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Proof step 45 from proof valid-wl-write-valid (previous step)

# apebs = # wlar, avhdr1.sqn = valid-sqns(v, m5, apebs).max, wlar[m] = wl-entry(m4, used), apebs[m] = mkpeb(aechdr(m4), avhdr1, buf2, m6, false), m < # wlar, valid-peb(mkpeb(aechdr(m4), avhdr1, buf, m3, false)), valid-peb(mkpeb(aechdr(m4), avhdr1, buf2, m6, false)), valid-peb(v, m5, mkpeb(aechdr(m4), avhdr1, buf2, m6, false)), valid-peb(v, m5, mkpeb(aechdr(m4), avhdr1, buf, m3, false)) ⊦ avhdr1.sqn = valid-sqns(v, m5, apebs[m, mkpeb(aechdr(m4), avhdr1, buf, m3, false)]).max

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: sqns name: valid-sqns-write seq: ⊦ n < # apebs → valid-sqns(v, m, apebs[n, apeb]) = ( valid-sqns(v, m, apebs) \ (valid-peb(v, m, apebs[n]) ⊃ {apebs[n].vidhdr.sqn} ;∅)) ∪ (valid-peb(v, m, apeb) ⊃ {apeb.vidhdr.sqn} ;∅) subst: [apeb, n, apebs, m, v] → [mkpeb(aechdr(m4), avhdr1, buf, m3, false), m, apebs, m5, v] flags: <all paths>

Additional information

Used simplifier rules: ⊦ valid-peb(v0, m0, apeb) → (valid-peb(v1, m1, apeb) ↔ v0 = v1 ∧ m0 = m1); (s)% ⊦ mkpeb(aehdr, avhdr, buf, m, boolvar).vidhdr = avhdr; (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]% ⊦ m = (bag ++ m).max ↔ ¬(bag ≠ ∅ ∧ m < bag.max); (s)%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic[encoding-nat]%lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic[encoding-nat]% ⊦ bag \ {n} = bag -- n; (s)% ⊦ bag -- n ≠ ∅ → ¬ bag.max < (bag -- n).max; (s)% ⊦ m < n ↔ ¬ n ≤ m; (ss)%fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat[encoding-nat]%fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat[encoding-nat]% ⊦ ¬ n < n; (s) Used terms: mkpeb(aechdr(m4), avhdr1, buf, m3, false), m, apebs, m5, v