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Proof step 1 from proof rely-PROP_FO (conclusion) (back to tree)

⊦ ( ctrok(gs.fctr, gs.fwriters) ∧ writersOk(gs.bq, gs.pq, gs.fwriters) ∧ pendingOk(gs.bq, gs.pq) ∧ SFTrailing(gs.bq) ∧ readFOsOk(gs.rfo, gs.fwriters)) ∧ (∃ n. n < # gs.bq(tid) ∧ isPropFO(n, l, gs.bq(tid))) ∧ ⟨fwriters := gs.fwriters, pq := gs.pq, bq := gs.bq, rfo := gs.rfo; PROP_FO(tid, l, fwriters; pq, bq, rfo); gs := ((gs.pq:= pq).bq:= bq).rfo:= rfo⟩ (gs0 = gs) → (∀ t0. relyws(t0, gs.freaders, gs0.freaders) ∧ relyws(t0, gs.fwriters, gs0.fwriters) ∧ rely(t0, gs.pq, gs.bq, gs0.pq, gs0.bq) ∧ gs.bq(t0) ⊒w gs0.bq(t0) ∧ (gs.rfo(t0) = [] → gs0.rfo(t0) = []) ∧ (gs.rfo(t0) = gs.bq(t0) → gs0.rfo(t0) = gs0.bq(t0)) ∧ (gs.rfo(t0) + SF ' = gs.bq(t0) → gs0.rfo(t0) + SF ' = gs0.bq(t0) ∨ gs0.rfo(t0) = gs0.bq(t0)))

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

Proof step 2 from proof rely-PROP_FO (previous step) (back to tree)

⟨fwriters := gs.fwriters, pq := gs.pq, bq := gs.bq, rfo := gs.rfo; PROP_FO(tid, l, fwriters; pq, bq, rfo); gs := ((gs.pq:= pq).bq:= bq).rfo:= rfo⟩ (gs0 = gs), SFTrailing(gs.bq), ctrok(gs.fctr, gs.fwriters), pendingOk(gs.bq, gs.pq), readFOsOk(gs.rfo, gs.fwriters), writersOk(gs.bq, gs.pq, gs.fwriters), ∃ n. n < # gs.bq(tid) ∧ isPropFO(n, l, gs.bq(tid)) ⊦ ∀ t0. relyws(t0, gs.freaders, gs0.freaders) ∧ relyws(t0, gs.fwriters, gs0.fwriters) ∧ rely(t0, gs.pq, gs.bq, gs0.pq, gs0.bq) ∧ gs.bq(t0) ⊒w gs0.bq(t0) ∧ (gs.rfo(t0) = [] → gs0.rfo(t0) = []) ∧ (gs.rfo(t0) = gs.bq(t0) → gs0.rfo(t0) = gs0.bq(t0)) ∧ (gs.rfo(t0) + SF ' = gs.bq(t0) → gs0.rfo(t0) + SF ' = gs0.bq(t0) ∨ gs0.rfo(t0) = gs0.bq(t0))

Heuristic symbolic execution applied rule assign left

And obtained the following new goals

Rule argument

Additional information

Proof step 3 from proof rely-PROP_FO (previous step) (back to tree)

⟨PROP_FO(tid, l, fwriters; pq, bq, rfo); gs := ((gs.pq:= pq).bq:= bq).rfo:= rfo⟩ (gs0 = gs), bq = gs.bq, fwriters = gs.fwriters, pq = gs.pq, rfo = gs.rfo, SFTrailing(gs.bq), ctrok(gs.fctr, gs.fwriters), pendingOk(gs.bq, gs.pq), readFOsOk(gs.rfo, gs.fwriters), writersOk(gs.bq, gs.pq, gs.fwriters), ∃ n. n < # gs.bq(tid) ∧ isPropFO(n, l, gs.bq(tid)) ⊦ ∀ t0. relyws(t0, gs.freaders, gs0.freaders) ∧ relyws(t0, gs.fwriters, gs0.fwriters) ∧ rely(t0, gs.pq, gs.bq, gs0.pq, gs0.bq) ∧ gs.bq(t0) ⊒w gs0.bq(t0) ∧ (gs.rfo(t0) = [] → gs0.rfo(t0) = []) ∧ (gs.rfo(t0) = gs.bq(t0) → gs0.rfo(t0) = gs0.bq(t0)) ∧ (gs.rfo(t0) + SF ' = gs.bq(t0) → gs0.rfo(t0) + SF ' = gs0.bq(t0) ∨ gs0.rfo(t0) = gs0.bq(t0))

