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Proof step 1 from proof step-CRASH (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)) ∧ FLITLInv(gs, lsf(t), pcf(t)) → ⟪pq := gs.pq, fctr := gs.fctr, bq := gs.bq, fwriters := gs.fwriters, freaders := gs.freaders, rfo := gs.rfo; CRASH(; pq, fctr, bq, fwriters, freaders, rfo, pcf); gs := (((((gs.pq:= pq).fctr:= fctr).bq:= bq).fwriters:= fwriters).freaders:= freaders).rfo:= rfo⟫ ( ( 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)) ∧ FLITLInv(gs, lsf(t), pcf(t)))

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

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

FLITLInv(gs, lsf(t), pcf(t)), 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) ⊦ ⟪pq := gs.pq, fctr := gs.fctr, bq := gs.bq, fwriters := gs.fwriters, freaders := gs.freaders, rfo := gs.rfo; CRASH(; pq, fctr, bq, fwriters, freaders, rfo, pcf); gs := (((((gs.pq:= pq).fctr:= fctr).bq:= bq).fwriters:= fwriters).freaders:= freaders).rfo:= rfo⟫ ( ( 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)) ∧ FLITLInv(gs, lsf(t), pcf(t)))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

FLITLInv(gs, lsf(t), pcf(t)), 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) ⊦ ⟪pq := gs.pq, fctr := gs.fctr, bq := gs.bq, fwriters := gs.fwriters, freaders := gs.freaders, rfo := gs.rfo; CRASH(; pq, fctr, bq, fwriters, freaders, rfo, pcf); gs := (((((gs.pq:= pq).fctr:= fctr).bq:= bq).fwriters:= fwriters).freaders:= freaders).rfo:= rfo⟫ ( 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) ∧ FLITLInv(gs, lsf(t), pcf(t)))

Heuristic symbolic execution applied rule assign right

And obtained the following new goals

Rule argument

Additional information

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

bq = gs.bq, fctr = gs.fctr, freaders = gs.freaders, fwriters = gs.fwriters, pq = gs.pq, rfo = gs.rfo, FLITLInv(gs, lsf(t), pcf(t)), 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) ⊦ ⟪CRASH(; pq, fctr, bq, fwriters, freaders, rfo, pcf); gs := (((((gs.pq:= pq).fctr:= fctr).bq:= bq).fwriters:= fwriters).freaders:= freaders).rfo:= rfo⟫ ( 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) ∧ FLITLInv(gs, lsf(t), pcf(t)))

Heuristic unfold applied rule call right

And obtained the following new goals

Rule argument

pos: right 1

Additional information

Used simplifier rules: decl: CRASH(; pq, fctr, bq, fwriters, freaders, rfo, pcf) {pcf := λ t. (pcf(t) = idle ⊃ idle;crashed), pq := λ l. [], fctr := λ l. 0, freaders := λ l. ∅, fwriters := λ l. ∅, bq := λ t. [], rfo := λ t. [] };

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

bq = gs.bq, fctr = gs.fctr, freaders = gs.freaders, fwriters = gs.fwriters, pq = gs.pq, rfo = gs.rfo, FLITLInv(gs, lsf(t), pcf(t)), 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) ⊦ ⟪pcf := λ t. (pcf(t) = idle ⊃ idle;crashed), pq := λ l. [], fctr := λ l. 0, freaders := λ l. ∅, fwriters := λ l. ∅, bq := λ t. [], rfo := λ t. []; gs := (((((gs.pq:= pq).fctr:= fctr).bq:= bq).fwriters:= fwriters).freaders:= freaders).rfo:= rfo⟫ ( 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) ∧ FLITLInv(gs, lsf(t), pcf(t)))

Heuristic symbolic execution applied rule assign right

And obtained the following new goals

Rule argument

Additional information

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

bq = gs.bq, fctr = gs.fctr, freaders = gs.freaders, fwriters = gs.fwriters, pq = gs.pq, rfo = gs.rfo, FLITLInv(gs, lsf(t), pcf(t)), 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) ⊦ ⟪gs := (((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])⟫ ( 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) ∧ FLITLInv(gs, lsf(t), (pcf(t) = idle ⊃ idle;crashed)))

Heuristic symbolic execution applied rule assign right

And obtained the following new goals

Rule argument

Additional information

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

bq = gs.bq, fctr = gs.fctr, freaders = gs.freaders, fwriters = gs.fwriters, pq = gs.pq, rfo = gs.rfo, FLITLInv(gs, lsf(t), pcf(t)), 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) ⊦ ctrok (((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).fctr, ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).fwriters) ∧ writersOk (((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).bq, ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).pq, ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).fwriters) ∧ pendingOk (((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).bq, ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).pq) ∧ SFTrailing(((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).bq) ∧ readFOsOk (((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).rfo, ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).fwriters) ∧ FLITLInv ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. []), lsf(t), (pcf(t) = idle ⊃ idle;crashed))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ (gs.fwriters:= fwriters).fwriters = fwriters; (s) ⊦ (gs.freaders:= fwriters0).fwriters = gs.fwriters; (s) ⊦ (gs.rfo:= bq0).fwriters = gs.fwriters; (s) ⊦ (gs.rfo:= bq0).rfo = bq0; (s) ⊦ (gs.bq:= bq).bq = bq; (s) ⊦ (gs.fwriters:= fwriters).bq = gs.bq; (s) ⊦ (gs.freaders:= fwriters0).bq = gs.bq; (s) ⊦ (gs.rfo:= bq0).bq = gs.bq; (s) ⊦ (gs.pq:= pq).pq = pq; (s) ⊦ (gs.fctr:= fctr).pq = gs.pq; (s) ⊦ (gs.bq:= bq).pq = gs.pq; (s) ⊦ (gs.fwriters:= fwriters).pq = gs.pq; (s) ⊦ (gs.freaders:= fwriters0).pq = gs.pq; (s) ⊦ (gs.rfo:= bq0).pq = gs.pq; (s) ⊦ (gs.fctr:= fctr).fctr = fctr; (s) ⊦ (gs.bq:= bq).fctr = gs.fctr; (s) ⊦ (gs.fwriters:= fwriters).fctr = gs.fctr; (s) ⊦ (gs.freaders:= fwriters0).fctr = gs.fctr; (s) ⊦ (gs.rfo:= bq0).fctr = gs.fctr; (s)

