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Proof step 1 from proof abs-addWriteFO (conclusion) (back to tree)

⊦ ((h | t) | false)|R = [] → (abs(h, rs, p(l := p(l) + FO(t) ')) ↔ abs(h, rs, p))

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

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

((h | t) | false)|R = [], ¬(abs(h, rs, p(l := p(l) + FO(t) ')) ↔ abs(h, rs, p)) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:abs-def, seq: ⊦ abs(h, rs, p) ↔ (∀ l. absloc(events((h | l) | false), rs(l), p(l))), subst:[h, rs, p] → [h, rs, p(l := p(l) + FO(t) ')], <all paths>

Additional information

Used terms: h, rs, p(l := p(l) + FO(t) ')

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

⊦ abs(h, rs, p) ↔ (∀ l. absloc(events((h | l) | false), rs(l), p(l)))

Lemma abs-def

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

((h | t) | false)|R = [] ⊦ (∀ l0. absloc(events((h | l0) | false), rs(l0), p(l := p(l) + FO(t) ')(l0))) ↔ abs(h, rs, p)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

((h | t) | false)|R = [] ⊦ (∀ l0. absloc(events((h | l0) | false), rs(l0), (l = l0 ⊃ p(l0) + FO(t) ';p(l0)))) ↔ abs(h, rs, p)

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:abs-def, seq: ⊦ abs(h, rs, p) ↔ (∀ l. absloc(events((h | l) | false), rs(l), p(l))), subst:[h, rs, p] → [h, rs, p], <all paths>

Additional information

Used terms: h, rs, p

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

⊦ abs(h, rs, p) ↔ (∀ l. absloc(events((h | l) | false), rs(l), p(l)))

Lemma abs-def

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

((h | t) | false)|R = [] ⊦ (∀ l0. absloc(events((h | l0) | false), rs(l0), (l = l0 ⊃ p(l0) + FO(t) ';p(l0)))) ↔ (∀ l. absloc(events((h | l) | false), rs(l), p(l)))

User applied rule reduce equation

And obtained the following new goals

Rule argument

pos: right 1

Additional information

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

((h | t) | false)|R = [], ¬(absloc(events((h | l0) | false), rs(l0), (l = l0 ⊃ p(l0) + FO(t) ';p(l0))) ↔ absloc(events((h | l0) | false), rs(l0), p(l0))) ⊦

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: l = l0

Additional information

Used formulas: l = l0

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

l = l0, ((h | t) | false)|R = [], ¬(absloc(events((h | l0) | false), rs(l0), (l = l0 ⊃ p(l0) + FO(t) ';p(l0))) ↔ absloc(events((h | l0) | false), rs(l0), p(l0))) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

((h | t) | false)|R = [], ¬(absloc(events((h | l0) | false), rs(l0), p(l0) + FO(t) ') ↔ absloc(events((h | l0) | false), rs(l0), p(l0))) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:absloc-addWriteFO, seq:⊦ (∀ e. e ∈ el ∧ e.R? → e.t ≠ t) → (absloc(el, rset, fowl + FO(t) ') ↔ absloc(el, rset, fowl)), subst:[el, t, rset, fowl] → [events((h | l0) | false), t, rs(l0), p(l0)], <all paths>

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: events((h | l0) | false), t, rs(l0), p(l0)

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

⊦ (∀ e. e ∈ el ∧ e.R? → e.t ≠ t) → (absloc(el, rset, fowl + FO(t) ') ↔ absloc(el, rset, fowl))

Lemma absloc-addWriteFO

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

((h | t) | false)|R = [], ¬(absloc(events((h | l0) | false), rs(l0), p(l0) + FO(t) ') ↔ absloc(events((h | l0) | false), rs(l0), p(l0))) ⊦ ∀ e. e ∈ events((h | l0) | false) ∧ e.R? → e.t ≠ t

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

((h | e.t) | false)|R = [], ¬(absloc(events((h | l0) | false), rs(l0), p(l0) + FO(e.t) ') ↔ absloc(events((h | l0) | false), rs(l0), p(l0))), e.R?, e ∈ events((h | l0) | false) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: PLIB-preds, name:restrict-events, seq: ⊦ events(h) = h.map(λ vpe. vpe._1), subst:[h] → [(h | l0) | false], <all paths>

Additional information

Used terms: (h | l0) | false

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

⊦ events(h) = h.map(λ vpe. vpe._1)

A spec-lemma

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

((h | e.t) | false)|R = [], ¬(absloc(((h | l0) | false).map(λ vpe. vpe._1), rs(l0), p(l0) + FO(e.t) ') ↔ absloc(((h | l0) | false).map(λ vpe. vpe._1), rs(l0), p(l0))), e.R?, e ∈ ((h | l0) | false).map(λ vpe. vpe._1) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ b ∈ x.map(f) ↔ (∃ a. b = f(a) ∧ a ∈ x); (as)