Heuristic split applied rule split left

And obtained the following new goals

Rule argument

pos: left 1

Additional information

Proof step 4 from proof rely-PROP_FO (previous step) (back to tree)

⟨gs := ((gs.pq:= pq0).bq:= bq0).rfo:= rfo0⟩ (gs0 = gs), ⟨PROP_FO(tid, l, fwriters; pq, bq, rfo)⟩ (pq = pq0 ∧ bq = bq0 ∧ rfo = rfo0), bq = gs.bq, fwriters = gs.fwriters, pq = gs.pq, rfo = gs.rfo, SFTrailing(gs.bq), ctrok(gs.fctr, gs.fwriters), pendingOk(gs.bq, gs.pq), readFOsOk(gs.rfo, gs.fwriters), writersOk(gs.bq, gs.pq, gs.fwriters), ∃ n. n < # gs.bq(tid) ∧ isPropFO(n, l, gs.bq(tid)) ⊦ ∀ t0. relyws(t0, gs.freaders, gs0.freaders) ∧ relyws(t0, gs.fwriters, gs0.fwriters) ∧ rely(t0, gs.pq, gs.bq, gs0.pq, gs0.bq) ∧ gs.bq(t0) ⊒w gs0.bq(t0) ∧ (gs.rfo(t0) = [] → gs0.rfo(t0) = []) ∧ (gs.rfo(t0) = gs.bq(t0) → gs0.rfo(t0) = gs0.bq(t0)) ∧ (gs.rfo(t0) + SF ' = gs.bq(t0) → gs0.rfo(t0) + SF ' = gs0.bq(t0) ∨ gs0.rfo(t0) = gs0.bq(t0))

Heuristic symbolic execution applied rule assign left

And obtained the following new goals

Rule argument

Additional information

Proof step 5 from proof rely-PROP_FO (previous step) (back to tree)

⟨PROP_FO(tid, l, fwriters; pq, bq, rfo)⟩ (pq = pq0 ∧ bq = bq0 ∧ rfo = rfo0), bq = gs.bq, fwriters = gs.fwriters, gs0 = ((gs.pq:= pq0).bq:= bq0).rfo:= rfo0, pq = gs.pq, rfo = gs.rfo, SFTrailing(gs.bq), ctrok(gs.fctr, gs.fwriters), pendingOk(gs.bq, gs.pq), readFOsOk(gs.rfo, gs.fwriters), writersOk(gs.bq, gs.pq, gs.fwriters), ∃ n. n < # gs.bq(tid) ∧ isPropFO(n, l, gs.bq(tid)) ⊦ ∀ t0. relyws(t0, gs.freaders, gs0.freaders) ∧ relyws(t0, gs.fwriters, gs0.fwriters) ∧ rely(t0, gs.pq, gs.bq, gs0.pq, gs0.bq) ∧ gs.bq(t0) ⊒w gs0.bq(t0) ∧ (gs.rfo(t0) = [] → gs0.rfo(t0) = []) ∧ (gs.rfo(t0) = gs.bq(t0) → gs0.rfo(t0) = gs0.bq(t0)) ∧ (gs.rfo(t0) + SF ' = gs.bq(t0) → gs0.rfo(t0) + SF ' = gs0.bq(t0) ∨ gs0.rfo(t0) = gs0.bq(t0))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ # x = 0 ↔ x = []; (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic ⊦ (gs.bq:= bq).bq = bq; (s) ⊦ (gs.rfo:= bq0).bq = gs.bq; (s) ⊦ (gs.rfo:= bq0).rfo = bq0; (s) ⊦ (gs.pq:= pq).pq = pq; (s) ⊦ (gs.bq:= bq).pq = gs.pq; (s) ⊦ (gs.rfo:= bq0).pq = gs.pq; (s) ⊦ relyws(t, tsf, tsf); (s) ⊦ (gs.pq:= pq).fwriters = gs.fwriters; (s) ⊦ (gs.bq:= bq).fwriters = gs.fwriters; (s) ⊦ (gs.rfo:= bq0).fwriters = gs.fwriters; (s) ⊦ (gs.pq:= pq).freaders = gs.freaders; (s) ⊦ (gs.bq:= bq).freaders = gs.freaders; (s) ⊦ (gs.rfo:= bq0).freaders = gs.freaders; (s)

