HOL html



module Lemma_fan  = struct

open Sphere;;
open Fan_defs;;
open Tactic_fan;;


(* ========================================================================== *)
(*                   COLLINEAR                          *)
(* ========================================================================== *)

let th3a12=prove(`!x v u.(~ collinear {x,v,u} ==> DISJOINT {x,u} {v})`,
   (let th=prove(`{x,v,u}={x,v,u}`, SET_TAC[]) in
REPEAT GEN_TAC THEN REWRITE_TAC[th;IN_DISJOINT] THEN MATCH_MP_TAC MONO_NOT THEN 
REWRITE_TAC[COLLINEAR_3;COLLINEAR_LEMMA; VECTOR_ARITH` a-b= vec 0 <=> a = b`; IN_SING] THEN STRIP_TAC 
  THEN REPEAT(POP_ASSUM MP_TAC) THEN DISCH_THEN(LABEL_TAC "a") THEN DISCH_TAC THEN REMOVE_THEN "a" MP_TAC 
THEN ASM_REWRITE_TAC[] THEN SET_TAC[]));; 


let th3a=prove(`!x v u.(~ collinear {x,v,u} ==> DISJOINT {x,v} {u})`,
   (let th=prove(`{x,v,u}={x,u,v}`, SET_TAC[]) in
REPEAT GEN_TAC THEN REWRITE_TAC[th;IN_DISJOINT] THEN MATCH_MP_TAC MONO_NOT THEN 
REWRITE_TAC[COLLINEAR_3;COLLINEAR_LEMMA; VECTOR_ARITH` a-b= vec 0 <=> a = b`; IN_SING] THEN STRIP_TAC 
  THEN REPEAT(POP_ASSUM MP_TAC) THEN DISCH_THEN(LABEL_TAC "a") THEN DISCH_TAC THEN REMOVE_THEN "a" MP_TAC 
THEN ASM_REWRITE_TAC[] THEN SET_TAC[]));; 
   let th3b=prove(`!x v u. ~ collinear {x,v,u} ==> ~(x=v) `,
REPEAT GEN_TAC THEN REWRITE_TAC[COLLINEAR_3;COLLINEAR_LEMMA; VECTOR_ARITH` a-b= vec 0 <=> a = b`; DE_MORGAN_THM] THEN SET_TAC[]);; 
let th3b1=prove(`!x v u. ~ collinear {x,v,u} ==> ~(x=u) `,
(let th=prove(`{x,v,u}={x,u,v}`, SET_TAC[]) in
REWRITE_TAC[th;th3b]));; 

   let th3c= prove(`!x v u. ~ collinear {x,v,u} ==> ~(u IN aff {x,v})`,
REPEAT GEN_TAC THEN MATCH_MP_TAC MONO_NOT 
THEN REWRITE_TAC[aff; AFFINE_HULL_2; IN_ELIM_THM;COLLINEAR_3;COLLINEAR_LEMMA; VECTOR_ARITH` a-b= vec 0 <=> a = b`; DE_MORGAN_THM] 
THEN STRIP_TAC THEN REPEAT(POP_ASSUM MP_TAC) THEN REWRITE_TAC[REAL_ARITH `u'+v'= &1 <=> v'= &1 -u'`] 
  THEN DISCH_TAC THEN ASM_REWRITE_TAC[] 
THEN REWRITE_TAC[VECTOR_ARITH`(u = u' % x + (&1 - u') % v) <=> (u - v = u' % (x-v))`] THEN SET_TAC[]);;   

let th3d=prove(`!x v u. ~(x=v)/\ ~(x=u) ==> DISJOINT {x} {v,u}`,
SET_TAC[]);;
let th3=prove(`!x v u. ~ collinear {x,v,u} ==> ~ (x=v) /\ ~(x=u) /\ DISJOINT {x,v} {u}/\ DISJOINT {x,u} {v} /\DISJOINT {x} {v,u} /\ ~(u IN aff {x,v})`, 
MESON_TAC[th3a;th3b;th3b1;th3c;th3d;th3a12]);;

let collinear1_fan=prove(`!x v u. ~ collinear {x,u,v} <=> ~(u IN aff {x,v})/\ ~ (x=v)`,
(let lem=prove(`!x v u. {x,v,u}= {x,u,v}`,SET_TAC[]) in
REPEAT GEN_TAC THEN EQ_TAC
THENL[
MESON_TAC[th3;lem];
REWRITE_TAC[SET_RULE`~(t1) /\ ~ t2<=> ~(t2\/ t1)`;COLLINEAR_3_EXPAND;aff; AFFINE_HULL_2;IN_ELIM_THM] 
THEN MATCH_MP_TAC MONO_NOT  THEN MATCH_MP_TAC MONO_OR THEN STRIP_TAC 
THENL[
REWRITE_TAC[];

STRIP_TAC THEN EXISTS_TAC`u':real` THEN EXISTS_TAC`&1- (u':real)` THEN ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC]]));;

let collinear_fan=prove(`!x v u. ~ collinear {x,v,u} <=> ~(u IN aff {x,v})/\ ~ (x=v)`,
(let lem=prove(`!x v u. {x,v,u}= {x,u,v}`,SET_TAC[]) in
MESON_TAC[collinear1_fan;lem]));;



let properties_inside_collinear0_fan=prove(`!(x:real^3)  (u:real^3) (w:real^3) a:real.
&0 <a /\ a< &1 
/\ ~collinear{x,w,u}
==> ~collinear{x,(&1 - a) % u + a % w,u}`,

REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[collinear1_fan]
THEN STRIP_TAC THEN ASM_REWRITE_TAC[]
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC 
THEN MATCH_MP_TAC MONO_NOT
THEN REWRITE_TAC[aff; AFFINE_HULL_2;IN_ELIM_THM]
THEN STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[VECTOR_ARITH`(&1 - a) % u + a % w = u' % x + v % u
<=> a % w = u' % x + (v+a- &1) % u`]
THEN MP_TAC(REAL_ARITH`&0< a ==> ~(a= &0)`)
THEN RESA_TAC
THEN MRESA1_TAC REAL_MUL_LINV`a:real`
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`
a % w = u' % x + (v+a- &1) % u:real^3
==> (inv ( a))%(a % w) = (inv (a))%(u' % x + (v+a- &1) % u)
`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN ASM_REWRITE_TAC[VECTOR_ARITH`A%B%C= (A*B)%C`;VECTOR_ARITH`A%(B+C)=A%B+A%C`]
THEN REDUCE_VECTOR_TAC
THEN STRIP_TAC
THEN EXISTS_TAC `(inv a * (u':real))`
THEN EXISTS_TAC `(inv a * (v +a - &1 :real))`
THEN ASM_REWRITE_TAC[REAL_ARITH`inv a * (u') + inv a * (v +a - &1)=inv a* (a+ (u'+v) - &1)`;REAL_ARITH`t3'+ &1 - &1=t3'`]);;

let properties_inside_collinear1_fan=prove(`!(x:real^3)  (u:real^3) (w:real^3) a:real.
&0 <a /\ a< &1 
/\ ~collinear{x,w,u}
==>  ~collinear{x,(&1 - a) % u + a % w,w}`,
REPEAT STRIP_TAC THEN
MRESAL_TAC properties_inside_collinear0_fan[`(x:real^3)`;` (w:real^3)`;`(u:real^3)`;`&1-a:real`][VECTOR_ARITH`(&1 - (&1 - a)) % w + (&1 - a) % u=(&1 - a) % u + a % w`;]
THENL[ ASM_TAC THEN REAL_ARITH_TAC;
STRIP_TAC THENL[ASM_TAC THEN REAL_ARITH_TAC;
ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={A,C,B}`]
THEN ASM_REWRITE_TAC[]]]);;


let properties_inside_collinear_fan=prove(`!(x:real^3)  (u:real^3) (w:real^3) a:real.
&0 <a /\ a< &1 
/\ ~collinear{x,u,w}
==>  ~collinear{x,(&1 - a) % u + a % w,u}
/\  ~collinear{x,(&1 - a) % u + a % w,w}`,

MESON_TAC[SET_RULE`{A,B,C}={A,C,B}`;properties_inside_collinear0_fan;properties_inside_collinear1_fan]
);;

let notcoplanar_imp_notcollinear_fan=prove(`!x:real^3 v:real^3 u:real^3 w:real^3.
~coplanar {x,v,u,w}==> ~collinear {x,u,w} /\  ~collinear {x,v,u}
/\ ~collinear {x,v,w}`,
REPEAT GEN_TAC THEN STRIP_TAC
THEN MRESA_TAC NOT_COPLANAR_NOT_COLLINEAR [`x:real^3`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN MRESA_TAC NOT_COPLANAR_NOT_COLLINEAR [`x:real^3`;`u:real^3`;`w:real^3`;`v:real^3`]
THEN POP_ASSUM MP_TAC THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C,D}={A,D,B,C}`]
THEN RESA_TAC
THEN MRESAL_TAC NOT_COPLANAR_NOT_COLLINEAR [`x:real^3`;`v:real^3`;`w:real^3`;`u:real^3`][SET_RULE`{A,B,C,D}={A,B,D,C}`]);;



(* ========================================================================== *)
(*                   COLLINEAR and CONTINUOUS                         *)
(* ========================================================================== *)



let collinear_continuous_fan=prove(`!x:real^3 v:real^3 u:real^3 w:real^3 c:real.
(\(t:real^1). (&1- drop(t))%u + drop(t) %w - (&1 -c)%x - c% v) continuous_on (:real^1)`,
  SIMP_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_AT;OPEN_UNIV;DIMINDEX_1]
THEN REPEAT STRIP_TAC
THEN REWRITE_TAC[drop]
THEN MATCH_MP_TAC CONTINUOUS_ADD 
THEN STRIP_TAC
THENL[
 MATCH_MP_TAC CONTINUOUS_VMUL 
THEN REWRITE_TAC[GSYM REAL_CONTINUOUS_CONTINUOUS1]
THEN MATCH_MP_TAC REAL_CONTINUOUS_SUB
THEN SIMP_TAC[REAL_CONTINUOUS_CONST;REAL_CONTINUOUS_AT_COMPONENT; DIMINDEX_1; ARITH];
REPEAT(MATCH_MP_TAC CONTINUOUS_SUB 
THEN SIMP_TAC[CONTINUOUS_CONST])
THEN MATCH_MP_TAC CONTINUOUS_VMUL 
THEN REWRITE_TAC[GSYM REAL_CONTINUOUS_CONTINUOUS1]
THEN SIMP_TAC[REAL_CONTINUOUS_AT_COMPONENT; DIMINDEX_1; ARITH]]);;


let collinear1_continuous_fan=prove(`!u:real^3 w:real^3 t:real^1.
(\(t:real^1). (&1- drop(t))%u + drop(t) %w) continuous at t`,
REPEAT STRIP_TAC
THEN REWRITE_TAC[drop]
THEN MATCH_MP_TAC CONTINUOUS_ADD 
THEN STRIP_TAC
THENL[
 MATCH_MP_TAC CONTINUOUS_VMUL 
THEN REWRITE_TAC[GSYM REAL_CONTINUOUS_CONTINUOUS1]
THEN MATCH_MP_TAC REAL_CONTINUOUS_SUB
THEN SIMP_TAC[REAL_CONTINUOUS_CONST;REAL_CONTINUOUS_AT_COMPONENT; DIMINDEX_1; ARITH];
 MATCH_MP_TAC CONTINUOUS_VMUL 
THEN REWRITE_TAC[GSYM REAL_CONTINUOUS_CONTINUOUS1]
THEN SIMP_TAC[REAL_CONTINUOUS_CONST;REAL_CONTINUOUS_AT_COMPONENT; DIMINDEX_1; ARITH]]);;


let CONTINUOUS_CLOSED_PREIMAGE_CONSTANT = prove
 (`!f:real^M->real^N s a.
      f continuous_on s /\ closed s ==> closed {x | x IN s /\ f(x) = a}`,
  REPEAT STRIP_TAC THEN
  ASM_CASES_TAC `{x | x IN s /\ (f:real^M->real^N)(x) = a} = {}` THEN
  ASM_REWRITE_TAC[CLOSED_EMPTY] THEN ONCE_REWRITE_TAC[SET_RULE
   `{x | x IN s /\ f(x) = a} = {x | x IN s /\ f(x) IN {a}}`] THEN
  MATCH_MP_TAC CONTINUOUS_CLOSED_PREIMAGE THEN
  ASM_REWRITE_TAC[CLOSED_SING] THEN  SET_TAC[]);;
let open_collinear_fan=prove(`!x:real^3 v:real^3 u:real^3 w:real^3 c:real.
open{t| ~((\(t:real^1). (&1- drop(t))%u + drop(t) %w - (&1 -c)%x - c% v)(t)= vec 0)}`,
REPEAT STRIP_TAC
THEN REWRITE_TAC[OPEN_CLOSED;DIFF; IN_ELIM_THM;]
THEN MP_TAC(ISPECL[`(\(t:real^1). (&1- drop(t))%u + drop(t) %w - (&1 -c)%x - c% v:real^3)`;`(:real^1)`;
`((vec 0):real^3)`]CONTINUOUS_CLOSED_PREIMAGE_CONSTANT)
THEN SIMP_TAC[CLOSED_UNIV; DIMINDEX_1; collinear_continuous_fan]);;

let open_vector_angle_fan=prove(`!x:real^3 v:real^3 u:real^3 w:real^3 c:real a:real.
(!t. ~((&1 - t) % u + t % w = x))
==>
open{t|  ~((\(t:real^1). vector_angle (v - x) (((&1 - drop(t)) % u + drop(t) % w) - x))(t) = a)}`,
REPEAT STRIP_TAC
THEN REWRITE_TAC[OPEN_CLOSED;DIFF; IN_ELIM_THM;]
THEN MP_TAC(ISPECL[`lift o (\(t:real^1). vector_angle (v - x:real^3) (((&1 - drop(t)) % u + drop(t) % w) - x))`;`(:real^1)`;
`lift (a:real)`]CONTINUOUS_CLOSED_PREIMAGE_CONSTANT)
THEN REWRITE_TAC[o_DEF;LIFT_EQ]
THEN DISCH_TAC
THEN POP_ASSUM MATCH_MP_TAC
THEN SIMP_TAC[CLOSED_UNIV; DIMINDEX_1;]
THEN REWRITE_TAC[GSYM o_DEF]
THEN REWRITE_TAC[GSYM FORALL_LIFT_FUN]
THEN MP_TAC(ISPECL[`x:real^3 `;`v:real^3 `;`u:real^3`;` w:real^3`;` &0`]collinear_continuous_fan)
THEN REDUCE_ARITH_TAC
THEN REDUCE_VECTOR_TAC
THEN SIMP_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_AT;OPEN_UNIV;DIMINDEX_1]
THEN MATCH_MP_TAC MONO_FORALL
THEN GEN_TAC
THEN DISCH_THEN (LABEL_TAC"MA")
THEN STRIP_TAC
THEN REMOVE_THEN "MA" MP_TAC
THEN ASM_REWRITE_TAC[]
THEN DISCH_TAC
THEN MP_TAC(ISPECL[`(\(t:real^1). ((&1- drop(t))%(u:real^3) + drop(t) %(w:real^3)) - (x:real^3) )`;`(\(t:real^3). lift (vector_angle ((v:real^3)-(x:real^3)) t))`;`x':real^1`] CONTINUOUS_AT_COMPOSE)
THEN ASM_REWRITE_TAC[GSYM o_ASSOC]
THEN REWRITE_TAC[o_DEF]
THEN DISCH_TAC
THEN POP_ASSUM MATCH_MP_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`(A+B)-C=A+B-C:real^3`;GSYM(o_DEF)]
THEN REWRITE_TAC[GSYM(REAL_CONTINUOUS_CONTINUOUS1);GSYM(I_DEF);I_O_ID]
THEN MATCH_MP_TAC(ISPECL[`(v:real^3)-(x:real^3)`;`(&1 - drop x') % u + drop x' % w - x:real^3
`]REAL_CONTINUOUS_AT_VECTOR_ANGLE)
THEN ASM_REWRITE_TAC[VECTOR_ARITH`A+B-C= vec 0<=> A+B=C:real^3`]);;




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

let subset_cyclic_set_fan=prove(`!x:real^3 v:real^3 V:real^3->bool W:real^3->bool.
V SUBSET W /\ cyclic_set W x v ==> cyclic_set V x v`,

REPEAT GEN_TAC THEN REWRITE_TAC[cyclic_set] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] 
  THEN MP_TAC(ISPECL[`V:real^3->bool`;`W:real^3->bool`]FINITE_SUBSET) THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN ASM_REWRITE_TAC[] THEN ASM_TAC THEN SET_TAC[]);;


let property_of_cyclic_set=prove(`!x:real^3 v:real^3 u:real^3 w1:real^3 w2:real^3.
 cyclic_set {u, w1, w2} x v
==> ~(v=x) /\ ~(u=x)/\ ~collinear {vec 0, v-x, u-x}`,

(let th= prove(`!x:real^3 v:real^3 u:real^3 w1:real^3 w2:real^3.
x IN {x,w1,w2}`, SET_TAC[]) in

(let th1=prove(`!x:real^3 v:real^3 u:real^3 w1:real^3 w2:real^3.
x IN  affine hull {x,v}
`,REPEAT GEN_TAC THEN REWRITE_TAC[AFFINE_HULL_2;INTER; IN_ELIM_THM] THEN EXISTS_TAC`&1` THEN EXISTS_TAC `&0` THEN
 MESON_TAC[REAL_ARITH`&1+ &0= &1`; VECTOR_ARITH`x= &1 % x + &0 % v`])
in

(let th2=prove(`!x:real^3 v:real^3 u:real^3 w1:real^3 w2:real^3.
x IN {x,w1,w2} INTER affine hull {x,v}
`, REWRITE_TAC[INTER;IN_ELIM_THM] THEN REWRITE_TAC[th;th1]) in


REPEAT GEN_TAC THEN 
REWRITE_TAC[COLLINEAR_LEMMA;DE_MORGAN_THM;VECTOR_ARITH`a-b=vec 0 <=> a=b`;cyclic_set;] THEN STRIP_TAC 
  THEN ASM_REWRITE_TAC[VECTOR_ARITH`(v:real^3)=(x:real^3) <=> x=v`] THEN STRIP_TAC
THENL[ STRIP_TAC  THEN
  POP_ASSUM MP_TAC  THEN POP_ASSUM MP_TAC THEN DISCH_THEN(LABEL_TAC"a")  THEN DISCH_TAC 
THEN REMOVE_THEN "a" MP_TAC THEN ASM_REWRITE_TAC[] THEN MP_TAC(ISPECL[`x:real^3`; `v:real^3`; `u:real^3`; `w1:real^3`; `w2:real^3`] th2) THEN ASM_TAC THEN SET_TAC[];

STRIP_TAC THENL[

STRIP_TAC  THEN
  POP_ASSUM MP_TAC  THEN POP_ASSUM MP_TAC THEN DISCH_THEN(LABEL_TAC"a")  THEN DISCH_TAC 
THEN REMOVE_THEN "a" MP_TAC THEN ASM_REWRITE_TAC[] THEN MP_TAC(ISPECL[`x:real^3`; `v:real^3`; `u:real^3`; `w1:real^3`; `w2:real^3`] th2) THEN ASM_TAC THEN SET_TAC[];

STRIP_TAC 
THEN POP_ASSUM MP_TAC  
THEN POP_ASSUM MP_TAC 
THEN DISCH_THEN(LABEL_TAC"a")  
THEN DISCH_TAC 
THEN REMOVE_THEN "a" MP_TAC 
THEN POP_ASSUM MP_TAC 
THEN REWRITE_TAC[VECTOR_ARITH`(c:real) % ((v:real^3)-(x:real^3))=(u:real^3)-x <=> u =  (&1 - c) % x+c % v`] 
THEN DISCH_TAC 
THEN SUBGOAL_THEN `(u:real^3) IN affine hull {(x:real^3),(v:real^3)}` ASSUME_TAC
THENL[
REWRITE_TAC[AFFINE_HULL_2; IN_ELIM_THM] 
THEN EXISTS_TAC `&1 - (c:real)` 
THEN EXISTS_TAC`c:real`
THEN ASM_REWRITE_TAC[REAL_ARITH`&1 - (c:real) +c= &1`;];

MP_TAC(ISPECL[`u:real^3`; `v:real^3`; `u:real^3`; `w1:real^3`; `w2:real^3`]th) 
THEN DISCH_TAC 
THEN ASM_TAC 
THEN SET_TAC[INTER]
]]]))));;
let property_of_cyclic_set1=prove(`!x:real^3 v:real^3 u:real^3 w1:real^3 w2:real^3.
 cyclic_set {u, w1, w2} x v ==> ~collinear {x, v, w1}`,
				 
(let th=prove(`{u,w1,w2}={w1,u,w2}`,SET_TAC[]) in

REPEAT GEN_TAC THEN DISCH_TAC 
THEN MP_TAC (SPECL [`x:real^3`; `v:real^3`; `w1:real^3`;`u:real^3`; `w2:real^3`] property_of_cyclic_set) THEN ASM_REWRITE_TAC[th] THEN STRIP_TAC THEN ASM_REWRITE_TAC[COLLINEAR_3]));;
let property_of_cyclic_set2=prove(`!x:real^3 v:real^3 u:real^3 w1:real^3 w2:real^3.
 cyclic_set {u, w1, w2} x v
==> ~collinear {x, v, w2}`,			 
( let th=prove(`{u,w1,w2}={w2,w1,u}`,SET_TAC[]) in
( let th1=prove(`{u,w1,w2}={w1,w2,u}`,SET_TAC[]) in

REPEAT GEN_TAC THEN DISCH_TAC 
THEN MP_TAC (SPECL [`x:real^3`; `v:real^3`; `w2:real^3`;`w1:real^3`; `u:real^3`] property_of_cyclic_set)
THEN ASM_REWRITE_TAC[th] THEN STRIP_TAC THEN POP_ASSUM MP_TAC THEN MESON_TAC[th1;COLLINEAR_3])));;
let property_of_cyclic_set3=prove(`!x:real^3 v:real^3 u:real^3 w1:real^3 w2:real^3.
 cyclic_set {u, w1, w2} x v
==> ~ collinear {x, v, u}`,
( let th=prove(`{u,w1,w2}={w1,u,w2}`,SET_TAC[]) in

REPEAT GEN_TAC THEN DISCH_TAC 
THEN MP_TAC (SPECL [`x:real^3`; `v:real^3`; `u:real^3`;`w1:real^3`; `w2:real^3`] property_of_cyclic_set)
THEN ASM_REWRITE_TAC[]
THEN STRIP_TAC THEN POP_ASSUM MP_TAC THEN ASM_MESON_TAC[COLLINEAR_3;th]));;


let properties_of_cyclic_set=prove(`!x:real^3 v:real^3 u:real^3 w1:real^3 w2:real^3.
 cyclic_set {u, w1, w2} x v
==> ~(v=x) /\ ~(u=x)/\ ~collinear {vec 0, v-x, u-x}
/\ ~collinear {x, v, u}
/\ ~collinear {x, v, w1}
/\  ~collinear {x, v, w2}`,

MESON_TAC[property_of_cyclic_set;property_of_cyclic_set2;property_of_cyclic_set1;property_of_cyclic_set3]);;






(* ========================================================================== *)
(*                   the properties in normal vector                    *)
(* ========================================================================== *)


 
let imp_norm_not_zero_fan=prove(`!v:real^3 x:real^3. ~(v = x) ==> ~(norm ( v - x) = &0)`,
REPEAT GEN_TAC THEN REPEAT STRIP_TAC THEN SUBGOAL_THEN `(v:real^3)-(x:real^3)= vec 0` ASSUME_TAC THENL
                   [POP_ASSUM MP_TAC THEN MESON_TAC[NORM_EQ_0];
                    SUBGOAL_THEN `(v:real^3) = (x:real^3)` ASSUME_TAC THENL
                     [POP_ASSUM MP_TAC THEN VECTOR_ARITH_TAC;
                     ASM_TAC THEN SET_TAC[]]]);;

let imp_norm_gl_zero_fan=prove(`!v:real^3 x:real^3. ~(v = x) ==> inv(norm ( v - x)) > &0`,
REPEAT GEN_TAC THEN STRIP_TAC THEN SUBGOAL_THEN `~(norm ( (v:real^3) - (x:real^3)) = &0)` ASSUME_TAC THENL
  [ASM_MESON_TAC[imp_norm_not_zero_fan];
   MP_TAC (ISPEC `(v:real^3)-(x:real^3)` NORM_POS_LE) THEN DISCH_TAC THEN
   SUBGOAL_THEN `norm((v:real^3)-(x:real^3))> &0` ASSUME_TAC THENL
     [POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC THEN REAL_ARITH_TAC;
      MP_TAC (ISPEC `norm((v:real^3)-(x:real^3))` REAL_LT_INV_EQ) THEN POP_ASSUM MP_TAC 
THEN REAL_ARITH_TAC]]);;

let imp_inv_norm_not_zero_fan=prove(`!v:real^3 x:real^3. ~(v = x) ==> ~(inv(norm ( v - x)) = &0)`,
REPEAT GEN_TAC THEN DISCH_TAC THEN SUBGOAL_THEN `inv(norm ((v:real^3) - (x:real^3))) > &0` ASSUME_TAC
THENL
  [ASM_MESON_TAC[imp_norm_gl_zero_fan]; 
   POP_ASSUM MP_TAC THEN REAL_ARITH_TAC]);;

let imp_norm_ge_zero_fan=prove(`!v:real^3 x:real^3. ~(v = x) ==> inv(norm ( v - x)) >= &0`,
REPEAT GEN_TAC THEN STRIP_TAC THEN SUBGOAL_THEN `~(norm ( (v:real^3) - (x:real^3)) = &0)` ASSUME_TAC THENL
  [ASM_MESON_TAC[imp_norm_not_zero_fan];
   MP_TAC (ISPEC `(v:real^3)-(x:real^3)` NORM_POS_LE) THEN DISCH_TAC THEN
   SUBGOAL_THEN `norm((v:real^3)-(x:real^3))> &0` ASSUME_TAC THENL
     [POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC THEN REAL_ARITH_TAC;
      MP_TAC (ISPEC `norm((v:real^3)-(x:real^3))` REAL_LT_INV_EQ) THEN POP_ASSUM MP_TAC THEN REAL_ARITH_TAC]]);;
let norm_of_normal_vector_is_unit_fan=prove(`!v:real^3 x:real^3. ~(v = x) ==> norm(inv(norm ( v - x))% (v-x))= &1`,
REPEAT GEN_TAC THEN DISCH_TAC THEN REWRITE_TAC[NORM_MUL] THEN SUBGOAL_THEN ` inv(norm ( (v:real^3) - (x:real^3))) >= &0` ASSUME_TAC THENL[ ASM_MESON_TAC[imp_norm_ge_zero_fan]; 
                SUBGOAL_THEN ` ~(norm ( (v:real^3) - (x:real^3))= &0)` ASSUME_TAC THENL
                   [ASM_MESON_TAC[imp_norm_not_zero_fan];
                    SUBGOAL_THEN ` abs(inv(norm ( (v:real^3) - (x:real^3))))= inv(norm ( (v:real^3) - (x:real^3)))` ASSUME_TAC THENL
                       [ASM_MESON_TAC[REAL_ABS_REFL;REAL_ARITH `(a:real)>= &0 <=> &0 <= a`; ];
                        MP_TAC(ISPEC `norm ( (v:real^3) - (x:real^3))` REAL_MUL_LINV)THEN ASM_REWRITE_TAC[]]]]);;


let norm_origin_fan=prove(`!x:real^3.
 (\(y:real^3). lift(norm(y-x))) continuous_on (:real^3) `,
GEN_TAC
THEN MP_TAC(ISPECL[`(\(y:real^3). y-(x:real^3))`;`(\(y:real^3). lift(norm(y)))`;`(:real^3)`]CONTINUOUS_ON_COMPOSE)
THEN REWRITE_TAC[o_DEF]
THEN DISCH_TAC
THEN POP_ASSUM MATCH_MP_TAC
THEN REWRITE_TAC[CONTINUOUS_ON_LIFT_NORM;GSYM(o_DEF)]
THEN MATCH_MP_TAC CONTINUOUS_ON_SUB 
THEN SIMP_TAC[CONTINUOUS_ON_CONST;CONTINUOUS_ON_ID]);;
let REAL_ABS_SUB_NORM = prove
 (`!x y. abs(norm(x) - norm(y)) <= norm(x - y)`,
  REWRITE_TAC[REAL_ARITH `abs(x - y) <= a <=> x <= y + a /\ y <= x + a`] THEN
  MESON_TAC[NORM_TRIANGLE_SUB; NORM_SUB]);;


let IMP_NORM_FAN=prove(`!va:real^3 vb:real^3. ~(va = vb)
==> ~(norm (va-vb) = &0) /\ &0 <= norm (va-vb) /\ &0 < norm (va-vb) /\ &0 <= inv (norm (va-vb)) 
/\ &0 < inv (norm (va-vb)) /\ inv (norm (va-vb)) * norm (va-vb) = &1`,
REPEAT GEN_TAC
THEN DISCH_TAC
THEN MRESA_TAC imp_norm_not_zero_fan[`va:real^3`;`vb:real^3`]
THEN ASSUME_TAC(ISPEC`va-vb:real^3`NORM_POS_LE)
THEN MP_TAC(REAL_ARITH`~(norm(va-vb:real^3)= &0) /\ &0 <= norm(va-vb:real^3)==> &0 <norm(va-vb:real^3)`)
THEN RESA_TAC
THEN MRESA1_TAC REAL_LE_INV `norm(va-vb:real^3)`
THEN MRESA1_TAC REAL_LT_INV `norm(va-vb:real^3)`
THEN MRESA1_TAC REAL_MUL_LINV `norm(va-vb:real^3)`
THEN ASM_REWRITE_TAC[]);;




(* ========================================================================== *)
(*   the normal coordinate is the definiton of frame in flyspeck                 *)
(* ========================================================================== *)


let e3_fan=new_definition`e3_fan  (x:real^3) (v:real^3) (u:real^3) = inv(norm((v:real^3)-(x:real^3))) % ((v:real^3)-(x:real^3))`;;



  let e2_fan=new_definition`e2_fan (x:real^3) (v:real^3) (u:real^3) = inv(norm((e3_fan (x:real^3) (v:real^3) (u:real^3)) cross ((u:real^3)-(x:real^3)))) % ((e3_fan (x:real^3) (v:real^3) (u:real^3)) cross ((u:real^3)-(x:real^3))) `;;  
let e1_fan=new_definition`e1_fan (x:real^3) (v:real^3) (u:real^3)=(e2_fan (x:real^3) (v:real^3) (u:real^3)) cross (e3_fan (x:real^3) (v:real^3) (u:real^3))`;;

  
  let e3_mul_dist_fan=prove(`!x:real^3 v:real^3 u:real^3. ~(v=x) ==> dist (v,x) % e3_fan x v u = v - x`,
REPEAT GEN_TAC THEN REWRITE_TAC[e3_fan; dist; VECTOR_ARITH `(a:real) % (b:real)% (v:real^3)=(a*b)%v`] THEN 
MESON_TAC[imp_norm_not_zero_fan; REAL_MUL_RINV; VECTOR_ARITH `&1 %(v:real^3)=v`]);;
let norm_dot_fan=prove(`!x:real^3. norm x = &1 ==> x dot x = &1`,
 ASM_MESON_TAC[NORM_POW_2; REAL_ARITH `&1 pow 2= &1`]);;

  let e3_is_normal_fan=prove(`!x:real^3 v:real^3 u:real^3. ~(v=x) ==> e3_fan x v u dot e3_fan x v u = &1`,
REPEAT GEN_TAC THEN REWRITE_TAC[e3_fan]THEN DISCH_TAC 
THEN SUBGOAL_THEN `norm(inv(norm((v:real^3)-(x:real^3))) %(v-x)) pow 2= &1 pow 2` ASSUME_TAC THENL
 [ASM_MESON_TAC[norm_of_normal_vector_is_unit_fan] ;
ASM_MESON_TAC[NORM_POW_2; REAL_ARITH `&1 pow 2= &1`]]);;
  let e2_is_normal_fan= prove(`!x:real^3 v:real^3 u:real^3. ~(v=x) /\ ~(u=x) /\ ~(collinear {vec 0, v-x, u-x}) ==> e2_fan x v u dot e2_fan x v u = &1`,
REPEAT GEN_TAC THEN STRIP_TAC THEN SUBGOAL_THEN `~((e3_fan (x:real^3) (v:real^3) (u:real^3)) cross ((u:real^3)-(x:real^3))= vec 0)` ASSUME_TAC 
THENL[
POP_ASSUM MP_TAC THEN MATCH_MP_TAC MONO_NOT THEN REWRITE_TAC[e3_fan;CROSS_LMUL] 
THEN DISCH_TAC THEN MP_TAC(ISPECL [`v:real^3`; `x:real^3`] imp_inv_norm_not_zero_fan) 
THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN 
MP_TAC(ISPECL [`inv(norm((v:real^3)-(x:real^3)))`; `((v:real^3) -(x:real^3)) cross ((u:real^3)-(x:real^3))`; `(vec 0):real^3`] VECTOR_MUL_LCANCEL_IMP) 
THEN ASM_REWRITE_TAC[VECTOR_MUL_RZERO;CROSS_EQ_0 ];

MP_TAC(ISPECL [`(e3_fan (x:real^3) (v:real^3) (u:real^3)) cross ((u:real^3)-(x:real^3))`; `((vec 0):real^3)`] norm_of_normal_vector_is_unit_fan) THEN 
ASM_REWRITE_TAC[] THEN REWRITE_TAC[e2_fan; VECTOR_ARITH`(v:real^3)- vec 0 = v`] THEN MESON_TAC[norm_dot_fan]]);; 

  let e2_orthogonal_e3_fan=prove(`!x:real^3 v:real^3 u:real^3. 
~(v=x) /\ ~(u=x) /\ ~(collinear {vec 0, v-x, u-x}) ==> (e2_fan x v u) dot (e3_fan x v u)= &0`,
REPEAT GEN_TAC THEN STRIP_TAC THEN REWRITE_TAC[e2_fan;e3_fan;CROSS_LMUL;DOT_RMUL;] THEN VEC3_TAC);;


  let e1_is_normal_fan=prove(`!x:real^3 v:real^3 u:real^3. 
~(v=x) /\ ~(u=x) /\ ~(collinear {vec 0, v-x, u-x}) ==> e1_fan x v u dot e1_fan x v u = &1`,
REPEAT GEN_TAC THEN STRIP_TAC THEN 
REWRITE_TAC[e1_fan;DOT_CROSS] THEN 
MP_TAC (ISPECL [`x:real^3`; `v:real^3`; `u:real^3` ] e2_orthogonal_e3_fan) 
THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN 
MP_TAC (ISPECL [`x:real^3`; `v:real^3`; `u:real^3` ] e2_is_normal_fan) 
THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN  
MP_TAC (ISPECL [`x:real^3`; `v:real^3`; `u:real^3` ] e3_is_normal_fan) 
THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC);;
  let e1_orthogonal_e3_fan=prove(`!x:real^3 v:real^3 u:real^3. ~(v=x) /\ ~(u=x) /\ ~(collinear {vec 0, v-x, u-x}) 
==> (e1_fan x v u) dot (e3_fan x v u)= &0`,
REPEAT GEN_TAC THEN STRIP_TAC THEN REWRITE_TAC[e1_fan;DOT_CROSS_SELF] );;

  let e1_orthogonal_e2_fan=prove(`!x:real^3 v:real^3 u:real^3. ~(v=x) /\ ~(u=x) /\ ~(collinear {vec 0, v-x, u-x}) 
==> (e1_fan x v u) dot (e2_fan x v u)= &0`,
REPEAT GEN_TAC THEN STRIP_TAC THEN REWRITE_TAC[e1_fan;DOT_CROSS_SELF] );;

  let e1_cross_e2_dot_e3_fan=prove(`!x:real^3 v:real^3 u:real^3. ~(v=x) /\ ~(u=x) /\ ~(collinear {vec 0, v-x, u-x}) ==>
&0 < (e1_fan x v u cross e2_fan x v u) dot e3_fan x v u`,
REPEAT GEN_TAC THEN STRIP_TAC THEN REWRITE_TAC[e1_fan;CROSS_TRIPLE] 
THEN MP_TAC (ISPECL [`x:real^3`; `v:real^3`; `u:real^3` ] e1_is_normal_fan) 
THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[e1_fan] THEN DISCH_TAC THEN ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC);;


  let orthonormal_e1_e2_e3_fan=prove(`!x:real^3 v:real^3 u:real^3. ~(v=x) /\ ~(u=x) /\ ~(collinear {vec 0, v-x, u-x}) ==>
(orthonormal (e1_fan x v u) (e2_fan x v u) (e3_fan x v u))`,
REPEAT GEN_TAC THEN REWRITE_TAC[orthonormal] THEN DISCH_TAC THEN 
ASM_MESON_TAC[e1_is_normal_fan;e2_is_normal_fan;e3_is_normal_fan;e1_orthogonal_e2_fan;
e1_orthogonal_e3_fan;e2_orthogonal_e3_fan;e1_cross_e2_dot_e3_fan]);;


  let dot_e2_fan=prove(`!x:real^3 v:real^3 u:real^3. ~(v=x) /\ ~(u=x) /\ ~(collinear {vec 0, v-x, u-x}) 
==> (u-x) dot e2_fan x v u = &0`,
REPEAT GEN_TAC THEN STRIP_TAC THEN REWRITE_TAC[e2_fan;DOT_RMUL;DOT_CROSS_SELF] THEN REAL_ARITH_TAC);;
let vdot_e2_fan=prove(`!x:real^3 v:real^3 u:real^3. ~(v=x) /\ ~(u=x) /\ ~(collinear {vec 0, v-x, u-x}) 
==> (v-x) dot e2_fan x v u = &0`,
REPEAT GEN_TAC THEN STRIP_TAC THEN REWRITE_TAC[e2_fan;e3_fan;CROSS_LMUL;DOT_RMUL;DOT_CROSS_SELF] THEN REAL_ARITH_TAC);;
let vcross_e3_fan=prove(`!x:real^3 v:real^3 u:real^3. ~(v=x) /\ ~(u=x) /\ ~(collinear {vec 0, v-x, u-x}) 
==>
(v - x) cross (e3_fan x v u) = vec 0`,

REPEAT GEN_TAC THEN STRIP_TAC THEN REWRITE_TAC[e3_fan;CROSS_RMUL;CROSS_REFL] THEN VECTOR_ARITH_TAC);;
let udot_e1_fan=prove(`!x:real^3 v:real^3 u:real^3. 
~(v=x) /\ ~(u=x) /\ ~(collinear {vec 0, v-x, u-x}) 
==> &0 < (u - x) dot e1_fan x v u `,
REPEAT GEN_TAC THEN STRIP_TAC THEN REWRITE_TAC[e1_fan; e2_fan;CROSS_LMUL;DOT_RMUL;DOT_SYM;DOT_LMUL;CROSS_TRIPLE]
THEN SUBGOAL_THEN `~((e3_fan (x:real^3) (v:real^3) (u:real^3)) cross ((u:real^3)-(x:real^3))= vec 0)` ASSUME_TAC
THENL[
POP_ASSUM MP_TAC THEN MATCH_MP_TAC MONO_NOT THEN REWRITE_TAC[e3_fan;CROSS_LMUL] 
THEN DISCH_TAC THEN MP_TAC(ISPECL [`v:real^3`; `x:real^3`] imp_inv_norm_not_zero_fan) 
THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN 
MP_TAC(ISPECL [`inv(norm((v:real^3)-(x:real^3)))`; `((v:real^3) -(x:real^3)) cross ((u:real^3)-(x:real^3))`; `(vec 0):real^3`] VECTOR_MUL_LCANCEL_IMP) 
THEN ASM_REWRITE_TAC[VECTOR_MUL_RZERO;CROSS_EQ_0 ];
MP_TAC(ISPECL [`(e3_fan (x:real^3) (v:real^3) (u:real^3)) cross ((u:real^3)-(x:real^3))`; `((vec 0):real^3)`]imp_norm_gl_zero_fan) THEN 
ASM_REWRITE_TAC[REAL_ARITH`(a:real)> &0 <=> &0 < (a:real)`;VECTOR_ARITH `(a:real^3)- vec 0=a`] THEN DISCH_TAC
  THEN MP_TAC(ISPEC `e3_fan (x:real^3) (v:real^3) (u:real^3) cross (u-x)`DOT_POS_LT) THEN ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[REAL_LT_MUL]]);;  

let udot_e1_fan1=prove(`!x:real^3 v:real^3 u:real^3. 
~(v=x) /\ ~(u=x) /\ ~(collinear {vec 0, v-x, u-x}) 
==> &0 <= (u - x) dot e1_fan x v u `,
REPEAT GEN_TAC THEN STRIP_TAC THEN 
MP_TAC(ISPECL[`x:real^3` ;`v:real^3` ;`u:real^3`]udot_e1_fan) THEN ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC);; 

let vdot_e1_fan=prove(`!x:real^3 v:real^3 u:real^3. 
~(v=x) /\ ~(u=x) /\ ~(collinear {vec 0, v-x, u-x}) 
==>  (v - x) dot e1_fan x v u = &0`,
REPEAT STRIP_TAC THEN MP_TAC(ISPECL[`x:real^3`;`v:real^3`;`u:real^3`]e3_mul_dist_fan) THEN RES_TAC THEN SYM_ASSUM_TAC THEN
REWRITE_TAC[DOT_SYM;DOT_LMUL] THEN MP_TAC(ISPECL[`x:real^3`;`v:real^3`;`u:real^3`]e1_orthogonal_e3_fan) THEN RESA_TAC THEN
REAL_ARITH_TAC);;

