fixpoint "--rewrite"

data Vec 1 = [
  | VNil  { }
  | VCons { mhead : @(0), mtail : Vec @(0)}
]

match isNil VNil       = true 
match isNil VCons x xs = false

match isCons VNil       = false 
match isCons VCons x xs = true


match head VCons x xs = (x)
match tail VCons x xs = (xs)

constant mall: (func(0, [func(0,[int;bool]); Vec int; bool]))
constant geq: (func(0, [int; int; bool]))
constant mand: (func(0, [bool; bool; bool]))
constant lte: (func(0, [int; int; bool]))
constant isCons: (func(1, [Vec @(0); bool]))
constant isNil:  (func(1, [Vec @(0); bool]))
constant head: (func(1, [Vec @(0); @(0)]))
constant tail: (func(1, [Vec @(0); Vec @(0)]))


define mall(f:func(0,[int;bool]), xs:Vec int) : bool = {
    if (isNil xs) then true 
    else (mand (f (head xs)) (mall f (tail xs)))
}

define mand(x:bool, y:bool) : bool = { 
  if x then y else false 
  }

define geq(x:int, y:int) : bool = { 
  x >= y 
  }

define lte(x:int, y:int) : bool = { 
  x <= y 
  }

expand [1 : True]

bind 0 xs  : {v: Vec int | true }
bind 1 x   : {v: int | true }
bind 2 y   : {v: int | true }

constraint:
  env [0; 1; 2]
  lhs {v : int | true }
  rhs {v : int | (mall (geq y) (VCons x xs)) = (mand (lte x y) (mall (geq y) xs))  }
  id 1 tag []