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(* ========================================================================= *)
(* The five Platonic solids exist and there are no others. *)
(* ========================================================================= *)
needs "100/polyhedron.ml";;
needs "Multivariate/cross.ml";;
prioritize_real();;
(* ------------------------------------------------------------------------- *)
(* Some standard regular polyhedra (vertex coordinates from Wikipedia). *)
(* ------------------------------------------------------------------------- *)
let std_tetrahedron = new_definition
`std_tetrahedron =
convex hull
{vector[&1;&1;&1],vector[-- &1;-- &1;&1],
vector[-- &1;&1;-- &1],vector[&1;-- &1;-- &1]}:real^3->bool`;;
let std_cube = new_definition
`std_cube =
convex hull
{vector[&1;&1;&1],vector[&1;&1;-- &1],
vector[&1;-- &1;&1],vector[&1;-- &1;-- &1],
vector[-- &1;&1;&1],vector[-- &1;&1;-- &1],
vector[-- &1;-- &1;&1],vector[-- &1;-- &1;-- &1]}:real^3->bool`;;
let std_octahedron = new_definition
`std_octahedron =
convex hull
{vector[&1;&0;&0],vector[-- &1;&0;&0],
vector[&0;&0;&1],vector[&0;&0;-- &1],
vector[&0;&1;&0],vector[&0;-- &1;&0]}:real^3->bool`;;
let std_dodecahedron = new_definition
`std_dodecahedron =
let p = (&1 + sqrt(&5)) / &2 in
convex hull
{vector[&1;&1;&1],vector[&1;&1;-- &1],
vector[&1;-- &1;&1],vector[&1;-- &1;-- &1],
vector[-- &1;&1;&1],vector[-- &1;&1;-- &1],
vector[-- &1;-- &1;&1],vector[-- &1;-- &1;-- &1],
vector[&0;inv p;p],vector[&0;inv p;--p],
vector[&0;--inv p;p],vector[&0;--inv p;--p],
vector[inv p;p;&0],vector[inv p;--p;&0],
vector[--inv p;p;&0],vector[--inv p;--p;&0],
vector[p;&0;inv p],vector[--p;&0;inv p],
vector[p;&0;--inv p],vector[--p;&0;--inv p]}:real^3->bool`;;
let std_icosahedron = new_definition
`std_icosahedron =
let p = (&1 + sqrt(&5)) / &2 in
convex hull
{vector[&0; &1; p],vector[&0; &1; --p],
vector[&0; -- &1; p],vector[&0; -- &1; --p],
vector[&1; p; &0],vector[&1; --p; &0],
vector[-- &1; p; &0],vector[-- &1; --p; &0],
vector[p; &0; &1],vector[--p; &0; &1],
vector[p; &0; -- &1],vector[--p; &0; -- &1]}:real^3->bool`;;
(* ------------------------------------------------------------------------- *)
(* Slightly ad hoc conversions for computation in Q[sqrt(5)]. *)
(* Numbers are canonically represented as either a rational constant r or an *)
(* expression r1 + r2 * sqrt(5) where r2 is nonzero but r1 may be zero and *)
(* must be present. *)
(* ------------------------------------------------------------------------- *)
let REAL_RAT5_OF_RAT_CONV =
let pth = prove
(`p = p + &0 * sqrt(&5)`,
REAL_ARITH_TAC) in
let conv = REWR_CONV pth in
fun tm -> if is_ratconst tm then conv tm else REFL tm;;
let REAL_RAT_OF_RAT5_CONV =
let pth = prove
(`p + &0 * sqrt(&5) = p`,
REAL_ARITH_TAC) in
GEN_REWRITE_CONV TRY_CONV [pth];;
let REAL_RAT5_ADD_CONV =
let pth = prove
(`(a1 + b1 * sqrt(&5)) + (a2 + b2 * sqrt(&5)) =
(a1 + a2) + (b1 + b2) * sqrt(&5)`,
REAL_ARITH_TAC) in
REAL_RAT_ADD_CONV ORELSEC
(BINOP_CONV REAL_RAT5_OF_RAT_CONV THENC
GEN_REWRITE_CONV I [pth] THENC
LAND_CONV REAL_RAT_ADD_CONV THENC
RAND_CONV(LAND_CONV REAL_RAT_ADD_CONV) THENC
REAL_RAT_OF_RAT5_CONV);;
let REAL_RAT5_SUB_CONV =
let pth = prove
(`(a1 + b1 * sqrt(&5)) - (a2 + b2 * sqrt(&5)) =
(a1 - a2) + (b1 - b2) * sqrt(&5)`,
REAL_ARITH_TAC) in
REAL_RAT_SUB_CONV ORELSEC
(BINOP_CONV REAL_RAT5_OF_RAT_CONV THENC
GEN_REWRITE_CONV I [pth] THENC
LAND_CONV REAL_RAT_SUB_CONV THENC
RAND_CONV(LAND_CONV REAL_RAT_SUB_CONV) THENC
REAL_RAT_OF_RAT5_CONV);;
let REAL_RAT5_MUL_CONV =
let pth = prove
(`(a1 + b1 * sqrt(&5)) * (a2 + b2 * sqrt(&5)) =
(a1 * a2 + &5 * b1 * b2) + (a1 * b2 + a2 * b1) * sqrt(&5)`,
MP_TAC(ISPEC `&5` SQRT_POW_2) THEN CONV_TAC REAL_FIELD) in
REAL_RAT_MUL_CONV ORELSEC
(BINOP_CONV REAL_RAT5_OF_RAT_CONV THENC
GEN_REWRITE_CONV I [pth] THENC
LAND_CONV(COMB_CONV (RAND_CONV REAL_RAT_MUL_CONV) THENC
RAND_CONV REAL_RAT_MUL_CONV THENC
REAL_RAT_ADD_CONV) THENC
RAND_CONV(LAND_CONV
(BINOP_CONV REAL_RAT_MUL_CONV THENC REAL_RAT_ADD_CONV)) THENC
REAL_RAT_OF_RAT5_CONV);;
let REAL_RAT5_INV_CONV =
let pth = prove
(`~(a pow 2 = &5 * b pow 2)
==> inv(a + b * sqrt(&5)) =
a / (a pow 2 - &5 * b pow 2) +
--b / (a pow 2 - &5 * b pow 2) * sqrt(&5)`,
REPEAT GEN_TAC THEN
GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [GSYM REAL_SUB_0] THEN
SUBGOAL_THEN
`a pow 2 - &5 * b pow 2 = (a + b * sqrt(&5)) * (a - b * sqrt(&5))`
SUBST1_TAC THENL
[MP_TAC(SPEC `&5` SQRT_POW_2) THEN CONV_TAC REAL_FIELD;
REWRITE_TAC[REAL_ENTIRE; DE_MORGAN_THM] THEN CONV_TAC REAL_FIELD]) in
fun tm ->
try REAL_RAT_INV_CONV tm with Failure _ ->
let th1 = PART_MATCH (lhs o rand) pth tm in
let th2 = MP th1 (EQT_ELIM(REAL_RAT_REDUCE_CONV(lhand(concl th1)))) in
let th3 = CONV_RULE(funpow 2 RAND_CONV (funpow 2 LAND_CONV
REAL_RAT_NEG_CONV)) th2 in
let th4 = CONV_RULE(RAND_CONV(RAND_CONV(LAND_CONV
(RAND_CONV(LAND_CONV REAL_RAT_POW_CONV THENC
RAND_CONV(RAND_CONV REAL_RAT_POW_CONV THENC
REAL_RAT_MUL_CONV) THENC
REAL_RAT_SUB_CONV) THENC
REAL_RAT_DIV_CONV)))) th3 in
let th5 = CONV_RULE(RAND_CONV(LAND_CONV
(RAND_CONV(LAND_CONV REAL_RAT_POW_CONV THENC
RAND_CONV(RAND_CONV REAL_RAT_POW_CONV THENC
REAL_RAT_MUL_CONV) THENC
REAL_RAT_SUB_CONV) THENC
REAL_RAT_DIV_CONV))) th4 in
th5;;
let REAL_RAT5_DIV_CONV =
GEN_REWRITE_CONV I [real_div] THENC
RAND_CONV REAL_RAT5_INV_CONV THENC
REAL_RAT5_MUL_CONV;;
let REAL_RAT5_LE_CONV =
let lemma = prove
(`!x y. x <= y * sqrt(&5) <=>
x <= &0 /\ &0 <= y \/
&0 <= x /\ &0 <= y /\ x pow 2 <= &5 * y pow 2 \/
x <= &0 /\ y <= &0 /\ &5 * y pow 2 <= x pow 2`,
REPEAT GEN_TAC THEN MP_TAC(ISPEC `&5` SQRT_POW_2) THEN
REWRITE_TAC[REAL_POS] THEN DISCH_THEN(fun th ->
GEN_REWRITE_TAC (RAND_CONV o ONCE_DEPTH_CONV) [SYM th]) THEN
REWRITE_TAC[GSYM REAL_POW_MUL; GSYM REAL_LE_SQUARE_ABS] THEN
MP_TAC(ISPECL [`sqrt(&5)`; `y:real`] (CONJUNCT1 REAL_LE_MUL_EQ)) THEN
SIMP_TAC[SQRT_POS_LT; REAL_OF_NUM_LT; ARITH] THEN REAL_ARITH_TAC) in
let pth = prove
(`(a1 + b1 * sqrt(&5)) <= (a2 + b2 * sqrt(&5)) <=>
a1 <= a2 /\ b1 <= b2 \/
a2 <= a1 /\ b1 <= b2 /\ (a1 - a2) pow 2 <= &5 * (b2 - b1) pow 2 \/
a1 <= a2 /\ b2 <= b1 /\ &5 * (b2 - b1) pow 2 <= (a1 - a2) pow 2`,
REWRITE_TAC[REAL_ARITH
`a + b * x <= a' + b' * x <=> a - a' <= (b' - b) * x`] THEN
REWRITE_TAC[lemma] THEN REAL_ARITH_TAC) in
REAL_RAT_LE_CONV ORELSEC
(BINOP_CONV REAL_RAT5_OF_RAT_CONV THENC
GEN_REWRITE_CONV I [pth] THENC
REAL_RAT_REDUCE_CONV);;
let REAL_RAT5_EQ_CONV =
GEN_REWRITE_CONV I [GSYM REAL_LE_ANTISYM] THENC
BINOP_CONV REAL_RAT5_LE_CONV THENC
GEN_REWRITE_CONV I [AND_CLAUSES];;
(* ------------------------------------------------------------------------- *)
(* Conversions for operations on 3D vectors with coordinates in Q[sqrt(5)] *)
(* ------------------------------------------------------------------------- *)
let VECTOR3_SUB_CONV =
let pth = prove
(`vector[x1;x2;x3] - vector[y1;y2;y3]:real^3 =
vector[x1-y1; x2-y2; x3-y3]`,
SIMP_TAC[CART_EQ; DIMINDEX_3; FORALL_3] THEN
REWRITE_TAC[VECTOR_3; VECTOR_SUB_COMPONENT]) in
GEN_REWRITE_CONV I [pth] THENC RAND_CONV(LIST_CONV REAL_RAT5_SUB_CONV);;
let VECTOR3_CROSS_CONV =
let pth = prove
(`(vector[x1;x2;x3]) cross (vector[y1;y2;y3]) =
vector[x2 * y3 - x3 * y2; x3 * y1 - x1 * y3; x1 * y2 - x2 * y1]`,
REWRITE_TAC[cross; VECTOR_3]) in
GEN_REWRITE_CONV I [pth] THENC
RAND_CONV(LIST_CONV(BINOP_CONV REAL_RAT5_MUL_CONV THENC REAL_RAT5_SUB_CONV));;
let VECTOR3_EQ_0_CONV =
let pth = prove
(`vector[x1;x2;x3]:real^3 = vec 0 <=>
x1 = &0 /\ x2 = &0 /\ x3 = &0`,
SIMP_TAC[CART_EQ; DIMINDEX_3; FORALL_3] THEN
REWRITE_TAC[VECTOR_3; VEC_COMPONENT]) in
GEN_REWRITE_CONV I [pth] THENC
DEPTH_BINOP_CONV `(/\)` REAL_RAT5_EQ_CONV THENC
REWRITE_CONV[];;
let VECTOR3_DOT_CONV =
let pth = prove
(`(vector[x1;x2;x3]:real^3) dot (vector[y1;y2;y3]) =
x1*y1 + x2*y2 + x3*y3`,
REWRITE_TAC[DOT_3; VECTOR_3]) in
GEN_REWRITE_CONV I [pth] THENC
DEPTH_BINOP_CONV `(+):real->real->real` REAL_RAT5_MUL_CONV THENC
RAND_CONV REAL_RAT5_ADD_CONV THENC
REAL_RAT5_ADD_CONV;;
(* ------------------------------------------------------------------------- *)
(* Put any irrational coordinates in our standard form. *)
(* ------------------------------------------------------------------------- *)
let STD_DODECAHEDRON = prove
(`std_dodecahedron =
convex hull
{ vector[&1; &1; &1],
vector[&1; &1; -- &1],
vector[&1; -- &1; &1],
vector[&1; -- &1; -- &1],
vector[-- &1; &1; &1],
vector[-- &1; &1; -- &1],
vector[-- &1; -- &1; &1],
vector[-- &1; -- &1; -- &1],
vector[&0; -- &1 / &2 + &1 / &2 * sqrt(&5); &1 / &2 + &1 / &2 * sqrt(&5)],
vector[&0; -- &1 / &2 + &1 / &2 * sqrt(&5); -- &1 / &2 + -- &1 / &2 * sqrt(&5)],
vector[&0; &1 / &2 + -- &1 / &2 * sqrt(&5); &1 / &2 + &1 / &2 * sqrt(&5)],
vector[&0; &1 / &2 + -- &1 / &2 * sqrt(&5); -- &1 / &2 + -- &1 / &2 * sqrt(&5)],
vector[-- &1 / &2 + &1 / &2 * sqrt(&5); &1 / &2 + &1 / &2 * sqrt(&5); &0],
vector[-- &1 / &2 + &1 / &2 * sqrt(&5); -- &1 / &2 + -- &1 / &2 * sqrt(&5); &0],
vector[&1 / &2 + -- &1 / &2 * sqrt(&5); &1 / &2 + &1 / &2 * sqrt(&5); &0],
vector[&1 / &2 + -- &1 / &2 * sqrt(&5); -- &1 / &2 + -- &1 / &2 * sqrt(&5); &0],
vector[&1 / &2 + &1 / &2 * sqrt(&5); &0; -- &1 / &2 + &1 / &2 * sqrt(&5)],
vector[-- &1 / &2 + -- &1 / &2 * sqrt(&5); &0; -- &1 / &2 + &1 / &2 * sqrt(&5)],
vector[&1 / &2 + &1 / &2 * sqrt(&5); &0; &1 / &2 + -- &1 / &2 * sqrt(&5)],
vector[-- &1 / &2 + -- &1 / &2 * sqrt(&5); &0; &1 / &2 + -- &1 / &2 * sqrt(&5)]}`,
let golden_inverse = prove
(`inv((&1 + sqrt(&5)) / &2) = -- &1 / &2 + &1 / &2 * sqrt(&5)`,
MP_TAC(ISPEC `&5` SQRT_POW_2) THEN CONV_TAC REAL_FIELD) in
REWRITE_TAC[std_dodecahedron] THEN
CONV_TAC(ONCE_DEPTH_CONV let_CONV) THEN
REWRITE_TAC[golden_inverse] THEN
REWRITE_TAC[REAL_ARITH `(&1 + s) / &2 = &1 / &2 + &1 / &2 * s`] THEN
REWRITE_TAC[REAL_ARITH `--(a + b * sqrt(&5)) = --a + --b * sqrt(&5)`] THEN
CONV_TAC REAL_RAT_REDUCE_CONV THEN REWRITE_TAC[]);;
let STD_ICOSAHEDRON = prove
(`std_icosahedron =
convex hull
{ vector[&0; &1; &1 / &2 + &1 / &2 * sqrt(&5)],
vector[&0; &1; -- &1 / &2 + -- &1 / &2 * sqrt(&5)],
vector[&0; -- &1; &1 / &2 + &1 / &2 * sqrt(&5)],
vector[&0; -- &1; -- &1 / &2 + -- &1 / &2 * sqrt(&5)],
vector[&1; &1 / &2 + &1 / &2 * sqrt(&5); &0],
vector[&1; -- &1 / &2 + -- &1 / &2 * sqrt(&5); &0],
vector[-- &1; &1 / &2 + &1 / &2 * sqrt(&5); &0],
vector[-- &1; -- &1 / &2 + -- &1 / &2 * sqrt(&5); &0],
vector[&1 / &2 + &1 / &2 * sqrt(&5); &0; &1],
vector[-- &1 / &2 + -- &1 / &2 * sqrt(&5); &0; &1],
vector[&1 / &2 + &1 / &2 * sqrt(&5); &0; -- &1],
vector[-- &1 / &2 + -- &1 / &2 * sqrt(&5); &0; -- &1]}`,
REWRITE_TAC[std_icosahedron] THEN
CONV_TAC(ONCE_DEPTH_CONV let_CONV) THEN
REWRITE_TAC[REAL_ARITH `(&1 + s) / &2 = &1 / &2 + &1 / &2 * s`] THEN
REWRITE_TAC[REAL_ARITH `--(a + b * sqrt(&5)) = --a + --b * sqrt(&5)`] THEN
CONV_TAC REAL_RAT_REDUCE_CONV THEN REWRITE_TAC[]);;
(* ------------------------------------------------------------------------- *)
(* Explicit computation of facets. *)
(* ------------------------------------------------------------------------- *)
let COMPUTE_FACES_2 = prove
(`!f s:real^3->bool.
