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Proof step 1 from proof step-absstep-M6-M7-doMEMBER-same (conclusion) (back to tree)

gs.ar[ls.n] = ls.e ∨ gs.ar[ls.n] = ⊥ ∨ (ls.n + 1) % ls.sz = ls.n0, # gs.ar ≠ 0 ∧ htok(gs.ar), # gs0.ar ≠ 0 ∧ htok(gs0.ar), lsf(tid) = ls, lsf0(tid) = ls0, alsf(tid) = als, alsf(tid).tid = tid, lsf(tid).tid = tid, ∀ tid0. tid0 ≠ tid → lsf(tid0).tid = tid0, ∀ tid0. tid0 ≠ tid → alsf(tid0).tid = tid0, pcf(tid) = M6, pcf0(tid) = M7, apcf(tid) = apc, ls.n < # gs.ar ∧ ls.e ≠ ⊥ ∧ ls.sz = # gs.ar ∧ (ls.n0 < # gs.ar ∧ ls.n0 = get_hash(ls.e, ls.sz)) ∧ ¬ ls.b ∧ allslotsfull(gs.ar, ls.n0, ls.n, ls.e, false), ls0.n < # gs0.ar ∧ ls0.e ≠ ⊥ ∧ ls0.sz = # gs0.ar ∧ (ls0.n0 < # gs0.ar ∧ ls0.n0 = get_hash(ls0.e, ls0.sz)) ∧ ¬ ls0.b ∧ (ls0.e0 ≠ ⊥ → ls0.e0 = gs0.ar[ls0.n]) ∧ allslotsfull(gs0.ar, ls0.n0, ls0.n, ls0.e, false), SetLInv(ags, als, apc), ∀ 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)), pre(ags, als, apc, ADoMember(ls.tid)) → ags0 = gstepf(ags, als, apc, ADoMember(ls.tid)) ∧ als0 = lstepf(ags, als, apc, ADoMember(ls.tid)) ∧ apc0 = pcstepf(ags, als, apc, ADoMember(ls.tid)), pre(gs, ls, M6, doMEMBER(ls.tid)) ∧ gs0 = gstepf(gs, ls, M6, doMEMBER(ls.tid)) ∧ ls0 = lstepf(gs, ls, M6, doMEMBER(ls.tid)) ∧ M7 = pcstepf(gs, ls, M6, doMEMBER(ls.tid)), absset(gs.ar, ags.set), ls.tid = als.tid ∧ apc = invMember ∧ (apc = invMember ∨ apc = invInsert → ls.e = als.le) ⊦ pre(ags, als, apc, ADoMember(ls.tid)) ∧ absset(gs0.ar, ags0.set) ∧ ls0.tid = als0.tid ∧ apc0 = (ls0.e0 ≠ ls0.e ∧ ls0.e0 ≠ ⊥ ∧ (ls0.n + 1) % # gs0.ar ≠ ls0.n0 ⊃ invMember;retMember) ∧ (ls0.e0 = ls0.e ∨ ls0.e0 = ⊥ ∨ (ls0.n + 1) % # gs0.ar = ls0.n0 → (ls0.e = ls0.e0 ↔ als0.lb)) ∧ (apc0 = invMember ∨ apc0 = invInsert → ls0.e = als0.le)

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

Proof step 2 from proof step-absstep-M6-M7-doMEMBER-same (previous step) (back to tree)

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, M7 = pcstepf(gs, ls, M6, doMEMBER(ls.tid)), alsf(tid) = als, apc = invMember, apcf(tid) = apc, gs0 = gstepf(gs, ls, M6, doMEMBER(ls.tid)), ls0 = lstepf(gs, ls, M6, doMEMBER(ls.tid)), lsf(tid) = ls, lsf0(tid) = ls0, pcf(tid) = M6, pcf0(tid) = M7, # 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, M6, doMEMBER(ls.tid)), ¬ ls.b, ¬ ls0.b, apc = invMember ∨ apc = invInsert → ls.e = als.le, pre(ags, als, apc, ADoMember(ls.tid)) → ags0 = gstepf(ags, als, apc, ADoMember(ls.tid)) ∧ als0 = lstepf(ags, als, apc, ADoMember(ls.tid)) ∧ apc0 = pcstepf(ags, als, apc, ADoMember(ls.tid)), ls0.e0 ≠ ⊥ → ls0.e0 = gs0.ar[ls0.n], ∀ 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, gs.ar[ls.n] = ls.e ∨ gs.ar[ls.n] = ⊥ ∨ (ls.n + 1) % ls.sz = ls.n0 ⊦ pre(ags, als, apc, ADoMember(ls.tid)) ∧ absset(gs0.ar, ags0.set) ∧ ls0.tid = als0.tid ∧ apc0 = (ls0.e0 ≠ ls0.e ∧ ls0.e0 ≠ ⊥ ∧ (ls0.n + 1) % # gs0.ar ≠ ls0.n0 ⊃ invMember;retMember) ∧ (ls0.e0 = ls0.e ∨ ls0.e0 = ⊥ ∨ (ls0.n + 1) % # gs0.ar = ls0.n0 → (ls0.e = ls0.e0 ↔ als0.lb)) ∧ (apc0 = invMember ∨ apc0 = invInsert → ls0.e = als0.le)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ (ls.lb:= lb).le = ls.le; (s) ⊦ (ls.lb:= lb).lb ↔ lb; (s) ⊦ invMember ≠ retMember; (s) ⊦ (ls.lb:= lb).tid = ls.tid; (s) ⊦ ADoMember(tid) = ADoMember(tid0) ↔ tid = tid0; (s) ⊦ pre(gs, ls, invMember, a) ↔ a = ADoMember(ls.tid); (s) ⊦ ls.tid = tid → pcstepf(gs, ls, invMember, ADoMember(tid)) = retMember; (s) ⊦ ls.tid = tid → lstepf(gs, ls, invMember, ADoMember(tid)) = ls.lb:= (ls.le ∈ gs.set); (s) ⊦ ls.tid = tid → gstepf(gs, ls, invMember, ADoMember(tid)) = 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 ≠ retMember; (s) ⊦ invInsert ≠ invMember; (s) ⊦ gstepf(gs, ls, M6, a) = gs; (s) ⊦ doMEMBER(tid) = doMEMBER(tid0) ↔ tid = tid0; (s) ⊦ pre(gs, ls, M6, a) ↔ a = (gs.ar[ls.n] = ls.e ∨ gs.ar[ls.n] = ⊥ ∨ (ls.n + 1) % ls.sz = ls.n0 ⊃ doMEMBER(ls.tid) ;τ ); (s) ⊦ pcstepf(gs, ls, M6, a) = M7; (s) ⊦ lstepf(gs, ls, M6, a) = ls.e0:= gs.ar[ls.n]; (s)

Proof step 3 from proof step-absstep-M6-M7-doMEMBER-same (previous step) (back to tree)

