HOL html


(* ========================================================================== *)
(* FLYSPECK - BOOK FORMALIZATION                                              *)
(*                                                                            *)
(* Chapter: Local Fan                                                   *)
(* Author: Nguyen Quang Truong                                          *)
(* Date: 2010-05-09                                                           *)
(* ========================================================================== *)

(* rho_fan interior_angle *)

flyspeck_needs "local/LDURDPN.hl";;


module type Local_lemmas_type = sig

end;;

(* ================================================================ *)
(* lemma 6.13 - KOMWBWC *)

module Local_lemmas = struct

parse_as_infix("has_orders",(12,"right"));;
parse_as_infix("cyclic_on",(13,"right"));;

open Ldurdpn;;
open Lvducxu;;
open Sphere;;
open Trigonometry1;;
open Trigonometry2;;
open Hypermap;;
open Fan;;
open Prove_by_refinement;;
open Wrgcvdr_cizmrrh;;
open Aff_sgn_tac;;


let rho_node1 = new_definition ` rho_node1 (FF:real^3 # real^3 -> bool) v = (@w. (v,w) IN FF)`;;

let ivs_rho_node1 = new_definition ` ivs_rho_node1 (FF:real^3 # real^3 -> bool) v = (@a. a,v IN FF )`;;

let SWITCH_TAC t = UNDISCH_TAC t THEN DISCH_THEN (ASSUME_TAC o GSYM);;







let LOCAL_FAN_IMP_IN_V = prove_by_refinement
(` local_fan (V,E,FF) /\ x,y IN FF ==> x IN V /\ y IN V `,
[REWRITE_TAC[local_fan];
NHANH lemma_face_subset;
NHANH ELMS_OF_HYPERMAP_HYP;
LET_TAC THEN STRIP_TAC;
DOWN;
UNDISCH_TAC ` face H (x':real^3 # real^3) SUBSET dart H`;
ASM_REWRITE_TAC[SUBSET];
DISCH_THEN NHANH;
DOWN_TAC;
REWRITE_TAC[FAN; darts_of_hyp; IN_UNION; Wrgcvdr_cizmrrh.IN_ORD_E_EQ_IN_E];
STRIP_TAC;
DOWN;
UNDISCH_TAC ` UNIONS E SUBSET (V:real^3 -> bool) `;
NHANH lemma_sub_support;
REWRITE_TAC[INSERT_SUBSET; SUBSET];
MESON_TAC[];
DOWN;
REWRITE_TAC[self_pairs; IN_ELIM_THM; PAIR_EQ];
MESON_TAC[]]);;


let LOCAL_FAN_RHO_NODE_PROS = prove_by_refinement
(`local_fan (V,E,FF)
         ==> (!x. x IN V ==> x, (rho_node1 FF x) IN FF) /\
              (!x. x IN FF ==> x = FST x,rho_node1 FF (FST x))`,
[ABBREV_TAC `k = (\x. FST (x:real^3 # real^3)) `;
SWITCH_TAC `(\x. FST (x:real^3 # real^3)) = k`;
STRIP_TAC;
MP_TAC2 Wrgcvdr_cizmrrh.BIJ_BETWEEN_FF_AND_V;
REWRITE_TAC[BIJ];
STRIP_TAC;
SUBGOAL_THEN ` (!x. x IN V ==> x,rho_node1 FF x IN (FF:real^3 # real^3 -> bool))` ASSUME_TAC;
REPEAT STRIP_TAC;
REWRITE_TAC[rho_node1];
CONV_TAC SELECT_CONV;
DOWN THEN DOWN;
REWRITE_TAC[SURJ];
STRIP_TAC;
FIRST_ASSUM NHANH;
ASM_REWRITE_TAC[];
STRIP_TAC;
EXISTS_TAC ` SND (y:real^3 # real^3) `;
FIRST_X_ASSUM (SUBST1_TAC o SYM);
ASM_REWRITE_TAC[];
DOWN THEN SIMP_TAC[];
DOWN;
REWRITE_TAC[SURJ];
STRIP_TAC THEN STRIP_TAC;
UNDISCH_TAC ` !x. x IN (FF:real^3 # real^3 -> bool) ==> k x IN (V:real^3 -> bool) ` ;
DISCH_THEN NHANH;
FIRST_X_ASSUM NHANH;
DOWN THEN DOWN;
ASM_REWRITE_TAC[INJ];
MESON_TAC[FST]]);;




let ORTHONORMAL_CYCLIC = prove_by_refinement
(`~( x:real^N = y ) /\ FINITE U /\ (! u1 u2. u1 IN U /\ u2 IN U 
==> (u1 - u2) dot (x - y ) = &0) /\
U INTER affine hull {x,y} = {} ==>
cyclic_set U x y `,
[SIMP_TAC[cyclic_set];
STRIP_TAC;
REWRITE_TAC[VECTOR_ARITH` p = q + h % x <=> p - q = h % x `];
REPEAT STRIP_TAC;
FIRST_X_ASSUM (ASSUME_TAC o (SPECL [`p:real^N`;` q:real^N`]));
FIRST_X_ASSUM MP_TAC2;
ASM_REWRITE_TAC[IN];
ASM_REWRITE_TAC[DOT_LMUL; REAL_ENTIRE; DOT_EQ_0];
DISCH_THEN DISJ_CASES_TAC;
ASM_REWRITE_TAC[VECTOR_MUL_LZERO; VECTOR_ADD_RID];
ASM_REWRITE_TAC[VECTOR_ADD_RID; VECTOR_MUL_RZERO]]);;




let FAN_SINGLETON_V_DARTS = prove_by_refinement
(` FAN (x:real^N,V,E) /\ V = {v} ==> darts_of_hyp E V = {(v,v)}`,
[REWRITE_TAC[FAN; darts_of_hyp; EXTENSION; IN_INSERT; NOT_IN_EMPTY; SUBSET; IN_UNION; ord_pairs; self_pairs; IN_ELIM_THM];
NHANH lemma_sub_support;
REWRITE_TAC[INSERT_SUBSET];
REPEAT STRIP_TAC;
EQ_TAC;
ASM_MESON_TAC[];
STRIP_TAC;
DISJ2_TAC;
EXISTS_TAC ` v:real^N`;
ASM_REWRITE_TAC[EE; IN_ELIM_THM];
DOWN_TAC THEN REWRITE_TAC[graph; IN];
STRIP_TAC;
FIRST_ASSUM NHANH;
REWRITE_TAC[HAS_SIZE; Geomdetail.CARD2];
REPEAT STRIP_TAC;
UNDISCH_TAC ` (E:(real^N -> bool) -> bool) {v,x'':real^N}`;
PAT_ONCE_REWRITE_TAC `\x. x ==> i ` [GSYM IN ];
PHA;
NHANH lemma_sub_support;
REWRITE_TAC[INSERT_SUBSET; IN];
ASM_MESON_TAC[]]);;





let FAN_IN_DARTS_FST_EQ_SND_SELF_PAIRS = prove_by_refinement
(` FAN (x:real^N,V,E) /\ y IN darts_of_hyp E V 
==> ( FST y = SND y <=> y IN self_pairs E V )`,
[REWRITE_TAC[FAN; graph; darts_of_hyp; IN_UNION; ord_pairs; self_pairs; IN_ELIM_THM];
STRIP_TAC;
FIRST_X_ASSUM (MP_TAC2 o (SPEC `{a,b:real^N}`));
ONCE_REWRITE_TAC[GSYM IN];
ASM_REWRITE_TAC[];
ASM_REWRITE_TAC[HAS_SIZE; Geomdetail.CARD2; PAIR_EQ];
MESON_TAC[];
ASM_REWRITE_TAC[];
EXISTS_TAC ` v:real^N`;
ASM_REWRITE_TAC[]]);;




let FAN_FST_EQ_SND_SUPPER_EQ = prove_by_refinement
(` FAN (x,V,E) /\ FST y = SND y ==>
ee_of_hyp (x,V,E) y = y /\
nn_of_hyp (x,V,E) y = y /\
ff_of_hyp (x,V,E) y = y `,
[STRIP_TAC;
ASM_CASES_TAC ` y IN darts_of_hyp E (V:real^3 -> bool) `;
ASM_SIMP_TAC[Lvducxu.EE_OF_HYP_IDE_FST_SND_EQ];
UNDISCH_TAC ` FST y = SND (y:real^3 # real^3) `; 
ASSUME_TAC2 (
ISPEC `x:real^3 ` (GEN `x:real^N `FAN_IN_DARTS_FST_EQ_SND_SELF_PAIRS));
ASM_SIMP_TAC[nn_of_hyp3; ff_of_hyp3];
ASM_REWRITE_TAC[nn_of_hyp2; ee_of_hyp2; ff_of_hyp2]]);;






let COLLINEAR_CROSS_0 = prove(
` collinear {x,y,z} <=> (y - x ) cross ( z - x ) = vec 0 `,
ONCE_REWRITE_TAC[Trigonometry2.COLLINEAR_TRANSABLE] THEN 
REWRITE_TAC[CROSS_EQ_0]);;




let DET_CROSS = prove(` det (vector [x;y;z]) = (x cross y ) dot z `,
REWRITE_TAC[Trigonometry2.DET_VEC3_EXPAND; Trigonometry2.DET_VEC3_AS_CROSS_DOT]);;



let COPLANAR_IFF_CROSS_DOT = prove(
` coplanar {x,y,z,t} <=> (( y - x ) cross ( z - x ) ) dot ( t - x ) = &0`,
REWRITE_TAC[COPLANAR_DET_EQ_0; DET_CROSS]);;





let CROSS_DOT_COPLANAR = prove(
 ` (x cross y) dot z = &0 <=> coplanar {vec 0, x, y, z}`,
REWRITE_TAC[COPLANAR_IFF_CROSS_DOT; VECTOR_SUB_RZERO]);;





let SUBSET_NOT_COLLINEAR_AFFINE_HULL_EQ = prove_by_refinement
(` {a:real^3,b,c} SUBSET affine hull {x,y,z} /\ ~( collinear {a,b,c})
==> affine hull {x,y,z} = affine hull {a,b,c} `,
[REWRITE_TAC[COLLINEAR_CROSS_0];
REWRITE_TAC[GSYM DOT_EQ_0];
REWRITE_TAC[CROSS_DOT_COPLANAR];
STRIP_TAC;
REWRITE_TAC[EXTENSION];
GEN_TAC;
EQ_TAC;
DOWN THEN NHANH (SMOOTH_GEN_ALL Trigonometry2.NONCOPLANAR_3_BASIS);
REWRITE_TAC[VECTOR_SUB_RZERO];
STRIP_TAC;
FIRST_X_ASSUM (MP_TAC o (SPEC ` x':real^3 - a`));
REPEAT STRIP_TAC;
SUBGOAL_THEN ` coplanar {a,b,c,x':real^3}` MP_TAC;
DOWN_TAC;
REWRITE_TAC[coplanar; INSERT_SUBSET; EMPTY_SUBSET];
MESON_TAC[];
ASM_REWRITE_TAC[COPLANAR_IFF_CROSS_DOT; DOT_RADD; DOT_RMUL; DOT_CROSS_SELF;
 Collect_geom.ZERO_NEUTRAL; REAL_ENTIRE; DOT_EQ_0; GSYM COLLINEAR_CROSS_0];
STRIP_TAC;
FIRST_X_ASSUM SUBST_ALL_TAC;
UNDISCH_TAC ` x' - (a:real^3) = t1 % (b - a) + t2 % (c - a) + &0 % ((b - a) cross (c - a)) `;
REWRITE_TAC[VECTOR_MUL_LZERO; VECTOR_ADD_RID; VECTOR_ARITH` x' - a = t1 % (b - a) + t2 % (c - a) <=> x' = (&1 - t1 - t2) % a + t1 % b + t2 % c `; AFFINE_HULL_3; IN_ELIM_THM];
MESON_TAC[REAL_ARITH` (&1 - a - b ) + a + b = &1 `];
DOWN_TAC;
NHANH NOT_COPLANAR_NOT_COLLINEAR;
ONCE_REWRITE_TAC[Trigonometry2.COLLINEAR_TRANSABLE];
REWRITE_TAC[VECTOR_SUB_RZERO];
MESON_TAC[];
DOWN_TAC;
NHANH (ISPEC ` affine ` HULL_MONO);
REWRITE_TAC[HULL_HULL; SUBSET];
MESON_TAC[]]);;




let THREE_NOT_COLL_DETER_PLANE = prove(
`plane P /\ {a:real^3,b,c} SUBSET P /\ ~collinear {a,b,c}
==> affine hull {a,b,c} = P `, REWRITE_TAC[plane] THEN 
MESON_TAC [SUBSET_NOT_COLLINEAR_AFFINE_HULL_EQ ]);;




let LOCAL_FAN_NOT_V_SING = prove_by_refinement
(` local_fan (V,E,FF) ==> ~(?v. V = {v}) `,
[REWRITE_TAC[local_fan; dih2k];
LET_TAC;
NHANH ELMS_OF_HYPERMAP_HYP;
REPEAT STRIP_TAC;
UNDISCH_TAC ` V = {v:real^3}`;
UNDISCH_TAC `FAN (vec 0:real^3,V,E) `;
PHA;
NHANH FAN_SINGLETON_V_DARTS;
NHANH (MESON[CARD_SINGLETON]` S = {c} ==> CARD S = 1 `);
STRIP_TAC;
UNDISCH_TAC ` CARD (dart (H:(real^3 # real^3) hypermap)) = 2 * CARD (FF:real^3 # real^3 -> bool)` ;
DOWN;
EXPAND_TAC "H";
ASM_SIMP_TAC[];
MESON_TAC[FACE_NOT_EMPTY; ARITH_RULE` 1 <= a ==> ~( 1 = 2 * a ) `]]);;





let LOCAL_FAN_NOT_SING_FF = prove_by_refinement
(` local_fan (V,E,FF) ==> ~(? x. FF = {x})`,
[STRIP_TAC;
ABBREV_TAC ` k = (\x. FST (x:real^3 # real^3)) `;
SWITCH_TAC ` (\x. FST (x:real^3 # real^3)) = k `;
ASSUME_TAC2 Wrgcvdr_cizmrrh.BIJ_BETWEEN_FF_AND_V;
DOWN;
REWRITE_TAC[BIJ];
NHANH Wrgcvdr_cizmrrh.SURJ_IMP_S2_EQ_IMAGE_S1;
REPEAT STRIP_TAC;
FIRST_X_ASSUM SUBST_ALL_TAC;
DOWN;
REWRITE_TAC[IMAGE; IN_INSERT; NOT_IN_EMPTY; SET_RULE` {y | ?x'. x' = x /\ y = k x'} = {k x }`];
DOWN_TAC THEN NHANH LOCAL_FAN_NOT_V_SING;
MESON_TAC[]]);;






let LOCAL_FAN_IN_FF_DISTINCT = prove_by_refinement
(` local_fan (V,E,FF) /\ x IN FF ==> ~( FST x = SND x ) `,
[NHANH LOCAL_FAN_NOT_SING_FF;
REWRITE_TAC[local_fan];
LET_TAC;
STRIP_TAC;
DOWN;
ASM_REWRITE_TAC[];
NHANH lemma_face_identity;
STRIP_TAC;
FIRST_X_ASSUM SUBST_ALL_TAC;
STRIP_TAC;
SUBGOAL_THEN ` face_map (H:(real^3 # real^3) hypermap) x = x ` ASSUME_TAC;
DOWN_TAC;
NHANH ELMS_OF_HYPERMAP_HYP;
IMP_TAC THEN SIMP_TAC[];
STRIP_TAC;
STRIP_TAC;
DOWN;
UNDISCH_TAC ` FAN (vec 0:real^3,V,E) `;
PHA;
SIMP_TAC[FAN_FST_EQ_SND_SUPPER_EQ];
DOWN THEN PHA THEN ONCE_REWRITE_TAC[orbit_one_point];
REWRITE_TAC[GSYM face; ETA_AX];
ASM_MESON_TAC[]]);;




let LOCAL_FAN_IN_FF_IN_ORD_PAIRS = prove_by_refinement
(` local_fan (V,E,FF) /\ x IN FF ==> x IN ord_pairs E `,
[NHANH LOCAL_FAN_IN_FF_DISTINCT;
REWRITE_TAC[local_fan];
NHANH ELMS_OF_HYPERMAP_HYP;
LET_TAC;
NHANH lemma_face_subset;
ASM_REWRITE_TAC[SUBSET];
STRIP_TAC;
UNDISCH_TAC ` (x:real^3 # real^3) IN FF `;
ASM_REWRITE_TAC[];
FIRST_X_ASSUM NHANH;
ASM_REWRITE_TAC[darts_of_hyp; IN_UNION];
STRIP_TAC;
DOWN;
REWRITE_TAC[self_pairs; IN_ELIM_THM];
STRIP_TAC;
FIRST_X_ASSUM SUBST_ALL_TAC;
REPLICATE_TAC 4 DOWN;
REWRITE_TAC[]]);;






let LOCAL_FAN_IN_FF_NOT_COLLINEAR = prove(
` local_fan (V,E,FF) /\ x IN FF ==> ~( collinear {vec 0, FST x, SND x }) `,
NHANH LOCAL_FAN_IN_FF_IN_ORD_PAIRS THEN 
REWRITE_TAC[Wrgcvdr_cizmrrh.IN_ORD_E_EQ_IN_E; local_fan; FAN; fan6; IN] THEN 
LET_TAC THEN STRIP_TAC THEN DOWN THEN FIRST_X_ASSUM NHANH THEN 
SIMP_TAC[SET_RULE` {a} UNION S = a INSERT S `]);;







let LOCAL_FAN_CHARACTER_OF_RHO_NODE = prove_by_refinement
(` local_fan (V,E,FF) ==> (! v. v IN V ==> ~( rho_node1 FF v = v ) /\
v, rho_node1 FF v IN ord_pairs E /\
~ (collinear {vec 0, v, rho_node1 FF v})) `,
[NHANH LOCAL_FAN_RHO_NODE_PROS;
STRIP_TAC;
FIRST_X_ASSUM NHANH;
FIRST_X_ASSUM NHANH;
GEN_TAC THEN IMP_TAC THEN REMOVE_TAC;
DOWN THEN PHA;
NHANH LOCAL_FAN_IN_FF_DISTINCT;
NHANH LOCAL_FAN_IN_FF_NOT_COLLINEAR;
NHANH LOCAL_FAN_IN_FF_IN_ORD_PAIRS;
SIMP_TAC[]]);;


let graph2 = prove(`graph E <=> (!e. e IN E ==> FINITE e /\ CARD e = 2)`,
REWRITE_TAC[graph; IN; HAS_SIZE]);;

let GRAPH_WITH_SET2 = prove(
` graph E ==> (! a b. {a,b} IN E ==> ~( a = b )) `,
REWRITE_TAC[graph2] THEN DISCH_THEN NHANH THEN 
SIMP_TAC[Geomdetail.CARD2]);;



let FAN_V_TWO_ELMS_IN_E_DARTS2 = prove_by_refinement
(` FAN (x:real^N,V,E) /\ V = {v1,v2} /\ {v1,v2} IN E ==>
darts_of_hyp E V = {(v1,v2), (v2,v1)}`,
[REWRITE_TAC[darts_of_hyp; FAN; EXTENSION; IN_UNION; ord_pairs; self_pairs;
 IN_ELIM_THM; SUBSET; graph2; IN_INSERT; NOT_IN_EMPTY];
STRIP_TAC THEN GEN_TAC;
EQ_TAC;
STRIP_TAC;
UNDISCH_TAC ` {a:real^N, b} IN E `;
NHANH lemma_sub_support;
REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET];
FIRST_X_ASSUM NHANH;
FIRST_X_ASSUM NHANH;
ASM_REWRITE_TAC[Geomdetail.CARD2; PAIR_EQ];
MESON_TAC[];
UNDISCH_TAC ` v:real^N IN V `;
UNDISCH_TAC ` !x. ~(x IN EE (v:real^N) E)`;
UNDISCH_TAC ` {v1, v2:real^N} IN E`;
ASM_REWRITE_TAC[EE; IN_ELIM_THM];
PHA;
MESON_TAC[INSERT_COMM];
STRIP_TAC;
DISJ1_TAC;
DOWN THEN DOWN;
MESON_TAC[];
DISJ1_TAC;
DOWN THEN DOWN THEN PHA;
MESON_TAC[INSERT_COMM]]);;



let FAN_IN_E_DIFF = prove(
` FAN (x,V,E) ==> (! x y. {x,y} IN E ==> ~( x,y = y,x)) `,
REWRITE_TAC[PAIR_EQ; FAN] THEN NHANH GRAPH_WITH_SET2 THEN MESON_TAC[]);;




let LOCAL_FAN_NOT_TWO_V_IN_E = prove_by_refinement
(` local_fan (V,E,FF) ==> ~(? v1 v2. V = {v1,v2} /\ {v1,v2} IN E )`,
[NHANH Wrgcvdr_cizmrrh.LOCAL_FAN_IMP_FAN;
STRIP_TAC THEN STRIP_TAC;
DOWN_TAC;
NHANH FAN_V_TWO_ELMS_IN_E_DARTS2;
NHANH FAN_IN_E_DIFF;
NHANH LOCAL_FAN_NOT_SING_FF;
REWRITE_TAC[GSYM Geomdetail.CARD2; local_fan; dih2k];
LET_TAC;
NHANH ELMS_OF_HYPERMAP_HYP;
ASM_REWRITE_TAC[];
STRIP_TAC;
UNDISCH_TAC` CARD (dart (H:(real^3 # real^3) hypermap)) = 2 * CARD (FF:real^3 # real^3 -> bool) `;
ASM_SIMP_TAC[ARITH_RULE` 2 = 2 * x <=> x = 1 `];
STRIP_TAC;
SUBGOAL_THEN ` face H (x:real^3 # real^3) = {x} ` ASSUME_TAC;
MATCH_MP_TAC set_one_point;
ASM_REWRITE_TAC[FACE_FINITE; face_refl];
UNDISCH_TAC ` ~(?x. FF = {x:real^3 # real^3})`;
ASM_REWRITE_TAC[];
MESON_TAC[]]);;






let LOCAL_FAN_ORBIT_MAP_V = prove(
` local_fan (V,E,FF) ==> (! v. v IN V ==> orbit_map (rho_node1 FF) v = V) `,
NHANH LOCAL_FAN_RHO_NODE_PROS THEN 
STRIP_TAC THEN 
ABBREV_TAC ` k = (\x. FST (x:real^3 # real^3)) ` THEN 
SWITCH_TAC `(\x. FST (x:real^3 # real^3)) = k ` THEN 
ASSUME_TAC2 (SMOOTH_GEN_ALL Wrgcvdr_cizmrrh.WRGCVDR) THEN 
FIRST_X_ASSUM (MP_TAC o (SPEC ` rho_node1 (FF:real^3 # real^3 -> bool) `)) THEN 
STRIP_TAC THEN 
FIRST_X_ASSUM MP_TAC2 THEN 
DISCH_THEN MP_TAC2 THEN 
SIMP_TAC[EQ_SYM_EQ]);;





let LOCAL_FAN_RHO_NODE_PROS2 = ONCE_REWRITE_RULE[TAUT` a /\ b <=> b /\ a `] LOCAL_FAN_RHO_NODE_PROS;;



let FINTE_OF_N_FIRST_ELMS2 = prove(`FINITE {ITER n f x | n < i}`,
REWRITE_TAC[SET_RULE` {ITER n f x | n < i} = {y | ?n. n < i /\ y = ITER n f x}`]
THEN REWRITE_TAC[Wrgcvdr_cizmrrh.FINITE_OF_N_FIRST_ELMS]);;




let PLANE_AFFINE_HUL_INTER_P = prove_by_refinement
(` plane P /\ {x,y,z} SUBSET P 
==> affine hull {x, x + ( y - x ) cross ( z - x )} INTER P = {x} `,
[REWRITE_TAC[AFFINE_HULL_2; EXTENSION; IN_INSERT; NOT_IN_EMPTY; IN_INTER];
STRIP_TAC;
GEN_TAC;
REWRITE_TAC[IN_ELIM_THM];
EQ_TAC;
STRIP_TAC;
SUBGOAL_THEN ` coplanar {x,y,z,x':real^3} ` ASSUME_TAC;
DOWN_TAC;
REWRITE_TAC[coplanar; plane; INSERT_SUBSET];
MESON_TAC[];
DOWN;
ASM_REWRITE_TAC[COPLANAR_IFF_CROSS_DOT; VECTOR_ARITH` u % x + v % ( x + y ) =
 ( u + v ) % x + v % y `; VECTOR_MUL_LID; VECTOR_ARITH ` (a + b:real^N) - a = 
b `; VECTOR_ARITH` a + b = a <=> b = vec 0 `; DOT_RMUL; REAL_ENTIRE; DOT_EQ_0;
 VECTOR_MUL_EQ_0];
DOWN THEN PHA;
SIMP_TAC[INSERT_SUBSET];
STRIP_TAC;
EXISTS_TAC` &1 `;
EXISTS_TAC ` &0 `;
REWRITE_TAC[REAL_ADD_RID; VECTOR_MUL_LZERO; VECTOR_MUL_LID; VECTOR_ADD_RID]]);;





let FAN_IMP_V_DIFF = prove( ` FAN (x,V,E) ==> (! v. v IN V ==> ~( v = x )) `,
REWRITE_TAC[FAN; fan2] THEN MESON_TAC[]);;






let LOCAL_FAN_IMP_CYCLIC_SET = prove_by_refinement
(` local_fan (V,E,FF) /\ v IN V /\
{ITER n (rho_node1 FF) v | n <= l } = U /\
plane P /\ vec 0 IN P /\
U SUBSET P /\ v cross (rho_node1 FF v ) = e 
==>  cyclic_set U (vec 0 ) e `,
[NHANH LOCAL_FAN_RHO_NODE_PROS;
STRIP_TAC;
MATCH_MP_TAC ORTHONORMAL_CYCLIC;
UNDISCH_TAC ` v:real^3 IN V `;
UNDISCH_TAC ` !x. x IN V ==> x,rho_node1 FF x IN (FF:real^3 # real^3 -> bool) `;
STRIP_TAC;
FIRST_ASSUM NHANH;
STRIP_TAC;
DOWN;
UNDISCH_TAC ` local_fan (V,E,FF) `;
PHA;
NHANH LOCAL_FAN_IN_FF_NOT_COLLINEAR;
REWRITE_TAC[ GSYM CROSS_EQ_0];
ASM_SIMP_TAC[];
EXPAND_TAC "U";
REWRITE_TAC[ARITH_RULE` a <= b <=> a < b + 1 `];
REWRITE_TAC[FINTE_OF_N_FIRST_ELMS2];
STRIP_TAC;
ASM_CASES_TAC ` l = 0 `;
ASM_REWRITE_TAC[ARITH_RULE ` x < 0 + 1 <=> x = 0`; IN_ELIM_THM];
CONJ_TAC;
REPEAT STRIP_TAC;
ASM_REWRITE_TAC[VECTOR_SUB_REFL; DOT_LZERO];
REWRITE_TAC[EXTENSION; NOT_IN_EMPTY; IN_INTER; IN_ELIM_THM];
STRIP_TAC;
STRIP_TAC;
DOWN;
ASM_REWRITE_TAC[ITER; AFFINE_HULL_2; IN_ELIM_THM; VECTOR_MUL_RZERO; VECTOR_ADD_LID];
STRIP_TAC;
SUBGOAL_THEN ` v dot (e:real^3) = &0` ASSUME_TAC;
EXPAND_TAC "e";
REWRITE_TAC[DOT_CROSS_SELF];
DOWN;
ASM_REWRITE_TAC[DOT_LMUL; REAL_ENTIRE; DOT_EQ_0];
STRIP_TAC;
FIRST_X_ASSUM SUBST_ALL_TAC;
DOWN;
REWRITE_TAC[VECTOR_MUL_LZERO];
MP_TAC2 Wrgcvdr_cizmrrh.LOCAL_FAN_IMP_FAN;
REWRITE_TAC[FAN; fan2];
UNDISCH_TAC ` v:real^3 IN V `;
MESON_TAC[];
SUBGOAL_THEN ` {vec 0, v, rho_node1 (FF:real^3 # real^3 -> bool) v} SUBSET P ` ASSUME_TAC;
ASM_REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET];
UNDISCH_TAC ` U SUBSET (P:real^3 -> bool) `;
MATCH_MP_TAC (SET_RULE` a IN S /\ b IN S ==> S SUBSET A ==> a IN A /\ b IN A `);
EXPAND_TAC "U";
REWRITE_TAC[IN_ELIM_THM];
CONJ_TAC;
EXISTS_TAC ` 0`;
DOWN;
REWRITE_TAC[ITER];
ARITH_TAC;
EXISTS_TAC ` 1 `;
DOWN;
REWRITE_TAC[ITER1];
ARITH_TAC;
DOWN;
UNDISCH_TAC ` plane (P:real^3 -> bool) `;
PHA;
NHANH PLANE_AFFINE_HUL_INTER_P;
ASM_REWRITE_TAC[VECTOR_SUB_RZERO; VECTOR_ADD_LID];
ASSUME_TAC2 LOCAL_FAN_ORBIT_MAP_V;
UNDISCH_TAC ` v:real^3 IN V `;
FIRST_ASSUM NHANH;
ASM_REWRITE_TAC[ARITH_RULE` a < b + 1 <=> a <= b `];
REPEAT STRIP_TAC;
SUBGOAL_THEN ` ! u. u IN U ==> (u:real^3) dot --e = &0 ` ASSUME_TAC;
REPEAT STRIP_TAC;
SUBGOAL_THEN ` {vec 0, v, rho_node1 FF v, u:real^3} SUBSET P` MP_TAC;
DOWN_TAC;
SIMP_TAC[INSERT_SUBSET];
STRIP_TAC;
DOWN;
UNDISCH_TAC ` U SUBSET (P:real^3 -> bool) `;
SET_TAC[];
UNDISCH_TAC `plane (P:real^3 -> bool) `;
REWRITE_TAC[plane];
STRIP_TAC;
ASM_REWRITE_TAC[];
STRIP_TAC;
SUBGOAL_THEN ` coplanar {vec 0, v, rho_node1 FF v, u:real^3} ` MP_TAC;
REWRITE_TAC[coplanar];
DOWN THEN MESON_TAC[];
ASM_REWRITE_TAC[COPLANAR_IFF_CROSS_DOT; VECTOR_SUB_RZERO; DOT_RNEG];
SIMP_TAC[DOT_SYM];
REAL_ARITH_TAC;
ASM_SIMP_TAC[DOT_LSUB; VECTOR_SUB_LZERO];
REAL_ARITH_TAC;
REWRITE_TAC[EXTENSION; NOT_IN_EMPTY; IN_INTER];
REPEAT STRIP_TAC;
SUBGOAL_THEN ` x IN affine hull {vec 0, e:real^3} INTER P` MP_TAC;
ASM_REWRITE_TAC[IN_INTER];
UNDISCH_TAC ` x IN U:real ^3 -> bool `;
UNDISCH_TAC ` U SUBSET P:real^3 -> bool `;
SET_TAC[];
ASM_REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY];
SUBGOAL_THEN ` U SUBSET (V:real^3 -> bool) ` MP_TAC;
EXPAND_TAC "U";
EXPAND_TAC "V";
REWRITE_TAC[GSYM POWER_TO_ITER; lemma_subset_orbit];
REWRITE_TAC[SUBSET];
STRIP_TAC;
UNDISCH_TAC ` x:real^3 IN U `;
FIRST_X_ASSUM NHANH;
MP_TAC2 Wrgcvdr_cizmrrh.LOCAL_FAN_IMP_FAN;
REWRITE_TAC[FAN; fan2];
MESON_TAC[]]);;




let LOCAL_FAN_ITER_RHO_NODE_IN_V = prove(
` local_fan (V,E,FF) /\ v IN V ==>
(! i. ITER i (rho_node1 FF ) v IN V )`,
NHANH LOCAL_FAN_ORBIT_MAP_V THEN STRIP_TAC THEN 
DOWN THEN FIRST_X_ASSUM (NHANH_PAT `\x. x ==> y `) THEN 
STRIP_TAC THEN EXPAND_TAC "V" THEN 
REWRITE_TAC[orbit_map; POWER_TO_ITER; IN_ELIM_THM; ARITH_RULE` a >= 0 `] THEN 
MESON_TAC[]);;



let ORD2_ORBIT_MAP = prove(` f ( f x ) = (x:A) ==> orbit_map f x = {x, f x} `,
STRIP_TAC THEN SUBGOAL_THEN ` 0 < 2 /\ ITER 2 f (x:A) = x ` MP_TAC THENL 
[ASM_REWRITE_TAC[ARITH_RULE ` 0 < 2 `; ITER12];
NHANH Lvducxu.ITER_CYCLIC_ORBIT] THEN SIMP_TAC[] THEN STRIP_TAC THEN 
REWRITE_TAC[EXTENSION; IN_INSERT; NOT_IN_EMPTY; IN_ELIM_THM; ARITH_RULE ` 
x < 2 <=> x = 0 \/ x = 1 `] THEN MESON_TAC[ITER; ITER1]);;


let LOCAL_FAN_IMP_NOT_SEMI_IDE = prove(
` local_fan (V,E,FF) ==> (!v. v IN V ==> ~( rho_node1 FF ( rho_node1 FF v ) = v ))`,
NHANH LOCAL_FAN_CHARACTER_OF_RHO_NODE THEN
NHANH LOCAL_FAN_ORBIT_MAP_V THEN 
NHANH LOCAL_FAN_NOT_TWO_V_IN_E THEN 
REWRITE_TAC[Wrgcvdr_cizmrrh.IN_ORD_E_EQ_IN_E] THEN 
MESON_TAC[ORD2_ORBIT_MAP]);;



open Fan_misc;;
open Fan_defs;;
open Planarity;;
open Topology;;




let RHO_NODE_SET_IN_A_PLANE_IMP_POS_DIRECT = prove_by_refinement
(`local_fan (V,E,FF) /\
 v IN V /\
 {ITER n (rho_node1 FF) v | n <= l} = U /\
 plane P /\
 vec 0 IN P /\
 U SUBSET P /\
 v cross rho_node1 FF v = e
 ==> (!i. i < l
          ==> &0 <
              (ITER i (rho_node1 FF) v cross ITER (i + 1) (rho_node1 FF) v) dot
              e)`,
[STRIP_TAC;
INDUCT_TAC;
REWRITE_TAC[ITER; ITER1; ARITH_RULE` 0 + 1 = 1 `];
MP_TAC2 LOCAL_FAN_CHARACTER_OF_RHO_NODE;
DISCH_THEN (MP_TAC2 o (SPEC `v:real^3`));
REWRITE_TAC[COLLINEAR_CROSS_0];
REWRITE_TAC[VECTOR_SUB_RZERO];
ASM_SIMP_TAC[DOT_POS_LT];
NHANH (ARITH_RULE` SUC a < b ==> a < b `);
FIRST_X_ASSUM NHANH;
STRIP_TAC;
MP_TAC2 LOCAL_FAN_CHARACTER_OF_RHO_NODE;





DISCH_THEN (MP_TAC2 o (SPEC ` ITER i (rho_node1 (FF:real^3 # real^3 -> bool)) v:real^3`));
MP_TAC2 LOCAL_FAN_ITER_RHO_NODE_IN_V;
STRIP_TAC;
ABBREV_TAC` px = ITER i (rho_node1 FF) v:real^3 `;
ABBREV_TAC` py = ITER (i + 1) (rho_node1 FF) v:real^3 `;
ABBREV_TAC` pz = ITER (SUC i + 1) (rho_node1 FF) v:real^3 `;
ASM_REWRITE_TAC[ADD1];
SUBGOAL_THEN ` {px,py,pz} SUBSET (U:real^3 -> bool) ` MP_TAC;
REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET];
EXPAND_TAC "U";
REWRITE_TAC[IN_ELIM_THM];
EXPAND_TAC "px";
EXPAND_TAC "py";
EXPAND_TAC "pz";

MP_TAC2 (ARITH_RULE` SUC i < l ==> SUC i + 1 <= l /\ i <= l /\ i + 1 <= l`);
MESON_TAC[];

STRIP_TAC;
SUBGOAL_THEN ` {px, py, pz} SUBSET (P:real^3 -> bool) ` ASSUME_TAC;
UNDISCH_TAC ` U SUBSET (P:real^3 -> bool) `;
DOWN;
PHA;
REWRITE_TAC[SUBSET_TRANS];
SUBGOAL_THEN ` rho_node1 (FF:real^3 # real^3 -> bool) px = py /\
rho_node1 (FF:real^3 # real^3 -> bool) py = pz` MP_TAC;
EXPAND_TAC "px";
EXPAND_TAC "py";
EXPAND_TAC "pz";
REWRITE_TAC[GSYM ITER; ADD1];
STRIP_TAC;
SUBGOAL_THEN ` {vec 0, px, rho_node1 FF px} SUBSET (P:real^3 -> bool) ` MP_TAC;
UNDISCH_TAC ` {px,py,pz:real^3} SUBSET P `;

ASM_SIMP_TAC[INSERT_SUBSET; EMPTY_SUBSET];
STRIP_TAC;
SUBGOAL_THEN ` affine hull {vec 0, px, py:real^3} = P ` MP_TAC;
MATCH_MP_TAC THREE_NOT_COLL_DETER_PLANE;
ASM_REWRITE_TAC[];
UNDISCH_TAC ` ~collinear {vec 0, px, rho_node1 FF (px:real^3)}`;
UNDISCH_TAC ` {vec 0, px, rho_node1 FF px} SUBSET (P:real^3 -> bool) `;
PHA;
ASM_SIMP_TAC[];

REWRITE_TAC[AFFINE_HULL_3];
STRIP_TAC;
UNDISCH_TAC ` {px, py, pz} SUBSET (P:real^3 -> bool) `;
REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET];
STRIP_TAC;
DOWN;
EXPAND_TAC "P";
REWRITE_TAC[IN_ELIM_THM; VECTOR_MUL_RZERO; VECTOR_ADD_LID];
STRIP_TAC;
ASM_CASES_TAC` v' < &0 `;
ASM_REWRITE_TAC[CROSS_RADD; CROSS_RMUL; CROSS_REFL; VECTOR_MUL_RZERO; VECTOR_ADD_RID];
ONCE_REWRITE_TAC[CROSS_SKEW];
REWRITE_TAC[ DOT_LMUL; DOT_LNEG; REAL_ARITH` a * -- b = (-- a ) * b `];
MATCH_MP_TAC REAL_LT_MUL;
ASM_SIMP_TAC[REAL_ARITH` &0 < -- a <=> a < &0 `];
DOWN;
REWRITE_TAC[REAL_ARITH` ~( a < &0 ) <=> &0 <= a `];
STRIP_TAC;
ASM_CASES_TAC ` &0 <= w `;
ASSUME_TAC2 LOCAL_FAN_ORBIT_MAP_V;
UNDISCH_TAC `v:real^3 IN V `;
FIRST_ASSUM NHANH;
STRIP_TAC;
SUBGOAL_THEN ` px:real^3 IN V ` MP_TAC;
EXPAND_TAC "V";
EXPAND_TAC "px";
REWRITE_TAC[lemma_in_orbit; GSYM POWER_TO_ITER];
MP_TAC2 LOCAL_FAN_IMP_NOT_SEMI_IDE;
DISCH_THEN NHANH;
SWITCH_TAC ` pz = v' % px + w % (py:real^3) `;
ASM_REWRITE_TAC[];
STRIP_TAC;
MP_TAC2 Wrgcvdr_cizmrrh.LOCAL_FAN_IMP_FAN;
REWRITE_TAC[FAN; fan7];
STRIP_TAC;
FIRST_X_ASSUM (MP_TAC2 o (SPECL [` {px, py:real^3}`;` {pz:real^3 }`]));
UNDISCH_TAC ` (px:real^3),rho_node1 FF px IN ord_pairs E`;
ASM_SIMP_TAC[Wrgcvdr_cizmrrh.IN_ORD_E_EQ_IN_E; IN_UNION];
STRIP_TAC;
DISJ2_TAC;
REWRITE_TAC[IN_ELIM_THM; EXTENSION; IN_INSERT; NOT_IN_EMPTY];

EXISTS_TAC ` pz:real^3 `;
ASM_REWRITE_TAC[];
UNDISCH_TAC ` ITER (SUC i + 1) (rho_node1 FF) v:real^3 = pz:real^3`;
STRIP_TAC;
EXPAND_TAC "pz";
EXPAND_TAC "V";
REWRITE_TAC[GSYM POWER_TO_ITER; lemma_in_orbit];
SUBGOAL_THEN ` {px,py} INTER {pz:real^3} = {} ` SUBST1_TAC;
ASM_REWRITE_TAC[SET_RULE ` {a,b} INTER {x} = {} <=> ~( x = a ) /\ ~( x = b )`];
ASSUME_TAC2 LOCAL_FAN_CHARACTER_OF_RHO_NODE;
SUBGOAL_THEN ` py:real^3 IN V ` MP_TAC;
EXPAND_TAC "py";
EXPAND_TAC "V";
EXPAND_TAC "px";
REWRITE_TAC[GSYM ITER];
REWRITE_TAC[GSYM POWER_TO_ITER; lemma_in_orbit];
FIRST_X_ASSUM NHANH;
ASM_SIMP_TAC[AFF_GE_EQ_AFFINE_HULL; AFFINE_HULL_SING];
SUBGOAL_THEN ` pz:real^3 IN V /\ py IN V` MP_TAC;
SWITCH_TAC ` ITER (SUC i + 1) (rho_node1 FF) v = pz:real^3 `;
ASM_REWRITE_TAC[GSYM ITER];
EXPAND_TAC "V";
EXPAND_TAC "py";
EXPAND_TAC "px";
REWRITE_TAC[GSYM ITER];
REWRITE_TAC[GSYM POWER_TO_ITER; lemma_in_orbit];
STRIP_TAC;
SUBGOAL_THEN ` DISJOINT {vec 0} {px, py:real^3} /\ ~( vec 0 = pz:real^3)` MP_TAC;
UNDISCH_TAC ` fan2 (vec 0:real^3, V:real^3 -> bool, E: (real^3 -> bool) -> bool) `;
REWRITE_TAC[fan2; SET_RULE `DISJOINT {x} {a,b} <=> ~( x = a ) /\ ~( x = b ) `];
ASM_MESON_TAC[];



REWRITE_TAC[AFF_GE_EQ_AFFINE_HULL; AFFINE_HULL_SING];
STRIP_TAC;
SUBGOAL_THEN ` pz:real^3 IN aff_ge {vec 0} {px, py} INTER aff_ge {vec 0} {pz} ` ASSUME_TAC;
ASM_SIMP_TAC[IN_INTER; Fan.AFF_GE_1_2; Fan.AFF_GE_1_1];
CONJ_TAC;
REWRITE_TAC[IN_ELIM_THM; VECTOR_MUL_RZERO; VECTOR_ADD_LID];
EXPAND_TAC "pz";
UNDISCH_TAC ` &0 <= v' `;
UNDISCH_TAC ` &0 <= w `;
MESON_TAC[REAL_ARITH` (&1 - a - b ) + a + b = &1 `];
REWRITE_TAC[IN_ELIM_THM];
REWRITE_TAC[ENDS_IN_HALFLINE];
DISCH_THEN SUBST_ALL_TAC;
DOWN;
REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY];
DOWN;
MESON_TAC[];
ASM_CASES_TAC ` v' = &0 `;
UNDISCH_TAC` pz = v' % px + w % (py:real^3) `;
ASM_REWRITE_TAC[VECTOR_MUL_LZERO; VECTOR_ADD_LID];
NHANH (MESON[COLLINEAR_LEMMA]` x = a % y ==> collinear {vec 0,y,x}`);
ASSUME_TAC2 LOCAL_FAN_CHARACTER_OF_RHO_NODE;
ASSUME_TAC2 LOCAL_FAN_ITER_RHO_NODE_IN_V;
FIRST_ASSUM (MP_TAC o (SPEC ` i + 1 `));
FIRST_ASSUM NHANH;
ASM_REWRITE_TAC[];
SIMP_TAC[];

ASSUME_TAC2 LOCAL_FAN_CHARACTER_OF_RHO_NODE;
ASSUME_TAC2 LOCAL_FAN_IMP_NOT_SEMI_IDE;
ASSUME_TAC2 LOCAL_FAN_ITER_RHO_NODE_IN_V;
FIRST_ASSUM (MP_TAC o (SPEC ` i: num`));
FIRST_X_ASSUM NHANH;
FIRST_ASSUM (MP_TAC o (SPEC ` i + 1`));
FIRST_ASSUM NHANH;

SWITCH_TAC ` pz = v' % px + w % (py:real^3) `;
ASM_REWRITE_TAC[Wrgcvdr_cizmrrh.IN_ORD_E_EQ_IN_E];
STRIP_TAC THEN STRIP_TAC;

MP_TAC2 Wrgcvdr_cizmrrh.LOCAL_FAN_IMP_FAN;
REWRITE_TAC[FAN; fan7];
STRIP_TAC;
FIRST_X_ASSUM (MP_TAC2 o (SPECL [` {py, pz:real^3}`;` {px:real^3}`]));
ASM_REWRITE_TAC[IN_UNION; IN_ELIM_THM; EXTENSION; IN_INSERT; NOT_IN_EMPTY];
DISJ2_TAC;
EXISTS_TAC ` px:real^3`;
UNDISCH_TAC ` ITER (i) (rho_node1 FF) v = (px:real^3)`;
DISCH_THEN (SUBST1_TAC o SYM);
ASM_REWRITE_TAC[];
SUBGOAL_THEN ` {py, pz} INTER {px:real^3} = {} ` SUBST1_TAC;
ASM_REWRITE_TAC[SET_RULE` {a,b} INTER {x} = {} <=> ~( a = x ) /\ ~( b = x )`; AFF_GE_EQ_AFFINE_HULL; AFFINE_HULL_SING];
SUBGOAL_THEN ` DISJOINT {vec 0} {py, pz:real^3} /\ ~( vec 0 = px:real^3)` MP_TAC;
REWRITE_TAC[SET_RULE` DISJOINT {x} {a,b} <=> ~( x = a ) /\ ~( x = b )`];
DOWN_TAC;
NHANH Collect_geom.NOT_COL3_IMP_DIFF;
SIMP_TAC[DE_MORGAN_THM];

