HOL html


(* ========================================================================== *)
(* FLYSPECK - BOOK FORMALIZATION                                              *)
(*                                                                            *)
(* Chapter: Fan                                              *)
(* Author: Hoang Le Truong                                        *)
(* Date: 2010-02-09                                                           *)
(* ========================================================================== *)



(*flyspeck_needs "general/sphere.hl";;*)


module Fan  = struct


(*
# use "/home/truong/Desktop/googlecode/hol_light/hol.ml";;
needs "/home/truong/Desktop/googlecode/hol_light/Multivariate/flyspeck.ml";;
needs "/home/truong/Desktop/googlecode/flyspeck/text_formalization/general/sphere.hl";;
*)

		
open Sphere;;
open Fan_defs;;
open Tactic_fan;;
open Lemma_fan;;		
open Hypermap;;


(* ========================================================================== *)
(*                                FAN                       *)
(* ========================================================================== *)



let graph = new_definition `graph E <=> (!e. E e ==> (e HAS_SIZE 2))`;;
let fan1 = new_definition`fan1(x,V,E):bool <=>  FINITE V /\ ~(V SUBSET {}) `;;
let fan2 = new_definition`fan2(x,V,E):bool <=>   ~(x IN V)`;;
let fan3=new_definition`fan3(x,V,E):bool <=> (!v. (v IN V) ==> cyclic_set {w | {v,w} IN E } x v)`;;
let fan4 = new_definition`fan4(x,V,E):bool<=>  (!e. (e IN E) ==> (aff_gt {x} e  INTER V={}))`;;
let fan5 = new_definition`fan5(x,V,E):bool<=> (!e f. (e IN E)/\ (f IN E) /\ ~(e=f) ==> (aff_gt {x} e INTER aff_gt {x} f ={}))`;;
let fan = new_definition`fan(x,V,E)<=>  ((UNIONS E) SUBSET V) /\ graph(E)/\ fan1(x,V,E)/\ fan2(x,V,E)/\ fan3(x,V,E)/\ fan4 (x,V,E) /\ fan5(x,V,E)`;;
let base_point_fan=new_definition`base_point_fan (x,V,E)=x`;;
let set_of_edge=new_definition`set_of_edge v V E={w|{v,w} IN E /\ w IN V}`;;

let fan6= new_definition`fan6(x,V,E):bool<=>(!e. (e IN E) ==> ~(collinear ({x} UNION e)))`;;
let fan7= new_definition`fan7(x,V,E):bool<=> (!e1 e2. (e1 IN E UNION {{v}| v IN V}) /\ (e2 IN E UNION {{v}| v IN V})
==> ((aff_ge {x} e1) INTER (aff_ge {x} e2) = aff_ge {x} (e1 INTER e2)))`;;


let FAN=new_definition`FAN(x,V,E) <=> ((UNIONS E) SUBSET V) /\ graph(E) /\ fan1(x,V,E) /\ fan2(x,V,E)/\
fan6(x,V,E)/\ fan7(x,V,E)`;;


(* ========================================================================== *)
(*                            Invariance theorems                  *)
(* ========================================================================== *)



let GRAPH = prove
 (`!E. graph E <=> !e. e IN E ==> e HAS_SIZE 2`,
  REWRITE_TAC[graph; IN]);;
let CYCLIC_SET = prove
 (`cyclic_set W v w <=>
         ~(v = w) /\
         FINITE W /\
         (!p q h. p IN W /\ q IN W /\ p - q = h % (v - w) ==> p = q) /\
         W INTER affine hull {v, w} = {}`,
  REWRITE_TAC[cyclic_set; IN; VECTOR_ARITH `p - q:real^N = r <=> p = q + r`]);;
let CYCLIC_SET_TRANSLATION_EQ = prove
 (`!a:real^N s x y.
    cyclic_set (IMAGE (\x. a + x) s) (a + x) (a + y) <=> cyclic_set s x y`,
  REWRITE_TAC[CYCLIC_SET] THEN GEOM_TRANSLATE_TAC[]);;
let CYCLIC_SET_LINEAR_IMAGE = prove
 (`!f:real^M->real^N s x y.
        linear f /\ (!x y. f x = f y ==> x = y)
        ==> (cyclic_set (IMAGE f s) (f x) (f y) <=> cyclic_set s x y)`,
  GEN_REWRITE_TAC (funpow 4 BINDER_CONV o RAND_CONV o TOP_DEPTH_CONV)
   [CYCLIC_SET; RIGHT_FORALL_IMP_THM; IMP_CONJ; FORALL_IN_IMAGE] THEN
  GEOM_TRANSFORM_TAC[]);;
add_translation_invariants [CYCLIC_SET_TRANSLATION_EQ];;

add_linear_invariants [CYCLIC_SET_LINEAR_IMAGE];;

let GRAPH_TRANSLATION_EQ = prove
 (`!a:real^N E. graph (IMAGE (IMAGE (\x. a + x)) E) <=> graph E`,
  REWRITE_TAC[GRAPH] THEN GEOM_TRANSLATE_TAC[]);;
let GRAPH_LINEAR_IMAGE_EQ = prove
 (`!f:real^M->real^N E.
    linear f /\ (!x y. f x = f y ==> x = y)
    ==> (graph(IMAGE (IMAGE f) E) <=> graph E)`,
  REWRITE_TAC[GRAPH] THEN GEOM_TRANSFORM_TAC[]);;
let FAN1_TRANSLATION_EQ = prove
 (`!a:real^N x V E.
        fan1(a + x,IMAGE (\x. a + x) V,IMAGE (IMAGE (\x. a + x)) E) <=>
        fan1(x,V,E)`,
  REWRITE_TAC[fan1] THEN GEOM_TRANSLATE_TAC[]);;
let FAN1_LINEAR_IMAGE_EQ = prove
 (`!f:real^M->real^N x V E.
    linear f /\ (!x y. f x = f y ==> x = y)
    ==> (fan1(f x,IMAGE f V,IMAGE (IMAGE f) E) <=> fan1(x,V,E))`,
  REWRITE_TAC[fan1] THEN GEOM_TRANSFORM_TAC[]);;
let FAN2_TRANSLATION_EQ = prove
 (`!a:real^N x V E.
        fan2(a + x,IMAGE (\x. a + x) V,IMAGE (IMAGE (\x. a + x)) E) <=>
        fan2(x,V,E)`,
  REWRITE_TAC[fan2] THEN GEOM_TRANSLATE_TAC[]);;
let FAN2_LINEAR_IMAGE_EQ = prove
 (`!f:real^M->real^N x V E.
    linear f /\ (!x y. f x = f y ==> x = y)
    ==> (fan2(f x,IMAGE f V,IMAGE (IMAGE f) E) <=> fan2(x,V,E))`,
  REWRITE_TAC[fan2] THEN GEOM_TRANSFORM_TAC[]);;
let FAN3_TRANSLATION_EQ = prove
 (`!a:real^N x V E.
        fan3(a + x,IMAGE (\x. a + x) V,IMAGE (IMAGE (\x. a + x)) E) <=>
        fan3(x,V,E)`,
  REWRITE_TAC[fan3] THEN GEOM_TRANSLATE_TAC[]);;
let FAN3_LINEAR_IMAGE_EQ = prove
 (`!f:real^M->real^N x V E.
    linear f /\ (!x y. f x = f y ==> x = y)
    ==> (fan3(f x,IMAGE f V,IMAGE (IMAGE f) E) <=> fan3(x,V,E))`,
  let lemma = prove
   (`{w | {a,w} IN IMAGE (IMAGE f) s} =
     IMAGE f {w |w| {a,f w} IN IMAGE (IMAGE f) s}`,
    MATCH_MP_TAC(SET_RULE
     `(!y. P y ==> ?x. y = f x) ==> {w | P w} = IMAGE f {w | P(f w)}`) THEN
    REWRITE_TAC[IN_IMAGE; EXTENSION; IN_IMAGE; IN_INSERT; NOT_IN_EMPTY] THEN
    MESON_TAC[]) in
  REWRITE_TAC[fan3; FORALL_IN_IMAGE; lemma] THEN GEOM_TRANSFORM_TAC[]);;
let FAN4_TRANSLATION_EQ = prove
 (`!a:real^N x V E.
        fan4(a + x,IMAGE (\x. a + x) V,IMAGE (IMAGE (\x. a + x)) E) <=>
        fan4(x,V,E)`,
  REWRITE_TAC[fan4] THEN GEOM_TRANSLATE_TAC[]);;
let FAN4_LINEAR_IMAGE_EQ = prove
 (`!f:real^M->real^N x V E.
    linear f /\ (!x y. f x = f y ==> x = y)
    ==> (fan4(f x,IMAGE f V,IMAGE (IMAGE f) E) <=> fan4(x,V,E))`,
  REWRITE_TAC[fan4] THEN GEOM_TRANSFORM_TAC[]);;
let FAN5_TRANSLATION_EQ = prove
 (`!a:real^N x V E.
        fan5(a + x,IMAGE (\x. a + x) V,IMAGE (IMAGE (\x. a + x)) E) <=>
        fan5(x,V,E)`,
  REWRITE_TAC[fan5] THEN GEOM_TRANSLATE_TAC[]);;
let FAN5_LINEAR_IMAGE_EQ = prove
 (`!f:real^M->real^N x V E.
    linear f /\ (!x y. f x = f y ==> x = y)
    ==> (fan5(f x,IMAGE f V,IMAGE (IMAGE f) E) <=> fan5(x,V,E))`,
  GEN_REWRITE_TAC (funpow 4 BINDER_CONV o RAND_CONV o TOP_DEPTH_CONV)
   [fan5; IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_IMAGE] THEN
  GEOM_TRANSFORM_TAC[]);;
let FAN6_TRANSLATION_EQ = prove
 (`!a:real^N x V E.
        fan6(a + x,IMAGE (\x. a + x) V,IMAGE (IMAGE (\x. a + x)) E) <=>
        fan6(x,V,E)`,
  REWRITE_TAC[fan6] THEN GEOM_TRANSLATE_TAC[]);;
let FAN6_LINEAR_IMAGE_EQ = prove
 (`!f:real^M->real^N x V E.
    linear f /\ (!x y. f x = f y ==> x = y)
    ==> (fan6(f x,IMAGE f V,IMAGE (IMAGE f) E) <=> fan6(x,V,E))`,
  REWRITE_TAC[fan6] THEN GEOM_TRANSFORM_TAC[]);;
let FAN7_TRANSLATION_EQ = prove
 (`!a:real^N x V E.
        fan7(a + x,IMAGE (\x. a + x) V,IMAGE (IMAGE (\x. a + x)) E) <=>
        fan7(x,V,E)`,
  REWRITE_TAC[fan7] THEN
  REWRITE_TAC[SET_RULE
   `e IN s UNION {f x | t x} <=> e IN s \/ ?x. t x /\ e = f x`] THEN
  GEOM_TRANSLATE_TAC[]);;
let FAN7_LINEAR_IMAGE_EQ = prove
 (`!f:real^M->real^N x V E.
    linear f /\ (!x y. f x = f y ==> x = y)
    ==> (fan7(f x,IMAGE f V,IMAGE (IMAGE f) E) <=> fan7(x,V,E))`,
  REWRITE_TAC[fan7; IN_UNION] THEN
  REWRITE_TAC[LEFT_OR_DISTRIB; RIGHT_OR_DISTRIB;
              TAUT `a \/ b ==> c <=> (a ==> c) /\ (b ==> c)`] THEN
  GEN_REWRITE_TAC (funpow 4 BINDER_CONV o RAND_CONV o TOP_DEPTH_CONV)
   [FORALL_AND_THM; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
  REWRITE_TAC[FORALL_IN_GSPEC] THEN REWRITE_TAC[IMP_IMP] THEN
  GEOM_TRANSFORM_TAC[]);;
add_translation_invariants
 [GRAPH_TRANSLATION_EQ;
  FAN1_TRANSLATION_EQ;
  FAN2_TRANSLATION_EQ;
  FAN3_TRANSLATION_EQ;
  FAN4_TRANSLATION_EQ;
  FAN5_TRANSLATION_EQ;
  FAN6_TRANSLATION_EQ;
  FAN7_TRANSLATION_EQ];;

add_linear_invariants
 [GRAPH_LINEAR_IMAGE_EQ;
  FAN1_LINEAR_IMAGE_EQ;
  FAN2_LINEAR_IMAGE_EQ;
  FAN3_LINEAR_IMAGE_EQ;
  FAN4_LINEAR_IMAGE_EQ;
  FAN5_LINEAR_IMAGE_EQ;
  FAN6_LINEAR_IMAGE_EQ;
  FAN7_LINEAR_IMAGE_EQ];;

let FAN_TRANSLATION_EQ = prove
 (`!a:real^N x V E.
        FAN(a + x,IMAGE (\x. a + x) V,IMAGE (IMAGE (\x. a + x)) E) <=>
        FAN(x,V,E)`,
  REWRITE_TAC[FAN] THEN GEOM_TRANSLATE_TAC[]);;
let FAN_LINEAR_IMAGE_EQ = prove
 (`!f:real^M->real^N x V E.
    linear f /\ (!x y. f x = f y ==> x = y)
    ==> (FAN(f x,IMAGE f V,IMAGE (IMAGE f) E) <=> FAN(x,V,E))`,
  REWRITE_TAC[FAN] THEN GEOM_TRANSFORM_TAC[]);;
add_translation_invariants [FAN_TRANSLATION_EQ];;
add_linear_invariants [FAN_LINEAR_IMAGE_EQ];;

