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

(* ========== snapshot 1631 ============= *)
flyspeck_needs "local/LVDUCXU.hl";; 


module type Ldurdpn_type = sig

end;;

   
module Ldurdpn = struct

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


let SUBSET_P_HULL = prove(` (S:A -> bool) SUBSET P hull S`,
REWRITE_TAC[HULL_SUBSET]);;

let IN_HULL_INSERT = prove(`!x:A. x IN P hull ( x INSERT S ) `,
GEN_TAC THEN 
MATCH_MP_TAC (SET_RULE` x IN S /\ S SUBSET P hull S ==> x IN P hull S `) THEN 
REWRITE_TAC[IN_INSERT; SUBSET_P_HULL]);;



let VECTOR_SCALE_CHANGE = prove(` ~( a = &0 ) ==> 
( a % x = y <=> x = &1 / a % y ) `,
DISCH_TAC THEN EQ_TAC THENL [
DISCH_THEN (SUBST1_TAC o SYM) THEN 
REWRITE_TAC[VECTOR_MUL_ASSOC] THEN 
ASM_SIMP_TAC[REAL_FIELD` ~( a = &0) ==> &1 / a * a = &1 `; VECTOR_MUL_LID];
SIMP_TAC[VECTOR_MUL_ASSOC] THEN 
ASM_SIMP_TAC[REAL_FIELD` ~( a = &0) ==> a * &1 / a = &1 `; VECTOR_MUL_LID]]);;



let AFF_CONV0_IN_AFF_LT = prove_by_refinement(
` ~collinear {vec 0, u, v} ==>
~(aff {vec 0, u} INTER conv0 {v, w} = {})
 ==> w IN aff_lt {vec 0, u} {v} `,
[NHANH th3a;
SIMP_TAC[AFF_LT_2_1; Geomdetail.CONV0_SET2; AFF2];
REWRITE_TAC[SET_RULE` ~( a INTER b = {}) <=> ? x. x IN a /\ x IN b`;
IN_ELIM_THM];
REPEAT STRIP_TAC;
DOWN;
ASM_REWRITE_TAC[VECTOR_MUL_RZERO; VECTOR_ADD_LID];
PAT_ONCE_REWRITE_TAC `\x. x ==> y` 
[VECTOR_ARITH` x = a + (b:real^N) <=> b = x + -- a `];
ASSUME_TAC2 (REAL_ARITH` &0 < b ==> ~( b = &0 ) `);
DOWN;
SIMP_TAC[VECTOR_SCALE_CHANGE];
REPEAT STRIP_TAC;
REWRITE_TAC[VECTOR_MUL_ASSOC; VECTOR_ADD_LDISTRIB; GSYM VECTOR_MUL_LNEG];
SUBGOAL_THEN ` (&1 / b * --a) < &0 ` MP_TAC;
REWRITE_TAC[REAL_ARITH` &1 / b * --a < &0 <=> &0 < a / b `];
MATCH_MP_TAC REAL_LT_DIV;
ASM_REWRITE_TAC[];
MESON_TAC[REAL_ARITH` (&1 - a - b ) + a + b = &1 `]]);;


let LDURDPN = prove_by_refinement
(` ~ collinear {vec 0, (u:real^3), v} /\ ~ collinear {vec 0, u, w}
==> ( azim (vec 0) u v w = pi <=> (? A. plane A /\
{vec 0, u, v, w} SUBSET A) /\ ~( aff {vec 0, u} INTER conv0 {v, w} = {}) )`,
[SIMP_TAC[AZIM_EQ_PI_ALT];
STRIP_TAC;
EQ_TAC;
UNDISCH_TAC ` ~collinear {vec 0, (u:real^3), v}`;
NHANH th3;
SIMP_TAC[AFF_LT_2_1; plane];
STRIP_TAC THEN STRIP_TAC;
DOWN;
REWRITE_TAC[IN_ELIM_THM];
STRIP_TAC;
CONJ_TAC;
EXISTS_TAC ` affine hull {vec 0, (u:real^3), v}`;
REWRITE_TAC[AFFINE_HULL_3; INSERT_SUBSET; EMPTY_SUBSET];
CONJ_TAC;
EXISTS_TAC `(vec 0):real^3 `;
EXISTS_TAC ` u:real^3 `;
EXISTS_TAC `v:real^3 `;
ASM_REWRITE_TAC[];
REWRITE_TAC[GSYM AFFINE_HULL_3];
NGOAC;
CONJ_TAC;
MESON_TAC[INSERT_COMM; IN_HULL_INSERT];
REWRITE_TAC[AFFINE_HULL_3; IN_ELIM_THM];
DOWN THEN DOWN THEN PHA;
MESON_TAC[];
REWRITE_TAC[SET_RULE` ~( x INTER y = {}) <=> ?a. a IN x /\ a IN y`];
EXISTS_TAC ` ( &1 / (&1 - t3)) % ((w:real^3) - t3 % v ) `;
REWRITE_TAC[AFF2; Geomdetail.CONV0_SET2; IN_ELIM_THM];
CONJ_TAC;
EXISTS_TAC ` ( &1 / (&1 - t3)) * t1 `;
REWRITE_TAC[GSYM VECTOR_MUL_ASSOC];
ASM_SIMP_TAC[
REAL_FIELD` t3 < &0 ==> &1 - &1 / (&1 - t3) * t1 = 
&1 / (&1 - t3) * (&1 - t3 - t1 ) `];
REWRITE_TAC[GSYM VECTOR_ADD_LDISTRIB; GSYM VECTOR_MUL_ASSOC];
MATCH_MP_TAC (MESON[]` x = y ==> f x = f y`);
ASM_SIMP_TAC[REAL_RING` t1 + t2 + t3 = &1 ==> &1 - t3 - t1 = t2 `];
VECTOR_ARITH_TAC;
EXISTS_TAC ` ( &1 / (&1 - t3)) * ( -- t3) `;
EXISTS_TAC ` ( &1 / (&1 - t3))  `;
CONJ_TAC;
MATCH_MP_TAC REAL_LT_MUL;
ASM_REWRITE_TAC[GSYM (REAL_FIELD` t3 < &0 <=> &0 < -- t3 `)];
MATCH_MP_TAC REAL_LT_DIV;
ASM_REAL_ARITH_TAC;
CONJ_TAC;
MATCH_MP_TAC REAL_LT_DIV;
ASM_REAL_ARITH_TAC;
ASM_SIMP_TAC[
REAL_FIELD` t3 < &0 ==> &1 / (&1 - t3) * --t3 + &1 / (&1 - t3) = &1 `];
VECTOR_ARITH_TAC;
ASM_SIMP_TAC[AFF_CONV0_IN_AFF_LT]]);;

end;;