Proof step 1 from proof w-in-propFOs (conclusion) (back to tree)

System applied rule normalize

And obtained the following new goals

• Proof step 2:

  W(v, t) ∈ propFOs(t0, l, bel) ⊦

Rule argument

Additional information

Proof step 2 from proof w-in-propFOs (previous step) (back to tree)

Heuristic structural induction applied rule structural induction

And obtained the following new goals

• Proof step 3:

  W(v, t) ∈ propFOs(t0, l, be + bel0), ∀ v, t, t0, l. ¬ W(v, t) ∈ propFOs(t0, l, bel0) ⊦

Rule argument

Additional information

Used simplifier rules: propFOs-base: ⊦ propFOs(t, l, []) = []; List('a) freely generated by [], +;; ⊦ ¬ a ∈ []; (s)

Proof step 3 from proof w-in-propFOs (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 4:

  ∀ v, t, t0, l. ¬ W(v, t) ∈ propFOs(t0, l, bel0), W(v, t) ∈ propFOs(t0, l, be ') ∨ W(v, t) ∈ propFOs(t0, l, bel0) ⊦

Rule argument

Additional information

Used simplifier rules: propFOs-append: ⊦ propFOs(t, l, bel0 + bel1) = propFOs(t, l, bel0) + propFOs(t, l, bel1); a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ a ∈ x + y ↔ a ∈ x ∨ a ∈ y; (s) ⊦ a + x = a ' + x; (s)

Proof step 4 from proof w-in-propFOs (previous step) (back to tree)

Heuristic pl case distinction applied rule case distinction

And obtained the following new goals

• Proof step 5:

  W(v, t) ∈ propFOs(t0, l, be '), ∀ v, t, t0, l. ¬ W(v, t) ∈ propFOs(t0, l, bel0) ⊦

• Proof step 8:

  W(v, t) ∈ propFOs(t0, l, bel0), ∀ v, t, t0, l. ¬ W(v, t) ∈ propFOs(t0, l, bel0) ⊦

Rule argument

Additional information

Proof step 5 from proof w-in-propFOs (previous step) (back to tree)

User applied rule constructor cut

And obtained the following new goals

• Proof step 6:

  be = FO(l0), W(v, t) ∈ propFOs(t0, l, be '), ∀ v, t, t0, l. ¬ W(v, t) ∈ propFOs(t0, l, bel0) ⊦

Rule argument

Additional information

Used simplifier rules: propFOs-one-sf: ⊦ propFOs(t, l, SF ') = []; propFOs-one-w: ⊦ propFOs(t, l, W(l0, v) ') = []; Bevent freely generated by SF, W, FO;; ⊦ ¬ a ∈ []; (s)

Proof step 6 from proof w-in-propFOs (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 7:

  W(v, t) ∈ propFOs(t0, l, FO(l0) '), ∀ v, t, t0, l. ¬ W(v, t) ∈ propFOs(t0, l, bel0) ⊦

Rule argument

Additional information

Proof step 7 from proof w-in-propFOs (previous step) (back to tree)

Heuristic cut applied rule cut formula

And closed the goal!

Rule argument

Additional information

Used simplifier rules: propFOs-one-fo-same: ⊦ propFOs(t, l, FO(l) ') = FO(t) '; propFOs-one-fo-other: ⊦ l0 ≠ l → propFOs(t, l, FO(l0) ') = []; ⊦ ¬ a ∈ []; (s) ⊦ a ∈ a0 ' ↔ a = a0; (s) ⊦ FO(tid) ≠ W(v, tid0); (s) Used formulas: l0 = l

Proof step 8 from proof w-in-propFOs (previous step) (back to tree)

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

Additional information

Used terms: v, t, t0, l