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Proof step 99 from proof lim-exists (previous step)

ii(0) = 0, # σσ = ∞, mon(σσ), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. σσ[n + 1] = σσ[n] + ⌜lf(n)⌝ , ∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ (nf from ii(n)) to # lf(n) = lf(n) ⊦ σσ[n] = σσ[0] + ⌜nf to ii(n)⌝ , σσ = [], ∃ n. # σσ[n] = ∞

User applied rule structural induction

And obtained the following new goals

Proof step 100:
∀ ii, σσ, lf, nf. ii(0) = 0 ∧ # σσ = ∞ ∧ mon(σσ) ∧ (∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (∀ n. σσ[n + 1] = σσ[n] + ⌜lf(n)⌝ ) ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ (nf from ii(n)) to # lf(n) = lf(n)) → σσ[n0] = σσ[0] + ⌜nf to ii(n0)⌝ ∨ σσ = [] ∨ (∃ n. # σσ[n] = ∞), ii(0) = 0, # σσ = ∞, mon(σσ), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. σσ[n + 1] = σσ[n] + ⌜lf(n)⌝ , ∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ (nf from ii(n)) to # lf(n) = lf(n) ⊦ σσ[n0 +1] = σσ[0] + ⌜nf to ii(n0 +1)⌝ , σσ = [], ∃ n. # σσ[n] = ∞

Rule argument

indvar: n

Additional information

Used simplifier rules: nat freely generated by 0, +1;;%appendend-equal: ⊦ σ + σ0 = σ ↔ ¬ ¬(σ0 = [] ∨ # σ = ∞); from specification stream%fin-empty: ⊦ ⌜x⌝ = [] ↔ x = []; from specification stream%to-empty-inf: ⊦ nf to n = [] ↔ n = 0; from specification stream%append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream