1 (* ========================================================================= *)
2 (* Results intended for Flyspeck. *)
3 (* ========================================================================= *)
5 needs "Multivariate/polytope.ml";;
6 needs "Multivariate/realanalysis.ml";;
7 needs "Multivariate/geom.ml";;
8 needs "Multivariate/cross.ml";;
12 (* ------------------------------------------------------------------------- *)
13 (* Not really Flyspeck-specific but needs both angles and cross products. *)
14 (* ------------------------------------------------------------------------- *)
16 let NORM_CROSS = prove
17 (`!x y. norm(x cross y) = norm(x) * norm(y) * sin(vector_angle x y)`,
19 MATCH_MP_TAC REAL_POW_EQ THEN EXISTS_TAC `2` THEN
20 SIMP_TAC[NORM_POS_LE; SIN_VECTOR_ANGLE_POS; REAL_LE_MUL; ARITH_EQ] THEN
21 MP_TAC(SPECL [`x:real^3`; `y:real^3`] NORM_CROSS_DOT) THEN
22 REWRITE_TAC[VECTOR_ANGLE] THEN
23 MP_TAC(SPEC `vector_angle (x:real^3) y` SIN_CIRCLE) THEN
26 (* ------------------------------------------------------------------------- *)
27 (* Other miscelleneous lemmas. *)
28 (* ------------------------------------------------------------------------- *)
30 let COPLANAR_INSERT_0_NEG = prove
31 (`coplanar(vec 0 INSERT --x INSERT s) <=> coplanar(vec 0 INSERT x INSERT s)`,
32 REWRITE_TAC[coplanar; INSERT_SUBSET] THEN
33 ONCE_REWRITE_TAC[TAUT `a /\ b /\ c <=> ~(a ==> ~(b /\ c))`] THEN
34 SIMP_TAC[AFFINE_HULL_EQ_SPAN; SPAN_NEG_EQ]);;
36 let COPLANAR_IMP_NEGLIGIBLE = prove
37 (`!s:real^3->bool. coplanar s ==> negligible s`,
38 REWRITE_TAC[coplanar] THEN
39 MESON_TAC[NEGLIGIBLE_AFFINE_HULL_3; NEGLIGIBLE_SUBSET]);;
41 let NOT_COPLANAR_0_4_IMP_INDEPENDENT = prove
42 (`!v1 v2 v3:real^N. ~coplanar {vec 0,v1,v2,v3} ==> independent {v1,v2,v3}`,
43 REPEAT GEN_TAC THEN REWRITE_TAC[independent; CONTRAPOS_THM] THEN
44 REWRITE_TAC[dependent] THEN
46 `!v1 v2 v3:real^N. v1 IN span {v2,v3} ==> coplanar{vec 0,v1,v2,v3}`
48 [REPEAT STRIP_TAC THEN REWRITE_TAC[coplanar] THEN
49 MAP_EVERY EXISTS_TAC [`vec 0:real^N`; `v2:real^N`; `v3:real^N`] THEN
50 SIMP_TAC[AFFINE_HULL_EQ_SPAN; HULL_INC; IN_INSERT] THEN
51 REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET] THEN
52 ASM_SIMP_TAC[SPAN_SUPERSET; IN_INSERT] THEN
53 POP_ASSUM MP_TAC THEN SPEC_TAC(`v1:real^N`,`v1:real^N`) THEN
54 REWRITE_TAC[GSYM SUBSET] THEN MATCH_MP_TAC SPAN_MONO THEN SET_TAC[];
55 REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY] THEN
56 STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
57 FIRST_X_ASSUM SUBST_ALL_TAC THENL
58 [FIRST_X_ASSUM(MP_TAC o SPECL [`v1:real^N`; `v2:real^N`; `v3:real^N`]);
59 FIRST_X_ASSUM(MP_TAC o SPECL [`v2:real^N`; `v3:real^N`; `v1:real^N`]);
60 FIRST_X_ASSUM(MP_TAC o SPECL [`v3:real^N`; `v1:real^N`; `v2:real^N`])]
61 THEN REWRITE_TAC[INSERT_AC] THEN DISCH_THEN MATCH_MP_TAC THEN
62 FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE
63 `a IN s ==> s SUBSET t ==> a IN t`)) THEN
64 MATCH_MP_TAC SPAN_MONO THEN SET_TAC[]]);;
66 let NOT_COPLANAR_NOT_COLLINEAR = prove
67 (`!v1 v2 v3 w:real^N. ~coplanar {v1, v2, v3, w} ==> ~collinear {v1, v2, v3}`,
69 REWRITE_TAC[COLLINEAR_AFFINE_HULL; coplanar; CONTRAPOS_THM] THEN
70 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `x:real^N` THEN
71 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `y:real^N` THEN
72 REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET] THEN STRIP_TAC THEN
73 EXISTS_TAC `w:real^N` THEN ASM_SIMP_TAC[HULL_INC; IN_INSERT] THEN
75 MATCH_MP_TAC(SET_RULE `!t. t SUBSET s /\ x IN t ==> x IN s`) THEN
76 EXISTS_TAC `affine hull {x:real^N,y}` THEN
77 ASM_REWRITE_TAC[] THEN MATCH_MP_TAC HULL_MONO THEN SET_TAC[]);;
79 (* ------------------------------------------------------------------------- *)
80 (* Some special scaling theorems. *)
81 (* ------------------------------------------------------------------------- *)
83 let SUBSET_AFFINE_HULL_SPECIAL_SCALE = prove
86 ==> (vec 0 INSERT (a % x) INSERT s SUBSET affine hull t <=>
87 vec 0 INSERT x INSERT s SUBSET affine hull t)`,
88 REPEAT STRIP_TAC THEN REWRITE_TAC[INSERT_SUBSET] THEN
89 MATCH_MP_TAC(TAUT `(a ==> (b <=> c)) ==> (a /\ b <=> a /\ c)`) THEN
90 ASM_SIMP_TAC[AFFINE_HULL_EQ_SPAN; SPAN_MUL_EQ]);;
92 let COLLINEAR_SPECIAL_SCALE = prove
93 (`!a x y. ~(a = &0) ==> (collinear {vec 0,a % x,y} <=> collinear{vec 0,x,y})`,
94 REPEAT STRIP_TAC THEN REWRITE_TAC[COLLINEAR_AFFINE_HULL] THEN
95 ASM_SIMP_TAC[SUBSET_AFFINE_HULL_SPECIAL_SCALE]);;
97 let COLLINEAR_SCALE_ALL = prove
98 (`!a b v w. ~(a = &0) /\ ~(b = &0)
99 ==> (collinear {vec 0,a % v,b % w} <=> collinear {vec 0,v,w})`,
100 REPEAT STRIP_TAC THEN ASM_SIMP_TAC[COLLINEAR_SPECIAL_SCALE] THEN
101 ONCE_REWRITE_TAC[SET_RULE `{a,b,c} = {a,c,b}`] THEN
102 ASM_SIMP_TAC[COLLINEAR_SPECIAL_SCALE]);;
104 let COPLANAR_SPECIAL_SCALE = prove
106 ~(a = &0) ==> (coplanar {vec 0,a % x,y,z} <=> coplanar {vec 0,x,y,z})`,
107 REPEAT STRIP_TAC THEN REWRITE_TAC[coplanar] THEN
108 ASM_SIMP_TAC[SUBSET_AFFINE_HULL_SPECIAL_SCALE]);;
110 let COPLANAR_SCALE_ALL = prove
112 ~(a = &0) /\ ~(b = &0) /\ ~(c = &0)
113 ==> (coplanar {vec 0,a % x,b % y,c % z} <=> coplanar {vec 0,x,y,z})`,
114 REPEAT STRIP_TAC THEN ASM_SIMP_TAC[COPLANAR_SPECIAL_SCALE] THEN
115 ONCE_REWRITE_TAC[SET_RULE `{a,b,c,d} = {a,c,d,b}`] THEN
116 ASM_SIMP_TAC[COPLANAR_SPECIAL_SCALE] THEN
117 ONCE_REWRITE_TAC[SET_RULE `{a,b,c,d} = {a,c,d,b}`] THEN
118 ASM_SIMP_TAC[COPLANAR_SPECIAL_SCALE]);;
120 (* ------------------------------------------------------------------------- *)
121 (* Specialized lemmas about "dropout". *)
122 (* ------------------------------------------------------------------------- *)
124 let DROPOUT_BASIS_3 = prove
125 (`(dropout 3:real^3->real^2) (basis 1) = basis 1 /\
126 (dropout 3:real^3->real^2) (basis 2) = basis 2 /\
127 (dropout 3:real^3->real^2) (basis 3) = vec 0`,
128 SIMP_TAC[LAMBDA_BETA; dropout; basis; CART_EQ; DIMINDEX_2; DIMINDEX_3; ARITH;
129 VEC_COMPONENT; LT_IMP_LE; ARITH_RULE `i <= 2 ==> i + 1 <= 3`;
130 ARITH_RULE `1 <= i + 1`] THEN
133 let COLLINEAR_BASIS_3 = prove
134 (`collinear {vec 0,basis 3,x} <=> (dropout 3:real^3->real^2) x = vec 0`,
135 SIMP_TAC[CART_EQ; FORALL_2; FORALL_3; DIMINDEX_2; DIMINDEX_3;
136 dropout; LAMBDA_BETA; BASIS_COMPONENT; ARITH; REAL_MUL_RID;
137 VECTOR_MUL_COMPONENT; VEC_COMPONENT; REAL_MUL_RZERO; UNWIND_THM1;
138 COLLINEAR_LEMMA] THEN
139 REWRITE_TAC[RIGHT_EXISTS_AND_THM; GSYM EXISTS_REFL] THEN REAL_ARITH_TAC);;
141 let OPEN_DROPOUT_3 = prove
142 (`!P. open {x | P x} ==> open {x | P((dropout 3:real^3->real^2) x)}`,
143 REPEAT STRIP_TAC THEN MP_TAC(ISPECL
144 [`dropout 3:real^3->real^2`; `{x:real^2 | P x}`]
145 CONTINUOUS_OPEN_PREIMAGE_UNIV) THEN
146 ASM_REWRITE_TAC[IN_ELIM_THM] THEN DISCH_THEN MATCH_MP_TAC THEN
147 GEN_TAC THEN MATCH_MP_TAC LINEAR_CONTINUOUS_AT THEN
148 SIMP_TAC[LINEAR_DROPOUT; DIMINDEX_2; DIMINDEX_3; ARITH]);;
150 let SLICE_DROPOUT_3 = prove
151 (`!P t. slice 3 t {x | P((dropout 3:real^3->real^2) x)} = {x | P x}`,
152 REPEAT GEN_TAC THEN REWRITE_TAC[slice] THEN
153 REWRITE_TAC[EXTENSION; IN_IMAGE; IN_ELIM_THM; IN_INTER] THEN
154 X_GEN_TAC `y:real^2` THEN EQ_TAC THENL [MESON_TAC[]; ALL_TAC] THEN
155 DISCH_TAC THEN EXISTS_TAC `(pushin 3 t:real^2->real^3) y` THEN
156 ASM_SIMP_TAC[DIMINDEX_2; DIMINDEX_3; DROPOUT_PUSHIN; ARITH] THEN
157 SIMP_TAC[pushin; LAMBDA_BETA; LT_REFL; DIMINDEX_3; ARITH]);;
159 let NOT_COPLANAR_IMP_NOT_COLLINEAR_DROPOUT_3 = prove
161 ~coplanar {vec 0,basis 3, x, y}
162 ==> ~collinear {vec 0,dropout 3 x:real^2,dropout 3 y}`,
163 REPEAT GEN_TAC THEN REWRITE_TAC[COLLINEAR_AFFINE_HULL; coplanar] THEN
164 REWRITE_TAC[CONTRAPOS_THM; INSERT_SUBSET; LEFT_IMP_EXISTS_THM] THEN
165 MAP_EVERY X_GEN_TAC [`u:real^2`; `v:real^2`] THEN
166 REWRITE_TAC[EMPTY_SUBSET] THEN
167 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
168 ASM_SIMP_TAC[AFFINE_HULL_EQ_SPAN] THEN STRIP_TAC THEN
169 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [AFFINE_HULL_2]) THEN
170 REWRITE_TAC[IN_ELIM_THM; LEFT_IMP_EXISTS_THM] THEN
171 MAP_EVERY X_GEN_TAC [`a:real`;`b:real`] THEN STRIP_TAC THEN
172 SUBGOAL_THEN `?r s. a * r + b * s = -- &1` STRIP_ASSUME_TAC THENL
173 [ASM_CASES_TAC `a = &0` THENL
174 [UNDISCH_TAC `a + b = &1` THEN
175 ASM_SIMP_TAC[REAL_MUL_LZERO; REAL_ADD_LID; REAL_MUL_LID; EXISTS_REFL];
176 ASM_SIMP_TAC[REAL_FIELD
177 `~(a = &0) ==> (a * r + x = y <=> r = (y - x) / a)`] THEN
180 EXISTS_TAC `vector[(u:real^2)$1; u$2; r]:real^3` THEN
181 EXISTS_TAC `vector[(v:real^2)$1; v$2; s]:real^3` THEN
182 EXISTS_TAC `basis 3:real^3` THEN
183 MATCH_MP_TAC(TAUT `a /\ (a ==> b) ==> a /\ b`) THEN CONJ_TAC THENL
184 [REWRITE_TAC[AFFINE_HULL_3; IN_ELIM_THM; LEFT_IMP_EXISTS_THM] THEN
185 MAP_EVERY EXISTS_TAC [`a / &2`;`b / &2`; `&1 / &2`] THEN
186 ASM_REWRITE_TAC[REAL_ARITH
187 `a / &2 + b / &2 + &1 / &2 = &1 <=> a + b = &1`] THEN
188 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [CART_EQ]) THEN
189 SIMP_TAC[CART_EQ; DIMINDEX_2; DIMINDEX_3; FORALL_2; FORALL_3;
190 VEC_COMPONENT; VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT;
191 VECTOR_3; BASIS_COMPONENT; ARITH] THEN
192 REPEAT(FIRST_X_ASSUM(MP_TAC o SYM)) THEN CONV_TAC REAL_RING;
194 SIMP_TAC[AFFINE_HULL_EQ_SPAN] THEN DISCH_TAC THEN
195 SIMP_TAC[SPAN_SUPERSET; IN_INSERT] THEN
197 `!x. (dropout 3:real^3->real^2) x IN span {u,v}
198 ==> x IN span {vector [u$1; u$2; r], vector [v$1; v$2; s], basis 3}`
199 (fun th -> ASM_MESON_TAC[th]) THEN
200 GEN_TAC THEN REWRITE_TAC[SPAN_2; SPAN_3] THEN
201 SIMP_TAC[IN_ELIM_THM; IN_UNIV; CART_EQ; DIMINDEX_2; DIMINDEX_3;
202 FORALL_2; FORALL_3; dropout; VECTOR_ADD_COMPONENT; LAMBDA_BETA;
203 VECTOR_MUL_COMPONENT; VECTOR_3; BASIS_COMPONENT; ARITH] THEN
204 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `c:real` THEN
205 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `d:real` THEN
206 STRIP_TAC THEN ASM_REWRITE_TAC[REAL_MUL_RZERO; REAL_ADD_RID] THEN
207 REWRITE_TAC[REAL_ARITH `x = a + b + c * &1 <=> c = x - a - b`] THEN
208 REWRITE_TAC[EXISTS_REFL]);;
210 let SLICE_312 = prove
211 (`!s:real^3->bool. slice 1 t s = {y:real^2 | vector[t;y$1;y$2] IN s}`,
212 SIMP_TAC[EXTENSION; IN_SLICE; DIMINDEX_2; DIMINDEX_3; ARITH] THEN
213 REPEAT GEN_TAC THEN REWRITE_TAC[IN_ELIM_THM] THEN
214 AP_THM_TAC THEN AP_TERM_TAC THEN
215 SIMP_TAC[CART_EQ; pushin; LAMBDA_BETA; FORALL_3; DIMINDEX_3; ARITH;
218 let SLICE_123 = prove
219 (`!s:real^3->bool. slice 3 t s = {y:real^2 | vector[y$1;y$2;t] IN s}`,
220 SIMP_TAC[EXTENSION; IN_SLICE; DIMINDEX_2; DIMINDEX_3; ARITH] THEN
221 REPEAT GEN_TAC THEN REWRITE_TAC[IN_ELIM_THM] THEN
222 AP_THM_TAC THEN AP_TERM_TAC THEN
223 SIMP_TAC[CART_EQ; pushin; LAMBDA_BETA; FORALL_3; DIMINDEX_3; ARITH;
226 (* ------------------------------------------------------------------------- *)
227 (* "Padding" injection from real^2 -> real^3 with zero last coordinate. *)
228 (* ------------------------------------------------------------------------- *)
230 let pad2d3d = new_definition
231 `(pad2d3d:real^2->real^3) x = lambda i. if i < 3 then x$i else &0`;;
233 let FORALL_PAD2D3D_THM = prove
234 (`!P. (!y:real^3. y$3 = &0 ==> P y) <=> (!x. P(pad2d3d x))`,
235 GEN_TAC THEN EQ_TAC THEN REPEAT STRIP_TAC THENL
236 [FIRST_X_ASSUM MATCH_MP_TAC THEN REWRITE_TAC[pad2d3d] THEN
237 SIMP_TAC[LAMBDA_BETA; DIMINDEX_3; ARITH; LT_REFL];
238 FIRST_X_ASSUM(MP_TAC o SPEC `(lambda i. (y:real^3)$i):real^2`) THEN
239 MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN
240 SIMP_TAC[CART_EQ; pad2d3d; DIMINDEX_3; ARITH; LAMBDA_BETA; DIMINDEX_2;
241 ARITH_RULE `i < 3 <=> i <= 2`] THEN
242 REWRITE_TAC[ARITH_RULE `i <= 3 <=> i <= 2 \/ i = 3`] THEN
245 let QUANTIFY_PAD2D3D_THM = prove
246 (`(!P. (!y:real^3. y$3 = &0 ==> P y) <=> (!x. P(pad2d3d x))) /\
247 (!P. (?y:real^3. y$3 = &0 /\ P y) <=> (?x. P(pad2d3d x)))`,
248 REWRITE_TAC[MESON[] `(?y. P y) <=> ~(!x. ~P x)`] THEN
249 REWRITE_TAC[GSYM FORALL_PAD2D3D_THM] THEN MESON_TAC[]);;
251 let LINEAR_PAD2D3D = prove
253 REWRITE_TAC[linear; pad2d3d] THEN
254 SIMP_TAC[CART_EQ; VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT;
255 LAMBDA_BETA; DIMINDEX_2; DIMINDEX_3; ARITH;
256 ARITH_RULE `i < 3 ==> i <= 2`] THEN
257 REPEAT STRIP_TAC THEN REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[]) THEN
260 let INJECTIVE_PAD2D3D = prove
261 (`!x y. pad2d3d x = pad2d3d y ==> x = y`,
262 SIMP_TAC[CART_EQ; pad2d3d; LAMBDA_BETA; DIMINDEX_3; DIMINDEX_2] THEN
263 REWRITE_TAC[ARITH_RULE `i < 3 <=> i <= 2`] THEN
264 MESON_TAC[ARITH_RULE `i <= 2 ==> i <= 3`]);;
266 let NORM_PAD2D3D = prove
267 (`!x. norm(pad2d3d x) = norm x`,
268 SIMP_TAC[NORM_EQ; DOT_2; DOT_3; pad2d3d; LAMBDA_BETA;
269 DIMINDEX_2; DIMINDEX_3; ARITH] THEN
272 (* ------------------------------------------------------------------------- *)
273 (* Apply 3D->2D conversion to a goal. Take care to preserve variable names. *)
274 (* ------------------------------------------------------------------------- *)
276 let PAD2D3D_QUANTIFY_CONV =
277 let gv = genvar `:real^2` in
278 let pth = CONV_RULE (BINOP_CONV(BINDER_CONV(RAND_CONV(GEN_ALPHA_CONV gv))))
279 QUANTIFY_PAD2D3D_THM in
280 let conv1 = GEN_REWRITE_CONV I [pth]
281 and dest_quant tm = try dest_forall tm with Failure _ -> dest_exists tm in
284 let name = fst(dest_var(fst(dest_quant tm))) in
285 let ty = snd(dest_var(fst(dest_quant(rand(concl th))))) in
286 CONV_RULE(RAND_CONV(GEN_ALPHA_CONV(mk_var(name,ty)))) th;;
289 let pad2d3d_tm = `pad2d3d`
290 and pths = [LINEAR_PAD2D3D; INJECTIVE_PAD2D3D; NORM_PAD2D3D]
292 (`{} = IMAGE pad2d3d {} /\
293 vec 0 = pad2d3d(vec 0)`,
294 REWRITE_TAC[IMAGE_CLAUSES] THEN MESON_TAC[LINEAR_PAD2D3D; LINEAR_0]) in
296 GEN_REWRITE_TAC REDEPTH_CONV [LINEAR_INVARIANTS pad2d3d_tm pths] in
297 fun gl -> (GEN_REWRITE_TAC ONCE_DEPTH_CONV [cth] THEN
298 CONV_TAC(DEPTH_CONV PAD2D3D_QUANTIFY_CONV) THEN
301 (* ------------------------------------------------------------------------- *)
302 (* The notion of a plane, and using it to characterize coplanarity. *)
303 (* ------------------------------------------------------------------------- *)
305 let plane = new_definition
306 `plane x = (?u v w. ~(collinear {u,v,w}) /\ x = affine hull {u,v,w})`;;
308 let PLANE_TRANSLATION_EQ = prove
309 (`!a:real^N s. plane(IMAGE (\x. a + x) s) <=> plane s`,
310 REWRITE_TAC[plane] THEN GEOM_TRANSLATE_TAC[]);;
312 let PLANE_TRANSLATION = prove
313 (`!a:real^N s. plane s ==> plane(IMAGE (\x. a + x) s)`,
314 REWRITE_TAC[PLANE_TRANSLATION_EQ]);;
316 add_translation_invariants [PLANE_TRANSLATION_EQ];;
318 let PLANE_LINEAR_IMAGE_EQ = prove
319 (`!f:real^M->real^N p.
320 linear f /\ (!x y. f x = f y ==> x = y)
321 ==> (plane(IMAGE f p) <=> plane p)`,
322 REPEAT STRIP_TAC THEN REWRITE_TAC[plane] THEN
323 MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC
324 `?u. u IN IMAGE f (:real^M) /\
325 ?v. v IN IMAGE f (:real^M) /\
326 ?w. w IN IMAGE (f:real^M->real^N) (:real^M) /\
327 ~collinear {u, v, w} /\ IMAGE f p = affine hull {u, v, w}` THEN
329 [REWRITE_TAC[RIGHT_AND_EXISTS_THM; IN_IMAGE; IN_UNIV] THEN
330 REPEAT(AP_TERM_TAC THEN ABS_TAC) THEN
331 EQ_TAC THEN DISCH_TAC THEN ASM_REWRITE_TAC[] THEN
332 SUBGOAL_THEN `{u,v,w} SUBSET IMAGE (f:real^M->real^N) p` MP_TAC THENL
333 [ASM_REWRITE_TAC[HULL_SUBSET]; SET_TAC[]];
334 REWRITE_TAC[EXISTS_IN_IMAGE; IN_UNIV] THEN
335 REWRITE_TAC[SET_RULE `{f a,f b,f c} = IMAGE f {a,b,c}`] THEN
336 ASM_SIMP_TAC[AFFINE_HULL_LINEAR_IMAGE] THEN
337 REPEAT(AP_TERM_TAC THEN ABS_TAC) THEN BINOP_TAC THENL
338 [ASM_MESON_TAC[COLLINEAR_LINEAR_IMAGE_EQ]; ASM SET_TAC[]]]);;
340 let PLANE_LINEAR_IMAGE = prove
341 (`!f:real^M->real^N p.
342 linear f /\ plane p /\ (!x y. f x = f y ==> x = y)
343 ==> plane(IMAGE f p)`,
344 MESON_TAC[PLANE_LINEAR_IMAGE_EQ]);;
346 add_linear_invariants [PLANE_LINEAR_IMAGE_EQ];;
348 let AFFINE_PLANE = prove
349 (`!p. plane p ==> affine p`,
350 SIMP_TAC[plane; LEFT_IMP_EXISTS_THM; AFFINE_AFFINE_HULL]);;
352 let ROTATION_PLANE_HORIZONTAL = prove
354 ==> ?a f. orthogonal_transformation f /\ det(matrix f) = &1 /\
355 IMAGE f (IMAGE (\x. a + x) s) = {z:real^3 | z$3 = &0}`,
357 (`span {z:real^3 | z$3 = &0} = {z:real^3 | z$3 = &0}`,
358 REWRITE_TAC[SPAN_EQ_SELF; subspace; IN_ELIM_THM] THEN
359 SIMP_TAC[VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT; VEC_COMPONENT;
360 DIMINDEX_3; ARITH] THEN REAL_ARITH_TAC) in
361 REPEAT STRIP_TAC THEN
362 FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [plane]) THEN
363 REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN
364 MAP_EVERY X_GEN_TAC [`a:real^3`; `b:real^3`; `c:real^3`] THEN
366 ASM_CASES_TAC t THENL [ASM_REWRITE_TAC[COLLINEAR_2; INSERT_AC];
368 [`a:real^3 = b`; `a:real^3 = c`; `b:real^3 = c`] THEN
369 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC SUBST1_TAC) THEN
370 ASM_SIMP_TAC[AFFINE_HULL_INSERT_SPAN; IN_INSERT; NOT_IN_EMPTY] THEN
371 EXISTS_TAC `--a:real^3` THEN
372 REWRITE_TAC[SET_RULE `IMAGE (\x:real^3. --a + x) {a + x | x | x IN s} =
373 IMAGE (\x. --a + a + x) s`] THEN
374 REWRITE_TAC[VECTOR_ARITH `--a + a + x:real^3 = x`; IMAGE_ID] THEN
375 REWRITE_TAC[SET_RULE `{x - a:real^x | x = b \/ x = c} = {b - a,c - a}`] THEN
376 MP_TAC(ISPEC `span{b - a:real^3,c - a}`
377 ROTATION_LOWDIM_HORIZONTAL) THEN
378 REWRITE_TAC[DIMINDEX_3] THEN ANTS_TAC THENL
379 [MATCH_MP_TAC LET_TRANS THEN
380 EXISTS_TAC `CARD{b - a:real^3,c - a}` THEN
381 SIMP_TAC[DIM_SPAN; DIM_LE_CARD; FINITE_RULES] THEN
382 SIMP_TAC[CARD_CLAUSES; FINITE_RULES] THEN ARITH_TAC;
384 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `f:real^3->real^3` THEN
385 STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
386 FIRST_ASSUM(ASSUME_TAC o MATCH_MP ORTHOGONAL_TRANSFORMATION_LINEAR) THEN
387 ASM_SIMP_TAC[GSYM SPAN_LINEAR_IMAGE] THEN
388 GEN_REWRITE_TAC RAND_CONV [GSYM lemma] THEN
389 MATCH_MP_TAC DIM_EQ_SPAN THEN CONJ_TAC THENL
390 [ASM_MESON_TAC[IMAGE_SUBSET; SPAN_INC; SUBSET_TRANS]; ALL_TAC] THEN
391 MATCH_MP_TAC LE_TRANS THEN EXISTS_TAC `2` THEN CONJ_TAC THENL
392 [MP_TAC(ISPECL [`{z:real^3 | z$3 = &0}`; `(:real^3)`] DIM_EQ_SPAN) THEN
393 REWRITE_TAC[SUBSET_UNIV; DIM_UNIV; DIMINDEX_3; lemma] THEN
394 MATCH_MP_TAC(TAUT `~r /\ (~p ==> q) ==> (q ==> r) ==> p`) THEN
395 REWRITE_TAC[ARITH_RULE `~(x <= 2) <=> 3 <= x`] THEN
396 REWRITE_TAC[EXTENSION; SPAN_UNIV; IN_ELIM_THM] THEN
397 DISCH_THEN(MP_TAC o SPEC `vector[&0;&0;&1]:real^3`) THEN
398 REWRITE_TAC[IN_UNIV; VECTOR_3] THEN REAL_ARITH_TAC;
400 MATCH_MP_TAC LE_TRANS THEN EXISTS_TAC `dim {b - a:real^3,c - a}` THEN
402 [ALL_TAC; ASM_MESON_TAC[LE_REFL; DIM_INJECTIVE_LINEAR_IMAGE;
403 ORTHOGONAL_TRANSFORMATION_INJECTIVE]] THEN
404 MP_TAC(ISPEC `{b - a:real^3,c - a}` INDEPENDENT_BOUND_GENERAL) THEN
405 SIMP_TAC[CARD_CLAUSES; FINITE_RULES; IN_SING; NOT_IN_EMPTY] THEN
406 ASM_REWRITE_TAC[VECTOR_ARITH `b - a:real^3 = c - a <=> b = c`; ARITH] THEN
407 DISCH_THEN MATCH_MP_TAC THEN
408 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE (RAND_CONV o RAND_CONV)
409 [SET_RULE `{a,b,c} = {b,a,c}`]) THEN
410 REWRITE_TAC[] THEN ONCE_REWRITE_TAC[COLLINEAR_3] THEN
411 REWRITE_TAC[independent; CONTRAPOS_THM; dependent] THEN
412 REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY; RIGHT_OR_DISTRIB] THEN
413 REWRITE_TAC[EXISTS_OR_THM; UNWIND_THM2] THEN
414 ASM_SIMP_TAC[SET_RULE `~(a = b) ==> {a,b} DELETE b = {a}`;
415 SET_RULE `~(a = b) ==> {a,b} DELETE a = {b}`;
416 VECTOR_ARITH `b - a:real^3 = c - a <=> b = c`] THEN
417 REWRITE_TAC[SPAN_BREAKDOWN_EQ; SPAN_EMPTY; IN_SING] THEN
418 ONCE_REWRITE_TAC[VECTOR_SUB_EQ] THEN MESON_TAC[COLLINEAR_LEMMA; INSERT_AC]);;
420 let ROTATION_HORIZONTAL_PLANE = prove
422 ==> ?a f. orthogonal_transformation f /\ det(matrix f) = &1 /\
423 IMAGE (\x. a + x) (IMAGE f {z:real^3 | z$3 = &0}) = p`,
424 REPEAT STRIP_TAC THEN
425 FIRST_X_ASSUM(MP_TAC o MATCH_MP ROTATION_PLANE_HORIZONTAL) THEN
426 DISCH_THEN(X_CHOOSE_THEN `a:real^3`
427 (X_CHOOSE_THEN `f:real^3->real^3` STRIP_ASSUME_TAC)) THEN
428 FIRST_ASSUM(X_CHOOSE_THEN `g:real^3->real^3` STRIP_ASSUME_TAC o MATCH_MP
429 ORTHOGONAL_TRANSFORMATION_INVERSE) THEN
430 MAP_EVERY EXISTS_TAC [`--a:real^3`; `g:real^3->real^3`] THEN
431 FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN
432 ASM_REWRITE_TAC[GSYM IMAGE_o; o_DEF; IMAGE_ID;
433 VECTOR_ARITH `--a + a + x:real^3 = x`] THEN
434 MATCH_MP_TAC(REAL_RING `!f. f * g = &1 /\ f = &1 ==> g = &1`) THEN
435 EXISTS_TAC `det(matrix(f:real^3->real^3))` THEN
436 REWRITE_TAC[GSYM DET_MUL] THEN
437 ASM_SIMP_TAC[GSYM MATRIX_COMPOSE; ORTHOGONAL_TRANSFORMATION_LINEAR] THEN
438 ASM_REWRITE_TAC[o_DEF; MATRIX_ID; DET_I]);;
442 ==> !s:real^N->bool. coplanar s <=> ?x. plane x /\ s SUBSET x`,
443 DISCH_TAC THEN GEN_TAC THEN REWRITE_TAC[coplanar; plane] THEN
444 CONV_TAC SYM_CONV THEN REWRITE_TAC[LEFT_AND_EXISTS_THM] THEN
445 ONCE_REWRITE_TAC[MESON[]
446 `(?x u v w. p x u v w) <=> (?u v w x. p x u v w)`] THEN
447 REWRITE_TAC[GSYM CONJ_ASSOC; RIGHT_EXISTS_AND_THM; UNWIND_THM2] THEN
448 EQ_TAC THENL [MESON_TAC[]; REWRITE_TAC[LEFT_IMP_EXISTS_THM]] THEN
449 MAP_EVERY X_GEN_TAC [`u:real^N`; `v:real^N`; `w:real^N`] THEN DISCH_TAC THEN
451 `s SUBSET {u + x:real^N | x | x IN span {y - u | y IN {v,w}}}`
453 [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP
454 (REWRITE_RULE[IMP_CONJ] SUBSET_TRANS)) THEN
455 REWRITE_TAC[AFFINE_HULL_INSERT_SUBSET_SPAN];
457 ONCE_REWRITE_TAC[SIMPLE_IMAGE_GEN] THEN
458 DISCH_THEN(MP_TAC o ISPEC `\x:real^N. x - u` o MATCH_MP IMAGE_SUBSET) THEN
459 REWRITE_TAC[GSYM IMAGE_o; o_DEF; VECTOR_ADD_SUB; IMAGE_ID; SIMPLE_IMAGE] THEN
460 REWRITE_TAC[IMAGE_CLAUSES] THEN
461 MP_TAC(ISPECL [`{v - u:real^N,w - u}`; `2`] LOWDIM_EXPAND_BASIS) THEN
463 [ASM_REWRITE_TAC[] THEN MATCH_MP_TAC LE_TRANS THEN
464 EXISTS_TAC `CARD{v - u:real^N,w - u}` THEN
465 SIMP_TAC[DIM_LE_CARD; FINITE_INSERT; FINITE_RULES] THEN
466 SIMP_TAC[CARD_CLAUSES; FINITE_RULES] THEN ARITH_TAC;
468 DISCH_THEN(X_CHOOSE_THEN `t:real^N->bool`
469 (CONJUNCTS_THEN2 MP_TAC STRIP_ASSUME_TAC)) THEN
470 CONV_TAC(LAND_CONV HAS_SIZE_CONV) THEN
471 REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN
472 MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real^N`] THEN
473 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC SUBST_ALL_TAC) THEN
474 UNDISCH_TAC `span {v - u, w - u} SUBSET span {a:real^N, b}` THEN
475 REWRITE_TAC[GSYM IMP_CONJ_ALT] THEN
476 DISCH_THEN(ASSUME_TAC o MATCH_MP SUBSET_TRANS) THEN
477 MAP_EVERY EXISTS_TAC [`u:real^N`; `u + a:real^N`; `u + b:real^N`] THEN
479 [REWRITE_TAC[COLLINEAR_3; COLLINEAR_LEMMA;
480 VECTOR_ARITH `--x = vec 0 <=> x = vec 0`;
481 VECTOR_ARITH `u - (u + a):real^N = --a`;
482 VECTOR_ARITH `(u + b) - (u + a):real^N = b - a`] THEN
483 REWRITE_TAC[DE_MORGAN_THM; VECTOR_SUB_EQ;
484 VECTOR_ARITH `b - a = c % -- a <=> (c - &1) % a + &1 % b = vec 0`] THEN
485 ASM_REWRITE_TAC[] THEN CONJ_TAC THENL
486 [ASM_MESON_TAC[IN_INSERT; INDEPENDENT_NONZERO]; ALL_TAC] THEN
487 DISCH_THEN(X_CHOOSE_TAC `u:real`) THEN
488 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [independent]) THEN
489 REWRITE_TAC[DEPENDENT_EXPLICIT] THEN
490 MAP_EVERY EXISTS_TAC [`{a:real^N,b}`;
491 `\x:real^N. if x = a then u - &1 else &1`] THEN
492 REWRITE_TAC[FINITE_INSERT; FINITE_RULES; SUBSET_REFL] THEN
494 [EXISTS_TAC `b:real^N` THEN ASM_REWRITE_TAC[IN_INSERT] THEN
497 SIMP_TAC[VSUM_CLAUSES; FINITE_RULES] THEN
498 ASM_REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY; VECTOR_ADD_RID];
500 W(MP_TAC o PART_MATCH (lhs o rand) AFFINE_HULL_INSERT_SPAN o rand o snd) THEN
502 [REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY] THEN
503 REWRITE_TAC[VECTOR_ARITH `u = u + a <=> a = vec 0`] THEN
504 ASM_MESON_TAC[INDEPENDENT_NONZERO; IN_INSERT];
506 DISCH_THEN SUBST1_TAC THEN
507 FIRST_ASSUM(MP_TAC o ISPEC `\x:real^N. u + x` o MATCH_MP IMAGE_SUBSET) THEN
508 REWRITE_TAC[GSYM IMAGE_o; o_DEF; IMAGE_ID;
509 ONCE_REWRITE_RULE[VECTOR_ADD_SYM] VECTOR_SUB_ADD] THEN
510 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] SUBSET_TRANS) THEN
511 REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN
512 REWRITE_TAC[SIMPLE_IMAGE; IMAGE_CLAUSES; VECTOR_ADD_SUB] THEN
515 let COPLANAR_DET_EQ_0 = prove
516 (`!v0 v1 (v2: real^3) v3.
517 coplanar {v0,v1,v2,v3} <=>
518 det(vector[v1 - v0; v2 - v0; v3 - v0]) = &0`,
519 REPEAT GEN_TAC THEN REWRITE_TAC[DET_EQ_0_RANK; RANK_ROW] THEN
520 REWRITE_TAC[rows; row; LAMBDA_ETA] THEN
521 ONCE_REWRITE_TAC[SIMPLE_IMAGE_GEN] THEN
522 REWRITE_TAC[GSYM numseg; DIMINDEX_3] THEN
523 CONV_TAC(ONCE_DEPTH_CONV NUMSEG_CONV) THEN
524 SIMP_TAC[IMAGE_CLAUSES; coplanar; VECTOR_3] THEN EQ_TAC THENL
525 [REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN
526 MAP_EVERY X_GEN_TAC [`a:real^3`; `b:real^3`; `c:real^3`] THEN
527 W(MP_TAC o PART_MATCH lhand AFFINE_HULL_INSERT_SUBSET_SPAN o
528 rand o lhand o snd) THEN
529 REWRITE_TAC[GSYM IMP_CONJ_ALT] THEN
530 DISCH_THEN(MP_TAC o MATCH_MP SUBSET_TRANS) THEN
531 DISCH_THEN(MP_TAC o ISPEC `\x:real^3. x - a` o MATCH_MP IMAGE_SUBSET) THEN
532 ONCE_REWRITE_TAC[SIMPLE_IMAGE_GEN] THEN
533 REWRITE_TAC[IMAGE_CLAUSES; GSYM IMAGE_o; o_DEF; VECTOR_ADD_SUB; IMAGE_ID;
535 REWRITE_TAC[INSERT_SUBSET] THEN STRIP_TAC THEN
536 GEN_REWRITE_TAC LAND_CONV [GSYM DIM_SPAN] THEN MATCH_MP_TAC LET_TRANS THEN
537 EXISTS_TAC `CARD {b - a:real^3,c - a}` THEN
539 [MATCH_MP_TAC SPAN_CARD_GE_DIM;
540 SIMP_TAC[CARD_CLAUSES; FINITE_RULES] THEN ARITH_TAC] THEN
541 REWRITE_TAC[FINITE_INSERT; FINITE_RULES] THEN
542 GEN_REWRITE_TAC RAND_CONV [GSYM SPAN_SPAN] THEN
543 MATCH_MP_TAC SPAN_MONO THEN REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET] THEN
544 MP_TAC(VECTOR_ARITH `!x y:real^3. x - y = (x - a) - (y - a)`) THEN
545 DISCH_THEN(fun th -> REPEAT CONJ_TAC THEN
546 GEN_REWRITE_TAC LAND_CONV [th]) THEN
547 MATCH_MP_TAC SPAN_SUB THEN ASM_REWRITE_TAC[];
549 MP_TAC(ISPECL [`{v1 - v0,v2 - v0,v3 - v0}:real^3->bool`; `2`]
550 LOWDIM_EXPAND_BASIS) THEN
551 ASM_REWRITE_TAC[ARITH_RULE `n <= 2 <=> n < 3`; DIMINDEX_3; ARITH] THEN
552 DISCH_THEN(X_CHOOSE_THEN `t:real^3->bool`
553 (CONJUNCTS_THEN2 MP_TAC STRIP_ASSUME_TAC)) THEN
554 CONV_TAC(LAND_CONV HAS_SIZE_CONV) THEN
555 REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN
556 MAP_EVERY X_GEN_TAC [`a:real^3`; `b:real^3`] THEN
557 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC SUBST_ALL_TAC) THEN
558 SIMP_TAC[COPLANAR; DIMINDEX_3; ARITH; plane] THEN
559 MAP_EVERY EXISTS_TAC [`v0:real^3`; `v0 + a:real^3`; `v0 + b:real^3`] THEN
560 W(MP_TAC o PART_MATCH (lhs o rand) AFFINE_HULL_INSERT_SPAN o
563 [REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY] THEN
564 REWRITE_TAC[VECTOR_ARITH `u = u + a <=> a = vec 0`] THEN
565 ASM_MESON_TAC[INDEPENDENT_NONZERO; IN_INSERT];
567 DISCH_THEN SUBST1_TAC THEN
568 ONCE_REWRITE_TAC[SIMPLE_IMAGE_GEN] THEN
569 REWRITE_TAC[SIMPLE_IMAGE; IMAGE_CLAUSES; IMAGE_ID; VECTOR_ADD_SUB] THEN
570 MATCH_MP_TAC SUBSET_TRANS THEN EXISTS_TAC
571 `IMAGE (\v:real^3. v0 + v) (span{v1 - v0, v2 - v0, v3 - v0})` THEN
572 ASM_SIMP_TAC[IMAGE_SUBSET] THEN
573 REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET; IN_IMAGE] THEN CONJ_TAC THENL
574 [EXISTS_TAC `vec 0:real^3` THEN REWRITE_TAC[SPAN_0] THEN VECTOR_ARITH_TAC;
575 REWRITE_TAC[VECTOR_ARITH `v1:real^N = v0 + x <=> x = v1 - v0`] THEN
576 REWRITE_TAC[UNWIND_THM2] THEN REPEAT CONJ_TAC THEN
577 MATCH_MP_TAC SPAN_SUPERSET THEN REWRITE_TAC[IN_INSERT]]]);;
579 let COPLANAR_CROSS_DOT = prove
580 (`!v w x y. coplanar {v,w,x,y} <=> ((w - v) cross (x - v)) dot (y - v) = &0`,
581 REWRITE_TAC[COPLANAR_DET_EQ_0; GSYM DOT_CROSS_DET] THEN
582 MESON_TAC[CROSS_TRIPLE; DOT_SYM]);;
584 let PLANE_AFFINE_HULL_3 = prove
585 (`!a b c:real^N. plane(affine hull {a,b,c}) <=> ~collinear{a,b,c}`,
586 REWRITE_TAC[plane] THEN MESON_TAC[COLLINEAR_AFFINE_HULL_COLLINEAR]);;
588 let AFFINE_HULL_3_GENERATED = prove
590 s SUBSET affine hull {u,v,w} /\ ~collinear s
591 ==> affine hull {u,v,w} = affine hull s`,
592 REWRITE_TAC[COLLINEAR_AFF_DIM; INT_NOT_LE] THEN REPEAT STRIP_TAC THEN
593 CONV_TAC SYM_CONV THEN
594 GEN_REWRITE_TAC RAND_CONV [GSYM HULL_HULL] THEN
595 MATCH_MP_TAC AFF_DIM_EQ_AFFINE_HULL THEN ASM_REWRITE_TAC[] THEN
596 MATCH_MP_TAC INT_LE_TRANS THEN EXISTS_TAC `&2:int` THEN
597 CONJ_TAC THENL [ALL_TAC; ASM_INT_ARITH_TAC] THEN
598 REWRITE_TAC[AFF_DIM_AFFINE_HULL] THEN
599 W(MP_TAC o PART_MATCH (lhand o rand) AFF_DIM_LE_CARD o lhand o snd) THEN
600 REWRITE_TAC[FINITE_INSERT; FINITE_EMPTY] THEN
601 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] INT_LE_TRANS) THEN
602 REWRITE_TAC[INT_LE_SUB_RADD; INT_OF_NUM_ADD; INT_OF_NUM_LE] THEN
603 SIMP_TAC[CARD_CLAUSES; FINITE_INSERT; FINITE_EMPTY] THEN ARITH_TAC);;
605 (* ------------------------------------------------------------------------- *)
606 (* Additional WLOG tactic to rotate any plane p to {z | z$3 = &0}. *)
607 (* ------------------------------------------------------------------------- *)
609 let GEOM_HORIZONTAL_PLANE_RULE =
611 (TAUT `(p ==> (x <=> x')) /\ (~p ==> (x <=> T)) ==> (x' ==> x)`)
613 (`!a f. orthogonal_transformation (f:real^N->real^N)
614 ==> ((!P. (!x. P x) <=> (!x. P (a + f x))) /\
615 (!P. (?x. P x) <=> (?x. P (a + f x))) /\
616 (!Q. (!s. Q s) <=> (!s. Q (IMAGE (\x. a + x) (IMAGE f s)))) /\
617 (!Q. (?s. Q s) <=> (?s. Q (IMAGE (\x. a + x) (IMAGE f s))))) /\
619 IMAGE (\x. a + x) (IMAGE f {x | P(a + f x)}))`,
620 REPEAT GEN_TAC THEN DISCH_TAC THEN
621 MP_TAC(ISPEC `(\x. a + x) o (f:real^N->real^N)`
622 QUANTIFY_SURJECTION_THM) THEN REWRITE_TAC[o_THM; IMAGE_o] THEN
623 DISCH_THEN MATCH_MP_TAC THEN
624 ASM_MESON_TAC[ORTHOGONAL_TRANSFORMATION_SURJECTIVE;
625 VECTOR_ARITH `a + (x - a:real^N) = x`])
627 (`!a f. {} = IMAGE (\x:real^3. a + x) (IMAGE f {})`,
628 REWRITE_TAC[IMAGE_CLAUSES])
631 orthogonal_transformation f /\ det(matrix f) = &1
633 (!x y. f x = f y ==> x = y) /\
635 (!x. norm(f x) = norm x) /\
636 (2 <= dimindex(:3) ==> det(matrix f) = &1)`,
637 GEN_TAC THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN REPEAT CONJ_TAC THENL
638 [ASM_SIMP_TAC[ORTHOGONAL_TRANSFORMATION_LINEAR];
639 ASM_MESON_TAC[ORTHOGONAL_TRANSFORMATION_INJECTIVE];
640 ASM_MESON_TAC[ORTHOGONAL_TRANSFORMATION_SURJECTIVE];
641 ASM_MESON_TAC[ORTHOGONAL_TRANSFORMATION]])
643 `(!a f. q a f ==> (p <=> p' a f))
644 ==> ((?a f. q a f) ==> (p <=> !a f. q a f ==> p' a f))` in
646 let x,bod = dest_forall tm in
647 let th1 = EXISTS_GENVAR_RULE
648 (UNDISCH(ISPEC x ROTATION_HORIZONTAL_PLANE)) in
649 let [a;f],tm1 = strip_exists(concl th1) in
650 let [th_orth;th_det;th_im] = CONJUNCTS(ASSUME tm1) in
651 let th2 = PROVE_HYP th_orth (UNDISCH(ISPECL [a;f] pth)) in
652 let th3 = (EXPAND_QUANTS_CONV(ASSUME(concl th2)) THENC
653 SUBS_CONV[GSYM th_im; ISPECL [a;f] cth]) bod in
654 let th4 = PROVE_HYP th2 th3 in
655 let th5 = TRANSLATION_INVARIANTS a in
656 let th6 = GEN_REWRITE_RULE (RAND_CONV o REDEPTH_CONV)
657 [ASSUME(concl th5)] th4 in
658 let th7 = PROVE_HYP th5 th6 in
659 let th8s = CONJUNCTS(MATCH_MP oth (CONJ th_orth th_det)) in
660 let th9 = LINEAR_INVARIANTS f th8s in
661 let th10 = GEN_REWRITE_RULE (RAND_CONV o REDEPTH_CONV) [th9] th7 in
662 let th11 = if intersect (frees(concl th10)) [a;f] = []
663 then PROVE_HYP th1 (itlist SIMPLE_CHOOSE [a;f] th10)
664 else MP (MATCH_MP fth (GENL [a;f] (DISCH_ALL th10))) th1 in
665 let th12 = REWRITE_CONV[ASSUME(mk_neg(hd(hyp th11)))] bod in
666 let th13 = ifn(CONJ (DISCH_ALL th11) (DISCH_ALL th12)) in
667 let th14 = MATCH_MP MONO_FORALL (GEN x th13) in
668 GEN_REWRITE_RULE (TRY_CONV o LAND_CONV) [FORALL_SIMP] th14;;
670 let GEOM_HORIZONTAL_PLANE_TAC p =
672 let avs,bod = strip_forall w
673 and avs' = subtract (frees w) (freesl(map (concl o snd) asl)) in
674 let avs,bod = strip_forall w in
675 MAP_EVERY X_GEN_TAC avs THEN
676 MAP_EVERY (fun t -> SPEC_TAC(t,t)) (rev(subtract (avs@avs') [p])) THEN
678 W(MATCH_MP_TAC o GEOM_HORIZONTAL_PLANE_RULE o snd));;
680 (* ------------------------------------------------------------------------- *)
681 (* Affsign and its special cases, with invariance theorems. *)
682 (* ------------------------------------------------------------------------- *)
684 let lin_combo = new_definition
685 `lin_combo V f = vsum V (\v. f v % (v:real^N))`;;
687 let affsign = new_definition
688 `affsign sgn s t (v:real^A) <=>
689 (?f. (v = lin_combo (s UNION t) f) /\
690 (!w. t w ==> sgn (f w)) /\
691 (sum (s UNION t) f = &1))`;;
693 let sgn_gt = new_definition `sgn_gt = (\t. (&0 < t))`;;
694 let sgn_ge = new_definition `sgn_ge = (\t. (&0 <= t))`;;
695 let sgn_lt = new_definition `sgn_lt = (\t. (t < &0))`;;
696 let sgn_le = new_definition `sgn_le = (\t. (t <= &0))`;;
698 let aff_gt_def = new_definition `aff_gt = affsign sgn_gt`;;
699 let aff_ge_def = new_definition `aff_ge = affsign sgn_ge`;;
700 let aff_lt_def = new_definition `aff_lt = affsign sgn_lt`;;
701 let aff_le_def = new_definition `aff_le = affsign sgn_le`;;
705 {y | ?f. y = vsum (s UNION t) (\v. f v % v) /\
706 (!w. w IN t ==> sgn(f w)) /\
707 sum (s UNION t) f = &1}`,
708 REWRITE_TAC[FUN_EQ_THM; affsign; lin_combo; IN_ELIM_THM] THEN
711 let AFFSIGN_ALT = prove
713 {y | ?f. (!w. w IN (s UNION t) ==> w IN t ==> sgn(f w)) /\
714 sum (s UNION t) f = &1 /\
715 vsum (s UNION t) (\v. f v % v) = y}`,
716 REWRITE_TAC[SET_RULE `(w IN (s UNION t) ==> w IN t ==> P w) <=>
717 (w IN t ==> P w)`] THEN
718 REWRITE_TAC[AFFSIGN; EXTENSION; IN_ELIM_THM] THEN MESON_TAC[]);;
720 let IN_AFFSIGN = prove
721 (`y IN affsign sgn s t <=>
722 ?u. (!x. x IN t ==> sgn(u x)) /\
723 sum (s UNION t) u = &1 /\
724 vsum (s UNION t) (\x. u(x) % x) = y`,
725 REWRITE_TAC[AFFSIGN; IN_ELIM_THM] THEN SET_TAC[]);;
727 let AFFSIGN_DISJOINT_DIFF = prove
728 (`!s t. affsign sgn s t = affsign sgn (s DIFF t) t`,
729 REWRITE_TAC[AFFSIGN; SET_RULE `(s DIFF t) UNION t = s UNION t`]);;
731 let AFF_GE_DISJOINT_DIFF = prove
732 (`!s t. aff_ge s t = aff_ge (s DIFF t) t`,
733 REWRITE_TAC[aff_ge_def] THEN MATCH_ACCEPT_TAC AFFSIGN_DISJOINT_DIFF);;
735 let AFFSIGN_INJECTIVE_LINEAR_IMAGE = prove
736 (`!f:real^M->real^N sgn s t v.
737 linear f /\ (!x y. f x = f y ==> x = y)
738 ==> (affsign sgn (IMAGE f s) (IMAGE f t) =
739 IMAGE f (affsign sgn s t))`,
741 (`vsum s (\x. u x % x) = vsum {x | x IN s /\ ~(u x = &0)} (\x. u x % x)`,
742 MATCH_MP_TAC VSUM_SUPERSET THEN SIMP_TAC[SUBSET; IN_ELIM_THM] THEN
743 REWRITE_TAC[TAUT `p /\ ~(p /\ ~q) <=> p /\ q`] THEN
744 SIMP_TAC[o_THM; VECTOR_MUL_LZERO]) in
746 (`!f:real^M->real^N s.
747 linear f /\ (!x y. f x = f y ==> x = y)
748 ==> (sum(IMAGE f s) u = &1 /\ vsum(IMAGE f s) (\x. u x % x) = y <=>
749 sum s (u o f) = &1 /\ f(vsum s (\x. (u o f) x % x)) = y)`,
750 REPEAT STRIP_TAC THEN
751 W(MP_TAC o PART_MATCH (lhs o rand) SUM_IMAGE o funpow 3 lhand o snd) THEN
752 ANTS_TAC THENL [ASM_MESON_TAC[]; DISCH_THEN SUBST1_TAC] THEN
753 MATCH_MP_TAC(MESON[] `(p ==> z = x) ==> (p /\ x = y <=> p /\ z = y)`) THEN
754 DISCH_TAC THEN ONCE_REWRITE_TAC[lemma0] THEN
756 `{y | y IN IMAGE (f:real^M->real^N) s /\ ~(u y = &0)} =
757 IMAGE f {x | x IN s /\ ~(u(f x) = &0)}`
758 SUBST1_TAC THENL [ASM SET_TAC[]; CONV_TAC SYM_CONV] THEN
759 SUBGOAL_THEN `FINITE {x | x IN s /\ ~(u((f:real^M->real^N) x) = &0)}`
761 [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE
762 (LAND_CONV o RATOR_CONV o RATOR_CONV) [sum]) THEN
763 ONCE_REWRITE_TAC[ITERATE_EXPAND_CASES] THEN
764 REWRITE_TAC[GSYM sum; support; NEUTRAL_REAL_ADD; o_THM] THEN
765 COND_CASES_TAC THEN ASM_REWRITE_TAC[REAL_OF_NUM_EQ; ARITH_EQ];
766 W(MP_TAC o PART_MATCH (lhs o rand) VSUM_IMAGE o lhand o snd) THEN
767 ANTS_TAC THENL [ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[]; ALL_TAC] THEN
768 DISCH_THEN SUBST1_TAC THEN REWRITE_TAC[o_DEF] THEN
769 ASM_SIMP_TAC[LINEAR_VSUM; o_DEF; GSYM LINEAR_CMUL]]) in
770 REPEAT GEN_TAC THEN DISCH_TAC THEN REWRITE_TAC[EXTENSION; IN_AFFSIGN] THEN
771 REWRITE_TAC[FORALL_IN_IMAGE] THEN REWRITE_TAC[IN_IMAGE; IN_AFFSIGN] THEN
772 REWRITE_TAC[GSYM IMAGE_UNION] THEN
773 FIRST_ASSUM(fun th -> REWRITE_TAC[MATCH_MP lemma1 th]) THEN
774 X_GEN_TAC `y:real^N` THEN EQ_TAC THENL
775 [DISCH_THEN(X_CHOOSE_THEN `u:real^N->real` STRIP_ASSUME_TAC) THEN
776 EXISTS_TAC `vsum (s UNION t) (\x. (u o (f:real^M->real^N)) x % x)` THEN
777 ASM_REWRITE_TAC[] THEN
778 EXISTS_TAC `(u:real^N->real) o (f:real^M->real^N)` THEN
779 ASM_REWRITE_TAC[] THEN ASM_REWRITE_TAC[o_THM];
780 MP_TAC(ISPEC `f:real^M->real^N` LINEAR_INJECTIVE_LEFT_INVERSE) THEN
781 ASM_REWRITE_TAC[FUN_EQ_THM; o_THM; I_THM] THEN
782 DISCH_THEN(X_CHOOSE_THEN `g:real^N->real^M` STRIP_ASSUME_TAC) THEN
783 DISCH_THEN(X_CHOOSE_THEN `x:real^M`
784 (CONJUNCTS_THEN2 SUBST1_TAC MP_TAC)) THEN
785 DISCH_THEN(X_CHOOSE_THEN `u:real^M->real` STRIP_ASSUME_TAC) THEN
786 EXISTS_TAC `(u:real^M->real) o (g:real^N->real^M)` THEN
787 ASM_REWRITE_TAC[o_DEF; ETA_AX]]);;
789 let AFF_GE_INJECTIVE_LINEAR_IMAGE = prove
790 (`!f:real^M->real^N s t.
791 linear f /\ (!x y. f x = f y ==> x = y)
792 ==> aff_ge (IMAGE f s) (IMAGE f t) = IMAGE f (aff_ge s t)`,
793 REWRITE_TAC[aff_ge_def; AFFSIGN_INJECTIVE_LINEAR_IMAGE]);;
795 let AFF_GT_INJECTIVE_LINEAR_IMAGE = prove
796 (`!f:real^M->real^N s t.
797 linear f /\ (!x y. f x = f y ==> x = y)
798 ==> aff_gt (IMAGE f s) (IMAGE f t) = IMAGE f (aff_gt s t)`,
799 REWRITE_TAC[aff_gt_def; AFFSIGN_INJECTIVE_LINEAR_IMAGE]);;
801 let AFF_LE_INJECTIVE_LINEAR_IMAGE = prove
802 (`!f:real^M->real^N s t.
803 linear f /\ (!x y. f x = f y ==> x = y)
804 ==> aff_le (IMAGE f s) (IMAGE f t) = IMAGE f (aff_le s t)`,
805 REWRITE_TAC[aff_le_def; AFFSIGN_INJECTIVE_LINEAR_IMAGE]);;
807 let AFF_LT_INJECTIVE_LINEAR_IMAGE = prove
808 (`!f:real^M->real^N s t.
809 linear f /\ (!x y. f x = f y ==> x = y)
810 ==> aff_lt (IMAGE f s) (IMAGE f t) = IMAGE f (aff_lt s t)`,
811 REWRITE_TAC[aff_lt_def; AFFSIGN_INJECTIVE_LINEAR_IMAGE]);;
813 add_linear_invariants
814 [AFFSIGN_INJECTIVE_LINEAR_IMAGE;
815 AFF_GE_INJECTIVE_LINEAR_IMAGE;
816 AFF_GT_INJECTIVE_LINEAR_IMAGE;
817 AFF_LE_INJECTIVE_LINEAR_IMAGE;
818 AFF_LT_INJECTIVE_LINEAR_IMAGE];;
820 let IN_AFFSIGN_TRANSLATION = prove
821 (`!sgn s t a v:real^N.
