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Proof step 12 from proof valid-global-sqn-add-valid (previous step)

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} ;∅) = ∅

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 13:
avhdr.sqn < (valid-sqns(v, m0, apebs) -- apebs[m].vidhdr.sqn).max, valid-sqns(v, m0, apebs).max < (valid-sqns(v, m0, apebs) -- apebs[m].vidhdr.sqn).max, avhdr.sqn < n, m < # apebs, valid-sqns(v, m0, apebs).max < n, avhdr.sqn < ((valid-sqns(v, m0, apebs) -- apebs[m].vidhdr.sqn) ++ avhdr.sqn).max, valid-peb(v, m0, apebs[m]), valid-peb(mkpeb(aehdr, avhdr, buf, m1, boolvar)), n ≤ (valid-sqns(v, m0, apebs) -- apebs[m].vidhdr.sqn).max ⊦ (valid-sqns(v, m0, apebs) -- apebs[m].vidhdr.sqn) ∪ (valid-peb(v, m0, mkpeb(aehdr, avhdr, buf, m1, boolvar)) ⊃ {avhdr.sqn} ;∅) = ∅

Rule argument

Additional information

Used simplifier rules: ⊦ m < (bag ++ m).max → (bag ++ m).max = bag.max; (s)% ⊦ mkpeb(aehdr, avhdr, buf, m, boolvar).vidhdr = avhdr; (s)%⊦ valid-peb(apeb) → (valid-peb(v, m, apeb) ↔ v = apeb.vidhdr.vol ∧ m = apeb.vidhdr.leb); (ss)% ⊦ m < n → n ≠ 0; (s)% ⊦ m > n ↔ n < m; (s)% ⊦ bag ≠ ∅ → (bag ++ m).max = (bag.max > m ⊃ bag.max;m); (s)% ⊦ bag ∪ {n} = bag ++ n; (s)% ⊦ bag -- n ≠ ∅ → ¬ bag.max < (bag -- n).max; (s)%union-com: ⊦ bag ∪ bag2 = bag2 ∪ bag; from specification bag-union[nat-bag]%union-ass: ⊦ bag ∪ bag3 ∪ bag4 = (bag ∪ bag3) ∪ bag4; from specification bag-union[nat-bag]% ⊦ ∅ ∪ bag0 = bag0; (s)%fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic%n < # apebs ⊦ valid-peb(v, m, apebs[n]) → valid-sqns(v, m, apebs) ≠ ∅; (s)% ⊦ m < n → (n ≤ m ↔ false); (s)% ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ bag \ {n} = bag -- n; (s)%⊦ valid-peb(v0, m0, apeb) → (valid-peb(v1, m1, apeb) ↔ v0 = v1 ∧ m0 = m1); (s)