namespace imp
open tactic

@[reducible]
def uname := string

inductive aexp
| val   : nat → aexp
| var   : uname → aexp
| plus  : aexp → aexp → aexp
| times : aexp → aexp → aexp

instance : decidable_eq aexp :=
by mk_dec_eq_instance

@[reducible]
def value := nat

def state := uname → value

open aexp

def aval : aexp → state → value
| (val n)      s := n
| (var x)      s := s x
| (plus a₁ a₂) s := aval a₁ s + aval a₂ s
| (times a₁ a₂) s := aval a₁ s * aval a₂ s

example : aval (plus (val 3) (var "x")) (λ x, 0) = 3 :=
rfl

def updt (s : state) (x : uname) (v : value) : state :=
λ y, if x = y then v else s y

def asimp_const : aexp → aexp
| (val n)      := val n
| (var x)      := var x
| (plus a₁ a₂) :=
  match asimp_const a₁, asimp_const a₂ with
  | val n₁, val n₂ := val (n₁ + n₂)
  | b₁,     b₂     := plus b₁ b₂
  end
| (times a₁ a₂) :=
  match asimp_const a₁, asimp_const a₂ with
  | val n₁, val n₂ := val (n₁ * n₂)
  | b₁,     b₂     := times b₁ b₂
  end

example : asimp_const (plus (plus (val 2) (val 3)) (var "x")) = plus (val 5) (var "x") :=
rfl

attribute [ematch] asimp_const aval

-- set_option trace.smt.ematch true

meta def not_done : tactic unit := fail_if_success now

lemma aval_asimp_const (a : aexp) (s : state) : aval (asimp_const a) s = aval a s :=
begin [smt]
 induction a,
 all_goals {destruct (asimp_const a_1), all_goals {destruct (asimp_const a_2), eblast}}
end

lemma ex2 (a : aexp) (s : state) : aval (asimp_const a) s = aval a s :=
begin [smt]
 induction a,
 all_goals {destruct (asimp_const a_1), all_goals {destruct (asimp_const a_2), eblast_using [asimp_const, aval]}}
end

end imp