46 lines
1.4 KiB
Text
46 lines
1.4 KiB
Text
new_frontend
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inductive InfTree.{u} (α : Type u)
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| leaf : α → InfTree α
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| node : (Nat → InfTree α) → InfTree α
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open InfTree
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def szn.{u} {α : Type u} (n : Nat) : InfTree α → InfTree α → Nat
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| leaf a, t2 => 1
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| node c, leaf b => 0
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| node c, node d => szn n (c n) (d n)
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universes u
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theorem ex1 {α : Type u} (n : Nat) (c : Nat → InfTree α) (d : Nat → InfTree α) : szn n (node c) (node d) = szn n (c n) (d n) :=
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rfl
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inductive BTree (α : Type u)
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| leaf : α → BTree α
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| node : (Bool → Bool → BTree α) → BTree α
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def BTree.sz {α : Type u} : BTree α → Nat
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| leaf a => 1
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| node c => sz (c true true) + sz (c true false) + sz (c false true) + sz (c false false) + 1
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theorem ex2 {α : Type u} (c : Bool → Bool → BTree α) : (BTree.node c).sz = (c true true).sz + (c true false).sz + (c false true).sz + (c false false).sz + 1 :=
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rfl
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inductive L (α : Type u)
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| nil : L α
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| cons : (Unit → α) → (Unit → L α) → L α
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def L.append {α} : L α → L α → L α
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| nil, bs => bs
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| cons a as, bs => cons a (fun _ => append (as ()) bs)
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theorem L.appendNil {α} : (as : L α) → append as nil = as
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| nil => rfl
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| cons a as =>
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show cons a (fun _ => append (as ()) nil) = cons a as from
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have ih : append (as ()) nil = as () from appendNil $ as ()
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have thunkAux : (fun _ => as ()) = as from funext fun x => by
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cases x
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exact rfl
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by rw [ih, thunkAux]; exact rfl
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