823 ==> affsign sgn (IMAGE (\x. a + x) s) (IMAGE (\x. a + x) t) (a + v)`,
824 REPEAT GEN_TAC THEN REWRITE_TAC[affsign; lin_combo] THEN
825 ONCE_REWRITE_TAC[SET_RULE `(!x. s x ==> p x) <=> (!x. x IN s ==> p x)`] THEN
826 DISCH_THEN(X_CHOOSE_THEN `f:real^N->real`
827 (CONJUNCTS_THEN2 SUBST_ALL_TAC STRIP_ASSUME_TAC)) THEN
828 EXISTS_TAC `\x. (f:real^N->real)(x - a)` THEN
829 ASM_REWRITE_TAC[GSYM IMAGE_UNION] THEN REPEAT CONJ_TAC THENL
831 ASM_REWRITE_TAC[FORALL_IN_IMAGE; ETA_AX;
832 VECTOR_ARITH `(a + x) - a:real^N = x`];
833 W(MP_TAC o PART_MATCH (lhs o rand) SUM_IMAGE o lhs o snd) THEN
834 SIMP_TAC[VECTOR_ARITH `a + x:real^N = a + y <=> x = y`] THEN
835 ASM_REWRITE_TAC[o_DEF; VECTOR_ADD_SUB; ETA_AX]] THEN
836 MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC
837 `a + vsum {x | x IN s UNION t /\ ~(f x = &0)} (\v:real^N. f v % v)` THEN
839 [AP_TERM_TAC THEN MATCH_MP_TAC VSUM_SUPERSET THEN
840 REWRITE_TAC[VECTOR_MUL_EQ_0; SUBSET; IN_ELIM_THM] THEN MESON_TAC[];
842 MATCH_MP_TAC EQ_TRANS THEN
843 EXISTS_TAC `vsum (IMAGE (\x:real^N. a + x)
844 {x | x IN s UNION t /\ ~(f x = &0)})
845 (\v. f(v - a) % v)` THEN
848 CONV_TAC SYM_CONV THEN MATCH_MP_TAC VSUM_SUPERSET THEN
849 CONJ_TAC THENL [SET_TAC[]; ALL_TAC] THEN
850 ASM_REWRITE_TAC[IMP_CONJ; FORALL_IN_IMAGE; VECTOR_MUL_EQ_0] THEN
851 REWRITE_TAC[VECTOR_ADD_SUB] THEN SET_TAC[]] THEN
852 SUBGOAL_THEN `FINITE {x:real^N | x IN s UNION t /\ ~(f x = &0)}`
854 [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE
855 (LAND_CONV o RATOR_CONV o RATOR_CONV) [sum]) THEN
856 ONCE_REWRITE_TAC[ITERATE_EXPAND_CASES] THEN
857 REWRITE_TAC[GSYM sum; support; NEUTRAL_REAL_ADD] THEN
858 COND_CASES_TAC THEN ASM_REWRITE_TAC[REAL_OF_NUM_EQ; ARITH_EQ];
860 W(MP_TAC o PART_MATCH (lhs o rand) VSUM_IMAGE o rhs o snd) THEN
861 ASM_SIMP_TAC[VECTOR_ARITH `a + x:real^N = a + y <=> x = y`] THEN
862 DISCH_THEN(K ALL_TAC) THEN REWRITE_TAC[o_DEF; VECTOR_ADD_SUB] THEN
863 ASM_SIMP_TAC[VECTOR_ADD_LDISTRIB; VSUM_ADD] THEN
864 AP_THM_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[VSUM_RMUL] THEN
865 GEN_REWRITE_TAC LAND_CONV [GSYM VECTOR_MUL_LID] THEN
866 AP_THM_TAC THEN AP_TERM_TAC THEN FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN
867 MATCH_MP_TAC SUM_SUPERSET THEN SET_TAC[]);;
869 let AFFSIGN_TRANSLATION = prove
871 affsign sgn (IMAGE (\x. a + x) s) (IMAGE (\x. a + x) t) =
872 IMAGE (\x. a + x) (affsign sgn s t)`,
873 REPEAT GEN_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL
874 [REWRITE_TAC[SUBSET; IN] THEN GEN_TAC THEN
875 DISCH_THEN(MP_TAC o SPEC `--a:real^N` o
876 MATCH_MP IN_AFFSIGN_TRANSLATION) THEN
877 REWRITE_TAC[GSYM IMAGE_o; o_DEF; VECTOR_ARITH `--a + a + x:real^N = x`;
879 DISCH_TAC THEN REWRITE_TAC[IMAGE; IN_ELIM_THM] THEN
880 EXISTS_TAC `--a + x:real^N` THEN ASM_REWRITE_TAC[IN] THEN VECTOR_ARITH_TAC;
881 REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN GEN_TAC THEN REWRITE_TAC[IN] THEN
882 DISCH_THEN(MP_TAC o SPEC `a:real^N` o MATCH_MP IN_AFFSIGN_TRANSLATION) THEN
885 let AFF_GE_TRANSLATION = prove
887 aff_ge (IMAGE (\x. a + x) s) (IMAGE (\x. a + x) t) =
888 IMAGE (\x. a + x) (aff_ge s t)`,
889 REWRITE_TAC[aff_ge_def; AFFSIGN_TRANSLATION]);;
891 let AFF_GT_TRANSLATION = prove
893 aff_gt (IMAGE (\x. a + x) s) (IMAGE (\x. a + x) t) =
894 IMAGE (\x. a + x) (aff_gt s t)`,
895 REWRITE_TAC[aff_gt_def; AFFSIGN_TRANSLATION]);;
897 let AFF_LE_TRANSLATION = prove
899 aff_le (IMAGE (\x. a + x) s) (IMAGE (\x. a + x) t) =
900 IMAGE (\x. a + x) (aff_le s t)`,
901 REWRITE_TAC[aff_le_def; AFFSIGN_TRANSLATION]);;
903 let AFF_LT_TRANSLATION = prove
905 aff_lt (IMAGE (\x. a + x) s) (IMAGE (\x. a + x) t) =
906 IMAGE (\x. a + x) (aff_lt s t)`,
907 REWRITE_TAC[aff_lt_def; AFFSIGN_TRANSLATION]);;
909 add_translation_invariants
910 [AFFSIGN_TRANSLATION;
914 AFF_LT_TRANSLATION];;
916 (* ------------------------------------------------------------------------- *)
917 (* Automate special cases of affsign. *)
918 (* ------------------------------------------------------------------------- *)
921 REWRITE_TAC[DISJOINT_INSERT; DISJOINT_EMPTY] THEN
922 REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY; DE_MORGAN_THM] THEN
923 REPEAT STRIP_TAC THEN
924 REWRITE_TAC[aff_ge_def; aff_gt_def; aff_le_def; aff_lt_def;
925 sgn_ge; sgn_gt; sgn_le; sgn_lt; AFFSIGN_ALT] THEN
926 REWRITE_TAC[SET_RULE `(x INSERT s) UNION t = x INSERT (s UNION t)`] THEN
927 REWRITE_TAC[UNION_EMPTY] THEN
928 SIMP_TAC[FINITE_INSERT; FINITE_UNION; AFFINE_HULL_FINITE_STEP_GEN;
929 FINITE_EMPTY; RIGHT_EXISTS_AND_THM; REAL_LT_ADD;
930 REAL_LE_ADD; REAL_ARITH `&0 <= a / &2 <=> &0 <= a`;
931 REAL_ARITH `&0 < a / &2 <=> &0 < a`;
932 REAL_ARITH `a / &2 <= &0 <=> a <= &0`;
933 REAL_ARITH `a / &2 < &0 <=> a < &0`;
934 REAL_ARITH `a < &0 /\ b < &0 ==> a + b < &0`;
935 REAL_ARITH `a < &0 /\ b <= &0 ==> a + b <= &0`] THEN
936 ASM_REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY; real_ge] THEN
937 REWRITE_TAC[REAL_ARITH `x - y:real = z <=> x = y + z`;
938 VECTOR_ARITH `x - y:real^N = z <=> x = y + z`] THEN
939 REWRITE_TAC[RIGHT_AND_EXISTS_THM; REAL_ADD_RID; VECTOR_ADD_RID] THEN
940 ONCE_REWRITE_TAC[REAL_ARITH `&1 = x <=> x = &1`] THEN
941 REWRITE_TAC[] THEN SET_TAC[];;
943 let AFF_GE_1_1 = prove
950 y = t1 % x + t2 % v }`,
953 let AFF_GE_1_2 = prove
956 ==> aff_ge {x} {v,w} =
959 &0 <= t2 /\ &0 <= t3 /\
962 y = t1 % x + t2 % v + t3 % w}`,
965 let AFF_GE_2_1 = prove
968 ==> aff_ge {x,v} {w} =
972 y = t1 % x + t2 % v + t3 % w}`,
975 let AFF_GT_1_1 = prove
982 y = t1 % x + t2 % v}`,
985 let AFF_GT_1_2 = prove
988 ==> aff_gt {x} {v,w} =
990 &0 < t2 /\ &0 < t3 /\
992 y = t1 % x + t2 % v + t3 % w}`,
995 let AFF_GT_2_1 = prove
998 ==> aff_gt {x,v} {w} =
1001 t1 + t2 + t3 = &1 /\
1002 y = t1 % x + t2 % v + t3 % w}`,
1005 let AFF_GT_3_1 = prove
1007 DISJOINT {v,w,x} {y}
1008 ==> aff_gt {v,w,x} {y} =
1011 t1 + t2 + t3 + t4 = &1 /\
1012 z = t1 % v + t2 % w + t3 % x + t4 % y}`,
1015 let AFF_LT_1_1 = prove
1018 ==> aff_lt {x} {v} =
1022 y = t1 % x + t2 % v}`,
1025 let AFF_LT_2_1 = prove
1028 ==> aff_lt {x,v} {w} =
1031 t1 + t2 + t3 = &1 /\
1032 y = t1 % x + t2 % v + t3 % w}`,
1035 let AFF_GE_1_2_0 = prove
1037 ~(v = vec 0) /\ ~(w = vec 0)
1038 ==> aff_ge {vec 0} {v,w} = {a % v + b % w | &0 <= a /\ &0 <= b}`,
1039 SIMP_TAC[AFF_GE_1_2;
1040 SET_RULE `DISJOINT {a} {b,c} <=> ~(b = a) /\ ~(c = a)`] THEN
1041 REWRITE_TAC[VECTOR_MUL_RZERO; VECTOR_ADD_LID] THEN
1042 ONCE_REWRITE_TAC[MESON[]
1043 `(?a b c. P b c /\ Q b c /\ R a b c /\ S b c) <=>
1044 (?b c. P b c /\ Q b c /\ S b c /\ ?a. R a b c)`] THEN
1045 REWRITE_TAC[REAL_ARITH `t + s:real = &1 <=> t = &1 - s`; EXISTS_REFL] THEN
1048 let AFF_GE_1_1_0 = prove
1049 (`!v. ~(v = vec 0) ==> aff_ge {vec 0} {v} = {a % v | &0 <= a}`,
1050 REPEAT STRIP_TAC THEN
1051 GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [SET_RULE `{a} = {a,a}`] THEN
1052 ASM_SIMP_TAC[AFF_GE_1_2_0; GSYM VECTOR_ADD_RDISTRIB] THEN
1053 REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN
1054 MESON_TAC[REAL_LE_ADD; REAL_ARITH
1055 `&0 <= a ==> &0 <= a / &2 /\ a / &2 + a / &2 = a`]);;
1057 let AFF_GE_2_1_0 = prove
1058 (`!v w. DISJOINT {vec 0, v} {w}
1059 ==> aff_ge {vec 0, v} {w} = {s % v + t % w |s,t| &0 <= t}`,
1060 SIMP_TAC[AFF_GE_2_1; VECTOR_MUL_RZERO; VECTOR_ADD_LID] THEN
1061 REPEAT STRIP_TAC THEN
1062 ONCE_REWRITE_TAC[TAUT `p /\ q /\ r <=> q /\ p /\ r`] THEN
1063 ONCE_REWRITE_TAC[MESON[] `(?a b c. P a b c) <=> (?c b a. P a b c)`] THEN
1064 REWRITE_TAC[REAL_ARITH `t + u = &1 <=> t = &1 - u`; UNWIND_THM2] THEN
1067 (* ------------------------------------------------------------------------- *)
1068 (* Properties of affsign variants. *)
1069 (* ------------------------------------------------------------------------- *)
1071 let CONVEX_AFFSIGN = prove
1072 (`!sgn. (!x y u. sgn(x) /\ sgn(y) /\ &0 <= u /\ u <= &1
1073 ==> sgn((&1 - u) * x + u * y))
1074 ==> !s t:real^N->bool. convex(affsign sgn s t)`,
1075 REPEAT STRIP_TAC THEN REWRITE_TAC[AFFSIGN; CONVEX_ALT] THEN
1076 MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`; `u:real`] THEN
1077 REWRITE_TAC[IN_ELIM_THM; IMP_CONJ; LEFT_IMP_EXISTS_THM] THEN
1078 REWRITE_TAC[IMP_IMP; GSYM CONJ_ASSOC] THEN
1079 X_GEN_TAC `f:real^N->real` THEN STRIP_TAC THEN
1080 X_GEN_TAC `g:real^N->real` THEN STRIP_TAC THEN
1081 EXISTS_TAC `\x:real^N. (&1 - u) * f x + u * g x` THEN
1082 ASM_REWRITE_TAC[VECTOR_ADD_RDISTRIB] THEN REPEAT CONJ_TAC THENL
1083 [CONV_TAC SYM_CONV THEN
1084 W(MP_TAC o PART_MATCH (lhs o rand) VSUM_ADD_GEN o lhand o snd) THEN
1085 REWRITE_TAC[GSYM VECTOR_MUL_ASSOC; VSUM_LMUL] THEN
1086 DISCH_THEN MATCH_MP_TAC;
1088 W(MP_TAC o PART_MATCH (lhs o rand) SUM_ADD_GEN o lhand o snd) THEN
1089 ASM_REWRITE_TAC[GSYM VECTOR_MUL_ASSOC; SUM_LMUL] THEN
1090 REWRITE_TAC[REAL_MUL_RID; REAL_SUB_ADD] THEN DISCH_THEN MATCH_MP_TAC] THEN
1092 [MP_TAC(ASSUME `sum (s UNION t:real^N->bool) f = &1`);
1093 MP_TAC(ASSUME `sum (s UNION t:real^N->bool) g = &1`)]) THEN
1094 GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [sum] THEN
1095 ONCE_REWRITE_TAC[iterate] THEN
1096 REWRITE_TAC[support; NEUTRAL_REAL_ADD] THEN
1097 COND_CASES_TAC THEN REWRITE_TAC[REAL_OF_NUM_EQ; ARITH_EQ] THEN
1098 DISCH_THEN(K ALL_TAC) THEN
1099 FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP
1100 (REWRITE_RULE[IMP_CONJ] FINITE_SUBSET)) THEN
1101 REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN GEN_TAC THEN
1102 MATCH_MP_TAC MONO_AND THEN REWRITE_TAC[CONTRAPOS_THM] THEN
1103 DISCH_THEN SUBST1_TAC THEN
1104 REWRITE_TAC[VECTOR_MUL_LZERO; VECTOR_MUL_RZERO; REAL_MUL_RZERO]);;
1106 let CONVEX_AFF_GE = prove
1107 (`!s t. convex(aff_ge s t)`,
1108 REWRITE_TAC[aff_ge_def; sgn_ge] THEN MATCH_MP_TAC CONVEX_AFFSIGN THEN
1109 SIMP_TAC[REAL_LE_MUL; REAL_LE_ADD; REAL_SUB_LE]);;
1111 let CONVEX_AFF_LE = prove
1112 (`!s t. convex(aff_le s t)`,
1113 REWRITE_TAC[aff_le_def; sgn_le] THEN MATCH_MP_TAC CONVEX_AFFSIGN THEN
1114 REWRITE_TAC[REAL_ARITH `x <= &0 <=> &0 <= --x`; REAL_NEG_ADD; GSYM
1116 SIMP_TAC[REAL_LE_MUL; REAL_LE_ADD; REAL_SUB_LE]);;
1118 let CONVEX_AFF_GT = prove
1119 (`!s t. convex(aff_gt s t)`,
1120 REWRITE_TAC[aff_gt_def; sgn_gt] THEN MATCH_MP_TAC CONVEX_AFFSIGN THEN
1121 REWRITE_TAC[REAL_ARITH `&0 <= x <=> x = &0 \/ &0 < x`;
1122 REAL_ARITH `x <= &1 <=> x = &1 \/ x < &1`] THEN
1123 REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
1124 CONV_TAC REAL_RAT_REDUCE_CONV THEN
1125 REWRITE_TAC[REAL_MUL_LZERO; REAL_ADD_LID; REAL_ADD_RID; REAL_MUL_LID] THEN
1126 ASM_SIMP_TAC[REAL_LT_ADD; REAL_LT_MUL; REAL_SUB_LT]);;
1128 let CONVEX_AFF_LT = prove
1129 (`!s t. convex(aff_lt s t)`,
1130 REWRITE_TAC[aff_lt_def; sgn_lt] THEN MATCH_MP_TAC CONVEX_AFFSIGN THEN
1131 REWRITE_TAC[REAL_ARITH `x < &0 <=> &0 < --x`; REAL_NEG_ADD; GSYM
1133 REWRITE_TAC[REAL_ARITH `&0 <= x <=> x = &0 \/ &0 < x`;
1134 REAL_ARITH `x <= &1 <=> x = &1 \/ x < &1`] THEN
1135 REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
1136 CONV_TAC REAL_RAT_REDUCE_CONV THEN
1137 REWRITE_TAC[REAL_MUL_LZERO; REAL_ADD_LID; REAL_ADD_RID; REAL_MUL_LID] THEN
1138 ASM_SIMP_TAC[REAL_LT_ADD; REAL_LT_MUL; REAL_SUB_LT]);;
1140 let AFFSIGN_SUBSET_AFFINE_HULL = prove
1141 (`!sgn s t. (affsign sgn s t) SUBSET (affine hull (s UNION t))`,
1142 REWRITE_TAC[AFFINE_HULL_FINITE; AFFSIGN] THEN SET_TAC[]);;
1144 let AFF_GE_SUBSET_AFFINE_HULL = prove
1145 (`!s t. (aff_ge s t) SUBSET (affine hull (s UNION t))`,
1146 REWRITE_TAC[aff_ge_def; AFFSIGN_SUBSET_AFFINE_HULL]);;
1148 let AFF_LE_SUBSET_AFFINE_HULL = prove
1149 (`!s t. (aff_le s t) SUBSET (affine hull (s UNION t))`,
1150 REWRITE_TAC[aff_le_def; AFFSIGN_SUBSET_AFFINE_HULL]);;
1152 let AFF_GT_SUBSET_AFFINE_HULL = prove
1153 (`!s t. (aff_gt s t) SUBSET (affine hull (s UNION t))`,
1154 REWRITE_TAC[aff_gt_def; AFFSIGN_SUBSET_AFFINE_HULL]);;
1156 let AFF_LT_SUBSET_AFFINE_HULL = prove
1157 (`!s t. (aff_lt s t) SUBSET (affine hull (s UNION t))`,
1158 REWRITE_TAC[aff_lt_def; AFFSIGN_SUBSET_AFFINE_HULL]);;
1160 let AFFSIGN_EQ_AFFINE_HULL = prove
1161 (`!sgn s t. affsign sgn s {} = affine hull s`,
1162 REWRITE_TAC[AFFSIGN; AFFINE_HULL_FINITE] THEN
1163 REWRITE_TAC[UNION_EMPTY; NOT_IN_EMPTY] THEN SET_TAC[]);;
1165 let AFF_GE_EQ_AFFINE_HULL = prove
1166 (`!s t. aff_ge s {} = affine hull s`,
1167 REWRITE_TAC[aff_ge_def; AFFSIGN_EQ_AFFINE_HULL]);;
1169 let AFF_LE_EQ_AFFINE_HULL = prove
1170 (`!s t. aff_le s {} = affine hull s`,
1171 REWRITE_TAC[aff_le_def; AFFSIGN_EQ_AFFINE_HULL]);;
1173 let AFF_GT_EQ_AFFINE_HULL = prove
1174 (`!s t. aff_gt s {} = affine hull s`,
1175 REWRITE_TAC[aff_gt_def; AFFSIGN_EQ_AFFINE_HULL]);;
1177 let AFF_LT_EQ_AFFINE_HULL = prove
1178 (`!s t. aff_lt s {} = affine hull s`,
1179 REWRITE_TAC[aff_lt_def; AFFSIGN_EQ_AFFINE_HULL]);;
1181 let AFFSIGN_SUBSET_AFFSIGN = prove
1183 (!x. sgn1 x ==> sgn2 x) ==> affsign sgn1 s t SUBSET affsign sgn2 s t`,
1184 REPEAT STRIP_TAC THEN REWRITE_TAC[AFFSIGN; SUBSET; IN_ELIM_THM] THEN
1185 GEN_TAC THEN MATCH_MP_TAC MONO_EXISTS THEN ASM_MESON_TAC[]);;
1187 let AFF_GT_SUBSET_AFF_GE = prove
1188 (`!s t. aff_gt s t SUBSET aff_ge s t`,
1189 REPEAT GEN_TAC THEN REWRITE_TAC[aff_gt_def; aff_ge_def] THEN
1190 MATCH_MP_TAC AFFSIGN_SUBSET_AFFSIGN THEN
1191 SIMP_TAC[sgn_gt; sgn_ge; REAL_LT_IMP_LE]);;
1193 let AFFSIGN_MONO_LEFT = prove
1194 (`!sgn s s' t:real^N->bool.
1195 s SUBSET s' ==> affsign sgn s t SUBSET affsign sgn s' t`,
1196 REPEAT STRIP_TAC THEN
1197 REWRITE_TAC[AFFSIGN; SUBSET; IN_ELIM_THM] THEN
1198 X_GEN_TAC `y:real^N` THEN
1199 DISCH_THEN(X_CHOOSE_THEN `u:real^N->real` STRIP_ASSUME_TAC) THEN
1200 EXISTS_TAC `\x:real^N. if x IN s UNION t then u x else &0` THEN
1201 REWRITE_TAC[COND_RAND; COND_RATOR; VECTOR_MUL_LZERO] THEN
1202 REWRITE_TAC[GSYM SUM_RESTRICT_SET; GSYM VSUM_RESTRICT_SET] THEN
1203 ASM_SIMP_TAC[SET_RULE
1204 `s SUBSET s' ==> {x | x IN s' UNION t /\ x IN s UNION t} = s UNION t`] THEN
1207 let AFFSIGN_MONO_SHUFFLE = prove
1209 s' UNION t' = s UNION t /\ t' SUBSET t
1210 ==> affsign sgn s t SUBSET affsign sgn s' t'`,
1211 REPEAT STRIP_TAC THEN REWRITE_TAC[AFFSIGN; SUBSET; IN_ELIM_THM] THEN
1212 GEN_TAC THEN MATCH_MP_TAC MONO_EXISTS THEN
1213 ASM_REWRITE_TAC[] THEN ASM SET_TAC[]);;
1215 let AFF_GT_MONO_LEFT = prove
1216 (`!s s' t. s SUBSET s' ==> aff_gt s t SUBSET aff_gt s' t`,
1217 REWRITE_TAC[aff_gt_def; AFFSIGN_MONO_LEFT]);;
1219 let AFF_GE_MONO_LEFT = prove
1220 (`!s s' t. s SUBSET s' ==> aff_ge s t SUBSET aff_ge s' t`,
1221 REWRITE_TAC[aff_ge_def; AFFSIGN_MONO_LEFT]);;
1223 let AFF_LT_MONO_LEFT = prove
1224 (`!s s' t. s SUBSET s' ==> aff_lt s t SUBSET aff_lt s' t`,
1225 REWRITE_TAC[aff_lt_def; AFFSIGN_MONO_LEFT]);;
1227 let AFF_LE_MONO_LEFT = prove
1228 (`!s s' t. s SUBSET s' ==> aff_le s t SUBSET aff_le s' t`,
1229 REWRITE_TAC[aff_le_def; AFFSIGN_MONO_LEFT]);;
1231 let AFFSIGN_MONO_RIGHT = prove
1232 (`!sgn s t t':real^N->bool.
1233 sgn(&0) /\ t SUBSET t' /\ DISJOINT s t'
1234 ==> affsign sgn s t SUBSET affsign sgn s t'`,
1235 REPEAT STRIP_TAC THEN
1236 REWRITE_TAC[AFFSIGN; SUBSET; IN_ELIM_THM] THEN
1237 X_GEN_TAC `y:real^N` THEN
1238 DISCH_THEN(X_CHOOSE_THEN `u:real^N->real` STRIP_ASSUME_TAC) THEN
1239 EXISTS_TAC `\x:real^N. if x IN s UNION t then u x else &0` THEN
1240 REWRITE_TAC[COND_RAND; COND_RATOR; VECTOR_MUL_LZERO] THEN
1241 REWRITE_TAC[GSYM SUM_RESTRICT_SET; GSYM VSUM_RESTRICT_SET] THEN
1242 ASM_SIMP_TAC[SET_RULE
1243 `t SUBSET t' ==> {x | x IN s UNION t' /\ x IN s UNION t} = s UNION t`] THEN
1246 let AFF_GE_MONO_RIGHT = prove
1247 (`!s t t'. t SUBSET t' /\ DISJOINT s t' ==> aff_ge s t SUBSET aff_ge s t'`,
1248 SIMP_TAC[aff_ge_def; AFFSIGN_MONO_RIGHT; sgn_ge; REAL_POS]);;
1250 let AFF_LE_MONO_RIGHT = prove
1251 (`!s t t'. t SUBSET t' /\ DISJOINT s t' ==> aff_le s t SUBSET aff_le s t'`,
1252 SIMP_TAC[aff_le_def; AFFSIGN_MONO_RIGHT; sgn_le; REAL_LE_REFL]);;
1254 let AFFINE_HULL_SUBSET_AFFSIGN = prove
1255 (`!sgn s t:real^N->bool.
1256 sgn(&0) /\ DISJOINT s t
1257 ==> affine hull s SUBSET affsign sgn s t`,
1258 REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_TRANS THEN
1259 EXISTS_TAC `affsign sgn (s:real^N->bool) {}` THEN CONJ_TAC THENL
1260 [REWRITE_TAC[AFFSIGN_EQ_AFFINE_HULL; SUBSET_REFL];
1261 MATCH_MP_TAC AFFSIGN_MONO_RIGHT THEN ASM SET_TAC[]]);;
1263 let AFFINE_HULL_SUBSET_AFF_GE = prove
1264 (`!s t. DISJOINT s t ==> affine hull s SUBSET aff_ge s t`,
1265 SIMP_TAC[aff_ge_def; sgn_ge; REAL_LE_REFL; AFFINE_HULL_SUBSET_AFFSIGN]);;
1267 let AFF_GE_AFF_GT_DECOMP = prove
1269 FINITE s /\ FINITE t /\ DISJOINT s t
1270 ==> aff_ge s t = aff_gt s t UNION
1271 UNIONS {aff_ge s (t DELETE a) | a | a IN t}`,
1272 REPEAT STRIP_TAC THEN
1273 MATCH_MP_TAC(SET_RULE
1274 `t' SUBSET t /\ (!a. a IN s ==> f(a) SUBSET t) /\
1275 (!y. y IN t ==> y IN t' \/ ?a. a IN s /\ y IN f(a))
1276 ==> t = t' UNION UNIONS {f a | a IN s}`) THEN
1277 REWRITE_TAC[AFF_GT_SUBSET_AFF_GE] THEN
1278 ASM_SIMP_TAC[DELETE_SUBSET; AFF_GE_MONO_RIGHT] THEN
1279 REWRITE_TAC[aff_ge_def; aff_gt_def; AFFSIGN; sgn_ge; sgn_gt] THEN
1280 X_GEN_TAC `y:real^N` THEN REWRITE_TAC[IN_ELIM_THM] THEN
1281 DISCH_THEN(X_CHOOSE_THEN `u:real^N->real` STRIP_ASSUME_TAC) THEN
1282 ASM_CASES_TAC `!x:real^N. x IN t ==> &0 < u x` THENL
1283 [DISJ1_TAC THEN EXISTS_TAC `u:real^N->real` THEN ASM_REWRITE_TAC[];
1285 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [NOT_FORALL_THM]) THEN
1286 ASM_SIMP_TAC[REAL_ARITH `&0 <= x ==> (&0 < x <=> ~(x = &0))`] THEN
1287 REWRITE_TAC[NOT_IMP] THEN MATCH_MP_TAC MONO_EXISTS THEN
1288 X_GEN_TAC `a:real^N` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
1289 EXISTS_TAC `u:real^N->real` THEN
1290 ASM_SIMP_TAC[SET_RULE
1291 `a IN t /\ DISJOINT s t
1292 ==> s UNION (t DELETE a) = (s UNION t) DELETE a`] THEN
1293 ASM_SIMP_TAC[IN_DELETE; SUM_DELETE; VSUM_DELETE; REAL_SUB_RZERO;
1294 FINITE_UNION; IN_UNION] THEN
1295 REWRITE_TAC[VECTOR_MUL_LZERO; VECTOR_SUB_RZERO]]);;
1297 let AFFSIGN_SPECIAL_SCALE = prove
1299 FINITE s /\ FINITE t /\
1300 ~(vec 0 IN t) /\ ~(v IN t) /\ ~((a % v) IN t) /\
1301 (!x. sgn x ==> sgn(x / &2)) /\
1302 (!x y. sgn x /\ sgn y ==> sgn(x + y)) /\
1304 ==> affsign sgn (vec 0 INSERT (a % v) INSERT s) t =
1305 affsign sgn (vec 0 INSERT v INSERT s) t`,
1306 REWRITE_TAC[EXTENSION] THEN REPEAT STRIP_TAC THEN
1307 REWRITE_TAC[AFFSIGN_ALT; IN_ELIM_THM; INSERT_UNION_EQ] THEN
1308 ASM_SIMP_TAC[FINITE_INSERT; FINITE_UNION; AFFINE_HULL_FINITE_STEP_GEN;
1309 RIGHT_EXISTS_AND_THM] THEN
1310 REWRITE_TAC[VECTOR_MUL_RZERO; VECTOR_SUB_RZERO] THEN
1311 GEN_REWRITE_TAC BINOP_CONV [SWAP_EXISTS_THM] THEN
1312 GEN_REWRITE_TAC (BINOP_CONV o BINDER_CONV) [SWAP_EXISTS_THM] THEN
1313 REWRITE_TAC[LEFT_EXISTS_AND_THM; RIGHT_EXISTS_AND_THM] THEN
1314 REWRITE_TAC[REAL_ARITH `x = &1 - v - v' <=> v = &1 - (x + v')`] THEN
1315 REWRITE_TAC[EXISTS_REFL] THEN
1316 FIRST_ASSUM(MP_TAC o MATCH_MP(MESON[REAL_LT_IMP_NZ; REAL_DIV_LMUL]
1317 `!a. &0 < a ==> (!y. ?x. a * x = y)`)) THEN
1318 DISCH_THEN(MP_TAC o MATCH_MP QUANTIFY_SURJECTION_THM) THEN
1319 DISCH_THEN(CONV_TAC o RAND_CONV o EXPAND_QUANTS_CONV) THEN
1320 REWRITE_TAC[VECTOR_MUL_ASSOC; REAL_MUL_SYM]);;
1322 let AFF_GE_SPECIAL_SCALE = prove
1324 FINITE s /\ FINITE t /\
1325 ~(vec 0 IN t) /\ ~(v IN t) /\ ~((a % v) IN t) /\
1327 ==> aff_ge (vec 0 INSERT (a % v) INSERT s) t =
1328 aff_ge (vec 0 INSERT v INSERT s) t`,
1329 REPEAT STRIP_TAC THEN REWRITE_TAC[aff_ge_def] THEN
1330 MATCH_MP_TAC AFFSIGN_SPECIAL_SCALE THEN
1331 ASM_REWRITE_TAC[sgn_ge] THEN REAL_ARITH_TAC);;
1333 let AFF_LE_SPECIAL_SCALE = prove
1335 FINITE s /\ FINITE t /\
1336 ~(vec 0 IN t) /\ ~(v IN t) /\ ~((a % v) IN t) /\
1338 ==> aff_le (vec 0 INSERT (a % v) INSERT s) t =
1339 aff_le (vec 0 INSERT v INSERT s) t`,
1340 REPEAT STRIP_TAC THEN REWRITE_TAC[aff_le_def] THEN
1341 MATCH_MP_TAC AFFSIGN_SPECIAL_SCALE THEN
1342 ASM_REWRITE_TAC[sgn_le] THEN REAL_ARITH_TAC);;
1344 let AFF_GT_SPECIAL_SCALE = prove
1346 FINITE s /\ FINITE t /\
1347 ~(vec 0 IN t) /\ ~(v IN t) /\ ~((a % v) IN t) /\
1349 ==> aff_gt (vec 0 INSERT (a % v) INSERT s) t =
1350 aff_gt (vec 0 INSERT v INSERT s) t`,
1351 REPEAT STRIP_TAC THEN REWRITE_TAC[aff_gt_def] THEN
1352 MATCH_MP_TAC AFFSIGN_SPECIAL_SCALE THEN
1353 ASM_REWRITE_TAC[sgn_gt] THEN REAL_ARITH_TAC);;
1355 let AFF_LT_SPECIAL_SCALE = prove
1357 FINITE s /\ FINITE t /\
1358 ~(vec 0 IN t) /\ ~(v IN t) /\ ~((a % v) IN t) /\
1360 ==> aff_lt (vec 0 INSERT (a % v) INSERT s) t =
1361 aff_lt (vec 0 INSERT v INSERT s) t`,
1362 REPEAT STRIP_TAC THEN REWRITE_TAC[aff_lt_def] THEN
1363 MATCH_MP_TAC AFFSIGN_SPECIAL_SCALE THEN
1364 ASM_REWRITE_TAC[sgn_lt] THEN REAL_ARITH_TAC);;
1366 let AFF_GE_SCALE_LEMMA = prove
1368 &0 < a /\ ~(v = vec 0)
1369 ==> aff_ge {vec 0} {a % u,v} = aff_ge {vec 0} {u,v}`,
1370 REPEAT STRIP_TAC THEN ASM_CASES_TAC `u:real^N = vec 0` THEN
1371 ASM_REWRITE_TAC[VECTOR_MUL_RZERO] THEN
1372 ASM_SIMP_TAC[AFF_GE_1_2_0; VECTOR_MUL_EQ_0; REAL_LT_IMP_NZ;
1373 SET_RULE `DISJOINT {a} {b,c} <=> ~(b = a) /\ ~(c = a)`] THEN
1374 REWRITE_TAC[GSYM SUBSET_ANTISYM_EQ; SUBSET; FORALL_IN_GSPEC] THEN
1375 CONJ_TAC THEN MAP_EVERY X_GEN_TAC [`b:real`; `c:real`] THEN
1376 REWRITE_TAC[IN_ELIM_THM] THEN STRIP_TAC THENL
1377 [EXISTS_TAC `a * b:real`; EXISTS_TAC `b / a:real`] THEN
1378 EXISTS_TAC `c:real` THEN
1379 ASM_SIMP_TAC[REAL_LE_DIV; REAL_LE_MUL; REAL_LT_IMP_LE] THEN
1380 REWRITE_TAC[VECTOR_MUL_ASSOC] THEN
1381 REPLICATE_TAC 2 (AP_THM_TAC THEN AP_TERM_TAC) THEN
1382 UNDISCH_TAC `&0 < a` THEN CONV_TAC REAL_FIELD);;
1384 let AFFSIGN_0 = prove
1386 FINITE s /\ FINITE t /\ (vec 0) IN (s DIFF t)
1387 ==> affsign sgn s t =
1388 { vsum (s UNION t) (\v. f v % v) |f|
1389 !x:real^N. x IN t ==> sgn(f x)}`,
1390 REPEAT STRIP_TAC THEN REWRITE_TAC[AFFSIGN] THEN
1391 FIRST_ASSUM(SUBST1_TAC o MATCH_MP (SET_RULE
1392 `x IN s DIFF t ==> s UNION t = x INSERT ((s UNION t) DELETE x)`)) THEN
1393 ASM_SIMP_TAC[SUM_CLAUSES; VSUM_CLAUSES; FINITE_UNION; FINITE_DELETE] THEN
1394 REWRITE_TAC[IN_DELETE; VECTOR_MUL_RZERO; VECTOR_ADD_LID] THEN
1395 MATCH_MP_TAC SUBSET_ANTISYM THEN
1396 REWRITE_TAC[FORALL_IN_GSPEC; SUBSET; LEFT_IMP_EXISTS_THM] THEN
1397 REWRITE_TAC[IN_ELIM_THM] THEN CONJ_TAC THENL
1398 [MAP_EVERY X_GEN_TAC [`y:real^N`; `f:real^N->real`] THEN
1399 STRIP_TAC THEN EXISTS_TAC `f:real^N->real` THEN ASM_REWRITE_TAC[];
1400 X_GEN_TAC `f:real^N->real` THEN DISCH_TAC THEN
1402 `\x:real^N. if x = vec 0
1403 then &1 - sum ((s UNION t) DELETE vec 0) (\x. f x)
1406 `!x:real^N. x IN (s UNION t) DELETE vec 0 ==> ~(x = vec 0)`) THEN
1407 SIMP_TAC[ETA_AX; REAL_SUB_ADD] THEN DISCH_THEN(K ALL_TAC) THEN
1410 let AFF_GE_0_AFFINE_MULTIPLE_CONVEX = prove
1411 (`!s t:real^N->bool.
1412 FINITE s /\ FINITE t /\ vec 0 IN (s DIFF t) /\ ~(t = {})
1414 {x + c % y | x IN affine hull (s DIFF t) /\
1415 y IN convex hull t /\ &0 <= c}`,
1416 REPEAT STRIP_TAC THEN ASM_SIMP_TAC[aff_ge_def; AFFSIGN_0; sgn_ge] THEN
1417 ONCE_REWRITE_TAC[SET_RULE `s UNION t = (s DIFF t) UNION t`] THEN
1418 ASM_SIMP_TAC[VSUM_UNION; FINITE_DIFF;
1419 SET_RULE `DISJOINT (s DIFF t) t`] THEN
1420 ASM_SIMP_TAC[AFFINE_HULL_EQ_SPAN; HULL_INC] THEN
1421 ASM_SIMP_TAC[SPAN_FINITE; FINITE_DIFF; CONVEX_HULL_FINITE] THEN
1422 MATCH_MP_TAC SUBSET_ANTISYM THEN REWRITE_TAC[SUBSET; FORALL_IN_GSPEC] THEN
1423 REWRITE_TAC[IN_ELIM_THM] THEN CONJ_TAC THENL
1424 [X_GEN_TAC `f:real^N->real` THEN DISCH_TAC THEN
1425 EXISTS_TAC `vsum (s DIFF t) (\x:real^N. f x % x)` THEN
1426 ASM_CASES_TAC `sum t (f:real^N->real) = &0` THENL
1427 [MP_TAC(ISPECL [`f:real^N->real`; `t:real^N->bool`] SUM_POS_EQ_0) THEN
1428 ASM_SIMP_TAC[VECTOR_MUL_LZERO; REAL_MUL_LZERO; VSUM_0] THEN
1429 DISCH_TAC THEN EXISTS_TAC `&0` THEN
1430 REWRITE_TAC[VECTOR_MUL_LZERO; REAL_LE_REFL] THEN
1431 REWRITE_TAC[RIGHT_EXISTS_AND_THM] THEN CONJ_TAC THENL
1432 [EXISTS_TAC `f:real^N->real` THEN REWRITE_TAC[]; ALL_TAC] THEN
1433 ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN
1434 REWRITE_TAC[RIGHT_EXISTS_AND_THM; GSYM EXISTS_REFL] THEN
1435 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
1436 DISCH_THEN(X_CHOOSE_TAC `a:real^N`) THEN
1437 EXISTS_TAC `\x:real^N. if x = a then &1 else &0` THEN
1438 ASM_REWRITE_TAC[SUM_DELTA] THEN MESON_TAC[REAL_POS];
1439 EXISTS_TAC `sum t (f:real^N->real)` THEN
1440 EXISTS_TAC `inv(sum t (f:real^N->real)) % vsum t (\v. f v % v)` THEN
1441 REPEAT CONJ_TAC THENL
1442 [EXISTS_TAC `f:real^N->real` THEN REWRITE_TAC[];
1443 EXISTS_TAC `\x:real^N. f x / sum t (f:real^N->real)` THEN
1444 ASM_SIMP_TAC[REAL_LE_DIV; SUM_POS_LE] THEN
1445 ONCE_REWRITE_TAC[REAL_ARITH `x / y:real = inv y * x`] THEN
1446 ASM_SIMP_TAC[GSYM VECTOR_MUL_ASSOC; SUM_LMUL; VSUM_LMUL] THEN
1447 ASM_SIMP_TAC[REAL_MUL_LINV];
1448 ASM_SIMP_TAC[SUM_POS_LE];
1449 AP_TERM_TAC THEN ASM_CASES_TAC `sum t (f:real^N->real) = &0` THEN
1450 ASM_SIMP_TAC[VECTOR_MUL_ASSOC; REAL_MUL_RINV; VECTOR_MUL_LID]]];
1451 MAP_EVERY X_GEN_TAC [`x:real^N`; `c:real`; `y:real^N`] THEN
1452 DISCH_THEN(CONJUNCTS_THEN2
1453 (X_CHOOSE_THEN `u:real^N->real` (SUBST1_TAC o SYM)) MP_TAC) THEN
1454 DISCH_THEN(CONJUNCTS_THEN2
1455 (X_CHOOSE_THEN `v:real^N->real`MP_TAC) ASSUME_TAC) THEN
1456 REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
1457 DISCH_THEN(SUBST1_TAC o SYM) THEN
1458 EXISTS_TAC `(\x. if x IN t then c * v x else u x):real^N->real` THEN
1459 ASM_SIMP_TAC[REAL_LE_MUL; VSUM_LMUL; GSYM VECTOR_MUL_ASSOC] THEN
1460 AP_THM_TAC THEN AP_TERM_TAC THEN MATCH_MP_TAC VSUM_EQ THEN
1461 SIMP_TAC[IN_DIFF]]);;
1463 let AFF_GE_0_MULTIPLE_AFFINE_CONVEX = prove
1464 (`!s t:real^N->bool.
1465 FINITE s /\ FINITE t /\ vec 0 IN (s DIFF t) /\ ~(t = {})
1467 affine hull (s DIFF t) UNION
1468 {c % (x + y) | x IN affine hull (s DIFF t) /\
1469 y IN convex hull t /\ &0 <= c}`,
1470 REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN
1471 REWRITE_TAC[UNION_SUBSET] THEN REPEAT CONJ_TAC THENL
1472 [ASM_SIMP_TAC[AFF_GE_0_AFFINE_MULTIPLE_CONVEX;
1473 AFFINE_HULL_EQ_SPAN; HULL_INC] THEN
1474 REWRITE_TAC[SUBSET; FORALL_IN_GSPEC] THEN
1475 MAP_EVERY X_GEN_TAC [`x:real^N`; `c:real`; `y:real^N`] THEN STRIP_TAC THEN
1476 REWRITE_TAC[IN_ELIM_THM; IN_UNION] THEN ASM_CASES_TAC `c = &0` THENL
1477 [DISJ1_TAC THEN ASM_REWRITE_TAC[VECTOR_MUL_LZERO; VECTOR_ADD_RID];
1478 DISJ2_TAC THEN MAP_EVERY EXISTS_TAC
1479 [`c:real`; `inv(c) % x:real^N`; `y:real^N`] THEN
1480 ASM_SIMP_TAC[SPAN_MUL; VECTOR_ADD_LDISTRIB; VECTOR_MUL_ASSOC;
1481 REAL_MUL_RINV; VECTOR_MUL_LID]];
1482 REWRITE_TAC[aff_ge_def] THEN ONCE_REWRITE_TAC[AFFSIGN_DISJOINT_DIFF] THEN
1483 REWRITE_TAC[GSYM aff_ge_def] THEN
1484 MATCH_MP_TAC AFFINE_HULL_SUBSET_AFF_GE THEN SET_TAC[];
1485 ASM_SIMP_TAC[AFF_GE_0_AFFINE_MULTIPLE_CONVEX;
1486 AFFINE_HULL_EQ_SPAN; HULL_INC] THEN
1487 REWRITE_TAC[SUBSET; FORALL_IN_GSPEC] THEN
1488 MAP_EVERY X_GEN_TAC [`c:real`; `x:real^N`; `y:real^N`] THEN STRIP_TAC THEN
1489 REWRITE_TAC[IN_ELIM_THM] THEN MAP_EVERY EXISTS_TAC
1490 [`c % x:real^N`; `c:real`; `y:real^N`] THEN
1491 ASM_SIMP_TAC[SPAN_MUL; VECTOR_ADD_LDISTRIB]]);;
1493 let AFF_GE_0_AFFINE_CONVEX_CONE = prove
1494 (`!s t:real^N->bool.
1495 FINITE s /\ FINITE t /\ vec 0 IN (s DIFF t)
1497 {x + y | x IN affine hull (s DIFF t) /\
1498 y IN convex_cone hull t}`,
1499 REPEAT STRIP_TAC THEN ASM_CASES_TAC `t:real^N->bool = {}` THENL
1500 [ASM_REWRITE_TAC[AFF_GE_EQ_AFFINE_HULL; CONVEX_CONE_HULL_EMPTY] THEN
1501 REWRITE_TAC[IN_SING; DIFF_EMPTY] THEN
1502 REWRITE_TAC[SET_RULE `{x + y:real^N | P x /\ y = a} = {x + a | P x}`] THEN
1503 REWRITE_TAC[VECTOR_ADD_RID] THEN SET_TAC[];
1504 ASM_SIMP_TAC[CONVEX_CONE_HULL_CONVEX_HULL_NONEMPTY;
1505 AFF_GE_0_AFFINE_MULTIPLE_CONVEX] THEN
1508 let AFF_GE_0_N = prove
1510 FINITE s /\ ~(vec 0 IN s)
1511 ==> aff_ge {vec 0} s =
1512 {y | ?u. (!x. x IN s ==> &0 <= u x) /\
1513 y = vsum s (\x. u x % x)}`,
1514 REPEAT STRIP_TAC THEN REWRITE_TAC[aff_ge_def] THEN
1515 ASM_SIMP_TAC[AFFSIGN_0; IN_DIFF; IN_INSERT; NOT_IN_EMPTY;
1516 FINITE_INSERT; FINITE_EMPTY] THEN
1517 ASM_SIMP_TAC[EXTENSION; sgn_ge; IN_ELIM_THM; INSERT_UNION; UNION_EMPTY] THEN
1518 ASM_SIMP_TAC[VSUM_CLAUSES; VECTOR_MUL_RZERO; VECTOR_ADD_LID]);;
1520 let AFF_GE_0_CONVEX_HULL = prove
1522 FINITE s /\ ~(s = {}) /\ ~(vec 0 IN s)
1523 ==> aff_ge {vec 0} s = {t % y | &0 <= t /\ y IN convex hull s}`,
1524 REPEAT STRIP_TAC THEN
1525 ASM_SIMP_TAC[AFF_GE_0_AFFINE_MULTIPLE_CONVEX; IN_DIFF;
1526 FINITE_INSERT; FINITE_EMPTY; IN_INSERT] THEN
1527 ASM_SIMP_TAC[SET_RULE `~(a IN s) ==> {a} DIFF s = {a}`] THEN
1528 REWRITE_TAC[AFFINE_HULL_SING; IN_SING] THEN
1529 REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN MESON_TAC[VECTOR_ADD_LID]);;
1531 let AFF_GE_0_CONVEX_HULL_ALT = prove
1533 FINITE s /\ ~(vec 0 IN s)
1534 ==> aff_ge {vec 0} s =
1535 vec 0 INSERT {t % y | &0 < t /\ y IN convex hull s}`,
1536 REPEAT STRIP_TAC THEN
1537 ASM_CASES_TAC `s:real^N->bool = {}` THENL
1538 [ASM_REWRITE_TAC[AFF_GE_EQ_AFFINE_HULL; CONVEX_HULL_EMPTY] THEN
1539 REWRITE_TAC[AFFINE_HULL_SING; NOT_IN_EMPTY] THEN SET_TAC[];
1540 ASM_SIMP_TAC[AFF_GE_0_CONVEX_HULL; EXTENSION; IN_ELIM_THM; IN_INSERT] THEN
1541 X_GEN_TAC `y:real^N` THEN ASM_CASES_TAC `y:real^N = vec 0` THEN
1542 ASM_REWRITE_TAC[] THENL
1543 [EXISTS_TAC `&0` THEN ASM_REWRITE_TAC[REAL_POS; VECTOR_MUL_LZERO] THEN
1544 ASM_REWRITE_TAC[MEMBER_NOT_EMPTY; CONVEX_HULL_EQ_EMPTY];
1545 AP_TERM_TAC THEN REWRITE_TAC[FUN_EQ_THM] THEN X_GEN_TAC `t:real` THEN
1546 AP_TERM_TAC THEN ABS_TAC THEN
1547 ASM_CASES_TAC `t = &0` THEN
1548 ASM_REWRITE_TAC[VECTOR_MUL_LZERO; REAL_LT_REFL] THEN
1549 ASM_REWRITE_TAC[REAL_LT_LE]]]);;
1551 let AFF_GE_0_CONVEX_CONE_NEGATIONS = prove
1552 (`!s t:real^N->bool.
1553 FINITE s /\ FINITE t /\ vec 0 IN (s DIFF t)
1555 convex_cone hull (s UNION t UNION IMAGE (--) (s DIFF t))`,
1556 REPEAT STRIP_TAC THEN ASM_SIMP_TAC[AFF_GE_0_AFFINE_CONVEX_CONE] THEN
1557 ASM_SIMP_TAC[AFFINE_HULL_EQ_SPAN; HULL_INC] THEN
1558 REWRITE_TAC[SPAN_CONVEX_CONE_ALLSIGNS; GSYM CONVEX_CONE_HULL_UNION] THEN
1559 AP_TERM_TAC THEN SET_TAC[]);;
1561 let CONVEX_HULL_AFF_GE = prove
1562 (`!s. convex hull s = aff_ge {} s`,
1563 SIMP_TAC[aff_ge_def; AFFSIGN; CONVEX_HULL_FINITE; sgn_ge; UNION_EMPTY] THEN
1564 REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN REPEAT GEN_TAC THEN
1565 AP_TERM_TAC THEN REWRITE_TAC[FUN_EQ_THM] THEN MESON_TAC[]);;
1567 let POLYHEDRON_AFF_GE = prove
1568 (`!s t:real^N->bool. FINITE s /\ FINITE t ==> polyhedron(aff_ge s t)`,
1569 REPEAT STRIP_TAC THEN REWRITE_TAC[aff_ge_def] THEN
1570 ONCE_REWRITE_TAC[AFFSIGN_DISJOINT_DIFF] THEN
1571 REWRITE_TAC[GSYM aff_ge_def] THEN
1572 SUBGOAL_THEN `FINITE(s DIFF t) /\ FINITE(t:real^N->bool) /\
1573 DISJOINT (s DIFF t) t`
1574 MP_TAC THENL [ASM_SIMP_TAC[FINITE_DIFF] THEN ASM SET_TAC[]; ALL_TAC] THEN
1575 POP_ASSUM_LIST(K ALL_TAC) THEN
1576 SPEC_TAC(`s DIFF t:real^N->bool`,`s:real^N->bool`) THEN
1577 MATCH_MP_TAC SET_PROVE_CASES THEN CONJ_TAC THENL
1578 [REPEAT STRIP_TAC THEN REWRITE_TAC[GSYM CONVEX_HULL_AFF_GE] THEN
1579 MATCH_MP_TAC POLYTOPE_IMP_POLYHEDRON THEN REWRITE_TAC[polytope] THEN
1582 MAP_EVERY X_GEN_TAC [`a:real^N`; `s:real^N->bool`] THEN
1583 GEOM_ORIGIN_TAC `a:real^N` THEN REPEAT STRIP_TAC THEN
1584 SUBGOAL_THEN `(vec 0:real^N) IN ((vec 0 INSERT s) DIFF t)` ASSUME_TAC THENL
1585 [ASM SET_TAC[]; ALL_TAC] THEN
1586 ASM_SIMP_TAC[AFF_GE_0_CONVEX_CONE_NEGATIONS; FINITE_INSERT] THEN
1587 MATCH_MP_TAC POLYHEDRON_CONVEX_CONE_HULL THEN
1588 ASM_SIMP_TAC[FINITE_INSERT; FINITE_UNION; FINITE_DIFF; FINITE_IMAGE]);;
1590 let CLOSED_AFF_GE = prove
1591 (`!s t:real^N->bool. FINITE s /\ FINITE t ==> closed(aff_ge s t)`,
1592 SIMP_TAC[POLYHEDRON_AFF_GE; POLYHEDRON_IMP_CLOSED]);;
1594 let CONIC_AFF_GE_0 = prove
1595 (`!s:real^N->bool. FINITE s /\ ~(vec 0 IN s) ==> conic(aff_ge {vec 0} s)`,
1596 REPEAT STRIP_TAC THEN ASM_SIMP_TAC[AFF_GE_0_N; conic] THEN
1597 REWRITE_TAC[IN_ELIM_THM] THEN GEN_TAC THEN X_GEN_TAC `c:real` THEN
1598 DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN
1599 DISCH_THEN(X_CHOOSE_THEN `u:real^N->real` STRIP_ASSUME_TAC) THEN
1600 EXISTS_TAC `\v. c * (u:real^N->real) v` THEN
1601 REWRITE_TAC[GSYM VECTOR_MUL_ASSOC; VSUM_LMUL] THEN
1602 ASM_MESON_TAC[REAL_LE_MUL]);;
1604 let ANGLES_ADD_AFF_GE = prove
1606 ~(v = u) /\ ~(w = u) /\ ~(x = u) /\ x IN aff_ge {u} {v,w}
1607 ==> angle(v,u,x) + angle(x,u,w) = angle(v,u,w)`,
1608 GEOM_ORIGIN_TAC `u:real^N` THEN REPEAT GEN_TAC THEN
1609 REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
1610 ASM_SIMP_TAC[AFF_GE_1_2_0] THEN
1611 REWRITE_TAC[IN_ELIM_THM; LEFT_IMP_EXISTS_THM] THEN
1612 MAP_EVERY X_GEN_TAC [`a:real`; `b:real`] THEN
1613 DISCH_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC MP_TAC) THEN
1614 SUBGOAL_THEN `a = &0 /\ b = &0 \/ &0 < a + b` STRIP_ASSUME_TAC THENL
1615 [ASM_REAL_ARITH_TAC;
1616 ASM_REWRITE_TAC[VECTOR_MUL_LZERO; VECTOR_ADD_LID];
1618 DISCH_TAC THEN MP_TAC(ISPECL
1619 [`v:real^N`; `w:real^N`; `inv(a + b) % x:real^N`; `vec 0:real^N`]
1620 ANGLES_ADD_BETWEEN) THEN
1621 ASM_REWRITE_TAC[angle; VECTOR_SUB_RZERO] THEN
1622 ASM_SIMP_TAC[VECTOR_ANGLE_RMUL; VECTOR_ANGLE_LMUL;
1623 REAL_INV_EQ_0; REAL_LE_INV_EQ; REAL_LT_IMP_NZ; REAL_LT_IMP_LE] THEN
1624 DISCH_THEN MATCH_MP_TAC THEN
1625 REWRITE_TAC[BETWEEN_IN_SEGMENT; CONVEX_HULL_2; SEGMENT_CONVEX_HULL] THEN
1626 REWRITE_TAC[IN_ELIM_THM] THEN
1627 MAP_EVERY EXISTS_TAC [`a / (a + b):real`; `b / (a + b):real`] THEN
1628 ASM_SIMP_TAC[REAL_LE_DIV; REAL_LT_IMP_LE; VECTOR_ADD_LDISTRIB] THEN
1629 REWRITE_TAC[VECTOR_MUL_ASSOC; real_div; REAL_MUL_AC] THEN
1630 UNDISCH_TAC `&0 < a + b` THEN CONV_TAC REAL_FIELD);;
1632 let AFF_GE_2_1_0_DROPOUT_3 = prove
1634 ~collinear{vec 0,basis 3,z}
1635 ==> (w IN aff_ge {vec 0,basis 3} {z} <=>
1636 (dropout 3 w) IN aff_ge {vec 0:real^2} {dropout 3 z})`,
1638 ASM_CASES_TAC `z:real^3 = vec 0` THENL
1639 [ASM_REWRITE_TAC[COLLINEAR_2; INSERT_AC]; ALL_TAC] THEN
1640 ASM_CASES_TAC `z:real^3 = basis 3` THENL
1641 [ASM_REWRITE_TAC[COLLINEAR_2; INSERT_AC]; ALL_TAC] THEN
1642 REWRITE_TAC[COLLINEAR_BASIS_3] THEN DISCH_TAC THEN
1643 ASM_SIMP_TAC[AFF_GE_2_1_0; SET_RULE `DISJOINT s {a} <=> ~(a IN s)`;
1644 IN_INSERT; NOT_IN_EMPTY; AFF_GE_1_1_0] THEN
1645 REWRITE_TAC[IN_ELIM_THM] THEN
1646 MATCH_MP_TAC(MESON[]
1647 `(!t. ((?s. P s t) <=> Q t)) ==> ((?s t. P s t) <=> (?t. Q t))`) THEN
1648 X_GEN_TAC `t:real` THEN EQ_TAC THENL
1650 ASM_REWRITE_TAC[DROPOUT_ADD; DROPOUT_MUL; DROPOUT_BASIS_3] THEN
1652 STRIP_TAC THEN EXISTS_TAC `(w:real^3)$3 - t * (z:real^3)$3` THEN
1653 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [CART_EQ]) THEN
1654 ASM_REWRITE_TAC[CART_EQ; FORALL_2; FORALL_3; DIMINDEX_2; DIMINDEX_3] THEN
1655 REWRITE_TAC[VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT] THEN
1656 SIMP_TAC[dropout; LAMBDA_BETA; DIMINDEX_2; ARITH; BASIS_COMPONENT;
1658 CONV_TAC REAL_RING]);;
1660 let AFF_GE_2_1_0_SEMIALGEBRAIC = prove
1662 ~collinear {vec 0,x,y} /\ ~collinear {vec 0,x,z}
1663 ==> (z IN aff_ge {vec 0,x} {y} <=>
1664 (x cross y) cross x cross z = vec 0 /\
1665 &0 <= (x cross z) dot (x cross y))`,
1667 (`~(y = vec 0) ==> ((?s. x = s % y) <=> y cross x = vec 0)`,
1668 REWRITE_TAC[CROSS_EQ_0] THEN SIMP_TAC[COLLINEAR_LEMMA_ALT])
1672 ==> ((?t. &0 <= t /\ x = t % y) <=>
1673 (?t. x = t % y) /\ &0 <= x dot y)`,
1674 REPEAT STRIP_TAC THEN REWRITE_TAC[LEFT_AND_EXISTS_THM] THEN
1675 AP_TERM_TAC THEN REWRITE_TAC[FUN_EQ_THM] THEN X_GEN_TAC `t:real` THEN
1676 ASM_CASES_TAC `x:real^N = t % y` THEN
1677 ASM_SIMP_TAC[DOT_LMUL; REAL_LE_MUL_EQ; DOT_POS_LT]) in
1679 MAP_EVERY (fun t -> ASM_CASES_TAC t THENL
1680 [ASM_REWRITE_TAC[COLLINEAR_2; INSERT_AC]; ALL_TAC])
1681 [`x:real^3 = vec 0`; `y:real^3 = vec 0`; `y:real^3 = x`] THEN
1683 ASM_SIMP_TAC[AFF_GE_2_1_0; IN_ELIM_THM; SET_RULE
1684 `DISJOINT {a,b} {c} <=> ~(a = c) /\ ~(b = c)`] THEN
1685 ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN
1686 REWRITE_TAC[RIGHT_EXISTS_AND_THM; VECTOR_ARITH
1687 `a:real^N = b + c <=> a - c = b`] THEN
1688 RULE_ASSUM_TAC(REWRITE_RULE[GSYM CROSS_EQ_0]) THEN
1689 ASM_SIMP_TAC[lemma0; lemma1; CROSS_RMUL; CROSS_RSUB; VECTOR_SUB_EQ]);;
1691 (* ------------------------------------------------------------------------- *)
1692 (* Special case of aff_ge {x} {y}, i.e. rays or half-lines. *)
1693 (* ------------------------------------------------------------------------- *)
1695 let HALFLINE_REFL = prove
1696 (`!x. aff_ge {x} {x} = {x}`,
1697 ONCE_REWRITE_TAC[AFF_GE_DISJOINT_DIFF] THEN
1698 ASM_REWRITE_TAC[DIFF_EQ_EMPTY; GSYM CONVEX_HULL_AFF_GE; CONVEX_HULL_SING]);;
1700 let HALFLINE_EXPLICIT = prove
1703 {z | ?t1 t2. &0 <= t2 /\ t1 + t2 = &1 /\ z = t1 % x + t2 % y}`,
1704 REPEAT GEN_TAC THEN ASM_CASES_TAC `x:real^N = y` THENL
1705 [ASM_REWRITE_TAC[HALFLINE_REFL]; AFF_TAC] THEN
1706 REWRITE_TAC[REAL_ARITH `x + y = &1 <=> x = &1 - y`] THEN
1707 REWRITE_TAC[VECTOR_ARITH `(&1 - x) % v + x % v:real^N = v`;
1708 MESON[] `(?x y. P y /\ x = f y /\ Q x y) <=> (?y. P y /\ Q (f y) y)`] THEN
1709 REWRITE_TAC[IN_ELIM_THM; IN_SING; EXTENSION] THEN MESON_TAC[REAL_POS]);;
1711 let HALFLINE = prove
1714 {z | ?t. &0 <= t /\ z = (&1 - t) % x + t % y}`,
1715 REWRITE_TAC[HALFLINE_EXPLICIT; REAL_ARITH `x + y = &1 <=> x = &1 - y`] THEN
1718 let CLOSED_HALFLINE = prove
1719 (`!x y. closed(aff_ge {x} {y})`,
1720 SIMP_TAC[CLOSED_AFF_GE; FINITE_SING]);;
1722 let SEGMENT_SUBSET_HALFLINE = prove
1723 (`!x y. segment[x,y] SUBSET aff_ge {x} {y}`,
1724 REWRITE_TAC[SEGMENT_CONVEX_HULL; CONVEX_HULL_2; HALFLINE_EXPLICIT] THEN
1727 let ENDS_IN_HALFLINE = prove
1728 (`(!x y. x IN aff_ge {x} {y}) /\ (!x y. y IN aff_ge {x} {y})`,
1729 MESON_TAC[SEGMENT_SUBSET_HALFLINE; SUBSET; ENDS_IN_SEGMENT]);;
1731 let HALFLINE_SUBSET_AFFINE_HULL = prove
1732 (`!x y. aff_ge {x} {y} SUBSET affine hull {x,y}`,
1733 REWRITE_TAC[AFF_GE_SUBSET_AFFINE_HULL; SET_RULE `{x,y} = {x} UNION {y}`]);;
1735 let HALFLINE_INTER_COMPACT_SEGMENT = prove
1737 compact s /\ convex s /\ a IN s
1738 ==> ?c. aff_ge {a} {b} INTER s = segment[a,c]`,
1739 REPEAT STRIP_TAC THEN ASM_CASES_TAC `b:real^N = a` THENL
1740 [EXISTS_TAC `a:real^N` THEN
1741 ASM_REWRITE_TAC[SEGMENT_REFL; HALFLINE_REFL] THEN ASM SET_TAC[];
1744 `?u v:real^N. aff_ge {a} {b} INTER s = segment[u,v]`
1745 STRIP_ASSUME_TAC THENL
1746 [MATCH_MP_TAC COMPACT_CONVEX_COLLINEAR_SEGMENT THEN
1747 ASM_SIMP_TAC[CLOSED_INTER_COMPACT; CLOSED_AFF_GE; FINITE_SING] THEN
1748 ASM_SIMP_TAC[CONVEX_INTER; CONVEX_AFF_GE] THEN CONJ_TAC THENL
1749 [REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_INTER] THEN
1750 ASM_MESON_TAC[ENDS_IN_HALFLINE];
1751 MATCH_MP_TAC COLLINEAR_SUBSET THEN
1752 EXISTS_TAC `affine hull {a:real^N,b}` THEN
1753 REWRITE_TAC[COLLINEAR_AFFINE_HULL_COLLINEAR; COLLINEAR_2] THEN
1754 MATCH_MP_TAC(SET_RULE `s SUBSET u ==> (s INTER t) SUBSET u`) THEN
1755 REWRITE_TAC[HALFLINE_SUBSET_AFFINE_HULL]];
1756 ASM_CASES_TAC `u:real^N = a` THENL [ASM_MESON_TAC[]; ALL_TAC] THEN
1757 ASM_CASES_TAC `v:real^N = a` THENL
1758 [ASM_MESON_TAC[SEGMENT_SYM]; ALL_TAC] THEN
1759 SUBGOAL_THEN `u IN aff_ge {a:real^N} {b} /\ v IN aff_ge {a} {b}`
1760 MP_TAC THENL [ASM_MESON_TAC[IN_INTER; ENDS_IN_SEGMENT]; ALL_TAC] THEN
1761 GEN_REWRITE_TAC (LAND_CONV o TOP_DEPTH_CONV) [HALFLINE; IN_ELIM_THM] THEN
1762 DISCH_THEN(CONJUNCTS_THEN2
1763 (X_CHOOSE_THEN `s:real` MP_TAC) (X_CHOOSE_THEN `t:real` MP_TAC)) THEN
1764 MAP_EVERY ASM_CASES_TAC [`s = &0`; `t = &0`] THEN
1765 ASM_REWRITE_TAC[REAL_SUB_RZERO; VECTOR_MUL_LID; VECTOR_MUL_LZERO;
1766 VECTOR_ADD_RID] THEN
1767 ASM_REWRITE_TAC[REAL_LE_LT] THEN REPEAT STRIP_TAC THEN
1768 SUBGOAL_THEN `(a:real^N) IN segment[u,v]` MP_TAC THENL
1769 [ASM_MESON_TAC[IN_INTER; ENDS_IN_HALFLINE]; ALL_TAC] THEN
1770 ASM_REWRITE_TAC[IN_SEGMENT; LEFT_IMP_EXISTS_THM] THEN
1771 X_GEN_TAC `u:real` THEN
1772 REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
1773 REWRITE_TAC[VECTOR_ARITH
1774 `a = (&1 - u) % ((&1 - s) % a + s % b) + u % ((&1 - t) % a + t % b) <=>
1775 ((&1 - u) * s + u * t) % (b - a):real^N = vec 0`] THEN
1776 ASM_REWRITE_TAC[VECTOR_MUL_EQ_0; VECTOR_SUB_EQ] THEN
1777 ASM_SIMP_TAC[REAL_LE_MUL; REAL_SUB_LE; REAL_LT_IMP_LE; REAL_ARITH
1778 `&0 <= x /\ &0 <= y ==> (x + y = &0 <=> x = &0 /\ y = &0)`] THEN
1779 ASM_SIMP_TAC[REAL_ENTIRE; REAL_LT_IMP_NZ] THEN REAL_ARITH_TAC]);;
1781 (* ------------------------------------------------------------------------- *)
1782 (* Definition and properties of conv0. *)
1783 (* ------------------------------------------------------------------------- *)
1785 let conv0 = new_definition `conv0 S:real^A->bool = affsign sgn_gt {} S`;;
1787 let CONV0_INJECTIVE_LINEAR_IMAGE = prove
1788 (`!f s. linear f /\ (!x y. f x = f y ==> x = y)
1789 ==> conv0(IMAGE f s) = IMAGE f (conv0 s)`,
1790 REPEAT GEN_TAC THEN DISCH_TAC THEN
1791 FIRST_ASSUM(ASSUME_TAC o GSYM o MATCH_MP AFFSIGN_INJECTIVE_LINEAR_IMAGE) THEN
1792 ASM_REWRITE_TAC[conv0; IMAGE_CLAUSES]);;
1794 add_linear_invariants [CONV0_INJECTIVE_LINEAR_IMAGE];;
1796 let CONV0_TRANSLATION = prove
1797 (`!a s. conv0(IMAGE (\x. a + x) s) = IMAGE (\x. a + x) (conv0 s)`,
1798 REWRITE_TAC[conv0; GSYM AFFSIGN_TRANSLATION; IMAGE_CLAUSES]);;
1800 add_translation_invariants [CONV0_TRANSLATION];;
1802 let CONV0_SUBSET_CONVEX_HULL = prove
1803 (`!s. conv0 s SUBSET convex hull s`,
1804 REWRITE_TAC[conv0; AFFSIGN; sgn_gt; CONVEX_HULL_FINITE; UNION_EMPTY] THEN
1805 REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN
1806 REPEAT GEN_TAC THEN MATCH_MP_TAC MONO_EXISTS THEN
1807 MESON_TAC[REAL_LT_IMP_LE]);;
1809 let CONV0_AFF_GT = prove
1810 (`!s. conv0 s = aff_gt {} s`,
1811 REWRITE_TAC[conv0; aff_gt_def]);;
1813 let CONVEX_HULL_CONV0_DECOMP = prove
1816 ==> convex hull s = conv0 s UNION
1817 UNIONS {convex hull (s DELETE a) | a | a IN s}`,
1818 REWRITE_TAC[CONV0_AFF_GT; CONVEX_HULL_AFF_GE] THEN
1819 REPEAT STRIP_TAC THEN MATCH_MP_TAC AFF_GE_AFF_GT_DECOMP THEN
1820 ASM_REWRITE_TAC[FINITE_EMPTY] THEN SET_TAC[]);;
1822 let CONVEX_CONV0 = prove
1823 (`!s. convex(conv0 s)`,
1824 REWRITE_TAC[CONV0_AFF_GT; CONVEX_AFF_GT]);;
1826 let BOUNDED_CONV0 = prove
1827 (`!s:real^N->bool. bounded s ==> bounded(conv0 s)`,
1828 REPEAT STRIP_TAC THEN MATCH_MP_TAC BOUNDED_SUBSET THEN
1829 EXISTS_TAC `convex hull s:real^N->bool` THEN
1830 ASM_SIMP_TAC[BOUNDED_CONVEX_HULL; CONV0_SUBSET_CONVEX_HULL]);;
1832 let MEASURABLE_CONV0 = prove
1833 (`!s. bounded s ==> measurable(conv0 s)`,
1834 REPEAT STRIP_TAC THEN MATCH_MP_TAC MEASURABLE_CONVEX THEN
1835 ASM_SIMP_TAC[CONVEX_CONV0; BOUNDED_CONV0]);;
1837 let NEGLIGIBLE_CONVEX_HULL_DIFF_CONV0 = prove
1839 FINITE s /\ CARD s <= dimindex(:N) + 1
1840 ==> negligible(convex hull s DIFF conv0 s)`,
1841 REPEAT STRIP_TAC THEN ASM_SIMP_TAC[CONVEX_HULL_CONV0_DECOMP] THEN
1842 REWRITE_TAC[SET_RULE `(s UNION t) DIFF s = t DIFF s`] THEN
1843 MATCH_MP_TAC NEGLIGIBLE_DIFF THEN MATCH_MP_TAC NEGLIGIBLE_UNIONS THEN
1844 ONCE_REWRITE_TAC[SIMPLE_IMAGE] THEN ASM_SIMP_TAC[FINITE_IMAGE] THEN
1845 REWRITE_TAC[FORALL_IN_IMAGE] THEN REPEAT STRIP_TAC THEN
1846 MATCH_MP_TAC NEGLIGIBLE_CONVEX_HULL THEN
1847 ASM_SIMP_TAC[FINITE_DELETE; CARD_DELETE] THEN ASM_ARITH_TAC);;
1849 let MEASURE_CONV0_CONVEX_HULL = prove
1851 FINITE s /\ CARD s <= dimindex(:N) + 1
1852 ==> measure(conv0 s) = measure(convex hull s)`,
1853 REPEAT STRIP_TAC THEN MATCH_MP_TAC MEASURE_NEGLIGIBLE_SYMDIFF THEN
1854 ASM_SIMP_TAC[MEASURABLE_CONVEX_HULL; FINITE_IMP_BOUNDED] THEN
1855 MATCH_MP_TAC NEGLIGIBLE_UNION THEN
1856 ASM_SIMP_TAC[NEGLIGIBLE_CONVEX_HULL_DIFF_CONV0] THEN
1857 ASM_SIMP_TAC[CONV0_SUBSET_CONVEX_HULL; NEGLIGIBLE_EMPTY;
1858 SET_RULE `s SUBSET t ==> s DIFF t = {}`]);;
1860 (* ------------------------------------------------------------------------- *)
1861 (* Orthonormal triples of vectors in 3D. *)
1862 (* ------------------------------------------------------------------------- *)
1864 let orthonormal = new_definition
1865 `orthonormal e1 e2 e3 <=>
1866 e1 dot e1 = &1 /\ e2 dot e2 = &1 /\ e3 dot e3 = &1 /\
1867 e1 dot e2 = &0 /\ e1 dot e3 = &0 /\ e2 dot e3 = &0 /\
1868 &0 < (e1 cross e2) dot e3`;;
1870 let ORTHONORMAL_LINEAR_IMAGE = prove
1871 (`!f. linear(f) /\ (!x. norm(f x) = norm x) /\
1872 (2 <= dimindex(:3) ==> det(matrix f) = &1)
1873 ==> !e1 e2 e3. orthonormal (f e1) (f e2) (f e3) <=>
1874 orthonormal e1 e2 e3`,
1875 SIMP_TAC[DIMINDEX_3; ARITH; CONJ_ASSOC; GSYM ORTHOGONAL_TRANSFORMATION] THEN
1876 SIMP_TAC[orthonormal; CROSS_ORTHOGONAL_TRANSFORMATION] THEN
1877 SIMP_TAC[orthogonal_transformation; VECTOR_MUL_LID]);;
1879 add_linear_invariants [ORTHONORMAL_LINEAR_IMAGE];;
1881 let ORTHONORMAL_PERMUTE = prove
1882 (`!e1 e2 e3. orthonormal e1 e2 e3 ==> orthonormal e2 e3 e1`,
1883 REWRITE_TAC[orthonormal] THEN
1884 REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
1885 ONCE_REWRITE_TAC[GSYM CROSS_TRIPLE] THEN ASM_REWRITE_TAC[] THEN
1886 ONCE_REWRITE_TAC[DOT_SYM] THEN ASM_REWRITE_TAC[]);;
1888 let ORTHONORMAL_CROSS = prove
1890 orthonormal e1 e2 e3
1891 ==> e2 cross e3 = e1 /\ e3 cross e1 = e2 /\ e1 cross e2 = e3`,
1893 `!e1 e2 e3. orthonormal e1 e2 e3 ==> e3 cross e1 = e2`
1894 (fun th -> MESON_TAC[th; ORTHONORMAL_PERMUTE]) THEN
1895 GEOM_BASIS_MULTIPLE_TAC 1 `e1:real^3` THEN X_GEN_TAC `k:real` THEN
1896 REWRITE_TAC[orthonormal; DOT_LMUL; DOT_RMUL] THEN
1897 SIMP_TAC[DOT_BASIS_BASIS; DIMINDEX_3; ARITH; REAL_MUL_RID] THEN
1898 REWRITE_TAC[REAL_RING `k * k = &1 <=> k = &1 \/ k = -- &1`] THEN
1899 ASM_CASES_TAC `k = -- &1` THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
1900 ASM_CASES_TAC `k = &1` THEN
1901 ASM_REWRITE_TAC[VECTOR_MUL_LID; REAL_MUL_LID; REAL_MUL_RID] THEN
1902 SIMP_TAC[cross; DOT_3; VECTOR_3; CART_EQ; FORALL_3; DIMINDEX_3;
1903 BASIS_COMPONENT; DIMINDEX_3; ARITH; REAL_POS] THEN
1904 REWRITE_TAC[REAL_MUL_LZERO; REAL_SUB_RZERO; REAL_ADD_RID;
1907 ASM_CASES_TAC `(e2:real^3)$1 = &0` THEN ASM_REWRITE_TAC[] THEN
1908 ASM_CASES_TAC `(e3:real^3)$1 = &0` THEN ASM_REWRITE_TAC[] THEN
1909 REWRITE_TAC[REAL_MUL_RZERO; REAL_SUB_RZERO; REAL_ADD_LID] THEN
1910 REWRITE_TAC[REAL_SUB_LZERO; REAL_MUL_RID] THEN
1911 MATCH_MP_TAC(REAL_ARITH
1912 `(u = &1 /\ v = &1 /\ w = &0 ==> a = b /\ --c = d \/ a = --b /\ c = d) /\
1913 (a = --b /\ c = d ==> x <= &0)
1914 ==> (u = &1 /\ v = &1 /\ w = &0 /\ &0 < x
1915 ==> a:real = b /\ --c:real = d)`) THEN
1916 CONJ_TAC THENL [CONV_TAC REAL_RING; ALL_TAC] THEN
1917 DISCH_THEN(CONJUNCTS_THEN SUBST1_TAC) THEN
1918 MATCH_MP_TAC(REAL_ARITH
1919 `&0 <= x * x /\ &0 <= y * y ==> --x * x + y * -- y <= &0`) THEN
1920 REWRITE_TAC[REAL_LE_SQUARE]);;
1922 let ORTHONORMAL_IMP_NONZERO = prove
1923 (`!e1 e2 e3. orthonormal e1 e2 e3
1924 ==> ~(e1 = vec 0) /\ ~(e2 = vec 0) /\ ~(e3 = vec 0)`,
1925 REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN
1926 REWRITE_TAC[DE_MORGAN_THM] THEN STRIP_TAC THEN
1927 ASM_REWRITE_TAC[orthonormal; DOT_LZERO] THEN REAL_ARITH_TAC);;
1929 let ORTHONORMAL_IMP_DISTINCT = prove
1930 (`!e1 e2 e3. orthonormal e1 e2 e3 ==> ~(e1 = e2) /\ ~(e1 = e3) /\ ~(e2 = e3)`,
1931 REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN
1932 REWRITE_TAC[DE_MORGAN_THM] THEN STRIP_TAC THEN
1933 ASM_REWRITE_TAC[orthonormal; DOT_LZERO] THEN REAL_ARITH_TAC);;
1935 let ORTHONORMAL_IMP_INDEPENDENT = prove
1936 (`!e1 e2 e3. orthonormal e1 e2 e3 ==> independent {e1,e2,e3}`,
1937 REPEAT STRIP_TAC THEN MATCH_MP_TAC PAIRWISE_ORTHOGONAL_INDEPENDENT THEN
1938 CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[ORTHONORMAL_IMP_NONZERO]] THEN
1939 RULE_ASSUM_TAC(REWRITE_RULE[orthonormal]) THEN
1940 REWRITE_TAC[pairwise; IN_INSERT; NOT_IN_EMPTY] THEN
1941 REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[orthogonal] THEN
1942 ASM_MESON_TAC[DOT_SYM]);;
1944 let ORTHONORMAL_IMP_SPANNING = prove
1945 (`!e1 e2 e3. orthonormal e1 e2 e3 ==> span {e1,e2,e3} = (:real^3)`,
1946 REPEAT STRIP_TAC THEN
1947 MP_TAC(ISPECL [`(:real^3)`; `{e1:real^3,e2,e3}`] CARD_EQ_DIM) THEN
1948 ASM_SIMP_TAC[ORTHONORMAL_IMP_INDEPENDENT; SUBSET_UNIV] THEN
1949 REWRITE_TAC[DIM_UNIV; DIMINDEX_3; HAS_SIZE; FINITE_INSERT; FINITE_EMPTY] THEN
1950 SIMP_TAC[CARD_CLAUSES; FINITE_INSERT; FINITE_EMPTY; IN_INSERT] THEN
1951 FIRST_ASSUM(ASSUME_TAC o MATCH_MP ORTHONORMAL_IMP_DISTINCT) THEN
1952 ASM_REWRITE_TAC[NOT_IN_EMPTY; ARITH] THEN SET_TAC[]);;
1954 let ORTHONORMAL_IMP_INDEPENDENT_EXPLICIT_0 = prove
1955 (`!e1 e2 e3 t1 t2 t3.