FINITE s
==> (f face_of (convex hull s) /\ aff_dim f = &2 <=>
?x y z. x IN s /\ y IN s /\ z IN s /\
let a = (z - x) cross (y - x) in
~(a = vec 0) /\
let b = a dot x in
((!w. w IN s ==> a dot w <= b) \/
(!w. w IN s ==> a dot w >= b)) /\
f = convex hull (s INTER {x | a dot x = b}))`,
REPEAT GEN_TAC THEN STRIP_TAC THEN EQ_TAC THENL
[STRIP_TAC THEN
SUBGOAL_THEN `?t:real^3->bool. t SUBSET s /\ f = convex hull t`
MP_TAC THENL
[MATCH_MP_TAC FACE_OF_CONVEX_HULL_SUBSET THEN
ASM_SIMP_TAC[FINITE_IMP_COMPACT];
DISCH_THEN(X_CHOOSE_THEN `t:real^3->bool` MP_TAC)] THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC SUBST_ALL_TAC) THEN
RULE_ASSUM_TAC(REWRITE_RULE[AFF_DIM_CONVEX_HULL]) THEN
MP_TAC(ISPEC `t:real^3->bool` AFFINE_BASIS_EXISTS) THEN
DISCH_THEN(X_CHOOSE_THEN `u:real^3->bool` STRIP_ASSUME_TAC) THEN
SUBGOAL_THEN `(u:real^3->bool) HAS_SIZE 3` MP_TAC THENL
[ASM_SIMP_TAC[HAS_SIZE; AFFINE_INDEPENDENT_IMP_FINITE] THEN
REWRITE_TAC[GSYM INT_OF_NUM_EQ] THEN MATCH_MP_TAC(INT_ARITH
`aff_dim(u:real^3->bool) = &2 /\ aff_dim u = &(CARD u) - &1
==> &(CARD u):int = &3`) THEN CONJ_TAC
THENL [ASM_MESON_TAC[AFF_DIM_AFFINE_HULL]; ASM_MESON_TAC[AFF_DIM_UNIQUE]];
ALL_TAC] THEN
CONV_TAC(LAND_CONV HAS_SIZE_CONV) THEN SIMP_TAC[LEFT_IMP_EXISTS_THM] THEN
MAP_EVERY X_GEN_TAC [`x:real^3`; `y:real^3`; `z:real^3`] THEN
REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
DISCH_THEN SUBST_ALL_TAC THEN
MAP_EVERY EXISTS_TAC [`x:real^3`; `y:real^3`; `z:real^3`] THEN
REPLICATE_TAC 3 (CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC]) THEN
REPEAT LET_TAC THEN
SUBGOAL_THEN `~collinear{x:real^3,y,z}` MP_TAC THENL
[ASM_REWRITE_TAC[COLLINEAR_3_EQ_AFFINE_DEPENDENT]; ALL_TAC] THEN
ONCE_REWRITE_TAC[SET_RULE `{x,y,z} = {z,x,y}`] THEN
ONCE_REWRITE_TAC[COLLINEAR_3] THEN ASM_REWRITE_TAC[GSYM CROSS_EQ_0] THEN
DISCH_TAC THEN ASM_REWRITE_TAC[] THEN
SUBGOAL_THEN `(a:real^3) dot y = b /\ (a:real^3) dot z = b`
STRIP_ASSUME_TAC THENL
[MAP_EVERY UNDISCH_TAC
[`(z - x) cross (y - x) = a`; `(a:real^3) dot x = b`] THEN VEC3_TAC;
ALL_TAC] THEN
MP_TAC(ISPECL [`convex hull s:real^3->bool`; `convex hull t:real^3->bool`]
EXPOSED_FACE_OF_POLYHEDRON) THEN
ASM_SIMP_TAC[POLYHEDRON_CONVEX_HULL; exposed_face_of] THEN
REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN
MAP_EVERY X_GEN_TAC [`a':real^3`; `b':real`] THEN
DISCH_THEN(STRIP_ASSUME_TAC o GSYM) THEN
SUBGOAL_THEN
`aff_dim(t:real^3->bool)
<= aff_dim({x:real^3 | a dot x = b} INTER {x | a' dot x = b'})`
MP_TAC THENL
[GEN_REWRITE_TAC LAND_CONV [GSYM AFF_DIM_AFFINE_HULL] THEN
FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC (LAND_CONV o RAND_CONV)
[SYM th]) THEN
REWRITE_TAC[AFF_DIM_AFFINE_HULL] THEN MATCH_MP_TAC AFF_DIM_SUBSET THEN
REWRITE_TAC[SUBSET_INTER] THEN CONJ_TAC THENL
[ASM SET_TAC[];
MATCH_MP_TAC SUBSET_TRANS THEN EXISTS_TAC `t:real^3->bool` THEN
ASM_REWRITE_TAC[] THEN MATCH_MP_TAC SUBSET_TRANS THEN
EXISTS_TAC `convex hull t:real^3->bool` THEN
REWRITE_TAC[HULL_SUBSET] THEN ASM SET_TAC[]];
ALL_TAC] THEN
ASM_SIMP_TAC[AFF_DIM_AFFINE_INTER_HYPERPLANE; AFF_DIM_HYPERPLANE;
AFFINE_HYPERPLANE; DIMINDEX_3] THEN
REPEAT(COND_CASES_TAC THEN CONV_TAC INT_REDUCE_CONV) THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I
[SUBSET_HYPERPLANES]) THEN
ASM_REWRITE_TAC[HYPERPLANE_EQ_EMPTY] THEN
DISCH_THEN(DISJ_CASES_THEN2 SUBST_ALL_TAC (MP_TAC o SYM)) THENL
[RULE_ASSUM_TAC(REWRITE_RULE[INTER_UNIV]) THEN
SUBGOAL_THEN `s SUBSET {x:real^3 | a dot x = b}` ASSUME_TAC THENL
[MATCH_MP_TAC SUBSET_TRANS THEN
EXISTS_TAC `convex hull s:real^3->bool` THEN
REWRITE_TAC[HULL_SUBSET] THEN ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC SUBSET_TRANS THEN
EXISTS_TAC `affine hull t:real^3->bool` THEN
REWRITE_TAC[CONVEX_HULL_SUBSET_AFFINE_HULL] THEN
FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC LAND_CONV [SYM th]) THEN
MATCH_MP_TAC HULL_MINIMAL THEN REWRITE_TAC[AFFINE_HYPERPLANE] THEN
ASM SET_TAC[];
ALL_TAC] THEN
CONJ_TAC THENL
[RULE_ASSUM_TAC(REWRITE_RULE[SUBSET; IN_ELIM_THM]) THEN
ASM_SIMP_TAC[real_ge; REAL_LE_REFL];
ASM_SIMP_TAC[SET_RULE `s SUBSET t ==> s INTER t = s`]];
ALL_TAC] THEN
DISCH_THEN(fun th -> SUBST_ALL_TAC th THEN ASSUME_TAC th) THEN
CONJ_TAC THENL
[MATCH_MP_TAC(TAUT `(~p /\ ~q ==> F) ==> p \/ q`) THEN
REWRITE_TAC[NOT_FORALL_THM; NOT_IMP; real_ge; REAL_NOT_LE] THEN
DISCH_THEN(CONJUNCTS_THEN2
(X_CHOOSE_TAC `u:real^3`) (X_CHOOSE_TAC `v:real^3`)) THEN
SUBGOAL_THEN `(a':real^3) dot u < b' /\ a' dot v < b'` ASSUME_TAC THENL
[REWRITE_TAC[REAL_LT_LE] THEN REWRITE_TAC
[SET_RULE `f x <= b /\ ~(f x = b) <=>
x IN {x | f x <= b} /\ ~(x IN {x | f x = b})`] THEN
ASM_REWRITE_TAC[] THEN ASM_SIMP_TAC[IN_ELIM_THM; REAL_LT_IMP_NE] THEN
SUBGOAL_THEN `(u:real^3) IN convex hull s /\ v IN convex hull s`
MP_TAC THENL [ASM_SIMP_TAC[HULL_INC]; ASM SET_TAC[]];
ALL_TAC] THEN
SUBGOAL_THEN `?w:real^3. w IN segment[u,v] /\ w IN {w | a' dot w = b'}`
MP_TAC THENL
[ASM_REWRITE_TAC[] THEN REWRITE_TAC[IN_ELIM_THM] THEN
MATCH_MP_TAC CONNECTED_IVT_HYPERPLANE THEN
MAP_EVERY EXISTS_TAC [`v:real^3`; `u:real^3`] THEN
ASM_SIMP_TAC[ENDS_IN_SEGMENT; CONNECTED_SEGMENT; REAL_LT_IMP_LE];
REWRITE_TAC[IN_SEGMENT; IN_ELIM_THM; LEFT_AND_EXISTS_THM] THEN
ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN
REWRITE_TAC[GSYM CONJ_ASSOC; RIGHT_EXISTS_AND_THM] THEN
REWRITE_TAC[UNWIND_THM2; DOT_RADD; DOT_RMUL; CONJ_ASSOC] THEN
DISCH_THEN(CHOOSE_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC MP_TAC)) THEN
MATCH_MP_TAC(REAL_ARITH `a < b ==> a = b ==> F`) THEN
MATCH_MP_TAC REAL_CONVEX_BOUND_LT THEN ASM_REAL_ARITH_TAC];
MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL
[MATCH_MP_TAC HULL_MONO THEN REWRITE_TAC[SUBSET_INTER] THEN
ASM_REWRITE_TAC[] THEN MATCH_MP_TAC SUBSET_TRANS THEN
EXISTS_TAC `convex hull t:real^3->bool` THEN
REWRITE_TAC[HULL_SUBSET] THEN ASM SET_TAC[];
FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC RAND_CONV [SYM th]) THEN
REWRITE_TAC[SUBSET_INTER] THEN
SIMP_TAC[HULL_MONO; INTER_SUBSET] THEN
ASM_REWRITE_TAC[] THEN MATCH_MP_TAC SUBSET_TRANS THEN
EXISTS_TAC `convex hull {x:real^3 | a dot x = b}` THEN
SIMP_TAC[HULL_MONO; INTER_SUBSET] THEN
MATCH_MP_TAC(SET_RULE `s = t ==> s SUBSET t`) THEN
REWRITE_TAC[CONVEX_HULL_EQ; CONVEX_HYPERPLANE]]];
REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN
MAP_EVERY X_GEN_TAC [`x:real^3`; `y:real^3`; `z:real^3`] THEN
REPEAT LET_TAC THEN
DISCH_THEN(REPEAT_TCL CONJUNCTS_THEN ASSUME_TAC) THEN CONJ_TAC THENL
[ASM_REWRITE_TAC[] THEN
SUBGOAL_THEN
`convex hull (s INTER {x:real^3 | a dot x = b}) =
(convex hull s) INTER {x | a dot x = b}`
SUBST1_TAC THENL
[MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL
[SIMP_TAC[SUBSET_INTER; HULL_MONO; INTER_SUBSET] THEN
MATCH_MP_TAC SUBSET_TRANS THEN
EXISTS_TAC `convex hull {x:real^3 | a dot x = b}` THEN
SIMP_TAC[HULL_MONO; INTER_SUBSET] THEN
MATCH_MP_TAC(SET_RULE `s = t ==> s SUBSET t`) THEN
REWRITE_TAC[CONVEX_HULL_EQ; CONVEX_HYPERPLANE];
ALL_TAC] THEN
ASM_CASES_TAC `s SUBSET {x:real^3 | a dot x = b}` THENL
[ASM_SIMP_TAC[SET_RULE `s SUBSET t ==> s INTER t = s`] THEN SET_TAC[];
ALL_TAC] THEN
MATCH_MP_TAC SUBSET_TRANS THEN EXISTS_TAC
`convex hull (convex hull (s INTER {x:real^3 | a dot x = b}) UNION
convex hull (s DIFF {x | a dot x = b})) INTER
{x | a dot x = b}` THEN
CONJ_TAC THENL
[MATCH_MP_TAC(SET_RULE
`s SUBSET t ==> (s INTER u) SUBSET (t INTER u)`) THEN
MATCH_MP_TAC HULL_MONO THEN MATCH_MP_TAC(SET_RULE
`s INTER t SUBSET (P hull (s INTER t)) /\
s DIFF t SUBSET (P hull (s DIFF t))
==> s SUBSET (P hull (s INTER t)) UNION (P hull (s DIFF t))`) THEN
REWRITE_TAC[HULL_SUBSET];
ALL_TAC] THEN
W(MP_TAC o PART_MATCH (lhs o rand) CONVEX_HULL_UNION_NONEMPTY_EXPLICIT o
lhand o lhand o snd) THEN
ANTS_TAC THENL
[SIMP_TAC[CONVEX_CONVEX_HULL; CONVEX_HULL_EQ_EMPTY] THEN ASM SET_TAC[];
DISCH_THEN SUBST1_TAC] THEN
REWRITE_TAC[SUBSET; IN_INTER; IMP_CONJ; FORALL_IN_GSPEC] THEN
MAP_EVERY X_GEN_TAC [`p:real^3`; `u:real`; `q:real^3`] THEN
REPLICATE_TAC 4 DISCH_TAC THEN ASM_CASES_TAC `u = &0` THEN
ASM_REWRITE_TAC[VECTOR_ARITH `(&1 - &0) % p + &0 % q:real^N = p`] THEN
MATCH_MP_TAC(TAUT `~p ==> p ==> q`) THEN REWRITE_TAC[IN_ELIM_THM] THEN
REWRITE_TAC[DOT_RADD; DOT_RMUL] THEN FIRST_X_ASSUM DISJ_CASES_TAC THENL
[MATCH_MP_TAC(REAL_ARITH `x < y ==> ~(x = y)`) THEN
MATCH_MP_TAC(REAL_ARITH
`(&1 - u) * p = (&1 - u) * b /\ u * q < u * b
==> (&1 - u) * p + u * q < b`) THEN
CONJ_TAC THENL
[SUBGOAL_THEN `p IN {x:real^3 | a dot x = b}` MP_TAC THENL
[FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE
`x IN s ==> s SUBSET t ==> x IN t`)) THEN
MATCH_MP_TAC HULL_MINIMAL THEN REWRITE_TAC[CONVEX_HYPERPLANE] THEN
SET_TAC[];
SIMP_TAC[IN_ELIM_THM]];
MATCH_MP_TAC REAL_LT_LMUL THEN CONJ_TAC THENL
[ASM_REAL_ARITH_TAC; ALL_TAC] THEN
ONCE_REWRITE_TAC[SET_RULE
`(a:real^3) dot q < b <=> q IN {x | a dot x < b}`] THEN
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE
`x IN s ==> s SUBSET t ==> x IN t`)) THEN
MATCH_MP_TAC HULL_MINIMAL THEN REWRITE_TAC[CONVEX_HALFSPACE_LT] THEN
ASM_SIMP_TAC[SUBSET; IN_DIFF; IN_ELIM_THM; REAL_LT_LE]];
MATCH_MP_TAC(REAL_ARITH `x > y ==> ~(x = y)`) THEN
MATCH_MP_TAC(REAL_ARITH
`(&1 - u) * p = (&1 - u) * b /\ u * b < u * q
==> (&1 - u) * p + u * q > b`) THEN
CONJ_TAC THENL
[SUBGOAL_THEN `p IN {x:real^3 | a dot x = b}` MP_TAC THENL
[FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE
`x IN s ==> s SUBSET t ==> x IN t`)) THEN
MATCH_MP_TAC HULL_MINIMAL THEN REWRITE_TAC[CONVEX_HYPERPLANE] THEN
SET_TAC[];
SIMP_TAC[IN_ELIM_THM]];
MATCH_MP_TAC REAL_LT_LMUL THEN CONJ_TAC THENL
[ASM_REAL_ARITH_TAC; REWRITE_TAC[GSYM real_gt]] THEN
ONCE_REWRITE_TAC[SET_RULE
`(a:real^3) dot q > b <=> q IN {x | a dot x > b}`] THEN
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE
`x IN s ==> s SUBSET t ==> x IN t`)) THEN
MATCH_MP_TAC HULL_MINIMAL THEN REWRITE_TAC[CONVEX_HALFSPACE_GT] THEN
RULE_ASSUM_TAC(REWRITE_RULE[real_ge]) THEN
ASM_SIMP_TAC[SUBSET; IN_DIFF; IN_ELIM_THM; real_gt; REAL_LT_LE]]];
ALL_TAC] THEN
FIRST_X_ASSUM DISJ_CASES_TAC THENL
[MATCH_MP_TAC FACE_OF_INTER_SUPPORTING_HYPERPLANE_LE THEN
REWRITE_TAC[CONVEX_CONVEX_HULL] THEN
SIMP_TAC[SET_RULE `(!x. x IN s ==> P x) <=> s SUBSET {x | P x}`] THEN
MATCH_MP_TAC HULL_MINIMAL THEN REWRITE_TAC[CONVEX_HALFSPACE_LE] THEN
ASM_SIMP_TAC[SUBSET; IN_ELIM_THM];
MATCH_MP_TAC FACE_OF_INTER_SUPPORTING_HYPERPLANE_GE THEN
REWRITE_TAC[CONVEX_CONVEX_HULL] THEN
SIMP_TAC[SET_RULE `(!x. x IN s ==> P x) <=> s SUBSET {x | P x}`] THEN
MATCH_MP_TAC HULL_MINIMAL THEN REWRITE_TAC[CONVEX_HALFSPACE_GE] THEN
ASM_SIMP_TAC[SUBSET; IN_ELIM_THM]];
REWRITE_TAC[GSYM INT_LE_ANTISYM] THEN CONJ_TAC THENL
[MATCH_MP_TAC INT_LE_TRANS THEN
EXISTS_TAC `aff_dim {x:real^3 | a dot x = b}` THEN CONJ_TAC THENL
[MATCH_MP_TAC AFF_DIM_SUBSET THEN ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC HULL_MINIMAL THEN REWRITE_TAC[CONVEX_HYPERPLANE] THEN
SET_TAC[];
ASM_SIMP_TAC[AFF_DIM_HYPERPLANE; DIMINDEX_3] THEN INT_ARITH_TAC];
MATCH_MP_TAC INT_LE_TRANS THEN EXISTS_TAC `aff_dim {x:real^3,y,z}` THEN
CONJ_TAC THENL
[SUBGOAL_THEN `~collinear{x:real^3,y,z}` MP_TAC THENL
[ONCE_REWRITE_TAC[SET_RULE `{x,y,z} = {z,x,y}`] THEN
ONCE_REWRITE_TAC[COLLINEAR_3] THEN
ASM_REWRITE_TAC[GSYM CROSS_EQ_0];
REWRITE_TAC[COLLINEAR_3_EQ_AFFINE_DEPENDENT; DE_MORGAN_THM] THEN
STRIP_TAC] THEN
ASM_SIMP_TAC[AFF_DIM_AFFINE_INDEPENDENT] THEN
SIMP_TAC[CARD_CLAUSES; FINITE_INSERT; FINITE_EMPTY] THEN
ASM_REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY; ARITH] THEN
CONV_TAC INT_REDUCE_CONV;
MATCH_MP_TAC AFF_DIM_SUBSET THEN ASM_REWRITE_TAC[INSERT_SUBSET] THEN
REWRITE_TAC[EMPTY_SUBSET] THEN REPEAT CONJ_TAC THEN
MATCH_MP_TAC HULL_INC THEN
ASM_REWRITE_TAC[IN_INTER; IN_ELIM_THM] THEN
MAP_EVERY UNDISCH_TAC
[`(z - x) cross (y - x) = a`; `(a:real^3) dot x = b`] THEN
VEC3_TAC]]]]);;
let COMPUTE_FACES_2_STEP_1 = prove
(`!f v s t:real^3->bool.
(?x y z. x IN (v INSERT s) /\ y IN (v INSERT s) /\ z IN (v INSERT s) /\
let a = (z - x) cross (y - x) in
~(a = vec 0) /\
let b = a dot x in
((!w. w IN t ==> a dot w <= b) \/
(!w. w IN t ==> a dot w >= b)) /\
f = convex hull (t INTER {x | a dot x = b})) <=>
(?y z. y IN s /\ z IN s /\
let a = (z - v) cross (y - v) in
~(a = vec 0) /\
let b = a dot v in
((!w. w IN t ==> a dot w <= b) \/
(!w. w IN t ==> a dot w >= b)) /\
f = convex hull (t INTER {x | a dot x = b})) \/
(?x y z. x IN s /\ y IN s /\ z IN s /\
let a = (z - x) cross (y - x) in
~(a = vec 0) /\
let b = a dot x in
((!w. w IN t ==> a dot w <= b) \/
(!w. w IN t ==> a dot w >= b)) /\
f = convex hull (t INTER {x | a dot x = b}))`,
REPEAT GEN_TAC THEN REWRITE_TAC[IN_INSERT] THEN MATCH_MP_TAC(MESON[]
`(!x y z. Q x y z ==> Q x z y) /\
(!x y z. Q x y z ==> Q y x z) /\
(!x z. ~(Q x x z))
==> ((?x y z. (x = v \/ P x) /\ (y = v \/ P y) /\ (z = v \/ P z) /\
Q x y z) <=>
(?y z. P y /\ P z /\ Q v y z) \/
(?x y z. P x /\ P y /\ P z /\ Q x y z))`) THEN
CONV_TAC(ONCE_DEPTH_CONV let_CONV) THEN
REWRITE_TAC[VECTOR_SUB_REFL; CROSS_0] THEN
CONJ_TAC THEN REPEAT GEN_TAC THEN
CONV_TAC(ONCE_DEPTH_CONV let_CONV) THEN
MAP_EVERY (SUBST1_TAC o VEC3_RULE)
[`(z - y) cross (x - y) = --((z - x) cross (y - x))`;
`(y - x) cross (z - x) = --((z - x) cross (y - x))`] THEN
REWRITE_TAC[VECTOR_NEG_EQ_0; DOT_LNEG; REAL_EQ_NEG2; REAL_LE_NEG2;
real_ge] THEN
REWRITE_TAC[DISJ_ACI] THEN
REWRITE_TAC[VEC3_RULE
`((z - x) cross (y - x)) dot y = ((z - x) cross (y - x)) dot x`]);;
let COMPUTE_FACES_2_STEP_2 = prove
(`!f u v s:real^3->bool.
