https://github.com/JacquesCarette/hol-light
Tip revision: b27a524086caf73530b7c2c5da1b237d3539f143 authored by Jacques Carette on 24 August 2020, 14:18:07 UTC
Merge pull request #35 from sjjs7/final-changes
Merge pull request #35 from sjjs7/final-changes
Tip revision: b27a524
flyspeck.ml
(* ========================================================================= *)
(* Results intended for Flyspeck. *)
(* ========================================================================= *)
needs "Multivariate/polytope.ml";;
needs "Multivariate/realanalysis.ml";;
needs "Multivariate/geom.ml";;
needs "Multivariate/cross.ml";;
prioritize_vector();;
(* ------------------------------------------------------------------------- *)
(* Not really Flyspeck-specific but needs both angles and cross products. *)
(* ------------------------------------------------------------------------- *)
let NORM_CROSS = prove
(`!x y. norm(x cross y) = norm(x) * norm(y) * sin(vector_angle x y)`,
REPEAT GEN_TAC THEN
MATCH_MP_TAC REAL_POW_EQ THEN EXISTS_TAC `2` THEN
SIMP_TAC[NORM_POS_LE; SIN_VECTOR_ANGLE_POS; REAL_LE_MUL; ARITH_EQ] THEN
MP_TAC(SPECL [`x:real^3`; `y:real^3`] NORM_CROSS_DOT) THEN
REWRITE_TAC[VECTOR_ANGLE] THEN
MP_TAC(SPEC `vector_angle (x:real^3) y` SIN_CIRCLE) THEN
CONV_TAC REAL_RING);;
(* ------------------------------------------------------------------------- *)
(* Other miscelleneous lemmas. *)
(* ------------------------------------------------------------------------- *)
let COPLANAR_INSERT_0_NEG = prove
(`coplanar(vec 0 INSERT --x INSERT s) <=> coplanar(vec 0 INSERT x INSERT s)`,
REWRITE_TAC[coplanar; INSERT_SUBSET] THEN
ONCE_REWRITE_TAC[TAUT `a /\ b /\ c <=> ~(a ==> ~(b /\ c))`] THEN
SIMP_TAC[AFFINE_HULL_EQ_SPAN; SPAN_NEG_EQ]);;
let COPLANAR_IMP_NEGLIGIBLE = prove
(`!s:real^3->bool. coplanar s ==> negligible s`,
REWRITE_TAC[coplanar] THEN
MESON_TAC[NEGLIGIBLE_AFFINE_HULL_3; NEGLIGIBLE_SUBSET]);;
let NOT_COPLANAR_0_4_IMP_INDEPENDENT = prove
(`!v1 v2 v3:real^N. ~coplanar {vec 0,v1,v2,v3} ==> independent {v1,v2,v3}`,
REPEAT GEN_TAC THEN REWRITE_TAC[independent; CONTRAPOS_THM] THEN
REWRITE_TAC[dependent] THEN
SUBGOAL_THEN
`!v1 v2 v3:real^N. v1 IN span {v2,v3} ==> coplanar{vec 0,v1,v2,v3}`
ASSUME_TAC THENL
[REPEAT STRIP_TAC THEN REWRITE_TAC[coplanar] THEN
MAP_EVERY EXISTS_TAC [`vec 0:real^N`; `v2:real^N`; `v3:real^N`] THEN
SIMP_TAC[AFFINE_HULL_EQ_SPAN; HULL_INC; IN_INSERT] THEN
REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET] THEN
ASM_SIMP_TAC[SPAN_SUPERSET; IN_INSERT] THEN
POP_ASSUM MP_TAC THEN SPEC_TAC(`v1:real^N`,`v1:real^N`) THEN
REWRITE_TAC[GSYM SUBSET] THEN MATCH_MP_TAC SPAN_MONO THEN SET_TAC[];
REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY] THEN
STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
FIRST_X_ASSUM SUBST_ALL_TAC THENL
[FIRST_X_ASSUM(MP_TAC o SPECL [`v1:real^N`; `v2:real^N`; `v3:real^N`]);
FIRST_X_ASSUM(MP_TAC o SPECL [`v2:real^N`; `v3:real^N`; `v1:real^N`]);
FIRST_X_ASSUM(MP_TAC o SPECL [`v3:real^N`; `v1:real^N`; `v2:real^N`])]
THEN REWRITE_TAC[INSERT_AC] THEN DISCH_THEN MATCH_MP_TAC THEN
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE
`a IN s ==> s SUBSET t ==> a IN t`)) THEN
MATCH_MP_TAC SPAN_MONO THEN SET_TAC[]]);;
let NOT_COPLANAR_NOT_COLLINEAR = prove
(`!v1 v2 v3 w:real^N. ~coplanar {v1, v2, v3, w} ==> ~collinear {v1, v2, v3}`,
REPEAT GEN_TAC THEN
REWRITE_TAC[COLLINEAR_AFFINE_HULL; coplanar; CONTRAPOS_THM] THEN
MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `x:real^N` THEN
MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `y:real^N` THEN
REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET] THEN STRIP_TAC THEN
EXISTS_TAC `w:real^N` THEN ASM_SIMP_TAC[HULL_INC; IN_INSERT] THEN
REPEAT CONJ_TAC THEN
MATCH_MP_TAC(SET_RULE `!t. t SUBSET s /\ x IN t ==> x IN s`) THEN
EXISTS_TAC `affine hull {x:real^N,y}` THEN
ASM_REWRITE_TAC[] THEN MATCH_MP_TAC HULL_MONO THEN SET_TAC[]);;
(* ------------------------------------------------------------------------- *)
(* Some special scaling theorems. *)
(* ------------------------------------------------------------------------- *)
let SUBSET_AFFINE_HULL_SPECIAL_SCALE = prove
(`!a x s t.
~(a = &0)
==> (vec 0 INSERT (a % x) INSERT s SUBSET affine hull t <=>
vec 0 INSERT x INSERT s SUBSET affine hull t)`,
REPEAT STRIP_TAC THEN REWRITE_TAC[INSERT_SUBSET] THEN
MATCH_MP_TAC(TAUT `(a ==> (b <=> c)) ==> (a /\ b <=> a /\ c)`) THEN
ASM_SIMP_TAC[AFFINE_HULL_EQ_SPAN; SPAN_MUL_EQ]);;
let COLLINEAR_SPECIAL_SCALE = prove
(`!a x y. ~(a = &0) ==> (collinear {vec 0,a % x,y} <=> collinear{vec 0,x,y})`,
REPEAT STRIP_TAC THEN REWRITE_TAC[COLLINEAR_AFFINE_HULL] THEN
ASM_SIMP_TAC[SUBSET_AFFINE_HULL_SPECIAL_SCALE]);;
let COLLINEAR_SCALE_ALL = prove
(`!a b v w. ~(a = &0) /\ ~(b = &0)
==> (collinear {vec 0,a % v,b % w} <=> collinear {vec 0,v,w})`,
REPEAT STRIP_TAC THEN ASM_SIMP_TAC[COLLINEAR_SPECIAL_SCALE] THEN
ONCE_REWRITE_TAC[SET_RULE `{a,b,c} = {a,c,b}`] THEN
ASM_SIMP_TAC[COLLINEAR_SPECIAL_SCALE]);;
let COPLANAR_SPECIAL_SCALE = prove
(`!a x y z.
~(a = &0) ==> (coplanar {vec 0,a % x,y,z} <=> coplanar {vec 0,x,y,z})`,
REPEAT STRIP_TAC THEN REWRITE_TAC[coplanar] THEN
ASM_SIMP_TAC[SUBSET_AFFINE_HULL_SPECIAL_SCALE]);;
let COPLANAR_SCALE_ALL = prove
(`!a b c x y z.
~(a = &0) /\ ~(b = &0) /\ ~(c = &0)
==> (coplanar {vec 0,a % x,b % y,c % z} <=> coplanar {vec 0,x,y,z})`,
REPEAT STRIP_TAC THEN ASM_SIMP_TAC[COPLANAR_SPECIAL_SCALE] THEN
ONCE_REWRITE_TAC[SET_RULE `{a,b,c,d} = {a,c,d,b}`] THEN
ASM_SIMP_TAC[COPLANAR_SPECIAL_SCALE] THEN
ONCE_REWRITE_TAC[SET_RULE `{a,b,c,d} = {a,c,d,b}`] THEN
ASM_SIMP_TAC[COPLANAR_SPECIAL_SCALE]);;
(* ------------------------------------------------------------------------- *)
(* Specialized lemmas about "dropout". *)
(* ------------------------------------------------------------------------- *)
let DROPOUT_BASIS_3 = prove
(`(dropout 3:real^3->real^2) (basis 1) = basis 1 /\
(dropout 3:real^3->real^2) (basis 2) = basis 2 /\
(dropout 3:real^3->real^2) (basis 3) = vec 0`,
SIMP_TAC[LAMBDA_BETA; dropout; basis; CART_EQ; DIMINDEX_2; DIMINDEX_3; ARITH;
VEC_COMPONENT; LT_IMP_LE; ARITH_RULE `i <= 2 ==> i + 1 <= 3`;
ARITH_RULE `1 <= i + 1`] THEN
ARITH_TAC);;
let COLLINEAR_BASIS_3 = prove
(`collinear {vec 0,basis 3,x} <=> (dropout 3:real^3->real^2) x = vec 0`,
SIMP_TAC[CART_EQ; FORALL_2; FORALL_3; DIMINDEX_2; DIMINDEX_3;
dropout; LAMBDA_BETA; BASIS_COMPONENT; ARITH; REAL_MUL_RID;
VECTOR_MUL_COMPONENT; VEC_COMPONENT; REAL_MUL_RZERO; UNWIND_THM1;
COLLINEAR_LEMMA] THEN
REWRITE_TAC[RIGHT_EXISTS_AND_THM; GSYM EXISTS_REFL] THEN REAL_ARITH_TAC);;
let OPEN_DROPOUT_3 = prove
(`!P. open {x | P x} ==> open {x | P((dropout 3:real^3->real^2) x)}`,
REPEAT STRIP_TAC THEN MP_TAC(ISPECL
[`dropout 3:real^3->real^2`; `{x:real^2 | P x}`]
CONTINUOUS_OPEN_PREIMAGE_UNIV) THEN
ASM_REWRITE_TAC[IN_ELIM_THM] THEN DISCH_THEN MATCH_MP_TAC THEN
GEN_TAC THEN MATCH_MP_TAC LINEAR_CONTINUOUS_AT THEN
SIMP_TAC[LINEAR_DROPOUT; DIMINDEX_2; DIMINDEX_3; ARITH]);;
let SLICE_DROPOUT_3 = prove
(`!P t. slice 3 t {x | P((dropout 3:real^3->real^2) x)} = {x | P x}`,
REPEAT GEN_TAC THEN REWRITE_TAC[slice] THEN
REWRITE_TAC[EXTENSION; IN_IMAGE; IN_ELIM_THM; IN_INTER] THEN
X_GEN_TAC `y:real^2` THEN EQ_TAC THENL [MESON_TAC[]; ALL_TAC] THEN
DISCH_TAC THEN EXISTS_TAC `(pushin 3 t:real^2->real^3) y` THEN
ASM_SIMP_TAC[DIMINDEX_2; DIMINDEX_3; DROPOUT_PUSHIN; ARITH] THEN
SIMP_TAC[pushin; LAMBDA_BETA; LT_REFL; DIMINDEX_3; ARITH]);;
let NOT_COPLANAR_IMP_NOT_COLLINEAR_DROPOUT_3 = prove
(`!x y:real^3.
~coplanar {vec 0,basis 3, x, y}
==> ~collinear {vec 0,dropout 3 x:real^2,dropout 3 y}`,
REPEAT GEN_TAC THEN REWRITE_TAC[COLLINEAR_AFFINE_HULL; coplanar] THEN
REWRITE_TAC[CONTRAPOS_THM; INSERT_SUBSET; LEFT_IMP_EXISTS_THM] THEN
MAP_EVERY X_GEN_TAC [`u:real^2`; `v:real^2`] THEN
REWRITE_TAC[EMPTY_SUBSET] THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
ASM_SIMP_TAC[AFFINE_HULL_EQ_SPAN] THEN STRIP_TAC THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [AFFINE_HULL_2]) THEN
REWRITE_TAC[IN_ELIM_THM; LEFT_IMP_EXISTS_THM] THEN
MAP_EVERY X_GEN_TAC [`a:real`;`b:real`] THEN STRIP_TAC THEN
SUBGOAL_THEN `?r s. a * r + b * s = -- &1` STRIP_ASSUME_TAC THENL
[ASM_CASES_TAC `a = &0` THENL
[UNDISCH_TAC `a + b = &1` THEN
ASM_SIMP_TAC[REAL_MUL_LZERO; REAL_ADD_LID; REAL_MUL_LID; EXISTS_REFL];
ASM_SIMP_TAC[REAL_FIELD
`~(a = &0) ==> (a * r + x = y <=> r = (y - x) / a)`] THEN
MESON_TAC[]];
ALL_TAC] THEN
EXISTS_TAC `vector[(u:real^2)$1; u$2; r]:real^3` THEN
EXISTS_TAC `vector[(v:real^2)$1; v$2; s]:real^3` THEN
EXISTS_TAC `basis 3:real^3` THEN
MATCH_MP_TAC(TAUT `a /\ (a ==> b) ==> a /\ b`) THEN CONJ_TAC THENL
[REWRITE_TAC[AFFINE_HULL_3; IN_ELIM_THM; LEFT_IMP_EXISTS_THM] THEN
MAP_EVERY EXISTS_TAC [`a / &2`;`b / &2`; `&1 / &2`] THEN
ASM_REWRITE_TAC[REAL_ARITH
`a / &2 + b / &2 + &1 / &2 = &1 <=> a + b = &1`] THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [CART_EQ]) THEN
SIMP_TAC[CART_EQ; DIMINDEX_2; DIMINDEX_3; FORALL_2; FORALL_3;
VEC_COMPONENT; VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT;
VECTOR_3; BASIS_COMPONENT; ARITH] THEN
REPEAT(FIRST_X_ASSUM(MP_TAC o SYM)) THEN CONV_TAC REAL_RING;
ALL_TAC] THEN
SIMP_TAC[AFFINE_HULL_EQ_SPAN] THEN DISCH_TAC THEN
SIMP_TAC[SPAN_SUPERSET; IN_INSERT] THEN
SUBGOAL_THEN
`!x. (dropout 3:real^3->real^2) x IN span {u,v}
==> x IN span {vector [u$1; u$2; r], vector [v$1; v$2; s], basis 3}`
(fun th -> ASM_MESON_TAC[th]) THEN
GEN_TAC THEN REWRITE_TAC[SPAN_2; SPAN_3] THEN
SIMP_TAC[IN_ELIM_THM; IN_UNIV; CART_EQ; DIMINDEX_2; DIMINDEX_3;
FORALL_2; FORALL_3; dropout; VECTOR_ADD_COMPONENT; LAMBDA_BETA;
VECTOR_MUL_COMPONENT; VECTOR_3; BASIS_COMPONENT; ARITH] THEN
MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `c:real` THEN
MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `d:real` THEN
STRIP_TAC THEN ASM_REWRITE_TAC[REAL_MUL_RZERO; REAL_ADD_RID] THEN
REWRITE_TAC[REAL_ARITH `x = a + b + c * &1 <=> c = x - a - b`] THEN
REWRITE_TAC[EXISTS_REFL]);;
let SLICE_312 = prove
(`!s:real^3->bool. slice 1 t s = {y:real^2 | vector[t;y$1;y$2] IN s}`,
SIMP_TAC[EXTENSION; IN_SLICE; DIMINDEX_2; DIMINDEX_3; ARITH] THEN
REPEAT GEN_TAC THEN REWRITE_TAC[IN_ELIM_THM] THEN
AP_THM_TAC THEN AP_TERM_TAC THEN
SIMP_TAC[CART_EQ; pushin; LAMBDA_BETA; FORALL_3; DIMINDEX_3; ARITH;
VECTOR_3]);;
let SLICE_123 = prove
(`!s:real^3->bool. slice 3 t s = {y:real^2 | vector[y$1;y$2;t] IN s}`,
SIMP_TAC[EXTENSION; IN_SLICE; DIMINDEX_2; DIMINDEX_3; ARITH] THEN
REPEAT GEN_TAC THEN REWRITE_TAC[IN_ELIM_THM] THEN
AP_THM_TAC THEN AP_TERM_TAC THEN
SIMP_TAC[CART_EQ; pushin; LAMBDA_BETA; FORALL_3; DIMINDEX_3; ARITH;
VECTOR_3]);;
(* ------------------------------------------------------------------------- *)
(* "Padding" injection from real^2 -> real^3 with zero last coordinate. *)
(* ------------------------------------------------------------------------- *)
let pad2d3d = new_definition
`(pad2d3d:real^2->real^3) x = lambda i. if i < 3 then x$i else &0`;;
let FORALL_PAD2D3D_THM = prove
(`!P. (!y:real^3. y$3 = &0 ==> P y) <=> (!x. P(pad2d3d x))`,
GEN_TAC THEN EQ_TAC THEN REPEAT STRIP_TAC THENL
[FIRST_X_ASSUM MATCH_MP_TAC THEN REWRITE_TAC[pad2d3d] THEN
SIMP_TAC[LAMBDA_BETA; DIMINDEX_3; ARITH; LT_REFL];
FIRST_X_ASSUM(MP_TAC o SPEC `(lambda i. (y:real^3)$i):real^2`) THEN
MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN
SIMP_TAC[CART_EQ; pad2d3d; DIMINDEX_3; ARITH; LAMBDA_BETA; DIMINDEX_2;
ARITH_RULE `i < 3 <=> i <= 2`] THEN
REWRITE_TAC[ARITH_RULE `i <= 3 <=> i <= 2 \/ i = 3`] THEN
ASM_MESON_TAC[]]);;
let QUANTIFY_PAD2D3D_THM = prove
(`(!P. (!y:real^3. y$3 = &0 ==> P y) <=> (!x. P(pad2d3d x))) /\
(!P. (?y:real^3. y$3 = &0 /\ P y) <=> (?x. P(pad2d3d x)))`,
REWRITE_TAC[MESON[] `(?y. P y) <=> ~(!x. ~P x)`] THEN
REWRITE_TAC[GSYM FORALL_PAD2D3D_THM] THEN MESON_TAC[]);;
let LINEAR_PAD2D3D = prove
(`linear pad2d3d`,
REWRITE_TAC[linear; pad2d3d] THEN
SIMP_TAC[CART_EQ; VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT;
LAMBDA_BETA; DIMINDEX_2; DIMINDEX_3; ARITH;
ARITH_RULE `i < 3 ==> i <= 2`] THEN
REPEAT STRIP_TAC THEN REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[]) THEN
REAL_ARITH_TAC);;
let INJECTIVE_PAD2D3D = prove
(`!x y. pad2d3d x = pad2d3d y ==> x = y`,
SIMP_TAC[CART_EQ; pad2d3d; LAMBDA_BETA; DIMINDEX_3; DIMINDEX_2] THEN
REWRITE_TAC[ARITH_RULE `i < 3 <=> i <= 2`] THEN
MESON_TAC[ARITH_RULE `i <= 2 ==> i <= 3`]);;
let NORM_PAD2D3D = prove
(`!x. norm(pad2d3d x) = norm x`,
SIMP_TAC[NORM_EQ; DOT_2; DOT_3; pad2d3d; LAMBDA_BETA;
DIMINDEX_2; DIMINDEX_3; ARITH] THEN
REAL_ARITH_TAC);;
(* ------------------------------------------------------------------------- *)
(* Apply 3D->2D conversion to a goal. Take care to preserve variable names. *)
(* ------------------------------------------------------------------------- *)
let PAD2D3D_QUANTIFY_CONV =
let gv = genvar `:real^2` in
let pth = CONV_RULE (BINOP_CONV(BINDER_CONV(RAND_CONV(GEN_ALPHA_CONV gv))))
QUANTIFY_PAD2D3D_THM in
let conv1 = GEN_REWRITE_CONV I [pth]
and dest_quant tm = try dest_forall tm with Failure _ -> dest_exists tm in
fun tm ->
let th = conv1 tm in
let name = fst(dest_var(fst(dest_quant tm))) in
let ty = snd(dest_var(fst(dest_quant(rand(concl th))))) in
CONV_RULE(RAND_CONV(GEN_ALPHA_CONV(mk_var(name,ty)))) th;;
let PAD2D3D_TAC =
let pad2d3d_tm = `pad2d3d`
and pths = [LINEAR_PAD2D3D; INJECTIVE_PAD2D3D; NORM_PAD2D3D]
and cth = prove
(`{} = IMAGE pad2d3d {} /\
vec 0 = pad2d3d(vec 0)`,
REWRITE_TAC[IMAGE_CLAUSES] THEN MESON_TAC[LINEAR_PAD2D3D; LINEAR_0]) in
let lasttac =
GEN_REWRITE_TAC REDEPTH_CONV [LINEAR_INVARIANTS pad2d3d_tm pths] in
fun gl -> (GEN_REWRITE_TAC ONCE_DEPTH_CONV [cth] THEN
CONV_TAC(DEPTH_CONV PAD2D3D_QUANTIFY_CONV) THEN
lasttac) gl;;
(* ------------------------------------------------------------------------- *)
(* The notion of a plane, and using it to characterize coplanarity. *)
(* ------------------------------------------------------------------------- *)
let plane = new_definition
`plane x = (?u v w. ~(collinear {u,v,w}) /\ x = affine hull {u,v,w})`;;
let PLANE_TRANSLATION_EQ = prove
(`!a:real^N s. plane(IMAGE (\x. a + x) s) <=> plane s`,
REWRITE_TAC[plane] THEN GEOM_TRANSLATE_TAC[]);;
let PLANE_TRANSLATION = prove
(`!a:real^N s. plane s ==> plane(IMAGE (\x. a + x) s)`,
REWRITE_TAC[PLANE_TRANSLATION_EQ]);;
add_translation_invariants [PLANE_TRANSLATION_EQ];;
let PLANE_LINEAR_IMAGE_EQ = prove
(`!f:real^M->real^N p.
linear f /\ (!x y. f x = f y ==> x = y)
==> (plane(IMAGE f p) <=> plane p)`,
REPEAT STRIP_TAC THEN REWRITE_TAC[plane] THEN
MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC
`?u. u IN IMAGE f (:real^M) /\
?v. v IN IMAGE f (:real^M) /\
?w. w IN IMAGE (f:real^M->real^N) (:real^M) /\
~collinear {u, v, w} /\ IMAGE f p = affine hull {u, v, w}` THEN
CONJ_TAC THENL
[REWRITE_TAC[RIGHT_AND_EXISTS_THM; IN_IMAGE; IN_UNIV] THEN
REPEAT(AP_TERM_TAC THEN ABS_TAC) THEN
EQ_TAC THEN DISCH_TAC THEN ASM_REWRITE_TAC[] THEN
SUBGOAL_THEN `{u,v,w} SUBSET IMAGE (f:real^M->real^N) p` MP_TAC THENL
[ASM_REWRITE_TAC[HULL_SUBSET]; SET_TAC[]];
REWRITE_TAC[EXISTS_IN_IMAGE; IN_UNIV] THEN
REWRITE_TAC[SET_RULE `{f a,f b,f c} = IMAGE f {a,b,c}`] THEN
ASM_SIMP_TAC[AFFINE_HULL_LINEAR_IMAGE] THEN
REPEAT(AP_TERM_TAC THEN ABS_TAC) THEN BINOP_TAC THENL
[ASM_MESON_TAC[COLLINEAR_LINEAR_IMAGE_EQ]; ASM SET_TAC[]]]);;
let PLANE_LINEAR_IMAGE = prove
(`!f:real^M->real^N p.
linear f /\ plane p /\ (!x y. f x = f y ==> x = y)
==> plane(IMAGE f p)`,
MESON_TAC[PLANE_LINEAR_IMAGE_EQ]);;
add_linear_invariants [PLANE_LINEAR_IMAGE_EQ];;
let AFFINE_PLANE = prove
(`!p. plane p ==> affine p`,
SIMP_TAC[plane; LEFT_IMP_EXISTS_THM; AFFINE_AFFINE_HULL]);;
let ROTATION_PLANE_HORIZONTAL = prove
(`!s. plane s
==> ?a f. orthogonal_transformation f /\ det(matrix f) = &1 /\
IMAGE f (IMAGE (\x. a + x) s) = {z:real^3 | z$3 = &0}`,
let lemma = prove
(`span {z:real^3 | z$3 = &0} = {z:real^3 | z$3 = &0}`,
REWRITE_TAC[SPAN_EQ_SELF; subspace; IN_ELIM_THM] THEN
SIMP_TAC[VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT; VEC_COMPONENT;
DIMINDEX_3; ARITH] THEN REAL_ARITH_TAC) in
REPEAT STRIP_TAC THEN
FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [plane]) THEN
REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN
MAP_EVERY X_GEN_TAC [`a:real^3`; `b:real^3`; `c:real^3`] THEN
MAP_EVERY (fun t ->
ASM_CASES_TAC t THENL [ASM_REWRITE_TAC[COLLINEAR_2; INSERT_AC];
ALL_TAC])
[`a:real^3 = b`; `a:real^3 = c`; `b:real^3 = c`] THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC SUBST1_TAC) THEN
ASM_SIMP_TAC[AFFINE_HULL_INSERT_SPAN; IN_INSERT; NOT_IN_EMPTY] THEN
EXISTS_TAC `--a:real^3` THEN
REWRITE_TAC[SET_RULE `IMAGE (\x:real^3. --a + x) {a + x | x | x IN s} =
IMAGE (\x. --a + a + x) s`] THEN
REWRITE_TAC[VECTOR_ARITH `--a + a + x:real^3 = x`; IMAGE_ID] THEN
REWRITE_TAC[SET_RULE `{x - a:real^x | x = b \/ x = c} = {b - a,c - a}`] THEN
MP_TAC(ISPEC `span{b - a:real^3,c - a}`
ROTATION_LOWDIM_HORIZONTAL) THEN
REWRITE_TAC[DIMINDEX_3] THEN ANTS_TAC THENL
[MATCH_MP_TAC LET_TRANS THEN
EXISTS_TAC `CARD{b - a:real^3,c - a}` THEN
SIMP_TAC[DIM_SPAN; DIM_LE_CARD; FINITE_RULES] THEN
SIMP_TAC[CARD_CLAUSES; FINITE_RULES] THEN ARITH_TAC;
ALL_TAC] THEN
MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `f:real^3->real^3` THEN
STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
FIRST_ASSUM(ASSUME_TAC o MATCH_MP ORTHOGONAL_TRANSFORMATION_LINEAR) THEN
ASM_SIMP_TAC[GSYM SPAN_LINEAR_IMAGE] THEN
GEN_REWRITE_TAC RAND_CONV [GSYM lemma] THEN
MATCH_MP_TAC DIM_EQ_SPAN THEN CONJ_TAC THENL
[ASM_MESON_TAC[IMAGE_SUBSET; SPAN_INC; SUBSET_TRANS]; ALL_TAC] THEN
MATCH_MP_TAC LE_TRANS THEN EXISTS_TAC `2` THEN CONJ_TAC THENL
[MP_TAC(ISPECL [`{z:real^3 | z$3 = &0}`; `(:real^3)`] DIM_EQ_SPAN) THEN
REWRITE_TAC[SUBSET_UNIV; DIM_UNIV; DIMINDEX_3; lemma] THEN
MATCH_MP_TAC(TAUT `~r /\ (~p ==> q) ==> (q ==> r) ==> p`) THEN
REWRITE_TAC[ARITH_RULE `~(x <= 2) <=> 3 <= x`] THEN
REWRITE_TAC[EXTENSION; SPAN_UNIV; IN_ELIM_THM] THEN
DISCH_THEN(MP_TAC o SPEC `vector[&0;&0;&1]:real^3`) THEN
REWRITE_TAC[IN_UNIV; VECTOR_3] THEN REAL_ARITH_TAC;
ALL_TAC] THEN
MATCH_MP_TAC LE_TRANS THEN EXISTS_TAC `dim {b - a:real^3,c - a}` THEN
CONJ_TAC THENL
[ALL_TAC; ASM_MESON_TAC[LE_REFL; DIM_INJECTIVE_LINEAR_IMAGE;
ORTHOGONAL_TRANSFORMATION_INJECTIVE]] THEN
MP_TAC(ISPEC `{b - a:real^3,c - a}` INDEPENDENT_BOUND_GENERAL) THEN
SIMP_TAC[CARD_CLAUSES; FINITE_RULES; IN_SING; NOT_IN_EMPTY] THEN
ASM_REWRITE_TAC[VECTOR_ARITH `b - a:real^3 = c - a <=> b = c`; ARITH] THEN
DISCH_THEN MATCH_MP_TAC THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE (RAND_CONV o RAND_CONV)
[SET_RULE `{a,b,c} = {b,a,c}`]) THEN
REWRITE_TAC[] THEN ONCE_REWRITE_TAC[COLLINEAR_3] THEN
REWRITE_TAC[independent; CONTRAPOS_THM; dependent] THEN
REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY; RIGHT_OR_DISTRIB] THEN
REWRITE_TAC[EXISTS_OR_THM; UNWIND_THM2] THEN
ASM_SIMP_TAC[SET_RULE `~(a = b) ==> {a,b} DELETE b = {a}`;
SET_RULE `~(a = b) ==> {a,b} DELETE a = {b}`;
VECTOR_ARITH `b - a:real^3 = c - a <=> b = c`] THEN
REWRITE_TAC[SPAN_BREAKDOWN_EQ; SPAN_EMPTY; IN_SING] THEN
ONCE_REWRITE_TAC[VECTOR_SUB_EQ] THEN MESON_TAC[COLLINEAR_LEMMA; INSERT_AC]);;
let ROTATION_HORIZONTAL_PLANE = prove
(`!p. plane p
==> ?a f. orthogonal_transformation f /\ det(matrix f) = &1 /\
IMAGE (\x. a + x) (IMAGE f {z:real^3 | z$3 = &0}) = p`,
REPEAT STRIP_TAC THEN
FIRST_X_ASSUM(MP_TAC o MATCH_MP ROTATION_PLANE_HORIZONTAL) THEN
DISCH_THEN(X_CHOOSE_THEN `a:real^3`
(X_CHOOSE_THEN `f:real^3->real^3` STRIP_ASSUME_TAC)) THEN
FIRST_ASSUM(X_CHOOSE_THEN `g:real^3->real^3` STRIP_ASSUME_TAC o MATCH_MP
ORTHOGONAL_TRANSFORMATION_INVERSE) THEN
MAP_EVERY EXISTS_TAC [`--a:real^3`; `g:real^3->real^3`] THEN
FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN
ASM_REWRITE_TAC[GSYM IMAGE_o; o_DEF; IMAGE_ID;
VECTOR_ARITH `--a + a + x:real^3 = x`] THEN
MATCH_MP_TAC(REAL_RING `!f. f * g = &1 /\ f = &1 ==> g = &1`) THEN
EXISTS_TAC `det(matrix(f:real^3->real^3))` THEN
REWRITE_TAC[GSYM DET_MUL] THEN
ASM_SIMP_TAC[GSYM MATRIX_COMPOSE; ORTHOGONAL_TRANSFORMATION_LINEAR] THEN
ASM_REWRITE_TAC[o_DEF; MATRIX_ID; DET_I]);;
let COPLANAR = prove
(`2 <= dimindex(:N)
==> !s:real^N->bool. coplanar s <=> ?x. plane x /\ s SUBSET x`,
DISCH_TAC THEN GEN_TAC THEN REWRITE_TAC[coplanar; plane] THEN
CONV_TAC SYM_CONV THEN REWRITE_TAC[LEFT_AND_EXISTS_THM] THEN
ONCE_REWRITE_TAC[MESON[]
`(?x u v w. p x u v w) <=> (?u v w x. p x u v w)`] THEN
REWRITE_TAC[GSYM CONJ_ASSOC; RIGHT_EXISTS_AND_THM; UNWIND_THM2] THEN
EQ_TAC THENL [MESON_TAC[]; REWRITE_TAC[LEFT_IMP_EXISTS_THM]] THEN
MAP_EVERY X_GEN_TAC [`u:real^N`; `v:real^N`; `w:real^N`] THEN DISCH_TAC THEN
SUBGOAL_THEN
`s SUBSET {u + x:real^N | x | x IN span {y - u | y IN {v,w}}}`
MP_TAC THENL
[FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP
(REWRITE_RULE[IMP_CONJ] SUBSET_TRANS)) THEN
REWRITE_TAC[AFFINE_HULL_INSERT_SUBSET_SPAN];
ALL_TAC] THEN
ONCE_REWRITE_TAC[SIMPLE_IMAGE_GEN] THEN
DISCH_THEN(MP_TAC o ISPEC `\x:real^N. x - u` o MATCH_MP IMAGE_SUBSET) THEN
REWRITE_TAC[GSYM IMAGE_o; o_DEF; VECTOR_ADD_SUB; IMAGE_ID; SIMPLE_IMAGE] THEN
REWRITE_TAC[IMAGE_CLAUSES] THEN
MP_TAC(ISPECL [`{v - u:real^N,w - u}`; `2`] LOWDIM_EXPAND_BASIS) THEN
ANTS_TAC THENL
[ASM_REWRITE_TAC[] THEN MATCH_MP_TAC LE_TRANS THEN
EXISTS_TAC `CARD{v - u:real^N,w - u}` THEN
SIMP_TAC[DIM_LE_CARD; FINITE_INSERT; FINITE_RULES] THEN
SIMP_TAC[CARD_CLAUSES; FINITE_RULES] THEN ARITH_TAC;
ALL_TAC] THEN
DISCH_THEN(X_CHOOSE_THEN `t:real^N->bool`
(CONJUNCTS_THEN2 MP_TAC STRIP_ASSUME_TAC)) THEN
CONV_TAC(LAND_CONV HAS_SIZE_CONV) THEN
REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN
MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real^N`] THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC SUBST_ALL_TAC) THEN
UNDISCH_TAC `span {v - u, w - u} SUBSET span {a:real^N, b}` THEN
REWRITE_TAC[GSYM IMP_CONJ_ALT] THEN
DISCH_THEN(ASSUME_TAC o MATCH_MP SUBSET_TRANS) THEN
MAP_EVERY EXISTS_TAC [`u:real^N`; `u + a:real^N`; `u + b:real^N`] THEN
CONJ_TAC THENL
[REWRITE_TAC[COLLINEAR_3; COLLINEAR_LEMMA;
VECTOR_ARITH `--x = vec 0 <=> x = vec 0`;
VECTOR_ARITH `u - (u + a):real^N = --a`;
VECTOR_ARITH `(u + b) - (u + a):real^N = b - a`] THEN
REWRITE_TAC[DE_MORGAN_THM; VECTOR_SUB_EQ;
VECTOR_ARITH `b - a = c % -- a <=> (c - &1) % a + &1 % b = vec 0`] THEN
ASM_REWRITE_TAC[] THEN CONJ_TAC THENL
[ASM_MESON_TAC[IN_INSERT; INDEPENDENT_NONZERO]; ALL_TAC] THEN
DISCH_THEN(X_CHOOSE_TAC `u:real`) THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [independent]) THEN
REWRITE_TAC[DEPENDENT_EXPLICIT] THEN
MAP_EVERY EXISTS_TAC [`{a:real^N,b}`;
`\x:real^N. if x = a then u - &1 else &1`] THEN
REWRITE_TAC[FINITE_INSERT; FINITE_RULES; SUBSET_REFL] THEN
CONJ_TAC THENL
[EXISTS_TAC `b:real^N` THEN ASM_REWRITE_TAC[IN_INSERT] THEN
REAL_ARITH_TAC;
ALL_TAC] THEN
SIMP_TAC[VSUM_CLAUSES; FINITE_RULES] THEN
ASM_REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY; VECTOR_ADD_RID];
ALL_TAC] THEN
W(MP_TAC o PART_MATCH (lhs o rand) AFFINE_HULL_INSERT_SPAN o rand o snd) THEN
ANTS_TAC THENL
[REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY] THEN
REWRITE_TAC[VECTOR_ARITH `u = u + a <=> a = vec 0`] THEN
ASM_MESON_TAC[INDEPENDENT_NONZERO; IN_INSERT];
ALL_TAC] THEN
DISCH_THEN SUBST1_TAC THEN
FIRST_ASSUM(MP_TAC o ISPEC `\x:real^N. u + x` o MATCH_MP IMAGE_SUBSET) THEN
REWRITE_TAC[GSYM IMAGE_o; o_DEF; IMAGE_ID;
ONCE_REWRITE_RULE[VECTOR_ADD_SYM] VECTOR_SUB_ADD] THEN
MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] SUBSET_TRANS) THEN
REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN
REWRITE_TAC[SIMPLE_IMAGE; IMAGE_CLAUSES; VECTOR_ADD_SUB] THEN
SET_TAC[]);;
let COPLANAR_DET_EQ_0 = prove
(`!v0 v1 (v2: real^3) v3.
coplanar {v0,v1,v2,v3} <=>
det(vector[v1 - v0; v2 - v0; v3 - v0]) = &0`,
REPEAT GEN_TAC THEN REWRITE_TAC[DET_EQ_0_RANK; RANK_ROW] THEN
REWRITE_TAC[rows; row; LAMBDA_ETA] THEN
ONCE_REWRITE_TAC[SIMPLE_IMAGE_GEN] THEN
REWRITE_TAC[GSYM numseg; DIMINDEX_3] THEN
CONV_TAC(ONCE_DEPTH_CONV NUMSEG_CONV) THEN
SIMP_TAC[IMAGE_CLAUSES; coplanar; VECTOR_3] THEN EQ_TAC THENL
[REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN
MAP_EVERY X_GEN_TAC [`a:real^3`; `b:real^3`; `c:real^3`] THEN
W(MP_TAC o PART_MATCH lhand AFFINE_HULL_INSERT_SUBSET_SPAN o
rand o lhand o snd) THEN
REWRITE_TAC[GSYM IMP_CONJ_ALT] THEN
DISCH_THEN(MP_TAC o MATCH_MP SUBSET_TRANS) THEN
DISCH_THEN(MP_TAC o ISPEC `\x:real^3. x - a` o MATCH_MP IMAGE_SUBSET) THEN
ONCE_REWRITE_TAC[SIMPLE_IMAGE_GEN] THEN
REWRITE_TAC[IMAGE_CLAUSES; GSYM IMAGE_o; o_DEF; VECTOR_ADD_SUB; IMAGE_ID;
SIMPLE_IMAGE] THEN
REWRITE_TAC[INSERT_SUBSET] THEN STRIP_TAC THEN
GEN_REWRITE_TAC LAND_CONV [GSYM DIM_SPAN] THEN MATCH_MP_TAC LET_TRANS THEN
EXISTS_TAC `CARD {b - a:real^3,c - a}` THEN
CONJ_TAC THENL
[MATCH_MP_TAC SPAN_CARD_GE_DIM;
SIMP_TAC[CARD_CLAUSES; FINITE_RULES] THEN ARITH_TAC] THEN
REWRITE_TAC[FINITE_INSERT; FINITE_RULES] THEN
GEN_REWRITE_TAC RAND_CONV [GSYM SPAN_SPAN] THEN
MATCH_MP_TAC SPAN_MONO THEN REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET] THEN
MP_TAC(VECTOR_ARITH `!x y:real^3. x - y = (x - a) - (y - a)`) THEN
DISCH_THEN(fun th -> REPEAT CONJ_TAC THEN
GEN_REWRITE_TAC LAND_CONV [th]) THEN
MATCH_MP_TAC SPAN_SUB THEN ASM_REWRITE_TAC[];
DISCH_TAC THEN
MP_TAC(ISPECL [`{v1 - v0,v2 - v0,v3 - v0}:real^3->bool`; `2`]
LOWDIM_EXPAND_BASIS) THEN
ASM_REWRITE_TAC[ARITH_RULE `n <= 2 <=> n < 3`; DIMINDEX_3; ARITH] THEN
DISCH_THEN(X_CHOOSE_THEN `t:real^3->bool`
(CONJUNCTS_THEN2 MP_TAC STRIP_ASSUME_TAC)) THEN
CONV_TAC(LAND_CONV HAS_SIZE_CONV) THEN
REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN
MAP_EVERY X_GEN_TAC [`a:real^3`; `b:real^3`] THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC SUBST_ALL_TAC) THEN
SIMP_TAC[COPLANAR; DIMINDEX_3; ARITH; plane] THEN
MAP_EVERY EXISTS_TAC [`v0:real^3`; `v0 + a:real^3`; `v0 + b:real^3`] THEN
W(MP_TAC o PART_MATCH (lhs o rand) AFFINE_HULL_INSERT_SPAN o
rand o snd) THEN
ANTS_TAC THENL
[REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY] THEN
REWRITE_TAC[VECTOR_ARITH `u = u + a <=> a = vec 0`] THEN
ASM_MESON_TAC[INDEPENDENT_NONZERO; IN_INSERT];
ALL_TAC] THEN
DISCH_THEN SUBST1_TAC THEN
ONCE_REWRITE_TAC[SIMPLE_IMAGE_GEN] THEN
REWRITE_TAC[SIMPLE_IMAGE; IMAGE_CLAUSES; IMAGE_ID; VECTOR_ADD_SUB] THEN
MATCH_MP_TAC SUBSET_TRANS THEN EXISTS_TAC
`IMAGE (\v:real^3. v0 + v) (span{v1 - v0, v2 - v0, v3 - v0})` THEN
ASM_SIMP_TAC[IMAGE_SUBSET] THEN
REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET; IN_IMAGE] THEN CONJ_TAC THENL
[EXISTS_TAC `vec 0:real^3` THEN REWRITE_TAC[SPAN_0] THEN VECTOR_ARITH_TAC;
REWRITE_TAC[VECTOR_ARITH `v1:real^N = v0 + x <=> x = v1 - v0`] THEN
REWRITE_TAC[UNWIND_THM2] THEN REPEAT CONJ_TAC THEN
MATCH_MP_TAC SPAN_SUPERSET THEN REWRITE_TAC[IN_INSERT]]]);;
let COPLANAR_CROSS_DOT = prove
(`!v w x y. coplanar {v,w,x,y} <=> ((w - v) cross (x - v)) dot (y - v) = &0`,
REWRITE_TAC[COPLANAR_DET_EQ_0; GSYM DOT_CROSS_DET] THEN
MESON_TAC[CROSS_TRIPLE; DOT_SYM]);;
let PLANE_AFFINE_HULL_3 = prove
(`!a b c:real^N. plane(affine hull {a,b,c}) <=> ~collinear{a,b,c}`,
REWRITE_TAC[plane] THEN MESON_TAC[COLLINEAR_AFFINE_HULL_COLLINEAR]);;
let AFFINE_HULL_3_GENERATED = prove
(`!s u v w:real^N.
s SUBSET affine hull {u,v,w} /\ ~collinear s
==> affine hull {u,v,w} = affine hull s`,
REWRITE_TAC[COLLINEAR_AFF_DIM; INT_NOT_LE] THEN REPEAT STRIP_TAC THEN
CONV_TAC SYM_CONV THEN
GEN_REWRITE_TAC RAND_CONV [GSYM HULL_HULL] THEN
MATCH_MP_TAC AFF_DIM_EQ_AFFINE_HULL THEN ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC INT_LE_TRANS THEN EXISTS_TAC `&2:int` THEN
CONJ_TAC THENL [ALL_TAC; ASM_INT_ARITH_TAC] THEN
REWRITE_TAC[AFF_DIM_AFFINE_HULL] THEN
W(MP_TAC o PART_MATCH (lhand o rand) AFF_DIM_LE_CARD o lhand o snd) THEN
REWRITE_TAC[FINITE_INSERT; FINITE_EMPTY] THEN
MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] INT_LE_TRANS) THEN
REWRITE_TAC[INT_LE_SUB_RADD; INT_OF_NUM_ADD; INT_OF_NUM_LE] THEN
SIMP_TAC[CARD_CLAUSES; FINITE_INSERT; FINITE_EMPTY] THEN ARITH_TAC);;
(* ------------------------------------------------------------------------- *)
(* Additional WLOG tactic to rotate any plane p to {z | z$3 = &0}. *)
(* ------------------------------------------------------------------------- *)
let GEOM_HORIZONTAL_PLANE_RULE =
let ifn = MATCH_MP
(TAUT `(p ==> (x <=> x')) /\ (~p ==> (x <=> T)) ==> (x' ==> x)`)
and pth = prove
(`!a f. orthogonal_transformation (f:real^N->real^N)
==> ((!P. (!x. P x) <=> (!x. P (a + f x))) /\
(!P. (?x. P x) <=> (?x. P (a + f x))) /\
(!Q. (!s. Q s) <=> (!s. Q (IMAGE (\x. a + x) (IMAGE f s)))) /\
(!Q. (?s. Q s) <=> (?s. Q (IMAGE (\x. a + x) (IMAGE f s))))) /\
(!P. {x | P x} =
IMAGE (\x. a + x) (IMAGE f {x | P(a + f x)}))`,
REPEAT GEN_TAC THEN DISCH_TAC THEN
MP_TAC(ISPEC `(\x. a + x) o (f:real^N->real^N)`
QUANTIFY_SURJECTION_THM) THEN REWRITE_TAC[o_THM; IMAGE_o] THEN
DISCH_THEN MATCH_MP_TAC THEN
ASM_MESON_TAC[ORTHOGONAL_TRANSFORMATION_SURJECTIVE;
VECTOR_ARITH `a + (x - a:real^N) = x`])
and cth = prove
(`!a f. {} = IMAGE (\x:real^3. a + x) (IMAGE f {})`,
REWRITE_TAC[IMAGE_CLAUSES])
and oth = prove
(`!f:real^3->real^3.
orthogonal_transformation f /\ det(matrix f) = &1
==> linear f /\
(!x y. f x = f y ==> x = y) /\
(!y. ?x. f x = y) /\
(!x. norm(f x) = norm x) /\
(2 <= dimindex(:3) ==> det(matrix f) = &1)`,
GEN_TAC THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN REPEAT CONJ_TAC THENL
[ASM_SIMP_TAC[ORTHOGONAL_TRANSFORMATION_LINEAR];
ASM_MESON_TAC[ORTHOGONAL_TRANSFORMATION_INJECTIVE];
ASM_MESON_TAC[ORTHOGONAL_TRANSFORMATION_SURJECTIVE];
ASM_MESON_TAC[ORTHOGONAL_TRANSFORMATION]])
and fth = MESON[]
`(!a f. q a f ==> (p <=> p' a f))
==> ((?a f. q a f) ==> (p <=> !a f. q a f ==> p' a f))` in
fun tm ->
let x,bod = dest_forall tm in
let th1 = EXISTS_GENVAR_RULE
(UNDISCH(ISPEC x ROTATION_HORIZONTAL_PLANE)) in
let [a;f],tm1 = strip_exists(concl th1) in
let [th_orth;th_det;th_im] = CONJUNCTS(ASSUME tm1) in
let th2 = PROVE_HYP th_orth (UNDISCH(ISPECL [a;f] pth)) in
let th3 = (EXPAND_QUANTS_CONV(ASSUME(concl th2)) THENC
SUBS_CONV[GSYM th_im; ISPECL [a;f] cth]) bod in
let th4 = PROVE_HYP th2 th3 in
let th5 = TRANSLATION_INVARIANTS a in
let th6 = GEN_REWRITE_RULE (RAND_CONV o REDEPTH_CONV)
[ASSUME(concl th5)] th4 in
let th7 = PROVE_HYP th5 th6 in
let th8s = CONJUNCTS(MATCH_MP oth (CONJ th_orth th_det)) in
let th9 = LINEAR_INVARIANTS f th8s in
let th10 = GEN_REWRITE_RULE (RAND_CONV o REDEPTH_CONV) [th9] th7 in
let th11 = if intersect (frees(concl th10)) [a;f] = []
then PROVE_HYP th1 (itlist SIMPLE_CHOOSE [a;f] th10)
else MP (MATCH_MP fth (GENL [a;f] (DISCH_ALL th10))) th1 in
let th12 = REWRITE_CONV[ASSUME(mk_neg(hd(hyp th11)))] bod in
let th13 = ifn(CONJ (DISCH_ALL th11) (DISCH_ALL th12)) in
let th14 = MATCH_MP MONO_FORALL (GEN x th13) in
GEN_REWRITE_RULE (TRY_CONV o LAND_CONV) [FORALL_SIMP] th14;;
let GEOM_HORIZONTAL_PLANE_TAC p =
W(fun (asl,w) ->
let avs,bod = strip_forall w
and avs' = subtract (frees w) (freesl(map (concl o snd) asl)) in
let avs,bod = strip_forall w in
MAP_EVERY X_GEN_TAC avs THEN
MAP_EVERY (fun t -> SPEC_TAC(t,t)) (rev(subtract (avs@avs') [p])) THEN
SPEC_TAC(p,p) THEN
W(MATCH_MP_TAC o GEOM_HORIZONTAL_PLANE_RULE o snd));;
(* ------------------------------------------------------------------------- *)
(* Affsign and its special cases, with invariance theorems. *)
(* ------------------------------------------------------------------------- *)
let lin_combo = new_definition
`lin_combo V f = vsum V (\v. f v % (v:real^N))`;;
let affsign = new_definition
`affsign sgn s t (v:real^A) <=>
(?f. (v = lin_combo (s UNION t) f) /\
(!w. t w ==> sgn (f w)) /\
(sum (s UNION t) f = &1))`;;
let sgn_gt = new_definition `sgn_gt = (\t. (&0 < t))`;;
let sgn_ge = new_definition `sgn_ge = (\t. (&0 <= t))`;;
let sgn_lt = new_definition `sgn_lt = (\t. (t < &0))`;;
let sgn_le = new_definition `sgn_le = (\t. (t <= &0))`;;
let aff_gt_def = new_definition `aff_gt = affsign sgn_gt`;;
let aff_ge_def = new_definition `aff_ge = affsign sgn_ge`;;
let aff_lt_def = new_definition `aff_lt = affsign sgn_lt`;;
let aff_le_def = new_definition `aff_le = affsign sgn_le`;;
let AFFSIGN = prove
(`affsign sgn s t =
{y | ?f. y = vsum (s UNION t) (\v. f v % v) /\
(!w. w IN t ==> sgn(f w)) /\
sum (s UNION t) f = &1}`,
REWRITE_TAC[FUN_EQ_THM; affsign; lin_combo; IN_ELIM_THM] THEN
REWRITE_TAC[IN]);;
let AFFSIGN_ALT = prove
(`affsign sgn s t =
{y | ?f. (!w. w IN (s UNION t) ==> w IN t ==> sgn(f w)) /\
sum (s UNION t) f = &1 /\
vsum (s UNION t) (\v. f v % v) = y}`,
REWRITE_TAC[SET_RULE `(w IN (s UNION t) ==> w IN t ==> P w) <=>
(w IN t ==> P w)`] THEN
REWRITE_TAC[AFFSIGN; EXTENSION; IN_ELIM_THM] THEN MESON_TAC[]);;
let IN_AFFSIGN = prove
(`y IN affsign sgn s t <=>
?u. (!x. x IN t ==> sgn(u x)) /\
sum (s UNION t) u = &1 /\
vsum (s UNION t) (\x. u(x) % x) = y`,
REWRITE_TAC[AFFSIGN; IN_ELIM_THM] THEN SET_TAC[]);;
let AFFSIGN_DISJOINT_DIFF = prove
(`!s t. affsign sgn s t = affsign sgn (s DIFF t) t`,
REWRITE_TAC[AFFSIGN; SET_RULE `(s DIFF t) UNION t = s UNION t`]);;
let AFF_GE_DISJOINT_DIFF = prove
(`!s t. aff_ge s t = aff_ge (s DIFF t) t`,
REWRITE_TAC[aff_ge_def] THEN MATCH_ACCEPT_TAC AFFSIGN_DISJOINT_DIFF);;
let AFFSIGN_INJECTIVE_LINEAR_IMAGE = prove
(`!f:real^M->real^N sgn s t v.
linear f /\ (!x y. f x = f y ==> x = y)
==> (affsign sgn (IMAGE f s) (IMAGE f t) =
IMAGE f (affsign sgn s t))`,
let lemma0 = prove
(`vsum s (\x. u x % x) = vsum {x | x IN s /\ ~(u x = &0)} (\x. u x % x)`,
MATCH_MP_TAC VSUM_SUPERSET THEN SIMP_TAC[SUBSET; IN_ELIM_THM] THEN
REWRITE_TAC[TAUT `p /\ ~(p /\ ~q) <=> p /\ q`] THEN
SIMP_TAC[o_THM; VECTOR_MUL_LZERO]) in
let lemma1 = prove
(`!f:real^M->real^N s.
linear f /\ (!x y. f x = f y ==> x = y)
==> (sum(IMAGE f s) u = &1 /\ vsum(IMAGE f s) (\x. u x % x) = y <=>
sum s (u o f) = &1 /\ f(vsum s (\x. (u o f) x % x)) = y)`,
REPEAT STRIP_TAC THEN
W(MP_TAC o PART_MATCH (lhs o rand) SUM_IMAGE o funpow 3 lhand o snd) THEN
ANTS_TAC THENL [ASM_MESON_TAC[]; DISCH_THEN SUBST1_TAC] THEN
MATCH_MP_TAC(MESON[] `(p ==> z = x) ==> (p /\ x = y <=> p /\ z = y)`) THEN
DISCH_TAC THEN ONCE_REWRITE_TAC[lemma0] THEN
SUBGOAL_THEN
`{y | y IN IMAGE (f:real^M->real^N) s /\ ~(u y = &0)} =
IMAGE f {x | x IN s /\ ~(u(f x) = &0)}`
SUBST1_TAC THENL [ASM SET_TAC[]; CONV_TAC SYM_CONV] THEN
SUBGOAL_THEN `FINITE {x | x IN s /\ ~(u((f:real^M->real^N) x) = &0)}`
ASSUME_TAC THENL
[FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE
(LAND_CONV o RATOR_CONV o RATOR_CONV) [sum]) THEN
ONCE_REWRITE_TAC[ITERATE_EXPAND_CASES] THEN
REWRITE_TAC[GSYM sum; support; NEUTRAL_REAL_ADD; o_THM] THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[REAL_OF_NUM_EQ; ARITH_EQ];
W(MP_TAC o PART_MATCH (lhs o rand) VSUM_IMAGE o lhand o snd) THEN
ANTS_TAC THENL [ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[]; ALL_TAC] THEN
DISCH_THEN SUBST1_TAC THEN REWRITE_TAC[o_DEF] THEN
ASM_SIMP_TAC[LINEAR_VSUM; o_DEF; GSYM LINEAR_CMUL]]) in
REPEAT GEN_TAC THEN DISCH_TAC THEN REWRITE_TAC[EXTENSION; IN_AFFSIGN] THEN
REWRITE_TAC[FORALL_IN_IMAGE] THEN REWRITE_TAC[IN_IMAGE; IN_AFFSIGN] THEN
REWRITE_TAC[GSYM IMAGE_UNION] THEN
FIRST_ASSUM(fun th -> REWRITE_TAC[MATCH_MP lemma1 th]) THEN
X_GEN_TAC `y:real^N` THEN EQ_TAC THENL
[DISCH_THEN(X_CHOOSE_THEN `u:real^N->real` STRIP_ASSUME_TAC) THEN
EXISTS_TAC `vsum (s UNION t) (\x. (u o (f:real^M->real^N)) x % x)` THEN
ASM_REWRITE_TAC[] THEN
EXISTS_TAC `(u:real^N->real) o (f:real^M->real^N)` THEN
ASM_REWRITE_TAC[] THEN ASM_REWRITE_TAC[o_THM];
MP_TAC(ISPEC `f:real^M->real^N` LINEAR_INJECTIVE_LEFT_INVERSE) THEN
ASM_REWRITE_TAC[FUN_EQ_THM; o_THM; I_THM] THEN
DISCH_THEN(X_CHOOSE_THEN `g:real^N->real^M` STRIP_ASSUME_TAC) THEN
DISCH_THEN(X_CHOOSE_THEN `x:real^M`
(CONJUNCTS_THEN2 SUBST1_TAC MP_TAC)) THEN
DISCH_THEN(X_CHOOSE_THEN `u:real^M->real` STRIP_ASSUME_TAC) THEN
EXISTS_TAC `(u:real^M->real) o (g:real^N->real^M)` THEN
ASM_REWRITE_TAC[o_DEF; ETA_AX]]);;
let AFF_GE_INJECTIVE_LINEAR_IMAGE = prove
(`!f:real^M->real^N s t.
linear f /\ (!x y. f x = f y ==> x = y)
==> aff_ge (IMAGE f s) (IMAGE f t) = IMAGE f (aff_ge s t)`,
REWRITE_TAC[aff_ge_def; AFFSIGN_INJECTIVE_LINEAR_IMAGE]);;
let AFF_GT_INJECTIVE_LINEAR_IMAGE = prove
(`!f:real^M->real^N s t.
linear f /\ (!x y. f x = f y ==> x = y)
==> aff_gt (IMAGE f s) (IMAGE f t) = IMAGE f (aff_gt s t)`,
REWRITE_TAC[aff_gt_def; AFFSIGN_INJECTIVE_LINEAR_IMAGE]);;
let AFF_LE_INJECTIVE_LINEAR_IMAGE = prove
(`!f:real^M->real^N s t.
linear f /\ (!x y. f x = f y ==> x = y)
==> aff_le (IMAGE f s) (IMAGE f t) = IMAGE f (aff_le s t)`,
REWRITE_TAC[aff_le_def; AFFSIGN_INJECTIVE_LINEAR_IMAGE]);;
let AFF_LT_INJECTIVE_LINEAR_IMAGE = prove
(`!f:real^M->real^N s t.
linear f /\ (!x y. f x = f y ==> x = y)
==> aff_lt (IMAGE f s) (IMAGE f t) = IMAGE f (aff_lt s t)`,
REWRITE_TAC[aff_lt_def; AFFSIGN_INJECTIVE_LINEAR_IMAGE]);;
add_linear_invariants
[AFFSIGN_INJECTIVE_LINEAR_IMAGE;
AFF_GE_INJECTIVE_LINEAR_IMAGE;
AFF_GT_INJECTIVE_LINEAR_IMAGE;
AFF_LE_INJECTIVE_LINEAR_IMAGE;
AFF_LT_INJECTIVE_LINEAR_IMAGE];;
let IN_AFFSIGN_TRANSLATION = prove
(`!sgn s t a v:real^N.
affsign sgn s t v
==> affsign sgn (IMAGE (\x. a + x) s) (IMAGE (\x. a + x) t) (a + v)`,
REPEAT GEN_TAC THEN REWRITE_TAC[affsign; lin_combo] THEN
ONCE_REWRITE_TAC[SET_RULE `(!x. s x ==> p x) <=> (!x. x IN s ==> p x)`] THEN
DISCH_THEN(X_CHOOSE_THEN `f:real^N->real`
(CONJUNCTS_THEN2 SUBST_ALL_TAC STRIP_ASSUME_TAC)) THEN
EXISTS_TAC `\x. (f:real^N->real)(x - a)` THEN
ASM_REWRITE_TAC[GSYM IMAGE_UNION] THEN REPEAT CONJ_TAC THENL
[ALL_TAC;
ASM_REWRITE_TAC[FORALL_IN_IMAGE; ETA_AX;
VECTOR_ARITH `(a + x) - a:real^N = x`];
W(MP_TAC o PART_MATCH (lhs o rand) SUM_IMAGE o lhs o snd) THEN
SIMP_TAC[VECTOR_ARITH `a + x:real^N = a + y <=> x = y`] THEN
ASM_REWRITE_TAC[o_DEF; VECTOR_ADD_SUB; ETA_AX]] THEN
MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC
`a + vsum {x | x IN s UNION t /\ ~(f x = &0)} (\v:real^N. f v % v)` THEN
CONJ_TAC THENL
[AP_TERM_TAC THEN MATCH_MP_TAC VSUM_SUPERSET THEN
REWRITE_TAC[VECTOR_MUL_EQ_0; SUBSET; IN_ELIM_THM] THEN MESON_TAC[];
ALL_TAC] THEN
MATCH_MP_TAC EQ_TRANS THEN
EXISTS_TAC `vsum (IMAGE (\x:real^N. a + x)
{x | x IN s UNION t /\ ~(f x = &0)})
(\v. f(v - a) % v)` THEN
CONJ_TAC THENL
[ALL_TAC;
CONV_TAC SYM_CONV THEN MATCH_MP_TAC VSUM_SUPERSET THEN
CONJ_TAC THENL [SET_TAC[]; ALL_TAC] THEN
ASM_REWRITE_TAC[IMP_CONJ; FORALL_IN_IMAGE; VECTOR_MUL_EQ_0] THEN
REWRITE_TAC[VECTOR_ADD_SUB] THEN SET_TAC[]] THEN
SUBGOAL_THEN `FINITE {x:real^N | x IN s UNION t /\ ~(f x = &0)}`
ASSUME_TAC THENL
[FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE
(LAND_CONV o RATOR_CONV o RATOR_CONV) [sum]) THEN
ONCE_REWRITE_TAC[ITERATE_EXPAND_CASES] THEN
REWRITE_TAC[GSYM sum; support; NEUTRAL_REAL_ADD] THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[REAL_OF_NUM_EQ; ARITH_EQ];
ALL_TAC] THEN
W(MP_TAC o PART_MATCH (lhs o rand) VSUM_IMAGE o rhs o snd) THEN
ASM_SIMP_TAC[VECTOR_ARITH `a + x:real^N = a + y <=> x = y`] THEN
DISCH_THEN(K ALL_TAC) THEN REWRITE_TAC[o_DEF; VECTOR_ADD_SUB] THEN
ASM_SIMP_TAC[VECTOR_ADD_LDISTRIB; VSUM_ADD] THEN
AP_THM_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[VSUM_RMUL] THEN
GEN_REWRITE_TAC LAND_CONV [GSYM VECTOR_MUL_LID] THEN
AP_THM_TAC THEN AP_TERM_TAC THEN FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN
MATCH_MP_TAC SUM_SUPERSET THEN SET_TAC[]);;
let AFFSIGN_TRANSLATION = prove
(`!a:real^N sgn s t.
affsign sgn (IMAGE (\x. a + x) s) (IMAGE (\x. a + x) t) =
IMAGE (\x. a + x) (affsign sgn s t)`,
REPEAT GEN_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL
[REWRITE_TAC[SUBSET; IN] THEN GEN_TAC THEN
DISCH_THEN(MP_TAC o SPEC `--a:real^N` o
MATCH_MP IN_AFFSIGN_TRANSLATION) THEN
REWRITE_TAC[GSYM IMAGE_o; o_DEF; VECTOR_ARITH `--a + a + x:real^N = x`;
IMAGE_ID] THEN
DISCH_TAC THEN REWRITE_TAC[IMAGE; IN_ELIM_THM] THEN
EXISTS_TAC `--a + x:real^N` THEN ASM_REWRITE_TAC[IN] THEN VECTOR_ARITH_TAC;
REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN GEN_TAC THEN REWRITE_TAC[IN] THEN
DISCH_THEN(MP_TAC o SPEC `a:real^N` o MATCH_MP IN_AFFSIGN_TRANSLATION) THEN
REWRITE_TAC[]]);;
let AFF_GE_TRANSLATION = prove
(`!a:real^N s t.
aff_ge (IMAGE (\x. a + x) s) (IMAGE (\x. a + x) t) =
IMAGE (\x. a + x) (aff_ge s t)`,
REWRITE_TAC[aff_ge_def; AFFSIGN_TRANSLATION]);;
let AFF_GT_TRANSLATION = prove
(`!a:real^N s t.
aff_gt (IMAGE (\x. a + x) s) (IMAGE (\x. a + x) t) =
IMAGE (\x. a + x) (aff_gt s t)`,
REWRITE_TAC[aff_gt_def; AFFSIGN_TRANSLATION]);;
let AFF_LE_TRANSLATION = prove
(`!a:real^N s t.
aff_le (IMAGE (\x. a + x) s) (IMAGE (\x. a + x) t) =
IMAGE (\x. a + x) (aff_le s t)`,
REWRITE_TAC[aff_le_def; AFFSIGN_TRANSLATION]);;
let AFF_LT_TRANSLATION = prove
(`!a:real^N s t.
aff_lt (IMAGE (\x. a + x) s) (IMAGE (\x. a + x) t) =
IMAGE (\x. a + x) (aff_lt s t)`,
REWRITE_TAC[aff_lt_def; AFFSIGN_TRANSLATION]);;
add_translation_invariants
[AFFSIGN_TRANSLATION;
AFF_GE_TRANSLATION;
AFF_GT_TRANSLATION;
AFF_LE_TRANSLATION;
AFF_LT_TRANSLATION];;
(* ------------------------------------------------------------------------- *)
(* Automate special cases of affsign. *)
(* ------------------------------------------------------------------------- *)
let AFF_TAC =
REWRITE_TAC[DISJOINT_INSERT; DISJOINT_EMPTY] THEN
REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY; DE_MORGAN_THM] THEN
REPEAT STRIP_TAC THEN
REWRITE_TAC[aff_ge_def; aff_gt_def; aff_le_def; aff_lt_def;
sgn_ge; sgn_gt; sgn_le; sgn_lt; AFFSIGN_ALT] THEN
REWRITE_TAC[SET_RULE `(x INSERT s) UNION t = x INSERT (s UNION t)`] THEN
REWRITE_TAC[UNION_EMPTY] THEN
SIMP_TAC[FINITE_INSERT; FINITE_UNION; AFFINE_HULL_FINITE_STEP_GEN;
FINITE_EMPTY; RIGHT_EXISTS_AND_THM; REAL_LT_ADD;
REAL_LE_ADD; REAL_ARITH `&0 <= a / &2 <=> &0 <= a`;
REAL_ARITH `&0 < a / &2 <=> &0 < a`;
REAL_ARITH `a / &2 <= &0 <=> a <= &0`;
REAL_ARITH `a / &2 < &0 <=> a < &0`;
REAL_ARITH `a < &0 /\ b < &0 ==> a + b < &0`;
REAL_ARITH `a < &0 /\ b <= &0 ==> a + b <= &0`] THEN
ASM_REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY; real_ge] THEN
REWRITE_TAC[REAL_ARITH `x - y:real = z <=> x = y + z`;
VECTOR_ARITH `x - y:real^N = z <=> x = y + z`] THEN
REWRITE_TAC[RIGHT_AND_EXISTS_THM; REAL_ADD_RID; VECTOR_ADD_RID] THEN
ONCE_REWRITE_TAC[REAL_ARITH `&1 = x <=> x = &1`] THEN
REWRITE_TAC[] THEN SET_TAC[];;
let AFF_GE_1_1 = prove
(`!x v w.