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) = invMember, lsf0(tid) = lsf(tid).e0:= gs.ar[lsf(tid).n], pcf(tid) = M6, pcf0(tid) = M7, lsf0(tid).e ≠ ⊥, alsf(tid).le ≠ ⊥, lsf(tid).n0 < # gs.ar, lsf(tid).n < # gs.ar, lsf0(tid).n0 < # gs.ar, lsf0(tid).n < # gs.ar, SetLInv(ags, alsf(tid), invMember), absset(gs.ar, ags.set), allslotsfull(gs.ar, lsf(tid).n0, lsf(tid).n, alsf(tid).le, false), allslotsfull(gs.ar, lsf0(tid).n0, lsf0(tid).n, lsf0(tid).e, false), htok(gs.ar), ¬ lsf(tid).b, ¬ lsf0(tid).b, lsf0(tid).e0 ≠ ⊥ → lsf0(tid).e0 = gs.ar[lsf0(tid).n], ∀ 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)), gs.ar[lsf(tid).n] = alsf(tid).le ∨ gs.ar[lsf(tid).n] = ⊥ ∨ (lsf(tid).n + 1) % # gs.ar = lsf(tid).n0 ⊦ lsf0(tid).tid = tid ∧ (lsf0(tid).e0 = lsf0(tid).e ∨ lsf0(tid).e0 = ⊥ ∨ (lsf0(tid).n + 1) % # gs.ar = lsf0(tid).n0) ∧ (lsf0(tid).e = lsf0(tid).e0 ↔ alsf(tid).le ∈ ags.set) ∧ (lsf0(tid).e0 ≠ lsf0(tid).e ∧ lsf0(tid).e0 ≠ ⊥ ∧ (lsf0(tid).n + 1) % # gs.ar ≠ lsf0(tid).n0 → lsf0(tid).e = alsf(tid).le)

Heuristic use patterns applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: Hashset, name:elim-mkls, seq:⊦ ( ((((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), subst:[ls, b, tid, sz0, e0, e, sz, n] → [lsf(tid), b, tid1, n0, e0, e, n, n1]

Additional information

Used simplifier rules: ⊦ (((((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

Proof step 4 from proof step-absstep-M6-M7-doMEMBER-same (previous step) (back to tree)

lsf0(tid).n0 = get_hash(lsf0(tid).e, # gs.ar), lsf0(tid).sz = # gs.ar, alsf(tid).tid = tid, apcf(tid) = invMember, ¬ b, e = alsf(tid).le, lsf(tid) = mkls(tid1, e, b, n, n0, n1, e0), lsf0(tid) = lsf(tid).e0:= gs.ar[n1], n = # gs.ar, n0 = get_hash(alsf(tid).le, # gs.ar), pcf(tid) = M6, pcf0(tid) = M7, tid1 = tid, lsf0(tid).e ≠ ⊥, alsf(tid).le ≠ ⊥, lsf0(tid).n < # gs.ar, lsf0(tid).n0 < # gs.ar, n1 < # gs.ar, 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, n0, n1, alsf(tid).le, false), htok(gs.ar), ¬ lsf0(tid).b, lsf0(tid).e0 ≠ ⊥ → lsf0(tid).e0 = gs.ar[lsf0(tid).n], ∀ 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, gs.ar[n1] = alsf(tid).le ∨ gs.ar[n1] = ⊥ ∨ (n1 + 1) % # gs.ar = n0 ⊦ lsf0(tid).tid = tid ∧ (lsf0(tid).e0 = lsf0(tid).e ∨ lsf0(tid).e0 = ⊥ ∨ (lsf0(tid).n + 1) % # gs.ar = lsf0(tid).n0) ∧ (lsf0(tid).e = lsf0(tid).e0 ↔ alsf(tid).le ∈ ags.set) ∧ (lsf0(tid).e0 ≠ lsf0(tid).e ∧ lsf0(tid).e0 ≠ ⊥ ∧ (lsf0(tid).n + 1) % # gs.ar ≠ lsf0(tid).n0 → lsf0(tid).e = alsf(tid).le)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ mkls(tid, e, b, sz, sz0, sz1, e0).e = e; (s) ⊦ mkls(tid, e, b, sz, sz0, sz1, e0).n0 = sz0; (s) ⊦ mkls(tid, e, b, sz, sz0, sz1, e0).n = sz1; (s) ⊦ mkls(tid, e, b, sz, sz0, sz1, e0).e0 = e0; (s) ⊦ mkls(tid, e, b, sz, sz0, sz1, e0).tid = tid; (s) ⊦ mkls(tid, e, b, sz, sz0, sz1, e0).b ↔ b; (s) ⊦ mkls(tid, e, b, sz, sz0, sz1, e0).sz = sz; (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 ⊦ mkls(tid, e, b, sz, sz0, sz1, e0).e0:= e2 = mkls(tid, e, b, sz, sz0, sz1, e2); (s)

Proof step 5 from proof step-absstep-M6-M7-doMEMBER-same (previous step) (back to tree)

alsf(tid).tid = tid, apcf(tid) = invMember, lsf(tid) = mkls(tid, alsf(tid).le, false, # gs.ar, get_hash(alsf(tid).le, # gs.ar), n1, e0), lsf0(tid) = mkls(tid, alsf(tid).le, false, # gs.ar, get_hash(alsf(tid).le, # gs.ar), n1, gs.ar[n1]), pcf(tid) = M6, pcf0(tid) = M7, alsf(tid).le ≠ ⊥, ¬(alsf(tid).le = gs.ar[n1] ↔ alsf(tid).le ∈ ags.set), get_hash(alsf(tid).le, # gs.ar) < # gs.ar, n1 < # gs.ar, SetLInv(ags, alsf(tid), invMember), absset(gs.ar, ags.set), allslotsfull(gs.ar, get_hash(alsf(tid).le, # gs.ar), n1, alsf(tid).le, false), htok(gs.ar), ∀ 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)), gs.ar[n1] = alsf(tid).le ∨ gs.ar[n1] = ⊥ ∨ (n1 + 1) % # gs.ar = get_hash(alsf(tid).le, # gs.ar) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: abs, name:absset-def, seq: ⊦ absset(ar, set) ↔ (∀ e. e ∈ set ↔ (∃ n. n < # ar ∧ e = ar[n] ∧ e ≠ ⊥)), subst:[ar, set] → [gs.ar, ags.set], <all paths>

Additional information

Used terms: gs.ar, ags.set

Proof step 6 from proof step-absstep-M6-M7-doMEMBER-same (previous step) (back to tree)

⊦ absset(ar, set) ↔ (∀ e. e ∈ set ↔ (∃ n. n < # ar ∧ e = ar[n] ∧ e ≠ ⊥))

