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Proof step 1 from proof stable-W5 (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)) ∧ notMember(ls.t, gs.freaders) ∧ oneMember(ls.t, ls.loc, gs.fwriters) ∧ FOFollowsW(ls.t, gs.bq(ls.t), gs.pq(ls.loc), ls.loc) ∧ ¬ SF ∈ gs.bq(ls.t) ∧ gs.rfo(ls.t) = [] ∧ (gs.bq(ls.t) = [] ∨ gs.bq(ls.t) = FO(ls.loc) ' ∨ gs.bq(ls.t) = W(ls.loc, ls.val) ' + FO(ls.loc) ') ∧ (gs.bq(ls.t) ≠ [] ∧ gs.bq(ls.t).head.W? → gs.bq(ls.t).head = W(ls.loc, ls.val))) ∧ ctrok(gs0.fctr, gs0.fwriters) ∧ writersOk(gs0.bq, gs0.pq, gs0.fwriters) ∧ pendingOk(gs0.bq, gs0.pq) ∧ SFTrailing(gs0.bq) ∧ readFOsOk(gs0.rfo, gs0.fwriters)) ∧ relyws(ls.t, gs.freaders, gs0.freaders) ∧ relyws(ls.t, gs.fwriters, gs0.fwriters) ∧ rely(ls.t, gs.pq, gs.bq, gs0.pq, gs0.bq) ∧ gs.bq(ls.t) ⊒w gs0.bq(ls.t) ∧ (gs.rfo(ls.t) = [] → gs0.rfo(ls.t) = []) ∧ (gs.rfo(ls.t) = gs.bq(ls.t) → gs0.rfo(ls.t) = gs0.bq(ls.t)) ∧ (gs.rfo(ls.t) + SF ' = gs.bq(ls.t) → gs0.rfo(ls.t) + SF ' = gs0.bq(ls.t) ∨ gs0.rfo(ls.t) = gs0.bq(ls.t)) → notMember(ls.t, gs0.freaders) ∧ oneMember(ls.t, ls.loc, gs0.fwriters) ∧ FOFollowsW(ls.t, gs0.bq(ls.t), gs0.pq(ls.loc), ls.loc) ∧ ¬ SF ∈ gs0.bq(ls.t) ∧ gs0.rfo(ls.t) = [] ∧ (gs0.bq(ls.t) = [] ∨ gs0.bq(ls.t) = FO(ls.loc) ' ∨ gs0.bq(ls.t) = W(ls.loc, ls.val) ' + FO(ls.loc) ') ∧ (gs0.bq(ls.t) ≠ [] ∧ gs0.bq(ls.t).head.W? → gs0.bq(ls.t).head = W(ls.loc, ls.val))

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

Proof step 2 from proof stable-W5 (previous step) (back to tree)

gs.rfo(ls.t) = [], FOFollowsW(ls.t, gs.bq(ls.t), gs.pq(ls.loc), ls.loc), SFTrailing(gs.bq), SFTrailing(gs0.bq), ctrok(gs.fctr, gs.fwriters), ctrok(gs0.fctr, gs0.fwriters), notMember(ls.t, gs.freaders), oneMember(ls.t, ls.loc, gs.fwriters), pendingOk(gs.bq, gs.pq), pendingOk(gs0.bq, gs0.pq), readFOsOk(gs.rfo, gs.fwriters), readFOsOk(gs0.rfo, gs0.fwriters), rely(ls.t, gs.pq, gs.bq, gs0.pq, gs0.bq), relyws(ls.t, gs.freaders, gs0.freaders), relyws(ls.t, gs.fwriters, gs0.fwriters), writersOk(gs.bq, gs.pq, gs.fwriters), writersOk(gs0.bq, gs0.pq, gs0.fwriters), gs.bq(ls.t) ⊒w gs0.bq(ls.t), ¬ SF ∈ gs.bq(ls.t), gs.bq(ls.t) ≠ [] ∧ gs.bq(ls.t).head.W? → gs.bq(ls.t).head = W(ls.loc, ls.val), gs.rfo(ls.t) = [] → gs0.rfo(ls.t) = [], gs.rfo(ls.t) = gs.bq(ls.t) → gs0.rfo(ls.t) = gs0.bq(ls.t), gs.rfo(ls.t) + SF ' = gs.bq(ls.t) → gs0.rfo(ls.t) + SF ' = gs0.bq(ls.t) ∨ gs0.rfo(ls.t) = gs0.bq(ls.t), gs.bq(ls.t) = [] ∨ gs.bq(ls.t) = FO(ls.loc) ' ∨ gs.bq(ls.t) = W(ls.loc, ls.val) ' + FO(ls.loc) ' ⊦ notMember(ls.t, gs0.freaders) ∧ oneMember(ls.t, ls.loc, gs0.fwriters) ∧ FOFollowsW(ls.t, gs0.bq(ls.t), gs0.pq(ls.loc), ls.loc) ∧ ¬ SF ∈ gs0.bq(ls.t) ∧ gs0.rfo(ls.t) = [] ∧ (gs0.bq(ls.t) = [] ∨ gs0.bq(ls.t) = FO(ls.loc) ' ∨ gs0.bq(ls.t) = W(ls.loc, ls.val) ' + FO(ls.loc) ') ∧ (gs0.bq(ls.t) ≠ [] ∧ gs0.bq(ls.t).head.W? → gs0.bq(ls.t).head = W(ls.loc, ls.val))

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 ⊦ [] + x = x; (s) ⊦ bel ⊒w bel0 ∧ ¬ be ∈ bel → ¬ be ∈ bel0; (as) ⊦ rely(t, pq, bq, pq0, bq0) ∧ bq(t) ⊒w bq0(t) ∧ FOFollowsW(t, bq(t), pq(l), l) → FOFollowsW(t, bq0(t), pq0(l), l); (ss) ⊦ relyws(t, tsf0, tsf1) ∧ oneMember(t, l, tsf0) → oneMember(t, l, tsf1); (s) ⊦ relyws(t, tsf0, tsf1) ∧ notMember(t, tsf0) → notMember(t, tsf1); (s)

Proof step 3 from proof stable-W5 (previous step) (back to tree)

gs0.rfo(ls.t) = [], gs.rfo(ls.t) = [], FOFollowsW(ls.t, gs.bq(ls.t), gs.pq(ls.loc), ls.loc), SFTrailing(gs.bq), SFTrailing(gs0.bq), ctrok(gs.fctr, gs.fwriters), ctrok(gs0.fctr, gs0.fwriters), notMember(ls.t, gs.freaders), oneMember(ls.t, ls.loc, gs.fwriters), pendingOk(gs.bq, gs.pq), pendingOk(gs0.bq, gs0.pq), readFOsOk(gs.rfo, gs.fwriters), readFOsOk(gs0.rfo, gs0.fwriters), rely(ls.t, gs.pq, gs.bq, gs0.pq, gs0.bq), relyws(ls.t, gs.freaders, gs0.freaders), relyws(ls.t, gs.fwriters, gs0.fwriters), writersOk(gs.bq, gs.pq, gs.fwriters), writersOk(gs0.bq, gs0.pq, gs0.fwriters), gs.bq(ls.t) ⊒w gs0.bq(ls.t), ¬ SF ∈ gs.bq(ls.t), [] = gs.bq(ls.t) → [] = gs0.bq(ls.t), gs.bq(ls.t) ≠ [] ∧ gs.bq(ls.t).head.W? → gs.bq(ls.t).head = W(ls.loc, ls.val), SF ' = gs.bq(ls.t) → SF ' = gs0.bq(ls.t) ∨ [] = gs0.bq(ls.t), gs.bq(ls.t) = [] ∨ gs.bq(ls.t) = FO(ls.loc) ' ∨ gs.bq(ls.t) = W(ls.loc, ls.val) ' + FO(ls.loc) ' ⊦ (gs0.bq(ls.t) = [] ∨ gs0.bq(ls.t) = FO(ls.loc) ' ∨ gs0.bq(ls.t) = W(ls.loc, ls.val) ' + FO(ls.loc) ') ∧ (gs0.bq(ls.t) ≠ [] ∧ gs0.bq(ls.t).head.W? → gs0.bq(ls.t).head = W(ls.loc, ls.val))

