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Proof step 16 from proof valid-sqns-write-h (previous step)

apebs[m1] = mkpeb(aehdr, avhdr, buf, m0, boolvar), valid-sqns(v, m, apebs[n, apeb], m1) = (n < m1 ⊃ (valid-sqns(v, m, apebs, m1) \ (valid-peb(v, m, apebs[n]) ⊃ {apebs[n].vidhdr.sqn} ;∅)) ∪ (valid-peb(v, m, apeb) ⊃ {apeb.vidhdr.sqn} ;∅) ;valid-sqns(v, m, apebs, m1) ), m1 < # apebs, valid-peb(v, m, mkpeb(aehdr, avhdr, buf, m0, boolvar)) ⊦ valid-sqns(v, m, apebs[n, apeb], m1) ++ avhdr.sqn = ((valid-sqns(v, m, apebs, m1) ++ avhdr.sqn) \ (valid-peb(v, m, apebs[n]) ⊃ {apebs[n].vidhdr.sqn} ;∅)) ∪ (valid-peb(v, m, apeb) ⊃ {apeb.vidhdr.sqn} ;∅), m1 = n, m1 < n

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 17:
apebs[m1] = mkpeb(aehdr, avhdr, buf, m0, boolvar), valid-sqns(v, m, apebs[n, apeb], m1) = (valid-sqns(v, m, apebs, m1) \ (valid-peb(v, m, apebs[n]) ⊃ {apebs[n].vidhdr.sqn} ;∅)) ∪ (valid-peb(v, m, apeb) ⊃ {apeb.vidhdr.sqn} ;∅), n < # apebs, n < m1, m1 < # apebs, n + 1 < # apebs, valid-peb(v, m, mkpeb(aehdr, avhdr, buf, m0, boolvar)) ⊦ valid-sqns(v, m, apebs[n, apeb], m1) ++ avhdr.sqn = ((valid-sqns(v, m, apebs, m1) ++ avhdr.sqn) \ (valid-peb(v, m, apebs[n]) ⊃ {apebs[n].vidhdr.sqn} ;∅)) ∪ (valid-peb(v, m, apeb) ⊃ {apeb.vidhdr.sqn} ;∅), m1 = n

Rule argument

Additional information

Used simplifier rules: ⊦ m +1 = m + 1; (s)% ⊦ m < n → n ≠ 0; (s)%trans: ⊦ n < n0 ∧ n0 < n1 → n < n1; (forward) from specification nat-basic%lf: ⊦ m < n ∧ n < n0 → m +1 < n0; (forward) from specification nat-basic%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic% ⊦ m ≠ n → (m < n ↔ ¬ n < m); (ss)