A spec-lemma

Proof step 7 from proof step-absstep-M6-M7-doMEMBER-same (previous step) (back to tree)

alsf(tid).tid = tid, apcf(tid) = invMember, lsf(tid) = mkls(tid, alsf(tid).le, false, # gs.ar, get_hash(alsf(tid).le, # gs.ar), n1, e0), lsf0(tid) = mkls(tid, alsf(tid).le, false, # gs.ar, get_hash(alsf(tid).le, # gs.ar), n1, gs.ar[n1]), pcf(tid) = M6, pcf0(tid) = M7, alsf(tid).le ≠ ⊥, ¬(alsf(tid).le = gs.ar[n1] ↔ alsf(tid).le ∈ ags.set), n1 < # gs.ar, get_hash(alsf(tid).le, # gs.ar) < # gs.ar, SetLInv(ags, alsf(tid), invMember), allslotsfull(gs.ar, get_hash(alsf(tid).le, # gs.ar), n1, alsf(tid).le, false), htok(gs.ar), ∀ 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, ∀ e. e ∈ ags.set ↔ (∃ n. n < # gs.ar ∧ e = gs.ar[n] ∧ e ≠ ⊥), gs.ar[n1] = alsf(tid).le ∨ gs.ar[n1] = ⊥ ∨ (n1 + 1) % # gs.ar = get_hash(alsf(tid).le, # gs.ar) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 8 from proof step-absstep-M6-M7-doMEMBER-same (previous step) (back to tree)

alsf(tid).tid = tid, apcf(tid) = invMember, lsf(tid) = mkls(tid, alsf(tid).le, false, # gs.ar, get_hash(alsf(tid).le, # gs.ar), n1, e0), lsf0(tid) = mkls(tid, alsf(tid).le, false, # gs.ar, get_hash(alsf(tid).le, # gs.ar), n1, gs.ar[n1]), pcf(tid) = M6, pcf0(tid) = M7, alsf(tid).le ≠ ⊥, ¬(alsf(tid).le = gs.ar[n1] ↔ alsf(tid).le ∈ ags.set), get_hash(alsf(tid).le, # gs.ar) < # gs.ar, n1 < # gs.ar, SetLInv(ags, alsf(tid), invMember), allslotsfull(gs.ar, get_hash(alsf(tid).le, # gs.ar), n1, alsf(tid).le, false), htok(gs.ar), ∀ 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)), ∀ e. e ∈ ags.set ↔ (∃ n. n < # gs.ar ∧ e = gs.ar[n] ∧ gs.ar[n] ≠ ⊥), gs.ar[n1] = alsf(tid).le ∨ gs.ar[n1] = ⊥ ∨ (n1 + 1) % # gs.ar = get_hash(alsf(tid).le, # gs.ar) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 17 terms: alsf(tid).le abort: F discard: F computed: T

Additional information

Used terms: alsf(tid).le

Proof step 9 from proof step-absstep-M6-M7-doMEMBER-same (previous step) (back to tree)

alsf(tid).tid = tid, apcf(tid) = invMember, lsf(tid) = mkls(tid, alsf(tid).le, false, # gs.ar, get_hash(alsf(tid).le, # gs.ar), n1, e0), lsf0(tid) = mkls(tid, alsf(tid).le, false, # gs.ar, get_hash(alsf(tid).le, # gs.ar), n1, gs.ar[n1]), pcf(tid) = M6, pcf0(tid) = M7, alsf(tid).le ≠ ⊥, ¬(alsf(tid).le = gs.ar[n1] ↔ alsf(tid).le ∈ ags.set), n1 < # gs.ar, get_hash(alsf(tid).le, # gs.ar) < # gs.ar, SetLInv(ags, alsf(tid), invMember), allslotsfull(gs.ar, get_hash(alsf(tid).le, # gs.ar), n1, alsf(tid).le, false), htok(gs.ar), alsf(tid).le ∈ ags.set ↔ (∃ n. n < # gs.ar ∧ alsf(tid).le = gs.ar[n] ∧ gs.ar[n] ≠ ⊥), ∀ e. e ∈ ags.set ↔ (∃ n. n < # gs.ar ∧ e = gs.ar[n] ∧ gs.ar[n] ≠ ⊥), ∀ 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, gs.ar[n1] = alsf(tid).le ∨ gs.ar[n1] = ⊥ ∨ (n1 + 1) % # gs.ar = get_hash(alsf(tid).le, # gs.ar) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 10 from proof step-absstep-M6-M7-doMEMBER-same (previous step) (back to tree)

alsf(tid).tid = tid, apcf(tid) = invMember, lsf(tid) = mkls(tid, alsf(tid).le, false, # gs.ar, get_hash(alsf(tid).le, # gs.ar), n1, e0), lsf0(tid) = mkls(tid, alsf(tid).le, false, # gs.ar, get_hash(alsf(tid).le, # gs.ar), n1, gs.ar[n1]), pcf(tid) = M6, pcf0(tid) = M7, alsf(tid).le ≠ ⊥, ¬(alsf(tid).le = gs.ar[n1] ↔ alsf(tid).le ∈ ags.set), get_hash(alsf(tid).le, # gs.ar) < # gs.ar, n1 < # gs.ar, SetLInv(ags, alsf(tid), invMember), allslotsfull(gs.ar, get_hash(alsf(tid).le, # gs.ar), n1, alsf(tid).le, false), htok(gs.ar), alsf(tid).le ∈ ags.set ↔ (∃ n. n < # gs.ar ∧ alsf(tid).le = gs.ar[n] ∧ gs.ar[n] ≠ ⊥), ∀ 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)), ∀ e. e ∈ ags.set ↔ (∃ n. n < # gs.ar ∧ e = gs.ar[n] ∧ gs.ar[n] ≠ ⊥), gs.ar[n1] = alsf(tid).le ∨ gs.ar[n1] = ⊥ ∨ (n1 + 1) % # gs.ar = get_hash(alsf(tid).le, # gs.ar) ⊦

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 19

Additional information

Proof step 11 from proof step-absstep-M6-M7-doMEMBER-same (previous step) (back to tree)

alsf(tid).tid = tid, gs.ar[n1] = alsf(tid).le, apcf(tid) = invMember, lsf(tid) = mkls(tid, alsf(tid).le, false, # gs.ar, get_hash(alsf(tid).le, # gs.ar), n1, e0), lsf0(tid) = mkls(tid, alsf(tid).le, false, # gs.ar, get_hash(alsf(tid).le, # gs.ar), n1, gs.ar[n1]), pcf(tid) = M6, pcf0(tid) = M7, alsf(tid).le ≠ ⊥, ¬(alsf(tid).le = gs.ar[n1] ↔ alsf(tid).le ∈ ags.set), n1 < # gs.ar, get_hash(alsf(tid).le, # gs.ar) < # gs.ar, SetLInv(ags, alsf(tid), invMember), allslotsfull(gs.ar, get_hash(alsf(tid).le, # gs.ar), n1, alsf(tid).le, false), htok(gs.ar), alsf(tid).le ∈ ags.set ↔ (∃ n. n < # gs.ar ∧ alsf(tid).le = gs.ar[n] ∧ gs.ar[n] ≠ ⊥), ∀ e. e ∈ ags.set ↔ (∃ n. n < # gs.ar ∧ e = gs.ar[n] ∧ gs.ar[n] ≠ ⊥), ∀ 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 ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 12 from proof step-absstep-M6-M7-doMEMBER-same (previous step) (back to tree)

