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

⊦ abs(h, rs, p) ∧ firstPersistW(p(l)) < # p(l) ∧ h[firstVolWrite(l, h)] = (e, bv) → p(l)[firstPersistW(p(l))].val = e.v

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

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

h[firstVolWrite(l, h)] = (e, bv), p(l)[firstPersistW(p(l))].val ≠ e.v, firstPersistW(p(l)) < # p(l), abs(h, rs, p) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

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

h[firstVolWrite(l, h)] = (e, bv), p(l)[firstPersistW(p(l))].val ≠ e.v, p(l) ≠ [], firstPersistW(p(l)) < # p(l), 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 4 from proof abs-firstPersistW-is-firstVolWrite (previous step) (back to tree)

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

Lemma abs-def

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

h[firstVolWrite(l, h)] = (e, bv), p(l)[firstPersistW(p(l))].val ≠ e.v, p(l) ≠ [], firstPersistW(p(l)) < # p(l), ∀ l. absloc(events((h | l) | false), rs(l), p(l)) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 5 terms: l abort: F discard: T computed: T

Additional information

Used terms: l

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

h[firstVolWrite(l, h)] = (e, bv), p(l)[firstPersistW(p(l))].val ≠ e.v, p(l) ≠ [], firstPersistW(p(l)) < # p(l), absloc(events((h | l) | false), rs(l), p(l)) ⊦

User applied rule structural induction

And obtained the following new goals

Rule argument

term: h

Additional information

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

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

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

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

(vpe ' + h0)[firstVolWrite(l, vpe ' + h0)] = (e, bv), p(l)[firstPersistW(p(l))].val ≠ e.v, p(l) ≠ [], firstPersistW(p(l)) < # p(l), absloc(events((vpe._1.l = l ⊃ vpe ';[]) | false) + events((h0 | l) | false), rs(l), p(l)), ∀ l0, e0, bv0, p0, rs0. h0[firstVolWrite(l0, h0)] ≠ (e0, bv0) ∨ p0(l0)[firstPersistW(p0(l0))].val = e0.v ∨ p0(l0) = [] ∨ ¬ firstPersistW(p0(l0)) < # p0(l0) ∨ ¬ absloc(events((h0 | l0) | false), rs0(l0), p0(l0)) ⊦

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

Rule argument

term: vpe._1

Additional information

Used terms: e0, bv0

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

(vpe ' + h0)[firstVolWrite(l, vpe ' + h0)] = (e, bv), vpe = (e0, bv0), p(l)[firstPersistW(p(l))].val ≠ e.v, p(l) ≠ [], firstPersistW(p(l)) < # p(l), absloc(events((e0.l = l ⊃ vpe ';[]) | false) + events((h0 | l) | false), rs(l), p(l)), ∀ l0, e0, bv0, p0, rs0. h0[firstVolWrite(l0, h0)] ≠ (e0, bv0) ∨ p0(l0)[firstPersistW(p0(l0))].val = e0.v ∨ p0(l0) = [] ∨ ¬ firstPersistW(p0(l0)) < # p0(l0) ∨ ¬ absloc(events((h0 | l0) | false), rs0(l0), p0(l0)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

((e0, bv0) ' + h0)[firstVolWrite(l, (e0, bv0) ' + h0)] = (e, bv), p(l)[firstPersistW(p(l))].val ≠ e.v, p(l) ≠ [], firstPersistW(p(l)) < # p(l), absloc(events((e0.l = l ⊃ (e0, bv0) ';[]) | false) + events((h0 | l) | false), rs(l), p(l)), ∀ l0, e0, bv0, p0, rs0. h0[firstVolWrite(l0, h0)] ≠ (e0, bv0) ∨ p0(l0)[firstPersistW(p0(l0))].val = e0.v ∨ p0(l0) = [] ∨ ¬ firstPersistW(p0(l0)) < # p0(l0) ∨ ¬ absloc(events((h0 | l0) | false), rs0(l0), p0(l0)) ⊦

Heuristic if-then-else split applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 5