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

FLITLInv(gs, lsf(t), pcf(t)), 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) ⊦ ctrok(λ l. 0, λ l. ∅) ∧ writersOk(λ t. [], λ l. [], λ l. ∅) ∧ pendingOk(λ t. [], λ l. []) ∧ SFTrailing(λ t. []) ∧ readFOsOk(λ t. [], λ l. ∅) ∧ FLITLInv ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. []), lsf(t), (pcf(t) = idle ⊃ idle;crashed))

Heuristic if-then-else split applied rule case distinction

And obtained the following new goals

Rule argument

pos: right 1

Additional information

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

FLITLInv(gs, lsf(t), pcf(t)), 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), ¬ ctrok(λ l. 0, λ l. ∅) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: FLITPSC-preds, name:ctrok-def, seq: ⊦ ctrok(ctr, writers) ↔ (∀ l. # writers(l) = ctr(l)), subst:[ctr, writers] → [λ l. 0, λ l. ∅], <all paths>

Additional information

Used simplifier rules: ⊦ # ∅ = 0; (s) Used terms: λ l. 0, λ l. ∅

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

⊦ ctrok(ctr, writers) ↔ (∀ l. # writers(l) = ctr(l))

A spec-lemma

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

FLITLInv(gs, lsf(t), pcf(t)), 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), ¬ writersOk(λ t. [], λ l. [], λ l. ∅) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: FLIT-preds, name:writersOk, seq:⊦ writersOk(bq, p, writers) ↔ (∀ l, t. (t ∈ writers(l) → (∀ l0. l0 ≠ l → ¬ t ∈ writers(l0))) ∧ (¬ t ∈ writers(l) → p(l) |W | t = [] ∧ bq(t) |WL l = [])), subst:[bq, p, writers] → [λ t. [], λ l. [], λ l. ∅], <all paths>

Additional information

Used simplifier rules: ⊦ ¬ a ∈ ∅; (s) ⊦ [] |W = []; (s) ⊦ [] | t = []; (s) ⊦ [] |WL l = []; (s) Used terms: λ t. [], λ l. [], λ l. ∅

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

⊦ writersOk(bq, p, writers) ↔ (∀ l, t. (t ∈ writers(l) → (∀ l0. l0 ≠ l → ¬ t ∈ writers(l0))) ∧ (¬ t ∈ writers(l) → p(l) |W | t = [] ∧ bq(t) |WL l = []))

A spec-lemma

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

FLITLInv(gs, lsf(t), pcf(t)), 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), ¬ pendingOk(λ t. [], λ l. []) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: FLIT-preds, name:pendingOk, seq: ⊦ pendingOk(bq, p) ↔ (∀ l, t. #(p(l) |W | t) + #(bq(t) |WL l) ≤ 1), subst:[bq, p] → [λ t. [], λ l. []], <all paths>

Additional information

Used simplifier rules: ⊦ [] |W = []; (s) ⊦ [] | t = []; (s) ⊦ [] |WL l = []; (s) ⊦ # [] = 0; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic Used terms: λ t. [], λ l. []

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

⊦ pendingOk(bq, p) ↔ (∀ l, t. #(p(l) |W | t) + #(bq(t) |WL l) ≤ 1)

A spec-lemma

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

FLITLInv(gs, lsf(t), pcf(t)), 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), ¬ SFTrailing(λ t. []) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: FLIT-preds, name:SFTrailing, seq: ⊦ SFTrailing(bq) ↔ (∀ t. bq(t) = [] ∨ ¬ SF ∈ bq(t).butlast), subst:[bq] → [λ t. []], <all paths>

Additional information

Used terms: λ t. []

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

⊦ SFTrailing(bq) ↔ (∀ t. bq(t) = [] ∨ ¬ SF ∈ bq(t).butlast)

A spec-lemma

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

FLITLInv(gs, lsf(t), pcf(t)), 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), ¬ readFOsOk(λ t. [], λ l. ∅) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: FLIT-preds, name:readFOsOk, seq: ⊦ readFOsOk(rfo, writers) ↔ rfo.FOs? ∧ (∀ t, l. t ∈ writers(l) → rfo(t) = []), subst:[rfo, writers] → [λ t. [], λ l. ∅], <all paths>

Additional information

Used terms: λ t. [], λ l. ∅

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

⊦ readFOsOk(rfo, writers) ↔ rfo.FOs? ∧ (∀ t, l. t ∈ writers(l) → rfo(t) = [])

A spec-lemma

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

FLITLInv(gs, lsf(t), pcf(t)), 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), ¬ (λ t. []).FOs? ∨ (∃ t, l. ¬(t ∈ ∅ → [] = [])) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ¬ a ∈ ∅; (s)

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

FLITLInv(gs, lsf(t), pcf(t)), 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), ¬ (λ t. []).FOs? ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: FLIT-preds, name:FOs, seq: ⊦ rfo.FOs? ↔ (∀ t. rfo(t).forall(.FO?)), subst:[rfo] → [λ t. []], <all paths>

Additional information

Used simplifier rules: ⊦ [].forall(p); (s) Used terms: λ t. []

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

⊦ rfo.FOs? ↔ (∀ t. rfo(t).forall(.FO?))

