48 lines
1.9 KiB
Text
48 lines
1.9 KiB
Text
inductive LazyList (α : Type u)
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| nil : LazyList α
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| cons (hd : α) (tl : LazyList α) : LazyList α
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| delayed (t : Thunk (LazyList α)) : LazyList α
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namespace LazyList
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def length : LazyList α → Nat
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| nil => 0
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| cons _ as => length as + 1
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| delayed as => length as.get
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def force : LazyList α → Option (α × LazyList α)
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| delayed as => force as.get
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| nil => none
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| cons a as => some (a,as)
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end LazyList
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def rotate (f : LazyList τ) (r : List τ) (a : LazyList τ)
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(h : f.length + 1 = r.length) : LazyList τ :=
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match r with
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| List.nil => False.elim (by simp_arith [LazyList.length] at h)
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| y::r' =>
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match f.force with
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| none => LazyList.cons y a
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| some (x, f') => LazyList.cons x (rotate f' r' (LazyList.cons y a) (sorry))
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theorem rotate_inv {F : LazyList τ} {R : List τ} : (h : F.length + 1 = R.length) → (rotate F R nil h).length = F.length + R.length := by
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match F with
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| LazyList.nil => intro h; unfold rotate; sorry
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| LazyList.cons Fh Ft => sorry
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| LazyList.delayed Ft => sorry
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theorem LazyList.ind {α : Type u} {motive : LazyList α → Sort v}
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(nil : motive LazyList.nil)
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(cons : (hd : α) → (tl : LazyList α) → motive tl → motive (LazyList.cons hd tl))
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(delayed : (t : Thunk (LazyList α)) → motive t.get → motive (LazyList.delayed t))
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(t : LazyList α) : motive t :=
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match t with
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| LazyList.nil => nil
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| LazyList.cons h t => cons h t (ind nil cons delayed t)
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| LazyList.delayed t => delayed t (ind nil cons delayed t.get)
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-- Remark: Lean used well-founded recursion behind the scenes to define LazyList.ind
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theorem rotate_inv' {F : LazyList τ} {R : List τ} : (h : F.length + 1 = R.length) → (rotate F R nil h).length = F.length + R.length := by
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induction F using LazyList.ind with
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| nil => intro h; unfold rotate; sorry
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| cons h t ih => trace_state; sorry
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| delayed t => trace_state; sorry
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