let BASE_POINT_FAN_TRANSLATION_EQ = prove
 (`!a x V E.
    base_point_fan(a + x,IMAGE (\x. a + x) V,IMAGE (IMAGE (\x. a + x)) E) =
    a + base_point_fan(x,V,E)`,
  REWRITE_TAC[base_point_fan]);;
let BASE_POINT_FAN_LINEAR_IMAGE_EQ = prove
 (`!f:real^M->real^N x V E.
        linear f
        ==> base_point_fan(f x,IMAGE f V,IMAGE (IMAGE f) E) =
            f(base_point_fan(x,V,E))`,
  REWRITE_TAC[base_point_fan]);;
let SET_OF_EDGE_TRANSLATION_EQ = prove
 (`!a:real^N x V E.
        set_of_edge (a + x) (IMAGE (\x. a + x) V)
                    (IMAGE (IMAGE (\x. a + x)) E) =
        IMAGE (\x. a + x) (set_of_edge x V E)`,
  REWRITE_TAC[set_of_edge] THEN GEOM_TRANSLATE_TAC[]);;
let SET_OF_EDGE_LINEAR_IMAGE_EQ = prove
 (`!f:real^M->real^N x V E.
        linear f /\ (!x y. f x = f y ==> x = y)
        ==> set_of_edge (f x) (IMAGE f V) (IMAGE (IMAGE f) E) =
            IMAGE f (set_of_edge x V E)`,
  let lemma = prove
   (`{w | {a,w} IN IMAGE (IMAGE f) s /\ P w} =
     {f w |w| {a,f w} IN IMAGE (IMAGE f) s /\ P(f w)}`,
    MATCH_MP_TAC(SET_RULE
     `(!y. P y ==> ?x. y = f x) ==> {w | P w} = {f w |w| P(f w)}`) THEN
    REWRITE_TAC[IN_IMAGE; EXTENSION; IN_IMAGE; IN_INSERT; NOT_IN_EMPTY] THEN
    MESON_TAC[]) in
  REWRITE_TAC[set_of_edge; lemma] THEN
  GEOM_TRANSFORM_TAC[] THEN SET_TAC[]);;
add_translation_invariants
 [BASE_POINT_FAN_TRANSLATION_EQ; SET_OF_EDGE_TRANSLATION_EQ];;

add_linear_invariants
 [BASE_POINT_FAN_LINEAR_IMAGE_EQ; SET_OF_EDGE_LINEAR_IMAGE_EQ];;







let CTVTAQA=prove(`!(x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (E1:(real^3->bool)->bool).
FAN(x,V,E) /\ E1 SUBSET E  
==>
FAN(x,V,E1)`,
REPEAT GEN_TAC
THEN REWRITE_TAC[FAN;fan1;fan2;fan6;fan7;graph]
THEN ASM_SET_TAC[]);;


(* ========================================================================== *)
(*                        SET_OF_EDGE                     *)
(* ========================================================================== *)


let set_of_edge_subset_edges=prove(`!(V:real^3->bool) (E:(real^3->bool)->bool) (v:real^3).
 set_of_edge v V E SUBSET V`,
REWRITE_TAC[set_of_edge] THEN SET_TAC[]);;


let remark_fan1=prove(`!v w V E. 
(v IN V) /\ (w IN V) ==>
((w IN set_of_edge v V E)<=>(v IN set_of_edge w V E))`, 

(let lemma=prove(`{w,v}={v,w}`, SET_TAC[]) in
REPEAT GEN_TAC THEN STRIP_TAC THEN REWRITE_TAC[set_of_edge;IN_ELIM_THM]  THEN ASM_REWRITE_TAC[] THEN MESON_TAC[lemma]));;
let remark_finite_fan1=prove(`!v:real^3 (V:real^3->bool) (E:(real^3->bool)->bool). FINITE V ==> FINITE (set_of_edge v V E)`,
REPEAT GEN_TAC THEN DISCH_TAC 
  THEN MP_TAC(ISPECL [`(set_of_edge (v:real^3) (V:real^3->bool) (E:(real^3->bool)->bool))`; `V:real^3->bool`] FINITE_SUBSET) 
  THEN ASM_REWRITE_TAC[]
  THEN REWRITE_TAC[set_of_edge] THEN SET_TAC[] );;
let properties_of_set_of_edge=prove(`!v:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) u:real^3.  
UNIONS E SUBSET V
==>
({v,u} IN E <=> u IN set_of_edge v V E)`,
REPEAT GEN_TAC THEN REWRITE_TAC[UNIONS; SUBSET; set_of_edge ; IN_ELIM_THM] 
THEN STRIP_TAC
 THEN POP_ASSUM(fun th-> MP_TAC(ISPEC `u:real^3`th)) THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN DISCH_TAC
  THEN POP_ASSUM(fun th-> MP_TAC(ISPEC `{(v:real^3),(u:real^3)}`th)) THEN SET_TAC[]);;



let expand_edge_graph_fan=prove(`!(x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (e:real^3->bool).
FAN(x,V,E) /\ e IN E
==> ?v:real^3 w:real^3. e={v,w}`,
REPEAT GEN_TAC
THEN REWRITE_TAC[FAN;IN]
THEN STRIP_TAC
THEN FIND_ASSUM MP_TAC `graph (E:(real^3->bool)->bool)`
THEN REWRITE_TAC[graph]
THEN DISCH_TAC
THEN POP_ASSUM(fun th-> MP_TAC(ISPEC`e:real^3->bool`th))
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[HAS_SIZE]
THEN STRIP_TAC
THEN SUBGOAL_THEN `~((e:real^3->bool)={})` ASSUME_TAC
THENL[
STRIP_TAC
THEN MP_TAC(ISPEC`(e:real^3->bool)`CARD_EQ_0)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC 
THEN ARITH_TAC;
MP_TAC(SET_RULE `~((e:real^3->bool)={})==> ?v:real^3. v IN e`)
THEN RESA_TAC
THEN SUBGOAL_THEN`~((e:real^3->bool) DELETE v={})` ASSUME_TAC
THENL[
STRIP_TAC
THEN MP_TAC(ISPECL[`v:real^3`;`(e:real^3->bool)`;]CARD_DELETE)
THEN RESA_TAC
THEN MP_TAC(ISPECL[`(e:real^3->bool)`;`v:real^3`] FINITE_DELETE)
THEN RESA_TAC
THEN MP_TAC(ISPEC`(e:real^3->bool) DELETE v`CARD_EQ_0)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN ARITH_TAC;
MP_TAC(SET_RULE `~((e:real^3->bool)DELETE v={})==> ?w:real^3. w IN (e:real^3->bool)DELETE v/\ w IN e`)
THEN RESA_TAC
THEN MP_TAC(SET_RULE `(v IN (e:real^3->bool))/\ (w IN (e:real^3->bool))==> {v,w} SUBSET e`)
THEN RESA_TAC
THEN MP_TAC(SET_RULE `(w IN (e:real^3->bool)DELETE v)==> ~(v=w)`)
THEN RESA_TAC
THEN MP_TAC(ISPECL [`{v:real^3,w:real^3}`;`(e:(real^3->bool))`] FINITE_SUBSET)
THEN RESA_TAC
THEN ASSUME_TAC(SET_RULE `v:real^3 IN {v:real^3,w:real^3} `)
THEN MP_TAC(ISPECL[`v:real^3`;`{v:real^3,w:real^3}`;]CARD_DELETE)
THEN RESA_TAC
THEN MP_TAC(SET_RULE `v IN {v,w}==>{v:real^3,w:real^3} DELETE v PSUBSET {v,w}`)
THEN RESA_TAC
THEN MP_TAC(ISPECL[`{v:real^3,w:real^3} DELETE v`;`{v:real^3,w:real^3}`]CARD_PSUBSET)
THEN POP_ASSUM (fun th->REWRITE_TAC[th])
THEN FIND_ASSUM MP_TAC`FINITE {v:real^3,w:real^3}`
THEN DISCH_TAC
THEN POP_ASSUM (fun th->REWRITE_TAC[th])
THEN DISCH_TAC
THEN MP_TAC(ARITH_RULE`CARD ({v, w} DELETE v) < CARD {v, w}/\ CARD ({v, w} DELETE v) = CARD {v, w}-1
<=>CARD ({v, w} DELETE v) +1= CARD {v:real^3, w:real^3}`)
THEN POP_ASSUM (fun th->REWRITE_TAC[th])
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th;])
THEN REWRITE_TAC[ARITH_RULE`A=A`]
THEN DISCH_TAC
THEN SUBGOAL_THEN `w:real^3 IN ({v:real^3,w:real^3} DELETE v)` ASSUME_TAC
THENL[
ASM_SET_TAC[];
MP_TAC(ISPECL[`{v:real^3,w:real^3}`;`v:real^3`] FINITE_DELETE)
THEN RESA_TAC
THEN MP_TAC(ISPECL[`w:real^3`;`{v:real^3,w:real^3} DELETE v`;]CARD_DELETE)
THEN RESA_TAC
THEN MP_TAC(SET_RULE `w IN ({v,w} DELETE v)==>{v:real^3,w:real^3} DELETE v DELETE w PSUBSET {v,w} DELETE v`)
THEN RESA_TAC
THEN MP_TAC(ISPECL[`{v:real^3,w:real^3} DELETE v DELETE w`;`{v:real^3,w:real^3} DELETE v`]CARD_PSUBSET)
THEN POP_ASSUM (fun th->REWRITE_TAC[th])
THEN FIND_ASSUM MP_TAC`FINITE ({v:real^3,w:real^3} DELETE v)`
THEN DISCH_TAC
THEN POP_ASSUM (fun th->REWRITE_TAC[th])
THEN DISCH_TAC
THEN MP_TAC(ARITH_RULE`CARD ({v, w} DELETE v DELETE w) < CARD ({v, w} DELETE v)/\ CARD ({v, w} DELETE v DELETE w) = CARD ({v, w} DELETE v)-1
<=>CARD ({v, w} DELETE v DELETE w) +1= CARD ({v:real^3, w:real^3} DELETE v)`)
THEN POP_ASSUM (fun th->REWRITE_TAC[th])
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th;])
THEN REWRITE_TAC[ARITH_RULE`A=A`]
THEN DISCH_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM (fun th->REWRITE_TAC[])
THEN POP_ASSUM (fun th->REWRITE_TAC[])
THEN ASSUME_TAC(SET_RULE `{v, w} DELETE v:real^3 DELETE w:real^3={}`)
THEN POP_ASSUM (fun th->REWRITE_TAC[th;CARD_CLAUSES; ARITH_RULE `0+1=1`])
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"B")
THEN DISCH_TAC
THEN REMOVE_THEN "B" MP_TAC
THEN POP_ASSUM (fun th->REWRITE_TAC[SYM(th);ARITH_RULE` 1+1=2`])
THEN FIND_ASSUM MP_TAC`CARD (e:real^3->bool)=2`
THEN DISCH_TAC
THEN POP_ASSUM (fun th->REWRITE_TAC[SYM(th)])
THEN DISCH_TAC
THEN MP_TAC(ISPECL[`{v:real^3,w:real^3}`;`e:real^3->bool`]CARD_SUBSET_EQ)
THEN POP_ASSUM (fun th->REWRITE_TAC[SYM(th)])
THEN RESA_TAC
THEN EXISTS_TAC`v:real^3`
THEN EXISTS_TAC`w:real^3`
THEN POP_ASSUM (fun th->REWRITE_TAC[SYM(th)])]]]);; 





let add_edge_graph_fan=prove(`!(V:A->bool) (E:(A->bool)->bool) v:A u:A.
 v IN V /\ u IN V /\ E1=E UNION {{v,u}} /\ UNIONS E SUBSET V==> UNIONS E1 SUBSET V`,
REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[UNIONS; SUBSET; UNION; IN_ELIM_THM;IN_SING]
THEN DISCH_THEN(LABEL_TAC "YEU")
THEN REPEAT STRIP_TAC
THENL[SUBGOAL_THEN`(?u. u IN E /\ x IN (u:A->bool))`ASSUME_TAC
THENL[
EXISTS_TAC`u':A->bool` THEN ASM_REWRITE_TAC[];
REMOVE_THEN "YEU" (fun th-> MRESA1_TAC th `x:A`)];
ASM_TAC THEN SET_TAC[]]);;

let garph_add_edge_is_garph=prove(`!(V:A->bool) (E:(A->bool)->bool) v:A u:A.
 E1=E UNION {{v,u}} /\ ~(v=u) /\ graph E==> graph E1`,
REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[GRAPH; UNION;IN_ELIM_THM;IN_SING]
THEN DISCH_THEN(LABEL_TAC"YEU")
THEN GEN_TAC THEN STRIP_TAC
THENL[REMOVE_THEN "YEU" (fun th-> MRESA1_TAC th `e:A->bool`);
ASM_REWRITE_TAC[HAS_SIZE] THEN STRIP_TAC
THEN MRESA_TAC CARD_2_FAN[`v:A`;`u:A`]
THEN SIMP_TAC[HAS_SIZE; CARD_CLAUSES; FINITE_INSERT; FINITE_EMPTY;
                 IN_INSERT; NOT_IN_EMPTY]]);;
let add_edge_into_collinear_fan=prove(`!x:real^3 (E:(real^3->bool)->bool) v:real^3 u:real^3.
~collinear {x,v,u} /\
(!e. e IN E ==> ~collinear ({x} UNION e))
==>
(!e. e IN E UNION {{v, u}} ==> ~collinear ({x} UNION e))`,
REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[]
THEN GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV)[UNION;]
THEN REWRITE_TAC[IN_ELIM_THM;IN_SING]
THEN POP_ASSUM MP_TAC THEN DISCH_THEN (LABEL_TAC"YEU")
THEN STRIP_TAC
THENL[REMOVE_THEN "YEU"(fun th-> MRESA1_TAC th `e:real^3->bool`);
ASM_REWRITE_TAC[SET_RULE`{X} UNION {A,B}={X,A,B}`]]);;

let properties_edges_eq_fan=prove(`!e:A-> bool v:A u:A.
FINITE e /\ ~(e={v,u}) /\ ~(v=u)  /\ CARD e=2  ==>  ~(v IN e)\/ ~(u IN e)`,
REPEAT GEN_TAC THEN STRIP_TAC
THEN POP_ASSUM MP_TAC THEN DISCH_THEN(LABEL_TAC "YEU")
THEN ONCE_REWRITE_TAC[SET_RULE`~A \/ ~B<=> ~(A/\B)`]
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`v IN e /\ u IN e ==> {v:A,u} SUBSET e`)
THEN RESA_TAC
THEN MRESA_TAC CARD_2_FAN[`v:A`;`u:A`]
THEN REMOVE_THEN "YEU" MP_TAC
THEN POP_ASSUM (fun th-> REWRITE_TAC[SYM(th)])
THEN STRIP_TAC
THEN MRESA_TAC CARD_SUBSET_EQ[`{v,u:A}`;`e:A->bool`]);;