STRIP_TAC;
SUBGOAL_THEN ` v' % (px:real^3) IN aff_ge {vec 0} {py, pz} INTER aff_ge {vec 0} {px} ` MP_TAC;
ASM_SIMP_TAC[Fan.AFF_GE_1_2; Fan.AFF_GE_1_1; IN_INTER; IN_ELIM_THM];
CONJ_TAC;
UNDISCH_TAC ` v' % px + w % py = (pz:real^3 )`;
UNDISCH_TAC` ~( &0 <= w ) `;
NHANH (REAL_ARITH` ~( &0 <= x ) ==> &0 <=  -- x `);
PHA;
SIMP_TAC[VECTOR_ARITH` a + b % y = z <=> a = ( -- b ) % y + z `; VECTOR_MUL_RZERO; VECTOR_ADD_LID];
MESON_TAC[REAL_ARITH` &0 <= &1 /\ a + -- a + &1 = &1 `; VECTOR_MUL_LID];
REWRITE_TAC[HALFLINE; IN_ELIM_THM];
EXISTS_TAC ` v':real` ;
ASM_REWRITE_TAC[VECTOR_MUL_RZERO; VECTOR_ADD_LID];
DISCH_TAC;
DISCH_THEN SUBST_ALL_TAC;
DOWN;
ASM_REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY; VECTOR_MUL_EQ_0; 
AFF_GE_EQ_AFFINE_HULL; AFFINE_HULL_SING]]);;




let AZIM_RANGE = prove(`! v w w1 w2. &0 <= azim v w w1 w2 /\
 azim v w w1 w2 < &2 * pi `,
REWRITE_TAC[azim]);;



let PI_TO_TWO_PI_NEG_SIN = prove_by_refinement
(`! x. pi < x /\ x < &2 * pi ==> sin x < &0 `, [GEN_TAC;
ONCE_REWRITE_TAC[MESON[REAL_ARITH` x = x - pi + pi `]` sin x = sin ( x - pi + pi )`];
ONCE_REWRITE_TAC[REAL_ARITH` x = x - pi + pi `];
REWRITE_TAC[SIN_PERIODIC_PI];
ABBREV_TAC ` xx = x - pi `;
REWRITE_TAC[REAL_ARITH` a + b < c + b <=> a < c `;
REAL_ARITH` a - b < c - b <=> a < c `; REAL_SUB_REFL;
REAL_ARITH` &2 * x - x = x `; REAL_ARITH` -- x < &0 <=> &0 < x `];
REWRITE_TAC[SIN_POS_PI]]);;



let MIXED_PROD_POS_IMP_RANGE_AZIM = prove_by_refinement
(`~collinear {vec 0, u, v} /\ ~collinear {vec 0, u, w} /\
&0 < (u cross v) dot w ==> &0 < azim (vec 0) u v w /\
azim (vec 0) u v w < pi `,
[STRIP_TAC;
MP_TAC2 (SPEC_ALL Planarity.JBDNJJB);
STRIP_TAC;
SUBGOAL_THEN` &0 < sin (azim (vec 0) u v w) ` ASSUME_TAC;
ASM_REWRITE_TAC[];
MATCH_MP_TAC REAL_LT_MUL;
ASM_REWRITE_TAC[];
MP_TAC (SPECL [`vec 0:real^3 `;` u:real^3`;` v:real^3 `;` w:real^3 `]
 AZIM_RANGE);
ASM_CASES_TAC ` azim (vec 0) u v w <= pi `;
MP_TAC PI_POS;
REWRITE_TAC[REAL_ARITH` &0 < x <=> -- x < &0 `];
PHA;
NGOAC;
NHANH (REAL_ARITH` a < b /\ b <= c ==> a < c `);
STRIP_TAC;
MP_TAC2 (SPEC ` azim (vec 0) u v w ` Trigonometry1.SIN_NEGPOS_PI);
STRIP_TAC;
FIRST_X_ASSUM SUBST_ALL_TAC;
ASM_REWRITE_TAC[REAL_ARITH` -- x < &0 <=> &0 < x `];
STRIP_TAC;
UNDISCH_TAC` &0 < sin (azim (vec 0) u v w) `;
MP_TAC2 (SPEC ` azim (vec 0) u v w ` PI_TO_TWO_PI_NEG_SIN);
ASM_REWRITE_TAC[REAL_ARITH` a < b <=> ~( b <= a ) `];
PHA;
REWRITE_TAC[REAL_ARITH` ~( a < b /\ b < a ) `]]);;






let COLLINEAR_ONCE_VEC_0 = prove(
` ~ ( x = vec 0 ) ==> (! y. collinear {vec 0, x, y} <=> ? t. y = t % x )`,
SIMP_TAC[COLLINEAR_LEMMA] THEN MESON_TAC[VECTOR_MUL_LZERO]);;







let AFF_SGN_TRULE tm = prove (tm, AFF_SGN_TAC);;
let AFF_GE11 = AFF_SGN_TRULE`!x v:real^N.
         ~(x = v)
         ==> aff_ge {x} {v} =
             {y | ?t1 t2. &0 <= t2 /\ t1 + t2 = &1 /\ y = t1 % x + t2 % v} `;;


let X_IN_AFF_GE11 = prove(`! x:real^N. ~( c = x ) ==> x IN aff_ge {c} {x}`,
SIMP_TAC[AFF_GE11; IN_ELIM_THM] THEN REPEAT STRIP_TAC THEN 
EXISTS_TAC `&0 ` THEN EXISTS_TAC ` &1 ` THEN 
REWRITE_TAC[REAL_ARITH` &0 <= &1 /\ &0 + &1 = &1 `] THEN VECTOR_ARITH_TAC);;


let FAN_IN_AFF_GE_IMP_EQ = prove_by_refinement
(` FAN (x:real^N,V,E) /\ v IN V /\ {a,b} IN E /\ v IN aff_ge {x} {a,b} ==> v = a \/ v = b `,
[REWRITE_TAC[fan7;FAN; fan2];
STRIP_TAC;
ASM_CASES_TAC ` v = a \/ v = (b:real^N) `;
FIRST_ASSUM ACCEPT_TAC;
DOWN;
REWRITE_TAC[SET_RULE` ~( x = a \/ x = b ) <=> {a,b} INTER {x} = {}`];
FIRST_X_ASSUM (MP_TAC2 o (SPECL [`{a,b:real^N} `;` {v:real^N}`]));
ASM_REWRITE_TAC[IN_UNION; IN_ELIM_THM];
DISJ2_TAC;
EXISTS_TAC `v:real^N`;
ASM_REWRITE_TAC[];
STRIP_TAC;
DISCH_THEN SUBST_ALL_TAC;
ASM_CASES_TAC ` v = x:real^N `;
FIRST_X_ASSUM SUBST_ALL_TAC;
ASM_MESON_TAC[];
DOWN;
ONCE_REWRITE_TAC[EQ_SYM_EQ];
NHANH X_IN_AFF_GE11;
STRIP_TAC;
UNDISCH_TAC ` aff_ge {x} {a, b} INTER aff_ge {x} {v} = aff_ge {x:real^N} {}`;
REWRITE_TAC[AFF_GE_EQ_AFFINE_HULL; AFFINE_HULL_SING; EXTENSION; IN_INTER; IN_INSERT; NOT_IN_EMPTY];
ASM_MESON_TAC[]]);;








let AFF_GE22 = 
AFF_SGN_TRULE`! a b x (y:real^N). DISJOINT {a,b} {x,y} ==> aff_ge {a,b} {x,y} =
 {z | ? aa bb xx yy. &0 <= xx /\ &0 <= yy 
/\ aa + bb + xx + yy = &1 /\ z = aa % a + bb % b + xx % x + yy % y }`;;

let RHO_NODE_IS_SUCCESEOR_AZIM = prove_by_refinement
(`local_fan (V,E,FF) /\
 v IN V /\
 {ITER n (rho_node1 FF) v | n <= l} = U /\
 plane P /\
 vec 0 IN P /\
 U SUBSET P /\
 v cross rho_node1 FF v = e
 ==> (!x n. x = ITER n ( rho_node1 FF) v /\ n < l
          ==> (!y. y IN U /\ ~(y = x) /\ ~(y = rho_node1 FF x)
                   ==> azim (vec 0) e x (rho_node1 FF x) < azim (vec 0) e x y))`,
[NHANH RHO_NODE_SET_IN_A_PLANE_IMP_POS_DIRECT;
REPEAT STRIP_TAC;
UNDISCH_TAC` n < l:num `;
FIRST_ASSUM NHANH;
SWITCH_TAC ` x = ITER n (rho_node1 FF) (v:real^3)`;
ASSUME_TAC2 LOCAL_FAN_ITER_RHO_NODE_IN_V;
NHANH_PAT `\x. x ==> y ` REAL_LT_IMP_NE;
SIMP_TAC[EQ_SYM_EQ; CROSS_DOT_COPLANAR];
ONCE_REWRITE_TAC[Collect_geom.PER_SET2];
NHANH NOT_COPLANAR_NOT_COLLINEAR;
ONCE_REWRITE_TAC[SET_RULE` {a,b,c} = {c,b,a}`];
NHANH NOT_COPLANAR_NOT_COLLINEAR;
SIMP_TAC[INSERT_COMM];
ONCE_REWRITE_TAC[GSYM CROSS_TRIPLE];
STRIP_TAC;
UNDISCH_TAC ` &0 < (e cross ITER n (rho_node1 FF) v) dot ITER (n + 1) (rho_node1 FF) (v:real^3) `;
UNDISCH_TAC ` ~collinear {e, ITER (n + 1) (rho_node1 FF) (v:real^3), vec 0} `;
UNDISCH_TAC` ~collinear {e, ITER n (rho_node1 FF) v:real^3, vec 0}`;
PHA;
NHANH (SIMP_RULE[INSERT_COMM] MIXED_PROD_POS_IMP_RANGE_AZIM);
ASM_REWRITE_TAC[];
STRIP_TAC;
ABBREV_TAC ` sx = ITER (n + 1 ) (rho_node1 FF) v:real^3 `;
EXPAND_TAC "x";
REWRITE_TAC[GSYM ITER; ADD1];
ASM_REWRITE_TAC[];
ASM_CASES_TAC ` y IN aff_ge {vec 0} {x , sx:real^3}`;
SUBGOAL_THEN ` {x,sx:real^3} IN  E ` ASSUME_TAC;
REWRITE_TAC[Wrgcvdr_cizmrrh.IN_E_IFF_IN_ORD_E];
MATCH_MP_TAC LOCAL_FAN_IN_FF_IN_ORD_PAIRS;
ASM_REWRITE_TAC[];
SUBGOAL_THEN ` x IN (V:real^3 -> bool) ` MP_TAC;
EXPAND_TAC "x";
ASM_REWRITE_TAC[];

MP_TAC2 LOCAL_FAN_RHO_NODE_PROS2;
STRIP_TAC;
FIRST_ASSUM NHANH;
EXPAND_TAC "x";
REWRITE_TAC[GSYM ITER; ADD1];
ASM_SIMP_TAC[];

SUBGOAL_THEN ` y:real^3 IN V ` ASSUME_TAC;
UNDISCH_TAC ` y IN U:real^3 -> bool ` ;
EXPAND_TAC "U";
REWRITE_TAC[IN_ELIM_THM];
STRIP_TAC;
ASM_REWRITE_TAC[];
MP_TAC2 (ISPECL [`V:real^3 -> bool`;` E: (real^3 -> bool) -> bool`;
 ` vec 0:real^3 `;` x:real^3 `;` y:real^3 `;` sx: real^3 `] (GEN_ALL
 FAN_IN_AFF_GE_IMP_EQ));
ASSUME_TAC2 Wrgcvdr_cizmrrh.LOCAL_FAN_IMP_FAN;
FIRST_X_ASSUM ACCEPT_TAC;

ASM_REWRITE_TAC[];
UNDISCH_TAC ` ~(y:real^3 = rho_node1 FF (x:real^3)) `;
EXPAND_TAC "x";
REWRITE_TAC[ADD1; GSYM ITER];
ASM_SIMP_TAC[];
SUBGOAL_THEN ` rho_node1 (FF:real^3 # real^3 -> bool) x = sx ` ASSUME_TAC;
EXPAND_TAC "x";
REWRITE_TAC[ADD1; GSYM ITER];
FIRST_ASSUM ACCEPT_TAC;

SUBGOAL_THEN ` wedge (vec 0) e x sx = aff_gt {vec 0, e} {x,sx} ` ASSUME_TAC;
MATCH_MP_TAC WEDGE_LUNE;
ASM_REWRITE_TAC[GSYM CROSS_DOT_COPLANAR];
ASM_SIMP_TAC[REAL_ARITH` &0 < x ==> ~( x = &0) `];

UNDISCH_TAC` &0 < (e cross x) dot sx `;
NHANH_PAT `\x. x ==> y ` REAL_LT_IMP_NE;
SIMP_TAC[EQ_SYM_EQ; CROSS_DOT_COPLANAR];
NHANH (SMOOTH_GEN_ALL Trigonometry2.NONCOPLANAR_3_BASIS);
STRIP_TAC;
FIRST_ASSUM (MP_TAC o (SPEC ` y: real^3 `));
REWRITE_TAC[VECTOR_SUB_RZERO];
STRIP_TAC;
UNDISCH_TAC ` ~(y IN aff_ge {vec 0} {x, sx:real^3}) `;
UNDISCH_TAC ` ~coplanar {vec 0, e, x, sx:real^3} `;
ONCE_REWRITE_TAC[SET_RULE` {a,b,c} = {b,c,a}`];
NHANH NOT_COPLANAR_NOT_COLLINEAR;
NHANH th3; 
SIMP_TAC[Fan.AFF_GE_1_2];
STRIP_TAC;
REWRITE_TAC[IN_ELIM_THM; VECTOR_MUL_RZERO; VECTOR_ADD_LID];
REWRITE_TAC[MESON[REAL_ARITH` (&1 - a - b) + a + b = &1`]` (?t1 t2 t3.
       &0 <= t2 /\ &0 <= t3 /\ t1 + t2 + t3 = &1 /\ y = t2 % x + t3 % sx)
<=> (? t2 t3. &0 <= t2 /\ &0 <= t3 /\ y = t2 % x + t3 % sx) `];
STRIP_TAC;
SUBGOAL_THEN ` ~( y:real^3 IN aff_ge {vec 0, e} {x, sx} )` ASSUME_TAC;
SUBGOAL_THEN ` DISJOINT {vec 0, e} {x, sx:real^3} ` MP_TAC;
REWRITE_TAC[DISJOINT; SET_RULE` (a INSERT A ) INTER ( b INSERT B) = {} <=>
~( a = b ) /\ ~ ( a IN B ) /\ ~( b IN A ) /\ A INTER B = {} `; NOT_IN_EMPTY;
IN_INSERT; INTER_EMPTY];
ASM_REWRITE_TAC[];
UNDISCH_TAC ` ~coplanar {vec 0, x, sx, e:real^3}`;
SIMP_TAC[INSERT_COMM];
NHANH NOT_COPLANAR_NOT_COLLINEAR;
NHANH th3;
SIMP_TAC[EQ_SYM_EQ];


SIMP_TAC[AFF_GE22; IN_ELIM_THM; VECTOR_MUL_RZERO; VECTOR_ADD_LID];
STRIP_TAC;
UNDISCH_TAC ` ~(?t2 t3. &0 <= t2 /\ &0 <= t3 /\ y = t2 % x + t3 % sx:real^3)`;
UNDISCH_TAC` !ta tb tc.
     (y:real^3) = ta % e + tb % x + tc % sx ==> ta = t1 /\ tb = t2 /\ tc = t3 `;
REWRITE_TAC[TAUT` ~ a ==> ~ b <=> b ==> a `];

SUBGOAL_THEN ` {vec 0,x,sx:real^3} SUBSET P ` ASSUME_TAC;
ASM_REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET];
UNDISCH_TAC ` U SUBSET (P:real^3 -> bool) `;
MATCH_MP_TAC (SET_RULE` x IN U /\ y IN U ==> U SUBSET P ==> x IN P /\ y IN P `);
EXPAND_TAC "U";
EXPAND_TAC "sx";
EXPAND_TAC "x";
REWRITE_TAC[IN_ELIM_THM; GSYM ITER; ADD1];
UNDISCH_TAC ` n < l:num `;
MESON_TAC[ARITH_RULE ` a < b:num ==> a <= b /\ a + 1 <= b `];
SUBGOAL_THEN ` affine hull {vec 0,x, sx:real^3} = P ` ASSUME_TAC;
MATCH_MP_TAC THREE_NOT_COLL_DETER_PLANE;
ASM_REWRITE_TAC[];
ASSUME_TAC2 (SET_RULE` y:real^3 IN U /\ U SUBSET P ==> y IN P `);
DOWN;
EXPAND_TAC "P";
REWRITE_TAC[AFFINE_HULL_3; IN_ELIM_THM; VECTOR_MUL_RZERO; VECTOR_ADD_LID];

REPEAT STRIP_TAC;
EXISTS_TAC ` xx:real ` ;
EXISTS_TAC `yy:real `;
ASM_REWRITE_TAC[];
FIRST_ASSUM (MP_TAC o (SPECL [`&0 `;` v': real`;` w:real `]));
ANTS_TAC;
DOWN THEN REMOVE_TAC;
UNDISCH_TAC ` y = t1 % e + t2 % x + t3 % (sx:real^3) `;
REMOVE_TAC;
ASM_REWRITE_TAC[VECTOR_MUL_LZERO; VECTOR_ADD_LID];
DOWN;
FIRST_ASSUM NHANH;
SIMP_TAC[];
STRIP_TAC;
STRIP_TAC;
UNDISCH_TAC ` &0 = t1 `;
DISCH_THEN (SUBST1_TAC o SYM);
VECTOR_ARITH_TAC;
ASSUME_TAC2 (SIMP_RULE[INSERT_COMM] (SPECL [`vec 0:real^3 `;` e:real^3 `;
` x:real^3 `;` sx: real^3 `] WEDGE_LUNE_GE));
UNDISCH_TAC ` ~(y IN aff_ge {vec 0, e} {x, sx:real^3})`;
SIMP_TAC[INSERT_COMM];
FIRST_X_ASSUM (SUBST1_TAC o SYM);
REWRITE_TAC[IN_ELIM_THM];
MP_TAC (SPECL [`vec 0:real^3 `;` e:real^3 `;` x:real^3 `;` y:real^3 `]
 AZIM_RANGE);
REAL_ARITH_TAC]);;





let FIRST_AZIM_CYCLE_EQ_RHO_NODE = prove_by_refinement
(`local_fan (V,E,FF) /\
     v IN V /\
     {ITER n (rho_node1 FF) v | n <= l} = U /\
     plane P /\
     vec 0 IN P /\
     U SUBSET P /\
     v cross rho_node1 FF v = e
     ==> (!x n.
              x = ITER n (rho_node1 FF) v /\ n < l
              ==> azim_cycle U (vec 0) e x = rho_node1 FF x ) `,
[NHANH RHO_NODE_IS_SUCCESEOR_AZIM;
NHANH LOCAL_FAN_IMP_CYCLIC_SET;
REPEAT STRIP_TAC;
MATCH_MP_TAC Wrgcvdr_cizmrrh.IDENTIFY_AZIM_CYCLE;
ASSUME_TAC2 LOCAL_FAN_CHARACTER_OF_RHO_NODE;
ASSUME_TAC2 LOCAL_FAN_ITER_RHO_NODE_IN_V;
SUBGOAL_THEN ` x IN (V:real^3 -> bool) ` MP_TAC;
ASM_REWRITE_TAC[];
FIRST_ASSUM NHANH;
SIMP_TAC[];
STRIP_TAC;
SUBGOAL_THEN ` rho_node1 (FF:real^3 # real^3 -> bool) x IN U ` MP_TAC;
ASM_REWRITE_TAC[GSYM ITER; ADD1];
EXPAND_TAC "U";
REWRITE_TAC[IN_ELIM_THM];
UNDISCH_TAC ` n < l:num `;
MESON_TAC[ARITH_RULE` a < l:num ==> a + 1 <= l `];
UNDISCH_TAC ` cyclic_set U (vec 0) (e:real^3) `;
PHA;
SIMP_TAC[IN];
REPEAT STRIP_TAC;
DOWN THEN DOWN;
UNDISCH_TAC ` ~(rho_node1 FF x = (x:real^3)) `;
SET_TAC[];

ASSUME_TAC2 RHO_NODE_SET_IN_A_PLANE_IMP_POS_DIRECT;
UNDISCH_TAC ` n < l:num `;
PHA;
FIRST_X_ASSUM NHANH;
NHANH REAL_LT_IMP_NE;
DOWN;
ASM_SIMP_TAC[CROSS_DOT_COPLANAR; EQ_SYM_EQ; INSERT_COMM];
ONCE_REWRITE_TAC[Collect_geom.PER_SET2];
NHANH NOT_COPLANAR_NOT_COLLINEAR;
SIMP_TAC[INSERT_COMM];

UNDISCH_TAC ` !(x:real^3) n.
          x = ITER n (rho_node1 FF) v /\ n < l
          ==> (!y. y IN U /\ ~(y = x) /\ ~(y = rho_node1 FF x)
                   ==> azim (vec 0) e x (rho_node1 FF x) < azim (vec 0) e x y)`;
DISCH_THEN ASSUME_TAC2;
ASM_CASES_TAC ` q:real^3 = rho_node1 FF (x:real^3) `;
DISJ2_TAC;
FIRST_X_ASSUM SUBST1_TAC;
REWRITE_TAC[REAL_LE_REFL];
DISJ1_TAC;
REPLICATE_TAC 4 DOWN THEN PHA;
REWRITE_TAC[IN];
MESON_TAC[]]);;






let SEQUENCE_OF_RHO_NODE_IS_SUC = prove_by_refinement
(`local_fan (V,E,FF) /\
       v IN V /\
       {ITER n (rho_node1 FF) v | n <= l} = U /\
       plane P /\
       vec 0 IN P /\
       U SUBSET P /\
       v cross rho_node1 FF v = e
       ==> (!x n.
                x = ITER n (rho_node1 FF) v /\ n < l
                ==> (!y. y IN U /\ ~(y = x) /\ ~(y = rho_node1 FF x)
                         ==> azim (vec 0) e y x  <
                             azim (vec 0) e y ( rho_node1 FF x)))`,
[NHANH RHO_NODE_IS_SUCCESEOR_AZIM;
NHANH LOCAL_FAN_IMP_CYCLIC_SET;
STRIP_TAC;
FIRST_X_ASSUM NHANH;
REPEAT GEN_TAC THEN STRIP_TAC;
FIRST_X_ASSUM NHANH;
GEN_TAC THEN STRIP_TAC;
DOWN;
SUBGOAL_THEN` cyclic_set {x, rho_node1 FF (x:real^3), y} (vec 0) e` MP_TAC;
MATCH_MP_TAC (SPEC ` U:real^3 -> bool` (GEN `W: real^3 -> bool`
 (SPEC_ALL Fan.subset_cyclic_set_fan)));
ASM_REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET; GSYM ITER; ADD1];
EXPAND_TAC "U";
REWRITE_TAC[IN_ELIM_THM];
UNDISCH_TAC ` n < l:num `;
MESON_TAC[ARITH_RULE` n < l:num ==> n <= l /\ n + 1 <= l `];
PHA;
NHANH_PAT `\x. x ==> y ` REAL_LT_IMP_LE;
ONCE_REWRITE_TAC[TAUT` a /\ b /\ c <=> (a /\c) /\ b`];
NHANH Fan.sum2_azim_fan;
STRIP_TAC;
ASM_CASES_TAC ` azim (vec 0) e x (rho_node1 FF x) = &0 `;
DOWN;
NHANH (MESON[AZIM_EQ_0_PI_IMP_COPLANAR]` azim v0 v1 w1 w2 = &0 
         ==> coplanar {v0, v1, w1, w2} `);
REWRITE_TAC[GSYM CROSS_DOT_COPLANAR];
STRIP_TAC;
ASSUME_TAC2 RHO_NODE_SET_IN_A_PLANE_IMP_POS_DIRECT;
UNDISCH_TAC ` n < l:num `;
FIRST_X_ASSUM NHANH;
DOWN;
ASM_REWRITE_TAC[GSYM ITER; ADD1];
PAT_ONCE_REWRITE_TAC `\x. x ==> ll ` [CROSS_TRIPLE];
SIMP_TAC[REAL_ARITH` a = &0 ==> ~( &0 < a ) `];
SUBGOAL_THEN ` azim (vec 0) e (rho_node1 FF x) y < azim (vec 0) e x y ` MP_TAC;
DOWN;
ASM_REWRITE_TAC[REAL_ARITH ` a < b + a <=> &0 < b `];
MESON_TAC[REWRITE_RULE[REAL_ARITH` a <= b <=> b = a \/ a < b `] AZIM_RANGE];
UNDISCH_TAC ` cyclic_set U (vec 0) (e:real^3) `;
NHANH Wrgcvdr_cizmrrh.CYCLIC_SET_IMP_NOT_COLLINEAR;
STRIP_TAC;
SUBGOAL_THEN` x IN (U:real^3 -> bool) /\ rho_node1 FF x IN U /\ y IN U ` MP_TAC;
ASM_REWRITE_TAC[GSYM ITER; ADD1];
EXPAND_TAC "U";
REWRITE_TAC[IN_ELIM_THM];
UNDISCH_TAC ` n < l:num `;
MESON_TAC[ARITH_RULE` a < n:num ==> a <= n /\ a + 1 <= n `];
FIRST_ASSUM NHANH;
STRIP_TAC;
ASM_CASES_TAC` azim (vec 0) e (rho_node1 (FF:real^3 # real^3 -> bool) x:real^3) y = &0 `;
REPLICATE_TAC 3 DOWN;
ONCE_REWRITE_TAC[SET_RULE` {a,b,c} = {b,c,a}`];
ONCE_REWRITE_TAC[TAUT` a ==> b ==> c <=> a /\ b ==> c `];
NHANH AZIM_EQ_0_ALT;
SIMP_TAC[];
STRIP_TAC;
UNDISCH_TAC ` ~collinear {vec 0, (e:real^3), rho_node1  (FF:real^3 # real^3 -> bool) x}`;
NHANH th3a;
SIMP_TAC[AFF_GT_2_1; IN_ELIM_THM; VECTOR_MUL_RZERO; VECTOR_ADD_LID];
REPEAT STRIP_TAC;
SUBGOAL_THEN ` {vec 0, x, rho_node1 FF (x:real^3)} SUBSET P ` ASSUME_TAC;
ASM_REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET];
UNDISCH_TAC ` U SUBSET (P:real^3 -> bool) `;
MATCH_MP_TAC (SET_RULE` a IN U /\ b IN U ==> U SUBSET P ==> a IN P /\ b IN P `);
UNDISCH_TAC ` x:real^3 IN U `;
UNDISCH_TAC ` rho_node1 (FF: real^3 # real^3 -> bool) x IN U `;
ASM_REWRITE_TAC[];
SIMP_TAC[];
ASSUME_TAC2 LOCAL_FAN_CHARACTER_OF_RHO_NODE;
SUBGOAL_THEN ` x IN (V:real^3 -> bool) ` ASSUME_TAC;
ASM_SIMP_TAC[];
MP_TAC2 LOCAL_FAN_ITER_RHO_NODE_IN_V;
DOWN;
FIRST_X_ASSUM NHANH;
STRIP_TAC;
SUBGOAL_THEN ` affine hull {vec 0, x, rho_node1 (FF:real^3 # real^3 -> bool) x} = P` MP_TAC;
MATCH_MP_TAC THREE_NOT_COLL_DETER_PLANE;
ASM_REWRITE_TAC[];
STRIP_TAC;
MP_TAC2 (SET_RULE` y IN U /\ U SUBSET (P:real^3 -> bool) ==> y IN P`);
EXPAND_TAC "P";
REWRITE_TAC[AFFINE_HULL_3; IN_ELIM_THM; VECTOR_MUL_RZERO; VECTOR_ADD_LID];
STRIP_TAC;
ASSUME_TAC2 RHO_NODE_SET_IN_A_PLANE_IMP_POS_DIRECT;
UNDISCH_TAC ` n < l:num`;
FIRST_X_ASSUM NHANH;
SWITCH_TAC ` x = ITER n (rho_node1 FF) (v:real^3) `;
ASM_REWRITE_TAC[GSYM ADD1; ITER];
NHANH_PAT `\x. x ==> y ` REAL_LT_IMP_NE;
SIMP_TAC[EQ_SYM_EQ; CROSS_DOT_COPLANAR];
NHANH (SMOOTH_GEN_ALL Trigonometry2.NONCOPLANAR_3_BASIS);
REWRITE_TAC[VECTOR_SUB_RZERO];
STRIP_TAC;
FIRST_X_ASSUM (MP_TAC o (SPEC ` y:real^3 `));
STRIP_TAC;
UNDISCH_TAC ` y = v' % x + w % rho_node1 (FF:real^3 # real^3 -> bool) x`;
PAT_ONCE_REWRITE_TAC`\x. x ==> y ` [VECTOR_ARITH` a + b % x = a + b % x + &0 % e `];
FIRST_ASSUM NHANH;
STRIP_TAC;
UNDISCH_TAC ` y = t2 % e + t3 % rho_node1 (FF:real^3 # real^3 -> bool) x`;
PAT_ONCE_REWRITE_TAC`\x. x ==> y ` [VECTOR_ARITH` a + b % xx = &0 % x + b % xx + a `];
FIRST_ASSUM NHANH;
FIRST_X_ASSUM (SUBST1_TAC o SYM);
STRIP_TAC;
FIRST_X_ASSUM SUBST_ALL_TAC;
UNDISCH_TAC ` y = &0 % x + t3 % rho_node1 FF x + &0 % (e:real^3)`;
REWRITE_TAC[VECTOR_MUL_LZERO; VECTOR_ADD_LID; VECTOR_ADD_RID];
MP_TAC2 Wrgcvdr_cizmrrh.LOCAL_FAN_IMP_FAN;
REWRITE_TAC[FAN; fan7];
REPEAT STRIP_TAC;
FIRST_X_ASSUM (MP_TAC o (SPECL [`{y:real^3}`;` {rho_node1  (FF:real^3 # real^3 -> bool) x }`]));
ANTS_TAC;
ASSUME_TAC2 LOCAL_FAN_ITER_RHO_NODE_IN_V;
REWRITE_TAC[IN_UNION; IN_ELIM_THM];
CONJ_TAC;
DISJ2_TAC;
EXISTS_TAC `y:real^3 `;
UNDISCH_TAC ` y IN U:real^3 -> bool`;
EXPAND_TAC "U";
REWRITE_TAC[IN_ELIM_THM];
STRIP_TAC;
REPLICATE_TAC 3 DOWN THEN PHA;
SIMP_TAC[];
DISJ2_TAC;
EXISTS_TAC ` rho_node1 (FF:real^3 # real^3 -> bool) x `;
EXPAND_TAC "x";
REWRITE_TAC[GSYM ITER];
DOWN THEN SIMP_TAC[];
UNDISCH_TAC ` ~( y:real^3 = rho_node1 FF (x:real^3) ) `;
SIMP_TAC[SET_RULE` ~( a = b ) ==> {a} INTER {b} = {} `; AFF_GE_EQ_AFFINE_HULL; AFFINE_HULL_SING];
SUBGOAL_THEN `! y. y:real^3 IN U ==> ~( vec 0 = y:real^3) ` MP_TAC;
UNDISCH_TAC ` !v. v IN U ==> ~collinear {v, vec 0, e:real^3} `;
DISCH_THEN NHANH;
NHANH th3b;
SIMP_TAC[EQ_SYM_EQ];
STRIP_TAC;
UNDISCH_TAC `y:real^3 IN U `;
UNDISCH_TAC ` rho_node1 FF (x:real^3) IN (U:real^3 -> bool) `;
FIRST_X_ASSUM NHANH;
STRIP_TAC THEN STRIP_TAC;
SUBGOAL_THEN ` y:real^3 IN aff_ge {vec 0} {y} INTER aff_ge {vec 0} {rho_node1 FF (x:real^3)} ` ASSUME_TAC;
DOWN THEN DOWN THEN DOWN;
PHA;
SIMP_TAC[IN_INTER; X_IN_AFF_GE11];
SIMP_TAC[AFF_GE11; IN_ELIM_THM; VECTOR_ARITH` t % vec 0 + x = x `];
STRIP_TAC;
EXISTS_TAC ` &1 - t3 `;
EXISTS_TAC ` t3:real `;
UNDISCH_TAC ` y = t3 % rho_node1  (FF:real^3 # real^3 -> bool) x`;
SIMP_TAC[];
REMOVE_TAC;
UNDISCH_TAC` &0 < t3`;
REAL_ARITH_TAC;
REMOVE_TAC;
DISCH_THEN SUBST_ALL_TAC;
DOWN THEN DOWN;
REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY];
MESON_TAC[];
MP_TAC (MESON[AZIM_RANGE]` &0 <= azim (vec 0) e (rho_node1 FF (x:real^3)) y`);
PHA;
NHANH (REAL_ARITH` &0 <= a /\ a < b ==> ~( b = &0 ) `);
STRIP_TAC;
SWITCH_TAC ` x = ITER n (rho_node1 FF) (v:real^3) `;
ASM_REWRITE_TAC[];
DOWN_TAC;
ONCE_REWRITE_TAC[SET_RULE` {a,b,c} = {b,c,a}`];
STRIP_TAC;
UNDISCH_TAC ` ~collinear {vec 0, e, (y:real^3) }`;
UNDISCH_TAC ` ~collinear {vec 0, e, x:real^3}`;
PHA;
NHANH AZIM_COMPL;
ASM_REWRITE_TAC[];
STRIP_TAC;
UNDISCH_TAC ` ~collinear {vec 0, e, y:real^3}`;
UNDISCH_TAC ` ~collinear {vec 0, e, rho_node1  (FF:real^3 # real^3 -> bool) x}`;
PHA;
NHANH AZIM_COMPL;
SWITCH_TAC ` azim (vec 0) e x y =
      azim (vec 0) e x (rho_node1  (FF:real^3 # real^3 -> bool) x) + azim (vec 0) e   (rho_node1 FF x) y `;
ASM_SIMP_TAC[REAL_ARITH` a - b < a - c <=> c < b `]]);;






let V_AZIM_SMALLEST_ELMS = prove_by_refinement
(`local_fan (V,E,FF) /\
 v IN V /\
 {ITER n (rho_node1 FF) v | n <= l} = U /\
 plane P /\
 vec 0 IN P /\
 U SUBSET P /\
 v cross rho_node1 FF v = e
 ==> ITER l (rho_node1 FF) v = ls /\
     (!i. i < l ==> ~(ls = ITER i (rho_node1 FF) v))
 ==> (!y. y IN U /\ ~(y = v) /\ ~(y = ls)
          ==> azim (vec 0) e ls v < azim (vec 0) e ls y)`,
[NHANH SEQUENCE_OF_RHO_NODE_IS_SUC;
STRIP_TAC THEN STRIP_TAC;
GEN_TAC;
EXPAND_TAC "U";
REWRITE_TAC[IN_ELIM_THM];
STRIP_TAC;
UNDISCH_TAC ` n <= l:num `;
SWITCH_TAC ` y = ITER n (rho_node1 FF) v:real^3 `;
ASM_REWRITE_TAC[];
EXPAND_TAC "y";
ASM_CASES_TAC ` n = 0 `;
FIRST_X_ASSUM SUBST_ALL_TAC;
DOWN;
ASM_REWRITE_TAC[ITER];
ASM_CASES_TAC` n = l:num `;
FIRST_X_ASSUM SUBST_ALL_TAC;
UNDISCH_TAC ` ITER l (rho_node1 FF) v = ls:real^3 `;
ASM_REWRITE_TAC[];
DOWN THEN DOWN;
SPEC_TAC (`n:num`,` n:num`);
INDUCT_TAC;
REWRITE_TAC[];
REPEAT STRIP_TAC;
ASM_CASES_TAC ` n' = 0 `;
ASM_REWRITE_TAC[ADD; ITER1];
UNDISCH_TAC ` !x n.
          x = ITER n (rho_node1 FF) (v:real^3) /\ n < l
          ==> (!y. y IN U /\ ~(y = x) /\ ~(y = rho_node1 FF x)
                   ==> azim (vec 0) e y x < azim (vec 0) e y (rho_node1 FF x))`;
DISCH_THEN (MP_TAC o (SPECL [`v:real^3 `;`0`]));
SIMP_TAC[ITER];
ANTS_TAC;
DOWN THEN DOWN THEN ARITH_TAC;
DISCH_THEN MATCH_MP_TAC;
FIRST_X_ASSUM SUBST_ALL_TAC;
SUBST_ALL_TAC (SYM ONE);
DOWN THEN DOWN;
PHA;
REWRITE_TAC[ARITH_RULE` ~( 1 = z ) /\ 1 <= z <=> 0 < z /\ 1 < z `];
UNDISCH_TAC ` !i. i < l ==> ~(ls = ITER i (rho_node1 FF) (v:real^3)) `;
DISCH_THEN NHANH;
SIMP_TAC[ITER1; ITER];
EXPAND_TAC "ls";
EXPAND_TAC "U";
STRIP_TAC;
REWRITE_TAC[IN_ELIM_THM];
MESON_TAC[LE_REFL];
UNDISCH_TAC ` ~(n' = 0)
      ==> ~(n' = l)
      ==> n' <= l
      ==> azim (vec 0) e ls v < azim (vec 0) e ls (ITER n' (rho_node1 FF) v)`;
PHA;
ANTS_TAC;
REPLICATE_TAC 3 DOWN THEN PHA;
ARITH_TAC;
STRIP_TAC;
UNDISCH_TAC ` !x n.
          x = ITER n (rho_node1 FF) v /\ n < l
          ==> (!y. y IN U /\ ~(y = x) /\ ~(y = rho_node1 FF x)
                   ==> azim (vec 0) e y x < azim (vec 0) e y (rho_node1 FF x))`;
DISCH_THEN (MP_TAC o (SPECL [`ITER n' (rho_node1 FF) v:real^3 `;` n':num`]));
ANTS_TAC;
UNDISCH_TAC` SUC n' <= l `;
REWRITE_TAC[ARITH_RULE` SUC n <= m <=> n < m `];
DISCH_THEN (MP_TAC o (SPEC ` ls:real^3 `));
ANTS_TAC;
UNDISCH_TAC ` SUC n' <= l `;
UNDISCH_TAC ` ~( SUC n' = l ) `;
PHA;
REWRITE_TAC[ARITH_RULE` ~(SUC n' = l) /\ SUC n' <= l <=> n' < l /\ SUC n' < l`];
FIRST_ASSUM NHANH;
SIMP_TAC[ITER];
STRIP_TAC;
EXPAND_TAC "ls";
EXPAND_TAC "U";
REWRITE_TAC[IN_ELIM_THM];
MESON_TAC[LE_REFL];
DOWN THEN PHA;
REWRITE_TAC[ITER];
REAL_ARITH_TAC]);;





let AZIM_LAST_POINT_IN_RHO_SET = prove_by_refinement
(`local_fan (V,E,FF) /\
 v IN V /\
 {ITER n (rho_node1 FF) v | n <= l} = U /\
 plane P /\
 vec 0 IN P /\
 U SUBSET P /\
 v cross rho_node1 FF v = e
 ==> ITER l (rho_node1 FF) v = ls /\
     (!i. i < l ==> ~(ls = ITER i (rho_node1 FF) v))
 ==> azim_cycle U (vec 0) e ls = v`,
[NHANH V_AZIM_SMALLEST_ELMS;
STRIP_TAC;
FIRST_X_ASSUM NHANH;
STRIP_TAC;
ASM_CASES_TAC ` l = 0 `;
UNDISCH_TAC ` {ITER n (rho_node1 FF) (v:real^3) | n <= l} = U`;
MP_TAC2 (ARITH_RULE` l = 0 ==> (!n. n <=l <=> n = 0 ) `);
REMOVE_TAC;
REWRITE_TAC[SET_RULE` {p n | n = 0 } = {p (0) }`; ITER];
DISCH_THEN (SUBST1_TAC o SYM);
EXPAND_TAC "ls";
FIRST_X_ASSUM SUBST1_TAC;
REWRITE_TAC[ITER];
MESON_TAC[Lvducxu.W_SUBSET_SINGLETON_IMP_IDE; SUBSET_REFL];


MATCH_MP_TAC Wrgcvdr_cizmrrh.IDENTIFY_AZIM_CYCLE;
ASSUME_TAC2 LOCAL_FAN_IMP_CYCLIC_SET;
UNDISCH_TAC ` ~( l = 0 ) `;
REWRITE_TAC[ARITH_RULE` ~( l = 0) <=> 0 < l `];
USE_FIRST ` !i. i < l ==> ~(ls = ITER i (rho_node1 FF) v:real^3) ` NHANH;
SUBGOAL_THEN ` ls:real^3 IN U /\ v IN U` ASSUME_TAC;
EXPAND_TAC "ls";
EXPAND_TAC "U";
REWRITE_TAC[IN_ELIM_THM];
MESON_TAC[LE_REFL; LE_0; ITER];

ASM_SIMP_TAC[ITER; LE_0];
STRIP_TAC;
CONJ_TAC;
DOWN THEN DOWN THEN DOWN THEN PHA;
SET_TAC[];
CONJ_TAC;
UNDISCH_TAC ` cyclic_set U (vec 0) (e:real^3) `;
NHANH Wrgcvdr_cizmrrh.CYCLIC_SET_IMP_NOT_COLLINEAR;
ASM_MESON_TAC[];
CONJ_TAC;
ONCE_REWRITE_TAC[GSYM IN];
ASM_REWRITE_TAC[];
REPEAT STRIP_TAC;
ASM_CASES_TAC ` q = v:real^3 `;
ASM_SIMP_TAC[REAL_LE_REFL];
DISJ1_TAC;
ASM_MESON_TAC[IN]]);;




let LOOP_MAP_IMP_DIFF_FIRST_ELMS = prove_by_refinement
(` (! v:A. v IN V ==> orbit_map f v = V) /\ v IN V /\ k < CARD V /\
l < k ==> ~( ITER k f v = ITER l f v ) `,
[REPEAT STRIP_TAC;
UNDISCH_TAC ` v:A IN V `;
REWRITE_TAC[];
FIRST_ASSUM NHANH;
STRIP_TAC;
SUBGOAL_THEN ` ITER l f (v:A) IN V ` ASSUME_TAC;
EXPAND_TAC "V";
REWRITE_TAC[orbit_map; IN_ELIM_THM; POWER_TO_ITER];
EXISTS_TAC `l:num`;
REWRITE_TAC[ARITH_RULE` x >= 0 `];
DOWN;
PHA;
FIRST_ASSUM NHANH;
SUBGOAL_THEN ` ITER ( k - l ) f ( ITER l f v ) = ITER l f (v:A) ` ASSUME_TAC;
ASM_SIMP_TAC[ITER_ADD; ARITH_RULE`l < k ==> ( k - l ) + l:num = k `];
DOWN;
UNDISCH_TAC ` l < k:num`;
PHA;
ONCE_REWRITE_TAC[ARITH_RULE` a < b <=> 0 < b - a `];
NHANH Lvducxu.ITER_CYCLIC_ORBIT;
SIMP_TAC[];
ASM_MESON_TAC[
ARITH_RULE` a < b:num /\ aa <= a - i ==> ~( aa = b ) `;
ISPECL [`f:A -> A `;` ITER l f (v:A) `; ` k - l:num` ] (GEN_ALL Wrgcvdr_cizmrrh.CARD_LE_K_OF_SET_K_FIRST_ELMS)]]);;




let CARD_IMAGE_INJ2 = prove(
` INJ f A B /\ FINITE A ==> CARD (IMAGE f A ) = CARD A `,
REWRITE_TAC[INJ] THEN STRIP_TAC THEN 
MATCH_MP_TAC CARD_IMAGE_INJ THEN ASM_REWRITE_TAC[]);;



let BIJ_IMP_CARD_EQ = prove(` BIJ f A B /\ FINITE A ==> CARD A = CARD B `,
REWRITE_TAC[BIJ] THEN 
NHANH Wrgcvdr_cizmrrh.SURJ_IMP_S2_EQ_IMAGE_S1 THEN MESON_TAC[CARD_IMAGE_INJ2]);;



let LOFA_IMP_DIS_ELMS = prove(
` local_fan (V,E,FF) /\ v IN V /\ l < CARD FF
==> (!i. i < l ==> ~(ITER l (rho_node1 FF) v = ITER i (rho_node1 FF) v))`,
STRIP_TAC THEN GEN_TAC THEN STRIP_TAC THEN MATCH_MP_TAC LOOP_MAP_IMP_DIFF_FIRST_ELMS THEN 
MP_TAC2 LOCAL_FAN_ORBIT_MAP_V THEN ASM_REWRITE_TAC[] THEN 
ABBREV_TAC ` k = (\x. FST (x:real^3 # real^3 )) ` THEN STRIP_TAC THEN 
MP_TAC2 Wrgcvdr_cizmrrh.BIJ_BETWEEN_FF_AND_V THEN 
ASSUME_TAC2 Wrgcvdr_cizmrrh.LOCAL_FAN_FINITE_FF THEN 
DOWN THEN REWRITE_TAC[TAUT` a ==> b ==> c <=> b /\ a ==> c `] THEN 
NHANH BIJ_IMP_CARD_EQ THEN PURE_ONCE_REWRITE_TAC[EQ_SYM_EQ] THEN 
ASM_SIMP_TAC[]);;




let AZIM_LAST_POINT_IN_RHO_SET2 = prove(
`    local_fan (V,E,FF) /\
       v IN V /\
       {ITER n (rho_node1 FF) v | n <= l} = U /\
       plane P /\
       vec 0 IN P /\
       U SUBSET P /\
       v cross rho_node1 FF v = e
       ==> ITER l (rho_node1 FF) v = ls /\
           l < CARD FF
       ==> azim_cycle U (vec 0) e ls = v `,
NHANH AZIM_LAST_POINT_IN_RHO_SET THEN STRIP_TAC THEN STRIP_TAC THEN 
FIRST_X_ASSUM MATCH_MP_TAC THEN EXPAND_TAC "ls" THEN 
REWRITE_TAC[] THEN MATCH_MP_TAC LOFA_IMP_DIS_ELMS THEN 
ASM_REWRITE_TAC[]);;




let KOMWBWC = prove_by_refinement(`!E V P l FF U e ls v.
 local_fan (V,E,FF) /\
       v IN V /\
       {ITER n (rho_node1 FF) v | n <= l} = U /\
 plane P /\
 vec 0 IN P /\
 U SUBSET P /\
 v cross rho_node1 FF v = e
 ==> cyclic_set U (vec 0) e /\
     (!i. i < l
          ==> &0 <
              (ITER i (rho_node1 FF) v cross ITER (i + 1) (rho_node1 FF) v) dot
              e) /\
     (!x n.
          x = ITER n (rho_node1 FF) v /\ n < l
          ==> azim_cycle U (vec 0) e x = rho_node1 FF x) /\
     (ITER l (rho_node1 FF) v = ls /\
l < CARD FF
      ==> azim_cycle U (vec 0) e ls = v)`,
[REPEAT GEN_TAC THEN STRIP_TAC; MP_TAC2 LOCAL_FAN_IMP_CYCLIC_SET;
STRIP_TAC; MP_TAC2 RHO_NODE_SET_IN_A_PLANE_IMP_POS_DIRECT;
REMOVE_TAC; MP_TAC2 AZIM_LAST_POINT_IN_RHO_SET2;
REMOVE_TAC; MP_TAC2 FIRST_AZIM_CYCLE_EQ_RHO_NODE]);;




(* -------------------------------- *)
(* ================================ *)
(* -------------------------------- *)
(*          lemma OZQVSFF           *)
(* ================================ *)

let interior_angle1 = new_definition
` interior_angle1 x FF v = azim x v (rho_node1 FF v) (@a. a,v IN FF)`;;



let WEDGE_GE_AZIM_LE = prove(
`! x v0 v1 w1 w2. x IN wedge_ge v0 v1 w1 w2 <=> azim v0 v1 w1 x <= azim v0 v1 w1 w2 `,
REWRITE_TAC[wedge_ge; IN_ELIM_THM; AZIM_RANGE]);;



let IN_WEDGE_IMP_AZIM_LE = prove(`! x y. y IN wedge_ge v0 v1 w1 w2 /\
 azim v0 v1 w1 x <= azim v0 v1 w1 y /\
 ~collinear {v0, v1, x} /\
 ~collinear {v0, v1, y} /\
 ~collinear {v0, v1, w1}
 ==> azim v0 v1 x y <= azim v0 v1 w1 w2`,
NHANH Fan.sum4_azim_fan THEN 
REWRITE_TAC[WEDGE_GE_AZIM_LE] THEN 
REPEAT GEN_TAC THEN 
MP_TAC (SPECL [`v0:real^3`;` v1:real^3`;`w1:real^3`;` x:real^3`] AZIM_RANGE) THEN 
REAL_ARITH_TAC);;




let LOFA_IMAGE_RHO_NODE_IDE = prove(
`local_fan (V,E,FF) ==> IMAGE (rho_node1 FF) V = V`,
STRIP_TAC THEN ASSUME_TAC2 LOCAL_FAN_ORBIT_MAP_V THEN 
ASM_SIMP_TAC[EQ_SYM_EQ; Lvducxu.W_EQ_ITS_ORBIT_IMP_EQ_ITS_IMAGE]);;



let EXISTS_INVERSE_OF_V = prove(
` local_fan (V,E,FF) /\ v IN V ==> ? vv. vv IN V /\ rho_node1 FF vv = v `,
NHANH LOFA_IMAGE_RHO_NODE_IDE THEN 
REWRITE_TAC[IMAGE] THEN SET_TAC[]);;



let LOFA_IN_V_SO_DO_RHO_NODE_V = prove(
` local_fan (V,E,FF) /\ v IN V ==> rho_node1 FF v IN V `,
NHANH LOCAL_FAN_ORBIT_MAP_V THEN STRIP_TAC THEN 
DOWN THEN FIRST_X_ASSUM (NHANH_PAT `\x. x ==> y `) THEN 
STRIP_TAC THEN EXPAND_TAC "V" THEN 
REWRITE_TAC[in_orbit_map1]);;


let HYP_MAPS_INVERSABLE = 
MESON[hypermap_lemma; PERMUTES_INVERSES_o]`! H:(A) hypermap.
inverse (face_map H) o (face_map H ) = I /\
inverse (node_map H) o (node_map H ) = I /\
inverse (edge_map H) o (edge_map H ) = I /\
(face_map H ) o inverse (face_map H)  = I /\
(node_map H ) o inverse (node_map H)  = I/\
(edge_map H ) o inverse (edge_map H)  = I`;;



let LOFA_DARTS_FF_UNION_SWITCH_FF = prove_by_refinement
(` local_fan (V,E,FF) ==> darts_of_hyp E V = FF UNION {v,w | w,v IN FF }`,
[REWRITE_TAC[local_fan];
LET_TAC;
REWRITE_TAC[dih2k; Lvducxu.HAS_ORD2_INTERPRET; Lvducxu.EDGE_MAP_RESO_INVERSE];
LET_TAC;
STRIP_TAC;
UNDISCH_TAC ` x':real^3 # real^3 IN dart H `;
FIRST_X_ASSUM NHANH;
ONCE_REWRITE_TAC[inverse2_hypermap_maps];
REWRITE_TAC[IMAGE_o];
ONCE_REWRITE_TAC[Lvducxu.ENF_IMAGE_ITSELF];
ASM_REWRITE_TAC[GSYM IMAGE_o; GSYM o_ASSOC; HYP_MAPS_INVERSABLE; I_O_ID;
GSYM Lvducxu.ENF_IMAGE_ITSELF];
UNDISCH_TAC ` FAN (vec 0:real^3, V, E) `;
NHANH ELMS_OF_HYPERMAP_HYP;
ASM_REWRITE_TAC[];
STRIP_TAC;
ASM_REWRITE_TAC[];
STRIP_TAC;
ASM_REWRITE_TAC[];
MATCH_MP_TAC (MESON[]` x = y ==> f x = f y`);
REWRITE_TAC[IMAGE; EXTENSION; IN_ELIM_THM; ee_of_hyp2];
SUBGOAL_THEN ` x' IN dart (H:(real^3 # real^3) hypermap) ` MP_TAC;
FIRST_X_ASSUM MP_TAC;
ASM_REWRITE_TAC[];
NHANH lemma_face_subset;
REWRITE_TAC[SUBSET];
STRIP_TAC;
GEN_TAC THEN EQ_TAC;
STRIP_TAC;
DOWN THEN DOWN;
FIRST_X_ASSUM NHANH;
ASM_SIMP_TAC[];
REPEAT STRIP_TAC;
EXISTS_TAC ` SND (x''':real^3 #real^3 ) `;
EXISTS_TAC ` FST (x''':real^3 #real^3 ) `;
ASM_REWRITE_TAC[];
STRIP_TAC;
EXISTS_TAC `w:real^3, v:real^3 `;
DOWN THEN DOWN THEN PHA;
FIRST_X_ASSUM NHANH;
ASM_SIMP_TAC[]]);;





let SELF_CYCLIC_IMP_FINITE = prove_by_refinement
(`(! x:A. x IN V ==> V = orbit_map f x ) ==> FINITE V `,
[ASM_CASES_TAC `V:A -> bool  = {} `;
ASM_REWRITE_TAC[FINITE_EMPTY];
DOWN;
REWRITE_TAC[SET_RULE` ~( x = {}) <=> ? a. a IN x `];
REPEAT STRIP_TAC;
UNDISCH_TAC` a:A IN V `;
FIRST_ASSUM NHANH;
STRIP_TAC;
MP_TAC (ISPECL [`f:A -> A `;` a:A `] in_orbit_map1);
FIRST_X_ASSUM (SUBST1_TAC o SYM);
FIRST_ASSUM NHANH;
STRIP_TAC;
SUBGOAL_THEN ` a:A IN V ` MP_TAC;
FIRST_X_ASSUM ACCEPT_TAC;
FIRST_ASSUM SUBST1_TAC;
REWRITE_TAC[orbit_map; IN_ELIM_THM; POWER_TO_ITER; GSYM ITER_ALT];
STRIP_TAC;
FIRST_X_ASSUM (MP_TAC o SYM);
MP_TAC (ARITH_RULE` 0 < SUC n `);
PHA;
NHANH Lvducxu.ITER_CYCLIC_ORBIT;
UNDISCH_TAC ` a:A IN V `;
FIRST_X_ASSUM NHANH;
ASM_SIMP_TAC[orbit_map; POWER_TO_ITER; GSYM ITER_ALT];
REPEAT STRIP_TAC;
REWRITE_TAC[
SET_RULE` {ITER n' f a | n' < n} = {y | ? n' . n' < n /\ y = ITER n' f a }`;
Wrgcvdr_cizmrrh.FINITE_OF_N_FIRST_ELMS]]);;






let SELF_CYCLIC_IMP_BIJ = prove(
` (!x. x IN S ==> S = orbit_map f x) ==> BIJ f S S `,
NHANH SELF_CYCLIC_IMP_FINITE THEN 
MESON_TAC[Lvducxu.FIN_LOOP_IMP_BIJ_ITSELF]);;




let LOFA_IN_E_IMP_IN_FF = prove(
` local_fan (V,E,FF) /\ {a,b} IN E ==> a,b IN FF \/ b,a IN FF `,
REWRITE_TAC[Wrgcvdr_cizmrrh.IN_E_IFF_IN_ORD_E] THEN 
NHANH LOFA_DARTS_FF_UNION_SWITCH_FF THEN 
REWRITE_TAC[darts_of_hyp] THEN PHA THEN 
NHANH (SET_RULE` a UNION b = x /\ xx IN a ==> xx IN x `) THEN 
REWRITE_TAC[IN_UNION; IN_ELIM_THM; PAIR_EQ] THEN 
STRIP_TAC THENL [ASM_REWRITE_TAC[];ASM_REWRITE_TAC[]]);;