alsf(tid).tid = tid, gs.ar[n1] = alsf(tid).le, apcf(tid) = invMember, lsf(tid) = mkls(tid, gs.ar[n1], false, # gs.ar, get_hash(gs.ar[n1], # gs.ar), n1, e0), lsf0(tid) = mkls(tid, gs.ar[n1], false, # gs.ar, get_hash(gs.ar[n1], # gs.ar), n1, gs.ar[n1]), pcf(tid) = M6, pcf0(tid) = M7, gs.ar[n1] ≠ ⊥, get_hash(gs.ar[n1], # gs.ar) < # gs.ar, n1 < # gs.ar, SetLInv(ags, alsf(tid), invMember), allslotsfull(gs.ar, get_hash(gs.ar[n1], # gs.ar), n1, gs.ar[n1], false), htok(gs.ar), ¬ gs.ar[n1] ∈ ags.set, ∀ 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)), ∀ e. e ∈ ags.set ↔ (∃ n. n < # gs.ar ∧ e = gs.ar[n] ∧ gs.ar[n] ≠ ⊥) ⊦ ∃ n. n < # gs.ar ∧ gs.ar[n1] = gs.ar[n] ∧ gs.ar[n] ≠ ⊥

Heuristic quantifier closing applied rule exists right

And closed the goal!

Rule argument

pos: right 1 terms: n1

Additional information

Used terms: n1

Proof step 13 from proof step-absstep-M6-M7-doMEMBER-same (previous step) (back to tree)

alsf(tid).tid = tid, gs.ar[n1] = ⊥, apcf(tid) = invMember, lsf(tid) = mkls(tid, alsf(tid).le, false, # gs.ar, get_hash(alsf(tid).le, # gs.ar), n1, e0), lsf0(tid) = mkls(tid, alsf(tid).le, false, # gs.ar, get_hash(alsf(tid).le, # gs.ar), n1, gs.ar[n1]), pcf(tid) = M6, pcf0(tid) = M7, alsf(tid).le ≠ ⊥, ¬(alsf(tid).le = gs.ar[n1] ↔ alsf(tid).le ∈ ags.set), n1 < # gs.ar, get_hash(alsf(tid).le, # gs.ar) < # gs.ar, SetLInv(ags, alsf(tid), invMember), allslotsfull(gs.ar, get_hash(alsf(tid).le, # gs.ar), n1, alsf(tid).le, false), htok(gs.ar), alsf(tid).le ∈ ags.set ↔ (∃ n. n < # gs.ar ∧ alsf(tid).le = gs.ar[n] ∧ gs.ar[n] ≠ ⊥), ∀ e. e ∈ ags.set ↔ (∃ n. n < # gs.ar ∧ e = gs.ar[n] ∧ gs.ar[n] ≠ ⊥), ∀ 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 ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: size-zero-01: ⊦ s ≠ ∅ → # s ≠ 0; (forward) from specification set-basic htok-returnfalse-f: n < # ar ⊦ htok(ar) ∧ ar[n] = ⊥ ∧ e ≠ ⊥ ∧ allslotsfull(ar, get_hash(e, # ar), n, e, false) → (∀ m. m < # ar → ar[m] ≠ e); (forward) from specification hashset-preds ⊦ m < n → (n ≤ m ↔ false); (s) ⊦ m < n ↔ ¬ n ≤ m; (ss) ne-01: ⊦ a ∈ s → s ≠ ∅; (forward) from specification set-basic

Proof step 14 from proof step-absstep-M6-M7-doMEMBER-same (previous step) (back to tree)

alsf(tid).le = gs.ar[n], alsf(tid).tid = tid, gs.ar[n1] = ⊥, apcf(tid) = invMember, lsf(tid) = mkls(tid, gs.ar[n], false, # gs.ar, get_hash(gs.ar[n], # gs.ar), n1, e0), lsf0(tid) = mkls(tid, gs.ar[n], false, # gs.ar, get_hash(gs.ar[n], # gs.ar), n1, ⊥), pcf(tid) = M6, pcf0(tid) = M7, # ags.set ≠ 0, ags.set ≠ ∅, gs.ar[n] ≠ ⊥, n < # gs.ar, get_hash(gs.ar[n], # gs.ar) < # gs.ar, n1 < # gs.ar, SetLInv(ags, alsf(tid), invMember), allslotsfull(gs.ar, get_hash(gs.ar[n], # gs.ar), n1, gs.ar[n], false), htok(gs.ar), gs.ar[n] ∈ ags.set, ∀ m. m < # gs.ar → gs.ar[m] ≠ gs.ar[n], ∀ 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)), ∀ e. e ∈ ags.set ↔ (∃ n. n < # gs.ar ∧ e = gs.ar[n] ∧ gs.ar[n] ≠ ⊥) ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 19 terms: n

Additional information

Used terms: n

Proof step 15 from proof step-absstep-M6-M7-doMEMBER-same (previous step) (back to tree)

(n1 + 1) % # gs.ar = get_hash(alsf(tid).le, # gs.ar), alsf(tid).tid = tid, apcf(tid) = invMember, lsf(tid) = mkls(tid, alsf(tid).le, false, # gs.ar, get_hash(alsf(tid).le, # gs.ar), n1, e0), lsf0(tid) = mkls(tid, alsf(tid).le, false, # gs.ar, get_hash(alsf(tid).le, # gs.ar), n1, gs.ar[n1]), pcf(tid) = M6, pcf0(tid) = M7, alsf(tid).le ≠ ⊥, ¬(alsf(tid).le = gs.ar[n1] ↔ alsf(tid).le ∈ ags.set), n1 < # gs.ar, get_hash(alsf(tid).le, # gs.ar) < # gs.ar, SetLInv(ags, alsf(tid), invMember), allslotsfull(gs.ar, get_hash(alsf(tid).le, # gs.ar), n1, alsf(tid).le, false), htok(gs.ar), alsf(tid).le ∈ ags.set ↔ (∃ n. n < # gs.ar ∧ alsf(tid).le = gs.ar[n] ∧ gs.ar[n] ≠ ⊥), ∀ e. e ∈ ags.set ↔ (∃ n. n < # gs.ar ∧ e = gs.ar[n] ∧ gs.ar[n] ≠ ⊥), ∀ 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 ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: n ≠ 0 ⊦ m % n < n; (s) ⊦ m < n → n ≠ 0; (s) lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic

Proof step 16 from proof step-absstep-M6-M7-doMEMBER-same (previous step) (back to tree)

