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|
(* ========================================================================= *)
(* Some geometric notions in real^N. *)
(* ========================================================================= *)
needs "Multivariate/realanalysis.ml";;
prioritize_vector();;
(* ------------------------------------------------------------------------- *)
(* Pythagoras's theorem is almost immediate. *)
(* ------------------------------------------------------------------------- *)
let PYTHAGORAS = prove
(`!A B C:real^N.
orthogonal (A - B) (C - B)
==> norm(C - A) pow 2 = norm(B - A) pow 2 + norm(C - B) pow 2`,
REWRITE_TAC[NORM_POW_2; orthogonal; DOT_LSUB; DOT_RSUB; DOT_SYM] THEN
CONV_TAC REAL_RING);;
(* ------------------------------------------------------------------------- *)
(* Angle between vectors (always 0 <= angle <= pi). *)
(* ------------------------------------------------------------------------- *)
let vector_angle = new_definition
`vector_angle x y = if x = vec 0 \/ y = vec 0 then pi / &2
else acs((x dot y) / (norm x * norm y))`;;
let VECTOR_ANGLE_LINEAR_IMAGE_EQ = prove
(`!f x y. linear f /\ (!x. norm(f x) = norm x)
==> (vector_angle (f x) (f y) = vector_angle x y)`,
REPEAT STRIP_TAC THEN ASM_SIMP_TAC[vector_angle; GSYM NORM_EQ_0] THEN
ASM_MESON_TAC[PRESERVES_NORM_PRESERVES_DOT]);;
add_linear_invariants [VECTOR_ANGLE_LINEAR_IMAGE_EQ];;
(* ------------------------------------------------------------------------- *)
(* Basic properties of vector angles. *)
(* ------------------------------------------------------------------------- *)
let VECTOR_ANGLE_REFL = prove
(`!x. vector_angle x x = if x = vec 0 then pi / &2 else &0`,
GEN_TAC THEN REWRITE_TAC[vector_angle; DISJ_ACI] THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[GSYM NORM_POW_2; REAL_POW_2] THEN
ASM_SIMP_TAC[REAL_DIV_REFL; REAL_ENTIRE; NORM_EQ_0; ACS_1]);;
let VECTOR_ANGLE_SYM = prove
(`!x y. vector_angle x y = vector_angle y x`,
REWRITE_TAC[vector_angle; DISJ_SYM; DOT_SYM; REAL_MUL_SYM]);;
let VECTOR_ANGLE_LMUL = prove
(`!a x y:real^N.
vector_angle (a % x) y =
if a = &0 then pi / &2
else if &0 <= a then vector_angle x y
else pi - vector_angle x y`,
REPEAT STRIP_TAC THEN COND_CASES_TAC THEN
ASM_REWRITE_TAC[vector_angle; VECTOR_MUL_EQ_0] THEN
ASM_CASES_TAC `x:real^N = vec 0 \/ y:real^N = vec 0` THEN
ASM_REWRITE_TAC[] THENL [REAL_ARITH_TAC; ALL_TAC] THEN
REWRITE_TAC[NORM_MUL; DOT_LMUL; real_div; REAL_INV_MUL; real_abs] THEN
COND_CASES_TAC THEN
ASM_REWRITE_TAC[REAL_INV_NEG; REAL_MUL_LNEG; REAL_MUL_RNEG] THEN
ASM_SIMP_TAC[REAL_FIELD
`~(a = &0) ==> (a * x) * (inv a * y) * z = x * y * z`] THEN
MATCH_MP_TAC ACS_NEG THEN
REWRITE_TAC[GSYM REAL_ABS_BOUNDS; GSYM REAL_INV_MUL] THEN
REWRITE_TAC[GSYM real_div; NORM_CAUCHY_SCHWARZ_DIV]);;
let VECTOR_ANGLE_RMUL = prove
(`!a x y:real^N.
vector_angle x (a % y) =
if a = &0 then pi / &2
else if &0 <= a then vector_angle x y
else pi - vector_angle x y`,
ONCE_REWRITE_TAC[VECTOR_ANGLE_SYM] THEN
REWRITE_TAC[VECTOR_ANGLE_LMUL]);;
let VECTOR_ANGLE_LNEG = prove
(`!x y. vector_angle (--x) y = pi - vector_angle x y`,
REWRITE_TAC[VECTOR_ARITH `--x = -- &1 % x`; VECTOR_ANGLE_LMUL] THEN
CONV_TAC REAL_RAT_REDUCE_CONV);;
let VECTOR_ANGLE_RNEG = prove
(`!x y. vector_angle x (--y) = pi - vector_angle x y`,
ONCE_REWRITE_TAC[VECTOR_ANGLE_SYM] THEN REWRITE_TAC[VECTOR_ANGLE_LNEG]);;
let VECTOR_ANGLE_NEG2 = prove
(`!x y. vector_angle (--x) (--y) = vector_angle x y`,
REWRITE_TAC[VECTOR_ANGLE_LNEG; VECTOR_ANGLE_RNEG] THEN REAL_ARITH_TAC);;
let VECTOR_ANGLE = prove
(`!x y:real^N. x dot y = norm(x) * norm(y) * cos(vector_angle x y)`,
REPEAT GEN_TAC THEN REWRITE_TAC[vector_angle] THEN
ASM_CASES_TAC `x:real^N = vec 0` THEN
ASM_REWRITE_TAC[DOT_LZERO; NORM_0; REAL_MUL_LZERO] THEN
ASM_CASES_TAC `y:real^N = vec 0` THEN
ASM_REWRITE_TAC[DOT_RZERO; NORM_0; REAL_MUL_LZERO; REAL_MUL_RZERO] THEN
ONCE_REWRITE_TAC[AC REAL_MUL_AC `a * b * c:real = c * a * b`] THEN
ASM_SIMP_TAC[GSYM REAL_EQ_LDIV_EQ; REAL_LT_MUL; NORM_POS_LT] THEN
MATCH_MP_TAC(GSYM COS_ACS) THEN
ASM_REWRITE_TAC[REAL_BOUNDS_LE; NORM_CAUCHY_SCHWARZ_DIV]);;
let VECTOR_ANGLE_RANGE = prove
(`!x y:real^N. &0 <= vector_angle x y /\ vector_angle x y <= pi`,
REPEAT GEN_TAC THEN REWRITE_TAC[vector_angle] THEN COND_CASES_TAC THENL
[MP_TAC PI_POS THEN REAL_ARITH_TAC; ALL_TAC] THEN
RULE_ASSUM_TAC(REWRITE_RULE[DE_MORGAN_THM]) THEN MATCH_MP_TAC ACS_BOUNDS THEN
ASM_REWRITE_TAC[REAL_BOUNDS_LE; NORM_CAUCHY_SCHWARZ_DIV]);;
let ORTHOGONAL_VECTOR_ANGLE = prove
(`!x y:real^N. orthogonal x y <=> vector_angle x y = pi / &2`,
REPEAT STRIP_TAC THEN REWRITE_TAC[orthogonal; vector_angle] THEN
ASM_CASES_TAC `x:real^N = vec 0` THEN ASM_REWRITE_TAC[DOT_LZERO] THEN
ASM_CASES_TAC `y:real^N = vec 0` THEN ASM_REWRITE_TAC[DOT_RZERO] THEN
EQ_TAC THENL
[SIMP_TAC[real_div; REAL_MUL_LZERO] THEN DISCH_TAC THEN
REWRITE_TAC[GSYM real_div; GSYM COS_PI2] THEN
MATCH_MP_TAC ACS_COS THEN MP_TAC PI_POS THEN REAL_ARITH_TAC;