Proof step 6 from proof rely-PROP_FO (previous step) (back to tree)

⟨PROP_FO(tid, l, gs.fwriters; pq, bq, rfo)⟩ (pq = pq0 ∧ bq = bq0 ∧ rfo = rfo0), bq = gs.bq, pq = gs.pq, rfo = gs.rfo, gs.bq(tid) ≠ [], n < # gs.bq(tid), SFTrailing(gs.bq), ctrok(gs.fctr, gs.fwriters), isPropFO(n, l, gs.bq(tid)), pendingOk(gs.bq, gs.pq), readFOsOk(gs.rfo, gs.fwriters), writersOk(gs.bq, gs.pq, gs.fwriters) ⊦ rely(t0, gs.pq, gs.bq, pq0, bq0) ∧ gs.bq(t0) ⊒w bq0(t0) ∧ (gs.rfo(t0) = [] → rfo0(t0) = []) ∧ (gs.rfo(t0) = gs.bq(t0) → rfo0(t0) = bq0(t0)) ∧ (gs.rfo(t0) + SF ' = gs.bq(t0) → rfo0(t0) + SF ' = bq0(t0) ∨ rfo0(t0) = bq0(t0))

Heuristic unfold applied rule call left

And obtained the following new goals

Rule argument

pos: left 1

Additional information

Used simplifier rules: decl: PROP_FO(tid, l, fwriters; pq, bq, rfo) {choose n with n < # bq(tid) ∧ isPropFO(n, l, bq(tid)) in pq(l) := pq(l) + FO(tid), bq(tid) := bq(tid) -p n, rfo(tid) := (tid ∈ fwriters(l) ⊃ rfo(tid);rfo(tid) -p n) ifnone skip };

Proof step 7 from proof rely-PROP_FO (previous step) (back to tree)

⟨choose n with n < # bq(tid) ∧ isPropFO(n, l, bq(tid)) in pq(l) := pq(l) + FO(tid), bq(tid) := bq(tid) -p n, rfo(tid) := (tid ∈ fwriters(l) ⊃ rfo(tid);rfo(tid) -p n) ifnone skip⟩ (pq = pq0 ∧ bq = bq0 ∧ rfo = rfo0), bq = gs.bq, fwriters = gs.fwriters, pq = gs.pq, rfo = gs.rfo, gs.bq(tid) ≠ [], n < # gs.bq(tid), SFTrailing(gs.bq), ctrok(gs.fctr, gs.fwriters), isPropFO(n, l, gs.bq(tid)), pendingOk(gs.bq, gs.pq), readFOsOk(gs.rfo, gs.fwriters), writersOk(gs.bq, gs.pq, gs.fwriters) ⊦ rely(t0, gs.pq, gs.bq, pq0, bq0) ∧ gs.bq(t0) ⊒w bq0(t0) ∧ (gs.rfo(t0) = [] → rfo0(t0) = []) ∧ (gs.rfo(t0) = gs.bq(t0) → rfo0(t0) = bq0(t0)) ∧ (gs.rfo(t0) + SF ' = gs.bq(t0) → rfo0(t0) + SF ' = bq0(t0) ∨ rfo0(t0) = bq0(t0))

