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Proof step 126 from proof infinite-append-butlast (previous step)

ii(2 + n1) = (m0 + 1) + # x0, ii(0) = 0, ii(n1 + 1) = m0 + 1, lf(2 + n1) = (σ[m0] ' + x0).last ' + x1, (σ from m0 + 1) to # x0 = ⌜x0⌝ , (σ[m0] ' + x0).last ∈ lf(2 + n1), σ[m0] ∈ σ[m0] ' + x0, (finp(σ) ⊃ # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → (m = 0 ⊃ lf(0);lf(m).tail) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. n ≠ 0 → ii(n) ≠ 0 ⊦ σ[m0] ' + x0 = [], σ[m0 + # x0] = (σ[m0] ' + x0).last, # σ < ⌜m0⌝ + ⌜1⌝ + ⌜# x0⌝

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 127:
ii(n1 + 1) = m0 + 1, ii(0) = 0, ii(2 + n1) = (m0 + 1) + # x0, lf(2 + n1) = (σ[m0] ' + x0).last ' + x1, (σ from m0 + 1) to # x0 = ⌜x0⌝ , (σ[m0] ' + x0).last ∈ lf(2 + n1), (finp(σ) ⊃ # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → (m = 0 ⊃ lf(0);lf(m).tail) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n. n ≠ 0 → ii(n) ≠ 0, ∀ m, n. m ≤ n → ii(m) ≤ ii(n) ⊦ σ[m0 + # x0] = (σ[m0] ' + x0).last, # σ < ⌜m0⌝ + ⌜1⌝ + ⌜# x0⌝

Rule argument

Additional information

Used simplifier rules: in-14: ⊦ a ' + x = y → a ∈ y; (forward) from specification list% ⊦ a ∈ b ' ↔ a = b; (s)% ⊦ a ∈ x + y ↔ a ∈ x ∨ a ∈ y; (s)%a: ⊦ (x + y) + z = x + y + z; from specification list% ⊦ a ' + x ≠ []; (s)