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Proof step 17 from proof valid-sqns-write-h (previous step)

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

Heuristic if-then-else split applied rule case distinction

And obtained the following new goals

Proof step 18:
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 + 1 < # apebs, m1 < # apebs, n < m1, n < # apebs, valid-peb(v, m, mkpeb(aehdr, avhdr, buf, m0, boolvar)), valid-peb(v, m, apebs[n]) ⊦ valid-sqns(v, m, apebs[n, apeb], m1) ++ avhdr.sqn = ((valid-sqns(v, m, apebs, m1) ++ avhdr.sqn) \ {apebs[n].vidhdr.sqn} ) ∪ (valid-peb(v, m, apeb) ⊃ {apeb.vidhdr.sqn} ;∅), m1 = n

Rule argument

pos: right 1

Additional information

Used simplifier rules: ⊦ bag \ ∅ = bag; (s)% ⊦ (bag ++ n) ∪ bag2 = (bag ∪ bag2) ++ n; (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]% ⊦ bag ++ n = bag2 ++ n ↔ bag = bag2; (s)