Proof step 1 from proof abs-firstPersistW-is-in-hist (conclusion) (back to tree)

System applied rule normalize

And obtained the following new goals

• Proof step 2:

  (h |W | l) | false = [], firstPersistW(p(l)) < # p(l), abs(h, rs, p) ⊦

Rule argument

Additional information

Proof step 2 from proof abs-firstPersistW-is-in-hist (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 3:

  (h |W | l) | false = [], p(l) ≠ [], firstPersistW(p(l)) < # p(l), abs(h, rs, p) ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ # x = 0 ↔ x = []; (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic

Proof step 3 from proof abs-firstPersistW-is-in-hist (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 4:

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

• Proof step 5:

  (h |W | l) | false = [], p(l) ≠ [], firstPersistW(p(l)) < # p(l), ∀ l. absloc(events((h | l) | false), rs(l), p(l)) ⊦

Rule argument

Additional information

Used terms: h, rs, p

Proof step 4 from proof abs-firstPersistW-is-in-hist (previous step) (back to tree)

Lemma abs-def

Proof step 5 from proof abs-firstPersistW-is-in-hist (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 6:

  (h |W | l) | false = [], p(l) ≠ [], firstPersistW(p(l)) < # p(l), absloc(events((h | l) | false), rs(l), p(l)) ⊦

Rule argument

Additional information

Used terms: l

Proof step 6 from proof abs-firstPersistW-is-in-hist (previous step) (back to tree)

User applied rule structural induction

And obtained the following new goals

• Proof step 7:

  ((vpe + h0)|W | l) | false = [], p(l) ≠ [], firstPersistW(p(l)) < # p(l), absloc(events((vpe + h0 | l) | false), rs(l), p(l)), ∀ l, p, rs. ¬((h0 |W | l) | false = [] ∧ p(l) ≠ [] ∧ firstPersistW(p(l)) < # p(l) ∧ absloc(events((h0 | l) | false), rs(l), p(l))) ⊦

Rule argument

Additional information

Used simplifier rules: absloc-base: ⊦ absloc([], rset, fowl) ↔ fowl |W = []; List('a) freely generated by [], +;; ⊦ [] |W = []; (s) ⊦ [] | l = []; (s) ⊦ [] | b = []; (s) ⊦ events([]) = []; (s) ⊦ fowl |W = [] → firstPersistW(fowl) = # fowl; (s) ⊦ ¬ n < n; (s)

Proof step 7 from proof abs-firstPersistW-is-in-hist (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 8:

  ((vpe._1.W? ⊃ vpe ';[]) | l) | false = [], (h0 |W | l) | false = [], p(l) ≠ [], firstPersistW(p(l)) < # p(l), absloc(events((vpe._1.l = l ⊃ vpe ';[]) | false) + events((h0 | l) | false), rs(l), p(l)), ∀ l0, p0, rs0. (h0 |W | l0) | false ≠ [] ∨ p0(l0) = [] ∨ ¬ firstPersistW(p0(l0)) < # p0(l0) ∨ ¬ absloc(events((h0 | l0) | false), rs0(l0), p0(l0)) ⊦

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ events(h1 + h2) = events(h1) + events(h2); (s) ⊦ h + h0 | b = (h | b) + (h0 | b); (s) ⊦ vpe ' | l = (vpe._1.l = l ⊃ vpe ';[]); (s) ⊦ h + h0 | l0 = (h | l0) + (h0 | l0); (s) ⊦ a + x = a ' + x; (s) ⊦ x + y = [] ↔ x = [] ∧ y = []; (s) ⊦ vpe ' |W = (vpe._1.W? ⊃ vpe ';[]); (s) ⊦ (h + h0)|W = h |W + h0 |W; (s)

Proof step 8 from proof abs-firstPersistW-is-in-hist (previous step) (back to tree)

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

• Proof step 9:

  vpe = (e, boolvar), (h0 |W | l) | false = [], ((e.W? ⊃ vpe ';[]) | l) | false = [], p(l) ≠ [], firstPersistW(p(l)) < # p(l), absloc(events((e.l = l ⊃ vpe ';[]) | false) + events((h0 | l) | false), rs(l), p(l)), ∀ l0, p0, rs0. (h0 |W | l0) | false ≠ [] ∨ p0(l0) = [] ∨ ¬ firstPersistW(p0(l0)) < # p0(l0) ∨ ¬ absloc(events((h0 | l0) | false), rs0(l0), p0(l0)) ⊦