 



let properties_coordinate=prove(`!x:real^3 v:real^3 u:real^3. 
~(collinear {x, v, u}) 
==> (orthonormal (e1_fan x v u) (e2_fan x v u) (e3_fan x v u))
/\ dist (v,x) % e3_fan x v u = v - x
/\ ((v - x) cross (e3_fan x v u) = vec 0)
/\  (v-x) dot e2_fan x v u = &0
/\ ((u-x) dot e2_fan x v u = &0)
/\  &0 <= (u - x) dot e1_fan x v u
/\ &0 < (u - x) dot e1_fan x v u
/\ (v - x) dot e1_fan x v u = &0`,
(	let lem=prove(`!a b c. {a,b,c}={b,a,c}`,SET_TAC[]) in
REPEAT GEN_TAC THEN DISCH_THEN(LABEL_TAC "a") THEN MP_TAC(ISPECL[`x:real^3`;`v:real^3`;`u:real^3`]th3)
  THEN RED_TAC THEN REMOVE_THEN "a" MP_TAC THEN REWRITE_TAC[lem;] THEN GEN_REWRITE_TAC(LAND_CONV o RAND_CONV o ONCE_DEPTH_CONV)[COLLINEAR_3]
THEN
ASM_MESON_TAC[orthonormal_e1_e2_e3_fan;e3_mul_dist_fan; dot_e2_fan;vdot_e2_fan;vcross_e3_fan;udot_e1_fan;udot_e1_fan1;vdot_e1_fan]));;

let module_of_vector =prove(`!x:real^3 v:real^3 u:real^3 w:real^3 r:real psi:real h:real.
 ~(v=x) /\ ~(u=x) /\ ~(collinear {vec 0, v-x, u-x}) 
/\ (&0 < r) /\ (w=(r * cos psi) % e1_fan x v u + (r * sin psi) % e2_fan x v u + h % (v-x))
==>
sqrt(((w cross (e3_fan x v u)) dot e1_fan x v u) pow 2 + ((w cross (e3_fan x v u)) dot e2_fan x v u) pow 2) = r`,
REPEAT GEN_TAC THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[CROSS_LADD;CROSS_LMUL;] THEN
MP_TAC(ISPECL [`x:real^3`; `v:real^3`; `u:real^3` ] orthonormal_e1_e2_e3_fan) THEN ASM_REWRITE_TAC[]
  THEN DISCH_THEN (LABEL_TAC "a") 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 ASM_REWRITE_TAC[] THEN DISCH_TAC
  THEN ASM_REWRITE_TAC[] 
THEN MP_TAC(ISPECL [`x:real^3`; `v:real^3`; `u:real^3` ]vcross_e3_fan) THEN ASM_REWRITE_TAC[] THEN DISCH_TAC 
  THEN ASM_REWRITE_TAC[]
THEN MP_TAC(ISPECL[`e1_fan (x:real^3) (v:real^3) (u:real^3)`;`e3_fan (x:real^3) (v:real^3) (u:real^3)`]CROSS_SKEW) 
  THEN DISCH_TAC THEN ASM_REWRITE_TAC[]
  THEN REWRITE_TAC[DOT_LADD;DOT_LMUL;DOT_LZERO;DOT_LNEG]
  THEN REMOVE_THEN "a" MP_TAC THEN REWRITE_TAC[orthonormal] THEN  DISCH_TAC THEN ASM_REWRITE_TAC[DOT_SYM]
  THEN REWRITE_TAC[REAL_ARITH `-- &0 = &0`; REAL_ARITH`(a:real)* &0 = &0`; REAL_ARITH `(a:real) * &1 = a`;
REAL_ARITH `(a:real) + &0 = a`;REAL_ARITH `&0 + (a:real) = a`;REAL_POW_MUL; REAL_ARITH `-- &1 pow 2 = &1`;
REAL_ARITH `(d:real) * (b:real) + d * (c:real) = d * ( b + c)`;SIN_CIRCLE; sqrt] THEN MATCH_MP_TAC SELECT_UNIQUE
THEN REWRITE_TAC[BETA_THM] THEN GEN_TAC THEN EQ_TAC

	   THENL[
              STRIP_TAC THEN SUBGOAL_THEN `((y:real) - (r:real))* (y + r) = &0` ASSUME_TAC
	       THENL[
		 REWRITE_TAC[REAL_ADD_LDISTRIB; REAL_ARITH `((a:real)- (b:real)) * (c:real)= a *c - b * c`; 
REAL_ARITH`(y:real) * (r:real)= r * y`; REAL_ARITH `((a:real) +(b:real)) - ((b:real) + (c:real))= a - c`; 
REAL_ARITH `(a:real)- (c:real)= &0 <=> a = c`] 
		   THEN POP_ASSUM MP_TAC THEN REAL_ARITH_TAC;
		 MP_TAC (ISPECL [`(y:real)- (r:real)`; `(y:real)+(r:real)` ]REAL_ENTIRE) THEN ASM_REWRITE_TAC[] 
		   THEN STRIP_TAC
		   THENL
		     [POP_ASSUM MP_TAC THEN REAL_ARITH_TAC;
		     REPEAT (POP_ASSUM MP_TAC) THEN REAL_ARITH_TAC]];

	     DISCH_TAC THEN ASM_REWRITE_TAC[] 
	       THEN REPEAT (POP_ASSUM MP_TAC) THEN REAL_ARITH_TAC]);;

(* ========================================================================== *)
(*                          COPLANAR   (^_^)                *)
(* ========================================================================== *)



let azim_line_fan=prove(`!x:real^3 v:real^3 u:real^3 w:real^3 t:real^1.
~coplanar {x,v,u,(&1-drop(t))%u+drop(t)%w}
==> (\(t:real^1). azim x v u ((&1 - drop(t)) % u + drop(t) % w)) real_continuous at t`,
REPEAT STRIP_TAC
THEN REWRITE_TAC[REAL_CONTINUOUS_CONTINUOUS1; o_DEF; CONTINUOUS_AT_LIFT_COMPONENT]
THEN MP_TAC(ISPECL[`(\(t:real^1). ((&1 - drop(t)) % (u:real^3) + drop(t) %(w:real^3)))`;`(\(w:real^3). lift(azim (x:real^3) (v:real^3) (u:real^3) w))`;`t:real^1`] CONTINUOUS_AT_COMPOSE)
THEN REWRITE_TAC[o_DEF]
THEN DISCH_TAC
THEN POP_ASSUM MATCH_MP_TAC
THEN REWRITE_TAC[ISPECL[`u:real^3`;`w:real^3`;`t:real^1`]collinear1_continuous_fan]
THEN MP_TAC(ISPECL[`x:real^3`;`v:real^3`;`u:real^3`;`((&1 - drop(t)) % (u:real^3) + drop(t) %(w:real^3))`]REAL_CONTINUOUS_AT_AZIM)
THEN REWRITE_TAC[REAL_CONTINUOUS_CONTINUOUS1; o_DEF; CONTINUOUS_AT_LIFT_COMPONENT]
THEN ASM_MESON_TAC[]);;






let continuous_coplanar_fan=prove(`!x:real^3  v:real^3 u:real^3 w:real^3.
 ~(coplanar{x,v,u,w})
==>(!t:real.  ~(t= &0) ==> ~coplanar {x,v,u,(&1-t)%u+t%w} )`,
REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[]
THEN FIND_ASSUM MP_TAC`~(coplanar{x,v,u,w:real^3})`
THEN MATCH_MP_TAC MONO_NOT
THEN REWRITE_TAC[COPLANAR_DET_EQ_0;VECTOR_ARITH`((&1 - t) % u + t % w) - x=(&1 - t) % (u-x) + t % (w - x):real^3`;DET_3;VECTOR_3;VECTOR_ADD_COMPONENT;VECTOR_MUL_COMPONENT
;REAL_ARITH`(v - x)$1 * (u - x)$2 * ((&1 - t) * (u - x)$3 + t * (w - x)$3) +
 (v - x)$2 * (u - x)$3 * ((&1 - t) * (u - x)$1 + t * (w - x)$1) +
 (v - x)$3 * (u - x)$1 * ((&1 - t) * (u - x)$2 + t * (w - x)$2) -
 (v - x)$1 * (u - x)$3 * ((&1 - t) * (u - x)$2 + t * (w - x)$2) -
 (v - x)$2 * (u - x)$1 * ((&1 - t) * (u - x)$3 + t * (w - x)$3) -
 (v - x)$3 * (u - x)$2 * ((&1 - t) * (u - x)$1 + t * (w - x)$1)=
t*((v - x)$1 * (u - x)$2 * ((w - x)$3) +
 (v - x)$2 * (u - x)$3 * ( (w - x)$1) +
 (v - x)$3 * (u - x)$1 * ((w - x)$2) -
 (v - x)$1 * (u - x)$3 * ((w - x)$2) -
 (v - x)$2 * (u - x)$1 * ( (w - x)$3) -
 (v - x)$3 * (u - x)$2 * ( (w - x)$1)):real`;REAL_ENTIRE]
THEN ASM_TAC
THEN REAL_ARITH_TAC);;




let open_is_not_zero_fan=prove(`open{y:real^1 | ?x. ~(x = &0) /\ y = lift x}`,
(let equality_real_fan=prove(`{y:real^1 | ?x. ~(x = &0) /\ y = lift x}={y:real^1 | ~(drop y = &0)}`,
REWRITE_TAC[EXTENSION;IN_ELIM_THM]
THEN GEN_TAC
THEN EQ_TAC
THENL[
STRIP_TAC
THEN ASM_REWRITE_TAC[LIFT_DROP];
STRIP_TAC
THEN EXISTS_TAC`drop (x:real^1)`
THEN ASM_REWRITE_TAC[LIFT_DROP]])in
(let ngu=prove(`{x | x IN (:real^1) /\ x = vec 0}={x | x IN (:real^1) /\ x$1 = &0}`,
REWRITE_TAC[EXTENSION;IN_ELIM_THM]
THEN GEN_TAC
THEN EQ_TAC
THENL[STRIP_TAC
THEN ASM_REWRITE_TAC[VEC_COMPONENT];
SIMP_TAC[IN_UNIV;CART_EQ;LAMBDA_BETA;VEC_COMPONENT;DIMINDEX_1;ARITH_RULE`1<=i /\ i<=1<=>i=1`]])
in
REWRITE_TAC[equality_real_fan]
THEN REWRITE_TAC[OPEN_CLOSED;DIFF; IN_ELIM_THM;]
THEN MP_TAC(ISPECL[`(\(t:real^1). t)`;`(:real^1)`;`(vec 0):real^1`]CONTINUOUS_CLOSED_PREIMAGE_CONSTANT)
THEN MP_TAC(ISPECL[`0`;`1`]VEC_COMPONENT)
THEN SIMP_TAC[CONTINUOUS_ON_ID;CLOSED_UNIV; DIMINDEX_1;drop;ngu])));;










let azim_continuous_when_not_coplanar=prove(`!x:real^3 v:real^3 u:real^3 w:real^3.
~coplanar {x,v,u,w}
==> (\(t:real). azim x v u ((&1 - t) % u + t % w)) real_continuous_on {t:real| ~(t= &0)}`,
REWRITE_TAC[REAL_CONTINUOUS_ON;o_DEF;IMAGE;IN_ELIM_THM]
THEN REPEAT STRIP_TAC
THEN ASSUME_TAC(open_is_not_zero_fan)
THEN MRESA_TAC CONTINUOUS_ON_EQ_CONTINUOUS_AT[`(\t:real^1. lift (azim x v u ((&1 - drop t) % u + drop t % w))):real^1->real^1`;`{y:real^1| ?x. ~(x = &0) /\ y = lift x}`]
THEN REWRITE_TAC[IN_ELIM_THM]
THEN REPEAT STRIP_TAC
THEN ASM_REWRITE_TAC[LIFT_DROP]
THEN MRESAL_TAC azim_line_fan[`x:real^3`;` v:real^3`;` u:real^3`;` w:real^3`;` (lift x''):real^1`][REAL_CONTINUOUS_CONTINUOUS1; o_DEF;LIFT_DROP]
THEN POP_ASSUM MATCH_MP_TAC
THEN MRESA_TAC continuous_coplanar_fan[`x:real^3 `;` v:real^3`;` u:real^3`;` w:real^3`]
THEN POP_ASSUM (fun th-> MRESA1_TAC th `x'':real`));;

let injective_azim_coplanar=prove(`!x:real^3 v:real^3 u:real^3 w:real^3.
~coplanar {x,v,u,w}
==> 
!a:real b:real. ~(a= &0) /\ ~(b= &0)/\ (\(t:real). azim x v u ((&1 - t) % u + t % w))a=(\(t:real). azim x v u ((&1 - t) % u + t % w))b==>a=b`,
REWRITE_TAC[]
THEN REPEAT STRIP_TAC
THEN MRESA_TAC continuous_coplanar_fan[`x:real^3`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN POP_ASSUM(fun th -> MP_TAC(ISPEC`a:real`th)THEN ASSUME_TAC(th))
THEN POP_ASSUM(fun th -> MP_TAC(ISPEC`b:real`th))
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C,D}={A,B,D,C}`]
THEN ASM_REWRITE_TAC[]
THEN REPEAT STRIP_TAC
THEN MRESA_TAC NOT_COPLANAR_NOT_COLLINEAR[`x:real^3`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN MRESA_TAC NOT_COPLANAR_NOT_COLLINEAR[`x:real^3`;`v:real^3`;`(&1 - a) % u + a % w:real^3`;`u:real^3`]
THEN MRESA_TAC NOT_COPLANAR_NOT_COLLINEAR[`x:real^3`;`v:real^3`;`(&1 - b) % u + b % w:real^3`;`u:real^3`]
THEN MRESA_TAC AZIM_EQ [`x:real^3`;`v:real^3`;`u:real^3`;`(&1 - b) % u + b % w:real^3`;`(&1 - a) % u + a % w:real^3`;]
THEN MRESA_TAC AZIM_EQ_0 [`x:real^3`;`v:real^3`;`(&1 - a) % u + a % w:real^3`;`(&1 - b) % u + b % w:real^3`;]
THEN MRESA_TAC AZIM_EQ_0_PI_IMP_COPLANAR [`x:real^3`;`v:real^3`;`(&1 - a) % u + a % w:real^3`;`(&1 - b) % u + b % w:real^3`;]
THEN POP_ASSUM MP_TAC
THEN FIND_ASSUM MP_TAC`~coplanar {x,v,u,w:real^3}`
THEN REWRITE_TAC[COPLANAR_DET_EQ_0;VECTOR_ARITH`((&1 - t) % u + t % w) - x=(&1 - t) % (u-x) + t % (w - x):real^3`;DET_3;VECTOR_3;VECTOR_ADD_COMPONENT;VECTOR_MUL_COMPONENT]
THEN STRIP_TAC
THEN REWRITE_TAC[REAL_ARITH`(v - x)$1 *
 ((&1 - a) * (u - x)$2 + a * (w - x)$2) *
 ((&1 - b) * (u - x)$3 + b * (w - x)$3) +
 (v - x)$2 *
 ((&1 - a) * (u - x)$3 + a * (w - x)$3) *
 ((&1 - b) * (u - x)$1 + b * (w - x)$1) +
 (v - x)$3 *
 ((&1 - a) * (u - x)$1 + a * (w - x)$1) *
 ((&1 - b) * (u - x)$2 + b * (w - x)$2) -
 (v - x)$1 *
 ((&1 - a) * (u - x)$3 + a * (w - x)$3) *
 ((&1 - b) * (u - x)$2 + b * (w - x)$2) -
 (v - x)$2 *
 ((&1 - a) * (u - x)$1 + a * (w - x)$1) *
 ((&1 - b) * (u - x)$3 + b * (w - x)$3) -
 (v - x)$3 *
 ((&1 - a) * (u - x)$2 + a * (w - x)$2) *
 ((&1 - b) * (u - x)$1 + b * (w - x)$1)=
(b-a)*((v - x)$1 * (u - x)$2 * (w - x)$3 +
   (v - x)$2 * (u - x)$3 * (w - x)$1 +
   (v - x)$3 * (u - x)$1 * (w - x)$2 -
   (v - x)$1 * (u - x)$3 * (w - x)$2 -
   (v - x)$2 * (u - x)$1 * (w - x)$3 -
   (v - x)$3 * (u - x)$2 * (w - x)$1)`;REAL_ENTIRE]
THEN ASM_REWRITE_TAC[]
THEN REAL_ARITH_TAC);;













(* ========================================================================== *)
(*   the sphere coordinate is the definiton of frame in flyspeck                 *)
(* ========================================================================== *)


let SINCOS_PRINCIPAL_VALUE_FAN = prove(
`!x:real. ?y:real. (&0<= y /\ y < &2* pi) /\ (sin(y) = sin(x) /\ cos(y) = cos(x))`,
  GEN_TAC THEN MP_TAC(SPECL [`x:real`] SINCOS_PRINCIPAL_VALUE) THEN STRIP_TAC THEN
DISJ_CASES_TAC(REAL_ARITH`((y:real) < &0)\/ (&0 <= y)`) THENL
[ EXISTS_TAC `(y:real)+ &2 * pi` THEN ASSUME_TAC(PI_POS) 
THEN ASM_REWRITE_TAC[SIN_PERIODIC;COS_PERIODIC] THEN REPEAT(POP_ASSUM MP_TAC) THEN REAL_ARITH_TAC;
 EXISTS_TAC `(y:real)` THEN ASSUME_TAC(PI_POS) 
THEN ASM_REWRITE_TAC[] THEN REPEAT(POP_ASSUM MP_TAC) THEN REAL_ARITH_TAC]);;
let sin_of_u_fan=prove(`!x:real^3 v:real^3 u:real^3 r1:real psi:real h1:real.
  ~collinear {u,x,v} /\ ~(v=x) /\ ~(u=x)/\ ~collinear {vec 0, v-x, u-x} /\ &0 < r1 
/\ u - x = (r1 * cos psi) % (e1_fan x v u) + (r1 * sin psi) % (e2_fan x v u) + h1 % (v-x)
==> sin psi = &0`,
REPEAT GEN_TAC THEN STRIP_TAC THEN  MP_TAC (ISPECL [`x:real^3`; `v:real^3`; `u:real^3`] dot_e2_fan) 
	  THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[DOT_LADD;DOT_LMUL] 
	  THEN MP_TAC (ISPECL [`x:real^3`; `v:real^3`; `u:real^3`] vdot_e2_fan) THEN ASM_REWRITE_TAC[] THEN DISCH_TAC 
	  THEN MP_TAC (ISPECL [`x:real^3`; `v:real^3`; `u:real^3`] e2_is_normal_fan) THEN ASM_REWRITE_TAC[] THEN DISCH_TAC 
	  THEN MP_TAC (ISPECL [`x:real^3`; `v:real^3`; `u:real^3`] e1_orthogonal_e2_fan) 
	  THEN ASM_REWRITE_TAC[] THEN DISCH_TAC 
	  THEN ASM_REWRITE_TAC[]
          THEN REWRITE_TAC[REAL_ARITH`(a:real)* &0 = &0`; REAL_ARITH`(a:real)+ &0= a`; REAL_ARITH`&0 + (a:real)= a`; 
               REAL_ARITH`(a:real) * &1= a`]
	  THEN DISCH_TAC
          THEN MATCH_MP_TAC(ISPECL [`sin (psi:real)`;`&0`; `r1:real`] REAL_EQ_LCANCEL_IMP)  THEN ASM_REWRITE_TAC[] 
	  THEN REPEAT(POP_ASSUM MP_TAC) THEN REAL_ARITH_TAC);;
let cos_of_u_fan=prove(`!x:real^3 v:real^3 u:real^3 r1:real psi:real h1:real.
  ~collinear {u,x,v} /\ ~(v=x) /\ ~(u=x)/\ ~collinear {vec 0, v-x, u-x} /\ &0 < r1 
/\ u - x = (r1 * cos psi) % (e1_fan x v u) + (r1 * sin psi) % (e2_fan x v u) + h1 % (v-x)
==> cos psi = &1`,
REPEAT GEN_TAC THEN STRIP_TAC 
THEN MP_TAC(ISPECL[`x:real^3`; `v:real^3`; `u:real^3`; `(u:real^3)-(x:real^3)`; `r1:real`; `psi:real`; `h1:real`]module_of_vector) THEN ASM_REWRITE_TAC[] THEN 
 POP_ASSUM (fun th-> REWRITE_TAC[SYM(th)] THEN ASSUME_TAC(th))
THEN POP_ASSUM MP_TAC
  THEN MP_TAC(ISPECL[`(u:real^3)-(x:real^3)`; `e3_fan (x:real^3) (v:real^3)(u:real^3)`;`e1_fan (x:real^3) (v:real^3)(u:real^3)`]CROSS_TRIPLE) THEN DISCH_TAC THEN POP_ASSUM (fun th-> REWRITE_TAC[th])
THEN MP_TAC(ISPECL[`(u:real^3)-(x:real^3)`; `e3_fan (x:real^3) (v:real^3)(u:real^3)`;`e2_fan (x:real^3) (v:real^3)(u:real^3)`]CROSS_TRIPLE) THEN DISCH_TAC THEN POP_ASSUM (fun th-> REWRITE_TAC[th])
 THEN MP_TAC (ISPECL [`x:real^3`; `v:real^3`; `u:real^3`] orthonormal_e1_e2_e3_fan) THEN ASM_REWRITE_TAC[] THEN DISCH_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 POP_ASSUM (fun th-> REWRITE_TAC[th]) THEN STRIP_TAC THEN 
POP_ASSUM (fun th-> REWRITE_TAC[th]) THEN POP_ASSUM (fun th-> REWRITE_TAC[th]) THEN
POP_ASSUM (fun th-> REWRITE_TAC[CROSS_SKEW;th])
  THEN MP_TAC(ISPECL[`x:real^3`; `v:real^3`; `u:real^3`] dot_e2_fan)THEN ASM_REWRITE_TAC[]
THEN DISCH_TAC THEN MP_TAC(ISPECL[ `e2_fan (x:real^3) (v:real^3)(u:real^3)`;`(u:real^3)-(x:real^3)`]DOT_SYM) THEN DISCH_TAC THEN ASM_REWRITE_TAC[REAL_ARITH`&0 pow 2 +(a:real)=a`] THEN
MP_TAC (ISPECL [`x:real^3`; `v:real^3`; `u:real^3`] udot_e1_fan1) THEN ASM_REWRITE_TAC[DOT_LNEG;] THEN DISCH_TAC 
  THEN  MP_TAC(ISPECL[ `e1_fan (x:real^3) (v:real^3)(u:real^3)`;`(u:real^3)-(x:real^3)`]DOT_SYM) THEN DISCH_TAC THEN
POP_ASSUM (fun th-> REWRITE_TAC[th]) THEN REWRITE_TAC[POW_2_SQRT_ABS;REAL_ABS_NEG] THEN
 MP_TAC(ISPECL[ `((u:real^3)-(x:real^3)) dot e1_fan (x:real^3) (v:real^3)(u:real^3)`]
  REAL_ABS_REFL) THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN ASM_REWRITE_TAC[] 
  THEN DISCH_THEN (LABEL_TAC "a") THEN DISCH_TAC
THEN
 MP_TAC(ISPECL[`x:real^3`; `v:real^3`; `u:real^3`; `r1:real`; `psi:real`; `h1:real`] sin_of_u_fan) THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN REMOVE_THEN "a" MP_TAC THEN ASM_REWRITE_TAC[VECTOR_ARITH`
 (r1 * cos psi) % e1_fan x v u + (r1 * &0) % e2_fan x v u + h1 % (v - x)=
 (r1 * cos psi) % e1_fan x v u + h1 % (v - x)`] THEN DISCH_TAC THEN
  SUBGOAL_THEN`((u:real^3) - (x:real^3)) dot e1_fan x (v:real^3) u = (((r1:real) * cos (psi:real)) % e1_fan x v u + (h1:real) % (v - x)) dot e1_fan x v u` ASSUME_TAC

THENL[ASM_MESON_TAC[];

POP_ASSUM MP_TAC THEN REWRITE_TAC[DOT_LADD;DOT_LMUL] THEN POP_ASSUM MP_TAC THEN  MP_TAC(ISPECL[`x:real^3`;`v:real^3`;`u:real^3`]e3_mul_dist_fan) THEN ASM_REWRITE_TAC[]
  THEN DISCH_TAC
THEN POP_ASSUM (fun th-> REWRITE_TAC[SYM(th)])
THEN MP_TAC(ISPECL[`x:real^3`;`v:real^3`;`u:real^3`]
e1_orthogonal_e3_fan) THEN ASM_REWRITE_TAC[]
  THEN DISCH_TAC THEN ASM_REWRITE_TAC[DOT_LMUL;DOT_SYM]
THEN MP_TAC(ISPECL[`x:real^3`;`v:real^3`;`u:real^3`]
e1_is_normal_fan) THEN ASM_REWRITE_TAC[]
  THEN DISCH_TAC THEN ASM_REWRITE_TAC[REAL_ARITH`(a:real)* &1+ (b:real)*(c:real)* &0= a`] THEN REPEAT DISCH_TAC 
  THEN MP_TAC(ISPECL[`&1`;`cos (psi:real)`; `r1:real`]REAL_EQ_LCANCEL_IMP) THEN REWRITE_TAC[REAL_ARITH`(a:real)* &1=a`; REAL_ARITH`&1 = (a:real) <=> a= &1`] THEN DISCH_TAC THEN POP_ASSUM MATCH_MP_TAC THEN REPEAT(POP_ASSUM MP_TAC) THEN REAL_ARITH_TAC]);;

 

let sincos_of_u_fan=prove(`!x:real^3 v:real^3 u:real^3 r1:real psi:real h1:real.
  ~collinear {u,x,v} /\ ~(v=x) /\ ~(u=x)/\ ~collinear {vec 0, v-x, u-x} /\ &0 < r1 
/\ u - x = (r1 * cos psi) % (e1_fan x v u) + (r1 * sin psi) % (e2_fan x v u) + h1 % (v-x)
==>  sin psi = &0 /\ cos psi = &1`,
MESON_TAC[cos_of_u_fan;sin_of_u_fan]);;



let sincos1_of_u_fan=prove(`!x:real^3 v:real^3 u:real^3 r1:real psi:real h1:real.
  ~collinear {x,v,u} /\ &0 < r1 
/\ u - x = (r1 * cos psi) % (e1_fan x v u) + (r1 * sin psi) % (e2_fan x v u) + h1 % (v-x)
==>  sin psi = &0 /\ cos psi = &1`,

REPEAT STRIP_TAC
THEN FIND_ASSUM MP_TAC`~collinear {x,v,u:real^3}`
THEN ONCE_REWRITE_TAC[SET_RULE`{X,V,U}={U,X,V}`]
THEN DISCH_TAC
THEN FIND_ASSUM MP_TAC`~collinear {x,v,u:real^3}`
THEN ONCE_REWRITE_TAC[SET_RULE`{X,V,U}={V,X,U}`]
THEN ONCE_REWRITE_TAC[COLLINEAR_3]
THEN DISCH_TAC
THEN MRESA_TAC th3[`(x:real^3)`;` (v:real^3)`;` (u:real^3)`]
THEN MRESA_TAC sincos_of_u_fan[`(x:real^3)`;`(v:real^3)`;` (u:real^3)`;`r1:real`; `psi:real`; `h1:real`])
;;








(*------------------------------------------------------------*)
(* change spherical coordinate in fan                         *)
(*------------------------------------------------------------*)



let change_spherical_coordinate_fan= new_definition`change_spherical_coordinate_fan (x:real^3) (v:real^3) (u:real^3) = ((\t. let r = t$1 and theta = t$2 and phi = t$3 in        
           x +(r * cos theta * sin phi) % e1_fan x v u +                       
           (r * sin theta * sin phi) % e2_fan x v u +                          
           (r * cos phi) % e3_fan x v u):real^3->real^3) ` ;;




(*---------------------------------------------------------------------------------------*)
(* the function of change coordinate is(spherecial) continuous                     *)
(*---------------------------------------------------------------------------------------*)


let REAL_CONTINUOUS_AT_COMPONENT = prove 
   (`!i a. 1 <= i /\ i <= dimindex(:N)        
           ==> (\x:real^N. x$i) real_continuous at a`,
    REWRITE_TAC[REAL_CONTINUOUS_CONTINUOUS1; o_DEF; 
                CONTINUOUS_AT_LIFT_COMPONENT]);;          
                                     

  let continuous_change_spherical_coordinate_fan = prove                              
   (`!x':real^3 v:real^3 u:real^3 x:real^3.
((\t. let r = t$1 and theta = t$2 and phi = t$3 in        
           (r * cos theta * sin phi) % e1_fan x' v u +                       
           (r * sin theta * sin phi) % e2_fan x' v u +                          
           (r * cos phi) % e3_fan x' v u))                    
     continuous at x`,                                                    
REPEAT STRIP_TAC THEN    
CONV_TAC(TOP_DEPTH_CONV let_CONV) THEN
    REPEAT(MATCH_MP_TAC CONTINUOUS_ADD THEN CONJ_TAC) THEN
    MATCH_MP_TAC CONTINUOUS_VMUL THEN
    REWRITE_TAC[GSYM REAL_CONTINUOUS_CONTINUOUS1] THEN
    REPEAT(MATCH_MP_TAC REAL_CONTINUOUS_MUL THEN CONJ_TAC) THEN
    SIMP_TAC[REAL_CONTINUOUS_AT_COMPONENT; DIMINDEX_3; ARITH] THEN
    MATCH_MP_TAC(REWRITE_RULE[o_DEF] REAL_CONTINUOUS_AT_COMPOSE) THEN
    SIMP_TAC[REAL_CONTINUOUS_AT_COMPONENT; DIMINDEX_3; ARITH] THEN
    REWRITE_TAC[REAL_CONTINUOUS_WITHIN_SIN; REAL_CONTINUOUS_WITHIN_COS]);;


(* ========================================================================== *)
(*                        AFFINE                    *)
(* ========================================================================== *)

(* local definitions *)

let complement_set= new_definition`complement_set {x:real^3, v:real^3} = {y:real^3| ~(y IN aff {x,v})} `;;


let AFF_GT_1_1 = prove
 (`!x v.                                                                     
        DISJOINT {x} {v}                                            
        ==> aff_gt {x} {v} =                                       
             {y | ?t1 t2.                                                     
                     &0 < t2 /\
                     t1 + t2 = &1 /\                      
                     y = t1 % x + t2 % v}`, 
  AFF_TAC);;                                                         


let AFF_LT_2_1 = prove
 (`!x v w.
        DISJOINT {x,v} {w}
        ==> aff_lt {x,v} {w} =
             {y | ?t1 t2 t3.
                      t3 < &0 /\
                     t1 + t2 + t3 = &1 /\
                     y = t1 % x + t2 % v + t3 % w}`,
  AFF_TAC);;


let AFF_GE_2_1 = prove
 (`!x v w.
        DISJOINT {x,v} {w}
        ==> aff_ge {x,v} {w} =
             {y | ?t1 t2 t3.
                     &0 <= t3 /\
                     t1 + t2 + t3 = &1 /\
                     y = t1 % x + t2 % v + t3 % w}`,
  AFF_TAC);;
let AFF_GE_1_2 = prove
 (`!x v w.
        DISJOINT {x} {v,w}
        ==> aff_ge {x} {v,w} =
             {y | ?t1 t2 t3.
                     &0 <= t2 /\ &0 <= t3 /\
                     t1 + t2 + t3 = &1 /\
                     y = t1 % x + t2 % v + t3 % w}`,
  AFF_TAC);;

let AFF_GT_1_2=prove(`!x v w.
         DISJOINT {x} {v, w}
         ==> aff_gt {x} {v, w} =
             {y | ?t1 t2 t3.
                      &0 < t2 /\
                      &0 < t3 /\
                      t1 + t2 + t3 = &1 /\
                      y = t1 % x + t2 % v + t3 % w}`,
AFF_TAC);;
let AFF_GT_2_2=prove(`!x u v w.
         DISJOINT {x,  u} {v, w}
         ==> aff_gt {x, u} {v, w} =
             {y | ?t1 t2 t3 t4.
                      &0 < t3 /\
                      &0 < t4 /\
                      t1 + t2 + t3 +t4= &1 /\
                      y = t1 % x + t2 %u + t3 % v + t4 % w}`,
AFF_TAC);;


let AFF_GE_1_10 = prove
 (`!x v w.
        DISJOINT {x} {v}
        ==> aff_ge {x} {v} =
             {y | ?t1 t2.
                     &0 <= t2 /\
                     t1 + t2 = &1 /\
                     y = t1 % x + t2 % v }`,
  AFF_TAC);;
let AFF_GE_1_1=prove(`!x:real^3 v:real^3. 
~(x=v)
==> aff_ge {x} {v} = {y:real^3 | ?t1:real t2:real. (&0 <= t2 ) /\ (t1 + t2 = &1) /\ (y = t1 % x + t2 % v )}`,
(let lemma=prove(`!x v.  ~(x=v) <=> DISJOINT {x} {v} `,

REWRITE_TAC[DISJOINT; INTER; IN_SING; EXTENSION; EMPTY; IN_ELIM_THM] THEN ASM_SET_TAC[]) in

REWRITE_TAC[lemma] THEN AFF_TAC));;




let affine_hull_2_fan= prove(`(!x:real^3 v:real^3. aff {x , v} = {y:real^3| ?t1:real t2:real. (t1 + t2 = &1 )/\ (y = t1 % x + t2 % v )})`,
REWRITE_TAC[aff;AFFINE_HULL_2] THEN ASM_SET_TAC[]);;



let aff_subset_aff_ge=prove(`!x:real^3 v:real^3 w:real^3.
DISJOINT {x,v} {w}
==> aff {x,v} SUBSET aff_ge {x,v} {w}`,

REPEAT GEN_TAC THEN STRIP_TAC THEN
MP_TAC(ISPECL[`(x:real^3) `;` (v:real^3)`;` (w:real^3)`]AFF_GE_2_1) THEN ASM_REWRITE_TAC[] 
  THEN DISCH_TAC
THEN ASM_REWRITE_TAC[aff; AFFINE_HULL_2; SUBSET; AFF_GE_2_1; IN_ELIM_THM]
  THEN GEN_TAC THEN STRIP_TAC
  THEN EXISTS_TAC`u:real` THEN EXISTS_TAC`v':real` THEN EXISTS_TAC`&0`
  THEN ASM_REWRITE_TAC[VECTOR_ARITH`a=b +c + &0 % d<=>a=b+c`]
  THEN REPEAT (POP_ASSUM MP_TAC) THEN REAL_ARITH_TAC);;



let COMPLEMENT_SET_FAN=prove(`!x:real^3 v:real^3 u:real^3 y:real^3 w:real^3 t1:real t2:real t3:real. 
~( w IN aff {x, v}) /\ ~(t3 = &0) /\ (t1 + t2 + t3 = &1)
==> t1 % x + t2 % v + t3 % w IN
 complement_set {x, v}`,
 REPEAT GEN_TAC THEN ASSUME_TAC(affine_hull_2_fan) THEN STRIP_TAC THEN 
REWRITE_TAC[complement_set; IN_ELIM_THM] THEN ASM_REWRITE_TAC[] THEN 
REWRITE_TAC[IN_ELIM_THM] THEN STRIP_TAC THEN REPEAT(POP_ASSUM MP_TAC) THEN DISCH_TAC THEN ASM_REWRITE_TAC[]
  THEN REWRITE_TAC[IN_ELIM_THM] THEN REPEAT DISCH_TAC THEN
  SUBGOAL_THEN  ` (t3:real) % w =((t1':real)- (t1:real)) % (x:real^3) + ((t2':real)- (t2:real)) % (v:real^3) ` ASSUME_TAC
 THENL
  [POP_ASSUM MP_TAC THEN VECTOR_ARITH_TAC;
   REPEAT(POP_ASSUM MP_TAC) THEN DISCH_THEN(LABEL_TAC "b") THEN DISCH_THEN(LABEL_TAC "c") THEN DISCH_THEN(LABEL_TAC "d")
     THEN REPEAT STRIP_TAC THEN USE_THEN "c" MP_TAC THEN REWRITE_TAC[CONTRAPOS_THM] THEN
     EXISTS_TAC `((t1':real) - (t1:real))/(t3:real)` THEN EXISTS_TAC `((t2':real) - (t2:real))/(t3:real)`
     THEN SUBGOAL_THEN  `((t1':real) - (t1:real))/(t3:real)+ ((t2':real) - (t2:real))/(t3:real) = &1` ASSUME_TAC  THENL
        [REWRITE_TAC[real_div] THEN REWRITE_TAC[REAL_ARITH `a*b+c*b=(a+c)*b`] THEN
        SUBGOAL_THEN `(t1':real) - (t1:real) + (t2':real) - (t2:real) - (t3:real) = &0` ASSUME_TAC THENL
           [REPEAT (POP_ASSUM MP_TAC) THEN REAL_ARITH_TAC;
            SUBGOAL_THEN `(t1':real) - (t1:real) + (t2':real) - (t2:real) = (t3:real)` ASSUME_TAC THENL
              [POP_ASSUM MP_TAC THEN REAL_ARITH_TAC;
               ASM_MESON_TAC[REAL_MUL_RINV]]]; 
       ASM_REWRITE_TAC[] THEN REWRITE_TAC[real_div] THEN
       REWRITE_TAC[VECTOR_ARITH ` (((t1':real) - (t1:real)) * inv (t3:real)) % (x:real^3) + (((t2':real) - (t2:real)) * inv t3) % (v:real^3) = inv t3 % ((t1' - t1) % x + (t2' - t2) % v)`] THEN 
       SUBGOAL_THEN `(t3:real) % (w:real^3) = t3 %( inv t3 % (((t1':real) - (t1:real)) % (x:real^3) + ((t2':real) - (t2:real)) % (v:real^3)))` ASSUME_TAC  THENL
                  [REWRITE_TAC[VECTOR_ARITH ` (t3:real) % (inv t3 % (((t1':real) - (t1:real)) % (x:real^3) + ((t2':real) - (t2:real)) % (v:real^3)))= (t3 * inv t3) % ((t1' - t1) % x + (t2' - t2)  % v) `] THEN
                   SUBGOAL_THEN `((t3:real) * inv (t3:real) = &1) ` ASSUME_TAC THENL
                                  [ASM_MESON_TAC[REAL_MUL_RINV]; 
                                    ASM_REWRITE_TAC[]  THEN VECTOR_ARITH_TAC];
                 ASM_MESON_TAC[VECTOR_MUL_LCANCEL_IMP]]]]);;


let aff_ge_inter_aff_ge=prove(`!(x:real^3) (v:real^3) (w:real^3).
~collinear {x,v,w}
==>
aff_ge {x} {v , w} = aff_ge {x , v} {w} INTER aff_ge {x , w} {v}`,

