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Proof step 10 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 ⊦ m1 = n, 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 simplifier applied rule simplifier

And obtained the following new goals

Proof step 11:
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 ⊦ m1 = n, valid-sqns(v, m, apebs[n, apeb], m1) ∪ (valid-peb(v, m, apebs[m1]) ⊃ {apebs[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} ;∅) )

Rule argument

Additional information

Used simplifier rules: ⊦ m ≠ n → apebs[n, apeb][m] = apebs[m]; (s)