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Proof step 12 from proof valid-eraseq-stronger-next-append-lref (previous step)

apebs[m] = mkpeb(aehdr, avhdr, buf, m0, boolvar), eqent.pnum < # apebs, eqent.pnum < m, m < # apebs, eqent.pnum + 1 < # apebs, valid-eraseq-stronger(subarray(apebs, 0, m), q0), ∀ v0, m1. (valid-peb(v0, m1, apebs[eqent.pnum]) ↔ eqent.lebref = v0 × m1) ∧ (eqent.lebref = v0 × m1 → apebs[eqent.pnum].vidhdr.sqn ≠ valid-sqns(v0, m1, apebs, m).max), ∀ apebs, m, aehdr, avhdr, buf, m0, boolvar, v, n. apebs[m] = mkpeb(aehdr, avhdr, buf, m0, boolvar) ∧ m < # apebs ∧ valid-eraseq-stronger(subarray(apebs, 0, m), q0) → ( valid-eraseq-stronger(subarray(apebs, 0, m + 1), q0 + eq-entry(m, v × n) ') ↔ valid-peb(v, n, mkpeb(aehdr, avhdr, buf, m0, boolvar)) ∧ valid-sqns(v, n, apebs, m) ≠ ∅ ∧ avhdr.sqn < valid-sqns(v, n, apebs, m).max) ⊦ ¬ eqent.pnum ∈ q0 + eq-entry(m, v × n) ' ∧ (∀ v0, m1. (valid-peb(v0, m1, subarray(apebs, 0, m + 1)[eqent.pnum]) ↔ eqent.lebref = v0 × m1) ∧ ( eqent.lebref = v0 × m1 → subarray(apebs, 0, m + 1)[eqent.pnum].vidhdr.sqn ≠ (valid-sqns(v0, m1, apebs, m) ∪ (valid-peb(v0, m1, mkpeb(aehdr, avhdr, buf, m0, boolvar)) ⊃ {avhdr.sqn} ;∅)).max)) ∧ valid-eraseq-stronger(subarray(apebs, 0, m + 1), q0 + eq-entry(m, v × n) ') ↔ valid-peb(v, n, mkpeb(aehdr, avhdr, buf, m0, boolvar)) ∧ valid-sqns(v, n, apebs, m) ≠ ∅ ∧ avhdr.sqn < valid-sqns(v, n, apebs, m).max, eqent.pnum ∈ q0

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 13:
apebs[m] = mkpeb(aehdr, avhdr, buf, m0, boolvar), eqent = eq-entry(m2, lref), m2 + 1 < # apebs, m < # apebs, m2 < m, m2 < # apebs, valid-eraseq-stronger(subarray(apebs, 0, m), q0), ∀ apebs, m, aehdr, avhdr, buf, m0, boolvar, v, n. apebs[m] = mkpeb(aehdr, avhdr, buf, m0, boolvar) ∧ m < # apebs ∧ valid-eraseq-stronger(subarray(apebs, 0, m), q0) → ( valid-eraseq-stronger(subarray(apebs, 0, m + 1), q0 + eq-entry(m, v × n) ') ↔ valid-peb(v, n, mkpeb(aehdr, avhdr, buf, m0, boolvar)) ∧ valid-sqns(v, n, apebs, m) ≠ ∅ ∧ avhdr.sqn < valid-sqns(v, n, apebs, m).max), ∀ v0, m1. (valid-peb(v0, m1, apebs[m2]) ↔ lref = v0 × m1) ∧ (lref = v0 × m1 → apebs[m2].vidhdr.sqn ≠ valid-sqns(v0, m1, apebs, m).max) ⊦ ¬ m2 ∈ q0 + eq-entry(m, v × n) ' ∧ (∀ v0, m1. (valid-peb(v0, m1, subarray(apebs, 0, m + 1)[m2]) ↔ lref = v0 × m1) ∧ ( lref = v0 × m1 → subarray(apebs, 0, m + 1)[m2].vidhdr.sqn ≠ (valid-sqns(v0, m1, apebs, m) ∪ (valid-peb(v0, m1, mkpeb(aehdr, avhdr, buf, m0, boolvar)) ⊃ {avhdr.sqn} ;∅)).max)) ∧ valid-eraseq-stronger(subarray(apebs, 0, m + 1), q0 + eq-entry(m, v × n) ') ↔ valid-peb(v, n, mkpeb(aehdr, avhdr, buf, m0, boolvar)) ∧ valid-sqns(v, n, apebs, m) ≠ ∅ ∧ avhdr.sqn < valid-sqns(v, n, apebs, m).max, m2 ∈ q0

Rule argument

spec: erasequeueentry name: elim seq: ⊦ m = eqent.pnum ∧ lref = eqent.lebref ↔ eqent = eq-entry(m, lref) subst: [eqent, m, lref] → [eqent, m2, lref]

Additional information

Used simplifier rules: ⊦ m = eqent.pnum ∧ lref = eqent.lebref ↔ eqent = eq-entry(m, lref); (elim) Used terms: eqent, m2, lref