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Proof step 7 from proof inv-get-inverse-between (previous step)

# nσ = ∞, ⌜n⌝ + ⌜1⌝ < leninv(nσ), stutf(nσ), max∞(λ n0. nσ[n0] = n).n + m + 1 < (leninv(nσ) = ∞ ⊃ ⌜λ m0. max∞(λ n0. nσ[n0] = m0).n⌝ ;⌜(λ m0. max∞(λ n0. nσ[n0] = m0).n) to leninv(nσ).n⌝ )[n + 1], ∀ m. ∃ m0. m < nσ[m0], ∀ m. ∃ m0. m < m0 ∧ nσ[m] < nσ[m0] ⊦ (leninv(nσ) = ∞ ⊃ ⌜λ m0. max∞(λ n0. nσ[n0] = m0).n⌝ ;⌜(λ m0. max∞(λ n0. nσ[n0] = m0).n) to leninv(nσ).n⌝ )[n + 1] = 0, nσ[(leninv(nσ) = ∞ ⊃ ⌜λ m0. max∞(λ n0. nσ[n0] = m0).n⌝ ;⌜(λ m0. max∞(λ n0. nσ[n0] = m0).n) to leninv(nσ).n⌝ )[n] + m + 1] = n + 1

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 8:
⊦ stutf(nσ) → (leninv(nσ) = ∞ ↔ # nσ = ∞ ∧ (∀ m. ∃ m0. m < m0 ∧ nσ[m] < nσ[m0]))
Proof step 9:
stutf(nσ), # nσ = ∞, ⌜n⌝ + ⌜1⌝ < leninv(nσ), stutf(nσ), max∞(λ n0. nσ[n0] = n).n + m + 1 < (# nσ = ∞ ∧ (∀ m. ∃ m0. m < m0 ∧ nσ[m] < nσ[m0]) ⊃ ⌜λ m0. max∞(λ n0. nσ[n0] = m0).n⌝ ;⌜(λ m0. max∞(λ n0. nσ[n0] = m0).n) to leninv(nσ).n⌝ ) [n + 1 ], ∀ m. ∃ m0. m < nσ[m0], ∀ m. ∃ m0. m < m0 ∧ nσ[m] < nσ[m0] ⊦ (# nσ = ∞ ∧ (∀ m. ∃ m0. m < m0 ∧ nσ[m] < nσ[m0]) ⊃ ⌜λ m0. max∞(λ n0. nσ[n0] = m0).n⌝ ;⌜(λ m0. max∞(λ n0. nσ[n0] = m0).n) to leninv(nσ).n⌝ ) [n + 1 ] = 0, nσ [ (# nσ = ∞ ∧ (∀ m. ∃ m0. m < m0 ∧ nσ[m] < nσ[m0]) ⊃ ⌜λ m0. max∞(λ n0. nσ[n0] = m0).n⌝ ;⌜(λ m0. max∞(λ n0. nσ[n0] = m0).n) to leninv(nσ).n⌝ ) [n ] + m + 1 ] = n + 1

Rule argument

spec: inst: name: leninv-inf seq: ⊦ stutf(nσ) → ( leninv(nσ) = ∞ ↔ # nσ = ∞ ∧ (∀ m. ∃ m0. m < m0 ∧ nσ[m] < nσ[m0])) subst: [nσ] → [nσ] flags: rewrite, current

Additional information

Used terms: nσ