DISCH_THEN(MP_TAC o AP_TERM `cos`) THEN
SIMP_TAC[COS_ACS; REAL_BOUNDS_LE; NORM_CAUCHY_SCHWARZ_DIV; COS_PI2] THEN
ASM_SIMP_TAC[REAL_EQ_LDIV_EQ; REAL_LT_MUL; NORM_POS_LT; REAL_MUL_LZERO]]);;
let VECTOR_ANGLE_EQ_0 = prove
(`!x y:real^N. vector_angle x y = &0 <=>
~(x = vec 0) /\ ~(y = vec 0) /\ norm(x) % y = norm(y) % x`,
REPEAT GEN_TAC THEN
MAP_EVERY ASM_CASES_TAC [`x:real^N = vec 0`; `y:real^N = vec 0`] THEN
ASM_SIMP_TAC[vector_angle; PI_NZ; REAL_ARITH `x / &2 = &0 <=> x = &0`] THEN
REWRITE_TAC[GSYM NORM_CAUCHY_SCHWARZ_EQ] THEN
GEN_REWRITE_TAC (RAND_CONV o RAND_CONV) [GSYM REAL_MUL_LID] THEN
ASM_SIMP_TAC[GSYM REAL_EQ_LDIV_EQ; NORM_POS_LT; REAL_LT_MUL] THEN
MESON_TAC[ACS_1; COS_ACS; REAL_BOUNDS_LE; NORM_CAUCHY_SCHWARZ_DIV; COS_0]);;
let VECTOR_ANGLE_EQ_PI = prove
(`!x y:real^N. vector_angle x y = pi <=>
~(x = vec 0) /\ ~(y = vec 0) /\
norm(x) % y + norm(y) % x = vec 0`,
REPEAT GEN_TAC THEN
MP_TAC(ISPECL [`x:real^N`; `--y:real^N`] VECTOR_ANGLE_EQ_0) THEN
SIMP_TAC[VECTOR_ANGLE_RNEG; REAL_ARITH `pi - x = &0 <=> x = pi`] THEN
STRIP_TAC THEN
REWRITE_TAC[NORM_NEG; VECTOR_ARITH `--x = vec 0 <=> x = vec 0`] THEN
AP_TERM_TAC THEN AP_TERM_TAC THEN VECTOR_ARITH_TAC);;
let VECTOR_ANGLE_EQ_0_DIST = prove
(`!x y:real^N. vector_angle x y = &0 <=>
~(x = vec 0) /\ ~(y = vec 0) /\ norm(x + y) = norm x + norm y`,
REWRITE_TAC[VECTOR_ANGLE_EQ_0; GSYM NORM_TRIANGLE_EQ]);;
let VECTOR_ANGLE_EQ_PI_DIST = prove
(`!x y:real^N. vector_angle x y = pi <=>
~(x = vec 0) /\ ~(y = vec 0) /\ norm(x - y) = norm x + norm y`,
REPEAT GEN_TAC THEN
MP_TAC(ISPECL [`x:real^N`; `--y:real^N`] VECTOR_ANGLE_EQ_0_DIST) THEN
SIMP_TAC[VECTOR_ANGLE_RNEG; REAL_ARITH `pi - x = &0 <=> x = pi`] THEN
STRIP_TAC THEN REWRITE_TAC[NORM_NEG] THEN NORM_ARITH_TAC);;
let SIN_VECTOR_ANGLE_POS = prove
(`!v w. &0 <= sin(vector_angle v w)`,
SIMP_TAC[SIN_POS_PI_LE; VECTOR_ANGLE_RANGE]);;
let SIN_VECTOR_ANGLE_EQ_0 = prove
(`!x y. sin(vector_angle x y) = &0 <=>
vector_angle x y = &0 \/ vector_angle x y = pi`,
MESON_TAC[SIN_POS_PI; VECTOR_ANGLE_RANGE; REAL_LT_LE; SIN_0; SIN_PI]);;
let ASN_SIN_VECTOR_ANGLE = prove
(`!x y:real^N.
asn(sin(vector_angle x y)) =
if vector_angle x y <= pi / &2 then vector_angle x y
else pi - vector_angle x y`,
REPEAT GEN_TAC THEN COND_CASES_TAC THENL
[ALL_TAC;
MATCH_MP_TAC EQ_TRANS THEN
EXISTS_TAC `asn(sin(pi - vector_angle (x:real^N) y))` THEN CONJ_TAC THENL
[AP_TERM_TAC THEN REWRITE_TAC[SIN_SUB; SIN_PI; COS_PI] THEN
REAL_ARITH_TAC;
ALL_TAC]] THEN
MATCH_MP_TAC ASN_SIN THEN
MP_TAC(ISPECL [`x:real^N`; `y:real^N`] VECTOR_ANGLE_RANGE) THEN
ASM_REAL_ARITH_TAC);;
let SIN_VECTOR_ANGLE_EQ = prove
(`!x y w z.
sin(vector_angle x y) = sin(vector_angle w z) <=>
vector_angle x y = vector_angle w z \/
vector_angle x y = pi - vector_angle w z`,
REPEAT GEN_TAC THEN EQ_TAC THEN
STRIP_TAC THEN ASM_REWRITE_TAC[SIN_SUB; SIN_PI; COS_PI] THENL
[ALL_TAC; REAL_ARITH_TAC] THEN
FIRST_X_ASSUM(MP_TAC o AP_TERM `asn`) THEN
REWRITE_TAC[ASN_SIN_VECTOR_ANGLE] THEN REAL_ARITH_TAC);;
let CONTINUOUS_WITHIN_CX_VECTOR_ANGLE_COMPOSE = prove
(`!f:real^M->real^N g x s.
~(f x = vec 0) /\ ~(g x = vec 0) /\
f continuous (at x within s) /\
g continuous (at x within s)
==> (\x. Cx(vector_angle (f x) (g x))) continuous (at x within s)`,
REPEAT STRIP_TAC THEN
ASM_CASES_TAC `trivial_limit(at (x:real^M) within s)` THEN
ASM_SIMP_TAC[CONTINUOUS_TRIVIAL_LIMIT; vector_angle] THEN
SUBGOAL_THEN
`(cacs o (\x. Cx(((f x:real^N) dot g x) / (norm(f x) * norm(g x)))))
continuous (at (x:real^M) within s)`
MP_TAC THENL
[MATCH_MP_TAC CONTINUOUS_WITHIN_COMPOSE THEN CONJ_TAC THENL
[REWRITE_TAC[CX_DIV; CX_MUL] THEN REWRITE_TAC[WITHIN_UNIV] THEN
MATCH_MP_TAC CONTINUOUS_COMPLEX_DIV THEN
ASM_SIMP_TAC[NETLIMIT_WITHIN; COMPLEX_ENTIRE; CX_INJ; NORM_EQ_0] THEN
REWRITE_TAC[CONTINUOUS_CX_LIFT; GSYM CX_MUL; LIFT_CMUL] THEN
ASM_SIMP_TAC[CONTINUOUS_LIFT_DOT2] THEN
MATCH_MP_TAC CONTINUOUS_MUL THEN
ASM_SIMP_TAC[CONTINUOUS_LIFT_NORM_COMPOSE; o_DEF];
MATCH_MP_TAC CONTINUOUS_WITHIN_SUBSET THEN
EXISTS_TAC `{z | real z /\ abs(Re z) <= &1}` THEN
REWRITE_TAC[CONTINUOUS_WITHIN_CACS_REAL] THEN
REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_UNIV; IN_ELIM_THM] THEN
REWRITE_TAC[REAL_CX; RE_CX; NORM_CAUCHY_SCHWARZ_DIV]];
ASM_SIMP_TAC[CONTINUOUS_WITHIN; CX_ACS; o_DEF;
NORM_CAUCHY_SCHWARZ_DIV] THEN
MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] LIM_TRANSFORM_EVENTUALLY) THEN
SUBGOAL_THEN
`eventually (\y. ~((f:real^M->real^N) y = vec 0) /\
~((g:real^M->real^N) y = vec 0))
(at x within s)`
MP_TAC THENL
[REWRITE_TAC[EVENTUALLY_AND] THEN CONJ_TAC THENL
[UNDISCH_TAC `(f:real^M->real^N) continuous (at x within s)`;
UNDISCH_TAC `(g:real^M->real^N) continuous (at x within s)`] THEN
REWRITE_TAC[CONTINUOUS_WITHIN; tendsto] THENL
[DISCH_THEN(MP_TAC o SPEC `norm((f:real^M->real^N) x)`);
DISCH_THEN(MP_TAC o SPEC `norm((g:real^M->real^N) x)`)] THEN
ASM_REWRITE_TAC[NORM_POS_LT] THEN
MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] EVENTUALLY_MONO) THEN
REWRITE_TAC[] THEN CONV_TAC NORM_ARITH;
MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] EVENTUALLY_MONO) THEN
SIMP_TAC[CX_ACS; NORM_CAUCHY_SCHWARZ_DIV]]]);;
let CONTINUOUS_AT_CX_VECTOR_ANGLE = prove
(`!c x:real^N. ~(x = vec 0) ==> (Cx o vector_angle c) continuous (at x)`,
REPEAT STRIP_TAC THEN REWRITE_TAC[o_DEF; vector_angle] THEN
ASM_CASES_TAC `c:real^N = vec 0` THEN ASM_REWRITE_TAC[CONTINUOUS_CONST] THEN
MATCH_MP_TAC CONTINUOUS_TRANSFORM_AT THEN
MAP_EVERY EXISTS_TAC [`\x:real^N. cacs(Cx((c dot x) / (norm c * norm x)))`;
`norm(x:real^N)`] THEN
ASM_REWRITE_TAC[NORM_POS_LT] THEN CONJ_TAC THENL
[X_GEN_TAC `z:real^N` THEN DISCH_TAC THEN COND_CASES_TAC THENL
[ASM_MESON_TAC[NORM_ARITH `~(dist(vec 0,x) < norm x)`]; ALL_TAC] THEN
MATCH_MP_TAC(GSYM CX_ACS) THEN REWRITE_TAC[NORM_CAUCHY_SCHWARZ_DIV];
ALL_TAC] THEN
ONCE_REWRITE_TAC[GSYM WITHIN_UNIV] THEN
MATCH_MP_TAC(REWRITE_RULE[o_DEF] CONTINUOUS_WITHIN_COMPOSE) THEN
CONJ_TAC THENL
[REWRITE_TAC[CX_DIV; CX_MUL] THEN REWRITE_TAC[WITHIN_UNIV] THEN
MATCH_MP_TAC CONTINUOUS_COMPLEX_DIV THEN
ASM_REWRITE_TAC[NETLIMIT_AT; COMPLEX_ENTIRE; CX_INJ; NORM_EQ_0] THEN
SIMP_TAC[CONTINUOUS_COMPLEX_MUL; CONTINUOUS_CONST;
CONTINUOUS_AT_CX_NORM; CONTINUOUS_AT_CX_DOT];
MATCH_MP_TAC CONTINUOUS_WITHIN_SUBSET THEN
EXISTS_TAC `{z | real z /\ abs(Re z) <= &1}` THEN
REWRITE_TAC[CONTINUOUS_WITHIN_CACS_REAL] THEN
REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_UNIV; IN_ELIM_THM] THEN
REWRITE_TAC[REAL_CX; RE_CX; NORM_CAUCHY_SCHWARZ_DIV]]);;
let CONTINUOUS_WITHIN_CX_VECTOR_ANGLE = prove
(`!c x:real^N s.
~(x = vec 0) ==> (Cx o vector_angle c) continuous (at x within s)`,
SIMP_TAC[CONTINUOUS_AT_WITHIN; CONTINUOUS_AT_CX_VECTOR_ANGLE]);;
let REAL_CONTINUOUS_AT_VECTOR_ANGLE = prove
(`!c x:real^N. ~(x = vec 0) ==> (vector_angle c) real_continuous (at x)`,
REWRITE_TAC[REAL_CONTINUOUS_CONTINUOUS; CONTINUOUS_AT_CX_VECTOR_ANGLE]);;
let REAL_CONTINUOUS_WITHIN_VECTOR_ANGLE = prove
(`!c s x:real^N. ~(x = vec 0)
==> (vector_angle c) real_continuous (at x within s)`,
REWRITE_TAC[REAL_CONTINUOUS_CONTINUOUS; CONTINUOUS_WITHIN_CX_VECTOR_ANGLE]);;
let CONTINUOUS_ON_CX_VECTOR_ANGLE = prove
(`!s. ~(vec 0 IN s) ==> (Cx o vector_angle c) continuous_on s`,
REPEAT STRIP_TAC THEN REWRITE_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN] THEN
ASM_MESON_TAC[CONTINUOUS_WITHIN_CX_VECTOR_ANGLE]);;
let VECTOR_ANGLE_EQ = prove
(`!u v x y. ~(u = vec 0) /\ ~(v = vec 0) /\ ~(x = vec 0) /\ ~(y = vec 0)
==> (vector_angle u v = vector_angle x y <=>
(x dot y) * norm(u) * norm(v) =
(u dot v) * norm(x) * norm(y))`,
SIMP_TAC[vector_angle; NORM_EQ_0; REAL_FIELD
`~(u = &0) /\ ~(v = &0) /\ ~(x = &0) /\ ~(y = &0)
==> (a * u * v = b * x * y <=> a / (x * y) = b / (u * v))`] THEN
REPEAT STRIP_TAC THEN EQ_TAC THEN SIMP_TAC[] THEN
DISCH_THEN(MP_TAC o AP_TERM `cos`) THEN
SIMP_TAC[COS_ACS; NORM_CAUCHY_SCHWARZ_DIV; REAL_BOUNDS_LE]);;
let COS_VECTOR_ANGLE_EQ = prove
(`!u v x y.
cos(vector_angle u v) = cos(vector_angle x y) <=>
vector_angle u v = vector_angle x y`,
MESON_TAC[ACS_COS; VECTOR_ANGLE_RANGE]);;
let COLLINEAR_VECTOR_ANGLE = prove
(`!x y. ~(x = vec 0) /\ ~(y = vec 0)
==> (collinear {vec 0,x,y} <=>
vector_angle x y = &0 \/ vector_angle x y = pi)`,
REWRITE_TAC[GSYM NORM_CAUCHY_SCHWARZ_EQUAL; NORM_CAUCHY_SCHWARZ_ABS_EQ;
VECTOR_ANGLE_EQ_0; VECTOR_ANGLE_EQ_PI] THEN
REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THEN BINOP_TAC THEN
VECTOR_ARITH_TAC);;
let COLLINEAR_SIN_VECTOR_ANGLE = prove
(`!x y. ~(x = vec 0) /\ ~(y = vec 0)
==> (collinear {vec 0,x,y} <=> sin(vector_angle x y) = &0)`,
REWRITE_TAC[SIN_VECTOR_ANGLE_EQ_0; COLLINEAR_VECTOR_ANGLE]);;
let COLLINEAR_SIN_VECTOR_ANGLE_IMP = prove
(`!x y. sin(vector_angle x y) = &0
==> ~(x = vec 0) /\ ~(y = vec 0) /\ collinear {vec 0,x,y}`,
MESON_TAC[COLLINEAR_SIN_VECTOR_ANGLE; SIN_VECTOR_ANGLE_EQ_0;
VECTOR_ANGLE_EQ_0_DIST; VECTOR_ANGLE_EQ_PI_DIST]);;
let VECTOR_ANGLE_EQ_0_RIGHT = prove
(`!x y z:real^N. vector_angle x y = &0
==> (vector_angle x z = vector_angle y z)`,
REWRITE_TAC[VECTOR_ANGLE_EQ_0] THEN REPEAT STRIP_TAC THEN
MATCH_MP_TAC EQ_TRANS THEN
EXISTS_TAC `vector_angle (norm(x:real^N) % y) (z:real^N)` THEN CONJ_TAC THENL
[ASM_REWRITE_TAC[] THEN
ASM_REWRITE_TAC[VECTOR_ANGLE_LMUL; NORM_EQ_0; NORM_POS_LE];
REWRITE_TAC[VECTOR_ANGLE_LMUL] THEN
ASM_REWRITE_TAC[NORM_EQ_0; NORM_POS_LE]]);;
let VECTOR_ANGLE_EQ_0_LEFT = prove
(`!x y z:real^N. vector_angle x y = &0
==> (vector_angle z x = vector_angle z y)`,
MESON_TAC[VECTOR_ANGLE_EQ_0_RIGHT; VECTOR_ANGLE_SYM]);;
let VECTOR_ANGLE_EQ_PI_RIGHT = prove
(`!x y z:real^N. vector_angle x y = pi
==> (vector_angle x z = pi - vector_angle y z)`,
REPEAT STRIP_TAC THEN
MP_TAC(ISPECL [`--x:real^N`; `y:real^N`; `z:real^N`]
VECTOR_ANGLE_EQ_0_RIGHT) THEN
ASM_REWRITE_TAC[VECTOR_ANGLE_LNEG] THEN REAL_ARITH_TAC);;
let VECTOR_ANGLE_EQ_PI_LEFT = prove
(`!x y z:real^N. vector_angle x y = pi
==> (vector_angle z x = pi - vector_angle z y)`,
MESON_TAC[VECTOR_ANGLE_EQ_PI_RIGHT; VECTOR_ANGLE_SYM]);;
let COS_VECTOR_ANGLE = prove
(`!x y:real^N.