DISJOINT {x} {v}
==> aff_ge {x} {v} =
{y | ?t1 t2.
&0 <= t2 /\
t1 + t2 = &1 /\
y = t1 % x + t2 % v }`,
AFF_TAC);;
let AFF_GE_1_2 = prove
(`!x v w.
DISJOINT {x} {v,w}
==> aff_ge {x} {v,w} =
{y | ?t1 t2 t3.
&0 <= t2 /\ &0 <= t3 /\
t1 + t2 + t3 = &1 /\
y = t1 % x + t2 % v + t3 % w}`,
AFF_TAC);;
let AFF_GE_2_1 = prove
(`!x v w.
DISJOINT {x,v} {w}
==> aff_ge {x,v} {w} =
{y | ?t1 t2 t3.
&0 <= t3 /\
t1 + t2 + t3 = &1 /\
y = t1 % x + t2 % v + t3 % w}`,
AFF_TAC);;
let AFF_GT_1_1 = prove
(`!x v w.
DISJOINT {x} {v}
==> aff_gt {x} {v} =
{y | ?t1 t2.
&0 < t2 /\
t1 + t2 = &1 /\
y = t1 % x + t2 % v}`,
AFF_TAC);;
let AFF_GT_1_2 = prove
(`!x v w.
DISJOINT {x} {v,w}
==> aff_gt {x} {v,w} =
{y | ?t1 t2 t3.
&0 < t2 /\ &0 < t3 /\
t1 + t2 + t3 = &1 /\
y = t1 % x + t2 % v + t3 % w}`,
AFF_TAC);;
let AFF_GT_2_1 = prove
(`!x v w.
DISJOINT {x,v} {w}
==> aff_gt {x,v} {w} =
{y | ?t1 t2 t3.
&0 < t3 /\
t1 + t2 + t3 = &1 /\
y = t1 % x + t2 % v + t3 % w}`,
AFF_TAC);;
let AFF_GT_3_1 = prove
(`!v w x y.
DISJOINT {v,w,x} {y}
==> aff_gt {v,w,x} {y} =
{z | ?t1 t2 t3 t4.
&0 < t4 /\
t1 + t2 + t3 + t4 = &1 /\
z = t1 % v + t2 % w + t3 % x + t4 % y}`,
AFF_TAC);;
let AFF_LT_1_1 = prove
(`!x v.
DISJOINT {x} {v}
==> aff_lt {x} {v} =
{y | ?t1 t2.
t2 < &0 /\
t1 + t2 = &1 /\
y = t1 % x + t2 % v}`,
AFF_TAC);;
let AFF_LT_2_1 = prove
(`!x v w.
DISJOINT {x,v} {w}
==> aff_lt {x,v} {w} =
{y | ?t1 t2 t3.
t3 < &0 /\
t1 + t2 + t3 = &1 /\
y = t1 % x + t2 % v + t3 % w}`,
AFF_TAC);;
let AFF_GE_1_2_0 = prove
(`!v w.
~(v = vec 0) /\ ~(w = vec 0)
==> aff_ge {vec 0} {v,w} = {a % v + b % w | &0 <= a /\ &0 <= b}`,
SIMP_TAC[AFF_GE_1_2;
SET_RULE `DISJOINT {a} {b,c} <=> ~(b = a) /\ ~(c = a)`] THEN
REWRITE_TAC[VECTOR_MUL_RZERO; VECTOR_ADD_LID] THEN
ONCE_REWRITE_TAC[MESON[]
`(?a b c. P b c /\ Q b c /\ R a b c /\ S b c) <=>
(?b c. P b c /\ Q b c /\ S b c /\ ?a. R a b c)`] THEN
REWRITE_TAC[REAL_ARITH `t + s:real = &1 <=> t = &1 - s`; EXISTS_REFL] THEN
SET_TAC[]);;
let AFF_GE_1_1_0 = prove
(`!v. ~(v = vec 0) ==> aff_ge {vec 0} {v} = {a % v | &0 <= a}`,
REPEAT STRIP_TAC THEN
GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [SET_RULE `{a} = {a,a}`] THEN
ASM_SIMP_TAC[AFF_GE_1_2_0; GSYM VECTOR_ADD_RDISTRIB] THEN
REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN
MESON_TAC[REAL_LE_ADD; REAL_ARITH
`&0 <= a ==> &0 <= a / &2 /\ a / &2 + a / &2 = a`]);;
let AFF_GE_2_1_0 = prove
(`!v w. DISJOINT {vec 0, v} {w}
==> aff_ge {vec 0, v} {w} = {s % v + t % w |s,t| &0 <= t}`,
SIMP_TAC[AFF_GE_2_1; VECTOR_MUL_RZERO; VECTOR_ADD_LID] THEN
REPEAT STRIP_TAC THEN
ONCE_REWRITE_TAC[TAUT `p /\ q /\ r <=> q /\ p /\ r`] THEN
ONCE_REWRITE_TAC[MESON[] `(?a b c. P a b c) <=> (?c b a. P a b c)`] THEN
REWRITE_TAC[REAL_ARITH `t + u = &1 <=> t = &1 - u`; UNWIND_THM2] THEN
SET_TAC[]);;
(* ------------------------------------------------------------------------- *)
(* Properties of affsign variants. *)
(* ------------------------------------------------------------------------- *)
let CONVEX_AFFSIGN = prove
(`!sgn. (!x y u. sgn(x) /\ sgn(y) /\ &0 <= u /\ u <= &1
==> sgn((&1 - u) * x + u * y))
==> !s t:real^N->bool. convex(affsign sgn s t)`,
REPEAT STRIP_TAC THEN REWRITE_TAC[AFFSIGN; CONVEX_ALT] THEN
MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`; `u:real`] THEN
REWRITE_TAC[IN_ELIM_THM; IMP_CONJ; LEFT_IMP_EXISTS_THM] THEN
REWRITE_TAC[IMP_IMP; GSYM CONJ_ASSOC] THEN
X_GEN_TAC `f:real^N->real` THEN STRIP_TAC THEN
X_GEN_TAC `g:real^N->real` THEN STRIP_TAC THEN
EXISTS_TAC `\x:real^N. (&1 - u) * f x + u * g x` THEN
ASM_REWRITE_TAC[VECTOR_ADD_RDISTRIB] THEN REPEAT CONJ_TAC THENL
[CONV_TAC SYM_CONV THEN
W(MP_TAC o PART_MATCH (lhs o rand) VSUM_ADD_GEN o lhand o snd) THEN
REWRITE_TAC[GSYM VECTOR_MUL_ASSOC; VSUM_LMUL] THEN
DISCH_THEN MATCH_MP_TAC;
ASM_MESON_TAC[];
W(MP_TAC o PART_MATCH (lhs o rand) SUM_ADD_GEN o lhand o snd) THEN
ASM_REWRITE_TAC[GSYM VECTOR_MUL_ASSOC; SUM_LMUL] THEN
REWRITE_TAC[REAL_MUL_RID; REAL_SUB_ADD] THEN DISCH_THEN MATCH_MP_TAC] THEN
(CONJ_TAC THENL
[MP_TAC(ASSUME `sum (s UNION t:real^N->bool) f = &1`);
MP_TAC(ASSUME `sum (s UNION t:real^N->bool) g = &1`)]) THEN
GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [sum] THEN
ONCE_REWRITE_TAC[iterate] THEN
REWRITE_TAC[support; NEUTRAL_REAL_ADD] THEN
COND_CASES_TAC THEN REWRITE_TAC[REAL_OF_NUM_EQ; ARITH_EQ] THEN
DISCH_THEN(K ALL_TAC) THEN
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP
(REWRITE_RULE[IMP_CONJ] FINITE_SUBSET)) THEN
REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN GEN_TAC THEN
MATCH_MP_TAC MONO_AND THEN REWRITE_TAC[CONTRAPOS_THM] THEN
DISCH_THEN SUBST1_TAC THEN
REWRITE_TAC[VECTOR_MUL_LZERO; VECTOR_MUL_RZERO; REAL_MUL_RZERO]);;
let CONVEX_AFF_GE = prove
(`!s t. convex(aff_ge s t)`,
REWRITE_TAC[aff_ge_def; sgn_ge] THEN MATCH_MP_TAC CONVEX_AFFSIGN THEN
SIMP_TAC[REAL_LE_MUL; REAL_LE_ADD; REAL_SUB_LE]);;
let CONVEX_AFF_LE = prove
(`!s t. convex(aff_le s t)`,
REWRITE_TAC[aff_le_def; sgn_le] THEN MATCH_MP_TAC CONVEX_AFFSIGN THEN
REWRITE_TAC[REAL_ARITH `x <= &0 <=> &0 <= --x`; REAL_NEG_ADD; GSYM
REAL_MUL_RNEG] THEN
SIMP_TAC[REAL_LE_MUL; REAL_LE_ADD; REAL_SUB_LE]);;
let CONVEX_AFF_GT = prove
(`!s t. convex(aff_gt s t)`,
REWRITE_TAC[aff_gt_def; sgn_gt] THEN MATCH_MP_TAC CONVEX_AFFSIGN THEN
REWRITE_TAC[REAL_ARITH `&0 <= x <=> x = &0 \/ &0 < x`;
REAL_ARITH `x <= &1 <=> x = &1 \/ x < &1`] THEN
REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
CONV_TAC REAL_RAT_REDUCE_CONV THEN
REWRITE_TAC[REAL_MUL_LZERO; REAL_ADD_LID; REAL_ADD_RID; REAL_MUL_LID] THEN
ASM_SIMP_TAC[REAL_LT_ADD; REAL_LT_MUL; REAL_SUB_LT]);;
let CONVEX_AFF_LT = prove
(`!s t. convex(aff_lt s t)`,
REWRITE_TAC[aff_lt_def; sgn_lt] THEN MATCH_MP_TAC CONVEX_AFFSIGN THEN
REWRITE_TAC[REAL_ARITH `x < &0 <=> &0 < --x`; REAL_NEG_ADD; GSYM
REAL_MUL_RNEG] THEN
REWRITE_TAC[REAL_ARITH `&0 <= x <=> x = &0 \/ &0 < x`;
REAL_ARITH `x <= &1 <=> x = &1 \/ x < &1`] THEN
REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
CONV_TAC REAL_RAT_REDUCE_CONV THEN
REWRITE_TAC[REAL_MUL_LZERO; REAL_ADD_LID; REAL_ADD_RID; REAL_MUL_LID] THEN
ASM_SIMP_TAC[REAL_LT_ADD; REAL_LT_MUL; REAL_SUB_LT]);;
let AFFSIGN_SUBSET_AFFINE_HULL = prove
(`!sgn s t. (affsign sgn s t) SUBSET (affine hull (s UNION t))`,
REWRITE_TAC[AFFINE_HULL_FINITE; AFFSIGN] THEN SET_TAC[]);;
let AFF_GE_SUBSET_AFFINE_HULL = prove
(`!s t. (aff_ge s t) SUBSET (affine hull (s UNION t))`,
REWRITE_TAC[aff_ge_def; AFFSIGN_SUBSET_AFFINE_HULL]);;
let AFF_LE_SUBSET_AFFINE_HULL = prove
(`!s t. (aff_le s t) SUBSET (affine hull (s UNION t))`,
REWRITE_TAC[aff_le_def; AFFSIGN_SUBSET_AFFINE_HULL]);;
let AFF_GT_SUBSET_AFFINE_HULL = prove
(`!s t. (aff_gt s t) SUBSET (affine hull (s UNION t))`,
REWRITE_TAC[aff_gt_def; AFFSIGN_SUBSET_AFFINE_HULL]);;
let AFF_LT_SUBSET_AFFINE_HULL = prove
(`!s t. (aff_lt s t) SUBSET (affine hull (s UNION t))`,
REWRITE_TAC[aff_lt_def; AFFSIGN_SUBSET_AFFINE_HULL]);;
let AFFSIGN_EQ_AFFINE_HULL = prove
(`!sgn s t. affsign sgn s {} = affine hull s`,
REWRITE_TAC[AFFSIGN; AFFINE_HULL_FINITE] THEN
REWRITE_TAC[UNION_EMPTY; NOT_IN_EMPTY] THEN SET_TAC[]);;
let AFF_GE_EQ_AFFINE_HULL = prove
(`!s t. aff_ge s {} = affine hull s`,
REWRITE_TAC[aff_ge_def; AFFSIGN_EQ_AFFINE_HULL]);;
let AFF_LE_EQ_AFFINE_HULL = prove
(`!s t. aff_le s {} = affine hull s`,
REWRITE_TAC[aff_le_def; AFFSIGN_EQ_AFFINE_HULL]);;
let AFF_GT_EQ_AFFINE_HULL = prove
(`!s t. aff_gt s {} = affine hull s`,
REWRITE_TAC[aff_gt_def; AFFSIGN_EQ_AFFINE_HULL]);;
let AFF_LT_EQ_AFFINE_HULL = prove
(`!s t. aff_lt s {} = affine hull s`,
REWRITE_TAC[aff_lt_def; AFFSIGN_EQ_AFFINE_HULL]);;
let AFFSIGN_SUBSET_AFFSIGN = prove
(`!sgn1 sgn2 s t.
(!x. sgn1 x ==> sgn2 x) ==> affsign sgn1 s t SUBSET affsign sgn2 s t`,
REPEAT STRIP_TAC THEN REWRITE_TAC[AFFSIGN; SUBSET; IN_ELIM_THM] THEN
GEN_TAC THEN MATCH_MP_TAC MONO_EXISTS THEN ASM_MESON_TAC[]);;
let AFF_GT_SUBSET_AFF_GE = prove
(`!s t. aff_gt s t SUBSET aff_ge s t`,
REPEAT GEN_TAC THEN REWRITE_TAC[aff_gt_def; aff_ge_def] THEN
MATCH_MP_TAC AFFSIGN_SUBSET_AFFSIGN THEN
SIMP_TAC[sgn_gt; sgn_ge; REAL_LT_IMP_LE]);;
let AFFSIGN_MONO_LEFT = prove
(`!sgn s s' t:real^N->bool.
s SUBSET s' ==> affsign sgn s t SUBSET affsign sgn s' t`,
REPEAT STRIP_TAC THEN
REWRITE_TAC[AFFSIGN; SUBSET; IN_ELIM_THM] THEN
X_GEN_TAC `y:real^N` THEN
DISCH_THEN(X_CHOOSE_THEN `u:real^N->real` STRIP_ASSUME_TAC) THEN
EXISTS_TAC `\x:real^N. if x IN s UNION t then u x else &0` THEN
REWRITE_TAC[COND_RAND; COND_RATOR; VECTOR_MUL_LZERO] THEN
REWRITE_TAC[GSYM SUM_RESTRICT_SET; GSYM VSUM_RESTRICT_SET] THEN
ASM_SIMP_TAC[SET_RULE
`s SUBSET s' ==> {x | x IN s' UNION t /\ x IN s UNION t} = s UNION t`] THEN
ASM SET_TAC[]);;
let AFFSIGN_MONO_SHUFFLE = prove
(`!sgn s t s' t'.
s' UNION t' = s UNION t /\ t' SUBSET t
==> affsign sgn s t SUBSET affsign sgn s' t'`,
REPEAT STRIP_TAC THEN REWRITE_TAC[AFFSIGN; SUBSET; IN_ELIM_THM] THEN
GEN_TAC THEN MATCH_MP_TAC MONO_EXISTS THEN
ASM_REWRITE_TAC[] THEN ASM SET_TAC[]);;
let AFF_GT_MONO_LEFT = prove
(`!s s' t. s SUBSET s' ==> aff_gt s t SUBSET aff_gt s' t`,
REWRITE_TAC[aff_gt_def; AFFSIGN_MONO_LEFT]);;
let AFF_GE_MONO_LEFT = prove
(`!s s' t. s SUBSET s' ==> aff_ge s t SUBSET aff_ge s' t`,
REWRITE_TAC[aff_ge_def; AFFSIGN_MONO_LEFT]);;
let AFF_LT_MONO_LEFT = prove
(`!s s' t. s SUBSET s' ==> aff_lt s t SUBSET aff_lt s' t`,
REWRITE_TAC[aff_lt_def; AFFSIGN_MONO_LEFT]);;
let AFF_LE_MONO_LEFT = prove
(`!s s' t. s SUBSET s' ==> aff_le s t SUBSET aff_le s' t`,
REWRITE_TAC[aff_le_def; AFFSIGN_MONO_LEFT]);;
let AFFSIGN_MONO_RIGHT = prove
(`!sgn s t t':real^N->bool.
sgn(&0) /\ t SUBSET t' /\ DISJOINT s t'
==> affsign sgn s t SUBSET affsign sgn s t'`,
REPEAT STRIP_TAC THEN
REWRITE_TAC[AFFSIGN; SUBSET; IN_ELIM_THM] THEN
X_GEN_TAC `y:real^N` THEN
DISCH_THEN(X_CHOOSE_THEN `u:real^N->real` STRIP_ASSUME_TAC) THEN
EXISTS_TAC `\x:real^N. if x IN s UNION t then u x else &0` THEN
REWRITE_TAC[COND_RAND; COND_RATOR; VECTOR_MUL_LZERO] THEN
REWRITE_TAC[GSYM SUM_RESTRICT_SET; GSYM VSUM_RESTRICT_SET] THEN
ASM_SIMP_TAC[SET_RULE
`t SUBSET t' ==> {x | x IN s UNION t' /\ x IN s UNION t} = s UNION t`] THEN
ASM SET_TAC[]);;
let AFF_GE_MONO_RIGHT = prove
(`!s t t'. t SUBSET t' /\ DISJOINT s t' ==> aff_ge s t SUBSET aff_ge s t'`,
SIMP_TAC[aff_ge_def; AFFSIGN_MONO_RIGHT; sgn_ge; REAL_POS]);;
let AFF_LE_MONO_RIGHT = prove
(`!s t t'. t SUBSET t' /\ DISJOINT s t' ==> aff_le s t SUBSET aff_le s t'`,
SIMP_TAC[aff_le_def; AFFSIGN_MONO_RIGHT; sgn_le; REAL_LE_REFL]);;
let AFFINE_HULL_SUBSET_AFFSIGN = prove
(`!sgn s t:real^N->bool.
sgn(&0) /\ DISJOINT s t
==> affine hull s SUBSET affsign sgn s t`,
REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_TRANS THEN
EXISTS_TAC `affsign sgn (s:real^N->bool) {}` THEN CONJ_TAC THENL
[REWRITE_TAC[AFFSIGN_EQ_AFFINE_HULL; SUBSET_REFL];
MATCH_MP_TAC AFFSIGN_MONO_RIGHT THEN ASM SET_TAC[]]);;
let AFFINE_HULL_SUBSET_AFF_GE = prove
(`!s t. DISJOINT s t ==> affine hull s SUBSET aff_ge s t`,
SIMP_TAC[aff_ge_def; sgn_ge; REAL_LE_REFL; AFFINE_HULL_SUBSET_AFFSIGN]);;
let AFF_GE_AFF_GT_DECOMP = prove
(`!s:real^N->bool.
FINITE s /\ FINITE t /\ DISJOINT s t
==> aff_ge s t = aff_gt s t UNION
UNIONS {aff_ge s (t DELETE a) | a | a IN t}`,
REPEAT STRIP_TAC THEN
MATCH_MP_TAC(SET_RULE
`t' SUBSET t /\ (!a. a IN s ==> f(a) SUBSET t) /\
(!y. y IN t ==> y IN t' \/ ?a. a IN s /\ y IN f(a))
==> t = t' UNION UNIONS {f a | a IN s}`) THEN
REWRITE_TAC[AFF_GT_SUBSET_AFF_GE] THEN
ASM_SIMP_TAC[DELETE_SUBSET; AFF_GE_MONO_RIGHT] THEN
REWRITE_TAC[aff_ge_def; aff_gt_def; AFFSIGN; sgn_ge; sgn_gt] THEN
X_GEN_TAC `y:real^N` THEN REWRITE_TAC[IN_ELIM_THM] THEN
DISCH_THEN(X_CHOOSE_THEN `u:real^N->real` STRIP_ASSUME_TAC) THEN
ASM_CASES_TAC `!x:real^N. x IN t ==> &0 < u x` THENL
[DISJ1_TAC THEN EXISTS_TAC `u:real^N->real` THEN ASM_REWRITE_TAC[];
DISJ2_TAC THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [NOT_FORALL_THM]) THEN
ASM_SIMP_TAC[REAL_ARITH `&0 <= x ==> (&0 < x <=> ~(x = &0))`] THEN
REWRITE_TAC[NOT_IMP] THEN MATCH_MP_TAC MONO_EXISTS THEN
X_GEN_TAC `a:real^N` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
EXISTS_TAC `u:real^N->real` THEN
ASM_SIMP_TAC[SET_RULE
`a IN t /\ DISJOINT s t
==> s UNION (t DELETE a) = (s UNION t) DELETE a`] THEN
ASM_SIMP_TAC[IN_DELETE; SUM_DELETE; VSUM_DELETE; REAL_SUB_RZERO;
FINITE_UNION; IN_UNION] THEN
REWRITE_TAC[VECTOR_MUL_LZERO; VECTOR_SUB_RZERO]]);;
let AFFSIGN_SPECIAL_SCALE = prove
(`!sgn s t a v.
FINITE s /\ FINITE t /\
~(vec 0 IN t) /\ ~(v IN t) /\ ~((a % v) IN t) /\
(!x. sgn x ==> sgn(x / &2)) /\
(!x y. sgn x /\ sgn y ==> sgn(x + y)) /\
&0 < a
==> affsign sgn (vec 0 INSERT (a % v) INSERT s) t =
affsign sgn (vec 0 INSERT v INSERT s) t`,
REWRITE_TAC[EXTENSION] THEN REPEAT STRIP_TAC THEN
REWRITE_TAC[AFFSIGN_ALT; IN_ELIM_THM; INSERT_UNION_EQ] THEN
ASM_SIMP_TAC[FINITE_INSERT; FINITE_UNION; AFFINE_HULL_FINITE_STEP_GEN;
RIGHT_EXISTS_AND_THM] THEN
REWRITE_TAC[VECTOR_MUL_RZERO; VECTOR_SUB_RZERO] THEN
GEN_REWRITE_TAC BINOP_CONV [SWAP_EXISTS_THM] THEN
GEN_REWRITE_TAC (BINOP_CONV o BINDER_CONV) [SWAP_EXISTS_THM] THEN
REWRITE_TAC[LEFT_EXISTS_AND_THM; RIGHT_EXISTS_AND_THM] THEN
REWRITE_TAC[REAL_ARITH `x = &1 - v - v' <=> v = &1 - (x + v')`] THEN
REWRITE_TAC[EXISTS_REFL] THEN
FIRST_ASSUM(MP_TAC o MATCH_MP(MESON[REAL_LT_IMP_NZ; REAL_DIV_LMUL]
`!a. &0 < a ==> (!y. ?x. a * x = y)`)) THEN
DISCH_THEN(MP_TAC o MATCH_MP QUANTIFY_SURJECTION_THM) THEN
DISCH_THEN(CONV_TAC o RAND_CONV o EXPAND_QUANTS_CONV) THEN
REWRITE_TAC[VECTOR_MUL_ASSOC; REAL_MUL_SYM]);;
let AFF_GE_SPECIAL_SCALE = prove
(`!s t a v.
FINITE s /\ FINITE t /\
~(vec 0 IN t) /\ ~(v IN t) /\ ~((a % v) IN t) /\
&0 < a
==> aff_ge (vec 0 INSERT (a % v) INSERT s) t =
aff_ge (vec 0 INSERT v INSERT s) t`,
REPEAT STRIP_TAC THEN REWRITE_TAC[aff_ge_def] THEN
MATCH_MP_TAC AFFSIGN_SPECIAL_SCALE THEN
ASM_REWRITE_TAC[sgn_ge] THEN REAL_ARITH_TAC);;
let AFF_LE_SPECIAL_SCALE = prove
(`!s t a v.
FINITE s /\ FINITE t /\
~(vec 0 IN t) /\ ~(v IN t) /\ ~((a % v) IN t) /\
&0 < a
==> aff_le (vec 0 INSERT (a % v) INSERT s) t =
aff_le (vec 0 INSERT v INSERT s) t`,
REPEAT STRIP_TAC THEN REWRITE_TAC[aff_le_def] THEN
MATCH_MP_TAC AFFSIGN_SPECIAL_SCALE THEN
ASM_REWRITE_TAC[sgn_le] THEN REAL_ARITH_TAC);;
let AFF_GT_SPECIAL_SCALE = prove
(`!s t a v.
FINITE s /\ FINITE t /\
~(vec 0 IN t) /\ ~(v IN t) /\ ~((a % v) IN t) /\
&0 < a
==> aff_gt (vec 0 INSERT (a % v) INSERT s) t =
aff_gt (vec 0 INSERT v INSERT s) t`,
REPEAT STRIP_TAC THEN REWRITE_TAC[aff_gt_def] THEN
MATCH_MP_TAC AFFSIGN_SPECIAL_SCALE THEN
ASM_REWRITE_TAC[sgn_gt] THEN REAL_ARITH_TAC);;
let AFF_LT_SPECIAL_SCALE = prove
(`!s t a v.
FINITE s /\ FINITE t /\
~(vec 0 IN t) /\ ~(v IN t) /\ ~((a % v) IN t) /\
&0 < a
==> aff_lt (vec 0 INSERT (a % v) INSERT s) t =
aff_lt (vec 0 INSERT v INSERT s) t`,
REPEAT STRIP_TAC THEN REWRITE_TAC[aff_lt_def] THEN
MATCH_MP_TAC AFFSIGN_SPECIAL_SCALE THEN
ASM_REWRITE_TAC[sgn_lt] THEN REAL_ARITH_TAC);;
let AFF_GE_SCALE_LEMMA = prove
(`!a u v:real^N.
&0 < a /\ ~(v = vec 0)
==> aff_ge {vec 0} {a % u,v} = aff_ge {vec 0} {u,v}`,
REPEAT STRIP_TAC THEN ASM_CASES_TAC `u:real^N = vec 0` THEN
ASM_REWRITE_TAC[VECTOR_MUL_RZERO] THEN
ASM_SIMP_TAC[AFF_GE_1_2_0; VECTOR_MUL_EQ_0; REAL_LT_IMP_NZ;
SET_RULE `DISJOINT {a} {b,c} <=> ~(b = a) /\ ~(c = a)`] THEN
REWRITE_TAC[GSYM SUBSET_ANTISYM_EQ; SUBSET; FORALL_IN_GSPEC] THEN
CONJ_TAC THEN MAP_EVERY X_GEN_TAC [`b:real`; `c:real`] THEN
REWRITE_TAC[IN_ELIM_THM] THEN STRIP_TAC THENL
[EXISTS_TAC `a * b:real`; EXISTS_TAC `b / a:real`] THEN
EXISTS_TAC `c:real` THEN
ASM_SIMP_TAC[REAL_LE_DIV; REAL_LE_MUL; REAL_LT_IMP_LE] THEN
REWRITE_TAC[VECTOR_MUL_ASSOC] THEN
REPLICATE_TAC 2 (AP_THM_TAC THEN AP_TERM_TAC) THEN
UNDISCH_TAC `&0 < a` THEN CONV_TAC REAL_FIELD);;
let AFFSIGN_0 = prove
(`!sgn s t.
FINITE s /\ FINITE t /\ (vec 0) IN (s DIFF t)
==> affsign sgn s t =
{ vsum (s UNION t) (\v. f v % v) |f|
!x:real^N. x IN t ==> sgn(f x)}`,
REPEAT STRIP_TAC THEN REWRITE_TAC[AFFSIGN] THEN
FIRST_ASSUM(SUBST1_TAC o MATCH_MP (SET_RULE
`x IN s DIFF t ==> s UNION t = x INSERT ((s UNION t) DELETE x)`)) THEN
ASM_SIMP_TAC[SUM_CLAUSES; VSUM_CLAUSES; FINITE_UNION; FINITE_DELETE] THEN
REWRITE_TAC[IN_DELETE; VECTOR_MUL_RZERO; VECTOR_ADD_LID] THEN
MATCH_MP_TAC SUBSET_ANTISYM THEN
REWRITE_TAC[FORALL_IN_GSPEC; SUBSET; LEFT_IMP_EXISTS_THM] THEN
REWRITE_TAC[IN_ELIM_THM] THEN CONJ_TAC THENL
[MAP_EVERY X_GEN_TAC [`y:real^N`; `f:real^N->real`] THEN
STRIP_TAC THEN EXISTS_TAC `f:real^N->real` THEN ASM_REWRITE_TAC[];
X_GEN_TAC `f:real^N->real` THEN DISCH_TAC THEN
EXISTS_TAC
`\x:real^N. if x = vec 0
then &1 - sum ((s UNION t) DELETE vec 0) (\x. f x)
else f x` THEN
MP_TAC(SET_RULE
`!x:real^N. x IN (s UNION t) DELETE vec 0 ==> ~(x = vec 0)`) THEN
SIMP_TAC[ETA_AX; REAL_SUB_ADD] THEN DISCH_THEN(K ALL_TAC) THEN
ASM SET_TAC[]]);;
let AFF_GE_0_AFFINE_MULTIPLE_CONVEX = prove
(`!s t:real^N->bool.
FINITE s /\ FINITE t /\ vec 0 IN (s DIFF t) /\ ~(t = {})
==> aff_ge s t =
{x + c % y | x IN affine hull (s DIFF t) /\
y IN convex hull t /\ &0 <= c}`,
REPEAT STRIP_TAC THEN ASM_SIMP_TAC[aff_ge_def; AFFSIGN_0; sgn_ge] THEN
ONCE_REWRITE_TAC[SET_RULE `s UNION t = (s DIFF t) UNION t`] THEN
ASM_SIMP_TAC[VSUM_UNION; FINITE_DIFF;
SET_RULE `DISJOINT (s DIFF t) t`] THEN
ASM_SIMP_TAC[AFFINE_HULL_EQ_SPAN; HULL_INC] THEN
ASM_SIMP_TAC[SPAN_FINITE; FINITE_DIFF; CONVEX_HULL_FINITE] THEN
MATCH_MP_TAC SUBSET_ANTISYM THEN REWRITE_TAC[SUBSET; FORALL_IN_GSPEC] THEN
REWRITE_TAC[IN_ELIM_THM] THEN CONJ_TAC THENL
[X_GEN_TAC `f:real^N->real` THEN DISCH_TAC THEN
EXISTS_TAC `vsum (s DIFF t) (\x:real^N. f x % x)` THEN
ASM_CASES_TAC `sum t (f:real^N->real) = &0` THENL
[MP_TAC(ISPECL [`f:real^N->real`; `t:real^N->bool`] SUM_POS_EQ_0) THEN
ASM_SIMP_TAC[VECTOR_MUL_LZERO; REAL_MUL_LZERO; VSUM_0] THEN
DISCH_TAC THEN EXISTS_TAC `&0` THEN
REWRITE_TAC[VECTOR_MUL_LZERO; REAL_LE_REFL] THEN
REWRITE_TAC[RIGHT_EXISTS_AND_THM] THEN CONJ_TAC THENL
[EXISTS_TAC `f:real^N->real` THEN REWRITE_TAC[]; ALL_TAC] THEN
ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN
REWRITE_TAC[RIGHT_EXISTS_AND_THM; GSYM EXISTS_REFL] THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
DISCH_THEN(X_CHOOSE_TAC `a:real^N`) THEN
EXISTS_TAC `\x:real^N. if x = a then &1 else &0` THEN
ASM_REWRITE_TAC[SUM_DELTA] THEN MESON_TAC[REAL_POS];
EXISTS_TAC `sum t (f:real^N->real)` THEN
EXISTS_TAC `inv(sum t (f:real^N->real)) % vsum t (\v. f v % v)` THEN
REPEAT CONJ_TAC THENL
[EXISTS_TAC `f:real^N->real` THEN REWRITE_TAC[];
EXISTS_TAC `\x:real^N. f x / sum t (f:real^N->real)` THEN
ASM_SIMP_TAC[REAL_LE_DIV; SUM_POS_LE] THEN
ONCE_REWRITE_TAC[REAL_ARITH `x / y:real = inv y * x`] THEN
ASM_SIMP_TAC[GSYM VECTOR_MUL_ASSOC; SUM_LMUL; VSUM_LMUL] THEN
ASM_SIMP_TAC[REAL_MUL_LINV];
ASM_SIMP_TAC[SUM_POS_LE];
AP_TERM_TAC THEN ASM_CASES_TAC `sum t (f:real^N->real) = &0` THEN
ASM_SIMP_TAC[VECTOR_MUL_ASSOC; REAL_MUL_RINV; VECTOR_MUL_LID]]];
MAP_EVERY X_GEN_TAC [`x:real^N`; `c:real`; `y:real^N`] THEN
DISCH_THEN(CONJUNCTS_THEN2
(X_CHOOSE_THEN `u:real^N->real` (SUBST1_TAC o SYM)) MP_TAC) THEN
DISCH_THEN(CONJUNCTS_THEN2
(X_CHOOSE_THEN `v:real^N->real`MP_TAC) ASSUME_TAC) THEN
REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
DISCH_THEN(SUBST1_TAC o SYM) THEN
EXISTS_TAC `(\x. if x IN t then c * v x else u x):real^N->real` THEN
ASM_SIMP_TAC[REAL_LE_MUL; VSUM_LMUL; GSYM VECTOR_MUL_ASSOC] THEN
AP_THM_TAC THEN AP_TERM_TAC THEN MATCH_MP_TAC VSUM_EQ THEN
SIMP_TAC[IN_DIFF]]);;
let AFF_GE_0_MULTIPLE_AFFINE_CONVEX = prove
(`!s t:real^N->bool.
FINITE s /\ FINITE t /\ vec 0 IN (s DIFF t) /\ ~(t = {})
==> aff_ge s t =
affine hull (s DIFF t) UNION
{c % (x + y) | x IN affine hull (s DIFF t) /\
y IN convex hull t /\ &0 <= c}`,
REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN
REWRITE_TAC[UNION_SUBSET] THEN REPEAT CONJ_TAC THENL
[ASM_SIMP_TAC[AFF_GE_0_AFFINE_MULTIPLE_CONVEX;
AFFINE_HULL_EQ_SPAN; HULL_INC] THEN
REWRITE_TAC[SUBSET; FORALL_IN_GSPEC] THEN
MAP_EVERY X_GEN_TAC [`x:real^N`; `c:real`; `y:real^N`] THEN STRIP_TAC THEN
REWRITE_TAC[IN_ELIM_THM; IN_UNION] THEN ASM_CASES_TAC `c = &0` THENL
[DISJ1_TAC THEN ASM_REWRITE_TAC[VECTOR_MUL_LZERO; VECTOR_ADD_RID];
DISJ2_TAC THEN MAP_EVERY EXISTS_TAC
[`c:real`; `inv(c) % x:real^N`; `y:real^N`] THEN
ASM_SIMP_TAC[SPAN_MUL; VECTOR_ADD_LDISTRIB; VECTOR_MUL_ASSOC;
REAL_MUL_RINV; VECTOR_MUL_LID]];
REWRITE_TAC[aff_ge_def] THEN ONCE_REWRITE_TAC[AFFSIGN_DISJOINT_DIFF] THEN
REWRITE_TAC[GSYM aff_ge_def] THEN
MATCH_MP_TAC AFFINE_HULL_SUBSET_AFF_GE THEN SET_TAC[];
ASM_SIMP_TAC[AFF_GE_0_AFFINE_MULTIPLE_CONVEX;
AFFINE_HULL_EQ_SPAN; HULL_INC] THEN
REWRITE_TAC[SUBSET; FORALL_IN_GSPEC] THEN
MAP_EVERY X_GEN_TAC [`c:real`; `x:real^N`; `y:real^N`] THEN STRIP_TAC THEN
REWRITE_TAC[IN_ELIM_THM] THEN MAP_EVERY EXISTS_TAC
[`c % x:real^N`; `c:real`; `y:real^N`] THEN
ASM_SIMP_TAC[SPAN_MUL; VECTOR_ADD_LDISTRIB]]);;
let AFF_GE_0_AFFINE_CONVEX_CONE = prove
(`!s t:real^N->bool.
FINITE s /\ FINITE t /\ vec 0 IN (s DIFF t)
==> aff_ge s t =
{x + y | x IN affine hull (s DIFF t) /\
y IN convex_cone hull t}`,
REPEAT STRIP_TAC THEN ASM_CASES_TAC `t:real^N->bool = {}` THENL
[ASM_REWRITE_TAC[AFF_GE_EQ_AFFINE_HULL; CONVEX_CONE_HULL_EMPTY] THEN
REWRITE_TAC[IN_SING; DIFF_EMPTY] THEN
REWRITE_TAC[SET_RULE `{x + y:real^N | P x /\ y = a} = {x + a | P x}`] THEN
REWRITE_TAC[VECTOR_ADD_RID] THEN SET_TAC[];
ASM_SIMP_TAC[CONVEX_CONE_HULL_CONVEX_HULL_NONEMPTY;
AFF_GE_0_AFFINE_MULTIPLE_CONVEX] THEN
SET_TAC[]]);;
let AFF_GE_0_N = prove
(`!s:real^N->bool.
FINITE s /\ ~(vec 0 IN s)
==> aff_ge {vec 0} s =
{y | ?u. (!x. x IN s ==> &0 <= u x) /\
y = vsum s (\x. u x % x)}`,
REPEAT STRIP_TAC THEN REWRITE_TAC[aff_ge_def] THEN
ASM_SIMP_TAC[AFFSIGN_0; IN_DIFF; IN_INSERT; NOT_IN_EMPTY;
FINITE_INSERT; FINITE_EMPTY] THEN
ASM_SIMP_TAC[EXTENSION; sgn_ge; IN_ELIM_THM; INSERT_UNION; UNION_EMPTY] THEN
ASM_SIMP_TAC[VSUM_CLAUSES; VECTOR_MUL_RZERO; VECTOR_ADD_LID]);;
let AFF_GE_0_CONVEX_HULL = prove
(`!s:real^N->bool.
FINITE s /\ ~(s = {}) /\ ~(vec 0 IN s)
==> aff_ge {vec 0} s = {t % y | &0 <= t /\ y IN convex hull s}`,
REPEAT STRIP_TAC THEN
ASM_SIMP_TAC[AFF_GE_0_AFFINE_MULTIPLE_CONVEX; IN_DIFF;
FINITE_INSERT; FINITE_EMPTY; IN_INSERT] THEN
ASM_SIMP_TAC[SET_RULE `~(a IN s) ==> {a} DIFF s = {a}`] THEN
REWRITE_TAC[AFFINE_HULL_SING; IN_SING] THEN
REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN MESON_TAC[VECTOR_ADD_LID]);;
let AFF_GE_0_CONVEX_HULL_ALT = prove
(`!s:real^N->bool.
FINITE s /\ ~(vec 0 IN s)
==> aff_ge {vec 0} s =
vec 0 INSERT {t % y | &0 < t /\ y IN convex hull s}`,
REPEAT STRIP_TAC THEN
ASM_CASES_TAC `s:real^N->bool = {}` THENL
[ASM_REWRITE_TAC[AFF_GE_EQ_AFFINE_HULL; CONVEX_HULL_EMPTY] THEN
REWRITE_TAC[AFFINE_HULL_SING; NOT_IN_EMPTY] THEN SET_TAC[];
ASM_SIMP_TAC[AFF_GE_0_CONVEX_HULL; EXTENSION; IN_ELIM_THM; IN_INSERT] THEN
X_GEN_TAC `y:real^N` THEN ASM_CASES_TAC `y:real^N = vec 0` THEN
ASM_REWRITE_TAC[] THENL
[EXISTS_TAC `&0` THEN ASM_REWRITE_TAC[REAL_POS; VECTOR_MUL_LZERO] THEN
ASM_REWRITE_TAC[MEMBER_NOT_EMPTY; CONVEX_HULL_EQ_EMPTY];
AP_TERM_TAC THEN REWRITE_TAC[FUN_EQ_THM] THEN X_GEN_TAC `t:real` THEN
AP_TERM_TAC THEN ABS_TAC THEN
ASM_CASES_TAC `t = &0` THEN
ASM_REWRITE_TAC[VECTOR_MUL_LZERO; REAL_LT_REFL] THEN
ASM_REWRITE_TAC[REAL_LT_LE]]]);;
let AFF_GE_0_CONVEX_CONE_NEGATIONS = prove
(`!s t:real^N->bool.
FINITE s /\ FINITE t /\ vec 0 IN (s DIFF t)
==> aff_ge s t =
convex_cone hull (s UNION t UNION IMAGE (--) (s DIFF t))`,
REPEAT STRIP_TAC THEN ASM_SIMP_TAC[AFF_GE_0_AFFINE_CONVEX_CONE] THEN
ASM_SIMP_TAC[AFFINE_HULL_EQ_SPAN; HULL_INC] THEN
REWRITE_TAC[SPAN_CONVEX_CONE_ALLSIGNS; GSYM CONVEX_CONE_HULL_UNION] THEN
AP_TERM_TAC THEN SET_TAC[]);;
let CONVEX_HULL_AFF_GE = prove
(`!s. convex hull s = aff_ge {} s`,
SIMP_TAC[aff_ge_def; AFFSIGN; CONVEX_HULL_FINITE; sgn_ge; UNION_EMPTY] THEN
REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN REPEAT GEN_TAC THEN
AP_TERM_TAC THEN REWRITE_TAC[FUN_EQ_THM] THEN MESON_TAC[]);;
let POLYHEDRON_AFF_GE = prove
(`!s t:real^N->bool. FINITE s /\ FINITE t ==> polyhedron(aff_ge s t)`,
REPEAT STRIP_TAC THEN REWRITE_TAC[aff_ge_def] THEN
ONCE_REWRITE_TAC[AFFSIGN_DISJOINT_DIFF] THEN
REWRITE_TAC[GSYM aff_ge_def] THEN
SUBGOAL_THEN `FINITE(s DIFF t) /\ FINITE(t:real^N->bool) /\
DISJOINT (s DIFF t) t`
MP_TAC THENL [ASM_SIMP_TAC[FINITE_DIFF] THEN ASM SET_TAC[]; ALL_TAC] THEN
POP_ASSUM_LIST(K ALL_TAC) THEN
SPEC_TAC(`s DIFF t:real^N->bool`,`s:real^N->bool`) THEN
MATCH_MP_TAC SET_PROVE_CASES THEN CONJ_TAC THENL
[REPEAT STRIP_TAC THEN REWRITE_TAC[GSYM CONVEX_HULL_AFF_GE] THEN
MATCH_MP_TAC POLYTOPE_IMP_POLYHEDRON THEN REWRITE_TAC[polytope] THEN
ASM_MESON_TAC[];
ALL_TAC] THEN
MAP_EVERY X_GEN_TAC [`a:real^N`; `s:real^N->bool`] THEN
GEOM_ORIGIN_TAC `a:real^N` THEN REPEAT STRIP_TAC THEN
SUBGOAL_THEN `(vec 0:real^N) IN ((vec 0 INSERT s) DIFF t)` ASSUME_TAC THENL
[ASM SET_TAC[]; ALL_TAC] THEN
ASM_SIMP_TAC[AFF_GE_0_CONVEX_CONE_NEGATIONS; FINITE_INSERT] THEN
MATCH_MP_TAC POLYHEDRON_CONVEX_CONE_HULL THEN
ASM_SIMP_TAC[FINITE_INSERT; FINITE_UNION; FINITE_DIFF; FINITE_IMAGE]);;
let CLOSED_AFF_GE = prove
(`!s t:real^N->bool. FINITE s /\ FINITE t ==> closed(aff_ge s t)`,
SIMP_TAC[POLYHEDRON_AFF_GE; POLYHEDRON_IMP_CLOSED]);;
let CONIC_AFF_GE_0 = prove
(`!s:real^N->bool. FINITE s /\ ~(vec 0 IN s) ==> conic(aff_ge {vec 0} s)`,
REPEAT STRIP_TAC THEN ASM_SIMP_TAC[AFF_GE_0_N; conic] THEN
REWRITE_TAC[IN_ELIM_THM] THEN GEN_TAC THEN X_GEN_TAC `c:real` THEN
DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN
DISCH_THEN(X_CHOOSE_THEN `u:real^N->real` STRIP_ASSUME_TAC) THEN
EXISTS_TAC `\v. c * (u:real^N->real) v` THEN
REWRITE_TAC[GSYM VECTOR_MUL_ASSOC; VSUM_LMUL] THEN
ASM_MESON_TAC[REAL_LE_MUL]);;
let ANGLES_ADD_AFF_GE = prove
(`!u v w x:real^N.
~(v = u) /\ ~(w = u) /\ ~(x = u) /\ x IN aff_ge {u} {v,w}
==> angle(v,u,x) + angle(x,u,w) = angle(v,u,w)`,
GEOM_ORIGIN_TAC `u:real^N` THEN REPEAT GEN_TAC THEN
REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
ASM_SIMP_TAC[AFF_GE_1_2_0] THEN
REWRITE_TAC[IN_ELIM_THM; LEFT_IMP_EXISTS_THM] THEN
MAP_EVERY X_GEN_TAC [`a:real`; `b:real`] THEN
DISCH_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC MP_TAC) THEN
SUBGOAL_THEN `a = &0 /\ b = &0 \/ &0 < a + b` STRIP_ASSUME_TAC THENL
[ASM_REAL_ARITH_TAC;
ASM_REWRITE_TAC[VECTOR_MUL_LZERO; VECTOR_ADD_LID];
ALL_TAC] THEN
DISCH_TAC THEN MP_TAC(ISPECL
[`v:real^N`; `w:real^N`; `inv(a + b) % x:real^N`; `vec 0:real^N`]
ANGLES_ADD_BETWEEN) THEN
ASM_REWRITE_TAC[angle; VECTOR_SUB_RZERO] THEN
ASM_SIMP_TAC[VECTOR_ANGLE_RMUL; VECTOR_ANGLE_LMUL;
REAL_INV_EQ_0; REAL_LE_INV_EQ; REAL_LT_IMP_NZ; REAL_LT_IMP_LE] THEN
DISCH_THEN MATCH_MP_TAC THEN
REWRITE_TAC[BETWEEN_IN_SEGMENT; CONVEX_HULL_2; SEGMENT_CONVEX_HULL] THEN
REWRITE_TAC[IN_ELIM_THM] THEN
MAP_EVERY EXISTS_TAC [`a / (a + b):real`; `b / (a + b):real`] THEN
ASM_SIMP_TAC[REAL_LE_DIV; REAL_LT_IMP_LE; VECTOR_ADD_LDISTRIB] THEN
REWRITE_TAC[VECTOR_MUL_ASSOC; real_div; REAL_MUL_AC] THEN
UNDISCH_TAC `&0 < a + b` THEN CONV_TAC REAL_FIELD);;
let AFF_GE_2_1_0_DROPOUT_3 = prove
(`!w z:real^3.
~collinear{vec 0,basis 3,z}
==> (w IN aff_ge {vec 0,basis 3} {z} <=>
(dropout 3 w) IN aff_ge {vec 0:real^2} {dropout 3 z})`,
REPEAT GEN_TAC THEN
ASM_CASES_TAC `z:real^3 = vec 0` THENL
[ASM_REWRITE_TAC[COLLINEAR_2; INSERT_AC]; ALL_TAC] THEN
ASM_CASES_TAC `z:real^3 = basis 3` THENL
[ASM_REWRITE_TAC[COLLINEAR_2; INSERT_AC]; ALL_TAC] THEN
REWRITE_TAC[COLLINEAR_BASIS_3] THEN DISCH_TAC THEN
ASM_SIMP_TAC[AFF_GE_2_1_0; SET_RULE `DISJOINT s {a} <=> ~(a IN s)`;
IN_INSERT; NOT_IN_EMPTY; AFF_GE_1_1_0] THEN
REWRITE_TAC[IN_ELIM_THM] THEN
MATCH_MP_TAC(MESON[]
`(!t. ((?s. P s t) <=> Q t)) ==> ((?s t. P s t) <=> (?t. Q t))`) THEN
X_GEN_TAC `t:real` THEN EQ_TAC THENL
[STRIP_TAC THEN
ASM_REWRITE_TAC[DROPOUT_ADD; DROPOUT_MUL; DROPOUT_BASIS_3] THEN
VECTOR_ARITH_TAC;
STRIP_TAC THEN EXISTS_TAC `(w:real^3)$3 - t * (z:real^3)$3` THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [CART_EQ]) THEN
ASM_REWRITE_TAC[CART_EQ; FORALL_2; FORALL_3; DIMINDEX_2; DIMINDEX_3] THEN
REWRITE_TAC[VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT] THEN
SIMP_TAC[dropout; LAMBDA_BETA; DIMINDEX_2; ARITH; BASIS_COMPONENT;
DIMINDEX_3] THEN
CONV_TAC REAL_RING]);;
let AFF_GE_2_1_0_SEMIALGEBRAIC = prove
(`!x y z:real^3.
~collinear {vec 0,x,y} /\ ~collinear {vec 0,x,z}
==> (z IN aff_ge {vec 0,x} {y} <=>
(x cross y) cross x cross z = vec 0 /\
&0 <= (x cross z) dot (x cross y))`,
let lemma0 = prove
(`~(y = vec 0) ==> ((?s. x = s % y) <=> y cross x = vec 0)`,
REWRITE_TAC[CROSS_EQ_0] THEN SIMP_TAC[COLLINEAR_LEMMA_ALT])
and lemma1 = prove
(`!x y:real^N.
~(y = vec 0)
==> ((?t. &0 <= t /\ x = t % y) <=>
(?t. x = t % y) /\ &0 <= x dot y)`,
REPEAT STRIP_TAC THEN REWRITE_TAC[LEFT_AND_EXISTS_THM] THEN
AP_TERM_TAC THEN REWRITE_TAC[FUN_EQ_THM] THEN X_GEN_TAC `t:real` THEN
ASM_CASES_TAC `x:real^N = t % y` THEN
ASM_SIMP_TAC[DOT_LMUL; REAL_LE_MUL_EQ; DOT_POS_LT]) in
REPEAT GEN_TAC THEN
MAP_EVERY (fun t -> ASM_CASES_TAC t THENL
[ASM_REWRITE_TAC[COLLINEAR_2; INSERT_AC]; ALL_TAC])
[`x:real^3 = vec 0`; `y:real^3 = vec 0`; `y:real^3 = x`] THEN
STRIP_TAC THEN
ASM_SIMP_TAC[AFF_GE_2_1_0; IN_ELIM_THM; SET_RULE
`DISJOINT {a,b} {c} <=> ~(a = c) /\ ~(b = c)`] THEN
ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN
REWRITE_TAC[RIGHT_EXISTS_AND_THM; VECTOR_ARITH
`a:real^N = b + c <=> a - c = b`] THEN
RULE_ASSUM_TAC(REWRITE_RULE[GSYM CROSS_EQ_0]) THEN
ASM_SIMP_TAC[lemma0; lemma1; CROSS_RMUL; CROSS_RSUB; VECTOR_SUB_EQ]);;
(* ------------------------------------------------------------------------- *)
(* Special case of aff_ge {x} {y}, i.e. rays or half-lines. *)
(* ------------------------------------------------------------------------- *)
let HALFLINE_REFL = prove
(`!x. aff_ge {x} {x} = {x}`,
ONCE_REWRITE_TAC[AFF_GE_DISJOINT_DIFF] THEN
ASM_REWRITE_TAC[DIFF_EQ_EMPTY; GSYM CONVEX_HULL_AFF_GE; CONVEX_HULL_SING]);;
let HALFLINE_EXPLICIT = prove
(`!x y:real^N.
aff_ge {x} {y} =
{z | ?t1 t2. &0 <= t2 /\ t1 + t2 = &1 /\ z = t1 % x + t2 % y}`,
REPEAT GEN_TAC THEN ASM_CASES_TAC `x:real^N = y` THENL
[ASM_REWRITE_TAC[HALFLINE_REFL]; AFF_TAC] THEN
REWRITE_TAC[REAL_ARITH `x + y = &1 <=> x = &1 - y`] THEN
REWRITE_TAC[VECTOR_ARITH `(&1 - x) % v + x % v:real^N = v`;
MESON[] `(?x y. P y /\ x = f y /\ Q x y) <=> (?y. P y /\ Q (f y) y)`] THEN
REWRITE_TAC[IN_ELIM_THM; IN_SING; EXTENSION] THEN MESON_TAC[REAL_POS]);;
let HALFLINE = prove
(`!x y:real^N.
aff_ge {x} {y} =
{z | ?t. &0 <= t /\ z = (&1 - t) % x + t % y}`,
REWRITE_TAC[HALFLINE_EXPLICIT; REAL_ARITH `x + y = &1 <=> x = &1 - y`] THEN
SET_TAC[]);;
let CLOSED_HALFLINE = prove
(`!x y. closed(aff_ge {x} {y})`,
SIMP_TAC[CLOSED_AFF_GE; FINITE_SING]);;
let SEGMENT_SUBSET_HALFLINE = prove
(`!x y. segment[x,y] SUBSET aff_ge {x} {y}`,
REWRITE_TAC[SEGMENT_CONVEX_HULL; CONVEX_HULL_2; HALFLINE_EXPLICIT] THEN
SET_TAC[]);;
let ENDS_IN_HALFLINE = prove
(`(!x y. x IN aff_ge {x} {y}) /\ (!x y. y IN aff_ge {x} {y})`,
MESON_TAC[SEGMENT_SUBSET_HALFLINE; SUBSET; ENDS_IN_SEGMENT]);;
let HALFLINE_SUBSET_AFFINE_HULL = prove
(`!x y. aff_ge {x} {y} SUBSET affine hull {x,y}`,
REWRITE_TAC[AFF_GE_SUBSET_AFFINE_HULL; SET_RULE `{x,y} = {x} UNION {y}`]);;
let HALFLINE_INTER_COMPACT_SEGMENT = prove
(`!s a b:real^N.
compact s /\ convex s /\ a IN s
==> ?c. aff_ge {a} {b} INTER s = segment[a,c]`,
REPEAT STRIP_TAC THEN ASM_CASES_TAC `b:real^N = a` THENL
[EXISTS_TAC `a:real^N` THEN
ASM_REWRITE_TAC[SEGMENT_REFL; HALFLINE_REFL] THEN ASM SET_TAC[];
ALL_TAC] THEN
SUBGOAL_THEN
`?u v:real^N. aff_ge {a} {b} INTER s = segment[u,v]`
STRIP_ASSUME_TAC THENL
[MATCH_MP_TAC COMPACT_CONVEX_COLLINEAR_SEGMENT THEN
ASM_SIMP_TAC[CLOSED_INTER_COMPACT; CLOSED_AFF_GE; FINITE_SING] THEN
ASM_SIMP_TAC[CONVEX_INTER; CONVEX_AFF_GE] THEN CONJ_TAC THENL
[REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_INTER] THEN
ASM_MESON_TAC[ENDS_IN_HALFLINE];
MATCH_MP_TAC COLLINEAR_SUBSET THEN
EXISTS_TAC `affine hull {a:real^N,b}` THEN
REWRITE_TAC[COLLINEAR_AFFINE_HULL_COLLINEAR; COLLINEAR_2] THEN
MATCH_MP_TAC(SET_RULE `s SUBSET u ==> (s INTER t) SUBSET u`) THEN
REWRITE_TAC[HALFLINE_SUBSET_AFFINE_HULL]];
ASM_CASES_TAC `u:real^N = a` THENL [ASM_MESON_TAC[]; ALL_TAC] THEN
ASM_CASES_TAC `v:real^N = a` THENL
[ASM_MESON_TAC[SEGMENT_SYM]; ALL_TAC] THEN
SUBGOAL_THEN `u IN aff_ge {a:real^N} {b} /\ v IN aff_ge {a} {b}`
MP_TAC THENL [ASM_MESON_TAC[IN_INTER; ENDS_IN_SEGMENT]; ALL_TAC] THEN
GEN_REWRITE_TAC (LAND_CONV o TOP_DEPTH_CONV) [HALFLINE; IN_ELIM_THM] THEN
DISCH_THEN(CONJUNCTS_THEN2
(X_CHOOSE_THEN `s:real` MP_TAC) (X_CHOOSE_THEN `t:real` MP_TAC)) THEN
MAP_EVERY ASM_CASES_TAC [`s = &0`; `t = &0`] THEN
ASM_REWRITE_TAC[REAL_SUB_RZERO; VECTOR_MUL_LID; VECTOR_MUL_LZERO;
VECTOR_ADD_RID] THEN
ASM_REWRITE_TAC[REAL_LE_LT] THEN REPEAT STRIP_TAC THEN
SUBGOAL_THEN `(a:real^N) IN segment[u,v]` MP_TAC THENL
[ASM_MESON_TAC[IN_INTER; ENDS_IN_HALFLINE]; ALL_TAC] THEN
ASM_REWRITE_TAC[IN_SEGMENT; LEFT_IMP_EXISTS_THM] THEN
X_GEN_TAC `u:real` THEN
REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
REWRITE_TAC[VECTOR_ARITH
`a = (&1 - u) % ((&1 - s) % a + s % b) + u % ((&1 - t) % a + t % b) <=>
((&1 - u) * s + u * t) % (b - a):real^N = vec 0`] THEN
ASM_REWRITE_TAC[VECTOR_MUL_EQ_0; VECTOR_SUB_EQ] THEN
ASM_SIMP_TAC[REAL_LE_MUL; REAL_SUB_LE; REAL_LT_IMP_LE; REAL_ARITH
`&0 <= x /\ &0 <= y ==> (x + y = &0 <=> x = &0 /\ y = &0)`] THEN
ASM_SIMP_TAC[REAL_ENTIRE; REAL_LT_IMP_NZ] THEN REAL_ARITH_TAC]);;
(* ------------------------------------------------------------------------- *)
(* Definition and properties of conv0. *)
(* ------------------------------------------------------------------------- *)
let conv0 = new_definition `conv0 S:real^A->bool = affsign sgn_gt {} S`;;
let CONV0_INJECTIVE_LINEAR_IMAGE = prove
(`!f s. linear f /\ (!x y. f x = f y ==> x = y)
==> conv0(IMAGE f s) = IMAGE f (conv0 s)`,
REPEAT GEN_TAC THEN DISCH_TAC THEN
FIRST_ASSUM(ASSUME_TAC o GSYM o MATCH_MP AFFSIGN_INJECTIVE_LINEAR_IMAGE) THEN
ASM_REWRITE_TAC[conv0; IMAGE_CLAUSES]);;
add_linear_invariants [CONV0_INJECTIVE_LINEAR_IMAGE];;
let CONV0_TRANSLATION = prove
(`!a s. conv0(IMAGE (\x. a + x) s) = IMAGE (\x. a + x) (conv0 s)`,
REWRITE_TAC[conv0; GSYM AFFSIGN_TRANSLATION; IMAGE_CLAUSES]);;
add_translation_invariants [CONV0_TRANSLATION];;
let CONV0_SUBSET_CONVEX_HULL = prove
(`!s. conv0 s SUBSET convex hull s`,
REWRITE_TAC[conv0; AFFSIGN; sgn_gt; CONVEX_HULL_FINITE; UNION_EMPTY] THEN
REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN
REPEAT GEN_TAC THEN MATCH_MP_TAC MONO_EXISTS THEN
MESON_TAC[REAL_LT_IMP_LE]);;
let CONV0_AFF_GT = prove
(`!s. conv0 s = aff_gt {} s`,
REWRITE_TAC[conv0; aff_gt_def]);;
let CONVEX_HULL_CONV0_DECOMP = prove
(`!s:real^N->bool.
FINITE s
==> convex hull s = conv0 s UNION
UNIONS {convex hull (s DELETE a) | a | a IN s}`,
REWRITE_TAC[CONV0_AFF_GT; CONVEX_HULL_AFF_GE] THEN
REPEAT STRIP_TAC THEN MATCH_MP_TAC AFF_GE_AFF_GT_DECOMP THEN
ASM_REWRITE_TAC[FINITE_EMPTY] THEN SET_TAC[]);;
let CONVEX_CONV0 = prove
(`!s. convex(conv0 s)`,
REWRITE_TAC[CONV0_AFF_GT; CONVEX_AFF_GT]);;
let BOUNDED_CONV0 = prove
(`!s:real^N->bool. bounded s ==> bounded(conv0 s)`,
REPEAT STRIP_TAC THEN MATCH_MP_TAC BOUNDED_SUBSET THEN
EXISTS_TAC `convex hull s:real^N->bool` THEN
ASM_SIMP_TAC[BOUNDED_CONVEX_HULL; CONV0_SUBSET_CONVEX_HULL]);;
let MEASURABLE_CONV0 = prove
(`!s. bounded s ==> measurable(conv0 s)`,
REPEAT STRIP_TAC THEN MATCH_MP_TAC MEASURABLE_CONVEX THEN
ASM_SIMP_TAC[CONVEX_CONV0; BOUNDED_CONV0]);;
let NEGLIGIBLE_CONVEX_HULL_DIFF_CONV0 = prove
(`!s:real^N->bool.
FINITE s /\ CARD s <= dimindex(:N) + 1
==> negligible(convex hull s DIFF conv0 s)`,
REPEAT STRIP_TAC THEN ASM_SIMP_TAC[CONVEX_HULL_CONV0_DECOMP] THEN
REWRITE_TAC[SET_RULE `(s UNION t) DIFF s = t DIFF s`] THEN
MATCH_MP_TAC NEGLIGIBLE_DIFF THEN MATCH_MP_TAC NEGLIGIBLE_UNIONS THEN
ONCE_REWRITE_TAC[SIMPLE_IMAGE] THEN ASM_SIMP_TAC[FINITE_IMAGE] THEN
REWRITE_TAC[FORALL_IN_IMAGE] THEN REPEAT STRIP_TAC THEN
MATCH_MP_TAC NEGLIGIBLE_CONVEX_HULL THEN
ASM_SIMP_TAC[FINITE_DELETE; CARD_DELETE] THEN ASM_ARITH_TAC);;
let MEASURE_CONV0_CONVEX_HULL = prove
(`!s:real^N->bool.