1956 orthonormal e1 e2 e3
1957 ==> (t1 % e1 + t2 % e2 + t3 % e3 = vec 0 <=>
1958 t1 = &0 /\ t2 = &0 /\ t3 = &0)`,
1959 REPEAT STRIP_TAC THEN MATCH_MP_TAC INDEPENDENT_3 THEN
1960 ASM_MESON_TAC[ORTHONORMAL_IMP_INDEPENDENT; ORTHONORMAL_IMP_DISTINCT]);;
1962 let ORTHONORMAL_IMP_INDEPENDENT_EXPLICIT = prove
1963 (`!e1 e2 e3 s1 s2 s3 t1 t2 t3.
1964 orthonormal e1 e2 e3
1965 ==> (s1 % e1 + s2 % e2 + s3 % e3 = t1 % e1 + t2 % e2 + t3 % e3 <=>
1966 s1 = t1 /\ s2 = t2 /\ s3 = t3)`,
1967 SIMP_TAC[ORTHONORMAL_IMP_INDEPENDENT_EXPLICIT_0; REAL_SUB_0; VECTOR_ARITH
1968 `a % x + b % y + c % z:real^3 = a' % x + b' % y + c' % z <=>
1969 (a - a') % x + (b - b') % y + (c - c') % z = vec 0`]);;
1971 (* ------------------------------------------------------------------------- *)
1972 (* Flyspeck arcV is the same as angle even in degenerate cases. *)
1973 (* ------------------------------------------------------------------------- *)
1975 let arcV = new_definition
1976 `arcV u v w = acs (( (v - u) dot (w - u))/((norm (v-u)) * (norm (w-u))))`;;
1978 let ARCV_ANGLE = prove
1979 (`!u v w:real^N. arcV u v w = angle(v,u,w)`,
1980 REPEAT GEN_TAC THEN REWRITE_TAC[arcV; angle; vector_angle] THEN
1981 REWRITE_TAC[VECTOR_SUB_EQ] THEN
1982 ASM_CASES_TAC `v:real^N = u` THEN
1983 ASM_REWRITE_TAC[VECTOR_SUB_REFL; NORM_0; DOT_LZERO] THEN
1984 REWRITE_TAC[real_div; REAL_MUL_LZERO; ACS_0] THEN
1985 ASM_CASES_TAC `w:real^N = u` THEN
1986 ASM_REWRITE_TAC[VECTOR_SUB_REFL; NORM_0; DOT_RZERO] THEN
1987 REWRITE_TAC[real_div; REAL_MUL_LZERO; ACS_0]);;
1989 let ARCV_LINEAR_IMAGE_EQ = prove
1991 linear f /\ (!x. norm(f x) = norm x)
1992 ==> arcV (f a) (f b) (f c) = arcV a b c`,
1993 REWRITE_TAC[ARCV_ANGLE; ANGLE_LINEAR_IMAGE_EQ]);;
1995 add_linear_invariants [ARCV_LINEAR_IMAGE_EQ];;
1997 let ARCV_TRANSLATION_EQ = prove
1998 (`!a b c d. arcV (a + b) (a + c) (a + d) = arcV b c d`,
1999 REWRITE_TAC[ARCV_ANGLE; ANGLE_TRANSLATION_EQ]);;
2001 add_translation_invariants [ARCV_TRANSLATION_EQ];;
2003 (* ------------------------------------------------------------------------- *)
2004 (* Azimuth angle. *)
2005 (* ------------------------------------------------------------------------- *)
2007 let AZIM_EXISTS = prove
2009 ?theta. &0 <= theta /\ theta < &2 * pi /\
2012 orthonormal e1 e2 e3 /\
2013 dist(w,v) % e3 = w - v /\
2016 w1 - v = (r1 * cos psi) % e1 +
2017 (r1 * sin psi) % e2 +
2019 w2 - v = (r2 * cos (psi + theta)) % e1 +
2020 (r2 * sin (psi + theta)) % e2 +
2022 (~collinear {v, w, w1} ==> &0 < r1) /\
2023 (~collinear {v, w, w2} ==> &0 < r2)`,
2025 (`cos(p) % e + sin(p) % rotate2d (pi / &2) e = rotate2d p e`,
2026 SIMP_TAC[CART_EQ; VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT;
2027 FORALL_2; rotate2d; LAMBDA_BETA; DIMINDEX_2; ARITH; VECTOR_2] THEN
2028 REWRITE_TAC[SIN_PI2; COS_PI2] THEN REAL_ARITH_TAC) in
2029 GEN_GEOM_ORIGIN_TAC `v:real^3` ["e1"; "e2"; "e3"] THEN
2030 REWRITE_TAC[VECTOR_SUB_RZERO; DIST_0] THEN
2031 REPEAT GEN_TAC THEN REWRITE_TAC[RIGHT_AND_EXISTS_THM] THEN
2032 GEN_REWRITE_TAC I [SWAP_EXISTS_THM] THEN
2033 EXISTS_TAC `(w dot (w1:real^3)) / (w dot w)` THEN
2034 GEN_REWRITE_TAC I [SWAP_EXISTS_THM] THEN
2035 EXISTS_TAC `(w dot (w2:real^3)) / (w dot w)` THEN
2036 GEOM_BASIS_MULTIPLE_TAC 3 `w:real^3` THEN
2037 X_GEN_TAC `w:real` THEN GEN_REWRITE_TAC LAND_CONV
2038 [REAL_ARITH `&0 <= w <=> w = &0 \/ &0 < w`] THEN
2040 [ASM_REWRITE_TAC[VECTOR_MUL_LZERO; VECTOR_MUL_RZERO; NORM_0] THEN
2041 EXISTS_TAC `&0` THEN MP_TAC PI_POS THEN REAL_ARITH_TAC;
2043 SIMP_TAC[DOT_LMUL; NORM_MUL; DIMINDEX_3; ARITH; DOT_RMUL; DOT_BASIS;
2044 VECTOR_MUL_COMPONENT; NORM_BASIS; BASIS_COMPONENT] THEN
2045 REWRITE_TAC[VECTOR_MUL_ASSOC; REAL_MUL_RID] THEN
2046 ASM_SIMP_TAC[REAL_FIELD `&0 < w ==> (w * x) / (w * w) * w = x`;
2047 REAL_ARITH `&0 < w ==> abs w = w`] THEN
2048 ASM_REWRITE_TAC[VECTOR_ARITH
2049 `a % x:real^3 = a % y <=> a % (x - y) = vec 0`] THEN
2050 ASM_SIMP_TAC[VECTOR_MUL_EQ_0; REAL_LT_IMP_NZ; BASIS_NONZERO;
2051 DIMINDEX_3; ARITH; VECTOR_SUB_EQ] THEN
2052 REWRITE_TAC[MESON[] `(!e3. p e3 /\ e3 = a ==> q e3) <=> p a ==> q a`] THEN
2053 ONCE_REWRITE_TAC[VECTOR_ARITH `x:real^3 = a + b + c <=> x - c = a + b`] THEN
2055 ABBREV_TAC `v1:real^3 = w1 - (w1$3) % basis 3` THEN
2056 ABBREV_TAC `v2:real^3 = w2 - (w2$3) % basis 3` THEN
2058 `(collinear{vec 0, w % basis 3, w1} <=>
2059 w1 - w1$3 % basis 3:real^3 = vec 0) /\
2060 (collinear{vec 0, w % basis 3, w2} <=>
2061 w2 - w2$3 % basis 3:real^3 = vec 0)`
2062 (fun th -> REWRITE_TAC[th])
2064 [ASM_SIMP_TAC[COLLINEAR_LEMMA; VECTOR_MUL_EQ_0; REAL_LT_IMP_NZ;
2065 BASIS_NONZERO; DIMINDEX_3; ARITH] THEN
2066 MAP_EVERY EXPAND_TAC ["v1"; "v2"] THEN
2067 SIMP_TAC[CART_EQ; VEC_COMPONENT; VECTOR_ADD_COMPONENT; FORALL_3;
2068 VECTOR_MUL_COMPONENT; BASIS_COMPONENT; DIMINDEX_3; ARITH;
2069 VECTOR_SUB_COMPONENT; REAL_MUL_RZERO; REAL_MUL_RID;
2070 REAL_SUB_RZERO] THEN
2071 REWRITE_TAC[RIGHT_EXISTS_AND_THM] THEN
2072 CONV_TAC(BINOP_CONV(BINOP_CONV(ONCE_DEPTH_CONV SYM_CONV))) THEN
2073 ASM_SIMP_TAC[GSYM REAL_EQ_RDIV_EQ; EXISTS_REFL] THEN REAL_ARITH_TAC;
2075 ASM_REWRITE_TAC[] THEN
2076 SUBGOAL_THEN `(v1:real^3)$3 = &0 /\ (v2:real^3)$3 = &0` MP_TAC THENL
2077 [MAP_EVERY EXPAND_TAC ["v1"; "v2"] THEN
2078 REWRITE_TAC[VECTOR_SUB_COMPONENT; VECTOR_MUL_COMPONENT; VECTOR_SUB_EQ] THEN
2079 SIMP_TAC[BASIS_COMPONENT; DIMINDEX_3; ARITH] THEN REAL_ARITH_TAC;
2081 MAP_EVERY (fun t -> SPEC_TAC(t,t)) [`v2:real^3`; `v1:real^3`] THEN
2082 POP_ASSUM_LIST(K ALL_TAC) THEN REWRITE_TAC[orthonormal] THEN
2083 SIMP_TAC[DOT_BASIS; BASIS_COMPONENT; DIMINDEX_3; ARITH] THEN
2084 ONCE_REWRITE_TAC[TAUT `a /\ b /\ c /\ d /\ e /\ f <=>
2085 d /\ e /\ a /\ b /\ c /\ f`] THEN
2086 REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
2087 PAD2D3D_TAC THEN REPEAT STRIP_TAC THEN
2088 SIMP_TAC[cross; VECTOR_3; pad2d3d; LAMBDA_BETA; DIMINDEX_3; ARITH] THEN
2089 REWRITE_TAC[IMP_IMP; GSYM CONJ_ASSOC] THEN
2090 ASM_CASES_TAC `v1:real^2 = vec 0` THEN ASM_REWRITE_TAC[NORM_POS_LT] THENL
2091 [MP_TAC(ISPECL [`basis 1:real^2`; `v2:real^2`]
2092 ROTATION_ROTATE2D_EXISTS_GEN) THEN
2093 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `t:real` THEN
2094 STRIP_TAC THEN ASM_REWRITE_TAC[] THEN REPEAT STRIP_TAC THEN
2095 MP_TAC(ISPECL [`e1:real^2`; `basis 1:real^2`]
2096 ROTATION_ROTATE2D_EXISTS_GEN) THEN
2097 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `p:real` THEN STRIP_TAC THEN
2098 MAP_EVERY EXISTS_TAC [`&0`; `norm(v2:real^2)`] THEN
2099 ASM_REWRITE_TAC[NORM_POS_LT] THEN
2100 REWRITE_TAC[REAL_MUL_LZERO; VECTOR_MUL_LZERO; VECTOR_ADD_RID] THEN
2101 SUBGOAL_THEN `norm(e1:real^2) = &1 /\ norm(e2:real^2) = &1`
2102 STRIP_ASSUME_TAC THENL [ASM_REWRITE_TAC[NORM_EQ_1]; ALL_TAC] THEN
2103 SUBGOAL_THEN `e2 = rotate2d (pi / &2) e1` SUBST1_TAC THENL
2104 [MATCH_MP_TAC ROTATION_ROTATE2D_EXISTS_ORTHOGONAL_ORIENTED THEN
2105 ASM_REWRITE_TAC[NORM_EQ_1; orthogonal];
2107 REWRITE_TAC[GSYM VECTOR_MUL_ASSOC; GSYM VECTOR_ADD_LDISTRIB] THEN
2108 REWRITE_TAC[lemma] THEN ONCE_REWRITE_TAC[REAL_ADD_SYM] THEN
2109 REWRITE_TAC[ROTATE2D_ADD] THEN ASM_REWRITE_TAC[VECTOR_MUL_LID] THEN
2110 MATCH_MP_TAC VECTOR_MUL_LCANCEL_IMP THEN
2111 EXISTS_TAC `norm(basis 1:real^2)` THEN
2112 ASM_SIMP_TAC[NORM_EQ_0; BASIS_NONZERO; DIMINDEX_2; ARITH] THEN
2113 FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC LAND_CONV [SYM th]) THEN
2114 ONCE_REWRITE_TAC[VECTOR_ARITH `a % b % x:real^2 = b % a % x`] THEN
2116 SIMP_TAC[GSYM(MATCH_MP LINEAR_CMUL (SPEC_ALL LINEAR_ROTATE2D))] THEN
2118 ASM_SIMP_TAC[LINEAR_CMUL; LINEAR_ROTATE2D; VECTOR_MUL_LID];
2119 MP_TAC(ISPECL [`v1:real^2`; `v2:real^2`] ROTATION_ROTATE2D_EXISTS_GEN) THEN
2120 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `t:real` THEN
2121 STRIP_TAC THEN ASM_REWRITE_TAC[] THEN REPEAT STRIP_TAC THEN
2122 MP_TAC(ISPECL [`e1:real^2`; `v1:real^2`] ROTATION_ROTATE2D_EXISTS_GEN) THEN
2123 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `p:real` THEN STRIP_TAC THEN
2124 MAP_EVERY EXISTS_TAC [`norm(v1:real^2)`; `norm(v2:real^2)`] THEN
2125 ASM_REWRITE_TAC[NORM_POS_LT] THEN
2126 SUBGOAL_THEN `norm(e1:real^2) = &1 /\ norm(e2:real^2) = &1`
2127 STRIP_ASSUME_TAC THENL [ASM_REWRITE_TAC[NORM_EQ_1]; ALL_TAC] THEN
2128 SUBGOAL_THEN `e2 = rotate2d (pi / &2) e1` SUBST1_TAC THENL
2129 [MATCH_MP_TAC ROTATION_ROTATE2D_EXISTS_ORTHOGONAL_ORIENTED THEN
2130 ASM_REWRITE_TAC[NORM_EQ_1; orthogonal];
2132 REWRITE_TAC[GSYM VECTOR_MUL_ASSOC; GSYM VECTOR_ADD_LDISTRIB] THEN
2133 REWRITE_TAC[lemma] THEN ONCE_REWRITE_TAC[REAL_ADD_SYM] THEN
2134 REWRITE_TAC[ROTATE2D_ADD] THEN ASM_REWRITE_TAC[VECTOR_MUL_LID] THEN
2135 MATCH_MP_TAC VECTOR_MUL_LCANCEL_IMP THEN EXISTS_TAC `norm(v1:real^2)` THEN
2136 ASM_REWRITE_TAC[NORM_EQ_0] THEN
2137 FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC LAND_CONV [SYM th]) THEN
2138 ONCE_REWRITE_TAC[VECTOR_ARITH `a % b % x:real^2 = b % a % x`] THEN
2140 SIMP_TAC[GSYM(MATCH_MP LINEAR_CMUL (SPEC_ALL LINEAR_ROTATE2D))] THEN
2142 ASM_SIMP_TAC[LINEAR_CMUL; LINEAR_ROTATE2D; VECTOR_MUL_LID]]);;
2145 (REWRITE_RULE[SKOLEM_THM]
2146 (REWRITE_RULE[RIGHT_EXISTS_IMP_THM] AZIM_EXISTS));;
2148 let azim_def = new_definition
2150 if collinear {v,w,w1} \/ collinear {v,w,w2} then &0
2151 else @theta. &0 <= theta /\ theta < &2 * pi /\
2154 orthonormal e1 e2 e3 /\
2155 dist(w,v) % e3 = w - v /\
2158 w1 - v = (r1 * cos psi) % e1 +
2159 (r1 * sin psi) % e2 +
2161 w2 - v = (r2 * cos (psi + theta)) % e1 +
2162 (r2 * sin (psi + theta)) % e2 +
2164 &0 < r1 /\ &0 < r2`;;
2167 (`!v w w1 w2:real^3.
2168 &0 <= azim v w w1 w2 /\ azim v w w1 w2 < &2 * pi /\
2171 orthonormal e1 e2 e3 /\
2172 dist(w,v) % e3 = w - v /\
2175 w1 - v = (r1 * cos psi) % e1 +
2176 (r1 * sin psi) % e2 +
2178 w2 - v = (r2 * cos (psi + azim v w w1 w2)) % e1 +
2179 (r2 * sin (psi + azim v w w1 w2)) % e2 +
2181 (~collinear {v, w, w1} ==> &0 < r1) /\
2182 (~collinear {v, w, w2} ==> &0 < r2)`,
2183 REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[azim_def] THEN
2184 COND_CASES_TAC THENL
2186 RULE_ASSUM_TAC(REWRITE_RULE[DE_MORGAN_THM]) THEN ASM_REWRITE_TAC[] THEN
2187 CONV_TAC SELECT_CONV THEN
2188 MP_TAC(ISPECL [`v:real^3`; `w:real^3`; `w1:real^3`; `w2:real^3`]
2190 ASM_REWRITE_TAC[]] THEN
2191 SIMP_TAC[PI_POS; REAL_LT_MUL; REAL_OF_NUM_LT; ARITH; REAL_LE_REFL] THEN
2192 FIRST_X_ASSUM DISJ_CASES_TAC THENL
2193 [MP_TAC(ISPECL [`v:real^3`; `w:real^3`; `w2:real^3`; `w1:real^3`]
2195 DISCH_THEN(CHOOSE_THEN(MP_TAC o CONJUNCT2 o CONJUNCT2)) THEN
2196 REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN
2197 MAP_EVERY X_GEN_TAC [`h2:real`; `h1:real`] THEN
2198 DISCH_TAC THEN MAP_EVERY EXISTS_TAC [`h1:real`; `h2:real`] THEN
2199 MAP_EVERY X_GEN_TAC [`e1:real^3`; `e2:real^3`; `e3:real^3`] THEN
2201 FIRST_X_ASSUM(MP_TAC o SPECL [`e1:real^3`; `e2:real^3`; `e3:real^3`]) THEN
2202 ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN
2203 X_GEN_TAC `psi:real` THEN
2204 REWRITE_TAC[LEFT_IMP_EXISTS_THM; REAL_ADD_RID] THEN
2205 MAP_EVERY X_GEN_TAC [`r2:real`; `r1:real`] THEN STRIP_TAC THEN
2206 MAP_EVERY EXISTS_TAC [`&0`; `r2:real`];
2207 MP_TAC(ISPECL [`v:real^3`; `w:real^3`; `w1:real^3`; `w2:real^3`]
2209 DISCH_THEN(CHOOSE_THEN(MP_TAC o CONJUNCT2 o CONJUNCT2)) THEN
2210 REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN
2211 MAP_EVERY X_GEN_TAC [`h1:real`; `h2:real`] THEN
2212 DISCH_TAC THEN MAP_EVERY EXISTS_TAC [`h1:real`; `h2:real`] THEN
2213 MAP_EVERY X_GEN_TAC [`e1:real^3`; `e2:real^3`; `e3:real^3`] THEN
2215 FIRST_X_ASSUM(MP_TAC o SPECL [`e1:real^3`; `e2:real^3`; `e3:real^3`]) THEN
2216 ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN
2217 X_GEN_TAC `psi:real` THEN
2218 REWRITE_TAC[LEFT_IMP_EXISTS_THM; REAL_ADD_RID] THEN
2219 MAP_EVERY X_GEN_TAC [`r1:real`; `r2:real`] THEN STRIP_TAC THEN
2220 MAP_EVERY EXISTS_TAC [`r1:real`; `&0`]] THEN
2221 ASM_REWRITE_TAC[] THEN
2222 FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV
2223 [SET_RULE `{v,w,x} = {w,v,x}`]) THEN
2224 ONCE_REWRITE_TAC[COLLINEAR_3] THEN ASM_REWRITE_TAC[] THEN
2225 UNDISCH_THEN `dist(w:real^3,v) % e3 = w - v` (SUBST1_TAC o SYM) THEN
2226 REWRITE_TAC[GSYM DOT_CAUCHY_SCHWARZ_EQUAL] THEN
2227 RULE_ASSUM_TAC(REWRITE_RULE[orthonormal]) THEN
2228 ASM_REWRITE_TAC[DOT_LADD; DOT_RADD; DOT_LMUL; DOT_RMUL; REAL_MUL_RZERO] THEN
2229 ONCE_REWRITE_TAC[DOT_SYM] THEN ASM_REWRITE_TAC[REAL_MUL_RZERO] THEN
2230 REWRITE_TAC[REAL_ADD_LID; REAL_ADD_RID; REAL_MUL_RID] THEN
2231 REWRITE_TAC[REAL_ARITH `(r * c) * (r * c):real = r pow 2 * c pow 2`] THEN
2232 REWRITE_TAC[REAL_ARITH `r * c + r * s + f:real = r * (s + c) + f`] THEN
2233 REWRITE_TAC[SIN_CIRCLE] THEN REWRITE_TAC[REAL_RING
2234 `(d * h * d) pow 2 = (d * d) * (r * &1 + h * d * h * d) <=>
2235 d = &0 \/ r = &0`] THEN
2236 ASM_REWRITE_TAC[DIST_EQ_0; REAL_POW_EQ_0; ARITH] THEN
2237 DISCH_THEN SUBST1_TAC THEN REWRITE_TAC[REAL_MUL_LZERO; DOT_LZERO]);;
2239 let AZIM_UNIQUE = prove
2240 (`!v w w1 w2 h1 h2 r1 r2 e1 e2 e3 psi theta.
2243 orthonormal e1 e2 e3 /\
2244 dist(w,v) % e3 = w - v /\
2246 &0 < r1 /\ &0 < r2 /\
2247 w1 - v = (r1 * cos psi) % e1 +
2248 (r1 * sin psi) % e2 +
2250 w2 - v = (r2 * cos (psi + theta)) % e1 +
2251 (r2 * sin (psi + theta)) % e2 +
2253 ==> azim v w w1 w2 = theta`,
2254 REPEAT STRIP_TAC THEN
2255 SUBGOAL_THEN `~collinear{v:real^3,w,w2} /\ ~collinear {v,w,w1}`
2256 STRIP_ASSUME_TAC THENL
2257 [ONCE_REWRITE_TAC[SET_RULE `{a,b,c} = {b,a,c}`] THEN
2258 ONCE_REWRITE_TAC[COLLINEAR_3] THEN REWRITE_TAC[COLLINEAR_LEMMA] THEN
2259 ASM_REWRITE_TAC[] THEN ASM_REWRITE_TAC[VECTOR_SUB_EQ] THEN
2260 UNDISCH_THEN `dist(w:real^3,v) % e3 = w - v` (SUBST1_TAC o SYM) THEN
2261 REWRITE_TAC[VECTOR_MUL_ASSOC; VECTOR_ARITH
2262 `a + b + c % x:real^N = d % x <=> a + b + (c - d) % x = vec 0`] THEN
2263 ASM_SIMP_TAC[ORTHONORMAL_IMP_INDEPENDENT_EXPLICIT_0] THEN
2264 ASM_SIMP_TAC[CONJ_ASSOC; REAL_LT_IMP_NZ; SIN_CIRCLE; REAL_RING
2265 `s pow 2 + c pow 2 = &1 ==> (r * c = &0 /\ r * s = &0 <=> r = &0)`];
2267 SUBGOAL_THEN `(azim v w w1 w2 - theta) / (&2 * pi) = &0` MP_TAC THENL
2268 [ALL_TAC; MP_TAC PI_POS THEN CONV_TAC REAL_FIELD] THEN
2269 MATCH_MP_TAC REAL_EQ_INTEGERS_IMP THEN
2270 ASM_SIMP_TAC[REAL_SUB_RZERO; REAL_ABS_DIV; REAL_ABS_MUL; REAL_ABS_NUM;
2271 REAL_ABS_PI; REAL_LT_LDIV_EQ; REAL_LT_MUL; REAL_OF_NUM_LT; ARITH;
2272 PI_POS; INTEGER_CLOSED; REAL_MUL_LID] THEN
2273 MP_TAC(ISPECL [`v:real^3`; `w:real^3`; `w1:real^3`; `w2:real^3`] azim) THEN
2274 ASM_REWRITE_TAC[] THEN
2275 REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
2276 ASM_SIMP_TAC[REAL_ARITH
2277 `&0 <= x /\ x < k /\ &0 <= y /\ y < k ==> abs(x - y) < k`] THEN
2278 REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN
2279 MAP_EVERY X_GEN_TAC [`k1:real`; `k2:real`] THEN
2280 DISCH_THEN(MP_TAC o SPECL [`e1:real^3`; `e2:real^3`; `e3:real^3`]) THEN
2281 ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN
2282 MAP_EVERY X_GEN_TAC [`phi:real`; `s1:real`; `s2:real`] THEN
2283 UNDISCH_THEN `dist(w:real^3,v) % e3 = w - v` (SUBST1_TAC o SYM) THEN
2284 REWRITE_TAC[VECTOR_MUL_ASSOC] THEN
2285 ASM_SIMP_TAC[ORTHONORMAL_IMP_INDEPENDENT_EXPLICIT] THEN
2286 ONCE_REWRITE_TAC[TAUT `a /\ b /\ c /\ d <=> (c /\ d) /\ a /\ b`] THEN
2287 DISCH_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC MP_TAC) THEN
2288 DISCH_THEN(CONJUNCTS_THEN (MP_TAC o MATCH_MP (REAL_FIELD
2289 `r * c = r' * c' /\ r * s = r' * s' /\ u:real = v
2290 ==> s pow 2 + c pow 2 = &1 /\ s' pow 2 + c' pow 2 = &1 /\
2291 &0 < r /\ (r pow 2 = r' pow 2 ==> r = r')
2292 ==> s = s' /\ c = c'`))) THEN
2293 ASM_REWRITE_TAC[SIN_CIRCLE; GSYM REAL_EQ_SQUARE_ABS] THEN
2294 ASM_SIMP_TAC[REAL_ARITH
2295 `&0 < x /\ &0 < y ==> (abs x = abs y <=> x = y)`] THEN
2296 REWRITE_TAC[SIN_COS_EQ] THEN
2297 REWRITE_TAC[REAL_ARITH
2298 `psi + theta = (phi + az) + x:real <=> psi = phi + x + (az - theta)`] THEN
2299 DISCH_THEN(X_CHOOSE_THEN `m:real` STRIP_ASSUME_TAC) THEN
2300 ASM_REWRITE_TAC[REAL_EQ_ADD_LCANCEL] THEN
2301 REWRITE_TAC[REAL_ARITH
2302 `&2 * m * pi + x = &2 * n * pi <=> x = (n - m) * &2 * pi`] THEN
2303 DISCH_THEN(X_CHOOSE_THEN `n:real` STRIP_ASSUME_TAC) THEN
2304 ASM_SIMP_TAC[PI_POS; REAL_FIELD `&0 < pi ==> (x * &2 * pi) / (&2 * pi) = x`;
2307 let AZIM_TRANSLATION = prove
2308 (`!a v w w1 w2. azim (a + v) (a + w) (a + w1) (a + w2) = azim v w w1 w2`,
2309 REPEAT GEN_TAC THEN REWRITE_TAC[azim_def] THEN
2310 REWRITE_TAC[VECTOR_ARITH `(a + w) - (a + v):real^3 = w - v`;
2311 VECTOR_ARITH `a + w:real^3 = a + v <=> w = v`;
2312 NORM_ARITH `dist(a + v,a + w) = dist(v,w)`] THEN
2313 REWRITE_TAC[SET_RULE
2314 `{a + x,a + y,a + z} = IMAGE (\x:real^3. a + x) {x,y,z}`] THEN
2315 REWRITE_TAC[COLLINEAR_TRANSLATION_EQ]);;
2317 add_translation_invariants [AZIM_TRANSLATION];;
2319 let AZIM_LINEAR_IMAGE = prove
2320 (`!f. linear f /\ (!x. norm(f x) = norm x) /\
2321 (2 <= dimindex(:3) ==> det(matrix f) = &1)
2322 ==> !v w w1 w2. azim (f v) (f w) (f w1) (f w2) = azim v w w1 w2`,
2323 REPEAT STRIP_TAC THEN REWRITE_TAC[azim_def] THEN
2324 ASM_SIMP_TAC[GSYM LINEAR_SUB; dist] THEN
2325 MP_TAC(ISPEC `f:real^3->real^3` QUANTIFY_SURJECTION_THM) THEN
2327 [ASM_MESON_TAC[ORTHOGONAL_TRANSFORMATION;
2328 ORTHOGONAL_TRANSFORMATION_SURJECTIVE];
2330 DISCH_THEN(CONV_TAC o LAND_CONV o EXPAND_QUANTS_CONV) THEN
2331 ASM_SIMP_TAC[ORTHONORMAL_LINEAR_IMAGE] THEN
2332 ASM_SIMP_TAC[GSYM LINEAR_CMUL; GSYM LINEAR_ADD] THEN
2333 SUBGOAL_THEN `!x y. (f:real^3->real^3) x = f y <=> x = y` ASSUME_TAC THENL
2334 [ASM_MESON_TAC[PRESERVES_NORM_INJECTIVE]; ALL_TAC] THEN
2335 ASM_REWRITE_TAC[] THEN
2336 REWRITE_TAC[SET_RULE `{f x,f y,f z} = IMAGE f {x,y,z}`] THEN
2337 ASM_SIMP_TAC[COLLINEAR_LINEAR_IMAGE_EQ]);;
2339 add_linear_invariants [AZIM_LINEAR_IMAGE];;
2341 let AZIM_DEGENERATE = prove
2342 (`(!v w w1 w2. v = w ==> azim v w w1 w2 = &0) /\
2343 (!v w w1 w2. collinear{v,w,w1} ==> azim v w w1 w2 = &0) /\
2344 (!v w w1 w2. collinear{v,w,w2} ==> azim v w w1 w2 = &0)`,
2345 REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[azim_def] THEN
2346 ASM_REWRITE_TAC[] THEN REWRITE_TAC[INSERT_AC; COLLINEAR_2]);;
2348 let AZIM_REFL_ALT = prove
2349 (`!v x y. azim v v x y = &0`,
2350 REPEAT GEN_TAC THEN MATCH_MP_TAC(last(CONJUNCTS AZIM_DEGENERATE)) THEN
2351 REWRITE_TAC[COLLINEAR_2; INSERT_AC]);;
2353 let AZIM_SPECIAL_SCALE = prove
2356 ==> azim (vec 0) (a % v) w1 w2 = azim (vec 0) v w1 w2`,
2357 REPEAT STRIP_TAC THEN REWRITE_TAC[azim_def] THEN
2358 REWRITE_TAC[VECTOR_SUB_RZERO; DIST_0] THEN
2359 FIRST_ASSUM(MP_TAC o MATCH_MP(MESON[REAL_LT_IMP_NZ; REAL_DIV_LMUL]
2360 `!a. &0 < a ==> (!y. ?x. a * x = y)`)) THEN
2361 DISCH_THEN(MP_TAC o MATCH_MP QUANTIFY_SURJECTION_THM) THEN
2362 DISCH_THEN(CONV_TAC o RAND_CONV o
2363 PARTIAL_EXPAND_QUANTS_CONV ["psi"; "r1"; "r2"]) THEN
2364 REWRITE_TAC[VECTOR_MUL_ASSOC; REAL_MUL_SYM] THEN
2365 ASM_SIMP_TAC[VECTOR_MUL_EQ_0; REAL_LT_IMP_NZ] THEN
2366 ASM_SIMP_TAC[NORM_MUL; REAL_ARITH `&0 < a ==> abs a = a`] THEN
2367 REWRITE_TAC[GSYM VECTOR_MUL_ASSOC] THEN
2368 REWRITE_TAC[VECTOR_ARITH `a % x:real^3 = a % y <=> a % (x - y) = vec 0`] THEN
2369 ASM_SIMP_TAC[REAL_LT_IMP_NZ; VECTOR_MUL_EQ_0] THEN
2370 REWRITE_TAC[VECTOR_SUB_EQ] THEN
2371 ASM_SIMP_TAC[COLLINEAR_SPECIAL_SCALE; REAL_LT_IMP_NZ]);;
2373 let AZIM_SCALE_ALL = prove
2375 &0 < a /\ &0 < b /\ &0 < c
2376 ==> azim (vec 0) (a % v) (b % w1) (c % w2) = azim (vec 0) v w1 w2`,
2377 let lemma = MESON[REAL_LT_IMP_NZ; REAL_DIV_LMUL]
2378 `!a. &0 < a ==> (!y. ?x. a * x = y)` in
2379 let SCALE_QUANT_TAC side asm avoid =
2380 MP_TAC(MATCH_MP lemma (ASSUME asm)) THEN
2381 DISCH_THEN(MP_TAC o MATCH_MP QUANTIFY_SURJECTION_THM) THEN
2382 DISCH_THEN(CONV_TAC o side o PARTIAL_EXPAND_QUANTS_CONV avoid) in
2383 REPEAT STRIP_TAC THEN
2384 ASM_SIMP_TAC[azim_def; COLLINEAR_SCALE_ALL; REAL_LT_IMP_NZ] THEN
2385 COND_CASES_TAC THEN ASM_REWRITE_TAC[VECTOR_SUB_RZERO] THEN
2386 ASM_SIMP_TAC[DIST_0; NORM_MUL; GSYM VECTOR_MUL_ASSOC] THEN
2387 ASM_SIMP_TAC[REAL_ARITH `&0 < a ==> abs a = a`; VECTOR_MUL_LCANCEL] THEN
2388 ASM_SIMP_TAC[VECTOR_MUL_EQ_0; REAL_LT_IMP_NZ] THEN
2389 SCALE_QUANT_TAC RAND_CONV `&0 < a` ["psi"; "r1"; "r2"] THEN
2390 SCALE_QUANT_TAC LAND_CONV `&0 < b` ["psi"; "h2"; "r2"] THEN
2391 SCALE_QUANT_TAC LAND_CONV `&0 < c` ["psi"; "h1"; "r1"] THEN
2392 ASM_SIMP_TAC[GSYM VECTOR_MUL_ASSOC; GSYM VECTOR_ADD_LDISTRIB;
2393 VECTOR_MUL_LCANCEL; REAL_LT_IMP_NZ; REAL_LT_MUL_EQ] THEN
2394 REWRITE_TAC[VECTOR_MUL_ASSOC; REAL_MUL_AC]);;
2396 let AZIM_ARG = prove
2397 (`!x y:real^3. azim (vec 0) (basis 3) x y = Arg(dropout 3 y / dropout 3 x)`,
2399 (`(r * cos t) % basis 1 + (r * sin t) % basis 2 = Cx r * cexp(ii * Cx t)`,
2400 REWRITE_TAC[CEXP_EULER; COMPLEX_BASIS; GSYM CX_SIN; GSYM CX_COS;
2401 COMPLEX_CMUL; CX_MUL] THEN
2402 CONV_TAC COMPLEX_RING) in
2403 REPEAT STRIP_TAC THEN
2404 ASM_CASES_TAC `collinear {vec 0:real^3,basis 3,x}` THENL
2405 [ASM_SIMP_TAC[AZIM_DEGENERATE] THEN
2406 RULE_ASSUM_TAC(REWRITE_RULE[COLLINEAR_BASIS_3]) THEN
2407 ASM_REWRITE_TAC[COMPLEX_VEC_0; complex_div; COMPLEX_INV_0;
2408 COMPLEX_MUL_RZERO; ARG_0];
2410 ASM_CASES_TAC `collinear {vec 0:real^3,basis 3,y}` THENL
2411 [ASM_SIMP_TAC[AZIM_DEGENERATE] THEN
2412 RULE_ASSUM_TAC(REWRITE_RULE[COLLINEAR_BASIS_3]) THEN
2413 ASM_REWRITE_TAC[COMPLEX_VEC_0; complex_div; COMPLEX_MUL_LZERO; ARG_0];
2415 MP_TAC(ISPECL [`vec 0:real^3`; `basis 3:real^3`; `x:real^3`; `y:real^3`]
2417 ABBREV_TAC `a = azim (vec 0) (basis 3) x (y:real^3)` THEN
2418 REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
2419 ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM; VECTOR_SUB_RZERO; DIST_0] THEN
2420 MAP_EVERY X_GEN_TAC [`h1:real`; `h2:real`] THEN
2421 DISCH_THEN(MP_TAC o SPECL
2422 [`basis 1:real^3`; `basis 2:real^3`; `basis 3:real^3`]) THEN
2423 SIMP_TAC[orthonormal; DOT_BASIS_BASIS; CROSS_BASIS; DIMINDEX_3; NORM_BASIS;
2424 ARITH; VECTOR_MUL_LID; BASIS_NONZERO; REAL_LT_01; LEFT_IMP_EXISTS_THM] THEN
2425 MAP_EVERY X_GEN_TAC [`psi:real`; `r1:real`; `r2:real`] THEN
2426 DISCH_THEN(STRIP_ASSUME_TAC o GSYM) THEN
2427 REWRITE_TAC[DROPOUT_ADD; DROPOUT_MUL; DROPOUT_BASIS_3] THEN
2428 REWRITE_TAC[VECTOR_MUL_RZERO; VECTOR_ADD_RID; lemma] THEN
2429 REWRITE_TAC[complex_div; COMPLEX_INV_MUL] THEN
2430 ONCE_REWRITE_TAC[COMPLEX_RING
2431 `(a * b) * (c * d):complex = (a * c) * b * d`] THEN
2432 REWRITE_TAC[GSYM complex_div; GSYM CX_DIV; GSYM CEXP_SUB] THEN
2433 CONV_TAC SYM_CONV THEN MATCH_MP_TAC ARG_UNIQUE THEN
2434 EXISTS_TAC `r2 / r1:real` THEN ASM_SIMP_TAC[REAL_LT_DIV] THEN
2435 AP_TERM_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[CX_ADD] THEN
2436 CONV_TAC COMPLEX_RING);;
2438 let REAL_CONTINUOUS_AT_AZIM_SHARP = prove
2440 ~collinear{v,w,w1} /\ ~(w2 IN aff_ge {v,w} {w1})
2441 ==> (azim v w w1) real_continuous at w2`,
2442 GEOM_ORIGIN_TAC `v:real^3` THEN
2443 GEOM_BASIS_MULTIPLE_TAC 3 `w:real^3` THEN
2444 X_GEN_TAC `w:real` THEN ASM_CASES_TAC `w = &0` THENL
2445 [ASM_REWRITE_TAC[VECTOR_MUL_LZERO; INSERT_AC; COLLINEAR_2]; ALL_TAC] THEN
2446 ASM_SIMP_TAC[REAL_LE_LT; COLLINEAR_SPECIAL_SCALE] THEN
2447 DISCH_TAC THEN REPEAT GEN_TAC THEN
2448 REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
2449 W(MP_TAC o PART_MATCH (lhs o rand) AFF_GE_SPECIAL_SCALE o
2450 rand o rand o lhand o snd) THEN
2451 ASM_REWRITE_TAC[FINITE_INSERT; FINITE_EMPTY; IN_SING] THEN ANTS_TAC THENL
2452 [POP_ASSUM_LIST(MP_TAC o end_itlist CONJ) THEN
2453 ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN REWRITE_TAC[DE_MORGAN_THM] THEN
2454 DISCH_THEN(STRIP_ASSUME_TAC o GSYM) THENL
2455 [ASM_REWRITE_TAC[COLLINEAR_2; INSERT_AC];
2456 ASM_REWRITE_TAC[COLLINEAR_2; INSERT_AC];
2457 ASM_SIMP_TAC[COLLINEAR_LEMMA_ALT; BASIS_NONZERO; DIMINDEX_3; ARITH] THEN
2459 DISCH_THEN SUBST1_TAC THEN DISCH_TAC] THEN
2460 ASM_SIMP_TAC[AZIM_SPECIAL_SCALE; AZIM_ARG] THEN
2461 MATCH_MP_TAC(REWRITE_RULE[o_DEF]
2462 REAL_CONTINUOUS_CONTINUOUS_AT_COMPOSE) THEN
2464 [REWRITE_TAC[complex_div] THEN MATCH_MP_TAC CONTINUOUS_COMPLEX_MUL THEN
2465 REWRITE_TAC[CONTINUOUS_CONST; ETA_AX] THEN
2466 SIMP_TAC[LINEAR_CONTINUOUS_AT; LINEAR_DROPOUT; DIMINDEX_3; DIMINDEX_2;
2469 MATCH_MP_TAC REAL_CONTINUOUS_AT_WITHIN THEN
2470 MATCH_MP_TAC REAL_CONTINUOUS_AT_ARG THEN
2471 MP_TAC(ISPECL [`w2:real^3`; `w1:real^3`] AFF_GE_2_1_0_DROPOUT_3) THEN
2472 ASM_REWRITE_TAC[] THEN
2473 REPEAT(FIRST_X_ASSUM(MP_TAC o
2474 GEN_REWRITE_RULE RAND_CONV [COLLINEAR_BASIS_3])) THEN
2475 SPEC_TAC(`(dropout 3:real^3->real^2) w2`,`v2:real^2`) THEN
2476 SPEC_TAC(`(dropout 3:real^3->real^2) w1`,`v1:real^2`) THEN
2477 POP_ASSUM_LIST(K ALL_TAC) THEN
2478 GEOM_BASIS_MULTIPLE_TAC 1 `v1:complex` THEN
2479 X_GEN_TAC `w:real` THEN ASM_CASES_TAC `w = &0` THEN
2480 ASM_REWRITE_TAC[VECTOR_MUL_LZERO] THEN
2481 GEN_REWRITE_TAC LAND_CONV [REAL_LE_LT] THEN ASM_REWRITE_TAC[] THEN
2482 DISCH_TAC THEN X_GEN_TAC `z:complex` THEN
2483 DISCH_THEN(K ALL_TAC) THEN
2484 REWRITE_TAC[CONTRAPOS_THM; COMPLEX_BASIS; COMPLEX_CMUL] THEN
2485 REWRITE_TAC[COMPLEX_MUL_RID; RE_DIV_CX; IM_DIV_CX; real] THEN
2486 ASM_SIMP_TAC[REAL_DIV_EQ_0; REAL_LE_RDIV_EQ; REAL_MUL_LZERO] THEN
2488 W(MP_TAC o PART_MATCH (lhs o rand) AFF_GE_1_1_0 o rand o snd) THEN
2489 ASM_REWRITE_TAC[COMPLEX_VEC_0; CX_INJ] THEN DISCH_THEN SUBST1_TAC THEN
2490 REWRITE_TAC[IN_ELIM_THM] THEN EXISTS_TAC `Re z / w` THEN
2491 ASM_SIMP_TAC[REAL_LE_DIV; REAL_LT_IMP_LE; COMPLEX_EQ] THEN
2492 ASM_SIMP_TAC[COMPLEX_CMUL; CX_DIV; COMPLEX_DIV_RMUL; CX_INJ] THEN
2493 REWRITE_TAC[RE_CX; IM_CX]);;
2495 let REAL_CONTINUOUS_AT_AZIM = prove
2496 (`!v w w1 w2. ~coplanar{v,w,w1,w2} ==> (azim v w w1) real_continuous at w2`,
2497 REPEAT STRIP_TAC THEN MATCH_MP_TAC REAL_CONTINUOUS_AT_AZIM_SHARP THEN
2499 [ASM_MESON_TAC[NOT_COPLANAR_NOT_COLLINEAR; INSERT_AC];
2500 DISCH_THEN(MP_TAC o MATCH_MP (REWRITE_RULE[SUBSET]
2501 AFF_GE_SUBSET_AFFINE_HULL)) THEN
2502 POP_ASSUM MP_TAC THEN REWRITE_TAC[coplanar; CONTRAPOS_THM] THEN
2503 REWRITE_TAC[SET_RULE `{a,b} UNION {c} = {a,b,c}`] THEN
2504 DISCH_TAC THEN MAP_EVERY EXISTS_TAC
2505 [`v:real^3`; `w:real^3`; `w1:real^3`] THEN
2506 SIMP_TAC[SET_RULE `{a,b,c,d} SUBSET s <=> {a,b,c} SUBSET s /\ d IN s`] THEN
2507 ASM_REWRITE_TAC[HULL_SUBSET]]);;
2509 let AZIM_REFL = prove
2510 (`!v0 v1 w. azim v0 v1 w w = &0`,
2511 GEOM_ORIGIN_TAC `v0:real^3` THEN
2512 GEOM_BASIS_MULTIPLE_TAC 3 `v1:real^3` THEN
2514 GEN_REWRITE_TAC LAND_CONV [REAL_ARITH `&0 <= x <=> x = &0 \/ &0 < x`] THEN
2515 STRIP_TAC THEN ASM_SIMP_TAC[VECTOR_MUL_LZERO; AZIM_DEGENERATE] THEN
2516 ASM_SIMP_TAC[AZIM_SPECIAL_SCALE; AZIM_ARG; ARG_EQ_0] THEN
2517 X_GEN_TAC `w:real^3` THEN
2518 ASM_CASES_TAC `(dropout 3 :real^3->real^2) w = Cx(&0)` THEN
2519 ASM_SIMP_TAC[COMPLEX_DIV_REFL; REAL_CX; RE_CX; REAL_POS] THEN
2520 ASM_SIMP_TAC[complex_div; COMPLEX_MUL_LZERO; REAL_CX; RE_CX; REAL_POS]);;
2524 ~collinear{v0,v1,w} /\ ~collinear{v0,v1,x} /\ ~collinear{v0,v1,y}
2525 ==> (azim v0 v1 w x = azim v0 v1 w y <=> y IN aff_gt {v0,v1} {x})`,
2526 GEOM_ORIGIN_TAC `v0:real^3` THEN
2527 GEOM_BASIS_MULTIPLE_TAC 3 `v1:real^3` THEN
2529 GEN_REWRITE_TAC LAND_CONV [REAL_ARITH `&0 <= x <=> x = &0 \/ &0 < x`] THEN
2531 [ASM_REWRITE_TAC[VECTOR_MUL_LZERO; INSERT_AC; COLLINEAR_2]; ALL_TAC] THEN
2532 ASM_SIMP_TAC[AZIM_SPECIAL_SCALE; REAL_LT_IMP_NZ; COLLINEAR_SPECIAL_SCALE] THEN
2533 REPEAT STRIP_TAC THEN
2534 W(MP_TAC o PART_MATCH (lhs o rand) AFF_GT_SPECIAL_SCALE o
2535 rand o rand o snd) THEN
2537 [ASM_REWRITE_TAC[IN_INSERT; FINITE_INSERT; FINITE_EMPTY; NOT_IN_EMPTY] THEN
2538 REPEAT CONJ_TAC THEN DISCH_THEN(SUBST_ALL_TAC o SYM) THEN
2539 TRY(RULE_ASSUM_TAC(REWRITE_RULE[INSERT_AC; COLLINEAR_2]) THEN
2540 FIRST_X_ASSUM CONTR_TAC) THEN
2541 UNDISCH_TAC `~collinear {vec 0:real^3, basis 3, v1 % basis 3}` THEN
2542 REWRITE_TAC[COLLINEAR_LEMMA] THEN MESON_TAC[];
2543 DISCH_THEN SUBST1_TAC] THEN
2544 REWRITE_TAC[AZIM_ARG] THEN CONV_TAC(LAND_CONV SYM_CONV) THEN
2545 W(MP_TAC o PART_MATCH (lhs o rand) ARG_EQ o lhand o snd) THEN
2546 RULE_ASSUM_TAC(REWRITE_RULE[COLLINEAR_BASIS_3]) THEN
2547 ASM_REWRITE_TAC[complex_div; COMPLEX_ENTIRE; COMPLEX_INV_EQ_0] THEN
2548 ASM_REWRITE_TAC[GSYM complex_div; GSYM COMPLEX_VEC_0] THEN
2549 DISCH_THEN SUBST1_TAC THEN
2550 ASM_SIMP_TAC[GSYM COMPLEX_VEC_0; COMPLEX_FIELD
2551 `~(w = Cx(&0)) ==> (y / w = x * u / w <=> y = x * u)`] THEN
2552 W(MP_TAC o PART_MATCH (lhs o rand) AFF_GT_2_1 o rand o rand o snd) THEN
2554 [REWRITE_TAC[SET_RULE `DISJOINT {a,b} {x} <=> ~(x = a) /\ ~(x = b)`] THEN
2555 ASM_MESON_TAC[DROPOUT_BASIS_3; DROPOUT_0];
2556 DISCH_THEN SUBST1_TAC] THEN
2557 REWRITE_TAC[IN_ELIM_THM; VECTOR_MUL_RZERO; VECTOR_ADD_LID] THEN
2558 ONCE_REWRITE_TAC[MESON[]
2559 `(?a b c. p c /\ q a b c /\ r b c) <=>
2560 (?c. p c /\ ?b. r b c /\ ?a. q a b c)`] THEN
2561 SIMP_TAC[REAL_ARITH `a + b + c = &1 <=> a = &1 - b - c`; EXISTS_REFL] THEN
2562 AP_TERM_TAC THEN GEN_REWRITE_TAC I [FUN_EQ_THM] THEN
2563 X_GEN_TAC `t:real` THEN REWRITE_TAC[] THEN AP_TERM_TAC THEN
2564 REWRITE_TAC[GSYM COMPLEX_CMUL] THEN
2565 SIMP_TAC[CART_EQ; FORALL_2; FORALL_3; DIMINDEX_2; DIMINDEX_3;
2566 dropout; LAMBDA_BETA; BASIS_COMPONENT; ARITH; REAL_MUL_RID;
2567 VECTOR_MUL_COMPONENT; VEC_COMPONENT; REAL_MUL_RZERO; UNWIND_THM1;
2568 VECTOR_ADD_COMPONENT; REAL_ADD_LID; RIGHT_EXISTS_AND_THM] THEN
2569 REWRITE_TAC[REAL_ARITH `y:real = t + z <=> t = y - z`; EXISTS_REFL]);;
2571 let AZIM_EQ_ALT = prove
2573 ~collinear{v0,v1,w} /\ ~collinear{v0,v1,x} /\ ~collinear{v0,v1,y}
2574 ==> (azim v0 v1 w x = azim v0 v1 w y <=> x IN aff_gt {v0,v1} {y})`,
2575 ASM_SIMP_TAC[GSYM AZIM_EQ] THEN MESON_TAC[]);;
2577 let AZIM_EQ_0 = prove
2579 ~collinear{v0,v1,w} /\ ~collinear{v0,v1,x}
2580 ==> (azim v0 v1 w x = &0 <=> w IN aff_gt {v0,v1} {x})`,
2581 REPEAT STRIP_TAC THEN MATCH_MP_TAC EQ_TRANS THEN
2582 EXISTS_TAC `azim v0 v1 w x = azim v0 v1 w w` THEN CONJ_TAC THENL
2583 [REWRITE_TAC[AZIM_REFL];
2584 ASM_SIMP_TAC[AZIM_EQ]]);;
2586 let AZIM_EQ_0_ALT = prove
2588 ~collinear{v0,v1,w} /\ ~collinear{v0,v1,x}
2589 ==> (azim v0 v1 w x = &0 <=> x IN aff_gt {v0,v1} {w})`,
2590 REPEAT STRIP_TAC THEN MATCH_MP_TAC EQ_TRANS THEN
2591 EXISTS_TAC `azim v0 v1 w x = azim v0 v1 w w` THEN CONJ_TAC THENL
2592 [REWRITE_TAC[AZIM_REFL];
2593 ASM_SIMP_TAC[AZIM_EQ_ALT]]);;
2595 let AZIM_EQ_0_GE = prove
2597 ~collinear{v0,v1,w} /\ ~collinear{v0,v1,x}
2598 ==> (azim v0 v1 w x = &0 <=> w IN aff_ge {v0,v1} {x})`,
2600 ASM_CASES_TAC `v1:real^3 = v0` THENL
2601 [ASM_REWRITE_TAC[INSERT_AC; COLLINEAR_2]; STRIP_TAC] THEN
2602 W(MP_TAC o PART_MATCH (lhs o rand) AFF_GE_AFF_GT_DECOMP o
2603 rand o rand o snd) THEN
2605 [SIMP_TAC[FINITE_INSERT; FINITE_EMPTY; DISJOINT_INSERT; DISJOINT_EMPTY] THEN
2606 REWRITE_TAC[IN_SING] THEN
2607 CONJ_TAC THEN DISCH_THEN SUBST_ALL_TAC THEN
2608 RULE_ASSUM_TAC(REWRITE_RULE[COLLINEAR_2; INSERT_AC]) THEN
2609 FIRST_ASSUM CONTR_TAC;
2610 DISCH_THEN SUBST1_TAC] THEN
2611 ASM_SIMP_TAC[AZIM_EQ_0] THEN
2612 REWRITE_TAC[SIMPLE_IMAGE; IMAGE_CLAUSES; UNIONS_1] THEN
2613 REWRITE_TAC[SET_RULE `{x} DELETE x = {}`] THEN
2614 REWRITE_TAC[AFF_GE_EQ_AFFINE_HULL; IN_UNION] THEN
2615 ASM_SIMP_TAC[GSYM COLLINEAR_3_AFFINE_HULL]);;
2617 let AZIM_COMPL_EQ_0 = prove
2619 ~collinear {z,w,w1} /\ ~collinear {z,w,w2} /\ azim z w w1 w2 = &0
2620 ==> azim z w w2 w1 = &0`,
2621 REWRITE_TAC[IMP_CONJ] THEN
2622 GEOM_ORIGIN_TAC `z:real^3` THEN
2623 GEOM_BASIS_MULTIPLE_TAC 3 `w:real^3` THEN
2624 X_GEN_TAC `w:real` THEN GEN_REWRITE_TAC LAND_CONV [REAL_LE_LT] THEN
2625 ASM_CASES_TAC `w = &0` THEN ASM_REWRITE_TAC[] THENL
2626 [ASM_REWRITE_TAC[VECTOR_MUL_LZERO; INSERT_AC; COLLINEAR_2]; ALL_TAC] THEN
2627 DISCH_TAC THEN REPEAT GEN_TAC THEN
2628 ASM_SIMP_TAC[AZIM_SPECIAL_SCALE; COLLINEAR_SPECIAL_SCALE; AZIM_ARG] THEN
2629 REWRITE_TAC[ARG_EQ_0; real; IM_COMPLEX_DIV_EQ_0; RE_COMPLEX_DIV_GE_0] THEN
2630 REWRITE_TAC[complex_mul; RE; IM; cnj] THEN REAL_ARITH_TAC);;
2632 let AZIM_COMPL = prove
2634 ~collinear {z,w,w1} /\ ~collinear {z,w,w2}
2635 ==> azim z w w2 w1 = if azim z w w1 w2 = &0 then &0
2636 else &2 * pi - azim z w w1 w2`,
2637 REPEAT GEN_TAC THEN COND_CASES_TAC THENL
2638 [ASM_MESON_TAC[AZIM_COMPL_EQ_0]; ALL_TAC] THEN
2639 DISCH_THEN(fun th -> POP_ASSUM MP_TAC THEN MP_TAC th) THEN
2640 GEOM_ORIGIN_TAC `z:real^3` THEN
2641 GEOM_BASIS_MULTIPLE_TAC 3 `w:real^3` THEN
2642 X_GEN_TAC `w:real` THEN GEN_REWRITE_TAC LAND_CONV [REAL_LE_LT] THEN
2643 ASM_CASES_TAC `w = &0` THEN ASM_REWRITE_TAC[] THENL
2644 [ASM_REWRITE_TAC[VECTOR_MUL_LZERO; INSERT_AC; COLLINEAR_2]; ALL_TAC] THEN
2645 DISCH_TAC THEN REPEAT GEN_TAC THEN
2646 ASM_SIMP_TAC[AZIM_SPECIAL_SCALE; COLLINEAR_SPECIAL_SCALE; AZIM_ARG] THEN
2647 REWRITE_TAC[COLLINEAR_BASIS_3] THEN REWRITE_TAC[ARG_EQ_0] THEN
2648 REPEAT STRIP_TAC THEN
2649 MP_TAC(ISPEC `(dropout 3:real^3->real^2) w2 /
2650 (dropout 3:real^3->real^2) w1` ARG_INV) THEN
2651 ASM_REWRITE_TAC[COMPLEX_INV_DIV]);;
2653 let AZIM_EQ_PI_SYM = prove
2655 ~collinear {z, w, w1} /\ ~collinear {z, w, w2}
2656 ==> (azim z w w1 w2 = pi <=> azim z w w2 w1 = pi)`,
2657 REPEAT STRIP_TAC THEN
2658 W(MP_TAC o PART_MATCH (lhs o rand) AZIM_COMPL o lhand o rand o snd) THEN
2659 ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC);;
2661 let AZIM_EQ_0_SYM = prove
2663 ~collinear {z, w, w1} /\ ~collinear {z, w, w2}
2664 ==> (azim z w w1 w2 = &0 <=> azim z w w2 w1 = &0)`,
2665 MESON_TAC[AZIM_COMPL_EQ_0]);;
2667 let AZIM_EQ_0_GE_ALT = prove
2669 ~collinear{v0,v1,w} /\ ~collinear{v0,v1,x}
2670 ==> (azim v0 v1 w x = &0 <=> x IN aff_ge {v0,v1} {w})`,
2671 ASM_MESON_TAC[AZIM_EQ_0_SYM; AZIM_EQ_0_GE]);;
2673 let AZIM_EQ_PI = prove
2675 ~collinear{v0,v1,w} /\ ~collinear{v0,v1,x}
2676 ==> (azim v0 v1 w x = pi <=> w IN aff_lt {v0,v1} {x})`,
2677 GEOM_ORIGIN_TAC `v0:real^3` THEN
2678 GEOM_BASIS_MULTIPLE_TAC 3 `v1:real^3` THEN
2680 GEN_REWRITE_TAC LAND_CONV [REAL_ARITH `&0 <= x <=> x = &0 \/ &0 < x`] THEN
2682 [ASM_REWRITE_TAC[VECTOR_MUL_LZERO; INSERT_AC; COLLINEAR_2]; ALL_TAC] THEN
2683 ASM_SIMP_TAC[AZIM_SPECIAL_SCALE; REAL_LT_IMP_NZ;
2684 COLLINEAR_SPECIAL_SCALE] THEN
2685 REPEAT STRIP_TAC THEN
2686 W(MP_TAC o PART_MATCH (lhs o rand) AFF_LT_SPECIAL_SCALE o
2687 rand o rand o snd) THEN
2689 [ASM_REWRITE_TAC[IN_INSERT; FINITE_INSERT; FINITE_EMPTY; NOT_IN_EMPTY] THEN
2690 REPEAT CONJ_TAC THEN DISCH_THEN(SUBST_ALL_TAC o SYM) THEN
2691 TRY(RULE_ASSUM_TAC(REWRITE_RULE[INSERT_AC; COLLINEAR_2]) THEN
2692 FIRST_X_ASSUM CONTR_TAC) THEN
2693 UNDISCH_TAC `~collinear {vec 0:real^3, basis 3, v1 % basis 3}` THEN
2694 REWRITE_TAC[COLLINEAR_LEMMA] THEN MESON_TAC[];
2695 DISCH_THEN SUBST1_TAC] THEN
2696 REWRITE_TAC[AZIM_ARG] THEN CONV_TAC(LAND_CONV SYM_CONV) THEN
2697 CONV_TAC(LAND_CONV SYM_CONV) THEN REWRITE_TAC[ARG_EQ_PI] THEN
2698 MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC
2699 `(dropout 3 (w:real^3)) IN aff_lt {vec 0:real^2} {dropout 3 (x:real^3)}` THEN
2701 [REPEAT(POP_ASSUM MP_TAC) THEN REWRITE_TAC[COLLINEAR_BASIS_3] THEN
2702 SPEC_TAC(`(dropout 3:real^3->real^2) x`,`y:complex`) THEN
2703 SPEC_TAC(`(dropout 3:real^3->real^2) w`,`v:complex`) THEN
2704 GEOM_BASIS_MULTIPLE_TAC 1 `v:complex` THEN
2705 X_GEN_TAC `v:real` THEN GEN_REWRITE_TAC LAND_CONV [REAL_LE_LT] THEN
2706 ASM_CASES_TAC `v = &0` THEN ASM_REWRITE_TAC[VECTOR_MUL_LZERO] THEN
2707 REWRITE_TAC[COMPLEX_CMUL; COMPLEX_BASIS; COMPLEX_VEC_0] THEN
2708 SIMP_TAC[ARG_DIV_CX; COMPLEX_MUL_RID] THEN
2709 REWRITE_TAC[real; RE_DIV_CX; IM_DIV_CX; CX_INJ] THEN
2710 ASM_SIMP_TAC[REAL_LT_LDIV_EQ; REAL_EQ_LDIV_EQ; REAL_MUL_LZERO] THEN
2711 REPEAT STRIP_TAC THEN
2712 W(MP_TAC o PART_MATCH (lhs o rand) AFF_LT_1_1 o rand o rand o snd) THEN
2713 ASM_REWRITE_TAC[DISJOINT_INSERT; DISJOINT_EMPTY; IN_SING] THEN
2714 DISCH_THEN SUBST1_TAC THEN
2715 REWRITE_TAC[COMPLEX_CMUL; IN_ELIM_THM; COMPLEX_MUL_RZERO] THEN
2716 ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN
2717 REWRITE_TAC[REAL_ARITH `t1 + t2 = &1 <=> t1 = &1 - t2`] THEN
2718 REWRITE_TAC[RIGHT_EXISTS_AND_THM; UNWIND_THM2; COMPLEX_ADD_LID] THEN
2720 [REWRITE_TAC[GSYM real; REAL] THEN
2721 DISCH_THEN(CONJUNCTS_THEN2 (SUBST1_TAC o SYM) ASSUME_TAC) THEN
2722 EXISTS_TAC `v / Re y` THEN REWRITE_TAC[GSYM CX_MUL; CX_INJ] THEN
2724 [ALL_TAC; REPEAT(POP_ASSUM MP_TAC) THEN CONV_TAC REAL_FIELD];
2725 DISCH_THEN(X_CHOOSE_THEN `t:real`
2726 (CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
2727 ASM_SIMP_TAC[CX_INJ; REAL_ARITH `x < &0 ==> ~(x = &0)`; COMPLEX_FIELD
2728 `~(t = Cx(&0)) ==> (v = t * y <=> y = v / t)`] THEN
2729 DISCH_THEN SUBST1_TAC THEN REWRITE_TAC[GSYM CX_DIV] THEN
2730 REWRITE_TAC[RE_CX; IM_CX]] THEN
2731 REWRITE_TAC[REAL_ARITH `x < &0 <=> &0 < --x`] THEN
2732 REWRITE_TAC[real_div; GSYM REAL_MUL_RNEG; GSYM REAL_INV_NEG] THEN
2733 MATCH_MP_TAC REAL_LT_MUL THEN ASM_REWRITE_TAC[REAL_LT_INV_EQ] THEN
2735 W(MP_TAC o PART_MATCH (lhs o rand) AFF_LT_2_1 o rand o rand o snd) THEN
2737 [REWRITE_TAC[SET_RULE `DISJOINT {a,b} {x} <=> ~(x = a) /\ ~(x = b)`] THEN
2738 CONJ_TAC THEN DISCH_THEN SUBST_ALL_TAC THEN
2739 RULE_ASSUM_TAC(REWRITE_RULE[COLLINEAR_2; INSERT_AC]) THEN
2740 FIRST_ASSUM CONTR_TAC;
2741 DISCH_THEN SUBST1_TAC] THEN
2742 W(MP_TAC o PART_MATCH (lhs o rand) AFF_LT_1_1 o rand o lhand o snd) THEN
2744 [REWRITE_TAC[SET_RULE `DISJOINT {a} {x} <=> ~(x = a)`] THEN
2745 ASM_MESON_TAC[COLLINEAR_BASIS_3];
2746 DISCH_THEN SUBST1_TAC] THEN
2747 REWRITE_TAC[VECTOR_MUL_RZERO; VECTOR_ADD_LID; IN_ELIM_THM] THEN
2748 ONCE_REWRITE_TAC[REAL_ARITH `s + t = &1 <=> s = &1- t`] THEN
2749 ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN
2750 REWRITE_TAC[RIGHT_EXISTS_AND_THM; UNWIND_THM2] THEN
2751 GEN_REWRITE_TAC (RAND_CONV o BINDER_CONV) [SWAP_EXISTS_THM] THEN
2752 REWRITE_TAC[RIGHT_EXISTS_AND_THM; UNWIND_THM2] THEN
2753 ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN AP_TERM_TAC THEN
2754 REWRITE_TAC[FUN_EQ_THM; RIGHT_EXISTS_AND_THM] THEN X_GEN_TAC `t:real` THEN
2756 SIMP_TAC[CART_EQ; FORALL_2; FORALL_3; DIMINDEX_2; DIMINDEX_3;
2757 dropout; LAMBDA_BETA; BASIS_COMPONENT; ARITH; REAL_MUL_RID;
2758 VECTOR_MUL_COMPONENT; VEC_COMPONENT; REAL_MUL_RZERO; UNWIND_THM1;
2759 VECTOR_ADD_COMPONENT; REAL_ADD_LID; RIGHT_EXISTS_AND_THM] THEN
2760 REWRITE_TAC[REAL_ARITH `x:real = t + y <=> t = x - y`] THEN
2761 REWRITE_TAC[EXISTS_REFL]]);;
2763 let AZIM_EQ_PI_ALT = prove
2765 ~collinear{v0,v1,w} /\ ~collinear{v0,v1,x}
2766 ==> (azim v0 v1 w x = pi <=> x IN aff_lt {v0,v1} {w})`,
2767 REPEAT GEN_TAC THEN DISCH_TAC THEN
2768 FIRST_ASSUM(SUBST1_TAC o MATCH_MP AZIM_EQ_PI_SYM) THEN
2769 ASM_SIMP_TAC[AZIM_EQ_PI]);;
2771 let AZIM_EQ_0_PI_IMP_COPLANAR = prove
2773 azim v0 v1 w1 w2 = &0 \/ azim v0 v1 w1 w2 = pi
2774 ==> coplanar {v0,v1,w1,w2}`,
2775 REPEAT GEN_TAC THEN ASM_CASES_TAC `collinear {v0:real^3,v1,w1}` THENL
2776 [MP_TAC(ISPECL [`v0:real^3`; `v1:real^3`; `w1:real^3`; `w2:real^3`]
2777 NOT_COPLANAR_NOT_COLLINEAR) THEN
2778 ASM_REWRITE_TAC[] THEN CONV_TAC TAUT;
2779 POP_ASSUM MP_TAC] THEN
2780 ASM_CASES_TAC `collinear {v0:real^3,v1,w2}` THENL
2781 [MP_TAC(ISPECL [`v0:real^3`; `v1:real^3`; `w2:real^3`; `w1:real^3`]
2782 NOT_COPLANAR_NOT_COLLINEAR) THEN
2783 ASM_REWRITE_TAC[] THEN REWRITE_TAC[INSERT_AC] THEN CONV_TAC TAUT;
2784 POP_ASSUM MP_TAC] THEN
2785 MAP_EVERY (fun t -> SPEC_TAC(t,t))
2786 [`w2:real^3`; `w1:real^3`; `v1:real^3`; `v0:real^3`] THEN
2787 GEOM_ORIGIN_TAC `v0:real^3` THEN
2788 GEOM_BASIS_MULTIPLE_TAC 3 `v1:real^3` THEN
2789 X_GEN_TAC `w:real` THEN GEN_REWRITE_TAC LAND_CONV [REAL_LE_LT] THEN
2790 ASM_CASES_TAC `w = &0` THEN ASM_REWRITE_TAC[] THENL
2791 [ASM_REWRITE_TAC[VECTOR_MUL_LZERO; INSERT_AC; COLLINEAR_2]; ALL_TAC] THEN
2792 SIMP_TAC[AZIM_SPECIAL_SCALE] THEN
2793 ASM_SIMP_TAC[AZIM_ARG; COLLINEAR_SPECIAL_SCALE] THEN
2794 REWRITE_TAC[COLLINEAR_BASIS_3; ARG_EQ_0_PI] THEN
2795 REWRITE_TAC[real; IM_COMPLEX_DIV_EQ_0] THEN
2796 REWRITE_TAC[complex_mul; cnj; IM; RE] THEN
2797 REWRITE_TAC[REAL_ARITH `x * --y + a * b = &0 <=> x * y = a * b`] THEN
2798 REWRITE_TAC[RE_DEF; IM_DEF] THEN DISCH_TAC THEN REPEAT GEN_TAC THEN
2799 DISCH_TAC THEN DISCH_TAC THEN
2800 SIMP_TAC[dropout; LAMBDA_BETA; DIMINDEX_3; ARITH; DIMINDEX_2] THEN
2801 DISCH_TAC THEN REWRITE_TAC[coplanar] THEN
2802 MAP_EVERY EXISTS_TAC [`vec 0:real^3`; `w % basis 3:real^3`; `w1:real^3`] THEN
2803 ONCE_REWRITE_TAC[SET_RULE `{a,b,c,d} = d INSERT {a,b,c}`] THEN
2804 ONCE_REWRITE_TAC[INSERT_SUBSET] THEN REWRITE_TAC[HULL_SUBSET] THEN
2805 SIMP_TAC[AFFINE_HULL_EQ_SPAN; IN_INSERT; HULL_INC] THEN
2806 REWRITE_TAC[SPAN_BREAKDOWN_EQ; SPAN_EMPTY; IN_SING] THEN
2807 REWRITE_TAC[VECTOR_MUL_RZERO; VECTOR_SUB_RZERO] THEN
2808 REPEAT(POP_ASSUM MP_TAC) THEN
2809 SIMP_TAC[CART_EQ; DIMINDEX_2; FORALL_2; FORALL_3; dropout; LAMBDA_BETA;
2810 DIMINDEX_2; DIMINDEX_3; ARITH; VEC_COMPONENT; ARITH;
2811 VECTOR_SUB_COMPONENT; VECTOR_MUL_COMPONENT; BASIS_COMPONENT] THEN
2812 REPEAT STRIP_TAC THEN
2813 REWRITE_TAC[REAL_MUL_RZERO; REAL_SUB_RZERO] THEN
2814 ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN
2815 REWRITE_TAC[RIGHT_EXISTS_AND_THM] THEN
2816 ASM_SIMP_TAC[EXISTS_REFL; REAL_FIELD
2817 `&0 < w ==> (x - k * w * &1 - y = &0 <=> k = (x - y) / w)`] THEN
2818 SUBGOAL_THEN `~((w1:real^3)$2 = &0) \/ ~((w2:real^3)$1 = &0)`
2819 STRIP_ASSUME_TAC THENL
2820 [REPEAT(POP_ASSUM MP_TAC) THEN CONV_TAC REAL_RING;
2821 EXISTS_TAC `(w2:real^3)$2 / (w1:real^3)$2` THEN
2822 REPEAT(POP_ASSUM MP_TAC) THEN CONV_TAC REAL_FIELD;
2823 EXISTS_TAC `(w2:real^3)$1 / (w1:real^3)$1` THEN
2824 REPEAT(POP_ASSUM MP_TAC) THEN CONV_TAC REAL_FIELD]);;
2826 let AZIM_SAME_WITHIN_AFF_GE = prove
2828 v IN aff_ge {a} {u,w} /\ ~collinear{a,u,v} /\ ~collinear{a,u,w}
2829 ==> azim a u v z = azim a u w z`,
2830 GEOM_ORIGIN_TAC `a:real^3` THEN
2831 GEOM_BASIS_MULTIPLE_TAC 3 `u:real^3` THEN
2832 X_GEN_TAC `u:real` THEN ASM_CASES_TAC `u = &0` THEN
2833 ASM_SIMP_TAC[AZIM_DEGENERATE; VECTOR_MUL_LZERO; REAL_LE_LT] THEN
2834 ASM_SIMP_TAC[AZIM_SPECIAL_SCALE; COLLINEAR_SPECIAL_SCALE] THEN
2835 DISCH_TAC THEN REPEAT GEN_TAC THEN
2836 ASM_CASES_TAC `w:real^3 = vec 0` THENL
2837 [ASM_REWRITE_TAC[COLLINEAR_2; INSERT_AC]; ALL_TAC] THEN
2838 ASM_SIMP_TAC[AFF_GE_SCALE_LEMMA] THEN
2839 REWRITE_TAC[COLLINEAR_BASIS_3; AZIM_ARG] THEN
2840 ASM_SIMP_TAC[AFF_GE_1_2_0; BASIS_NONZERO; ARITH; DIMINDEX_3;
2841 SET_RULE `DISJOINT {a} {b,c} <=> ~(b = a) /\ ~(c = a)`] THEN
2842 REWRITE_TAC[IMP_CONJ; LEFT_IMP_EXISTS_THM; IN_ELIM_THM] THEN
2843 MAP_EVERY X_GEN_TAC [`a:real`; `b:real`] THEN DISCH_TAC THEN DISCH_TAC THEN
2844 DISCH_THEN(MP_TAC o AP_TERM `dropout 3:real^3->real^2`) THEN
2845 REWRITE_TAC[DROPOUT_ADD; DROPOUT_MUL; DROPOUT_BASIS_3] THEN
2846 REWRITE_TAC[VECTOR_MUL_RZERO; VECTOR_ADD_LID] THEN
2847 DISCH_THEN SUBST1_TAC THEN REPEAT DISCH_TAC THEN
2848 REWRITE_TAC[COMPLEX_CMUL] THEN
2849 REWRITE_TAC[complex_div; COMPLEX_INV_MUL; GSYM CX_INV] THEN
2850 ONCE_REWRITE_TAC[COMPLEX_RING `a * b * c:complex = b * a * c`] THEN
2851 MATCH_MP_TAC ARG_MUL_CX THEN REWRITE_TAC[REAL_LT_INV_EQ] THEN
2852 ASM_REWRITE_TAC[REAL_LT_LE] THEN ASM_MESON_TAC[VECTOR_MUL_LZERO]);;
2854 let AZIM_SAME_WITHIN_AFF_GE_ALT = prove
2856 v IN aff_ge {a} {u,w} /\ ~collinear{a,u,v} /\ ~collinear{a,u,w}
2857 ==> azim a u z v = azim a u z w`,
2858 REPEAT GEN_TAC THEN DISCH_TAC THEN
2859 FIRST_ASSUM(ASSUME_TAC o MATCH_MP AZIM_SAME_WITHIN_AFF_GE) THEN
2860 ASM_CASES_TAC `collinear {a:real^3,u,z}` THEN
2861 ASM_SIMP_TAC[AZIM_DEGENERATE] THEN
2862 W(MP_TAC o PART_MATCH (lhs o rand) AZIM_COMPL o lhand o snd) THEN
2863 ASM_REWRITE_TAC[] THEN DISCH_THEN SUBST1_TAC THEN
2864 W(MP_TAC o PART_MATCH (lhs o rand) AZIM_COMPL o rand o snd) THEN
2867 let COLLINEAR_WITHIN_AFF_GE_COLLINEAR = prove
2869 v IN aff_ge {a} {u,w} /\ collinear{a,u,w} ==> collinear{a,v,w}`,
2870 GEOM_ORIGIN_TAC `a:real^N` THEN REPEAT GEN_TAC THEN
2871 ASM_CASES_TAC `w:real^N = vec 0` THENL
2872 [ASM_REWRITE_TAC[COLLINEAR_2; INSERT_AC]; ALL_TAC] THEN
2873 ASM_CASES_TAC `u:real^N = vec 0` THENL
2874 [ONCE_REWRITE_TAC[AFF_GE_DISJOINT_DIFF] THEN
2875 ASM_REWRITE_TAC[SET_RULE `{a} DIFF {a,b} = {}`] THEN
2876 REWRITE_TAC[GSYM CONVEX_HULL_AFF_GE] THEN
2877 ONCE_REWRITE_TAC[SET_RULE `{z,v,w} = {z,w,v}`] THEN
2878 ASM_SIMP_TAC[COLLINEAR_3_AFFINE_HULL] THEN
2879 MESON_TAC[CONVEX_HULL_SUBSET_AFFINE_HULL; SUBSET];
2880 ONCE_REWRITE_TAC[SET_RULE `{z,v,w} = {z,w,v}`] THEN
2881 ASM_REWRITE_TAC[COLLINEAR_LEMMA_ALT] THEN
2882 DISCH_THEN(CONJUNCTS_THEN2 MP_TAC (X_CHOOSE_TAC `a:real`)) THEN
2883 ASM_SIMP_TAC[AFF_GE_1_2_0; SET_RULE
2884 `DISJOINT {a} {b,c} <=> ~(b = a) /\ ~(c = a)`] THEN
2885 REWRITE_TAC[IN_ELIM_THM; LEFT_IMP_EXISTS_THM] THEN
2886 MAP_EVERY X_GEN_TAC [`b:real`; `c:real`] THEN STRIP_TAC THEN
2887 ASM_REWRITE_TAC[GSYM VECTOR_ADD_RDISTRIB; VECTOR_MUL_ASSOC] THEN
2890 let AZIM_EQ_IMP = prove
2892 ~collinear {v0, v1, w} /\
2893 ~collinear {v0, v1, y} /\
2894 x IN aff_gt {v0, v1} {y}
2895 ==> azim v0 v1 w x = azim v0 v1 w y`,
2896 REPEAT GEN_TAC THEN ASM_CASES_TAC `v1:real^3 = v0` THENL
2897 [ASM_REWRITE_TAC[INSERT_AC; COLLINEAR_2]; ALL_TAC] THEN
2898 ASM_CASES_TAC `collinear {v0:real^3,v1,x}` THENL
2899 [ALL_TAC; ASM_SIMP_TAC[AZIM_EQ_ALT]] THEN
2900 UNDISCH_TAC `collinear {v0:real^3,v1,x}` THEN
2902 `(s /\ p ==> r) ==> p ==> ~q /\ ~r /\ s ==> t`) THEN
2903 ASM_SIMP_TAC[COLLINEAR_3_IN_AFFINE_HULL] THEN
2904 ASM_CASES_TAC `y:real^3 = v0` THEN
2905 ASM_SIMP_TAC[HULL_INC; IN_INSERT] THEN
2906 ASM_CASES_TAC `y:real^3 = v1` THEN
2907 ASM_SIMP_TAC[HULL_INC; IN_INSERT] THEN
2908 ASM_SIMP_TAC[AFF_GT_2_1; SET_RULE
2909 `DISJOINT {a,b} {c} <=> ~(c = a) /\ ~(c = b)`] THEN
2910 REWRITE_TAC[AFFINE_HULL_2; IN_ELIM_THM; LEFT_AND_EXISTS_THM] THEN
2911 REWRITE_TAC[RIGHT_AND_EXISTS_THM; LEFT_IMP_EXISTS_THM] THEN
2913 [`t1:real`; `t2:real`; `t3:real`; `s1:real`; `s2:real`] THEN
2914 DISCH_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC MP_TAC) THEN
2915 FIRST_X_ASSUM(MP_TAC o AP_TERM `(%) (inv t3) :real^3->real^3`) THEN
2916 ASM_SIMP_TAC[VECTOR_ADD_LDISTRIB; VECTOR_MUL_ASSOC; REAL_MUL_LINV;
2917 REAL_LT_IMP_NZ; VECTOR_ARITH
2918 `x:real^N = y + z + &1 % w <=> w = x - (y + z)`] THEN
2919 REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
2920 EXISTS_TAC `inv t3 * s1 - inv t3 * t1:real` THEN
2921 EXISTS_TAC `inv t3 * s2 - inv t3 * t2:real` THEN CONJ_TAC THENL
2922 [ASM_SIMP_TAC[REAL_FIELD
2923 `&0 < t ==> (inv t * a - inv t * b + inv t * c - inv t * d = &1 <=>
2924 (a + c) - (b + d) = t)`] THEN
2926 VECTOR_ARITH_TAC]);;
2928 let AZIM_EQ_0_GE_IMP = prove
2929 (`!v0 v1 w x. x IN aff_ge {v0, v1} {w} ==> azim v0 v1 w x = &0`,
2930 REPEAT STRIP_TAC THEN ASM_CASES_TAC `collinear {v0:real^3,v1,w}` THEN
2931 ASM_SIMP_TAC[AZIM_DEGENERATE] THEN
2932 ASM_CASES_TAC `collinear {v0:real^3,v1,x}` THEN
2933 ASM_SIMP_TAC[AZIM_DEGENERATE] THEN ASM_MESON_TAC[AZIM_EQ_0_GE_ALT]);;
2935 let REAL_SGN_SIN_AZIM = prove
2936 (`!v w x y. real_sgn(sin(azim v w x y)) =
2937 real_sgn(((w - v) cross (x - v)) dot (y - v))`,
2938 GEOM_ORIGIN_TAC `v:real^3` THEN REWRITE_TAC[VECTOR_SUB_RZERO] THEN
2939 GEOM_BASIS_MULTIPLE_TAC 3 `w:real^3` THEN
2940 X_GEN_TAC `w:real` THEN ASM_CASES_TAC `w = &0` THEN
2941 ASM_REWRITE_TAC[VECTOR_MUL_LZERO; CROSS_LZERO; DOT_LZERO; REAL_SGN_0;
2942 AZIM_REFL_ALT; SIN_0] THEN
2943 ASM_REWRITE_TAC[REAL_LE_LT] THEN REPEAT STRIP_TAC THEN
2944 ASM_SIMP_TAC[AZIM_SPECIAL_SCALE; CROSS_LMUL; DOT_LMUL] THEN
2945 REWRITE_TAC[REAL_SGN_MUL] THEN
2946 GEN_REWRITE_TAC (RAND_CONV o LAND_CONV) [real_sgn] THEN
2947 ASM_REWRITE_TAC[REAL_MUL_LID; AZIM_ARG] THEN
2948 MATCH_MP_TAC EQ_TRANS THEN
2949 EXISTS_TAC `real_sgn(Im(dropout 3 (y:real^3) / dropout 3 (x:real^3)))` THEN
2952 REWRITE_TAC[REAL_SGN_IM_COMPLEX_DIV] THEN AP_TERM_TAC THEN
2953 SIMP_TAC[CART_EQ; DIMINDEX_3; FORALL_3; cross; VECTOR_3; DOT_3; dropout;
2954 LAMBDA_BETA; ARITH; cnj; complex_mul; RE_DEF; IM_DEF; DIMINDEX_2;
2955 complex; VECTOR_2; BASIS_COMPONENT] THEN REAL_ARITH_TAC] THEN
2957 SPEC_TAC(`(dropout 3:real^3->real^2) x`,`z:complex`) THEN
2958 SPEC_TAC(`(dropout 3:real^3->real^2) y`,`w:complex`) THEN
2959 POP_ASSUM_LIST(K ALL_TAC) THEN GEOM_BASIS_MULTIPLE_TAC 1 `z:complex` THEN
2960 REWRITE_TAC[COMPLEX_CMUL; COMPLEX_BASIS; COMPLEX_MUL_RID] THEN
2961 X_GEN_TAC `x:real` THEN DISCH_TAC THEN X_GEN_TAC `z:complex` THEN
2962 ASM_CASES_TAC `x = &0` THENL
2963 [ASM_REWRITE_TAC[complex_div; COMPLEX_INV_0; COMPLEX_MUL_RZERO] THEN
2964 REWRITE_TAC[ARG_0; SIN_0; IM_CX; REAL_SGN_0];
2965 SUBGOAL_THEN `&0 < x` ASSUME_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC]] THEN
2966 ASM_SIMP_TAC[ARG_DIV_CX; IM_DIV_CX; REAL_SGN_DIV] THEN
2967 GEN_REWRITE_TAC (RAND_CONV o RAND_CONV) [real_sgn] THEN
2968 ASM_REWRITE_TAC[REAL_DIV_1] THEN ASM_CASES_TAC `z = Cx(&0)` THEN
2969 ASM_REWRITE_TAC[IM_CX; ARG_0; SIN_0] THEN
2970 GEN_REWRITE_TAC (funpow 3 RAND_CONV) [ARG] THEN
2971 REWRITE_TAC[IM_MUL_CX; REAL_SGN_MUL] THEN
2972 GEN_REWRITE_TAC (RAND_CONV o LAND_CONV) [real_sgn] THEN
2973 ASM_REWRITE_TAC[COMPLEX_NORM_NZ; REAL_MUL_LID] THEN
2974 REWRITE_TAC[IM_CEXP; RE_MUL_II; IM_MUL_II; RE_CX; REAL_SGN_MUL] THEN
2975 GEN_REWRITE_TAC (RAND_CONV o LAND_CONV) [real_sgn] THEN
2976 REWRITE_TAC[REAL_EXP_POS_LT; REAL_MUL_LID]);;
2978 let AZIM_IN_UPPER_HALFSPACE = prove
2979 (`!v w x y. azim v w x y <= pi <=>
2980 &0 <= ((w - v) cross (x - v)) dot (y - v)`,
2981 REPEAT STRIP_TAC THEN MATCH_MP_TAC EQ_TRANS THEN
2982 EXISTS_TAC `&0 <= sin(azim v w x y)` THEN CONJ_TAC THENL
2983 [EQ_TAC THEN SIMP_TAC[SIN_POS_PI_LE; azim] THEN
2984 MP_TAC(ISPEC `azim v w x y - pi` SIN_POS_PI) THEN
2985 REWRITE_TAC[SIN_SUB; SIN_PI; COS_PI; azim;
2986 REAL_ARITH `x - pi < pi <=> x < &2 * pi`] THEN
2988 ONCE_REWRITE_TAC[GSYM REAL_SGN_INEQS] THEN
2989 REWRITE_TAC[REAL_SGN_SIN_AZIM]]);;
2991 (* ------------------------------------------------------------------------- *)
2992 (* Dihedral angle and relation to azimuth angle. *)
2993 (* ------------------------------------------------------------------------- *)
2995 let dihV = new_definition
3000 let vap = ( vc dot vc) % va - ( va dot vc) % vc in
3001 let vbp = ( vc dot vc) % vb - ( vb dot vc) % vc in
3002 arcV (vec 0) vap vbp`;;
3005 (`dihV (w0:real^N) w1 w2 w3 =
3009 let vap = (vc dot vc) % va - (va dot vc) % vc in
3010 let vbp = (vc dot vc) % vb - (vb dot vc) % vc in
3011 angle(vap,vec 0,vbp)`,
3012 REWRITE_TAC[dihV; ARCV_ANGLE]);;
3014 let DIHV_TRANSLATION_EQ = prove
3015 (`!a w0 w1 w2 w3:real^N.