let LOFA_IMP_EE_TWO_ELMS = prove_by_refinement
(` local_fan (V,E,FF) /\ vv IN V /\ rho_node1 FF vv = v
==> EE v E = {rho_node1 FF v, vv }`,
[NHANH LOCAL_FAN_RHO_NODE_PROS2;
IMP_TAC THEN STRIP_TAC;
STRIP_TAC;
SUBGOAL_THEN ` v:real^3 IN V ` ASSUME_TAC;
ASM_MESON_TAC[LOFA_IN_V_SO_DO_RHO_NODE_V];
DOWN;
UNDISCH_TAC ` vv:real^3 IN V `;
FIRST_ASSUM NHANH;
ASSUME_TAC2 (
REWRITE_RULE[TAUT` a /\ b ==> c <=> a ==> b ==> c `; 
RIGHT_FORALL_IMP_THM] (GEN `x:real^3 # real^3 ` LOCAL_FAN_IN_FF_IN_ORD_PAIRS));
FIRST_ASSUM NHANH;
REPEAT STRIP_TAC;
REWRITE_TAC[EXTENSION; EE; IN_ELIM_THM];
GEN_TAC THEN EQ_TAC;
UNDISCH_TAC ` local_fan (V,E,FF) `;
PHA;
NHANH LOFA_IN_E_IMP_IN_FF;
UNDISCH_TAC ` !x. x IN FF ==> x = FST x,rho_node1 FF (FST (x:real^3 # real^3))`;
DISCH_THEN NHANH;
STRIP_TAC;
DOWN;
SIMP_TAC[PAIR_EQ; IN_INSERT];
DOWN;
SIMP_TAC[PAIR_EQ];
ASSUME_TAC2 (SPEC `v:real^3 ` (GEN `y:real^3 ` LOCAL_FAN_IMP_IN_V));
MP_TAC2 LOCAL_FAN_ORBIT_MAP_V;
SIMP_TAC[EQ_SYM_EQ];
NHANH SELF_CYCLIC_IMP_BIJ;
REWRITE_TAC[BIJ; INJ];
STRIP_TAC;
UNDISCH_TAC ` x IN V /\ v:real^3 IN V`;
PHA;
EXPAND_TAC "v";
STRIP_TAC;
DOWN;
UNDISCH_TAC ` x:real^3 IN V `;
UNDISCH_TAC ` vv:real^3 IN V `;
PHA;
FIRST_X_ASSUM NHANH;
SIMP_TAC[IN_INSERT];
REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY];
STRIP_TAC;
UNDISCH_TAC `v, rho_node1 FF v IN FF `;
UNDISCH_TAC ` local_fan (V,E,FF) `;
PHA;
NHANH LOCAL_FAN_IN_FF_IN_ORD_PAIRS;
ASM_SIMP_TAC[Wrgcvdr_cizmrrh.IN_E_IFF_IN_ORD_E];
UNDISCH_TAC ` vv,rho_node1 FF vv IN FF `;
FIRST_X_ASSUM NHANH;
ASM_REWRITE_TAC[];
UNDISCH_TAC `local_fan (V,E,FF) `;
PHA;
NHANH LOCAL_FAN_IN_FF_IN_ORD_PAIRS;
SIMP_TAC[Wrgcvdr_cizmrrh.IN_ORD_E_EQ_IN_E; INSERT_COMM]]);;


let LOFA_CARD_EE_V_2 = prove(
`local_fan (V,E,FF) /\ vv IN V /\ rho_node1 FF vv = v
==> CARD (EE v E ) = 2 `, NHANH LOFA_IMP_EE_TWO_ELMS THEN 
SIMP_TAC[Geomdetail.CARD_SET2] THEN NHANH LOCAL_FAN_IMP_NOT_SEMI_IDE THEN 
MESON_TAC[]);;



let LOFA_CARD_EE_V_1 = prove(
`local_fan (V,E,FF) /\ v IN V ==> CARD (EE v E) = 2 `,
NHANH EXISTS_INVERSE_OF_V THEN 
MESON_TAC[ LOFA_CARD_EE_V_2]);;





let RHO_NODE_INVERSE_POINT = MESON[]`
 w,v IN FF /\ (! ww. ww,v IN FF ==> ww = w )
==> (@a. a,v IN FF ) = w `;;



let AZIM_CYCLE_TWO_POINT_SET = prove_by_refinement
(`azim_cycle {a,b} v w a = b `,
[ASM_CASES_TAC ` a = b:real^3`;
ASM_REWRITE_TAC[INSERT_INSERT];
REWRITE_TAC[ MATCH_MP Lvducxu.W_SUBSET_SINGLETON_IMP_IDE (SET_RULE` {p:real^3} SUBSET {p}`)];
ABBREV_TAC ` W = {a,b:real^3} `;
ABBREV_TAC ` u = azim_cycle W v w a `;
ABBREV_TAC ` p = a:real^3`;
SUBGOAL_THEN ` ~(u = p) /\
                   W u /\
                   (!q. ~(q = p) /\ W q
                        ==> azim v w p u < azim v w p q \/
                            azim v w p u = azim v w p q /\
                            norm (projection (w - v) (u - v)) <=
                            norm (projection (w - v) (q - v))) ` ASSUME_TAC;
EXPAND_TAC "u";
SUBGOAL_THEN ` ~(W SUBSET {p:real^3}) ` MP_TAC;
ASM SET_TAC[];
REWRITE_TAC[azim_cycle];
SIMP_TAC[];
DISCH_TAC;
CONV_TAC SELECT_CONV;
EXISTS_TAC `b:real^3 `;
ASM_REWRITE_TAC[];
EXPAND_TAC "W";
CONJ_TAC;
SET_TAC[];
REWRITE_TAC[SET_RULE` ~(q = p) /\ {p, b} q <=> q = b /\ ~( q = p)`];
SIMP_TAC[REAL_LE_REFL];
ASM SET_TAC[]]);;




let LOFA_IMP_BIJ_VV = prove(
` local_fan (V,E,FF) ==> BIJ (rho_node1 FF ) V V `,
NHANH LOCAL_FAN_ORBIT_MAP_V THEN STRIP_TAC THEN 
MATCH_MP_TAC SELF_CYCLIC_IMP_BIJ THEN DOWN THEN 
SIMP_TAC[EQ_SYM_EQ]);;






let MOST_EXPAND_IN_WEDGE_GE = prove_by_refinement
(` ~collinear {v0,v1, w1} /\
   ~collinear {v0,v1, w2} /\
   ~collinear {v0,v1, x} /\ ~ collinear {v0, v1, y} /\
y IN wedge_ge v0 v1 w1 w2 /\
azim v0 v1 w1 x <= azim v0 v1 w1 y /\
azim v0 v1 x y = azim v0 v1 w1 w2 ==>
azim v0 v1 w1 x = &0 /\
azim v0 v1 y w2 = &0 `,
[REWRITE_TAC[wedge_ge; IN_ELIM_THM];
STRIP_TAC;
SUBGOAL_THEN ` azim v0 v1 w1 w2 = azim v0 v1 w1 y + azim v0 v1 y w2` ASSUME_TAC;
MATCH_MP_TAC Fan.sum4_azim_fan;
ASM_REWRITE_TAC[];
SUBGOAL_THEN ` azim v0 v1 w1 y = azim v0 v1 w1 x + azim v0 v1 x y` ASSUME_TAC;
MATCH_MP_TAC Fan.sum4_azim_fan;
ASM_REWRITE_TAC[];
UNDISCH_TAC` azim v0 v1 w1 w2 = azim v0 v1 w1 y + azim v0 v1 y w2 `;
ASM_REWRITE_TAC[];
MP_TAC (
SPECL [`v0:real^3`; ` v1:real^3 `;`w1:real^3 `;` x:real^3 `] AZIM_RANGE);
MP_TAC (
SPECL [`v0:real^3`; ` v1:real^3 `;`y:real^3 `;` w2:real^3 `] AZIM_RANGE);
REAL_ARITH_TAC]);;





let OZQVSFF = prove_by_refinement
(`! u w. convex_local_fan (V,E,FF) /\ 
{u,v,w} SUBSET V /\
plane P /\
{vec 0, u, v, w} SUBSET P /\
{u,w} INTER aff {vec 0, v} = {} /\
~( aff {v, vec 0} INTER conv0 {w,u} = {}  ) ==>
interior_angle1 (vec 0) FF v = pi /\
rho_node1 FF v IN P /\ ivs_rho_node1 FF v IN P`,
[MATCH_MP_TAC (MESON[REAL_ARITH` a <= b \/ b <= a `]`
(! x y. P x y ==> P y x ) /\ (! x y. azim (vec 0) v (rho_node1 FF v) y <=
azim (vec 0) v (rho_node1 FF v) x ==> P x y ) ==>
(! x y. P x y ) `);
CONJ_TAC;
SIMP_TAC[INSERT_COMM];
REWRITE_TAC[convex_local_fan; INSERT_SUBSET; EMPTY_SUBSET; Trigonometry2.INSERT_INTER_EMPTY];
NHANH Wrgcvdr_cizmrrh.LOCAL_FAN_IMP_FAN;
NHANH FAN_IMP_V_DIFF;
REPLICATE_TAC 4 STRIP_TAC;
UNDISCH_TAC ` !v:real^3. v IN V ==> ~(v = vec 0)`;
DISCH_TAC;
UNDISCH_TAC` v:real^3 IN V `;
FIRST_ASSUM NHANH;
STRIP_TAC;
UNDISCH_TAC ` ~( u IN aff {vec 0, v:real^3})`;
UNDISCH_TAC ` ~( w:real^3 IN aff {vec 0, v})`;
ASM_SIMP_TAC[INSERT_COMM; Trigonometry2.NOT_EQ_IMP_AFF_AND_COLL3];
REPEAT DISCH_TAC;
SUBGOAL_THEN ` azim (vec 0) v w u = pi ` ASSUME_TAC;
DOWN THEN DOWN THEN PHA;
PAT_ONCE_REWRITE_TAC `\x. y /\ x ==> k ` [INSERT_COMM];
NHANH (SIMP_RULE[INSERT_COMM] Ldurdpn.LDURDPN);
STRIP_TAC;
FIRST_X_ASSUM SUBST1_TAC;
CONJ_TAC;
EXISTS_TAC `P:real^3 -> bool`;
ASM_REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET];
FIRST_ASSUM ACCEPT_TAC;
UNDISCH_TAC ` v:real^3 IN V `;
ASSUME_TAC2 LOCAL_FAN_RHO_NODE_PROS2;
DOWN THEN IMP_TAC;
REMOVE_TAC;
DISCH_THEN NHANH;
UNDISCH_TAC ` !x:real^3 #real^3. x IN FF ==> azim_in_fan x E <= pi /\ V SUBSET wedge_in_fan_ge x E`;
DISCH_TAC;
FIRST_ASSUM NHANH;
REWRITE_TAC[SUBSET];
STRIP_TAC;
UNDISCH_TAC` w:real^3 IN V `;
UNDISCH_TAC ` u:real^3 IN V `;
FIRST_ASSUM NHANH;
REWRITE_TAC[wedge_in_fan_ge];
SUBGOAL_THEN ` CARD (EE (v:real^3) E ) = 2 ` MP_TAC;
MATCH_MP_TAC LOFA_CARD_EE_V_1;
ASM_REWRITE_TAC[];
NHANH (ARITH_RULE ` a = 2 ==> a > 1 `);
SIMP_TAC[];
REPLICATE_TAC 3 STRIP_TAC;
UNDISCH_TAC ` v:real^3 IN V `;
UNDISCH_TAC ` local_fan (V,E,FF) `;
PHA;
NHANH EXISTS_INVERSE_OF_V;
STRIP_TAC;
ASSUME_TAC2 LOFA_IMP_EE_TWO_ELMS;
UNDISCH_TAC ` azim_in_fan (v,rho_node1 FF v) E <= pi `;
REWRITE_TAC[azim_in_fan];
LET_TAC;
ASM_REWRITE_TAC[];
SUBGOAL_THEN` azim (vec 0) v w u <= azim (vec 0) v (rho_node1 FF v) d ` ASSUME_TAC;
MATCH_MP_TAC IN_WEDGE_IMP_AZIM_LE;
ASM_SIMP_TAC[INSERT_COMM];
UNDISCH_TAC ` v,rho_node1 FF v IN FF `;
UNDISCH_TAC ` local_fan (V,E,FF) `;
PHA;
NHANH LOCAL_FAN_IN_FF_NOT_COLLINEAR;
SIMP_TAC[INSERT_COMM];
DOWN;
UNDISCH_TAC ` azim (vec 0) v w u = pi `;
PHA;
DISCH_TAC;
REWRITE_TAC[interior_angle1];
SUBGOAL_THEN `(@a. a,v IN (FF:real^3 # real^3 -> bool)) = vv ` ASSUME_TAC;
MATCH_MP_TAC RHO_NODE_INVERSE_POINT;
MP_TAC2 LOCAL_FAN_RHO_NODE_PROS2;
STRIP_TAC;
UNDISCH_TAC ` vv:real^3 IN V `;
FIRST_X_ASSUM NHANH;
ASM_SIMP_TAC[];
REPEAT STRIP_TAC;
DOWN;
FIRST_ASSUM NHANH;
REWRITE_TAC[PAIR_EQ];
UNDISCH_TAC `local_fan (V,E,FF) `;
PHA;
NGOAC;
NHANH LOCAL_FAN_IMP_IN_V;
UNDISCH_TAC ` vv:real^3 IN V `;
NHANH LOFA_IMP_BIJ_VV;
REWRITE_TAC[BIJ; INJ];
EXPAND_TAC "v";
MESON_TAC[];
SUBGOAL_THEN ` azim_cycle   (EE v E) (vec 0) v (rho_node1 FF v) = vv ` ASSUME_TAC;
UNDISCH_TAC ` EE v E = {rho_node1 FF v, vv} `;
EXPAND_TAC "d";
SIMP_TAC[AZIM_CYCLE_TWO_POINT_SET];
FIRST_X_ASSUM SUBST_ALL_TAC;
ONCE_REWRITE_TAC[TAUT` a/\ b <=> a /\ (a ==> b ) `];
CONJ_TAC;
FIRST_X_ASSUM SUBST1_TAC;
DOWN;
ASM_REWRITE_TAC[];
REAL_ARITH_TAC;
FIRST_ASSUM SUBST1_TAC;
STRIP_TAC;

SUBGOAL_THEN ` ~collinear {v, rho_node1 FF v, vec 0} /\ ~collinear {d, v, vec 0} ` ASSUME_TAC;


ASSUME_TAC2 LOCAL_FAN_CHARACTER_OF_RHO_NODE;
UNDISCH_TAC ` v:real^3 IN V `;
UNDISCH_TAC ` vv:real^3 IN V `;

FIRST_X_ASSUM NHANH;
ASM_SIMP_TAC[INSERT_COMM];



SUBGOAL_THEN ` azim (vec 0) v (rho_node1 FF v) w = &0 /\
azim (vec 0) v u vv = &0 ` MP_TAC;
MATCH_MP_TAC MOST_EXPAND_IN_WEDGE_GE;
ASM_SIMP_TAC[INSERT_COMM];



DOWN THEN STRIP_TAC;
UNDISCH_TAC `~ collinear {v, w, vec 0:real^3 }`;
UNDISCH_TAC `~ collinear {v, rho_node1 FF v, vec 0:real^3 }`;
ONCE_REWRITE_TAC[TAUT` a ==> b ==> c <=> b /\a ==> c `];
ONCE_REWRITE_TAC[SET_RULE` {a,b,c} = {c,a,b} `];
NHANH AZIM_EQ_0_ALT;
NHANH AZIM_EQ_0_SYM;
STRIP_TAC;
ASSUME_TAC2 (SIMP_RULE[INSERT_COMM] (SPECL [` vec 0:real^3`;` v:real^3 `;`u:real^3 `;` d:real^3 `] AZIM_EQ_0_ALT));
DOWN;
ASM_SIMP_TAC[];
DOWN;
FIRST_X_ASSUM (SUBST1_TAC o SYM);
ASM_SIMP_TAC[ivs_rho_node1];
REMOVE_TAC THEN REMOVE_TAC;
MATCH_MP_TAC (
SET_RULE` A SUBSET S /\ B SUBSET S ==> a IN A /\ b IN B ==> a IN S /\ b IN S `);
CONJ_TAC;
SUBGOAL_THEN ` affine hull {vec 0,v,w:real^3} = P ` MP_TAC;
MATCH_MP_TAC THREE_NOT_COLL_DETER_PLANE;
ASM_REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET];
DISCH_THEN (SUBST1_TAC o SYM);
REWRITE_TAC[SET_RULE` {a,b,c} = {a,b} UNION {c}`];
REWRITE_TAC[INSERT_UNION;AFF_GT_SUBSET_AFFINE_HULL; UNION_EMPTY];
SUBGOAL_THEN ` affine hull {u,v,vec 0:real^3} = P ` MP_TAC;
MATCH_MP_TAC THREE_NOT_COLL_DETER_PLANE;
ASM_REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET];
DISCH_THEN (SUBST1_TAC o SYM);
REWRITE_TAC[SET_RULE` {c,a,b} = {a,b} UNION {c}`];
REWRITE_TAC[INSERT_UNION;AFF_GT_SUBSET_AFFINE_HULL; UNION_EMPTY]]);;



(* ============================================= *)
(* ============================================= *)
(* ============================================= *)
(* ============================================= *)
(*              lemma KCHMAMG                    *)


let REAL_LT_DIV_NEG = prove(` a < &0 /\ b < &0 ==> &0 < a /b `,
SIMP_TAC [REAL_FIELD` b < &0 ==> a / b = ( -- a) / ( -- b ) `; REAL_ARITH ` 
b < &0 <=> &0 < -- b `; REAL_LT_DIV]);;


let IN_CONV0 = prove_by_refinement
(` &0 < a /\ &0 < b ==> &1 / ( a + b ) % ( a % x + b % (y:real^N)) IN conv0 {x,y}`,
[REWRITE_TAC[Geomdetail.CONV0_SET2; IN_ELIM_THM];
DISCH_TAC;
EXISTS_TAC ` a / ( a + b ) `;
EXISTS_TAC ` b / ( a + b ) `;
CONJ_TAC;
MATCH_MP_TAC REAL_LT_DIV;
DOWN;
REAL_ARITH_TAC;
CONJ_TAC;
MATCH_MP_TAC REAL_LT_DIV;
DOWN;
REAL_ARITH_TAC;
CONJ_TAC;
DOWN;
CONV_TAC REAL_FIELD;
REWRITE_TAC[VECTOR_ADD_LDISTRIB; VECTOR_MUL_ASSOC;
REAL_ARITH` &1 / x * y = y / x `]]);;


(* lemma about intersection between conv0 {x,y} and
aff {w,v} in affine hull {a,b,c} *)



let INTERSECTION_LEMMA = prove_by_refinement
(` ~ ( z = x ) /\ DISJOINT {x:real^N} {v,w} /\ ~( x = u ) /\
 ~(aff_gt {x} {v, w} INTER aff_lt {x} {u} = {}) /\
z IN affine hull {x,v,w} ==>
(? a b t. {a,b} SUBSET {u,v,w} /\ t IN aff {x,z} INTER conv0 {a,b})`,
[REWRITE_TAC[AFFINE_HULL_3; IN_ELIM_THM; REAL_ARITH` a + b = &1 <=> &1 - b = a `];
STRIP_TAC;
DOWN;
ASM_CASES_TAC ` &0 < v' /\ &0 < w' `;
ASM_REWRITE_TAC[VECTOR_ARITH` a = x % b + c + d <=> a - x % b = c + d`];
STRIP_TAC;
EXISTS_TAC ` v:real^N `;
EXISTS_TAC `w:real^N `;
EXISTS_TAC ` (&1 / (v' + w') ) % ( z - u' % x:real^N ) `;
REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET; IN_INSERT; AFF2;
 Geomdetail.CONV0_SET2; IN_INTER; IN_ELIM_THM];
CONJ_TAC;
EXISTS_TAC ` ( -- u') / (v' + w' ) `;
REWRITE_TAC[VECTOR_SUB_LDISTRIB; VECTOR_MUL_ASSOC; REAL_ARITH` (&1 / a ) * b = b / a `];
ASM_SIMP_TAC [REAL_FIELD` &0 < v' /\ &0 < w' ==> &1 - --u' / (v' + w') = (u' + v' + w' ) / (v' + w') `];
EXPAND_TAC "u'";
REWRITE_TAC[REAL_ARITH` &1 - ( a + b) + a + b = &1 `];
VECTOR_ARITH_TAC;
EXISTS_TAC ` v' / ( v' + w') `;
EXISTS_TAC ` w' / ( v' + w') `;
CONJ_TAC;
MATCH_MP_TAC REAL_LT_DIV;
ASM_REWRITE_TAC[];
MATCH_MP_TAC REAL_LT_ADD;
FIRST_X_ASSUM ACCEPT_TAC;
CONJ_TAC;
MATCH_MP_TAC REAL_LT_DIV;
ASM_REWRITE_TAC[];
MATCH_MP_TAC REAL_LT_ADD;
FIRST_X_ASSUM ACCEPT_TAC;
ASM_SIMP_TAC[REAL_FIELD` &0 < v' /\ &0 < w' ==> v' / (v' + w') + w' / (v' + w') = &1`];
REWRITE_TAC[VECTOR_ADD_LDISTRIB; VECTOR_MUL_ASSOC];
ASM_SIMP_TAC[REAL_FIELD` &0 < v' /\ &0 < w' ==> &1 / (v' + w') * v' = v' / ( v' + w') /\ (&1 / (v' + w') * w') = w' / ( v' + w') `];

(* CASES v' < &0 and w' < &0 *)
ASM_CASES_TAC ` v' < &0 /\ w' < &0 `;
ASM_REWRITE_TAC[VECTOR_ARITH` a = x % b + c + d <=> a - x % b = c + d`];
STRIP_TAC;
EXISTS_TAC ` v:real^N `;
EXISTS_TAC `w:real^N `;
EXISTS_TAC ` (&1 / (v' + w') ) % ( z - u' % x:real^N ) `;
REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET; IN_INSERT; AFF2;
 Geomdetail.CONV0_SET2; IN_INTER; IN_ELIM_THM];
CONJ_TAC;
EXISTS_TAC ` ( -- u') / (v' + w' ) `;
REWRITE_TAC[VECTOR_SUB_LDISTRIB; VECTOR_MUL_ASSOC; REAL_ARITH` (&1 / a ) * b = b / a `];
ASM_SIMP_TAC [REAL_FIELD`  v' < &0 /\ w' < &0 ==> &1 - --u' / (v' + w') = (u' + v' + w' ) / (v' + w') `];
EXPAND_TAC "u'";
REWRITE_TAC[REAL_ARITH` &1 - ( a + b) + a + b = &1 `];
VECTOR_ARITH_TAC;
EXISTS_TAC ` v' / ( v' + w') `;
EXISTS_TAC ` w' / ( v' + w') `;
CONJ_TAC;
MATCH_MP_TAC REAL_LT_DIV_NEG;
ASM_SIMP_TAC[REAL_ARITH` a < &0 /\ b < &0 ==> a + b < &0`];


CONJ_TAC;
MATCH_MP_TAC REAL_LT_DIV_NEG;
ASM_SIMP_TAC[REAL_ARITH` a < &0 /\ b < &0 ==> a + b < &0`];

ASM_SIMP_TAC[REAL_FIELD` v' < &0 /\ w' < &0 ==> v' / (v' + w') + w' / (v' + w') = &1`];
REWRITE_TAC[VECTOR_ADD_LDISTRIB; VECTOR_MUL_ASSOC];
ASM_SIMP_TAC[REAL_FIELD` v' < &0 /\ w' < &0 ==> &1 / (v' + w') * v' = v' / ( v' + w') /\ (&1 / (v' + w') * w') = w' / ( v' + w') `];


UNDISCH_TAC ` ~(aff_gt {x} {v, w} INTER aff_lt {x} {u:real^N} = {})`;
UNDISCH_TAC ` ~ ( x = u:real^N) `;

ASM_SIMP_TAC[Planarity.AFF_GT_1_2; AFF_LT_1_1; SET_RULE` ~( a INTER b = {} ) <=> ? x. x IN a /\ x IN b `; IN_ELIM_THM; SET_RULE` ~( a = b ) <=> DISJOINT {a} {b} `];
REPEAT STRIP_TAC;
FIRST_ASSUM (SUBST1_TAC o SYM);
DOWN_TAC;

ASM_CASES_TAC ` v' = &0 \/ w' = &0`;
DOWN;
SPEC_TAC (`v:real^N`,`v:real^N `);
SPEC_TAC (`w:real^N`,`w:real^N `);

SPEC_TAC (`v':real`,`v':real `);
SPEC_TAC (`w':real`,`w':real `);
SPEC_TAC (`t2:real`,`t2:real `);
SPEC_TAC (`t3:real`,`t3:real `);
MATCH_MP_TAC (MESON[]` (! t3 t2 x y a b. P x y a b t3 t2 ==> P y x b a t2 t3 ) /\ (! x y a b t3 t2 . y = &0 ==> P x y a b t3 t2)
==> (! t3 t2 x y a b. (y = &0 \/ x = &0 ) ==> P x y a b t3 t2) `);

CONJ_TAC;
ONCE_REWRITE_TAC[TAUT` a ==> b ==> c <=> b ==> a ==> c `];
REPEAT GEN_TAC;
DISCH_TAC;
ANTS_TAC;
DOWN;
SIMP_TAC[INSERT_COMM; REAL_ADD_SYM; CONJ_SYM; VECTOR_ADD_SYM];
ONCE_REWRITE_TAC[EQ_SYM_EQ];
SIMP_TAC[];
SIMP_TAC[INSERT_COMM];

REPEAT GEN_TAC;


DISCH_THEN SUBST_ALL_TAC;
STRIP_TAC;
DOWN;
EXPAND_TAC "u'";
REWRITE_TAC[VECTOR_ARITH` z = (&1 - (&0 + w')) % x + &0 % v + w' % w <=>
z - x = w' % ( w - x ) `];
DOWN;
ASM_REWRITE_TAC[];
ONCE_REWRITE_TAC[VECTOR_ARITH ` t1 % x + t2 % v + t3 % w = t1' % x + t2' % u
<=> (t1 + t2 + t3 ) % x + t2 % ( v - x ) + t3 % ( w - x ) = ( t1' + t2' ) % x + t2' % (u - x ) `];
ASM_REWRITE_TAC[VECTOR_ARITH` x + a = x + b <=> a = b:real^N `; VECTOR_ARITH `
a % z + b % x = y <=> b % x = y + -- a % z `];



ASM_CASES_TAC ` w' = &0 `;
ASM_REWRITE_TAC[VECTOR_MUL_LZERO; VECTOR_SUB_EQ];
MP_TAC2 (REAL_ARITH ` &0 < t3 ==> ~( t3 = &0 ) `);
SIMP_TAC[Ldurdpn.VECTOR_SCALE_CHANGE];
REWRITE_TAC[VECTOR_MUL_ASSOC; VECTOR_ADD_LDISTRIB;
VECTOR_SUB_LDISTRIB; VECTOR_ARITH` a = x - y + b % d - t <=> a + y + t = x + b % d `];
REPEAT STRIP_TAC;
EXISTS_TAC `u:real^N`;
EXISTS_TAC `v:real^N `;

EXISTS_TAC ` (t3 / (w' * (t2' - t2 ))) % ( z - x + ((w' * &1 / t3) * t2') % x + ((w' * &1 / t3) * --t2) % (x:real^N) ) `;
REWRITE_TAC[INSERT_SUBSET; IN_INSERT; EMPTY_SUBSET; IN_INTER;AFF2; IN_ELIM_THM];

CONJ_TAC;
EXISTS_TAC ` &1 - t3 / (w' * (t2' - t2)) `;
DOWN THEN DOWN;
REWRITE_TAC[REAL_ARITH` (w' * &1 / t3) * t2' = t2' * w' / t3 `; GSYM VECTOR_ADD_RDISTRIB; VECTOR_ADD_LDISTRIB; VECTOR_SUB_LDISTRIB];
REWRITE_TAC[REAL_ARITH` a - ( a - b ) = b `];

REWRITE_TAC[VECTOR_SUB_RDISTRIB;
VECTOR_ARITH` x - t + z = zz - t + x:real^N <=> z = zz `];
STRIP_TAC THEN STRIP_TAC;
REWRITE_TAC[VECTOR_MUL_ASSOC; GSYM VECTOR_ADD_RDISTRIB];
MATCH_MP_TAC (MESON[]` a = b ==> a % x = b % x `);
UNDISCH_TAC ` &0 < t3 `;
UNDISCH_TAC ` ~ ( w' = &0) `;
UNDISCH_TAC ` t2' < &0 `;
UNDISCH_TAC ` &0 < t2 `;
CONV_TAC REAL_FIELD;

ASM_REWRITE_TAC[];
REWRITE_TAC[Geomdetail.CONV0_SET2; IN_ELIM_THM; VECTOR_ADD_LDISTRIB;
VECTOR_MUL_ASSOC];
EXISTS_TAC ` t3 / (w' * (t2' - t2)) * (w' * &1 / t3) * t2' `;
EXISTS_TAC` t3 / (w' * (t2' - t2)) * (w' * &1 / t3) * --t2 `;

CONJ_TAC;
ASM_SIMP_TAC[REAL_FIELD` ~(t3 = &0) ==> ~( w' = &0 )
==> t3 / (w' * (t2' - t2)) * (w' * &1 / t3) * t2' = t2' / ( t2' - t2) `];
MATCH_MP_TAC REAL_LT_DIV_NEG;
ASM_REAL_ARITH_TAC;
ASSUME_TAC2 (
REAL_FIELD` ~(t3 = &0) /\ ~( w' = &0 ) /\ t2' < &0 /\ &0 < t2 
==> t3 / (w' * (t2' - t2)) * (w' * &1 / t3) * --t2 = ( -- t2 ) / ( t2' - t2 ) /\
t3 / (w' * (t2' - t2)) * (w' * &1 / t3) * t2' +
 t3 / (w' * (t2' - t2)) * (w' * &1 / t3) * --t2 =
 &1 `);
ASM_REWRITE_TAC[];

MATCH_MP_TAC REAL_LT_DIV_NEG;
UNDISCH_TAC ` &0 < t2 `;
UNDISCH_TAC ` t2' < &0 `;
REAL_ARITH_TAC;
DOWN;
SIMP_TAC[DE_MORGAN_THM; REAL_ARITH` ~( a < b ) <=> a = b \/ b < a `];
REWRITE_TAC[TAUT` (a \/ b) /\ c /\ d <=> ((a \/ b) /\ c ) /\ d`; REAL_ARITH` (v' < &0 \/ w' < &0) /\
     (&0 < v' \/ &0 < w') <=> v' < &0 /\ &0 < w' \/ w' < &0 /\ &0 < v' `];
STRIP_TAC;
PHA;
REWRITE_TAC[TAUT` a /\ ( b \/ c ) /\ d <=> ( b \/ c) /\ a /\ d `];
SPEC_TAC (`v':real`,`v':real`);
SPEC_TAC (`t2:real`,`t2:real`);
SPEC_TAC (`v:real^N`,`v:real^N`);
SPEC_TAC (`w':real`,`w':real`);
SPEC_TAC (`t3:real`,`t3:real`);
SPEC_TAC (`w:real^N`,`w:real^N`);
IMP_TAC;
MATCH_MP_TAC (MESON[]`(! a b c x y z. P a b c x y z ==> P x y z a b c ) /\
(! a b c x y z. R c z ==> P a b c x y z ) ==> (! a b c x y z. R c z \/ R z c 
==> P a b c x y z ) `);
CONJ_TAC;
ONCE_REWRITE_TAC[TAUT` a ==> b ==> c <=> b ==> a ==> c `];
REPEAT GEN_TAC THEN STRIP_TAC;
ANTS_TAC;
DOWN_TAC;
SIMP_TAC[INSERT_COMM; REAL_ADD_SYM; VECTOR_ADD_SYM];
ONCE_REWRITE_TAC[EQ_SYM_EQ];
SIMP_TAC[];
SIMP_TAC[VECTOR_ADD_SYM; INSERT_COMM];
REPEAT STRIP_TAC;
SWITCH_TAC ` x' = t1 % x + t2 % v + t3 % w:real^N`;
SWITCH_TAC `z = u' % x + v'' % v + w'' % w:real^N `;
ASM_REWRITE_TAC[];
UNDISCH_TAC ` x' = t1' % x + t2' % u:real^N `;


EXPAND_TAC "x'";
REWRITE_TAC[VECTOR_ARITH` a + b % x + c = d <=> b % x = d - a - c `];
MP_TAC2 (REAL_ARITH` &0 < t2 ==> ~( t2 = &0 ) `);
ABBREV_TAC ` set = {v,w:real^N} `;

SIMP_TAC[Ldurdpn.VECTOR_SCALE_CHANGE];
REPEAT STRIP_TAC;
UNDISCH_TAC ` u' % x + v'' % v + w'' % w = z:real^N `;
ASM_REWRITE_TAC[VECTOR_MUL_ASSOC; VECTOR_SUB_LDISTRIB; VECTOR_ADD_LDISTRIB];
REWRITE_TAC[REAL_ARITH` a * &1 / b * c = ( a * c ) / b `;
VECTOR_ARITH ` u' % x +
 (tt % x + vv % u) -
 t22 % x -
 ww % w +
 w'' % w =
 z <=> z + ( t22 - u' - tt ) % x = vv % u + (w'' - ww) % w `];

SUBGOAL_THEN ` &1 + ((v'' * t1) / t2 - u' - (v'' * t1') / t2) =
(v'' * t2') / t2 + (w'' - (v'' * t3) / t2) ` ASSUME_TAC;
ASM_SIMP_TAC[
REAL_FIELD` ~ ( m = &0 ) ==> a / m - b - c / m = ( a - m * b - c ) / m /\
&1 + a / m = ( m + a ) / m /\ a / m + b - c / m = (a + m * b - c ) / m`];
MATCH_MP_TAC (MESON[]` a = b ==> a / x = b / x`);
UNDISCH_TAC ` t1' + t2' = &1 `;
UNDISCH_TAC ` t1 + t2 + t3 = &1 `;
EXPAND_TAC "u'";
SIMP_TAC[REAL_ARITH` a + b = c <=> a = c - b `];
REAL_ARITH_TAC;
ABBREV_TAC ` tu = (v'' * t2') / t2 `;
ABBREV_TAC ` tw = (w'' - (v'' * t3) / t2) `;
STRIP_TAC;
EXISTS_TAC `u:real^N `;
EXISTS_TAC ` w:real^N `;
EXISTS_TAC ` ( &1 / (tu + tw) ) % ( tu % u + tw % w:real^N ) `;
EXPAND_TAC "set";
REWRITE_TAC[INSERT_SUBSET; IN_INSERT; EMPTY_SUBSET; IN_INTER];
SUBGOAL_THEN ` &0 < tu /\ &0 < tw ` ASSUME_TAC;
CONJ_TAC;
EXPAND_TAC "tu";
MATCH_MP_TAC REAL_LT_DIV;
ASM_REWRITE_TAC[REAL_MUL_POS_LT];
EXPAND_TAC "tw";
MATCH_MP_TAC (REAL_ARITH` &0 < a /\ b < &0 ==> &0 < a - b `);
ASM_REWRITE_TAC[ REAL_ARITH` a / b < &0 <=> &0 < ( -- a ) / b `];
MATCH_MP_TAC REAL_LT_DIV;
ASM_REWRITE_TAC[REAL_ARITH` -- ( a * b ) = ( -- a ) * b `; REAL_MUL_POS_LT];
ASM_REWRITE_TAC[REAL_ARITH ` &0 < -- a <=> a < &0 `];
CONJ_TAC;
REWRITE_TAC[AFF2; IN_ELIM_THM];
FIRST_X_ASSUM (SUBST1_TAC o SYM);
EXISTS_TAC ` (&1 / (tu + tw)) * (((v'' * t1) / t2 - u' - (v'' * t1') / t2) ) `;
REWRITE_TAC[VECTOR_ADD_LDISTRIB; VECTOR_MUL_ASSOC;
 VECTOR_ARITH` a + z % x = z % x + b <=> a = b `];
MATCH_MP_TAC (MESON[]` a = b ==> a % x = b % x `);
UNDISCH_TAC` &1 + (v'' * t1) / t2 - u' - (v'' * t1') / t2 = tu + tw `;
DOWN;
NHANH (REAL_ARITH` &0 < a /\ &0 < b ==> ~( a + b = &0) `);
CONV_TAC REAL_FIELD;
DOWN;
REWRITE_TAC[IN_CONV0]]);;




let CVX_LO_IMP_LO = prove(` convex_local_fan (V,E,FF) ==> local_fan (V,E,FF)`,
SIMP_TAC[convex_local_fan]);;



let S_SUBSET_IMP_AFF_S_TOO = prove(
` S SUBSET aff SS ==> aff S SUBSET aff SS `,
NHANH_PAT `\x. x ==> y ` (ISPEC ` affine ` HULL_MONO)
THEN SIMP_TAC[aff; HULL_HULL]);;


let AFF2_DET_BY_TWO_POINTS = prove_by_refinement
(` {x:real^N,y} SUBSET aff {a,b} /\ ~( x = y ) ==>
aff {a,b} = aff {x,y} `,
[STRIP_TAC;
MATCH_MP_TAC Collect_geom.RCEABUJ;
ASM_REWRITE_TAC[line];
EXISTS_TAC ` a:real^N`;
EXISTS_TAC `b:real^N`;
REWRITE_TAC[aff];
STRIP_TAC;
FIRST_X_ASSUM SUBST_ALL_TAC;
DOWN;
PHA;
DOWN;
REWRITE_TAC[INSERT_INSERT; AFFINE_HULL_SING; aff; INSERT_SUBSET; IN_INSERT;
NOT_IN_EMPTY];
SIMP_TAC[]]);;

let IN_CONV0_EQ_EQ = prove(
`! x:real^N. x IN conv0 {a,b} ==> ( x = a <=> a = b ) `,
REWRITE_TAC[Geomdetail.CONV0_SET2; IN_ELIM_THM] THEN 
REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THEN 
UNDISCH_TAC` a' + b' = &1 ` THEN 
SIMP_TAC[REAL_ARITH` a + b = &1 <=> a = &1 - b `] THEN 
DISCH_TAC THEN 
REWRITE_TAC[VECTOR_ARITH` (&1 - b') % a + b' % b = a + b' % (b - a ) `;
VECTOR_ARITH` a + x = a <=> x = vec 0`; VECTOR_MUL_EQ_0] THEN 
ASM_SIMP_TAC[REAL_ARITH` &0 < a ==> ~( a = &0 ) `; VECTOR_SUB_EQ; EQ_SYM_EQ]);;




let CONV02_SUBSET_AFF2 = prove(` conv0 {a,b} SUBSET aff {a,b} `,
REWRITE_TAC[Geomdetail.CONV0_SET2; AFF2; SUBSET; IN_ELIM_THM] THEN
MESON_TAC[REAL_ARITH` a + b = &1 ==> b = &1 - a `]);;



let IN_CONV0_IMP_AFF_EQ = prove_by_refinement
(` a:real^N IN conv0 {x,y} ==> aff {x,y} = aff {x,a} `,
[NHANH IN_CONV0_EQ_EQ;
STRIP_TAC;
ASM_CASES_TAC` a = x :real^N `;
ASM_REWRITE_TAC[];
DOWN;
ASM_REWRITE_TAC[];
DISCH_THEN (SUBST1_TAC o SYM);
REWRITE_TAC[];
MATCH_MP_TAC AFF2_DET_BY_TWO_POINTS;
DOWN;
SIMP_TAC[INSERT_SUBSET; EMPTY_SUBSET; EQ_SYM_EQ; aff; Ldurdpn.IN_HULL_INSERT];
ASM_SIMP_TAC[REWRITE_RULE[SUBSET; aff] CONV02_SUBSET_AFF2]]);;






let IN_CONV0_AFF_SUBSET = prove(`! t:real^N. t IN aff {a,b} INTER conv0 {x,y} /\
x IN aff {a,b} ==> aff {x,y} SUBSET aff {a,b} `,
REWRITE_TAC[IN_INTER] THEN NHANH IN_CONV0_EQ_EQ THEN 
NHANH IN_CONV0_IMP_AFF_EQ THEN GEN_TAC THEN STRIP_TAC THEN 
ASM_REWRITE_TAC[] THEN 
MATCH_MP_TAC Collect_geom.AFFINE_CONTAIN_LINE THEN
ASM_REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET; aff; AFFINE_AFFINE_HULL]);;




let CONDS_FOR_INTER_AFF_CONV0 = prove_by_refinement
(`&0 < t2 /\
 &0 < t3 /\
 t1 + t2 + t3 = ss /\
 t = t1 % x + t2 % v + t3 % (w:real^N) /\
 t1' + t2' = ss /\
 t = t1' % x + t2' % u
 ==> (?tt. tt IN aff {x, u} /\ tt IN conv0 {v, w})`,
[STRIP_TAC;
DOWN;
ASM_REWRITE_TAC[VECTOR_ARITH` a % x + y = z <=> z - a % x = y `];
REWRITE_TAC[ VECTOR_ARITH ` (t1' % x + u) - t1 % x = (t1' - t1 ) % x + u `];
STRIP_TAC;
EXISTS_TAC` &1 / (t2 + t3 ) % ( (t1' - t1) % x + t2' % (u:real^N) )`;
CONJ_TAC;
REWRITE_TAC[VECTOR_ADD_LDISTRIB; VECTOR_MUL_ASSOC; REAL_ARITH` &1 / a * b = b / a `;
AFF2; IN_ELIM_THM];
EXISTS_TAC ` (t1' - t1) / (t2 + t3) `;
MATCH_MP_TAC (MESON[]` a = b ==> f a = f b `);
MATCH_MP_TAC (MESON[]` a = b ==> a % u = b % u `);
DOWN;
REMOVE_TAC;
DOWN;
DOWN THEN REMOVE_TAC;
DOWN_TAC;
CONV_TAC REAL_FIELD;
ASM_REWRITE_TAC[];
MATCH_MP_TAC IN_CONV0;
ASM_REWRITE_TAC[]]);;




let INTER_AFF_GT_LT_IMP_INTER_AFF_CONV0 = prove_by_refinement
(`DISJOINT {x} {v,w} /\ ~(u = x ) ==>
t IN aff_gt {x} {v,w} INTER aff_lt {x} {u} ==>
(? tt. tt IN aff {x,u} INTER conv0 {v,w}) `,
[SIMP_TAC[Planarity.AFF_GT_1_2; AFF_LT_1_1; SET_RULE` ~(x = y) <=> DISJOINT {y} {x}`];
REWRITE_TAC[IN_INTER; IN_ELIM_THM];
REPEAT STRIP_TAC;
ABBREV_TAC ` ss = &1 `;
MATCH_MP_TAC CONDS_FOR_INTER_AFF_CONV0;
DOWN THEN DOWN;
ASM_SIMP_TAC[]]);;


let SUBSET_AFF2_IMP_COLL = prove_by_refinement
(` S SUBSET aff {a:real^N, b} ==> collinear S `,
[REWRITE_TAC[collinear; SUBSET; AFF2; IN_ELIM_THM];
STRIP_TAC;
EXISTS_TAC ` a - b:real^N `;
FIRST_X_ASSUM NHANH;
REPEAT STRIP_TAC;
EXISTS_TAC ` t - t':real `;
ASM_REWRITE_TAC[];
CONV_TAC VECTOR_ARITH]);;



let COLLINEAR_SUBSET_AFF2 = prove_by_refinement
(` a:real^N IN S /\ b IN S /\ ~( a = b ) /\ collinear S 
==> S SUBSET aff {a,b} `,
[REWRITE_TAC[collinear];
STRIP_TAC;
SUBGOAL_THEN ` ? c. a - b:real^N = c % u ` ASSUME_TAC;
ASM_SIMP_TAC[];

REWRITE_TAC[SUBSET];
GEN_TAC;
UNDISCH_TAC ` b:real^N  IN S `;
PHA;
ONCE_REWRITE_TAC[CONJ_SYM];
FIRST_X_ASSUM NHANH;
FIRST_X_ASSUM CHOOSE_TAC;
ASM_CASES_TAC ` c = &0 `;
FIRST_X_ASSUM SUBST_ALL_TAC;
DOWN THEN DOWN;
SIMP_TAC[VECTOR_MUL_LZERO; VECTOR_SUB_EQ];
UNDISCH_TAC `  a - b = c % u:real^N `;
ONCE_REWRITE_TAC[EQ_SYM_EQ];
ASM_SIMP_TAC[Ldurdpn.VECTOR_SCALE_CHANGE];
REWRITE_TAC[VECTOR_ARITH` a = x - b:real^N <=> x = a + b `];
STRIP_TAC THEN STRIP_TAC;
ASM_REWRITE_TAC[AFF2; IN_ELIM_THM; VECTOR_MUL_ASSOC; VECTOR_ARITH` a % ( x - y ) + y = a % x + ( &1 - a ) % y `];
MESON_TAC[]]);;




let AFF_CONV0_COLL4 = prove_by_refinement
(` t:real^N IN aff {a, b} INTER conv0 {x, y} /\ x IN aff {a, b}
==> collinear {a,b,x,y} `,
[NHANH IN_CONV0_AFF_SUBSET;
STRIP_TAC;
MATCH_MP_TAC (GEN_ALL SUBSET_AFF2_IMP_COLL);
EXISTS_TAC ` a:real^N `;
EXISTS_TAC `b:real^N `;
REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET; Planarity.POINT_IN_LINE];
DOWN;
ONCE_REWRITE_TAC[INSERT_COMM];
REWRITE_TAC[SUBSET; Planarity.POINT_IN_LINE];
DISCH_TAC;
CONJ_TAC;
FIRST_ASSUM MATCH_MP_TAC;
PURE_ONCE_REWRITE_TAC[INSERT_COMM];
PURE_REWRITE_TAC[Planarity.POINT_IN_LINE];
FIRST_ASSUM MATCH_MP_TAC;
PURE_REWRITE_TAC[Planarity.POINT_IN_LINE]]);;




let CONDS_IN_HAFL_LINE = prove(
 `&0 <= t /\ a - x = t % (b - x) ==> a IN aff_ge {x} {b} `,
REWRITE_TAC[HALFLINE; IN_ELIM_THM] THEN STRIP_TAC THEN 
EXISTS_TAC ` t:real ` THEN ASM_REWRITE_TAC[] THEN DOWN
THEN CONV_TAC VECTOR_ARITH);;

let PRESERABLE_AFF_GE_SUBSET = prove(
` a IN aff_ge {x} {b} ==> aff_ge {x} {a} SUBSET aff_ge {x} {b} `,
REWRITE_TAC[HALFLINE; SUBSET; IN_ELIM_THM] THEN REPEAT STRIP_TAC THEN 
EXISTS_TAC ` t * t':real ` THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ 
MATCH_MP_TAC REAL_LE_MUL THEN ASM_REWRITE_TAC[]; VECTOR_ARITH_TAC]);;





let AFF_GE_EQ = prove_by_refinement
(`&0 < t /\ a - x:real^N = t % (b - x) ==>
aff_ge {x} {a} = aff_ge {x} {b} `,
[STRIP_TAC;
SUBGOAL_THEN ` a:real^N IN aff_ge {x} {b} ` ASSUME_TAC;
MATCH_MP_TAC CONDS_IN_HAFL_LINE;
ASM_REWRITE_TAC[];
ASM_REAL_ARITH_TAC;
SUBGOAL_THEN ` b:real^N IN aff_ge {x} {a} ` ASSUME_TAC;
MATCH_MP_TAC (GEN_ALL CONDS_IN_HAFL_LINE);
EXISTS_TAC ` &1 / t `;
ASSUME_TAC2 (REAL_ARITH` &0 < t ==> ~( t = &0 ) `);
SWITCH_TAC `a - x = t % (b - x:real^N)`;
DOWN;
ASM_SIMP_TAC[Ldurdpn.VECTOR_SCALE_CHANGE];
STRIP_TAC;
MATCH_MP_TAC REAL_LE_DIV;
UNDISCH_TAC ` &0 < t `;
REAL_ARITH_TAC;
DOWN THEN DOWN;
NHANH PRESERABLE_AFF_GE_SUBSET;
SET_TAC[]]);;




let FAN_INTERSECTION_PRO_EXPRESS = prove_by_refinement
(` FAN (x:real^N,V,E) /\ a IN V /\ b IN V /\ ~( a = b ) 
==> ~( ? t. &0 < t /\
a - x = t % ( b - x ) )`,
[NHANH FAN_IMP_V_DIFF;
REWRITE_TAC[FAN; fan7];
STRIP_TAC;
SUBGOAL_THEN ` aff_ge {x} {a:real^N} INTER aff_ge {x} {b} = aff_ge {x} ({a} INTER {b})` ASSUME_TAC;
FIRST_X_ASSUM MATCH_MP_TAC;
REWRITE_TAC[IN_UNION; IN_ELIM_THM; SET_RULE` {a} = {b} <=> a = b `];
ASM_MESON_TAC[];
DOWN THEN DOWN;
SIMP_TAC[SET_RULE` ~(a = b) <=> {a} INTER {b} = {} `; AFF_GE_EQ_AFFINE_HULL];
REWRITE_TAC[AFFINE_HULL_SING];
NHANH AFF_GE_EQ;
REPEAT STRIP_TAC;
FIRST_X_ASSUM SUBST_ALL_TAC;
SUBGOAL_THEN ` b IN aff_ge {x} {b} INTER aff_ge {x} {b:real^N} ` ASSUME_TAC;
REWRITE_TAC[ENDS_IN_HALFLINE; IN_INTER];
DOWN;
ASM_SIMP_TAC[IN_INSERT; NOT_IN_EMPTY]]);;





let FAN_INTERSECTION_PRO_EXPRESS2 = 
MESON[FAN_INTERSECTION_PRO_EXPRESS]`FAN (x:real^N,V,E) /\ a IN V /\ b IN V 
/\ ~(a = b) /\ &0 < t ==> ~( a - x = t % (b - x))`;;





let FAN_AFF2_INTER_CONV0_IMP_NO_IN_AF = prove_by_refinement
(` FAN (x:real^N,V,E) /\ {u,v,w} SUBSET V /\ t IN aff {x,v}  INTER conv0 {u,w} 
/\ CARD {u,v,w} = 3 ==> ~( u IN aff {x,v} )`,
[NHANH FAN_IMP_V_DIFF;
STRIP_TAC;
STRIP_TAC;
ASSUME_TAC2 (
ISPECL [` t:real^N `;` x:real^N`;` v:real^N `;`u:real^N `;` w:real^N `] (GEN_ALL AFF_CONV0_COLL4));
SUBGOAL_THEN ` {x, v, u, w} SUBSET aff { x,v:real^N } ` ASSUME_TAC;
MATCH_MP_TAC COLLINEAR_SUBSET_AFF2;
ASM_REWRITE_TAC[IN_INSERT];
DOWN_TAC;
SIMP_TAC[EQ_SYM_EQ; INSERT_SUBSET];
DOWN;
REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET; Planarity.POINT_IN_LINE];
ONCE_REWRITE_TAC[INSERT_COMM];
REWRITE_TAC[Planarity.POINT_IN_LINE];
REWRITE_TAC[AFF2; IN_ELIM_THM];
STRIP_TAC;
ASM_CASES_TAC ` &0 < t' \/ &0 < t'' `;

(* very smart way *)

REPLICATE_TAC 5 DOWN;
REMOVE_TAC;
DOWN_TAC;
DAO;
SPEC_TAC (`t':real`,`t':real`);
SPEC_TAC (`u:real^N `,`u:real^N `);
SPEC_TAC (`t'':real`,`t'':real`);
SPEC_TAC (`w:real^N `,`w:real^N`);
MATCH_MP_TAC (
MESON[]` (! x a y b. P x a y b ==> P y b x a ) /\ (! x a y b. ~( Q b /\ P x a y b )) ==> (! x a y b. ~(( Q b \/ Q a ) /\ P x a y b )) `);
CONJ_TAC;
ONCE_REWRITE_TAC[EQ_SYM_EQ];
SIMP_TAC[INSERT_COMM];