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: right 1

Additional information

Proof step 4 from proof stable-W5 (previous step) (back to tree)

gs.rfo(ls.t) = [], gs0.rfo(ls.t) = [], gs0.bq(ls.t) ≠ [], gs0.bq(ls.t) ≠ FO(ls.loc) ', gs0.bq(ls.t) ≠ W(ls.loc, ls.val) ' + FO(ls.loc) ', FOFollowsW(ls.t, gs.bq(ls.t), gs.pq(ls.loc), ls.loc), SFTrailing(gs0.bq), SFTrailing(gs.bq), ctrok(gs0.fctr, gs0.fwriters), ctrok(gs.fctr, gs.fwriters), notMember(ls.t, gs.freaders), oneMember(ls.t, ls.loc, gs.fwriters), pendingOk(gs0.bq, gs0.pq), pendingOk(gs.bq, gs.pq), readFOsOk(gs0.rfo, gs0.fwriters), readFOsOk(gs.rfo, gs.fwriters), rely(ls.t, gs.pq, gs.bq, gs0.pq, gs0.bq), relyws(ls.t, gs.fwriters, gs0.fwriters), relyws(ls.t, gs.freaders, gs0.freaders), writersOk(gs0.bq, gs0.pq, gs0.fwriters), writersOk(gs.bq, gs.pq, gs.fwriters), gs.bq(ls.t) ⊒w gs0.bq(ls.t), ¬ SF ∈ gs.bq(ls.t), SF ' = gs.bq(ls.t) → SF ' = gs0.bq(ls.t) ∨ [] = gs0.bq(ls.t), gs.bq(ls.t) ≠ [] ∧ gs.bq(ls.t).head.W? → gs.bq(ls.t).head = W(ls.loc, ls.val), [] = gs.bq(ls.t) → [] = gs0.bq(ls.t), gs.bq(ls.t) = [] ∨ gs.bq(ls.t) = FO(ls.loc) ' ∨ gs.bq(ls.t) = W(ls.loc, ls.val) ' + FO(ls.loc) ' ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 5 from proof stable-W5 (previous step) (back to tree)

gs0.rfo(ls.t) = [], gs.rfo(ls.t) = [], gs0.bq(ls.t) ≠ [], gs0.bq(ls.t) ≠ FO(ls.loc) ', gs0.bq(ls.t) ≠ W(ls.loc, ls.val) ' + FO(ls.loc) ', [] ≠ gs.bq(ls.t), FOFollowsW(ls.t, gs.bq(ls.t), gs.pq(ls.loc), ls.loc), SFTrailing(gs.bq), SFTrailing(gs0.bq), ctrok(gs.fctr, gs.fwriters), ctrok(gs0.fctr, gs0.fwriters), notMember(ls.t, gs.freaders), oneMember(ls.t, ls.loc, gs.fwriters), pendingOk(gs.bq, gs.pq), pendingOk(gs0.bq, gs0.pq), readFOsOk(gs.rfo, gs.fwriters), readFOsOk(gs0.rfo, gs0.fwriters), rely(ls.t, gs.pq, gs.bq, gs0.pq, gs0.bq), relyws(ls.t, gs.freaders, gs0.freaders), relyws(ls.t, gs.fwriters, gs0.fwriters), writersOk(gs.bq, gs.pq, gs.fwriters), writersOk(gs0.bq, gs0.pq, gs0.fwriters), gs.bq(ls.t) ⊒w gs0.bq(ls.t), ¬ SF ∈ gs.bq(ls.t), gs.bq(ls.t).head.W? → gs.bq(ls.t).head = W(ls.loc, ls.val), SF ' = gs.bq(ls.t) → SF ' = gs0.bq(ls.t), gs.bq(ls.t) = FO(ls.loc) ' ∨ gs.bq(ls.t) = W(ls.loc, ls.val) ' + FO(ls.loc) ' ⊦

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: list-data, name:elim-add, seq: ⊦ x ≠ [] → (head = x.head ∧ tail = x.tail ↔ x = head + tail), subst:[x, head, tail] → [gs.bq(ls.t), be, bel]; ['a] → [Bevent]

Additional information

Used simplifier rules: ⊦ x ≠ [] → (head = x.head ∧ tail = x.tail ↔ x = head + tail); (elim) Used terms: gs.bq(ls.t), be, bel

Proof step 6 from proof stable-W5 (previous step) (back to tree)

gs.bq(ls.t) = be + bel, gs.rfo(ls.t) = [], gs0.rfo(ls.t) = [], gs0.bq(ls.t) ≠ [], gs0.bq(ls.t) ≠ FO(ls.loc) ', gs0.bq(ls.t) ≠ W(ls.loc, ls.val) ' + FO(ls.loc) ', [] ≠ gs.bq(ls.t), FOFollowsW(ls.t, gs.bq(ls.t), gs.pq(ls.loc), ls.loc), SFTrailing(gs0.bq), SFTrailing(gs.bq), ctrok(gs0.fctr, gs0.fwriters), ctrok(gs.fctr, gs.fwriters), notMember(ls.t, gs.freaders), oneMember(ls.t, ls.loc, gs.fwriters), pendingOk(gs0.bq, gs0.pq), pendingOk(gs.bq, gs.pq), readFOsOk(gs0.rfo, gs0.fwriters), readFOsOk(gs.rfo, gs.fwriters), rely(ls.t, gs.pq, gs.bq, gs0.pq, gs0.bq), relyws(ls.t, gs.fwriters, gs0.fwriters), relyws(ls.t, gs.freaders, gs0.freaders), writersOk(gs0.bq, gs0.pq, gs0.fwriters), writersOk(gs.bq, gs.pq, gs.fwriters), gs.bq(ls.t) ⊒w gs0.bq(ls.t), ¬ SF ∈ gs.bq(ls.t), SF ' = gs.bq(ls.t) → SF ' = gs0.bq(ls.t), be.W? → be = W(ls.loc, ls.val), gs.bq(ls.t) = FO(ls.loc) ' ∨ gs.bq(ls.t) = W(ls.loc, ls.val) ' + FO(ls.loc) ' ⊦

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 ⊦ a ∈ a0 ' + x ↔ a = a0 ∨ a ∈ x; (s) ⊦ a ' = a0 ' + x ↔ a = a0 ∧ x = []; (ws) ⊦ a ∈ a0 ' + x ↔ ¬(a ≠ a0 ∧ ¬ a ∈ x); (ss) ⊦ a ' ≠ []; (s) ⊦ x + y = [] ↔ x = [] ∧ y = []; (s) in-14: ⊦ a ' + x = y → a ∈ y; (forward) from specification list ⊦ a + x = a ' + x; (s)

Proof step 7 from proof stable-W5 (previous step) (back to tree)

gs.bq(ls.t) = be ' + bel, gs0.rfo(ls.t) = [], gs.rfo(ls.t) = [], SF ' ≠ be ' + bel, gs0.bq(ls.t) ≠ [], gs0.bq(ls.t) ≠ FO(ls.loc) ', gs0.bq(ls.t) ≠ W(ls.loc, ls.val) ' + FO(ls.loc) ', SF ≠ be, FOFollowsW(ls.t, be ' + bel, gs.pq(ls.loc), ls.loc), SFTrailing(gs.bq), SFTrailing(gs0.bq), ctrok(gs.fctr, gs.fwriters), ctrok(gs0.fctr, gs0.fwriters), notMember(ls.t, gs.freaders), oneMember(ls.t, ls.loc, gs.fwriters), pendingOk(gs.bq, gs.pq), pendingOk(gs0.bq, gs0.pq), readFOsOk(gs.rfo, gs.fwriters), readFOsOk(gs0.rfo, gs0.fwriters), rely(ls.t, gs.pq, gs.bq, gs0.pq, gs0.bq), relyws(ls.t, gs.freaders, gs0.freaders), relyws(ls.t, gs.fwriters, gs0.fwriters), writersOk(gs.bq, gs.pq, gs.fwriters), writersOk(gs0.bq, gs0.pq, gs0.fwriters), be ' + bel ⊒w gs0.bq(ls.t), ¬ SF ∈ bel, be.W? → be = W(ls.loc, ls.val), be ' + bel = FO(ls.loc) ' ∨ be ' + bel = W(ls.loc, ls.val) ' + FO(ls.loc) ' ⊦

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 28

Additional information

Used simplifier rules: ⊦ a ' = a0 ' + x ↔ a = a0 ∧ x = []; (ws) ⊦ FO(l) ≠ SF; (s) ⊦ x + [] = x; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ be ' ⊒w bel ↔ bel = be ' ∨ bel = []; (ws)