Heuristic symbolic execution applied rule choose left

And obtained the following new goals

Rule argument

Additional information

Proof step 8 from proof rely-PROP_FO (previous step) (back to tree)

⟨pq(l) := pq(l) + FO(tid), bq(tid) := bq(tid) -p n, rfo(tid) := (tid ∈ fwriters(l) ⊃ rfo(tid);rfo(tid) -p n)⟩ (pq = pq0 ∧ bq = bq0 ∧ rfo = rfo0), bq = gs.bq, fwriters = gs.fwriters, pq = gs.pq, rfo = gs.rfo, gs.bq(tid) ≠ [], n0 < # gs.bq(tid), n < # bq(tid), SFTrailing(gs.bq), ctrok(gs.fctr, gs.fwriters), isPropFO(n0, l, gs.bq(tid)), isPropFO(n, l, bq(tid)), pendingOk(gs.bq, gs.pq), readFOsOk(gs.rfo, gs.fwriters), writersOk(gs.bq, gs.pq, gs.fwriters) ⊦ rely(t0, gs.pq, gs.bq, pq0, bq0) ∧ gs.bq(t0) ⊒w bq0(t0) ∧ (gs.rfo(t0) = [] → rfo0(t0) = []) ∧ (gs.rfo(t0) = gs.bq(t0) → rfo0(t0) = bq0(t0)) ∧ (gs.rfo(t0) + SF ' = gs.bq(t0) → rfo0(t0) + SF ' = bq0(t0) ∨ rfo0(t0) = bq0(t0))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ x + a = x + a '; (s)

Proof step 9 from proof rely-PROP_FO (previous step) (back to tree)

⟨pq(l) := pq(l) + FO(tid) ', bq(tid) := bq(tid) -p n, rfo(tid) := (tid ∈ fwriters(l) ⊃ rfo(tid);rfo(tid) -p n)⟩ (pq = pq0 ∧ bq = bq0 ∧ rfo = rfo0), bq = gs.bq, fwriters = gs.fwriters, pq = gs.pq, rfo = gs.rfo, gs.bq(tid) ≠ [], n < # gs.bq(tid), n0 < # gs.bq(tid), SFTrailing(gs.bq), ctrok(gs.fctr, gs.fwriters), isPropFO(n, l, gs.bq(tid)), isPropFO(n0, l, gs.bq(tid)), pendingOk(gs.bq, gs.pq), readFOsOk(gs.rfo, gs.fwriters), writersOk(gs.bq, gs.pq, gs.fwriters) ⊦ rely(t0, gs.pq, gs.bq, pq0, bq0) ∧ gs.bq(t0) ⊒w bq0(t0) ∧ (gs.rfo(t0) = [] → rfo0(t0) = []) ∧ (gs.rfo(t0) = gs.bq(t0) → rfo0(t0) = bq0(t0)) ∧ (gs.rfo(t0) + SF ' = gs.bq(t0) → rfo0(t0) + SF ' = bq0(t0) ∨ rfo0(t0) = bq0(t0))

Heuristic symbolic execution applied rule assign left

And obtained the following new goals

Rule argument

Additional information

Proof step 10 from proof rely-PROP_FO (previous step) (back to tree)