(?y z. y IN (u INSERT s) /\ z IN (u INSERT s) /\
let a = (z - v) cross (y - v) in
~(a = vec 0) /\
let b = a dot v in
((!w. w IN t ==> a dot w <= b) \/
(!w. w IN t ==> a dot w >= b)) /\
f = convex hull (t INTER {x | a dot x = b})) <=>
(?z. z IN s /\
let a = (z - v) cross (u - v) in
~(a = vec 0) /\
let b = a dot v in
((!w. w IN t ==> a dot w <= b) \/
(!w. w IN t ==> a dot w >= b)) /\
f = convex hull (t INTER {x | a dot x = b})) \/
(?y z. y IN s /\ z IN s /\
let a = (z - v) cross (y - v) in
~(a = vec 0) /\
let b = a dot v in
((!w. w IN t ==> a dot w <= b) \/
(!w. w IN t ==> a dot w >= b)) /\
f = convex hull (t INTER {x | a dot x = b}))`,
REPEAT GEN_TAC THEN REWRITE_TAC[IN_INSERT] THEN MATCH_MP_TAC(MESON[]
`(!x y. Q x y ==> Q y x) /\
(!x. ~(Q x x))
==> ((?y z. (y = u \/ P y) /\ (z = u \/ P z) /\
Q y z) <=>
(?z. P z /\ Q u z) \/
(?y z. P y /\ P z /\ Q y z))`) THEN
CONV_TAC(ONCE_DEPTH_CONV let_CONV) THEN
REWRITE_TAC[CROSS_REFL] THEN REPEAT GEN_TAC THEN
CONV_TAC(ONCE_DEPTH_CONV let_CONV) THEN SUBST1_TAC
(VEC3_RULE `(x - v) cross (y - v) = --((y - v) cross (x - v))`) THEN
REWRITE_TAC[VECTOR_NEG_EQ_0; DOT_LNEG; REAL_EQ_NEG2; REAL_LE_NEG2;
real_ge] THEN REWRITE_TAC[DISJ_ACI]);;
let COMPUTE_FACES_TAC =
let lemma = prove
(`(x INSERT s) INTER {x | P x} =
if P x then x INSERT (s INTER {x | P x})
else s INTER {x | P x}`,
COND_CASES_TAC THEN ASM SET_TAC[]) in
SIMP_TAC[COMPUTE_FACES_2; FINITE_INSERT; FINITE_EMPTY] THEN
REWRITE_TAC[COMPUTE_FACES_2_STEP_1] THEN
REWRITE_TAC[COMPUTE_FACES_2_STEP_2] THEN
REWRITE_TAC[NOT_IN_EMPTY] THEN
REWRITE_TAC[EXISTS_IN_INSERT; NOT_IN_EMPTY] THEN
REWRITE_TAC[FORALL_IN_INSERT; NOT_IN_EMPTY] THEN
CONV_TAC(ONCE_DEPTH_CONV VECTOR3_SUB_CONV) THEN
CONV_TAC(ONCE_DEPTH_CONV VECTOR3_CROSS_CONV) THEN
CONV_TAC(ONCE_DEPTH_CONV let_CONV) THEN
CONV_TAC(ONCE_DEPTH_CONV VECTOR3_EQ_0_CONV) THEN
REWRITE_TAC[real_ge] THEN
CONV_TAC(ONCE_DEPTH_CONV VECTOR3_DOT_CONV) THEN
CONV_TAC(ONCE_DEPTH_CONV let_CONV) THEN
CONV_TAC(ONCE_DEPTH_CONV REAL_RAT5_LE_CONV) THEN
REWRITE_TAC[INSERT_AC] THEN REWRITE_TAC[DISJ_ACI] THEN
REPEAT(CHANGED_TAC
(ONCE_REWRITE_TAC[lemma] THEN
CONV_TAC(ONCE_DEPTH_CONV
(LAND_CONV VECTOR3_DOT_CONV THENC REAL_RAT5_EQ_CONV))) THEN
REWRITE_TAC[]) THEN
REWRITE_TAC[INTER_EMPTY] THEN
REWRITE_TAC[INSERT_AC] THEN REWRITE_TAC[DISJ_ACI];;
(* ------------------------------------------------------------------------- *)
(* Apply this to our standard Platonic solids to derive facets. *)
(* Note: this is quite slow and can take a couple of hours. *)
(* ------------------------------------------------------------------------- *)
let TETRAHEDRON_FACETS = time prove
(`!f:real^3->bool.
f face_of std_tetrahedron /\ aff_dim f = &2 <=>
f = convex hull {vector[-- &1; -- &1; &1], vector[-- &1; &1; -- &1], vector[&1; -- &1; -- &1]} \/
f = convex hull {vector[-- &1; -- &1; &1], vector[-- &1; &1; -- &1], vector[&1; &1; &1]} \/
f = convex hull {vector[-- &1; -- &1; &1], vector[&1; -- &1; -- &1], vector[&1; &1; &1]} \/
f = convex hull {vector[-- &1; &1; -- &1], vector[&1; -- &1; -- &1], vector[&1; &1; &1]}`,
GEN_TAC THEN REWRITE_TAC[std_tetrahedron] THEN COMPUTE_FACES_TAC);;
let CUBE_FACETS = time prove
(`!f:real^3->bool.
f face_of std_cube /\ aff_dim f = &2 <=>
f = convex hull {vector[-- &1; -- &1; -- &1], vector[-- &1; -- &1; &1], vector[-- &1; &1; -- &1], vector[-- &1; &1; &1]} \/
f = convex hull {vector[-- &1; -- &1; -- &1], vector[-- &1; -- &1; &1], vector[&1; -- &1; -- &1], vector[&1; -- &1; &1]} \/
f = convex hull {vector[-- &1; -- &1; -- &1], vector[-- &1; &1; -- &1], vector[&1; -- &1; -- &1], vector[&1; &1; -- &1]} \/
f = convex hull {vector[-- &1; -- &1; &1], vector[-- &1; &1; &1], vector[&1; -- &1; &1], vector[&1; &1; &1]} \/
f = convex hull {vector[-- &1; &1; -- &1], vector[-- &1; &1; &1], vector[&1; &1; -- &1], vector[&1; &1; &1]} \/
f = convex hull {vector[&1; -- &1; -- &1], vector[&1; -- &1; &1], vector[&1; &1; -- &1], vector[&1; &1; &1]}`,
GEN_TAC THEN REWRITE_TAC[std_cube] THEN COMPUTE_FACES_TAC);;
let OCTAHEDRON_FACETS = time prove
(`!f:real^3->bool.
f face_of std_octahedron /\ aff_dim f = &2 <=>
f = convex hull {vector[-- &1; &0; &0], vector[&0; -- &1; &0], vector[&0; &0; -- &1]} \/
f = convex hull {vector[-- &1; &0; &0], vector[&0; -- &1; &0], vector[&0; &0; &1]} \/
f = convex hull {vector[-- &1; &0; &0], vector[&0; &1; &0], vector[&0; &0; -- &1]} \/
f = convex hull {vector[-- &1; &0; &0], vector[&0; &1; &0], vector[&0; &0; &1]} \/
f = convex hull {vector[&1; &0; &0], vector[&0; -- &1; &0], vector[&0; &0; -- &1]} \/
f = convex hull {vector[&1; &0; &0], vector[&0; -- &1; &0], vector[&0; &0; &1]} \/
f = convex hull {vector[&1; &0; &0], vector[&0; &1; &0], vector[&0; &0; -- &1]} \/
f = convex hull {vector[&1; &0; &0], vector[&0; &1; &0], vector[&0; &0; &1]}`,
GEN_TAC THEN REWRITE_TAC[std_octahedron] THEN COMPUTE_FACES_TAC);;
let ICOSAHEDRON_FACETS = time prove
(`!f:real^3->bool.
f face_of std_icosahedron /\ aff_dim f = &2 <=>
f = convex hull {vector[-- &1 / &2 + -- &1 / &2 * sqrt(&5); &0; -- &1], vector[-- &1 / &2 + -- &1 / &2 * sqrt(&5); &0; &1], vector[-- &1; -- &1 / &2 + -- &1 / &2 * sqrt(&5); &0]} \/
f = convex hull {vector[-- &1 / &2 + -- &1 / &2 * sqrt(&5); &0; -- &1], vector[-- &1 / &2 + -- &1 / &2 * sqrt(&5); &0; &1], vector[-- &1; &1 / &2 + &1 / &2 * sqrt(&5); &0]} \/
f = convex hull {vector[-- &1 / &2 + -- &1 / &2 * sqrt(&5); &0; -- &1], vector[-- &1; -- &1 / &2 + -- &1 / &2 * sqrt(&5); &0], vector[&0; -- &1; -- &1 / &2 + -- &1 / &2 * sqrt(&5)]} \/
f = convex hull {vector[-- &1 / &2 + -- &1 / &2 * sqrt(&5); &0; -- &1], vector[-- &1; &1 / &2 + &1 / &2 * sqrt(&5); &0], vector[&0; &1; -- &1 / &2 + -- &1 / &2 * sqrt(&5)]} \/
f = convex hull {vector[-- &1 / &2 + -- &1 / &2 * sqrt(&5); &0; -- &1], vector[&0; -- &1; -- &1 / &2 + -- &1 / &2 * sqrt(&5)], vector[&0; &1; -- &1 / &2 + -- &1 / &2 * sqrt(&5)]} \/
f = convex hull {vector[-- &1 / &2 + -- &1 / &2 * sqrt(&5); &0; &1], vector[-- &1; -- &1 / &2 + -- &1 / &2 * sqrt(&5); &0], vector[&0; -- &1; &1 / &2 + &1 / &2 * sqrt(&5)]} \/
f = convex hull {vector[-- &1 / &2 + -- &1 / &2 * sqrt(&5); &0; &1], vector[-- &1; &1 / &2 + &1 / &2 * sqrt(&5); &0], vector[&0; &1; &1 / &2 + &1 / &2 * sqrt(&5)]} \/
f = convex hull {vector[-- &1 / &2 + -- &1 / &2 * sqrt(&5); &0; &1], vector[&0; -- &1; &1 / &2 + &1 / &2 * sqrt(&5)], vector[&0; &1; &1 / &2 + &1 / &2 * sqrt(&5)]} \/
f = convex hull {vector[&1 / &2 + &1 / &2 * sqrt(&5); &0; -- &1], vector[&1 / &2 + &1 / &2 * sqrt(&5); &0; &1], vector[&1; -- &1 / &2 + -- &1 / &2 * sqrt(&5); &0]} \/
f = convex hull {vector[&1 / &2 + &1 / &2 * sqrt(&5); &0; -- &1], vector[&1 / &2 + &1 / &2 * sqrt(&5); &0; &1], vector[&1; &1 / &2 + &1 / &2 * sqrt(&5); &0]} \/
f = convex hull {vector[&1 / &2 + &1 / &2 * sqrt(&5); &0; -- &1], vector[&1; -- &1 / &2 + -- &1 / &2 * sqrt(&5); &0], vector[&0; -- &1; -- &1 / &2 + -- &1 / &2 * sqrt(&5)]} \/
f = convex hull {vector[&1 / &2 + &1 / &2 * sqrt(&5); &0; -- &1], vector[&1; &1 / &2 + &1 / &2 * sqrt(&5); &0], vector[&0; &1; -- &1 / &2 + -- &1 / &2 * sqrt(&5)]} \/
f = convex hull {vector[&1 / &2 + &1 / &2 * sqrt(&5); &0; -- &1], vector[&0; -- &1; -- &1 / &2 + -- &1 / &2 * sqrt(&5)], vector[&0; &1; -- &1 / &2 + -- &1 / &2 * sqrt(&5)]} \/
f = convex hull {vector[&1 / &2 + &1 / &2 * sqrt(&5); &0; &1], vector[&1; -- &1 / &2 + -- &1 / &2 * sqrt(&5); &0], vector[&0; -- &1; &1 / &2 + &1 / &2 * sqrt(&5)]} \/
f = convex hull {vector[&1 / &2 + &1 / &2 * sqrt(&5); &0; &1], vector[&1; &1 / &2 + &1 / &2 * sqrt(&5); &0], vector[&0; &1; &1 / &2 + &1 / &2 * sqrt(&5)]} \/
f = convex hull {vector[&1 / &2 + &1 / &2 * sqrt(&5); &0; &1], vector[&0; -- &1; &1 / &2 + &1 / &2 * sqrt(&5)], vector[&0; &1; &1 / &2 + &1 / &2 * sqrt(&5)]} \/
f = convex hull {vector[-- &1; -- &1 / &2 + -- &1 / &2 * sqrt(&5); &0], vector[&1; -- &1 / &2 + -- &1 / &2 * sqrt(&5); &0], vector[&0; -- &1; -- &1 / &2 + -- &1 / &2 * sqrt(&5)]} \/
f = convex hull {vector[-- &1; -- &1 / &2 + -- &1 / &2 * sqrt(&5); &0], vector[&1; -- &1 / &2 + -- &1 / &2 * sqrt(&5); &0], vector[&0; -- &1; &1 / &2 + &1 / &2 * sqrt(&5)]} \/
f = convex hull {vector[-- &1; &1 / &2 + &1 / &2 * sqrt(&5); &0], vector[&1; &1 / &2 + &1 / &2 * sqrt(&5); &0], vector[&0; &1; -- &1 / &2 + -- &1 / &2 * sqrt(&5)]} \/
f = convex hull {vector[-- &1; &1 / &2 + &1 / &2 * sqrt(&5); &0], vector[&1; &1 / &2 + &1 / &2 * sqrt(&5); &0], vector[&0; &1; &1 / &2 + &1 / &2 * sqrt(&5)]}`,
GEN_TAC THEN REWRITE_TAC[STD_ICOSAHEDRON] THEN COMPUTE_FACES_TAC);;
let DODECAHEDRON_FACETS = time prove
(`!f:real^3->bool.
f face_of std_dodecahedron /\ aff_dim f = &2 <=>
f = convex hull {vector[-- &1 / &2 + -- &1 / &2 * sqrt(&5); &0; -- &1 / &2 + &1 / &2 * sqrt(&5)], vector[-- &1 / &2 + -- &1 / &2 * sqrt(&5); &0; &1 / &2 + -- &1 / &2 * sqrt(&5)], vector[&1 / &2 + -- &1 / &2 * sqrt(&5); -- &1 / &2 + -- &1 / &2 * sqrt(&5); &0], vector[-- &1; -- &1; -- &1], vector[-- &1; -- &1; &1]} \/
f = convex hull {vector[-- &1 / &2 + -- &1 / &2 * sqrt(&5); &0; -- &1 / &2 + &1 / &2 * sqrt(&5)], vector[-- &1 / &2 + -- &1 / &2 * sqrt(&5); &0; &1 / &2 + -- &1 / &2 * sqrt(&5)], vector[&1 / &2 + -- &1 / &2 * sqrt(&5); &1 / &2 + &1 / &2 * sqrt(&5); &0], vector[-- &1; &1; -- &1], vector[-- &1; &1; &1]} \/
f = convex hull {vector[-- &1 / &2 + -- &1 / &2 * sqrt(&5); &0; -- &1 / &2 + &1 / &2 * sqrt(&5)], vector[-- &1; -- &1; &1], vector[-- &1; &1; &1], vector[&0; -- &1 / &2 + &1 / &2 * sqrt(&5); &1 / &2 + &1 / &2 * sqrt(&5)], vector[&0; &1 / &2 + -- &1 / &2 * sqrt(&5); &1 / &2 + &1 / &2 * sqrt(&5)]} \/
f = convex hull {vector[-- &1 / &2 + -- &1 / &2 * sqrt(&5); &0; &1 / &2 + -- &1 / &2 * sqrt(&5)], vector[-- &1; -- &1; -- &1], vector[-- &1; &1; -- &1], vector[&0; -- &1 / &2 + &1 / &2 * sqrt(&5); -- &1 / &2 + -- &1 / &2 * sqrt(&5)], vector[&0; &1 / &2 + -- &1 / &2 * sqrt(&5); -- &1 / &2 + -- &1 / &2 * sqrt(&5)]} \/
f = convex hull {vector[-- &1 / &2 + &1 / &2 * sqrt(&5); -- &1 / &2 + -- &1 / &2 * sqrt(&5); &0], vector[&1 / &2 + -- &1 / &2 * sqrt(&5); -- &1 / &2 + -- &1 / &2 * sqrt(&5); &0], vector[-- &1; -- &1; -- &1], vector[&1; -- &1; -- &1], vector[&0; &1 / &2 + -- &1 / &2 * sqrt(&5); -- &1 / &2 + -- &1 / &2 * sqrt(&5)]} \/
f = convex hull {vector[-- &1 / &2 + &1 / &2 * sqrt(&5); -- &1 / &2 + -- &1 / &2 * sqrt(&5); &0], vector[&1 / &2 + -- &1 / &2 * sqrt(&5); -- &1 / &2 + -- &1 / &2 * sqrt(&5); &0], vector[-- &1; -- &1; &1], vector[&1; -- &1; &1], vector[&0; &1 / &2 + -- &1 / &2 * sqrt(&5); &1 / &2 + &1 / &2 * sqrt(&5)]} \/
f = convex hull {vector[-- &1 / &2 + &1 / &2 * sqrt(&5); -- &1 / &2 + -- &1 / &2 * sqrt(&5); &0], vector[&1 / &2 + &1 / &2 * sqrt(&5); &0; -- &1 / &2 + &1 / &2 * sqrt(&5)], vector[&1 / &2 + &1 / &2 * sqrt(&5); &0; &1 / &2 + -- &1 / &2 * sqrt(&5)], vector[&1; -- &1; -- &1], vector[&1; -- &1; &1]} \/
f = convex hull {vector[-- &1 / &2 + &1 / &2 * sqrt(&5); &1 / &2 + &1 / &2 * sqrt(&5); &0], vector[&1 / &2 + -- &1 / &2 * sqrt(&5); &1 / &2 + &1 / &2 * sqrt(&5); &0], vector[-- &1; &1; -- &1], vector[&1; &1; -- &1], vector[&0; -- &1 / &2 + &1 / &2 * sqrt(&5); -- &1 / &2 + -- &1 / &2 * sqrt(&5)]} \/
f = convex hull {vector[-- &1 / &2 + &1 / &2 * sqrt(&5); &1 / &2 + &1 / &2 * sqrt(&5); &0], vector[&1 / &2 + -- &1 / &2 * sqrt(&5); &1 / &2 + &1 / &2 * sqrt(&5); &0], vector[-- &1; &1; &1], vector[&1; &1; &1], vector[&0; -- &1 / &2 + &1 / &2 * sqrt(&5); &1 / &2 + &1 / &2 * sqrt(&5)]} \/
f = convex hull {vector[-- &1 / &2 + &1 / &2 * sqrt(&5); &1 / &2 + &1 / &2 * sqrt(&5); &0], vector[&1 / &2 + &1 / &2 * sqrt(&5); &0; -- &1 / &2 + &1 / &2 * sqrt(&5)], vector[&1 / &2 + &1 / &2 * sqrt(&5); &0; &1 / &2 + -- &1 / &2 * sqrt(&5)], vector[&1; &1; -- &1], vector[&1; &1; &1]} \/
f = convex hull {vector[&1 / &2 + &1 / &2 * sqrt(&5); &0; -- &1 / &2 + &1 / &2 * sqrt(&5)], vector[&1; -- &1; &1], vector[&1; &1; &1], vector[&0; -- &1 / &2 + &1 / &2 * sqrt(&5); &1 / &2 + &1 / &2 * sqrt(&5)], vector[&0; &1 / &2 + -- &1 / &2 * sqrt(&5); &1 / &2 + &1 / &2 * sqrt(&5)]} \/
f = convex hull {vector[&1 / &2 + &1 / &2 * sqrt(&5); &0; &1 / &2 + -- &1 / &2 * sqrt(&5)], vector[&1; -- &1; -- &1], vector[&1; &1; -- &1], vector[&0; -- &1 / &2 + &1 / &2 * sqrt(&5); -- &1 / &2 + -- &1 / &2 * sqrt(&5)], vector[&0; &1 / &2 + -- &1 / &2 * sqrt(&5); -- &1 / &2 + -- &1 / &2 * sqrt(&5)]}`,
GEN_TAC THEN REWRITE_TAC[STD_DODECAHEDRON] THEN COMPUTE_FACES_TAC);;
(* ------------------------------------------------------------------------- *)
(* Given a coplanar set, return a hyperplane containing it. *)
(* Maps term s to theorem |- !x. x IN s ==> n dot x = d *)
(* Currently assumes |s| >= 3 but it would be trivial to do other cases. *)
(* ------------------------------------------------------------------------- *)
let COPLANAR_HYPERPLANE_RULE =
let rec allsets m l =
if m = 0 then [[]] else
match l with
[] -> []
| h::t -> map (fun g -> h::g) (allsets (m - 1) t) @ allsets m t in
let mk_sub = mk_binop `(-):real^3->real^3->real^3`
and mk_cross = mk_binop `cross`
and mk_dot = mk_binop `(dot):real^3->real^3->real`
and zerovec_tm = `vector[&0;&0;&0]:real^3`
and template = `(!x:real^3. x IN s ==> n dot x = d)`
and s_tm = `s:real^3->bool`
and n_tm = `n:real^3`
and d_tm = `d:real` in
let mk_normal [x;y;z] = mk_cross (mk_sub y x) (mk_sub z x) in
let eval_normal t =
(BINOP_CONV VECTOR3_SUB_CONV THENC VECTOR3_CROSS_CONV) (mk_normal t) in
let check_normal t =
let th = eval_normal t in
let n = rand(concl th) in
if n = zerovec_tm then failwith "check_normal" else n in
fun tm ->
let s = dest_setenum tm in
if length s < 3 then failwith "COPLANAR_HYPERPLANE_RULE: trivial" else
let n = tryfind check_normal (allsets 3 s) in
let d = rand(concl(VECTOR3_DOT_CONV(mk_dot n (hd s)))) in
let ptm = vsubst [tm,s_tm; n,n_tm; d,d_tm] template in
EQT_ELIM
((REWRITE_CONV[FORALL_IN_INSERT; NOT_IN_EMPTY] THENC
DEPTH_BINOP_CONV `/\`
(LAND_CONV VECTOR3_DOT_CONV THENC REAL_RAT5_EQ_CONV) THENC
GEN_REWRITE_CONV DEPTH_CONV [AND_CLAUSES]) ptm);;
(* ------------------------------------------------------------------------- *)
(* Explicit computation of edges, assuming hyperplane containing the set. *)
(* ------------------------------------------------------------------------- *)
let COMPUTE_FACES_1 = prove
(`!s:real^3->bool n d.