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

((h | vpe._1.t) | false)|R = [], ¬(absloc(((h | l0) | false).map(._1), rs(l0), p(l0) + FO(vpe._1.t) ') ↔ absloc(((h | l0) | false).map(._1), rs(l0), p(l0))), vpe._1.R?, vpe ∈ (h | l0) | false ⊦

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

Rule argument

term: vpe._1

Additional information

Used terms: e, boolvar

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

vpe = (e, boolvar), ((h | e.t) | false)|R = [], ¬(absloc(((h | l0) | false).map(._1), rs(l0), p(l0) + FO(e.t) ') ↔ absloc(((h | l0) | false).map(._1), rs(l0), p(l0))), e.R?, vpe ∈ (h | l0) | false ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

((h | e.t) | false)|R = [], ¬(absloc(((h | l0) | false).map(._1), rs(l0), p(l0) + FO(e.t) ') ↔ absloc(((h | l0) | false).map(._1), rs(l0), p(l0))), e.R?, (e, boolvar) ∈ (h | l0) | false ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: PLIB-preds, name:restrict-vp, seq: ⊦ h | b = h.filter(λ vpe. vpe._2 ↔ b), subst:[h, b] → [h | l0, false], <all paths>

Additional information

Used terms: h | l0, false

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

⊦ h | b = h.filter(λ vpe. vpe._2 ↔ b)

A spec-lemma

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

((h | e.t) | false)|R = [], ¬ ( absloc((h | l0).filter(λ vpe. vpe._2 ↔ false).map(._1), rs(l0), p(l0) + FO(e.t) ') ↔ absloc((h | l0).filter(λ vpe. vpe._2 ↔ false).map(._1), rs(l0), p(l0))), e.R?, (e, boolvar) ∈ (h | l0).filter(λ vpe. vpe._2 ↔ false) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ a ∈ x.filter(p) ↔ a ∈ x ∧ p(a); (s)

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

((h | e.t) | false)|R = [], ¬(absloc((h | l0).filter(λ vpe. ¬ vpe._2).map(._1), rs(l0), p(l0) + FO(e.t) ') ↔ absloc((h | l0).filter(λ vpe. ¬ vpe._2).map(._1), rs(l0), p(l0))), e.R?, (e, false) ∈ h | l0 ⊦

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: e

Additional information

Used simplifier rules: RWevent freely generated by R, W;; ⊦ W(t, l, v).t = t; (s) ⊦ ¬ W(t, l, v).R?; (s)

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

e = R(t, l, v), ((h | e.t) | false)|R = [], ¬(absloc((h | l0).filter(λ vpe. ¬ vpe._2).map(._1), rs(l0), p(l0) + FO(e.t) ') ↔ absloc((h | l0).filter(λ vpe. ¬ vpe._2).map(._1), rs(l0), p(l0))), e.R?, (e, false) ∈ h | l0 ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ R(t, l, v).R?; (s) ⊦ R(t, l, v).t = t; (s)

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

((h | t) | false)|R = [], ¬(absloc((h | l0).filter(λ vpe. ¬ vpe._2).map(._1), rs(l0), p(l0) + FO(t) ') ↔ absloc((h | l0).filter(λ vpe. ¬ vpe._2).map(._1), rs(l0), p(l0))), (R(t, l, v), false) ∈ h | l0 ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: PLIB-preds, name:restrict-loc, seq: ⊦ h | l = h.filter(λ vpe. vpe._1.l = l), subst:[h, l] → [h, l0], <all paths>

Additional information

Used terms: h, l0

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

⊦ h | l = h.filter(λ vpe. vpe._1.l = l)

A spec-lemma

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

((h | t) | false)|R = [], ¬ ( absloc(h.filter(λ vpe. vpe._1.l = l0).filter(λ vpe. ¬ vpe._2).map(._1), rs(l0), p(l0) + FO(t) ') ↔ absloc(h.filter(λ vpe. vpe._1.l = l0).filter(λ vpe. ¬ vpe._2).map(._1), rs(l0), p(l0))), (R(t, l, v), false) ∈ h.filter(λ vpe. vpe._1.l = l0) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ R(t, l, v).l = l; (s) ⊦ a ∈ x.filter(p) ↔ a ∈ x ∧ p(a); (s)

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

((h | t) | false)|R = [], ¬ ( absloc(h.filter(λ vpe. vpe._1.l = l0).filter(λ vpe. ¬ vpe._2).map(._1), rs(l0), p(l0) + FO(t) ') ↔ absloc(h.filter(λ vpe. vpe._1.l = l0).filter(λ vpe. ¬ vpe._2).map(._1), rs(l0), p(l0))), (R(t, l0, v), false) ∈ h ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: list, name:some-element, seq: ⊦ x = [] ↔ ¬(∃ a. a ∈ x), subst:[x] → [((h | t) | false)|R]; ['a] → [(RWevent, Bool)], <all paths>