(* ========================================================================== *)
(*                        BASIC FAN                     *)
(* ========================================================================== *)


let set_edges_is_finite_fan=prove(`!(x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool).
FAN(x,V,E)==> FINITE E`,
REPEAT GEN_TAC
THEN REWRITE_TAC[FAN;fan1;graph]
THEN STRIP_TAC
THEN MP_TAC(ISPEC `V:real^3->bool`FINITE_POWERSET)
THEN RESA_TAC
THEN MP_TAC(SET_RULE`UNIONS (E:(real^3->bool)->bool) SUBSET (V:real^3->bool)==> E SUBSET {t| t SUBSET V}`) 
THEN RESA_TAC
  THEN MP_TAC(ISPECL [`(E:(real^3->bool)->bool)`;`{t| t SUBSET (V:real^3->bool)}`] FINITE_SUBSET)
THEN ASM_SET_TAC[]);;


let properties_of_set_of_edge_fan=prove(`!x:real^3  (V:real^3->bool) (E:(real^3->bool)->bool) v:real^3 u:real^3.  
FAN(x,V,E)
==>
({v,u} IN E <=> u IN set_of_edge v V E)`,
REWRITE_TAC[FAN] THEN REPEAT GEN_TAC THEN REWRITE_TAC[UNIONS; SUBSET; set_of_edge ; IN_ELIM_THM] 
THEN STRIP_TAC THEN REPEAT (POP_ASSUM MP_TAC) THEN DISCH_THEN (LABEL_TAC"a") THEN STRIP_TAC 
 THEN REMOVE_THEN "a"(fun th-> MP_TAC(ISPEC `u:real^3`th)) THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN DISCH_TAC
  THEN POP_ASSUM(fun th-> MP_TAC(ISPEC `{(v:real^3),(u:real^3)}`th)) THEN SET_TAC[]);;


let properties_of_graph=prove(`!(x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (u:real^3) (v:real^3). 
FAN(x,V,E) /\{v,u} IN E==> v IN V`,

REPEAT GEN_TAC THEN REWRITE_TAC[FAN] THEN STRIP_TAC THEN REPEAT (POP_ASSUM MP_TAC) THEN DISCH_THEN(LABEL_TAC "a") THEN REPEAT STRIP_TAC THEN
REMOVE_THEN "a" MP_TAC THEN REWRITE_TAC[UNIONS; SUBSET; IN_ELIM_THM] THEN DISCH_TAC 
  THEN POP_ASSUM(fun th-> MP_TAC(ISPEC`v:real^3` th)) THEN DISCH_TAC
THEN POP_ASSUM MATCH_MP_TAC THEN EXISTS_TAC `{(v:real^3),(u:real^3)}` THEN ASM_TAC THEN SET_TAC[]);;

let properties_of_graph1=prove(`!(x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (u:real^3) (v:real^3). 
FAN(x,V,E) /\{v,u} IN E==> u IN V`,
ONCE_REWRITE_TAC[SET_RULE`{A,B}={B,A}`] THEN SET_TAC[properties_of_graph]);;

let th4=prove(`!(x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (u:real^3) (v:real^3). 
FAN(x,V,E) /\{v,u} IN E==> ~(x=v)`,
REPEAT GEN_TAC THEN STRIP_TAC THEN MP_TAC(ISPECL[`(x:real^3)`;` (V:real^3->bool)`;` (E:(real^3->bool)->bool)`;` (u:real^3)`; `(v:real^3)`]properties_of_graph) THEN ASM_REWRITE_TAC[] THEN REPEAT(POP_ASSUM MP_TAC) THEN REWRITE_TAC[FAN;fan2] THEN SET_TAC[]);;
let remark4_fan=prove(`!(x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (u:real^3) (v:real^3). 
FAN(x,V,E) /\ {v,u} IN E==> ~ (x=v) /\ ~(x=u) /\ DISJOINT {x,v} {u}/\ DISJOINT {x,u} {v} /\ DISJOINT {x} {v,u} /\ ~(u IN aff {x,v})`,

REPEAT GEN_TAC THEN REWRITE_TAC[FAN;fan6] THEN STRIP_TAC THEN POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC THEN DISCH_THEN(LABEL_TAC"a") THEN REPEAT DISCH_TAC THEN REMOVE_THEN"a"(fun th->MP_TAC(ISPEC`{(v:real^3),(u:real^3)}`th)) THEN ASM_REWRITE_TAC[SET_RULE`{a} UNION {b,c}={a,b,c}`] THEN REWRITE_TAC[th3]);;

let collinears_fan=prove(`!(x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (u:real^3) (v:real^3). 
FAN(x,V,E) /\{v,u} IN E==>( ~ collinear {x,v,u} <=> ~(u IN aff {x,v}))`,
MESON_TAC[remark4_fan;collinear_fan]);;
let remark1_fan=prove(`!(x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (u:real^3) (v:real^3). 
FAN(x,V,E) ==>
FINITE (set_of_edge v V E) 
/\({v,u} IN E <=> u IN set_of_edge v V E)/\
({v,u} IN E==>
~ collinear {x,v,u}/\
~ (x=v) /\ ~(x=u)/\ ~(v=u) /\ DISJOINT {x,v} {u}/\ DISJOINT {x,u} {v} /\DISJOINT {x} {v,u}  /\ ~(u IN aff {x,v}) /\ v IN V /\ u IN V/\ ( ~ collinear {x,v,u} <=> ~(u IN aff {x,v})))`,
MESON_TAC[FAN;fan1;remark_finite_fan1;remark4_fan; properties_of_graph; SET_RULE`DISJOINT {x,v} {u} ==> ~(v=u)`;collinears_fan;properties_of_set_of_edge_fan;properties_of_graph1]);;





(* ========================================================================== *)
(*                       EXISTS  SIGMA_FAN                     *)
(* ========================================================================== *)



let sigma_fan=new_definition`sigma_fan (x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (v:real^3) (u:real^3)=  
(if (set_of_edge v V E = {u} ) then u 
    else (@(w:real^3).  (w IN (set_of_edge v V E)) /\ ~(w=u) /\
(!(w1:real^3). (w1 IN (set_of_edge v V E))/\ ~(w1=u) ==> azim x v u w <= azim x v u w1)))`;;

(* This extends sigma to have a larger domain
   of R^3 rather than just V.  It is the
   identity outside V (compare the definition
   of `permutes` ).   *)




let extension_sigma_fan=new_definition`extension_sigma_fan (x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (v:real^3) (u:real^3)=  
(if ~(u IN set_of_edge v V E ) then u 
    else (sigma_fan x V E v u))`;;




let exists_sigma_fan=prove(`
(!(x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (v:real^3) (u:real^3).
~ (set_of_edge v V E= {u} ) 
/\ FAN(x,V,E) /\ (u IN (set_of_edge v V E))
==>
(?(w:real^3). (w IN (set_of_edge v V E)) /\ ~(w=u) /\
(!(w1:real^3). (w1 IN (set_of_edge v V E))/\ ~(w1=u) ==> azim x v u w <= azim x v u w1)))`,
(let lemma = prove
   (`!X:real->bool. 
          FINITE X /\ ~(X = {}) 
          ==> ?a. a IN X /\ !x. x IN X ==> a <= x`,
    MESON_TAC[INF_FINITE]) in


MP_TAC(lemma) THEN DISCH_THEN(LABEL_TAC "a") THEN REPEAT GEN_TAC THEN REWRITE_TAC[FAN;fan1] THEN STRIP_TAC THEN MP_TAC (ISPECL[`(v:real^3)` ;`(V:real^3->bool)`; `(E:(real^3->bool)->bool)`]remark_finite_fan1) THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN
  
SUBGOAL_THEN `FINITE ((set_of_edge (v:real^3) (V:real^3->bool) (E:(real^3->bool)->bool)) DELETE  (u:real^3))` ASSUME_TAC
THENL[(*2*)
ASM_MESON_TAC[FINITE_DELETE_IMP];(*2*)

DISJ_CASES_TAC(SET_RULE`((set_of_edge (v:real^3) (V:real^3->bool) (E:(real^3->bool)->bool)) DELETE  (u:real^3)={})\/
 ~((set_of_edge (v:real^3) (V:real^3->bool) (E:(real^3->bool)->bool)) DELETE  (u:real^3)={})`)
THENL[(*3*)
POP_ASSUM MP_TAC THEN REWRITE_TAC[DELETE;EXTENSION;IN_ELIM_THM] THEN DISCH_TAC
  THEN SUBGOAL_THEN `set_of_edge (v:real^3) (V:real^3->bool) (E:(real^3->bool)->bool)={u:real^3}` ASSUME_TAC
THENL[(*4*)
ASM_TAC THEN SET_TAC[];(*4*)
ASM_TAC THEN SET_TAC[]](*4*);(*3*)
SUBGOAL_THEN`~(IMAGE (azim x v u) ((set_of_edge (v:real^3) (V:real^3->bool) (E:(real^3->bool)->bool)) DELETE  (u:real^3))={})` ASSUME_TAC
THENL[(*4*)
REWRITE_TAC[IMAGE_EQ_EMPTY] THEN ASM_MESON_TAC[];(*4*)

SUBGOAL_THEN` FINITE (IMAGE (azim x v u) ((set_of_edge (v:real^3) (V:real^3->bool) (E:(real^3->bool)->bool)) DELETE  (u:real^3)))` ASSUME_TAC
THENL[(*5*)
ASM_MESON_TAC[FINITE_IMAGE];(*5*)
REMOVE_THEN "a" (fun th ->MP_TAC(ISPEC `(IMAGE (azim x v u) ((set_of_edge (v:real^3) (V:real^3->bool) (E:(real^3->bool)->bool)) DELETE  (u:real^3)))` th)) 
  THEN ASM_REWRITE_TAC[IMAGE;DELETE;IN_ELIM_THM] THEN STRIP_TAC THEN EXISTS_TAC`x':real^3`
  THEN ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[]
](*5*)](*4*)](*3*)](*2*)(*1*)));;


let SIGMA_FAN=prove(`!(x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (v:real^3) (u:real^3).
~ (set_of_edge v V E= {u} ) 
/\ FAN(x,V,E) /\ (u IN (set_of_edge v V E))
==>
((sigma_fan x V E v u) IN (set_of_edge v V E)) /\ ~((sigma_fan x V E v u)=u) /\
(!(w1:real^3). (w1 IN (set_of_edge v V E))/\ ~(w1=u) ==> azim x v u (sigma_fan x V E v u) <= azim x v u w1)`,

REPEAT GEN_TAC THEN STRIP_TAC THEN ONCE_REWRITE_TAC[sigma_fan]
  THEN MP_TAC(ISPECL[`(x:real^3)`; `(V:real^3->bool)`; `(E:(real^3->bool)->bool)`; `(v:real^3)`; `(u:real^3)`]exists_sigma_fan)  THEN ASM_REWRITE_TAC[] THEN STRIP_TAC THEN SELECT_ELIM_TAC THEN EXISTS_TAC`w:real^3` THEN ASM_REWRITE_TAC[]);;




let sigma_fan_in_set_of_edge=prove(`!(x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (v:real^3) (u:real^3).
FAN(x,V,E) /\ (u IN (set_of_edge v V E))
==>
((sigma_fan x V E v u) IN (set_of_edge v V E))`,

REPEAT STRIP_TAC THEN DISJ_CASES_TAC(SET_RULE`~ (set_of_edge v V E= {u:real^3} )\/ (set_of_edge v V E= {u} )`)
THENL[
 MP_TAC(ISPECL[`(x:real^3)`; `(V:real^3->bool)`; `(E:(real^3->bool)->bool)`; `(v:real^3)`; `(u:real^3)`]SIGMA_FAN) 
THEN ASM_TAC 
THEN SET_TAC[];
ASM_REWRITE_TAC[sigma_fan;IN_SING]]);;