pq(l := pq(l) + FO(tid) ') = pq0, bq(tid := bq(tid) -p n) = bq0, rfo(tid := (tid ∈ fwriters(l) ⊃ rfo(tid);rfo(tid) -p n)) = rfo0, bq = gs.bq, fwriters = gs.fwriters, pq = gs.pq, rfo = gs.rfo, gs.bq(tid) ≠ [], n0 < # gs.bq(tid), n < # gs.bq(tid), SFTrailing(gs.bq), ctrok(gs.fctr, gs.fwriters), isPropFO(n0, l, gs.bq(tid)), isPropFO(n, l, gs.bq(tid)), pendingOk(gs.bq, gs.pq), readFOsOk(gs.rfo, gs.fwriters), writersOk(gs.bq, gs.pq, gs.fwriters) ⊦ rely(t0, gs.pq, gs.bq, pq0, bq0) ∧ gs.bq(t0) ⊒w bq0(t0) ∧ (gs.rfo(t0) = [] → rfo0(t0) = []) ∧ (gs.rfo(t0) = gs.bq(t0) → rfo0(t0) = bq0(t0)) ∧ (gs.rfo(t0) + SF ' = gs.bq(t0) → rfo0(t0) + SF ' = bq0(t0) ∨ rfo0(t0) = bq0(t0))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ isPropFO(n, l, bq(t0)) → rely(t, pq, bq, pq(l := pq(l) + FO(t0) '), bq(t0 := bq(t0) -p n)); (ss)

Proof step 11 from proof rely-PROP_FO (previous step) (back to tree)

gs.bq(tid) ≠ [], n < # gs.bq(tid), n0 < # gs.bq(tid), SFTrailing(gs.bq), ctrok(gs.fctr, gs.fwriters), isPropFO(n, l, gs.bq(tid)), isPropFO(n0, l, gs.bq(tid)), pendingOk(gs.bq, gs.pq), readFOsOk(gs.rfo, gs.fwriters), writersOk(gs.bq, gs.pq, gs.fwriters) ⊦ gs.bq(t0) ⊒w (tid = t0 ⊃ gs.bq(t0) -p n;gs.bq(t0)) ∧ (gs.rfo(t0) = [] → (tid = t0 ⊃ (t0 ∈ gs.fwriters(l) ⊃ gs.rfo(t0);gs.rfo(t0) -p n);gs.rfo(t0)) = []) ∧ (gs.rfo(t0) = gs.bq(t0) → (tid = t0 ⊃ (t0 ∈ gs.fwriters(l) ⊃ gs.rfo(t0);gs.rfo(t0) -p n);gs.rfo(t0)) = (tid = t0 ⊃ gs.bq(t0) -p n;gs.bq(t0))) ∧ ( gs.rfo(t0) + SF ' = gs.bq(t0) → (tid = t0 ⊃ (t0 ∈ gs.fwriters(l) ⊃ gs.rfo(t0);gs.rfo(t0) -p n);gs.rfo(t0)) + SF ' = (tid = t0 ⊃ gs.bq(t0) -p n;gs.bq(t0)) ∨ (tid = t0 ⊃ (t0 ∈ gs.fwriters(l) ⊃ gs.rfo(t0);gs.rfo(t0) -p n);gs.rfo(t0)) = (tid = t0 ⊃ gs.bq(t0) -p n;gs.bq(t0)))

Heuristic nonsplitting cut applied rule cut formula

And obtained the following new goals

Rule argument

fma: tid = t0

Additional information

Used simplifier rules: ⊦ bel ⊒w bel; (s) Used formulas: tid = t0

Proof step 12 from proof rely-PROP_FO (previous step) (back to tree)

