Proof step 1 from proof step-absstep-W1-W2-tau-same (conclusion) (back to tree)

System applied rule normalize

And obtained the following new goals

• Proof step 2:

  ls.l = als.il, ags.pm = gs.mem, ags.pm = ∅ ⊕ (ags.h | true), ls.t = als.t, lsf(t).t = t, alsf(t).t = t, ls.v = als.iv, ags.vm = gs.mem ⊕ gs.p, ags.vm = ∅ ⊕ ags.h, W2 = pcstepf(gs, ls, W1, sfencePSC(t)), alsf(t) = als, apc = invWrite, apcf(t) = apc, gs0 = gstepf(gs, ls, W1, sfencePSC(t)), ls0 = lstepf(gs, ls, W1, sfencePSC(t)), lsf(t) = ls, lsf0(t) = ls0, pcf(t) = W1, pcf0(t) = W2, PLIBLInv(ags, als, apc), abs(ags.h, gs.readers, gs.p), allReadsOK(ags.h), allWritesOK(ags.h), ctrok(gs0.ctr, gs0.writers), ctrok(gs.ctr, gs.writers), noPending(t, gs0.p), noPendingW(t, gs0.p), noPendingW(t, gs.p), notMember(t, gs0.readers), notMember(t, gs0.writers), notMember(t, gs.readers), notMember(t, gs.writers), pendingOk(gs0.p), pendingOk(gs.p), pre(gs, ls, W1, sfencePSC(t)), writersOk(gs0.p, gs0.writers), writersOk(gs.p, gs.writers), ∀ l. ¬ FO(ls.t) ∈ gs.p(l) → ((ags.h | l) | ls.t) | false = [], ∀ t0. t0 ≠ t → FLITPSCLInv(gs, lsf(t0), pcf(t0)) ∧ PLIBLInv(ags, alsf(t0), apcf(t0)) ∧ labs(gs, lsf(t0), pcf(t0), ags, alsf(t0), apcf(t0)), ∀ t0. t0 ≠ t → alsf(t0).t = t0, ∀ t0. t0 ≠ t → lsf(t0).t = t0 ⊦ (ags.pm = gs0.mem ∧ ags.vm = gs0.mem ⊕ gs0.p ∧ abs(ags.h, gs0.readers, gs0.p)) ∧ ls0.t = als.t ∧ apc = invWrite ∧ (ls0.l = als.il ∧ ls0.v = als.iv) ∧ (∀ l. ((ags.h | l) | ls0.t) | false = [])

Rule argument

Additional information

Proof step 2 from proof step-absstep-W1-W2-tau-same (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 3:

  lsf(t).l = alsf(t).il, gs.mem = ∅ ⊕ (ags.h | true), ags.pm = gs.mem, lsf(t).t = t, alsf(t).t = t, lsf(t).v = alsf(t).iv, ags.vm = gs.mem ⊕ gs.p, ags.vm = ∅ ⊕ ags.h, apcf(t) = invWrite, lsf0(t) = lsf(t), pcf(t) = W1, pcf0(t) = W2, ((ags.h | l) | t) | false ≠ [], PLIBLInv(ags, alsf(t), invWrite), abs(ags.h, gs.readers, gs.p), allReadsOK(ags.h), allWritesOK(ags.h), ctrok(gs.ctr, gs.writers), noPending(t, gs.p), noPendingW(t, gs.p), notMember(t, gs.readers), notMember(t, gs.writers), pendingOk(gs.p), writersOk(gs.p, gs.writers), ∀ t0. alsf(t0).t = t0, ∀ t0. lsf(t0).t = t0, ∀ l. ¬ FO(t) ∈ gs.p(l), ∀ l. ¬ FO(t) ∈ gs.p(l) → ((ags.h | l) | t) | false = [], ∀ t0. t0 ≠ t → FLITPSCLInv(gs, lsf(t0), pcf(t0)) ∧ PLIBLInv(ags, alsf(t0), apcf(t0)) ∧ labs(gs, lsf(t0), pcf(t0), ags, alsf(t0), apcf(t0)) ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ gstepf(gs, ls, W1, a) = gs; (s) ⊦ sfencePSC(t) = sfencePSC(t0) ↔ t = t0; (s) ⊦ pre(gs, ls, W1, a) ↔ (∀ l. ¬ FO(ls.t) ∈ gs.p(l)) ∧ a = sfencePSC(ls.t); (s) ⊦ pcstepf(gs, ls, W1, a) = W2; (s) ⊦ lstepf(gs, ls, W1, a) = ls; (s)

Proof step 3 from proof step-absstep-W1-W2-tau-same (previous step) (back to tree)