REWRITE_TAC[Geomdetail.CARD3];
REPEAT STRIP_TAC;
UNDISCH_TAC` u = t' % v + (&1 - t') % x:real^N `;
REWRITE_TAC[VECTOR_ARITH` u = t' % v + (&1 - t') % x <=> u - x = t' % (v - x )`];

MATCH_MP_TAC (
MESON[FAN_INTERSECTION_PRO_EXPRESS]`FAN (x,V,E) /\ a IN V /\ b IN V /\ ~(a = b)
   /\ &0 < t ==> ~( a - x = t % (b - x))`);
DOWN_TAC;
SIMP_TAC[INSERT_SUBSET; DE_MORGAN_THM];

ASM_CASES_TAC` t' = &0 `;
FIRST_X_ASSUM SUBST_ALL_TAC;
UNDISCH_TAC ` u = &0 % v + (&1 - &0) % x:real^N `;
REWRITE_TAC[VECTOR_MUL_LZERO; VECTOR_SUB_RDISTRIB; VECTOR_ADD_LID; VECTOR_SUB_RZERO; VECTOR_MUL_LID];
UNDISCH_TAC ` {u, v, w} SUBSET V:real^N -> bool `;
ASM_SIMP_TAC[INSERT_SUBSET];

ASM_CASES_TAC ` t'' = &0 `;
FIRST_X_ASSUM SUBST_ALL_TAC;
UNDISCH_TAC ` w = &0 % v + (&1 - &0) % x:real^N `;
REWRITE_TAC[VECTOR_MUL_LZERO; VECTOR_SUB_RDISTRIB; VECTOR_ADD_LID; VECTOR_SUB_RZERO; VECTOR_MUL_LID];
UNDISCH_TAC ` {u, v, w} SUBSET V:real^N -> bool `;
ASM_SIMP_TAC[INSERT_SUBSET];
DOWN THEN DOWN THEN DOWN THEN PHA;
REWRITE_TAC[REAL_ARITH` ~(&0 < t' \/ &0 < t'') /\ ~(t' = &0) /\ ~(t'' = &0)
<=> t' < &0 /\ t'' < &0 `];
DOWN THEN DOWN;
REWRITE_TAC[VECTOR_ARITH` u = t' % v + (&1 - t') % x <=> 
u - x = t'% ( v - x ) `];
REPEAT STRIP_TAC;
ASSUME_TAC2 (REAL_ARITH` t' < &0 ==> ~( t' = &0) `);
SWITCH_TAC ` u - x = t' % (v - x:real^N)`;
DOWN;
ASM_SIMP_TAC[Ldurdpn.VECTOR_SCALE_CHANGE];
STRIP_TAC;
UNDISCH_TAC ` w - x = t'' % (v - x:real^N)`;
ASM_REWRITE_TAC[VECTOR_MUL_ASSOC];
MATCH_MP_TAC FAN_INTERSECTION_PRO_EXPRESS2;
DOWN_TAC;
SIMP_TAC[INSERT_SUBSET; Geomdetail.CARD3; DE_MORGAN_THM; REAL_ARITH ` a * &1 / b = a / b `];
STRIP_TAC;
MATCH_MP_TAC REAL_LT_DIV_NEG;
ASM_REWRITE_TAC[]]);;




let AFF2_ITR_CONV0_IMP_SAME_ENDS = prove(
`! a b. t IN aff {x,y} INTER conv0 {a,b} ==>
(a IN aff {x,y} <=> b IN aff {x,y} ) `,
MATCH_MP_TAC (
MESON[]` (! a b. P a b ==> P b a ) /\ (! a b. P a b /\ Q a ==> Q b )
==> (! a b. P a b ==> (Q a <=> Q b )) `) THEN 
SIMP_TAC[INSERT_COMM; INTER_COMM] THEN 
NHANH IN_CONV0_AFF_SUBSET THEN REWRITE_TAC[SUBSET] THEN 
REPEAT STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN 
REWRITE_TAC[Planarity.POINT_IN_LINE1]);;






let AFF_XX_CASES = prove_by_refinement
(`!x: real^N. aff_lt {x} {x} = {} /\ aff_le {x} {x} = {} /\
aff_gt {x} {x} = {x}`,
[REWRITE_TAC[aff_le_def; sgn_le; aff_lt_def; aff_gt_def; affsign; FUN_EQ_THM; UNION_IDEMPOT; SET_RULE`~( {} x)`; sgn_gt; lin_combo; VSUM_SING; SUM_SING; sgn_lt] THEN 
REWRITE_TAC[SET_RULE` {x} a <=> x = a `];
GEN_TAC;
CONJ_TAC;
MESON_TAC[REAL_ARITH` ~( &1 < &0 ) `];
CONJ_TAC;
MESON_TAC[REAL_ARITH` ~( &1 <= &0 ) `];
GEN_TAC;
EQ_TAC;
STRIP_TAC;
FIRST_X_ASSUM SUBST_ALL_TAC;
DOWN_TAC;
SIMP_TAC[VECTOR_MUL_LID];
STRIP_TAC;
EXISTS_TAC `(\x:real^N. &1 ) `;
ASM_REWRITE_TAC[VECTOR_MUL_LID];
GEN_TAC THEN DISCH_TAC;
REAL_ARITH_TAC]);;




let NOT_X_IN_AFF_X_A = prove_by_refinement
(` ~ ( x:real^N IN aff_lt {x} {a} )`,
[ASM_CASES_TAC ` x = a:real^N `;
ASM_REWRITE_TAC[AFF_XX_CASES; NOT_IN_EMPTY];
DOWN;
SIMP_TAC[SET_RULE` ~( a = x ) <=> DISJOINT {a} {x} `; AFF_LT_1_1];
REWRITE_TAC[GSYM (SET_RULE` ~( a = x ) <=> DISJOINT {a} {x} `); IN_ELIM_THM;
VECTOR_ARITH` x = t1 % x + t2 % a  <=> t2 % ( a - x )  = x - (t1 + t2 ) % x  `];
REPEAT STRIP_TAC;
DOWN;
ASM_REWRITE_TAC[VECTOR_ARITH` x - &1 % x = vec 0 `; VECTOR_MUL_EQ_0; VECTOR_SUB_EQ];
ASM_REAL_ARITH_TAC]);;




let INTER_EQ_EM_EXPAND = SET_RULE` A INTER B = {} <=> ~( ?x. x IN A /\ x IN B )`;;





let NOT_INTER_EQ_EM_IMP_AFF_SUBSET = prove_by_refinement
(` x' IN aff_gt {x:real^N} {v, w} INTER aff_lt {x} {u}
==> aff {x,u} SUBSET affine hull {x,v,w} `, [REWRITE_TAC[IN_INTER]; 
STRIP_TAC; SUBGOAL_THEN ` ~ (x' = x:real^N) ` ASSUME_TAC;
ASM_MESON_TAC[NOT_X_IN_AFF_X_A];
SUBGOAL_THEN ` {x,x':real^N} SUBSET aff {x,u} ` ASSUME_TAC;
REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET; Planarity.POINT_IN_LINE];
MP_TAC (ISPECL [`{x:real^N }`;` {u:real^N }`] AFF_LT_SUBSET_AFFINE_HULL);
REWRITE_TAC[GSYM aff; SET_RULE` {x} UNION S = x INSERT S `; SUBSET];
ASM_SIMP_TAC[];
SUBGOAL_THEN ` aff {x,u:real^N} = aff {x,x'} ` ASSUME_TAC;
MATCH_MP_TAC AFF2_DET_BY_TWO_POINTS;
ASM_SIMP_TAC[];
ASM_REWRITE_TAC[];
MATCH_MP_TAC Collect_geom.AFFINE_CONTAIN_LINE;
REWRITE_TAC[AFFINE_AFFINE_HULL];
MP_TAC (ISPECL [`{x:real^N }`;` {v,w:real^N }`] AFF_GT_SUBSET_AFFINE_HULL);
REWRITE_TAC[aff; SET_RULE` {x} UNION S = x INSERT S `; INSERT_SUBSET;
EMPTY_SUBSET; Ldurdpn.IN_HULL_INSERT];
REWRITE_TAC[SUBSET];
DISCH_THEN MATCH_MP_TAC;
FIRST_ASSUM ACCEPT_TAC]);;


let IN_AFF_HULL_3 = prove(` x' IN aff_gt {x} {v, w} INTER aff_lt {x} {u} 
==> u IN affine hull {x, v, w} `,
NHANH NOT_INTER_EQ_EM_IMP_AFF_SUBSET THEN 
REWRITE_TAC[AFF2; SUBSET; IN_ELIM_THM] THEN 
STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN 
EXISTS_TAC ` &0 ` THEN VECTOR_ARITH_TAC);;




let LOCAL_FAN_ORBIT_MAP_VITER = prove(
` local_fan (V,E,FF) ==> (! v n. v IN V ==> ITER n (rho_node1 FF) v IN V ) `,
NHANH LOCAL_FAN_ORBIT_MAP_V THEN STRIP_TAC THEN REPEAT GEN_TAC THEN 
FIRST_X_ASSUM (NHANH_PAT `\x. x ==> y `) THEN STRIP_TAC THEN EXPAND_TAC "V"
THEN REWRITE_TAC[REWRITE_RULE[POWER_TO_ITER] Hypermap.lemma_in_orbit]);;





let INTER_AFF_GT_LT_IMP_INTER_AFF_CONV0 = REWRITE_RULE[TAUT` a ==> b ==> c 
<=> a /\ b ==> c `] (ISPEC ` t:real^N ` (GEN_ALL
 INTER_AFF_GT_LT_IMP_INTER_AFF_CONV0));;


let AFF_GT_AFF_LT_INTERPRET = prove_by_refinement
( `! x:real^N. DISJOINT {x} {v, w} /\ DISJOINT {x} {u}
         ==> ((?a b c.
                   a < &0 /\
                   &0 < b /\
                   &0 < c /\
                   a % (u - x) = b % (v - x) + c % (w - x)) <=>
              (?tt. tt IN aff_gt {x} {v, w} INTER aff_lt {x} {u})) `,
[GEN_TAC THEN STRIP_TAC;
EQ_TAC;
ASM_SIMP_TAC[Planarity.AFF_GT_1_2; AFF_LT_1_1; IN_INTER; IN_ELIM_THM];
STRIP_TAC;
EXISTS_TAC ` (x:real^N) + a % (u - x) `;
CONJ_TAC;
ASM_REWRITE_TAC[];
EXISTS_TAC ` &1 - b - c `;
EXISTS_TAC `b:real `;
EXISTS_TAC `c:real `;
ASM_REWRITE_TAC[];
CONJ_TAC;
REAL_ARITH_TAC;
VECTOR_ARITH_TAC;
EXISTS_TAC ` &1 - a `;
EXISTS_TAC `a:real `;
ASM_REWRITE_TAC[];
CONJ_TAC;
REAL_ARITH_TAC;
FIRST_X_ASSUM (SUBST1_TAC o SYM);
VECTOR_ARITH_TAC;
ASM_SIMP_TAC[Planarity.AFF_GT_1_2; AFF_LT_1_1; IN_INTER; IN_ELIM_THM];
STRIP_TAC;
EXISTS_TAC ` t2':real `;
EXISTS_TAC ` t2:real ` ;
EXISTS_TAC ` t3:real `;
ASM_REWRITE_TAC[];
DOWN;
DOWN;
SIMP_TAC[REAL_ARITH` a + b = &1 <=> a = &1 - b `];
REMOVE_TAC;
DOWN;
REMOVE_TAC;
DOWN;
DOWN;
SIMP_TAC[REAL_ARITH` a + b = &1 <=> a = &1 - b `];
REMOVE_TAC;
REMOVE_TAC;
VECTOR_ARITH_TAC]);;




let AFF_GT_AFF_LT_INTERPRET2 = REWRITE_RULE[SET_RULE`DISJOINT {x} {v, w} /\
 DISJOINT {x} {u} <=> DISJOINT {x} {u,v,w} `] AFF_GT_AFF_LT_INTERPRET;;
let EXISTS_IN = SET_RULE` ~( a = {} ) <=> ? x. x IN a `;;


let AFF_GT_LT_INTER_SYM = prove_by_refinement
(`! x:real^N. DISJOINT {x} {u, v, w} ==>
~( aff_gt {x} {v,w} INTER aff_lt {x} {u} = {} ) ==>
~( aff_gt {x} {u,v} INTER aff_lt {x} {w} = {} )`,
[REWRITE_TAC[EXISTS_IN];
SIMP_TAC[GSYM AFF_GT_AFF_LT_INTERPRET2];
ONCE_REWRITE_TAC[SET_RULE` {a,b,c} = {c,a,b} `];
SIMP_TAC[GSYM AFF_GT_AFF_LT_INTERPRET2];
REPEAT STRIP_TAC;
EXISTS_TAC ` -- c:real `;
EXISTS_TAC ` -- a :real `;
EXISTS_TAC ` b:real `;
ASM_SIMP_TAC[VECTOR_ARITH` a % (u - x) = b % (v - x) + c % (w - x) 
==> --c % (w - x) = --a % (u - x) + b % (v - x) `];
ASM_REAL_ARITH_TAC]);;





let EXISTS_INTERSECTION_PROPERPLY = prove_by_refinement
(` (~(z:real^N = x) /\
       DISJOINT {x} {v, w} /\
       ~(x = u) /\
       ~(aff_gt {x} {v, w} INTER aff_lt {x} {u} = {}) /\
       z IN affine hull {x, v, w}) /\
       ~ collinear {x,v,w} 
       ==> (?a b t.
                {a, b} SUBSET {u, v, w} /\
                t IN aff {x, z} INTER conv0 {a, b} /\
                ~(a IN aff {x,z}))`,
[NHANH INTERSECTION_LEMMA;
STRIP_TAC;
ASM_CASES_TAC` ~( a:real^N IN aff {x,z}) `;
EXISTS_TAC ` a:real^N `;
EXISTS_TAC ` b:real^N `;
EXISTS_TAC ` t:real^N `;
ASM_REWRITE_TAC[];
UNDISCH_TAC` {a, b} SUBSET {u, v, w:real^N } `;
ONCE_REWRITE_TAC[INSERT_SUBSET];
ONCE_REWRITE_TAC[IN_INSERT];
STRIP_TAC;
EXISTS_TAC` v:real^N `;
EXISTS_TAC `w:real^N `;
REWRITE_TAC[IN_INSERT; INSERT_SUBSET; EMPTY_SUBSET];
SUBGOAL_THEN` (?tt. tt IN aff {x, u} INTER conv0 {v, w:real^N})` MP_TAC;
MATCH_MP_TAC (GEN_ALL INTER_AFF_GT_LT_IMP_INTER_AFF_CONV0);
ASM_SIMP_TAC[];
UNDISCH_TAC ` ~(aff_gt {x} {v, w} INTER aff_lt {x} {u:real^N} = {}) `;
SET_TAC[];
STRIP_TAC;
EXISTS_TAC `tt:real^N `;
SUBGOAL_THEN` aff {x,z} = aff {x,u:real^N} ` SUBST_ALL_TAC;
MATCH_MP_TAC Planarity.sym_line_fan1;
UNDISCH_TAC ` ~ ~(a IN aff {x, z:real^N}) `;
ASM_SIMP_TAC[SET_RULE` DISJOINT {x} {u,z} <=> ~( z = x ) /\ ~ (x = u )`];
ASM_REWRITE_TAC[];
STRIP_TAC;
SUBGOAL_THEN ` w IN aff {x, u:real^N} ` MP_TAC;
DOWN THEN DOWN;
NHANH AFF2_ITR_CONV0_IMP_SAME_ENDS;
SIMP_TAC[];
STRIP_TAC;
SUBGOAL_THEN ` {x,v,w:real^N} SUBSET aff {x,u} ` MP_TAC;
ASM_REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET; Planarity.POINT_IN_LINE];
REWRITE_TAC[];
NHANH SUBSET_AFF2_IMP_COLL;
ASM_REWRITE_TAC[];
UNDISCH_TAC` a IN {v,w:real^N} ` ;
REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY];
DOWN_TAC;
DAO;
SPEC_TAC (`v:real^N`,`v:real^N`);
SPEC_TAC (`w:real^N`,`w:real^N`);
MATCH_MP_TAC (MESON[]` (! x y. (P x y ==> P y x) /\ ( L x y ==> L y x )) /\
(! x y. a = x /\ P x y ==> L x y ) ==>
 (! y x. (a = x \/ a = y) /\ P x y ==> L x y ) `);
CONJ_TAC;
SIMP_TAC[INSERT_COMM; DISJ_SYM];
REPEAT STRIP_TAC;
EXISTS_TAC ` w:real^N`;
EXISTS_TAC `u:real^N`;
REWRITE_TAC[INSERT_SUBSET; IN_INSERT; EMPTY_SUBSET; RIGHT_EXISTS_AND_THM];
CONJ_TAC;
STRIP_TAC;
SUBGOAL_THEN ` {x:real^N,v,w} SUBSET aff {x,z} ` MP_TAC;
FIRST_X_ASSUM SUBST_ALL_TAC;
ASM_REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET; Planarity.POINT_IN_LINE];
UNDISCH_TAC` ~collinear {x, v, w:real^N}`;
MESON_TAC[SUBSET_AFF2_IMP_COLL];
SUBGOAL_THEN ` aff {x, z} = aff {x,v:real^N}` SUBST1_TAC;
MATCH_MP_TAC AFF2_DET_BY_TWO_POINTS;
REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET; Planarity.POINT_IN_LINE];
FIRST_X_ASSUM SUBST_ALL_TAC;
ASM_SIMP_TAC[EQ_SYM_EQ];
UNDISCH_TAC` DISJOINT {x} {v, w:real^N}` ;
SET_TAC[];
MATCH_MP_TAC (GEN_ALL INTER_AFF_GT_LT_IMP_INTER_AFF_CONV0);
ASM_SIMP_TAC[INSERT_COMM; RIGHT_EXISTS_AND_THM];
CONJ_TAC;
DOWN THEN DOWN THEN DOWN;
SET_TAC[];
REWRITE_TAC[GSYM EXISTS_IN];
MATCH_MP_TAC (
REWRITE_RULE[TAUT` a ==> b ==> c <=> a /\ b ==> c `] AFF_GT_LT_INTER_SYM);
ASM_SIMP_TAC[INSERT_COMM];
DOWN THEN DOWN THEN SET_TAC[]]);;






let LOFA_V_SUBSET_AFF_HULL = prove_by_refinement
(`!w v.
     v,w IN FF /\
     convex_local_fan (V,E,FF) /\
     u IN V /\
     ~(aff_gt {vec 0} {v, w} INTER aff_lt {vec 0} {u} = {})
==> V SUBSET affine hull {v,w,vec 0} /\
~ collinear {vec 0,v,w} `,
[NHANH CVX_LO_IMP_LO;
PURE_ONCE_REWRITE_TAC[TAUT` a /\ (b /\c)/\d <=>(c/\a)/\b/\d `];
NHANH LOCAL_FAN_IN_FF_NOT_COLLINEAR;
SIMP_TAC[];
NHANH LOCAL_FAN_ORBIT_MAP_V;
PHA;
ONCE_REWRITE_TAC[TAUT` a /\b/\c/\d <=> (a/\c)/\b/\d`];
NHANH LOCAL_FAN_IMP_IN_V;
REPEAT STRIP_TAC;
UNDISCH_TAC ` v:real^3 IN V`;
FIRST_ASSUM NHANH;
STRIP_TAC;
FIRST_X_ASSUM (SUBST1_TAC o SYM);
REWRITE_TAC[orbit_map; SUBSET; IN_ELIM_THM; ARITH_RULE` a >= 0`];
REPEAT STRIP_TAC;
ASM_REWRITE_TAC[];
SPEC_TAC (`n:num`,`n:num`);
INDUCT_TAC;
REWRITE_TAC[POWER_0; I_THM; Trigonometry2.IN_P_HULL_INSERT];
MP_TAC (ISPECL [`u:real^3 `;`v:real^3`;`w:real^3 `;` vec 0: real^3 `;
` (rho_node1 FF POWER n') v  `] (GEN_ALL EXISTS_INTERSECTION_PROPERPLY));
ANTS_TAC;
ASM_SIMP_TAC[INSERT_COMM; DISJOINT_INSERT; DISJOINT_EMPTY; IN_INSERT; DE_MORGAN_THM; NOT_IN_EMPTY];
SUBGOAL_THEN ` (rho_node1 FF POWER n') v IN V ` ASSUME_TAC;
ABBREV_TAC `conjl <=> (rho_node1 FF POWER n') v IN V `;
UNDISCH_TAC ` v:real^3 IN V `;
FIRST_X_ASSUM NHANH;
FIRST_X_ASSUM (SUBST1_TAC o SYM);
STRIP_TAC;
EXPAND_TAC "V";
REWRITE_TAC[orbit_map; IN_ELIM_THM; ARITH_RULE` a >= 0`];
MESON_TAC[];
MP_TAC2 Wrgcvdr_cizmrrh.LOCAL_FAN_IMP_FAN;
NHANH FAN_IMP_V_DIFF;
ASM_SIMP_TAC[EQ_SYM_EQ];
STRIP_TAC;
SUBGOAL_THEN ` ~(b IN aff {vec 0, (rho_node1 FF POWER n') v})` MP_TAC;
DOWN;
DOWN;
NHANH AFF2_ITR_CONV0_IMP_SAME_ENDS;
SIMP_TAC[];
DOWN THEN PHA;
REWRITE_TAC[SET_RULE` ~( x IN S ) /\ ~(y IN S ) <=> {x,y} INTER S = {} `];
STRIP_TAC;
SUBGOAL_THEN ` interior_angle1 (vec 0) FF ((rho_node1 FF POWER n') v) = pi /\
             rho_node1 FF ((rho_node1 FF POWER n') v) IN (affine hull {v, w, vec 0}) /\
             ivs_rho_node1 FF ((rho_node1 FF POWER n') v) IN ( affine hull {v, w, vec 0})` ASSUME_TAC;
MATCH_MP_TAC (GEN_ALL OZQVSFF);
EXISTS_TAC `E:(real^3 -> bool) -> bool `;
EXISTS_TAC `V:real^3 -> bool `;
EXISTS_TAC ` a:real^3 `;
EXISTS_TAC ` b:real^3 `;
ASM_REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET];
SUBGOAL_THEN ` {a,b:real^3} SUBSET V` MP_TAC;
UNDISCH_TAC `{a, b} SUBSET {u, v, w:real^3}`;
MATCH_MP_TAC (SET_RULE` A SUBSET B ==> c SUBSET A ==> c SUBSET B `);
ASM_REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET];
SIMP_TAC[INSERT_SUBSET];
SUBGOAL_THEN ` {u, v, w} SUBSET affine hull {v,w,vec 0:real^3}` ASSUME_TAC;
REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET; Ldurdpn.IN_HULL_INSERT];
ONCE_REWRITE_TAC[INSERT_COMM];
REWRITE_TAC[Ldurdpn.IN_HULL_INSERT];
ONCE_REWRITE_TAC[SET_RULE` {a,b,c} = {c,b,a} `];
MATCH_MP_TAC (GEN_ALL IN_AFF_HULL_3);
UNDISCH_TAC` ~(aff_gt {vec 0} {v:real^3, w} INTER aff_lt {vec 0} {u} = {}) `;
SET_TAC[];
ASSUME_TAC2 (ISPECL [`{a,b:real^3}`;` {u,v,w:real^3}`; 
`affine hull {v,w, vec 0:real^3 }` ] SUBSET_TRANS);
DOWN;
SIMP_TAC[INSERT_SUBSET];
STRIP_TAC THEN STRIP_TAC;
ONCE_REWRITE_TAC[SET_RULE`{a,b,c} = {c,a,b} `];
REWRITE_TAC[Ldurdpn.IN_HULL_INSERT; POWER_TO_ITER];
ASSUME_TAC2 LOCAL_FAN_ORBIT_MAP_VITER;
UNDISCH_TAC `v:real^3 IN V `;
FIRST_X_ASSUM (MP_TAC o (SPECL [`v:real^3`;` n': num `]));
DISCH_THEN NHANH;
SIMP_TAC[];
STRIP_TAC;
CONJ_TAC;
REWRITE_TAC[plane];
EXISTS_TAC ` vec 0:real^3 `;
EXISTS_TAC `v:real^3 `;
EXISTS_TAC `w:real^3 `;
ASM_REWRITE_TAC[];
UNDISCH_TAC ` t IN aff {vec 0, (rho_node1 FF POWER n') v} INTER conv0 {a, b:real^3}`;
SIMP_TAC[INSERT_COMM; POWER_TO_ITER];
SET_TAC[];
DOWN;
SIMP_TAC[COM_POWER; o_THM]]);;





let DETER_RHO_NODE = prove(` local_fan (V,E,FF) /\ v,w IN FF 
==> rho_node1 FF v = w `, NHANH LOCAL_FAN_RHO_NODE_PROS THEN 
STRIP_TAC THEN DOWN THEN FIRST_X_ASSUM NHANH THEN 
SIMP_TAC[PAIR_EQ]);;




let CARD_RECUSIVE_EQ = prove_by_refinement
(` !k f. 0 < k ==> 
           (CARD {ITER n f x:A | n < k} = k
           <=> CARD {ITER n f x | n < k - 1} = k - 1 /\
               (!i. i < k - 1 ==> ~(ITER i f x = ITER (k - 1) f x))) `,
[REPEAT GEN_TAC;
STRIP_TAC;
EQ_TAC;
REWRITE_TAC[Wrgcvdr_cizmrrh.CARD_K_FIRST_ELMS_EQ_K];
SUBGOAL_THEN ` {ITER n f x:A | n < k} = ITER (k - 1) f x INSERT {ITER n f x | n < k - 1} ` SUBST1_TAC;
REWRITE_TAC[EXTENSION; IN_ELIM_THM; IN_INSERT];
ASM_SIMP_TAC[ARITH_RULE`0 < b ==> ( a < b <=> a = b - 1 \/ a < b - 1 )`];
MESON_TAC[];
STRIP_TAC;
SUBGOAL_THEN ` FINITE {ITER n f x:A | n < k - 1} /\
~( ITER (k - 1) f x IN {ITER n f x | n < k - 1}) ` ASSUME_TAC;
REWRITE_TAC[IN_ELIM_THM; SET_RULE` {ITER n f x | n < k } = {y | ?n. n < k /\ 
y = ITER n f x}`; Wrgcvdr_cizmrrh.FINITE_OF_N_FIRST_ELMS];
DOWN THEN MESON_TAC[];
DOWN THEN STRIP_TAC;
ASM_SIMP_TAC[Geomdetail.CARD_CLAUSES_IMP];
UNDISCH_TAC ` 0 < k `;
ARITH_TAC]);;





let LE_CARDV_IMP_CARD_DETERED = prove_by_refinement
(`(! v. v:A IN V ==> orbit_map f v = V ) /\ v IN V ==>
(! l. l <= CARD V ==> CARD {ITER n f v | n < l} = l)`,
[STRIP_TAC;
INDUCT_TAC;
REWRITE_TAC[LT; SET_RULE` {ITER n f v | n | F} = {} `; CARD_CLAUSES];
NHANH (ARITH_RULE` SUC n <= c ==> n <= c `);
FIRST_X_ASSUM NHANH;
STRIP_TAC;
MP_TAC (ARITH_RULE` 0 < SUC l `);
ASM_SIMP_TAC[ CARD_RECUSIVE_EQ; ARITH_RULE` SUC l - 1 = l `];
REPEAT STRIP_TAC;
DOWN THEN PHA;
ONCE_REWRITE_TAC[EQ_SYM_EQ];
MATCH_MP_TAC LOOP_MAP_IMP_DIFF_FIRST_ELMS;
ASM_REWRITE_TAC[];
UNDISCH_TAC ` SUC l <= CARD (V:A -> bool) `;
ARITH_TAC]);;


let LEMMA_SUBSET_ORBIT_MAP = REWRITE_RULE[POWER_TO_ITER] Hypermap.lemma_subset_orbit;;



let LEMMA_SUBSET_ORBIT_MAP_LT = prove(
` {ITER i p x | i < n} SUBSET orbit_map p x`,
ASM_CASES_TAC ` n = 0 ` THENL [ASM_REWRITE_TAC[LT] THEN SET_TAC[];
ASM_SIMP_TAC[ARITH_RULE`~( b = 0 ) ==> ( a < b  <=> a <= b - 1) `;
LEMMA_SUBSET_ORBIT_MAP]]);;



let LOOP_SET_DETER_FIRTS_ELMS = prove_by_refinement
(` (!v. v:A IN V ==> orbit_map f v = V) ==>
(! v. v IN V ==> {ITER n f v | n < CARD V } = V ) `,
[REPEAT STRIP_TAC;
SUBGOAL_THEN ` CARD {ITER n f v:A | n < CARD V} = CARD (V:A -> bool) ` MP_TAC;
ASSUME_TAC2 LE_CARDV_IMP_CARD_DETERED;
FIRST_X_ASSUM MATCH_MP_TAC;
ARITH_TAC;
SUBGOAL_THEN ` {ITER n f v:A | n < CARD V} SUBSET V ` ASSUME_TAC;
DOWN;
FIRST_X_ASSUM NHANH;
STRIP_TAC;
ABBREV_TAC ` nn = CARD (V:A -> bool) `;
EXPAND_TAC "V";
REWRITE_TAC[ARITH_RULE ` a < b ==> a <= b - 1 `; LEMMA_SUBSET_ORBIT_MAP_LT];
DOWN_TAC;
SIMP_TAC[EQ_SYM_EQ];
NHANH SELF_CYCLIC_IMP_FINITE;
ONCE_REWRITE_TAC[EQ_SYM_EQ];
STRIP_TAC;
ASM_SIMP_TAC[GSYM SUBSET_CARD_EQ]]);;




let lemma_in_orbit_iter = REWRITE_RULE[POWER_TO_ITER] Hypermap.lemma_in_orbit;;




let SIN_SUB_PERIODIC = prove(` sin x = -- ( sin ( x - pi )) `,
REWRITE_TAC[GSYM SIN_PERIODIC_PI; REAL_ARITH` a - b + b = a `]);;


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

let KCHMAMG = prove_by_refinement
(`convex_local_fan (V,E,FF) /\ circular V E
 ==> (!v. v IN V ==> interior_angle1 (vec 0) FF v = pi) /\
     (?A. plane A /\
          vec 0 IN A /\
          V SUBSET A /\
          (?e. (!x. x IN A ==> e dot x = &0) /\
               cyclic_set V (vec 0) e /\
               (!v. v IN V
                    ==> azim_cycle V (vec 0) e v = rho_node1 FF v /\
                        azim (vec 0) e v (rho_node1 FF v) =
                        dihV (vec 0) e v (rho_node1 FF v) /\
                        azim (vec 0) e v (rho_node1 FF v) =
                        arcV (vec 0) v (rho_node1 FF v) /\
azim (vec 0) e v (rho_node1 FF v) < pi)))`,
[REWRITE_TAC[circular];
STRIP_TAC;
DOWN_TAC;
NGOAC;
NHANH CVX_LO_IMP_LO;
ONCE_REWRITE_TAC[TAUT` ((a/\b)/\c)/\ x IN S <=>a /\(b/\c)/\x IN S`];
NHANH LOFA_IN_E_IMP_IN_FF;
PHA;
SPEC_TAC (`v:real^3`,`v:real^3`);
SPEC_TAC (`w:real^3`,`w:real^3 `);
ONCE_REWRITE_TAC[TAUT` a /\a1 /\a2 /\a3/\a4 <=> a3 /\ a/\a1/\a2/\a4`];
MATCH_MP_TAC (
MESON[]`(! x y. P x y ==> P y x) /\ (! x y. Q x y /\ P x y ==> L)
==> (! x y. (Q x y \/ Q y x) /\ P x y ==> L)`);
CONJ_TAC;
SIMP_TAC[INSERT_COMM];
REPEAT GEN_TAC THEN STRIP_TAC;
ONCE_REWRITE_TAC[TAUT` a /\ b <=> b /\ ( b ==> a ) `];
CONJ_TAC;
EXISTS_TAC ` affine hull {vec 0, v, w:real^3}`;
REWRITE_TAC[plane];
UNDISCH_TAC ` v,w IN FF:real^3 # real^3 -> bool `;
UNDISCH_TAC ` local_fan (V,E,FF) `;
PHA;
NHANH LOCAL_FAN_IN_FF_NOT_COLLINEAR;
STRIP_TAC;
CONJ_TAC;
EXISTS_TAC ` vec 0:real^3 `;
EXISTS_TAC ` v:real^3 `;
EXISTS_TAC ` w:real^3 `;
DOWN;
SIMP_TAC[];

REWRITE_TAC[Ldurdpn.IN_HULL_INSERT];
ASSUME_TAC2 LOFA_V_SUBSET_AFF_HULL;
ASM_SIMP_TAC[INSERT_COMM];
ABBREV_TAC ` e = v cross (rho_node1 FF v ) `;
EXISTS_TAC ` e:real^3 `;
SUBGOAL_THEN ` v:real^3 IN V /\ w IN V ` MP_TAC;
MATCH_MP_TAC LOCAL_FAN_IMP_IN_V;
ASM_REWRITE_TAC[];
STRIP_TAC;
ASSUME_TAC2 LOFA_IN_V_SO_DO_RHO_NODE_V;
DOWN THEN DOWN;
UNDISCH_TAC ` V SUBSET affine hull {v, w, vec 0} /\ ~collinear {vec 0, v, w:real^3}`;
REWRITE_TAC[SUBSET];
STRIP_TAC;
PHA;
FIRST_ASSUM (NHANH_PAT`\x. x ==> y ` );
STRIP_TAC;
ASSUME_TAC2 DETER_RHO_NODE;
CONJ_TAC;
REWRITE_TAC[AFFINE_HULL_3; IN_ELIM_THM];
REPEAT STRIP_TAC;
ASM_REWRITE_TAC[];
EXPAND_TAC "e";
UNDISCH_TAC ` rho_node1 FF v = w `;
SIMP_TAC[VECTOR_MUL_RZERO; VECTOR_ADD_RID; DOT_RADD; DOT_RMUL; DOT_CROSS_SELF;
 Collect_geom.ZERO_NEUTRAL];
REWRITE_TAC[DOT_RZERO; Collect_geom.ZERO_NEUTRAL];

ASSUME_TAC2 LOCAL_FAN_ORBIT_MAP_V;
DOWN THEN NHANH LOOP_SET_DETER_FIRTS_ELMS;
STRIP_TAC;
UNDISCH_TAC` v:real^3 IN V `;
FIRST_X_ASSUM (NHANH_PAT`\x. x ==> y`);

FIRST_ASSUM (NHANH_PAT`\x. x ==> y`);
STRIP_TAC;
SUBGOAL_THEN` ITER (CARD (V:real^3 -> bool)) (rho_node1 FF) v IN V ` ASSUME_TAC;
DOWN;
EXPAND_TAC "V";
REWRITE_TAC[lemma_in_orbit_iter];
SUBGOAL_THEN`ITER (CARD (V:real^3 -> bool)) (rho_node1 FF) v INSERT {ITER n (rho_node1 FF) v | n < CARD V} = {ITER n (rho_node1 FF) v | n <= CARD V} ` ASSUME_TAC;
REWRITE_TAC[EXTENSION; IN_INSERT; IN_ELIM_THM];
MESON_TAC[ARITH_RULE` a <= b <=> a = b \/ a < b:num `];
REPLICATE_TAC 3 DOWN THEN PHA;
NHANH (SET_RULE` a = b /\ x IN b /\ x INSERT a = h ==> h = b `);
STRIP_TAC;

MP_TAC2 (
SPECL [`V:real^3 -> bool`;`affine hull {v, w, vec 0:real^3} `;` CARD (V:real^3 
-> bool)`] (GENL [`U:real^3 -> bool`;` P:real^3 -> bool `;`l:num`] 
(SPEC_ALL KOMWBWC)));
ASSUME_TAC2 LOFA_V_SUBSET_AFF_HULL;
ASM_REWRITE_TAC[plane];
CONJ_TAC;
EXISTS_TAC` vec 0:real^3 `;
EXISTS_TAC `v:real^3 `;
EXISTS_TAC ` w:real^3 `;
ASM_REWRITE_TAC[];
SIMP_TAC[INSERT_COMM];
ONCE_REWRITE_TAC[SET_RULE` {a,b,c} = {c,a,b}`];
REWRITE_TAC[Ldurdpn.IN_HULL_INSERT];
STRIP_TAC;




GEN_TAC THEN STRIP_TAC;
CONJ_TAC;
DOWN;
UNDISCH_TAC` {ITER n (rho_node1 FF) v | n < CARD V} = V`;
STRIP_TAC;
EXPAND_TAC "V";
REWRITE_TAC[IN_ELIM_THM];
ONCE_REWRITE_TAC[CONJ_SYM];
STRIP_TAC;
DOWN THEN DOWN THEN PHA;
ASM_SIMP_TAC[];
UNDISCH_TAC `!x n.
          x = ITER n (rho_node1 FF) v /\ n < CARD V
          ==> azim_cycle V (vec 0) e (ITER n (rho_node1 FF) v) =
              rho_node1 FF (ITER n (rho_node1 FF) v) `;
MESON_TAC[];
SUBGOAL_THEN` &0 < sin (azim (vec 0) e v' (rho_node1 FF v')) ` ASSUME_TAC;
MP_TAC (SPECL [`e:real^3 `;` v':real^3 `;` rho_node1 FF v' `]
 Trigonometry2.JBDNJJB);
REWRITE_TAC[re_eqvl];
STRIP_TAC;
FIRST_X_ASSUM SUBST1_TAC;
MATCH_MP_TAC REAL_LT_MUL;
UNDISCH_TAC ` v':real^3 IN V `;
UNDISCH_TAC `{ITER n (rho_node1 FF) v | n < CARD V} = V `;
DISCH_THEN (SUBST1_TAC o SYM);
REWRITE_TAC[IN_ELIM_THM];
UNDISCH_TAC ` !i. i < CARD (V:real^3 -> bool)
          ==> &0 <
              (ITER i (rho_node1 FF) v cross ITER (i + 1) (rho_node1 FF) v) dot
              e `;
DISCH_THEN NHANH;
STRIP_TAC;
DOWN;
ASM_SIMP_TAC[GSYM ITER; ADD1];
ONCE_REWRITE_TAC[CROSS_TRIPLE];
ASM_REWRITE_TAC[];
MP_TAC (SPECL [`vec 0:real^3 `;` e:real^3 `;` v':real^3 `;
`rho_node1 FF v' `] AZIM_RANGE);

ASM_CASES_TAC ` pi <= azim (vec 0) e v' (rho_node1 FF v') `;
STRIP_TAC;
ABBREV_TAC ` a = azim (vec 0) e v' (rho_node1 FF v') `;
SUBGOAL_THEN ` sin a <= &0 ` MP_TAC;
ONCE_REWRITE_TAC[SIN_SUB_PERIODIC];
REWRITE_TAC[REAL_ARITH` -- a <= &0 <=> &0 <= a `];
MATCH_MP_TAC SIN_POS_PI_LE;
UNDISCH_TAC ` pi <= a `;
UNDISCH_TAC ` a < &2 * pi `;
REAL_ARITH_TAC;
UNDISCH_TAC ` &0 < sin a `;
MESON_TAC[REAL_ARITH` &0 < a ==> ~(a <= &0 ) `];
DOWN;
REWRITE_TAC[REAL_ARITH` ~( a <= b ) <=> b < a `];
SUBGOAL_THEN `~ collinear {vec 0, e, v'} /\ ~ collinear {vec 0, e, rho_node1 FF v'}` ASSUME_TAC;
SUBGOAL_THEN ` rho_node1 FF v' IN V ` ASSUME_TAC;
MATCH_MP_TAC LOFA_IN_V_SO_DO_RHO_NODE_V;
ASM_REWRITE_TAC[];

ONCE_REWRITE_TAC[SET_RULE`{a,b,c} = {c,a,b}`];
UNDISCH_TAC` cyclic_set V (vec 0) (e:real^3) `;
NHANH Wrgcvdr_cizmrrh.CYCLIC_SET_IMP_NOT_COLLINEAR;
ASM_SIMP_TAC[];
DOWN;

SIMP_TAC[AZIM_DIHV_SAME];

REPEAT STRIP_TAC;
REWRITE_TAC[Trigonometry2.DIHV_FORMULAR; VECTOR_SUB_RZERO];

SUBGOAL_THEN` v' dot e = &0 /\ rho_node1 FF v' dot (e:real^3) = &0 ` MP_TAC;

SUBGOAL_THEN ` rho_node1 FF v' IN V ` ASSUME_TAC;
MATCH_MP_TAC LOFA_IN_V_SO_DO_RHO_NODE_V;
ASM_REWRITE_TAC[];
DOWN;
UNDISCH_TAC ` v':real^3 IN V `;
UNDISCH_TAC` !x. x IN V ==> x IN affine hull {v, w, vec 0:real^3}`;
DISCH_THEN NHANH;
REWRITE_TAC[AFFINE_HULL_3; IN_ELIM_THM];
STRIP_TAC THEN STRIP_TAC;
ASM_REWRITE_TAC[];
EXPAND_TAC "e";
USE_FIRST ` rho_node1 FF v = w ` SUBST1_TAC;
REWRITE_TAC[VECTOR_MUL_RZERO; VECTOR_ADD_RID];
REWRITE_TAC[DOT_LADD; DOT_LMUL; DOT_CROSS_SELF];
REAL_ARITH_TAC;
SIMP_TAC[VECTOR_MUL_LZERO; VECTOR_SUB_RZERO];
SUBGOAL_THEN` &0 < e dot (e:real^3)` ASSUME_TAC;
UNDISCH_TAC `cyclic_set V (vec 0) (e:real^3) `;
SIMP_TAC[DOT_POS_LT; cyclic_set; EQ_SYM_EQ];
REMOVE_TAC;
DOWN;
MESON_TAC[Trigonometry2.WHEN_A_B_POS_ARCV_STABLE];
STRIP_TAC;

REPEAT STRIP_TAC;
ASSUME_TAC2 (SPEC `v:real^3,w:real^3 ` (GEN `x:real^3# real^3 `
 LOCAL_FAN_IN_FF_NOT_COLLINEAR));
SUBGOAL_THEN` affine hull {vec 0, v, w:real^3} = A ` ASSUME_TAC;
MATCH_MP_TAC THREE_NOT_COLL_DETER_PLANE;
ASM_REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET];
UNDISCH_TAC` V:real^3 -> bool SUBSET A `;
MATCH_MP_TAC (SET_RULE` x IN A /\ y IN A ==> A SUBSET B ==> x IN B /\ y IN B`);
MATCH_MP_TAC LOCAL_FAN_IMP_IN_V;
ASM_REWRITE_TAC[];
MP_TAC (ISPECL [` vec 0:real^3 `;` v':real^3 ` ] (GENL [` x:real^N `;
` z:real^N `] EXISTS_INTERSECTION_PROPERPLY));
ANTS_TAC;
ASM_REWRITE_TAC[];
MP_TAC2 (SET_RULE` v' :real^3 IN V /\ V SUBSET A ==> v' IN A`);
SIMP_TAC[];
DISCH_TAC;
SUBGOAL_THEN ` v IN V /\ w:real^3 IN V ` MP_TAC;
MATCH_MP_TAC LOCAL_FAN_IMP_IN_V;
ASM_REWRITE_TAC[];
MP_TAC2 Wrgcvdr_cizmrrh.LOCAL_FAN_IMP_FAN;
NHANH FAN_IMP_V_DIFF;
STRIP_TAC;
UNDISCH_TAC ` v':real^3 IN V `;
UNDISCH_TAC ` u:real^3 IN V `;
FIRST_X_ASSUM NHANH;
SET_TAC[];



STRIP_TAC;
SUBGOAL_THEN ` ~(b IN aff {vec 0, v':real^3}) ` MP_TAC;
DOWN THEN DOWN;
NHANH AFF2_ITR_CONV0_IMP_SAME_ENDS;
SIMP_TAC[];



DOWN THEN PHA;
REWRITE_TAC[SET_RULE` ~( a IN X) /\ ~( b IN X ) <=> {a,b} INTER X = {}`];
STRIP_TAC;
MP_TAC (SPECL [`E:(real^3 -> bool) -> bool `;` V:real^3 -> bool`;
` FF: real^3 #real^3 -> bool `;` v':real^3 ` ;`A:real^3 -> bool`;` a:real^3 `;
` b:real^3 `] (GEN_ALL OZQVSFF));
ANTS_TAC;
SUBGOAL_THEN` {a,b} SUBSET (V:real^3 -> bool) ` MP_TAC;



UNDISCH_TAC` {a, b} SUBSET {u, v, w:real^3}`;
MATCH_MP_TAC (SET_RULE` B SUBSET C ==> A SUBSET B ==> A SUBSET C `);
ASM_REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET];
MATCH_MP_TAC LOCAL_FAN_IMP_IN_V;
ASM_REWRITE_TAC[];
UNDISCH_TAC` v':real^3 IN V `;
UNDISCH_TAC ` (V:real^3 -> bool) SUBSET A `;
SIMP_TAC[INSERT_SUBSET];
REWRITE_TAC[SUBSET];
DISCH_THEN NHANH;
ASM_SIMP_TAC[];
DOWN THEN PHA THEN REMOVE_TAC;
DOWN THEN SIMP_TAC[INSERT_COMM];
SET_TAC[];
SIMP_TAC[]]);;


(* ========================================================= *)
(* JULY *)

let CONVEX_SOME_WEDGE = prove(`~collinear {v0, v1, w1} /\
         ~collinear {v0, v1, w2} /\
         &0 < azim v0 v1 w1 w2 /\
         azim v0 v1 w1 w2 < pi 
==> convex ( wedge v0 v1 w1 w2 ) `,
SIMP_TAC[WEDGE_LUNE_GT; CONVEX_AFF_GT]);;




let AZIM_EQ_0_GE_ALT2 = prove_by_refinement
(` !v0 v1 w x.
           ~collinear {v0, v1, w} 
           ==> (azim v0 v1 w x = &0 <=> x IN aff_ge {v0, v1} {w}) `,
[REPEAT STRIP_TAC;
ASM_CASES_TAC` ~ collinear {v0,v1,x:real^3} `;
DOWN_TAC;
REWRITE_TAC[AZIM_EQ_0_GE_ALT];
DOWN;
SIMP_TAC[AZIM_DEGENERATE];
DOWN;
NHANH Fan.th3a;
NHANH Topology.aff_subset_aff_ge;
NHANH Fan.th3b;
SIMP_TAC[GSYM Trigonometry2.NOT_EQ_IMP_AFF_AND_COLL3];
SET_TAC[]]);;





let WEDGE_GE_EQ_AFF_GE = prove_by_refinement
(` azim v0 v1 w1 w2 < pi /\ ~ collinear {v0,v1,w1} /\ ~ collinear {v0,v1,w2} 
 ==> wedge_ge v0 v1 w1 w2 = aff_ge {v0, v1} {w1,w2} `,
[ASM_SIMP_TAC[wedge_ge];
NHANH Fan.th3b1;
ONCE_REWRITE_TAC[INSERT_COMM];
NHANH Fan.th3b1;
STRIP_TAC;
SUBGOAL_THEN` DISJOINT {v1,v0} {w2,w1:real^3}` ASSUME_TAC;
ASM SET_TAC[];
SUBGOAL_THEN ` aff_ge {v0,v1} {w1} SUBSET aff_ge {v0,v1} {w1,w2:real^3}` ASSUME_TAC;
MATCH_MP_TAC AFF_GE_MONO_RIGHT;
DOWN THEN SET_TAC[];



ASM_CASES_TAC` azim v0 v1 w1 w2 = &0 `;

ASM_SIMP_TAC[IN_ELIM_THM; REAL_LE_ANTISYM; REAL_ARITH` &0 = a <=> a = &0`];
DOWN;
UNDISCH_TAC` ~ collinear {v1,v0,w1:real^3 }`;
ONCE_REWRITE_TAC[INSERT_COMM];
ASM_SIMP_TAC[AZIM_EQ_0_GE_ALT2; GSYM SUBSET_ANTISYM_EQ; SET_RULE` {x| x IN S } = S `];


(* *)
SUBGOAL_THEN` DISJOINT {v0,v1} {w1: real^3}` MP_TAC;
DOWN THEN SET_TAC[];
DOWN THEN DOWN;
SIMP_TAC[Collect_geom.simp_def2; AFF_GE22; INSERT_COMM];
REWRITE_TAC[IN_ELIM_THM; SUBSET];
REPEAT STRIP_TAC;
DOWN;
ASM_REWRITE_TAC[];
SIMP_TAC[];
STRIP_TAC;
EXISTS_TAC ` aa + yy * ta `;
EXISTS_TAC ` bb + yy * tb `;
EXISTS_TAC ` xx + yy * t `;
ASM_REWRITE_TAC[REAL_ARITH` (aa + yy * ta) + (bb + yy * tb) + xx + yy * t =
aa + bb + xx + yy *  ( ta + tb + t ) `; REAL_MUL_RID];


CONJ_TAC;
MATCH_MP_TAC REAL_LE_ADD;
ASM_REWRITE_TAC[];
MATCH_MP_TAC REAL_LE_MUL;
ASM_REWRITE_TAC[];
CONV_TAC VECTOR_ARITH;
MP_TAC (SPECL [`v0:real^3 `;` v1:real^3 `;`w1:real^3 `;` w2:real^3 `] AZIM_RANGE);
ASM_REWRITE_TAC[REAL_ARITH` &0 <= a <=> a = &0 \/ &0 < a `];
STRIP_TAC;
DOWN_TAC THEN SIMP_TAC[INSERT_COMM];
STRIP_TAC;
ASSUME_TAC2 WEDGE_LUNE_GT;



REWRITE_TAC[EXTENSION; IN_ELIM_THM];
GEN_TAC;
EQ_TAC;
ASM_SIMP_TAC[AZIM_EQ_0_GE_ALT2];
STRIP_TAC;
UNDISCH_TAC ` x IN aff_ge {v0, v1} {w1:real^3}`;
UNDISCH_TAC ` aff_ge {v0, v1} {w1} SUBSET aff_ge {v0, v1} {w1, w2:real^3}`;
REWRITE_TAC[SUBSET];
DISCH_THEN NHANH;
SIMP_TAC[];

DOWN;
REWRITE_TAC[REAL_ARITH` a <= b <=> a = b \/ a < b `];
SUBGOAL_THEN` ~ collinear {v0,v1,x:real^3} ` ASSUME_TAC;
STRIP_TAC;
UNDISCH_TAC ` &0 < azim v0 v1 w1 x `;
DOWN;
SIMP_TAC[AZIM_DEGENERATE];
DISCH_TAC;
REAL_ARITH_TAC;



STRIP_TAC;
DOWN;
ASM_SIMP_TAC[AZIM_EQ_ALT];
STRIP_TAC;

SUBGOAL_THEN ` aff_ge {v0,v1} {w2} SUBSET aff_ge {v0,v1} {w1,w2:real^3} ` ASSUME_TAC;
MATCH_MP_TAC AFF_GE_MONO_RIGHT;
ASM_REWRITE_TAC[];
SET_TAC[];
DOWN;
REWRITE_TAC[SUBSET];
DISCH_THEN MATCH_MP_TAC;
MP_TAC (ISPECL [`{v0,v1:real^3} `;` {w2:real^3} `] AFF_GT_SUBSET_AFF_GE);
REWRITE_TAC[SUBSET];
DISCH_THEN MATCH_MP_TAC;
DOWN THEN SIMP_TAC[];