Proof step 8 from proof stable-W5 (previous step) (back to tree)

be ' + bel = W(ls.loc, ls.val) ' + FO(ls.loc) ', gs.bq(ls.t) = be ' + bel, gs.rfo(ls.t) = [], gs0.rfo(ls.t) = [], SF ' ≠ be ' + bel, gs0.bq(ls.t) ≠ [], gs0.bq(ls.t) ≠ FO(ls.loc) ', gs0.bq(ls.t) ≠ W(ls.loc, ls.val) ' + FO(ls.loc) ', SF ≠ be, FOFollowsW(ls.t, be ' + bel, gs.pq(ls.loc), ls.loc), SFTrailing(gs0.bq), SFTrailing(gs.bq), ctrok(gs0.fctr, gs0.fwriters), ctrok(gs.fctr, gs.fwriters), notMember(ls.t, gs.freaders), oneMember(ls.t, ls.loc, gs.fwriters), pendingOk(gs0.bq, gs0.pq), pendingOk(gs.bq, gs.pq), readFOsOk(gs0.rfo, gs0.fwriters), readFOsOk(gs.rfo, gs.fwriters), rely(ls.t, gs.pq, gs.bq, gs0.pq, gs0.bq), relyws(ls.t, gs.fwriters, gs0.fwriters), relyws(ls.t, gs.freaders, gs0.freaders), writersOk(gs0.bq, gs0.pq, gs0.fwriters), writersOk(gs.bq, gs.pq, gs.fwriters), be ' + bel ⊒w gs0.bq(ls.t), ¬ SF ∈ bel, be.W? → be = W(ls.loc, ls.val) ⊦

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 in-14: ⊦ a ' + x = y → a ∈ y; (forward) from specification list ⊦ FO(l) ≠ SF; (s) ⊦ a ∈ a0 ' ↔ a = a0; (s) ⊦ W(l, v) ≠ SF; (s) ⊦ a ' + x = a0 ' + y ↔ a = a0 ∧ x = y; (ws)

Proof step 9 from proof stable-W5 (previous step) (back to tree)

gs.bq(ls.t) = W(ls.loc, ls.val) ' + FO(ls.loc) ', gs.rfo(ls.t) = [], gs0.rfo(ls.t) = [], SF ' ≠ gs.bq(ls.t), gs0.bq(ls.t) ≠ gs.bq(ls.t), gs0.bq(ls.t) ≠ FO(ls.loc) ', gs0.bq(ls.t) ≠ [], FOFollowsW(ls.t, gs.bq(ls.t), gs.pq(ls.loc), ls.loc), SFTrailing(gs0.bq), SFTrailing(gs.bq), ctrok(gs0.fctr, gs0.fwriters), ctrok(gs.fctr, gs.fwriters), notMember(ls.t, gs.freaders), oneMember(ls.t, ls.loc, gs.fwriters), pendingOk(gs0.bq, gs0.pq), pendingOk(gs.bq, gs.pq), readFOsOk(gs0.rfo, gs0.fwriters), readFOsOk(gs.rfo, gs.fwriters), rely(ls.t, gs.pq, gs.bq, gs0.pq, gs0.bq), relyws(ls.t, gs.fwriters, gs0.fwriters), relyws(ls.t, gs.freaders, gs0.freaders), writersOk(gs0.bq, gs0.pq, gs0.fwriters), writersOk(gs.bq, gs.pq, gs.fwriters), FO(ls.loc) ∈ gs.bq(ls.t), W(ls.loc, ls.val) ∈ gs.bq(ls.t), gs.bq(ls.t) ⊒w gs0.bq(ls.t) ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: FLIT-preds, name:B-postfix-weak-same-loc-postfix, seq: ⊦ bel ⊒w bel0 ∧ bel.forall(λ be. be = SF ∨ be.l = l) → bel ⊒ bel0, subst:[bel, bel0, l] → [gs.bq(ls.t), gs0.bq(ls.t), ls.loc]

Additional information

Used terms: gs.bq(ls.t), gs0.bq(ls.t), ls.loc

Proof step 10 from proof stable-W5 (previous step) (back to tree)

⊦ bel ⊒w bel0 ∧ bel.forall(λ be. be = SF ∨ be.l = l) → bel ⊒ bel0

A spec-lemma

Proof step 11 from proof stable-W5 (previous step) (back to tree)

gs.bq(ls.t) = W(ls.loc, ls.val) ' + FO(ls.loc) ', gs0.rfo(ls.t) = [], gs.rfo(ls.t) = [], SF ' ≠ gs.bq(ls.t), gs0.bq(ls.t) ≠ gs.bq(ls.t), gs0.bq(ls.t) ≠ FO(ls.loc) ', gs0.bq(ls.t) ≠ [], FOFollowsW(ls.t, gs.bq(ls.t), gs.pq(ls.loc), ls.loc), SFTrailing(gs.bq), SFTrailing(gs0.bq), ctrok(gs.fctr, gs.fwriters), ctrok(gs0.fctr, gs0.fwriters), notMember(ls.t, gs.freaders), oneMember(ls.t, ls.loc, gs.fwriters), pendingOk(gs.bq, gs.pq), pendingOk(gs0.bq, gs0.pq), readFOsOk(gs.rfo, gs.fwriters), readFOsOk(gs0.rfo, gs0.fwriters), rely(ls.t, gs.pq, gs.bq, gs0.pq, gs0.bq), relyws(ls.t, gs.freaders, gs0.freaders), relyws(ls.t, gs.fwriters, gs0.fwriters), writersOk(gs.bq, gs.pq, gs.fwriters), writersOk(gs0.bq, gs0.pq, gs0.fwriters), W(ls.loc, ls.val) ∈ gs.bq(ls.t), FO(ls.loc) ∈ gs.bq(ls.t), gs.bq(ls.t) ⊒w gs0.bq(ls.t), gs.bq(ls.t) ⊒w gs0.bq(ls.t) ∧ gs.bq(ls.t).forall(λ be. be = SF ∨ be.l = ls.loc) → gs.bq(ls.t) ⊒ gs0.bq(ls.t) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 12 from proof stable-W5 (previous step) (back to tree)

gs.bq(ls.t) = W(ls.loc, ls.val) ' + FO(ls.loc) ', gs.rfo(ls.t) = [], gs0.rfo(ls.t) = [], SF ' ≠ gs.bq(ls.t), gs0.bq(ls.t) ≠ gs.bq(ls.t), gs0.bq(ls.t) ≠ FO(ls.loc) ', gs0.bq(ls.t) ≠ [], FOFollowsW(ls.t, gs.bq(ls.t), gs.pq(ls.loc), ls.loc), SFTrailing(gs0.bq), SFTrailing(gs.bq), ctrok(gs0.fctr, gs0.fwriters), ctrok(gs.fctr, gs.fwriters), notMember(ls.t, gs.freaders), oneMember(ls.t, ls.loc, gs.fwriters), pendingOk(gs0.bq, gs0.pq), pendingOk(gs.bq, gs.pq), readFOsOk(gs0.rfo, gs0.fwriters), readFOsOk(gs.rfo, gs.fwriters), rely(ls.t, gs.pq, gs.bq, gs0.pq, gs0.bq), relyws(ls.t, gs.fwriters, gs0.fwriters), relyws(ls.t, gs.freaders, gs0.freaders), writersOk(gs0.bq, gs0.pq, gs0.fwriters), writersOk(gs.bq, gs.pq, gs.fwriters), FO(ls.loc) ∈ gs.bq(ls.t), W(ls.loc, ls.val) ∈ gs.bq(ls.t), gs.bq(ls.t) ⊒w gs0.bq(ls.t), gs.bq(ls.t).forall(λ be. be = SF ∨ be.l = ls.loc) → gs.bq(ls.t) ⊒ gs0.bq(ls.t) ⊦

User applied rule subst equation

And obtained the following new goals

Rule argument

pos: left 1, flags:

Additional information

Proof step 13 from proof stable-W5 (previous step) (back to tree)

gs.bq(ls.t) = W(ls.loc, ls.val) ' + FO(ls.loc) ', gs0.rfo(ls.t) = [], gs.rfo(ls.t) = [], SF ' ≠ W(ls.loc, ls.val) ' + FO(ls.loc) ', gs0.bq(ls.t) ≠ W(ls.loc, ls.val) ' + FO(ls.loc) ', gs0.bq(ls.t) ≠ FO(ls.loc) ', gs0.bq(ls.t) ≠ [], FOFollowsW(ls.t, W(ls.loc, ls.val) ' + FO(ls.loc) ', gs.pq(ls.loc), ls.loc), SFTrailing(gs.bq), SFTrailing(gs0.bq), ctrok(gs.fctr, gs.fwriters), ctrok(gs0.fctr, gs0.fwriters), notMember(ls.t, gs.freaders), oneMember(ls.t, ls.loc, gs.fwriters), pendingOk(gs.bq, gs.pq), pendingOk(gs0.bq, gs0.pq), readFOsOk(gs.rfo, gs.fwriters), readFOsOk(gs0.rfo, gs0.fwriters), rely(ls.t, gs.pq, gs.bq, gs0.pq, gs0.bq), relyws(ls.t, gs.freaders, gs0.freaders), relyws(ls.t, gs.fwriters, gs0.fwriters), writersOk(gs.bq, gs.pq, gs.fwriters), writersOk(gs0.bq, gs0.pq, gs0.fwriters), W(ls.loc, ls.val) ∈ W(ls.loc, ls.val) ' + FO(ls.loc) ', FO(ls.loc) ∈ W(ls.loc, ls.val) ' + FO(ls.loc) ', W(ls.loc, ls.val) ' + FO(ls.loc) ' ⊒w gs0.bq(ls.t), (W(ls.loc, ls.val) ' + FO(ls.loc) ').forall(λ be. be = SF ∨ be.l = ls.loc) → W(ls.loc, ls.val) ' + FO(ls.loc) ' ⊒ gs0.bq(ls.t) ⊦

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 27

Additional information

Used simplifier rules: ⊦ a ∈ a0 ' + x ↔ a = a0 ∨ a ∈ x; (s) ⊦ W(l, v) ≠ FO(l0); (s) ⊦ a ∈ a0 ' ↔ a = a0; (s) ⊦ (a ' + x).forall(p) ↔ p(a) ∧ x.forall(p); (s) ⊦ W(l, v) ≠ SF; (s) ⊦ W(l, v).l = l; (s) ⊦ a '.forall(p) ↔ p(a); (s) ⊦ FO(l) ≠ SF; (s) ⊦ FO(l).l = l; (s) a: ⊦ (x + y) + z = x + y + z; from specification list

Proof step 14 from proof stable-W5 (previous step) (back to tree)

gs.bq(ls.t) = W(ls.loc, ls.val) ' + FO(ls.loc) ', gs.rfo(ls.t) = [], gs0.rfo(ls.t) = [], SF ' ≠ W(ls.loc, ls.val) ' + FO(ls.loc) ', gs0.bq(ls.t) ≠ W(ls.loc, ls.val) ' + FO(ls.loc) ', gs0.bq(ls.t) ≠ FO(ls.loc) ', gs0.bq(ls.t) ≠ [], FOFollowsW(ls.t, W(ls.loc, ls.val) ' + FO(ls.loc) ', gs.pq(ls.loc), ls.loc), SFTrailing(gs0.bq), SFTrailing(gs.bq), ctrok(gs0.fctr, gs0.fwriters), ctrok(gs.fctr, gs.fwriters), notMember(ls.t, gs.freaders), oneMember(ls.t, ls.loc, gs.fwriters), pendingOk(gs0.bq, gs0.pq), pendingOk(gs.bq, gs.pq), readFOsOk(gs0.rfo, gs0.fwriters), readFOsOk(gs.rfo, gs.fwriters), rely(ls.t, gs.pq, gs.bq, gs0.pq, gs0.bq), relyws(ls.t, gs.fwriters, gs0.fwriters), relyws(ls.t, gs.freaders, gs0.freaders), writersOk(gs0.bq, gs0.pq, gs0.fwriters), writersOk(gs.bq, gs.pq, gs.fwriters), FO(ls.loc) ∈ W(ls.loc, ls.val) ' + FO(ls.loc) ', W(ls.loc, ls.val) ∈ W(ls.loc, ls.val) ' + FO(ls.loc) ', W(ls.loc, ls.val) ' + FO(ls.loc) ' ⊒ gs0.bq(ls.t), W(ls.loc, ls.val) ' + FO(ls.loc) ' ⊒w gs0.bq(ls.t) ⊦

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 ⊦ a ∈ a0 ' ↔ a = a0; (s) ⊦ W(l, v) ≠ FO(l0); (s) ⊦ a ∈ a0 ' + x ↔ a = a0 ∨ a ∈ x; (s)

Proof step 15 from proof stable-W5 (previous step) (back to tree)

gs.bq(ls.t) = W(ls.loc, ls.val) ' + FO(ls.loc) ', gs0.rfo(ls.t) = [], gs.rfo(ls.t) = [], SF ' ≠ gs.bq(ls.t), gs0.bq(ls.t) ≠ gs.bq(ls.t), gs0.bq(ls.t) ≠ FO(ls.loc) ', gs0.bq(ls.t) ≠ [], FOFollowsW(ls.t, gs.bq(ls.t), gs.pq(ls.loc), ls.loc), SFTrailing(gs.bq), SFTrailing(gs0.bq), ctrok(gs.fctr, gs.fwriters), ctrok(gs0.fctr, gs0.fwriters), notMember(ls.t, gs.freaders), oneMember(ls.t, ls.loc, gs.fwriters), pendingOk(gs.bq, gs.pq), pendingOk(gs0.bq, gs0.pq), readFOsOk(gs.rfo, gs.fwriters), readFOsOk(gs0.rfo, gs0.fwriters), rely(ls.t, gs.pq, gs.bq, gs0.pq, gs0.bq), relyws(ls.t, gs.freaders, gs0.freaders), relyws(ls.t, gs.fwriters, gs0.fwriters), writersOk(gs.bq, gs.pq, gs.fwriters), writersOk(gs0.bq, gs0.pq, gs0.fwriters), gs.bq(ls.t) ⊒ gs0.bq(ls.t), gs.bq(ls.t) ⊒w gs0.bq(ls.t) ⊦

User applied rule subst equation

And obtained the following new goals

Rule argument

pos: left 1, flags: discard

Additional information

Proof step 16 from proof stable-W5 (previous step) (back to tree)

gs.rfo(ls.t) = [], gs0.rfo(ls.t) = [], SF ' ≠ W(ls.loc, ls.val) ' + FO(ls.loc) ', gs0.bq(ls.t) ≠ W(ls.loc, ls.val) ' + FO(ls.loc) ', gs0.bq(ls.t) ≠ FO(ls.loc) ', gs0.bq(ls.t) ≠ [], FOFollowsW(ls.t, W(ls.loc, ls.val) ' + FO(ls.loc) ', gs.pq(ls.loc), ls.loc), SFTrailing(gs0.bq), SFTrailing(gs.bq), ctrok(gs0.fctr, gs0.fwriters), ctrok(gs.fctr, gs.fwriters), notMember(ls.t, gs.freaders), oneMember(ls.t, ls.loc, gs.fwriters), pendingOk(gs0.bq, gs0.pq), pendingOk(gs.bq, gs.pq), readFOsOk(gs0.rfo, gs0.fwriters), readFOsOk(gs.rfo, gs.fwriters), rely(ls.t, gs.pq, gs.bq, gs0.pq, gs0.bq), relyws(ls.t, gs.fwriters, gs0.fwriters), relyws(ls.t, gs.freaders, gs0.freaders), writersOk(gs0.bq, gs0.pq, gs0.fwriters), writersOk(gs.bq, gs.pq, gs.fwriters), W(ls.loc, ls.val) ' + FO(ls.loc) ' ⊒ gs0.bq(ls.t), W(ls.loc, ls.val) ' + FO(ls.loc) ' ⊒w gs0.bq(ls.t) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: list-last, name:postfix, seq: ⊦ x ⊒ y ↔ (∃ z. z + y = x), subst:[x, y] → [W(ls.loc, ls.val) ' + FO(ls.loc) ', gs0.bq(ls.t)]; ['a] → [Bevent], <all paths>

Additional information

Used terms: W(ls.loc, ls.val) ' + FO(ls.loc) ', gs0.bq(ls.t)

Proof step 17 from proof stable-W5 (previous step) (back to tree)

⊦ x ⊒ y ↔ (∃ z. z + y = x)