3016 dihV (a + w0) (a + w1) (a + w2) (a + w3) = dihV w0 w1 w2 w3`,
3017 REWRITE_TAC[DIHV; VECTOR_ARITH `(a + x) - (a + y):real^N = x - y`]);;
3019 add_translation_invariants [DIHV_TRANSLATION_EQ];;
3021 let DIHV_LINEAR_IMAGE = prove
3022 (`!f:real^M->real^N w0 w1 w2 w3.
3023 linear f /\ (!x. norm(f x) = norm x)
3024 ==> dihV (f w0) (f w1) (f w2) (f w3) = dihV w0 w1 w2 w3`,
3025 REPEAT STRIP_TAC THEN REWRITE_TAC[DIHV] THEN
3026 ASM_SIMP_TAC[GSYM LINEAR_SUB] THEN
3027 CONV_TAC(TOP_DEPTH_CONV let_CONV) THEN
3028 ASM_SIMP_TAC[PRESERVES_NORM_PRESERVES_DOT] THEN
3029 ASM_SIMP_TAC[GSYM LINEAR_CMUL; GSYM LINEAR_SUB] THEN
3030 REWRITE_TAC[angle; VECTOR_SUB_RZERO] THEN
3031 ASM_SIMP_TAC[VECTOR_ANGLE_LINEAR_IMAGE_EQ]);;
3033 add_linear_invariants [DIHV_LINEAR_IMAGE];;
3035 let DIHV_SPECIAL_SCALE = prove
3036 (`!a v w1 w2:real^N.
3038 ==> dihV (vec 0) (a % v) w1 w2 = dihV (vec 0) v w1 w2`,
3039 REPEAT STRIP_TAC THEN REWRITE_TAC[DIHV; VECTOR_SUB_RZERO] THEN
3040 CONV_TAC(TOP_DEPTH_CONV let_CONV) THEN
3041 REWRITE_TAC[DOT_LMUL; DOT_RMUL; GSYM VECTOR_MUL_ASSOC] THEN
3042 REWRITE_TAC[VECTOR_ARITH `a % a % x - a % b % a % y:real^N =
3043 (a * a) % (x - b % y)`] THEN
3044 REWRITE_TAC[angle; VECTOR_SUB_RZERO] THEN
3045 REWRITE_TAC[VECTOR_ANGLE_LMUL; VECTOR_ANGLE_RMUL] THEN
3046 ASM_REWRITE_TAC[REAL_LE_SQUARE; REAL_ENTIRE]);;
3048 let DIHV_RANGE = prove
3049 (`!w0 w1 w2 w3. &0 <= dihV w0 w1 w2 w3 /\ dihV w0 w1 w2 w3 <= pi`,
3050 REWRITE_TAC[DIHV] THEN CONV_TAC(TOP_DEPTH_CONV let_CONV) THEN
3051 REWRITE_TAC[ANGLE_RANGE]);;
3053 let COS_AZIM_DIHV = prove
3054 (`!v w v1 v2:real^3.
3055 ~collinear {v,w,v1} /\ ~collinear {v,w,v2}
3056 ==> cos(azim v w v1 v2) = cos(dihV v w v1 v2)`,
3057 REPEAT GEN_TAC THEN ASM_CASES_TAC `w:real^3 = v` THENL
3058 [ASM_REWRITE_TAC[INSERT_AC; COLLINEAR_2]; POP_ASSUM MP_TAC] THEN
3059 REWRITE_TAC[IMP_IMP; GSYM CONJ_ASSOC] THEN
3060 GEOM_ORIGIN_TAC `v:real^3` THEN GEOM_BASIS_MULTIPLE_TAC 3 `w:real^3` THEN
3061 X_GEN_TAC `w:real` THEN GEN_REWRITE_TAC LAND_CONV [REAL_LE_LT] THEN
3062 ASM_CASES_TAC `w = &0` THEN ASM_REWRITE_TAC[VECTOR_MUL_LZERO] THEN
3063 ASM_SIMP_TAC[AZIM_SPECIAL_SCALE; DIHV_SPECIAL_SCALE; REAL_LT_IMP_NZ;
3064 COLLINEAR_SPECIAL_SCALE; COLLINEAR_BASIS_3] THEN
3065 DISCH_TAC THEN POP_ASSUM_LIST(K ALL_TAC) THEN
3066 MAP_EVERY X_GEN_TAC [`w1:real^3`; `w2:real^3`] THEN
3067 DISCH_THEN(STRIP_ASSUME_TAC o CONJUNCT2) THEN
3068 REWRITE_TAC[DIHV; VECTOR_SUB_RZERO] THEN
3069 CONV_TAC(TOP_DEPTH_CONV let_CONV) THEN
3070 SIMP_TAC[DOT_BASIS_BASIS; DIMINDEX_3; ARITH] THEN
3071 SIMP_TAC[DOT_BASIS; DIMINDEX_3; ARITH; VECTOR_MUL_LID] THEN
3072 MP_TAC(ISPECL [`vec 0:real^3`; `basis 3:real^3`; `w1:real^3`; `w2:real^3`]
3074 ABBREV_TAC `a = azim (vec 0) (basis 3) w1 (w2:real^3)` THEN
3075 REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
3076 ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM; VECTOR_SUB_RZERO; DIST_0] THEN
3077 MAP_EVERY X_GEN_TAC [`h1:real`; `h2:real`] THEN
3078 DISCH_THEN(MP_TAC o SPECL
3079 [`basis 1:real^3`; `basis 2:real^3`; `basis 3:real^3`]) THEN
3080 SIMP_TAC[orthonormal; DOT_BASIS_BASIS; CROSS_BASIS; DIMINDEX_3; NORM_BASIS;
3081 ARITH; VECTOR_MUL_LID; BASIS_NONZERO; REAL_LT_01; LEFT_IMP_EXISTS_THM] THEN
3082 ASM_REWRITE_TAC[COLLINEAR_BASIS_3] THEN
3083 MAP_EVERY X_GEN_TAC [`psi:real`; `r1:real`; `r2:real`] THEN
3084 DISCH_THEN(STRIP_ASSUME_TAC o GSYM) THEN
3085 REWRITE_TAC[VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT] THEN
3086 SIMP_TAC[BASIS_COMPONENT; DIMINDEX_3; ARITH; REAL_MUL_RZERO] THEN
3087 REWRITE_TAC[REAL_MUL_RID; REAL_ADD_LID] THEN
3088 REWRITE_TAC[VECTOR_ARITH `(a + b + c) - c:real^N = a + b`] THEN
3089 REWRITE_TAC[COS_ANGLE; VECTOR_SUB_RZERO] THEN
3090 REWRITE_TAC[vector_norm; GSYM DOT_EQ_0; DIMINDEX_3; FORALL_3; DOT_3] THEN
3091 REWRITE_TAC[VEC_COMPONENT; VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT] THEN
3092 SIMP_TAC[BASIS_COMPONENT; DIMINDEX_3; ARITH; REAL_MUL_RZERO] THEN
3093 REWRITE_TAC[REAL_MUL_RID; REAL_ADD_LID; REAL_ADD_RID; REAL_MUL_RZERO] THEN
3094 REWRITE_TAC[REAL_ARITH `(r * c) * (r * c) + (r * s) * (r * s):real =
3095 r pow 2 * (s pow 2 + c pow 2)`] THEN
3096 ASM_SIMP_TAC[SIN_CIRCLE; REAL_MUL_RID; REAL_POW_EQ_0; REAL_LT_IMP_NZ] THEN
3097 ASM_SIMP_TAC[POW_2_SQRT; REAL_LT_IMP_LE] THEN
3098 REWRITE_TAC[REAL_ARITH `(r1 * c1) * (r2 * c2) + (r1 * s1) * (r2 * s2):real =
3099 (r1 * r2) * (c1 * c2 + s1 * s2)`] THEN
3100 ASM_SIMP_TAC[REAL_FIELD
3101 `&0 < r1 /\ &0 < r2 ==> ((r1 * r2) * x) / (r1 * r2) = x`] THEN
3102 ONCE_REWRITE_TAC[REAL_ARITH `a:real = b + c * d <=> b - --c * d = a`] THEN
3103 GEN_REWRITE_TAC (funpow 3 LAND_CONV) [GSYM COS_NEG] THEN
3104 REWRITE_TAC[GSYM SIN_NEG; GSYM COS_ADD] THEN AP_TERM_TAC THEN
3107 let AZIM_DIHV_SAME = prove
3108 (`!v w v1 v2:real^3.
3109 ~collinear {v,w,v1} /\ ~collinear {v,w,v2} /\
3111 ==> azim v w v1 v2 = dihV v w v1 v2`,
3112 REPEAT STRIP_TAC THEN MATCH_MP_TAC COS_INJ_PI THEN
3113 ASM_SIMP_TAC[COS_AZIM_DIHV; azim; REAL_LT_IMP_LE; DIHV_RANGE]);;
3115 let AZIM_DIHV_COMPL = prove
3116 (`!v w v1 v2:real^3.
3117 ~collinear {v,w,v1} /\ ~collinear {v,w,v2} /\
3118 pi <= azim v w v1 v2
3119 ==> azim v w v1 v2 = &2 * pi - dihV v w v1 v2`,
3120 REPEAT STRIP_TAC THEN
3121 ONCE_REWRITE_TAC[REAL_ARITH `x = &2 * pi - y <=> y = &2 * pi - x`] THEN
3122 MATCH_MP_TAC COS_INJ_PI THEN
3123 REWRITE_TAC[COS_SUB; SIN_NPI; COS_NPI; REAL_MUL_LZERO] THEN
3124 CONV_TAC REAL_RAT_REDUCE_CONV THEN
3125 ASM_SIMP_TAC[COS_AZIM_DIHV; REAL_ADD_RID; REAL_MUL_LID] THEN
3126 ASM_REWRITE_TAC[DIHV_RANGE] THEN MATCH_MP_TAC(REAL_ARITH
3127 `p <= x /\ x < &2 * p ==> &0 <= &2 * p - x /\ &2 * p - x <= p`) THEN
3128 ASM_SIMP_TAC[azim]);;
3130 let AZIM_DIVH = prove
3131 (`!v w v1 v2:real^3.
3132 ~collinear {v,w,v1} /\ ~collinear {v,w,v2}
3133 ==> azim v w v1 v2 = if azim v w v1 v2 < pi then dihV v w v1 v2
3134 else &2 * pi - dihV v w v1 v2`,
3135 REPEAT STRIP_TAC THEN COND_CASES_TAC THEN
3136 RULE_ASSUM_TAC(REWRITE_RULE[REAL_NOT_LT]) THEN
3137 ASM_SIMP_TAC[AZIM_DIHV_SAME; AZIM_DIHV_COMPL]);;
3139 let AZIM_DIHV_EQ_0 = prove
3141 ~collinear {v0,v1,w1} /\ ~collinear {v0,v1,w2}
3142 ==> (azim v0 v1 w1 w2 = &0 <=> dihV v0 v1 w1 w2 = &0)`,
3143 REPEAT STRIP_TAC THEN
3144 W(MP_TAC o PART_MATCH (lhs o rand) AZIM_DIVH o lhs o lhs o snd) THEN
3145 ASM_REWRITE_TAC[] THEN COND_CASES_TAC THEN ASM_SIMP_TAC[] THEN
3146 ONCE_REWRITE_TAC[REAL_ARITH `a:real = p - b <=> b = p - a`] THEN
3147 DISCH_THEN SUBST1_TAC THEN
3148 REWRITE_TAC[REAL_ARITH `&2 * p - (&2 * p - a) = &0 <=> a = &0`] THEN
3149 MATCH_MP_TAC(REAL_ARITH
3150 `a < &2 * pi /\ ~(a < pi) ==> (a = &0 <=> &2 * pi - a = &0)`) THEN
3151 ASM_REWRITE_TAC[azim]);;
3153 let AZIM_DIHV_EQ_PI = prove
3155 ~collinear {v0,v1,w1} /\ ~collinear {v0,v1,w2}
3156 ==> (azim v0 v1 w1 w2 = pi <=> dihV v0 v1 w1 w2 = pi)`,
3157 REPEAT STRIP_TAC THEN
3158 W(MP_TAC o PART_MATCH (lhs o rand) AZIM_DIVH o lhs o lhs o snd) THEN
3159 ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC);;
3161 let AZIM_EQ_0_PI_EQ_COPLANAR = prove
3163 ~collinear {v0,v1,w1} /\ ~collinear {v0,v1,w2}
3164 ==> (azim v0 v1 w1 w2 = &0 \/ azim v0 v1 w1 w2 = pi <=>
3165 coplanar {v0,v1,w1,w2})`,
3166 REWRITE_TAC[TAUT `(a <=> b) <=> (a ==> b) /\ (b ==> a)`] THEN
3167 REWRITE_TAC[AZIM_EQ_0_PI_IMP_COPLANAR] THEN
3168 SIMP_TAC[GSYM IMP_CONJ_ALT; COPLANAR; DIMINDEX_3; ARITH] THEN
3169 REWRITE_TAC[IMP_CONJ; LEFT_IMP_EXISTS_THM] THEN
3171 [`v0:real^3`; `v1:real^3`; `v2:real^3`; `v3:real^3`; `p:real^3->bool`] THEN
3172 GEOM_HORIZONTAL_PLANE_TAC `p:real^3->bool` THEN
3173 REWRITE_TAC[INSERT_SUBSET; IN_ELIM_THM; IMP_CONJ; RIGHT_FORALL_IMP_THM;
3175 SIMP_TAC[AZIM_DIHV_EQ_0; AZIM_DIHV_EQ_PI] THEN
3176 REWRITE_TAC[DIHV] THEN
3177 CONV_TAC(TOP_DEPTH_CONV let_CONV) THEN
3178 DISCH_THEN(K ALL_TAC) THEN PAD2D3D_TAC THEN
3179 REWRITE_TAC[angle; VECTOR_SUB_RZERO] THEN
3180 GEOM_ORIGIN_TAC `v0:real^2` THEN REWRITE_TAC[VECTOR_SUB_RZERO] THEN
3181 REPEAT STRIP_TAC THEN
3182 W(MP_TAC o PART_MATCH (rand o rand) COLLINEAR_VECTOR_ANGLE o snd) THEN
3184 [REPEAT(POP_ASSUM MP_TAC); DISCH_THEN(SUBST1_TAC o SYM)] THEN
3185 REWRITE_TAC[GSYM DOT_CAUCHY_SCHWARZ_EQUAL] THEN
3186 REWRITE_TAC[DOT_2; CART_EQ; FORALL_2; DIMINDEX_2; VEC_COMPONENT;
3187 VECTOR_SUB_COMPONENT; VECTOR_MUL_COMPONENT] THEN
3188 CONV_TAC REAL_RING);;
3190 let DIHV_EQ_0_PI_EQ_COPLANAR = prove
3191 (`!v0 v1 w1 w2:real^3.
3192 ~collinear {v0,v1,w1} /\ ~collinear {v0,v1,w2}
3193 ==> (dihV v0 v1 w1 w2 = &0 \/ dihV v0 v1 w1 w2 = pi <=>
3194 coplanar {v0,v1,w1,w2})`,
3195 SIMP_TAC[GSYM AZIM_DIHV_EQ_0; GSYM AZIM_DIHV_EQ_PI;
3196 AZIM_EQ_0_PI_EQ_COPLANAR]);;
3198 let DIHV_SYM = prove
3199 (`!v0 v1 v2 v3:real^N.
3200 dihV v0 v1 v3 v2 = dihV v0 v1 v2 v3`,
3201 REPEAT GEN_TAC THEN REWRITE_TAC[DIHV] THEN
3202 CONV_TAC(TOP_DEPTH_CONV let_CONV) THEN
3203 REWRITE_TAC[DOT_SYM; ANGLE_SYM]);;
3205 let DIHV_NEG = prove
3206 (`!v0 v1 v2 v3. dihV (--v0) (--v1) (--v2) (--v3) = dihV v0 v1 v2 v3`,
3207 REWRITE_TAC[DIHV; VECTOR_ARITH `--a - --b:real^N = --(a - b)`] THEN
3208 CONV_TAC(TOP_DEPTH_CONV let_CONV) THEN
3209 REWRITE_TAC[DOT_RNEG; DOT_LNEG; REAL_NEG_NEG] THEN
3210 REWRITE_TAC[VECTOR_MUL_RNEG] THEN
3211 REWRITE_TAC[angle; VECTOR_ARITH `--a - --b:real^N = --(a - b)`] THEN
3212 REWRITE_TAC[VECTOR_SUB_RZERO; VECTOR_ANGLE_NEG2]);;
3214 let DIHV_NEG_0 = prove
3215 (`!v1 v2 v3. dihV (vec 0) (--v1) (--v2) (--v3) = dihV (vec 0) v1 v2 v3`,
3217 GEN_REWRITE_TAC RAND_CONV [GSYM DIHV_NEG] THEN
3218 REWRITE_TAC[VECTOR_NEG_0]);;
3220 let DIHV_ARCV = prove
3222 orthogonal (e - u) (v - u) /\ orthogonal (e - u) (w - u) /\ ~(e = u)
3223 ==> dihV u e v w = arcV u v w`,
3224 GEOM_ORIGIN_TAC `u:real^N` THEN
3225 REWRITE_TAC[dihV; orthogonal; VECTOR_SUB_RZERO] THEN
3226 CONV_TAC(TOP_DEPTH_CONV let_CONV) THEN
3227 SIMP_TAC[DOT_SYM; VECTOR_MUL_LZERO; VECTOR_SUB_RZERO] THEN
3228 REWRITE_TAC[ARCV_ANGLE; angle; VECTOR_SUB_RZERO] THEN
3229 REWRITE_TAC[VECTOR_ANGLE_LMUL; VECTOR_ANGLE_RMUL] THEN
3230 SIMP_TAC[DOT_POS_LE; DOT_EQ_0]);;
3232 let AZIM_DIHV_SAME_STRONG = prove
3233 (`!v w v1 v2:real^3.
3234 ~collinear {v,w,v1} /\ ~collinear {v,w,v2} /\
3235 azim v w v1 v2 <= pi
3236 ==> azim v w v1 v2 = dihV v w v1 v2`,
3237 REWRITE_TAC[REAL_LE_LT] THEN
3238 MESON_TAC[AZIM_DIHV_SAME; AZIM_DIHV_EQ_PI]);;
3240 let AZIM_ARCV = prove
3242 orthogonal (e - u) (v - u) /\ orthogonal (e - u) (w - u) /\
3243 ~collinear{u,e,v} /\ ~collinear{u,e,w} /\
3245 ==> azim u e v w = arcV u v w`,
3246 REPEAT GEN_TAC THEN ASM_CASES_TAC `u:real^3 = e` THENL
3247 [ASM_REWRITE_TAC[INSERT_AC; COLLINEAR_2]; ALL_TAC] THEN
3248 STRIP_TAC THEN ASM_SIMP_TAC[GSYM DIHV_ARCV] THEN
3249 MATCH_MP_TAC AZIM_DIHV_SAME_STRONG THEN ASM_REWRITE_TAC[]);;
3251 let COLLINEAR_AZIM_0_OR_PI = prove
3252 (`!u e v w. collinear {u,v,w} ==> azim u e v w = &0 \/ azim u e v w = pi`,
3253 REPEAT STRIP_TAC THEN
3254 ASM_CASES_TAC `collinear{u:real^3,e,v}` THEN
3255 ASM_SIMP_TAC[AZIM_DEGENERATE] THEN
3256 ASM_CASES_TAC `collinear{u:real^3,e,w}` THEN
3257 ASM_SIMP_TAC[AZIM_DEGENERATE] THEN
3258 ASM_SIMP_TAC[AZIM_EQ_0_PI_EQ_COPLANAR] THEN
3259 ONCE_REWRITE_TAC[SET_RULE `{u,e,v,w} = {u,v,w,e}`] THEN
3260 ASM_MESON_TAC[NOT_COPLANAR_NOT_COLLINEAR]);;
3262 let REAL_CONTINUOUS_WITHIN_DIHV_COMPOSE = prove
3263 (`!f:real^M->real^N g h k x s.
3264 ~collinear {f x,g x,h x} /\ ~collinear {f x,g x,k x} /\
3265 f continuous (at x within s) /\ g continuous (at x within s) /\
3266 h continuous (at x within s) /\ k continuous (at x within s)
3267 ==> (\x. dihV (f x) (g x) (h x) (k x)) real_continuous (at x within s)`,
3268 REPEAT STRIP_TAC THEN REWRITE_TAC[dihV] THEN
3269 CONV_TAC(TOP_DEPTH_CONV let_CONV) THEN
3270 REWRITE_TAC[ARCV_ANGLE; angle; REAL_CONTINUOUS_CONTINUOUS; o_DEF] THEN
3271 REWRITE_TAC[VECTOR_SUB_RZERO] THEN
3272 MATCH_MP_TAC CONTINUOUS_WITHIN_CX_VECTOR_ANGLE_COMPOSE THEN
3273 ASM_REWRITE_TAC[VECTOR_SUB_EQ; GSYM COLLINEAR_3_DOT_MULTIPLES] THEN
3274 CONJ_TAC THEN MATCH_MP_TAC CONTINUOUS_SUB THEN CONJ_TAC THEN
3275 MATCH_MP_TAC CONTINUOUS_MUL THEN REWRITE_TAC[o_DEF] THEN
3276 ASM_SIMP_TAC[CONTINUOUS_LIFT_DOT2; o_DEF; CONTINUOUS_SUB]);;
3278 let REAL_CONTINUOUS_AT_DIHV_COMPOSE = prove
3279 (`!f:real^M->real^N g h k x.
3280 ~collinear {f x,g x,h x} /\ ~collinear {f x,g x,k x} /\
3281 f continuous (at x) /\ g continuous (at x) /\
3282 h continuous (at x) /\ k continuous (at x)
3283 ==> (\x. dihV (f x) (g x) (h x) (k x)) real_continuous (at x)`,
3284 ONCE_REWRITE_TAC[GSYM WITHIN_UNIV] THEN
3285 REWRITE_TAC[REAL_CONTINUOUS_WITHIN_DIHV_COMPOSE]);;
3287 let REAL_CONTINUOUS_WITHINREAL_DIHV_COMPOSE = prove
3288 (`!f:real->real^N g h k x s.
3289 ~collinear {f x,g x,h x} /\ ~collinear {f x,g x,k x} /\
3290 f continuous (atreal x within s) /\ g continuous (atreal x within s) /\
3291 h continuous (atreal x within s) /\ k continuous (atreal x within s)
3292 ==> (\x. dihV (f x) (g x) (h x) (k x)) real_continuous
3293 (atreal x within s)`,
3294 REWRITE_TAC[CONTINUOUS_CONTINUOUS_WITHINREAL;
3295 REAL_CONTINUOUS_REAL_CONTINUOUS_WITHINREAL] THEN
3296 SIMP_TAC[o_DEF; REAL_CONTINUOUS_WITHIN_DIHV_COMPOSE; LIFT_DROP]);;
3298 let REAL_CONTINUOUS_ATREAL_DIHV_COMPOSE = prove
3299 (`!f:real->real^N g h k x.
3300 ~collinear {f x,g x,h x} /\ ~collinear {f x,g x,k x} /\
3301 f continuous (atreal x) /\ g continuous (atreal x) /\
3302 h continuous (atreal x) /\ k continuous (atreal x)
3303 ==> (\x. dihV (f x) (g x) (h x) (k x)) real_continuous (atreal x)`,
3304 ONCE_REWRITE_TAC[GSYM WITHINREAL_UNIV] THEN
3305 REWRITE_TAC[REAL_CONTINUOUS_WITHINREAL_DIHV_COMPOSE]);;
3307 let REAL_CONTINUOUS_AT_DIHV = prove
3308 (`!v w w1 w2:real^N.
3309 ~collinear {v, w, w2} ==> dihV v w w1 real_continuous at w2`,
3310 REPEAT STRIP_TAC THEN GEN_REWRITE_TAC LAND_CONV [GSYM ETA_AX] THEN
3311 REWRITE_TAC[dihV] THEN CONV_TAC(TOP_DEPTH_CONV let_CONV) THEN
3312 GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN
3313 MATCH_MP_TAC REAL_CONTINUOUS_CONTINUOUS_AT_COMPOSE THEN CONJ_TAC THENL
3314 [MATCH_MP_TAC CONTINUOUS_SUB THEN CONJ_TAC THEN
3315 MATCH_MP_TAC CONTINUOUS_MUL THEN
3316 SIMP_TAC[CONTINUOUS_CONST; o_DEF; CONTINUOUS_SUB; CONTINUOUS_AT_ID;
3317 CONTINUOUS_LIFT_DOT2];
3318 GEN_REWRITE_TAC LAND_CONV [GSYM ETA_AX] THEN
3319 REWRITE_TAC[ARCV_ANGLE; angle] THEN
3320 REWRITE_TAC[VECTOR_SUB_RZERO; ETA_AX] THEN
3321 MATCH_MP_TAC REAL_CONTINUOUS_WITHIN_VECTOR_ANGLE THEN
3322 POP_ASSUM MP_TAC THEN GEOM_ORIGIN_TAC `v:real^N` THEN
3323 REWRITE_TAC[VECTOR_SUB_RZERO; CONTRAPOS_THM; VECTOR_SUB_EQ] THEN
3324 MAP_EVERY X_GEN_TAC [`z:real^N`; `w:real^N`] THEN
3325 ASM_CASES_TAC `w:real^N = vec 0` THEN
3326 ASM_REWRITE_TAC[COLLINEAR_LEMMA_ALT] THEN DISCH_THEN(MP_TAC o AP_TERM
3327 `(%) (inv((w:real^N) dot w)):real^N->real^N`) THEN
3328 ASM_SIMP_TAC[VECTOR_MUL_ASSOC; REAL_MUL_LINV; DOT_EQ_0] THEN
3329 MESON_TAC[VECTOR_MUL_LID]]);;
3331 let REAL_CONTINUOUS_WITHIN_AZIM_COMPOSE = prove
3332 (`!f:real^M->real^3 g h k x s.
3333 ~collinear {f x,g x,h x} /\ ~collinear {f x,g x,k x} /\
3334 ~(k x IN aff_ge {f x,g x} {h x}) /\
3335 f continuous (at x within s) /\ g continuous (at x within s) /\
3336 h continuous (at x within s) /\ k continuous (at x within s)
3337 ==> (\x. azim (f x) (g x) (h x) (k x)) real_continuous (at x within s)`,
3339 (`!s t u f:real^M->real^N g h.
3340 (closed s /\ closed t) /\ s UNION t = UNIV /\
3341 (g continuous_on (u INTER s) /\ h continuous_on (u INTER t)) /\
3342 (!x. x IN u INTER s ==> g x = f x) /\
3343 (!x. x IN u INTER t ==> h x = f x)
3344 ==> f continuous_on u`,
3345 REPEAT STRIP_TAC THEN
3346 SUBGOAL_THEN `u:real^M->bool = (u INTER s) UNION (u INTER t)`
3347 SUBST1_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
3348 MATCH_MP_TAC CONTINUOUS_ON_UNION_LOCAL THEN
3349 REWRITE_TAC[CLOSED_IN_CLOSED] THEN REPEAT CONJ_TAC THENL
3350 [EXISTS_TAC `s:real^M->bool` THEN ASM SET_TAC[];
3351 EXISTS_TAC `t:real^M->bool` THEN ASM SET_TAC[];
3352 ASM_MESON_TAC[CONTINUOUS_ON_EQ];
3353 ASM_MESON_TAC[CONTINUOUS_ON_EQ]]) in
3354 REPEAT STRIP_TAC THEN REWRITE_TAC[REAL_CONTINUOUS_CONTINUOUS; o_DEF] THEN
3356 `(\x:real^M. Cx(azim (f x) (g x) (h x) (k x))) =
3357 (\z. Cx(azim (vec 0) (fstcart z)
3358 (fstcart(sndcart z)) (sndcart(sndcart z)))) o
3359 (\x. pastecart (g x - f x) (pastecart (h x - f x) (k x - f x)))`
3361 [REWRITE_TAC[FUN_EQ_THM; o_THM; FSTCART_PASTECART; SNDCART_PASTECART] THEN
3362 X_GEN_TAC `y:real^M` THEN
3363 SUBST1_TAC(VECTOR_ARITH `vec 0 = (f:real^M->real^3) y - f y`) THEN
3364 SIMP_TAC[ONCE_REWRITE_RULE[VECTOR_ADD_SYM] AZIM_TRANSLATION; VECTOR_SUB];
3365 MATCH_MP_TAC CONTINUOUS_WITHIN_COMPOSE THEN
3366 ASM_SIMP_TAC[CONTINUOUS_PASTECART; CONTINUOUS_SUB]] THEN
3367 MATCH_MP_TAC CONTINUOUS_AT_WITHIN THEN
3369 `!z. ~collinear {vec 0,fstcart z,fstcart(sndcart z)} /\
3370 ~collinear {vec 0,fstcart z,sndcart(sndcart z)} /\
3371 ~(sndcart(sndcart z) IN aff_ge {vec 0,fstcart z} {fstcart(sndcart z)})
3372 ==> (\z. Cx(azim (vec 0) (fstcart z) (fstcart(sndcart z))
3373 (sndcart(sndcart z))))
3377 ASM_SIMP_TAC[FSTCART_PASTECART; SNDCART_PASTECART; GSYM COLLINEAR_3] THEN
3378 REPEAT(CONJ_TAC THENL [ASM_MESON_TAC[INSERT_AC]; ALL_TAC]) THEN
3379 SUBST1_TAC(VECTOR_ARITH `vec 0 = (f:real^M->real^3) x - f x`) THEN
3380 ONCE_REWRITE_TAC[SET_RULE `{a,b} = {a} UNION {b}`] THEN
3381 REWRITE_TAC[GSYM IMAGE_UNION; SET_RULE
3382 `{a - b:real^3} = IMAGE (\x. x - b) {a}`] THEN
3383 REWRITE_TAC[ONCE_REWRITE_RULE[VECTOR_ADD_SYM] AFF_GE_TRANSLATION;
3385 ASM_REWRITE_TAC[IN_IMAGE; VECTOR_ARITH `a + x:real^3 = b + x <=> a = b`;
3386 UNWIND_THM1; SET_RULE `{a} UNION {b} = {a,b}`]] THEN
3387 ONCE_REWRITE_TAC[SET_RULE
3388 `(!x. ~P x /\ ~Q x /\ ~R x ==> J x) <=>
3389 (!x. x IN UNIV DIFF (({x | P x} UNION {x | Q x}) UNION {x | R x})
3391 MATCH_MP_TAC(MESON[CONTINUOUS_ON_EQ_CONTINUOUS_AT]
3392 `open s /\ f continuous_on s ==> !z. z IN s ==> f continuous at z`) THEN
3394 [REWRITE_TAC[GSYM closed] THEN
3395 MATCH_MP_TAC(MESON[]
3396 `!t'. s UNION t = s UNION t' /\ closed(s UNION t')
3397 ==> closed(s UNION t)`) THEN
3399 `{z | (fstcart z cross fstcart(sndcart z)) cross
3400 fstcart z cross sndcart(sndcart z) = vec 0 /\
3401 &0 <= (fstcart z cross sndcart(sndcart z)) dot
3402 (fstcart z cross fstcart(sndcart z))}` THEN
3404 [MATCH_MP_TAC(SET_RULE
3405 `(!x. ~(x IN s) ==> (x IN t <=> x IN t'))
3406 ==> s UNION t = s UNION t'`) THEN
3407 REWRITE_TAC[AFF_GE_2_1_0_SEMIALGEBRAIC; IN_UNION; IN_ELIM_THM;
3410 MATCH_MP_TAC CLOSED_UNION THEN CONJ_TAC THENL
3411 [MATCH_MP_TAC CLOSED_UNION THEN CONJ_TAC THEN
3412 REWRITE_TAC[GSYM DOT_CAUCHY_SCHWARZ_EQUAL] THEN
3413 ONCE_REWRITE_TAC[GSYM REAL_SUB_0] THEN
3414 REWRITE_TAC[GSYM LIFT_EQ; LIFT_NUM] THEN
3415 SIMP_TAC[SET_RULE `{x | f x = a} = {x | x IN UNIV /\ f x IN {a}}`] THEN
3416 MATCH_MP_TAC CONTINUOUS_CLOSED_PREIMAGE THEN
3417 SIMP_TAC[CLOSED_UNIV; CLOSED_SING; LIFT_SUB; REAL_POW_2; LIFT_CMUL] THEN
3418 MATCH_MP_TAC CONTINUOUS_ON_SUB THEN
3419 CONJ_TAC THEN MATCH_MP_TAC CONTINUOUS_ON_MUL THEN
3420 REWRITE_TAC[o_DEF] THEN REPEAT CONJ_TAC THEN
3421 MATCH_MP_TAC CONTINUOUS_ON_LIFT_DOT2 THEN CONJ_TAC;
3422 ONCE_REWRITE_TAC[MESON[LIFT_DROP; real_ge]
3423 `&0 <= x <=> drop(lift x) >= &0`] THEN
3424 REWRITE_TAC[SET_RULE
3425 `{z | f z = vec 0 /\ drop(g z) >= &0} =
3426 {z | z IN UNIV /\ f z IN {vec 0}} INTER
3427 {z | z IN UNIV /\ g z IN {k | drop(k) >= &0}}`] THEN
3428 MATCH_MP_TAC CLOSED_INTER THEN
3429 CONJ_TAC THEN MATCH_MP_TAC CONTINUOUS_CLOSED_PREIMAGE THEN
3430 REWRITE_TAC[CLOSED_SING; drop; CLOSED_UNIV;
3431 CLOSED_HALFSPACE_COMPONENT_GE] THEN
3432 REPEAT((MATCH_MP_TAC CONTINUOUS_ON_CROSS ORELSE
3433 MATCH_MP_TAC CONTINUOUS_ON_LIFT_DOT2) THEN CONJ_TAC)] THEN
3434 TRY(GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF]) THEN
3435 SIMP_TAC[CONTINUOUS_ON_COMPOSE; LINEAR_CONTINUOUS_ON;
3436 LINEAR_FSTCART; LINEAR_SNDCART];
3437 MATCH_MP_TAC lemma THEN
3438 MAP_EVERY EXISTS_TAC
3439 [`{z | z IN UNIV /\ lift((fstcart z cross (fstcart(sndcart z))) dot
3440 (sndcart(sndcart z))) IN {x | x$1 >= &0}}`;
3441 `{z | z IN UNIV /\ lift((fstcart z cross (fstcart(sndcart z))) dot
3442 (sndcart(sndcart z))) IN {x | x$1 <= &0}}`;
3443 `\z. Cx(dihV (vec 0:real^3) (fstcart z)
3444 (fstcart(sndcart z)) (sndcart(sndcart z)))`;
3445 `\z. Cx(&2 * pi - dihV (vec 0:real^3) (fstcart z)
3446 (fstcart(sndcart z)) (sndcart(sndcart z)))`] THEN
3448 [CONJ_TAC THEN MATCH_MP_TAC CONTINUOUS_CLOSED_PREIMAGE THEN
3449 REWRITE_TAC[CLOSED_UNIV; CLOSED_HALFSPACE_COMPONENT_GE;
3450 CLOSED_HALFSPACE_COMPONENT_LE] THEN
3451 MATCH_MP_TAC CONTINUOUS_ON_LIFT_DOT2 THEN
3452 (CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_CROSS; ALL_TAC]) THEN
3453 ONCE_REWRITE_TAC[GSYM o_DEF] THEN
3454 SIMP_TAC[CONTINUOUS_ON_COMPOSE; LINEAR_CONTINUOUS_ON;
3455 LINEAR_FSTCART; LINEAR_SNDCART];
3458 [REWRITE_TAC[EXTENSION; IN_UNION; IN_UNIV; IN_ELIM_THM] THEN
3462 [CONJ_TAC THEN MATCH_MP_TAC CONTINUOUS_AT_IMP_CONTINUOUS_ON THEN
3463 REWRITE_TAC[FORALL_PASTECART; IN_DIFF; IN_UNIV; IN_UNION; IN_INTER;
3464 FSTCART_PASTECART; SNDCART_PASTECART; IN_ELIM_THM; DE_MORGAN_THM] THEN
3465 MAP_EVERY X_GEN_TAC [`x:real^3`; `y:real^3`; `z:real^3`] THEN
3466 REPEAT STRIP_TAC THEN REWRITE_TAC[CX_SUB] THEN
3467 TRY(MATCH_MP_TAC CONTINUOUS_SUB THEN REWRITE_TAC[CONTINUOUS_CONST]) THEN
3468 GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN
3469 REWRITE_TAC[GSYM REAL_CONTINUOUS_CONTINUOUS] THEN
3470 MATCH_MP_TAC REAL_CONTINUOUS_AT_DIHV_COMPOSE THEN
3471 ASM_REWRITE_TAC[FSTCART_PASTECART; SNDCART_PASTECART;
3472 CONTINUOUS_CONST] THEN
3473 ONCE_REWRITE_TAC[GSYM o_DEF] THEN
3474 SIMP_TAC[CONTINUOUS_AT_COMPOSE; LINEAR_CONTINUOUS_AT;
3475 LINEAR_FSTCART; LINEAR_SNDCART];
3477 REWRITE_TAC[FORALL_PASTECART; IN_DIFF; IN_UNIV; IN_UNION; IN_INTER; CX_INJ;
3478 FSTCART_PASTECART; SNDCART_PASTECART; IN_ELIM_THM; DE_MORGAN_THM] THEN
3480 [REWRITE_TAC[GSYM drop; LIFT_DROP; real_ge] THEN
3481 MAP_EVERY X_GEN_TAC [`x:real^3`; `y:real^3`; `z:real^3`] THEN
3482 REPEAT STRIP_TAC THEN MATCH_MP_TAC(GSYM AZIM_DIHV_SAME_STRONG) THEN
3483 ASM_REWRITE_TAC[AZIM_IN_UPPER_HALFSPACE; VECTOR_SUB_RZERO];
3484 REWRITE_TAC[GSYM drop; LIFT_DROP] THEN
3485 MAP_EVERY X_GEN_TAC [`x:real^3`; `y:real^3`; `z:real^3`] THEN
3486 REPEAT STRIP_TAC THEN MATCH_MP_TAC(GSYM AZIM_DIHV_COMPL) THEN
3487 ASM_REWRITE_TAC[] THEN
3488 MATCH_MP_TAC(REAL_ARITH
3489 `(x <= pi ==> x = pi) ==> pi <= x`) THEN
3490 ASM_REWRITE_TAC[AZIM_IN_UPPER_HALFSPACE; VECTOR_SUB_RZERO] THEN
3491 ASM_SIMP_TAC[REAL_ARITH `x <= &0 ==> (&0 <= x <=> x = &0)`] THEN
3492 REWRITE_TAC[REWRITE_RULE[VECTOR_SUB_RZERO]
3493 (SPEC `vec 0:real^3` (GSYM COPLANAR_CROSS_DOT))] THEN
3494 ASM_SIMP_TAC[GSYM AZIM_EQ_0_PI_EQ_COPLANAR; AZIM_EQ_0_GE_ALT]]]);;
3496 let REAL_CONTINUOUS_AT_AZIM_COMPOSE = prove
3497 (`!f:real^M->real^3 g h k x.
3498 ~collinear {f x,g x,h x} /\ ~collinear {f x,g x,k x} /\
3499 ~(k x IN aff_ge {f x,g x} {h x}) /\
3500 f continuous (at x) /\ g continuous (at x) /\
3501 h continuous (at x) /\ k continuous (at x)
3502 ==> (\x. azim (f x) (g x) (h x) (k x)) real_continuous (at x)`,
3503 ONCE_REWRITE_TAC[GSYM WITHIN_UNIV] THEN
3504 REWRITE_TAC[REAL_CONTINUOUS_WITHIN_AZIM_COMPOSE]);;
3506 let REAL_CONTINUOUS_WITHINREAL_AZIM_COMPOSE = prove
3507 (`!f:real->real^3 g h k x s.
3508 ~collinear {f x,g x,h x} /\ ~collinear {f x,g x,k x} /\
3509 ~(k x IN aff_ge {f x,g x} {h x}) /\
3510 f continuous (atreal x within s) /\ g continuous (atreal x within s) /\
3511 h continuous (atreal x within s) /\ k continuous (atreal x within s)
3512 ==> (\x. azim (f x) (g x) (h x) (k x)) real_continuous
3513 (atreal x within s)`,
3514 REWRITE_TAC[CONTINUOUS_CONTINUOUS_WITHINREAL;
3515 REAL_CONTINUOUS_REAL_CONTINUOUS_WITHINREAL] THEN
3516 SIMP_TAC[o_DEF; REAL_CONTINUOUS_WITHIN_AZIM_COMPOSE; LIFT_DROP]);;
3518 let REAL_CONTINUOUS_ATREAL_AZIM_COMPOSE = prove
3519 (`!f:real->real^3 g h k x.
3520 ~collinear {f x,g x,h x} /\ ~collinear {f x,g x,k x} /\
3521 ~(k x IN aff_ge {f x,g x} {h x}) /\
3522 f continuous (atreal x) /\ g continuous (atreal x) /\
3523 h continuous (atreal x) /\ k continuous (atreal x)
3524 ==> (\x. azim (f x) (g x) (h x) (k x)) real_continuous (atreal x)`,
3525 ONCE_REWRITE_TAC[GSYM WITHINREAL_UNIV] THEN
3526 REWRITE_TAC[REAL_CONTINUOUS_WITHINREAL_AZIM_COMPOSE]);;
3528 (* ------------------------------------------------------------------------- *)
3529 (* Can consider angle as defined by arcV a zenith angle. *)
3530 (* ------------------------------------------------------------------------- *)
3532 let ZENITH_EXISTS = prove
3534 ~(u = v) /\ ~(w = v)
3538 dist(w,v) % e3 = w - v /\
3540 u = v + u' + (r * cos phi) % e3)`,
3541 ONCE_REWRITE_TAC[VECTOR_ARITH
3542 `u:real^3 = v + u' + x <=> u - v = u' + x`] THEN
3543 GEN_GEOM_ORIGIN_TAC `v:real^3` ["u'"; "e3"] THEN
3544 REWRITE_TAC[VECTOR_SUB_RZERO; DIST_0] THEN
3545 ONCE_REWRITE_TAC[VECTOR_ARITH
3546 `u:real^3 = u' + x <=> u - u' = x`] THEN
3547 GEOM_BASIS_MULTIPLE_TAC 3 `w:real^3` THEN
3548 X_GEN_TAC `w:real` THEN ASM_CASES_TAC `w = &0` THEN
3549 ASM_REWRITE_TAC[VECTOR_MUL_LZERO; REAL_LE_LT] THEN DISCH_TAC THEN
3550 SIMP_TAC[NORM_MUL; NORM_BASIS; DIMINDEX_3; ARITH] THEN
3551 ASM_SIMP_TAC[REAL_ARITH `&0 < w ==> abs w * &1 = w`] THEN
3552 ASM_SIMP_TAC[VECTOR_MUL_LCANCEL; REAL_LT_IMP_NZ] THEN
3553 REWRITE_TAC[RIGHT_EXISTS_AND_THM; UNWIND_THM2] THEN
3554 REWRITE_TAC[ARCV_ANGLE; angle; VECTOR_SUB_RZERO] THEN
3555 ASM_SIMP_TAC[VECTOR_MUL_EQ_0; REAL_LT_IMP_NZ] THEN REPEAT STRIP_TAC THEN
3556 MP_TAC(ISPECL [`u:real^3`; `w % basis 3:real^3`] VECTOR_ANGLE) THEN
3557 REWRITE_TAC[DOT_RMUL; NORM_MUL] THEN
3558 ASM_SIMP_TAC[REAL_ARITH
3559 `&0 < w ==> n * ((abs w) * x) * y = w * n * x * y`] THEN
3560 ASM_REWRITE_TAC[REAL_EQ_MUL_LCANCEL] THEN
3561 SIMP_TAC[NORM_BASIS; DIMINDEX_3; ARITH; REAL_MUL_LID] THEN
3562 DISCH_THEN(SUBST1_TAC o SYM) THEN
3563 REWRITE_TAC[VECTOR_ARITH `u - u':real^3 = x <=> u' = u - x`] THEN
3564 ONCE_REWRITE_TAC[CONJ_SYM] THEN REWRITE_TAC[UNWIND_THM2] THEN
3565 REWRITE_TAC[DOT_LSUB; DOT_RMUL; DOT_LMUL] THEN
3566 SIMP_TAC[DOT_BASIS_BASIS; DIMINDEX_3; ARITH] THEN REAL_ARITH_TAC);;
3568 (* ------------------------------------------------------------------------- *)
3569 (* Spherical coordinates. *)
3570 (* ------------------------------------------------------------------------- *)
3572 let SPHERICAL_COORDINATES = prove
3573 (`!u v w u' e1 e2 e3 r phi theta.
