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Proof step 43 from proof infinite-append-prefixnextli (previous step)

lf(0) = [], ii(0) = 0, nextli0(lf(n0)) = [], ii(n0 + 1) = ii(n0), n0 < m, (finp(σ) ⊃ # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → nextli0(lf(m)) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ x. nextli(x) = x + nextli0(x), ∀ n. lf(n + 1) = nextli(lf(n)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # nextli0(lf(n)) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # nextli0(lf(n)) = ⌜nextli0(lf(n))⌝ , ∀ n0. lf(n0) = (σ to ii(n0)).l ⊦ # σ = ⌜ii(n0)⌝ , m = 0, # σ < ⌜ii(n0)⌝ , ii(m) = ii(n0) ∧ lf(m) = lf(n0)

User applied rule structural induction

And obtained the following new goals

Proof step 44:
∀ lf, ii, nextli0, n0, σ, n, nextli. lf(0) = [] ∧ ii(0) = 0 ∧ nextli0(lf(n0)) = [] ∧ ii(n0 + 1) = ii(n0) ∧ n0 < m0 ∧ (finp(σ) ⊃ # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → nextli0(lf(m)) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)) ∧ (∀ x. nextli(x) = x + nextli0(x)) ∧ (∀ n. lf(n + 1) = nextli(lf(n))) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (∀ n. ii(n + 1) = ii(n) + # nextli0(lf(n)) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # nextli0(lf(n)) = ⌜nextli0(lf(n))⌝ ) ∧ (∀ n0. lf(n0) = (σ to ii(n0)).l) → # σ = ⌜ii(n0)⌝ ∨ m0 = 0 ∨ # σ < ⌜ii(n0)⌝ ∨ ii(m0) = ii(n0) ∧ lf(m0) = lf(n0), lf(0) = [], ii(0) = 0, nextli0(lf(n0)) = [], ii(n0 + 1) = ii(n0), n0 < m0 +1, (finp(σ) ⊃ # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → nextli0(lf(m)) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ x. nextli(x) = x + nextli0(x), ∀ n. lf(n + 1) = nextli(lf(n)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # nextli0(lf(n)) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # nextli0(lf(n)) = ⌜nextli0(lf(n))⌝ , ∀ n0. lf(n0) = (σ to ii(n0)).l ⊦ # σ = ⌜ii(n0)⌝ , m0 +1 = 0, # σ < ⌜ii(n0)⌝ , ii(m0 +1) = ii(n0) ∧ lf(m0 +1) = lf(n0)

Rule argument

indvar: m

Additional information

Used simplifier rules: nat freely generated by 0, +1;;