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Proof step 33 from proof valid-eraseq-stronger-prev-append-lref (previous step)

avhdr0.sqn = avhdr.sqn, apebs[m3] = mkpeb(aehdr0, avhdr0, buf0, m4, false), apebs[m] = mkpeb(aehdr, avhdr, buf, m1, boolvar), m3 + 1 < # apebs, m3 < m, m0 < # apebs, valid-sqns(v0, m2, apebs, m).max < avhdr.sqn, m0 < m, m < # apebs, m0 + 1 < # apebs, m3 < # apebs, valid-eraseq-stronger(subarray(apebs, 0, m + 1), q0 + eq-entry(m0, v0 × m2) '), valid-eraseq-stronger(subarray(apebs, 0, m), q0), valid-peb(v0, m2, mkpeb(aehdr, avhdr, buf, m1, boolvar)), valid-peb(v0, m2, apebs[m0]), valid-peb(v0, m2, mkpeb(aehdr0, avhdr0, buf0, m4, false)) ⊦ avhdr.sqn = valid-sqns(v0, m2, apebs, m).max, m0 = m3, m3 ∈ q0, m0 ∈ q0

User applied rule apply lemma

And obtained the following new goals

Proof step 34:
⊦ n0 < m0 ∧ n1 < m0 ∧ m0 ≤ # apebs ∧ n0 ≠ n1 ∧ valid-peb(v, m, apebs[n0]) ∧ valid-peb(v, m, apebs[n1]) → (∃ bag. valid-sqns(v, m, apebs, m0) = bag ++ apebs[n0].vidhdr.sqn ++ apebs[n1].vidhdr.sqn)
Proof step 35:
avhdr0.sqn = avhdr.sqn, apebs[m] = mkpeb(aehdr, avhdr, buf, m1, boolvar), apebs[m3] = mkpeb(aehdr0, avhdr0, buf0, m4, false), m3 < # apebs, m0 + 1 < # apebs, m < # apebs, m0 < m, valid-sqns(v0, m2, apebs, m).max < avhdr.sqn, m0 < # apebs, m3 < m, m3 + 1 < # apebs, valid-eraseq-stronger(subarray(apebs, 0, m), q0), valid-eraseq-stronger(subarray(apebs, 0, m + 1), q0 + eq-entry(m0, v0 × m2) '), valid-peb(v0, m2, mkpeb(aehdr0, avhdr0, buf0, m4, false)), valid-peb(v0, m2, apebs[m0]), valid-peb(v0, m2, mkpeb(aehdr, avhdr, buf, m1, boolvar)), m0 < m ∧ m3 < m ∧ m ≤ # apebs ∧ m0 ≠ m3 ∧ valid-peb(v0, m2, apebs[m0]) ∧ valid-peb(v0, m2, apebs[m3]) → (∃ bag. valid-sqns(v0, m2, apebs, m) = bag ++ apebs[m0].vidhdr.sqn ++ apebs[m3].vidhdr.sqn) ⊦ avhdr.sqn = valid-sqns(v0, m2, apebs, m).max, m0 = m3, m0 ∈ q0, m3 ∈ q0

Rule argument

spec: sqns name: valid-sqns-extract2-sqn-h seq: ⊦ n0 < m0 ∧ n1 < m0 ∧ m0 ≤ # apebs ∧ n0 ≠ n1 ∧ valid-peb(v, m, apebs[n0]) ∧ valid-peb(v, m, apebs[n1]) → (∃ bag. valid-sqns(v, m, apebs, m0) = bag ++ apebs[n0].vidhdr.sqn ++ apebs[n1].vidhdr.sqn) subst: [n0, m0, n1, apebs, v, m] → [m0, m, m3, apebs, v0, m2]

Additional information

Used terms: m0, m, m3, apebs, v0, m2