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Proof step 20 from proof infinite-append-butlast (previous step)

ii(0) = 0, (σ from ii(n0)) to # lf(n0).rest = ⌜lf(n0).rest⌝ , ii(n0 + 1) = ii(n0) + # lf(n0).rest, ii(n0 + 1) = ((ii(n0) - 1) + # lf(n0) - 1) + 1, ⌜(ii(n0) - 1) + # lf(n0) - 1⌝ < # 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)) ⊦ (σ from (ii(n0) - 1)) to # lf(n0) = ⌜lf(n0)⌝ , n0 = 0

User applied rule insert elim lemma

And obtained the following new goals

Proof step 21:
ii(0) = 0, (σ from ii(n0)) to # lf(n0).rest = ⌜lf(n0).rest⌝ , ii(n0 + 1) = ii(n0) + # lf(n0).rest, ii(n0 + 1) = ((ii(n0) - 1) + # lf(n0) - 1) + 1, ⌜(ii(n0) - 1) + # lf(n0) - 1⌝ < # 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)) ⊦ ii(n0) ≠ 0, (σ from (ii(n0) - 1)) to # lf(n0) = ⌜lf(n0)⌝ , n0 = 0
Proof step 23:
ii(n0) = m0 + 1 ∧ ii(n0) ≠ 0, ii(0) = 0, (σ from ii(n0)) to # lf(n0).rest = ⌜lf(n0).rest⌝ , ii(n0 + 1) = ii(n0) + # lf(n0).rest, ii(n0 + 1) = (m0 + # lf(n0) - 1) + 1, ⌜m0 + # lf(n0) - 1⌝ < # 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)) ⊦ (σ from m0) to # lf(n0) = ⌜lf(n0)⌝ , n0 = 0

Rule argument

spec: nat inst: name: elim-pred-c seq: n ≠ 0 ⊦ m = n - 1 ↔ n = m + 1 subst: [n, m] → [ii(n0), m0] flags:

Additional information

Used simplifier rules: n ≠ 0 ⊦ m = n - 1 ↔ n = m + 1; (elim) Used terms: ii(n0), m0