SUBGOAL_THEN` x IN wedge v0 v1 w1 w2 ` ASSUME_TAC;
REWRITE_TAC[wedge; IN_ELIM_THM];
DOWN THEN DOWN THEN DOWN THEN PHA;
SIMP_TAC[];
DOWN;
MATCH_MP_TAC (SET_RULE` a SUBSET b ==> x IN a ==> x IN b `);
ASM_REWRITE_TAC[AFF_GT_SUBSET_AFF_GE];

ASM_SIMP_TAC[AFF_GE22; IN_ELIM_THM];
STRIP_TAC;
ASM_CASES_TAC ` yy = &0 `;
FIRST_X_ASSUM SUBST_ALL_TAC;

UNDISCH_TAC` ~collinear {v0, v1, w1:real^3}`;
NHANH AZIM_EQ_0_GE_ALT2;
STRIP_TAC;
SUBGOAL_THEN ` azim v0 v1 w1 x = &0 ` ASSUME_TAC;
FIRST_X_ASSUM SUBST1_TAC;
ASSUME_TAC2 (SET_RULE` DISJOINT {v0, v1} {w1, w2} ==> DISJOINT {v0,v1} {w1:real^3} `);
DOWN;
SIMP_TAC[Collect_geom.simp_def2];
ASM_REWRITE_TAC[IN_ELIM_THM];
STRIP_TAC;
EXISTS_TAC` aa:real `;
EXISTS_TAC ` bb:real `;
EXISTS_TAC ` xx:real `;
ASM_REWRITE_TAC[VECTOR_MUL_LZERO; VECTOR_ADD_RID];
UNDISCH_TAC ` aa + bb + xx + &0 = &1 `;
REAL_ARITH_TAC;
FIRST_X_ASSUM SUBST_ALL_TAC;
REWRITE_TAC[AZIM_RANGE];



ASM_CASES_TAC` collinear {v0,v1,x:real^3} `;
DOWN;
SIMP_TAC[AZIM_DEGENERATE; AZIM_RANGE];


ASM_CASES_TAC ` xx = &0 `;
SUBGOAL_THEN ` azim v0 v1 w1 x = azim v0 v1 w1 w2 ` ASSUME_TAC;
SUBGOAL_THEN ` azim v0 v1 w1 x = azim v0 v1 w1 w2 <=> x IN aff_gt {v0,v1} {w2}` SUBST1_TAC;
MATCH_MP_TAC AZIM_EQ_ALT;
ASM_REWRITE_TAC[];
MP_TAC2 (SET_RULE` DISJOINT {v0,v1} {w1,w2} ==> DISJOINT {v0,v1} {w2:real^3}`);
SIMP_TAC[AFF_GT_2_1; IN_ELIM_THM];
REMOVE_TAC;
EXISTS_TAC ` aa: real`;
EXISTS_TAC` bb:real `;
EXISTS_TAC ` yy:real `;
ASM_REWRITE_TAC[VECTOR_MUL_LZERO; VECTOR_ADD_LID];
FIRST_X_ASSUM SUBST_ALL_TAC;
UNDISCH_TAC ` &0 <= yy `;
UNDISCH_TAC ` ~( yy = &0 ) `;
UNDISCH_TAC ` aa + bb + &0 + yy = &1 `;
REAL_ARITH_TAC;
ASM_REWRITE_TAC[REAL_LE_REFL];
SUBGOAL_THEN` x IN wedge v0 v1 w1 w2 ` ASSUME_TAC;
ASM_SIMP_TAC[Planarity.AFF_GT_2_2; IN_ELIM_THM];
EXISTS_TAC ` aa:real ` ;
EXISTS_TAC ` bb:real`;
EXISTS_TAC ` xx:real ` ;
EXISTS_TAC` yy:real`;
ASM_REWRITE_TAC[REAL_ARITH` a < b <=> ~( b = a ) /\ a <= b `];
DOWN;
REWRITE_TAC[wedge; IN_ELIM_THM];
IMP_TAC;
REMOVE_TAC;
REAL_ARITH_TAC]);;







let AZIM_PI_WEDGE_GE_SIN = prove_by_refinement(` azim u v w ww = pi ==>
wedge_ge u v w ww = {x| &0 <= sin (azim u v w x ) } `,
[REWRITE_TAC[wedge_ge; EXTENSION; IN_ELIM_THM];

ONCE_REWRITE_TAC[GSYM (SPEC ` -- (u:real^3) ` AZIM_TRANSLATION)];

REWRITE_TAC[VECTOR_ARITH` -- a + (a:real^N) = vec 0`;VECTOR_ARITH` -- a + b = b - a:real^N `; re_eqvl];
REPEAT STRIP_TAC;
ASM_REWRITE_TAC[];
EQ_TAC;
REWRITE_TAC[SIN_POS_PI_LE];
MP_TAC (SPECL [`u - u:real^3 `;` v - u:real^3 `;` w - u:real^3 `;
` x - u:real^3 `] AZIM_RANGE);
STRIP_TAC;
ASM_CASES_TAC ` azim (u - u) (v - u) (w - u) (x - u) <= pi `;
ASM_REWRITE_TAC[];
ABBREV_TAC ` goc = azim (u - u) (v - u) (w - u) (x - u) `;
STRIP_TAC;
SUBGOAL_THEN` sin goc < &0 ` MP_TAC;
ONCE_REWRITE_TAC[SIN_SUB_PERIODIC];
REWRITE_TAC[REAL_ARITH` -- a < &0 <=> &0 < a `];
MATCH_MP_TAC SIN_POS_PI;
ASM_REWRITE_TAC[REAL_ARITH` a - b < b <=> a < &2 * b `;REAL_ARITH` &0 < a - pi <=> ~( a <= pi ) `];
DOWN;
REAL_ARITH_TAC]);;





let AZIM_PI_WEDGE_GE_CROSS_DOT = prove_by_refinement(` azim u v w ww = pi ==> 
wedge_ge u v w ww = {x | &0 <= ((v - u) cross ( w - u )) dot ( x - u )}`,
[SIMP_TAC[AZIM_PI_WEDGE_GE_SIN; EXTENSION; IN_ELIM_THM];
REPEAT STRIP_TAC;
ONCE_REWRITE_TAC[GSYM (SPEC ` -- (u:real^3) ` AZIM_TRANSLATION)];
REWRITE_TAC[VECTOR_ARITH` -- a + a:real^N = vec 0 `; VECTOR_ARITH` -- a + x = x - a:real^N `];
MP_TAC (SPECL [` v - u:real^3 `;` w - u:real^3 `;` x - u:real^3 `] Trigonometry2.JBDNJJB);
REWRITE_TAC[re_eqvl];
STRIP_TAC;
ASM_REWRITE_TAC[VECTOR_SUB_REFL];
ASM_SIMP_TAC[REAL_LE_MUL_EQ]]);;




let AZIM_PI_CONVEX_WEDGE = prove_by_refinement(` azim u v w ww = pi
     ==> convex (wedge_ge u v w ww) `,
[SIMP_TAC[AZIM_PI_WEDGE_GE_CROSS_DOT; convex; IN_ELIM_THM];
REPEAT STRIP_TAC;
ONCE_REWRITE_TAC[VECTOR_ARITH` (u' % x + v' % y) = u' % ( x - u ) + 
v' % (y - u ) + (u' + v') % u `];
ASM_REWRITE_TAC[VECTOR_MUL_LID; VECTOR_ARITH ` ( a + c + b ) - b = (a:real^N) + c `; DOT_RADD; DOT_RMUL];
MATCH_MP_TAC REAL_LE_ADD;
CONJ_TAC;
MATCH_MP_TAC REAL_LE_MUL;
ASM_REWRITE_TAC[];
MATCH_MP_TAC REAL_LE_MUL;
ASM_REWRITE_TAC[]]);;





let CONVEX_WEDGE_LE_PI = prove(` azim v0 v1 w1 w2 <= pi /\
       ~collinear {v0, v1, w1} /\
       ~collinear {v0, v1, w2}
       ==> convex ( wedge_ge v0 v1 w1 w2 )`,
ASM_CASES_TAC ` azim v0 v1 w1 w2 = pi ` THENL [
ASM_SIMP_TAC[AZIM_PI_CONVEX_WEDGE];
ASM_SIMP_TAC[REAL_ARITH` ~( a = b ) ==> ( a <= b <=> a < b )`]] THEN 
NHANH WEDGE_GE_EQ_AFF_GE THEN SIMP_TAC[CONVEX_AFF_GE]);;






let wedge_in_fan_ge2 = prove(`  wedge_in_fan_ge x E =
         (if CARD (EE (FST x) E) > 1
          then wedge_ge (vec 0) (FST x ) (SND x ) (azim_cycle (EE (FST x) E) (vec 0) (FST x ) (SND x ) )
          else {x | T}) `,
PAT_ONCE_REWRITE_TAC `\x. x = y ` [GSYM PAIR]
THEN REWRITE_TAC[wedge_in_fan_ge]);;





let azim_in_fan2 = prove(` azim_in_fan x E =
         (let d = azim_cycle (EE ( FST x ) E) (vec 0) (FST x ) (SND x ) in
          if CARD (EE (FST x ) E) > 1 then azim (vec 0) (FST x ) (SND x) d else &2 * pi) `,
PAT_ONCE_REWRITE_TAC `\x. x = y ` [GSYM PAIR]
THEN REWRITE_TAC[azim_in_fan]);;




let LOFA_IMP_NOT_INCLUDE_VEC0 = prove(
` local_fan (V,E,FF) ==> ~( vec 0 IN V ) `,
NHANH Wrgcvdr_cizmrrh.LOCAL_FAN_IMP_FAN THEN NHANH FAN_IMP_V_DIFF
THEN STRIP_TAC THEN DOWN THEN SET_TAC[]);;





let AZIM_SPEC_DEGENERATE = prove(
` azim v0 v1 w1 v0 = &0 /\ azim v0 v1 w1 v1 = &0 `,
SUBGOAL_THEN ` collinear {v0, v1, v0} /\ collinear {v0,v1,v1:real^3} ` MP_TAC
THENL [SIMP_TAC[INSERT_INSERT; INSERT_COMM; COLLINEAR_2];
SIMP_TAC[AZIM_DEGENERATE]]);;






let CONDS_IN_CONV2 = prove_by_refinement
(` &0 <= t2 /\ &0 <= t3 /\ ~( t2 = &0 /\ t3 = &0) 
==> t2 / (t2 + t3) % v + t3 / (t2 + t3) % w IN conv {v, w} `,
[STRIP_TAC;
SUBGOAL_THEN ` ~( t2 + t3 = &0 ) ` ASSUME_TAC;
DOWN_TAC;
REAL_ARITH_TAC;
REWRITE_TAC[Collect_geom.CONV_SET2; IN_ELIM_THM];
EXISTS_TAC ` t2 / (t2 + t3 ) `;
EXISTS_TAC ` t3 / (t2 + t3 ) `;
CONJ_TAC;
MATCH_MP_TAC REAL_LE_DIV;
DOWN_TAC THEN REAL_ARITH_TAC;
CONJ_TAC;
MATCH_MP_TAC REAL_LE_DIV;
DOWN_TAC THEN REAL_ARITH_TAC;
ASM_SIMP_TAC[GSYM Geomdetail.SUM_TWO_RATIO]]);;









let PGSQVBL = prove_by_refinement
(` convex_local_fan (V,E,FF) /\ {v,w} SUBSET V /\ x IN FF ==>
aff_ge {vec 0} {v,w} SUBSET wedge_in_fan_ge x E `,
[REWRITE_TAC[convex_local_fan];
STRIP_TAC;
DOWN;
FIRST_X_ASSUM NHANH;
REWRITE_TAC[azim_in_fan2; wedge_in_fan_ge2];
LET_TAC;
STRIP_TAC;
SUBGOAL_THEN ` FST (x:real^3 # real^3) IN V /\ SND x IN V ` MP_TAC;
MATCH_MP_TAC LOCAL_FAN_IMP_IN_V;
ASM_REWRITE_TAC[];
STRIP_TAC;
ASSUME_TAC2 (SPEC `FST (x:real^3 # real^3 ) ` (GEN `v:real^3 ` LOFA_CARD_EE_V_1));
UNDISCH_TAC` (if CARD (EE (FST x) E) > 1
       then azim (vec 0) (FST x) (SND x) d
       else &2 * pi) <=
      pi `;
UNDISCH_TAC` V SUBSET
      (if CARD (EE (FST x) E) > 1
       then wedge_ge (vec 0) (FST x) (SND x) d
       else {x | T}) `;
PHA;
ASM_SIMP_TAC[ARITH_RULE` a = 2 ==> a > 1 `];
ASSUME_TAC2 (SPEC ` FST (x:real^3 #real^3 ) ` (GEN `v:real^3 ` EXISTS_INVERSE_OF_V));
DOWN THEN STRIP_TAC;
MP_TAC2 (SPEC ` FST (x:real^3 # real^3) ` (GEN `v:real^3 ` LOFA_IMP_EE_TWO_ELMS));
DISCH_THEN SUBST_ALL_TAC;
SUBGOAL_THEN ` rho_node1 FF (FST x) = SND (x:real^3 # real^3) ` ASSUME_TAC;
MATCH_MP_TAC DETER_RHO_NODE;
ASM_REWRITE_TAC[];
UNDISCH_TAC ` azim_cycle {rho_node1 FF (FST x), vv} (vec 0) (FST x) (SND x) = d `;
ASM_REWRITE_TAC[AZIM_CYCLE_TWO_POINT_SET];
MP_TAC2 LOCAL_FAN_RHO_NODE_PROS2;
REPEAT STRIP_TAC;
UNDISCH_TAC` vv:real^3 IN V `;
FIRST_X_ASSUM NHANH;
STRIP_TAC;
DOWN;
UNDISCH_TAC ` local_fan (V,E,FF) `;
PHA;
NHANH LOCAL_FAN_IN_FF_NOT_COLLINEAR;
ASM_REWRITE_TAC[];
STRIP_TAC;
ASSUME_TAC2 LOCAL_FAN_IN_FF_NOT_COLLINEAR;
SUBGOAL_THEN ` conv {v,w:real^3} SUBSET wedge_ge  (vec 0) (FST x) (SND x) d ` ASSUME_TAC;
SUBGOAL_THEN ` convex (wedge_ge (vec 0) (FST x ) (SND x ) d) ` ASSUME_TAC;
MATCH_MP_TAC CONVEX_WEDGE_LE_PI;
ASM_SIMP_TAC[];
UNDISCH_TAC` ~ collinear {vec 0, d, FST (x:real^3 # real^3)}`;
SIMP_TAC[INSERT_COMM];
MATCH_MP_TAC Geomdetail.CONVEX_IM_CONV2_SU;
ASM_REWRITE_TAC[SET_RULE` a IN S /\ b IN S <=> {a,b} SUBSET S `];
MATCH_MP_TAC SUBSET_TRANS;
EXISTS_TAC` V:real^3 -> bool `;
ASM_REWRITE_TAC[];
SUBGOAL_THEN ` DISJOINT {vec 0} {v,w:real^3} ` MP_TAC;
MP_TAC2 LOFA_IMP_NOT_INCLUDE_VEC0;
UNDISCH_TAC` {v,w:real^3} SUBSET V `;
SET_TAC[];

SIMP_TAC[Fan.AFF_GE_1_2; SUBSET; IN_ELIM_THM];
REPEAT STRIP_TAC;
ASM_CASES_TAC ` t2 = &0 /\ t3 = &0 `;
ASM_REWRITE_TAC[VECTOR_MUL_LZERO; VECTOR_MUL_RZERO; VECTOR_ADD_LID; wedge_ge; IN_ELIM_THM];
REWRITE_TAC[AZIM_SPEC_DEGENERATE; AZIM_RANGE; REAL_LE_REFL];
SUBGOAL_THEN ` (t2 / ( t2 + t3)) % v + ( t3 / ( t2 + t3 )) % w IN conv {v,w:real^3}` ASSUME_TAC;
MATCH_MP_TAC CONDS_IN_CONV2;
ASM_REWRITE_TAC[];
DOWN;
UNDISCH_TAC ` conv {v, w} SUBSET wedge_ge (vec 0) (FST x) (SND x) d `;
REWRITE_TAC[SUBSET];
DISCH_THEN NHANH;
ABBREV_TAC `p = t2 / (t2 + t3) % v + t3 / (t2 + t3) % (w:real^3) `;
SUBGOAL_THEN ` x' = t1 % (vec 0) + (&0) % (FST (x:real^3 # real^3)) + (t2 + t3 ) % (p:real^3) ` ASSUME_TAC;
ASM_REWRITE_TAC[VECTOR_MUL_LZERO; VECTOR_MUL_RZERO; VECTOR_ADD_LID];
EXPAND_TAC "p";
REWRITE_TAC[VECTOR_ADD_LDISTRIB; VECTOR_MUL_ASSOC];
MP_TAC2 (REAL_ARITH` &0 <= t2 /\ &0 <= t3 /\ ~( t2 = &0 /\ t3 = &0) 
==> ~( t2 + t3 = &0) `);
SIMP_TAC[REAL_FIELD` ~( t = &0 ) ==> t * a / t = a `];
ASM_CASES_TAC ` collinear {vec 0, FST (x:real^3 # real^3), p}`;
DOWN;
ASSUME_TAC2 LOFA_IMP_NOT_INCLUDE_VEC0;
SUBGOAL_THEN` ~( FST (x:real^3 # real^3 ) = vec 0 ) ` ASSUME_TAC;
UNDISCH_TAC ` FST (x:real^3 # real^3 ) IN V `;
DOWN;
SET_TAC[];
ASM_SIMP_TAC[COLLINEAR_LEMMA_ALT];
STRIP_TAC;
SUBGOAL_THEN ` collinear {vec 0, FST (x:real^3 # real^3), x'}` MP_TAC;
SIMP_TAC[COLLINEAR_LEMMA_ALT];
DISJ2_TAC;
EXISTS_TAC ` (t2 + t3) * c `;
UNDISCH_TAC` x' = t1 % vec 0 + &0 % FST (x:real^3 # real^3) + (t2 + t3) % (p:real^3)`;
SIMP_TAC[];
ASM_REWRITE_TAC[];
DISCH_TAC;
VECTOR_ARITH_TAC;

UNDISCH_TAC` x' = t1 % vec 0 + t2 % v + t3 % (w:real^3) `;
DISCH_THEN (SUBST1_TAC o SYM);
SIMP_TAC[AZIM_DEGENERATE; wedge_ge; IN_ELIM_THM; AZIM_RANGE; REAL_LE_REFL];

REWRITE_TAC[wedge_ge; IN_ELIM_THM];
SUBGOAL_THEN` azim (vec 0) (FST x) (SND x) p = azim (vec 0) (FST x) (SND x) x'` SUBST1_TAC;
UNDISCH_TAC` x' = t1 % vec 0 + &0 % FST (x:real^3 # real^3) + (t2 + t3) % p`;
SIMP_TAC[];
DISCH_TAC;
MATCH_MP_TAC Topology.th1;
ASSUME_TAC2 (REAL_ARITH` &0 <= t2 /\ &0 <= t3 /\ ~( t2 = &0 /\ t3 = &0 ) ==> t2 + t3 > &0 `);

ASM_REWRITE_TAC[REAL_ADD_LID];
ASM_SIMP_TAC[Fan.th3a];
SIMP_TAC[]]);;

(* begin new lemma *)
(* =============== *)
(* =============== *)




let FST_EQ_IF_SAME_SND = prove_by_refinement
(` local_fan (V,E,FF) /\ (w1,v) IN FF /\ (w2,v) IN FF 
==> w1 = w2`,
[NHANH LOFA_IMP_BIJ_VV;
STRIP_TAC;
SUBGOAL_THEN` {w1,w2,v} SUBSET (V:real^3 -> bool) ` ASSUME_TAC;
REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET];
ASM_MESON_TAC[LOCAL_FAN_IMP_IN_V];

SUBGOAL_THEN` rho_node1 FF w1 = v /\ rho_node1 FF w2 = v ` ASSUME_TAC;
ASM_MESON_TAC[DETER_RHO_NODE];
DOWN_TAC;
REWRITE_TAC[BIJ; INJ; INSERT_SUBSET];
MESON_TAC[]]);;





let PRE_IVS_RHO_NODE1_DETE = prove_by_refinement
(` local_fan (V,E,FF) /\ (vv,v) IN FF 
==> (@a. a,v IN FF) = vv `,
[STRIP_TAC; MATCH_MP_TAC RHO_NODE_INVERSE_POINT;
ASM_REWRITE_TAC[]; REPEAT STRIP_TAC; 
MATCH_MP_TAC  FST_EQ_IF_SAME_SND; ASM_REWRITE_TAC[]]);;


let IVS_RHO_NODE1_DETE = prove(`  local_fan (V,E,FF) /\ (vv,v) IN FF 
==> ivs_rho_node1 FF v = vv`,
REWRITE_TAC[PRE_IVS_RHO_NODE1_DETE; ivs_rho_node1]);;




let AZIM_EQ_0_SYM2 = prove(`! w1 w2. azim v0 v1 w1 w2 = &0 <=> azim v0 v1 w2 w1 = &0 `,
REWRITE_TAC[MESON[]` (! x y. P x y <=> P y x ) <=> (! x y. P x y ==> P y x )`] THEN 
MESON_TAC[AZIM_EQ_0_SYM; AZIM_DEGENERATE]);;



let LOCAL_FAN_IN_FF_IN_ORD_PAIRS2 =
prove(`local_fan (V,E,FF) /\ x,y IN FF ==> {x,y} IN E `,
NHANH LOCAL_FAN_IN_FF_IN_ORD_PAIRS THEN 
SIMP_TAC[Wrgcvdr_cizmrrh.IN_E_IFF_IN_ORD_E]);;






let INTERIOR_ANGLE1_POS = prove_by_refinement
(` local_fan (V,E,FF) /\ v IN V ==> &0 < interior_angle1 (vec 0) FF v `,
[ASM_CASES_TAC` interior_angle1 (vec 0) FF v = &0 `;
DOWN;
REWRITE_TAC[interior_angle1];
NHANH EXISTS_INVERSE_OF_V;
REPEAT STRIP_TAC;
MP_TAC2 LOCAL_FAN_RHO_NODE_PROS2;
STRIP_TAC;
UNDISCH_TAC` vv:real^3 IN V `;
UNDISCH_TAC ` v:real^3 IN V `;
FIRST_X_ASSUM NHANH;
STRIP_TAC THEN STRIP_TAC;
SUBGOAL_THEN` (@a. a,v IN (FF:real^3 # real^3 -> bool)) = vv ` ASSUME_TAC;
MATCH_MP_TAC PRE_IVS_RHO_NODE1_DETE;
DOWN;
ASM_REWRITE_TAC[];
FIRST_X_ASSUM SUBST_ALL_TAC;
ABBREV_TAC ` vc = rho_node1 FF v `;
SUBGOAL_THEN ` vv = vc:real^3 ` ASSUME_TAC;
MATCH_MP_TAC Fan.UNIQUE_AZIM_POINT_FAN;
EXISTS_TAC ` vec 0:real^3 `;
EXISTS_TAC `V:real^3 -> bool` ;
EXISTS_TAC ` E: (real^3 -> bool) -> bool `;
EXISTS_TAC ` v:real^3 `;
EXISTS_TAC `vv:real^3 `;
UNDISCH_TAC` azim (vec 0) v vc vv = &0 `;
ONCE_REWRITE_TAC[AZIM_EQ_0_SYM2];
SIMP_TAC[AZIM_REFL];

MP_TAC2 Wrgcvdr_cizmrrh.LOCAL_FAN_IMP_FAN;
STRIP_TAC THEN STRIP_TAC;

ASM_MESON_TAC[LOCAL_FAN_IN_FF_IN_ORD_PAIRS2; INSERT_COMM];
ASM_MESON_TAC[LOCAL_FAN_IMP_NOT_SEMI_IDE];
REMOVE_TAC;
DOWN;
REWRITE_TAC[interior_angle1];
MP_TAC (SPECL [`vec 0:real^3 `;` v:real^3 `;` rho_node1 FF v `;
` (@a. a,v IN (FF:real^3 # real^3 -> bool))`] AZIM_RANGE);
REAL_ARITH_TAC]);;

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


let FAN_IMP_NOT_IN_AFF_GE = prove_by_refinement
(` FAN (x:real^N,V,E) /\ {v, w} SUBSET V /\ ~( v = w)
==> ~ ( v IN aff_ge {x} {w} ) `,
[NHANH FAN_IMP_V_DIFF;
REWRITE_TAC[FAN; fan7];
STRIP_TAC;
FIRST_X_ASSUM (MP_TAC o (SPECL [`{v:real^N}`;`{w:real^N}`]));
ANTS_TAC;
REWRITE_TAC[IN_UNION; IN_ELIM_THM];
UNDISCH_TAC` {v, w:real^N} SUBSET V`;
SET_TAC[];
ASM_SIMP_TAC[SET_RULE` ~( a = b ) ==> {a} INTER {b} = {} `;
AFFINE_HULL_SING; AFF_GE_EQ_AFFINE_HULL];
STRIP_TAC;
MP_TAC (ISPECL [`x:real^N`;` v:real^N`] (CONJUNCT2 ENDS_IN_HALFLINE));
DOWN THEN DOWN;
REMOVE_TAC;
DOWN;
REWRITE_TAC[INSERT_SUBSET];
FIRST_ASSUM NHANH;
SET_TAC[]]);;






let IN_AFF_LT_IMP_IN_CONV = prove_by_refinement
(`DISJOINT {x:real^N} {b} /\ a IN aff_lt {x} {b} ==> x IN conv0 {a,b} `,
[IMP_TAC;
SIMP_TAC[AFF_LT_1_1; IN_ELIM_THM];
REPEAT STRIP_TAC;
DOWN;
ONCE_REWRITE_TAC[VECTOR_ARITH` a = x % b + y % c <=> x % b = a + (-- y) % c`];
ASSUME_TAC2 (REAL_ARITH` t2 < &0 /\ t1 + t2 = &1 ==> ~(t1 = &0 )`);
ASM_SIMP_TAC[Ldurdpn.VECTOR_SCALE_CHANGE];
ASSUME_TAC2 (REAL_ARITH` t2 < &0 /\ t1 + t2 = &1 ==> &0 < t1`);
REWRITE_TAC[VECTOR_ADD_LDISTRIB;REAL_ARITH` &1 / a * b = b / a `;
 VECTOR_MUL_ASSOC; Collect_geom.CONV0_SET2; IN_ELIM_THM];
STRIP_TAC;
EXISTS_TAC` &1 / t1 `;
EXISTS_TAC ` ( -- t2 ) / t1 `;
CONJ_TAC;
MATCH_MP_TAC REAL_LT_DIV;
ASM_REWRITE_TAC[];
REAL_ARITH_TAC;
CONJ_TAC;
MATCH_MP_TAC REAL_LT_DIV;
ASM_REWRITE_TAC[REAL_ARITH` &0 < -- a <=> a < &0 `];
REWRITE_TAC[];
UNDISCH_TAC` t1 + t2 = &1 `;
UNDISCH_TAC ` ~( t1 = &0 ) `;
CONV_TAC REAL_FIELD]);;



let FAN_SUB_NOT_EQ_COLL_IN_CONV0 = prove_by_refinement
(` FAN (x,V,E) /\ {v:real^N, w} SUBSET V /\ ~(v = w) /\
 collinear {x, v, w} ==> x IN (conv0 {v,w}) `,
[REWRITE_TAC[COLLINEAR_3_EXPAND];
NHANH FAN_IMP_V_DIFF;
STRIP_TAC;
UNDISCH_TAC ` {v, w:real^N} SUBSET V`;
REWRITE_TAC[INSERT_SUBSET];
ASM_MESON_TAC[];
ASM_CASES_TAC ` &0 <= &1 - u `;
SUBGOAL_THEN` v IN aff_ge {x} {w:real^N}` ASSUME_TAC;
REWRITE_TAC[HALFLINE; IN_ELIM_THM];
EXISTS_TAC ` &1 - u `;
ASM_REWRITE_TAC[REAL_ARITH` a - ( a - b ) = b `];
MP_TAC2 FAN_IMP_NOT_IN_AFF_GE;
ASM_REWRITE_TAC[];
MATCH_MP_TAC IN_AFF_LT_IMP_IN_CONV;
SUBGOAL_THEN` DISJOINT {x} {w:real^N}` ASSUME_TAC;
ASM SET_TAC[];

ASM_SIMP_TAC[AFF_LT_1_1; IN_ELIM_THM];
EXISTS_TAC` u:real`;
EXISTS_TAC` &1 - u `;
ASM_REWRITE_TAC[REAL_ARITH` a + b - a = b `;REAL_ARITH`b < a <=> ~( a <= b )  `]]);;





let IN_CONV_LINE_SEPERATABLE = prove_by_refinement
(` x IN conv0 {a,b:real^N} ==> aff {a,b} = aff_ge {x} {a} UNION aff_ge {x} {b} `,
[REWRITE_TAC[Collect_geom.CONV0_SET2; IN_ELIM_THM; EXTENSION];
STRIP_TAC THEN GEN_TAC;
EQ_TAC;
REWRITE_TAC[Collect_geom.AFF_2POINTS_INTERPRET; IN_ELIM_THM];
STRIP_TAC;
ASSUME_TAC2 (REAL_ARITH` ta + tb = &1 /\ a' + b' = &1 ==> ta <= a' \/ tb <= b' `);
DOWN_TAC;
DAO;
SPEC_TAC (`ta:real`,`ta:real`);
SPEC_TAC (`tb:real`,`tb:real`);
SPEC_TAC (`a':real`,`a':real`);
SPEC_TAC (`b':real`,`b':real`);
SPEC_TAC (`a:real^N`,`a:real^N`);
SPEC_TAC (`b:real^N`,`b:real^N`);
MATCH_MP_TAC (MESON[]`(! x y a b aa bb. P x y a b aa bb ==> P y x b a bb aa) /\
(! x y. L x y ==> L y x ) /\ (! x y a b aa bb. bb <= b /\ P x y a b aa bb ==> L x y ) ==> (! x y a b aa bb. (bb <= b \/ aa <= a ) /\ P x y a b aa bb ==> L x y ) `);

SIMP_TAC[UNION_COMM; VECTOR_ADD_SYM; REAL_ADD_SYM];
ONCE_REWRITE_TAC[EQ_SYM_EQ];
SIMP_TAC[];
REPEAT STRIP_TAC;


REWRITE_TAC[IN_UNION; IN_ELIM_THM; HALFLINE];
DISJ2_TAC;
EXISTS_TAC` tb - ( ta / a' ) * b' `;
CONJ_TAC;
SUBGOAL_THEN` ta / a' <= &1 ` ASSUME_TAC;
MATCH_MP_TAC Trigonometry2.REAL_LE_LDIV;
ASM_REWRITE_TAC[];
SUBGOAL_THEN` ta / a' * b' <= b' ` ASSUME_TAC;
REWRITE_TAC[REAL_ARITH` a * b <= b <=> &0 <= ( &1 - a ) * b `];
MATCH_MP_TAC REAL_LE_MUL;
DOWN THEN DOWN THEN REAL_ARITH_TAC;
ASM_REAL_ARITH_TAC;
EXPAND_TAC "x";
EXPAND_TAC "x'";
REWRITE_TAC[VECTOR_ADD_LDISTRIB; VECTOR_MUL_ASSOC; VECTOR_ARITH` ( a + b % x ) + c % x = a + ( b + c ) % x `];
MATCH_MP_TAC (MESON[]` a = aa /\ b = bb ==> a % x + b % y = aa % x + bb % y `);
CONJ_TAC;
REPLICATE_TAC 4 DOWN;
REMOVE_TAC;
DOWN THEN DOWN;
REMOVE_TAC;
DOWN;
PHA;
CONV_TAC REAL_FIELD;


REPLICATE_TAC 4 DOWN;
REMOVE_TAC;
DOWN THEN DOWN;
REMOVE_TAC;
DOWN;
PHA;
CONV_TAC REAL_FIELD;

SPEC_TAC (`x':real^N`,`x':real^N`);
REWRITE_TAC[GSYM SUBSET; UNION_SUBSET];
SUBGOAL_THEN ` x:real^N IN aff {a,b}` ASSUME_TAC;
REWRITE_TAC[Collect_geom.AFF_2POINTS_INTERPRET; IN_ELIM_THM];
EXISTS_TAC` a':real`;
EXISTS_TAC `b':real`;
ASM_REWRITE_TAC[];
ASSUME_TAC (ISPECL [` a:real^N`;`b:real^N`] Planarity.POINT_IN_LINE);
ASSUME_TAC (ISPECL [` b:real^N`;`a:real^N`] Planarity.POINT_IN_LINE);
MP_TAC (ISPECL [`x:real^N`;` a:real^N`] HALFLINE_SUBSET_AFFINE_HULL);
MP_TAC (ISPECL [`x:real^N`;` b:real^N`] HALFLINE_SUBSET_AFFINE_HULL);
SUBGOAL_THEN ` aff {x,a:real^N} SUBSET aff {a,b} ` MP_TAC;
MATCH_MP_TAC S_SUBSET_IMP_AFF_S_TOO;
ASM_REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET];
SUBGOAL_THEN ` aff {x,b:real^N} SUBSET aff {a,b} ` MP_TAC;
MATCH_MP_TAC S_SUBSET_IMP_AFF_S_TOO;
DOWN;
ASM_SIMP_TAC[INSERT_SUBSET; EMPTY_SUBSET];
ONCE_REWRITE_TAC[INSERT_COMM];
ASM_REWRITE_TAC[];

REWRITE_TAC[GSYM aff];
PHA;
SET_TAC[]]);;








let CVLF_LF_F = prove(
` convex_local_fan (V,E,FF) ==> local_fan (V,E,FF) /\ FAN (vec 0,V,E)`,
SIMP_TAC[convex_local_fan] THEN NHANH Wrgcvdr_cizmrrh.LOCAL_FAN_IMP_FAN THEN
SIMP_TAC[]);;



let EMPTY_NOT_EXISTS_IN = SET_RULE` a = {} <=> ~(? x. x IN a ) `;;




let LUNAR_IMP_INTERIOR_ANGLE1_EQ_PI = prove_by_refinement
(`convex_local_fan (V,E,FF) /\ lunar (v,w) V E
 ==> vec 0 IN conv0 {v,w} /\
(!u. u IN V DIFF {v, w} ==> interior_angle1 (vec 0) FF u = pi /\
             rho_node1 FF u IN aff {u,v,w} /\
             ivs_rho_node1 FF u IN aff {u,v,w})`,
[REWRITE_TAC[lunar];
STRIP_TAC;
SUBGOAL_THEN` vec 0 IN conv0 {v,w:real^3} ` ASSUME_TAC;
MATCH_MP_TAC FAN_SUB_NOT_EQ_COLL_IN_CONV0;
ASSUME_TAC2 CVLF_LF_F;
ASM_REWRITE_TAC[];
ASM_REWRITE_TAC[];
GEN_TAC THEN DISCH_TAC;
ABBREV_TAC` conclusion <=> interior_angle1 (vec 0) FF u = pi /\
 rho_node1 FF u IN aff {u, v, w} /\
 ivs_rho_node1 FF u IN aff {u, v, w} `;
SUBGOAL_THEN` vec 0 IN aff {u:real^3, vec 0} ` ASSUME_TAC;
ONCE_REWRITE_TAC[INSERT_COMM];
ASM_REWRITE_TAC[Planarity.POINT_IN_LINE];
SUBGOAL_THEN` ~(aff {u, vec 0} INTER conv0 {v,w:real^3} = {})` ASSUME_TAC;
REWRITE_TAC[INTER_EQ_EM_EXPAND];
EXISTS_TAC` vec 0:real^3 `;
ASM_REWRITE_TAC[];
ASM_CASES_TAC ` v IN aff {u, vec 0:real^3} `;
SUBGOAL_THEN`aff {u, vec 0} = aff {v,w:real^3} ` ASSUME_TAC;
MATCH_MP_TAC AFF2_DET_BY_TWO_POINTS;
ASM_REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET];
DOWN THEN DOWN;
SIMP_TAC[INTER_EQ_EM_EXPAND; GSYM IN_INTER];
MESON_TAC[AFF2_ITR_CONV0_IMP_SAME_ENDS];
SUBGOAL_THEN` u IN aff {u:real^3, vec 0} ` MP_TAC;
REWRITE_TAC[Planarity.POINT_IN_LINE];
ASM_REWRITE_TAC[];
UNDISCH_TAC` vec 0 IN conv0 {v, w:real^3}`;
NHANH IN_CONV_LINE_SEPERATABLE;
SIMP_TAC[IN_UNION];
REPEAT STRIP_TAC;
SUBGOAL_THEN` ~( u:real^3 IN aff_ge {vec 0} {v} )` MP_TAC;
MATCH_MP_TAC FAN_IMP_NOT_IN_AFF_GE;
ASSUME_TAC2 CVLF_LF_F;
ASM_REWRITE_TAC[];
UNDISCH_TAC` {v,w:real^3} SUBSET V `;
UNDISCH_TAC `u:real^3 IN V DIFF {v,w} `;
SET_TAC[];
ASM_REWRITE_TAC[];
SUBGOAL_THEN` ~( u IN aff_ge {vec 0} {w:real^3}) ` MP_TAC;
MATCH_MP_TAC FAN_IMP_NOT_IN_AFF_GE;
ASSUME_TAC2 CVLF_LF_F;
ASM_REWRITE_TAC[];
UNDISCH_TAC` {v,w:real^3} SUBSET V `;
UNDISCH_TAC `u:real^3 IN V DIFF {v,w} `;
SET_TAC[];
ASM_REWRITE_TAC[];
SUBGOAL_THEN` {v,w:real^3} INTER aff {u, vec 0} = {} ` ASSUME_TAC;
ASM_REWRITE_TAC[Trigonometry2.INSERT_INTER_EMPTY];
DOWN THEN DOWN;
REWRITE_TAC[EMPTY_NOT_EXISTS_IN];
MESON_TAC[AFF2_ITR_CONV0_IMP_SAME_ENDS];
EXPAND_TAC "conclusion";
MATCH_MP_TAC OZQVSFF;
EXISTS_TAC ` v:real^3 `;
EXISTS_TAC ` w:real^3 `;
UNDISCH_TAC` {v,w:real^3} SUBSET V `;
UNDISCH_TAC` u IN V DIFF {v,w:real^3}`;
ASM_REWRITE_TAC[INSERT_SUBSET; IN_DIFF];
SIMP_TAC[];
STRIP_TAC THEN STRIP_TAC;
CONJ_TAC;
REWRITE_TAC[plane];
SUBGOAL_THEN` ~ collinear {u,v,w:real^3} ` ASSUME_TAC;
ASM_SIMP_TAC[Collect_geom.NOT_TWO_EQ_IMP_COL_EQUAVALENT];
UNDISCH_TAC` vec 0 IN conv0 {v, w:real^3}`;
NHANH IN_CONV_LINE_SEPERATABLE;
SIMP_TAC[];
STRIP_TAC;
REWRITE_TAC[IN_UNION; DE_MORGAN_THM];
CONJ_TAC;
MATCH_MP_TAC FAN_IMP_NOT_IN_AFF_GE;
ASSUME_TAC2 CVLF_LF_F;
ASM_REWRITE_TAC[];
UNDISCH_TAC` ~( u:real^3 IN {v,w})`;
ASM_REWRITE_TAC[INSERT_SUBSET; IN_INSERT];
MESON_TAC[];
MATCH_MP_TAC FAN_IMP_NOT_IN_AFF_GE;
ASSUME_TAC2 CVLF_LF_F;
ASM_REWRITE_TAC[];
UNDISCH_TAC` ~( u:real^3 IN {v,w})`;
ASM_REWRITE_TAC[INSERT_SUBSET; IN_INSERT];
MESON_TAC[];
DOWN;
REWRITE_TAC[aff];
MESON_TAC[];
CONJ_TAC;
MP_TAC (ISPECL [` v:real^3`;` w:real^3`] (GEN_ALL CONV02_SUBSET_AFF2));
SUBGOAL_THEN` aff {v,w} SUBSET aff {u,v,w:real^3}` ASSUME_TAC;
REWRITE_TAC[aff];
MATCH_MP_TAC HULL_MONO;
REWRITE_TAC[INSERT_SUBSET; IN_INSERT; EMPTY_SUBSET];
REWRITE_TAC[EMPTY_SUBSET];
STRIP_TAC;
CONJ_TAC;
UNDISCH_TAC` aff {v, w} SUBSET aff {u, v, w:real^3}`;
REWRITE_TAC[SUBSET];
DISCH_THEN MATCH_MP_TAC;
DOWN;
REWRITE_TAC[SUBSET];
DISCH_THEN MATCH_MP_TAC;
FIRST_ASSUM ACCEPT_TAC;
REWRITE_TAC[aff];
MESON_TAC[INSERT_COMM; Trigonometry2.IN_P_HULL_INSERT];
ASM_SIMP_TAC[INSERT_COMM]]);;


let AFF_GE_MONO_TRANS = prove(
` S SUBSET X ==> aff_ge (X DIFF S ) ( Y UNION S) SUBSET aff_ge X Y `,
STRIP_TAC THEN REWRITE_TAC[aff_ge_def; SUBSET; IN; affsign] THEN 
ASM_SIMP_TAC[SET_RULE` S SUBSET X ==> (X DIFF S UNION Y UNION S) = X UNION Y `]
THEN SET_TAC[]);;


let AFF_GT_MONO_TRANS = prove(
` S SUBSET X ==> aff_gt (X DIFF S ) ( Y UNION S) SUBSET aff_gt X Y `,
STRIP_TAC THEN REWRITE_TAC[aff_gt_def; SUBSET; IN; affsign] THEN 
ASM_SIMP_TAC[SET_RULE` S SUBSET X ==> (X DIFF S UNION Y UNION S) = X UNION Y `]
THEN SET_TAC[]);;

let CONV_UNION_SUB_AFF_GE = prove(
` conv (X:(real^N -> bool) UNION Y) SUBSET aff_ge X Y `,
REWRITE_TAC[Collect_geom.conv; GSYM aff_ge_def] THEN 
MP_TAC (ISPECL [`X:real^N -> bool`;`X:real^N -> bool`] (GEN_ALL AFF_GE_MONO_TRANS))
THEN SIMP_TAC[SUBSET_REFL; DIFF_EQ_EMPTY; UNION_COMM]);;



let CONV_MONO = prove(` S SUBSET SS ==> conv S SUBSET conv SS `,
REWRITE_TAC[Geomdetail.conv; GSYM aff_ge_def] THEN 
STRIP_TAC THEN MATCH_MP_TAC AFF_GE_MONO_RIGHT THEN 
ASM_REWRITE_TAC[DISJOINT_EMPTY]);;




let CONV3_SUBSET_AFF_GE_3S = prove(` conv {a,b,c:real^N} SUBSET aff_ge {a,b,c} S `,
MP_TAC (ISPECL [` {a,b,c:real^N}`;` S:real^N -> bool`] (GEN_ALL CONV_UNION_SUB_AFF_GE)) THEN 
MATCH_MP_TAC (SET_RULE` a SUBSET b ==> b SUBSET c ==> a SUBSET c `) THEN 
MATCH_MP_TAC CONV_MONO THEN SET_TAC[]);;


let P_SET3_IMP_SET2 = 
MESON[INSERT_INSERT]` (! (a:A) b c. P ({a,b,c}) ) ==> (! (a:A) b. (P {a,b})) `;;


let CONV2_SUBSET_AFF_GE_2S = 
MATCH_MP P_SET3_IMP_SET2 (GEN_ALL CONV3_SUBSET_AFF_GE_3S);;








(* ------------------------------------------------------------------------- *)
(* Search (automatically updates)                                            *)
(* ------------------------------------------------------------------------- *)

let full t = mk_comb(mk_var("<full term>",W mk_fun_ty (type_of t)),t);;

(* very rough measure of the size of a printed term *)
let rec term_length tm = match tm with
  | Abs(s,x) -> 1 + term_length x
  | Comb(s,x) -> if ((s = `NUMERAL`) or (s = `DECIMAL`)) then 2
    else ( (term_length s) + term_length x)
  | _ -> 1;;

let sortlength_thml thml =  
  let ltml = map 
   (function (s,t) as r -> (term_length (concl t),r)) thml in
  let stml = sort (fun (a,_) (b,_) -> (a < b)) ltml in
    map snd stml;;

let search_thml  =
  let rec immediatesublist l1 l2 =
    match (l1,l2) with
      [],_ -> true
    | _,[] -> false
    | (h1::t1,h2::t2) -> h1 = h2 & immediatesublist t1 t2 in
  let rec sublist l1 l2 =
    match (l1,l2) with
      [],_ -> true
    | _,[] -> false
    | (h1::t1,h2::t2) -> immediatesublist l1 l2 or sublist l1 t2 in
  let exists_subterm_satisfying p (n,th) = can (find_term p) (concl th) in
  let exists_fullterm_satisfying p (n,th) =  p (concl th) in
  let name_contains s (n,th) = sublist (explode s) (explode n) in
  let rec filterpred tm =
    match tm with
      Comb(Var("<omit this pattern>",_),t) -> not o filterpred t
    | Comb(Var("<match theorem name>",_),Var(pat,_)) -> name_contains pat
    | Comb(Var("<match aconv>",_),pat) -> exists_subterm_satisfying (aconv pat)
    | Comb(Var("<full term>",_),pat) -> exists_fullterm_satisfying (can (term_match [] pat)) 
    | pat -> exists_subterm_satisfying (can (term_match [] pat)) in
  fun pats thml -> update_database ();
    if (pats = []) then failwith "keywords: omit, name, exactly, full" else
        (itlist (filter o filterpred) pats thml);;



let (PAT_TAC:term list -> (string * thm) -> tactic)  = 
  fun pat sth -> if (search_thml pat [sth] = []) then NO_TAC else ALL_TAC;;

let (FIRST_PAT_ASSUM: term list -> thm_tactic -> tactic) =
  fun pat ttac gl -> tryfind (fun sth -> (PAT_TAC pat sth THEN ttac (snd sth)) gl) (goal_asms gl);;

let (FIRST_PAT_X_ASSUM:term list -> thm_tactic -> tactic) =
  fun pat ttac ->
    FIRST_PAT_ASSUM pat (fun th -> UNDISCH_THEN (concl th) ttac);;
(* ================================================================== *)






let X_IN_AFF_GE = prove_by_refinement
(` x IN aff_ge S {x:real^N} `,
[REWRITE_TAC[aff_ge_def; IN; affsign; sgn_ge; lin_combo];
EXISTS_TAC` (\a. if a = (x:real^N) then &1 else &0 ) `;
CONJ_TAC;
REWRITE_TAC[];
SUBGOAL_THEN` vsum (S UNION {x:real^N}) (\v. (if v = x then &1 else &0) % v)
= vsum ({x}) (\v. (if v = x then &1 else &0) % v) ` ASSUME_TAC;
MATCH_MP_TAC VSUM_UNION_LZERO;
REWRITE_TAC[FINITE_SINGLETON; IN_INSERT; NOT_IN_EMPTY];
SIMP_TAC[VECTOR_MUL_LZERO];
ASM_REWRITE_TAC[VSUM_SING; VECTOR_MUL_LID];

SIMP_TAC[SET_RULE` {a} x <=> x = a `; REAL_ARITH` &0 <= &1 `];
SUBGOAL_THEN` sum (S UNION {x:real^N}) (\a. if a = x then &1 else &0)
= sum ({x}) (\a. if a = x then &1 else &0) ` ASSUME_TAC;
MATCH_MP_TAC SUM_UNION_LZERO;
SIMP_TAC[FINITE_SING; IN_INSERT; DE_MORGAN_THM];
ASM_REWRITE_TAC[SUM_SING]]);;







let LOFA_IMP_BIJ_FF_V = MESON[Wrgcvdr_cizmrrh.BIJ_BETWEEN_FF_AND_V]
` local_fan (V,E,FF) ==> BIJ (\x. FST x) FF V `;;

let LOFA_IMP_CARD_FF_V_EQ = prove(
` local_fan (V,E,FF) ==> CARD FF = CARD V `,
NHANH Wrgcvdr_cizmrrh.LOCAL_FAN_FINITE_FF THEN 
NHANH LOFA_IMP_BIJ_FF_V THEN PHA THEN 
NHANH BIJ_IMP_CARD_EQ THEN SIMP_TAC[]);;



let FIRST_IN_AFF = prove(` a IN aff ( a INSERT S ) `,
REWRITE_TAC[aff; Trigonometry2.IN_P_HULL_INSERT]);;






let HALF_CIRCULAR_IN_PLANE = prove_by_refinement
(` (convex_local_fan (V,E,FF) /\ lunar (v,w) V E ) /\ 
n < CARD V /\ w = ITER n (rho_node1 FF) v 
 ==> { ITER l (rho_node1 FF) v | l <= n } SUBSET aff {vec 0, v, rho_node1 FF v}`,
[NHANH LUNAR_IMP_INTERIOR_ANGLE1_EQ_PI;
REWRITE_TAC[SUBSET; IN_ELIM_THM; lunar];
REPEAT STRIP_TAC;
ASM_REWRITE_TAC[];
UNDISCH_TAC ` l <= n:num`;
SPEC_TAC (`l:num`,`l:num`);
INDUCT_TAC;
REWRITE_TAC[ITER];
DISCH_TAC;
MP_TAC (ISPECL [`affine: (real^3 -> bool) -> bool `;
` {vec 0, v, rho_node1 FF v } `] HULL_SUBSET);
REWRITE_TAC[aff; SUBSET];
DISCH_THEN MATCH_MP_TAC;
REWRITE_TAC[IN_INSERT];
NHANH (ARITH_RULE` SUC a <= b ==> a <= b `);
FIRST_X_ASSUM NHANH;
STRIP_TAC;
FIRST_X_ASSUM (MP_TAC o (SPEC` ITER l' (rho_node1 FF) v `));
ASM_CASES_TAC` l' = 0 `;
ASM_REWRITE_TAC[ITER];
REPEAT STRIP_TAC;
MP_TAC (ISPECL [`affine: (real^3 -> bool) -> bool `;
` {vec 0, v, rho_node1 FF v } `] HULL_SUBSET);
REWRITE_TAC[aff; SUBSET];
DISCH_THEN MATCH_MP_TAC;
REWRITE_TAC[IN_INSERT];
ANTS_TAC;
ASSUME_TAC2 CVX_LO_IMP_LO;
ASSUME_TAC2 LOFA_IMP_CARD_FF_V_EQ;
MP_TAC2 (SPEC `n:num` (GEN `l:num` LOFA_IMP_DIS_ELMS));
UNDISCH_TAC` !x. x IN {v, w} ==> x IN (V:real^3 -> bool) `;
SIMP_TAC[IN_INSERT];
DISCH_THEN (MP_TAC o (SPEC`l': num`));
ANTS_TAC;
UNDISCH_TAC` SUC l' <= n `;
ARITH_TAC;


MP_TAC2 (SPEC `l':num` (GEN `l:num` LOFA_IMP_DIS_ELMS));
CONJ_TAC;
UNDISCH_TAC` !x. x IN {v, w} ==> x IN (V:real^3 -> bool) `;
SIMP_TAC[IN_INSERT];
UNDISCH_TAC` n < CARD (V:real^3 -> bool) `;
UNDISCH_TAC` l' <= n:num `;
ARITH_TAC;
DISCH_THEN (MP_TAC o (SPEC`0`));
ANTS_TAC;
UNDISCH_TAC` ~( l' = 0)`;
ARITH_TAC;
ASM_SIMP_TAC[IN_DIFF; IN_INSERT; NOT_IN_EMPTY; DE_MORGAN_THM; ITER];
REPEAT STRIP_TAC;

