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Require Export Morphisms Setoid.
Class lops := lmk_ops {
car: Type;
weq: relation car
}.
Implicit Arguments car [].
Coercion car: lops >-> Sortclass.
Instance weq_Equivalence `{lops}: Equivalence weq.
Proof.
Admitted.
Module lset.
Canonical Structure lset_ops A := lmk_ops (list A) (fun h k => True).
End lset.
Class ops := mk_ops {
ob: Type;
mor: ob -> ob -> lops;
dot: forall n m p, mor n m -> mor m p -> mor n p
}.
Coercion mor: ops >-> Funclass.
Implicit Arguments ob [].
Instance dot_weq `{ops} n m p: Proper (weq ==> weq ==> weq) (dot n m p).
Proof.
Admitted.
Section s.
Import lset.
Context `{X:lops} {I: Type}.
Axiom sup : forall (f: I -> X) (J : list I), X.
Global Instance sup_weq: Proper (pointwise_relation _ weq ==> weq ==> weq) sup.
Proof.
Admitted.
End s.
Axiom ord : forall (n : nat), Type.
Axiom seq : forall n, list (ord n).
Infix "==" := weq (at level 79).
Infix "*" := (dot _ _ _) (left associativity, at level 40).
Notation "∑_ ( i ∈ l ) f" := (@sup (mor _ _) _ (fun i => f) l)
(at level 41, f at level 41, i, l at level 50).
Axiom dotxsum : forall `{X : ops} I J n m p (f: I -> X m n) (x: X p m) y,
x * (∑_(i∈ J) f i) == y.
Definition mx X n m := ord n -> ord m -> X.
Section bsl.
Context `{X : ops} {u: ob X}.
Notation U := (car (@mor X u u)).
Lemma toto n m p q (M : mx U n m) N (P : mx U p q) Q i j : ∑_(j0 ∈ seq m) M i j0 * (∑_(j1 ∈ seq p) N j0 j1 * P j1 j) == Q.
Proof.
Fail setoid_rewrite dotxsum.
(* Toplevel input, characters 0-22:
Error:
Tactic failure:setoid rewrite failed: Unable to satisfy the rewriting constraints.
Unable to satisfy the following constraints:
UNDEFINED EVARS:
?101==[X u n m p q M N P Q i j j0 |- U] (goal evar)
?106==[X u n m p q M N P Q i j |- relation (X u u)] (internal placeholder)
?107==[X u n m p q M N P Q i j |- relation (list (ord m))]
(internal placeholder)
?108==[X u n m p q M N P Q i j (do_subrelation:=do_subrelation)
|- Proper (pointwise_relation (ord m) weq ==> ?107 ==> ?106) sup]
(internal placeholder)
?109==[X u n m p q M N P Q i j |- ProperProxy ?107 (seq m)]
(internal placeholder)
?110==[X u n m p q M N P Q i j |- relation (X u u)] (internal placeholder)
?111==[X u n m p q M N P Q i j (do_subrelation:=do_subrelation)
|- Proper (?106 ==> ?110 ==> Basics.flip Basics.impl) weq]
(internal placeholder)
?112==[X u n m p q M N P Q i j |- ProperProxy ?110 Q] (internal placeholder)UNIVERSES:
{} |= Top.14 <= Top.37
Top.25 <= Top.24
Top.25 <= Top.32
ALGEBRAIC UNIVERSES:{}
UNDEFINED UNIVERSES:METAS:
470[y] := ?101 : car (?99 ?467 ?465)
469[x] := M i _UNBOUND_REL_1 : car (?99 ?467 ?466) [type is checked]
468[f] := fun i : ?463 => N _UNBOUND_REL_2 i * P i j :
?463 -> ?99 ?466 ?465 [type is checked]
467[p] := u : ob ?99 [type is checked]
466[m] := u : ob ?99 [type is checked]
465[n] := u : ob ?99 [type is checked]
464[J] := seq p : list ?463 [type is checked]
463[I] := ord p : Type [type is checked]
*)
Abort.
End bsl.
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