A spec-lemma

Proof step 18 from proof stable-W5 (previous step) (back to tree)

gs0.rfo(ls.t) = [], gs.rfo(ls.t) = [], SF ' ≠ W(ls.loc, ls.val) ' + FO(ls.loc) ', gs0.bq(ls.t) ≠ W(ls.loc, ls.val) ' + FO(ls.loc) ', gs0.bq(ls.t) ≠ FO(ls.loc) ', gs0.bq(ls.t) ≠ [], FOFollowsW(ls.t, W(ls.loc, ls.val) ' + FO(ls.loc) ', gs.pq(ls.loc), ls.loc), SFTrailing(gs.bq), SFTrailing(gs0.bq), ctrok(gs.fctr, gs.fwriters), ctrok(gs0.fctr, gs0.fwriters), notMember(ls.t, gs.freaders), oneMember(ls.t, ls.loc, gs.fwriters), pendingOk(gs.bq, gs.pq), pendingOk(gs0.bq, gs0.pq), readFOsOk(gs.rfo, gs.fwriters), readFOsOk(gs0.rfo, gs0.fwriters), rely(ls.t, gs.pq, gs.bq, gs0.pq, gs0.bq), relyws(ls.t, gs.freaders, gs0.freaders), relyws(ls.t, gs.fwriters, gs0.fwriters), writersOk(gs.bq, gs.pq, gs.fwriters), writersOk(gs0.bq, gs0.pq, gs0.fwriters), W(ls.loc, ls.val) ' + FO(ls.loc) ' ⊒w gs0.bq(ls.t), ∃ z. z + gs0.bq(ls.t) = W(ls.loc, ls.val) ' + FO(ls.loc) ' ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ a ∈ a0 ' ↔ a = a0; (s) a: ⊦ (x + y) + z = x + y + z; from specification list inapp-12: ⊦ x + y = z + a ' ∧ a0 ∈ x ∧ y ≠ [] → a0 ∈ z; (forward) from specification list ⊦ bel0 + bel ⊒w bel; (s) ni: ⊦ x + a ' = y + z ∧ ¬ a ∈ z → z = []; (forward) from specification list ⊦ W(l, v) ≠ FO(l0); (s) ⊦ a ∈ x + y ↔ a ∈ x ∨ a ∈ y; (s) ⊦ a ' + x = x0 + x1 → (a ∈ x0 ↔ x0 ≠ []); (ss) ⊦ y ≠ [] → (a ' = x + y ↔ x = [] ∧ a ' = y); (s) ⊦ x = y + x ↔ y = []; (s) in-15: ⊦ x + a ' = y → a ∈ y; (forward) from specification list in-14: ⊦ a ' + x = y → a ∈ y; (forward) from specification list

Proof step 19 from proof stable-W5 (previous step) (back to tree)

bel + gs0.bq(ls.t) = W(ls.loc, ls.val) ' + FO(ls.loc) ', gs.rfo(ls.t) = [], gs0.rfo(ls.t) = [], gs0.bq(ls.t) ≠ FO(ls.loc) ', gs0.bq(ls.t) ≠ [], bel ≠ [], FOFollowsW(ls.t, bel + gs0.bq(ls.t), gs.pq(ls.loc), ls.loc), SFTrailing(gs0.bq), SFTrailing(gs.bq), ctrok(gs0.fctr, gs0.fwriters), ctrok(gs.fctr, gs.fwriters), notMember(ls.t, gs.freaders), oneMember(ls.t, ls.loc, gs.fwriters), pendingOk(gs0.bq, gs0.pq), pendingOk(gs.bq, gs.pq), readFOsOk(gs0.rfo, gs0.fwriters), readFOsOk(gs.rfo, gs.fwriters), rely(ls.t, gs.pq, gs.bq, gs0.pq, gs0.bq), relyws(ls.t, gs.fwriters, gs0.fwriters), relyws(ls.t, gs.freaders, gs0.freaders), writersOk(gs0.bq, gs0.pq, gs0.fwriters), writersOk(gs.bq, gs.pq, gs.fwriters), FO(ls.loc) ∈ gs0.bq(ls.t), W(ls.loc, ls.val) ∈ bel, ¬ FO(ls.loc) ∈ bel ⊦

User applied rule constructor cut

And closed the goal!

Rule argument

var: bel

Additional information

Used simplifier rules: List('a) freely generated by [], +;; ⊦ a + x = a ' + x; (s) ⊦ a ' + x = a0 ' + y ↔ a = a0 ∧ x = y; (ws) ⊦ y ≠ [] → (a ' = x + y ↔ x = [] ∧ a ' = y); (s) a: ⊦ (x + y) + z = x + y + z; from specification list

Proof step 20 from proof stable-W5 (previous step) (back to tree)

gs.rfo(ls.t) = [], gs0.rfo(ls.t) = [], gs0.bq(ls.t) ≠ [], gs0.bq(ls.t).head ≠ W(ls.loc, ls.val), gs0.bq(ls.t).head.W?, FOFollowsW(ls.t, gs.bq(ls.t), gs.pq(ls.loc), ls.loc), SFTrailing(gs0.bq), SFTrailing(gs.bq), ctrok(gs0.fctr, gs0.fwriters), ctrok(gs.fctr, gs.fwriters), notMember(ls.t, gs.freaders), oneMember(ls.t, ls.loc, gs.fwriters), pendingOk(gs0.bq, gs0.pq), pendingOk(gs.bq, gs.pq), readFOsOk(gs0.rfo, gs0.fwriters), readFOsOk(gs.rfo, gs.fwriters), rely(ls.t, gs.pq, gs.bq, gs0.pq, gs0.bq), relyws(ls.t, gs.fwriters, gs0.fwriters), relyws(ls.t, gs.freaders, gs0.freaders), writersOk(gs0.bq, gs0.pq, gs0.fwriters), writersOk(gs.bq, gs.pq, gs.fwriters), gs.bq(ls.t) ⊒w gs0.bq(ls.t), ¬ SF ∈ gs.bq(ls.t), SF ' = gs.bq(ls.t) → SF ' = gs0.bq(ls.t) ∨ [] = gs0.bq(ls.t), gs.bq(ls.t) ≠ [] ∧ gs.bq(ls.t).head.W? → gs.bq(ls.t).head = W(ls.loc, ls.val), [] = gs.bq(ls.t) → [] = gs0.bq(ls.t), gs.bq(ls.t) = [] ∨ gs.bq(ls.t) = FO(ls.loc) ' ∨ gs.bq(ls.t) = W(ls.loc, ls.val) ' + FO(ls.loc) ' ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 21 from proof stable-W5 (previous step) (back to tree)

gs0.rfo(ls.t) = [], gs.rfo(ls.t) = [], gs0.bq(ls.t) ≠ [], gs0.bq(ls.t).head ≠ W(ls.loc, ls.val), [] ≠ gs.bq(ls.t), gs0.bq(ls.t).head.W?, FOFollowsW(ls.t, gs.bq(ls.t), gs.pq(ls.loc), ls.loc), SFTrailing(gs.bq), SFTrailing(gs0.bq), ctrok(gs.fctr, gs.fwriters), ctrok(gs0.fctr, gs0.fwriters), notMember(ls.t, gs.freaders), oneMember(ls.t, ls.loc, gs.fwriters), pendingOk(gs.bq, gs.pq), pendingOk(gs0.bq, gs0.pq), readFOsOk(gs.rfo, gs.fwriters), readFOsOk(gs0.rfo, gs0.fwriters), rely(ls.t, gs.pq, gs.bq, gs0.pq, gs0.bq), relyws(ls.t, gs.freaders, gs0.freaders), relyws(ls.t, gs.fwriters, gs0.fwriters), writersOk(gs.bq, gs.pq, gs.fwriters), writersOk(gs0.bq, gs0.pq, gs0.fwriters), gs.bq(ls.t) ⊒w gs0.bq(ls.t), ¬ SF ∈ gs.bq(ls.t), gs.bq(ls.t).head.W? → gs.bq(ls.t).head = W(ls.loc, ls.val), SF ' = gs.bq(ls.t) → SF ' = gs0.bq(ls.t), gs.bq(ls.t) = FO(ls.loc) ' ∨ gs.bq(ls.t) = W(ls.loc, ls.val) ' + FO(ls.loc) ' ⊦

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: list-data, name:elim-add, seq: ⊦ x ≠ [] → (head = x.head ∧ tail = x.tail ↔ x = head + tail), subst:[x, head, tail] → [gs0.bq(ls.t), be, bel]; ['a] → [Bevent]

Additional information

Used simplifier rules: ⊦ x ≠ [] → (head = x.head ∧ tail = x.tail ↔ x = head + tail); (elim) Used terms: gs0.bq(ls.t), be, bel

