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Proof step 2 from proof absexpt-vols-abs (previous step)

abs○(avols, apebs, vols, q, v) ⊦ abs(avols[v, fill(mkabstractvolume(n), erased)], apebs, vols[v, fill(mkebatbl(n), unmapped)], q - v), v ∈ vols

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 3:
∀ 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)))

Rule argument

Additional information

Used simplifier rules: absexpt-v-def: ⊦ abs○(avols, apebs, vols, q, v0) ↔ (∀ v. (v ∈ vols ↔ v ∈ avols) ∧ ( v ∈ vols → # vols[v] = # avols[v] ∧ ( v ≠ v0 → (∀ m. m < # vols[v] → (vols[v][m] = unmapped → ¬ avols[v][m].mapped?) ∧ ( vols[v][m] ≠ unmapped → avols[v][m].mapped? ∧ avols[v][m].data = apebs[vols[v][m].pnum].data ∧ apebs[vols[v][m].pnum].fill ≤ avols[v][m].fill) ∧ (avols[v][m] = erased → ¬ v × m ∈ q)))));%abs-def: ⊦ abs(avols, apebs, vols, q) ↔ (∀ 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 = apebs[vols[v][m].pnum].data ∧ apebs[vols[v][m].pnum].fill ≤ avols[v][m].fill) ∧ (avols[v][m] = erased → ¬ v × m ∈ q))));