FINITE s /\ CARD s <= dimindex(:N) + 1
==> measure(conv0 s) = measure(convex hull s)`,
REPEAT STRIP_TAC THEN MATCH_MP_TAC MEASURE_NEGLIGIBLE_SYMDIFF THEN
ASM_SIMP_TAC[MEASURABLE_CONVEX_HULL; FINITE_IMP_BOUNDED] THEN
MATCH_MP_TAC NEGLIGIBLE_UNION THEN
ASM_SIMP_TAC[NEGLIGIBLE_CONVEX_HULL_DIFF_CONV0] THEN
ASM_SIMP_TAC[CONV0_SUBSET_CONVEX_HULL; NEGLIGIBLE_EMPTY;
SET_RULE `s SUBSET t ==> s DIFF t = {}`]);;
(* ------------------------------------------------------------------------- *)
(* Orthonormal triples of vectors in 3D. *)
(* ------------------------------------------------------------------------- *)
let orthonormal = new_definition
`orthonormal e1 e2 e3 <=>
e1 dot e1 = &1 /\ e2 dot e2 = &1 /\ e3 dot e3 = &1 /\
e1 dot e2 = &0 /\ e1 dot e3 = &0 /\ e2 dot e3 = &0 /\
&0 < (e1 cross e2) dot e3`;;
let ORTHONORMAL_LINEAR_IMAGE = prove
(`!f. linear(f) /\ (!x. norm(f x) = norm x) /\
(2 <= dimindex(:3) ==> det(matrix f) = &1)
==> !e1 e2 e3. orthonormal (f e1) (f e2) (f e3) <=>
orthonormal e1 e2 e3`,
SIMP_TAC[DIMINDEX_3; ARITH; CONJ_ASSOC; GSYM ORTHOGONAL_TRANSFORMATION] THEN
SIMP_TAC[orthonormal; CROSS_ORTHOGONAL_TRANSFORMATION] THEN
SIMP_TAC[orthogonal_transformation; VECTOR_MUL_LID]);;
add_linear_invariants [ORTHONORMAL_LINEAR_IMAGE];;
let ORTHONORMAL_PERMUTE = prove
(`!e1 e2 e3. orthonormal e1 e2 e3 ==> orthonormal e2 e3 e1`,
REWRITE_TAC[orthonormal] THEN
REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
ONCE_REWRITE_TAC[GSYM CROSS_TRIPLE] THEN ASM_REWRITE_TAC[] THEN
ONCE_REWRITE_TAC[DOT_SYM] THEN ASM_REWRITE_TAC[]);;
let ORTHONORMAL_CROSS = prove
(`!e1 e2 e3.
orthonormal e1 e2 e3
==> e2 cross e3 = e1 /\ e3 cross e1 = e2 /\ e1 cross e2 = e3`,
SUBGOAL_THEN
`!e1 e2 e3. orthonormal e1 e2 e3 ==> e3 cross e1 = e2`
(fun th -> MESON_TAC[th; ORTHONORMAL_PERMUTE]) THEN
GEOM_BASIS_MULTIPLE_TAC 1 `e1:real^3` THEN X_GEN_TAC `k:real` THEN
REWRITE_TAC[orthonormal; DOT_LMUL; DOT_RMUL] THEN
SIMP_TAC[DOT_BASIS_BASIS; DIMINDEX_3; ARITH; REAL_MUL_RID] THEN
REWRITE_TAC[REAL_RING `k * k = &1 <=> k = &1 \/ k = -- &1`] THEN
ASM_CASES_TAC `k = -- &1` THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
ASM_CASES_TAC `k = &1` THEN
ASM_REWRITE_TAC[VECTOR_MUL_LID; REAL_MUL_LID; REAL_MUL_RID] THEN
SIMP_TAC[cross; DOT_3; VECTOR_3; CART_EQ; FORALL_3; DIMINDEX_3;
BASIS_COMPONENT; DIMINDEX_3; ARITH; REAL_POS] THEN
REWRITE_TAC[REAL_MUL_LZERO; REAL_SUB_RZERO; REAL_ADD_RID;
REAL_MUL_LID] THEN
REPEAT GEN_TAC THEN
ASM_CASES_TAC `(e2:real^3)$1 = &0` THEN ASM_REWRITE_TAC[] THEN
ASM_CASES_TAC `(e3:real^3)$1 = &0` THEN ASM_REWRITE_TAC[] THEN
REWRITE_TAC[REAL_MUL_RZERO; REAL_SUB_RZERO; REAL_ADD_LID] THEN
REWRITE_TAC[REAL_SUB_LZERO; REAL_MUL_RID] THEN
MATCH_MP_TAC(REAL_ARITH
`(u = &1 /\ v = &1 /\ w = &0 ==> a = b /\ --c = d \/ a = --b /\ c = d) /\
(a = --b /\ c = d ==> x <= &0)
==> (u = &1 /\ v = &1 /\ w = &0 /\ &0 < x
==> a:real = b /\ --c:real = d)`) THEN
CONJ_TAC THENL [CONV_TAC REAL_RING; ALL_TAC] THEN
DISCH_THEN(CONJUNCTS_THEN SUBST1_TAC) THEN
MATCH_MP_TAC(REAL_ARITH
`&0 <= x * x /\ &0 <= y * y ==> --x * x + y * -- y <= &0`) THEN
REWRITE_TAC[REAL_LE_SQUARE]);;
let ORTHONORMAL_IMP_NONZERO = prove
(`!e1 e2 e3. orthonormal e1 e2 e3
==> ~(e1 = vec 0) /\ ~(e2 = vec 0) /\ ~(e3 = vec 0)`,
REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN
REWRITE_TAC[DE_MORGAN_THM] THEN STRIP_TAC THEN
ASM_REWRITE_TAC[orthonormal; DOT_LZERO] THEN REAL_ARITH_TAC);;
let ORTHONORMAL_IMP_DISTINCT = prove
(`!e1 e2 e3. orthonormal e1 e2 e3 ==> ~(e1 = e2) /\ ~(e1 = e3) /\ ~(e2 = e3)`,
REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN
REWRITE_TAC[DE_MORGAN_THM] THEN STRIP_TAC THEN
ASM_REWRITE_TAC[orthonormal; DOT_LZERO] THEN REAL_ARITH_TAC);;
let ORTHONORMAL_IMP_INDEPENDENT = prove
(`!e1 e2 e3. orthonormal e1 e2 e3 ==> independent {e1,e2,e3}`,
REPEAT STRIP_TAC THEN MATCH_MP_TAC PAIRWISE_ORTHOGONAL_INDEPENDENT THEN
CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[ORTHONORMAL_IMP_NONZERO]] THEN
RULE_ASSUM_TAC(REWRITE_RULE[orthonormal]) THEN
REWRITE_TAC[pairwise; IN_INSERT; NOT_IN_EMPTY] THEN
REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[orthogonal] THEN
ASM_MESON_TAC[DOT_SYM]);;
let ORTHONORMAL_IMP_SPANNING = prove
(`!e1 e2 e3. orthonormal e1 e2 e3 ==> span {e1,e2,e3} = (:real^3)`,
REPEAT STRIP_TAC THEN
MP_TAC(ISPECL [`(:real^3)`; `{e1:real^3,e2,e3}`] CARD_EQ_DIM) THEN
ASM_SIMP_TAC[ORTHONORMAL_IMP_INDEPENDENT; SUBSET_UNIV] THEN
REWRITE_TAC[DIM_UNIV; DIMINDEX_3; HAS_SIZE; FINITE_INSERT; FINITE_EMPTY] THEN
SIMP_TAC[CARD_CLAUSES; FINITE_INSERT; FINITE_EMPTY; IN_INSERT] THEN
FIRST_ASSUM(ASSUME_TAC o MATCH_MP ORTHONORMAL_IMP_DISTINCT) THEN
ASM_REWRITE_TAC[NOT_IN_EMPTY; ARITH] THEN SET_TAC[]);;
let ORTHONORMAL_IMP_INDEPENDENT_EXPLICIT_0 = prove
(`!e1 e2 e3 t1 t2 t3.
orthonormal e1 e2 e3
==> (t1 % e1 + t2 % e2 + t3 % e3 = vec 0 <=>
t1 = &0 /\ t2 = &0 /\ t3 = &0)`,
REPEAT STRIP_TAC THEN MATCH_MP_TAC INDEPENDENT_3 THEN
ASM_MESON_TAC[ORTHONORMAL_IMP_INDEPENDENT; ORTHONORMAL_IMP_DISTINCT]);;
let ORTHONORMAL_IMP_INDEPENDENT_EXPLICIT = prove
(`!e1 e2 e3 s1 s2 s3 t1 t2 t3.
orthonormal e1 e2 e3
==> (s1 % e1 + s2 % e2 + s3 % e3 = t1 % e1 + t2 % e2 + t3 % e3 <=>
s1 = t1 /\ s2 = t2 /\ s3 = t3)`,
SIMP_TAC[ORTHONORMAL_IMP_INDEPENDENT_EXPLICIT_0; REAL_SUB_0; VECTOR_ARITH
`a % x + b % y + c % z:real^3 = a' % x + b' % y + c' % z <=>
(a - a') % x + (b - b') % y + (c - c') % z = vec 0`]);;
(* ------------------------------------------------------------------------- *)
(* Flyspeck arcV is the same as angle even in degenerate cases. *)
(* ------------------------------------------------------------------------- *)
let arcV = new_definition
`arcV u v w = acs (( (v - u) dot (w - u))/((norm (v-u)) * (norm (w-u))))`;;
let ARCV_ANGLE = prove
(`!u v w:real^N. arcV u v w = angle(v,u,w)`,
REPEAT GEN_TAC THEN REWRITE_TAC[arcV; angle; vector_angle] THEN
REWRITE_TAC[VECTOR_SUB_EQ] THEN
ASM_CASES_TAC `v:real^N = u` THEN
ASM_REWRITE_TAC[VECTOR_SUB_REFL; NORM_0; DOT_LZERO] THEN
REWRITE_TAC[real_div; REAL_MUL_LZERO; ACS_0] THEN
ASM_CASES_TAC `w:real^N = u` THEN
ASM_REWRITE_TAC[VECTOR_SUB_REFL; NORM_0; DOT_RZERO] THEN
REWRITE_TAC[real_div; REAL_MUL_LZERO; ACS_0]);;
let ARCV_LINEAR_IMAGE_EQ = prove
(`!f a b c.
linear f /\ (!x. norm(f x) = norm x)
==> arcV (f a) (f b) (f c) = arcV a b c`,
REWRITE_TAC[ARCV_ANGLE; ANGLE_LINEAR_IMAGE_EQ]);;
add_linear_invariants [ARCV_LINEAR_IMAGE_EQ];;
let ARCV_TRANSLATION_EQ = prove
(`!a b c d. arcV (a + b) (a + c) (a + d) = arcV b c d`,
REWRITE_TAC[ARCV_ANGLE; ANGLE_TRANSLATION_EQ]);;
add_translation_invariants [ARCV_TRANSLATION_EQ];;
(* ------------------------------------------------------------------------- *)
(* Azimuth angle. *)
(* ------------------------------------------------------------------------- *)
let AZIM_EXISTS = prove
(`!v w w1 w2.
?theta. &0 <= theta /\ theta < &2 * pi /\
?h1 h2.
!e1 e2 e3.
orthonormal e1 e2 e3 /\
dist(w,v) % e3 = w - v /\
~(w = v)
==> ?psi r1 r2.
w1 - v = (r1 * cos psi) % e1 +
(r1 * sin psi) % e2 +
h1 % (w - v) /\
w2 - v = (r2 * cos (psi + theta)) % e1 +
(r2 * sin (psi + theta)) % e2 +
h2 % (w - v) /\
(~collinear {v, w, w1} ==> &0 < r1) /\
(~collinear {v, w, w2} ==> &0 < r2)`,
let lemma = prove
(`cos(p) % e + sin(p) % rotate2d (pi / &2) e = rotate2d p e`,
SIMP_TAC[CART_EQ; VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT;
FORALL_2; rotate2d; LAMBDA_BETA; DIMINDEX_2; ARITH; VECTOR_2] THEN
REWRITE_TAC[SIN_PI2; COS_PI2] THEN REAL_ARITH_TAC) in
GEN_GEOM_ORIGIN_TAC `v:real^3` ["e1"; "e2"; "e3"] THEN
REWRITE_TAC[VECTOR_SUB_RZERO; DIST_0] THEN
REPEAT GEN_TAC THEN REWRITE_TAC[RIGHT_AND_EXISTS_THM] THEN
GEN_REWRITE_TAC I [SWAP_EXISTS_THM] THEN
EXISTS_TAC `(w dot (w1:real^3)) / (w dot w)` THEN
GEN_REWRITE_TAC I [SWAP_EXISTS_THM] THEN
EXISTS_TAC `(w dot (w2:real^3)) / (w dot w)` THEN
GEOM_BASIS_MULTIPLE_TAC 3 `w:real^3` THEN
X_GEN_TAC `w:real` THEN GEN_REWRITE_TAC LAND_CONV
[REAL_ARITH `&0 <= w <=> w = &0 \/ &0 < w`] THEN
STRIP_TAC THENL
[ASM_REWRITE_TAC[VECTOR_MUL_LZERO; VECTOR_MUL_RZERO; NORM_0] THEN
EXISTS_TAC `&0` THEN MP_TAC PI_POS THEN REAL_ARITH_TAC;
ALL_TAC] THEN
SIMP_TAC[DOT_LMUL; NORM_MUL; DIMINDEX_3; ARITH; DOT_RMUL; DOT_BASIS;
VECTOR_MUL_COMPONENT; NORM_BASIS; BASIS_COMPONENT] THEN
REWRITE_TAC[VECTOR_MUL_ASSOC; REAL_MUL_RID] THEN
ASM_SIMP_TAC[REAL_FIELD `&0 < w ==> (w * x) / (w * w) * w = x`;
REAL_ARITH `&0 < w ==> abs w = w`] THEN
ASM_REWRITE_TAC[VECTOR_ARITH
`a % x:real^3 = a % y <=> a % (x - y) = vec 0`] THEN
ASM_SIMP_TAC[VECTOR_MUL_EQ_0; REAL_LT_IMP_NZ; BASIS_NONZERO;
DIMINDEX_3; ARITH; VECTOR_SUB_EQ] THEN
REWRITE_TAC[MESON[] `(!e3. p e3 /\ e3 = a ==> q e3) <=> p a ==> q a`] THEN
ONCE_REWRITE_TAC[VECTOR_ARITH `x:real^3 = a + b + c <=> x - c = a + b`] THEN
REPEAT GEN_TAC THEN
ABBREV_TAC `v1:real^3 = w1 - (w1$3) % basis 3` THEN
ABBREV_TAC `v2:real^3 = w2 - (w2$3) % basis 3` THEN
SUBGOAL_THEN
`(collinear{vec 0, w % basis 3, w1} <=>
w1 - w1$3 % basis 3:real^3 = vec 0) /\
(collinear{vec 0, w % basis 3, w2} <=>
w2 - w2$3 % basis 3:real^3 = vec 0)`
(fun th -> REWRITE_TAC[th])
THENL
[ASM_SIMP_TAC[COLLINEAR_LEMMA; VECTOR_MUL_EQ_0; REAL_LT_IMP_NZ;
BASIS_NONZERO; DIMINDEX_3; ARITH] THEN
MAP_EVERY EXPAND_TAC ["v1"; "v2"] THEN
SIMP_TAC[CART_EQ; VEC_COMPONENT; VECTOR_ADD_COMPONENT; FORALL_3;
VECTOR_MUL_COMPONENT; BASIS_COMPONENT; DIMINDEX_3; ARITH;
VECTOR_SUB_COMPONENT; REAL_MUL_RZERO; REAL_MUL_RID;
REAL_SUB_RZERO] THEN
REWRITE_TAC[RIGHT_EXISTS_AND_THM] THEN
CONV_TAC(BINOP_CONV(BINOP_CONV(ONCE_DEPTH_CONV SYM_CONV))) THEN
ASM_SIMP_TAC[GSYM REAL_EQ_RDIV_EQ; EXISTS_REFL] THEN REAL_ARITH_TAC;
ALL_TAC] THEN
ASM_REWRITE_TAC[] THEN
SUBGOAL_THEN `(v1:real^3)$3 = &0 /\ (v2:real^3)$3 = &0` MP_TAC THENL
[MAP_EVERY EXPAND_TAC ["v1"; "v2"] THEN
REWRITE_TAC[VECTOR_SUB_COMPONENT; VECTOR_MUL_COMPONENT; VECTOR_SUB_EQ] THEN
SIMP_TAC[BASIS_COMPONENT; DIMINDEX_3; ARITH] THEN REAL_ARITH_TAC;
ALL_TAC] THEN
MAP_EVERY (fun t -> SPEC_TAC(t,t)) [`v2:real^3`; `v1:real^3`] THEN
POP_ASSUM_LIST(K ALL_TAC) THEN REWRITE_TAC[orthonormal] THEN
SIMP_TAC[DOT_BASIS; BASIS_COMPONENT; DIMINDEX_3; ARITH] THEN
ONCE_REWRITE_TAC[TAUT `a /\ b /\ c /\ d /\ e /\ f <=>
d /\ e /\ a /\ b /\ c /\ f`] THEN
REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
PAD2D3D_TAC THEN REPEAT STRIP_TAC THEN
SIMP_TAC[cross; VECTOR_3; pad2d3d; LAMBDA_BETA; DIMINDEX_3; ARITH] THEN
REWRITE_TAC[IMP_IMP; GSYM CONJ_ASSOC] THEN
ASM_CASES_TAC `v1:real^2 = vec 0` THEN ASM_REWRITE_TAC[NORM_POS_LT] THENL
[MP_TAC(ISPECL [`basis 1:real^2`; `v2:real^2`]
ROTATION_ROTATE2D_EXISTS_GEN) THEN
MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `t:real` THEN
STRIP_TAC THEN ASM_REWRITE_TAC[] THEN REPEAT STRIP_TAC THEN
MP_TAC(ISPECL [`e1:real^2`; `basis 1:real^2`]
ROTATION_ROTATE2D_EXISTS_GEN) THEN
MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `p:real` THEN STRIP_TAC THEN
MAP_EVERY EXISTS_TAC [`&0`; `norm(v2:real^2)`] THEN
ASM_REWRITE_TAC[NORM_POS_LT] THEN
REWRITE_TAC[REAL_MUL_LZERO; VECTOR_MUL_LZERO; VECTOR_ADD_RID] THEN
SUBGOAL_THEN `norm(e1:real^2) = &1 /\ norm(e2:real^2) = &1`
STRIP_ASSUME_TAC THENL [ASM_REWRITE_TAC[NORM_EQ_1]; ALL_TAC] THEN
SUBGOAL_THEN `e2 = rotate2d (pi / &2) e1` SUBST1_TAC THENL
[MATCH_MP_TAC ROTATION_ROTATE2D_EXISTS_ORTHOGONAL_ORIENTED THEN
ASM_REWRITE_TAC[NORM_EQ_1; orthogonal];
ALL_TAC] THEN
REWRITE_TAC[GSYM VECTOR_MUL_ASSOC; GSYM VECTOR_ADD_LDISTRIB] THEN
REWRITE_TAC[lemma] THEN ONCE_REWRITE_TAC[REAL_ADD_SYM] THEN
REWRITE_TAC[ROTATE2D_ADD] THEN ASM_REWRITE_TAC[VECTOR_MUL_LID] THEN
MATCH_MP_TAC VECTOR_MUL_LCANCEL_IMP THEN
EXISTS_TAC `norm(basis 1:real^2)` THEN
ASM_SIMP_TAC[NORM_EQ_0; BASIS_NONZERO; DIMINDEX_2; ARITH] THEN
FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC LAND_CONV [SYM th]) THEN
ONCE_REWRITE_TAC[VECTOR_ARITH `a % b % x:real^2 = b % a % x`] THEN
AP_TERM_TAC THEN
SIMP_TAC[GSYM(MATCH_MP LINEAR_CMUL (SPEC_ALL LINEAR_ROTATE2D))] THEN
AP_TERM_TAC THEN
ASM_SIMP_TAC[LINEAR_CMUL; LINEAR_ROTATE2D; VECTOR_MUL_LID];
MP_TAC(ISPECL [`v1:real^2`; `v2:real^2`] ROTATION_ROTATE2D_EXISTS_GEN) THEN
MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `t:real` THEN
STRIP_TAC THEN ASM_REWRITE_TAC[] THEN REPEAT STRIP_TAC THEN
MP_TAC(ISPECL [`e1:real^2`; `v1:real^2`] ROTATION_ROTATE2D_EXISTS_GEN) THEN
MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `p:real` THEN STRIP_TAC THEN
MAP_EVERY EXISTS_TAC [`norm(v1:real^2)`; `norm(v2:real^2)`] THEN
ASM_REWRITE_TAC[NORM_POS_LT] THEN
SUBGOAL_THEN `norm(e1:real^2) = &1 /\ norm(e2:real^2) = &1`
STRIP_ASSUME_TAC THENL [ASM_REWRITE_TAC[NORM_EQ_1]; ALL_TAC] THEN
SUBGOAL_THEN `e2 = rotate2d (pi / &2) e1` SUBST1_TAC THENL
[MATCH_MP_TAC ROTATION_ROTATE2D_EXISTS_ORTHOGONAL_ORIENTED THEN
ASM_REWRITE_TAC[NORM_EQ_1; orthogonal];
ALL_TAC] THEN
REWRITE_TAC[GSYM VECTOR_MUL_ASSOC; GSYM VECTOR_ADD_LDISTRIB] THEN
REWRITE_TAC[lemma] THEN ONCE_REWRITE_TAC[REAL_ADD_SYM] THEN
REWRITE_TAC[ROTATE2D_ADD] THEN ASM_REWRITE_TAC[VECTOR_MUL_LID] THEN
MATCH_MP_TAC VECTOR_MUL_LCANCEL_IMP THEN EXISTS_TAC `norm(v1:real^2)` THEN
ASM_REWRITE_TAC[NORM_EQ_0] THEN
FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC LAND_CONV [SYM th]) THEN
ONCE_REWRITE_TAC[VECTOR_ARITH `a % b % x:real^2 = b % a % x`] THEN
AP_TERM_TAC THEN
SIMP_TAC[GSYM(MATCH_MP LINEAR_CMUL (SPEC_ALL LINEAR_ROTATE2D))] THEN
AP_TERM_TAC THEN
ASM_SIMP_TAC[LINEAR_CMUL; LINEAR_ROTATE2D; VECTOR_MUL_LID]]);;
let azim_spec =
(REWRITE_RULE[SKOLEM_THM]
(REWRITE_RULE[RIGHT_EXISTS_IMP_THM] AZIM_EXISTS));;
let azim_def = new_definition
`azim v w w1 w2 =
if collinear {v,w,w1} \/ collinear {v,w,w2} then &0
else @theta. &0 <= theta /\ theta < &2 * pi /\
?h1 h2.
!e1 e2 e3.
orthonormal e1 e2 e3 /\
dist(w,v) % e3 = w - v /\
~(w = v)
==> ?psi r1 r2.
w1 - v = (r1 * cos psi) % e1 +
(r1 * sin psi) % e2 +
h1 % (w - v) /\
w2 - v = (r2 * cos (psi + theta)) % e1 +
(r2 * sin (psi + theta)) % e2 +
h2 % (w - v) /\
&0 < r1 /\ &0 < r2`;;
let azim = prove
(`!v w w1 w2:real^3.
&0 <= azim v w w1 w2 /\ azim v w w1 w2 < &2 * pi /\
?h1 h2.
!e1 e2 e3.
orthonormal e1 e2 e3 /\
dist(w,v) % e3 = w - v /\
~(w = v)
==> ?psi r1 r2.
w1 - v = (r1 * cos psi) % e1 +
(r1 * sin psi) % e2 +
h1 % (w - v) /\
w2 - v = (r2 * cos (psi + azim v w w1 w2)) % e1 +
(r2 * sin (psi + azim v w w1 w2)) % e2 +
h2 % (w - v) /\
(~collinear {v, w, w1} ==> &0 < r1) /\
(~collinear {v, w, w2} ==> &0 < r2)`,
REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[azim_def] THEN
COND_CASES_TAC THENL
[ALL_TAC;
RULE_ASSUM_TAC(REWRITE_RULE[DE_MORGAN_THM]) THEN ASM_REWRITE_TAC[] THEN
CONV_TAC SELECT_CONV THEN
MP_TAC(ISPECL [`v:real^3`; `w:real^3`; `w1:real^3`; `w2:real^3`]
AZIM_EXISTS) THEN
ASM_REWRITE_TAC[]] THEN
SIMP_TAC[PI_POS; REAL_LT_MUL; REAL_OF_NUM_LT; ARITH; REAL_LE_REFL] THEN
FIRST_X_ASSUM DISJ_CASES_TAC THENL
[MP_TAC(ISPECL [`v:real^3`; `w:real^3`; `w2:real^3`; `w1:real^3`]
AZIM_EXISTS) THEN
DISCH_THEN(CHOOSE_THEN(MP_TAC o CONJUNCT2 o CONJUNCT2)) THEN
REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN
MAP_EVERY X_GEN_TAC [`h2:real`; `h1:real`] THEN
DISCH_TAC THEN MAP_EVERY EXISTS_TAC [`h1:real`; `h2:real`] THEN
MAP_EVERY X_GEN_TAC [`e1:real^3`; `e2:real^3`; `e3:real^3`] THEN
STRIP_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPECL [`e1:real^3`; `e2:real^3`; `e3:real^3`]) THEN
ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN
X_GEN_TAC `psi:real` THEN
REWRITE_TAC[LEFT_IMP_EXISTS_THM; REAL_ADD_RID] THEN
MAP_EVERY X_GEN_TAC [`r2:real`; `r1:real`] THEN STRIP_TAC THEN
MAP_EVERY EXISTS_TAC [`&0`; `r2:real`];
MP_TAC(ISPECL [`v:real^3`; `w:real^3`; `w1:real^3`; `w2:real^3`]
AZIM_EXISTS) THEN
DISCH_THEN(CHOOSE_THEN(MP_TAC o CONJUNCT2 o CONJUNCT2)) THEN
REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN
MAP_EVERY X_GEN_TAC [`h1:real`; `h2:real`] THEN
DISCH_TAC THEN MAP_EVERY EXISTS_TAC [`h1:real`; `h2:real`] THEN
MAP_EVERY X_GEN_TAC [`e1:real^3`; `e2:real^3`; `e3:real^3`] THEN
STRIP_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPECL [`e1:real^3`; `e2:real^3`; `e3:real^3`]) THEN
ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN
X_GEN_TAC `psi:real` THEN
REWRITE_TAC[LEFT_IMP_EXISTS_THM; REAL_ADD_RID] THEN
MAP_EVERY X_GEN_TAC [`r1:real`; `r2:real`] THEN STRIP_TAC THEN
MAP_EVERY EXISTS_TAC [`r1:real`; `&0`]] THEN
ASM_REWRITE_TAC[] THEN
FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV
[SET_RULE `{v,w,x} = {w,v,x}`]) THEN
ONCE_REWRITE_TAC[COLLINEAR_3] THEN ASM_REWRITE_TAC[] THEN
UNDISCH_THEN `dist(w:real^3,v) % e3 = w - v` (SUBST1_TAC o SYM) THEN
REWRITE_TAC[GSYM DOT_CAUCHY_SCHWARZ_EQUAL] THEN
RULE_ASSUM_TAC(REWRITE_RULE[orthonormal]) THEN
ASM_REWRITE_TAC[DOT_LADD; DOT_RADD; DOT_LMUL; DOT_RMUL; REAL_MUL_RZERO] THEN
ONCE_REWRITE_TAC[DOT_SYM] THEN ASM_REWRITE_TAC[REAL_MUL_RZERO] THEN
REWRITE_TAC[REAL_ADD_LID; REAL_ADD_RID; REAL_MUL_RID] THEN
REWRITE_TAC[REAL_ARITH `(r * c) * (r * c):real = r pow 2 * c pow 2`] THEN
REWRITE_TAC[REAL_ARITH `r * c + r * s + f:real = r * (s + c) + f`] THEN
REWRITE_TAC[SIN_CIRCLE] THEN REWRITE_TAC[REAL_RING
`(d * h * d) pow 2 = (d * d) * (r * &1 + h * d * h * d) <=>
d = &0 \/ r = &0`] THEN
ASM_REWRITE_TAC[DIST_EQ_0; REAL_POW_EQ_0; ARITH] THEN
DISCH_THEN SUBST1_TAC THEN REWRITE_TAC[REAL_MUL_LZERO; DOT_LZERO]);;
let AZIM_UNIQUE = prove
(`!v w w1 w2 h1 h2 r1 r2 e1 e2 e3 psi theta.
&0 <= theta /\
theta < &2 * pi /\
orthonormal e1 e2 e3 /\
dist(w,v) % e3 = w - v /\
~(w = v) /\
&0 < r1 /\ &0 < r2 /\
w1 - v = (r1 * cos psi) % e1 +
(r1 * sin psi) % e2 +
h1 % (w - v) /\
w2 - v = (r2 * cos (psi + theta)) % e1 +
(r2 * sin (psi + theta)) % e2 +
h2 % (w - v)
==> azim v w w1 w2 = theta`,
REPEAT STRIP_TAC THEN
SUBGOAL_THEN `~collinear{v:real^3,w,w2} /\ ~collinear {v,w,w1}`
STRIP_ASSUME_TAC THENL
[ONCE_REWRITE_TAC[SET_RULE `{a,b,c} = {b,a,c}`] THEN
ONCE_REWRITE_TAC[COLLINEAR_3] THEN REWRITE_TAC[COLLINEAR_LEMMA] THEN
ASM_REWRITE_TAC[] THEN ASM_REWRITE_TAC[VECTOR_SUB_EQ] THEN
UNDISCH_THEN `dist(w:real^3,v) % e3 = w - v` (SUBST1_TAC o SYM) THEN
REWRITE_TAC[VECTOR_MUL_ASSOC; VECTOR_ARITH
`a + b + c % x:real^N = d % x <=> a + b + (c - d) % x = vec 0`] THEN
ASM_SIMP_TAC[ORTHONORMAL_IMP_INDEPENDENT_EXPLICIT_0] THEN
ASM_SIMP_TAC[CONJ_ASSOC; REAL_LT_IMP_NZ; SIN_CIRCLE; REAL_RING
`s pow 2 + c pow 2 = &1 ==> (r * c = &0 /\ r * s = &0 <=> r = &0)`];
ALL_TAC] THEN
SUBGOAL_THEN `(azim v w w1 w2 - theta) / (&2 * pi) = &0` MP_TAC THENL
[ALL_TAC; MP_TAC PI_POS THEN CONV_TAC REAL_FIELD] THEN
MATCH_MP_TAC REAL_EQ_INTEGERS_IMP THEN
ASM_SIMP_TAC[REAL_SUB_RZERO; REAL_ABS_DIV; REAL_ABS_MUL; REAL_ABS_NUM;
REAL_ABS_PI; REAL_LT_LDIV_EQ; REAL_LT_MUL; REAL_OF_NUM_LT; ARITH;
PI_POS; INTEGER_CLOSED; REAL_MUL_LID] THEN
MP_TAC(ISPECL [`v:real^3`; `w:real^3`; `w1:real^3`; `w2:real^3`] azim) THEN
ASM_REWRITE_TAC[] THEN
REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
ASM_SIMP_TAC[REAL_ARITH
`&0 <= x /\ x < k /\ &0 <= y /\ y < k ==> abs(x - y) < k`] THEN
REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN
MAP_EVERY X_GEN_TAC [`k1:real`; `k2:real`] THEN
DISCH_THEN(MP_TAC o SPECL [`e1:real^3`; `e2:real^3`; `e3:real^3`]) THEN
ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN
MAP_EVERY X_GEN_TAC [`phi:real`; `s1:real`; `s2:real`] THEN
UNDISCH_THEN `dist(w:real^3,v) % e3 = w - v` (SUBST1_TAC o SYM) THEN
REWRITE_TAC[VECTOR_MUL_ASSOC] THEN
ASM_SIMP_TAC[ORTHONORMAL_IMP_INDEPENDENT_EXPLICIT] THEN
ONCE_REWRITE_TAC[TAUT `a /\ b /\ c /\ d <=> (c /\ d) /\ a /\ b`] THEN
DISCH_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC MP_TAC) THEN
DISCH_THEN(CONJUNCTS_THEN (MP_TAC o MATCH_MP (REAL_FIELD
`r * c = r' * c' /\ r * s = r' * s' /\ u:real = v
==> s pow 2 + c pow 2 = &1 /\ s' pow 2 + c' pow 2 = &1 /\
&0 < r /\ (r pow 2 = r' pow 2 ==> r = r')
==> s = s' /\ c = c'`))) THEN
ASM_REWRITE_TAC[SIN_CIRCLE; GSYM REAL_EQ_SQUARE_ABS] THEN
ASM_SIMP_TAC[REAL_ARITH
`&0 < x /\ &0 < y ==> (abs x = abs y <=> x = y)`] THEN
REWRITE_TAC[SIN_COS_EQ] THEN
REWRITE_TAC[REAL_ARITH
`psi + theta = (phi + az) + x:real <=> psi = phi + x + (az - theta)`] THEN
DISCH_THEN(X_CHOOSE_THEN `m:real` STRIP_ASSUME_TAC) THEN
ASM_REWRITE_TAC[REAL_EQ_ADD_LCANCEL] THEN
REWRITE_TAC[REAL_ARITH
`&2 * m * pi + x = &2 * n * pi <=> x = (n - m) * &2 * pi`] THEN
DISCH_THEN(X_CHOOSE_THEN `n:real` STRIP_ASSUME_TAC) THEN
ASM_SIMP_TAC[PI_POS; REAL_FIELD `&0 < pi ==> (x * &2 * pi) / (&2 * pi) = x`;
INTEGER_CLOSED]);;
let AZIM_TRANSLATION = prove
(`!a v w w1 w2. azim (a + v) (a + w) (a + w1) (a + w2) = azim v w w1 w2`,
REPEAT GEN_TAC THEN REWRITE_TAC[azim_def] THEN
REWRITE_TAC[VECTOR_ARITH `(a + w) - (a + v):real^3 = w - v`;
VECTOR_ARITH `a + w:real^3 = a + v <=> w = v`;
NORM_ARITH `dist(a + v,a + w) = dist(v,w)`] THEN
REWRITE_TAC[SET_RULE
`{a + x,a + y,a + z} = IMAGE (\x:real^3. a + x) {x,y,z}`] THEN
REWRITE_TAC[COLLINEAR_TRANSLATION_EQ]);;
add_translation_invariants [AZIM_TRANSLATION];;
let AZIM_LINEAR_IMAGE = prove
(`!f. linear f /\ (!x. norm(f x) = norm x) /\
(2 <= dimindex(:3) ==> det(matrix f) = &1)
==> !v w w1 w2. azim (f v) (f w) (f w1) (f w2) = azim v w w1 w2`,
REPEAT STRIP_TAC THEN REWRITE_TAC[azim_def] THEN
ASM_SIMP_TAC[GSYM LINEAR_SUB; dist] THEN
MP_TAC(ISPEC `f:real^3->real^3` QUANTIFY_SURJECTION_THM) THEN
ANTS_TAC THENL
[ASM_MESON_TAC[ORTHOGONAL_TRANSFORMATION;
ORTHOGONAL_TRANSFORMATION_SURJECTIVE];
ALL_TAC] THEN
DISCH_THEN(CONV_TAC o LAND_CONV o EXPAND_QUANTS_CONV) THEN
ASM_SIMP_TAC[ORTHONORMAL_LINEAR_IMAGE] THEN
ASM_SIMP_TAC[GSYM LINEAR_CMUL; GSYM LINEAR_ADD] THEN
SUBGOAL_THEN `!x y. (f:real^3->real^3) x = f y <=> x = y` ASSUME_TAC THENL
[ASM_MESON_TAC[PRESERVES_NORM_INJECTIVE]; ALL_TAC] THEN
ASM_REWRITE_TAC[] THEN
REWRITE_TAC[SET_RULE `{f x,f y,f z} = IMAGE f {x,y,z}`] THEN
ASM_SIMP_TAC[COLLINEAR_LINEAR_IMAGE_EQ]);;
add_linear_invariants [AZIM_LINEAR_IMAGE];;
let AZIM_DEGENERATE = prove
(`(!v w w1 w2. v = w ==> azim v w w1 w2 = &0) /\
(!v w w1 w2. collinear{v,w,w1} ==> azim v w w1 w2 = &0) /\
(!v w w1 w2. collinear{v,w,w2} ==> azim v w w1 w2 = &0)`,
REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[azim_def] THEN
ASM_REWRITE_TAC[] THEN REWRITE_TAC[INSERT_AC; COLLINEAR_2]);;
let AZIM_REFL_ALT = prove
(`!v x y. azim v v x y = &0`,
REPEAT GEN_TAC THEN MATCH_MP_TAC(last(CONJUNCTS AZIM_DEGENERATE)) THEN
REWRITE_TAC[COLLINEAR_2; INSERT_AC]);;
let AZIM_SPECIAL_SCALE = prove
(`!a v w1 w2.
&0 < a
==> azim (vec 0) (a % v) w1 w2 = azim (vec 0) v w1 w2`,
REPEAT STRIP_TAC THEN REWRITE_TAC[azim_def] THEN
REWRITE_TAC[VECTOR_SUB_RZERO; DIST_0] THEN
FIRST_ASSUM(MP_TAC o MATCH_MP(MESON[REAL_LT_IMP_NZ; REAL_DIV_LMUL]
`!a. &0 < a ==> (!y. ?x. a * x = y)`)) THEN
DISCH_THEN(MP_TAC o MATCH_MP QUANTIFY_SURJECTION_THM) THEN
DISCH_THEN(CONV_TAC o RAND_CONV o
PARTIAL_EXPAND_QUANTS_CONV ["psi"; "r1"; "r2"]) THEN
REWRITE_TAC[VECTOR_MUL_ASSOC; REAL_MUL_SYM] THEN
ASM_SIMP_TAC[VECTOR_MUL_EQ_0; REAL_LT_IMP_NZ] THEN
ASM_SIMP_TAC[NORM_MUL; REAL_ARITH `&0 < a ==> abs a = a`] THEN
REWRITE_TAC[GSYM VECTOR_MUL_ASSOC] THEN
REWRITE_TAC[VECTOR_ARITH `a % x:real^3 = a % y <=> a % (x - y) = vec 0`] THEN
ASM_SIMP_TAC[REAL_LT_IMP_NZ; VECTOR_MUL_EQ_0] THEN
REWRITE_TAC[VECTOR_SUB_EQ] THEN
ASM_SIMP_TAC[COLLINEAR_SPECIAL_SCALE; REAL_LT_IMP_NZ]);;
let AZIM_SCALE_ALL = prove
(`!a v w1 w2.
&0 < a /\ &0 < b /\ &0 < c
==> azim (vec 0) (a % v) (b % w1) (c % w2) = azim (vec 0) v w1 w2`,
let lemma = MESON[REAL_LT_IMP_NZ; REAL_DIV_LMUL]
`!a. &0 < a ==> (!y. ?x. a * x = y)` in
let SCALE_QUANT_TAC side asm avoid =
MP_TAC(MATCH_MP lemma (ASSUME asm)) THEN
DISCH_THEN(MP_TAC o MATCH_MP QUANTIFY_SURJECTION_THM) THEN
DISCH_THEN(CONV_TAC o side o PARTIAL_EXPAND_QUANTS_CONV avoid) in
REPEAT STRIP_TAC THEN
ASM_SIMP_TAC[azim_def; COLLINEAR_SCALE_ALL; REAL_LT_IMP_NZ] THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[VECTOR_SUB_RZERO] THEN
ASM_SIMP_TAC[DIST_0; NORM_MUL; GSYM VECTOR_MUL_ASSOC] THEN
ASM_SIMP_TAC[REAL_ARITH `&0 < a ==> abs a = a`; VECTOR_MUL_LCANCEL] THEN
ASM_SIMP_TAC[VECTOR_MUL_EQ_0; REAL_LT_IMP_NZ] THEN
SCALE_QUANT_TAC RAND_CONV `&0 < a` ["psi"; "r1"; "r2"] THEN
SCALE_QUANT_TAC LAND_CONV `&0 < b` ["psi"; "h2"; "r2"] THEN
SCALE_QUANT_TAC LAND_CONV `&0 < c` ["psi"; "h1"; "r1"] THEN
ASM_SIMP_TAC[GSYM VECTOR_MUL_ASSOC; GSYM VECTOR_ADD_LDISTRIB;
VECTOR_MUL_LCANCEL; REAL_LT_IMP_NZ; REAL_LT_MUL_EQ] THEN
REWRITE_TAC[VECTOR_MUL_ASSOC; REAL_MUL_AC]);;
let AZIM_ARG = prove
(`!x y:real^3. azim (vec 0) (basis 3) x y = Arg(dropout 3 y / dropout 3 x)`,
let lemma = prove
(`(r * cos t) % basis 1 + (r * sin t) % basis 2 = Cx r * cexp(ii * Cx t)`,
REWRITE_TAC[CEXP_EULER; COMPLEX_BASIS; GSYM CX_SIN; GSYM CX_COS;
COMPLEX_CMUL; CX_MUL] THEN
CONV_TAC COMPLEX_RING) in
REPEAT STRIP_TAC THEN
ASM_CASES_TAC `collinear {vec 0:real^3,basis 3,x}` THENL
[ASM_SIMP_TAC[AZIM_DEGENERATE] THEN
RULE_ASSUM_TAC(REWRITE_RULE[COLLINEAR_BASIS_3]) THEN
ASM_REWRITE_TAC[COMPLEX_VEC_0; complex_div; COMPLEX_INV_0;
COMPLEX_MUL_RZERO; ARG_0];
ALL_TAC] THEN
ASM_CASES_TAC `collinear {vec 0:real^3,basis 3,y}` THENL
[ASM_SIMP_TAC[AZIM_DEGENERATE] THEN
RULE_ASSUM_TAC(REWRITE_RULE[COLLINEAR_BASIS_3]) THEN
ASM_REWRITE_TAC[COMPLEX_VEC_0; complex_div; COMPLEX_MUL_LZERO; ARG_0];
ALL_TAC] THEN
MP_TAC(ISPECL [`vec 0:real^3`; `basis 3:real^3`; `x:real^3`; `y:real^3`]
azim) THEN
ABBREV_TAC `a = azim (vec 0) (basis 3) x (y:real^3)` THEN
REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM; VECTOR_SUB_RZERO; DIST_0] THEN
MAP_EVERY X_GEN_TAC [`h1:real`; `h2:real`] THEN
DISCH_THEN(MP_TAC o SPECL
[`basis 1:real^3`; `basis 2:real^3`; `basis 3:real^3`]) THEN
SIMP_TAC[orthonormal; DOT_BASIS_BASIS; CROSS_BASIS; DIMINDEX_3; NORM_BASIS;
ARITH; VECTOR_MUL_LID; BASIS_NONZERO; REAL_LT_01; LEFT_IMP_EXISTS_THM] THEN
MAP_EVERY X_GEN_TAC [`psi:real`; `r1:real`; `r2:real`] THEN
DISCH_THEN(STRIP_ASSUME_TAC o GSYM) THEN
REWRITE_TAC[DROPOUT_ADD; DROPOUT_MUL; DROPOUT_BASIS_3] THEN
REWRITE_TAC[VECTOR_MUL_RZERO; VECTOR_ADD_RID; lemma] THEN
REWRITE_TAC[complex_div; COMPLEX_INV_MUL] THEN
ONCE_REWRITE_TAC[COMPLEX_RING
`(a * b) * (c * d):complex = (a * c) * b * d`] THEN
REWRITE_TAC[GSYM complex_div; GSYM CX_DIV; GSYM CEXP_SUB] THEN
CONV_TAC SYM_CONV THEN MATCH_MP_TAC ARG_UNIQUE THEN
EXISTS_TAC `r2 / r1:real` THEN ASM_SIMP_TAC[REAL_LT_DIV] THEN
AP_TERM_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[CX_ADD] THEN
CONV_TAC COMPLEX_RING);;
let REAL_CONTINUOUS_AT_AZIM_SHARP = prove
(`!v w w1 w2.
~collinear{v,w,w1} /\ ~(w2 IN aff_ge {v,w} {w1})
==> (azim v w w1) real_continuous at w2`,
GEOM_ORIGIN_TAC `v:real^3` THEN
GEOM_BASIS_MULTIPLE_TAC 3 `w:real^3` THEN
X_GEN_TAC `w:real` THEN ASM_CASES_TAC `w = &0` THENL
[ASM_REWRITE_TAC[VECTOR_MUL_LZERO; INSERT_AC; COLLINEAR_2]; ALL_TAC] THEN
ASM_SIMP_TAC[REAL_LE_LT; COLLINEAR_SPECIAL_SCALE] THEN
DISCH_TAC THEN REPEAT GEN_TAC THEN
REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
W(MP_TAC o PART_MATCH (lhs o rand) AFF_GE_SPECIAL_SCALE o
rand o rand o lhand o snd) THEN
ASM_REWRITE_TAC[FINITE_INSERT; FINITE_EMPTY; IN_SING] THEN ANTS_TAC THENL
[POP_ASSUM_LIST(MP_TAC o end_itlist CONJ) THEN
ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN REWRITE_TAC[DE_MORGAN_THM] THEN
DISCH_THEN(STRIP_ASSUME_TAC o GSYM) THENL
[ASM_REWRITE_TAC[COLLINEAR_2; INSERT_AC];
ASM_REWRITE_TAC[COLLINEAR_2; INSERT_AC];
ASM_SIMP_TAC[COLLINEAR_LEMMA_ALT; BASIS_NONZERO; DIMINDEX_3; ARITH] THEN
MESON_TAC[]];
DISCH_THEN SUBST1_TAC THEN DISCH_TAC] THEN
ASM_SIMP_TAC[AZIM_SPECIAL_SCALE; AZIM_ARG] THEN
MATCH_MP_TAC(REWRITE_RULE[o_DEF]
REAL_CONTINUOUS_CONTINUOUS_AT_COMPOSE) THEN
CONJ_TAC THENL
[REWRITE_TAC[complex_div] THEN MATCH_MP_TAC CONTINUOUS_COMPLEX_MUL THEN
REWRITE_TAC[CONTINUOUS_CONST; ETA_AX] THEN
SIMP_TAC[LINEAR_CONTINUOUS_AT; LINEAR_DROPOUT; DIMINDEX_3; DIMINDEX_2;
ARITH];
ALL_TAC] THEN
MATCH_MP_TAC REAL_CONTINUOUS_AT_WITHIN THEN
MATCH_MP_TAC REAL_CONTINUOUS_AT_ARG THEN
MP_TAC(ISPECL [`w2:real^3`; `w1:real^3`] AFF_GE_2_1_0_DROPOUT_3) THEN
ASM_REWRITE_TAC[] THEN
REPEAT(FIRST_X_ASSUM(MP_TAC o
GEN_REWRITE_RULE RAND_CONV [COLLINEAR_BASIS_3])) THEN
SPEC_TAC(`(dropout 3:real^3->real^2) w2`,`v2:real^2`) THEN
SPEC_TAC(`(dropout 3:real^3->real^2) w1`,`v1:real^2`) THEN
POP_ASSUM_LIST(K ALL_TAC) THEN
GEOM_BASIS_MULTIPLE_TAC 1 `v1:complex` THEN
X_GEN_TAC `w:real` THEN ASM_CASES_TAC `w = &0` THEN
ASM_REWRITE_TAC[VECTOR_MUL_LZERO] THEN
GEN_REWRITE_TAC LAND_CONV [REAL_LE_LT] THEN ASM_REWRITE_TAC[] THEN
DISCH_TAC THEN X_GEN_TAC `z:complex` THEN
DISCH_THEN(K ALL_TAC) THEN
REWRITE_TAC[CONTRAPOS_THM; COMPLEX_BASIS; COMPLEX_CMUL] THEN
REWRITE_TAC[COMPLEX_MUL_RID; RE_DIV_CX; IM_DIV_CX; real] THEN
ASM_SIMP_TAC[REAL_DIV_EQ_0; REAL_LE_RDIV_EQ; REAL_MUL_LZERO] THEN
STRIP_TAC THEN
W(MP_TAC o PART_MATCH (lhs o rand) AFF_GE_1_1_0 o rand o snd) THEN
ASM_REWRITE_TAC[COMPLEX_VEC_0; CX_INJ] THEN DISCH_THEN SUBST1_TAC THEN
REWRITE_TAC[IN_ELIM_THM] THEN EXISTS_TAC `Re z / w` THEN
ASM_SIMP_TAC[REAL_LE_DIV; REAL_LT_IMP_LE; COMPLEX_EQ] THEN
ASM_SIMP_TAC[COMPLEX_CMUL; CX_DIV; COMPLEX_DIV_RMUL; CX_INJ] THEN
REWRITE_TAC[RE_CX; IM_CX]);;
let REAL_CONTINUOUS_AT_AZIM = prove
(`!v w w1 w2. ~coplanar{v,w,w1,w2} ==> (azim v w w1) real_continuous at w2`,
REPEAT STRIP_TAC THEN MATCH_MP_TAC REAL_CONTINUOUS_AT_AZIM_SHARP THEN
CONJ_TAC THENL
[ASM_MESON_TAC[NOT_COPLANAR_NOT_COLLINEAR; INSERT_AC];
DISCH_THEN(MP_TAC o MATCH_MP (REWRITE_RULE[SUBSET]
AFF_GE_SUBSET_AFFINE_HULL)) THEN
POP_ASSUM MP_TAC THEN REWRITE_TAC[coplanar; CONTRAPOS_THM] THEN
REWRITE_TAC[SET_RULE `{a,b} UNION {c} = {a,b,c}`] THEN
DISCH_TAC THEN MAP_EVERY EXISTS_TAC
[`v:real^3`; `w:real^3`; `w1:real^3`] THEN
SIMP_TAC[SET_RULE `{a,b,c,d} SUBSET s <=> {a,b,c} SUBSET s /\ d IN s`] THEN
ASM_REWRITE_TAC[HULL_SUBSET]]);;
let AZIM_REFL = prove
(`!v0 v1 w. azim v0 v1 w w = &0`,
GEOM_ORIGIN_TAC `v0:real^3` THEN
GEOM_BASIS_MULTIPLE_TAC 3 `v1:real^3` THEN
GEN_TAC THEN
GEN_REWRITE_TAC LAND_CONV [REAL_ARITH `&0 <= x <=> x = &0 \/ &0 < x`] THEN
STRIP_TAC THEN ASM_SIMP_TAC[VECTOR_MUL_LZERO; AZIM_DEGENERATE] THEN
ASM_SIMP_TAC[AZIM_SPECIAL_SCALE; AZIM_ARG; ARG_EQ_0] THEN
X_GEN_TAC `w:real^3` THEN
ASM_CASES_TAC `(dropout 3 :real^3->real^2) w = Cx(&0)` THEN
ASM_SIMP_TAC[COMPLEX_DIV_REFL; REAL_CX; RE_CX; REAL_POS] THEN
ASM_SIMP_TAC[complex_div; COMPLEX_MUL_LZERO; REAL_CX; RE_CX; REAL_POS]);;
let AZIM_EQ = prove
(`!v0 v1 w x y.
~collinear{v0,v1,w} /\ ~collinear{v0,v1,x} /\ ~collinear{v0,v1,y}
==> (azim v0 v1 w x = azim v0 v1 w y <=> y IN aff_gt {v0,v1} {x})`,
GEOM_ORIGIN_TAC `v0:real^3` THEN
GEOM_BASIS_MULTIPLE_TAC 3 `v1:real^3` THEN
GEN_TAC THEN
GEN_REWRITE_TAC LAND_CONV [REAL_ARITH `&0 <= x <=> x = &0 \/ &0 < x`] THEN
STRIP_TAC THENL
[ASM_REWRITE_TAC[VECTOR_MUL_LZERO; INSERT_AC; COLLINEAR_2]; ALL_TAC] THEN
ASM_SIMP_TAC[AZIM_SPECIAL_SCALE; REAL_LT_IMP_NZ; COLLINEAR_SPECIAL_SCALE] THEN
REPEAT STRIP_TAC THEN
W(MP_TAC o PART_MATCH (lhs o rand) AFF_GT_SPECIAL_SCALE o
rand o rand o snd) THEN
ANTS_TAC THENL
[ASM_REWRITE_TAC[IN_INSERT; FINITE_INSERT; FINITE_EMPTY; NOT_IN_EMPTY] THEN
REPEAT CONJ_TAC THEN DISCH_THEN(SUBST_ALL_TAC o SYM) THEN
TRY(RULE_ASSUM_TAC(REWRITE_RULE[INSERT_AC; COLLINEAR_2]) THEN
FIRST_X_ASSUM CONTR_TAC) THEN
UNDISCH_TAC `~collinear {vec 0:real^3, basis 3, v1 % basis 3}` THEN
REWRITE_TAC[COLLINEAR_LEMMA] THEN MESON_TAC[];
DISCH_THEN SUBST1_TAC] THEN
REWRITE_TAC[AZIM_ARG] THEN CONV_TAC(LAND_CONV SYM_CONV) THEN
W(MP_TAC o PART_MATCH (lhs o rand) ARG_EQ o lhand o snd) THEN
RULE_ASSUM_TAC(REWRITE_RULE[COLLINEAR_BASIS_3]) THEN
ASM_REWRITE_TAC[complex_div; COMPLEX_ENTIRE; COMPLEX_INV_EQ_0] THEN
ASM_REWRITE_TAC[GSYM complex_div; GSYM COMPLEX_VEC_0] THEN
DISCH_THEN SUBST1_TAC THEN
ASM_SIMP_TAC[GSYM COMPLEX_VEC_0; COMPLEX_FIELD
`~(w = Cx(&0)) ==> (y / w = x * u / w <=> y = x * u)`] THEN
W(MP_TAC o PART_MATCH (lhs o rand) AFF_GT_2_1 o rand o rand o snd) THEN
ANTS_TAC THENL
[REWRITE_TAC[SET_RULE `DISJOINT {a,b} {x} <=> ~(x = a) /\ ~(x = b)`] THEN
ASM_MESON_TAC[DROPOUT_BASIS_3; DROPOUT_0];
DISCH_THEN SUBST1_TAC] THEN
REWRITE_TAC[IN_ELIM_THM; VECTOR_MUL_RZERO; VECTOR_ADD_LID] THEN
ONCE_REWRITE_TAC[MESON[]
`(?a b c. p c /\ q a b c /\ r b c) <=>
(?c. p c /\ ?b. r b c /\ ?a. q a b c)`] THEN
SIMP_TAC[REAL_ARITH `a + b + c = &1 <=> a = &1 - b - c`; EXISTS_REFL] THEN
AP_TERM_TAC THEN GEN_REWRITE_TAC I [FUN_EQ_THM] THEN
X_GEN_TAC `t:real` THEN REWRITE_TAC[] THEN AP_TERM_TAC THEN
REWRITE_TAC[GSYM COMPLEX_CMUL] THEN
SIMP_TAC[CART_EQ; FORALL_2; FORALL_3; DIMINDEX_2; DIMINDEX_3;
dropout; LAMBDA_BETA; BASIS_COMPONENT; ARITH; REAL_MUL_RID;
VECTOR_MUL_COMPONENT; VEC_COMPONENT; REAL_MUL_RZERO; UNWIND_THM1;
VECTOR_ADD_COMPONENT; REAL_ADD_LID; RIGHT_EXISTS_AND_THM] THEN
REWRITE_TAC[REAL_ARITH `y:real = t + z <=> t = y - z`; EXISTS_REFL]);;
let AZIM_EQ_ALT = prove
(`!v0 v1 w x y.
~collinear{v0,v1,w} /\ ~collinear{v0,v1,x} /\ ~collinear{v0,v1,y}
==> (azim v0 v1 w x = azim v0 v1 w y <=> x IN aff_gt {v0,v1} {y})`,
ASM_SIMP_TAC[GSYM AZIM_EQ] THEN MESON_TAC[]);;
let AZIM_EQ_0 = prove
(`!v0 v1 w x.
~collinear{v0,v1,w} /\ ~collinear{v0,v1,x}
==> (azim v0 v1 w x = &0 <=> w IN aff_gt {v0,v1} {x})`,
REPEAT STRIP_TAC THEN MATCH_MP_TAC EQ_TRANS THEN
EXISTS_TAC `azim v0 v1 w x = azim v0 v1 w w` THEN CONJ_TAC THENL
[REWRITE_TAC[AZIM_REFL];
ASM_SIMP_TAC[AZIM_EQ]]);;
let AZIM_EQ_0_ALT = prove
(`!v0 v1 w x.
~collinear{v0,v1,w} /\ ~collinear{v0,v1,x}
==> (azim v0 v1 w x = &0 <=> x IN aff_gt {v0,v1} {w})`,
REPEAT STRIP_TAC THEN MATCH_MP_TAC EQ_TRANS THEN
EXISTS_TAC `azim v0 v1 w x = azim v0 v1 w w` THEN CONJ_TAC THENL
[REWRITE_TAC[AZIM_REFL];
ASM_SIMP_TAC[AZIM_EQ_ALT]]);;
let AZIM_EQ_0_GE = prove
(`!v0 v1 w x.
~collinear{v0,v1,w} /\ ~collinear{v0,v1,x}
==> (azim v0 v1 w x = &0 <=> w IN aff_ge {v0,v1} {x})`,
REPEAT GEN_TAC THEN
ASM_CASES_TAC `v1:real^3 = v0` THENL
[ASM_REWRITE_TAC[INSERT_AC; COLLINEAR_2]; STRIP_TAC] THEN
W(MP_TAC o PART_MATCH (lhs o rand) AFF_GE_AFF_GT_DECOMP o
rand o rand o snd) THEN
ANTS_TAC THENL
[SIMP_TAC[FINITE_INSERT; FINITE_EMPTY; DISJOINT_INSERT; DISJOINT_EMPTY] THEN
REWRITE_TAC[IN_SING] THEN
CONJ_TAC THEN DISCH_THEN SUBST_ALL_TAC THEN
RULE_ASSUM_TAC(REWRITE_RULE[COLLINEAR_2; INSERT_AC]) THEN
FIRST_ASSUM CONTR_TAC;
DISCH_THEN SUBST1_TAC] THEN
ASM_SIMP_TAC[AZIM_EQ_0] THEN
REWRITE_TAC[SIMPLE_IMAGE; IMAGE_CLAUSES; UNIONS_1] THEN
REWRITE_TAC[SET_RULE `{x} DELETE x = {}`] THEN
REWRITE_TAC[AFF_GE_EQ_AFFINE_HULL; IN_UNION] THEN
ASM_SIMP_TAC[GSYM COLLINEAR_3_AFFINE_HULL]);;
let AZIM_COMPL_EQ_0 = prove
(`!z w w1 w2.
~collinear {z,w,w1} /\ ~collinear {z,w,w2} /\ azim z w w1 w2 = &0
==> azim z w w2 w1 = &0`,
REWRITE_TAC[IMP_CONJ] THEN
GEOM_ORIGIN_TAC `z:real^3` THEN
GEOM_BASIS_MULTIPLE_TAC 3 `w:real^3` THEN
X_GEN_TAC `w:real` THEN GEN_REWRITE_TAC LAND_CONV [REAL_LE_LT] THEN
ASM_CASES_TAC `w = &0` THEN ASM_REWRITE_TAC[] THENL
[ASM_REWRITE_TAC[VECTOR_MUL_LZERO; INSERT_AC; COLLINEAR_2]; ALL_TAC] THEN
DISCH_TAC THEN REPEAT GEN_TAC THEN
ASM_SIMP_TAC[AZIM_SPECIAL_SCALE; COLLINEAR_SPECIAL_SCALE; AZIM_ARG] THEN
REWRITE_TAC[ARG_EQ_0; real; IM_COMPLEX_DIV_EQ_0; RE_COMPLEX_DIV_GE_0] THEN
REWRITE_TAC[complex_mul; RE; IM; cnj] THEN REAL_ARITH_TAC);;
let AZIM_COMPL = prove
(`!z w w1 w2.
~collinear {z,w,w1} /\ ~collinear {z,w,w2}
==> azim z w w2 w1 = if azim z w w1 w2 = &0 then &0
else &2 * pi - azim z w w1 w2`,
REPEAT GEN_TAC THEN COND_CASES_TAC THENL
[ASM_MESON_TAC[AZIM_COMPL_EQ_0]; ALL_TAC] THEN
DISCH_THEN(fun th -> POP_ASSUM MP_TAC THEN MP_TAC th) THEN
GEOM_ORIGIN_TAC `z:real^3` THEN
GEOM_BASIS_MULTIPLE_TAC 3 `w:real^3` THEN
X_GEN_TAC `w:real` THEN GEN_REWRITE_TAC LAND_CONV [REAL_LE_LT] THEN
ASM_CASES_TAC `w = &0` THEN ASM_REWRITE_TAC[] THENL
[ASM_REWRITE_TAC[VECTOR_MUL_LZERO; INSERT_AC; COLLINEAR_2]; ALL_TAC] THEN
DISCH_TAC THEN REPEAT GEN_TAC THEN
ASM_SIMP_TAC[AZIM_SPECIAL_SCALE; COLLINEAR_SPECIAL_SCALE; AZIM_ARG] THEN
REWRITE_TAC[COLLINEAR_BASIS_3] THEN REWRITE_TAC[ARG_EQ_0] THEN
REPEAT STRIP_TAC THEN
MP_TAC(ISPEC `(dropout 3:real^3->real^2) w2 /
(dropout 3:real^3->real^2) w1` ARG_INV) THEN
ASM_REWRITE_TAC[COMPLEX_INV_DIV]);;
let AZIM_EQ_PI_SYM = prove
(`!z w w1 w2.
~collinear {z, w, w1} /\ ~collinear {z, w, w2}
==> (azim z w w1 w2 = pi <=> azim z w w2 w1 = pi)`,
REPEAT STRIP_TAC THEN
W(MP_TAC o PART_MATCH (lhs o rand) AZIM_COMPL o lhand o rand o snd) THEN
ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC);;
let AZIM_EQ_0_SYM = prove
(`!z w w1 w2.
~collinear {z, w, w1} /\ ~collinear {z, w, w2}
==> (azim z w w1 w2 = &0 <=> azim z w w2 w1 = &0)`,
MESON_TAC[AZIM_COMPL_EQ_0]);;
let AZIM_EQ_0_GE_ALT = prove
(`!v0 v1 w x.
~collinear{v0,v1,w} /\ ~collinear{v0,v1,x}
==> (azim v0 v1 w x = &0 <=> x IN aff_ge {v0,v1} {w})`,
ASM_MESON_TAC[AZIM_EQ_0_SYM; AZIM_EQ_0_GE]);;
let AZIM_EQ_PI = prove
(`!v0 v1 w x.
~collinear{v0,v1,w} /\ ~collinear{v0,v1,x}
==> (azim v0 v1 w x = pi <=> w IN aff_lt {v0,v1} {x})`,
GEOM_ORIGIN_TAC `v0:real^3` THEN
GEOM_BASIS_MULTIPLE_TAC 3 `v1:real^3` THEN
GEN_TAC THEN
GEN_REWRITE_TAC LAND_CONV [REAL_ARITH `&0 <= x <=> x = &0 \/ &0 < x`] THEN
STRIP_TAC THENL
[ASM_REWRITE_TAC[VECTOR_MUL_LZERO; INSERT_AC; COLLINEAR_2]; ALL_TAC] THEN
ASM_SIMP_TAC[AZIM_SPECIAL_SCALE; REAL_LT_IMP_NZ;
COLLINEAR_SPECIAL_SCALE] THEN
REPEAT STRIP_TAC THEN
W(MP_TAC o PART_MATCH (lhs o rand) AFF_LT_SPECIAL_SCALE o
rand o rand o snd) THEN
ANTS_TAC THENL
[ASM_REWRITE_TAC[IN_INSERT; FINITE_INSERT; FINITE_EMPTY; NOT_IN_EMPTY] THEN
REPEAT CONJ_TAC THEN DISCH_THEN(SUBST_ALL_TAC o SYM) THEN
TRY(RULE_ASSUM_TAC(REWRITE_RULE[INSERT_AC; COLLINEAR_2]) THEN
FIRST_X_ASSUM CONTR_TAC) THEN
UNDISCH_TAC `~collinear {vec 0:real^3, basis 3, v1 % basis 3}` THEN
REWRITE_TAC[COLLINEAR_LEMMA] THEN MESON_TAC[];
DISCH_THEN SUBST1_TAC] THEN
REWRITE_TAC[AZIM_ARG] THEN CONV_TAC(LAND_CONV SYM_CONV) THEN
CONV_TAC(LAND_CONV SYM_CONV) THEN REWRITE_TAC[ARG_EQ_PI] THEN
MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC
`(dropout 3 (w:real^3)) IN aff_lt {vec 0:real^2} {dropout 3 (x:real^3)}` THEN
CONJ_TAC THENL
[REPEAT(POP_ASSUM MP_TAC) THEN REWRITE_TAC[COLLINEAR_BASIS_3] THEN
SPEC_TAC(`(dropout 3:real^3->real^2) x`,`y:complex`) THEN
SPEC_TAC(`(dropout 3:real^3->real^2) w`,`v:complex`) THEN
GEOM_BASIS_MULTIPLE_TAC 1 `v:complex` THEN
X_GEN_TAC `v:real` THEN GEN_REWRITE_TAC LAND_CONV [REAL_LE_LT] THEN
ASM_CASES_TAC `v = &0` THEN ASM_REWRITE_TAC[VECTOR_MUL_LZERO] THEN
REWRITE_TAC[COMPLEX_CMUL; COMPLEX_BASIS; COMPLEX_VEC_0] THEN
SIMP_TAC[ARG_DIV_CX; COMPLEX_MUL_RID] THEN
REWRITE_TAC[real; RE_DIV_CX; IM_DIV_CX; CX_INJ] THEN
ASM_SIMP_TAC[REAL_LT_LDIV_EQ; REAL_EQ_LDIV_EQ; REAL_MUL_LZERO] THEN
REPEAT STRIP_TAC THEN
W(MP_TAC o PART_MATCH (lhs o rand) AFF_LT_1_1 o rand o rand o snd) THEN
ASM_REWRITE_TAC[DISJOINT_INSERT; DISJOINT_EMPTY; IN_SING] THEN
DISCH_THEN SUBST1_TAC THEN
REWRITE_TAC[COMPLEX_CMUL; IN_ELIM_THM; COMPLEX_MUL_RZERO] THEN
ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN
REWRITE_TAC[REAL_ARITH `t1 + t2 = &1 <=> t1 = &1 - t2`] THEN
REWRITE_TAC[RIGHT_EXISTS_AND_THM; UNWIND_THM2; COMPLEX_ADD_LID] THEN
EQ_TAC THENL
[REWRITE_TAC[GSYM real; REAL] THEN
DISCH_THEN(CONJUNCTS_THEN2 (SUBST1_TAC o SYM) ASSUME_TAC) THEN
EXISTS_TAC `v / Re y` THEN REWRITE_TAC[GSYM CX_MUL; CX_INJ] THEN
CONJ_TAC THENL
[ALL_TAC; REPEAT(POP_ASSUM MP_TAC) THEN CONV_TAC REAL_FIELD];
DISCH_THEN(X_CHOOSE_THEN `t:real`
(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
ASM_SIMP_TAC[CX_INJ; REAL_ARITH `x < &0 ==> ~(x = &0)`; COMPLEX_FIELD
`~(t = Cx(&0)) ==> (v = t * y <=> y = v / t)`] THEN
DISCH_THEN SUBST1_TAC THEN REWRITE_TAC[GSYM CX_DIV] THEN
REWRITE_TAC[RE_CX; IM_CX]] THEN
REWRITE_TAC[REAL_ARITH `x < &0 <=> &0 < --x`] THEN
REWRITE_TAC[real_div; GSYM REAL_MUL_RNEG; GSYM REAL_INV_NEG] THEN
MATCH_MP_TAC REAL_LT_MUL THEN ASM_REWRITE_TAC[REAL_LT_INV_EQ] THEN
ASM_REAL_ARITH_TAC;
W(MP_TAC o PART_MATCH (lhs o rand) AFF_LT_2_1 o rand o rand o snd) THEN
ANTS_TAC THENL
[REWRITE_TAC[SET_RULE `DISJOINT {a,b} {x} <=> ~(x = a) /\ ~(x = b)`] THEN
CONJ_TAC THEN DISCH_THEN SUBST_ALL_TAC THEN
RULE_ASSUM_TAC(REWRITE_RULE[COLLINEAR_2; INSERT_AC]) THEN
FIRST_ASSUM CONTR_TAC;
DISCH_THEN SUBST1_TAC] THEN
W(MP_TAC o PART_MATCH (lhs o rand) AFF_LT_1_1 o rand o lhand o snd) THEN
ANTS_TAC THENL
[REWRITE_TAC[SET_RULE `DISJOINT {a} {x} <=> ~(x = a)`] THEN
ASM_MESON_TAC[COLLINEAR_BASIS_3];
DISCH_THEN SUBST1_TAC] THEN
REWRITE_TAC[VECTOR_MUL_RZERO; VECTOR_ADD_LID; IN_ELIM_THM] THEN
ONCE_REWRITE_TAC[REAL_ARITH `s + t = &1 <=> s = &1- t`] THEN
ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN
REWRITE_TAC[RIGHT_EXISTS_AND_THM; UNWIND_THM2] THEN
GEN_REWRITE_TAC (RAND_CONV o BINDER_CONV) [SWAP_EXISTS_THM] THEN
REWRITE_TAC[RIGHT_EXISTS_AND_THM; UNWIND_THM2] THEN
ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN AP_TERM_TAC THEN
REWRITE_TAC[FUN_EQ_THM; RIGHT_EXISTS_AND_THM] THEN X_GEN_TAC `t:real` THEN
AP_TERM_TAC THEN
SIMP_TAC[CART_EQ; FORALL_2; FORALL_3; DIMINDEX_2; DIMINDEX_3;
dropout; LAMBDA_BETA; BASIS_COMPONENT; ARITH; REAL_MUL_RID;
VECTOR_MUL_COMPONENT; VEC_COMPONENT; REAL_MUL_RZERO; UNWIND_THM1;
VECTOR_ADD_COMPONENT; REAL_ADD_LID; RIGHT_EXISTS_AND_THM] THEN
REWRITE_TAC[REAL_ARITH `x:real = t + y <=> t = x - y`] THEN
REWRITE_TAC[EXISTS_REFL]]);;
let AZIM_EQ_PI_ALT = prove
(`!v0 v1 w x.