MP_TAC2 LOCAL_FAN_ORBIT_MAP_VITER;
DISCH_THEN MATCH_MP_TAC;
UNDISCH_TAC ` !x. x IN {v:real^3, w} ==> x IN V`;
SIMP_TAC[IN_INSERT];

SUBGOAL_THEN` aff {ITER l' (rho_node1 FF) v, v, w} SUBSET aff {vec 0, v, rho_node1 FF v}` MP_TAC;
MATCH_MP_TAC S_SUBSET_IMP_AFF_S_TOO;
ASM_REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET];
CONJ_TAC;
MESON_TAC[FIRST_IN_AFF; INSERT_COMM];

SUBGOAL_THEN` aff {vec 0, v:real^3} SUBSET aff {vec 0, v, rho_node1 FF v }` MP_TAC;
REWRITE_TAC[aff];
MATCH_MP_TAC HULL_MONO;
SET_TAC[];
REWRITE_TAC[SUBSET];
DISCH_THEN MATCH_MP_TAC;
UNDISCH_TAC ` vec 0 IN conv0 {v,w:real^3}`;
NHANH IN_CONV0_IMP_AFF_EQ;
SIMP_TAC[INSERT_COMM];
STRIP_TAC;
FIRST_X_ASSUM (SUBST1_TAC o SYM);
ASM_REWRITE_TAC[Planarity.POINT_IN_LINE1];
SIMP_TAC[SUBSET; ITER]]);;






let LOFA_IMP_AZIM_RHO_NODE_ST = prove_by_refinement
(` local_fan (V,E,FF) /\ v IN V /\  v cross rho_node1 FF v = e ==>
~(azim ( vec 0 ) e v (rho_node1 FF v ) = &0 \/ azim (vec 0) e v (rho_node1 FF v ) = pi ) `,
[NHANH (MESON[Trigonometry.YIXJNJQ1; SIN_PI]` a = &0 \/ a = pi ==> sin a = &0 `);
MP_TAC (SPECL [`e:real^3`;` v:real^3`;` rho_node1 FF v `] Trigonometry2.JBDNJJB);
REWRITE_TAC[re_eqvl];
STRIP_TAC;
STRIP_TAC;
ASM_SIMP_TAC[REAL_ENTIRE; DE_MORGAN_THM];
DISJ2_TAC;
CONJ_TAC;
UNDISCH_TAC` &0 < t `;
REAL_ARITH_TAC;
ONCE_REWRITE_TAC[CROSS_TRIPLE];
ASM_REWRITE_TAC[DOT_EQ_0];
FIRST_X_ASSUM (SUBST1_TAC o SYM);
REWRITE_TAC[CROSS_EQ_0];
UNDISCH_TAC `v:real^3 IN V `;
ASSUME_TAC2 LOCAL_FAN_CHARACTER_OF_RHO_NODE;
FIRST_X_ASSUM NHANH;
SIMP_TAC[]]);;








let LOFA_IMP_DIS_ELMS2 = 
MESON[LOFA_IMP_DIS_ELMS]` local_fan (V,E,FF) /\ v IN V 
       ==> (!i l. i < l /\ l < CARD FF
                ==> ~(ITER l (rho_node1 FF) v = ITER i (rho_node1 FF) v))`;;







let RHO_NODE1_MONO_WITH_AZIM = prove_by_refinement
(` local_fan (V,E,FF) /\
     v IN V /\
     {ITER n (rho_node1 FF) v | n <= l} = U /\
     plane P /\
     vec 0 IN P /\
     U SUBSET P /\
     v cross rho_node1 FF v = e
==> (! n m. n < m /\ m < CARD V /\ m <= l
            ==> azim (vec 0) e v ( ITER n (rho_node1 FF) v) <
                azim (vec 0) e v ( ITER m (rho_node1 FF) v) )`,
[STRIP_TAC;
GEN_TAC;
INDUCT_TAC;
REWRITE_TAC[LT];
ASM_CASES_TAC ` m = n:num`;
FIRST_X_ASSUM (SUBST_ALL_TAC);
ASSUME_TAC2 SEQUENCE_OF_RHO_NODE_IS_SUC;
FIRST_X_ASSUM (MP_TAC o (SPECL [` ITER n (rho_node1 FF) v`;`n:num`]));
PHA;
NHANH (ARITH_RULE` SUC a <= b ==> a < b `);
IMP_TAC;
DISCH_THEN NHANH;
STRIP_TAC;
FIRST_X_ASSUM (MP_TAC o (SPEC`v:real^3 `));
REWRITE_TAC[ITER];
ASM_CASES_TAC` n = 0 `;
ASM_REWRITE_TAC[ITER; AZIM_REFL];
ASSUME_TAC2 LOFA_IMP_AZIM_RHO_NODE_ST;
MP_TAC (SPECL [`vec 0:real^3 `;` e:real^3 `;` v:real^3 `;` rho_node1 FF v `] AZIM_RANGE);
DOWN;
REAL_ARITH_TAC;





(* 7777777 *)
DISCH_THEN MATCH_MP_TAC;
ASSUME_TAC2 LOFA_IMP_DIS_ELMS2;
EXPAND_TAC "U";
REWRITE_TAC[IN_ELIM_THM];
CONJ_TAC;
EXISTS_TAC ` 0`;
REWRITE_TAC[ITER; LE_0];

FIRST_ASSUM (MP_TAC o (SPECL [`0`;` n:num`]));
FIRST_ASSUM (MP_TAC o (SPECL [`0`;` n + 1`]));
PHA;
ASM_REWRITE_TAC[GSYM ADD1];

IMP_TAC;
ANTS_TAC;
REWRITE_TAC[LT_0];
ASSUME_TAC2 LOFA_IMP_CARD_FF_V_EQ;
ASM_REWRITE_TAC[];
SIMP_TAC[ITER];

ASSUME_TAC2 LOFA_IMP_CARD_FF_V_EQ;
STRIP_TAC;
ANTS_TAC;
UNDISCH_TAC` ~( n = 0 ) `;
UNDISCH_TAC` SUC n < CARD (V:real^3 -> bool) `;
ASM_REWRITE_TAC[];
ARITH_TAC;
SIMP_TAC[];

STRIP_TAC;
UNDISCH_TAC` n < m /\ m < CARD (V:real^3 -> bool) /\ m <= l
      ==> azim (vec 0) e v (ITER n (rho_node1 FF) v) <
          azim (vec 0) e v (ITER m (rho_node1 FF) v) ` ;
ANTS_TAC;
REPLICATE_TAC 4 DOWN THEN PHA;
ARITH_TAC;
ASSUME_TAC2 SEQUENCE_OF_RHO_NODE_IS_SUC;

FIRST_X_ASSUM (MP_TAC o (SPECL [` ITER m (rho_node1 FF) v`;` m:num`]));
REWRITE_TAC[];
ANTS_TAC;
DOWN;
ARITH_TAC;

DISCH_THEN (MP_TAC o (SPEC ` v:real^3 `));
ANTS_TAC;
EXPAND_TAC "U";
REWRITE_TAC[IN_ELIM_THM];
EXISTS_TAC `0`;
REWRITE_TAC[ITER; LE_0];
ASM_CASES_TAC ` m = 0 `;
FIRST_X_ASSUM SUBST_ALL_TAC;
REWRITE_TAC[ITER; AZIM_REFL];
MP_TAC (SPECL [` vec 0:real^3`;` e:real^3 `;` v:real^3 `;` rho_node1 FF v `] AZIM_RANGE);
REAL_ARITH_TAC;

ASSUME_TAC2 LOFA_IMP_DIS_ELMS2;
ANTS_TAC;
FIRST_X_ASSUM (MP_TAC o (SPECL [`0` ;` m:num `]));
PHA;
ANTS_TAC;
REPLICATE_TAC 3 DOWN THEN PHA;
ASSUME_TAC2 LOFA_IMP_CARD_FF_V_EQ;
ASM_REWRITE_TAC[];
ARITH_TAC;
SIMP_TAC[ITER];
ANTS_TAC;

FIRST_X_ASSUM (MP_TAC o (SPECL [`0` ;`SUC m`]));
PHA;
ANTS_TAC;

ASSUME_TAC2 LOFA_IMP_CARD_FF_V_EQ;
ASM_REWRITE_TAC[];
ARITH_TAC;
SIMP_TAC[ITER];

REWRITE_TAC[ITER];
REAL_ARITH_TAC]);;








let DISJOINT_IMP_Z_IN_AFF_GT =
prove(` DISJOINT {x,y} {z} ==> z IN aff_gt {x,y} {z} `,
SIMP_TAC[AFF_GT_2_1; IN_ELIM_THM] THEN DISCH_TAC THEN EXISTS_TAC` &0` THEN 
EXISTS_TAC ` &0 ` THEN EXISTS_TAC ` &1 ` THEN 
REWRITE_TAC[VECTOR_MUL_LZERO; VECTOR_ADD_LID; VECTOR_MUL_LID] THEN 
REAL_ARITH_TAC);;



let LOFA_IMP_DIS_ELMS23 = MESON[LOFA_IMP_CARD_FF_V_EQ; LOFA_IMP_DIS_ELMS2]
` local_fan (V,E,FF) /\ v IN V
     ==> (!i l.
              i < l /\ l < CARD V
              ==> ~(ITER l (rho_node1 FF) v = ITER i (rho_node1 FF) v))`;;





let NOT_COLL_IMP_COPL = prove(
` ~ collinear {vec 0,v,w} ==> ~ coplanar {vec 0, v,w, v cross w } `,
REWRITE_TAC[GSYM CROSS_DOT_COPLANAR; GSYM CROSS_EQ_0; DOT_EQ_0]);;



let COLL_IFF_COLL_CROSS = prove_by_refinement
(` collinear {vec 0, v, w} <=>  collinear {vec 0, v, v cross w} `,
[EQ_TAC; SIMP_TAC[GSYM CROSS_EQ_0; CROSS_0];
ASM_CASES_TAC` collinear {vec 0,v,w:real^3}`;
ASM_REWRITE_TAC[]; MP_TAC2 NOT_COLL_IMP_COPL;
MESON_TAC[NOT_COPLANAR_NOT_COLLINEAR; INSERT_COMM]]);;






let LOCAL_FAN_CHARACTER_OF_RHO_NODE2 =
prove(` local_fan (V,E,FF) /\ v IN V ==>  ~collinear {vec 0, v, rho_node1 FF v} `,
NHANH LOCAL_FAN_CHARACTER_OF_RHO_NODE THEN SIMP_TAC[]);;




let LUNAR_IMP_HALF_CIRCLE_SUBSET_AFF_GT = prove_by_refinement
(`convex_local_fan (V,E,FF) /\ lunar (v,w) V E
 ==> (?i. i < CARD V /\
          w = ITER i (rho_node1 FF) v /\
          {ITER l (rho_node1 FF) v | 0 < l /\ l < i} SUBSET
          aff_gt {vec 0, v} {rho_node1 FF v})`,
[NHANH LUNAR_IMP_INTERIOR_ANGLE1_EQ_PI;

REWRITE_TAC[convex_local_fan; lunar; INSERT_SUBSET];
NHANH LOCAL_FAN_ORBIT_MAP_V;
NHANH LOOP_SET_DETER_FIRTS_ELMS;
STRIP_TAC;
SUBGOAL_THEN` orbit_map (rho_node1 FF) v = (V:real^3 -> bool) `ASSUME_TAC;
ASM_SIMP_TAC[];
SUBGOAL_THEN` {ITER n (rho_node1 FF) v | n < CARD V} = V `ASSUME_TAC;
ASM_SIMP_TAC[];
UNDISCH_TAC` w:real^3 IN V `;
EXPAND_TAC "V";
REWRITE_TAC[IN_ELIM_THM];
STRIP_TAC;
EXISTS_TAC` n:num`;
ASM_REWRITE_TAC[SUBSET; IN_ELIM_THM];
REPEAT STRIP_TAC;

ASM_REWRITE_TAC[];
MP_TAC2 HALF_CIRCULAR_IN_PLANE;
SWITCH_TAC ` w = ITER n (rho_node1 FF) v `;
ASM_SIMP_TAC[convex_local_fan; lunar];
ASM_REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET];
EXPAND_TAC "w";
EXPAND_TAC "V";
REWRITE_TAC[IN_ELIM_THM];
EXISTS_TAC` n:num`;
ASM_REWRITE_TAC[];
ABBREV_TAC` U = {ITER l (rho_node1 FF) v | l <= n} `;
ABBREV_TAC` P = aff {vec 0, v, rho_node1 FF v} `;
ABBREV_TAC` e = v cross rho_node1 FF v `;
STRIP_TAC;
MP_TAC2 (SPEC `n:num` (GEN `l:num` FIRST_AZIM_CYCLE_EQ_RHO_NODE));
EXPAND_TAC "P";
REWRITE_TAC[plane; FIRST_IN_AFF; aff];
EXISTS_TAC` vec 0:real^3 `;
EXISTS_TAC` v:real^3`;
EXISTS_TAC` rho_node1 FF v `;
ASSUME_TAC2 LOCAL_FAN_CHARACTER_OF_RHO_NODE;
UNDISCH_TAC` v:real^3 IN V`;
FIRST_X_ASSUM NHANH;
SIMP_TAC[];
MP_TAC2 (SPEC `n:num ` (GEN ` l:num` RHO_NODE1_MONO_WITH_AZIM));
EXPAND_TAC "P";
REWRITE_TAC[plane; FIRST_IN_AFF];
EXISTS_TAC` vec 0:real^3 `;
EXISTS_TAC` v:real^3 `;
EXISTS_TAC ` rho_node1 FF v `;
REWRITE_TAC[aff];
ASSUME_TAC2 LOCAL_FAN_CHARACTER_OF_RHO_NODE;
ASM_SIMP_TAC[];
STRIP_TAC;
FIRST_ASSUM (ASSUME_TAC o (SPECL [`0`;` l:num`]));
DOWN;
ANTS_TAC;
ASM_REWRITE_TAC[];
UNDISCH_TAC` l < n:num`;
UNDISCH_TAC` n < CARD (V:real^3 -> bool) `;
ARITH_TAC;


FIRST_ASSUM (ASSUME_TAC o (SPECL [` l:num`;`n:num`]));
DOWN;
ANTS_TAC;
ASM_REWRITE_TAC[LE_REFL];


REWRITE_TAC[ITER; AZIM_REFL];
SUBGOAL_THEN` azim (vec 0) e v (ITER n (rho_node1 FF) v) = pi <=>
v IN aff_lt {vec 0, e:real^3} {ITER n (rho_node1 FF) v}` ASSUME_TAC;
MATCH_MP_TAC AZIM_EQ_PI;

UNDISCH_TAC` vec 0 IN conv0 {v,w:real^3}`;
REWRITE_TAC[Geomdetail.CONV0_SET2; IN_ELIM_THM];
NHANH (REAL_ARITH` &0 < a ==> ~( a = &0 ) `);
REWRITE_TAC[VECTOR_ARITH` vec 0 = a + b <=> b = -- a `];
STRIP_TAC;
SWITCH_TAC` w = ITER n (rho_node1 FF) v`;
ASM_REWRITE_TAC[];
SUBGOAL_THEN` collinear {vec 0, &1 % e, b % w} <=> collinear {vec 0, e, w:real^3}` ASSUME_TAC;
MATCH_MP_TAC COLLINEAR_SCALE_ALL;
ASM_REWRITE_TAC[];
REAL_ARITH_TAC;
FIRST_X_ASSUM (SUBST1_TAC o SYM);
ASM_REWRITE_TAC[GSYM VECTOR_MUL_LNEG];

SUBGOAL_THEN` collinear {vec 0, &1 % e, ( -- a ) % v} <=> collinear {vec 0, e, v:real^3}` SUBST1_TAC;
MATCH_MP_TAC COLLINEAR_SCALE_ALL;
UNDISCH_TAC` ~( a = &0 )`;
REAL_ARITH_TAC;
REWRITE_TAC[];
EXPAND_TAC "e";
ONCE_REWRITE_TAC[SET_RULE` {a,b} = {b,a} `];
ONCE_REWRITE_TAC[GSYM COLL_IFF_COLL_CROSS];
ASSUME_TAC2 LOCAL_FAN_CHARACTER_OF_RHO_NODE;
UNDISCH_TAC` v:real^3 IN V`;
FIRST_X_ASSUM NHANH;
SIMP_TAC[];


SUBGOAL_THEN ` azim (vec 0) e v (ITER n (rho_node1 FF) v) = pi ` ASSUME_TAC;
FIRST_X_ASSUM SUBST1_TAC;
SUBGOAL_THEN` ~( aff {vec 0, e:real^3} INTER conv0 {w,v} = {} ) ` MP_TAC;
REWRITE_TAC[EMPTY_NOT_EXISTS_IN];
EXISTS_TAC` vec 0:real^3 `;
SWITCH_TAC` w = ITER n (rho_node1 FF) v `;
ASM_SIMP_TAC[IN_INTER; Planarity.POINT_IN_LINE; INSERT_COMM];

ASM_REWRITE_TAC[];
MATCH_MP_TAC Ldurdpn.AFF_CONV0_IN_AFF_LT;
SWITCH_TAC` w = ITER n (rho_node1 FF ) v `;





UNDISCH_TAC` vec 0 IN conv0 {v,w:real^3}`;
REWRITE_TAC[Geomdetail.CONV0_SET2; IN_ELIM_THM];
NHANH (REAL_ARITH` &0 < a ==> ~( a = &0 ) `);
REWRITE_TAC[VECTOR_ARITH` vec 0 = a + b <=> b = -- a `];
STRIP_TAC;
ASM_REWRITE_TAC[];
SUBGOAL_THEN` collinear {vec 0, &1 % e, b % w} <=> collinear {vec 0, e, w:real^3}` ASSUME_TAC;
MATCH_MP_TAC COLLINEAR_SCALE_ALL;
ASM_REWRITE_TAC[];
REAL_ARITH_TAC;
FIRST_X_ASSUM (SUBST1_TAC o SYM);
ASM_REWRITE_TAC[GSYM VECTOR_MUL_LNEG];

SUBGOAL_THEN` collinear {vec 0, &1 % e, ( -- a ) % v} <=> collinear {vec 0, e, v:real^3}` SUBST1_TAC;
MATCH_MP_TAC COLLINEAR_SCALE_ALL;
UNDISCH_TAC` ~( a = &0 )`;
REAL_ARITH_TAC;
REWRITE_TAC[];
EXPAND_TAC "e";
ONCE_REWRITE_TAC[SET_RULE` {a,b} = {b,a} `];
ONCE_REWRITE_TAC[GSYM COLL_IFF_COLL_CROSS];
ASSUME_TAC2 LOCAL_FAN_CHARACTER_OF_RHO_NODE;
UNDISCH_TAC` v:real^3 IN V`;
FIRST_X_ASSUM NHANH;
SIMP_TAC[];

ASM_REWRITE_TAC[];
SWITCH_TAC` w = ITER n (rho_node1 FF) v `;
ASM_REWRITE_TAC[];
REPEAT STRIP_TAC;
MP_TAC2 (SPEC `  azim (vec 0) e v (ITER l (rho_node1 FF) v) ` SIN_POS_PI);
MP_TAC (SPECL [`e:real^3`;`v:real^3`;` ITER l (rho_node1 FF) v `] Trigonometry2.JBDNJJB);
REWRITE_TAC[re_eqvl];
STRIP_TAC;


SUBGOAL_THEN` ITER l (rho_node1 FF) v IN aff {vec 0, v, rho_node1 FF v}` MP_TAC;
ASM_REWRITE_TAC[];
UNDISCH_TAC` U SUBSET (P:real^3 -> bool) `;
REWRITE_TAC[SUBSET];
DISCH_THEN MATCH_MP_TAC;
EXPAND_TAC "U";
REWRITE_TAC[IN_ELIM_THM];
EXISTS_TAC `l:num`;
ASM_REWRITE_TAC[];
UNDISCH_TAC` l < n:num`;
ARITH_TAC;


REWRITE_TAC[aff; AFFINE_HULL_3; IN_ELIM_THM];
STRIP_TAC;
ASM_REWRITE_TAC[];

ONCE_REWRITE_TAC[CROSS_TRIPLE];
REWRITE_TAC[CROSS_RADD; CROSS_RMUL; CROSS_0; CROSS_REFL; VECTOR_MUL_RZERO; VECTOR_ADD_LID];
ASM_REWRITE_TAC[DOT_LMUL];


SUBGOAL_THEN` &0 < e dot (e:real^3)` ASSUME_TAC;
REWRITE_TAC[DOT_POS_LT];
EXPAND_TAC "e";
REWRITE_TAC[CROSS_EQ_0];
MATCH_MP_TAC LOCAL_FAN_CHARACTER_OF_RHO_NODE2;
ASM_REWRITE_TAC[];

ASM_SIMP_TAC[REAL_LT_MUL_EQ];
MP_TAC2 LOCAL_FAN_CHARACTER_OF_RHO_NODE2;
NHANH Fan.th3a;
SIMP_TAC[AFF_GT_2_1; IN_ELIM_THM];
REPEAT STRIP_TAC;
EXISTS_TAC` u:real `;
EXISTS_TAC` v':real `;
EXISTS_TAC` w':real`;
ASM_REWRITE_TAC[];
VECTOR_ARITH_TAC]);;







let LOOP_SET_ITER_CARD_ID = prove_by_refinement
(` (!v:A. v IN V ==> orbit_map f v = V) ==>
(! v. v IN V ==> ITER (CARD V) f v = v )`,
[NHANH LOOP_SET_DETER_FIRTS_ELMS;
NHANH (ONCE_REWRITE_RULE[EQ_SYM_EQ] SELF_CYCLIC_IMP_BIJ);

STRIP_TAC;
FIRST_ASSUM NHANH;
GEN_TAC THEN STRIP_TAC;
SUBGOAL_THEN` ITER (CARD (V:A -> bool)) (f:A -> A) (v:A) IN V` MP_TAC;
DOWN;
REMOVE_TAC;
DOWN;
DOWN;
REMOVE_TAC;
FIRST_X_ASSUM (NHANH_PAT`\x. x ==> y`);
STRIP_TAC;
EXPAND_TAC "V";
REWRITE_TAC[lemma_in_orbit_iter];
EXPAND_TAC "V";
REWRITE_TAC[IN_ELIM_THM];
STRIP_TAC;
DOWN;
ASM_CASES_TAC` n = 0 `;
ASM_REWRITE_TAC[ITER];

ASM_REWRITE_TAC[];
ABBREV_TAC` cc = CARD (V:A -> bool) `;
ASSUME_TAC2 (ARITH_RULE` n < cc ==> cc = (cc - n ) + (n:num) `);
ABBREV_TAC ` m = cc - (n:num)`;
ASM_REWRITE_TAC[GSYM ITER_ADD];

SUBGOAL_THEN` 0 < m ` MP_TAC;
UNDISCH_TAC` n < cc:num`;
EXPAND_TAC "m";
ARITH_TAC;

PHA;
NHANH Lvducxu.ITER_CYCLIC_ORBIT;

SUBGOAL_THEN` ITER n f (v:A) IN V ` MP_TAC;
EXPAND_TAC "V";
REWRITE_TAC[IN_ELIM_THM];
EXISTS_TAC `n:num`;
ASM_REWRITE_TAC[];

ASM_SIMP_TAC[];

REPEAT STRIP_TAC;
SUBGOAL_THEN` CARD (V:A -> bool) <= m ` MP_TAC;
FIRST_X_ASSUM SUBST1_TAC;
REWRITE_TAC[Wrgcvdr_cizmrrh.CARD_LE_K_OF_SET_K_FIRST_ELMS];
ASM_REWRITE_TAC[];

UNDISCH_TAC` ~ ( n = 0 ) `;
ARITH_TAC]);;





let LOFA_IMP_ITER_RHO_NODE_ID = prove(
`local_fan (V,E,FF) ==> (! v. v IN V ==> ITER (CARD V) (rho_node1 FF) v = v )`,
NHANH LOCAL_FAN_ORBIT_MAP_V THEN 
SIMP_TAC[LOOP_SET_ITER_CARD_ID]);;




let CONV_SUBSET_AFF_GE = prove(` conv SS SUBSET aff_ge S SS`,
REWRITE_TAC[Collect_geom.conv; GSYM aff_ge_def] THEN 
MATCH_MP_TAC AFF_GE_MONO_LEFT THEN SET_TAC[]);;

let CONV0_SUBSET_AFF_GT = prove(` conv0 SS SUBSET aff_gt S SS`,
REWRITE_TAC[conv0; GSYM aff_gt_def] THEN 
MATCH_MP_TAC AFF_GT_MONO_LEFT THEN SET_TAC[]);;





let collinear_fan22 = 
MESON[Fan.collinear_fan]`collinear {x, v,u} <=> u IN aff {x, v} \/ (x = v)`;;

let IN_CONV0_IMP_COLL_IFF = prove(
 ` x IN conv0 {a,b} ==> (collinear {a,x,v} <=> collinear {a,b,v} )`,
NHANH IN_CONV0_IMP_AFF_EQ THEN NHANH IN_CONV0_EQ_EQ THEN 
SIMP_TAC[collinear_fan22; EQ_SYM_EQ]);;

let IN_CONV0_IMP_COLL_ENDS_AFF = prove(
` x IN conv0 {a:real^N,b} ==> (collinear {a,x,v} <=> collinear {b,x,v})`,
NHANH IN_CONV0_IMP_COLL_IFF THEN ONCE_REWRITE_TAC[INSERT_COMM] THEN 
NHANH IN_CONV0_IMP_COLL_IFF THEN SIMP_TAC[INSERT_COMM]);;








let IN_CONV0_IMP_AZIM_PI = prove_by_refinement
(` ~ collinear {x,a,e} /\ x IN conv0 {a,b}
==> azim x e a b = pi `,
[STRIP_TAC;
SUBGOAL_THEN` azim x e a b = pi <=> b IN aff_lt {x,e} {a}` MP_TAC;
MATCH_MP_TAC AZIM_EQ_PI_ALT;
DOWN;
NHANH (SPEC `e:real^N` (GEN `v:real^N` IN_CONV0_IMP_COLL_ENDS_AFF));
DOWN;
ASM_SIMP_TAC[INSERT_COMM];
SIMP_TAC[];
DISCH_TAC;
DOWN_TAC;
SPEC_TAC (`x:real^3`,`x:real^3`);
GEOM_ORIGIN_TAC `x:real^3`;
REPEAT STRIP_TAC;
SUBGOAL_THEN` ~(aff {vec 0, e} INTER conv0 {a,b:real^3} = {})` MP_TAC;
REWRITE_TAC[INTER_EQ_EM_EXPAND];
EXISTS_TAC` vec 0:real^3 `;
ASM_REWRITE_TAC[Planarity.POINT_IN_LINE];

MATCH_MP_TAC Ldurdpn.AFF_CONV0_IN_AFF_LT;
DOWN_TAC;
SIMP_TAC[INSERT_COMM]]);;









let AFF_GT_MONO = prove(`!S. (S:real^N -> bool) SUBSET Y ==> aff_gt X Y SUBSET aff_gt (X UNION S) ( Y DIFF S)`, GEN_TAC THEN
DISCH_TAC THEN REWRITE_TAC[SUBSET; IN; aff_gt_def; affsign] THEN 
ASM_SIMP_TAC[SET_RULE`S SUBSET Y ==> (X UNION S) UNION Y DIFF S = X UNION Y`] THEN 
SET_TAC[]);;






(* aff_gt X Y SUBSET aff (X UNION Y) *)
(* ================================= *)
let AFF_GT_SUB_AFF_UNION = 
REWRITE_RULE[SUBSET_REFL; DIFF_EQ_EMPTY; AFF_GT_EQ_AFFINE_HULL; GSYM aff]
 (SPEC `Y:real^N -> bool` AFF_GT_MONO);;






let SIN_AZIM_NEG_PI_LT = prove_by_refinement
(` sin ( azim x y u v ) < &0 <=> pi < azim x y u v `,
[MP_TAC (SPECL [`x:real^3`;`y:real^3`;`u:real^3`;` v:real^3 `] AZIM_RANGE);
STRIP_TAC;
EQ_TAC;
ASM_CASES_TAC` azim x y u v <= pi `;
SUBGOAL_THEN` &0 <= sin ( azim x y u v )` MP_TAC;
ASM_SIMP_TAC[Trigonometry.WIBGJRR];
REAL_ARITH_TAC;
DOWN;
REAL_ARITH_TAC;
ASM_SIMP_TAC [ PI_TO_TWO_PI_NEG_SIN]]);;








let NEXT_OPOSITE_POINT_IS_NOT_IN_AFF_GT = prove_by_refinement
(`( convex_local_fan (V,E,FF) /\ lunar (v,w) V E)
==> ? i.  w = ITER i (rho_node1 FF) v /\
 i + 1 < CARD V /\
 ~(ITER ( i + 1 ) (rho_node1 FF) v IN aff_gt {vec 0, v} {rho_node1 FF v})`,
[NHANH LUNAR_IMP_HALF_CIRCLE_SUBSET_AFF_GT;
STRIP_TAC;
SUBGOAL_THEN` ~ collinear {vec 0, v, rho_node1 FF v}` ASSUME_TAC;
MATCH_MP_TAC LOCAL_FAN_CHARACTER_OF_RHO_NODE2;
UNDISCH_TAC `lunar (v:real^3,w) V E `;
ASM_SIMP_TAC[lunar; INSERT_SUBSET; CVX_LO_IMP_LO];

EXISTS_TAC `i:num`;
ASSUME_TAC2 (SPEC `i:num` (GEN `n:num` HALF_CIRCULAR_IN_PLANE));
ASM_REWRITE_TAC[];
ASM_CASES_TAC` i + 1 = CARD (V:real^3 -> bool) `;
SUBGOAL_THEN` rho_node1 FF w = v ` ASSUME_TAC;
ASM_REWRITE_TAC[GSYM ITER; ADD1];
ASSUME_TAC2 CVX_LO_IMP_LO;
MP_TAC2 LOFA_IMP_ITER_RHO_NODE_ID;
UNDISCH_TAC` lunar (v:real^3 ,w) V E `;
REWRITE_TAC[lunar; INSERT_SUBSET];
SIMP_TAC[];

UNDISCH_TAC` lunar (v:real^3,w) V E `;
REWRITE_TAC[lunar; INSERT_SUBSET];
STRIP_TAC;
ASSUME_TAC2 CVX_LO_IMP_LO;
SUBGOAL_THEN` ~collinear {vec 0, w, rho_node1 FF w}` MP_TAC;
MATCH_MP_TAC LOCAL_FAN_CHARACTER_OF_RHO_NODE2;
ASM_REWRITE_TAC[];

DOWN THEN DOWN;
EXPAND_TAC "v";
ASM_SIMP_TAC[INSERT_COMM];
ASM_REWRITE_TAC[ARITH_RULE` a + 1 < b <=> a < b /\ ~(a + 1 = b )`];
STRIP_TAC;
SUBGOAL_THEN` {ITER l (rho_node1 FF) v | l <= i + 1} SUBSET
      aff {vec 0, v, rho_node1 FF v}` ASSUME_TAC;
REWRITE_TAC[ARITH_RULE` a <= b + 1 <=> a = b + 1 \/ a <= b `;
 SET_RULE` { f x | x = a \/ Q x } = f a INSERT { f x | Q x }`; INSERT_SUBSET];
ASM_REWRITE_TAC[];
MP_TAC (ISPECL [`{vec 0,v:real^3}`;`{rho_node1 FF v}`] 
(GEN_ALL AFF_GT_SUB_AFF_UNION));
REWRITE_TAC[SET_RULE` {a,b} UNION {c} = {a,b,c}`; SUBSET];
DISCH_THEN MATCH_MP_TAC;
FIRST_X_ASSUM ACCEPT_TAC;
ABBREV_TAC` P = aff {vec 0, v, rho_node1 FF v} `;
ABBREV_TAC` U = {ITER l (rho_node1 FF) v | l <= i + 1} `;
ABBREV_TAC` l = i + 1 `;
ABBREV_TAC ` e = v cross rho_node1 FF v `;
MP_TAC2 RHO_NODE1_MONO_WITH_AZIM;
UNDISCH_TAC` lunar (v:real^3,w) V E `;
REWRITE_TAC[lunar; INSERT_SUBSET];
STRIP_TAC;
ASSUME_TAC2 CVX_LO_IMP_LO;
ASM_REWRITE_TAC[plane];
EXPAND_TAC "P";
REWRITE_TAC[FIRST_IN_AFF];
EXISTS_TAC `vec 0:real^3 `;
EXISTS_TAC` v:real^3 `;
EXISTS_TAC ` rho_node1 FF v `;
REWRITE_TAC[aff];
MP_TAC2 LOCAL_FAN_CHARACTER_OF_RHO_NODE2;



STRIP_TAC;
FIRST_X_ASSUM (ASSUME_TAC o (SPECL [`i:num`;` i + 1`]));
DOWN;
ANTS_TAC;
EXPAND_TAC "l";
REWRITE_TAC[ARITH_RULE` a < a + 1`; LE_REFL; ARITH_RULE` a + 1 < b <=> a < b /\ ~( a + 1 = b )`];
ASM_REWRITE_TAC[];

SUBGOAL_THEN` azim (vec 0) e v (ITER i (rho_node1 FF) v) = pi` SUBST1_TAC;
MATCH_MP_TAC IN_CONV0_IMP_AZIM_PI;
MP_TAC2 LUNAR_IMP_INTERIOR_ANGLE1_EQ_PI;
ASM_SIMP_TAC[];
EXPAND_TAC "e";
STRIP_TAC;

REWRITE_TAC[GSYM COLL_IFF_COLL_CROSS];
MP_TAC2 LOCAL_FAN_CHARACTER_OF_RHO_NODE2;
UNDISCH_TAC `lunar (v:real^3,w) V E `;
ASM_SIMP_TAC[lunar; INSERT_SUBSET; CVX_LO_IMP_LO];

REWRITE_TAC[GSYM SIN_AZIM_NEG_PI_LT];
UNDISCH_TAC` ~collinear {vec 0, v, rho_node1 FF v} `;
NHANH Fan.th3a;
NHANH AFF_GT_2_1;
STRIP_TAC;
UNDISCH_TAC` ITER l (rho_node1 FF) v IN aff_gt {vec 0, v} {rho_node1 FF v} `;
ASM_REWRITE_TAC[IN_ELIM_THM];
STRIP_TAC;
MP_TAC (
SPECL [`e:real^3`;` v:real^3`;` ITER l (rho_node1 FF) v`] Trigonometry2.JBDNJJB);
REWRITE_TAC[re_eqvl];
STRIP_TAC;
ASM_REWRITE_TAC[];
ONCE_REWRITE_TAC[CROSS_TRIPLE];
ASM_REWRITE_TAC[CROSS_RADD; CROSS_RMUL; CROSS_REFL; CROSS_0; VECTOR_MUL_RZERO; VECTOR_ADD_LID; DOT_LMUL];
SUBGOAL_THEN ` &0 < e dot (e:real^3) ` ASSUME_TAC;
REWRITE_TAC[DOT_POS_LT];
UNDISCH_TAC` ~collinear {vec 0, v, rho_node1 FF v} `;

ASM_SIMP_TAC[GSYM CROSS_EQ_0];

ASM_SIMP_TAC[REAL_ARITH` a * b * c < &0 <=> &0 < a * ( --b ) * c `; REAL_LT_MUL_EQ];
UNDISCH_TAC` &0 < t3 `;
REAL_ARITH_TAC]);;










let FOR_AFF_GT_NOT_INTERSECTION = prove_by_refinement
(`a1 % (x:real^N) + b1 % y + t % u = a2 % x + b2 % y + tt % v /\
 &0 < t /\
 &0 < tt /\
 a1 + b1 + t = &1 /\
 a2 + b2 + tt = &1
 ==> u = (a2 - a1) / t % x + (b2 - b1) / t % y + tt / t % v /\
     (a2 - a1) / t + (b2 - b1) / t + tt / t = &1 /\
     &0 < tt / t`,
[REWRITE_TAC[VECTOR_ARITH`a + (b:real^N) = c <=> b = c - a `;
REAL_ARITH` a / t + b / t = ( a + b ) / t`];
NHANH_PAT`\x. x ==> y` REAL_POS_NZ;
STRIP_TAC;
UNDISCH_TAC` t % u = (a2 % x + b2 % y + tt % v) - a1 % x - b1 % (y:real^N)`;
ASM_SIMP_TAC[Ldurdpn.VECTOR_SCALE_CHANGE; REAL_ARITH` ( a - b ) / c = a / c - b / c`;
VECTOR_SUB_LDISTRIB; VECTOR_ADD_LDISTRIB; VECTOR_SUB_RDISTRIB];
STRIP_TAC;
CONJ_TAC;
VECTOR_ARITH_TAC;
CONJ_TAC;
UNDISCH_TAC` a1 + b1 + t = &1 `;
UNDISCH_TAC` a2 + b2 + tt = &1 `;
SIMP_TAC[ARITH_RULE` a + b = c <=> a = c - b `];
REWRITE_TAC[REAL_POLY_CONV`&1 - (b2 + tt) - (&1 - (b1 + t)) + b2 - b1 + tt`];
REPEAT STRIP_TAC;
UNDISCH_TAC` &0 < t `;
CONV_TAC REAL_FIELD;
MATCH_MP_TAC REAL_LT_DIV;
ASM_REWRITE_TAC[]]);;










let NOT_COLL_RHONODE_SND_POINT = prove_by_refinement
(` convex_local_fan (V,E,FF) /\ lunar (v,w) V E
==> ~ collinear {vec 0, v, rho_node1 FF w }`,
[NHANH LUNAR_IMP_INTERIOR_ANGLE1_EQ_PI;
REWRITE_TAC[collinear_fan22; DE_MORGAN_THM];
STRIP_TAC;
UNDISCH_TAC` vec 0 IN conv0 {v,w:real^3}`;

NHANH IN_CONV0_IMP_AFF_EQ;
NHANH IN_CONV0_EQ_EQ;
ONCE_REWRITE_TAC[INSERT_COMM];
NHANH IN_CONV0_IMP_AFF_EQ;

NHANH IN_CONV0_EQ_EQ;
SIMP_TAC[INSERT_COMM];
STRIP_TAC;
FIRST_X_ASSUM (SUBST1_TAC o SYM);
ASM_REWRITE_TAC[];
UNDISCH_TAC` vec 0 = w <=> w = (v:real^3)`;
SIMP_TAC[EQ_SYM_EQ; GSYM collinear_fan22; GSYM DE_MORGAN_THM];

ONCE_REWRITE_TAC[INSERT_COMM];
STRIP_TAC;
MATCH_MP_TAC LOCAL_FAN_CHARACTER_OF_RHO_NODE2;
DOWN_TAC;
SIMP_TAC[convex_local_fan; lunar; INSERT_SUBSET]]);;




let NOT_INTERSECTION_BWT_AFF_GTS = prove_by_refinement
(` convex_local_fan (V,E,FF) /\ lunar (v,w) V E
==> aff_gt {vec 0, v} {rho_node1 FF w} INTER aff_gt {vec 0, v} {rho_node1 FF v} = {}`,
[NHANH NOT_COLL_RHONODE_SND_POINT;
NHANH NEXT_OPOSITE_POINT_IS_NOT_IN_AFF_GT;

REWRITE_TAC[convex_local_fan; lunar; INSERT_SUBSET];
STRIP_TAC;
ASSUME_TAC2 LOCAL_FAN_CHARACTER_OF_RHO_NODE2;
DOWN THEN DOWN THEN PHA;
NHANH Fan.th3a;
SIMP_TAC[AFF_GT_2_1; INTER_EQ_EM_EXPAND; IN_ELIM_THM];
REPEAT STRIP_TAC;
DOWN THEN ASM_REWRITE_TAC[GSYM ITER; ADD1];
STRIP_TAC;
ASSUME_TAC2 (ISPECL [` ITER (i + 1) (rho_node1 FF) v `;` vec 0:real^3`;` v:real^3`;`rho_node1 FF v`;` t1':real` ;` t1:real`;` t2':real`;` t2:real`;`t3':real`;` t3:real`] (GEN_ALL FOR_AFF_GT_NOT_INTERSECTION));
DOWN THEN STRIP_TAC;


UNDISCH_TAC` ~(ITER (i + 1) (rho_node1 FF) v IN aff_gt {vec 0, v} {rho_node1 FF v}) `;
ASM_SIMP_TAC[AFF_GT_2_1; IN_ELIM_THM];
EXISTS_TAC` (t1' - t1) / t3 `;
EXISTS_TAC` (t2' - t2) / t3 `;
EXISTS_TAC` t3' / t3 `;
ASM_REWRITE_TAC[]]);;


let LUNAR_COMM = prove(` lunar (v,w) V E <=> lunar (w,v) V E `,
SIMP_TAC[EQ_SYM_EQ; lunar; INSERT_COMM]);;




let CONV0_AFF_GT_EQ = prove_by_refinement
(` x:real^N IN conv0 {a,b} /\ ~( collinear {a,x,v}) 
==> aff_gt {x,a} {v} = aff_gt {x,a,b} {v}`,
[NHANH IN_CONV0_IMP_COLL_ENDS_AFF;
NHANH IN_CONV0_IMP_AFF_EQ;
REWRITE_TAC[IN_ELIM_THM];
NHANH Fan.th3a;
REPEAT STRIP_TAC;
REWRITE_TAC[GSYM SUBSET_ANTISYM_EQ];
CONJ_TAC;
MATCH_MP_TAC AFF_GT_MONO_LEFT;
SET_TAC[];
SUBGOAL_THEN` b IN aff {a,b:real^N}` MP_TAC;
REWRITE_TAC[Planarity.POINT_IN_LINE1];
SUBGOAL_THEN` ~ collinear {b,x,v:real^N}` MP_TAC;
FIRST_X_ASSUM (SUBST1_TAC o SYM);
ASM_REWRITE_TAC[];
NHANH Fan.th3a;
STRIP_TAC;
ASSUME_TAC2 (SET_RULE` DISJOINT {a, x} {v} /\ DISJOINT {b,x} {v:real^N}
==> DISJOINT {x,a,b} {v}`);
DOWN;
UNDISCH_TAC` DISJOINT {a, x} {v:real^N}`;
SIMP_TAC[AFF_GT_2_1; INSERT_COMM; AFF_GT_3_1; SUBSET; IN_ELIM_THM];
REPEAT STRIP_TAC;
ASM_REWRITE_TAC[];
UNDISCH_TAC` b IN aff {a,b:real^N}`;
ASM_REWRITE_TAC[];
REWRITE_TAC[AFF2; IN_ELIM_THM];
STRIP_TAC;
ASM_REWRITE_TAC[];
EXISTS_TAC` t1 + t3 * ( &1 - t ) `;
EXISTS_TAC` t2 + t3 * t `;
EXISTS_TAC` t4:real`;
ASM_REWRITE_TAC[];
CONJ_TAC;
UNDISCH_TAC` t1 + t2 + t3 + t4 = &1 `;
REAL_ARITH_TAC;
VECTOR_ARITH_TAC]);;








let AFF_GT_SAME_WITH_ENDS = prove_by_refinement
(` convex_local_fan (V,E,FF) /\ lunar (v,w) V E
==> aff_gt {vec 0, v} {rho_node1 FF w} = aff_gt {vec 0, w} {rho_node1 FF w }`,
[NHANH LUNAR_IMP_INTERIOR_ANGLE1_EQ_PI;
NHANH NOT_COLL_RHONODE_SND_POINT;
REWRITE_TAC[lunar; INSERT_SUBSET; convex_local_fan];
STRIP_TAC;
SUBGOAL_THEN` ~ collinear {vec 0, w, rho_node1 FF w }` ASSUME_TAC;
MATCH_MP_TAC LOCAL_FAN_CHARACTER_OF_RHO_NODE2;
ASM_REWRITE_TAC[];
SUBGOAL_THEN` aff_gt {vec 0, v} {rho_node1 FF w} = aff_gt {vec 0, v, w} {rho_node1 FF w}` SUBST1_TAC;
MATCH_MP_TAC CONV0_AFF_GT_EQ;
UNDISCH_TAC` ~collinear {vec 0, v, rho_node1 FF w} `;
ASM_SIMP_TAC[INSERT_COMM];
SUBGOAL_THEN` aff_gt {vec 0, w} {rho_node1 FF w} = aff_gt {vec 0, w, v} {rho_node1 FF w}` SUBST1_TAC;
MATCH_MP_TAC CONV0_AFF_GT_EQ;
UNDISCH_TAC` ~collinear {vec 0, w, rho_node1 FF w} `;
ASM_SIMP_TAC[INSERT_COMM];
SIMP_TAC[INSERT_COMM]]);;




let CRAZY_THMMM = prove_by_refinement
(`  w:real^N IN aff {b,c} /\ z IN aff_gt {b,c} {w}
==> z IN aff {b,c} `,
[ASM_CASES_TAC` DISJOINT {b,c} {w:real^N}`;
ASM_SIMP_TAC[AFF_GT_2_1; IN_ELIM_THM; AFF2];
STRIP_TAC;
ASM_REWRITE_TAC[];
EXISTS_TAC` t1 + t3 * t `;
UNDISCH_TAC` t1 + t2 + t3 = &1 `;
SIMP_TAC[REAL_ARITH` a + b = &1 <=> a = &1 - b `];
DISCH_TAC;
VECTOR_ARITH_TAC;
DOWN;
REWRITE_TAC[SET_RULE`~DISJOINT {b, c} {w} <=> w = b \/ w = c`];
STRIP_TAC;
ASM_REWRITE_TAC[];
MP_TAC (SPECL [`{b,c:real^N}`;` {b:real^N}`] (GEN_ALL (REWRITE_RULE[SUBSET_REFL;
 DIFF_EQ_EMPTY; AFF_GT_EQ_AFFINE_HULL] (SPEC `Y:real^N -> bool` AFF_GT_MONO))));
REWRITE_TAC[aff; SET_RULE` {a,b} UNION {a} = {a,b}`];
SET_TAC[];
ASM_REWRITE_TAC[];
MP_TAC (SPECL [`{b,c:real^N}`;` {c:real^N}`] (GEN_ALL (REWRITE_RULE[SUBSET_REFL;
 DIFF_EQ_EMPTY; AFF_GT_EQ_AFFINE_HULL] (SPEC `Y:real^N -> bool` AFF_GT_MONO))));
REWRITE_TAC[aff; SET_RULE` {a,b} UNION {b} = {a,b}`];
SET_TAC[]]);;




let USEFULL_THHM = prove_by_refinement
(`(z:real^N) IN aff {b, c} /\ z IN aff_gt {b, c} {w} ==> w IN aff {b, c}`,
[ASM_CASES_TAC` DISJOINT {b,c} {w:real^N}`;
ASM_SIMP_TAC[AFF_GT_2_1; IN_ELIM_THM; AFF2];
STRIP_TAC;
ASM_REWRITE_TAC[];
DOWN;
ASM_REWRITE_TAC[VECTOR_ARITH` a = x + y + c % z <=> c % z = a - x - y`];
ASSUME_TAC2 (REAL_ARITH` &0 < t3 ==> ~(t3 = &0) `);
ASM_SIMP_TAC[Ldurdpn.VECTOR_SCALE_CHANGE; VECTOR_ADD_LDISTRIB; VECTOR_SUB_LDISTRIB; VECTOR_MUL_ASSOC; REAL_ARITH` &1 / a * b = b / a `];
STRIP_TAC;
EXISTS_TAC` t / t3 - t1 / t3 `;
UNDISCH_TAC` t1 +  t2 + t3 = &1 `;
SIMP_TAC[REAL_ARITH` a + b = &1 <=> a = &1 - b `; REAL_ARITH` (a - ( b + c )) / t = a / t - b / t - c / t `];
ASM_SIMP_TAC[REAL_FIELD` ~( a = &0 ) ==> a / a = &1 `];
DISCH_TAC;
VECTOR_ARITH_TAC;
DOWN;
REWRITE_TAC[SET_RULE` ~ DISJOINT {a,b} {x} <=> x = a \/ x = b `];
MESON_TAC[Planarity.POINT_IN_LINE; Planarity.POINT_IN_LINE1]]);;





let COLL_IN_AFF_GT_TOO = prove(` ~ collinear {x,y,z} /\ a IN aff_gt {x,y} {z}
==> ~ collinear {x,y,a}`,
REWRITE_TAC[collinear_fan22; DE_MORGAN_THM] THEN 
SIMP_TAC[] THEN MESON_TAC[USEFULL_THHM]);;




let AFF_GT_IN_IMP_SUBSET = prove_by_refinement
(`~collinear {x, y, z:real^N} /\ a IN aff_gt {x, y} {z}
==> aff_gt {x,y} {a} SUBSET aff_gt {x,y} {z}`,
[NHANH COLL_IN_AFF_GT_TOO;
NHANH Fan.th3a;
STRIP_TAC;
UNDISCH_TAC` a IN aff_gt {x,y} {z:real^N} `;
ASM_SIMP_TAC[AFF_GT_2_1; IN_ELIM_THM; SUBSET];
REPEAT STRIP_TAC;
ASM_REWRITE_TAC[];
EXISTS_TAC`t1' + t3' * t1 `;
EXISTS_TAC` t2' + t3' * t2 `;
EXISTS_TAC` t3' * t3 `;
CONJ_TAC;
MATCH_MP_TAC REAL_LT_MUL;
ASM_REWRITE_TAC[];
CONJ_TAC;
ASM_REWRITE_TAC[REAL_ARITH` (t1' + t3' * t1) + (t2' + t3' * t2) + t3' * t3 
= t1' + t2' + t3' * ( t1 + t2 + t3 ) `; REAL_MUL_RID];
VECTOR_ARITH_TAC]);;




let FOR_AFF_GT_NOT_INTERSECTION2 = 
let t = SPECL [` &0:real`;` &0`;`&1 `] (GENL [`a2:real`;` b2:real`;`tt:real`]
 FOR_AFF_GT_NOT_INTERSECTION) in REWRITE_RULE[VECTOR_MUL_LZERO; VECTOR_ADD_LID;
 VECTOR_MUL_LID; REAL_ADD_LID; REAL_ARITH` &0 < &1 /\ a - &0 = a`] t;;




let INVS_IN_AFF_GT = prove_by_refinement
(` ~collinear {x:real^N, y, z} /\ a IN aff_gt {x, y} {z} 
==> z IN aff_gt {x,y} {a} `,
[NHANH COLL_IN_AFF_GT_TOO;
PHA THEN IMP_TAC;
NHANH Fan.th3a;
SIMP_TAC[AFF_GT_2_1; IN_ELIM_THM];
REPEAT STRIP_TAC;
UNDISCH_TAC` t1 + t2 + t3 = &1 `;
UNDISCH_TAC` &0 < t3 `;
SWITCH_TAC` a = t1 % x + t2 % y + t3 % (z:real^N)`;
DOWN;
PHA;
NHANH FOR_AFF_GT_NOT_INTERSECTION2;
MESON_TAC[]]);;






let COLL_IN_AFF_GT_AFF_GT_EQ = prove(
` ~collinear {x:real^N, y, z} /\ a IN aff_gt {x, y} {z} 
==> aff_gt {x,y} {z} = aff_gt {x,y} {a} `,
NHANH INVS_IN_AFF_GT THEN NHANH COLL_IN_AFF_GT_TOO THEN 
NHANH AFF_GT_IN_IMP_SUBSET THEN PHA THEN 
NHANH AFF_GT_IN_IMP_SUBSET THEN SIMP_TAC[GSYM SUBSET_ANTISYM_EQ]);;








let NEXT_OPOSITE_POINT_IS_NOT_IN_AFF_GT2 = prove_by_refinement
(`convex_local_fan (V,E,FF) /\ lunar (v,w) V E 
==> (?i. w = ITER i (rho_node1 FF) v /\
                i + 1 < CARD V /\
 ~( i = 0 ) /\ ~( i = 1 ) /\
                ~(ITER (i + 1) (rho_node1 FF) v IN
                  aff_gt {vec 0, v} {rho_node1 FF v}))`,
[NHANH NEXT_OPOSITE_POINT_IS_NOT_IN_AFF_GT;
STRIP_TAC;
EXISTS_TAC` i:num`;
ASM_REWRITE_TAC[];
CONJ_TAC;
STRIP_TAC;
FIRST_X_ASSUM SUBST_ALL_TAC;
DOWN_TAC;
REWRITE_TAC[lunar; ITER];
STRIP_TAC;
UNDISCH_TAC` ~( v = w:real^3)`;
ASM_REWRITE_TAC[];
STRIP_TAC;
FIRST_X_ASSUM SUBST_ALL_TAC;
DOWN_TAC;
REWRITE_TAC[lunar; ITER12; INSERT_SUBSET; convex_local_fan];
STRIP_TAC;
UNDISCH_TAC` collinear {vec 0, v, w:real^3}`;
ASM_REWRITE_TAC[];
MATCH_MP_TAC LOCAL_FAN_CHARACTER_OF_RHO_NODE2;
ASM_REWRITE_TAC[]]);;




let LUNAR_IMP_HALF_CIRCLE_SUBSET_AFF_GT100 = prove_by_refinement
(`convex_local_fan (V,E,FF) /\ lunar (v,w) V E
       ==> (?i. i < CARD V /\ ~( i = 0 ) /\ ~ ( i = 1 ) /\
                w = ITER i (rho_node1 FF) v /\
                {ITER l (rho_node1 FF) v | 0 < l /\ l < i} SUBSET
                aff_gt {vec 0, v} {rho_node1 FF v})`,
[NHANH LUNAR_IMP_HALF_CIRCLE_SUBSET_AFF_GT;
STRIP_TAC;
EXISTS_TAC` i:num`;
ASM_REWRITE_TAC[];
CONJ_TAC;
STRIP_TAC;
FIRST_X_ASSUM SUBST_ALL_TAC;
DOWN_TAC;
REWRITE_TAC[lunar; ITER];
STRIP_TAC;
UNDISCH_TAC` ~( v = w:real^3)`;
ASM_REWRITE_TAC[];
STRIP_TAC;
FIRST_X_ASSUM SUBST_ALL_TAC;
DOWN_TAC;
REWRITE_TAC[lunar; ITER12; INSERT_SUBSET; convex_local_fan];
STRIP_TAC;
UNDISCH_TAC` collinear {vec 0, v, w:real^3}`;
ASM_REWRITE_TAC[];
MATCH_MP_TAC LOCAL_FAN_CHARACTER_OF_RHO_NODE2;
ASM_REWRITE_TAC[]]);;







let IVS_RHO_IDD = prove(` local_fan (V,E,FF) /\ v IN V
==> ivs_rho_node1 FF ( rho_node1 FF v ) = v `,
NHANH LOCAL_FAN_RHO_NODE_PROS2 THEN STRIP_TAC THEN 
DOWN THEN FIRST_X_ASSUM NHANH THEN REWRITE_TAC[ivs_rho_node1] THEN 
ASM_MESON_TAC[PRE_IVS_RHO_NODE1_DETE]);;





let AFF_IVS_RHO_NODE_EQQ = prove_by_refinement
(`convex_local_fan (V,E,FF) /\ lunar (v,w) V E 
==> aff_gt {vec 0, w} {rho_node1 FF w} = aff_gt {vec 0, v} {ivs_rho_node1 FF v}`,
[NHANH AFF_GT_SAME_WITH_ENDS;
ONCE_REWRITE_TAC[LUNAR_COMM];
NHANH LUNAR_IMP_HALF_CIRCLE_SUBSET_AFF_GT100;
STRIP_TAC;
FIRST_ASSUM (SUBST1_TAC o SYM);
MATCH_MP_TAC COLL_IN_AFF_GT_AFF_GT_EQ;