(n1 + 1) % # gs.ar = get_hash(alsf(tid).le, # gs.ar), alsf(tid).tid = tid, apcf(tid) = invMember, lsf(tid) = mkls(tid, alsf(tid).le, false, # gs.ar, get_hash(alsf(tid).le, # gs.ar), n1, e0), lsf0(tid) = mkls(tid, alsf(tid).le, false, # gs.ar, get_hash(alsf(tid).le, # gs.ar), n1, gs.ar[n1]), pcf(tid) = M6, pcf0(tid) = M7, alsf(tid).le ≠ ⊥, ¬(alsf(tid).le = gs.ar[n1] ↔ alsf(tid).le ∈ ags.set), n1 < # gs.ar, get_hash(alsf(tid).le, # gs.ar) < # gs.ar, SetLInv(ags, alsf(tid), invMember), allslotsfull(gs.ar, get_hash(alsf(tid).le, # gs.ar), n1, alsf(tid).le, false), htok(gs.ar), alsf(tid).le ∈ ags.set ↔ (∃ n. n < # gs.ar ∧ alsf(tid).le = gs.ar[n] ∧ gs.ar[n] ≠ ⊥), ∀ e. e ∈ ags.set ↔ (∃ n. n < # gs.ar ∧ e = gs.ar[n] ∧ gs.ar[n] ≠ ⊥), ∀ 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 ⊦

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 9

Additional information

Proof step 17 from proof step-absstep-M6-M7-doMEMBER-same (previous step) (back to tree)

(n1 + 1) % # gs.ar = get_hash(alsf(tid).le, # gs.ar), alsf(tid).le = gs.ar[n1], alsf(tid).tid = tid, apcf(tid) = invMember, lsf(tid) = mkls(tid, alsf(tid).le, false, # gs.ar, get_hash(alsf(tid).le, # gs.ar), n1, e0), lsf0(tid) = mkls(tid, alsf(tid).le, false, # gs.ar, get_hash(alsf(tid).le, # gs.ar), n1, gs.ar[n1]), pcf(tid) = M6, pcf0(tid) = M7, alsf(tid).le ≠ ⊥, get_hash(alsf(tid).le, # gs.ar) < # gs.ar, n1 < # gs.ar, SetLInv(ags, alsf(tid), invMember), allslotsfull(gs.ar, get_hash(alsf(tid).le, # gs.ar), n1, alsf(tid).le, false), htok(gs.ar), ¬ alsf(tid).le ∈ ags.set, alsf(tid).le ∈ ags.set ↔ (∃ n. n < # gs.ar ∧ alsf(tid).le = gs.ar[n] ∧ gs.ar[n] ≠ ⊥), ∀ 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)), ∀ e. e ∈ ags.set ↔ (∃ n. n < # gs.ar ∧ e = gs.ar[n] ∧ gs.ar[n] ≠ ⊥) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 18 from proof step-absstep-M6-M7-doMEMBER-same (previous step) (back to tree)

(n1 + 1) % # gs.ar = get_hash(gs.ar[n1], # gs.ar), alsf(tid).le = gs.ar[n1], alsf(tid).tid = tid, apcf(tid) = invMember, lsf(tid) = mkls(tid, gs.ar[n1], false, # gs.ar, get_hash(gs.ar[n1], # gs.ar), n1, e0), lsf0(tid) = mkls(tid, gs.ar[n1], false, # gs.ar, get_hash(gs.ar[n1], # gs.ar), n1, gs.ar[n1]), pcf(tid) = M6, pcf0(tid) = M7, gs.ar[n1] ≠ ⊥, n1 < # gs.ar, get_hash(gs.ar[n1], # gs.ar) < # gs.ar, SetLInv(ags, alsf(tid), invMember), allslotsfull(gs.ar, get_hash(gs.ar[n1], # gs.ar), n1, gs.ar[n1], false), htok(gs.ar), ¬ gs.ar[n1] ∈ ags.set, ∀ e. e ∈ ags.set ↔ (∃ n. n < # gs.ar ∧ e = gs.ar[n] ∧ gs.ar[n] ≠ ⊥), ∀ 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 ⊦ ∃ n. n < # gs.ar ∧ gs.ar[n1] = gs.ar[n] ∧ gs.ar[n] ≠ ⊥

Heuristic quantifier closing applied rule exists right

And closed the goal!

Rule argument

pos: right 1 terms: n1

Additional information

Used terms: n1

Proof step 19 from proof step-absstep-M6-M7-doMEMBER-same (previous step) (back to tree)

(n1 + 1) % # gs.ar = get_hash(alsf(tid).le, # gs.ar), alsf(tid).tid = tid, apcf(tid) = invMember, lsf(tid) = mkls(tid, alsf(tid).le, false, # gs.ar, get_hash(alsf(tid).le, # gs.ar), n1, e0), lsf0(tid) = mkls(tid, alsf(tid).le, false, # gs.ar, get_hash(alsf(tid).le, # gs.ar), n1, gs.ar[n1]), pcf(tid) = M6, pcf0(tid) = M7, alsf(tid).le ≠ gs.ar[n1], alsf(tid).le ≠ ⊥, get_hash(alsf(tid).le, # gs.ar) < # gs.ar, n1 < # gs.ar, SetLInv(ags, alsf(tid), invMember), allslotsfull(gs.ar, get_hash(alsf(tid).le, # gs.ar), n1, alsf(tid).le, false), htok(gs.ar), alsf(tid).le ∈ ags.set, alsf(tid).le ∈ ags.set ↔ (∃ n. n < # gs.ar ∧ alsf(tid).le = gs.ar[n] ∧ gs.ar[n] ≠ ⊥), ∀ 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)), ∀ e. e ∈ ags.set ↔ (∃ n. n < # gs.ar ∧ e = gs.ar[n] ∧ gs.ar[n] ≠ ⊥) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: size-zero-01: ⊦ s ≠ ∅ → # s ≠ 0; (forward) from specification set-basic ne-01: ⊦ a ∈ s → s ≠ ∅; (forward) from specification set-basic

Proof step 20 from proof step-absstep-M6-M7-doMEMBER-same (previous step) (back to tree)