3574 ~collinear {v, w, u} /\
3575 ~collinear {v, w, u'} /\
3576 orthonormal e1 e2 e3 /\
3577 dist(w,v) % e3 = w - v /\
3578 (v + e1) IN aff_gt {v, w} {u} /\
3580 phi = arcV v u' w /\
3581 theta = azim v w u u'
3582 ==> u' = v + (r * cos theta * sin phi) % e1 +
3583 (r * sin theta * sin phi) % e2 +
3584 (r * cos phi) % e3`,
3585 ONCE_REWRITE_TAC[VECTOR_ARITH
3586 `u':real^3 = u + v + w <=> u' - u = v + w`] THEN
3587 GEN_GEOM_ORIGIN_TAC `v:real^3` ["e1"; "e2"; "e3"] THEN
3588 REWRITE_TAC[VECTOR_ADD_RID; VECTOR_ADD_LID] THEN
3589 REWRITE_TAC[TRANSLATION_INVARIANTS `v:real^3`] THEN
3590 GEOM_BASIS_MULTIPLE_TAC 3 `w:real^3` THEN
3591 REWRITE_TAC[VECTOR_SUB_RZERO; DIST_0] THEN
3592 X_GEN_TAC `w:real` THEN ASM_CASES_TAC `w = &0` THENL
3593 [ASM_REWRITE_TAC[VECTOR_MUL_LZERO; INSERT_AC; COLLINEAR_2]; ALL_TAC] THEN
3594 ASM_REWRITE_TAC[REAL_LE_LT] THEN DISCH_TAC THEN
3596 [`u:real^3`; `v:real^3`; `e1:real^3`; `e2:real^3`; `e3:real^3`;
3597 `r:real`; `phi:real`; `theta:real`] THEN
3598 ASM_CASES_TAC `u:real^3 = w % basis 3` THENL
3599 [ASM_REWRITE_TAC[INSERT_AC; COLLINEAR_2]; ALL_TAC] THEN
3600 ASM_CASES_TAC `v:real^3 = w % basis 3` THENL
3601 [ASM_REWRITE_TAC[INSERT_AC; COLLINEAR_2]; ALL_TAC] THEN
3602 DISCH_THEN(MP_TAC o GSYM) THEN
3603 ASM_SIMP_TAC[AZIM_SPECIAL_SCALE; COLLINEAR_SPECIAL_SCALE] THEN
3604 SIMP_TAC[NORM_MUL; NORM_BASIS; DIMINDEX_3; ARITH] THEN
3605 ASM_SIMP_TAC[REAL_ARITH `&0 < w ==> abs w * &1 = w`] THEN
3606 ASM_REWRITE_TAC[VECTOR_MUL_LCANCEL] THEN
3607 ASM_CASES_TAC `e3:real^3 = basis 3` THEN ASM_REWRITE_TAC[] THEN
3608 REWRITE_TAC[ARCV_ANGLE; angle; VECTOR_SUB_RZERO] THEN
3609 ASM_SIMP_TAC[VECTOR_ANGLE_RMUL; REAL_LT_IMP_LE] THEN
3610 ASM_CASES_TAC `u:real^3 = vec 0` THENL
3611 [ASM_REWRITE_TAC[INSERT_AC; COLLINEAR_2]; ALL_TAC] THEN
3612 ASM_CASES_TAC `v:real^3 = vec 0` THENL
3613 [ASM_REWRITE_TAC[INSERT_AC; COLLINEAR_2]; ALL_TAC] THEN
3614 ASM_CASES_TAC `u:real^3 = basis 3` THENL
3615 [ASM_REWRITE_TAC[INSERT_AC; COLLINEAR_2]; ALL_TAC] THEN
3616 ASM_CASES_TAC `v:real^3 = basis 3` THENL
3617 [ASM_REWRITE_TAC[INSERT_AC; COLLINEAR_2]; ALL_TAC] THEN
3619 MP_TAC(ISPECL [`v:real^3`; `basis 3:real^3`] VECTOR_ANGLE) THEN
3620 ASM_SIMP_TAC[DOT_BASIS; NORM_BASIS; DIMINDEX_3; ARITH; REAL_MUL_LID] THEN
3623 [`vec 0:real^3`; `w % basis 3:real^3`; `u:real^3`; `e1:real^3`]
3625 ASM_SIMP_TAC[AZIM_SPECIAL_SCALE; COLLINEAR_SPECIAL_SCALE] THEN
3627 [SIMP_TAC[COLLINEAR_LEMMA; BASIS_NONZERO; DIMINDEX_3; ARITH] THEN
3628 STRIP_TAC THEN UNDISCH_TAC `orthonormal e1 e2 (basis 3)` THEN
3629 ASM_REWRITE_TAC[orthonormal; DOT_LZERO; REAL_OF_NUM_EQ; ARITH_EQ] THEN
3630 ASM_CASES_TAC `c = &0` THEN
3631 ASM_SIMP_TAC[VECTOR_MUL_LZERO; CROSS_LZERO; DOT_LZERO; REAL_LT_REFL;
3632 DOT_LMUL; DOT_BASIS_BASIS; DIMINDEX_3; ARITH; REAL_MUL_RID];
3635 `dropout 3 (v:real^3):real^2 =
3636 norm(dropout 3 (v:real^3):real^2) %
3637 (cos theta % (dropout 3 (e1:real^3)) +
3638 sin theta % (dropout 3 (e2:real^3)))`
3641 SUBGOAL_THEN `norm((dropout 3:real^3->real^2) v) = r * sin phi`
3643 [REWRITE_TAC[NORM_EQ_SQUARE] THEN CONJ_TAC THENL
3644 [ASM_MESON_TAC[REAL_LE_MUL; NORM_POS_LE; SIN_VECTOR_ANGLE_POS];
3646 UNDISCH_TAC `(v:real^3)$3 = r * cos phi` THEN
3647 MATCH_MP_TAC(REAL_RING
3648 `x + a pow 2 = y + b pow 2 ==> a:real = b ==> x = y`) THEN
3649 REWRITE_TAC[REAL_POW_MUL; GSYM REAL_ADD_LDISTRIB] THEN
3650 REWRITE_TAC[SIN_CIRCLE; REAL_MUL_RID] THEN
3651 UNDISCH_THEN `norm(v:real^3) = r` (SUBST1_TAC o SYM) THEN
3652 REWRITE_TAC[NORM_POW_2; DOT_2; DOT_3] THEN
3653 SIMP_TAC[dropout; LAMBDA_BETA; DIMINDEX_2; ARITH] THEN
3656 REWRITE_TAC[CART_EQ; DIMINDEX_3; DIMINDEX_2; FORALL_3; FORALL_2] THEN
3657 SIMP_TAC[dropout; LAMBDA_BETA; DIMINDEX_2; ARITH; VECTOR_ADD_COMPONENT;
3658 VECTOR_MUL_COMPONENT; BASIS_COMPONENT; DIMINDEX_3] THEN
3659 REPEAT STRIP_TAC THEN TRY REAL_ARITH_TAC THEN
3660 ASM_REWRITE_TAC[] THEN
3661 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [orthonormal]) THEN
3662 SIMP_TAC[DOT_BASIS; DIMINDEX_3; ARITH] THEN CONV_TAC REAL_RING] THEN
3663 REPEAT(FIRST_X_ASSUM(MP_TAC o
3664 GEN_REWRITE_RULE LAND_CONV [AZIM_ARG])) THEN
3665 REPEAT(FIRST_X_ASSUM(MP_TAC o
3666 GEN_REWRITE_RULE RAND_CONV [COLLINEAR_BASIS_3])) THEN
3667 SUBGOAL_THEN `norm((dropout 3:real^3->real^2) e1) = &1 /\
3668 norm((dropout 3:real^3->real^2) e2) = &1 /\
3669 dropout 3 (e2:real^3) / dropout 3 (e1:real^3) = ii`
3671 [MATCH_MP_TAC(TAUT `(a /\ b) /\ (a /\ b ==> c) ==> a /\ b /\ c`) THEN
3673 [REWRITE_TAC[NORM_EQ_1] THEN
3674 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [orthonormal]) THEN
3675 SIMP_TAC[DOT_BASIS; DIMINDEX_3; ARITH; dropout; LAMBDA_BETA;
3676 DOT_2; DIMINDEX_2; DOT_3] THEN
3679 ASM_CASES_TAC `dropout 3 (e1:real^3) = Cx(&0)` THEN
3680 ASM_SIMP_TAC[COMPLEX_NORM_CX; REAL_OF_NUM_EQ; ARITH_EQ; REAL_ABS_NUM] THEN
3681 ASM_SIMP_TAC[COMPLEX_FIELD
3682 `~(x = Cx(&0)) ==> (y / x = ii <=> y = ii * x)`] THEN
3683 DISCH_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP ORTHONORMAL_CROSS) THEN
3684 SIMP_TAC[CART_EQ; DIMINDEX_2; DIMINDEX_3; FORALL_2; FORALL_3;
3685 cross; VECTOR_3; BASIS_COMPONENT; ARITH; dropout; LAMBDA_BETA;
3686 complex_mul; ii; complex; RE_DEF; IM_DEF; VECTOR_2] THEN
3689 SPEC_TAC(`(dropout 3:real^3->real^2) e2`,`d2:real^2`) THEN
3690 SPEC_TAC(`(dropout 3:real^3->real^2) e1`,`d1:real^2`) THEN
3691 SPEC_TAC(`(dropout 3:real^3->real^2) v`,`z:real^2`) THEN
3692 SPEC_TAC(`(dropout 3:real^3->real^2) u`,`w:real^2`) THEN
3693 POP_ASSUM_LIST(K ALL_TAC) THEN
3694 GEOM_BASIS_MULTIPLE_TAC 1 `w:real^2` THEN
3695 X_GEN_TAC `k:real` THEN GEN_REWRITE_TAC LAND_CONV [REAL_LE_LT] THEN
3696 ASM_CASES_TAC `k = &0` THEN ASM_REWRITE_TAC[VECTOR_MUL_LZERO] THEN
3697 REWRITE_TAC[COMPLEX_CMUL; COMPLEX_BASIS; COMPLEX_VEC_0] THEN
3698 SIMP_TAC[ARG_DIV_CX; COMPLEX_MUL_RID] THEN DISCH_TAC THEN REPEAT GEN_TAC THEN
3699 ASM_CASES_TAC `d1 = Cx(&1)` THENL
3700 [ASM_SIMP_TAC[COMPLEX_DIV_1; COMPLEX_MUL_LID] THEN
3701 REPEAT STRIP_TAC THEN MP_TAC(SPEC `z:complex` ARG) THEN
3702 ASM_REWRITE_TAC[CEXP_EULER; CX_SIN; CX_COS; COMPLEX_MUL_RID] THEN
3703 CONV_TAC COMPLEX_RING;
3704 ASM_REWRITE_TAC[ARG_EQ_0] THEN
3705 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [COMPLEX_EQ]) THEN
3706 REWRITE_TAC[RE_CX; IM_CX;real] THEN
3707 ASM_CASES_TAC `Im d1 = &0` THEN ASM_REWRITE_TAC[] THEN
3708 ASM_SIMP_TAC[REAL_NORM; real] THEN REAL_ARITH_TAC]);;
3710 (* ------------------------------------------------------------------------- *)
3711 (* Definition of a wedge and invariance theorems. *)
3712 (* ------------------------------------------------------------------------- *)
3714 let wedge = new_definition
3715 `wedge v0 v1 w1 w2 = {y | ~collinear {v0,v1,y} /\
3716 &0 < azim v0 v1 w1 y /\
3717 azim v0 v1 w1 y < azim v0 v1 w1 w2}`;;
3719 let WEDGE_ALT = prove
3722 ==> wedge v0 v1 w1 w2 = {y | ~(y IN affine hull {v0,v1}) /\
3723 &0 < azim v0 v1 w1 y /\
3724 azim v0 v1 w1 y < azim v0 v1 w1 w2}`,
3725 SIMP_TAC[wedge; COLLINEAR_3_AFFINE_HULL]);;
3727 let WEDGE_TRANSLATION = prove
3728 (`!a v w w1 w2. wedge (a + v) (a + w) (a + w1) (a + w2) =
3729 IMAGE (\x. a + x) (wedge v w w1 w2)`,
3730 REPEAT GEN_TAC THEN CONV_TAC SYM_CONV THEN
3731 MATCH_MP_TAC SURJECTIVE_IMAGE_EQ THEN CONJ_TAC THENL
3732 [MESON_TAC[VECTOR_ARITH `a + (x - a):real^3 = x`]; ALL_TAC] THEN
3733 REWRITE_TAC[wedge; IN_ELIM_THM; AZIM_TRANSLATION] THEN
3734 REWRITE_TAC[SET_RULE
3735 `{a + x,a + y,a + z} = IMAGE (\x:real^N. a + x) {x,y,z}`] THEN
3736 REWRITE_TAC[COLLINEAR_TRANSLATION_EQ]);;
3738 add_translation_invariants [WEDGE_TRANSLATION];;
3740 let WEDGE_LINEAR_IMAGE = prove
3741 (`!f. linear f /\ (!x. norm(f x) = norm x) /\
3742 (2 <= dimindex(:3) ==> det(matrix f) = &1)
3743 ==> !v w w1 w2. wedge (f v) (f w) (f w1) (f w2) =
3744 IMAGE f (wedge v w w1 w2)`,
3745 REPEAT STRIP_TAC THEN CONV_TAC SYM_CONV THEN
3746 MATCH_MP_TAC SURJECTIVE_IMAGE_EQ THEN CONJ_TAC THENL
3747 [ASM_MESON_TAC[ORTHOGONAL_TRANSFORMATION_SURJECTIVE;
3748 ORTHOGONAL_TRANSFORMATION];
3750 X_GEN_TAC `y:real^3` THEN REWRITE_TAC[wedge; IN_ELIM_THM] THEN
3751 BINOP_TAC THEN ASM_SIMP_TAC[AZIM_LINEAR_IMAGE] THEN
3752 SUBST1_TAC(SET_RULE `{f v,f w,f y} = IMAGE (f:real^3->real^3) {v,w,y}`) THEN
3753 ASM_MESON_TAC[COLLINEAR_LINEAR_IMAGE_EQ; PRESERVES_NORM_INJECTIVE]);;
3755 add_linear_invariants [WEDGE_LINEAR_IMAGE];;
3757 let WEDGE_SPECIAL_SCALE = prove
3760 ~collinear{vec 0,a % v,w1} /\
3761 ~collinear{vec 0,a % v,w2}
3762 ==> wedge (vec 0) (a % v) w1 w2 = wedge (vec 0) v w1 w2`,
3763 SIMP_TAC[wedge; AZIM_SPECIAL_SCALE; COLLINEAR_SPECIAL_SCALE;
3766 let WEDGE_DEGENERATE = prove
3767 (`(!z w w1 w2. z = w ==> wedge z w w1 w2 = {}) /\
3768 (!z w w1 w2. collinear{z,w,w1} ==> wedge z w w1 w2 = {}) /\
3769 (!z w w1 w2. collinear{z,w,w2} ==> wedge z w w1 w2 = {})`,
3770 REWRITE_TAC[wedge] THEN SIMP_TAC[AZIM_DEGENERATE] THEN
3771 REWRITE_TAC[REAL_LT_REFL; REAL_LT_ANTISYM; EMPTY_GSPEC]);;
3773 (* ------------------------------------------------------------------------- *)
3774 (* Basic relation between wedge and aff, so Tarski-type characterization. *)
3775 (* ------------------------------------------------------------------------- *)
3777 let AFF_GT_LEMMA = prove
3779 &0 < t1 /\ ~(v2 = vec 0)
3780 ==> aff_gt {vec 0} {t1 % basis 1, v2} =
3781 {a % basis 1 + b % v2 | &0 < a /\ &0 < b}`,
3782 REWRITE_TAC[AFFSIGN_ALT; aff_gt_def; sgn_gt; IN_ELIM_THM] THEN
3783 REWRITE_TAC[SET_RULE `{a} UNION {b,c} = {a,b,c}`] THEN
3784 REWRITE_TAC[SET_RULE `x IN {a} <=> a = x`] THEN
3785 ASM_SIMP_TAC[FINITE_INSERT; FINITE_UNION; AFFINE_HULL_FINITE_STEP_GEN;
3786 RIGHT_EXISTS_AND_THM; REAL_LT_ADD; REAL_HALF; FINITE_EMPTY] THEN
3787 ASM_REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY] THEN
3788 REWRITE_TAC[IN_INSERT; VECTOR_ARITH `vec 0 = a % x <=> a % x = vec 0`] THEN
3789 ASM_SIMP_TAC[VECTOR_MUL_EQ_0; REAL_LT_IMP_NZ; BASIS_NONZERO;
3790 DIMINDEX_GE_1; LE_REFL] THEN
3791 REPEAT STRIP_TAC THEN REWRITE_TAC[VECTOR_MUL_RZERO; VECTOR_SUB_RZERO] THEN
3792 REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN X_GEN_TAC `y:real^N` THEN
3793 REWRITE_TAC[RIGHT_AND_EXISTS_THM] THEN
3794 REWRITE_TAC[REAL_ARITH `&1 - v - v' - v'' = &0 <=> v = &1 - v' - v''`] THEN
3795 ONCE_REWRITE_TAC[MESON[] `(?a b c. P a b c) <=> (?b c a. P a b c)`] THEN
3796 REWRITE_TAC[RIGHT_EXISTS_AND_THM; UNWIND_THM2] THEN
3797 REWRITE_TAC[RIGHT_AND_EXISTS_THM] THEN ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN
3798 AP_TERM_TAC THEN REWRITE_TAC[FUN_EQ_THM] THEN X_GEN_TAC `b:real` THEN
3799 REWRITE_TAC[VECTOR_ARITH `y - a - b:real^N = vec 0 <=> y = a + b`] THEN
3800 EQ_TAC THEN DISCH_THEN(X_CHOOSE_THEN `a:real` STRIP_ASSUME_TAC) THENL
3801 [EXISTS_TAC `a * t1:real`; EXISTS_TAC `a / t1:real`] THEN
3802 ASM_SIMP_TAC[REAL_LT_DIV; REAL_LT_MUL; VECTOR_MUL_ASSOC] THEN
3803 ASM_SIMP_TAC[REAL_DIV_RMUL; REAL_LT_IMP_NZ]);;
3805 let WEDGE_LUNE_GT = prove
3807 ~collinear {v0,v1,w1} /\ ~collinear {v0,v1,w2} /\
3808 &0 < azim v0 v1 w1 w2 /\ azim v0 v1 w1 w2 < pi
3809 ==> wedge v0 v1 w1 w2 = aff_gt {v0,v1} {w1,w2}`,
3811 (`!a x:real^3. (?a. x = a % basis 3) <=> dropout 3 x:real^2 = vec 0`,
3812 SIMP_TAC[CART_EQ; FORALL_2; FORALL_3; DIMINDEX_2; DIMINDEX_3;
3813 dropout; LAMBDA_BETA; BASIS_COMPONENT; ARITH; REAL_MUL_RID;
3814 VECTOR_MUL_COMPONENT; VEC_COMPONENT; REAL_MUL_RZERO; UNWIND_THM1] THEN
3816 REWRITE_TAC[wedge] THEN GEOM_ORIGIN_TAC `v0:real^3` THEN
3817 GEOM_BASIS_MULTIPLE_TAC 3 `v1:real^3` THEN
3818 X_GEN_TAC `w:real` THEN GEN_REWRITE_TAC LAND_CONV [REAL_LE_LT] THEN
3819 ASM_CASES_TAC `w = &0` THEN ASM_REWRITE_TAC[] THENL
3820 [ASM_REWRITE_TAC[VECTOR_MUL_LZERO; INSERT_AC; COLLINEAR_2]; ALL_TAC] THEN
3821 ASM_SIMP_TAC[AZIM_SPECIAL_SCALE; COLLINEAR_SPECIAL_SCALE] THEN
3822 POP_ASSUM_LIST(K ALL_TAC) THEN DISCH_TAC THEN
3823 MAP_EVERY X_GEN_TAC [`w1:real^3`; `w2:real^3`] THEN
3824 REPLICATE_TAC 2 (DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
3825 ONCE_REWRITE_TAC[TAUT `~a /\ b /\ c <=> ~(~a ==> ~(b /\ c))`] THEN
3826 ASM_SIMP_TAC[AZIM_ARG] THEN REWRITE_TAC[COLLINEAR_BASIS_3] THEN
3827 RULE_ASSUM_TAC(REWRITE_RULE[COLLINEAR_BASIS_3]) THEN STRIP_TAC THEN
3828 REWRITE_TAC[NOT_IMP; GSYM CONJ_ASSOC] THEN
3829 W(MP_TAC o PART_MATCH (lhs o rand) AFF_GT_SPECIAL_SCALE o rand o snd) THEN
3831 `~(w1:real^3 = vec 0) /\ ~(w2:real^3 = vec 0) /\
3832 ~(w1 = basis 3) /\ ~(w2 = basis 3)`
3833 STRIP_ASSUME_TAC THENL
3834 [REPEAT STRIP_TAC THEN
3835 REPEAT(FIRST_X_ASSUM(MP_TAC o check (is_neg o concl))) THEN
3836 ASM_REWRITE_TAC[DROPOUT_BASIS_3; DROPOUT_0; DROPOUT_MUL; VECTOR_MUL_RZERO];
3839 [ASM_REWRITE_TAC[FINITE_INSERT; FINITE_EMPTY; IN_INSERT; NOT_IN_EMPTY] THEN
3840 DISCH_THEN(DISJ_CASES_THEN (SUBST_ALL_TAC o SYM)) THEN
3841 REPEAT(FIRST_X_ASSUM(MP_TAC o check (is_neg o concl))) THEN
3842 ASM_REWRITE_TAC[DROPOUT_BASIS_3; DROPOUT_0; DROPOUT_MUL; VECTOR_MUL_RZERO];
3843 DISCH_THEN SUBST1_TAC] THEN
3844 REWRITE_TAC[AFFSIGN_ALT; aff_gt_def; sgn_gt; IN_ELIM_THM] THEN
3845 REWRITE_TAC[SET_RULE `{a,b} UNION {c,d} = {a,b,d,c}`] THEN
3846 REWRITE_TAC[SET_RULE `x IN {a} <=> a = x`] THEN
3847 ASM_SIMP_TAC[FINITE_INSERT; FINITE_UNION; AFFINE_HULL_FINITE_STEP_GEN;
3848 RIGHT_EXISTS_AND_THM; REAL_LT_ADD; REAL_HALF; FINITE_EMPTY] THEN
3849 ASM_REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY] THEN
3850 REWRITE_TAC[VECTOR_MUL_RZERO; VECTOR_SUB_RZERO] THEN
3851 MATCH_MP_TAC EQ_TRANS THEN
3852 EXISTS_TAC `{y | (dropout 3:real^3->real^2) y IN
3854 {dropout 3 (w1:real^3),dropout 3 (w2:real^3)}}` THEN
3857 REWRITE_TAC[AFFSIGN_ALT; aff_gt_def; sgn_gt; IN_ELIM_THM] THEN
3858 REWRITE_TAC[SET_RULE `{a} UNION {b,c} = {a,b,c}`] THEN
3859 REWRITE_TAC[SET_RULE `x IN {a} <=> a = x`] THEN
3860 ASM_SIMP_TAC[FINITE_INSERT; FINITE_UNION; AFFINE_HULL_FINITE_STEP_GEN;
3861 RIGHT_EXISTS_AND_THM; REAL_LT_ADD; REAL_HALF; FINITE_EMPTY] THEN
3862 ASM_REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY] THEN
3863 REWRITE_TAC[VECTOR_MUL_RZERO; VECTOR_SUB_RZERO] THEN
3864 REWRITE_TAC[REAL_EQ_SUB_RADD; RIGHT_AND_EXISTS_THM] THEN
3865 REWRITE_TAC[REAL_ARITH `&1 = x + v <=> v = &1 - x`] THEN
3866 ONCE_REWRITE_TAC[TAUT `a /\ b /\ c /\ d <=> c /\ d /\ a /\ b`] THEN
3867 ONCE_REWRITE_TAC[MESON[]
3868 `(?a b c d. P a b c d) <=> (?b c d a. P a b c d)`] THEN
3869 REWRITE_TAC[UNWIND_THM2] THEN
3870 ONCE_REWRITE_TAC[MESON[]
3871 `(?a b c. P a b c) <=> (?c b a. P a b c)`] THEN
3872 REWRITE_TAC[UNWIND_THM2] THEN REWRITE_TAC[VECTOR_ARITH
3873 `y - a - b - c:real^N = vec 0 <=> y - b - c = a`] THEN
3874 REWRITE_TAC[LEFT_EXISTS_AND_THM; lemma] THEN
3875 REWRITE_TAC[DROPOUT_SUB; DROPOUT_MUL] THEN
3876 REWRITE_TAC[VECTOR_ARITH `y - a - b:real^2 = vec 0 <=> y = a + b`] THEN
3877 REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN MESON_TAC[VECTOR_ADD_SYM]] THEN
3878 MATCH_MP_TAC(SET_RULE
3879 `{x | P x} = s ==> {y | P(dropout 3 y)} = {y | dropout 3 y IN s}`) THEN
3880 MP_TAC(CONJ (ASSUME `~((dropout 3:real^3->real^2) w1 = vec 0)`)
3881 (ASSUME `~((dropout 3:real^3->real^2) w2 = vec 0)`)) THEN
3882 UNDISCH_TAC `Arg(dropout 3 (w2:real^3) / dropout 3 (w1:real^3)) < pi` THEN
3883 UNDISCH_TAC `&0 < Arg(dropout 3 (w2:real^3) / dropout 3 (w1:real^3))` THEN
3884 SPEC_TAC(`(dropout 3:real^3->real^2) w2`,`v2:complex`) THEN
3885 SPEC_TAC(`(dropout 3:real^3->real^2) w1`,`v1:complex`) THEN
3886 POP_ASSUM_LIST(K ALL_TAC) THEN GEOM_BASIS_MULTIPLE_TAC 1 `v1:complex` THEN
3887 X_GEN_TAC `v1:real` THEN GEN_REWRITE_TAC LAND_CONV [REAL_LE_LT] THEN
3888 ASM_CASES_TAC `v1 = &0` THEN ASM_REWRITE_TAC[VECTOR_MUL_LZERO] THEN
3889 SIMP_TAC[AFF_GT_LEMMA] THEN
3890 REWRITE_TAC[COMPLEX_CMUL; COMPLEX_BASIS; COMPLEX_VEC_0] THEN
3891 ASM_SIMP_TAC[ARG_DIV_CX; COMPLEX_MUL_RID; CX_INJ] THEN DISCH_TAC THEN
3892 POP_ASSUM_LIST(K ALL_TAC) THEN X_GEN_TAC `z:complex` THEN
3893 REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN
3894 REWRITE_TAC[SUBSET; FORALL_IN_GSPEC; IN_ELIM_THM] THEN CONJ_TAC THENL
3895 [X_GEN_TAC `w:complex` THEN STRIP_TAC THEN
3896 MP_TAC(SPECL [`\t. Arg(Cx t + Cx(&1 - t) * z)`;
3897 `&0`; `&1`; `Arg w`] REAL_IVT_DECREASING) THEN
3898 REWRITE_TAC[REAL_POS; REAL_SUB_REFL; COMPLEX_MUL_LZERO] THEN
3899 REWRITE_TAC[REAL_SUB_RZERO; COMPLEX_ADD_LID; COMPLEX_MUL_LID] THEN
3900 ASM_SIMP_TAC[COMPLEX_ADD_RID; ARG_NUM; REAL_LT_IMP_LE] THEN ANTS_TAC THENL
3901 [REWRITE_TAC[REAL_CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN] THEN
3902 REWRITE_TAC[REAL_CONTINUOUS_CONTINUOUS; IN_REAL_INTERVAL] THEN
3903 X_GEN_TAC `t:real` THEN STRIP_TAC THEN
3904 ONCE_REWRITE_TAC[GSYM o_DEF] THEN REWRITE_TAC[o_ASSOC] THEN
3905 MATCH_MP_TAC CONTINUOUS_WITHINREAL_COMPOSE THEN
3906 REWRITE_TAC[] THEN CONJ_TAC THENL
3907 [MATCH_MP_TAC CONTINUOUS_ADD THEN CONJ_TAC THENL
3908 [GEN_REWRITE_TAC LAND_CONV [SYM(CONJUNCT2(SPEC_ALL I_O_ID))] THEN
3909 REWRITE_TAC[GSYM REAL_CONTINUOUS_CONTINUOUS] THEN
3910 REWRITE_TAC[I_DEF; REAL_CONTINUOUS_WITHIN_ID];
3911 MATCH_MP_TAC CONTINUOUS_COMPLEX_MUL THEN
3912 REWRITE_TAC[CONTINUOUS_CONST] THEN ONCE_REWRITE_TAC[GSYM o_DEF] THEN
3913 REWRITE_TAC[GSYM REAL_CONTINUOUS_CONTINUOUS] THEN
3914 SIMP_TAC[REAL_CONTINUOUS_SUB; REAL_CONTINUOUS_CONST;
3915 REAL_CONTINUOUS_WITHIN_ID]];
3916 MATCH_MP_TAC CONTINUOUS_WITHIN_SUBSET THEN
3917 EXISTS_TAC `{z | &0 <= Im z}` THEN CONJ_TAC THENL
3918 [MATCH_MP_TAC CONTINUOUS_WITHIN_UPPERHALF_ARG THEN
3919 ASM_CASES_TAC `t = &1` THENL
3920 [ASM_REWRITE_TAC[REAL_SUB_REFL] THEN CONV_TAC COMPLEX_RING;
3922 DISCH_THEN(MP_TAC o AP_TERM `Im`) THEN
3923 REWRITE_TAC[IM_ADD; IM_CX; IM_MUL_CX; REAL_ADD_LID; REAL_ENTIRE] THEN
3924 ASM_REWRITE_TAC[REAL_SUB_0] THEN
3925 ASM_MESON_TAC[ARG_LT_PI; REAL_LT_IMP_NZ; REAL_LT_TRANS];
3926 REWRITE_TAC[FORALL_IN_IMAGE; SUBSET; IN_REAL_INTERVAL] THEN
3927 REWRITE_TAC[IN_ELIM_THM; IM_ADD; IM_CX; IM_MUL_CX] THEN
3928 REPEAT STRIP_TAC THEN REWRITE_TAC[REAL_ADD_LID] THEN
3929 MATCH_MP_TAC REAL_LE_MUL THEN REWRITE_TAC[GSYM ARG_LE_PI] THEN
3930 ASM_REAL_ARITH_TAC]];
3931 REWRITE_TAC[IN_REAL_INTERVAL] THEN
3932 DISCH_THEN(X_CHOOSE_THEN `t:real` MP_TAC) THEN
3933 ASM_CASES_TAC `t = &0` THENL
3934 [ASM_REWRITE_TAC[REAL_SUB_RZERO; COMPLEX_ADD_LID; COMPLEX_MUL_LID] THEN
3935 ASM_MESON_TAC[REAL_LT_REFL];
3937 ASM_CASES_TAC `t = &1` THENL
3938 [ASM_REWRITE_TAC[REAL_SUB_REFL; COMPLEX_MUL_LZERO] THEN
3939 REWRITE_TAC[COMPLEX_ADD_RID; ARG_NUM] THEN ASM_MESON_TAC[REAL_LT_REFL];
3941 GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [REAL_LE_LT] THEN
3942 ASM_REWRITE_TAC[] THEN ABBREV_TAC `u = Cx t + Cx(&1 - t) * z` THEN
3943 ASM_CASES_TAC `u = Cx(&0)` THENL
3944 [ASM_MESON_TAC[ARG_0; REAL_LT_REFL]; ALL_TAC] THEN
3946 EXISTS_TAC `norm(w:complex) / norm(u:complex) * t` THEN
3947 EXISTS_TAC `norm(w:complex) / norm(u:complex) * (&1 - t)` THEN
3948 ASM_SIMP_TAC[REAL_LT_MUL; REAL_LT_DIV; COMPLEX_NORM_NZ; REAL_SUB_LT] THEN
3949 SIMP_TAC[CX_MUL; GSYM COMPLEX_MUL_ASSOC; GSYM COMPLEX_ADD_LDISTRIB] THEN
3950 ASM_REWRITE_TAC[CX_DIV] THEN
3951 ASM_SIMP_TAC[CX_INJ; COMPLEX_NORM_ZERO; COMPLEX_FIELD
3952 `~(nu = Cx(&0)) ==> (w = nw / nu * u <=> nu * w = nw * u)`] THEN
3953 GEN_REWRITE_TAC (BINOP_CONV o RAND_CONV) [ARG] THEN
3954 ASM_REWRITE_TAC[COMPLEX_MUL_AC]];
3955 MAP_EVERY X_GEN_TAC [`a:real`; `b:real`] THEN STRIP_TAC THEN
3956 SUBGOAL_THEN `Cx a + Cx b * z = complex(a + b * Re z,b * Im z)`
3958 [REWRITE_TAC[COMPLEX_EQ; RE; IM; RE_ADD; IM_ADD; RE_CX; IM_CX;
3959 RE_MUL_CX; IM_MUL_CX] THEN
3962 REWRITE_TAC[COMPLEX_EQ; IM; IM_CX] THEN
3963 SUBGOAL_THEN `&0 < Im z` ASSUME_TAC THENL
3964 [ASM_REWRITE_TAC[GSYM ARG_LT_PI]; ALL_TAC] THEN
3965 ASM_SIMP_TAC[ARG_ATAN_UPPERHALF; REAL_LT_MUL; REAL_LT_IMP_NZ; IM] THEN
3966 REWRITE_TAC[RE; REAL_SUB_LT; ATN_BOUNDS] THEN
3967 REWRITE_TAC[REAL_ARITH `pi / &2 - x < pi / &2 - y <=> y < x`] THEN
3968 REWRITE_TAC[ATN_MONO_LT_EQ] THEN
3969 ASM_SIMP_TAC[REAL_LT_RDIV_EQ; REAL_LT_MUL] THEN
3970 ASM_SIMP_TAC[REAL_FIELD `&0 < z ==> w / z * b * z = b * w`] THEN
3971 ASM_REAL_ARITH_TAC]);;
3973 let WEDGE_LUNE_GE = prove
3975 ~collinear {v0,v1,w1} /\ ~collinear {v0,v1,w2} /\
3976 &0 < azim v0 v1 w1 w2 /\ azim v0 v1 w1 w2 < pi
3977 ==> {x | &0 <= azim v0 v1 w1 x /\
3978 azim v0 v1 w1 x <= azim v0 v1 w1 w2} =
3979 aff_ge {v0,v1} {w1,w2}`,
3981 MAP_EVERY (fun t -> ASM_CASES_TAC t THENL
3982 [ASM_REWRITE_TAC[COLLINEAR_2; INSERT_AC] THEN NO_TAC; ALL_TAC])
3983 [`v1:real^3 = v0`; `w1:real^3 = v0`; `w2:real^3 = v0`;
3984 `w1:real^3 = v1`; `w2:real^3 = v1`] THEN
3985 ASM_CASES_TAC `w1:real^3 = w2` THEN
3986 ASM_REWRITE_TAC[AZIM_REFL; REAL_LT_REFL] THEN
3987 STRIP_TAC THEN ASM_SIMP_TAC[REAL_ARITH
3989 ==> (&0 <= x /\ x <= a <=> x = &0 \/ x = a \/ &0 < x /\ x < a)`] THEN
3990 MATCH_MP_TAC(SET_RULE
3991 `!c. c SUBSET {x | p x} /\ c SUBSET s /\
3992 ({x | ~(~c x ==> ~p x)} UNION {x | ~(~c x ==> ~q x)} UNION
3993 ({x | ~c x /\ r x} DIFF c) = s DIFF c)
3994 ==> {x | p x \/ q x \/ r x} = s`) THEN
3995 EXISTS_TAC `{x:real^3 | collinear {v0,v1,x}}` THEN
3996 ASM_SIMP_TAC[IN_ELIM_THM; AZIM_EQ_ALT; AZIM_EQ_0_ALT;
3997 GSYM wedge; WEDGE_LUNE_GT] THEN
3998 REPEAT CONJ_TAC THENL
3999 [ASM_SIMP_TAC[SUBSET; IN_ELIM_THM; AZIM_DEGENERATE];
4000 ASM_SIMP_TAC[COLLINEAR_3_AFFINE_HULL] THEN
4001 REWRITE_TAC[SET_RULE `{x | x IN s} = s`] THEN
4002 MATCH_MP_TAC AFFINE_HULL_SUBSET_AFF_GE THEN
4003 ASM_REWRITE_TAC[DISJOINT_INSERT; IN_INSERT; NOT_IN_EMPTY; DISJOINT_EMPTY];
4005 REWRITE_TAC[NOT_IMP] THEN MATCH_MP_TAC(SET_RULE
4006 `(!x. ~c x ==> (p x \/ q x \/ x IN t <=> x IN e))
4007 ==> {x | ~c x /\ p x} UNION {x | ~c x /\ q x} UNION (t DIFF {x | c x}) =
4008 e DIFF {x | c x}`) THEN
4009 X_GEN_TAC `y:real^3` THEN DISCH_TAC THEN
4010 W(MP_TAC o PART_MATCH (lhs o rand) AFF_GE_AFF_GT_DECOMP o rand o
4013 [ASM_REWRITE_TAC[FINITE_INSERT; FINITE_EMPTY] THEN
4014 ASM_REWRITE_TAC[DISJOINT_INSERT; IN_INSERT; NOT_IN_EMPTY; DISJOINT_EMPTY];
4015 DISCH_THEN SUBST1_TAC] THEN
4016 REWRITE_TAC[IN_UNION] THEN
4017 REWRITE_TAC[SIMPLE_IMAGE; IMAGE_CLAUSES; UNIONS_2] THEN
4018 ASM_SIMP_TAC[SET_RULE `~(w1 = w2) ==> {w1,w2} DELETE w1 = {w2}`;
4019 SET_RULE `~(w1 = w2) ==> {w1,w2} DELETE w2 = {w1}`] THEN
4020 REWRITE_TAC[IN_UNION; DISJ_ACI] THEN
4021 W(MP_TAC o PART_MATCH (lhs o rand) AFF_GE_AFF_GT_DECOMP o rand o lhand o
4024 [ASM_REWRITE_TAC[FINITE_INSERT; FINITE_EMPTY] THEN
4025 ASM_REWRITE_TAC[DISJOINT_INSERT; IN_INSERT; NOT_IN_EMPTY; DISJOINT_EMPTY];
4026 DISCH_THEN SUBST1_TAC] THEN
4027 W(MP_TAC o PART_MATCH (lhs o rand) AFF_GE_AFF_GT_DECOMP o rand o lhand o
4028 rand o rand o snd) THEN
4030 [ASM_REWRITE_TAC[FINITE_INSERT; FINITE_EMPTY] THEN
4031 ASM_REWRITE_TAC[DISJOINT_INSERT; IN_INSERT; NOT_IN_EMPTY; DISJOINT_EMPTY];
4032 DISCH_THEN SUBST1_TAC] THEN
4033 REWRITE_TAC[IN_UNION] THEN
4034 REWRITE_TAC[SIMPLE_IMAGE; IMAGE_CLAUSES; UNIONS_1] THEN
4035 REWRITE_TAC[SET_RULE `{a} DELETE a = {}`; AFF_GE_EQ_AFFINE_HULL] THEN
4036 ASM_MESON_TAC[COLLINEAR_3_AFFINE_HULL]);;
4038 let WEDGE_LUNE = prove
4040 ~coplanar{v0,v1,w1,w2} /\ azim v0 v1 w1 w2 < pi
4041 ==> wedge v0 v1 w1 w2 = aff_gt {v0,v1} {w1,w2}`,
4042 REPEAT STRIP_TAC THEN MATCH_MP_TAC WEDGE_LUNE_GT THEN
4043 ASM_REWRITE_TAC[] THEN REPEAT CONJ_TAC THENL
4044 [MP_TAC(ISPECL [`v0:real^3`; `v1:real^3`; `w1:real^3`; `w2:real^3`]
4045 NOT_COPLANAR_NOT_COLLINEAR) THEN
4047 MP_TAC(ISPECL [`v0:real^3`; `v1:real^3`; `w2:real^3`; `w1:real^3`]
4048 NOT_COPLANAR_NOT_COLLINEAR) THEN
4049 ONCE_REWRITE_TAC[SET_RULE `{a,b,c,d} = {a,b,d,c}`] THEN
4051 REWRITE_TAC[azim; REAL_LT_LE] THEN
4052 ASM_MESON_TAC[AZIM_EQ_0_PI_IMP_COPLANAR]]);;
4055 (`wedge v1 v2 w1 w2 =
4056 if collinear{v1,v2,w1} \/ collinear{v1,v2,w2} then {}
4061 let n = z cross u1 in
4063 if w2 IN (aff_ge {v1,v2} {w1}) then {}
4064 else if w2 IN (aff_lt {v1,v2} {w1}) then aff_gt {v1,v2,w1} {v1 + n}
4065 else if d > &0 then aff_gt {v1,v2} {w1,w2}
4066 else (:real^3) DIFF aff_ge {v1,v2} {w1,w2}`,
4067 REPEAT GEN_TAC THEN COND_CASES_TAC THENL
4068 [FIRST_X_ASSUM DISJ_CASES_TAC THEN
4069 ASM_SIMP_TAC[WEDGE_DEGENERATE];
4070 POP_ASSUM MP_TAC THEN REWRITE_TAC[DE_MORGAN_THM] THEN STRIP_TAC] THEN
4071 ASM_SIMP_TAC[GSYM AZIM_EQ_0_GE_ALT] THEN
4072 ASM_CASES_TAC `azim v1 v2 w1 w2 = &0` THENL
4073 [ASM_REWRITE_TAC[wedge] THEN
4074 ASM_REWRITE_TAC[REAL_LT_ANTISYM; LET_DEF; LET_END_DEF; EMPTY_GSPEC];
4076 ASM_SIMP_TAC[GSYM AZIM_EQ_PI_ALT] THEN
4077 ASM_CASES_TAC `azim v1 v2 w1 w2 = pi` THEN ASM_REWRITE_TAC[] THEN
4078 REWRITE_TAC[LET_DEF; LET_END_DEF] THEN
4079 POP_ASSUM_LIST(MP_TAC o end_itlist CONJ) THEN
4080 GEOM_ORIGIN_TAC `v1:real^3` THEN
4081 REWRITE_TAC[VECTOR_ADD_RID; TRANSLATION_INVARIANTS `v1:real^3`] THEN
4082 REWRITE_TAC[VECTOR_SUB_RZERO; VECTOR_ADD_LID] THEN
4083 GEOM_BASIS_MULTIPLE_TAC 3 `v2:real^3` THEN
4084 X_GEN_TAC `v2:real` THEN
4085 GEN_REWRITE_TAC LAND_CONV [REAL_ARITH `&0 <= x <=> x = &0 \/ &0 < x`] THEN
4087 [ASM_REWRITE_TAC[VECTOR_MUL_LZERO; INSERT_AC; COLLINEAR_2]; ALL_TAC]) THEN
4088 ASM_SIMP_TAC[AZIM_SPECIAL_SCALE; COLLINEAR_SPECIAL_SCALE; REAL_LT_IMP_NZ;
4089 WEDGE_SPECIAL_SCALE] THEN
4090 (REPEAT GEN_TAC THEN
4091 MAP_EVERY (fun t -> ASM_CASES_TAC t THENL
4092 [ASM_REWRITE_TAC[COLLINEAR_2; INSERT_AC] THEN NO_TAC; ALL_TAC])
4093 [`w1:real^3 = vec 0`; `w2:real^3 = vec 0`; `w1:real^3 = basis 3`;
4094 `w2:real^3 = basis 3`] THEN
4095 ASM_CASES_TAC `w1:real^3 = v2 % basis 3` THENL
4096 [ASM_REWRITE_TAC[COLLINEAR_LEMMA] THEN MESON_TAC[]; ALL_TAC] THEN
4097 ASM_CASES_TAC `w2:real^3 = v2 % basis 3` THENL
4098 [ASM_REWRITE_TAC[COLLINEAR_LEMMA] THEN MESON_TAC[]; ALL_TAC])
4100 [REPEAT STRIP_TAC THEN REWRITE_TAC[EXTENSION] THEN X_GEN_TAC `y:real^3` THEN
4101 MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC
4102 `(dropout 3 (y:real^3)) IN
4103 aff_gt {vec 0:real^2,dropout 3 (w1:real^3)}
4104 {rotate2d (pi / &2) (dropout 3 (w1:real^3))}` THEN
4106 [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE LAND_CONV [AZIM_ARG]) THEN
4107 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE (RAND_CONV o LAND_CONV)
4109 REPEAT(FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV
4110 [COLLINEAR_BASIS_3])) THEN
4111 POP_ASSUM_LIST(K ALL_TAC) THEN
4112 REWRITE_TAC[wedge; IN_ELIM_THM; AZIM_ARG; COLLINEAR_BASIS_3] THEN
4113 SPEC_TAC(`(dropout 3:real^3->real^2) y`,`x:real^2`) THEN
4114 SPEC_TAC(`(dropout 3:real^3->real^2) w2`,`v2:real^2`) THEN
4115 SPEC_TAC(`(dropout 3:real^3->real^2) w1`,`v1:real^2`) THEN
4116 GEOM_BASIS_MULTIPLE_TAC 1 `v1:complex` THEN
4117 X_GEN_TAC `v:real` THEN GEN_REWRITE_TAC LAND_CONV [REAL_LE_LT] THEN
4118 ASM_CASES_TAC `v = &0` THEN ASM_REWRITE_TAC[VECTOR_MUL_LZERO] THEN
4119 REWRITE_TAC[COMPLEX_CMUL; COMPLEX_BASIS; COMPLEX_VEC_0] THEN
4120 SIMP_TAC[ARG_DIV_CX; COMPLEX_MUL_RID] THEN
4121 REWRITE_TAC[real; RE_DIV_CX; IM_DIV_CX; CX_INJ] THEN
4122 ASM_SIMP_TAC[REAL_LT_LDIV_EQ; REAL_EQ_LDIV_EQ; REAL_MUL_LZERO] THEN
4123 REPEAT STRIP_TAC THEN REWRITE_TAC[ARG_LT_PI; ROTATE2D_PI2] THEN
4124 W(MP_TAC o PART_MATCH (lhs o rand) AFF_GT_2_1 o rand o rand o snd) THEN
4125 ASM_REWRITE_TAC[DISJOINT_INSERT; DISJOINT_EMPTY; IN_SING] THEN
4127 [CONV_TAC(ONCE_DEPTH_CONV SYM_CONV) THEN
4128 ASM_REWRITE_TAC[COMPLEX_ENTIRE; II_NZ; CX_INJ] THEN
4129 DISCH_THEN(MP_TAC o AP_TERM `Re`) THEN
4130 REWRITE_TAC[RE_MUL_II; RE_CX; IM_CX] THEN ASM_REAL_ARITH_TAC;
4131 DISCH_THEN SUBST1_TAC] THEN
4132 REWRITE_TAC[COMPLEX_CMUL; IN_ELIM_THM; COMPLEX_MUL_RZERO] THEN
4133 ONCE_REWRITE_TAC[MESON[] `(?a b c. P a b c) <=> (?b c a. P a b c)`] THEN
4134 REWRITE_TAC[REAL_ARITH `t1 + t2 = &1 <=> t1 = &1 - t2`] THEN
4135 REWRITE_TAC[RIGHT_EXISTS_AND_THM; UNWIND_THM2; COMPLEX_ADD_LID] THEN
4138 MAP_EVERY EXISTS_TAC [`Re x / v`; `Im x / v`] THEN
4139 ASM_SIMP_TAC[REAL_LT_DIV; COMPLEX_EQ; IM_ADD; RE_ADD] THEN
4140 REWRITE_TAC[RE_MUL_CX; IM_MUL_CX; RE_CX; IM_CX; RE_II; IM_II] THEN
4141 UNDISCH_TAC `~(v = &0)` THEN CONV_TAC REAL_FIELD;
4142 REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN
4143 MAP_EVERY X_GEN_TAC [`s:real`; `t:real`] THEN
4144 STRIP_TAC THEN ASM_REWRITE_TAC[COMPLEX_EQ; IM_ADD; RE_ADD] THEN
4145 REWRITE_TAC[RE_MUL_CX; IM_MUL_CX; RE_CX; IM_CX; RE_II; IM_II] THEN
4146 ASM_SIMP_TAC[REAL_MUL_RZERO; REAL_MUL_LID; REAL_LT_MUL; REAL_ADD_LID;
4147 REAL_MUL_LZERO] THEN
4148 MAP_EVERY UNDISCH_TAC [`&0 < v`; `&0 < t`] THEN
4149 CONV_TAC REAL_FIELD];
4151 W(MP_TAC o PART_MATCH (lhs o rand) AFF_GT_3_1 o rand o rand o snd) THEN
4153 [REWRITE_TAC[SET_RULE
4154 `DISJOINT {a,b,c} {x} <=> ~(x = a) /\ ~(x = b) /\ ~(x = c)`] THEN
4155 ASM_SIMP_TAC[CROSS_EQ_0; CROSS_EQ_SELF; VECTOR_MUL_EQ_0; REAL_LT_IMP_NZ;
4156 REAL_LT_IMP_NZ; BASIS_NONZERO; DIMINDEX_3;
4157 ARITH; COLLINEAR_SPECIAL_SCALE];
4158 DISCH_THEN SUBST1_TAC] THEN
4159 W(MP_TAC o PART_MATCH (lhs o rand) AFF_GT_2_1 o rand o lhand o snd) THEN
4160 REWRITE_TAC[ROTATE2D_PI2] THEN ANTS_TAC THENL
4161 [REWRITE_TAC[SET_RULE `DISJOINT {a,b} {x} <=> ~(x = a) /\ ~(x = b)`] THEN
4162 REWRITE_TAC[COMPLEX_ENTIRE; COMPLEX_RING `ii * x = x <=> x = Cx(&0)`;
4163 COMPLEX_VEC_0; II_NZ] THEN
4164 ASM_REWRITE_TAC[GSYM COMPLEX_VEC_0; GSYM COLLINEAR_BASIS_3];
4165 DISCH_THEN SUBST1_TAC] THEN
4166 REWRITE_TAC[IN_ELIM_THM; VECTOR_MUL_RZERO; VECTOR_ADD_LID] THEN
4167 ONCE_REWRITE_TAC[MESON[]
4168 `(?a b c d. P a b c d) <=> (?d c b a. P a b c d)`] THEN
4169 ONCE_REWRITE_TAC[REAL_ARITH `s + t = &1 <=> s = &1 - t`] THEN
4170 REWRITE_TAC[UNWIND_THM2; RIGHT_EXISTS_AND_THM] THEN
4171 ONCE_REWRITE_TAC[MESON[] `(?a b c. P a b c) <=> (?c b a. P a b c)`] THEN
4172 REWRITE_TAC[UNWIND_THM2; RIGHT_EXISTS_AND_THM] THEN
4173 REWRITE_TAC[RIGHT_AND_EXISTS_THM] THEN
4174 SIMP_TAC[CART_EQ; FORALL_2; FORALL_3; DIMINDEX_2; DIMINDEX_3;
4175 dropout; LAMBDA_BETA; BASIS_COMPONENT; ARITH; REAL_MUL_RID;
4176 VECTOR_MUL_COMPONENT; VEC_COMPONENT; REAL_MUL_RZERO; UNWIND_THM1;
4177 VECTOR_ADD_COMPONENT; cross; VECTOR_3;
4178 REWRITE_RULE[RE_DEF; IM_DEF] RE_MUL_II;
4179 REWRITE_RULE[RE_DEF; IM_DEF] IM_MUL_II;
4180 REAL_ADD_LID; REAL_MUL_LZERO; REAL_SUB_REFL; REAL_ADD_RID;
4181 REAL_SUB_LZERO; REAL_SUB_RZERO] THEN
4182 ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN
4183 AP_TERM_TAC THEN REWRITE_TAC[FUN_EQ_THM] THEN X_GEN_TAC `s:real` THEN
4184 REWRITE_TAC[RIGHT_EXISTS_AND_THM] THEN
4185 ASM_SIMP_TAC[EXISTS_REFL; REAL_FIELD
4186 `&0 < v ==> (x = a * v + b <=> a = (x - b) / v)`] THEN
4187 REWRITE_TAC[REAL_MUL_RNEG; REAL_MUL_ASSOC] THEN EQ_TAC THEN
4188 DISCH_THEN(X_CHOOSE_THEN `t:real` STRIP_ASSUME_TAC) THENL
4189 [EXISTS_TAC `t / v2:real`; EXISTS_TAC `t * v2:real`] THEN
4190 ASM_SIMP_TAC[REAL_DIV_RMUL; REAL_LT_DIV; REAL_LT_IMP_NZ; REAL_LT_MUL];
4192 REWRITE_TAC[CROSS_LMUL] THEN
4193 SIMP_TAC[cross; BASIS_COMPONENT; DIMINDEX_3; ARITH; DOT_3; VECTOR_3;
4194 VECTOR_MUL_COMPONENT; REAL_MUL_LZERO; REAL_SUB_RZERO; REAL_NEG_0;
4195 REAL_MUL_RZERO; REAL_SUB_LZERO; REAL_MUL_LID; REAL_ADD_RID] THEN
4196 REWRITE_TAC[REAL_ARITH
4197 `(v * --x2) * y1 + (v * x1) * y2 > &0 <=> &0 < v * (x1 * y2 - x2 * y1)`] THEN
4198 ASM_SIMP_TAC[REAL_LT_MUL_EQ; REAL_SUB_LT] THEN
4199 REWRITE_TAC[AZIM_ARG; COLLINEAR_BASIS_3] THEN STRIP_TAC THEN
4201 `w1$2 * w2$1 < w1$1 * w2$2 <=>
4202 Arg(dropout 3 (w2:real^3) / dropout 3 (w1:real^3)) < pi`
4204 [MATCH_MP_TAC EQ_TRANS THEN
4205 EXISTS_TAC `&0 < Im(dropout 3 (w2:real^3) / dropout 3 (w1:real^3))` THEN
4207 [REWRITE_TAC[IM_COMPLEX_DIV_GT_0] THEN
4208 REWRITE_TAC[complex_mul; cnj; RE_DEF; IM_DEF; complex] THEN
4209 SIMP_TAC[dropout; VECTOR_2; LAMBDA_BETA; DIMINDEX_3; ARITH;
4212 REWRITE_TAC[GSYM ARG_LT_PI] THEN ASM_MESON_TAC[ARG_LT_NZ]];
4214 COND_CASES_TAC THENL
4215 [W(MP_TAC o PART_MATCH (lhs o rand) AFF_GT_SPECIAL_SCALE o rand o snd) THEN
4216 ASM_REWRITE_TAC[FINITE_INSERT; FINITE_EMPTY; IN_INSERT; NOT_IN_EMPTY] THEN
4217 DISCH_THEN SUBST1_TAC THEN MATCH_MP_TAC WEDGE_LUNE THEN
4218 ASM_SIMP_TAC[GSYM AZIM_EQ_0_PI_EQ_COPLANAR; COLLINEAR_BASIS_3] THEN
4219 ASM_REWRITE_TAC[AZIM_ARG];
4221 REWRITE_TAC[wedge] THEN
4222 GEN_REWRITE_TAC (funpow 3 RAND_CONV) [SET_RULE `{a,b} = {b,a}`] THEN
4223 W(MP_TAC o PART_MATCH (rand o rand) WEDGE_LUNE_GE o rand o rand o snd) THEN
4224 ASM_SIMP_TAC[COLLINEAR_SPECIAL_SCALE; REAL_LT_IMP_NZ; AZIM_SPECIAL_SCALE] THEN
4225 ASM_REWRITE_TAC[AZIM_ARG; COLLINEAR_BASIS_3] THEN ANTS_TAC THENL
4226 [ASM_REWRITE_TAC[ARG_LT_NZ] THEN
4227 ONCE_REWRITE_TAC[GSYM ARG_INV_EQ_0] THEN
4228 ASM_REWRITE_TAC[COMPLEX_INV_DIV] THEN
4229 ONCE_REWRITE_TAC[GSYM COMPLEX_INV_DIV] THEN
4230 ASM_SIMP_TAC[ARG_INV; GSYM ARG_EQ_0] THEN
4233 DISCH_THEN(SUBST1_TAC o SYM) THEN
4234 REWRITE_TAC[EXTENSION; IN_DIFF; IN_UNIV; IN_ELIM_THM; ARG] THEN
4235 REWRITE_TAC[REAL_NOT_LE] THEN
4236 POP_ASSUM_LIST(MP_TAC o end_itlist CONJ) THEN
4237 SPEC_TAC(`(dropout 3:real^3->real^2) w1`,`w:complex`) THEN
4238 SPEC_TAC(`(dropout 3:real^3->real^2) w2`,`z:complex`) THEN
4239 REPEAT GEN_TAC THEN STRIP_TAC THEN X_GEN_TAC `x3:real^3` THEN
4240 SPEC_TAC(`(dropout 3:real^3->real^2) x3`,`x:complex`) THEN
4241 GEN_TAC THEN REWRITE_TAC[COMPLEX_VEC_0] THEN
4242 RULE_ASSUM_TAC(REWRITE_RULE[COMPLEX_VEC_0]) THEN
4243 ASM_CASES_TAC `x = Cx(&0)` THEN ASM_REWRITE_TAC[] THENL
4244 [ASM_REWRITE_TAC[complex_div; COMPLEX_MUL_LZERO; REAL_NOT_LT; ARG; ARG_0];
4246 ASM_REWRITE_TAC[ARG_LT_NZ] THEN
4247 MAP_EVERY UNDISCH_TAC
4248 [`~(Arg (z / w) < pi)`;
4249 `~(Arg (z / w) = pi)`;
4250 `~(Arg (z / w) = &0)`;
4253 `~(z = Cx (&0))`] THEN
4254 POP_ASSUM_LIST(K ALL_TAC) THEN REWRITE_TAC[GSYM COMPLEX_VEC_0] THEN
4255 GEOM_BASIS_MULTIPLE_TAC 1 `w:complex` THEN
4256 X_GEN_TAC `w:real` THEN GEN_REWRITE_TAC LAND_CONV [REAL_LE_LT] THEN
4257 ASM_CASES_TAC `w = &0` THEN ASM_REWRITE_TAC[VECTOR_MUL_LZERO] THEN
4258 REWRITE_TAC[COMPLEX_CMUL; COMPLEX_BASIS; COMPLEX_VEC_0] THEN
4259 SIMP_TAC[ARG_DIV_CX; COMPLEX_MUL_RID] THEN
4260 REWRITE_TAC[real; RE_DIV_CX; IM_DIV_CX; CX_INJ] THEN
4261 SIMP_TAC[complex_div; ARG_MUL_CX] THEN
4262 SIMP_TAC[ARG_INV; GSYM ARG_EQ_0; ARG_INV_EQ_0] THEN
4263 DISCH_TAC THEN REPEAT GEN_TAC THEN REWRITE_TAC[GSYM complex_div] THEN
4264 ASM_CASES_TAC `Arg x = &0` THEN ASM_REWRITE_TAC[] THENL
4265 [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [ARG_EQ_0]) THEN
4266 DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN
4267 REWRITE_TAC[REAL] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN
4268 REWRITE_TAC[complex_div; CX_INJ] THEN
4269 ASM_SIMP_TAC[ARG_MUL_CX; REAL_LT_LE] THEN
4270 ASM_SIMP_TAC[ARG_INV; GSYM ARG_EQ_0];
4272 REWRITE_TAC[IMP_IMP; GSYM CONJ_ASSOC] THEN
4273 SIMP_TAC[PI_POS; REAL_ARITH
4274 `&0 < pi ==> (~(z = &0) /\ ~(z = pi) /\ ~(z < pi) <=> pi < z)`] THEN
4275 STRIP_TAC THEN REWRITE_TAC[REAL_LT_SUB_RADD] THEN
4276 DISJ_CASES_TAC(REAL_ARITH `Arg z <= Arg x \/ Arg x < Arg z`) THENL
4277 [ASM_REWRITE_TAC[GSYM REAL_NOT_LE] THEN
4278 ONCE_REWRITE_TAC[REAL_ADD_SYM] THEN
4279 ASM_SIMP_TAC[GSYM ARG_LE_DIV_SUM] THEN
4280 SIMP_TAC[ARG; REAL_LT_IMP_LE];
4282 ASM_REWRITE_TAC[] THEN
4283 MP_TAC(ISPECL [`x:complex`; `z:complex`] ARG_LE_DIV_SUM) THEN
4284 ASM_SIMP_TAC[REAL_LT_IMP_LE] THEN DISCH_THEN SUBST1_TAC THEN
4285 MATCH_MP_TAC(REAL_ARITH
4286 `&0 <= x /\ ~(x = &0) /\ y = k - z ==> k < y + x + z`) THEN
4287 ASM_REWRITE_TAC[ARG] THEN
4288 GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [GSYM COMPLEX_INV_DIV] THEN
4289 MATCH_MP_TAC ARG_INV THEN REWRITE_TAC[REAL] THEN
4290 DISCH_THEN(STRIP_ASSUME_TAC o GSYM) THEN
4291 ABBREV_TAC `t = Re(z / x)` THEN UNDISCH_TAC `Arg x < Arg z` THEN
4292 UNDISCH_TAC `z / x = Cx t` THEN
4293 ASM_SIMP_TAC[COMPLEX_FIELD
4294 `~(x = Cx(&0)) ==> (z / x = t <=> z = t * x)`] THEN
4295 ASM_CASES_TAC `t = &0` THEN ASM_REWRITE_TAC[COMPLEX_MUL_LZERO] THEN
4296 ASM_SIMP_TAC[ARG_MUL_CX; REAL_LT_LE]);;
4298 let OPEN_WEDGE = prove
4299 (`!z:real^3 w w1 w2. open(wedge z w w1 w2)`,
4301 ASM_CASES_TAC `z:real^3 = w \/ collinear{z,w,w1} \/ collinear{z,w,w2}` THENL
4302 [FIRST_X_ASSUM STRIP_ASSUME_TAC THEN
4303 ASM_SIMP_TAC[WEDGE_DEGENERATE; OPEN_EMPTY];
4304 FIRST_X_ASSUM MP_TAC THEN REWRITE_TAC[DE_MORGAN_THM]] THEN
4305 REWRITE_TAC[wedge] THEN GEOM_ORIGIN_TAC `z:real^3` THEN
4306 GEOM_BASIS_MULTIPLE_TAC 3 `w:real^3` THEN
4307 X_GEN_TAC `w:real` THEN GEN_REWRITE_TAC LAND_CONV [REAL_LE_LT] THEN
4308 ASM_CASES_TAC `w = &0` THEN ASM_REWRITE_TAC[VECTOR_MUL_LZERO] THEN
4309 ASM_SIMP_TAC[AZIM_SPECIAL_SCALE; COLLINEAR_SPECIAL_SCALE] THEN
4310 REPEAT STRIP_TAC THEN
4311 ONCE_REWRITE_TAC[TAUT `~a /\ b /\ c <=> ~(~a ==> ~(b /\ c))`] THEN
4312 ASM_SIMP_TAC[AZIM_ARG] THEN REWRITE_TAC[COLLINEAR_BASIS_3] THEN
4313 RULE_ASSUM_TAC(REWRITE_RULE[COLLINEAR_BASIS_3]) THEN
4314 REWRITE_TAC[NOT_IMP; GSYM CONJ_ASSOC; DROPOUT_0] THEN
4315 MATCH_MP_TAC OPEN_DROPOUT_3 THEN
4316 UNDISCH_TAC `~((dropout 3:real^3->real^2) w1 = vec 0)` THEN
4317 UNDISCH_TAC `~((dropout 3:real^3->real^2) w2 = vec 0)` THEN
4318 SPEC_TAC(`(dropout 3:real^3->real^2) w2`,`v2:complex`) THEN
4319 SPEC_TAC(`(dropout 3:real^3->real^2) w1`,`v1:complex`) THEN
4320 POP_ASSUM_LIST(K ALL_TAC) THEN GEOM_BASIS_MULTIPLE_TAC 1 `v1:complex` THEN
4321 X_GEN_TAC `v1:real` THEN GEN_REWRITE_TAC LAND_CONV [REAL_LE_LT] THEN
4322 ASM_CASES_TAC `v1 = &0` THEN ASM_REWRITE_TAC[VECTOR_MUL_LZERO] THEN
4323 REWRITE_TAC[COMPLEX_CMUL; COMPLEX_BASIS; COMPLEX_VEC_0] THEN
4324 SIMP_TAC[ARG_DIV_CX; COMPLEX_MUL_RID] THEN REPEAT STRIP_TAC THEN
4325 REWRITE_TAC[SET_RULE `{x | ~(x = a) /\ P x} = {x | P x} DIFF {a}`] THEN
4326 MATCH_MP_TAC OPEN_DIFF THEN REWRITE_TAC[CLOSED_SING] THEN
4327 MATCH_MP_TAC OPEN_ARG_LTT THEN
4328 SIMP_TAC[REAL_LT_IMP_LE; REAL_LE_REFL; ARG]);;
4330 let ARG_EQ_SUBSET_HALFLINE = prove
4331 (`!a. ?b. ~(b = vec 0) /\ {z | Arg z = a} SUBSET aff_ge {vec 0} {b}`,
4332 GEN_TAC THEN ASM_CASES_TAC `{z | Arg z = a} SUBSET {vec 0}` THENL
4333 [EXISTS_TAC `basis 1:real^2` THEN
4334 SIMP_TAC[BASIS_NONZERO; DIMINDEX_2; ARITH] THEN
4335 FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ]
4336 SUBSET_TRANS)) THEN SIMP_TAC[SUBSET; IN_SING; ENDS_IN_HALFLINE];
4338 FIRST_X_ASSUM(MP_TAC o MATCH_MP (SET_RULE
4339 `~(s SUBSET {a}) ==> ?z. ~(a = z) /\ z IN s`)) THEN
4340 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `z:complex` THEN
4341 REWRITE_TAC[IN_ELIM_THM] THEN DISCH_THEN(STRIP_ASSUME_TAC o GSYM) THEN
4342 ASM_REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN
4343 X_GEN_TAC `x:complex` THEN
4344 ASM_CASES_TAC `x:complex = vec 0` THEN ASM_REWRITE_TAC[ENDS_IN_HALFLINE] THEN
4345 RULE_ASSUM_TAC(REWRITE_RULE[COMPLEX_VEC_0]) THEN ASM_SIMP_TAC[ARG_EQ] THEN
4346 DISCH_THEN(X_CHOOSE_THEN `u:real` STRIP_ASSUME_TAC) THEN
4347 ASM_REWRITE_TAC[GSYM COMPLEX_CMUL] THEN
4348 REWRITE_TAC[HALFLINE_EXPLICIT; IN_ELIM_THM; VECTOR_MUL_RZERO] THEN
4349 MAP_EVERY EXISTS_TAC [`&1 - u`; `u:real`] THEN
4350 ASM_SIMP_TAC[VECTOR_ADD_LID; REAL_LT_IMP_LE] THEN ASM_REAL_ARITH_TAC);;
4352 let ARG_DIV_EQ_SUBSET_HALFLINE = prove
4353 (`!w a. ~(w = vec 0)
4354 ==> ?b. ~(b = vec 0) /\
4355 {z | Arg(z / w) = a} SUBSET aff_ge {vec 0} {b}`,
4356 REPEAT GEN_TAC THEN GEOM_BASIS_MULTIPLE_TAC 1 `w:complex` THEN
4357 X_GEN_TAC `w:real` THEN ASM_CASES_TAC `w = &0` THEN
4358 ASM_REWRITE_TAC[VECTOR_MUL_LZERO; REAL_LE_LT] THEN DISCH_TAC THEN
4359 X_GEN_TAC `a:real` THEN DISCH_THEN(K ALL_TAC) THEN
4360 ASM_SIMP_TAC[ARG_DIV_CX; COMPLEX_CMUL; COMPLEX_BASIS; GSYM CX_MUL;
4361 REAL_MUL_RID; ARG_EQ_SUBSET_HALFLINE]);;
4363 let COPLANAR_AZIM_EQ = prove
4365 (collinear{v0,v1,w1} ==> ~(a = &0))
4366 ==> coplanar {z | azim v0 v1 w1 z = a}`,
4367 REPEAT GEN_TAC THEN ASM_CASES_TAC `collinear{v0:real^3,v1,w1}` THENL
4368 [ASM_SIMP_TAC[azim_def; EMPTY_GSPEC; COPLANAR_EMPTY]; ALL_TAC] THEN
4369 ASM_REWRITE_TAC[] THEN POP_ASSUM MP_TAC THEN
4370 GEOM_ORIGIN_TAC `v0:real^3` THEN
4371 GEOM_BASIS_MULTIPLE_TAC 3 `v1:real^3` THEN
4372 X_GEN_TAC `v1:real` THEN ASM_CASES_TAC `v1 = &0` THENL
4373 [ASM_REWRITE_TAC[VECTOR_MUL_LZERO; INSERT_AC; COLLINEAR_2]; ALL_TAC] THEN
4374 ASM_SIMP_TAC[REAL_LE_LT; COLLINEAR_SPECIAL_SCALE] THEN REPEAT STRIP_TAC THEN
4375 ASM_SIMP_TAC[AZIM_SPECIAL_SCALE; AZIM_ARG] THEN
4376 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [COLLINEAR_BASIS_3]) THEN
4377 POP_ASSUM_LIST(K ALL_TAC) THEN DISCH_THEN(X_CHOOSE_THEN `b:real^2`
4378 STRIP_ASSUME_TAC o SPEC `a:real` o MATCH_MP ARG_DIV_EQ_SUBSET_HALFLINE) THEN
4379 REWRITE_TAC[coplanar] THEN MAP_EVERY EXISTS_TAC
4380 [`vec 0:real^3`; `pushin 3 (&0) (b:real^2):real^3`; `basis 3:real^3`] THEN
4381 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [SUBSET]) THEN
4382 REWRITE_TAC[AFFINE_HULL_3; HALFLINE; SUBSET; IN_ELIM_THM] THEN
4383 DISCH_THEN(fun th -> X_GEN_TAC `x:real^3` THEN DISCH_TAC THEN
4384 MP_TAC(SPEC `(dropout 3:real^3->real^2) x` th)) THEN
4385 ASM_REWRITE_TAC[VECTOR_MUL_RZERO; VECTOR_ADD_LID] THEN
4386 DISCH_THEN(X_CHOOSE_THEN `v:real` STRIP_ASSUME_TAC) THEN
4387 MAP_EVERY EXISTS_TAC [`&1 - v - (x:real^3)$3`; `v:real`; `(x:real^3)$3`] THEN
4388 CONJ_TAC THENL [REAL_ARITH_TAC; ALL_TAC] THEN
4389 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [CART_EQ]) THEN
4390 SIMP_TAC[CART_EQ; DIMINDEX_2; DIMINDEX_3; FORALL_2; FORALL_3; LAMBDA_BETA;
4391 dropout; pushin; VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT; ARITH;
4392 BASIS_COMPONENT] THEN
4395 (* ------------------------------------------------------------------------- *)
4396 (* Volume of a tetrahedron defined by conv0. *)
4397 (* ------------------------------------------------------------------------- *)
4399 let delta_x = new_definition
4400 `delta_x x1 x2 x3 x4 x5 x6 =
4401 x1*x4*(--x1 + x2 + x3 -x4 + x5 + x6) +
4402 x2*x5*(x1 - x2 + x3 + x4 -x5 + x6) +
4403 x3*x6*(x1 + x2 - x3 + x4 + x5 - x6)
4404 -x2*x3*x4 - x1*x3*x5 - x1*x2*x6 -x4*x5*x6:real`;;
4406 let VOLUME_OF_CLOSED_TETRAHEDRON = prove
4407 (`!x1 x2 x3 x4:real^3.
4408 measure(convex hull {x1,x2,x3,x4}) =
4409 sqrt(delta_x (dist(x1,x2) pow 2) (dist(x1,x3) pow 2) (dist(x1,x4) pow 2)
4410 (dist(x3,x4) pow 2) (dist(x2,x4) pow 2) (dist(x2,x3) pow 2))
4412 REPEAT GEN_TAC THEN REWRITE_TAC[LET_DEF; LET_END_DEF] THEN
4413 REWRITE_TAC[MEASURE_TETRAHEDRON] THEN
4414 REWRITE_TAC[REAL_ARITH `x / &6 = y / &12 <=> y = &2 * x`] THEN
4415 MATCH_MP_TAC SQRT_UNIQUE THEN
4416 SIMP_TAC[REAL_LE_MUL; REAL_ABS_POS; REAL_POS] THEN
4417 REWRITE_TAC[REAL_POW_MUL; REAL_POW2_ABS; delta_x] THEN
4418 REWRITE_TAC[dist; NORM_POW_2] THEN
4419 SIMP_TAC[DOT_3; VECTOR_SUB_COMPONENT; DIMINDEX_3; ARITH] THEN
4420 CONV_TAC REAL_RING);;
4422 let VOLUME_OF_TETRAHEDRON = prove
4423 (`!v1 v2 v3 v4:real^3.