(* ========================================================================== *)
(*                   CONDITION UNIQUE FOINT AND SIGMA_FAN                     *)
(* ========================================================================== *)





let UNIQUE_FOINT_FAN=prove(`!(x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (v:real^3) (u:real^3) w:real^3.
FAN(x,V,E) /\ ({v,u} IN E) /\ ({v,w} IN E)/\
 ~(aff_ge {x} {v,u} INTER aff_ge {x} {v,w}=aff_ge {x} {v}) 
==> u=w`,
REPEAT GEN_TAC THEN REWRITE_TAC[FAN;fan7;fan6] THEN STRIP_TAC 
  THEN POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC
  THEN DISCH_THEN (LABEL_TAC "a") THEN REPEAT STRIP_TAC
  THEN REMOVE_THEN "a" (fun th-> MP_TAC(ISPECL[`{(v:real^3),(u:real^3)}`;`{(v:real^3),(w:real^3)}`]th)) 
  THEN ASM_REWRITE_TAC[UNION;IN_ELIM_THM;]
  THEN DISJ_CASES_TAC(SET_RULE`~((u:real^3)=(w:real^3))\/ ((u:real^3)=(w:real^3))`)
THENL
[  MP_TAC(SET_RULE`~((u:real^3)=(w:real^3))==> {(v:real^3),(u:real^3)} INTER {(v:real^3),(w:real^3)}={v}`) THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN ASM_REWRITE_TAC[];
ASM_TAC THEN SET_TAC[]]);;
 







let UNIQUE1_POINT_FAN=prove(`!(x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (v:real^3) (u:real^3) w:real^3.
FAN(x,V,E) /\ ({v,u} IN E) /\ ({v,w} IN E)/\ w IN aff_gt {x,v} {u} 
==> u=w`,
REPEAT GEN_TAC THEN STRIP_TAC THEN MATCH_MP_TAC UNIQUE_FOINT_FAN THEN 
EXISTS_TAC `x:real^3` THEN EXISTS_TAC `V:real^3->bool` THEN EXISTS_TAC `E:(real^3->bool)->bool` THEN EXISTS_TAC `v:real^3`
	   THEN ASM_REWRITE_TAC[] THEN REPEAT(POP_ASSUM MP_TAC)
	   THEN REWRITE_TAC[FAN;fan1;fan2;fan6;fan7] THEN REPEAT STRIP_TAC THEN REPEAT(POP_ASSUM MP_TAC)
	   THEN DISCH_TAC THEN  DISCH_TAC THEN DISCH_TAC THEN DISCH_TAC THEN DISCH_TAC
	   THEN DISCH_THEN (LABEL_TAC "a")
	   THEN DISCH_THEN (LABEL_TAC "b") THEN REPEAT DISCH_TAC THEN POP_ASSUM MP_TAC THEN ASM_REWRITE_TAC[]
THEN REMOVE_THEN "a" (fun th-> MP_TAC(ISPEC `{(v:real^3),(w:real^3)}`th) THEN ASSUME_TAC(th)) 
  THEN POP_ASSUM (fun th-> MP_TAC(ISPEC `{(v:real^3),(u:real^3)}`th))
  THEN ASM_REWRITE_TAC[SET_RULE`{a} UNION {b,c}={a,b,c}`]
	   THEN POP_ASSUM MP_TAC THEN DISCH_THEN(LABEL_TAC "c")
	   THEN DISCH_TAC THEN DISCH_TAC
  THEN MP_TAC(ISPECL[`(x:real^3)`;` (v:real^3)`;` (w:real^3)`]th3) THEN ASM_REWRITE_TAC[]
  THEN MP_TAC(ISPECL[`(x:real^3)`;` (v:real^3)`;` (u:real^3)`]th3) THEN ASM_REWRITE_TAC[]
	   THEN DISCH_TAC THEN DISCH_TAC
  THEN MP_TAC(ISPECL[`(x:real^3)`;` (v:real^3)`;` (u:real^3)`]AFF_GT_2_1) THEN ASM_REWRITE_TAC[]
	   THEN DISCH_TAC THEN REMOVE_THEN "c" MP_TAC THEN ASM_REWRITE_TAC[IN_ELIM_THM] THEN STRIP_TAC
	   THEN SUBGOAL_THEN` aff_ge {(x:real^3)} {(v:real^3)} SUBSET aff {x,v}` ASSUME_TAC
THENL (*1*)[
MP_TAC(ISPECL[`(x:real^3)`;` (v:real^3)`;` (u:real^3)`]AFF_GE_1_10) 
THEN MP_TAC(SET_RULE`~((x:real^3) = (v:real^3))==> DISJOINT {x} {v}`)
  THEN ASM_REWRITE_TAC[] THEN DISCH_TAC
  THEN ASM_REWRITE_TAC[]
  THEN DISCH_TAC THEN ASM_REWRITE_TAC[aff;AFFINE_HULL_2] THEN ASM_TAC THEN SET_TAC[];(*1*)
	   
SUBGOAL_THEN `~((w:real^3) IN aff_ge {x} {v})` ASSUME_TAC 
THENL (*2*)[
ASM_TAC THEN SET_TAC[];(*2*)

POP_ASSUM MP_TAC THEN DISCH_THEN (LABEL_TAC "d") THEN DISJ_CASES_TAC(REAL_ARITH `(&0 <= (t2:real))\/ (&0 <= (-- (t2:real)))`)
THENL (*3*)[
 SUBGOAL_THEN `(w:real^3) IN aff_ge {(x:real^3)} {(v:real^3),(u:real^3)}` ASSUME_TAC
THENL (*4*)[
 MP_TAC(ISPECL[`(x:real^3)`;` (v:real^3)`;` (u:real^3)`]AFF_GE_1_2) 
   THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN ASM_REWRITE_TAC[IN_ELIM_THM]
THEN EXISTS_TAC `t1:real` 
  THEN EXISTS_TAC `t2:real` THEN EXISTS_TAC `t3:real` THEN ASM_REWRITE_TAC[]
    THEN REPEAT (POP_ASSUM MP_TAC) THEN REAL_ARITH_TAC;(*4*)

 SUBGOAL_THEN `(w:real^3) IN aff_ge {(x:real^3)} {(v:real^3),(w:real^3)}` ASSUME_TAC
THENL (*5*)[
 MP_TAC(ISPECL[`(x:real^3)`;` (v:real^3)`;` (w:real^3)`]AFF_GE_1_2) 
   THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN ASM_REWRITE_TAC[IN_ELIM_THM]
THEN EXISTS_TAC `&0:real` 
  THEN EXISTS_TAC `&0:real` THEN EXISTS_TAC `&1:real`  THEN REWRITE_TAC[VECTOR_ARITH`w= &0 % x+ &0 % v + &1 % w `] THEN REAL_ARITH_TAC;(*5*)

ASM_TAC THEN SET_TAC[]](*5*)](*4*);(*3*)

 SUBGOAL_THEN `(u:real^3) IN aff_ge {(x:real^3)} {(v:real^3),(w:real^3)}` ASSUME_TAC
THENL (*4*)[
 MP_TAC(ISPECL[`(x:real^3)`;` (v:real^3)`;` (w:real^3)`]AFF_GE_1_2) 
   THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN ASM_REWRITE_TAC[IN_ELIM_THM]
THEN EXISTS_TAC `inv(t3:real) *(--(t1:real))` 
  THEN EXISTS_TAC `inv(t3:real)*(--(t2:real))` THEN EXISTS_TAC `inv(t3:real)` 
THEN MP_TAC(ISPEC `t3:real` REAL_LT_INV) THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN MP_TAC(REAL_ARITH`&0< inv(t3:real)==> (&0 <= inv(t3:real))`) THEN ASM_REWRITE_TAC[] THEN DISCH_TAC
THEN MP_TAC(ISPECL [`inv (t3:real)`;`(--(t2:real))`] REAL_LE_MUL) THEN ASM_REWRITE_TAC[] THEN DISCH_TAC
THEN ASM_REWRITE_TAC[REAL_ARITH`inv(t3:real) * -- (t1:real)+ inv(t3) * --(t2:real) +inv (t3)=inv (t3)*(&1-t1-t2)`] THEN
MP_TAC(REAL_ARITH`(t1:real)+(t2:real)+(t3:real)= &1 ==> &1 - t1- t2=t3`) THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN ASM_REWRITE_TAC[] THEN
MP_TAC(REAL_ARITH`&0<(t3:real)==> ~(t3= &0)`) THEN ASM_REWRITE_TAC[VECTOR_ARITH`
 (inv t3 * --t1) % x +
 (inv t3 * --t2) % v +
 inv t3 % (t1 % x + t2 % v + t3 % u)= (inv t3 * t3) % u `
] THEN DISCH_TAC
THEN MP_TAC(ISPEC `t3:real` REAL_MUL_LINV) THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN ASM_REWRITE_TAC[VECTOR_ARITH ` v= &1 % v`];(*4*)

 SUBGOAL_THEN `(u:real^3) IN aff_ge {(x:real^3)} {(v:real^3),(u:real^3)}` ASSUME_TAC
THENL (*5*)[
 MP_TAC(ISPECL[`(x:real^3)`;` (v:real^3)`;` (u:real^3)`]AFF_GE_1_2) 
   THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN ASM_REWRITE_TAC[IN_ELIM_THM]
THEN EXISTS_TAC `&0:real` 
  THEN EXISTS_TAC `&0:real` THEN EXISTS_TAC `&1:real`  THEN REWRITE_TAC[VECTOR_ARITH`u= &0 % x+ &0 % v + &1 % u `] THEN REAL_ARITH_TAC;(*5*)
ASM_TAC THEN SET_TAC[]](*5*)](*4*)](*3*)](*2*)](*1*));;

let UNIQUE_AZIM_POINT_FAN=prove(`!(x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (v:real^3) (u:real^3) w:real^3 w1:real^3.
FAN(x,V,E) /\ ({v,u} IN E) /\ ({v,w} IN E) /\ { v, w1} IN E /\ (azim x v u w =azim x v u w1) 
==> w=w1`,

REPEAT GEN_TAC THEN STRIP_TAC THEN REPEAT (POP_ASSUM MP_TAC) THEN DISCH_TAC THEN POP_ASSUM(fun th-> MP_TAC(th) THEN ASSUME_TAC(th)) THEN REWRITE_TAC[FAN;fan6] THEN REPEAT STRIP_TAC THEN  REPEAT (POP_ASSUM MP_TAC) THEN DISCH_TAC
  THEN DISCH_TAC THEN DISCH_TAC THEN DISCH_TAC THEN DISCH_TAC THEN DISCH_THEN (LABEL_TAC "a") THEN REPEAT DISCH_TAC
  THEN REMOVE_THEN "a"(fun th ->MP_TAC(ISPEC`{(v:real^3),(u:real^3)}`th) THEN ASSUME_TAC(th))
  THEN POP_ASSUM(fun th ->MP_TAC(ISPEC `{(v:real^3),(w:real^3)}`th) THEN ASSUME_TAC(th))
  THEN POP_ASSUM(fun th ->MP_TAC(ISPEC `{(v:real^3),(w1:real^3)}`th)) THEN ASM_REWRITE_TAC[SET_RULE`{a} UNION {b,c}={a,b,c}`]
  THEN REPEAT STRIP_TAC
  THEN MP_TAC(ISPECL[`(x:real^3)`;`(v:real^3) `;`(u:real^3) `;`w:real^3`;` w1:real^3`] AZIM_EQ) THEN 
ASM_REWRITE_TAC[] THEN DISCH_TAC THEN 
MP_TAC(ISPECL[`(x:real^3)`;` (V:real^3->bool)`;` (E:(real^3->bool)->bool)`;` (v:real^3)`;` (w:real^3)`;` w1:real^3`]UNIQUE1_POINT_FAN) THEN ASM_REWRITE_TAC[]      );;

let UNIQUE_AZIM_0_POINT_FAN=prove(`!(x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (v:real^3) (u:real^3) w:real^3.
FAN(x,V,E) /\ ({v,u} IN E) /\ ({v,w} IN E) /\ (azim x v u w = &0) 
==> u=w`,

REPEAT GEN_TAC THEN STRIP_TAC THEN REPEAT (POP_ASSUM MP_TAC) THEN DISCH_TAC THEN POP_ASSUM(fun th-> MP_TAC(th) THEN ASSUME_TAC(th)) THEN REWRITE_TAC[FAN;fan6] THEN REPEAT STRIP_TAC THEN  REPEAT (POP_ASSUM MP_TAC) THEN DISCH_TAC
  THEN DISCH_TAC THEN DISCH_TAC THEN DISCH_TAC THEN DISCH_TAC THEN DISCH_THEN (LABEL_TAC "a") THEN REPEAT DISCH_TAC
  THEN REMOVE_THEN "a"(fun th ->MP_TAC(ISPEC`{(v:real^3),(u:real^3)}`th) THEN ASSUME_TAC(th))
  THEN POP_ASSUM(fun th ->MP_TAC(ISPEC `{(v:real^3),(w:real^3)}`th)) THEN ASM_REWRITE_TAC[SET_RULE`{a} UNION {b,c}={a,b,c}`]
  THEN REPEAT STRIP_TAC
  THEN MP_TAC(ISPECL[`(x:real^3)`;`(v:real^3) `;`(u:real^3) `;`w:real^3`] AZIM_EQ_0_ALT) THEN 
ASM_REWRITE_TAC[] THEN DISCH_TAC THEN 
MP_TAC(ISPECL[`(x:real^3)`;` (V:real^3->bool)`;` (E:(real^3->bool)->bool)`;` (v:real^3)`;` (u:real^3)`;` w:real^3`]UNIQUE1_POINT_FAN) THEN ASM_REWRITE_TAC[]	       );;



let UNIQUE_SIGMA_FAN=prove(`
(!(x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (v:real^3) (u:real^3) (w:real^3).
~ (set_of_edge v V E= {u} ) 
/\ FAN(x,V,E) /\ (u IN (set_of_edge v V E))
/\ (w IN (set_of_edge v V E)) /\ ~(w=u) /\
(!(w1:real^3). (w1 IN (set_of_edge v V E))/\ ~(w1=u) ==> azim x v u w <= azim x v u w1)
==> sigma_fan x V E v u=w)`,

	(   let th=prove(`!(x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (v:real^3) (u:real^3) w:real^3.
FAN(x,V,E) /\ ({v,u} IN E) /\ ({v,w} IN E)
==> (w IN aff_gt {x,v} {u} ==> u=w)`, MESON_TAC[UNIQUE1_POINT_FAN]
) in

