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(* 02110-1301 USA                                                     *)


(*****************************************************************************)
(*          Projet Formel - Calculus of Inductive Constructions V5.10        *)
(*****************************************************************************)
(*									     *)
(*  Free Monoid functor is the Left adjoint of the Forgetful functor         *)
(*									     *)
(*****************************************************************************)
(*									     *)
(*                     A. SAIBI	  May 95                  		     *)
(*									     *)
(*****************************************************************************)


Require Export FunFreeMon.
Require Export FunForget_UA.
Require Export Th_Adjoint. 

Set Implicit Arguments.
Unset Strict Implicit.

Definition LeftAdj_FunForget := LeftAdjUA UA_FM.

Lemma Adj_FunFreeMon_FunForget : FunFreeMon =_F Adjoint LeftAdj_FunForget.
Proof.
(* Goal: @Equal_Functor SET MON FunFreeMon (@Adjoint MON SET FunForget LeftAdj_FunForget) *)
unfold Equal_Functor in |- *; intros A B f.
(* Goal: @Equal_hom MON (@FOb SET MON FunFreeMon A) (@FOb SET MON FunFreeMon B) (@FMor SET MON FunFreeMon A B f) (@FOb SET MON (@Adjoint MON SET FunForget LeftAdj_FunForget) A) (@FOb SET MON (@Adjoint MON SET FunForget LeftAdj_FunForget) B) (@FMor SET MON (@Adjoint MON SET FunForget LeftAdj_FunForget) A B f) *)
apply Build_Equal_hom; simpl in |- *.
(* Goal: @Equal_MonMor (@AdjointUA_ob MON SET FunForget UA_FM A) (@AdjointUA_ob MON SET FunForget UA_FM B) (@FMor SET MON FunFreeMon A B f) (@FMor SET MON (@AdjointUA MON SET FunForget UA_FM) A B f) *)
unfold Equal_MonMor in |- *; simpl in |- *.
(* Goal: @Ext (Tlist_setoid A) (Tlist_setoid B) (@Dist_map A B f) (@UA_FM_diese_map A (FreeMonoid B) (@Comp SET A B (Tlist_setoid B) f (UA_FM_mor B))) *)
unfold Ext in |- *; simpl in |- *; intro l.
(* Goal: @Equal_Tlist B (@Dist_fun A B f l) (@UA_FM_diese_fun A (FreeMonoid B) (@Comp SET A B (Tlist_setoid B) f (UA_FM_mor B)) l) *)
elim l; simpl in |- *.
(* Goal: @Equal_Tlist B (@Dist_fun A B f t) (@UA_FM_diese_fun A (FreeMonoid B) (@Comp SET A B (Tlist_setoid B) f (UA_FM_mor B)) t) *)
trivial.
(* Goal: forall (c : Carrier A) (t : Tlist A) (_ : @Equal_Tlist B (@Dist_fun A B f t) (@UA_FM_diese_fun A (FreeMonoid B) (@Comp SET A B (Tlist_setoid B) f (UA_FM_mor B)) t)), and (@Equal B (@Ap A B f c) (@Ap A B f c)) (@Equal_Tlist B (@Dist_fun A B f t) (@UA_FM_diese_fun A (FreeMonoid B) (@Comp SET A B (Tlist_setoid B) f (UA_FM_mor B)) t)) *)
intros c t H; split.
(* Goal: @Equal B (@Ap A B f c) (@Ap A B f c) *)
(* Goal: @Equal_Tlist B (@Dist_fun A B f t) (@UA_FM_diese_fun A (FreeMonoid B) (@Comp SET A B (Tlist_setoid B) f (UA_FM_mor B)) t) *)
apply Refl.
(* Goal: @Equal_Tlist B (@Dist_fun A B f t) (@UA_FM_diese_fun A (FreeMonoid B) (@Comp SET A B (Tlist_setoid B) f (UA_FM_mor B)) t) *)
trivial.
Qed.