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Proof step 14 from proof infinite-append-butlast (previous step)

ii(0) = 0, ii(n0 + 1) = m1 + 1, (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), ii(n0 + 1) = ii(n0) + # (n0 = 0 ⊃ lf(0);lf(n0).rest) ∧ ¬ # σ < ⌜ii(n0 + 1)⌝ ∧ (σ from ii(n0)) to # (n0 = 0 ⊃ lf(0);lf(n0).rest) = ⌜(n0 = 0 ⊃ lf(0);lf(n0).rest)⌝ , ∀ 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 ⊦ m1 = (n0 = 0 ⊃ 0;ii(n0) - 1) + # lf(n0) - 1 ∧ ⌜(n0 = 0 ⊃ 0;ii(n0) - 1) + # lf(n0) - 1⌝ < # σ ∧ (σ from (n0 = 0 ⊃ 0;ii(n0) - 1)) to # lf(n0) = ⌜lf(n0)⌝ ∧ (m ≤ n1 → (m = 0 ⊃ 0;ii(m) - 1) ≤ (n1 = 0 ⊃ 0;ii(n1) - 1)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜(n = 0 ⊃ 0;ii(n) - 1)⌝ + ⌜1⌝ ∧ (∀ m. n ≤ m → lf(m).rest = [] ∧ (m = 0 ⊃ 0;ii(m) - 1) = (n = 0 ⊃ 0;ii(n) - 1)) ;∃ m0. (m0 = 0 ⊃ 0;ii(m0) - 1) ≤ n2 ∧ n2 < ii(m0 + 1) - 1 )

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 15:
(σ from ii(n0)) to # (n0 = 0 ⊃ lf(0);lf(n0).rest) = ⌜(n0 = 0 ⊃ lf(0);lf(n0).rest)⌝ , m1 + 1 = ii(n0) + # (n0 = 0 ⊃ lf(0);lf(n0).rest), ii(n0 + 1) = m1 + 1, ii(0) = 0, ⌜m1⌝ < # σ, ∀ n. σ[ii(n + 1) - 1] = lf(n + 1).first, ∀ n. n ≠ 0 → ii(n) ≠ 0, ∀ 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)⌝ , ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. lf(n).last = lf(n + 1).first ∧ lf(n) ≠ [], (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)) ⊦ m1 = (n0 = 0 ⊃ 0;ii(n0) - 1) + # lf(n0) - 1 ∧ (σ from (n0 = 0 ⊃ 0;ii(n0) - 1)) to # lf(n0) = ⌜lf(n0)⌝ ∧ (m ≤ n1 → (m = 0 ⊃ 0;ii(m) - 1) ≤ (n1 = 0 ⊃ 0;ii(n1) - 1)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜(n = 0 ⊃ 0;ii(n) - 1)⌝ + ⌜1⌝ ∧ (∀ m. n ≤ m → lf(m).rest = [] ∧ (m = 0 ⊃ 0;ii(m) - 1) = (n = 0 ⊃ 0;ii(n) - 1)) ;∃ m0. (m0 = 0 ⊃ 0;ii(m0) - 1) ≤ n2 ∧ n2 < ii(m0 + 1) - 1 )

Rule argument

Additional information

Used simplifier rules: add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf%add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf%size-succ: ⊦ # σ < ⌜n⌝ + ⌜1⌝ ↔ ¬ ⌜n⌝ < # σ; from specification stream%c: ⊦ m + n = n + m; from specification nat-basic2%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic2%add-fin: ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; from specification nat-inf