This is a listing of all goals defined over the current specification. Axioms are yellow, proof obligations purple and theorems green. Click on individual names to get more information on the proof.
| Name: | Sequent: | Type: |
|---|---|---|
| infinite-append-prefixnextli | ∀ x. x ⊑ nextli(x) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ¬ # σ < ⌜ii(n + 1)⌝ ∧ σ to ii(n + 1) = ⌜nextli((σ to ii(n)).l)⌝ ∧ (# σ = ⌜ii(n)⌝ ⊃ ii(n) = ii(n + 1);ii(n) < ii(n + 1))) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ ⌜nextli(σ.l)⌝ = σ ∧ (∀ m. (n ≤ m ⊃ ii(m) = ii(n);ii(m) < ii(m + 1))) ;∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1) ) | A User Theorem |
| infinite-append-prefixnextli-weak | ∀ x. x ⊑ nextli(x) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n) ≤ ii(n + 1) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ σ to ii(n + 1) = ⌜nextli((σ to ii(n)).l)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ ⌜nextli(σ.l)⌝ = σ ∧ (∀ m. n ≤ m → ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)) | A User Theorem |
| infinite-append-prefixnextnli | ∀ x, n. x ⊑ nextnli(x, n) ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n) ≤ ii(n + 1) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ σ to ii(n + 1) = ⌜nextnli((σ to ii(n)).l, n)⌝ ) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ ⌜nextnli(σ.l, n)⌝ = σ ∧ (∀ m. n ≤ m → ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)) | A User Theorem |
| nextli-lemma | ⊦ ∃ lf. lf(0) = [] ∧ (∀ n. lf(n + 1) = nextli(lf(n))) | A User Theorem |
| nextnli-lemma | ⊦ ∃ lf. lf(0) = x ∧ (∀ n. lf(n + 1) = nextnli(lf(n), n)) | A User Theorem |