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Proof step 22 from proof infinite-append-prefixnextli-weak (previous step)

nextli(lf(n0)) = (σ to ii(n0 + 1)).l, ii(0) = 0, lf(0) = [], lf(n0 + 1) = nextli(lf(n0)), lf(n0) = (σ to ii(n0)).l, ∀ n0. lf(n0) = (σ to ii(n0)).l, ∀ n. ii(n + 1) = ii(n) + # nextli0(lf(n)) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # nextli0(lf(n)) = ⌜nextli0(lf(n))⌝ , ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. lf(n + 1) = nextli(lf(n)), ∀ x. nextli(x) = x + nextli0(x), (finp(σ) ⊃ # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → nextli0(lf(m)) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)) ⊦ σ to ii(n0 + 1) = ⌜nextli(lf(n0))⌝

User applied rule case distinction

And closed the goal!

Rule argument

pos: left 11

Additional information

Used simplifier rules: notfin-sizeinf: ⊦ ¬ finp(σ) → # σ = ∞; from specification stream%less-inf1: ⊦ ¬ ∞ < n∞; from specification nat-inf%stream-inf-forward: ⊦ ¬ finp(σ) → (∃ nf. ⌜nf⌝ = σ); (forward) from specification stream-basic%to-inf: ⊦ ⌜nf⌝ to n = ⌜nf to n⌝ ; from specification stream%sel-02: ⊦ ⌜x⌝ .l = x; from specification stream-basic%inj-02: ⊦ ⌜x⌝ = ⌜x0⌝ ↔ x = x0; from specification stream-basic%stream-fin-forward: ⊦ finp(σ) → (∃ x. ⌜x⌝ = σ); (forward) from specification stream-basic%size-fin: ⊦ # ⌜x⌝ = ⌜# x⌝ ; from specification stream%inj: ⊦ ⌜m⌝ = ⌜m0⌝ ↔ m = m0; from specification nat-inf-basic%to-fin: ⊦ ⌜x⌝ to n = ⌜x to n⌝ ; from specification stream