A spec-lemma

Proof step 22 from proof step-CRASH (previous step) (back to tree)

FLITLInv(gs, lsf(t), pcf(t)), 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), ¬ FLITLInv ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. []), lsf(t), (pcf(t) = idle ⊃ idle;crashed)) ⊦

Heuristic if-then-else split applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 7

Additional information

Proof step 23 from proof step-CRASH (previous step) (back to tree)

pcf(t) = idle, FLITLInv(gs, lsf(t), pcf(t)), 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), ¬ FLITLInv((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. []), lsf(t), idle) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 24 from proof step-CRASH (previous step) (back to tree)

pcf(t) = idle, FLITLInv(gs, lsf(t), idle), 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), ¬ FLITLInv((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. []), lsf(t), idle) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: FLIT, name:FLITLInv, seq:⊦ FLITLInv(gs, ls, pc) ↔ ( pc = W1 ∨ pc = W2 ∨ pc = W3 ∨ pc = W4 ∨ pc = W5 ∨ pc = W6 ∨ pc = W7 → notMember(ls.t, gs.freaders)) ∧ (pc = W1 ∨ pc = W2 ∨ pc = W7 → notMember(ls.t, gs.fwriters)) ∧ (pc = W3 ∨ pc = W4 ∨ pc = W5 ∨ pc = W6 → oneMember(ls.t, ls.loc, gs.fwriters)) ∧ (pc = W1 ∨ pc = W2 → noPendingW(ls.t, gs.bq, gs.pq)) ∧ (pc = W3 ∨ pc = W7 → noPending(ls.t, gs.bq, gs.pq)) ∧ (pc = W5 ∨ pc = W6 → FOFollowsW(ls.t, gs.bq(ls.t), gs.pq(ls.loc), ls.loc)) ∧ (pc = W1 ∨ pc = W3 ∨ pc = W4 ∨ pc = W5 ∨ pc = W7 → ¬ SF ∈ gs.bq(ls.t)) ∧ (pc = W3 ∨ pc = W4 ∨ pc = W5 ∨ pc = W6 ∨ pc = W7 → gs.rfo(ls.t) = []) ∧ (pc = W3 → gs.bq(ls.t) = []) ∧ (pc = W4 → gs.bq(ls.t) = [] ∨ gs.bq(ls.t) = W(ls.loc, ls.val) ') ∧ ( pc = W5 → gs.bq(ls.t) = [] ∨ gs.bq(ls.t) = FO(ls.loc) ' ∨ gs.bq(ls.t) = W(ls.loc, ls.val) ' + FO(ls.loc) ') ∧ ( pc = W6 → gs.bq(ls.t) = [] ∨ gs.bq(ls.t) = SF ' ∨ gs.bq(ls.t) = FO(ls.loc) ' + SF ' ∨ gs.bq(ls.t) = W(ls.loc, ls.val) ' + FO(ls.loc) ' + SF ') ∧ ( pc = W4 ∨ pc = W5 ∨ pc = W6 → gs.bq(ls.t) ≠ [] ∧ gs.bq(ls.t).head.W? → gs.bq(ls.t).head = W(ls.loc, ls.val)) ∧ (pc = W1 ∨ pc = W7 → gs.rfo(ls.t) = gs.bq(ls.t)) ∧ (pc = W2 → gs.rfo(ls.t) = gs.bq(ls.t) ∨ gs.rfo(ls.t) + SF ' = gs.bq(ls.t)) ∧ (pc = R1 ∨ pc = R2 ∨ pc = R3 ∨ pc = R4 → noPendingW(ls.t, gs.bq, gs.pq)) ∧ (pc = R1 ∨ pc = R2 ∨ pc = R3 ∨ pc = R4 → notMember(ls.t, gs.fwriters)) ∧ (pc = R1 ∨ pc = R4 → notMember(ls.t, gs.freaders)) ∧ (pc = R2 ∨ pc = R3 → oneMember(ls.t, ls.loc, gs.freaders)) ∧ (pc = R1 ∨ pc = R2 ∨ pc = R3 ∨ pc = R4 → ¬ SF ∈ gs.bq(ls.t)) ∧ (pc = R1 ∨ pc = R2 ∨ pc = R3 ∨ pc = R4 → gs.rfo(ls.t) = gs.bq(ls.t)) ∧ (pc = idle ∨ pc = ready ∨ pc = crashed → notMember(ls.t, gs.fwriters)) ∧ (pc = idle ∨ pc = ready ∨ pc = crashed → notMember(ls.t, gs.freaders)) ∧ (pc = idle ∨ pc = ready ∨ pc = crashed → noPendingW(ls.t, gs.bq, gs.pq)) ∧ (pc = idle ∨ pc = crashed → noPending(ls.t, gs.bq, gs.pq)) ∧ (pc = idle ∨ pc = ready ∨ pc = crashed → ¬ SF ∈ gs.bq(ls.t)) ∧ (pc = idle ∨ pc = ready ∨ pc = crashed → gs.rfo(ls.t) = gs.bq(ls.t)), subst:[gs, ls, pc] → [ ( ( ( ((gs.pq:= (λ l. [])).fctr:= (λ l. 0)) .bq:= (λ t. [])) .fwriters:= (λ l. ∅)) .freaders:= (λ l. ∅)) .rfo:= (λ t. []), lsf(t), idle], <all paths>

Additional information

Used terms: ( ( (((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])) .fwriters:= (λ l. ∅)) .freaders:= (λ l. ∅)) .rfo:= (λ t. []), lsf(t), idle

Proof step 25 from proof step-CRASH (previous step) (back to tree)

