Proof step 1 from proof step-absstep-I18-I19-doINSERT-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, I19 = pcstepf(gs, ls, I18, doINSERT(ls.tid, false)), alsf(tid) = als, apc = invInsert, apcf(tid) = apc, gs0 = gstepf(gs, ls, I18, doINSERT(ls.tid, false)), ls0 = lstepf(gs, ls, I18, doINSERT(ls.tid, false)), lsf(tid) = ls, lsf0(tid) = ls0, pcf(tid) = I18, pcf0(tid) = I19, # 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.n0, ls0.e, true), allslotsfull(gs.ar, ls.n0, ls.n0, ls.e, true), htok(gs0.ar), htok(gs.ar), pre(gs, ls, I18, doINSERT(ls.tid, false)), ¬ ls.b, ¬ ls0.b, apc = invMember ∨ apc = invInsert → ls.e = als.le, pre(ags, als, apc, ADoInsert(ls.tid, false)) → ags0 = gstepf(ags, als, apc, ADoInsert(ls.tid, false)) ∧ als0 = lstepf(ags, als, apc, ADoInsert(ls.tid, false)) ∧ apc0 = pcstepf(ags, als, apc, ADoInsert(ls.tid, false)), ∀ 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 ⊦ pre(ags, als, apc, ADoInsert(ls.tid, false)) ∧ absset(gs0.ar, ags0.set) ∧ ls0.tid = als0.tid ∧ apc0 = retInsert ∧ (als0.lb ↔ ls0.b) ∧ (apc0 = invMember ∨ apc0 = invInsert → ls0.e = als0.le)

Rule argument

Additional information

Proof step 2 from proof step-absstep-I18-I19-doINSERT-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).n0 = get_hash(alsf(tid).le, # gs.ar), lsf0(tid).n0 = get_hash(lsf0(tid).e, # gs.ar), lsf(tid).sz = # gs.ar, lsf0(tid).sz = # gs.ar, alsf(tid).tid = tid, lsf(tid).tid = tid, apcf(tid) = invInsert, lsf0(tid) = lsf(tid).b:= false, pcf(tid) = I18, pcf0(tid) = I19, lsf0(tid).e ≠ ⊥, alsf(tid).le ≠ ⊥, lsf0(tid).tid ≠ tid, lsf(tid).n0 < # gs.ar, lsf(tid).n < # gs.ar, lsf0(tid).n0 < # gs.ar, lsf0(tid).n < # gs.ar, SetLInv(ags, alsf(tid), invInsert), absset(gs.ar, ags.set), allslotsfull(gs.ar, lsf(tid).n0, lsf(tid).n0, alsf(tid).le, true), allslotsfull(gs.ar, lsf0(tid).n0, lsf0(tid).n0, lsf0(tid).e, true), htok(gs.ar), ¬ lsf(tid).b, ¬ lsf0(tid).b, ∀ tid0. lsf(tid0).tid = tid0, ∀ tid0. alsf(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: ⊦ (ls.lb:= lb).lb ↔ lb; (s) ⊦ (ls.lb:= lb).tid = ls.tid; (s) ⊦ ADoInsert(tid, b) = ADoInsert(tid0, b0) ↔ tid = tid0 ∧ (b ↔ b0); (ws) ⊦ pre(gs, ls, invInsert, a) ↔ (∃ b. a = ADoInsert(ls.tid, b)); (s) ⊦ ls.tid = tid → pcstepf(gs, ls, invInsert, ADoInsert(tid, b)) = retInsert; (s) ⊦ ls.tid = tid → lstepf(gs, ls, invInsert, ADoInsert(tid, b)) = (b ⊃ ls.lb:= true;ls.lb:= false); (s) ⊦ ls.tid = tid → gstepf(gs, ls, invInsert, ADoInsert(tid, b)) = (b ⊃ gs.set:= (gs.set ++ ls.le);gs); (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 ⊦ invInsert ≠ retInsert; (s) ⊦ invMember ≠ retInsert; (s) ⊦ gstepf(gs, ls, I18, a) = gs; (s) ⊦ pre(gs, ls, I18, a) ↔ a = doINSERT(ls.tid, false); (s) ⊦ pcstepf(gs, ls, I18, a) = I19; (s) ⊦ lstepf(gs, ls, I18, a) = ls.b:= false; (s)

Proof step 3 from proof step-absstep-I18-I19-doINSERT-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).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