cos(vector_angle x y) = if x = vec 0 \/ y = vec 0 then &0
else (x dot y) / (norm x * norm y)`,
REPEAT GEN_TAC THEN
ASM_CASES_TAC `x:real^N = vec 0` THENL
[ASM_REWRITE_TAC[vector_angle; COS_PI2]; ALL_TAC] THEN
ASM_CASES_TAC `y:real^N = vec 0` THENL
[ASM_REWRITE_TAC[vector_angle; COS_PI2]; ALL_TAC] THEN
ASM_SIMP_TAC[REAL_EQ_RDIV_EQ; REAL_LT_MUL; NORM_POS_LT; VECTOR_ANGLE] THEN
REAL_ARITH_TAC);;
let SIN_VECTOR_ANGLE = prove
(`!x y:real^N.
sin(vector_angle x y) =
if x = vec 0 \/ y = vec 0 then &1
else sqrt(&1 - ((x dot y) / (norm x * norm y)) pow 2)`,
SIMP_TAC[SIN_COS_SQRT; SIN_VECTOR_ANGLE_POS; COS_VECTOR_ANGLE] THEN
REPEAT GEN_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN
CONV_TAC REAL_RAT_REDUCE_CONV THEN REWRITE_TAC[SQRT_1]);;
let SIN_SQUARED_VECTOR_ANGLE = prove
(`!x y:real^N.
sin(vector_angle x y) pow 2 =
if x = vec 0 \/ y = vec 0 then &1
else &1 - ((x dot y) / (norm x * norm y)) pow 2`,
REPEAT GEN_TAC THEN REWRITE_TAC
[REWRITE_RULE [REAL_ARITH `s + c = &1 <=> s = &1 - c`] SIN_CIRCLE] THEN
REWRITE_TAC[COS_VECTOR_ANGLE] THEN REAL_ARITH_TAC);;
let VECTOR_ANGLE_COMPLEX_LMUL = prove
(`!a. ~(a = Cx(&0))
==> vector_angle (a * x) (a * y) = vector_angle x y`,
REPEAT STRIP_TAC THEN
ASM_CASES_TAC `x = Cx(&0)` THENL
[ASM_REWRITE_TAC[COMPLEX_MUL_RZERO; vector_angle; COMPLEX_VEC_0];
ALL_TAC] THEN
ASM_CASES_TAC `y = Cx(&0)` THENL
[ASM_REWRITE_TAC[COMPLEX_MUL_RZERO; vector_angle; COMPLEX_VEC_0];
ALL_TAC] THEN
MP_TAC(ISPECL
[`a * x:complex`; `a * y:complex`; `x:complex`; `y:complex`]
VECTOR_ANGLE_EQ) THEN
ASM_REWRITE_TAC[COMPLEX_VEC_0; COMPLEX_ENTIRE] THEN
DISCH_THEN SUBST1_TAC THEN
REWRITE_TAC[COMPLEX_NORM_MUL] THEN MATCH_MP_TAC(REAL_RING
`a pow 2 * xy:real = d ==> xy * (a * x) * (a * y) = d * x * y`) THEN
REWRITE_TAC[NORM_POW_2] THEN
REWRITE_TAC[DOT_2; complex_mul; GSYM RE_DEF; GSYM IM_DEF; RE; IM] THEN
REAL_ARITH_TAC);;
let VECTOR_ANGLE_1 = prove
(`!x. vector_angle x (Cx(&1)) = acs(Re x / norm x)`,
GEN_TAC THEN
SIMP_TAC[vector_angle; COMPLEX_VEC_0; CX_INJ; REAL_OF_NUM_EQ; ARITH_EQ] THEN
COND_CASES_TAC THENL
[ASM_REWRITE_TAC[real_div; RE_CX; ACS_0; REAL_MUL_LZERO]; ALL_TAC] THEN
REWRITE_TAC[COMPLEX_NORM_CX; REAL_ABS_NUM; REAL_MUL_RID] THEN
REWRITE_TAC[DOT_2; GSYM RE_DEF; GSYM IM_DEF; RE_CX; IM_CX] THEN
AP_TERM_TAC THEN REAL_ARITH_TAC);;
let ARG_EQ_VECTOR_ANGLE_1 = prove
(`!z. ~(z = Cx(&0)) /\ &0 <= Im z ==> Arg z = vector_angle z (Cx(&1))`,
REPEAT STRIP_TAC THEN REWRITE_TAC[VECTOR_ANGLE_1] THEN
GEN_REWRITE_TAC (RAND_CONV o RAND_CONV o LAND_CONV o RAND_CONV) [ARG] THEN
REWRITE_TAC[RE_MUL_CX; RE_CEXP; RE_II; IM_MUL_II; IM_CX; RE_CX] THEN
REWRITE_TAC[REAL_MUL_LZERO; REAL_EXP_0; REAL_MUL_LID] THEN
ASM_SIMP_TAC[COMPLEX_NORM_ZERO; REAL_FIELD
`~(z = &0) ==> (z * x) / z = x`] THEN
CONV_TAC SYM_CONV THEN MATCH_MP_TAC ACS_COS THEN
ASM_REWRITE_TAC[ARG; ARG_LE_PI]);;
let VECTOR_ANGLE_ARG = prove
(`!w z. ~(w = Cx(&0)) /\ ~(z = Cx(&0))
==> vector_angle w z = if &0 <= Im(z / w) then Arg(z / w)
else &2 * pi - Arg(z / w)`,
REPEAT STRIP_TAC THEN COND_CASES_TAC THENL
[SUBGOAL_THEN `z = w * (z / w) /\ w = w * Cx(&1)` MP_TAC THENL
[REPEAT(POP_ASSUM MP_TAC) THEN CONV_TAC COMPLEX_FIELD; ALL_TAC];
SUBGOAL_THEN `w = z * (w / z) /\ z = z * Cx(&1)` MP_TAC THENL
[REPEAT(POP_ASSUM MP_TAC) THEN CONV_TAC COMPLEX_FIELD; ALL_TAC]] THEN
DISCH_THEN(fun th -> GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [th]) THEN
ASM_SIMP_TAC[VECTOR_ANGLE_COMPLEX_LMUL] THENL
[ONCE_REWRITE_TAC[VECTOR_ANGLE_SYM] THEN CONV_TAC SYM_CONV THEN
MATCH_MP_TAC ARG_EQ_VECTOR_ANGLE_1 THEN ASM_REWRITE_TAC[] THEN
REPEAT(POP_ASSUM MP_TAC) THEN CONV_TAC COMPLEX_FIELD;
MP_TAC(ISPEC `z / w:complex` ARG_INV) THEN ANTS_TAC THENL
[ASM_MESON_TAC[real; REAL_LE_REFL]; DISCH_THEN(SUBST1_TAC o SYM)] THEN
REWRITE_TAC[COMPLEX_INV_DIV] THEN CONV_TAC SYM_CONV THEN
MATCH_MP_TAC ARG_EQ_VECTOR_ANGLE_1 THEN CONJ_TAC THENL
[REPEAT(POP_ASSUM MP_TAC) THEN CONV_TAC COMPLEX_FIELD;
ONCE_REWRITE_TAC[GSYM COMPLEX_INV_DIV] THEN
REWRITE_TAC[IM_COMPLEX_INV_GE_0] THEN ASM_REAL_ARITH_TAC]]);;
(* ------------------------------------------------------------------------- *)
(* Traditional geometric notion of angle (always 0 <= theta <= pi). *)
(* ------------------------------------------------------------------------- *)
let angle = new_definition
`angle(a,b,c) = vector_angle (a - b) (c - b)`;;
let ANGLE_LINEAR_IMAGE_EQ = prove
(`!f a b c.