 REPEAT STRIP_TAC THEN MRESA_TAC th3 [`x:real^3`;`v:real^3`;`w:real^3`]
   THEN MP_TAC(ISPECL[`x:real^3`;`v:real^3`;`w:real^3`]AFF_GE_1_2)
		THEN ASM_REWRITE_TAC[] THEN DISCH_TAC 
		THEN MP_TAC(ISPECL[`x:real^3`;`v:real^3`;`w:real^3`]AFF_GE_2_1)
		THEN ASM_REWRITE_TAC[] THEN DISCH_TAC 
		THEN MP_TAC(ISPECL[`x:real^3`;`w:real^3`;`v:real^3`]AFF_GE_2_1)
		THEN ASM_REWRITE_TAC[] THEN DISCH_TAC 
   THEN ASM_REWRITE_TAC[INTER;IN_ELIM_THM;EXTENSION]THEN GEN_TAC THEN EQ_TAC 
THENL(*1*)[
STRIP_TAC THEN STRIP_TAC
THENL(*2*)[
    EXISTS_TAC `t1:real` THEN EXISTS_TAC `t2:real` THEN EXISTS_TAC `t3:real` THEN ASM_MESON_TAC[];
EXISTS_TAC `(t1:real)` THEN
         EXISTS_TAC `(t3:real)` THEN EXISTS_TAC `(t2:real)` 
THEN
         ASM_MESON_TAC[REAL_ARITH `(t1:real)+ (t3:real) +(t2:real)=t1 + t2 + t3`;VECTOR_ARITH ` t1 % x + t2 % v + t3 % w = (t1:real) % (x:real^3) + (t3:real) % (w:real^3) + (t2:real) % (v:real^3)`]](*2*);

         STRIP_TAC THEN POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC 
	   THEN POP_ASSUM(fun th-> GEN_REWRITE_TAC(PATH_CONV "rrlr" o ONCE_DEPTH_CONV )[th] THEN ASSUME_TAC(th)) THEN POP_ASSUM MP_TAC 
	   THEN POP_ASSUM(fun th-> GEN_REWRITE_TAC(PATH_CONV "rrlr" o ONCE_DEPTH_CONV )[SYM(th)] THEN ASSUME_TAC(th))
THEN DISJ_CASES_TAC(SET_RULE`t3 - t2' = &0 \/ ~((t3:real) - (t2':real) = &0) `)
THENL[POP_ASSUM MP_TAC
THEN REWRITE_TAC[REAL_ARITH`A-B= &0 <=> A=B`]
THEN REPEAT STRIP_TAC
THEN EXISTS_TAC`t1':real`
THEN EXISTS_TAC`t3':real`
THEN EXISTS_TAC`t2':real`
THEN ASM_REWRITE_TAC[VECTOR_ARITH`t1' % x + t2' % w + t3' % v = t1' % x + t3' % v + t2' % w`;
REAL_ARITH`t1' + t3' + t2'=t1' + t2' + t3'`]
THEN ASM_TAC THEN REAL_ARITH_TAC;

REWRITE_TAC[VECTOR_ARITH
 `a % x + b % y + c % z= a1 % x + b1 % z + c1 % y <=> (c-b1) % z = (a1-a) % x + (c1-b)% y`]	   
	   THEN REWRITE_TAC[REAL_ARITH`a+b+c=a1+b1+c1<=> c1-b=(a-a1)+(c-b1)`]
THEN MRESA1_TAC REAL_MUL_LINV`t3 - t2'`
	   THEN DISCH_TAC THEN DISCH_TAC 	   THEN DISCH_TAC THEN DISCH_TAC 
THEN MP_TAC(SET_RULE`
(t3 - t2') % w = (t1' - t1) % x + (t3' - t2) %  v:real^3
==> (inv (t3 - t2'))%((t3 - t2') % w ) = (inv (t3 - t2'))%((t1' - t1) % x + (t3' - t2) %  v:real^3)`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN POP_ASSUM(fun th-> ASM_REWRITE_TAC[VECTOR_ARITH`A%B%C= (A*B)%C`;VECTOR_ARITH`&1 %A=A`;VECTOR_ARITH`A%(B+C)=A%B+A%C`] THEN ASSUME_TAC(SYM(th)))
THEN STRIP_TAC
THEN SUBGOAL_THEN`w IN aff{(x:real^3),v}` ASSUME_TAC
THENL[REWRITE_TAC[aff;AFFINE_HULL_2;IN_ELIM_THM;] THEN EXISTS_TAC`inv(t3-t2') *(t1'-t1)`
THEN EXISTS_TAC`inv(t3-t2') *(t3'-t2)` THEN ASM_REWRITE_TAC[REAL_ARITH`A*B+A*C=A*(B+C)`];

ASM_SET_TAC[]]]]);;

let SCALE_AFF_TAC th=REPEAT GEN_TAC
THEN DISCH_TAC
THEN REPEAT GEN_TAC
THEN MRESAL_TAC  th[`x:real^3`;`v:real^3`;`u:real^3`][IN_ELIM_THM]
THEN STRIP_TAC
THEN EXISTS_TAC`&1 - a* t2- a* t3:real`
THEN EXISTS_TAC`a* t2:real`
THEN EXISTS_TAC`a* t3:real`
THEN ONCE_REWRITE_TAC[TAUT`A/\B/\C/\D<=>D/\ A/\B/\C`]
THEN STRIP_TAC
THENL[
ASM_REWRITE_TAC[]
THEN FIND_ASSUM MP_TAC `t1+t2+t3= &1:real`
THEN REWRITE_TAC[REAL_ARITH`A+B+C= &1<=>A= &1- B -C:real`]
THEN DISCH_TAC
THEN ASM_REWRITE_TAC[]
THEN VECTOR_ARITH_TAC;

MP_TAC(ISPECL[`a:real`;`(t2:real)`]REAL_LT_MUL)
THEN RESA_TAC
THEN MRESA_TAC REAL_LT_MUL[`a:real`;`(t3:real)`]
THEN MRESA_TAC REAL_LE_MUL[`a:real`;`(t2:real)`]
THEN MRESA_TAC REAL_LE_MUL[`a:real`;`(t3:real)`]
THEN ASM_TAC
THEN REAL_ARITH_TAC];;


let scale_aff_ge_fan=prove(`!x:real^3 v:real^3 u:real^3.
     DISJOINT {x} {v,u}
==>
(!y:real^3 a:real. y IN  aff_ge {x} {v,u} /\ &0 <= a==> a%(y-x)+x IN aff_ge{x} {v,u})`,
SCALE_AFF_TAC AFF_GE_1_2);;
let scale_aff_gt_fan=prove(`!x:real^3 v:real^3 u:real^3.
     DISJOINT {x} {v,u}
==>
(!y:real^3 a:real. y IN  aff_gt {x} {v,u} /\ &0 < a==> a%(y-x)+x IN aff_gt{x} {v,u})`,
SCALE_AFF_TAC AFF_GT_1_2);;


let origin_is_not_aff_gt_fan=prove(`!x:real^3 v:real^3 u:real^3.
   ~(u IN aff {x,v}) /\ DISJOINT {x} {v,u}==> ~(x IN  aff_gt {x} {v,u})`,
REPEAT GEN_TAC
THEN STRIP_TAC
THEN MRESAL_TAC  AFF_GT_1_2[`x:real^3`;`v:real^3`;`u:real^3`][IN_ELIM_THM]
THEN REWRITE_TAC[GSYM FORALL_NOT_THM;DE_MORGAN_THM]
THEN REPEAT STRIP_TAC 
THEN DISJ_CASES_TAC(REAL_ARITH`~( &0< (t2:real))\/ ( &0< (t2:real))`)
THENL[ASM_MESON_TAC[];
DISJ_CASES_TAC(REAL_ARITH`~( &0< (t3:real))\/ ( &0< (t3:real))`)
THENL[ASM_MESON_TAC[];
DISJ_CASES_TAC(REAL_ARITH`~( t1+t2+(t3:real)= &1)\/ (  t1+t2+(t3:real)= &1)`)
THENL[ASM_MESON_TAC[];
ASM_REWRITE_TAC[]
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[REAL_ARITH`A+B+C= &1<=>A= &1- B -C:real`]
THEN DISCH_TAC
THEN MP_TAC(REAL_ARITH`&0<t3:real==> ~(t3= &0)`)
THEN RESA_TAC
THEN MP_TAC(ISPEC`(t3:real)`REAL_MUL_LINV)
THEN RESA_TAC
THEN ASM_REWRITE_TAC[]
THEN REWRITE_TAC[VECTOR_ARITH`A=( &1-B-C) %A+B%E+C%D <=> C%(D-A)= (--B)%(E-A)`]
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`t3 % (u - x) = (--t2) % (v - x):real^3 ==> (inv (t3))%(t3 % (u - x)) = (inv (t3))%((--t2) % (v - x))`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN ASM_REWRITE_TAC[VECTOR_ARITH`A%B%C=(A*B)%C`]
THEN REDUCE_VECTOR_TAC
THEN REWRITE_TAC[VECTOR_ARITH`A-B=C%(V-B)<=>A=( &1-C)%B+C%V`]
THEN FIND_ASSUM MP_TAC `~(u IN aff {x,v}:real^3->bool)`
THEN MATCH_MP_TAC MONO_NOT
THEN STRIP_TAC
THEN REWRITE_TAC[aff; AFFINE_HULL_2; IN_ELIM_THM]
THEN EXISTS_TAC`&1 - inv t3 * --t2:real`
THEN EXISTS_TAC`inv t3 * --t2:real`
THEN ASM_REWRITE_TAC[]
THEN REAL_ARITH_TAC]]]);;






let properties_of_collinear4_points_fan=prove(`!x:real^3  v:real^3 u:real^3 v1:real^3.
~collinear{x,v,u}
/\ v1 IN aff_gt {x} {v,u}
==> ~collinear{x,v1,v}`,
REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[]
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC th3[`(x:real^3)` ;` (v:real^3)`;`(u:real^3) `;]
THEN MRESA_TAC AFF_GT_1_2[`(x:real^3)` ;` (v:real^3)`;`(u:real^3) `;]
THEN POP_ASSUM(fun th-> REWRITE_TAC[th;IN_ELIM_THM])
THEN STRIP_TAC 
THEN ASM_REWRITE_TAC[collinear1_fan;]
THEN FIND_ASSUM MP_TAC`~(u IN aff {x, v:real^3})`
THEN MATCH_MP_TAC MONO_NOT
THEN REWRITE_TAC[aff;AFFINE_HULL_2;IN_ELIM_THM]
THEN STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`t1 % x + t2 % v + t3 % u = u' % x + v' % v <=> t3 % u = (u'-t1) % x + (v'-t2) % v`;]
THEN MP_TAC (REAL_ARITH`&0< t3==> ~(t3= &0)`) THEN RESA_TAC
THEN MRESA1_TAC REAL_MUL_LINV `(t3:real)`
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`t3 % u = (u'-t1) % x + (v'-t2) % v ==> (inv (t3)) % (t3) % ( u) = (inv (t3)) % ( (u'-t1) % x + (v'-t2) % v:real^3)`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th] 
THEN ASM_REWRITE_TAC[VECTOR_ARITH`(A%B%C=(A*B)%C:real^3)`;VECTOR_ARITH`A%(B+C)=A%B+A%C`])
THEN REDUCE_VECTOR_TAC
THEN STRIP_TAC
THEN EXISTS_TAC`inv t3 * ((u' - t1):real)`
THEN EXISTS_TAC`inv t3 * ((v' - t2):real)`
THEN ASM_REWRITE_TAC[REAL_ARITH`inv t3 * (u' - t1) + inv t3 * (v' - t2)=inv t3 *(t3+ (u'+v') -( t1+ t2+t3))`;REAL_ARITH`A+ &1- &1=A`]);;



let properties_of_coplanar=prove(`!x:real^3 v:real^3 u:real^3 v1:real^3.
~collinear{x,v,u}
/\ v1 IN aff_gt {x} {v,u}
==> coplanar{x,v1,v,u}`,
REPEAT STRIP_TAC
THEN REWRITE_TAC[coplanar]
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC th3[`(x:real^3)` ;` (v:real^3)`;`(u:real^3) `;]
THEN MRESA_TAC AFF_GT_1_2[`(x:real^3)` ;` (v:real^3)`;`(u:real^3) `;]
THEN POP_ASSUM(fun th-> REWRITE_TAC[th;IN_ELIM_THM])
THEN STRIP_TAC
THEN EXISTS_TAC`x:real^3` 
THEN EXISTS_TAC`v:real^3`
THEN EXISTS_TAC`u:real^3`
THEN SUBGOAL_THEN`(x:real^3)IN affine hull {x,v,u:real^3}` ASSUME_TAC
THENL[REWRITE_TAC[ AFFINE_HULL_3;IN_ELIM_THM]
THEN EXISTS_TAC`&1`
THEN EXISTS_TAC`&0`
THEN EXISTS_TAC`&0`
THEN REDUCE_ARITH_TAC
THEN VECTOR_ARITH_TAC;
SUBGOAL_THEN`(v:real^3)IN affine hull {x,v,u:real^3}` ASSUME_TAC
THENL[ REWRITE_TAC[ AFFINE_HULL_3;IN_ELIM_THM]
THEN EXISTS_TAC`&0`
THEN EXISTS_TAC`&1`
THEN EXISTS_TAC`&0`
THEN REDUCE_ARITH_TAC
THEN VECTOR_ARITH_TAC;
SUBGOAL_THEN`(u:real^3)IN affine hull {x,v,u:real^3}` ASSUME_TAC
THENL[
REWRITE_TAC[ AFFINE_HULL_3;IN_ELIM_THM]
THEN EXISTS_TAC`&0`
THEN EXISTS_TAC`&0`
THEN EXISTS_TAC`&1`
THEN REDUCE_ARITH_TAC
THEN VECTOR_ARITH_TAC;
SUBGOAL_THEN`(v1:real^3)IN affine hull {x,v,u:real^3}` ASSUME_TAC
THENL[
REWRITE_TAC[ AFFINE_HULL_3;IN_ELIM_THM]
THEN EXISTS_TAC`t1:real`
THEN EXISTS_TAC`t2:real`
THEN EXISTS_TAC`t3:real`
THEN ASM_REWRITE_TAC[];
ASM_TAC
THEN SET_TAC[]]]]]);;



let aff_gt_subset_aff_ge=prove(`!x:real^3 v:real^3 u:real^3.
DISJOINT {x} {v,u}
==>
aff_gt {x} {v,u} SUBSET aff_ge {x} {v,u}`,
REPEAT STRIP_TAC
THEN MRESAL_TAC  AFF_GE_1_2[`x:real^3`;`v:real^3`;`u:real^3`][IN_ELIM_THM]
THEN MRESAL_TAC  AFF_GT_1_2[`x:real^3`;`v:real^3`;`u:real^3`][IN_ELIM_THM;SUBSET]
THEN REPEAT STRIP_TAC
THEN EXISTS_TAC`t1:real`
THEN EXISTS_TAC`t2:real`
THEN EXISTS_TAC`t3:real`
THEN ASM_REWRITE_TAC[]
THEN ASM_TAC
THEN REAL_ARITH_TAC);;


let aff_gt1_subset_aff_ge=prove(`!x:real^3 v:real^3 u:real^3 v1:real^3.
DISJOINT {x} {v,u} /\ ~collinear {x,v1,u}
/\ v1 IN aff_ge {x} {v,u}
==>
aff_gt {x} {v1,u} SUBSET aff_ge {x} {v,u}`,

REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC th3[`x:real^3`;`v1:real^3`;`u:real^3`]
THEN MRESAL_TAC  AFF_GE_1_2[`x:real^3`;`v:real^3`;`u:real^3`][IN_ELIM_THM]
THEN MRESAL_TAC  AFF_GT_1_2[`x:real^3`;`v1:real^3`;`u:real^3`][IN_ELIM_THM;SUBSET]
THEN REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`t1' % x + t2' % (t1 % x + t2 % v + t3 % u) + t3' % u
=(t1'+ t2'*t1) % x + (t2'* t2) % v + (t2' * t3  + t3') % u:real^3`]
THEN STRIP_TAC
THEN EXISTS_TAC`t1' + t2' * t1:real`
THEN EXISTS_TAC`t2' * t2:real`
THEN EXISTS_TAC`t2' * t3 +t3':real`
THEN MP_TAC(REAL_ARITH`&0<t2'==> &0<= t2'`)
THEN RESA_TAC
THEN MRESA_TAC REAL_LE_MUL[`t2':real`;`t2:real`]
THEN MRESA_TAC REAL_LE_MUL[`t2':real`;`t3:real`]
THEN ASM_REWRITE_TAC[REAL_ARITH`(t1' + t2' * t1) + t2' * t2 + t2' * t3 + t3'=
t1'+t2'*(t1+t2+t3)+t3'`; REAL_ARITH`A* &1=A`]
THEN ASM_TAC
THEN REAL_ARITH_TAC);;

let aff_gt12_subset_aff_ge=prove(`!x:real^3 v:real^3 u:real^3 v1:real^3.
DISJOINT {x} {v,u} /\ ~collinear {x,v1,v}
/\ v1 IN aff_ge {x} {v,u}
==>
aff_gt {x} {v1,v} SUBSET aff_ge {x} {v,u}`,

REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC th3[`x:real^3`;`v1:real^3`;`v:real^3`]
THEN MRESAL_TAC  AFF_GE_1_2[`x:real^3`;`v:real^3`;`u:real^3`][IN_ELIM_THM]
THEN MRESAL_TAC  AFF_GT_1_2[`x:real^3`;`v1:real^3`;`v:real^3`][IN_ELIM_THM;SUBSET]
THEN REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`t1' % x + t2' % (t1 % x + t2 % v + t3 % u) + t3' % v
=(t1'+ t2'*t1) % x + (t2'* t2+t3') % v + (t2' * t3 ) % u:real^3`]
THEN STRIP_TAC
THEN EXISTS_TAC`t1' + t2' * t1:real`
THEN EXISTS_TAC`t2' * t2 +t3':real`
THEN EXISTS_TAC`t2' * t3 :real`
THEN MP_TAC(REAL_ARITH`&0<t2'==> &0<= t2'`)
THEN RESA_TAC
THEN MRESA_TAC REAL_LE_MUL[`t2':real`;`t2:real`]
THEN MRESA_TAC REAL_LE_MUL[`t2':real`;`t3:real`]
THEN ASM_REWRITE_TAC[REAL_ARITH`(t1' + t2' * t1) + (t2' * t2 +t3')+ t2' * t3 =
t1'+t2'*(t1+t2+t3)+t3'`; REAL_ARITH`A* &1=A`]
THEN ASM_TAC
THEN REAL_ARITH_TAC);;

let aff_gt2_subset_aff_ge=prove(`!x:real^3 v:real^3 u:real^3 v1:real^3.
DISJOINT {x} {v,u} /\ ~collinear {x,v1,u} /\ ~collinear {x,v1,v}
/\ v1 IN aff_gt {x} {v,u}
==>
azim x v1 v u= pi`,

REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC th3[`x:real^3`;`v1:real^3`;`u:real^3`]
THEN MRESAL_TAC  AFF_GT_1_2[`x:real^3`;`v:real^3`;`u:real^3`][IN_ELIM_THM]
THEN MRESA_TAC 	AZIM_EQ_PI[`x:real^3`;`v1:real^3`;`v:real^3`;`u:real^3`]
THEN REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`v1 = t1 % x + t2 % v + t3 % u
<=> t2 % v = (--t1) % x + v1 + (--t3) % u`]
THEN MP_TAC(REAL_ARITH`&0<t2==> ~( t2= &0)`)
THEN RESA_TAC
THEN MP_TAC(REAL_ARITH`&0<t3==> -- t3< &0`)
THEN RESA_TAC
THEN MRESA1_TAC REAL_MUL_LINV`t2:real`
THEN MRESA1_TAC REAL_LT_INV`t2:real`
THEN MRESAL_TAC REAL_LT_LMUL[`inv t2:real`;`-- t3:real`;`&0`][REAL_ARITH`A * &0= &0`]
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`t2 % v = --t1 % x + v1 + --t3 % u:real^3 ==> (inv (t2))%(t2 % v ) = (inv (t2))%( --t1 % x + v1 + --t3 % u)`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN ASM_REWRITE_TAC[VECTOR_ARITH`A%B%C=(A*B)%C`; VECTOR_ARITH`A%(B+C+D)=A%B+A%C+A%D`]
THEN REDUCE_VECTOR_TAC
THEN STRIP_TAC
THEN MRESAL_TAC  AFF_LT_2_1[`x:real^3`;`v1:real^3`;`u:real^3`][IN_ELIM_THM;SUBSET]
THEN EXISTS_TAC`(inv t2 * --t1):real`
THEN EXISTS_TAC`inv t2:real`
THEN EXISTS_TAC`(inv t2 * --t3):real`
THEN ASM_REWRITE_TAC[REAL_ARITH`inv t2 * (--t1) + inv t2 + inv t2 * (--t3)=
inv t2 * (t2+ &1 -(t1 +t2 +t3))`; REAL_ARITH`A+ &1- &1=A`]);;


let remove_variable_fan=prove(`!x:real^3 v:real^3 u:real^3 w:real^3 t1:real t2:real t3:real.
 &0 < t3 
/\ w = t1 % x + t2 % v + t3 % u
==> u= inv(t3) % w - (inv(t3)*t1) % x- (inv(t3) * t2) % v`,
REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN GEN_REWRITE_TAC(LAND_CONV o ONCE_DEPTH_CONV)[VECTOR_ARITH`w = t1'' % x + t2'' % v + t3'' % u <=> t3'' % u = w-t1'' % x - t2'' % (v:real^3)`]
THEN MP_TAC(REAL_ARITH `&0 < (t3:real) ==>  ~(t3 = &0)`)
THEN MP_TAC(ISPEC`(t3:real)`REAL_LT_INV)
THEN POP_ASSUM(fun th-> REWRITE_TAC[th] THEN ASSUME_TAC(th))
THEN STRIP_TAC THEN STRIP_TAC 
THEN MP_TAC(ISPEC`(t3:real)`REAL_MUL_LINV)
THEN POP_ASSUM(fun th-> REWRITE_TAC[th] THEN ASSUME_TAC(th))
THEN STRIP_TAC
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`
t3 % u = w-t1 % x - t2 % v:real^3
==> (inv (t3))%(t3 % u) = (inv (t3))%( w-t1 % x - t2 % v:real^3)`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN POP_ASSUM(fun th-> REWRITE_TAC[th;VECTOR_ARITH`A%B%C= (A*B)%C`;VECTOR_ARITH`&1 %A=A`;VECTOR_ARITH`A%(B-C-D)=A%B-A%C-A%D`] THEN ASSUME_TAC(th)));;


let aff_gt_inter_aff_gt=prove(`!(x:real^3) (v:real^3) (w:real^3).
~collinear {x,v,w}
==>
aff_gt {x} {v , w} = aff_gt {x , v} {w} INTER aff_gt {x , w} {v}`,

 REPEAT STRIP_TAC THEN MRESA_TAC th3 [`x:real^3`;`v:real^3`;`w:real^3`]
   THEN MP_TAC(ISPECL[`x:real^3`;`v:real^3`;`w:real^3`]AFF_GT_1_2)
		THEN ASM_REWRITE_TAC[] THEN DISCH_TAC 
		THEN MP_TAC(ISPECL[`x:real^3`;`v:real^3`;`w:real^3`]AFF_GT_2_1)
		THEN ASM_REWRITE_TAC[] THEN DISCH_TAC 
		THEN MP_TAC(ISPECL[`x:real^3`;`w:real^3`;`v:real^3`]AFF_GT_2_1)
		THEN ASM_REWRITE_TAC[] THEN DISCH_TAC 
   THEN ASM_REWRITE_TAC[INTER;IN_ELIM_THM;EXTENSION]THEN GEN_TAC THEN EQ_TAC 
THENL(*1*)[
STRIP_TAC THEN STRIP_TAC
THENL(*2*)[
    EXISTS_TAC `t1:real` THEN EXISTS_TAC `t2:real` THEN EXISTS_TAC `t3:real` THEN ASM_MESON_TAC[];
EXISTS_TAC `(t1:real)` THEN
         EXISTS_TAC `(t3:real)` THEN EXISTS_TAC `(t2:real)` 
THEN
         ASM_MESON_TAC[REAL_ARITH `(t1:real)+ (t3:real) +(t2:real)=t1 + t2 + t3`;VECTOR_ARITH ` t1 % x + t2 % v + t3 % w = (t1:real) % (x:real^3) + (t3:real) % (w:real^3) + (t2:real) % (v:real^3)`]](*2*);

         STRIP_TAC THEN POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC 
	   THEN POP_ASSUM(fun th-> GEN_REWRITE_TAC(PATH_CONV "rrlr" o ONCE_DEPTH_CONV )[th] THEN ASSUME_TAC(th)) THEN POP_ASSUM MP_TAC 
	   THEN POP_ASSUM(fun th-> GEN_REWRITE_TAC(PATH_CONV "rrlr" o ONCE_DEPTH_CONV )[SYM(th)] THEN ASSUME_TAC(th))
THEN DISJ_CASES_TAC(SET_RULE`t3 - t2' = &0 \/ ~((t3:real) - (t2':real) = &0) `)
THENL[POP_ASSUM MP_TAC
THEN REWRITE_TAC[REAL_ARITH`A-B= &0 <=> A=B`]
THEN REPEAT STRIP_TAC
THEN EXISTS_TAC`t1':real`
THEN EXISTS_TAC`t3':real`
THEN EXISTS_TAC`t2':real`
THEN ASM_REWRITE_TAC[VECTOR_ARITH`t1' % x + t2' % w + t3' % v = t1' % x + t3' % v + t2' % w`;
REAL_ARITH`t1' + t3' + t2'=t1' + t2' + t3'`]
THEN ASM_TAC THEN REAL_ARITH_TAC;

REWRITE_TAC[VECTOR_ARITH
 `a % x + b % y + c % z= a1 % x + b1 % z + c1 % y <=> (c-b1) % z = (a1-a) % x + (c1-b)% y`]	   
	   THEN REWRITE_TAC[REAL_ARITH`a+b+c=a1+b1+c1<=> c1-b=(a-a1)+(c-b1)`]
THEN MRESA1_TAC REAL_MUL_LINV`t3 - t2'`
	   THEN DISCH_TAC THEN DISCH_TAC 	   THEN DISCH_TAC THEN DISCH_TAC 
THEN MP_TAC(SET_RULE`
(t3 - t2') % w = (t1' - t1) % x + (t3' - t2) %  v:real^3
==> (inv (t3 - t2'))%((t3 - t2') % w ) = (inv (t3 - t2'))%((t1' - t1) % x + (t3' - t2) %  v:real^3)`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN POP_ASSUM(fun th-> ASM_REWRITE_TAC[VECTOR_ARITH`A%B%C= (A*B)%C`;VECTOR_ARITH`&1 %A=A`;VECTOR_ARITH`A%(B+C)=A%B+A%C`] THEN ASSUME_TAC(SYM(th)))
THEN STRIP_TAC
THEN SUBGOAL_THEN`w IN aff{(x:real^3),v}` ASSUME_TAC
THENL[REWRITE_TAC[aff;AFFINE_HULL_2;IN_ELIM_THM;] THEN EXISTS_TAC`inv(t3-t2') *(t1'-t1)`
THEN EXISTS_TAC`inv(t3-t2') *(t3'-t2)` THEN ASM_REWRITE_TAC[REAL_ARITH`A*B+A*C=A*(B+C)`];

ASM_SET_TAC[]]]]);;


let aff_gt3_subset_aff_gt=prove(`!x:real^3 v:real^3 u:real^3 v1:real^3.
DISJOINT {x} {v,u} /\ ~collinear{x,v,v1}
/\ v1 IN aff_gt {x} {v,u}
==>
aff_gt {x} {v,v1} SUBSET aff_gt {x} {v,u}`,
REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC th3[`x:real^3`;`v:real^3`;`v1:real^3`]
THEN MRESAL_TAC  AFF_GT_1_2[`x:real^3`;`v:real^3`;`u:real^3`][IN_ELIM_THM]
THEN MRESAL_TAC  AFF_GT_1_2[`x:real^3`;`v:real^3`;`v1:real^3`][IN_ELIM_THM;SUBSET]
THEN REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`t1' % x + t2' % v + t3' % (t1 % x + t2 % v + t3 % u)
=(t1'+ t3'*t1) % x + (t2'+ t3' * t2) % v + (t3' * t3) % u:real^3`]
THEN STRIP_TAC
THEN EXISTS_TAC`t1' + t3' * t1:real`
THEN EXISTS_TAC`t2' + t3' * t2:real`
THEN EXISTS_TAC`t3' * t3:real`
THEN MRESA_TAC REAL_LT_MUL[`t3':real`;`t2:real`]
THEN MRESA_TAC REAL_LT_MUL[`t3':real`;`t3:real`]
THEN ASM_REWRITE_TAC[REAL_ARITH`(t1' + t3' * t1) + (t2' + t3' * t2) + t3' * t3=
t1'+ t2' + t3'*(t1+t2+t3)`; REAL_ARITH`A* &1=A`]
THEN ASM_TAC
THEN REAL_ARITH_TAC);;



let aff_ge1_subset_aff_ge=prove(`!x:real^3 v:real^3 u:real^3 v1:real^3.
DISJOINT {x} {v,u} /\ ~collinear{x,v1,u}
/\ v1 IN aff_ge {x} {v,u}
==>
aff_ge {x} {v1,u} SUBSET aff_ge {x} {v,u}`,

REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC th3[`x:real^3`;`v1:real^3`;`u:real^3`]
THEN MRESAL_TAC  AFF_GE_1_2[`x:real^3`;`v:real^3`;`u:real^3`][IN_ELIM_THM]
THEN MRESAL_TAC  AFF_GE_1_2[`x:real^3`;`v1:real^3`;`u:real^3`][IN_ELIM_THM;SUBSET]
THEN REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`t1' % x + t2' % (t1 % x + t2 % v + t3 % u) + t3' % u
=(t1'+ t2'*t1) % x + (t2'* t2) % v + (t2' * t3  + t3') % u:real^3`]
THEN STRIP_TAC
THEN EXISTS_TAC`t1' + t2' * t1:real`
THEN EXISTS_TAC`t2' * t2:real`
THEN EXISTS_TAC`t2' * t3 +t3':real`
THEN MP_TAC(REAL_ARITH`&0<t2'==> &0<= t2'`)
THEN RESA_TAC
THEN MRESA_TAC REAL_LE_MUL[`t2':real`;`t2:real`]
THEN MRESA_TAC REAL_LE_MUL[`t2':real`;`t3:real`]
THEN ASM_REWRITE_TAC[REAL_ARITH`(t1' + t2' * t1) + t2' * t2 + t2' * t3 + t3'=
t1'+t2'*(t1+t2+t3)+t3'`; REAL_ARITH`A* &1=A`]
THEN ASM_TAC
THEN REAL_ARITH_TAC);;

let aff_ge_1_1_subset_aff_ge_fan=prove(`!x:real^3 v:real^3 u:real^3 v1:real^3.
DISJOINT {x} {v,u} /\ ~(x=v1)
/\ v1 IN aff_ge {x} {v,u}
==>
aff_ge {x} {v1} SUBSET aff_ge {x} {v,u}`,

REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN MRESAL_TAC  AFF_GE_1_2[`x:real^3`;`v:real^3`;`u:real^3`][IN_ELIM_THM]
THEN MRESAL_TAC  AFF_GE_1_1[`x:real^3`;`v1:real^3`][IN_ELIM_THM;SUBSET]
THEN REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`t1' % x + t2' % (t1 % x + t2 % v + t3 % u)
=(t1'+ t2'*t1) % x + (t2'* t2) % v + (t2' * t3 ) % u:real^3`]
THEN STRIP_TAC
THEN EXISTS_TAC`t1' + t2' * t1:real`
THEN EXISTS_TAC`t2' * t2:real`
THEN EXISTS_TAC`t2' * t3 :real`
THEN MP_TAC(REAL_ARITH`&0<t2'==> &0<= t2'`)
THEN RESA_TAC
THEN MRESA_TAC REAL_LE_MUL[`t2':real`;`t2:real`]
THEN MRESA_TAC REAL_LE_MUL[`t2':real`;`t3:real`]
THEN ASM_REWRITE_TAC[REAL_ARITH`(t1' + t2' * t1) + t2' * t2 + t2' * t3 =
t1'+t2'*(t1+t2+t3)`; REAL_ARITH`A* &1=A`]);;


let decomposition_planar_by_angle_fan=prove(`!x:real^3 v:real^3 u:real^3 w:real^3.
~ collinear {x,v,u} /\ ~collinear {x,v,w}
/\ w IN aff_ge {x,v} {u}
==> u IN aff_gt {x} {v,w} \/ w IN aff_ge {x} {v,u}`,
REPEAT STRIP_TAC
THEN MRESAL_TAC aff_ge_inter_aff_ge[`(x:real^3)`;`(v:real^3)`;`(u:real^3)`][INTER; IN_ELIM_THM]
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC AZIM_EQ_0_GE[`x:real^3`;`v:real^3`;`w:real^3`; `u:real^3`]
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM(th)])
THEN MRESA_TAC AZIM_EQ_0_ALT[`x:real^3`;`v:real^3`;`w:real^3`; `u:real^3`]
THEN MRESA_TAC th3[`x:real^3`;`v:real^3`;`w:real^3`]
THEN MRESAL_TAC  AFF_GT_2_1[`x:real^3`;`v:real^3`;`w:real^3`][IN_ELIM_THM]
THEN REPEAT STRIP_TAC
THEN DISJ_CASES_TAC(REAL_ARITH`(&0 < t2) \/   &0 <= --(t2:real)`)
THENL[
SUBGOAL_THEN `u IN aff_gt {x} {v,w:real^3}` ASSUME_TAC
THENL[
MRESAL_TAC  AFF_GT_1_2[`x:real^3`;`v:real^3`;`w:real^3`][IN_ELIM_THM]
THEN EXISTS_TAC`t1:real`
THEN EXISTS_TAC`t2:real`
THEN EXISTS_TAC`t3:real`
THEN MP_TAC(REAL_ARITH`&0< t3==> &0 <= t3:real`)
THEN ASM_REWRITE_TAC[];
POP_ASSUM MP_TAC THEN SET_TAC[]];
MRESA_TAC remove_variable_fan[`x:real^3`; `v:real^3`; `w:real^3`;`u:real^3`;`t1:real`;`t2:real`;`t3:real`]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM(th);])
THEN DISCH_TAC
THEN REWRITE_TAC[VECTOR_ARITH`inv t3 % u - (inv t3 * t1) % x - (inv t3 * t2) % v
=(--inv t3 * t1) % x + inv t3 % u + (inv t3 * (--t2)) % v`]
THEN MP_TAC(REAL_ARITH`&0<t3==> ~( t3= &0)`)
THEN RESA_TAC
THEN MRESA1_TAC REAL_MUL_LINV`t3:real`
THEN MRESA1_TAC REAL_LT_INV`t3:real`
THEN MP_TAC(REAL_ARITH`&0< inv t3==> &0 <= inv t3`)
THEN RESA_TAC
THEN MRESA_TAC REAL_LE_MUL[`inv t3:real`;`-- (t2:real)`]
THEN DISCH_TAC
THEN SUBGOAL_THEN `w IN aff_ge {x, u} {v:real^3}` ASSUME_TAC
THENL[ 
MRESA_TAC th3[`x:real^3`;`v:real^3`;`u:real^3`]
THEN MRESAL_TAC  AFF_GE_2_1[`x:real^3`;`u:real^3`;`v:real^3`][IN_ELIM_THM]
THEN EXISTS_TAC`(--inv t3 * t1):real`
THEN EXISTS_TAC`inv t3:real`
THEN EXISTS_TAC`(inv t3 * --t2):real`
THEN ASM_REWRITE_TAC[REAL_ARITH`--inv t3 * t1 + inv t3 + inv t3 * --t2=
inv t3 * (t3+ &1- (t1 +t2 + t3))`; REAL_ARITH`a + &1 - &1 =a`];
POP_ASSUM MP_TAC THEN SET_TAC[]]]);;

let point_in_aff_ge=prove(`!(x:real^3) (v:real^3) (w:real^3).
~collinear {x,v,w}
==>
x IN aff_ge {x} {v,w}
/\ v IN aff_ge {x} {v,w}
/\ w IN aff_ge {x} {v,w}`,

REPEAT GEN_TAC THEN STRIP_TAC THEN MRESA_TAC th3 [`x:real^3`;`v:real^3`;`w:real^3`]
  THEN MP_TAC(ISPECL[`x:real^3`;`v:real^3`;`w:real^3`]AFF_GE_1_2)
THEN RESA_TAC
		THEN ASM_REWRITE_TAC[IN_ELIM_THM]
THEN STRIP_TAC
THENL[ EXISTS_TAC`&1:real`
THEN EXISTS_TAC`&0:real`
THEN EXISTS_TAC`&0:real`
THEN REDUCE_VECTOR_TAC
THEN REAL_ARITH_TAC;  
STRIP_TAC
THENL[ EXISTS_TAC`&0:real`
THEN EXISTS_TAC`&1:real`
THEN EXISTS_TAC`&0:real`
THEN REDUCE_VECTOR_TAC
THEN REAL_ARITH_TAC;

EXISTS_TAC`&0:real`
THEN EXISTS_TAC`&0:real`
THEN EXISTS_TAC`&1:real`
THEN REDUCE_VECTOR_TAC
THEN REAL_ARITH_TAC]]);;  


let aff_ge_subset_aff_gt_union_aff_ge=prove(`!(x:real^3) (v:real^3) (w:real^3).
~collinear {x,v,w}
==>
aff_ge {x} {v , w} SUBSET  (aff_gt {x , v} {w}) UNION (aff_ge {x} {v})`,
REPEAT STRIP_TAC THEN MRESA_TAC th3 [`x:real^3`;`v:real^3`;`w:real^3`]
  THEN MP_TAC(ISPECL[`x:real^3`;`v:real^3`;`w:real^3`]AFF_GE_1_2)
		THEN ASM_REWRITE_TAC[] THEN DISCH_TAC 
		THEN MP_TAC(ISPECL[`x:real^3`;`v:real^3`;`w:real^3`]AFF_GT_2_1)
		THEN ASM_REWRITE_TAC[] THEN DISCH_TAC 
		THEN MP_TAC(ISPECL[`x:real^3`;`v:real^3`]AFF_GE_1_1)
		THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN ASM_REWRITE_TAC[SUBSET; UNION;IN_ELIM_THM]
  THEN GEN_TAC THEN
REWRITE_TAC[REAL_ARITH `(&0 <= (t3:real)) <=> (&0 < t3) \/ ( t3 = &0)`; TAUT `(a \/ b) /\ (c \/ d) /\ e /\ f <=> ((a \/ b)/\ c /\ e /\ f) \/ ((a \/ b) /\ d /\ e /\ f)`; EXISTS_OR_THM] THEN
MATCH_MP_TAC MONO_OR THEN
SUBGOAL_THEN `((?t1:real t2:real t3:real.
       (&0 < t2 \/ t2 = &0) /\
        &0< t3 /\
       t1 + t2 + t3 = &1 /\
       (x':real^3) = t1 % x + t2 % v + t3 % w)
  ==> (?t1 t2 t3.
            &0< t3 /\ t1 + t2 + t3 = &1 /\ x' = t1 % x + t2 % v + t3 % w))` ASSUME_TAC
THENL  
[MESON_TAC[];
     ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC THEN MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC THEN
     REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN GEN_TAC THEN
     REWRITE_TAC[REAL_ARITH `(&0< (t2:real) \/ (t2 = &0)) <=> ( &0<= t2)`] THEN STRIP_TAC THEN POP_ASSUM MP_TAC THEN
     POP_ASSUM MP_TAC THEN ASM_REWRITE_TAC[] THEN
     REWRITE_TAC [REAL_ARITH `(a:real)+ &0 = a`; VECTOR_ARITH  `&0 % (w:real^3) = vec 0`; 
     VECTOR_ARITH `  ((x':real^3) = (t1:real) % (x:real^3) + (t2:real) % (v:real^3) + vec 0)<=> ( x' = t1 % x + t2 % v )` ]
       THEN MESON_TAC[]]);;	

let pos_in_aff_ge_fan=prove(`!x:real^3 v:real^3 u:real^3 a:real.
DISJOINT {x} {v,u}
/\ &0<a /\ a< &1
==>
(&1-a)%v + a % u IN aff_ge {x} {v,u:real^3}`,

REPEAT STRIP_TAC
THEN MRESAL_TAC AFF_GE_1_2[`x:real^3`;`v:real^3`;`u:real^3`][IN_ELIM_THM]
THEN EXISTS_TAC`&0`
THEN EXISTS_TAC`&1 -a:real`
THEN EXISTS_TAC`a:real`
THEN MP_TAC(REAL_ARITH`&0< a /\ a  < &1 ==> &0 <= &1 - a /\ &0 <= a`)
THEN RESA_TAC
THEN ASM_REWRITE_TAC[]
THEN REDUCE_VECTOR_TAC
THEN REAL_ARITH_TAC);;


let aff_gt1_subset_aff_gt=prove(`!x:real^3 v:real^3 u:real^3 v1:real^3.
DISJOINT {x} {v,u} /\ ~collinear {x,v1,u}
/\ v1 IN aff_gt {x} {v,u}
==>
aff_gt {x} {v1,u} SUBSET aff_gt {x} {v,u}`,

REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC th3[`x:real^3`;`v1:real^3`;`u:real^3`]
THEN MRESAL_TAC  AFF_GT_1_2[`x:real^3`;`v:real^3`;`u:real^3`][IN_ELIM_THM]
THEN MRESAL_TAC  AFF_GT_1_2[`x:real^3`;`v1:real^3`;`u:real^3`][IN_ELIM_THM;SUBSET]
THEN REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`t1' % x + t2' % (t1 % x + t2 % v + t3 % u) + t3' % u
=(t1'+ t2'*t1) % x + (t2'* t2) % v + (t2' * t3  + t3') % u:real^3`]
THEN STRIP_TAC
THEN EXISTS_TAC`t1' + t2' * t1:real`
THEN EXISTS_TAC`t2' * t2:real`
THEN EXISTS_TAC`t2' * t3 +t3':real`
THEN MP_TAC(REAL_ARITH`&0<t2'==> &0<= t2'`)
THEN RESA_TAC
THEN MRESA_TAC REAL_LT_MUL[`t2':real`;`t2:real`]
THEN MRESA_TAC REAL_LT_MUL[`t2':real`;`t3:real`]
THEN ASM_REWRITE_TAC[REAL_ARITH`(t1' + t2' * t1) + t2' * t2 + t2' * t3 + t3'=
t1'+t2'*(t1+t2+t3)+t3'`; REAL_ARITH`A* &1=A`]
THEN ASM_TAC
THEN REAL_ARITH_TAC);;

let aff_ge_eq_aff_gt_union_aff_ge=prove(`!(x:real^3) (v:real^3) (w:real^3).
~collinear {x,v,w}
==>
aff_ge {x} {v , w} =  (aff_gt {x} {v,w}) UNION (aff_ge {x} {v}) UNION (aff_ge {x} {w})`,

REPEAT STRIP_TAC THEN MRESA_TAC th3 [`x:real^3`;`v:real^3`;`w:real^3`]
  THEN MP_TAC(ISPECL[`x:real^3`;`v:real^3`;`w:real^3`]AFF_GE_1_2)
		THEN ASM_REWRITE_TAC[] THEN DISCH_TAC 
		THEN MP_TAC(ISPECL[`x:real^3`;`v:real^3`;`w:real^3`]AFF_GT_1_2)
		THEN ASM_REWRITE_TAC[] THEN DISCH_TAC 
		THEN MP_TAC(ISPECL[`x:real^3`;`v:real^3`]AFF_GE_1_1)
		THEN MP_TAC(ISPECL[`x:real^3`;`w:real^3`]AFF_GE_1_1) THEN RESA_TAC THEN RESA_TAC
		THEN ASM_REWRITE_TAC[EXTENSION;UNION;IN_ELIM_THM]
THEN GEN_TAC THEN EQ_TAC
THENL[STRIP_TAC
THEN MP_TAC(REAL_ARITH`&0<= t2/\ &0<=t3==> (t2= &0)\/ (t3= &0)\/ (&0<t2 /\ &0<t3)`)
THEN RESA_TAC
THENL[
SUBGOAL_THEN `(?t1' t2'.
      &0 <= t2' /\
      t1' + t2' = &1 /\
      t1 % x + &0 % v + t3 % w = t1' % x + t2' % w:real^3)`ASSUME_TAC
THENL[
EXISTS_TAC`t1:real`
THEN EXISTS_TAC`t3:real`
THEN REDUCE_VECTOR_TAC
THEN ASM_REWRITE_TAC[]
THEN ASM_TAC THEN REAL_ARITH_TAC;
POP_ASSUM MP_TAC THEN SET_TAC[]];

SUBGOAL_THEN `(?t1' t2'.
      &0 <= t2' /\
      t1' + t2' = &1 /\
      t1 % x + t2 % v + &0 % w = t1' % x + t2' % v:real^3)`ASSUME_TAC
THENL[
EXISTS_TAC`t1:real`
THEN EXISTS_TAC`t2:real`
THEN REDUCE_VECTOR_TAC
THEN ASM_REWRITE_TAC[]
THEN ASM_TAC THEN REAL_ARITH_TAC;
POP_ASSUM MP_TAC THEN SET_TAC[]];

SUBGOAL_THEN `(?t1' t2' t3'.
      &0 < t2' /\
      &0 < t3' /\
      t1' + t2' + t3' = &1 /\
      t1 % x + t2 % v + t3 % w = t1' % x + t2' % v + t3' % w:real^3)`ASSUME_TAC
THENL[
EXISTS_TAC`t1:real`
THEN EXISTS_TAC`t2:real`
THEN EXISTS_TAC`t3:real`
THEN REDUCE_VECTOR_TAC
THEN ASM_REWRITE_TAC[]
THEN ASM_TAC THEN REAL_ARITH_TAC;
POP_ASSUM MP_TAC THEN SET_TAC[]]];

STRIP_TAC
THENL[
EXISTS_TAC`t1:real`
THEN EXISTS_TAC`t2:real`
THEN EXISTS_TAC`t3:real`
THEN REDUCE_VECTOR_TAC
THEN ASM_REWRITE_TAC[]
THEN ASM_TAC THEN REAL_ARITH_TAC;
EXISTS_TAC`t1:real`
THEN EXISTS_TAC`t2:real`
THEN EXISTS_TAC`&0:real`
THEN REDUCE_VECTOR_TAC
THEN ASM_REWRITE_TAC[]
THEN ASM_TAC THEN REAL_ARITH_TAC;
EXISTS_TAC`t1:real`
THEN EXISTS_TAC`&0:real`
THEN EXISTS_TAC`t2:real`
THEN REDUCE_VECTOR_TAC
THEN ASM_REWRITE_TAC[]
THEN ASM_TAC THEN REAL_ARITH_TAC]]);;
let AFFINE_HULL_1=prove(`!a. affine hull {a} ={u % a| u = &1}`, 
  SIMP_TAC[AFFINE_HULL_FINITE; FINITE_INSERT; FINITE_RULES] THEN
  SIMP_TAC[AFFINE_HULL_FINITE_STEP; FINITE_INSERT; FINITE_RULES] THEN
  REWRITE_TAC[REAL_ARITH `x - y = z:real <=> x = y + z`;
              VECTOR_ARITH `x - y = z:real^N <=> x = y + z`] THEN
  REWRITE_TAC[VECTOR_ADD_RID; REAL_ADD_RID] THEN SET_TAC[]);;

let aff_ge1_1_subset_aff_ge=prove(`!x:real^3 v:real^3 u:real^3 v1:real^3.
DISJOINT {x} {v,u} /\ ~(x=v1)
/\ v1 IN aff_ge {x} {v,u}
==>
aff_ge {x} {v1} SUBSET aff_ge {x} {v,u}`,
REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN MRESAL_TAC  AFF_GE_1_2[`x:real^3`;`v:real^3`;`u:real^3`][IN_ELIM_THM]
THEN MRESAL_TAC  AFF_GE_1_1[`x:real^3`;`v1:real^3`;][IN_ELIM_THM;SUBSET]
THEN REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`t1' % x + t2' % (t1 % x + t2 % v + t3 % u) 
=(t1'+ t2'*t1) % x + (t2'* t2) % v + (t2' * t3 ) % u:real^3`]
THEN STRIP_TAC
THEN EXISTS_TAC`t1' + t2' * t1:real`
THEN EXISTS_TAC`t2' * t2:real`
THEN EXISTS_TAC`t2' * t3:real`
THEN MP_TAC(REAL_ARITH`&0<t2'==> &0<= t2'`)
THEN RESA_TAC
THEN MRESA_TAC REAL_LE_MUL[`t2':real`;`t2:real`]
THEN MRESA_TAC REAL_LE_MUL[`t2':real`;`t3:real`]
THEN ASM_REWRITE_TAC[REAL_ARITH`(t1' + t2' * t1) + t2' * t2 + t2' * t3 =
t1'+t2'*(t1+t2+t3)`; REAL_ARITH`A* &1=A`]
THEN ASM_TAC
THEN REAL_ARITH_TAC);;


let properties1_inside_fan=prove(`!x:real^3 u:real^3 w:real^3.
DISJOINT {x} {u,w}
/\ &0<a /\ a< &1
==> (&1-a)%u+ a%w IN aff_ge {x} {u,w:real^3}`,
REPEAT STRIP_TAC
THEN  MRESAL_TAC AFF_GE_1_2[`x:real^3`;`u:real^3`;`w:real^3`][IN_ELIM_THM]
THEN EXISTS_TAC`&0`
THEN EXISTS_TAC`&1 -a:real`
THEN EXISTS_TAC`a:real`
THEN MP_TAC(REAL_ARITH`&0< a /\ a  < &1 ==> &0 <= &1 - a /\ &0 <= a`)
THEN RESA_TAC
THEN ASM_REWRITE_TAC[]
THEN REDUCE_VECTOR_TAC
THEN REAL_ARITH_TAC);;



let properties_inside_collinear1_fan=prove(`!x:real^3 u:real^3 w:real^3.
~collinear{x,u,w}
/\ &0<a /\ a< &1
==>  aff_ge {x} {u} INTER aff_ge {x} {(&1-a)%u+ a%w,w:real^3} SUBSET aff_ge {x} {}`,

REPEAT STRIP_TAC
THEN MRESA_TAC properties_inside_collinear_fan[`(x:real^3)`;`(u:real^3)`;`(w:real^3)`;`a:real`]
THEN MRESA_TAC th3[`(x:real^3)`;`((&1-a)%u+ a%w:real^3)`;`(w:real^3)`]
THEN MRESA_TAC th3[`(x:real^3)`;`u:real^3`;`(w:real^3)`]
THEN  MRESAL_TAC AFF_GE_1_2[`x:real^3`;`(&1-a)%u+ a%w:real^3`;`w:real^3`][IN_ELIM_THM]
THEN  MRESAL_TAC AFF_GE_1_1[`x:real^3`;`u:real^3`][IN_ELIM_THM;INTER;SUBSET;AFF_GE_EQ_AFFINE_HULL;AFFINE_HULL_1]
THEN REPEAT STRIP_TAC
THEN EXISTS_TAC`&1`
THEN REDUCE_VECTOR_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[]
THEN POP_ASSUM MP_TAC
THEN MP_TAC(REAL_ARITH`&0<= t2' /\ &0 <= t3==> (t2'= &0 /\ t3 = &0)\/ (&0< t2' \/ &0 <t3)  `)
THEN RESA_TAC
THENL[REDUCE_ARITH_TAC THEN DISCH_TAC THEN ASM_REWRITE_TAC[]
THEN REDUCE_VECTOR_TAC;
STRIP_TAC THEN 
REWRITE_TAC[VECTOR_ARITH`t1 % x + t2 % u = t1' % x + t2' % ((&1 - a) % u + a % w) + t3 % w
<=>  (t2'* a + t3) % w = (t1-t1') % x +( t2-t2' * (&1-a))% u   :real^3`]
THEN MRESA_TAC REAL_LT_MUL[`t2':real`;`a:real`]
THEN MP_TAC(REAL_ARITH` &0< t2'*(a) /\ &0<= t3 ==> &0 < t2'*(a)+t3 /\ ~(t2'*(a)+t3:real= &0)`) THEN RESA_TAC
THEN MRESA1_TAC REAL_MUL_LINV`t2'*(a)+t3:real`
THEN MRESA1_TAC REAL_LT_INV`t2'*(a)+t3:real`
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`(t2'* a + t3) % w = (t1-t1') % x +( t2-t2' * (&1-a))% u:real^3
 ==> (inv (t2' * a+t3))%((t2'* a + t3) % w) = (inv (t2' * a+t3))%( (t1-t1') % x +( t2-t2' * (&1-a))% u)`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN ASM_REWRITE_TAC[VECTOR_ARITH`A%B%C=(A*B)%C`; VECTOR_ARITH`A%(B+C)=A%B+A%C`]
THEN REDUCE_VECTOR_TAC
THEN STRIP_TAC
THEN SUBGOAL_THEN`w IN aff {x,u:real^3}`ASSUME_TAC
THENL[REWRITE_TAC[aff;AFFINE_HULL_2;IN_ELIM_THM]
THEN EXISTS_TAC`(inv (t2' * a+t3)) * (t1 - t1'):real`
THEN EXISTS_TAC`(inv (t2' *  a+t3) * (t2 - t2' *(&1- a))):real`
THEN ASM_REWRITE_TAC[REAL_ARITH`inv (t2' * a + t3) * (t1 - t1') + inv (t2' * a + t3) * (t2 - t2' * (&1 - a))
= inv (t2' * a + t3) * ((t2'*a +t3)+(t1 + t2) -(t1'+ t2' +t3) ):real`; REAL_ARITH`A+ &1- &1= A`];
ASM_MESON_TAC[]];

STRIP_TAC THEN 
REWRITE_TAC[VECTOR_ARITH`t1 % x + t2 % u = t1' % x + t2' % ((&1 - a) % u + a % w) + t3 % w
<=>  (t2'* a + t3) % w = (t1-t1') % x +( t2-t2' * (&1-a))% u   :real^3`]
THEN MP_TAC(REAL_ARITH`&0<a==> &0<=a`) THEN RESA_TAC
THEN MRESA_TAC REAL_LE_MUL[`t2':real`;`a:real`]
THEN MP_TAC(REAL_ARITH` &0<= t2'*(a) /\ &0< t3 ==> &0 < t2'*(a)+t3 /\ ~(t2'*(a)+t3:real= &0)`) THEN RESA_TAC
THEN MRESA1_TAC REAL_MUL_LINV`t2'*(a)+t3:real`
THEN MRESA1_TAC REAL_LT_INV`t2'*(a)+t3:real`
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`(t2'* a + t3) % w = (t1-t1') % x +( t2-t2' * (&1-a))% u:real^3
 ==> (inv (t2' * a+t3))%((t2'* a + t3) % w) = (inv (t2' * a+t3))%( (t1-t1') % x +( t2-t2' * (&1-a))% u)`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN ASM_REWRITE_TAC[VECTOR_ARITH`A%B%C=(A*B)%C`; VECTOR_ARITH`A%(B+C)=A%B+A%C`]
THEN REDUCE_VECTOR_TAC
THEN STRIP_TAC
THEN SUBGOAL_THEN`w IN aff {x,u:real^3}`ASSUME_TAC
THENL[
REWRITE_TAC[aff;AFFINE_HULL_2;IN_ELIM_THM]
THEN EXISTS_TAC`(inv (t2' * a+t3)) * (t1 - t1'):real`
THEN EXISTS_TAC`(inv (t2' *  a+t3) * (t2 - t2' *(&1- a))):real`
THEN ASM_REWRITE_TAC[REAL_ARITH`inv (t2' * a + t3) * (t1 - t1') + inv (t2' * a + t3) * (t2 - t2' * (&1 - a))
= inv (t2' * a + t3) * ((t2'*a +t3)+(t1 + t2) -(t1'+ t2' +t3) ):real`; REAL_ARITH`A+ &1- &1= A`];
ASM_MESON_TAC[]]]);;




let exists_in_aff_gt=prove(`!x:real^3  v:real^3 u:real^3.
~collinear {x,v,u} ==> ?y:real^3. y IN aff_gt {x} {v, u}`,
REPEAT STRIP_TAC
THEN MRESA_TAC th3[`x:real^3`;`v:real^3`;`u:real^3`]
THEN MRESAL_TAC  AFF_GT_1_2[`x:real^3`;`v:real^3`;`u:real^3`][IN_ELIM_THM]
THEN EXISTS_TAC`&0 % x+ &1 / &2 % v+ &1/ &2 %u:real^3 `
THEN EXISTS_TAC`&0`
THEN EXISTS_TAC`&1/ &2`
THEN EXISTS_TAC`&1/ &2`
THEN ASM_REWRITE_TAC[]
THEN REAL_ARITH_TAC);;


let in_aff_2_2_fan=prove(`!x:real^3 v:real^3 u:real^3 w:real^3.
~coplanar {x,v,u,w}
==> 
(!t:real. &0< t /\ t< &1
==> (!t1:real t2:real t3:real. &0<t3  /\ t1+t2+t3= &1
==>t1 % x + t2 % v + t3 % ((&1 - t) % u + t % w) IN aff_gt {x,v} {u,w}))`,


REPEAT STRIP_TAC
THEN MRESA_TAC notcoplanar_imp_notcollinear_fan[`x:real^3`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN MRESA_TAC th3[`x:real^3`;`v:real^3`;`u:real^3`]
THEN MRESA_TAC th3[`x:real^3`;`u:real^3`;`w:real^3`]
THEN MRESA_TAC th3[`x:real^3`;`v:real^3`;`w:real^3`]
THEN SUBGOAL_THEN `DISJOINT {x,v:real^3} {u,w:real^3}` ASSUME_TAC
THENL[
REWRITE_TAC[DISJOINT_SYM;SET_RULE`{v:real^3,w:real^3}= {v} UNION {w}`;DISJOINT_UNION]
THEN REWRITE_TAC[SET_RULE`{v} UNION {w}={v:real^3,w:real^3}`]
THEN ONCE_REWRITE_TAC[DISJOINT_SYM]
THEN ASM_REWRITE_TAC[];
MRESAL_TAC AFF_GT_2_2[`x:real^3`;`v:real^3`;`u:real^3`;`w:real^3`][IN_ELIM_THM]
THEN EXISTS_TAC`t1:real`
THEN EXISTS_TAC`t2:real`
THEN EXISTS_TAC`t3*(&1-t):real`
THEN EXISTS_TAC`t3*(t):real`
THEN ASM_REWRITE_TAC[REAL_ARITH`t1 +t2+  t3 * (&1 - t) + t3 * t = t1+t2+t3:real`;VECTOR_ARITH`t1 % x + t2 % v + t3 % ((&1 - t) % u + t % w) =
 t1 % x + t2 % v + (t3 * (&1 - t)) % u  + (t3 * t) % w:real^3`]
THEN STRIP_TAC THEN MATCH_MP_TAC REAL_LT_MUL
THEN ASM_TAC
THEN REAL_ARITH_TAC]);;






let condition_to_in_aff_gt_by_angle=prove(`!x:real^3 v:real^3 u:real^3 s1:real.
~collinear {x,v,u} /\  &0< (v - x) dot (u - x) /\ &0< s1 
/\ s1< atn ((norm ((v - x) cross (u - x))) * inv((v - x) dot (u - x)))
==>
sin s1 % e1_fan x v u + cos s1 % e3_fan x v u + x IN aff_gt {x} {v, u}`,
REPEAT STRIP_TAC
THEN ASSUME_TAC(ISPEC`(norm ((v - x) cross (u - x)) * inv ((v - x) dot (u - x))):real`ATN_BOUNDS)
THEN MP_TAC (REAL_ARITH`s1< atn ((norm ((v - x) cross (u - x))) * inv((v - x) dot (u - x)))
/\ atn ((norm ((v - x) cross (u - x))) * inv((v - x) dot (u - x))) < pi/ &2
==> s1< pi / &2`) THEN RESA_TAC
THEN MRESA_TAC th3[`x:real^3`;`v:real^3`;`u:real^3`]
THEN MRESAL_TAC AFF_GT_1_2[`x:real^3`;`v:real^3`;`u:real^3`][IN_ELIM_THM]
THEN REWRITE_TAC[VECTOR_ARITH`sin s1 % e1_fan x v u + cos s1 % e3_fan x v u + x =
     t1 % x + t2 % v + t3 % u
<=> sin s1 % e1_fan x v u + cos s1 % e3_fan x v u  =
     (t1- &1) % x + t2 % v + t3 % u`;e1_fan;e2_fan;e3_fan;CROSS_LMUL;CROSS_RMUL]
THEN ONCE_REWRITE_TAC[CROSS_SKEW;]
THEN REWRITE_TAC[CROSS_LAGRANGE;VECTOR_ARITH`-- (A-B) = B-A:real^3`]
THEN SUBGOAL_THEN`~(norm((v:real^3)-(x:real^3))= &0)` ASSUME_TAC
THENL[
ASM_REWRITE_TAC[NORM_EQ_0;VECTOR_ARITH`v-x=vec 0<=> x=v`];
MP_TAC(ISPEC`norm((v:real^3)-(x:real^3))`REAL_MUL_LINV)
THEN RESA_TAC
THEN REWRITE_TAC[VECTOR_ARITH`A%B%C=(A*B)%C`; REAL_ARITH`(((A*B)*D)*D)=A*B*(D pow 2)`;]
THEN ONCE_REWRITE_TAC[VECTOR_ARITH`A%(B-C)=A%B-A%C`]
THEN ASM_REWRITE_TAC[VECTOR_ARITH`A%B%C=(A*B)%C`;DOT_SQUARE_NORM;REAL_ARITH`(A*B*C pow 2) * D pow 2=A*B*(C*D) pow 2`;REAL_ARITH`A* &1 pow 2=A`;NORM_MUL;REAL_ABS_INV;REAL_ABS_NORM;REAL_INV_MUL;REAL_INV_INV
; ]
THEN ASM_REWRITE_TAC[REAL_ARITH`A*(B*C) * D pow 2= A*C * D*(D*B)`;REAL_ARITH`A*B* &1= A*B`;VECTOR_ARITH`A-B+C%D-C%E=A-B+C%(D-E)`;VECTOR_ARITH`A-C%D+B%D=A+(B-C)%D`;
REAL_ARITH`A*B-(C*D*B)*E=B*(A-C*D*E)`]
THEN ONCE_REWRITE_TAC[GSYM CROSS_SKEW]
THEN SUBGOAL_THEN`~(norm((v - x) cross (u - x:real^3))= &0)` ASSUME_TAC
THENL[
ASM_REWRITE_TAC[NORM_EQ_0]
THEN MP_TAC(ISPECL[`v-x:real^3`;`u-x:real^3`]CROSS_EQ_0)
THEN ONCE_REWRITE_TAC[GSYM COLLINEAR_3;]
THEN RESA_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={B,A,C}`]
THEN ASM_REWRITE_TAC[];
MP_TAC(ISPEC`norm((v - x) cross (u - x:real^3))`REAL_MUL_LINV)
THEN RESA_TAC
THEN ASSUME_TAC(ISPEC`(v - x) cross (u - x:real^3)`NORM_POS_LE)
THEN MP_TAC(REAL_ARITH`~(norm ((v - x) cross (u - x:real^3)) = &0)/\ &0 <= norm ((v - x) cross (u - x:real^3))==> &0< norm ((v - x) cross (u - x:real^3)) `)
THEN RESA_TAC
THEN MRESA1_TAC REAL_LT_INV`norm((v - x) cross (u - x:real^3))`

THEN ASSUME_TAC(ISPEC`(v - x:real^3)`NORM_POS_LE)
THEN MP_TAC(REAL_ARITH`~(norm ((v - x:real^3)) = &0)/\ &0 <= norm ((v - x:real^3))==> &0< norm ((v - x:real^3)) `)
THEN RESA_TAC
THEN MRESA1_TAC REAL_LT_INV`norm((v - x:real^3))`
THEN MRESA1_TAC COS_POS_PI2`s1:real`
THEN MRESA1_TAC SIN_POS_PI2`s1:real`
THEN EXISTS_TAC`&1-(sin s1 * norm (v - x) * inv (norm ((v - x) cross (u - x))))
-(inv (norm (v - x)) *
      (cos s1 -
       sin s1 * inv (norm ((v - x) cross (u - x))) * ((v - x) dot (u - x:real^3))))`
THEN EXISTS_TAC `(inv (norm (v - x)) *
      (cos s1 -
       sin s1 * inv (norm ((v - x) cross (u - x))) * ((v - x) dot (u - x:real^3))))`
THEN EXISTS_TAC`(sin s1 * norm (v - x) * inv (norm ((v - x) cross (u - x:real^3))))`
THEN STRIP_TAC
THENL[
MATCH_MP_TAC REAL_LT_MUL
THEN ASM_REWRITE_TAC[REAL_ARITH`&0<A-B<=> B<A`]
THEN ASSUME_TAC(PI_WORKS)
THEN MP_TAC(REAL_ARITH`&0< pi /\ &0< s1 ==> --(pi / &2) < s1`)
THEN RESA_TAC
THEN MRESAL_TAC  TAN_MONO_LT[`s1:real`;`atn (norm ((v - x) cross (u - x)) * inv ((v - x) dot (u - x))):real`][ATN_TAN]
THEN MRESAL_TAC REAL_LT_LMUL[`inv (norm ((v - x) cross (u - x:real^3)))`;`tan s1:real`;`norm ((v - x) cross (u - x)) * inv ((v - x) dot (u - x:real^3))`][REAL_ARITH`A*B*C=(A*B)*C`;REAL_ARITH`&1*A=A`]
THEN MP_TAC(REAL_ARITH`&0<(v - x) dot (u - x:real^3)==> ~((v - x) dot (u - x:real^3)= &0)`)
THEN RESA_TAC
THEN MP_TAC(ISPEC`(v - x) dot (u - x:real^3)`REAL_MUL_LINV)
THEN RESA_TAC
THEN MP_TAC(REAL_ARITH`&0<cos s1 ==> ~(cos s1= &0)`)
THEN RESA_TAC
THEN MP_TAC(ISPEC`cos s1`REAL_MUL_LINV)
THEN RESA_TAC
THEN MRESAL_TAC REAL_LT_RMUL[`inv (norm ((v - x) cross (u - x:real^3)))* tan s1:real`;`inv ((v - x) dot (u - x:real^3))`;`(v - x) dot (u - x:real^3)`][REAL_ARITH`(A*B)*C=A*C*B`;tan]

THEN MRESAL_TAC REAL_LT_RMUL[`inv (norm ((v - x) cross (u - x:real^3)))* ((v - x) dot (u - x:real^3))* sin s1 / cos s1`;`&1`;`cos s1`][REAL_ARITH`&1* A=A`;real_div;REAL_ARITH`(A*B*C*D)*E=(C*A*B)*(D*E)`;REAL_ARITH`A* &1=A`]
THEN ASM_TAC THEN REAL_ARITH_TAC;
STRIP_TAC
THENL[
MATCH_MP_TAC REAL_LT_MUL
THEN ASM_REWRITE_TAC[]
THEN MATCH_MP_TAC REAL_LT_MUL
THEN ASM_REWRITE_TAC[];

STRIP_TAC
THENL[
REAL_ARITH_TAC;
VECTOR_ARITH_TAC]]]]]);;







let condition1_to_in_aff_gt_by_angle=prove(`!x:real^3 v:real^3 u:real^3 s1:real.
~collinear {x,v,u} /\ &0< s1 /\ s1< pi/ &2
/\ (v - x) dot (u - x:real^3) <= &0
==>
sin s1 % e1_fan x v u + cos s1 % e3_fan x v u + x IN aff_gt {x} {v, u}`,

REPEAT STRIP_TAC
THEN MRESA_TAC th3[`x:real^3`;`v:real^3`;`u:real^3`]
THEN MRESAL_TAC AFF_GT_1_2[`x:real^3`;`v:real^3`;`u:real^3`][IN_ELIM_THM]
THEN REWRITE_TAC[VECTOR_ARITH`sin s1 % e1_fan x v u + cos s1 % e3_fan x v u + x =
     t1 % x + t2 % v + t3 % u
<=> sin s1 % e1_fan x v u + cos s1 % e3_fan x v u  =
     (t1- &1) % x + t2 % v + t3 % u`;e1_fan;e2_fan;e3_fan;CROSS_LMUL;CROSS_RMUL]
THEN ONCE_REWRITE_TAC[CROSS_SKEW;]
THEN REWRITE_TAC[CROSS_LAGRANGE;VECTOR_ARITH`-- (A-B) = B-A:real^3`]
THEN SUBGOAL_THEN`~(norm((v:real^3)-(x:real^3))= &0)` ASSUME_TAC
THENL[
ASM_REWRITE_TAC[NORM_EQ_0;VECTOR_ARITH`v-x=vec 0<=> x=v`];
MP_TAC(ISPEC`norm((v:real^3)-(x:real^3))`REAL_MUL_LINV)
THEN RESA_TAC
THEN REWRITE_TAC[VECTOR_ARITH`A%B%C=(A*B)%C`; REAL_ARITH`(((A*B)*D)*D)=A*B*(D pow 2)`;]
THEN ONCE_REWRITE_TAC[VECTOR_ARITH`A%(B-C)=A%B-A%C`]
THEN ASM_REWRITE_TAC[VECTOR_ARITH`A%B%C=(A*B)%C`;DOT_SQUARE_NORM;REAL_ARITH`(A*B*C pow 2) * D pow 2=A*B*(C*D) pow 2`;REAL_ARITH`A* &1 pow 2=A`;NORM_MUL;REAL_ABS_INV;REAL_ABS_NORM;REAL_INV_MUL;REAL_INV_INV
; ]
THEN ASM_REWRITE_TAC[REAL_ARITH`A*(B*C) * D pow 2= A*C * D*(D*B)`;REAL_ARITH`A*B* &1= A*B`;VECTOR_ARITH`A-B+C%D-C%E=A-B+C%(D-E)`;VECTOR_ARITH`A-C%D+B%D=A+(B-C)%D`;
REAL_ARITH`A*B-(C*D*B)*E=B*(A-C*D*E)`]
THEN ONCE_REWRITE_TAC[GSYM CROSS_SKEW]
THEN SUBGOAL_THEN`~(norm((v - x) cross (u - x:real^3))= &0)` ASSUME_TAC
THENL[
ASM_REWRITE_TAC[NORM_EQ_0]
THEN MP_TAC(ISPECL[`v-x:real^3`;`u-x:real^3`]CROSS_EQ_0)
THEN ONCE_REWRITE_TAC[GSYM COLLINEAR_3;]
THEN RESA_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={B,A,C}`]
THEN ASM_REWRITE_TAC[];
MP_TAC(ISPEC`norm((v - x) cross (u - x:real^3))`REAL_MUL_LINV)
THEN RESA_TAC
THEN ASSUME_TAC(ISPEC`(v - x) cross (u - x:real^3)`NORM_POS_LE)
THEN MP_TAC(REAL_ARITH`~(norm ((v - x) cross (u - x:real^3)) = &0)/\ &0 <= norm ((v - x) cross (u - x:real^3))==> &0< norm ((v - x) cross (u - x:real^3)) `)
THEN RESA_TAC
THEN MRESA1_TAC REAL_LT_INV`norm((v - x) cross (u - x:real^3))`

THEN ASSUME_TAC(ISPEC`(v - x:real^3)`NORM_POS_LE)
THEN MP_TAC(REAL_ARITH`~(norm ((v - x:real^3)) = &0)/\ &0 <= norm ((v - x:real^3))==> &0< norm ((v - x:real^3)) `)
THEN RESA_TAC
THEN MRESA1_TAC REAL_LT_INV`norm((v - x:real^3))`
THEN MRESA1_TAC COS_POS_PI2`s1:real`
THEN MRESA1_TAC SIN_POS_PI2`s1:real`
THEN EXISTS_TAC`&1-(sin s1 * norm (v - x) * inv (norm ((v - x) cross (u - x))))
-(inv (norm (v - x)) *
      (cos s1 -
       sin s1 * inv (norm ((v - x) cross (u - x))) * ((v - x) dot (u - x:real^3))))`
THEN EXISTS_TAC `(inv (norm (v - x)) *
      (cos s1 -
       sin s1 * inv (norm ((v - x) cross (u - x))) * ((v - x) dot (u - x:real^3))))`
THEN EXISTS_TAC`(sin s1 * norm (v - x) * inv (norm ((v - x) cross (u - x:real^3))))`
THEN STRIP_TAC
THENL[
MATCH_MP_TAC REAL_LT_MUL
THEN ASM_REWRITE_TAC[REAL_ARITH`&0<A-B<=> B<A`]
THEN MATCH_MP_TAC(REAL_ARITH`&0< cos s1 /\ sin s1 * inv (norm ((v - x) cross (u - x))) * ((v - x) dot (u - x)) <= &0==> sin s1 * inv (norm ((v - x) cross (u - x))) * ((v - x) dot (u - x)) < cos s1`)
THEN ASM_REWRITE_TAC[REAL_ARITH`A *B *C<= &0<=> &0<= A*B*(-- C)`]
THEN MATCH_MP_TAC REAL_LE_MUL
THEN MP_TAC(REAL_ARITH`&0< sin s1==> &0<= sin s1`) THEN RESA_TAC
THEN ASM_REWRITE_TAC[]
THEN MATCH_MP_TAC REAL_LE_MUL
THEN MP_TAC(REAL_ARITH`&0< inv (norm ((v - x) cross (u - x)))==> &0<= inv (norm ((v - x) cross (u - x)))`) THEN RESA_TAC
THEN ASM_REWRITE_TAC[]
THEN ASM_TAC THEN REAL_ARITH_TAC;
STRIP_TAC
THENL[
MATCH_MP_TAC REAL_LT_MUL
THEN ASM_REWRITE_TAC[]
THEN MATCH_MP_TAC REAL_LT_MUL
THEN ASM_REWRITE_TAC[];
STRIP_TAC
THENL[
REAL_ARITH_TAC;
VECTOR_ARITH_TAC]]]]]);;





let scale_in_edges_fan=prove(`!(x:real^3) (v:real^3) (u:real^3) (w:real^3). 
DISJOINT {x} {v,u}
/\ w IN aff_gt {x} {v,u}
==> 
(?a t:real. &0<a /\ &0<t /\ t< &1
/\ a%(w-x) = (&1-t)% v+ t%u-x)`,

REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN MRESAL_TAC AFF_GT_1_2[`x:real^3`;`v:real^3`;`u:real^3`][IN_ELIM_THM]
THEN STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[VECTOR_ARITH`w = t1 % x + t2 % v + t3 % u 
<=> w-x = ((t1+t2+t3)- &1) % x + (((t1+t2+t3) -t1)- t3)% (v-x) + t3 % (u-x):real^3`]
THEN ASM_REWRITE_TAC[REAL_ARITH`&1- &1= &0`] THEN REDUCE_VECTOR_TAC
THEN MP_TAC(REAL_ARITH`&0< t2 /\ &0< t3 /\ t1+t2+t3= &1 ==> ~(&1-t1= &0)/\ &0< &1- t1`)
THEN RESA_TAC
THEN MRESA1_TAC REAL_MUL_LINV `(&1-t1:real)`
THEN MRESA1_TAC REAL_LT_INV `(&1-t1:real)`
THEN MRESA_TAC REAL_LT_MUL [`inv(&1-t1:real)`;`t3:real`]
THEN MRESA_TAC REAL_LT_MUL [`inv(&1-t1:real)`;`t2:real`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[REAL_ARITH`inv (&1 - t1) * t2=inv (&1 - t1) * ((t1+t2+t3)-t1)- inv(&1-t1)*t3:real`]
THEN RESA_TAC
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`
w - x = (&1 - t1 - t3) % (v - x) + t3 % (u - x):real^3
==> (inv (&1- t1))%(w - x ) = (inv (&1-t1))%((&1 - t1 - t3) % (v - x) + t3 % (u - x))
`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN ASM_REWRITE_TAC[VECTOR_ARITH`A%B%C= (A*B)%C`;VECTOR_ARITH`A%(B+C)=A%B+A%C`;REAL_ARITH`inv (&1 - t1) * (&1 - t1 - t3)=inv (&1 - t1) * (&1 - t1) - inv (&1 - t1) * (t3)`;VECTOR_ARITH`(&1-A)%(U-X)+A%(V-X)=(&1-A)%U+A%V-X`]
THEN REDUCE_VECTOR_TAC
THEN STRIP_TAC
THEN EXISTS_TAC`inv(&1- t1:real)`
THEN EXISTS_TAC`inv(&1-t1) *t3:real`
THEN ASM_REWRITE_TAC[REAL_ARITH`A< &1<=> &0< &1- A`]);;



let aff_gt_imp_not_collinear=prove(`!x u v w:real^3.
~collinear{x,v,u}/\ w IN aff_gt{x,v} {u}==> ~collinear{x,v,w}`,
REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC th3[`(x:real^3)` ;` (v:real^3)`;`(u:real^3) `;]
THEN POP_ASSUM MP_TAC
THEN MRESAL_TAC  AFF_GT_2_1[`x:real^3`;`v:real^3`;`u:real^3`][IN_ELIM_THM]
THEN ASM_REWRITE_TAC[collinear_fan;aff;AFFINE_HULL_2;IN_ELIM_THM;]
THEN DISCH_THEN(LABEL_TAC"A")
THEN REPEAT STRIP_TAC
THEN REMOVE_THEN "A" MP_TAC
THEN ASM_REWRITE_TAC[]
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`t1 % x + t2 % v + t3 % u = u' % x + v' % v<=>
t3 % u = (u'-t1) % x + (v'-t2) % v`]
THEN MP_TAC(REAL_ARITH`&0<t3==> ~(t3= &0)`) THEN RESA_TAC
THEN MRESA1_TAC REAL_MUL_LINV`t3:real`
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`t3 % u = (u' - t1) % x + (v' - t2) % v:real^3
 ==> (inv (t3))%(t3 % u ) = (inv (t3))%( (u' - t1) % x + (v' - t2) % v)`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN ASM_REWRITE_TAC[VECTOR_ARITH`A%B%C=(A*B)%C`; VECTOR_ARITH`A%(B+C)=A%B+A%C`]
THEN REDUCE_VECTOR_TAC
THEN STRIP_TAC
THEN EXISTS_TAC`(inv (t3)) * (u' - t1):real`
THEN EXISTS_TAC`(inv (t3)) * (v' -t2):real`
THEN ASM_REWRITE_TAC[REAL_ARITH`inv (t3) * (u' - t1) + inv (t3) * (v'- t2)
= inv (t3) * (t3+ (u'+v')- (t1 + t2 +t3)):real`; REAL_ARITH`A+ &1- &1= A`]);;

let aff_gt_1_2_scale_fan=prove(`!x:real^3 v:real^3 u:real^3 w:real^3 a:real.
&0< a /\ a % (u-x)= w-x /\ ~collinear {x,w,v}
==> aff_gt {x} {u,v} =aff_gt {x} {w,v}`,
REPEAT GEN_TAC
THEN GEOM_ORIGIN_TAC `x:real^3`
THEN REDUCE_VECTOR_TAC
THEN REPEAT STRIP_TAC 
THEN MP_TAC(REAL_ARITH`&0<a:real==> ~(a= &0)`) THEN RESA_TAC
THEN MRESA1_TAC REAL_MUL_LINV `a:real`
THEN MRESA1_TAC REAL_LT_INV`a:real`
THEN FIND_ASSUM(MP_TAC)`a %u=w:real^3`
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`
a%u=w:real^3
==> (inv (a))%(a%u) = (inv (a))%(w)`)
THEN ASM_REWRITE_TAC[VECTOR_ARITH`A%B%C=(A*B)%C:real^3`]
THEN REWRITE_TAC[VECTOR_ARITH`&1 % u= inv a % w<=> inv a % w= u`]
THEN STRIP_TAC
THEN MRESA_TAC COLLINEAR_SPECIAL_SCALE[`a:real`;`u:real^3`;`v:real^3`]
THEN MRESA_TAC th3[`((vec 0):real^3)` ;` (u:real^3)`;`(v:real^3) `;]
THEN MRESA_TAC th3[`((vec 0):real^3)` ;` (w:real^3)`;`(v:real^3) `;]
THEN MRESAL_TAC  AFF_GT_1_2[`(vec 0):real^3`;`u:real^3`;`v:real^3`][IN_ELIM_THM;EXTENSION]
THEN MRESAL_TAC  AFF_GT_1_2[`(vec 0):real^3`;`w:real^3`;`v:real^3`][IN_ELIM_THM]
THEN REDUCE_VECTOR_TAC
THEN GEN_TAC THEN EQ_TAC
THENL[ STRIP_TAC
THEN EXISTS_TAC`&1- inv a * t2-t3:real`
THEN EXISTS_TAC `inv a * t2:real`
THEN EXISTS_TAC `t3:real`
THEN ASM_REWRITE_TAC[REAL_ARITH`&1 - inv a * t2 - t3 + inv a * t2 + t3 = &1`;VECTOR_ARITH`(A*B)%C=B%(A%C)`]
THEN MATCH_MP_TAC REAL_LT_MUL
THEN ASM_REWRITE_TAC[];
STRIP_TAC
THEN EXISTS_TAC`&1- a * t2-t3:real`
THEN EXISTS_TAC `a * t2:real`
THEN EXISTS_TAC `t3:real`
THEN ASM_REWRITE_TAC[REAL_ARITH`&1 - a * t2 - t3 + a * t2 + t3 = &1`;VECTOR_ARITH`(A*B)%C=B%(A%C)`]
THEN MATCH_MP_TAC REAL_LT_MUL
THEN ASM_REWRITE_TAC[]]);;