⊦ FLITLInv(gs, ls, pc) ↔ (pc = W1 ∨ pc = W2 ∨ pc = W3 ∨ pc = W4 ∨ pc = W5 ∨ pc = W6 ∨ pc = W7 → notMember(ls.t, gs.freaders)) ∧ (pc = W1 ∨ pc = W2 ∨ pc = W7 → notMember(ls.t, gs.fwriters)) ∧ (pc = W3 ∨ pc = W4 ∨ pc = W5 ∨ pc = W6 → oneMember(ls.t, ls.loc, gs.fwriters)) ∧ (pc = W1 ∨ pc = W2 → noPendingW(ls.t, gs.bq, gs.pq)) ∧ (pc = W3 ∨ pc = W7 → noPending(ls.t, gs.bq, gs.pq)) ∧ (pc = W5 ∨ pc = W6 → FOFollowsW(ls.t, gs.bq(ls.t), gs.pq(ls.loc), ls.loc)) ∧ (pc = W1 ∨ pc = W3 ∨ pc = W4 ∨ pc = W5 ∨ pc = W7 → ¬ SF ∈ gs.bq(ls.t)) ∧ (pc = W3 ∨ pc = W4 ∨ pc = W5 ∨ pc = W6 ∨ pc = W7 → gs.rfo(ls.t) = []) ∧ (pc = W3 → gs.bq(ls.t) = []) ∧ (pc = W4 → gs.bq(ls.t) = [] ∨ gs.bq(ls.t) = W(ls.loc, ls.val) ') ∧ (pc = W5 → gs.bq(ls.t) = [] ∨ gs.bq(ls.t) = FO(ls.loc) ' ∨ gs.bq(ls.t) = W(ls.loc, ls.val) ' + FO(ls.loc) ') ∧ (pc = W6 → gs.bq(ls.t) = [] ∨ gs.bq(ls.t) = SF ' ∨ gs.bq(ls.t) = FO(ls.loc) ' + SF ' ∨ gs.bq(ls.t) = W(ls.loc, ls.val) ' + FO(ls.loc) ' + SF ') ∧ (pc = W4 ∨ pc = W5 ∨ pc = W6 → gs.bq(ls.t) ≠ [] ∧ gs.bq(ls.t).head.W? → gs.bq(ls.t).head = W(ls.loc, ls.val)) ∧ (pc = W1 ∨ pc = W7 → gs.rfo(ls.t) = gs.bq(ls.t)) ∧ (pc = W2 → gs.rfo(ls.t) = gs.bq(ls.t) ∨ gs.rfo(ls.t) + SF ' = gs.bq(ls.t)) ∧ (pc = R1 ∨ pc = R2 ∨ pc = R3 ∨ pc = R4 → noPendingW(ls.t, gs.bq, gs.pq)) ∧ (pc = R1 ∨ pc = R2 ∨ pc = R3 ∨ pc = R4 → notMember(ls.t, gs.fwriters)) ∧ (pc = R1 ∨ pc = R4 → notMember(ls.t, gs.freaders)) ∧ (pc = R2 ∨ pc = R3 → oneMember(ls.t, ls.loc, gs.freaders)) ∧ (pc = R1 ∨ pc = R2 ∨ pc = R3 ∨ pc = R4 → ¬ SF ∈ gs.bq(ls.t)) ∧ (pc = R1 ∨ pc = R2 ∨ pc = R3 ∨ pc = R4 → gs.rfo(ls.t) = gs.bq(ls.t)) ∧ (pc = idle ∨ pc = ready ∨ pc = crashed → notMember(ls.t, gs.fwriters)) ∧ (pc = idle ∨ pc = ready ∨ pc = crashed → notMember(ls.t, gs.freaders)) ∧ (pc = idle ∨ pc = ready ∨ pc = crashed → noPendingW(ls.t, gs.bq, gs.pq)) ∧ (pc = idle ∨ pc = crashed → noPending(ls.t, gs.bq, gs.pq)) ∧ (pc = idle ∨ pc = ready ∨ pc = crashed → ¬ SF ∈ gs.bq(ls.t)) ∧ (pc = idle ∨ pc = ready ∨ pc = crashed → gs.rfo(ls.t) = gs.bq(ls.t))

A spec-lemma

Proof step 26 from proof step-CRASH (previous step) (back to tree)

