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Proof step 3 from proof absexpt-vols-abs (previous step)

∀ v0. (v0 ∈ vols ↔ v0 ∈ avols) ∧ ( v0 ∈ vols → # vols[v0] = # avols[v0] ∧ (∀ m. m < # vols[v0] → (vols[v0][m] = unmapped → ¬ avols[v0][m].mapped?) ∧ ( vols[v0][m] ≠ unmapped → avols[v0][m].mapped? ∧ avols[v0][m].data = apebs[vols[v0][m].pnum].data ∧ apebs[vols[v0][m].pnum].fill ≤ avols[v0][m].fill) ∧ (avols[v0][m] = erased → ¬ v0 × m ∈ q))) ⊦ v ∈ 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 = apebs[vols[v, fill(mkebatbl(n), unmapped)][v0][m].pnum].data ∧ apebs[vols[v, fill(mkebatbl(n), unmapped)][v0][m].pnum].fill ≤ avols[v, fill(mkabstractvolume(n), erased)][v0][m].fill) ∧ (avols[v, fill(mkabstractvolume(n), erased)][v0][m] = erased → ¬ v0 × m ∈ q - v)))

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 4:
apebs[vols[v, fill(mkebatbl(n), unmapped)][v0][m].pnum] = mkpeb(aehdr, avhdr, buf, m0, boolvar), ∀ v0. (v0 ∈ vols ↔ v0 ∈ avols) ∧ ( v0 ∈ vols → # vols[v0] = # avols[v0] ∧ (∀ m. m < # vols[v0] → (vols[v0][m] = unmapped → ¬ avols[v0][m].mapped?) ∧ ( vols[v0][m] ≠ unmapped → avols[v0][m].mapped? ∧ avols[v0][m].data = apebs[vols[v0][m].pnum].data ∧ apebs[vols[v0][m].pnum].fill ≤ avols[v0][m].fill) ∧ (avols[v0][m] = erased → ¬ v0 × m ∈ q))) ⊦ v ∈ 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 ∈ q - v)))

Rule argument

spec: apeb name: elim seq: ⊦ ( ( (aehdr = apeb.echdr ∧ avhdr = apeb.vidhdr) ∧ buf = apeb.data) ∧ m = apeb.fill) ∧ (boolvar ↔ apeb.bad) ↔ apeb = mkpeb(aehdr, avhdr, buf, m, boolvar) subst: [apeb, aehdr, avhdr, buf, m, boolvar] → [apebs[vols[v, fill(mkebatbl(n), unmapped)][v0][m].pnum], aehdr, avhdr, buf, m0, boolvar]

Additional information

Used simplifier rules: ⊦ ( ((aehdr = apeb.echdr ∧ avhdr = apeb.vidhdr) ∧ buf = apeb.data) ∧ m = apeb.fill) ∧ (boolvar ↔ apeb.bad) ↔ apeb = mkpeb(aehdr, avhdr, buf, m, boolvar); (elim) Used terms: apebs[vols[v, fill(mkebatbl(n), unmapped)][v0][m].pnum], aehdr, avhdr, buf, m0, boolvar