linear f /\ (!x. norm(f x) = norm x)
==> angle(f a,f b,f c) = angle(a,b,c)`,
SIMP_TAC[angle; GSYM LINEAR_SUB; VECTOR_ANGLE_LINEAR_IMAGE_EQ]);;
add_linear_invariants [ANGLE_LINEAR_IMAGE_EQ];;
let ANGLE_TRANSLATION_EQ = prove
(`!a b c d. angle(a + b,a + c,a + d) = angle(b,c,d)`,
REPEAT GEN_TAC THEN REWRITE_TAC[angle] THEN
BINOP_TAC THEN VECTOR_ARITH_TAC);;
add_translation_invariants [ANGLE_TRANSLATION_EQ];;
let VECTOR_ANGLE_ANGLE = prove
(`vector_angle x y = angle(x,vec 0,y)`,
REWRITE_TAC[angle; VECTOR_SUB_RZERO]);;
let ANGLE_EQ_PI_DIST = prove
(`!A B C:real^N.
angle(A,B,C) = pi <=>
~(A = B) /\ ~(C = B) /\ dist(A,C) = dist(A,B) + dist(B,C)`,
REWRITE_TAC[angle; VECTOR_ANGLE_EQ_PI_DIST] THEN NORM_ARITH_TAC);;
let SIN_ANGLE_POS = prove
(`!A B C. &0 <= sin(angle(A,B,C))`,
REWRITE_TAC[angle; SIN_VECTOR_ANGLE_POS]);;
let ANGLE = prove
(`!A B C. (A - C) dot (B - C) = dist(A,C) * dist(B,C) * cos(angle(A,C,B))`,
REWRITE_TAC[angle; dist; GSYM VECTOR_ANGLE]);;
let ANGLE_REFL = prove
(`!A B. angle(A,A,B) = pi / &2 /\ angle(B,A,A) = pi / &2`,
REWRITE_TAC[angle; vector_angle; VECTOR_SUB_REFL]);;
let ANGLE_REFL_MID = prove
(`!A B. ~(A = B) ==> angle(A,B,A) = &0`,
SIMP_TAC[angle; vector_angle; VECTOR_SUB_EQ; GSYM NORM_POW_2; ARITH;
GSYM REAL_POW_2; REAL_DIV_REFL; ACS_1; REAL_POW_EQ_0; NORM_EQ_0]);;
let ANGLE_SYM = prove
(`!A B C. angle(A,B,C) = angle(C,B,A)`,
REWRITE_TAC[angle; vector_angle; VECTOR_SUB_EQ; DISJ_SYM;
REAL_MUL_SYM; DOT_SYM]);;
let ANGLE_RANGE = prove
(`!A B C. &0 <= angle(A,B,C) /\ angle(A,B,C) <= pi`,
REWRITE_TAC[angle; VECTOR_ANGLE_RANGE]);;
let COS_ANGLE_EQ = prove
(`!a b c a' b' c'.
cos(angle(a,b,c)) = cos(angle(a',b',c')) <=>
angle(a,b,c) = angle(a',b',c')`,
REWRITE_TAC[angle; COS_VECTOR_ANGLE_EQ]);;
let ANGLE_EQ = prove
(`!a b c a' b' c'.
~(a = b) /\ ~(c = b) /\ ~(a' = b') /\ ~(c' = b')
==> (angle(a,b,c) = angle(a',b',c') <=>
((a' - b') dot (c' - b')) * norm (a - b) * norm (c - b) =
((a - b) dot (c - b)) * norm (a' - b') * norm (c' - b'))`,
SIMP_TAC[angle; VECTOR_ANGLE_EQ; VECTOR_SUB_EQ]);;
let SIN_ANGLE_EQ_0 = prove
(`!A B C. sin(angle(A,B,C)) = &0 <=> angle(A,B,C) = &0 \/ angle(A,B,C) = pi`,
REWRITE_TAC[angle; SIN_VECTOR_ANGLE_EQ_0]);;
let SIN_ANGLE_EQ = prove
(`!A B C A' B' C'. sin(angle(A,B,C)) = sin(angle(A',B',C')) <=>
angle(A,B,C) = angle(A',B',C') \/
angle(A,B,C) = pi - angle(A',B',C')`,
REWRITE_TAC[angle; SIN_VECTOR_ANGLE_EQ]);;
let COLLINEAR_ANGLE = prove
(`!A B C. ~(A = B) /\ ~(B = C)
==> (collinear {A,B,C} <=> angle(A,B,C) = &0 \/ angle(A,B,C) = pi)`,
ONCE_REWRITE_TAC[COLLINEAR_3] THEN
SIMP_TAC[COLLINEAR_VECTOR_ANGLE; VECTOR_SUB_EQ; angle]);;
let COLLINEAR_SIN_ANGLE = prove
(`!A B C. ~(A = B) /\ ~(B = C)
==> (collinear {A,B,C} <=> sin(angle(A,B,C)) = &0)`,
REWRITE_TAC[SIN_ANGLE_EQ_0; COLLINEAR_ANGLE]);;
let COLLINEAR_SIN_ANGLE_IMP = prove
(`!A B C. sin(angle(A,B,C)) = &0
==> ~(A = B) /\ ~(B = C) /\ collinear {A,B,C}`,
REPEAT GEN_TAC THEN
ONCE_REWRITE_TAC[COLLINEAR_3] THEN REWRITE_TAC[angle] THEN
DISCH_THEN(MP_TAC o MATCH_MP COLLINEAR_SIN_VECTOR_ANGLE_IMP) THEN
SIMP_TAC[VECTOR_SUB_EQ]);;
let ANGLE_EQ_0_RIGHT = prove
(`!A B C. angle(A,B,C) = &0 ==> angle(A,B,D) = angle(C,B,D)`,
REWRITE_TAC[VECTOR_ANGLE_EQ_0_RIGHT; angle]);;
let ANGLE_EQ_0_LEFT = prove
(`!A B C. angle(A,B,C) = &0 ==> angle(D,B,A) = angle(D,B,C)`,
MESON_TAC[ANGLE_EQ_0_RIGHT; ANGLE_SYM]);;
let ANGLE_EQ_PI_RIGHT = prove
(`!A B C. angle(A,B,C) = pi ==> angle(A,B,D) = pi - angle(C,B,D)`,
REWRITE_TAC[VECTOR_ANGLE_EQ_PI_RIGHT; angle]);;
let ANGLE_EQ_PI_LEFT = prove
(`!A B C. angle(A,B,C) = pi ==> angle(A,B,D) = pi - angle(C,B,D)`,
MESON_TAC[ANGLE_EQ_PI_RIGHT; ANGLE_SYM]);;
let COS_ANGLE = prove
(`!a b c. cos(angle(a,b,c)) = if a = b \/ c = b then &0
else ((a - b) dot (c - b)) /
(norm(a - b) * norm(c - b))`,
REWRITE_TAC[angle; COS_VECTOR_ANGLE; VECTOR_SUB_EQ]);;
let SIN_ANGLE = prove
(`!a b c. sin(angle(a,b,c)) =
if a = b \/ c = b then &1
else sqrt(&1 - (((a - b) dot (c - b)) /
(norm(a - b) * norm(c - b))) pow 2)`,
REWRITE_TAC[angle; SIN_VECTOR_ANGLE; VECTOR_SUB_EQ]);;
let SIN_SQUARED_ANGLE = prove
(`!a b c. sin(angle(a,b,c)) pow 2 =
if a = b \/ c = b then &1
else &1 - (((a - b) dot (c - b)) /
(norm(a - b) * norm(c - b))) pow 2`,
REWRITE_TAC[angle; SIN_SQUARED_VECTOR_ANGLE; VECTOR_SUB_EQ]);;
(* ------------------------------------------------------------------------- *)
(* The law of cosines. *)
(* ------------------------------------------------------------------------- *)
let LAW_OF_COSINES = prove
(`!A B C:real^N.
dist(B,C) pow 2 = (dist(A,B) pow 2 + dist(A,C) pow 2) -
&2 * dist(A,B) * dist(A,C) * cos(angle(B,A,C))`,
REPEAT GEN_TAC THEN
REWRITE_TAC[angle; ONCE_REWRITE_RULE[NORM_SUB] dist; GSYM VECTOR_ANGLE;
NORM_POW_2] THEN
VECTOR_ARITH_TAC);;
(* ------------------------------------------------------------------------- *)
(* The law of sines. *)
(* ------------------------------------------------------------------------- *)
let LAW_OF_SINES = prove
(`!A B C:real^N.
