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Proof step 6 from proof inv-get-inverse-between (previous step)

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), ⌜n⌝ + ⌜1⌝ < leninv(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, (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] = 0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 7:
# 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

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]));%less-inf2: ⊦ ⌜m⌝ < ∞; from specification nat-inf%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic1%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic2%sel-inf: ⊦ ⌜nnf⌝ [n] = nnf(n); from specification stream[natstream]