let in_aff_gt_1_2=prove(`!x:real^3 v:real^3 u:real^3 t:real.
DISJOINT {x} {v,u} /\ &0< t /\ t< &1==>  (&1-t)% v+ t% u IN aff_gt {x} {v,u}`,
REPEAT STRIP_TAC
THEN MRESAL_TAC  AFF_GT_1_2[`x:real^3`;`v:real^3`;`u:real^3`][IN_ELIM_THM;]
THEN EXISTS_TAC`&0`
THEN EXISTS_TAC`&1 - t:real`
THEN EXISTS_TAC`t:real`
THEN ASM_REWRITE_TAC[REAL_ARITH`&0< &1 - t<=> t< &1`;REAL_ARITH`&0 + &1 - t + t= &1`]
THEN VECTOR_ARITH_TAC);;

let sym_line1_fan=prove(`!x y z:real^N. x IN aff {y, z} /\ ~(x=y)
==> z IN aff {x,y}`,
REPEAT GEN_TAC
THEN REWRITE_TAC[aff;AFFINE_HULL_2;IN_ELIM_THM]
THEN STRIP_TAC
THEN ASM_TAC
THEN DISJ_CASES_TAC(REAL_ARITH`(v= &0)\/ ~(v= &0)`)
THENL[ ASM_REWRITE_TAC[]
THEN REDUCE_ARITH_TAC
THEN RESA_TAC
THEN REDUCE_VECTOR_TAC
THEN RESA_TAC
THEN SET_TAC[];
MP_TAC(ISPEC`(v:real)`REAL_MUL_LINV)
THEN RESA_TAC
THEN REPEAT STRIP_TAC
THEN EXISTS_TAC`inv(v:real)`
THEN EXISTS_TAC`-- inv(v:real) *u`
THEN ASM_REWRITE_TAC[REAL_ARITH` inv v + --inv v * u = inv v * (v+ &1- (u+v)) `;REAL_ARITH`A+ &1 - &1= A`;VECTOR_ARITH`inv v % (u % y + v % z) + (--inv v * u) % y=(inv v * v) % z `]
THEN REDUCE_VECTOR_TAC]);;


let POINT_IN_LINE=prove(`!x y:real^N. x IN  aff {x,y}`,
REPEAT GEN_TAC
THEN REWRITE_TAC[aff;AFFINE_HULL_2;IN_ELIM_THM]
THEN EXISTS_TAC`&1`
THEN EXISTS_TAC`&0`
THEN REDUCE_VECTOR_TAC
THEN REAL_ARITH_TAC);;
let POINT_IN_LINE1=prove(`!x y:real^N. y IN  aff {x,y}`,
REPEAT GEN_TAC
THEN REWRITE_TAC[aff;AFFINE_HULL_2;IN_ELIM_THM]
THEN EXISTS_TAC`&0`
THEN EXISTS_TAC`&1`
THEN REDUCE_VECTOR_TAC
THEN REAL_ARITH_TAC);;

let AFFINE_HULL_AFFINE_EQ =prove(`!s:real^N->bool. affine hull (affine hull s)= affine  hull s`,
STRIP_TAC THEN MATCH_MP_TAC AFFINE_HULLS_EQ 
THEN ASSUME_TAC(ISPEC `s:real^N->bool` AFFINE_AFFINE_HULL)
THEN MRESA1_TAC AFFINE_HULL_EQ`affine hull s:real^N->bool`
THEN MRESA_TAC HULL_SUBSET[`affine:(real^N->bool)->bool`;` s:real^N->bool`;]
THEN SET_TAC[]);;
let sym_line0_fan=prove( `!x y z:real^N. x IN aff {y, z} /\ DISJOINT {x} {y,z}
==> aff {x,z} SUBSET aff {x,y}`,
REPEAT GEN_TAC
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`DISJOINT {x} {y,z}==> ~(x=y:real^N)`)
THEN RESA_TAC
THEN MRESA_TAC sym_line1_fan[`x:real^N`;`y:real^N`;`z:real^N`]
THEN MP_TAC(SET_RULE`x IN aff {x, y} /\ z IN aff {x, y} ==> {x, z:real^N} SUBSET aff {x, y}`)
THEN REWRITE_TAC[POINT_IN_LINE;]
THEN RESA_TAC
THEN MRESA_TAC HULL_MONO[`affine:(real^N->bool)->bool`;` {x, z:real^N}`;`aff {x, y:real^N}`]
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[aff;AFFINE_HULL_AFFINE_EQ ]);;

let sym_line_fan=prove(`!x y z:real^N. x IN aff {y, z} /\ DISJOINT {x} {y,z}
==> aff {x,z} = aff {x,y}`,
REPEAT STRIP_TAC
THEN MRESA_TAC sym_line0_fan[`x:real^N`;`y:real^N`;`z:real^N`]
THEN MRESA_TAC sym_line0_fan[`x:real^N`;`z:real^N`;`y:real^N`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B}={B,A}`]
THEN ASM_REWRITE_TAC[]
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B}={B,A}`]
THEN POP_ASSUM MP_TAC
THEN SET_TAC[]);;
let sym_line01_fan=prove(`!x y z:real^N. x IN aff {y, z} /\ DISJOINT {y} {x,z}
==> aff {y,x} SUBSET aff {y,z}`,
REPEAT GEN_TAC
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`y IN aff {y, z} /\ x IN aff {y, z} ==> {y,x:real^N} SUBSET aff {y,z}`)
THEN ASM_REWRITE_TAC[POINT_IN_LINE;]
THEN STRIP_TAC
THEN MRESA_TAC HULL_MONO[`affine:(real^N->bool)->bool`;` {y,x:real^N}`;`aff { y,z:real^N}`]
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[aff;AFFINE_HULL_AFFINE_EQ ]);;

let sym_line02_fan=prove(`!x y z:real^N. x IN aff {y, z} /\ DISJOINT {y} {x,z}
==> aff {y,z} SUBSET aff {y,x}`,
REPEAT GEN_TAC
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`DISJOINT {y} {x,z}==> ~(x=y:real^N)`)
THEN RESA_TAC
THEN MRESA_TAC sym_line1_fan[`x:real^N`;`y:real^N`;`z:real^N`]
THEN MP_TAC(SET_RULE`y IN aff {y, x} /\ z IN aff {y, x} ==> {y,z:real^N} SUBSET aff {y,x}`)
THEN ASM_REWRITE_TAC[POINT_IN_LINE;]
THEN STRIP_TAC
THEN MRESA_TAC HULL_MONO[`affine:(real^N->bool)->bool`;` {y,z:real^N}`;`aff { y,x:real^N}`]
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[aff;AFFINE_HULL_AFFINE_EQ ]
THEN POP_ASSUM MP_TAC
THEN ASSUME_TAC(SET_RULE`{y,x}={x,y}`)
THEN POP_ASSUM (fun th-> ASM_REWRITE_TAC[th;])
THEN REWRITE_TAC[aff]
THEN RESA_TAC);;


let sym_line_fan0=prove(`!x y z:real^N. x IN aff {y, z} /\ DISJOINT {x} {y,z} /\ DISJOINT {y} {x,z}
==> aff {x,z} = aff {y,z}`,
REPEAT STRIP_TAC
THEN MRESA_TAC sym_line_fan[`x:real^N`;`y:real^N`;`z:real^N`]
THEN SUBGOAL_THEN `y IN aff {x,z:real^N}` ASSUME_TAC
THENL[ASM_REWRITE_TAC[POINT_IN_LINE1];
MRESA_TAC sym_line_fan[`y:real^N`;`x:real^N`;`z:real^N`]
THEN ASSUME_TAC(SET_RULE`{x,y}={y,x:real^N}`)
THEN ASM_REWRITE_TAC[]]);;

let sym_line_fan1=prove(`!x y z:real^N. x IN aff {y, z} /\ DISJOINT {y} {x,z}
==> aff {y,z} = aff {y,x}`,
REPEAT STRIP_TAC
THEN MRESA_TAC sym_line01_fan[`x:real^N`;`y:real^N`;`z:real^N`]
THEN MRESA_TAC sym_line02_fan[`x:real^N`;`y:real^N`;`z:real^N`]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN SET_TAC[]);;

let aff_ge_1_1_subset_aff_fan=prove(`!x y z:real^3. ~(y=z) /\ x IN aff_ge {y} {z} ==> x IN aff {y,z} `,
REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN MRESAL_TAC AFF_GE_1_1[`y:real^3`;`z:real^3`][IN_ELIM_THM]
THEN REWRITE_TAC[aff;AFFINE_HULL_2;IN_ELIM_THM]
THEN STRIP_TAC
THEN EXISTS_TAC`t1:real`
THEN EXISTS_TAC`t2:real`
THEN ASM_REWRITE_TAC[]);;

let place_there_point_line_fan=prove(`!x:real^3 y:real^3 z:real^3.
~(x=y)/\ z IN aff {x,y}==> ?t:real. &0<t /\ t< &1 /\ (&1-t)%y+t % z IN aff_ge {x} {y}`,
REPEAT GEN_TAC
THEN REWRITE_TAC[aff; AFFINE_HULL_2;IN_ELIM_THM]
THEN RESA_TAC
THEN MRESAL_TAC AFF_GE_1_1[`x:real^3`;`y:real^3`][IN_ELIM_THM]
THEN DISJ_CASES_TAC(REAL_ARITH`&0<= v \/ v< &0`)
THENL[EXISTS_TAC`&1/ &2`
THEN ASM_REWRITE_TAC[REAL_ARITH`&0< &1/ &2 /\ &1/ &2< &1`]
THEN EXISTS_TAC`&1/ &2 *u`
THEN EXISTS_TAC`&1/ &2*(&1+v)`
THEN ASM_REWRITE_TAC[REAL_ARITH`&1/ &2 *u + &1/ &2 * (&1 + v)= &1/ &2 *(&1 +(u+v))`;REAL_ARITH`&1 / &2 * (&1 + &1) = &1`;VECTOR_ARITH`(&1 - &1 / &2) % y + &1 / &2 % (u % x + v % y) =
 (&1 / &2 * u) % x + (&1 / &2 * (&1 + v)) % y`]
THEN MATCH_MP_TAC REAL_LE_MUL
THEN ASM_REWRITE_TAC[REAL_ARITH`&0<= &1/ &2`]
THEN MATCH_MP_TAC REAL_LE_ADD
THEN ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC;
EXISTS_TAC`inv u`
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[REAL_ARITH`v< &0 <=> (u+v)< u `]
THEN STRIP_TAC
THEN MP_TAC(REAL_ARITH`&1< u==> &0< u /\ ~(u= &0)`)
THEN RESA_TAC
THEN MRESA1_TAC REAL_LT_INV `u:real`
THEN MRESA1_TAC REAL_INV_LT_1 `u:real`
THEN MRESA1_TAC REAL_MUL_LINV `u:real`
THEN EXISTS_TAC`&1`
THEN EXISTS_TAC `&0`
THEN ASM_REWRITE_TAC[REAL_ARITH`&0<= &0 /\ &1 + &0 = &1/\ u+ &1 - &1 =u`;VECTOR_ARITH`(&1 - inv u) % y + inv u % (u % x + v % y)=(&1 - inv u*(u+ &1- (u+v))) % y + (inv u * u) % x /\ (&1 - &1) % y + &1 % x = &1 % x + &0 % y`]]);;



let permutes_4points_collinear=prove(`!x y z w:real^N.
~(x=y)/\ ~(x=z) /\ y IN aff {x,z}/\ ~collinear{x,y,w}==> ~collinear{x,z,w}`,
REWRITE_TAC[collinear_fan]
THEN REPEAT STRIP_TAC
THEN ASM_REWRITE_TAC[]
THEN POP_ASSUM MP_TAC
THEN MP_TAC(SET_RULE`~(x=y)/\ ~(x=z) ==> DISJOINT {x} {y, z:real^N}`)
THEN RESA_TAC
THEN MRESA_TAC sym_line_fan1[`y:real^N`;`x:real^N`;`z:real^N`]);;

let permutes_4points_collinear1=prove(`!x y z w:real^N.
~(x=y)/\ ~(x=z) /\  y IN aff {x,z}/\ ~collinear{x,z,w}==> ~collinear{x,y,w}`,
REWRITE_TAC[collinear_fan]
THEN REPEAT STRIP_TAC
THEN ASM_REWRITE_TAC[]
THEN POP_ASSUM MP_TAC
THEN MP_TAC(SET_RULE`~(x=y)/\ ~(x=z) ==>DISJOINT {x} {y, z:real^N}`)
THEN RESA_TAC
THEN MRESA_TAC sym_line01_fan[`y:real^N`;`x:real^N`;`z:real^N`]
THEN ASM_TAC  THEN SET_TAC[]);;

let in_aff_gt_eq_azim=prove(`!x y z w0 w1:real^3.
~(x=z) /\ y IN aff_gt {x} {z}==> azim x y w0 w1=azim x z w0 w1`,
REPEAT STRIP_TAC 
THEN POP_ASSUM MP_TAC
THEN MRESAL_TAC AFF_GT_1_1[`x:real^3`;`z:real^3`][IN_ELIM_THM;SET_RULE`DISJOINT {x} {z}<=> ~(x=z)`]
THEN ASM_TAC
THEN GEOM_ORIGIN_TAC `x:real^3`
THEN REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`u % (x + vec 0) + v%(x+ z)= (u+v)%x+v %z`;VECTOR_ARITH`(x + y = &1 % x + v % z)<=> y = v % z`]
THEN RESA_TAC
THEN ASM_TAC THEN SET_TAC[AZIM_SPECIAL_SCALE]);;
let no_origin_aff_ge_is_aff_gt=prove(`!x y z:real^3. 
~(x=y) /\ ~(x=z) /\ z IN aff_ge {x} {y}==> z IN aff_gt {x} {y}`,
REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN MRESAL_TAC AFF_GT_1_1[`x:real^3`;`y:real^3`][IN_ELIM_THM;SET_RULE`DISJOINT {x} {z}<=> ~(x=z)`]
THEN MRESAL_TAC AFF_GE_1_1[`x:real^3`;`y:real^3`][IN_ELIM_THM;SET_RULE`DISJOINT {x} {z}<=> ~(x=z)`]
THEN STRIP_TAC
THEN POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC
THEN MP_TAC(REAL_ARITH`&0<= t2==> t2= &0 \/ &0< t2`)
THEN RESA_TAC
THEN REDUCE_ARITH_TAC
THEN RESA_TAC
THEN REDUCE_VECTOR_TAC
THEN ASM_REWRITE_TAC[]
THEN STRIP_TAC
THEN EXISTS_TAC`t1:real`
THEN EXISTS_TAC`t2:real`
THEN ASM_REWRITE_TAC[]);;



let aff_ge_2_1_is_exists_point_inaff_ge_1_2=prove(`!x:real^3 y:real^3 z:real^3 w:real^3.
DISJOINT {x} {y,w} /\ DISJOINT {x,y} {w}/\ z IN aff_ge {x,y} {w}==> ?t. &0<t /\ t< &1 /\ (&1-t) %y+ t%z IN aff_ge {x} {y,w}`,

REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN MRESAL_TAC AFF_GE_2_1[`x:real^3`;`y:real^3`;`w:real^3`][IN_ELIM_THM]
THEN STRIP_TAC
THEN MRESAL_TAC AFF_GE_1_2[`x:real^3`;`y:real^3`;`w:real^3`][IN_ELIM_THM]
THEN DISJ_CASES_TAC(REAL_ARITH`&0<= t2 \/ t2 < &0`)
THENL[EXISTS_TAC`&1/ &2`
THEN ASM_REWRITE_TAC[REAL_ARITH`&0< &1/ &2/\ &1/ &2 < &1`]
THEN EXISTS_TAC`&1/ &2 * t1:real`
THEN EXISTS_TAC`&1/ &2 * (t2+ &1):real`
THEN EXISTS_TAC`&1/ &2 * t3:real`
THEN ASM_REWRITE_TAC[REAL_ARITH`&1 / &2 * t1 + &1 / &2 * (t2 + &1) + &1 / &2 * t3 = &1/ &2 *(&1 +(t1+t2+t3))`;VECTOR_ARITH`(&1 - &1 / &2) % y + &1 / &2 % (t1 % x + t2 % y + t3 % w) =
 (&1 / &2 * t1) % x + (&1 / &2 * (t2 + &1)) % y + (&1 / &2 * t3) % w`;REAL_ARITH`&1/ &2 *(&1 + &1)= &1`]
THEN STRIP_TAC
THEN MATCH_MP_TAC REAL_LE_MUL
THEN ASM_REWRITE_TAC[REAL_ARITH`&0<= &1/ &2`]
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC;

EXISTS_TAC`inv(&1 - t2):real`
THEN MP_TAC(REAL_ARITH`t2< &0 ==> &0< &1 -t2 /\ &1< &1 -t2 /\ &0<= &1 -t2 /\ ~(&1- t2= &0) `)
THEN RESA_TAC
THEN MRESA1_TAC REAL_LT_INV`&1-t2`
THEN MRESA1_TAC REAL_LE_INV`&1-t2`
THEN MRESA1_TAC REAL_MUL_LINV `&1- t2:real`
THEN MRESA1_TAC REAL_INV_LT_1`&1- t2`
THEN EXISTS_TAC`inv(&1 - t2)* t1:real`
THEN EXISTS_TAC`inv(&1 - t2)* t2 + &1 - inv(&1 - t2):real`
THEN EXISTS_TAC`inv(&1 - t2)* t3`
THEN ASM_REWRITE_TAC[REAL_ARITH`inv (&1 - t2) * t1 +
 (inv (&1 - t2) * t2 + &1 - inv (&1 - t2)) +
 inv (&1 - t2) * t3= inv (&1 - t2) * (t1 +t2+t3)+ &1 - inv (&1 - t2) /\ inv (&1 - t2) * &1 + &1 - inv (&1 - t2) = &1`;VECTOR_ARITH`(&1 - inv (&1 - t2)) % y + inv (&1 - t2) % (t1 % x + t2 % y + t3 % w) =
 (inv (&1 - t2) * t1) % x +
 (inv (&1 - t2) * t2 + &1 - inv (&1 - t2)) % y +
 (inv (&1 - t2) * t3) % w`;REAL_ARITH`inv (&1 - t2) * t2 + &1 - inv (&1 - t2)= &1 - inv (&1 - t2) *(&1-t2)`;REAL_ARITH`&0<= &1 - &1`]
THEN MATCH_MP_TAC REAL_LE_MUL
THEN ASM_REWRITE_TAC[]]);;



let point_in_aff_gt_2_1_change_point_in_aff_gt_1_2=prove(` !x:real^3 v:real^3 u:real^3 y:real^3.
 ~collinear {x,v,u}
/\  y IN aff_gt {x} {v,u}
==> u IN aff_gt {x,v} {y}`,
REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"YEU")
THEN MRESA_TAC properties_of_collinear4_points_fan[`x:real^3`;`v:real^3`;`u:real^3`;`y:real^3`]
THEN REMOVE_THEN "YEU" MP_TAC
THEN MRESA_TAC th3[`x:real^3`;`v:real^3`;`u:real^3`;]
THEN MRESA_TAC th3[`x:real^3`;`y:real^3`;`v:real^3`;]
THEN MRESA_TAC AFF_GT_1_2[`x:real^3`;`v:real^3`;`u:real^3`;]
THEN MRESAL_TAC AFF_GT_2_1[`x:real^3`;`v:real^3`;`y:real^3`;][IN_ELIM_THM]
THEN RESA_TAC
THEN MP_TAC(REAL_ARITH`&0< t3==> ~(t3= &0)`)
THEN RESA_TAC
THEN MRESA1_TAC REAL_LT_INV`t3:real`
THEN MRESA1_TAC REAL_MUL_LINV`t3:real`
THEN EXISTS_TAC`-- inv t3 * t1:real`
THEN EXISTS_TAC`-- inv t3 * t2:real`
THEN EXISTS_TAC`inv t3 :real`
THEN ASM_REWRITE_TAC[REAL_ARITH`--inv t3 * t1 + --inv t3 * t2 + inv t3= inv t3 *( t3 + &1- (t1+t2+t3))`;REAL_ARITH`A+ &1- &1 =A`;VECTOR_ARITH`(--inv t3 * t1) % x +
 (--inv t3 * t2) % v +
 inv t3 % (t1 % x + t2 % v + t3 % u)= (inv t3 * t3) % u`]
THEN VECTOR_ARITH_TAC);;
let pos_in_aff_gt_fan=prove(`!x:real^3 v:real^3 u:real^3 a:real.
DISJOINT {x} {v,u}
/\ &0<a /\ a< &1
==>
(&1-a)%v + a % u IN aff_gt {x} {v,u:real^3}`,

REPEAT STRIP_TAC
THEN MRESAL_TAC AFF_GT_1_2[`x:real^3`;`v:real^3`;`u:real^3`][IN_ELIM_THM]
THEN EXISTS_TAC`&0`
THEN EXISTS_TAC`&1 -a:real`
THEN EXISTS_TAC`a:real`
THEN MP_TAC(REAL_ARITH`&0< a /\ a  < &1 ==> &0 < &1 - a /\ &0 < a`)
THEN RESA_TAC
THEN ASM_REWRITE_TAC[]
THEN REDUCE_VECTOR_TAC
THEN REAL_ARITH_TAC);;

let pos_in_aff_gt_2_1_fan=prove(`!x:real^3 v:real^3 u:real^3 a:real.
DISJOINT {x,v} {u}
/\ &0<a /\ a< &1
==>
(&1-a)%v + a % u IN aff_gt {x,v} {u:real^3}`,

REPEAT STRIP_TAC
THEN MRESAL_TAC AFF_GT_2_1[`x:real^3`;`v:real^3`;`u:real^3`][IN_ELIM_THM]
THEN EXISTS_TAC`&0`
THEN EXISTS_TAC`&1 -a:real`
THEN EXISTS_TAC`a:real`
THEN MP_TAC(REAL_ARITH`&0< a /\ a  < &1 ==> &0 < &1 - a /\ &0 < a`)
THEN RESA_TAC
THEN ASM_REWRITE_TAC[]
THEN REDUCE_VECTOR_TAC
THEN REAL_ARITH_TAC);;





(* ========================================================================== *)
(*                        SEGMENT       (^_^)              *)
(* ========================================================================== *)


let segment_in_segment=prove(`!x y z:real^N. z IN segment [x,y]==>  (!t. &0<= t /\ t<= &1 ==> (&1-t) %z +t %y IN segment[x,y])`,
REWRITE_TAC[segment;IN_ELIM_THM]
THEN REPEAT STRIP_TAC
THEN ASM_REWRITE_TAC[]
THEN EXISTS_TAC`(&1- t)* u+t:real`
THEN REWRITE_TAC[VECTOR_ARITH`(&1 - t) % ((&1 - u) % x + u % y) + t % y =
 (&1 - ((&1 - t) * u + t)) % x + ((&1 - t) * u + t) % y:real^N`]
THEN STRIP_TAC
THENL[MATCH_MP_TAC REAL_LE_ADD
THEN ASM_REWRITE_TAC[]
THEN MATCH_MP_TAC REAL_LE_MUL
THEN ASM_REWRITE_TAC[REAL_ARITH`&0<= &1- t<=> t<= &1`];

REWRITE_TAC[REAL_ARITH`(&1 - t) * u + t <= &1<=> &0<= (&1 - t) * (&1-u)`]
THEN MATCH_MP_TAC REAL_LE_MUL
THEN ASM_REWRITE_TAC[REAL_ARITH`&0<= &1- t<=> t<= &1`]]);;
let segmentsubset_aff_gt=prove(`!x y z w:real^N.
DISJOINT {x} {y,z}/\ w IN aff_gt {x} {y,z}
==> !t. &0<= t /\ t< &1 ==> (&1-t) %w+t%z IN aff_gt {x} {y,z}`,
REPEAT GEN_TAC THEN STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN MRESAL_TAC AFF_GT_1_2[`x:real^N`;`y:real^N`;`z:real^N`][IN_ELIM_THM]
THEN REPEAT STRIP_TAC
THEN ASM_REWRITE_TAC[]
THEN EXISTS_TAC`(&1-t)*t1:real`
THEN EXISTS_TAC`(&1-t)*t2:real`
THEN EXISTS_TAC`(&1-t)*t3+t:real`
THEN ASM_REWRITE_TAC[VECTOR_ARITH`(&1 - t) % (t1 % x + t2 % y + t3 % z) + t % z =
 ((&1 - t) * t1) % x + ((&1 - t) * t2) % y + ((&1 - t) * t3 + t) % z:real^N`;REAL_ARITH`(&1 - t) * t1 + (&1 - t) * t2 + (&1 - t) * t3 + t=(&1 - t) * (t1 + t2 +t3) + t/\ (&1 - t) * &1 + t = &1`]
THEN STRIP_TAC
THENL[MATCH_MP_TAC REAL_LT_MUL
THEN ASM_REWRITE_TAC[REAL_ARITH`&0< &1- t<=> t< &1`];

MATCH_MP_TAC (REAL_ARITH`&0<A /\ &0<=B==> &0< A+B`)
THEN ASM_REWRITE_TAC[]
THEN MATCH_MP_TAC REAL_LT_MUL
THEN ASM_REWRITE_TAC[REAL_ARITH`&0< &1- t<=> t< &1`]]);;
(* ========================================================================== *)
(*                        SOME LINEAR FUNCTIONS       (^_^)              *)
(* ========================================================================== *)


let linear_aff_fan=prove(`!x:real^3 v:real^3 u:real^3.
linear (\(t:real^2). t$1 %(v-x)+t$2 %(u-x))`,
REPEAT STRIP_TAC
THEN MATCH_MP_TAC LINEAR_COMPOSE_ADD 
THEN STRIP_TAC
THEN MATCH_MP_TAC LINEAR_VMUL_COMPONENT
THEN SIMP_TAC[LINEAR_ID; DIMINDEX_2; ARITH]);;
let linear1_aff_fan=prove(`!x:real^3 v:real^3 u:real^3 w:real^3.
linear (\(t:real^3). t$1 %(v-x)+t$2 %(u-x)+t$3 %(w-u))`,
REPEAT STRIP_TAC
THEN REPEAT(MATCH_MP_TAC LINEAR_COMPOSE_ADD THEN STRIP_TAC)
THEN MATCH_MP_TAC LINEAR_VMUL_COMPONENT
THEN SIMP_TAC[LINEAR_ID; DIMINDEX_3; ARITH]);;




let linear_inj_fan=prove(`!x:real^3 v:real^3 u:real^3.
~collinear{x,v,u}
==>(!(a:real^2) (b:real^2). (\(t:real^2). t$1 %(v-x)+t$2 %(u-x))(a)=(\(t:real^2). t$1 %(v-x)+t$2 %(u-x))(b) ==>a=b)`,

REPEAT GEN_TAC
THEN DISCH_TAC
THEN ASSUME_TAC(ISPECL[`x:real^3`;` v:real^3`;`u:real^3`]linear_aff_fan)
THEN MP_TAC(ISPEC`(\(t:real^2). t$1 %(v-x)+t$2 %(u-x):real^3)`LINEAR_INJECTIVE_0)
THEN RESP_TAC
THEN REMOVE_ASSUM_TAC
THEN GEN_TAC
THEN DISJ_CASES_TAC(REAL_ARITH`(a:real^2)$2= &0 \/ ~(a$2= &0)`)
THENL[
ASM_REWRITE_TAC[]
THEN REDUCE_VECTOR_TAC
THEN REWRITE_TAC[VECTOR_MUL_EQ_0;VECTOR_ARITH`A-B=vec 0<=> B=A`]
THEN MP_TAC(ISPECL[`x:real^3`;` v:real^3`;`u:real^3`]th3)
THEN RESA_TAC
THEN ASM_TAC
THEN SIMP_TAC[ LAMBDA_BETA;CART_EQ; DIMINDEX_2; FORALL_2; VEC_COMPONENT; ARITH];

REWRITE_TAC[VECTOR_ARITH`A+B=vec 0<=>B= --A`]
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`a$2 % (u - x) = --((a:real^2)$1 % (v - x:real^3)) ==> (inv (a$2)) % a$2 % (u - x) = (inv (a$2)) % (--(a$1 % (v - x)))`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th] 
THEN ASM_REWRITE_TAC[VECTOR_ARITH`(A%B%C=(A*B)%C:real^3)`])
THEN MP_TAC(ISPEC`(a:real^2)$2`REAL_MUL_LINV)
THEN RESA_TAC
THEN REDUCE_VECTOR_TAC
THEN REWRITE_TAC[VECTOR_ARITH`A-B=C%(--(D%(U-B)))<=> A= (&1+C*D)%B+(--C*D)%U:real^3`]
THEN MP_TAC(ISPECL[`x:real^3`;` v:real^3`;`u:real^3`]th3)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[aff; AFFINE_HULL_2;IN_ELIM_THM]
THEN DISCH_THEN(LABEL_TAC"A")
THEN DISCH_TAC
THEN SUBGOAL_THEN `F`ASSUME_TAC
THENL[
REMOVE_THEN "A" MP_TAC
THEN ASM_REWRITE_TAC[]
THEN EXISTS_TAC`(&1 + inv ((a:real^2)$2) * a$1)`
THEN EXISTS_TAC`(--inv ((a:real^2)$2) * a$1)`
THEN ASM_REWRITE_TAC[]
THEN REAL_ARITH_TAC;
ASM_MESON_TAC[]]]);;




(* ========================================================================== *)
(*                        AFFINE AND DOT                   *)
(* ========================================================================== *)


let exp_aff_ge_by_dot=prove(`!x:real^3 v:real^3 u:real^3.
~collinear {x,v,u}
==> aff_ge {x,v} {u}={w:real^3| (w-x) dot (e2_fan x v u)= &0 /\ &0 <= (w-x) dot (e1_fan x v u)  }`,
(let CROSS_LAGRANGE1 = prove
 (`!x y z. (x cross y) cross z = (x dot z) % y - (z dot y) % x`,
  VEC3_TAC) in

REPEAT STRIP_TAC THEN MP_TAC(ISPECL[`x:real^3`;`v:real^3`;`u:real^3`]th3) THEN RES_TAC
  THEN MP_TAC(ISPECL[`x:real^3`;`v:real^3`;`u:real^3`]AFF_GE_2_1) THEN RESA_TAC
  THEN MP_TAC(ISPECL[`x:real^3`;`v:real^3`;`u:real^3`]properties_coordinate) THEN RESA_TAC
  THEN REWRITE_TAC[EXTENSION;IN_ELIM_THM] THEN GEN_TAC THEN EQ_TAC
THENL[
STRIP_TAC THEN ASM_REWRITE_TAC[VECTOR_ARITH`(a % x + b +c) -x= (a- &1)% x + b + c `] THEN 
REMOVE_ASSUM_TAC THEN SYM_ASSUM_TAC THEN REWRITE_TAC[VECTOR_ARITH`((a-(a+b+c)) % x + b % v +c % u)=  b % (v-x) + c % (u-x)`] 
THEN ASM_REWRITE_TAC[DOT_LADD;DOT_LMUL]
   THEN REDUCE_ARITH_TAC
  THEN ASM_MESON_TAC[REAL_LE_MUL] ; 

STRIP_TAC THEN POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC THEN DISCH_THEN(LABEL_TAC"a")
  THEN DISCH_THEN(LABEL_TAC"b")
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_IMP_SPANNING) THEN ASM_REWRITE_TAC[SPAN_3;EXTENSION] 
  THEN DISCH_TAC THEN POP_ASSUM(fun th-> MP_TAC(ISPEC`(x':real^3)-(x:real^3)`th)) THEN REWRITE_TAC[SET_RULE`(a:real^3) IN (:real^3)`;IN_ELIM_THM] THEN RES_TAC THEN REMOVE_THEN "a" MP_TAC THEN ASM_REWRITE_TAC[DOT_LADD;DOT_LMUL]
  THEN POP_ASSUM MP_TAC THEN DISCH_THEN(LABEL_TAC"c")
  THEN FIND_ASSUM(MP_TAC)`orthonormal (e1_fan (x:real^3) (v:real^3) (u:real^3)) (e2_fan x v u) (e3_fan x v u)`
  THEN REWRITE_TAC[orthonormal] THEN STRIP_TAC THEN ASM_REWRITE_TAC[]  THEN ASM_REWRITE_TAC[DOT_SYM]
  THEN REDUCE_ARITH_TAC
  THEN DISCH_TAC THEN REMOVE_THEN "c" MP_TAC THEN ASM_REWRITE_TAC[] THEN REDUCE_VECTOR_TAC THEN DISCH_THEN (LABEL_TAC"a")
  THEN REMOVE_THEN "b" MP_TAC THEN ASM_REWRITE_TAC[DOT_LADD;DOT_LMUL;] THEN REWRITE_TAC[DOT_SYM] THEN ASM_REWRITE_TAC[]
  THEN REDUCE_ARITH_TAC
  THEN DISCH_TAC THEN REMOVE_THEN "a" MP_TAC
  THEN ASM_REWRITE_TAC[e1_fan;e2_fan;CROSS_LMUL;VECTOR_ARITH`a% b% v=(a*b)%v`;CROSS_LAGRANGE1] 
  THEN REDUCE_VECTOR_TAC THEN REWRITE_TAC[VECTOR_ARITH`a%(x- b % v)+ c % v=(c- a* b) % v+  a % x `;
e3_fan;VECTOR_ARITH`a% b% v=(a*b)%v`]
  THEN STRIP_TAC THEN
EXISTS_TAC
`&1 - ((((w:real) -
   ((u':real) * inv (norm (inv (norm ((v:real^3) - (x:real^3))) % (v - x) cross ((u:real^3) - x)))) *
   (inv (norm (v - x)) % (v - x) dot (u - x))) *
  inv (norm (v - x)))+
((u':real) * inv (norm (e3_fan (x:real^3) (v:real^3) (u:real^3) cross (u - x)))))`
  THEN EXISTS_TAC
`(((w:real) -
   ((u':real) * inv (norm (inv (norm ((v:real^3) - (x:real^3))) % (v - x) cross ((u:real^3) - x)))) *
   (inv (norm (v - x)) % (v - x) dot (u - x))) *
  inv (norm (v - x)))`
  THEN EXISTS_TAC
` ((u':real) * inv (norm (e3_fan (x:real^3) (v:real^3) (u:real^3) cross (u - x))))`
THEN
STRIP_TAC
THENL[

SUBGOAL_THEN `~(collinear {vec 0, v-x, u-x})==> ~((e3_fan (x:real^3) (v:real^3) (u:real^3)) cross ((u:real^3)-(x:real^3))= vec 0)` ASSUME_TAC
THENL[
 MATCH_MP_TAC MONO_NOT THEN REWRITE_TAC[e3_fan;CROSS_LMUL] 
THEN DISCH_TAC THEN MP_TAC(ISPECL [`v:real^3`; `x:real^3`] imp_inv_norm_not_zero_fan) 
THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN 
MP_TAC(ISPECL [`inv(norm((v:real^3)-(x:real^3)))`; `((v:real^3) -(x:real^3)) cross ((u:real^3)-(x:real^3))`; `(vec 0):real^3`] VECTOR_MUL_LCANCEL_IMP) 
THEN ASM_REWRITE_TAC[VECTOR_MUL_RZERO;CROSS_EQ_0 ];

POP_ASSUM MP_TAC THEN REWRITE_TAC[GSYM COLLINEAR_3] 
THEN GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o RAND_CONV o ONCE_DEPTH_CONV)[SET_RULE`{a,b,c}={b,a,c}`] THEN RED_TAC
THEN
MP_TAC(ISPECL [`(e3_fan (x:real^3) (v:real^3) (u:real^3)) cross ((u:real^3)-(x:real^3))`; `((vec 0):real^3)`] imp_norm_ge_zero_fan)
  THEN REDUCE_VECTOR_TAC THEN RES_TAC THEN 
MP_TAC(ISPECL[`u':real`;`inv (norm ((e3_fan (x:real^3) (v:real^3) (u:real^3)) cross ((u:real^3)-(x:real^3))))`] 
REAL_LE_MUL) THEN RES_TAC THEN POP_ASSUM MATCH_MP_TAC THEN POP_ASSUM MP_TAC
 THEN REAL_ARITH_TAC];

STRIP_TAC THENL[REAL_ARITH_TAC;
REWRITE_TAC[e3_fan] THEN POP_ASSUM MP_TAC THEN VECTOR_ARITH_TAC]]]));;





let exp_aff_ge_by_dot_1_1=prove(`!x:real^3 v:real^3 u:real^3.
~collinear {x,v,u}
==>
aff_ge {x} {v}={w:real^3| (w-x) dot (e2_fan x v u)= &0 /\ &0 <= (w-x) dot (e3_fan x v u) 
/\ (w-x) dot (e1_fan x v u)= &0 }`,

REPEAT STRIP_TAC THEN MP_TAC(ISPECL[`x:real^3`;`v:real^3`;`u:real^3`]th3) 
  THEN RES_TAC THEN MP_TAC(ISPECL[`x:real^3`;`v:real^3`]AFF_GE_1_1) THEN RESA_TAC
  THEN MP_TAC(ISPECL[`x:real^3`;`v:real^3`;`u:real^3`]properties_coordinate) THEN RESA_TAC
  THEN REWRITE_TAC[EXTENSION;IN_ELIM_THM] THEN GEN_TAC THEN EQ_TAC
THENL[
STRIP_TAC THEN ASM_REWRITE_TAC[VECTOR_ARITH`(a % x + b) -x= (a- &1)% x + b  `] 
THEN 
REMOVE_ASSUM_TAC THEN SYM_ASSUM_TAC THEN REWRITE_TAC[VECTOR_ARITH`((a-(a+b)) % x + b % v)=  b % (v-x)`] 
THEN ASM_REWRITE_TAC[DOT_LADD;DOT_LMUL]
   THEN REDUCE_ARITH_TAC THEN POP_ASSUM MP_TAC
  THEN FIND_ASSUM(fun th-> REWRITE_TAC[SYM(th)])`dist (v,x) % e3_fan x v u = v- x:real^3`
  THEN REWRITE_TAC[DOT_LMUL] 
    THEN FIND_ASSUM(MP_TAC)`orthonormal (e1_fan (x:real^3) (v:real^3) (u:real^3)) (e2_fan x v u) (e3_fan x v u)`
  THEN REWRITE_TAC[orthonormal] THEN RESA_TAC THEN REDUCE_ARITH_TAC THEN MP_TAC(ISPECL[`v:real^3`;`x:real^3`]DIST_POS_LE)
  THEN MESON_TAC[REAL_LE_MUL];

STRIP_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 DISCH_THEN (LABEL_TAC "c")
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_IMP_SPANNING) THEN ASM_REWRITE_TAC[SPAN_3;EXTENSION] 
  THEN DISCH_TAC THEN POP_ASSUM(fun th-> MP_TAC(ISPEC`(x':real^3)-(x:real^3)`th)) THEN REWRITE_TAC[SET_RULE`(a:real^3) IN (:real^3)`;IN_ELIM_THM] THEN RES_TAC THEN REMOVE_THEN "a" MP_TAC THEN ASM_REWRITE_TAC[DOT_LADD;DOT_LMUL]
  THEN POP_ASSUM MP_TAC THEN DISCH_THEN(LABEL_TAC"d")
  THEN FIND_ASSUM(MP_TAC)`orthonormal (e1_fan (x:real^3) (v:real^3) (u:real^3)) (e2_fan x v u) (e3_fan x v u)`
  THEN REWRITE_TAC[orthonormal] THEN STRIP_TAC THEN ASM_REWRITE_TAC[]  THEN ASM_REWRITE_TAC[DOT_SYM]
  THEN REDUCE_ARITH_TAC
  THEN REMOVE_THEN "c" MP_TAC THEN ASM_REWRITE_TAC[DOT_LADD;DOT_LMUL]
  THEN ASM_REWRITE_TAC[DOT_SYM]
  THEN REDUCE_ARITH_TAC THEN DISCH_TAC
  THEN DISCH_TAC THEN REMOVE_THEN "d" MP_TAC THEN ASM_REWRITE_TAC[] THEN REDUCE_VECTOR_TAC 
  THEN  DISCH_TAC
  THEN REMOVE_THEN "b" MP_TAC THEN ASM_REWRITE_TAC[DOT_LADD;DOT_LMUL;] THEN REWRITE_TAC[DOT_SYM] THEN ASM_REWRITE_TAC[]
  THEN REDUCE_ARITH_TAC THEN POP_ASSUM MP_TAC THEN REWRITE_TAC[e3_fan;VECTOR_ARITH`a% b% v=(a*b)%v`;
VECTOR_ARITH`a-b=c %(v-b)<=> a= (&1-c) % b + c % v`] THEN DISCH_THEN (LABEL_TAC"a")
  THEN STRIP_TAC THEN
EXISTS_TAC
`&1 - (w:real) * (inv (norm ((v:real^3) - (x:real^3))))`
  THEN EXISTS_TAC
`(w:real) * (inv (norm ((v:real^3) - (x:real^3))))`
THEN
STRIP_TAC
THENL[
MP_TAC(ISPECL[`v:real^3`;`x:real^3`]imp_norm_ge_zero_fan) THEN RES_TAC THEN MATCH_MP_TAC REAL_LE_MUL 
THEN ASM_REWRITE_TAC[]  THEN POP_ASSUM MP_TAC THEN REAL_ARITH_TAC;
STRIP_TAC THENL[REAL_ARITH_TAC;
ASM_REWRITE_TAC[]]]]);;