sin(angle(A,B,C)) * dist(B,C) = sin(angle(B,A,C)) * dist(A,C)`,
REPEAT GEN_TAC THEN MATCH_MP_TAC REAL_POW_EQ THEN EXISTS_TAC `2` THEN
SIMP_TAC[SIN_ANGLE_POS; DIST_POS_LE; REAL_LE_MUL; ARITH] THEN
REWRITE_TAC[REAL_POW_MUL; MATCH_MP
(REAL_ARITH `x + y = &1 ==> x = &1 - y`) (SPEC_ALL SIN_CIRCLE)] THEN
ASM_CASES_TAC `A:real^N = B` THEN ASM_REWRITE_TAC[ANGLE_REFL; COS_PI2] THEN
RULE_ASSUM_TAC(ONCE_REWRITE_RULE[GSYM VECTOR_SUB_EQ]) THEN
RULE_ASSUM_TAC(REWRITE_RULE[GSYM NORM_EQ_0]) THEN
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_RING
`~(a = &0) ==> a pow 2 * x = a pow 2 * y ==> x = y`)) THEN
ONCE_REWRITE_TAC[DIST_SYM] THEN REWRITE_TAC[GSYM dist] THEN
GEN_REWRITE_TAC (RAND_CONV o LAND_CONV o ONCE_DEPTH_CONV) [DIST_SYM] THEN
REWRITE_TAC[REAL_RING
`a * (&1 - x) * b = c * (&1 - y) * d <=>
a * b - a * b * x = c * d - c * d * y`] THEN
REWRITE_TAC[GSYM REAL_POW_MUL; GSYM ANGLE] THEN
REWRITE_TAC[REAL_POW_MUL; dist; NORM_POW_2] THEN
REWRITE_TAC[DOT_LSUB; DOT_RSUB; DOT_SYM] THEN CONV_TAC REAL_RING);;
(* ------------------------------------------------------------------------- *)
(* The sum of the angles of a triangle. *)
(* ------------------------------------------------------------------------- *)
let TRIANGLE_ANGLE_SUM_LEMMA = prove
(`!A B C:real^N. ~(A = B) /\ ~(A = C) /\ ~(B = C)
==> cos(angle(B,A,C) + angle(A,B,C) + angle(B,C,A)) = -- &1`,
REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[GSYM VECTOR_SUB_EQ] THEN
REWRITE_TAC[GSYM NORM_EQ_0] THEN
MP_TAC(ISPECL [`A:real^N`; `B:real^N`; `C:real^N`] LAW_OF_COSINES) THEN
MP_TAC(ISPECL [`B:real^N`; `A:real^N`; `C:real^N`] LAW_OF_COSINES) THEN
MP_TAC(ISPECL [`C:real^N`; `B:real^N`; `A:real^N`] LAW_OF_COSINES) THEN
MP_TAC(ISPECL [`A:real^N`; `B:real^N`; `C:real^N`] LAW_OF_SINES) THEN
MP_TAC(ISPECL [`B:real^N`; `A:real^N`; `C:real^N`] LAW_OF_SINES) THEN
MP_TAC(ISPECL [`B:real^N`; `C:real^N`; `A:real^N`] LAW_OF_SINES) THEN
REWRITE_TAC[COS_ADD; SIN_ADD; dist; NORM_SUB] THEN
MAP_EVERY (fun t -> MP_TAC(SPEC t SIN_CIRCLE))
[`angle(B:real^N,A,C)`; `angle(A:real^N,B,C)`; `angle(B:real^N,C,A)`] THEN
REWRITE_TAC[COS_ADD; SIN_ADD; ANGLE_SYM] THEN CONV_TAC REAL_RING);;
let COS_MINUS1_LEMMA = prove
(`!x. cos(x) = -- &1 /\ &0 <= x /\ x < &3 * pi ==> x = pi`,
REPEAT STRIP_TAC THEN
SUBGOAL_THEN `?n. integer n /\ x = n * pi`
(X_CHOOSE_THEN `nn:real` (CONJUNCTS_THEN2 ASSUME_TAC SUBST_ALL_TAC)) THEN
REWRITE_TAC[GSYM SIN_EQ_0] THENL
[MP_TAC(SPEC `x:real` SIN_CIRCLE) THEN ASM_REWRITE_TAC[] THEN
CONV_TAC REAL_RING;
ALL_TAC] THEN
SUBGOAL_THEN `?n. nn = &n` (X_CHOOSE_THEN `n:num` SUBST_ALL_TAC) THENL
[FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [REAL_MUL_POS_LE]) THEN
SIMP_TAC[PI_POS; REAL_ARITH `&0 < p ==> ~(p < &0) /\ ~(p = &0)`] THEN
ASM_MESON_TAC[INTEGER_POS; REAL_LT_LE];
ALL_TAC] THEN
MATCH_MP_TAC(REAL_RING `n = &1 ==> n * p = p`) THEN
REWRITE_TAC[REAL_OF_NUM_EQ] THEN
MATCH_MP_TAC(ARITH_RULE `n < 3 /\ ~(n = 0) /\ ~(n = 2) ==> n = 1`) THEN
RULE_ASSUM_TAC(SIMP_RULE[REAL_LT_RMUL_EQ; PI_POS; REAL_OF_NUM_LT]) THEN
ASM_REWRITE_TAC[] THEN CONJ_TAC THEN DISCH_THEN SUBST_ALL_TAC THEN
REPEAT(POP_ASSUM MP_TAC) THEN SIMP_TAC[COS_0; REAL_MUL_LZERO; COS_NPI] THEN
REAL_ARITH_TAC);;
let TRIANGLE_ANGLE_SUM = prove
(`!A B C:real^N. ~(A = B /\ B = C /\ A = C)
==> angle(B,A,C) + angle(A,B,C) + angle(B,C,A) = pi`,
REPEAT GEN_TAC THEN MAP_EVERY ASM_CASES_TAC
[`A:real^N = B`; `B:real^N = C`; `A:real^N = C`] THEN
ASM_SIMP_TAC[ANGLE_REFL_MID; ANGLE_REFL; REAL_HALF; REAL_ADD_RID] THEN
REPEAT(FIRST_X_ASSUM SUBST_ALL_TAC) THEN
REPEAT(POP_ASSUM MP_TAC) THEN REWRITE_TAC[REAL_ADD_LID; REAL_HALF] THEN
REPEAT STRIP_TAC THEN MATCH_MP_TAC COS_MINUS1_LEMMA THEN
ASM_SIMP_TAC[TRIANGLE_ANGLE_SUM_LEMMA; REAL_LE_ADD; ANGLE_RANGE] THEN
MATCH_MP_TAC(REAL_ARITH
`&0 <= x /\ x <= p /\ &0 <= y /\ y <= p /\ &0 <= z /\ z <= p /\
~(x = p /\ y = p /\ z = p)
==> x + y + z < &3 * p`) THEN
ASM_SIMP_TAC[ANGLE_RANGE] THEN REPEAT STRIP_TAC THEN
REPEAT(FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [ANGLE_EQ_PI_DIST])) THEN
REPEAT(FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV
[GSYM VECTOR_SUB_EQ])) THEN
REWRITE_TAC[GSYM NORM_EQ_0; dist; NORM_SUB] THEN REAL_ARITH_TAC);;
(* ------------------------------------------------------------------------- *)
(* A few more lemmas about angles. *)
(* ------------------------------------------------------------------------- *)
let ANGLE_EQ_PI_OTHERS = prove
(`!A B C:real^N.
