(** A magic (but costly) tactic that goes from [int] back to the [Z] friendly world ... *)
Ltac i2z := match goal with
| H : ?a = ?b |- _ =>
generalize (f_equal i2z H);
try autorewrite with i2z; clear H; intro H; i2z
| |- ?a = ?b =>
apply (i2z_eq a b); try autorewrite with i2z; i2z
| H : _ |- _ => progress autorewrite with i2z in H; i2z
| _ => try autorewrite with i2z
end.
(** A reflexive version of the [i2z] tactic *)
(** this [i2z_refl] is actually weaker than [i2z]. For instance, if a [i2z] is buried deep inside a subterm, [i2z_refl] may miss it. See also the limitation about [Set] or [Type] part below. Anyhow, [i2z_refl] is enough for applying [romega]. *)
Ltac i2z_gen := match goal with
| |- ?a = ?b => apply (i2z_eq a b); i2z_gen
| H : ?a = ?b |- _ =>
generalize (f_equal i2z H); clear H; i2z_gen
| H : eq (A:=Z) ?a ?b |- _ => revert H; i2z_gen
| H : Z.lt ?a ?b |- _ => revert H; i2z_gen
| H : Z.le ?a ?b |- _ => revert H; i2z_gen
| H : Z.gt ?a ?b |- _ => revert H; i2z_gen
| H : Z.ge ?a ?b |- _ => revert H; i2z_gen
| H : _ -> ?X |- _ =>
(* A [Set] or [Type] part cannot be dealt with easily using the [ExprP] datatype. So we forget it, leaving a goal that can be weaker than the original. *)
match type of X with
| Type => clear H; i2z_gen
| Prop => revert H; i2z_gen
end
| H : _ <-> _ |- _ => revert H; i2z_gen
| H : _ /\ _ |- _ => revert H; i2z_gen
| H : _ \/ _ |- _ => revert H; i2z_gen
| H : ~ _ |- _ => revert H; i2z_gen
| _ => idtac
end.
Inductive ExprI : Set :=
| EI0 : ExprI
| EI1 : ExprI
| EI2 : ExprI
| EI3 : ExprI
| EIadd : ExprI -> ExprI -> ExprI
| EIopp : ExprI -> ExprI
| EIsub : ExprI -> ExprI -> ExprI
| EImul : ExprI -> ExprI -> ExprI
| EImax : ExprI -> ExprI -> ExprI
| EIraw : int -> ExprI.
Inductive ExprZ : Set :=
| EZadd : ExprZ -> ExprZ -> ExprZ
| EZopp : ExprZ -> ExprZ
| EZsub : ExprZ -> ExprZ -> ExprZ
| EZmul : ExprZ -> ExprZ -> ExprZ
| EZmax : ExprZ -> ExprZ -> ExprZ
| EZofI : ExprI -> ExprZ
| EZraw : Z -> ExprZ.
Inductive ExprP : Type :=
| EPeq : ExprZ -> ExprZ -> ExprP
| EPlt : ExprZ -> ExprZ -> ExprP
| EPle : ExprZ -> ExprZ -> ExprP
| EPgt : ExprZ -> ExprZ -> ExprP
| EPge : ExprZ -> ExprZ -> ExprP
| EPimpl : ExprP -> ExprP -> ExprP
| EPequiv : ExprP -> ExprP -> ExprP
| EPand : ExprP -> ExprP -> ExprP
| EPor : ExprP -> ExprP -> ExprP
| EPneg : ExprP -> ExprP
| EPraw : Prop -> ExprP.
(** [int] to [ExprI] *)
Ltac i2ei trm :=
match constr:(trm) with
| 0 => constr:(EI0)
| 1 => constr:(EI1)
| 2 => constr:(EI2)
| 3 => constr:(EI3)
| ?x + ?y => let ex := i2ei x with ey := i2ei y in constr:(EIadd ex ey)
| ?x - ?y => let ex := i2ei x with ey := i2ei y in constr:(EIsub ex ey)
| ?x * ?y => let ex := i2ei x with ey := i2ei y in constr:(EImul ex ey)
| max ?x ?y => let ex := i2ei x with ey := i2ei y in constr:(EImax ex ey)
| - ?x => let ex := i2ei x in constr:(EIopp ex)
| ?x => constr:(EIraw x)
end
(** [Z] to [ExprZ] *)
with z2ez trm :=
match constr:(trm) with
| (?x + ?y)%Z => let ex := z2ez x with ey := z2ez y in constr:(EZadd ex ey)
| (?x - ?y)%Z => let ex := z2ez x with ey := z2ez y in constr:(EZsub ex ey)
| (?x * ?y)%Z => let ex := z2ez x with ey := z2ez y in constr:(EZmul ex ey)
| (Z.max ?x ?y) => let ex := z2ez x with ey := z2ez y in constr:(EZmax ex ey)
| (- ?x)%Z =>  let ex := z2ez x in constr:(EZopp ex)
| i2z ?x => let ex := i2ei x in constr:(EZofI ex)
| ?x => constr:(EZraw x)
end.