Additional information

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

e0.l = l, ((e0, bv0) ' + h0)[firstVolWrite(l, (e0, bv0) ' + h0)] = (e, bv), p(l)[firstPersistW(p(l))].val ≠ e.v, p(l) ≠ [], firstPersistW(p(l)) < # p(l), absloc(events((e0, bv0) ' | false) + events((h0 | l) | false), rs(l), p(l)), ∀ l0, e0, bv0, p0, rs0. h0[firstVolWrite(l0, h0)] ≠ (e0, bv0) ∨ p0(l0)[firstPersistW(p0(l0))].val = e0.v ∨ p0(l0) = [] ∨ ¬ firstPersistW(p0(l0)) < # p0(l0) ∨ ¬ absloc(events((h0 | l0) | false), rs0(l0), p0(l0)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

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

((e0, bv0) ' + h0)[firstVolWrite(e0.l, (e0, bv0) ' + h0)] = (e, bv), p(e0.l)[firstPersistW(p(e0.l))].val ≠ e.v, p(e0.l) ≠ [], firstPersistW(p(e0.l)) < # p(e0.l), absloc(events((¬ bv0 ⊃ (e0, false) ';[])) + events((h0 | e0.l) | false), rs(e0.l), p(e0.l)), ∀ l0, e0, bv0, p0, rs0. h0[firstVolWrite(l0, h0)] ≠ (e0, bv0) ∨ p0(l0)[firstPersistW(p0(l0))].val = e0.v ∨ p0(l0) = [] ∨ ¬ firstPersistW(p0(l0)) < # p0(l0) ∨ ¬ absloc(events((h0 | l0) | false), rs0(l0), p0(l0)) ⊦

Heuristic if-then-else split applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 5

Additional information

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

((e0, bv0) ' + h0)[firstVolWrite(e0.l, (e0, bv0) ' + h0)] = (e, bv), ¬ bv0, p(e0.l)[firstPersistW(p(e0.l))].val ≠ e.v, p(e0.l) ≠ [], firstPersistW(p(e0.l)) < # p(e0.l), absloc(events((e0, false) ') + events((h0 | e0.l) | false), rs(e0.l), p(e0.l)), ∀ l0, e0, bv0, p0, rs0. h0[firstVolWrite(l0, h0)] ≠ (e0, bv0) ∨ p0(l0)[firstPersistW(p0(l0))].val = e0.v ∨ p0(l0) = [] ∨ ¬ firstPersistW(p0(l0)) < # p0(l0) ∨ ¬ absloc(events((h0 | l0) | false), rs0(l0), p0(l0)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

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

((e0, false) ' + h0)[firstVolWrite(e0.l, (e0, false) ' + h0)] = (e, bv), p(e0.l)[firstPersistW(p(e0.l))].val ≠ e.v, p(e0.l) ≠ [], firstPersistW(p(e0.l)) < # p(e0.l), absloc(e0 ' + events((h0 | e0.l) | false), rs(e0.l), p(e0.l)), ∀ l0, e0, bv0, p0, rs0. h0[firstVolWrite(l0, h0)] ≠ (e0, bv0) ∨ p0(l0)[firstPersistW(p0(l0))].val = e0.v ∨ p0(l0) = [] ∨ ¬ firstPersistW(p0(l0)) < # p0(l0) ∨ ¬ absloc(events((h0 | l0) | false), rs0(l0), p0(l0)) ⊦

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: e0

Additional information

Used simplifier rules: RWevent freely generated by R, W;;

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

((e0, false) ' + h0)[firstVolWrite(e0.l, (e0, false) ' + h0)] = (e, bv), e0 = R(t, l, v), p(e0.l)[firstPersistW(p(e0.l))].val ≠ e.v, p(e0.l) ≠ [], firstPersistW(p(e0.l)) < # p(e0.l), absloc(e0 ' + events((h0 | e0.l) | false), rs(e0.l), p(e0.l)), ∀ l0, e0, bv0, p0, rs0. h0[firstVolWrite(l0, h0)] ≠ (e0, bv0) ∨ p0(l0)[firstPersistW(p0(l0))].val = e0.v ∨ p0(l0) = [] ∨ ¬ firstPersistW(p0(l0)) < # p0(l0) ∨ ¬ absloc(events((h0 | l0) | false), rs0(l0), p0(l0)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

