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

⊦ ¬ i∞ < i∞0 → (i∞0 < i∞ ↔ i∞ ≠ i∞0)

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

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

¬(i∞0 < i∞ ↔ i∞ ≠ i∞0), ¬ i∞ < i∞0 ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

i∞ ≠ i∞0, ¬ i∞ < i∞0, ¬ i∞0 < i∞ ⊦

User applied rule structural induction

And obtained the following new goals

Rule argument

indvar: i∞

Additional information

Used simplifier rules: neginf-less: ⊦ -∞ < i∞ ↔ i∞ ≠ -∞; not-inf-less: ⊦ ¬ ∞ < i∞; less-inf: ⊦ i∞ < ∞ ↔ i∞ ≠ ∞; intinf freely generated by -∞, ∞, ⌜ ⌝;;

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

⌜i⌝ ≠ i∞0, ¬ ⌜i⌝ < i∞0, ¬ i∞0 < ⌜i⌝ ⊦

User applied rule structural induction

And closed the goal!

Rule argument

indvar: i∞0

Additional information

Used simplifier rules: neginf-less: ⊦ -∞ < i∞ ↔ i∞ ≠ -∞; less-inf: ⊦ i∞ < ∞ ↔ i∞ ≠ ∞; less-int: ⊦ ⌜i⌝ < ⌜i0⌝ ↔ i < i0; intinf freely generated by -∞, ∞, ⌜ ⌝;; ⊦ ⌜i⌝ = ⌜i0⌝ ↔ i = i0; (s) ⊦ ¬ i < j → (i = j ↔ ¬ j < i); (ss) ⊦ ¬ i < j → (j < i ↔ i ≠ j); (ss) ⊦ ⌜i⌝ ≠ ∞; (s) ⊦ ⌜i⌝ ≠ -∞; (s)

Proof Tree for antisym (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