pcf(t) = idle, FLITLInv(gs, lsf(t), idle), 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), ¬ ( idle = W1 ∨ idle = W2 ∨ idle = W3 ∨ idle = W4 ∨ idle = W5 ∨ idle = W6 ∨ idle = W7 → notMember (lsf(t).t, ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).freaders)) ∨ ¬ ( idle = W1 ∨ idle = W2 ∨ idle = W7 → notMember (lsf(t).t, ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).fwriters)) ∨ ¬ ( idle = W3 ∨ idle = W4 ∨ idle = W5 ∨ idle = W6 → oneMember (lsf(t).t, lsf(t).loc, ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).fwriters)) ∨ ¬ ( idle = W1 ∨ idle = W2 → noPendingW (lsf(t).t, ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).bq, ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).pq)) ∨ ¬ ( idle = W3 ∨ idle = W7 → noPending (lsf(t).t, ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).bq, ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).pq)) ∨ ¬ ( idle = W5 ∨ idle = W6 → FOFollowsW (lsf(t).t, ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).bq(lsf(t).t), ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).pq(lsf(t).loc), lsf(t).loc)) ∨ ¬ ( idle = W1 ∨ idle = W3 ∨ idle = W4 ∨ idle = W5 ∨ idle = W7 → ¬ SF ∈ ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).bq(lsf(t).t)) ∨ ¬ ( idle = W3 ∨ idle = W4 ∨ idle = W5 ∨ idle = W6 ∨ idle = W7 → ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).rfo(lsf(t).t) = []) ∨ ¬ (idle = W3 → ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).bq(lsf(t).t) = []) ∨ ¬ ( idle = W4 → ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).bq(lsf(t).t) = [] ∨ ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).bq(lsf(t).t) = W(lsf(t).loc, lsf(t).val) ') ∨ ¬ ( idle = W5 → ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).bq(lsf(t).t) = [] ∨ ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).bq(lsf(t).t) = FO(lsf(t).loc) ' ∨ ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).bq(lsf(t).t) = W(lsf(t).loc, lsf(t).val) ' + FO(lsf(t).loc) ') ∨ ¬ ( idle = W6 → ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).bq(lsf(t).t) = [] ∨ ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).bq(lsf(t).t) = SF ' ∨ ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).bq(lsf(t).t) = FO(lsf(t).loc) ' + SF ' ∨ ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).bq(lsf(t).t) = W(lsf(t).loc, lsf(t).val) ' + FO(lsf(t).loc) ' + SF ') ∨ ¬ ( idle = W4 ∨ idle = W5 ∨ idle = W6 → ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).bq(lsf(t).t) ≠ [] ∧ ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).bq(lsf(t).t).head.W? → ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).bq(lsf(t).t).head = W(lsf(t).loc, lsf(t).val)) ∨ ¬ ( idle = W1 ∨ idle = W7 → ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).rfo(lsf(t).t) = ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).bq(lsf(t).t)) ∨ ¬ ( idle = W2 → ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).rfo(lsf(t).t) = ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).bq(lsf(t).t) ∨ ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).rfo(lsf(t).t) + SF ' = ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).bq(lsf(t).t)) ∨ ¬ ( idle = R1 ∨ idle = R2 ∨ idle = R3 ∨ idle = R4 → noPendingW (lsf(t).t, ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).bq, ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).pq)) ∨ ¬ ( idle = R1 ∨ idle = R2 ∨ idle = R3 ∨ idle = R4 → notMember (lsf(t).t, ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).fwriters)) ∨ ¬ ( idle = R1 ∨ idle = R4 → notMember (lsf(t).t, ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).freaders)) ∨ ¬ ( idle = R2 ∨ idle = R3 → oneMember (lsf(t).t, lsf(t).loc, ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).freaders)) ∨ ¬ ( idle = R1 ∨ idle = R2 ∨ idle = R3 ∨ idle = R4 → ¬ SF ∈ ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).bq(lsf(t).t)) ∨ ¬ ( idle = R1 ∨ idle = R2 ∨ idle = R3 ∨ idle = R4 → ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).rfo(lsf(t).t) = ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).bq(lsf(t).t)) ∨ ¬ ( idle = idle ∨ idle = ready ∨ idle = crashed → notMember (lsf(t).t, ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).fwriters)) ∨ ¬ ( idle = idle ∨ idle = ready ∨ idle = crashed → notMember (lsf(t).t, ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).freaders)) ∨ ¬ ( idle = idle ∨ idle = ready ∨ idle = crashed → noPendingW (lsf(t).t, ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).bq, ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).pq)) ∨ ¬ ( idle = idle ∨ idle = crashed → noPending (lsf(t).t, ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).bq, ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).pq)) ∨ ¬ ( idle = idle ∨ idle = ready ∨ idle = crashed → ¬ SF ∈ ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).bq(lsf(t).t)) ∨ ¬ ( idle = idle ∨ idle = ready ∨ idle = crashed → ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).rfo(lsf(t).t) = ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).bq(lsf(t).t)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ noPending(t, bq, pq) → noPendingW(t, bq, pq); (ss) ⊦ (gs.bq:= bq).bq = bq; (s) ⊦ (gs.fwriters:= fwriters).bq = gs.bq; (s) ⊦ (gs.freaders:= fwriters0).bq = gs.bq; (s) ⊦ (gs.rfo:= bq0).bq = gs.bq; (s) ⊦ (gs.rfo:= bq0).rfo = bq0; (s) ⊦ ¬ a ∈ []; (s) ⊦ (gs.pq:= pq).pq = pq; (s) ⊦ (gs.fctr:= fctr).pq = gs.pq; (s) ⊦ (gs.bq:= bq).pq = gs.pq; (s) ⊦ (gs.fwriters:= fwriters).pq = gs.pq; (s) ⊦ (gs.freaders:= fwriters0).pq = gs.pq; (s) ⊦ (gs.rfo:= bq0).pq = gs.pq; (s) ⊦ (gs.freaders:= fwriters0).freaders = fwriters0; (s) ⊦ (gs.rfo:= bq0).freaders = gs.freaders; (s) ⊦ (gs.fwriters:= fwriters).fwriters = fwriters; (s) ⊦ (gs.freaders:= fwriters0).fwriters = gs.fwriters; (s) ⊦ (gs.rfo:= bq0).fwriters = gs.fwriters; (s) ⊦ R4 ≠ idle; (s) ⊦ R3 ≠ idle; (s) ⊦ R2 ≠ idle; (s) ⊦ R1 ≠ idle; (s) ⊦ W2 ≠ idle; (s) ⊦ W7 ≠ idle; (s) ⊦ W1 ≠ idle; (s) ⊦ W6 ≠ idle; (s) ⊦ W5 ≠ idle; (s) ⊦ W4 ≠ idle; (s) ⊦ W3 ≠ idle; (s)

Proof step 27 from proof step-CRASH (previous step) (back to tree)

pcf(t) = idle, FLITLInv(gs, lsf(t), idle), 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), ¬ notMember(lsf(t).t, λ l. ∅) ∨ ¬ noPending(lsf(t).t, λ t. [], λ l. []) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: FLIT-preds, name:noPending, seq: ⊦ noPending(t, bq, p) ↔ (∀ l. p(l) | t = [] ∧ bq(t) = []), subst:[t, bq, p] → [lsf(t).t, λ t. [], λ l. []], <all paths>

Additional information

Used terms: lsf(t).t, λ t. [], λ l. []

Proof step 28 from proof step-CRASH (previous step) (back to tree)

⊦ noPending(t, bq, p) ↔ (∀ l. p(l) | t = [] ∧ bq(t) = [])

A spec-lemma

Proof step 29 from proof step-CRASH (previous step) (back to tree)

pcf(t) = idle, FLITLInv(gs, lsf(t), idle), 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), ¬ notMember(lsf(t).t, λ l. ∅) ∨ ¬(∀ l. [] | lsf(t).t = [] ∧ [] = []) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ [] | t = []; (s)

Proof step 30 from proof step-CRASH (previous step) (back to tree)

pcf(t) = idle, FLITLInv(gs, lsf(t), idle), 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), ¬ notMember(lsf(t).t, λ l. ∅) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: FLITPSC-preds, name:noWriter, seq: ⊦ notMember(t, tsf) ↔ (∀ l. ¬ t ∈ tsf(l)), subst:[t, tsf] → [lsf(t).t, λ l. ∅], <all paths>

Additional information

Used simplifier rules: ⊦ ¬ a ∈ ∅; (s) Used terms: lsf(t).t, λ l. ∅

Proof step 31 from proof step-CRASH (previous step) (back to tree)

⊦ notMember(t, tsf) ↔ (∀ l. ¬ t ∈ tsf(l))