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).l = l; (s) a: ⊦ (x + y) + z = x + y + z; from specification list c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ (a ' + x)[n + 1] = x[n]; (s) ⊦ firstVolWrite(l, (R(t, l0, v), b) ' + h) = firstVolWrite(l, h) + 1; (s)

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

h0[firstVolWrite(l, h0)] = (e, bv), p(l)[firstPersistW(p(l))].val ≠ e.v, p(l) ≠ [], firstPersistW(p(l)) < # p(l), absloc(events((h0 | l) | false), rs(l), p(l)), ¬ t ∈ rs(l) → FO(t) ∈ p(l), ∀ l0, e0, bv0, p0, rs0. h0[firstVolWrite(l0, h0)] ≠ (e0, bv0) ∨ p0(l0)[firstPersistW(p0(l0))].val = e0.v ∨ p0(l0) = [] ∨ ¬ firstPersistW(p0(l0)) < # p0(l0) ∨ ¬ absloc(events((h0 | l0) | false), rs0(l0), p0(l0)) ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 7 terms: l, e, bv, p, rs

Additional information

Used terms: l, e, bv, p, rs

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

((e0, false) ' + h0)[firstVolWrite(e0.l, (e0, false) ' + h0)] = (e, bv), e0 = W(t, l, v), p(e0.l)[firstPersistW(p(e0.l))].val ≠ e.v, p(e0.l) ≠ [], firstPersistW(p(e0.l)) < # p(e0.l), absloc(e0 ' + events((h0 | e0.l) | false), rs(e0.l), p(e0.l)), ∀ l0, e0, bv0, p0, rs0. h0[firstVolWrite(l0, h0)] ≠ (e0, bv0) ∨ p0(l0)[firstPersistW(p0(l0))].val = e0.v ∨ p0(l0) = [] ∨ ¬ firstPersistW(p0(l0)) < # p0(l0) ∨ ¬ absloc(events((h0 | l0) | false), rs0(l0), p0(l0)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ W(t, l, v).v = v; (s) ⊦ # x = 0 ↔ x = []; (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic ⊦ W(t, l, v).l = l; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ (a ' + x)[0] = a; (s) ⊦ firstVolWrite(l, (W(t, l, v), false) ' + h) = 0; (s)

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

p(l)[firstPersistW(p(l))].val ≠ v, p(l) ≠ [], firstPersistW(p(l)) < # p(l), absloc(W(t, l, v) ' + events((h0 | l) | false), rs(l), p(l)), ∀ l0, e0, bv0, p0, rs0. h0[firstVolWrite(l0, h0)] ≠ (e0, bv0) ∨ p0(l0)[firstPersistW(p0(l0))].val = e0.v ∨ p0(l0) = [] ∨ ¬ firstPersistW(p0(l0)) < # p0(l0) ∨ ¬ absloc(events((h0 | l0) | false), rs0(l0), p0(l0)) ⊦

User applied rule weakening

And obtained the following new goals

Rule argument

poslist: left 5

Additional information

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

p(l)[firstPersistW(p(l))].val ≠ v, p(l) ≠ [], firstPersistW(p(l)) < # p(l), absloc(W(t, l, v) ' + events((h0 | l) | false), rs(l), p(l)) ⊦

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: ∃ fowl. fowl = p(l)

Additional information

Used formulas: ∃ fowl. fowl = p(l)

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

p(l)[firstPersistW(p(l))].val ≠ v, p(l) ≠ [], firstPersistW(p(l)) < # p(l), absloc(W(t, l, v) ' + events((h0 | l) | false), rs(l), p(l)), ∃ fowl. fowl = p(l) ⊦

User applied rule exists left

And obtained the following new goals

Rule argument

pos: left 5

Additional information

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

fowl = p(l), p(l)[firstPersistW(p(l))].val ≠ v, p(l) ≠ [], firstPersistW(p(l)) < # p(l), absloc(W(t, l, v) ' + events((h0 | l) | false), rs(l), p(l)) ⊦