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

• Proof step 4:

  lsf(t).l = alsf(t).il, gs.mem = ∅ ⊕ (h | true), alsf(t).t = t, lsf(t).t = t, lsf(t).v = alsf(t).iv, ags = mkPLIBgs(mem, mem0, h), apcf(t) = invWrite, lsf0(t) = lsf(t), mem = ∅ ⊕ h, mem = gs.mem ⊕ gs.p, mem0 = gs.mem, pcf(t) = W1, pcf0(t) = W2, ((h | l) | t) | false ≠ [], PLIBLInv(ags, alsf(t), invWrite), abs(h, gs.readers, gs.p), allReadsOK(h), allWritesOK(h), ctrok(gs.ctr, gs.writers), noPending(t, gs.p), noPendingW(t, gs.p), notMember(t, gs.writers), notMember(t, gs.readers), pendingOk(gs.p), writersOk(gs.p, gs.writers), ∀ t0. t0 ≠ t → FLITPSCLInv(gs, lsf(t0), pcf(t0)) ∧ PLIBLInv(ags, alsf(t0), apcf(t0)) ∧ labs(gs, lsf(t0), pcf(t0), ags, alsf(t0), apcf(t0)), ∀ l. ¬ FO(t) ∈ gs.p(l) → ((h | l) | t) | false = [], ∀ l. ¬ FO(t) ∈ gs.p(l), ∀ t0. lsf(t0).t = t0, ∀ t0. alsf(t0).t = t0 ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ (vm = gs.vm ∧ pm = gs.pm) ∧ h = gs.h ↔ gs = mkPLIBgs(vm, pm, h); (elim) Used terms: ags, mem, mem0, h

Proof step 4 from proof step-absstep-W1-W2-tau-same (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 5:

  lsf(t).l = alsf(t).il, gs.mem = ∅ ⊕ (h | true), lsf(t).t = t, alsf(t).t = t, lsf(t).v = alsf(t).iv, apcf(t) = invWrite, lsf0(t) = lsf(t), pcf(t) = W1, pcf0(t) = W2, gs.mem ⊕ gs.p = ∅ ⊕ h, ((h | l) | t) | false ≠ [], PLIBLInv(mkPLIBgs(∅ ⊕ h, gs.mem, h), alsf(t), invWrite), abs(h, gs.readers, gs.p), allReadsOK(h), allWritesOK(h), ctrok(gs.ctr, gs.writers), noPending(t, gs.p), noPendingW(t, gs.p), notMember(t, gs.readers), notMember(t, gs.writers), pendingOk(gs.p), writersOk(gs.p, gs.writers), ∀ t0. alsf(t0).t = t0, ∀ t0. lsf(t0).t = t0, ∀ l. ¬ FO(t) ∈ gs.p(l), ∀ l. ¬ FO(t) ∈ gs.p(l) → ((h | l) | t) | false = [], ∀ t0. t0 ≠ t → FLITPSCLInv(gs, lsf(t0), pcf(t0)) ∧ PLIBLInv(mkPLIBgs(∅ ⊕ h, gs.mem, h), alsf(t0), apcf(t0)) ∧ labs(gs, lsf(t0), pcf(t0), mkPLIBgs(∅ ⊕ h, gs.mem, h), alsf(t0), apcf(t0)) ⊦

Rule argument

Additional information

Proof step 5 from proof step-absstep-W1-W2-tau-same (previous step) (back to tree)

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

• Proof step 6:

  lsf(t).l = alsf(t).il, alsf(t).t = t, lsf(t).t = t, lsf(t).v = alsf(t).iv, apcf(t) = invWrite, gs = mkFLITPSCgs(mem, p, ctr, writers, writers0), lsf0(t) = lsf(t), mem = ∅ ⊕ (h | true), pcf(t) = W1, pcf0(t) = W2, mem ⊕ p = ∅ ⊕ h, ((h | l) | t) | false ≠ [], PLIBLInv(mkPLIBgs(∅ ⊕ h, mem, h), alsf(t), invWrite), abs(h, writers0, p), allReadsOK(h), allWritesOK(h), ctrok(ctr, writers), noPending(t, p), noPendingW(t, p), notMember(t, writers), notMember(t, writers0), pendingOk(p), writersOk(p, writers), ∀ t0. t0 ≠ t → FLITPSCLInv(gs, lsf(t0), pcf(t0)) ∧ PLIBLInv(mkPLIBgs(∅ ⊕ h, mem, h), alsf(t0), apcf(t0)) ∧ labs(gs, lsf(t0), pcf(t0), mkPLIBgs(∅ ⊕ h, mem, h), alsf(t0), apcf(t0)), ∀ l. ¬ FO(t) ∈ p(l) → ((h | l) | t) | false = [], ∀ l. ¬ FO(t) ∈ p(l), ∀ t0. lsf(t0).t = t0, ∀ t0. alsf(t0).t = t0 ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ (((mem = gs.mem ∧ p = gs.p) ∧ ctr = gs.ctr) ∧ writers = gs.writers) ∧ readers = gs.readers ↔ gs = mkFLITPSCgs(mem, p, ctr, writers, readers); (elim) Used terms: gs, mem, ctr, writers, writers0, p

