2 header {* Meson test cases *}
9 WARNING: there are many potential conflicts between variables used
10 below and constants declared in HOL!
13 hide_const (open) implies union inter subset sum quotient
16 Test data for the MESON proof procedure
17 (Excludes the equality problems 51, 52, 56, 58)
21 subsection {* Interactive examples *}
24 "(\<exists>x. P x) & (\<forall>x. L x --> ~ (M x & R x)) & (\<forall>x. P x --> (M x & L x)) & ((\<forall>x. P x --> Q x) | (\<exists>x. P x & R x)) --> (\<exists>x. Q x & P x)"
27 val prem25 = Thm.assume @{cprop "\<not> ?thesis"};
28 val nnf25 = Meson.make_nnf @{context} prem25;
29 val xsko25 = Meson.skolemize @{context} nnf25;
31 apply (tactic {* cut_facts_tac [xsko25] 1 THEN REPEAT (etac exE 1) *})
33 val [_, sko25] = #prems (#1 (Subgoal.focus @{context} 1 (#goal @{Isar.goal})));
34 val clauses25 = Meson.make_clauses [sko25]; (*7 clauses*)
35 val horns25 = Meson.make_horns clauses25; (*16 Horn clauses*)
36 val go25 :: _ = Meson.gocls clauses25;
38 Goal.prove @{context} [] [] @{prop False} (fn _ =>
40 Meson.depth_prolog_tac horns25);
45 "((\<exists>x. p x) = (\<exists>x. q x)) & (\<forall>x. \<forall>y. p x & q y --> (r x = s y)) --> ((\<forall>x. p x --> r x) = (\<forall>x. q x --> s x))"
48 val prem26 = Thm.assume @{cprop "\<not> ?thesis"}
49 val nnf26 = Meson.make_nnf @{context} prem26;
50 val xsko26 = Meson.skolemize @{context} nnf26;
52 apply (tactic {* cut_facts_tac [xsko26] 1 THEN REPEAT (etac exE 1) *})
54 val [_, sko26] = #prems (#1 (Subgoal.focus @{context} 1 (#goal @{Isar.goal})));
55 val clauses26 = Meson.make_clauses [sko26]; (*9 clauses*)
56 val horns26 = Meson.make_horns clauses26; (*24 Horn clauses*)
57 val go26 :: _ = Meson.gocls clauses26;
59 Goal.prove @{context} [] [] @{prop False} (fn _ =>
61 Meson.depth_prolog_tac horns26); (*7 ms*)
62 (*Proof is of length 107!!*)
66 lemma problem_43: -- "NOW PROVED AUTOMATICALLY!!" (*16 Horn clauses*)
67 "(\<forall>x. \<forall>y. q x y = (\<forall>z. p z x = (p z y::bool))) --> (\<forall>x. (\<forall>y. q x y = (q y x::bool)))"
70 val prem43 = Thm.assume @{cprop "\<not> ?thesis"};
71 val nnf43 = Meson.make_nnf @{context} prem43;
72 val xsko43 = Meson.skolemize @{context} nnf43;
74 apply (tactic {* cut_facts_tac [xsko43] 1 THEN REPEAT (etac exE 1) *})
76 val [_, sko43] = #prems (#1 (Subgoal.focus @{context} 1 (#goal @{Isar.goal})));
77 val clauses43 = Meson.make_clauses [sko43]; (*6*)
78 val horns43 = Meson.make_horns clauses43; (*16*)
79 val go43 :: _ = Meson.gocls clauses43;
81 Goal.prove @{context} [] [] @{prop False} (fn _ =>
83 Meson.best_prolog_tac Meson.size_of_subgoals horns43); (*7ms*)
88 #1 (q x xa ==> ~ q x xa) ==> q xa x
89 #2 (q xa x ==> ~ q xa x) ==> q x xa
90 #3 (~ q x xa ==> q x xa) ==> ~ q xa x
91 #4 (~ q xa x ==> q xa x) ==> ~ q x xa
92 #5 [| ~ q ?U ?V ==> q ?U ?V; ~ p ?W ?U ==> p ?W ?U |] ==> p ?W ?V
93 #6 [| ~ p ?W ?U ==> p ?W ?U; p ?W ?V ==> ~ p ?W ?V |] ==> ~ q ?U ?V
94 #7 [| p ?W ?V ==> ~ p ?W ?V; ~ q ?U ?V ==> q ?U ?V |] ==> ~ p ?W ?U
95 #8 [| ~ q ?U ?V ==> q ?U ?V; ~ p ?W ?V ==> p ?W ?V |] ==> p ?W ?U
96 #9 [| ~ p ?W ?V ==> p ?W ?V; p ?W ?U ==> ~ p ?W ?U |] ==> ~ q ?U ?V
97 #10 [| p ?W ?U ==> ~ p ?W ?U; ~ q ?U ?V ==> q ?U ?V |] ==> ~ p ?W ?V
98 #11 [| p (xb ?U ?V) ?U ==> ~ p (xb ?U ?V) ?U;
99 p (xb ?U ?V) ?V ==> ~ p (xb ?U ?V) ?V |] ==> q ?U ?V
100 #12 [| p (xb ?U ?V) ?V ==> ~ p (xb ?U ?V) ?V; q ?U ?V ==> ~ q ?U ?V |] ==>
102 #13 [| q ?U ?V ==> ~ q ?U ?V; p (xb ?U ?V) ?U ==> ~ p (xb ?U ?V) ?U |] ==>
104 #14 [| ~ p (xb ?U ?V) ?U ==> p (xb ?U ?V) ?U;
105 ~ p (xb ?U ?V) ?V ==> p (xb ?U ?V) ?V |] ==> q ?U ?V
106 #15 [| ~ p (xb ?U ?V) ?V ==> p (xb ?U ?V) ?V; q ?U ?V ==> ~ q ?U ?V |] ==>
108 #16 [| q ?U ?V ==> ~ q ?U ?V; ~ p (xb ?U ?V) ?U ==> p (xb ?U ?V) ?U |] ==>
111 And here is the proof! (Unkn is the start state after use of goal clause)
112 [Unkn, Res ([Thm "#14"], false, 1), Res ([Thm "#5"], false, 1),
113 Res ([Thm "#1"], false, 1), Asm 1, Res ([Thm "#13"], false, 1), Asm 2,
114 Asm 1, Res ([Thm "#13"], false, 1), Asm 1, Res ([Thm "#10"], false, 1),
115 Res ([Thm "#16"], false, 1), Asm 2, Asm 1, Res ([Thm "#1"], false, 1),
116 Asm 1, Res ([Thm "#14"], false, 1), Res ([Thm "#5"], false, 1),
117 Res ([Thm "#2"], false, 1), Asm 1, Res ([Thm "#13"], false, 1), Asm 2,
118 Asm 1, Res ([Thm "#8"], false, 1), Res ([Thm "#2"], false, 1), Asm 1,
119 Res ([Thm "#12"], false, 1), Asm 2, Asm 1] : lderiv list
124 MORE and MUCH HARDER test data for the MESON proof procedure
125 (courtesy John Harrison).
128 (* ========================================================================= *)
129 (* 100 problems selected from the TPTP library *)
130 (* ========================================================================= *)
133 * Original timings for John Harrison's MESON_TAC.
134 * Timings below on a 600MHz Pentium III (perch)
135 * Some timings below refer to griffon, which is a dual 2.5GHz Power Mac G5.
137 * A few variable names have been changed to avoid clashing with constants.
139 * Changed numeric constants e.g. 0, 1, 2... to num0, num1, num2...
141 * Here's a list giving typical CPU times, as well as common names and
142 * literature references.
144 * BOO003-1 34.6 B2 part 1 [McCharen, et al., 1976]; Lemma proved [Overbeek, et al., 1976]; prob2_part1.ver1.in [ANL]
145 * BOO004-1 36.7 B2 part 2 [McCharen, et al., 1976]; Lemma proved [Overbeek, et al., 1976]; prob2_part2.ver1 [ANL]
146 * BOO005-1 47.4 B3 part 1 [McCharen, et al., 1976]; B5 [McCharen, et al., 1976]; Lemma proved [Overbeek, et al., 1976]; prob3_part1.ver1.in [ANL]
147 * BOO006-1 48.4 B3 part 2 [McCharen, et al., 1976]; B6 [McCharen, et al., 1976]; Lemma proved [Overbeek, et al., 1976]; prob3_part2.ver1 [ANL]
148 * BOO011-1 19.0 B7 [McCharen, et al., 1976]; prob7.ver1 [ANL]
149 * CAT001-3 45.2 C1 [McCharen, et al., 1976]; p1.ver3.in [ANL]
150 * CAT003-3 10.5 C3 [McCharen, et al., 1976]; p3.ver3.in [ANL]
151 * CAT005-1 480.1 C5 [McCharen, et al., 1976]; p5.ver1.in [ANL]
152 * CAT007-1 11.9 C7 [McCharen, et al., 1976]; p7.ver1.in [ANL]
153 * CAT018-1 81.3 p18.ver1.in [ANL]
154 * COL001-2 16.0 C1 [Wos & McCune, 1988]
155 * COL023-1 5.1 [McCune & Wos, 1988]
156 * COL032-1 15.8 [McCune & Wos, 1988]
157 * COL052-2 13.2 bird4.ver2.in [ANL]
158 * COL075-2 116.9 [Jech, 1994]
159 * COM001-1 1.7 shortburst [Wilson & Minker, 1976]
160 * COM002-1 4.4 burstall [Wilson & Minker, 1976]
162 * COM003-2 22.1 [Brushi, 1991]
164 * GEO003-1 71.7 T3 [McCharen, et al., 1976]; t3.ver1.in [ANL]
165 * GEO017-2 78.8 D4.1 [Quaife, 1989]
166 * GEO027-3 181.5 D10.1 [Quaife, 1989]
167 * GEO058-2 104.0 R4 [Quaife, 1989]
168 * GEO079-1 2.4 GEOMETRY THEOREM [Slagle, 1967]
169 * GRP001-1 47.8 CADE-11 Competition 1 [Overbeek, 1990]; G1 [McCharen, et al., 1976]; THEOREM 1 [Lusk & McCune, 1993]; wos10 [Wilson & Minker, 1976]; xsquared.ver1.in [ANL]; [Robinson, 1963]
170 * GRP008-1 50.4 Problem 4 [Wos, 1965]; wos4 [Wilson & Minker, 1976]
171 * GRP013-1 40.2 Problem 11 [Wos, 1965]; wos11 [Wilson & Minker, 1976]
172 * GRP037-3 43.8 Problem 17 [Wos, 1965]; wos17 [Wilson & Minker, 1976]
173 * GRP031-2 3.2 ls23 [Lawrence & Starkey, 1974]; ls23 [Wilson & Minker, 1976]
174 * GRP034-4 2.5 ls26 [Lawrence & Starkey, 1974]; ls26 [Wilson & Minker, 1976]
175 * GRP047-2 11.7 [Veroff, 1992]
176 * GRP130-1 170.6 Bennett QG8 [TPTP]; QG8 [Slaney, 1993]
177 * GRP156-1 48.7 ax_mono1c [Schulz, 1995]
178 * GRP168-1 159.1 p01a [Schulz, 1995]
179 * HEN003-3 39.9 HP3 [McCharen, et al., 1976]
180 * HEN007-2 125.7 H7 [McCharen, et al., 1976]
181 * HEN008-4 62.0 H8 [McCharen, et al., 1976]
182 * HEN009-5 136.3 H9 [McCharen, et al., 1976]; hp9.ver3.in [ANL]
183 * HEN012-3 48.5 new.ver2.in [ANL]
184 * LCL010-1 370.9 EC-73 [McCune & Wos, 1992]; ec_yq.in [OTTER]
185 * LCL077-2 51.6 morgan.two.ver1.in [ANL]
186 * LCL082-1 14.6 IC-1.1 [Wos, et al., 1990]; IC-65 [McCune & Wos, 1992]; ls2 [SETHEO]; S1 [Pfenning, 1988]
187 * LCL111-1 585.6 CADE-11 Competition 6 [Overbeek, 1990]; mv25.in [OTTER]; MV-57 [McCune & Wos, 1992]; mv.in part 2 [OTTER]; ovb6 [SETHEO]; THEOREM 6 [Lusk & McCune, 1993]
188 * LCL143-1 10.9 Lattice structure theorem 2 [Bonacina, 1991]
189 * LCL182-1 271.6 Problem 2.16 [Whitehead & Russell, 1927]
190 * LCL200-1 12.0 Problem 2.46 [Whitehead & Russell, 1927]
191 * LCL215-1 214.4 Problem 2.62 [Whitehead & Russell, 1927]; Problem 2.63 [Whitehead & Russell, 1927]
192 * LCL230-2 0.2 Pelletier 5 [Pelletier, 1986]
193 * LDA003-1 68.5 Problem 3 [Jech, 1993]
194 * MSC002-1 9.2 DBABHP [Michie, et al., 1972]; DBABHP [Wilson & Minker, 1976]
195 * MSC003-1 3.2 HASPARTS-T1 [Wilson & Minker, 1976]
196 * MSC004-1 9.3 HASPARTS-T2 [Wilson & Minker, 1976]
197 * MSC005-1 1.8 Problem 5.1 [Plaisted, 1982]
198 * MSC006-1 39.0 nonob.lop [SETHEO]
199 * NUM001-1 14.0 Chang-Lee-10a [Chang, 1970]; ls28 [Lawrence & Starkey, 1974]; ls28 [Wilson & Minker, 1976]
200 * NUM021-1 52.3 ls65 [Lawrence & Starkey, 1974]; ls65 [Wilson & Minker, 1976]
201 * NUM024-1 64.6 ls75 [Lawrence & Starkey, 1974]; ls75 [Wilson & Minker, 1976]
202 * NUM180-1 621.2 LIM2.1 [Quaife]
203 * NUM228-1 575.9 TRECDEF4 cor. [Quaife]
204 * PLA002-1 37.4 Problem 5.7 [Plaisted, 1982]
205 * PLA006-1 7.2 [Segre & Elkan, 1994]
206 * PLA017-1 484.8 [Segre & Elkan, 1994]
207 * PLA022-1 19.1 [Segre & Elkan, 1994]
208 * PLA022-2 19.7 [Segre & Elkan, 1994]
209 * PRV001-1 10.3 PV1 [McCharen, et al., 1976]
210 * PRV003-1 3.9 E2 [McCharen, et al., 1976]; v2.lop [SETHEO]
211 * PRV005-1 4.3 E4 [McCharen, et al., 1976]; v4.lop [SETHEO]
212 * PRV006-1 6.0 E5 [McCharen, et al., 1976]; v5.lop [SETHEO]
213 * PRV009-1 2.2 Hoares FIND [Bledsoe, 1977]; Problem 5.5 [Plaisted, 1982]
214 * PUZ012-1 3.5 Boxes-of-fruit [Wos, 1988]; Boxes-of-fruit [Wos, et al., 1992]; boxes.ver1.in [ANL]
215 * PUZ020-1 56.6 knightknave.in [ANL]
216 * PUZ025-1 58.4 Problem 35 [Smullyan, 1978]; tandl35.ver1.in [ANL]
217 * PUZ029-1 5.1 pigs.ver1.in [ANL]
218 * RNG001-3 82.4 EX6-T? [Wilson & Minker, 1976]; ex6.lop [SETHEO]; Example 6a [Fleisig, et al., 1974]; FEX6T1 [SPRFN]; FEX6T2 [SPRFN]
219 * RNG001-5 399.8 Problem 21 [Wos, 1965]; wos21 [Wilson & Minker, 1976]
220 * RNG011-5 8.4 CADE-11 Competition Eq-10 [Overbeek, 1990]; PROBLEM 10 [Zhang, 1993]; THEOREM EQ-10 [Lusk & McCune, 1993]
221 * RNG023-6 9.1 [Stevens, 1987]
222 * RNG028-2 9.3 PROOF III [Anantharaman & Hsiang, 1990]
223 * RNG038-2 16.2 Problem 27 [Wos, 1965]; wos27 [Wilson & Minker, 1976]
224 * RNG040-2 180.5 Problem 29 [Wos, 1965]; wos29 [Wilson & Minker, 1976]
225 * RNG041-1 35.8 Problem 30 [Wos, 1965]; wos30 [Wilson & Minker, 1976]
226 * ROB010-1 205.0 Lemma 3.3 [Winker, 1990]; RA2 [Lusk & Wos, 1992]
227 * ROB013-1 23.6 Lemma 3.5 [Winker, 1990]
228 * ROB016-1 15.2 Corollary 3.7 [Winker, 1990]
229 * ROB021-1 230.4 [McCune, 1992]
230 * SET005-1 192.2 ls108 [Lawrence & Starkey, 1974]; ls108 [Wilson & Minker, 1976]
231 * SET009-1 10.5 ls116 [Lawrence & Starkey, 1974]; ls116 [Wilson & Minker, 1976]
232 * SET025-4 694.7 Lemma 10 [Boyer, et al, 1986]
233 * SET046-5 2.3 p42.in [ANL]; Pelletier 42 [Pelletier, 1986]
234 * SET047-5 3.7 p43.in [ANL]; Pelletier 43 [Pelletier, 1986]
235 * SYN034-1 2.8 QW [Michie, et al., 1972]; QW [Wilson & Minker, 1976]
236 * SYN071-1 1.9 Pelletier 48 [Pelletier, 1986]
237 * SYN349-1 61.7 Ch17N5 [Tammet, 1994]
238 * SYN352-1 5.5 Ch18N4 [Tammet, 1994]
239 * TOP001-2 61.1 Lemma 1a [Wick & McCune, 1989]
240 * TOP002-2 0.4 Lemma 1b [Wick & McCune, 1989]
241 * TOP004-1 181.6 Lemma 1d [Wick & McCune, 1989]
242 * TOP004-2 9.0 Lemma 1d [Wick & McCune, 1989]
243 * TOP005-2 139.8 Lemma 1e [Wick & McCune, 1989]
246 abbreviation "EQU001_0_ax equal \<equiv> (\<forall>X. equal(X::'a,X)) &
247 (\<forall>Y X. equal(X::'a,Y) --> equal(Y::'a,X)) &
248 (\<forall>Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z))"
250 abbreviation "BOO002_0_ax equal INVERSE multiplicative_identity
251 additive_identity multiply product add sum \<equiv>
252 (\<forall>X Y. sum(X::'a,Y,add(X::'a,Y))) &
253 (\<forall>X Y. product(X::'a,Y,multiply(X::'a,Y))) &
254 (\<forall>Y X Z. sum(X::'a,Y,Z) --> sum(Y::'a,X,Z)) &
255 (\<forall>Y X Z. product(X::'a,Y,Z) --> product(Y::'a,X,Z)) &
256 (\<forall>X. sum(additive_identity::'a,X,X)) &
257 (\<forall>X. sum(X::'a,additive_identity,X)) &
258 (\<forall>X. product(multiplicative_identity::'a,X,X)) &
259 (\<forall>X. product(X::'a,multiplicative_identity,X)) &
260 (\<forall>Y Z X V3 V1 V2 V4. product(X::'a,Y,V1) & product(X::'a,Z,V2) & sum(Y::'a,Z,V3) & product(X::'a,V3,V4) --> sum(V1::'a,V2,V4)) &
261 (\<forall>Y Z V1 V2 X V3 V4. product(X::'a,Y,V1) & product(X::'a,Z,V2) & sum(Y::'a,Z,V3) & sum(V1::'a,V2,V4) --> product(X::'a,V3,V4)) &
262 (\<forall>Y Z V3 X V1 V2 V4. product(Y::'a,X,V1) & product(Z::'a,X,V2) & sum(Y::'a,Z,V3) & product(V3::'a,X,V4) --> sum(V1::'a,V2,V4)) &
263 (\<forall>Y Z V1 V2 V3 X V4. product(Y::'a,X,V1) & product(Z::'a,X,V2) & sum(Y::'a,Z,V3) & sum(V1::'a,V2,V4) --> product(V3::'a,X,V4)) &
264 (\<forall>Y Z X V3 V1 V2 V4. sum(X::'a,Y,V1) & sum(X::'a,Z,V2) & product(Y::'a,Z,V3) & sum(X::'a,V3,V4) --> product(V1::'a,V2,V4)) &
265 (\<forall>Y Z V1 V2 X V3 V4. sum(X::'a,Y,V1) & sum(X::'a,Z,V2) & product(Y::'a,Z,V3) & product(V1::'a,V2,V4) --> sum(X::'a,V3,V4)) &
266 (\<forall>Y Z V3 X V1 V2 V4. sum(Y::'a,X,V1) & sum(Z::'a,X,V2) & product(Y::'a,Z,V3) & sum(V3::'a,X,V4) --> product(V1::'a,V2,V4)) &