Additional information

Used terms: ((h | t) | false)|R

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

⊦ x = [] ↔ ¬(∃ a. a ∈ x)

A spec-lemma

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

¬ ( absloc(h.filter(λ vpe. vpe._1.l = l0).filter(λ vpe. ¬ vpe._2).map(._1), rs(l0), p(l0) + FO(t) ') ↔ absloc(h.filter(λ vpe. vpe._1.l = l0).filter(λ vpe. ¬ vpe._2).map(._1), rs(l0), p(l0))), (R(t, l0, v), false) ∈ h, ∀ a. ¬ a ∈ ((h | t) | false)|R ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 3 terms: (R(t, l0, v), false) abort: F discard: T computed: T

Additional information

Used terms: (R(t, l0, v), false)

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

¬ ( absloc(h.filter(λ vpe. vpe._1.l = l0).filter(λ vpe. ¬ vpe._2).map(._1), rs(l0), p(l0) + FO(t) ') ↔ absloc(h.filter(λ vpe. vpe._1.l = l0).filter(λ vpe. ¬ vpe._2).map(._1), rs(l0), p(l0))), (R(t, l0, v), false) ∈ h, ¬ (R(t, l0, v), false) ∈ ((h | t) | false)|R ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: PLIB-preds, name:restrict-R, seq: ⊦ h |R = h.filter(λ vpe. vpe._1.R?), subst:[h] → [(h | t) | false], <all paths>

Additional information

Used terms: (h | t) | false

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

⊦ h |R = h.filter(λ vpe. vpe._1.R?)

A spec-lemma

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

¬ ( absloc(h.filter(λ vpe. vpe._1.l = l0).filter(λ vpe. ¬ vpe._2).map(._1), rs(l0), p(l0) + FO(t) ') ↔ absloc(h.filter(λ vpe. vpe._1.l = l0).filter(λ vpe. ¬ vpe._2).map(._1), rs(l0), p(l0))), (R(t, l0, v), false) ∈ h, ¬ (R(t, l0, v), false) ∈ ((h | t) | false).filter(λ vpe. vpe._1.R?) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ R(t, l, v).R?; (s) ⊦ a ∈ x.filter(p) ↔ a ∈ x ∧ p(a); (s)

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

¬ ( absloc(h.filter(λ vpe. vpe._1.l = l0).filter(λ vpe. ¬ vpe._2).map(._1), rs(l0), p(l0) + FO(t) ') ↔ absloc(h.filter(λ vpe. vpe._1.l = l0).filter(λ vpe. ¬ vpe._2).map(._1), rs(l0), p(l0))), (R(t, l0, v), false) ∈ h, ¬ (R(t, l0, v), false) ∈ (h | t) | false ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: PLIB-preds, name:restrict-vp, seq: ⊦ h | b = h.filter(λ vpe. vpe._2 ↔ b), subst:[h, b] → [h | t, false], <all paths>

Additional information

Used terms: h | t, false

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

⊦ h | b = h.filter(λ vpe. vpe._2 ↔ b)

A spec-lemma

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

¬ ( absloc(h.filter(λ vpe. vpe._1.l = l0).filter(λ vpe. ¬ vpe._2).map(._1), rs(l0), p(l0) + FO(t) ') ↔ absloc(h.filter(λ vpe. vpe._1.l = l0).filter(λ vpe. ¬ vpe._2).map(._1), rs(l0), p(l0))), (R(t, l0, v), false) ∈ h, ¬ (R(t, l0, v), false) ∈ (h | t).filter(λ vpe. vpe._2 ↔ false) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ a ∈ x.filter(p) ↔ a ∈ x ∧ p(a); (s)

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

¬ ( absloc(h.filter(λ vpe. vpe._1.l = l0).filter(λ vpe. ¬ vpe._2).map(._1), rs(l0), p(l0) + FO(t) ') ↔ absloc(h.filter(λ vpe. vpe._1.l = l0).filter(λ vpe. ¬ vpe._2).map(._1), rs(l0), p(l0))), (R(t, l0, v), false) ∈ h, ¬ (R(t, l0, v), false) ∈ h | t ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: PLIB-preds, name:restrict-thread, seq: ⊦ h | t = h.filter(λ vpe. vpe._1.t = t), subst:[h, t] → [h, t], <all paths>

Additional information

Used simplifier rules: ⊦ a ∈ x.filter(p) ↔ a ∈ x ∧ p(a); (s) ⊦ R(t, l, v).t = t; (s) Used terms: h, t

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

⊦ h | t = h.filter(λ vpe. vpe._1.t = t)

A spec-lemma

Proof Tree for abs-addWriteFO (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