[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 9 from proof from-append-tofrom (previous step)

#(σ from n) < ⌜n1⌝ ⊦ # σ < ⌜n⌝ + ⌜n1⌝ , #((σ from n) to n1) = ⌜n1⌝ ∧ (⌜n0⌝ < ⌜n⌝ + #((σ from n) to n1) → ((σ to n) + ((σ from n) to n1))[n0] = (σ to n + n1)[n0])

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 10:
⊦ ⌜n⌝ < # σ ∧ finp(σ) → #(σ from n) = ⌜(# σ).n - n⌝
Proof step 11:
#(σ from n) < ⌜n1⌝ ⊦ # σ < ⌜n⌝ + ⌜n1⌝ , #((σ from n) to n1) = ⌜n1⌝ ∧ (⌜n0⌝ < ⌜n⌝ + #((σ from n) to n1) → ((σ to n) + ((σ from n) to n1))[n0] = (σ to n + n1)[n0]), ⌜n⌝ < # σ ∧ finp(σ)

Rule argument

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

Additional information

Used simplifier rules: size-fin: ⊦ # ⌜x⌝ = ⌜# x⌝ ;% ⊦ ⌜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% ⊦ m < n → n ≠ 0; (s)% ⊦ finp(⌜x⌝ ); (s)% ⊦ ⌜m⌝ .n = m; (s)% ⊦ m ≤ n ↔ ¬ n < m; (ss)% ⊦ m < n → ¬ n < m; (s)%n0 ≤ n ⊦ n - n0 < m ↔ n < m + n0; (s)% ⊦ ⌜n⌝ < ⌜m⌝ + ⌜n0⌝ ↔ n < m + n0; (s)% ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ m < n → (n ≤ 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 Used terms: n, σ