267 (\<forall>Y Z V1 V2 V3 X V4. sum(Y::'a,X,V1) & sum(Z::'a,X,V2) & product(Y::'a,Z,V3) & product(V1::'a,V2,V4) --> sum(V3::'a,X,V4)) &
268 (\<forall>X. sum(INVERSE(X),X,multiplicative_identity)) &
269 (\<forall>X. sum(X::'a,INVERSE(X),multiplicative_identity)) &
270 (\<forall>X. product(INVERSE(X),X,additive_identity)) &
271 (\<forall>X. product(X::'a,INVERSE(X),additive_identity)) &
272 (\<forall>X Y U V. sum(X::'a,Y,U) & sum(X::'a,Y,V) --> equal(U::'a,V)) &
273 (\<forall>X Y U V. product(X::'a,Y,U) & product(X::'a,Y,V) --> equal(U::'a,V))"
275 abbreviation "BOO002_0_eq INVERSE multiply add product sum equal \<equiv>
276 (\<forall>X Y W Z. equal(X::'a,Y) & sum(X::'a,W,Z) --> sum(Y::'a,W,Z)) &
277 (\<forall>X W Y Z. equal(X::'a,Y) & sum(W::'a,X,Z) --> sum(W::'a,Y,Z)) &
278 (\<forall>X W Z Y. equal(X::'a,Y) & sum(W::'a,Z,X) --> sum(W::'a,Z,Y)) &
279 (\<forall>X Y W Z. equal(X::'a,Y) & product(X::'a,W,Z) --> product(Y::'a,W,Z)) &
280 (\<forall>X W Y Z. equal(X::'a,Y) & product(W::'a,X,Z) --> product(W::'a,Y,Z)) &
281 (\<forall>X W Z Y. equal(X::'a,Y) & product(W::'a,Z,X) --> product(W::'a,Z,Y)) &
282 (\<forall>X Y W. equal(X::'a,Y) --> equal(add(X::'a,W),add(Y::'a,W))) &
283 (\<forall>X W Y. equal(X::'a,Y) --> equal(add(W::'a,X),add(W::'a,Y))) &
284 (\<forall>X Y W. equal(X::'a,Y) --> equal(multiply(X::'a,W),multiply(Y::'a,W))) &
285 (\<forall>X W Y. equal(X::'a,Y) --> equal(multiply(W::'a,X),multiply(W::'a,Y))) &
286 (\<forall>X Y. equal(X::'a,Y) --> equal(INVERSE(X),INVERSE(Y)))"
288 (*51194 inferences so far. Searching to depth 13. 232.9 secs*)
291 BOO002_0_ax equal INVERSE multiplicative_identity additive_identity multiply product add sum &
292 BOO002_0_eq INVERSE multiply add product sum equal &
293 (~product(x::'a,x,x)) --> False"
296 (*51194 inferences so far. Searching to depth 13. 204.6 secs
297 Strange! The previous problem also has 51194 inferences at depth 13. They
298 must be very similar!*)
301 BOO002_0_ax equal INVERSE multiplicative_identity additive_identity multiply product add sum &
302 BOO002_0_eq INVERSE multiply add product sum equal &
303 (~sum(x::'a,x,x)) --> False"
306 (*74799 inferences so far. Searching to depth 13. 290.0 secs*)
309 BOO002_0_ax equal INVERSE multiplicative_identity additive_identity multiply product add sum &
310 BOO002_0_eq INVERSE multiply add product sum equal &
311 (~sum(x::'a,multiplicative_identity,multiplicative_identity)) --> False"
314 (*74799 inferences so far. Searching to depth 13. 314.6 secs*)
317 BOO002_0_ax equal INVERSE multiplicative_identity additive_identity multiply product add sum &
318 BOO002_0_eq INVERSE multiply add product sum equal &
319 (~product(x::'a,additive_identity,additive_identity)) --> False"
322 (*5 inferences so far. Searching to depth 5. 1.3 secs*)
325 BOO002_0_ax equal INVERSE multiplicative_identity additive_identity multiply product add sum &
326 BOO002_0_eq INVERSE multiply add product sum equal &
327 (~equal(INVERSE(additive_identity),multiplicative_identity)) --> False"
330 abbreviation "CAT003_0_ax f1 compos codomain domain equal there_exists equivalent \<equiv>
331 (\<forall>Y X. equivalent(X::'a,Y) --> there_exists(X)) &
332 (\<forall>X Y. equivalent(X::'a,Y) --> equal(X::'a,Y)) &
333 (\<forall>X Y. there_exists(X) & equal(X::'a,Y) --> equivalent(X::'a,Y)) &
334 (\<forall>X. there_exists(domain(X)) --> there_exists(X)) &
335 (\<forall>X. there_exists(codomain(X)) --> there_exists(X)) &
336 (\<forall>Y X. there_exists(compos(X::'a,Y)) --> there_exists(domain(X))) &
337 (\<forall>X Y. there_exists(compos(X::'a,Y)) --> equal(domain(X),codomain(Y))) &
338 (\<forall>X Y. there_exists(domain(X)) & equal(domain(X),codomain(Y)) --> there_exists(compos(X::'a,Y))) &
339 (\<forall>X Y Z. equal(compos(X::'a,compos(Y::'a,Z)),compos(compos(X::'a,Y),Z))) &
340 (\<forall>X. equal(compos(X::'a,domain(X)),X)) &
341 (\<forall>X. equal(compos(codomain(X),X),X)) &
342 (\<forall>X Y. equivalent(X::'a,Y) --> there_exists(Y)) &
343 (\<forall>X Y. there_exists(X) & there_exists(Y) & equal(X::'a,Y) --> equivalent(X::'a,Y)) &
344 (\<forall>Y X. there_exists(compos(X::'a,Y)) --> there_exists(codomain(X))) &
345 (\<forall>X Y. there_exists(f1(X::'a,Y)) | equal(X::'a,Y)) &
346 (\<forall>X Y. equal(X::'a,f1(X::'a,Y)) | equal(Y::'a,f1(X::'a,Y)) | equal(X::'a,Y)) &
347 (\<forall>X Y. equal(X::'a,f1(X::'a,Y)) & equal(Y::'a,f1(X::'a,Y)) --> equal(X::'a,Y))"
349 abbreviation "CAT003_0_eq f1 compos codomain domain equivalent there_exists equal \<equiv>
350 (\<forall>X Y. equal(X::'a,Y) & there_exists(X) --> there_exists(Y)) &
351 (\<forall>X Y Z. equal(X::'a,Y) & equivalent(X::'a,Z) --> equivalent(Y::'a,Z)) &
352 (\<forall>X Z Y. equal(X::'a,Y) & equivalent(Z::'a,X) --> equivalent(Z::'a,Y)) &
353 (\<forall>X Y. equal(X::'a,Y) --> equal(domain(X),domain(Y))) &
354 (\<forall>X Y. equal(X::'a,Y) --> equal(codomain(X),codomain(Y))) &
355 (\<forall>X Y Z. equal(X::'a,Y) --> equal(compos(X::'a,Z),compos(Y::'a,Z))) &
356 (\<forall>X Z Y. equal(X::'a,Y) --> equal(compos(Z::'a,X),compos(Z::'a,Y))) &
357 (\<forall>A B C. equal(A::'a,B) --> equal(f1(A::'a,C),f1(B::'a,C))) &
358 (\<forall>D F' E. equal(D::'a,E) --> equal(f1(F'::'a,D),f1(F'::'a,E)))"
360 (*4007 inferences so far. Searching to depth 9. 13 secs*)
363 CAT003_0_ax f1 compos codomain domain equal there_exists equivalent &
364 CAT003_0_eq f1 compos codomain domain equivalent there_exists equal &
365 (there_exists(compos(a::'a,b))) &
366 (\<forall>Y X Z. equal(compos(compos(a::'a,b),X),Y) & equal(compos(compos(a::'a,b),Z),Y) --> equal(X::'a,Z)) &
367 (there_exists(compos(b::'a,h))) &
368 (equal(compos(b::'a,h),compos(b::'a,g))) &
369 (~equal(h::'a,g)) --> False"
372 (*245 inferences so far. Searching to depth 7. 1.0 secs*)
375 CAT003_0_ax f1 compos codomain domain equal there_exists equivalent &
376 CAT003_0_eq f1 compos codomain domain equivalent there_exists equal &
377 (there_exists(compos(a::'a,b))) &
378 (\<forall>Y X Z. equal(compos(X::'a,compos(a::'a,b)),Y) & equal(compos(Z::'a,compos(a::'a,b)),Y) --> equal(X::'a,Z)) &
380 (equal(compos(h::'a,a),compos(g::'a,a))) &
381 (~equal(g::'a,h)) --> False"
384 abbreviation "CAT001_0_ax equal codomain domain identity_map compos product defined \<equiv>
385 (\<forall>X Y. defined(X::'a,Y) --> product(X::'a,Y,compos(X::'a,Y))) &
386 (\<forall>Z X Y. product(X::'a,Y,Z) --> defined(X::'a,Y)) &
387 (\<forall>X Xy Y Z. product(X::'a,Y,Xy) & defined(Xy::'a,Z) --> defined(Y::'a,Z)) &
388 (\<forall>Y Xy Z X Yz. product(X::'a,Y,Xy) & product(Y::'a,Z,Yz) & defined(Xy::'a,Z) --> defined(X::'a,Yz)) &
389 (\<forall>Xy Y Z X Yz Xyz. product(X::'a,Y,Xy) & product(Xy::'a,Z,Xyz) & product(Y::'a,Z,Yz) --> product(X::'a,Yz,Xyz)) &
390 (\<forall>Z Yz X Y. product(Y::'a,Z,Yz) & defined(X::'a,Yz) --> defined(X::'a,Y)) &
391 (\<forall>Y X Yz Xy Z. product(Y::'a,Z,Yz) & product(X::'a,Y,Xy) & defined(X::'a,Yz) --> defined(Xy::'a,Z)) &
392 (\<forall>Yz X Y Xy Z Xyz. product(Y::'a,Z,Yz) & product(X::'a,Yz,Xyz) & product(X::'a,Y,Xy) --> product(Xy::'a,Z,Xyz)) &
393 (\<forall>Y X Z. defined(X::'a,Y) & defined(Y::'a,Z) & identity_map(Y) --> defined(X::'a,Z)) &
394 (\<forall>X. identity_map(domain(X))) &
395 (\<forall>X. identity_map(codomain(X))) &
396 (\<forall>X. defined(X::'a,domain(X))) &
397 (\<forall>X. defined(codomain(X),X)) &
398 (\<forall>X. product(X::'a,domain(X),X)) &
399 (\<forall>X. product(codomain(X),X,X)) &
400 (\<forall>X Y. defined(X::'a,Y) & identity_map(X) --> product(X::'a,Y,Y)) &
401 (\<forall>Y X. defined(X::'a,Y) & identity_map(Y) --> product(X::'a,Y,X)) &
402 (\<forall>X Y Z W. product(X::'a,Y,Z) & product(X::'a,Y,W) --> equal(Z::'a,W))"
404 abbreviation "CAT001_0_eq compos defined identity_map codomain domain product equal \<equiv>
405 (\<forall>X Y Z W. equal(X::'a,Y) & product(X::'a,Z,W) --> product(Y::'a,Z,W)) &
406 (\<forall>X Z Y W. equal(X::'a,Y) & product(Z::'a,X,W) --> product(Z::'a,Y,W)) &
407 (\<forall>X Z W Y. equal(X::'a,Y) & product(Z::'a,W,X) --> product(Z::'a,W,Y)) &
408 (\<forall>X Y. equal(X::'a,Y) --> equal(domain(X),domain(Y))) &
409 (\<forall>X Y. equal(X::'a,Y) --> equal(codomain(X),codomain(Y))) &
410 (\<forall>X Y. equal(X::'a,Y) & identity_map(X) --> identity_map(Y)) &
411 (\<forall>X Y Z. equal(X::'a,Y) & defined(X::'a,Z) --> defined(Y::'a,Z)) &
412 (\<forall>X Z Y. equal(X::'a,Y) & defined(Z::'a,X) --> defined(Z::'a,Y)) &
413 (\<forall>X Z Y. equal(X::'a,Y) --> equal(compos(Z::'a,X),compos(Z::'a,Y))) &
414 (\<forall>X Y Z. equal(X::'a,Y) --> equal(compos(X::'a,Z),compos(Y::'a,Z)))"
416 (*54288 inferences so far. Searching to depth 14. 118.0 secs*)
419 CAT001_0_ax equal codomain domain identity_map compos product defined &
420 CAT001_0_eq compos defined identity_map codomain domain product equal &
423 (~equal(domain(a),d)) --> False"
426 (*1728 inferences so far. Searching to depth 10. 5.8 secs*)
429 CAT001_0_ax equal codomain domain identity_map compos product defined &
430 CAT001_0_eq compos defined identity_map codomain domain product equal &
431 (equal(domain(a),codomain(b))) &
432 (~defined(a::'a,b)) --> False"
435 (*82895 inferences so far. Searching to depth 13. 355 secs*)
438 CAT001_0_ax equal codomain domain identity_map compos product defined &
439 CAT001_0_eq compos defined identity_map codomain domain product equal &
442 (~defined(a::'a,compos(b::'a,c))) --> False"
445 (*1118 inferences so far. Searching to depth 8. 2.3 secs*)
448 (\<forall>X Y Z. equal(apply(apply(apply(s::'a,X),Y),Z),apply(apply(X::'a,Z),apply(Y::'a,Z)))) &
449 (\<forall>Y X. equal(apply(apply(k::'a,X),Y),X)) &
450 (\<forall>X Y Z. equal(apply(apply(apply(b::'a,X),Y),Z),apply(X::'a,apply(Y::'a,Z)))) &
451 (\<forall>X. equal(apply(i::'a,X),X)) &
452 (\<forall>A B C. equal(A::'a,B) --> equal(apply(A::'a,C),apply(B::'a,C))) &
453 (\<forall>D F' E. equal(D::'a,E) --> equal(apply(F'::'a,D),apply(F'::'a,E))) &
454 (\<forall>X. equal(apply(apply(apply(s::'a,apply(b::'a,X)),i),apply(apply(s::'a,apply(b::'a,X)),i)),apply(x::'a,apply(apply(apply(s::'a,apply(b::'a,X)),i),apply(apply(s::'a,apply(b::'a,X)),i))))) &
455 (\<forall>Y. ~equal(Y::'a,apply(combinator::'a,Y))) --> False"
458 (*500 inferences so far. Searching to depth 8. 0.9 secs*)
461 (\<forall>X Y Z. equal(apply(apply(apply(b::'a,X),Y),Z),apply(X::'a,apply(Y::'a,Z)))) &
462 (\<forall>X Y Z. equal(apply(apply(apply(n::'a,X),Y),Z),apply(apply(apply(X::'a,Z),Y),Z))) &
463 (\<forall>A B C. equal(A::'a,B) --> equal(apply(A::'a,C),apply(B::'a,C))) &
464 (\<forall>D F' E. equal(D::'a,E) --> equal(apply(F'::'a,D),apply(F'::'a,E))) &
465 (\<forall>Y. ~equal(Y::'a,apply(combinator::'a,Y))) --> False"
468 (*3018 inferences so far. Searching to depth 10. 4.3 secs*)
471 (\<forall>X. equal(apply(m::'a,X),apply(X::'a,X))) &
472 (\<forall>Y X Z. equal(apply(apply(apply(q::'a,X),Y),Z),apply(Y::'a,apply(X::'a,Z)))) &
473 (\<forall>A B C. equal(A::'a,B) --> equal(apply(A::'a,C),apply(B::'a,C))) &
474 (\<forall>D F' E. equal(D::'a,E) --> equal(apply(F'::'a,D),apply(F'::'a,E))) &
475 (\<forall>G H. equal(G::'a,H) --> equal(f(G),f(H))) &
476 (\<forall>Y. ~equal(apply(Y::'a,f(Y)),apply(f(Y),apply(Y::'a,f(Y))))) --> False"
479 (*381878 inferences so far. Searching to depth 13. 670.4 secs*)
482 (\<forall>X Y W. equal(response(compos(X::'a,Y),W),response(X::'a,response(Y::'a,W)))) &
483 (\<forall>X Y. agreeable(X) --> equal(response(X::'a,common_bird(Y)),response(Y::'a,common_bird(Y)))) &
484 (\<forall>Z X. equal(response(X::'a,Z),response(compatible(X),Z)) --> agreeable(X)) &
485 (\<forall>A B. equal(A::'a,B) --> equal(common_bird(A),common_bird(B))) &
486 (\<forall>C D. equal(C::'a,D) --> equal(compatible(C),compatible(D))) &
487 (\<forall>Q R. equal(Q::'a,R) & agreeable(Q) --> agreeable(R)) &
488 (\<forall>A B C. equal(A::'a,B) --> equal(compos(A::'a,C),compos(B::'a,C))) &
489 (\<forall>D F' E. equal(D::'a,E) --> equal(compos(F'::'a,D),compos(F'::'a,E))) &
490 (\<forall>G H I'. equal(G::'a,H) --> equal(response(G::'a,I'),response(H::'a,I'))) &
491 (\<forall>J L K'. equal(J::'a,K') --> equal(response(L::'a,J),response(L::'a,K'))) &
494 (equal(c::'a,compos(a::'a,b))) --> False"
497 (*13201 inferences so far. Searching to depth 11. 31.9 secs*)
500 (\<forall>Y X. equal(apply(apply(k::'a,X),Y),X)) &
501 (\<forall>X Y Z. equal(apply(apply(apply(abstraction::'a,X),Y),Z),apply(apply(X::'a,apply(k::'a,Z)),apply(Y::'a,Z)))) &
502 (\<forall>D E F'. equal(D::'a,E) --> equal(apply(D::'a,F'),apply(E::'a,F'))) &
503 (\<forall>G I' H. equal(G::'a,H) --> equal(apply(I'::'a,G),apply(I'::'a,H))) &
504 (\<forall>A B. equal(A::'a,B) --> equal(b(A),b(B))) &
505 (\<forall>C D. equal(C::'a,D) --> equal(c(C),c(D))) &
506 (\<forall>Y. ~equal(apply(apply(Y::'a,b(Y)),c(Y)),apply(b(Y),b(Y)))) --> False"
509 (*33 inferences so far. Searching to depth 7. 0.1 secs*)
511 "(\<forall>Goal_state Start_state. follows(Goal_state::'a,Start_state) --> succeeds(Goal_state::'a,Start_state)) &
512 (\<forall>Goal_state Intermediate_state Start_state. succeeds(Goal_state::'a,Intermediate_state) & succeeds(Intermediate_state::'a,Start_state) --> succeeds(Goal_state::'a,Start_state)) &
513 (\<forall>Start_state Label Goal_state. has(Start_state::'a,goto(Label)) & labels(Label::'a,Goal_state) --> succeeds(Goal_state::'a,Start_state)) &
514 (\<forall>Start_state Condition Goal_state. has(Start_state::'a,ifthen(Condition::'a,Goal_state)) --> succeeds(Goal_state::'a,Start_state)) &