(!x. x IN s ==> n dot x = d)
==> FINITE s /\ ~(n = vec 0)
==> !f. f face_of (convex hull s) /\ aff_dim f = &1 <=>
?x y. x IN s /\ y IN s /\
let a = n cross (y - x) in
~(a = vec 0) /\
let b = a dot x in
((!w. w IN s ==> a dot w <= b) \/
(!w. w IN s ==> a dot w >= b)) /\
f = convex hull (s INTER {x | a dot x = b})`,
REPEAT GEN_TAC THEN STRIP_TAC THEN STRIP_TAC THEN GEN_TAC THEN EQ_TAC THENL
[STRIP_TAC THEN
SUBGOAL_THEN `?t:real^3->bool. t SUBSET s /\ f = convex hull t`
MP_TAC THENL
[MATCH_MP_TAC FACE_OF_CONVEX_HULL_SUBSET THEN
ASM_SIMP_TAC[FINITE_IMP_COMPACT];
DISCH_THEN(X_CHOOSE_THEN `t:real^3->bool` MP_TAC)] THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC SUBST_ALL_TAC) THEN
RULE_ASSUM_TAC(REWRITE_RULE[AFF_DIM_CONVEX_HULL]) THEN
MP_TAC(ISPEC `t:real^3->bool` AFFINE_BASIS_EXISTS) THEN
DISCH_THEN(X_CHOOSE_THEN `u:real^3->bool` STRIP_ASSUME_TAC) THEN
SUBGOAL_THEN `(u:real^3->bool) HAS_SIZE 2` MP_TAC THENL
[ASM_SIMP_TAC[HAS_SIZE; AFFINE_INDEPENDENT_IMP_FINITE] THEN
REWRITE_TAC[GSYM INT_OF_NUM_EQ] THEN MATCH_MP_TAC(INT_ARITH
`aff_dim(u:real^3->bool) = &1 /\ aff_dim u = &(CARD u) - &1
==> &(CARD u):int = &2`) THEN CONJ_TAC
THENL [ASM_MESON_TAC[AFF_DIM_AFFINE_HULL]; ASM_MESON_TAC[AFF_DIM_UNIQUE]];
ALL_TAC] THEN
CONV_TAC(LAND_CONV HAS_SIZE_CONV) THEN SIMP_TAC[LEFT_IMP_EXISTS_THM] THEN
MAP_EVERY X_GEN_TAC [`x:real^3`; `y:real^3`] THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC SUBST_ALL_TAC) THEN
MAP_EVERY EXISTS_TAC [`x:real^3`; `y:real^3`] THEN
REPLICATE_TAC 2 (CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC]) THEN
SUBGOAL_THEN `(x:real^3) IN s /\ y IN s` STRIP_ASSUME_TAC THENL
[ASM SET_TAC[]; ALL_TAC] THEN
REPEAT LET_TAC THEN
MP_TAC(ISPECL [`n:real^3`; `y - x:real^3`] NORM_AND_CROSS_EQ_0) THEN
ASM_SIMP_TAC[DOT_RSUB; VECTOR_SUB_EQ; REAL_SUB_0] THEN DISCH_TAC THEN
SUBGOAL_THEN `(a:real^3) dot y = b` ASSUME_TAC THENL
[MAP_EVERY UNDISCH_TAC
[`n cross (y - x) = a`; `(a:real^3) dot x = b`] THEN VEC3_TAC;
ALL_TAC] THEN
MP_TAC(ISPECL [`convex hull s:real^3->bool`; `convex hull t:real^3->bool`]
EXPOSED_FACE_OF_POLYHEDRON) THEN
ASM_SIMP_TAC[POLYHEDRON_CONVEX_HULL; EXPOSED_FACE_OF_PARALLEL] THEN
REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN
MAP_EVERY X_GEN_TAC [`a':real^3`; `b':real`] THEN
SUBGOAL_THEN `~(convex hull t:real^3->bool = {})` ASSUME_TAC THENL
[REWRITE_TAC[GSYM MEMBER_NOT_EMPTY] THEN EXISTS_TAC `x:real^3` THEN
MATCH_MP_TAC HULL_INC THEN ASM SET_TAC[];
ASM_REWRITE_TAC[]] THEN
ASM_CASES_TAC `convex hull t:real^3->bool = convex hull s` THEN
ASM_REWRITE_TAC[] THENL
[FIRST_X_ASSUM(ASSUME_TAC o GEN_REWRITE_RULE RAND_CONV
[GSYM AFFINE_HULL_CONVEX_HULL]) THEN
UNDISCH_THEN `convex hull t:real^3->bool = convex hull s`
(fun th -> SUBST_ALL_TAC th THEN ASSUME_TAC th) THEN
RULE_ASSUM_TAC(REWRITE_RULE[AFFINE_HULL_CONVEX_HULL]) THEN
REWRITE_TAC[SET_RULE `s = s INTER t <=> s SUBSET t`] THEN STRIP_TAC THEN
SUBGOAL_THEN `s SUBSET {x:real^3 | a dot x = b}` ASSUME_TAC THENL
[MATCH_MP_TAC SUBSET_TRANS THEN
EXISTS_TAC `affine hull s:real^3->bool` THEN
REWRITE_TAC[HULL_SUBSET] THEN
FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC LAND_CONV [SYM th]) THEN
MATCH_MP_TAC HULL_MINIMAL THEN REWRITE_TAC[AFFINE_HYPERPLANE] THEN
ASM SET_TAC[];
CONJ_TAC THENL
[RULE_ASSUM_TAC(REWRITE_RULE[SUBSET; IN_ELIM_THM]) THEN
ASM_SIMP_TAC[real_ge; REAL_LE_REFL];
AP_TERM_TAC THEN ASM SET_TAC[]]];
STRIP_TAC] THEN
RULE_ASSUM_TAC(REWRITE_RULE[AFFINE_HULL_CONVEX_HULL]) THEN
SUBGOAL_THEN
`aff_dim(t:real^3->bool)
<= aff_dim(({x:real^3 | a dot x = b} INTER {x:real^3 | a' dot x = b'})
INTER {x | n dot x = d})`
MP_TAC THENL
[GEN_REWRITE_TAC LAND_CONV [GSYM AFF_DIM_AFFINE_HULL] THEN
FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC (LAND_CONV o RAND_CONV)
[SYM th]) THEN
REWRITE_TAC[AFF_DIM_AFFINE_HULL] THEN MATCH_MP_TAC AFF_DIM_SUBSET THEN
REWRITE_TAC[SUBSET_INTER; INSERT_SUBSET; EMPTY_SUBSET; IN_ELIM_THM] THEN
ASM_SIMP_TAC[] THEN
SUBGOAL_THEN `(x:real^3) IN convex hull t /\ y IN convex hull t`
MP_TAC THENL
[CONJ_TAC THEN MATCH_MP_TAC HULL_INC THEN ASM SET_TAC[];
ASM SET_TAC[]];
ALL_TAC] THEN
ASM_SIMP_TAC[AFF_DIM_AFFINE_INTER_HYPERPLANE; AFF_DIM_HYPERPLANE;
AFFINE_HYPERPLANE; DIMINDEX_3; AFFINE_INTER] THEN
ASM_CASES_TAC `{x:real^3 | a dot x = b} SUBSET {v | a' dot v = b'}` THEN
ASM_REWRITE_TAC[] THENL
[ALL_TAC;
REPEAT(COND_CASES_TAC THEN CONV_TAC INT_REDUCE_CONV) THEN
FIRST_X_ASSUM(MP_TAC o MATCH_MP (SET_RULE
`s INTER t SUBSET u ==> !x. x IN s /\ x IN t ==> x IN u`)) THEN
DISCH_THEN(MP_TAC o SPEC `x + n:real^3`) THEN
MATCH_MP_TAC(TAUT `p /\ q /\ ~r ==> (p /\ q ==> r) ==> s`) THEN
ASM_SIMP_TAC[IN_ELIM_THM; DOT_RADD] THEN REPEAT CONJ_TAC THENL
[EXPAND_TAC "a" THEN VEC3_TAC;
ALL_TAC;
ASM_REWRITE_TAC[REAL_EQ_ADD_LCANCEL_0; DOT_EQ_0]] THEN
SUBGOAL_THEN `a' dot (x:real^3) = b'` SUBST1_TAC THENL
[SUBGOAL_THEN `(x:real^3) IN convex hull t` MP_TAC THENL
[MATCH_MP_TAC HULL_INC THEN ASM SET_TAC[]; ASM SET_TAC[]];
ALL_TAC] THEN
SUBGOAL_THEN `(n:real^3) dot (x + a') = n dot x` MP_TAC THENL
[ALL_TAC;
SIMP_TAC[DOT_RADD] THEN REWRITE_TAC[DOT_SYM] THEN REAL_ARITH_TAC] THEN
MATCH_MP_TAC(REAL_ARITH `x:real = d /\ y = d ==> x = y`) THEN
SUBGOAL_THEN
`affine hull s SUBSET {x:real^3 | n dot x = d}`
MP_TAC THENL
[MATCH_MP_TAC HULL_MINIMAL THEN REWRITE_TAC[AFFINE_HYPERPLANE] THEN
ASM_SIMP_TAC[SUBSET; IN_ELIM_THM];
REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN ASM_SIMP_TAC[HULL_INC]]] THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [SUBSET_HYPERPLANES]) THEN
ASM_REWRITE_TAC[HYPERPLANE_EQ_EMPTY; HYPERPLANE_EQ_UNIV] THEN
DISCH_THEN(fun th -> DISCH_THEN(K ALL_TAC) THEN MP_TAC(SYM th)) THEN
DISCH_THEN(fun th -> SUBST_ALL_TAC th THEN ASSUME_TAC th) THEN
CONJ_TAC THENL
[MATCH_MP_TAC(TAUT `(~p /\ ~q ==> F) ==> p \/ q`) THEN
REWRITE_TAC[NOT_FORALL_THM; NOT_IMP; real_ge; REAL_NOT_LE] THEN
DISCH_THEN(CONJUNCTS_THEN2
(X_CHOOSE_TAC `u:real^3`) (X_CHOOSE_TAC `v:real^3`)) THEN
SUBGOAL_THEN `(a':real^3) dot u < b' /\ a' dot v < b'` ASSUME_TAC THENL
[REWRITE_TAC[REAL_LT_LE] THEN REWRITE_TAC
[SET_RULE `f x <= b /\ ~(f x = b) <=>
x IN {x | f x <= b} /\ ~(x IN {x | f x = b})`] THEN
ASM_REWRITE_TAC[] THEN ASM_SIMP_TAC[IN_ELIM_THM; REAL_LT_IMP_NE] THEN
SUBGOAL_THEN `(u:real^3) IN convex hull s /\ v IN convex hull s`
MP_TAC THENL [ASM_SIMP_TAC[HULL_INC]; ASM SET_TAC[]];
ALL_TAC] THEN
SUBGOAL_THEN `?w:real^3. w IN segment[u,v] /\ w IN {w | a' dot w = b'}`
MP_TAC THENL
[ASM_REWRITE_TAC[] THEN REWRITE_TAC[IN_ELIM_THM] THEN
MATCH_MP_TAC CONNECTED_IVT_HYPERPLANE THEN
MAP_EVERY EXISTS_TAC [`v:real^3`; `u:real^3`] THEN
ASM_SIMP_TAC[ENDS_IN_SEGMENT; CONNECTED_SEGMENT; REAL_LT_IMP_LE];
REWRITE_TAC[IN_SEGMENT; IN_ELIM_THM; LEFT_AND_EXISTS_THM] THEN
ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN
REWRITE_TAC[GSYM CONJ_ASSOC; RIGHT_EXISTS_AND_THM] THEN
REWRITE_TAC[UNWIND_THM2; DOT_RADD; DOT_RMUL; CONJ_ASSOC] THEN
DISCH_THEN(CHOOSE_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC MP_TAC)) THEN
MATCH_MP_TAC(REAL_ARITH `a < b ==> a = b ==> F`) THEN
MATCH_MP_TAC REAL_CONVEX_BOUND_LT THEN ASM_REAL_ARITH_TAC];
FIRST_ASSUM(fun th -> GEN_REWRITE_TAC LAND_CONV [SYM th]) THEN
MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL
[MATCH_MP_TAC HULL_MONO THEN REWRITE_TAC[SUBSET_INTER] THEN
ASM_REWRITE_TAC[] THEN MATCH_MP_TAC SUBSET_TRANS THEN
EXISTS_TAC `convex hull t:real^3->bool` THEN
REWRITE_TAC[HULL_SUBSET] THEN ASM SET_TAC[];
ASM_REWRITE_TAC[SUBSET_INTER] THEN
SIMP_TAC[HULL_MONO; INTER_SUBSET] THEN
ASM_REWRITE_TAC[] THEN MATCH_MP_TAC SUBSET_TRANS THEN
EXISTS_TAC `convex hull {x:real^3 | a dot x = b}` THEN
SIMP_TAC[HULL_MONO; INTER_SUBSET] THEN
MATCH_MP_TAC(SET_RULE `s = t ==> s SUBSET t`) THEN
REWRITE_TAC[CONVEX_HULL_EQ; CONVEX_HYPERPLANE]]];
REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN
MAP_EVERY X_GEN_TAC [`x:real^3`; `y:real^3`] THEN
REPEAT LET_TAC THEN
DISCH_THEN(REPEAT_TCL CONJUNCTS_THEN ASSUME_TAC) THEN CONJ_TAC THENL
[ASM_REWRITE_TAC[] THEN
SUBGOAL_THEN
`convex hull (s INTER {x:real^3 | a dot x = b}) =
(convex hull s) INTER {x | a dot x = b}`
SUBST1_TAC THENL
[MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL
[SIMP_TAC[SUBSET_INTER; HULL_MONO; INTER_SUBSET] THEN
MATCH_MP_TAC SUBSET_TRANS THEN
EXISTS_TAC `convex hull {x:real^3 | a dot x = b}` THEN
SIMP_TAC[HULL_MONO; INTER_SUBSET] THEN
MATCH_MP_TAC(SET_RULE `s = t ==> s SUBSET t`) THEN
REWRITE_TAC[CONVEX_HULL_EQ; CONVEX_HYPERPLANE];
ALL_TAC] THEN
ASM_CASES_TAC `s SUBSET {x:real^3 | a dot x = b}` THENL
[ASM_SIMP_TAC[SET_RULE `s SUBSET t ==> s INTER t = s`] THEN SET_TAC[];
ALL_TAC] THEN
MATCH_MP_TAC SUBSET_TRANS THEN EXISTS_TAC
`convex hull (convex hull (s INTER {x:real^3 | a dot x = b}) UNION
convex hull (s DIFF {x | a dot x = b})) INTER
{x | a dot x = b}` THEN
CONJ_TAC THENL
[MATCH_MP_TAC(SET_RULE
`s SUBSET t ==> (s INTER u) SUBSET (t INTER u)`) THEN
MATCH_MP_TAC HULL_MONO THEN MATCH_MP_TAC(SET_RULE
`s INTER t SUBSET (P hull (s INTER t)) /\
s DIFF t SUBSET (P hull (s DIFF t))
==> s SUBSET (P hull (s INTER t)) UNION (P hull (s DIFF t))`) THEN
REWRITE_TAC[HULL_SUBSET];
ALL_TAC] THEN
W(MP_TAC o PART_MATCH (lhs o rand) CONVEX_HULL_UNION_NONEMPTY_EXPLICIT o
lhand o lhand o snd) THEN
ANTS_TAC THENL
[SIMP_TAC[CONVEX_CONVEX_HULL; CONVEX_HULL_EQ_EMPTY] THEN ASM SET_TAC[];
DISCH_THEN SUBST1_TAC] THEN
REWRITE_TAC[SUBSET; IN_INTER; IMP_CONJ; FORALL_IN_GSPEC] THEN
MAP_EVERY X_GEN_TAC [`p:real^3`; `u:real`; `q:real^3`] THEN
REPLICATE_TAC 4 DISCH_TAC THEN ASM_CASES_TAC `u = &0` THEN
ASM_REWRITE_TAC[VECTOR_ARITH `(&1 - &0) % p + &0 % q:real^N = p`] THEN
MATCH_MP_TAC(TAUT `~p ==> p ==> q`) THEN REWRITE_TAC[IN_ELIM_THM] THEN
REWRITE_TAC[DOT_RADD; DOT_RMUL] THEN FIRST_X_ASSUM DISJ_CASES_TAC THENL
[MATCH_MP_TAC(REAL_ARITH `x < y ==> ~(x = y)`) THEN
MATCH_MP_TAC(REAL_ARITH
`(&1 - u) * p = (&1 - u) * b /\ u * q < u * b
==> (&1 - u) * p + u * q < b`) THEN
CONJ_TAC THENL
[SUBGOAL_THEN `p IN {x:real^3 | a dot x = b}` MP_TAC THENL
[FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE
`x IN s ==> s SUBSET t ==> x IN t`)) THEN
MATCH_MP_TAC HULL_MINIMAL THEN REWRITE_TAC[CONVEX_HYPERPLANE] THEN
SET_TAC[];
SIMP_TAC[IN_ELIM_THM]];
MATCH_MP_TAC REAL_LT_LMUL THEN CONJ_TAC THENL
[ASM_REAL_ARITH_TAC; ALL_TAC] THEN
ONCE_REWRITE_TAC[SET_RULE
`(a:real^3) dot q < b <=> q IN {x | a dot x < b}`] THEN
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE
`x IN s ==> s SUBSET t ==> x IN t`)) THEN
MATCH_MP_TAC HULL_MINIMAL THEN REWRITE_TAC[CONVEX_HALFSPACE_LT] THEN
ASM_SIMP_TAC[SUBSET; IN_DIFF; IN_ELIM_THM; REAL_LT_LE]];
MATCH_MP_TAC(REAL_ARITH `x > y ==> ~(x = y)`) THEN
MATCH_MP_TAC(REAL_ARITH
`(&1 - u) * p = (&1 - u) * b /\ u * b < u * q
==> (&1 - u) * p + u * q > b`) THEN
CONJ_TAC THENL
[SUBGOAL_THEN `p IN {x:real^3 | a dot x = b}` MP_TAC THENL
[FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE
`x IN s ==> s SUBSET t ==> x IN t`)) THEN
MATCH_MP_TAC HULL_MINIMAL THEN REWRITE_TAC[CONVEX_HYPERPLANE] THEN
SET_TAC[];
SIMP_TAC[IN_ELIM_THM]];
MATCH_MP_TAC REAL_LT_LMUL THEN CONJ_TAC THENL
[ASM_REAL_ARITH_TAC; REWRITE_TAC[GSYM real_gt]] THEN
ONCE_REWRITE_TAC[SET_RULE
`(a:real^3) dot q > b <=> q IN {x | a dot x > b}`] THEN
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE
`x IN s ==> s SUBSET t ==> x IN t`)) THEN
MATCH_MP_TAC HULL_MINIMAL THEN REWRITE_TAC[CONVEX_HALFSPACE_GT] THEN
RULE_ASSUM_TAC(REWRITE_RULE[real_ge]) THEN
ASM_SIMP_TAC[SUBSET; IN_DIFF; IN_ELIM_THM; real_gt; REAL_LT_LE]]];
ALL_TAC] THEN
FIRST_X_ASSUM DISJ_CASES_TAC THENL
[MATCH_MP_TAC FACE_OF_INTER_SUPPORTING_HYPERPLANE_LE THEN
REWRITE_TAC[CONVEX_CONVEX_HULL] THEN
SIMP_TAC[SET_RULE `(!x. x IN s ==> P x) <=> s SUBSET {x | P x}`] THEN
MATCH_MP_TAC HULL_MINIMAL THEN REWRITE_TAC[CONVEX_HALFSPACE_LE] THEN
ASM_SIMP_TAC[SUBSET; IN_ELIM_THM];
MATCH_MP_TAC FACE_OF_INTER_SUPPORTING_HYPERPLANE_GE THEN
REWRITE_TAC[CONVEX_CONVEX_HULL] THEN
SIMP_TAC[SET_RULE `(!x. x IN s ==> P x) <=> s SUBSET {x | P x}`] THEN
MATCH_MP_TAC HULL_MINIMAL THEN REWRITE_TAC[CONVEX_HALFSPACE_GE] THEN
ASM_SIMP_TAC[SUBSET; IN_ELIM_THM]];
ASM_REWRITE_TAC[GSYM INT_LE_ANTISYM] THEN CONJ_TAC THENL
[ALL_TAC;
MATCH_MP_TAC INT_LE_TRANS THEN EXISTS_TAC `aff_dim{x:real^3,y}` THEN
CONJ_TAC THENL
[ASM_REWRITE_TAC[AFF_DIM_2] THEN
ASM_MESON_TAC[CROSS_0; VECTOR_SUB_REFL; INT_LE_REFL];
MATCH_MP_TAC AFF_DIM_SUBSET THEN
REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET] THEN
CONJ_TAC THEN MATCH_MP_TAC HULL_INC THEN
ASM_REWRITE_TAC[IN_INTER; IN_ELIM_THM] THEN
MAP_EVERY UNDISCH_TAC
[`n cross (y - x) = a`; `(a:real^3) dot x = b`] THEN
VEC3_TAC]] THEN
REWRITE_TAC[AFF_DIM_CONVEX_HULL] THEN MATCH_MP_TAC INT_LE_TRANS THEN
EXISTS_TAC
`aff_dim({x:real^3 | a dot x = b} INTER {x | n dot x = d})` THEN
CONJ_TAC THENL
[MATCH_MP_TAC AFF_DIM_SUBSET THEN ASM SET_TAC[]; ALL_TAC] THEN
ASM_SIMP_TAC[AFF_DIM_AFFINE_INTER_HYPERPLANE; AFFINE_HYPERPLANE;
AFF_DIM_HYPERPLANE; DIMINDEX_3] THEN
REPEAT(COND_CASES_TAC THEN CONV_TAC INT_REDUCE_CONV) THEN
FIRST_X_ASSUM(MP_TAC o SPEC `x + n:real^3` o
GEN_REWRITE_RULE I [SUBSET]) THEN
ASM_SIMP_TAC[IN_ELIM_THM; DOT_RADD; REAL_EQ_ADD_LCANCEL_0; DOT_EQ_0] THEN
EXPAND_TAC "a" THEN VEC3_TAC]]);;
(* ------------------------------------------------------------------------- *)
(* Given a coplanar set, return exhaustive edge case theorem. *)
(* ------------------------------------------------------------------------- *)
let COMPUTE_EDGES_CONV =
let lemma = prove
(`(x INSERT s) INTER {x | P x} =
if P x then x INSERT (s INTER {x | P x})
else s INTER {x | P x}`,
COND_CASES_TAC THEN ASM SET_TAC[]) in
fun tm ->
let th1 = MATCH_MP COMPUTE_FACES_1 (COPLANAR_HYPERPLANE_RULE tm) in
let th2 = MP (CONV_RULE(LAND_CONV
(COMB2_CONV (RAND_CONV(PURE_REWRITE_CONV[FINITE_INSERT; FINITE_EMPTY]))
(RAND_CONV VECTOR3_EQ_0_CONV THENC
GEN_REWRITE_CONV I [NOT_CLAUSES]) THENC
GEN_REWRITE_CONV I [AND_CLAUSES])) th1) TRUTH in