ASSUME_TAC(th) THEN
REPEAT GEN_TAC  THEN STRIP_TAC THEN REPEAT(POP_ASSUM MP_TAC) THEN DISCH_THEN (LABEL_TAC"123") THEN 
DISCH_TAC  THEN DISCH_THEN(LABEL_TAC "1") 
  THEN DISCH_THEN(LABEL_TAC "2") THEN DISCH_THEN(LABEL_TAC "3") THEN DISCH_THEN(LABEL_TAC "4") 
  THEN DISCH_THEN(LABEL_TAC"a") 
  THEN MP_TAC(ISPECL[`(x:real^3)`; `(V:real^3->bool)`; `(E:(real^3->bool)->bool)` ;`(v:real^3)`; `(u:real^3)`]SIGMA_FAN) 
  THEN ASM_REWRITE_TAC[] THEN STRIP_TAC
  THEN REMOVE_THEN "a" (fun th->MP_TAC(ISPEC `sigma_fan (x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (v:real^3) (u:real^3)` th)) 
  THEN ASM_REWRITE_TAC[]
  THEN POP_ASSUM (fun th-> MP_TAC(ISPEC `w:real^3` th)) THEN ASM_REWRITE_TAC[] 
  THEN POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC THEN DISCH_THEN (LABEL_TAC"b")
  THEN REPEAT DISCH_TAC
  THEN SUBGOAL_THEN `azim x v u (sigma_fan (x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (v:real^3) (u:real^3)) = azim x v u (w:real^3)` ASSUME_TAC
THENL[
REPEAT (POP_ASSUM MP_TAC) THEN REAL_ARITH_TAC;
REMOVE_THEN "123" (fun th->MP_TAC (ISPECL[`(x:real^3)`; `(V:real^3->bool)` ;`(E:(real^3->bool)->bool)`; `(v:real^3)` ; `(sigma_fan (x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (v:real^3) (u:real^3))`;`(w:real^3)`]th)) 
THEN ASM_REWRITE_TAC[] THEN DISCH_TAC
THEN
REMOVE_THEN"1" MP_TAC 
THEN  REWRITE_TAC[FAN;fan6;fan7] THEN STRIP_TAC THEN
POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC THEN DISCH_THEN(LABEL_TAC "c") THEN DISCH_THEN (LABEL_TAC "d")
  THEN REMOVE_THEN"2" MP_TAC THEN REMOVE_THEN"3" MP_TAC THEN REMOVE_THEN"b" MP_TAC 
  THEN REWRITE_TAC[set_of_edge;IN_ELIM_THM] THEN REPEAT STRIP_TAC
  THEN REMOVE_THEN "c" MP_TAC THEN DISCH_TAC
  THEN POP_ASSUM (fun th->MP_TAC (ISPEC`{(v:real^3),(w:real^3)}`th) THEN ASSUME_TAC(th))
  THEN POP_ASSUM (fun th->MP_TAC (ISPEC`{(v:real^3),(u:real^3)}`th) THEN ASSUME_TAC(th))
  THEN POP_ASSUM (fun th->MP_TAC (ISPEC`{(v:real^3),(sigma_fan (x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (v:real^3) (u:real^3))}`th) THEN ASSUME_TAC(th))
  THEN REWRITE_TAC[SET_RULE`{a} UNION {b,c}={a,b,c}`] THEN REPEAT STRIP_TAC  
  THEN MP_TAC(ISPECL[`(x:real^3)`; `(v:real^3)`; `(u:real^3)` ;`(sigma_fan (x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (v:real^3) (u:real^3))`; `(w:real^3)`]AZIM_EQ )
  THEN ASM_REWRITE_TAC[]
  THEN ASM_MESON_TAC[]]));; 



(* ========================================================================== *)
(*                   CYCLIC SET FAN                    *)
(* ========================================================================== *)





let CYCLIC_SET_EDGE_FAN=prove(`!(x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (v:real^3).
FAN(x,V,E) /\ v IN V 
==> cyclic_set (set_of_edge v V E) x v`,
REPEAT GEN_TAC 
THEN STRIP_TAC 
THEN REPEAT (POP_ASSUM MP_TAC) 
THEN DISCH_TAC THEN POP_ASSUM(fun th-> MP_TAC(th) 
THEN ASSUME_TAC(th)) THEN REWRITE_TAC[FAN; cyclic_set;fan1;fan2;fan6;fan7] THEN STRIP_TAC THEN STRIP_TAC THEN STRIP_TAC
  THENL(*1*)[ASM_TAC THEN SET_TAC[];(*1*)
STRIP_TAC
  THENL (*2*)[ASM_TAC THEN SET_TAC[remark_finite_fan1];(*2*)
 STRIP_TAC THENL(*3*)[

ASM_REWRITE_TAC[set_of_edge;IN_ELIM_THM] THEN
REPEAT GEN_TAC THEN POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC THEN
  DISCH_THEN(LABEL_TAC "a") THEN   DISCH_THEN(LABEL_TAC "b")
THEN STRIP_TAC THEN STRIP_TAC THEN
REMOVE_THEN "a" (fun th-> MP_TAC(ISPEC `{(v:real^3), (p:real^3)}` th) THEN ASSUME_TAC(th))
  THEN POP_ASSUM (fun th-> MP_TAC(ISPEC `{(v:real^3), (q:real^3)}` th))
  THEN ASM_REWRITE_TAC[SET_RULE`{a} UNION {b,c}={a,b,c}`] THEN REPEAT STRIP_TAC
  THEN MP_TAC(ISPECL[`(x:real^3)`;` (v:real^3)`;` (p:real^3)`]th3) THEN ASM_REWRITE_TAC[]
  THEN MP_TAC(ISPECL[`(x:real^3)`;` (v:real^3)`;` (q:real^3)`]th3) THEN ASM_REWRITE_TAC[]
  THEN POP_ASSUM (fun th-> REWRITE_TAC[])
  THEN POP_ASSUM (fun th-> REWRITE_TAC[])
  THEN POP_ASSUM (fun th-> REWRITE_TAC[SYM(th)] THEN ASSUME_TAC(th))
  THEN REPEAT STRIP_TAC THEN SUBGOAL_THEN `(q:real^3) IN aff_gt {(x:real^3),(v:real^3)} {(p:real^3)}` ASSUME_TAC

THENL (*4*)[
 MP_TAC(ISPECL[`(x:real^3)`;` (v:real^3)`;` (p:real^3)`]AFF_GT_2_1) THEN ASM_REWRITE_TAC[]
	   THEN DISCH_TAC THEN ASM_REWRITE_TAC[IN_ELIM_THM]
   THEN EXISTS_TAC `--(h:real)` THEN EXISTS_TAC `(h:real)` THEN EXISTS_TAC `&1` THEN REWRITE_TAC[VECTOR_ARITH`q= (-- (h:real)) % x + (h:real) % v + &1 % (q+ h% (x-v))`] THEN REAL_ARITH_TAC;(*4*)

ASM_MESON_TAC[UNIQUE1_POINT_FAN]](*4*);(*3*)

REWRITE_TAC[INTER;EXTENSION;IN_ELIM_THM] THEN GEN_TAC 
THEN SUBGOAL_THEN`(x':real^3) IN set_of_edge (v:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) ==> ~(x' IN aff {x,v})`
ASSUME_TAC
THENL(*4*)[
ASM_REWRITE_TAC[set_of_edge;IN_ELIM_THM] THEN
REPEAT GEN_TAC THEN POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC THEN
  DISCH_THEN(LABEL_TAC "a") THEN   DISCH_THEN(LABEL_TAC "b")
THEN STRIP_TAC THEN STRIP_TAC THEN
REMOVE_THEN "a" (fun th-> MP_TAC(ISPEC `{(v:real^3), (x':real^3)}` th) )
  THEN ASM_REWRITE_TAC[SET_RULE`{a} UNION {b,c}={a,b,c}`] THEN STRIP_TAC
  THEN MP_TAC(ISPECL[`(x:real^3)`;` (v:real^3)`;` (x':real^3)`]th3) THEN ASM_MESON_TAC[]; (*4*)

EQ_TAC THENL(*5*)[

POP_ASSUM MP_TAC THEN DISCH_THEN (LABEL_TAC "a") THEN STRIP_TAC THEN REMOVE_THEN "a" MP_TAC THEN ASM_REWRITE_TAC[aff];(*5*)

ASM_TAC THEN SET_TAC[]](*5*)](*4*)] (*3*)]]);; 

let XOHLED=prove(`!(x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (v:real^3).
FAN(x,V,E) /\ v IN V 
==> cyclic_set (set_of_edge v V E) x v`,
MESON_TAC[CYCLIC_SET_EDGE_FAN]);;

(* ========================================================================== *)
(*                   SIGMA_FAN IS PERMUTES     (^_^)                          *)
(* ========================================================================== *)


let azim1=new_definition`azim1 (x:real^3) (v:real^3) (u:real^3) (w:real^3)= &2 * pi- azim x v u w`;;

let exists_inverse_sigma_fan_alt=prove(`
(!(x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (v:real^3) (u:real^3).
~ (set_of_edge v V E= {u} ) 
/\ FAN(x,V,E) /\ (u IN (set_of_edge v V E))
==>
(?(w:real^3). (w IN (set_of_edge v V E)) /\ ~(w=u) /\
(!(w1:real^3). (w1 IN (set_of_edge v V E))/\ ~(w1=u) ==> azim1 x v u w <=  azim1 x v u w1)))
`,
(let lemma = prove
   (`!X:real->bool. 
          FINITE X /\ ~(X = {}) 
          ==> ?a. a IN X /\ !x. x IN X ==> a <= x`,
    MESON_TAC[INF_FINITE]) in
MP_TAC(lemma) THEN DISCH_THEN(LABEL_TAC "a") THEN REPEAT GEN_TAC THEN REWRITE_TAC[FAN;fan1] THEN STRIP_TAC THEN MP_TAC (ISPECL[`(v:real^3)` ;`(V:real^3->bool)`; `(E:(real^3->bool)->bool)`]remark_finite_fan1) THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN
SUBGOAL_THEN `FINITE ((set_of_edge (v:real^3) (V:real^3->bool) (E:(real^3->bool)->bool)) DELETE  (u:real^3))` ASSUME_TAC
THENL[(*2*)

ASM_MESON_TAC[FINITE_DELETE_IMP];(*2*)
DISJ_CASES_TAC(SET_RULE`((set_of_edge (v:real^3) (V:real^3->bool) (E:(real^3->bool)->bool)) DELETE  (u:real^3)={})\/
 ~((set_of_edge (v:real^3) (V:real^3->bool) (E:(real^3->bool)->bool)) DELETE  (u:real^3)={})`)
THENL[(*3*)
POP_ASSUM MP_TAC THEN REWRITE_TAC[DELETE;EXTENSION;IN_ELIM_THM] THEN DISCH_TAC
  THEN SUBGOAL_THEN `set_of_edge (v:real^3) (V:real^3->bool) (E:(real^3->bool)->bool)={u:real^3}` ASSUME_TAC
THENL[(*4*)
ASM_TAC THEN SET_TAC[];(*4*)
ASM_TAC THEN SET_TAC[](*4*)];

SUBGOAL_THEN`~(IMAGE ( azim1 x v u) ((set_of_edge (v:real^3) (V:real^3->bool) (E:(real^3->bool)->bool)) DELETE  (u:real^3))={})` ASSUME_TAC
THENL[(*4*)
REWRITE_TAC[IMAGE_EQ_EMPTY] THEN ASM_MESON_TAC[];(*4*)
SUBGOAL_THEN` FINITE (IMAGE (azim1 x v u) ((set_of_edge (v:real^3) (V:real^3->bool) (E:(real^3->bool)->bool)) DELETE  (u:real^3)))` ASSUME_TAC
THENL[(*5*)
ASM_MESON_TAC[FINITE_IMAGE];(*5*)
REMOVE_THEN "a" (fun th ->MP_TAC(ISPEC `(IMAGE (azim1 x v u) ((set_of_edge (v:real^3) (V:real^3->bool) (E:(real^3->bool)->bool)) DELETE  (u:real^3)))` th)) 
  THEN ASM_REWRITE_TAC[IMAGE;DELETE;IN_ELIM_THM]THEN STRIP_TAC
THEN EXISTS_TAC`x':real^3`
  THEN ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[]
](*5*)](*4*)](*3*)](*2*)(*1*)));;



let  SUR_SIGMA_FAN=prove(`!(x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (v:real^3) (u:real^3).
FAN(x,V,E) /\ ({v,u} IN E)
==> ?w.  {v,w} IN E /\sigma_fan x V E v w= u
`,
REPEAT GEN_TAC THEN STRIP_TAC 
THEN DISJ_CASES_TAC(SET_RULE `(set_of_edge (v:real^3) (V:real^3->bool) (E:(real^3->bool)->bool)={(u:real^3)})\/ ~(set_of_edge (v:real^3) (V:real^3->bool) (E:(real^3->bool)->bool)={(u:real^3)})`)
THENL(*1*)[
EXISTS_TAC`u:real^3` THEN ASM_REWRITE_TAC[sigma_fan];(*1*)

MP_TAC(ISPECL[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool)`;` (v:real^3)`;` (u:real^3)`]properties_of_set_of_edge_fan) THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN
MP_TAC(ISPECL[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool)`;` (v:real^3)`;` (u:real^3)`]exists_inverse_sigma_fan_alt) THEN ASM_REWRITE_TAC[azim1;] THEN STRIP_TAC THEN
MP_TAC(ISPECL[`(x:real^3) `;`(V:real^3->bool)`;` (E:(real^3->bool)->bool)`;` (v:real^3)`;` (w:real^3)`]properties_of_set_of_edge_fan) THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN EXISTS_TAC `w:real^3` THEN ASM_REWRITE_TAC[] 
  THEN MATCH_MP_TAC(ISPECL[`(x:real^3)`;` (V:real^3->bool)`;` (E:(real^3->bool)->bool)`;` (v:real^3)`;` (w:real^3)`;` (u:real^3)`] UNIQUE_SIGMA_FAN)
  THEN ASM_REWRITE_TAC[] 
  THEN POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC THEN DISCH_THEN(LABEL_TAC "be") THEN DISCH_TAC
THEN DISJ_CASES_TAC(SET_RULE`set_of_edge (v:real^3) (V:real^3->bool) (E:(real^3->bool)->bool)={w} \/ ~(set_of_edge (v:real^3) (V:real^3->bool) (E:(real^3->bool)->bool)={w})`)
THENL(*2*)[
FIND_ASSUM(MP_TAC)`u:real^3 IN set_of_edge v V E` THEN POP_ASSUM(fun th->REWRITE_TAC[th;IN_SING]) THEN ASM_TAC THEN SET_TAC[] ;(*2*)

