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Proof step 44 from proof infinite-append-prefixnextli (previous step)

∀ 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)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 45:
lf(0) = [], ii(0) = 0, nextli0(lf(n0)) = [], ii(n0 + 1) = ii(n0), n0 ≤ m0, ∀ 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), (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)⌝ , # σ < ⌜ii(n0)⌝ , ii(m0 + 1) = ii(n0) ∧ lf(m0 + 1) = lf(n0)

Rule argument

Additional information

Used simplifier rules: lels: ⊦ ¬ m < n ↔ n ≤ m; from specification nat%s: ⊦ m < n + 1 ↔ ¬ n < m; from specification nat-basic2%c: ⊦ m + n = n + m; from specification nat-basic2%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic2%oz: ⊦ m + 1 ≠ 0; from specification nat-basic2%succ: ⊦ m +1 = m + 1; from specification nat-basic2