(************************************************************************)
(*         *      The Rocq Prover / The Rocq Development Team           *)
(*  v      *         Copyright INRIA, CNRS and contributors             *)
(* <O___,, * (see version control and CREDITS file for authors & dates) *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
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(** An alternative to [zify] in ML parametrised by user-provided classes instances.  The framework has currently several limitations that are in place for simplicity. For instance, we only consider binary operators of type [Op: S -> S -> S]. Another limitation is that our injection theorems e.g. [TBOpInj], are using Leibniz equality; the payoff is that there is no need for morphisms... *)
(** An injection [InjTyp S T] declares an injection from source type S to target type T. *)
Class InjTyp (S : Type) (T : Type) :=
mkinj {
(* [inj] is the injection function *)
inj  : S -> T;
pred : T -> Prop;
(* [cstr] states that [pred] holds for any injected element. [cstr (inj x)] is introduced in the goal for any leaf term of the form [inj x] *)
cstr : forall x, pred (inj x)
}.
(** [BinOp Op] declares a source operator [Op: S1 -> S2 -> S3]. *)
Class BinOp {S1 S2 S3 T1 T2 T3:Type} (Op : S1 -> S2 -> S3) {I1 : InjTyp S1 T1} {I2 : InjTyp S2 T2} {I3 : InjTyp S3 T3} :=
mkbop {
(* [TBOp] is the target operator after injection of operands. *)
TBOp : T1 -> T2 -> T3;
(* [TBOpInj] states the correctness of the injection. *)
TBOpInj : forall (n:S1) (m:S2), inj (Op n m) = TBOp (inj n) (inj m)
}.
(** [Unop Op] declares a source operator [Op : S1 -> S2]. *)
Class UnOp {S1 S2 T1 T2:Type} (Op : S1 -> S2) {I1 : InjTyp S1 T1} {I2 : InjTyp S2 T2}  :=
mkuop {
(* [TUOp] is the target operator after injection of operands. *)
TUOp : T1 -> T2;
(* [TUOpInj] states the correctness of the injection. *)
TUOpInj : forall (x:S1), inj (Op x) = TUOp (inj x)
}.
(** [CstOp Op] declares a source constant [Op : S]. *)
Class CstOp {S T:Type} (Op : S) {I : InjTyp S T} :=
mkcst {
(* [TCst] is the target constant. *)
TCst : T;
(* [TCstInj] states the correctness of the injection. *)
TCstInj : inj Op = TCst
}.
(** In the framework, [Prop] is mapped to [Prop] and the injection is phrased in terms  of [=] instead of [<->]. *)
(** [BinRel R] declares the injection of a binary relation. *)
Class BinRel {S:Type} {T:Type} (R : S -> S -> Prop) {I : InjTyp S T}  :=
mkbrel {
TR : T -> T -> Prop;
TRInj : forall n m : S, R n m <->  TR (@inj _ _ I n) (inj m)
}.
(** [PropOp Op] declares morphisms for [<->]. This will be used to deal with e.g. [and], [or],... *)
Class PropOp (Op : Prop -> Prop -> Prop) :=
mkprop {
op_iff : forall (p1 p2 q1 q2:Prop), (p1 <-> q1) -> (p2 <-> q2) -> (Op p1 p2 <-> Op q1 q2)
}.
Class PropUOp (Op : Prop -> Prop) :=
mkuprop {
uop_iff :   forall (p1 q1 :Prop), (p1 <-> q1) -> (Op p1 <-> Op q1)
}.
(** Once the term is injected, terms can be replaced by their specification. NB1: The Ltac code is currently limited to (Op: Z -> Z -> Z) NB2: This is not sufficient to cope with [Z.div] or [Z.mod] *)
Class BinOpSpec {T1 T2 T3: Type} (Op : T1 -> T2 -> T3)  :=
mkbspec {
BPred : T1 -> T2 -> T3 -> Prop;
BSpec : forall x y, BPred x y (Op x y)
}.
Class UnOpSpec {T1 T2: Type} (Op : T1 -> T2)  :=
mkuspec {
UPred : T1 -> T2 -> Prop;
USpec : forall x, UPred x (Op x)
}.
(** After injections, e.g. nat -> Z, the fact that Z.of_nat x * Z.of_nat y is positive is lost. This information can be recovered using instance of the  [Saturate] class. *)
Class Saturate {T: Type} (Op : T -> T -> T) :=
mksat {
(** Given [Op x y], - [PArg1] is the pre-condition of x - [PArg2] is the pre-condition of y - [PRes]  is the pos-condition of (Op x y) *)
PArg1 : T -> Prop;
PArg2 : T -> Prop;
PRes  : T -> T -> T -> Prop;
(** [SatOk] states the correctness of the reasoning *)
SatOk : forall x y, PArg1 x -> PArg2 y -> PRes x y (Op x y)
}.
(* )Arguments PRes {_ _} _. *)
(* The [ZifyInst.saturate] iterates over all the instances and for every pattern of the form [H1 : PArg1 ?x , H2 : PArg2 ?y , T : context[Op ?x ?y] |- _ ] [H1 : PArg1 ?x , H2 : PArg2 ?y |- context[Op ?x ?y] ] asserts  (SatOK x y H1 H2) *)
(** The rest of the file is for internal use by the ML tactic. There are  data-structures and lemmas used to inductively construct the injected terms. *)
(** The data-structures [injterm] and [injected_prop] are used to store source and target expressions together with a correctness proof. *)
Record injterm {S T: Type} (I : S -> T) :=
mkinjterm { source : S ; target : T ; inj_ok : I source = target}.
Record injprop  :=
mkinjprop {
source_prop : Prop ; target_prop : Prop ;
injprop_ok : source_prop <-> target_prop}.
(** Lemmas for building rewrite rules. *)
Definition PropOp_iff (Op : Prop -> Prop -> Prop) :=
forall (p1 p2 q1 q2:Prop), (p1 <-> q1) -> (p2 <-> q2) -> (Op p1 p2 <-> Op q1 q2).
Definition PropUOp_iff (Op : Prop -> Prop) :=
forall (p1 q1 :Prop), (p1 <-> q1) -> (Op p1 <-> Op q1).

(* Specifications: *)
Lemma mkapp2 (S1 S2 S3 T1 T2 T3 : Type) (Op : S1 -> S2 -> S3)
(I1 : S1 -> T1) (I2 : S2 -> T2) (I3 : S3 -> T3)
(TBOP : T1 -> T2 -> T3)
(TBOPINJ : forall n m, I3 (Op n m) = TBOP (I1 n) (I2 m))
(s1 : S1) (t1 : T1) (P1: I1 s1 = t1)
(s2 : S2) (t2 : T2) (P2: I2 s2 = t2):  I3 (Op s1 s2) = TBOP t1 t2.
Lemma mkapp (S1 S2 T1 T2 : Type) (OP : S1 -> S2)
(I1 : S1 -> T1)
(I2 : S2 -> T2)
(TUOP : T1 -> T2)
(TUOPINJ : forall n, I2 (OP n) = TUOP (I1 n))
(s1: S1) (t1: T1) (P1: I1 s1 = t1): I2 (OP s1) = TUOP t1.
Lemma mkrel (S T : Type) (R : S -> S -> Prop)
(I : S -> T)
(TR : T -> T -> Prop)
(TRINJ : forall n m : S, R n m <->  TR (I n) (I m))
(s1 : S) (t1 : T) (P1 : I s1 = t1)
(s2 : S) (t2 : T) (P2 : I s2 = t2):
R s1 s2 <-> TR t1 t2.