ASM_REWRITE_TAC[] THEN GEN_TAC THEN STRIP_TAC THEN
DISJ_CASES_TAC(SET_RULE`~(azim (x:real^3) v w u <= azim x v w w1)\/ (azim (x:real^3) v w u <= azim x v w w1)`)
THENL(*3*)[
ASM_REWRITE_TAC[] THEN SUBGOAL_THEN`azim (x:real^3) v w w1 <= azim x v w u` ASSUME_TAC
THENL(*4*)[
POP_ASSUM MP_TAC THEN REAL_ARITH_TAC;(*4*)
 SUBGOAL_THEN `{(w:real^3),(w1:real^3),(u:real^3)} SUBSET set_of_edge v V E` ASSUME_TAC
THENL(*5*)[
ASM_TAC THEN SET_TAC[];(*5*)
FIND_ASSUM(MP_TAC)`FAN((x:real^3),V,E)` THEN REWRITE_TAC[FAN;fan6] THEN STRIP_TAC THEN POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC THEN DISCH_THEN (LABEL_TAC "b") THEN STRIP_TAC THEN
MP_TAC(ISPECL[`(x:real^3)`;` (V:real^3->bool)`;` (E:(real^3->bool)->bool)`;` (u:real^3)`;` (v:real^3)`]properties_of_graph) THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN
MP_TAC(ISPECL[`(x:real^3)`;` (V:real^3->bool)`;` (E:(real^3->bool)->bool)`;` (v:real^3)`]CYCLIC_SET_EDGE_FAN) 
THEN ASM_REWRITE_TAC[] THEN DISCH_TAC
  THEN MP_TAC(ISPECL[`(x:real^3)`;` (v:real^3)`;`{(w:real^3), (w1:real^3),(u:real^3)}`;`set_of_edge(v:real^3) (V:real^3->bool) (E:(real^3->bool)->bool)` ]subset_cyclic_set_fan)
		THEN ASM_REWRITE_TAC[] THEN DISCH_TAC
  THEN MP_TAC(ISPECL[`x:real^3`;`v:real^3`;`w:real^3`;`w1:real^3`;`u:real^3`]sum2_azim_fan) THEN ASM_REWRITE_TAC[]
  THEN MP_TAC(ISPECL[`x:real^3`;`v:real^3`;`w:real^3`;`w1:real^3`]azim) THEN STRIP_TAC THEN STRIP_TAC
  THEN SUBGOAL_THEN`azim (x:real^3) v w1 u <= azim x v w u` ASSUME_TAC
THENL(*6*)[ REPEAT (POP_ASSUM MP_TAC) THEN REAL_ARITH_TAC;(*6*)
POP_ASSUM MP_TAC THEN POP_ASSUM(fun th->REWRITE_TAC[]) THEN ASM_REWRITE_TAC[] THEN
MP_TAC (ISPECL[`(v:real^3)`;` (V:real^3->bool)`;` (E:(real^3->bool)->bool)`;`(w1:real^3)`]properties_of_set_of_edge)
THEN  ASM_REWRITE_TAC[] THEN DISCH_TAC 
 THEN REMOVE_THEN "b" (fun th->MP_TAC (ISPEC`{(v:real^3),(w:real^3)}`th) THEN ASSUME_TAC(th))
  THEN POP_ASSUM (fun th->MP_TAC (ISPEC`{(v:real^3),(u:real^3)}`th) THEN ASSUME_TAC(th))
  THEN POP_ASSUM (fun th->MP_TAC (ISPEC`{(v:real^3),w1}`th) THEN ASSUME_TAC(th))
  THEN REWRITE_TAC[SET_RULE`{a} UNION {b,c}={a,b,c}`] THEN ASM_REWRITE_TAC[]
  THEN DISCH_TAC THEN DISCH_TAC THEN DISCH_TAC
THEN
  DISJ_CASES_TAC(REAL_ARITH `(azim (x:real^3)  (v:real^3) (u:real^3) (w1:real^3)= &0) \/ ~(azim (x:real^3)  (v:real^3) (u:real^3) (w1:real^3) = &0)`)
THENL(*7*)[
MP_TAC(ISPECL[`(x:real^3)`;` (V:real^3->bool)`;` (E:(real^3->bool)->bool)`;` (v:real^3)`;` (u:real^3)`;`w1:real^3`]UNIQUE_AZIM_0_POINT_FAN) 
THEN ASM_TAC THEN SET_TAC[];(*7*)

  DISJ_CASES_TAC(REAL_ARITH `(azim (x:real^3)  (v:real^3) (u:real^3) (w:real^3)= &0) \/ ~(azim (x:real^3)  (v:real^3) (u:real^3) (w:real^3) = &0)`)
THENL(*8*)[
MP_TAC(ISPECL[`(x:real^3)`;` (V:real^3->bool)`;` (E:(real^3->bool)->bool)`;` (v:real^3)`;` (u:real^3)`;`w:real^3`]UNIQUE_AZIM_0_POINT_FAN) 
THEN ASM_TAC THEN SET_TAC[];(*8*)
 MP_TAC(ISPECL[`(x:real^3)`;` (v:real^3)`;` (u:real^3)`;` (w1:real^3)`]AZIM_COMPL) THEN ASM_REWRITE_TAC[]
   THEN  MP_TAC(ISPECL[`(x:real^3)`;` (v:real^3)`;` (u:real^3)`;` (w:real^3)`]AZIM_COMPL) THEN ASM_REWRITE_TAC[]
   THEN DISCH_TAC THEN DISCH_TAC THEN ASM_REWRITE_TAC[]
   THEN FIND_ASSUM(MP_TAC o (SPEC`w1:real^3`))`!w1:real^3.  w1 IN set_of_edge v V E /\ ~(w1 = u)
           ==> &2 * pi - azim x v u w <= &2 * pi - azim x v u w1` 
 THEN REMOVE_THEN "be" MP_TAC THEN ASM_REWRITE_TAC[] THEN DISCH_TAC
THEN ASM_REWRITE_TAC[] THEN DISJ_CASES_TAC(SET_RULE`~(w1:real^3=u)\/ (w1=u)`)
THENL(*9*)[
ASM_REWRITE_TAC[REAL_ARITH`&2 * pi -(a:real)<= &2 *pi - b <=> b <= a`] THEN DISJ_CASES_TAC(SET_RULE `(azim(x:real^3) v u w =azim x v u w1)\/ ~(azim(x:real^3) v u w =azim x v u w1)`) 
THENL(*10*)[
MP_TAC(ISPECL[`(x:real^3)`;` (V:real^3->bool)`;` (E:(real^3->bool)->bool)`;` (v:real^3)`;` (u:real^3)`;` (w:real^3)`;`(w1:real^3)`]UNIQUE_AZIM_POINT_FAN) THEN ASM_REWRITE_TAC[];(*10*)
POP_ASSUM MP_TAC THEN REAL_ARITH_TAC](*10*);(*9*)
SUBGOAL_THEN `azim (x:real^3) v w u= azim x v w w1` ASSUME_TAC
THENL(*10*)[POP_ASSUM(fun th->REWRITE_TAC[th]);(*10*)
REPEAT (POP_ASSUM MP_TAC) THEN REAL_ARITH_TAC](*10*)(*9*)](*8*)](*7*)](*6*)](*5*)](*4*)](*3*);
ASM_TAC THEN SET_TAC[]]]]);;





let MONO_SIGMA_FAN=prove(`!(x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (v:real^3) (u:real^3) (w:real^3).
FAN(x,V,E) /\ ({v,u} IN E)
/\ ({v,w} IN E)/\
 (sigma_fan x V E v u= sigma_fan x V E v w)
==> u=w
`,
REPEAT GEN_TAC THEN STRIP_TAC THEN REPEAT(POP_ASSUM MP_TAC) THEN DISCH_THEN (LABEL_TAC"1") 
THEN USE_THEN "1" MP_TAC THEN REWRITE_TAC[FAN;fan6] 
THEN REPEAT STRIP_TAC THEN REPEAT(POP_ASSUM MP_TAC) THEN DISCH_THEN (LABEL_TAC"1") THEN 
DISCH_THEN (LABEL_TAC"a") THEN DISCH_TAC THEN DISCH_TAC THEN DISCH_TAC THEN DISCH_THEN(LABEL_TAC "b") THEN
REPEAT STRIP_TAC THEN MP_TAC(ISPECL[`(x:real^3)`;` (V:real^3->bool)`;` (E:(real^3->bool)->bool)`;` (u:real^3)`;` (v:real^3)`]properties_of_graph) THEN ASM_REWRITE_TAC[] THEN DISCH_TAC
THEN DISJ_CASES_TAC(SET_RULE`(set_of_edge (v:real^3) (V:real^3->bool) (E:(real^3->bool)->bool)={(u:real^3)})\/ ~((set_of_edge v V E={u}))`)
THENL(*1*)[
POP_ASSUM MP_TAC THEN ASM_REWRITE_TAC[set_of_edge; EXTENSION;IN_ELIM_THM] 
THEN DISCH_TAC THEN POP_ASSUM(fun th-> MP_TAC(ISPEC`w:real^3`th)) THEN ASM_REWRITE_TAC[] 
THEN REMOVE_THEN "a" MP_TAC THEN REWRITE_TAC[UNIONS;SUBSET; IN_ELIM_THM]
THEN DISCH_TAC THEN POP_ASSUM (fun th->MP_TAC(ISPEC`w:real^3`th)) 
THEN ASM_REWRITE_TAC[IN_SING;LEFT_IMP_EXISTS_THM]
THEN STRIP_TAC THEN POP_ASSUM (fun th->MP_TAC(ISPEC`{(v:real^3),(w:real^3)}`th)) 
  THEN ASM_TAC THEN SET_TAC[];(*1*)

DISJ_CASES_TAC(SET_RULE`(set_of_edge (v:real^3) (V:real^3->bool) (E:(real^3->bool)->bool)={(w:real^3)})\/ ~((set_of_edge v V E={w}))`)
THENL(*2*)[

POP_ASSUM MP_TAC THEN ASM_REWRITE_TAC[set_of_edge; EXTENSION;IN_ELIM_THM] 
THEN DISCH_TAC THEN POP_ASSUM(fun th-> MP_TAC(ISPEC`u:real^3`th)) THEN ASM_REWRITE_TAC[] 
THEN REMOVE_THEN "a" MP_TAC THEN REWRITE_TAC[UNIONS;SUBSET; IN_ELIM_THM]
THEN DISCH_TAC THEN POP_ASSUM (fun th->MP_TAC(ISPEC`u:real^3`th)) 
THEN ASM_REWRITE_TAC[IN_SING;LEFT_IMP_EXISTS_THM]
THEN STRIP_TAC THEN POP_ASSUM (fun th->MP_TAC(ISPEC`{(v:real^3),(u:real^3)}`th)) 
  THEN ASM_TAC THEN SET_TAC[];(*2*)


MP_TAC(ISPECL[`(v:real^3)`;` (V:real^3->bool)`;` (E:(real^3->bool)->bool)`;` (u:real^3)`]properties_of_set_of_edge)
  THEN ASM_REWRITE_TAC[] THEN STRIP_TAC THEN
MP_TAC(ISPECL[`(v:real^3)`;` (V:real^3->bool)`;` (E:(real^3->bool)->bool)`;` (w:real^3)`]properties_of_set_of_edge)
  THEN ASM_REWRITE_TAC[] THEN STRIP_TAC
  THEN MP_TAC(ISPECL[`(x:real^3)`;` (V:real^3->bool)`;` (E:(real^3->bool)->bool)`;` (v:real^3)`;` (u:real^3)`]SIGMA_FAN)
THEN ASM_REWRITE_TAC[] THEN STRIP_TAC
  THEN POP_ASSUM(fun th-> MP_TAC(ISPEC`(w:real^3)`th))
THEN MP_TAC(ISPECL[`(x:real^3)`;` (V:real^3->bool)`;` (E:(real^3->bool)->bool)`;` (v:real^3)`;` (w:real^3)`]SIGMA_FAN)
THEN ASM_REWRITE_TAC[] THEN STRIP_TAC 
  THEN POP_ASSUM(fun th-> MP_TAC(ISPEC`(u:real^3)`th))
  THEN ASM_REWRITE_TAC[] 
THEN MP_TAC (ISPECL[`(v:real^3)`;` (V:real^3->bool)`;` (E:(real^3->bool)->bool)`;`(sigma_fan (x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (v:real^3) (w:real^3))`]properties_of_set_of_edge)
THEN  ASM_REWRITE_TAC[] THEN DISCH_TAC 
 THEN REMOVE_THEN "b" (fun th->MP_TAC (ISPEC`{(v:real^3),(w:real^3)}`th) THEN ASSUME_TAC(th))
  THEN POP_ASSUM (fun th->MP_TAC (ISPEC`{(v:real^3),(u:real^3)}`th) THEN ASSUME_TAC(th))
  THEN POP_ASSUM (fun th->MP_TAC (ISPEC`{(v:real^3),(sigma_fan (x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (v:real^3) (u:real^3))}`th) THEN ASSUME_TAC(th))
  THEN REWRITE_TAC[SET_RULE`{a} UNION {b,c}={a,b,c}`] THEN ASM_REWRITE_TAC[] 
  THEN DISCH_TAC THEN DISCH_TAC THEN DISCH_TAC THEN
DISJ_CASES_TAC(SET_RULE`~((u:real^3)=(w:real^3))\/ u=w`)

