Proof step 1 from proof step-absstep-M11-M12-tau-same (conclusion) (back to tree)

System applied rule normalize

And obtained the following new goals

• Proof step 2:

  ls0.n0 = get_hash(ls0.e, ls0.sz), ls.n0 = get_hash(ls.e, ls.sz), ls0.sz = # gs0.ar, ls.sz = # gs.ar, ls.tid = als.tid, lsf(tid).tid = tid, alsf(tid).tid = tid, M12 = pcstepf(gs, ls, M11, τ), alsf(tid) = als, apc = retMember, apcf(tid) = apc, gs0 = gstepf(gs, ls, M11, τ), ls0 = lstepf(gs, ls, M11, τ), lsf(tid) = ls, lsf0(tid) = ls0, pcf(tid) = M11, pcf0(tid) = M12, # gs.ar ≠ 0, # gs0.ar ≠ 0, ls.e ≠ ls.e0, ls.e ≠ ⊥, ls0.e ≠ ls0.e0, ls0.e ≠ ⊥, ls0.n < # gs0.ar, ls0.n0 < # gs0.ar, ls.n < # gs.ar, ls.n0 < # gs.ar, SetLInv(ags, als, apc), absset(gs.ar, ags.set), htok(gs0.ar), htok(gs.ar), pre(gs, ls, M11, τ), ¬ ls.b, ¬ ls0.b, ¬ als.lb, apc = invMember ∨ apc = invInsert → ls.e = als.le, ∀ tid0. tid0 ≠ tid → HashsetLInv(gs, lsf(tid0), pcf(tid0)) ∧ SetLInv(ags, alsf(tid0), apcf(tid0)) ∧ labs(gs, lsf(tid0), pcf(tid0), ags, alsf(tid0), apcf(tid0)), ∀ tid0. tid0 ≠ tid → alsf(tid0).tid = tid0, ∀ tid0. tid0 ≠ tid → lsf(tid0).tid = tid0 ⊦ absset(gs0.ar, ags.set) ∧ ls0.tid = als.tid ∧ apc = retMember ∧ (als.lb ↔ ls0.b) ∧ (apc = invMember ∨ apc = invInsert → ls0.e = als.le)

Rule argument

Additional information

Proof step 2 from proof step-absstep-M11-M12-tau-same (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 3:

  lsf0(tid).n0 = get_hash(lsf0(tid).e, # gs.ar), lsf(tid).n0 = get_hash(lsf(tid).e, # gs.ar), lsf0(tid).sz = # gs.ar, lsf(tid).sz = # gs.ar, lsf(tid).tid = tid, alsf(tid).tid = tid, apcf(tid) = retMember, lsf0(tid) = lsf(tid).b:= false, pcf(tid) = M11, pcf0(tid) = M12, lsf0(tid).e ≠ ⊥, lsf0(tid).e ≠ lsf0(tid).e0, lsf(tid).e ≠ ⊥, lsf(tid).e ≠ lsf(tid).e0, lsf0(tid).tid ≠ tid, lsf0(tid).n < # gs.ar, lsf0(tid).n0 < # gs.ar, lsf(tid).n < # gs.ar, lsf(tid).n0 < # gs.ar, SetLInv(ags, alsf(tid), retMember), absset(gs.ar, ags.set), htok(gs.ar), ¬ lsf0(tid).b, ¬ lsf(tid).b, ¬ alsf(tid).lb, ∀ tid0. alsf(tid0).tid = tid0, ∀ tid0. lsf(tid0).tid = tid0, ∀ tid0. tid0 ≠ tid → HashsetLInv(gs, lsf(tid0), pcf(tid0)) ∧ SetLInv(ags, alsf(tid0), apcf(tid0)) ∧ labs(gs, lsf(tid0), pcf(tid0), ags, alsf(tid0), apcf(tid0)) ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ invInsert ≠ retMember; (s) ⊦ invMember ≠ retMember; (s) ⊦ m < n → n ≠ 0; (s) lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic ⊦ gstepf(gs, ls, M11, τ) = gs; (s) ⊦ pre(gs, ls, M11, a) ↔ a = τ; (s) ⊦ pcstepf(gs, ls, M11, a) = M12; (s) ⊦ lstepf(gs, ls, M11, τ) = ls.b:= false; (s)

Proof step 3 from proof step-absstep-M11-M12-tau-same (previous step) (back to tree)

Heuristic use patterns applied rule apply elim lemma

And closed the goal!

Rule argument

Additional information

Used simplifier rules: ⊦ mkls(tid, e, b, sz, sz0, sz1, e0).b:= b0 = mkls(tid, e, b0, sz, sz0, sz1, e0); (s) lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic ⊦ mkls(tid, e, b, sz, sz0, sz1, e0).sz = sz; (s) ⊦ mkls(tid, e, b, sz, sz0, sz1, e0).n0 = sz0; (s) ⊦ mkls(tid, e, b, sz, sz0, sz1, e0).e = e; (s) ⊦ mkls(tid, e, b, sz, sz0, sz1, e0).e0 = e0; (s) ⊦ mkls(tid, e, b, sz, sz0, sz1, e0).tid = tid; (s) ⊦ (((((tid = ls.tid ∧ e = ls.e) ∧ (b ↔ ls.b)) ∧ sz = ls.sz) ∧ sz0 = ls.n0) ∧ n = ls.n) ∧ e0 = ls.e0 ↔ ls = mkls(tid, e, b, sz, sz0, n, e0); (elim) Used terms: lsf(tid), b, tid1, n0, e0, e, n, n1