User applied rule subst equation

And obtained the following new goals

Rule argument

pos: left 1, flags: rotate, discard

Additional information

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

fowl[firstPersistW(fowl)].val ≠ v, fowl ≠ [], firstPersistW(fowl) < # fowl, absloc(W(t, l, v) ' + events((h0 | l) | false), rs(l), fowl) ⊦

User applied rule structural induction

And obtained the following new goals

Rule argument

term: fowl

Additional information

Used simplifier rules: List('a) freely generated by [], +;;

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

fow + fowl0 ≠ [], (fow + fowl0)[firstPersistW(fow + fowl0)].val ≠ v, firstPersistW(fow + fowl0) < #(fow + fowl0), absloc(W(t, l, v) ' + events((h0 | l) | false), rs(l), fow + fowl0), ∀ v, t, l, h0, rs. ¬ ( fowl0[firstPersistW(fowl0)].val ≠ v ∧ fowl0 ≠ [] ∧ firstPersistW(fowl0) < # fowl0 ∧ absloc(W(t, l, v) ' + events((h0 | l) | false), rs(l), fowl0)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ m < n ↔ ¬ n ≤ m; (ss) c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ m < n + 1 ↔ ¬ n < m; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ m +1 = m + 1; (s) ⊦ #(a ' + x) = (# x)+1; (s) ⊦ a + x = a ' + x; (s) ⊦ a ' ≠ []; (s) ⊦ x + y = [] ↔ x = [] ∧ y = []; (s)

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

(fow ' + fowl0)[firstPersistW(fow ' + fowl0)].val ≠ v, absloc(W(t, l, v) ' + events((h0 | l) | false), rs(l), fow ' + fowl0), firstPersistW(fow ' + fowl0) ≤ # fowl0, ∀ v0, t0, l0, h, rs0. fowl0[firstPersistW(fowl0)].val = v0 ∨ fowl0 = [] ∨ ¬ firstPersistW(fowl0) < # fowl0 ∨ ¬ absloc(W(t0, l0, v0) ' + events((h | l0) | false), rs0(l0), fowl0) ⊦

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: fow

Additional information

Used simplifier rules: absloc-recW-W: ⊦ absloc(W(t, l, v) ' + el, rset, W(v0, t0) ' + fowl) ↔ t0 = t ∧ v0 = v ∧ absloc(el, rset, fowl); FOW freely generated by FO, W;; ⊦ (a ' + x)[0] = a; (s) ⊦ W(v, tid).val = v; (s) ⊦ firstPersistW(W(v, t) ' + fowl) = 0; (s) ⊦ 0 ≤ m; (s) a: ⊦ (x + y) + z = x + y + z; from specification list

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

fow = FO(t1), (fow ' + fowl0)[firstPersistW(fow ' + fowl0)].val ≠ v, absloc(W(t, l, v) ' + events((h0 | l) | false), rs(l), fow ' + fowl0), firstPersistW(fow ' + fowl0) ≤ # fowl0, ∀ v0, t0, l0, h, rs0. fowl0[firstPersistW(fowl0)].val = v0 ∨ fowl0 = [] ∨ ¬ firstPersistW(fowl0) < # fowl0 ∨ ¬ absloc(W(t0, l0, v0) ' + events((h | l0) | false), rs0(l0), fowl0) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: absloc-recW-FO: ⊦ absloc(W(t, l, v) ' + el, rset, FO(t0) ' + fowl) ↔ absloc(W(t, l, v) ' + el, rset, fowl); ⊦ # x = 0 ↔ x = []; (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic a: ⊦ (x + y) + z = x + y + z; from specification list c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ m + 1 ≤ n ↔ m < n; (s) ⊦ firstPersistW(FO(t) ' + fowl) = firstPersistW(fowl) + 1; (s) ⊦ (a ' + x)[n + 1] = x[n]; (s)

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

fowl0[firstPersistW(fowl0)].val ≠ v, fowl0 ≠ [], firstPersistW(fowl0) < # fowl0, absloc(W(t, l, v) ' + events((h0 | l) | false), rs(l), fowl0), ∀ v0, t0, l0, h, rs0. fowl0[firstPersistW(fowl0)].val = v0 ∨ ¬ absloc(W(t0, l0, v0) ' + events((h | l0) | false), rs0(l0), fowl0) ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 5 terms: v, t, l, h0, rs