(****************************************************************************)
(*        the conditions to add azim                  *)
(****************************************************************************)





let sum1_azim_fan=prove(`!x:real^3 v:real^3 u:real^3 w1:real^3 w2:real^3.
 cyclic_set {u, w1, w2} x v /\ (azim x v u w1 + azim x v w1 w2) < &2 * pi
==> 
azim x v u w2 = azim x v u w1+ azim x v w1 w2
`,
( let th=prove(`!x v u. {v,x,u}={x,v,u}/\{v,x,u}={u,x,v}`,SET_TAC[]) in  


REPEAT GEN_TAC THEN STRIP_TAC 
THEN MP_TAC (SPECL [`x:real^3`; `v:real^3`; `u:real^3`;`w1:real^3`; `w2:real^3`] property_of_cyclic_set) 
THEN ASM_REWRITE_TAC[] THEN DISCH_TAC
THEN
MP_TAC (SPECL [`x:real^3`; `v:real^3`; `u:real^3`;`w1:real^3`; `w2:real^3`] property_of_cyclic_set1) 
THEN ASM_REWRITE_TAC[] THEN DISCH_TAC
  THEN MP_TAC (SPECL [`x:real^3`; `v:real^3`; `u:real^3`;`w1:real^3`; `w2:real^3`] property_of_cyclic_set2) 
THEN ASM_REWRITE_TAC[] THEN DISCH_TAC
  THEN MP_TAC(ISPECL[`v:real^3`; `x:real^3`; `u:real^3`]COLLINEAR_3) THEN ASM_REWRITE_TAC[] THEN
DISCH_TAC THEN  SUBGOAL_THEN `~collinear {(x:real^3),(v:real^3),(u:real^3)}/\ ~collinear {(u:real^3),(x:real^3),(v:real^3)}` ASSUME_TAC
THENL[ASM_MESON_TAC[th];

MP_TAC (SPECL [`x:real^3`; `v:real^3`; `w1:real^3`; `w2:real^3`] azim) 
  THEN STRIP_TAC 
THEN POP_ASSUM(MP_TAC o SPECL [`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)`]) 
  THEN MP_TAC(ISPECL[`(x:real^3)`; `(v:real^3)`; `(u:real^3)`] orthonormal_e1_e2_e3_fan) 
THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN
MP_TAC(ISPECL[`(x:real^3)`; `(v:real^3)`; `(u:real^3)`] e3_mul_dist_fan) THEN ASM_REWRITE_TAC[] THEN DISCH_TAC 
  THEN ASM_REWRITE_TAC[] THEN STRIP_TAC
  THEN MP_TAC(SPEC `psi:real` SINCOS_PRINCIPAL_VALUE_FAN ) THEN STRIP_TAC THEN
MP_TAC(ISPECL[`x:real^3`;`v:real^3`;`u:real^3`;`u:real^3`]AZIM_EXISTS) THEN STRIP_TAC 
THEN
POP_ASSUM (fun th-> 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)`]th)) THEN ASM_REWRITE_TAC[] THEN STRIP_TAC
THEN POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC THEN ASM_REWRITE_TAC[] THEN REPEAT DISCH_TAC
  THEN MP_TAC(ISPECL[`x:real^3`; `v:real^3`; `u:real^3` ;`r1':real`; `psi':real`; `h1':real`]sincos_of_u_fan)
  THEN ASM_REWRITE_TAC[] THEN DISCH_TAC 

THEN MP_TAC (ISPECL [`x:real^3`; `v:real^3`; `u:real^3`; `w1:real^3`; `h1':real`; `h1:real`; `r1':real`; `r1: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)`; `psi':real`; `y:real`  ] AZIM_UNIQUE)
  THEN ASM_REWRITE_TAC[COS_ADD;SIN_ADD;REAL_ARITH` &1 * (a:real) - &0 * (b:real)=a`;REAL_ARITH`&0 * (a:real) + &1 * (b:real)=b`]
THEN DISCH_TAC THEN MP_TAC (ISPECL [`x:real^3`; `v:real^3`; `u:real^3`; `w2:real^3`; `h1':real`; `h2:real`; `r1':real`; `r2: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)`; `psi':real`; `azim (x:real^3) (v:real^3) (u:real^3) (w1:real^3) + azim x v w1 (w2:real^3)`  ] AZIM_UNIQUE) 
THEN DISCH_TAC THEN POP_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN STRIP_TAC
THENL[ ASM_MESON_TAC[REAL_LE_ADD];


ASM_REWRITE_TAC[COS_ADD;SIN_ADD;REAL_ARITH` &1 * (a:real) - &0 * (b:real)=a`;REAL_ARITH`&0 * (a:real) + &1 * (b:real)=b`]
     ]]));;






let sum3_azim_fan=prove(`!x:real^3 v:real^3 u:real^3 w1:real^3 w2:real^3.
((azim x v u w1 + azim x v w1 w2) < &2 * pi)
/\
(~collinear {(x:real^3),(v:real^3),(w1:real^3)})
/\(~collinear {(x:real^3),(v:real^3),(w2:real^3)})
/\ (~collinear {(x:real^3),(v:real^3),(u:real^3)})
==> 
azim x v u w2 = azim x v u w1+ azim x v w1 w2
`, (let th=prove(`!x v u. {x,v,u}={v,x,u}`,SET_TAC[]) in
 (let th1=prove(`!x v u. {x,v,u}={u,x,v}`,SET_TAC[]) in

REPEAT GEN_TAC THEN STRIP_TAC  THEN POP_ASSUM MP_TAC THEN DISCH_THEN(LABEL_TAC "a") 

THEN USE_THEN "a" MP_TAC THEN GEN_REWRITE_TAC ( LAND_CONV  o ONCE_DEPTH_CONV)[th] THEN DISCH_THEN(LABEL_TAC "b")

THEN USE_THEN "a" MP_TAC THEN GEN_REWRITE_TAC ( LAND_CONV  o ONCE_DEPTH_CONV)[th1] THEN DISCH_TAC

THEN USE_THEN "b" MP_TAC THEN
GEN_REWRITE_TAC ( LAND_CONV  o ONCE_DEPTH_CONV)[COLLINEAR_3] THEN STRIP_TAC
THEN  
MP_TAC(ISPECL[`x:real^3`;`v:real^3`;`u:real^3`]th3) THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN
MP_TAC (SPECL [`x:real^3`; `v:real^3`; `w1:real^3`; `w2:real^3`] azim) 
  THEN STRIP_TAC 
THEN POP_ASSUM(MP_TAC o SPECL [`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)`]) 
  THEN MP_TAC(ISPECL[`(x:real^3)`; `(v:real^3)`; `(u:real^3)`] orthonormal_e1_e2_e3_fan) 
THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN
MP_TAC(ISPECL[`(x:real^3)`; `(v:real^3)`; `(u:real^3)`] e3_mul_dist_fan) THEN ASM_REWRITE_TAC[] THEN DISCH_TAC 
  THEN ASM_REWRITE_TAC[] THEN STRIP_TAC
  THEN MP_TAC(SPEC `psi:real` SINCOS_PRINCIPAL_VALUE_FAN ) THEN STRIP_TAC THEN
MP_TAC(ISPECL[`x:real^3`;`v:real^3`;`u:real^3`;`u:real^3`]AZIM_EXISTS) THEN STRIP_TAC 
THEN
POP_ASSUM (fun th-> 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)`]th)) THEN ASM_REWRITE_TAC[] THEN STRIP_TAC
THEN POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC THEN ASM_REWRITE_TAC[] THEN REPEAT DISCH_TAC
  THEN MP_TAC(ISPECL[`x:real^3`; `v:real^3`; `u:real^3` ;`r1':real`; `psi':real`; `h1':real`]sincos_of_u_fan)
  THEN ASM_REWRITE_TAC[] THEN DISCH_TAC 

THEN MP_TAC (ISPECL [`x:real^3`; `v:real^3`; `u:real^3`; `w1:real^3`; `h1':real`; `h1:real`; `r1':real`; `r1: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)`; `psi':real`; `y:real`  ] AZIM_UNIQUE)
  THEN ASM_REWRITE_TAC[COS_ADD;SIN_ADD;REAL_ARITH` &1 * (a:real) - &0 * (b:real)=a`;REAL_ARITH`&0 * (a:real) + &1 * (b:real)=b`]
THEN DISCH_TAC THEN MP_TAC (ISPECL [`x:real^3`; `v:real^3`; `u:real^3`; `w2:real^3`; `h1':real`; `h2:real`; `r1':real`; `r2: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)`; `psi':real`; `azim (x:real^3) (v:real^3) (u:real^3) (w1:real^3) + azim x v w1 (w2:real^3)`  ] AZIM_UNIQUE) 
THEN DISCH_TAC THEN POP_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN STRIP_TAC
THENL[
 ASM_MESON_TAC[REAL_LE_ADD];

ASM_REWRITE_TAC[COS_ADD;SIN_ADD;REAL_ARITH` &1 * (a:real) - &0 * (b:real)=a`;REAL_ARITH`&0 * (a:real) + &1 * (b:real)=b`]])));;


let sum2_azim_fan=prove(`!x:real^3 v:real^3 u:real^3 w1:real^3 w2:real^3.
 cyclic_set {u, w1, w2} x v /\ azim x v u w1 <= azim x v u w2
==> 
azim x v u w2 = azim x v u w1 + azim x v w1 w2
`,

(let th=prove(`!x v u. {v,x,u}={x,v,u}/\{v,x,u}={u,x,v}`,SET_TAC[]) in

REWRITE_TAC[REAL_ARITH`(a:real)=(b:real)+(c:real) <=> c=a-b`] THEN
REPEAT GEN_TAC THEN STRIP_TAC 
THEN MP_TAC (SPECL [`x:real^3`; `v:real^3`; `u:real^3`;`w1:real^3`; `w2:real^3`] property_of_cyclic_set) 
THEN ASM_REWRITE_TAC[] THEN DISCH_TAC
THEN
MP_TAC (SPECL [`x:real^3`; `v:real^3`; `u:real^3`;`w1:real^3`; `w2:real^3`] property_of_cyclic_set1) 
THEN ASM_REWRITE_TAC[] THEN DISCH_TAC
  THEN MP_TAC (SPECL [`x:real^3`; `v:real^3`; `u:real^3`;`w1:real^3`; `w2:real^3`] property_of_cyclic_set2) 
THEN ASM_REWRITE_TAC[] THEN DISCH_TAC
  THEN MP_TAC(ISPECL[`v:real^3`; `x:real^3`; `u:real^3`]COLLINEAR_3) THEN ASM_REWRITE_TAC[] THEN
DISCH_TAC THEN  SUBGOAL_THEN `~collinear {(x:real^3),(v:real^3),(u:real^3)}/\ ~collinear {(u:real^3),(x:real^3),(v:real^3)}` ASSUME_TAC
THENL[ASM_MESON_TAC[th];
MP_TAC (SPECL [`x:real^3`; `v:real^3`; `u:real^3`; `w1:real^3`] azim) 
  THEN STRIP_TAC 
THEN POP_ASSUM(MP_TAC o SPECL [`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)`]) 
THEN MP_TAC (SPECL [`x:real^3`; `v:real^3`; `u:real^3`; `w2:real^3`] azim) THEN STRIP_TAC 
THEN POP_ASSUM(MP_TAC o SPECL [`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)`]) 
THEN MP_TAC(ISPECL[`(x:real^3)`; `(v:real^3)`; `(u:real^3)`] orthonormal_e1_e2_e3_fan) 
THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN
MP_TAC(ISPECL[`(x:real^3)`; `(v:real^3)`; `(u:real^3)`] e3_mul_dist_fan) THEN ASM_REWRITE_TAC[] THEN DISCH_TAC 
  THEN ASM_REWRITE_TAC[]
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 DISCH_THEN (LABEL_TAC"b")
THEN REPEAT STRIP_TAC
   THEN POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC
  THEN DISCH_THEN (LABEL_TAC"c") THEN REPEAT STRIP_TAC 
  THEN MP_TAC(ISPECL[`x:real^3`; `v:real^3`; `u:real^3` ;`r1:real`; `psi:real`; `h1':real`]sincos_of_u_fan)
  THEN ASM_REWRITE_TAC[] THEN DISCH_TAC
THEN MP_TAC(ISPECL[`x:real^3`; `v:real^3`; `u:real^3` ;`r1':real`; `psi':real`; `h1:real`]sincos_of_u_fan)
  THEN REMOVE_THEN "a" MP_TAC THEN ASM_REWRITE_TAC[] THEN  DISCH_TAC THEN DISCH_TAC
  THEN REMOVE_THEN "b" MP_TAC THEN REMOVE_THEN "c" MP_TAC
  THEN ASM_REWRITE_TAC[COS_ADD;SIN_ADD;REAL_ARITH` &1 * (a:real) - &0 * (b:real)=a`;REAL_ARITH`&0 * (a:real) + &1 * (b:real)=b`]
  THEN REPEAT STRIP_TAC 
  THEN MP_TAC (ISPECL [`x:real^3`; `v:real^3`; `w1:real^3`; `w2:real^3`; `h2:real`; `h2':real`; `r2':real`; `r2: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)`; `azim (x:real^3) (v:real^3) (u:real^3) (w1:real^3)`; `azim (x:real^3) (v:real^3) (u:real^3) (w2:real^3) - azim x v u (w1:real^3)`  ] AZIM_UNIQUE) 
THEN DISCH_TAC THEN POP_ASSUM MATCH_MP_TAC 
THEN ASM_REWRITE_TAC[REAL_ARITH`(a:real)+(b:real)-a=b`; REAL_ARITH`(&0 <= (a:real)-(b:real))<=> b<= a`] THEN MP_TAC(ISPEC `azim (x:real^3) (v:real^3) (u:real^3) (w1:real^3)` REAL_NEG_LE0) 
THEN ASM_REWRITE_TAC[] THEN DISCH_TAC 
THEN MP_TAC(ISPECL[`azim (x:real^3) (v:real^3) (u:real^3) (w2:real^3)`;`&2 * pi`;`--azim (x:real^3) (v:real^3) (u:real^3) (w1:real^3)`;`&0`]REAL_LTE_ADD2 ) 
THEN ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC]));;



let sum4_azim_fan=prove(`!x:real^3 v:real^3 u:real^3 w1:real^3 w2:real^3.
 azim x v u w1 <= azim x v u w2
/\ (~collinear {(x:real^3),(v:real^3),(w1:real^3)})
/\(~collinear {(x:real^3),(v:real^3),(w2:real^3)})
/\ (~collinear {(x:real^3),(v:real^3),(u:real^3)})

==> 
azim x v u w2 = azim x v u w1 + azim x v w1 w2
`,(let th=prove(`!x v u. {x,v,u}={v,x,u}`,SET_TAC[]) in
(let th1=prove(`!x v u. {x,v,u}={u,x,v}`,SET_TAC[]) in

REWRITE_TAC[REAL_ARITH`(a:real)=(b:real)+(c:real) <=> c=a-b`] THEN

REPEAT GEN_TAC THEN STRIP_TAC  THEN POP_ASSUM MP_TAC THEN DISCH_THEN(LABEL_TAC "a1") 

THEN USE_THEN "a1" MP_TAC THEN GEN_REWRITE_TAC ( LAND_CONV  o ONCE_DEPTH_CONV)[th] THEN DISCH_THEN(LABEL_TAC "b1")

THEN USE_THEN "a1" MP_TAC THEN GEN_REWRITE_TAC ( LAND_CONV  o ONCE_DEPTH_CONV)[th1] THEN DISCH_TAC

THEN USE_THEN "b1" MP_TAC THEN
GEN_REWRITE_TAC ( LAND_CONV  o ONCE_DEPTH_CONV)[COLLINEAR_3] THEN STRIP_TAC
THEN
MP_TAC(ISPECL[`x:real^3`;`v:real^3`;`u:real^3`]th3) THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN
MP_TAC (SPECL [`x:real^3`; `v:real^3`; `u:real^3`; `w1:real^3`] azim) 
  THEN STRIP_TAC 
THEN POP_ASSUM(MP_TAC o SPECL [`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)`]) 
THEN MP_TAC (SPECL [`x:real^3`; `v:real^3`; `u:real^3`; `w2:real^3`] azim) THEN STRIP_TAC 
THEN POP_ASSUM(MP_TAC o SPECL [`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)`]) 
THEN MP_TAC(ISPECL[`(x:real^3)`; `(v:real^3)`; `(u:real^3)`] orthonormal_e1_e2_e3_fan) 
THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN
MP_TAC(ISPECL[`(x:real^3)`; `(v:real^3)`; `(u:real^3)`] e3_mul_dist_fan) THEN ASM_REWRITE_TAC[] THEN DISCH_TAC 
  THEN ASM_REWRITE_TAC[]
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 DISCH_THEN (LABEL_TAC"b")
THEN REPEAT STRIP_TAC
   THEN POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC
  THEN DISCH_THEN (LABEL_TAC"c") THEN REPEAT STRIP_TAC 
  THEN MP_TAC(ISPECL[`x:real^3`; `v:real^3`; `u:real^3` ;`r1:real`; `psi:real`; `h1':real`]sincos_of_u_fan)
  THEN ASM_REWRITE_TAC[] THEN DISCH_TAC
THEN MP_TAC(ISPECL[`x:real^3`; `v:real^3`; `u:real^3` ;`r1':real`; `psi':real`; `h1:real`]sincos_of_u_fan)
  THEN REMOVE_THEN "a" MP_TAC THEN ASM_REWRITE_TAC[] THEN  DISCH_TAC THEN DISCH_TAC
  THEN REMOVE_THEN "b" MP_TAC THEN REMOVE_THEN "c" MP_TAC
  THEN ASM_REWRITE_TAC[COS_ADD;SIN_ADD;REAL_ARITH` &1 * (a:real) - &0 * (b:real)=a`;REAL_ARITH`&0 * (a:real) + &1 * (b:real)=b`]
  THEN REPEAT STRIP_TAC
  THEN MP_TAC (ISPECL [`x:real^3`; `v:real^3`; `w1:real^3`; `w2:real^3`; `h2:real`; `h2':real`; `r2':real`; `r2: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)`; `azim (x:real^3) (v:real^3) (u:real^3) (w1:real^3)`; `azim (x:real^3) (v:real^3) (u:real^3) (w2:real^3) - azim x v u (w1:real^3)`  ] AZIM_UNIQUE) 
THEN DISCH_TAC
THEN POP_ASSUM MATCH_MP_TAC
THEN ASM_REWRITE_TAC[REAL_ARITH`(a:real)+(b:real)-a=b`; REAL_ARITH`(&0 <= (a:real)-(b:real))<=> b<= a`] THEN MP_TAC(ISPEC `azim (x:real^3) (v:real^3) (u:real^3) (w1:real^3)` REAL_NEG_LE0) 
THEN ASM_REWRITE_TAC[] THEN DISCH_TAC 
THEN MP_TAC(ISPECL[`azim (x:real^3) (v:real^3) (u:real^3) (w2:real^3)`;`&2 * pi`;`--azim (x:real^3) (v:real^3) (u:real^3) (w1:real^3)`;`&0`]REAL_LTE_ADD2 ) 
THEN ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC
)));;



let sum5_azim_fan=prove(`!x:real^3 v:real^3 u:real^3 w1:real^3 w2:real^3.
  azim x v  w1 w2 <= azim x v u w2
/\ (~collinear {(x:real^3),(v:real^3),(w1:real^3)})
/\(~collinear {(x:real^3),(v:real^3),(w2:real^3)})
/\ (~collinear {(x:real^3),(v:real^3),(u:real^3)})

==> 
azim x v u w2 = azim x v u w1 + azim x v w1 w2
`,
REPEAT STRIP_TAC THEN REPEAT (POP_ASSUM MP_TAC) THEN DISCH_THEN(LABEL_TAC"1") THEN REPEAT STRIP_TAC
  THEN DISJ_CASES_TAC(REAL_ARITH`(azim x v u w2)= &0 \/ ~(azim x v u w2 = &0)`)
THENL(*1*)[
SUBGOAL_THEN`azim x v w1 w2 = &0` ASSUME_TAC
THENL(*2*)[
REPEAT(POP_ASSUM MP_TAC) THEN MP_TAC(ISPECL[`x:real^3`;`v:real^3`;`w1:real^3`;`w2:real^3`]azim) THEN REAL_ARITH_TAC;
(*2*)
MP_TAC(ISPECL[`x:real^3`;`v:real^3`;`w1:real^3`;`w2:real^3`]AZIM_EQ_0_SYM) THEN ASM_REWRITE_TAC[] THEN DISCH_TAC
THEN
MP_TAC(ISPECL[`x:real^3`;`v:real^3`;`u:real^3`;`w2:real^3`]AZIM_EQ_0_SYM) THEN ASM_REWRITE_TAC[] THEN DISCH_TAC
THEN
SUBGOAL_THEN`azim x v w2 w1 = azim x v w2 u` ASSUME_TAC
THENL(*3*)[ASM_MESON_TAC[];(*3*)
REWRITE_TAC[REAL_ARITH`&0 = a + &0 <=> a= &0`] THEN
MP_TAC(ISPECL[`x:real^3`;`v:real^3`;`w2:real^3`;`u:real^3`;`w1:real^3`]AZIM_EQ_ALT) THEN ASM_REWRITE_TAC[] THEN DISCH_TAC
  THEN MP_TAC(ISPECL[`x:real^3`;`v:real^3`;`u:real^3`;`w1:real^3`]AZIM_EQ_0) THEN ASM_REWRITE_TAC[]]];
 DISJ_CASES_TAC(REAL_ARITH`(azim x v w1 w2)= &0 \/ ~(azim x v w1 w2 = &0)`)
THENL(*4*)[
ASM_REWRITE_TAC[REAL_ARITH`b = a + &0 <=> b= a`] THEN
MP_TAC(ISPECL[`x:real^3`;`v:real^3`;`w1:real^3`;`w2:real^3`]AZIM_EQ_0_ALT) THEN ASM_REWRITE_TAC[]THEN ASM_MESON_TAC[AZIM_EQ_ALT]  ;(*4*)
REMOVE_THEN"1" MP_TAC THEN
MP_TAC(ISPECL[`x:real^3`;`v:real^3`;`w1:real^3`;`w2:real^3`]AZIM_COMPL
) THEN ASM_REWRITE_TAC[]  
  THEN MP_TAC(ISPECL[`x:real^3`;`v:real^3`;`u:real^3`;`w2:real^3`]AZIM_COMPL
) THEN ASM_REWRITE_TAC[REAL_ARITH`a=b-c <=> c= b-a`] THEN DISCH_TAC THEN DISCH_TAC
  THEN ASM_REWRITE_TAC[REAL_ARITH`a-b=c+a-d<=> d=b+c`;REAL_ARITH`a-b<=a-d<=> d<=b`] THEN ASM_MESON_TAC[sum4_azim_fan]]
]);;



(* ========================================================================== *)
(*                   AZIM                          *)
(* ========================================================================== *)




let th = prove
 (`!x:real^3 v:real^3 u:real^3 w:real^3.
        ~collinear {x,v,u} /\ ~collinear{x,v,w}
        ==> {y:real^3 | ~collinear {x,v,y} /\ azim x v u w = azim x v u y} = 
            aff_gt {x , v} {w}`,
  GEOM_ORIGIN_TAC `x:real^3` THEN
  GEOM_BASIS_MULTIPLE_TAC 3 `v:real^3` THEN                         
  X_GEN_TAC `v:real` THEN ASM_CASES_TAC `v = &0` THENL
   [ASM_REWRITE_TAC[VECTOR_MUL_LZERO; INSERT_AC; COLLINEAR_2]; ALL_TAC] THEN
  ASM_REWRITE_TAC[VECTOR_MUL_LZERO; REAL_LE_LT] THEN DISCH_TAC THEN
  MAP_EVERY X_GEN_TAC [`u:real^3`; `w:real^3`] THEN
  ASM_CASES_TAC `w:real^3 = vec 0` THENL
   [ASM_REWRITE_TAC[INSERT_AC; COLLINEAR_2]; ALL_TAC] THEN
  ASM_CASES_TAC `w:real^3 = v % basis 3` THENL
   [ASM_REWRITE_TAC[INSERT_AC; COLLINEAR_2]; ALL_TAC] THEN
  ASM_SIMP_TAC[AZIM_SPECIAL_SCALE; COLLINEAR_SPECIAL_SCALE] THEN
  ASM_CASES_TAC `w:real^3 = basis 3` THENL
   [ASM_REWRITE_TAC[INSERT_AC; COLLINEAR_2]; ALL_TAC] THEN
  ASM_SIMP_TAC[AFF_GT_SPECIAL_SCALE; IN_SING; FINITE_INSERT; FINITE_EMPTY] THEN
  POP_ASSUM_LIST(K ALL_TAC) THEN REWRITE_TAC[COLLINEAR_BASIS_3; AZIM_ARG] THEN
  DISCH_TAC THEN MATCH_MP_TAC EQ_TRANS THEN 
  EXISTS_TAC `{y:real^3 | (dropout 3 y:real^2) IN 
                          aff_gt {vec 0} {dropout 3 (w:real^3)}}` THEN
  CONJ_TAC THENL
   [REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN X_GEN_TAC `y:real^3` THEN
    POP_ASSUM MP_TAC THEN
    MAP_EVERY SPEC_TAC
     [`(dropout 3:real^3->real^2) u`,`u:real^2`;
      `(dropout 3:real^3->real^2) v`,`v:real^2`;
      `(dropout 3:real^3->real^2) w`,`w:real^2`;
      `(dropout 3:real^3->real^2) y`,`y:real^2`] THEN
    SIMP_TAC[AFF_GT_1_1; SET_RULE `DISJOINT {x} {y} <=> ~(x = y)`] THEN
    REWRITE_TAC[VECTOR_MUL_RZERO; VECTOR_ADD_LID; IN_ELIM_THM] THEN
    REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN 
    ONCE_REWRITE_TAC[TAUT `a /\ b /\ c <=> a /\ c /\ b`] THEN
    REWRITE_TAC[REAL_ARITH `t1 + t2 = &1 <=> t1 = &1 - t2`] THEN
    REWRITE_TAC[RIGHT_EXISTS_AND_THM; EXISTS_REFL] THEN
    ASM_CASES_TAC `y:real^2 = vec 0` THEN ASM_REWRITE_TAC[] THENL
     [ASM_MESON_TAC[VECTOR_MUL_EQ_0; REAL_LT_IMP_NZ]; ALL_TAC] THEN
    RULE_ASSUM_TAC(REWRITE_RULE[COMPLEX_VEC_0]) THEN
    GEN_REWRITE_TAC LAND_CONV [EQ_SYM_EQ] THEN
    ASM_SIMP_TAC[ARG_EQ; COMPLEX_CMUL; COMPLEX_FIELD 
     `~(w = Cx(&0)) /\ ~(z = Cx(&0)) ==> ~(w / z = Cx(&0))`] THEN
    ASM_SIMP_TAC[COMPLEX_FIELD
     `~(u = Cx(&0)) ==> (w / u = x * y / u <=> w = x * y)`];
    SUBGOAL_THEN `~(w:real^3 = vec 0) /\ ~(w = basis 3)` ASSUME_TAC THENL
     [ASM_MESON_TAC[DROPOUT_BASIS_3; DROPOUT_0]; ALL_TAC] THEN
    ASM_SIMP_TAC[AFF_GT_1_1; AFF_GT_2_1; DISJOINT_INSERT; IN_INSERT;
                 DISJOINT_EMPTY; NOT_IN_EMPTY] THEN
    REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN X_GEN_TAC `y:real^3` THEN
    REWRITE_TAC[VECTOR_MUL_RZERO; VECTOR_ADD_LID] THEN
    ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN
    GEN_REWRITE_TAC (RAND_CONV o BINDER_CONV) [SWAP_EXISTS_THM] THEN
    ONCE_REWRITE_TAC[TAUT `a /\ b /\ c <=> a /\ c /\ b`] THEN
    REWRITE_TAC[REAL_ARITH `t1 + t2 = &1 <=> t1 = &1 - t2`] THEN
    REWRITE_TAC[RIGHT_EXISTS_AND_THM; EXISTS_REFL] THEN      
    SIMP_TAC[CART_EQ; DIMINDEX_3; FORALL_3; VECTOR_ADD_COMPONENT;
             VECTOR_MUL_COMPONENT; BASIS_COMPONENT; ARITH; DIMINDEX_2;
             DROPOUT_BASIS_3; FORALL_2; dropout; LAMBDA_BETA] THEN
    ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN
    REWRITE_TAC[REAL_MUL_RZERO; REAL_ADD_LID; RIGHT_EXISTS_AND_THM] THEN
    REWRITE_TAC[REAL_ARITH `y = t * &1 + s <=> t = y - s`; EXISTS_REFL]]);;


let th1=prove(`(!x:real^3 v:real^3 u:real^3 w:real^3 t1:real t2:real t3:real. (t3 > &0) /\ (t1 + t2 + t3 = &1)
/\ DISJOINT {x,v} {w} /\ ~collinear {x,v,u}/\ ~collinear {x,v,w}
 ==> azim x v u w =
 azim x v u (t1 % x + t2 % v + t3 % w))`,
REPEAT GEN_TAC THEN STRIP_TAC THEN ASSUME_TAC(AFF_GT_2_1) 
THEN POP_ASSUM(MP_TAC o ISPECL [`x:real^3`;`v:real^3`;`w:real^3`]) 
  THEN ASM_REWRITE_TAC[] THEN DISCH_TAC 
THEN ABBREV_TAC `(y:real^3)= (t1:real)  % (x:real^3) + (t2:real) % (v:real^3) + (t3:real) % (w:real^3)`
      THEN SUBGOAL_THEN `(y:real^3) IN aff_gt {(x:real^3),(v:real^3)} {w:real^3}` ASSUME_TAC
THENL[
ASM_REWRITE_TAC[IN_ELIM_THM] THEN EXISTS_TAC `t1:real` 
THEN EXISTS_TAC `t2:real` THEN EXISTS_TAC `t3:real`
THEN EXPAND_TAC "y" THEN ASM_MESON_TAC[REAL_ARITH`(a:real)> &0 <=> &0 < a ` ];
						
 POP_ASSUM MP_TAC THEN
ASSUME_TAC(th) THEN POP_ASSUM(MP_TAC o ISPECL [`x:real^3`;`v:real^3`;`u:real^3`;`w:real^3`]) 
  THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN POP_ASSUM (fun th-> REWRITE_TAC[SYM(th)]) 
THEN REWRITE_TAC[IN_ELIM_THM] THEN ASM_SET_TAC[]]);;
  let th2= prove(`!x:real^3 v:real^3 w:real^3. ~(x=v)==>  (w IN complement_set {x,v}==> ~ collinear {x,v,w})`,   
REPEAT GEN_TAC THEN DISCH_TAC THEN REWRITE_TAC[CONTRAPOS_THM; COLLINEAR_3;COLLINEAR_LEMMA; complement_set; IN_ELIM_THM;affine_hull_2_fan] THEN STRIP_TAC
THENL[
ASM_MESON_TAC[VECTOR_ARITH`(x-v= vec 0)<=> (x=v)`];
 EXISTS_TAC `&0` THEN EXISTS_TAC `&1` THEN REWRITE_TAC[REAL_ARITH`&0+ &1 = &1`; VECTOR_ARITH`&0 % x= vec 0`; VECTOR_ARITH`w=vec 0 + &1 % v <=> w - v = vec 0`] THEN ASM_SET_TAC[];
EXISTS_TAC `c:real` THEN EXISTS_TAC `&1 - (c:real)` THEN REWRITE_TAC[REAL_ARITH`c+ &1 - c = &1`; VECTOR_ARITH`w=c % x  + (&1 - c) % v <=> w - v = c % (x-v)`] THEN ASM_SET_TAC[]]);;



(* ========================================================================== *)
(*                   CARD                         *)
(* ========================================================================== *)


let CARD_SING=prove(`!x:real^3 s:real^3->bool. 
 FINITE s 
/\ s={x}
==>
CARD s = 1`,
REPEAT STRIP_TAC THEN 
MP_TAC(SET_RULE`(s:real^3->bool)={(x:real^3)} ==> ~(s={}) /\ x IN s /\ s DELETE x ={}`)
  THEN ASM_REWRITE_TAC[] THEN DISCH_TAC
  THEN MP_TAC(ISPEC`s:real^3->bool`CARD_EQ_0) THEN ASM_REWRITE_TAC[]
  THEN MP_TAC(ISPECL[`x:real^3`;`s:real^3->bool`]CARD_DELETE) THEN ASM_REWRITE_TAC[CARD_CLAUSES]
  THEN ARITH_TAC);;


(* ========================================================================== *)
(*                   CLOSED AFF                         *)
(* ========================================================================== *)







let closed_aff_ge_2_1=prove(`!x:real^3 v:real^3 u:real^3.
~collinear {x,v,u}
==>
closed (aff_ge {x,v} {u})`,
(let lemma=prove(`!x:real^3 v:real^3 u:real^3.
{w:real^3| (w-x) dot (e2_fan x v u)= &0 /\ &0 <= (w-x) dot (e1_fan x v u)  }
={w:real^3| (w-x) dot (e2_fan x v u)= &0} INTER {w:real^3| (w-x) dot (e1_fan x v u) >= &0 }`,
REWRITE_TAC[INTER; IN_ELIM_THM;REAL_ARITH`&0<=a <=> a >= &0`]) in
(
let lemma1=prove(`!x:real^3 v:real^3 u:real^3.
closed {w:real^3| (w-x) dot (e2_fan x v u)= &0}`,
REWRITE_TAC[ DOT_SYM] THEN REWRITE_TAC[DOT_RSUB;REAL_ARITH`a-b= &0<=> a=b`;] 
 THEN REPEAT GEN_TAC THEN MP_TAC(ISPECL[`e2_fan (x:real^3) (v:real^3) (u:real^3)`;
` e2_fan (x:real^3) (v:real^3) (u:real^3) dot x`]CLOSED_HYPERPLANE) THEN ASM_SET_TAC[]) in
 (
let lemma2=prove(`!x:real^3 v:real^3 u:real^3.
closed {w:real^3| (w-x) dot (e1_fan x v u) >= &0 }`,
REWRITE_TAC[ DOT_SYM] THEN REWRITE_TAC[DOT_RSUB;REAL_ARITH`a-b>= &0<=> a>=b`;] 
 THEN REPEAT GEN_TAC THEN MP_TAC(ISPECL[`e1_fan (x:real^3) (v:real^3) (u:real^3)`;
` e1_fan (x:real^3) (v:real^3) (u:real^3) dot x`]CLOSED_HALFSPACE_GE) THEN ASM_SET_TAC[]) in

REPEAT STRIP_TAC THEN
ASM_MESON_TAC[exp_aff_ge_by_dot;lemma;lemma1;lemma2;CLOSED_INTER]))));;




let closed_aff_ge_1_2=prove(`!(x:real^3)  (v:real^3) (w:real^3).
~collinear {x, v, w}
==>
closed (aff_ge {x} {v , w})`,
REPEAT STRIP_TAC 
THEN POP_ASSUM (fun th-> MP_TAC (th) THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={A,C,B}`] THEN ASSUME_TAC(th))
THEN MRESA_TAC aff_ge_inter_aff_ge[`x:real^3`;`v:real^3`;`w:real^3`]
THEN  MRESA_TAC closed_aff_ge_2_1[`x:real^3`;`w:real^3`;`v:real^3`]
THEN  MRESA_TAC closed_aff_ge_2_1[`x:real^3`;`v:real^3`;`w:real^3`]
THEN ASM_MESON_TAC[CLOSED_INTER]);;




let closed_halfline_fan=prove(`!(x:real^3) (v:real^3) (u:real^3). 
~collinear {x,v,u}
==>
closed (aff_ge {x} { v})`,


(let lemma=prove(`!x v u :real^3.
{w:real^3| (w-x) dot (e2_fan x v u)= &0 /\ &0 <= (w-x) dot (e3_fan x v u) 
/\ (w-x) dot (e1_fan x v u)= &0 }= {w:real^3| (w-x) dot (e2_fan x v u)= &0} INTER
({w:real^3| (w-x) dot (e1_fan x v u)= &0} INTER {w:real^3| &0 <= (w-x) dot (e3_fan x v u)})`,
REWRITE_TAC[INTER;IN_ELIM_THM] THEN ASM_SET_TAC[]) in


(let lemma1=prove(`!x:real^3 v:real^3 u:real^3.
closed {w:real^3| (w-x) dot (e2_fan x v u)= &0}`,
REWRITE_TAC[ DOT_SYM] THEN REWRITE_TAC[DOT_RSUB;REAL_ARITH`a-b= &0<=> a=b`;] 
 THEN REPEAT GEN_TAC THEN MP_TAC(ISPECL[`e2_fan (x:real^3) (v:real^3) (u:real^3)`;
` e2_fan (x:real^3) (v:real^3) (u:real^3) dot x`]CLOSED_HYPERPLANE) THEN ASM_SET_TAC[]) in

(let lemma3=prove(`!x:real^3 v:real^3 u:real^3.
closed {w:real^3| (w-x) dot (e1_fan x v u)= &0}`,
REWRITE_TAC[ DOT_SYM] THEN REWRITE_TAC[DOT_RSUB;REAL_ARITH`a-b= &0<=> a=b`;] 
 THEN REPEAT GEN_TAC THEN MP_TAC(ISPECL[`e1_fan (x:real^3) (v:real^3) (u:real^3)`;
` e1_fan (x:real^3) (v:real^3) (u:real^3) dot x`]CLOSED_HYPERPLANE) THEN ASM_SET_TAC[]) in
 
(let lemma2=prove(`!x:real^3 v:real^3 u:real^3.
closed {w:real^3| &0 <= (w-x) dot (e3_fan x v u)  }`,
REWRITE_TAC[ DOT_SYM] THEN REWRITE_TAC[DOT_RSUB;REAL_ARITH`&0 <= a-b<=> a>=b`;] 
 THEN REPEAT GEN_TAC THEN MP_TAC(ISPECL[`e3_fan (x:real^3) (v:real^3) (u:real^3)`;
` e3_fan (x:real^3) (v:real^3) (u:real^3) dot x`]CLOSED_HALFSPACE_GE) THEN ASM_SET_TAC[]) in
REPEAT STRIP_TAC THEN MP_TAC(ISPECL[`x:real^3`;` v:real^3`; `u:real^3`]exp_aff_ge_by_dot_1_1)
  THEN REWRITE_TAC[lemma]
THEN RESA_TAC THEN ASSUME_TAC(ISPECL[`x:real^3`;` v:real^3`; `u:real^3`]lemma1) THEN ASSUME_TAC(ISPECL[`x:real^3`;` v:real^3`; `u:real^3`]lemma2) THEN ASSUME_TAC(ISPECL[`x:real^3`;` v:real^3`; `u:real^3`]lemma3)
  THEN SUBGOAL_THEN`closed({w:real^3| (w-x) dot (e1_fan x v u)= &0} INTER {w:real^3| &0 <= (w-x) dot (e3_fan x v u)})`
ASSUME_TAC
THENL[ASM_MESON_TAC[CLOSED_INTER];
ASM_MESON_TAC[CLOSED_INTER]])))));;




