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Proof step 1 from proof absloc-persistFO (conclusion) (back to tree)

⊦ (el = [] ∨ ¬ el.head.R?) ∧ absloc(el, rset, fowl) ∧ isPersistFO(n, fowl) → absloc(el, rset, fowl -p n)

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

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

absloc(el, rset, fowl), isPersistFO(n, fowl), ¬ absloc(el, rset, fowl -p n), el = [] ∨ ¬ el.head.R? ⊦

User applied rule structural induction

And obtained the following new goals

Rule argument

term: el

Additional information

Used simplifier rules: absloc-base: ⊦ absloc([], rset, fowl) ↔ fowl |W = []; List('a) freely generated by [], +;; ⊦ isPersistFO(n, fowl) → (fowl -p n)|W = fowl |W; (s)

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

absloc(e + el0, rset, fowl), isPersistFO(n, fowl), ¬ absloc(e + el0, rset, fowl -p n), ∀ rset, fowl, n. ¬(absloc(el0, rset, fowl) ∧ isPersistFO(n, fowl) ∧ ¬ absloc(el0, rset, fowl -p n) ∧ (el0 = [] ∨ ¬ el0.head.R?)), e + el0 = [] ∨ ¬ (e + el0).head.R? ⊦

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 ' + x).head = a; (s) ⊦ a + x = a ' + x; (s) ⊦ a ' ≠ []; (s) ⊦ x + y = [] ↔ x = [] ∧ y = []; (s)

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

absloc(e ' + el0, rset, fowl), isPersistFO(n, fowl), ¬ e.R?, ¬ absloc(e ' + el0, rset, fowl -p n), ∀ rset0, fowl0, n0. ¬ absloc(el0, rset0, fowl0) ∨ ¬ isPersistFO(n0, fowl0) ∨ absloc(el0, rset0, fowl0 -p n0) ∨ el0 ≠ [] ∧ el0.head.R? ⊦

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: e

Additional information

Used simplifier rules: absloc-recR: ⊦ absloc(R(t, l, v) ' + el, rset, fowl) ↔ (¬ t ∈ rset → FO(t) ∈ fowl) ∧ absloc(el, rset, fowl); RWevent freely generated by R, W;; a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ R(t, l, v).R?; (s)

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

e = W(t, l, v), absloc(e ' + el0, rset, fowl), isPersistFO(n, fowl), ¬ e.R?, ¬ absloc(e ' + el0, rset, fowl -p n), ∀ rset0, fowl0, n0. ¬ absloc(el0, rset0, fowl0) ∨ ¬ isPersistFO(n0, fowl0) ∨ absloc(el0, rset0, fowl0 -p n0) ∨ el0 ≠ [] ∧ el0.head.R? ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

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

absloc(W(t, l, v) ' + el0, rset, fowl), isPersistFO(n, fowl), ¬ absloc(W(t, l, v) ' + el0, rset, fowl -p n), ∀ rset0, fowl0, n0. ¬ absloc(el0, rset0, fowl0) ∨ ¬ isPersistFO(n0, fowl0) ∨ absloc(el0, rset0, fowl0 -p n0) ∨ el0 ≠ [] ∧ el0.head.R? ⊦

User applied rule structural induction

And obtained the following new goals

Rule argument

term: fowl

Additional information

Used simplifier rules: absloc-recW-empty: ⊦ ¬ absloc(W(t, l, v) ' + el, rset, []); List('a) freely generated by [], +;; a: ⊦ (x + y) + z = x + y + z; from specification list

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

absloc(W(t, l, v) ' + el0, rset, fow + fowl0), isPersistFO(n, fow + fowl0), ¬ absloc(W(t, l, v) ' + el0, rset, (fow + fowl0) -p n), ∀ rset0, fowl0, n0. ¬ absloc(el0, rset0, fowl0) ∨ ¬ isPersistFO(n0, fowl0) ∨ absloc(el0, rset0, fowl0 -p n0) ∨ el0 ≠ [] ∧ el0.head.R?, ∀ t, l, v, el0, rset, n. ¬ ( absloc(W(t, l, v) ' + el0, rset, fowl0) ∧ isPersistFO(n, fowl0) ∧ ¬ absloc(W(t, l, v) ' + el0, rset, fowl0 -p n) ∧ (∀ rset0, fowl0, n0. ¬ absloc(el0, rset0, fowl0) ∨ ¬ isPersistFO(n0, fowl0) ∨ absloc(el0, rset0, fowl0 -p n0) ∨ el0 ≠ [] ∧ el0.head.R?)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ a + x = a ' + x; (s)

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

absloc(W(t, l, v) ' + el0, rset, fow ' + fowl0), isPersistFO(n, fow ' + fowl0), ¬ absloc(W(t, l, v) ' + el0, rset, (fow ' + fowl0) -p n), ∀ rset0, fowl0, n0. ¬ absloc(el0, rset0, fowl0) ∨ ¬ isPersistFO(n0, fowl0) ∨ absloc(el0, rset0, fowl0 -p n0) ∨ el0 ≠ [] ∧ el0.head.R?, ∀ t0, l0, v0, el, rset0, n0. ¬ absloc(W(t0, l0, v0) ' + el, rset0, fowl0) ∨ ¬ isPersistFO(n0, fowl0) ∨ absloc(W(t0, l0, v0) ' + el, rset0, fowl0 -p n0) ∨ ¬(∀ rset0, fowl0, n0. ¬ absloc(el, rset0, fowl0) ∨ ¬ isPersistFO(n0, fowl0) ∨ absloc(el, rset0, fowl0 -p n0) ∨ el ≠ [] ∧ el.head.R?) ⊦