4424 measure(conv0 {v1,v2,v3,v4}) =
4425 let x12 = dist(v1,v2) pow 2 in
4426 let x13 = dist(v1,v3) pow 2 in
4427 let x14 = dist(v1,v4) pow 2 in
4428 let x23 = dist(v2,v3) pow 2 in
4429 let x24 = dist(v2,v4) pow 2 in
4430 let x34 = dist(v3,v4) pow 2 in
4431 sqrt(delta_x x12 x13 x14 x34 x24 x23)/(&12)`,
4432 REPEAT GEN_TAC THEN CONV_TAC(TOP_DEPTH_CONV let_CONV) THEN
4433 ASM_SIMP_TAC[GSYM VOLUME_OF_CLOSED_TETRAHEDRON] THEN
4434 MATCH_MP_TAC MEASURE_CONV0_CONVEX_HULL THEN
4435 SIMP_TAC[DIMINDEX_3; FINITE_INSERT; FINITE_EMPTY; CARD_CLAUSES] THEN
4438 (* ------------------------------------------------------------------------- *)
4439 (* Circle area. Should maybe extend WLOG tactics for such scaling. *)
4440 (* ------------------------------------------------------------------------- *)
4442 let AREA_UNIT_CBALL = prove
4443 (`measure(cball(vec 0:real^2,&1)) = pi`,
4444 REPEAT STRIP_TAC THEN
4445 MATCH_MP_TAC(INST_TYPE[`:1`,`:M`; `:2`,`:N`] FUBINI_SIMPLE_COMPACT) THEN
4447 SIMP_TAC[DIMINDEX_1; DIMINDEX_2; ARITH; COMPACT_CBALL; SLICE_CBALL] THEN
4448 REWRITE_TAC[VEC_COMPONENT; DROPOUT_0; REAL_SUB_RZERO] THEN
4449 ONCE_REWRITE_TAC[COND_RAND] THEN REWRITE_TAC[MEASURE_EMPTY] THEN
4450 SUBGOAL_THEN `!t. abs(t) <= &1 <=> t IN real_interval[-- &1,&1]`
4451 (fun th -> REWRITE_TAC[th])
4452 THENL [REWRITE_TAC[IN_REAL_INTERVAL] THEN REAL_ARITH_TAC; ALL_TAC] THEN
4453 REWRITE_TAC[HAS_REAL_INTEGRAL_RESTRICT_UNIV; BALL_1] THEN
4454 MATCH_MP_TAC HAS_REAL_INTEGRAL_EQ THEN
4455 EXISTS_TAC `\t. &2 * sqrt(&1 - t pow 2)` THEN CONJ_TAC THENL
4456 [X_GEN_TAC `t:real` THEN SIMP_TAC[IN_REAL_INTERVAL; MEASURE_INTERVAL] THEN
4457 REWRITE_TAC[REAL_BOUNDS_LE; VECTOR_ADD_LID; VECTOR_SUB_LZERO] THEN
4459 W(MP_TAC o PART_MATCH (lhs o rand) CONTENT_1 o rand o snd) THEN
4460 REWRITE_TAC[LIFT_DROP; DROP_NEG] THEN
4461 ANTS_TAC THENL [ALL_TAC; SIMP_TAC[REAL_POW_ONE] THEN REAL_ARITH_TAC] THEN
4462 MATCH_MP_TAC(REAL_ARITH `&0 <= x ==> --x <= x`) THEN
4463 ASM_SIMP_TAC[SQRT_POS_LE; REAL_SUB_LE; GSYM REAL_LE_SQUARE_ABS;
4467 [`\x. asn(x) + x * sqrt(&1 - x pow 2)`;
4468 `\x. &2 * sqrt(&1 - x pow 2)`;
4469 `-- &1`; `&1`] REAL_FUNDAMENTAL_THEOREM_OF_CALCULUS_INTERIOR) THEN
4470 REWRITE_TAC[ASN_1; ASN_NEG_1] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN
4471 REWRITE_TAC[SQRT_0; REAL_MUL_RZERO; REAL_ADD_RID] THEN
4472 REWRITE_TAC[REAL_ARITH `x / &2 - --(x / &2) = x`] THEN
4473 DISCH_THEN MATCH_MP_TAC THEN CONJ_TAC THENL
4474 [MATCH_MP_TAC REAL_CONTINUOUS_ON_ADD THEN
4475 SIMP_TAC[REAL_CONTINUOUS_ON_ASN; IN_REAL_INTERVAL; REAL_BOUNDS_LE] THEN
4476 MATCH_MP_TAC REAL_CONTINUOUS_ON_MUL THEN
4477 REWRITE_TAC[REAL_CONTINUOUS_ON_ID] THEN
4478 GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN
4479 MATCH_MP_TAC REAL_CONTINUOUS_ON_COMPOSE THEN
4480 SIMP_TAC[REAL_CONTINUOUS_ON_SUB; REAL_CONTINUOUS_ON_POW;
4481 REAL_CONTINUOUS_ON_ID; REAL_CONTINUOUS_ON_CONST] THEN
4482 MATCH_MP_TAC REAL_CONTINUOUS_ON_SQRT THEN
4483 REWRITE_TAC[FORALL_IN_IMAGE; IN_REAL_INTERVAL] THEN
4484 REWRITE_TAC[REAL_ARITH `&0 <= &1 - x <=> x <= &1 pow 2`] THEN
4485 REWRITE_TAC[GSYM REAL_LE_SQUARE_ABS; REAL_ABS_NUM] THEN
4487 REWRITE_TAC[IN_REAL_INTERVAL; REAL_BOUNDS_LT] THEN REPEAT STRIP_TAC THEN
4489 CONV_TAC NUM_REDUCE_CONV THEN
4490 REWRITE_TAC[REAL_MUL_LID; REAL_POW_1; REAL_MUL_RID] THEN
4491 REWRITE_TAC[REAL_SUB_LZERO; REAL_MUL_RNEG; REAL_INV_MUL] THEN
4492 ASM_REWRITE_TAC[REAL_SUB_LT; ABS_SQUARE_LT_1] THEN
4493 MATCH_MP_TAC(REAL_FIELD
4494 `s pow 2 = &1 - x pow 2 /\ x pow 2 < &1
4495 ==> (inv s + x * --(&2 * x) * inv (&2) * inv s + s) = &2 * s`) THEN
4496 ASM_SIMP_TAC[ABS_SQUARE_LT_1; SQRT_POW_2; REAL_SUB_LE; REAL_LT_IMP_LE]]);;
4498 let AREA_CBALL = prove
4499 (`!z:real^2 r. &0 <= r ==> measure(cball(z,r)) = pi * r pow 2`,
4500 REPEAT STRIP_TAC THEN ASM_CASES_TAC `r = &0` THENL
4501 [ASM_SIMP_TAC[CBALL_SING; REAL_POW_2; REAL_MUL_RZERO] THEN
4502 MATCH_MP_TAC MEASURE_UNIQUE THEN
4503 REWRITE_TAC[HAS_MEASURE_0; NEGLIGIBLE_SING];
4505 SUBGOAL_THEN `&0 < r` ASSUME_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
4506 MP_TAC(ISPECL [`cball(vec 0:real^2,&1)`; `r:real`; `z:real^2`; `pi`]
4507 HAS_MEASURE_AFFINITY) THEN
4508 REWRITE_TAC[HAS_MEASURE_MEASURABLE_MEASURE; MEASURABLE_CBALL;
4509 AREA_UNIT_CBALL] THEN
4510 ASM_REWRITE_TAC[real_abs; DIMINDEX_2] THEN
4511 DISCH_THEN(MP_TAC o CONJUNCT2) THEN
4512 GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [REAL_MUL_SYM] THEN
4513 DISCH_THEN(SUBST1_TAC o SYM) THEN AP_TERM_TAC THEN
4514 MATCH_MP_TAC SUBSET_ANTISYM THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN
4515 REWRITE_TAC[IN_CBALL_0; IN_IMAGE] THEN REWRITE_TAC[IN_CBALL] THEN
4516 REWRITE_TAC[NORM_ARITH `dist(z,a + z) = norm a`; NORM_MUL] THEN
4517 ONCE_REWRITE_TAC[REAL_ARITH `abs r * x <= r <=> abs r * x <= r * &1`] THEN
4518 ASM_SIMP_TAC[real_abs; REAL_LE_LMUL; dist] THEN X_GEN_TAC `w:real^2` THEN
4519 DISCH_TAC THEN EXISTS_TAC `inv(r) % (w - z):real^2` THEN
4520 ASM_SIMP_TAC[VECTOR_MUL_ASSOC; REAL_MUL_RINV] THEN
4521 CONJ_TAC THENL [NORM_ARITH_TAC; ALL_TAC] THEN
4522 REWRITE_TAC[NORM_MUL; REAL_ABS_INV] THEN ASM_REWRITE_TAC[real_abs] THEN
4523 ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN
4524 ASM_SIMP_TAC[GSYM real_div; REAL_LE_LDIV_EQ; REAL_MUL_LID] THEN
4525 ONCE_REWRITE_TAC[NORM_SUB] THEN ASM_REWRITE_TAC[]);;
4527 let AREA_BALL = prove
4528 (`!z:real^2 r. &0 <= r ==> measure(ball(z,r)) = pi * r pow 2`,
4529 SIMP_TAC[GSYM INTERIOR_CBALL; GSYM AREA_CBALL] THEN
4530 REPEAT STRIP_TAC THEN MATCH_MP_TAC MEASURE_INTERIOR THEN
4531 SIMP_TAC[BOUNDED_CBALL; NEGLIGIBLE_CONVEX_FRONTIER; CONVEX_CBALL]);;
4533 (* ------------------------------------------------------------------------- *)
4534 (* Volume of a ball. *)
4535 (* ------------------------------------------------------------------------- *)
4537 let VOLUME_CBALL = prove
4538 (`!z:real^3 r. &0 <= r ==> measure(cball(z,r)) = &4 / &3 * pi * r pow 3`,
4539 GEOM_ORIGIN_TAC `z:real^3` THEN REPEAT STRIP_TAC THEN
4540 MATCH_MP_TAC(INST_TYPE[`:2`,`:M`; `:3`,`:N`] FUBINI_SIMPLE_COMPACT) THEN
4542 SIMP_TAC[DIMINDEX_2; DIMINDEX_3; ARITH; COMPACT_CBALL; SLICE_CBALL] THEN
4543 REWRITE_TAC[VEC_COMPONENT; DROPOUT_0; REAL_SUB_RZERO] THEN
4544 ONCE_REWRITE_TAC[COND_RAND] THEN REWRITE_TAC[MEASURE_EMPTY] THEN
4545 SUBGOAL_THEN `!t. abs(t) <= r <=> t IN real_interval[--r,r]`
4546 (fun th -> REWRITE_TAC[th])
4547 THENL [REWRITE_TAC[IN_REAL_INTERVAL] THEN REAL_ARITH_TAC; ALL_TAC] THEN
4548 REWRITE_TAC[HAS_REAL_INTEGRAL_RESTRICT_UNIV] THEN
4549 MATCH_MP_TAC HAS_REAL_INTEGRAL_EQ THEN
4550 EXISTS_TAC `\t. pi * (r pow 2 - t pow 2)` THEN CONJ_TAC THENL
4551 [X_GEN_TAC `t:real` THEN REWRITE_TAC[IN_REAL_INTERVAL; REAL_BOUNDS_LE] THEN
4552 SIMP_TAC[AREA_CBALL; SQRT_POS_LE; REAL_SUB_LE; GSYM REAL_LE_SQUARE_ABS;
4553 SQRT_POW_2; REAL_ARITH `abs x <= r ==> abs x <= abs r`];
4556 [`\t. pi * (r pow 2 * t - &1 / &3 * t pow 3)`;
4557 `\t. pi * (r pow 2 - t pow 2)`;
4558 `--r:real`; `r:real`] REAL_FUNDAMENTAL_THEOREM_OF_CALCULUS) THEN
4559 REWRITE_TAC[] THEN ANTS_TAC THENL
4560 [CONJ_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
4561 REPEAT STRIP_TAC THEN REAL_DIFF_TAC THEN
4562 CONV_TAC NUM_REDUCE_CONV THEN CONV_TAC REAL_RING;
4563 MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN
4564 CONV_TAC REAL_RING]);;
4566 let VOLUME_BALL = prove
4567 (`!z:real^3 r. &0 <= r ==> measure(ball(z,r)) = &4 / &3 * pi * r pow 3`,
4568 SIMP_TAC[GSYM INTERIOR_CBALL; GSYM VOLUME_CBALL] THEN
4569 REPEAT STRIP_TAC THEN MATCH_MP_TAC MEASURE_INTERIOR THEN
4570 SIMP_TAC[BOUNDED_CBALL; NEGLIGIBLE_CONVEX_FRONTIER; CONVEX_CBALL]);;
4572 (* ------------------------------------------------------------------------- *)
4574 (* ------------------------------------------------------------------------- *)
4576 let rconesgn = new_definition
4577 `rconesgn sgn v w h =
4578 {x:real^A | sgn ((x-v) dot (w-v)) (dist(x,v)*dist(w,v)*h)}`;;
4580 let rcone_gt = new_definition `rcone_gt = rconesgn ( > )`;;
4582 let rcone_ge = new_definition `rcone_ge = rconesgn ( >= )`;;
4584 let rcone_eq = new_definition `rcone_eq = rconesgn ( = )`;;
4586 let frustum = new_definition
4587 `frustum v0 v1 h1 h2 a =
4588 { y:real^N | rcone_gt v0 v1 a y /\
4589 let d = (y - v0) dot (v1 - v0) in
4590 let n = norm(v1 - v0) in
4591 (h1*n < d /\ d < h2*n)}`;;
4593 let frustt = new_definition `frustt v0 v1 h a = frustum v0 v1 (&0) h a`;;
4595 let FRUSTUM_DEGENERATE = prove
4596 (`!v0 h1 h2 a. frustum v0 v0 h1 h2 a = {}`,
4597 REWRITE_TAC[frustum; VECTOR_SUB_REFL; NORM_0; DOT_RZERO] THEN
4598 CONV_TAC(TOP_DEPTH_CONV let_CONV) THEN
4599 REWRITE_TAC[REAL_MUL_RZERO; REAL_LT_REFL] THEN SET_TAC[]);;
4601 let CONVEX_RCONE_GT = prove
4602 (`!v0 v1:real^N a. &0 <= a ==> convex(rcone_gt v0 v1 a)`,
4603 REWRITE_TAC[rcone_gt; rconesgn] THEN
4604 GEOM_ORIGIN_TAC `v0:real^N` THEN REPEAT GEN_TAC THEN
4605 REWRITE_TAC[VECTOR_SUB_RZERO; DIST_0] THEN
4606 REWRITE_TAC[CONVEX_ALT; IN_ELIM_THM; real_gt; DOT_LADD; DOT_LMUL] THEN
4607 DISCH_TAC THEN MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`; `t:real`] THEN
4608 REPEAT STRIP_TAC THEN MATCH_MP_TAC REAL_LET_TRANS THEN
4609 EXISTS_TAC `(&1 - t) * norm(x:real^N) * norm v1 * a +
4610 t * norm(y:real^N) * norm(v1:real^N) * a` THEN
4612 [REWRITE_TAC[GSYM REAL_ADD_RDISTRIB; REAL_MUL_ASSOC] THEN
4613 MATCH_MP_TAC REAL_LE_RMUL THEN ASM_REWRITE_TAC[] THEN
4614 MATCH_MP_TAC REAL_LE_RMUL THEN REWRITE_TAC[NORM_POS_LE] THEN
4615 MATCH_MP_TAC(NORM_ARITH
4616 `norm(x:real^N) = a /\ norm(y) = b ==> norm(x + y) <= a + b`) THEN
4617 REWRITE_TAC[NORM_MUL] THEN CONJ_TAC THEN AP_THM_TAC THEN AP_TERM_TAC THEN
4619 MATCH_MP_TAC REAL_CONVEX_BOUND2_LT THEN ASM_REAL_ARITH_TAC]);;
4621 let OPEN_RCONE_GT = prove
4622 (`!v0 v1:real^N a. open(rcone_gt v0 v1 a)`,
4623 REWRITE_TAC[rcone_gt; rconesgn] THEN
4624 GEOM_ORIGIN_TAC `v0:real^N` THEN REPEAT GEN_TAC THEN
4625 REWRITE_TAC[VECTOR_SUB_RZERO; DIST_0] THEN
4626 MP_TAC(ISPECL [`\x:real^N. lift(x dot v1 - norm x * norm v1 * a)`;
4627 `{x:real^1 | x$1 > &0}`]
4628 CONTINUOUS_OPEN_PREIMAGE_UNIV) THEN
4629 REWRITE_TAC[OPEN_HALFSPACE_COMPONENT_GT] THEN REWRITE_TAC[GSYM drop] THEN
4630 REWRITE_TAC[IN_ELIM_THM; real_gt; REAL_SUB_LT; LIFT_DROP] THEN
4631 DISCH_THEN MATCH_MP_TAC THEN GEN_TAC THEN REWRITE_TAC[LIFT_SUB] THEN
4632 ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN REWRITE_TAC[LIFT_CMUL] THEN
4633 MATCH_MP_TAC CONTINUOUS_SUB THEN ONCE_REWRITE_TAC[DOT_SYM] THEN
4634 REWRITE_TAC[REWRITE_RULE[o_DEF] CONTINUOUS_AT_LIFT_DOT] THEN
4635 MATCH_MP_TAC CONTINUOUS_CMUL THEN
4636 REWRITE_TAC[REWRITE_RULE[o_DEF] CONTINUOUS_AT_LIFT_NORM]);;
4638 let RCONE_GT_NEG = prove
4640 rcone_gt v0 v1 (--a) =
4641 IMAGE (\x. &2 % v0 - x) ((:real^N) DIFF rcone_ge v0 v1 a)`,
4642 REPEAT GEN_TAC THEN CONV_TAC SYM_CONV THEN
4643 MATCH_MP_TAC SURJECTIVE_IMAGE_EQ THEN REWRITE_TAC[] THEN CONJ_TAC THENL
4644 [MESON_TAC[VECTOR_ARITH `a - (a - b):real^N = b`];
4645 REWRITE_TAC[rcone_gt; rconesgn; rcone_ge;
4646 IN_ELIM_THM; IN_DIFF; IN_UNIV] THEN
4647 REWRITE_TAC[NORM_ARITH `dist(&2 % x - y,x) = dist(y,x)`] THEN
4648 REWRITE_TAC[VECTOR_ARITH `&2 % v - x - v:real^N = --(x - v)`] THEN
4649 REWRITE_TAC[DOT_LNEG] THEN REAL_ARITH_TAC]);;
4651 let VOLUME_FRUSTT_STRONG = prove
4652 (`!v0 v1:real^3 h a.
4654 ==> bounded(frustt v0 v1 h a) /\
4655 convex(frustt v0 v1 h a) /\
4656 measurable(frustt v0 v1 h a) /\
4657 measure(frustt v0 v1 h a) =
4658 if v1 = v0 \/ &1 <= a \/ h < &0 then &0
4659 else pi * ((h / a) pow 2 - h pow 2) * h / &3`,
4660 REPEAT GEN_TAC THEN DISCH_TAC THEN
4661 REWRITE_TAC[frustt; frustum; rcone_gt; rconesgn; IN_ELIM_THM] THEN
4662 CONV_TAC(TOP_DEPTH_CONV let_CONV) THEN GEOM_ORIGIN_TAC `v0:real^3` THEN
4663 REWRITE_TAC[VECTOR_SUB_RZERO; REAL_MUL_LZERO; DIST_0; real_gt] THEN
4664 GEOM_BASIS_MULTIPLE_TAC 1 `v1:real^3` THEN
4665 X_GEN_TAC `b:real` THEN REPEAT(GEN_TAC ORELSE DISCH_TAC) THEN
4666 FIRST_ASSUM(DISJ_CASES_TAC o MATCH_MP (REAL_ARITH
4667 `&0 <= x ==> x = &0 \/ &0 < x`)) THEN
4668 ASM_REWRITE_TAC[DOT_RZERO; REAL_MUL_LZERO; REAL_MUL_RZERO; REAL_LT_REFL;
4669 MEASURABLE_EMPTY; MEASURE_EMPTY; EMPTY_GSPEC; VECTOR_MUL_LZERO;
4670 BOUNDED_EMPTY; CONVEX_EMPTY] THEN
4671 ASM_CASES_TAC `&1 <= a` THEN ASM_REWRITE_TAC[] THENL
4673 `!y:real^3. ~(norm(y) * norm(b % basis 1:real^3) * a
4674 < y dot (b % basis 1))`
4675 (fun th -> REWRITE_TAC[th; EMPTY_GSPEC; MEASURABLE_EMPTY;
4676 BOUNDED_EMPTY; CONVEX_EMPTY; MEASURE_EMPTY]) THEN
4677 REWRITE_TAC[REAL_NOT_LT] THEN X_GEN_TAC `y:real^3` THEN
4678 MATCH_MP_TAC(REAL_ARITH `abs(x) <= a ==> x <= a`) THEN
4679 SIMP_TAC[DOT_RMUL; NORM_MUL; REAL_ABS_MUL; DOT_BASIS; NORM_BASIS;
4680 DIMINDEX_3; ARITH] THEN
4681 REWRITE_TAC[REAL_ARITH
4682 `b * y <= n * (b * &1) * a <=> b * &1 * y <= b * a * n`] THEN
4683 MATCH_MP_TAC REAL_LE_LMUL THEN REWRITE_TAC[REAL_ABS_POS] THEN
4684 MATCH_MP_TAC REAL_LE_MUL2 THEN
4685 ASM_SIMP_TAC[REAL_POS; REAL_ABS_POS; COMPONENT_LE_NORM; DIMINDEX_3; ARITH];
4687 RULE_ASSUM_TAC(REWRITE_RULE[REAL_NOT_LE]) THEN
4688 SIMP_TAC[NORM_MUL; NORM_BASIS; DOT_BASIS; DOT_RMUL; DIMINDEX_3; ARITH] THEN
4689 ONCE_REWRITE_TAC[REAL_ARITH `n * x * a:real = x * n * a`] THEN
4690 ASM_REWRITE_TAC[real_abs; REAL_MUL_RID] THEN
4691 ASM_SIMP_TAC[REAL_MUL_RID; REAL_LT_LMUL_EQ; REAL_LT_MUL_EQ; NORM_POS_LT] THEN
4692 ASM_SIMP_TAC[VECTOR_MUL_EQ_0; BASIS_NONZERO; DIMINDEX_3; ARITH;
4693 REAL_LT_IMP_NZ] THEN
4694 ASM_SIMP_TAC[GSYM REAL_LT_RDIV_EQ; NORM_LT_SQUARE] THEN
4695 ASM_SIMP_TAC[REAL_POW_DIV; REAL_POW_LT; REAL_LT_RDIV_EQ] THEN
4696 REWRITE_TAC[REAL_ARITH `(&0 * x < y /\ u < v) /\ &0 < y /\ y < h <=>
4697 &0 < y /\ y < h /\ u < v`] THEN
4698 MATCH_MP_TAC(TAUT `a /\ b /\ (a /\ b ==> c) ==> a /\ b /\ c`) THEN
4699 REPEAT CONJ_TAC THENL
4700 [MATCH_MP_TAC BOUNDED_SUBSET THEN
4701 EXISTS_TAC `ball(vec 0:real^3,h / a)` THEN
4702 REWRITE_TAC[BOUNDED_BALL; IN_BALL_0; SUBSET; IN_ELIM_THM] THEN
4703 REWRITE_TAC[NORM_LT_SQUARE] THEN
4704 ASM_SIMP_TAC[REAL_POW_DIV; REAL_LT_RDIV_EQ; REAL_POW_LT] THEN
4705 X_GEN_TAC `x:real^3` THEN STRIP_TAC THEN
4706 CONJ_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
4707 FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP
4708 (REWRITE_RULE[IMP_CONJ] REAL_LTE_TRANS)) THEN
4709 MATCH_MP_TAC REAL_POW_LE2 THEN ASM_REAL_ARITH_TAC;
4710 REWRITE_TAC[SET_RULE `{x | P x /\ Q x /\ R x} =
4711 {x | Q x} INTER {x | P x /\ R x}`] THEN
4712 REWRITE_TAC[REAL_ARITH `&0 < y <=> y > &0`] THEN
4713 MATCH_MP_TAC CONVEX_INTER THEN
4714 REWRITE_TAC[CONVEX_HALFSPACE_COMPONENT_LT] THEN
4715 MP_TAC(ISPECL [`vec 0:real^3`; `basis 1:real^3`; `a:real`]
4716 CONVEX_RCONE_GT) THEN
4717 ASM_SIMP_TAC[REAL_LT_IMP_LE; rcone_gt; rconesgn] THEN
4718 REWRITE_TAC[VECTOR_SUB_RZERO; DIST_0] THEN
4719 SIMP_TAC[DOT_BASIS; NORM_BASIS; DIMINDEX_3; ARITH] THEN
4720 REWRITE_TAC[real_gt; REAL_MUL_LID] THEN
4721 ASM_SIMP_TAC[GSYM REAL_LT_RDIV_EQ] THEN
4722 REWRITE_TAC[NORM_LT_SQUARE] THEN
4723 ASM_SIMP_TAC[REAL_POW_DIV; REAL_LT_RDIV_EQ; REAL_POW_LT] THEN
4724 REWRITE_TAC[REAL_MUL_LZERO];
4727 MATCH_MP_TAC(INST_TYPE [`:2`,`:M`] FUBINI_SIMPLE_CONVEX_STRONG) THEN
4728 EXISTS_TAC `1` THEN REWRITE_TAC[DIMINDEX_2; DIMINDEX_3; ARITH] THEN
4729 ASM_REWRITE_TAC[] THEN
4730 SIMP_TAC[SLICE_312; DIMINDEX_2; DIMINDEX_3; ARITH; IN_ELIM_THM;
4731 VECTOR_3; DOT_3; GSYM DOT_2] THEN
4732 SUBGOAL_THEN `&0 < inv(a pow 2) - &1` ASSUME_TAC THENL
4733 [REWRITE_TAC[REAL_SUB_LT] THEN MATCH_MP_TAC REAL_INV_1_LT THEN
4734 ASM_SIMP_TAC[REAL_POW_1_LT; REAL_LT_IMP_LE; ARITH; REAL_POW_LT];
4736 MATCH_MP_TAC HAS_REAL_INTEGRAL_EQ THEN
4737 EXISTS_TAC `\t. if &0 < t /\ t < h then pi * (inv(a pow 2) - &1) * t pow 2
4740 [X_GEN_TAC `t:real` THEN DISCH_TAC THEN REWRITE_TAC[] THEN
4742 ASM_REWRITE_TAC[EMPTY_GSPEC; CONJ_ASSOC;
4743 MEASURE_EMPTY; MEASURABLE_EMPTY] THEN
4744 MATCH_MP_TAC EQ_TRANS THEN
4745 EXISTS_TAC `measure(ball(vec 0:real^2,sqrt(inv(a pow 2) - &1) * t))` THEN
4747 [W(MP_TAC o PART_MATCH (lhs o rand) AREA_BALL o rand o snd) THEN
4748 ASM_SIMP_TAC[REAL_LT_IMP_LE; SQRT_POS_LT; REAL_LT_MUL] THEN
4749 ASM_SIMP_TAC[SQRT_POW_2; REAL_LT_IMP_LE; REAL_POW_MUL];
4750 AP_TERM_TAC THEN REWRITE_TAC[IN_BALL_0; EXTENSION; IN_ELIM_THM] THEN
4751 REWRITE_TAC[NORM_LT_SQUARE] THEN
4752 ASM_SIMP_TAC[SQRT_POS_LT; SQRT_POW_2; REAL_LT_IMP_LE; REAL_LT_MUL;
4753 REAL_POW_MUL; GSYM REAL_LT_RDIV_EQ; REAL_POW_LT] THEN
4756 REWRITE_TAC[GSYM IN_REAL_INTERVAL; HAS_REAL_INTEGRAL_RESTRICT_UNIV] THEN
4757 REWRITE_TAC[HAS_REAL_INTEGRAL_OPEN_INTERVAL] THEN
4758 COND_CASES_TAC THENL
4759 [ASM_MESON_TAC[REAL_INTERVAL_EQ_EMPTY; HAS_REAL_INTEGRAL_EMPTY];
4760 RULE_ASSUM_TAC(REWRITE_RULE[REAL_NOT_LT])] THEN
4762 [`\t. pi / &3 * (inv (a pow 2) - &1) * t pow 3`;
4763 `\t. pi * (inv (a pow 2) - &1) * t pow 2`;
4764 `&0`; `h:real`] REAL_FUNDAMENTAL_THEOREM_OF_CALCULUS) THEN
4765 REWRITE_TAC[] THEN ANTS_TAC THENL
4766 [ASM_REWRITE_TAC[] THEN
4767 REPEAT STRIP_TAC THEN REAL_DIFF_TAC THEN
4768 CONV_TAC NUM_REDUCE_CONV THEN CONV_TAC REAL_RING;
4769 MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN
4770 UNDISCH_TAC `&0 < a` THEN CONV_TAC REAL_FIELD]);;
4772 let VOLUME_FRUSTT = prove
4773 (`!v0 v1:real^3 h a.
4775 ==> measurable(frustt v0 v1 h a) /\
4776 measure(frustt v0 v1 h a) =
4777 if v1 = v0 \/ &1 <= a \/ h < &0 then &0
4778 else pi * ((h / a) pow 2 - h pow 2) * h / &3`,
4779 SIMP_TAC[VOLUME_FRUSTT_STRONG]);;
4781 (* ------------------------------------------------------------------------- *)
4783 (* ------------------------------------------------------------------------- *)
4785 let scale = new_definition
4786 `scale (t:real^3) (u:real^3):real^3 =
4787 vector[t$1 * u$1; t$2 * u$2; t$3 * u$3]`;;
4789 let normball = new_definition `normball x r = { y:real^A | dist(y,x) < r}`;;
4791 let ellipsoid = new_definition
4792 `ellipsoid t r = IMAGE (scale t) (normball(vec 0) r)`;;
4794 let NORMBALL_BALL = prove
4795 (`!z r. normball z r = ball(z,r)`,
4796 REWRITE_TAC[normball; ball; DIST_SYM]);;
4798 let MEASURE_SCALE = prove
4800 ==> measurable(IMAGE (scale t) s) /\
4801 measure(IMAGE (scale t) s) = abs(t$1 * t$2 * t$3) * measure s`,
4802 GEN_TAC THEN GEN_REWRITE_TAC LAND_CONV [HAS_MEASURE_MEASURE] THEN
4803 DISCH_THEN(MP_TAC o SPEC `\i. (t:real^3)$i` o
4804 MATCH_MP HAS_MEASURE_STRETCH) THEN
4805 REWRITE_TAC[DIMINDEX_3; PRODUCT_3] THEN
4806 SUBGOAL_THEN `(\x:real^3. (lambda k. t$k * x$k):real^3) = scale t`
4808 [SIMP_TAC[CART_EQ; FUN_EQ_THM; scale; LAMBDA_BETA; DIMINDEX_3;
4809 VECTOR_3; ARITH; FORALL_3];
4810 MESON_TAC[measurable; MEASURE_UNIQUE]]);;
4812 let MEASURE_ELLIPSOID = prove
4814 ==> measurable(ellipsoid t r) /\
4815 measure(ellipsoid t r) =
4816 abs(t$1 * t$2 * t$3) * &4 / &3 * pi * r pow 3`,
4817 REPEAT GEN_TAC THEN DISCH_TAC THEN
4818 FIRST_X_ASSUM(SUBST1_TAC o SYM o
4819 SPEC `vec 0:real^3` o MATCH_MP VOLUME_BALL) THEN
4820 REWRITE_TAC[normball; ellipsoid] THEN ONCE_REWRITE_TAC[DIST_SYM] THEN
4821 REWRITE_TAC[GSYM ball] THEN MATCH_MP_TAC MEASURE_SCALE THEN
4822 REWRITE_TAC[MEASURABLE_BALL]);;
4824 let MEASURABLE_ELLIPSOID = prove
4825 (`!t r. measurable(ellipsoid t r)`,
4827 ASM_CASES_TAC `&0 <= r` THEN ASM_SIMP_TAC[MEASURE_ELLIPSOID] THEN
4828 REWRITE_TAC[ellipsoid; NORMBALL_BALL; IMAGE; IN_BALL_0] THEN
4829 ASM_SIMP_TAC[NORM_ARITH `~(&0 <= r) ==> ~(norm(x:real^3) < r)`] THEN
4830 REWRITE_TAC[EMPTY_GSPEC; MEASURABLE_EMPTY]);;
4832 (* ------------------------------------------------------------------------- *)
4834 (* ------------------------------------------------------------------------- *)
4836 let conic_cap = new_definition
4837 `conic_cap v0 v1 r a = normball v0 r INTER rcone_gt v0 v1 a`;;
4839 let CONIC_CAP_DEGENERATE = prove
4840 (`!v0 r a. conic_cap v0 v0 r a = {}`,
4841 REWRITE_TAC[conic_cap; rcone_gt; rconesgn; VECTOR_SUB_REFL] THEN
4842 REWRITE_TAC[DIST_REFL; DOT_RZERO; REAL_MUL_RZERO; REAL_MUL_LZERO] THEN
4843 REWRITE_TAC[real_gt; REAL_LT_REFL] THEN SET_TAC[]);;
4845 let BOUNDED_CONIC_CAP = prove
4846 (`!v0 v1:real^3 r a. bounded(conic_cap v0 v1 r a)`,
4847 REPEAT GEN_TAC THEN REWRITE_TAC[conic_cap; NORMBALL_BALL] THEN
4848 MATCH_MP_TAC BOUNDED_SUBSET THEN EXISTS_TAC `ball(v0:real^3,r)` THEN
4849 REWRITE_TAC[BOUNDED_BALL] THEN SET_TAC[]);;
4851 let MEASURABLE_CONIC_CAP = prove
4852 (`!v0 v1:real^3 r a. measurable(conic_cap v0 v1 r a)`,
4853 REPEAT GEN_TAC THEN REWRITE_TAC[conic_cap; NORMBALL_BALL] THEN
4854 MATCH_MP_TAC MEASURABLE_OPEN THEN
4855 SIMP_TAC[OPEN_INTER; OPEN_RCONE_GT; OPEN_BALL] THEN
4856 MATCH_MP_TAC BOUNDED_SUBSET THEN EXISTS_TAC `ball(v0:real^3,r)` THEN
4857 REWRITE_TAC[BOUNDED_BALL] THEN SET_TAC[]);;
4859 let VOLUME_CONIC_CAP_STRONG = prove
4860 (`!v0 v1:real^3 r a.
4862 ==> bounded(conic_cap v0 v1 r a) /\
4863 convex(conic_cap v0 v1 r a) /\
4864 measurable(conic_cap v0 v1 r a) /\
4865 measure(conic_cap v0 v1 r a) =
4866 if v1 = v0 \/ &1 <= a \/ r < &0 then &0
4867 else &2 / &3 * pi * (&1 - a) * r pow 3`,
4868 REPEAT GEN_TAC THEN DISCH_TAC THEN
4869 REWRITE_TAC[conic_cap; rcone_gt; rconesgn; IN_ELIM_THM] THEN
4870 REWRITE_TAC[ONCE_REWRITE_RULE[DIST_SYM] normball; GSYM ball] THEN
4871 CONV_TAC(TOP_DEPTH_CONV let_CONV) THEN GEOM_ORIGIN_TAC `v0:real^3` THEN
4872 REWRITE_TAC[VECTOR_SUB_RZERO; REAL_MUL_LZERO; DIST_0; real_gt] THEN
4873 GEOM_BASIS_MULTIPLE_TAC 1 `v1:real^3` THEN
4874 X_GEN_TAC `b:real` THEN REPEAT(GEN_TAC ORELSE DISCH_TAC) THEN
4875 FIRST_ASSUM(DISJ_CASES_TAC o MATCH_MP (REAL_ARITH
4876 `&0 <= x ==> x = &0 \/ &0 < x`))
4878 [ASM_REWRITE_TAC[VECTOR_MUL_LZERO; GSYM REAL_NOT_LE; DOT_RZERO] THEN
4879 ASM_SIMP_TAC[REAL_LE_MUL; REAL_LT_IMP_LE; NORM_POS_LE] THEN
4880 REWRITE_TAC[EMPTY_GSPEC; INTER_EMPTY; MEASURE_EMPTY; MEASURABLE_EMPTY;
4881 CONVEX_EMPTY; BOUNDED_EMPTY];
4883 ASM_CASES_TAC `&1 <= a` THEN ASM_REWRITE_TAC[] THENL
4885 `!y:real^3. ~(norm(y) * norm(b % basis 1:real^3) * a
4886 < y dot (b % basis 1))`
4887 (fun th -> REWRITE_TAC[th; EMPTY_GSPEC; INTER_EMPTY; MEASURE_EMPTY;
4888 MEASURABLE_EMPTY; BOUNDED_EMPTY; CONVEX_EMPTY]) THEN
4889 REWRITE_TAC[REAL_NOT_LT] THEN X_GEN_TAC `y:real^3` THEN
4890 MATCH_MP_TAC(REAL_ARITH `abs(x) <= a ==> x <= a`) THEN
4891 SIMP_TAC[DOT_RMUL; NORM_MUL; REAL_ABS_MUL; DOT_BASIS; NORM_BASIS;
4892 DIMINDEX_3; ARITH] THEN
4893 REWRITE_TAC[REAL_ARITH
4894 `b * y <= n * (b * &1) * a <=> b * &1 * y <= b * a * n`] THEN
4895 MATCH_MP_TAC REAL_LE_LMUL THEN REWRITE_TAC[REAL_ABS_POS] THEN
4896 MATCH_MP_TAC REAL_LE_MUL2 THEN
4897 ASM_SIMP_TAC[REAL_POS; REAL_ABS_POS; COMPONENT_LE_NORM; DIMINDEX_3; ARITH];
4899 RULE_ASSUM_TAC(REWRITE_RULE[REAL_NOT_LE]) THEN
4900 SIMP_TAC[DOT_RMUL; NORM_MUL; REAL_ABS_NORM; DOT_BASIS;
4901 DIMINDEX_3; ARITH; NORM_BASIS] THEN
4902 ONCE_REWRITE_TAC[REAL_ARITH `n * x * a:real = x * n * a`] THEN
4903 ASM_REWRITE_TAC[real_abs; REAL_MUL_RID] THEN
4904 ASM_SIMP_TAC[REAL_MUL_RID; REAL_LT_LMUL_EQ; REAL_LT_MUL_EQ; NORM_POS_LT] THEN
4905 ASM_SIMP_TAC[GSYM REAL_LT_RDIV_EQ; NORM_LT_SQUARE] THEN
4906 ASM_SIMP_TAC[REAL_POW_DIV; REAL_POW_LT; REAL_LT_RDIV_EQ] THEN
4907 REWRITE_TAC[INTER; REAL_MUL_LZERO; IN_BALL_0; IN_ELIM_THM] THEN
4908 ASM_SIMP_TAC[VECTOR_MUL_EQ_0; BASIS_NONZERO; DIMINDEX_3; ARITH;
4909 REAL_LT_IMP_NZ] THEN
4910 COND_CASES_TAC THENL
4911 [ASM_SIMP_TAC[NORM_ARITH `r < &0 ==> ~(norm x < r)`] THEN
4912 REWRITE_TAC[EMPTY_GSPEC; MEASURE_EMPTY; MEASURABLE_EMPTY;
4913 BOUNDED_EMPTY; CONVEX_EMPTY];
4914 RULE_ASSUM_TAC(ONCE_REWRITE_RULE[REAL_NOT_LT])] THEN
4915 MATCH_MP_TAC(TAUT `a /\ b /\ (a /\ b ==> c /\ d) ==> a /\ b /\ c /\ d`) THEN
4916 REPEAT CONJ_TAC THENL
4917 [MATCH_MP_TAC BOUNDED_SUBSET THEN
4918 EXISTS_TAC `ball(vec 0:real^3,r)` THEN
4919 SIMP_TAC[BOUNDED_BALL; IN_BALL_0; SUBSET; IN_ELIM_THM];
4920 ONCE_REWRITE_TAC[SET_RULE
4921 `{x | P x /\ Q x} = {x | P x} INTER {x | Q x}`] THEN
4922 MATCH_MP_TAC CONVEX_INTER THEN
4923 REWRITE_TAC[GSYM IN_BALL_0; CONVEX_BALL; SIMPLE_IMAGE; IMAGE_ID] THEN
4924 MP_TAC(ISPECL [`vec 0:real^3`; `basis 1:real^3`; `a:real`]
4925 CONVEX_RCONE_GT) THEN
4926 ASM_SIMP_TAC[REAL_LT_IMP_LE; rcone_gt; rconesgn] THEN
4927 REWRITE_TAC[VECTOR_SUB_RZERO; DIST_0] THEN
4928 SIMP_TAC[DOT_BASIS; NORM_BASIS; DIMINDEX_3; ARITH] THEN
4929 REWRITE_TAC[real_gt; REAL_MUL_LID] THEN
4930 ASM_SIMP_TAC[GSYM REAL_LT_RDIV_EQ] THEN
4931 REWRITE_TAC[NORM_LT_SQUARE] THEN
4932 ASM_SIMP_TAC[REAL_POW_DIV; REAL_LT_RDIV_EQ; REAL_POW_LT] THEN
4933 REWRITE_TAC[REAL_MUL_LZERO];
4935 MATCH_MP_TAC(INST_TYPE [`:2`,`:M`] FUBINI_SIMPLE_CONVEX_STRONG) THEN
4936 EXISTS_TAC `1` THEN ASM_REWRITE_TAC[DIMINDEX_2; DIMINDEX_3; ARITH] THEN
4937 SIMP_TAC[SLICE_312; DIMINDEX_2; DIMINDEX_3; ARITH; IN_ELIM_THM;
4938 VECTOR_3; DOT_3; GSYM DOT_2] THEN
4939 SUBGOAL_THEN `&0 < inv(a pow 2) - &1` ASSUME_TAC THENL
4940 [REWRITE_TAC[REAL_SUB_LT] THEN MATCH_MP_TAC REAL_INV_1_LT THEN
4941 ASM_SIMP_TAC[REAL_POW_1_LT; REAL_LT_IMP_LE; ARITH; REAL_POW_LT];
4943 MATCH_MP_TAC HAS_REAL_INTEGRAL_EQ THEN
4944 EXISTS_TAC `\t. if &0 < t /\ t < r
4946 {y:real^2 | norm(vector[t; y$1; y$2]:real^3) pow 2
4948 (t * t + y dot y) * a pow 2 < t pow 2}
4951 [X_GEN_TAC `t:real` THEN DISCH_TAC THEN REWRITE_TAC[] THEN
4952 ASM_CASES_TAC `&0 < t` THEN
4953 ASM_REWRITE_TAC[EMPTY_GSPEC; MEASURE_EMPTY; MEASURABLE_EMPTY] THEN
4954 ASM_CASES_TAC `t:real < r` THEN ASM_REWRITE_TAC[] THENL
4955 [REWRITE_TAC[NORM_LT_SQUARE] THEN
4956 SUBGOAL_THEN `&0 < r` (fun th -> REWRITE_TAC[th; NORM_POW_2]) THEN
4959 SUBGOAL_THEN `!y. ~(norm(vector[t; (y:real^2)$1; y$2]:real^3) < r)`
4960 (fun th -> REWRITE_TAC[th; EMPTY_GSPEC; MEASURE_EMPTY;
4961 MEASURABLE_EMPTY]) THEN
4962 ASM_REWRITE_TAC[NORM_LT_SQUARE; DOT_3; VECTOR_3] THEN
4964 MATCH_MP_TAC(REAL_ARITH `&0 <= a /\ &0 <= b /\ c <= d
4965 ==> ~(&0 < r /\ d + a + b < c)`) THEN
4966 REWRITE_TAC[REAL_LE_SQUARE] THEN
4967 REWRITE_TAC[REAL_POW_2] THEN MATCH_MP_TAC REAL_LE_MUL2 THEN
4970 REWRITE_TAC[GSYM IN_REAL_INTERVAL; HAS_REAL_INTEGRAL_RESTRICT_UNIV] THEN
4971 REWRITE_TAC[HAS_REAL_INTEGRAL_OPEN_INTERVAL] THEN
4972 REWRITE_TAC[NORM_POW_2; DOT_3; VECTOR_3; DOT_2] THEN
4973 ONCE_REWRITE_TAC[REAL_ARITH
4974 `pi * &2 / &3 * (&1 - a) * r pow 3 =
4975 pi / &3 * (inv (a pow 2) - &1) * (a * r) pow 3 +
4976 (pi * &2 / &3 * (&1 - a) * r pow 3 -
4977 pi / &3 * (inv (a pow 2) - &1) * (a * r) pow 3)`] THEN
4978 MATCH_MP_TAC HAS_REAL_INTEGRAL_COMBINE THEN
4979 EXISTS_TAC `a * r:real` THEN
4980 REWRITE_TAC[REAL_ARITH `a * r <= r <=> &0 <= r * (&1 - a)`] THEN
4981 ASM_SIMP_TAC[REAL_LE_MUL; REAL_SUB_LE; REAL_LT_IMP_LE] THEN CONJ_TAC THENL
4982 [MATCH_MP_TAC HAS_REAL_INTEGRAL_EQ THEN EXISTS_TAC
4983 `\t. measure(ball(vec 0:real^2,sqrt(inv(a pow 2) - &1) * t))` THEN
4985 [X_GEN_TAC `t:real` THEN REWRITE_TAC[IN_REAL_INTERVAL] THEN
4986 STRIP_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN
4987 REWRITE_TAC[IN_BALL_0; NORM_LT_SQUARE_ALT] THEN
4988 ASM_SIMP_TAC[SQRT_POS_LE; REAL_LE_MUL; SQRT_POW_2; REAL_LT_IMP_LE;
4990 REWRITE_TAC[REAL_ARITH `x < (a - &1) * t <=> t + x < t * a`] THEN
4991 ASM_SIMP_TAC[GSYM real_div; REAL_LT_RDIV_EQ; REAL_POW_LT] THEN
4992 X_GEN_TAC `x:real^2` THEN REWRITE_TAC[DOT_2] THEN
4993 ASM_SIMP_TAC[GSYM REAL_POW_2; GSYM REAL_LT_RDIV_EQ; REAL_POW_LT] THEN
4994 MATCH_MP_TAC(REAL_ARITH `b <= a ==> (x < b <=> x < a /\ x < b)`) THEN
4995 ASM_SIMP_TAC[REAL_LE_LDIV_EQ; REAL_POW_LT; GSYM REAL_POW_MUL] THEN
4996 ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN
4997 REWRITE_TAC[GSYM REAL_LE_SQUARE_ABS] THEN ASM_REAL_ARITH_TAC;
4999 MATCH_MP_TAC HAS_REAL_INTEGRAL_EQ THEN
5000 EXISTS_TAC `\t. pi * (inv(a pow 2) - &1) * t pow 2` THEN
5002 [X_GEN_TAC `t:real` THEN REWRITE_TAC[IN_REAL_INTERVAL] THEN
5004 W(MP_TAC o PART_MATCH (lhs o rand) AREA_BALL o rand o snd) THEN
5005 ASM_SIMP_TAC[REAL_POW_MUL; REAL_LT_IMP_LE; SQRT_POS_LT; REAL_LE_MUL;
5009 [`\t. pi / &3 * (inv (a pow 2) - &1) * t pow 3`;
5010 `\t. pi * (inv (a pow 2) - &1) * t pow 2`;
5011 `&0`; `a * r:real`] REAL_FUNDAMENTAL_THEOREM_OF_CALCULUS) THEN
5012 ASM_SIMP_TAC[REAL_LE_MUL; REAL_LT_IMP_LE] THEN ANTS_TAC THENL
5013 [ASM_REWRITE_TAC[] THEN
5014 REPEAT STRIP_TAC THEN REAL_DIFF_TAC THEN
5015 CONV_TAC NUM_REDUCE_CONV THEN CONV_TAC REAL_RING;
5016 MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN
5017 UNDISCH_TAC `&0 < a` THEN CONV_TAC REAL_FIELD];
5018 MATCH_MP_TAC HAS_REAL_INTEGRAL_EQ THEN EXISTS_TAC
5019 `\t. measure(ball(vec 0:real^2,sqrt(r pow 2 - t pow 2)))` THEN
5021 [X_GEN_TAC `t:real` THEN REWRITE_TAC[IN_REAL_INTERVAL] THEN
5022 STRIP_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN
5023 REWRITE_TAC[IN_BALL_0; NORM_LT_SQUARE_ALT] THEN
5024 SUBGOAL_THEN `&0 <= t` ASSUME_TAC THENL
5025 [MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `a * r:real` THEN
5026 ASM_SIMP_TAC[REAL_LE_MUL; REAL_LT_IMP_LE];
5028 ASM_SIMP_TAC[SQRT_POS_LE; SQRT_POW_2; REAL_SUB_LE; REAL_POW_LE2] THEN
5029 X_GEN_TAC `x:real^2` THEN REWRITE_TAC[DOT_2] THEN
5030 REWRITE_TAC[REAL_ARITH `x < r - t <=> t + x < r`] THEN
5031 ASM_SIMP_TAC[GSYM REAL_POW_2; GSYM REAL_LT_RDIV_EQ; REAL_POW_LT] THEN
5032 MATCH_MP_TAC(REAL_ARITH `a <= b ==> (x < a <=> x < a /\ x < b)`) THEN
5033 ASM_SIMP_TAC[REAL_LE_RDIV_EQ; REAL_POW_LT; GSYM REAL_POW_MUL] THEN
5034 ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN
5035 ASM_SIMP_TAC[REAL_POW_LE2; REAL_LE_MUL; REAL_LT_IMP_LE];
5037 MATCH_MP_TAC HAS_REAL_INTEGRAL_EQ THEN
5038 EXISTS_TAC `\t. pi * (r pow 2 - t pow 2)` THEN
5040 [X_GEN_TAC `t:real` THEN REWRITE_TAC[IN_REAL_INTERVAL] THEN
5042 W(MP_TAC o PART_MATCH (lhs o rand) AREA_BALL o rand o snd) THEN
5043 SUBGOAL_THEN `&0 <= t` ASSUME_TAC THENL
5044 [MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `a * r:real` THEN
5045 ASM_SIMP_TAC[REAL_LE_MUL; REAL_LT_IMP_LE];
5047 ASM_SIMP_TAC[SQRT_POS_LE; SQRT_POW_2; REAL_SUB_LE; REAL_POW_LE2];
5050 [`\t. pi * (r pow 2 * t - t pow 3 / &3)`;
5051 `\t. pi * (r pow 2 - t pow 2)`;
5052 `a * r:real`; `r:real`] REAL_FUNDAMENTAL_THEOREM_OF_CALCULUS) THEN
5053 ASM_SIMP_TAC[REAL_LE_MUL; REAL_LT_IMP_LE] THEN ANTS_TAC THENL
5054 [ASM_REWRITE_TAC[REAL_ARITH `a * r <= r <=> &0 <= r * (&1 - a)`] THEN
5055 ASM_SIMP_TAC[REAL_LE_MUL; REAL_LT_IMP_LE; REAL_SUB_LE] THEN
5056 REPEAT STRIP_TAC THEN REAL_DIFF_TAC THEN
5057 CONV_TAC NUM_REDUCE_CONV THEN CONV_TAC REAL_RING;
5058 MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN
5059 UNDISCH_TAC `&0 < a` THEN CONV_TAC REAL_FIELD]]);;
5061 let VOLUME_CONIC_CAP = prove
5062 (`!v0 v1:real^3 r a.
5064 ==> measurable(conic_cap v0 v1 r a) /\ measure(conic_cap v0 v1 r a) =
5065 if v1 = v0 \/ &1 <= a \/ r < &0 then &0
5066 else &2 / &3 * pi * (&1 - a) * r pow 3`,
5067 SIMP_TAC[VOLUME_CONIC_CAP_STRONG]);;
5069 (* ------------------------------------------------------------------------- *)
5070 (* Negligibility of a circular cone. *)
5071 (* This isn't exactly using the Flyspeck definition of "cone" but we use it *)
5072 (* to get that later on. Could now simplify this using WLOG tactics. *)
5073 (* ------------------------------------------------------------------------- *)
5075 let NEGLIGIBLE_CIRCULAR_CONE_0_NONPARALLEL = prove
5076 (`!c:real^N k. ~(c = vec 0) /\ ~(k = &0) /\ ~(k = pi)
5077 ==> negligible {x | vector_angle c x = k}`,
5078 REPEAT STRIP_TAC THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN
5079 EXISTS_TAC `(vec 0:real^N) INSERT
5080 UNIONS { {x | x IN ((:real^N) DIFF ball(vec 0,inv(&n + &1))) /\
5081 Cx(vector_angle c x) = Cx k} |
5085 REWRITE_TAC[SUBSET; IN_INSERT; IN_UNIONS; IN_ELIM_THM; CX_INJ] THEN
5086 X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN REWRITE_TAC[IN_UNIV] THEN
5087 ASM_CASES_TAC `x:real^N = vec 0` THEN ASM_REWRITE_TAC[] THEN
5088 REWRITE_TAC[LEFT_AND_EXISTS_THM; IN_DIFF; IN_UNIV] THEN
5089 ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN REWRITE_TAC[UNWIND_THM2] THEN
5090 ASM_REWRITE_TAC[IN_ELIM_THM] THEN
5091 MP_TAC(SPEC `norm(x:real^N)` REAL_ARCH_INV) THEN
5092 ASM_REWRITE_TAC[NORM_POS_LT; IN_BALL_0; REAL_NOT_LT; REAL_LT_INV_EQ] THEN
5093 MATCH_MP_TAC MONO_EXISTS THEN REPEAT STRIP_TAC THEN
5094 MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `inv(&n)` THEN
5095 ASM_SIMP_TAC[REAL_LT_IMP_LE] THEN MATCH_MP_TAC REAL_LE_INV2 THEN
5096 ASM_REAL_ARITH_TAC] THEN
5097 REWRITE_TAC[NEGLIGIBLE_INSERT] THEN
5098 MATCH_MP_TAC NEGLIGIBLE_COUNTABLE_UNIONS THEN X_GEN_TAC `n:num` THEN
5099 MATCH_MP_TAC STARLIKE_NEGLIGIBLE_STRONG THEN EXISTS_TAC `c:real^N` THEN
5101 [MATCH_MP_TAC CONTINUOUS_CLOSED_PREIMAGE_CONSTANT THEN
5102 SIMP_TAC[CLOSED_DIFF; CLOSED_UNIV; OPEN_BALL] THEN
5103 MATCH_MP_TAC(REWRITE_RULE[o_DEF] CONTINUOUS_ON_CX_VECTOR_ANGLE) THEN
5104 REWRITE_TAC[IN_DIFF; IN_BALL_0; NORM_0; IN_UNIV] THEN
5105 REWRITE_TAC[REAL_LT_INV_EQ] THEN REAL_ARITH_TAC;
5107 MAP_EVERY X_GEN_TAC [`a:real`; `x:real^N`] THEN
5108 SIMP_TAC[IN_ELIM_THM; IN_UNIV; IN_DIFF; IN_BALL_0; REAL_NOT_LT; CX_INJ] THEN
5109 REWRITE_TAC[DE_MORGAN_THM] THEN ASM_CASES_TAC `(c + x:real^N) = vec 0` THENL
5110 [ASM_REWRITE_TAC[GSYM REAL_NOT_LT; REAL_LT_INV_EQ; NORM_0] THEN
5113 ASM_CASES_TAC `c + a % x:real^N = vec 0` THENL
5114 [ASM_REWRITE_TAC[GSYM REAL_NOT_LT; REAL_LT_INV_EQ; NORM_0] THEN
5117 ASM_CASES_TAC `x:real^N = vec 0` THENL
5118 [ASM_REWRITE_TAC[VECTOR_ADD_RID; VECTOR_ANGLE_REFL];
5120 ASM_CASES_TAC `a = &0` THENL
5121 [ASM_REWRITE_TAC[VECTOR_MUL_LZERO; VECTOR_ADD_RID; VECTOR_ANGLE_REFL];
5123 REWRITE_TAC[TAUT `~a \/ ~b <=> a ==> ~b`] THEN REPEAT STRIP_TAC THEN
5124 MP_TAC(ISPECL [`vec 0:real^N`; `c:real^N`; `c + a % x:real^N`;
5125 `vec 0:real^N`; `c:real^N`; `c + x:real^N`]
5126 CONGRUENT_TRIANGLES_ASA_FULL) THEN
5127 REWRITE_TAC[angle; VECTOR_ADD_SUB] THEN ASM_SIMP_TAC[VECTOR_SUB_RZERO] THEN
5128 REWRITE_TAC[NORM_ARITH `dist(x,x + a) = norm(a)`; NORM_MUL] THEN
5129 REWRITE_TAC[REAL_FIELD `a * x = x <=> a = &1 \/ x = &0`] THEN
5130 ASM_SIMP_TAC[REAL_ARITH `&0 <= a /\ a < &1 ==> ~(abs a = &1)`] THEN
5131 ASM_REWRITE_TAC[NORM_EQ_0; VECTOR_ANGLE_RMUL; COLLINEAR_LEMMA] THEN
5132 DISCH_THEN(X_CHOOSE_THEN `u:real` MP_TAC) THEN
5133 DISCH_THEN(MP_TAC o AP_TERM `\x:real^N. inv(a) % x`) THEN
5134 ASM_SIMP_TAC[VECTOR_MUL_ASSOC; VECTOR_ADD_LDISTRIB;
5135 VECTOR_MUL_LID; REAL_MUL_LINV] THEN
5136 REWRITE_TAC[VECTOR_ARITH `a % c + x = b % c <=> x = (b - a) % c`] THEN
5137 DISCH_THEN SUBST_ALL_TAC THEN
5138 RULE_ASSUM_TAC(REWRITE_RULE[VECTOR_ARITH `c + a % c = (a + &1) % c`]) THEN
5139 UNDISCH_TAC `vector_angle c ((inv a * u - inv a + &1) % c:real^N) = k` THEN
5140 RULE_ASSUM_TAC(REWRITE_RULE
5141 [VECTOR_ANGLE_RMUL; VECTOR_MUL_EQ_0; DE_MORGAN_THM]) THEN
5142 ASM_REWRITE_TAC[VECTOR_ANGLE_RMUL; VECTOR_ANGLE_REFL] THEN
5143 ASM_REAL_ARITH_TAC);;
5145 let NEGLIGIBLE_CIRCULAR_CONE_0 = prove
5146 (`!c:real^N k. 2 <= dimindex(:N) /\ ~(c = vec 0)
5147 ==> negligible {x | vector_angle c x = k}`,
5148 REPEAT STRIP_TAC THEN
5149 SUBGOAL_THEN `orthogonal (basis 1:real^N) (basis 2)` ASSUME_TAC THENL
5150 [ASM_SIMP_TAC[ORTHOGONAL_BASIS_BASIS; ARITH;
5151 ARITH_RULE `2 <= d ==> 1 <= d`];
5153 ASM_CASES_TAC `k = &0 \/ k = pi` THENL
5154 [ALL_TAC; ASM_MESON_TAC[NEGLIGIBLE_CIRCULAR_CONE_0_NONPARALLEL]] THEN
5156 `?b:real^N. ~(b = vec 0) /\
5157 ~(vector_angle c b = &0) /\
5158 ~(vector_angle c b = pi)`
5159 STRIP_ASSUME_TAC THENL
5160 [MATCH_MP_TAC(MESON[] `!a b. P a \/ P b ==> ?x. P x`) THEN
5161 MAP_EVERY EXISTS_TAC [`basis 1:real^N`; `basis 2:real^N`] THEN
5162 REWRITE_TAC[BASIS_EQ_0] THEN
5163 ASM_SIMP_TAC[ARITH_RULE `2 <= d ==> 1 <= d`; IN_NUMSEG; ARITH] THEN
5164 REWRITE_TAC[GSYM DE_MORGAN_THM] THEN STRIP_TAC THEN
5165 REPEAT(FIRST_X_ASSUM(MP_TAC o SPEC `basis 1:real^N` o
5166 MATCH_MP VECTOR_ANGLE_EQ_0_LEFT)) THEN
5167 REPEAT(FIRST_X_ASSUM(MP_TAC o SPEC `basis 1:real^N` o
5168 MATCH_MP VECTOR_ANGLE_EQ_PI_LEFT)) THEN
5169 DISCH_THEN SUBST1_TAC THEN REWRITE_TAC[VECTOR_ANGLE_REFL; BASIS_EQ_0] THEN
5170 ASM_SIMP_TAC[ARITH_RULE `2 <= d ==> 1 <= d`; IN_NUMSEG; ARITH] THEN
5171 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [ORTHOGONAL_VECTOR_ANGLE]) THEN
5172 REWRITE_TAC[VECTOR_ANGLE_SYM] THEN MP_TAC PI_POS THEN REAL_ARITH_TAC;
5174 ASM_CASES_TAC `k = &0 \/ k = pi` THENL
5175 [ALL_TAC; ASM_MESON_TAC[NEGLIGIBLE_CIRCULAR_CONE_0_NONPARALLEL]] THEN
5176 MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN
5177 FIRST_X_ASSUM(DISJ_CASES_THEN SUBST_ALL_TAC) THENL
5178 [EXISTS_TAC `{x:real^N | vector_angle b x = vector_angle c b}` THEN
5179 ASM_SIMP_TAC[NEGLIGIBLE_CIRCULAR_CONE_0_NONPARALLEL] THEN
5180 REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN
5181 MESON_TAC[VECTOR_ANGLE_EQ_0_RIGHT; VECTOR_ANGLE_SYM];
5182 EXISTS_TAC `{x:real^N | vector_angle b x = pi - vector_angle c b}` THEN
5183 ASM_SIMP_TAC[NEGLIGIBLE_CIRCULAR_CONE_0_NONPARALLEL;
5184 REAL_SUB_0; REAL_ARITH `p - x = p <=> x = &0`] THEN
5185 REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN
5186 MESON_TAC[VECTOR_ANGLE_EQ_PI_RIGHT; VECTOR_ANGLE_SYM]]);;
5188 let NEGLIGIBLE_CIRCULAR_CONE = prove
5190 2 <= dimindex(:N) /\ ~(c = vec 0)
5191 ==> negligible(a INSERT {x | vector_angle c (x - a) = k})`,
5192 REPEAT STRIP_TAC THEN REWRITE_TAC[NEGLIGIBLE_INSERT] THEN
5193 MATCH_MP_TAC NEGLIGIBLE_TRANSLATION_REV THEN EXISTS_TAC `--a:real^N` THEN
5194 MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN
5195 EXISTS_TAC `{x:real^N | vector_angle c x = k}` THEN
5196 ASM_SIMP_TAC[NEGLIGIBLE_CIRCULAR_CONE_0] THEN
5197 REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_ELIM_THM] THEN
5198 REWRITE_TAC[VECTOR_ARITH `--a + x:real^N = x - a`]);;
5200 let NEGLIGIBLE_RCONE_EQ = prove
5201 (`!w z:real^3 h. ~(w = z) ==> negligible(rcone_eq z w h)`,
5202 REWRITE_TAC[rcone_eq; rconesgn] THEN GEOM_ORIGIN_TAC `z:real^3` THEN
5203 REPEAT STRIP_TAC THEN REWRITE_TAC[DIST_0; VECTOR_SUB_RZERO] THEN
5204 ASM_CASES_TAC `abs(h) <= &1` THENL
5205 [MP_TAC(ISPECL [`w:real^3`; `acs h`] NEGLIGIBLE_CIRCULAR_CONE_0) THEN
5206 ASM_REWRITE_TAC[DIMINDEX_3; ARITH] THEN
5207 REWRITE_TAC[GSYM HAS_MEASURE_0] THEN
5208 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT]
5209 HAS_MEASURE_NEGLIGIBLE_SYMDIFF) THEN
5210 MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `{vec 0:real^3}` THEN
5211 REWRITE_TAC[NEGLIGIBLE_SING] THEN MATCH_MP_TAC(SET_RULE
5212 `(!x. ~(x = a) ==> (x IN s <=> x IN t))
5213 ==> (s DIFF t) UNION (t DIFF s) SUBSET {a}`) THEN
5214 X_GEN_TAC `x:real^3` THEN DISCH_TAC THEN REWRITE_TAC[IN_ELIM_THM] THEN
5215 ASM_SIMP_TAC[vector_angle] THEN ASM_SIMP_TAC[NORM_EQ_0; REAL_FIELD
5216 `~(x = &0) /\ ~(w = &0) ==> (a = x * w * b <=> a / (w * x) = b)`] THEN
5217 GEN_REWRITE_TAC (RAND_CONV o ONCE_DEPTH_CONV) [DOT_SYM] THEN
5218 MATCH_MP_TAC ACS_INJ THEN ASM_REWRITE_TAC[NORM_CAUCHY_SCHWARZ_DIV];
5219 MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN
5220 EXISTS_TAC `{vec 0}:real^3->bool` THEN
5221 REWRITE_TAC[NEGLIGIBLE_SING] THEN
5222 REWRITE_TAC[SET_RULE `{x | P x} SUBSET {a} <=> !x. ~(x = a) ==> ~P x`] THEN
5223 X_GEN_TAC `x:real^3` THEN REPEAT DISCH_TAC THEN
5224 MP_TAC(ISPECL [`x:real^3`; `w:real^3`] NORM_CAUCHY_SCHWARZ_ABS) THEN
5225 ASM_REWRITE_TAC[REAL_ABS_MUL; REAL_ABS_NORM; REAL_ARITH
5226 `~(x * w * h <= x * w) <=> &0 < x * w * (h - &1)`] THEN
5227 REPEAT(MATCH_MP_TAC REAL_LT_MUL THEN ASM_REWRITE_TAC[NORM_POS_LT]) THEN
5228 ASM_REAL_ARITH_TAC]);;
5230 (* ------------------------------------------------------------------------- *)
5231 (* Area of sector of a circle delimited by Arg values. *)
5232 (* ------------------------------------------------------------------------- *)
5234 let NEGLIGIBLE_ARG_EQ = prove
5235 (`!t. negligible {z | Arg z = t}`,
5236 GEN_TAC THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN
5237 EXISTS_TAC `{z | cexp(ii * Cx(pi / &2 + t)) dot z = &0}` THEN
5238 SIMP_TAC[NEGLIGIBLE_HYPERPLANE; COMPLEX_VEC_0; CEXP_NZ] THEN
5239 REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN X_GEN_TAC `z:complex` THEN
5240 DISCH_TAC THEN MP_TAC(SPEC `z:complex` ARG) THEN ASM_REWRITE_TAC[] THEN
5241 REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
5242 DISCH_THEN SUBST1_TAC THEN
5243 REWRITE_TAC[GSYM COMPLEX_CMUL; DOT_RMUL; REAL_ENTIRE] THEN
5244 DISJ2_TAC THEN REWRITE_TAC[CEXP_EULER] THEN
5245 REWRITE_TAC[DOT_2; GSYM RE_DEF; GSYM IM_DEF] THEN
5246 REWRITE_TAC[GSYM CX_SIN; GSYM CX_COS; RE_ADD; IM_ADD;
5247 RE_MUL_II; IM_MUL_II; RE_CX; IM_CX] THEN
5248 REWRITE_TAC[SIN_ADD; COS_ADD; SIN_PI2; COS_PI2] THEN
5251 let MEASURABLE_CLOSED_SECTOR_LE = prove
5252 (`!r t. measurable {z | norm(z) <= r /\ Arg z <= t}`,
5253 REPEAT GEN_TAC THEN MATCH_MP_TAC MEASURABLE_COMPACT THEN
5254 REWRITE_TAC[SET_RULE `{z | P z /\ Q z} = {z | P z} INTER {z | Q z}`] THEN
5255 MATCH_MP_TAC COMPACT_INTER_CLOSED THEN REWRITE_TAC[CLOSED_ARG_LE] THEN
5256 REWRITE_TAC[NORM_ARITH `norm z = dist(vec 0,z)`; GSYM cball] THEN
5257 REWRITE_TAC[COMPACT_CBALL]);;
5259 let MEASURABLE_CLOSED_SECTOR_LT = prove
5260 (`!r t. measurable {z | norm(z) <= r /\ Arg z < t}`,
5261 REPEAT GEN_TAC THEN MATCH_MP_TAC MEASURABLE_NEGLIGIBLE_SYMDIFF THEN
5262 EXISTS_TAC `{z | norm(z) <= r /\ Arg z <= t}` THEN
5263 REWRITE_TAC[MEASURABLE_CLOSED_SECTOR_LE] THEN
5264 MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN
5265 EXISTS_TAC `{z | Arg z = t}` THEN
5266 REWRITE_TAC[NEGLIGIBLE_ARG_EQ; NEGLIGIBLE_UNION_EQ] THEN
5267 REWRITE_TAC[SUBSET; IN_DIFF; IN_UNION; IN_ELIM_THM] THEN REAL_ARITH_TAC);;
5269 let MEASURABLE_CLOSED_SECTOR_LTE = prove
5270 (`!r s t. measurable {z | norm(z) <= r /\ s < Arg z /\ Arg z <= t}`,
5271 REPEAT GEN_TAC THEN REWRITE_TAC[SET_RULE
5272 `{z | P z /\ Q z /\ R z} = {z | P z /\ R z} DIFF {z | P z /\ ~Q z}`] THEN
5273 SIMP_TAC[MEASURABLE_DIFF; REAL_NOT_LT; MEASURABLE_CLOSED_SECTOR_LE]);;
5275 let MEASURE_CLOSED_SECTOR_LE = prove
5276 (`!t r. &0 <= r /\ &0 <= t /\ t <= &2 * pi
5277 ==> measure {x:real^2 | norm(x) <= r /\ Arg(x) <= t} =
5279 REPEAT STRIP_TAC THEN
5281 [`\t. measure {z:real^2 | norm(z) <= r /\ Arg(z) <= t}`;
5282 `&2 * pi`] REAL_CONTINUOUS_ADDITIVE_IMP_LINEAR_INTERVAL) THEN
5285 DISCH_THEN(MP_TAC o SPECL [`t / (&2 * pi)`; `&2 * pi`]) THEN
5286 MP_TAC(SPECL [`vec 0:real^2`; `r:real`] AREA_CBALL) THEN
5287 ASM_REWRITE_TAC[cball; NORM_ARITH `dist(vec 0,z) = norm z`] THEN
5288 SIMP_TAC[ARG; REAL_LT_IMP_LE] THEN DISCH_THEN(K ALL_TAC) THEN
5289 SIMP_TAC[PI_POS; REAL_FIELD `&0 < p ==> t / (&2 * p) * p * r = t * r / &2`;
5290 REAL_FIELD `&0 < p ==> t / (&2 * p) * &2 * p = t`] THEN
5291 DISCH_THEN MATCH_MP_TAC THEN MP_TAC PI_POS THEN ASM_REAL_ARITH_TAC] THEN
5292 REWRITE_TAC[] THEN REPEAT STRIP_TAC THENL
5293 [MATCH_MP_TAC REALLIM_TRANSFORM_BOUND THEN
5294 EXISTS_TAC `\t. r pow 2 * sin(t)` THEN REWRITE_TAC[] THEN CONJ_TAC THENL
5295 [REWRITE_TAC[EVENTUALLY_WITHINREAL] THEN EXISTS_TAC `pi / &2` THEN
5296 SIMP_TAC[PI_POS; REAL_LT_DIV; IN_ELIM_THM; REAL_OF_NUM_LT; ARITH] THEN
5297 X_GEN_TAC `x:real` THEN REWRITE_TAC[REAL_SUB_RZERO] THEN
5298 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
5299 ASM_SIMP_TAC[real_abs; MEASURE_POS_LE; MEASURABLE_CLOSED_SECTOR_LE] THEN
5300 STRIP_TAC THEN MATCH_MP_TAC REAL_LE_TRANS THEN
5301 EXISTS_TAC `measure(interval[vec 0,complex(r,r * sin x)])` THEN
5303 [MATCH_MP_TAC MEASURE_SUBSET THEN
5304 REWRITE_TAC[MEASURABLE_CLOSED_SECTOR_LE; MEASURABLE_INTERVAL] THEN
5305 REWRITE_TAC[SUBSET; IN_ELIM_THM; IN_INTERVAL] THEN
5306 X_GEN_TAC `z:complex` THEN STRIP_TAC THEN
5307 REWRITE_TAC[DIMINDEX_2; FORALL_2; VEC_COMPONENT] THEN
5308 REWRITE_TAC[GSYM IM_DEF; GSYM RE_DEF; IM; RE] THEN
5309 SUBST1_TAC(last(CONJUNCTS(SPEC `z:complex` ARG))) THEN
5310 REWRITE_TAC[RE_MUL_CX; IM_MUL_CX; CEXP_EULER] THEN
5311 REWRITE_TAC[RE_ADD; GSYM CX_COS; GSYM CX_SIN; RE_CX; IM_CX;
5312 RE_MUL_II; IM_MUL_II; IM_ADD] THEN
5313 REWRITE_TAC[REAL_NEG_0; REAL_ADD_LID; REAL_ADD_RID] THEN
5314 SUBGOAL_THEN `&0 <= Arg z /\ Arg z < pi / &2 /\ Arg z <= pi / &2`
5315 STRIP_ASSUME_TAC THENL
5316 [REWRITE_TAC[ARG] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN
5317 REPEAT CONJ_TAC THENL
5318 [MATCH_MP_TAC REAL_LE_MUL THEN REWRITE_TAC[NORM_POS_LE] THEN
5319 MATCH_MP_TAC COS_POS_PI_LE THEN ASM_REAL_ARITH_TAC;
5320 MATCH_MP_TAC(REAL_ARITH `abs(a * b) <= c * &1 ==> a * b <= c`) THEN
5321 REWRITE_TAC[REAL_ABS_MUL; REAL_ABS_NORM] THEN
5322 MATCH_MP_TAC REAL_LE_MUL2 THEN
5323 ASM_REWRITE_TAC[NORM_POS_LE; REAL_ABS_POS; COS_BOUND];
5324 MATCH_MP_TAC REAL_LE_MUL THEN REWRITE_TAC[NORM_POS_LE] THEN
5325 MATCH_MP_TAC SIN_POS_PI_LE THEN ASM_REAL_ARITH_TAC;
5326 MATCH_MP_TAC REAL_LE_MUL2 THEN ASM_REWRITE_TAC[NORM_POS_LE] THEN
5328 [MATCH_MP_TAC SIN_POS_PI_LE THEN ASM_REAL_ARITH_TAC;
5329 MATCH_MP_TAC SIN_MONO_LE THEN ASM_REAL_ARITH_TAC]];
5330 REWRITE_TAC[MEASURE_INTERVAL; CONTENT_CLOSED_INTERVAL_CASES] THEN
5331 REWRITE_TAC[FORALL_2; PRODUCT_2; DIMINDEX_2; VEC_COMPONENT] THEN
5332 REWRITE_TAC[GSYM IM_DEF; GSYM RE_DEF; IM; RE] THEN
5333 REWRITE_TAC[REAL_SUB_RZERO; REAL_POW_2; REAL_MUL_ASSOC] THEN
5334 SUBGOAL_THEN `&0 <= sin x` (fun th ->
5335 ASM_SIMP_TAC[REAL_LE_MUL; REAL_LE_REFL; REAL_LE_MUL; th]) THEN
5336 MATCH_MP_TAC SIN_POS_PI_LE THEN ASM_REAL_ARITH_TAC];
5337 MATCH_MP_TAC REALLIM_ATREAL_WITHINREAL THEN
5338 SUBGOAL_THEN `(\t. r pow 2 * sin t) real_continuous atreal (&0)`
5340 [MATCH_MP_TAC REAL_CONTINUOUS_LMUL THEN
5341 REWRITE_TAC[ETA_AX; REAL_CONTINUOUS_AT_SIN];
5342 REWRITE_TAC[REAL_CONTINUOUS_ATREAL; SIN_0; REAL_MUL_RZERO]]];
5343 ASM_SIMP_TAC[REAL_ARITH
5345 ==> (norm z <= r /\ Arg z <= x + y <=>
5346 norm z <= r /\ Arg z <= x \/
5347 norm z <= r /\ x < Arg z /\ Arg z <= x + y)`] THEN
5348 REWRITE_TAC[SET_RULE `{z | Q z \/ R z} = {z | Q z} UNION {z | R z}`] THEN
5349 SIMP_TAC[MEASURE_UNION; MEASURABLE_CLOSED_SECTOR_LE;
5350 MEASURABLE_CLOSED_SECTOR_LTE] THEN
5351 REWRITE_TAC[GSYM REAL_NOT_LE; SET_RULE
5352 `{z | P z /\ Q z} INTER {z | P z /\ ~Q z /\ R z} = {}`] THEN
5353 REWRITE_TAC[MEASURE_EMPTY; REAL_SUB_RZERO; REAL_EQ_ADD_LCANCEL] THEN
5354 REWRITE_TAC[REAL_NOT_LE] THEN MATCH_MP_TAC EQ_TRANS THEN
5355 EXISTS_TAC `measure {z | norm z <= r /\ x < Arg z /\ Arg z < x + y}` THEN
5357 [MATCH_MP_TAC MEASURE_NEGLIGIBLE_SYMDIFF THEN
5358 REWRITE_TAC[MEASURABLE_CLOSED_SECTOR_LTE] THEN
5359 MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN
5360 EXISTS_TAC `{z | Arg z = x + y}` THEN
5361 REWRITE_TAC[NEGLIGIBLE_ARG_EQ; NEGLIGIBLE_UNION_EQ] THEN
5362 REWRITE_TAC[SUBSET; IN_DIFF; IN_UNION; IN_ELIM_THM] THEN REAL_ARITH_TAC;
5364 MATCH_MP_TAC EQ_TRANS THEN
5365 EXISTS_TAC `measure {z | norm z <= r /\ &0 < Arg z /\ Arg z < y}` THEN
5368 MATCH_MP_TAC MEASURE_NEGLIGIBLE_SYMDIFF THEN
5369 REWRITE_TAC[MEASURABLE_CLOSED_SECTOR_LE] THEN
5370 MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN
5371 EXISTS_TAC `{z | Arg z = &0} UNION {z | Arg z = y}` THEN
5372 REWRITE_TAC[NEGLIGIBLE_ARG_EQ; NEGLIGIBLE_UNION_EQ] THEN
5373 REWRITE_TAC[SUBSET; IN_DIFF; IN_UNION; IN_ELIM_THM] THEN
5374 MP_TAC ARG THEN MATCH_MP_TAC MONO_FORALL THEN REAL_ARITH_TAC] THEN
5375 MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC
5376 `measure (IMAGE (rotate2d x)
5377 {z | norm z <= r /\ &0 < Arg z /\ Arg z < y})` THEN
5380 ASM_SIMP_TAC[MEASURE_ORTHOGONAL_IMAGE_EQ;
5381 ORTHOGONAL_TRANSFORMATION_ROTATE2D]] THEN
5382 AP_TERM_TAC THEN CONV_TAC SYM_CONV THEN
5383 MATCH_MP_TAC SURJECTIVE_IMAGE_EQ THEN CONJ_TAC THENL
5384 [ASM_MESON_TAC[ORTHOGONAL_TRANSFORMATION_SURJECTIVE;
5385 ORTHOGONAL_TRANSFORMATION_ROTATE2D]; ALL_TAC] THEN
5386 X_GEN_TAC `z:complex` THEN REWRITE_TAC[IN_ELIM_THM] THEN
5387 ASM_CASES_TAC `z = Cx(&0)` THENL
5388 [ASM_REWRITE_TAC[Arg_DEF; ROTATE2D_0] THEN
5391 REWRITE_TAC[NORM_ROTATE2D] THEN AP_TERM_TAC THEN EQ_TAC THENL
5393 SUBGOAL_THEN `z = rotate2d (--x) (rotate2d x z)` SUBST1_TAC THENL
5394 [REWRITE_TAC[GSYM ROTATE2D_ADD; REAL_ADD_LINV; ROTATE2D_ZERO];
5396 MP_TAC(ISPECL [`--x:real`; `rotate2d x z`] ARG_ROTATE2D) THEN
5397 ASM_REWRITE_TAC[ROTATE2D_EQ_0] THEN
5398 ANTS_TAC THENL [ASM_REAL_ARITH_TAC; DISCH_THEN SUBST1_TAC] THEN
5401 MP_TAC(ISPECL [`x:real`; `z:complex`] ARG_ROTATE2D) THEN
5402 ASM_REWRITE_TAC[ROTATE2D_EQ_0] THEN
5403 ANTS_TAC THENL [ASM_REAL_ARITH_TAC; DISCH_THEN SUBST1_TAC] THEN
5404 ASM_REAL_ARITH_TAC]]);;
5406 let HAS_MEASURE_OPEN_SECTOR_LT = prove
5407 (`!t r. &0 <= t /\ t <= &2 * pi
5408 ==> {x:real^2 | norm(x) < r /\ &0 < Arg x /\ Arg x < t}
5409 has_measure (if &0 <= r then t * r pow 2 / &2 else &0)`,
5410 REPEAT STRIP_TAC THEN COND_CASES_TAC THEN
5411 ASM_SIMP_TAC[NORM_ARITH `~(&0 <= r) ==> ~(norm x < r)`;
5412 EMPTY_GSPEC; HAS_MEASURE_EMPTY] THEN
5413 MATCH_MP_TAC HAS_MEASURE_NEGLIGIBLE_SYMDIFF THEN
5414 EXISTS_TAC `{x | norm x <= r /\ Arg x <= t}` THEN
5415 REWRITE_TAC[HAS_MEASURE_MEASURABLE_MEASURE] THEN
5416 ASM_SIMP_TAC[MEASURE_CLOSED_SECTOR_LE; MEASURABLE_CLOSED_SECTOR_LE] THEN
5417 MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN
5418 EXISTS_TAC `{x | dist(vec 0,x) = r} UNION
5419 {z | Arg z = &0} UNION {z | Arg z = t}` THEN
5420 REWRITE_TAC[NEGLIGIBLE_ARG_EQ; REWRITE_RULE[sphere] NEGLIGIBLE_SPHERE;
5421 NEGLIGIBLE_UNION_EQ] THEN
5422 REWRITE_TAC[DIST_0; SUBSET; IN_DIFF; IN_UNION; IN_ELIM_THM] THEN
5423 MP_TAC ARG THEN MATCH_MP_TAC MONO_FORALL THEN REAL_ARITH_TAC);;
5425 let MEASURE_OPEN_SECTOR_LT = prove
5426 (`!t r. &0 <= t /\ t <= &2 * pi
5427 ==> measure {x:real^2 | norm(x) < r /\ &0 < Arg x /\ Arg x < t} =
5428 if &0 <= r then t * r pow 2 / &2 else &0`,
5429 SIMP_TAC[REWRITE_RULE[HAS_MEASURE_MEASURABLE_MEASURE]
5430 HAS_MEASURE_OPEN_SECTOR_LT]);;
5432 let HAS_MEASURE_OPEN_SECTOR_LT_GEN = prove
5435 ==> {x | norm(x) < r /\ &0 < Arg(x / w) /\ Arg(x / w) < Arg(z / w)}
5436 has_measure (if &0 <= r then Arg(z / w) * r pow 2 / &2 else &0)`,
5437 GEOM_BASIS_MULTIPLE_TAC 1 `w:complex` THEN
5438 X_GEN_TAC `w:real` THEN GEN_REWRITE_TAC LAND_CONV [REAL_LE_LT] THEN
5439 ASM_CASES_TAC `w = &0` THEN ASM_REWRITE_TAC[VECTOR_MUL_LZERO] THEN
5440 REWRITE_TAC[COMPLEX_CMUL; COMPLEX_BASIS; COMPLEX_VEC_0] THEN
5441 SIMP_TAC[ARG_DIV_CX; COMPLEX_MUL_RID] THEN ASM_REWRITE_TAC[CX_INJ] THEN
5442 REPEAT STRIP_TAC THEN MATCH_MP_TAC HAS_MEASURE_OPEN_SECTOR_LT THEN
5443 SIMP_TAC[ARG; REAL_LT_IMP_LE]);;
5445 (* ------------------------------------------------------------------------- *)
5446 (* Hence volume of a wedge of a ball. *)
5447 (* ------------------------------------------------------------------------- *)
5449 let MEASURABLE_BALL_WEDGE = prove
5450 (`!z:real^3 w w1 w2. measurable(ball(z,r) INTER wedge z w w1 w2)`,
5451 REPEAT STRIP_TAC THEN MATCH_MP_TAC MEASURABLE_OPEN THEN CONJ_TAC THENL
5452 [MATCH_MP_TAC BOUNDED_INTER THEN REWRITE_TAC[BOUNDED_BALL];
5453 MATCH_MP_TAC OPEN_INTER THEN REWRITE_TAC[OPEN_BALL] THEN
5454 ASM_SIMP_TAC[OPEN_WEDGE]]);;
5456 let VOLUME_BALL_WEDGE = prove
5457 (`!z:real^3 w r w1 w2.
