/
pick.ml
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pick.ml
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(* ========================================================================= *)
(* Pick's theorem. *)
(* ========================================================================= *)
needs "Multivariate/polytope.ml";;
needs "Multivariate/measure.ml";;
needs "Multivariate/moretop.ml";;
prioritize_real();;
(* ------------------------------------------------------------------------- *)
(* Misc lemmas. *)
(* ------------------------------------------------------------------------- *)
let COLLINEAR_IMP_NEGLIGIBLE = prove
(`!s:real^2->bool. collinear s ==> negligible s`,
REWRITE_TAC[COLLINEAR_AFFINE_HULL] THEN
MESON_TAC[NEGLIGIBLE_AFFINE_HULL_2; NEGLIGIBLE_SUBSET]);;
let CONVEX_HULL_3_0 = prove
(`!a b:real^N.
convex hull {vec 0,a,b} =
{x % a + y % b | &0 <= x /\ &0 <= y /\ x + y <= &1}`,
REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[SET_RULE `{c,a,b} = {a,b,c}`] THEN
REWRITE_TAC[CONVEX_HULL_3; EXTENSION; IN_ELIM_THM] THEN
X_GEN_TAC `y:real^N` THEN
AP_TERM_TAC THEN REWRITE_TAC[FUN_EQ_THM] THEN X_GEN_TAC `x:real` THEN
AP_TERM_TAC THEN REWRITE_TAC[FUN_EQ_THM] THEN X_GEN_TAC `y:real` THEN
REWRITE_TAC[VECTOR_MUL_RZERO; VECTOR_ADD_RID] THEN
EQ_TAC THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THENL
[ASM_ARITH_TAC; EXISTS_TAC `&1 - x - y` THEN ASM_ARITH_TAC]);;
let INTERIOR_CONVEX_HULL_3_0 = prove
(`!a b:real^2.
~(collinear {vec 0,a,b})
==> interior(convex hull {vec 0,a,b}) =
{x % a + y % b | &0 < x /\ &0 < y /\ x + y < &1}`,
REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[SET_RULE `{c,a,b} = {a,b,c}`] THEN
STRIP_TAC THEN ASM_SIMP_TAC[INTERIOR_CONVEX_HULL_3] THEN
REWRITE_TAC[TAUT `a /\ x = &1 /\ b <=> x = &1 /\ a /\ b`] THEN
REWRITE_TAC[VECTOR_MUL_RZERO; VECTOR_ADD_RID] THEN
REWRITE_TAC[REAL_ARITH `x + y + z = &1 <=> &1 - x - y = z`; UNWIND_THM1] THEN
REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN
GEN_TAC THEN REPEAT(AP_TERM_TAC THEN ABS_TAC) THEN
EQ_TAC THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
ASM_REAL_ARITH_TAC);;
let MEASURE_CONVEX_HULL_2_TRIVIAL = prove
(`(!a:real^2. measure(convex hull {a}) = &0) /\
(!a b:real^2. measure(convex hull {a,b}) = &0)`,
REPEAT STRIP_TAC THEN
MATCH_MP_TAC MEASURE_EQ_0 THEN
MATCH_MP_TAC COLLINEAR_IMP_NEGLIGIBLE THEN
REWRITE_TAC[GSYM SEGMENT_CONVEX_HULL; CONVEX_HULL_SING] THEN
REWRITE_TAC[COLLINEAR_SING; COLLINEAR_SEGMENT]);;
let NEGLIGIBLE_SEGMENT_2 = prove
(`!a b:real^2. negligible(segment[a,b])`,
SIMP_TAC[COLLINEAR_IMP_NEGLIGIBLE; COLLINEAR_SEGMENT]);;
(* ------------------------------------------------------------------------- *)
(* Decomposing an additive function on a triangle. *)
(* ------------------------------------------------------------------------- *)
let TRIANGLE_DECOMPOSITION = prove
(`!a b c d:real^2.
d IN convex hull {a,b,c}
==> (convex hull {a,b,c} =
convex hull {d,b,c} UNION
convex hull {d,a,c} UNION
convex hull {d,a,b})`,
REPEAT STRIP_TAC THEN
MATCH_MP_TAC SUBSET_ANTISYM THEN REWRITE_TAC[UNION_SUBSET] THEN
CONJ_TAC THENL
[REWRITE_TAC[SUBSET] THEN X_GEN_TAC `x:real^2` THEN DISCH_TAC THEN
MP_TAC(ISPECL [`{a:real^2,b,c}`; `d:real^2`; `x:real^2`]
IN_CONVEX_HULL_EXCHANGE) THEN
ASM_REWRITE_TAC[EXISTS_IN_INSERT; NOT_IN_EMPTY; IN_UNION] THEN
REPEAT(MATCH_MP_TAC MONO_OR THEN CONJ_TAC) THEN
SPEC_TAC(`x:real^2`,`x:real^2`) THEN REWRITE_TAC[GSYM SUBSET] THEN
MATCH_MP_TAC HULL_MONO THEN SET_TAC[];
SIMP_TAC[SUBSET_HULL; CONVEX_CONVEX_HULL] THEN
REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET] THEN
ASM_SIMP_TAC[HULL_INC; IN_INSERT]]);;
let TRIANGLE_ADDITIVE_DECOMPOSITION = prove
(`!f:(real^2->bool)->real a b c d.
(!s t. compact s /\ compact t
==> f(s UNION t) = f(s) + f(t) - f(s INTER t)) /\
~(a = b) /\ ~(a = c) /\ ~(b = c) /\
~affine_dependent {a,b,c} /\ d IN convex hull {a,b,c}
==> f(convex hull {a,b,c}) =
(f(convex hull {a,b,d}) +
f(convex hull {a,c,d}) +
f(convex hull {b,c,d})) -
(f(convex hull {a,d}) +
f(convex hull {b,d}) +
f(convex hull {c,d})) +
f(convex hull {d})`,
REPEAT STRIP_TAC THEN
FIRST_ASSUM(SUBST1_TAC o MATCH_MP TRIANGLE_DECOMPOSITION) THEN
ASM (CONV_TAC o GEN_SIMPLIFY_CONV TOP_DEPTH_SQCONV (basic_ss []) 5)
[COMPACT_UNION; COMPACT_INTER; COMPACT_CONVEX_HULL;
FINITE_IMP_COMPACT; FINITE_INSERT; FINITE_EMPTY;
UNION_OVER_INTER] THEN
MP_TAC(ISPECL [`{a:real^2,b,c}`; `d:real^2`]
CONVEX_HULL_EXCHANGE_INTER) THEN
ASM_REWRITE_TAC[] THEN
SIMP_TAC[INSERT_SUBSET; EMPTY_SUBSET; IN_INSERT; NOT_IN_EMPTY;
SET_RULE `s SUBSET u /\ t SUBSET u ==> (s INTER t) SUBSET u`] THEN
ASM_REWRITE_TAC[INSERT_INTER; IN_INSERT; NOT_IN_EMPTY; INTER_EMPTY] THEN
DISCH_TAC THEN REWRITE_TAC[INSERT_AC] THEN REAL_ARITH_TAC);;
(* ------------------------------------------------------------------------- *)
(* Vectors all of whose coordinates are integers. *)
(* ------------------------------------------------------------------------- *)
let integral_vector = define
`integral_vector(x:real^N) <=>
!i. 1 <= i /\ i <= dimindex(:N) ==> integer(x$i)`;;
let INTEGRAL_VECTOR_VEC = prove
(`!n. integral_vector(vec n)`,
REWRITE_TAC[integral_vector; VEC_COMPONENT; INTEGER_CLOSED]);;
let INTEGRAL_VECTOR_STDBASIS = prove
(`!i. integral_vector(basis i:real^N)`,
REWRITE_TAC[integral_vector] THEN
REPEAT STRIP_TAC THEN ASM_SIMP_TAC[BASIS_COMPONENT] THEN
COND_CASES_TAC THEN REWRITE_TAC[INTEGER_CLOSED]);;
let INTEGRAL_VECTOR_ADD = prove
(`!x y:real^N.
integral_vector x /\ integral_vector y ==> integral_vector(x + y)`,
SIMP_TAC[integral_vector; VECTOR_ADD_COMPONENT; INTEGER_CLOSED]);;
let INTEGRAL_VECTOR_SUB = prove
(`!x y:real^N.
integral_vector x /\ integral_vector y ==> integral_vector(x - y)`,
SIMP_TAC[integral_vector; VECTOR_SUB_COMPONENT; INTEGER_CLOSED]);;
let INTEGRAL_VECTOR_ADD_LCANCEL = prove
(`!x y:real^N.
integral_vector x ==> (integral_vector(x + y) <=> integral_vector y)`,
MESON_TAC[INTEGRAL_VECTOR_ADD; INTEGRAL_VECTOR_SUB;
VECTOR_ARITH `(x + y) - x:real^N = y`]);;
let FINITE_BOUNDED_INTEGER_POINTS = prove
(`!s:real^N->bool. bounded s ==> FINITE {x | x IN s /\ integral_vector x}`,
REPEAT STRIP_TAC THEN
FIRST_ASSUM(MP_TAC o MATCH_MP BOUNDED_SUBSET_CLOSED_INTERVAL) THEN
REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN
MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real^N`] THEN
REWRITE_TAC[SUBSET; IN_INTERVAL; integral_vector] THEN DISCH_TAC THEN
MATCH_MP_TAC FINITE_SUBSET THEN
EXISTS_TAC `{x:real^N | !i. 1 <= i /\ i <= dimindex(:N)
==> integer(x$i) /\
(a:real^N)$i <= x$i /\ x$i <= (b:real^N)$i}` THEN
CONJ_TAC THENL
[MATCH_MP_TAC FINITE_CART THEN REWRITE_TAC[FINITE_INTSEG];
ASM SET_TAC[]]);;
let FINITE_TRIANGLE_INTEGER_POINTS = prove
(`!a b c:real^N. FINITE {x | x IN convex hull {a,b,c} /\ integral_vector x}`,
REPEAT GEN_TAC THEN MATCH_MP_TAC FINITE_BOUNDED_INTEGER_POINTS THEN
SIMP_TAC[FINITE_IMP_BOUNDED_CONVEX_HULL; FINITE_INSERT; FINITE_EMPTY]);;
(* ------------------------------------------------------------------------- *)
(* Properties of a basis for the integer lattice. *)
(* ------------------------------------------------------------------------- *)
let LINEAR_INTEGRAL_VECTOR = prove
(`!f:real^N->real^N.
linear f
==> ((!x. integral_vector x ==> integral_vector(f x)) <=>
(!i j. 1 <= i /\ i <= dimindex(:N) /\
1 <= j /\ j <= dimindex(:N)
==> integer(matrix f$i$j)))`,
REPEAT STRIP_TAC THEN
FIRST_ASSUM(fun th -> ONCE_REWRITE_TAC[GSYM(MATCH_MP MATRIX_WORKS th)]) THEN
ABBREV_TAC `M = matrix(f:real^N->real^N)` THEN
SIMP_TAC[integral_vector; matrix_vector_mul; LAMBDA_BETA] THEN
EQ_TAC THEN REPEAT GEN_TAC THEN DISCH_TAC THENL
[MAP_EVERY X_GEN_TAC [`i:num`; `j:num`] THEN STRIP_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPEC `basis j:real^N`) THEN
REWRITE_TAC[GSYM integral_vector; INTEGRAL_VECTOR_STDBASIS] THEN
DISCH_THEN(MP_TAC o SPEC `i:num`) THEN ASM_REWRITE_TAC[] THEN
ASM_SIMP_TAC[BASIS_COMPONENT; COND_RAND; COND_RATOR] THEN
ASM_REWRITE_TAC[REAL_MUL_RZERO; SUM_DELTA; IN_NUMSEG; REAL_MUL_RID];
X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
X_GEN_TAC `i:num` THEN STRIP_TAC THEN
MATCH_MP_TAC INTEGER_SUM THEN
ASM_SIMP_TAC[INTEGER_CLOSED; IN_NUMSEG]]);;
let INTEGRAL_BASIS_UNIMODULAR = prove
(`!f:real^N->real^N.
linear f /\ IMAGE f integral_vector = integral_vector
==> abs(det(matrix f)) = &1`,
REWRITE_TAC[GSYM SUBSET_ANTISYM_EQ; SUBSET; FORALL_IN_IMAGE] THEN
REWRITE_TAC[IN_IMAGE] THEN REWRITE_TAC[IN] THEN REPEAT STRIP_TAC THEN
SUBGOAL_THEN
`!i j. 1 <= i /\ i <= dimindex(:N) /\
1 <= j /\ j <= dimindex(:N)
==> integer(matrix(f:real^N->real^N)$i$j)`
ASSUME_TAC THENL [ASM_SIMP_TAC[GSYM LINEAR_INTEGRAL_VECTOR]; ALL_TAC] THEN
SUBGOAL_THEN
`?g:real^N->real^N. linear g /\ (!x. g(f x) = x) /\ (!y. f(g y) = y)`
STRIP_ASSUME_TAC THENL
[MATCH_MP_TAC LINEAR_BIJECTIVE_LEFT_RIGHT_INVERSE THEN ASM_SIMP_TAC[] THEN
MATCH_MP_TAC(TAUT `(b ==> a) /\ b ==> a /\ b`) THEN CONJ_TAC THENL
[ASM_MESON_TAC[LINEAR_SURJECTIVE_IMP_INJECTIVE]; ALL_TAC] THEN
SUBGOAL_THEN `!y. y:real^N IN span(IMAGE f (:real^N))` MP_TAC THENL
[ALL_TAC; ASM_SIMP_TAC[SPAN_LINEAR_IMAGE; SPAN_UNIV] THEN SET_TAC[]] THEN
GEN_TAC THEN GEN_REWRITE_TAC LAND_CONV [GSYM BASIS_EXPANSION] THEN
MATCH_MP_TAC SPAN_VSUM THEN REWRITE_TAC[FINITE_NUMSEG; IN_NUMSEG] THEN
X_GEN_TAC `k:num` THEN STRIP_TAC THEN MATCH_MP_TAC SPAN_MUL THEN
MATCH_MP_TAC SPAN_SUPERSET THEN REWRITE_TAC[IN_IMAGE; IN_UNIV] THEN
ASM_MESON_TAC[INTEGRAL_VECTOR_STDBASIS];
ALL_TAC] THEN
SUBGOAL_THEN
`!i j. 1 <= i /\ i <= dimindex(:N) /\
1 <= j /\ j <= dimindex(:N)
==> integer(matrix(g:real^N->real^N)$i$j)`
ASSUME_TAC THENL
[ASM_SIMP_TAC[GSYM LINEAR_INTEGRAL_VECTOR] THEN ASM_MESON_TAC[];
ALL_TAC] THEN
SUBGOAL_THEN
`det(matrix(f:real^N->real^N)) * det(matrix(g:real^N->real^N)) =
det(matrix(I:real^N->real^N))`
MP_TAC THENL
[ASM_SIMP_TAC[GSYM DET_MUL; GSYM MATRIX_COMPOSE] THEN
REPEAT AP_TERM_TAC THEN ASM_REWRITE_TAC[FUN_EQ_THM; o_THM; I_THM];
ALL_TAC] THEN
DISCH_THEN(MP_TAC o AP_TERM `abs:real->real`) THEN
REWRITE_TAC[MATRIX_I; DET_I; REAL_ABS_NUM] THEN
ASM_SIMP_TAC[INTEGER_DET; INTEGER_ABS_MUL_EQ_1]);;
(* ------------------------------------------------------------------------- *)
(* Pick's theorem for an elementary triangle. *)
(* ------------------------------------------------------------------------- *)
let PICK_ELEMENTARY_TRIANGLE_0 = prove
(`!a b:real^2.
{x | x IN convex hull {vec 0,a,b} /\ integral_vector x} = {vec 0,a,b}
==> measure(convex hull {vec 0,a,b}) =
if collinear {vec 0,a,b} then &0 else &1 / &2`,
REPEAT GEN_TAC THEN COND_CASES_TAC THEN
ASM_SIMP_TAC[MEASURE_EQ_0; COLLINEAR_IMP_NEGLIGIBLE;
COLLINEAR_CONVEX_HULL_COLLINEAR] THEN
POP_ASSUM MP_TAC THEN
MAP_EVERY (fun t ->
ASM_CASES_TAC t THENL [ASM_REWRITE_TAC[INSERT_AC; COLLINEAR_2]; ALL_TAC])
[`a:real^2 = vec 0`; `b:real^2 = vec 0`; `a:real^2 = b`] THEN
DISCH_TAC THEN SUBGOAL_THEN `independent {a:real^2,b}` ASSUME_TAC THENL
[UNDISCH_TAC `~collinear{vec 0:real^2, a, b}` THEN
REWRITE_TAC[independent; CONTRAPOS_THM] THEN
REWRITE_TAC[dependent; EXISTS_IN_INSERT; NOT_IN_EMPTY] THEN STRIP_TAC THENL
[ONCE_REWRITE_TAC[SET_RULE `{c,a,b} = {c,b,a}`]; ALL_TAC] THEN
ASM_SIMP_TAC[COLLINEAR_3_AFFINE_HULL] THEN
ASM_SIMP_TAC[AFFINE_HULL_EQ_SPAN; HULL_INC; IN_INSERT] THEN
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP
(SET_RULE `a IN s ==> s SUBSET t ==> a IN t`)) THEN
MATCH_MP_TAC SPAN_MONO THEN SET_TAC[];
ALL_TAC] THEN
SUBGOAL_THEN `span{a,b} = (:real^2)` ASSUME_TAC THENL
[MP_TAC(ISPECL [`(:real^2)`; `{a:real^2,b}`] CARD_EQ_DIM) THEN
ASM_REWRITE_TAC[SUBSET_UNIV; SUBSET; EXTENSION; IN_ELIM_THM; IN_UNIV] THEN
DISCH_THEN MATCH_MP_TAC THEN
REWRITE_TAC[HAS_SIZE; FINITE_INSERT; FINITE_EMPTY] THEN
SIMP_TAC[CARD_CLAUSES; FINITE_INSERT; FINITE_EMPTY; IN_INSERT] THEN
ASM_REWRITE_TAC[NOT_IN_EMPTY; DIM_UNIV; DIMINDEX_2; ARITH];
ALL_TAC] THEN
REWRITE_TAC[GSYM SUBSET_ANTISYM_EQ; SUBSET; FORALL_IN_INSERT;
FORALL_IN_GSPEC] THEN
REWRITE_TAC[IN_ELIM_THM; NOT_IN_EMPTY; IN_INSERT] THEN STRIP_TAC THEN
MP_TAC(ISPEC `\x:real^2. transp(vector[a;b]:real^2^2) ** x`
INTEGRAL_BASIS_UNIMODULAR) THEN
REWRITE_TAC[MATRIX_OF_MATRIX_VECTOR_MUL; MATRIX_VECTOR_MUL_LINEAR] THEN
REWRITE_TAC[DET_2; MEASURE_TRIANGLE; VECTOR_2; DET_TRANSP; VEC_COMPONENT] THEN
ANTS_TAC THENL [ALL_TAC; REAL_ARITH_TAC] THEN
MATCH_MP_TAC SUBSET_ANTISYM THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN
CONJ_TAC THENL
[REWRITE_TAC[IN] THEN
SIMP_TAC[LINEAR_INTEGRAL_VECTOR; MATRIX_VECTOR_MUL_LINEAR; LAMBDA_BETA;
MATRIX_OF_MATRIX_VECTOR_MUL; transp; DIMINDEX_2; ARITH] THEN
MAP_EVERY UNDISCH_TAC
[`integral_vector(a:real^2)`; `integral_vector(b:real^2)`] THEN
REWRITE_TAC[integral_vector; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
REWRITE_TAC[IMP_IMP; FORALL_2; DIMINDEX_2; VECTOR_2] THEN
REWRITE_TAC[CONJ_ACI];
ALL_TAC] THEN
REWRITE_TAC[IN_IMAGE] THEN REWRITE_TAC[IN] THEN
X_GEN_TAC `x:real^2` THEN DISCH_TAC THEN REWRITE_TAC[EXISTS_VECTOR_2] THEN
REWRITE_TAC[MATRIX_VECTOR_COLUMN; TRANSP_TRANSP] THEN
REWRITE_TAC[DIMINDEX_2; VSUM_2; VECTOR_2; integral_vector; FORALL_2] THEN
SUBGOAL_THEN `(x:real^2) IN span{a,b}` MP_TAC THENL
[ASM_REWRITE_TAC[IN_UNIV]; ALL_TAC] THEN
REWRITE_TAC[SPAN_2; IN_UNIV; IN_ELIM_THM] THEN
MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `u:real` THEN
MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `v:real` THEN
DISCH_TAC THEN ASM_REWRITE_TAC[] THEN
FIRST_ASSUM(MP_TAC o SPEC `frac u % a + frac v % b:real^2`) THEN
FIRST_X_ASSUM(MP_TAC o SPEC
`(&1 - frac u) % a + (&1 - frac v) % b:real^2`) THEN
MATCH_MP_TAC(TAUT
`b' /\ (b' ==> b) /\ (a \/ a') /\ (c \/ c' ==> x)
==> (a /\ b ==> c) ==> (a' /\ b' ==> c') ==> x`) THEN
REPEAT CONJ_TAC THENL
[SUBGOAL_THEN `integral_vector(floor u % a + floor v % b:real^2)`
MP_TAC THENL
[MAP_EVERY UNDISCH_TAC
[`integral_vector(a:real^2)`; `integral_vector(b:real^2)`] THEN
SIMP_TAC[integral_vector; DIMINDEX_2; FORALL_2;
VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT] THEN
SIMP_TAC[FLOOR; INTEGER_CLOSED];
UNDISCH_TAC `integral_vector(x:real^2)` THEN REWRITE_TAC[IMP_IMP] THEN
DISCH_THEN(MP_TAC o MATCH_MP INTEGRAL_VECTOR_SUB) THEN
ASM_REWRITE_TAC[VECTOR_ARITH
`(x % a + y % b) - (u % a + v % b) = (x - u) % a + (y - v) % b`] THEN
MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN BINOP_TAC THEN
AP_THM_TAC THEN AP_TERM_TAC THEN
REWRITE_TAC[REAL_ARITH `u - x:real = y <=> u = x + y`] THEN
REWRITE_TAC[GSYM FLOOR_FRAC]];
REWRITE_TAC[VECTOR_ARITH
`(&1 - u) % a + (&1 - v) % b = (a + b) - (u % a + v % b)`] THEN
ASM_SIMP_TAC[INTEGRAL_VECTOR_ADD; INTEGRAL_VECTOR_SUB];
REWRITE_TAC[CONVEX_HULL_3_0; IN_ELIM_THM] THEN
SUBGOAL_THEN
`&0 <= frac u /\ &0 <= frac v /\ frac u + frac v <= &1 \/
&0 <= &1 - frac u /\ &0 <= &1 - frac v /\
(&1 - frac u) + (&1 - frac v) <= &1`
MP_TAC THENL
[MP_TAC(SPEC `u:real` FLOOR_FRAC) THEN
MP_TAC(SPEC `v:real` FLOOR_FRAC) THEN REAL_ARITH_TAC;
MESON_TAC[]];
REWRITE_TAC
[VECTOR_ARITH `x % a + y % b = a <=> (x - &1) % a + y % b = vec 0`;
VECTOR_ARITH `x % a + y % b = b <=> x % a + (y - &1) % b = vec 0`] THEN
ASM_SIMP_TAC[INDEPENDENT_2; GSYM REAL_FRAC_EQ_0] THEN
MP_TAC(SPEC `u:real` FLOOR_FRAC) THEN
MP_TAC(SPEC `v:real` FLOOR_FRAC) THEN REAL_ARITH_TAC]);;
let PICK_ELEMENTARY_TRIANGLE = prove
(`!a b c:real^2.
{x | x IN convex hull {a,b,c} /\ integral_vector x} = {a,b,c}
==> measure(convex hull {a,b,c}) =
if collinear {a,b,c} then &0 else &1 / &2`,
REPEAT GEN_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP (SET_RULE
`s = t ==> (!x. x IN s <=> x IN t) /\ s = t`)) THEN
REWRITE_TAC[IMP_CONJ] THEN DISCH_THEN(MP_TAC o SPEC `a:real^2`) THEN
REWRITE_TAC[IN_INSERT; IN_ELIM_THM] THEN
GEOM_ORIGIN_TAC `a:real^2`THEN
SIMP_TAC[INTEGRAL_VECTOR_ADD_LCANCEL; VECTOR_ADD_RID] THEN
REWRITE_TAC[PICK_ELEMENTARY_TRIANGLE_0]);;
(* ------------------------------------------------------------------------- *)
(* Our form of Pick's theorem holds degenerately for a flat triangle. *)
(* ------------------------------------------------------------------------- *)
let PICK_TRIANGLE_FLAT = prove
(`!a b c:real^2.
integral_vector a /\ integral_vector b /\ integral_vector c /\
c IN segment[a,b]
==> measure(convex hull {a,b,c}) =
&(CARD {x | x IN convex hull {a,b,c} /\ integral_vector x}) -
(&(CARD {x | x IN convex hull {b,c} /\ integral_vector x}) +
&(CARD {x | x IN convex hull {a,c} /\ integral_vector x}) +
&(CARD {x | x IN convex hull {a,b} /\ integral_vector x})) / &2 +
&1 / &2`,
REPEAT STRIP_TAC THEN REWRITE_TAC[GSYM SEGMENT_CONVEX_HULL] THEN
SUBGOAL_THEN `convex hull {a:real^2,b,c} = segment[a,b]` SUBST1_TAC THENL
[REWRITE_TAC[SEGMENT_CONVEX_HULL] THEN MATCH_MP_TAC CONVEX_HULLS_EQ THEN
ASM_REWRITE_TAC[GSYM SEGMENT_CONVEX_HULL; INSERT_SUBSET; EMPTY_SUBSET] THEN
SIMP_TAC[ENDS_IN_SEGMENT; HULL_INC; IN_INSERT];
ALL_TAC] THEN
SUBGOAL_THEN `measure(segment[a:real^2,b]) = &0` SUBST1_TAC THENL
[MATCH_MP_TAC MEASURE_EQ_0 THEN
MATCH_MP_TAC COLLINEAR_IMP_NEGLIGIBLE THEN
REWRITE_TAC[COLLINEAR_SEGMENT];
ALL_TAC] THEN
REWRITE_TAC[REAL_ARITH
`&0 = c - (a + b + c) / &2 + &1 / &2 <=> a + b = c + &1`] THEN
REWRITE_TAC[REAL_OF_NUM_ADD; REAL_OF_NUM_EQ] THEN
SUBGOAL_THEN
`segment[a:real^2,b] = segment[b,c] UNION segment[a,c]`
SUBST1_TAC THENL [ASM_MESON_TAC[SEGMENT_SYM; UNION_SEGMENT]; ALL_TAC] THEN
REWRITE_TAC[SET_RULE
`{x | x IN (s UNION t) /\ P x} =
{x | x IN s /\ P x} UNION {x | x IN t /\ P x}`] THEN
SIMP_TAC[CARD_UNION_GEN; FINITE_BOUNDED_INTEGER_POINTS; BOUNDED_SEGMENT] THEN
MATCH_MP_TAC(ARITH_RULE
`z:num <= x /\ z = 1 ==> x + y = (x + y) - z + 1`) THEN
CONJ_TAC THENL
[MATCH_MP_TAC CARD_SUBSET THEN
SIMP_TAC[FINITE_BOUNDED_INTEGER_POINTS; BOUNDED_SEGMENT] THEN SET_TAC[];
REWRITE_TAC[SET_RULE `{x | x IN s /\ P x} INTER {x | x IN t /\ P x} =
{x | x IN (s INTER t) /\ P x}`] THEN
SUBGOAL_THEN
`segment[b:real^2,c] INTER segment[a,c] = {c}`
SUBST1_TAC THENL [ASM_MESON_TAC[INTER_SEGMENT; SEGMENT_SYM]; ALL_TAC] THEN
SUBGOAL_THEN `{x:real^2 | x IN {c} /\ integral_vector x} = {c}`
SUBST1_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
SIMP_TAC[CARD_CLAUSES; FINITE_EMPTY; ARITH; NOT_IN_EMPTY]]);;
(* ------------------------------------------------------------------------- *)
(* Pick's theorem for a triangle. *)
(* ------------------------------------------------------------------------- *)
let PICK_TRIANGLE_ALT = prove
(`!a b c:real^2.
integral_vector a /\ integral_vector b /\ integral_vector c
==> measure(convex hull {a,b,c}) =
&(CARD {x | x IN convex hull {a,b,c} /\ integral_vector x}) -
(&(CARD {x | x IN convex hull {b,c} /\ integral_vector x}) +
&(CARD {x | x IN convex hull {a,c} /\ integral_vector x}) +
&(CARD {x | x IN convex hull {a,b} /\ integral_vector x})) / &2 +
&1 / &2`,
let tac a bc =
MATCH_MP_TAC CARD_PSUBSET THEN
REWRITE_TAC[FINITE_TRIANGLE_INTEGER_POINTS] THEN
REWRITE_TAC[PSUBSET] THEN CONJ_TAC THENL
[MATCH_MP_TAC(SET_RULE
`s SUBSET t ==> {x | x IN s /\ P x} SUBSET {x | x IN t /\ P x}`) THEN
MATCH_MP_TAC HULL_MINIMAL THEN REWRITE_TAC[CONVEX_CONVEX_HULL] THEN
ASM_SIMP_TAC[INSERT_SUBSET; EMPTY_SUBSET; IN_INSERT; HULL_INC];
DISCH_TAC] THEN
SUBGOAL_THEN(subst[bc,`bc:real^2->bool`]
`convex hull {a:real^2,b,c} = convex hull bc`)
ASSUME_TAC THENL
[MATCH_MP_TAC CONVEX_HULLS_EQ THEN
ASM_SIMP_TAC[HULL_INC; IN_INSERT; INSERT_SUBSET; EMPTY_SUBSET] THEN
SUBGOAL_THEN(subst [a,`x:real^2`] `x IN convex hull {a:real^2,b,c}`)
MP_TAC THENL
[SIMP_TAC[HULL_INC; IN_INSERT]; ASM SET_TAC[]];
ALL_TAC] THEN
MP_TAC(ISPECL [`{a:real^2,b,c}`; a]
EXTREME_POINT_OF_CONVEX_HULL_AFFINE_INDEPENDENT) THEN
ASM_REWRITE_TAC[IN_INSERT] THEN
DISCH_THEN(MP_TAC o MATCH_MP EXTREME_POINT_OF_CONVEX_HULL) THEN
ASM_REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY] in
REPEAT GEN_TAC THEN
WF_INDUCT_TAC `CARD {x:real^2 | x IN convex hull {a,b,c} /\
integral_vector x}` THEN
ASM_CASES_TAC `collinear{a:real^2,b,c}` THENL
[FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [COLLINEAR_BETWEEN_CASES]) THEN
REWRITE_TAC[BETWEEN_IN_SEGMENT] THEN REPEAT STRIP_TAC THENL
[MP_TAC(ISPECL [`b:real^2`; `c:real^2`; `a:real^2`] PICK_TRIANGLE_FLAT);
MP_TAC(ISPECL [`a:real^2`; `c:real^2`; `b:real^2`] PICK_TRIANGLE_FLAT);
MP_TAC(ISPECL [`a:real^2`; `b:real^2`; `c:real^2`]
PICK_TRIANGLE_FLAT)] THEN
(ANTS_TAC THENL [ASM_MESON_TAC[SEGMENT_SYM]; ALL_TAC] THEN
REWRITE_TAC[SET_RULE `{x | x IN s /\ P x} = s INTER P`] THEN
REWRITE_TAC[INSERT_AC; REAL_ADD_AC]);
ALL_TAC] THEN
UNDISCH_TAC `~collinear{a:real^2,b,c}` THEN
MAP_EVERY
(fun t -> ASM_CASES_TAC t THENL
[ASM_REWRITE_TAC[INSERT_AC; COLLINEAR_2]; ALL_TAC])
[`a:real^2 = b`; `a:real^2 = c`; `b:real^2 = c`] THEN
DISCH_TAC THEN STRIP_TAC THEN
ASM_CASES_TAC
`{x:real^2 | x IN convex hull {a, b, c} /\ integral_vector x} =
{a,b,c}`
THENL
[ASM_SIMP_TAC[PICK_ELEMENTARY_TRIANGLE] THEN
SUBGOAL_THEN
`{x | x IN convex hull {b,c} /\ integral_vector x} = {b,c} /\
{x | x IN convex hull {a,c} /\ integral_vector x} = {a,c} /\
{x | x IN convex hull {a,b} /\ integral_vector x} = {a:real^2,b}`
(REPEAT_TCL CONJUNCTS_THEN SUBST1_TAC) THENL
[REPEAT CONJ_TAC THEN
(FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE
`{x | x IN cs /\ P x} = s
==> t SUBSET s /\ t SUBSET ct /\ ct SUBSET cs /\
(s DIFF t) INTER ct = {}
==> {x | x IN ct /\ P x} = t`)) THEN
REPEAT CONJ_TAC THENL
[SET_TAC[];
MATCH_ACCEPT_TAC HULL_SUBSET;
MATCH_MP_TAC HULL_MONO THEN SET_TAC[];
ASM_REWRITE_TAC[INSERT_DIFF; IN_INSERT; NOT_IN_EMPTY; EMPTY_DIFF] THEN
MATCH_MP_TAC(SET_RULE `~(x IN s) ==> {x} INTER s = {}`) THEN
REWRITE_TAC[GSYM SEGMENT_CONVEX_HULL; GSYM BETWEEN_IN_SEGMENT] THEN
DISCH_THEN(MP_TAC o MATCH_MP BETWEEN_IMP_COLLINEAR) THEN
UNDISCH_TAC `~collinear{a:real^2,b,c}` THEN REWRITE_TAC[INSERT_AC]]);
SIMP_TAC[CARD_CLAUSES; FINITE_INSERT; FINITE_EMPTY] THEN
ASM_REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY] THEN
CONV_TAC NUM_REDUCE_CONV THEN CONV_TAC REAL_RAT_REDUCE_CONV];
ALL_TAC] THEN
SUBGOAL_THEN
`?d:real^2. d IN convex hull {a, b, c} /\ integral_vector d /\
~(d = a) /\ ~(d = b) /\ ~(d = c)`
STRIP_ASSUME_TAC THENL
[FIRST_X_ASSUM(MP_TAC o MATCH_MP (SET_RULE
`~(s = t) ==> t SUBSET s ==> ?d. d IN s /\ ~(d IN t)`)) THEN
REWRITE_TAC[SUBSET; FORALL_IN_INSERT; IN_ELIM_THM] THEN
ASM_SIMP_TAC[IN_INSERT; NOT_IN_EMPTY; DE_MORGAN_THM; GSYM CONJ_ASSOC] THEN
DISCH_THEN MATCH_MP_TAC THEN SIMP_TAC[HULL_INC; IN_INSERT];
ALL_TAC] THEN
FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV
[COLLINEAR_3_EQ_AFFINE_DEPENDENT]) THEN
ASM_REWRITE_TAC[] THEN DISCH_TAC THEN
MP_TAC(ISPECL
[`measure:(real^2->bool)->real`;
`a:real^2`; `b:real^2`; `c:real^2`; `d:real^2`]
TRIANGLE_ADDITIVE_DECOMPOSITION) THEN
SIMP_TAC[MEASURE_UNION; MEASURABLE_COMPACT] THEN
ASM_REWRITE_TAC[] THEN DISCH_THEN SUBST1_TAC THEN
REWRITE_TAC[MEASURE_CONVEX_HULL_2_TRIVIAL; REAL_ADD_RID; REAL_SUB_RZERO] THEN
MP_TAC(ISPECL
[`\s. &(CARD {x:real^2 | x IN s /\ integral_vector x})`;
`a:real^2`; `b:real^2`; `c:real^2`; `d:real^2`]
TRIANGLE_ADDITIVE_DECOMPOSITION) THEN
ASM_REWRITE_TAC[] THEN ANTS_TAC THENL
[REWRITE_TAC[SET_RULE `{x | x IN (s UNION t) /\ P x} =
{x | x IN s /\ P x} UNION {x | x IN t /\ P x}`;
SET_RULE `{x | x IN (s INTER t) /\ P x} =
{x | x IN s /\ P x} INTER {x | x IN t /\ P x}`] THEN
REPEAT STRIP_TAC THEN
REWRITE_TAC[REAL_ARITH `x:real = y + z - w <=> x + w = y + z`] THEN
REWRITE_TAC[REAL_OF_NUM_ADD; REAL_OF_NUM_EQ] THEN
MATCH_MP_TAC(ARITH_RULE
`x:num = (y + z) - w /\ w <= z ==> x + w = y + z`) THEN
CONJ_TAC THENL
[MATCH_MP_TAC CARD_UNION_GEN;
MATCH_MP_TAC CARD_SUBSET THEN REWRITE_TAC[INTER_SUBSET]] THEN
ASM_SIMP_TAC[FINITE_BOUNDED_INTEGER_POINTS; COMPACT_IMP_BOUNDED];
DISCH_THEN SUBST1_TAC] THEN
FIRST_X_ASSUM(fun th ->
MP_TAC(ISPECL [`a:real^2`; `b:real^2`; `d:real^2`] th) THEN
MP_TAC(ISPECL [`a:real^2`; `c:real^2`; `d:real^2`] th) THEN
MP_TAC(ISPECL [`b:real^2`; `c:real^2`; `d:real^2`] th)) THEN
ASM_REWRITE_TAC[] THEN
ANTS_TAC THENL [tac `a:real^2` `{b:real^2,c,d}`; DISCH_THEN SUBST1_TAC] THEN
ANTS_TAC THENL [tac `b:real^2` `{a:real^2,c,d}`; DISCH_THEN SUBST1_TAC] THEN
ANTS_TAC THENL [tac `c:real^2` `{a:real^2,b,d}`; DISCH_THEN SUBST1_TAC] THEN
SUBGOAL_THEN `{x:real^2 | x IN convex hull {d} /\ integral_vector x} = {d}`
SUBST1_TAC THENL
[REWRITE_TAC[CONVEX_HULL_SING] THEN ASM SET_TAC[]; ALL_TAC] THEN
SIMP_TAC[CARD_CLAUSES; FINITE_RULES; NOT_IN_EMPTY] THEN
CONV_TAC NUM_REDUCE_CONV THEN
REWRITE_TAC[SET_RULE `{x | x IN s /\ P x} = s INTER P`] THEN
REWRITE_TAC[INSERT_AC] THEN REAL_ARITH_TAC);;
let PICK_TRIANGLE = prove
(`!a b c:real^2.
integral_vector a /\ integral_vector b /\ integral_vector c
==> measure(convex hull {a,b,c}) =
if collinear {a,b,c} then &0
else &(CARD {x | x IN interior(convex hull {a,b,c}) /\
integral_vector x}) +
&(CARD {x | x IN frontier(convex hull {a,b,c}) /\
integral_vector x}) / &2 - &1`,
REPEAT STRIP_TAC THEN COND_CASES_TAC THEN
ASM_SIMP_TAC[MEASURE_EQ_0; COLLINEAR_IMP_NEGLIGIBLE;
COLLINEAR_CONVEX_HULL_COLLINEAR] THEN
ASM_SIMP_TAC[PICK_TRIANGLE_ALT] THEN
REWRITE_TAC[INTERIOR_OF_TRIANGLE; FRONTIER_OF_TRIANGLE] THEN
REWRITE_TAC[SET_RULE
`{x | x IN (s DIFF t) /\ P x} =
{x | x IN s /\ P x} DIFF {x | x IN t /\ P x}`] THEN
MATCH_MP_TAC(REAL_ARITH
`i + c = s /\ ccc = c + &3
==> s - ccc / &2 + &1 / &2 = i + c / &2 - &1`) THEN
CONJ_TAC THENL
[REWRITE_TAC[REAL_OF_NUM_ADD; REAL_OF_NUM_EQ] THEN
MATCH_MP_TAC(ARITH_RULE `y:num <= x /\ x - y = z ==> z + y = x`) THEN
CONJ_TAC THENL
[MATCH_MP_TAC CARD_SUBSET; MATCH_MP_TAC(GSYM CARD_DIFF)] THEN
ASM_SIMP_TAC[FINITE_BOUNDED_INTEGER_POINTS;
FINITE_IMP_BOUNDED_CONVEX_HULL; FINITE_INSERT; FINITE_EMPTY] THEN
MATCH_MP_TAC(SET_RULE
`s SUBSET t ==> {x | x IN s /\ P x} SUBSET {x | x IN t /\ P x}`) THEN
REWRITE_TAC[UNION_SUBSET; SEGMENT_CONVEX_HULL] THEN
REPEAT CONJ_TAC THEN MATCH_MP_TAC HULL_MONO THEN SET_TAC[];
REWRITE_TAC[SET_RULE
`{x | x IN (s UNION t) /\ P x} =
{x | x IN s /\ P x} UNION {x | x IN t /\ P x}`] THEN
SIMP_TAC[CARD_UNION_GEN; FINITE_BOUNDED_INTEGER_POINTS;
FINITE_INTER; FINITE_UNION; BOUNDED_SEGMENT; UNION_OVER_INTER] THEN
REWRITE_TAC[SET_RULE
`{x | x IN s /\ P x} INTER {x | x IN t /\ P x} =
{x | x IN (s INTER t) /\ P x}`] THEN
SUBGOAL_THEN
`segment[b:real^2,c] INTER segment [c,a] = {c} /\
segment[a,b] INTER segment [b,c] = {b} /\
segment[a,b] INTER segment [c,a] = {a}`
(REPEAT_TCL CONJUNCTS_THEN SUBST1_TAC) THENL
[ASM_MESON_TAC[INTER_SEGMENT; SEGMENT_SYM; INSERT_AC]; ALL_TAC] THEN
ASM_SIMP_TAC[SET_RULE `P a ==> {x | x IN {a} /\ P x} = {a}`] THEN
ASM_CASES_TAC `b:real^2 = a` THENL
[ASM_MESON_TAC[COLLINEAR_2; INSERT_AC]; ALL_TAC] THEN
ASM_SIMP_TAC[SET_RULE `~(a = b) ==> {b} INTER {a} = {}`] THEN
REWRITE_TAC[REAL_OF_NUM_ADD; REAL_OF_NUM_EQ] THEN
REWRITE_TAC[NOT_IN_EMPTY; EMPTY_GSPEC; CARD_CLAUSES; SUB_0] THEN
MATCH_MP_TAC(ARITH_RULE
`c:num <= ca /\ a <= ab /\ b <= bc /\
bc' + ac' + ab' + a + b + c = ab + bc + ca + 3
==> bc' + ac' + ab' = (ab + (bc + ca) - c) - (b + a) + 3`) THEN
ASM_SIMP_TAC[CARD_SUBSET; SING_SUBSET; IN_ELIM_THM; ENDS_IN_SEGMENT;
FINITE_BOUNDED_INTEGER_POINTS; BOUNDED_SEGMENT] THEN
SIMP_TAC[NOT_IN_EMPTY; EMPTY_GSPEC; CARD_CLAUSES; FINITE_INSERT;
FINITE_EMPTY] THEN CONV_TAC NUM_REDUCE_CONV THEN
REWRITE_TAC[SET_RULE `{x | x IN s /\ P x} = s INTER P`] THEN
REWRITE_TAC[SEGMENT_CONVEX_HULL; INSERT_AC] THEN ARITH_TAC]);;
(* ------------------------------------------------------------------------- *)
(* Parity lemma for segment crossing a polygon. *)
(* ------------------------------------------------------------------------- *)
let PARITY_LEMMA = prove
(`!a b c d p x:real^2.
simple_path(p ++ linepath(a,b)) /\
pathstart p = b /\ pathfinish p = a /\
segment(a,b) INTER segment(c,d) = {x} /\
segment[c,d] INTER path_image p = {}
==> (c IN inside(path_image(p ++ linepath(a,b))) <=>
d IN outside(path_image(p ++ linepath(a,b))))`,
let lemma = prove
(`!a b x y:real^N.
collinear{y,a,b} /\ between x (a,b) /\
dist(y,x) < dist(x,b) /\ dist(y,x) < dist(x,a)
==> y IN segment(a,b)`,
REPEAT STRIP_TAC THEN MATCH_MP_TAC COLLINEAR_DIST_IN_OPEN_SEGMENT THEN
ASM_REWRITE_TAC[] THEN
REPEAT(POP_ASSUM MP_TAC) THEN REWRITE_TAC[between; DIST_SYM] THEN
NORM_ARITH_TAC)
and symlemma = prove
(`(!n. P(--n) <=> P (n)) /\ (!n. &0 < n dot x ==> P n)
==> !n:real^N. ~(n dot x = &0) ==> P n`,
STRIP_TAC THEN GEN_TAC THEN
REWRITE_TAC[REAL_ARITH `~(x = &0) <=> &0 < x \/ &0 < --x`] THEN
REWRITE_TAC[GSYM DOT_LNEG] THEN ASM_MESON_TAC[]) in
REPEAT STRIP_TAC THEN
MP_TAC(ISPECL [`p:real^1->real^2`; `linepath(a:real^2,b)`]
SIMPLE_PATH_JOIN_LOOP_EQ) THEN
FIRST_ASSUM(MP_TAC o MATCH_MP SIMPLE_PATH_IMP_PATH) THEN
ASM_SIMP_TAC[PATH_JOIN; PATHSTART_LINEPATH; PATHFINISH_LINEPATH] THEN
DISCH_THEN(ASSUME_TAC o CONJUNCT1) THEN STRIP_TAC THEN
MP_TAC(ISPECL [`(a:real^2) INSERT b INSERT c INSERT d INSERT path_image p`;
`x:real^2`]
DISTANCE_ATTAINS_INF) THEN
REWRITE_TAC[FORALL_IN_INSERT] THEN
ONCE_REWRITE_TAC[SET_RULE `a INSERT b INSERT c INSERT d INSERT s =
{a,b,c,d} UNION s`] THEN
ASM_SIMP_TAC[CLOSED_UNION; FINITE_IMP_CLOSED; CLOSED_PATH_IMAGE;
FINITE_INSERT; FINITE_EMPTY] THEN
ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
DISCH_THEN(X_CHOOSE_THEN `cp:real^2` MP_TAC) THEN
DISJ_CASES_TAC(NORM_ARITH `cp = x \/ &0 < dist(x:real^2,cp)`) THENL
[FIRST_X_ASSUM SUBST_ALL_TAC THEN
MATCH_MP_TAC(TAUT `~a ==> a /\ b ==> c`) THEN
FIRST_ASSUM(MP_TAC o MATCH_MP (SET_RULE `a = {x} ==> x IN a`)) THEN
REWRITE_TAC[open_segment; IN_DIFF; IN_UNION; IN_INSERT; NOT_IN_EMPTY;
IN_INTER; DE_MORGAN_THM] THEN
STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE
`p INTER s SUBSET u ==> x IN (s DIFF u) ==> ~(x IN p)`)) THEN
ASM_REWRITE_TAC[IN_DIFF; IN_INSERT; NOT_IN_EMPTY; PATH_IMAGE_LINEPATH];
ALL_TAC] THEN
ABBREV_TAC `e = dist(x:real^2,cp)` THEN FIRST_X_ASSUM(K ALL_TAC o SYM) THEN
DISCH_THEN(STRIP_ASSUME_TAC o CONJUNCT2) THEN
RULE_ASSUM_TAC(REWRITE_RULE[ARC_LINEPATH_EQ]) THEN
MP_TAC(ISPECL [`a:real^2`; `b:real^2`; `c:real^2`; `d:real^2`]
FINITE_INTER_COLLINEAR_OPEN_SEGMENTS) THEN
MP_TAC(ISPECL [`a:real^2`; `b:real^2`; `d:real^2`; `c:real^2`]
FINITE_INTER_COLLINEAR_OPEN_SEGMENTS) THEN
SUBST1_TAC(MESON[SEGMENT_SYM] `segment(d:real^2,c) = segment(c,d)`) THEN
ASM_REWRITE_TAC[FINITE_SING; NOT_INSERT_EMPTY] THEN REPEAT DISCH_TAC THEN
SUBGOAL_THEN `~(a IN segment[c:real^2,d]) /\ ~(b IN segment[c,d])`
STRIP_ASSUME_TAC THENL
[ASM_MESON_TAC[PATHSTART_IN_PATH_IMAGE; PATHFINISH_IN_PATH_IMAGE;
IN_INTER; NOT_IN_EMPTY];
ALL_TAC] THEN
SUBGOAL_THEN `~(c:real^2 = a) /\ ~(c = b) /\ ~(d = a) /\ ~(d = b)`
STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[ENDS_IN_SEGMENT]; ALL_TAC] THEN
SUBGOAL_THEN `x IN segment(a:real^2,b) /\ x IN segment(c,d)` MP_TAC THENL
[ASM SET_TAC[]; ALL_TAC] THEN
REWRITE_TAC[IN_OPEN_SEGMENT_ALT] THEN STRIP_TAC THEN
SUBGOAL_THEN
`{c,d} INTER path_image(p ++ linepath(a:real^2,b)) = {}`
ASSUME_TAC THENL
[ASM_SIMP_TAC[PATH_IMAGE_JOIN; PATH_LINEPATH; PATHSTART_LINEPATH] THEN
REWRITE_TAC[SET_RULE
`{c,d} INTER (s UNION t) = {} <=>
(~(c IN s) /\ ~(d IN s)) /\ ~(c IN t) /\ ~(d IN t)`] THEN
CONJ_TAC THENL
[ASM_MESON_TAC[ENDS_IN_SEGMENT; IN_INTER; NOT_IN_EMPTY];
REWRITE_TAC[PATH_IMAGE_LINEPATH; GSYM BETWEEN_IN_SEGMENT] THEN
CONJ_TAC THEN DISCH_THEN(ASSUME_TAC o MATCH_MP BETWEEN_IMP_COLLINEAR) THEN
RULE_ASSUM_TAC(REWRITE_RULE[INSERT_AC]) THEN ASM_MESON_TAC[]];
ALL_TAC] THEN
MP_TAC(ISPEC `b - x:real^2` ORTHOGONAL_TO_VECTOR_EXISTS) THEN
REWRITE_TAC[DIMINDEX_2; LE_REFL; LEFT_IMP_EXISTS_THM] THEN
X_GEN_TAC `n:real^2` THEN STRIP_TAC THEN
SUBGOAL_THEN `(x:real^2) IN segment(a,b) /\ x IN segment(c,d)` MP_TAC THENL
[ASM SET_TAC[];
SIMP_TAC[IN_OPEN_SEGMENT_ALT; GSYM BETWEEN_IN_SEGMENT] THEN STRIP_TAC] THEN
SUBGOAL_THEN `~collinear{a:real^2, b, c, d}` ASSUME_TAC THENL
[UNDISCH_TAC `~collinear{a:real^2,b,c}` THEN REWRITE_TAC[CONTRAPOS_THM] THEN
MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] COLLINEAR_SUBSET) THEN SET_TAC[];
ALL_TAC] THEN
SUBGOAL_THEN `~(n dot (d - x:real^2) = &0)` MP_TAC THENL
[REWRITE_TAC[GSYM orthogonal] THEN DISCH_TAC THEN
MP_TAC(SPECL [`n:real^2`; `d - x:real^2`; `b - x:real^2`]
ORTHOGONAL_TO_ORTHOGONAL_2D) THEN
ANTS_TAC THENL [ASM_MESON_TAC[ORTHOGONAL_SYM]; ALL_TAC] THEN
REWRITE_TAC[GSYM COLLINEAR_3] THEN DISCH_TAC THEN
UNDISCH_TAC `~collinear{a:real^2, b, c, d}` THEN ASM_REWRITE_TAC[] THEN
ONCE_REWRITE_TAC[SET_RULE `{a,b,c,d} = {b,d,a,c}`] THEN
ASM_SIMP_TAC[COLLINEAR_4_3] THEN CONJ_TAC THENL
[MATCH_MP_TAC COLLINEAR_SUBSET THEN EXISTS_TAC `{b:real^2,x,a,d}` THEN
CONJ_TAC THENL [ASM_SIMP_TAC[COLLINEAR_4_3]; SET_TAC[]] THEN
ONCE_REWRITE_TAC[SET_RULE `{a,b,c} = {c,b,a}`] THEN
ASM_SIMP_TAC[BETWEEN_IMP_COLLINEAR];
MATCH_MP_TAC COLLINEAR_SUBSET THEN EXISTS_TAC `{d:real^2,x,b,c}` THEN
CONJ_TAC THENL [ASM_SIMP_TAC[COLLINEAR_4_3]; SET_TAC[]] THEN
ONCE_REWRITE_TAC[SET_RULE `{a,b,c} = {c,b,a}`] THEN
ASM_SIMP_TAC[BETWEEN_IMP_COLLINEAR]];
ALL_TAC] THEN
DISCH_THEN(fun th -> POP_ASSUM_LIST(MP_TAC o end_itlist CONJ) THEN
MP_TAC th) THEN
SPEC_TAC(`n:real^2`,`n:real^2`) THEN
MATCH_MP_TAC symlemma THEN CONJ_TAC THENL
[REWRITE_TAC[ORTHOGONAL_RNEG; VECTOR_NEG_EQ_0]; ALL_TAC] THEN
GEN_TAC THEN DISCH_TAC THEN STRIP_TAC THEN
SUBGOAL_THEN `n dot (c - x:real^2) < &0` ASSUME_TAC THENL
[UNDISCH_TAC `&0 < n dot (d - x:real^2)` THEN
SUBGOAL_THEN `(x:real^2) IN segment(c,d)` MP_TAC THENL
[ASM SET_TAC[]; ALL_TAC] THEN
ASM_REWRITE_TAC[IN_SEGMENT] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
REWRITE_TAC[VECTOR_ARITH
`d - ((&1 - u) % c + u % d):real^N = (&1 - u) % (d - c) /\
c - ((&1 - u) % c + u % d) = --u % (d - c)`] THEN
REWRITE_TAC[DOT_RMUL; REAL_MUL_LNEG; REAL_ARITH `--x < &0 <=> &0 < x`] THEN
ASM_SIMP_TAC[REAL_LT_MUL_EQ; REAL_SUB_LT];
ALL_TAC] THEN
SUBGOAL_THEN
`!y. y IN ball(x:real^2,e)
==> y IN segment(a,b) \/
&0 < n dot (y - x) \/
n dot (y - x) < &0`
ASSUME_TAC THENL
[REWRITE_TAC[IN_BALL] THEN REPEAT STRIP_TAC THEN
MATCH_MP_TAC(TAUT `(~c /\ ~b ==> a) ==> a \/ b \/ c`) THEN
REWRITE_TAC[REAL_ARITH `~(x < &0) /\ ~(&0 < x) <=> x = &0`] THEN
REWRITE_TAC[GSYM orthogonal] THEN DISCH_TAC THEN
MP_TAC(SPECL [`n:real^2`; `y - x:real^2`; `b - x:real^2`]
ORTHOGONAL_TO_ORTHOGONAL_2D) THEN
ANTS_TAC THENL [ASM_MESON_TAC[ORTHOGONAL_SYM]; ALL_TAC] THEN
REWRITE_TAC[GSYM COLLINEAR_3] THEN DISCH_TAC THEN
MATCH_MP_TAC lemma THEN EXISTS_TAC `x:real^2` THEN
ASM_REWRITE_TAC[] THEN CONJ_TAC THENL
[ALL_TAC; ASM_MESON_TAC[REAL_LTE_TRANS; DIST_SYM]] THEN
ONCE_REWRITE_TAC[SET_RULE `{y,a,b} = {a,b,y}`] THEN
MATCH_MP_TAC COLLINEAR_3_TRANS THEN EXISTS_TAC `x:real^2` THEN
ASM_REWRITE_TAC[] THEN UNDISCH_TAC `collinear{y:real^2, x, b}` THEN
MP_TAC(MATCH_MP BETWEEN_IMP_COLLINEAR (ASSUME
`between (x:real^2) (a,b)`)) THEN
SIMP_TAC[INSERT_AC];
ALL_TAC] THEN
MP_TAC(SPEC `p ++ linepath(a:real^2,b)` JORDAN_INSIDE_OUTSIDE) THEN
ASM_REWRITE_TAC[PATHFINISH_JOIN; PATHSTART_JOIN; PATHFINISH_LINEPATH] THEN
STRIP_TAC THEN
SUBGOAL_THEN
`~(connected_component((:real^2) DIFF path_image(p ++ linepath (a,b))) c d)`
MP_TAC THENL
[DISCH_TAC;
ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN
DISCH_THEN(MP_TAC o SPEC `path_image(p ++ linepath(a:real^2,b))` o
MATCH_MP (SET_RULE
`~(x IN s <=> y IN t)
==> !p. s UNION t = (:real^2) DIFF p /\ {x,y} INTER p = {}
==> x IN s /\ y IN s \/ x IN t /\ y IN t`)) THEN
ASM_REWRITE_TAC[connected_component] THEN
ASM_REWRITE_TAC[SET_RULE `t SUBSET UNIV DIFF s <=> t INTER s = {}`] THEN
ASM_MESON_TAC[INSIDE_NO_OVERLAP; OUTSIDE_NO_OVERLAP]] THEN
MP_TAC(SPEC `p ++ linepath(a:real^2,b)` JORDAN_DISCONNECTED) THEN
ASM_REWRITE_TAC[PATHFINISH_JOIN; PATHSTART_JOIN; PATHFINISH_LINEPATH] THEN
REWRITE_TAC[CONNECTED_IFF_CONNECTED_COMPONENT] THEN
SUBGOAL_THEN
`!u v. u IN inside(path_image(p ++ linepath(a,b))) /\
v IN outside(path_image(p ++ linepath(a,b)))
==> connected_component
((:real^2) DIFF path_image (p ++ linepath (a,b))) u v`
ASSUME_TAC THENL
[ALL_TAC;
MAP_EVERY X_GEN_TAC [`u:real^2`; `v:real^2`] THEN
GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV)
[SYM(ASSUME `inside (path_image (p ++ linepath (a,b))) UNION
outside (path_image (p ++ linepath (a,b))) =
(:real^2) DIFF path_image (p ++ linepath (a,b))`)] THEN
REWRITE_TAC[IN_UNION; CONNECTED_IFF_CONNECTED_COMPONENT] THEN
STRIP_TAC THENL
[REWRITE_TAC[connected_component] THEN
EXISTS_TAC `inside(path_image(p ++ linepath(a:real^2,b)))`;
ASM_MESON_TAC[];
ASM_MESON_TAC[CONNECTED_COMPONENT_SYM];
REWRITE_TAC[connected_component] THEN
EXISTS_TAC `outside(path_image(p ++ linepath(a:real^2,b)))`] THEN
ASM_REWRITE_TAC[SET_RULE `s SUBSET UNIV DIFF t <=> s INTER t = {}`] THEN
REWRITE_TAC[OUTSIDE_NO_OVERLAP; INSIDE_NO_OVERLAP]] THEN
SUBGOAL_THEN `(x:real^2) IN path_image(p ++ linepath(a,b))` ASSUME_TAC THENL
[ASM_SIMP_TAC[PATHSTART_LINEPATH; PATH_IMAGE_JOIN; PATH_LINEPATH] THEN
REWRITE_TAC[IN_UNION; PATH_IMAGE_LINEPATH] THEN DISJ2_TAC THEN
RULE_ASSUM_TAC(REWRITE_RULE[open_segment]) THEN ASM SET_TAC[];
ALL_TAC] THEN
MAP_EVERY X_GEN_TAC [`u:real^2`; `v:real^2`] THEN STRIP_TAC THEN
UNDISCH_TAC
`frontier(inside(path_image(p ++ linepath(a:real^2,b)))) =
path_image(p ++ linepath(a,b))` THEN
REWRITE_TAC[EXTENSION] THEN
DISCH_THEN(MP_TAC o SPEC `x:real^2`) THEN ASM_REWRITE_TAC[frontier] THEN
REWRITE_TAC[IN_DIFF; CLOSURE_APPROACHABLE] THEN
DISCH_THEN(MP_TAC o SPEC `e:real` o CONJUNCT1) THEN
ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN
X_GEN_TAC `w:real^2` THEN STRIP_TAC THEN
MATCH_MP_TAC CONNECTED_COMPONENT_TRANS THEN EXISTS_TAC `w:real^2` THEN
CONJ_TAC THENL
[REWRITE_TAC[connected_component] THEN
EXISTS_TAC `inside(path_image(p ++ linepath(a:real^2,b)))` THEN
ASM_REWRITE_TAC[SET_RULE `s SUBSET UNIV DIFF t <=> s INTER t = {}`] THEN
REWRITE_TAC[INSIDE_NO_OVERLAP];
ALL_TAC] THEN
UNDISCH_TAC
`frontier(outside(path_image(p ++ linepath(a:real^2,b)))) =
path_image(p ++ linepath(a,b))` THEN
REWRITE_TAC[EXTENSION] THEN
DISCH_THEN(MP_TAC o SPEC `x:real^2`) THEN ASM_REWRITE_TAC[frontier] THEN
REWRITE_TAC[IN_DIFF; CLOSURE_APPROACHABLE] THEN
DISCH_THEN(MP_TAC o SPEC `e:real` o CONJUNCT1) THEN
ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN
X_GEN_TAC `z:real^2` THEN STRIP_TAC THEN
MATCH_MP_TAC CONNECTED_COMPONENT_TRANS THEN EXISTS_TAC `z:real^2` THEN
CONJ_TAC THENL
[ALL_TAC;
REWRITE_TAC[connected_component] THEN
EXISTS_TAC `outside(path_image(p ++ linepath(a:real^2,b)))` THEN
ASM_REWRITE_TAC[SET_RULE `s SUBSET UNIV DIFF t <=> s INTER t = {}`] THEN
REWRITE_TAC[OUTSIDE_NO_OVERLAP]] THEN
SUBGOAL_THEN
`!y. dist(y,x) < e /\ ~(y IN path_image(p ++ linepath (a,b)))
==> connected_component
((:real^2) DIFF path_image(p ++ linepath(a,b))) c y`
ASSUME_TAC THENL
[ALL_TAC;
MATCH_MP_TAC CONNECTED_COMPONENT_TRANS THEN EXISTS_TAC `c:real^2` THEN
CONJ_TAC THENL [MATCH_MP_TAC CONNECTED_COMPONENT_SYM; ALL_TAC] THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN
ASM_MESON_TAC[INSIDE_NO_OVERLAP; OUTSIDE_NO_OVERLAP; IN_INTER;
NOT_IN_EMPTY]] THEN
X_GEN_TAC `y:real^2` THEN STRIP_TAC THEN
SUBGOAL_THEN `segment[c,d] INTER path_image(p ++ linepath(a,b)) = {x:real^2}`
ASSUME_TAC THENL
[MATCH_MP_TAC(SET_RULE
`{c,d} INTER p = {} /\ (segment[c,d] DIFF {c,d}) INTER p = {x}
==> segment[c,d] INTER p = {x}`) THEN
ASM_SIMP_TAC[PATH_IMAGE_JOIN; PATHSTART_LINEPATH; PATH_LINEPATH] THEN
MATCH_MP_TAC(SET_RULE
`cd INTER p = {} /\ l INTER (cd DIFF {c,d}) = {x}
==> (cd DIFF {c,d}) INTER (p UNION l) = {x}`) THEN
ASM_REWRITE_TAC[GSYM open_segment; PATH_IMAGE_LINEPATH] THEN
MATCH_MP_TAC(SET_RULE
`~(a IN segment[c,d]) /\ ~(b IN segment[c,d]) /\
segment(a,b) INTER segment(c,d) = {x} /\
segment(a,b) = segment[a,b] DIFF {a,b} /\
segment(c,d) = segment[c,d] DIFF {c,d}
==> segment[a,b] INTER segment(c,d) = {x}`) THEN
ASM_REWRITE_TAC[] THEN REWRITE_TAC[open_segment];
ALL_TAC] THEN
UNDISCH_THEN
`!y. y IN ball(x:real^2,e)
==> y IN segment(a,b) \/ &0 < n dot (y - x) \/ n dot (y - x) < &0`
(MP_TAC o SPEC `y:real^2`) THEN
REWRITE_TAC[IN_BALL] THEN ONCE_REWRITE_TAC[DIST_SYM] THEN
ASM_REWRITE_TAC[] THEN
DISCH_THEN(REPEAT_TCL DISJ_CASES_THEN MP_TAC) THENL
[MATCH_MP_TAC(TAUT `~p ==> p ==> q`) THEN
UNDISCH_TAC `~(y IN path_image(p ++ linepath(a:real^2,b)))` THEN
ASM_SIMP_TAC[PATHSTART_LINEPATH; PATH_IMAGE_JOIN; PATH_LINEPATH] THEN
SIMP_TAC[CONTRAPOS_THM; open_segment; IN_DIFF; IN_UNION;
PATH_IMAGE_LINEPATH];
DISCH_TAC THEN MATCH_MP_TAC CONNECTED_COMPONENT_TRANS THEN
EXISTS_TAC `d:real^2` THEN ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC CONNECTED_COMPONENT_TRANS THEN
EXISTS_TAC `x + min (&1 / &2) (e / &2 / norm(d - x)) % (d - x):real^2` THEN
REWRITE_TAC[connected_component] THEN CONJ_TAC THENL
[EXISTS_TAC `segment[x:real^2,d] DELETE x` THEN
SIMP_TAC[CONVEX_SEMIOPEN_SEGMENT; CONVEX_CONNECTED] THEN
ASM_REWRITE_TAC[IN_DELETE; ENDS_IN_SEGMENT] THEN REPEAT CONJ_TAC THENL
[FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE
`cd INTER p = {x}
==> xd SUBSET cd
==> (xd DELETE x) SUBSET (UNIV DIFF p)`)) THEN
REWRITE_TAC[SUBSET_SEGMENT; ENDS_IN_SEGMENT] THEN
UNDISCH_TAC `segment (a,b) INTER segment (c,d) = {x:real^2}` THEN
REWRITE_TAC[open_segment] THEN SET_TAC[];
REWRITE_TAC[IN_SEGMENT; VECTOR_ARITH
`x + a % (y - x):real^N = (&1 - a) % x + a % y`] THEN
EXISTS_TAC `min (&1 / &2) (e / &2 / norm(d - x:real^2))` THEN
REWRITE_TAC[] THEN CONJ_TAC THENL [ALL_TAC; REAL_ARITH_TAC] THEN
REWRITE_TAC[REAL_LE_MIN] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN
ASM_SIMP_TAC[REAL_LE_DIV; REAL_POS; NORM_POS_LE; REAL_LT_IMP_LE];
ASM_REWRITE_TAC[VECTOR_MUL_EQ_0; VECTOR_SUB_EQ;
VECTOR_ARITH `x + a:real^N = x <=> a = vec 0`] THEN
MATCH_MP_TAC(REAL_ARITH `&0 < x ==> ~(min (&1 / &2) x = &0)`) THEN
MATCH_MP_TAC REAL_LT_DIV THEN ASM_REWRITE_TAC[REAL_HALF] THEN
ASM_REWRITE_TAC[NORM_POS_LT; VECTOR_SUB_EQ]];
EXISTS_TAC `ball(x,e) INTER {w:real^2 | &0 < n dot (w - x)}` THEN
REPEAT CONJ_TAC THENL
[MATCH_MP_TAC CONVEX_CONNECTED THEN MATCH_MP_TAC CONVEX_INTER THEN
REWRITE_TAC[CONVEX_BALL; DOT_RSUB; REAL_SUB_LT] THEN
REWRITE_TAC[GSYM real_gt; CONVEX_HALFSPACE_GT];
ASM_SIMP_TAC[PATHSTART_LINEPATH; PATH_IMAGE_JOIN; PATH_LINEPATH] THEN
MATCH_MP_TAC(SET_RULE
`p SUBSET (UNIV DIFF b) /\ l INTER w = {}
==> (b INTER w) SUBSET (UNIV DIFF (p UNION l))`) THEN
ASM_REWRITE_TAC[SUBSET; IN_DIFF; IN_UNIV; IN_BALL; REAL_NOT_LT] THEN
MATCH_MP_TAC(SET_RULE
`!t. t INTER u = {} /\ s SUBSET t ==> s INTER u = {}`) THEN
EXISTS_TAC `affine hull {x:real^2,b}` THEN CONJ_TAC THENL
[REWRITE_TAC[AFFINE_HULL_2; FORALL_IN_GSPEC; SET_RULE
`s INTER t = {} <=> !x. x IN s ==> ~(x IN t)`] THEN
REWRITE_TAC[IN_ELIM_THM] THEN
SIMP_TAC[REAL_ARITH `u + v = &1 <=> u = &1 - v`] THEN
REWRITE_TAC[DOT_RMUL; VECTOR_ARITH
`((&1 - v) % x + v % b) - x:real^N = v % (b - x)`] THEN
RULE_ASSUM_TAC(REWRITE_RULE[orthogonal]) THEN
ONCE_REWRITE_TAC[DOT_SYM] THEN
ASM_REWRITE_TAC[REAL_MUL_RZERO; REAL_LT_REFL];
REWRITE_TAC[PATH_IMAGE_LINEPATH; SEGMENT_CONVEX_HULL] THEN
SIMP_TAC[SUBSET_HULL; AFFINE_IMP_CONVEX; AFFINE_AFFINE_HULL] THEN
REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET] THEN
SIMP_TAC[HULL_INC; IN_INSERT] THEN
ASM_SIMP_TAC[GSYM COLLINEAR_3_AFFINE_HULL] THEN
ONCE_REWRITE_TAC[SET_RULE `{x,b,a} = {a,x,b}`] THEN
MATCH_MP_TAC BETWEEN_IMP_COLLINEAR THEN ASM_REWRITE_TAC[]];
REWRITE_TAC[IN_BALL; IN_INTER; IN_ELIM_THM; dist] THEN
REWRITE_TAC[NORM_ARITH `norm(x - (x + a):real^N) = norm a`] THEN
REWRITE_TAC[VECTOR_ARITH `(x + a) - x:real^N = a`] THEN CONJ_TAC THENL
[ASM_SIMP_TAC[NORM_MUL; GSYM REAL_LT_RDIV_EQ; NORM_POS_LT;
VECTOR_SUB_EQ] THEN
MATCH_MP_TAC(REAL_ARITH
`&0 < x /\ x < e ==> abs(min (&1 / &2) x) < e`) THEN
ASM_SIMP_TAC[REAL_LT_DIV; REAL_HALF; NORM_POS_LT; VECTOR_SUB_EQ;
REAL_LT_DIV2_EQ] THEN ASM_REAL_ARITH_TAC;
REWRITE_TAC[DOT_RMUL] THEN MATCH_MP_TAC REAL_LT_MUL THEN
ASM_REWRITE_TAC[REAL_LT_MIN] THEN
ASM_SIMP_TAC[REAL_LT_DIV; REAL_HALF; NORM_POS_LT; VECTOR_SUB_EQ;
REAL_LT_01]];
REWRITE_TAC[IN_BALL; IN_INTER; IN_ELIM_THM] THEN
ONCE_REWRITE_TAC[DIST_SYM] THEN ASM_REWRITE_TAC[]]];
DISCH_TAC THEN MATCH_MP_TAC CONNECTED_COMPONENT_TRANS THEN
EXISTS_TAC `x + min (&1 / &2) (e / &2 / norm(c - x)) % (c - x):real^2` THEN
REWRITE_TAC[connected_component] THEN CONJ_TAC THENL
[EXISTS_TAC `segment[x:real^2,c] DELETE x` THEN
SIMP_TAC[CONVEX_SEMIOPEN_SEGMENT; CONVEX_CONNECTED] THEN
ASM_REWRITE_TAC[IN_DELETE; ENDS_IN_SEGMENT] THEN REPEAT CONJ_TAC THENL
[FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE
`cd INTER p = {x}
==> xd SUBSET cd
==> (xd DELETE x) SUBSET (UNIV DIFF p)`)) THEN
REWRITE_TAC[SUBSET_SEGMENT; ENDS_IN_SEGMENT] THEN
UNDISCH_TAC `segment (a,b) INTER segment (c,d) = {x:real^2}` THEN
REWRITE_TAC[open_segment] THEN SET_TAC[];
REWRITE_TAC[IN_SEGMENT; VECTOR_ARITH
`x + a % (y - x):real^N = (&1 - a) % x + a % y`] THEN
EXISTS_TAC `min (&1 / &2) (e / &2 / norm(c - x:real^2))` THEN
REWRITE_TAC[] THEN CONJ_TAC THENL [ALL_TAC; REAL_ARITH_TAC] THEN
REWRITE_TAC[REAL_LE_MIN] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN
ASM_SIMP_TAC[REAL_LE_DIV; REAL_POS; NORM_POS_LE; REAL_LT_IMP_LE];
ASM_REWRITE_TAC[VECTOR_MUL_EQ_0; VECTOR_SUB_EQ;
VECTOR_ARITH `x + a:real^N = x <=> a = vec 0`] THEN
MATCH_MP_TAC(REAL_ARITH `&0 < x ==> ~(min (&1 / &2) x = &0)`) THEN
MATCH_MP_TAC REAL_LT_DIV THEN ASM_REWRITE_TAC[REAL_HALF] THEN
ASM_REWRITE_TAC[NORM_POS_LT; VECTOR_SUB_EQ]];
EXISTS_TAC `ball(x,e) INTER {w:real^2 | n dot (w - x) < &0}` THEN
REPEAT CONJ_TAC THENL
[MATCH_MP_TAC CONVEX_CONNECTED THEN MATCH_MP_TAC CONVEX_INTER THEN
REWRITE_TAC[CONVEX_BALL; DOT_RSUB; REAL_ARITH `a - b < &0 <=> a < b`;
CONVEX_HALFSPACE_LT];
ASM_SIMP_TAC[PATHSTART_LINEPATH; PATH_IMAGE_JOIN; PATH_LINEPATH] THEN
MATCH_MP_TAC(SET_RULE
`p SUBSET (UNIV DIFF b) /\ l INTER w = {}
==> (b INTER w) SUBSET (UNIV DIFF (p UNION l))`) THEN
ASM_REWRITE_TAC[SUBSET; IN_DIFF; IN_UNIV; IN_BALL; REAL_NOT_LT] THEN
MATCH_MP_TAC(SET_RULE
`!t. t INTER u = {} /\ s SUBSET t ==> s INTER u = {}`) THEN
EXISTS_TAC `affine hull {x:real^2,b}` THEN CONJ_TAC THENL
[REWRITE_TAC[AFFINE_HULL_2; FORALL_IN_GSPEC; SET_RULE
`s INTER t = {} <=> !x. x IN s ==> ~(x IN t)`] THEN
REWRITE_TAC[IN_ELIM_THM] THEN
SIMP_TAC[REAL_ARITH `u + v = &1 <=> u = &1 - v`] THEN
REWRITE_TAC[DOT_RMUL; VECTOR_ARITH
`((&1 - v) % x + v % b) - x:real^N = v % (b - x)`] THEN
RULE_ASSUM_TAC(REWRITE_RULE[orthogonal]) THEN
ONCE_REWRITE_TAC[DOT_SYM] THEN
ASM_REWRITE_TAC[REAL_MUL_RZERO; REAL_LT_REFL];
REWRITE_TAC[PATH_IMAGE_LINEPATH; SEGMENT_CONVEX_HULL] THEN
SIMP_TAC[SUBSET_HULL; AFFINE_IMP_CONVEX; AFFINE_AFFINE_HULL] THEN