lean2/tests/lean/run/is_nil.lean

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import logic
open tactic
inductive list (A : Type) : Type :=
| nil {} : list A
| cons : A → list A → list A
namespace list end list open list
open eq
definition is_nil {A : Type} (l : list A) : Prop
:= list.rec true (fun h t r, false) l
theorem is_nil_nil (A : Type) : is_nil (@nil A)
:= of_eq_true (refl true)
theorem cons_ne_nil {A : Type} (a : A) (l : list A) : ¬ cons a l = nil
:= not.intro (assume H : cons a l = nil,
absurd
(calc true = is_nil (@nil A) : refl _
... = is_nil (cons a l) : { symm H }
... = false : refl _)
true_ne_false)
theorem T : is_nil (@nil Prop)
:= by apply is_nil_nil
(*
local list = Const("list", {1})(Prop)
local isNil = Const("is_nil", {1})(Prop)
local Nil = Const({"list", "nil"}, {1})(Prop)
local m = mk_metavar("m", list)
print(isNil(Nil))
print(isNil(m))
function test_unify(env, m, lhs, rhs, num_s)
print(tostring(lhs) .. " =?= " .. tostring(rhs) .. ", expected: " .. tostring(num_s))
local ss = unify(env, lhs, rhs, name_generator(), substitution())
local n = 0
for s in ss do
print("solution: " .. tostring(s:instantiate(m)))
n = n + 1
end
if num_s ~= n then print("n: " .. n) end
assert(num_s == n)
end
print("=====================")
test_unify(get_env(), m, isNil(Nil), isNil(m), 2)
*)