A spec-lemma

Proof step 32 from proof step-CRASH (previous step) (back to tree)

pcf(t) ≠ idle, FLITLInv(gs, lsf(t), pcf(t)), 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), ¬ FLITLInv((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. []), lsf(t), crashed) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: FLIT, name:FLITLInv, seq:⊦ FLITLInv(gs, ls, pc) ↔ ( pc = W1 ∨ pc = W2 ∨ pc = W3 ∨ pc = W4 ∨ pc = W5 ∨ pc = W6 ∨ pc = W7 → notMember(ls.t, gs.freaders)) ∧ (pc = W1 ∨ pc = W2 ∨ pc = W7 → notMember(ls.t, gs.fwriters)) ∧ (pc = W3 ∨ pc = W4 ∨ pc = W5 ∨ pc = W6 → oneMember(ls.t, ls.loc, gs.fwriters)) ∧ (pc = W1 ∨ pc = W2 → noPendingW(ls.t, gs.bq, gs.pq)) ∧ (pc = W3 ∨ pc = W7 → noPending(ls.t, gs.bq, gs.pq)) ∧ (pc = W5 ∨ pc = W6 → FOFollowsW(ls.t, gs.bq(ls.t), gs.pq(ls.loc), ls.loc)) ∧ (pc = W1 ∨ pc = W3 ∨ pc = W4 ∨ pc = W5 ∨ pc = W7 → ¬ SF ∈ gs.bq(ls.t)) ∧ (pc = W3 ∨ pc = W4 ∨ pc = W5 ∨ pc = W6 ∨ pc = W7 → gs.rfo(ls.t) = []) ∧ (pc = W3 → gs.bq(ls.t) = []) ∧ (pc = W4 → gs.bq(ls.t) = [] ∨ gs.bq(ls.t) = W(ls.loc, ls.val) ') ∧ ( pc = W5 → gs.bq(ls.t) = [] ∨ gs.bq(ls.t) = FO(ls.loc) ' ∨ gs.bq(ls.t) = W(ls.loc, ls.val) ' + FO(ls.loc) ') ∧ ( pc = W6 → gs.bq(ls.t) = [] ∨ gs.bq(ls.t) = SF ' ∨ gs.bq(ls.t) = FO(ls.loc) ' + SF ' ∨ gs.bq(ls.t) = W(ls.loc, ls.val) ' + FO(ls.loc) ' + SF ') ∧ ( pc = W4 ∨ pc = W5 ∨ pc = W6 → gs.bq(ls.t) ≠ [] ∧ gs.bq(ls.t).head.W? → gs.bq(ls.t).head = W(ls.loc, ls.val)) ∧ (pc = W1 ∨ pc = W7 → gs.rfo(ls.t) = gs.bq(ls.t)) ∧ (pc = W2 → gs.rfo(ls.t) = gs.bq(ls.t) ∨ gs.rfo(ls.t) + SF ' = gs.bq(ls.t)) ∧ (pc = R1 ∨ pc = R2 ∨ pc = R3 ∨ pc = R4 → noPendingW(ls.t, gs.bq, gs.pq)) ∧ (pc = R1 ∨ pc = R2 ∨ pc = R3 ∨ pc = R4 → notMember(ls.t, gs.fwriters)) ∧ (pc = R1 ∨ pc = R4 → notMember(ls.t, gs.freaders)) ∧ (pc = R2 ∨ pc = R3 → oneMember(ls.t, ls.loc, gs.freaders)) ∧ (pc = R1 ∨ pc = R2 ∨ pc = R3 ∨ pc = R4 → ¬ SF ∈ gs.bq(ls.t)) ∧ (pc = R1 ∨ pc = R2 ∨ pc = R3 ∨ pc = R4 → gs.rfo(ls.t) = gs.bq(ls.t)) ∧ (pc = idle ∨ pc = ready ∨ pc = crashed → notMember(ls.t, gs.fwriters)) ∧ (pc = idle ∨ pc = ready ∨ pc = crashed → notMember(ls.t, gs.freaders)) ∧ (pc = idle ∨ pc = ready ∨ pc = crashed → noPendingW(ls.t, gs.bq, gs.pq)) ∧ (pc = idle ∨ pc = crashed → noPending(ls.t, gs.bq, gs.pq)) ∧ (pc = idle ∨ pc = ready ∨ pc = crashed → ¬ SF ∈ gs.bq(ls.t)) ∧ (pc = idle ∨ pc = ready ∨ pc = crashed → gs.rfo(ls.t) = gs.bq(ls.t)), subst:[gs, ls, pc] → [ ( ( ( ((gs.pq:= (λ l. [])).fctr:= (λ l. 0)) .bq:= (λ t. [])) .fwriters:= (λ l. ∅)) .freaders:= (λ l. ∅)) .rfo:= (λ t. []), lsf(t), crashed], <all paths>

Additional information

Used terms: ( ( (((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])) .fwriters:= (λ l. ∅)) .freaders:= (λ l. ∅)) .rfo:= (λ t. []), lsf(t), crashed

Proof step 33 from proof step-CRASH (previous step) (back to tree)