Additional information

Used terms: v, t, l, h0, rs

Proof step 27 from proof abs-firstPersistW-is-firstVolWrite (previous step) (back to tree)

((e0, bv0) ' + h0)[firstVolWrite(e0.l, (e0, bv0) ' + h0)] = (e, bv), bv0, p(e0.l)[firstPersistW(p(e0.l))].val ≠ e.v, p(e0.l) ≠ [], firstPersistW(p(e0.l)) < # p(e0.l), absloc(events([]) + events((h0 | e0.l) | false), rs(e0.l), p(e0.l)), ∀ l0, e0, bv0, p0, rs0. h0[firstVolWrite(l0, h0)] ≠ (e0, bv0) ∨ p0(l0)[firstPersistW(p0(l0))].val = e0.v ∨ p0(l0) = [] ∨ ¬ firstPersistW(p0(l0)) < # p0(l0) ∨ ¬ absloc(events((h0 | l0) | false), rs0(l0), p0(l0)) ⊦

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) ⊦ events([]) = []; (s)

Proof step 28 from proof abs-firstPersistW-is-firstVolWrite (previous step) (back to tree)

((e0, true) ' + h0)[firstVolWrite(e0.l, (e0, true) ' + h0)] = (e, bv), p(e0.l)[firstPersistW(p(e0.l))].val ≠ e.v, p(e0.l) ≠ [], firstPersistW(p(e0.l)) < # p(e0.l), absloc(events((h0 | e0.l) | false), rs(e0.l), p(e0.l)), ∀ l0, e0, bv0, p0, rs0. h0[firstVolWrite(l0, h0)] ≠ (e0, bv0) ∨ p0(l0)[firstPersistW(p0(l0))].val = e0.v ∨ p0(l0) = [] ∨ ¬ firstPersistW(p0(l0)) < # p0(l0) ∨ ¬ absloc(events((h0 | l0) | false), rs0(l0), p0(l0)) ⊦

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: e0

Additional information

Used simplifier rules: RWevent freely generated by R, W;;

Proof step 29 from proof abs-firstPersistW-is-firstVolWrite (previous step) (back to tree)

((e0, true) ' + h0)[firstVolWrite(e0.l, (e0, true) ' + h0)] = (e, bv), e0 = R(t, l, v), p(e0.l)[firstPersistW(p(e0.l))].val ≠ e.v, p(e0.l) ≠ [], firstPersistW(p(e0.l)) < # p(e0.l), absloc(events((h0 | e0.l) | false), rs(e0.l), p(e0.l)), ∀ l0, e0, bv0, p0, rs0. h0[firstVolWrite(l0, h0)] ≠ (e0, bv0) ∨ p0(l0)[firstPersistW(p0(l0))].val = e0.v ∨ p0(l0) = [] ∨ ¬ firstPersistW(p0(l0)) < # p0(l0) ∨ ¬ absloc(events((h0 | l0) | false), rs0(l0), p0(l0)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ # x = 0 ↔ x = []; (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic ⊦ R(t, l, v).l = l; (s) a: ⊦ (x + y) + z = x + y + z; from specification list c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ (a ' + x)[n + 1] = x[n]; (s) ⊦ firstVolWrite(l, (R(t, l0, v), b) ' + h) = firstVolWrite(l, h) + 1; (s)

Proof step 30 from proof abs-firstPersistW-is-firstVolWrite (previous step) (back to tree)

h0[firstVolWrite(l, h0)] = (e, bv), p(l)[firstPersistW(p(l))].val ≠ e.v, p(l) ≠ [], firstPersistW(p(l)) < # p(l), absloc(events((h0 | l) | false), rs(l), p(l)), ∀ l0, e0, bv0, p0, rs0. h0[firstVolWrite(l0, h0)] ≠ (e0, bv0) ∨ p0(l0)[firstPersistW(p0(l0))].val = e0.v ∨ p0(l0) = [] ∨ ¬ firstPersistW(p0(l0)) < # p0(l0) ∨ ¬ absloc(events((h0 | l0) | false), rs0(l0), p0(l0)) ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 6 terms: l, e, bv, p, rs