Rule argument

Additional information

Used terms: e, boolvar

Proof step 9 from proof abs-firstPersistW-is-in-hist (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 10:

  ((e.W? ⊃ (e, boolvar) ';[]) | l) | false = [], (h0 |W | l) | false = [], p(l) ≠ [], firstPersistW(p(l)) < # p(l), absloc(events((e.l = l ⊃ (e, boolvar) ';[]) | false) + events((h0 | l) | false), rs(l), p(l)), ∀ l0, p0, rs0. (h0 |W | l0) | false ≠ [] ∨ p0(l0) = [] ∨ ¬ firstPersistW(p0(l0)) < # p0(l0) ∨ ¬ absloc(events((h0 | l0) | false), rs0(l0), p0(l0)) ⊦

Rule argument

Additional information

Proof step 10 from proof abs-firstPersistW-is-in-hist (previous step) (back to tree)

Heuristic if-then-else split applied rule case distinction

And obtained the following new goals

• Proof step 11:

  (h0 |W | l) | false = [], ((e, boolvar) ' | l) | false = [], p(l) ≠ [], e.W?, firstPersistW(p(l)) < # p(l), absloc(events((e.l = l ⊃ (e, boolvar) ';[]) | false) + events((h0 | l) | false), rs(l), p(l)), ∀ l0, p0, rs0. (h0 |W | l0) | false ≠ [] ∨ p0(l0) = [] ∨ ¬ firstPersistW(p0(l0)) < # p0(l0) ∨ ¬ absloc(events((h0 | l0) | false), rs0(l0), p0(l0)) ⊦

• Proof step 15:

  (h0 |W | l) | false = [], ([] | l) | false = [], p(l) ≠ [], firstPersistW(p(l)) < # p(l), absloc(events((e.l = l ⊃ (e, boolvar) ';[]) | false) + events((h0 | l) | false), rs(l), p(l)), ¬ e.W?, ∀ l0, p0, rs0. (h0 |W | l0) | false ≠ [] ∨ p0(l0) = [] ∨ ¬ firstPersistW(p0(l0)) < # p0(l0) ∨ ¬ absloc(events((h0 | l0) | false), rs0(l0), p0(l0)) ⊦

Rule argument

Additional information

Proof step 11 from proof abs-firstPersistW-is-in-hist (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 12:

  (e.l = l ⊃ (e, boolvar) ';[]) | false = [], (h0 |W | l) | false = [], p(l) ≠ [], e.W?, firstPersistW(p(l)) < # p(l), absloc(events((h0 | l) | false), rs(l), p(l)), ∀ l0, p0, rs0. (h0 |W | l0) | false ≠ [] ∨ p0(l0) = [] ∨ ¬ firstPersistW(p0(l0)) < # p0(l0) ∨ ¬ absloc(events((h0 | l0) | false), rs0(l0), p0(l0)) ⊦

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ [] + x = x; (s) ⊦ events([]) = []; (s) ⊦ vpe ' | l = (vpe._1.l = l ⊃ vpe ';[]); (s)

Proof step 12 from proof abs-firstPersistW-is-in-hist (previous step) (back to tree)

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

• Proof step 13:

  e = W(t, l1, v), (h0 |W | l) | false = [], (l1 = l ⊃ (e, boolvar) ';[]) | false = [], p(l) ≠ [], e.W?, firstPersistW(p(l)) < # p(l), absloc(events((h0 | l) | false), rs(l), p(l)), ∀ l0, p0, rs0. (h0 |W | l0) | false ≠ [] ∨ p0(l0) = [] ∨ ¬ firstPersistW(p0(l0)) < # p0(l0) ∨ ¬ absloc(events((h0 | l0) | false), rs0(l0), p0(l0)) ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ e.W? → ((t = e.t ∧ l = e.l) ∧ v = e.v ↔ e = W(t, l, v)); (elim) Used terms: e, t, l1, v

Proof step 13 from proof abs-firstPersistW-is-in-hist (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 14:

  (h0 |W | l) | false = [], (l1 = l ⊃ (W(t, l, v), boolvar) ';[]) | false = [], p(l) ≠ [], firstPersistW(p(l)) < # p(l), absloc(events((h0 | l) | false), rs(l), p(l)), ∀ l0, p0, rs0. (h0 |W | l0) | false ≠ [] ∨ p0(l0) = [] ∨ ¬ firstPersistW(p0(l0)) < # p0(l0) ∨ ¬ absloc(events((h0 | l0) | false), rs0(l0), p0(l0)) ⊦