~collinear{v0,v1,w} /\ ~collinear{v0,v1,x}
==> (azim v0 v1 w x = pi <=> x IN aff_lt {v0,v1} {w})`,
REPEAT GEN_TAC THEN DISCH_TAC THEN
FIRST_ASSUM(SUBST1_TAC o MATCH_MP AZIM_EQ_PI_SYM) THEN
ASM_SIMP_TAC[AZIM_EQ_PI]);;
let AZIM_EQ_0_PI_IMP_COPLANAR = prove
(`!v0 v1 w1 w2.
azim v0 v1 w1 w2 = &0 \/ azim v0 v1 w1 w2 = pi
==> coplanar {v0,v1,w1,w2}`,
REPEAT GEN_TAC THEN ASM_CASES_TAC `collinear {v0:real^3,v1,w1}` THENL
[MP_TAC(ISPECL [`v0:real^3`; `v1:real^3`; `w1:real^3`; `w2:real^3`]
NOT_COPLANAR_NOT_COLLINEAR) THEN
ASM_REWRITE_TAC[] THEN CONV_TAC TAUT;
POP_ASSUM MP_TAC] THEN
ASM_CASES_TAC `collinear {v0:real^3,v1,w2}` THENL
[MP_TAC(ISPECL [`v0:real^3`; `v1:real^3`; `w2:real^3`; `w1:real^3`]
NOT_COPLANAR_NOT_COLLINEAR) THEN
ASM_REWRITE_TAC[] THEN REWRITE_TAC[INSERT_AC] THEN CONV_TAC TAUT;
POP_ASSUM MP_TAC] THEN
MAP_EVERY (fun t -> SPEC_TAC(t,t))
[`w2:real^3`; `w1:real^3`; `v1:real^3`; `v0:real^3`] THEN
GEOM_ORIGIN_TAC `v0:real^3` THEN
GEOM_BASIS_MULTIPLE_TAC 3 `v1:real^3` THEN
X_GEN_TAC `w:real` THEN GEN_REWRITE_TAC LAND_CONV [REAL_LE_LT] THEN
ASM_CASES_TAC `w = &0` THEN ASM_REWRITE_TAC[] THENL
[ASM_REWRITE_TAC[VECTOR_MUL_LZERO; INSERT_AC; COLLINEAR_2]; ALL_TAC] THEN
SIMP_TAC[AZIM_SPECIAL_SCALE] THEN
ASM_SIMP_TAC[AZIM_ARG; COLLINEAR_SPECIAL_SCALE] THEN
REWRITE_TAC[COLLINEAR_BASIS_3; ARG_EQ_0_PI] THEN
REWRITE_TAC[real; IM_COMPLEX_DIV_EQ_0] THEN
REWRITE_TAC[complex_mul; cnj; IM; RE] THEN
REWRITE_TAC[REAL_ARITH `x * --y + a * b = &0 <=> x * y = a * b`] THEN
REWRITE_TAC[RE_DEF; IM_DEF] THEN DISCH_TAC THEN REPEAT GEN_TAC THEN
DISCH_TAC THEN DISCH_TAC THEN
SIMP_TAC[dropout; LAMBDA_BETA; DIMINDEX_3; ARITH; DIMINDEX_2] THEN
DISCH_TAC THEN REWRITE_TAC[coplanar] THEN
MAP_EVERY EXISTS_TAC [`vec 0:real^3`; `w % basis 3:real^3`; `w1:real^3`] THEN
ONCE_REWRITE_TAC[SET_RULE `{a,b,c,d} = d INSERT {a,b,c}`] THEN
ONCE_REWRITE_TAC[INSERT_SUBSET] THEN REWRITE_TAC[HULL_SUBSET] THEN
SIMP_TAC[AFFINE_HULL_EQ_SPAN; IN_INSERT; HULL_INC] THEN
REWRITE_TAC[SPAN_BREAKDOWN_EQ; SPAN_EMPTY; IN_SING] THEN
REWRITE_TAC[VECTOR_MUL_RZERO; VECTOR_SUB_RZERO] THEN
REPEAT(POP_ASSUM MP_TAC) THEN
SIMP_TAC[CART_EQ; DIMINDEX_2; FORALL_2; FORALL_3; dropout; LAMBDA_BETA;
DIMINDEX_2; DIMINDEX_3; ARITH; VEC_COMPONENT; ARITH;
VECTOR_SUB_COMPONENT; VECTOR_MUL_COMPONENT; BASIS_COMPONENT] THEN
REPEAT STRIP_TAC THEN
REWRITE_TAC[REAL_MUL_RZERO; REAL_SUB_RZERO] THEN
ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN
REWRITE_TAC[RIGHT_EXISTS_AND_THM] THEN
ASM_SIMP_TAC[EXISTS_REFL; REAL_FIELD
`&0 < w ==> (x - k * w * &1 - y = &0 <=> k = (x - y) / w)`] THEN
SUBGOAL_THEN `~((w1:real^3)$2 = &0) \/ ~((w2:real^3)$1 = &0)`
STRIP_ASSUME_TAC THENL
[REPEAT(POP_ASSUM MP_TAC) THEN CONV_TAC REAL_RING;
EXISTS_TAC `(w2:real^3)$2 / (w1:real^3)$2` THEN
REPEAT(POP_ASSUM MP_TAC) THEN CONV_TAC REAL_FIELD;
EXISTS_TAC `(w2:real^3)$1 / (w1:real^3)$1` THEN
REPEAT(POP_ASSUM MP_TAC) THEN CONV_TAC REAL_FIELD]);;
let AZIM_SAME_WITHIN_AFF_GE = prove
(`!a u v w z.
v IN aff_ge {a} {u,w} /\ ~collinear{a,u,v} /\ ~collinear{a,u,w}
==> azim a u v z = azim a u w z`,
GEOM_ORIGIN_TAC `a:real^3` THEN
GEOM_BASIS_MULTIPLE_TAC 3 `u:real^3` THEN
X_GEN_TAC `u:real` THEN ASM_CASES_TAC `u = &0` THEN
ASM_SIMP_TAC[AZIM_DEGENERATE; VECTOR_MUL_LZERO; REAL_LE_LT] THEN
ASM_SIMP_TAC[AZIM_SPECIAL_SCALE; COLLINEAR_SPECIAL_SCALE] THEN
DISCH_TAC THEN REPEAT GEN_TAC THEN
ASM_CASES_TAC `w:real^3 = vec 0` THENL
[ASM_REWRITE_TAC[COLLINEAR_2; INSERT_AC]; ALL_TAC] THEN
ASM_SIMP_TAC[AFF_GE_SCALE_LEMMA] THEN
REWRITE_TAC[COLLINEAR_BASIS_3; AZIM_ARG] THEN
ASM_SIMP_TAC[AFF_GE_1_2_0; BASIS_NONZERO; ARITH; DIMINDEX_3;
SET_RULE `DISJOINT {a} {b,c} <=> ~(b = a) /\ ~(c = a)`] THEN
REWRITE_TAC[IMP_CONJ; LEFT_IMP_EXISTS_THM; IN_ELIM_THM] THEN
MAP_EVERY X_GEN_TAC [`a:real`; `b:real`] THEN DISCH_TAC THEN DISCH_TAC THEN
DISCH_THEN(MP_TAC o AP_TERM `dropout 3:real^3->real^2`) THEN
REWRITE_TAC[DROPOUT_ADD; DROPOUT_MUL; DROPOUT_BASIS_3] THEN
REWRITE_TAC[VECTOR_MUL_RZERO; VECTOR_ADD_LID] THEN
DISCH_THEN SUBST1_TAC THEN REPEAT DISCH_TAC THEN
REWRITE_TAC[COMPLEX_CMUL] THEN
REWRITE_TAC[complex_div; COMPLEX_INV_MUL; GSYM CX_INV] THEN
ONCE_REWRITE_TAC[COMPLEX_RING `a * b * c:complex = b * a * c`] THEN
MATCH_MP_TAC ARG_MUL_CX THEN REWRITE_TAC[REAL_LT_INV_EQ] THEN
ASM_REWRITE_TAC[REAL_LT_LE] THEN ASM_MESON_TAC[VECTOR_MUL_LZERO]);;
let AZIM_SAME_WITHIN_AFF_GE_ALT = prove
(`!a u v w z.
v IN aff_ge {a} {u,w} /\ ~collinear{a,u,v} /\ ~collinear{a,u,w}
==> azim a u z v = azim a u z w`,
REPEAT GEN_TAC THEN DISCH_TAC THEN
FIRST_ASSUM(ASSUME_TAC o MATCH_MP AZIM_SAME_WITHIN_AFF_GE) THEN
ASM_CASES_TAC `collinear {a:real^3,u,z}` THEN
ASM_SIMP_TAC[AZIM_DEGENERATE] THEN
W(MP_TAC o PART_MATCH (lhs o rand) AZIM_COMPL o lhand o snd) THEN
ASM_REWRITE_TAC[] THEN DISCH_THEN SUBST1_TAC THEN
W(MP_TAC o PART_MATCH (lhs o rand) AZIM_COMPL o rand o snd) THEN
ASM_SIMP_TAC[]);;
let COLLINEAR_WITHIN_AFF_GE_COLLINEAR = prove
(`!a u v w:real^N.
v IN aff_ge {a} {u,w} /\ collinear{a,u,w} ==> collinear{a,v,w}`,
GEOM_ORIGIN_TAC `a:real^N` THEN REPEAT GEN_TAC THEN
ASM_CASES_TAC `w:real^N = vec 0` THENL
[ASM_REWRITE_TAC[COLLINEAR_2; INSERT_AC]; ALL_TAC] THEN
ASM_CASES_TAC `u:real^N = vec 0` THENL
[ONCE_REWRITE_TAC[AFF_GE_DISJOINT_DIFF] THEN
ASM_REWRITE_TAC[SET_RULE `{a} DIFF {a,b} = {}`] THEN
REWRITE_TAC[GSYM CONVEX_HULL_AFF_GE] THEN
ONCE_REWRITE_TAC[SET_RULE `{z,v,w} = {z,w,v}`] THEN
ASM_SIMP_TAC[COLLINEAR_3_AFFINE_HULL] THEN
MESON_TAC[CONVEX_HULL_SUBSET_AFFINE_HULL; SUBSET];
ONCE_REWRITE_TAC[SET_RULE `{z,v,w} = {z,w,v}`] THEN
ASM_REWRITE_TAC[COLLINEAR_LEMMA_ALT] THEN
DISCH_THEN(CONJUNCTS_THEN2 MP_TAC (X_CHOOSE_TAC `a:real`)) THEN
ASM_SIMP_TAC[AFF_GE_1_2_0; SET_RULE
`DISJOINT {a} {b,c} <=> ~(b = a) /\ ~(c = a)`] THEN
REWRITE_TAC[IN_ELIM_THM; LEFT_IMP_EXISTS_THM] THEN
MAP_EVERY X_GEN_TAC [`b:real`; `c:real`] THEN STRIP_TAC THEN
ASM_REWRITE_TAC[GSYM VECTOR_ADD_RDISTRIB; VECTOR_MUL_ASSOC] THEN
MESON_TAC[]]);;
let AZIM_EQ_IMP = prove
(`!v0 v1 w x y.
~collinear {v0, v1, w} /\
~collinear {v0, v1, y} /\
x IN aff_gt {v0, v1} {y}
==> azim v0 v1 w x = azim v0 v1 w y`,
REPEAT GEN_TAC THEN ASM_CASES_TAC `v1:real^3 = v0` THENL
[ASM_REWRITE_TAC[INSERT_AC; COLLINEAR_2]; ALL_TAC] THEN
ASM_CASES_TAC `collinear {v0:real^3,v1,x}` THENL
[ALL_TAC; ASM_SIMP_TAC[AZIM_EQ_ALT]] THEN
UNDISCH_TAC `collinear {v0:real^3,v1,x}` THEN
MATCH_MP_TAC(TAUT
`(s /\ p ==> r) ==> p ==> ~q /\ ~r /\ s ==> t`) THEN
ASM_SIMP_TAC[COLLINEAR_3_IN_AFFINE_HULL] THEN
ASM_CASES_TAC `y:real^3 = v0` THEN
ASM_SIMP_TAC[HULL_INC; IN_INSERT] THEN
ASM_CASES_TAC `y:real^3 = v1` THEN
ASM_SIMP_TAC[HULL_INC; IN_INSERT] THEN
ASM_SIMP_TAC[AFF_GT_2_1; SET_RULE
`DISJOINT {a,b} {c} <=> ~(c = a) /\ ~(c = b)`] THEN
REWRITE_TAC[AFFINE_HULL_2; IN_ELIM_THM; LEFT_AND_EXISTS_THM] THEN
REWRITE_TAC[RIGHT_AND_EXISTS_THM; LEFT_IMP_EXISTS_THM] THEN
MAP_EVERY X_GEN_TAC
[`t1:real`; `t2:real`; `t3:real`; `s1:real`; `s2:real`] THEN
DISCH_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC MP_TAC) THEN
FIRST_X_ASSUM(MP_TAC o AP_TERM `(%) (inv t3) :real^3->real^3`) THEN
ASM_SIMP_TAC[VECTOR_ADD_LDISTRIB; VECTOR_MUL_ASSOC; REAL_MUL_LINV;
REAL_LT_IMP_NZ; VECTOR_ARITH
`x:real^N = y + z + &1 % w <=> w = x - (y + z)`] THEN
REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
EXISTS_TAC `inv t3 * s1 - inv t3 * t1:real` THEN
EXISTS_TAC `inv t3 * s2 - inv t3 * t2:real` THEN CONJ_TAC THENL
[ASM_SIMP_TAC[REAL_FIELD
`&0 < t ==> (inv t * a - inv t * b + inv t * c - inv t * d = &1 <=>
(a + c) - (b + d) = t)`] THEN
ASM_REAL_ARITH_TAC;
VECTOR_ARITH_TAC]);;
let AZIM_EQ_0_GE_IMP = prove
(`!v0 v1 w x. x IN aff_ge {v0, v1} {w} ==> azim v0 v1 w x = &0`,
REPEAT STRIP_TAC THEN ASM_CASES_TAC `collinear {v0:real^3,v1,w}` THEN
ASM_SIMP_TAC[AZIM_DEGENERATE] THEN
ASM_CASES_TAC `collinear {v0:real^3,v1,x}` THEN
ASM_SIMP_TAC[AZIM_DEGENERATE] THEN ASM_MESON_TAC[AZIM_EQ_0_GE_ALT]);;
let REAL_SGN_SIN_AZIM = prove
(`!v w x y. real_sgn(sin(azim v w x y)) =
real_sgn(((w - v) cross (x - v)) dot (y - v))`,
GEOM_ORIGIN_TAC `v:real^3` THEN REWRITE_TAC[VECTOR_SUB_RZERO] THEN
GEOM_BASIS_MULTIPLE_TAC 3 `w:real^3` THEN
X_GEN_TAC `w:real` THEN ASM_CASES_TAC `w = &0` THEN
ASM_REWRITE_TAC[VECTOR_MUL_LZERO; CROSS_LZERO; DOT_LZERO; REAL_SGN_0;
AZIM_REFL_ALT; SIN_0] THEN
ASM_REWRITE_TAC[REAL_LE_LT] THEN REPEAT STRIP_TAC THEN
ASM_SIMP_TAC[AZIM_SPECIAL_SCALE; CROSS_LMUL; DOT_LMUL] THEN
REWRITE_TAC[REAL_SGN_MUL] THEN
GEN_REWRITE_TAC (RAND_CONV o LAND_CONV) [real_sgn] THEN
ASM_REWRITE_TAC[REAL_MUL_LID; AZIM_ARG] THEN
MATCH_MP_TAC EQ_TRANS THEN
EXISTS_TAC `real_sgn(Im(dropout 3 (y:real^3) / dropout 3 (x:real^3)))` THEN
CONJ_TAC THENL
[ALL_TAC;
REWRITE_TAC[REAL_SGN_IM_COMPLEX_DIV] THEN AP_TERM_TAC THEN
SIMP_TAC[CART_EQ; DIMINDEX_3; FORALL_3; cross; VECTOR_3; DOT_3; dropout;
LAMBDA_BETA; ARITH; cnj; complex_mul; RE_DEF; IM_DEF; DIMINDEX_2;
complex; VECTOR_2; BASIS_COMPONENT] THEN REAL_ARITH_TAC] THEN
SPEC_TAC(`(dropout 3:real^3->real^2) x`,`z:complex`) THEN
SPEC_TAC(`(dropout 3:real^3->real^2) y`,`w:complex`) THEN
POP_ASSUM_LIST(K ALL_TAC) THEN GEOM_BASIS_MULTIPLE_TAC 1 `z:complex` THEN
REWRITE_TAC[COMPLEX_CMUL; COMPLEX_BASIS; COMPLEX_MUL_RID] THEN
X_GEN_TAC `x:real` THEN DISCH_TAC THEN X_GEN_TAC `z:complex` THEN
ASM_CASES_TAC `x = &0` THENL
[ASM_REWRITE_TAC[complex_div; COMPLEX_INV_0; COMPLEX_MUL_RZERO] THEN
REWRITE_TAC[ARG_0; SIN_0; IM_CX; REAL_SGN_0];
SUBGOAL_THEN `&0 < x` ASSUME_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC]] THEN
ASM_SIMP_TAC[ARG_DIV_CX; IM_DIV_CX; REAL_SGN_DIV] THEN
GEN_REWRITE_TAC (RAND_CONV o RAND_CONV) [real_sgn] THEN
ASM_REWRITE_TAC[REAL_DIV_1] THEN ASM_CASES_TAC `z = Cx(&0)` THEN
ASM_REWRITE_TAC[IM_CX; ARG_0; SIN_0] THEN
GEN_REWRITE_TAC (funpow 3 RAND_CONV) [ARG] THEN
REWRITE_TAC[IM_MUL_CX; REAL_SGN_MUL] THEN
GEN_REWRITE_TAC (RAND_CONV o LAND_CONV) [real_sgn] THEN
ASM_REWRITE_TAC[COMPLEX_NORM_NZ; REAL_MUL_LID] THEN
REWRITE_TAC[IM_CEXP; RE_MUL_II; IM_MUL_II; RE_CX; REAL_SGN_MUL] THEN
GEN_REWRITE_TAC (RAND_CONV o LAND_CONV) [real_sgn] THEN
REWRITE_TAC[REAL_EXP_POS_LT; REAL_MUL_LID]);;
let AZIM_IN_UPPER_HALFSPACE = prove
(`!v w x y. azim v w x y <= pi <=>
&0 <= ((w - v) cross (x - v)) dot (y - v)`,
REPEAT STRIP_TAC THEN MATCH_MP_TAC EQ_TRANS THEN
EXISTS_TAC `&0 <= sin(azim v w x y)` THEN CONJ_TAC THENL
[EQ_TAC THEN SIMP_TAC[SIN_POS_PI_LE; azim] THEN
MP_TAC(ISPEC `azim v w x y - pi` SIN_POS_PI) THEN
REWRITE_TAC[SIN_SUB; SIN_PI; COS_PI; azim;
REAL_ARITH `x - pi < pi <=> x < &2 * pi`] THEN
REAL_ARITH_TAC;
ONCE_REWRITE_TAC[GSYM REAL_SGN_INEQS] THEN
REWRITE_TAC[REAL_SGN_SIN_AZIM]]);;
(* ------------------------------------------------------------------------- *)
(* Dihedral angle and relation to azimuth angle. *)
(* ------------------------------------------------------------------------- *)
let dihV = new_definition
`dihV w0 w1 w2 w3 =
let va = w2 - w0 in
let vb = w3 - w0 in
let vc = w1 - w0 in
let vap = ( vc dot vc) % va - ( va dot vc) % vc in
let vbp = ( vc dot vc) % vb - ( vb dot vc) % vc in
arcV (vec 0) vap vbp`;;
let DIHV = prove
(`dihV (w0:real^N) w1 w2 w3 =
let va = w2 - w0 in
let vb = w3 - w0 in
let vc = w1 - w0 in
let vap = (vc dot vc) % va - (va dot vc) % vc in
let vbp = (vc dot vc) % vb - (vb dot vc) % vc in
angle(vap,vec 0,vbp)`,
REWRITE_TAC[dihV; ARCV_ANGLE]);;
let DIHV_TRANSLATION_EQ = prove
(`!a w0 w1 w2 w3:real^N.
dihV (a + w0) (a + w1) (a + w2) (a + w3) = dihV w0 w1 w2 w3`,
REWRITE_TAC[DIHV; VECTOR_ARITH `(a + x) - (a + y):real^N = x - y`]);;
add_translation_invariants [DIHV_TRANSLATION_EQ];;
let DIHV_LINEAR_IMAGE = prove
(`!f:real^M->real^N w0 w1 w2 w3.
linear f /\ (!x. norm(f x) = norm x)
==> dihV (f w0) (f w1) (f w2) (f w3) = dihV w0 w1 w2 w3`,
REPEAT STRIP_TAC THEN REWRITE_TAC[DIHV] THEN
ASM_SIMP_TAC[GSYM LINEAR_SUB] THEN
CONV_TAC(TOP_DEPTH_CONV let_CONV) THEN
ASM_SIMP_TAC[PRESERVES_NORM_PRESERVES_DOT] THEN
ASM_SIMP_TAC[GSYM LINEAR_CMUL; GSYM LINEAR_SUB] THEN
REWRITE_TAC[angle; VECTOR_SUB_RZERO] THEN
ASM_SIMP_TAC[VECTOR_ANGLE_LINEAR_IMAGE_EQ]);;
add_linear_invariants [DIHV_LINEAR_IMAGE];;
let DIHV_SPECIAL_SCALE = prove
(`!a v w1 w2:real^N.
~(a = &0)
==> dihV (vec 0) (a % v) w1 w2 = dihV (vec 0) v w1 w2`,
REPEAT STRIP_TAC THEN REWRITE_TAC[DIHV; VECTOR_SUB_RZERO] THEN
CONV_TAC(TOP_DEPTH_CONV let_CONV) THEN
REWRITE_TAC[DOT_LMUL; DOT_RMUL; GSYM VECTOR_MUL_ASSOC] THEN
REWRITE_TAC[VECTOR_ARITH `a % a % x - a % b % a % y:real^N =
(a * a) % (x - b % y)`] THEN
REWRITE_TAC[angle; VECTOR_SUB_RZERO] THEN
REWRITE_TAC[VECTOR_ANGLE_LMUL; VECTOR_ANGLE_RMUL] THEN
ASM_REWRITE_TAC[REAL_LE_SQUARE; REAL_ENTIRE]);;
let DIHV_RANGE = prove
(`!w0 w1 w2 w3. &0 <= dihV w0 w1 w2 w3 /\ dihV w0 w1 w2 w3 <= pi`,
REWRITE_TAC[DIHV] THEN CONV_TAC(TOP_DEPTH_CONV let_CONV) THEN
REWRITE_TAC[ANGLE_RANGE]);;
let COS_AZIM_DIHV = prove
(`!v w v1 v2:real^3.
~collinear {v,w,v1} /\ ~collinear {v,w,v2}
==> cos(azim v w v1 v2) = cos(dihV v w v1 v2)`,
REPEAT GEN_TAC THEN ASM_CASES_TAC `w:real^3 = v` THENL
[ASM_REWRITE_TAC[INSERT_AC; COLLINEAR_2]; POP_ASSUM MP_TAC] THEN
REWRITE_TAC[IMP_IMP; GSYM CONJ_ASSOC] THEN
GEOM_ORIGIN_TAC `v:real^3` THEN GEOM_BASIS_MULTIPLE_TAC 3 `w:real^3` THEN
X_GEN_TAC `w:real` THEN GEN_REWRITE_TAC LAND_CONV [REAL_LE_LT] THEN
ASM_CASES_TAC `w = &0` THEN ASM_REWRITE_TAC[VECTOR_MUL_LZERO] THEN
ASM_SIMP_TAC[AZIM_SPECIAL_SCALE; DIHV_SPECIAL_SCALE; REAL_LT_IMP_NZ;
COLLINEAR_SPECIAL_SCALE; COLLINEAR_BASIS_3] THEN
DISCH_TAC THEN POP_ASSUM_LIST(K ALL_TAC) THEN
MAP_EVERY X_GEN_TAC [`w1:real^3`; `w2:real^3`] THEN
DISCH_THEN(STRIP_ASSUME_TAC o CONJUNCT2) THEN
REWRITE_TAC[DIHV; VECTOR_SUB_RZERO] THEN
CONV_TAC(TOP_DEPTH_CONV let_CONV) THEN
SIMP_TAC[DOT_BASIS_BASIS; DIMINDEX_3; ARITH] THEN
SIMP_TAC[DOT_BASIS; DIMINDEX_3; ARITH; VECTOR_MUL_LID] THEN
MP_TAC(ISPECL [`vec 0:real^3`; `basis 3:real^3`; `w1:real^3`; `w2:real^3`]
azim) THEN
ABBREV_TAC `a = azim (vec 0) (basis 3) w1 (w2:real^3)` THEN
REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM; VECTOR_SUB_RZERO; DIST_0] THEN
MAP_EVERY X_GEN_TAC [`h1:real`; `h2:real`] THEN
DISCH_THEN(MP_TAC o SPECL
[`basis 1:real^3`; `basis 2:real^3`; `basis 3:real^3`]) THEN
SIMP_TAC[orthonormal; DOT_BASIS_BASIS; CROSS_BASIS; DIMINDEX_3; NORM_BASIS;
ARITH; VECTOR_MUL_LID; BASIS_NONZERO; REAL_LT_01; LEFT_IMP_EXISTS_THM] THEN
ASM_REWRITE_TAC[COLLINEAR_BASIS_3] THEN
MAP_EVERY X_GEN_TAC [`psi:real`; `r1:real`; `r2:real`] THEN
DISCH_THEN(STRIP_ASSUME_TAC o GSYM) THEN
REWRITE_TAC[VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT] THEN
SIMP_TAC[BASIS_COMPONENT; DIMINDEX_3; ARITH; REAL_MUL_RZERO] THEN
REWRITE_TAC[REAL_MUL_RID; REAL_ADD_LID] THEN
REWRITE_TAC[VECTOR_ARITH `(a + b + c) - c:real^N = a + b`] THEN
REWRITE_TAC[COS_ANGLE; VECTOR_SUB_RZERO] THEN
REWRITE_TAC[vector_norm; GSYM DOT_EQ_0; DIMINDEX_3; FORALL_3; DOT_3] THEN
REWRITE_TAC[VEC_COMPONENT; VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT] THEN
SIMP_TAC[BASIS_COMPONENT; DIMINDEX_3; ARITH; REAL_MUL_RZERO] THEN
REWRITE_TAC[REAL_MUL_RID; REAL_ADD_LID; REAL_ADD_RID; REAL_MUL_RZERO] THEN
REWRITE_TAC[REAL_ARITH `(r * c) * (r * c) + (r * s) * (r * s):real =
r pow 2 * (s pow 2 + c pow 2)`] THEN
ASM_SIMP_TAC[SIN_CIRCLE; REAL_MUL_RID; REAL_POW_EQ_0; REAL_LT_IMP_NZ] THEN
ASM_SIMP_TAC[POW_2_SQRT; REAL_LT_IMP_LE] THEN
REWRITE_TAC[REAL_ARITH `(r1 * c1) * (r2 * c2) + (r1 * s1) * (r2 * s2):real =
(r1 * r2) * (c1 * c2 + s1 * s2)`] THEN
ASM_SIMP_TAC[REAL_FIELD
`&0 < r1 /\ &0 < r2 ==> ((r1 * r2) * x) / (r1 * r2) = x`] THEN
ONCE_REWRITE_TAC[REAL_ARITH `a:real = b + c * d <=> b - --c * d = a`] THEN
GEN_REWRITE_TAC (funpow 3 LAND_CONV) [GSYM COS_NEG] THEN
REWRITE_TAC[GSYM SIN_NEG; GSYM COS_ADD] THEN AP_TERM_TAC THEN
REAL_ARITH_TAC);;
let AZIM_DIHV_SAME = prove
(`!v w v1 v2:real^3.
~collinear {v,w,v1} /\ ~collinear {v,w,v2} /\
azim v w v1 v2 < pi
==> azim v w v1 v2 = dihV v w v1 v2`,
REPEAT STRIP_TAC THEN MATCH_MP_TAC COS_INJ_PI THEN
ASM_SIMP_TAC[COS_AZIM_DIHV; azim; REAL_LT_IMP_LE; DIHV_RANGE]);;
let AZIM_DIHV_COMPL = prove
(`!v w v1 v2:real^3.
~collinear {v,w,v1} /\ ~collinear {v,w,v2} /\
pi <= azim v w v1 v2
==> azim v w v1 v2 = &2 * pi - dihV v w v1 v2`,
REPEAT STRIP_TAC THEN
ONCE_REWRITE_TAC[REAL_ARITH `x = &2 * pi - y <=> y = &2 * pi - x`] THEN
MATCH_MP_TAC COS_INJ_PI THEN
REWRITE_TAC[COS_SUB; SIN_NPI; COS_NPI; REAL_MUL_LZERO] THEN
CONV_TAC REAL_RAT_REDUCE_CONV THEN
ASM_SIMP_TAC[COS_AZIM_DIHV; REAL_ADD_RID; REAL_MUL_LID] THEN
ASM_REWRITE_TAC[DIHV_RANGE] THEN MATCH_MP_TAC(REAL_ARITH
`p <= x /\ x < &2 * p ==> &0 <= &2 * p - x /\ &2 * p - x <= p`) THEN
ASM_SIMP_TAC[azim]);;
let AZIM_DIVH = prove
(`!v w v1 v2:real^3.
~collinear {v,w,v1} /\ ~collinear {v,w,v2}
==> azim v w v1 v2 = if azim v w v1 v2 < pi then dihV v w v1 v2
else &2 * pi - dihV v w v1 v2`,
REPEAT STRIP_TAC THEN COND_CASES_TAC THEN
RULE_ASSUM_TAC(REWRITE_RULE[REAL_NOT_LT]) THEN
ASM_SIMP_TAC[AZIM_DIHV_SAME; AZIM_DIHV_COMPL]);;
let AZIM_DIHV_EQ_0 = prove
(`!v0 v1 w1 w2.
~collinear {v0,v1,w1} /\ ~collinear {v0,v1,w2}
==> (azim v0 v1 w1 w2 = &0 <=> dihV v0 v1 w1 w2 = &0)`,
REPEAT STRIP_TAC THEN
W(MP_TAC o PART_MATCH (lhs o rand) AZIM_DIVH o lhs o lhs o snd) THEN
ASM_REWRITE_TAC[] THEN COND_CASES_TAC THEN ASM_SIMP_TAC[] THEN
ONCE_REWRITE_TAC[REAL_ARITH `a:real = p - b <=> b = p - a`] THEN
DISCH_THEN SUBST1_TAC THEN
REWRITE_TAC[REAL_ARITH `&2 * p - (&2 * p - a) = &0 <=> a = &0`] THEN
MATCH_MP_TAC(REAL_ARITH
`a < &2 * pi /\ ~(a < pi) ==> (a = &0 <=> &2 * pi - a = &0)`) THEN
ASM_REWRITE_TAC[azim]);;
let AZIM_DIHV_EQ_PI = prove
(`!v0 v1 w1 w2.
~collinear {v0,v1,w1} /\ ~collinear {v0,v1,w2}
==> (azim v0 v1 w1 w2 = pi <=> dihV v0 v1 w1 w2 = pi)`,
REPEAT STRIP_TAC THEN
W(MP_TAC o PART_MATCH (lhs o rand) AZIM_DIVH o lhs o lhs o snd) THEN
ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC);;
let AZIM_EQ_0_PI_EQ_COPLANAR = prove
(`!v0 v1 w1 w2.
~collinear {v0,v1,w1} /\ ~collinear {v0,v1,w2}
==> (azim v0 v1 w1 w2 = &0 \/ azim v0 v1 w1 w2 = pi <=>
coplanar {v0,v1,w1,w2})`,
REWRITE_TAC[TAUT `(a <=> b) <=> (a ==> b) /\ (b ==> a)`] THEN
REWRITE_TAC[AZIM_EQ_0_PI_IMP_COPLANAR] THEN
SIMP_TAC[GSYM IMP_CONJ_ALT; COPLANAR; DIMINDEX_3; ARITH] THEN
REWRITE_TAC[IMP_CONJ; LEFT_IMP_EXISTS_THM] THEN
MAP_EVERY X_GEN_TAC
[`v0:real^3`; `v1:real^3`; `v2:real^3`; `v3:real^3`; `p:real^3->bool`] THEN
GEOM_HORIZONTAL_PLANE_TAC `p:real^3->bool` THEN
REWRITE_TAC[INSERT_SUBSET; IN_ELIM_THM; IMP_CONJ; RIGHT_FORALL_IMP_THM;
EMPTY_SUBSET] THEN
SIMP_TAC[AZIM_DIHV_EQ_0; AZIM_DIHV_EQ_PI] THEN
REWRITE_TAC[DIHV] THEN
CONV_TAC(TOP_DEPTH_CONV let_CONV) THEN
DISCH_THEN(K ALL_TAC) THEN PAD2D3D_TAC THEN
REWRITE_TAC[angle; VECTOR_SUB_RZERO] THEN
GEOM_ORIGIN_TAC `v0:real^2` THEN REWRITE_TAC[VECTOR_SUB_RZERO] THEN
REPEAT STRIP_TAC THEN
W(MP_TAC o PART_MATCH (rand o rand) COLLINEAR_VECTOR_ANGLE o snd) THEN
ANTS_TAC THENL
[REPEAT(POP_ASSUM MP_TAC); DISCH_THEN(SUBST1_TAC o SYM)] THEN
REWRITE_TAC[GSYM DOT_CAUCHY_SCHWARZ_EQUAL] THEN
REWRITE_TAC[DOT_2; CART_EQ; FORALL_2; DIMINDEX_2; VEC_COMPONENT;
VECTOR_SUB_COMPONENT; VECTOR_MUL_COMPONENT] THEN
CONV_TAC REAL_RING);;
let DIHV_EQ_0_PI_EQ_COPLANAR = prove
(`!v0 v1 w1 w2:real^3.
~collinear {v0,v1,w1} /\ ~collinear {v0,v1,w2}
==> (dihV v0 v1 w1 w2 = &0 \/ dihV v0 v1 w1 w2 = pi <=>
coplanar {v0,v1,w1,w2})`,
SIMP_TAC[GSYM AZIM_DIHV_EQ_0; GSYM AZIM_DIHV_EQ_PI;
AZIM_EQ_0_PI_EQ_COPLANAR]);;
let DIHV_SYM = prove
(`!v0 v1 v2 v3:real^N.
dihV v0 v1 v3 v2 = dihV v0 v1 v2 v3`,
REPEAT GEN_TAC THEN REWRITE_TAC[DIHV] THEN
CONV_TAC(TOP_DEPTH_CONV let_CONV) THEN
REWRITE_TAC[DOT_SYM; ANGLE_SYM]);;
let DIHV_NEG = prove
(`!v0 v1 v2 v3. dihV (--v0) (--v1) (--v2) (--v3) = dihV v0 v1 v2 v3`,
REWRITE_TAC[DIHV; VECTOR_ARITH `--a - --b:real^N = --(a - b)`] THEN
CONV_TAC(TOP_DEPTH_CONV let_CONV) THEN
REWRITE_TAC[DOT_RNEG; DOT_LNEG; REAL_NEG_NEG] THEN
REWRITE_TAC[VECTOR_MUL_RNEG] THEN
REWRITE_TAC[angle; VECTOR_ARITH `--a - --b:real^N = --(a - b)`] THEN
REWRITE_TAC[VECTOR_SUB_RZERO; VECTOR_ANGLE_NEG2]);;
let DIHV_NEG_0 = prove
(`!v1 v2 v3. dihV (vec 0) (--v1) (--v2) (--v3) = dihV (vec 0) v1 v2 v3`,
REPEAT GEN_TAC THEN
GEN_REWRITE_TAC RAND_CONV [GSYM DIHV_NEG] THEN
REWRITE_TAC[VECTOR_NEG_0]);;
let DIHV_ARCV = prove
(`!e u v w:real^N.
orthogonal (e - u) (v - u) /\ orthogonal (e - u) (w - u) /\ ~(e = u)
==> dihV u e v w = arcV u v w`,
GEOM_ORIGIN_TAC `u:real^N` THEN
REWRITE_TAC[dihV; orthogonal; VECTOR_SUB_RZERO] THEN
CONV_TAC(TOP_DEPTH_CONV let_CONV) THEN
SIMP_TAC[DOT_SYM; VECTOR_MUL_LZERO; VECTOR_SUB_RZERO] THEN
REWRITE_TAC[ARCV_ANGLE; angle; VECTOR_SUB_RZERO] THEN
REWRITE_TAC[VECTOR_ANGLE_LMUL; VECTOR_ANGLE_RMUL] THEN
SIMP_TAC[DOT_POS_LE; DOT_EQ_0]);;
let AZIM_DIHV_SAME_STRONG = prove
(`!v w v1 v2:real^3.
~collinear {v,w,v1} /\ ~collinear {v,w,v2} /\
azim v w v1 v2 <= pi
==> azim v w v1 v2 = dihV v w v1 v2`,
REWRITE_TAC[REAL_LE_LT] THEN
MESON_TAC[AZIM_DIHV_SAME; AZIM_DIHV_EQ_PI]);;
let AZIM_ARCV = prove
(`!e u v w:real^3.
orthogonal (e - u) (v - u) /\ orthogonal (e - u) (w - u) /\
~collinear{u,e,v} /\ ~collinear{u,e,w} /\
azim u e v w <= pi
==> azim u e v w = arcV u v w`,
REPEAT GEN_TAC THEN ASM_CASES_TAC `u:real^3 = e` THENL
[ASM_REWRITE_TAC[INSERT_AC; COLLINEAR_2]; ALL_TAC] THEN
STRIP_TAC THEN ASM_SIMP_TAC[GSYM DIHV_ARCV] THEN
MATCH_MP_TAC AZIM_DIHV_SAME_STRONG THEN ASM_REWRITE_TAC[]);;
let COLLINEAR_AZIM_0_OR_PI = prove
(`!u e v w. collinear {u,v,w} ==> azim u e v w = &0 \/ azim u e v w = pi`,
REPEAT STRIP_TAC THEN
ASM_CASES_TAC `collinear{u:real^3,e,v}` THEN
ASM_SIMP_TAC[AZIM_DEGENERATE] THEN
ASM_CASES_TAC `collinear{u:real^3,e,w}` THEN
ASM_SIMP_TAC[AZIM_DEGENERATE] THEN
ASM_SIMP_TAC[AZIM_EQ_0_PI_EQ_COPLANAR] THEN
ONCE_REWRITE_TAC[SET_RULE `{u,e,v,w} = {u,v,w,e}`] THEN
ASM_MESON_TAC[NOT_COPLANAR_NOT_COLLINEAR]);;
let REAL_CONTINUOUS_WITHIN_DIHV_COMPOSE = prove
(`!f:real^M->real^N g h k x s.
~collinear {f x,g x,h x} /\ ~collinear {f x,g x,k x} /\
f continuous (at x within s) /\ g continuous (at x within s) /\
h continuous (at x within s) /\ k continuous (at x within s)
==> (\x. dihV (f x) (g x) (h x) (k x)) real_continuous (at x within s)`,
REPEAT STRIP_TAC THEN REWRITE_TAC[dihV] THEN
CONV_TAC(TOP_DEPTH_CONV let_CONV) THEN
REWRITE_TAC[ARCV_ANGLE; angle; REAL_CONTINUOUS_CONTINUOUS; o_DEF] THEN
REWRITE_TAC[VECTOR_SUB_RZERO] THEN
MATCH_MP_TAC CONTINUOUS_WITHIN_CX_VECTOR_ANGLE_COMPOSE THEN
ASM_REWRITE_TAC[VECTOR_SUB_EQ; GSYM COLLINEAR_3_DOT_MULTIPLES] THEN
CONJ_TAC THEN MATCH_MP_TAC CONTINUOUS_SUB THEN CONJ_TAC THEN
MATCH_MP_TAC CONTINUOUS_MUL THEN REWRITE_TAC[o_DEF] THEN
ASM_SIMP_TAC[CONTINUOUS_LIFT_DOT2; o_DEF; CONTINUOUS_SUB]);;
let REAL_CONTINUOUS_AT_DIHV_COMPOSE = prove
(`!f:real^M->real^N g h k x.
~collinear {f x,g x,h x} /\ ~collinear {f x,g x,k x} /\
f continuous (at x) /\ g continuous (at x) /\
h continuous (at x) /\ k continuous (at x)
==> (\x. dihV (f x) (g x) (h x) (k x)) real_continuous (at x)`,
ONCE_REWRITE_TAC[GSYM WITHIN_UNIV] THEN
REWRITE_TAC[REAL_CONTINUOUS_WITHIN_DIHV_COMPOSE]);;
let REAL_CONTINUOUS_WITHINREAL_DIHV_COMPOSE = prove
(`!f:real->real^N g h k x s.
~collinear {f x,g x,h x} /\ ~collinear {f x,g x,k x} /\
f continuous (atreal x within s) /\ g continuous (atreal x within s) /\
h continuous (atreal x within s) /\ k continuous (atreal x within s)
==> (\x. dihV (f x) (g x) (h x) (k x)) real_continuous
(atreal x within s)`,
REWRITE_TAC[CONTINUOUS_CONTINUOUS_WITHINREAL;
REAL_CONTINUOUS_REAL_CONTINUOUS_WITHINREAL] THEN
SIMP_TAC[o_DEF; REAL_CONTINUOUS_WITHIN_DIHV_COMPOSE; LIFT_DROP]);;
let REAL_CONTINUOUS_ATREAL_DIHV_COMPOSE = prove
(`!f:real->real^N g h k x.
~collinear {f x,g x,h x} /\ ~collinear {f x,g x,k x} /\
f continuous (atreal x) /\ g continuous (atreal x) /\
h continuous (atreal x) /\ k continuous (atreal x)
==> (\x. dihV (f x) (g x) (h x) (k x)) real_continuous (atreal x)`,
ONCE_REWRITE_TAC[GSYM WITHINREAL_UNIV] THEN
REWRITE_TAC[REAL_CONTINUOUS_WITHINREAL_DIHV_COMPOSE]);;
let REAL_CONTINUOUS_AT_DIHV = prove
(`!v w w1 w2:real^N.
~collinear {v, w, w2} ==> dihV v w w1 real_continuous at w2`,
REPEAT STRIP_TAC THEN GEN_REWRITE_TAC LAND_CONV [GSYM ETA_AX] THEN
REWRITE_TAC[dihV] THEN CONV_TAC(TOP_DEPTH_CONV let_CONV) THEN
GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN
MATCH_MP_TAC REAL_CONTINUOUS_CONTINUOUS_AT_COMPOSE THEN CONJ_TAC THENL
[MATCH_MP_TAC CONTINUOUS_SUB THEN CONJ_TAC THEN
MATCH_MP_TAC CONTINUOUS_MUL THEN
SIMP_TAC[CONTINUOUS_CONST; o_DEF; CONTINUOUS_SUB; CONTINUOUS_AT_ID;
CONTINUOUS_LIFT_DOT2];
GEN_REWRITE_TAC LAND_CONV [GSYM ETA_AX] THEN
REWRITE_TAC[ARCV_ANGLE; angle] THEN
REWRITE_TAC[VECTOR_SUB_RZERO; ETA_AX] THEN
MATCH_MP_TAC REAL_CONTINUOUS_WITHIN_VECTOR_ANGLE THEN
POP_ASSUM MP_TAC THEN GEOM_ORIGIN_TAC `v:real^N` THEN
REWRITE_TAC[VECTOR_SUB_RZERO; CONTRAPOS_THM; VECTOR_SUB_EQ] THEN
MAP_EVERY X_GEN_TAC [`z:real^N`; `w:real^N`] THEN
ASM_CASES_TAC `w:real^N = vec 0` THEN
ASM_REWRITE_TAC[COLLINEAR_LEMMA_ALT] THEN DISCH_THEN(MP_TAC o AP_TERM
`(%) (inv((w:real^N) dot w)):real^N->real^N`) THEN
ASM_SIMP_TAC[VECTOR_MUL_ASSOC; REAL_MUL_LINV; DOT_EQ_0] THEN
MESON_TAC[VECTOR_MUL_LID]]);;
let REAL_CONTINUOUS_WITHIN_AZIM_COMPOSE = prove
(`!f:real^M->real^3 g h k x s.
~collinear {f x,g x,h x} /\ ~collinear {f x,g x,k x} /\
~(k x IN aff_ge {f x,g x} {h x}) /\
f continuous (at x within s) /\ g continuous (at x within s) /\
h continuous (at x within s) /\ k continuous (at x within s)
==> (\x. azim (f x) (g x) (h x) (k x)) real_continuous (at x within s)`,
let lemma = prove
(`!s t u f:real^M->real^N g h.
(closed s /\ closed t) /\ s UNION t = UNIV /\
(g continuous_on (u INTER s) /\ h continuous_on (u INTER t)) /\
(!x. x IN u INTER s ==> g x = f x) /\
(!x. x IN u INTER t ==> h x = f x)
==> f continuous_on u`,
REPEAT STRIP_TAC THEN
SUBGOAL_THEN `u:real^M->bool = (u INTER s) UNION (u INTER t)`
SUBST1_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
MATCH_MP_TAC CONTINUOUS_ON_UNION_LOCAL THEN
REWRITE_TAC[CLOSED_IN_CLOSED] THEN REPEAT CONJ_TAC THENL
[EXISTS_TAC `s:real^M->bool` THEN ASM SET_TAC[];
EXISTS_TAC `t:real^M->bool` THEN ASM SET_TAC[];
ASM_MESON_TAC[CONTINUOUS_ON_EQ];
ASM_MESON_TAC[CONTINUOUS_ON_EQ]]) in
REPEAT STRIP_TAC THEN REWRITE_TAC[REAL_CONTINUOUS_CONTINUOUS; o_DEF] THEN
SUBGOAL_THEN
`(\x:real^M. Cx(azim (f x) (g x) (h x) (k x))) =
(\z. Cx(azim (vec 0) (fstcart z)
(fstcart(sndcart z)) (sndcart(sndcart z)))) o
(\x. pastecart (g x - f x) (pastecart (h x - f x) (k x - f x)))`
SUBST1_TAC THENL
[REWRITE_TAC[FUN_EQ_THM; o_THM; FSTCART_PASTECART; SNDCART_PASTECART] THEN
X_GEN_TAC `y:real^M` THEN
SUBST1_TAC(VECTOR_ARITH `vec 0 = (f:real^M->real^3) y - f y`) THEN
SIMP_TAC[ONCE_REWRITE_RULE[VECTOR_ADD_SYM] AZIM_TRANSLATION; VECTOR_SUB];
MATCH_MP_TAC CONTINUOUS_WITHIN_COMPOSE THEN
ASM_SIMP_TAC[CONTINUOUS_PASTECART; CONTINUOUS_SUB]] THEN
MATCH_MP_TAC CONTINUOUS_AT_WITHIN THEN
SUBGOAL_THEN
`!z. ~collinear {vec 0,fstcart z,fstcart(sndcart z)} /\
~collinear {vec 0,fstcart z,sndcart(sndcart z)} /\
~(sndcart(sndcart z) IN aff_ge {vec 0,fstcart z} {fstcart(sndcart z)})
==> (\z. Cx(azim (vec 0) (fstcart z) (fstcart(sndcart z))
(sndcart(sndcart z))))
continuous (at z)`
MATCH_MP_TAC THENL
[ALL_TAC;
ASM_SIMP_TAC[FSTCART_PASTECART; SNDCART_PASTECART; GSYM COLLINEAR_3] THEN
REPEAT(CONJ_TAC THENL [ASM_MESON_TAC[INSERT_AC]; ALL_TAC]) THEN
SUBST1_TAC(VECTOR_ARITH `vec 0 = (f:real^M->real^3) x - f x`) THEN
ONCE_REWRITE_TAC[SET_RULE `{a,b} = {a} UNION {b}`] THEN
REWRITE_TAC[GSYM IMAGE_UNION; SET_RULE
`{a - b:real^3} = IMAGE (\x. x - b) {a}`] THEN
REWRITE_TAC[ONCE_REWRITE_RULE[VECTOR_ADD_SYM] AFF_GE_TRANSLATION;
VECTOR_SUB] THEN
ASM_REWRITE_TAC[IN_IMAGE; VECTOR_ARITH `a + x:real^3 = b + x <=> a = b`;
UNWIND_THM1; SET_RULE `{a} UNION {b} = {a,b}`]] THEN
ONCE_REWRITE_TAC[SET_RULE
`(!x. ~P x /\ ~Q x /\ ~R x ==> J x) <=>
(!x. x IN UNIV DIFF (({x | P x} UNION {x | Q x}) UNION {x | R x})
==> J x)`] THEN
MATCH_MP_TAC(MESON[CONTINUOUS_ON_EQ_CONTINUOUS_AT]
`open s /\ f continuous_on s ==> !z. z IN s ==> f continuous at z`) THEN
CONJ_TAC THENL
[REWRITE_TAC[GSYM closed] THEN
MATCH_MP_TAC(MESON[]
`!t'. s UNION t = s UNION t' /\ closed(s UNION t')
==> closed(s UNION t)`) THEN
EXISTS_TAC
`{z | (fstcart z cross fstcart(sndcart z)) cross
fstcart z cross sndcart(sndcart z) = vec 0 /\
&0 <= (fstcart z cross sndcart(sndcart z)) dot
(fstcart z cross fstcart(sndcart z))}` THEN
CONJ_TAC THENL
[MATCH_MP_TAC(SET_RULE
`(!x. ~(x IN s) ==> (x IN t <=> x IN t'))
==> s UNION t = s UNION t'`) THEN
REWRITE_TAC[AFF_GE_2_1_0_SEMIALGEBRAIC; IN_UNION; IN_ELIM_THM;
DE_MORGAN_THM];
ALL_TAC] THEN
MATCH_MP_TAC CLOSED_UNION THEN CONJ_TAC THENL
[MATCH_MP_TAC CLOSED_UNION THEN CONJ_TAC THEN
REWRITE_TAC[GSYM DOT_CAUCHY_SCHWARZ_EQUAL] THEN
ONCE_REWRITE_TAC[GSYM REAL_SUB_0] THEN
REWRITE_TAC[GSYM LIFT_EQ; LIFT_NUM] THEN
SIMP_TAC[SET_RULE `{x | f x = a} = {x | x IN UNIV /\ f x IN {a}}`] THEN
MATCH_MP_TAC CONTINUOUS_CLOSED_PREIMAGE THEN
SIMP_TAC[CLOSED_UNIV; CLOSED_SING; LIFT_SUB; REAL_POW_2; LIFT_CMUL] THEN
MATCH_MP_TAC CONTINUOUS_ON_SUB THEN
CONJ_TAC THEN MATCH_MP_TAC CONTINUOUS_ON_MUL THEN
REWRITE_TAC[o_DEF] THEN REPEAT CONJ_TAC THEN
MATCH_MP_TAC CONTINUOUS_ON_LIFT_DOT2 THEN CONJ_TAC;
ONCE_REWRITE_TAC[MESON[LIFT_DROP; real_ge]
`&0 <= x <=> drop(lift x) >= &0`] THEN
REWRITE_TAC[SET_RULE
`{z | f z = vec 0 /\ drop(g z) >= &0} =
{z | z IN UNIV /\ f z IN {vec 0}} INTER
{z | z IN UNIV /\ g z IN {k | drop(k) >= &0}}`] THEN
MATCH_MP_TAC CLOSED_INTER THEN
CONJ_TAC THEN MATCH_MP_TAC CONTINUOUS_CLOSED_PREIMAGE THEN
REWRITE_TAC[CLOSED_SING; drop; CLOSED_UNIV;
CLOSED_HALFSPACE_COMPONENT_GE] THEN
REPEAT((MATCH_MP_TAC CONTINUOUS_ON_CROSS ORELSE
MATCH_MP_TAC CONTINUOUS_ON_LIFT_DOT2) THEN CONJ_TAC)] THEN
TRY(GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF]) THEN
SIMP_TAC[CONTINUOUS_ON_COMPOSE; LINEAR_CONTINUOUS_ON;
LINEAR_FSTCART; LINEAR_SNDCART];
MATCH_MP_TAC lemma THEN
MAP_EVERY EXISTS_TAC
[`{z | z IN UNIV /\ lift((fstcart z cross (fstcart(sndcart z))) dot
(sndcart(sndcart z))) IN {x | x$1 >= &0}}`;
`{z | z IN UNIV /\ lift((fstcart z cross (fstcart(sndcart z))) dot
(sndcart(sndcart z))) IN {x | x$1 <= &0}}`;
`\z. Cx(dihV (vec 0:real^3) (fstcart z)
(fstcart(sndcart z)) (sndcart(sndcart z)))`;
`\z. Cx(&2 * pi - dihV (vec 0:real^3) (fstcart z)
(fstcart(sndcart z)) (sndcart(sndcart z)))`] THEN
CONJ_TAC THENL
[CONJ_TAC THEN MATCH_MP_TAC CONTINUOUS_CLOSED_PREIMAGE THEN
REWRITE_TAC[CLOSED_UNIV; CLOSED_HALFSPACE_COMPONENT_GE;
CLOSED_HALFSPACE_COMPONENT_LE] THEN
MATCH_MP_TAC CONTINUOUS_ON_LIFT_DOT2 THEN
(CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_CROSS; ALL_TAC]) THEN
ONCE_REWRITE_TAC[GSYM o_DEF] THEN
SIMP_TAC[CONTINUOUS_ON_COMPOSE; LINEAR_CONTINUOUS_ON;
LINEAR_FSTCART; LINEAR_SNDCART];
ALL_TAC] THEN
CONJ_TAC THENL
[REWRITE_TAC[EXTENSION; IN_UNION; IN_UNIV; IN_ELIM_THM] THEN
REAL_ARITH_TAC;
ALL_TAC] THEN
CONJ_TAC THENL
[CONJ_TAC THEN MATCH_MP_TAC CONTINUOUS_AT_IMP_CONTINUOUS_ON THEN
REWRITE_TAC[FORALL_PASTECART; IN_DIFF; IN_UNIV; IN_UNION; IN_INTER;
FSTCART_PASTECART; SNDCART_PASTECART; IN_ELIM_THM; DE_MORGAN_THM] THEN
MAP_EVERY X_GEN_TAC [`x:real^3`; `y:real^3`; `z:real^3`] THEN
REPEAT STRIP_TAC THEN REWRITE_TAC[CX_SUB] THEN
TRY(MATCH_MP_TAC CONTINUOUS_SUB THEN REWRITE_TAC[CONTINUOUS_CONST]) THEN
GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN
REWRITE_TAC[GSYM REAL_CONTINUOUS_CONTINUOUS] THEN
MATCH_MP_TAC REAL_CONTINUOUS_AT_DIHV_COMPOSE THEN
ASM_REWRITE_TAC[FSTCART_PASTECART; SNDCART_PASTECART;
CONTINUOUS_CONST] THEN
ONCE_REWRITE_TAC[GSYM o_DEF] THEN
SIMP_TAC[CONTINUOUS_AT_COMPOSE; LINEAR_CONTINUOUS_AT;
LINEAR_FSTCART; LINEAR_SNDCART];
ALL_TAC] THEN
REWRITE_TAC[FORALL_PASTECART; IN_DIFF; IN_UNIV; IN_UNION; IN_INTER; CX_INJ;
FSTCART_PASTECART; SNDCART_PASTECART; IN_ELIM_THM; DE_MORGAN_THM] THEN
CONJ_TAC THENL
[REWRITE_TAC[GSYM drop; LIFT_DROP; real_ge] THEN
MAP_EVERY X_GEN_TAC [`x:real^3`; `y:real^3`; `z:real^3`] THEN
REPEAT STRIP_TAC THEN MATCH_MP_TAC(GSYM AZIM_DIHV_SAME_STRONG) THEN
ASM_REWRITE_TAC[AZIM_IN_UPPER_HALFSPACE; VECTOR_SUB_RZERO];
REWRITE_TAC[GSYM drop; LIFT_DROP] THEN
MAP_EVERY X_GEN_TAC [`x:real^3`; `y:real^3`; `z:real^3`] THEN
REPEAT STRIP_TAC THEN MATCH_MP_TAC(GSYM AZIM_DIHV_COMPL) THEN
ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC(REAL_ARITH
`(x <= pi ==> x = pi) ==> pi <= x`) THEN
ASM_REWRITE_TAC[AZIM_IN_UPPER_HALFSPACE; VECTOR_SUB_RZERO] THEN
ASM_SIMP_TAC[REAL_ARITH `x <= &0 ==> (&0 <= x <=> x = &0)`] THEN
REWRITE_TAC[REWRITE_RULE[VECTOR_SUB_RZERO]
(SPEC `vec 0:real^3` (GSYM COPLANAR_CROSS_DOT))] THEN
ASM_SIMP_TAC[GSYM AZIM_EQ_0_PI_EQ_COPLANAR; AZIM_EQ_0_GE_ALT]]]);;
let REAL_CONTINUOUS_AT_AZIM_COMPOSE = prove
(`!f:real^M->real^3 g h k x.
~collinear {f x,g x,h x} /\ ~collinear {f x,g x,k x} /\
~(k x IN aff_ge {f x,g x} {h x}) /\
f continuous (at x) /\ g continuous (at x) /\
h continuous (at x) /\ k continuous (at x)
==> (\x. azim (f x) (g x) (h x) (k x)) real_continuous (at x)`,
ONCE_REWRITE_TAC[GSYM WITHIN_UNIV] THEN
REWRITE_TAC[REAL_CONTINUOUS_WITHIN_AZIM_COMPOSE]);;
let REAL_CONTINUOUS_WITHINREAL_AZIM_COMPOSE = prove
(`!f:real->real^3 g h k x s.
~collinear {f x,g x,h x} /\ ~collinear {f x,g x,k x} /\
~(k x IN aff_ge {f x,g x} {h x}) /\
f continuous (atreal x within s) /\ g continuous (atreal x within s) /\
h continuous (atreal x within s) /\ k continuous (atreal x within s)
==> (\x. azim (f x) (g x) (h x) (k x)) real_continuous
(atreal x within s)`,
REWRITE_TAC[CONTINUOUS_CONTINUOUS_WITHINREAL;
REAL_CONTINUOUS_REAL_CONTINUOUS_WITHINREAL] THEN
SIMP_TAC[o_DEF; REAL_CONTINUOUS_WITHIN_AZIM_COMPOSE; LIFT_DROP]);;
let REAL_CONTINUOUS_ATREAL_AZIM_COMPOSE = prove
(`!f:real->real^3 g h k x.
~collinear {f x,g x,h x} /\ ~collinear {f x,g x,k x} /\
~(k x IN aff_ge {f x,g x} {h x}) /\
f continuous (atreal x) /\ g continuous (atreal x) /\
h continuous (atreal x) /\ k continuous (atreal x)
==> (\x. azim (f x) (g x) (h x) (k x)) real_continuous (atreal x)`,
ONCE_REWRITE_TAC[GSYM WITHINREAL_UNIV] THEN
REWRITE_TAC[REAL_CONTINUOUS_WITHINREAL_AZIM_COMPOSE]);;
(* ------------------------------------------------------------------------- *)
(* Can consider angle as defined by arcV a zenith angle. *)
(* ------------------------------------------------------------------------- *)
let ZENITH_EXISTS = prove
(`!u v w:real^3.
~(u = v) /\ ~(w = v)
==> (?u' r phi e3.
phi = arcV v u w /\
r = dist(u,v) /\
dist(w,v) % e3 = w - v /\
u' dot e3 = &0 /\
u = v + u' + (r * cos phi) % e3)`,
ONCE_REWRITE_TAC[VECTOR_ARITH
`u:real^3 = v + u' + x <=> u - v = u' + x`] THEN
GEN_GEOM_ORIGIN_TAC `v:real^3` ["u'"; "e3"] THEN
REWRITE_TAC[VECTOR_SUB_RZERO; DIST_0] THEN
ONCE_REWRITE_TAC[VECTOR_ARITH
`u:real^3 = u' + x <=> u - u' = x`] THEN
GEOM_BASIS_MULTIPLE_TAC 3 `w:real^3` THEN
X_GEN_TAC `w:real` THEN ASM_CASES_TAC `w = &0` THEN
ASM_REWRITE_TAC[VECTOR_MUL_LZERO; REAL_LE_LT] THEN DISCH_TAC THEN
SIMP_TAC[NORM_MUL; NORM_BASIS; DIMINDEX_3; ARITH] THEN
ASM_SIMP_TAC[REAL_ARITH `&0 < w ==> abs w * &1 = w`] THEN
ASM_SIMP_TAC[VECTOR_MUL_LCANCEL; REAL_LT_IMP_NZ] THEN
REWRITE_TAC[RIGHT_EXISTS_AND_THM; UNWIND_THM2] THEN
REWRITE_TAC[ARCV_ANGLE; angle; VECTOR_SUB_RZERO] THEN
ASM_SIMP_TAC[VECTOR_MUL_EQ_0; REAL_LT_IMP_NZ] THEN REPEAT STRIP_TAC THEN
MP_TAC(ISPECL [`u:real^3`; `w % basis 3:real^3`] VECTOR_ANGLE) THEN
REWRITE_TAC[DOT_RMUL; NORM_MUL] THEN
ASM_SIMP_TAC[REAL_ARITH
`&0 < w ==> n * ((abs w) * x) * y = w * n * x * y`] THEN
ASM_REWRITE_TAC[REAL_EQ_MUL_LCANCEL] THEN
SIMP_TAC[NORM_BASIS; DIMINDEX_3; ARITH; REAL_MUL_LID] THEN
DISCH_THEN(SUBST1_TAC o SYM) THEN
REWRITE_TAC[VECTOR_ARITH `u - u':real^3 = x <=> u' = u - x`] THEN
ONCE_REWRITE_TAC[CONJ_SYM] THEN REWRITE_TAC[UNWIND_THM2] THEN
REWRITE_TAC[DOT_LSUB; DOT_RMUL; DOT_LMUL] THEN
SIMP_TAC[DOT_BASIS_BASIS; DIMINDEX_3; ARITH] THEN REAL_ARITH_TAC);;
(* ------------------------------------------------------------------------- *)
(* Spherical coordinates. *)
(* ------------------------------------------------------------------------- *)
let SPHERICAL_COORDINATES = prove
(`!u v w u' e1 e2 e3 r phi theta.