515 (labels(loop::'a,p3)) &
516 (has(p3::'a,ifthen(equal(register_j::'a,n),p4))) &
517 (has(p4::'a,goto(out))) &
518 (follows(p5::'a,p4)) &
519 (follows(p8::'a,p3)) &
520 (has(p8::'a,goto(loop))) &
521 (~succeeds(p3::'a,p3)) --> False"
524 (*533 inferences so far. Searching to depth 13. 0.3 secs*)
526 "(\<forall>Goal_state Start_state. follows(Goal_state::'a,Start_state) --> succeeds(Goal_state::'a,Start_state)) &
527 (\<forall>Goal_state Intermediate_state Start_state. succeeds(Goal_state::'a,Intermediate_state) & succeeds(Intermediate_state::'a,Start_state) --> succeeds(Goal_state::'a,Start_state)) &
528 (\<forall>Start_state Label Goal_state. has(Start_state::'a,goto(Label)) & labels(Label::'a,Goal_state) --> succeeds(Goal_state::'a,Start_state)) &
529 (\<forall>Start_state Condition Goal_state. has(Start_state::'a,ifthen(Condition::'a,Goal_state)) --> succeeds(Goal_state::'a,Start_state)) &
530 (has(p1::'a,assign(register_j::'a,num0))) &
531 (follows(p2::'a,p1)) &
532 (has(p2::'a,assign(register_k::'a,num1))) &
533 (labels(loop::'a,p3)) &
534 (follows(p3::'a,p2)) &
535 (has(p3::'a,ifthen(equal(register_j::'a,n),p4))) &
536 (has(p4::'a,goto(out))) &
537 (follows(p5::'a,p4)) &
538 (follows(p6::'a,p3)) &
539 (has(p6::'a,assign(register_k::'a,mtimes(num2::'a,register_k)))) &
540 (follows(p7::'a,p6)) &
541 (has(p7::'a,assign(register_j::'a,mplus(register_j::'a,num1)))) &
542 (follows(p8::'a,p7)) &
543 (has(p8::'a,goto(loop))) &
544 (~succeeds(p3::'a,p3)) --> False"
547 (*4821 inferences so far. Searching to depth 14. 1.3 secs*)
549 "(\<forall>Goal_state Start_state. ~(fails(Goal_state::'a,Start_state) & follows(Goal_state::'a,Start_state))) &
550 (\<forall>Goal_state Intermediate_state Start_state. fails(Goal_state::'a,Start_state) --> fails(Goal_state::'a,Intermediate_state) | fails(Intermediate_state::'a,Start_state)) &
551 (\<forall>Start_state Label Goal_state. ~(fails(Goal_state::'a,Start_state) & has(Start_state::'a,goto(Label)) & labels(Label::'a,Goal_state))) &
552 (\<forall>Start_state Condition Goal_state. ~(fails(Goal_state::'a,Start_state) & has(Start_state::'a,ifthen(Condition::'a,Goal_state)))) &
553 (has(p1::'a,assign(register_j::'a,num0))) &
554 (follows(p2::'a,p1)) &
555 (has(p2::'a,assign(register_k::'a,num1))) &
556 (labels(loop::'a,p3)) &
557 (follows(p3::'a,p2)) &
558 (has(p3::'a,ifthen(equal(register_j::'a,n),p4))) &
559 (has(p4::'a,goto(out))) &
560 (follows(p5::'a,p4)) &
561 (follows(p6::'a,p3)) &
562 (has(p6::'a,assign(register_k::'a,mtimes(num2::'a,register_k)))) &
563 (follows(p7::'a,p6)) &
564 (has(p7::'a,assign(register_j::'a,mplus(register_j::'a,num1)))) &
565 (follows(p8::'a,p7)) &
566 (has(p8::'a,goto(loop))) &
567 (fails(p3::'a,p3)) --> False"
570 (*98 inferences so far. Searching to depth 10. 1.1 secs*)
572 "(\<forall>X Y Z. program_decides(X) & program(Y) --> decides(X::'a,Y,Z)) &
573 (\<forall>X. program_decides(X) | program(f2(X))) &
574 (\<forall>X. decides(X::'a,f2(X),f1(X)) --> program_decides(X)) &
575 (\<forall>X. program_program_decides(X) --> program(X)) &
576 (\<forall>X. program_program_decides(X) --> program_decides(X)) &
577 (\<forall>X. program(X) & program_decides(X) --> program_program_decides(X)) &
578 (\<forall>X. algorithm_program_decides(X) --> algorithm(X)) &
579 (\<forall>X. algorithm_program_decides(X) --> program_decides(X)) &
580 (\<forall>X. algorithm(X) & program_decides(X) --> algorithm_program_decides(X)) &
581 (\<forall>Y X. program_halts2(X::'a,Y) --> program(X)) &
582 (\<forall>X Y. program_halts2(X::'a,Y) --> halts2(X::'a,Y)) &
583 (\<forall>X Y. program(X) & halts2(X::'a,Y) --> program_halts2(X::'a,Y)) &
584 (\<forall>W X Y Z. halts3_outputs(X::'a,Y,Z,W) --> halts3(X::'a,Y,Z)) &
585 (\<forall>Y Z X W. halts3_outputs(X::'a,Y,Z,W) --> outputs(X::'a,W)) &
586 (\<forall>Y Z X W. halts3(X::'a,Y,Z) & outputs(X::'a,W) --> halts3_outputs(X::'a,Y,Z,W)) &
587 (\<forall>Y X. program_not_halts2(X::'a,Y) --> program(X)) &
588 (\<forall>X Y. ~(program_not_halts2(X::'a,Y) & halts2(X::'a,Y))) &
589 (\<forall>X Y. program(X) --> program_not_halts2(X::'a,Y) | halts2(X::'a,Y)) &
590 (\<forall>W X Y. halts2_outputs(X::'a,Y,W) --> halts2(X::'a,Y)) &
591 (\<forall>Y X W. halts2_outputs(X::'a,Y,W) --> outputs(X::'a,W)) &
592 (\<forall>Y X W. halts2(X::'a,Y) & outputs(X::'a,W) --> halts2_outputs(X::'a,Y,W)) &
593 (\<forall>X W Y Z. program_halts2_halts3_outputs(X::'a,Y,Z,W) --> program_halts2(Y::'a,Z)) &
594 (\<forall>X Y Z W. program_halts2_halts3_outputs(X::'a,Y,Z,W) --> halts3_outputs(X::'a,Y,Z,W)) &
595 (\<forall>X Y Z W. program_halts2(Y::'a,Z) & halts3_outputs(X::'a,Y,Z,W) --> program_halts2_halts3_outputs(X::'a,Y,Z,W)) &
596 (\<forall>X W Y Z. program_not_halts2_halts3_outputs(X::'a,Y,Z,W) --> program_not_halts2(Y::'a,Z)) &
597 (\<forall>X Y Z W. program_not_halts2_halts3_outputs(X::'a,Y,Z,W) --> halts3_outputs(X::'a,Y,Z,W)) &
598 (\<forall>X Y Z W. program_not_halts2(Y::'a,Z) & halts3_outputs(X::'a,Y,Z,W) --> program_not_halts2_halts3_outputs(X::'a,Y,Z,W)) &
599 (\<forall>X W Y. program_halts2_halts2_outputs(X::'a,Y,W) --> program_halts2(Y::'a,Y)) &
600 (\<forall>X Y W. program_halts2_halts2_outputs(X::'a,Y,W) --> halts2_outputs(X::'a,Y,W)) &
601 (\<forall>X Y W. program_halts2(Y::'a,Y) & halts2_outputs(X::'a,Y,W) --> program_halts2_halts2_outputs(X::'a,Y,W)) &
602 (\<forall>X W Y. program_not_halts2_halts2_outputs(X::'a,Y,W) --> program_not_halts2(Y::'a,Y)) &
603 (\<forall>X Y W. program_not_halts2_halts2_outputs(X::'a,Y,W) --> halts2_outputs(X::'a,Y,W)) &
604 (\<forall>X Y W. program_not_halts2(Y::'a,Y) & halts2_outputs(X::'a,Y,W) --> program_not_halts2_halts2_outputs(X::'a,Y,W)) &
605 (\<forall>X. algorithm_program_decides(X) --> program_program_decides(c1)) &
606 (\<forall>W Y Z. program_program_decides(W) --> program_halts2_halts3_outputs(W::'a,Y,Z,good)) &
607 (\<forall>W Y Z. program_program_decides(W) --> program_not_halts2_halts3_outputs(W::'a,Y,Z,bad)) &
608 (\<forall>W. program(W) & program_halts2_halts3_outputs(W::'a,f3(W),f3(W),good) & program_not_halts2_halts3_outputs(W::'a,f3(W),f3(W),bad) --> program(c2)) &
609 (\<forall>W Y. program(W) & program_halts2_halts3_outputs(W::'a,f3(W),f3(W),good) & program_not_halts2_halts3_outputs(W::'a,f3(W),f3(W),bad) --> program_halts2_halts2_outputs(c2::'a,Y,good)) &
610 (\<forall>W Y. program(W) & program_halts2_halts3_outputs(W::'a,f3(W),f3(W),good) & program_not_halts2_halts3_outputs(W::'a,f3(W),f3(W),bad) --> program_not_halts2_halts2_outputs(c2::'a,Y,bad)) &
611 (\<forall>V. program(V) & program_halts2_halts2_outputs(V::'a,f4(V),good) & program_not_halts2_halts2_outputs(V::'a,f4(V),bad) --> program(c3)) &
612 (\<forall>V Y. program(V) & program_halts2_halts2_outputs(V::'a,f4(V),good) & program_not_halts2_halts2_outputs(V::'a,f4(V),bad) & program_halts2(Y::'a,Y) --> halts2(c3::'a,Y)) &
613 (\<forall>V Y. program(V) & program_halts2_halts2_outputs(V::'a,f4(V),good) & program_not_halts2_halts2_outputs(V::'a,f4(V),bad) --> program_not_halts2_halts2_outputs(c3::'a,Y,bad)) &
614 (algorithm_program_decides(c4)) --> False"
617 (*2100398 inferences so far. Searching to depth 12.
618 1256s (21 mins) on griffon*)
621 (\<forall>C D P Q X Y. failure_node(X::'a,or(C::'a,P)) & failure_node(Y::'a,or(D::'a,Q)) & contradictory(P::'a,Q) & siblings(X::'a,Y) --> failure_node(parent_of(X::'a,Y),or(C::'a,D))) &
622 (\<forall>X. contradictory(negate(X),X)) &
623 (\<forall>X. contradictory(X::'a,negate(X))) &
624 (\<forall>X. siblings(left_child_of(X),right_child_of(X))) &
625 (\<forall>D E. equal(D::'a,E) --> equal(left_child_of(D),left_child_of(E))) &
626 (\<forall>F' G. equal(F'::'a,G) --> equal(negate(F'),negate(G))) &
627 (\<forall>H I' J. equal(H::'a,I') --> equal(or(H::'a,J),or(I'::'a,J))) &
628 (\<forall>K' M L. equal(K'::'a,L) --> equal(or(M::'a,K'),or(M::'a,L))) &
629 (\<forall>N O' P. equal(N::'a,O') --> equal(parent_of(N::'a,P),parent_of(O'::'a,P))) &
630 (\<forall>Q S' R. equal(Q::'a,R) --> equal(parent_of(S'::'a,Q),parent_of(S'::'a,R))) &
631 (\<forall>T' U. equal(T'::'a,U) --> equal(right_child_of(T'),right_child_of(U))) &
632 (\<forall>V W X. equal(V::'a,W) & contradictory(V::'a,X) --> contradictory(W::'a,X)) &
633 (\<forall>Y A1 Z. equal(Y::'a,Z) & contradictory(A1::'a,Y) --> contradictory(A1::'a,Z)) &
634 (\<forall>B1 C1 D1. equal(B1::'a,C1) & failure_node(B1::'a,D1) --> failure_node(C1::'a,D1)) &
635 (\<forall>E1 G1 F1. equal(E1::'a,F1) & failure_node(G1::'a,E1) --> failure_node(G1::'a,F1)) &
636 (\<forall>H1 I1 J1. equal(H1::'a,I1) & siblings(H1::'a,J1) --> siblings(I1::'a,J1)) &
637 (\<forall>K1 M1 L1. equal(K1::'a,L1) & siblings(M1::'a,K1) --> siblings(M1::'a,L1)) &
638 (failure_node(n_left::'a,or(EMPTY::'a,atom))) &
639 (failure_node(n_right::'a,or(EMPTY::'a,negate(atom)))) &
640 (equal(n_left::'a,left_child_of(n))) &
641 (equal(n_right::'a,right_child_of(n))) &
642 (\<forall>Z. ~failure_node(Z::'a,or(EMPTY::'a,EMPTY))) --> False"
645 abbreviation "GEO001_0_ax continuous lower_dimension_point_3 lower_dimension_point_2
646 lower_dimension_point_1 extension euclid2 euclid1 outer_pasch equidistant equal between \<equiv>
647 (\<forall>X Y. between(X::'a,Y,X) --> equal(X::'a,Y)) &
648 (\<forall>V X Y Z. between(X::'a,Y,V) & between(Y::'a,Z,V) --> between(X::'a,Y,Z)) &
649 (\<forall>Y X V Z. between(X::'a,Y,Z) & between(X::'a,Y,V) --> equal(X::'a,Y) | between(X::'a,Z,V) | between(X::'a,V,Z)) &
650 (\<forall>Y X. equidistant(X::'a,Y,Y,X)) &
651 (\<forall>Z X Y. equidistant(X::'a,Y,Z,Z) --> equal(X::'a,Y)) &
652 (\<forall>X Y Z V V2 W. equidistant(X::'a,Y,Z,V) & equidistant(X::'a,Y,V2,W) --> equidistant(Z::'a,V,V2,W)) &
653 (\<forall>W X Z V Y. between(X::'a,W,V) & between(Y::'a,V,Z) --> between(X::'a,outer_pasch(W::'a,X,Y,Z,V),Y)) &
654 (\<forall>W X Y Z V. between(X::'a,W,V) & between(Y::'a,V,Z) --> between(Z::'a,W,outer_pasch(W::'a,X,Y,Z,V))) &
655 (\<forall>W X Y Z V. between(X::'a,V,W) & between(Y::'a,V,Z) --> equal(X::'a,V) | between(X::'a,Z,euclid1(W::'a,X,Y,Z,V))) &
656 (\<forall>W X Y Z V. between(X::'a,V,W) & between(Y::'a,V,Z) --> equal(X::'a,V) | between(X::'a,Y,euclid2(W::'a,X,Y,Z,V))) &
657 (\<forall>W X Y Z V. between(X::'a,V,W) & between(Y::'a,V,Z) --> equal(X::'a,V) | between(euclid1(W::'a,X,Y,Z,V),W,euclid2(W::'a,X,Y,Z,V))) &
658 (\<forall>X1 Y1 X Y Z V Z1 V1. equidistant(X::'a,Y,X1,Y1) & equidistant(Y::'a,Z,Y1,Z1) & equidistant(X::'a,V,X1,V1) & equidistant(Y::'a,V,Y1,V1) & between(X::'a,Y,Z) & between(X1::'a,Y1,Z1) --> equal(X::'a,Y) | equidistant(Z::'a,V,Z1,V1)) &
659 (\<forall>X Y W V. between(X::'a,Y,extension(X::'a,Y,W,V))) &
660 (\<forall>X Y W V. equidistant(Y::'a,extension(X::'a,Y,W,V),W,V)) &
661 (~between(lower_dimension_point_1::'a,lower_dimension_point_2,lower_dimension_point_3)) &
662 (~between(lower_dimension_point_2::'a,lower_dimension_point_3,lower_dimension_point_1)) &
663 (~between(lower_dimension_point_3::'a,lower_dimension_point_1,lower_dimension_point_2)) &
664 (\<forall>Z X Y W V. equidistant(X::'a,W,X,V) & equidistant(Y::'a,W,Y,V) & equidistant(Z::'a,W,Z,V) --> between(X::'a,Y,Z) | between(Y::'a,Z,X) | between(Z::'a,X,Y) | equal(W::'a,V)) &
665 (\<forall>X Y Z X1 Z1 V. equidistant(V::'a,X,V,X1) & equidistant(V::'a,Z,V,Z1) & between(V::'a,X,Z) & between(X::'a,Y,Z) --> equidistant(V::'a,Y,Z,continuous(X::'a,Y,Z,X1,Z1,V))) &
666 (\<forall>X Y Z X1 V Z1. equidistant(V::'a,X,V,X1) & equidistant(V::'a,Z,V,Z1) & between(V::'a,X,Z) & between(X::'a,Y,Z) --> between(X1::'a,continuous(X::'a,Y,Z,X1,Z1,V),Z1))"
668 abbreviation "GEO001_0_eq continuous extension euclid2 euclid1 outer_pasch equidistant
669 between equal \<equiv>
670 (\<forall>X Y W Z. equal(X::'a,Y) & between(X::'a,W,Z) --> between(Y::'a,W,Z)) &
671 (\<forall>X W Y Z. equal(X::'a,Y) & between(W::'a,X,Z) --> between(W::'a,Y,Z)) &
672 (\<forall>X W Z Y. equal(X::'a,Y) & between(W::'a,Z,X) --> between(W::'a,Z,Y)) &
673 (\<forall>X Y V W Z. equal(X::'a,Y) & equidistant(X::'a,V,W,Z) --> equidistant(Y::'a,V,W,Z)) &
674 (\<forall>X V Y W Z. equal(X::'a,Y) & equidistant(V::'a,X,W,Z) --> equidistant(V::'a,Y,W,Z)) &
675 (\<forall>X V W Y Z. equal(X::'a,Y) & equidistant(V::'a,W,X,Z) --> equidistant(V::'a,W,Y,Z)) &
676 (\<forall>X V W Z Y. equal(X::'a,Y) & equidistant(V::'a,W,Z,X) --> equidistant(V::'a,W,Z,Y)) &
677 (\<forall>X Y V1 V2 V3 V4. equal(X::'a,Y) --> equal(outer_pasch(X::'a,V1,V2,V3,V4),outer_pasch(Y::'a,V1,V2,V3,V4))) &
678 (\<forall>X V1 Y V2 V3 V4. equal(X::'a,Y) --> equal(outer_pasch(V1::'a,X,V2,V3,V4),outer_pasch(V1::'a,Y,V2,V3,V4))) &
679 (\<forall>X V1 V2 Y V3 V4. equal(X::'a,Y) --> equal(outer_pasch(V1::'a,V2,X,V3,V4),outer_pasch(V1::'a,V2,Y,V3,V4))) &
680 (\<forall>X V1 V2 V3 Y V4. equal(X::'a,Y) --> equal(outer_pasch(V1::'a,V2,V3,X,V4),outer_pasch(V1::'a,V2,V3,Y,V4))) &
681 (\<forall>X V1 V2 V3 V4 Y. equal(X::'a,Y) --> equal(outer_pasch(V1::'a,V2,V3,V4,X),outer_pasch(V1::'a,V2,V3,V4,Y))) &
682 (\<forall>A B C D E F'. equal(A::'a,B) --> equal(euclid1(A::'a,C,D,E,F'),euclid1(B::'a,C,D,E,F'))) &
683 (\<forall>G I' H J K' L. equal(G::'a,H) --> equal(euclid1(I'::'a,G,J,K',L),euclid1(I'::'a,H,J,K',L))) &
684 (\<forall>M O' P N Q R. equal(M::'a,N) --> equal(euclid1(O'::'a,P,M,Q,R),euclid1(O'::'a,P,N,Q,R))) &
685 (\<forall>S' U V W T' X. equal(S'::'a,T') --> equal(euclid1(U::'a,V,W,S',X),euclid1(U::'a,V,W,T',X))) &
