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

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 8:
# 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], ⌜n⌝ + ⌜1⌝ < leninv(nσ), stutf(nσ), ∀ m. ∃ m0. m < m0 ∧ nσ[m] < nσ[m0], ∀ m. ∃ m0. m < nσ[m0] ⊦ 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

Rule argument

Additional information

Used simplifier rules: leninv-inf-strong: ⊦ stutf(nσ) → ( leninv(nσ) = ∞ ↔ # nσ = ∞ ∧ (∀ m. ∃ m0. m < m0 ∧ nσ[m] < nσ[m0]) ∧ (∀ m. ∃ m0. m < nσ[m0]));% ⊦ m < n → n ≠ 0; (s)% ⊦ ⌜m⌝ < ∞; (s)%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ ⌜nnf⌝ [n] = nnf(n); (s)