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Proof step 81 from proof infinite-append-butlast (previous step)

(σ from m2 + 1) to # x0 = ⌜x0⌝ , ii(0) = 0, ii(n0 + 1) = (m2 + # x0) + 1, m2 + # lf(n0) = m0 + 1, ii(n0) = m2 + 1, lf(n0) = a ' + x0, ⌜m2⌝ + ⌜# x0⌝ < # σ, a ∈ lf(n0), (finp(σ) ⊃ # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → (m = 0 ⊃ lf(0);lf(m).rest) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n. lf(n).last = lf(n + 1).first ∧ lf(n) ≠ [], ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # (n = 0 ⊃ lf(0);lf(n).rest) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # (n = 0 ⊃ lf(0);lf(n).rest) = ⌜(n = 0 ⊃ lf(0);lf(n).rest)⌝ , ∀ n. n ≠ 0 → ii(n) ≠ 0, ∀ n. σ[ii(n + 1) - 1] = lf(n + 1).first ⊦ m2 + # x0 = m0, lf(n0) = []

User applied rule insert equation

And closed the goal!

Rule argument

pos: left 6 flags: discard

Additional information

Used simplifier rules: len-05: ⊦ #(a ' + x) = (# x)+1; from specification list%succ: ⊦ m +1 = m + 1; from specification nat-basic2%a: ⊦ (x + y) + z = x + y + z; from specification list%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic2%eq-01: ⊦ m + n = m + n0 ↔ n = n0; from specification nat-basic2%c: ⊦ m + n = n + m; from specification nat-basic2