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Proof step 31 from proof inv-ls-succ (previous step)

# nσ = ∞, stutf(nσ), ∀ n. nσ[n] ≠ 1, ∀ n0, nσ. n0 < 0 → # nσ = ∞ ∧ stutf(nσ) ∧ (∀ n1. nσ[n1] ≠ n0 + 1) ∧ (∀ m. ∃ m0. m < m0 ∧ nσ[m] < nσ[m0]) → false ⊦ nσ[m] = 0, m = 0

User applied rule insert lemma

And obtained the following new goals

Proof step 32:
⊦ stutf(nσ) ∧ ⌜m⌝ < # nσ ∧ n ≤ nσ[m] → (∃ m0. m0 ≤ m ∧ nσ[m0] = n)
Proof step 33:
# nσ = ∞, stutf(nσ), stutf(nσ) ∧ ⌜m⌝ < # nσ ∧ 1 ≤ nσ[m] → (∃ m0. m0 ≤ m ∧ nσ[m0] = 1), ∀ n. nσ[n] ≠ 1, ∀ n0, nσ. n0 < 0 → # nσ = ∞ ∧ stutf(nσ) ∧ (∀ n1. nσ[n1] ≠ n0 + 1) ∧ (∀ m. ∃ m0. m < m0 ∧ nσ[m] < nσ[m0]) → false ⊦ nσ[m] = 0, m = 0

Rule argument

spec: <toplevel> name: stutf-alltaken seq: ⊦ stutf(nσ) ∧ ⌜m⌝ < # nσ ∧ n ≤ nσ[m] → (∃ m0. m0 ≤ m ∧ nσ[m0] = n) subst: [nσ, m, n] → [nσ, m, 1]

Additional information

Used terms: nσ, m, 1