(*--------------------------------------------------------------------------------------------*)
(*       The properties of      ballnorm_fan (x:real^3)={y:real^3 | dist(x,y) = &1}           *)
(*--------------------------------------------------------------------------------------------*) 





let ballnorm_fan=new_definition`ballnorm_fan (x:real^3)={y:real^3 | dist(x,y) = &1}`;;

let closed_ballnorm_fan=prove(`!x:real^3. closed (ballnorm_fan x)`,
GEN_TAC THEN REWRITE_TAC[ballnorm_fan] THEN
SUBGOAL_THEN  `{y:real^3 | dist((x:real^3),(y:real^3)) = &1} = frontier( ball((x:real^3), &1))` ASSUME_TAC
 THENL [ASSUME_TAC(REAL_ARITH `&0 < &1`) THEN POP_ASSUM MP_TAC THEN
        SIMP_TAC[frontier; CLOSURE_BALL; INTERIOR_OPEN; OPEN_BALL;
           REAL_LT_IMP_LE] THEN
  REWRITE_TAC[EXTENSION; IN_DIFF; IN_ELIM_THM; IN_BALL; IN_CBALL] THEN
  REAL_ARITH_TAC;
   ASM_REWRITE_TAC[] THEN MESON_TAC[FRONTIER_CLOSED]]);;
let bounded_ballnorm_fan=prove(`!x:real^3 . bounded(ballnorm_fan x)`,
REPEAT GEN_TAC THEN REWRITE_TAC[ballnorm_fan;bounded] THEN
 EXISTS_TAC `norm(x:real^3) + &1`  THEN REWRITE_TAC[ dist; IN_ELIM_THM] 
  THEN GEN_TAC THEN STRIP_TAC THEN ASSUME_TAC(NORM_TRIANGLE_SUB) THEN
POP_ASSUM (MP_TAC o ISPECL [`(x':real^3)`; `(x:real^3)`]  o INST_TYPE [`:real^3`,`:real^3`])
  THEN REWRITE_TAC[NORM_SUB] THEN ASM_REWRITE_TAC[]);;
let bounded_ballnorm_fans=prove(`!x:real^3 v:real^3 w:real^3. bounded (aff_ge {x} {v, w} INTER ballnorm_fan x)`,
REPEAT GEN_TAC THEN ASSUME_TAC (bounded_ballnorm_fan) THEN
POP_ASSUM (MP_TAC o ISPEC `x:real^3`) THEN DISCH_TAC THEN
SUBGOAL_THEN `aff_ge {x} {(v:real^3), (w:real^3)} INTER ballnorm_fan x SUBSET ballnorm_fan (x:real^3)` ASSUME_TAC THENL
[ASM_SET_TAC[];
ASM_MESON_TAC[BOUNDED_SUBSET ]]);;


(*--------------------------------------------------------------------------------------------*)
(*       The properties of fan in norm ball                                                  *)
(*--------------------------------------------------------------------------------------------*) 




let closed_aff_ge_ballnorm_fan=prove(`!(x:real^3)   (v:real^3) (w:real^3).
~collinear{x,v,w}
==>
closed (aff_ge {x} {v, w} INTER ballnorm_fan x)`,
ASM_MESON_TAC[closed_aff_ge_1_2; closed_ballnorm_fan;CLOSED_INTER]);;




let compact_aff_ge_ballnorm_fan=prove(`
!(x:real^3) (v:real^3) (w:real^3).
~collinear{x,v,w}
==>
compact (aff_ge {x} {v, w} INTER ballnorm_fan x)`,
REPEAT GEN_TAC THEN DISCH_TAC THEN
SUBGOAL_THEN `closed (aff_ge {x} {v, w} INTER ballnorm_fan x)` ASSUME_TAC 
THENL
  [ASM_MESON_TAC[closed_aff_ge_ballnorm_fan];
    ASSUME_TAC(bounded_ballnorm_fans) 
    THEN
   POP_ASSUM (MP_TAC o ISPECL [`x:real^3`; `v:real^3`; `w:real^3`]) THEN
    ASM_MESON_TAC[BOUNDED_CLOSED_IMP_COMPACT ]]);;






let closed_point_fan=prove(`
(!x:real^3 v:real^3 u:real^3. 
~collinear {x,v,u}
==>
closed (aff_ge {x} {v} INTER ballnorm_fan x) )`,
REPEAT GEN_TAC THEN DISCH_TAC THEN
SUBGOAL_THEN `closed (aff_ge {(x:real^3)} {(v:real^3)})` ASSUME_TAC THENL
   [ASM_MESON_TAC[ closed_halfline_fan]; 
    SUBGOAL_THEN `closed (ballnorm_fan (x:real^3))` ASSUME_TAC THENL
       [ASM_MESON_TAC[closed_ballnorm_fan];
        ASM_MESON_TAC[CLOSED_INTER]]]);;


(* ========================================================================== *)
(*                   RCONE                         *)
(* ========================================================================== *)


(* rcone^0(x,v,h) *)

let rcone_fan = new_definition `rcone_fan  (x:real^3) (v:real^3) (h:real) = {y:real^3 | (y-x) dot (v-x) >(dist(y,x)*dist(v,x)*h)}`;;

let origin_not_in_rcone_fan=prove(`!(x:real^3) (v:real^3) (h:real). ~(x IN rcone_fan x v h)`,
REPEAT GEN_TAC 
THEN REWRITE_TAC[rcone_fan; IN_ELIM_THM; VECTOR_ARITH`x-x= vec 0`; DOT_LZERO;DIST_REFL] 
THEN REDUCE_ARITH_TAC 
THEN REAL_ARITH_TAC);; 

let conditions_in_rcone_fan=prove(`!x v u w:real^3 s:real.
~collinear {x,v,u}/\ w IN aff_gt {x} {v,u} /\ &0<s /\ s< pi/ &2 /\u IN rcone_fan x v (cos s)==> w IN rcone_fan x v (cos s) `,

REWRITE_TAC[rcone_fan;IN_ELIM_THM]
THEN REPEAT STRIP_TAC
THEN ASM_TAC
THEN DISCH_TAC
THEN MRESA_TAC th3[`(x:real^3)` ;` (v:real^3)`;`(u:real^3)`;]
THEN MRESAL_TAC AFF_GT_1_2[`(x:real^3)` ;` (v:real^3)`;` (u:real^3)  `;][IN_ELIM_THM]
THEN REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[dist;VECTOR_ARITH`(t1 % x + t2 % v + t3 % u) - x=((t1+t2+t3)- &1) % x + t2 % (v-x) + t3 % (u-x)`;REAL_ARITH`&1 - &1= &0`;REAL_ARITH`A>B<=> B<A`;DOT_LADD;DOT_LMUL]
THEN REDUCE_VECTOR_TAC
THEN REDUCE_ARITH_TAC
THEN STRIP_TAC
THEN MP_TAC(REAL_ARITH`&0< t3==> &0<= t3`) THEN RESA_TAC
THEN MRESA1_TAC REAL_ABS_REFL`t3:real`
THEN MRESA_TAC NORM_MUL[`t3:real`;`u-x:real^3`]
THEN MP_TAC(REAL_ARITH`&0< t2==> &0<= t2`) THEN RESA_TAC
THEN MRESA1_TAC REAL_ABS_REFL`t2:real`
THEN MRESA_TAC NORM_MUL[`t2:real`;`v-x:real^3`]
THEN MRESA_TAC REAL_LT_LMUL[`t3:real`;`norm (u - x) * norm (v - x:real^3) * (cos s):real`;`(u - x) dot (v - x:real^3)`]
THEN MRESA1_TAC COS_BOUNDS`s:real`
THEN MRESA1_TAC DOT_POS_LE`(v - x):real^3`
THEN MRESA_TAC REAL_LE_MUL[`t2:real`;`(v-x) dot (v-x:real^3)`;]
THEN MRESA_TAC REAL_LE_LMUL[`t2*((v - x:real^3) dot (v-x)):real`;`cos s:real`;`&1`]
THEN POP_ASSUM MP_TAC THEN REDUCE_ARITH_TAC 
THEN GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[DOT_SQUARE_NORM;]
THEN STRIP_TAC
THEN MP_TAC(REAL_ARITH` t3 * norm (u - x) * norm (v - x) * cos s< t3 * ((u - x) dot (v - x))
/\(t2 * norm (v - x) pow 2) * cos s <= t2 * ((v - x) dot (v - x))
==> (t2 * norm (v - x:real^3)  + t3 * norm (u - x)) * norm (v - x) * cos s< t2 * ((v - x) dot (v - x))+ t3 * ((u - x) dot (v - x))`)
THEN RESA_TAC
THEN MRESA_TAC NORM_TRIANGLE[`t2 %(v - x:real^3)`;`t3 % (u - x:real^3)`]
THEN MRESA1_TAC NORM_POS_LE`(v - x):real^3`
THEN MRESA1_TAC COS_POS_PI2`(s):real`
THEN MP_TAC(REAL_ARITH`&0< cos s:real ==> &0<= cos s`) THEN RESA_TAC
THEN MRESA_TAC REAL_LE_MUL[`norm  (v-x:real^3)`;`cos s:real`]
THEN MRESA_TAC REAL_LE_RMUL[`norm (t2 % (v - x) + t3 % (u - x):real^3):real`;`t2 * norm (v - x:real^3) + t3 * norm (u - x)`;`norm (v - x:real^3) * cos s`]
THEN POP_ASSUM MP_TAC
THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC THEN REMOVE_ASSUM_TAC 
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC);;


let not_empty_rcone_fan_inter_aff_gt=prove(`!x v u:real^3 h:real.
~collinear {x,v,u} /\ &0< h /\ h<= pi==>
~(rcone_fan x v (cos h) INTER aff_gt {x} {v, u}={})`,
REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC th3[`x:real^3`;`v:real^3`;`u:real^3`]
THEN SUBGOAL_THEN`~(norm((v:real^3)-(x:real^3))= &0)` ASSUME_TAC
THENL(*1*)[ASM_REWRITE_TAC[NORM_EQ_0;VECTOR_ARITH`v-x=vec 0<=> x=v`];
ASM_REWRITE_TAC[rcone_fan;SET_RULE`~(A={})<=> ?y. y IN A`;INTER;IN_ELIM_THM]
THEN DISJ_CASES_TAC(REAL_ARITH`(v - x) dot (u - x:real^3) <= &0 \/ &0< (v - x) dot (u - x)`)
THENL(*2*)[ABBREV_TAC`s1= min h (pi / &2) / &2:real`
THEN MP_TAC(REAL_ARITH` &0< pi /\ min h (pi / &2) / &2 =s1 /\ &0< h:real ==> &0<= s1 /\ &0<s1 /\ s1< h/\ s1<pi/ &2`)
THEN ASM_REWRITE_TAC[PI_WORKS]
THEN STRIP_TAC
THEN EXISTS_TAC`sin (s1) % (e1_fan x v u) + (cos s1) %(e3_fan x v u)+x :real^3 `
THEN REWRITE_TAC[dist;vector_norm;VECTOR_ARITH`(B+C+A)-A=(B+C:real^3)`; DOT_LADD;DOT_RADD;DOT_RMUL;DOT_LMUL;]
THEN MRESAL_TAC properties_coordinate[`x:real^3`;`v:real^3`;`u:real^3`][orthonormal]
THEN ONCE_REWRITE_TAC[DOT_SYM]
THEN ASSUME_TAC(ISPEC`s1:real`SIN_CIRCLE)
THEN ASM_REWRITE_TAC[]
THEN REDUCE_ARITH_TAC
THEN ASM_REWRITE_TAC[REAL_ARITH`sin s * sin s + cos s * cos s =(sin s) pow 2 + (cos s) pow 2`;e3_fan;DOT_RMUL; ]
THEN ONCE_REWRITE_TAC[GSYM vector_norm;]
THEN REWRITE_TAC[DOT_SQUARE_NORM;]
THEN MRESAL1_TAC  SQRT_POW_2`&1`[REAL_ARITH`a pow 2 = a * a`;REAL_ARITH`&0<= &1`;]
THEN POP_ASSUM MP_TAC
THEN MRESAL_TAC SQRT_MUL[`&1`;`&1`][REAL_ARITH`&0<= &1`;]
THEN POP_ASSUM (fun th-> REWRITE_TAC[SYM(th); REAL_ARITH`&1* &1= &1`])
THEN RESA_TAC
THEN REDUCE_ARITH_TAC
THEN MP_TAC(ISPEC`norm((v:real^3)-(x:real^3))`REAL_MUL_LINV)
THEN REWRITE_TAC[REAL_ARITH`A*B*D *D=(B*D) *D *A`;REAL_ARITH`A>B<=> B<A`]
THEN RESA_TAC
THEN REDUCE_ARITH_TAC
THEN ASSUME_TAC(ISPEC`v-x:real^3`NORM_POS_LE)
THEN STRIP_TAC
THENL(*3*)[ MATCH_MP_TAC REAL_LT_LMUL
THEN MP_TAC(REAL_ARITH`~(norm (v - x:real^3) = &0)/\ &0 <= norm (v - x)==> &0< norm (v - x) `)
THEN RESA_TAC
THEN MATCH_MP_TAC COS_MONO_LT
THEN ASM_REWRITE_TAC[];(*3*)
MRESA_TAC condition1_to_in_aff_gt_by_angle[`x:real^3`;`v:real^3`;`u:real^3`;`s1:real`]
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[e3_fan]](*3*);(*2*)
SUBGOAL_THEN`&0<(atn ((norm ((v - x) cross (u - x))) * inv((v - x) dot (u - x:real^3))))` ASSUME_TAC
THENL(*3*)[MP_TAC(ISPEC`(v - x) dot (u - x:real^3)`REAL_LT_INV)
THEN RESA_TAC
THEN ASSUME_TAC(PI_WORKS)
THEN MP_TAC(REAL_ARITH`&0< pi ==> --(pi / &2) < &0`)
THEN RESA_TAC
THEN MRESAL_TAC  ATN_MONO_LT[`&0:real`;` (norm ((v - x) cross (u - x)) * inv ((v - x) dot (u - x))):real`][ ATN_0]
THEN POP_ASSUM MATCH_MP_TAC
THEN MATCH_MP_TAC REAL_LT_MUL
THEN ASM_REWRITE_TAC[]
THEN SUBGOAL_THEN`~(norm((v - x) cross (u - x:real^3))= &0)` ASSUME_TAC
THENL(*4*)[
ASM_REWRITE_TAC[NORM_EQ_0]
THEN MP_TAC(ISPECL[`v-x:real^3`;`u-x:real^3`]CROSS_EQ_0)
THEN ONCE_REWRITE_TAC[GSYM COLLINEAR_3;]
THEN RESA_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={B,A,C}`]
THEN ASM_REWRITE_TAC[];(*4*)

POP_ASSUM MP_TAC
THEN MP_TAC(ISPEC`(v - x) cross (u - x:real^3)`NORM_POS_LE)
THEN REAL_ARITH_TAC](*4*);(*3*)

 ASSUME_TAC(ISPEC`(norm ((v - x) cross (u - x)) * inv ((v - x) dot (u - x))):real`ATN_BOUNDS)
THEN ABBREV_TAC`s2= atn ((norm ((v - x) cross (u - x))) * inv((v - x) dot (u - x))):real`
THEN ABBREV_TAC`s1= min (h:real) (s2:real) / &2`
THEN MP_TAC(REAL_ARITH`&0< h /\ s1= min (h:real) (s2) / &2  /\ &0< pi /\  &0< s2 /\ s2 < pi/ &2==> &0<= s1 /\ &0<s1 /\ s1<pi/ &2 /\ s1<h/\  s1< s2
`)
THEN REWRITE_TAC[PI_WORKS]
THEN RESA_TAC
THEN EXISTS_TAC`sin (s1) % (e1_fan x v u) + (cos s1) %(e3_fan x v u)+x :real^3 `
THEN REWRITE_TAC[dist;vector_norm;VECTOR_ARITH`(B+C+A)-A=(B+C:real^3)`; DOT_LADD;DOT_RADD;DOT_RMUL;DOT_LMUL;]
THEN MRESAL_TAC properties_coordinate[`x:real^3`;`v:real^3`;`u:real^3`][orthonormal]
THEN ONCE_REWRITE_TAC[DOT_SYM]
THEN ASSUME_TAC(ISPEC`s1:real`SIN_CIRCLE)
THEN ASM_REWRITE_TAC[]
THEN REDUCE_ARITH_TAC
THEN ASM_REWRITE_TAC[REAL_ARITH`sin s * sin s + cos s * cos s =(sin s) pow 2 + (cos s) pow 2`;e3_fan;DOT_RMUL; ]
THEN ONCE_REWRITE_TAC[GSYM vector_norm;]
THEN REWRITE_TAC[DOT_SQUARE_NORM;]
THEN MRESAL1_TAC  SQRT_POW_2`&1`[REAL_ARITH`a pow 2 = a * a`;REAL_ARITH`&0<= &1`;]
THEN POP_ASSUM MP_TAC
THEN MRESAL_TAC SQRT_MUL[`&1`;`&1`][REAL_ARITH`&0<= &1`;]
THEN POP_ASSUM (fun th-> REWRITE_TAC[SYM(th); REAL_ARITH`&1* &1= &1`])
THEN RESA_TAC
THEN REDUCE_ARITH_TAC
THEN MP_TAC(ISPEC`norm((v:real^3)-(x:real^3))`REAL_MUL_LINV)
THEN REWRITE_TAC[REAL_ARITH`A*B*D *D=(B*D) *D *A`;REAL_ARITH`A>B<=> B<A`]
THEN RESA_TAC
THEN REDUCE_ARITH_TAC
THEN ASSUME_TAC(ISPEC`v-x:real^3`NORM_POS_LE)
THEN STRIP_TAC
THENL(*4*)[ MATCH_MP_TAC REAL_LT_LMUL
THEN MP_TAC(REAL_ARITH`~(norm (v - x:real^3) = &0)/\ &0 <= norm (v - x)==> &0< norm (v - x) `)
THEN RESA_TAC
THEN MATCH_MP_TAC COS_MONO_LT
THEN ASM_REWRITE_TAC[];(*4*)
MRESA_TAC condition_to_in_aff_gt_by_angle[`x:real^3`;`v:real^3`;`u:real^3`;`s1:real`]
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[e3_fan]]]]]);;



(* ========================================================================== *)
(*                   TOPOLOGY COMPONENT YFAN                         *)
(* ========================================================================== *)




let in_topological_component_yfan_is_connected=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) U:real^3->bool.
U IN topological_component_yfan (x,V,E)
==> connected U`,
REWRITE_TAC[topological_component_yfan;IN_ELIM_THM]
THEN REPEAT STRIP_TAC
THEN ASM_REWRITE_TAC[CONNECTED_CONNECTED_COMPONENT]);;


let exists_point_in_component_yfan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) U:real^3->bool.
U IN topological_component_yfan (x,V,E)
==> ?z. z IN U`,
REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[topological_component_yfan;IN_ELIM_THM]
THEN STRIP_TAC
THEN EXISTS_TAC`y:real^3`
THEN ASM_REWRITE_TAC[CONNECTED_COMPONENT_SET;IN_ELIM_THM]
THEN EXISTS_TAC`{y:real^3}`
THEN ASM_REWRITE_TAC[CONNECTED_SING;IN_SING]
THEN ASM_TAC THEN SET_TAC[]);;
let in_topological_component_yfan_is_connected=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) U:real^3->bool.
U IN topological_component_yfan (x,V,E)
==> connected U`,
REWRITE_TAC[topological_component_yfan;IN_ELIM_THM]
THEN REPEAT STRIP_TAC
THEN ASM_REWRITE_TAC[CONNECTED_CONNECTED_COMPONENT]);;

let expand_element_in_topological_component_yfan=prove(`!x:real^3 (V:real^3->bool) (E:(real^3->bool)->bool) U:real^3->bool z:real^3.
 U IN topological_component_yfan (x,V,E)
/\ z IN U
==> U=connected_component (yfan(x,V,E)) z`,
REWRITE_TAC[topological_component_yfan;IN_ELIM_THM]
THEN REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[]
THEN STRIP_TAC 
THEN MRESA_TAC CONNECTED_COMPONENT_EQ[`yfan(x:real^3, (V:real^3->bool) ,E)`;`y:real^3`;`z:real^3`]);;


(* ========================================================================== *)
(*                 BASIC PROPERTIES OF CONVEX                         *)
(* ========================================================================== *)

let expansion_convex_fan=prove(`!(v:real^3) (u:real^3) (w:real^3) (t:real) s:real.
&0 <= t /\ t<= &1
/\ &0 <=s /\ s <= &1 
==> (&1-s)%v+s%((&1-t)%u+ t%w) IN convex hull{v,u,w}`,

REWRITE_TAC[REAL_ARITH`A<= &1 <=> &0<= &1 -A`]
THEN REPEAT STRIP_TAC
THEN REWRITE_TAC[CONVEX_HULL_3; IN_ELIM_THM;]
THEN EXISTS_TAC`&1 - (s:real)`
THEN EXISTS_TAC`(s:real)*(&1 - (t:real))`
THEN EXISTS_TAC`(s:real)*(t:real)`
THEN ASM_REWRITE_TAC[VECTOR_ARITH`s%((&1-t)%u+ t%w)= (s*(&1-t))%u+ (s*t)%w:real^3`]
THEN STRIP_TAC
THENL[MATCH_MP_TAC REAL_LE_MUL
THEN ASM_TAC
THEN REAL_ARITH_TAC;
STRIP_TAC
THENL[MATCH_MP_TAC REAL_LE_MUL
THEN ASM_TAC
THEN REAL_ARITH_TAC;
REAL_ARITH_TAC]]);;

let expansion1_convex_fan=prove(`!(v:real^3) (u:real^3) s:real.
&0 <=s /\ s <= &1 
==> (&1-s)%v+s%u IN convex hull{v,u}`,
REPEAT STRIP_TAC
THEN MP_TAC(ISPECL[`(v:real^3)`;` (u:real^3)`;` (u:real^3)`;` &0`;`s:real`]expansion_convex_fan)
THEN ASM_REWRITE_TAC[SET_RULE`{A,B,B}={A,B}`]
THEN REDUCE_ARITH_TAC
THEN REDUCE_VECTOR_TAC
THEN DISCH_TAC
THEN POP_ASSUM MATCH_MP_TAC
THEN REAL_ARITH_TAC);;


(* ========================================================================== *)
(*       CROSS_DOT       (^_^)                 *)
(* ========================================================================== *)


let JBDNJJB=prove(`!u:real^3 v:real^3 w:real^3.
~collinear {vec 0, u, v} /\ ~collinear {vec 0, u, w}
==>
?t:real. &0< t /\ sin(azim (vec 0) u v w)=t *(u cross v) dot w`,

REPEAT STRIP_TAC
THEN MRESA_TAC th3[`((vec 0):real^3)`;` (u:real^3)`;` (v:real^3)`]
THEN MRESA_TAC properties_coordinate[`((vec 0):real^3)`;` (u:real^3)`;` (v:real^3)`]
THEN MRESA_TAC azim[`((vec 0):real^3)`;` (u:real^3)`;` (v:real^3)`;`(w:real^3)`]
THEN POP_ASSUM (fun th->MRESA_TAC th [`e1_fan ((vec 0):real^3) (u:real^3) (v:real^3)`;`e2_fan ((vec 0):real^3) (u:real^3) (v:real^3)`;`e3_fan ((vec 0):real^3) (u:real^3) (v:real^3)`])
THEN POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC
THEN DISCH_THEN (LABEL_TAC"YEU EM")
THEN DISCH_TAC THEN DISCH_TAC
THEN MRESA_TAC sincos1_of_u_fan[`((vec 0):real^3)`;` (u:real^3)`;` (v:real^3)`;`r1:real`; `psi:real`; `h1:real`]
THEN REMOVE_THEN "YEU EM" MP_TAC
THEN ASM_REWRITE_TAC[COS_ADD;SIN_ADD;]
THEN REDUCE_ARITH_TAC
THEN REDUCE_VECTOR_TAC
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`w =
      (r2 * cos (azim (vec 0) u v w)) % e1_fan (vec 0) u v +
      (r2 * sin (azim (vec 0) u v w)) % e2_fan (vec 0) u v +
      h2 % u ==>
(u cross v) dot w =
(u cross v) dot ((r2 * cos (azim (vec 0) u v w)) % e1_fan (vec 0) u v +
      (r2 * sin (azim (vec 0) u v w)) % e2_fan (vec 0) u v +
      h2 % u)`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN REWRITE_TAC[DOT_LADD; DOT_RADD; DOT_LMUL; DOT_RMUL;DOT_CROSS_SELF; e2_fan;e1_fan;e3_fan;
VECTOR_ARITH`A- vec 0= A`;CROSS_LADD; CROSS_RADD; CROSS_LMUL; CROSS_RMUL;CROSS_REFL;CROSS_RNEG;CROSS_LNEG;]
THEN REDUCE_ARITH_TAC
THEN REWRITE_TAC[NORM_MUL;REAL_INV_MUL; REAL_ABS_INV;REAL_INV_INV;REAL_ABS_NORM;DOT_SQUARE_NORM
;REAL_ARITH`(r2 * sin (azim (vec 0) u v w)) *
 (norm u * inv (norm (u cross v))) *
 inv (norm u) *
 norm (u cross v) pow 2 =(r2* norm(u cross v)) * sin (azim (vec 0) u v w) *
 ( inv (norm u) * norm u)*
( inv (norm (u cross v))* norm (u cross (v:real^3)))`
]
THEN MP_TAC(ISPECL[`u:real^3`;`(vec 0) :real^3`]imp_norm_not_zero_fan)
THEN REDUCE_VECTOR_TAC
THEN RESA_TAC
THEN MP_TAC(ISPEC`(norm(u:real^3))`REAL_MUL_LINV)
THEN RESA_TAC
THEN ASSUME_TAC(ISPEC`u:real^3`NORM_POS_LE)
THEN MP_TAC(REAL_ARITH`~(&0 =norm(u:real^3)) /\ &0 <= norm(u:real^3)==> &0 <norm(u:real^3)`)
THEN RESA_TAC
THEN SUBGOAL_THEN`~(u cross v = vec 0)` ASSUME_TAC
THENL[ASM_REWRITE_TAC[CROSS_EQ_0];

MP_TAC(ISPECL[`u cross v :real^3`;`(vec 0) :real^3`]imp_norm_not_zero_fan)
THEN REDUCE_VECTOR_TAC
THEN RESA_TAC
THEN MP_TAC(ISPEC`(norm(u cross v:real^3))`REAL_MUL_LINV)
THEN RESA_TAC
THEN ASSUME_TAC(ISPEC`u cross v:real^3`NORM_POS_LE)
THEN MP_TAC(REAL_ARITH`~(&0 =norm(u cross v:real^3)) /\ &0 <= norm(u cross v:real^3)==> &0 <norm(u cross v:real^3)`)
THEN RESA_TAC
THEN MRESA_TAC REAL_LT_MUL[`r2:real`;`norm(u cross v:real^3)`]
THEN MP_TAC(REAL_ARITH`&0<(r2:real)*norm(u cross v:real^3)==> ~((r2:real)*norm(u cross v:real^3)= &0)`)
THEN REDUCE_VECTOR_TAC
THEN RESA_TAC
THEN MP_TAC(ISPEC`(r2 * norm(u cross v:real^3))`REAL_MUL_LINV)
THEN RESA_TAC
THEN MP_TAC(ISPEC`(r2 * norm(u cross v:real^3))`REAL_LT_INV)
THEN RESA_TAC
THEN REDUCE_ARITH_TAC
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`(u cross v) dot w = (r2 * norm (u cross v)) * sin (azim (vec 0) u v w) ==>
inv (r2 * norm (u cross v))*(r2 * norm (u cross v)) * sin (azim (vec 0) u v w)= inv (r2 * norm (u cross v)) *((u cross v) dot w)`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN ASM_REWRITE_TAC[REAL_ARITH`inv (r2 * norm (u cross v)) * (r2 * norm (u cross v)) *
 sin (azim (vec 0) u v w)=(inv (r2 * norm (u cross v)) * (r2 * norm (u cross v)))*
 sin (azim (vec 0) u v w)`]
THEN REDUCE_ARITH_TAC
THEN STRIP_TAC
THEN EXISTS_TAC`inv (r2 * norm (u cross v)):real`
THEN ASM_REWRITE_TAC[]]);;





let cross_dot_fully_surrounded_fan=prove(`!x:real^3 v1:real^3 u1:real^3 v:real^3.
~collinear{x,v1,u1}
/\ ~collinear{x,v1,v}
/\ &0< azim x v1 v u1
/\  azim x v1 v u1 < pi
==> &0 < ((v1 - x) cross (v - x)) dot (u1 - x)`,

REPEAT STRIP_TAC
THEN MRESA1_TAC SIN_POS_PI`azim x v1 v (u1:real^3)`
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC AZIM_TRANSLATION[`-- x:real^3`;`x:real^3`;`v1:real^3`;` v:real^3`;`u1:real^3`]
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`x-x= vec 0`;VECTOR_ARITH`(-- X)+A= A-X:real^3`]
THEN DISCH_TAC
THEN POP_ASSUM (fun th-> REWRITE_TAC[SYM(th)])
THEN MRESA_TAC JBDNJJB[`(v1-x):real^3`;`v-x:real^3`;`u1-x:real^3`]
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[GSYM COLLINEAR_3;]
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={B,A,C}`]
THEN RESA_TAC
THEN MRESAL_TAC REAL_LT_LMUL_EQ [` &0:real `;`(((v1 - x) cross (v - x)) dot ((u1 - x):real^3)):real`;`t:real`][REAL_ARITH`a * &0 = &0`]);;

let cross_dot_fully_surrounded_ge_fan=prove(`!x:real^3 v1:real^3 u1:real^3 v:real^3.
~collinear{x,v1,u1}
/\ ~collinear{x,v1,v}
/\ &0<= azim x v1 v u1
/\  azim x v1 v u1 <= pi
==> &0 <= ((v1 - x) cross (v - x)) dot (u1 - x)`,
REPEAT STRIP_TAC
THEN MRESA1_TAC SIN_POS_PI_LE`azim x v1 v (u1:real^3)`
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC AZIM_TRANSLATION[`-- x:real^3`;`x:real^3`;`v1:real^3`;` v:real^3`;`u1:real^3`]
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`x-x= vec 0`;VECTOR_ARITH`(-- X)+A= A-X:real^3`]
THEN DISCH_TAC
THEN POP_ASSUM (fun th-> REWRITE_TAC[SYM(th)])
THEN MRESA_TAC JBDNJJB[`(v1-x):real^3`;`v-x:real^3`;`u1-x:real^3`]
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[GSYM COLLINEAR_3;]
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={B,A,C}`]
THEN RESA_TAC
THEN MRESAL_TAC REAL_LE_LMUL_EQ [` &0:real `;`(((v1 - x) cross (v - x)) dot ((u1 - x):real^3)):real`;`t:real`][REAL_ARITH`a * &0 = &0`]);;

let coplanar_is_cross_fan=prove(`!x:real^3 v:real^3 u:real^3 v1:real^3.
~collinear{x,v,u}
/\ v1 IN aff_gt {x} {v,u}
==> ((v-x) cross (u-x)) dot (v1-x)= &0`,
REPEAT STRIP_TAC
THEN MRESA_TAC properties_of_coplanar[`x:real^3`;`v:real^3`;`u:real^3`;`v1:real^3`]
THEN ONCE_REWRITE_TAC[DOT_SYM;]
THEN REWRITE_TAC[DOT_CROSS_DET]
THEN ONCE_REWRITE_TAC[GSYM COPLANAR_DET_EQ_0]
THEN ASM_REWRITE_TAC[]);;





let cut_inside_fan=prove(`!x:real^3 v:real^3 u:real^3 w:real^3 w1:real^3.
~collinear {x,v,w1} /\ ~collinear {x,u,w} /\  ~collinear {x,v,u}
/\ ~collinear {x,v,w}
/\ &0< azim x u w v /\ azim x u w v < pi 
/\ &0< azim x v u w1 /\ azim x v u w1 < pi 
/\ &0< azim x v w1 w /\ azim x v w1 w < pi 
==> ~(aff_ge {x,v} {w1} INTER aff_gt {x} {u,w:real^3}={})`,
REWRITE_TAC[SET_RULE`~(A={})<=> ?x. x IN A`;IN_ELIM_THM; INTER] 
THEN REPEAT STRIP_TAC
THEN MRESA_TAC th3[`x:real^3`;`v:real^3`;`w1:real^3`]
THEN MRESA_TAC th3[`x:real^3`;`u:real^3`;`w:real^3`]
THEN MRESAL_TAC  AFF_GE_2_1[`x:real^3`;`v:real^3`;`w1:real^3`][IN_ELIM_THM]
THEN MRESAL_TAC  AFF_GT_1_2[`x:real^3`;`u:real^3`;`w:real^3`][IN_ELIM_THM]
THEN MRESA_TAC cross_dot_fully_surrounded_fan[`x:real^3`;`u:real^3`;`v:real^3`;`w:real^3`]
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN (LABEL_TAC"CON")
THEN MRESA_TAC cross_dot_fully_surrounded_fan[`x:real^3`;`v:real^3`;`w1:real^3`;`u:real^3`]
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN (LABEL_TAC"CON BE")
THEN MRESA_TAC cross_dot_fully_surrounded_fan[`x:real^3`;`v:real^3`;`w:real^3`;`w1:real^3`]
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN (LABEL_TAC"CON EM")
THEN ABBREV_TAC`a1=(v-x):real^3`
THEN ABBREV_TAC`a2=(w1-x):real^3`
THEN ABBREV_TAC`a3=(w-x) :real^3`
THEN ABBREV_TAC`a4=(u-x):real^3`
THEN ABBREV_TAC`va=a1 cross a2:real^3`
THEN ABBREV_TAC`vb=a3 cross a4:real^3`
THEN EXISTS_TAC`(vb:real^3) cross (va:real^3)+(x:real^3)`
THEN STRIP_TAC
THENL(*2*)[
EXISTS_TAC`&1-(vb:real^3) dot (a2:real^3)+ vb dot (a1:real^3)`
THEN EXISTS_TAC`(vb:real^3) dot (a2:real^3)`
THEN EXISTS_TAC`--((vb:real^3) dot (a1:real^3))`
THEN ASM_REWRITE_TAC[REAL_ARITH`(&1 - vb dot a2 + vb dot a1) + vb dot a2 + --(vb dot a1) = &1`]
THEN SUBGOAL_THEN `&0<= --((vb:real^3) dot (a1:real^3))` ASSUME_TAC
THENL(*3*)[
EXPAND_TAC"vb"
THEN ONCE_REWRITE_TAC[CROSS_SKEW;]
THEN REWRITE_TAC[DOT_LNEG]
THEN REMOVE_THEN "CON"MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={A,C,B}`]
THEN ASM_REWRITE_TAC[] 
THEN REAL_ARITH_TAC;
ASM_REWRITE_TAC[]
THEN EXPAND_TAC"va"
THEN REWRITE_TAC[CROSS_LAGRANGE;VECTOR_MUL_LNEG]
THEN EXPAND_TAC"a1"
THEN EXPAND_TAC"a2"
THEN VECTOR_ARITH_TAC];(*2*)
ONCE_REWRITE_TAC[CROSS_SKEW]
THEN EXPAND_TAC"vb"
THEN REWRITE_TAC[CROSS_LAGRANGE;]
THEN EXISTS_TAC`&1+(va:real^3) dot (a4:real^3)- va dot (a3:real^3)`
THEN EXISTS_TAC`((va:real^3) dot (a3:real^3))`
THEN EXISTS_TAC`--(va:real^3) dot (a4:real^3)`
THEN ASM_REWRITE_TAC[DOT_LNEG;VECTOR_MUL_LNEG;REAL_ARITH`(&1 + va dot a4 - va dot a3) + va dot a3 + --(va dot a4) = &1`;]
THEN STRIP_TAC
THENL(*3*)[
EXPAND_TAC"va"
THEN ONCE_REWRITE_TAC[CROSS_SKEW;]
THEN REWRITE_TAC[DOT_LNEG;CROSS_TRIPLE]
THEN ASM_REWRITE_TAC[] 
THEN ASM_TAC 
THEN REAL_ARITH_TAC;
EXPAND_TAC"a3"
THEN EXPAND_TAC"a4"
THEN REWRITE_TAC[VECTOR_ARITH`(&1+A-B)%X+B%U+ --(A%V)=X-(A%(V-X)-B%(U-X))`]
THEN VECTOR_ARITH_TAC]]);;



let exists_cut_in_edge_fan=prove(`!x:real^3 v:real^3 u:real^3 w:real^3 w1:real^3.
~collinear {x,v,w1} /\ ~collinear {x,u,w} /\  ~collinear {x,v,u}
/\ ~collinear {x,v,w}
/\ &0< azim x u w v /\ azim x u w v < pi 
/\ &0< azim x v u w1 /\ azim x v u w1 < pi 
/\ &0< azim x v w1 w /\ azim x v w1 w < pi 
==> ?a. &0< a /\ a< &1
/\ (&1-a) %u + a % w IN aff_ge {x,v} {w1:real^3}`,

REPEAT STRIP_TAC 
THEN MRESA_TAC th3[`x:real^3`;`v:real^3`;`w1:real^3`]
THEN MRESA_TAC th3[`x:real^3`;`u:real^3`;`w:real^3`]
THEN MRESAL_TAC  AFF_GE_2_1[`x:real^3`;`v:real^3`;`w1:real^3`][IN_ELIM_THM]
THEN MRESAL_TAC  AFF_GT_1_2[`x:real^3`;`u:real^3`;`w:real^3`][IN_ELIM_THM]
THEN MRESA_TAC cut_inside_fan[`x:real^3`;`v:real^3`;`u:real^3`;`w:real^3`;`w1:real^3`]
THEN POP_ASSUM MP_TAC
THEN GEN_REWRITE_TAC(LAND_CONV o DEPTH_CONV)[SET_RULE`~(A={})<=> ?x. x IN A`;INTER;]
THEN REWRITE_TAC[IN_ELIM_THM]
THEN STRIP_TAC
THEN EXISTS_TAC`inv(&1-t1')*t3'`
THEN MP_TAC(REAL_ARITH`&0< t2' /\ &0< t3' /\ t1'+t2'+t3'= &1==>  &0 < &1- t1' /\ ~(&1- t1' = &0)/\ t2'+t3'= &1- t1'`)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN MP_TAC(ISPEC`&1- (t1':real)`REAL_LT_INV)
THEN RESA_TAC
THEN MP_TAC(ISPEC`&1- (t1':real)`REAL_MUL_LINV)
THEN RESA_TAC
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`
t2' + t3' = &1 - t1':real
==> (inv ( &1 - t1'))*(t2' + t3') = (inv ( &1 - t1'))*( &1 - t1':real)
`)
THEN POP_ASSUM (fun th->GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[th])
THEN ASM_REWRITE_TAC[REAL_ARITH`A*(B+C)= &1<=>  &1 -A*C=A*B`;REAL_ARITH`A< &1 <=> &0< &1-A`]
THEN STRIP_TAC
THEN MRESA_TAC REAL_LT_MUL[`inv (&1- t1'):real`;`t2':real`;]
THEN MRESA_TAC REAL_LT_MUL[`inv (&1- t1'):real`;`t3':real`;]
THEN REWRITE_TAC[VECTOR_ARITH`(A*B)%X+(A*C)%Y=A%(B%X+C%Y)`; VECTOR_ARITH`A%(t2' % u + t3' % w)= A%((t1'%x +t2' % u + t3' % w) - t1' %x) :real^3`]
THEN FIND_ASSUM MP_TAC`x' = t1' % x + t2' % u + t3' % w:real^3`
THEN DISCH_TAC
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM(th)])
THEN ASM_REWRITE_TAC[VECTOR_ARITH`(t1 % x + t2 % v + t3 % w1) - t1' % x=(t1-t1') % x + t2 % v + t3 % w1`;
VECTOR_ARITH`A%(B+C+D)=A%B+A%C+A%D`;VECTOR_ARITH`A%B%C=(A*B)%C`]
THEN EXISTS_TAC`(inv (&1 - t1') * (t1 - t1')):real`
THEN EXISTS_TAC`(inv (&1 - t1') * t2):real`
THEN EXISTS_TAC`(inv (&1 - t1') * t3):real`
THEN ASM_REWRITE_TAC[REAL_ARITH`inv (&1 - t1') * (t1 - t1') + inv (&1 - t1') * t2 + inv (&1 - t1') * t3
=inv (&1 - t1') * ((t1 +t2 + t3)-t1')`]
THEN MATCH_MP_TAC REAL_LE_MUL
THEN ASM_REWRITE_TAC[]
THEN ASM_TAC THEN REAL_ARITH_TAC);;


let properties_of_fully_surrounded1_fan=prove(`!x:real^3 v:real^3 u:real^3 w:real^3 w1:real^3.
~coplanar {x,v,u,w}/\ &0< azim x u w v /\ azim x u w v < pi 
==> &0 < azim x v u w /\ azim x v u w < pi`,
REPEAT STRIP_TAC
THEN MRESA_TAC azim[`(x:real^3)`;` (v:real^3)`;` (u:real^3)`;`(w:real^3)`]
THEN MRESA_TAC notcoplanar_imp_notcollinear_fan[`x:real^3`;`v:real^3`;`u:real^3`;`w:real^3`]
THENL[
 MP_TAC(REAL_ARITH` &0<= azim x v u (w:real^3) ==> azim x v u w = &0 \/ &0< azim x v u w`)
THEN RESA_TAC
THEN MRESA_TAC AZIM_EQ_0_PI_IMP_COPLANAR[`x:real^3`;`v:real^3`;`u:real^3`;`w:real^3`];

