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Proof step 1 from proof leninv-less (conclusion) (back to tree)

⊦ leninv(nσ) < n∞ ↔ (∃ n. leninv(nσ) = ⌜n⌝ ∧ ⌜n⌝ < n∞)

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: right 1

Additional information

Used simplifier rules: lsle-trans: ⊦ n∞ < n∞1 ∧ ¬ n∞2 < n∞1 → n∞ < n∞2; (forward) from specification nat-inf lels-trans: ⊦ ¬ n∞1 < n∞ ∧ n∞1 < n∞2 → n∞ < n∞2; (forward) from specification nat-inf

Proof step 2 from proof leninv-less (previous step) (back to tree)

leninv(nσ) < n∞ ⊦ ∃ n. leninv(nσ) = ⌜n⌝ ∧ ⌜n⌝ < n∞

User applied rule exists right

And obtained the following new goals

Rule argument

pos: right 1 terms: leninv(nσ).n abort: F discard: F computed: F

Additional information

Used terms: leninv(nσ).n

Proof step 3 from proof leninv-less (previous step) (back to tree)

leninv(nσ) < n∞ ⊦ ∃ n. leninv(nσ) = ⌜n⌝ ∧ ⌜n⌝ < n∞, leninv(nσ) = ⌜leninv(nσ).n⌝ ∧ ⌜leninv(nσ).n⌝ < n∞

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: nat-inf, name:fin-inject-embed, seq:n∞ ≠ ∞ ⊦ ⌜n∞.n⌝ = n∞, subst:[n∞] → [leninv(nσ)], <all paths>

Additional information

Used simplifier rules: ⊦ ¬ ∞ < n∞; (s) Used terms: leninv(nσ)

Proof step 4 from proof leninv-less (previous step) (back to tree)

n∞ ≠ ∞ ⊦ ⌜n∞.n⌝ = n∞

A spec-lemma

Proof Tree for leninv-less (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