angle(A,B,C) = pi
==> angle(B,C,A) = &0 /\ angle(A,C,B) = &0 /\
angle(B,A,C) = &0 /\ angle(C,A,B) = &0`,
REPEAT GEN_TAC THEN DISCH_TAC THEN
FIRST_ASSUM(STRIP_ASSUME_TAC o GEN_REWRITE_RULE I [ANGLE_EQ_PI_DIST]) THEN
MP_TAC(ISPECL [`A:real^N`; `B:real^N`; `C:real^N`] TRIANGLE_ANGLE_SUM) THEN
ASM_REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o MATCH_MP (REAL_ARITH
`x + p + y = p ==> &0 <= x /\ &0 <= y ==> x = &0 /\ y = &0`)) THEN
SIMP_TAC[ANGLE_RANGE; ANGLE_SYM]);;
let ANGLE_EQ_0_DIST = prove
(`!A B C:real^N. angle(A,B,C) = &0 <=>
~(A = B) /\ ~(C = B) /\
(dist(A,B) = dist(A,C) + dist(C,B) \/
dist(B,C) = dist(A,C) + dist(A,B))`,
REPEAT GEN_TAC THEN
ASM_CASES_TAC `A:real^N = B` THENL
[ASM_REWRITE_TAC[angle; VECTOR_ANGLE_EQ_0; VECTOR_SUB_EQ]; ALL_TAC] THEN
ASM_CASES_TAC `B:real^N = C` THENL
[ASM_REWRITE_TAC[angle; VECTOR_ANGLE_EQ_0; VECTOR_SUB_EQ]; ALL_TAC] THEN
ASM_CASES_TAC `A:real^N = C` THENL
[ASM_SIMP_TAC[ANGLE_REFL_MID; DIST_REFL; REAL_ADD_LID]; ALL_TAC] THEN
EQ_TAC THENL
[ALL_TAC;
STRIP_TAC THENL
[MP_TAC(ISPECL[`A:real^N`; `C:real^N`; `B:real^N`] ANGLE_EQ_PI_DIST);
MP_TAC(ISPECL[`B:real^N`; `A:real^N`; `C:real^N`] ANGLE_EQ_PI_DIST)] THEN
ASM_REWRITE_TAC[] THEN REWRITE_TAC[DIST_SYM; REAL_ADD_AC] THEN
DISCH_THEN(MP_TAC o MATCH_MP ANGLE_EQ_PI_OTHERS) THEN SIMP_TAC[]] THEN
ASM_REWRITE_TAC[angle; VECTOR_ANGLE_EQ_0; VECTOR_SUB_EQ] THEN
REPEAT_TCL DISJ_CASES_THEN ASSUME_TAC
(ISPECL [`norm(A - B:real^N)`; `norm(C - B:real^N)`]
REAL_LT_TOTAL)
THENL
[ASM_REWRITE_TAC[VECTOR_MUL_LCANCEL; NORM_EQ_0; VECTOR_SUB_EQ;
VECTOR_ARITH `c - b:real^N = a - b <=> a = c`];
ONCE_REWRITE_TAC[VECTOR_ARITH
`norm(A - B) % (C - B) = norm(C - B) % (A - B) <=>
(norm(C - B) - norm(A - B)) % (A - B) = norm(A - B) % (C - A)`];
ONCE_REWRITE_TAC[VECTOR_ARITH
`norm(A - B) % (C - B) = norm(C - B) % (A - B) <=>
(norm(A - B) - norm(C - B)) % (C - B) = norm(C - B) % (A - C)`]] THEN
DISCH_THEN(MP_TAC o MATCH_MP
(REWRITE_RULE[IMP_CONJ] NORM_CROSS_MULTIPLY)) THEN
ASM_SIMP_TAC[REAL_SUB_LT; NORM_POS_LT; VECTOR_SUB_EQ] THEN
SIMP_TAC[GSYM DIST_TRIANGLE_EQ] THEN SIMP_TAC[DIST_SYM]);;
let ANGLE_EQ_0_DIST_ABS = prove
(`!A B C:real^N. angle(A,B,C) = &0 <=>
~(A = B) /\ ~(C = B) /\
dist(A,C) = abs(dist(A,B) - dist(C,B))`,
REPEAT GEN_TAC THEN REWRITE_TAC[ANGLE_EQ_0_DIST] THEN
AP_TERM_TAC THEN AP_TERM_TAC THEN
MP_TAC(ISPECL [`A:real^N`; `C:real^N`] DIST_POS_LE) THEN
REWRITE_TAC[DIST_SYM] THEN REAL_ARITH_TAC);;
(* ------------------------------------------------------------------------- *)
(* Some rules for congruent triangles (not necessarily in the same real^N). *)
(* ------------------------------------------------------------------------- *)
let CONGRUENT_TRIANGLES_SSS = prove
(`!A B C:real^M A' B' C':real^N.
dist(A,B) = dist(A',B') /\
dist(B,C) = dist(B',C') /\
dist(C,A) = dist(C',A')
==> angle(A,B,C) = angle(A',B',C')`,
REPEAT GEN_TAC THEN MAP_EVERY ASM_CASES_TAC
[`dist(A':real^N,B') = &0`; `dist(B':real^N,C') = &0`] THEN
ASM_REWRITE_TAC[] THEN STRIP_TAC THEN
RULE_ASSUM_TAC(REWRITE_RULE[DIST_EQ_0]) THEN
ASM_SIMP_TAC[ANGLE_REFL_MID; ANGLE_REFL] THEN
ONCE_REWRITE_TAC[GSYM COS_ANGLE_EQ] THEN
MP_TAC(ISPECL [`B:real^M`; `A:real^M`; `C:real^M`] LAW_OF_COSINES) THEN
MP_TAC(ISPECL [`B':real^N`; `A':real^N`; `C':real^N`] LAW_OF_COSINES) THEN
REPEAT(POP_ASSUM MP_TAC) THEN REWRITE_TAC[GSYM DIST_EQ_0; DIST_SYM] THEN
CONV_TAC REAL_FIELD);;
let CONGRUENT_TRIANGLES_SAS = prove
(`!A B C:real^M A' B' C':real^N.
dist(A,B) = dist(A',B') /\
angle(A,B,C) = angle(A',B',C') /\
dist(B,C) = dist(B',C')
==> dist(A,C) = dist(A',C')`,
REPEAT STRIP_TAC THEN REWRITE_TAC[DIST_EQ] THEN
MP_TAC(ISPECL [`B:real^M`; `A:real^M`; `C:real^M`] LAW_OF_COSINES) THEN
MP_TAC(ISPECL [`B':real^N`; `A':real^N`; `C':real^N`] LAW_OF_COSINES) THEN
REPEAT(DISCH_THEN SUBST1_TAC) THEN
REPEAT BINOP_TAC THEN ASM_MESON_TAC[DIST_SYM]);;
let CONGRUENT_TRIANGLES_AAS = prove
(`!A B C:real^M A' B' C':real^N.
angle(A,B,C) = angle(A',B',C') /\
angle(B,C,A) = angle(B',C',A') /\
dist(A,B) = dist(A',B') /\
~(collinear {A,B,C})
==> dist(A,C) = dist(A',C') /\ dist(B,C) = dist(B',C')`,
REPEAT GEN_TAC THEN ASM_CASES_TAC `A:real^M = B` THENL
[FIRST_X_ASSUM SUBST_ALL_TAC THEN REWRITE_TAC[INSERT_AC; COLLINEAR_2];
ALL_TAC] THEN
DISCH_TAC THEN SUBGOAL_THEN `~(A':real^N = B')` ASSUME_TAC THENL
[ASM_MESON_TAC[DIST_EQ_0]; ALL_TAC] THEN
SUBGOAL_THEN `angle(C:real^M,A,B) = angle(C':real^N,A',B')` ASSUME_TAC THENL
[MP_TAC(ISPECL [`A:real^M`; `B:real^M`; `C:real^M`] TRIANGLE_ANGLE_SUM) THEN
MP_TAC(ISPECL [`A':real^N`; `B':real^N`; `C':real^N`]
TRIANGLE_ANGLE_SUM) THEN ASM_REWRITE_TAC[IMP_IMP] THEN
REWRITE_TAC[ANGLE_SYM] THEN REAL_ARITH_TAC;
ALL_TAC] THEN
MATCH_MP_TAC(TAUT `a /\ (a ==> b) ==> a /\ b`) THEN CONJ_TAC THENL
[MP_TAC(ISPECL [`C:real^M`; `B:real^M`; `A:real^M`] LAW_OF_SINES) THEN
MP_TAC(ISPECL [`C':real^N`; `B':real^N`; `A':real^N`] LAW_OF_SINES) THEN
SUBGOAL_THEN `~(sin(angle(B':real^N,C',A')) = &0)` MP_TAC THENL
[ASM_MESON_TAC[COLLINEAR_SIN_ANGLE_IMP; INSERT_AC];
ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[ANGLE_SYM; DIST_SYM] THEN
ASM_REWRITE_TAC[] THEN REWRITE_TAC[ANGLE_SYM; DIST_SYM] THEN
CONV_TAC REAL_FIELD];
ASM_MESON_TAC[CONGRUENT_TRIANGLES_SAS; DIST_SYM; ANGLE_SYM]]);;
let CONGRUENT_TRIANGLES_ASA = prove
(`!A B C:real^M A' B' C':real^N.