686 (\<forall>Y A1 B1 C1 D1 Z. equal(Y::'a,Z) --> equal(euclid1(A1::'a,B1,C1,D1,Y),euclid1(A1::'a,B1,C1,D1,Z))) &
687 (\<forall>E1 F1 G1 H1 I1 J1. equal(E1::'a,F1) --> equal(euclid2(E1::'a,G1,H1,I1,J1),euclid2(F1::'a,G1,H1,I1,J1))) &
688 (\<forall>K1 M1 L1 N1 O1 P1. equal(K1::'a,L1) --> equal(euclid2(M1::'a,K1,N1,O1,P1),euclid2(M1::'a,L1,N1,O1,P1))) &
689 (\<forall>Q1 S1 T1 R1 U1 V1. equal(Q1::'a,R1) --> equal(euclid2(S1::'a,T1,Q1,U1,V1),euclid2(S1::'a,T1,R1,U1,V1))) &
690 (\<forall>W1 Y1 Z1 A2 X1 B2. equal(W1::'a,X1) --> equal(euclid2(Y1::'a,Z1,A2,W1,B2),euclid2(Y1::'a,Z1,A2,X1,B2))) &
691 (\<forall>C2 E2 F2 G2 H2 D2. equal(C2::'a,D2) --> equal(euclid2(E2::'a,F2,G2,H2,C2),euclid2(E2::'a,F2,G2,H2,D2))) &
692 (\<forall>X Y V1 V2 V3. equal(X::'a,Y) --> equal(extension(X::'a,V1,V2,V3),extension(Y::'a,V1,V2,V3))) &
693 (\<forall>X V1 Y V2 V3. equal(X::'a,Y) --> equal(extension(V1::'a,X,V2,V3),extension(V1::'a,Y,V2,V3))) &
694 (\<forall>X V1 V2 Y V3. equal(X::'a,Y) --> equal(extension(V1::'a,V2,X,V3),extension(V1::'a,V2,Y,V3))) &
695 (\<forall>X V1 V2 V3 Y. equal(X::'a,Y) --> equal(extension(V1::'a,V2,V3,X),extension(V1::'a,V2,V3,Y))) &
696 (\<forall>X Y V1 V2 V3 V4 V5. equal(X::'a,Y) --> equal(continuous(X::'a,V1,V2,V3,V4,V5),continuous(Y::'a,V1,V2,V3,V4,V5))) &
697 (\<forall>X V1 Y V2 V3 V4 V5. equal(X::'a,Y) --> equal(continuous(V1::'a,X,V2,V3,V4,V5),continuous(V1::'a,Y,V2,V3,V4,V5))) &
698 (\<forall>X V1 V2 Y V3 V4 V5. equal(X::'a,Y) --> equal(continuous(V1::'a,V2,X,V3,V4,V5),continuous(V1::'a,V2,Y,V3,V4,V5))) &
699 (\<forall>X V1 V2 V3 Y V4 V5. equal(X::'a,Y) --> equal(continuous(V1::'a,V2,V3,X,V4,V5),continuous(V1::'a,V2,V3,Y,V4,V5))) &
700 (\<forall>X V1 V2 V3 V4 Y V5. equal(X::'a,Y) --> equal(continuous(V1::'a,V2,V3,V4,X,V5),continuous(V1::'a,V2,V3,V4,Y,V5))) &
701 (\<forall>X V1 V2 V3 V4 V5 Y. equal(X::'a,Y) --> equal(continuous(V1::'a,V2,V3,V4,V5,X),continuous(V1::'a,V2,V3,V4,V5,Y)))"
704 (*179 inferences so far. Searching to depth 7. 3.9 secs*)
707 GEO001_0_ax continuous lower_dimension_point_3 lower_dimension_point_2
708 lower_dimension_point_1 extension euclid2 euclid1 outer_pasch equidistant equal between &
709 GEO001_0_eq continuous extension euclid2 euclid1 outer_pasch equidistant between equal &
710 (~between(a::'a,b,b)) --> False"
713 abbreviation "GEO002_ax_eq continuous euclid2 euclid1 lower_dimension_point_3
714 lower_dimension_point_2 lower_dimension_point_1 inner_pasch extension
715 between equal equidistant \<equiv>
716 (\<forall>Y X. equidistant(X::'a,Y,Y,X)) &
717 (\<forall>X Y Z V V2 W. equidistant(X::'a,Y,Z,V) & equidistant(X::'a,Y,V2,W) --> equidistant(Z::'a,V,V2,W)) &
718 (\<forall>Z X Y. equidistant(X::'a,Y,Z,Z) --> equal(X::'a,Y)) &
719 (\<forall>X Y W V. between(X::'a,Y,extension(X::'a,Y,W,V))) &
720 (\<forall>X Y W V. equidistant(Y::'a,extension(X::'a,Y,W,V),W,V)) &
721 (\<forall>X1 Y1 X Y Z V Z1 V1. equidistant(X::'a,Y,X1,Y1) & equidistant(Y::'a,Z,Y1,Z1) & equidistant(X::'a,V,X1,V1) & equidistant(Y::'a,V,Y1,V1) & between(X::'a,Y,Z) & between(X1::'a,Y1,Z1) --> equal(X::'a,Y) | equidistant(Z::'a,V,Z1,V1)) &
722 (\<forall>X Y. between(X::'a,Y,X) --> equal(X::'a,Y)) &
723 (\<forall>U V W X Y. between(U::'a,V,W) & between(Y::'a,X,W) --> between(V::'a,inner_pasch(U::'a,V,W,X,Y),Y)) &
724 (\<forall>V W X Y U. between(U::'a,V,W) & between(Y::'a,X,W) --> between(X::'a,inner_pasch(U::'a,V,W,X,Y),U)) &
725 (~between(lower_dimension_point_1::'a,lower_dimension_point_2,lower_dimension_point_3)) &
726 (~between(lower_dimension_point_2::'a,lower_dimension_point_3,lower_dimension_point_1)) &
727 (~between(lower_dimension_point_3::'a,lower_dimension_point_1,lower_dimension_point_2)) &
728 (\<forall>Z X Y W V. equidistant(X::'a,W,X,V) & equidistant(Y::'a,W,Y,V) & equidistant(Z::'a,W,Z,V) --> between(X::'a,Y,Z) | between(Y::'a,Z,X) | between(Z::'a,X,Y) | equal(W::'a,V)) &
729 (\<forall>U V W X Y. between(U::'a,W,Y) & between(V::'a,W,X) --> equal(U::'a,W) | between(U::'a,V,euclid1(U::'a,V,W,X,Y))) &
730 (\<forall>U V W X Y. between(U::'a,W,Y) & between(V::'a,W,X) --> equal(U::'a,W) | between(U::'a,X,euclid2(U::'a,V,W,X,Y))) &
731 (\<forall>U V W X Y. between(U::'a,W,Y) & between(V::'a,W,X) --> equal(U::'a,W) | between(euclid1(U::'a,V,W,X,Y),Y,euclid2(U::'a,V,W,X,Y))) &
732 (\<forall>U V V1 W X X1. equidistant(U::'a,V,U,V1) & equidistant(U::'a,X,U,X1) & between(U::'a,V,X) & between(V::'a,W,X) --> between(V1::'a,continuous(U::'a,V,V1,W,X,X1),X1)) &
733 (\<forall>U V V1 W X X1. equidistant(U::'a,V,U,V1) & equidistant(U::'a,X,U,X1) & between(U::'a,V,X) & between(V::'a,W,X) --> equidistant(U::'a,W,U,continuous(U::'a,V,V1,W,X,X1))) &
734 (\<forall>X Y W Z. equal(X::'a,Y) & between(X::'a,W,Z) --> between(Y::'a,W,Z)) &
735 (\<forall>X W Y Z. equal(X::'a,Y) & between(W::'a,X,Z) --> between(W::'a,Y,Z)) &
736 (\<forall>X W Z Y. equal(X::'a,Y) & between(W::'a,Z,X) --> between(W::'a,Z,Y)) &
737 (\<forall>X Y V W Z. equal(X::'a,Y) & equidistant(X::'a,V,W,Z) --> equidistant(Y::'a,V,W,Z)) &
738 (\<forall>X V Y W Z. equal(X::'a,Y) & equidistant(V::'a,X,W,Z) --> equidistant(V::'a,Y,W,Z)) &
739 (\<forall>X V W Y Z. equal(X::'a,Y) & equidistant(V::'a,W,X,Z) --> equidistant(V::'a,W,Y,Z)) &
740 (\<forall>X V W Z Y. equal(X::'a,Y) & equidistant(V::'a,W,Z,X) --> equidistant(V::'a,W,Z,Y)) &
741 (\<forall>X Y V1 V2 V3 V4. equal(X::'a,Y) --> equal(inner_pasch(X::'a,V1,V2,V3,V4),inner_pasch(Y::'a,V1,V2,V3,V4))) &
742 (\<forall>X V1 Y V2 V3 V4. equal(X::'a,Y) --> equal(inner_pasch(V1::'a,X,V2,V3,V4),inner_pasch(V1::'a,Y,V2,V3,V4))) &
743 (\<forall>X V1 V2 Y V3 V4. equal(X::'a,Y) --> equal(inner_pasch(V1::'a,V2,X,V3,V4),inner_pasch(V1::'a,V2,Y,V3,V4))) &
744 (\<forall>X V1 V2 V3 Y V4. equal(X::'a,Y) --> equal(inner_pasch(V1::'a,V2,V3,X,V4),inner_pasch(V1::'a,V2,V3,Y,V4))) &
745 (\<forall>X V1 V2 V3 V4 Y. equal(X::'a,Y) --> equal(inner_pasch(V1::'a,V2,V3,V4,X),inner_pasch(V1::'a,V2,V3,V4,Y))) &
746 (\<forall>A B C D E F'. equal(A::'a,B) --> equal(euclid1(A::'a,C,D,E,F'),euclid1(B::'a,C,D,E,F'))) &
747 (\<forall>G I' H J K' L. equal(G::'a,H) --> equal(euclid1(I'::'a,G,J,K',L),euclid1(I'::'a,H,J,K',L))) &
748 (\<forall>M O' P N Q R. equal(M::'a,N) --> equal(euclid1(O'::'a,P,M,Q,R),euclid1(O'::'a,P,N,Q,R))) &
749 (\<forall>S' U V W T' X. equal(S'::'a,T') --> equal(euclid1(U::'a,V,W,S',X),euclid1(U::'a,V,W,T',X))) &
750 (\<forall>Y A1 B1 C1 D1 Z. equal(Y::'a,Z) --> equal(euclid1(A1::'a,B1,C1,D1,Y),euclid1(A1::'a,B1,C1,D1,Z))) &
751 (\<forall>E1 F1 G1 H1 I1 J1. equal(E1::'a,F1) --> equal(euclid2(E1::'a,G1,H1,I1,J1),euclid2(F1::'a,G1,H1,I1,J1))) &
752 (\<forall>K1 M1 L1 N1 O1 P1. equal(K1::'a,L1) --> equal(euclid2(M1::'a,K1,N1,O1,P1),euclid2(M1::'a,L1,N1,O1,P1))) &
753 (\<forall>Q1 S1 T1 R1 U1 V1. equal(Q1::'a,R1) --> equal(euclid2(S1::'a,T1,Q1,U1,V1),euclid2(S1::'a,T1,R1,U1,V1))) &
754 (\<forall>W1 Y1 Z1 A2 X1 B2. equal(W1::'a,X1) --> equal(euclid2(Y1::'a,Z1,A2,W1,B2),euclid2(Y1::'a,Z1,A2,X1,B2))) &
755 (\<forall>C2 E2 F2 G2 H2 D2. equal(C2::'a,D2) --> equal(euclid2(E2::'a,F2,G2,H2,C2),euclid2(E2::'a,F2,G2,H2,D2))) &
756 (\<forall>X Y V1 V2 V3. equal(X::'a,Y) --> equal(extension(X::'a,V1,V2,V3),extension(Y::'a,V1,V2,V3))) &
757 (\<forall>X V1 Y V2 V3. equal(X::'a,Y) --> equal(extension(V1::'a,X,V2,V3),extension(V1::'a,Y,V2,V3))) &
758 (\<forall>X V1 V2 Y V3. equal(X::'a,Y) --> equal(extension(V1::'a,V2,X,V3),extension(V1::'a,V2,Y,V3))) &
759 (\<forall>X V1 V2 V3 Y. equal(X::'a,Y) --> equal(extension(V1::'a,V2,V3,X),extension(V1::'a,V2,V3,Y))) &
760 (\<forall>X Y V1 V2 V3 V4 V5. equal(X::'a,Y) --> equal(continuous(X::'a,V1,V2,V3,V4,V5),continuous(Y::'a,V1,V2,V3,V4,V5))) &
761 (\<forall>X V1 Y V2 V3 V4 V5. equal(X::'a,Y) --> equal(continuous(V1::'a,X,V2,V3,V4,V5),continuous(V1::'a,Y,V2,V3,V4,V5))) &
762 (\<forall>X V1 V2 Y V3 V4 V5. equal(X::'a,Y) --> equal(continuous(V1::'a,V2,X,V3,V4,V5),continuous(V1::'a,V2,Y,V3,V4,V5))) &
763 (\<forall>X V1 V2 V3 Y V4 V5. equal(X::'a,Y) --> equal(continuous(V1::'a,V2,V3,X,V4,V5),continuous(V1::'a,V2,V3,Y,V4,V5))) &
764 (\<forall>X V1 V2 V3 V4 Y V5. equal(X::'a,Y) --> equal(continuous(V1::'a,V2,V3,V4,X,V5),continuous(V1::'a,V2,V3,V4,Y,V5))) &
765 (\<forall>X V1 V2 V3 V4 V5 Y. equal(X::'a,Y) --> equal(continuous(V1::'a,V2,V3,V4,V5,X),continuous(V1::'a,V2,V3,V4,V5,Y)))"
767 (*4272 inferences so far. Searching to depth 10. 29.4 secs*)
770 GEO002_ax_eq continuous euclid2 euclid1 lower_dimension_point_3
771 lower_dimension_point_2 lower_dimension_point_1 inner_pasch extension
772 between equal equidistant &
773 (equidistant(u::'a,v,w,x)) &
774 (~equidistant(u::'a,v,x,w)) --> False"
777 (*382903 inferences so far. Searching to depth 9. 1022s (17 mins) on griffon*)
780 GEO002_ax_eq continuous euclid2 euclid1 lower_dimension_point_3
781 lower_dimension_point_2 lower_dimension_point_1 inner_pasch extension
782 between equal equidistant &
783 (\<forall>U V. equal(reflection(U::'a,V),extension(U::'a,V,U,V))) &
784 (\<forall>X Y Z. equal(X::'a,Y) --> equal(reflection(X::'a,Z),reflection(Y::'a,Z))) &
785 (\<forall>A1 C1 B1. equal(A1::'a,B1) --> equal(reflection(C1::'a,A1),reflection(C1::'a,B1))) &
786 (\<forall>U V. equidistant(U::'a,V,U,V)) &
787 (\<forall>W X U V. equidistant(U::'a,V,W,X) --> equidistant(W::'a,X,U,V)) &
788 (\<forall>V U W X. equidistant(U::'a,V,W,X) --> equidistant(V::'a,U,W,X)) &
789 (\<forall>U V X W. equidistant(U::'a,V,W,X) --> equidistant(U::'a,V,X,W)) &
790 (\<forall>V U X W. equidistant(U::'a,V,W,X) --> equidistant(V::'a,U,X,W)) &
791 (\<forall>W X V U. equidistant(U::'a,V,W,X) --> equidistant(W::'a,X,V,U)) &
792 (\<forall>X W U V. equidistant(U::'a,V,W,X) --> equidistant(X::'a,W,U,V)) &
793 (\<forall>X W V U. equidistant(U::'a,V,W,X) --> equidistant(X::'a,W,V,U)) &
794 (\<forall>W X U V Y Z. equidistant(U::'a,V,W,X) & equidistant(W::'a,X,Y,Z) --> equidistant(U::'a,V,Y,Z)) &
795 (\<forall>U V W. equal(V::'a,extension(U::'a,V,W,W))) &
796 (\<forall>W X U V Y. equal(Y::'a,extension(U::'a,V,W,X)) --> between(U::'a,V,Y)) &
797 (\<forall>U V. between(U::'a,V,reflection(U::'a,V))) &
798 (\<forall>U V. equidistant(V::'a,reflection(U::'a,V),U,V)) &
799 (\<forall>U V. equal(U::'a,V) --> equal(V::'a,reflection(U::'a,V))) &
800 (\<forall>U. equal(U::'a,reflection(U::'a,U))) &
801 (\<forall>U V. equal(V::'a,reflection(U::'a,V)) --> equal(U::'a,V)) &
802 (\<forall>U V. equidistant(U::'a,U,V,V)) &
803 (\<forall>V V1 U W U1 W1. equidistant(U::'a,V,U1,V1) & equidistant(V::'a,W,V1,W1) & between(U::'a,V,W) & between(U1::'a,V1,W1) --> equidistant(U::'a,W,U1,W1)) &
804 (\<forall>U V W X. between(U::'a,V,W) & between(U::'a,V,X) & equidistant(V::'a,W,V,X) --> equal(U::'a,V) | equal(W::'a,X)) &
805 (between(u::'a,v,w)) &
807 (~equal(w::'a,extension(u::'a,v,v,w))) --> False"
810 (*313884 inferences so far. Searching to depth 10. 887 secs: 15 mins.*)
813 GEO002_ax_eq continuous euclid2 euclid1 lower_dimension_point_3
814 lower_dimension_point_2 lower_dimension_point_1 inner_pasch extension
815 between equal equidistant &
816 (\<forall>U V. equal(reflection(U::'a,V),extension(U::'a,V,U,V))) &
817 (\<forall>X Y Z. equal(X::'a,Y) --> equal(reflection(X::'a,Z),reflection(Y::'a,Z))) &
818 (\<forall>A1 C1 B1. equal(A1::'a,B1) --> equal(reflection(C1::'a,A1),reflection(C1::'a,B1))) &
819 (equal(v::'a,reflection(u::'a,v))) &
820 (~equal(u::'a,v)) --> False"
823 (*0 inferences so far. Searching to depth 0. 0.2 secs*)
825 "(\<forall>U V W X Y Z. right_angle(U::'a,V,W) & right_angle(X::'a,Y,Z) --> eq(U::'a,V,W,X,Y,Z)) &
826 (\<forall>U V W X Y Z. CONGRUENT(U::'a,V,W,X,Y,Z) --> eq(U::'a,V,W,X,Y,Z)) &
827 (\<forall>V W U X. trapezoid(U::'a,V,W,X) --> parallel(V::'a,W,U,X)) &
828 (\<forall>U V X Y. parallel(U::'a,V,X,Y) --> eq(X::'a,V,U,V,X,Y)) &
829 (trapezoid(a::'a,b,c,d)) &
830 (~eq(a::'a,c,b,c,a,d)) --> False"
833 abbreviation "GRP003_0_ax equal multiply INVERSE identity product \<equiv>
834 (\<forall>X. product(identity::'a,X,X)) &
835 (\<forall>X. product(X::'a,identity,X)) &
836 (\<forall>X. product(INVERSE(X),X,identity)) &
837 (\<forall>X. product(X::'a,INVERSE(X),identity)) &
838 (\<forall>X Y. product(X::'a,Y,multiply(X::'a,Y))) &
839 (\<forall>X Y Z W. product(X::'a,Y,Z) & product(X::'a,Y,W) --> equal(Z::'a,W)) &
840 (\<forall>Y U Z X V W. product(X::'a,Y,U) & product(Y::'a,Z,V) & product(U::'a,Z,W) --> product(X::'a,V,W)) &
841 (\<forall>Y X V U Z W. product(X::'a,Y,U) & product(Y::'a,Z,V) & product(X::'a,V,W) --> product(U::'a,Z,W))"
843 abbreviation "GRP003_0_eq product multiply INVERSE equal \<equiv>
844 (\<forall>X Y. equal(X::'a,Y) --> equal(INVERSE(X),INVERSE(Y))) &
845 (\<forall>X Y W. equal(X::'a,Y) --> equal(multiply(X::'a,W),multiply(Y::'a,W))) &
846 (\<forall>X W Y. equal(X::'a,Y) --> equal(multiply(W::'a,X),multiply(W::'a,Y))) &
847 (\<forall>X Y W Z. equal(X::'a,Y) & product(X::'a,W,Z) --> product(Y::'a,W,Z)) &
848 (\<forall>X W Y Z. equal(X::'a,Y) & product(W::'a,X,Z) --> product(W::'a,Y,Z)) &
849 (\<forall>X W Z Y. equal(X::'a,Y) & product(W::'a,Z,X) --> product(W::'a,Z,Y))"
851 (*2032008 inferences so far. Searching to depth 16. 658s (11 mins) on griffon*)
854 GRP003_0_ax equal multiply INVERSE identity product &
855 GRP003_0_eq product multiply INVERSE equal &
856 (\<forall>X. product(X::'a,X,identity)) &
857 (product(a::'a,b,c)) &
858 (~product(b::'a,a,c)) --> False"
861 (*2386 inferences so far. Searching to depth 11. 8.7 secs*)
864 GRP003_0_ax equal multiply INVERSE identity product &