5458 &0 <= r ==> measure(ball(z,r) INTER wedge z w w1 w2) =
5459 azim z w w1 w2 * &2 * r pow 3 / &3`,
5461 ASM_CASES_TAC `z:real^3 = w \/ collinear{z,w,w1} \/ collinear{z,w,w2}` THENL
5462 [FIRST_X_ASSUM STRIP_ASSUME_TAC THEN
5463 ASM_SIMP_TAC[WEDGE_DEGENERATE; AZIM_DEGENERATE; INTER_EMPTY; REAL_MUL_LZERO;
5465 FIRST_X_ASSUM MP_TAC THEN REWRITE_TAC[IMP_IMP; DE_MORGAN_THM]] THEN
5466 REWRITE_TAC[wedge] THEN GEOM_ORIGIN_TAC `z:real^3` THEN
5467 GEOM_BASIS_MULTIPLE_TAC 3 `w:real^3` THEN
5468 X_GEN_TAC `w:real` THEN GEN_REWRITE_TAC LAND_CONV [REAL_LE_LT] THEN
5469 ASM_CASES_TAC `w = &0` THEN ASM_REWRITE_TAC[VECTOR_MUL_LZERO] THEN
5470 ASM_SIMP_TAC[AZIM_SPECIAL_SCALE; COLLINEAR_SPECIAL_SCALE] THEN
5471 REPEAT STRIP_TAC THEN
5472 MATCH_MP_TAC(INST_TYPE[`:2`,`:M`; `:3`,`:N`] FUBINI_SIMPLE_OPEN) THEN
5473 EXISTS_TAC `3` THEN REWRITE_TAC[DIMINDEX_2; DIMINDEX_3; ARITH] THEN
5474 REPEAT CONJ_TAC THENL
5475 [MESON_TAC[BOUNDED_SUBSET; INTER_SUBSET; BOUNDED_BALL];
5476 REWRITE_TAC[GSYM wedge] THEN MATCH_MP_TAC OPEN_INTER THEN
5477 ASM_REWRITE_TAC[OPEN_BALL; OPEN_WEDGE];
5478 SIMP_TAC[SLICE_INTER; DIMINDEX_2; DIMINDEX_3; ARITH; SLICE_BALL]] THEN
5479 ONCE_REWRITE_TAC[TAUT `~a /\ b /\ c <=> ~(~a ==> ~(b /\ c))`] THEN
5480 ASM_SIMP_TAC[AZIM_ARG] THEN REWRITE_TAC[COLLINEAR_BASIS_3] THEN
5481 RULE_ASSUM_TAC(REWRITE_RULE[COLLINEAR_BASIS_3]) THEN
5482 REWRITE_TAC[NOT_IMP; GSYM CONJ_ASSOC; DROPOUT_0] THEN
5483 MAP_EVERY ABBREV_TAC
5484 [`v1:real^2 = dropout 3 (w1:real^3)`;
5485 `v2:real^2 = dropout 3 (w2:real^3)`] THEN
5486 REWRITE_TAC[SLICE_DROPOUT_3; VEC_COMPONENT; REAL_SUB_RZERO] THEN
5487 ONCE_REWRITE_TAC[COND_RAND] THEN
5488 ONCE_REWRITE_TAC[COND_RATOR] THEN
5489 REWRITE_TAC[INTER_EMPTY] THEN REWRITE_TAC[INTER; IN_BALL_0; IN_ELIM_THM] THEN
5490 ONCE_REWRITE_TAC[COND_RAND] THEN REWRITE_TAC[MEASURE_EMPTY] THEN
5491 MAP_EVERY UNDISCH_TAC
5492 [`~(v1:complex = vec 0)`; `~(v2:complex = vec 0)`] THEN
5493 MAP_EVERY (fun t -> SPEC_TAC(t,t)) [`v2:complex`; `v1:complex`] THEN
5494 UNDISCH_TAC `&0 <= r` THEN SPEC_TAC(`r:real`,`r:real`) THEN
5495 REWRITE_TAC[RIGHT_IMP_FORALL_THM; IMP_IMP; GSYM CONJ_ASSOC] THEN
5496 POP_ASSUM_LIST(K ALL_TAC) THEN GEOM_BASIS_MULTIPLE_TAC 1 `v1:complex` THEN
5497 X_GEN_TAC `v1:real` THEN GEN_REWRITE_TAC LAND_CONV [REAL_LE_LT] THEN
5498 ASM_CASES_TAC `v1 = &0` THEN ASM_REWRITE_TAC[VECTOR_MUL_LZERO] THEN
5499 REWRITE_TAC[COMPLEX_CMUL; COMPLEX_BASIS; COMPLEX_VEC_0] THEN
5500 SIMP_TAC[ARG_DIV_CX; COMPLEX_MUL_RID; CX_INJ] THEN REPEAT STRIP_TAC THEN
5502 `!t z. ~(z = Cx(&0)) /\ &0 < Arg z /\ Arg z < t <=>
5503 &0 < Arg z /\ Arg z < t`
5504 (fun th -> REWRITE_TAC[th])
5505 THENL [MESON_TAC[ARG_0; REAL_LT_REFL]; ALL_TAC] THEN
5506 ASM_SIMP_TAC[MEASURE_OPEN_SECTOR_LT; REAL_LE_REFL; ARG; REAL_LT_IMP_LE] THEN
5507 SUBGOAL_THEN `!t. abs(t) < r <=> t IN real_interval(--r,r)`
5508 (fun th -> REWRITE_TAC[th])
5509 THENL [REWRITE_TAC[IN_REAL_INTERVAL] THEN REAL_ARITH_TAC; ALL_TAC] THEN
5510 REWRITE_TAC[HAS_REAL_INTEGRAL_RESTRICT_UNIV] THEN
5511 REWRITE_TAC[HAS_REAL_INTEGRAL_OPEN_INTERVAL] THEN
5512 MATCH_MP_TAC HAS_REAL_INTEGRAL_EQ THEN
5513 EXISTS_TAC `\t. Arg v2 * (r pow 2 - t pow 2) / &2` THEN CONJ_TAC THENL
5514 [X_GEN_TAC `t:real` THEN REWRITE_TAC[IN_REAL_INTERVAL; REAL_BOUNDS_LE] THEN
5515 SIMP_TAC[AREA_CBALL; SQRT_POS_LE; REAL_SUB_LE; GSYM REAL_LE_SQUARE_ABS;
5516 SQRT_POW_2; REAL_ARITH `abs x <= r ==> abs x <= abs r`];
5519 [`\t. Arg v2 * (r pow 2 * t - &1 / &3 * t pow 3) / &2`;
5520 `\t. Arg v2 * (r pow 2 - t pow 2) / &2`;
5521 `--r:real`; `r:real`] REAL_FUNDAMENTAL_THEOREM_OF_CALCULUS) THEN
5522 REWRITE_TAC[] THEN ANTS_TAC THENL
5523 [CONJ_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
5524 REPEAT STRIP_TAC THEN REAL_DIFF_TAC THEN
5525 CONV_TAC NUM_REDUCE_CONV THEN CONV_TAC REAL_RING;
5526 MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN
5527 CONV_TAC REAL_RING]);;
5529 (* ------------------------------------------------------------------------- *)
5530 (* Hence volume of lune. *)
5531 (* ------------------------------------------------------------------------- *)
5533 let HAS_MEASURE_LUNE = prove
5534 (`!z:real^3 w r w1 w2.
5535 &0 <= r /\ ~(w = z) /\
5536 ~collinear {z,w,w1} /\ ~collinear {z,w,w2} /\ ~(dihV z w w1 w2 = pi)
5537 ==> (ball(z,r) INTER aff_gt {z,w} {w1,w2})
5538 has_measure (dihV z w w1 w2 * &2 * r pow 3 / &3)`,
5539 GEOM_ORIGIN_TAC `z:real^3` THEN
5540 GEOM_BASIS_MULTIPLE_TAC 3 `w:real^3` THEN
5541 X_GEN_TAC `w:real` THEN GEN_REWRITE_TAC LAND_CONV [REAL_LE_LT] THEN
5542 ASM_CASES_TAC `w = &0` THEN ASM_REWRITE_TAC[] THENL
5543 [ASM_REWRITE_TAC[VECTOR_MUL_LZERO; INSERT_AC; COLLINEAR_2]; ALL_TAC] THEN
5544 DISCH_TAC THEN REPEAT GEN_TAC THEN
5545 ASM_SIMP_TAC[DIHV_SPECIAL_SCALE] THEN
5546 MP_TAC(ISPECL [`{}:real^3->bool`; `{w1:real^3,w2:real^3}`;
5547 `w:real`; `basis 3:real^3`] AFF_GT_SPECIAL_SCALE) THEN
5548 ASM_CASES_TAC `w1:real^3 = vec 0` THENL
5549 [ASM_REWRITE_TAC[INSERT_AC; COLLINEAR_2]; ALL_TAC] THEN
5550 ASM_CASES_TAC `w2:real^3 = vec 0` THENL
5551 [ASM_REWRITE_TAC[INSERT_AC; COLLINEAR_2]; ALL_TAC] THEN
5552 ASM_REWRITE_TAC[FINITE_INSERT; FINITE_EMPTY; IN_INSERT; NOT_IN_EMPTY] THEN
5553 ASM_CASES_TAC `w1:real^3 = w % basis 3` THENL
5554 [ASM_REWRITE_TAC[INSERT_AC; COLLINEAR_2]; ALL_TAC] THEN
5555 ASM_CASES_TAC `w2:real^3 = w % basis 3` THENL
5556 [ASM_REWRITE_TAC[INSERT_AC; COLLINEAR_2]; ALL_TAC] THEN
5557 ASM_SIMP_TAC[COLLINEAR_SPECIAL_SCALE] THEN
5558 ASM_CASES_TAC `w1:real^3 = basis 3` THENL
5559 [ASM_REWRITE_TAC[INSERT_AC; COLLINEAR_2]; ALL_TAC] THEN
5560 ASM_CASES_TAC `w2:real^3 = basis 3` THENL
5561 [ASM_REWRITE_TAC[INSERT_AC; COLLINEAR_2]; ALL_TAC] THEN
5562 ASM_REWRITE_TAC[] THEN DISCH_THEN SUBST1_TAC THEN
5563 ASM_SIMP_TAC[VECTOR_MUL_EQ_0; REAL_LT_IMP_NZ] THEN STRIP_TAC THEN
5564 ASM_CASES_TAC `azim (vec 0) (basis 3) w1 w2 = &0` THENL
5565 [MP_TAC(ASSUME `azim (vec 0) (basis 3) w1 w2 = &0`) THEN
5566 W(MP_TAC o PART_MATCH (lhs o rand) AZIM_DIVH o lhs o lhand o snd) THEN
5567 ASM_REWRITE_TAC[PI_POS] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN
5568 REWRITE_TAC[REAL_MUL_LZERO; HAS_MEASURE_0] THEN
5569 MATCH_MP_TAC COPLANAR_IMP_NEGLIGIBLE THEN
5570 MATCH_MP_TAC COPLANAR_SUBSET THEN
5571 EXISTS_TAC `affine hull {vec 0:real^3,basis 3,w1,w2}` THEN
5573 [ASM_MESON_TAC[COPLANAR_AFFINE_HULL_COPLANAR; AZIM_EQ_0_PI_IMP_COPLANAR];
5575 MATCH_MP_TAC(SET_RULE `t SUBSET u ==> (s INTER t) SUBSET u`) THEN
5576 SIMP_TAC[aff_gt_def; AFFSIGN; sgn_gt; AFFINE_HULL_FINITE;
5577 FINITE_INSERT; FINITE_EMPTY] THEN
5578 REWRITE_TAC[SET_RULE `{a,b} UNION {c,d} = {a,b,c,d}`] THEN
5579 REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN GEN_TAC THEN
5580 MATCH_MP_TAC MONO_EXISTS THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[];
5582 SUBGOAL_THEN `&0 < azim (vec 0) (basis 3) w1 w2` ASSUME_TAC THENL
5583 [ASM_REWRITE_TAC[REAL_LT_LE; azim]; ALL_TAC] THEN
5584 ASM_CASES_TAC `azim (vec 0) (basis 3) w1 w2 < pi` THENL
5585 [ASM_SIMP_TAC[GSYM AZIM_DIHV_SAME; GSYM WEDGE_LUNE_GT] THEN
5586 ASM_SIMP_TAC[HAS_MEASURE_MEASURABLE_MEASURE; MEASURABLE_BALL_WEDGE;
5589 ASM_CASES_TAC `azim (vec 0) (basis 3) w1 w2 = pi` THENL
5590 [MP_TAC(ISPECL [`vec 0:real^3`; `basis 3:real^3`; `w1:real^3`; `w2:real^3`]
5592 ASM_REWRITE_TAC[REAL_LT_REFL] THEN ASM_REAL_ARITH_TAC;
5595 `dihV (vec 0) (basis 3) w1 w2 = azim (vec 0) (basis 3) w2 w1`
5597 [W(MP_TAC o PART_MATCH (lhs o rand) AZIM_COMPL o rand o snd) THEN
5598 ASM_REWRITE_TAC[] THEN DISCH_THEN SUBST1_TAC THEN
5599 ONCE_REWRITE_TAC[REAL_ARITH `x:real = y - z <=> z = y - x`] THEN
5600 MATCH_MP_TAC AZIM_DIHV_COMPL THEN
5601 ASM_REWRITE_TAC[GSYM REAL_NOT_LT];
5603 SUBGOAL_THEN `&0 < azim (vec 0) (basis 3) w2 w1 /\
5604 azim (vec 0) (basis 3) w2 w1 < pi`
5606 [W(MP_TAC o PART_MATCH (lhs o rand) AZIM_COMPL o lhand o rand o snd) THEN
5607 ASM_REWRITE_TAC[] THEN DISCH_THEN SUBST1_TAC THEN
5608 MP_TAC(ISPECL [`vec 0:real^3`; `basis 3:real^3`; `w1:real^3`; `w2:real^3`]
5610 REWRITE_TAC[CONJ_ASSOC] THEN DISCH_THEN(MP_TAC o CONJUNCT1) THEN
5613 SUBST1_TAC(SET_RULE `{w1:real^3,w2} = {w2,w1}`) THEN
5614 ASM_SIMP_TAC[GSYM AZIM_DIHV_SAME; GSYM WEDGE_LUNE_GT] THEN
5615 ASM_SIMP_TAC[HAS_MEASURE_MEASURABLE_MEASURE; MEASURABLE_BALL_WEDGE;
5616 VOLUME_BALL_WEDGE]);;
5618 let HAS_MEASURE_LUNE_SIMPLE = prove
5619 (`!z:real^3 w r w1 w2.
5620 &0 <= r /\ ~coplanar{z,w,w1,w2}
5621 ==> (ball(z,r) INTER aff_gt {z,w} {w1,w2})
5622 has_measure (dihV z w w1 w2 * &2 * r pow 3 / &3)`,
5624 ASM_CASES_TAC `w:real^3 = z` THENL
5625 [ASM_REWRITE_TAC[INSERT_AC; COPLANAR_3]; ALL_TAC] THEN
5626 REPEAT STRIP_TAC THEN MATCH_MP_TAC HAS_MEASURE_LUNE THEN
5627 ASM_REWRITE_TAC[] THEN
5628 MATCH_MP_TAC(TAUT `a /\ b /\ (a /\ b ==> c) ==> a /\ b /\ c`) THEN
5629 REPEAT(CONJ_TAC THENL
5630 [ASM_MESON_TAC[NOT_COPLANAR_NOT_COLLINEAR; INSERT_AC]; ALL_TAC]) THEN
5631 REPEAT STRIP_TAC THEN
5632 MP_TAC(ISPECL [`z:real^3`; `w:real^3`; `w1:real^3`; `w2:real^3`]
5634 ASM_REWRITE_TAC[REAL_ARITH `&2 * pi - pi = pi`; COND_ID] THEN
5635 ASM_MESON_TAC[AZIM_EQ_0_PI_IMP_COPLANAR]);;
5637 (* ------------------------------------------------------------------------- *)
5638 (* Now the volume of a solid triangle. *)
5639 (* ------------------------------------------------------------------------- *)
5641 let MEASURABLE_BALL_AFF_GT = prove
5642 (`!z r s t. measurable(ball(z,r) INTER aff_gt s t)`,
5643 MESON_TAC[MEASURABLE_CONVEX; CONVEX_INTER; CONVEX_AFF_GT; CONVEX_BALL;
5644 BOUNDED_INTER; BOUNDED_BALL]);;
5646 let AFF_GT_SHUFFLE = prove
5648 FINITE s /\ FINITE t /\
5649 vec 0 IN s /\ ~(vec 0 IN t) /\
5650 ~(v IN s) /\ ~(--v IN s) /\ ~(v IN t)
5651 ==> aff_gt (v INSERT s) t =
5652 aff_gt s (v INSERT t) UNION
5653 aff_gt s (--v INSERT t) UNION
5655 REPEAT STRIP_TAC THEN
5656 REWRITE_TAC[aff_gt_def; AFFSIGN_ALT; sgn_gt] THEN
5657 REWRITE_TAC[SET_RULE `(v INSERT s) UNION t = v INSERT (s UNION t)`;
5658 SET_RULE `s UNION (v INSERT t) = v INSERT (s UNION t)`] THEN
5659 ASM_SIMP_TAC[FINITE_INSERT; FINITE_UNION; AFFINE_HULL_FINITE_STEP_GEN;
5660 RIGHT_EXISTS_AND_THM; REAL_LT_ADD; REAL_HALF; FINITE_EMPTY] THEN
5661 REWRITE_TAC[IN_INSERT] THEN
5662 ASM_SIMP_TAC[SET_RULE
5664 ==> ((w IN s UNION t ==> w = a \/ w IN t ==> P w) <=>
5665 (w IN t ==> P w))`] THEN
5666 REWRITE_TAC[SET_RULE `x IN (s UNION t)
5667 ==> x IN t ==> P x <=> x IN t ==> P x`] THEN
5668 REWRITE_TAC[EXTENSION; IN_UNION; IN_ELIM_THM] THEN
5669 X_GEN_TAC `y:real^N` THEN EQ_TAC THENL
5670 [DISCH_THEN(X_CHOOSE_THEN `v:real` ASSUME_TAC) THEN
5671 ASM_CASES_TAC `&0 < v` THENL
5672 [DISJ1_TAC THEN EXISTS_TAC `v:real` THEN ASM_REWRITE_TAC[];
5674 ASM_CASES_TAC `v = &0` THENL
5676 FIRST_ASSUM(fun th -> MP_TAC th THEN MATCH_MP_TAC MONO_EXISTS) THEN
5677 ASM_REWRITE_TAC[REAL_SUB_RZERO; VECTOR_MUL_LZERO; VECTOR_SUB_RZERO];
5679 EXISTS_TAC `--v:real` THEN CONJ_TAC THENL
5680 [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
5681 FIRST_X_ASSUM(X_CHOOSE_THEN `f:real^N->real` STRIP_ASSUME_TAC) THEN
5682 EXISTS_TAC `\x:real^N. if x = vec 0 then f(x) + &2 * v else f(x)` THEN
5683 REWRITE_TAC[] THEN REPEAT CONJ_TAC THENL
5685 ASM_SIMP_TAC[SUM_CASES_1; FINITE_UNION; IN_UNION] THEN REAL_ARITH_TAC;
5686 REWRITE_TAC[VECTOR_ARITH `--a % --x:real^N = a % x`] THEN
5687 FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC RAND_CONV [GSYM th]) THEN
5688 MATCH_MP_TAC VSUM_EQ THEN REWRITE_TAC[] THEN
5689 REPEAT STRIP_TAC THEN COND_CASES_TAC THEN
5690 ASM_REWRITE_TAC[VECTOR_MUL_RZERO]];
5691 DISCH_THEN(DISJ_CASES_THEN MP_TAC) THENL [MESON_TAC[]; ALL_TAC] THEN
5692 DISCH_THEN(DISJ_CASES_THEN MP_TAC) THENL
5693 [DISCH_THEN(X_CHOOSE_THEN `a:real`
5694 (CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
5695 DISCH_THEN(X_CHOOSE_THEN `f:real^N->real` STRIP_ASSUME_TAC) THEN
5696 EXISTS_TAC `--a:real` THEN
5697 EXISTS_TAC `\x:real^N. if x = vec 0 then &2 * a + f(vec 0) else f x` THEN
5698 ASM_SIMP_TAC[SUM_CASES_1; FINITE_UNION; IN_UNION] THEN
5699 CONJ_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN
5700 CONJ_TAC THENL [REAL_ARITH_TAC; ALL_TAC] THEN
5701 ONCE_REWRITE_TAC[VECTOR_ARITH `y - --a % v:real^N = y - a % --v`] THEN
5702 FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC RAND_CONV [SYM th]) THEN
5703 MATCH_MP_TAC VSUM_EQ THEN REPEAT GEN_TAC THEN REWRITE_TAC[] THEN
5704 DISCH_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[VECTOR_MUL_RZERO];
5705 GEN_REWRITE_TAC RAND_CONV [SWAP_EXISTS_THM] THEN
5706 MATCH_MP_TAC MONO_EXISTS THEN REPEAT STRIP_TAC THEN
5707 EXISTS_TAC `&0` THEN ASM_REWRITE_TAC[REAL_SUB_RZERO] THEN
5708 VECTOR_ARITH_TAC]]);;
5710 let MEASURE_BALL_AFF_GT_SHUFFLE_LEMMA = prove
5713 independent(v INSERT((s DELETE vec 0) UNION t)) /\
5714 FINITE s /\ FINITE t /\ CARD(s UNION t) <= dimindex(:N) /\
5715 vec 0 IN s /\ ~(vec 0 IN t) /\
5716 ~(v IN s) /\ ~(--v IN s) /\ ~(v IN t)
5717 ==> measure(ball(vec 0,r) INTER aff_gt (v INSERT s) t) =
5718 measure(ball(vec 0,r) INTER aff_gt s (v INSERT t)) +
5719 measure(ball(vec 0,r) INTER aff_gt s (--v INSERT t))`,
5721 (`!s t u:real^N->bool.
5722 measurable s /\ measurable t /\ s INTER t = {} /\ negligible u
5723 ==> measure(s UNION t UNION u) = measure s + measure t`,
5724 REPEAT STRIP_TAC THEN REWRITE_TAC[UNION_ASSOC] THEN
5725 ASM_SIMP_TAC[GSYM MEASURE_DISJOINT_UNION; DISJOINT] THEN
5726 MATCH_MP_TAC MEASURE_NEGLIGIBLE_SYMDIFF THEN
5727 ASM_SIMP_TAC[MEASURABLE_UNION] THEN
5728 FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP
5729 (REWRITE_RULE[IMP_CONJ] NEGLIGIBLE_SUBSET)) THEN SET_TAC[]) in
5730 REPEAT STRIP_TAC THEN
5731 W(MP_TAC o PART_MATCH (lhs o rand) AFF_GT_SHUFFLE o
5732 rand o rand o lhand o snd) THEN
5733 ASM_REWRITE_TAC[] THEN DISCH_THEN SUBST1_TAC THEN
5734 REWRITE_TAC[UNION_OVER_INTER] THEN MATCH_MP_TAC lemma THEN
5735 ASM_REWRITE_TAC[MEASURABLE_BALL_AFF_GT] THEN CONJ_TAC THENL
5736 [MATCH_MP_TAC(SET_RULE
5737 `t INTER u = {} ==> (s INTER t) INTER (s INTER u) = {}`) THEN
5738 REWRITE_TAC[aff_gt_def; AFFSIGN_ALT; sgn_gt] THEN
5739 REWRITE_TAC[SET_RULE `(v INSERT s) UNION t = v INSERT (s UNION t)`;
5740 SET_RULE `s UNION (v INSERT t) = v INSERT (s UNION t)`] THEN
5741 ASM_SIMP_TAC[FINITE_INSERT; FINITE_UNION; AFFINE_HULL_FINITE_STEP_GEN;
5742 RIGHT_EXISTS_AND_THM; REAL_LT_ADD;
5743 REAL_HALF; FINITE_EMPTY] THEN
5744 REWRITE_TAC[IN_INSERT] THEN
5745 ASM_SIMP_TAC[SET_RULE
5746 `~(a IN s) ==> ((w IN s UNION t ==> w = a \/ w IN t ==> P w) <=>
5747 (w IN t ==> P w))`] THEN
5748 GEN_REWRITE_TAC I [EXTENSION] THEN
5749 REWRITE_TAC[IN_INTER; NOT_IN_EMPTY; IN_ELIM_THM] THEN
5750 X_GEN_TAC `y:real^N` THEN
5751 DISCH_THEN(CONJUNCTS_THEN2
5752 (X_CHOOSE_THEN `a:real`
5753 (CONJUNCTS_THEN2 ASSUME_TAC
5754 (X_CHOOSE_THEN `f:real^N->real` STRIP_ASSUME_TAC)))
5755 (X_CHOOSE_THEN `b:real`
5756 (CONJUNCTS_THEN2 ASSUME_TAC
5757 (X_CHOOSE_THEN `g:real^N->real` STRIP_ASSUME_TAC)))) THEN
5758 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [INDEPENDENT_EXPLICIT]) THEN
5759 REWRITE_TAC[FINITE_INSERT; FINITE_DELETE; FINITE_UNION] THEN
5760 DISCH_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC MP_TAC) THEN
5761 DISCH_THEN(MP_TAC o SPEC
5762 `\x. if x = v then a + b else (f:real^N->real) x - g x`) THEN
5763 ASM_SIMP_TAC[VSUM_CLAUSES; FINITE_DELETE; FINITE_UNION] THEN
5764 ASM_REWRITE_TAC[IN_DELETE; IN_UNION] THEN
5765 REWRITE_TAC[NOT_IMP] THEN CONJ_TAC THENL
5766 [ALL_TAC; DISCH_THEN(MP_TAC o SPEC `v:real^N`) THEN
5767 REWRITE_TAC[IN_INSERT] THEN ASM_REAL_ARITH_TAC] THEN
5768 ASM_SIMP_TAC[SET_RULE
5769 `~(a IN t) ==> (s DELETE a) UNION t = (s UNION t) DELETE a`] THEN
5770 ASM_SIMP_TAC[VSUM_DELETE_CASES; FINITE_UNION; IN_UNION] THEN
5771 REWRITE_TAC[VECTOR_MUL_RZERO; VECTOR_SUB_RZERO] THEN
5773 `!x:real^N. (if x = v then a + b else f x - g x) % x =
5774 (if x = v then a else f x) % x -
5775 (if x = v then --b else g x) % x`
5776 (fun th -> REWRITE_TAC[th])
5778 [GEN_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN VECTOR_ARITH_TAC;
5779 ASM_SIMP_TAC[VSUM_SUB; FINITE_UNION]] THEN
5780 MATCH_MP_TAC EQ_TRANS THEN
5781 EXISTS_TAC `(a + b) % v + (y - a % v) - (y - b % --v):real^N` THEN
5782 CONJ_TAC THENL [ALL_TAC; VECTOR_ARITH_TAC] THEN
5783 AP_TERM_TAC THEN BINOP_TAC THEN
5784 FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC RAND_CONV [GSYM th]) THEN
5785 MATCH_MP_TAC VSUM_EQ THEN GEN_TAC THEN REWRITE_TAC[] THEN
5786 COND_CASES_TAC THEN ASM_REWRITE_TAC[IN_UNION];
5787 MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN
5788 EXISTS_TAC `aff_gt s t :real^N->bool` THEN
5789 REWRITE_TAC[INTER_SUBSET] THEN
5790 MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN
5791 EXISTS_TAC `affine hull (s UNION t:real^N->bool)` THEN
5792 REWRITE_TAC[AFF_GT_SUBSET_AFFINE_HULL] THEN
5793 ASM_SIMP_TAC[AFFINE_HULL_EQ_SPAN; IN_UNION; HULL_INC] THEN
5794 ONCE_REWRITE_TAC[GSYM SPAN_DELETE_0] THEN
5795 MATCH_MP_TAC NEGLIGIBLE_LOWDIM THEN
5796 MATCH_MP_TAC LET_TRANS THEN
5797 EXISTS_TAC `CARD((s UNION t) DELETE (vec 0:real^N))` THEN
5798 ASM_SIMP_TAC[DIM_LE_CARD; FINITE_DELETE; FINITE_UNION; DIM_SPAN] THEN
5799 ASM_SIMP_TAC[CARD_DELETE; IN_UNION; FINITE_UNION] THEN
5800 MATCH_MP_TAC(ARITH_RULE `1 <= n /\ x <= n ==> x - 1 < n`) THEN
5801 ASM_REWRITE_TAC[DIMINDEX_GE_1]]);;
5803 let MEASURE_BALL_AFF_GT_SHUFFLE = prove
5805 &0 <= r /\ ~(v IN (s UNION t)) /\
5806 independent(v INSERT (s UNION t))
5807 ==> measure(ball(vec 0,r) INTER aff_gt (vec 0 INSERT v INSERT s) t) =
5808 measure(ball(vec 0,r) INTER aff_gt (vec 0 INSERT s) (v INSERT t)) +
5809 measure(ball(vec 0,r) INTER
5810 aff_gt (vec 0 INSERT s) (--v INSERT t))`,
5811 REWRITE_TAC[IN_UNION; DE_MORGAN_THM] THEN REPEAT STRIP_TAC THEN
5812 MP_TAC(ISPECL [`r:real`; `(vec 0:real^N) INSERT s`;
5813 `t:real^N->bool`; `v:real^N`]
5814 MEASURE_BALL_AFF_GT_SHUFFLE_LEMMA) THEN
5815 ANTS_TAC THENL [ALL_TAC; REWRITE_TAC[INSERT_AC]] THEN
5816 ASM_REWRITE_TAC[IN_INSERT; FINITE_INSERT] THEN
5817 FIRST_ASSUM(MP_TAC o MATCH_MP INDEPENDENT_NONZERO) THEN
5818 REWRITE_TAC[IN_INSERT; IN_UNION; DE_MORGAN_THM] THEN
5819 STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
5820 FIRST_ASSUM(MP_TAC o MATCH_MP INDEPENDENT_BOUND) THEN
5821 REWRITE_TAC[FINITE_INSERT; FINITE_UNION] THEN
5822 DISCH_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC MP_TAC) THEN
5823 REWRITE_TAC[SET_RULE `(a INSERT s) UNION t = a INSERT (s UNION t)`] THEN
5824 ASM_SIMP_TAC[CARD_CLAUSES; FINITE_UNION; IN_UNION; FINITE_INSERT] THEN
5825 DISCH_TAC THEN ASM_REWRITE_TAC[VECTOR_NEG_EQ_0] THEN CONJ_TAC THENL
5826 [FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP
5827 (REWRITE_RULE[IMP_CONJ] INDEPENDENT_MONO)) THEN
5829 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [independent]) THEN
5830 REWRITE_TAC[dependent; CONTRAPOS_THM] THEN DISCH_TAC THEN
5831 EXISTS_TAC `v:real^N` THEN REWRITE_TAC[IN_INSERT] THEN
5832 GEN_REWRITE_TAC LAND_CONV [GSYM VECTOR_NEG_NEG] THEN
5833 MATCH_MP_TAC SPAN_NEG THEN MATCH_MP_TAC SPAN_SUPERSET THEN
5834 ASM_REWRITE_TAC[IN_DELETE; VECTOR_ARITH `--v:real^N = v <=> v = vec 0`;
5835 IN_INSERT; IN_UNION]]);;
5837 let MEASURE_LUNE_DECOMPOSITION = prove
5839 &0 <= r /\ ~coplanar {vec 0, v1, v2, v3}
5840 ==> measure(ball(vec 0,r) INTER aff_gt {vec 0} {v1,v2,v3}) +
5841 measure(ball(vec 0,r) INTER aff_gt {vec 0} {--v1,v2,v3}) =
5842 dihV (vec 0) v1 v2 v3 * &2 * r pow 3 / &3`,
5843 let rec distinctpairs l =
5845 x::t -> itlist (fun y a -> (x,y) :: a) t (distinctpairs t)
5847 REPEAT GEN_TAC THEN MAP_EVERY
5848 (fun t -> ASM_CASES_TAC t THENL
5849 [ASM_REWRITE_TAC[INSERT_AC; COPLANAR_3]; ALL_TAC])
5850 (map mk_eq (distinctpairs
5851 [`v3:real^3`; `v2:real^3`; `v1:real^3`; `vec 0:real^3`])) THEN
5852 REPEAT STRIP_TAC THEN
5853 ASM_SIMP_TAC[GSYM(REWRITE_RULE[HAS_MEASURE_MEASURABLE_MEASURE]
5854 HAS_MEASURE_LUNE_SIMPLE)] THEN
5855 CONV_TAC SYM_CONV THEN MATCH_MP_TAC MEASURE_BALL_AFF_GT_SHUFFLE THEN
5856 ASM_REWRITE_TAC[UNION_EMPTY; IN_INSERT; NOT_IN_EMPTY] THEN
5857 ASM_SIMP_TAC[NOT_COPLANAR_0_4_IMP_INDEPENDENT]);;
5859 let SOLID_TRIANGLE_CONGRUENT_NEG = prove
5860 (`!r v1 v2 v3:real^N.
5861 measure(ball(vec 0,r) INTER aff_gt {vec 0} {--v1, --v2, --v3}) =
5862 measure(ball(vec 0,r) INTER aff_gt {vec 0} {v1, v2, v3})`,
5865 `ball(vec 0:real^N,r) INTER aff_gt {vec 0} {--v1, --v2, --v3} =
5867 (ball(vec 0,r) INTER aff_gt {vec 0} {v1, v2, v3})`
5870 MATCH_MP_TAC MEASURE_ORTHOGONAL_IMAGE_EQ THEN
5871 REWRITE_TAC[ORTHOGONAL_TRANSFORMATION; linear; NORM_NEG] THEN
5872 CONJ_TAC THEN VECTOR_ARITH_TAC] THEN
5873 CONV_TAC SYM_CONV THEN MATCH_MP_TAC SURJECTIVE_IMAGE_EQ THEN
5874 CONJ_TAC THENL [MESON_TAC[VECTOR_NEG_NEG]; ALL_TAC] THEN
5875 REWRITE_TAC[IN_INTER; IN_BALL_0; NORM_NEG] THEN
5876 REWRITE_TAC[AFFSIGN_ALT; aff_gt_def; sgn_gt; IN_ELIM_THM] THEN
5877 REWRITE_TAC[SET_RULE `{a} UNION {b,c,d} = {a,b,d,c}`] THEN
5878 REWRITE_TAC[SET_RULE `x IN {a} <=> a = x`] THEN
5879 ASM_SIMP_TAC[FINITE_INSERT; FINITE_UNION; AFFINE_HULL_FINITE_STEP_GEN;
5880 RIGHT_EXISTS_AND_THM; REAL_LT_ADD; REAL_HALF; FINITE_EMPTY] THEN
5881 ASM_REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY] THEN
5882 REWRITE_TAC[VECTOR_MUL_RZERO; VECTOR_SUB_RZERO] THEN
5883 REWRITE_TAC[VECTOR_ARITH `vec 0:real^N = --x <=> vec 0 = x`] THEN
5884 REWRITE_TAC[VECTOR_ARITH `--x - a % --w:real^N = --(x - a % w)`] THEN
5885 REWRITE_TAC[VECTOR_NEG_EQ_0]);;
5887 let VOLUME_SOLID_TRIANGLE = prove
5888 (`!r v0 v1 v2 v3:real^3.
5889 &0 < r /\ ~coplanar{v0, v1, v2, v3}
5890 ==> measure(ball(v0,r) INTER aff_gt {v0} {v1,v2,v3}) =
5891 let a123 = dihV v0 v1 v2 v3 in
5892 let a231 = dihV v0 v2 v3 v1 in
5893 let a312 = dihV v0 v3 v1 v2 in
5894 (a123 + a231 + a312 - pi) * r pow 3 / &3`,
5896 W(MP_TAC o PART_MATCH (lhs o rand) MEASURE_BALL_AFF_GT_SHUFFLE o
5897 convl o lhand o lhand o snd) THEN
5898 ASM_REWRITE_TAC[UNION_EMPTY; IN_INSERT; IN_UNION; NOT_IN_EMPTY] THEN
5899 REWRITE_TAC[SET_RULE `(a INSERT s) UNION t = a INSERT (s UNION t)`] THEN
5900 ASM_SIMP_TAC[UNION_EMPTY; REAL_LT_IMP_LE] THEN ANTS_TAC THENL
5902 [DISCH_THEN(STRIP_THM_THEN SUBST_ALL_TAC) THEN
5903 RULE_ASSUM_TAC(REWRITE_RULE[INSERT_AC]) THEN
5904 RULE_ASSUM_TAC(REWRITE_RULE[COPLANAR_3]) THEN
5905 FIRST_ASSUM CONTR_TAC;
5906 MATCH_MP_TAC NOT_COPLANAR_0_4_IMP_INDEPENDENT THEN
5907 RULE_ASSUM_TAC(REWRITE_RULE[INSERT_AC]) THEN
5908 ASM_REWRITE_TAC[INSERT_AC]];
5909 DISCH_THEN SUBST1_TAC] in
5910 GEN_TAC THEN GEOM_ORIGIN_TAC `v0:real^3` THEN
5911 CONV_TAC(TOP_DEPTH_CONV let_CONV) THEN REPEAT STRIP_TAC THEN
5913 `measure(ball(vec 0:real^3,r) INTER aff_gt {vec 0,v1,v2,v3} {}) =
5914 &4 / &3 * pi * r pow 3`
5916 [MP_TAC(SPECL [`vec 0:real^3`; `r:real`] VOLUME_BALL) THEN
5917 ASM_SIMP_TAC[REAL_LT_IMP_LE] THEN
5918 DISCH_THEN(SUBST1_TAC o SYM) THEN AP_TERM_TAC THEN
5919 MATCH_MP_TAC(SET_RULE `t = UNIV ==> s INTER t = s`) THEN
5920 REWRITE_TAC[AFF_GT_EQ_AFFINE_HULL] THEN
5921 SIMP_TAC[AFFINE_HULL_EQ_SPAN; HULL_INC; IN_INSERT; SPAN_INSERT_0] THEN
5922 REWRITE_TAC[SET_RULE `s = UNIV <=> UNIV SUBSET s`] THEN
5923 MATCH_MP_TAC CARD_GE_DIM_INDEPENDENT THEN
5924 ASM_SIMP_TAC[DIM_UNIV; DIMINDEX_3; SUBSET_UNIV] THEN
5925 ASM_SIMP_TAC[NOT_COPLANAR_0_4_IMP_INDEPENDENT] THEN
5926 SIMP_TAC[CARD_CLAUSES; FINITE_INSERT; FINITE_EMPTY] THEN
5927 REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY] THEN
5929 ASM_CASES_TAC t THENL
5930 [FIRST_X_ASSUM SUBST_ALL_TAC THEN
5931 RULE_ASSUM_TAC(REWRITE_RULE[INSERT_AC; COPLANAR_3]) THEN
5934 [`v3:real^3 = v2`; `v3:real^3 = v1`; `v2:real^3 = v1`] THEN
5935 CONV_TAC NUM_REDUCE_CONV;
5938 `~(coplanar {vec 0:real^3,v1,v2,v3}) /\
5939 ~(coplanar {vec 0,--v1,v2,v3}) /\
5940 ~(coplanar {vec 0,v1,--v2,v3}) /\
5941 ~(coplanar {vec 0,--v1,--v2,v3}) /\
5942 ~(coplanar {vec 0,--v1,--v2,v3}) /\
5943 ~(coplanar {vec 0,--v1,v2,--v3}) /\
5944 ~(coplanar {vec 0,v1,--v2,--v3}) /\
5945 ~(coplanar {vec 0,--v1,--v2,--v3}) /\
5946 ~(coplanar {vec 0,--v1,--v2,--v3})`
5947 STRIP_ASSUME_TAC THENL
5949 (REWRITE_TAC[COPLANAR_INSERT_0_NEG] THEN
5950 ONCE_REWRITE_TAC[SET_RULE `{vec 0,a,b,c} = {vec 0,b,c,a}`]) THEN
5955 lhand o lhand; rand o lhand; lhand o rand; rand o rand] THEN
5956 MP_TAC(ISPECL [`v1:real^3`; `v2:real^3`; `v3:real^3`]
5957 MEASURE_LUNE_DECOMPOSITION) THEN
5958 MP_TAC(ISPECL [`v2:real^3`; `v3:real^3`; `v1:real^3`]
5959 MEASURE_LUNE_DECOMPOSITION) THEN
5960 MP_TAC(ISPECL [`v3:real^3`; `v1:real^3`; `v2:real^3`]
5961 MEASURE_LUNE_DECOMPOSITION) THEN
5962 MP_TAC(ISPECL [`--v1:real^3`; `--v2:real^3`; `--v3:real^3`]
5963 MEASURE_LUNE_DECOMPOSITION) THEN
5964 MP_TAC(ISPECL [`--v2:real^3`; `--v3:real^3`; `--v1:real^3`]
5965 MEASURE_LUNE_DECOMPOSITION) THEN
5966 MP_TAC(ISPECL [`--v3:real^3`; `--v1:real^3`; `--v2:real^3`]
5967 MEASURE_LUNE_DECOMPOSITION) THEN
5968 ASM_REWRITE_TAC[VECTOR_NEG_NEG] THEN
5969 ASM_SIMP_TAC[REAL_LT_IMP_LE; INSERT_AC] THEN
5970 RULE_ASSUM_TAC(REWRITE_RULE[INSERT_AC]) THEN ASM_REWRITE_TAC[] THEN
5971 REWRITE_TAC[DIHV_NEG_0] THEN
5972 REWRITE_TAC[SOLID_TRIANGLE_CONGRUENT_NEG] THEN
5973 REWRITE_TAC[INSERT_AC] THEN REAL_ARITH_TAC);;
5975 (* ------------------------------------------------------------------------- *)
5976 (* Volume of wedge of a frustum. *)
5977 (* ------------------------------------------------------------------------- *)
5979 let MEASURABLE_BOUNDED_INTER_OPEN = prove
5980 (`!s t:real^N->bool.
5981 measurable s /\ bounded s /\ open t ==> measurable(s INTER t)`,
5982 REPEAT STRIP_TAC THEN
5983 FIRST_X_ASSUM(MP_TAC o MATCH_MP BOUNDED_SUBSET_OPEN_INTERVAL) THEN
5984 REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN
5985 MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real^N`] THEN
5986 DISCH_THEN(SUBST1_TAC o MATCH_MP (SET_RULE
5987 `s SUBSET i ==> s INTER t = s INTER (t INTER i)`)) THEN
5988 MATCH_MP_TAC MEASURABLE_INTER THEN ASM_REWRITE_TAC[] THEN
5989 MATCH_MP_TAC MEASURABLE_OPEN THEN
5990 ASM_SIMP_TAC[OPEN_INTER; OPEN_INTERVAL; BOUNDED_INTER; BOUNDED_INTERVAL]);;
5992 let SLICE_SPECIAL_WEDGE = prove
5994 ~collinear {vec 0, basis 3, w1} /\ ~collinear {vec 0, basis 3, w2}
5995 ==> slice 3 t (wedge (vec 0) (basis 3) w1 w2) =
5996 {z | &0 < Arg(z / dropout 3 w1) /\
5997 Arg(z / dropout 3 w1) < Arg(dropout 3 w2 / dropout 3 w1)}`,
5998 REWRITE_TAC[wedge] THEN
5999 ONCE_REWRITE_TAC[TAUT `~a /\ b /\ c <=> ~(~a ==> ~(b /\ c))`] THEN
6000 ASM_SIMP_TAC[AZIM_ARG] THEN REWRITE_TAC[COLLINEAR_BASIS_3] THEN
6001 REWRITE_TAC[NOT_IMP; GSYM CONJ_ASSOC; DROPOUT_0] THEN
6002 MAP_EVERY ABBREV_TAC
6003 [`v1:real^2 = dropout 3 (w1:real^3)`;
6004 `v2:real^2 = dropout 3 (w2:real^3)`] THEN
6005 REWRITE_TAC[SLICE_DROPOUT_3; VEC_COMPONENT; REAL_SUB_RZERO] THEN
6006 REPEAT STRIP_TAC THEN
6007 REWRITE_TAC[EXTENSION; IN_ELIM_THM; COMPLEX_VEC_0] THEN
6008 X_GEN_TAC `w:complex` THEN
6009 ASM_CASES_TAC `w = Cx(&0)` THEN ASM_REWRITE_TAC[] THEN
6010 ASM_REWRITE_TAC[complex_div; COMPLEX_MUL_LZERO; ARG_0; REAL_LT_REFL]);;
6012 let VOLUME_FRUSTT_WEDGE = prove
6013 (`!v0 v1:real^3 w1 w2 h a.
6014 &0 < a /\ ~collinear {v0,v1,w1} /\ ~collinear {v0,v1,w2}
6015 ==> bounded(frustt v0 v1 h a INTER wedge v0 v1 w1 w2) /\
6016 measurable(frustt v0 v1 h a INTER wedge v0 v1 w1 w2) /\
6017 measure(frustt v0 v1 h a INTER wedge v0 v1 w1 w2) =
6018 if &1 <= a \/ h < &0 then &0
6019 else azim v0 v1 w1 w2 * ((h / a) pow 2 - h pow 2) * h / &6`,
6020 REPEAT GEN_TAC THEN ASM_CASES_TAC `v1:real^3 = v0` THENL
6021 [ASM_REWRITE_TAC[INSERT_AC; COLLINEAR_2]; STRIP_TAC] THEN
6022 MATCH_MP_TAC(TAUT `a /\ b /\ (a /\ b ==> c) ==> a /\ b /\ c`) THEN
6023 REPEAT CONJ_TAC THENL
6024 [MATCH_MP_TAC BOUNDED_INTER THEN ASM_SIMP_TAC[VOLUME_FRUSTT_STRONG];
6025 MATCH_MP_TAC MEASURABLE_BOUNDED_INTER_OPEN THEN
6026 ASM_SIMP_TAC[VOLUME_FRUSTT_STRONG; OPEN_WEDGE];
6028 REWRITE_TAC[frustt; frustum; rcone_gt; rconesgn; IN_ELIM_THM] THEN
6029 CONV_TAC(TOP_DEPTH_CONV let_CONV) THEN
6030 POP_ASSUM_LIST(MP_TAC o end_itlist CONJ) THEN
6031 GEOM_ORIGIN_TAC `v0:real^3` THEN
6032 REWRITE_TAC[VECTOR_SUB_RZERO; REAL_MUL_LZERO; DIST_0; real_gt] THEN
6033 GEOM_BASIS_MULTIPLE_TAC 3 `v1:real^3` THEN
6034 X_GEN_TAC `b:real` THEN GEN_REWRITE_TAC LAND_CONV [REAL_LE_LT] THEN
6035 ASM_CASES_TAC `b = &0` THEN ASM_REWRITE_TAC[VECTOR_MUL_LZERO] THEN
6036 ASM_SIMP_TAC[COLLINEAR_SPECIAL_SCALE; WEDGE_SPECIAL_SCALE] THEN
6037 ASM_REWRITE_TAC[VECTOR_MUL_EQ_0] THEN
6038 DISCH_TAC THEN REPEAT GEN_TAC THEN STRIP_TAC THEN
6039 ASM_CASES_TAC `&1 <= a` THEN ASM_REWRITE_TAC[] THENL
6041 `!y:real^3. ~(norm(y) * norm(b % basis 3:real^3) * a
6042 < y dot (b % basis 3))`
6043 (fun th -> REWRITE_TAC[th; EMPTY_GSPEC; MEASURABLE_EMPTY;
6044 INTER_EMPTY; MEASURE_EMPTY]) THEN
6045 REWRITE_TAC[REAL_NOT_LT] THEN X_GEN_TAC `y:real^3` THEN
6046 MATCH_MP_TAC(REAL_ARITH `abs(x) <= a ==> x <= a`) THEN
6047 SIMP_TAC[DOT_RMUL; NORM_MUL; REAL_ABS_MUL; DOT_BASIS; NORM_BASIS;
6048 DIMINDEX_3; ARITH] THEN
6049 REWRITE_TAC[REAL_ARITH
6050 `b * y <= n * (b * &1) * a <=> b * &1 * y <= b * a * n`] THEN
6051 MATCH_MP_TAC REAL_LE_LMUL THEN REWRITE_TAC[REAL_ABS_POS] THEN
6052 MATCH_MP_TAC REAL_LE_MUL2 THEN
6053 ASM_SIMP_TAC[REAL_POS; REAL_ABS_POS; COMPONENT_LE_NORM; DIMINDEX_3; ARITH];
6055 RULE_ASSUM_TAC(REWRITE_RULE[REAL_NOT_LE]) THEN
6056 SIMP_TAC[NORM_MUL; NORM_BASIS; DOT_BASIS; DOT_RMUL; DIMINDEX_3; ARITH] THEN
6057 ONCE_REWRITE_TAC[REAL_ARITH `n * x * a:real = x * n * a`] THEN
6058 ASM_REWRITE_TAC[real_abs; REAL_MUL_RID] THEN
6059 ASM_SIMP_TAC[REAL_MUL_RID; REAL_LT_LMUL_EQ; REAL_LT_MUL_EQ; NORM_POS_LT] THEN
6060 ASM_SIMP_TAC[VECTOR_MUL_EQ_0; BASIS_NONZERO; DIMINDEX_3; ARITH;
6061 REAL_LT_IMP_NZ] THEN
6062 ASM_SIMP_TAC[GSYM REAL_LT_RDIV_EQ; NORM_LT_SQUARE] THEN
6063 ASM_SIMP_TAC[REAL_POW_DIV; REAL_POW_LT; REAL_LT_RDIV_EQ] THEN
6064 REWRITE_TAC[REAL_ARITH `(&0 * x < y /\ u < v) /\ &0 < y /\ y < h <=>
6065 &0 < y /\ y < h /\ u < v`] THEN
6066 DISCH_TAC THEN MATCH_MP_TAC(INST_TYPE [`:2`,`:M`] FUBINI_SIMPLE_ALT) THEN
6067 EXISTS_TAC `3` THEN ASM_REWRITE_TAC[DIMINDEX_2; DIMINDEX_3; ARITH] THEN
6068 ASM_SIMP_TAC[WEDGE_SPECIAL_SCALE; REAL_LT_IMP_LE] THEN
6069 ASM_SIMP_TAC[REAL_LT_LMUL_EQ; SLICE_INTER; DIMINDEX_2;
6070 DIMINDEX_3; ARITH] THEN
6072 `!t. slice 3 t {y:real^3 | norm y * a < y$3 /\ &0 < y$3 /\ y$3 < h} =
6074 then ball(vec 0:real^2,sqrt(inv(a pow 2) - &1) * t)
6076 (fun th -> ASM_SIMP_TAC[th; SLICE_SPECIAL_WEDGE])
6078 [REWRITE_TAC[EXTENSION] THEN
6079 MAP_EVERY X_GEN_TAC [`t:real`; `z:real^2`] THEN
6080 SIMP_TAC[SLICE_123; DIMINDEX_2; DIMINDEX_3; ARITH; IN_ELIM_THM;
6081 VECTOR_3; DOT_3; GSYM DOT_2] THEN
6082 ASM_CASES_TAC `t < h` THEN ASM_REWRITE_TAC[NOT_IN_EMPTY] THEN
6083 REWRITE_TAC[IN_BALL_0; IN_DELETE] THEN
6084 MATCH_MP_TAC(REAL_ARITH
6085 `&0 <= a /\ (a < t <=> u < v) ==> (a < t /\ &0 < t <=> u < v)`) THEN
6086 ASM_SIMP_TAC[REAL_LE_MUL; NORM_POS_LE; REAL_LT_IMP_LE] THEN
6087 ASM_SIMP_TAC[GSYM REAL_LT_RDIV_EQ; NORM_LT_SQUARE] THEN
6088 SUBGOAL_THEN `&0 < inv(a pow 2) - &1` ASSUME_TAC THENL
6089 [REWRITE_TAC[REAL_SUB_LT] THEN MATCH_MP_TAC REAL_INV_1_LT THEN
6090 ASM_SIMP_TAC[REAL_POW_1_LT; REAL_LT_IMP_LE; ARITH; REAL_POW_LT];
6092 ASM_SIMP_TAC[REAL_LT_MUL; SQRT_POS_LT; REAL_POW_MUL; SQRT_POW_2;
6093 REAL_LT_IMP_LE; REAL_LT_MUL_EQ] THEN
6094 ASM_SIMP_TAC[real_div; REAL_LT_MUL_EQ; REAL_LT_INV_EQ] THEN
6095 ASM_CASES_TAC `&0 < t` THEN ASM_REWRITE_TAC[] THEN
6096 REWRITE_TAC[DOT_3; DOT_2; VECTOR_3; REAL_INV_POW] THEN
6099 GEN_REWRITE_TAC (LAND_CONV o TOP_DEPTH_CONV) [COND_RATOR; COND_RAND] THEN
6100 GEN_REWRITE_TAC (RAND_CONV o RATOR_CONV o LAND_CONV o TOP_DEPTH_CONV)
6101 [COND_RATOR; COND_RAND] THEN
6102 REWRITE_TAC[INTER_EMPTY; MEASURABLE_EMPTY; MEASURE_EMPTY] THEN
6103 REWRITE_TAC[INTER; IN_BALL_0; IN_ELIM_THM] THEN
6104 RULE_ASSUM_TAC(REWRITE_RULE[COLLINEAR_BASIS_3]) THEN
6105 ASM_SIMP_TAC[REWRITE_RULE[HAS_MEASURE_MEASURABLE_MEASURE]
6106 HAS_MEASURE_OPEN_SECTOR_LT_GEN] THEN
6107 REWRITE_TAC[COND_ID] THEN
6108 SUBGOAL_THEN `&0 < inv(a pow 2) - &1` ASSUME_TAC THENL
6109 [REWRITE_TAC[REAL_SUB_LT] THEN MATCH_MP_TAC REAL_INV_1_LT THEN
6110 ASM_SIMP_TAC[REAL_POW_1_LT; REAL_LT_IMP_LE; ARITH; REAL_POW_LT];
6112 ASM_SIMP_TAC[REAL_LE_MUL_EQ; SQRT_POS_LT] THEN
6113 ASM_SIMP_TAC[AZIM_SPECIAL_SCALE; AZIM_ARG; COLLINEAR_BASIS_3] THEN
6114 MATCH_MP_TAC HAS_REAL_INTEGRAL_EQ THEN
6116 `\t. if &0 < t /\ t < h
6117 then Arg(dropout 3 (w2:real^3) / dropout 3 (w1:real^3)) / &2 *
6118 (inv(a pow 2) - &1) * t pow 2
6121 [X_GEN_TAC `t:real` THEN DISCH_TAC THEN REWRITE_TAC[] THEN
6122 ASM_CASES_TAC `t < h` THEN ASM_REWRITE_TAC[] THEN
6123 REWRITE_TAC[REAL_ARITH `&0 <= t <=> t = &0 \/ &0 < t`] THEN
6124 ASM_CASES_TAC `t = &0` THEN
6125 ASM_REWRITE_TAC[REAL_LT_REFL; REAL_MUL_RZERO; SQRT_0] THEN
6126 CONV_TAC REAL_RAT_REDUCE_CONV THEN REWRITE_TAC[REAL_MUL_RZERO] THEN
6127 ASM_SIMP_TAC[REAL_POW_MUL; SQRT_POW_2; REAL_LT_IMP_LE] THEN
6130 REWRITE_TAC[GSYM IN_REAL_INTERVAL; HAS_REAL_INTEGRAL_RESTRICT_UNIV] THEN
6131 REWRITE_TAC[HAS_REAL_INTEGRAL_OPEN_INTERVAL] THEN
6132 COND_CASES_TAC THENL
6133 [ASM_MESON_TAC[REAL_INTERVAL_EQ_EMPTY; HAS_REAL_INTEGRAL_EMPTY];
6134 RULE_ASSUM_TAC(REWRITE_RULE[REAL_NOT_LT])] THEN
6135 ABBREV_TAC `g = Arg(dropout 3 (w2:real^3) / dropout 3 (w1:real^3))` THEN
6137 [`\t. g / &6 * (inv (a pow 2) - &1) * t pow 3`;
6138 `\t. g / &2 * (inv (a pow 2) - &1) * t pow 2`;
6139 `&0`; `h:real`] REAL_FUNDAMENTAL_THEOREM_OF_CALCULUS) THEN
6140 REWRITE_TAC[] THEN ANTS_TAC THENL
6141 [ASM_REWRITE_TAC[] THEN
6142 REPEAT STRIP_TAC THEN REAL_DIFF_TAC THEN
6143 CONV_TAC NUM_REDUCE_CONV THEN CONV_TAC REAL_RING;
6144 MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN
6145 UNDISCH_TAC `&0 < a` THEN CONV_TAC REAL_FIELD]);;
6147 (* ------------------------------------------------------------------------- *)
6148 (* Wedge of a conic cap. *)
6149 (* ------------------------------------------------------------------------- *)
6151 let VOLUME_CONIC_CAP_WEDGE_WEAK = prove
6152 (`!v0 v1:real^3 w1 w2 r a.