Rule argument

Additional information

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

Proof step 14 from proof abs-firstPersistW-is-in-hist (previous step) (back to tree)

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

Additional information

Used terms: l, p, rs

Proof step 15 from proof abs-firstPersistW-is-in-hist (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 16:

  (h0 |W | l) | false = [], p(l) ≠ [], firstPersistW(p(l)) < # p(l), absloc(events((e.l = l ⊃ (e, boolvar) ';[]) | false) + events((h0 | l) | false), rs(l), p(l)), ¬ e.W?, ∀ l0, p0, rs0. (h0 |W | l0) | false ≠ [] ∨ p0(l0) = [] ∨ ¬ firstPersistW(p0(l0)) < # p0(l0) ∨ ¬ absloc(events((h0 | l0) | false), rs0(l0), p0(l0)) ⊦

Rule argument

Additional information

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

Proof step 16 from proof abs-firstPersistW-is-in-hist (previous step) (back to tree)

Heuristic if-then-else split applied rule case distinction

And obtained the following new goals

• Proof step 17:

  e.l = l, (h0 |W | l) | false = [], p(l) ≠ [], firstPersistW(p(l)) < # p(l), absloc(events((e, boolvar) ' | false) + events((h0 | l) | false), rs(l), p(l)), ¬ e.W?, ∀ l0, p0, rs0. (h0 |W | l0) | false ≠ [] ∨ p0(l0) = [] ∨ ¬ firstPersistW(p0(l0)) < # p0(l0) ∨ ¬ absloc(events((h0 | l0) | false), rs0(l0), p0(l0)) ⊦

• Proof step 25:

  (h0 |W | l) | false = [], e.l ≠ l, p(l) ≠ [], firstPersistW(p(l)) < # p(l), absloc(events([] | false) + events((h0 | l) | false), rs(l), p(l)), ¬ e.W?, ∀ l0, p0, rs0. (h0 |W | l0) | false ≠ [] ∨ p0(l0) = [] ∨ ¬ firstPersistW(p0(l0)) < # p0(l0) ∨ ¬ absloc(events((h0 | l0) | false), rs0(l0), p0(l0)) ⊦

Rule argument

Additional information

Proof step 17 from proof abs-firstPersistW-is-in-hist (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 18:

  (h0 |W | e.l) | false = [], p(e.l) ≠ [], firstPersistW(p(e.l)) < # p(e.l), absloc(events((¬ boolvar ⊃ (e, false) ';[])) + events((h0 | e.l) | false), rs(e.l), p(e.l)), ¬ e.W?, ∀ l0, p0, rs0. (h0 |W | l0) | false ≠ [] ∨ p0(l0) = [] ∨ ¬ firstPersistW(p0(l0)) < # p0(l0) ∨ ¬ absloc(events((h0 | l0) | false), rs0(l0), p0(l0)) ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ # x = 0 ↔ x = []; (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic ⊦ (e, b) ' | b0 = (b ↔ b0 ⊃ (e, b) ';[]); (s)

Proof step 18 from proof abs-firstPersistW-is-in-hist (previous step) (back to tree)

Heuristic if-then-else split applied rule case distinction

And obtained the following new goals

• Proof step 19:

  ¬ boolvar, (h0 |W | e.l) | false = [], p(e.l) ≠ [], firstPersistW(p(e.l)) < # p(e.l), absloc(events((e, false) ') + events((h0 | e.l) | false), rs(e.l), p(e.l)), ¬ e.W?, ∀ l0, p0, rs0. (h0 |W | l0) | false ≠ [] ∨ p0(l0) = [] ∨ ¬ firstPersistW(p0(l0)) < # p0(l0) ∨ ¬ absloc(events((h0 | l0) | false), rs0(l0), p0(l0)) ⊦

• Proof step 23:

  boolvar, (h0 |W | e.l) | false = [], p(e.l) ≠ [], firstPersistW(p(e.l)) < # p(e.l), absloc(events([]) + events((h0 | e.l) | false), rs(e.l), p(e.l)), ¬ e.W?, ∀ l0, p0, rs0. (h0 |W | l0) | false ≠ [] ∨ p0(l0) = [] ∨ ¬ firstPersistW(p0(l0)) < # p0(l0) ∨ ¬ absloc(events((h0 | l0) | false), rs0(l0), p0(l0)) ⊦