ASSUME_TAC2 (ONCE_REWRITE_RULE[LUNAR_COMM] NOT_COLL_RHONODE_SND_POINT);
DOWN;
SIMP_TAC[];
STRIP_TAC;
UNDISCH_TAC` {ITER l (rho_node1 FF) w | 0 < l /\ l < i} SUBSET
      aff_gt {vec 0, w} {rho_node1 FF w}`;
ASM_REWRITE_TAC[SUBSET];
DISCH_THEN MATCH_MP_TAC;
REWRITE_TAC[IN_ELIM_THM];
EXISTS_TAC` i - 1 `;
MP_TAC2 (ARITH_RULE` ~( i = 0 ) /\ ~( i = 1 ) ==> 0 < i - 1 /\ i - 1 < i /\
i = SUC ( i - 1 ) `);
SIMP_TAC[];
STRIP_TAC;
FIRST_ASSUM SUBST1_TAC;
REWRITE_TAC[ITER; ARITH_RULE` SUC a - 1 = a `];
MATCH_MP_TAC IVS_RHO_IDD;
UNDISCH_TAC` convex_local_fan (V,E,FF) `;
SIMP_TAC[convex_local_fan];
NHANH LOCAL_FAN_ORBIT_MAP_V;
STRIP_TAC;
UNDISCH_TAC` lunar (w,v:real^3) V E `;
REWRITE_TAC[lunar; INSERT_SUBSET];
STRIP_TAC;
UNDISCH_TAC` w:real^3 IN V `;
FIRST_X_ASSUM NHANH;
FIRST_X_ASSUM (NHANH_PAT`\x. x ==> y `);
STRIP_TAC;
EXPAND_TAC "V";
REWRITE_TAC[lemma_in_orbit_iter]]);;









let LOFA_IMP_LT_CARD_SET_V = prove(
`! v. local_fan (V,E,FF) /\ v IN V ==> {ITER n (rho_node1 FF) v | n < CARD V} = V`,
NHANH LOCAL_FAN_ORBIT_MAP_V THEN NHANH LOOP_SET_DETER_FIRTS_ELMS THEN 
SIMP_TAC[]);;









let NOT_COLL_IMP_NOT_AFF_SUB = prove(
`!v:real^N. ~ collinear {x,y,z} /\ v IN aff {x,y} ==> ~( v IN aff_gt {x,y} {z})`,
REPEAT STRIP_TAC THEN 
ASSUME_TAC2 (ISPECL [`z:real^N`;`x:real^N`;`y:real^N`;` v:real^N`] (GEN_ALL 
COLL_IN_AFF_GT_TOO)) THEN DOWN THEN
ASM_REWRITE_TAC[collinear_fan22]);;







let HALP_CIRCLE_IS_INTERSECTION = prove_by_refinement
(` convex_local_fan (V,E,FF) /\ lunar (v,w) V E
       ==> (?i. i < CARD V /\
                ~(i = 0) /\
                ~(i = 1) /\
                w = ITER i (rho_node1 FF) v /\
                {ITER l (rho_node1 FF) v | 0 < l /\ l < i} = 
                aff_gt {vec 0, v} {rho_node1 FF v} INTER V)`,
[NHANH LUNAR_IMP_HALF_CIRCLE_SUBSET_AFF_GT100;
ONCE_REWRITE_TAC[LUNAR_COMM];
NHANH LUNAR_IMP_HALF_CIRCLE_SUBSET_AFF_GT100;
NHANH AFF_GT_SAME_WITH_ENDS;
REWRITE_TAC[convex_local_fan];
NHANH LOFA_IMP_ITER_RHO_NODE_ID;
STRIP_TAC;
EXISTS_TAC` i':num`;
CONJ_TAC;
ASM_REWRITE_TAC[];
CONJ_TAC;
ASM_REWRITE_TAC[];
CONJ_TAC;
ASM_REWRITE_TAC[];
CONJ_TAC;
FIRST_ASSUM ACCEPT_TAC;
REWRITE_TAC[GSYM SUBSET_ANTISYM_EQ; SUBSET_INTER];
CONJ_TAC;
DOWN THEN SIMP_TAC[];
SIMP_TAC[SUBSET; IN_ELIM_THM];
REPEAT STRIP_TAC;
DOWN THEN SIMP_TAC[];
ASSUME_TAC2 LOCAL_FAN_ORBIT_MAP_V;
UNDISCH_TAC` lunar (w,v:real^3) V E `;
REWRITE_TAC[lunar; IN_ELIM_THM; INSERT_SUBSET];
FIRST_X_ASSUM (NHANH_PAT`\x. x ==> y`);
STRIP_TAC THEN DISCH_TAC;
EXPAND_TAC "V";
REWRITE_TAC[lemma_in_orbit_iter];
MP_TAC NOT_INTERSECTION_BWT_AFF_GTS;
ANTS_TAC;
REWRITE_TAC[convex_local_fan];
CONJ_TAC;
CONJ_TAC;
FIRST_ASSUM ACCEPT_TAC;
FIRST_ASSUM ACCEPT_TAC;
ONCE_REWRITE_TAC[LUNAR_COMM];
FIRST_ASSUM ACCEPT_TAC;
STRIP_TAC;

SUBGOAL_THEN` v IN V /\ w:real^3 IN V ` MP_TAC;
UNDISCH_TAC` lunar (w,v:real^3) V E `;
SIMP_TAC[lunar; INSERT_SUBSET];
STRIP_TAC;

ASSUME_TAC2 LOFA_IMP_LT_CARD_SET_V;
REWRITE_TAC[SUBSET; IN_INTER];
GEN_TAC THEN STRIP_TAC;
DOWN;
EXPAND_TAC "V";
SIMP_TAC[IN_ELIM_THM];
ASSUME_TAC2 LOCAL_FAN_CHARACTER_OF_RHO_NODE2;

SUBGOAL_THEN` ~( v IN aff_gt {vec 0, v} {rho_node1 FF v})` ASSUME_TAC;
MATCH_MP_TAC NOT_COLL_IMP_NOT_AFF_SUB;

DOWN;
SIMP_TAC[Planarity.POINT_IN_LINE1];

SUBGOAL_THEN ` convex_local_fan (V,E,FF) ` ASSUME_TAC;
REWRITE_TAC[convex_local_fan];
CONJ_TAC;
FIRST_ASSUM ACCEPT_TAC;
FIRST_ASSUM ACCEPT_TAC;


SUBGOAL_THEN` ~collinear {vec 0, w, rho_node1 FF v}` ASSUME_TAC;
MATCH_MP_TAC NOT_COLL_RHONODE_SND_POINT;
ASM_REWRITE_TAC[];


SUBGOAL_THEN` ~( w IN aff_gt {vec 0, w:real^3} {rho_node1 FF v})` ASSUME_TAC;
MATCH_MP_TAC NOT_COLL_IMP_NOT_AFF_SUB;
DOWN THEN SIMP_TAC[Planarity.POINT_IN_LINE1];

STRIP_TAC;
ASM_CASES_TAC` n = 0 `;
FIRST_X_ASSUM SUBST_ALL_TAC;
DOWN;
REWRITE_TAC[ITER];
DISCH_THEN SUBST_ALL_TAC;

UNDISCH_TAC` ~(v IN aff_gt {vec 0, v} {rho_node1 FF v})`;
UNDISCH_TAC` (v IN aff_gt {vec 0, v} {rho_node1 FF v})`;
SIMP_TAC[];


ASM_CASES_TAC` n = i':num`;
FIRST_X_ASSUM SUBST_ALL_TAC;
SWITCH_TAC` x = ITER i' (rho_node1 FF) v `;
SWITCH_TAC` w = ITER i' (rho_node1 FF) v `;
SWITCH_TAC` v = ITER i (rho_node1 FF) w `;
UNDISCH_TAC` ~(w IN aff_gt {vec 0, w} {rho_node1 FF v}) `;

ASM_REWRITE_TAC[];
EXPAND_TAC "w";
ASM_REWRITE_TAC[];



SWITCH_TAC` w = ITER i' (rho_node1 FF) v `;
SWITCH_TAC` v = ITER i (rho_node1 FF) w `;
UNDISCH_TAC` ~( n = i':num)`;
REWRITE_TAC[ARITH_RULE` ~( a = n:num) <=> a < n \/ n < a `];
STRIP_TAC;
EXISTS_TAC` n:num`;
ASM_REWRITE_TAC[ARITH_RULE` 0 < n <=> ~( n = 0 ) `];
SUBGOAL_THEN` x IN aff_gt {vec 0, v} {rho_node1 FF w}` ASSUME_TAC;
UNDISCH_TAC` {ITER l (rho_node1 FF) w | 0 < l /\ l < i} SUBSET
      aff_gt {vec 0, w} {rho_node1 FF w} `;
ASM_REWRITE_TAC[];
ASSUME_TAC2 (ONCE_REWRITE_RULE[LUNAR_COMM] AFF_GT_SAME_WITH_ENDS);
ASM_REWRITE_TAC[SUBSET];
DISCH_THEN MATCH_MP_TAC;
REWRITE_TAC[IN_ELIM_THM];
EXISTS_TAC` n:num - i'`;
CONJ_TAC;
CONJ_TAC;
UNDISCH_TAC` i' < n:num`;
ARITH_TAC;
ASM_CASES_TAC` n - i' < i:num`;
DOWN THEN REWRITE_TAC[];
DOWN;
ASM_SIMP_TAC[ARITH_RULE` c < a ==> ( ~( a - c < b:num ) <=> b + c <= a )`];
UNDISCH_TAC` n < CARD (V:real^3 -> bool) `;
PHA;
NHANH (ARITH_RULE` a < b /\ c <= a:num ==> c < b `);

STRIP_TAC;
MP_TAC2 LOFA_IMP_DIS_ELMS23;
DISCH_THEN (MP_TAC o (SPECL [`0`;` (i:num) + i' `]));
ANTS_TAC;
ASM_SIMP_TAC[ARITH_RULE` ~ (a = 0 ) ==> 0 < a + b `];

ASM_REWRITE_TAC[ITER; GSYM ITER_ADD];
ASSUME_TAC2 (ARITH_RULE` i' < n:num ==> n = n - i' + i'`);
ABBREV_TAC` ll = n - i':num`;
ASM_REWRITE_TAC[GSYM ITER_ADD];

UNDISCH_TAC` aff_gt {vec 0, v} {rho_node1 FF w} INTER
      aff_gt {vec 0, v} {rho_node1 FF v} =
      {} `;
REWRITE_TAC[INTER_EQ_EM_EXPAND; NOT_EXISTS_THM];
DISCH_THEN (MP_TAC o (SPEC `x:real^3`));
ASM_REWRITE_TAC[]]);;












let CONVEX_LOFA_IMP_INANGLE_LE_PI = prove_by_refinement
(`convex_local_fan (V,E,FF) /\ v IN V ==> interior_angle1 (vec 0) FF v <= pi `,
[REWRITE_TAC[convex_local_fan; azim_in_fan2];
STRIP_TAC;
ASSUME_TAC2 EXISTS_INVERSE_OF_V;
DOWN THEN STRIP_TAC;
ASSUME_TAC2 LOFA_IMP_EE_TWO_ELMS;
ASSUME_TAC2 LOFA_CARD_EE_V_1;
ASSUME_TAC2 LOCAL_FAN_RHO_NODE_PROS2;
DOWN THEN STRIP_TAC;
UNDISCH_TAC` v:real^3 IN V `;
FIRST_ASSUM NHANH;
UNDISCH_TAC` !x. x IN FF
          ==> (let d = azim_cycle  (EE (FST x) E) (vec 0) (FST x) (SND x) in
               if CARD (EE (FST x) E) > 1
               then azim (vec 0) (FST x) (SND x) d
               else &2 * pi) <=
              pi /\
              V SUBSET wedge_in_fan_ge x E`;
DISCH_THEN NHANH;
LET_TAC;
SWITCH_TAC` EE v E = {rho_node1 FF v, vv} `;
ASM_SIMP_TAC[ARITH_RULE` a = 2 ==> a > 1 `];
STRIP_TAC;
DOWN THEN DOWN THEN PHA;

ASSUME_TAC2 (SPEC `vv:real^3 ` (GEN` v:real^3 ` IVS_RHO_IDD));
EXPAND_TAC "d";
SIMP_TAC[];
UNDISCH_TAC` {rho_node1 FF v, vv} = EE v E `;
DISCH_THEN (SUBST1_TAC o SYM);
EXPAND_TAC "v";
DOWN;
SIMP_TAC[interior_angle1; GSYM ivs_rho_node1; AZIM_CYCLE_TWO_POINT_SET]]);;







(* x IN aff_gt S {x} *)
let X_IN_AFF_GT_X = 
let t = ISPECL [`S:real^N -> bool`;` {x:real^N}`] (GEN_ALL CONV0_SUBSET_AFF_GT)
in REWRITE_RULE[Geomdetail.CONV0_SING; INSERT_SUBSET; EMPTY_SUBSET] t;;





let IVS_RNODE_IN_AFF_V = prove(`convex_local_fan (V,E,FF) /\ lunar (v,w) V E
==> ivs_rho_node1 FF w IN aff_gt {vec 0, v} {rho_node1 FF v}`,
ONCE_REWRITE_TAC[LUNAR_COMM] THEN NHANH AFF_IVS_RHO_NODE_EQQ THEN 
MESON_TAC[X_IN_AFF_GT_X]);;








let AZIM_LE_PI_EQ_DIHV = prove(`~( collinear {a,b,x}) /\ ~ (collinear {a,b,y})
==> azim a b x y <= pi ==> azim a b x y = dihV a b x y `,
DISCH_TAC THEN REWRITE_TAC[REAL_ARITH` a <= b <=> a = b \/ a < b `] THEN 
STRIP_TAC THENL [DOWN THEN ASM_SIMP_TAC[AZIM_DIHV_EQ_PI];
ASM_SIMP_TAC[AZIM_DIHV_SAME]]);;





let LOFA_IMP_NOT_COLL_IVS = prove(
`!v. local_fan (V,E,FF) /\ v IN V ==> ~ collinear {vec 0,v,ivs_rho_node1 FF v}`,
NHANH EXISTS_INVERSE_OF_V THEN REPEAT STRIP_TAC THEN 
DOWN THEN ASSUME_TAC2 (SPEC`vv:real^3 ` (GEN`v:real^3` LOCAL_FAN_CHARACTER_OF_RHO_NODE2)) THEN EXPAND_TAC "v" THEN
ASSUME_TAC2 (SPEC`vv:real^3 ` (GEN`v:real^3` IVS_RHO_IDD)) THEN 
DOWN THEN DOWN THEN ASM_SIMP_TAC[INSERT_COMM]);;







let DIHV_NOT_CHANGE = prove(
` &0 < c /\ a + b + c = &1 ==> dihV x y ( a % x + b % y + c % v ) w = dihV x y v w `,
SIMP_TAC[Trigonometry2.DIHV_FORMULAR; REAL_ARITH` a + b = &1 <=> a = &1 - b`;
VECTOR_ARITH` ((&1 - (b + c)) % x + b % y + c % v) - x = b % ( y - x ) + c % (v - x )`] THEN 
SIMP_TAC[VECTOR_ADD_RDISTRIB; DOT_LADD; DOT_LMUL; VECTOR_ADD_LDISTRIB; VECTOR_MUL_ASSOC; REAL_MUL_SYM] THEN 
REWRITE_TAC[VECTOR_ARITH` (a + b ) - ( a + c ) = b - (c:real^N)`; GSYM VECTOR_MUL_ASSOC; GSYM VECTOR_SUB_LDISTRIB] THEN 
ONCE_REWRITE_TAC[Trigonometry2.ARC_SYM] THEN 
SIMP_TAC [GSYM Trigonometry2.WHEN_K_POS_ARCV_STABLE]);;




let LUNAR_IMP_INTERIOR_ANGLE_EQQ = prove_by_refinement
(`convex_local_fan (V,E,FF) /\ lunar (v,w) V E
 ==> interior_angle1 (vec 0) FF v = interior_angle1 (vec 0) FF w `,
[NHANH IVS_RNODE_IN_AFF_V;
NHANH NOT_COLL_RHONODE_SND_POINT;
ONCE_REWRITE_TAC[LUNAR_COMM];
NHANH IVS_RNODE_IN_AFF_V;
NHANH NOT_COLL_RHONODE_SND_POINT;


NHANH LUNAR_IMP_INTERIOR_ANGLE1_EQ_PI;
REWRITE_TAC[lunar; INSERT_SUBSET];
STRIP_TAC;
MP_TAC2 CONVEX_LOFA_IMP_INANGLE_LE_PI;
MP_TAC2 (
SPEC `w:real^3 ` (GEN`v:real^3 ` CONVEX_LOFA_IMP_INANGLE_LE_PI));

REWRITE_TAC[interior_angle1; GSYM ivs_rho_node1];


ASSUME_TAC2 CVX_LO_IMP_LO;
ASSUME_TAC2 LOCAL_FAN_CHARACTER_OF_RHO_NODE2;
ASSUME_TAC2 (SPEC `w:real^3 ` (GEN`v:real^3 ` LOCAL_FAN_CHARACTER_OF_RHO_NODE2));
REPEAT DISCH_TAC;
SUBGOAL_THEN` azim (vec 0) v (rho_node1 FF v) (ivs_rho_node1 FF v) =
dihV (vec 0) v (rho_node1 FF v) (ivs_rho_node1 FF v)` MP_TAC;
DOWN;
MATCH_MP_TAC AZIM_LE_PI_EQ_DIHV;
ASM_SIMP_TAC[];
MATCH_MP_TAC LOFA_IMP_NOT_COLL_IVS;
ASM_REWRITE_TAC[];

SUBGOAL_THEN` azim (vec 0) w (rho_node1 FF w) (ivs_rho_node1 FF w) =
dihV (vec 0) w (rho_node1 FF w) (ivs_rho_node1 FF w)` MP_TAC;
DOWN;
REMOVE_TAC THEN DOWN;
MATCH_MP_TAC AZIM_LE_PI_EQ_DIHV;
ASM_SIMP_TAC[];
MATCH_MP_TAC LOFA_IMP_NOT_COLL_IVS;
ASM_REWRITE_TAC[];
DISCH_THEN SUBST1_TAC;
DISCH_THEN SUBST1_TAC;
UNDISCH_TAC` ~collinear {vec 0, v, rho_node1 FF v} `;
UNDISCH_TAC` ~collinear {vec 0, w, rho_node1 FF w} `;
PHA;
NHANH Fan.th3a;
NHANH AFF_GT_2_1;
STRIP_TAC;
UNDISCH_TAC` ivs_rho_node1 FF w IN aff_gt {vec 0, v} {rho_node1 FF v} `;
UNDISCH_TAC` ivs_rho_node1 FF v IN aff_gt {vec 0, w} {rho_node1 FF w} `;
ASM_REWRITE_TAC[IN_ELIM_THM];
PHA THEN STRIP_TAC;
UNDISCH_TAC` vec 0 IN conv0 {w,v:real^3}`;
REWRITE_TAC[Collect_geom.CONV0_SET2; IN_ELIM_THM; VECTOR_ARITH` vec 0 = a + b % y <=> a = ( -- b ) % y `];
STRIP_TAC;

ONCE_REWRITE_TAC[Trigonometry2.DIHV_SYM];



SUBGOAL_THEN` dihV (vec 0) ( -- b % v ) (ivs_rho_node1 FF v) (rho_node1 FF v) =
dihV (vec 0) ( v ) (ivs_rho_node1 FF v) (rho_node1 FF v)` MP_TAC;
MATCH_MP_TAC DIHV_SPECIAL_SCALE;
ASM_SIMP_TAC[REAL_ARITH` &0 < b ==> ~( -- b = &0)`];


SUBGOAL_THEN` dihV (vec 0) ( a % w ) (ivs_rho_node1 FF w) (rho_node1 FF w) =
dihV (vec 0) w (ivs_rho_node1 FF w) (rho_node1 FF w)` MP_TAC;
MATCH_MP_TAC DIHV_SPECIAL_SCALE;
ASM_SIMP_TAC[REAL_ARITH` &0 < b ==> ~(b = &0)`];

DISCH_THEN (SUBST1_TAC o SYM);
FIRST_ASSUM SUBST1_TAC;

SUBGOAL_THEN` dihV (vec 0) ( -- b % v ) (ivs_rho_node1 FF w) (rho_node1 FF w) =
dihV (vec 0) ( v ) (ivs_rho_node1 FF w) (rho_node1 FF w)` MP_TAC;
MATCH_MP_TAC DIHV_SPECIAL_SCALE;
ASM_SIMP_TAC[REAL_ARITH` &0 < b ==> ~( -- b = &0)`];
SIMP_TAC[];
REMOVE_TAC;


DISCH_THEN (SUBST1_TAC o SYM);
FIRST_ASSUM (SUBST1_TAC o SYM);

SUBGOAL_THEN` dihV (vec 0) ( a % w ) (ivs_rho_node1 FF v) (rho_node1 FF v) =
dihV (vec 0) ( w ) (ivs_rho_node1 FF v) (rho_node1 FF v)` MP_TAC;
MATCH_MP_TAC DIHV_SPECIAL_SCALE;
ASM_SIMP_TAC[REAL_ARITH` &0 < b ==> ~( b = &0)`];

SIMP_TAC[];


ASM_SIMP_TAC[REWRITE_RULE[TAUT` a /\ b ==> c <=> a ==> b ==> c`] DIHV_NOT_CHANGE];

REMOVE_TAC;


SUBGOAL_THEN` dihV (vec 0) (a % w ) (rho_node1 FF w) (rho_node1 FF v) = 
dihV (vec 0) w (rho_node1 FF w) (rho_node1 FF v)` MP_TAC;
MATCH_MP_TAC DIHV_SPECIAL_SCALE;
ASM_SIMP_TAC[REAL_ARITH` &0 < b ==> ~(b = &0)`];
DISCH_THEN (SUBST1_TAC o SYM);
FIRST_X_ASSUM SUBST1_TAC;

MP_TAC2 (REAL_ARITH` &0 < b ==> ~( -- b = &0 ) `);
SIMP_TAC[DIHV_SPECIAL_SCALE; Trigonometry2.DIHV_SYM]]);;





let HKIRPEP = prove_by_refinement
(`convex_local_fan (V,E,FF) /\ lunar (v,w) V E
 ==> (!u. u IN V DIFF {v, w} ==> interior_angle1 (vec 0) FF u = pi) /\
     &0 < interior_angle1 (vec 0) FF v /\
     interior_angle1 (vec 0) FF v <= pi /\
     interior_angle1 (vec 0) FF v = interior_angle1 (vec 0) FF w /\
     (?i. i < CARD V /\
                ~(i = 0) /\
                ~(i = 1) /\
                w = ITER i (rho_node1 FF) v /\
                {ITER l (rho_node1 FF) v | 0 < l /\ l < i} =
                aff_gt {vec 0, v} {rho_node1 FF v} INTER V)/\
    (?j. j < CARD V /\
                ~(j = 0) /\
                ~(j = 1) /\
                v = ITER j (rho_node1 FF) w /\
                {ITER l (rho_node1 FF) w | 0 < l /\ l < j} =
                aff_gt {vec 0, v} {ivs_rho_node1 FF v} INTER V)`,
[NHANH HALP_CIRCLE_IS_INTERSECTION;
NHANH LUNAR_IMP_INTERIOR_ANGLE_EQQ;
NHANH (GSYM AFF_IVS_RHO_NODE_EQQ);
ONCE_REWRITE_TAC[LUNAR_COMM];
NHANH HALP_CIRCLE_IS_INTERSECTION;
NHANH LUNAR_IMP_INTERIOR_ANGLE1_EQ_PI;
SIMP_TAC[INSERT_COMM; LUNAR_IMP_INTERIOR_ANGLE_EQQ];
REWRITE_TAC[lunar; INSERT_SUBSET; INSERT_COMM];
STRIP_TAC;
ASSUME_TAC2 CONVEX_LOFA_IMP_INANGLE_LE_PI;
ASSUME_TAC2 CVX_LO_IMP_LO;
MP_TAC2 INTERIOR_ANGLE1_POS;
STRIP_TAC;
UNDISCH_TAC` interior_angle1 (vec 0) FF v = interior_angle1 (vec 0) FF w `;
DISCH_THEN (SUBST1_TAC o SYM);
ASM_REWRITE_TAC[]]);;




(* ============================ END ========================== *)
(*                      begin lemma EGHNAVX                    *)


let NEXT_PRO_IMP_ALLS = prove_by_refinement(
`!(le: A -> A -> bool). (! i. le  (f i) ( f  (i + 1) ))/\
(! a b c. le a b /\ le b c ==> le a c )
==> (! i j. i < j ==> le  (f i ) ( f j )) `,
[GEN_TAC THEN STRIP_TAC;
GEN_TAC THEN INDUCT_TAC;
REWRITE_TAC[LT];
REWRITE_TAC[ARITH_RULE` a < SUC b <=> a = b \/ a < b `];
STRIP_TAC;
ASM_REWRITE_TAC[ADD1];
DOWN;
FIRST_X_ASSUM NHANH;
REWRITE_TAC[ADD1];
ASM_MESON_TAC[]]);;



let FINITE_CARD1_IMP_SINGLETON = prove
(` FINITE (S:A -> bool) /\ CARD S = 1 ==> (? x. S = {x}) `,
ASM_CASES_TAC` S:A -> bool = {} ` THENL [
ASM_REWRITE_TAC[CARD_CLAUSES; ARITH_RULE`~( 0 = 1 )`]; DOWN THEN
REWRITE_TAC[EMPTY_NOT_EXISTS_IN] THEN REPEAT STRIP_TAC THEN 
EXISTS_TAC`x:A` THEN MATCH_MP_TAC Hypermap.set_one_point THEN 
ASM_REWRITE_TAC[]]);;








let SURJ_IMP_FINITE = prove(` SURJ (f:A -> B) A B /\ FINITE A ==> FINITE B `,
NHANH_PAT`\x. x ==> y` FINITE_IMAGE THEN 
NHANH Wrgcvdr_cizmrrh.SURJ_IMP_S2_EQ_IMAGE_S1 THEN 
STRIP_TAC THEN DOWN THEN ASM_REWRITE_TAC[]);;



let BIJ_FINITE_TOO = prove(` BIJ (f:A -> B) X Y /\ FINITE X ==> FINITE Y `,
REWRITE_TAC[BIJ] THEN PHA THEN NHANH SURJ_IMP_FINITE THEN SIMP_TAC[]);;




let EQ_IFF_IMP = MESON[]`(! a b. P a b <=> P b a ) <=> (! a b. P a b ==> P b a)`;;




let NOT_EMP_INJ_IMP_SURJ = prove_by_refinement
(`let f1 = (\y. if ?x. x IN X /\ f x = y then (@x. x IN X /\ f x = y )
 else  (@x. x IN X)) 
in
 ~( X = {} ) /\ INJ (f:A -> B) X Y ==> SURJ f1 Y X`,
[LET_TAC;
REWRITE_TAC[INJ; SURJ; EXISTS_IN];
STRIP_TAC;
CONJ_TAC;
REPEAT STRIP_TAC;
EXPAND_TAC "f1";
ASM_CASES_TAC` ?x. x IN X /\ (f: A -> B) x = x' `;
ASM_REWRITE_TAC[];
DOWN;
MESON_TAC[];
ASM_MESON_TAC[];
REPEAT STRIP_TAC;
EXISTS_TAC `(f: A -> B) x' `;
EXPAND_TAC "f1";
ASM_MESON_TAC[]]);;





let LOFA_V_NOT_EMP = prove(`local_fan (V,E, FF) ==> ~( V = {} )`,
REWRITE_TAC[local_fan; FAN; fan1] THEN LET_TAC THEN SET_TAC[]);;




let  LOCAL_FAN_FINITE_V = prove(`local_fan (V,E,FF) ==> FINITE V`,
NHANH Wrgcvdr_cizmrrh.LOCAL_FAN_FINITE_FF THEN 
NHANH LOFA_IMP_BIJ_FF_V THEN REWRITE_TAC[BIJ] THEN 
PHA THEN NHANH SURJ_IMP_FINITE THEN SIMP_TAC[]);;






let ITER_CARD_MINUS1_EQ_IVS_RN1 = prove_by_refinement
(` local_fan (V,E,FF) ==> 
(! v. v IN V ==> ITER (CARD V - 1) (rho_node1 FF) v = ivs_rho_node1 FF v)`,
[NHANH LOFA_IMP_ITER_RHO_NODE_ID;
STRIP_TAC;
FIRST_X_ASSUM NHANH;
REPEAT STRIP_TAC;
EXPAND_TAC "v";
ASSUME_TAC2 LOFA_V_NOT_EMP;
ASSUME_TAC2 LOCAL_FAN_FINITE_V;
DOWN THEN NHANH CARD_EQ_0;
STRIP_TAC;
UNDISCH_TAC` ~(V:real^3 -> bool = {} )`;
FIRST_ASSUM (SUBST1_TAC o SYM);

NHANH (ARITH_RULE`~( x = 0 ) ==> x = x - 1 + 1 `);
ABBREV_TAC` u = CARD (V:real^3 -> bool) -1`;
SIMP_TAC[GSYM ADD1; ITER];
ASSUME_TAC2 LOCAL_FAN_ORBIT_MAP_VITER;
SUBGOAL_THEN` ITER u (rho_node1 FF) v IN V ` ASSUME_TAC;
FIRST_X_ASSUM MATCH_MP_TAC;
FIRST_ASSUM ACCEPT_TAC;
STRIP_TAC;
UNDISCH_TAC` ITER u (rho_node1 FF) v IN V`;
UNDISCH_TAC` local_fan (V,E,FF)`;
PHA;
NHANH IVS_RHO_IDD;
SIMP_TAC[];
REWRITE_TAC[GSYM ITER];
FIRST_X_ASSUM (SUBST1_TAC o SYM);
ASM_REWRITE_TAC[]]);;








let FIRST_EQ0_LAST_LT_PI = prove_by_refinement
(`convex_local_fan (V,E,FF) /\
 v0 IN V /\
 CARD V = k /\
 (!i. ITER i (rho_node1 FF) v0 = vv i) /\
 (!i. azim (vec 0) v0 (vv 1) (vv i) = bta i)
 ==> bta 1 = &0 /\
     bta (k - 1) <= pi `,
[REWRITE_TAC[convex_local_fan];
NHANH ITER_CARD_MINUS1_EQ_IVS_RN1;
STRIP_TAC;
DOWN;
DOWN;
ONCE_REWRITE_TAC[GSYM EQ_SYM_EQ];
SIMP_TAC[];
REPEAT STRIP_TAC;
REWRITE_TAC[ITER12; AZIM_REFL];
REWRITE_TAC[ITER12];
EXPAND_TAC "k";
ASM_SIMP_TAC[];
ASSUME_TAC2 (REWRITE_RULE[convex_local_fan] (SPEC `v0:real^3 ` 
(GEN `v:real^3 ` CONVEX_LOFA_IMP_INANGLE_LE_PI)));
DOWN;
REWRITE_TAC[interior_angle1; ivs_rho_node1]]);;







let PROJEC_EQ_0_IFF_COLL = prove_by_refinement
(` projection e  (x:real^N) = vec 0 <=> (?t. x = t % e ) `,
[EQ_TAC;
REWRITE_TAC[Wrgcvdr_cizmrrh.PROJECT_EQ_VEC0_IMP_PARALLED];
STRIP_TAC;
ASM_REWRITE_TAC[projection];
ASM_CASES_TAC`(e:real^N) = vec 0 `;
ASM_REWRITE_TAC[VECTOR_MUL_RZERO; VECTOR_SUB_REFL];
DOWN;
SIMP_TAC[GSYM DOT_EQ_0; DOT_LMUL; REAL_FIELD` ~( x = &0) ==> (a * x ) / x = a `; VECTOR_SUB_REFL; DOT_LZERO]]);;







let DETERMINE_WEDGE_IN_FAN = prove(` local_fan (V,E,FF) /\ x IN FF 
==> wedge_in_fan_ge x E = wedge_ge (vec 0) (FST x) (SND x)
                 (azim_cycle (EE (FST x) E) (vec 0) (FST x) (SND x)) `,
REWRITE_TAC[wedge_in_fan_ge2] THEN COND_CASES_TAC THENL [
REWRITE_TAC[]; PAT_ONCE_REWRITE_TAC`\x. y /\ x ==> z ` [GSYM PAIR] THEN 
NHANH LOCAL_FAN_IMP_IN_V THEN ONCE_REWRITE_TAC[TAUT`(a/\b)/\c/\d <=> (a/\c)/\b/\d`] THEN 
NHANH LOFA_CARD_EE_V_1 THEN 
NHANH (ARITH_RULE` a = 2 ==> a > 1 `) THEN ASM_MESON_TAC[]]);;





let LOCAL_FAN_IMP_IN_V2 = prove(
`local_fan (V,E,FF) /\ x IN FF ==> FST x IN V /\ SND x IN V`,
PAT_ONCE_REWRITE_TAC`\x. y /\ x ==> l` [GSYM PAIR]
THEN PURE_REWRITE_TAC[LOCAL_FAN_IMP_IN_V]);;









let LOFA_DETERMINE_AZIM_IN_FA = prove(` local_fan (V,E,FF) /\ x IN FF 
==> azim_in_fan x E = azim (vec 0) (FST x) (SND x) ( azim_cycle (EE (FST x) E)
 (vec 0) (FST x) (SND x) ) `,
NHANH LOCAL_FAN_IMP_IN_V2 THEN STRIP_TAC THEN 
UNDISCH_TAC` FST (x:real^3 # real^3) IN V` THEN 
UNDISCH_TAC` local_fan (V,E,FF)` THEN PHA THEN 
NHANH LOFA_CARD_EE_V_1 THEN 
NHANH (ARITH_RULE` a = 2 ==> a > 1 `) THEN 
REWRITE_TAC[azim_in_fan2] THEN LET_TAC THEN SIMP_TAC[]);;








let PRIOR_TO_LESS_THAN_PI_LEMMA = prove_by_refinement
(` convex_local_fan (V,E,FF) /\ v IN V 
==> (! w. w IN V ==> azim (vec 0) v (rho_node1 FF v ) w <= 
azim (vec 0) v (rho_node1 FF v)
          (azim_cycle (EE v E) (vec 0) v (rho_node1 FF v)) ) `,
[REWRITE_TAC[convex_local_fan];
NHANH LOCAL_FAN_RHO_NODE_PROS2;
STRIP_TAC;
SUBGOAL_THEN` v, rho_node1 FF v IN FF` ASSUME_TAC;
DOWN;
ASM_REWRITE_TAC[];
DOWN;
UNDISCH_TAC` local_fan (V,E,FF)`;
PHA;
NHANH DETERMINE_WEDGE_IN_FAN;
FIRST_X_ASSUM NHANH;
STRIP_TAC;
UNDISCH_TAC` V SUBSET wedge_in_fan_ge (v,rho_node1 FF v) E `;
REWRITE_TAC[SUBSET];
STRIP_TAC;
FIRST_ASSUM NHANH;
ASM_REWRITE_TAC[wedge_ge; IN_ELIM_THM];
SIMP_TAC[]]);;








let IN_V_IMP_AZIM_LESS_PI = prove_by_refinement
(` convex_local_fan (V,E,FF) /\ v IN V 
==> (! w. w IN V ==> azim (vec 0) v (rho_node1 FF v ) w <= pi ) `,
[NHANH PRIOR_TO_LESS_THAN_PI_LEMMA;
REWRITE_TAC[convex_local_fan];
NHANH LOCAL_FAN_RHO_NODE_PROS2;
STRIP_TAC;
SUBGOAL_THEN` v, rho_node1 FF v IN FF` ASSUME_TAC;
DOWN;
REMOVE_TAC;
DOWN;
ASM_REWRITE_TAC[];
DOWN;
UNDISCH_TAC` local_fan (V,E,FF)`;
PHA;
NHANH LOFA_DETERMINE_AZIM_IN_FA;
FIRST_X_ASSUM NHANH;
DOWN;
FIRST_X_ASSUM NHANH;
REPEAT STRIP_TAC;
ASM_REAL_ARITH_TAC]);;







let NEXT_PRO_IMP_ALLS_STRICT = prove
(`! le. (! i. i + 1 < k ==> le ((f: num -> A) i) (f (i + 1))) /\ (! a b c. le a b /\ le b c ==> le a c ) 
==> (! i j. i < j /\ j < k ==> le ( f i ) ( f j )) `,
GEN_TAC THEN STRIP_TAC THEN GEN_TAC THEN INDUCT_TAC THENL [
REWRITE_TAC[LT]; REWRITE_TAC[ADD1]] THEN 
UNDISCH_TAC` !i. i + 1 < k ==> le ((f:num -> A) i) (f (i + 1)) ` THEN
DISCH_THEN NHANH THEN REWRITE_TAC[ARITH_RULE` a < b + 1 <=> a < b \/ a = b `]
THEN NHANH (ARITH_RULE` a + 1 < b ==> a < b `) THEN ASM_MESON_TAC[]);;






let LOFA_IMP_V_DIFF = prove(` local_fan (V,E,FF) ==>
(! v. v IN V ==> ~( v = vec 0 )) `,
NHANH Wrgcvdr_cizmrrh.LOCAL_FAN_IMP_FAN THEN 
NHANH FAN_IMP_V_DIFF THEN SIMP_TAC[]);;





let SIN_AZIM_POS_PI_LT = prove(
` &0 <= sin (azim x y u v) <=> azim x y u v <= pi `,
REWRITE_TAC[REAL_ARITH` a <= b <=> ~( b < a )`; SIN_AZIM_NEG_PI_LT]);;










let SIN_AZIM_MUTUAL_SROSS = prove(
` ( sin (azim (vec 0) u v w) < &0 <=> (u cross v) dot w < &0 ) /\
( &0 < sin (azim (vec 0) u v w) <=> &0 < (u cross v) dot w )`,
MP_TAC (SPEC_ALL Trigonometry2.JBDNJJB) THEN REWRITE_TAC[re_eqvl] THEN 
STRIP_TAC THEN DOWN_TAC THEN 
SIMP_TAC[REAL_LT_MUL_EQ; REAL_ARITH` a * b < &0 <=> &0 < a * --b `] THEN 
REWRITE_TAC[REAL_ARITH` &0 < -- a <=> a < &0 `]);;



let VECTOR_ADD_LDISTRIB1 = CONJ VECTOR_ADD_LDISTRIB VECTOR_MUL_ASSOC;;


let OPPOSITE_SIDES_IMP_INTER = prove_by_refinement (
` (a:real^N) IN aff_gt {va, vb} {vc} /\ b IN aff_lt {va, vb} {vc}
/\ DISJOINT {va, vb} {vc} ==> ~(conv0 {a,b} INTER aff {va, vb} = {})`,
[NHANH Planarity.AFF_LT_2_1;
NHANH AFF_GT_2_1;
STRIP_TAC;
UNDISCH_TAC` a IN aff_gt {va, vb} {vc: real^N}`;
UNDISCH_TAC` b IN aff_lt {va, vb} {vc: real^N}`;
ASM_REWRITE_TAC[IN_ELIM_THM; Geomdetail.CONV0_SET2;
 Collect_geom.AFF_2POINTS_INTERPRET; INTER_EQ_EM_EXPAND];
REPEAT STRIP_TAC;
EXISTS_TAC` ( -- t3 ) / ( -- t3 + t3') % (a:real^N) + t3' / ( --t3 + t3') % b `;
CONJ_TAC;
EXISTS_TAC` --t3 / (--t3 + t3') `;
EXISTS_TAC` t3' / ( -- t3 + t3' )`;
REWRITE_TAC[ GSYM Geomdetail.SUM_TWO_RATIO];
CONJ_TAC;
MATCH_MP_TAC REAL_LT_DIV;
UNDISCH_TAC` t3 < &0 `;
UNDISCH_TAC` &0 < t3' `;
PHA;
REAL_ARITH_TAC;

CONJ_TAC;
MATCH_MP_TAC REAL_LT_DIV;
UNDISCH_TAC` t3 < &0 `;
UNDISCH_TAC` &0 < t3' `;
PHA;
REAL_ARITH_TAC;


UNDISCH_TAC` t3 < &0 `;
UNDISCH_TAC` &0 < t3' `;
PHA;
REAL_ARITH_TAC;


ASM_REWRITE_TAC[VECTOR_ADD_LDISTRIB; VECTOR_MUL_ASSOC; GSYM (REAL_ARITH` 
( x * a ) / b = ( a / b ) * x `);  REAL_ARITH` -- a * b = -- (a * b)`;
VECTOR_ARITH` ( -- a / b ) % x = -- ((a / b ) % x ) `];
SIMP_TAC[REAL_MUL_SYM];
REWRITE_TAC[VECTOR_ARITH` -- (( a % va) + (b % vb) + x ) + 
( aa % va + bb % vb + x ) = (-- a + aa ) % va + (-- b + bb ) % vb `];



EXISTS_TAC` (--((t1' * t3) / (--t3 + t3')) + (t1 * t3') / (--t3 + t3'))`;
EXISTS_TAC` (--((t2' * t3) / (--t3 + t3')) + (t2 * t3') / (--t3 + t3')) `;
REWRITE_TAC[];


REWRITE_TAC[REAL_POLY_CONV `(--((t1' * t3) / (--t3 + t3')) + 
(t1 * t3') / (--t3 + t3')) + --((t2' * t3) / (--t3 + t3')) +
 (t2 * t3') / (--t3 + t3')`];
REWRITE_TAC[ REAL_ARITH` -- &1 * x * y = ( -- x ) * y `; 
REAL_ARITH` a * b * c = ( a * b ) * c `;
REAL_ARITH` a * x + b * x = (a + b) * x`];
UNDISCH_TAC` t1 + t2 + t3 = &1 `;
UNDISCH_TAC` t1' + t2' + t3' = &1 `;
SIMP_TAC[ REAL_ARITH` a + b = &1 <=> a = &1 - b `; REAL_POLY_CONV` (&1 - (t2 + t3)) * t3' +
      --(&1 - (t2' + t3')) * t3 +
      t2 * t3' +
      --t2' * t3 `];
REPEAT STRIP_TAC;
MATCH_MP_TAC REAL_MUL_RINV;
ASM_REAL_ARITH_TAC]);;




let DISJOINT_DOUBLE_SING = SET_RULE` DISJOINT {a} {b} <=> ~( a = b ) `;;



let AFF_IN_TWO_PARTS = prove_by_refinement (
` aff {x,y:real^N} = aff_ge {x} {y} UNION aff_lt {x} {y}`,
[REWRITE_TAC[HALFLINE];
ASM_CASES_TAC` x = y:real^N`;
ASM_REWRITE_TAC[AFF_XX_CASES; AFF2; UNION_EMPTY; FUN_EQ_THM; IN_ELIM_THM];
GEN_TAC THEN EQ_TAC;
STRIP_TAC;
EXISTS_TAC `&1 `;
ASM_REWRITE_TAC[REAL_ARITH` &0 <= &1 `];
CONV_TAC VECTOR_ARITH;
STRIP_TAC;
EXISTS_TAC` t:real`;
ASM_REWRITE_TAC[VECTOR_ADD_SYM];
DOWN;
REWRITE_TAC[ GSYM DISJOINT_DOUBLE_SING];
SIMP_TAC[AFF_LT_1_1; AFF2; EXTENSION; IN_UNION; IN_ELIM_THM];
STRIP_TAC THEN GEN_TAC;
EQ_TAC;
STRIP_TAC;
ASM_CASES_TAC` &0 <= &1 - t`;
DISJ1_TAC;
EXISTS_TAC` &1 - t `;
ASM_REWRITE_TAC[REAL_ARITH` a - ( a - b ) = b `];
DISJ2_TAC;
EXISTS_TAC` t:real`;
EXISTS_TAC` &1 - t `;
ASM_REWRITE_TAC[];
ASM_REAL_ARITH_TAC;
MESON_TAC[REAL_ARITH` t = a - ( a - t ) `;REAL_ARITH` a + b = c <=> b = c - a `]]);;






let INTER_UNION_EMPTY = SET_RULE` X INTER (A UNION B) = {} <=>
X INTER A = {} /\ X INTER B = {}`




let CONV0_SUB_CONV = prove (
` x IN conv0 S ==> x IN conv S`,
REWRITE_TAC[conv0; Collect_geom.conv; affsign; IN; sgn_gt; sgn_ge]
THEN MESON_TAC[REAL_ARITH` a < b ==> a <= b `]);;





let MONO_AZIM_AS_BTA_I = prove_by_refinement
(`convex_local_fan (V,E,FF) /\
 v0 IN V /\
 CARD V = k /\
 (!v. v IN V /\ ~(v = v0) ==> ~collinear {vec 0, v0, v}) /\
 (!i. ITER i (rho_node1 FF) v0 = vv i) /\
 (!i. azim (vec 0) v0 (vv 1) (vv i) = bta i)
==> (!i j. i < j /\ j < k ==> bta i <= bta j) `,
[STRIP_TAC;
MATCH_MP_TAC NEXT_PRO_IMP_ALLS_STRICT;
REWRITE_TAC[REAL_LE_TRANS];
DOWN_TAC THEN REWRITE_TAC[convex_local_fan];
REPEAT STRIP_TAC;
ASSUME_TAC2 LOFA_IMP_V_DIFF;
UNDISCH_TAC` v0:real^3 IN V `;
FIRST_ASSUM NHANH;
ONCE_REWRITE_TAC[EQ_SYM_EQ];
NHANH Trigonometry2.NOT_EQ_IMP_EXISTS_BASIC;
STRIP_TAC;
MP_TAC (SPECL [` vec 0:real^3 `;` v0:real^3 `;` (vv 1):real^3 `;
`(vv: num -> real^3) i `] azim);
STRIP_TAC;
FIRST_X_ASSUM MP_TAC;
SWITCH_TAC` ~( vec 0 = v0:real^3 )`;
DISCH_THEN ASSUME_TAC2;
DOWN;
STRIP_TAC;
MP_TAC (SPECL [` vec 0:real^3 `;` v0:real^3 `;` (vv 1):real^3 `;
`(vv: num -> real^3) (i +1) `] azim);
STRIP_TAC;
FIRST_X_ASSUM ASSUME_TAC2;
DOWN THEN STRIP_TAC;
DOWN_TAC;
REWRITE_TAC[VECTOR_SUB_RZERO];
STRIP_TAC;


UNDISCH_TAC` vv 1 = (r1' * cos psi') % e1 + (r1' * sin psi') % e2 + h1' % (v0:real^3)`;
UNDISCH_TAC` vv 1 = (r1 * cos psi) % e1 + (r1 * sin psi) % e2 + h1 % v0:real^3`;
UNDISCH_TAC` orthonormal e1 e2 e3`;
PHA;
EXPAND_TAC "v0";
REWRITE_TAC[VECTOR_MUL_ASSOC];
NHANH (MESON[Trigonometry2.th]` orthonormal e1 e2 e3 /\
                      x = t1 % e1 + t2 % e2 + t3 % e3 /\
                      x = tt1 % e1 + tt2 % e2 + tt3 % e3
                           ==> tt1 = t1 /\ tt2 = t2 /\ tt3 = t3`);
STRIP_TAC;



