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+(* -*- mode: coq; mode: visual-line -*- *)
+Set Universe Polymorphism.
+Set Primitive Projections.
+Close Scope nat_scope.
+
+Record prod (A B : Type) := pair { fst : A ; snd : B }.
+Arguments pair {A B} _ _.
+Arguments fst {A B} _ / .
+Arguments snd {A B} _ / .
+Notation "x * y" := (prod x y) : type_scope.
+Notation "( x , y , .. , z )" := (pair .. (pair x y) .. z) : core_scope.
+
+Definition Type1 := Eval hnf in let gt := (Set : Type@{i}) in Type@{i}.
+Definition Type2 := Eval hnf in let gt := (Type1 : Type@{i}) in Type@{i}.
+
+Inductive paths {A : Type} (a : A) : A -> Type :=
+  idpath : paths a a.
+Arguments idpath {A a} , [A] a.
+Notation "x = y" := (@paths _ x y) : type_scope.
+Definition concat {A} {x y z : A} (p : x = y) (q : y = z) : x = z
+  := match p, q with idpath, idpath => idpath end.
+
+Definition path_prod {A B : Type} (z z' : A * B)
+: (fst z = fst z') -> (snd z = snd z') -> (z = z').
+Proof.
+  destruct z, z'; simpl; intros [] []; reflexivity.
+Defined.  
+
+Module Type TypeM.
+  Parameter m : Type2.
+End TypeM.
+
+Module ProdM (XM : TypeM) (YM : TypeM) <: TypeM.
+  Definition m := XM.m * YM.m.
+End ProdM.
+
+Module Type FunctionM (XM YM : TypeM).
+  Parameter m : XM.m -> YM.m.
+End FunctionM.
+
+Module IdmapM (XM : TypeM) <: FunctionM XM XM.
+  Definition m := (fun x => x) : XM.m -> XM.m.
+End IdmapM.
+
+Module Type HomotopyM (XM YM : TypeM) (fM gM : FunctionM XM YM).
+  Parameter m : forall x, fM.m x = gM.m x.
+End HomotopyM.
+
+Module ComposeM (XM YM ZM : TypeM)
+       (gM : FunctionM YM ZM) (fM : FunctionM XM YM)
+       <: FunctionM XM ZM.
+  Definition m := (fun x => gM.m (fM.m x)).
+End ComposeM.
+
+Module Type CorecM (YM ZM : TypeM) (fM : FunctionM YM ZM)
+       (XM : TypeM) (gM : FunctionM XM ZM).
+  Parameter m : XM.m -> YM.m.
+  Parameter m_beta : forall x, fM.m (m x) = gM.m x.
+End CorecM.
+
+Module Type CoindpathsM (YM ZM : TypeM) (fM : FunctionM YM ZM)
+       (XM : TypeM) (hM kM : FunctionM XM YM).
+  Module fhM := ComposeM XM YM ZM fM hM.
+  Module fkM := ComposeM XM YM ZM fM kM.
+  Declare Module mM (pM : HomotopyM XM ZM fhM fkM)
+    : HomotopyM XM YM hM kM.
+End CoindpathsM.
+
+Module Type Comodality (XM : TypeM).
+  Parameter m : Type2.
+  Module mM <: TypeM.
+    Definition m := m.
+  End mM.
+  Parameter from : m -> XM.m.
+  Module fromM <: FunctionM mM XM.
+    Definition m := from.
+  End fromM.
+  Declare Module corecM : CorecM mM XM fromM.
+  Declare Module coindpathsM : CoindpathsM mM XM fromM.
+End Comodality.
+
+Module Comodality_Theory (F : Comodality).
+
+  Module F_functor_M (XM YM : TypeM) (fM : FunctionM XM YM)
+         (FXM : Comodality XM) (FYM : Comodality YM).
+    Module f_o_from_M <: FunctionM FXM.mM YM.
+      Definition m := fun x => fM.m (FXM.from x).
+    End f_o_from_M.
+    Module mM := FYM.corecM FXM.mM f_o_from_M.
+    Definition m := mM.m.
+  End F_functor_M.
+
+  Module F_prod_cmp_M (XM YM : TypeM)
+         (FXM : Comodality XM) (FYM : Comodality YM).
+    Module PM := ProdM XM YM.
+    Module PFM := ProdM FXM FYM.
+    Module fstM <: FunctionM PM XM.
+      Definition m := @fst XM.m YM.m.
+    End fstM.
+    Module sndM <: FunctionM PM YM.
+      Definition m := @snd XM.m YM.m.
+    End sndM.
+    Module FPM := F PM.
+    Module FfstM := F_functor_M PM XM fstM FPM FXM.
+    Module FsndM := F_functor_M PM YM sndM FPM FYM.
+    Definition m : FPM.m -> PFM.m
+      := fun z => (FfstM.m z , FsndM.m z).
+  End F_prod_cmp_M.
+
+  Module isequiv_F_prod_cmp_M
+         (XM YM : TypeM)
+         (FXM : Comodality XM) (FYM : Comodality YM).
+    (** The comparison map *)
+    Module cmpM := F_prod_cmp_M XM YM FXM FYM.
+    Module FPM := cmpM.FPM.
+    (** We construct an inverse to it using corecursion. *)
+    Module prod_from_M <: FunctionM cmpM.PFM cmpM.PM.
+      Definition m : cmpM.PFM.m -> cmpM.PM.m
+        := fun z => ( FXM.from (fst z) , FYM.from (snd z) ).
+    End prod_from_M.
+    Module cmpinvM <: FunctionM cmpM.PFM FPM
+      := FPM.corecM cmpM.PFM prod_from_M.
+    (** We prove the first homotopy *)
+    Module cmpinv_o_cmp_M <: FunctionM FPM FPM
+      := ComposeM FPM cmpM.PFM FPM cmpinvM cmpM.
+    Module idmap_FPM <: FunctionM FPM FPM
+      := IdmapM FPM.
+    Module cip_FPM := FPM.coindpathsM FPM cmpinv_o_cmp_M idmap_FPM.
+    Module cip_FPHM <: HomotopyM FPM cmpM.PM cip_FPM.fhM cip_FPM.fkM.
+      Definition m : forall x, cip_FPM.fhM.m@{i j} x = cip_FPM.fkM.m@{i j} x.
+      Proof.
+        intros x.
+        refine (concat (cmpinvM.m_beta@{i j} (cmpM.m@{i j} x)) _).
+        apply path_prod@{i i i}; simpl.
+        - exact (cmpM.FfstM.mM.m_beta@{i j} x).
+        - exact (cmpM.FsndM.mM.m_beta@{i j} x).
+      Defined.
+    Fail End cip_FPHM.
+(*  End isequiv_F_prod_cmp_M.
+
+End Comodality_Theory.*)