Proof step 6 from proof step-absstep-W1-W2-tau-same (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 7:

  lsf(t).l = alsf(t).il, lsf(t).t = t, alsf(t).t = t, lsf(t).v = alsf(t).iv, apcf(t) = invWrite, lsf0(t) = lsf(t), pcf(t) = W1, pcf0(t) = W2, (∅ ⊕ (h | true)) ⊕ p = ∅ ⊕ h, ((h | l) | t) | false ≠ [], PLIBLInv(mkPLIBgs(∅ ⊕ h, ∅ ⊕ (h | true), h), alsf(t), invWrite), abs(h, writers0, p), allReadsOK(h), allWritesOK(h), ctrok(ctr, writers), noPending(t, p), noPendingW(t, p), notMember(t, writers0), notMember(t, writers), pendingOk(p), writersOk(p, writers), ∀ t0. alsf(t0).t = t0, ∀ t0. lsf(t0).t = t0, ∀ l. ¬ FO(t) ∈ p(l), ∀ l. ¬ FO(t) ∈ p(l) → ((h | l) | t) | false = [], ∀ t0. t0 ≠ t → FLITPSCLInv(mkFLITPSCgs(∅ ⊕ (h | true), p, ctr, writers, writers0), lsf(t0), pcf(t0)) ∧ PLIBLInv(mkPLIBgs(∅ ⊕ h, ∅ ⊕ (h | true), h), alsf(t0), apcf(t0)) ∧ labs(mkFLITPSCgs(∅ ⊕ (h | true), p, ctr, writers, writers0), lsf(t0), pcf(t0), mkPLIBgs(∅ ⊕ h, ∅ ⊕ (h | true), h), alsf(t0), apcf(t0)) ⊦

Rule argument

Additional information

Proof step 7 from proof step-absstep-W1-W2-tau-same (previous step) (back to tree)

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

• Proof step 8:

  lsf(t).l = il, lsf(t).t = t, lsf(t).v = iv, alsf(t) = mkPLIBls(t1, il, iv, ov), apcf(t) = invWrite, lsf0(t) = lsf(t), pcf(t) = W1, pcf0(t) = W2, t1 = t, (∅ ⊕ (h | true)) ⊕ p = ∅ ⊕ h, ((h | l) | t) | false ≠ [], PLIBLInv(mkPLIBgs(∅ ⊕ h, ∅ ⊕ (h | true), h), alsf(t), invWrite), abs(h, writers0, p), allReadsOK(h), allWritesOK(h), ctrok(ctr, writers), noPending(t, p), noPendingW(t, p), notMember(t, writers), notMember(t, writers0), pendingOk(p), writersOk(p, writers), ∀ t0. t0 ≠ t → FLITPSCLInv(mkFLITPSCgs(∅ ⊕ (h | true), p, ctr, writers, writers0), lsf(t0), pcf(t0)) ∧ PLIBLInv(mkPLIBgs(∅ ⊕ h, ∅ ⊕ (h | true), h), alsf(t0), apcf(t0)) ∧ labs(mkFLITPSCgs(∅ ⊕ (h | true), p, ctr, writers, writers0), lsf(t0), pcf(t0), mkPLIBgs(∅ ⊕ h, ∅ ⊕ (h | true), h), alsf(t0), apcf(t0)), ∀ l. ¬ FO(t) ∈ p(l) → ((h | l) | t) | false = [], ∀ l. ¬ FO(t) ∈ p(l), ∀ t0. lsf(t0).t = t0, ∀ t0. alsf(t0).t = t0 ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ ((t = ls.t ∧ il = ls.il) ∧ iv = ls.iv) ∧ ov = ls.ov ↔ ls = mkPLIBls(t, il, iv, ov); (elim) Used terms: alsf(t), iv, t1, il, ov

