(* ========================================================================== *)
(* FLYSPECK - BOOK FORMALIZATION                                              *)
(*                                                                            *)
(* Chapter: Fan                                                               *)
(* Author: Alexey Solovyev                                                    *)
(* Date: 2010-04-06                                                           *)
(*                                                                            *)
(* Equivalence of inverse_sigma_fan and inverse1_sigma_fan                    *)
(* ========================================================================== *)

(* TODO: results from this file should be added into introduction.hl *)

(* flyspeck_needs "fan/fan_defs.hl";;*)

module Fan_misc = struct


let inverse_sigma_fan = new_definition `inverse_sigma_fan x V E v = inverse(extension_sigma_fan x V E v)`;;

let inverse1_sigma_fan=new_definition`inverse1_sigma_fan (x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (v:real^3)= @g. (!w:real^3. {v,w} IN E==> {v, g w} IN E)
/\ (!w:real^3. {v,w} IN E==> (sigma_fan x V E v)( g w) =w)
/\ (!w:real^3. {v,w} IN E==> g (sigma_fan x V E v w) =w)`;;

(* We are using this definition from tame_defs.hl *)
let dart1_of_fan = new_definition
  `dart1_of_fan ((V:A->bool),(E:(A->bool)->bool)) =  { (v,w) | {v,w} IN E }`;;



let REMOVE_ASSUM = POP_ASSUM (fun th -> ALL_TAC);;


let EXTENSION_SIGMA_FAN_EQ_RES = prove(`!x V E v. extension_sigma_fan x V E v = res (sigma_fan x V E v) (set_of_edge v V E)`,
   REPEAT GEN_TAC THEN REWRITE_TAC[FUN_EQ_THM; Fan.extension_sigma_fan; Sphere.res] THEN
     MESON_TAC[]);;


(* ------------------------------------------------------------------------- *)
(* inverse_sigma_fan_bis is the inverse of extension_sigma_fan                   *)
(* ------------------------------------------------------------------------- *)

let INVERSE_SIGMA_FAN = prove(`!x V E v. FAN (x,V,E) 
			      ==> (extension_sigma_fan x V E v) o (inverse_sigma_fan x V E v) = I
                                  /\ (inverse_sigma_fan x V E v) o (extension_sigma_fan x V E v) = I`,
   REPEAT GEN_TAC THEN REWRITE_TAC[inverse_sigma_fan] THEN
     DISCH_TAC THEN
     MATCH_MP_TAC PERMUTES_INVERSES_o THEN
     DISJ_CASES_TAC (MESON[] `(?u:real^3. {v, u} IN E) \/ (!u. ~({v,u} IN E))`) THENL
     [
       ASM_MESON_TAC[Fan.permutes_sigma_fan];
       ALL_TAC
     ] THEN

     EXISTS_TAC `{}:real^3->bool` THEN
     REWRITE_TAC[PERMUTES_EMPTY; FUN_EQ_THM] THEN
     X_GEN_TAC `w:real^3` THEN
     REWRITE_TAC[I_THM; Fan.extension_sigma_fan] THEN
     REWRITE_TAC[Fan.set_of_edge; IN_ELIM_THM] THEN
     ASM_MESON_TAC[]);;
			       


let EXTENSION_SIGMA_FAN_INJECTIVE = prove(`!x V E v. FAN (x,V,E) ==> (!u w. extension_sigma_fan x V E v u = extension_sigma_fan x V E v w ==> u = w)`,
    REPEAT GEN_TAC THEN
      DISCH_THEN (MP_TAC o (fun th -> CONJUNCT2 (SPEC `v:real^3` (MATCH_MP INVERSE_SIGMA_FAN th)))) THEN
      ABBREV_TAC `f = extension_sigma_fan (x:real^3) V E v` THEN
      ABBREV_TAC `g = inverse_sigma_fan (x:real^3) V E v` THEN
      REPEAT STRIP_TAC THEN
      POP_ASSUM (MP_TAC o (fun th -> AP_TERM `g:real^3->real^3` th)) THEN
      POP_ASSUM MP_TAC THEN
      REWRITE_TAC[FUN_EQ_THM; o_THM; I_THM] THEN
      MESON_TAC[]);;
     


(* ------------------------------------------------------------------------- *)
(* Connection between dart1_of_fan and set_of_edge                           *)
(* ------------------------------------------------------------------------- *)
	 
