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Proof step 9 from proof valid-global-sqn-add-valid (previous step)

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

Heuristic nonsplitting cut applied rule cut formula

And obtained the following new goals

Proof step 10:
m < # apebs, avhdr.sqn < n, avhdr.sqn < ( (valid-sqns(v, m0, apebs) \ (valid-peb(v, m0, apebs[m]) ⊃ {apebs[m].vidhdr.sqn} ;∅)) ∪ (valid-peb(v, m0, mkpeb(aehdr, avhdr, buf, m1, boolvar)) ⊃ {avhdr.sqn} ;∅) ).max, valid-peb(mkpeb(aehdr, avhdr, buf, m1, boolvar)), n ≤ ( (valid-sqns(v, m0, apebs) \ (valid-peb(v, m0, apebs[m]) ⊃ {apebs[m].vidhdr.sqn} ;∅)) ∪ (valid-peb(v, m0, mkpeb(aehdr, avhdr, buf, m1, boolvar)) ⊃ {avhdr.sqn} ;∅) ).max, valid-sqns(v, m0, apebs) ≠ ∅ → valid-sqns(v, m0, apebs).max < n ⊦ valid-sqns(v, m0, apebs) = ∅, (valid-sqns(v, m0, apebs) \ (valid-peb(v, m0, apebs[m]) ⊃ {apebs[m].vidhdr.sqn} ;∅)) ∪ (valid-peb(v, m0, mkpeb(aehdr, avhdr, buf, m1, boolvar)) ⊃ {avhdr.sqn} ;∅) = ∅

Rule argument

fma: valid-sqns(v, m0, apebs) = ∅

Additional information

Used simplifier rules: n < # apebs ⊦ valid-peb(v, m, apebs[n]) → valid-sqns(v, m, apebs) ≠ ∅; (s)% ⊦ bag0 ⊆ ∅ ↔ bag0 = ∅; (s)%bag ⊆ bag3 ⊦ (bag3 \ bag) ∪ bag4 = (bag3 ∪ bag4) \ 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]%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic%fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat% ⊦ ∅ ∪ bag0 = bag0; (s)% ⊦ ∅ \ bag0 = ∅; (s)% ⊦ bag \ ∅ = bag; (s)%⊦ valid-peb(v0, m0, apeb) → (valid-peb(v1, m1, apeb) ↔ v0 = v1 ∧ m0 = m1); (s)% ⊦ {m} .max = m; (s)% ⊦ ¬ n < n; (s)% ⊦ bag \ {n} = bag -- n; (s) Used formulas: valid-sqns(v, m0, apebs) = ∅