tid = t0, gs.bq(tid) ≠ [], n0 < # gs.bq(tid), n < # gs.bq(tid), SFTrailing(gs.bq), ctrok(gs.fctr, gs.fwriters), isPropFO(n0, l, gs.bq(tid)), isPropFO(n, l, gs.bq(tid)), pendingOk(gs.bq, gs.pq), readFOsOk(gs.rfo, gs.fwriters), writersOk(gs.bq, gs.pq, gs.fwriters) ⊦ gs.bq(t0) ⊒w (tid = t0 ⊃ gs.bq(t0) -p n;gs.bq(t0)) ∧ (gs.rfo(t0) = [] → (tid = t0 ⊃ (t0 ∈ gs.fwriters(l) ⊃ gs.rfo(t0);gs.rfo(t0) -p n);gs.rfo(t0)) = []) ∧ (gs.rfo(t0) = gs.bq(t0) → (tid = t0 ⊃ (t0 ∈ gs.fwriters(l) ⊃ gs.rfo(t0);gs.rfo(t0) -p n);gs.rfo(t0)) = (tid = t0 ⊃ gs.bq(t0) -p n;gs.bq(t0))) ∧ ( gs.rfo(t0) + SF ' = gs.bq(t0) → (tid = t0 ⊃ (t0 ∈ gs.fwriters(l) ⊃ gs.rfo(t0);gs.rfo(t0) -p n);gs.rfo(t0)) + SF ' = (tid = t0 ⊃ gs.bq(t0) -p n;gs.bq(t0)) ∨ (tid = t0 ⊃ (t0 ∈ gs.fwriters(l) ⊃ gs.rfo(t0);gs.rfo(t0) -p n);gs.rfo(t0)) = (tid = t0 ⊃ gs.bq(t0) -p n;gs.bq(t0)))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ # x = 0 ↔ x = []; (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic ⊦ isPropFO(n, l, bel) → bel ⊒w bel -p n; (ss)

Proof step 13 from proof rely-PROP_FO (previous step) (back to tree)

gs.bq(t0) ≠ [], n0 < # gs.bq(t0), n < # gs.bq(t0), SFTrailing(gs.bq), ctrok(gs.fctr, gs.fwriters), isPropFO(n0, l, gs.bq(t0)), isPropFO(n, l, gs.bq(t0)), pendingOk(gs.bq, gs.pq), readFOsOk(gs.rfo, gs.fwriters), writersOk(gs.bq, gs.pq, gs.fwriters) ⊦ (gs.rfo(t0) = [] → (t0 ∈ gs.fwriters(l) ⊃ gs.rfo(t0);gs.rfo(t0) -p n) = []) ∧ (gs.rfo(t0) = gs.bq(t0) → (t0 ∈ gs.fwriters(l) ⊃ gs.rfo(t0);gs.rfo(t0) -p n) = gs.bq(t0) -p n) ∧ ( gs.rfo(t0) + SF ' = gs.bq(t0) → (t0 ∈ gs.fwriters(l) ⊃ gs.rfo(t0);gs.rfo(t0) -p n) + SF ' = gs.bq(t0) -p n ∨ (t0 ∈ gs.fwriters(l) ⊃ gs.rfo(t0);gs.rfo(t0) -p n) = gs.bq(t0) -p n)

Heuristic nonsplitting cut applied rule cut formula

And obtained the following new goals

Rule argument

fma: t0 ∈ gs.fwriters(l)

Additional information

Used simplifier rules: readFOsOk-empty-rfo-while-writing: ⊦ readFOsOk(rfo, writers) ∧ t ∈ writers(l) → rfo(t) = []; (forward) from specification FLIT-preds ne-01: ⊦ a ∈ s → s ≠ ∅; (forward) from specification set-basic size-zero-01: ⊦ s ≠ ∅ → # s ≠ 0; (forward) from specification set-basic ⊦ ctrok(ctr, writers) → # writers(l) = ctr(l); (s) ⊦ [] + x = x; (s) a: ⊦ (x + y) + z = x + y + z; from specification list f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic ⊦ # a ' = 0 +1; (s) ⊦ m < 1 ↔ m = 0; (s) ⊦ a ' ≠ []; (s) ⊦ a ' -p 0 = []; (s) Used formulas: t0 ∈ gs.fwriters(l)

Proof step 14 from proof rely-PROP_FO (previous step) (back to tree)

