Proof step 1 from proof step-absstep-M4-M5-tau-same (conclusion) (back to tree)

System applied rule normalize

And obtained the following new goals

• Proof step 2:

  ls.n = ls.n0, 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, M5 = pcstepf(gs, ls, M4, τ), alsf(tid) = als, apc = invMember, apcf(tid) = apc, gs0 = gstepf(gs, ls, M4, τ), ls0 = lstepf(gs, ls, M4, τ), lsf(tid) = ls, lsf0(tid) = ls0, pcf(tid) = M4, pcf0(tid) = M5, # gs.ar ≠ 0, # gs0.ar ≠ 0, ls.e ≠ ⊥, 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), allslotsfull(gs0.ar, ls0.n0, ls0.n, ls0.e, false), allslotsfull(gs.ar, ls.n0, ls.n, ls.e, false), htok(gs0.ar), htok(gs.ar), pre(gs, ls, M4, τ), ¬ ls0.b, 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 = invMember ∧ (apc = invMember ∨ apc = invInsert → ls0.e = als.le)

Rule argument

Additional information

Proof step 2 from proof step-absstep-M4-M5-tau-same (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 3:

  lsf(tid).e = alsf(tid).le, lsf(tid).n = lsf(tid).n0, lsf0(tid).n0 = get_hash(lsf0(tid).e, # gs.ar), lsf(tid).n0 = get_hash(alsf(tid).le, # gs.ar), lsf0(tid).sz = # gs.ar, lsf(tid).sz = # gs.ar, lsf(tid).tid = tid, alsf(tid).tid = tid, apcf(tid) = invMember, lsf0(tid) = lsf(tid).b:= false, pcf(tid) = M4, pcf0(tid) = M5, lsf0(tid).e ≠ ⊥, alsf(tid).le ≠ ⊥, lsf0(tid).n < # gs.ar, lsf0(tid).n0 < # gs.ar, lsf(tid).n0 < # gs.ar, SetLInv(ags, alsf(tid), invMember), absset(gs.ar, ags.set), allslotsfull(gs.ar, lsf0(tid).n0, lsf0(tid).n, lsf0(tid).e, false), allslotsfull(gs.ar, lsf(tid).n0, lsf(tid).n0, alsf(tid).le, false), htok(gs.ar), ¬ lsf0(tid).b, ∀ 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. alsf(tid0).tid = tid0, ∀ tid0. lsf(tid0).tid = tid0 ⊦ lsf0(tid).tid = tid ∧ lsf0(tid).e = alsf(tid).le

Rule argument

Additional information

Used simplifier rules: ⊦ 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, M4, τ) = gs; (s) ⊦ pre(gs, ls, M4, a) ↔ a = τ; (s) ⊦ pcstepf(gs, ls, M4, a) = M5; (s) ⊦ lstepf(gs, ls, M4, τ) = ls.b:= false; (s)

Proof step 3 from proof step-absstep-M4-M5-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).n = sz1; (s) ⊦ mkls(tid, e, b, sz, sz0, sz1, e0).n0 = sz0; (s) ⊦ mkls(tid, e, b, sz, sz0, sz1, e0).b ↔ b; (s) ⊦ mkls(tid, e, b, sz, sz0, sz1, e0).tid = tid; (s) ⊦ mkls(tid, e, b, sz, sz0, sz1, e0).e = e; (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