1 (* ========================================================================== *)
2 (* FLYSPECK - BOOK FORMALIZATION *)
3 (* Section: Appendix *)
4 (* Chapter: Local Fan *)
5 (* Author: John Harrison *)
7 (* ========================================================================== *)
9 module Pqcsxwg = struct
12 let mk_simplex1 = new_definition `mk_simplex1 v0 v1 v2 x1 x2 x3 x4 x5 x6 =
13 (let uinv = &1 / ups_x x1 x2 x6 in
14 let d = delta_x x1 x2 x3 x4 x5 x6 in
15 let d5 = delta_x5 x1 x2 x3 x4 x5 x6 in
16 let d6 = delta_x4 x1 x2 x3 x4 x5 x6 in
17 let vcross = (v1 - v0) cross (v2 - v0) in
18 v0 + uinv % ((&2 * sqrt d) % vcross + d5 % (v1 - v0) + d6 % (v2 - v0)))`;;
20 let PQCSXWG1_concl = `!v0 v1 v2 v3 x1 x2 x3 x4 x5 x6.
21 &0 < x1 /\ &0 < x2 /\ &0 < x3 /\ &0 < x4 /\ &0 < x5 /\ &0 < x6 /\
22 ~collinear {v0,v1,v2} /\
23 x1 = dist(v1,v0) pow 2 /\
24 x2 = dist(v2,v0) pow 2 /\
25 x6 = dist(v1,v2) pow 2 /\
26 &0 < delta_x x1 x2 x3 x4 x5 x6 /\
27 v3 = mk_simplex1 v0 v1 v2 x1 x2 x3 x4 x5 x6 ==>
28 (x3 = dist(v3,v0) pow 2 /\
29 x5 = dist(v3,v1) pow 2 /\
30 x4 = dist(v3,v2) pow 2 /\
31 (v1 - v0) dot ((v2 - v0) cross (v3 - v0)) > &0)`;;
33 let PQCSXWG2_concl = `!(v0:real^3) v1 v2 v3 x1 x2 x3 x4 x5 x6.
34 &0 < x1 /\ &0 < x2 /\ &0 < x3 /\ &0 < x4 /\ &0 < x5 /\ &0 < x6 /\
35 ~collinear {v0,v1,v2} /\
36 x1 = dist(v1,v0) pow 2 /\
37 x2 = dist(v2,v0) pow 2 /\
38 x6 = dist(v1,v2) pow 2 /\
39 &0 < delta_x x1 x2 x3 x4 x5 x6 /\
40 v3 = mk_simplex1 v0 v1 v2 x1 x2 x3 x4 x5 x6 ==>
41 (\q. mk_simplex1 v0 v1 v2 x1 x2 x3 x4 q x6) continuous atreal x5`;;
43 (* ------------------------------------------------------------------------- *)
44 (* The main result. *)
45 (* ------------------------------------------------------------------------- *)
47 let MK_SIMPLEX_TRANSLATION = prove
48 (`!a v0 v1 v2 x1 x2 x3 x4 x5 x6.
49 mk_simplex1 (a + v0) (a + v1) (a + v2) x1 x2 x3 x4 x5 x6 =
50 a + mk_simplex1 v0 v1 v2 x1 x2 x3 x4 x5 x6`,
51 REPEAT GEN_TAC THEN REWRITE_TAC[mk_simplex1] THEN
52 CONV_TAC(TOP_DEPTH_CONV let_CONV) THEN
53 REWRITE_TAC[VECTOR_ARITH `(a + x) - (a + y):real^N = x - y`] THEN
54 REWRITE_TAC[GSYM VECTOR_ADD_ASSOC]);;
56 add_translation_invariants [MK_SIMPLEX_TRANSLATION];;
60 GEOM_ORIGIN_TAC `v0:real^3` THEN REPEAT GEN_TAC THEN
61 REWRITE_TAC[mk_simplex1; VECTOR_SUB_RZERO; VECTOR_ADD_LID] THEN
62 REPEAT LET_TAC THEN REWRITE_TAC[CONJ_ASSOC] THEN
63 DISCH_THEN(CONJUNCTS_THEN2 MP_TAC SUBST1_TAC) THEN
64 REWRITE_TAC[GSYM CONJ_ASSOC] THEN
65 DISCH_THEN(STRIP_ASSUME_TAC o GSYM) THEN
66 REWRITE_TAC[dist; VECTOR_SUB_RZERO] THEN
67 REWRITE_TAC[VECTOR_ADD_LDISTRIB; VECTOR_MUL_ASSOC] THEN
68 REWRITE_TAC[CROSS_RADD; CROSS_RMUL;
69 VECTOR_ARITH `(a + x % b + c) - b:real^N = a + (x - &1) % b + c`;
70 VECTOR_ARITH `(a + b + x % c) - c:real^N = a + b + (x - &1) % c`] THEN
72 `!a b c. norm(a % vcross + b % v1 + c % v2:real^3) pow 2 =
73 norm(a % vcross) pow 2 + norm(b % v1 + c % v2) pow 2`
74 (fun th -> REWRITE_TAC[th])
76 [REPEAT GEN_TAC THEN MATCH_MP_TAC NORM_ADD_PYTHAGOREAN THEN
77 EXPAND_TAC "vcross" THEN REWRITE_TAC[orthogonal] THEN VEC3_TAC;