SUBGOAL_THEN` ~collinear {vec 0, v0, vv 1:real^3}` ASSUME_TAC;
UNDISCH_TAC` !v. v IN V /\ ~(v = v0) ==> ~collinear {vec 0, v0, v:real^3} `;
DISCH_THEN MATCH_MP_TAC;
SWITCH_TAC` !i. ITER i (rho_node1 FF) v0 = vv i`;
FIRST_ASSUM (fun x -> REWRITE_TAC[x]);
ASSUME_TAC2 LOCAL_FAN_ORBIT_MAP_VITER;
CONJ_TAC;
FIRST_X_ASSUM MATCH_MP_TAC;
FIRST_ASSUM ACCEPT_TAC;

ASSUME_TAC2 LOCAL_FAN_CHARACTER_OF_RHO_NODE;
REWRITE_TAC[ITER12];
UNDISCH_TAC` v0 IN (V:real^3 -> bool)`;
FIRST_X_ASSUM NHANH;
SIMP_TAC[];

USE_FIRST` ~collinear {vec 0, v0, vv 1:real^3} ==> &0 < r1 ` ASSUME_TAC2;
USE_FIRST` ~collinear {vec 0, v0, vv 1:real^3} ==> &0 < r1' ` ASSUME_TAC2;

ASSUME_TAC2 (SPECL [` r1': real`;` r1: real`;` psi': real`; `psi:real` ]
 Trigonometry2.R_POS_SIN_COS_IDENT);
DOWN;
REWRITE_TAC[Trigonometry2.SIN_COS_IDEN_IFF_DIFFER_PERS];
STRIP_TAC;



SUBGOAL_THEN `vv (i:num) IN (V:real^3 -> bool) ` ASSUME_TAC;
SWITCH_TAC` !i. ITER i (rho_node1 FF) v0 = vv i `;
ASM_REWRITE_TAC[];
ASSUME_TAC2 LOCAL_FAN_ORBIT_MAP_VITER;
FIRST_X_ASSUM MATCH_MP_TAC;
FIRST_ASSUM ACCEPT_TAC;

DOWN;
SUBGOAL_THEN` convex_local_fan (V,E,FF) ` MP_TAC;
ASM_REWRITE_TAC[convex_local_fan];





PHA;
NHANH IN_V_IMP_AZIM_LESS_PI;
STRIP_TAC;
UNDISCH_TAC` v0:real^3 IN V`;
FIRST_ASSUM NHANH;



SUBGOAL_THEN` rho_node1 FF (vv (i:num)) = vv ( i + 1 ) ` ASSUME_TAC;
USE_FIRST` !i. ITER i (rho_node1 FF) v0 = vv i ` (fun x -> REWRITE_TAC[GSYM x]);
REWRITE_TAC[GSYM ADD1; ITER];
FIRST_ASSUM SUBST1_TAC;
REWRITE_TAC[SYM SIN_AZIM_POS_PI_LT];
REWRITE_TAC[REAL_ARITH`  &0 <= a <=> ~( a < &0 ) `;SIN_AZIM_MUTUAL_SROSS ];

ABBREV_TAC` va = azim (vec 0) v0 (vv 1) (vv i) `;
ABBREV_TAC` vb = azim (vec 0) v0 (vv 1) (vv (i + 1 )) `;
ASM_REWRITE_TAC[];

ONCE_REWRITE_TAC[REAL_ADD_SYM];
REWRITE_TAC[Trigonometry2.SIN_COS_PERIODIC_IN_WHOLE; REAL_ARITH` a + b + c = (a + b ) + c `];
EXPAND_TAC "v0";
REWRITE_TAC[CROSS_LADD; CROSS_RADD; VECTOR_MUL_ASSOC; CROSS_LMUL; CROSS_RMUL;
 CROSS_REFL; VECTOR_MUL_RZERO; VECTOR_ADD_RID; DOT_LADD; DOT_LMUL;
 DOT_RMUL;DOT_CROSS_SELF; Collect_geom.ZERO_NEUTRAL];
STRIP_TAC;
DOWN;
REWRITE_TAC[DOT_LZERO; Collect_geom.ZERO_NEUTRAL];
ASSUME_TAC2 ORTHONORMAL_CROSS;
ASM_REWRITE_TAC[];
ONCE_REWRITE_TAC[CROSS_SKEW];
ASM_REWRITE_TAC[];

SUBGOAL_THEN` e3 dot (e3: real^3) = &1 ` ASSUME_TAC;
UNDISCH_TAC` orthonormal e1 e2 e3 `;
SIMP_TAC[orthonormal];


ASM_REWRITE_TAC[DOT_LNEG; REAL_RING` (r2 * a) * (r2' * b) * &1 +
    (r2 * aa) * (r2' * bb) * -- &1
= r2 * r2' * (b * a - bb * aa ) `; GSYM SIN_SUB];
REWRITE_TAC[REAL_ARITH` (a + b ) - ( c + b ) = a - c `];





SUBGOAL_THEN` &0 < dist (v0, vec 0:real^3)` ASSUME_TAC;
ASM_REWRITE_TAC[GSYM DIST_NZ];
ASM_CASES_TAC` i = 0 `;
USE_FIRST` !i. azim (vec 0) v0 (vv 1) (vv i) = bta (i:num)` 
(fun x -> REWRITE_TAC[GSYM x]);

USE_FIRST` i = 0 ` SUBST1_TAC;
USE_FIRST` !i. ITER i (rho_node1 FF) v0 = vv i` (fun x -> REWRITE_TAC[GSYM x]);
REWRITE_TAC[ITER; ITER12; GSYM ADD1; AZIM_SPEC_DEGENERATE; AZIM_REFL; REAL_LE_REFL];



SWITCH_TAC` vv (i:num) =
      (r2 * cos (psi + va)) % e1 + (r2 * sin (psi + va)) % e2 + h2 % (v0:real^3)`;
SWITCH_TAC` vv (i + 1) =
      (r2' * cos (psi' + vb)) % e1 + (r2' * sin (psi' + vb)) % e2 + h2' % (v0:real^3)`;
SWITCH_TAC` psi' = psi + real_of_int k' * &2 * pi `;


SUBGOAL_THEN` ~( vv i = v0:real^3 ) /\ ~(vv (i + 1 ) = v0 ) ` ASSUME_TAC;
UNDISCH_TAC` dist (v0:real^3,vec 0) % e3 IN (V:real^3 -> bool)`;
ASM_REWRITE_TAC[];
STRIP_TAC;
ASSUME_TAC2 (SPEC `v0: real^3 ` (GEN `v:real^3 ` LOFA_IMP_DIS_ELMS23));

FIRST_ASSUM (MP_TAC o (SPECL [`0`;` i:num`]));

PHA;
ANTS_TAC;
ASM_REWRITE_TAC[ARITH_RULE` 0 < i <=> ~( i = 0 ) `];
UNDISCH_TAC` i + 1 < k `;
ARITH_TAC;



FIRST_ASSUM (MP_TAC o (SPECL [`0`;` i + 1`]));
REWRITE_TAC[ARITH_RULE` 0 < i + 1 `];
ASM_REWRITE_TAC[];
USE_FIRST` !i. ITER i (rho_node1 FF) v0 = vv i ` (fun x -> REWRITE_TAC[GSYM x]);
SIMP_TAC[ITER];


STRIP_TAC;
UNDISCH_TAC` (vv:num -> real^3) i IN V `;
UNDISCH_TAC` local_fan (V,E,FF) `;
PHA;
NHANH LOFA_IN_V_SO_DO_RHO_NODE_V;
ASM_REWRITE_TAC[];
DOWN THEN DOWN THEN PHA THEN STRIP_TAC;
USE_FIRST` !v. v IN V /\ ~(v = v0) ==> ~collinear {vec 0, v0, v:real^3}` 
(ASSUME_TAC2 o (SPEC` (vv:num -> real^3) i `));

USE_FIRST` !v. v IN V /\ ~(v = v0) ==> ~collinear {vec 0, v0, v:real^3}` 
(ASSUME_TAC2 o (SPEC` (vv:num -> real^3) (i + 1) `));
USE_FIRST` ~collinear {vec 0, v0, (vv: num -> real^3) i} ==> &0 < r2 ` ASSUME_TAC2;
USE_FIRST` ~collinear {vec 0, v0, (vv: num -> real^3) (i + 1 )} ==> &0 < r2' ` ASSUME_TAC2;

UNDISCH_TAC` ~(dist (v0:real^3,vec 0) * r2 * r2' * sin (vb - va) < &0) `;
ASM_SIMP_TAC[REAL_LE_MUL_EQ; REAL_ARITH` ~( a < b ) <=> b <= a `];



MP_TAC (SPEC` v0:real^3 ` (GEN` v:real^3 ` IN_V_IMP_AZIM_LESS_PI));
ANTS_TAC;
ASM_REWRITE_TAC[];
UNDISCH_TAC` dist (v0:real^3,vec 0) % (e3:real^3) IN V`;
ASM_REWRITE_TAC[];
DISCH_TAC;
FIRST_ASSUM (MP_TAC2 o (SPEC` (vv:num -> real^3) i `));
FIRST_ASSUM (MP_TAC2 o (SPEC` (vv:num -> real^3) (i + 1 ) `));
ASM_REWRITE_TAC[];

SUBGOAL_THEN` rho_node1 FF v0 = vv 1` ASSUME_TAC;
SWITCH_TAC ` !i. ITER i (rho_node1 FF) v0 = vv i `;
DOWN;
SIMP_TAC[ITER12];


FIRST_ASSUM (fun x -> REWRITE_TAC[x]);
USE_FIRST` !i. azim (vec 0) v0 (vv 1) (vv i) = bta i ` (fun x -> REWRITE_TAC[GSYM x]);
MP_TAC (SPECL [`vec 0:real^3 `;` v0:real^3 `;` (vv:num -> real^3) 1`;
` (vv:num -> real^3) i `] AZIM_RANGE);

MP_TAC (SPECL [`vec 0:real^3 `;` v0:real^3 `;` (vv:num -> real^3) 1`;
` (vv:num -> real^3) (i + 1) `] AZIM_RANGE);
REWRITE_TAC[ASSUME` azim (vec 0) v0 (vv 1) (vv (i + 1)) = vb `; ASSUME` azim (vec 0) v0 (vv 1) (vv i) = va `];

PHA;
NHANH (REAL_ARITH` &0 <= vb /\
 vb < &2 * pi /\
 &0 <= va /\
 va < &2 * pi /\
 vb <= pi /\
 va <= pi /\
 &0 <= sin (vb - va) ==> vb - va <= pi /\ -- pi <= vb - va `);
STRIP_TAC;



ASM_CASES_TAC ` ~(vb - va = -- pi) `;
ASM_CASES_TAC`  vb - va < &0 `;
REPLICATE_TAC 3 DOWN THEN PHA;
REWRITE_TAC[REAL_ARITH` a <= b /\ ~( b = a ) /\ x < &0 <=> a < b /\ x < &0`];
STRIP_TAC;
ASSUME_TAC2 (SPEC ` vb - (va:real)` Trigonometry1.SIN_NEGPOS_PI);
SUBGOAL_THEN` sin (vb - va ) < &0 ` ASSUME_TAC;
ASM_REWRITE_TAC[];
ABBREV_TAC` xx = vb - va:real`;
UNDISCH_TAC` sin xx < &0 `;
UNDISCH_TAC` &0 <= sin xx `;
REAL_ARITH_TAC;

DOWN;
REAL_ARITH_TAC;

UNDISCH_TAC` &0 <= vb `;
UNDISCH_TAC` va <= pi `;
DOWN THEN PHA;
REWRITE_TAC[REAL_ARITH` vb - va = --pi /\ va <= pi /\ &0 <= vb <=> 
vb = &0 /\ va = pi`];
STRIP_TAC;

UNDISCH_TAC` ~ collinear {vec 0, v0, vv (i + 1):real^3}`;
UNDISCH_TAC` ~ collinear {vec 0, v0, vv 1:real^3}`;
PHA;
NHANH AZIM_EQ_0_ALT;
STRIP_TAC;
UNDISCH_TAC` ~ collinear {vec 0, v0, vv (i: num) :real^3}`;
UNDISCH_TAC` ~ collinear {vec 0, v0, vv 1:real^3}`;
PHA;
NHANH AZIM_EQ_PI_ALT;
STRIP_TAC;
SWITCH_TAC` vv 1:real^3 =
      (r1 * cos psi) % e1 + (r1 * sin psi) % e2 + (h1 * dist (v0:real^3,vec 0)) % e3`;
SWITCH_TAC`vv 1: real^3 =
      (r1' * cos psi') % e1 +
      (r1' * sin psi') % e2 +
      (h1' * dist (v0: real^3,vec 0)) % e3`;
UNDISCH_TAC` va = pi `;
UNDISCH_TAC` vb = &0 `;
UNDISCH_TAC` azim (vec 0) v0 (vv 1) (vv (i + 1)) = &0 <=>
      vv (i + 1) IN aff_gt {vec 0, v0} {vv 1}`;
UNDISCH_TAC`azim (vec 0) v0 (vv 1) (vv i) = pi <=>
      vv i IN aff_lt {vec 0, v0} {vv 1} `;
ASM_SIMP_TAC[];
ASSUME_TAC2 (
ISPECL [` vec 0: real^3`;` v0: real^3`;` (vv:num -> real^3) 1` ] Fan.th3a) ;

REPEAT STRIP_TAC;
ASSUME_TAC2 (
ISPECL [` (vv:num -> real^3) 1 `;`  (vv:num -> real^3) (i + 1)`;
`  (vv:num -> real^3) i`;`vec 0: real^3`; ` v0: real^3`] (GEN_ALL OPPOSITE_SIDES_IMP_INTER));
DOWN;
REWRITE_TAC[AFF_IN_TWO_PARTS; INTER_UNION_EMPTY; DE_MORGAN_THM];
STRIP_TAC;
ASSUME_TAC (ISPECL [`{vec 0:real^3}`;`{(vv: num -> real^3) (i + 1), vv i }`]

 (GEN_ALL CONV_SUBSET_AFF_GE));
(* ****************** *)

ASSUME_TAC (
ISPECL [`(vv: num -> real^3) (i + 1)`;` (vv:num -> real^3) i `]
Geomdetail.CONV02_SU_CONV2);
ASSUME_TAC2 LOCAL_FAN_RHO_NODE_PROS2;
DOWN THEN STRIP_TAC;
UNDISCH_TAC` (vv:num -> real^3) i IN V`;
FIRST_ASSUM NHANH;
STRIP_TAC;
DOWN;
UNDISCH_TAC ` local_fan (V,E,FF)`;
PHA;
NHANH LOCAL_FAN_IN_FF_IN_ORD_PAIRS2;
ASM_REWRITE_TAC[];
NHANH Wrgcvdr_cizmrrh.LOCAL_FAN_IMP_FAN;
REWRITE_TAC[FAN; fan7];
STRIP_TAC;
DOWN THEN DOWN;
FIRST_ASSUM (ASSUME_TAC o (ISPECL [`{v0:real^3}`;`{(vv: num -> real^3) i, 
vv (i + 1)}`]));
STRIP_TAC;
SUBGOAL_THEN `{v0: real^3} IN E UNION {{v} | v IN V}` MP_TAC;

UNDISCH_TAC`dist (v0: real^3,vec 0) % (e3:real^3) IN V`;
ASM_REWRITE_TAC[IN_UNION; IN_ELIM_THM];
MESON_TAC[];


REPEAT STRIP_TAC;
UNDISCH_TAC` {v0:real^3} IN E UNION {{v} | v IN V} /\
      {vv i, vv (i + 1)} IN E UNION {{v} | v IN V}
      ==> aff_ge {vec 0} {v0} INTER aff_ge {vec 0} {vv i, vv (i + 1)} =
          aff_ge {vec 0} ({v0} INTER {vv i, vv (i + 1)}) `;
ANTS_TAC;
DOWN THEN DOWN;
PHA;
SIMP_TAC[IN_UNION];

SUBGOAL_THEN `{v0: real^3} INTER {vv i, vv (i + 1)} = {}` SUBST1_TAC;
UNDISCH_TAC` ~(vv (i:num) = v0:real^3)`;
UNDISCH_TAC` ~(vv (i + 1) = v0:real^3)`;
REWRITE_TAC[Trigonometry2.INSERT_INTER_EMPTY; IN_INSERT; NOT_IN_EMPTY];
MESON_TAC[];

REWRITE_TAC[AFF_GE_EQ_AFFINE_HULL; AFFINE_HULL_SING];
UNDISCH_TAC` conv {vv (i + 1), (vv: num -> real^3) i} SUBSET 
aff_ge {vec 0} {vv (i + 1), vv i}`;
UNDISCH_TAC` conv0 {vv (i + 1), (vv: num -> real^3) i} SUBSET
conv {vv (i + 1), (vv: num -> real^3) i}`;
PHA;
SIMP_TAC[INSERT_COMM];
NHANH (
SET_RULE`a SUBSET b /\ b SUBSET c /\ x INTER c = S ==> x INTER a SUBSET S`);
STRIP_TAC;
DOWN;
UNDISCH_TAC` ~(conv0 {vv (i + 1), vv i} INTER aff_ge {vec 0} {v0:real^3} = {})`;
PHA;
SIMP_TAC[INSERT_COMM; INTER_COMM; SET_RULE` ~( s = {}) /\ s SUBSET {x} <=> 
s = {x}`];

NHANH (SET_RULE` A INTER B = {x} ==> x IN B`);
NHANH CONV0_SUB_CONV;
STRIP_TAC;
SUBGOAL_THEN` collinear {vec 0, (vv:num -> real^3) i, vv (i + 1)}` ASSUME_TAC;
ASM_REWRITE_TAC[Collect_geom.COLLINEAR_AS_IN_CONV2];
ASSUME_TAC2 (SPEC `(vv:num -> real^3 ) i ` (GEN `v: real^3`
 LOCAL_FAN_CHARACTER_OF_RHO_NODE2));
DOWN;
ASM_REWRITE_TAC[];

SUBGOAL_THEN` circular (V:real^3 -> bool) E ` MP_TAC;
REWRITE_TAC[circular];
EXISTS_TAC` (vv: num -> real^3) (i + 1) `;
EXISTS_TAC` (vv: num -> real^3) i`;
EXISTS_TAC` v0: real^3`;



ASSUME_TAC (ISPECL [`{vec 0:real^3}`;` {(vv:num -> real^3) (i + 1), vv i}`]
 (GEN_ALL CONV0_SUBSET_AFF_GT));
ASSUME_TAC2 LOCAL_FAN_CHARACTER_OF_RHO_NODE;
UNDISCH_TAC` (vv: num -> real^3) i IN V `;
FIRST_ASSUM (NHANH_PAT`\x. x ==> l`);
ASM_SIMP_TAC[ GSYM Wrgcvdr_cizmrrh.IN_E_IFF_IN_ORD_E; INSERT_COMM];
STRIP_TAC;
UNDISCH_TAC` dist (v0:real^3,vec 0) % (e3:real^3) IN V`;
ASM_SIMP_TAC[];
STRIP_TAC;
UNDISCH_TAC` conv0 {vv (i + 1), (vv i):real^3} SUBSET aff_gt {vec 0} {vv (i + 1), vv i}`;
UNDISCH_TAC` ~(conv0 {vv (i + 1), vv i} INTER aff_lt {vec 0} {v0:real^3} = {})`;
PHA;
SIMP_TAC[INSERT_COMM];
CONV_TAC SET_RULE;

STRIP_TAC;
ASSUME_TAC2 KCHMAMG;
DOWN THEN STRIP_TAC;



ABBREV_TAC `U = {ITER n (rho_node1 FF) v0 | n <= i + 1} `;
ABBREV_TAC` ee = v0 cross rho_node1 FF v0`;
MP_TAC (SPECL [`v0: real^3`;`A:real^3 -> bool`;` ee:real^3`;`i + 1`]
 (GENL [` v:real^3 `;`P:real^3 -> bool`;`e:real^3`;`l: num`]
 SEQUENCE_OF_RHO_NODE_IS_SUC));
ANTS_TAC;
ASM_REWRITE_TAC[];
UNDISCH_TAC` dist (v0:real^3,vec 0) % (e3:real^3) IN V`;
ASM_SIMP_TAC[];
STRIP_TAC;
MATCH_MP_TAC SUBSET_TRANS;
EXISTS_TAC`V:real^3 -> bool`;
ASM_REWRITE_TAC[];
EXPAND_TAC "U";
REWRITE_TAC[SUBSET; IN_ELIM_THM];
ASSUME_TAC2 LOCAL_FAN_ORBIT_MAP_VITER;
DOWN THEN DOWN;
MESON_TAC[];



DISCH_THEN (MP_TAC o (SPECL [`ITER i (rho_node1 FF) v0`;` i:num`]));
ANTS_TAC;
REWRITE_TAC[ARITH_RULE` a < a + 1 `];
DISCH_THEN (MP_TAC o (SPEC` v0:real^3`));
ANTS_TAC;
EXPAND_TAC "U";
ASM_REWRITE_TAC[IN_ELIM_THM];
EXISTS_TAC`0`;
UNDISCH_TAC` ! i. ITER i (rho_node1 FF) v0 = vv i`;
PAT_ONCE_REWRITE_TAC`\x. x ==> y ` [EQ_SYM_EQ];
SIMP_TAC[ITER; LE_0];

ASM_REWRITE_TAC[];
SUBGOAL_THEN` &0 < sin ( azim (vec 0) ee v0 (vv (i + 1)))` ASSUME_TAC;
REWRITE_TAC[SIN_AZIM_MUTUAL_SROSS];
UNDISCH_TAC` vv (i + 1) IN aff_gt {vec 0, v0: real^3} {vv 1}`;
ASM_SIMP_TAC[AFF_GT_2_1; IN_ELIM_THM];
STRIP_TAC;
ONCE_REWRITE_TAC[CROSS_TRIPLE];
ASM_REWRITE_TAC[CROSS_RADD; CROSS_RMUL; CROSS_REFL; CROSS_0; VECTOR_MUL_RZERO;
 VECTOR_ADD_LID];
UNDISCH_TAC` v0 cross rho_node1 FF v0 = ee`;
ASM_SIMP_TAC[DOT_LMUL];
STRIP_TAC;
MATCH_MP_TAC REAL_LT_MUL;
ASM_REWRITE_TAC[DOT_POS_LT];
EXPAND_TAC "ee";
ASM_REWRITE_TAC[CROSS_EQ_0];
DOWN;
NHANH (REAL_ARITH` &0 < x ==> &0 <= x `);
REWRITE_TAC[SIN_AZIM_POS_PI_LT];
PHA;
NHANH (REAL_ARITH` a <= h /\ b < a ==> b <= h `);
REWRITE_TAC[GSYM SIN_AZIM_POS_PI_LT];
REWRITE_TAC[REAL_ARITH` a <= b <=> ~( b < a)`; SIN_AZIM_MUTUAL_SROSS];
STRIP_TAC;
DOWN;
UNDISCH_TAC` (vv:num -> real^3) i IN aff_lt {vec 0, v0} {vv 1}`;
ASM_SIMP_TAC[Planarity.AFF_LT_2_1; IN_ELIM_THM];
STRIP_TAC;
ONCE_REWRITE_TAC[CROSS_TRIPLE];
ASM_REWRITE_TAC[CROSS_RADD; CROSS_RMUL; CROSS_0; CROSS_REFL; VECTOR_MUL_RZERO;
 VECTOR_ADD_LID; DOT_LMUL];
UNDISCH_TAC` v0 cross rho_node1 FF v0 = ee`;
ASM_SIMP_TAC[];
STRIP_TAC;
SUBGOAL_THEN` &0 < ee dot (ee:real^3)` MP_TAC;
REWRITE_TAC[DOT_POS_LT];
EXPAND_TAC "ee";
REWRITE_TAC[CROSS_EQ_0];
FIRST_X_ASSUM ACCEPT_TAC;
SIMP_TAC[REAL_LE_MUL_EQ; REAL_ARITH` ~( a < b) <=> b <= a`];
ASM_SIMP_TAC[REAL_ARITH` a <= b <=> ~(b < a )`]]);;







let SUCCESSIVE_RHO_NODE1_AFF_LT = prove_by_refinement
(`  local_fan (V,E,FF) /\
     v IN V /\
     {ITER n (rho_node1 FF) v | n <= l} = U /\
     plane P /\
     vec 0 IN P /\
     U SUBSET P
==> ! i. 0 < i /\ i < l 
         ==> ITER (i - 1) (rho_node1 FF) v IN aff_lt {vec 0, ITER i (rho_node1 FF) v}  {ITER (i + 1 ) (rho_node1 FF) v }`,
[STRIP_TAC;
ABBREV_TAC` e = v cross rho_node1 FF v `;
ASSUME_TAC2 RHO_NODE_SET_IN_A_PLANE_IMP_POS_DIRECT;
GEN_TAC;
FIRST_ASSUM (MP_TAC o (SPEC`i - 1 `));
FIRST_ASSUM (MP_TAC o (SPEC`i:num `));
REPEAT STRIP_TAC;
ASSUME_TAC2 LOCAL_FAN_ORBIT_MAP_VITER;
FIRST_ASSUM (MP_TAC o (SPECL [`v:real^3`;` i:num`]));
ANTS_TAC;
FIRST_X_ASSUM ACCEPT_TAC;
STRIP_TAC;
ASSUME_TAC2 
(SPEC` ITER i (rho_node1 FF) v ` 
(GEN` v: real^3` LOCAL_FAN_CHARACTER_OF_RHO_NODE2));
DOWN;
SUBGOAL_THEN` {vec 0, ITER i (rho_node1 FF) v, rho_node1 FF (ITER i (rho_node1 FF) v)} SUBSET P` MP_TAC;
ASM_REWRITE_TAC[INSERT_SUBSET];
REWRITE_TAC[GSYM INSERT_SUBSET];
MATCH_MP_TAC SUBSET_TRANS;
EXISTS_TAC` U:real^3 -> bool`;
ASM_REWRITE_TAC[];
EXPAND_TAC "U";
REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET; IN_ELIM_THM; GSYM ITER];
UNDISCH_TAC` i < l:num`;
MESON_TAC[ARITH_RULE` a < b:num ==> a <= b /\ SUC a <= b `];
UNDISCH_TAC` plane  (P:real^3 -> bool)`;
PHA;
NHANH THREE_NOT_COLL_DETER_PLANE;
STRIP_TAC;
SUBGOAL_THEN` ITER (i - 1) (rho_node1 FF) v IN P` ASSUME_TAC;
UNDISCH_TAC` U SUBSET (P:real^3 -> bool)`;
REWRITE_TAC[SUBSET];
DISCH_THEN MATCH_MP_TAC;
EXPAND_TAC "U";
REWRITE_TAC[IN_ELIM_THM];
EXISTS_TAC` i - 1 `;
UNDISCH_TAC` i < l:num`;
REWRITE_TAC[];
CONV_TAC ARITH_RULE;

DOWN;
EXPAND_TAC "P";
REWRITE_TAC[AFFINE_HULL_3; IN_ELIM_THM];
STRIP_TAC;
SUBGOAL_THEN` w < &0 ` ASSUME_TAC;
UNDISCH_TAC` i < l:num`;
NHANH (ARITH_RULE` a < b ==> a - 1 < b `);
UNDISCH_TAC` i < l
      ==> &0 <
          (ITER i (rho_node1 FF) v cross ITER (i + 1) (rho_node1 FF) v) dot e`;
DISCH_THEN NHANH;
UNDISCH_TAC` i - 1 < l
      ==> &0 <
          (ITER (i - 1) (rho_node1 FF) v cross
           ITER (i - 1 + 1) (rho_node1 FF) v) dot
          e`;
DISCH_THEN NHANH;
ASM_SIMP_TAC[ARITH_RULE` 0 < a ==> a - 1 + 1 = a `; GSYM ITER; ADD1;
 CROSS_LADD; CROSS_LMUL; CROSS_0; CROSS_REFL];
REWRITE_TAC[VECTOR_MUL_RZERO; VECTOR_ADD_LID];
STRIP_TAC;
DOWN;
ONCE_REWRITE_TAC[CROSS_SKEW];
REWRITE_TAC[VECTOR_MUL_RNEG];
REWRITE_TAC[GSYM VECTOR_MUL_LNEG; DOT_LMUL];
ASM_SIMP_TAC[REAL_LT_MUL_EQ];
REAL_ARITH_TAC;
UNDISCH_TAC`~collinear
       {vec 0, ITER i (rho_node1 FF) v, rho_node1 FF
                                        (ITER i (rho_node1 FF) v )}`;
NHANH Fan.th3a;
NHANH AFF_LT_2_1;
SIMP_TAC[ITER; GSYM ADD1; IN_ELIM_THM];
STRIP_TAC;
EXISTS_TAC` u:real`;
EXISTS_TAC` v': real`;
EXISTS_TAC`w:real`;
ASM_REWRITE_TAC[]]);;







let IN_OPOSITE_SIDE_IMP_INTER = prove(
` b IN aff_lt {va, vb} {vc} /\
     DISJOINT {va, vb} {vc}
     ==> ~(conv0 {vc, b} INTER aff {va, vb} = {})`,
NHANH DISJOINT_IMP_Z_IN_AFF_GT THEN STRIP_TAC THEN 
MATCH_MP_TAC OPPOSITE_SIDES_IMP_INTER THEN ASM_REWRITE_TAC[]);;







let TWO_SIDES_SUCESSIVE = prove_by_refinement
(`  local_fan (V,E,FF) /\
     v IN V /\
     {ITER n (rho_node1 FF) v | n <= l} = U /\
     plane P /\
     vec 0 IN P /\
     U SUBSET P
==> ! i. 0 < i /\ i < l 
         ==> ~( conv0 { ITER (i - 1) (rho_node1 FF) v, ITER (i + 1 ) (rho_node1 FF) v } INTER aff {vec 0, ITER i (rho_node1 FF) v} = {} )`,
[NHANH SUCCESSIVE_RHO_NODE1_AFF_LT;
STRIP_TAC;
FIRST_ASSUM NHANH;
ASSUME_TAC2 LOCAL_FAN_ORBIT_MAP_VITER;
GEN_TAC;
FIRST_ASSUM (ASSUME_TAC2 o (SPECL[`v:real^3`;`i:num`]));
ASSUME_TAC2 (SPEC`ITER i (rho_node1 FF) v ` (GEN `v:real^3 `
 LOCAL_FAN_CHARACTER_OF_RHO_NODE2));
DOWN THEN NHANH Fan.th3a;
STRIP_TAC THEN STRIP_TAC;
ONCE_REWRITE_TAC[INSERT_COMM];
MATCH_MP_TAC IN_OPOSITE_SIDE_IMP_INTER;
ONCE_REWRITE_TAC[INSERT_COMM];
ASM_REWRITE_TAC[GSYM ADD1; ITER]]);;






let LOCAL_FAN_CHARACTER_OF_RHO_NODE3 = prove(
`local_fan (V,E,FF) /\ v IN V ==>
! i. ~ collinear {vec 0, ITER i (rho_node1 FF) v, ITER (i + 1) (rho_node1 FF) v}`,
NHANH LOCAL_FAN_ORBIT_MAP_VITER THEN STRIP_TAC THEN GEN_TAC THEN 
FIRST_X_ASSUM (ASSUME_TAC2 o (SPECL [`v:real^3`;` i:num`])) THEN 
REWRITE_TAC[GSYM ADD1; ITER] THEN 
MATCH_MP_TAC LOCAL_FAN_CHARACTER_OF_RHO_NODE2 THEN 
ASM_REWRITE_TAC[]);;







let CNVX_IMP_INTERIOR_ANGLE_PI = prove_by_refinement
(` convex_local_fan (V,E,FF) /\
     v IN V /\
     {ITER n (rho_node1 FF) v | n <= l} = U /\
     plane P /\
     vec 0 IN P /\
     U SUBSET P
==> ! i. 0 < i /\ i < l 
         ==> interior_angle1 (vec 0) FF ( ITER i (rho_node1 FF) v) = pi `,
[STRIP_TAC;
ABBREV_TAC` e = v cross rho_node1 FF v `;
ASSUME_TAC2 CVX_LO_IMP_LO;
ASSUME_TAC2 TWO_SIDES_SUCESSIVE;
GEN_TAC;
DOWN THEN PHA;
STRIP_TAC;
DOWN THEN DOWN THEN PHA;
FIRST_ASSUM NHANH;
ONCE_REWRITE_TAC[INTER_COMM];
STRIP_TAC;
SUBGOAL_THEN` {ITER (i + 1) (rho_node1 FF) v, ITER (i - 1) (rho_node1 FF) v}
 INTER aff {vec 0, ITER i (rho_node1 FF) v} = {} ` ASSUME_TAC;
ASSUME_TAC2 LOCAL_FAN_ORBIT_MAP_VITER;
FIRST_ASSUM (MP_TAC o (SPECL [`v:real^3`;`i - 1 `]));
FIRST_ASSUM (MP_TAC o (SPECL [`v:real^3`;`i:num `]));
REPLICATE_TAC 2 (DISCH_THEN ASSUME_TAC2);
ASSUME_TAC2 LOCAL_FAN_CHARACTER_OF_RHO_NODE3;
FIRST_ASSUM (MP_TAC o (SPEC`i - 1 `));
FIRST_ASSUM (MP_TAC o (SPEC`i:num`));
STRIP_TAC;
ASSUME_TAC2 (ARITH_RULE`0 < i ==> i - 1 + 1 = i`);
DOWN;
SIMP_TAC[Collect_geom.PER_SET2];
DOWN;
NHANH Fan.th3c;
SIMP_TAC[Collect_geom.PER_SET2];
REPEAT STRIP_TAC;
ABBREV_TAC` a1 = ITER (i - 1) (rho_node1 FF) v`;
ABBREV_TAC` a2 = ITER i (rho_node1 FF) v`;
ABBREV_TAC` a3 = ITER (i + 1) (rho_node1 FF) v`;
UNDISCH_TAC` ~(a3 IN aff {a2, vec 0:real^3})`;
UNDISCH_TAC` ~(a1 IN aff {a2, vec 0: real^3})`;
SET_TAC[];

ABBREV_TAC` w = ITER (i - 1) (rho_node1 FF) v`;
ABBREV_TAC` vv = ITER i (rho_node1 FF) v`;
ABBREV_TAC` u = ITER (i + 1) (rho_node1 FF) v`;
SUBGOAL_THEN` interior_angle1 (vec 0) FF vv = pi /\
             rho_node1 FF vv IN P /\
             ivs_rho_node1 FF vv IN P ` MP_TAC;
MATCH_MP_TAC OZQVSFF;
EXISTS_TAC` u: real^3`;
EXISTS_TAC` w:real^3`;
DOWN_TAC;
SIMP_TAC[Collect_geom.PER_SET2];
STRIP_TAC;
ASSUME_TAC2 LOCAL_FAN_ORBIT_MAP_VITER;
CONJ_TAC;
REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET];
FIRST_ASSUM (ASSUME_TAC2 o (SPECL [`v:real^3`;` i - 1 `]));
FIRST_ASSUM (ASSUME_TAC2 o (SPECL [`v:real^3`;` i: num `]));
FIRST_ASSUM (ASSUME_TAC2 o (SPECL [`v:real^3`;` i + 1 `]));
REPLICATE_TAC 3 DOWN THEN PHA;
EXPAND_TAC "u";
EXPAND_TAC "vv";
EXPAND_TAC "w";
SIMP_TAC[];

ONCE_REWRITE_TAC[INSERT_SUBSET];
ASM_REWRITE_TAC[];
MATCH_MP_TAC SUBSET_TRANS;
EXISTS_TAC`U:real^3 -> bool`;
ASM_REWRITE_TAC[];
REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET];
EXPAND_TAC "U";
EXPAND_TAC "u";
EXPAND_TAC "vv";
EXPAND_TAC "w";
REWRITE_TAC[IN_ELIM_THM];
UNDISCH_TAC` i < l:num`;
NHANH (ARITH_RULE` a < b:num ==> a <= b /\ a - 1 <= b /\ a + 1 <= b`);
MESON_TAC[];
SIMP_TAC[]]);;


let INSERT_UNION2 = SET_RULE` (x INSERT S) UNION U = S UNION (x INSERT U)`;;




let EGHNAVX = prove_by_refinement
(`convex_local_fan (V,E,FF) /\
 v0 IN V /\
 CARD V = k /\
 (!v. v IN V /\ ~(v = v0) ==> ~collinear {vec 0, v0, v}) /\
 (!i. ITER i (rho_node1 FF) v0 = vv i) /\
 (!i. azim (vec 0) v0 (vv 1) (vv i) = bta i)
 ==> bta 1 = &0 /\
     bta (k - 1) <= pi /\
     (!i j. i < j /\ j < k ==> bta i <= bta j) /\
     (bta i = &0 /\ i < k
      ==> (!j. 0 < j /\ j < i ==> interior_angle1 (vec 0) FF (vv j) = pi) /\
          {vv k | 0 < k /\ k <= i} SUBSET aff_gt {vec 0, v0} {vv 1}) /\
     (bta i = bta (k - 1) /\ 0 < i /\ i < k - 1
      ==> (!j. i < j /\ j < k ==> interior_angle1 (vec 0) FF (vv j) = pi) /\
          {vv n | i <= n /\ n < k} SUBSET aff_gt {vec 0, v0} {vv (k - 1)})`,
[NHANH MONO_AZIM_AS_BTA_I;
STRIP_TAC;
ASSUME_TAC2 FIRST_EQ0_LAST_LT_PI;
ASM_REWRITE_TAC[];
ASSUME_TAC2 CVX_LO_IMP_LO;
ASSUME_TAC2 LOCAL_FAN_ORBIT_MAP_VITER;
ASSUME_TAC2 (SPEC` v0:real^3` (GEN`v:real^3` LOFA_IMP_DIS_ELMS23));

CONJ_TAC;
STRIP_TAC;
SUBGOAL_THEN` {vv k | 0 < k /\ k <= i} SUBSET aff_gt {vec 0, v0:real^3} {vv 1}` ASSUME_TAC;
REWRITE_TAC[SUBSET; IN_ELIM_THM];
REPEAT STRIP_TAC;
SUBGOAL_THEN` azim (vec 0) v0 (vv 1) (vv k') = &0` ASSUME_TAC;
ASM_CASES_TAC` k' = i:num`;
ASM_REWRITE_TAC[];
UNDISCH_TAC` !i j. i < j /\ j < k ==> bta i <= (bta:num -> real) j`;

DISCH_THEN (MP_TAC o (SPECL [`k': num`;` i: num`]));
ANTS_TAC;
ASM_REWRITE_TAC[];
UNDISCH_TAC` k' <= i:num`;
UNDISCH_TAC` ~( k' = i:num)`;
PHA;
ARITH_TAC;

ASM_REWRITE_TAC[];
SUBGOAL_THEN` &0 <= azim (vec 0) v0 (vv 1) (vv k')` MP_TAC;
REWRITE_TAC[AZIM_RANGE];

ASM_REWRITE_TAC[];
REAL_ARITH_TAC;




ASSUME_TAC2 LOCAL_FAN_CHARACTER_OF_RHO_NODE;
FIRST_X_ASSUM (ASSUME_TAC2 o (SPEC` v0:real^3`));
FIRST_ASSUM (MP_TAC o (SPECL [`0`;`k': num`]));
PHA;
ANTS_TAC;
ASM_REWRITE_TAC[];
UNDISCH_TAC` k' <= i:num`;
UNDISCH_TAC` i < k:num`;
ARITH_TAC;

REWRITE_TAC[ITER];
DOWN;
SUBGOAL_THEN` rho_node1 FF v0 = ITER 1 (rho_node1 FF) v0` MP_TAC;
REWRITE_TAC[ITER12];
ASM_SIMP_TAC[];


FIRST_ASSUM (ASSUME_TAC2 o (SPECL [`v0:real^3`;`1`]));
FIRST_ASSUM (ASSUME_TAC2 o (SPECL [`v0:real^3`;`k':num`]));
DOWN THEN DOWN;
ASM_REWRITE_TAC[];
REPEAT STRIP_TAC;
UNDISCH_TAC` !v. v IN V /\ ~(v = v0) ==> ~collinear {vec 0, v0, v:real^3}`;
STRIP_TAC;
FIRST_ASSUM (ASSUME_TAC2 o (SPEC`(vv:num -> real^3) 1 `));
FIRST_ASSUM (ASSUME_TAC2 o (SPEC`(vv:num -> real^3) k' `));
DOWN THEN DOWN THEN PHA; 

NHANH (GSYM AZIM_EQ_0_ALT);
SIMP_TAC[];
ASM_REWRITE_TAC[];

ASSUME_TAC (ISPECL [`{vec 0, v0:real^3}`;` {(vv:num -> real^3) 1}`]
 AFF_GT_SUBSET_AFFINE_HULL);
ASM_REWRITE_TAC[];
UNDISCH_TAC` !i. ITER i (rho_node1 FF) v0 = vv i`;
STRIP_TAC;
FIRST_ASSUM (fun x -> REWRITE_TAC[GSYM x]);
MATCH_MP_TAC (GEN_ALL CNVX_IMP_INTERIOR_ANGLE_PI);
EXISTS_TAC`E:(real^3 -> bool) -> bool`;
EXISTS_TAC` V:real^3 -> bool`;
EXISTS_TAC`{ITER n (rho_node1 FF) v0 | n <= i}`;
EXISTS_TAC` affine hull ({vec 0, v0:real^3} UNION {vv 1})`;
ASM_REWRITE_TAC[];
ASSUME_TAC2 (SPEC` v0:real^3` (GEN `v:real^3`
 LOCAL_FAN_CHARACTER_OF_RHO_NODE2));
CONJ_TAC;
REWRITE_TAC[INSERT_UNION2; UNION_EMPTY];

DOWN;
SUBGOAL_THEN` rho_node1 FF v0 = ITER 1 (rho_node1 FF) v0` ASSUME_TAC;
REWRITE_TAC[ITER12];
ASM_REWRITE_TAC[plane];
MESON_TAC[INSERT_COMM];

CONJ_TAC;
REWRITE_TAC[INSERT_UNION2; UNION_EMPTY];
MESON_TAC[Trigonometry2.IN_P_HULL_INSERT; INSERT_COMM];


REWRITE_TAC[SUBSET; IN_ELIM_THM];
REPEAT STRIP_TAC;
ASM_CASES_TAC` n = 0 `;
ASM_REWRITE_TAC[];

SUBGOAL_THEN` vv 0 = ITER 0 (rho_node1 FF) v0` MP_TAC;
ASM_REWRITE_TAC[];
SIMP_TAC[ITER; Trigonometry2.IN_P_HULL_INSERT; INSERT_UNION2; UNION_EMPTY];


SUBGOAL_THEN` x IN {(vv:num ->real^3) k | 0 < k /\ k <= i}` MP_TAC;
REWRITE_TAC[IN_ELIM_THM];
EXISTS_TAC`n:num`;
ASM_REWRITE_TAC[];
DOWN;
ARITH_TAC;
UNDISCH_TAC` {vv k | 0 < k /\ k <= i} SUBSET aff_gt {vec 0, v0:real^3} {vv 1}`;
REWRITE_TAC[SUBSET];
DISCH_THEN NHANH;
UNDISCH_TAC` aff_gt {vec 0, v0:real^3} {vv 1} SUBSET affine hull ({vec 0, v0} UNION {vv  1})`;
REWRITE_TAC[SUBSET];
DISCH_THEN NHANH;
SIMP_TAC[INSERT_UNION2; UNION_EMPTY];







STRIP_TAC;
SUBGOAL_THEN` {vv n | i <= n /\ n < k} SUBSET aff_gt {vec 0, v0:real^3} {vv (k - 1)}` ASSUME_TAC;
REWRITE_TAC[SUBSET; IN_ELIM_THM];
REPEAT STRIP_TAC;
SUBGOAL_THEN` azim (vec 0) v0 (vv 1) x = azim (vec 0) v0 (vv 1) (vv (k - 1))` ASSUME_TAC;
ASM_CASES_TAC` n = i:num`;
ASM_REWRITE_TAC[];
ASM_CASES_TAC` n = k - 1 `;
ASM_REWRITE_TAC[];
UNDISCH_TAC` !i j. i < j /\ j < k ==> bta i <= (bta:num -> real) j`;
STRIP_TAC;
FIRST_ASSUM (MP_TAC o (SPECL [`i:num`;` n:num`]));
ANTS_TAC;
ASM_ARITH_TAC;
FIRST_ASSUM (MP_TAC o (SPECL [`n:num`;`k - 1`]));
ANTS_TAC;
ASM_ARITH_TAC;
ASM_REWRITE_TAC[];
REAL_ARITH_TAC;

FIRST_ASSUM (ASSUME_TAC2 o (SPECL [`v0:real^3`;`1`]));
FIRST_ASSUM (ASSUME_TAC2 o (SPECL [`v0:real^3`;`n:num`]));
FIRST_ASSUM (ASSUME_TAC2 o (SPECL [`v0:real^3`;`k - 1`]));
FIRST_ASSUM (MP_TAC o (SPECL [`0`;`n:num`]));
PHA;
ANTS_TAC;
ASM_REWRITE_TAC[];
ASM_ARITH_TAC;
FIRST_ASSUM (MP_TAC o (SPECL [`0`;` k - 1`]));
ANTS_TAC;
ASM_REWRITE_TAC[];
ASM_ARITH_TAC;
ANTS_TAC;
ASM_REWRITE_TAC[];
ASM_ARITH_TAC;
REWRITE_TAC[ITER];
ASM_REWRITE_TAC[];
ASSUME_TAC2 LOCAL_FAN_CHARACTER_OF_RHO_NODE;
FIRST_X_ASSUM (ASSUME_TAC2 o (SPEC`v0: real^3`));
SUBGOAL_THEN` rho_node1 FF v0 = ITER 1 (rho_node1 FF) v0` MP_TAC;
REWRITE_TAC[ITER12];
ASM_REWRITE_TAC[];
REPEAT STRIP_TAC;
UNDISCH_TAC` ITER 1 (rho_node1 FF) v0 IN V `;
UNDISCH_TAC` ITER n (rho_node1 FF) v0 IN V `;
UNDISCH_TAC` ITER (k - 1) (rho_node1 FF) v0 IN V `;
UNDISCH_TAC` ~(rho_node1 FF v0 = v0) /\
      v0,rho_node1 FF v0 IN ord_pairs E /\
      ~collinear {vec 0, v0, rho_node1 FF v0}`;
ASM_REWRITE_TAC[];
REPEAT STRIP_TAC;
UNDISCH_TAC` !v. v IN V /\ ~(v = v0) ==> ~collinear {vec 0, v0, v:real^3}`;
STRIP_TAC;
FIRST_ASSUM (ASSUME_TAC2 o (SPEC`(vv:num -> real^3) 1 `));
FIRST_ASSUM (ASSUME_TAC2 o (SPEC`(vv:num -> real^3) n `));
FIRST_ASSUM (ASSUME_TAC2 o (SPEC`(vv:num -> real^3) (k - 1) `));
REPLICATE_TAC 3 DOWN THEN PHA;
NHANH AZIM_EQ_ALT;


UNDISCH_TAC` azim (vec 0) v0 (vv 1) x = azim (vec 0) v0 (vv 1) (vv (k - 1))`;
ASM_SIMP_TAC[];





ASM_REWRITE_TAC[];
ABBREV_TAC` v00 = ITER i (rho_node1 FF) v0`;
MP_TAC (SPECL [`E:(real^3 -> bool) -> bool`;` V:real^3 -> bool`;
` {ITER n (rho_node1 FF) v00 | n <= k - i}`;` affine hull {vec 0, v0:real^3, vv (k - 1)}`;
` k - i:num`;` FF:real^3 # real^3 -> bool`;` v00: real^3`] 
(GEN_ALL CNVX_IMP_INTERIOR_ANGLE_PI));
ANTS_TAC;
ASM_REWRITE_TAC[];

FIRST_ASSUM (ASSUME_TAC2 o (SPECL [`v0:real^3`;`i:num`]));
DOWN;
FIRST_ASSUM SUBST1_TAC;
SIMP_TAC[Trigonometry2.IN_P_HULL_INSERT];

FIRST_ASSUM (ASSUME_TAC2 o (SPECL [`v0:real^3`;`k - 1`]));
FIRST_ASSUM (MP_TAC o (SPECL [`0`;`k - 1`]));
ANTS_TAC;
MATCH_MP_TAC LT_TRANS;
EXISTS_TAC `i:num`;
ASM_REWRITE_TAC[];
ANTS_TAC;
ASM_REWRITE_TAC[];
UNDISCH_TAC` i < k - 1`;
ARITH_TAC;
REWRITE_TAC[ITER];
DOWN;
ASM_REWRITE_TAC[];
STRIP_TAC THEN STRIP_TAC;
SUBGOAL_THEN` ~ collinear {vec 0, v0:real^3, vv (k - 1)}` ASSUME_TAC;
DOWN THEN DOWN THEN PHA;
ASM_REWRITE_TAC[];
STRIP_TAC THEN CONJ_TAC;
REWRITE_TAC[plane];
DOWN THEN DOWN;
MESON_TAC[];
ASSUME_TAC (ISPECL [`{vec 0, v0:real^3}`;`{(vv:num -> real^3) (k - 1)}`]
AFF_GT_SUBSET_AFFINE_HULL);
REWRITE_TAC[SUBSET; IN_ELIM_THM];
REPEAT STRIP_TAC;
ASM_CASES_TAC` n = k - (i:num)`;
ASM_REWRITE_TAC[];
EXPAND_TAC "v00";
ASSUME_TAC2 (ARITH_RULE` i < k - 1 ==> k - i + i = (k:num) `);
ASM_REWRITE_TAC[ITER_ADD];
ASSUME_TAC2 LOFA_IMP_ITER_RHO_NODE_ID;
FIRST_ASSUM (ASSUME_TAC2 o (SPEC`v0:real^3`));
DOWN;
ONCE_REWRITE_TAC[INSERT_COMM];
ASM_REWRITE_TAC[];
ASM_SIMP_TAC[Trigonometry2.IN_P_HULL_INSERT];
UNDISCH_TAC` aff_gt {vec 0, v0:real^3} {vv (k - 1)} SUBSET
      affine hull ({vec 0, v0} UNION {vv (k - 1)})`;
REWRITE_TAC[INSERT_UNION2; UNION_EMPTY; INSERT_COMM; SUBSET];
DISCH_THEN MATCH_MP_TAC;
UNDISCH_TAC` {vv n | i <= n /\ n < k} SUBSET aff_gt {vec 0, v0:real^3} {vv (k - 1)}`;
REWRITE_TAC[SUBSET; IN_ELIM_THM; INSERT_COMM];
DISCH_THEN MATCH_MP_TAC;
EXISTS_TAC `n + i:num`;
ASM_REWRITE_TAC[];
EXPAND_TAC "v00";
REWRITE_TAC[ITER_ADD];
ASM_REWRITE_TAC[];
DOWN;
UNDISCH_TAC` n <= k - (i:num)`;
ARITH_TAC;
STRIP_TAC THEN GEN_TAC;
STRIP_TAC;
FIRST_X_ASSUM (MP_TAC o (SPEC` j - (i:num)`));
ANTS_TAC;
DOWN THEN DOWN;
ARITH_TAC;
EXPAND_TAC "v00";
REWRITE_TAC[ITER_ADD];
UNDISCH_TAC` i < j:num`;
NHANH (ARITH_RULE` a < b:num ==> b - a + a = b `);
ASM_SIMP_TAC[]]);;








end;;