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Proof step 9 from proof abs-unmap (previous step)

# vols[v0] = # avols[v0], v0 ∈ vols ↔ v0 ∈ avols, apebs[vols[v, vols[v][m, unmapped]][v0][m0].pnum] = mkpeb(aehdr, avhdr, buf, m1, boolvar), vols[v][m] = ⌜n⌝ , m < # vols[v], v ∈ vols, m0 < # avols[v0] → (vols[v0][m0] = unmapped → ¬ avols[v0][m0].mapped?) ∧ ( vols[v0][m0] ≠ unmapped → avols[v0][m0].mapped? ∧ avols[v0][m0].data = apebs[vols[v0][m0].pnum].data ∧ apebs[vols[v0][m0].pnum].fill ≤ avols[v0][m0].fill) ∧ (avols[v0][m0] = erased → ¬ v0 × m0 ∈ q) ⊦ (v0 ∈ vols[v, vols[v][m, unmapped]] ↔ v0 ∈ avols[v, avols[v][m, unmapped]]) ∧ ( v0 ∈ vols[v, vols[v][m, unmapped]] → # vols[v, vols[v][m, unmapped]][v0] = # avols[v, avols[v][m, unmapped]][v0] ∧ ( m0 < # vols[v, vols[v][m, unmapped]][v0] → (vols[v, vols[v][m, unmapped]][v0][m0] = unmapped → ¬ avols[v, avols[v][m, unmapped]][v0][m0].mapped?) ∧ ( vols[v, vols[v][m, unmapped]][v0][m0] ≠ unmapped → avols[v, avols[v][m, unmapped]][v0][m0].mapped? ∧ avols[v, avols[v][m, unmapped]][v0][m0].data = buf ∧ m1 ≤ avols[v, avols[v][m, unmapped]][v0][m0].fill) ∧ (avols[v, avols[v][m, unmapped]][v0][m0] = erased → ¬ v0 × m0 ∈ q + eq-entry(n, v × m) ')))

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 10:
# vols[v0] = # avols[v0], v0 ∈ vols ↔ v0 ∈ avols, vols[v][m] = ⌜n⌝ , apebs[vols[v, vols[v][m, unmapped]][v0][m0].pnum] = mkpeb(aehdr, avhdr, buf, m1, boolvar), apebs[vols[v0][m0].pnum] = mkpeb(aehdr0, avhdr0, buf0, m2, boolvar0), m < # vols[v], v ∈ vols, m0 < # avols[v0] → (vols[v0][m0] = unmapped → ¬ avols[v0][m0].mapped?) ∧ (vols[v0][m0] ≠ unmapped → avols[v0][m0].mapped? ∧ avols[v0][m0].data = buf0 ∧ m2 ≤ avols[v0][m0].fill) ∧ (avols[v0][m0] = erased → ¬ v0 × m0 ∈ q) ⊦ (v0 ∈ vols[v, vols[v][m, unmapped]] ↔ v0 ∈ avols[v, avols[v][m, unmapped]]) ∧ ( v0 ∈ vols[v, vols[v][m, unmapped]] → # vols[v, vols[v][m, unmapped]][v0] = # avols[v, avols[v][m, unmapped]][v0] ∧ ( m0 < # vols[v, vols[v][m, unmapped]][v0] → (vols[v, vols[v][m, unmapped]][v0][m0] = unmapped → ¬ avols[v, avols[v][m, unmapped]][v0][m0].mapped?) ∧ ( vols[v, vols[v][m, unmapped]][v0][m0] ≠ unmapped → avols[v, avols[v][m, unmapped]][v0][m0].mapped? ∧ avols[v, avols[v][m, unmapped]][v0][m0].data = buf ∧ m1 ≤ avols[v, avols[v][m, unmapped]][v0][m0].fill) ∧ (avols[v, avols[v][m, unmapped]][v0][m0] = erased → ¬ v0 × m0 ∈ q + eq-entry(n, v × m) ')))

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[v0][m0].pnum], aehdr0, avhdr0, buf0, m2, boolvar0]

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[v0][m0].pnum], aehdr0, avhdr0, buf0, m2, boolvar0