Proof step 22 from proof stable-W5 (previous step) (back to tree)

gs0.bq(ls.t) = be + bel, gs.rfo(ls.t) = [], gs0.rfo(ls.t) = [], gs0.bq(ls.t) ≠ [], [] ≠ gs.bq(ls.t), be ≠ W(ls.loc, ls.val), be.W?, FOFollowsW(ls.t, gs.bq(ls.t), gs.pq(ls.loc), ls.loc), SFTrailing(gs0.bq), SFTrailing(gs.bq), ctrok(gs0.fctr, gs0.fwriters), ctrok(gs.fctr, gs.fwriters), notMember(ls.t, gs.freaders), oneMember(ls.t, ls.loc, gs.fwriters), pendingOk(gs0.bq, gs0.pq), pendingOk(gs.bq, gs.pq), readFOsOk(gs0.rfo, gs0.fwriters), readFOsOk(gs.rfo, gs.fwriters), rely(ls.t, gs.pq, gs.bq, gs0.pq, gs0.bq), relyws(ls.t, gs.fwriters, gs0.fwriters), relyws(ls.t, gs.freaders, gs0.freaders), writersOk(gs0.bq, gs0.pq, gs0.fwriters), writersOk(gs.bq, gs.pq, gs.fwriters), gs.bq(ls.t) ⊒w gs0.bq(ls.t), ¬ SF ∈ gs.bq(ls.t), SF ' = gs.bq(ls.t) → SF ' = gs0.bq(ls.t), gs.bq(ls.t).head.W? → gs.bq(ls.t).head = W(ls.loc, ls.val), gs.bq(ls.t) = FO(ls.loc) ' ∨ gs.bq(ls.t) = W(ls.loc, ls.val) ' + FO(ls.loc) ' ⊦

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 ⊦ a ∈ a0 ' + x ↔ a = a0 ∨ a ∈ x; (s) ⊦ a ' ≠ []; (s) ⊦ x + y = [] ↔ x = [] ∧ y = []; (s) in-14: ⊦ a ' + x = y → a ∈ y; (forward) from specification list ⊦ a + x = a ' + x; (s)

Proof step 23 from proof stable-W5 (previous step) (back to tree)

gs0.bq(ls.t) = be ' + bel, gs0.rfo(ls.t) = [], gs.rfo(ls.t) = [], [] ≠ gs.bq(ls.t), be ≠ W(ls.loc, ls.val), be.W?, FOFollowsW(ls.t, gs.bq(ls.t), gs.pq(ls.loc), ls.loc), SFTrailing(gs.bq), SFTrailing(gs0.bq), ctrok(gs.fctr, gs.fwriters), ctrok(gs0.fctr, gs0.fwriters), notMember(ls.t, gs.freaders), oneMember(ls.t, ls.loc, gs.fwriters), pendingOk(gs.bq, gs.pq), pendingOk(gs0.bq, gs0.pq), readFOsOk(gs.rfo, gs.fwriters), readFOsOk(gs0.rfo, gs0.fwriters), rely(ls.t, gs.pq, gs.bq, gs0.pq, gs0.bq), relyws(ls.t, gs.freaders, gs0.freaders), relyws(ls.t, gs.fwriters, gs0.fwriters), writersOk(gs.bq, gs.pq, gs.fwriters), writersOk(gs0.bq, gs0.pq, gs0.fwriters), gs.bq(ls.t) ⊒w be ' + bel, ¬ SF ∈ gs.bq(ls.t), gs.bq(ls.t).head.W? → gs.bq(ls.t).head = W(ls.loc, ls.val), SF ' = gs.bq(ls.t) → SF ' = be ' + bel, gs.bq(ls.t) = FO(ls.loc) ' ∨ gs.bq(ls.t) = W(ls.loc, ls.val) ' + FO(ls.loc) ' ⊦

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: list-data, name:elim-add, seq: ⊦ x ≠ [] → (head = x.head ∧ tail = x.tail ↔ x = head + tail), subst:[x, head, tail] → [gs.bq(ls.t), be0, bel0]; ['a] → [Bevent]

Additional information

Used simplifier rules: ⊦ x ≠ [] → (head = x.head ∧ tail = x.tail ↔ x = head + tail); (elim) Used terms: gs.bq(ls.t), be0, bel0

Proof step 24 from proof stable-W5 (previous step) (back to tree)

gs0.bq(ls.t) = be ' + bel, gs.bq(ls.t) = be0 + bel0, gs.rfo(ls.t) = [], gs0.rfo(ls.t) = [], [] ≠ gs.bq(ls.t), be ≠ W(ls.loc, ls.val), be.W?, FOFollowsW(ls.t, gs.bq(ls.t), gs.pq(ls.loc), ls.loc), SFTrailing(gs0.bq), SFTrailing(gs.bq), ctrok(gs0.fctr, gs0.fwriters), ctrok(gs.fctr, gs.fwriters), notMember(ls.t, gs.freaders), oneMember(ls.t, ls.loc, gs.fwriters), pendingOk(gs0.bq, gs0.pq), pendingOk(gs.bq, gs.pq), readFOsOk(gs0.rfo, gs0.fwriters), readFOsOk(gs.rfo, gs.fwriters), rely(ls.t, gs.pq, gs.bq, gs0.pq, gs0.bq), relyws(ls.t, gs.fwriters, gs0.fwriters), relyws(ls.t, gs.freaders, gs0.freaders), writersOk(gs0.bq, gs0.pq, gs0.fwriters), writersOk(gs.bq, gs.pq, gs.fwriters), gs.bq(ls.t) ⊒w be ' + bel, ¬ SF ∈ gs.bq(ls.t), SF ' = gs.bq(ls.t) → SF ' = be ' + bel, be0.W? → be0 = W(ls.loc, ls.val), gs.bq(ls.t) = FO(ls.loc) ' ∨ gs.bq(ls.t) = W(ls.loc, ls.val) ' + FO(ls.loc) ' ⊦

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 ⊦ a ∈ a0 ' + x ↔ a = a0 ∨ a ∈ x; (s) ⊦ a ' = a0 ' + x ↔ a = a0 ∧ x = []; (ws) ⊦ a ∈ a0 ' + x ↔ ¬(a ≠ a0 ∧ ¬ a ∈ x); (ss) ⊦ a ' ≠ []; (s) ⊦ x + y = [] ↔ x = [] ∧ y = []; (s) in-14: ⊦ a ' + x = y → a ∈ y; (forward) from specification list ⊦ a + x = a ' + x; (s)

Proof step 25 from proof stable-W5 (previous step) (back to tree)

gs0.bq(ls.t) = be ' + bel, gs.bq(ls.t) = be0 ' + bel0, gs0.rfo(ls.t) = [], gs.rfo(ls.t) = [], SF ' ≠ be0 ' + bel0, SF ≠ be0, be ≠ W(ls.loc, ls.val), be.W?, FOFollowsW(ls.t, be0 ' + bel0, gs.pq(ls.loc), ls.loc), SFTrailing(gs.bq), SFTrailing(gs0.bq), ctrok(gs.fctr, gs.fwriters), ctrok(gs0.fctr, gs0.fwriters), notMember(ls.t, gs.freaders), oneMember(ls.t, ls.loc, gs.fwriters), pendingOk(gs.bq, gs.pq), pendingOk(gs0.bq, gs0.pq), readFOsOk(gs.rfo, gs.fwriters), readFOsOk(gs0.rfo, gs0.fwriters), rely(ls.t, gs.pq, gs.bq, gs0.pq, gs0.bq), relyws(ls.t, gs.freaders, gs0.freaders), relyws(ls.t, gs.fwriters, gs0.fwriters), writersOk(gs.bq, gs.pq, gs.fwriters), writersOk(gs0.bq, gs0.pq, gs0.fwriters), be0 ' + bel0 ⊒w be ' + bel, ¬ SF ∈ bel0, be0.W? → be0 = W(ls.loc, ls.val), be0 ' + bel0 = FO(ls.loc) ' ∨ be0 ' + bel0 = W(ls.loc, ls.val) ' + FO(ls.loc) ' ⊦

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 28

Additional information

Used simplifier rules: ⊦ a ' = a0 ' + x ↔ a = a0 ∧ x = []; (ws) ⊦ FO(l) ≠ SF; (s) ⊦ be ' ⊒w be0 ' + bel ↔ be0 = be ∧ bel = []; (s) ⊦ ¬ a ∈ []; (s) ⊦ x + [] = x; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ W(l, v) ≠ FO(l0); (s) ⊦ ¬ FO(l).W?; (s)

