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Proof step 1 from proof finremtau-empty-equiv (conclusion) (back to tree)

⊦ ext !(al) = [] ↔ (∀ n. n < # al → ext(al[n]) = None)

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:remtau-empty-equiv, seq: ⊦ ext !(α) = [] ↔ (∀ n. ⌜n⌝ < # α → ext(α[n]) = None), subst:[ext, α] → [ext, ⌜al⌝ ]; ['a, 'e] → ['a, 'e]

Additional information

Used simplifier rules: finremtau-def: ⊦ ext !(⌜al⌝ ) = ⌜ext !(al)⌝ ; ⊦ ⌜x⌝ = [] ↔ x = []; (s) ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s) ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; (s) ⊦ ⌜x⌝ [n] = x[n]; (s) Used terms: ext, ⌜al⌝

Proof step 2 from proof finremtau-empty-equiv (previous step) (back to tree)

⊦ ext !(α) = [] ↔ (∀ n. ⌜n⌝ < # α → ext(α[n]) = None)

Lemma remtau-empty-equiv

Proof Tree for finremtau-empty-equiv (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