865 GRP003_0_eq product multiply INVERSE equal &
866 (\<forall>A B. equal(A::'a,B) --> equal(h(A),h(B))) &
867 (\<forall>C D. equal(C::'a,D) --> equal(j(C),j(D))) &
868 (\<forall>A B. equal(A::'a,B) & q(A) --> q(B)) &
869 (\<forall>B A C. q(A) & product(A::'a,B,C) --> product(B::'a,A,C)) &
870 (\<forall>A. product(j(A),A,h(A)) | product(A::'a,j(A),h(A)) | q(A)) &
871 (\<forall>A. product(j(A),A,h(A)) & product(A::'a,j(A),h(A)) --> q(A)) &
872 (~q(identity)) --> False"
875 (*8625 inferences so far. Searching to depth 11. 20 secs*)
878 GRP003_0_ax equal multiply INVERSE identity product &
879 GRP003_0_eq product multiply INVERSE equal &
880 (\<forall>A. product(A::'a,A,identity)) &
881 (product(a::'a,b,c)) &
882 (product(INVERSE(a),INVERSE(b),d)) &
883 (\<forall>A C B. product(INVERSE(A),INVERSE(B),C) --> product(A::'a,C,B)) &
884 (~product(c::'a,d,identity)) --> False"
887 (*2448 inferences so far. Searching to depth 10. 7.2 secs*)
890 GRP003_0_ax equal multiply INVERSE identity product &
891 GRP003_0_eq product multiply INVERSE equal &
892 (\<forall>A B C. subgroup_member(A) & subgroup_member(B) & product(A::'a,INVERSE(B),C) --> subgroup_member(C)) &
893 (\<forall>A B. equal(A::'a,B) & subgroup_member(A) --> subgroup_member(B)) &
894 (\<forall>A. subgroup_member(A) --> product(Gidentity::'a,A,A)) &
895 (\<forall>A. subgroup_member(A) --> product(A::'a,Gidentity,A)) &
896 (\<forall>A. subgroup_member(A) --> product(A::'a,Ginverse(A),Gidentity)) &
897 (\<forall>A. subgroup_member(A) --> product(Ginverse(A),A,Gidentity)) &
898 (\<forall>A. subgroup_member(A) --> subgroup_member(Ginverse(A))) &
899 (\<forall>A B. equal(A::'a,B) --> equal(Ginverse(A),Ginverse(B))) &
900 (\<forall>A C D B. product(A::'a,B,C) & product(A::'a,D,C) --> equal(D::'a,B)) &
901 (\<forall>B C D A. product(A::'a,B,C) & product(D::'a,B,C) --> equal(D::'a,A)) &
902 (subgroup_member(a)) &
903 (subgroup_member(Gidentity)) &
904 (~equal(INVERSE(a),Ginverse(a))) --> False"
907 (*163 inferences so far. Searching to depth 11. 0.3 secs*)
909 "(\<forall>X Y. product(X::'a,Y,multiply(X::'a,Y))) &
910 (\<forall>X Y Z W. product(X::'a,Y,Z) & product(X::'a,Y,W) --> equal(Z::'a,W)) &
911 (\<forall>Y U Z X V W. product(X::'a,Y,U) & product(Y::'a,Z,V) & product(U::'a,Z,W) --> product(X::'a,V,W)) &
912 (\<forall>Y X V U Z W. product(X::'a,Y,U) & product(Y::'a,Z,V) & product(X::'a,V,W) --> product(U::'a,Z,W)) &
913 (\<forall>A. product(A::'a,INVERSE(A),identity)) &
914 (\<forall>A. product(A::'a,identity,A)) &
915 (\<forall>A. ~product(A::'a,a,identity)) --> False"
918 (*47 inferences so far. Searching to depth 11. 0.2 secs*)
920 "(\<forall>X Y. product(X::'a,Y,multiply(X::'a,Y))) &
921 (\<forall>X. product(identity::'a,X,X)) &
922 (\<forall>X. product(X::'a,identity,X)) &
923 (\<forall>X. product(X::'a,INVERSE(X),identity)) &
924 (\<forall>Y U Z X V W. product(X::'a,Y,U) & product(Y::'a,Z,V) & product(U::'a,Z,W) --> product(X::'a,V,W)) &
925 (\<forall>Y X V U Z W. product(X::'a,Y,U) & product(Y::'a,Z,V) & product(X::'a,V,W) --> product(U::'a,Z,W)) &
926 (\<forall>B A C. subgroup_member(A) & subgroup_member(B) & product(B::'a,INVERSE(A),C) --> subgroup_member(C)) &
927 (subgroup_member(a)) &
928 (~subgroup_member(INVERSE(a))) --> False"
931 (*3287 inferences so far. Searching to depth 14. 3.5 secs*)
933 "(\<forall>X. product(identity::'a,X,X)) &
934 (\<forall>X. product(INVERSE(X),X,identity)) &
935 (\<forall>X Y. product(X::'a,Y,multiply(X::'a,Y))) &
936 (\<forall>X Y Z W. product(X::'a,Y,Z) & product(X::'a,Y,W) --> equal(Z::'a,W)) &
937 (\<forall>Y U Z X V W. product(X::'a,Y,U) & product(Y::'a,Z,V) & product(U::'a,Z,W) --> product(X::'a,V,W)) &
938 (\<forall>Y X V U Z W. product(X::'a,Y,U) & product(Y::'a,Z,V) & product(X::'a,V,W) --> product(U::'a,Z,W)) &
939 (\<forall>X W Z Y. equal(X::'a,Y) & product(W::'a,Z,X) --> product(W::'a,Z,Y)) &
941 (~equal(multiply(c::'a,a),multiply(c::'a,b))) --> False"
944 (*25559 inferences so far. Searching to depth 19. 16.9 secs*)
946 "(group_element(e_1)) &
947 (group_element(e_2)) &
948 (~equal(e_1::'a,e_2)) &
949 (~equal(e_2::'a,e_1)) &
950 (\<forall>X Y. group_element(X) & group_element(Y) --> product(X::'a,Y,e_1) | product(X::'a,Y,e_2)) &
951 (\<forall>X Y W Z. product(X::'a,Y,W) & product(X::'a,Y,Z) --> equal(W::'a,Z)) &
952 (\<forall>X Y W Z. product(X::'a,W,Y) & product(X::'a,Z,Y) --> equal(W::'a,Z)) &
953 (\<forall>Y X W Z. product(W::'a,Y,X) & product(Z::'a,Y,X) --> equal(W::'a,Z)) &
954 (\<forall>Z1 Z2 Y X. product(X::'a,Y,Z1) & product(X::'a,Z1,Z2) --> product(Z2::'a,Y,X)) --> False"
957 abbreviation "GRP004_0_ax INVERSE identity multiply equal \<equiv>
958 (\<forall>X. equal(multiply(identity::'a,X),X)) &
959 (\<forall>X. equal(multiply(INVERSE(X),X),identity)) &
960 (\<forall>X Y Z. equal(multiply(multiply(X::'a,Y),Z),multiply(X::'a,multiply(Y::'a,Z)))) &
961 (\<forall>A B. equal(A::'a,B) --> equal(INVERSE(A),INVERSE(B))) &
962 (\<forall>C D E. equal(C::'a,D) --> equal(multiply(C::'a,E),multiply(D::'a,E))) &
963 (\<forall>F' H G. equal(F'::'a,G) --> equal(multiply(H::'a,F'),multiply(H::'a,G)))"
965 abbreviation "GRP004_2_ax multiply least_upper_bound greatest_lower_bound equal \<equiv>
966 (\<forall>Y X. equal(greatest_lower_bound(X::'a,Y),greatest_lower_bound(Y::'a,X))) &
967 (\<forall>Y X. equal(least_upper_bound(X::'a,Y),least_upper_bound(Y::'a,X))) &
968 (\<forall>X Y Z. equal(greatest_lower_bound(X::'a,greatest_lower_bound(Y::'a,Z)),greatest_lower_bound(greatest_lower_bound(X::'a,Y),Z))) &
969 (\<forall>X Y Z. equal(least_upper_bound(X::'a,least_upper_bound(Y::'a,Z)),least_upper_bound(least_upper_bound(X::'a,Y),Z))) &
970 (\<forall>X. equal(least_upper_bound(X::'a,X),X)) &
971 (\<forall>X. equal(greatest_lower_bound(X::'a,X),X)) &
972 (\<forall>Y X. equal(least_upper_bound(X::'a,greatest_lower_bound(X::'a,Y)),X)) &
973 (\<forall>Y X. equal(greatest_lower_bound(X::'a,least_upper_bound(X::'a,Y)),X)) &
974 (\<forall>Y X Z. equal(multiply(X::'a,least_upper_bound(Y::'a,Z)),least_upper_bound(multiply(X::'a,Y),multiply(X::'a,Z)))) &
975 (\<forall>Y X Z. equal(multiply(X::'a,greatest_lower_bound(Y::'a,Z)),greatest_lower_bound(multiply(X::'a,Y),multiply(X::'a,Z)))) &
976 (\<forall>Y Z X. equal(multiply(least_upper_bound(Y::'a,Z),X),least_upper_bound(multiply(Y::'a,X),multiply(Z::'a,X)))) &
977 (\<forall>Y Z X. equal(multiply(greatest_lower_bound(Y::'a,Z),X),greatest_lower_bound(multiply(Y::'a,X),multiply(Z::'a,X)))) &
978 (\<forall>A B C. equal(A::'a,B) --> equal(greatest_lower_bound(A::'a,C),greatest_lower_bound(B::'a,C))) &
979 (\<forall>A C B. equal(A::'a,B) --> equal(greatest_lower_bound(C::'a,A),greatest_lower_bound(C::'a,B))) &
980 (\<forall>A B C. equal(A::'a,B) --> equal(least_upper_bound(A::'a,C),least_upper_bound(B::'a,C))) &
981 (\<forall>A C B. equal(A::'a,B) --> equal(least_upper_bound(C::'a,A),least_upper_bound(C::'a,B))) &
982 (\<forall>A B C. equal(A::'a,B) --> equal(multiply(A::'a,C),multiply(B::'a,C))) &
983 (\<forall>A C B. equal(A::'a,B) --> equal(multiply(C::'a,A),multiply(C::'a,B)))"
985 (*3468 inferences so far. Searching to depth 10. 9.1 secs*)
988 GRP004_0_ax INVERSE identity multiply equal &
989 GRP004_2_ax multiply least_upper_bound greatest_lower_bound equal &
990 (equal(least_upper_bound(a::'a,b),b)) &
991 (~equal(greatest_lower_bound(multiply(a::'a,c),multiply(b::'a,c)),multiply(a::'a,c))) --> False"
994 (*4394 inferences so far. Searching to depth 10. 8.2 secs*)
997 GRP004_0_ax INVERSE identity multiply equal &
998 GRP004_2_ax multiply least_upper_bound greatest_lower_bound equal &
999 (equal(least_upper_bound(a::'a,b),b)) &
1000 (~equal(least_upper_bound(multiply(INVERSE(c),multiply(a::'a,c)),multiply(INVERSE(c),multiply(b::'a,c))),multiply(INVERSE(c),multiply(b::'a,c)))) --> False"
1003 abbreviation "HEN002_0_ax identity Zero Divide equal mless_equal \<equiv>
1004 (\<forall>X Y. mless_equal(X::'a,Y) --> equal(Divide(X::'a,Y),Zero)) &
1005 (\<forall>X Y. equal(Divide(X::'a,Y),Zero) --> mless_equal(X::'a,Y)) &
1006 (\<forall>Y X. mless_equal(Divide(X::'a,Y),X)) &
1007 (\<forall>X Y Z. mless_equal(Divide(Divide(X::'a,Z),Divide(Y::'a,Z)),Divide(Divide(X::'a,Y),Z))) &
1008 (\<forall>X. mless_equal(Zero::'a,X)) &
1009 (\<forall>X Y. mless_equal(X::'a,Y) & mless_equal(Y::'a,X) --> equal(X::'a,Y)) &
1010 (\<forall>X. mless_equal(X::'a,identity))"
1012 abbreviation "HEN002_0_eq mless_equal Divide equal \<equiv>
1013 (\<forall>A B C. equal(A::'a,B) --> equal(Divide(A::'a,C),Divide(B::'a,C))) &
1014 (\<forall>D F' E. equal(D::'a,E) --> equal(Divide(F'::'a,D),Divide(F'::'a,E))) &
1015 (\<forall>G H I'. equal(G::'a,H) & mless_equal(G::'a,I') --> mless_equal(H::'a,I')) &
1016 (\<forall>J L K'. equal(J::'a,K') & mless_equal(L::'a,J) --> mless_equal(L::'a,K'))"
1018 (*250258 inferences so far. Searching to depth 16. 406.2 secs*)
1020 "EQU001_0_ax equal &
1021 HEN002_0_ax identity Zero Divide equal mless_equal &
1022 HEN002_0_eq mless_equal Divide equal &
1023 (~equal(Divide(a::'a,a),Zero)) --> False"
1026 (*202177 inferences so far. Searching to depth 14. 451 secs*)
1028 "EQU001_0_ax equal &
1029 (\<forall>X Y. mless_equal(X::'a,Y) --> quotient(X::'a,Y,Zero)) &
1030 (\<forall>X Y. quotient(X::'a,Y,Zero) --> mless_equal(X::'a,Y)) &
1031 (\<forall>Y Z X. quotient(X::'a,Y,Z) --> mless_equal(Z::'a,X)) &
1032 (\<forall>Y X V3 V2 V1 Z V4 V5. quotient(X::'a,Y,V1) & quotient(Y::'a,Z,V2) & quotient(X::'a,Z,V3) & quotient(V3::'a,V2,V4) & quotient(V1::'a,Z,V5) --> mless_equal(V4::'a,V5)) &
1033 (\<forall>X. mless_equal(Zero::'a,X)) &
1034 (\<forall>X Y. mless_equal(X::'a,Y) & mless_equal(Y::'a,X) --> equal(X::'a,Y)) &
1035 (\<forall>X. mless_equal(X::'a,identity)) &
1036 (\<forall>X Y. quotient(X::'a,Y,Divide(X::'a,Y))) &
1037 (\<forall>X Y Z W. quotient(X::'a,Y,Z) & quotient(X::'a,Y,W) --> equal(Z::'a,W)) &
1038 (\<forall>X Y W Z. equal(X::'a,Y) & quotient(X::'a,W,Z) --> quotient(Y::'a,W,Z)) &
1039 (\<forall>X W Y Z. equal(X::'a,Y) & quotient(W::'a,X,Z) --> quotient(W::'a,Y,Z)) &
1040 (\<forall>X W Z Y. equal(X::'a,Y) & quotient(W::'a,Z,X) --> quotient(W::'a,Z,Y)) &
1041 (\<forall>X Z Y. equal(X::'a,Y) & mless_equal(Z::'a,X) --> mless_equal(Z::'a,Y)) &
1042 (\<forall>X Y Z. equal(X::'a,Y) & mless_equal(X::'a,Z) --> mless_equal(Y::'a,Z)) &
1043 (\<forall>X Y W. equal(X::'a,Y) --> equal(Divide(X::'a,W),Divide(Y::'a,W))) &
1044 (\<forall>X W Y. equal(X::'a,Y) --> equal(Divide(W::'a,X),Divide(W::'a,Y))) &
1045 (\<forall>X. quotient(X::'a,identity,Zero)) &
1046 (\<forall>X. quotient(Zero::'a,X,Zero)) &
1047 (\<forall>X. quotient(X::'a,X,Zero)) &
1048 (\<forall>X. quotient(X::'a,Zero,X)) &
1049 (\<forall>Y X Z. mless_equal(X::'a,Y) & mless_equal(Y::'a,Z) --> mless_equal(X::'a,Z)) &
1050 (\<forall>W1 X Z W2 Y. quotient(X::'a,Y,W1) & mless_equal(W1::'a,Z) & quotient(X::'a,Z,W2) --> mless_equal(W2::'a,Y)) &
1051 (mless_equal(x::'a,y)) &
1052 (quotient(z::'a,y,zQy)) &
1053 (quotient(z::'a,x,zQx)) &
1054 (~mless_equal(zQy::'a,zQx)) --> False"
1057 (*60026 inferences so far. Searching to depth 12. 42.2 secs*)
1059 "EQU001_0_ax equal &
1060 HEN002_0_ax identity Zero Divide equal mless_equal &
1061 HEN002_0_eq mless_equal Divide equal &
1062 (\<forall>X. equal(Divide(X::'a,identity),Zero)) &
1063 (\<forall>X. equal(Divide(Zero::'a,X),Zero)) &
1064 (\<forall>X. equal(Divide(X::'a,X),Zero)) &
1065 (equal(Divide(a::'a,Zero),a)) &
1066 (\<forall>Y X Z. mless_equal(X::'a,Y) & mless_equal(Y::'a,Z) --> mless_equal(X::'a,Z)) &
1067 (\<forall>X Z Y. mless_equal(Divide(X::'a,Y),Z) --> mless_equal(Divide(X::'a,Z),Y)) &
1068 (\<forall>Y Z X. mless_equal(X::'a,Y) --> mless_equal(Divide(Z::'a,Y),Divide(Z::'a,X))) &
1069 (mless_equal(a::'a,b)) &
1070 (~mless_equal(Divide(a::'a,c),Divide(b::'a,c))) --> False"
1073 (*3160 inferences so far. Searching to depth 11. 3.5 secs*)
1075 "EQU001_0_ax equal &
1076 (\<forall>Y X. equal(Divide(Divide(X::'a,Y),X),Zero)) &
1077 (\<forall>X Y Z. equal(Divide(Divide(Divide(X::'a,Z),Divide(Y::'a,Z)),Divide(Divide(X::'a,Y),Z)),Zero)) &
1078 (\<forall>X. equal(Divide(Zero::'a,X),Zero)) &
1079 (\<forall>X Y. equal(Divide(X::'a,Y),Zero) & equal(Divide(Y::'a,X),Zero) --> equal(X::'a,Y)) &
1080 (\<forall>X. equal(Divide(X::'a,identity),Zero)) &
1081 (\<forall>A B C. equal(A::'a,B) --> equal(Divide(A::'a,C),Divide(B::'a,C))) &
1082 (\<forall>D F' E. equal(D::'a,E) --> equal(Divide(F'::'a,D),Divide(F'::'a,E))) &
1083 (\<forall>Y X Z. equal(Divide(X::'a,Y),Zero) & equal(Divide(Y::'a,Z),Zero) --> equal(Divide(X::'a,Z),Zero)) &
1084 (\<forall>X Z Y. equal(Divide(Divide(X::'a,Y),Z),Zero) --> equal(Divide(Divide(X::'a,Z),Y),Zero)) &
1085 (\<forall>Y Z X. equal(Divide(X::'a,Y),Zero) --> equal(Divide(Divide(Z::'a,Y),Divide(Z::'a,X)),Zero)) &
1086 (~equal(Divide(identity::'a,a),Divide(identity::'a,Divide(identity::'a,Divide(identity::'a,a))))) &
1087 (equal(Divide(identity::'a,a),b)) &
1088 (equal(Divide(identity::'a,b),c)) &
1089 (equal(Divide(identity::'a,c),d)) &
1090 (~equal(b::'a,d)) --> False"
1093 (*970373 inferences so far. Searching to depth 17. 890.0 secs*)
1095 "EQU001_0_ax equal &
1096 HEN002_0_ax identity Zero Divide equal mless_equal &
1097 HEN002_0_eq mless_equal Divide equal &
1098 (~mless_equal(a::'a,a)) --> False"
1102 (*1063 inferences so far. Searching to depth 20. 1.0 secs*)
1104 "(\<forall>X Y. is_a_theorem(equivalent(X::'a,Y)) & is_a_theorem(X) --> is_a_theorem(Y)) &
