no goals
α : Type ?u.35974
β : Type ?u.35977
s : Set ℕ
inst✝ : Infinite ↑s
x : ↑s
h : ¬∃ n, ↑x + n + 1 ∈ s
a : ℕ
ha : a ∈ s
hax : a ≥ succ ↑x
⊢ ↑x + (a - (↑x + 1)) + 1 ∈ s
rwa [add_right_comm, add_tsub_cancel_of_le hax]
no goals
α : Type ?u.35974
β : Type ?u.35977
s : Set ℕ
inst✝ : Infinite ↑s
x : ↑s
h : ¬∃ n, ↑x + n + 1 ∈ s
this : ∀ (a : ℕ), a ∈ s → a < succ ↑x
⊢ ∀ (a : ℕ), a ∈ Multiset.filter (fun x => x ∈ s) (Multiset.range (succ ↑x)) → a ∈ s
simp [-Multiset.range_succ]
no goals
α : Type ?u.35974
β : Type ?u.35977
s : Set ℕ
inst✝ : Infinite ↑s
x : ↑s
h : ¬∃ n, ↑x + n + 1 ∈ s
this : ∀ (a : ℕ), a ∈ s → a < succ ↑x
⊢ ∀ (x_1 : { y // y ∈ s }),
    x_1 ∈
      Multiset.toFinset
        (Multiset.pmap (fun y hy => { val := y, property := hy })
          (Multiset.filter (fun x => x ∈ s) (Multiset.range (succ ↑x)))
          (_ : ∀ (a : ℕ), a ∈ Multiset.filter (fun x => x ∈ s) (Multiset.range (succ ↑x)) → a ∈ s))
simpa [Subtype.ext_iff_val, Multiset.mem_filter, -Multiset.range_succ]