79 REWRITE_TAC[CROSS_REFL; VECTOR_MUL_RZERO; VECTOR_ADD_RID; real_gt] THEN
80 REWRITE_TAC[DOT_RADD; DOT_RMUL; DOT_CROSS_SELF] THEN
81 REWRITE_TAC[REAL_MUL_RZERO; REAL_ADD_RID] THEN
82 REWRITE_TAC[VEC3_RULE `v1 dot (v2 cross v) = (v1 cross v2) dot v`] THEN
83 SUBGOAL_THEN `~(vcross:real^3 = vec 0)` ASSUME_TAC THENL
84 [EXPAND_TAC "vcross" THEN REWRITE_TAC[CROSS_EQ_0] THEN ASM_REWRITE_TAC[];
85 ASM_SIMP_TAC[GSYM NORM_POW_2; NORM_POS_LT; REAL_POW_LT; REAL_LT_MUL_EQ;
86 REAL_ARITH `&0 < x * &2 * y <=> &0 < x * y`; SQRT_POS_LT]] THEN
87 SUBGOAL_THEN `&0 < ups_x x1 x2 x6` ASSUME_TAC THENL
88 [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I
89 [Collect_geom2.NOT_COL_EQ_UPS_X_POS]) THEN
90 MAP_EVERY EXPAND_TAC ["x1"; "x2"; "x6"] THEN REWRITE_TAC[DIST_SYM];
91 EXPAND_TAC "uinv" THEN
92 ASM_SIMP_TAC[REAL_LT_DIV; REAL_OF_NUM_LT; ARITH]] THEN
93 REWRITE_TAC[NORM_MUL; REAL_POW_MUL; REAL_POW2_ABS] THEN
94 ASM_SIMP_TAC[SQRT_POW_2; REAL_LT_IMP_LE] THEN
95 REWRITE_TAC[REAL_ARITH `x * &2 pow 2 * y = &4 * x * y`] THEN
96 REWRITE_TAC[NORM_POW_2; VECTOR_ARITH
97 `(a + b) dot (a + b:real^3) = a dot a + b dot b + &2 * a dot b`] THEN
98 REWRITE_TAC[DOT_LMUL] THEN REWRITE_TAC[DOT_RMUL] THEN
99 ONCE_REWRITE_TAC[REAL_ARITH
100 `x3:real = a + b /\ x5 = a + c /\ x4 = a + d <=>
101 x3 = a + b /\ x3 - x5 = b - c /\ x3 - x4 = b - d`] THEN
102 REWRITE_TAC[REAL_ARITH
103 `(b * b * x + c * c * y + &2 * b * c * z) -
104 ((b - &1) * (b - &1) * x + c * c * y + &2 * (b - &1) * c * z) =
105 (&2 * b - &1) * x + &2 * c * z /\
106 (b * b * x + c * c * y + &2 * b * c * z) -
107 (b * b * x + (c - &1) * (c - &1) * y + &2 * b * (c - &1) * z) =
108 (&2 * c - &1) * y + &2 * b * z`] THEN
109 RULE_ASSUM_TAC(REWRITE_RULE[DIST_0]) THEN
110 ASM_REWRITE_TAC[GSYM NORM_POW_2; REAL_ARITH
111 `x = (&2 * b - &1) * y + &2 * c * z <=>
112 b * y + c * z = (y + x) / &2`] THEN
113 EXPAND_TAC "vcross" THEN REWRITE_TAC[NORM_POW_2; DOT_CROSS] THEN
114 ASM_REWRITE_TAC[GSYM NORM_POW_2] THEN
115 SUBST1_TAC(VECTOR_ARITH `(v2:real^3) dot v1 = v1 dot v2`) THEN
116 REWRITE_TAC[GSYM REAL_POW_2] THEN
117 SUBGOAL_THEN `(v1:real^3) dot v2 = ((x1 + x2) - x6) / &2` SUBST1_TAC THENL
118 [MAP_EVERY EXPAND_TAC ["x1"; "x2"; "x6"] THEN
119 REWRITE_TAC[dist; NORM_POW_2; DOT_RSUB; DOT_LSUB] THEN
120 REWRITE_TAC[DOT_SYM] THEN REAL_ARITH_TAC;
122 REWRITE_TAC[REAL_ARITH
123 `(&4 * u pow 2 * d) * x + (u * e) * (u * e) * y + (u * f) * (u * f) * z +
124 &2 * (u * e) * (u * f) * j =
125 u pow 2 * (&4 * d * x + e pow 2 * y + f pow 2 * z + &2 * e * f * j)`] THEN
126 REWRITE_TAC[REAL_ARITH
127 `(u * d) * x + (u * e) * y:real = z <=> u * (d * x + e * y) = z`] THEN
128 EXPAND_TAC "uinv" THEN MATCH_MP_TAC(REAL_FIELD
130 u pow 2 * x = y /\ u * a = b /\ u * c = d
131 ==> x = (&1 / u) pow 2 * y /\
132 (&1 / u) * b = a /\ (&1 / u) * d = c`) THEN
133 ASM_REWRITE_TAC[] THEN MAP_EVERY EXPAND_TAC ["uinv"; "d"; "d5"; "d6"] THEN
134 REPEAT(FIRST_X_ASSUM(MP_TAC o MATCH_MP REAL_LT_IMP_NZ)) THEN
135 REWRITE_TAC[Nonlin_def.delta_x5; Nonlin_def.delta_x4] THEN
136 REWRITE_TAC[Sphere.ups_x; Sphere.delta_x] THEN CONV_TAC REAL_RING);;