224 lines
5.3 KiB
Text
224 lines
5.3 KiB
Text
inductive PList (α : Type) : Prop
|
||
| nil
|
||
| cons : α → PList α → PList α
|
||
|
||
infixr:67 " ::: " => PList.cons
|
||
|
||
def map {α β} (f : α → β) : List α → List β
|
||
| [] => []
|
||
| a::as => f a :: map f as
|
||
|
||
def pmap {α β} (f : α → β) : PList α → PList β
|
||
| PList.nil => PList.nil
|
||
| a:::as => f a ::: pmap f as
|
||
|
||
theorem ex1 : map Nat.succ [1] = [2] :=
|
||
rfl
|
||
|
||
theorem ex2 : map Nat.succ [] = [] :=
|
||
rfl
|
||
|
||
theorem ex3 (a : Nat) : map Nat.succ [a] = [Nat.succ a] :=
|
||
rfl
|
||
|
||
theorem ex4 {α β} (f : α → β) (a : α) (as : List α) : map f (a::as) = (f a) :: map f as :=
|
||
rfl
|
||
|
||
theorem ex5 {α β} (f : α → β) : map f [] = [] :=
|
||
rfl
|
||
|
||
def map2 {α β} (f : α → β) (as : List α) : List β :=
|
||
let rec loop : List α → List β
|
||
| [] => []
|
||
| a::as => f a :: loop as;
|
||
loop as
|
||
|
||
def pmap2 {α β} (f : α → β) (as : PList α) : PList β :=
|
||
let rec loop : PList α → PList β
|
||
| PList.nil => PList.nil
|
||
| a:::as => f a ::: loop as;
|
||
loop as
|
||
|
||
|
||
theorem ex6 {α β} (f : α → β) (a : α) (as : List α) : map2 f (a::as) = (f a) :: map2 f as :=
|
||
rfl
|
||
|
||
def foo (xs : List Nat) : List Nat :=
|
||
match xs with
|
||
| [] => []
|
||
| x::xs =>
|
||
let y := 2 * x;
|
||
match xs with
|
||
| [] => []
|
||
| x::xs => (y + x) :: foo xs
|
||
|
||
def pfoo (xs : PList Nat) : PList Nat :=
|
||
match xs with
|
||
| PList.nil => PList.nil
|
||
| x:::xs =>
|
||
let y := 2 * x;
|
||
match xs with
|
||
| PList.nil => PList.nil
|
||
| x:::xs => (y + x) ::: pfoo xs
|
||
|
||
#eval foo [1, 2, 3, 4]
|
||
|
||
theorem fooEq (x y : Nat) (xs : List Nat) : foo (x::y::xs) = (2*x + y) :: foo xs :=
|
||
rfl
|
||
|
||
def bla (x : Nat) (ys : List Nat) : List Nat :=
|
||
if x % 2 == 0 then
|
||
match ys with
|
||
| [] => []
|
||
| y::ys => (y + x/2) :: bla (x/2) ys
|
||
else
|
||
match ys with
|
||
| [] => []
|
||
| y::ys => (y + x/2 + 1) :: bla (x/2) ys
|
||
|
||
def pbla (x : Nat) (ys : PList Nat) : PList Nat :=
|
||
if x % 2 == 0 then
|
||
match ys with
|
||
| PList.nil => PList.nil
|
||
| y:::ys => (y + x/2) ::: pbla (x/2) ys
|
||
else
|
||
match ys with
|
||
| PList.nil => PList.nil
|
||
| y:::ys => (y + x/2 + 1) ::: pbla (x/2) ys
|
||
|
||
theorem blaEq (y : Nat) (ys : List Nat) : bla 4 (y::ys) = (y+2) :: bla 2 ys :=
|
||
rfl
|
||
|
||
inductive PNat : Prop
|
||
| zero
|
||
| succ : PNat → PNat
|
||
|
||
def f : Nat → Nat → Nat
|
||
| 0, y => y
|
||
| x+1, y =>
|
||
match f x y with
|
||
| 0 => f x y
|
||
| v => f x v + 1
|
||
|
||
def pf : PNat → PNat → PNat
|
||
| PNat.zero, y => y
|
||
| PNat.succ x, y =>
|
||
match pf x y with
|
||
| PNat.zero => pf x y
|
||
| v => PNat.succ $ pf x v
|
||
|
||
def g (xs : List Nat) : Nat :=
|
||
match xs with
|
||
| [] => 0
|
||
| y::ys =>
|
||
match ys with
|
||
| [] => 1
|
||
| _ => g ys + 1
|
||
|
||
def pg (xs : PList Nat) : True :=
|
||
match xs with
|
||
| PList.nil => True.intro
|
||
| y:::ys =>
|
||
match ys with