(** [Prop] to [ExprP] *)
Ltac p2ep trm :=
match constr:(trm) with
| (?x <-> ?y) => let ex := p2ep x with ey := p2ep y in constr:(EPequiv ex ey)
| (?x -> ?y) => let ex := p2ep x with ey := p2ep y in constr:(EPimpl ex ey)
| (?x /\ ?y) => let ex := p2ep x with ey := p2ep y in constr:(EPand ex ey)
| (?x \/ ?y) => let ex := p2ep x with ey := p2ep y in constr:(EPor ex ey)
| (~ ?x) => let ex := p2ep x in constr:(EPneg ex)
| (eq (A:=Z) ?x ?y) => let ex := z2ez x with ey := z2ez y in constr:(EPeq ex ey)
| (?x < ?y)%Z => let ex := z2ez x with ey := z2ez y in constr:(EPlt ex ey)
| (?x <= ?y)%Z => let ex := z2ez x with ey := z2ez y in constr:(EPle ex ey)
| (?x > ?y)%Z => let ex := z2ez x with ey := z2ez y in constr:(EPgt ex ey)
| (?x >= ?y)%Z => let ex := z2ez x with ey := z2ez y in constr:(EPge ex ey)
| ?x =>  constr:(EPraw x)
end.
(** [ExprI] to [int] *)
Fixpoint ei2i (e:ExprI) : int :=
match e with
| EI0 => 0
| EI1 => 1
| EI2 => 2
| EI3 => 3
| EIadd e1 e2 => (ei2i e1)+(ei2i e2)
| EIsub e1 e2 => (ei2i e1)-(ei2i e2)
| EImul e1 e2 => (ei2i e1)*(ei2i e2)
| EImax e1 e2 => max (ei2i e1) (ei2i e2)
| EIopp e => -(ei2i e)
| EIraw i => i
end.
(** [ExprZ] to [Z] *)
Fixpoint ez2z (e:ExprZ) : Z :=
match e with
| EZadd e1 e2 => ((ez2z e1)+(ez2z e2))%Z
| EZsub e1 e2 => ((ez2z e1)-(ez2z e2))%Z
| EZmul e1 e2 => ((ez2z e1)*(ez2z e2))%Z
| EZmax e1 e2 => Z.max (ez2z e1) (ez2z e2)
| EZopp e => (-(ez2z e))%Z
| EZofI e => i2z (ei2i e)
| EZraw z => z
end.
(** [ExprP] to [Prop] *)
Fixpoint ep2p (e:ExprP) : Prop :=
match e with
| EPeq e1 e2 => (ez2z e1) = (ez2z e2)
| EPlt e1 e2 => ((ez2z e1)<(ez2z e2))%Z
| EPle e1 e2 => ((ez2z e1)<=(ez2z e2))%Z
| EPgt e1 e2 => ((ez2z e1)>(ez2z e2))%Z
| EPge e1 e2 => ((ez2z e1)>=(ez2z e2))%Z
| EPimpl e1 e2 => (ep2p e1) -> (ep2p e2)
| EPequiv e1 e2 => (ep2p e1) <-> (ep2p e2)
| EPand e1 e2 => (ep2p e1) /\ (ep2p e2)
| EPor e1 e2 => (ep2p e1) \/ (ep2p e2)
| EPneg e => ~ (ep2p e)
| EPraw p => p
end.
(** [ExprI] (supposed under a [i2z]) to a simplified [ExprZ] *)
Fixpoint norm_ei (e:ExprI) : ExprZ :=
match e with
| EI0 => EZraw (0%Z)
| EI1 => EZraw (1%Z)
| EI2 => EZraw (2%Z)
| EI3 => EZraw (3%Z)
| EIadd e1 e2 => EZadd (norm_ei e1) (norm_ei e2)
| EIsub e1 e2 => EZsub (norm_ei e1) (norm_ei e2)
| EImul e1 e2 => EZmul (norm_ei e1) (norm_ei e2)
| EImax e1 e2 => EZmax (norm_ei e1) (norm_ei e2)
| EIopp e => EZopp (norm_ei e)
| EIraw i => EZofI (EIraw i)
end.
(** [ExprZ] to a simplified [ExprZ] *)
Fixpoint norm_ez (e:ExprZ) : ExprZ :=
match e with
| EZadd e1 e2 => EZadd (norm_ez e1) (norm_ez e2)
| EZsub e1 e2 => EZsub (norm_ez e1) (norm_ez e2)
| EZmul e1 e2 => EZmul (norm_ez e1) (norm_ez e2)
| EZmax e1 e2 => EZmax (norm_ez e1) (norm_ez e2)
| EZopp e => EZopp (norm_ez e)
| EZofI e =>  norm_ei e
| EZraw z => EZraw z
end.
(** [ExprP] to a simplified [ExprP] *)
Fixpoint norm_ep (e:ExprP) : ExprP :=
match e with
| EPeq e1 e2 => EPeq (norm_ez e1) (norm_ez e2)
| EPlt e1 e2 => EPlt (norm_ez e1) (norm_ez e2)
| EPle e1 e2 => EPle (norm_ez e1) (norm_ez e2)
| EPgt e1 e2 => EPgt (norm_ez e1) (norm_ez e2)
| EPge e1 e2 => EPge (norm_ez e1) (norm_ez e2)
| EPimpl e1 e2 => EPimpl (norm_ep e1) (norm_ep e2)
| EPequiv e1 e2 => EPequiv (norm_ep e1) (norm_ep e2)
| EPand e1 e2 => EPand (norm_ep e1) (norm_ep e2)
| EPor e1 e2 => EPor (norm_ep e1) (norm_ep e2)
| EPneg e => EPneg (norm_ep e)
| EPraw p => EPraw p
end.

Lemma norm_ep_correct2 (e:ExprP) : ep2p (norm_ep e) -> ep2p e.
Ltac i2z_refl :=
i2z_gen;
match goal with |- ?t =>
let e := p2ep t in
change (ep2p e); apply norm_ep_correct2; simpl
end.