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: fow

Additional information

Used simplifier rules: FOW freely generated by FO, W;; ⊦ ¬ isPersistFO(n, W(v, t) ' + fowl); (s) a: ⊦ (x + y) + z = x + y + z; from specification list

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

fow = FO(t1), absloc(W(t, l, v) ' + el0, rset, fow ' + fowl0), isPersistFO(n, fow ' + fowl0), ¬ absloc(W(t, l, v) ' + el0, rset, (fow ' + fowl0) -p n), ∀ t0, l0, v0, el, rset0, n0. ¬ absloc(W(t0, l0, v0) ' + el, rset0, fowl0) ∨ ¬ isPersistFO(n0, fowl0) ∨ absloc(W(t0, l0, v0) ' + el, rset0, fowl0 -p n0) ∨ ¬(∀ rset0, fowl0, n0. ¬ absloc(el, rset0, fowl0) ∨ ¬ isPersistFO(n0, fowl0) ∨ absloc(el, rset0, fowl0 -p n0) ∨ el ≠ [] ∧ el.head.R?), ∀ rset0, fowl0, n0. ¬ absloc(el0, rset0, fowl0) ∨ ¬ isPersistFO(n0, fowl0) ∨ absloc(el0, rset0, fowl0 -p n0) ∨ el0 ≠ [] ∧ el0.head.R? ⊦

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

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

absloc(W(t, l, v) ' + el0, rset, fowl0), isPersistFO(n, FO(t1) ' + fowl0), ¬ absloc(W(t, l, v) ' + el0, rset, (FO(t1) ' + fowl0) -p n), ∀ rset0, fowl0, n0. ¬ absloc(el0, rset0, fowl0) ∨ ¬ isPersistFO(n0, fowl0) ∨ absloc(el0, rset0, fowl0 -p n0) ∨ el0 ≠ [] ∧ el0.head.R?, ∀ t0, l0, v0, el, rset0, n0. ¬ absloc(W(t0, l0, v0) ' + el, rset0, fowl0) ∨ ¬ isPersistFO(n0, fowl0) ∨ absloc(W(t0, l0, v0) ' + el, rset0, fowl0 -p n0) ∨ ¬(∀ rset0, fowl0, n0. ¬ absloc(el, rset0, fowl0) ∨ ¬ isPersistFO(n0, fowl0) ∨ absloc(el, rset0, fowl0 -p n0) ∨ el ≠ [] ∧ el.head.R?) ⊦

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: n

Additional information

Used simplifier rules: Nat freely generated by 0, +1;; ⊦ (a ' + x) -p 0 = x; (s) a: ⊦ (x + y) + z = x + y + z; from specification list

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

n = n1 +1, absloc(W(t, l, v) ' + el0, rset, fowl0), isPersistFO(n, FO(t1) ' + fowl0), ¬ absloc(W(t, l, v) ' + el0, rset, (FO(t1) ' + fowl0) -p n), ∀ t0, l0, v0, el, rset0, n0. ¬ absloc(W(t0, l0, v0) ' + el, rset0, fowl0) ∨ ¬ isPersistFO(n0, fowl0) ∨ absloc(W(t0, l0, v0) ' + el, rset0, fowl0 -p n0) ∨ ¬(∀ rset0, fowl0, n0. ¬ absloc(el, rset0, fowl0) ∨ ¬ isPersistFO(n0, fowl0) ∨ absloc(el, rset0, fowl0 -p n0) ∨ el ≠ [] ∧ el.head.R?), ∀ rset0, fowl0, n0. ¬ absloc(el0, rset0, fowl0) ∨ ¬ isPersistFO(n0, fowl0) ∨ absloc(el0, rset0, fowl0 -p n0) ∨ el0 ≠ [] ∧ el0.head.R? ⊦

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); 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 = a ' + x; (s) ⊦ (a ' + x) -p (n + 1) = a + (x -p n); (s) ⊦ m +1 = m + 1; (s) ⊦ FO(tid).tid = tid; (s) ⊦ FO(tid).FO?; (s) ⊦ isPersistFO(n + 1, fow ' + fowl) ↔ isPersistFO(n, fowl) ∧ fow.FO? ∧ fowl[n].tid ≠ fow.tid; (s)

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

fowl0[n1].tid ≠ t1, absloc(W(t, l, v) ' + el0, rset, fowl0), isPersistFO(n1, fowl0), ¬ absloc(W(t, l, v) ' + el0, rset, fowl0 -p n1), ∀ rset0, fowl0, n0. ¬ absloc(el0, rset0, fowl0) ∨ ¬ isPersistFO(n0, fowl0) ∨ absloc(el0, rset0, fowl0 -p n0) ∨ el0 ≠ [] ∧ el0.head.R?, ∀ t0, l0, v0, el, rset0, n0. ¬ absloc(W(t0, l0, v0) ' + el, rset0, fowl0) ∨ ¬ isPersistFO(n0, fowl0) ∨ absloc(W(t0, l0, v0) ' + el, rset0, fowl0 -p n0) ∨ ¬(∀ rset0, fowl0, n0. ¬ absloc(el, rset0, fowl0) ∨ ¬ isPersistFO(n0, fowl0) ∨ absloc(el, rset0, fowl0 -p n0) ∨ el ≠ [] ∧ el.head.R?) ⊦

User applied rule all left

And closed the goal!

Rule argument

pos: left 6 terms: t, l, v, el0, rset, n1 abort: F discard: T computed: T

Additional information

Used terms: t, l, v, el0, rset, n1

Proof Tree for absloc-persistFO (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