~collinear {v, w, u} /\
~collinear {v, w, u'} /\
orthonormal e1 e2 e3 /\
dist(w,v) % e3 = w - v /\
(v + e1) IN aff_gt {v, w} {u} /\
r = dist(v,u') /\
phi = arcV v u' w /\
theta = azim v w u u'
==> u' = v + (r * cos theta * sin phi) % e1 +
(r * sin theta * sin phi) % e2 +
(r * cos phi) % e3`,
ONCE_REWRITE_TAC[VECTOR_ARITH
`u':real^3 = u + v + w <=> u' - u = v + w`] THEN
GEN_GEOM_ORIGIN_TAC `v:real^3` ["e1"; "e2"; "e3"] THEN
REWRITE_TAC[VECTOR_ADD_RID; VECTOR_ADD_LID] THEN
REWRITE_TAC[TRANSLATION_INVARIANTS `v:real^3`] THEN
GEOM_BASIS_MULTIPLE_TAC 3 `w:real^3` THEN
REWRITE_TAC[VECTOR_SUB_RZERO; DIST_0] THEN
X_GEN_TAC `w:real` THEN ASM_CASES_TAC `w = &0` THENL
[ASM_REWRITE_TAC[VECTOR_MUL_LZERO; INSERT_AC; COLLINEAR_2]; ALL_TAC] THEN
ASM_REWRITE_TAC[REAL_LE_LT] THEN DISCH_TAC THEN
MAP_EVERY X_GEN_TAC
[`u:real^3`; `v:real^3`; `e1:real^3`; `e2:real^3`; `e3:real^3`;
`r:real`; `phi:real`; `theta:real`] THEN
ASM_CASES_TAC `u:real^3 = w % basis 3` THENL
[ASM_REWRITE_TAC[INSERT_AC; COLLINEAR_2]; ALL_TAC] THEN
ASM_CASES_TAC `v:real^3 = w % basis 3` THENL
[ASM_REWRITE_TAC[INSERT_AC; COLLINEAR_2]; ALL_TAC] THEN
DISCH_THEN(MP_TAC o GSYM) THEN
ASM_SIMP_TAC[AZIM_SPECIAL_SCALE; COLLINEAR_SPECIAL_SCALE] THEN
SIMP_TAC[NORM_MUL; NORM_BASIS; DIMINDEX_3; ARITH] THEN
ASM_SIMP_TAC[REAL_ARITH `&0 < w ==> abs w * &1 = w`] THEN
ASM_REWRITE_TAC[VECTOR_MUL_LCANCEL] THEN
ASM_CASES_TAC `e3:real^3 = basis 3` THEN ASM_REWRITE_TAC[] THEN
REWRITE_TAC[ARCV_ANGLE; angle; VECTOR_SUB_RZERO] THEN
ASM_SIMP_TAC[VECTOR_ANGLE_RMUL; REAL_LT_IMP_LE] THEN
ASM_CASES_TAC `u:real^3 = vec 0` THENL
[ASM_REWRITE_TAC[INSERT_AC; COLLINEAR_2]; ALL_TAC] THEN
ASM_CASES_TAC `v:real^3 = vec 0` THENL
[ASM_REWRITE_TAC[INSERT_AC; COLLINEAR_2]; ALL_TAC] THEN
ASM_CASES_TAC `u:real^3 = basis 3` THENL
[ASM_REWRITE_TAC[INSERT_AC; COLLINEAR_2]; ALL_TAC] THEN
ASM_CASES_TAC `v:real^3 = basis 3` THENL
[ASM_REWRITE_TAC[INSERT_AC; COLLINEAR_2]; ALL_TAC] THEN
STRIP_TAC THEN
MP_TAC(ISPECL [`v:real^3`; `basis 3:real^3`] VECTOR_ANGLE) THEN
ASM_SIMP_TAC[DOT_BASIS; NORM_BASIS; DIMINDEX_3; ARITH; REAL_MUL_LID] THEN
DISCH_TAC THEN
MP_TAC(ISPECL
[`vec 0:real^3`; `w % basis 3:real^3`; `u:real^3`; `e1:real^3`]
AZIM_EQ_0_ALT) THEN
ASM_SIMP_TAC[AZIM_SPECIAL_SCALE; COLLINEAR_SPECIAL_SCALE] THEN
ANTS_TAC THENL
[SIMP_TAC[COLLINEAR_LEMMA; BASIS_NONZERO; DIMINDEX_3; ARITH] THEN
STRIP_TAC THEN UNDISCH_TAC `orthonormal e1 e2 (basis 3)` THEN
ASM_REWRITE_TAC[orthonormal; DOT_LZERO; REAL_OF_NUM_EQ; ARITH_EQ] THEN
ASM_CASES_TAC `c = &0` THEN
ASM_SIMP_TAC[VECTOR_MUL_LZERO; CROSS_LZERO; DOT_LZERO; REAL_LT_REFL;
DOT_LMUL; DOT_BASIS_BASIS; DIMINDEX_3; ARITH; REAL_MUL_RID];
DISCH_TAC] THEN
SUBGOAL_THEN
`dropout 3 (v:real^3):real^2 =
norm(dropout 3 (v:real^3):real^2) %
(cos theta % (dropout 3 (e1:real^3)) +
sin theta % (dropout 3 (e2:real^3)))`
MP_TAC THENL
[ALL_TAC;
SUBGOAL_THEN `norm((dropout 3:real^3->real^2) v) = r * sin phi`
SUBST1_TAC THENL
[REWRITE_TAC[NORM_EQ_SQUARE] THEN CONJ_TAC THENL
[ASM_MESON_TAC[REAL_LE_MUL; NORM_POS_LE; SIN_VECTOR_ANGLE_POS];
ALL_TAC] THEN
UNDISCH_TAC `(v:real^3)$3 = r * cos phi` THEN
MATCH_MP_TAC(REAL_RING
`x + a pow 2 = y + b pow 2 ==> a:real = b ==> x = y`) THEN
REWRITE_TAC[REAL_POW_MUL; GSYM REAL_ADD_LDISTRIB] THEN
REWRITE_TAC[SIN_CIRCLE; REAL_MUL_RID] THEN
UNDISCH_THEN `norm(v:real^3) = r` (SUBST1_TAC o SYM) THEN
REWRITE_TAC[NORM_POW_2; DOT_2; DOT_3] THEN
SIMP_TAC[dropout; LAMBDA_BETA; DIMINDEX_2; DIMINDEX_3; ARITH] THEN
REAL_ARITH_TAC;
ALL_TAC] THEN
REWRITE_TAC[CART_EQ; DIMINDEX_3; DIMINDEX_2; FORALL_3; FORALL_2] THEN
SIMP_TAC[dropout; LAMBDA_BETA; DIMINDEX_2; ARITH; VECTOR_ADD_COMPONENT;
VECTOR_MUL_COMPONENT; BASIS_COMPONENT; DIMINDEX_3] THEN
REPEAT STRIP_TAC THEN TRY REAL_ARITH_TAC THEN
ASM_REWRITE_TAC[] THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [orthonormal]) THEN
SIMP_TAC[DOT_BASIS; DIMINDEX_3; ARITH] THEN CONV_TAC REAL_RING] THEN
REPEAT(FIRST_X_ASSUM(MP_TAC o
GEN_REWRITE_RULE LAND_CONV [AZIM_ARG])) THEN
REPEAT(FIRST_X_ASSUM(MP_TAC o
GEN_REWRITE_RULE RAND_CONV [COLLINEAR_BASIS_3])) THEN
SUBGOAL_THEN `norm((dropout 3:real^3->real^2) e1) = &1 /\
norm((dropout 3:real^3->real^2) e2) = &1 /\
dropout 3 (e2:real^3) / dropout 3 (e1:real^3) = ii`
MP_TAC THENL
[MATCH_MP_TAC(TAUT `(a /\ b) /\ (a /\ b ==> c) ==> a /\ b /\ c`) THEN
CONJ_TAC THENL
[REWRITE_TAC[NORM_EQ_1] THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [orthonormal]) THEN
SIMP_TAC[DOT_BASIS; DIMINDEX_3; ARITH; dropout; LAMBDA_BETA;
DOT_2; DIMINDEX_2; DOT_3] THEN
CONV_TAC REAL_RING;
ALL_TAC] THEN
ASM_CASES_TAC `dropout 3 (e1:real^3) = Cx(&0)` THEN
ASM_SIMP_TAC[COMPLEX_NORM_CX; REAL_OF_NUM_EQ; ARITH_EQ; REAL_ABS_NUM] THEN
ASM_SIMP_TAC[COMPLEX_FIELD
`~(x = Cx(&0)) ==> (y / x = ii <=> y = ii * x)`] THEN
DISCH_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP ORTHONORMAL_CROSS) THEN
SIMP_TAC[CART_EQ; DIMINDEX_2; DIMINDEX_3; FORALL_2; FORALL_3;
cross; VECTOR_3; BASIS_COMPONENT; ARITH; dropout; LAMBDA_BETA;
complex_mul; ii; complex; RE_DEF; IM_DEF; VECTOR_2] THEN
CONV_TAC REAL_RING;
ALL_TAC] THEN
SPEC_TAC(`(dropout 3:real^3->real^2) e2`,`d2:real^2`) THEN
SPEC_TAC(`(dropout 3:real^3->real^2) e1`,`d1:real^2`) THEN
SPEC_TAC(`(dropout 3:real^3->real^2) v`,`z:real^2`) THEN
SPEC_TAC(`(dropout 3:real^3->real^2) u`,`w:real^2`) THEN
POP_ASSUM_LIST(K ALL_TAC) THEN
GEOM_BASIS_MULTIPLE_TAC 1 `w:real^2` THEN
X_GEN_TAC `k:real` THEN GEN_REWRITE_TAC LAND_CONV [REAL_LE_LT] THEN
ASM_CASES_TAC `k = &0` THEN ASM_REWRITE_TAC[VECTOR_MUL_LZERO] THEN
REWRITE_TAC[COMPLEX_CMUL; COMPLEX_BASIS; COMPLEX_VEC_0] THEN
SIMP_TAC[ARG_DIV_CX; COMPLEX_MUL_RID] THEN DISCH_TAC THEN REPEAT GEN_TAC THEN
ASM_CASES_TAC `d1 = Cx(&1)` THENL
[ASM_SIMP_TAC[COMPLEX_DIV_1; COMPLEX_MUL_LID] THEN
REPEAT STRIP_TAC THEN MP_TAC(SPEC `z:complex` ARG) THEN
ASM_REWRITE_TAC[CEXP_EULER; CX_SIN; CX_COS; COMPLEX_MUL_RID] THEN
CONV_TAC COMPLEX_RING;
ASM_REWRITE_TAC[ARG_EQ_0] THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [COMPLEX_EQ]) THEN
REWRITE_TAC[RE_CX; IM_CX;real] THEN
ASM_CASES_TAC `Im d1 = &0` THEN ASM_REWRITE_TAC[] THEN
ASM_SIMP_TAC[REAL_NORM; real] THEN REAL_ARITH_TAC]);;
(* ------------------------------------------------------------------------- *)
(* Definition of a wedge and invariance theorems. *)
(* ------------------------------------------------------------------------- *)
let wedge = new_definition
`wedge v0 v1 w1 w2 = {y | ~collinear {v0,v1,y} /\
&0 < azim v0 v1 w1 y /\
azim v0 v1 w1 y < azim v0 v1 w1 w2}`;;
let WEDGE_ALT = prove
(`!v0 v1 w1 w2.
~(v0 = v1)
==> wedge v0 v1 w1 w2 = {y | ~(y IN affine hull {v0,v1}) /\
&0 < azim v0 v1 w1 y /\
azim v0 v1 w1 y < azim v0 v1 w1 w2}`,
SIMP_TAC[wedge; COLLINEAR_3_AFFINE_HULL]);;
let WEDGE_TRANSLATION = prove
(`!a v w w1 w2. wedge (a + v) (a + w) (a + w1) (a + w2) =
IMAGE (\x. a + x) (wedge v w w1 w2)`,
REPEAT GEN_TAC THEN CONV_TAC SYM_CONV THEN
MATCH_MP_TAC SURJECTIVE_IMAGE_EQ THEN CONJ_TAC THENL
[MESON_TAC[VECTOR_ARITH `a + (x - a):real^3 = x`]; ALL_TAC] THEN
REWRITE_TAC[wedge; IN_ELIM_THM; AZIM_TRANSLATION] THEN
REWRITE_TAC[SET_RULE
`{a + x,a + y,a + z} = IMAGE (\x:real^N. a + x) {x,y,z}`] THEN
REWRITE_TAC[COLLINEAR_TRANSLATION_EQ]);;
add_translation_invariants [WEDGE_TRANSLATION];;
let WEDGE_LINEAR_IMAGE = prove
(`!f. linear f /\ (!x. norm(f x) = norm x) /\
(2 <= dimindex(:3) ==> det(matrix f) = &1)
==> !v w w1 w2. wedge (f v) (f w) (f w1) (f w2) =
IMAGE f (wedge v w w1 w2)`,
REPEAT STRIP_TAC THEN CONV_TAC SYM_CONV THEN
MATCH_MP_TAC SURJECTIVE_IMAGE_EQ THEN CONJ_TAC THENL
[ASM_MESON_TAC[ORTHOGONAL_TRANSFORMATION_SURJECTIVE;
ORTHOGONAL_TRANSFORMATION];
ALL_TAC] THEN
X_GEN_TAC `y:real^3` THEN REWRITE_TAC[wedge; IN_ELIM_THM] THEN
BINOP_TAC THEN ASM_SIMP_TAC[AZIM_LINEAR_IMAGE] THEN
SUBST1_TAC(SET_RULE `{f v,f w,f y} = IMAGE (f:real^3->real^3) {v,w,y}`) THEN
ASM_MESON_TAC[COLLINEAR_LINEAR_IMAGE_EQ; PRESERVES_NORM_INJECTIVE]);;
add_linear_invariants [WEDGE_LINEAR_IMAGE];;
let WEDGE_SPECIAL_SCALE = prove
(`!a v w1 w2.
&0 < a /\
~collinear{vec 0,a % v,w1} /\
~collinear{vec 0,a % v,w2}
==> wedge (vec 0) (a % v) w1 w2 = wedge (vec 0) v w1 w2`,
SIMP_TAC[wedge; AZIM_SPECIAL_SCALE; COLLINEAR_SPECIAL_SCALE;
REAL_LT_IMP_NZ]);;
let WEDGE_DEGENERATE = prove
(`(!z w w1 w2. z = w ==> wedge z w w1 w2 = {}) /\
(!z w w1 w2. collinear{z,w,w1} ==> wedge z w w1 w2 = {}) /\
(!z w w1 w2. collinear{z,w,w2} ==> wedge z w w1 w2 = {})`,
REWRITE_TAC[wedge] THEN SIMP_TAC[AZIM_DEGENERATE] THEN
REWRITE_TAC[REAL_LT_REFL; REAL_LT_ANTISYM; EMPTY_GSPEC]);;
(* ------------------------------------------------------------------------- *)
(* Basic relation between wedge and aff, so Tarski-type characterization. *)
(* ------------------------------------------------------------------------- *)
let AFF_GT_LEMMA = prove
(`!v1 v2:real^N.
&0 < t1 /\ ~(v2 = vec 0)
==> aff_gt {vec 0} {t1 % basis 1, v2} =
{a % basis 1 + b % v2 | &0 < a /\ &0 < b}`,
REWRITE_TAC[AFFSIGN_ALT; aff_gt_def; sgn_gt; IN_ELIM_THM] THEN
REWRITE_TAC[SET_RULE `{a} UNION {b,c} = {a,b,c}`] THEN
REWRITE_TAC[SET_RULE `x IN {a} <=> a = x`] THEN
ASM_SIMP_TAC[FINITE_INSERT; FINITE_UNION; AFFINE_HULL_FINITE_STEP_GEN;
RIGHT_EXISTS_AND_THM; REAL_LT_ADD; REAL_HALF; FINITE_EMPTY] THEN
ASM_REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY] THEN
REWRITE_TAC[IN_INSERT; VECTOR_ARITH `vec 0 = a % x <=> a % x = vec 0`] THEN
ASM_SIMP_TAC[VECTOR_MUL_EQ_0; REAL_LT_IMP_NZ; BASIS_NONZERO;
DIMINDEX_GE_1; LE_REFL] THEN
REPEAT STRIP_TAC THEN REWRITE_TAC[VECTOR_MUL_RZERO; VECTOR_SUB_RZERO] THEN
REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN X_GEN_TAC `y:real^N` THEN
REWRITE_TAC[RIGHT_AND_EXISTS_THM] THEN
REWRITE_TAC[REAL_ARITH `&1 - v - v' - v'' = &0 <=> v = &1 - v' - v''`] THEN
ONCE_REWRITE_TAC[MESON[] `(?a b c. P a b c) <=> (?b c a. P a b c)`] THEN
REWRITE_TAC[RIGHT_EXISTS_AND_THM; UNWIND_THM2] THEN
REWRITE_TAC[RIGHT_AND_EXISTS_THM] THEN ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN
AP_TERM_TAC THEN REWRITE_TAC[FUN_EQ_THM] THEN X_GEN_TAC `b:real` THEN
REWRITE_TAC[VECTOR_ARITH `y - a - b:real^N = vec 0 <=> y = a + b`] THEN
EQ_TAC THEN DISCH_THEN(X_CHOOSE_THEN `a:real` STRIP_ASSUME_TAC) THENL
[EXISTS_TAC `a * t1:real`; EXISTS_TAC `a / t1:real`] THEN
ASM_SIMP_TAC[REAL_LT_DIV; REAL_LT_MUL; VECTOR_MUL_ASSOC] THEN
ASM_SIMP_TAC[REAL_DIV_RMUL; REAL_LT_IMP_NZ]);;
let WEDGE_LUNE_GT = prove
(`!v0 v1 w1 w2.
~collinear {v0,v1,w1} /\ ~collinear {v0,v1,w2} /\
&0 < azim v0 v1 w1 w2 /\ azim v0 v1 w1 w2 < pi
==> wedge v0 v1 w1 w2 = aff_gt {v0,v1} {w1,w2}`,
let lemma = prove
(`!a x:real^3. (?a. x = a % basis 3) <=> dropout 3 x:real^2 = vec 0`,
SIMP_TAC[CART_EQ; FORALL_2; FORALL_3; DIMINDEX_2; DIMINDEX_3;
dropout; LAMBDA_BETA; BASIS_COMPONENT; ARITH; REAL_MUL_RID;
VECTOR_MUL_COMPONENT; VEC_COMPONENT; REAL_MUL_RZERO; UNWIND_THM1] THEN
MESON_TAC[]) in
REWRITE_TAC[wedge] THEN GEOM_ORIGIN_TAC `v0:real^3` THEN
GEOM_BASIS_MULTIPLE_TAC 3 `v1:real^3` THEN
X_GEN_TAC `w:real` THEN GEN_REWRITE_TAC LAND_CONV [REAL_LE_LT] THEN
ASM_CASES_TAC `w = &0` THEN ASM_REWRITE_TAC[] THENL
[ASM_REWRITE_TAC[VECTOR_MUL_LZERO; INSERT_AC; COLLINEAR_2]; ALL_TAC] THEN
ASM_SIMP_TAC[AZIM_SPECIAL_SCALE; COLLINEAR_SPECIAL_SCALE] THEN
POP_ASSUM_LIST(K ALL_TAC) THEN DISCH_TAC THEN
MAP_EVERY X_GEN_TAC [`w1:real^3`; `w2:real^3`] THEN
REPLICATE_TAC 2 (DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
ONCE_REWRITE_TAC[TAUT `~a /\ b /\ c <=> ~(~a ==> ~(b /\ c))`] THEN
ASM_SIMP_TAC[AZIM_ARG] THEN REWRITE_TAC[COLLINEAR_BASIS_3] THEN
RULE_ASSUM_TAC(REWRITE_RULE[COLLINEAR_BASIS_3]) THEN STRIP_TAC THEN
REWRITE_TAC[NOT_IMP; GSYM CONJ_ASSOC] THEN
W(MP_TAC o PART_MATCH (lhs o rand) AFF_GT_SPECIAL_SCALE o rand o snd) THEN
SUBGOAL_THEN
`~(w1:real^3 = vec 0) /\ ~(w2:real^3 = vec 0) /\
~(w1 = basis 3) /\ ~(w2 = basis 3)`
STRIP_ASSUME_TAC THENL
[REPEAT STRIP_TAC THEN
REPEAT(FIRST_X_ASSUM(MP_TAC o check (is_neg o concl))) THEN
ASM_REWRITE_TAC[DROPOUT_BASIS_3; DROPOUT_0; DROPOUT_MUL; VECTOR_MUL_RZERO];
ALL_TAC] THEN
ANTS_TAC THENL
[ASM_REWRITE_TAC[FINITE_INSERT; FINITE_EMPTY; IN_INSERT; NOT_IN_EMPTY] THEN
DISCH_THEN(DISJ_CASES_THEN (SUBST_ALL_TAC o SYM)) THEN
REPEAT(FIRST_X_ASSUM(MP_TAC o check (is_neg o concl))) THEN
ASM_REWRITE_TAC[DROPOUT_BASIS_3; DROPOUT_0; DROPOUT_MUL; VECTOR_MUL_RZERO];
DISCH_THEN SUBST1_TAC] THEN
REWRITE_TAC[AFFSIGN_ALT; aff_gt_def; sgn_gt; IN_ELIM_THM] THEN
REWRITE_TAC[SET_RULE `{a,b} UNION {c,d} = {a,b,d,c}`] THEN
REWRITE_TAC[SET_RULE `x IN {a} <=> a = x`] THEN
ASM_SIMP_TAC[FINITE_INSERT; FINITE_UNION; AFFINE_HULL_FINITE_STEP_GEN;
RIGHT_EXISTS_AND_THM; REAL_LT_ADD; REAL_HALF; FINITE_EMPTY] THEN
ASM_REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY] THEN
REWRITE_TAC[VECTOR_MUL_RZERO; VECTOR_SUB_RZERO] THEN
MATCH_MP_TAC EQ_TRANS THEN
EXISTS_TAC `{y | (dropout 3:real^3->real^2) y IN
aff_gt {vec 0}
{dropout 3 (w1:real^3),dropout 3 (w2:real^3)}}` THEN
CONJ_TAC THENL
[ALL_TAC;
REWRITE_TAC[AFFSIGN_ALT; aff_gt_def; sgn_gt; IN_ELIM_THM] THEN
REWRITE_TAC[SET_RULE `{a} UNION {b,c} = {a,b,c}`] THEN
REWRITE_TAC[SET_RULE `x IN {a} <=> a = x`] THEN
ASM_SIMP_TAC[FINITE_INSERT; FINITE_UNION; AFFINE_HULL_FINITE_STEP_GEN;
RIGHT_EXISTS_AND_THM; REAL_LT_ADD; REAL_HALF; FINITE_EMPTY] THEN
ASM_REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY] THEN
REWRITE_TAC[VECTOR_MUL_RZERO; VECTOR_SUB_RZERO] THEN
REWRITE_TAC[REAL_EQ_SUB_RADD; RIGHT_AND_EXISTS_THM] THEN
REWRITE_TAC[REAL_ARITH `&1 = x + v <=> v = &1 - x`] THEN
ONCE_REWRITE_TAC[TAUT `a /\ b /\ c /\ d <=> c /\ d /\ a /\ b`] THEN
ONCE_REWRITE_TAC[MESON[]
`(?a b c d. P a b c d) <=> (?b c d a. P a b c d)`] THEN
REWRITE_TAC[UNWIND_THM2] THEN
ONCE_REWRITE_TAC[MESON[]
`(?a b c. P a b c) <=> (?c b a. P a b c)`] THEN
REWRITE_TAC[UNWIND_THM2] THEN REWRITE_TAC[VECTOR_ARITH
`y - a - b - c:real^N = vec 0 <=> y - b - c = a`] THEN
REWRITE_TAC[LEFT_EXISTS_AND_THM; lemma] THEN
REWRITE_TAC[DROPOUT_SUB; DROPOUT_MUL] THEN
REWRITE_TAC[VECTOR_ARITH `y - a - b:real^2 = vec 0 <=> y = a + b`] THEN
REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN MESON_TAC[VECTOR_ADD_SYM]] THEN
MATCH_MP_TAC(SET_RULE
`{x | P x} = s ==> {y | P(dropout 3 y)} = {y | dropout 3 y IN s}`) THEN
MP_TAC(CONJ (ASSUME `~((dropout 3:real^3->real^2) w1 = vec 0)`)
(ASSUME `~((dropout 3:real^3->real^2) w2 = vec 0)`)) THEN
UNDISCH_TAC `Arg(dropout 3 (w2:real^3) / dropout 3 (w1:real^3)) < pi` THEN
UNDISCH_TAC `&0 < Arg(dropout 3 (w2:real^3) / dropout 3 (w1:real^3))` THEN
SPEC_TAC(`(dropout 3:real^3->real^2) w2`,`v2:complex`) THEN
SPEC_TAC(`(dropout 3:real^3->real^2) w1`,`v1:complex`) THEN
POP_ASSUM_LIST(K ALL_TAC) THEN GEOM_BASIS_MULTIPLE_TAC 1 `v1:complex` THEN
X_GEN_TAC `v1:real` THEN GEN_REWRITE_TAC LAND_CONV [REAL_LE_LT] THEN
ASM_CASES_TAC `v1 = &0` THEN ASM_REWRITE_TAC[VECTOR_MUL_LZERO] THEN
SIMP_TAC[AFF_GT_LEMMA] THEN
REWRITE_TAC[COMPLEX_CMUL; COMPLEX_BASIS; COMPLEX_VEC_0] THEN
ASM_SIMP_TAC[ARG_DIV_CX; COMPLEX_MUL_RID; CX_INJ] THEN DISCH_TAC THEN
POP_ASSUM_LIST(K ALL_TAC) THEN X_GEN_TAC `z:complex` THEN
REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN
REWRITE_TAC[SUBSET; FORALL_IN_GSPEC; IN_ELIM_THM] THEN CONJ_TAC THENL
[X_GEN_TAC `w:complex` THEN STRIP_TAC THEN
MP_TAC(SPECL [`\t. Arg(Cx t + Cx(&1 - t) * z)`;
`&0`; `&1`; `Arg w`] REAL_IVT_DECREASING) THEN
REWRITE_TAC[REAL_POS; REAL_SUB_REFL; COMPLEX_MUL_LZERO] THEN
REWRITE_TAC[REAL_SUB_RZERO; COMPLEX_ADD_LID; COMPLEX_MUL_LID] THEN
ASM_SIMP_TAC[COMPLEX_ADD_RID; ARG_NUM; REAL_LT_IMP_LE] THEN ANTS_TAC THENL
[REWRITE_TAC[REAL_CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN] THEN
REWRITE_TAC[REAL_CONTINUOUS_CONTINUOUS; IN_REAL_INTERVAL] THEN
X_GEN_TAC `t:real` THEN STRIP_TAC THEN
ONCE_REWRITE_TAC[GSYM o_DEF] THEN REWRITE_TAC[o_ASSOC] THEN
MATCH_MP_TAC CONTINUOUS_WITHINREAL_COMPOSE THEN
REWRITE_TAC[] THEN CONJ_TAC THENL
[MATCH_MP_TAC CONTINUOUS_ADD THEN CONJ_TAC THENL
[GEN_REWRITE_TAC LAND_CONV [SYM(CONJUNCT2(SPEC_ALL I_O_ID))] THEN
REWRITE_TAC[GSYM REAL_CONTINUOUS_CONTINUOUS] THEN
REWRITE_TAC[I_DEF; REAL_CONTINUOUS_WITHIN_ID];
MATCH_MP_TAC CONTINUOUS_COMPLEX_MUL THEN
REWRITE_TAC[CONTINUOUS_CONST] THEN ONCE_REWRITE_TAC[GSYM o_DEF] THEN
REWRITE_TAC[GSYM REAL_CONTINUOUS_CONTINUOUS] THEN
SIMP_TAC[REAL_CONTINUOUS_SUB; REAL_CONTINUOUS_CONST;
REAL_CONTINUOUS_WITHIN_ID]];
MATCH_MP_TAC CONTINUOUS_WITHIN_SUBSET THEN
EXISTS_TAC `{z | &0 <= Im z}` THEN CONJ_TAC THENL
[MATCH_MP_TAC CONTINUOUS_WITHIN_UPPERHALF_ARG THEN
ASM_CASES_TAC `t = &1` THENL
[ASM_REWRITE_TAC[REAL_SUB_REFL] THEN CONV_TAC COMPLEX_RING;
ALL_TAC] THEN
DISCH_THEN(MP_TAC o AP_TERM `Im`) THEN
REWRITE_TAC[IM_ADD; IM_CX; IM_MUL_CX; REAL_ADD_LID; REAL_ENTIRE] THEN
ASM_REWRITE_TAC[REAL_SUB_0] THEN
ASM_MESON_TAC[ARG_LT_PI; REAL_LT_IMP_NZ; REAL_LT_TRANS];
REWRITE_TAC[FORALL_IN_IMAGE; SUBSET; IN_REAL_INTERVAL] THEN
REWRITE_TAC[IN_ELIM_THM; IM_ADD; IM_CX; IM_MUL_CX] THEN
REPEAT STRIP_TAC THEN REWRITE_TAC[REAL_ADD_LID] THEN
MATCH_MP_TAC REAL_LE_MUL THEN REWRITE_TAC[GSYM ARG_LE_PI] THEN
ASM_REAL_ARITH_TAC]];
REWRITE_TAC[IN_REAL_INTERVAL] THEN
DISCH_THEN(X_CHOOSE_THEN `t:real` MP_TAC) THEN
ASM_CASES_TAC `t = &0` THENL
[ASM_REWRITE_TAC[REAL_SUB_RZERO; COMPLEX_ADD_LID; COMPLEX_MUL_LID] THEN
ASM_MESON_TAC[REAL_LT_REFL];
ALL_TAC] THEN
ASM_CASES_TAC `t = &1` THENL
[ASM_REWRITE_TAC[REAL_SUB_REFL; COMPLEX_MUL_LZERO] THEN
REWRITE_TAC[COMPLEX_ADD_RID; ARG_NUM] THEN ASM_MESON_TAC[REAL_LT_REFL];
ALL_TAC] THEN
GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [REAL_LE_LT] THEN
ASM_REWRITE_TAC[] THEN ABBREV_TAC `u = Cx t + Cx(&1 - t) * z` THEN
ASM_CASES_TAC `u = Cx(&0)` THENL
[ASM_MESON_TAC[ARG_0; REAL_LT_REFL]; ALL_TAC] THEN
STRIP_TAC THEN
EXISTS_TAC `norm(w:complex) / norm(u:complex) * t` THEN
EXISTS_TAC `norm(w:complex) / norm(u:complex) * (&1 - t)` THEN
ASM_SIMP_TAC[REAL_LT_MUL; REAL_LT_DIV; COMPLEX_NORM_NZ; REAL_SUB_LT] THEN
SIMP_TAC[CX_MUL; GSYM COMPLEX_MUL_ASSOC; GSYM COMPLEX_ADD_LDISTRIB] THEN
ASM_REWRITE_TAC[CX_DIV] THEN
ASM_SIMP_TAC[CX_INJ; COMPLEX_NORM_ZERO; COMPLEX_FIELD
`~(nu = Cx(&0)) ==> (w = nw / nu * u <=> nu * w = nw * u)`] THEN
GEN_REWRITE_TAC (BINOP_CONV o RAND_CONV) [ARG] THEN
ASM_REWRITE_TAC[COMPLEX_MUL_AC]];
MAP_EVERY X_GEN_TAC [`a:real`; `b:real`] THEN STRIP_TAC THEN
SUBGOAL_THEN `Cx a + Cx b * z = complex(a + b * Re z,b * Im z)`
SUBST1_TAC THENL
[REWRITE_TAC[COMPLEX_EQ; RE; IM; RE_ADD; IM_ADD; RE_CX; IM_CX;
RE_MUL_CX; IM_MUL_CX] THEN
REAL_ARITH_TAC;
ALL_TAC] THEN
REWRITE_TAC[COMPLEX_EQ; IM; IM_CX] THEN
SUBGOAL_THEN `&0 < Im z` ASSUME_TAC THENL
[ASM_REWRITE_TAC[GSYM ARG_LT_PI]; ALL_TAC] THEN
ASM_SIMP_TAC[ARG_ATAN_UPPERHALF; REAL_LT_MUL; REAL_LT_IMP_NZ; IM] THEN
REWRITE_TAC[RE; REAL_SUB_LT; ATN_BOUNDS] THEN
REWRITE_TAC[REAL_ARITH `pi / &2 - x < pi / &2 - y <=> y < x`] THEN
REWRITE_TAC[ATN_MONO_LT_EQ] THEN
ASM_SIMP_TAC[REAL_LT_RDIV_EQ; REAL_LT_MUL] THEN
ASM_SIMP_TAC[REAL_FIELD `&0 < z ==> w / z * b * z = b * w`] THEN
ASM_REAL_ARITH_TAC]);;
let WEDGE_LUNE_GE = prove
(`!v0 v1 w1 w2.
~collinear {v0,v1,w1} /\ ~collinear {v0,v1,w2} /\
&0 < azim v0 v1 w1 w2 /\ azim v0 v1 w1 w2 < pi
==> {x | &0 <= azim v0 v1 w1 x /\
azim v0 v1 w1 x <= azim v0 v1 w1 w2} =
aff_ge {v0,v1} {w1,w2}`,
REPEAT GEN_TAC THEN
MAP_EVERY (fun t -> ASM_CASES_TAC t THENL
[ASM_REWRITE_TAC[COLLINEAR_2; INSERT_AC] THEN NO_TAC; ALL_TAC])
[`v1:real^3 = v0`; `w1:real^3 = v0`; `w2:real^3 = v0`;
`w1:real^3 = v1`; `w2:real^3 = v1`] THEN
ASM_CASES_TAC `w1:real^3 = w2` THEN
ASM_REWRITE_TAC[AZIM_REFL; REAL_LT_REFL] THEN
STRIP_TAC THEN ASM_SIMP_TAC[REAL_ARITH
`&0 < a
==> (&0 <= x /\ x <= a <=> x = &0 \/ x = a \/ &0 < x /\ x < a)`] THEN
MATCH_MP_TAC(SET_RULE
`!c. c SUBSET {x | p x} /\ c SUBSET s /\
({x | ~(~c x ==> ~p x)} UNION {x | ~(~c x ==> ~q x)} UNION
({x | ~c x /\ r x} DIFF c) = s DIFF c)
==> {x | p x \/ q x \/ r x} = s`) THEN
EXISTS_TAC `{x:real^3 | collinear {v0,v1,x}}` THEN
ASM_SIMP_TAC[IN_ELIM_THM; AZIM_EQ_ALT; AZIM_EQ_0_ALT;
GSYM wedge; WEDGE_LUNE_GT] THEN
REPEAT CONJ_TAC THENL
[ASM_SIMP_TAC[SUBSET; IN_ELIM_THM; AZIM_DEGENERATE];
ASM_SIMP_TAC[COLLINEAR_3_AFFINE_HULL] THEN
REWRITE_TAC[SET_RULE `{x | x IN s} = s`] THEN
MATCH_MP_TAC AFFINE_HULL_SUBSET_AFF_GE THEN
ASM_REWRITE_TAC[DISJOINT_INSERT; IN_INSERT; NOT_IN_EMPTY; DISJOINT_EMPTY];
ALL_TAC] THEN
REWRITE_TAC[NOT_IMP] THEN MATCH_MP_TAC(SET_RULE
`(!x. ~c x ==> (p x \/ q x \/ x IN t <=> x IN e))
==> {x | ~c x /\ p x} UNION {x | ~c x /\ q x} UNION (t DIFF {x | c x}) =
e DIFF {x | c x}`) THEN
X_GEN_TAC `y:real^3` THEN DISCH_TAC THEN
W(MP_TAC o PART_MATCH (lhs o rand) AFF_GE_AFF_GT_DECOMP o rand o
rand o snd) THEN
ANTS_TAC THENL
[ASM_REWRITE_TAC[FINITE_INSERT; FINITE_EMPTY] THEN
ASM_REWRITE_TAC[DISJOINT_INSERT; IN_INSERT; NOT_IN_EMPTY; DISJOINT_EMPTY];
DISCH_THEN SUBST1_TAC] THEN
REWRITE_TAC[IN_UNION] THEN
REWRITE_TAC[SIMPLE_IMAGE; IMAGE_CLAUSES; UNIONS_2] THEN
ASM_SIMP_TAC[SET_RULE `~(w1 = w2) ==> {w1,w2} DELETE w1 = {w2}`;
SET_RULE `~(w1 = w2) ==> {w1,w2} DELETE w2 = {w1}`] THEN
REWRITE_TAC[IN_UNION; DISJ_ACI] THEN
W(MP_TAC o PART_MATCH (lhs o rand) AFF_GE_AFF_GT_DECOMP o rand o lhand o
rand o snd) THEN
ANTS_TAC THENL
[ASM_REWRITE_TAC[FINITE_INSERT; FINITE_EMPTY] THEN
ASM_REWRITE_TAC[DISJOINT_INSERT; IN_INSERT; NOT_IN_EMPTY; DISJOINT_EMPTY];
DISCH_THEN SUBST1_TAC] THEN
W(MP_TAC o PART_MATCH (lhs o rand) AFF_GE_AFF_GT_DECOMP o rand o lhand o
rand o rand o snd) THEN
ANTS_TAC THENL
[ASM_REWRITE_TAC[FINITE_INSERT; FINITE_EMPTY] THEN
ASM_REWRITE_TAC[DISJOINT_INSERT; IN_INSERT; NOT_IN_EMPTY; DISJOINT_EMPTY];
DISCH_THEN SUBST1_TAC] THEN
REWRITE_TAC[IN_UNION] THEN
REWRITE_TAC[SIMPLE_IMAGE; IMAGE_CLAUSES; UNIONS_1] THEN
REWRITE_TAC[SET_RULE `{a} DELETE a = {}`; AFF_GE_EQ_AFFINE_HULL] THEN
ASM_MESON_TAC[COLLINEAR_3_AFFINE_HULL]);;
let WEDGE_LUNE = prove
(`!v0 v1 w1 w2.
~coplanar{v0,v1,w1,w2} /\ azim v0 v1 w1 w2 < pi
==> wedge v0 v1 w1 w2 = aff_gt {v0,v1} {w1,w2}`,
REPEAT STRIP_TAC THEN MATCH_MP_TAC WEDGE_LUNE_GT THEN
ASM_REWRITE_TAC[] THEN REPEAT CONJ_TAC THENL
[MP_TAC(ISPECL [`v0:real^3`; `v1:real^3`; `w1:real^3`; `w2:real^3`]
NOT_COPLANAR_NOT_COLLINEAR) THEN
ASM_REWRITE_TAC[];
MP_TAC(ISPECL [`v0:real^3`; `v1:real^3`; `w2:real^3`; `w1:real^3`]
NOT_COPLANAR_NOT_COLLINEAR) THEN
ONCE_REWRITE_TAC[SET_RULE `{a,b,c,d} = {a,b,d,c}`] THEN
ASM_REWRITE_TAC[];
REWRITE_TAC[azim; REAL_LT_LE] THEN
ASM_MESON_TAC[AZIM_EQ_0_PI_IMP_COPLANAR]]);;
let WEDGE = prove
(`wedge v1 v2 w1 w2 =
if collinear{v1,v2,w1} \/ collinear{v1,v2,w2} then {}
else
let z = v2 - v1 in
let u1 = w1 - v1 in
let u2 = w2 - v1 in
let n = z cross u1 in
let d = n dot u2 in
if w2 IN (aff_ge {v1,v2} {w1}) then {}
else if w2 IN (aff_lt {v1,v2} {w1}) then aff_gt {v1,v2,w1} {v1 + n}
else if d > &0 then aff_gt {v1,v2} {w1,w2}
else (:real^3) DIFF aff_ge {v1,v2} {w1,w2}`,
REPEAT GEN_TAC THEN COND_CASES_TAC THENL
[FIRST_X_ASSUM DISJ_CASES_TAC THEN
ASM_SIMP_TAC[WEDGE_DEGENERATE];
POP_ASSUM MP_TAC THEN REWRITE_TAC[DE_MORGAN_THM] THEN STRIP_TAC] THEN
ASM_SIMP_TAC[GSYM AZIM_EQ_0_GE_ALT] THEN
ASM_CASES_TAC `azim v1 v2 w1 w2 = &0` THENL
[ASM_REWRITE_TAC[wedge] THEN
ASM_REWRITE_TAC[REAL_LT_ANTISYM; LET_DEF; LET_END_DEF; EMPTY_GSPEC];
ALL_TAC] THEN
ASM_SIMP_TAC[GSYM AZIM_EQ_PI_ALT] THEN
ASM_CASES_TAC `azim v1 v2 w1 w2 = pi` THEN ASM_REWRITE_TAC[] THEN
REWRITE_TAC[LET_DEF; LET_END_DEF] THEN
POP_ASSUM_LIST(MP_TAC o end_itlist CONJ) THEN
GEOM_ORIGIN_TAC `v1:real^3` THEN
REWRITE_TAC[VECTOR_ADD_RID; TRANSLATION_INVARIANTS `v1:real^3`] THEN
REWRITE_TAC[VECTOR_SUB_RZERO; VECTOR_ADD_LID] THEN
GEOM_BASIS_MULTIPLE_TAC 3 `v2:real^3` THEN
X_GEN_TAC `v2:real` THEN
GEN_REWRITE_TAC LAND_CONV [REAL_ARITH `&0 <= x <=> x = &0 \/ &0 < x`] THEN
(STRIP_TAC THENL
[ASM_REWRITE_TAC[VECTOR_MUL_LZERO; INSERT_AC; COLLINEAR_2]; ALL_TAC]) THEN
ASM_SIMP_TAC[AZIM_SPECIAL_SCALE; COLLINEAR_SPECIAL_SCALE; REAL_LT_IMP_NZ;
WEDGE_SPECIAL_SCALE] THEN
(REPEAT GEN_TAC THEN
MAP_EVERY (fun t -> ASM_CASES_TAC t THENL
[ASM_REWRITE_TAC[COLLINEAR_2; INSERT_AC] THEN NO_TAC; ALL_TAC])
[`w1:real^3 = vec 0`; `w2:real^3 = vec 0`; `w1:real^3 = basis 3`;
`w2:real^3 = basis 3`] THEN
ASM_CASES_TAC `w1:real^3 = v2 % basis 3` THENL
[ASM_REWRITE_TAC[COLLINEAR_LEMMA] THEN MESON_TAC[]; ALL_TAC] THEN
ASM_CASES_TAC `w2:real^3 = v2 % basis 3` THENL
[ASM_REWRITE_TAC[COLLINEAR_LEMMA] THEN MESON_TAC[]; ALL_TAC])
THENL
[REPEAT STRIP_TAC THEN REWRITE_TAC[EXTENSION] THEN X_GEN_TAC `y:real^3` THEN
MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC
`(dropout 3 (y:real^3)) IN
aff_gt {vec 0:real^2,dropout 3 (w1:real^3)}
{rotate2d (pi / &2) (dropout 3 (w1:real^3))}` THEN
CONJ_TAC THENL
[FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE LAND_CONV [AZIM_ARG]) THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE (RAND_CONV o LAND_CONV)
[AZIM_ARG]) THEN
REPEAT(FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV
[COLLINEAR_BASIS_3])) THEN
POP_ASSUM_LIST(K ALL_TAC) THEN
REWRITE_TAC[wedge; IN_ELIM_THM; AZIM_ARG; COLLINEAR_BASIS_3] THEN
SPEC_TAC(`(dropout 3:real^3->real^2) y`,`x:real^2`) THEN
SPEC_TAC(`(dropout 3:real^3->real^2) w2`,`v2:real^2`) THEN
SPEC_TAC(`(dropout 3:real^3->real^2) w1`,`v1:real^2`) THEN
GEOM_BASIS_MULTIPLE_TAC 1 `v1:complex` THEN
X_GEN_TAC `v:real` THEN GEN_REWRITE_TAC LAND_CONV [REAL_LE_LT] THEN
ASM_CASES_TAC `v = &0` THEN ASM_REWRITE_TAC[VECTOR_MUL_LZERO] THEN
REWRITE_TAC[COMPLEX_CMUL; COMPLEX_BASIS; COMPLEX_VEC_0] THEN
SIMP_TAC[ARG_DIV_CX; COMPLEX_MUL_RID] THEN
REWRITE_TAC[real; RE_DIV_CX; IM_DIV_CX; CX_INJ] THEN
ASM_SIMP_TAC[REAL_LT_LDIV_EQ; REAL_EQ_LDIV_EQ; REAL_MUL_LZERO] THEN
REPEAT STRIP_TAC THEN REWRITE_TAC[ARG_LT_PI; ROTATE2D_PI2] THEN
W(MP_TAC o PART_MATCH (lhs o rand) AFF_GT_2_1 o rand o rand o snd) THEN
ASM_REWRITE_TAC[DISJOINT_INSERT; DISJOINT_EMPTY; IN_SING] THEN
ANTS_TAC THENL
[CONV_TAC(ONCE_DEPTH_CONV SYM_CONV) THEN
ASM_REWRITE_TAC[COMPLEX_ENTIRE; II_NZ; CX_INJ] THEN
DISCH_THEN(MP_TAC o AP_TERM `Re`) THEN
REWRITE_TAC[RE_MUL_II; RE_CX; IM_CX] THEN ASM_REAL_ARITH_TAC;
DISCH_THEN SUBST1_TAC] THEN
REWRITE_TAC[COMPLEX_CMUL; IN_ELIM_THM; COMPLEX_MUL_RZERO] THEN
ONCE_REWRITE_TAC[MESON[] `(?a b c. P a b c) <=> (?b c a. P a b c)`] THEN
REWRITE_TAC[REAL_ARITH `t1 + t2 = &1 <=> t1 = &1 - t2`] THEN
REWRITE_TAC[RIGHT_EXISTS_AND_THM; UNWIND_THM2; COMPLEX_ADD_LID] THEN
EQ_TAC THENL
[DISCH_TAC THEN
MAP_EVERY EXISTS_TAC [`Re x / v`; `Im x / v`] THEN
ASM_SIMP_TAC[REAL_LT_DIV; COMPLEX_EQ; IM_ADD; RE_ADD] THEN
REWRITE_TAC[RE_MUL_CX; IM_MUL_CX; RE_CX; IM_CX; RE_II; IM_II] THEN
UNDISCH_TAC `~(v = &0)` THEN CONV_TAC REAL_FIELD;
REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN
MAP_EVERY X_GEN_TAC [`s:real`; `t:real`] THEN
STRIP_TAC THEN ASM_REWRITE_TAC[COMPLEX_EQ; IM_ADD; RE_ADD] THEN
REWRITE_TAC[RE_MUL_CX; IM_MUL_CX; RE_CX; IM_CX; RE_II; IM_II] THEN
ASM_SIMP_TAC[REAL_MUL_RZERO; REAL_MUL_LID; REAL_LT_MUL; REAL_ADD_LID;
REAL_MUL_LZERO] THEN
MAP_EVERY UNDISCH_TAC [`&0 < v`; `&0 < t`] THEN
CONV_TAC REAL_FIELD];
ALL_TAC] THEN
W(MP_TAC o PART_MATCH (lhs o rand) AFF_GT_3_1 o rand o rand o snd) THEN
ANTS_TAC THENL
[REWRITE_TAC[SET_RULE
`DISJOINT {a,b,c} {x} <=> ~(x = a) /\ ~(x = b) /\ ~(x = c)`] THEN
ASM_SIMP_TAC[CROSS_EQ_0; CROSS_EQ_SELF; VECTOR_MUL_EQ_0; REAL_LT_IMP_NZ;
REAL_LT_IMP_NZ; BASIS_NONZERO; DIMINDEX_3;
ARITH; COLLINEAR_SPECIAL_SCALE];
DISCH_THEN SUBST1_TAC] THEN
W(MP_TAC o PART_MATCH (lhs o rand) AFF_GT_2_1 o rand o lhand o snd) THEN
REWRITE_TAC[ROTATE2D_PI2] THEN ANTS_TAC THENL
[REWRITE_TAC[SET_RULE `DISJOINT {a,b} {x} <=> ~(x = a) /\ ~(x = b)`] THEN
REWRITE_TAC[COMPLEX_ENTIRE; COMPLEX_RING `ii * x = x <=> x = Cx(&0)`;
COMPLEX_VEC_0; II_NZ] THEN
ASM_REWRITE_TAC[GSYM COMPLEX_VEC_0; GSYM COLLINEAR_BASIS_3];
DISCH_THEN SUBST1_TAC] THEN
REWRITE_TAC[IN_ELIM_THM; VECTOR_MUL_RZERO; VECTOR_ADD_LID] THEN
ONCE_REWRITE_TAC[MESON[]
`(?a b c d. P a b c d) <=> (?d c b a. P a b c d)`] THEN
ONCE_REWRITE_TAC[REAL_ARITH `s + t = &1 <=> s = &1 - t`] THEN
REWRITE_TAC[UNWIND_THM2; RIGHT_EXISTS_AND_THM] THEN
ONCE_REWRITE_TAC[MESON[] `(?a b c. P a b c) <=> (?c b a. P a b c)`] THEN
REWRITE_TAC[UNWIND_THM2; RIGHT_EXISTS_AND_THM] THEN
REWRITE_TAC[RIGHT_AND_EXISTS_THM] THEN
SIMP_TAC[CART_EQ; FORALL_2; FORALL_3; DIMINDEX_2; DIMINDEX_3;
dropout; LAMBDA_BETA; BASIS_COMPONENT; ARITH; REAL_MUL_RID;
VECTOR_MUL_COMPONENT; VEC_COMPONENT; REAL_MUL_RZERO; UNWIND_THM1;
VECTOR_ADD_COMPONENT; cross; VECTOR_3;
REWRITE_RULE[RE_DEF; IM_DEF] RE_MUL_II;
REWRITE_RULE[RE_DEF; IM_DEF] IM_MUL_II;
REAL_ADD_LID; REAL_MUL_LZERO; REAL_SUB_REFL; REAL_ADD_RID;
REAL_SUB_LZERO; REAL_SUB_RZERO] THEN
ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN
AP_TERM_TAC THEN REWRITE_TAC[FUN_EQ_THM] THEN X_GEN_TAC `s:real` THEN
REWRITE_TAC[RIGHT_EXISTS_AND_THM] THEN
ASM_SIMP_TAC[EXISTS_REFL; REAL_FIELD
`&0 < v ==> (x = a * v + b <=> a = (x - b) / v)`] THEN
REWRITE_TAC[REAL_MUL_RNEG; REAL_MUL_ASSOC] THEN EQ_TAC THEN
DISCH_THEN(X_CHOOSE_THEN `t:real` STRIP_ASSUME_TAC) THENL
[EXISTS_TAC `t / v2:real`; EXISTS_TAC `t * v2:real`] THEN
ASM_SIMP_TAC[REAL_DIV_RMUL; REAL_LT_DIV; REAL_LT_IMP_NZ; REAL_LT_MUL];
ALL_TAC] THEN
REWRITE_TAC[CROSS_LMUL] THEN
SIMP_TAC[cross; BASIS_COMPONENT; DIMINDEX_3; ARITH; DOT_3; VECTOR_3;
VECTOR_MUL_COMPONENT; REAL_MUL_LZERO; REAL_SUB_RZERO; REAL_NEG_0;
REAL_MUL_RZERO; REAL_SUB_LZERO; REAL_MUL_LID; REAL_ADD_RID] THEN
REWRITE_TAC[REAL_ARITH
`(v * --x2) * y1 + (v * x1) * y2 > &0 <=> &0 < v * (x1 * y2 - x2 * y1)`] THEN
ASM_SIMP_TAC[REAL_LT_MUL_EQ; REAL_SUB_LT] THEN
REWRITE_TAC[AZIM_ARG; COLLINEAR_BASIS_3] THEN STRIP_TAC THEN
SUBGOAL_THEN
`w1$2 * w2$1 < w1$1 * w2$2 <=>
Arg(dropout 3 (w2:real^3) / dropout 3 (w1:real^3)) < pi`
SUBST1_TAC THENL
[MATCH_MP_TAC EQ_TRANS THEN
EXISTS_TAC `&0 < Im(dropout 3 (w2:real^3) / dropout 3 (w1:real^3))` THEN
CONJ_TAC THENL
[REWRITE_TAC[IM_COMPLEX_DIV_GT_0] THEN
REWRITE_TAC[complex_mul; cnj; RE_DEF; IM_DEF; complex] THEN
SIMP_TAC[dropout; VECTOR_2; LAMBDA_BETA; DIMINDEX_3; ARITH;
DIMINDEX_2] THEN
REAL_ARITH_TAC;
REWRITE_TAC[GSYM ARG_LT_PI] THEN ASM_MESON_TAC[ARG_LT_NZ]];
ALL_TAC] THEN
COND_CASES_TAC THENL
[W(MP_TAC o PART_MATCH (lhs o rand) AFF_GT_SPECIAL_SCALE o rand o snd) THEN
ASM_REWRITE_TAC[FINITE_INSERT; FINITE_EMPTY; IN_INSERT; NOT_IN_EMPTY] THEN
DISCH_THEN SUBST1_TAC THEN MATCH_MP_TAC WEDGE_LUNE THEN
ASM_SIMP_TAC[GSYM AZIM_EQ_0_PI_EQ_COPLANAR; COLLINEAR_BASIS_3] THEN
ASM_REWRITE_TAC[AZIM_ARG];
ALL_TAC] THEN
REWRITE_TAC[wedge] THEN
GEN_REWRITE_TAC (funpow 3 RAND_CONV) [SET_RULE `{a,b} = {b,a}`] THEN
W(MP_TAC o PART_MATCH (rand o rand) WEDGE_LUNE_GE o rand o rand o snd) THEN
ASM_SIMP_TAC[COLLINEAR_SPECIAL_SCALE; REAL_LT_IMP_NZ; AZIM_SPECIAL_SCALE] THEN
ASM_REWRITE_TAC[AZIM_ARG; COLLINEAR_BASIS_3] THEN ANTS_TAC THENL
[ASM_REWRITE_TAC[ARG_LT_NZ] THEN
ONCE_REWRITE_TAC[GSYM ARG_INV_EQ_0] THEN
ASM_REWRITE_TAC[COMPLEX_INV_DIV] THEN
ONCE_REWRITE_TAC[GSYM COMPLEX_INV_DIV] THEN
ASM_SIMP_TAC[ARG_INV; GSYM ARG_EQ_0] THEN
ASM_REAL_ARITH_TAC;
ALL_TAC] THEN
DISCH_THEN(SUBST1_TAC o SYM) THEN
REWRITE_TAC[EXTENSION; IN_DIFF; IN_UNIV; IN_ELIM_THM; ARG] THEN
REWRITE_TAC[REAL_NOT_LE] THEN
POP_ASSUM_LIST(MP_TAC o end_itlist CONJ) THEN
SPEC_TAC(`(dropout 3:real^3->real^2) w1`,`w:complex`) THEN
SPEC_TAC(`(dropout 3:real^3->real^2) w2`,`z:complex`) THEN
REPEAT GEN_TAC THEN STRIP_TAC THEN X_GEN_TAC `x3:real^3` THEN
SPEC_TAC(`(dropout 3:real^3->real^2) x3`,`x:complex`) THEN
GEN_TAC THEN REWRITE_TAC[COMPLEX_VEC_0] THEN
RULE_ASSUM_TAC(REWRITE_RULE[COMPLEX_VEC_0]) THEN
ASM_CASES_TAC `x = Cx(&0)` THEN ASM_REWRITE_TAC[] THENL
[ASM_REWRITE_TAC[complex_div; COMPLEX_MUL_LZERO; REAL_NOT_LT; ARG; ARG_0];
ALL_TAC] THEN
ASM_REWRITE_TAC[ARG_LT_NZ] THEN
MAP_EVERY UNDISCH_TAC
[`~(Arg (z / w) < pi)`;
`~(Arg (z / w) = pi)`;
`~(Arg (z / w) = &0)`;
`~(x = Cx (&0))`;
`~(w = Cx (&0))`;
`~(z = Cx (&0))`] THEN
POP_ASSUM_LIST(K ALL_TAC) THEN REWRITE_TAC[GSYM COMPLEX_VEC_0] THEN
GEOM_BASIS_MULTIPLE_TAC 1 `w:complex` THEN
X_GEN_TAC `w:real` THEN GEN_REWRITE_TAC LAND_CONV [REAL_LE_LT] THEN
ASM_CASES_TAC `w = &0` THEN ASM_REWRITE_TAC[VECTOR_MUL_LZERO] THEN
REWRITE_TAC[COMPLEX_CMUL; COMPLEX_BASIS; COMPLEX_VEC_0] THEN
SIMP_TAC[ARG_DIV_CX; COMPLEX_MUL_RID] THEN
REWRITE_TAC[real; RE_DIV_CX; IM_DIV_CX; CX_INJ] THEN
SIMP_TAC[complex_div; ARG_MUL_CX] THEN
SIMP_TAC[ARG_INV; GSYM ARG_EQ_0; ARG_INV_EQ_0] THEN
DISCH_TAC THEN REPEAT GEN_TAC THEN REWRITE_TAC[GSYM complex_div] THEN
ASM_CASES_TAC `Arg x = &0` THEN ASM_REWRITE_TAC[] THENL
[FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [ARG_EQ_0]) THEN
DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN
REWRITE_TAC[REAL] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN
REWRITE_TAC[complex_div; CX_INJ] THEN
ASM_SIMP_TAC[ARG_MUL_CX; REAL_LT_LE] THEN
ASM_SIMP_TAC[ARG_INV; GSYM ARG_EQ_0];
ALL_TAC] THEN
REWRITE_TAC[IMP_IMP; GSYM CONJ_ASSOC] THEN
SIMP_TAC[PI_POS; REAL_ARITH
`&0 < pi ==> (~(z = &0) /\ ~(z = pi) /\ ~(z < pi) <=> pi < z)`] THEN
STRIP_TAC THEN REWRITE_TAC[REAL_LT_SUB_RADD] THEN
DISJ_CASES_TAC(REAL_ARITH `Arg z <= Arg x \/ Arg x < Arg z`) THENL
[ASM_REWRITE_TAC[GSYM REAL_NOT_LE] THEN
ONCE_REWRITE_TAC[REAL_ADD_SYM] THEN
ASM_SIMP_TAC[GSYM ARG_LE_DIV_SUM] THEN
SIMP_TAC[ARG; REAL_LT_IMP_LE];
ALL_TAC] THEN
ASM_REWRITE_TAC[] THEN
MP_TAC(ISPECL [`x:complex`; `z:complex`] ARG_LE_DIV_SUM) THEN
ASM_SIMP_TAC[REAL_LT_IMP_LE] THEN DISCH_THEN SUBST1_TAC THEN
MATCH_MP_TAC(REAL_ARITH
`&0 <= x /\ ~(x = &0) /\ y = k - z ==> k < y + x + z`) THEN
ASM_REWRITE_TAC[ARG] THEN
GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [GSYM COMPLEX_INV_DIV] THEN
MATCH_MP_TAC ARG_INV THEN REWRITE_TAC[REAL] THEN
DISCH_THEN(STRIP_ASSUME_TAC o GSYM) THEN
ABBREV_TAC `t = Re(z / x)` THEN UNDISCH_TAC `Arg x < Arg z` THEN
UNDISCH_TAC `z / x = Cx t` THEN
ASM_SIMP_TAC[COMPLEX_FIELD
`~(x = Cx(&0)) ==> (z / x = t <=> z = t * x)`] THEN
ASM_CASES_TAC `t = &0` THEN ASM_REWRITE_TAC[COMPLEX_MUL_LZERO] THEN
ASM_SIMP_TAC[ARG_MUL_CX; REAL_LT_LE]);;
let OPEN_WEDGE = prove
(`!z:real^3 w w1 w2. open(wedge z w w1 w2)`,
REPEAT GEN_TAC THEN
ASM_CASES_TAC `z:real^3 = w \/ collinear{z,w,w1} \/ collinear{z,w,w2}` THENL
[FIRST_X_ASSUM STRIP_ASSUME_TAC THEN
ASM_SIMP_TAC[WEDGE_DEGENERATE; OPEN_EMPTY];
FIRST_X_ASSUM MP_TAC THEN REWRITE_TAC[DE_MORGAN_THM]] THEN
REWRITE_TAC[wedge] THEN GEOM_ORIGIN_TAC `z:real^3` THEN
GEOM_BASIS_MULTIPLE_TAC 3 `w:real^3` THEN
X_GEN_TAC `w:real` THEN GEN_REWRITE_TAC LAND_CONV [REAL_LE_LT] THEN
ASM_CASES_TAC `w = &0` THEN ASM_REWRITE_TAC[VECTOR_MUL_LZERO] THEN
ASM_SIMP_TAC[AZIM_SPECIAL_SCALE; COLLINEAR_SPECIAL_SCALE] THEN
REPEAT STRIP_TAC THEN
ONCE_REWRITE_TAC[TAUT `~a /\ b /\ c <=> ~(~a ==> ~(b /\ c))`] THEN
ASM_SIMP_TAC[AZIM_ARG] THEN REWRITE_TAC[COLLINEAR_BASIS_3] THEN
RULE_ASSUM_TAC(REWRITE_RULE[COLLINEAR_BASIS_3]) THEN
REWRITE_TAC[NOT_IMP; GSYM CONJ_ASSOC; DROPOUT_0] THEN
MATCH_MP_TAC OPEN_DROPOUT_3 THEN
UNDISCH_TAC `~((dropout 3:real^3->real^2) w1 = vec 0)` THEN
UNDISCH_TAC `~((dropout 3:real^3->real^2) w2 = vec 0)` THEN
SPEC_TAC(`(dropout 3:real^3->real^2) w2`,`v2:complex`) THEN
SPEC_TAC(`(dropout 3:real^3->real^2) w1`,`v1:complex`) THEN
POP_ASSUM_LIST(K ALL_TAC) THEN GEOM_BASIS_MULTIPLE_TAC 1 `v1:complex` THEN
X_GEN_TAC `v1:real` THEN GEN_REWRITE_TAC LAND_CONV [REAL_LE_LT] THEN
ASM_CASES_TAC `v1 = &0` THEN ASM_REWRITE_TAC[VECTOR_MUL_LZERO] THEN
REWRITE_TAC[COMPLEX_CMUL; COMPLEX_BASIS; COMPLEX_VEC_0] THEN
SIMP_TAC[ARG_DIV_CX; COMPLEX_MUL_RID] THEN REPEAT STRIP_TAC THEN
REWRITE_TAC[SET_RULE `{x | ~(x = a) /\ P x} = {x | P x} DIFF {a}`] THEN
MATCH_MP_TAC OPEN_DIFF THEN REWRITE_TAC[CLOSED_SING] THEN
MATCH_MP_TAC OPEN_ARG_LTT THEN
SIMP_TAC[REAL_LT_IMP_LE; REAL_LE_REFL; ARG]);;
let ARG_EQ_SUBSET_HALFLINE = prove
(`!a. ?b. ~(b = vec 0) /\ {z | Arg z = a} SUBSET aff_ge {vec 0} {b}`,
GEN_TAC THEN ASM_CASES_TAC `{z | Arg z = a} SUBSET {vec 0}` THENL
[EXISTS_TAC `basis 1:real^2` THEN
SIMP_TAC[BASIS_NONZERO; DIMINDEX_2; ARITH] THEN
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ]
SUBSET_TRANS)) THEN SIMP_TAC[SUBSET; IN_SING; ENDS_IN_HALFLINE];
ALL_TAC] THEN
FIRST_X_ASSUM(MP_TAC o MATCH_MP (SET_RULE
`~(s SUBSET {a}) ==> ?z. ~(a = z) /\ z IN s`)) THEN
MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `z:complex` THEN
REWRITE_TAC[IN_ELIM_THM] THEN DISCH_THEN(STRIP_ASSUME_TAC o GSYM) THEN
ASM_REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN
X_GEN_TAC `x:complex` THEN
ASM_CASES_TAC `x:complex = vec 0` THEN ASM_REWRITE_TAC[ENDS_IN_HALFLINE] THEN
RULE_ASSUM_TAC(REWRITE_RULE[COMPLEX_VEC_0]) THEN ASM_SIMP_TAC[ARG_EQ] THEN
DISCH_THEN(X_CHOOSE_THEN `u:real` STRIP_ASSUME_TAC) THEN
ASM_REWRITE_TAC[GSYM COMPLEX_CMUL] THEN
REWRITE_TAC[HALFLINE_EXPLICIT; IN_ELIM_THM; VECTOR_MUL_RZERO] THEN
MAP_EVERY EXISTS_TAC [`&1 - u`; `u:real`] THEN
ASM_SIMP_TAC[VECTOR_ADD_LID; REAL_LT_IMP_LE] THEN ASM_REAL_ARITH_TAC);;
let ARG_DIV_EQ_SUBSET_HALFLINE = prove
(`!w a. ~(w = vec 0)
==> ?b. ~(b = vec 0) /\
{z | Arg(z / w) = a} SUBSET aff_ge {vec 0} {b}`,
REPEAT GEN_TAC THEN GEOM_BASIS_MULTIPLE_TAC 1 `w:complex` THEN
X_GEN_TAC `w:real` THEN ASM_CASES_TAC `w = &0` THEN
ASM_REWRITE_TAC[VECTOR_MUL_LZERO; REAL_LE_LT] THEN DISCH_TAC THEN
X_GEN_TAC `a:real` THEN DISCH_THEN(K ALL_TAC) THEN
ASM_SIMP_TAC[ARG_DIV_CX; COMPLEX_CMUL; COMPLEX_BASIS; GSYM CX_MUL;
REAL_MUL_RID; ARG_EQ_SUBSET_HALFLINE]);;
let COPLANAR_AZIM_EQ = prove
(`!v0 v1 w1 a.