(n1 + 1) % # gs.ar = get_hash(gs.ar[n], # gs.ar), alsf(tid).le = gs.ar[n], alsf(tid).tid = tid, apcf(tid) = invMember, lsf(tid) = mkls(tid, gs.ar[n], false, # gs.ar, get_hash(gs.ar[n], # gs.ar), n1, e0), lsf0(tid) = mkls(tid, gs.ar[n], false, # gs.ar, get_hash(gs.ar[n], # gs.ar), n1, gs.ar[n1]), pcf(tid) = M6, pcf0(tid) = M7, # ags.set ≠ 0, ags.set ≠ ∅, gs.ar[n] ≠ ⊥, gs.ar[n] ≠ gs.ar[n1], n < # gs.ar, get_hash(gs.ar[n], # gs.ar) < # gs.ar, n1 < # gs.ar, SetLInv(ags, alsf(tid), invMember), allslotsfull(gs.ar, get_hash(gs.ar[n], # gs.ar), n1, gs.ar[n], false), htok(gs.ar), gs.ar[n] ∈ ags.set, ∀ 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)), ∀ e. e ∈ ags.set ↔ (∃ n. n < # gs.ar ∧ e = gs.ar[n] ∧ gs.ar[n] ≠ ⊥) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: hashset-preds, name:allslotsfull-def, seq:⊦ allslotsfull(ar, n0, n, e, b) ↔ (∀ m. between(n0, m, n, b) ∧ m < # ar → ar[m] ≠ e ∧ ar[m] ≠ ⊥), subst:[n0, b, ar, e, n] → [get_hash(gs.ar[n], # gs.ar), false, gs.ar, gs.ar[n], n1], <all paths>

Additional information

Used terms: get_hash(gs.ar[n], # gs.ar), false, gs.ar, gs.ar[n], n1

Proof step 21 from proof step-absstep-M6-M7-doMEMBER-same (previous step) (back to tree)

⊦ allslotsfull(ar, n0, n, e, b) ↔ (∀ m. between(n0, m, n, b) ∧ m < # ar → ar[m] ≠ e ∧ ar[m] ≠ ⊥)

A spec-lemma

Proof step 22 from proof step-absstep-M6-M7-doMEMBER-same (previous step) (back to tree)

(n1 + 1) % # gs.ar = get_hash(gs.ar[n], # gs.ar), alsf(tid).le = gs.ar[n], alsf(tid).tid = tid, apcf(tid) = invMember, lsf(tid) = mkls(tid, gs.ar[n], false, # gs.ar, get_hash(gs.ar[n], # gs.ar), n1, e0), lsf0(tid) = mkls(tid, gs.ar[n], false, # gs.ar, get_hash(gs.ar[n], # gs.ar), n1, gs.ar[n1]), pcf(tid) = M6, pcf0(tid) = M7, # ags.set ≠ 0, ags.set ≠ ∅, gs.ar[n] ≠ ⊥, gs.ar[n] ≠ gs.ar[n1], n1 < # gs.ar, get_hash(gs.ar[n], # gs.ar) < # gs.ar, n < # gs.ar, SetLInv(ags, alsf(tid), invMember), htok(gs.ar), gs.ar[n] ∈ ags.set, ∀ e. e ∈ ags.set ↔ (∃ n. n < # gs.ar ∧ e = gs.ar[n] ∧ gs.ar[n] ≠ ⊥), ∀ 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, ∀ m. between(get_hash(gs.ar[n], # gs.ar), m, n1, false) ∧ m < # gs.ar → gs.ar[m] ≠ gs.ar[n] ∧ gs.ar[m] ≠ ⊥ ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 23 terms: n abort: F discard: F computed: T

Additional information

Used terms: n

Proof step 23 from proof step-absstep-M6-M7-doMEMBER-same (previous step) (back to tree)

(n1 + 1) % # gs.ar = get_hash(gs.ar[n], # gs.ar), alsf(tid).le = gs.ar[n], alsf(tid).tid = tid, apcf(tid) = invMember, lsf(tid) = mkls(tid, gs.ar[n], false, # gs.ar, get_hash(gs.ar[n], # gs.ar), n1, e0), lsf0(tid) = mkls(tid, gs.ar[n], false, # gs.ar, get_hash(gs.ar[n], # gs.ar), n1, gs.ar[n1]), pcf(tid) = M6, pcf0(tid) = M7, # ags.set ≠ 0, ags.set ≠ ∅, gs.ar[n] ≠ ⊥, gs.ar[n] ≠ gs.ar[n1], n < # gs.ar, get_hash(gs.ar[n], # gs.ar) < # gs.ar, n1 < # gs.ar, SetLInv(ags, alsf(tid), invMember), htok(gs.ar), gs.ar[n] ∈ ags.set, between(get_hash(gs.ar[n], # gs.ar), n, n1, false) ∧ n < # gs.ar → gs.ar[n] ≠ gs.ar[n] ∧ gs.ar[n] ≠ ⊥, ∀ m. between(get_hash(gs.ar[n], # gs.ar), m, n1, false) ∧ m < # gs.ar → gs.ar[m] ≠ gs.ar[n] ∧ gs.ar[m] ≠ ⊥, ∀ 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)), ∀ e. e ∈ ags.set ↔ (∃ n. n < # gs.ar ∧ e = gs.ar[n] ∧ gs.ar[n] ≠ ⊥) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 24 from proof step-absstep-M6-M7-doMEMBER-same (previous step) (back to tree)

(n1 + 1) % # gs.ar = get_hash(gs.ar[n], # gs.ar), alsf(tid).le = gs.ar[n], alsf(tid).tid = tid, apcf(tid) = invMember, lsf(tid) = mkls(tid, gs.ar[n], false, # gs.ar, get_hash(gs.ar[n], # gs.ar), n1, e0), lsf0(tid) = mkls(tid, gs.ar[n], false, # gs.ar, get_hash(gs.ar[n], # gs.ar), n1, gs.ar[n1]), pcf(tid) = M6, pcf0(tid) = M7, # ags.set ≠ 0, ags.set ≠ ∅, gs.ar[n] ≠ ⊥, gs.ar[n] ≠ gs.ar[n1], n1 < # gs.ar, get_hash(gs.ar[n], # gs.ar) < # gs.ar, n < # gs.ar, SetLInv(ags, alsf(tid), invMember), htok(gs.ar), gs.ar[n] ∈ ags.set, ¬ between(get_hash(gs.ar[n], # gs.ar), n, n1, false), ∀ e. e ∈ ags.set ↔ (∃ n. n < # gs.ar ∧ e = gs.ar[n] ∧ gs.ar[n] ≠ ⊥), ∀ 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, ∀ m. between(get_hash(gs.ar[n], # gs.ar), m, n1, false) ∧ m < # gs.ar → gs.ar[m] ≠ gs.ar[n] ∧ gs.ar[m] ≠ ⊥ ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: hashset-preds, name:between-def, seq:⊦ between(n0, m, n, b) ↔ n0 = n ∧ b ∨ (n < n0 ⊃ m < n ∨ n0 ≤ m;n0 ≤ m ∧ m < n), subst:[n0, m, n, b] → [get_hash(gs.ar[n], # gs.ar), n, n1, false], <all paths>

Additional information

Used terms: get_hash(gs.ar[n], # gs.ar), n, n1, false

Proof step 25 from proof step-absstep-M6-M7-doMEMBER-same (previous step) (back to tree)

⊦ between(n0, m, n, b) ↔ n0 = n ∧ b ∨ (n < n0 ⊃ m < n ∨ n0 ≤ m;n0 ≤ m ∧ m < n)