Proof step 26 from proof stable-W5 (previous step) (back to tree)

be0 ' + bel0 = W(ls.loc, ls.val) ' + FO(ls.loc) ', gs.bq(ls.t) = be0 ' + bel0, gs0.bq(ls.t) = be ' + bel, gs.rfo(ls.t) = [], gs0.rfo(ls.t) = [], SF ' ≠ be0 ' + bel0, SF ≠ be0, be ≠ W(ls.loc, ls.val), be.W?, FOFollowsW(ls.t, be0 ' + bel0, gs.pq(ls.loc), ls.loc), SFTrailing(gs0.bq), SFTrailing(gs.bq), ctrok(gs0.fctr, gs0.fwriters), ctrok(gs.fctr, gs.fwriters), notMember(ls.t, gs.freaders), oneMember(ls.t, ls.loc, gs.fwriters), pendingOk(gs0.bq, gs0.pq), pendingOk(gs.bq, gs.pq), readFOsOk(gs0.rfo, gs0.fwriters), readFOsOk(gs.rfo, gs.fwriters), rely(ls.t, gs.pq, gs.bq, gs0.pq, gs0.bq), relyws(ls.t, gs.fwriters, gs0.fwriters), relyws(ls.t, gs.freaders, gs0.freaders), writersOk(gs0.bq, gs0.pq, gs0.fwriters), writersOk(gs.bq, gs.pq, gs.fwriters), be0 ' + bel0 ⊒w be ' + bel, ¬ SF ∈ bel0, be0.W? → be0 = W(ls.loc, ls.val) ⊦

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 in-14: ⊦ a ' + x = y → a ∈ y; (forward) from specification list ⊦ FO(l) ≠ SF; (s) ⊦ a ∈ a0 ' ↔ a = a0; (s) ⊦ W(l, v) ≠ SF; (s) ⊦ a ' + x = a0 ' + y ↔ a = a0 ∧ x = y; (ws)

Proof step 27 from proof stable-W5 (previous step) (back to tree)

gs.bq(ls.t) = W(ls.loc, ls.val) ' + FO(ls.loc) ', gs0.bq(ls.t) = be ' + bel, gs.rfo(ls.t) = [], gs0.rfo(ls.t) = [], SF ' ≠ gs.bq(ls.t), be ≠ W(ls.loc, ls.val), be.W?, FOFollowsW(ls.t, gs.bq(ls.t), gs.pq(ls.loc), ls.loc), SFTrailing(gs0.bq), SFTrailing(gs.bq), ctrok(gs0.fctr, gs0.fwriters), ctrok(gs.fctr, gs.fwriters), notMember(ls.t, gs.freaders), oneMember(ls.t, ls.loc, gs.fwriters), pendingOk(gs0.bq, gs0.pq), pendingOk(gs.bq, gs.pq), readFOsOk(gs0.rfo, gs0.fwriters), readFOsOk(gs.rfo, gs.fwriters), rely(ls.t, gs.pq, gs.bq, gs0.pq, gs0.bq), relyws(ls.t, gs.fwriters, gs0.fwriters), relyws(ls.t, gs.freaders, gs0.freaders), writersOk(gs0.bq, gs0.pq, gs0.fwriters), writersOk(gs.bq, gs.pq, gs.fwriters), FO(ls.loc) ∈ gs.bq(ls.t), W(ls.loc, ls.val) ∈ gs.bq(ls.t), gs.bq(ls.t) ⊒w be ' + bel ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: FLIT-preds, name:B-postfix-weak-same-loc-postfix, seq: ⊦ bel ⊒w bel0 ∧ bel.forall(λ be. be = SF ∨ be.l = l) → bel ⊒ bel0, subst:[bel, bel0, l] → [gs.bq(ls.t), gs0.bq(ls.t), ls.loc]

Additional information

Used terms: gs.bq(ls.t), gs0.bq(ls.t), ls.loc

Proof step 28 from proof stable-W5 (previous step) (back to tree)

⊦ bel ⊒w bel0 ∧ bel.forall(λ be. be = SF ∨ be.l = l) → bel ⊒ bel0

A spec-lemma

Proof step 29 from proof stable-W5 (previous step) (back to tree)

gs0.bq(ls.t) = be ' + bel, gs.bq(ls.t) = W(ls.loc, ls.val) ' + FO(ls.loc) ', gs0.rfo(ls.t) = [], gs.rfo(ls.t) = [], SF ' ≠ gs.bq(ls.t), be ≠ W(ls.loc, ls.val), be.W?, FOFollowsW(ls.t, gs.bq(ls.t), gs.pq(ls.loc), ls.loc), SFTrailing(gs.bq), SFTrailing(gs0.bq), ctrok(gs.fctr, gs.fwriters), ctrok(gs0.fctr, gs0.fwriters), notMember(ls.t, gs.freaders), oneMember(ls.t, ls.loc, gs.fwriters), pendingOk(gs.bq, gs.pq), pendingOk(gs0.bq, gs0.pq), readFOsOk(gs.rfo, gs.fwriters), readFOsOk(gs0.rfo, gs0.fwriters), rely(ls.t, gs.pq, gs.bq, gs0.pq, gs0.bq), relyws(ls.t, gs.freaders, gs0.freaders), relyws(ls.t, gs.fwriters, gs0.fwriters), writersOk(gs.bq, gs.pq, gs.fwriters), writersOk(gs0.bq, gs0.pq, gs0.fwriters), W(ls.loc, ls.val) ∈ gs.bq(ls.t), FO(ls.loc) ∈ gs.bq(ls.t), gs.bq(ls.t) ⊒w be ' + bel, gs.bq(ls.t) ⊒w gs0.bq(ls.t) ∧ gs.bq(ls.t).forall(λ be. be = SF ∨ be.l = ls.loc) → gs.bq(ls.t) ⊒ gs0.bq(ls.t) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 30 from proof stable-W5 (previous step) (back to tree)

gs0.bq(ls.t) = be ' + bel, gs.bq(ls.t) = W(ls.loc, ls.val) ' + FO(ls.loc) ', gs0.rfo(ls.t) = [], gs.rfo(ls.t) = [], SF ' ≠ gs.bq(ls.t), be ≠ W(ls.loc, ls.val), be.W?, FOFollowsW(ls.t, gs.bq(ls.t), gs.pq(ls.loc), ls.loc), SFTrailing(gs.bq), SFTrailing(gs0.bq), ctrok(gs.fctr, gs.fwriters), ctrok(gs0.fctr, gs0.fwriters), notMember(ls.t, gs.freaders), oneMember(ls.t, ls.loc, gs.fwriters), pendingOk(gs.bq, gs.pq), pendingOk(gs0.bq, gs0.pq), readFOsOk(gs.rfo, gs.fwriters), readFOsOk(gs0.rfo, gs0.fwriters), rely(ls.t, gs.pq, gs.bq, gs0.pq, gs0.bq), relyws(ls.t, gs.freaders, gs0.freaders), relyws(ls.t, gs.fwriters, gs0.fwriters), writersOk(gs.bq, gs.pq, gs.fwriters), writersOk(gs0.bq, gs0.pq, gs0.fwriters), W(ls.loc, ls.val) ∈ gs.bq(ls.t), FO(ls.loc) ∈ gs.bq(ls.t), gs.bq(ls.t) ⊒w be ' + bel, gs.bq(ls.t).forall(λ be. be = SF ∨ be.l = ls.loc) → gs.bq(ls.t) ⊒ be ' + bel ⊦

User applied rule subst equation

And obtained the following new goals

Rule argument

pos: left 2, flags:

Additional information

Proof step 31 from proof stable-W5 (previous step) (back to tree)

gs.bq(ls.t) = W(ls.loc, ls.val) ' + FO(ls.loc) ', gs0.bq(ls.t) = be ' + bel, gs.rfo(ls.t) = [], gs0.rfo(ls.t) = [], SF ' ≠ W(ls.loc, ls.val) ' + FO(ls.loc) ', be ≠ W(ls.loc, ls.val), be.W?, FOFollowsW(ls.t, W(ls.loc, ls.val) ' + FO(ls.loc) ', gs.pq(ls.loc), ls.loc), SFTrailing(gs0.bq), SFTrailing(gs.bq), ctrok(gs0.fctr, gs0.fwriters), ctrok(gs.fctr, gs.fwriters), notMember(ls.t, gs.freaders), oneMember(ls.t, ls.loc, gs.fwriters), pendingOk(gs0.bq, gs0.pq), pendingOk(gs.bq, gs.pq), readFOsOk(gs0.rfo, gs0.fwriters), readFOsOk(gs.rfo, gs.fwriters), rely(ls.t, gs.pq, gs.bq, gs0.pq, gs0.bq), relyws(ls.t, gs.fwriters, gs0.fwriters), relyws(ls.t, gs.freaders, gs0.freaders), writersOk(gs0.bq, gs0.pq, gs0.fwriters), writersOk(gs.bq, gs.pq, gs.fwriters), FO(ls.loc) ∈ W(ls.loc, ls.val) ' + FO(ls.loc) ', W(ls.loc, ls.val) ∈ W(ls.loc, ls.val) ' + FO(ls.loc) ', W(ls.loc, ls.val) ' + FO(ls.loc) ' ⊒w be ' + bel, (W(ls.loc, ls.val) ' + FO(ls.loc) ').forall(λ be. be = SF ∨ be.l = ls.loc) → W(ls.loc, ls.val) ' + FO(ls.loc) ' ⊒ be ' + bel ⊦