Rule argument

Additional information

Proof step 19 from proof abs-firstPersistW-is-in-hist (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 20:

  (h0 |W | e.l) | false = [], p(e.l) ≠ [], firstPersistW(p(e.l)) < # p(e.l), absloc(e ' + events((h0 | e.l) | false), rs(e.l), p(e.l)), ¬ e.W?, ∀ l0, p0, rs0. (h0 |W | l0) | false ≠ [] ∨ p0(l0) = [] ∨ ¬ firstPersistW(p0(l0)) < # p0(l0) ∨ ¬ absloc(events((h0 | l0) | false), rs0(l0), p0(l0)) ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ events(vpe ') = vpe._1 '; (s)

Proof step 20 from proof abs-firstPersistW-is-in-hist (previous step) (back to tree)

User applied rule constructor cut

And obtained the following new goals

• Proof step 21:

  e = R(t, l, v), (h0 |W | e.l) | false = [], p(e.l) ≠ [], firstPersistW(p(e.l)) < # p(e.l), absloc(e ' + events((h0 | e.l) | false), rs(e.l), p(e.l)), ¬ e.W?, ∀ l0, p0, rs0. (h0 |W | l0) | false ≠ [] ∨ p0(l0) = [] ∨ ¬ firstPersistW(p0(l0)) < # p0(l0) ∨ ¬ absloc(events((h0 | l0) | false), rs0(l0), p0(l0)) ⊦

Rule argument

Additional information

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

Proof step 21 from proof abs-firstPersistW-is-in-hist (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 22:

  (h0 |W | l) | false = [], p(l) ≠ [], firstPersistW(p(l)) < # p(l), absloc(events((h0 | l) | false), rs(l), p(l)), ¬ t ∈ rs(l) → FO(t) ∈ p(l), ∀ l0, p0, rs0. (h0 |W | l0) | false ≠ [] ∨ p0(l0) = [] ∨ ¬ firstPersistW(p0(l0)) < # p0(l0) ∨ ¬ absloc(events((h0 | l0) | false), rs0(l0), p0(l0)) ⊦

Rule argument

Additional information

Used simplifier rules: absloc-recR: ⊦ absloc(R(t, l, v) ' + el, rset, fowl) ↔ (¬ t ∈ rset → FO(t) ∈ fowl) ∧ absloc(el, rset, fowl); ⊦ # x = 0 ↔ x = []; (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic ⊦ ¬ R(t, l, v).W?; (s) ⊦ R(t, l, v).l = l; (s) a: ⊦ (x + y) + z = x + y + z; from specification list

Proof step 22 from proof abs-firstPersistW-is-in-hist (previous step) (back to tree)

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

Additional information

Used terms: l, p, rs

Proof step 23 from proof abs-firstPersistW-is-in-hist (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 24:

  (h0 |W | e.l) | false = [], p(e.l) ≠ [], firstPersistW(p(e.l)) < # p(e.l), absloc(events((h0 | e.l) | false), rs(e.l), p(e.l)), ¬ e.W?, ∀ l0, p0, rs0. (h0 |W | l0) | false ≠ [] ∨ p0(l0) = [] ∨ ¬ firstPersistW(p0(l0)) < # p0(l0) ∨ ¬ absloc(events((h0 | l0) | false), rs0(l0), p0(l0)) ⊦

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ [] + x = x; (s) ⊦ events([]) = []; (s)

Proof step 24 from proof abs-firstPersistW-is-in-hist (previous step) (back to tree)

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

Additional information

Used terms: e.l, p, rs

Proof step 25 from proof abs-firstPersistW-is-in-hist (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 26:

  (h0 |W | l) | false = [], e.l ≠ l, p(l) ≠ [], firstPersistW(p(l)) < # p(l), absloc(events((h0 | l) | false), rs(l), p(l)), ¬ e.W?, ∀ l0, p0, rs0. (h0 |W | l0) | false ≠ [] ∨ p0(l0) = [] ∨ ¬ firstPersistW(p0(l0)) < # p0(l0) ∨ ¬ absloc(events((h0 | l0) | false), rs0(l0), p0(l0)) ⊦

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ [] + x = x; (s) ⊦ events([]) = []; (s) ⊦ [] | b = []; (s)

Proof step 26 from proof abs-firstPersistW-is-in-hist (previous step) (back to tree)

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

Additional information

Used terms: l, p, rs