angle(A,B,C) = angle(A',B',C') /\
dist(A,B) = dist(A',B') /\
angle(B,A,C) = angle(B',A',C') /\
~(collinear {A,B,C})
==> dist(A,C) = dist(A',C')`,
REPEAT GEN_TAC THEN ASM_CASES_TAC `A:real^M = B` THENL
[FIRST_X_ASSUM SUBST_ALL_TAC THEN REWRITE_TAC[INSERT_AC; COLLINEAR_2];
ALL_TAC] THEN
REPEAT STRIP_TAC THEN SUBGOAL_THEN `~(A':real^N = B')` ASSUME_TAC THENL
[ASM_MESON_TAC[DIST_EQ_0]; ALL_TAC] THEN
MP_TAC(ISPECL [`A:real^M`; `B:real^M`; `C:real^M`] TRIANGLE_ANGLE_SUM) THEN
MP_TAC(ISPECL [`A':real^N`; `B':real^N`; `C':real^N`]
TRIANGLE_ANGLE_SUM) THEN
ASM_REWRITE_TAC[IMP_IMP] THEN
DISCH_THEN(MP_TAC o MATCH_MP (REAL_ARITH
`a + b + x = pi /\ a + b + y = pi ==> x = y`)) THEN
ASM_MESON_TAC[CONGRUENT_TRIANGLES_AAS; DIST_SYM; ANGLE_SYM]);;
(* ------------------------------------------------------------------------- *)
(* Full versions where we deduce everything from the conditions. *)
(* ------------------------------------------------------------------------- *)
let CONGRUENT_TRIANGLES_SSS_FULL = prove
(`!A B C:real^M A' B' C':real^N.
dist(A,B) = dist(A',B') /\
dist(B,C) = dist(B',C') /\
dist(C,A) = dist(C',A')
==> dist(A,B) = dist(A',B') /\
dist(B,C) = dist(B',C') /\
dist(C,A) = dist(C',A') /\
angle(A,B,C) = angle(A',B',C') /\
angle(B,C,A) = angle(B',C',A') /\
angle(C,A,B) = angle(C',A',B')`,
MESON_TAC[CONGRUENT_TRIANGLES_SSS; DIST_SYM; ANGLE_SYM]);;
let CONGRUENT_TRIANGLES_SAS_FULL = prove
(`!A B C:real^M A' B' C':real^N.
dist(A,B) = dist(A',B') /\
angle(A,B,C) = angle(A',B',C') /\
dist(B,C) = dist(B',C')
==> dist(A,B) = dist(A',B') /\
dist(B,C) = dist(B',C') /\
dist(C,A) = dist(C',A') /\
angle(A,B,C) = angle(A',B',C') /\
angle(B,C,A) = angle(B',C',A') /\
angle(C,A,B) = angle(C',A',B')`,
MESON_TAC[CONGRUENT_TRIANGLES_SSS; DIST_SYM; ANGLE_SYM;
CONGRUENT_TRIANGLES_SAS]);;
let CONGRUENT_TRIANGLES_AAS_FULL = prove
(`!A B C:real^M A' B' C':real^N.
angle(A,B,C) = angle(A',B',C') /\
angle(B,C,A) = angle(B',C',A') /\
dist(A,B) = dist(A',B') /\
~(collinear {A,B,C})
==> dist(A,B) = dist(A',B') /\
dist(B,C) = dist(B',C') /\
dist(C,A) = dist(C',A') /\
angle(A,B,C) = angle(A',B',C') /\
angle(B,C,A) = angle(B',C',A') /\
angle(C,A,B) = angle(C',A',B')`,
MESON_TAC[CONGRUENT_TRIANGLES_SSS; DIST_SYM; ANGLE_SYM;
CONGRUENT_TRIANGLES_AAS]);;
let CONGRUENT_TRIANGLES_ASA_FULL = prove
(`!A B C:real^M A' B' C':real^N.
angle(A,B,C) = angle(A',B',C') /\
dist(A,B) = dist(A',B') /\
angle(B,A,C) = angle(B',A',C') /\
~(collinear {A,B,C})
==> dist(A,B) = dist(A',B') /\
dist(B,C) = dist(B',C') /\
dist(C,A) = dist(C',A') /\
angle(A,B,C) = angle(A',B',C') /\
angle(B,C,A) = angle(B',C',A') /\
angle(C,A,B) = angle(C',A',B')`,
MESON_TAC[CONGRUENT_TRIANGLES_ASA; CONGRUENT_TRIANGLES_SAS_FULL;
DIST_SYM; ANGLE_SYM]);;
(* ------------------------------------------------------------------------- *)
(* Between-ness. *)
(* ------------------------------------------------------------------------- *)
let ANGLE_BETWEEN = prove
(`!a b x. angle(a,x,b) = pi <=> ~(x = a) /\ ~(x = b) /\ between x (a,b)`,
REPEAT GEN_TAC THEN REWRITE_TAC[between; ANGLE_EQ_PI_DIST] THEN
REWRITE_TAC[EQ_SYM_EQ]);;
let BETWEEN_ANGLE = prove
(`!a b x. between x (a,b) <=> x = a \/ x = b \/ angle(a,x,b) = pi`,
REPEAT GEN_TAC THEN REWRITE_TAC[ANGLE_BETWEEN] THEN
MESON_TAC[BETWEEN_REFL]);;
let ANGLES_ALONG_LINE = prove
(`!A B C D:real^N.
~(C = A) /\ ~(C = B) /\ between C (A,B)
==> angle(A,C,D) + angle(B,C,D) = pi`,
REPEAT GEN_TAC THEN REWRITE_TAC[GSYM ANGLE_BETWEEN] THEN
DISCH_THEN(SUBST1_TAC o MATCH_MP ANGLE_EQ_PI_LEFT) THEN REAL_ARITH_TAC);;
let ANGLES_ADD_BETWEEN = prove
(`!A B C D:real^N.
between C (A,B) /\ ~(D = A) /\ ~(D = B)
==> angle(A,D,C) + angle(C,D,B) = angle(A,D,B)`,
REPEAT GEN_TAC THEN ASM_CASES_TAC `A:real^N = B` THENL
[ASM_SIMP_TAC[BETWEEN_REFL_EQ] THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
ASM_SIMP_TAC[ANGLE_REFL_MID; REAL_ADD_LID];
ALL_TAC] THEN
ASM_CASES_TAC `C:real^N = A` THEN
ASM_SIMP_TAC[ANGLE_REFL_MID; REAL_ADD_LID] THEN
ASM_CASES_TAC `C:real^N = B` THEN
ASM_SIMP_TAC[ANGLE_REFL_MID; REAL_ADD_RID] THEN
STRIP_TAC THEN
MP_TAC(ISPECL [`A:real^N`; `B:real^N`; `C:real^N`; `D:real^N`]
ANGLES_ALONG_LINE) THEN
MP_TAC(ISPECL [`A:real^N`; `B:real^N`; `D:real^N`] TRIANGLE_ANGLE_SUM) THEN
MP_TAC(ISPECL [`A:real^N`; `C:real^N`; `D:real^N`] TRIANGLE_ANGLE_SUM) THEN
MP_TAC(ISPECL [`B:real^N`; `C:real^N`; `D:real^N`] TRIANGLE_ANGLE_SUM) THEN
ASM_REWRITE_TAC[] THEN
SUBGOAL_THEN `angle(C:real^N,A,D) = angle(B,A,D) /\
angle(A,B,D) = angle(C,B,D)`
(CONJUNCTS_THEN SUBST1_TAC)
THENL [ALL_TAC; REWRITE_TAC[ANGLE_SYM] THEN REAL_ARITH_TAC] THEN
CONJ_TAC THEN MATCH_MP_TAC ANGLE_EQ_0_RIGHT THEN
ASM_MESON_TAC[ANGLE_EQ_PI_OTHERS; BETWEEN_ANGLE]);;
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