let IN_SET_OF_EDGE = prove(`!V E (v:A) w. UNIONS E SUBSET V /\ (v, w) IN dart1_of_fan (V,E) ==> v IN V /\ w IN V /\ w IN set_of_edge v V E /\ v IN set_of_edge w V E`,
   REPEAT GEN_TAC THEN STRIP_TAC THEN
     SUBGOAL_THEN `v:A IN V /\ w IN V` STRIP_ASSUME_TAC THENL
     [
       POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC THEN
	 REWRITE_TAC[SUBSET; IN_UNIONS; dart1_of_fan; IN_ELIM_THM; PAIR_EQ] THEN
	 STRIP_TAC THEN STRIP_TAC THEN
	 FIRST_ASSUM (MP_TAC o SPEC `v:A`) THEN
	 FIRST_X_ASSUM (MP_TAC o SPEC `w:A`) THEN
	 ANTS_TAC THENL [EXISTS_TAC `{v':A,w'}` THEN ASM SET_TAC[]; ALL_TAC] THEN
	 DISCH_THEN (fun th -> REWRITE_TAC[th]) THEN
	 ANTS_TAC THENL [EXISTS_TAC `{v':A,w'}` THEN ASM SET_TAC[]; ALL_TAC] THEN
	 REWRITE_TAC[];
       ALL_TAC
     ] THEN
     ASM_REWRITE_TAC[] THEN
     POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC THEN
     REWRITE_TAC[dart1_of_fan; Fan.set_of_edge; IN_ELIM_THM; PAIR_EQ] THEN
     MESON_TAC[SET_RULE `{w, v} = {v, w}`]);;


let FAN_IN_SET_OF_EDGE = prove(`!x V E v w. FAN (x,V,E) /\ {v,w} IN E ==> v IN V /\ w IN V /\ w IN set_of_edge v V E /\ v IN set_of_edge w V E`,
   REPEAT GEN_TAC THEN STRIP_TAC THEN
     MATCH_MP_TAC IN_SET_OF_EDGE THEN
     REWRITE_TAC[dart1_of_fan; IN_ELIM_THM; PAIR_EQ] THEN
     CONJ_TAC THENL
     [
       REMOVE_ASSUM THEN POP_ASSUM MP_TAC THEN
	 SIMP_TAC[Fan.FAN];
       ASM_MESON_TAC[]
     ] );;
	 
	 
(* ------------------------------------------------------------------------- *)
(* For fans, inverse1_sigma_fan and inverse_sigma_fan_bis are equivalent         *)
(* ------------------------------------------------------------------------- *)

let INVERSE_SIGMA_FAN_EQ_INVERSE1_SIGMA_FAN = prove(`!x V E v w. FAN (x,V,E) /\ {v, w} IN E ==> 
						      inverse1_sigma_fan x V E v w = inverse_sigma_fan x V E v w`,
    REPEAT GEN_TAC THEN
      DISCH_TAC THEN
      FIRST_ASSUM (STRIP_ASSUME_TAC o fun th -> (MATCH_MP FAN_IN_SET_OF_EDGE th)) THEN
      MP_TAC (SPECL [`x:real^3`; `V:real^3->bool`; `E:(real^3->bool)->bool`; `v:real^3`] EXTENSION_SIGMA_FAN_INJECTIVE) THEN
      ASM_REWRITE_TAC[] THEN
      ABBREV_TAC `f = extension_sigma_fan (x:real^3) V E v` THEN
      ABBREV_TAC `g = inverse1_sigma_fan (x:real^3) V E v` THEN
      ABBREV_TAC `h = inverse_sigma_fan (x:real^3) V E v` THEN
      DISCH_TAC THEN
      FIRST_X_ASSUM (MP_TAC o SPECL [`(g:real^3->real^3) w`; `(h:real^3->real^3) w`]) THEN
      ANTS_TAC THEN REWRITE_TAC[] THEN

      SUBGOAL_THEN `(f:real^3->real^3) ((g:real^3->real^3) w) = w` ASSUME_TAC THENL
      [
	REMOVE_ASSUM THEN POP_ASSUM (fun th -> REWRITE_TAC[SYM th]) THEN
	  POP_ASSUM (fun th -> REWRITE_TAC[SYM th]) THEN
	  REWRITE_TAC[Fan.extension_sigma_fan] THEN
	  MP_TAC (SPECL [`x:real^3`; `V:real^3->bool`; `E:(real^3->bool)->bool`; `v:real^3`] Fan.INVERSE1_SIGMA_FAN) THEN
	  ASM_REWRITE_TAC[] THEN
	  STRIP_TAC THEN
	  REMOVE_ASSUM THEN
	  REPLICATE_TAC 2 (FIRST_X_ASSUM (MP_TAC o SPEC `w:real^3`)) THEN
	  ASM_REWRITE_TAC[Fan.set_of_edge; IN_ELIM_THM] THEN
	  SIMP_TAC[] THEN
	  ABBREV_TAC `u:real^3 = inverse1_sigma_fan x V E v w` THEN
          DISCH_TAC THEN
          SUBGOAL_THEN `u:real^3 IN V` (fun th -> SIMP_TAC[th]) THEN
	  ASM_MESON_TAC[FAN_IN_SET_OF_EDGE];
	ALL_TAC
      ] THEN

      POP_ASSUM (fun th -> REWRITE_TAC[th]) THEN
      REPLICATE_TAC 3 (POP_ASSUM (fun th -> REWRITE_TAC[SYM th])) THEN
      ASM_MESON_TAC[INVERSE_SIGMA_FAN; FUN_EQ_THM; I_THM; o_THM]);;


end;;