|
||
| PList.nil => True.intro
|
||
| _ => pg ys
|
||
|
||
def aux : Nat → Nat → Nat
|
||
| 0, y => y
|
||
| x+1, y =>
|
||
match f x y with
|
||
| 0 => f x y
|
||
| v => f x v + 1
|
||
|
||
def paux : PNat → PNat → PNat
|
||
| PNat.zero, y => y
|
||
| PNat.succ x, y =>
|
||
match pf x y with
|
||
| PNat.zero => pf x y
|
||
| v => PNat.succ (pf x v)
|
||
|
||
theorem ex (x y : Nat) : f x y = aux x y := by
|
||
cases x
|
||
rfl
|
||
rfl
|
||
|
||
axiom F : Nat → Nat
|
||
|
||
inductive is_nat : Nat -> Prop
|
||
| Z : is_nat 0
|
||
| S {n} : is_nat n → is_nat (F n)
|
||
|
||
inductive is_nat_T : Nat -> Type
|
||
| Z : is_nat_T 0
|
||
| S {n} : is_nat_T n → is_nat_T (F n)
|
||
|
||
axiom P : Nat → Prop
|
||
axiom F0 : P 0
|
||
axiom F1 : P (F 0)
|
||
axiom FS {n : Nat} : P n → P (F (F n))
|
||
|
||
axiom T : Nat → Prop
|
||
axiom TF0 : T 0
|
||
axiom TF1 : T (F 0)
|
||
axiom TFS {n : Nat} : T n → T (F (F n))
|
||
|
||
-- set_option trace.Elab.definition.structural true in
|
||
theorem «nested recursion» : ∀ {n}, is_nat n → P n
|
||
| _, is_nat.Z => F0
|
||
| _, is_nat.S is_nat.Z => F1
|
||
| _, is_nat.S (is_nat.S h) => FS («nested recursion» h)
|
||
|
||
/-- forbidding this, because it shouldn't exist in the first place.
|
||
We don't expect this kind of inconsistent inaccessible patterns. -/
|
||
-- theorem «nested recursion, inaccessible» : ∀ {n}, is_nat n → P n
|
||
-- | _, .(is_nat.Z) => F0
|
||
-- | _, is_nat.S .(is_nat.Z) => F1
|
||
-- | _, is_nat.S (is_nat.S h) => FS («nested recursion, inaccessible» h)
|
||
|
||
theorem «reordered discriminants, type» : ∀ n, is_nat_T n → Nat → T n := fun n hn m =>
|
||
match n, m, hn with
|
||
| _, _, is_nat_T.Z => TF0
|
||
| _, _, is_nat_T.S is_nat_T.Z => TF1
|
||
| _, m, is_nat_T.S (is_nat_T.S h) => TFS («reordered discriminants, type» _ h m)
|
||
|
||
theorem «reordered discriminants» : ∀ n, is_nat n → Nat → P n := fun n hn m =>
|
||
match n, m, hn with
|
||
| _, _, is_nat.Z => F0
|
||
| _, _, is_nat.S is_nat.Z => F1
|
||
| _, m, is_nat.S (is_nat.S h) => FS («reordered discriminants» _ h m)
|
||
|
||
/-- known unsupported case for types, just here for reference. -/
|
||
-- def «unsupported nesting» (xs : List Nat) : True :=
|
||
-- match xs with
|
||
-- | List.nil => True.intro
|
||
-- | y::ys =>
|
||
-- match ys with
|
||
-- | List.nil => True.intro
|
||
-- | _::_::zs => «unsupported nesting» zs
|
||
-- | zs => «unsupported nesting» ys
|
||
|
||
def «unsupported nesting, predicate» (xs : PList Nat) : True :=
|
||
match xs with
|
||
| PList.nil => True.intro
|
||
| y:::ys =>
|
||
match ys with
|
||
| PList.nil => True.intro
|
||
| _:::_:::zs => «unsupported nesting, predicate» zs
|
||
| zs => «unsupported nesting, predicate» ys
|
||
|
||
|
||
def f1 (xs : List Nat) : Nat :=
|
||
match xs with
|
||
| [] => 0
|
||
| x::xs =>
|
||
match xs with
|
||
| [] => 0
|
||
| _ => f1 xs + 1
|
||
|
||
def pf1 (xs : PList Nat) : True :=
|
||
match xs with
|
||
| PList.nil => True.intro
|
||
| x:::xs =>
|
||
match xs with
|
||
| PList.nil => True.intro
|
||
| _ => pf1 xs
|