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Section group_morphism.
(* An example with default canonical structures *)
Variable A B : Type.
Variable plusA : A -> A -> A.
Variable plusB : B -> B -> B.
Variable zeroA : A.
Variable zeroB : B.
Variable eqA : A -> A -> Prop.
Variable eqB : B -> B -> Prop.
Variable phi : A -> B.
Record img := {
ia : A;
ib :> B;
prf : phi ia = ib
}.
Parameter eq_img : forall (i1:img) (i2:img),
eqB (ib i1) (ib i2) -> eqA (ia i1) (ia i2).
Lemma phi_img (a:A) : img.
intro a.
exists a (phi a).
refine ( refl_equal _).
Defined.
Canonical Structure phi_img.
Lemma zero_img : img.
exists zeroA zeroB.
admit.
Defined.
Canonical Structure zero_img.
Lemma plus_img : img -> img -> img.
intros i1 i2.
exists (plusA (ia i1) (ia i2)) (plusB (ib i1) (ib i2)).
admit.
Defined.
Canonical Structure plus_img.
(* Print Canonical Projections. *)
Goal forall a1 a2, eqA (plusA a1 zeroA) a2.
intros a1 a2.
refine (eq_img _ _ _).
change (eqB (plusB (phi a1) zeroB) (phi a2)).
Admitted.
End group_morphism.
Open Scope type_scope.
Section type_reification.
Inductive term :Type :=
Fun : term -> term -> term
| Prod : term -> term -> term
| Bool : term
| SET :term
| PROP :term
| TYPE :term
| Var : Type -> term.
Fixpoint interp (t:term) :=
match t with
Bool => bool
| SET => Set
| PROP => Prop
| TYPE => Type
| Fun a b => interp a -> interp b
| Prod a b => interp a * interp b
| Var x => x
end.
Record interp_pair :Type :=
{ repr:>term;
abs:>Type;
link: abs = interp repr }.
Lemma prod_interp :forall (a b:interp_pair),a * b = interp (Prod a b) .
proof.
let a:interp_pair,b:interp_pair.
reconsider thesis as (a * b = interp a * interp b).
thus thesis by (link a),(link b).
end proof.
Qed.
Lemma fun_interp :forall (a b:interp_pair), (a -> b) = interp (Fun a b).
proof.
let a:interp_pair,b:interp_pair.
reconsider thesis as ((a -> b) = (interp a -> interp b)).
thus thesis using rewrite (link a);rewrite (link b);reflexivity.
end proof.
Qed.
Canonical Structure ProdCan (a b:interp_pair) :=
Build_interp_pair (Prod a b) (a * b) (prod_interp a b).
Canonical Structure FunCan (a b:interp_pair) :=
Build_interp_pair (Fun a b) (a -> b) (fun_interp a b).
Canonical Structure BoolCan :=
Build_interp_pair Bool bool (refl_equal _).
Canonical Structure VarCan (x:Type) :=
Build_interp_pair (Var x) x (refl_equal _).
Canonical Structure SetCan :=
Build_interp_pair SET Set (refl_equal _).
Canonical Structure PropCan :=
Build_interp_pair PROP Prop (refl_equal _).
Canonical Structure TypeCan :=
Build_interp_pair TYPE Type (refl_equal _).
(* Print Canonical Projections. *)
Variable A:Type.
Variable Inhabited: term -> Prop.
Variable Inhabited_correct: forall p, Inhabited (repr p) -> abs p.
Lemma L : Prop * A -> bool * (Type -> Set) .
refine (Inhabited_correct _ _).
change (Inhabited (Fun (Prod PROP (Var A)) (Prod Bool (Fun TYPE SET)))).
Admitted.
Check L : abs _ .
End type_reification.
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