MP_TAC(REAL_ARITH` (azim x v u w = pi) \/ (pi < azim x v u w) \/  azim x v u w< pi`)
THEN RESA_TAC
THENL[
MRESA_TAC AZIM_EQ_0_PI_IMP_COPLANAR[`x:real^3`;`v:real^3`;`u:real^3`;`w:real^3`];

MP_TAC(REAL_ARITH`pi< azim x v u w /\ azim x v u w < &2 * pi ==> &0< azim x v u w - pi /\ azim x v u w - pi< pi`)
THEN RESA_TAC
THEN MRESAL1_TAC SIN_POS_PI`azim x v u (w:real^3) -pi`[SIN_SUB; SIN_PI; COS_PI;REAL_ARITH`&0< A * -- &1 -B * &0 <=> A < &0`]
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC AZIM_TRANSLATION[`-- x:real^3`;`x:real^3`;`v:real^3`;` u:real^3`;`w:real^3`]
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`x-x= vec 0`;VECTOR_ARITH`(-- X)+A= A-X:real^3`]
THEN DISCH_TAC
THEN MRESA_TAC JBDNJJB[`(v-x):real^3`;`u-x:real^3`;`w-x:real^3`]
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[GSYM COLLINEAR_3;]
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={B,A,C}`]
THEN RESA_TAC
THEN STRIP_TAC
THEN MRESAL_TAC REAL_LT_LMUL_EQ[`(((v - x) cross (u - x)) dot (w - x:real^3))`;`&0`;`t:real`][REAL_ARITH`A * &0= &0`]
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[DOT_LNEG;CROSS_TRIPLE]
THEN MRESA1_TAC SIN_POS_PI`azim x u  (w:real^3) v`
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC AZIM_TRANSLATION[`-- x:real^3`;`x:real^3`;`u:real^3`;` w:real^3`;`v:real^3`]
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`x-x= vec 0`;VECTOR_ARITH`(-- X)+A= A-X:real^3`]
THEN DISCH_TAC
THEN MRESA_TAC JBDNJJB[`(u-x):real^3`;`w-x:real^3`;`v-x:real^3`]
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[GSYM COLLINEAR_3;]
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={B,A,C}`]
THEN ASM_REWRITE_TAC[]
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={A,C,B}`]
THEN RESA_TAC
THEN STRIP_TAC
THEN MRESAL_TAC REAL_LT_LMUL_EQ[`&0`;`(((u - x) cross (w - x)) dot (v - x:real^3))`;`t':real`][REAL_ARITH`A * &0= &0`]
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC]])
;;



let inequality4_aim_in_convex_fan=prove(`!x:real^3 v:real^3 u:real^3 w:real^3 a:real.
~coplanar {x,v,u,w}/\ &0< azim x u w v /\ azim x u w v < pi 
/\ &0< a /\ a < &1
==> 
&0< azim x v u ((&1 - a) % u + a % w)
/\ azim x v u ((&1 - a) % u + a % w)< azim x v u w `,

REPEAT STRIP_TAC
THEN MRESA_TAC notcoplanar_imp_notcollinear_fan[`x:real^3`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN MRESA_TAC th3[`x:real^3`;`v:real^3`;`u:real^3`]
THEN MRESA_TAC th3[`x:real^3`;`u:real^3`;`w:real^3`]
THEN MRESA_TAC th3[`x:real^3`;`v:real^3`;`w:real^3`]
THEN MRESA_TAC properties_of_fully_surrounded1_fan[`x:real^3`;`v:real^3`;`u:real^3`;`w:real^3`;`w1:real^3`]
THEN MRESA_TAC WEDGE_LUNE_GT[`x:real^3`;` v:real^3`;`u:real^3`;`w:real^3`]
THEN POP_ASSUM (fun th-> MP_TAC(SYM(th)))
THEN DISCH_TAC
THEN MRESA_TAC in_aff_2_2_fan[`x:real^3`;` v:real^3`;`u:real^3`;`w:real^3`]
THEN POP_ASSUM(fun th-> MRESA1_TAC th `a:real`)
THEN POP_ASSUM(fun th-> MRESAL_TAC th [`&0:real`;`&0`;`&1`][REAL_ARITH`&0< &1/\ &0+ &0 + &1 = &1`;wedge;IN_ELIM_THM])
THEN POP_ASSUM MP_TAC  THEN POP_ASSUM MP_TAC  THEN REDUCE_VECTOR_TAC
THEN DISCH_TAC THEN DISCH_TAC
THEN ASM_REWRITE_TAC[]);;




let cut_in_angle_fan=prove(`!x:real^3 v:real^3 u:real^3 w:real^3 y:real^3.
 ~coplanar {x,v,u,w} /\ ~collinear {x,u,y}
/\ &0< azim x u w v /\ azim x u w v< pi
/\ azim x u w y< azim x u w v /\ &0< azim x u w y
==> let a1=(v-x):real^3 in
    let a2=w-x:real^3 in
    let a3=(y-x):real^3 in
    let a4=(u-x) :real^3 in
        let va=a1 cross a2:real^3 in
    let vb=a3 cross a4:real^3 in
    let v3= (vb:real^3) cross (va:real^3)+(x:real^3)
in v3 IN aff_gt {x} {v,w:real^3}`,

REPEAT STRIP_TAC
THEN CONV_TAC(TOP_DEPTH_CONV let_CONV) 
THEN ABBREV_TAC`a1=(v-x):real^3`
THEN ABBREV_TAC`a2=(y-x):real^3`
THEN ABBREV_TAC`a3=(u-x) :real^3`
THEN ABBREV_TAC`a4=w-x:real^3`
THEN ABBREV_TAC`va=a1 cross a4:real^3`
THEN ABBREV_TAC`vb=a2 cross a3:real^3`
THEN ABBREV_TAC`v3= (vb:real^3) cross (va:real^3)+(x:real^3)`
THEN MRESA_TAC notcoplanar_imp_notcollinear_fan[`x:real^3`;`v:real^3`;`u:real^3`;`w:real^3`]
THEN MRESA_TAC th3[`(x:real^3)` ;` (v:real^3)`;`(w:real^3) `;]
THEN MRESAL_TAC  AFF_GT_1_2[`x:real^3`;`v:real^3`;`w:real^3`][IN_ELIM_THM]
THEN EXISTS_TAC`&1-(vb:real^3) dot (a4:real^3)+vb dot (a1:real^3)`
THEN EXISTS_TAC`((vb:real^3) dot (a4:real^3))`
THEN EXISTS_TAC`(--((vb:real^3) dot (a1:real^3)))`
THEN ASM_REWRITE_TAC[REAL_ARITH`(&1 - vb dot a4 + vb dot a1) + (vb dot a4) + --(vb dot a1) = &1`;VECTOR_ARITH`(&1-A+B)%X+ (A)%U+ (--B) %V=A%(U-X)- B%(V-X)+X`]
THEN EXPAND_TAC"v3"
THEN EXPAND_TAC"va" 
THEN REWRITE_TAC[CROSS_LAGRANGE;VECTOR_ARITH`--(A-B)+C=B-A+C:real^3`]
THEN MP_TAC(REAL_ARITH`azim x u w v< pi
/\ azim x u w y< azim x u w v==>azim x u w (y:real^3)< pi`)
THEN RESA_TAC
THEN MRESA_TAC cross_dot_fully_surrounded_fan[`x:real^3`;`u:real^3`;`y:real^3`;`w:real^3`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[CROSS_SKEW; ]
THEN ASM_REWRITE_TAC[DOT_LNEG] 
THEN ONCE_REWRITE_TAC[CROSS_TRIPLE;] THEN ONCE_REWRITE_TAC[CROSS_SKEW;]
THEN ASM_REWRITE_TAC[DOT_LNEG;REAL_ARITH`--(--A)=A`]
THEN DISCH_TAC
THEN MP_TAC(REAL_ARITH`azim x u w y< azim x u w v==> azim x u w y<= azim x u w (v:real^3)`)
THEN RESA_TAC
THEN MRESA_TAC sum4_azim_fan[`x:real^3`;`u:real^3`;`w:real^3`;`y:real^3`;`v:real^3`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={A,C,B}`]
THEN RESA_TAC
THEN MP_TAC(REAL_ARITH`azim x u w y< azim x u w v/\ &0< azim x u w y/\ azim x u w v < pi /\ azim x u w v = azim x u w y + azim x u y v==>  ~(azim x u y (v:real^3)= &0)/\ &0< azim x u y v/\ azim x u y v < pi/\ ~(azim x u y (v:real^3)= pi)`)
THEN RESA_TAC
THEN MRESA_TAC cross_dot_fully_surrounded_fan[`x:real^3`;`u:real^3`;`v:real^3`;`y:real^3`]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={A,C,B}`]
THEN ONCE_REWRITE_TAC[CROSS_SKEW; ]
THEN ASM_REWRITE_TAC[DOT_LNEG] );;


(* ========================================================================== *)
(*            GRAPH    (^_^)                 *)
(* ========================================================================== *)









let GRAPH = prove
 (`!E. graph E <=> !e. e IN E ==> e HAS_SIZE 2`,
  REWRITE_TAC[graph; IN]);;



let CARD_2_FAN=prove(`!v:A w:A. ~(v=w) 
==> CARD {v,w}=2`,
REPEAT STRIP_TAC
THEN SUBGOAL_THEN`FINITE {v,w:A}`ASSUME_TAC
THENL[      SIMP_TAC[HAS_SIZE; CARD_CLAUSES; FINITE_INSERT; FINITE_EMPTY;
                 IN_INSERT; NOT_IN_EMPTY];

ASSUME_TAC(SET_RULE `v:A IN {v:A,w:A} `)
THEN MP_TAC(ISPECL[`v:A`;`{v:A,w:A}`;]CARD_DELETE)
THEN RESA_TAC
THEN MP_TAC(SET_RULE `v IN {v,w}==>{v:A,w:A} DELETE v PSUBSET {v,w}`)
THEN RESA_TAC
THEN MP_TAC(ISPECL[`{v:A,w:A} DELETE v`;`{v:A,w:A}`]CARD_PSUBSET)
THEN POP_ASSUM (fun th->REWRITE_TAC[th])
THEN FIND_ASSUM MP_TAC`FINITE {v:A,w:A}`
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:A, w:A}`)
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:A IN ({v:A,w:A} DELETE v)` ASSUME_TAC
THENL[
ASM_SET_TAC[];
MP_TAC(ISPECL[`{v:A,w:A}`;`v:A`] FINITE_DELETE)
THEN RESA_TAC
THEN MP_TAC(ISPECL[`w:A`;`{v:A,w:A} DELETE v`;]CARD_DELETE)
THEN RESA_TAC
THEN MP_TAC(SET_RULE `w IN ({v,w} DELETE v)==>{v:A,w:A} DELETE v DELETE w PSUBSET {v,w} DELETE v`)
THEN RESA_TAC
THEN MP_TAC(ISPECL[`{v:A,w:A} DELETE v DELETE w`;`{v:A,w:A} DELETE v`]CARD_PSUBSET)
THEN POP_ASSUM (fun th->REWRITE_TAC[th])
THEN FIND_ASSUM MP_TAC`FINITE ({v:A,w:A} 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:A, w:A} 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:A DELETE w:A={}`)
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 SET_TAC[]]]);; 



(* ========================================================================== *)
(*            CONDITION OF CROSS DOT 4 POINT    (^_^)                 *)
(* ========================================================================== *)


let condition_cross_dot_4point=prove(`!x:real^3 y:real^3 z:real^3 v:real^3 u:real^3.
              let a1 = y - x in
              let a2 = z - x in
              let a3 = v - x in
              let a4 = u - x in
              let va = a1 cross a2 in
              let vb = a3 cross a4 in
              let v3 = va cross vb + x in 
 ~collinear {x,v,u}
/\ &0<(a1 cross a2) dot a4 /\  &0 < --((a1 cross a2) dot a3)
==> v3 IN aff_gt {x} {v,u}`,
REPEAT GEN_TAC
THEN CONV_TAC(TOP_DEPTH_CONV let_CONV) 
THEN ABBREV_TAC`a1=(y-x):real^3`
THEN ABBREV_TAC`a2=(z-x):real^3`
THEN ABBREV_TAC`a3=(v-x) :real^3`
THEN ABBREV_TAC`a4=u-x:real^3`
THEN ABBREV_TAC`va=a1 cross a2:real^3`
THEN ABBREV_TAC`vb=a3 cross a4:real^3`
THEN ABBREV_TAC`v3= (va:real^3) cross (vb:real^3)+(x:real^3)`
THEN REPEAT STRIP_TAC
THEN MRESA_TAC th3[`(x:real^3)` ;` (v:real^3)`;`(u:real^3) `;]
THEN MRESAL_TAC  AFF_GT_1_2[`x:real^3`;`v:real^3`;`u:real^3`][IN_ELIM_THM]
THEN EXISTS_TAC`&1-(va:real^3) dot (a4:real^3)+va dot (a3:real^3)`
THEN EXISTS_TAC`((va:real^3) dot (a4:real^3))`
THEN EXISTS_TAC`(--((va:real^3) dot (a3:real^3)))`
THEN ASM_REWRITE_TAC[REAL_ARITH` (&1 - va dot a4 + va dot a3) + (va dot a4) + --(va dot a3) = &1`]
THEN EXPAND_TAC"v3"
THEN EXPAND_TAC"vb" 
THEN REWRITE_TAC[CROSS_LAGRANGE;VECTOR_ARITH`A+ B + --U%C=A +B-U%C:real^3`]
THEN EXPAND_TAC"a3" 
THEN EXPAND_TAC"a4" 
THEN VECTOR_ARITH_TAC);;



let aff_gt_2_1_crossr_dot_4point=prove(`!x:real^3 y:real^3 z:real^3 v:real^3 u:real^3.
              let a1 = y - x in
              let a2 = z - x in
              let a3 = v - x in
              let a4 = u - x in
 ~collinear {x,y,z}
/\ u IN aff_gt {x,y} {z}
/\ &0<(a1 cross a2) dot a3 
==> &0<(a1 cross a4) dot a3 `,
REPEAT GEN_TAC
THEN CONV_TAC(TOP_DEPTH_CONV let_CONV) 
THEN REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC th3[`x:real^3`;`y:real^3`;`z:real^3`]
THEN MRESAL_TAC AFF_GT_2_1[`x:real^3`;`y:real^3`;`z:real^3`][IN_ELIM_THM]
THEN REPEAT STRIP_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`(u' % x + v' % y + w % z) - x=((u'+v'+w) - &1) % x + v' % (y-x) + w % (z - x)`;REAL_ARITH`&1- &1= &0`]
THEN REDUCE_VECTOR_TAC
THEN REWRITE_TAC[CROSS_RMUL;CROSS_RADD;CROSS_REFL;]
THEN REDUCE_VECTOR_TAC
THEN REWRITE_TAC[DOT_LMUL]
THEN MATCH_MP_TAC REAL_LT_MUL
THEN ASM_REWRITE_TAC[]);;

let aff_gt_2_1_rcross_dot_4pointl=prove(`!x:real^3 y:real^3 z:real^3 v:real^3 u:real^3.
              let a1 = y - x in
              let a2 = z - x in
              let a3 = v - x in
              let a4 = u - x in
 ~collinear {x,y,z}
/\ u IN aff_gt {x,z} {y}
/\ &0<(a1 cross a2) dot a3 
==> &0<(a4 cross a2) dot a3 `,
REPEAT GEN_TAC
THEN CONV_TAC(TOP_DEPTH_CONV let_CONV) 
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={A,C,B}`]
THEN REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC th3[`x:real^3`;`z:real^3`;`y:real^3`]
THEN MRESAL_TAC AFF_GT_2_1[`x:real^3`;`z:real^3`;`y:real^3`][IN_ELIM_THM]
THEN REPEAT STRIP_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`(u' % x + v' % z + w % y) - x=((u'+v'+w) - &1) % x + v' % (z-x) + w % (y - x)`;REAL_ARITH`&1- &1= &0`]
THEN REDUCE_VECTOR_TAC
THEN REWRITE_TAC[CROSS_LMUL;CROSS_LADD;CROSS_REFL;]
THEN REDUCE_VECTOR_TAC
THEN REWRITE_TAC[DOT_LMUL]
THEN MATCH_MP_TAC REAL_LT_MUL
THEN ASM_REWRITE_TAC[]);;



let aff_gt_2_1_cross_dotr_4point=prove(`!x:real^3 y:real^3 z:real^3 v:real^3 u:real^3.
              let a1 = y - x in
              let a2 = z - x in
              let a3 = v - x in
              let a4 = u - x in
 ~collinear {x,y,v}
/\ u IN aff_gt {x,y} {v}
/\ &0<(a1 cross a2) dot a3 
==> &0<(a1 cross a2) dot a4 `,
REPEAT GEN_TAC
THEN CONV_TAC(TOP_DEPTH_CONV let_CONV) 
THEN REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC th3[`x:real^3`;`y:real^3`;`v:real^3`]
THEN MRESAL_TAC AFF_GT_2_1[`x:real^3`;`y:real^3`;`v:real^3`][IN_ELIM_THM]
THEN REPEAT STRIP_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`(u' % x + v' % z + w % y) - x=((u'+v'+w) - &1) % x + v' % (z-x) + w % (y - x)`;REAL_ARITH`&1- &1= &0`]
THEN REDUCE_VECTOR_TAC
THEN REWRITE_TAC[DOT_RMUL;DOT_RADD;DOT_CROSS_SELF]
THEN REDUCE_ARITH_TAC
THEN MATCH_MP_TAC REAL_LT_MUL
THEN ASM_REWRITE_TAC[]);;

let aff_gt_2_1_cross_dotl_4point=prove(`!x:real^3 y:real^3 z:real^3 v:real^3 u:real^3.
              let a1 = y - x in
              let a2 = z - x in
              let a3 = v - x in
              let a4 = u - x in
 ~collinear {x,z,v}
/\ u IN aff_gt {x,z} {v}
/\ &0<(a1 cross a2) dot a3 
==> &0<(a1 cross a2) dot a4 `,
REPEAT GEN_TAC
THEN CONV_TAC(TOP_DEPTH_CONV let_CONV) 
THEN REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC th3[`x:real^3`;`z:real^3`;`v:real^3`]
THEN MRESAL_TAC AFF_GT_2_1[`x:real^3`;`z:real^3`;`v:real^3`][IN_ELIM_THM]
THEN REPEAT STRIP_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`(u' % x + v' % z + w % y) - x=((u'+v'+w) - &1) % x + v' % (z-x) + w % (y - x)`;REAL_ARITH`&1- &1= &0`]
THEN REDUCE_VECTOR_TAC
THEN REWRITE_TAC[DOT_RMUL;DOT_RADD;DOT_CROSS_SELF]
THEN REDUCE_ARITH_TAC
THEN MATCH_MP_TAC REAL_LT_MUL
THEN ASM_REWRITE_TAC[]);;




let aff_gt_2_1r_rcross_dotl_4point=prove(`!x:real^3 y:real^3 z:real^3 v:real^3 u:real^3.
              let a1 = y - x in
              let a2 = z - x in
              let a3 = v - x in
              let a4 = u - x in
 ~collinear {x,v,y}
/\ u IN aff_gt {x,v} {y}
/\ &0<(a1 cross a2) dot a3 
==> &0<(a4 cross a2) dot a3 `,
REPEAT GEN_TAC
THEN CONV_TAC(TOP_DEPTH_CONV let_CONV) 
THEN REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC th3[`x:real^3`;`v:real^3`;`y:real^3`]
THEN MRESAL_TAC AFF_GT_2_1[`x:real^3`;`v:real^3`;`y:real^3`][IN_ELIM_THM]
THEN REPEAT STRIP_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`(u' % x + v' % z + w % y) - x=((u'+v'+w) - &1) % x + v' % (z-x) + w % (y - x)`;REAL_ARITH`&1- &1= &0`]
THEN REDUCE_VECTOR_TAC
THEN REWRITE_TAC[CROSS_LMUL;CROSS_LADD;DOT_LMUL;DOT_LADD;DOT_CROSS_SELF]
THEN REDUCE_ARITH_TAC
THEN MATCH_MP_TAC REAL_LT_MUL
THEN ASM_REWRITE_TAC[]);;




let aff_gt_1_2_cross_dotr_4point=prove(`!x:real^3 y:real^3 z:real^3 v:real^3 u:real^3.
              let a1 = y - x in
              let a2 = z - x in
              let a3 = v - x in
              let a4 = u - x in
 ~collinear {x,v,u}
/\ y IN aff_gt {x} {v,u}
/\ &0<(a1 cross a2) dot a3
 ==> &0< --((a1 cross a2) dot a4)`,

REPEAT GEN_TAC
THEN CONV_TAC(TOP_DEPTH_CONV let_CONV) 
THEN REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC th3[`x:real^3`;`v:real^3`;`u:real^3`]
THEN MRESAL_TAC AFF_GT_1_2[`x:real^3`;`v:real^3`;`u:real^3`][IN_ELIM_THM]
THEN STRIP_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`(u' % x + v' % z + w % y) - x=((u'+v'+w) - &1) % x + v' % (z-x) + w % (y - x)`;REAL_ARITH`&1- &1= &0`]
THEN REDUCE_VECTOR_TAC
THEN REWRITE_TAC[CROSS_LNEG;CROSS_LMUL;CROSS_LADD;CROSS_REFL;DOT_LMUL;DOT_LADD;]
THEN ONCE_REWRITE_TAC[CROSS_SKEW;CROSS_TRIPLE]
THEN REWRITE_TAC[CROSS_TRIPLE;CROSS_REFL;DOT_LZERO]
THEN REDUCE_ARITH_TAC
THEN STRIP_TAC
THEN ONCE_REWRITE_TAC[CROSS_SKEW;]
THEN REWRITE_TAC[DOT_LNEG;REAL_ARITH`--(A* (--B))=A*B`]
THEN ONCE_REWRITE_TAC[CROSS_TRIPLE;]
THEN MATCH_MP_TAC REAL_LT_MUL
THEN ASM_REWRITE_TAC[]
THEN MRESAL_TAC REAL_LT_RCANCEL_IMP[`&0`;`((u - x) cross (z - x)) dot (v - x:real^3)`;`t3:real`;][REAL_ARITH`&0 * A= &0`]
THEN POP_ASSUM MATCH_MP_TAC
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC);;




let aff_gt_1_2_cross_dotr_4point_neg=prove(`!x:real^3 y:real^3 z:real^3 v:real^3 u:real^3.
              let a1 = y - x in
              let a2 = z - x in
              let a3 = v - x in
              let a4 = u - x in
 ~collinear {x,v,u}
/\ y IN aff_gt {x} {v,u}
/\ &0< --((a1 cross a2) dot a3)
 ==> &0< ((a1 cross a2) dot a4)`,
REPEAT GEN_TAC
THEN CONV_TAC(TOP_DEPTH_CONV let_CONV) 
THEN REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC th3[`x:real^3`;`v:real^3`;`u:real^3`]
THEN MRESAL_TAC AFF_GT_1_2[`x:real^3`;`v:real^3`;`u:real^3`][IN_ELIM_THM]
THEN STRIP_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`(u' % x + v' % z + w % y) - x=((u'+v'+w) - &1) % x + v' % (z-x) + w % (y - x)`;REAL_ARITH`&1- &1= &0`]
THEN REDUCE_VECTOR_TAC
THEN REWRITE_TAC[CROSS_LNEG;CROSS_LMUL;CROSS_LADD;CROSS_REFL;DOT_LMUL;DOT_LADD;]
THEN ONCE_REWRITE_TAC[CROSS_SKEW;CROSS_TRIPLE]
THEN REWRITE_TAC[CROSS_TRIPLE;CROSS_REFL;DOT_LZERO]
THEN REDUCE_ARITH_TAC
THEN ONCE_REWRITE_TAC[CROSS_SKEW;]
THEN REWRITE_TAC[DOT_LNEG;REAL_ARITH`--(A* (--B))=A*B`]
THEN STRIP_TAC
THEN ONCE_REWRITE_TAC[CROSS_SKEW;]
THEN REWRITE_TAC[DOT_LNEG;REAL_ARITH`-- --A=A`]
THEN ONCE_REWRITE_TAC[CROSS_TRIPLE;]
THEN ONCE_REWRITE_TAC[CROSS_TRIPLE;]
THEN MATCH_MP_TAC REAL_LT_MUL
THEN ASM_REWRITE_TAC[]
THEN MRESAL_TAC REAL_LT_RCANCEL_IMP[`&0`;`((z - x) cross (u - x)) dot (v - x):real^3`;`t3:real`;][REAL_ARITH`&0 * A= &0`]
THEN POP_ASSUM MATCH_MP_TAC
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC);;


let aff_gt_1_2_cross_dotr_4point_zero=prove(`!x:real^3 y:real^3 z:real^3 v:real^3 u:real^3.
              let a1 = y - x in
              let a2 = z - x in
              let a3 = v - x in
              let a4 = u - x in
 ~collinear {x,v,u}
/\ y IN aff_gt {x} {v,u}
/\ (a1 cross a2) dot a3= &0
 ==> ((a1 cross a2) dot a4)= &0`,

REPEAT GEN_TAC
THEN CONV_TAC(TOP_DEPTH_CONV let_CONV) 
THEN REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC th3[`x:real^3`;`v:real^3`;`u:real^3`]
THEN MRESAL_TAC AFF_GT_1_2[`x:real^3`;`v:real^3`;`u:real^3`][IN_ELIM_THM]
THEN STRIP_TAC
THEN ASM_REWRITE_TAC[VECTOR_ARITH`(u' % x + v' % z + w % y) - x=((u'+v'+w) - &1) % x + v' % (z-x) + w % (y - x)`;REAL_ARITH`&1- &1= &0`]
THEN REDUCE_VECTOR_TAC
THEN REWRITE_TAC[CROSS_LNEG;CROSS_LMUL;CROSS_LADD;CROSS_REFL;DOT_LMUL;DOT_LADD;]
THEN ONCE_REWRITE_TAC[CROSS_SKEW;CROSS_TRIPLE]
THEN REWRITE_TAC[CROSS_TRIPLE;CROSS_REFL;DOT_LZERO]
THEN REDUCE_ARITH_TAC
THEN STRIP_TAC
THEN ONCE_REWRITE_TAC[CROSS_SKEW;]
THEN REWRITE_TAC[DOT_LNEG;REAL_ARITH`--(A* (--B))=A*B`]
THEN ONCE_REWRITE_TAC[CROSS_TRIPLE;]
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[REAL_ENTIRE]
THEN STRIP_TAC
THEN ASM_TAC THEN REAL_ARITH_TAC);;

let exists_esilon_real=prove(`!a:real b:real.
&0<a ==> ?t. &0< t /\ t< &1 /\ 
(!h. &0< h /\ h< t==> &0< a- h * b)`,
REPEAT STRIP_TAC
THEN DISJ_CASES_TAC(REAL_ARITH`b <= &0 \/ &0< b`)
THENL[ EXISTS_TAC`&1/ &2`
THEN REWRITE_TAC[REAL_ARITH`&0< &1/ &2 /\ &1/ &2< &1`;]
THEN REPEAT STRIP_TAC
THEN MATCH_MP_TAC(REAL_ARITH`&0<a /\ &0<= h*(-- b)==> &0< a-h*b`)
THEN ASM_REWRITE_TAC[]
THEN MATCH_MP_TAC REAL_LE_MUL
THEN ASM_REWRITE_TAC[REAL_ARITH`&0<= --B<=> B<= &0`]
THEN ASM_TAC THEN REAL_ARITH_TAC;
ABBREV_TAC`t1= (min (inv (b:real) * a) (&1)) / &2`
THEN MRESA1_TAC REAL_LT_INV`b:real`
THEN MRESA_TAC REAL_LT_MUL[`inv b:real`;`a:real`]
THEN MP_TAC(REAL_ARITH`&0 < inv b * a /\ t1= (min (inv (b:real) * a) (&1)) / &2
==> &0< t1 /\ t1< &1 /\ t1< inv b * a`)
THEN RESA_TAC
THEN EXISTS_TAC `t1:real`
THEN ASM_REWRITE_TAC[]
THEN REPEAT STRIP_TAC
THEN MP_TAC(REAL_ARITH`h<t1 /\ t1< inv b *a==> &0< inv b *a- h`)
THEN RESA_TAC
THEN MRESA_TAC REAL_LT_MUL[`b:real`;`inv b *a- h:real`]
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[REAL_ARITH`b * (inv b * a - h)= (inv b * b) * a- h *b `]
THEN MP_TAC(REAL_ARITH`&0<b==> ~(b= &0)`)
THEN RESA_TAC
THEN MRESA1_TAC REAL_MUL_LINV`b:real`
THEN REAL_ARITH_TAC]);;


let invariant_cross_dotr_esilon_3piont=prove(`!x:real^3 y:real^3 z:real^3 v:real^3 u:real^3.
              let a1 = y - x in
              let a2 = z - x in
              let a3 = v - x in
              let a4 = u - x in
 &0<(a1 cross a2) dot a3
 ==>
?t. &0< t /\ t< &1 /\ 
(!h. &0< h /\ h< t==>
 &0< ((a1 cross a2) dot ((&1 - h) % v + h % u-x)))`,

REPEAT GEN_TAC
THEN CONV_TAC(TOP_DEPTH_CONV let_CONV) 
THEN REPEAT STRIP_TAC
THEN REWRITE_TAC[VECTOR_ARITH`(&1 - h) % v + h % u-x=(&1 - h) % (v-x) + h % (u-x)`;]
THEN REWRITE_TAC[DOT_RMUL;DOT_RADD;]
THEN REWRITE_TAC[REAL_ARITH`(&1-h)*A+h*B=A-h*(A-B)`]
THEN MATCH_MP_TAC exists_esilon_real
THEN ASM_REWRITE_TAC[]);;


let invariant_rcross_dot_esilon_3piont=prove(`!x:real^3 y:real^3 z:real^3 v:real^3 u:real^3.
              let a1 = y - x in
              let a2 = z - x in
              let a3 = v - x in
              let a4 = u - x in
 &0<(a1 cross a2) dot a3
 ==>
?t. &0< t /\ t< &1 /\ 
(!h. &0< h /\ h< t==>
 &0< (((&1 - h) % y + h % u-x) cross a2) dot a3)`,
REPEAT GEN_TAC
THEN CONV_TAC(TOP_DEPTH_CONV let_CONV) 
THEN REPEAT STRIP_TAC
THEN REWRITE_TAC[VECTOR_ARITH`(&1 - h) % v + h % u-x=(&1 - h) % (v-x) + h % (u-x)`;]
THEN REWRITE_TAC[CROSS_LMUL;CROSS_LADD;DOT_LMUL;DOT_LADD;]
THEN REWRITE_TAC[REAL_ARITH`(&1-h)*A+h*B=A-h*(A-B)`]
THEN MATCH_MP_TAC exists_esilon_real
THEN ASM_REWRITE_TAC[]);;

let invariant_crossr_dot_esilon_3piont=prove(`!x:real^3 y:real^3 z:real^3 v:real^3 u:real^3.
              let a1 = y - x in
              let a2 = z - x in
              let a3 = v - x in
              let a4 = u - x in
 &0<(a1 cross a2) dot a3
 ==>
?t. &0< t /\ t< &1 /\ 
(!h. &0< h /\ h< t==>
 &0< (a1 cross ((&1 - h) % z + h % u-x)) dot a3)`,
REPEAT GEN_TAC
THEN CONV_TAC(TOP_DEPTH_CONV let_CONV) 
THEN REPEAT STRIP_TAC
THEN REWRITE_TAC[VECTOR_ARITH`(&1 - h) % v + h % u-x=(&1 - h) % (v-x) + h % (u-x)`;]
THEN REWRITE_TAC[CROSS_RMUL;CROSS_RADD;DOT_LMUL;DOT_LADD;]
THEN REWRITE_TAC[REAL_ARITH`(&1-h)*A+h*B=A-h*(A-B)`]
THEN MATCH_MP_TAC exists_esilon_real
THEN ASM_REWRITE_TAC[]);;


let condition_4point_aff_gt_1_2inter_aff_gt_1_2=prove(`!x:real^3 y:real^3 z:real^3 v:real^3 u:real^3 w:real^3 a:real.
              let a1 = y - x in
              let a2 = z - x in
              let a3 = v - x in
              let a4 = u - x in
              let a5 = w - x in

 ~collinear {x,v,u}
/\ ~collinear {x,u,w}
/\ ~collinear {x,y,z}
/\  &0< a /\ a< &1 
/\ y IN aff_gt {x} {v,u}
/\ &0<(a3 cross a4) dot a5
/\ (!h. &0< h /\ h< a==> ~collinear {x,v,(&1-h)%u+h%w})
/\ &0<(a3 cross a1) dot a2
 ==> ?t. &0< t /\ t< &1 /\ 
(!h. &0< h /\ h< t==> ~(aff_gt {x} {y,z} INTER aff_gt {x} {v,(&1-h)%u+h%w}={}))`,

REPEAT STRIP_TAC
THEN CONV_TAC(TOP_DEPTH_CONV let_CONV) 
THEN REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"LINH")
THEN ONCE_REWRITE_TAC[CROSS_TRIPLE]
THEN STRIP_TAC 
THEN MRESA_TAC aff_gt_1_2_cross_dotr_4point[`x:real^3`;`y:real^3`;`z:real^3`;`v:real^3`;`u:real^3`;]
THEN POP_ASSUM MP_TAC
THEN CONV_TAC(TOP_DEPTH_CONV let_CONV) 
THEN RESA_TAC
THEN MRESA_TAC invariant_cross_dotr_esilon_3piont[`x:real^3`; `z:real^3`;`y:real^3`;`u:real^3`;`w:real^3`]
THEN POP_ASSUM MP_TAC
THEN CONV_TAC(TOP_DEPTH_CONV let_CONV) 
THEN ONCE_REWRITE_TAC[CROSS_SKEW]
THEN ASM_REWRITE_TAC[DOT_LNEG]
THEN STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"YEU")
THEN MRESA_TAC properties_of_collinear4_points_fan[`x:real^3`;`v:real^3`;`u:real^3`;`y:real^3`]
THEN MRESA_TAC point_in_aff_gt_2_1_change_point_in_aff_gt_1_2[`x:real^3`;`v:real^3`;`u:real^3`;`y:real^3`]
THEN MRESA_TAC aff_gt_2_1r_rcross_dotl_4point[`x:real^3`;`y:real^3`;`z:real^3`;`v:real^3`;`u:real^3`]
THEN POP_ASSUM MP_TAC
THEN CONV_TAC(TOP_DEPTH_CONV let_CONV) 
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={A,C,B}`]
THEN RESA_TAC
THEN MRESA_TAC invariant_rcross_dot_esilon_3piont[`x:real^3`; `u:real^3`;`z:real^3`;`v:real^3`;`w:real^3`]
THEN POP_ASSUM MP_TAC
THEN CONV_TAC(TOP_DEPTH_CONV let_CONV) 
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"EM")
THEN ABBREV_TAC`t1= min (min t t') a:real`
THEN MP_TAC(REAL_ARITH`&0<t/\ t< &1 /\ &0< t' /\ t'< &1 /\ &0< a /\ a < &1 /\ t1=min (min t t') a ==> &0< t1 /\ t1 < &1`)
THEN RESA_TAC
THEN EXISTS_TAC`t1:real`
THEN ASM_REWRITE_TAC[SET_RULE`~(A={})<=> ?y1. y1 IN A`]
THEN REPEAT STRIP_TAC
THEN ABBREV_TAC`a1=(y-x):real^3`
THEN ABBREV_TAC`a2=(z-x):real^3`
THEN ABBREV_TAC`a3=(v-x) :real^3`
THEN ABBREV_TAC`a4=(&1 - h) % u + h % w-x:real^3`
THEN ABBREV_TAC`va=a1 cross a2:real^3`
THEN ABBREV_TAC`vb=a3 cross a4:real^3`
THEN ABBREV_TAC`v3= (vb:real^3) cross (va:real^3)+(x:real^3)`
THEN EXISTS_TAC `v3:real^3`
THEN MP_TAC(REAL_ARITH`h<t1 /\ t1< &1 /\ t1=min (min t t') a==> h<t' /\ h< &1 /\ h< t /\ h<a`)
THEN RESA_TAC
THEN REMOVE_THEN "EM" (fun th-> MRESA1_TAC th `h:real`)
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[CROSS_SKEW]
THEN REWRITE_TAC[DOT_LNEG]
THEN ONCE_REWRITE_TAC[CROSS_TRIPLE]
THEN REWRITE_TAC[GSYM DOT_LNEG]
THEN ONCE_REWRITE_TAC[GSYM CROSS_SKEW]
THEN RESA_TAC
THEN MRESA_TAC th3[`x:real^3`;`u:real^3`;`w:real^3`;]
THEN MRESA_TAC pos_in_aff_gt_2_1_fan [`x:real^3`;`u:real^3`;`w:real^3`;`h:real`]
THEN MRESAL_TAC aff_gt_2_1_cross_dotl_4point[`x:real^3`;`v:real^3`;`u:real^3`;`w:real^3`;`(&1 - h) % u + h % w:real^3`][VECTOR_ARITH`((&1 - h) % u + h % w) - x=(&1 - h) % u + h % w - x`]
THEN POP_ASSUM MP_TAC
THEN CONV_TAC(TOP_DEPTH_CONV let_CONV) 
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[CROSS_TRIPLE]
THEN STRIP_TAC
THEN MRESA_TAC aff_gt_inter_aff_gt [`(x:real^3)`;`(v:real^3)`;`(u:real^3)`]
THEN MP_TAC(SET_RULE`y IN aff_gt {x} {v, u} /\ aff_gt {x} {v, u} = aff_gt {x, v} {u} INTER aff_gt {x, u} {v}
==> y IN aff_gt {x, v} {u:real^3}`)
THEN RESA_TAC
THEN MRESAL_TAC aff_gt_2_1r_rcross_dotl_4point[`x:real^3`;`u:real^3`;`(&1 - h) % u + h % w:real^3`;`v:real^3`;`y:real^3`][VECTOR_ARITH`((&1 - h) % u + h % w) - x=(&1 - h) % u + h % w - x`]
THEN POP_ASSUM MP_TAC
THEN CONV_TAC(TOP_DEPTH_CONV let_CONV) 
THEN ASM_REWRITE_TAC[]
THEN ONCE_REWRITE_TAC[CROSS_TRIPLE]
THEN ONCE_REWRITE_TAC[CROSS_SKEW]
THEN ASM_REWRITE_TAC[DOT_LNEG]
THEN STRIP_TAC
THEN MRESAL_TAC condition_cross_dot_4point[`x:real^3`;`v:real^3`;`(&1 - h) % u + h % w:real^3`;`y:real^3`;`z:real^3` ][VECTOR_ARITH`((&1 - h) % u + h % w) - x=(&1 - h) % u + h % w - x`]
THEN POP_ASSUM MP_TAC
THEN CONV_TAC(TOP_DEPTH_CONV let_CONV) 
THEN ASM_REWRITE_TAC[]
THEN MRESAL_TAC condition_cross_dot_4point[`x:real^3`; `z:real^3`;`y:real^3` ;`v:real^3`;`(&1 - h) % u + h % w:real^3`][VECTOR_ARITH`((&1 - h) % u + h % w) - x=(&1 - h) % u + h % w - x`]
THEN POP_ASSUM MP_TAC
THEN CONV_TAC(TOP_DEPTH_CONV let_CONV) 
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[CROSS_SKEW]
THEN ASM_REWRITE_TAC[]
THEN ONCE_REWRITE_TAC[GSYM CROSS_RNEG]
THEN ONCE_REWRITE_TAC[GSYM CROSS_SKEW]
THEN REWRITE_TAC[DOT_LNEG;REAL_ARITH`--(--A)=A`]
THEN REMOVE_THEN "YEU" (fun th-> MRESA1_TAC th`h:real`)
THEN REMOVE_THEN "LINH" (fun th-> MRESA1_TAC th`h:real`)
THEN SET_TAC[]);;






end;;