1105 (\<forall>X Z Y. is_a_theorem(equivalent(equivalent(X::'a,Y),equivalent(equivalent(X::'a,Z),equivalent(Z::'a,Y))))) &
1106 (~is_a_theorem(equivalent(equivalent(a::'a,b),equivalent(equivalent(c::'a,b),equivalent(a::'a,c))))) --> False"
1109 (*2549 inferences so far. Searching to depth 12. 1.4 secs*)
1111 "(\<forall>X Y. is_a_theorem(implies(X,Y)) & is_a_theorem(X) --> is_a_theorem(Y)) &
1112 (\<forall>Y X. is_a_theorem(implies(X,implies(Y,X)))) &
1113 (\<forall>Y X Z. is_a_theorem(implies(implies(X,implies(Y,Z)),implies(implies(X,Y),implies(X,Z))))) &
1114 (\<forall>Y X. is_a_theorem(implies(implies(not(X),not(Y)),implies(Y,X)))) &
1115 (\<forall>X2 X1 X3. is_a_theorem(implies(X1,X2)) & is_a_theorem(implies(X2,X3)) --> is_a_theorem(implies(X1,X3))) &
1116 (~is_a_theorem(implies(not(not(a)),a))) --> False"
1119 (*2036 inferences so far. Searching to depth 20. 1.5 secs*)
1121 "(\<forall>X Y. is_a_theorem(implies(X::'a,Y)) & is_a_theorem(X) --> is_a_theorem(Y)) &
1122 (\<forall>Y Z U X. is_a_theorem(implies(implies(implies(X::'a,Y),Z),implies(implies(Z::'a,X),implies(U::'a,X))))) &
1123 (~is_a_theorem(implies(a::'a,implies(b::'a,a)))) --> False"
1126 (*1100 inferences so far. Searching to depth 13. 1.0 secs*)
1128 "(\<forall>X Y. is_a_theorem(implies(X,Y)) & is_a_theorem(X) --> is_a_theorem(Y)) &
1129 (\<forall>Y X. is_a_theorem(implies(X,implies(Y,X)))) &
1130 (\<forall>Y X Z. is_a_theorem(implies(implies(X,Y),implies(implies(Y,Z),implies(X,Z))))) &
1131 (\<forall>Y X. is_a_theorem(implies(implies(implies(X,Y),Y),implies(implies(Y,X),X)))) &
1132 (\<forall>Y X. is_a_theorem(implies(implies(not(X),not(Y)),implies(Y,X)))) &
1133 (~is_a_theorem(implies(implies(a,b),implies(implies(c,a),implies(c,b))))) --> False"
1136 (*667 inferences so far. Searching to depth 9. 1.4 secs*)
1138 "(\<forall>X. equal(X,X)) &
1139 (\<forall>Y X. equal(X,Y) --> equal(Y,X)) &
1140 (\<forall>Y X Z. equal(X,Y) & equal(Y,Z) --> equal(X,Z)) &
1141 (\<forall>X. equal(implies(true,X),X)) &
1142 (\<forall>Y X Z. equal(implies(implies(X,Y),implies(implies(Y,Z),implies(X,Z))),true)) &
1143 (\<forall>Y X. equal(implies(implies(X,Y),Y),implies(implies(Y,X),X))) &
1144 (\<forall>Y X. equal(implies(implies(not(X),not(Y)),implies(Y,X)),true)) &
1145 (\<forall>A B C. equal(A,B) --> equal(implies(A,C),implies(B,C))) &
1146 (\<forall>D F' E. equal(D,E) --> equal(implies(F',D),implies(F',E))) &
1147 (\<forall>G H. equal(G,H) --> equal(not(G),not(H))) &
1148 (\<forall>X Y. equal(big_V(X,Y),implies(implies(X,Y),Y))) &
1149 (\<forall>X Y. equal(big_hat(X,Y),not(big_V(not(X),not(Y))))) &
1150 (\<forall>X Y. ordered(X,Y) --> equal(implies(X,Y),true)) &
1151 (\<forall>X Y. equal(implies(X,Y),true) --> ordered(X,Y)) &
1152 (\<forall>A B C. equal(A,B) --> equal(big_V(A,C),big_V(B,C))) &
1153 (\<forall>D F' E. equal(D,E) --> equal(big_V(F',D),big_V(F',E))) &
1154 (\<forall>G H I'. equal(G,H) --> equal(big_hat(G,I'),big_hat(H,I'))) &
1155 (\<forall>J L K'. equal(J,K') --> equal(big_hat(L,J),big_hat(L,K'))) &
1156 (\<forall>M N O'. equal(M,N) & ordered(M,O') --> ordered(N,O')) &
1157 (\<forall>P R Q. equal(P,Q) & ordered(R,P) --> ordered(R,Q)) &
1159 (~ordered(implies(z,x),implies(z,y))) --> False"
1162 (*5245 inferences so far. Searching to depth 12. 4.6 secs*)
1164 "(\<forall>A. axiom(or(not(or(A,A)),A))) &
1165 (\<forall>B A. axiom(or(not(A),or(B,A)))) &
1166 (\<forall>B A. axiom(or(not(or(A,B)),or(B,A)))) &
1167 (\<forall>B A C. axiom(or(not(or(A,or(B,C))),or(B,or(A,C))))) &
1168 (\<forall>A C B. axiom(or(not(or(not(A),B)),or(not(or(C,A)),or(C,B))))) &
1169 (\<forall>X. axiom(X) --> theorem(X)) &
1170 (\<forall>X Y. axiom(or(not(Y),X)) & theorem(Y) --> theorem(X)) &
1171 (\<forall>X Y Z. axiom(or(not(X),Y)) & theorem(or(not(Y),Z)) --> theorem(or(not(X),Z))) &
1172 (~theorem(or(not(or(not(p),q)),or(not(not(q)),not(p))))) --> False"
1175 (*410 inferences so far. Searching to depth 10. 0.3 secs*)
1177 "(\<forall>A. axiom(or(not(or(A,A)),A))) &
1178 (\<forall>B A. axiom(or(not(A),or(B,A)))) &
1179 (\<forall>B A. axiom(or(not(or(A,B)),or(B,A)))) &
1180 (\<forall>B A C. axiom(or(not(or(A,or(B,C))),or(B,or(A,C))))) &
1181 (\<forall>A C B. axiom(or(not(or(not(A),B)),or(not(or(C,A)),or(C,B))))) &
1182 (\<forall>X. axiom(X) --> theorem(X)) &
1183 (\<forall>X Y. axiom(or(not(Y),X)) & theorem(Y) --> theorem(X)) &
1184 (\<forall>X Y Z. axiom(or(not(X),Y)) & theorem(or(not(Y),Z)) --> theorem(or(not(X),Z))) &
1185 (~theorem(or(not(not(or(p,q))),not(q)))) --> False"
1188 (*5849 inferences so far. Searching to depth 12. 5.6 secs*)
1190 "(\<forall>A. axiom(or(not(or(A,A)),A))) &
1191 (\<forall>B A. axiom(or(not(A),or(B,A)))) &
1192 (\<forall>B A. axiom(or(not(or(A,B)),or(B,A)))) &
1193 (\<forall>B A C. axiom(or(not(or(A,or(B,C))),or(B,or(A,C))))) &
1194 (\<forall>A C B. axiom(or(not(or(not(A),B)),or(not(or(C,A)),or(C,B))))) &
1195 (\<forall>X. axiom(X) --> theorem(X)) &
1196 (\<forall>X Y. axiom(or(not(Y),X)) & theorem(Y) --> theorem(X)) &
1197 (\<forall>X Y Z. axiom(or(not(X),Y)) & theorem(or(not(Y),Z)) --> theorem(or(not(X),Z))) &
1198 (~theorem(or(not(or(not(p),q)),or(not(or(p,q)),q)))) --> False"
1201 (*0 secs. Not sure that a search even starts!*)
1209 (*119585 inferences so far. Searching to depth 14. 262.4 secs*)
1211 "EQU001_0_ax equal &
1212 (\<forall>Y X Z. equal(f(X::'a,f(Y::'a,Z)),f(f(X::'a,Y),f(X::'a,Z)))) &
1213 (\<forall>X Y. left(X::'a,f(X::'a,Y))) &
1214 (\<forall>Y X Z. left(X::'a,Y) & left(Y::'a,Z) --> left(X::'a,Z)) &
1215 (equal(num2::'a,f(num1::'a,num1))) &
1216 (equal(num3::'a,f(num2::'a,num1))) &
1217 (equal(u::'a,f(num2::'a,num2))) &
1218 (\<forall>A B C. equal(A::'a,B) --> equal(f(A::'a,C),f(B::'a,C))) &
1219 (\<forall>D F' E. equal(D::'a,E) --> equal(f(F'::'a,D),f(F'::'a,E))) &
1220 (\<forall>G H I'. equal(G::'a,H) & left(G::'a,I') --> left(H::'a,I')) &
1221 (\<forall>J L K'. equal(J::'a,K') & left(L::'a,J) --> left(L::'a,K')) &
1222 (~left(num3::'a,u)) --> False"
1226 (*2392 inferences so far. Searching to depth 12. 2.2 secs*)
1228 "(at(something::'a,here,now)) &
1229 (\<forall>Place Situation. hand_at(Place::'a,Situation) --> hand_at(Place::'a,let_go(Situation))) &
1230 (\<forall>Place Another_place Situation. hand_at(Place::'a,Situation) --> hand_at(Another_place::'a,go(Another_place::'a,Situation))) &
1231 (\<forall>Thing Situation. ~held(Thing::'a,let_go(Situation))) &
1232 (\<forall>Situation Thing. at(Thing::'a,here,Situation) --> red(Thing)) &
1233 (\<forall>Thing Place Situation. at(Thing::'a,Place,Situation) --> at(Thing::'a,Place,let_go(Situation))) &
1234 (\<forall>Thing Place Situation. at(Thing::'a,Place,Situation) --> at(Thing::'a,Place,pick_up(Situation))) &
1235 (\<forall>Thing Place Situation. at(Thing::'a,Place,Situation) --> grabbed(Thing::'a,pick_up(go(Place::'a,let_go(Situation))))) &
1236 (\<forall>Thing Situation. red(Thing) & put(Thing::'a,there,Situation) --> answer(Situation)) &
1237 (\<forall>Place Thing Another_place Situation. at(Thing::'a,Place,Situation) & grabbed(Thing::'a,Situation) --> put(Thing::'a,Another_place,go(Another_place::'a,Situation))) &
1238 (\<forall>Thing Place Another_place Situation. at(Thing::'a,Place,Situation) --> held(Thing::'a,Situation) | at(Thing::'a,Place,go(Another_place::'a,Situation))) &
1239 (\<forall>One_place Thing Place Situation. hand_at(One_place::'a,Situation) & held(Thing::'a,Situation) --> at(Thing::'a,Place,go(Place::'a,Situation))) &
1240 (\<forall>Place Thing Situation. hand_at(Place::'a,Situation) & at(Thing::'a,Place,Situation) --> held(Thing::'a,pick_up(Situation))) &
1241 (\<forall>Situation. ~answer(Situation)) --> False"
1244 (*73 inferences so far. Searching to depth 10. 0.2 secs*)
1246 "(\<forall>Number_of_small_parts Small_part Big_part Number_of_mid_parts Mid_part. has_parts(Big_part::'a,Number_of_mid_parts,Mid_part) --> in'(object_in(Big_part::'a,Mid_part,Small_part,Number_of_mid_parts,Number_of_small_parts),Mid_part) | has_parts(Big_part::'a,mtimes(Number_of_mid_parts::'a,Number_of_small_parts),Small_part)) &
1247 (\<forall>Big_part Mid_part Number_of_mid_parts Number_of_small_parts Small_part. has_parts(Big_part::'a,Number_of_mid_parts,Mid_part) & has_parts(object_in(Big_part::'a,Mid_part,Small_part,Number_of_mid_parts,Number_of_small_parts),Number_of_small_parts,Small_part) --> has_parts(Big_part::'a,mtimes(Number_of_mid_parts::'a,Number_of_small_parts),Small_part)) &
1248 (in'(john::'a,boy)) &
1249 (\<forall>X. in'(X::'a,boy) --> in'(X::'a,human)) &
1250 (\<forall>X. in'(X::'a,hand) --> has_parts(X::'a,num5,fingers)) &
1251 (\<forall>X. in'(X::'a,human) --> has_parts(X::'a,num2,arm)) &
1252 (\<forall>X. in'(X::'a,arm) --> has_parts(X::'a,num1,hand)) &
1253 (~has_parts(john::'a,mtimes(num2::'a,num1),hand)) --> False"
1256 (*1486 inferences so far. Searching to depth 20. 1.2 secs*)
1258 "(\<forall>Number_of_small_parts Small_part Big_part Number_of_mid_parts Mid_part. has_parts(Big_part::'a,Number_of_mid_parts,Mid_part) --> in'(object_in(Big_part::'a,Mid_part,Small_part,Number_of_mid_parts,Number_of_small_parts),Mid_part) | has_parts(Big_part::'a,mtimes(Number_of_mid_parts::'a,Number_of_small_parts),Small_part)) &
1259 (\<forall>Big_part Mid_part Number_of_mid_parts Number_of_small_parts Small_part. has_parts(Big_part::'a,Number_of_mid_parts,Mid_part) & has_parts(object_in(Big_part::'a,Mid_part,Small_part,Number_of_mid_parts,Number_of_small_parts),Number_of_small_parts,Small_part) --> has_parts(Big_part::'a,mtimes(Number_of_mid_parts::'a,Number_of_small_parts),Small_part)) &
1260 (in'(john::'a,boy)) &
1261 (\<forall>X. in'(X::'a,boy) --> in'(X::'a,human)) &
1262 (\<forall>X. in'(X::'a,hand) --> has_parts(X::'a,num5,fingers)) &
1263 (\<forall>X. in'(X::'a,human) --> has_parts(X::'a,num2,arm)) &
1264 (\<forall>X. in'(X::'a,arm) --> has_parts(X::'a,num1,hand)) &
1265 (~has_parts(john::'a,mtimes(mtimes(num2::'a,num1),num5),fingers)) --> False"
1268 (*100 inferences so far. Searching to depth 12. 0.1 secs*)
1270 "(value(truth::'a,truth)) &
1271 (value(falsity::'a,falsity)) &
1272 (\<forall>X Y. value(X::'a,truth) & value(Y::'a,truth) --> value(xor(X::'a,Y),falsity)) &
1273 (\<forall>X Y. value(X::'a,truth) & value(Y::'a,falsity) --> value(xor(X::'a,Y),truth)) &
1274 (\<forall>X Y. value(X::'a,falsity) & value(Y::'a,truth) --> value(xor(X::'a,Y),truth)) &
1275 (\<forall>X Y. value(X::'a,falsity) & value(Y::'a,falsity) --> value(xor(X::'a,Y),falsity)) &
1276 (\<forall>Value. ~value(xor(xor(xor(xor(truth::'a,falsity),falsity),truth),falsity),Value)) --> False"
1279 (*19116 inferences so far. Searching to depth 16. 15.9 secs*)
1281 "(\<forall>Y X Z. p(X::'a,Y) & p(Y::'a,Z) --> p(X::'a,Z)) &
1282 (\<forall>Y X Z. q(X::'a,Y) & q(Y::'a,Z) --> q(X::'a,Z)) &
1283 (\<forall>Y X. q(X::'a,Y) --> q(Y::'a,X)) &
1284 (\<forall>X Y. p(X::'a,Y) | q(X::'a,Y)) &
1286 (~q(c::'a,d)) --> False"
1289 (*1713 inferences so far. Searching to depth 10. 2.8 secs*)
1291 "(\<forall>A. equal(A::'a,A)) &
1292 (\<forall>B A C. equal(A::'a,B) & equal(B::'a,C) --> equal(A::'a,C)) &
1293 (\<forall>B A. equal(add(A::'a,B),add(B::'a,A))) &
1294 (\<forall>A B C. equal(add(A::'a,add(B::'a,C)),add(add(A::'a,B),C))) &
1295 (\<forall>B A. equal(subtract(add(A::'a,B),B),A)) &
1296 (\<forall>A B. equal(A::'a,subtract(add(A::'a,B),B))) &
1297 (\<forall>A C B. equal(add(subtract(A::'a,B),C),subtract(add(A::'a,C),B))) &
1298 (\<forall>A C B. equal(subtract(add(A::'a,B),C),add(subtract(A::'a,C),B))) &
1299 (\<forall>A C B D. equal(A::'a,B) & equal(C::'a,add(A::'a,D)) --> equal(C::'a,add(B::'a,D))) &
1300 (\<forall>A C D B. equal(A::'a,B) & equal(C::'a,add(D::'a,A)) --> equal(C::'a,add(D::'a,B))) &
1301 (\<forall>A C B D. equal(A::'a,B) & equal(C::'a,subtract(A::'a,D)) --> equal(C::'a,subtract(B::'a,D))) &
1302 (\<forall>A C D B. equal(A::'a,B) & equal(C::'a,subtract(D::'a,A)) --> equal(C::'a,subtract(D::'a,B))) &
1303 (~equal(add(add(a::'a,b),c),add(a::'a,add(b::'a,c)))) --> False"
1306 abbreviation "NUM001_0_ax multiply successor num0 add equal \<equiv>
1307 (\<forall>A. equal(add(A::'a,num0),A)) &
1308 (\<forall>A B. equal(add(A::'a,successor(B)),successor(add(A::'a,B)))) &
1309 (\<forall>A. equal(multiply(A::'a,num0),num0)) &
1310 (\<forall>B A. equal(multiply(A::'a,successor(B)),add(multiply(A::'a,B),A))) &
1311 (\<forall>A B. equal(successor(A),successor(B)) --> equal(A::'a,B)) &
1312 (\<forall>A B. equal(A::'a,B) --> equal(successor(A),successor(B)))"
1314 abbreviation "NUM001_1_ax predecessor_of_1st_minus_2nd successor add equal mless \<equiv>
1315 (\<forall>A C B. mless(A::'a,B) & mless(C::'a,A) --> mless(C::'a,B)) &
1316 (\<forall>A B C. equal(add(successor(A),B),C) --> mless(B::'a,C)) &
1317 (\<forall>A B. mless(A::'a,B) --> equal(add(successor(predecessor_of_1st_minus_2nd(B::'a,A)),A),B))"
1319 abbreviation "NUM001_2_ax equal mless divides \<equiv>
1320 (\<forall>A B. divides(A::'a,B) --> mless(A::'a,B) | equal(A::'a,B)) &
1321 (\<forall>A B. mless(A::'a,B) --> divides(A::'a,B)) &
1322 (\<forall>A B. equal(A::'a,B) --> divides(A::'a,B))"
1324 (*20717 inferences so far. Searching to depth 11. 13.7 secs*)
1326 "EQU001_0_ax equal &
1327 NUM001_0_ax multiply successor num0 add equal &
1328 NUM001_1_ax predecessor_of_1st_minus_2nd successor add equal mless &
1329 NUM001_2_ax equal mless divides &
1332 (divides(c::'a,a)) &
