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Proof step 31 from proof valid-wl-use (previous step)

# apebs = # wlar, avhdr1.sqn = valid-sqns(v, m, apebs).max, apebs[n0] = mkpeb(aechdr(m3), avhdr1, buf1, m2, false), wlar[n0] = wl-entry(m3, used), n < # wlar, n0 < # wlar, valid-peb(mkpeb(aehdr, avhdr, buf, m0, boolvar)), valid-peb(v, m, mkpeb(aechdr(m3), avhdr1, buf1, m2, false)), valid-sqns(avhdr.vol, avhdr.leb, apebs) ≠ ∅ → valid-sqns(avhdr.vol, avhdr.leb, apebs).max < avhdr.sqn, ∀ m. m < # wlar ∧ m ≠ n ∧ wlar[m].status = used → apebs[m].vidhdr.vol ≠ avhdr.vol ∨ apebs[m].vidhdr.leb ≠ avhdr.leb ⊦ avhdr1.sqn = valid-sqns(v, m, apebs[n, mkpeb(aehdr, avhdr, buf, m0, boolvar)]).max, n0 = n

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 32:
⊦ n < # apebs → valid-sqns(v, m, apebs[n, apeb]) = (valid-sqns(v, m, apebs) \ (valid-peb(v, m, apebs[n]) ⊃ {apebs[n].vidhdr.sqn} ;∅)) ∪ (valid-peb(v, m, apeb) ⊃ {apeb.vidhdr.sqn} ;∅)
Proof step 33:
# apebs = # wlar, avhdr1.sqn = valid-sqns(v, m, apebs).max, wlar[n0] = wl-entry(m3, used), apebs[n0] = mkpeb(aechdr(m3), avhdr1, buf1, m2, false), n0 < # wlar, n < # wlar, n < # apebs, valid-peb(v, m, mkpeb(aechdr(m3), avhdr1, buf1, m2, false)), valid-peb(mkpeb(aehdr, avhdr, buf, m0, boolvar)), valid-sqns(avhdr.vol, avhdr.leb, apebs) ≠ ∅ → valid-sqns(avhdr.vol, avhdr.leb, apebs).max < avhdr.sqn, ∀ m. m < # wlar ∧ m ≠ n ∧ wlar[m].status = used → apebs[m].vidhdr.vol ≠ avhdr.vol ∨ apebs[m].vidhdr.leb ≠ avhdr.leb ⊦ avhdr1.sqn = ( (valid-sqns(v, m, apebs) \ (valid-peb(v, m, apebs[n]) ⊃ {apebs[n].vidhdr.sqn} ;∅)) ∪ (valid-peb(v, m, mkpeb(aehdr, avhdr, buf, m0, boolvar)) ⊃ {mkpeb(aehdr, avhdr, buf, m0, boolvar).vidhdr.sqn} ;∅) ).max, n0 = n

Rule argument

spec: sqns name: valid-sqns-write seq: ⊦ n < # apebs → valid-sqns(v, m, apebs[n, apeb]) = ( valid-sqns(v, m, apebs) \ (valid-peb(v, m, apebs[n]) ⊃ {apebs[n].vidhdr.sqn} ;∅)) ∪ (valid-peb(v, m, apeb) ⊃ {apeb.vidhdr.sqn} ;∅) subst: [apeb, n, apebs, m, v] → [mkpeb(aehdr, avhdr, buf, m0, boolvar), n, apebs, m, v] flags: <all paths>

Additional information

Used simplifier rules: ⊦ m < n ↔ ¬ n ≤ m; (ss)%fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat%fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat% ⊦ ¬ n < n; (s) Used terms: mkpeb(aehdr, avhdr, buf, m0, boolvar), n, apebs, m, v