[Imprint][Data Privacy Statement][E-Mail]

View long theorem info. View the proof. Back to main index. View the specification. View theorem base summary, View theorem base listing, the used properties by spec, the dependency list, the symbol table,

Proof step 38 from proof infinite-append-butlast (previous step)

(σ from ii(n0)) to # (n0 = 0 ⊃ lf(0);lf(n0).rest) = ⌜(n0 = 0 ⊃ lf(0);lf(n0).rest)⌝ , m0 + 1 = ii(n0) + # (n0 = 0 ⊃ lf(0);lf(n0).rest), ii(n0 + 1) = m0 + 1, ii(0) = 0, (n0 = 0 ⊃ 0;ii(n0) - 1) + # lf(n0) = m0 + 1, ii(m) ≤ ii(n1), (n1 = 0 ⊃ 0;ii(n1) - 1) < ii(m) - 1, m ≤ n1, ⌜m0⌝ < # σ, ∀ 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)), ∀ 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 = 0, 0 = ii(m) - 1

User applied rule case distinction

And obtained the following new goals

Proof step 39:
n1 ≠ 0, ii(n1) - 1 < ii(m) - 1, (σ from ii(n0)) to # (n0 = 0 ⊃ lf(0);lf(n0).rest) = ⌜(n0 = 0 ⊃ lf(0);lf(n0).rest)⌝ , m0 + 1 = ii(n0) + # (n0 = 0 ⊃ lf(0);lf(n0).rest), ii(n0 + 1) = m0 + 1, ii(0) = 0, (n0 = 0 ⊃ 0;ii(n0) - 1) + # lf(n0) = m0 + 1, ii(m) ≤ ii(n1), m ≤ n1, ⌜m0⌝ < # σ, ∀ 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)), ∀ 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 = 0, 0 = ii(m) - 1

Rule argument

pos: left 7

Additional information

Used simplifier rules: zero: ⊦ 0 < m ↔ m ≠ 0; from specification nat-basic1%lez-01: ⊦ m ≤ 0 ↔ m = 0; from specification nat