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 27

Additional information

Used simplifier rules: ⊦ W(l, v) ≠ FO(l0); (s) ⊦ a ∈ a0 ' ↔ a = a0; (s) ⊦ a ∈ a0 ' + x ↔ a = a0 ∨ a ∈ x; (s) ⊦ (a ' + x).forall(p) ↔ p(a) ∧ x.forall(p); (s) ⊦ W(l, v) ≠ SF; (s) ⊦ W(l, v).l = l; (s) ⊦ a '.forall(p) ↔ p(a); (s) ⊦ FO(l) ≠ SF; (s) ⊦ FO(l).l = l; (s) a: ⊦ (x + y) + z = x + y + z; from specification list

Proof step 32 from proof stable-W5 (previous step) (back to tree)

gs0.bq(ls.t) = be ' + bel, gs.bq(ls.t) = W(ls.loc, ls.val) ' + FO(ls.loc) ', gs0.rfo(ls.t) = [], gs.rfo(ls.t) = [], SF ' ≠ W(ls.loc, ls.val) ' + FO(ls.loc) ', be ≠ W(ls.loc, ls.val), be.W?, FOFollowsW(ls.t, W(ls.loc, ls.val) ' + FO(ls.loc) ', gs.pq(ls.loc), ls.loc), SFTrailing(gs.bq), SFTrailing(gs0.bq), ctrok(gs.fctr, gs.fwriters), ctrok(gs0.fctr, gs0.fwriters), notMember(ls.t, gs.freaders), oneMember(ls.t, ls.loc, gs.fwriters), pendingOk(gs.bq, gs.pq), pendingOk(gs0.bq, gs0.pq), readFOsOk(gs.rfo, gs.fwriters), readFOsOk(gs0.rfo, gs0.fwriters), rely(ls.t, gs.pq, gs.bq, gs0.pq, gs0.bq), relyws(ls.t, gs.freaders, gs0.freaders), relyws(ls.t, gs.fwriters, gs0.fwriters), writersOk(gs.bq, gs.pq, gs.fwriters), writersOk(gs0.bq, gs0.pq, gs0.fwriters), W(ls.loc, ls.val) ∈ W(ls.loc, ls.val) ' + FO(ls.loc) ', FO(ls.loc) ∈ W(ls.loc, ls.val) ' + FO(ls.loc) ', W(ls.loc, ls.val) ' + FO(ls.loc) ' ⊒ be ' + bel, W(ls.loc, ls.val) ' + FO(ls.loc) ' ⊒w be ' + bel ⊦

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 ⊦ a ∈ a0 ' + x ↔ a = a0 ∨ a ∈ x; (s) ⊦ a ∈ a0 ' ↔ a = a0; (s) ⊦ W(l, v) ≠ FO(l0); (s)

Proof step 33 from proof stable-W5 (previous step) (back to tree)

gs.bq(ls.t) = W(ls.loc, ls.val) ' + FO(ls.loc) ', gs0.bq(ls.t) = be ' + bel, gs.rfo(ls.t) = [], gs0.rfo(ls.t) = [], SF ' ≠ gs.bq(ls.t), be ≠ W(ls.loc, ls.val), be.W?, FOFollowsW(ls.t, gs.bq(ls.t), gs.pq(ls.loc), ls.loc), SFTrailing(gs0.bq), SFTrailing(gs.bq), ctrok(gs0.fctr, gs0.fwriters), ctrok(gs.fctr, gs.fwriters), notMember(ls.t, gs.freaders), oneMember(ls.t, ls.loc, gs.fwriters), pendingOk(gs0.bq, gs0.pq), pendingOk(gs.bq, gs.pq), readFOsOk(gs0.rfo, gs0.fwriters), readFOsOk(gs.rfo, gs.fwriters), rely(ls.t, gs.pq, gs.bq, gs0.pq, gs0.bq), relyws(ls.t, gs.fwriters, gs0.fwriters), relyws(ls.t, gs.freaders, gs0.freaders), writersOk(gs0.bq, gs0.pq, gs0.fwriters), writersOk(gs.bq, gs.pq, gs.fwriters), gs.bq(ls.t) ⊒ be ' + bel, gs.bq(ls.t) ⊒w be ' + bel ⊦

User applied rule subst equation

And obtained the following new goals

Rule argument

pos: left 1, flags: discard

Additional information

Proof step 34 from proof stable-W5 (previous step) (back to tree)

gs0.bq(ls.t) = be ' + bel, gs0.rfo(ls.t) = [], gs.rfo(ls.t) = [], SF ' ≠ W(ls.loc, ls.val) ' + FO(ls.loc) ', be ≠ W(ls.loc, ls.val), be.W?, FOFollowsW(ls.t, W(ls.loc, ls.val) ' + FO(ls.loc) ', gs.pq(ls.loc), ls.loc), SFTrailing(gs.bq), SFTrailing(gs0.bq), ctrok(gs.fctr, gs.fwriters), ctrok(gs0.fctr, gs0.fwriters), notMember(ls.t, gs.freaders), oneMember(ls.t, ls.loc, gs.fwriters), pendingOk(gs.bq, gs.pq), pendingOk(gs0.bq, gs0.pq), readFOsOk(gs.rfo, gs.fwriters), readFOsOk(gs0.rfo, gs0.fwriters), rely(ls.t, gs.pq, gs.bq, gs0.pq, gs0.bq), relyws(ls.t, gs.freaders, gs0.freaders), relyws(ls.t, gs.fwriters, gs0.fwriters), writersOk(gs.bq, gs.pq, gs.fwriters), writersOk(gs0.bq, gs0.pq, gs0.fwriters), W(ls.loc, ls.val) ' + FO(ls.loc) ' ⊒ be ' + bel, W(ls.loc, ls.val) ' + FO(ls.loc) ' ⊒w be ' + bel ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: list-last, name:postfix, seq: ⊦ x ⊒ y ↔ (∃ z. z + y = x), subst:[x, y] → [W(ls.loc, ls.val) ' + FO(ls.loc) ', be ' + bel]; ['a] → [Bevent], <all paths>

Additional information

Used simplifier rules: in-14: ⊦ a ' + x = y → a ∈ y; (forward) from specification list in-15: ⊦ x + a ' = y → a ∈ y; (forward) from specification list ⊦ a ≠ a0 → (a ∈ a0 ' + x ↔ a ∈ x); (s) ⊦ ¬ a ∈ y → (y + a0 ' + z = a ' + z0 ↔ y = [] ∧ a0 = a ∧ z = z0); (ws) ⊦ a ∈ x + y ↔ a ∈ x ∨ a ∈ y; (s) ⊦ a ∈ a0 ' + x ↔ a = a0 ∨ a ∈ x; (s) ⊦ W(l, v) ≠ FO(l0); (s) ⊦ ¬ FO(l).W?; (s) ⊦ bel0 + bel ⊒w bel; (s) ni: ⊦ x + a ' = y + z ∧ ¬ a ∈ z → z = []; (forward) from specification list in-16: ⊦ x + a ' = y + a0 ' + z ∧ a ≠ a0 → a ∈ z; (forward) from specification list inex: ⊦ x + a ' = y + a0 ' + z ∧ a ≠ a0 → (∃ z0. z = z0 + a '); (forward) from specification list ⊦ # y = # y0 → (x + y = z + y0 ↔ x = z ∧ y = y0); (ws) ⊦ x + a ' + y = a0 ' ↔ x = [] ∧ a = a0 ∧ y = []; (ws) a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: W(ls.loc, ls.val) ' + FO(ls.loc) ', be ' + bel

Proof step 35 from proof stable-W5 (previous step) (back to tree)

⊦ x ⊒ y ↔ (∃ z. z + y = x)

A spec-lemma

Proof Tree for stable-W5 (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