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Proof step 138 from proof infinite-append-butlast (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

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

Additional information

Used simplifier rules: add-fin: ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; from specification nat-inf%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic2%c: ⊦ m + n = n + m; from specification nat-basic2%less-fin: ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; from specification nat-inf%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic1%stream-fin-forward: ⊦ finp(σ) → (∃ x. ⌜x⌝ = σ); (forward) from specification stream-basic%l: ⊦ # x = 0 ↔ x = []; from specification list-data%sel: ⊦ ⌜m⌝ .n = m; from specification nat-inf-basic%succ-01: ⊦ m + 1 ≤ n ↔ m < n; from specification nat%lels: ⊦ ¬ m < n ↔ n ≤ m; from specification nat%leq-notgradd: ⊦ m ≤ n → (⌜n⌝ + n∞0 < ⌜m⌝ ↔ false); from specification nat-inf%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%ls-01: n0 ≤ n ⊦ n - n0 < m ↔ n < m + n0; from specification nat%s: ⊦ m < n + 1 ↔ ¬ n < m; from specification nat-basic2%ls-natinf-to-nat-2-1: ⊦ ⌜n⌝ + ⌜m⌝ < ⌜n0⌝ ↔ n + m < n0; from specification nat-inf%ls-natinf-to-nat-3-1: ⊦ ⌜n1⌝ + ⌜n⌝ + ⌜m⌝ < ⌜n0⌝ ↔ n1 + n + m < n0; from specification nat-inf%nls-fst: ⊦ ¬ m < n → ¬ m + m0 < n; from specification nat-basic2%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%add-ls-cancel-num-04: ⊦ m ≤ n → (⌜(* m)⌝ + n∞2 < ⌜(* n)⌝ + n∞1 ↔ n∞2 < ⌜n - m⌝ + n∞1); from specification nat-inf%add-all-zero: ⊦ n∞ + ⌜0⌝ = n∞; from specification nat-inf%l-02: ⊦ ¬ m + n < m; from specification nat-basic2%notfin-sizeinf: ⊦ ¬ finp(σ) → # σ = ∞; from specification stream%inf-less: ⊦ n∞ < ∞ ↔ n∞ ≠ ∞; from specification nat-inf%addeqinf-cancel: ⊦ ⌜m⌝ + n∞ = ∞ ↔ n∞ = ∞; from specification nat-inf%disj: ⊦ ⌜m⌝ ≠ ∞; from specification nat-inf-basic%less-inf1: ⊦ ¬ ∞ < n∞; from specification nat-inf%stream-inf-forward: ⊦ ¬ finp(σ) → (∃ nf. ⌜nf⌝ = σ); (forward) from specification stream-basic%from-inf: ⊦ ⌜nf⌝ from n = ⌜nf from n⌝ ; from specification stream%size-inf: ⊦ # ⌜nf⌝ = ∞; from specification stream%size-to: ¬ # σ < ⌜n⌝ ⊦ #(σ to n) = ⌜n⌝ ; from specification stream%inj: ⊦ ⌜m⌝ = ⌜m0⌝ ↔ m = m0; from specification nat-inf-basic Used terms: m0 + 1, σ