Library Coq.micromega.ZifyClasses

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 : S -> T;
      pred : T -> Prop;

      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 : T1 -> T2 -> T3;

      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 : T1 -> T2;

      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 : T;

      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 -> Prop;

SatOk states the correctness of the reasoning

SatOk : forall x y, PArg1 x -> PArg2 y -> PRes (Op x y)
}.

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 ).

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.

Hardcoded support and lemma for propositional logic

Lemma and_morph : forall (s1 s2 t1 t2:Prop), s1 <-> t1 -> s2 <-> t2 -> ((s1 /\ s2 ) <-> (t1 /\ t2 )).

Lemma or_morph : forall (s1 s2 t1 t2:Prop), s1 <-> t1 -> s2 <-> t2 -> ((s1 \/ s2 ) <-> (t1 \/ t2 )).

Lemma impl_morph : forall (s1 s2 t1 t2:Prop), s1 <-> t1 -> s2 <-> t2 -> ((s1 -> s2 ) <-> (t1 -> t2 )).

Lemma iff_morph : forall (s1 s2 t1 t2:Prop), s1 <-> t1 -> s2 <-> t2 -> ((s1 <-> s2 ) <-> (t1 <-> t2 )).

Lemma not_morph : forall (s1 t1:Prop), s1 <-> t1 -> (not s1 ) <-> (not t1 ).

Lemma eq_iff : forall (P Q : Prop), P = Q -> (P <-> Q ).

Lemma rew_iff (P Q : Prop) (IFF : P <-> Q) : P -> Q.

Registering constants for use by the plugin

Propositional logic

Library Coq.micromega.ZifyClasses

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