(collinear{v0,v1,w1} ==> ~(a = &0))
==> coplanar {z | azim v0 v1 w1 z = a}`,
REPEAT GEN_TAC THEN ASM_CASES_TAC `collinear{v0:real^3,v1,w1}` THENL
[ASM_SIMP_TAC[azim_def; EMPTY_GSPEC; COPLANAR_EMPTY]; ALL_TAC] THEN
ASM_REWRITE_TAC[] THEN POP_ASSUM MP_TAC THEN
GEOM_ORIGIN_TAC `v0:real^3` THEN
GEOM_BASIS_MULTIPLE_TAC 3 `v1:real^3` THEN
X_GEN_TAC `v1:real` THEN ASM_CASES_TAC `v1 = &0` THENL
[ASM_REWRITE_TAC[VECTOR_MUL_LZERO; INSERT_AC; COLLINEAR_2]; ALL_TAC] THEN
ASM_SIMP_TAC[REAL_LE_LT; COLLINEAR_SPECIAL_SCALE] THEN REPEAT STRIP_TAC THEN
ASM_SIMP_TAC[AZIM_SPECIAL_SCALE; AZIM_ARG] THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [COLLINEAR_BASIS_3]) THEN
POP_ASSUM_LIST(K ALL_TAC) THEN DISCH_THEN(X_CHOOSE_THEN `b:real^2`
STRIP_ASSUME_TAC o SPEC `a:real` o MATCH_MP ARG_DIV_EQ_SUBSET_HALFLINE) THEN
REWRITE_TAC[coplanar] THEN MAP_EVERY EXISTS_TAC
[`vec 0:real^3`; `pushin 3 (&0) (b:real^2):real^3`; `basis 3:real^3`] THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [SUBSET]) THEN
REWRITE_TAC[AFFINE_HULL_3; HALFLINE; SUBSET; IN_ELIM_THM] THEN
DISCH_THEN(fun th -> X_GEN_TAC `x:real^3` THEN DISCH_TAC THEN
MP_TAC(SPEC `(dropout 3:real^3->real^2) x` th)) THEN
ASM_REWRITE_TAC[VECTOR_MUL_RZERO; VECTOR_ADD_LID] THEN
DISCH_THEN(X_CHOOSE_THEN `v:real` STRIP_ASSUME_TAC) THEN
MAP_EVERY EXISTS_TAC [`&1 - v - (x:real^3)$3`; `v:real`; `(x:real^3)$3`] THEN
CONJ_TAC THENL [REAL_ARITH_TAC; ALL_TAC] THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [CART_EQ]) THEN
SIMP_TAC[CART_EQ; DIMINDEX_2; DIMINDEX_3; FORALL_2; FORALL_3; LAMBDA_BETA;
dropout; pushin; VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT; ARITH;
BASIS_COMPONENT] THEN
REAL_ARITH_TAC);;
(* ------------------------------------------------------------------------- *)
(* Volume of a tetrahedron defined by conv0. *)
(* ------------------------------------------------------------------------- *)
let delta_x = new_definition
`delta_x x1 x2 x3 x4 x5 x6 =
x1*x4*(--x1 + x2 + x3 -x4 + x5 + x6) +
x2*x5*(x1 - x2 + x3 + x4 -x5 + x6) +
x3*x6*(x1 + x2 - x3 + x4 + x5 - x6)
-x2*x3*x4 - x1*x3*x5 - x1*x2*x6 -x4*x5*x6:real`;;
let VOLUME_OF_CLOSED_TETRAHEDRON = prove
(`!x1 x2 x3 x4:real^3.
measure(convex hull {x1,x2,x3,x4}) =
sqrt(delta_x (dist(x1,x2) pow 2) (dist(x1,x3) pow 2) (dist(x1,x4) pow 2)
(dist(x3,x4) pow 2) (dist(x2,x4) pow 2) (dist(x2,x3) pow 2))
/ &12`,
REPEAT GEN_TAC THEN REWRITE_TAC[LET_DEF; LET_END_DEF] THEN
REWRITE_TAC[MEASURE_TETRAHEDRON] THEN
REWRITE_TAC[REAL_ARITH `x / &6 = y / &12 <=> y = &2 * x`] THEN
MATCH_MP_TAC SQRT_UNIQUE THEN
SIMP_TAC[REAL_LE_MUL; REAL_ABS_POS; REAL_POS] THEN
REWRITE_TAC[REAL_POW_MUL; REAL_POW2_ABS; delta_x] THEN
REWRITE_TAC[dist; NORM_POW_2] THEN
SIMP_TAC[DOT_3; VECTOR_SUB_COMPONENT; DIMINDEX_3; ARITH] THEN
CONV_TAC REAL_RING);;
let VOLUME_OF_TETRAHEDRON = prove
(`!v1 v2 v3 v4:real^3.
measure(conv0 {v1,v2,v3,v4}) =
let x12 = dist(v1,v2) pow 2 in
let x13 = dist(v1,v3) pow 2 in
let x14 = dist(v1,v4) pow 2 in
let x23 = dist(v2,v3) pow 2 in
let x24 = dist(v2,v4) pow 2 in
let x34 = dist(v3,v4) pow 2 in
sqrt(delta_x x12 x13 x14 x34 x24 x23)/(&12)`,
REPEAT GEN_TAC THEN CONV_TAC(TOP_DEPTH_CONV let_CONV) THEN
ASM_SIMP_TAC[GSYM VOLUME_OF_CLOSED_TETRAHEDRON] THEN
MATCH_MP_TAC MEASURE_CONV0_CONVEX_HULL THEN
SIMP_TAC[DIMINDEX_3; FINITE_INSERT; FINITE_EMPTY; CARD_CLAUSES] THEN
ARITH_TAC);;
(* ------------------------------------------------------------------------- *)
(* Circle area. Should maybe extend WLOG tactics for such scaling. *)
(* ------------------------------------------------------------------------- *)
let AREA_UNIT_CBALL = prove
(`measure(cball(vec 0:real^2,&1)) = pi`,
REPEAT STRIP_TAC THEN
MATCH_MP_TAC(INST_TYPE[`:1`,`:M`; `:2`,`:N`] FUBINI_SIMPLE_COMPACT) THEN
EXISTS_TAC `1` THEN
SIMP_TAC[DIMINDEX_1; DIMINDEX_2; ARITH; COMPACT_CBALL; SLICE_CBALL] THEN
REWRITE_TAC[VEC_COMPONENT; DROPOUT_0; REAL_SUB_RZERO] THEN
ONCE_REWRITE_TAC[COND_RAND] THEN REWRITE_TAC[MEASURE_EMPTY] THEN
SUBGOAL_THEN `!t. abs(t) <= &1 <=> t IN real_interval[-- &1,&1]`
(fun th -> REWRITE_TAC[th])
THENL [REWRITE_TAC[IN_REAL_INTERVAL] THEN REAL_ARITH_TAC; ALL_TAC] THEN
REWRITE_TAC[HAS_REAL_INTEGRAL_RESTRICT_UNIV; BALL_1] THEN
MATCH_MP_TAC HAS_REAL_INTEGRAL_EQ THEN
EXISTS_TAC `\t. &2 * sqrt(&1 - t pow 2)` THEN CONJ_TAC THENL
[X_GEN_TAC `t:real` THEN SIMP_TAC[IN_REAL_INTERVAL; MEASURE_INTERVAL] THEN
REWRITE_TAC[REAL_BOUNDS_LE; VECTOR_ADD_LID; VECTOR_SUB_LZERO] THEN
DISCH_TAC THEN
W(MP_TAC o PART_MATCH (lhs o rand) CONTENT_1 o rand o snd) THEN
REWRITE_TAC[LIFT_DROP; DROP_NEG] THEN
ANTS_TAC THENL [ALL_TAC; SIMP_TAC[REAL_POW_ONE] THEN REAL_ARITH_TAC] THEN
MATCH_MP_TAC(REAL_ARITH `&0 <= x ==> --x <= x`) THEN
ASM_SIMP_TAC[SQRT_POS_LE; REAL_SUB_LE; GSYM REAL_LE_SQUARE_ABS;
REAL_ABS_NUM];
ALL_TAC] THEN
MP_TAC(ISPECL
[`\x. asn(x) + x * sqrt(&1 - x pow 2)`;
`\x. &2 * sqrt(&1 - x pow 2)`;
`-- &1`; `&1`] REAL_FUNDAMENTAL_THEOREM_OF_CALCULUS_INTERIOR) THEN
REWRITE_TAC[ASN_1; ASN_NEG_1] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN
REWRITE_TAC[SQRT_0; REAL_MUL_RZERO; REAL_ADD_RID] THEN
REWRITE_TAC[REAL_ARITH `x / &2 - --(x / &2) = x`] THEN
DISCH_THEN MATCH_MP_TAC THEN CONJ_TAC THENL
[MATCH_MP_TAC REAL_CONTINUOUS_ON_ADD THEN
SIMP_TAC[REAL_CONTINUOUS_ON_ASN; IN_REAL_INTERVAL; REAL_BOUNDS_LE] THEN
MATCH_MP_TAC REAL_CONTINUOUS_ON_MUL THEN
REWRITE_TAC[REAL_CONTINUOUS_ON_ID] THEN
GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN
MATCH_MP_TAC REAL_CONTINUOUS_ON_COMPOSE THEN
SIMP_TAC[REAL_CONTINUOUS_ON_SUB; REAL_CONTINUOUS_ON_POW;
REAL_CONTINUOUS_ON_ID; REAL_CONTINUOUS_ON_CONST] THEN
REWRITE_TAC[REAL_CONTINUOUS_ON_SQRT];
REWRITE_TAC[IN_REAL_INTERVAL; REAL_BOUNDS_LT] THEN REPEAT STRIP_TAC THEN
REAL_DIFF_TAC THEN
CONV_TAC NUM_REDUCE_CONV THEN
REWRITE_TAC[REAL_MUL_LID; REAL_POW_1; REAL_MUL_RID] THEN
REWRITE_TAC[REAL_SUB_LZERO; REAL_MUL_RNEG; REAL_INV_MUL] THEN
ASM_REWRITE_TAC[REAL_SUB_LT; ABS_SQUARE_LT_1] THEN
MATCH_MP_TAC(REAL_FIELD
`s pow 2 = &1 - x pow 2 /\ x pow 2 < &1
==> (inv s + x * --(&2 * x) * inv (&2) * inv s + s) = &2 * s`) THEN
ASM_SIMP_TAC[ABS_SQUARE_LT_1; SQRT_POW_2; REAL_SUB_LE; REAL_LT_IMP_LE]]);;
let AREA_CBALL = prove
(`!z:real^2 r. &0 <= r ==> measure(cball(z,r)) = pi * r pow 2`,
REPEAT STRIP_TAC THEN ASM_CASES_TAC `r = &0` THENL
[ASM_SIMP_TAC[CBALL_SING; REAL_POW_2; REAL_MUL_RZERO] THEN
MATCH_MP_TAC MEASURE_UNIQUE THEN
REWRITE_TAC[HAS_MEASURE_0; NEGLIGIBLE_SING];
ALL_TAC] THEN
SUBGOAL_THEN `&0 < r` ASSUME_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
MP_TAC(ISPECL [`cball(vec 0:real^2,&1)`; `r:real`; `z:real^2`; `pi`]
HAS_MEASURE_AFFINITY) THEN
REWRITE_TAC[HAS_MEASURE_MEASURABLE_MEASURE; MEASURABLE_CBALL;
AREA_UNIT_CBALL] THEN
ASM_REWRITE_TAC[real_abs; DIMINDEX_2] THEN
DISCH_THEN(MP_TAC o CONJUNCT2) THEN
GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [REAL_MUL_SYM] THEN
DISCH_THEN(SUBST1_TAC o SYM) THEN AP_TERM_TAC THEN
MATCH_MP_TAC SUBSET_ANTISYM THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN
REWRITE_TAC[IN_CBALL_0; IN_IMAGE] THEN REWRITE_TAC[IN_CBALL] THEN
REWRITE_TAC[NORM_ARITH `dist(z,a + z) = norm a`; NORM_MUL] THEN
ONCE_REWRITE_TAC[REAL_ARITH `abs r * x <= r <=> abs r * x <= r * &1`] THEN
ASM_SIMP_TAC[real_abs; REAL_LE_LMUL; dist] THEN X_GEN_TAC `w:real^2` THEN
DISCH_TAC THEN EXISTS_TAC `inv(r) % (w - z):real^2` THEN
ASM_SIMP_TAC[VECTOR_MUL_ASSOC; REAL_MUL_RINV] THEN
CONJ_TAC THENL [NORM_ARITH_TAC; ALL_TAC] THEN
REWRITE_TAC[NORM_MUL; REAL_ABS_INV] THEN ASM_REWRITE_TAC[real_abs] THEN
ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN
ASM_SIMP_TAC[GSYM real_div; REAL_LE_LDIV_EQ; REAL_MUL_LID] THEN
ONCE_REWRITE_TAC[NORM_SUB] THEN ASM_REWRITE_TAC[]);;
let AREA_BALL = prove
(`!z:real^2 r. &0 <= r ==> measure(ball(z,r)) = pi * r pow 2`,
SIMP_TAC[GSYM INTERIOR_CBALL; GSYM AREA_CBALL] THEN
REPEAT STRIP_TAC THEN MATCH_MP_TAC MEASURE_INTERIOR THEN
SIMP_TAC[BOUNDED_CBALL; NEGLIGIBLE_CONVEX_FRONTIER; CONVEX_CBALL]);;
(* ------------------------------------------------------------------------- *)
(* Volume of a ball. *)
(* ------------------------------------------------------------------------- *)
let VOLUME_CBALL = prove
(`!z:real^3 r. &0 <= r ==> measure(cball(z,r)) = &4 / &3 * pi * r pow 3`,
GEOM_ORIGIN_TAC `z:real^3` THEN REPEAT STRIP_TAC THEN
MATCH_MP_TAC(INST_TYPE[`:2`,`:M`; `:3`,`:N`] FUBINI_SIMPLE_COMPACT) THEN
EXISTS_TAC `1` THEN
SIMP_TAC[DIMINDEX_2; DIMINDEX_3; ARITH; COMPACT_CBALL; SLICE_CBALL] THEN
REWRITE_TAC[VEC_COMPONENT; DROPOUT_0; REAL_SUB_RZERO] THEN
ONCE_REWRITE_TAC[COND_RAND] THEN REWRITE_TAC[MEASURE_EMPTY] THEN
SUBGOAL_THEN `!t. abs(t) <= r <=> t IN real_interval[--r,r]`
(fun th -> REWRITE_TAC[th])
THENL [REWRITE_TAC[IN_REAL_INTERVAL] THEN REAL_ARITH_TAC; ALL_TAC] THEN
REWRITE_TAC[HAS_REAL_INTEGRAL_RESTRICT_UNIV] THEN
MATCH_MP_TAC HAS_REAL_INTEGRAL_EQ THEN
EXISTS_TAC `\t. pi * (r pow 2 - t pow 2)` THEN CONJ_TAC THENL
[X_GEN_TAC `t:real` THEN REWRITE_TAC[IN_REAL_INTERVAL; REAL_BOUNDS_LE] THEN
SIMP_TAC[AREA_CBALL; SQRT_POS_LE; REAL_SUB_LE; GSYM REAL_LE_SQUARE_ABS;
SQRT_POW_2; REAL_ARITH `abs x <= r ==> abs x <= abs r`];
ALL_TAC] THEN
MP_TAC(ISPECL
[`\t. pi * (r pow 2 * t - &1 / &3 * t pow 3)`;
`\t. pi * (r pow 2 - t pow 2)`;
`--r:real`; `r:real`] REAL_FUNDAMENTAL_THEOREM_OF_CALCULUS) THEN
REWRITE_TAC[] THEN ANTS_TAC THENL
[CONJ_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
REPEAT STRIP_TAC THEN REAL_DIFF_TAC THEN
CONV_TAC NUM_REDUCE_CONV THEN CONV_TAC REAL_RING;
MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN
CONV_TAC REAL_RING]);;
let VOLUME_BALL = prove
(`!z:real^3 r. &0 <= r ==> measure(ball(z,r)) = &4 / &3 * pi * r pow 3`,
SIMP_TAC[GSYM INTERIOR_CBALL; GSYM VOLUME_CBALL] THEN
REPEAT STRIP_TAC THEN MATCH_MP_TAC MEASURE_INTERIOR THEN
SIMP_TAC[BOUNDED_CBALL; NEGLIGIBLE_CONVEX_FRONTIER; CONVEX_CBALL]);;
(* ------------------------------------------------------------------------- *)
(* Frustum. *)
(* ------------------------------------------------------------------------- *)
let rconesgn = new_definition
`rconesgn sgn v w h =
{x:real^A | sgn ((x-v) dot (w-v)) (dist(x,v)*dist(w,v)*h)}`;;
let rcone_gt = new_definition `rcone_gt = rconesgn ( > )`;;
let rcone_ge = new_definition `rcone_ge = rconesgn ( >= )`;;
let rcone_eq = new_definition `rcone_eq = rconesgn ( = )`;;
let frustum = new_definition
`frustum v0 v1 h1 h2 a =
{ y:real^N | rcone_gt v0 v1 a y /\
let d = (y - v0) dot (v1 - v0) in
let n = norm(v1 - v0) in
(h1*n < d /\ d < h2*n)}`;;
let frustt = new_definition `frustt v0 v1 h a = frustum v0 v1 (&0) h a`;;
let FRUSTUM_DEGENERATE = prove
(`!v0 h1 h2 a. frustum v0 v0 h1 h2 a = {}`,
REWRITE_TAC[frustum; VECTOR_SUB_REFL; NORM_0; DOT_RZERO] THEN
CONV_TAC(TOP_DEPTH_CONV let_CONV) THEN
REWRITE_TAC[REAL_MUL_RZERO; REAL_LT_REFL] THEN SET_TAC[]);;
let CONVEX_RCONE_GT = prove
(`!v0 v1:real^N a. &0 <= a ==> convex(rcone_gt v0 v1 a)`,
REWRITE_TAC[rcone_gt; rconesgn] THEN
GEOM_ORIGIN_TAC `v0:real^N` THEN REPEAT GEN_TAC THEN
REWRITE_TAC[VECTOR_SUB_RZERO; DIST_0] THEN
REWRITE_TAC[CONVEX_ALT; IN_ELIM_THM; real_gt; DOT_LADD; DOT_LMUL] THEN
DISCH_TAC THEN MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`; `t:real`] THEN
REPEAT STRIP_TAC THEN MATCH_MP_TAC REAL_LET_TRANS THEN
EXISTS_TAC `(&1 - t) * norm(x:real^N) * norm v1 * a +
t * norm(y:real^N) * norm(v1:real^N) * a` THEN
CONJ_TAC THENL
[REWRITE_TAC[GSYM REAL_ADD_RDISTRIB; REAL_MUL_ASSOC] THEN
MATCH_MP_TAC REAL_LE_RMUL THEN ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC REAL_LE_RMUL THEN REWRITE_TAC[NORM_POS_LE] THEN
MATCH_MP_TAC(NORM_ARITH
`norm(x:real^N) = a /\ norm(y) = b ==> norm(x + y) <= a + b`) THEN
REWRITE_TAC[NORM_MUL] THEN CONJ_TAC THEN AP_THM_TAC THEN AP_TERM_TAC THEN
ASM_REAL_ARITH_TAC;
MATCH_MP_TAC REAL_CONVEX_BOUND2_LT THEN ASM_REAL_ARITH_TAC]);;
let OPEN_RCONE_GT = prove
(`!v0 v1:real^N a. open(rcone_gt v0 v1 a)`,
REWRITE_TAC[rcone_gt; rconesgn] THEN
GEOM_ORIGIN_TAC `v0:real^N` THEN REPEAT GEN_TAC THEN
REWRITE_TAC[VECTOR_SUB_RZERO; DIST_0] THEN
MP_TAC(ISPECL [`\x:real^N. lift(x dot v1 - norm x * norm v1 * a)`;
`{x:real^1 | x$1 > &0}`]
CONTINUOUS_OPEN_PREIMAGE_UNIV) THEN
REWRITE_TAC[OPEN_HALFSPACE_COMPONENT_GT] THEN REWRITE_TAC[GSYM drop] THEN
REWRITE_TAC[IN_ELIM_THM; real_gt; REAL_SUB_LT; LIFT_DROP] THEN
DISCH_THEN MATCH_MP_TAC THEN GEN_TAC THEN REWRITE_TAC[LIFT_SUB] THEN
ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN REWRITE_TAC[LIFT_CMUL] THEN
MATCH_MP_TAC CONTINUOUS_SUB THEN ONCE_REWRITE_TAC[DOT_SYM] THEN
REWRITE_TAC[REWRITE_RULE[o_DEF] CONTINUOUS_AT_LIFT_DOT] THEN
MATCH_MP_TAC CONTINUOUS_CMUL THEN
REWRITE_TAC[REWRITE_RULE[o_DEF] CONTINUOUS_AT_LIFT_NORM]);;
let RCONE_GT_NEG = prove
(`!v0 v1:real^N a.
rcone_gt v0 v1 (--a) =
IMAGE (\x. &2 % v0 - x) ((:real^N) DIFF rcone_ge v0 v1 a)`,
REPEAT GEN_TAC THEN CONV_TAC SYM_CONV THEN
MATCH_MP_TAC SURJECTIVE_IMAGE_EQ THEN REWRITE_TAC[] THEN CONJ_TAC THENL
[MESON_TAC[VECTOR_ARITH `a - (a - b):real^N = b`];
REWRITE_TAC[rcone_gt; rconesgn; rcone_ge;
IN_ELIM_THM; IN_DIFF; IN_UNIV] THEN
REWRITE_TAC[NORM_ARITH `dist(&2 % x - y,x) = dist(y,x)`] THEN
REWRITE_TAC[VECTOR_ARITH `&2 % v - x - v:real^N = --(x - v)`] THEN
REWRITE_TAC[DOT_LNEG] THEN REAL_ARITH_TAC]);;
let VOLUME_FRUSTT_STRONG = prove
(`!v0 v1:real^3 h a.
&0 < a
==> bounded(frustt v0 v1 h a) /\
convex(frustt v0 v1 h a) /\
measurable(frustt v0 v1 h a) /\
measure(frustt v0 v1 h a) =
if v1 = v0 \/ &1 <= a \/ h < &0 then &0
else pi * ((h / a) pow 2 - h pow 2) * h / &3`,
REPEAT GEN_TAC THEN DISCH_TAC THEN
REWRITE_TAC[frustt; frustum; rcone_gt; rconesgn; IN_ELIM_THM] THEN
CONV_TAC(TOP_DEPTH_CONV let_CONV) THEN GEOM_ORIGIN_TAC `v0:real^3` THEN
REWRITE_TAC[VECTOR_SUB_RZERO; REAL_MUL_LZERO; DIST_0; real_gt] THEN
GEOM_BASIS_MULTIPLE_TAC 1 `v1:real^3` THEN
X_GEN_TAC `b:real` THEN REPEAT(GEN_TAC ORELSE DISCH_TAC) THEN
FIRST_ASSUM(DISJ_CASES_TAC o MATCH_MP (REAL_ARITH
`&0 <= x ==> x = &0 \/ &0 < x`)) THEN
ASM_REWRITE_TAC[DOT_RZERO; REAL_MUL_LZERO; REAL_MUL_RZERO; REAL_LT_REFL;
MEASURABLE_EMPTY; MEASURE_EMPTY; EMPTY_GSPEC; VECTOR_MUL_LZERO;
BOUNDED_EMPTY; CONVEX_EMPTY] THEN
ASM_CASES_TAC `&1 <= a` THEN ASM_REWRITE_TAC[] THENL
[SUBGOAL_THEN
`!y:real^3. ~(norm(y) * norm(b % basis 1:real^3) * a
< y dot (b % basis 1))`
(fun th -> REWRITE_TAC[th; EMPTY_GSPEC; MEASURABLE_EMPTY;
BOUNDED_EMPTY; CONVEX_EMPTY; MEASURE_EMPTY]) THEN
REWRITE_TAC[REAL_NOT_LT] THEN X_GEN_TAC `y:real^3` THEN
MATCH_MP_TAC(REAL_ARITH `abs(x) <= a ==> x <= a`) THEN
SIMP_TAC[DOT_RMUL; NORM_MUL; REAL_ABS_MUL; DOT_BASIS; NORM_BASIS;
DIMINDEX_3; ARITH] THEN
REWRITE_TAC[REAL_ARITH
`b * y <= n * (b * &1) * a <=> b * &1 * y <= b * a * n`] THEN
MATCH_MP_TAC REAL_LE_LMUL THEN REWRITE_TAC[REAL_ABS_POS] THEN
MATCH_MP_TAC REAL_LE_MUL2 THEN
ASM_SIMP_TAC[REAL_POS; REAL_ABS_POS; COMPONENT_LE_NORM; DIMINDEX_3; ARITH];
ALL_TAC] THEN
RULE_ASSUM_TAC(REWRITE_RULE[REAL_NOT_LE]) THEN
SIMP_TAC[NORM_MUL; NORM_BASIS; DOT_BASIS; DOT_RMUL; DIMINDEX_3; ARITH] THEN
ONCE_REWRITE_TAC[REAL_ARITH `n * x * a:real = x * n * a`] THEN
ASM_REWRITE_TAC[real_abs; REAL_MUL_RID] THEN
ASM_SIMP_TAC[REAL_MUL_RID; REAL_LT_LMUL_EQ; REAL_LT_MUL_EQ; NORM_POS_LT] THEN
ASM_SIMP_TAC[VECTOR_MUL_EQ_0; BASIS_NONZERO; DIMINDEX_3; ARITH;
REAL_LT_IMP_NZ] THEN
ASM_SIMP_TAC[GSYM REAL_LT_RDIV_EQ; NORM_LT_SQUARE] THEN
ASM_SIMP_TAC[REAL_POW_DIV; REAL_POW_LT; REAL_LT_RDIV_EQ] THEN
REWRITE_TAC[REAL_ARITH `(&0 * x < y /\ u < v) /\ &0 < y /\ y < h <=>
&0 < y /\ y < h /\ u < v`] THEN
MATCH_MP_TAC(TAUT `a /\ b /\ (a /\ b ==> c) ==> a /\ b /\ c`) THEN
REPEAT CONJ_TAC THENL
[MATCH_MP_TAC BOUNDED_SUBSET THEN
EXISTS_TAC `ball(vec 0:real^3,h / a)` THEN
REWRITE_TAC[BOUNDED_BALL; IN_BALL_0; SUBSET; IN_ELIM_THM] THEN
REWRITE_TAC[NORM_LT_SQUARE] THEN
ASM_SIMP_TAC[REAL_POW_DIV; REAL_LT_RDIV_EQ; REAL_POW_LT] THEN
X_GEN_TAC `x:real^3` THEN STRIP_TAC THEN
CONJ_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP
(REWRITE_RULE[IMP_CONJ] REAL_LTE_TRANS)) THEN
MATCH_MP_TAC REAL_POW_LE2 THEN ASM_REAL_ARITH_TAC;
REWRITE_TAC[SET_RULE `{x | P x /\ Q x /\ R x} =
{x | Q x} INTER {x | P x /\ R x}`] THEN
REWRITE_TAC[REAL_ARITH `&0 < y <=> y > &0`] THEN
MATCH_MP_TAC CONVEX_INTER THEN
REWRITE_TAC[CONVEX_HALFSPACE_COMPONENT_LT] THEN
MP_TAC(ISPECL [`vec 0:real^3`; `basis 1:real^3`; `a:real`]
CONVEX_RCONE_GT) THEN
ASM_SIMP_TAC[REAL_LT_IMP_LE; rcone_gt; rconesgn] THEN
REWRITE_TAC[VECTOR_SUB_RZERO; DIST_0] THEN
SIMP_TAC[DOT_BASIS; NORM_BASIS; DIMINDEX_3; ARITH] THEN
REWRITE_TAC[real_gt; REAL_MUL_LID] THEN
ASM_SIMP_TAC[GSYM REAL_LT_RDIV_EQ] THEN
REWRITE_TAC[NORM_LT_SQUARE] THEN
ASM_SIMP_TAC[REAL_POW_DIV; REAL_LT_RDIV_EQ; REAL_POW_LT] THEN
REWRITE_TAC[REAL_MUL_LZERO];
ALL_TAC] THEN
STRIP_TAC THEN
MATCH_MP_TAC(INST_TYPE [`:2`,`:M`] FUBINI_SIMPLE_CONVEX_STRONG) THEN
EXISTS_TAC `1` THEN REWRITE_TAC[DIMINDEX_2; DIMINDEX_3; ARITH] THEN
ASM_REWRITE_TAC[] THEN
SIMP_TAC[SLICE_312; DIMINDEX_2; DIMINDEX_3; ARITH; IN_ELIM_THM;
VECTOR_3; DOT_3; GSYM DOT_2] THEN
SUBGOAL_THEN `&0 < inv(a pow 2) - &1` ASSUME_TAC THENL
[REWRITE_TAC[REAL_SUB_LT] THEN MATCH_MP_TAC REAL_INV_1_LT THEN
ASM_SIMP_TAC[REAL_POW_1_LT; REAL_LT_IMP_LE; ARITH; REAL_POW_LT];
ALL_TAC] THEN
MATCH_MP_TAC HAS_REAL_INTEGRAL_EQ THEN
EXISTS_TAC `\t. if &0 < t /\ t < h then pi * (inv(a pow 2) - &1) * t pow 2
else &0` THEN
CONJ_TAC THENL
[X_GEN_TAC `t:real` THEN DISCH_TAC THEN REWRITE_TAC[] THEN
COND_CASES_TAC THEN
ASM_REWRITE_TAC[EMPTY_GSPEC; CONJ_ASSOC;
MEASURE_EMPTY; MEASURABLE_EMPTY] THEN
MATCH_MP_TAC EQ_TRANS THEN
EXISTS_TAC `measure(ball(vec 0:real^2,sqrt(inv(a pow 2) - &1) * t))` THEN
CONJ_TAC THENL
[W(MP_TAC o PART_MATCH (lhs o rand) AREA_BALL o rand o snd) THEN
ASM_SIMP_TAC[REAL_LT_IMP_LE; SQRT_POS_LT; REAL_LT_MUL] THEN
ASM_SIMP_TAC[SQRT_POW_2; REAL_LT_IMP_LE; REAL_POW_MUL];
AP_TERM_TAC THEN REWRITE_TAC[IN_BALL_0; EXTENSION; IN_ELIM_THM] THEN
REWRITE_TAC[NORM_LT_SQUARE] THEN
ASM_SIMP_TAC[SQRT_POS_LT; SQRT_POW_2; REAL_LT_IMP_LE; REAL_LT_MUL;
REAL_POW_MUL; GSYM REAL_LT_RDIV_EQ; REAL_POW_LT] THEN
REAL_ARITH_TAC];
ALL_TAC] THEN
REWRITE_TAC[GSYM IN_REAL_INTERVAL; HAS_REAL_INTEGRAL_RESTRICT_UNIV] THEN
REWRITE_TAC[HAS_REAL_INTEGRAL_OPEN_INTERVAL] THEN
COND_CASES_TAC THENL
[ASM_MESON_TAC[REAL_INTERVAL_EQ_EMPTY; HAS_REAL_INTEGRAL_EMPTY];
RULE_ASSUM_TAC(REWRITE_RULE[REAL_NOT_LT])] THEN
MP_TAC(ISPECL
[`\t. pi / &3 * (inv (a pow 2) - &1) * t pow 3`;
`\t. pi * (inv (a pow 2) - &1) * t pow 2`;
`&0`; `h:real`] REAL_FUNDAMENTAL_THEOREM_OF_CALCULUS) THEN
REWRITE_TAC[] THEN ANTS_TAC THENL
[ASM_REWRITE_TAC[] THEN
REPEAT STRIP_TAC THEN REAL_DIFF_TAC THEN
CONV_TAC NUM_REDUCE_CONV THEN CONV_TAC REAL_RING;
MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN
UNDISCH_TAC `&0 < a` THEN CONV_TAC REAL_FIELD]);;
let VOLUME_FRUSTT = prove
(`!v0 v1:real^3 h a.
&0 < a
==> measurable(frustt v0 v1 h a) /\
measure(frustt v0 v1 h a) =
if v1 = v0 \/ &1 <= a \/ h < &0 then &0
else pi * ((h / a) pow 2 - h pow 2) * h / &3`,
SIMP_TAC[VOLUME_FRUSTT_STRONG]);;
(* ------------------------------------------------------------------------- *)
(* Ellipsoid. *)
(* ------------------------------------------------------------------------- *)
let scale = new_definition
`scale (t:real^3) (u:real^3):real^3 =
vector[t$1 * u$1; t$2 * u$2; t$3 * u$3]`;;
let normball = new_definition `normball x r = { y:real^A | dist(y,x) < r}`;;
let ellipsoid = new_definition
`ellipsoid t r = IMAGE (scale t) (normball(vec 0) r)`;;
let NORMBALL_BALL = prove
(`!z r. normball z r = ball(z,r)`,
REWRITE_TAC[normball; ball; DIST_SYM]);;
let MEASURE_SCALE = prove
(`!s. measurable s
==> measurable(IMAGE (scale t) s) /\
measure(IMAGE (scale t) s) = abs(t$1 * t$2 * t$3) * measure s`,
GEN_TAC THEN GEN_REWRITE_TAC LAND_CONV [HAS_MEASURE_MEASURE] THEN
DISCH_THEN(MP_TAC o SPEC `\i. (t:real^3)$i` o
MATCH_MP HAS_MEASURE_STRETCH) THEN
REWRITE_TAC[DIMINDEX_3; PRODUCT_3] THEN
SUBGOAL_THEN `(\x:real^3. (lambda k. t$k * x$k):real^3) = scale t`
SUBST1_TAC THENL
[SIMP_TAC[CART_EQ; FUN_EQ_THM; scale; LAMBDA_BETA; DIMINDEX_3;
VECTOR_3; ARITH; FORALL_3];
MESON_TAC[measurable; MEASURE_UNIQUE]]);;
let MEASURE_ELLIPSOID = prove
(`!t r. &0 <= r
==> measurable(ellipsoid t r) /\
measure(ellipsoid t r) =
abs(t$1 * t$2 * t$3) * &4 / &3 * pi * r pow 3`,
REPEAT GEN_TAC THEN DISCH_TAC THEN
FIRST_X_ASSUM(SUBST1_TAC o SYM o
SPEC `vec 0:real^3` o MATCH_MP VOLUME_BALL) THEN
REWRITE_TAC[normball; ellipsoid] THEN ONCE_REWRITE_TAC[DIST_SYM] THEN
REWRITE_TAC[GSYM ball] THEN MATCH_MP_TAC MEASURE_SCALE THEN
REWRITE_TAC[MEASURABLE_BALL]);;
let MEASURABLE_ELLIPSOID = prove
(`!t r. measurable(ellipsoid t r)`,
REPEAT GEN_TAC THEN
ASM_CASES_TAC `&0 <= r` THEN ASM_SIMP_TAC[MEASURE_ELLIPSOID] THEN
REWRITE_TAC[ellipsoid; NORMBALL_BALL; IMAGE; IN_BALL_0] THEN
ASM_SIMP_TAC[NORM_ARITH `~(&0 <= r) ==> ~(norm(x:real^3) < r)`] THEN
REWRITE_TAC[EMPTY_GSPEC; MEASURABLE_EMPTY]);;
(* ------------------------------------------------------------------------- *)
(* Conic cap. *)
(* ------------------------------------------------------------------------- *)
let conic_cap = new_definition
`conic_cap v0 v1 r a = normball v0 r INTER rcone_gt v0 v1 a`;;
let CONIC_CAP_DEGENERATE = prove
(`!v0 r a. conic_cap v0 v0 r a = {}`,
REWRITE_TAC[conic_cap; rcone_gt; rconesgn; VECTOR_SUB_REFL] THEN
REWRITE_TAC[DIST_REFL; DOT_RZERO; REAL_MUL_RZERO; REAL_MUL_LZERO] THEN
REWRITE_TAC[real_gt; REAL_LT_REFL] THEN SET_TAC[]);;
let BOUNDED_CONIC_CAP = prove
(`!v0 v1:real^3 r a. bounded(conic_cap v0 v1 r a)`,
REPEAT GEN_TAC THEN REWRITE_TAC[conic_cap; NORMBALL_BALL] THEN
MATCH_MP_TAC BOUNDED_SUBSET THEN EXISTS_TAC `ball(v0:real^3,r)` THEN
REWRITE_TAC[BOUNDED_BALL] THEN SET_TAC[]);;
let MEASURABLE_CONIC_CAP = prove
(`!v0 v1:real^3 r a. measurable(conic_cap v0 v1 r a)`,
REPEAT GEN_TAC THEN REWRITE_TAC[conic_cap; NORMBALL_BALL] THEN
MATCH_MP_TAC MEASURABLE_OPEN THEN
SIMP_TAC[OPEN_INTER; OPEN_RCONE_GT; OPEN_BALL] THEN
MATCH_MP_TAC BOUNDED_SUBSET THEN EXISTS_TAC `ball(v0:real^3,r)` THEN
REWRITE_TAC[BOUNDED_BALL] THEN SET_TAC[]);;
let VOLUME_CONIC_CAP_STRONG = prove
(`!v0 v1:real^3 r a.
&0 < a
==> bounded(conic_cap v0 v1 r a) /\
convex(conic_cap v0 v1 r a) /\
measurable(conic_cap v0 v1 r a) /\
measure(conic_cap v0 v1 r a) =
if v1 = v0 \/ &1 <= a \/ r < &0 then &0
else &2 / &3 * pi * (&1 - a) * r pow 3`,
REPEAT GEN_TAC THEN DISCH_TAC THEN
REWRITE_TAC[conic_cap; rcone_gt; rconesgn; IN_ELIM_THM] THEN
REWRITE_TAC[ONCE_REWRITE_RULE[DIST_SYM] normball; GSYM ball] THEN
CONV_TAC(TOP_DEPTH_CONV let_CONV) THEN GEOM_ORIGIN_TAC `v0:real^3` THEN
REWRITE_TAC[VECTOR_SUB_RZERO; REAL_MUL_LZERO; DIST_0; real_gt] THEN
GEOM_BASIS_MULTIPLE_TAC 1 `v1:real^3` THEN
X_GEN_TAC `b:real` THEN REPEAT(GEN_TAC ORELSE DISCH_TAC) THEN
FIRST_ASSUM(DISJ_CASES_TAC o MATCH_MP (REAL_ARITH
`&0 <= x ==> x = &0 \/ &0 < x`))
THENL
[ASM_REWRITE_TAC[VECTOR_MUL_LZERO; GSYM REAL_NOT_LE; DOT_RZERO] THEN
ASM_SIMP_TAC[REAL_LE_MUL; REAL_LT_IMP_LE; NORM_POS_LE] THEN
REWRITE_TAC[EMPTY_GSPEC; INTER_EMPTY; MEASURE_EMPTY; MEASURABLE_EMPTY;
CONVEX_EMPTY; BOUNDED_EMPTY];
ALL_TAC] THEN
ASM_CASES_TAC `&1 <= a` THEN ASM_REWRITE_TAC[] THENL
[SUBGOAL_THEN
`!y:real^3. ~(norm(y) * norm(b % basis 1:real^3) * a
< y dot (b % basis 1))`
(fun th -> REWRITE_TAC[th; EMPTY_GSPEC; INTER_EMPTY; MEASURE_EMPTY;
MEASURABLE_EMPTY; BOUNDED_EMPTY; CONVEX_EMPTY]) THEN
REWRITE_TAC[REAL_NOT_LT] THEN X_GEN_TAC `y:real^3` THEN
MATCH_MP_TAC(REAL_ARITH `abs(x) <= a ==> x <= a`) THEN
SIMP_TAC[DOT_RMUL; NORM_MUL; REAL_ABS_MUL; DOT_BASIS; NORM_BASIS;
DIMINDEX_3; ARITH] THEN
REWRITE_TAC[REAL_ARITH
`b * y <= n * (b * &1) * a <=> b * &1 * y <= b * a * n`] THEN
MATCH_MP_TAC REAL_LE_LMUL THEN REWRITE_TAC[REAL_ABS_POS] THEN
MATCH_MP_TAC REAL_LE_MUL2 THEN
ASM_SIMP_TAC[REAL_POS; REAL_ABS_POS; COMPONENT_LE_NORM; DIMINDEX_3; ARITH];
ALL_TAC] THEN
RULE_ASSUM_TAC(REWRITE_RULE[REAL_NOT_LE]) THEN
SIMP_TAC[DOT_RMUL; NORM_MUL; REAL_ABS_NORM; DOT_BASIS;
DIMINDEX_3; ARITH; NORM_BASIS] THEN
ONCE_REWRITE_TAC[REAL_ARITH `n * x * a:real = x * n * a`] THEN
ASM_REWRITE_TAC[real_abs; REAL_MUL_RID] THEN
ASM_SIMP_TAC[REAL_MUL_RID; REAL_LT_LMUL_EQ; REAL_LT_MUL_EQ; NORM_POS_LT] THEN
ASM_SIMP_TAC[GSYM REAL_LT_RDIV_EQ; NORM_LT_SQUARE] THEN
ASM_SIMP_TAC[REAL_POW_DIV; REAL_POW_LT; REAL_LT_RDIV_EQ] THEN
REWRITE_TAC[INTER; REAL_MUL_LZERO; IN_BALL_0; IN_ELIM_THM] THEN
ASM_SIMP_TAC[VECTOR_MUL_EQ_0; BASIS_NONZERO; DIMINDEX_3; ARITH;
REAL_LT_IMP_NZ] THEN
COND_CASES_TAC THENL
[ASM_SIMP_TAC[NORM_ARITH `r < &0 ==> ~(norm x < r)`] THEN
REWRITE_TAC[EMPTY_GSPEC; MEASURE_EMPTY; MEASURABLE_EMPTY;
BOUNDED_EMPTY; CONVEX_EMPTY];
RULE_ASSUM_TAC(ONCE_REWRITE_RULE[REAL_NOT_LT])] THEN
MATCH_MP_TAC(TAUT `a /\ b /\ (a /\ b ==> c /\ d) ==> a /\ b /\ c /\ d`) THEN
REPEAT CONJ_TAC THENL
[MATCH_MP_TAC BOUNDED_SUBSET THEN
EXISTS_TAC `ball(vec 0:real^3,r)` THEN
SIMP_TAC[BOUNDED_BALL; IN_BALL_0; SUBSET; IN_ELIM_THM];
ONCE_REWRITE_TAC[SET_RULE
`{x | P x /\ Q x} = {x | P x} INTER {x | Q x}`] THEN
MATCH_MP_TAC CONVEX_INTER THEN
REWRITE_TAC[GSYM IN_BALL_0; CONVEX_BALL; SIMPLE_IMAGE; IMAGE_ID] THEN
MP_TAC(ISPECL [`vec 0:real^3`; `basis 1:real^3`; `a:real`]
CONVEX_RCONE_GT) THEN
ASM_SIMP_TAC[REAL_LT_IMP_LE; rcone_gt; rconesgn] THEN
REWRITE_TAC[VECTOR_SUB_RZERO; DIST_0] THEN
SIMP_TAC[DOT_BASIS; NORM_BASIS; DIMINDEX_3; ARITH] THEN
REWRITE_TAC[real_gt; REAL_MUL_LID] THEN
ASM_SIMP_TAC[GSYM REAL_LT_RDIV_EQ] THEN
REWRITE_TAC[NORM_LT_SQUARE] THEN
ASM_SIMP_TAC[REAL_POW_DIV; REAL_LT_RDIV_EQ; REAL_POW_LT] THEN
REWRITE_TAC[REAL_MUL_LZERO];
STRIP_TAC] THEN
MATCH_MP_TAC(INST_TYPE [`:2`,`:M`] FUBINI_SIMPLE_CONVEX_STRONG) THEN
EXISTS_TAC `1` THEN ASM_REWRITE_TAC[DIMINDEX_2; DIMINDEX_3; ARITH] THEN
SIMP_TAC[SLICE_312; DIMINDEX_2; DIMINDEX_3; ARITH; IN_ELIM_THM;
VECTOR_3; DOT_3; GSYM DOT_2] THEN
SUBGOAL_THEN `&0 < inv(a pow 2) - &1` ASSUME_TAC THENL
[REWRITE_TAC[REAL_SUB_LT] THEN MATCH_MP_TAC REAL_INV_1_LT THEN
ASM_SIMP_TAC[REAL_POW_1_LT; REAL_LT_IMP_LE; ARITH; REAL_POW_LT];
ALL_TAC] THEN
MATCH_MP_TAC HAS_REAL_INTEGRAL_EQ THEN
EXISTS_TAC `\t. if &0 < t /\ t < r
then measure
{y:real^2 | norm(vector[t; y$1; y$2]:real^3) pow 2
< r pow 2 /\
(t * t + y dot y) * a pow 2 < t pow 2}
else &0` THEN
CONJ_TAC THENL
[X_GEN_TAC `t:real` THEN DISCH_TAC THEN REWRITE_TAC[] THEN
ASM_CASES_TAC `&0 < t` THEN
ASM_REWRITE_TAC[EMPTY_GSPEC; MEASURE_EMPTY; MEASURABLE_EMPTY] THEN
ASM_CASES_TAC `t:real < r` THEN ASM_REWRITE_TAC[] THENL
[REWRITE_TAC[NORM_LT_SQUARE] THEN
SUBGOAL_THEN `&0 < r` (fun th -> REWRITE_TAC[th; NORM_POW_2]) THEN
ASM_REAL_ARITH_TAC;
ALL_TAC] THEN
SUBGOAL_THEN `!y. ~(norm(vector[t; (y:real^2)$1; y$2]:real^3) < r)`
(fun th -> REWRITE_TAC[th; EMPTY_GSPEC; MEASURE_EMPTY;
MEASURABLE_EMPTY]) THEN
ASM_REWRITE_TAC[NORM_LT_SQUARE; DOT_3; VECTOR_3] THEN
GEN_TAC THEN
MATCH_MP_TAC(REAL_ARITH `&0 <= a /\ &0 <= b /\ c <= d
==> ~(&0 < r /\ d + a + b < c)`) THEN
REWRITE_TAC[REAL_LE_SQUARE] THEN
REWRITE_TAC[REAL_POW_2] THEN MATCH_MP_TAC REAL_LE_MUL2 THEN
ASM_REAL_ARITH_TAC;
ALL_TAC] THEN
REWRITE_TAC[GSYM IN_REAL_INTERVAL; HAS_REAL_INTEGRAL_RESTRICT_UNIV] THEN
REWRITE_TAC[HAS_REAL_INTEGRAL_OPEN_INTERVAL] THEN
REWRITE_TAC[NORM_POW_2; DOT_3; VECTOR_3; DOT_2] THEN
ONCE_REWRITE_TAC[REAL_ARITH
`pi * &2 / &3 * (&1 - a) * r pow 3 =
pi / &3 * (inv (a pow 2) - &1) * (a * r) pow 3 +
(pi * &2 / &3 * (&1 - a) * r pow 3 -
pi / &3 * (inv (a pow 2) - &1) * (a * r) pow 3)`] THEN
MATCH_MP_TAC HAS_REAL_INTEGRAL_COMBINE THEN
EXISTS_TAC `a * r:real` THEN
REWRITE_TAC[REAL_ARITH `a * r <= r <=> &0 <= r * (&1 - a)`] THEN
ASM_SIMP_TAC[REAL_LE_MUL; REAL_SUB_LE; REAL_LT_IMP_LE] THEN CONJ_TAC THENL
[MATCH_MP_TAC HAS_REAL_INTEGRAL_EQ THEN EXISTS_TAC
`\t. measure(ball(vec 0:real^2,sqrt(inv(a pow 2) - &1) * t))` THEN
CONJ_TAC THENL
[X_GEN_TAC `t:real` THEN REWRITE_TAC[IN_REAL_INTERVAL] THEN
STRIP_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN
REWRITE_TAC[IN_BALL_0; NORM_LT_SQUARE_ALT] THEN
ASM_SIMP_TAC[SQRT_POS_LE; REAL_LE_MUL; SQRT_POW_2; REAL_LT_IMP_LE;
REAL_POW_MUL] THEN
REWRITE_TAC[REAL_ARITH `x < (a - &1) * t <=> t + x < t * a`] THEN
ASM_SIMP_TAC[GSYM real_div; REAL_LT_RDIV_EQ; REAL_POW_LT] THEN
X_GEN_TAC `x:real^2` THEN REWRITE_TAC[DOT_2] THEN
ASM_SIMP_TAC[GSYM REAL_POW_2; GSYM REAL_LT_RDIV_EQ; REAL_POW_LT] THEN
MATCH_MP_TAC(REAL_ARITH `b <= a ==> (x < b <=> x < a /\ x < b)`) THEN
ASM_SIMP_TAC[REAL_LE_LDIV_EQ; REAL_POW_LT; GSYM REAL_POW_MUL] THEN
ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN
REWRITE_TAC[GSYM REAL_LE_SQUARE_ABS] THEN ASM_REAL_ARITH_TAC;
ALL_TAC] THEN
MATCH_MP_TAC HAS_REAL_INTEGRAL_EQ THEN
EXISTS_TAC `\t. pi * (inv(a pow 2) - &1) * t pow 2` THEN
CONJ_TAC THENL
[X_GEN_TAC `t:real` THEN REWRITE_TAC[IN_REAL_INTERVAL] THEN
STRIP_TAC THEN
W(MP_TAC o PART_MATCH (lhs o rand) AREA_BALL o rand o snd) THEN
ASM_SIMP_TAC[REAL_POW_MUL; REAL_LT_IMP_LE; SQRT_POS_LT; REAL_LE_MUL;
SQRT_POW_2];
ALL_TAC] THEN
MP_TAC(ISPECL
[`\t. pi / &3 * (inv (a pow 2) - &1) * t pow 3`;
`\t. pi * (inv (a pow 2) - &1) * t pow 2`;
`&0`; `a * r:real`] REAL_FUNDAMENTAL_THEOREM_OF_CALCULUS) THEN
ASM_SIMP_TAC[REAL_LE_MUL; REAL_LT_IMP_LE] THEN ANTS_TAC THENL
[ASM_REWRITE_TAC[] THEN
REPEAT STRIP_TAC THEN REAL_DIFF_TAC THEN
CONV_TAC NUM_REDUCE_CONV THEN CONV_TAC REAL_RING;
MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN
UNDISCH_TAC `&0 < a` THEN CONV_TAC REAL_FIELD];
MATCH_MP_TAC HAS_REAL_INTEGRAL_EQ THEN EXISTS_TAC
`\t. measure(ball(vec 0:real^2,sqrt(r pow 2 - t pow 2)))` THEN
CONJ_TAC THENL
[X_GEN_TAC `t:real` THEN REWRITE_TAC[IN_REAL_INTERVAL] THEN
STRIP_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN
REWRITE_TAC[IN_BALL_0; NORM_LT_SQUARE_ALT] THEN
SUBGOAL_THEN `&0 <= t` ASSUME_TAC THENL
[MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `a * r:real` THEN
ASM_SIMP_TAC[REAL_LE_MUL; REAL_LT_IMP_LE];
ALL_TAC] THEN
ASM_SIMP_TAC[SQRT_POS_LE; SQRT_POW_2; REAL_SUB_LE; REAL_POW_LE2] THEN
X_GEN_TAC `x:real^2` THEN REWRITE_TAC[DOT_2] THEN
REWRITE_TAC[REAL_ARITH `x < r - t <=> t + x < r`] THEN
ASM_SIMP_TAC[GSYM REAL_POW_2; GSYM REAL_LT_RDIV_EQ; REAL_POW_LT] THEN
MATCH_MP_TAC(REAL_ARITH `a <= b ==> (x < a <=> x < a /\ x < b)`) THEN
ASM_SIMP_TAC[REAL_LE_RDIV_EQ; REAL_POW_LT; GSYM REAL_POW_MUL] THEN
ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN
ASM_SIMP_TAC[REAL_POW_LE2; REAL_LE_MUL; REAL_LT_IMP_LE];
ALL_TAC] THEN
MATCH_MP_TAC HAS_REAL_INTEGRAL_EQ THEN
EXISTS_TAC `\t. pi * (r pow 2 - t pow 2)` THEN
CONJ_TAC THENL
[X_GEN_TAC `t:real` THEN REWRITE_TAC[IN_REAL_INTERVAL] THEN
STRIP_TAC THEN
W(MP_TAC o PART_MATCH (lhs o rand) AREA_BALL o rand o snd) THEN
SUBGOAL_THEN `&0 <= t` ASSUME_TAC THENL
[MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `a * r:real` THEN
ASM_SIMP_TAC[REAL_LE_MUL; REAL_LT_IMP_LE];
ALL_TAC] THEN
ASM_SIMP_TAC[SQRT_POS_LE; SQRT_POW_2; REAL_SUB_LE; REAL_POW_LE2];
ALL_TAC] THEN
MP_TAC(ISPECL
[`\t. pi * (r pow 2 * t - t pow 3 / &3)`;
`\t. pi * (r pow 2 - t pow 2)`;
`a * r:real`; `r:real`] REAL_FUNDAMENTAL_THEOREM_OF_CALCULUS) THEN
ASM_SIMP_TAC[REAL_LE_MUL; REAL_LT_IMP_LE] THEN ANTS_TAC THENL
[ASM_REWRITE_TAC[REAL_ARITH `a * r <= r <=> &0 <= r * (&1 - a)`] THEN
ASM_SIMP_TAC[REAL_LE_MUL; REAL_LT_IMP_LE; REAL_SUB_LE] THEN
REPEAT STRIP_TAC THEN REAL_DIFF_TAC THEN
CONV_TAC NUM_REDUCE_CONV THEN CONV_TAC REAL_RING;
MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN
UNDISCH_TAC `&0 < a` THEN CONV_TAC REAL_FIELD]]);;
let VOLUME_CONIC_CAP = prove
(`!v0 v1:real^3 r a.
&0 < a
==> measurable(conic_cap v0 v1 r a) /\ measure(conic_cap v0 v1 r a) =
if v1 = v0 \/ &1 <= a \/ r < &0 then &0
else &2 / &3 * pi * (&1 - a) * r pow 3`,
SIMP_TAC[VOLUME_CONIC_CAP_STRONG]);;
(* ------------------------------------------------------------------------- *)
(* Negligibility of a circular cone. *)
(* This isn't exactly using the Flyspeck definition of "cone" but we use it *)
(* to get that later on. Could now simplify this using WLOG tactics. *)
(* ------------------------------------------------------------------------- *)
let NEGLIGIBLE_CIRCULAR_CONE_0_NONPARALLEL = prove
(`!c:real^N k. ~(c = vec 0) /\ ~(k = &0) /\ ~(k = pi)
==> negligible {x | vector_angle c x = k}`,
REPEAT STRIP_TAC THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN
EXISTS_TAC `(vec 0:real^N) INSERT
UNIONS { {x | x IN ((:real^N) DIFF ball(vec 0,inv(&n + &1))) /\
Cx(vector_angle c x) = Cx k} |
n IN (:num) }` THEN
CONJ_TAC THENL
[ALL_TAC;
REWRITE_TAC[SUBSET; IN_INSERT; IN_UNIONS; IN_ELIM_THM; CX_INJ] THEN
X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN REWRITE_TAC[IN_UNIV] THEN
ASM_CASES_TAC `x:real^N = vec 0` THEN ASM_REWRITE_TAC[] THEN
REWRITE_TAC[LEFT_AND_EXISTS_THM; IN_DIFF; IN_UNIV] THEN
ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN REWRITE_TAC[UNWIND_THM2] THEN
ASM_REWRITE_TAC[IN_ELIM_THM] THEN
MP_TAC(SPEC `norm(x:real^N)` REAL_ARCH_INV) THEN
ASM_REWRITE_TAC[NORM_POS_LT; IN_BALL_0; REAL_NOT_LT; REAL_LT_INV_EQ] THEN
MATCH_MP_TAC MONO_EXISTS THEN REPEAT STRIP_TAC THEN
MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `inv(&n)` THEN
ASM_SIMP_TAC[REAL_LT_IMP_LE] THEN MATCH_MP_TAC REAL_LE_INV2 THEN
ASM_REAL_ARITH_TAC] THEN
REWRITE_TAC[NEGLIGIBLE_INSERT] THEN
MATCH_MP_TAC NEGLIGIBLE_COUNTABLE_UNIONS THEN X_GEN_TAC `n:num` THEN
MATCH_MP_TAC STARLIKE_NEGLIGIBLE_STRONG THEN EXISTS_TAC `c:real^N` THEN
CONJ_TAC THENL
[MATCH_MP_TAC CONTINUOUS_CLOSED_PREIMAGE_CONSTANT THEN
SIMP_TAC[CLOSED_DIFF; CLOSED_UNIV; OPEN_BALL] THEN
MATCH_MP_TAC(REWRITE_RULE[o_DEF] CONTINUOUS_ON_CX_VECTOR_ANGLE) THEN
REWRITE_TAC[IN_DIFF; IN_BALL_0; NORM_0; IN_UNIV] THEN
REWRITE_TAC[REAL_LT_INV_EQ] THEN REAL_ARITH_TAC;
ALL_TAC] THEN
MAP_EVERY X_GEN_TAC [`a:real`; `x:real^N`] THEN
SIMP_TAC[IN_ELIM_THM; IN_UNIV; IN_DIFF; IN_BALL_0; REAL_NOT_LT; CX_INJ] THEN
REWRITE_TAC[DE_MORGAN_THM] THEN ASM_CASES_TAC `(c + x:real^N) = vec 0` THENL
[ASM_REWRITE_TAC[GSYM REAL_NOT_LT; REAL_LT_INV_EQ; NORM_0] THEN
REAL_ARITH_TAC;
ALL_TAC] THEN
ASM_CASES_TAC `c + a % x:real^N = vec 0` THENL
[ASM_REWRITE_TAC[GSYM REAL_NOT_LT; REAL_LT_INV_EQ; NORM_0] THEN
REAL_ARITH_TAC;
ALL_TAC] THEN
ASM_CASES_TAC `x:real^N = vec 0` THENL
[ASM_REWRITE_TAC[VECTOR_ADD_RID; VECTOR_ANGLE_REFL];
ALL_TAC] THEN
ASM_CASES_TAC `a = &0` THENL
[ASM_REWRITE_TAC[VECTOR_MUL_LZERO; VECTOR_ADD_RID; VECTOR_ANGLE_REFL];
ALL_TAC] THEN
REWRITE_TAC[TAUT `~a \/ ~b <=> a ==> ~b`] THEN REPEAT STRIP_TAC THEN
MP_TAC(ISPECL [`vec 0:real^N`; `c:real^N`; `c + a % x:real^N`;
`vec 0:real^N`; `c:real^N`; `c + x:real^N`]
CONGRUENT_TRIANGLES_ASA_FULL) THEN
REWRITE_TAC[angle; VECTOR_ADD_SUB] THEN ASM_SIMP_TAC[VECTOR_SUB_RZERO] THEN
REWRITE_TAC[NORM_ARITH `dist(x,x + a) = norm(a)`; NORM_MUL] THEN
REWRITE_TAC[REAL_FIELD `a * x = x <=> a = &1 \/ x = &0`] THEN
ASM_SIMP_TAC[REAL_ARITH `&0 <= a /\ a < &1 ==> ~(abs a = &1)`] THEN
ASM_REWRITE_TAC[NORM_EQ_0; VECTOR_ANGLE_RMUL; COLLINEAR_LEMMA] THEN
DISCH_THEN(X_CHOOSE_THEN `u:real` MP_TAC) THEN
DISCH_THEN(MP_TAC o AP_TERM `\x:real^N. inv(a) % x`) THEN
ASM_SIMP_TAC[VECTOR_MUL_ASSOC; VECTOR_ADD_LDISTRIB;
VECTOR_MUL_LID; REAL_MUL_LINV] THEN
REWRITE_TAC[VECTOR_ARITH `a % c + x = b % c <=> x = (b - a) % c`] THEN
DISCH_THEN SUBST_ALL_TAC THEN
RULE_ASSUM_TAC(REWRITE_RULE[VECTOR_ARITH `c + a % c = (a + &1) % c`]) THEN
UNDISCH_TAC `vector_angle c ((inv a * u - inv a + &1) % c:real^N) = k` THEN
RULE_ASSUM_TAC(REWRITE_RULE
[VECTOR_ANGLE_RMUL; VECTOR_MUL_EQ_0; DE_MORGAN_THM]) THEN
ASM_REWRITE_TAC[VECTOR_ANGLE_RMUL; VECTOR_ANGLE_REFL] THEN
ASM_REAL_ARITH_TAC);;
let NEGLIGIBLE_CIRCULAR_CONE_0 = prove
(`!c:real^N k. 2 <= dimindex(:N) /\ ~(c = vec 0)
==> negligible {x | vector_angle c x = k}`,
REPEAT STRIP_TAC THEN
SUBGOAL_THEN `orthogonal (basis 1:real^N) (basis 2)` ASSUME_TAC THENL
[ASM_SIMP_TAC[ORTHOGONAL_BASIS_BASIS; ARITH;
ARITH_RULE `2 <= d ==> 1 <= d`];
ALL_TAC] THEN
ASM_CASES_TAC `k = &0 \/ k = pi` THENL
[ALL_TAC; ASM_MESON_TAC[NEGLIGIBLE_CIRCULAR_CONE_0_NONPARALLEL]] THEN
SUBGOAL_THEN
`?b:real^N. ~(b = vec 0) /\
~(vector_angle c b = &0) /\
~(vector_angle c b = pi)`
STRIP_ASSUME_TAC THENL
[MATCH_MP_TAC(MESON[] `!a b. P a \/ P b ==> ?x. P x`) THEN
MAP_EVERY EXISTS_TAC [`basis 1:real^N`; `basis 2:real^N`] THEN
REWRITE_TAC[BASIS_EQ_0] THEN
ASM_SIMP_TAC[ARITH_RULE `2 <= d ==> 1 <= d`; IN_NUMSEG; ARITH] THEN
REWRITE_TAC[GSYM DE_MORGAN_THM] THEN STRIP_TAC THEN
REPEAT(FIRST_X_ASSUM(MP_TAC o SPEC `basis 1:real^N` o
MATCH_MP VECTOR_ANGLE_EQ_0_LEFT)) THEN
REPEAT(FIRST_X_ASSUM(MP_TAC o SPEC `basis 1:real^N` o
MATCH_MP VECTOR_ANGLE_EQ_PI_LEFT)) THEN
DISCH_THEN SUBST1_TAC THEN REWRITE_TAC[VECTOR_ANGLE_REFL; BASIS_EQ_0] THEN
ASM_SIMP_TAC[ARITH_RULE `2 <= d ==> 1 <= d`; IN_NUMSEG; ARITH] THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [ORTHOGONAL_VECTOR_ANGLE]) THEN
REWRITE_TAC[VECTOR_ANGLE_SYM] THEN MP_TAC PI_POS THEN REAL_ARITH_TAC;
ALL_TAC] THEN
ASM_CASES_TAC `k = &0 \/ k = pi` THENL
[ALL_TAC; ASM_MESON_TAC[NEGLIGIBLE_CIRCULAR_CONE_0_NONPARALLEL]] THEN
MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN
FIRST_X_ASSUM(DISJ_CASES_THEN SUBST_ALL_TAC) THENL
[EXISTS_TAC `{x:real^N | vector_angle b x = vector_angle c b}` THEN
ASM_SIMP_TAC[NEGLIGIBLE_CIRCULAR_CONE_0_NONPARALLEL] THEN
REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN
MESON_TAC[VECTOR_ANGLE_EQ_0_RIGHT; VECTOR_ANGLE_SYM];
EXISTS_TAC `{x:real^N | vector_angle b x = pi - vector_angle c b}` THEN
ASM_SIMP_TAC[NEGLIGIBLE_CIRCULAR_CONE_0_NONPARALLEL;
REAL_SUB_0; REAL_ARITH `p - x = p <=> x = &0`] THEN
REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN
MESON_TAC[VECTOR_ANGLE_EQ_PI_RIGHT; VECTOR_ANGLE_SYM]]);;
let NEGLIGIBLE_CIRCULAR_CONE = prove
(`!a:real^N c k.