CONV_RULE
(BINDER_CONV(RAND_CONV
(REWRITE_CONV[RIGHT_EXISTS_AND_THM] THENC
REWRITE_CONV[EXISTS_IN_INSERT; NOT_IN_EMPTY] THENC
REWRITE_CONV[FORALL_IN_INSERT; NOT_IN_EMPTY] THENC
ONCE_DEPTH_CONV VECTOR3_SUB_CONV THENC
ONCE_DEPTH_CONV VECTOR3_CROSS_CONV THENC
ONCE_DEPTH_CONV let_CONV THENC
ONCE_DEPTH_CONV VECTOR3_EQ_0_CONV THENC
REWRITE_CONV[real_ge] THENC
ONCE_DEPTH_CONV VECTOR3_DOT_CONV THENC
ONCE_DEPTH_CONV let_CONV THENC
ONCE_DEPTH_CONV REAL_RAT5_LE_CONV THENC
REWRITE_CONV[INSERT_AC] THENC REWRITE_CONV[DISJ_ACI] THENC
REPEATC(CHANGED_CONV
(ONCE_REWRITE_CONV[lemma] THENC
ONCE_DEPTH_CONV(LAND_CONV VECTOR3_DOT_CONV THENC
REAL_RAT5_EQ_CONV) THENC
REWRITE_CONV[])) THENC
REWRITE_CONV[INTER_EMPTY] THENC
REWRITE_CONV[INSERT_AC] THENC REWRITE_CONV[DISJ_ACI]
))) th2;;
(* ------------------------------------------------------------------------- *)
(* Use this to prove the number of edges per face for each Platonic solid. *)
(* ------------------------------------------------------------------------- *)
let CARD_EQ_LEMMA = prove
(`!x s n. 0 < n /\ ~(x IN s) /\ s HAS_SIZE (n - 1)
==> (x INSERT s) HAS_SIZE n`,
REWRITE_TAC[HAS_SIZE] THEN REPEAT STRIP_TAC THEN
ASM_SIMP_TAC[CARD_CLAUSES; FINITE_INSERT] THEN ASM_ARITH_TAC);;
let EDGES_PER_FACE_TAC th =
REPEAT STRIP_TAC THEN MATCH_MP_TAC EQ_TRANS THEN
EXISTS_TAC `CARD {e:real^3->bool | e face_of f /\ aff_dim(e) = &1}` THEN
CONJ_TAC THENL
[AP_TERM_TAC THEN GEN_REWRITE_TAC I [EXTENSION] THEN
REWRITE_TAC[IN_ELIM_THM] THEN
ASM_MESON_TAC[FACE_OF_FACE; FACE_OF_TRANS; FACE_OF_IMP_SUBSET];
ALL_TAC] THEN
MP_TAC(ISPEC `f:real^3->bool` th) THEN ASM_REWRITE_TAC[] THEN
DISCH_THEN(REPEAT_TCL DISJ_CASES_THEN SUBST1_TAC) THEN
W(fun (_,w) -> REWRITE_TAC[COMPUTE_EDGES_CONV(find_term is_setenum w)]) THEN
REWRITE_TAC[SET_RULE `x = a \/ x = b <=> x IN {a,b}`] THEN
REWRITE_TAC[GSYM IN_INSERT; SET_RULE `{x | x IN s} = s`] THEN
REWRITE_TAC[GSYM SEGMENT_CONVEX_HULL] THEN MATCH_MP_TAC
(MESON[HAS_SIZE] `s HAS_SIZE n ==> CARD s = n`) THEN
REPEAT
(MATCH_MP_TAC CARD_EQ_LEMMA THEN REPEAT CONJ_TAC THENL
[CONV_TAC NUM_REDUCE_CONV THEN NO_TAC;
REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY; SEGMENT_EQ; DE_MORGAN_THM] THEN
REPEAT CONJ_TAC THEN MATCH_MP_TAC(SET_RULE
`~(a = c /\ b = d) /\ ~(a = d /\ b = c) /\ ~(a = b /\ c = d)
==> ~({a,b} = {c,d})`) THEN
PURE_ONCE_REWRITE_TAC[GSYM VECTOR_SUB_EQ] THEN
CONV_TAC(ONCE_DEPTH_CONV VECTOR3_SUB_CONV) THEN
CONV_TAC(ONCE_DEPTH_CONV VECTOR3_EQ_0_CONV) THEN
REWRITE_TAC[] THEN NO_TAC;
ALL_TAC]) THEN
CONV_TAC NUM_REDUCE_CONV THEN REWRITE_TAC[CONJUNCT1 HAS_SIZE_CLAUSES];;
let TETRAHEDRON_EDGES_PER_FACE = prove
(`!f. f face_of std_tetrahedron /\ aff_dim(f) = &2
==> CARD {e | e face_of std_tetrahedron /\ aff_dim(e) = &1 /\
e SUBSET f} = 3`,
EDGES_PER_FACE_TAC TETRAHEDRON_FACETS);;
let CUBE_EDGES_PER_FACE = prove
(`!f. f face_of std_cube /\ aff_dim(f) = &2
==> CARD {e | e face_of std_cube /\ aff_dim(e) = &1 /\
e SUBSET f} = 4`,
EDGES_PER_FACE_TAC CUBE_FACETS);;
let OCTAHEDRON_EDGES_PER_FACE = prove
(`!f. f face_of std_octahedron /\ aff_dim(f) = &2
==> CARD {e | e face_of std_octahedron /\ aff_dim(e) = &1 /\
e SUBSET f} = 3`,
EDGES_PER_FACE_TAC OCTAHEDRON_FACETS);;
let DODECAHEDRON_EDGES_PER_FACE = prove
(`!f. f face_of std_dodecahedron /\ aff_dim(f) = &2
==> CARD {e | e face_of std_dodecahedron /\ aff_dim(e) = &1 /\
e SUBSET f} = 5`,
EDGES_PER_FACE_TAC DODECAHEDRON_FACETS);;
let ICOSAHEDRON_EDGES_PER_FACE = prove
(`!f. f face_of std_icosahedron /\ aff_dim(f) = &2
==> CARD {e | e face_of std_icosahedron /\ aff_dim(e) = &1 /\
e SUBSET f} = 3`,
EDGES_PER_FACE_TAC ICOSAHEDRON_FACETS);;
(* ------------------------------------------------------------------------- *)
(* Show that the Platonic solids are all full-dimensional. *)
(* ------------------------------------------------------------------------- *)
let POLYTOPE_3D_LEMMA = prove
(`(let a = (z - x) cross (y - x) in
~(a = vec 0) /\ ?w. w IN s /\ ~(a dot w = a dot x))
==> aff_dim(convex hull (x INSERT y INSERT z INSERT s:real^3->bool)) = &3`,
REPEAT GEN_TAC THEN LET_TAC THEN STRIP_TAC THEN
REWRITE_TAC[GSYM INT_LE_ANTISYM] THEN CONJ_TAC THENL
[REWRITE_TAC[GSYM DIMINDEX_3; AFF_DIM_LE_UNIV]; ALL_TAC] THEN
REWRITE_TAC[AFF_DIM_CONVEX_HULL] THEN MATCH_MP_TAC INT_LE_TRANS THEN
EXISTS_TAC `aff_dim {w:real^3,x,y,z}` THEN CONJ_TAC THENL
[ALL_TAC; MATCH_MP_TAC AFF_DIM_SUBSET THEN ASM SET_TAC[]] THEN
ONCE_REWRITE_TAC[AFF_DIM_INSERT] THEN COND_CASES_TAC THENL
[SUBGOAL_THEN `w IN {w:real^3 | a dot w = a dot x}` MP_TAC THENL
[FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE
`x IN s ==> s SUBSET t ==> x IN t`)) THEN
MATCH_MP_TAC HULL_MINIMAL THEN REWRITE_TAC[AFFINE_HYPERPLANE] THEN
REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET; IN_ELIM_THM] THEN
UNDISCH_TAC `~(a:real^3 = vec 0)` THEN EXPAND_TAC "a" THEN VEC3_TAC;
ASM_REWRITE_TAC[IN_ELIM_THM]];
UNDISCH_TAC `~(a:real^3 = vec 0)` THEN EXPAND_TAC "a" THEN
REWRITE_TAC[CROSS_EQ_0; GSYM COLLINEAR_3] THEN
REWRITE_TAC[COLLINEAR_3_EQ_AFFINE_DEPENDENT; INSERT_AC; DE_MORGAN_THM] THEN
STRIP_TAC THEN ASM_SIMP_TAC[AFF_DIM_AFFINE_INDEPENDENT] THEN
SIMP_TAC[CARD_CLAUSES; FINITE_INSERT; FINITE_EMPTY] THEN
ASM_REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY; ARITH] THEN INT_ARITH_TAC]);;
let POLYTOPE_FULLDIM_TAC =
MATCH_MP_TAC POLYTOPE_3D_LEMMA THEN
CONV_TAC(ONCE_DEPTH_CONV VECTOR3_SUB_CONV) THEN
CONV_TAC(ONCE_DEPTH_CONV VECTOR3_CROSS_CONV) THEN
CONV_TAC(ONCE_DEPTH_CONV let_CONV) THEN CONJ_TAC THENL
[CONV_TAC(RAND_CONV VECTOR3_EQ_0_CONV) THEN REWRITE_TAC[];
CONV_TAC(ONCE_DEPTH_CONV VECTOR3_DOT_CONV) THEN
REWRITE_TAC[EXISTS_IN_INSERT; NOT_IN_EMPTY] THEN
CONV_TAC(ONCE_DEPTH_CONV VECTOR3_DOT_CONV) THEN
CONV_TAC(ONCE_DEPTH_CONV REAL_RAT5_EQ_CONV) THEN
REWRITE_TAC[]];;
let STD_TETRAHEDRON_FULLDIM = prove
(`aff_dim std_tetrahedron = &3`,
REWRITE_TAC[std_tetrahedron] THEN POLYTOPE_FULLDIM_TAC);;
let STD_CUBE_FULLDIM = prove
(`aff_dim std_cube = &3`,
REWRITE_TAC[std_cube] THEN POLYTOPE_FULLDIM_TAC);;
let STD_OCTAHEDRON_FULLDIM = prove
(`aff_dim std_octahedron = &3`,
REWRITE_TAC[std_octahedron] THEN POLYTOPE_FULLDIM_TAC);;
let STD_DODECAHEDRON_FULLDIM = prove
(`aff_dim std_dodecahedron = &3`,
REWRITE_TAC[STD_DODECAHEDRON] THEN POLYTOPE_FULLDIM_TAC);;
let STD_ICOSAHEDRON_FULLDIM = prove
(`aff_dim std_icosahedron = &3`,
REWRITE_TAC[STD_ICOSAHEDRON] THEN POLYTOPE_FULLDIM_TAC);;
(* ------------------------------------------------------------------------- *)
(* Complete list of edges for each Platonic solid. *)
(* ------------------------------------------------------------------------- *)
let COMPUTE_EDGES_TAC defn fulldim facets =
GEN_TAC THEN MATCH_MP_TAC EQ_TRANS THEN
EXISTS_TAC
(vsubst[lhs(concl defn),`p:real^3->bool`]
`?f:real^3->bool. (f face_of p /\ aff_dim f = &2) /\
(e face_of f /\ aff_dim e = &1)`) THEN
CONJ_TAC THENL
[EQ_TAC THENL [STRIP_TAC; MESON_TAC[FACE_OF_TRANS]] THEN
MP_TAC(ISPECL [lhs(concl defn); `e:real^3->bool`]
FACE_OF_POLYHEDRON_SUBSET_FACET) THEN
ANTS_TAC THENL
[ASM_REWRITE_TAC[] THEN CONJ_TAC THENL
[REWRITE_TAC[defn] THEN
MATCH_MP_TAC POLYHEDRON_CONVEX_HULL THEN
REWRITE_TAC[FINITE_INSERT; FINITE_EMPTY];
CONJ_TAC THEN
DISCH_THEN(MP_TAC o AP_TERM `aff_dim:(real^3->bool)->int`) THEN
ASM_REWRITE_TAC[fulldim; AFF_DIM_EMPTY] THEN
CONV_TAC INT_REDUCE_CONV];
MATCH_MP_TAC MONO_EXISTS THEN REWRITE_TAC[facet_of] THEN
REWRITE_TAC[fulldim] THEN CONV_TAC INT_REDUCE_CONV THEN
ASM_MESON_TAC[FACE_OF_FACE]];
REWRITE_TAC[facets] THEN
REWRITE_TAC[TAUT `(a \/ b) /\ c <=> a /\ c \/ b /\ c`] THEN
REWRITE_TAC[EXISTS_OR_THM; UNWIND_THM2] THEN
CONV_TAC(LAND_CONV(DEPTH_BINOP_CONV `\/`
(fun tm -> REWR_CONV (COMPUTE_EDGES_CONV(rand(rand(lhand tm)))) tm))) THEN
REWRITE_TAC[INSERT_AC] THEN REWRITE_TAC[DISJ_ACI]];;
let TETRAHEDRON_EDGES = prove
(`!e. e face_of std_tetrahedron /\ aff_dim e = &1 <=>
e = convex hull {vector[-- &1; -- &1; &1], vector[-- &1; &1; -- &1]} \/
e = convex hull {vector[-- &1; -- &1; &1], vector[&1; -- &1; -- &1]} \/
e = convex hull {vector[-- &1; -- &1; &1], vector[&1; &1; &1]} \/
e = convex hull {vector[-- &1; &1; -- &1], vector[&1; -- &1; -- &1]} \/
e = convex hull {vector[-- &1; &1; -- &1], vector[&1; &1; &1]} \/
e = convex hull {vector[&1; -- &1; -- &1], vector[&1; &1; &1]}`,
COMPUTE_EDGES_TAC
std_tetrahedron STD_TETRAHEDRON_FULLDIM TETRAHEDRON_FACETS);;
let CUBE_EDGES = prove
(`!e. e face_of std_cube /\ aff_dim e = &1 <=>
e = convex hull {vector[-- &1; -- &1; -- &1], vector[-- &1; -- &1; &1]} \/
e = convex hull {vector[-- &1; -- &1; -- &1], vector[-- &1; &1; -- &1]} \/
e = convex hull {vector[-- &1; -- &1; -- &1], vector[&1; -- &1; -- &1]} \/
e = convex hull {vector[-- &1; -- &1; &1], vector[-- &1; &1; &1]} \/
e = convex hull {vector[-- &1; -- &1; &1], vector[&1; -- &1; &1]} \/
e = convex hull {vector[-- &1; &1; -- &1], vector[-- &1; &1; &1]} \/
e = convex hull {vector[-- &1; &1; -- &1], vector[&1; &1; -- &1]} \/
e = convex hull {vector[-- &1; &1; &1], vector[&1; &1; &1]} \/
e = convex hull {vector[&1; -- &1; -- &1], vector[&1; -- &1; &1]} \/
e = convex hull {vector[&1; -- &1; -- &1], vector[&1; &1; -- &1]} \/
e = convex hull {vector[&1; -- &1; &1], vector[&1; &1; &1]} \/
e = convex hull {vector[&1; &1; -- &1], vector[&1; &1; &1]}`,
COMPUTE_EDGES_TAC
std_cube STD_CUBE_FULLDIM CUBE_FACETS);;
let OCTAHEDRON_EDGES = prove
(`!e. e face_of std_octahedron /\ aff_dim e = &1 <=>
e = convex hull {vector[-- &1; &0; &0], vector[&0; -- &1; &0]} \/
e = convex hull {vector[-- &1; &0; &0], vector[&0; &1; &0]} \/
e = convex hull {vector[-- &1; &0; &0], vector[&0; &0; -- &1]} \/
e = convex hull {vector[-- &1; &0; &0], vector[&0; &0; &1]} \/
e = convex hull {vector[&1; &0; &0], vector[&0; -- &1; &0]} \/
e = convex hull {vector[&1; &0; &0], vector[&0; &1; &0]} \/
e = convex hull {vector[&1; &0; &0], vector[&0; &0; -- &1]} \/
e = convex hull {vector[&1; &0; &0], vector[&0; &0; &1]} \/
e = convex hull {vector[&0; -- &1; &0], vector[&0; &0; -- &1]} \/
e = convex hull {vector[&0; -- &1; &0], vector[&0; &0; &1]} \/
e = convex hull {vector[&0; &1; &0], vector[&0; &0; -- &1]} \/
e = convex hull {vector[&0; &1; &0], vector[&0; &0; &1]}`,
COMPUTE_EDGES_TAC
std_octahedron STD_OCTAHEDRON_FULLDIM OCTAHEDRON_FACETS);;
let DODECAHEDRON_EDGES = prove
(`!e. e face_of std_dodecahedron /\ aff_dim e = &1 <=>
e = convex hull {vector[-- &1 / &2 + -- &1 / &2 * sqrt (&5); &0; -- &1 / &2 + &1 / &2 * sqrt (&5)], vector[-- &1 / &2 + -- &1 / &2 * sqrt (&5); &0; &1 / &2 + -- &1 / &2 * sqrt (&5)]} \/
e = convex hull {vector[-- &1 / &2 + -- &1 / &2 * sqrt (&5); &0; -- &1 / &2 + &1 / &2 * sqrt (&5)], vector[-- &1; -- &1; &1]} \/
e = convex hull {vector[-- &1 / &2 + -- &1 / &2 * sqrt (&5); &0; -- &1 / &2 + &1 / &2 * sqrt (&5)], vector[-- &1; &1; &1]} \/
e = convex hull {vector[-- &1 / &2 + -- &1 / &2 * sqrt (&5); &0; &1 / &2 + -- &1 / &2 * sqrt (&5)], vector[-- &1; -- &1; -- &1]} \/
e = convex hull {vector[-- &1 / &2 + -- &1 / &2 * sqrt (&5); &0; &1 / &2 + -- &1 / &2 * sqrt (&5)], vector[-- &1; &1; -- &1]} \/
e = convex hull {vector[-- &1 / &2 + &1 / &2 * sqrt (&5); -- &1 / &2 + -- &1 / &2 * sqrt (&5); &0], vector[&1 / &2 + -- &1 / &2 * sqrt (&5); -- &1 / &2 + -- &1 / &2 * sqrt (&5); &0]} \/
e = convex hull {vector[-- &1 / &2 + &1 / &2 * sqrt (&5); -- &1 / &2 + -- &1 / &2 * sqrt (&5); &0], vector[&1; -- &1; -- &1]} \/
e = convex hull {vector[-- &1 / &2 + &1 / &2 * sqrt (&5); -- &1 / &2 + -- &1 / &2 * sqrt (&5); &0], vector[&1; -- &1; &1]} \/
e = convex hull {vector[-- &1 / &2 + &1 / &2 * sqrt (&5); &1 / &2 + &1 / &2 * sqrt (&5); &0], vector[&1 / &2 + -- &1 / &2 * sqrt (&5); &1 / &2 + &1 / &2 * sqrt (&5); &0]} \/
e = convex hull {vector[-- &1 / &2 + &1 / &2 * sqrt (&5); &1 / &2 + &1 / &2 * sqrt (&5); &0], vector[&1; &1; -- &1]} \/
e = convex hull {vector[-- &1 / &2 + &1 / &2 * sqrt (&5); &1 / &2 + &1 / &2 * sqrt (&5); &0], vector[&1; &1; &1]} \/
e = convex hull {vector[&1 / &2 + -- &1 / &2 * sqrt (&5); -- &1 / &2 + -- &1 / &2 * sqrt (&5); &0], vector[-- &1; -- &1; -- &1]} \/
e = convex hull {vector[&1 / &2 + -- &1 / &2 * sqrt (&5); -- &1 / &2 + -- &1 / &2 * sqrt (&5); &0], vector[-- &1; -- &1; &1]} \/
e = convex hull {vector[&1 / &2 + -- &1 / &2 * sqrt (&5); &1 / &2 + &1 / &2 * sqrt (&5); &0], vector[-- &1; &1; -- &1]} \/
e = convex hull {vector[&1 / &2 + -- &1 / &2 * sqrt (&5); &1 / &2 + &1 / &2 * sqrt (&5); &0], vector[-- &1; &1; &1]} \/
e = convex hull {vector[&1 / &2 + &1 / &2 * sqrt (&5); &0; -- &1 / &2 + &1 / &2 * sqrt (&5)], vector[&1 / &2 + &1 / &2 * sqrt (&5); &0; &1 / &2 + -- &1 / &2 * sqrt (&5)]} \/
e = convex hull {vector[&1 / &2 + &1 / &2 * sqrt (&5); &0; -- &1 / &2 + &1 / &2 * sqrt (&5)], vector[&1; -- &1; &1]} \/
e = convex hull {vector[&1 / &2 + &1 / &2 * sqrt (&5); &0; -- &1 / &2 + &1 / &2 * sqrt (&5)], vector[&1; &1; &1]} \/
e = convex hull {vector[&1 / &2 + &1 / &2 * sqrt (&5); &0; &1 / &2 + -- &1 / &2 * sqrt (&5)], vector[&1; -- &1; -- &1]} \/
e = convex hull {vector[&1 / &2 + &1 / &2 * sqrt (&5); &0; &1 / &2 + -- &1 / &2 * sqrt (&5)], vector[&1; &1; -- &1]} \/
e = convex hull {vector[-- &1; -- &1; -- &1], vector[&0; &1 / &2 + -- &1 / &2 * sqrt (&5); -- &1 / &2 + -- &1 / &2 * sqrt (&5)]} \/
e = convex hull {vector[-- &1; -- &1; &1], vector[&0; &1 / &2 + -- &1 / &2 * sqrt (&5); &1 / &2 + &1 / &2 * sqrt (&5)]} \/
e = convex hull {vector[-- &1; &1; -- &1], vector[&0; -- &1 / &2 + &1 / &2 * sqrt (&5); -- &1 / &2 + -- &1 / &2 * sqrt (&5)]} \/
e = convex hull {vector[-- &1; &1; &1], vector[&0; -- &1 / &2 + &1 / &2 * sqrt (&5); &1 / &2 + &1 / &2 * sqrt (&5)]} \/
e = convex hull {vector[&1; -- &1; -- &1], vector[&0; &1 / &2 + -- &1 / &2 * sqrt (&5); -- &1 / &2 + -- &1 / &2 * sqrt (&5)]} \/
e = convex hull {vector[&1; -- &1; &1], vector[&0; &1 / &2 + -- &1 / &2 * sqrt (&5); &1 / &2 + &1 / &2 * sqrt (&5)]} \/