THENL(*3*)[
ASM_REWRITE_TAC[] THEN SUBGOAL_THEN `{(w:real^3),(sigma_fan (x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (v:real^3) (w:real^3)),(u:real^3)}SUBSET set_of_edge v V E` ASSUME_TAC
THENL(*4*)[ ASM_TAC THEN SET_TAC[];(*4*)

MP_TAC(ISPECL[`(x:real^3)`;` (V:real^3->bool)`;` (E:(real^3->bool)->bool)`;` (v:real^3)`]CYCLIC_SET_EDGE_FAN) 
THEN ASM_REWRITE_TAC[] THEN DISCH_TAC
  THEN MP_TAC(ISPECL[`(x:real^3)`;` (v:real^3)`;`{(w:real^3), (sigma_fan (x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (v:real^3) (w:real^3)),(u:real^3)}`;`set_of_edge(v:real^3) (V:real^3->bool) (E:(real^3->bool)->bool)` ]subset_cyclic_set_fan)
		THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN DISCH_TAC
  THEN MP_TAC(ISPECL[`(x:real^3)`;` (v:real^3)`;` (w:real^3)`;`(sigma_fan (x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (v:real^3) (w:real^3))`;` (u:real^3)`]sum2_azim_fan) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN (LABEL_TAC "c") THEN
  DISJ_CASES_TAC(REAL_ARITH `(azim (x:real^3)  (v:real^3) (w:real^3) (sigma_fan (x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (v:real^3) (w:real^3))= &0) \/ ~(azim (x:real^3)  (v:real^3) (w:real^3) (sigma_fan (x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (v:real^3) (w:real^3)) = &0)`)

THENL(*5*)[
MP_TAC(ISPECL[`(x:real^3)`;` (V:real^3->bool)`;` (E:(real^3->bool)->bool)`;` (v:real^3)`;` (w:real^3)`;`(sigma_fan (x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (v:real^3) (w:real^3))`]UNIQUE_AZIM_0_POINT_FAN) 
THEN ASM_TAC THEN SET_TAC[];(*5*)

DISJ_CASES_TAC(REAL_ARITH `(azim (x:real^3)  (v:real^3) (w:real^3) (u:real^3)= &0) \/ ~(azim (x:real^3)  (v:real^3) (w:real^3) (u:real^3) = &0)`)

THENL(*6*)[

MP_TAC(ISPECL[`(x:real^3)`;` (V:real^3->bool)`;` (E:(real^3->bool)->bool)`;` (v:real^3)`;` (w:real^3)`;`(u:real^3)`]UNIQUE_AZIM_0_POINT_FAN) 
THEN ASM_TAC THEN SET_TAC[];(*6*)

  DISJ_CASES_TAC(REAL_ARITH `(azim (x:real^3)  (v:real^3)  (sigma_fan (x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (v:real^3) (w:real^3)) (u:real^3)= &0) \/ ~(azim (x:real^3)  (v:real^3) (sigma_fan (x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (v:real^3) (w:real^3)) (u:real^3) = &0)`)

THENL(*7*)[
MP_TAC(ISPECL[`(x:real^3)`;` (V:real^3->bool)`;` (E:(real^3->bool)->bool)`;` (v:real^3)`;`(sigma_fan (x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (v:real^3) (w:real^3))`;` (u:real^3)`]UNIQUE_AZIM_0_POINT_FAN) 
THEN ASM_TAC THEN SET_TAC[];(*7*)

MP_TAC(ISPECL[`(x:real^3)`;` (v:real^3)`;`(sigma_fan (x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (v:real^3) (w:real^3))`;` (u:real^3)`]AZIM_COMPL) 
THEN ASM_REWRITE_TAC[] THEN DISCH_TAC
THEN MP_TAC(ISPECL[`(x:real^3)`;` (v:real^3)`;` (w:real^3)`;` (u:real^3)`]AZIM_COMPL) THEN 
ASM_REWRITE_TAC[REAL_ARITH`&2 * pi - ((a:real)+(b:real))= --(a:real)+ (&2 * pi - b)`] THEN DISCH_TAC 
THEN ASM_REWRITE_TAC[] THEN POP_ASSUM MP_TAC THEN REWRITE_TAC[REAL_ARITH`b <= (a:real)+(b:real)<=> &0 <= a`]
  THEN MP_TAC(ISPECL[`(x:real^3)`;` (v:real^3)`;`(w:real^3)`;`(sigma_fan (x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (v:real^3) (w:real^3))`] azim)THEN REPEAT(POP_ASSUM MP_TAC) THEN REAL_ARITH_TAC
(*7*)](*6*)](*5*)](*4*)];(*3*)

ASM_REWRITE_TAC[](*3*)]]]);;



let permutes_sigma_fan=prove(`!(x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (v:real^3) (u:real^3).
FAN(x,V,E) /\ ({v,u} IN E)
==>
(extension_sigma_fan x V E v) permutes (set_of_edge v V E)`,
REPEAT GEN_TAC THEN STRIP_TAC THEN FIND_ASSUM(MP_TAC)`FAN((x:real^3), V, E)` THEN REWRITE_TAC[FAN;fan1] THEN STRIP_TAC
THEN MP_TAC(ISPECL[`(v:real^3)`;` (V:real^3->bool)`;` (E:(real^3->bool)->bool)`]remark_finite_fan1)
  THEN ASM_REWRITE_TAC[] THEN DISCH_TAC
  THEN MP_TAC(ISPECL[`set_of_edge (v:real^3) (V:real^3->bool) (E:(real^3->bool)->bool)`;`extension_sigma_fan (x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (v:real^3)`]PERMUTES_FINITE_INJECTIVE)
  THEN ASM_REWRITE_TAC[] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN STRIP_TAC
THENL[
GEN_TAC THEN STRIP_TAC THEN ASM_REWRITE_TAC[extension_sigma_fan];
STRIP_TAC
THENL[
GEN_TAC THEN STRIP_TAC THEN ASM_REWRITE_TAC[extension_sigma_fan] THEN DISJ_CASES_TAC(SET_RULE`set_of_edge (v:real^3) (V:real^3->bool) (E:(real^3->bool)->bool)={(x':real^3)}\/ ~(set_of_edge (v:real^3) (V:real^3->bool) (E:(real^3->bool)->bool)={(x':real^3)})`)
THENL[ASM_REWRITE_TAC[sigma_fan;IN_SING];
ASM_MESON_TAC[SIGMA_FAN]];
REPEAT GEN_TAC THEN STRIP_TAC THEN POP_ASSUM MP_TAC THEN ASM_REWRITE_TAC[extension_sigma_fan] THEN DISCH_TAC
  THEN MP_TAC(ISPECL[`(v:real^3)`;` (V:real^3->bool)`;` (E:(real^3->bool)->bool)`;` (x':real^3)`]properties_of_set_of_edge)
  THEN ASM_REWRITE_TAC[] THEN STRIP_TAC THEN
MP_TAC(ISPECL[`(v:real^3)`;` (V:real^3->bool)`;` (E:(real^3->bool)->bool)`;` (y:real^3)`]properties_of_set_of_edge)
  THEN ASM_REWRITE_TAC[] THEN STRIP_TAC THEN
ASM_MESON_TAC[MONO_SIGMA_FAN]]]);;



(* ========================================================================== *)
(*                   INVERSE OF SIGMA_FAN                   *)
(* ========================================================================== *)




let azim1=new_definition`azim1 (x:real^3) (v:real^3) (u:real^3) (w:real^3)= &2 * pi- azim x v u w`;;





let exists_function_inverse_sigma_fan_alt=prove(`!(x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (v:real^3).
FAN(x,V,E) ==> 
(?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))
`,
REPEAT STRIP_TAC THEN
MP_TAC(ISPECL[`(x:real^3)`;` (V:real^3->bool)`;` (E:(real^3->bool)->bool)`;` (v:real^3)`] sigma_fan_in_set_of_edge) THEN 
 ASM_REWRITE_TAC[] THEN DISCH_TAC
THEN MP_TAC(ISPECL[`sigma_fan (x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (v:real^3)`;`set_of_edge (v:real^3) (V:real^3->bool) (E:(real^3->bool)->bool)`; `set_of_edge (v:real^3) (V:real^3->bool) (E:(real^3->bool)->bool)`]BIJECTIVE_ON_LEFT_RIGHT_INVERSE)
  THEN MP_TAC(ISPECL[`(x:real^3)`;` (V:real^3->bool)`;` (E:(real^3->bool)->bool)`;` (v:real^3)`] properties_of_set_of_edge_fan) THEN 
 ASM_REWRITE_TAC[] THEN DISCH_TAC THEN ASM_REWRITE_TAC[] 

THEN
MP_TAC(ISPECL[`(x:real^3)`;` (V:real^3->bool)`;` (E:(real^3->bool)->bool)`;` (v:real^3)`] MONO_SIGMA_FAN) THEN 
 ASM_REWRITE_TAC[] THEN DISCH_TAC
THEN
MP_TAC(ISPECL[`(x:real^3)`;` (V:real^3->bool)`;` (E:(real^3->bool)->bool)`;` (v:real^3)`] SUR_SIGMA_FAN) THEN 
 ASM_REWRITE_TAC[] THEN DISCH_TAC THEN ASM_REWRITE_TAC[]);;
(*
This is the same as inverse_sigma_fan_alt,
but easier to use.
*)



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)`;;


let INVERSE1_SIGMA_FAN=prove(`!(x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (v:real^3).
FAN(x,V,E) ==> 
( (!w:real^3. {v,w} IN E==> {v, inverse1_sigma_fan (x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (v:real^3) w} IN E)
/\ (!w:real^3. {v,w} IN E==> (sigma_fan x V E v)( inverse1_sigma_fan (x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (v:real^3) w) =w)
/\ (!w:real^3. {v,w} IN E==> inverse1_sigma_fan (x:real^3) (V:real^3->bool) (E:(real^3->bool)->bool) (v:real^3) (sigma_fan x V E v w) =w))`,
REPEAT GEN_TAC THEN STRIP_TAC THEN ONCE_REWRITE_TAC[inverse1_sigma_fan] THEN MP_TAC(ISPECL[`(x:real^3)`;` (V:real^3->bool)`;` (E:(real^3->bool)->bool)`;` (v:real^3) `]exists_function_inverse_sigma_fan_alt)
  THEN ASM_REWRITE_TAC[] THEN STRIP_TAC THEN SELECT_ELIM_TAC  THEN EXISTS_TAC`g:real^3->real^3` THEN ASM_REWRITE_TAC[]);;




(* ========================================================================== *)
(*                   PROPERTIES OF FAN                   *)
(* ========================================================================== *)




let properties_of_fan7=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) v:real^3 u:real^3 v1:real^3  u1:real^3.
FAN(x,V,E)/\ {v,u} IN E /\ {v1,u1} IN E /\ v IN aff_ge {x} {v1,u1}
==> v = v1 \/ v = u1`,

REPEAT STRIP_TAC
THEN FIND_ASSUM MP_TAC `FAN(x:real^3,V,E)`
THEN REWRITE_TAC[FAN; fan6; fan7]
THEN MRESA_TAC remark1_fan[`x:real^3 `;`(V:real^3->bool) `;`(E:(real^3->bool)->bool)`;` u:real^3`;`v:real^3`]
THEN STRIP_TAC 
  THEN POP_ASSUM  (fun th-> MP_TAC(ISPECL[`{(v:real^3),(u:real^3)}`;`{(v1:real^3),(u1:real^3)}`]th)) 
  THEN ASM_REWRITE_TAC[UNION;IN_ELIM_THM;]
THEN SUBGOAL_THEN `(v:real^3) IN aff_ge {(x:real^3)} {(v:real^3),(u:real^3)}` ASSUME_TAC
THENL(*1*)[
 MP_TAC(ISPECL[`(x:real^3)`;` (v:real^3)`;` (u:real^3)`]AFF_GE_1_2) 
   THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN ASM_REWRITE_TAC[IN_ELIM_THM]
THEN EXISTS_TAC `&0:real` 
  THEN EXISTS_TAC `&1:real` THEN EXISTS_TAC `&0:real`  
THEN REWRITE_TAC[VECTOR_ARITH`w= &0 % x+ &1 % w + &0 % v `] THEN REAL_ARITH_TAC;

DISCH_TAC
THEN MP_TAC(SET_RULE`v IN aff_ge {x} {v, u} /\ v IN aff_ge {x} {v1, u1} ==> v IN aff_ge {x} {v, u} INTER aff_ge {x:real^3} {v1, u1:real^3}`)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN DISJ_CASES_TAC(SET_RULE`{v, u} INTER {v1, u1:real^3}={} \/ ~({v, u} INTER {v1, u1}={})`)
THENL(*2*)[
ASM_REWRITE_TAC[]
THEN SUBGOAL_THEN`aff_ge {x:real^3} {}= {y| ?t1. t1= &1 /\ y= t1 %x}` ASSUME_TAC
THENL(*3*)[
ASSUME_TAC(SET_RULE `DISJOINT {x:real^3} {}`)
THEN AFF_TAC;

ASM_REWRITE_TAC[IN_ELIM_THM]
THEN STRIP_TAC THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[] THEN REDUCE_VECTOR_TAC
THEN ASM_REWRITE_TAC[]];(*2*)

DISJ_CASES_TAC(SET_RULE`~(v IN {v, u} INTER {v1, u1:real^3}) \/ (v IN {v, u} INTER {v1, u1})`)
THENL(*3*)[
SUBGOAL_THEN`({v:real^3, u} INTER {v1, u1}={u})` ASSUME_TAC
THENL(*4*)[
 ASM_TAC
THEN SET_TAC[];(*4*)