2 <= dimindex(:N) /\ ~(c = vec 0)
==> negligible(a INSERT {x | vector_angle c (x - a) = k})`,
REPEAT STRIP_TAC THEN REWRITE_TAC[NEGLIGIBLE_INSERT] THEN
MATCH_MP_TAC NEGLIGIBLE_TRANSLATION_REV THEN EXISTS_TAC `--a:real^N` THEN
MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN
EXISTS_TAC `{x:real^N | vector_angle c x = k}` THEN
ASM_SIMP_TAC[NEGLIGIBLE_CIRCULAR_CONE_0] THEN
REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_ELIM_THM] THEN
REWRITE_TAC[VECTOR_ARITH `--a + x:real^N = x - a`]);;
let NEGLIGIBLE_RCONE_EQ = prove
(`!w z:real^3 h. ~(w = z) ==> negligible(rcone_eq z w h)`,
REWRITE_TAC[rcone_eq; rconesgn] THEN GEOM_ORIGIN_TAC `z:real^3` THEN
REPEAT STRIP_TAC THEN REWRITE_TAC[DIST_0; VECTOR_SUB_RZERO] THEN
ASM_CASES_TAC `abs(h) <= &1` THENL
[MP_TAC(ISPECL [`w:real^3`; `acs h`] NEGLIGIBLE_CIRCULAR_CONE_0) THEN
ASM_REWRITE_TAC[DIMINDEX_3; ARITH] THEN
REWRITE_TAC[GSYM HAS_MEASURE_0] THEN
MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT]
HAS_MEASURE_NEGLIGIBLE_SYMDIFF) THEN
MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `{vec 0:real^3}` THEN
REWRITE_TAC[NEGLIGIBLE_SING] THEN MATCH_MP_TAC(SET_RULE
`(!x. ~(x = a) ==> (x IN s <=> x IN t))
==> (s DIFF t) UNION (t DIFF s) SUBSET {a}`) THEN
X_GEN_TAC `x:real^3` THEN DISCH_TAC THEN REWRITE_TAC[IN_ELIM_THM] THEN
ASM_SIMP_TAC[vector_angle] THEN ASM_SIMP_TAC[NORM_EQ_0; REAL_FIELD
`~(x = &0) /\ ~(w = &0) ==> (a = x * w * b <=> a / (w * x) = b)`] THEN
GEN_REWRITE_TAC (RAND_CONV o ONCE_DEPTH_CONV) [DOT_SYM] THEN
MATCH_MP_TAC ACS_INJ THEN ASM_REWRITE_TAC[NORM_CAUCHY_SCHWARZ_DIV];
MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN
EXISTS_TAC `{vec 0}:real^3->bool` THEN
REWRITE_TAC[NEGLIGIBLE_SING] THEN
REWRITE_TAC[SET_RULE `{x | P x} SUBSET {a} <=> !x. ~(x = a) ==> ~P x`] THEN
X_GEN_TAC `x:real^3` THEN REPEAT DISCH_TAC THEN
MP_TAC(ISPECL [`x:real^3`; `w:real^3`] NORM_CAUCHY_SCHWARZ_ABS) THEN
ASM_REWRITE_TAC[REAL_ABS_MUL; REAL_ABS_NORM; REAL_ARITH
`~(x * w * h <= x * w) <=> &0 < x * w * (h - &1)`] THEN
REPEAT(MATCH_MP_TAC REAL_LT_MUL THEN ASM_REWRITE_TAC[NORM_POS_LT]) THEN
ASM_REAL_ARITH_TAC]);;
(* ------------------------------------------------------------------------- *)
(* Area of sector of a circle delimited by Arg values. *)
(* ------------------------------------------------------------------------- *)
let NEGLIGIBLE_ARG_EQ = prove
(`!t. negligible {z | Arg z = t}`,
GEN_TAC THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN
EXISTS_TAC `{z | cexp(ii * Cx(pi / &2 + t)) dot z = &0}` THEN
SIMP_TAC[NEGLIGIBLE_HYPERPLANE; COMPLEX_VEC_0; CEXP_NZ] THEN
REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN X_GEN_TAC `z:complex` THEN
DISCH_TAC THEN MP_TAC(SPEC `z:complex` ARG) THEN ASM_REWRITE_TAC[] THEN
REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
DISCH_THEN SUBST1_TAC THEN
REWRITE_TAC[GSYM COMPLEX_CMUL; DOT_RMUL; REAL_ENTIRE] THEN
DISJ2_TAC THEN REWRITE_TAC[CEXP_EULER] THEN
REWRITE_TAC[DOT_2; GSYM RE_DEF; GSYM IM_DEF] THEN
REWRITE_TAC[GSYM CX_SIN; GSYM CX_COS; RE_ADD; IM_ADD;
RE_MUL_II; IM_MUL_II; RE_CX; IM_CX] THEN
REWRITE_TAC[SIN_ADD; COS_ADD; SIN_PI2; COS_PI2] THEN
REAL_ARITH_TAC);;
let MEASURABLE_CLOSED_SECTOR_LE = prove
(`!r t. measurable {z | norm(z) <= r /\ Arg z <= t}`,
REPEAT GEN_TAC THEN MATCH_MP_TAC MEASURABLE_COMPACT THEN
REWRITE_TAC[SET_RULE `{z | P z /\ Q z} = {z | P z} INTER {z | Q z}`] THEN
MATCH_MP_TAC COMPACT_INTER_CLOSED THEN REWRITE_TAC[CLOSED_ARG_LE] THEN
REWRITE_TAC[NORM_ARITH `norm z = dist(vec 0,z)`; GSYM cball] THEN
REWRITE_TAC[COMPACT_CBALL]);;
let MEASURABLE_CLOSED_SECTOR_LT = prove
(`!r t. measurable {z | norm(z) <= r /\ Arg z < t}`,
REPEAT GEN_TAC THEN MATCH_MP_TAC MEASURABLE_NEGLIGIBLE_SYMDIFF THEN
EXISTS_TAC `{z | norm(z) <= r /\ Arg z <= t}` THEN
REWRITE_TAC[MEASURABLE_CLOSED_SECTOR_LE] THEN
MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN
EXISTS_TAC `{z | Arg z = t}` THEN
REWRITE_TAC[NEGLIGIBLE_ARG_EQ; NEGLIGIBLE_UNION_EQ] THEN
REWRITE_TAC[SUBSET; IN_DIFF; IN_UNION; IN_ELIM_THM] THEN REAL_ARITH_TAC);;
let MEASURABLE_CLOSED_SECTOR_LTE = prove
(`!r s t. measurable {z | norm(z) <= r /\ s < Arg z /\ Arg z <= t}`,
REPEAT GEN_TAC THEN REWRITE_TAC[SET_RULE
`{z | P z /\ Q z /\ R z} = {z | P z /\ R z} DIFF {z | P z /\ ~Q z}`] THEN
SIMP_TAC[MEASURABLE_DIFF; REAL_NOT_LT; MEASURABLE_CLOSED_SECTOR_LE]);;
let MEASURE_CLOSED_SECTOR_LE = prove
(`!t r. &0 <= r /\ &0 <= t /\ t <= &2 * pi
==> measure {x:real^2 | norm(x) <= r /\ Arg(x) <= t} =
t * r pow 2 / &2`,
REPEAT STRIP_TAC THEN
MP_TAC(ISPECL
[`\t. measure {z:real^2 | norm(z) <= r /\ Arg(z) <= t}`;
`&2 * pi`] REAL_CONTINUOUS_ADDITIVE_IMP_LINEAR_INTERVAL) THEN
ANTS_TAC THENL
[ALL_TAC;
DISCH_THEN(MP_TAC o SPECL [`t / (&2 * pi)`; `&2 * pi`]) THEN
MP_TAC(SPECL [`vec 0:real^2`; `r:real`] AREA_CBALL) THEN
ASM_REWRITE_TAC[cball; NORM_ARITH `dist(vec 0,z) = norm z`] THEN
SIMP_TAC[ARG; REAL_LT_IMP_LE] THEN DISCH_THEN(K ALL_TAC) THEN
SIMP_TAC[PI_POS; REAL_FIELD `&0 < p ==> t / (&2 * p) * p * r = t * r / &2`;
REAL_FIELD `&0 < p ==> t / (&2 * p) * &2 * p = t`] THEN
DISCH_THEN MATCH_MP_TAC THEN MP_TAC PI_POS THEN ASM_REAL_ARITH_TAC] THEN
REWRITE_TAC[] THEN REPEAT STRIP_TAC THENL
[MATCH_MP_TAC REALLIM_NULL_COMPARISON THEN
EXISTS_TAC `\t. r pow 2 * sin(t)` THEN REWRITE_TAC[] THEN CONJ_TAC THENL
[REWRITE_TAC[EVENTUALLY_WITHINREAL] THEN EXISTS_TAC `pi / &2` THEN
SIMP_TAC[PI_POS; REAL_LT_DIV; IN_ELIM_THM; REAL_OF_NUM_LT; ARITH] THEN
X_GEN_TAC `x:real` THEN REWRITE_TAC[REAL_SUB_RZERO] THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
ASM_SIMP_TAC[real_abs; MEASURE_POS_LE; MEASURABLE_CLOSED_SECTOR_LE] THEN
STRIP_TAC THEN MATCH_MP_TAC REAL_LE_TRANS THEN
EXISTS_TAC `measure(interval[vec 0,complex(r,r * sin x)])` THEN
CONJ_TAC THENL
[MATCH_MP_TAC MEASURE_SUBSET THEN
REWRITE_TAC[MEASURABLE_CLOSED_SECTOR_LE; MEASURABLE_INTERVAL] THEN
REWRITE_TAC[SUBSET; IN_ELIM_THM; IN_INTERVAL] THEN
X_GEN_TAC `z:complex` THEN STRIP_TAC THEN
REWRITE_TAC[DIMINDEX_2; FORALL_2; VEC_COMPONENT] THEN
REWRITE_TAC[GSYM IM_DEF; GSYM RE_DEF; IM; RE] THEN
SUBST1_TAC(last(CONJUNCTS(SPEC `z:complex` ARG))) THEN
REWRITE_TAC[RE_MUL_CX; IM_MUL_CX; CEXP_EULER] THEN
REWRITE_TAC[RE_ADD; GSYM CX_COS; GSYM CX_SIN; RE_CX; IM_CX;
RE_MUL_II; IM_MUL_II; IM_ADD] THEN
REWRITE_TAC[REAL_NEG_0; REAL_ADD_LID; REAL_ADD_RID] THEN
SUBGOAL_THEN `&0 <= Arg z /\ Arg z < pi / &2 /\ Arg z <= pi / &2`
STRIP_ASSUME_TAC THENL
[REWRITE_TAC[ARG] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN
REPEAT CONJ_TAC THENL
[MATCH_MP_TAC REAL_LE_MUL THEN REWRITE_TAC[NORM_POS_LE] THEN
MATCH_MP_TAC COS_POS_PI_LE THEN ASM_REAL_ARITH_TAC;
MATCH_MP_TAC(REAL_ARITH `abs(a * b) <= c * &1 ==> a * b <= c`) THEN
REWRITE_TAC[REAL_ABS_MUL; REAL_ABS_NORM] THEN
MATCH_MP_TAC REAL_LE_MUL2 THEN
ASM_REWRITE_TAC[NORM_POS_LE; REAL_ABS_POS; COS_BOUND];
MATCH_MP_TAC REAL_LE_MUL THEN REWRITE_TAC[NORM_POS_LE] THEN
MATCH_MP_TAC SIN_POS_PI_LE THEN ASM_REAL_ARITH_TAC;
MATCH_MP_TAC REAL_LE_MUL2 THEN ASM_REWRITE_TAC[NORM_POS_LE] THEN
CONJ_TAC THENL
[MATCH_MP_TAC SIN_POS_PI_LE THEN ASM_REAL_ARITH_TAC;
MATCH_MP_TAC SIN_MONO_LE THEN ASM_REAL_ARITH_TAC]];
REWRITE_TAC[MEASURE_INTERVAL; CONTENT_CLOSED_INTERVAL_CASES] THEN
REWRITE_TAC[FORALL_2; PRODUCT_2; DIMINDEX_2; VEC_COMPONENT] THEN
REWRITE_TAC[GSYM IM_DEF; GSYM RE_DEF; IM; RE] THEN
REWRITE_TAC[REAL_SUB_RZERO; REAL_POW_2; REAL_MUL_ASSOC] THEN
SUBGOAL_THEN `&0 <= sin x` (fun th ->
ASM_SIMP_TAC[REAL_LE_MUL; REAL_LE_REFL; REAL_LE_MUL; th]) THEN
MATCH_MP_TAC SIN_POS_PI_LE THEN ASM_REAL_ARITH_TAC];
MATCH_MP_TAC REALLIM_ATREAL_WITHINREAL THEN
SUBGOAL_THEN `(\t. r pow 2 * sin t) real_continuous atreal (&0)`
MP_TAC THENL
[MATCH_MP_TAC REAL_CONTINUOUS_LMUL THEN
REWRITE_TAC[ETA_AX; REAL_CONTINUOUS_AT_SIN];
REWRITE_TAC[REAL_CONTINUOUS_ATREAL; SIN_0; REAL_MUL_RZERO]]];
ASM_SIMP_TAC[REAL_ARITH
`&0 <= x /\ &0 <= y
==> (norm z <= r /\ Arg z <= x + y <=>
norm z <= r /\ Arg z <= x \/
norm z <= r /\ x < Arg z /\ Arg z <= x + y)`] THEN
REWRITE_TAC[SET_RULE `{z | Q z \/ R z} = {z | Q z} UNION {z | R z}`] THEN
SIMP_TAC[MEASURE_UNION; MEASURABLE_CLOSED_SECTOR_LE;
MEASURABLE_CLOSED_SECTOR_LTE] THEN
REWRITE_TAC[GSYM REAL_NOT_LE; SET_RULE
`{z | P z /\ Q z} INTER {z | P z /\ ~Q z /\ R z} = {}`] THEN
REWRITE_TAC[MEASURE_EMPTY; REAL_SUB_RZERO; REAL_EQ_ADD_LCANCEL] THEN
REWRITE_TAC[REAL_NOT_LE] THEN MATCH_MP_TAC EQ_TRANS THEN
EXISTS_TAC `measure {z | norm z <= r /\ x < Arg z /\ Arg z < x + y}` THEN
CONJ_TAC THENL
[MATCH_MP_TAC MEASURE_NEGLIGIBLE_SYMDIFF THEN
REWRITE_TAC[MEASURABLE_CLOSED_SECTOR_LTE] THEN
MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN
EXISTS_TAC `{z | Arg z = x + y}` THEN
REWRITE_TAC[NEGLIGIBLE_ARG_EQ; NEGLIGIBLE_UNION_EQ] THEN
REWRITE_TAC[SUBSET; IN_DIFF; IN_UNION; IN_ELIM_THM] THEN REAL_ARITH_TAC;
ALL_TAC] THEN
MATCH_MP_TAC EQ_TRANS THEN
EXISTS_TAC `measure {z | norm z <= r /\ &0 < Arg z /\ Arg z < y}` THEN
CONJ_TAC THENL
[ALL_TAC;
MATCH_MP_TAC MEASURE_NEGLIGIBLE_SYMDIFF THEN
REWRITE_TAC[MEASURABLE_CLOSED_SECTOR_LE] THEN
MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN
EXISTS_TAC `{z | Arg z = &0} UNION {z | Arg z = y}` THEN
REWRITE_TAC[NEGLIGIBLE_ARG_EQ; NEGLIGIBLE_UNION_EQ] THEN
REWRITE_TAC[SUBSET; IN_DIFF; IN_UNION; IN_ELIM_THM] THEN
MP_TAC ARG THEN MATCH_MP_TAC MONO_FORALL THEN REAL_ARITH_TAC] THEN
MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC
`measure (IMAGE (rotate2d x)
{z | norm z <= r /\ &0 < Arg z /\ Arg z < y})` THEN
CONJ_TAC THENL
[ALL_TAC;
ASM_SIMP_TAC[MEASURE_ORTHOGONAL_IMAGE_EQ;
ORTHOGONAL_TRANSFORMATION_ROTATE2D]] THEN
AP_TERM_TAC THEN CONV_TAC SYM_CONV THEN
MATCH_MP_TAC SURJECTIVE_IMAGE_EQ THEN CONJ_TAC THENL
[ASM_MESON_TAC[ORTHOGONAL_TRANSFORMATION_SURJECTIVE;
ORTHOGONAL_TRANSFORMATION_ROTATE2D]; ALL_TAC] THEN
X_GEN_TAC `z:complex` THEN REWRITE_TAC[IN_ELIM_THM] THEN
ASM_CASES_TAC `z = Cx(&0)` THENL
[ASM_REWRITE_TAC[Arg_DEF; ROTATE2D_0] THEN
ASM_REAL_ARITH_TAC;
ALL_TAC] THEN
REWRITE_TAC[NORM_ROTATE2D] THEN AP_TERM_TAC THEN EQ_TAC THENL
[STRIP_TAC THEN
SUBGOAL_THEN `z = rotate2d (--x) (rotate2d x z)` SUBST1_TAC THENL
[REWRITE_TAC[GSYM ROTATE2D_ADD; REAL_ADD_LINV; ROTATE2D_ZERO];
ALL_TAC] THEN
MP_TAC(ISPECL [`--x:real`; `rotate2d x z`] ARG_ROTATE2D) THEN
ASM_REWRITE_TAC[ROTATE2D_EQ_0] THEN
ANTS_TAC THENL [ASM_REAL_ARITH_TAC; DISCH_THEN SUBST1_TAC] THEN
ASM_REAL_ARITH_TAC;
STRIP_TAC THEN
MP_TAC(ISPECL [`x:real`; `z:complex`] ARG_ROTATE2D) THEN
ASM_REWRITE_TAC[ROTATE2D_EQ_0] THEN
ANTS_TAC THENL [ASM_REAL_ARITH_TAC; DISCH_THEN SUBST1_TAC] THEN
ASM_REAL_ARITH_TAC]]);;
let HAS_MEASURE_OPEN_SECTOR_LT = prove
(`!t r. &0 <= t /\ t <= &2 * pi
==> {x:real^2 | norm(x) < r /\ &0 < Arg x /\ Arg x < t}
has_measure (if &0 <= r then t * r pow 2 / &2 else &0)`,
REPEAT STRIP_TAC THEN COND_CASES_TAC THEN
ASM_SIMP_TAC[NORM_ARITH `~(&0 <= r) ==> ~(norm x < r)`;
EMPTY_GSPEC; HAS_MEASURE_EMPTY] THEN
MATCH_MP_TAC HAS_MEASURE_NEGLIGIBLE_SYMDIFF THEN
EXISTS_TAC `{x | norm x <= r /\ Arg x <= t}` THEN
REWRITE_TAC[HAS_MEASURE_MEASURABLE_MEASURE] THEN
ASM_SIMP_TAC[MEASURE_CLOSED_SECTOR_LE; MEASURABLE_CLOSED_SECTOR_LE] THEN
MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN
EXISTS_TAC `{x | dist(vec 0,x) = r} UNION
{z | Arg z = &0} UNION {z | Arg z = t}` THEN
REWRITE_TAC[NEGLIGIBLE_ARG_EQ; REWRITE_RULE[sphere] NEGLIGIBLE_SPHERE;
NEGLIGIBLE_UNION_EQ] THEN
REWRITE_TAC[DIST_0; SUBSET; IN_DIFF; IN_UNION; IN_ELIM_THM] THEN
MP_TAC ARG THEN MATCH_MP_TAC MONO_FORALL THEN REAL_ARITH_TAC);;
let MEASURE_OPEN_SECTOR_LT = prove
(`!t r. &0 <= t /\ t <= &2 * pi
==> measure {x:real^2 | norm(x) < r /\ &0 < Arg x /\ Arg x < t} =
if &0 <= r then t * r pow 2 / &2 else &0`,
SIMP_TAC[REWRITE_RULE[HAS_MEASURE_MEASURABLE_MEASURE]
HAS_MEASURE_OPEN_SECTOR_LT]);;
let HAS_MEASURE_OPEN_SECTOR_LT_GEN = prove
(`!w z.
~(w = vec 0)
==> {x | norm(x) < r /\ &0 < Arg(x / w) /\ Arg(x / w) < Arg(z / w)}
has_measure (if &0 <= r then Arg(z / w) * r pow 2 / &2 else &0)`,
GEOM_BASIS_MULTIPLE_TAC 1 `w:complex` THEN
X_GEN_TAC `w:real` THEN GEN_REWRITE_TAC LAND_CONV [REAL_LE_LT] THEN
ASM_CASES_TAC `w = &0` THEN ASM_REWRITE_TAC[VECTOR_MUL_LZERO] THEN
REWRITE_TAC[COMPLEX_CMUL; COMPLEX_BASIS; COMPLEX_VEC_0] THEN
SIMP_TAC[ARG_DIV_CX; COMPLEX_MUL_RID] THEN ASM_REWRITE_TAC[CX_INJ] THEN
REPEAT STRIP_TAC THEN MATCH_MP_TAC HAS_MEASURE_OPEN_SECTOR_LT THEN
SIMP_TAC[ARG; REAL_LT_IMP_LE]);;
(* ------------------------------------------------------------------------- *)
(* Hence volume of a wedge of a ball. *)
(* ------------------------------------------------------------------------- *)
let MEASURABLE_BALL_WEDGE = prove
(`!z:real^3 w w1 w2. measurable(ball(z,r) INTER wedge z w w1 w2)`,
REPEAT STRIP_TAC THEN MATCH_MP_TAC MEASURABLE_OPEN THEN CONJ_TAC THENL
[MATCH_MP_TAC BOUNDED_INTER THEN REWRITE_TAC[BOUNDED_BALL];
MATCH_MP_TAC OPEN_INTER THEN REWRITE_TAC[OPEN_BALL] THEN
ASM_SIMP_TAC[OPEN_WEDGE]]);;
let VOLUME_BALL_WEDGE = prove
(`!z:real^3 w r w1 w2.
&0 <= r ==> measure(ball(z,r) INTER wedge z w w1 w2) =
azim z w w1 w2 * &2 * r pow 3 / &3`,
REPEAT GEN_TAC THEN
ASM_CASES_TAC `z:real^3 = w \/ collinear{z,w,w1} \/ collinear{z,w,w2}` THENL
[FIRST_X_ASSUM STRIP_ASSUME_TAC THEN
ASM_SIMP_TAC[WEDGE_DEGENERATE; AZIM_DEGENERATE; INTER_EMPTY; REAL_MUL_LZERO;
MEASURE_EMPTY];
FIRST_X_ASSUM MP_TAC THEN REWRITE_TAC[IMP_IMP; DE_MORGAN_THM]] THEN
REWRITE_TAC[wedge] THEN GEOM_ORIGIN_TAC `z:real^3` THEN
GEOM_BASIS_MULTIPLE_TAC 3 `w:real^3` THEN
X_GEN_TAC `w:real` THEN GEN_REWRITE_TAC LAND_CONV [REAL_LE_LT] THEN
ASM_CASES_TAC `w = &0` THEN ASM_REWRITE_TAC[VECTOR_MUL_LZERO] THEN
ASM_SIMP_TAC[AZIM_SPECIAL_SCALE; COLLINEAR_SPECIAL_SCALE] THEN
REPEAT STRIP_TAC THEN
MATCH_MP_TAC(INST_TYPE[`:2`,`:M`; `:3`,`:N`] FUBINI_SIMPLE_OPEN) THEN
EXISTS_TAC `3` THEN REWRITE_TAC[DIMINDEX_2; DIMINDEX_3; ARITH] THEN
REPEAT CONJ_TAC THENL
[MESON_TAC[BOUNDED_SUBSET; INTER_SUBSET; BOUNDED_BALL];
REWRITE_TAC[GSYM wedge] THEN MATCH_MP_TAC OPEN_INTER THEN
ASM_REWRITE_TAC[OPEN_BALL; OPEN_WEDGE];
SIMP_TAC[SLICE_INTER; DIMINDEX_2; DIMINDEX_3; ARITH; SLICE_BALL]] THEN
ONCE_REWRITE_TAC[TAUT `~a /\ b /\ c <=> ~(~a ==> ~(b /\ c))`] THEN
ASM_SIMP_TAC[AZIM_ARG] THEN REWRITE_TAC[COLLINEAR_BASIS_3] THEN
RULE_ASSUM_TAC(REWRITE_RULE[COLLINEAR_BASIS_3]) THEN
REWRITE_TAC[NOT_IMP; GSYM CONJ_ASSOC; DROPOUT_0] THEN
MAP_EVERY ABBREV_TAC
[`v1:real^2 = dropout 3 (w1:real^3)`;
`v2:real^2 = dropout 3 (w2:real^3)`] THEN
REWRITE_TAC[SLICE_DROPOUT_3; VEC_COMPONENT; REAL_SUB_RZERO] THEN
ONCE_REWRITE_TAC[COND_RAND] THEN
ONCE_REWRITE_TAC[COND_RATOR] THEN
REWRITE_TAC[INTER_EMPTY] THEN REWRITE_TAC[INTER; IN_BALL_0; IN_ELIM_THM] THEN
ONCE_REWRITE_TAC[COND_RAND] THEN REWRITE_TAC[MEASURE_EMPTY] THEN
MAP_EVERY UNDISCH_TAC
[`~(v1:complex = vec 0)`; `~(v2:complex = vec 0)`] THEN
MAP_EVERY (fun t -> SPEC_TAC(t,t)) [`v2:complex`; `v1:complex`] THEN
UNDISCH_TAC `&0 <= r` THEN SPEC_TAC(`r:real`,`r:real`) THEN
REWRITE_TAC[RIGHT_IMP_FORALL_THM; IMP_IMP; GSYM CONJ_ASSOC] THEN
POP_ASSUM_LIST(K ALL_TAC) THEN GEOM_BASIS_MULTIPLE_TAC 1 `v1:complex` THEN
X_GEN_TAC `v1:real` THEN GEN_REWRITE_TAC LAND_CONV [REAL_LE_LT] THEN
ASM_CASES_TAC `v1 = &0` THEN ASM_REWRITE_TAC[VECTOR_MUL_LZERO] THEN
REWRITE_TAC[COMPLEX_CMUL; COMPLEX_BASIS; COMPLEX_VEC_0] THEN
SIMP_TAC[ARG_DIV_CX; COMPLEX_MUL_RID; CX_INJ] THEN REPEAT STRIP_TAC THEN
SUBGOAL_THEN
`!t z. ~(z = Cx(&0)) /\ &0 < Arg z /\ Arg z < t <=>
&0 < Arg z /\ Arg z < t`
(fun th -> REWRITE_TAC[th])
THENL [MESON_TAC[ARG_0; REAL_LT_REFL]; ALL_TAC] THEN
ASM_SIMP_TAC[MEASURE_OPEN_SECTOR_LT; REAL_LE_REFL; ARG; REAL_LT_IMP_LE] THEN
SUBGOAL_THEN `!t. abs(t) < r <=> t IN real_interval(--r,r)`
(fun th -> REWRITE_TAC[th])
THENL [REWRITE_TAC[IN_REAL_INTERVAL] THEN REAL_ARITH_TAC; ALL_TAC] THEN
REWRITE_TAC[HAS_REAL_INTEGRAL_RESTRICT_UNIV] THEN
REWRITE_TAC[HAS_REAL_INTEGRAL_OPEN_INTERVAL] THEN
MATCH_MP_TAC HAS_REAL_INTEGRAL_EQ THEN
EXISTS_TAC `\t. Arg v2 * (r pow 2 - t pow 2) / &2` THEN CONJ_TAC THENL
[X_GEN_TAC `t:real` THEN REWRITE_TAC[IN_REAL_INTERVAL; REAL_BOUNDS_LE] THEN
SIMP_TAC[AREA_CBALL; SQRT_POS_LE; REAL_SUB_LE; GSYM REAL_LE_SQUARE_ABS;
SQRT_POW_2; REAL_ARITH `abs x <= r ==> abs x <= abs r`];
ALL_TAC] THEN
MP_TAC(ISPECL
[`\t. Arg v2 * (r pow 2 * t - &1 / &3 * t pow 3) / &2`;
`\t. Arg v2 * (r pow 2 - t pow 2) / &2`;
`--r:real`; `r:real`] REAL_FUNDAMENTAL_THEOREM_OF_CALCULUS) THEN
REWRITE_TAC[] THEN ANTS_TAC THENL
[CONJ_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
REPEAT STRIP_TAC THEN REAL_DIFF_TAC THEN
CONV_TAC NUM_REDUCE_CONV THEN CONV_TAC REAL_RING;
MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN
CONV_TAC REAL_RING]);;
(* ------------------------------------------------------------------------- *)
(* Hence volume of lune. *)
(* ------------------------------------------------------------------------- *)
let HAS_MEASURE_LUNE = prove
(`!z:real^3 w r w1 w2.
&0 <= r /\ ~(w = z) /\
~collinear {z,w,w1} /\ ~collinear {z,w,w2} /\ ~(dihV z w w1 w2 = pi)
==> (ball(z,r) INTER aff_gt {z,w} {w1,w2})
has_measure (dihV z w w1 w2 * &2 * r pow 3 / &3)`,
GEOM_ORIGIN_TAC `z:real^3` THEN
GEOM_BASIS_MULTIPLE_TAC 3 `w:real^3` THEN
X_GEN_TAC `w:real` THEN GEN_REWRITE_TAC LAND_CONV [REAL_LE_LT] THEN
ASM_CASES_TAC `w = &0` THEN ASM_REWRITE_TAC[] THENL
[ASM_REWRITE_TAC[VECTOR_MUL_LZERO; INSERT_AC; COLLINEAR_2]; ALL_TAC] THEN
DISCH_TAC THEN REPEAT GEN_TAC THEN
ASM_SIMP_TAC[DIHV_SPECIAL_SCALE] THEN
MP_TAC(ISPECL [`{}:real^3->bool`; `{w1:real^3,w2:real^3}`;
`w:real`; `basis 3:real^3`] AFF_GT_SPECIAL_SCALE) THEN
ASM_CASES_TAC `w1:real^3 = vec 0` THENL
[ASM_REWRITE_TAC[INSERT_AC; COLLINEAR_2]; ALL_TAC] THEN
ASM_CASES_TAC `w2:real^3 = vec 0` THENL
[ASM_REWRITE_TAC[INSERT_AC; COLLINEAR_2]; ALL_TAC] THEN
ASM_REWRITE_TAC[FINITE_INSERT; FINITE_EMPTY; IN_INSERT; NOT_IN_EMPTY] THEN
ASM_CASES_TAC `w1:real^3 = w % basis 3` THENL
[ASM_REWRITE_TAC[INSERT_AC; COLLINEAR_2]; ALL_TAC] THEN
ASM_CASES_TAC `w2:real^3 = w % basis 3` THENL
[ASM_REWRITE_TAC[INSERT_AC; COLLINEAR_2]; ALL_TAC] THEN
ASM_SIMP_TAC[COLLINEAR_SPECIAL_SCALE] THEN
ASM_CASES_TAC `w1:real^3 = basis 3` THENL
[ASM_REWRITE_TAC[INSERT_AC; COLLINEAR_2]; ALL_TAC] THEN
ASM_CASES_TAC `w2:real^3 = basis 3` THENL
[ASM_REWRITE_TAC[INSERT_AC; COLLINEAR_2]; ALL_TAC] THEN
ASM_REWRITE_TAC[] THEN DISCH_THEN SUBST1_TAC THEN
ASM_SIMP_TAC[VECTOR_MUL_EQ_0; REAL_LT_IMP_NZ] THEN STRIP_TAC THEN
ASM_CASES_TAC `azim (vec 0) (basis 3) w1 w2 = &0` THENL
[MP_TAC(ASSUME `azim (vec 0) (basis 3) w1 w2 = &0`) THEN
W(MP_TAC o PART_MATCH (lhs o rand) AZIM_DIVH o lhs o lhand o snd) THEN
ASM_REWRITE_TAC[PI_POS] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN
REWRITE_TAC[REAL_MUL_LZERO; HAS_MEASURE_0] THEN
MATCH_MP_TAC COPLANAR_IMP_NEGLIGIBLE THEN
MATCH_MP_TAC COPLANAR_SUBSET THEN
EXISTS_TAC `affine hull {vec 0:real^3,basis 3,w1,w2}` THEN
CONJ_TAC THENL
[ASM_MESON_TAC[COPLANAR_AFFINE_HULL_COPLANAR; AZIM_EQ_0_PI_IMP_COPLANAR];
ALL_TAC] THEN
MATCH_MP_TAC(SET_RULE `t SUBSET u ==> (s INTER t) SUBSET u`) THEN
SIMP_TAC[aff_gt_def; AFFSIGN; sgn_gt; AFFINE_HULL_FINITE;
FINITE_INSERT; FINITE_EMPTY] THEN
REWRITE_TAC[SET_RULE `{a,b} UNION {c,d} = {a,b,c,d}`] THEN
REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN GEN_TAC THEN
MATCH_MP_TAC MONO_EXISTS THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[];
ALL_TAC] THEN
SUBGOAL_THEN `&0 < azim (vec 0) (basis 3) w1 w2` ASSUME_TAC THENL
[ASM_REWRITE_TAC[REAL_LT_LE; azim]; ALL_TAC] THEN
ASM_CASES_TAC `azim (vec 0) (basis 3) w1 w2 < pi` THENL
[ASM_SIMP_TAC[GSYM AZIM_DIHV_SAME; GSYM WEDGE_LUNE_GT] THEN
ASM_SIMP_TAC[HAS_MEASURE_MEASURABLE_MEASURE; MEASURABLE_BALL_WEDGE;
VOLUME_BALL_WEDGE];
ALL_TAC] THEN
ASM_CASES_TAC `azim (vec 0) (basis 3) w1 w2 = pi` THENL
[MP_TAC(ISPECL [`vec 0:real^3`; `basis 3:real^3`; `w1:real^3`; `w2:real^3`]
AZIM_DIVH) THEN
ASM_REWRITE_TAC[REAL_LT_REFL] THEN ASM_REAL_ARITH_TAC;
ALL_TAC] THEN
SUBGOAL_THEN
`dihV (vec 0) (basis 3) w1 w2 = azim (vec 0) (basis 3) w2 w1`
SUBST1_TAC THENL
[W(MP_TAC o PART_MATCH (lhs o rand) AZIM_COMPL o rand o snd) THEN
ASM_REWRITE_TAC[] THEN DISCH_THEN SUBST1_TAC THEN
ONCE_REWRITE_TAC[REAL_ARITH `x:real = y - z <=> z = y - x`] THEN
MATCH_MP_TAC AZIM_DIHV_COMPL THEN
ASM_REWRITE_TAC[GSYM REAL_NOT_LT];
ALL_TAC] THEN
SUBGOAL_THEN `&0 < azim (vec 0) (basis 3) w2 w1 /\
azim (vec 0) (basis 3) w2 w1 < pi`
ASSUME_TAC THENL
[W(MP_TAC o PART_MATCH (lhs o rand) AZIM_COMPL o lhand o rand o snd) THEN
ASM_REWRITE_TAC[] THEN DISCH_THEN SUBST1_TAC THEN
MP_TAC(ISPECL [`vec 0:real^3`; `basis 3:real^3`; `w1:real^3`; `w2:real^3`]
azim) THEN
REWRITE_TAC[CONJ_ASSOC] THEN DISCH_THEN(MP_TAC o CONJUNCT1) THEN
ASM_REAL_ARITH_TAC;
ALL_TAC] THEN
SUBST1_TAC(SET_RULE `{w1:real^3,w2} = {w2,w1}`) THEN
ASM_SIMP_TAC[GSYM AZIM_DIHV_SAME; GSYM WEDGE_LUNE_GT] THEN
ASM_SIMP_TAC[HAS_MEASURE_MEASURABLE_MEASURE; MEASURABLE_BALL_WEDGE;
VOLUME_BALL_WEDGE]);;
let HAS_MEASURE_LUNE_SIMPLE = prove
(`!z:real^3 w r w1 w2.
&0 <= r /\ ~coplanar{z,w,w1,w2}
==> (ball(z,r) INTER aff_gt {z,w} {w1,w2})
has_measure (dihV z w w1 w2 * &2 * r pow 3 / &3)`,
REPEAT GEN_TAC THEN
ASM_CASES_TAC `w:real^3 = z` THENL
[ASM_REWRITE_TAC[INSERT_AC; COPLANAR_3]; ALL_TAC] THEN
REPEAT STRIP_TAC THEN MATCH_MP_TAC HAS_MEASURE_LUNE THEN
ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC(TAUT `a /\ b /\ (a /\ b ==> c) ==> a /\ b /\ c`) THEN
REPEAT(CONJ_TAC THENL
[ASM_MESON_TAC[NOT_COPLANAR_NOT_COLLINEAR; INSERT_AC]; ALL_TAC]) THEN
REPEAT STRIP_TAC THEN
MP_TAC(ISPECL [`z:real^3`; `w:real^3`; `w1:real^3`; `w2:real^3`]
AZIM_DIVH) THEN
ASM_REWRITE_TAC[REAL_ARITH `&2 * pi - pi = pi`; COND_ID] THEN
ASM_MESON_TAC[AZIM_EQ_0_PI_IMP_COPLANAR]);;
(* ------------------------------------------------------------------------- *)
(* Now the volume of a solid triangle. *)
(* ------------------------------------------------------------------------- *)
let MEASURABLE_BALL_AFF_GT = prove
(`!z r s t. measurable(ball(z,r) INTER aff_gt s t)`,
MESON_TAC[MEASURABLE_CONVEX; CONVEX_INTER; CONVEX_AFF_GT; CONVEX_BALL;
BOUNDED_INTER; BOUNDED_BALL]);;
let AFF_GT_SHUFFLE = prove
(`!s t v:real^N.
FINITE s /\ FINITE t /\
vec 0 IN s /\ ~(vec 0 IN t) /\
~(v IN s) /\ ~(--v IN s) /\ ~(v IN t)
==> aff_gt (v INSERT s) t =
aff_gt s (v INSERT t) UNION
aff_gt s (--v INSERT t) UNION
aff_gt s t`,
REPEAT STRIP_TAC THEN
REWRITE_TAC[aff_gt_def; AFFSIGN_ALT; sgn_gt] THEN
REWRITE_TAC[SET_RULE `(v INSERT s) UNION t = v INSERT (s UNION t)`;
SET_RULE `s UNION (v INSERT t) = v INSERT (s UNION t)`] THEN
ASM_SIMP_TAC[FINITE_INSERT; FINITE_UNION; AFFINE_HULL_FINITE_STEP_GEN;
RIGHT_EXISTS_AND_THM; REAL_LT_ADD; REAL_HALF; FINITE_EMPTY] THEN
REWRITE_TAC[IN_INSERT] THEN
ASM_SIMP_TAC[SET_RULE
`~(a IN s)
==> ((w IN s UNION t ==> w = a \/ w IN t ==> P w) <=>
(w IN t ==> P w))`] THEN
REWRITE_TAC[SET_RULE `x IN (s UNION t)
==> x IN t ==> P x <=> x IN t ==> P x`] THEN
REWRITE_TAC[EXTENSION; IN_UNION; IN_ELIM_THM] THEN
X_GEN_TAC `y:real^N` THEN EQ_TAC THENL
[DISCH_THEN(X_CHOOSE_THEN `v:real` ASSUME_TAC) THEN
ASM_CASES_TAC `&0 < v` THENL
[DISJ1_TAC THEN EXISTS_TAC `v:real` THEN ASM_REWRITE_TAC[];
DISJ2_TAC] THEN
ASM_CASES_TAC `v = &0` THENL
[DISJ2_TAC THEN
FIRST_ASSUM(fun th -> MP_TAC th THEN MATCH_MP_TAC MONO_EXISTS) THEN
ASM_REWRITE_TAC[REAL_SUB_RZERO; VECTOR_MUL_LZERO; VECTOR_SUB_RZERO];
DISJ1_TAC] THEN
EXISTS_TAC `--v:real` THEN CONJ_TAC THENL
[ASM_REAL_ARITH_TAC; ALL_TAC] THEN
FIRST_X_ASSUM(X_CHOOSE_THEN `f:real^N->real` STRIP_ASSUME_TAC) THEN
EXISTS_TAC `\x:real^N. if x = vec 0 then f(x) + &2 * v else f(x)` THEN
REWRITE_TAC[] THEN REPEAT CONJ_TAC THENL
[ASM_MESON_TAC[];
ASM_SIMP_TAC[SUM_CASES_1; FINITE_UNION; IN_UNION] THEN REAL_ARITH_TAC;
REWRITE_TAC[VECTOR_ARITH `--a % --x:real^N = a % x`] THEN
FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC RAND_CONV [GSYM th]) THEN
MATCH_MP_TAC VSUM_EQ THEN REWRITE_TAC[] THEN
REPEAT STRIP_TAC THEN COND_CASES_TAC THEN
ASM_REWRITE_TAC[VECTOR_MUL_RZERO]];
DISCH_THEN(DISJ_CASES_THEN MP_TAC) THENL [MESON_TAC[]; ALL_TAC] THEN
DISCH_THEN(DISJ_CASES_THEN MP_TAC) THENL
[DISCH_THEN(X_CHOOSE_THEN `a:real`
(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
DISCH_THEN(X_CHOOSE_THEN `f:real^N->real` STRIP_ASSUME_TAC) THEN
EXISTS_TAC `--a:real` THEN
EXISTS_TAC `\x:real^N. if x = vec 0 then &2 * a + f(vec 0) else f x` THEN
ASM_SIMP_TAC[SUM_CASES_1; FINITE_UNION; IN_UNION] THEN
CONJ_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN
CONJ_TAC THENL [REAL_ARITH_TAC; ALL_TAC] THEN
ONCE_REWRITE_TAC[VECTOR_ARITH `y - --a % v:real^N = y - a % --v`] THEN
FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC RAND_CONV [SYM th]) THEN
MATCH_MP_TAC VSUM_EQ THEN REPEAT GEN_TAC THEN REWRITE_TAC[] THEN
DISCH_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[VECTOR_MUL_RZERO];
GEN_REWRITE_TAC RAND_CONV [SWAP_EXISTS_THM] THEN
MATCH_MP_TAC MONO_EXISTS THEN REPEAT STRIP_TAC THEN
EXISTS_TAC `&0` THEN ASM_REWRITE_TAC[REAL_SUB_RZERO] THEN
VECTOR_ARITH_TAC]]);;
let MEASURE_BALL_AFF_GT_SHUFFLE_LEMMA = prove
(`!r s t v:real^N.
&0 <= r /\
independent(v INSERT((s DELETE vec 0) UNION t)) /\
FINITE s /\ FINITE t /\ CARD(s UNION t) <= dimindex(:N) /\
vec 0 IN s /\ ~(vec 0 IN t) /\
~(v IN s) /\ ~(--v IN s) /\ ~(v IN t)
==> measure(ball(vec 0,r) INTER aff_gt (v INSERT s) t) =
measure(ball(vec 0,r) INTER aff_gt s (v INSERT t)) +
measure(ball(vec 0,r) INTER aff_gt s (--v INSERT t))`,
let lemma = prove
(`!s t u:real^N->bool.
measurable s /\ measurable t /\ s INTER t = {} /\ negligible u
==> measure(s UNION t UNION u) = measure s + measure t`,
REPEAT STRIP_TAC THEN REWRITE_TAC[UNION_ASSOC] THEN
ASM_SIMP_TAC[GSYM MEASURE_DISJOINT_UNION; DISJOINT] THEN
MATCH_MP_TAC MEASURE_NEGLIGIBLE_SYMDIFF THEN
ASM_SIMP_TAC[MEASURABLE_UNION] THEN
FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP
(REWRITE_RULE[IMP_CONJ] NEGLIGIBLE_SUBSET)) THEN SET_TAC[]) in
REPEAT STRIP_TAC THEN
W(MP_TAC o PART_MATCH (lhs o rand) AFF_GT_SHUFFLE o
rand o rand o lhand o snd) THEN
ASM_REWRITE_TAC[] THEN DISCH_THEN SUBST1_TAC THEN
REWRITE_TAC[UNION_OVER_INTER] THEN MATCH_MP_TAC lemma THEN
ASM_REWRITE_TAC[MEASURABLE_BALL_AFF_GT] THEN CONJ_TAC THENL
[MATCH_MP_TAC(SET_RULE
`t INTER u = {} ==> (s INTER t) INTER (s INTER u) = {}`) THEN
REWRITE_TAC[aff_gt_def; AFFSIGN_ALT; sgn_gt] THEN
REWRITE_TAC[SET_RULE `(v INSERT s) UNION t = v INSERT (s UNION t)`;
SET_RULE `s UNION (v INSERT t) = v INSERT (s UNION t)`] THEN
ASM_SIMP_TAC[FINITE_INSERT; FINITE_UNION; AFFINE_HULL_FINITE_STEP_GEN;
RIGHT_EXISTS_AND_THM; REAL_LT_ADD;
REAL_HALF; FINITE_EMPTY] THEN
REWRITE_TAC[IN_INSERT] THEN
ASM_SIMP_TAC[SET_RULE
`~(a IN s) ==> ((w IN s UNION t ==> w = a \/ w IN t ==> P w) <=>
(w IN t ==> P w))`] THEN
GEN_REWRITE_TAC I [EXTENSION] THEN
REWRITE_TAC[IN_INTER; NOT_IN_EMPTY; IN_ELIM_THM] THEN
X_GEN_TAC `y:real^N` THEN
DISCH_THEN(CONJUNCTS_THEN2
(X_CHOOSE_THEN `a:real`
(CONJUNCTS_THEN2 ASSUME_TAC
(X_CHOOSE_THEN `f:real^N->real` STRIP_ASSUME_TAC)))
(X_CHOOSE_THEN `b:real`
(CONJUNCTS_THEN2 ASSUME_TAC
(X_CHOOSE_THEN `g:real^N->real` STRIP_ASSUME_TAC)))) THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [INDEPENDENT_EXPLICIT]) THEN
REWRITE_TAC[FINITE_INSERT; FINITE_DELETE; FINITE_UNION] THEN
DISCH_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC MP_TAC) THEN
DISCH_THEN(MP_TAC o SPEC
`\x. if x = v then a + b else (f:real^N->real) x - g x`) THEN
ASM_SIMP_TAC[VSUM_CLAUSES; FINITE_DELETE; FINITE_UNION] THEN
ASM_REWRITE_TAC[IN_DELETE; IN_UNION] THEN
REWRITE_TAC[NOT_IMP] THEN CONJ_TAC THENL
[ALL_TAC; DISCH_THEN(MP_TAC o SPEC `v:real^N`) THEN
REWRITE_TAC[IN_INSERT] THEN ASM_REAL_ARITH_TAC] THEN
ASM_SIMP_TAC[SET_RULE
`~(a IN t) ==> (s DELETE a) UNION t = (s UNION t) DELETE a`] THEN
ASM_SIMP_TAC[VSUM_DELETE_CASES; FINITE_UNION; IN_UNION] THEN
REWRITE_TAC[VECTOR_MUL_RZERO; VECTOR_SUB_RZERO] THEN
SUBGOAL_THEN
`!x:real^N. (if x = v then a + b else f x - g x) % x =
(if x = v then a else f x) % x -
(if x = v then --b else g x) % x`
(fun th -> REWRITE_TAC[th])
THENL
[GEN_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN VECTOR_ARITH_TAC;
ASM_SIMP_TAC[VSUM_SUB; FINITE_UNION]] THEN
MATCH_MP_TAC EQ_TRANS THEN
EXISTS_TAC `(a + b) % v + (y - a % v) - (y - b % --v):real^N` THEN
CONJ_TAC THENL [ALL_TAC; VECTOR_ARITH_TAC] THEN
AP_TERM_TAC THEN BINOP_TAC THEN
FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC RAND_CONV [GSYM th]) THEN
MATCH_MP_TAC VSUM_EQ THEN GEN_TAC THEN REWRITE_TAC[] THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[IN_UNION];
MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN
EXISTS_TAC `aff_gt s t :real^N->bool` THEN
REWRITE_TAC[INTER_SUBSET] THEN
MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN
EXISTS_TAC `affine hull (s UNION t:real^N->bool)` THEN
REWRITE_TAC[AFF_GT_SUBSET_AFFINE_HULL] THEN
ASM_SIMP_TAC[AFFINE_HULL_EQ_SPAN; IN_UNION; HULL_INC] THEN
ONCE_REWRITE_TAC[GSYM SPAN_DELETE_0] THEN
MATCH_MP_TAC NEGLIGIBLE_LOWDIM THEN
MATCH_MP_TAC LET_TRANS THEN
EXISTS_TAC `CARD((s UNION t) DELETE (vec 0:real^N))` THEN
ASM_SIMP_TAC[DIM_LE_CARD; FINITE_DELETE; FINITE_UNION; DIM_SPAN] THEN
ASM_SIMP_TAC[CARD_DELETE; IN_UNION; FINITE_UNION] THEN
MATCH_MP_TAC(ARITH_RULE `1 <= n /\ x <= n ==> x - 1 < n`) THEN
ASM_REWRITE_TAC[DIMINDEX_GE_1]]);;
let MEASURE_BALL_AFF_GT_SHUFFLE = prove
(`!r s t v:real^N.
&0 <= r /\ ~(v IN (s UNION t)) /\
independent(v INSERT (s UNION t))
==> measure(ball(vec 0,r) INTER aff_gt (vec 0 INSERT v INSERT s) t) =
measure(ball(vec 0,r) INTER aff_gt (vec 0 INSERT s) (v INSERT t)) +
measure(ball(vec 0,r) INTER
aff_gt (vec 0 INSERT s) (--v INSERT t))`,
REWRITE_TAC[IN_UNION; DE_MORGAN_THM] THEN REPEAT STRIP_TAC THEN
MP_TAC(ISPECL [`r:real`; `(vec 0:real^N) INSERT s`;
`t:real^N->bool`; `v:real^N`]
MEASURE_BALL_AFF_GT_SHUFFLE_LEMMA) THEN
ANTS_TAC THENL [ALL_TAC; REWRITE_TAC[INSERT_AC]] THEN
ASM_REWRITE_TAC[IN_INSERT; FINITE_INSERT] THEN
FIRST_ASSUM(MP_TAC o MATCH_MP INDEPENDENT_NONZERO) THEN
REWRITE_TAC[IN_INSERT; IN_UNION; DE_MORGAN_THM] THEN
STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
FIRST_ASSUM(MP_TAC o MATCH_MP INDEPENDENT_BOUND) THEN
REWRITE_TAC[FINITE_INSERT; FINITE_UNION] THEN
DISCH_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC MP_TAC) THEN
REWRITE_TAC[SET_RULE `(a INSERT s) UNION t = a INSERT (s UNION t)`] THEN
ASM_SIMP_TAC[CARD_CLAUSES; FINITE_UNION; IN_UNION; FINITE_INSERT] THEN
DISCH_TAC THEN ASM_REWRITE_TAC[VECTOR_NEG_EQ_0] THEN CONJ_TAC THENL
[FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP
(REWRITE_RULE[IMP_CONJ] INDEPENDENT_MONO)) THEN
SET_TAC[];
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [independent]) THEN
REWRITE_TAC[dependent; CONTRAPOS_THM] THEN DISCH_TAC THEN
EXISTS_TAC `v:real^N` THEN REWRITE_TAC[IN_INSERT] THEN
GEN_REWRITE_TAC LAND_CONV [GSYM VECTOR_NEG_NEG] THEN
MATCH_MP_TAC SPAN_NEG THEN MATCH_MP_TAC SPAN_SUPERSET THEN
ASM_REWRITE_TAC[IN_DELETE; VECTOR_ARITH `--v:real^N = v <=> v = vec 0`;
IN_INSERT; IN_UNION]]);;
let MEASURE_LUNE_DECOMPOSITION = prove
(`!v1 v2 v3:real^3.
&0 <= r /\ ~coplanar {vec 0, v1, v2, v3}
==> measure(ball(vec 0,r) INTER aff_gt {vec 0} {v1,v2,v3}) +
measure(ball(vec 0,r) INTER aff_gt {vec 0} {--v1,v2,v3}) =
dihV (vec 0) v1 v2 v3 * &2 * r pow 3 / &3`,
let rec distinctpairs l =
match l with
x::t -> itlist (fun y a -> (x,y) :: a) t (distinctpairs t)
| [] -> [] in
REPEAT GEN_TAC THEN MAP_EVERY
(fun t -> ASM_CASES_TAC t THENL
[ASM_REWRITE_TAC[INSERT_AC; COPLANAR_3]; ALL_TAC])
(map mk_eq (distinctpairs
[`v3:real^3`; `v2:real^3`; `v1:real^3`; `vec 0:real^3`])) THEN
REPEAT STRIP_TAC THEN
ASM_SIMP_TAC[GSYM(REWRITE_RULE[HAS_MEASURE_MEASURABLE_MEASURE]
HAS_MEASURE_LUNE_SIMPLE)] THEN
CONV_TAC SYM_CONV THEN MATCH_MP_TAC MEASURE_BALL_AFF_GT_SHUFFLE THEN
ASM_REWRITE_TAC[UNION_EMPTY; IN_INSERT; NOT_IN_EMPTY] THEN
ASM_SIMP_TAC[NOT_COPLANAR_0_4_IMP_INDEPENDENT]);;
let SOLID_TRIANGLE_CONGRUENT_NEG = prove
(`!r v1 v2 v3:real^N.
measure(ball(vec 0,r) INTER aff_gt {vec 0} {--v1, --v2, --v3}) =
measure(ball(vec 0,r) INTER aff_gt {vec 0} {v1, v2, v3})`,
REPEAT GEN_TAC THEN
SUBGOAL_THEN
`ball(vec 0:real^N,r) INTER aff_gt {vec 0} {--v1, --v2, --v3} =
IMAGE (--)
(ball(vec 0,r) INTER aff_gt {vec 0} {v1, v2, v3})`
SUBST1_TAC THENL
[ALL_TAC;
MATCH_MP_TAC MEASURE_ORTHOGONAL_IMAGE_EQ THEN
REWRITE_TAC[ORTHOGONAL_TRANSFORMATION; linear; NORM_NEG] THEN
CONJ_TAC THEN VECTOR_ARITH_TAC] THEN
CONV_TAC SYM_CONV THEN MATCH_MP_TAC SURJECTIVE_IMAGE_EQ THEN
CONJ_TAC THENL [MESON_TAC[VECTOR_NEG_NEG]; ALL_TAC] THEN
REWRITE_TAC[IN_INTER; IN_BALL_0; NORM_NEG] THEN
REWRITE_TAC[AFFSIGN_ALT; aff_gt_def; sgn_gt; IN_ELIM_THM] THEN
REWRITE_TAC[SET_RULE `{a} UNION {b,c,d} = {a,b,d,c}`] THEN
REWRITE_TAC[SET_RULE `x IN {a} <=> a = x`] THEN
ASM_SIMP_TAC[FINITE_INSERT; FINITE_UNION; AFFINE_HULL_FINITE_STEP_GEN;
RIGHT_EXISTS_AND_THM; REAL_LT_ADD; REAL_HALF; FINITE_EMPTY] THEN
ASM_REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY] THEN
REWRITE_TAC[VECTOR_MUL_RZERO; VECTOR_SUB_RZERO] THEN
REWRITE_TAC[VECTOR_ARITH `vec 0:real^N = --x <=> vec 0 = x`] THEN
REWRITE_TAC[VECTOR_ARITH `--x - a % --w:real^N = --(x - a % w)`] THEN
REWRITE_TAC[VECTOR_NEG_EQ_0]);;
let VOLUME_SOLID_TRIANGLE = prove
(`!r v0 v1 v2 v3:real^3.
&0 < r /\ ~coplanar{v0, v1, v2, v3}
==> measure(ball(v0,r) INTER aff_gt {v0} {v1,v2,v3}) =
let a123 = dihV v0 v1 v2 v3 in
let a231 = dihV v0 v2 v3 v1 in
let a312 = dihV v0 v3 v1 v2 in
(a123 + a231 + a312 - pi) * r pow 3 / &3`,
let tac convl =
W(MP_TAC o PART_MATCH (lhs o rand) MEASURE_BALL_AFF_GT_SHUFFLE o
convl o lhand o lhand o snd) THEN
ASM_REWRITE_TAC[UNION_EMPTY; IN_INSERT; IN_UNION; NOT_IN_EMPTY] THEN
REWRITE_TAC[SET_RULE `(a INSERT s) UNION t = a INSERT (s UNION t)`] THEN
ASM_SIMP_TAC[UNION_EMPTY; REAL_LT_IMP_LE] THEN ANTS_TAC THENL
[CONJ_TAC THENL
[DISCH_THEN(STRIP_THM_THEN SUBST_ALL_TAC) THEN
RULE_ASSUM_TAC(REWRITE_RULE[INSERT_AC]) THEN
RULE_ASSUM_TAC(REWRITE_RULE[COPLANAR_3]) THEN
FIRST_ASSUM CONTR_TAC;
MATCH_MP_TAC NOT_COPLANAR_0_4_IMP_INDEPENDENT THEN
RULE_ASSUM_TAC(REWRITE_RULE[INSERT_AC]) THEN
ASM_REWRITE_TAC[INSERT_AC]];
DISCH_THEN SUBST1_TAC] in
GEN_TAC THEN GEOM_ORIGIN_TAC `v0:real^3` THEN
CONV_TAC(TOP_DEPTH_CONV let_CONV) THEN REPEAT STRIP_TAC THEN
SUBGOAL_THEN
`measure(ball(vec 0:real^3,r) INTER aff_gt {vec 0,v1,v2,v3} {}) =
&4 / &3 * pi * r pow 3`
MP_TAC THENL
[MP_TAC(SPECL [`vec 0:real^3`; `r:real`] VOLUME_BALL) THEN
ASM_SIMP_TAC[REAL_LT_IMP_LE] THEN
DISCH_THEN(SUBST1_TAC o SYM) THEN AP_TERM_TAC THEN
MATCH_MP_TAC(SET_RULE `t = UNIV ==> s INTER t = s`) THEN
REWRITE_TAC[AFF_GT_EQ_AFFINE_HULL] THEN
SIMP_TAC[AFFINE_HULL_EQ_SPAN; HULL_INC; IN_INSERT; SPAN_INSERT_0] THEN
REWRITE_TAC[SET_RULE `s = UNIV <=> UNIV SUBSET s`] THEN
MATCH_MP_TAC CARD_GE_DIM_INDEPENDENT THEN
ASM_SIMP_TAC[DIM_UNIV; DIMINDEX_3; SUBSET_UNIV] THEN
ASM_SIMP_TAC[NOT_COPLANAR_0_4_IMP_INDEPENDENT] THEN
SIMP_TAC[CARD_CLAUSES; FINITE_INSERT; FINITE_EMPTY] THEN
REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY] THEN
MAP_EVERY (fun t ->
ASM_CASES_TAC t THENL
[FIRST_X_ASSUM SUBST_ALL_TAC THEN
RULE_ASSUM_TAC(REWRITE_RULE[INSERT_AC; COPLANAR_3]) THEN
ASM_MESON_TAC[];
ASM_REWRITE_TAC[]])
[`v3:real^3 = v2`; `v3:real^3 = v1`; `v2:real^3 = v1`] THEN
CONV_TAC NUM_REDUCE_CONV;
ALL_TAC] THEN
SUBGOAL_THEN
`~(coplanar {vec 0:real^3,v1,v2,v3}) /\
~(coplanar {vec 0,--v1,v2,v3}) /\
~(coplanar {vec 0,v1,--v2,v3}) /\
~(coplanar {vec 0,--v1,--v2,v3}) /\
~(coplanar {vec 0,--v1,--v2,v3}) /\
~(coplanar {vec 0,--v1,v2,--v3}) /\
~(coplanar {vec 0,v1,--v2,--v3}) /\
~(coplanar {vec 0,--v1,--v2,--v3}) /\
~(coplanar {vec 0,--v1,--v2,--v3})`
STRIP_ASSUME_TAC THENL
[REPLICATE_TAC 3
(REWRITE_TAC[COPLANAR_INSERT_0_NEG] THEN
ONCE_REWRITE_TAC[SET_RULE `{vec 0,a,b,c} = {vec 0,b,c,a}`]) THEN
ASM_REWRITE_TAC[];
ALL_TAC] THEN
MAP_EVERY tac
[I; lhand; rand;
lhand o lhand; rand o lhand; lhand o rand; rand o rand] THEN
MP_TAC(ISPECL [`v1:real^3`; `v2:real^3`; `v3:real^3`]
MEASURE_LUNE_DECOMPOSITION) THEN
MP_TAC(ISPECL [`v2:real^3`; `v3:real^3`; `v1:real^3`]
MEASURE_LUNE_DECOMPOSITION) THEN
MP_TAC(ISPECL [`v3:real^3`; `v1:real^3`; `v2:real^3`]
MEASURE_LUNE_DECOMPOSITION) THEN
MP_TAC(ISPECL [`--v1:real^3`; `--v2:real^3`; `--v3:real^3`]
MEASURE_LUNE_DECOMPOSITION) THEN
MP_TAC(ISPECL [`--v2:real^3`; `--v3:real^3`; `--v1:real^3`]
MEASURE_LUNE_DECOMPOSITION) THEN
MP_TAC(ISPECL [`--v3:real^3`; `--v1:real^3`; `--v2:real^3`]
MEASURE_LUNE_DECOMPOSITION) THEN
ASM_REWRITE_TAC[VECTOR_NEG_NEG] THEN
ASM_SIMP_TAC[REAL_LT_IMP_LE; INSERT_AC] THEN
RULE_ASSUM_TAC(REWRITE_RULE[INSERT_AC]) THEN ASM_REWRITE_TAC[] THEN
REWRITE_TAC[DIHV_NEG_0] THEN
REWRITE_TAC[SOLID_TRIANGLE_CONGRUENT_NEG] THEN
REWRITE_TAC[INSERT_AC] THEN REAL_ARITH_TAC);;
(* ------------------------------------------------------------------------- *)
(* Volume of wedge of a frustum. *)
(* ------------------------------------------------------------------------- *)
let MEASURABLE_BOUNDED_INTER_OPEN = prove
(`!s t:real^N->bool.
measurable s /\ bounded s /\ open t ==> measurable(s INTER t)`,
REPEAT STRIP_TAC THEN
FIRST_X_ASSUM(MP_TAC o MATCH_MP BOUNDED_SUBSET_OPEN_INTERVAL) THEN
REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN
MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real^N`] THEN
DISCH_THEN(SUBST1_TAC o MATCH_MP (SET_RULE
`s SUBSET i ==> s INTER t = s INTER (t INTER i)`)) THEN
MATCH_MP_TAC MEASURABLE_INTER THEN ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC MEASURABLE_OPEN THEN
ASM_SIMP_TAC[OPEN_INTER; OPEN_INTERVAL; BOUNDED_INTER; BOUNDED_INTERVAL]);;
let SLICE_SPECIAL_WEDGE = prove
(`!w1 w2.
~collinear {vec 0, basis 3, w1} /\ ~collinear {vec 0, basis 3, w2}
==> slice 3 t (wedge (vec 0) (basis 3) w1 w2) =
{z | &0 < Arg(z / dropout 3 w1) /\
Arg(z / dropout 3 w1) < Arg(dropout 3 w2 / dropout 3 w1)}`,
REWRITE_TAC[wedge] THEN
ONCE_REWRITE_TAC[TAUT `~a /\ b /\ c <=> ~(~a ==> ~(b /\ c))`] THEN
ASM_SIMP_TAC[AZIM_ARG] THEN REWRITE_TAC[COLLINEAR_BASIS_3] THEN
REWRITE_TAC[NOT_IMP; GSYM CONJ_ASSOC; DROPOUT_0] THEN
MAP_EVERY ABBREV_TAC
[`v1:real^2 = dropout 3 (w1:real^3)`;
`v2:real^2 = dropout 3 (w2:real^3)`] THEN
REWRITE_TAC[SLICE_DROPOUT_3; VEC_COMPONENT; REAL_SUB_RZERO] THEN
REPEAT STRIP_TAC THEN
REWRITE_TAC[EXTENSION; IN_ELIM_THM; COMPLEX_VEC_0] THEN
X_GEN_TAC `w:complex` THEN
ASM_CASES_TAC `w = Cx(&0)` THEN ASM_REWRITE_TAC[] THEN
ASM_REWRITE_TAC[complex_div; COMPLEX_MUL_LZERO; ARG_0; REAL_LT_REFL]);;
let VOLUME_FRUSTT_WEDGE = prove
(`!v0 v1:real^3 w1 w2 h a.
&0 < a /\ ~collinear {v0,v1,w1} /\ ~collinear {v0,v1,w2}
==> bounded(frustt v0 v1 h a INTER wedge v0 v1 w1 w2) /\
measurable(frustt v0 v1 h a INTER wedge v0 v1 w1 w2) /\
measure(frustt v0 v1 h a INTER wedge v0 v1 w1 w2) =
if &1 <= a \/ h < &0 then &0
else azim v0 v1 w1 w2 * ((h / a) pow 2 - h pow 2) * h / &6`,
REPEAT GEN_TAC THEN ASM_CASES_TAC `v1:real^3 = v0` THENL
[ASM_REWRITE_TAC[INSERT_AC; COLLINEAR_2]; STRIP_TAC] THEN
MATCH_MP_TAC(TAUT `a /\ b /\ (a /\ b ==> c) ==> a /\ b /\ c`) THEN
REPEAT CONJ_TAC THENL
[MATCH_MP_TAC BOUNDED_INTER THEN ASM_SIMP_TAC[VOLUME_FRUSTT_STRONG];
MATCH_MP_TAC MEASURABLE_BOUNDED_INTER_OPEN THEN
ASM_SIMP_TAC[VOLUME_FRUSTT_STRONG; OPEN_WEDGE];
ALL_TAC] THEN
REWRITE_TAC[frustt; frustum; rcone_gt; rconesgn; IN_ELIM_THM] THEN
CONV_TAC(TOP_DEPTH_CONV let_CONV) THEN
POP_ASSUM_LIST(MP_TAC o end_itlist CONJ) THEN
GEOM_ORIGIN_TAC `v0:real^3` THEN
REWRITE_TAC[VECTOR_SUB_RZERO; REAL_MUL_LZERO; DIST_0; real_gt] THEN
GEOM_BASIS_MULTIPLE_TAC 3 `v1:real^3` THEN
X_GEN_TAC `b:real` THEN GEN_REWRITE_TAC LAND_CONV [REAL_LE_LT] THEN
ASM_CASES_TAC `b = &0` THEN ASM_REWRITE_TAC[VECTOR_MUL_LZERO] THEN
ASM_SIMP_TAC[COLLINEAR_SPECIAL_SCALE; WEDGE_SPECIAL_SCALE] THEN
ASM_REWRITE_TAC[VECTOR_MUL_EQ_0] THEN
DISCH_TAC THEN REPEAT GEN_TAC THEN STRIP_TAC THEN
ASM_CASES_TAC `&1 <= a` THEN ASM_REWRITE_TAC[] THENL
[SUBGOAL_THEN
`!y:real^3. ~(norm(y) * norm(b % basis 3:real^3) * a
< y dot (b % basis 3))`
(fun th -> REWRITE_TAC[th; EMPTY_GSPEC; MEASURABLE_EMPTY;
INTER_EMPTY; MEASURE_EMPTY]) THEN
REWRITE_TAC[REAL_NOT_LT] THEN X_GEN_TAC `y:real^3` THEN
MATCH_MP_TAC(REAL_ARITH `abs(x) <= a ==> x <= a`) THEN
SIMP_TAC[DOT_RMUL; NORM_MUL; REAL_ABS_MUL; DOT_BASIS; NORM_BASIS;
DIMINDEX_3; ARITH] THEN
REWRITE_TAC[REAL_ARITH
`b * y <= n * (b * &1) * a <=> b * &1 * y <= b * a * n`] THEN
MATCH_MP_TAC REAL_LE_LMUL THEN REWRITE_TAC[REAL_ABS_POS] THEN
MATCH_MP_TAC REAL_LE_MUL2 THEN
ASM_SIMP_TAC[REAL_POS; REAL_ABS_POS; COMPONENT_LE_NORM; DIMINDEX_3; ARITH];
ALL_TAC] THEN
RULE_ASSUM_TAC(REWRITE_RULE[REAL_NOT_LE]) THEN
SIMP_TAC[NORM_MUL; NORM_BASIS; DOT_BASIS; DOT_RMUL; DIMINDEX_3; ARITH] THEN
ONCE_REWRITE_TAC[REAL_ARITH `n * x * a:real = x * n * a`] THEN
ASM_REWRITE_TAC[real_abs; REAL_MUL_RID] THEN
ASM_SIMP_TAC[REAL_MUL_RID; REAL_LT_LMUL_EQ; REAL_LT_MUL_EQ; NORM_POS_LT] THEN
ASM_SIMP_TAC[VECTOR_MUL_EQ_0; BASIS_NONZERO; DIMINDEX_3; ARITH;
REAL_LT_IMP_NZ] THEN
ASM_SIMP_TAC[GSYM REAL_LT_RDIV_EQ; NORM_LT_SQUARE] THEN
ASM_SIMP_TAC[REAL_POW_DIV; REAL_POW_LT; REAL_LT_RDIV_EQ] THEN
REWRITE_TAC[REAL_ARITH `(&0 * x < y /\ u < v) /\ &0 < y /\ y < h <=>
&0 < y /\ y < h /\ u < v`] THEN
DISCH_TAC THEN MATCH_MP_TAC(INST_TYPE [`:2`,`:M`] FUBINI_SIMPLE_ALT) THEN
EXISTS_TAC `3` THEN ASM_REWRITE_TAC[DIMINDEX_2; DIMINDEX_3; ARITH] THEN
ASM_SIMP_TAC[WEDGE_SPECIAL_SCALE; REAL_LT_IMP_LE] THEN
ASM_SIMP_TAC[REAL_LT_LMUL_EQ; SLICE_INTER; DIMINDEX_2;
DIMINDEX_3; ARITH] THEN
SUBGOAL_THEN
`!t. slice 3 t {y:real^3 | norm y * a < y$3 /\ &0 < y$3 /\ y$3 < h} =
if t < h
then ball(vec 0:real^2,sqrt(inv(a pow 2) - &1) * t)
else {}`
(fun th -> ASM_SIMP_TAC[th; SLICE_SPECIAL_WEDGE])
THENL
[REWRITE_TAC[EXTENSION] THEN
MAP_EVERY X_GEN_TAC [`t:real`; `z:real^2`] THEN
SIMP_TAC[SLICE_123; DIMINDEX_2; DIMINDEX_3; ARITH; IN_ELIM_THM;
VECTOR_3; DOT_3; GSYM DOT_2] THEN
ASM_CASES_TAC `t < h` THEN ASM_REWRITE_TAC[NOT_IN_EMPTY] THEN
REWRITE_TAC[IN_BALL_0; IN_DELETE] THEN
MATCH_MP_TAC(REAL_ARITH
`&0 <= a /\ (a < t <=> u < v) ==> (a < t /\ &0 < t <=> u < v)`) THEN
ASM_SIMP_TAC[REAL_LE_MUL; NORM_POS_LE; REAL_LT_IMP_LE] THEN
ASM_SIMP_TAC[GSYM REAL_LT_RDIV_EQ; NORM_LT_SQUARE] THEN
SUBGOAL_THEN `&0 < inv(a pow 2) - &1` ASSUME_TAC THENL
[REWRITE_TAC[REAL_SUB_LT] THEN MATCH_MP_TAC REAL_INV_1_LT THEN
ASM_SIMP_TAC[REAL_POW_1_LT; REAL_LT_IMP_LE; ARITH; REAL_POW_LT];
ALL_TAC] THEN
ASM_SIMP_TAC[REAL_LT_MUL; SQRT_POS_LT; REAL_POW_MUL; SQRT_POW_2;
REAL_LT_IMP_LE; REAL_LT_MUL_EQ] THEN
ASM_SIMP_TAC[real_div; REAL_LT_MUL_EQ; REAL_LT_INV_EQ] THEN
ASM_CASES_TAC `&0 < t` THEN ASM_REWRITE_TAC[] THEN
REWRITE_TAC[DOT_3; DOT_2; VECTOR_3; REAL_INV_POW] THEN
REAL_ARITH_TAC;
ALL_TAC] THEN
GEN_REWRITE_TAC (LAND_CONV o TOP_DEPTH_CONV) [COND_RATOR; COND_RAND] THEN
GEN_REWRITE_TAC (RAND_CONV o RATOR_CONV o LAND_CONV o TOP_DEPTH_CONV)
[COND_RATOR; COND_RAND] THEN
REWRITE_TAC[INTER_EMPTY; MEASURABLE_EMPTY; MEASURE_EMPTY] THEN
REWRITE_TAC[INTER; IN_BALL_0; IN_ELIM_THM] THEN
RULE_ASSUM_TAC(REWRITE_RULE[COLLINEAR_BASIS_3]) THEN
ASM_SIMP_TAC[REWRITE_RULE[HAS_MEASURE_MEASURABLE_MEASURE]
HAS_MEASURE_OPEN_SECTOR_LT_GEN] THEN
REWRITE_TAC[COND_ID] THEN
SUBGOAL_THEN `&0 < inv(a pow 2) - &1` ASSUME_TAC THENL
[REWRITE_TAC[REAL_SUB_LT] THEN MATCH_MP_TAC REAL_INV_1_LT THEN
ASM_SIMP_TAC[REAL_POW_1_LT; REAL_LT_IMP_LE; ARITH; REAL_POW_LT];
ALL_TAC] THEN
ASM_SIMP_TAC[REAL_LE_MUL_EQ; SQRT_POS_LT] THEN
ASM_SIMP_TAC[AZIM_SPECIAL_SCALE; AZIM_ARG; COLLINEAR_BASIS_3] THEN
MATCH_MP_TAC HAS_REAL_INTEGRAL_EQ THEN
EXISTS_TAC
`\t. if &0 < t /\ t < h
then Arg(dropout 3 (w2:real^3) / dropout 3 (w1:real^3)) / &2 *
(inv(a pow 2) - &1) * t pow 2
else &0` THEN
CONJ_TAC THENL
[X_GEN_TAC `t:real` THEN DISCH_TAC THEN REWRITE_TAC[] THEN
ASM_CASES_TAC `t < h` THEN ASM_REWRITE_TAC[] THEN
REWRITE_TAC[REAL_ARITH `&0 <= t <=> t = &0 \/ &0 < t`] THEN
ASM_CASES_TAC `t = &0` THEN
ASM_REWRITE_TAC[REAL_LT_REFL; REAL_MUL_RZERO; SQRT_0] THEN
CONV_TAC REAL_RAT_REDUCE_CONV THEN REWRITE_TAC[REAL_MUL_RZERO] THEN
ASM_SIMP_TAC[REAL_POW_MUL; SQRT_POW_2; REAL_LT_IMP_LE] THEN
REAL_ARITH_TAC;
ALL_TAC] THEN
REWRITE_TAC[GSYM IN_REAL_INTERVAL; HAS_REAL_INTEGRAL_RESTRICT_UNIV] THEN
REWRITE_TAC[HAS_REAL_INTEGRAL_OPEN_INTERVAL] THEN
COND_CASES_TAC THENL
[ASM_MESON_TAC[REAL_INTERVAL_EQ_EMPTY; HAS_REAL_INTEGRAL_EMPTY];
RULE_ASSUM_TAC(REWRITE_RULE[REAL_NOT_LT])] THEN
ABBREV_TAC `g = Arg(dropout 3 (w2:real^3) / dropout 3 (w1:real^3))` THEN
MP_TAC(ISPECL
[`\t. g / &6 * (inv (a pow 2) - &1) * t pow 3`;
`\t. g / &2 * (inv (a pow 2) - &1) * t pow 2`;
`&0`; `h:real`] REAL_FUNDAMENTAL_THEOREM_OF_CALCULUS) THEN
REWRITE_TAC[] THEN ANTS_TAC THENL
[ASM_REWRITE_TAC[] THEN
REPEAT STRIP_TAC THEN REAL_DIFF_TAC THEN
CONV_TAC NUM_REDUCE_CONV THEN CONV_TAC REAL_RING;
MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN
UNDISCH_TAC `&0 < a` THEN CONV_TAC REAL_FIELD]);;
(* ------------------------------------------------------------------------- *)
(* Wedge of a conic cap. *)
(* ------------------------------------------------------------------------- *)
let VOLUME_CONIC_CAP_WEDGE_WEAK = prove
(`!v0 v1:real^3 w1 w2 r a.
