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Proof step 10 from proof swap-from-to (previous step)

#((σ from m) to n) = #((σ to m + n) from m), #(σ from m) < ⌜n⌝ , ⌜n0⌝ < #((σ from m) to n) ⊦ ((σ from m) to n)[n0] = ((σ to m + n) from m)[n0], # σ < ⌜m⌝ + ⌜n⌝

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

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

Additional information

Used simplifier rules: from-fin: ⊦ ⌜x⌝ from n = ⌜x from n⌝ ;%to-fin: ⊦ ⌜x⌝ to n = ⌜x to n⌝ ;%size-fin: ⊦ # ⌜x⌝ = ⌜# x⌝ ;%sel-fin: ⊦ ⌜x⌝ [n] = x[n];%length-from: ¬ # x < n ⊦ #(x from n) = # x - n;%length-to: ¬ # x < n ⊦ #(x to n) = n;%get-finto: ¬ # x < n, m < n ⊦ (x to n)[m] = x[m];%get-from-01: m + m0 < # x ⊦ (x from m)[m0] = x[m + m0];%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%ls-02: ⊦ m < n → ¬ n < m; from specification nat-basic1%lels-01: ⊦ ¬ m ≤ n ↔ n < m; from specification nat%ls-01: n0 ≤ n ⊦ n - n0 < m ↔ n < m + n0; from specification nat%test-02: ⊦ finp(⌜x⌝ ); from specification stream-basic%sel: ⊦ ⌜m⌝ .n = m; from specification nat-inf-basic%ls-natinf-to-nat-1-2: ⊦ ⌜n⌝ < ⌜m⌝ + ⌜n0⌝ ↔ n < m + n0; from specification nat-inf%lels: ⊦ ¬ m < n ↔ n ≤ m; from specification nat%ls-le-simp: ⊦ m < n → (n ≤ m ↔ false); from specification nat%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%c: ⊦ m + n = n + m; from specification nat-basic2%l-02: ⊦ ¬ m + n < m; from specification nat-basic2%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic2%eq-01: ⊦ m + n - m = n; from specification nat%inj: ⊦ ⌜m⌝ = ⌜m0⌝ ↔ m = m0; from specification nat-inf-basic%let: ⊦ n ≤ n0 ∧ n0 ≤ n1 → n ≤ n1; (forward) from specification nat%fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat%l: ⊦ # x = 0 ↔ x = []; from specification list-data%lep-02: ⊦ m + n ≤ m + n0 ↔ n ≤ n0; from specification nat%l: ⊦ m + n < m + n0 ↔ n < n0; from specification nat-basic2 Used terms: m, σ