ vols → # vols[v] = # avols[v] ∧ (∀ m. m < # vols[v] → (vols[v][m] = unmapped → ¬ avols[v][m].mapped?) ∧ ( vols[v][m] ≠ unmapped → avols[v][m].mapped? ∧ avols[v][m].data = apebs0[vols[v][m].pnum].data ∧ apebs0[vols[v][m].pnum].fill ≤ avols[v][m].fill) ∧ (avols[v][m] = erased → ¬ v × m ∈ q))) ⊦ VTBL_VOLID = v, v ∈ vols, v ∈ avols, VTBL_VOLID ∈ vols, (v0 ∈ vols[v, fill(mkebatbl(n), unmapped)] ↔ v0 ∈ avols[v, fill(mkabstractvolume(n), erased)]) ∧ ( v0 ∈ vols[v, fill(mkebatbl(n), unmapped)] → # vols[v, fill(mkebatbl(n), unmapped)][v0] = # avols[v, fill(mkabstractvolume(n), erased)][v0] ∧ ( m < # vols[v, fill(mkebatbl(n), unmapped)][v0] → (vols[v, fill(mkebatbl(n), unmapped)][v0][m] = unmapped → ¬ avols[v, fill(mkabstractvolume(n), erased)][v0][m].mapped?) ∧ ( vols[v, fill(mkebatbl(n), unmapped)][v0][m] ≠ unmapped → avols[v, fill(mkabstractvolume(n), erased)][v0][m].mapped? ∧ avols[v, fill(mkabstractvolume(n), erased)][v0][m].data = buf ∧ m0 ≤ avols[v, fill(mkabstractvolume(n), erased)][v0][m].fill) ∧ (avols[v, fill(mkabstractvolume(n), erased)][v0][m] = erased → ¬ v0 × m ∈ q0)))

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 25:
⊦ v ∈ vols[v0, ebatbl] ↔ v = v0 ∨ v ∈ vols
Proof step 26:
# apebs0 = # wlar0, # apebs = # wlar, apebs0[vols[v, fill(mkebatbl(n), unmapped)][v0][m].pnum] = mkpeb(aehdr, avhdr, buf, m0, boolvar), abs(avols, apebs, vols, q), bflipset-cons(bflipset0, wlar0), bflipset-cons(bflipset, wlar), ebm-cons(avols), id(apebs0, apebs, mapped-pebs(vols)), ubi-io-inv(apebs0), ubi-io-inv(apebs), unique-sqns(apebs0), unique-sqns(apebs), valid-ebatbl(apebs0, wlar0, vols[v, fill(mkebatbl(n), unmapped)], vtblpnum0), valid-ebatbl(apebs, wlar, vols, vtblpnum), valid-eraseq(q0), valid-eraseq(wlar0, q0), valid-eraseq(apebs0, q0), valid-eraseq(q), valid-eraseq(wlar, q), valid-eraseq(apebs, q), valid-global-sqn(apebs0, sqnum0), valid-global-sqn(apebs, sqnum), valid-vtbl(apebs0, vols[v, fill(mkebatbl(n), unmapped)], wlar0, vtblpnum0), valid-vtbl(apebs, vols, wlar, vtblpnum), valid-wl(apebs0, wlar0), valid-wl(apebs, wlar), q0.lebrefs ⊆ ∅ ++ (VTBL_VOLID × VTBL_LNUM), ∀ v. (v ∈ vols ↔ v ∈ avols) ∧ ( v ∈ vols → # vols[v] = # avols[v] ∧ (∀ m. m < # vols[v] → (vols[v][m] = unmapped → ¬ avols[v][m].mapped?) ∧ ( vols[v][m] ≠ unmapped → avols[v][m].mapped? ∧ avols[v][m].data = apebs0[vols[v][m].pnum].data ∧ apebs0[vols[v][m].pnum].fill ≤ avols[v][m].fill) ∧ (avols[v][m] = erased → ¬ v × m ∈ q))) ⊦ VTBL_VOLID = v, VTBL_VOLID ∈ vols, v ∈ avols, v ∈ vols, (v0 = v ∨ v0 ∈ vols ↔ v0 ∈ avols[v, fill(mkabstractvolume(n), erased)]) ∧ ( v0 = v ∨ v0 ∈ vols → # vols[v, fill(mkebatbl(n), unmapped)][v0] = # avols[v, fill(mkabstractvolume(n), erased)][v0] ∧ ( m < # vols[v, fill(mkebatbl(n), unmapped)][v0] → (vols[v, fill(mkebatbl(n), unmapped)][v0][m] = unmapped → ¬ avols[v, fill(mkabstractvolume(n), erased)][v0][m].mapped?) ∧ ( vols[v, fill(mkebatbl(n), unmapped)][v0][m] ≠ unmapped → avols[v, fill(mkabstractvolume(n), erased)][v0][m].mapped? ∧ avols[v, fill(mkabstractvolume(n), erased)][v0][m].data = buf ∧ m0 ≤ avols[v, fill(mkabstractvolume(n), erased)][v0][m].fill) ∧ (avols[v, fill(mkabstractvolume(n), erased)][v0][m] = erased → ¬ v0 × m ∈ q0)))

Rule argument

spec: store, inst: volumes name: In-insert seq: ⊦ v ∈ vols[v0, ebatbl] ↔ v = v0 ∨ v ∈ vols subst: [ebatbl, v0, vols, v] → [fill(mkebatbl(n), unmapped), v, vols, v0] flags: <all paths>

Additional information

Used terms: fill(mkebatbl(n), unmapped), v, vols, v0