&0 < a /\ ~collinear {v0,v1,w1} /\ ~collinear {v0,v1,w2}
==> bounded(conic_cap v0 v1 r a INTER wedge v0 v1 w1 w2) /\
measurable(conic_cap v0 v1 r a INTER wedge v0 v1 w1 w2) /\
measure(conic_cap v0 v1 r a INTER wedge v0 v1 w1 w2) =
if &1 <= a \/ r < &0 then &0
else azim v0 v1 w1 w2 / &3 * (&1 - a) * r pow 3`,
REPEAT GEN_TAC THEN ASM_CASES_TAC `v1:real^3 = v0` THENL
[ASM_REWRITE_TAC[COLLINEAR_2; INSERT_AC]; STRIP_TAC] THEN
MATCH_MP_TAC(TAUT `a /\ b /\ (a /\ b ==> c) ==> a /\ b /\ c`) THEN
REPEAT CONJ_TAC THENL
[MATCH_MP_TAC BOUNDED_INTER THEN ASM_SIMP_TAC[VOLUME_CONIC_CAP_STRONG];
MATCH_MP_TAC MEASURABLE_BOUNDED_INTER_OPEN THEN
ASM_SIMP_TAC[VOLUME_CONIC_CAP_STRONG; OPEN_WEDGE];
ALL_TAC] THEN
REWRITE_TAC[conic_cap; rcone_gt; rconesgn; IN_ELIM_THM] THEN
REWRITE_TAC[ONCE_REWRITE_RULE[DIST_SYM] normball; GSYM ball] THEN
CONV_TAC(TOP_DEPTH_CONV let_CONV) THEN
POP_ASSUM_LIST(MP_TAC o end_itlist CONJ) THEN
GEOM_ORIGIN_TAC `v0:real^3` THEN
REWRITE_TAC[VECTOR_SUB_RZERO; REAL_MUL_LZERO; DIST_0; real_gt] THEN
GEOM_BASIS_MULTIPLE_TAC 3 `v1:real^3` THEN
X_GEN_TAC `b:real` THEN
ASM_CASES_TAC `b = &0` THEN ASM_REWRITE_TAC[VECTOR_MUL_LZERO] THEN
SIMP_TAC[COLLINEAR_SPECIAL_SCALE; WEDGE_SPECIAL_SCALE] THEN
ASM_REWRITE_TAC[VECTOR_MUL_EQ_0] THEN
DISCH_TAC THEN REPEAT GEN_TAC THEN STRIP_TAC THEN
ASM_CASES_TAC `&1 <= a` THEN ASM_REWRITE_TAC[] THENL
[SUBGOAL_THEN
`!y:real^3. ~(norm(y) * norm(b % basis 3:real^3) * a
< y dot (b % basis 3))`
(fun th -> REWRITE_TAC[th; EMPTY_GSPEC; INTER_EMPTY; MEASURE_EMPTY;
MEASURABLE_EMPTY; BOUNDED_EMPTY; CONVEX_EMPTY]) THEN
REWRITE_TAC[REAL_NOT_LT] THEN X_GEN_TAC `y:real^3` THEN
MATCH_MP_TAC(REAL_ARITH `abs(x) <= a ==> x <= a`) THEN
SIMP_TAC[DOT_RMUL; NORM_MUL; REAL_ABS_MUL; DOT_BASIS; NORM_BASIS;
DIMINDEX_3; ARITH] THEN
REWRITE_TAC[REAL_ARITH
`b * y <= n * (b * &1) * a <=> b * &1 * y <= b * a * n`] THEN
MATCH_MP_TAC REAL_LE_LMUL THEN REWRITE_TAC[REAL_ABS_POS] THEN
MATCH_MP_TAC REAL_LE_MUL2 THEN
ASM_SIMP_TAC[REAL_POS; REAL_ABS_POS; COMPONENT_LE_NORM; DIMINDEX_3; ARITH];
ALL_TAC] THEN
RULE_ASSUM_TAC(REWRITE_RULE[REAL_NOT_LE]) THEN
SIMP_TAC[DOT_RMUL; NORM_MUL; REAL_ABS_NORM; DOT_BASIS;
DIMINDEX_3; ARITH; NORM_BASIS] THEN
ONCE_REWRITE_TAC[REAL_ARITH `n * x * a:real = x * n * a`] THEN
ASM_REWRITE_TAC[real_abs; REAL_MUL_RID] THEN
ASM_SIMP_TAC[REAL_MUL_RID; REAL_LT_LMUL_EQ; REAL_LT_MUL_EQ; NORM_POS_LT] THEN
ASM_SIMP_TAC[GSYM REAL_LT_RDIV_EQ; NORM_LT_SQUARE] THEN
ASM_SIMP_TAC[REAL_POW_DIV; REAL_POW_LT; REAL_LT_RDIV_EQ] THEN
REWRITE_TAC[INTER; REAL_MUL_LZERO; IN_BALL_0; IN_ELIM_THM] THEN
ASM_SIMP_TAC[VECTOR_MUL_EQ_0; BASIS_NONZERO; DIMINDEX_3; ARITH;
REAL_LT_IMP_NZ] THEN
COND_CASES_TAC THENL
[ASM_SIMP_TAC[NORM_ARITH `r < &0 ==> ~(norm x < r)`] THEN
REWRITE_TAC[EMPTY_GSPEC; MEASURE_EMPTY; MEASURABLE_EMPTY;
BOUNDED_EMPTY; CONVEX_EMPTY];
RULE_ASSUM_TAC(ONCE_REWRITE_RULE[REAL_NOT_LT])] THEN
STRIP_TAC THEN MATCH_MP_TAC(INST_TYPE [`:2`,`:M`] FUBINI_SIMPLE_ALT) THEN
EXISTS_TAC `3` THEN ASM_REWRITE_TAC[DIMINDEX_2; DIMINDEX_3; ARITH] THEN
SUBGOAL_THEN `&0 < b` ASSUME_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
ASM_SIMP_TAC[WEDGE_SPECIAL_SCALE; AZIM_SPECIAL_SCALE] THEN
ONCE_REWRITE_TAC[SET_RULE `{x | P x /\ x IN s} = {x | P x} INTER s`] THEN
ASM_SIMP_TAC[REAL_LT_LMUL_EQ; SLICE_INTER; DIMINDEX_2;
DIMINDEX_3; ARITH] THEN
RULE_ASSUM_TAC
(REWRITE_RULE[MATCH_MP COLLINEAR_SPECIAL_SCALE (ASSUME `~(b = &0)`)]) THEN
SUBGOAL_THEN `&0 < inv(a pow 2) - &1` ASSUME_TAC THENL
[REWRITE_TAC[REAL_SUB_LT] THEN MATCH_MP_TAC REAL_INV_1_LT THEN
ASM_SIMP_TAC[REAL_POW_1_LT; REAL_LT_IMP_LE; ARITH; REAL_POW_LT];
ALL_TAC] THEN
SUBGOAL_THEN
`!t. slice 3 t {y:real^3 | norm y < r /\ norm y * a < y$3} =
if &0 < t /\ t < r
then ball(vec 0:real^2,min (sqrt(r pow 2 - t pow 2))
(t * sqrt(inv(a pow 2) - &1)))
else {}`
(fun th -> ASM_SIMP_TAC[th; SLICE_SPECIAL_WEDGE])
THENL
[REWRITE_TAC[EXTENSION] THEN
MAP_EVERY X_GEN_TAC [`t:real`; `z:real^2`] THEN
SIMP_TAC[SLICE_123; DIMINDEX_2; DIMINDEX_3; ARITH; IN_ELIM_THM;
VECTOR_3; DOT_3; GSYM DOT_2] THEN
ASM_CASES_TAC `&0 < t` THEN ASM_REWRITE_TAC[] THENL
[ALL_TAC;
REWRITE_TAC[NOT_IN_EMPTY; DE_MORGAN_THM] THEN DISJ2_TAC THEN
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH
`~(&0 < t) ==> &0 <= a ==> ~(a < t)`)) THEN
ASM_SIMP_TAC[REAL_LE_MUL; NORM_POS_LE; REAL_LT_IMP_LE]] THEN
ASM_CASES_TAC `t < r` THEN ASM_REWRITE_TAC[] THENL
[ALL_TAC;
REWRITE_TAC[NOT_IN_EMPTY; DE_MORGAN_THM] THEN DISJ1_TAC THEN
REWRITE_TAC[NORM_LT_SQUARE; DE_MORGAN_THM] THEN DISJ2_TAC THEN
REWRITE_TAC[DOT_3; VECTOR_3] THEN
MATCH_MP_TAC(REAL_ARITH
`r <= t /\ &0 <= a /\ &0 <= b ==> ~(a + b + t < r)`) THEN
REWRITE_TAC[REAL_LE_SQUARE; REAL_POW_2] THEN
MATCH_MP_TAC REAL_LE_MUL2 THEN ASM_REAL_ARITH_TAC] THEN
REWRITE_TAC[IN_BALL_0; REAL_LT_MIN] THEN
ASM_SIMP_TAC[GSYM REAL_LT_RDIV_EQ] THEN REWRITE_TAC[NORM_LT_SQUARE] THEN
SUBGOAL_THEN `&0 < r` ASSUME_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
SUBGOAL_THEN `t pow 2 < r pow 2` ASSUME_TAC THENL
[MATCH_MP_TAC REAL_POW_LT2 THEN REWRITE_TAC[ARITH] THEN
ASM_REAL_ARITH_TAC;
ALL_TAC] THEN
ASM_SIMP_TAC[REAL_LT_DIV; SQRT_POS_LT; REAL_LT_MUL; REAL_SUB_LT;
SQRT_POW_2; REAL_LT_IMP_LE; REAL_POW_MUL] THEN
REWRITE_TAC[DOT_2; DOT_3; VECTOR_3] THEN
ONCE_REWRITE_TAC[REAL_ARITH `a + b + c < d <=> a + b < d - c`] THEN
BINOP_TAC THEN AP_TERM_TAC THEN
UNDISCH_TAC `&0 < a` THEN CONV_TAC REAL_FIELD;
ALL_TAC] THEN
GEN_REWRITE_TAC (LAND_CONV o TOP_DEPTH_CONV) [COND_RATOR; COND_RAND] THEN
GEN_REWRITE_TAC (RAND_CONV o RATOR_CONV o LAND_CONV o TOP_DEPTH_CONV)
[COND_RATOR; COND_RAND] THEN
REWRITE_TAC[INTER_EMPTY; MEASURABLE_EMPTY; MEASURE_EMPTY] THEN
REWRITE_TAC[INTER; IN_BALL_0; IN_ELIM_THM] THEN
RULE_ASSUM_TAC(REWRITE_RULE[COLLINEAR_BASIS_3]) THEN
ASM_SIMP_TAC[REWRITE_RULE[HAS_MEASURE_MEASURABLE_MEASURE]
HAS_MEASURE_OPEN_SECTOR_LT_GEN] THEN
REWRITE_TAC[COND_ID] THEN
ASM_SIMP_TAC[REAL_LE_MIN; SQRT_POS_LE; REAL_LT_IMP_LE; REAL_LE_MUL;
REAL_POW_LE2; ARITH; REAL_SUB_LE; REAL_LT_MUL; SQRT_POS_LT] THEN
REWRITE_TAC[GSYM IN_REAL_INTERVAL; HAS_REAL_INTEGRAL_RESTRICT_UNIV] THEN
REWRITE_TAC[HAS_REAL_INTEGRAL_OPEN_INTERVAL] THEN
REWRITE_TAC[NORM_POW_2; DOT_3; VECTOR_3; DOT_2] THEN
ASM_SIMP_TAC[AZIM_ARG; COLLINEAR_BASIS_3] THEN
ONCE_REWRITE_TAC[REAL_ARITH
`(&1 - a) * az / &3 * r pow 3 =
az / &6 * (inv (a pow 2) - &1) * (a * r) pow 3 +
(az * &1 / &3 * (&1 - a) * r pow 3 -
az / &6 * (inv (a pow 2) - &1) * (a * r) pow 3)`] THEN
MATCH_MP_TAC HAS_REAL_INTEGRAL_COMBINE THEN
EXISTS_TAC `a * r:real` THEN
REWRITE_TAC[REAL_ARITH `a * r <= r <=> &0 <= r * (&1 - a)`] THEN
ASM_SIMP_TAC[REAL_LE_MUL; REAL_SUB_LE; REAL_LT_IMP_LE] THEN
ABBREV_TAC `k = Arg(dropout 3 (w2:real^3) / dropout 3 (w1:real^3))` THEN
CONJ_TAC THENL
[MATCH_MP_TAC HAS_REAL_INTEGRAL_EQ THEN EXISTS_TAC
`\t. k * t pow 2 * (inv(a pow 2) - &1) / &2` THEN
CONJ_TAC THENL
[X_GEN_TAC `t:real` THEN REWRITE_TAC[IN_REAL_INTERVAL] THEN
STRIP_TAC THEN AP_TERM_TAC THEN
SUBGOAL_THEN `t pow 2 * (inv(a pow 2) - &1) <= r pow 2 - t pow 2`
ASSUME_TAC THENL
[REWRITE_TAC[REAL_ARITH `t * (a - &1) <= r - t <=> t * a <= r`] THEN
ASM_SIMP_TAC[GSYM real_div; REAL_LE_LDIV_EQ; REAL_POW_LT] THEN
REWRITE_TAC[GSYM REAL_POW_MUL] THEN MATCH_MP_TAC REAL_POW_LE2 THEN
ASM_REAL_ARITH_TAC;
ALL_TAC] THEN
SUBGOAL_THEN `t * sqrt(inv(a pow 2) - &1) <= sqrt(r pow 2 - t pow 2)`
(fun th -> SIMP_TAC[th; REAL_ARITH `a <= b ==> min b a = a`])
THENL
[MATCH_MP_TAC REAL_POW_LE2_REV THEN EXISTS_TAC `2` THEN
REWRITE_TAC[ARITH] THEN
SUBGOAL_THEN `&0 <= r pow 2 - t pow 2` ASSUME_TAC THENL
[FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH
`a <= x ==> &0 <= a ==> &0 <= x`)) THEN
ASM_SIMP_TAC[REAL_POW_2; REAL_LE_MUL; REAL_LE_SQUARE; REAL_LT_IMP_LE];
ASM_SIMP_TAC[SQRT_POS_LE; REAL_POW_MUL; SQRT_POW_2;
REAL_LT_IMP_LE]];
ASM_SIMP_TAC[REAL_POW_MUL; SQRT_POW_2; SQRT_POW_2; REAL_LT_IMP_LE] THEN
REAL_ARITH_TAC];
MP_TAC(ISPECL
[`\t. k / &6 * (inv (a pow 2) - &1) * t pow 3`;
`\t. k * t pow 2 * (inv (a pow 2) - &1) / &2`;
`&0`; `a * r:real`] REAL_FUNDAMENTAL_THEOREM_OF_CALCULUS) THEN
ASM_SIMP_TAC[REAL_LE_MUL; REAL_LT_IMP_LE] THEN ANTS_TAC THENL
[ASM_REWRITE_TAC[] THEN
REPEAT STRIP_TAC THEN REAL_DIFF_TAC THEN
CONV_TAC NUM_REDUCE_CONV THEN CONV_TAC REAL_RING;
MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN
UNDISCH_TAC `&0 < a` THEN CONV_TAC REAL_FIELD]];
MATCH_MP_TAC HAS_REAL_INTEGRAL_EQ THEN EXISTS_TAC
`\t:real. k * (r pow 2 - t pow 2) / &2` THEN
CONJ_TAC THENL
[X_GEN_TAC `t:real` THEN REWRITE_TAC[IN_REAL_INTERVAL] THEN
STRIP_TAC THEN AP_TERM_TAC THEN
SUBGOAL_THEN `&0 <= t` ASSUME_TAC THENL
[MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `a * r:real` THEN
ASM_SIMP_TAC[REAL_LE_MUL; REAL_LT_IMP_LE];
ALL_TAC] THEN
MATCH_MP_TAC(REAL_ARITH
`a <= b /\ a pow 2 = x ==> x / &2 = (min a b pow 2) / &2`) THEN
SUBGOAL_THEN `&0 <= r pow 2 - t pow 2` ASSUME_TAC THENL
[REWRITE_TAC[GSYM REAL_LE_SQUARE_ABS; REAL_SUB_LE] THEN
ASM_REAL_ARITH_TAC;
ALL_TAC] THEN
ASM_SIMP_TAC[SQRT_POW_2] THEN MATCH_MP_TAC REAL_POW_LE2_REV THEN
EXISTS_TAC `2` THEN REWRITE_TAC[ARITH] THEN
ASM_SIMP_TAC[SQRT_POW_2; REAL_POW_MUL; REAL_LE_MUL; SQRT_POS_LT;
REAL_LT_MUL; REAL_LT_IMP_LE; SQRT_POS_LE] THEN
REWRITE_TAC[REAL_ARITH `r - t <= t * (a - &1) <=> r <= t * a`] THEN
REWRITE_TAC[REAL_INV_POW; GSYM REAL_POW_MUL] THEN
MATCH_MP_TAC REAL_POW_LE2 THEN
ASM_SIMP_TAC[GSYM real_div; REAL_LE_RDIV_EQ] THEN
ASM_REAL_ARITH_TAC;
MP_TAC(ISPECL
[`\t. k / &2 * (r pow 2 * t - t pow 3 / &3)`;
`\t. k * (r pow 2 - t pow 2) / &2`;
`a * r:real`; `r:real`] REAL_FUNDAMENTAL_THEOREM_OF_CALCULUS) THEN
ASM_SIMP_TAC[REAL_LE_MUL; REAL_LT_IMP_LE] THEN ANTS_TAC THENL
[ASM_REWRITE_TAC[REAL_ARITH `a * r <= r <=> &0 <= r * (&1 - a)`] THEN
ASM_SIMP_TAC[REAL_LE_MUL; REAL_LT_IMP_LE; REAL_SUB_LE] THEN
REPEAT STRIP_TAC THEN REAL_DIFF_TAC THEN
CONV_TAC NUM_REDUCE_CONV THEN CONV_TAC REAL_RING;
MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN
UNDISCH_TAC `&0 < a` THEN CONV_TAC REAL_FIELD]]]);;
let BOUNDED_CONIC_CAP_WEDGE = prove
(`!v0 v1:real^3 w1 w2 r a.
bounded(conic_cap v0 v1 r a INTER wedge v0 v1 w1 w2)`,
REPEAT GEN_TAC THEN MATCH_MP_TAC BOUNDED_SUBSET THEN
EXISTS_TAC `conic_cap (v0:real^3) v1 r a` THEN
REWRITE_TAC[BOUNDED_CONIC_CAP] THEN SET_TAC[]);;
let MEASURABLE_CONIC_CAP_WEDGE = prove
(`!v0 v1:real^3 w1 w2 r a.
measurable(conic_cap v0 v1 r a INTER wedge v0 v1 w1 w2)`,
REPEAT GEN_TAC THEN MATCH_MP_TAC MEASURABLE_BOUNDED_INTER_OPEN THEN
REWRITE_TAC[BOUNDED_CONIC_CAP; MEASURABLE_CONIC_CAP; OPEN_WEDGE]);;
let VOLUME_CONIC_CAP_COMPL = prove
(`!v0 v1:real^3 w1 w2 r a.
&0 <= r
==> measure(conic_cap v0 v1 r a INTER wedge v0 v1 w1 w2) +
measure(conic_cap v0 v1 r (--a) INTER wedge v0 v1 w1 w2) =
azim v0 v1 w1 w2 * &2 * r pow 3 / &3`,
let lemma = prove
(`!f:real^N->real^N s t t' u.
measurable(s) /\ measurable(t) /\ measurable(u) /\
orthogonal_transformation f /\
s SUBSET u /\ t' SUBSET u /\ s INTER t' = {} /\
negligible(u DIFF (s UNION t')) /\
((!y. ?x. f x = y) ==> IMAGE f t = t')
==> measure s + measure t = measure u`,
REPEAT GEN_TAC THEN
ASM_CASES_TAC `orthogonal_transformation(f:real^N->real^N)` THEN
ASM_SIMP_TAC[ORTHOGONAL_TRANSFORMATION_SURJECTIVE] THEN
REPEAT STRIP_TAC THEN MATCH_MP_TAC EQ_TRANS THEN
EXISTS_TAC
`measure(s:real^N->bool) + measure(t':real^N->bool)` THEN
CONJ_TAC THENL [ASM_MESON_TAC[MEASURE_ORTHOGONAL_IMAGE_EQ]; ALL_TAC] THEN
W(MP_TAC o PART_MATCH (rhs o rand) MEASURE_DISJOINT_UNION o
lhand o snd) THEN
ASM_REWRITE_TAC[DISJOINT] THEN ANTS_TAC THENL
[ASM_MESON_TAC[MEASURABLE_LINEAR_IMAGE; ORTHOGONAL_TRANSFORMATION_LINEAR];
DISCH_THEN(SUBST1_TAC o SYM)] THEN
MATCH_MP_TAC MEASURE_NEGLIGIBLE_SYMDIFF THEN ASM_REWRITE_TAC[] THEN
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ]
NEGLIGIBLE_SUBSET)) THEN
REPEAT(POP_ASSUM MP_TAC) THEN SET_TAC[]) in
REWRITE_TAC[conic_cap; rcone_gt; NORMBALL_BALL; rconesgn] THEN
GEOM_ORIGIN_TAC `v0:real^3` THEN
REWRITE_TAC[VECTOR_SUB_RZERO; DIST_0; real_gt] THEN
GEOM_BASIS_MULTIPLE_TAC 3 `v1:real^3` THEN
X_GEN_TAC `v1:real` THEN
GEN_REWRITE_TAC LAND_CONV [REAL_ARITH `&0 <= x <=> x = &0 \/ &0 < x`] THEN
STRIP_TAC THENL
[ASM_SIMP_TAC[VECTOR_MUL_LZERO; WEDGE_DEGENERATE; AZIM_DEGENERATE] THEN
REWRITE_TAC[INTER_EMPTY; MEASURE_EMPTY] THEN REAL_ARITH_TAC;
ALL_TAC] THEN
ASM_SIMP_TAC[GSYM VOLUME_BALL_WEDGE] THEN REPEAT STRIP_TAC THEN
ASM_CASES_TAC `collinear {vec 0:real^3,v1 % basis 3,w1}` THENL
[ASM_SIMP_TAC[WEDGE_DEGENERATE; AZIM_DEGENERATE] THEN
REWRITE_TAC[INTER_EMPTY; MEASURE_EMPTY] THEN REAL_ARITH_TAC;
ALL_TAC] THEN
ASM_SIMP_TAC[GSYM VOLUME_BALL_WEDGE] THEN REPEAT STRIP_TAC THEN
ASM_CASES_TAC `collinear {vec 0:real^3,v1 % basis 3,w2}` THENL
[ASM_SIMP_TAC[WEDGE_DEGENERATE; AZIM_DEGENERATE] THEN
REWRITE_TAC[INTER_EMPTY; MEASURE_EMPTY] THEN REAL_ARITH_TAC;
ALL_TAC] THEN
ASM_SIMP_TAC[WEDGE_SPECIAL_SCALE] THEN
MAP_EVERY UNDISCH_TAC
[`~collinear{vec 0:real^3,v1 % basis 3,w1}`;
`~collinear{vec 0:real^3,v1 % basis 3,w2}`] THEN
ASM_SIMP_TAC[COLLINEAR_SPECIAL_SCALE] THEN REPEAT DISCH_TAC THEN
REWRITE_TAC[NORM_MUL; DOT_RMUL] THEN
ASM_SIMP_TAC[REAL_LT_LMUL_EQ; REAL_ARITH
`&0 < v1 ==> n * (abs v1 * y) * a = v1 * n * y * a`] THEN
MATCH_MP_TAC lemma THEN
MP_TAC(ISPECL
[`vec 0:real^3`; `basis 3:real^3`; `w1:real^3`; `w2:real^3`;
`r:real`; `a:real`] MEASURABLE_CONIC_CAP_WEDGE) THEN
MP_TAC(ISPECL
[`vec 0:real^3`; `basis 3:real^3`; `w1:real^3`; `w2:real^3`;
`r:real`; `--a:real`] MEASURABLE_CONIC_CAP_WEDGE) THEN
REWRITE_TAC[conic_cap; rcone_gt; NORMBALL_BALL; rconesgn] THEN
REWRITE_TAC[VECTOR_SUB_RZERO; DIST_0; real_gt] THEN
REPEAT DISCH_TAC THEN ASM_REWRITE_TAC[MEASURABLE_BALL_WEDGE] THEN
SIMP_TAC[NORM_BASIS; DOT_BASIS; DIMINDEX_3; ARITH; REAL_MUL_LID] THEN
EXISTS_TAC `(\x. vector[x$1; x$2; --(x$3)]):real^3->real^3` THEN
EXISTS_TAC `(ball(vec 0,r) INTER {x | norm x * a > x$3}) INTER
wedge (vec 0:real^3) (basis 3) w1 w2` THEN
CONJ_TAC THENL
[REWRITE_TAC[ORTHOGONAL_TRANSFORMATION; linear] THEN
REWRITE_TAC[CART_EQ; DIMINDEX_3; FORALL_3; VECTOR_3; vector_norm; DOT_3;
VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT] THEN
REPEAT(GEN_TAC ORELSE CONJ_TAC ORELSE AP_TERM_TAC) THEN
REAL_ARITH_TAC;
ALL_TAC] THEN
CONJ_TAC THENL [SET_TAC[]; ALL_TAC] THEN
CONJ_TAC THENL [SET_TAC[]; ALL_TAC] THEN
CONJ_TAC THENL
[REWRITE_TAC[EXTENSION; NOT_IN_EMPTY; IN_INTER; IN_ELIM_THM; real_gt] THEN
MESON_TAC[REAL_LT_ANTISYM];
ALL_TAC] THEN
CONJ_TAC THENL
[MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN
EXISTS_TAC `rcone_eq (vec 0:real^3) (basis 3) a` THEN
SIMP_TAC[NEGLIGIBLE_RCONE_EQ; BASIS_NONZERO; DIMINDEX_3; ARITH] THEN
REWRITE_TAC[SUBSET; rcone_eq; rconesgn; VECTOR_SUB_RZERO; DIST_0] THEN
SIMP_TAC[DOT_BASIS; NORM_BASIS; DIMINDEX_3; ARITH] THEN
REWRITE_TAC[IN_DIFF; IN_ELIM_THM; IN_INTER; IN_UNION] THEN
GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC;
ALL_TAC] THEN
REWRITE_TAC[] THEN DISCH_TAC THEN MATCH_MP_TAC SURJECTIVE_IMAGE_EQ THEN
ASM_REWRITE_TAC[] THEN
REWRITE_TAC[IN_INTER; IN_BALL_0; IN_ELIM_THM; VECTOR_3] THEN
X_GEN_TAC `x:real^3` THEN
SUBGOAL_THEN `norm(vector [x$1; x$2; --(x$3)]:real^3) = norm(x:real^3)`
SUBST1_TAC THENL
[REWRITE_TAC[NORM_EQ; DOT_3; VECTOR_3] THEN REAL_ARITH_TAC;
ALL_TAC] THEN
REWRITE_TAC[REAL_ARITH `n * a > --x <=> n * --a < x`] THEN
MATCH_MP_TAC(TAUT `(a ==> (b <=> b')) ==> (a /\ b <=> a /\ b')`) THEN
STRIP_TAC THEN
REWRITE_TAC[COLLINEAR_BASIS_3; wedge; AZIM_ARG] THEN
REWRITE_TAC[IN_ELIM_THM] THEN
SUBGOAL_THEN `(dropout 3 :real^3->real^2) (vector [x$1; x$2; --(x$3)]) =
(dropout 3 :real^3->real^2) x`
(fun th -> REWRITE_TAC[th]) THEN
SIMP_TAC[CART_EQ; DIMINDEX_2; FORALL_2; dropout; LAMBDA_BETA; ARITH;
DIMINDEX_3; VECTOR_3]);;
let VOLUME_CONIC_CAP_WEDGE_MEDIUM = prove
(`!v0 v1:real^3 w1 w2 r a.
&0 <= a /\ ~collinear {v0,v1,w1} /\ ~collinear {v0,v1,w2}
==> bounded(conic_cap v0 v1 r a INTER wedge v0 v1 w1 w2) /\
measurable(conic_cap v0 v1 r a INTER wedge v0 v1 w1 w2) /\
measure(conic_cap v0 v1 r a INTER wedge v0 v1 w1 w2) =
if &1 < abs a \/ r < &0 then &0
else azim v0 v1 w1 w2 / &3 * (&1 - a) * r pow 3`,
REWRITE_TAC[BOUNDED_CONIC_CAP_WEDGE; MEASURABLE_CONIC_CAP_WEDGE] THEN
REPEAT STRIP_TAC THEN FIRST_X_ASSUM(DISJ_CASES_TAC o MATCH_MP (REAL_ARITH
`&0 <= a ==> &0 < a \/ a = &0`))
THENL
[ASM_SIMP_TAC[VOLUME_CONIC_CAP_WEDGE_WEAK] THEN
REWRITE_TAC[REAL_LE_LT] THEN
ASM_CASES_TAC `a = &1` THEN ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC;
ALL_TAC] THEN
ASM_REWRITE_TAC[] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN
COND_CASES_TAC THENL
[REWRITE_TAC[conic_cap; NORMBALL_BALL] THEN
SUBGOAL_THEN `ball(v0:real^3,r) = {}`
(fun th -> SIMP_TAC[th; INTER_EMPTY; MEASURE_EMPTY]) THEN
REWRITE_TAC[BALL_EQ_EMPTY] THEN ASM_REAL_ARITH_TAC;
MP_TAC(ISPECL [`v0:real^3`; `v1:real^3`; `w1:real^3`; `w2:real^3`;
`r:real`; `&0`] VOLUME_CONIC_CAP_COMPL) THEN
REWRITE_TAC[REAL_NEG_0] THEN ASM_REAL_ARITH_TAC]);;
let VOLUME_CONIC_CAP_WEDGE = prove
(`!v0 v1:real^3 w1 w2 r a.
~collinear {v0,v1,w1} /\ ~collinear {v0,v1,w2}
==> bounded(conic_cap v0 v1 r a INTER wedge v0 v1 w1 w2) /\
measurable(conic_cap v0 v1 r a INTER wedge v0 v1 w1 w2) /\
measure(conic_cap v0 v1 r a INTER wedge v0 v1 w1 w2) =
if &1 < a \/ r < &0 then &0
else azim v0 v1 w1 w2 / &3 * (&1 - max a (-- &1)) * r pow 3`,
REWRITE_TAC[BOUNDED_CONIC_CAP_WEDGE; MEASURABLE_CONIC_CAP_WEDGE] THEN
REPEAT STRIP_TAC THEN
ASM_CASES_TAC `&0 <= a` THEN
ASM_SIMP_TAC[VOLUME_CONIC_CAP_WEDGE_MEDIUM;
REAL_ARITH `&0 <= a ==> abs a = a /\ max a (-- &1) = a`] THEN
MP_TAC(ISPECL [`v0:real^3`; `v1:real^3`; `w1:real^3`; `w2:real^3`;
`r:real`; `--a:real`] VOLUME_CONIC_CAP_WEDGE_MEDIUM) THEN
ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
STRIP_TAC THEN
MP_TAC(ISPECL [`v0:real^3`; `v1:real^3`; `w1:real^3`; `w2:real^3`;
`r:real`; `a:real`] VOLUME_CONIC_CAP_COMPL) THEN
ASM_CASES_TAC `r < &0` THENL
[REWRITE_TAC[conic_cap; NORMBALL_BALL] THEN
SUBGOAL_THEN `ball(v0:real^3,r) = {}`
(fun th -> SIMP_TAC[th; INTER_EMPTY; MEASURE_EMPTY]) THEN
REWRITE_TAC[BALL_EQ_EMPTY] THEN ASM_REAL_ARITH_TAC;
ALL_TAC] THEN
ASM_REWRITE_TAC[GSYM REAL_NOT_LT; REAL_ABS_NEG] THEN
ASM_SIMP_TAC[REAL_ARITH `~(&0 <= a) ==> ~(&1 < a) /\ abs a = --a`] THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[] THENL
[ASM_SIMP_TAC[REAL_ARITH `&1 < --a ==> max a (-- &1) = -- &1`] THEN
REAL_ARITH_TAC;
ASM_SIMP_TAC[REAL_ARITH `~(&1 < --a) ==> max a (-- &1) = a`] THEN
REAL_ARITH_TAC]);;
(* ------------------------------------------------------------------------- *)
(* Precise formulation of Flyspeck volume properties. *)
(* ------------------------------------------------------------------------- *)
(*** Might be preferable to switch
***
*** normball z r -> ball(z,r)
*** rect a b -> interval(a,b)
***
*** to fit existing libraries. But I left this alone for now,
*** to be absolutely sure I didn't introduce new errors.
*** I also maintain
***
*** NULLSET -> negligible
*** vol -> measure
***
*** as interface maps for the real^3 case.
***)
let cone = new_definition `cone v S:real^A->bool = affsign sgn_ge {v} S`;;
(*** JRH: should we exclude v for S = {}? Make it always open ***)
let cone0 = new_definition `cone0 v S:real^A->bool = affsign sgn_gt {v} S`;;
(*** JRH changed from cone to cone0 ***)
let solid_triangle = new_definition
`solid_triangle v0 S r = normball v0 r INTER cone0 v0 S`;;
let rect = new_definition
`rect (a:real^3) (b:real^3) =
{(v:real^3) | !i. (a$i < v$i /\ v$i < b$i )}`;;
let RECT_INTERVAL = prove
(`!a b. rect a b = interval(a,b)`,
REWRITE_TAC[rect; EXTENSION; IN_INTERVAL; IN_ELIM_THM] THEN
MESON_TAC[FINITE_INDEX_INRANGE]);;
let RCONE_GE_GT = prove
(`rcone_ge z w h =
rcone_gt z w h UNION
{ x | (x - z) dot (w - z) = norm(x - z) * norm(w - z) * h}`,
REWRITE_TAC[rcone_ge; rcone_gt; rconesgn] THEN
REWRITE_TAC[dist; EXTENSION; IN_UNION; NORM_SUB; IN_ELIM_THM] THEN
REAL_ARITH_TAC);;
let RCONE_GT_GE = prove
(`rcone_gt z w h =
rcone_ge z w h DIFF
{ x | (x - z) dot (w - z) = norm(x - z) * norm(w - z) * h}`,
REWRITE_TAC[rcone_ge; rcone_gt; rconesgn] THEN
REWRITE_TAC[dist; EXTENSION; IN_DIFF; NORM_SUB; IN_ELIM_THM] THEN
REAL_ARITH_TAC);;
override_interface("NULLSET",`negligible:(real^3->bool)->bool`);;
override_interface("vol",`measure:(real^3->bool)->real`);;
let is_sphere= new_definition
`is_sphere x=(?(v:real^3)(r:real). (r> &0)/\ (x={w:real^3 | norm (w-v)= r}))`;;
let c_cone = new_definition
`c_cone (v,w:real^3, r:real)=
{x:real^3 | ((x-v) dot w = norm (x-v)* norm w* r)}`;;
(*** JRH added the condition ~(w = 0), or the cone is all of space ***)
let circular_cone =new_definition
`circular_cone (V:real^3-> bool)=
(? (v,w:real^3)(r:real). ~(w = vec 0) /\ V = c_cone (v,w,r))`;;
let NULLSET_RULES = prove
(`(!P. ((plane P)\/ (is_sphere P) \/ (circular_cone P)) ==> NULLSET P) /\
(!(s:real^3->bool) t. (NULLSET s /\ NULLSET t) ==> NULLSET (s UNION t))`,
SIMP_TAC[NEGLIGIBLE_UNION] THEN
X_GEN_TAC `s:real^3->bool` THEN STRIP_TAC THENL
[MATCH_MP_TAC COPLANAR_IMP_NEGLIGIBLE THEN
SIMP_TAC[COPLANAR; DIMINDEX_3; ARITH] THEN ASM_MESON_TAC[SUBSET_REFL];
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [is_sphere]) THEN
STRIP_TAC THEN ASM_REWRITE_TAC[GSYM dist] THEN
ONCE_REWRITE_TAC[DIST_SYM] THEN
REWRITE_TAC[REWRITE_RULE[sphere] NEGLIGIBLE_SPHERE];
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [circular_cone]) THEN
REWRITE_TAC[EXISTS_PAIRED_THM; c_cone] THEN STRIP_TAC THEN
ASM_REWRITE_TAC[] THEN
MP_TAC(ISPECL [`w + v:real^3`; `v:real^3`; `r:real`]
NEGLIGIBLE_RCONE_EQ) THEN
ASM_REWRITE_TAC[rcone_eq; rconesgn] THEN
REWRITE_TAC[dist; VECTOR_ARITH `(w + v) - v:real^N = w`] THEN
ASM_REWRITE_TAC[VECTOR_ARITH `w + v:real^N = v <=> w = vec 0`]]);;
(*** JRH added &0 < a for frustum; otherwise it's in general unbounded ***)
let primitive = new_definition `primitive (C:real^3->bool) =
((?v0 v1 v2 v3 r. (C = solid_triangle v0 {v1,v2,v3} r)) \/
(?v0 v1 v2 v3. (C = conv0 {v0,v1,v2,v3})) \/
(?v0 v1 v2 v3 h a. &0 < a /\
(C = frustt v0 v1 h a INTER wedge v0 v1 v2 v3)) \/
(?v0 v1 v2 v3 r c. (C = conic_cap v0 v1 r c INTER wedge v0 v1 v2 v3)) \/
(?a b. (C = rect a b)) \/
(?t r. (C = ellipsoid t r)) \/
(?v0 v1 v2 v3 r. (C = normball v0 r INTER wedge v0 v1 v2 v3)))`;;
let MEASURABLE_RULES = prove
(`(!C. primitive C ==> measurable C) /\
(!Z. NULLSET Z ==> measurable Z) /\
(!X t. measurable X ==> (measurable (IMAGE (scale t) X))) /\
(!X v. measurable X ==> (measurable (IMAGE ((+) v) X))) /\
(!(s:real^3->bool) t. (measurable s /\ measurable t)
==> measurable (s UNION t)) /\
(!(s:real^3->bool) t. (measurable s /\ measurable t)
==> measurable (s INTER t)) /\
(!(s:real^3->bool) t. (measurable s /\ measurable t)
==> measurable (s DIFF t))`,
SIMP_TAC[MEASURABLE_UNION; MEASURABLE_INTER; MEASURABLE_DIFF] THEN
REWRITE_TAC[REWRITE_RULE[ETA_AX] MEASURABLE_TRANSLATION] THEN
SIMP_TAC[GSYM HAS_MEASURE_0; HAS_MEASURE_MEASURABLE_MEASURE] THEN
CONJ_TAC THENL
[ALL_TAC;
MAP_EVERY X_GEN_TAC [`X:real^3->bool`; `t:real^3`] THEN
REWRITE_TAC[HAS_MEASURE_MEASURE] THEN
DISCH_THEN(MP_TAC o MATCH_MP HAS_MEASURE_STRETCH) THEN
DISCH_THEN(MP_TAC o SPEC `\i. (t:real^3)$i`) THEN
REWRITE_TAC[HAS_MEASURE_MEASURABLE_MEASURE] THEN
DISCH_THEN(MP_TAC o CONJUNCT1) THEN MATCH_MP_TAC EQ_IMP THEN
AP_TERM_TAC THEN AP_THM_TAC THEN AP_TERM_TAC THEN
SIMP_TAC[FUN_EQ_THM; scale; CART_EQ; LAMBDA_BETA;
DIMINDEX_3; VECTOR_3; FORALL_3]] THEN
X_GEN_TAC `C:real^3->bool` THEN REWRITE_TAC[primitive] THEN
REWRITE_TAC[NORMBALL_BALL; RECT_INTERVAL] THEN
DISCH_THEN(REPEAT_TCL DISJ_CASES_THEN MP_TAC) THEN
REWRITE_TAC[LEFT_IMP_EXISTS_THM] THENL
[REPEAT STRIP_TAC THEN
ASM_REWRITE_TAC[solid_triangle; NORMBALL_BALL; cone0; GSYM aff_gt_def] THEN
REWRITE_TAC[MEASURABLE_BALL_AFF_GT];
REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC MEASURABLE_CONV0 THEN MATCH_MP_TAC FINITE_IMP_BOUNDED THEN
REWRITE_TAC[FINITE_INSERT; FINITE_EMPTY];
MAP_EVERY X_GEN_TAC
[`v0:real^3`; `v1:real^3`; `v2:real^3`; `v3:real^3`;
`h:real`; `a:real`] THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC SUBST1_TAC) THEN
ASM_CASES_TAC `collinear {v0:real^3, v1, v2}` THENL
[ASM_SIMP_TAC[WEDGE_DEGENERATE; INTER_EMPTY; MEASURABLE_EMPTY];
ALL_TAC] THEN
ASM_CASES_TAC `collinear {v0:real^3, v1, v3}` THENL
[ASM_SIMP_TAC[WEDGE_DEGENERATE; INTER_EMPTY; MEASURABLE_EMPTY];
ALL_TAC] THEN
ASM_SIMP_TAC[VOLUME_FRUSTT_WEDGE];
REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC MEASURABLE_BOUNDED_INTER_OPEN THEN
REWRITE_TAC[MEASURABLE_CONIC_CAP; BOUNDED_CONIC_CAP; OPEN_WEDGE];
SIMP_TAC[MEASURABLE_INTERVAL];
SIMP_TAC[MEASURABLE_ELLIPSOID];
SIMP_TAC[MEASURABLE_BALL_WEDGE]]);;
let vol_solid_triangle = new_definition `vol_solid_triangle v0 v1 v2 v3 r =
let a123 = dihV v0 v1 v2 v3 in
let a231 = dihV v0 v2 v3 v1 in
let a312 = dihV v0 v3 v1 v2 in
(a123 + a231 + a312 - pi)*(r pow 3)/(&3)`;;
let vol_frustt_wedge = new_definition `vol_frustt_wedge v0 v1 v2 v3 h a =
(azim v0 v1 v2 v3)*(h pow 3)*(&1/(a*a) - &1)/(&6)`;;
let vol_conic_cap_wedge = new_definition `vol_conic_cap_wedge v0 v1 v2 v3 r c =
(azim v0 v1 v2 v3)*(&1 - c)*(r pow 3)/(&3)`;;
(*** JRH corrected delta_x x12 x13 x14 x34 x24 x34 ***)
(*** to delta_x x12 x13 x14 x34 x24 x23 ***)
let vol_conv = new_definition `vol_conv v1 v2 v3 v4 =
let x12 = dist(v1,v2) pow 2 in
let x13 = dist(v1,v3) pow 2 in
let x14 = dist(v1,v4) pow 2 in
let x23 = dist(v2,v3) pow 2 in
let x24 = dist(v2,v4) pow 2 in
let x34 = dist(v3,v4) pow 2 in
sqrt(delta_x x12 x13 x14 x34 x24 x23)/(&12)`;;
let vol_rect = new_definition `vol_rect a b =
if (a$1 < b$1) /\ (a$2 < b$2) /\ (a$3 < b$3) then
(b$3-a$3)*(b$2-a$2)*(b$1-a$1) else &0`;;
let vol_ball_wedge = new_definition `vol_ball_wedge v0 v1 v2 v3 r =
(azim v0 v1 v2 v3)*(&2)*(r pow 3)/(&3)`;;
let SDIFF = new_definition `SDIFF X Y = (X DIFF Y) UNION (Y DIFF X)`;;
(*** JRH added the hypothesis "measurable" to the first one ***)
(*** Could change the definition to make this hold anyway ***)
(*** JRH changed solid triangle hypothesis to ~coplanar{...} ***)
(*** since the current condition is not enough in general ***)
let volume_props = prove
(`(!C. measurable C ==> vol C >= &0) /\
(!Z. NULLSET Z ==> (vol Z = &0)) /\
(!X Y. measurable X /\ measurable Y /\ NULLSET (SDIFF X Y)
==> (vol X = vol Y)) /\
(!X t. (measurable X) /\ (measurable (IMAGE (scale t) X))
==> (vol (IMAGE (scale t) X) = abs(t$1 * t$2 * t$3)*vol(X))) /\
(!X v. measurable X ==> (vol (IMAGE ((+) v) X) = vol X)) /\
(!v0 v1 v2 v3 r. (r > &0) /\ ~coplanar{v0,v1,v2,v3}
==> vol (solid_triangle v0 {v1,v2,v3} r) =
vol_solid_triangle v0 v1 v2 v3 r) /\
(!v0 v1 v2 v3. vol(conv0 {v0,v1,v2,v3}) = vol_conv v0 v1 v2 v3) /\
(!v0 v1 v2 v3 h a. ~(collinear {v0,v1,v2}) /\ ~(collinear {v0,v1,v3}) /\
(h >= &0) /\ (a > &0) /\ (a <= &1)
==> vol(frustt v0 v1 h a INTER wedge v0 v1 v2 v3) =
vol_frustt_wedge v0 v1 v2 v3 h a) /\
(!v0 v1 v2 v3 r c. ~(collinear {v0,v1,v2}) /\ ~(collinear {v0,v1,v3}) /\
(r >= &0) /\ (c >= -- (&1)) /\ (c <= &1)
==> (vol(conic_cap v0 v1 r c INTER wedge v0 v1 v2 v3) =
vol_conic_cap_wedge v0 v1 v2 v3 r c)) /\
(!(a:real^3) (b:real^3). vol(rect a b) = vol_rect a b) /\
(!v0 v1 v2 v3 r. ~(collinear {v0,v1,v2}) /\ ~(collinear {v0,v1,v3}) /\
(r >= &0)
==> (vol(normball v0 r INTER wedge v0 v1 v2 v3) =
vol_ball_wedge v0 v1 v2 v3 r))`,
SIMP_TAC[MEASURE_POS_LE; real_ge; real_gt] THEN REPEAT CONJ_TAC THENL
[SIMP_TAC[GSYM HAS_MEASURE_0; HAS_MEASURE_MEASURABLE_MEASURE];
MAP_EVERY X_GEN_TAC [`s:real^3->bool`; `t:real^3->bool`] THEN
STRIP_TAC THEN MATCH_MP_TAC MEASURE_NEGLIGIBLE_SYMDIFF THEN
ASM_REWRITE_TAC[] THEN
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ]
NEGLIGIBLE_SUBSET)) THEN
REWRITE_TAC[SDIFF] THEN SET_TAC[];
MAP_EVERY X_GEN_TAC [`X:real^3->bool`; `t:real^3`] THEN
REWRITE_TAC[HAS_MEASURE_MEASURE] THEN
DISCH_THEN(MP_TAC o MATCH_MP HAS_MEASURE_STRETCH o CONJUNCT1) THEN
DISCH_THEN(MP_TAC o SPEC `\i. (t:real^3)$i`) THEN
REWRITE_TAC[HAS_MEASURE_MEASURABLE_MEASURE] THEN
DISCH_THEN(MP_TAC o CONJUNCT2) THEN
REWRITE_TAC[DIMINDEX_3; PRODUCT_3] THEN
DISCH_THEN(SUBST1_TAC o SYM) THEN
AP_TERM_TAC THEN AP_THM_TAC THEN AP_TERM_TAC THEN
SIMP_TAC[FUN_EQ_THM; scale; CART_EQ; LAMBDA_BETA;
DIMINDEX_3; VECTOR_3; FORALL_3];
REWRITE_TAC[REWRITE_RULE[ETA_AX] MEASURE_TRANSLATION];
REPEAT STRIP_TAC THEN
REWRITE_TAC[solid_triangle; vol_solid_triangle; NORMBALL_BALL] THEN
REWRITE_TAC[cone0; GSYM aff_gt_def] THEN
MATCH_MP_TAC VOLUME_SOLID_TRIANGLE THEN ASM_REWRITE_TAC[];
REWRITE_TAC[vol_conv; VOLUME_OF_TETRAHEDRON];
SIMP_TAC[VOLUME_FRUSTT_WEDGE; vol_frustt_wedge] THEN
SIMP_TAC[REAL_ARITH `&0 <= h ==> ~(h < &0)`] THEN
SIMP_TAC[REAL_ARITH `a <= &1 ==> (&1 <= a <=> a = &1)`] THEN
REPEAT STRIP_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN
REPEAT(POP_ASSUM MP_TAC) THEN CONV_TAC REAL_FIELD;
SIMP_TAC[VOLUME_CONIC_CAP_WEDGE; vol_conic_cap_wedge] THEN
SIMP_TAC[REAL_ARITH `&0 <= r ==> ~(r < &0)`] THEN
SIMP_TAC[REAL_ARITH `c <= &1 ==> ~(&1 < c)`] THEN
ASM_SIMP_TAC[REAL_ARITH `-- &1 <= c ==> max c (-- &1) = c`] THEN
REPEAT STRIP_TAC THEN REAL_ARITH_TAC;
REWRITE_TAC[vol_rect; RECT_INTERVAL; MEASURE_INTERVAL] THEN
REWRITE_TAC[CONTENT_CLOSED_INTERVAL_CASES] THEN
REWRITE_TAC[DIMINDEX_3; FORALL_3; PRODUCT_3] THEN
MAP_EVERY X_GEN_TAC [`a:real^3`; `b:real^3`] THEN
REWRITE_TAC[REAL_LE_LT] THEN
ASM_CASES_TAC `(a:real^3)$1 = (b:real^3)$1` THEN
ASM_REWRITE_TAC[REAL_LT_REFL; REAL_MUL_LZERO; REAL_MUL_RZERO;
REAL_SUB_REFL; COND_ID] THEN
ASM_CASES_TAC `(a:real^3)$2 = (b:real^3)$2` THEN
ASM_REWRITE_TAC[REAL_LT_REFL; REAL_MUL_LZERO; REAL_MUL_RZERO;
REAL_SUB_REFL; COND_ID] THEN
ASM_CASES_TAC `(a:real^3)$3 = (b:real^3)$3` THEN
ASM_REWRITE_TAC[REAL_LT_REFL; REAL_MUL_LZERO; REAL_MUL_RZERO;
REAL_SUB_REFL; COND_ID] THEN
REWRITE_TAC[REAL_MUL_AC];
SIMP_TAC[VOLUME_BALL_WEDGE; NORMBALL_BALL; vol_ball_wedge]]);;
(* ------------------------------------------------------------------------- *)
(* Additional results on polyhedra and relation to fans. *)
(* ------------------------------------------------------------------------- *)
let POLYHEDRON_COLLINEAR_FACES_STRONG = prove
(`!P:real^N->bool f f' p q s t.
polyhedron P /\ vec 0 IN relative_interior P /\
f face_of P /\ ~(f = P) /\ f' face_of P /\ ~(f' = P) /\
p IN f /\ q IN f' /\ s > &0 /\ t > &0 /\ s % p = t % q
==> s = t`,
ONCE_REWRITE_TAC[MESON[]
`(!P f f' p q s t. Q P f f' p q s t) <=>
(!s t P f f' p q. Q P f f' p q s t)`] THEN
MATCH_MP_TAC REAL_WLOG_LT THEN
REWRITE_TAC[real_gt] THEN CONJ_TAC THENL [MESON_TAC[]; ALL_TAC] THEN
REPEAT STRIP_TAC THEN MATCH_MP_TAC(TAUT `F ==> p`) THEN
FIRST_X_ASSUM(MP_TAC o AP_TERM `(%) (inv s):real^N->real^N`) THEN
ASM_SIMP_TAC[VECTOR_MUL_ASSOC; REAL_MUL_LINV; REAL_LT_IMP_NZ] THEN
ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN
REWRITE_TAC[VECTOR_MUL_LID; GSYM real_div] THEN
ABBREV_TAC `u:real = t / s` THEN
SUBGOAL_THEN `&0 < u /\ &1 < u` MP_TAC THENL
[EXPAND_TAC "u" THEN ASM_SIMP_TAC[REAL_LT_RDIV_EQ] THEN
ASM_REWRITE_TAC[REAL_MUL_LZERO; REAL_MUL_LID];
ALL_TAC] THEN
MAP_EVERY (C UNDISCH_THEN (K ALL_TAC))
[`s < t`; `&0 < s`; `&0 < t`; `t:real / s = u`] THEN
SPEC_TAC(`u:real`,`t:real`) THEN GEN_TAC THEN STRIP_TAC THEN
DISCH_THEN(ASSUME_TAC o SYM) THEN
SUBGOAL_THEN `?g:real^N->bool. g facet_of P /\ f' SUBSET g`
STRIP_ASSUME_TAC THENL
[MATCH_MP_TAC FACE_OF_POLYHEDRON_SUBSET_FACET THEN ASM SET_TAC[];
ALL_TAC] THEN
SUBGOAL_THEN `~((vec 0:real^N) IN g)` ASSUME_TAC THENL
[DISCH_TAC THEN
MP_TAC(ISPECL [`P:real^N->bool`; `g:real^N->bool`; `P:real^N->bool`]
SUBSET_OF_FACE_OF) THEN
ASM_REWRITE_TAC[SUBSET_REFL; NOT_IMP] THEN CONJ_TAC THENL
[CONJ_TAC THENL [ASM_MESON_TAC[facet_of]; ASM SET_TAC[]];
ASM_MESON_TAC[facet_of; FACET_OF_REFL;
SUBSET_ANTISYM; FACE_OF_IMP_SUBSET]];
ALL_TAC] THEN
SUBGOAL_THEN `(g:real^N->bool) face_of P` MP_TAC THENL
[ASM_MESON_TAC[facet_of]; ALL_TAC] THEN
REWRITE_TAC[face_of] THEN DISCH_THEN(MP_TAC o last o CONJUNCTS) THEN
DISCH_THEN(MP_TAC o SPECL [`vec 0:real^N`; `t % q:real^N`; `q:real^N`]) THEN
ASM_REWRITE_TAC[] THEN REPEAT CONJ_TAC THENL
[ASM_MESON_TAC[RELATIVE_INTERIOR_SUBSET; SUBSET];
ASM_MESON_TAC[FACE_OF_IMP_SUBSET; SUBSET];
ASM_MESON_TAC[FACE_OF_IMP_SUBSET; SUBSET];
ALL_TAC] THEN
EXPAND_TAC "p" THEN REWRITE_TAC[IN_SEGMENT] THEN CONJ_TAC THENL
[CONV_TAC(RAND_CONV SYM_CONV) THEN
ASM_SIMP_TAC[VECTOR_MUL_EQ_0; REAL_LT_IMP_NZ] THEN ASM SET_TAC[];
EXISTS_TAC `inv t:real` THEN
ASM_SIMP_TAC[REAL_LT_INV_EQ; REAL_INV_LT_1] THEN
EXPAND_TAC "p" THEN REWRITE_TAC[VECTOR_MUL_ASSOC] THEN
ASM_SIMP_TAC[REAL_MUL_LINV; REAL_LT_IMP_NZ] THEN VECTOR_ARITH_TAC]);;
let POLYHEDRON_COLLINEAR_FACES = prove
(`!P:real^N->bool f f' p q s t.
polyhedron P /\ vec 0 IN interior P /\
f face_of P /\ ~(f = P) /\ f' face_of P /\ ~(f' = P) /\
p IN f /\ q IN f' /\ s > &0 /\ t > &0 /\ s % p = t % q
==> s = t`,
MESON_TAC[POLYHEDRON_COLLINEAR_FACES_STRONG;
INTERIOR_SUBSET_RELATIVE_INTERIOR; SUBSET]);;
let vertices = new_definition
`vertices s = {x:real^N | x extreme_point_of s}`;;
let edges = new_definition
`edges s = {{v,w} | segment[v,w] edge_of s}`;;
let VERTICES_TRANSLATION = prove
(`!a s. vertices (IMAGE (\x. a + x) s) = IMAGE (\x. a + x) (vertices s)`,
REWRITE_TAC[vertices] THEN GEOM_TRANSLATE_TAC[]);;
let VERTICES_LINEAR_IMAGE = prove
(`!f s. linear f /\ (!x y. f x = f y ==> x = y)
==> vertices(IMAGE f s) = IMAGE f (vertices s)`,
REWRITE_TAC[vertices; EXTREME_POINTS_OF_LINEAR_IMAGE]);;
let EDGES_TRANSLATION = prove
(`!a s. edges (IMAGE (\x. a + x) s) = IMAGE (IMAGE (\x. a + x)) (edges s)`,
REWRITE_TAC[edges] THEN GEOM_TRANSLATE_TAC[] THEN SET_TAC[]);;
let EDGES_LINEAR_IMAGE = prove
(`!f:real^M->real^N s.
linear f /\ (!x y. f x = f y ==> x = y)
==> edges(IMAGE f s) = IMAGE (IMAGE f) (edges s)`,
REPEAT GEN_TAC THEN DISCH_TAC THEN REWRITE_TAC[edges] THEN
MATCH_MP_TAC SUBSET_ANTISYM THEN
REWRITE_TAC[SUBSET; FORALL_IN_GSPEC; FORALL_IN_IMAGE] THEN CONJ_TAC THENL
[MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`] THEN STRIP_TAC THEN
REWRITE_TAC[IN_IMAGE] THEN ONCE_REWRITE_TAC[CONJ_SYM] THEN
REWRITE_TAC[EXISTS_IN_GSPEC] THEN
SUBGOAL_THEN `?v w. x = (f:real^M->real^N) v /\ y = f w` MP_TAC THENL
[ASM_MESON_TAC[ENDS_IN_SEGMENT; EDGE_OF_IMP_SUBSET; SUBSET; IN_IMAGE];
REPEAT(MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC) THEN
DISCH_THEN(CONJUNCTS_THEN SUBST_ALL_TAC)];
MAP_EVERY X_GEN_TAC [`v:real^M`; `w:real^M`] THEN DISCH_TAC THEN
REWRITE_TAC[IN_ELIM_THM] THEN
MAP_EVERY EXISTS_TAC [`(f:real^M->real^N) v`; `(f:real^M->real^N) w`]] THEN
REWRITE_TAC[IMAGE_CLAUSES] THEN
ASM_MESON_TAC[EDGE_OF_LINEAR_IMAGE; CLOSED_SEGMENT_LINEAR_IMAGE]);;
add_translation_invariants [VERTICES_TRANSLATION; EDGES_TRANSLATION];;
add_linear_invariants [VERTICES_LINEAR_IMAGE; EDGES_LINEAR_IMAGE];;
(*** Correspondence with Flypaper:
Definition 4.5: IS_AFFINE_HULL
affine / hull
aff_dim
AFF_DIM_EMPTY
Definition 4.6 : IN_INTERIOR
IN_RELATIVE_INTERIOR
CLOSURE_APPROACHABLE
(Don't have definition of relative boundary, but several
theorems use closure s DIFF relative_interior s.)
Definition 4.7: face_of
extreme_point_of (presumably it's meant to be the point not
the singleton set, which the definition literally gives)
facet_of
edge_of
(Don't have a definition of "proper"; I just use
~(f = {}) and/or ~(f = P) as needed.)
Lemma 4.18: KREIN_MILMAN_MINKOWSKI
Definition 4.8: polyhedron
vertices
Lemma 4.19: AFFINE_IMP_POLYHEDRON
Lemma 4.20 (i): FACET_OF_POLYHEDRON_EXPLICIT
Lemma 4.20 (ii): Direct consequence of RELATIVE_INTERIOR_POLYHEDRON
Lemma 4.20 (iii): FACE_OF_POLYHEDRON_EXPLICIT / FACE_OF_POLYHEDRON
Lemma 4.20 (iv): FACE_OF_TRANS
Lemma 4.20 (v): EXTREME_POINT_OF_FACE
Lemma 4.20 (vi): FACE_OF_EQ
Corr. 4.7: FACE_OF_POLYHEDRON_POLYHEDRON
Lemma 4.21: POLYHEDRON_COLLINEAR_FACES
Def 4.9: vertices
edges
****)
(* ------------------------------------------------------------------------- *)
(* Fix the congruence rules as expected in Flyspeck. *)
(* Should exclude 6 recent mixed real/vector limit results. *)
(* ------------------------------------------------------------------------- *)
let bcs =
[`(p <=> p') ==> (p' ==> (q <=> q')) ==> (p ==> q <=> p' ==> q')`;
`(g <=> g')
==> (g' ==> t = t')
==> (~g' ==> e = e')
==> (if g then t else e) = (if g' then t' else e')`;
`(!x. p x ==> f x = g x) ==> nsum {y | p y} (\i. f i) = nsum {y | p y} g`;
`(!i. a <= i /\ i <= b ==> f i = g i)
==> nsum (a..b) (\i. f i) = nsum (a..b) g`;
`(!x. x IN s ==> f x = g x) ==> nsum s (\i. f i) = nsum s g`;
`(!x. p x ==> f x = g x) ==> sum {y | p y} (\i. f i) = sum {y | p y} g`;
`(!i. a <= i /\ i <= b ==> f i = g i)
==> sum (a..b) (\i. f i) = sum (a..b) g`;
`(!x. x IN s ==> f x = g x) ==> sum s (\i. f i) = sum s g`;
`(!x. p x ==> f x = g x) ==> vsum {y | p y} (\i. f i) = vsum {y | p y} g`;
`(!i. a <= i /\ i <= b ==> f i = g i)
==> vsum (a..b) (\i. f i) = vsum (a..b) g`;
`(!x. x IN s ==> f x = g x) ==> vsum s (\i. f i) = vsum s g`;
`(!x. p x ==> f x = g x)
==> product {y | p y} (\i. f i) = product {y | p y} g`;
`(!i. a <= i /\ i <= b ==> f i = g i)
==> product (a..b) (\i. f i) = product (a..b) g`;
`(!x. x IN s ==> f x = g x) ==> product s (\i. f i) = product s g`;
`(!x. ~(x = a) ==> f x = g x)
==> (((\x. f x) --> l) (at a) <=> (g --> l) (at a))`;
`(!x. ~(x = a) ==> f x = g x)
==> (((\x. f x) --> l) (at a within s) <=> (g --> l) (at a within s))`]
and equiv t1 t2 = can (term_match [] t1) t2 && can (term_match [] t2) t1 in
let congs' =
filter (fun th -> exists (equiv (concl th)) bcs) (basic_congs()) in
set_basic_congs congs';;