A spec-lemma

Proof step 26 from proof step-absstep-M6-M7-doMEMBER-same (previous step) (back to tree)

(n1 + 1) % # gs.ar = get_hash(gs.ar[n], # gs.ar), alsf(tid).le = gs.ar[n], alsf(tid).tid = tid, apcf(tid) = invMember, lsf(tid) = mkls(tid, gs.ar[n], false, # gs.ar, get_hash(gs.ar[n], # gs.ar), n1, e0), lsf0(tid) = mkls(tid, gs.ar[n], false, # gs.ar, get_hash(gs.ar[n], # gs.ar), n1, gs.ar[n1]), pcf(tid) = M6, pcf0(tid) = M7, # ags.set ≠ 0, ags.set ≠ ∅, gs.ar[n] ≠ ⊥, gs.ar[n] ≠ gs.ar[n1], n < # gs.ar, get_hash(gs.ar[n], # gs.ar) < # gs.ar, n1 < # gs.ar, SetLInv(ags, alsf(tid), invMember), htok(gs.ar), gs.ar[n] ∈ ags.set, (n1 < get_hash(gs.ar[n], # gs.ar) ⊃ ¬ n < n1 ∧ ¬ get_hash(gs.ar[n], # gs.ar) ≤ n;¬ get_hash(gs.ar[n], # gs.ar) ≤ n ∨ ¬ n < n1), ∀ m. between(get_hash(gs.ar[n], # gs.ar), m, n1, false) ∧ m < # gs.ar → gs.ar[m] ≠ gs.ar[n] ∧ gs.ar[m] ≠ ⊥, ∀ 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)), ∀ e. e ∈ ags.set ↔ (∃ n. n < # gs.ar ∧ e = gs.ar[n] ∧ gs.ar[n] ≠ ⊥), get_hash(gs.ar[n], # gs.ar) ≠ n1 ∨ true ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 27 from proof step-absstep-M6-M7-doMEMBER-same (previous step) (back to tree)

(n1 + 1) % # gs.ar = get_hash(gs.ar[n], # gs.ar), alsf(tid).le = gs.ar[n], alsf(tid).tid = tid, apcf(tid) = invMember, lsf(tid) = mkls(tid, gs.ar[n], false, # gs.ar, get_hash(gs.ar[n], # gs.ar), n1, e0), lsf0(tid) = mkls(tid, gs.ar[n], false, # gs.ar, get_hash(gs.ar[n], # gs.ar), n1, gs.ar[n1]), pcf(tid) = M6, pcf0(tid) = M7, # ags.set ≠ 0, ags.set ≠ ∅, gs.ar[n] ≠ ⊥, gs.ar[n] ≠ gs.ar[n1], n1 < # gs.ar, get_hash(gs.ar[n], # gs.ar) < # gs.ar, n < # gs.ar, SetLInv(ags, alsf(tid), invMember), htok(gs.ar), gs.ar[n] ∈ ags.set, (n1 < get_hash(gs.ar[n], # gs.ar) ⊃ ¬ n < n1 ∧ ¬ get_hash(gs.ar[n], # gs.ar) ≤ n;¬ get_hash(gs.ar[n], # gs.ar) ≤ n ∨ ¬ n < n1), ∀ e. e ∈ ags.set ↔ (∃ n. n < # gs.ar ∧ e = gs.ar[n] ∧ gs.ar[n] ≠ ⊥), ∀ 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, ∀ m. between(get_hash(gs.ar[n], # gs.ar), m, n1, false) ∧ m < # gs.ar → gs.ar[m] ≠ gs.ar[n] ∧ gs.ar[m] ≠ ⊥ ⊦

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 19

Additional information

Used simplifier rules: ⊦ m ≤ n ↔ ¬ n < m; (ss) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic trans-Nat: ⊦ n < n0 ∧ n0 < n1 → n < n1; (forward) from specification nat-basic fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat lf: ⊦ m < n ∧ n < n0 → m +1 < n0; (forward) from specification nat-basic lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic ⊦ m < n → n ≠ 0; (s) ⊦ m +1 = m + 1; (s) ⊦ ¬ n < n; (s) ⊦ m < n → m % n = m; (s) ⊦ m < n + 1 ↔ ¬ n < m; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic ⊦ m < n ↔ ¬ n ≤ m; (ss) ⊦ m ≤ n → (n ≤ m ↔ m = n); (s)

Proof step 28 from proof step-absstep-M6-M7-doMEMBER-same (previous step) (back to tree)

(n1 + 1) % # gs.ar = get_hash(gs.ar[n], # gs.ar), alsf(tid).le = gs.ar[n], alsf(tid).tid = tid, apcf(tid) = invMember, lsf(tid) = mkls(tid, gs.ar[n], false, # gs.ar, get_hash(gs.ar[n], # gs.ar), n1, e0), lsf0(tid) = mkls(tid, gs.ar[n], false, # gs.ar, get_hash(gs.ar[n], # gs.ar), n1, gs.ar[n1]), pcf(tid) = M6, pcf0(tid) = M7, # ags.set ≠ 0, ags.set ≠ ∅, gs.ar[n] ≠ ⊥, gs.ar[n] ≠ gs.ar[n1], n < # gs.ar, get_hash(gs.ar[n], # gs.ar) < # gs.ar, n1 < # gs.ar, SetLInv(ags, alsf(tid), invMember), htok(gs.ar), gs.ar[n] ∈ ags.set, ¬ n1 < get_hash(gs.ar[n], # gs.ar), ∀ m. between(get_hash(gs.ar[n], # gs.ar), m, n1, false) ∧ m < # gs.ar → gs.ar[m] ≠ gs.ar[n] ∧ gs.ar[m] ≠ ⊥, ∀ 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)), ∀ e. e ∈ ags.set ↔ (∃ n. n < # gs.ar ∧ e = gs.ar[n] ∧ gs.ar[n] ≠ ⊥), ¬ get_hash(gs.ar[n], # gs.ar) ≤ n ∨ ¬ n < n1 ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat ⊦ m < n ↔ ¬ n ≤ m; (ss)

Proof step 29 from proof step-absstep-M6-M7-doMEMBER-same (previous step) (back to tree)

