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Proof step 1 from proof app (conclusion) (back to tree)

⊦ disj(x, y + z) → disj(x, y)

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

Proof step 2 from proof app (previous step) (back to tree)

disj(x, y + z), ¬ disj(x, y) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 3 from proof app (previous step) (back to tree)

disj(x, y + z), ¬ disj(x, y) ⊦

Heuristic structural induction applied rule structural induction

And obtained the following new goals

Rule argument

term: y

Additional information

Used simplifier rules: e: ⊦ disj(x, []); list('a) freely generated by [], +;; ⊦ [] + x = x; (s) a: ⊦ (x + y) + z = x + y + z; from specification list

Proof step 4 from proof app (previous step) (back to tree)

disj(x, (a + y0) + z), ¬ disj(x, a + y0), ∀ x, z. ¬(disj(x, y0 + z) ∧ ¬ disj(x, y0)) ⊦

Heuristic simplifier applied rule simplifier

And closed the goal!

Rule argument

Additional information

Used simplifier rules: app-02: ⊦ disj(x, y + z) ↔ disj(x, y) ∧ disj(x, z); one-01: ⊦ disj(x, a ') ↔ ¬ a ∈ x; a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ a + x = a ' + x; (s)

Proof Tree for app (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