Datasets:
Tasks:
Text Generation
Modalities:
Text
Sub-tasks:
language-modeling
Languages:
English
Size:
100K - 1M
License:
File size: 3,354 Bytes
4365a98 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 |
(* ========================================================================= *)
(* Ptolemy's theorem. *)
(* ========================================================================= *)
needs "Multivariate/transcendentals.ml";;
(* ------------------------------------------------------------------------- *)
(* Some 2-vector special cases. *)
(* ------------------------------------------------------------------------- *)
let DOT_VECTOR = prove
(`(vector [x1;y1] :real^2) dot (vector [x2;y2]) = x1 * x2 + y1 * y2`,
REWRITE_TAC[dot; DIMINDEX_2; SUM_2; VECTOR_2]);;
(* ------------------------------------------------------------------------- *)
(* Lemma about distance between points with polar coordinates. *)
(* ------------------------------------------------------------------------- *)
let DIST_SEGMENT_LEMMA = prove
(`!a1 a2. &0 <= a1 /\ a1 <= a2 /\ a2 <= &2 * pi /\ &0 <= radius
==> dist(centre + radius % vector [cos(a1);sin(a1)] :real^2,
centre + radius % vector [cos(a2);sin(a2)]) =
&2 * radius * sin((a2 - a1) / &2)`,
REPEAT STRIP_TAC THEN REWRITE_TAC[dist; vector_norm] THEN
MATCH_MP_TAC SQRT_UNIQUE THEN CONJ_TAC THENL
[MATCH_MP_TAC REAL_LE_MUL THEN REWRITE_TAC[REAL_POS] THEN
MATCH_MP_TAC REAL_LE_MUL THEN ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC SIN_POS_PI_LE THEN
ASM_REAL_ARITH_TAC;
ALL_TAC] THEN
REWRITE_TAC[VECTOR_ARITH `(c + r % x) - (c + r % y) = r % (x - y)`] THEN
REWRITE_TAC[VECTOR_ARITH `(r % x) dot (r % x) = (r pow 2) * (x dot x)`] THEN
REWRITE_TAC[DOT_LSUB; DOT_RSUB; DOT_VECTOR] THEN
SUBST1_TAC(REAL_ARITH `a1 = &2 * a1 / &2`) THEN
SUBST1_TAC(REAL_ARITH `a2 = &2 * a2 / &2`) THEN
REWRITE_TAC[REAL_ARITH `(&2 * x - &2 * y) / &2 = x - y`] THEN
REWRITE_TAC[SIN_SUB; SIN_DOUBLE; COS_DOUBLE] THEN
MP_TAC(SPEC `a1 / &2` SIN_CIRCLE) THEN MP_TAC(SPEC `a2 / &2` SIN_CIRCLE) THEN
CONV_TAC REAL_RING);;
(* ------------------------------------------------------------------------- *)
(* Hence the overall theorem. *)
(* ------------------------------------------------------------------------- *)
let PTOLEMY = prove
(`!A B C D:real^2 a b c d centre radius.
A = centre + radius % vector [cos(a);sin(a)] /\
B = centre + radius % vector [cos(b);sin(b)] /\
C = centre + radius % vector [cos(c);sin(c)] /\
D = centre + radius % vector [cos(d);sin(d)] /\
&0 <= radius /\
&0 <= a /\ a <= b /\ b <= c /\ c <= d /\ d <= &2 * pi
==> dist(A,C) * dist(B,D) =
dist(A,B) * dist(C,D) + dist(A,D) * dist(B,C)`,
REPEAT STRIP_TAC THEN
REPEAT(FIRST_X_ASSUM(SUBST1_TAC o check (is_var o lhs o concl))) THEN
REPEAT
(W(fun (asl,w) ->
let t = find_term
(fun t -> can (PART_MATCH (lhs o rand) DIST_SEGMENT_LEMMA) t) w in
MP_TAC (PART_MATCH (lhs o rand) DIST_SEGMENT_LEMMA t) THEN
ANTS_TAC THENL
[ASM_REAL_ARITH_TAC;
DISCH_THEN SUBST1_TAC])) THEN
REWRITE_TAC[REAL_ARITH `(x - y) / &2 = x / &2 - y / &2`] THEN
MAP_EVERY (fun t -> MP_TAC(SPEC t SIN_CIRCLE))
[`a / &2`; `b / &2`; `c / &2`; `d / &2`] THEN
REWRITE_TAC[SIN_SUB; SIN_ADD; COS_ADD; SIN_PI; COS_PI] THEN
CONV_TAC REAL_RING);;
|