(n1 + 1) % # gs.ar = get_hash(gs.ar[n], # gs.ar), alsf(tid).le = gs.ar[n], alsf(tid).tid = tid, apcf(tid) = invMember, lsf(tid) = mkls(tid, gs.ar[n], false, # gs.ar, get_hash(gs.ar[n], # gs.ar), n1, e0), lsf0(tid) = mkls(tid, gs.ar[n], false, # gs.ar, get_hash(gs.ar[n], # gs.ar), n1, gs.ar[n1]), pcf(tid) = M6, pcf0(tid) = M7, # ags.set ≠ 0, ags.set ≠ ∅, gs.ar[n] ≠ ⊥, gs.ar[n] ≠ gs.ar[n1], n1 < # gs.ar, n < # gs.ar, get_hash(gs.ar[n], # gs.ar) < # gs.ar, SetLInv(ags, alsf(tid), invMember), htok(gs.ar), gs.ar[n] ∈ ags.set, get_hash(gs.ar[n], # gs.ar) ≤ n1, ∀ e. e ∈ ags.set ↔ (∃ n. n < # gs.ar ∧ e = gs.ar[n] ∧ gs.ar[n] ≠ ⊥), ∀ 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, ∀ m. between(get_hash(gs.ar[n], # gs.ar), m, n1, false) ∧ m < # gs.ar → gs.ar[m] ≠ gs.ar[n] ∧ gs.ar[m] ≠ ⊥, ¬ get_hash(gs.ar[n], # gs.ar) ≤ n ∨ ¬ n < n1 ⊦

User applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: nat-div, name:div-elim, seq:n ≠ 0 ⊦ m0 = m / n ∧ n0 = m % n ↔ m = m0 * n + n0 ∧ n0 < n, subst:[m, n, m0, n0] → [n1 + 1, # gs.ar, m0, n0]

Additional information

Used simplifier rules: ⊦ m < n → n ≠ 0; (s) n ≠ 0 ⊦ m0 = m / n ∧ n0 = m % n ↔ m = m0 * n + n0 ∧ n0 < n; (elim) Used terms: n1 + 1, # gs.ar, m0, n0

Proof step 30 from proof step-absstep-M6-M7-doMEMBER-same (previous step) (back to tree)

n1 + 1 = m0 * # gs.ar + n0, alsf(tid).le = gs.ar[n], alsf(tid).tid = tid, apcf(tid) = invMember, lsf(tid) = mkls(tid, gs.ar[n], false, # gs.ar, get_hash(gs.ar[n], # gs.ar), n1, e0), lsf0(tid) = mkls(tid, gs.ar[n], false, # gs.ar, get_hash(gs.ar[n], # gs.ar), n1, gs.ar[n1]), n0 = get_hash(gs.ar[n], # gs.ar), pcf(tid) = M6, pcf0(tid) = M7, # gs.ar ≠ 0, # ags.set ≠ 0, ags.set ≠ ∅, gs.ar[n] ≠ ⊥, gs.ar[n] ≠ gs.ar[n1], get_hash(gs.ar[n], # gs.ar) < # gs.ar, n < # gs.ar, n1 < # gs.ar, n0 < # gs.ar, SetLInv(ags, alsf(tid), invMember), htok(gs.ar), gs.ar[n] ∈ ags.set, get_hash(gs.ar[n], # gs.ar) ≤ n1, ∀ m. between(get_hash(gs.ar[n], # gs.ar), m, n1, false) ∧ m < # gs.ar → gs.ar[m] ≠ gs.ar[n] ∧ gs.ar[m] ≠ ⊥, ∀ 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)), ∀ e. e ∈ ags.set ↔ (∃ n. n < # gs.ar ∧ e = gs.ar[n] ∧ gs.ar[n] ≠ ⊥), ¬ get_hash(gs.ar[n], # gs.ar) ≤ n ∨ ¬ n < n1 ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ m + 1 ≤ n ↔ m < n; (s) eqle-one: ⊦ m = n + 1 ∧ n < n0 → m ≤ n0; (forward) from specification nat ⊦ m < n → n ≠ 0; (s)

Proof step 31 from proof step-absstep-M6-M7-doMEMBER-same (previous step) (back to tree)

n1 + 1 = m0 * # gs.ar + get_hash(gs.ar[n], # gs.ar), alsf(tid).le = gs.ar[n], alsf(tid).tid = tid, apcf(tid) = invMember, lsf(tid) = mkls(tid, gs.ar[n], false, # gs.ar, get_hash(gs.ar[n], # gs.ar), n1, e0), lsf0(tid) = mkls(tid, gs.ar[n], false, # gs.ar, get_hash(gs.ar[n], # gs.ar), n1, gs.ar[n1]), pcf(tid) = M6, pcf0(tid) = M7, # ags.set ≠ 0, ags.set ≠ ∅, gs.ar[n] ≠ ⊥, gs.ar[n] ≠ gs.ar[n1], n < # gs.ar, get_hash(gs.ar[n], # gs.ar) < # gs.ar, n1 < # gs.ar, SetLInv(ags, alsf(tid), invMember), htok(gs.ar), gs.ar[n] ∈ ags.set, get_hash(gs.ar[n], # gs.ar) ≤ n1, ∀ e. e ∈ ags.set ↔ (∃ n. n < # gs.ar ∧ e = gs.ar[n] ∧ gs.ar[n] ≠ ⊥), ∀ 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, ∀ m. between(get_hash(gs.ar[n], # gs.ar), m, n1, false) ∧ m < # gs.ar → gs.ar[m] ≠ gs.ar[n] ∧ gs.ar[m] ≠ ⊥, ¬ get_hash(gs.ar[n], # gs.ar) ≤ n ∨ ¬ n < n1 ⊦

User applied rule constructor cut

And closed the goal!

Rule argument

var: m0

Additional information

Used simplifier rules: Nat freely generated by 0, +1;; ⊦ m +1 = m + 1; (s) ⊦ m * (n + 1) = m * n + m; (s) ⊦ m + m0 ≤ m ↔ m0 = 0; (s) ⊦ m + n = 0 ↔ m = 0 ∧ n = 0; (ws) ⊦ m ≤ n ↔ ¬ n < m; (ss) let: ⊦ n ≤ n0 ∧ n0 ≤ n1 → n ≤ n1; (forward) from specification nat lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic ⊦ n + 0 = n; (s) ⊦ m ≤ m; (s) ⊦ 0 < m ↔ m ≠ 0; (s) ⊦ m0 * m < m ↔ m0 = 0 ∧ m ≠ 0; (ws) ⊦ m < n → n ≠ 0; (s) ⊦ 0 ≤ m; (s) eqle-one: ⊦ m = n + 1 ∧ n < n0 → m ≤ n0; (forward) from specification nat f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat ⊦ 0 * n = 0; (s) mult-a: ⊦ m0 * m * n = (m0 * m) * n; from specification nat-mult mult-c: ⊦ m * n = n * m; from specification nat-mult ⊦ m + 1 ≠ 0; (s) ⊦ ¬ n < n; (s) ⊦ m < n + 1 ↔ ¬ n < m; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic ⊦ m < n ↔ ¬ n ≤ m; (ss) ⊦ m ≤ n → (n ≤ m ↔ m = n); (s) ⊦ m = n + 1 → ¬ m ≤ n; (s)

Proof Tree for step-absstep-M6-M7-doMEMBER-same (View conclusion)

The following image shows the proof tree of sequent calculus. The proved goal is at the bottom. You can click on the nodes to view additional rules. See here for details on colors in a proof tree.

the proof tree