Additional information

Used terms: l, e, bv, p, rs

Proof step 31 from proof abs-firstPersistW-is-firstVolWrite (previous step) (back to tree)

((e0, true) ' + h0)[firstVolWrite(e0.l, (e0, true) ' + h0)] = (e, bv), e0 = W(t, l, v), p(e0.l)[firstPersistW(p(e0.l))].val ≠ e.v, p(e0.l) ≠ [], firstPersistW(p(e0.l)) < # p(e0.l), absloc(events((h0 | e0.l) | false), rs(e0.l), p(e0.l)), ∀ l0, e0, bv0, p0, rs0. h0[firstVolWrite(l0, h0)] ≠ (e0, bv0) ∨ p0(l0)[firstPersistW(p0(l0))].val = e0.v ∨ p0(l0) = [] ∨ ¬ firstPersistW(p0(l0)) < # p0(l0) ∨ ¬ absloc(events((h0 | l0) | false), rs0(l0), p0(l0)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ # x = 0 ↔ x = []; (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic ⊦ W(t, l, v).l = l; (s) a: ⊦ (x + y) + z = x + y + z; from specification list c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ (a ' + x)[n + 1] = x[n]; (s) ⊦ firstVolWrite(l, (W(t, l0, v), true) ' + h) = firstVolWrite(l, h) + 1; (s)

Proof step 32 from proof abs-firstPersistW-is-firstVolWrite (previous step) (back to tree)

h0[firstVolWrite(l, h0)] = (e, bv), p(l)[firstPersistW(p(l))].val ≠ e.v, p(l) ≠ [], firstPersistW(p(l)) < # p(l), absloc(events((h0 | l) | false), rs(l), p(l)), ∀ l0, e0, bv0, p0, rs0. h0[firstVolWrite(l0, h0)] ≠ (e0, bv0) ∨ p0(l0)[firstPersistW(p0(l0))].val = e0.v ∨ p0(l0) = [] ∨ ¬ firstPersistW(p0(l0)) < # p0(l0) ∨ ¬ absloc(events((h0 | l0) | false), rs0(l0), p0(l0)) ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 6 terms: l, e, bv, p, rs

Additional information

Used terms: l, e, bv, p, rs

Proof step 33 from proof abs-firstPersistW-is-firstVolWrite (previous step) (back to tree)

((e0, bv0) ' + h0)[firstVolWrite(l, (e0, bv0) ' + h0)] = (e, bv), e0.l ≠ l, p(l)[firstPersistW(p(l))].val ≠ e.v, p(l) ≠ [], firstPersistW(p(l)) < # p(l), absloc(events([] | false) + events((h0 | l) | false), rs(l), p(l)), ∀ l0, e0, bv0, p0, rs0. h0[firstVolWrite(l0, h0)] ≠ (e0, bv0) ∨ p0(l0)[firstPersistW(p0(l0))].val = e0.v ∨ p0(l0) = [] ∨ ¬ firstPersistW(p0(l0)) < # p0(l0) ∨ ¬ absloc(events((h0 | l0) | false), rs0(l0), p0(l0)) ⊦

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) ⊦ events([]) = []; (s) ⊦ [] | b = []; (s)

Proof step 34 from proof abs-firstPersistW-is-firstVolWrite (previous step) (back to tree)

((e0, bv0) ' + h0)[firstVolWrite(l, (e0, bv0) ' + h0)] = (e, bv), e0.l ≠ l, p(l)[firstPersistW(p(l))].val ≠ e.v, p(l) ≠ [], firstPersistW(p(l)) < # p(l), absloc(events((h0 | l) | false), rs(l), p(l)), ∀ l0, e0, bv0, p0, rs0. h0[firstVolWrite(l0, h0)] ≠ (e0, bv0) ∨ p0(l0)[firstPersistW(p0(l0))].val = e0.v ∨ p0(l0) = [] ∨ ¬ firstPersistW(p0(l0)) < # p0(l0) ∨ ¬ absloc(events((h0 | l0) | false), rs0(l0), p0(l0)) ⊦

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: e0

Additional information

Used simplifier rules: RWevent freely generated by R, W;;

