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Proof step 31 from proof valid-eraseq-stronger-prev-append-lref (previous step)

avhdr0.sqn = (valid-sqns(v0, m2, apebs, m) ∪ (v = v0 ∧ n = m2 ⊃ {avhdr.sqn} ;∅)).max, 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(v, n, 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, v × n) '), valid-peb(v0, m2, mkpeb(aehdr0, avhdr0, buf0, m4, false)), valid-peb(v, n, apebs[m0]), valid-peb(v, n, mkpeb(aehdr, avhdr, buf, m1, boolvar)) ⊦ avhdr0.sqn = valid-sqns(v0, m2, apebs, m).max, m0 = m3, m0 ∈ q0, m3 ∈ q0

Heuristic if-then-else split applied rule case distinction

And obtained the following new goals

Proof step 32:
avhdr0.sqn = (valid-sqns(v0, m2, apebs, m) ∪ {avhdr.sqn} ).max, apebs[m3] = mkpeb(aehdr0, avhdr0, buf0, m4, false), apebs[m] = mkpeb(aehdr, avhdr, buf, m1, boolvar), n = m2, v = v0, m3 + 1 < # apebs, m3 < m, m0 < # apebs, valid-sqns(v, n, 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, v × n) '), valid-eraseq-stronger(subarray(apebs, 0, m), q0), valid-peb(v, n, mkpeb(aehdr, avhdr, buf, m1, boolvar)), valid-peb(v, n, apebs[m0]), valid-peb(v0, m2, mkpeb(aehdr0, avhdr0, buf0, m4, false)) ⊦ avhdr0.sqn = valid-sqns(v0, m2, apebs, m).max, m0 = m3, m3 ∈ q0, m0 ∈ q0

Rule argument

pos: left 1

Additional information

Used simplifier rules: ⊦ ∅ ∪ bag0 = bag0; (s)%union-ass: ⊦ bag ∪ bag3 ∪ bag4 = (bag ∪ bag3) ∪ bag4; from specification bag-union[nat-bag]%union-com: ⊦ bag ∪ bag2 = bag2 ∪ bag; from specification bag-union[nat-bag]