e = convex hull {vector[&1; &1; -- &1], vector[&0; -- &1 / &2 + &1 / &2 * sqrt (&5); -- &1 / &2 + -- &1 / &2 * sqrt (&5)]} \/
e = convex hull {vector[&1; &1; &1], vector[&0; -- &1 / &2 + &1 / &2 * sqrt (&5); &1 / &2 + &1 / &2 * sqrt (&5)]} \/
e = convex hull {vector[&0; -- &1 / &2 + &1 / &2 * sqrt (&5); -- &1 / &2 + -- &1 / &2 * sqrt (&5)], vector[&0; &1 / &2 + -- &1 / &2 * sqrt (&5); -- &1 / &2 + -- &1 / &2 * sqrt (&5)]} \/
e = convex hull {vector[&0; -- &1 / &2 + &1 / &2 * sqrt (&5); &1 / &2 + &1 / &2 * sqrt (&5)], vector[&0; &1 / &2 + -- &1 / &2 * sqrt (&5); &1 / &2 + &1 / &2 * sqrt (&5)]}`,
COMPUTE_EDGES_TAC
STD_DODECAHEDRON STD_DODECAHEDRON_FULLDIM DODECAHEDRON_FACETS);;
let ICOSAHEDRON_EDGES = prove
(`!e. e face_of std_icosahedron /\ aff_dim e = &1 <=>
e = convex hull {vector[-- &1 / &2 + -- &1 / &2 * sqrt (&5); &0; -- &1], vector[-- &1 / &2 + -- &1 / &2 * sqrt (&5); &0; &1]} \/
e = convex hull {vector[-- &1 / &2 + -- &1 / &2 * sqrt (&5); &0; -- &1], vector[-- &1; -- &1 / &2 + -- &1 / &2 * sqrt (&5); &0]} \/
e = convex hull {vector[-- &1 / &2 + -- &1 / &2 * sqrt (&5); &0; -- &1], vector[-- &1; &1 / &2 + &1 / &2 * sqrt (&5); &0]} \/
e = convex hull {vector[-- &1 / &2 + -- &1 / &2 * sqrt (&5); &0; -- &1], vector[&0; -- &1; -- &1 / &2 + -- &1 / &2 * sqrt (&5)]} \/
e = convex hull {vector[-- &1 / &2 + -- &1 / &2 * sqrt (&5); &0; -- &1], vector[&0; &1; -- &1 / &2 + -- &1 / &2 * sqrt (&5)]} \/
e = convex hull {vector[-- &1 / &2 + -- &1 / &2 * sqrt (&5); &0; &1], vector[-- &1; -- &1 / &2 + -- &1 / &2 * sqrt (&5); &0]} \/
e = convex hull {vector[-- &1 / &2 + -- &1 / &2 * sqrt (&5); &0; &1], vector[-- &1; &1 / &2 + &1 / &2 * sqrt (&5); &0]} \/
e = convex hull {vector[-- &1 / &2 + -- &1 / &2 * sqrt (&5); &0; &1], vector[&0; -- &1; &1 / &2 + &1 / &2 * sqrt (&5)]} \/
e = convex hull {vector[-- &1 / &2 + -- &1 / &2 * sqrt (&5); &0; &1], vector[&0; &1; &1 / &2 + &1 / &2 * sqrt (&5)]} \/
e = convex hull {vector[&1 / &2 + &1 / &2 * sqrt (&5); &0; -- &1], vector[&1 / &2 + &1 / &2 * sqrt (&5); &0; &1]} \/
e = convex hull {vector[&1 / &2 + &1 / &2 * sqrt (&5); &0; -- &1], vector[&1; -- &1 / &2 + -- &1 / &2 * sqrt (&5); &0]} \/
e = convex hull {vector[&1 / &2 + &1 / &2 * sqrt (&5); &0; -- &1], vector[&1; &1 / &2 + &1 / &2 * sqrt (&5); &0]} \/
e = convex hull {vector[&1 / &2 + &1 / &2 * sqrt (&5); &0; -- &1], vector[&0; -- &1; -- &1 / &2 + -- &1 / &2 * sqrt (&5)]} \/
e = convex hull {vector[&1 / &2 + &1 / &2 * sqrt (&5); &0; -- &1], vector[&0; &1; -- &1 / &2 + -- &1 / &2 * sqrt (&5)]} \/
e = convex hull {vector[&1 / &2 + &1 / &2 * sqrt (&5); &0; &1], vector[&1; -- &1 / &2 + -- &1 / &2 * sqrt (&5); &0]} \/
e = convex hull {vector[&1 / &2 + &1 / &2 * sqrt (&5); &0; &1], vector[&1; &1 / &2 + &1 / &2 * sqrt (&5); &0]} \/
e = convex hull {vector[&1 / &2 + &1 / &2 * sqrt (&5); &0; &1], vector[&0; -- &1; &1 / &2 + &1 / &2 * sqrt (&5)]} \/
e = convex hull {vector[&1 / &2 + &1 / &2 * sqrt (&5); &0; &1], vector[&0; &1; &1 / &2 + &1 / &2 * sqrt (&5)]} \/
e = convex hull {vector[-- &1; -- &1 / &2 + -- &1 / &2 * sqrt (&5); &0], vector[&1; -- &1 / &2 + -- &1 / &2 * sqrt (&5); &0]} \/
e = convex hull {vector[-- &1; -- &1 / &2 + -- &1 / &2 * sqrt (&5); &0], vector[&0; -- &1; -- &1 / &2 + -- &1 / &2 * sqrt (&5)]} \/
e = convex hull {vector[-- &1; -- &1 / &2 + -- &1 / &2 * sqrt (&5); &0], vector[&0; -- &1; &1 / &2 + &1 / &2 * sqrt (&5)]} \/
e = convex hull {vector[-- &1; &1 / &2 + &1 / &2 * sqrt (&5); &0], vector[&1; &1 / &2 + &1 / &2 * sqrt (&5); &0]} \/
e = convex hull {vector[-- &1; &1 / &2 + &1 / &2 * sqrt (&5); &0], vector[&0; &1; -- &1 / &2 + -- &1 / &2 * sqrt (&5)]} \/
e = convex hull {vector[-- &1; &1 / &2 + &1 / &2 * sqrt (&5); &0], vector[&0; &1; &1 / &2 + &1 / &2 * sqrt (&5)]} \/
e = convex hull {vector[&1; -- &1 / &2 + -- &1 / &2 * sqrt (&5); &0], vector[&0; -- &1; -- &1 / &2 + -- &1 / &2 * sqrt (&5)]} \/
e = convex hull {vector[&1; -- &1 / &2 + -- &1 / &2 * sqrt (&5); &0], vector[&0; -- &1; &1 / &2 + &1 / &2 * sqrt (&5)]} \/
e = convex hull {vector[&1; &1 / &2 + &1 / &2 * sqrt (&5); &0], vector[&0; &1; -- &1 / &2 + -- &1 / &2 * sqrt (&5)]} \/
e = convex hull {vector[&1; &1 / &2 + &1 / &2 * sqrt (&5); &0], vector[&0; &1; &1 / &2 + &1 / &2 * sqrt (&5)]} \/
e = convex hull {vector[&0; -- &1; -- &1 / &2 + -- &1 / &2 * sqrt (&5)], vector[&0; &1; -- &1 / &2 + -- &1 / &2 * sqrt (&5)]} \/
e = convex hull {vector[&0; -- &1; &1 / &2 + &1 / &2 * sqrt (&5)], vector[&0; &1; &1 / &2 + &1 / &2 * sqrt (&5)]}`,
COMPUTE_EDGES_TAC
STD_ICOSAHEDRON STD_ICOSAHEDRON_FULLDIM ICOSAHEDRON_FACETS);;
(* ------------------------------------------------------------------------- *)
(* Enumerate all the vertices. *)
(* ------------------------------------------------------------------------- *)
let COMPUTE_VERTICES_TAC defn fulldim edges =
GEN_TAC THEN MATCH_MP_TAC EQ_TRANS THEN
EXISTS_TAC
(vsubst[lhs(concl defn),`p:real^3->bool`]
`?e:real^3->bool. (e face_of p /\ aff_dim e = &1) /\
(v face_of e /\ aff_dim v = &0)`) THEN
CONJ_TAC THENL
[EQ_TAC THENL [STRIP_TAC; MESON_TAC[FACE_OF_TRANS]] THEN
MP_TAC(ISPECL [lhs(concl defn); `v:real^3->bool`]
FACE_OF_POLYHEDRON_SUBSET_FACET) THEN
ANTS_TAC THENL
[ASM_REWRITE_TAC[] THEN CONJ_TAC THENL
[REWRITE_TAC[defn] THEN
MATCH_MP_TAC POLYHEDRON_CONVEX_HULL THEN
REWRITE_TAC[FINITE_INSERT; FINITE_EMPTY];
CONJ_TAC THEN
DISCH_THEN(MP_TAC o AP_TERM `aff_dim:(real^3->bool)->int`) THEN
ASM_REWRITE_TAC[fulldim; AFF_DIM_EMPTY] THEN
CONV_TAC INT_REDUCE_CONV];
REWRITE_TAC[facet_of] THEN
DISCH_THEN(X_CHOOSE_THEN `f:real^3->bool` STRIP_ASSUME_TAC)] THEN
MP_TAC(ISPECL [`f:real^3->bool`; `v:real^3->bool`]
FACE_OF_POLYHEDRON_SUBSET_FACET) THEN
ANTS_TAC THENL
[REPEAT CONJ_TAC THENL
[MATCH_MP_TAC FACE_OF_POLYHEDRON_POLYHEDRON THEN
FIRST_ASSUM(fun th ->
EXISTS_TAC (rand(concl th)) THEN
CONJ_TAC THENL [ALL_TAC; ACCEPT_TAC th]) THEN
REWRITE_TAC[defn] THEN
MATCH_MP_TAC POLYHEDRON_CONVEX_HULL THEN
REWRITE_TAC[FINITE_INSERT; FINITE_EMPTY];
ASM_MESON_TAC[FACE_OF_FACE];
DISCH_THEN(MP_TAC o AP_TERM `aff_dim:(real^3->bool)->int`) THEN
ASM_REWRITE_TAC[fulldim; AFF_DIM_EMPTY] THEN
CONV_TAC INT_REDUCE_CONV;
DISCH_THEN(MP_TAC o AP_TERM `aff_dim:(real^3->bool)->int`) THEN
ASM_REWRITE_TAC[fulldim; AFF_DIM_EMPTY] THEN
CONV_TAC INT_REDUCE_CONV];
MATCH_MP_TAC MONO_EXISTS THEN REWRITE_TAC[facet_of] THEN
ASM_REWRITE_TAC[fulldim] THEN CONV_TAC INT_REDUCE_CONV THEN
ASM_MESON_TAC[FACE_OF_FACE; FACE_OF_TRANS]];
REWRITE_TAC[edges] THEN
REWRITE_TAC[TAUT `(a \/ b) /\ c <=> a /\ c \/ b /\ c`] THEN
REWRITE_TAC[EXISTS_OR_THM; UNWIND_THM2] THEN
REWRITE_TAC[AFF_DIM_EQ_0; RIGHT_AND_EXISTS_THM] THEN
ONCE_REWRITE_TAC[MESON[]
`v face_of s /\ v = {a} <=> {a} face_of s /\ v = {a}`] THEN
REWRITE_TAC[GSYM SEGMENT_CONVEX_HULL; FACE_OF_SING] THEN
REWRITE_TAC[EXTREME_POINT_OF_SEGMENT] THEN
REWRITE_TAC[TAUT `(a \/ b) /\ c <=> a /\ c \/ b /\ c`] THEN
REWRITE_TAC[EXISTS_OR_THM; UNWIND_THM2] THEN
REWRITE_TAC[DISJ_ACI]];;
let TETRAHEDRON_VERTICES = prove
(`!v. v face_of std_tetrahedron /\ aff_dim v = &0 <=>
v = {vector [-- &1; -- &1; &1]} \/
v = {vector [-- &1; &1; -- &1]} \/
v = {vector [&1; -- &1; -- &1]} \/
v = {vector [&1; &1; &1]}`,
COMPUTE_VERTICES_TAC
std_tetrahedron STD_TETRAHEDRON_FULLDIM TETRAHEDRON_EDGES);;
let CUBE_VERTICES = prove
(`!v. v face_of std_cube /\ aff_dim v = &0 <=>
v = {vector [-- &1; -- &1; -- &1]} \/
v = {vector [-- &1; -- &1; &1]} \/
v = {vector [-- &1; &1; -- &1]} \/
v = {vector [-- &1; &1; &1]} \/
v = {vector [&1; -- &1; -- &1]} \/
v = {vector [&1; -- &1; &1]} \/
v = {vector [&1; &1; -- &1]} \/
v = {vector [&1; &1; &1]}`,
COMPUTE_VERTICES_TAC
std_cube STD_CUBE_FULLDIM CUBE_EDGES);;
let OCTAHEDRON_VERTICES = prove
(`!v. v face_of std_octahedron /\ aff_dim v = &0 <=>
v = {vector [-- &1; &0; &0]} \/
v = {vector [&1; &0; &0]} \/
v = {vector [&0; -- &1; &0]} \/
v = {vector [&0; &1; &0]} \/
v = {vector [&0; &0; -- &1]} \/
v = {vector [&0; &0; &1]}`,
COMPUTE_VERTICES_TAC
std_octahedron STD_OCTAHEDRON_FULLDIM OCTAHEDRON_EDGES);;
let DODECAHEDRON_VERTICES = prove
(`!v. v face_of std_dodecahedron /\ aff_dim v = &0 <=>
v = {vector[-- &1 / &2 + -- &1 / &2 * sqrt (&5); &0; -- &1 / &2 + &1 / &2 * sqrt (&5)]} \/
v = {vector[-- &1 / &2 + -- &1 / &2 * sqrt (&5); &0; &1 / &2 + -- &1 / &2 * sqrt (&5)]} \/
v = {vector[-- &1 / &2 + &1 / &2 * sqrt (&5); -- &1 / &2 + -- &1 / &2 * sqrt (&5); &0]} \/
v = {vector[-- &1 / &2 + &1 / &2 * sqrt (&5); &1 / &2 + &1 / &2 * sqrt (&5); &0]} \/
v = {vector[&1 / &2 + -- &1 / &2 * sqrt (&5); -- &1 / &2 + -- &1 / &2 * sqrt (&5); &0]} \/
v = {vector[&1 / &2 + -- &1 / &2 * sqrt (&5); &1 / &2 + &1 / &2 * sqrt (&5); &0]} \/
v = {vector[&1 / &2 + &1 / &2 * sqrt (&5); &0; -- &1 / &2 + &1 / &2 * sqrt (&5)]} \/
v = {vector[&1 / &2 + &1 / &2 * sqrt (&5); &0; &1 / &2 + -- &1 / &2 * sqrt (&5)]} \/
v = {vector[-- &1; -- &1; -- &1]} \/
v = {vector[-- &1; -- &1; &1]} \/
v = {vector[-- &1; &1; -- &1]} \/
v = {vector[-- &1; &1; &1]} \/
v = {vector[&1; -- &1; -- &1]} \/
v = {vector[&1; -- &1; &1]} \/
v = {vector[&1; &1; -- &1]} \/
v = {vector[&1; &1; &1]} \/
v = {vector[&0; -- &1 / &2 + &1 / &2 * sqrt (&5); -- &1 / &2 + -- &1 / &2 * sqrt (&5)]} \/
v = {vector[&0; -- &1 / &2 + &1 / &2 * sqrt (&5); &1 / &2 + &1 / &2 * sqrt (&5)]} \/
v = {vector[&0; &1 / &2 + -- &1 / &2 * sqrt (&5); -- &1 / &2 + -- &1 / &2 * sqrt (&5)]} \/
v = {vector[&0; &1 / &2 + -- &1 / &2 * sqrt (&5); &1 / &2 + &1 / &2 * sqrt (&5)]}`,
COMPUTE_VERTICES_TAC
STD_DODECAHEDRON STD_DODECAHEDRON_FULLDIM DODECAHEDRON_EDGES);;
let ICOSAHEDRON_VERTICES = prove
(`!v. v face_of std_icosahedron /\ aff_dim v = &0 <=>
v = {vector [-- &1 / &2 + -- &1 / &2 * sqrt (&5); &0; -- &1]} \/
v = {vector [-- &1 / &2 + -- &1 / &2 * sqrt (&5); &0; &1]} \/
v = {vector [&1 / &2 + &1 / &2 * sqrt (&5); &0; -- &1]} \/
v = {vector [&1 / &2 + &1 / &2 * sqrt (&5); &0; &1]} \/
v = {vector [-- &1; -- &1 / &2 + -- &1 / &2 * sqrt (&5); &0]} \/
v = {vector [-- &1; &1 / &2 + &1 / &2 * sqrt (&5); &0]} \/
v = {vector [&1; -- &1 / &2 + -- &1 / &2 * sqrt (&5); &0]} \/
v = {vector [&1; &1 / &2 + &1 / &2 * sqrt (&5); &0]} \/
v = {vector [&0; -- &1; -- &1 / &2 + -- &1 / &2 * sqrt (&5)]} \/
v = {vector [&0; -- &1; &1 / &2 + &1 / &2 * sqrt (&5)]} \/
v = {vector [&0; &1; -- &1 / &2 + -- &1 / &2 * sqrt (&5)]} \/
v = {vector [&0; &1; &1 / &2 + &1 / &2 * sqrt (&5)]}`,
COMPUTE_VERTICES_TAC
STD_ICOSAHEDRON STD_ICOSAHEDRON_FULLDIM ICOSAHEDRON_EDGES);;
(* ------------------------------------------------------------------------- *)
(* Number of edges meeting at each vertex. *)
(* ------------------------------------------------------------------------- *)
let EDGES_PER_VERTEX_TAC defn edges verts =
REPEAT STRIP_TAC THEN MATCH_MP_TAC EQ_TRANS THEN
EXISTS_TAC
(vsubst[lhs(concl defn),`p:real^3->bool`]
`CARD {e | (e face_of p /\ aff_dim(e) = &1) /\
(v:real^3->bool) face_of e}`) THEN
CONJ_TAC THENL
[AP_TERM_TAC THEN REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN
ASM_MESON_TAC[FACE_OF_FACE];
ALL_TAC] THEN
MP_TAC(ISPEC `v:real^3->bool` verts) THEN
ASM_REWRITE_TAC[] THEN
DISCH_THEN(REPEAT_TCL DISJ_CASES_THEN SUBST1_TAC) THEN
REWRITE_TAC[edges] THEN
REWRITE_TAC[SET_RULE
`{e | (P e \/ Q e) /\ R e} =
{e | P e /\ R e} UNION {e | Q e /\ R e}`] THEN
REWRITE_TAC[MESON[FACE_OF_SING]
`e = a /\ {x} face_of e <=> e = a /\ x extreme_point_of a`] THEN
REWRITE_TAC[GSYM SEGMENT_CONVEX_HULL; EXTREME_POINT_OF_SEGMENT] THEN
ONCE_REWRITE_TAC[GSYM VECTOR_SUB_EQ] THEN
CONV_TAC(ONCE_DEPTH_CONV VECTOR3_SUB_CONV) THEN
CONV_TAC(ONCE_DEPTH_CONV VECTOR3_EQ_0_CONV) THEN
REWRITE_TAC[EMPTY_GSPEC; UNION_EMPTY] THEN
REWRITE_TAC[SET_RULE `{x | x = a} = {a}`] THEN
REWRITE_TAC[SET_RULE `{x} UNION s = x INSERT s`] THEN MATCH_MP_TAC
(MESON[HAS_SIZE] `s HAS_SIZE n ==> CARD s = n`) THEN
REPEAT
(MATCH_MP_TAC CARD_EQ_LEMMA THEN REPEAT CONJ_TAC THENL
[CONV_TAC NUM_REDUCE_CONV THEN NO_TAC;
REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY; DE_MORGAN_THM; SEGMENT_EQ] THEN
REPEAT CONJ_TAC THEN MATCH_MP_TAC(SET_RULE
`~(a = c /\ b = d) /\ ~(a = d /\ b = c) /\ ~(a = b /\ c = d)
==> ~({a,b} = {c,d})`) THEN
PURE_ONCE_REWRITE_TAC[GSYM VECTOR_SUB_EQ] THEN
CONV_TAC(ONCE_DEPTH_CONV VECTOR3_SUB_CONV) THEN
CONV_TAC(ONCE_DEPTH_CONV VECTOR3_EQ_0_CONV) THEN
REWRITE_TAC[] THEN NO_TAC;
ALL_TAC]) THEN
CONV_TAC NUM_REDUCE_CONV THEN REWRITE_TAC[CONJUNCT1 HAS_SIZE_CLAUSES];;
let TETRAHEDRON_EDGES_PER_VERTEX = prove
(`!v. v face_of std_tetrahedron /\ aff_dim(v) = &0
==> CARD {e | e face_of std_tetrahedron /\ aff_dim(e) = &1 /\
v SUBSET e} = 3`,
EDGES_PER_VERTEX_TAC
std_tetrahedron TETRAHEDRON_EDGES TETRAHEDRON_VERTICES);;
let CUBE_EDGES_PER_VERTEX = prove
(`!v. v face_of std_cube /\ aff_dim(v) = &0
==> CARD {e | e face_of std_cube /\ aff_dim(e) = &1 /\
v SUBSET e} = 3`,
EDGES_PER_VERTEX_TAC
std_cube CUBE_EDGES CUBE_VERTICES);;
let OCTAHEDRON_EDGES_PER_VERTEX = prove
(`!v. v face_of std_octahedron /\ aff_dim(v) = &0
==> CARD {e | e face_of std_octahedron /\ aff_dim(e) = &1 /\
v SUBSET e} = 4`,
EDGES_PER_VERTEX_TAC
std_octahedron OCTAHEDRON_EDGES OCTAHEDRON_VERTICES);;
let DODECAHEDRON_EDGES_PER_VERTEX = prove
(`!v. v face_of std_dodecahedron /\ aff_dim(v) = &0
==> CARD {e | e face_of std_dodecahedron /\ aff_dim(e) = &1 /\
v SUBSET e} = 3`,
EDGES_PER_VERTEX_TAC
STD_DODECAHEDRON DODECAHEDRON_EDGES DODECAHEDRON_VERTICES);;
let ICOSAHEDRON_EDGES_PER_VERTEX = prove
(`!v. v face_of std_icosahedron /\ aff_dim(v) = &0
==> CARD {e | e face_of std_icosahedron /\ aff_dim(e) = &1 /\
v SUBSET e} = 5`,
EDGES_PER_VERTEX_TAC
STD_ICOSAHEDRON ICOSAHEDRON_EDGES ICOSAHEDRON_VERTICES);;
(* ------------------------------------------------------------------------- *)
(* Number of Platonic solids. *)
(* ------------------------------------------------------------------------- *)
let MULTIPLE_COUNTING_LEMMA = prove
(`!R:A->B->bool s t.