Proof step 35 from proof abs-firstPersistW-is-firstVolWrite (previous step) (back to tree)

((e0, bv0) ' + h0)[firstVolWrite(l, (e0, bv0) ' + h0)] = (e, bv), e0 = R(t, l1, v), e0.l ≠ l, p(l)[firstPersistW(p(l))].val ≠ e.v, p(l) ≠ [], firstPersistW(p(l)) < # p(l), absloc(events((h0 | l) | false), rs(l), p(l)), ∀ l0, e0, bv0, p0, rs0. h0[firstVolWrite(l0, h0)] ≠ (e0, bv0) ∨ p0(l0)[firstPersistW(p0(l0))].val = e0.v ∨ p0(l0) = [] ∨ ¬ firstPersistW(p0(l0)) < # p0(l0) ∨ ¬ absloc(events((h0 | l0) | false), rs0(l0), p0(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 + y) + z = x + y + z; from specification list c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ (a ' + x)[n + 1] = x[n]; (s) ⊦ firstVolWrite(l, (R(t, l0, v), b) ' + h) = firstVolWrite(l, h) + 1; (s)

Proof step 36 from proof abs-firstPersistW-is-firstVolWrite (previous step) (back to tree)

h0[firstVolWrite(l, h0)] = (e, bv), p(l)[firstPersistW(p(l))].val ≠ e.v, l1 ≠ l, p(l) ≠ [], firstPersistW(p(l)) < # p(l), absloc(events((h0 | l) | false), rs(l), p(l)), ∀ l0, e0, bv0, p0, rs0. h0[firstVolWrite(l0, h0)] ≠ (e0, bv0) ∨ p0(l0)[firstPersistW(p0(l0))].val = e0.v ∨ p0(l0) = [] ∨ ¬ firstPersistW(p0(l0)) < # p0(l0) ∨ ¬ absloc(events((h0 | l0) | false), rs0(l0), p0(l0)) ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 7 terms: l, e, bv, p, rs

Additional information

Used terms: l, e, bv, p, rs

Proof step 37 from proof abs-firstPersistW-is-firstVolWrite (previous step) (back to tree)

((e0, bv0) ' + h0)[firstVolWrite(l, (e0, bv0) ' + h0)] = (e, bv), e0 = W(t, l1, v), e0.l ≠ l, p(l)[firstPersistW(p(l))].val ≠ e.v, p(l) ≠ [], firstPersistW(p(l)) < # p(l), absloc(events((h0 | l) | false), rs(l), p(l)), ∀ l0, e0, bv0, p0, rs0. h0[firstVolWrite(l0, h0)] ≠ (e0, bv0) ∨ p0(l0)[firstPersistW(p0(l0))].val = e0.v ∨ p0(l0) = [] ∨ ¬ firstPersistW(p0(l0)) < # p0(l0) ∨ ¬ absloc(events((h0 | l0) | false), rs0(l0), p0(l0)) ⊦

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 c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ (a ' + x)[n + 1] = x[n]; (s) ⊦ l0 ≠ l → firstVolWrite(l, (W(t, l0, v), b) ' + h) = firstVolWrite(l, h) + 1; (s) ⊦ W(t, l, v).l = l; (s)

Proof step 38 from proof abs-firstPersistW-is-firstVolWrite (previous step) (back to tree)

h0[firstVolWrite(l, h0)] = (e, bv), p(l)[firstPersistW(p(l))].val ≠ e.v, l1 ≠ l, p(l) ≠ [], firstPersistW(p(l)) < # p(l), absloc(events((h0 | l) | false), rs(l), p(l)), ∀ l0, e0, bv0, p0, rs0. h0[firstVolWrite(l0, h0)] ≠ (e0, bv0) ∨ p0(l0)[firstPersistW(p0(l0))].val = e0.v ∨ p0(l0) = [] ∨ ¬ firstPersistW(p0(l0)) < # p0(l0) ∨ ¬ absloc(events((h0 | l0) | false), rs0(l0), p0(l0)) ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 7 terms: l, e, bv, p, rs

Additional information

Used terms: l, e, bv, p, rs

Proof Tree for abs-firstPersistW-is-firstVolWrite (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