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Proof step 11 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-sqns(v, m0, apebs).max < n, valid-sqns(v, m0, apebs).max < ( (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) \ (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 12:
valid-sqns(v, m0, apebs).max < ( (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).max < n, 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)), valid-peb(v, m0, apebs[m]), 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) \ (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-peb(v, m0, apebs[m])

Additional information

Used simplifier rules: ⊦ bag \ ∅ = bag; (s)%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic%fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat% ⊦ ∅ ∪ bag0 = bag0; (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]% ⊦ ¬ n < n; (s)% ⊦ bag ∪ {n} = bag ++ n; (s)% ⊦ bag ≠ ∅ → (bag ++ m).max = (bag.max > m ⊃ bag.max;m); (s)% ⊦ m > n ↔ n < m; (s)% ⊦ ∅ ≠ bag ++ n; (s)% ⊦ m < n → n ≠ 0; (s)% ⊦ m < n → (n ≤ m ↔ false); (s) Used formulas: valid-peb(v, m0, apebs[m])