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Proof step 141 from proof infinite-append-butlast (previous step)

((σ from m0 + 1) to # y) + ⌜(σ from m0 + 1)[# y] '⌝ = ⌜y⌝ + ⌜a '⌝ , ii(n1 + 1) = m0 + 1, ii(0) = 0, ii(2 + n1) = 2 + m0 + # y, lf(2 + n1) = a ' + x1, ⌜# y⌝ + ⌜m0⌝ + ⌜1⌝ < # σ, #(σ from m0 + 1) < ⌜# y⌝ , (finp(σ) ⊃ # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → (m = 0 ⊃ lf(0);lf(m).tail) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n. n ≠ 0 → ii(n) ≠ 0, ∀ m, n. m ≤ n → ii(m) ≤ ii(n) ⊦ #((σ from m0 + 1) to # y) = ⌜# y⌝ , σ[m0 + # y + 1] = a, # σ < ⌜2⌝ + ⌜m0⌝ + ⌜# y⌝

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: stream name: fin-size-from seq: ⊦ ⌜n⌝ < # σ ∧ finp(σ) → #(σ from n) = ⌜(# σ).n - n⌝ subst: [n, σ] → [m0 + 1, σ] flags: <all paths>

Additional information

Used simplifier rules: ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; (s)%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic%stream-fin-forward: ⊦ finp(σ) → (∃ x. ⌜x⌝ = σ); (forward) from specification stream-basic% ⊦ # x = 0 ↔ x = []; (s)% ⊦ ⌜m⌝ .n = m; (s)% ⊦ m + 1 ≤ n ↔ m < n; (s)% ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ m ≤ n → (⌜n⌝ + n∞0 < ⌜m⌝ ↔ false); (s)%add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf%add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf%n0 ≤ n ⊦ n - n0 < m ↔ n < m + n0; (s)% ⊦ m < n + 1 ↔ ¬ n < m; (s)% ⊦ ⌜n1⌝ + ⌜n⌝ + ⌜m⌝ < ⌜n0⌝ ↔ n1 + n + m < n0; (s)% ⊦ ¬ m < n → ¬ m + m0 < n; (s)%lsle-trans: ⊦ n∞ < n∞1 ∧ ¬ n∞2 < n∞1 → n∞ < n∞2; (forward) from specification nat-inf%lele-trans: ⊦ ¬ n∞1 < n∞ ∧ ¬ n∞2 < n∞1 → ¬ n∞2 < n∞; (forward) from specification nat-inf% ⊦ ⌜n⌝ + n∞1 < ⌜n⌝ + n∞2 ↔ n∞1 < n∞2; (s)% ⊦ m ≤ n → (⌜(* m)⌝ + n∞2 < ⌜(* n)⌝ ↔ n∞2 < ⌜n - m⌝ ); (s)% ⊦ ¬ m < 0; (s)% ⊦ ¬ finp(σ) → # σ = ∞; (s)% ⊦ n∞ < ∞ ↔ n∞ ≠ ∞; (s)% ⊦ ⌜n⌝ + n∞1 = ∞ ↔ n∞1 = ∞; (s)% ⊦ ⌜m⌝ ≠ ∞; (s)% ⊦ ¬ ∞ < n∞; (s)%stream-inf-forward: ⊦ ¬ finp(σ) → (∃ nf. ⌜nf⌝ = σ); (forward) from specification stream-basic% ⊦ ⌜nf⌝ from n = ⌜nf from n⌝ ; (s)% ⊦ # ⌜nf⌝ = ∞; (s)%¬ # σ < ⌜n⌝ ⊦ #(σ to n) = ⌜n⌝ ; (s)% ⊦ ⌜m⌝ = ⌜m0⌝ ↔ m = m0; (s) Used terms: m0 + 1, σ