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Proof step 5 from proof valid-sqns-write-h (previous step)

m1 < # apebs, ∀ apebs, v, m, n, apeb. m1 ≤ # apebs → 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) ) ⊦ valid-sqns(v, m, apebs[n, apeb], m1 + 1) = (¬ 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} ;∅) )

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel> name: valid-sqns-rec seq: ⊦ n < # apebs → valid-sqns(v, m, apebs, n + 1) = valid-sqns(v, m, apebs, n) ∪ (valid-peb(v, m, apebs[n]) ⊃ {apebs[n].vidhdr.sqn} ;∅) subst: [n, apebs, m, v] → [m1, apebs[n, apeb], m, v] flags: <all paths>

Additional information

Used simplifier rules: ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ # apebs[n, apeb] = # apebs; (s)% ⊦ m < n → (n ≤ m ↔ false); (s)% ⊦ bag ∪ {n} = bag ++ 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 ⊆ bag3 ⊦ (bag3 \ bag) ∪ bag4 = (bag3 ∪ bag4) \ bag; (s) Used terms: m1, apebs[n, apeb], m, v