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Proof step 9 from proof valid-sqns-write-h (previous step)

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-sqns(v, m, apebs[n, apeb], m1) ∪ (valid-peb(v, m, apebs[n, apeb][m1]) ⊃ {apebs[n, apeb][m1].vidhdr.sqn} ;∅) = (¬ m1 < n ⊃ ( (valid-sqns(v, m, apebs, m1) ∪ (valid-peb(v, m, apebs[m1]) ⊃ {apebs[m1].vidhdr.sqn} ;∅)) \ (valid-peb(v, m, apebs[n]) ⊃ {apebs[n].vidhdr.sqn} ;∅)) ∪ (valid-peb(v, m, apeb) ⊃ {apeb.vidhdr.sqn} ;∅) ;valid-sqns(v, m, apebs, m1) ∪ (valid-peb(v, m, apebs[m1]) ⊃ {apebs[m1].vidhdr.sqn} ;∅) )

Heuristic weak cut applied rule cut formula

And obtained the following new goals

Rule argument

fma: m1 = n

Additional information

Used simplifier rules: ⊦ ¬ n < n; (s)% ⊦ m < # apebs → apebs[m, apeb][m] = apeb; (s)% ⊦ (bag ∪ bag2) \ bag2 = bag; (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]% ⊦ bag0 ∪ bag3 = bag0 ∪ bag4 ↔ bag3 = bag4; (s) Used formulas: m1 = n