⊦ FLITLInv(gs, ls, pc) ↔ (pc = W1 ∨ pc = W2 ∨ pc = W3 ∨ pc = W4 ∨ pc = W5 ∨ pc = W6 ∨ pc = W7 → notMember(ls.t, gs.freaders)) ∧ (pc = W1 ∨ pc = W2 ∨ pc = W7 → notMember(ls.t, gs.fwriters)) ∧ (pc = W3 ∨ pc = W4 ∨ pc = W5 ∨ pc = W6 → oneMember(ls.t, ls.loc, gs.fwriters)) ∧ (pc = W1 ∨ pc = W2 → noPendingW(ls.t, gs.bq, gs.pq)) ∧ (pc = W3 ∨ pc = W7 → noPending(ls.t, gs.bq, gs.pq)) ∧ (pc = W5 ∨ pc = W6 → FOFollowsW(ls.t, gs.bq(ls.t), gs.pq(ls.loc), ls.loc)) ∧ (pc = W1 ∨ pc = W3 ∨ pc = W4 ∨ pc = W5 ∨ pc = W7 → ¬ SF ∈ gs.bq(ls.t)) ∧ (pc = W3 ∨ pc = W4 ∨ pc = W5 ∨ pc = W6 ∨ pc = W7 → gs.rfo(ls.t) = []) ∧ (pc = W3 → gs.bq(ls.t) = []) ∧ (pc = W4 → gs.bq(ls.t) = [] ∨ gs.bq(ls.t) = W(ls.loc, ls.val) ') ∧ (pc = W5 → gs.bq(ls.t) = [] ∨ gs.bq(ls.t) = FO(ls.loc) ' ∨ gs.bq(ls.t) = W(ls.loc, ls.val) ' + FO(ls.loc) ') ∧ (pc = W6 → gs.bq(ls.t) = [] ∨ gs.bq(ls.t) = SF ' ∨ gs.bq(ls.t) = FO(ls.loc) ' + SF ' ∨ gs.bq(ls.t) = W(ls.loc, ls.val) ' + FO(ls.loc) ' + SF ') ∧ (pc = W4 ∨ pc = W5 ∨ pc = W6 → gs.bq(ls.t) ≠ [] ∧ gs.bq(ls.t).head.W? → gs.bq(ls.t).head = W(ls.loc, ls.val)) ∧ (pc = W1 ∨ pc = W7 → gs.rfo(ls.t) = gs.bq(ls.t)) ∧ (pc = W2 → gs.rfo(ls.t) = gs.bq(ls.t) ∨ gs.rfo(ls.t) + SF ' = gs.bq(ls.t)) ∧ (pc = R1 ∨ pc = R2 ∨ pc = R3 ∨ pc = R4 → noPendingW(ls.t, gs.bq, gs.pq)) ∧ (pc = R1 ∨ pc = R2 ∨ pc = R3 ∨ pc = R4 → notMember(ls.t, gs.fwriters)) ∧ (pc = R1 ∨ pc = R4 → notMember(ls.t, gs.freaders)) ∧ (pc = R2 ∨ pc = R3 → oneMember(ls.t, ls.loc, gs.freaders)) ∧ (pc = R1 ∨ pc = R2 ∨ pc = R3 ∨ pc = R4 → ¬ SF ∈ gs.bq(ls.t)) ∧ (pc = R1 ∨ pc = R2 ∨ pc = R3 ∨ pc = R4 → gs.rfo(ls.t) = gs.bq(ls.t)) ∧ (pc = idle ∨ pc = ready ∨ pc = crashed → notMember(ls.t, gs.fwriters)) ∧ (pc = idle ∨ pc = ready ∨ pc = crashed → notMember(ls.t, gs.freaders)) ∧ (pc = idle ∨ pc = ready ∨ pc = crashed → noPendingW(ls.t, gs.bq, gs.pq)) ∧ (pc = idle ∨ pc = crashed → noPending(ls.t, gs.bq, gs.pq)) ∧ (pc = idle ∨ pc = ready ∨ pc = crashed → ¬ SF ∈ gs.bq(ls.t)) ∧ (pc = idle ∨ pc = ready ∨ pc = crashed → gs.rfo(ls.t) = gs.bq(ls.t))

A spec-lemma

Proof step 34 from proof step-CRASH (previous step) (back to tree)

