Hugging Face's logo

Datasets:
hoskinson-center
/
proof-pile

Tasks:
Text Generation
Modalities:
Text
Sub-tasks:
language-modeling
Languages:
English
Size:
100K - 1M
Tags:
math
mathematics
formal-mathematics
Libraries:
Datasets
License:
Dataset card Viewer Files Community
3
proof-pile
File size: 11,083 Bytes 
4365a98
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
 
(* ========================================================================= *)
(* Isosceles triangle theorem.                                               *)
(* ========================================================================= *)

needs "Multivariate/geom.ml";;

(* ------------------------------------------------------------------------- *)
(* The theorem, according to Wikipedia.                                      *)
(* ------------------------------------------------------------------------- *)

let ISOSCELES_TRIANGLE_THEOREM = prove
 (`!A B C:real^N. dist(A,C) = dist(B,C) ==> angle(C,A,B) = angle(A,B,C)`,
  MP_TAC(INST_TYPE [`:N`,`:M`] CONGRUENT_TRIANGLES_SSS) THEN
  MESON_TAC[DIST_SYM; ANGLE_SYM]);;

(* ------------------------------------------------------------------------- *)
(* The obvious converse.                                                     *)
(* ------------------------------------------------------------------------- *)

let ISOSCELES_TRIANGLE_CONVERSE = prove
 (`!A B C:real^N. angle(C,A,B) = angle(A,B,C) /\ ~(collinear {A,B,C})
                  ==> dist(A,C) = dist(B,C)`,
  MP_TAC(INST_TYPE [`:N`,`:M`] CONGRUENT_TRIANGLES_ASA_FULL) THEN
  MESON_TAC[DIST_SYM; ANGLE_SYM]);;

(* ------------------------------------------------------------------------- *)
(* Some other equivalents sometimes called the ITT (see the Web page         *)
(* http://www.sonoma.edu/users/w/wilsonst/Courses/Math_150/Theorems/itt.html *)
(* ------------------------------------------------------------------------- *)

let lemma = prove
 (`!A B C D:real^N.
        between D (A,B)
        ==> (orthogonal (A - B) (C - D) <=>
                angle(A,D,C) = pi / &2 /\ angle(B,D,C) = pi / &2)`,
  REPEAT GEN_TAC THEN
  ASM_CASES_TAC `D:real^N = A` THENL
   [DISCH_TAC THEN ASM_SIMP_TAC[ANGLE_REFL] THEN
    GEN_REWRITE_TAC LAND_CONV [GSYM ORTHOGONAL_LNEG] THEN
    REWRITE_TAC[VECTOR_NEG_SUB; ORTHOGONAL_VECTOR_ANGLE; angle];
    ALL_TAC] THEN
  ASM_CASES_TAC `D:real^N = B` THENL
   [DISCH_TAC THEN ASM_SIMP_TAC[ANGLE_REFL] THEN
    REWRITE_TAC[ORTHOGONAL_VECTOR_ANGLE; angle];
    ALL_TAC] THEN
  DISCH_TAC THEN
  MP_TAC(ISPECL [`A:real^N`; `B:real^N`; `D:real^N`; `C:real^N`]
      ANGLES_ALONG_LINE) THEN
  ASM_REWRITE_TAC[ORTHOGONAL_VECTOR_ANGLE] THEN
  MATCH_MP_TAC(REAL_ARITH
   `x = z ==> x + y = p ==> (z = p / &2 <=> x = p / &2 /\ y = p / &2)`) THEN
  REWRITE_TAC[angle] THEN MATCH_MP_TAC VECTOR_ANGLE_EQ_0_RIGHT THEN
  ONCE_REWRITE_TAC[GSYM VECTOR_ANGLE_NEG2] THEN
  REWRITE_TAC[VECTOR_NEG_SUB; GSYM angle] THEN
  ASM_MESON_TAC[ANGLE_EQ_PI_OTHERS; BETWEEN_ANGLE]);;

let ISOSCELES_TRIANGLE_1 = prove
 (`!A B C D:real^N.
        dist(A,C) = dist(B,C) /\ D = midpoint(A,B)
        ==> angle(A,C,D) = angle(B,C,D)`,
  REPEAT STRIP_TAC THEN
  MP_TAC(ISPECL [`A:real^N`; `D:real^N`; `C:real^N`;
                 `B:real^N`; `D:real^N`; `C:real^N`]
        CONGRUENT_TRIANGLES_SSS_FULL) THEN
  ASM_REWRITE_TAC[DIST_MIDPOINT] THEN ASM_MESON_TAC[DIST_SYM; ANGLE_SYM]);;