FINITE s /\ FINITE t /\
(!x. x IN s ==> CARD {y | y IN t /\ R x y} = m) /\
(!y. y IN t ==> CARD {x | x IN s /\ R x y} = n)
==> m * CARD s = n * CARD t`,
REPEAT STRIP_TAC THEN
MP_TAC(ISPECL [`R:A->B->bool`; `\x:A y:B. 1`; `s:A->bool`; `t:B->bool`]
NSUM_NSUM_RESTRICT) THEN
ASM_SIMP_TAC[NSUM_CONST; FINITE_RESTRICT] THEN ARITH_TAC);;
let SIZE_ZERO_DIMENSIONAL_FACES = prove
(`!s:real^N->bool.
polyhedron s
==> CARD {f | f face_of s /\ aff_dim f = &0} =
CARD {v | v extreme_point_of s} /\
(FINITE {f | f face_of s /\ aff_dim f = &0} <=>
FINITE {v | v extreme_point_of s}) /\
(!n. {f | f face_of s /\ aff_dim f = &0} HAS_SIZE n <=>
{v | v extreme_point_of s} HAS_SIZE n)`,
REWRITE_TAC[RIGHT_AND_FORALL_THM] THEN REPEAT GEN_TAC THEN DISCH_TAC THEN
SUBGOAL_THEN `{f | f face_of s /\ aff_dim f = &0} =
IMAGE (\v:real^N. {v}) {v | v extreme_point_of s}`
SUBST1_TAC THENL
[CONV_TAC SYM_CONV THEN MATCH_MP_TAC SURJECTIVE_IMAGE_EQ THEN
REWRITE_TAC[AFF_DIM_SING; FACE_OF_SING; IN_ELIM_THM] THEN
REWRITE_TAC[AFF_DIM_EQ_0] THEN MESON_TAC[];
REPEAT STRIP_TAC THENL
[MATCH_MP_TAC CARD_IMAGE_INJ;
MATCH_MP_TAC FINITE_IMAGE_INJ_EQ;
MATCH_MP_TAC HAS_SIZE_IMAGE_INJ_EQ] THEN
ASM_SIMP_TAC[FINITE_POLYHEDRON_EXTREME_POINTS] THEN SET_TAC[]]);;
let PLATONIC_SOLIDS_LIMITS = prove
(`!p:real^3->bool m n.
polytope p /\ aff_dim p = &3 /\
(!f. f face_of p /\ aff_dim(f) = &2
==> CARD {e | e face_of p /\ aff_dim(e) = &1 /\ e SUBSET f} = m) /\
(!v. v face_of p /\ aff_dim(v) = &0
==> CARD {e | e face_of p /\ aff_dim(e) = &1 /\ v SUBSET e} = n)
==> m = 3 /\ n = 3 \/ // Tetrahedron
m = 4 /\ n = 3 \/ // Cube
m = 3 /\ n = 4 \/ // Octahedron
m = 5 /\ n = 3 \/ // Dodecahedron
m = 3 /\ n = 5 // Icosahedron`,
REPEAT STRIP_TAC THEN
MP_TAC(ISPEC `p:real^3->bool` EULER_RELATION) THEN
ASM_REWRITE_TAC[] THEN
SUBGOAL_THEN
`m * CARD {f:real^3->bool | f face_of p /\ aff_dim f = &2} =
2 * CARD {e | e face_of p /\ aff_dim e = &1} /\
n * CARD {v | v face_of p /\ aff_dim v = &0} =
2 * CARD {e | e face_of p /\ aff_dim e = &1}`
MP_TAC THENL
[CONJ_TAC THEN MATCH_MP_TAC MULTIPLE_COUNTING_LEMMA THENL
[EXISTS_TAC `\(f:real^3->bool) (e:real^3->bool). e SUBSET f`;
EXISTS_TAC `\(v:real^3->bool) (e:real^3->bool). v SUBSET e`] THEN
ONCE_REWRITE_TAC[SET_RULE `f face_of s <=> f IN {f | f face_of s}`] THEN
ASM_SIMP_TAC[FINITE_POLYTOPE_FACES; FINITE_RESTRICT] THEN
ASM_REWRITE_TAC[IN_ELIM_THM; GSYM CONJ_ASSOC] THEN
X_GEN_TAC `e:real^3->bool` THEN STRIP_TAC THENL
[MP_TAC(ISPECL [`p:real^3->bool`; `e:real^3->bool`]
POLYHEDRON_RIDGE_TWO_FACETS) THEN
ASM_SIMP_TAC[POLYTOPE_IMP_POLYHEDRON] THEN ANTS_TAC THENL
[CONV_TAC INT_REDUCE_CONV THEN DISCH_THEN SUBST_ALL_TAC THEN
RULE_ASSUM_TAC(REWRITE_RULE[AFF_DIM_EMPTY]) THEN ASM_INT_ARITH_TAC;
CONV_TAC INT_REDUCE_CONV THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN
MAP_EVERY X_GEN_TAC [`f1:real^3->bool`; `f2:real^3->bool`] THEN
STRIP_TAC THEN MATCH_MP_TAC EQ_TRANS THEN
EXISTS_TAC `CARD {f1:real^3->bool,f2}` THEN CONJ_TAC THENL
[AP_TERM_TAC THEN GEN_REWRITE_TAC I [EXTENSION] THEN
REWRITE_TAC[IN_ELIM_THM; IN_INSERT; NOT_IN_EMPTY] THEN
ASM_MESON_TAC[];
ASM_SIMP_TAC[CARD_CLAUSES; IN_INSERT; FINITE_RULES;
NOT_IN_EMPTY; ARITH]]];
SUBGOAL_THEN `?a b:real^3. e = segment[a,b]` STRIP_ASSUME_TAC THENL
[MATCH_MP_TAC COMPACT_CONVEX_COLLINEAR_SEGMENT THEN
REPEAT CONJ_TAC THENL
[DISCH_THEN SUBST_ALL_TAC THEN
RULE_ASSUM_TAC(REWRITE_RULE[AFF_DIM_EMPTY]) THEN ASM_INT_ARITH_TAC;
MATCH_MP_TAC FACE_OF_IMP_COMPACT THEN
EXISTS_TAC `p:real^3->bool` THEN
ASM_SIMP_TAC[POLYTOPE_IMP_CONVEX; POLYTOPE_IMP_COMPACT];
ASM_MESON_TAC[FACE_OF_IMP_CONVEX];
MP_TAC(ISPEC `e:real^3->bool` AFF_DIM) THEN
DISCH_THEN(X_CHOOSE_THEN `b:real^3->bool` MP_TAC) THEN
ASM_REWRITE_TAC[INT_ARITH `&1:int = b - &1 <=> b = &2`] THEN
DISCH_THEN(CONJUNCTS_THEN2 (ASSUME_TAC o SYM) MP_TAC) THEN
ASM_CASES_TAC `FINITE(b:real^3->bool)` THENL
[ALL_TAC; ASM_MESON_TAC[AFFINE_INDEPENDENT_IMP_FINITE]] THEN
REWRITE_TAC[INT_OF_NUM_EQ] THEN STRIP_TAC THEN
SUBGOAL_THEN `(b:real^3->bool) HAS_SIZE 2` MP_TAC THENL
[ASM_REWRITE_TAC[HAS_SIZE]; CONV_TAC(LAND_CONV HAS_SIZE_CONV)] THEN
REWRITE_TAC[COLLINEAR_AFFINE_HULL] THEN
REPEAT(MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC) THEN
ASM_MESON_TAC[HULL_SUBSET]];
ASM_CASES_TAC `a:real^3 = b` THENL
[UNDISCH_TAC `aff_dim(e:real^3->bool) = &1` THEN
ASM_REWRITE_TAC[SEGMENT_REFL; AFF_DIM_SING; INT_OF_NUM_EQ; ARITH_EQ];
ALL_TAC] THEN
MATCH_MP_TAC EQ_TRANS THEN
EXISTS_TAC `CARD {v:real^3 | v extreme_point_of segment[a,b]}` THEN
CONJ_TAC THENL
[MATCH_MP_TAC CARD_IMAGE_INJ_EQ THEN
EXISTS_TAC `\v:real^3. {v}` THEN
REWRITE_TAC[IN_ELIM_THM; FACE_OF_SING; AFF_DIM_SING] THEN
REPEAT CONJ_TAC THENL
[ASM_REWRITE_TAC[EXTREME_POINT_OF_SEGMENT] THEN
REWRITE_TAC[SET_RULE `{x | x = a \/ x = b} = {a,b}`] THEN
REWRITE_TAC[FINITE_INSERT; FINITE_EMPTY];
X_GEN_TAC `v:real^3` THEN REWRITE_TAC[GSYM FACE_OF_SING] THEN
ASM_MESON_TAC[FACE_OF_TRANS; FACE_OF_IMP_SUBSET];
X_GEN_TAC `s:real^3->bool` THEN REWRITE_TAC[AFF_DIM_EQ_0] THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN
DISCH_THEN(X_CHOOSE_THEN `v:real^3` SUBST_ALL_TAC) THEN
REWRITE_TAC[EXISTS_UNIQUE] THEN EXISTS_TAC `v:real^3` THEN
ASM_REWRITE_TAC[GSYM FACE_OF_SING] THEN CONJ_TAC THENL
[ASM_MESON_TAC[FACE_OF_FACE]; SET_TAC[]]];
ASM_REWRITE_TAC[EXTREME_POINT_OF_SEGMENT] THEN
REWRITE_TAC[SET_RULE `{x | x = a \/ x = b} = {a,b}`] THEN
ASM_SIMP_TAC[FINITE_INSERT; CARD_CLAUSES; FINITE_EMPTY] THEN
ASM_REWRITE_TAC[IN_SING; NOT_IN_EMPTY; ARITH]]]];
ALL_TAC] THEN
STRIP_TAC THEN
DISCH_THEN(ASSUME_TAC o MATCH_MP (ARITH_RULE
`(a + b) - c = 2 ==> a + b = c + 2`)) THEN
SUBGOAL_THEN `4 <= CARD {v:real^3->bool | v face_of p /\ aff_dim v = &0}`
ASSUME_TAC THENL
[ASM_SIMP_TAC[SIZE_ZERO_DIMENSIONAL_FACES; POLYTOPE_IMP_POLYHEDRON] THEN
MP_TAC(ISPEC `p:real^3->bool` POLYTOPE_VERTEX_LOWER_BOUND) THEN
ASM_REWRITE_TAC[] THEN CONV_TAC INT_REDUCE_CONV THEN
REWRITE_TAC[INT_OF_NUM_LE];
ALL_TAC] THEN
SUBGOAL_THEN `4 <= CARD {f:real^3->bool | f face_of p /\ aff_dim f = &2}`
ASSUME_TAC THENL
[MP_TAC(ISPEC `p:real^3->bool` POLYTOPE_FACET_LOWER_BOUND) THEN
ASM_REWRITE_TAC[] THEN CONV_TAC INT_REDUCE_CONV THEN
ASM_REWRITE_TAC[INT_OF_NUM_LE; facet_of] THEN
MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN AP_TERM_TAC THEN
GEN_REWRITE_TAC I [EXTENSION] THEN REWRITE_TAC[IN_ELIM_THM] THEN
CONV_TAC INT_REDUCE_CONV THEN GEN_TAC THEN EQ_TAC THEN
STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
ASM_MESON_TAC[INT_ARITH `~(&2:int = -- &1)`; AFF_DIM_EMPTY];
ALL_TAC] THEN
FIRST_ASSUM(MP_TAC o MATCH_MP (ARITH_RULE
`v + f = e + 2 ==> 4 <= v /\ 4 <= f ==> 6 <= e`)) THEN
ASM_REWRITE_TAC[] THEN
ASM_CASES_TAC
`CARD {e:real^3->bool | e face_of p /\ aff_dim e = &1} = 0` THEN
ASM_REWRITE_TAC[ARITH] THEN DISCH_TAC THEN
SUBGOAL_THEN `3 <= m` ASSUME_TAC THENL
[ASM_CASES_TAC `{f:real^3->bool | f face_of p /\ aff_dim f = &2} = {}` THENL
[UNDISCH_TAC
`4 <= CARD {f:real^3->bool | f face_of p /\ aff_dim f = &2}` THEN
ASM_REWRITE_TAC[CARD_CLAUSES] THEN ARITH_TAC;
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY])] THEN
REWRITE_TAC[IN_ELIM_THM] THEN
DISCH_THEN(X_CHOOSE_THEN `f:real^3->bool` MP_TAC) THEN
DISCH_THEN(fun th -> STRIP_ASSUME_TAC th THEN
FIRST_X_ASSUM(SUBST1_TAC o SYM o C MATCH_MP th)) THEN
MP_TAC(ISPEC `f:real^3->bool` POLYTOPE_FACET_LOWER_BOUND) THEN
ASM_REWRITE_TAC[facet_of] THEN CONV_TAC INT_REDUCE_CONV THEN
ANTS_TAC THENL [ASM_MESON_TAC[FACE_OF_POLYTOPE_POLYTOPE]; ALL_TAC] THEN
REWRITE_TAC[INT_OF_NUM_LE] THEN MATCH_MP_TAC EQ_IMP THEN
AP_TERM_TAC THEN AP_TERM_TAC THEN
GEN_REWRITE_TAC I [EXTENSION] THEN REWRITE_TAC[IN_ELIM_THM] THEN
CONV_TAC INT_REDUCE_CONV THEN X_GEN_TAC `e:real^3->bool` THEN
EQ_TAC THEN ASM_CASES_TAC `e:real^3->bool = {}` THEN
ASM_SIMP_TAC[AFF_DIM_EMPTY] THEN CONV_TAC INT_REDUCE_CONV THENL
[ASM_MESON_TAC[FACE_OF_TRANS; FACE_OF_IMP_SUBSET];
ASM_MESON_TAC[FACE_OF_FACE]];
ALL_TAC] THEN
FIRST_ASSUM(ASSUME_TAC o MATCH_MP (ARITH_RULE `3 <= m ==> ~(m = 0)`)) THEN
ASM_CASES_TAC `n = 0` THENL
[UNDISCH_THEN `n = 0` SUBST_ALL_TAC THEN
FIRST_X_ASSUM(MP_TAC o MATCH_MP (ARITH_RULE
`0 * x = 2 * e ==> e = 0`)) THEN ASM_REWRITE_TAC[];
ALL_TAC] THEN
FIRST_ASSUM(MP_TAC o MATCH_MP (NUM_RING
`v + f = e + 2 ==> !m n. m * n * v + n * m * f = m * n * (e + 2)`)) THEN
DISCH_THEN(MP_TAC o SPECL [`m:num`; `n:num`]) THEN ASM_REWRITE_TAC[] THEN
REWRITE_TAC[ARITH_RULE `m * 2 * e + n * 2 * e = m * n * (e + 2) <=>
e * 2 * (m + n) = m * n * (e + 2)`] THEN
ABBREV_TAC `E = CARD {e:real^3->bool | e face_of p /\ aff_dim e = &1}` THEN
ASM_CASES_TAC `n = 1` THENL
[ASM_REWRITE_TAC[MULT_CLAUSES; ARITH_RULE
`E * 2 * (n + 1) = n * (E + 2) <=> E * (n + 2) = 2 * n`] THEN
MATCH_MP_TAC(TAUT `~p ==> p ==> q`) THEN
MATCH_MP_TAC(ARITH_RULE `n:num < m ==> ~(m = n)`) THEN
MATCH_MP_TAC LTE_TRANS THEN EXISTS_TAC `2 * (m + 2)` THEN
CONJ_TAC THENL [ARITH_TAC; MATCH_MP_TAC LE_MULT2 THEN ASM_ARITH_TAC];
ALL_TAC] THEN
ASM_CASES_TAC `n = 2` THENL
[ASM_REWRITE_TAC[ARITH_RULE `E * 2 * (n + 2) = n * 2 * (E + 2) <=>
E = n`] THEN
DISCH_THEN(SUBST_ALL_TAC o SYM) THEN
FIRST_X_ASSUM(MP_TAC o MATCH_MP (NUM_RING
`E * c = 2 * E ==> E = 0 \/ c = 2`)) THEN
ASM_ARITH_TAC;
ALL_TAC] THEN
SUBGOAL_THEN `3 <= n` ASSUME_TAC THENL [ASM_ARITH_TAC; ALL_TAC] THEN
ASM_CASES_TAC `m * n < 2 * (m + n)` THENL
[DISCH_TAC;
DISCH_TAC THEN RULE_ASSUM_TAC(REWRITE_RULE[NOT_LT]) THEN
SUBGOAL_THEN `E * 2 * (m + n) <= E * m * n` MP_TAC THENL
[REWRITE_TAC[LE_MULT_LCANCEL] THEN ASM_ARITH_TAC;
ASM_REWRITE_TAC[ARITH_RULE `m * n * (E + 2) <= E * m * n <=>
2 * m * n = 0`] THEN
MATCH_MP_TAC(TAUT `~p ==> p ==> q`) THEN
REWRITE_TAC[MULT_EQ_0] THEN ASM_ARITH_TAC]] THEN
SUBGOAL_THEN `&m - &2:real < &4 /\ &n - &2 < &4` MP_TAC THENL
[CONJ_TAC THENL
[MATCH_MP_TAC REAL_LT_RCANCEL_IMP THEN EXISTS_TAC `&n - &2`;
MATCH_MP_TAC REAL_LT_LCANCEL_IMP THEN EXISTS_TAC `&m - &2`] THEN
ASM_SIMP_TAC[REAL_SUB_LT; REAL_OF_NUM_LT;
ARITH_RULE `2 < n <=> 3 <= n`] THEN
MATCH_MP_TAC REAL_LTE_TRANS THEN EXISTS_TAC `&4` THEN
REWRITE_TAC[REAL_ARITH `(m - &2) * (n - &2) < &4 <=>
m * n < &2 * (m + n)`] THEN
ASM_REWRITE_TAC[REAL_OF_NUM_MUL; REAL_OF_NUM_ADD; REAL_OF_NUM_LT] THEN
REWRITE_TAC[REAL_SUB_LDISTRIB; REAL_SUB_RDISTRIB; REAL_LE_SUB_LADD] THEN
REWRITE_TAC[REAL_OF_NUM_MUL; REAL_OF_NUM_ADD; REAL_OF_NUM_LE] THEN
ASM_ARITH_TAC;
ALL_TAC] THEN
REWRITE_TAC[REAL_LT_SUB_RADD; REAL_OF_NUM_ADD; REAL_OF_NUM_LT] THEN
REWRITE_TAC[ARITH_RULE `m < 4 + 2 <=> m <= 5`] THEN
ASM_SIMP_TAC[ARITH_RULE
`3 <= m ==> (m <= 5 <=> m = 3 \/ m = 4 \/ m = 5)`] THEN
STRIP_TAC THEN UNDISCH_TAC `E * 2 * (m + n) = m * n * (E + 2)` THEN
ASM_REWRITE_TAC[] THEN CONV_TAC NUM_REDUCE_CONV THEN ASM_ARITH_TAC);;
(* ------------------------------------------------------------------------- *)
(* If-and-only-if version. *)
(* ------------------------------------------------------------------------- *)
let PLATONIC_SOLIDS = prove
(`!m n.
