Definition state := total_map.Definition empty_state : state := t_empty 0.(* Zoe: Changing the name of this to aexp because the name clash breaks derivation*)
(* TODO: Fix the above at some point *)
Inductive aexp : Type :=
  | ANum : nat -> aexp
  | AId : id -> aexp               (* <----- NEW *)
  | APlus : aexp -> aexp -> aexp
  | AMinus : aexp -> aexp -> aexp
  | AMult : aexp -> aexp -> aexp.(* Zoe: Changing the name of this to aexp because the name clash breaks derivation*)
(* TODO: Fix the above at some point *)
Inductive aexp : Type :=
  | ANum : nat -> aexp
  | AId : id -> aexp               (* <----- NEW *)
  | APlus : aexp -> aexp -> aexp
  | AMinus : aexp -> aexp -> aexp
  | AMult : aexp -> aexp -> aexp.(* Zoe: Changing the name of this to aexp because the name clash breaks derivation*)
(* TODO: Fix the above at some point *)
Inductive aexp : Type :=
  | ANum : nat -> aexp
  | AId : id -> aexp               (* <----- NEW *)
  | APlus : aexp -> aexp -> aexp
  | AMinus : aexp -> aexp -> aexp
  | AMult : aexp -> aexp -> aexp.(* Zoe: Changing the name of this to aexp because the name clash breaks derivation*)
(* TODO: Fix the above at some point *)
Inductive aexp : Type :=
  | ANum : nat -> aexp
  | AId : id -> aexp               (* <----- NEW *)
  | APlus : aexp -> aexp -> aexp
  | AMinus : aexp -> aexp -> aexp
  | AMult : aexp -> aexp -> aexp.(* Zoe: Changing the name of this to aexp because the name clash breaks derivation*)
(* TODO: Fix the above at some point *)
Inductive aexp : Type :=
  | ANum : nat -> aexp
  | AId : id -> aexp               (* <----- NEW *)
  | APlus : aexp -> aexp -> aexp
  | AMinus : aexp -> aexp -> aexp
  | AMult : aexp -> aexp -> aexp.(* Zoe: Changing the name of this to aexp because the name clash breaks derivation*)
(* TODO: Fix the above at some point *)
Inductive aexp : Type :=
  | ANum : nat -> aexp
  | AId : id -> aexp               (* <----- NEW *)
  | APlus : aexp -> aexp -> aexp
  | AMinus : aexp -> aexp -> aexp
  | AMult : aexp -> aexp -> aexp.Definition W : id := Id "W".Definition X : id := Id "X".Definition Y : id := Id "Y".Definition Z : id := Id "Z".Instance dec_eq_aex (x y : aexp) : Dec (x = y).Inductive bexp : Type :=
  | BTrue : bexp
  | BFalse : bexp
  | BEq : aexp -> aexp -> bexp
  | BLe : aexp -> aexp -> bexp
  | BNot : bexp -> bexp
  | BAnd : bexp -> bexp -> bexp.Inductive bexp : Type :=
  | BTrue : bexp
  | BFalse : bexp
  | BEq : aexp -> aexp -> bexp
  | BLe : aexp -> aexp -> bexp
  | BNot : bexp -> bexp
  | BAnd : bexp -> bexp -> bexp.Inductive bexp : Type :=
  | BTrue : bexp
  | BFalse : bexp
  | BEq : aexp -> aexp -> bexp
  | BLe : aexp -> aexp -> bexp
  | BNot : bexp -> bexp
  | BAnd : bexp -> bexp -> bexp.Inductive bexp : Type :=
  | BTrue : bexp
  | BFalse : bexp
  | BEq : aexp -> aexp -> bexp
  | BLe : aexp -> aexp -> bexp
  | BNot : bexp -> bexp
  | BAnd : bexp -> bexp -> bexp.Inductive bexp : Type :=
  | BTrue : bexp
  | BFalse : bexp
  | BEq : aexp -> aexp -> bexp
  | BLe : aexp -> aexp -> bexp
  | BNot : bexp -> bexp
  | BAnd : bexp -> bexp -> bexp.Inductive bexp : Type :=
  | BTrue : bexp
  | BFalse : bexp
  | BEq : aexp -> aexp -> bexp
  | BLe : aexp -> aexp -> bexp
  | BNot : bexp -> bexp
  | BAnd : bexp -> bexp -> bexp.Inductive bexp : Type :=
  | BTrue : bexp
  | BFalse : bexp
  | BEq : aexp -> aexp -> bexp
  | BLe : aexp -> aexp -> bexp
  | BNot : bexp -> bexp
  | BAnd : bexp -> bexp -> bexp.Instance dec_eq_bexp (x y : bexp) : Dec (x = y).Fixpoint aeval (st : state) (a : aexp) : nat :=
  match a with
  | ANum n => n
  | AId x => t_lookup st x                                (* <----- NEW *)
  | APlus a1 a2 => (aeval st a1) + (aeval st a2)
  | AMinus a1 a2  => (aeval st a1) - (aeval st a2)
  | AMult a1 a2 => (aeval st a1) * (aeval st a2)
  end.Fixpoint beval (st : state) (b : bexp) : bool :=
  match b with
  | BTrue       => true
  | BFalse      => false
  | BEq a1 a2   => beq_nat (aeval st a1) (aeval st a2)
  | BLe a1 a2   => leb (aeval st a1) (aeval st a2)
  | BNot b1     => negb (beval st b1)
  | BAnd b1 b2  => andb (beval st b1) (beval st b2)
  end.Inductive com : Type :=
  | CSkip : com
  | CAss : id -> aexp -> com
  | CSeq : com -> com -> com
  | CIf : bexp -> com -> com -> com
  | CWhile : bexp -> com -> com.Inductive com : Type :=
  | CSkip : com
  | CAss : id -> aexp -> com
  | CSeq : com -> com -> com
  | CIf : bexp -> com -> com -> com
  | CWhile : bexp -> com -> com.Inductive com : Type :=
  | CSkip : com
  | CAss : id -> aexp -> com
  | CSeq : com -> com -> com
  | CIf : bexp -> com -> com -> com
  | CWhile : bexp -> com -> com.Inductive com : Type :=
  | CSkip : com
  | CAss : id -> aexp -> com
  | CSeq : com -> com -> com
  | CIf : bexp -> com -> com -> com
  | CWhile : bexp -> com -> com.Inductive com : Type :=
  | CSkip : com
  | CAss : id -> aexp -> com
  | CSeq : com -> com -> com
  | CIf : bexp -> com -> com -> com
  | CWhile : bexp -> com -> com.Inductive com : Type :=
  | CSkip : com
  | CAss : id -> aexp -> com
  | CSeq : com -> com -> com
  | CIf : bexp -> com -> com -> com
  | CWhile : bexp -> com -> com.Instance dec_eq_com (x y : com) : Dec (x = y).Notation "'IFB' c1 'THEN' c2 'ELSE' c3 'FI'" :=
  (CIf c1 c2 c3) (at level 80, right associativity).Definition subtract_slowly : com :=
  WHILE BNot (BEq (AId X) (ANum 0)) DO
    subtract_slowly_body
  END.Definition loop : com :=
  WHILE BTrue DO
    SKIP
  END.

Theorem loop_never_stops : forall st st', ~(loop / st \\ st').