ASM_REWRITE_TAC[]
THEN SUBGOAL_THEN` aff_ge {(x:real^3)} {(u:real^3)} SUBSET aff {x,u}` ASSUME_TAC
THENL(*5*)[
MP_TAC(ISPECL[`(x:real^3)`;` (u:real^3)`;]AFF_GE_1_1) 
THEN MP_TAC(SET_RULE`~((x:real^3) = (u:real^3))==> DISJOINT {x} {u}`)
  THEN ASM_REWRITE_TAC[] THEN DISCH_TAC
  THEN ASM_REWRITE_TAC[]
  THEN DISCH_TAC THEN ASM_REWRITE_TAC[aff;AFFINE_HULL_2] THEN ASM_TAC THEN SET_TAC[];(*5*)

FIND_ASSUM MP_TAC`{v,u:real^3} IN E`
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B}={B,A}`]
THEN DISCH_TAC
THEN MRESA_TAC remark1_fan[`x:real^3 `;`(V:real^3->bool) `;`(E:(real^3->bool)->bool)`;` v:real^3`;`u:real^3`]
THEN ASM_TAC
THEN SET_TAC[]]];(*3*)

POP_ASSUM MP_TAC
THEN SET_TAC[]]]])  ;;





let properties1_of_fan7=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) v:real^3 u:real^3 v1:real^3  u1:real^3.
FAN(x,V,E)/\ {v,u} IN E /\ {v1,u1} IN E /\ v IN aff_ge {x} {v1}
==> v = v1`,

REPEAT STRIP_TAC
THEN FIND_ASSUM MP_TAC `FAN(x:real^3,V,E)`
THEN REWRITE_TAC[FAN; fan6; fan7]
THEN MRESA_TAC remark1_fan[`x:real^3 `;`(V:real^3->bool) `;`(E:(real^3->bool)->bool)`;` u:real^3`;`v:real^3`]
THEN MRESA_TAC remark1_fan[`x:real^3 `;`(V:real^3->bool) `;`(E:(real^3->bool)->bool)`;` u1:real^3`;`v1:real^3`]
THEN MP_TAC(SET_RULE`(v1:real^3) IN V==> (?v. v IN V /\ {v1} = {v})`)
THEN REDA_TAC
THEN STRIP_TAC 
  THEN POP_ASSUM  (fun th-> MP_TAC(ISPECL[`{(v:real^3),(u:real^3)}`;`{(v1:real^3)}`]th)) 
  THEN ASM_REWRITE_TAC[UNION;IN_ELIM_THM;]
THEN SUBGOAL_THEN `(v:real^3) IN aff_ge {(x:real^3)} {(v:real^3),(u:real^3)}` ASSUME_TAC
THENL(*1*)[
 MP_TAC(ISPECL[`(x:real^3)`;` (v:real^3)`;` (u:real^3)`]AFF_GE_1_2) 
   THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN ASM_REWRITE_TAC[IN_ELIM_THM]
THEN EXISTS_TAC `&0:real` 
  THEN EXISTS_TAC `&1:real` THEN EXISTS_TAC `&0:real`  
THEN REWRITE_TAC[VECTOR_ARITH`w= &0 % x+ &1 % w + &0 % v `] THEN REAL_ARITH_TAC;
DISCH_TAC
THEN MP_TAC(SET_RULE`v IN aff_ge {x} {v, u:real^3} /\ v IN aff_ge {x} {v1} ==> v IN aff_ge {x} {v, u} INTER aff_ge {x:real^3} {v1}`)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN DISJ_CASES_TAC(SET_RULE`{v, u:real^3} INTER {v1}={} \/ ~({v, u} INTER {v1}={})`)
THENL(*2*)[
ASM_REWRITE_TAC[]
THEN SUBGOAL_THEN`aff_ge {x:real^3} {}= {y| ?t1. t1= &1 /\ y= t1 %x}` ASSUME_TAC
THENL(*3*)[
ASSUME_TAC(SET_RULE `DISJOINT {x:real^3} {}`)
THEN AFF_TAC;
ASM_REWRITE_TAC[IN_ELIM_THM]
THEN STRIP_TAC THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[] THEN REDUCE_VECTOR_TAC
THEN ASM_REWRITE_TAC[]];(*2*)

DISJ_CASES_TAC(SET_RULE`~(v IN {v, u} INTER {v1:real^3}) \/ (v IN {v, u} INTER {v1})`)
THENL(*3*)[
SUBGOAL_THEN`({v:real^3, u} INTER {v1:real^3}={u})` ASSUME_TAC
THENL(*4*)[
 ASM_TAC
THEN SET_TAC[];
ASM_REWRITE_TAC[]
THEN SUBGOAL_THEN` aff_ge {(x:real^3)} {(u:real^3)} SUBSET aff {x,u}` ASSUME_TAC
THENL(*5*)[
MP_TAC(ISPECL[`(x:real^3)`;` (u:real^3)`;]AFF_GE_1_1) 
THEN MP_TAC(SET_RULE`~((x:real^3) = (u:real^3))==> DISJOINT {x} {u}`)
  THEN ASM_REWRITE_TAC[] THEN DISCH_TAC
  THEN ASM_REWRITE_TAC[]
  THEN DISCH_TAC THEN ASM_REWRITE_TAC[aff;AFFINE_HULL_2] THEN ASM_TAC THEN SET_TAC[];
FIND_ASSUM MP_TAC`{v,u:real^3} IN E`
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B}={B,A}`]
THEN DISCH_TAC
THEN MRESA_TAC remark1_fan[`x:real^3 `;`(V:real^3->bool) `;`(E:(real^3->bool)->bool)`;`v:real^3`;`u:real^3`]
THEN ASM_TAC
THEN SET_TAC[]]];(*3*)

POP_ASSUM MP_TAC
THEN SET_TAC[]]]])  ;;




let expand_elements_by_azim_fan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) v:real^3 u:real^3 x1:real x2:real x3:real.
FAN(x,V,E)/\ {v,u} IN E /\ &0 < x1 /\ &0<= x2
/\ x2 < &2 * pi
/\ &0< x3 
/\ x3 < pi/ &2
==>
azim x v u
 (x +
  (x1 * cos (x2) * sin (x3)) % e1_fan x v u +
  (x1 * sin (x2) * sin (x3)) % e2_fan x v u +
  (x1 * cos (x3)) % e3_fan x v u) = x2`,

(let lem=prove(`!x v u. {x,v,u}= {x,u,v}`,SET_TAC[]) in
 (let lem1=prove(`!x v u. {x,v,u}= {v,x,u}`,SET_TAC[]) in

REPEAT STRIP_TAC
THEN MP_TAC(ISPECL[`x:real^3 `;`(V:real^3->bool) `;
`(E:(real^3->bool)->bool)`;` u:real^3`;` v:real^3`]remark1_fan)
THEN RESA_TAC
THEN FIND_ASSUM MP_TAC`~collinear {(x:real^3),(v:real^3),(u:real^3)}`
THEN GEN_REWRITE_TAC( LAND_CONV  o ONCE_DEPTH_CONV)[lem1]
THEN ONCE_REWRITE_TAC[COLLINEAR_3]
THEN RESA_TAC
THEN MP_TAC(ISPECL[`x:real^3`;`v:real^3`;`u:real^3`]orthonormal_e1_e2_e3_fan)
THEN RESA_TAC
THEN POP_ASSUM(fun th-> MP_TAC(th) THEN ASSUME_TAC(th))
THEN REWRITE_TAC[orthonormal]
THEN STRIP_TAC
THEN MP_TAC(ISPECL[`x:real^3`;`v:real^3`;`u:real^3`]properties_coordinate)
THEN RESA_TAC
THEN MP_TAC(ISPEC`x3:real`SIN_POS_PI2)
THEN RESA_TAC
THEN MP_TAC(ISPECL[`x:real^3`;`v:real^3`;`u:real^3`;`(x +
  (x1 * cos (x2) * sin (x3)) % e1_fan x v u +
  (x1 * sin (x2) * sin (x3)) % e2_fan x v u +
  (x1 * cos (x3)) % e3_fan (x:real^3) (v:real^3) (u:real^3))`;
  `((u-x) dot (e3_fan (x:real^3) (v:real^3) (u:real^3))) *inv (norm((v:real^3)-(x:real^3)))`;
  `(x1 * cos (x3)) * (inv (norm((v:real^3)-(x:real^3))))`;
`((u-x) dot (e1_fan (x:real^3) (v:real^3) (u:real^3)))`;
`x1 * sin (x3:real)`;
`e1_fan (x:real^3) (v:real^3) (u:real^3)`;`e2_fan (x:real^3) (v:real^3) (u:real^3)`;`e3_fan (x:real^3) (v:real^3) (u:real^3)`;`&0`;`x2:real`]AZIM_UNIQUE)
THEN DISCH_TAC
THEN POP_ASSUM MATCH_MP_TAC
THEN ASM_REWRITE_TAC[REAL_ARITH`&0+a=a`;]
THEN STRIP_TAC
THENL[
MATCH_MP_TAC REAL_LT_MUL
THEN REPEAT(POP_ASSUM MP_TAC)
THEN REAL_ARITH_TAC;

STRIP_TAC
THENL[

REWRITE_TAC[SIN_0;COS_0;VECTOR_ARITH`(A*B)%C=A%(B%C)`]
THEN REDUCE_ARITH_TAC
THEN REDUCE_VECTOR_TAC
THEN ONCE_REWRITE_TAC[GSYM e3_fan]
THEN MATCH_MP_TAC(ISPECL[`e3_fan (x:real^3) (v:real^3) (u:real^3)`;`(u:real^3)-(x:real^3)`;`
((u - x) dot e1_fan x v u) % (e1_fan (x:real^3) (v:real^3) (u:real^3)) +
 ((u - x) dot e3_fan x v u) % (e3_fan x v u)`]CROSS_DOT_CANCEL)
THEN ASM_REWRITE_TAC[DOT_RADD;DOT_RMUL;DOT_SYM]
THEN REDUCE_ARITH_TAC
THEN REWRITE_TAC[CROSS_RADD;CROSS_RMUL;CROSS_REFL]
THEN REDUCE_VECTOR_TAC
THEN MP_TAC(ISPECL[`e1_fan (x:real^3) (v:real^3) (u:real^3)`;`e2_fan (x:real^3) (v:real^3) (u:real^3)`;`e3_fan (x:real^3) (v:real^3) (u:real^3)`]ORTHONORMAL_CROSS)
THEN RESA_TAC
THEN ASM_REWRITE_TAC[]
THEN REWRITE_TAC[e1_fan]
THEN ONCE_REWRITE_TAC[CROSS_TRIPLE;e2_fan]
THEN REWRITE_TAC[VECTOR_ARITH`A%(B%C)=(B*A)%C`]
THEN ONCE_REWRITE_TAC[DOT_SYM]
THEN ONCE_REWRITE_TAC[GSYM DOT_RMUL]
THEN ONCE_REWRITE_TAC[GSYM e2_fan]
THEN ASM_REWRITE_TAC[]
THEN REDUCE_VECTOR_TAC
THEN STRIP_TAC
THEN FIND_ASSUM MP_TAC`&0<(e1_fan x v u cross e2_fan x v u) dot e3_fan (x:real^3) (v:real^3) (u:real^3)`
THEN POP_ASSUM(fun th-> REWRITE_TAC[th])
THEN REWRITE_TAC[DOT_RZERO]
THEN REAL_ARITH_TAC;
REWRITE_TAC[e3_fan]
THEN VECTOR_ARITH_TAC]])));;

let one_edge_fan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) v:real^3 u:real^3.
FAN(x,V,E)/\ {v,u} IN E /\ ~(CARD (set_of_edge v V E) > 1)
==> set_of_edge v V E={u}`,
REPEAT STRIP_TAC
THEN MP_TAC(ISPECL[`x:real^3 `;`(V:real^3->bool) `;`(E:(real^3->bool)->bool) `;`u:real^3 `;`v:real^3`]remark1_fan)
THEN RESA_TAC
THEN MP_TAC(SET_RULE`(u:real^3) IN set_of_edge v V E==> ~(set_of_edge (v:real^3) (V:real^3->bool) (E:(real^3->bool)->bool)={})/\ {(u:real^3)} SUBSET set_of_edge v V E`)
THEN RESA_TAC
THEN MP_TAC(ISPECL [`{(u:real^3)}`;`set_of_edge (v:real^3) (V:real^3->bool) (E:(real^3->bool)->bool)`] FINITE_SUBSET)
THEN RESA_TAC
THEN ASSUME_TAC(SET_RULE`u IN {(u:real^3)} /\ {(u:real^3)} DELETE u= {} /\ {} PSUBSET {(u:real^3)}`)
THEN MP_TAC(ISPECL[`(u:real^3)`;`{(u:real^3)}`;]CARD_DELETE)
THEN RESA_TAC
THEN MP_TAC(ISPECL[`{}:real^3->bool`;`{(u:real^3)}`]CARD_PSUBSET)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM(th)] THEN ASSUME_TAC(th))
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[CARD_CLAUSES]
THEN REPEAT DISCH_TAC
THEN MP_TAC(ARITH_RULE` 0 = CARD ({(u:real^3)}) - 1 /\  0 < CARD ({u}) <=>  1= CARD {u}`)
THEN RESA_TAC
THEN MP_TAC(ARITH_RULE`~(CARD (set_of_edge (v:real^3) (V:real^3->bool) (E:(real^3->bool)->bool)) > 1)==> CARD (set_of_edge v V E) <= 1`)
THEN RESA_TAC
THEN MP_TAC(ISPECL[`{u:real^3}`;`set_of_edge (v:real^3) (V:real^3->bool) (E:(real^3->bool)->bool)`]CARD_SUBSET_LE)
THEN ASM_SET_TAC[]);;





end;;