gs.bq(t0) ≠ [], n < # gs.bq(t0), n0 < # gs.bq(t0), SFTrailing(gs.bq), ctrok(gs.fctr, gs.fwriters), isPropFO(n, l, gs.bq(t0)), isPropFO(n0, l, gs.bq(t0)), pendingOk(gs.bq, gs.pq), readFOsOk(gs.rfo, gs.fwriters), writersOk(gs.bq, gs.pq, gs.fwriters), ¬ t0 ∈ gs.fwriters(l) ⊦ (gs.rfo(t0) = [] → (t0 ∈ gs.fwriters(l) ⊃ gs.rfo(t0);gs.rfo(t0) -p n) = []) ∧ (gs.rfo(t0) = gs.bq(t0) → (t0 ∈ gs.fwriters(l) ⊃ gs.rfo(t0);gs.rfo(t0) -p n) = gs.bq(t0) -p n) ∧ ( gs.rfo(t0) + SF ' = gs.bq(t0) → (t0 ∈ gs.fwriters(l) ⊃ gs.rfo(t0);gs.rfo(t0) -p n) + SF ' = gs.bq(t0) -p n ∨ (t0 ∈ gs.fwriters(l) ⊃ gs.rfo(t0);gs.rfo(t0) -p n) = gs.bq(t0) -p n)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 15 from proof rely-PROP_FO (previous step) (back to tree)

gs.bq(t0) ≠ [], n0 < # gs.bq(t0), n < # gs.bq(t0), SFTrailing(gs.bq), ctrok(gs.fctr, gs.fwriters), isPropFO(n0, l, gs.bq(t0)), isPropFO(n, l, gs.bq(t0)), pendingOk(gs.bq, gs.pq), readFOsOk(gs.rfo, gs.fwriters), writersOk(gs.bq, gs.pq, gs.fwriters), ¬ t0 ∈ gs.fwriters(l) ⊦ (gs.rfo(t0) = [] → gs.rfo(t0) -p n = []) ∧ (gs.rfo(t0) = gs.bq(t0) → gs.rfo(t0) -p n = gs.bq(t0) -p n) ∧ (gs.rfo(t0) + SF ' = gs.bq(t0) → (gs.rfo(t0) -p n) + SF ' = gs.bq(t0) -p n ∨ gs.rfo(t0) -p n = gs.bq(t0) -p n)

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: right 1

Additional information

Used simplifier rules: ⊦ [] -p n = []; (s)

Proof step 16 from proof rely-PROP_FO (previous step) (back to tree)

gs.rfo(t0) + SF ' = gs.bq(t0), (gs.rfo(t0) -p n) + SF ' ≠ gs.bq(t0) -p n, gs.rfo(t0) -p n ≠ gs.bq(t0) -p n, gs.bq(t0) ≠ [], n < # gs.bq(t0), n0 < # gs.bq(t0), SFTrailing(gs.bq), ctrok(gs.fctr, gs.fwriters), isPropFO(n, l, gs.bq(t0)), isPropFO(n0, l, gs.bq(t0)), pendingOk(gs.bq, gs.pq), readFOsOk(gs.rfo, gs.fwriters), writersOk(gs.bq, gs.pq, gs.fwriters), ¬ t0 ∈ gs.fwriters(l) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list in-15: ⊦ x + a ' = y → a ∈ y; (forward) from specification list

Proof step 17 from proof rely-PROP_FO (previous step) (back to tree)

gs.rfo(t0) + SF ' = gs.bq(t0), (gs.rfo(t0) -p n) + SF ' ≠ gs.bq(t0) -p n, gs.rfo(t0) -p n ≠ gs.bq(t0) -p n, gs.bq(t0) ≠ [], n0 < # gs.bq(t0), n < # gs.bq(t0), SFTrailing(gs.bq), ctrok(gs.fctr, gs.fwriters), isPropFO(n0, l, gs.bq(t0)), isPropFO(n, l, gs.bq(t0)), pendingOk(gs.bq, gs.pq), readFOsOk(gs.rfo, gs.fwriters), writersOk(gs.bq, gs.pq, gs.fwriters), SF ∈ gs.bq(t0), ¬ t0 ∈ gs.fwriters(l) ⊦