pcf(t) ≠ idle, FLITLInv(gs, lsf(t), pcf(t)), 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), ¬ ( crashed = W1 ∨ crashed = W2 ∨ crashed = W3 ∨ crashed = W4 ∨ crashed = W5 ∨ crashed = W6 ∨ crashed = W7 → notMember (lsf(t).t, ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).freaders)) ∨ ¬ ( crashed = W1 ∨ crashed = W2 ∨ crashed = W7 → notMember (lsf(t).t, ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).fwriters)) ∨ ¬ ( crashed = W3 ∨ crashed = W4 ∨ crashed = W5 ∨ crashed = W6 → oneMember (lsf(t).t, lsf(t).loc, ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).fwriters)) ∨ ¬ ( crashed = W1 ∨ crashed = W2 → noPendingW (lsf(t).t, ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).bq, ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).pq)) ∨ ¬ ( crashed = W3 ∨ crashed = W7 → noPending (lsf(t).t, ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).bq, ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).pq)) ∨ ¬ ( crashed = W5 ∨ crashed = W6 → FOFollowsW (lsf(t).t, ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).bq(lsf(t).t), ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).pq(lsf(t).loc), lsf(t).loc)) ∨ ¬ ( crashed = W1 ∨ crashed = W3 ∨ crashed = W4 ∨ crashed = W5 ∨ crashed = W7 → ¬ SF ∈ ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).bq(lsf(t).t)) ∨ ¬ ( crashed = W3 ∨ crashed = W4 ∨ crashed = W5 ∨ crashed = W6 ∨ crashed = W7 → ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).rfo(lsf(t).t) = []) ∨ ¬ ( crashed = W3 → ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).bq(lsf(t).t) = []) ∨ ¬ ( crashed = W4 → ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).bq(lsf(t).t) = [] ∨ ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).bq(lsf(t).t) = W(lsf(t).loc, lsf(t).val) ') ∨ ¬ ( crashed = W5 → ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).bq(lsf(t).t) = [] ∨ ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).bq(lsf(t).t) = FO(lsf(t).loc) ' ∨ ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).bq(lsf(t).t) = W(lsf(t).loc, lsf(t).val) ' + FO(lsf(t).loc) ') ∨ ¬ ( crashed = W6 → ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).bq(lsf(t).t) = [] ∨ ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).bq(lsf(t).t) = SF ' ∨ ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).bq(lsf(t).t) = FO(lsf(t).loc) ' + SF ' ∨ ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).bq(lsf(t).t) = W(lsf(t).loc, lsf(t).val) ' + FO(lsf(t).loc) ' + SF ') ∨ ¬ ( crashed = W4 ∨ crashed = W5 ∨ crashed = W6 → ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).bq(lsf(t).t) ≠ [] ∧ ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).bq(lsf(t).t).head.W? → ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).bq(lsf(t).t).head = W(lsf(t).loc, lsf(t).val)) ∨ ¬ ( crashed = W1 ∨ crashed = W7 → ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).rfo(lsf(t).t) = ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).bq(lsf(t).t)) ∨ ¬ ( crashed = W2 → ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).rfo(lsf(t).t) = ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).bq(lsf(t).t) ∨ ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).rfo(lsf(t).t) + SF ' = ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).bq(lsf(t).t)) ∨ ¬ ( crashed = R1 ∨ crashed = R2 ∨ crashed = R3 ∨ crashed = R4 → noPendingW (lsf(t).t, ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).bq, ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).pq)) ∨ ¬ ( crashed = R1 ∨ crashed = R2 ∨ crashed = R3 ∨ crashed = R4 → notMember (lsf(t).t, ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).fwriters)) ∨ ¬ ( crashed = R1 ∨ crashed = R4 → notMember (lsf(t).t, ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).freaders)) ∨ ¬ ( crashed = R2 ∨ crashed = R3 → oneMember (lsf(t).t, lsf(t).loc, ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).freaders)) ∨ ¬ ( crashed = R1 ∨ crashed = R2 ∨ crashed = R3 ∨ crashed = R4 → ¬ SF ∈ ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).bq(lsf(t).t)) ∨ ¬ ( crashed = R1 ∨ crashed = R2 ∨ crashed = R3 ∨ crashed = R4 → ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).rfo(lsf(t).t) = ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).bq(lsf(t).t)) ∨ ¬ ( crashed = idle ∨ crashed = ready ∨ crashed = crashed → notMember (lsf(t).t, ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).fwriters)) ∨ ¬ ( crashed = idle ∨ crashed = ready ∨ crashed = crashed → notMember (lsf(t).t, ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).freaders)) ∨ ¬ ( crashed = idle ∨ crashed = ready ∨ crashed = crashed → noPendingW (lsf(t).t, ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).bq, ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).pq)) ∨ ¬ ( crashed = idle ∨ crashed = crashed → noPending (lsf(t).t, ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).bq, ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).pq)) ∨ ¬ ( crashed = idle ∨ crashed = ready ∨ crashed = crashed → ¬ SF ∈ ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).bq(lsf(t).t)) ∨ ¬ ( crashed = idle ∨ crashed = ready ∨ crashed = crashed → ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).rfo(lsf(t).t) = ((((((gs.pq:= (λ l. [])).fctr:= (λ l. 0)).bq:= (λ t. [])).fwriters:= (λ l. ∅)).freaders:= (λ l. ∅)).rfo:= (λ t. [])).bq(lsf(t).t)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ noPending(t, bq, pq) → noPendingW(t, bq, pq); (ss) ⊦ (gs.bq:= bq).bq = bq; (s) ⊦ (gs.fwriters:= fwriters).bq = gs.bq; (s) ⊦ (gs.freaders:= fwriters0).bq = gs.bq; (s) ⊦ (gs.rfo:= bq0).bq = gs.bq; (s) ⊦ (gs.rfo:= bq0).rfo = bq0; (s) ⊦ crashed ≠ ready; (s) ⊦ crashed ≠ idle; (s) ⊦ ¬ a ∈ []; (s) ⊦ (gs.pq:= pq).pq = pq; (s) ⊦ (gs.fctr:= fctr).pq = gs.pq; (s) ⊦ (gs.bq:= bq).pq = gs.pq; (s) ⊦ (gs.fwriters:= fwriters).pq = gs.pq; (s) ⊦ (gs.freaders:= fwriters0).pq = gs.pq; (s) ⊦ (gs.rfo:= bq0).pq = gs.pq; (s) ⊦ (gs.freaders:= fwriters0).freaders = fwriters0; (s) ⊦ (gs.rfo:= bq0).freaders = gs.freaders; (s) ⊦ (gs.fwriters:= fwriters).fwriters = fwriters; (s) ⊦ (gs.freaders:= fwriters0).fwriters = gs.fwriters; (s) ⊦ (gs.rfo:= bq0).fwriters = gs.fwriters; (s) ⊦ R4 ≠ crashed; (s) ⊦ R3 ≠ crashed; (s) ⊦ R2 ≠ crashed; (s) ⊦ R1 ≠ crashed; (s) ⊦ W2 ≠ crashed; (s) ⊦ W7 ≠ crashed; (s) ⊦ W1 ≠ crashed; (s) ⊦ W6 ≠ crashed; (s) ⊦ W5 ≠ crashed; (s) ⊦ W4 ≠ crashed; (s) ⊦ W3 ≠ crashed; (s)

Proof step 35 from proof step-CRASH (previous step) (back to tree)

pcf(t) ≠ idle, FLITLInv(gs, lsf(t), pcf(t)), 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), ¬ notMember(lsf(t).t, λ l. ∅) ∨ ¬ noPending(lsf(t).t, λ t. [], λ l. []) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: FLIT-preds, name:noPending, seq: ⊦ noPending(t, bq, p) ↔ (∀ l. p(l) | t = [] ∧ bq(t) = []), subst:[t, bq, p] → [lsf(t).t, λ t. [], λ l. []], <all paths>

Additional information

Used terms: lsf(t).t, λ t. [], λ l. []

Proof step 36 from proof step-CRASH (previous step) (back to tree)

⊦ noPending(t, bq, p) ↔ (∀ l. p(l) | t = [] ∧ bq(t) = [])

A spec-lemma

Proof step 37 from proof step-CRASH (previous step) (back to tree)

pcf(t) ≠ idle, FLITLInv(gs, lsf(t), pcf(t)), 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), ¬ notMember(lsf(t).t, λ l. ∅) ∨ ¬(∀ l. [] | lsf(t).t = [] ∧ [] = []) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ [] | t = []; (s)

Proof step 38 from proof step-CRASH (previous step) (back to tree)

pcf(t) ≠ idle, FLITLInv(gs, lsf(t), pcf(t)), 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), ¬ notMember(lsf(t).t, λ l. ∅) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: FLITPSC-preds, name:noWriter, seq: ⊦ notMember(t, tsf) ↔ (∀ l. ¬ t ∈ tsf(l)), subst:[t, tsf] → [lsf(t).t, λ l. ∅], <all paths>

Additional information

Used simplifier rules: ⊦ ¬ a ∈ ∅; (s) Used terms: lsf(t).t, λ l. ∅

Proof step 39 from proof step-CRASH (previous step) (back to tree)

⊦ notMember(t, tsf) ↔ (∀ l. ¬ t ∈ tsf(l))

A spec-lemma

Proof Tree for step-CRASH (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