Proof step 8 from proof step-absstep-W1-W2-tau-same (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 9:

  lsf(t).t = t, alsf(t) = mkPLIBls(t, lsf(t).l, lsf(t).v, ov), apcf(t) = invWrite, lsf0(t) = lsf(t), pcf(t) = W1, pcf0(t) = W2, (∅ ⊕ (h | true)) ⊕ p = ∅ ⊕ h, ((h | l) | t) | false ≠ [], PLIBLInv(mkPLIBgs(∅ ⊕ h, ∅ ⊕ (h | true), h), mkPLIBls(t, lsf(t).l, lsf(t).v, ov), invWrite), abs(h, writers0, p), allReadsOK(h), allWritesOK(h), ctrok(ctr, writers), noPending(t, p), noPendingW(t, p), notMember(t, writers0), notMember(t, writers), pendingOk(p), writersOk(p, writers), ∀ t0. alsf(t0).t = t0, ∀ t0. lsf(t0).t = t0, ∀ l. ¬ FO(t) ∈ p(l), ∀ l. ¬ FO(t) ∈ p(l) → ((h | l) | t) | false = [], ∀ t0. t0 ≠ t → FLITPSCLInv(mkFLITPSCgs(∅ ⊕ (h | true), p, ctr, writers, writers0), lsf(t0), pcf(t0)) ∧ PLIBLInv(mkPLIBgs(∅ ⊕ h, ∅ ⊕ (h | true), h), alsf(t0), apcf(t0)) ∧ labs(mkFLITPSCgs(∅ ⊕ (h | true), p, ctr, writers, writers0), lsf(t0), pcf(t0), mkPLIBgs(∅ ⊕ h, ∅ ⊕ (h | true), h), alsf(t0), apcf(t0)) ⊦

Rule argument

Additional information

Proof step 9 from proof step-absstep-W1-W2-tau-same (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 10:

  lsf(t).t = t, alsf(t) = mkPLIBls(t, lsf(t).l, lsf(t).v, ov), apcf(t) = invWrite, lsf0(t) = lsf(t), pcf(t) = W1, pcf0(t) = W2, (∅ ⊕ (h | true)) ⊕ p = ∅ ⊕ h, ((h | l) | t) | false ≠ [], PLIBLInv(mkPLIBgs(∅ ⊕ h, ∅ ⊕ (h | true), h), mkPLIBls(t, lsf(t).l, lsf(t).v, ov), invWrite), abs(h, writers0, p), allReadsOK(h), allWritesOK(h), ctrok(ctr, writers), noPending(t, p), noPendingW(t, p), notMember(t, writers), notMember(t, writers0), pendingOk(p), writersOk(p, writers), ¬ FO(t) ∈ p(l) → ((h | l) | t) | false = [], ∀ t0. t0 ≠ t → FLITPSCLInv(mkFLITPSCgs(∅ ⊕ (h | true), p, ctr, writers, writers0), lsf(t0), pcf(t0)) ∧ PLIBLInv(mkPLIBgs(∅ ⊕ h, ∅ ⊕ (h | true), h), alsf(t0), apcf(t0)) ∧ labs(mkFLITPSCgs(∅ ⊕ (h | true), p, ctr, writers, writers0), lsf(t0), pcf(t0), mkPLIBgs(∅ ⊕ h, ∅ ⊕ (h | true), h), alsf(t0), apcf(t0)), ∀ l. ¬ FO(t) ∈ p(l), ∀ t0. lsf(t0).t = t0, ∀ t0. alsf(t0).t = t0 ⊦

Rule argument

Additional information

Used terms: l

Proof step 10 from proof step-absstep-W1-W2-tau-same (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 11:

  lsf(t).t = t, alsf(t) = mkPLIBls(t, lsf(t).l, lsf(t).v, ov), apcf(t) = invWrite, lsf0(t) = lsf(t), pcf(t) = W1, pcf0(t) = W2, (∅ ⊕ (h | true)) ⊕ p = ∅ ⊕ h, ((h | l) | t) | false ≠ [], PLIBLInv(mkPLIBgs(∅ ⊕ h, ∅ ⊕ (h | true), h), mkPLIBls(t, lsf(t).l, lsf(t).v, ov), invWrite), abs(h, writers0, p), allReadsOK(h), allWritesOK(h), ctrok(ctr, writers), noPending(t, p), noPendingW(t, p), notMember(t, writers0), notMember(t, writers), pendingOk(p), writersOk(p, writers), FO(t) ∈ p(l), ∀ t0. alsf(t0).t = t0, ∀ t0. lsf(t0).t = t0, ∀ l. ¬ FO(t) ∈ p(l), ∀ t0. t0 ≠ t → FLITPSCLInv(mkFLITPSCgs(∅ ⊕ (h | true), p, ctr, writers, writers0), lsf(t0), pcf(t0)) ∧ PLIBLInv(mkPLIBgs(∅ ⊕ h, ∅ ⊕ (h | true), h), alsf(t0), apcf(t0)) ∧ labs(mkFLITPSCgs(∅ ⊕ (h | true), p, ctr, writers, writers0), lsf(t0), pcf(t0), mkPLIBgs(∅ ⊕ h, ∅ ⊕ (h | true), h), alsf(t0), apcf(t0)) ⊦

Rule argument

Additional information

Proof step 11 from proof step-absstep-W1-W2-tau-same (previous step) (back to tree)

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

Additional information

Used terms: l