1333 (\<forall>A. ~equal(successor(A),num0)) --> False"
1336 (*26320 inferences so far. Searching to depth 10. 26.4 secs*)
1338 "EQU001_0_ax equal &
1339 NUM001_0_ax multiply successor num0 add equal &
1340 NUM001_1_ax predecessor_of_1st_minus_2nd successor add equal mless &
1341 (\<forall>B A. equal(add(A::'a,B),add(B::'a,A))) &
1342 (\<forall>B A C. equal(add(A::'a,B),add(C::'a,B)) --> equal(A::'a,C)) &
1344 (\<forall>A. ~equal(successor(A),num0)) --> False"
1347 abbreviation "SET004_0_ax not_homomorphism2 not_homomorphism1
1348 homomorphism compatible operation cantor diagonalise subset_relation
1349 one_to_one choice apply regular function identity_relation
1350 single_valued_class compos powerClass sum_class omega inductive
1351 successor_relation successor image' rng domain range_of INVERSE flip
1352 rot domain_of null_class restrct difference union complement
1353 intersection element_relation second first cross_product ordered_pair
1354 singleton unordered_pair equal universal_class not_subclass_element
1355 member subclass \<equiv>
1356 (\<forall>X U Y. subclass(X::'a,Y) & member(U::'a,X) --> member(U::'a,Y)) &
1357 (\<forall>X Y. member(not_subclass_element(X::'a,Y),X) | subclass(X::'a,Y)) &
1358 (\<forall>X Y. member(not_subclass_element(X::'a,Y),Y) --> subclass(X::'a,Y)) &
1359 (\<forall>X. subclass(X::'a,universal_class)) &
1360 (\<forall>X Y. equal(X::'a,Y) --> subclass(X::'a,Y)) &
1361 (\<forall>Y X. equal(X::'a,Y) --> subclass(Y::'a,X)) &
1362 (\<forall>X Y. subclass(X::'a,Y) & subclass(Y::'a,X) --> equal(X::'a,Y)) &
1363 (\<forall>X U Y. member(U::'a,unordered_pair(X::'a,Y)) --> equal(U::'a,X) | equal(U::'a,Y)) &
1364 (\<forall>X Y. member(X::'a,universal_class) --> member(X::'a,unordered_pair(X::'a,Y))) &
1365 (\<forall>X Y. member(Y::'a,universal_class) --> member(Y::'a,unordered_pair(X::'a,Y))) &
1366 (\<forall>X Y. member(unordered_pair(X::'a,Y),universal_class)) &
1367 (\<forall>X. equal(unordered_pair(X::'a,X),singleton(X))) &
1368 (\<forall>X Y. equal(unordered_pair(singleton(X),unordered_pair(X::'a,singleton(Y))),ordered_pair(X::'a,Y))) &
1369 (\<forall>V Y U X. member(ordered_pair(U::'a,V),cross_product(X::'a,Y)) --> member(U::'a,X)) &
1370 (\<forall>U X V Y. member(ordered_pair(U::'a,V),cross_product(X::'a,Y)) --> member(V::'a,Y)) &
1371 (\<forall>U V X Y. member(U::'a,X) & member(V::'a,Y) --> member(ordered_pair(U::'a,V),cross_product(X::'a,Y))) &
1372 (\<forall>X Y Z. member(Z::'a,cross_product(X::'a,Y)) --> equal(ordered_pair(first(Z),second(Z)),Z)) &
1373 (subclass(element_relation::'a,cross_product(universal_class::'a,universal_class))) &
1374 (\<forall>X Y. member(ordered_pair(X::'a,Y),element_relation) --> member(X::'a,Y)) &
1375 (\<forall>X Y. member(ordered_pair(X::'a,Y),cross_product(universal_class::'a,universal_class)) & member(X::'a,Y) --> member(ordered_pair(X::'a,Y),element_relation)) &
1376 (\<forall>Y Z X. member(Z::'a,intersection(X::'a,Y)) --> member(Z::'a,X)) &
1377 (\<forall>X Z Y. member(Z::'a,intersection(X::'a,Y)) --> member(Z::'a,Y)) &
1378 (\<forall>Z X Y. member(Z::'a,X) & member(Z::'a,Y) --> member(Z::'a,intersection(X::'a,Y))) &
1379 (\<forall>Z X. ~(member(Z::'a,complement(X)) & member(Z::'a,X))) &
1380 (\<forall>Z X. member(Z::'a,universal_class) --> member(Z::'a,complement(X)) | member(Z::'a,X)) &
1381 (\<forall>X Y. equal(complement(intersection(complement(X),complement(Y))),union(X::'a,Y))) &
1382 (\<forall>X Y. equal(intersection(complement(intersection(X::'a,Y)),complement(intersection(complement(X),complement(Y)))),difference(X::'a,Y))) &
1383 (\<forall>Xr X Y. equal(intersection(Xr::'a,cross_product(X::'a,Y)),restrct(Xr::'a,X,Y))) &
1384 (\<forall>Xr X Y. equal(intersection(cross_product(X::'a,Y),Xr),restrct(Xr::'a,X,Y))) &
1385 (\<forall>Z X. ~(equal(restrct(X::'a,singleton(Z),universal_class),null_class) & member(Z::'a,domain_of(X)))) &
1386 (\<forall>Z X. member(Z::'a,universal_class) --> equal(restrct(X::'a,singleton(Z),universal_class),null_class) | member(Z::'a,domain_of(X))) &
1387 (\<forall>X. subclass(rot(X),cross_product(cross_product(universal_class::'a,universal_class),universal_class))) &
1388 (\<forall>V W U X. member(ordered_pair(ordered_pair(U::'a,V),W),rot(X)) --> member(ordered_pair(ordered_pair(V::'a,W),U),X)) &
1389 (\<forall>U V W X. member(ordered_pair(ordered_pair(V::'a,W),U),X) & member(ordered_pair(ordered_pair(U::'a,V),W),cross_product(cross_product(universal_class::'a,universal_class),universal_class)) --> member(ordered_pair(ordered_pair(U::'a,V),W),rot(X))) &
1390 (\<forall>X. subclass(flip(X),cross_product(cross_product(universal_class::'a,universal_class),universal_class))) &
1391 (\<forall>V U W X. member(ordered_pair(ordered_pair(U::'a,V),W),flip(X)) --> member(ordered_pair(ordered_pair(V::'a,U),W),X)) &
1392 (\<forall>U V W X. member(ordered_pair(ordered_pair(V::'a,U),W),X) & member(ordered_pair(ordered_pair(U::'a,V),W),cross_product(cross_product(universal_class::'a,universal_class),universal_class)) --> member(ordered_pair(ordered_pair(U::'a,V),W),flip(X))) &
1393 (\<forall>Y. equal(domain_of(flip(cross_product(Y::'a,universal_class))),INVERSE(Y))) &
1394 (\<forall>Z. equal(domain_of(INVERSE(Z)),range_of(Z))) &
1395 (\<forall>Z X Y. equal(first(not_subclass_element(restrct(Z::'a,X,singleton(Y)),null_class)),domain(Z::'a,X,Y))) &
1396 (\<forall>Z X Y. equal(second(not_subclass_element(restrct(Z::'a,singleton(X),Y),null_class)),rng(Z::'a,X,Y))) &
1397 (\<forall>Xr X. equal(range_of(restrct(Xr::'a,X,universal_class)),image'(Xr::'a,X))) &
1398 (\<forall>X. equal(union(X::'a,singleton(X)),successor(X))) &
1399 (subclass(successor_relation::'a,cross_product(universal_class::'a,universal_class))) &
1400 (\<forall>X Y. member(ordered_pair(X::'a,Y),successor_relation) --> equal(successor(X),Y)) &
1401 (\<forall>X Y. equal(successor(X),Y) & member(ordered_pair(X::'a,Y),cross_product(universal_class::'a,universal_class)) --> member(ordered_pair(X::'a,Y),successor_relation)) &
1402 (\<forall>X. inductive(X) --> member(null_class::'a,X)) &
1403 (\<forall>X. inductive(X) --> subclass(image'(successor_relation::'a,X),X)) &
1404 (\<forall>X. member(null_class::'a,X) & subclass(image'(successor_relation::'a,X),X) --> inductive(X)) &
1405 (inductive(omega)) &
1406 (\<forall>Y. inductive(Y) --> subclass(omega::'a,Y)) &
1407 (member(omega::'a,universal_class)) &
1408 (\<forall>X. equal(domain_of(restrct(element_relation::'a,universal_class,X)),sum_class(X))) &
1409 (\<forall>X. member(X::'a,universal_class) --> member(sum_class(X),universal_class)) &
1410 (\<forall>X. equal(complement(image'(element_relation::'a,complement(X))),powerClass(X))) &
1411 (\<forall>U. member(U::'a,universal_class) --> member(powerClass(U),universal_class)) &
1412 (\<forall>Yr Xr. subclass(compos(Yr::'a,Xr),cross_product(universal_class::'a,universal_class))) &
1413 (\<forall>Z Yr Xr Y. member(ordered_pair(Y::'a,Z),compos(Yr::'a,Xr)) --> member(Z::'a,image'(Yr::'a,image'(Xr::'a,singleton(Y))))) &
1414 (\<forall>Y Z Yr Xr. member(Z::'a,image'(Yr::'a,image'(Xr::'a,singleton(Y)))) & member(ordered_pair(Y::'a,Z),cross_product(universal_class::'a,universal_class)) --> member(ordered_pair(Y::'a,Z),compos(Yr::'a,Xr))) &
1415 (\<forall>X. single_valued_class(X) --> subclass(compos(X::'a,INVERSE(X)),identity_relation)) &
1416 (\<forall>X. subclass(compos(X::'a,INVERSE(X)),identity_relation) --> single_valued_class(X)) &
1417 (\<forall>Xf. function(Xf) --> subclass(Xf::'a,cross_product(universal_class::'a,universal_class))) &
1418 (\<forall>Xf. function(Xf) --> subclass(compos(Xf::'a,INVERSE(Xf)),identity_relation)) &
1419 (\<forall>Xf. subclass(Xf::'a,cross_product(universal_class::'a,universal_class)) & subclass(compos(Xf::'a,INVERSE(Xf)),identity_relation) --> function(Xf)) &
1420 (\<forall>Xf X. function(Xf) & member(X::'a,universal_class) --> member(image'(Xf::'a,X),universal_class)) &
1421 (\<forall>X. equal(X::'a,null_class) | member(regular(X),X)) &
1422 (\<forall>X. equal(X::'a,null_class) | equal(intersection(X::'a,regular(X)),null_class)) &
1423 (\<forall>Xf Y. equal(sum_class(image'(Xf::'a,singleton(Y))),apply(Xf::'a,Y))) &
1424 (function(choice)) &
1425 (\<forall>Y. member(Y::'a,universal_class) --> equal(Y::'a,null_class) | member(apply(choice::'a,Y),Y)) &
1426 (\<forall>Xf. one_to_one(Xf) --> function(Xf)) &
1427 (\<forall>Xf. one_to_one(Xf) --> function(INVERSE(Xf))) &
1428 (\<forall>Xf. function(INVERSE(Xf)) & function(Xf) --> one_to_one(Xf)) &
1429 (equal(intersection(cross_product(universal_class::'a,universal_class),intersection(cross_product(universal_class::'a,universal_class),complement(compos(complement(element_relation),INVERSE(element_relation))))),subset_relation)) &
1430 (equal(intersection(INVERSE(subset_relation),subset_relation),identity_relation)) &
1431 (\<forall>Xr. equal(complement(domain_of(intersection(Xr::'a,identity_relation))),diagonalise(Xr))) &
1432 (\<forall>X. equal(intersection(domain_of(X),diagonalise(compos(INVERSE(element_relation),X))),cantor(X))) &
1433 (\<forall>Xf. operation(Xf) --> function(Xf)) &
1434 (\<forall>Xf. operation(Xf) --> equal(cross_product(domain_of(domain_of(Xf)),domain_of(domain_of(Xf))),domain_of(Xf))) &
1435 (\<forall>Xf. operation(Xf) --> subclass(range_of(Xf),domain_of(domain_of(Xf)))) &
1436 (\<forall>Xf. function(Xf) & equal(cross_product(domain_of(domain_of(Xf)),domain_of(domain_of(Xf))),domain_of(Xf)) & subclass(range_of(Xf),domain_of(domain_of(Xf))) --> operation(Xf)) &
1437 (\<forall>Xf1 Xf2 Xh. compatible(Xh::'a,Xf1,Xf2) --> function(Xh)) &
1438 (\<forall>Xf2 Xf1 Xh. compatible(Xh::'a,Xf1,Xf2) --> equal(domain_of(domain_of(Xf1)),domain_of(Xh))) &
1439 (\<forall>Xf1 Xh Xf2. compatible(Xh::'a,Xf1,Xf2) --> subclass(range_of(Xh),domain_of(domain_of(Xf2)))) &
1440 (\<forall>Xh Xh1 Xf1 Xf2. function(Xh) & equal(domain_of(domain_of(Xf1)),domain_of(Xh)) & subclass(range_of(Xh),domain_of(domain_of(Xf2))) --> compatible(Xh1::'a,Xf1,Xf2)) &
1441 (\<forall>Xh Xf2 Xf1. homomorphism(Xh::'a,Xf1,Xf2) --> operation(Xf1)) &
1442 (\<forall>Xh Xf1 Xf2. homomorphism(Xh::'a,Xf1,Xf2) --> operation(Xf2)) &
1443 (\<forall>Xh Xf1 Xf2. homomorphism(Xh::'a,Xf1,Xf2) --> compatible(Xh::'a,Xf1,Xf2)) &
1444 (\<forall>Xf2 Xh Xf1 X Y. homomorphism(Xh::'a,Xf1,Xf2) & member(ordered_pair(X::'a,Y),domain_of(Xf1)) --> equal(apply(Xf2::'a,ordered_pair(apply(Xh::'a,X),apply(Xh::'a,Y))),apply(Xh::'a,apply(Xf1::'a,ordered_pair(X::'a,Y))))) &
1445 (\<forall>Xh Xf1 Xf2. operation(Xf1) & operation(Xf2) & compatible(Xh::'a,Xf1,Xf2) --> member(ordered_pair(not_homomorphism1(Xh::'a,Xf1,Xf2),not_homomorphism2(Xh::'a,Xf1,Xf2)),domain_of(Xf1)) | homomorphism(Xh::'a,Xf1,Xf2)) &
1446 (\<forall>Xh Xf1 Xf2. operation(Xf1) & operation(Xf2) & compatible(Xh::'a,Xf1,Xf2) & equal(apply(Xf2::'a,ordered_pair(apply(Xh::'a,not_homomorphism1(Xh::'a,Xf1,Xf2)),apply(Xh::'a,not_homomorphism2(Xh::'a,Xf1,Xf2)))),apply(Xh::'a,apply(Xf1::'a,ordered_pair(not_homomorphism1(Xh::'a,Xf1,Xf2),not_homomorphism2(Xh::'a,Xf1,Xf2))))) --> homomorphism(Xh::'a,Xf1,Xf2))"
1448 abbreviation "SET004_0_eq subclass single_valued_class operation
1449 one_to_one member inductive homomorphism function compatible
1450 unordered_pair union sum_class successor singleton second rot restrct
1451 regular range_of rng powerClass ordered_pair not_subclass_element
1452 not_homomorphism2 not_homomorphism1 INVERSE intersection image' flip
1453 first domain_of domain difference diagonalise cross_product compos
1454 complement cantor apply equal \<equiv>
1455 (\<forall>D E F'. equal(D::'a,E) --> equal(apply(D::'a,F'),apply(E::'a,F'))) &
1456 (\<forall>G I' H. equal(G::'a,H) --> equal(apply(I'::'a,G),apply(I'::'a,H))) &
1457 (\<forall>J K'. equal(J::'a,K') --> equal(cantor(J),cantor(K'))) &
1458 (\<forall>L M. equal(L::'a,M) --> equal(complement(L),complement(M))) &
1459 (\<forall>N O' P. equal(N::'a,O') --> equal(compos(N::'a,P),compos(O'::'a,P))) &
1460 (\<forall>Q S' R. equal(Q::'a,R) --> equal(compos(S'::'a,Q),compos(S'::'a,R))) &
1461 (\<forall>T' U V. equal(T'::'a,U) --> equal(cross_product(T'::'a,V),cross_product(U::'a,V))) &
1462 (\<forall>W Y X. equal(W::'a,X) --> equal(cross_product(Y::'a,W),cross_product(Y::'a,X))) &
1463 (\<forall>Z A1. equal(Z::'a,A1) --> equal(diagonalise(Z),diagonalise(A1))) &
1464 (\<forall>B1 C1 D1. equal(B1::'a,C1) --> equal(difference(B1::'a,D1),difference(C1::'a,D1))) &
1465 (\<forall>E1 G1 F1. equal(E1::'a,F1) --> equal(difference(G1::'a,E1),difference(G1::'a,F1))) &
1466 (\<forall>H1 I1 J1 K1. equal(H1::'a,I1) --> equal(domain(H1::'a,J1,K1),domain(I1::'a,J1,K1))) &
1467 (\<forall>L1 N1 M1 O1. equal(L1::'a,M1) --> equal(domain(N1::'a,L1,O1),domain(N1::'a,M1,O1))) &
1468 (\<forall>P1 R1 S1 Q1. equal(P1::'a,Q1) --> equal(domain(R1::'a,S1,P1),domain(R1::'a,S1,Q1))) &
1469 (\<forall>T1 U1. equal(T1::'a,U1) --> equal(domain_of(T1),domain_of(U1))) &
1470 (\<forall>V1 W1. equal(V1::'a,W1) --> equal(first(V1),first(W1))) &
1471 (\<forall>X1 Y1. equal(X1::'a,Y1) --> equal(flip(X1),flip(Y1))) &
1472 (\<forall>Z1 A2 B2. equal(Z1::'a,A2) --> equal(image'(Z1::'a,B2),image'(A2::'a,B2))) &
1473 (\<forall>C2 E2 D2. equal(C2::'a,D2) --> equal(image'(E2::'a,C2),image'(E2::'a,D2))) &
1474 (\<forall>F2 G2 H2. equal(F2::'a,G2) --> equal(intersection(F2::'a,H2),intersection(G2::'a,H2))) &