(?p:real^3->bool.
polytope p /\ aff_dim p = &3 /\
(!f. f face_of p /\ aff_dim(f) = &2
==> CARD {e | e face_of p /\ aff_dim(e) = &1 /\ e SUBSET f} = m) /\
(!v. v face_of p /\ aff_dim(v) = &0
==> CARD {e | e face_of p /\ aff_dim(e) = &1 /\ v SUBSET e} = n)) <=>
m = 3 /\ n = 3 \/ // Tetrahedron
m = 4 /\ n = 3 \/ // Cube
m = 3 /\ n = 4 \/ // Octahedron
m = 5 /\ n = 3 \/ // Dodecahedron
m = 3 /\ n = 5 // Icosahedron`,
REPEAT GEN_TAC THEN EQ_TAC THEN
REWRITE_TAC[LEFT_IMP_EXISTS_THM; PLATONIC_SOLIDS_LIMITS] THEN
STRIP_TAC THENL
[EXISTS_TAC `std_tetrahedron` THEN
ASM_REWRITE_TAC[TETRAHEDRON_EDGES_PER_VERTEX; TETRAHEDRON_EDGES_PER_FACE;
STD_TETRAHEDRON_FULLDIM] THEN
REWRITE_TAC[std_tetrahedron] THEN MATCH_MP_TAC POLYTOPE_CONVEX_HULL THEN
REWRITE_TAC[FINITE_INSERT; FINITE_EMPTY];
EXISTS_TAC `std_cube` THEN
ASM_REWRITE_TAC[CUBE_EDGES_PER_VERTEX; CUBE_EDGES_PER_FACE;
STD_CUBE_FULLDIM] THEN
REWRITE_TAC[std_cube] THEN MATCH_MP_TAC POLYTOPE_CONVEX_HULL THEN
REWRITE_TAC[FINITE_INSERT; FINITE_EMPTY];
EXISTS_TAC `std_octahedron` THEN
ASM_REWRITE_TAC[OCTAHEDRON_EDGES_PER_VERTEX; OCTAHEDRON_EDGES_PER_FACE;
STD_OCTAHEDRON_FULLDIM] THEN
REWRITE_TAC[std_octahedron] THEN MATCH_MP_TAC POLYTOPE_CONVEX_HULL THEN
REWRITE_TAC[FINITE_INSERT; FINITE_EMPTY];
EXISTS_TAC `std_dodecahedron` THEN
ASM_REWRITE_TAC[DODECAHEDRON_EDGES_PER_VERTEX; DODECAHEDRON_EDGES_PER_FACE;
STD_DODECAHEDRON_FULLDIM] THEN
REWRITE_TAC[STD_DODECAHEDRON] THEN MATCH_MP_TAC POLYTOPE_CONVEX_HULL THEN
REWRITE_TAC[FINITE_INSERT; FINITE_EMPTY];
EXISTS_TAC `std_icosahedron` THEN
ASM_REWRITE_TAC[ICOSAHEDRON_EDGES_PER_VERTEX; ICOSAHEDRON_EDGES_PER_FACE;
STD_ICOSAHEDRON_FULLDIM] THEN
REWRITE_TAC[STD_ICOSAHEDRON] THEN MATCH_MP_TAC POLYTOPE_CONVEX_HULL THEN
REWRITE_TAC[FINITE_INSERT; FINITE_EMPTY]]);;
(* ------------------------------------------------------------------------- *)
(* Show that the regular polyhedra do have all edges and faces congruent. *)
(* ------------------------------------------------------------------------- *)
parse_as_infix("congruent",(12,"right"));;
let congruent = new_definition
`(s:real^N->bool) congruent (t:real^N->bool) <=>
?c f. orthogonal_transformation f /\ t = IMAGE (\x. c + f x) s`;;
let CONGRUENT_SIMPLE = prove
(`(?A:real^3^3. orthogonal_matrix A /\ IMAGE (\x:real^3. A ** x) s = t)
==> (convex hull s) congruent (convex hull t)`,
REPEAT GEN_TAC THEN
DISCH_THEN(CHOOSE_THEN (CONJUNCTS_THEN2 ASSUME_TAC (SUBST1_TAC o SYM))) THEN
SIMP_TAC[CONVEX_HULL_LINEAR_IMAGE; MATRIX_VECTOR_MUL_LINEAR] THEN
REWRITE_TAC[congruent] THEN EXISTS_TAC `vec 0:real^3` THEN
EXISTS_TAC `\x:real^3. (A:real^3^3) ** x` THEN
REWRITE_TAC[VECTOR_ADD_LID; ORTHOGONAL_TRANSFORMATION_MATRIX] THEN
ASM_SIMP_TAC[MATRIX_OF_MATRIX_VECTOR_MUL; MATRIX_VECTOR_MUL_LINEAR]);;
let CONGRUENT_SEGMENTS = prove
(`!a b c d:real^N.
dist(a,b) = dist(c,d)
==> segment[a,b] congruent segment[c,d]`,
REPEAT STRIP_TAC THEN
MP_TAC(ISPECL [`b - a:real^N`; `d - c:real^N`]
ORTHOGONAL_TRANSFORMATION_EXISTS) THEN
ANTS_TAC THENL [POP_ASSUM MP_TAC THEN NORM_ARITH_TAC; ALL_TAC] THEN
DISCH_THEN(X_CHOOSE_THEN `f:real^N->real^N` STRIP_ASSUME_TAC) THEN
REWRITE_TAC[congruent] THEN
EXISTS_TAC `c - (f:real^N->real^N) a` THEN
EXISTS_TAC `f:real^N->real^N` THEN
FIRST_ASSUM(ASSUME_TAC o MATCH_MP ORTHOGONAL_TRANSFORMATION_LINEAR) THEN
SUBGOAL_THEN
`(\x. (c - f a) + (f:real^N->real^N) x) = (\x. (c - f a) + x) o f`
SUBST1_TAC THENL [REWRITE_TAC[FUN_EQ_THM; o_THM]; ALL_TAC] THEN
ASM_SIMP_TAC[GSYM CONVEX_HULL_LINEAR_IMAGE; SEGMENT_CONVEX_HULL; IMAGE_o;
GSYM CONVEX_HULL_TRANSLATION] THEN
REWRITE_TAC[IMAGE_CLAUSES] THEN
AP_TERM_TAC THEN BINOP_TAC THENL
[VECTOR_ARITH_TAC; AP_THM_TAC THEN AP_TERM_TAC] THEN
REWRITE_TAC[VECTOR_ARITH `d:real^N = c - a + b <=> b - a = d - c`] THEN
ASM_MESON_TAC[LINEAR_SUB]);;
let compute_dist =
let mk_sub = mk_binop `(-):real^3->real^3->real^3`
and dot_tm = `(dot):real^3->real^3->real` in
fun v1 v2 -> let vth = VECTOR3_SUB_CONV(mk_sub v1 v2) in
let dth = CONV_RULE(RAND_CONV VECTOR3_DOT_CONV)
(MK_COMB(AP_TERM dot_tm vth,vth)) in
rand(concl dth);;
let le_rat5 =
let mk_le = mk_binop `(<=):real->real->bool` and t_tm = `T` in
fun r1 r2 -> rand(concl(REAL_RAT5_LE_CONV(mk_le r1 r2))) = t_tm;;
let three_adjacent_points s =
match s with
| x::t -> let (y,_)::(z,_)::_ =
sort (fun (_,r1) (_,r2) -> le_rat5 r1 r2)
(map (fun y -> y,compute_dist x y) t) in
x,y,z
| _ -> failwith "three_adjacent_points: no points";;
let mk_33matrix =
let a11_tm = `a11:real`
and a12_tm = `a12:real`
and a13_tm = `a13:real`
and a21_tm = `a21:real`
and a22_tm = `a22:real`
and a23_tm = `a23:real`
and a31_tm = `a31:real`
and a32_tm = `a32:real`
and a33_tm = `a33:real`
and pat_tm =
`vector[vector[a11; a12; a13];
vector[a21; a22; a23];
vector[a31; a32; a33]]:real^3^3` in
fun [a11;a12;a13;a21;a22;a23;a31;a32;a33] ->
vsubst[a11,a11_tm;
a12,a12_tm;
a13,a13_tm;
a21,a21_tm;
a22,a22_tm;
a23,a23_tm;
a31,a31_tm;
a32,a32_tm;
a33,a33_tm] pat_tm;;
let MATRIX_VECTOR_MUL_3 = prove
(`(vector[vector[a11;a12;a13];
vector[a21; a22; a23];
vector[a31; a32; a33]]:real^3^3) **
(vector[x1;x2;x3]:real^3) =
vector[a11 * x1 + a12 * x2 + a13 * x3;
a21 * x1 + a22 * x2 + a23 * x3;
a31 * x1 + a32 * x2 + a33 * x3]`,
SIMP_TAC[CART_EQ; matrix_vector_mul; LAMBDA_BETA] THEN
REWRITE_TAC[DIMINDEX_3; FORALL_3; SUM_3; VECTOR_3]);;
let MATRIX_LEMMA = prove
(`!A:real^3^3.
A ** x1 = x2 /\
A ** y1 = y2 /\
A ** z1 = z2 <=>
(vector [x1; y1; z1]:real^3^3) ** (row 1 A:real^3) =
vector [x2$1; y2$1; z2$1] /\
(vector [x1; y1; z1]:real^3^3) ** (row 2 A:real^3) =
vector [x2$2; y2$2; z2$2] /\
(vector [x1; y1; z1]:real^3^3) ** (row 3 A:real^3) =
vector [x2$3; y2$3; z2$3]`,
SIMP_TAC[CART_EQ; transp; matrix_vector_mul; row; VECTOR_3; LAMBDA_BETA] THEN
REWRITE_TAC[FORALL_3; DIMINDEX_3; VECTOR_3; SUM_3] THEN REAL_ARITH_TAC);;
let MATRIX_BY_CRAMER_LEMMA = prove
(`!A:real^3^3.
~(det(vector[x1; y1; z1]:real^3^3) = &0)
==> (A ** x1 = x2 /\
A ** y1 = y2 /\
A ** z1 = z2 <=>
A = lambda m k. det((lambda i j.
if j = k
then (vector[x2$m; y2$m; z2$m]:real^3)$i
else (vector[x1; y1; z1]:real^3^3)$i$j)
:real^3^3) /
det(vector[x1;y1;z1]:real^3^3))`,
REPEAT STRIP_TAC THEN GEN_REWRITE_TAC LAND_CONV [MATRIX_LEMMA] THEN
ASM_SIMP_TAC[CRAMER] THEN REWRITE_TAC[CART_EQ; row] THEN
SIMP_TAC[LAMBDA_BETA] THEN REWRITE_TAC[DIMINDEX_3; FORALL_3]);;
let MATRIX_BY_CRAMER = prove
(`!A:real^3^3 x1 y1 z1 x2 y2 z2.
let d = det(vector[x1; y1; z1]:real^3^3) in
~(d = &0)
==> (A ** x1 = x2 /\
A ** y1 = y2 /\
A ** z1 = z2 <=>
A$1$1 =
(x2$1 * y1$2 * z1$3 +
x1$2 * y1$3 * z2$1 +
x1$3 * y2$1 * z1$2 -
x2$1 * y1$3 * z1$2 -
x1$2 * y2$1 * z1$3 -
x1$3 * y1$2 * z2$1) / d /\
A$1$2 =
(x1$1 * y2$1 * z1$3 +
x2$1 * y1$3 * z1$1 +
x1$3 * y1$1 * z2$1 -
x1$1 * y1$3 * z2$1 -
x2$1 * y1$1 * z1$3 -
x1$3 * y2$1 * z1$1) / d /\
A$1$3 =
(x1$1 * y1$2 * z2$1 +
x1$2 * y2$1 * z1$1 +
x2$1 * y1$1 * z1$2 -
x1$1 * y2$1 * z1$2 -
x1$2 * y1$1 * z2$1 -
x2$1 * y1$2 * z1$1) / d /\
A$2$1 =
(x2$2 * y1$2 * z1$3 +
x1$2 * y1$3 * z2$2 +
x1$3 * y2$2 * z1$2 -
x2$2 * y1$3 * z1$2 -
x1$2 * y2$2 * z1$3 -
x1$3 * y1$2 * z2$2) / d /\
A$2$2 =
(x1$1 * y2$2 * z1$3 +
x2$2 * y1$3 * z1$1 +
x1$3 * y1$1 * z2$2 -
x1$1 * y1$3 * z2$2 -
x2$2 * y1$1 * z1$3 -
x1$3 * y2$2 * z1$1) / d /\
A$2$3 =
(x1$1 * y1$2 * z2$2 +
x1$2 * y2$2 * z1$1 +
x2$2 * y1$1 * z1$2 -
x1$1 * y2$2 * z1$2 -
x1$2 * y1$1 * z2$2 -
x2$2 * y1$2 * z1$1) / d /\
A$3$1 =
(x2$3 * y1$2 * z1$3 +
x1$2 * y1$3 * z2$3 +
x1$3 * y2$3 * z1$2 -
x2$3 * y1$3 * z1$2 -
x1$2 * y2$3 * z1$3 -
x1$3 * y1$2 * z2$3) / d /\
A$3$2 =
(x1$1 * y2$3 * z1$3 +
x2$3 * y1$3 * z1$1 +
x1$3 * y1$1 * z2$3 -
x1$1 * y1$3 * z2$3 -
x2$3 * y1$1 * z1$3 -
x1$3 * y2$3 * z1$1) / d /\
A$3$3 =
(x1$1 * y1$2 * z2$3 +
x1$2 * y2$3 * z1$1 +
x2$3 * y1$1 * z1$2 -
x1$1 * y2$3 * z1$2 -
x1$2 * y1$1 * z2$3 -
x2$3 * y1$2 * z1$1) / d)`,
REPEAT GEN_TAC THEN CONV_TAC let_CONV THEN DISCH_TAC THEN
ASM_SIMP_TAC[MATRIX_BY_CRAMER_LEMMA] THEN
REWRITE_TAC[DET_3; CART_EQ] THEN
SIMP_TAC[LAMBDA_BETA; DIMINDEX_3; ARITH; VECTOR_3] THEN
REWRITE_TAC[FORALL_3; ARITH; VECTOR_3] THEN REWRITE_TAC[CONJ_ACI]);;
let CONGRUENT_EDGES_TAC edges =
REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN REWRITE_TAC[IMP_IMP] THEN
REWRITE_TAC[edges] THEN
REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[GSYM SEGMENT_CONVEX_HULL] THEN
MATCH_MP_TAC CONGRUENT_SEGMENTS THEN REWRITE_TAC[DIST_EQ] THEN
REWRITE_TAC[dist; NORM_POW_2] THEN
CONV_TAC(ONCE_DEPTH_CONV VECTOR3_SUB_CONV) THEN
CONV_TAC(ONCE_DEPTH_CONV VECTOR3_DOT_CONV) THEN
CONV_TAC(ONCE_DEPTH_CONV REAL_RAT5_EQ_CONV) THEN
REWRITE_TAC[];;
let CONGRUENT_FACES_TAC facets =
REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN REWRITE_TAC[IMP_IMP] THEN
REWRITE_TAC[facets] THEN
REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
W(fun (asl,w) ->
let t1 = rand(lhand w) and t2 = rand(rand w) in
let (x1,y1,z1) = three_adjacent_points (dest_setenum t1)
and (x2,y2,z2) = three_adjacent_points (dest_setenum t2) in
let th1 = SPECL [`A:real^3^3`;x1;y1;z1;x2;y2;z2] MATRIX_BY_CRAMER in
let th2 = REWRITE_RULE[VECTOR_3; DET_3] th1 in
let th3 = CONV_RULE (DEPTH_CONV REAL_RAT5_MUL_CONV) th2 in
let th4 = CONV_RULE (DEPTH_CONV
(REAL_RAT5_ADD_CONV ORELSEC REAL_RAT5_SUB_CONV)) th3 in
let th5 = CONV_RULE let_CONV th4 in
let th6 = CONV_RULE(ONCE_DEPTH_CONV REAL_RAT5_DIV_CONV) th5 in
let th7 = CONV_RULE(ONCE_DEPTH_CONV REAL_RAT5_EQ_CONV) th6 in
let th8 = MP th7 (EQT_ELIM(REWRITE_CONV[] (lhand(concl th7)))) in
let tms = map rhs (conjuncts(rand(concl th8))) in
let matt = mk_33matrix tms in
MATCH_MP_TAC CONGRUENT_SIMPLE THEN EXISTS_TAC matt THEN CONJ_TAC THENL
[REWRITE_TAC[ORTHOGONAL_MATRIX; CART_EQ] THEN
SIMP_TAC[transp; LAMBDA_BETA; matrix_mul; mat] THEN
REWRITE_TAC[DIMINDEX_3; SUM_3; FORALL_3; VECTOR_3; ARITH] THEN
CONV_TAC(ONCE_DEPTH_CONV REAL_RAT5_MUL_CONV) THEN
CONV_TAC(DEPTH_CONV REAL_RAT5_ADD_CONV) THEN
CONV_TAC(ONCE_DEPTH_CONV REAL_RAT5_EQ_CONV) THEN
REWRITE_TAC[] THEN NO_TAC;
REWRITE_TAC[IMAGE_CLAUSES; MATRIX_VECTOR_MUL_3] THEN
CONV_TAC(ONCE_DEPTH_CONV REAL_RAT5_MUL_CONV) THEN
CONV_TAC(DEPTH_CONV REAL_RAT5_ADD_CONV) THEN
REWRITE_TAC[INSERT_AC]]);;
let TETRAHEDRON_CONGRUENT_EDGES = prove
(`!e1 e2. e1 face_of std_tetrahedron /\ aff_dim e1 = &1 /\
e2 face_of std_tetrahedron /\ aff_dim e2 = &1
==> e1 congruent e2`,
CONGRUENT_EDGES_TAC TETRAHEDRON_EDGES);;
let TETRAHEDRON_CONGRUENT_FACETS = prove
(`!f1 f2. f1 face_of std_tetrahedron /\ aff_dim f1 = &2 /\
f2 face_of std_tetrahedron /\ aff_dim f2 = &2
==> f1 congruent f2`,
CONGRUENT_FACES_TAC TETRAHEDRON_FACETS);;
let CUBE_CONGRUENT_EDGES = prove
(`!e1 e2. e1 face_of std_cube /\ aff_dim e1 = &1 /\
e2 face_of std_cube /\ aff_dim e2 = &1
==> e1 congruent e2`,
CONGRUENT_EDGES_TAC CUBE_EDGES);;
let CUBE_CONGRUENT_FACETS = prove
(`!f1 f2. f1 face_of std_cube /\ aff_dim f1 = &2 /\
f2 face_of std_cube /\ aff_dim f2 = &2
==> f1 congruent f2`,
CONGRUENT_FACES_TAC CUBE_FACETS);;
let OCTAHEDRON_CONGRUENT_EDGES = prove
(`!e1 e2. e1 face_of std_octahedron /\ aff_dim e1 = &1 /\
e2 face_of std_octahedron /\ aff_dim e2 = &1
==> e1 congruent e2`,
CONGRUENT_EDGES_TAC OCTAHEDRON_EDGES);;
let OCTAHEDRON_CONGRUENT_FACETS = prove
(`!f1 f2. f1 face_of std_octahedron /\ aff_dim f1 = &2 /\
f2 face_of std_octahedron /\ aff_dim f2 = &2
==> f1 congruent f2`,
CONGRUENT_FACES_TAC OCTAHEDRON_FACETS);;
let DODECAHEDRON_CONGRUENT_EDGES = prove
(`!e1 e2. e1 face_of std_dodecahedron /\ aff_dim e1 = &1 /\
e2 face_of std_dodecahedron /\ aff_dim e2 = &1
==> e1 congruent e2`,
CONGRUENT_EDGES_TAC DODECAHEDRON_EDGES);;
let DODECAHEDRON_CONGRUENT_FACETS = prove
(`!f1 f2. f1 face_of std_dodecahedron /\ aff_dim f1 = &2 /\
f2 face_of std_dodecahedron /\ aff_dim f2 = &2
==> f1 congruent f2`,
CONGRUENT_FACES_TAC DODECAHEDRON_FACETS);;
let ICOSAHEDRON_CONGRUENT_EDGES = prove
(`!e1 e2. e1 face_of std_icosahedron /\ aff_dim e1 = &1 /\
e2 face_of std_icosahedron /\ aff_dim e2 = &1
==> e1 congruent e2`,
CONGRUENT_EDGES_TAC ICOSAHEDRON_EDGES);;
let ICOSAHEDRON_CONGRUENT_FACETS = prove
(`!f1 f2. f1 face_of std_icosahedron /\ aff_dim f1 = &2 /\
f2 face_of std_icosahedron /\ aff_dim f2 = &2
==> f1 congruent f2`,
CONGRUENT_FACES_TAC ICOSAHEDRON_FACETS);;
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