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Proof step 32 from proof lim-exists (previous step)

∀ m. n ≤ m → σσ[m] = σσ[n], # σσ[n] = ⌜n0⌝ , ii(0) = 0, # σσ = ∞, # x = ii(n), mon(σσ), ∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. σσ[n + 1] = σσ[n] + ⌜lf(n)⌝ , ∀ n0. ii(n0 + 1) = ii(n0) + # lf(n0) ∧ ¬ ii(n) < ii(n0 + 1) ∧ (x from ii(n0)) to # lf(n0) = lf(n0) ⊦ σσ = [], ∃ n. # σσ[n] = ∞, (∀ n0. ∃ σ1. σσ[n0] + σ1 = σσ[n]) ∧ (∀ m. ¬ # σσ[n] < ⌜m⌝ → (∃ n0, σ1. (σσ[n] to m) + σ1 = σσ[n0]))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 33:
# x = ii(n), # σσ = ∞, ii(0) = 0, # σσ[n] = ⌜n0⌝ , mon(σσ), ∀ n0. ii(n0 + 1) = ii(n0) + # lf(n0) ∧ ¬ ii(n) < ii(n0 + 1) ∧ (x from ii(n0)) to # lf(n0) = lf(n0), ∀ n. σσ[n + 1] = σσ[n] + ⌜lf(n)⌝ , ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n), ∀ m. n ≤ m → σσ[m] = σσ[n] ⊦ σσ = [], ∃ n. # σσ[n] = ∞, (∃ σ1. σσ[n1] + σ1 = σσ[n]) ∧ (¬ n0 < m → (∃ n0, σ1. (σσ[n] to m) + σ1 = σσ[n0]))

Rule argument

Additional information

Used simplifier rules: less-fin: ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; from specification nat-inf