let ISOSCELES_TRIANGLE_2 = prove
 (`!A B C D:real^N.
        between D (A,B) /\
        dist(A,C) = dist(B,C) /\ angle(A,C,D) = angle(B,C,D)
        ==> orthogonal (A - B) (C - D)`,
  REPEAT STRIP_TAC THEN
  FIRST_ASSUM(ASSUME_TAC o MATCH_MP ISOSCELES_TRIANGLE_THEOREM) THEN
  MP_TAC(ISPECL [`D:real^N`; `C:real^N`; `A:real^N`;
                 `D:real^N`; `C:real^N`; `B:real^N`]
        CONGRUENT_TRIANGLES_SAS_FULL) THEN
  ANTS_TAC THENL [ASM_MESON_TAC[DIST_SYM; ANGLE_SYM]; ALL_TAC] THEN
  ASM_CASES_TAC `D:real^N = B` THEN
  ASM_SIMP_TAC[DIST_EQ_0; DIST_REFL; VECTOR_SUB_REFL; ORTHOGONAL_0] THEN
  ASM_CASES_TAC `D:real^N = A` THENL [ASM_MESON_TAC[DIST_EQ_0]; ALL_TAC] THEN
  ASM_SIMP_TAC[lemma] THEN
  MP_TAC(ISPECL [`A:real^N`; `B:real^N`; `D:real^N`; `C:real^N`]
      ANGLES_ALONG_LINE) THEN ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC);;

let ISOSCELES_TRIANGLE_3 = prove
 (`!A B C D:real^N.
        between D (A,B) /\
        dist(A,C) = dist(B,C) /\ orthogonal (A - B) (C - D)
        ==> D = midpoint(A,B)`,
  REPEAT GEN_TAC THEN ASM_CASES_TAC `A:real^N = B` THEN
  ASM_SIMP_TAC[BETWEEN_REFL_EQ; MIDPOINT_REFL] THEN
  ASM_CASES_TAC `D:real^N = A` THENL
   [ASM_REWRITE_TAC[] THEN STRIP_TAC THEN
    MP_TAC(ISPECL [`B:real^N`; `A:real^N`; `C:real^N`] PYTHAGORAS) THEN
    ANTS_TAC THENL
     [ASM_MESON_TAC[ORTHOGONAL_LNEG; VECTOR_NEG_SUB]; ALL_TAC] THEN
    ONCE_REWRITE_TAC[NORM_SUB] THEN ASM_REWRITE_TAC[GSYM dist] THEN
    ASM_REWRITE_TAC[REAL_RING `a = x pow 2 + a <=> x = &0`; DIST_EQ_0];
    ALL_TAC] THEN
  ASM_CASES_TAC `D:real^N = B` THENL
   [ASM_REWRITE_TAC[] THEN STRIP_TAC THEN
    MP_TAC(ISPECL [`A:real^N`; `B:real^N`; `C:real^N`] PYTHAGORAS) THEN
    ANTS_TAC THENL
     [ASM_MESON_TAC[ORTHOGONAL_LNEG; VECTOR_NEG_SUB]; ALL_TAC] THEN
    ONCE_REWRITE_TAC[NORM_SUB] THEN ASM_REWRITE_TAC[GSYM dist] THEN
    ASM_REWRITE_TAC[REAL_RING `a = x pow 2 + a <=> x = &0`; DIST_EQ_0];
    ALL_TAC] THEN
  DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
  ASM_SIMP_TAC[lemma; MIDPOINT_COLLINEAR; BETWEEN_IMP_COLLINEAR] THEN
  STRIP_TAC THEN
  FIRST_ASSUM(ASSUME_TAC o MATCH_MP ISOSCELES_TRIANGLE_THEOREM) THEN
  MP_TAC(ISPECL
   [`A:real^N`; `C:real^N`; `D:real^N`;
    `B:real^N`; `C:real^N`; `D:real^N`]
        CONGRUENT_TRIANGLES_SAS) THEN
  ANTS_TAC THENL [ALL_TAC; MESON_TAC[DIST_SYM]] THEN
  ASM_REWRITE_TAC[] THEN
  MP_TAC(ISPECL [`A:real^N`; `C:real^N`; `D:real^N`] TRIANGLE_ANGLE_SUM) THEN
  ANTS_TAC THENL [ASM_MESON_TAC[DIST_EQ_0]; ALL_TAC] THEN
  MP_TAC(ISPECL [`B:real^N`; `C:real^N`; `D:real^N`] TRIANGLE_ANGLE_SUM) THEN
  ANTS_TAC THENL [ASM_MESON_TAC[DIST_EQ_0]; ALL_TAC] THEN
  MATCH_MP_TAC(REAL_ARITH
   `a:real = a' /\ b = b'
    ==> a + x + b = p ==> a' + x' + b' = p ==> x' = x`) THEN
  CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[ANGLE_SYM]] THEN
  CONV_TAC SYM_CONV THEN
  UNDISCH_TAC `angle(C:real^N,A,B) = angle (A,B,C)` THEN
  MATCH_MP_TAC EQ_IMP THEN BINOP_TAC THENL
   [MATCH_MP_TAC ANGLE_EQ_0_LEFT;
    GEN_REWRITE_TAC RAND_CONV [ANGLE_SYM] THEN
    MATCH_MP_TAC ANGLE_EQ_0_RIGHT] THEN
  ASM_MESON_TAC[ANGLE_EQ_PI_OTHERS; BETWEEN_ANGLE]);;

(* ------------------------------------------------------------------------- *)
(* Now the converses to those as well.                                       *)
(* ------------------------------------------------------------------------- *)

let ISOSCELES_TRIANGLE_4 = prove
 (`!A B C D:real^N.
        D = midpoint(A,B) /\ orthogonal (A - B) (C - D)
        ==> dist(A,C) = dist(B,C)`,
  REPEAT GEN_TAC THEN ASM_SIMP_TAC[IMP_CONJ; BETWEEN_MIDPOINT; lemma] THEN
  DISCH_THEN(ASSUME_TAC o SYM) THEN ASM_REWRITE_TAC[] THEN
  REPEAT DISCH_TAC THEN MATCH_MP_TAC CONGRUENT_TRIANGLES_SAS THEN
  MAP_EVERY EXISTS_TAC [`D:real^N`; `D:real^N`] THEN
  ASM_REWRITE_TAC[] THEN EXPAND_TAC "D" THEN REWRITE_TAC[DIST_MIDPOINT]);;

let ISOSCELES_TRIANGLE_5 = prove
 (`!A B C D:real^N.
        ~collinear{D,C,A} /\ between D (A,B) /\
        angle(A,C,D) = angle(B,C,D) /\ orthogonal (A - B) (C - D)
        ==> dist(A,C) = dist(B,C)`,
  REPEAT GEN_TAC THEN
  ASM_CASES_TAC `C:real^N = D` THENL
   [ASM_REWRITE_TAC[INSERT_AC; COLLINEAR_2]; ALL_TAC] THEN
  DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
  UNDISCH_TAC `~(C:real^N = D)` THEN
  REWRITE_TAC[GSYM IMP_CONJ_ALT; GSYM CONJ_ASSOC] THEN
  ASM_CASES_TAC `A:real^N = B` THEN ASM_REWRITE_TAC[] THEN
  ASM_CASES_TAC `C:real^N = A` THENL
   [DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
    ASM_REWRITE_TAC[ANGLE_REFL] THEN
    FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [BETWEEN_ANGLE]) THEN
    ASM_CASES_TAC `D:real^N = A` THEN ASM_REWRITE_TAC[] THEN
    ASM_CASES_TAC `D:real^N = B` THEN ASM_REWRITE_TAC[] THEN
    ASM_SIMP_TAC[ANGLE_REFL_MID; REAL_ARITH `x / &2 = &0 <=> x = &0`;
                 PI_NZ] THEN
    DISCH_THEN(MP_TAC o MATCH_MP ANGLE_EQ_PI_OTHERS) THEN
    MP_TAC PI_NZ THEN REAL_ARITH_TAC;
    ALL_TAC] THEN
  ASM_CASES_TAC `C:real^N = B` THENL
   [DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
    ASM_REWRITE_TAC[ANGLE_REFL] THEN
    FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [BETWEEN_ANGLE]) THEN
    ASM_CASES_TAC `D:real^N = B` THEN ASM_REWRITE_TAC[] THEN
    ASM_CASES_TAC `D:real^N = A` THEN ASM_REWRITE_TAC[] THEN
    ASM_SIMP_TAC[ANGLE_REFL_MID; REAL_ARITH `&0 = x / &2 <=> x = &0`;
                 PI_NZ] THEN
    DISCH_THEN(MP_TAC o MATCH_MP ANGLE_EQ_PI_OTHERS) THEN
    MP_TAC PI_NZ THEN REAL_ARITH_TAC;
    ALL_TAC] THEN
  ASM_SIMP_TAC[IMP_CONJ; lemma] THEN
  REPEAT DISCH_TAC THEN MP_TAC(
    ISPECL [`D:real^N`; `C:real^N`; `A:real^N`;
            `D:real^N`; `C:real^N`; `B:real^N`]
     CONGRUENT_TRIANGLES_ASA_FULL) THEN
  ANTS_TAC THENL [ALL_TAC; MESON_TAC[DIST_SYM]] THEN
  ONCE_REWRITE_TAC[ANGLE_SYM] THEN ASM_REWRITE_TAC[]);;

let ISOSCELES_TRIANGLE_6 = prove
 (`!A B C D:real^N.
        ~collinear{D,C,A} /\ D = midpoint(A,B) /\ angle(A,C,D) = angle(B,C,D)
        ==> dist(A,C) = dist(B,C)`,
  REPEAT GEN_TAC THEN DISCH_THEN(STRIP_ASSUME_TAC o GSYM) THEN
  ASM_CASES_TAC `A:real^N = B` THEN ASM_REWRITE_TAC[] THEN
  MP_TAC(ISPECL [`A:real^N`; `C:real^N`; `D:real^N`] LAW_OF_SINES) THEN
  MP_TAC(ISPECL [`B:real^N`; `C:real^N`; `D:real^N`] LAW_OF_SINES) THEN
  ASM_REWRITE_TAC[] THEN DISCH_THEN SUBST1_TAC THEN
  EXPAND_TAC "D" THEN REWRITE_TAC[DIST_MIDPOINT] THEN
  ASM_SIMP_TAC[REAL_EQ_MUL_RCANCEL; REAL_LT_IMP_NZ; REAL_HALF; DIST_POS_LT;
               SIN_ANGLE_EQ] THEN
  STRIP_TAC THENL
   [MP_TAC(ISPECL [`D:real^N`; `C:real^N`; `A:real^N`;
                   `D:real^N`; `C:real^N`; `B:real^N`]
       CONGRUENT_TRIANGLES_AAS) THEN
    ANTS_TAC THENL [ALL_TAC; MESON_TAC[DIST_SYM]] THEN
    ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[ANGLE_SYM] THEN
    ASM_REWRITE_TAC[];
    MP_TAC(ISPECL [`A:real^N`; `B:real^N`; `C:real^N`]
                TRIANGLE_ANGLE_SUM) THEN
    ASM_REWRITE_TAC[] THEN
    SUBGOAL_THEN `angle(A:real^N,B,C) = angle(C,B,D) /\
                  angle(B,A,C) = angle(C,A,D)`
     (CONJUNCTS_THEN SUBST1_TAC)
    THENL
     [CONJ_TAC THEN GEN_REWRITE_TAC LAND_CONV [ANGLE_SYM] THEN
      MATCH_MP_TAC ANGLE_EQ_0_LEFT THEN
      MP_TAC(ISPECL [`A:real^N`; `B:real^N`] BETWEEN_MIDPOINT) THEN
      ASM_REWRITE_TAC[BETWEEN_ANGLE] THEN EXPAND_TAC "D" THEN
      REWRITE_TAC[MIDPOINT_EQ_ENDPOINT] THEN ASM_REWRITE_TAC[] THEN
      MESON_TAC[ANGLE_EQ_PI_OTHERS];
      ALL_TAC] THEN
    ASM_REWRITE_TAC[REAL_ARITH `a + pi - a + x = pi <=> x = &0`] THEN
    MAP_EVERY ASM_CASES_TAC
     [`B:real^N = C`; `A:real^N = C`] THEN
    ASM_REWRITE_TAC[ANGLE_REFL; REAL_ARITH `p / &2 = &0 <=> p = &0`] THEN
    ASM_REWRITE_TAC[PI_NZ] THEN DISCH_TAC THEN
    MP_TAC(ISPECL [`B:real^N`; `C:real^N`; `A:real^N`] COLLINEAR_ANGLE) THEN
    ASM_REWRITE_TAC[] THEN
    UNDISCH_TAC `~collinear{D:real^N,C,A}` THEN
    MATCH_MP_TAC(TAUT `(q ==> p) ==> ~p ==> q ==> r`) THEN
    ONCE_REWRITE_TAC[SET_RULE `{bd,c,a} = {c,a,bd}`] THEN
    ONCE_REWRITE_TAC[COLLINEAR_3] THEN
    REWRITE_TAC[COLLINEAR_LEMMA] THEN ASM_REWRITE_TAC[VECTOR_SUB_EQ] THEN
    EXPAND_TAC "D" THEN REWRITE_TAC[midpoint] THEN
    REWRITE_TAC[VECTOR_ARITH `inv(&2) % (A + B) - A = inv(&2) % (B - A)`] THEN
    MESON_TAC[VECTOR_MUL_ASSOC]]);;