User applied rule subst equation

And obtained the following new goals

Rule argument

pos: left 1, flags: rotate

Additional information

Proof step 18 from proof rely-PROP_FO (previous step) (back to tree)

gs.rfo(t0) + SF ' = gs.bq(t0), (gs.rfo(t0) -p n) + SF ' ≠ (gs.rfo(t0) + SF ') -p n, gs.rfo(t0) + SF ' ≠ [], gs.rfo(t0) -p n ≠ (gs.rfo(t0) + SF ') -p n, n < #(gs.rfo(t0) + SF '), n0 < #(gs.rfo(t0) + SF '), SFTrailing(gs.bq), ctrok(gs.fctr, gs.fwriters), isPropFO(n, l, gs.rfo(t0) + SF '), isPropFO(n0, l, gs.rfo(t0) + SF '), pendingOk(gs.bq, gs.pq), readFOsOk(gs.rfo, gs.fwriters), writersOk(gs.bq, gs.pq, gs.fwriters), SF ∈ gs.rfo(t0) + SF ', ¬ t0 ∈ gs.fwriters(l) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: FLIT-preds, name:isPropFO-delpos-last-SF, seq: ⊦ isPropFO(n, l, bel + SF ') → (bel + SF ') -p n = (bel -p n) + SF ', subst:[l, n, bel] → [l, n, gs.rfo(t0)], <all paths>

Additional information

Used terms: l, n, gs.rfo(t0)

Proof step 19 from proof rely-PROP_FO (previous step) (back to tree)

⊦ isPropFO(n, l, bel + SF ') → (bel + SF ') -p n = (bel -p n) + SF '

A spec-lemma

Proof step 20 from proof rely-PROP_FO (previous step) (back to tree)

⟨skip⟩ (pq = pq0 ∧ bq = bq0 ∧ rfo = rfo0), bq = gs.bq, fwriters = gs.fwriters, pq = gs.pq, rfo = gs.rfo, gs.bq(tid) ≠ [], n < # gs.bq(tid), SFTrailing(gs.bq), ctrok(gs.fctr, gs.fwriters), isPropFO(n, l, gs.bq(tid)), pendingOk(gs.bq, gs.pq), readFOsOk(gs.rfo, gs.fwriters), writersOk(gs.bq, gs.pq, gs.fwriters), ∀ n. ¬(n < # bq(tid) ∧ isPropFO(n, l, bq(tid))) ⊦ rely(t0, gs.pq, gs.bq, pq0, bq0) ∧ gs.bq(t0) ⊒w bq0(t0) ∧ (gs.rfo(t0) = [] → rfo0(t0) = []) ∧ (gs.rfo(t0) = gs.bq(t0) → rfo0(t0) = bq0(t0)) ∧ (gs.rfo(t0) + SF ' = gs.bq(t0) → rfo0(t0) + SF ' = bq0(t0) ∨ rfo0(t0) = bq0(t0))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 21 from proof rely-PROP_FO (previous step) (back to tree)

⟨skip⟩ (pq = pq0 ∧ bq = bq0 ∧ rfo = rfo0), bq = gs.bq, pq = gs.pq, rfo = gs.rfo, gs.bq(tid) ≠ [], n < # gs.bq(tid), SFTrailing(gs.bq), ctrok(gs.fctr, gs.fwriters), isPropFO(n, l, gs.bq(tid)), pendingOk(gs.bq, gs.pq), readFOsOk(gs.rfo, gs.fwriters), writersOk(gs.bq, gs.pq, gs.fwriters), ∀ n0. ¬ n0 < # gs.bq(tid) ∨ ¬ isPropFO(n0, l, gs.bq(tid)) ⊦ rely(t0, gs.pq, gs.bq, pq0, bq0) ∧ gs.bq(t0) ⊒w bq0(t0) ∧ (gs.rfo(t0) = [] → rfo0(t0) = []) ∧ (gs.rfo(t0) = gs.bq(t0) → rfo0(t0) = bq0(t0)) ∧ (gs.rfo(t0) + SF ' = gs.bq(t0) → rfo0(t0) + SF ' = bq0(t0) ∨ rfo0(t0) = bq0(t0))

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 13 terms: n

Additional information

Used terms: n

Proof Tree for rely-PROP_FO (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