6153 &0 < a /\ ~collinear {v0,v1,w1} /\ ~collinear {v0,v1,w2}
6154 ==> bounded(conic_cap v0 v1 r a INTER wedge v0 v1 w1 w2) /\
6155 measurable(conic_cap v0 v1 r a INTER wedge v0 v1 w1 w2) /\
6156 measure(conic_cap v0 v1 r a INTER wedge v0 v1 w1 w2) =
6157 if &1 <= a \/ r < &0 then &0
6158 else azim v0 v1 w1 w2 / &3 * (&1 - a) * r pow 3`,
6159 REPEAT GEN_TAC THEN ASM_CASES_TAC `v1:real^3 = v0` THENL
6160 [ASM_REWRITE_TAC[COLLINEAR_2; INSERT_AC]; STRIP_TAC] THEN
6161 MATCH_MP_TAC(TAUT `a /\ b /\ (a /\ b ==> c) ==> a /\ b /\ c`) THEN
6162 REPEAT CONJ_TAC THENL
6163 [MATCH_MP_TAC BOUNDED_INTER THEN ASM_SIMP_TAC[VOLUME_CONIC_CAP_STRONG];
6164 MATCH_MP_TAC MEASURABLE_BOUNDED_INTER_OPEN THEN
6165 ASM_SIMP_TAC[VOLUME_CONIC_CAP_STRONG; OPEN_WEDGE];
6167 REWRITE_TAC[conic_cap; rcone_gt; rconesgn; IN_ELIM_THM] THEN
6168 REWRITE_TAC[ONCE_REWRITE_RULE[DIST_SYM] normball; GSYM ball] THEN
6169 CONV_TAC(TOP_DEPTH_CONV let_CONV) THEN
6170 POP_ASSUM_LIST(MP_TAC o end_itlist CONJ) THEN
6171 GEOM_ORIGIN_TAC `v0:real^3` THEN
6172 REWRITE_TAC[VECTOR_SUB_RZERO; REAL_MUL_LZERO; DIST_0; real_gt] THEN
6173 GEOM_BASIS_MULTIPLE_TAC 3 `v1:real^3` THEN
6174 X_GEN_TAC `b:real` THEN
6175 ASM_CASES_TAC `b = &0` THEN ASM_REWRITE_TAC[VECTOR_MUL_LZERO] THEN
6176 SIMP_TAC[COLLINEAR_SPECIAL_SCALE; WEDGE_SPECIAL_SCALE] THEN
6177 ASM_REWRITE_TAC[VECTOR_MUL_EQ_0] THEN
6178 DISCH_TAC THEN REPEAT GEN_TAC THEN STRIP_TAC THEN
6179 ASM_CASES_TAC `&1 <= a` THEN ASM_REWRITE_TAC[] THENL
6181 `!y:real^3. ~(norm(y) * norm(b % basis 3:real^3) * a
6182 < y dot (b % basis 3))`
6183 (fun th -> REWRITE_TAC[th; EMPTY_GSPEC; INTER_EMPTY; MEASURE_EMPTY;
6184 MEASURABLE_EMPTY; BOUNDED_EMPTY; CONVEX_EMPTY]) THEN
6185 REWRITE_TAC[REAL_NOT_LT] THEN X_GEN_TAC `y:real^3` THEN
6186 MATCH_MP_TAC(REAL_ARITH `abs(x) <= a ==> x <= a`) THEN
6187 SIMP_TAC[DOT_RMUL; NORM_MUL; REAL_ABS_MUL; DOT_BASIS; NORM_BASIS;
6188 DIMINDEX_3; ARITH] THEN
6189 REWRITE_TAC[REAL_ARITH
6190 `b * y <= n * (b * &1) * a <=> b * &1 * y <= b * a * n`] THEN
6191 MATCH_MP_TAC REAL_LE_LMUL THEN REWRITE_TAC[REAL_ABS_POS] THEN
6192 MATCH_MP_TAC REAL_LE_MUL2 THEN
6193 ASM_SIMP_TAC[REAL_POS; REAL_ABS_POS; COMPONENT_LE_NORM; DIMINDEX_3; ARITH];
6195 RULE_ASSUM_TAC(REWRITE_RULE[REAL_NOT_LE]) THEN
6196 SIMP_TAC[DOT_RMUL; NORM_MUL; REAL_ABS_NORM; DOT_BASIS;
6197 DIMINDEX_3; ARITH; NORM_BASIS] THEN
6198 ONCE_REWRITE_TAC[REAL_ARITH `n * x * a:real = x * n * a`] THEN
6199 ASM_REWRITE_TAC[real_abs; REAL_MUL_RID] THEN
6200 ASM_SIMP_TAC[REAL_MUL_RID; REAL_LT_LMUL_EQ; REAL_LT_MUL_EQ; NORM_POS_LT] THEN
6201 ASM_SIMP_TAC[GSYM REAL_LT_RDIV_EQ; NORM_LT_SQUARE] THEN
6202 ASM_SIMP_TAC[REAL_POW_DIV; REAL_POW_LT; REAL_LT_RDIV_EQ] THEN
6203 REWRITE_TAC[INTER; REAL_MUL_LZERO; IN_BALL_0; IN_ELIM_THM] THEN
6204 ASM_SIMP_TAC[VECTOR_MUL_EQ_0; BASIS_NONZERO; DIMINDEX_3; ARITH;
6205 REAL_LT_IMP_NZ] THEN
6206 COND_CASES_TAC THENL
6207 [ASM_SIMP_TAC[NORM_ARITH `r < &0 ==> ~(norm x < r)`] THEN
6208 REWRITE_TAC[EMPTY_GSPEC; MEASURE_EMPTY; MEASURABLE_EMPTY;
6209 BOUNDED_EMPTY; CONVEX_EMPTY];
6210 RULE_ASSUM_TAC(ONCE_REWRITE_RULE[REAL_NOT_LT])] THEN
6211 STRIP_TAC THEN MATCH_MP_TAC(INST_TYPE [`:2`,`:M`] FUBINI_SIMPLE_ALT) THEN
6212 EXISTS_TAC `3` THEN ASM_REWRITE_TAC[DIMINDEX_2; DIMINDEX_3; ARITH] THEN
6213 SUBGOAL_THEN `&0 < b` ASSUME_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
6214 ASM_SIMP_TAC[WEDGE_SPECIAL_SCALE; AZIM_SPECIAL_SCALE] THEN
6215 ONCE_REWRITE_TAC[SET_RULE `{x | P x /\ x IN s} = {x | P x} INTER s`] THEN
6216 ASM_SIMP_TAC[REAL_LT_LMUL_EQ; SLICE_INTER; DIMINDEX_2;
6217 DIMINDEX_3; ARITH] THEN
6219 (REWRITE_RULE[MATCH_MP COLLINEAR_SPECIAL_SCALE (ASSUME `~(b = &0)`)]) THEN
6220 SUBGOAL_THEN `&0 < inv(a pow 2) - &1` ASSUME_TAC THENL
6221 [REWRITE_TAC[REAL_SUB_LT] THEN MATCH_MP_TAC REAL_INV_1_LT THEN
6222 ASM_SIMP_TAC[REAL_POW_1_LT; REAL_LT_IMP_LE; ARITH; REAL_POW_LT];
6225 `!t. slice 3 t {y:real^3 | norm y < r /\ norm y * a < y$3} =
6227 then ball(vec 0:real^2,min (sqrt(r pow 2 - t pow 2))
6228 (t * sqrt(inv(a pow 2) - &1)))
6230 (fun th -> ASM_SIMP_TAC[th; SLICE_SPECIAL_WEDGE])
6232 [REWRITE_TAC[EXTENSION] THEN
6233 MAP_EVERY X_GEN_TAC [`t:real`; `z:real^2`] THEN
6234 SIMP_TAC[SLICE_123; DIMINDEX_2; DIMINDEX_3; ARITH; IN_ELIM_THM;
6235 VECTOR_3; DOT_3; GSYM DOT_2] THEN
6236 ASM_CASES_TAC `&0 < t` THEN ASM_REWRITE_TAC[] THENL
6238 REWRITE_TAC[NOT_IN_EMPTY; DE_MORGAN_THM] THEN DISJ2_TAC THEN
6239 FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH
6240 `~(&0 < t) ==> &0 <= a ==> ~(a < t)`)) THEN
6241 ASM_SIMP_TAC[REAL_LE_MUL; NORM_POS_LE; REAL_LT_IMP_LE]] THEN
6242 ASM_CASES_TAC `t < r` THEN ASM_REWRITE_TAC[] THENL
6244 REWRITE_TAC[NOT_IN_EMPTY; DE_MORGAN_THM] THEN DISJ1_TAC THEN
6245 REWRITE_TAC[NORM_LT_SQUARE; DE_MORGAN_THM] THEN DISJ2_TAC THEN
6246 REWRITE_TAC[DOT_3; VECTOR_3] THEN
6247 MATCH_MP_TAC(REAL_ARITH
6248 `r <= t /\ &0 <= a /\ &0 <= b ==> ~(a + b + t < r)`) THEN
6249 REWRITE_TAC[REAL_LE_SQUARE; REAL_POW_2] THEN
6250 MATCH_MP_TAC REAL_LE_MUL2 THEN ASM_REAL_ARITH_TAC] THEN
6251 REWRITE_TAC[IN_BALL_0; REAL_LT_MIN] THEN
6252 ASM_SIMP_TAC[GSYM REAL_LT_RDIV_EQ] THEN REWRITE_TAC[NORM_LT_SQUARE] THEN
6253 SUBGOAL_THEN `&0 < r` ASSUME_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
6254 SUBGOAL_THEN `t pow 2 < r pow 2` ASSUME_TAC THENL
6255 [MATCH_MP_TAC REAL_POW_LT2 THEN REWRITE_TAC[ARITH] THEN
6258 ASM_SIMP_TAC[REAL_LT_DIV; SQRT_POS_LT; REAL_LT_MUL; REAL_SUB_LT;
6259 SQRT_POW_2; REAL_LT_IMP_LE; REAL_POW_MUL] THEN
6260 REWRITE_TAC[DOT_2; DOT_3; VECTOR_3] THEN
6261 ONCE_REWRITE_TAC[REAL_ARITH `a + b + c < d <=> a + b < d - c`] THEN
6262 BINOP_TAC THEN AP_TERM_TAC THEN
6263 UNDISCH_TAC `&0 < a` THEN CONV_TAC REAL_FIELD;
6265 GEN_REWRITE_TAC (LAND_CONV o TOP_DEPTH_CONV) [COND_RATOR; COND_RAND] THEN
6266 GEN_REWRITE_TAC (RAND_CONV o RATOR_CONV o LAND_CONV o TOP_DEPTH_CONV)
6267 [COND_RATOR; COND_RAND] THEN
6268 REWRITE_TAC[INTER_EMPTY; MEASURABLE_EMPTY; MEASURE_EMPTY] THEN
6269 REWRITE_TAC[INTER; IN_BALL_0; IN_ELIM_THM] THEN
6270 RULE_ASSUM_TAC(REWRITE_RULE[COLLINEAR_BASIS_3]) THEN
6271 ASM_SIMP_TAC[REWRITE_RULE[HAS_MEASURE_MEASURABLE_MEASURE]
6272 HAS_MEASURE_OPEN_SECTOR_LT_GEN] THEN
6273 REWRITE_TAC[COND_ID] THEN
6274 ASM_SIMP_TAC[REAL_LE_MIN; SQRT_POS_LE; REAL_LT_IMP_LE; REAL_LE_MUL;
6275 REAL_POW_LE2; ARITH; REAL_SUB_LE; REAL_LT_MUL; SQRT_POS_LT] THEN
6276 REWRITE_TAC[GSYM IN_REAL_INTERVAL; HAS_REAL_INTEGRAL_RESTRICT_UNIV] THEN
6277 REWRITE_TAC[HAS_REAL_INTEGRAL_OPEN_INTERVAL] THEN
6278 REWRITE_TAC[NORM_POW_2; DOT_3; VECTOR_3; DOT_2] THEN
6279 ASM_SIMP_TAC[AZIM_ARG; COLLINEAR_BASIS_3] THEN
6280 ONCE_REWRITE_TAC[REAL_ARITH
6281 `(&1 - a) * az / &3 * r pow 3 =
6282 az / &6 * (inv (a pow 2) - &1) * (a * r) pow 3 +
6283 (az * &1 / &3 * (&1 - a) * r pow 3 -
6284 az / &6 * (inv (a pow 2) - &1) * (a * r) pow 3)`] THEN
6285 MATCH_MP_TAC HAS_REAL_INTEGRAL_COMBINE THEN
6286 EXISTS_TAC `a * r:real` THEN
6287 REWRITE_TAC[REAL_ARITH `a * r <= r <=> &0 <= r * (&1 - a)`] THEN
6288 ASM_SIMP_TAC[REAL_LE_MUL; REAL_SUB_LE; REAL_LT_IMP_LE] THEN
6289 ABBREV_TAC `k = Arg(dropout 3 (w2:real^3) / dropout 3 (w1:real^3))` THEN
6291 [MATCH_MP_TAC HAS_REAL_INTEGRAL_EQ THEN EXISTS_TAC
6292 `\t. k * t pow 2 * (inv(a pow 2) - &1) / &2` THEN
6294 [X_GEN_TAC `t:real` THEN REWRITE_TAC[IN_REAL_INTERVAL] THEN
6295 STRIP_TAC THEN AP_TERM_TAC THEN
6296 SUBGOAL_THEN `t pow 2 * (inv(a pow 2) - &1) <= r pow 2 - t pow 2`
6298 [REWRITE_TAC[REAL_ARITH `t * (a - &1) <= r - t <=> t * a <= r`] THEN
6299 ASM_SIMP_TAC[GSYM real_div; REAL_LE_LDIV_EQ; REAL_POW_LT] THEN
6300 REWRITE_TAC[GSYM REAL_POW_MUL] THEN MATCH_MP_TAC REAL_POW_LE2 THEN
6303 SUBGOAL_THEN `t * sqrt(inv(a pow 2) - &1) <= sqrt(r pow 2 - t pow 2)`
6304 (fun th -> SIMP_TAC[th; REAL_ARITH `a <= b ==> min b a = a`])
6306 [MATCH_MP_TAC REAL_POW_LE2_REV THEN EXISTS_TAC `2` THEN
6307 REWRITE_TAC[ARITH] THEN
6308 SUBGOAL_THEN `&0 <= r pow 2 - t pow 2` ASSUME_TAC THENL
6309 [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH
6310 `a <= x ==> &0 <= a ==> &0 <= x`)) THEN
6311 ASM_SIMP_TAC[REAL_POW_2; REAL_LE_MUL; REAL_LE_SQUARE; REAL_LT_IMP_LE];
6312 ASM_SIMP_TAC[SQRT_POS_LE; REAL_POW_MUL; SQRT_POW_2;
6314 ASM_SIMP_TAC[REAL_POW_MUL; SQRT_POW_2; SQRT_POW_2; REAL_LT_IMP_LE] THEN
6317 [`\t. k / &6 * (inv (a pow 2) - &1) * t pow 3`;
6318 `\t. k * t pow 2 * (inv (a pow 2) - &1) / &2`;
6319 `&0`; `a * r:real`] REAL_FUNDAMENTAL_THEOREM_OF_CALCULUS) THEN
6320 ASM_SIMP_TAC[REAL_LE_MUL; REAL_LT_IMP_LE] THEN ANTS_TAC THENL
6321 [ASM_REWRITE_TAC[] THEN
6322 REPEAT STRIP_TAC THEN REAL_DIFF_TAC THEN
6323 CONV_TAC NUM_REDUCE_CONV THEN CONV_TAC REAL_RING;
6324 MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN
6325 UNDISCH_TAC `&0 < a` THEN CONV_TAC REAL_FIELD]];
6326 MATCH_MP_TAC HAS_REAL_INTEGRAL_EQ THEN EXISTS_TAC
6327 `\t:real. k * (r pow 2 - t pow 2) / &2` THEN
6329 [X_GEN_TAC `t:real` THEN REWRITE_TAC[IN_REAL_INTERVAL] THEN
6330 STRIP_TAC THEN AP_TERM_TAC THEN
6331 SUBGOAL_THEN `&0 <= t` ASSUME_TAC THENL
6332 [MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `a * r:real` THEN
6333 ASM_SIMP_TAC[REAL_LE_MUL; REAL_LT_IMP_LE];
6335 MATCH_MP_TAC(REAL_ARITH
6336 `a <= b /\ a pow 2 = x ==> x / &2 = (min a b pow 2) / &2`) THEN
6337 SUBGOAL_THEN `&0 <= r pow 2 - t pow 2` ASSUME_TAC THENL
6338 [REWRITE_TAC[GSYM REAL_LE_SQUARE_ABS; REAL_SUB_LE] THEN
6341 ASM_SIMP_TAC[SQRT_POW_2] THEN MATCH_MP_TAC REAL_POW_LE2_REV THEN
6342 EXISTS_TAC `2` THEN REWRITE_TAC[ARITH] THEN
6343 ASM_SIMP_TAC[SQRT_POW_2; REAL_POW_MUL; REAL_LE_MUL; SQRT_POS_LT;
6344 REAL_LT_MUL; REAL_LT_IMP_LE; SQRT_POS_LE] THEN
6345 REWRITE_TAC[REAL_ARITH `r - t <= t * (a - &1) <=> r <= t * a`] THEN
6346 REWRITE_TAC[REAL_INV_POW; GSYM REAL_POW_MUL] THEN
6347 MATCH_MP_TAC REAL_POW_LE2 THEN
6348 ASM_SIMP_TAC[GSYM real_div; REAL_LE_RDIV_EQ] THEN
6351 [`\t. k / &2 * (r pow 2 * t - t pow 3 / &3)`;
6352 `\t. k * (r pow 2 - t pow 2) / &2`;
6353 `a * r:real`; `r:real`] REAL_FUNDAMENTAL_THEOREM_OF_CALCULUS) THEN
6354 ASM_SIMP_TAC[REAL_LE_MUL; REAL_LT_IMP_LE] THEN ANTS_TAC THENL
6355 [ASM_REWRITE_TAC[REAL_ARITH `a * r <= r <=> &0 <= r * (&1 - a)`] THEN
6356 ASM_SIMP_TAC[REAL_LE_MUL; REAL_LT_IMP_LE; REAL_SUB_LE] THEN
6357 REPEAT STRIP_TAC THEN REAL_DIFF_TAC THEN
6358 CONV_TAC NUM_REDUCE_CONV THEN CONV_TAC REAL_RING;
6359 MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN
6360 UNDISCH_TAC `&0 < a` THEN CONV_TAC REAL_FIELD]]]);;
6362 let BOUNDED_CONIC_CAP_WEDGE = prove
6363 (`!v0 v1:real^3 w1 w2 r a.
6364 bounded(conic_cap v0 v1 r a INTER wedge v0 v1 w1 w2)`,
6365 REPEAT GEN_TAC THEN MATCH_MP_TAC BOUNDED_SUBSET THEN
6366 EXISTS_TAC `conic_cap (v0:real^3) v1 r a` THEN
6367 REWRITE_TAC[BOUNDED_CONIC_CAP] THEN SET_TAC[]);;
6369 let MEASURABLE_CONIC_CAP_WEDGE = prove
6370 (`!v0 v1:real^3 w1 w2 r a.
6371 measurable(conic_cap v0 v1 r a INTER wedge v0 v1 w1 w2)`,
6372 REPEAT GEN_TAC THEN MATCH_MP_TAC MEASURABLE_BOUNDED_INTER_OPEN THEN
6373 REWRITE_TAC[BOUNDED_CONIC_CAP; MEASURABLE_CONIC_CAP; OPEN_WEDGE]);;
6375 let VOLUME_CONIC_CAP_COMPL = prove
6376 (`!v0 v1:real^3 w1 w2 r a.
6378 ==> measure(conic_cap v0 v1 r a INTER wedge v0 v1 w1 w2) +
6379 measure(conic_cap v0 v1 r (--a) INTER wedge v0 v1 w1 w2) =
6380 azim v0 v1 w1 w2 * &2 * r pow 3 / &3`,
6382 (`!f:real^N->real^N s t t' u.
6383 measurable(s) /\ measurable(t) /\ measurable(u) /\
6384 orthogonal_transformation f /\
6385 s SUBSET u /\ t' SUBSET u /\ s INTER t' = {} /\
6386 negligible(u DIFF (s UNION t')) /\
6387 ((!y. ?x. f x = y) ==> IMAGE f t = t')
6388 ==> measure s + measure t = measure u`,
6390 ASM_CASES_TAC `orthogonal_transformation(f:real^N->real^N)` THEN
6391 ASM_SIMP_TAC[ORTHOGONAL_TRANSFORMATION_SURJECTIVE] THEN
6392 REPEAT STRIP_TAC THEN MATCH_MP_TAC EQ_TRANS THEN
6394 `measure(s:real^N->bool) + measure(t':real^N->bool)` THEN
6395 CONJ_TAC THENL [ASM_MESON_TAC[MEASURE_ORTHOGONAL_IMAGE_EQ]; ALL_TAC] THEN
6396 W(MP_TAC o PART_MATCH (rhs o rand) MEASURE_DISJOINT_UNION o
6398 ASM_REWRITE_TAC[DISJOINT] THEN ANTS_TAC THENL
6399 [ASM_MESON_TAC[MEASURABLE_LINEAR_IMAGE; ORTHOGONAL_TRANSFORMATION_LINEAR];
6400 DISCH_THEN(SUBST1_TAC o SYM)] THEN
6401 MATCH_MP_TAC MEASURE_NEGLIGIBLE_SYMDIFF THEN ASM_REWRITE_TAC[] THEN
6402 FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ]
6403 NEGLIGIBLE_SUBSET)) THEN
6404 REPEAT(POP_ASSUM MP_TAC) THEN SET_TAC[]) in
6405 REWRITE_TAC[conic_cap; rcone_gt; NORMBALL_BALL; rconesgn] THEN
6406 GEOM_ORIGIN_TAC `v0:real^3` THEN
6407 REWRITE_TAC[VECTOR_SUB_RZERO; DIST_0; real_gt] THEN
6408 GEOM_BASIS_MULTIPLE_TAC 3 `v1:real^3` THEN
6409 X_GEN_TAC `v1:real` THEN
6410 GEN_REWRITE_TAC LAND_CONV [REAL_ARITH `&0 <= x <=> x = &0 \/ &0 < x`] THEN
6412 [ASM_SIMP_TAC[VECTOR_MUL_LZERO; WEDGE_DEGENERATE; AZIM_DEGENERATE] THEN
6413 REWRITE_TAC[INTER_EMPTY; MEASURE_EMPTY] THEN REAL_ARITH_TAC;
6415 ASM_SIMP_TAC[GSYM VOLUME_BALL_WEDGE] THEN REPEAT STRIP_TAC THEN
6416 ASM_CASES_TAC `collinear {vec 0:real^3,v1 % basis 3,w1}` THENL
6417 [ASM_SIMP_TAC[WEDGE_DEGENERATE; AZIM_DEGENERATE] THEN
6418 REWRITE_TAC[INTER_EMPTY; MEASURE_EMPTY] THEN REAL_ARITH_TAC;
6420 ASM_SIMP_TAC[GSYM VOLUME_BALL_WEDGE] THEN REPEAT STRIP_TAC THEN
6421 ASM_CASES_TAC `collinear {vec 0:real^3,v1 % basis 3,w2}` THENL
6422 [ASM_SIMP_TAC[WEDGE_DEGENERATE; AZIM_DEGENERATE] THEN
6423 REWRITE_TAC[INTER_EMPTY; MEASURE_EMPTY] THEN REAL_ARITH_TAC;
6425 ASM_SIMP_TAC[WEDGE_SPECIAL_SCALE] THEN
6426 MAP_EVERY UNDISCH_TAC
6427 [`~collinear{vec 0:real^3,v1 % basis 3,w1}`;
6428 `~collinear{vec 0:real^3,v1 % basis 3,w2}`] THEN
6429 ASM_SIMP_TAC[COLLINEAR_SPECIAL_SCALE] THEN REPEAT DISCH_TAC THEN
6430 REWRITE_TAC[NORM_MUL; DOT_RMUL] THEN
6431 ASM_SIMP_TAC[REAL_LT_LMUL_EQ; REAL_ARITH
6432 `&0 < v1 ==> n * (abs v1 * y) * a = v1 * n * y * a`] THEN
6433 MATCH_MP_TAC lemma THEN
6435 [`vec 0:real^3`; `basis 3:real^3`; `w1:real^3`; `w2:real^3`;
6436 `r:real`; `a:real`] MEASURABLE_CONIC_CAP_WEDGE) THEN
6438 [`vec 0:real^3`; `basis 3:real^3`; `w1:real^3`; `w2:real^3`;
6439 `r:real`; `--a:real`] MEASURABLE_CONIC_CAP_WEDGE) THEN
6440 REWRITE_TAC[conic_cap; rcone_gt; NORMBALL_BALL; rconesgn] THEN
6441 REWRITE_TAC[VECTOR_SUB_RZERO; DIST_0; real_gt] THEN
6442 REPEAT DISCH_TAC THEN ASM_REWRITE_TAC[MEASURABLE_BALL_WEDGE] THEN
6443 SIMP_TAC[NORM_BASIS; DOT_BASIS; DIMINDEX_3; ARITH; REAL_MUL_LID] THEN
6444 EXISTS_TAC `(\x. vector[x$1; x$2; --(x$3)]):real^3->real^3` THEN
6445 EXISTS_TAC `(ball(vec 0,r) INTER {x | norm x * a > x$3}) INTER
6446 wedge (vec 0:real^3) (basis 3) w1 w2` THEN
6448 [REWRITE_TAC[ORTHOGONAL_TRANSFORMATION; linear] THEN
6449 REWRITE_TAC[CART_EQ; DIMINDEX_3; FORALL_3; VECTOR_3; vector_norm; DOT_3;
6450 VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT] THEN
6451 REPEAT(GEN_TAC ORELSE CONJ_TAC ORELSE AP_TERM_TAC) THEN
6454 CONJ_TAC THENL [SET_TAC[]; ALL_TAC] THEN
6455 CONJ_TAC THENL [SET_TAC[]; ALL_TAC] THEN
6457 [REWRITE_TAC[EXTENSION; NOT_IN_EMPTY; IN_INTER; IN_ELIM_THM; real_gt] THEN
6458 MESON_TAC[REAL_LT_ANTISYM];
6461 [MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN
6462 EXISTS_TAC `rcone_eq (vec 0:real^3) (basis 3) a` THEN
6463 SIMP_TAC[NEGLIGIBLE_RCONE_EQ; BASIS_NONZERO; DIMINDEX_3; ARITH] THEN
6464 REWRITE_TAC[SUBSET; rcone_eq; rconesgn; VECTOR_SUB_RZERO; DIST_0] THEN
6465 SIMP_TAC[DOT_BASIS; NORM_BASIS; DIMINDEX_3; ARITH] THEN
6466 REWRITE_TAC[IN_DIFF; IN_ELIM_THM; IN_INTER; IN_UNION] THEN
6467 GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
6468 ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC;
6470 REWRITE_TAC[] THEN DISCH_TAC THEN MATCH_MP_TAC SURJECTIVE_IMAGE_EQ THEN
6471 ASM_REWRITE_TAC[] THEN
6472 REWRITE_TAC[IN_INTER; IN_BALL_0; IN_ELIM_THM; VECTOR_3] THEN
6473 X_GEN_TAC `x:real^3` THEN
6474 SUBGOAL_THEN `norm(vector [x$1; x$2; --(x$3)]:real^3) = norm(x:real^3)`
6476 [REWRITE_TAC[NORM_EQ; DOT_3; VECTOR_3] THEN REAL_ARITH_TAC;
6478 REWRITE_TAC[REAL_ARITH `n * a > --x <=> n * --a < x`] THEN
6479 MATCH_MP_TAC(TAUT `(a ==> (b <=> b')) ==> (a /\ b <=> a /\ b')`) THEN
6481 REWRITE_TAC[COLLINEAR_BASIS_3; wedge; AZIM_ARG] THEN
6482 REWRITE_TAC[IN_ELIM_THM] THEN
6483 SUBGOAL_THEN `(dropout 3 :real^3->real^2) (vector [x$1; x$2; --(x$3)]) =
6484 (dropout 3 :real^3->real^2) x`
6485 (fun th -> REWRITE_TAC[th]) THEN
6486 SIMP_TAC[CART_EQ; DIMINDEX_2; FORALL_2; dropout; LAMBDA_BETA; ARITH;
6489 let VOLUME_CONIC_CAP_WEDGE_MEDIUM = prove
6490 (`!v0 v1:real^3 w1 w2 r a.
6491 &0 <= a /\ ~collinear {v0,v1,w1} /\ ~collinear {v0,v1,w2}
6492 ==> bounded(conic_cap v0 v1 r a INTER wedge v0 v1 w1 w2) /\
6493 measurable(conic_cap v0 v1 r a INTER wedge v0 v1 w1 w2) /\
6494 measure(conic_cap v0 v1 r a INTER wedge v0 v1 w1 w2) =
6495 if &1 < abs a \/ r < &0 then &0
6496 else azim v0 v1 w1 w2 / &3 * (&1 - a) * r pow 3`,
6497 REWRITE_TAC[BOUNDED_CONIC_CAP_WEDGE; MEASURABLE_CONIC_CAP_WEDGE] THEN
6498 REPEAT STRIP_TAC THEN FIRST_X_ASSUM(DISJ_CASES_TAC o MATCH_MP (REAL_ARITH
6499 `&0 <= a ==> &0 < a \/ a = &0`))
6501 [ASM_SIMP_TAC[VOLUME_CONIC_CAP_WEDGE_WEAK] THEN
6502 REWRITE_TAC[REAL_LE_LT] THEN
6503 ASM_CASES_TAC `a = &1` THEN ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC;
6505 ASM_REWRITE_TAC[] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN
6506 COND_CASES_TAC THENL
6507 [REWRITE_TAC[conic_cap; NORMBALL_BALL] THEN
6508 SUBGOAL_THEN `ball(v0:real^3,r) = {}`
6509 (fun th -> SIMP_TAC[th; INTER_EMPTY; MEASURE_EMPTY]) THEN
6510 REWRITE_TAC[BALL_EQ_EMPTY] THEN ASM_REAL_ARITH_TAC;
6511 MP_TAC(ISPECL [`v0:real^3`; `v1:real^3`; `w1:real^3`; `w2:real^3`;
6512 `r:real`; `&0`] VOLUME_CONIC_CAP_COMPL) THEN
6513 REWRITE_TAC[REAL_NEG_0] THEN ASM_REAL_ARITH_TAC]);;
6515 let VOLUME_CONIC_CAP_WEDGE = prove
6516 (`!v0 v1:real^3 w1 w2 r a.
6517 ~collinear {v0,v1,w1} /\ ~collinear {v0,v1,w2}
6518 ==> bounded(conic_cap v0 v1 r a INTER wedge v0 v1 w1 w2) /\
6519 measurable(conic_cap v0 v1 r a INTER wedge v0 v1 w1 w2) /\
6520 measure(conic_cap v0 v1 r a INTER wedge v0 v1 w1 w2) =
6521 if &1 < a \/ r < &0 then &0
6522 else azim v0 v1 w1 w2 / &3 * (&1 - max a (-- &1)) * r pow 3`,
6523 REWRITE_TAC[BOUNDED_CONIC_CAP_WEDGE; MEASURABLE_CONIC_CAP_WEDGE] THEN
6524 REPEAT STRIP_TAC THEN
6525 ASM_CASES_TAC `&0 <= a` THEN
6526 ASM_SIMP_TAC[VOLUME_CONIC_CAP_WEDGE_MEDIUM;
6527 REAL_ARITH `&0 <= a ==> abs a = a /\ max a (-- &1) = a`] THEN
6528 MP_TAC(ISPECL [`v0:real^3`; `v1:real^3`; `w1:real^3`; `w2:real^3`;
6529 `r:real`; `--a:real`] VOLUME_CONIC_CAP_WEDGE_MEDIUM) THEN
6530 ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
6532 MP_TAC(ISPECL [`v0:real^3`; `v1:real^3`; `w1:real^3`; `w2:real^3`;
6533 `r:real`; `a:real`] VOLUME_CONIC_CAP_COMPL) THEN
6534 ASM_CASES_TAC `r < &0` THENL
6535 [REWRITE_TAC[conic_cap; NORMBALL_BALL] THEN
6536 SUBGOAL_THEN `ball(v0:real^3,r) = {}`
6537 (fun th -> SIMP_TAC[th; INTER_EMPTY; MEASURE_EMPTY]) THEN
6538 REWRITE_TAC[BALL_EQ_EMPTY] THEN ASM_REAL_ARITH_TAC;
6540 ASM_REWRITE_TAC[GSYM REAL_NOT_LT; REAL_ABS_NEG] THEN
6541 ASM_SIMP_TAC[REAL_ARITH `~(&0 <= a) ==> ~(&1 < a) /\ abs a = --a`] THEN
6542 COND_CASES_TAC THEN ASM_REWRITE_TAC[] THENL
6543 [ASM_SIMP_TAC[REAL_ARITH `&1 < --a ==> max a (-- &1) = -- &1`] THEN
6545 ASM_SIMP_TAC[REAL_ARITH `~(&1 < --a) ==> max a (-- &1) = a`] THEN
6548 (* ------------------------------------------------------------------------- *)
6549 (* Precise formulation of Flyspeck volume properties. *)
6550 (* ------------------------------------------------------------------------- *)
6552 (*** Might be preferable to switch
6554 *** normball z r -> ball(z,r)
6555 *** rect a b -> interval(a,b)
6557 *** to fit existing libraries. But I left this alone for now,
6558 *** to be absolutely sure I didn't introduce new errors.
6561 *** NULLSET -> negligible
6564 *** as interface maps for the real^3 case.
6567 let cone = new_definition `cone v S:real^A->bool = affsign sgn_ge {v} S`;;
6569 (*** JRH: should we exclude v for S = {}? Make it always open ***)
6571 let cone0 = new_definition `cone0 v S:real^A->bool = affsign sgn_gt {v} S`;;
6573 (*** JRH changed from cone to cone0 ***)
6575 let solid_triangle = new_definition
6576 `solid_triangle v0 S r = normball v0 r INTER cone0 v0 S`;;
6578 let rect = new_definition
6579 `rect (a:real^3) (b:real^3) =
6580 {(v:real^3) | !i. (a$i < v$i /\ v$i < b$i )}`;;
6582 let RECT_INTERVAL = prove
6583 (`!a b. rect a b = interval(a,b)`,
6584 REWRITE_TAC[rect; EXTENSION; IN_INTERVAL; IN_ELIM_THM] THEN
6585 MESON_TAC[FINITE_INDEX_INRANGE]);;
6587 let RCONE_GE_GT = prove
6589 rcone_gt z w h UNION
6590 { x | (x - z) dot (w - z) = norm(x - z) * norm(w - z) * h}`,
6591 REWRITE_TAC[rcone_ge; rcone_gt; rconesgn] THEN
6592 REWRITE_TAC[dist; EXTENSION; IN_UNION; NORM_SUB; IN_ELIM_THM] THEN
6595 let RCONE_GT_GE = prove
6598 { x | (x - z) dot (w - z) = norm(x - z) * norm(w - z) * h}`,
6599 REWRITE_TAC[rcone_ge; rcone_gt; rconesgn] THEN
6600 REWRITE_TAC[dist; EXTENSION; IN_DIFF; NORM_SUB; IN_ELIM_THM] THEN
6603 override_interface("NULLSET",`negligible:(real^3->bool)->bool`);;
6604 override_interface("vol",`measure:(real^3->bool)->real`);;
6606 let is_sphere= new_definition
6607 `is_sphere x=(?(v:real^3)(r:real). (r> &0)/\ (x={w:real^3 | norm (w-v)= r}))`;;
6609 let c_cone = new_definition
6610 `c_cone (v,w:real^3, r:real)=
6611 {x:real^3 | ((x-v) dot w = norm (x-v)* norm w* r)}`;;
6613 (*** JRH added the condition ~(w = 0), or the cone is all of space ***)
6615 let circular_cone =new_definition
6616 `circular_cone (V:real^3-> bool)=
6617 (? (v,w:real^3)(r:real). ~(w = vec 0) /\ V = c_cone (v,w,r))`;;
6619 let NULLSET_RULES = prove
6620 (`(!P. ((plane P)\/ (is_sphere P) \/ (circular_cone P)) ==> NULLSET P) /\
6621 (!(s:real^3->bool) t. (NULLSET s /\ NULLSET t) ==> NULLSET (s UNION t))`,
6622 SIMP_TAC[NEGLIGIBLE_UNION] THEN
6623 X_GEN_TAC `s:real^3->bool` THEN STRIP_TAC THENL
6624 [MATCH_MP_TAC COPLANAR_IMP_NEGLIGIBLE THEN
6625 SIMP_TAC[COPLANAR; DIMINDEX_3; ARITH] THEN ASM_MESON_TAC[SUBSET_REFL];
6626 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [is_sphere]) THEN
6627 STRIP_TAC THEN ASM_REWRITE_TAC[GSYM dist] THEN
6628 ONCE_REWRITE_TAC[DIST_SYM] THEN
6629 REWRITE_TAC[REWRITE_RULE[sphere] NEGLIGIBLE_SPHERE];
6630 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [circular_cone]) THEN
6631 REWRITE_TAC[EXISTS_PAIRED_THM; c_cone] THEN STRIP_TAC THEN
6632 ASM_REWRITE_TAC[] THEN
6633 MP_TAC(ISPECL [`w + v:real^3`; `v:real^3`; `r:real`]
6634 NEGLIGIBLE_RCONE_EQ) THEN
6635 ASM_REWRITE_TAC[rcone_eq; rconesgn] THEN
6636 REWRITE_TAC[dist; VECTOR_ARITH `(w + v) - v:real^N = w`] THEN
6637 ASM_REWRITE_TAC[VECTOR_ARITH `w + v:real^N = v <=> w = vec 0`]]);;
6639 (*** JRH added &0 < a for frustum; otherwise it's in general unbounded ***)
6641 let primitive = new_definition `primitive (C:real^3->bool) =
6642 ((?v0 v1 v2 v3 r. (C = solid_triangle v0 {v1,v2,v3} r)) \/
6643 (?v0 v1 v2 v3. (C = conv0 {v0,v1,v2,v3})) \/
6644 (?v0 v1 v2 v3 h a. &0 < a /\
6645 (C = frustt v0 v1 h a INTER wedge v0 v1 v2 v3)) \/
6646 (?v0 v1 v2 v3 r c. (C = conic_cap v0 v1 r c INTER wedge v0 v1 v2 v3)) \/
6647 (?a b. (C = rect a b)) \/
6648 (?t r. (C = ellipsoid t r)) \/
6649 (?v0 v1 v2 v3 r. (C = normball v0 r INTER wedge v0 v1 v2 v3)))`;;
6651 let MEASURABLE_RULES = prove
6652 (`(!C. primitive C ==> measurable C) /\
6653 (!Z. NULLSET Z ==> measurable Z) /\
6654 (!X t. measurable X ==> (measurable (IMAGE (scale t) X))) /\
6655 (!X v. measurable X ==> (measurable (IMAGE ((+) v) X))) /\
6656 (!(s:real^3->bool) t. (measurable s /\ measurable t)
6657 ==> measurable (s UNION t)) /\
6658 (!(s:real^3->bool) t. (measurable s /\ measurable t)
6659 ==> measurable (s INTER t)) /\
6660 (!(s:real^3->bool) t. (measurable s /\ measurable t)
6661 ==> measurable (s DIFF t))`,
6662 SIMP_TAC[MEASURABLE_UNION; MEASURABLE_INTER; MEASURABLE_DIFF] THEN
6663 REWRITE_TAC[REWRITE_RULE[ETA_AX] MEASURABLE_TRANSLATION] THEN
6664 SIMP_TAC[GSYM HAS_MEASURE_0; HAS_MEASURE_MEASURABLE_MEASURE] THEN
6667 MAP_EVERY X_GEN_TAC [`X:real^3->bool`; `t:real^3`] THEN
6668 REWRITE_TAC[HAS_MEASURE_MEASURE] THEN
6669 DISCH_THEN(MP_TAC o MATCH_MP HAS_MEASURE_STRETCH) THEN
6670 DISCH_THEN(MP_TAC o SPEC `\i. (t:real^3)$i`) THEN
6671 REWRITE_TAC[HAS_MEASURE_MEASURABLE_MEASURE] THEN
6672 DISCH_THEN(MP_TAC o CONJUNCT1) THEN MATCH_MP_TAC EQ_IMP THEN
6673 AP_TERM_TAC THEN AP_THM_TAC THEN AP_TERM_TAC THEN
6674 SIMP_TAC[FUN_EQ_THM; scale; CART_EQ; LAMBDA_BETA;
6675 DIMINDEX_3; VECTOR_3; FORALL_3]] THEN
6676 X_GEN_TAC `C:real^3->bool` THEN REWRITE_TAC[primitive] THEN
6677 REWRITE_TAC[NORMBALL_BALL; RECT_INTERVAL] THEN
6678 DISCH_THEN(REPEAT_TCL DISJ_CASES_THEN MP_TAC) THEN
6679 REWRITE_TAC[LEFT_IMP_EXISTS_THM] THENL
6680 [REPEAT STRIP_TAC THEN
6681 ASM_REWRITE_TAC[solid_triangle; NORMBALL_BALL; cone0; GSYM aff_gt_def] THEN
6682 REWRITE_TAC[MEASURABLE_BALL_AFF_GT];
6683 REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
6684 MATCH_MP_TAC MEASURABLE_CONV0 THEN MATCH_MP_TAC FINITE_IMP_BOUNDED THEN
6685 REWRITE_TAC[FINITE_INSERT; FINITE_EMPTY];
6687 [`v0:real^3`; `v1:real^3`; `v2:real^3`; `v3:real^3`;
6688 `h:real`; `a:real`] THEN
6689 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC SUBST1_TAC) THEN
6690 ASM_CASES_TAC `collinear {v0:real^3, v1, v2}` THENL
6691 [ASM_SIMP_TAC[WEDGE_DEGENERATE; INTER_EMPTY; MEASURABLE_EMPTY];
6693 ASM_CASES_TAC `collinear {v0:real^3, v1, v3}` THENL
6694 [ASM_SIMP_TAC[WEDGE_DEGENERATE; INTER_EMPTY; MEASURABLE_EMPTY];
6696 ASM_SIMP_TAC[VOLUME_FRUSTT_WEDGE];
6697 REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
6698 MATCH_MP_TAC MEASURABLE_BOUNDED_INTER_OPEN THEN
6699 REWRITE_TAC[MEASURABLE_CONIC_CAP; BOUNDED_CONIC_CAP; OPEN_WEDGE];
6700 SIMP_TAC[MEASURABLE_INTERVAL];
6701 SIMP_TAC[MEASURABLE_ELLIPSOID];
6702 SIMP_TAC[MEASURABLE_BALL_WEDGE]]);;
6704 let vol_solid_triangle = new_definition `vol_solid_triangle v0 v1 v2 v3 r =
6705 let a123 = dihV v0 v1 v2 v3 in
6706 let a231 = dihV v0 v2 v3 v1 in
6707 let a312 = dihV v0 v3 v1 v2 in
6708 (a123 + a231 + a312 - pi)*(r pow 3)/(&3)`;;
6710 let vol_frustt_wedge = new_definition `vol_frustt_wedge v0 v1 v2 v3 h a =
6711 (azim v0 v1 v2 v3)*(h pow 3)*(&1/(a*a) - &1)/(&6)`;;
6713 let vol_conic_cap_wedge = new_definition `vol_conic_cap_wedge v0 v1 v2 v3 r c =
6714 (azim v0 v1 v2 v3)*(&1 - c)*(r pow 3)/(&3)`;;
6716 (*** JRH corrected delta_x x12 x13 x14 x34 x24 x34 ***)
6717 (*** to delta_x x12 x13 x14 x34 x24 x23 ***)
6719 let vol_conv = new_definition `vol_conv v1 v2 v3 v4 =
6720 let x12 = dist(v1,v2) pow 2 in
6721 let x13 = dist(v1,v3) pow 2 in
6722 let x14 = dist(v1,v4) pow 2 in
6723 let x23 = dist(v2,v3) pow 2 in
6724 let x24 = dist(v2,v4) pow 2 in
6725 let x34 = dist(v3,v4) pow 2 in
6726 sqrt(delta_x x12 x13 x14 x34 x24 x23)/(&12)`;;
6728 let vol_rect = new_definition `vol_rect a b =
6729 if (a$1 < b$1) /\ (a$2 < b$2) /\ (a$3 < b$3) then
6730 (b$3-a$3)*(b$2-a$2)*(b$1-a$1) else &0`;;
6732 let vol_ball_wedge = new_definition `vol_ball_wedge v0 v1 v2 v3 r =
6733 (azim v0 v1 v2 v3)*(&2)*(r pow 3)/(&3)`;;
6735 let SDIFF = new_definition `SDIFF X Y = (X DIFF Y) UNION (Y DIFF X)`;;
6737 (*** JRH added the hypothesis "measurable" to the first one ***)
6738 (*** Could change the definition to make this hold anyway ***)
6740 (*** JRH changed solid triangle hypothesis to ~coplanar{...} ***)
6741 (*** since the current condition is not enough in general ***)
6743 let volume_props = prove
6744 (`(!C. measurable C ==> vol C >= &0) /\
6745 (!Z. NULLSET Z ==> (vol Z = &0)) /\
6746 (!X Y. measurable X /\ measurable Y /\ NULLSET (SDIFF X Y)
6747 ==> (vol X = vol Y)) /\
6748 (!X t. (measurable X) /\ (measurable (IMAGE (scale t) X))
6749 ==> (vol (IMAGE (scale t) X) = abs(t$1 * t$2 * t$3)*vol(X))) /\
6750 (!X v. measurable X ==> (vol (IMAGE ((+) v) X) = vol X)) /\
6751 (!v0 v1 v2 v3 r. (r > &0) /\ ~coplanar{v0,v1,v2,v3}
6752 ==> vol (solid_triangle v0 {v1,v2,v3} r) =
6753 vol_solid_triangle v0 v1 v2 v3 r) /\
6754 (!v0 v1 v2 v3. vol(conv0 {v0,v1,v2,v3}) = vol_conv v0 v1 v2 v3) /\
6755 (!v0 v1 v2 v3 h a. ~(collinear {v0,v1,v2}) /\ ~(collinear {v0,v1,v3}) /\
6756 (h >= &0) /\ (a > &0) /\ (a <= &1)
6757 ==> vol(frustt v0 v1 h a INTER wedge v0 v1 v2 v3) =
6758 vol_frustt_wedge v0 v1 v2 v3 h a) /\
6759 (!v0 v1 v2 v3 r c. ~(collinear {v0,v1,v2}) /\ ~(collinear {v0,v1,v3}) /\
6760 (r >= &0) /\ (c >= -- (&1)) /\ (c <= &1)
6761 ==> (vol(conic_cap v0 v1 r c INTER wedge v0 v1 v2 v3) =
6762 vol_conic_cap_wedge v0 v1 v2 v3 r c)) /\
6763 (!(a:real^3) (b:real^3). vol(rect a b) = vol_rect a b) /\
6764 (!v0 v1 v2 v3 r. ~(collinear {v0,v1,v2}) /\ ~(collinear {v0,v1,v3}) /\
6766 ==> (vol(normball v0 r INTER wedge v0 v1 v2 v3) =
6767 vol_ball_wedge v0 v1 v2 v3 r))`,
6768 SIMP_TAC[MEASURE_POS_LE; real_ge; real_gt] THEN REPEAT CONJ_TAC THENL
6769 [SIMP_TAC[GSYM HAS_MEASURE_0; HAS_MEASURE_MEASURABLE_MEASURE];
6770 MAP_EVERY X_GEN_TAC [`s:real^3->bool`; `t:real^3->bool`] THEN
6771 STRIP_TAC THEN MATCH_MP_TAC MEASURE_NEGLIGIBLE_SYMDIFF THEN
6772 ASM_REWRITE_TAC[] THEN
6773 FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ]
6774 NEGLIGIBLE_SUBSET)) THEN
6775 REWRITE_TAC[SDIFF] THEN SET_TAC[];
6776 MAP_EVERY X_GEN_TAC [`X:real^3->bool`; `t:real^3`] THEN
6777 REWRITE_TAC[HAS_MEASURE_MEASURE] THEN
6778 DISCH_THEN(MP_TAC o MATCH_MP HAS_MEASURE_STRETCH o CONJUNCT1) THEN
6779 DISCH_THEN(MP_TAC o SPEC `\i. (t:real^3)$i`) THEN
6780 REWRITE_TAC[HAS_MEASURE_MEASURABLE_MEASURE] THEN
6781 DISCH_THEN(MP_TAC o CONJUNCT2) THEN
6782 REWRITE_TAC[DIMINDEX_3; PRODUCT_3] THEN
6783 DISCH_THEN(SUBST1_TAC o SYM) THEN
6784 AP_TERM_TAC THEN AP_THM_TAC THEN AP_TERM_TAC THEN
6785 SIMP_TAC[FUN_EQ_THM; scale; CART_EQ; LAMBDA_BETA;
6786 DIMINDEX_3; VECTOR_3; FORALL_3];
6787 REWRITE_TAC[REWRITE_RULE[ETA_AX] MEASURE_TRANSLATION];
6788 REPEAT STRIP_TAC THEN
6789 REWRITE_TAC[solid_triangle; vol_solid_triangle; NORMBALL_BALL] THEN
6790 REWRITE_TAC[cone0; GSYM aff_gt_def] THEN
6791 MATCH_MP_TAC VOLUME_SOLID_TRIANGLE THEN ASM_REWRITE_TAC[];
6792 REWRITE_TAC[vol_conv; VOLUME_OF_TETRAHEDRON];
6793 SIMP_TAC[VOLUME_FRUSTT_WEDGE; vol_frustt_wedge] THEN
6794 SIMP_TAC[REAL_ARITH `&0 <= h ==> ~(h < &0)`] THEN
6795 SIMP_TAC[REAL_ARITH `a <= &1 ==> (&1 <= a <=> a = &1)`] THEN
6796 REPEAT STRIP_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN
6797 REPEAT(POP_ASSUM MP_TAC) THEN CONV_TAC REAL_FIELD;
6798 SIMP_TAC[VOLUME_CONIC_CAP_WEDGE; vol_conic_cap_wedge] THEN
6799 SIMP_TAC[REAL_ARITH `&0 <= r ==> ~(r < &0)`] THEN
6800 SIMP_TAC[REAL_ARITH `c <= &1 ==> ~(&1 < c)`] THEN
6801 ASM_SIMP_TAC[REAL_ARITH `-- &1 <= c ==> max c (-- &1) = c`] THEN
6802 REPEAT STRIP_TAC THEN REAL_ARITH_TAC;
6803 REWRITE_TAC[vol_rect; RECT_INTERVAL; MEASURE_INTERVAL] THEN
6804 REWRITE_TAC[CONTENT_CLOSED_INTERVAL_CASES] THEN
6805 REWRITE_TAC[DIMINDEX_3; FORALL_3; PRODUCT_3] THEN
6806 MAP_EVERY X_GEN_TAC [`a:real^3`; `b:real^3`] THEN
6807 REWRITE_TAC[REAL_LE_LT] THEN
6808 ASM_CASES_TAC `(a:real^3)$1 = (b:real^3)$1` THEN
6809 ASM_REWRITE_TAC[REAL_LT_REFL; REAL_MUL_LZERO; REAL_MUL_RZERO;
6810 REAL_SUB_REFL; COND_ID] THEN
6811 ASM_CASES_TAC `(a:real^3)$2 = (b:real^3)$2` THEN
6812 ASM_REWRITE_TAC[REAL_LT_REFL; REAL_MUL_LZERO; REAL_MUL_RZERO;
6813 REAL_SUB_REFL; COND_ID] THEN
6814 ASM_CASES_TAC `(a:real^3)$3 = (b:real^3)$3` THEN
6815 ASM_REWRITE_TAC[REAL_LT_REFL; REAL_MUL_LZERO; REAL_MUL_RZERO;
6816 REAL_SUB_REFL; COND_ID] THEN
6817 REWRITE_TAC[REAL_MUL_AC];
6818 SIMP_TAC[VOLUME_BALL_WEDGE; NORMBALL_BALL; vol_ball_wedge]]);;
6820 (* ------------------------------------------------------------------------- *)
6821 (* Additional results on polyhedra and relation to fans. *)
6822 (* ------------------------------------------------------------------------- *)
6824 let POLYHEDRON_COLLINEAR_FACES_STRONG = prove
6825 (`!P:real^N->bool f f' p q s t.
6826 polyhedron P /\ vec 0 IN relative_interior P /\
6827 f face_of P /\ ~(f = P) /\ f' face_of P /\ ~(f' = P) /\
6828 p IN f /\ q IN f' /\ s > &0 /\ t > &0 /\ s % p = t % q
6830 ONCE_REWRITE_TAC[MESON[]
6831 `(!P f f' p q s t. Q P f f' p q s t) <=>
6832 (!s t P f f' p q. Q P f f' p q s t)`] THEN
6833 MATCH_MP_TAC REAL_WLOG_LT THEN
6834 REWRITE_TAC[real_gt] THEN CONJ_TAC THENL [MESON_TAC[]; ALL_TAC] THEN
6835 REPEAT STRIP_TAC THEN MATCH_MP_TAC(TAUT `F ==> p`) THEN
6836 FIRST_X_ASSUM(MP_TAC o AP_TERM `(%) (inv s):real^N->real^N`) THEN
6837 ASM_SIMP_TAC[VECTOR_MUL_ASSOC; REAL_MUL_LINV; REAL_LT_IMP_NZ] THEN
6838 ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN
6839 REWRITE_TAC[VECTOR_MUL_LID; GSYM real_div] THEN
6840 ABBREV_TAC `u:real = t / s` THEN
6841 SUBGOAL_THEN `&0 < u /\ &1 < u` MP_TAC THENL
6842 [EXPAND_TAC "u" THEN ASM_SIMP_TAC[REAL_LT_RDIV_EQ] THEN
6843 ASM_REWRITE_TAC[REAL_MUL_LZERO; REAL_MUL_LID];
6845 MAP_EVERY (C UNDISCH_THEN (K ALL_TAC))
6846 [`s < t`; `&0 < s`; `&0 < t`; `t:real / s = u`] THEN
6847 SPEC_TAC(`u:real`,`t:real`) THEN GEN_TAC THEN STRIP_TAC THEN
6848 DISCH_THEN(ASSUME_TAC o SYM) THEN
6849 SUBGOAL_THEN `?g:real^N->bool. g facet_of P /\ f' SUBSET g`
6850 STRIP_ASSUME_TAC THENL
6851 [MATCH_MP_TAC FACE_OF_POLYHEDRON_SUBSET_FACET THEN ASM SET_TAC[];
6853 SUBGOAL_THEN `~((vec 0:real^N) IN g)` ASSUME_TAC THENL
6855 MP_TAC(ISPECL [`P:real^N->bool`; `g:real^N->bool`; `P:real^N->bool`]
6856 SUBSET_OF_FACE_OF) THEN
6857 ASM_REWRITE_TAC[SUBSET_REFL; NOT_IMP] THEN CONJ_TAC THENL
6858 [CONJ_TAC THENL [ASM_MESON_TAC[facet_of]; ASM SET_TAC[]];
6859 ASM_MESON_TAC[facet_of; FACET_OF_REFL;
6860 SUBSET_ANTISYM; FACE_OF_IMP_SUBSET]];
6862 SUBGOAL_THEN `(g:real^N->bool) face_of P` MP_TAC THENL
6863 [ASM_MESON_TAC[facet_of]; ALL_TAC] THEN
6864 REWRITE_TAC[face_of] THEN DISCH_THEN(MP_TAC o last o CONJUNCTS) THEN
6865 DISCH_THEN(MP_TAC o SPECL [`vec 0:real^N`; `t % q:real^N`; `q:real^N`]) THEN
6866 ASM_REWRITE_TAC[] THEN REPEAT CONJ_TAC THENL
6867 [ASM_MESON_TAC[RELATIVE_INTERIOR_SUBSET; SUBSET];
6868 ASM_MESON_TAC[FACE_OF_IMP_SUBSET; SUBSET];
6869 ASM_MESON_TAC[FACE_OF_IMP_SUBSET; SUBSET];
6871 EXPAND_TAC "p" THEN REWRITE_TAC[IN_SEGMENT] THEN CONJ_TAC THENL
6872 [CONV_TAC(RAND_CONV SYM_CONV) THEN
6873 ASM_SIMP_TAC[VECTOR_MUL_EQ_0; REAL_LT_IMP_NZ] THEN ASM SET_TAC[];
6874 EXISTS_TAC `inv t:real` THEN
6875 ASM_SIMP_TAC[REAL_LT_INV_EQ; REAL_INV_LT_1] THEN
6876 EXPAND_TAC "p" THEN REWRITE_TAC[VECTOR_MUL_ASSOC] THEN
6877 ASM_SIMP_TAC[REAL_MUL_LINV; REAL_LT_IMP_NZ] THEN VECTOR_ARITH_TAC]);;
6879 let POLYHEDRON_COLLINEAR_FACES = prove
6880 (`!P:real^N->bool f f' p q s t.
6881 polyhedron P /\ vec 0 IN interior P /\
6882 f face_of P /\ ~(f = P) /\ f' face_of P /\ ~(f' = P) /\
6883 p IN f /\ q IN f' /\ s > &0 /\ t > &0 /\ s % p = t % q
6885 MESON_TAC[POLYHEDRON_COLLINEAR_FACES_STRONG;
6886 INTERIOR_SUBSET_RELATIVE_INTERIOR; SUBSET]);;
6888 let vertices = new_definition
6889 `vertices s = {x:real^N | x extreme_point_of s}`;;
6891 let edges = new_definition
6892 `edges s = {{v,w} | segment[v,w] edge_of s}`;;
6894 let VERTICES_TRANSLATION = prove
6895 (`!a s. vertices (IMAGE (\x. a + x) s) = IMAGE (\x. a + x) (vertices s)`,
6896 REWRITE_TAC[vertices] THEN GEOM_TRANSLATE_TAC[]);;
6898 let VERTICES_LINEAR_IMAGE = prove
6899 (`!f s. linear f /\ (!x y. f x = f y ==> x = y)
6900 ==> vertices(IMAGE f s) = IMAGE f (vertices s)`,
6901 REWRITE_TAC[vertices; EXTREME_POINTS_OF_LINEAR_IMAGE]);;
6903 let EDGES_TRANSLATION = prove
6904 (`!a s. edges (IMAGE (\x. a + x) s) = IMAGE (IMAGE (\x. a + x)) (edges s)`,
6905 REWRITE_TAC[edges] THEN GEOM_TRANSLATE_TAC[] THEN SET_TAC[]);;
6907 let EDGES_LINEAR_IMAGE = prove
6908 (`!f:real^M->real^N s.
6909 linear f /\ (!x y. f x = f y ==> x = y)
6910 ==> edges(IMAGE f s) = IMAGE (IMAGE f) (edges s)`,
6911 REPEAT GEN_TAC THEN DISCH_TAC THEN REWRITE_TAC[edges] THEN
6912 MATCH_MP_TAC SUBSET_ANTISYM THEN
6913 REWRITE_TAC[SUBSET; FORALL_IN_GSPEC; FORALL_IN_IMAGE] THEN CONJ_TAC THENL
6914 [MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`] THEN STRIP_TAC THEN
6915 REWRITE_TAC[IN_IMAGE] THEN ONCE_REWRITE_TAC[CONJ_SYM] THEN
6916 REWRITE_TAC[EXISTS_IN_GSPEC] THEN
6917 SUBGOAL_THEN `?v w. x = (f:real^M->real^N) v /\ y = f w` MP_TAC THENL
6918 [ASM_MESON_TAC[ENDS_IN_SEGMENT; EDGE_OF_IMP_SUBSET; SUBSET; IN_IMAGE];
6919 REPEAT(MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC) THEN
6920 DISCH_THEN(CONJUNCTS_THEN SUBST_ALL_TAC)];
6921 MAP_EVERY X_GEN_TAC [`v:real^M`; `w:real^M`] THEN DISCH_TAC THEN
6922 REWRITE_TAC[IN_ELIM_THM] THEN
6923 MAP_EVERY EXISTS_TAC [`(f:real^M->real^N) v`; `(f:real^M->real^N) w`]] THEN
6924 REWRITE_TAC[IMAGE_CLAUSES] THEN
6925 ASM_MESON_TAC[EDGE_OF_LINEAR_IMAGE; CLOSED_SEGMENT_LINEAR_IMAGE]);;
6927 add_translation_invariants [VERTICES_TRANSLATION; EDGES_TRANSLATION];;
6928 add_linear_invariants [VERTICES_LINEAR_IMAGE; EDGES_LINEAR_IMAGE];;
6930 (*** Correspondence with Flypaper:
6932 Definition 4.5: IS_AFFINE_HULL
6937 Definition 4.6 : IN_INTERIOR
6938 IN_RELATIVE_INTERIOR
6939 CLOSURE_APPROACHABLE
6940 (Don't have definition of relative boundary, but several
6941 theorems use closure s DIFF relative_interior s.)
6943 Definition 4.7: face_of
6944 extreme_point_of (presumably it's meant to be the point not
6945 the singleton set, which the definition literally gives)
6948 (Don't have a definition of "proper"; I just use
6949 ~(f = {}) and/or ~(f = P) as needed.)
6951 Lemma 4.18: KREIN_MILMAN_MINKOWSKI
6953 Definition 4.8: polyhedron
6956 Lemma 4.19: AFFINE_IMP_POLYHEDRON
6958 Lemma 4.20 (i): FACET_OF_POLYHEDRON_EXPLICIT
6960 Lemma 4.20 (ii): Direct consequence of RELATIVE_INTERIOR_POLYHEDRON
6962 Lemma 4.20 (iii): FACE_OF_POLYHEDRON_EXPLICIT / FACE_OF_POLYHEDRON
6964 Lemma 4.20 (iv): FACE_OF_TRANS
6966 Lemma 4.20 (v): EXTREME_POINT_OF_FACE
6968 Lemma 4.20 (vi): FACE_OF_EQ
6970 Corr. 4.7: FACE_OF_POLYHEDRON_POLYHEDRON
6972 Lemma 4.21: POLYHEDRON_COLLINEAR_FACES
6979 (* ------------------------------------------------------------------------- *)
6980 (* Temporary backwards-compatible fix for introduction of "sphere" and *)
6981 (* "relative_frontier". *)
6982 (* ------------------------------------------------------------------------- *)
6984 let COMPACT_SPHERE =
6985 REWRITE_RULE[sphere; NORM_ARITH `dist(a:real^N,b) = norm(b - a)`]
6988 let FRONTIER_CBALL = REWRITE_RULE[sphere] FRONTIER_CBALL;;
6990 let NEGLIGIBLE_SPHERE = REWRITE_RULE[sphere] NEGLIGIBLE_SPHERE;;
6992 let RELATIVE_FRONTIER_OF_POLYHEDRON = RELATIVE_BOUNDARY_OF_POLYHEDRON;;
6994 (* ------------------------------------------------------------------------- *)
6995 (* Fix the congruence rules as expected in Flyspeck. *)
6996 (* Should exclude 6 recent mixed real/vector limit results. *)
6997 (* ------------------------------------------------------------------------- *)
7000 [`(p <=> p') ==> (p' ==> (q <=> q')) ==> (p ==> q <=> p' ==> q')`;
7003 ==> (~g' ==> e = e')
7004 ==> (if g then t else e) = (if g' then t' else e')`;
7005 `(!x. p x ==> f x = g x) ==> nsum {y | p y} (\i. f i) = nsum {y | p y} g`;
7006 `(!i. a <= i /\ i <= b ==> f i = g i)
7007 ==> nsum (a..b) (\i. f i) = nsum (a..b) g`;
7008 `(!x. x IN s ==> f x = g x) ==> nsum s (\i. f i) = nsum s g`;
7009 `(!x. p x ==> f x = g x) ==> sum {y | p y} (\i. f i) = sum {y | p y} g`;
7010 `(!i. a <= i /\ i <= b ==> f i = g i)
7011 ==> sum (a..b) (\i. f i) = sum (a..b) g`;
7012 `(!x. x IN s ==> f x = g x) ==> sum s (\i. f i) = sum s g`;
7013 `(!x. p x ==> f x = g x) ==> vsum {y | p y} (\i. f i) = vsum {y | p y} g`;
7014 `(!i. a <= i /\ i <= b ==> f i = g i)
7015 ==> vsum (a..b) (\i. f i) = vsum (a..b) g`;
7016 `(!x. x IN s ==> f x = g x) ==> vsum s (\i. f i) = vsum s g`;
7017 `(!x. p x ==> f x = g x)
7018 ==> product {y | p y} (\i. f i) = product {y | p y} g`;
7019 `(!i. a <= i /\ i <= b ==> f i = g i)
7020 ==> product (a..b) (\i. f i) = product (a..b) g`;
7021 `(!x. x IN s ==> f x = g x) ==> product s (\i. f i) = product s g`;
7022 `(!x. ~(x = a) ==> f x = g x)
7023 ==> (((\x. f x) --> l) (at a) <=> (g --> l) (at a))`;
7024 `(!x. ~(x = a) ==> f x = g x)
7025 ==> (((\x. f x) --> l) (at a within s) <=> (g --> l) (at a within s))`]
7026 and equiv t1 t2 = can (term_match [] t1) t2 & can (term_match [] t2) t1 in
7028 filter (fun th -> exists (equiv (concl th)) bcs) (basic_congs()) in
7029 set_basic_congs congs';;