1475 (\<forall>I2 K2 J2. equal(I2::'a,J2) --> equal(intersection(K2::'a,I2),intersection(K2::'a,J2))) &
1476 (\<forall>L2 M2. equal(L2::'a,M2) --> equal(INVERSE(L2),INVERSE(M2))) &
1477 (\<forall>N2 O2 P2 Q2. equal(N2::'a,O2) --> equal(not_homomorphism1(N2::'a,P2,Q2),not_homomorphism1(O2::'a,P2,Q2))) &
1478 (\<forall>R2 T2 S2 U2. equal(R2::'a,S2) --> equal(not_homomorphism1(T2::'a,R2,U2),not_homomorphism1(T2::'a,S2,U2))) &
1479 (\<forall>V2 X2 Y2 W2. equal(V2::'a,W2) --> equal(not_homomorphism1(X2::'a,Y2,V2),not_homomorphism1(X2::'a,Y2,W2))) &
1480 (\<forall>Z2 A3 B3 C3. equal(Z2::'a,A3) --> equal(not_homomorphism2(Z2::'a,B3,C3),not_homomorphism2(A3::'a,B3,C3))) &
1481 (\<forall>D3 F3 E3 G3. equal(D3::'a,E3) --> equal(not_homomorphism2(F3::'a,D3,G3),not_homomorphism2(F3::'a,E3,G3))) &
1482 (\<forall>H3 J3 K3 I3. equal(H3::'a,I3) --> equal(not_homomorphism2(J3::'a,K3,H3),not_homomorphism2(J3::'a,K3,I3))) &
1483 (\<forall>L3 M3 N3. equal(L3::'a,M3) --> equal(not_subclass_element(L3::'a,N3),not_subclass_element(M3::'a,N3))) &
1484 (\<forall>O3 Q3 P3. equal(O3::'a,P3) --> equal(not_subclass_element(Q3::'a,O3),not_subclass_element(Q3::'a,P3))) &
1485 (\<forall>R3 S3 T3. equal(R3::'a,S3) --> equal(ordered_pair(R3::'a,T3),ordered_pair(S3::'a,T3))) &
1486 (\<forall>U3 W3 V3. equal(U3::'a,V3) --> equal(ordered_pair(W3::'a,U3),ordered_pair(W3::'a,V3))) &
1487 (\<forall>X3 Y3. equal(X3::'a,Y3) --> equal(powerClass(X3),powerClass(Y3))) &
1488 (\<forall>Z3 A4 B4 C4. equal(Z3::'a,A4) --> equal(rng(Z3::'a,B4,C4),rng(A4::'a,B4,C4))) &
1489 (\<forall>D4 F4 E4 G4. equal(D4::'a,E4) --> equal(rng(F4::'a,D4,G4),rng(F4::'a,E4,G4))) &
1490 (\<forall>H4 J4 K4 I4. equal(H4::'a,I4) --> equal(rng(J4::'a,K4,H4),rng(J4::'a,K4,I4))) &
1491 (\<forall>L4 M4. equal(L4::'a,M4) --> equal(range_of(L4),range_of(M4))) &
1492 (\<forall>N4 O4. equal(N4::'a,O4) --> equal(regular(N4),regular(O4))) &
1493 (\<forall>P4 Q4 R4 S4. equal(P4::'a,Q4) --> equal(restrct(P4::'a,R4,S4),restrct(Q4::'a,R4,S4))) &
1494 (\<forall>T4 V4 U4 W4. equal(T4::'a,U4) --> equal(restrct(V4::'a,T4,W4),restrct(V4::'a,U4,W4))) &
1495 (\<forall>X4 Z4 A5 Y4. equal(X4::'a,Y4) --> equal(restrct(Z4::'a,A5,X4),restrct(Z4::'a,A5,Y4))) &
1496 (\<forall>B5 C5. equal(B5::'a,C5) --> equal(rot(B5),rot(C5))) &
1497 (\<forall>D5 E5. equal(D5::'a,E5) --> equal(second(D5),second(E5))) &
1498 (\<forall>F5 G5. equal(F5::'a,G5) --> equal(singleton(F5),singleton(G5))) &
1499 (\<forall>H5 I5. equal(H5::'a,I5) --> equal(successor(H5),successor(I5))) &
1500 (\<forall>J5 K5. equal(J5::'a,K5) --> equal(sum_class(J5),sum_class(K5))) &
1501 (\<forall>L5 M5 N5. equal(L5::'a,M5) --> equal(union(L5::'a,N5),union(M5::'a,N5))) &
1502 (\<forall>O5 Q5 P5. equal(O5::'a,P5) --> equal(union(Q5::'a,O5),union(Q5::'a,P5))) &
1503 (\<forall>R5 S5 T5. equal(R5::'a,S5) --> equal(unordered_pair(R5::'a,T5),unordered_pair(S5::'a,T5))) &
1504 (\<forall>U5 W5 V5. equal(U5::'a,V5) --> equal(unordered_pair(W5::'a,U5),unordered_pair(W5::'a,V5))) &
1505 (\<forall>X5 Y5 Z5 A6. equal(X5::'a,Y5) & compatible(X5::'a,Z5,A6) --> compatible(Y5::'a,Z5,A6)) &
1506 (\<forall>B6 D6 C6 E6. equal(B6::'a,C6) & compatible(D6::'a,B6,E6) --> compatible(D6::'a,C6,E6)) &
1507 (\<forall>F6 H6 I6 G6. equal(F6::'a,G6) & compatible(H6::'a,I6,F6) --> compatible(H6::'a,I6,G6)) &
1508 (\<forall>J6 K6. equal(J6::'a,K6) & function(J6) --> function(K6)) &
1509 (\<forall>L6 M6 N6 O6. equal(L6::'a,M6) & homomorphism(L6::'a,N6,O6) --> homomorphism(M6::'a,N6,O6)) &
1510 (\<forall>P6 R6 Q6 S6. equal(P6::'a,Q6) & homomorphism(R6::'a,P6,S6) --> homomorphism(R6::'a,Q6,S6)) &
1511 (\<forall>T6 V6 W6 U6. equal(T6::'a,U6) & homomorphism(V6::'a,W6,T6) --> homomorphism(V6::'a,W6,U6)) &
1512 (\<forall>X6 Y6. equal(X6::'a,Y6) & inductive(X6) --> inductive(Y6)) &
1513 (\<forall>Z6 A7 B7. equal(Z6::'a,A7) & member(Z6::'a,B7) --> member(A7::'a,B7)) &
1514 (\<forall>C7 E7 D7. equal(C7::'a,D7) & member(E7::'a,C7) --> member(E7::'a,D7)) &
1515 (\<forall>F7 G7. equal(F7::'a,G7) & one_to_one(F7) --> one_to_one(G7)) &
1516 (\<forall>H7 I7. equal(H7::'a,I7) & operation(H7) --> operation(I7)) &
1517 (\<forall>J7 K7. equal(J7::'a,K7) & single_valued_class(J7) --> single_valued_class(K7)) &
1518 (\<forall>L7 M7 N7. equal(L7::'a,M7) & subclass(L7::'a,N7) --> subclass(M7::'a,N7)) &
1519 (\<forall>O7 Q7 P7. equal(O7::'a,P7) & subclass(Q7::'a,O7) --> subclass(Q7::'a,P7))"
1521 abbreviation "SET004_1_ax range_of function maps apply
1522 application_function singleton_relation element_relation complement
1523 intersection single_valued3 singleton image' domain single_valued2
1524 second single_valued1 identity_relation INVERSE not_subclass_element
1525 first domain_of domain_relation composition_function compos equal
1526 ordered_pair member universal_class cross_product compose_class
1528 (\<forall>X. subclass(compose_class(X),cross_product(universal_class::'a,universal_class))) &
1529 (\<forall>X Y Z. member(ordered_pair(Y::'a,Z),compose_class(X)) --> equal(compos(X::'a,Y),Z)) &
1530 (\<forall>Y Z X. member(ordered_pair(Y::'a,Z),cross_product(universal_class::'a,universal_class)) & equal(compos(X::'a,Y),Z) --> member(ordered_pair(Y::'a,Z),compose_class(X))) &
1531 (subclass(composition_function::'a,cross_product(universal_class::'a,cross_product(universal_class::'a,universal_class)))) &
1532 (\<forall>X Y Z. member(ordered_pair(X::'a,ordered_pair(Y::'a,Z)),composition_function) --> equal(compos(X::'a,Y),Z)) &
1533 (\<forall>X Y. member(ordered_pair(X::'a,Y),cross_product(universal_class::'a,universal_class)) --> member(ordered_pair(X::'a,ordered_pair(Y::'a,compos(X::'a,Y))),composition_function)) &
1534 (subclass(domain_relation::'a,cross_product(universal_class::'a,universal_class))) &
1535 (\<forall>X Y. member(ordered_pair(X::'a,Y),domain_relation) --> equal(domain_of(X),Y)) &
1536 (\<forall>X. member(X::'a,universal_class) --> member(ordered_pair(X::'a,domain_of(X)),domain_relation)) &
1537 (\<forall>X. equal(first(not_subclass_element(compos(X::'a,INVERSE(X)),identity_relation)),single_valued1(X))) &
1538 (\<forall>X. equal(second(not_subclass_element(compos(X::'a,INVERSE(X)),identity_relation)),single_valued2(X))) &
1539 (\<forall>X. equal(domain(X::'a,image'(INVERSE(X),singleton(single_valued1(X))),single_valued2(X)),single_valued3(X))) &
1540 (equal(intersection(complement(compos(element_relation::'a,complement(identity_relation))),element_relation),singleton_relation)) &
1541 (subclass(application_function::'a,cross_product(universal_class::'a,cross_product(universal_class::'a,universal_class)))) &
1542 (\<forall>Z Y X. member(ordered_pair(X::'a,ordered_pair(Y::'a,Z)),application_function) --> member(Y::'a,domain_of(X))) &
1543 (\<forall>X Y Z. member(ordered_pair(X::'a,ordered_pair(Y::'a,Z)),application_function) --> equal(apply(X::'a,Y),Z)) &
1544 (\<forall>Z X Y. member(ordered_pair(X::'a,ordered_pair(Y::'a,Z)),cross_product(universal_class::'a,cross_product(universal_class::'a,universal_class))) & member(Y::'a,domain_of(X)) --> member(ordered_pair(X::'a,ordered_pair(Y::'a,apply(X::'a,Y))),application_function)) &
1545 (\<forall>X Y Xf. maps(Xf::'a,X,Y) --> function(Xf)) &
1546 (\<forall>Y Xf X. maps(Xf::'a,X,Y) --> equal(domain_of(Xf),X)) &
1547 (\<forall>X Xf Y. maps(Xf::'a,X,Y) --> subclass(range_of(Xf),Y)) &
1548 (\<forall>Xf Y. function(Xf) & subclass(range_of(Xf),Y) --> maps(Xf::'a,domain_of(Xf),Y))"
1550 abbreviation "SET004_1_eq maps single_valued3 single_valued2 single_valued1 compose_class equal \<equiv>
1551 (\<forall>L M. equal(L::'a,M) --> equal(compose_class(L),compose_class(M))) &
1552 (\<forall>N2 O2. equal(N2::'a,O2) --> equal(single_valued1(N2),single_valued1(O2))) &
1553 (\<forall>P2 Q2. equal(P2::'a,Q2) --> equal(single_valued2(P2),single_valued2(Q2))) &
1554 (\<forall>R2 S2. equal(R2::'a,S2) --> equal(single_valued3(R2),single_valued3(S2))) &
1555 (\<forall>X2 Y2 Z2 A3. equal(X2::'a,Y2) & maps(X2::'a,Z2,A3) --> maps(Y2::'a,Z2,A3)) &
1556 (\<forall>B3 D3 C3 E3. equal(B3::'a,C3) & maps(D3::'a,B3,E3) --> maps(D3::'a,C3,E3)) &
1557 (\<forall>F3 H3 I3 G3. equal(F3::'a,G3) & maps(H3::'a,I3,F3) --> maps(H3::'a,I3,G3))"
1559 abbreviation "NUM004_0_ax integer_of omega ordinal_multiply
1560 add_relation ordinal_add recursion apply range_of union_of_range_map
1561 function recursion_equation_functions rest_relation rest_of
1562 limit_ordinals kind_1_ordinals successor_relation image'
1563 universal_class sum_class element_relation ordinal_numbers section
1564 not_well_ordering ordered_pair least member well_ordering singleton
1565 domain_of segment null_class intersection asymmetric compos transitive
1566 cross_product connected identity_relation complement restrct subclass
1567 irreflexive symmetrization_of INVERSE union equal \<equiv>
1568 (\<forall>X. equal(union(X::'a,INVERSE(X)),symmetrization_of(X))) &
1569 (\<forall>X Y. irreflexive(X::'a,Y) --> subclass(restrct(X::'a,Y,Y),complement(identity_relation))) &
1570 (\<forall>X Y. subclass(restrct(X::'a,Y,Y),complement(identity_relation)) --> irreflexive(X::'a,Y)) &
1571 (\<forall>Y X. connected(X::'a,Y) --> subclass(cross_product(Y::'a,Y),union(identity_relation::'a,symmetrization_of(X)))) &
1572 (\<forall>X Y. subclass(cross_product(Y::'a,Y),union(identity_relation::'a,symmetrization_of(X))) --> connected(X::'a,Y)) &
1573 (\<forall>Xr Y. transitive(Xr::'a,Y) --> subclass(compos(restrct(Xr::'a,Y,Y),restrct(Xr::'a,Y,Y)),restrct(Xr::'a,Y,Y))) &
1574 (\<forall>Xr Y. subclass(compos(restrct(Xr::'a,Y,Y),restrct(Xr::'a,Y,Y)),restrct(Xr::'a,Y,Y)) --> transitive(Xr::'a,Y)) &
1575 (\<forall>Xr Y. asymmetric(Xr::'a,Y) --> equal(restrct(intersection(Xr::'a,INVERSE(Xr)),Y,Y),null_class)) &
1576 (\<forall>Xr Y. equal(restrct(intersection(Xr::'a,INVERSE(Xr)),Y,Y),null_class) --> asymmetric(Xr::'a,Y)) &
1577 (\<forall>Xr Y Z. equal(segment(Xr::'a,Y,Z),domain_of(restrct(Xr::'a,Y,singleton(Z))))) &
1578 (\<forall>X Y. well_ordering(X::'a,Y) --> connected(X::'a,Y)) &
1579 (\<forall>Y Xr U. well_ordering(Xr::'a,Y) & subclass(U::'a,Y) --> equal(U::'a,null_class) | member(least(Xr::'a,U),U)) &
1580 (\<forall>Y V Xr U. well_ordering(Xr::'a,Y) & subclass(U::'a,Y) & member(V::'a,U) --> member(least(Xr::'a,U),U)) &
1581 (\<forall>Y Xr U. well_ordering(Xr::'a,Y) & subclass(U::'a,Y) --> equal(segment(Xr::'a,U,least(Xr::'a,U)),null_class)) &
1582 (\<forall>Y V U Xr. ~(well_ordering(Xr::'a,Y) & subclass(U::'a,Y) & member(V::'a,U) & member(ordered_pair(V::'a,least(Xr::'a,U)),Xr))) &
1583 (\<forall>Xr Y. connected(Xr::'a,Y) & equal(not_well_ordering(Xr::'a,Y),null_class) --> well_ordering(Xr::'a,Y)) &
1584 (\<forall>Xr Y. connected(Xr::'a,Y) --> subclass(not_well_ordering(Xr::'a,Y),Y) | well_ordering(Xr::'a,Y)) &
1585 (\<forall>V Xr Y. member(V::'a,not_well_ordering(Xr::'a,Y)) & equal(segment(Xr::'a,not_well_ordering(Xr::'a,Y),V),null_class) & connected(Xr::'a,Y) --> well_ordering(Xr::'a,Y)) &
1586 (\<forall>Xr Y Z. section(Xr::'a,Y,Z) --> subclass(Y::'a,Z)) &
1587 (\<forall>Xr Z Y. section(Xr::'a,Y,Z) --> subclass(domain_of(restrct(Xr::'a,Z,Y)),Y)) &
1588 (\<forall>Xr Y Z. subclass(Y::'a,Z) & subclass(domain_of(restrct(Xr::'a,Z,Y)),Y) --> section(Xr::'a,Y,Z)) &
1589 (\<forall>X. member(X::'a,ordinal_numbers) --> well_ordering(element_relation::'a,X)) &
1590 (\<forall>X. member(X::'a,ordinal_numbers) --> subclass(sum_class(X),X)) &
1591 (\<forall>X. well_ordering(element_relation::'a,X) & subclass(sum_class(X),X) & member(X::'a,universal_class) --> member(X::'a,ordinal_numbers)) &
1592 (\<forall>X. well_ordering(element_relation::'a,X) & subclass(sum_class(X),X) --> member(X::'a,ordinal_numbers) | equal(X::'a,ordinal_numbers)) &
1593 (equal(union(singleton(null_class),image'(successor_relation::'a,ordinal_numbers)),kind_1_ordinals)) &
1594 (equal(intersection(complement(kind_1_ordinals),ordinal_numbers),limit_ordinals)) &
1595 (\<forall>X. subclass(rest_of(X),cross_product(universal_class::'a,universal_class))) &
1596 (\<forall>V U X. member(ordered_pair(U::'a,V),rest_of(X)) --> member(U::'a,domain_of(X))) &
1597 (\<forall>X U V. member(ordered_pair(U::'a,V),rest_of(X)) --> equal(restrct(X::'a,U,universal_class),V)) &
1598 (\<forall>U V X. member(U::'a,domain_of(X)) & equal(restrct(X::'a,U,universal_class),V) --> member(ordered_pair(U::'a,V),rest_of(X))) &
1599 (subclass(rest_relation::'a,cross_product(universal_class::'a,universal_class))) &
1600 (\<forall>X Y. member(ordered_pair(X::'a,Y),rest_relation) --> equal(rest_of(X),Y)) &
1601 (\<forall>X. member(X::'a,universal_class) --> member(ordered_pair(X::'a,rest_of(X)),rest_relation)) &
1602 (\<forall>X Z. member(X::'a,recursion_equation_functions(Z)) --> function(Z)) &
1603 (\<forall>Z X. member(X::'a,recursion_equation_functions(Z)) --> function(X)) &
1604 (\<forall>Z X. member(X::'a,recursion_equation_functions(Z)) --> member(domain_of(X),ordinal_numbers)) &
1605 (\<forall>Z X. member(X::'a,recursion_equation_functions(Z)) --> equal(compos(Z::'a,rest_of(X)),X)) &
1606 (\<forall>X Z. function(Z) & function(X) & member(domain_of(X),ordinal_numbers) & equal(compos(Z::'a,rest_of(X)),X) --> member(X::'a,recursion_equation_functions(Z))) &
1607 (subclass(union_of_range_map::'a,cross_product(universal_class::'a,universal_class))) &
1608 (\<forall>X Y. member(ordered_pair(X::'a,Y),union_of_range_map) --> equal(sum_class(range_of(X)),Y)) &
1609 (\<forall>X Y. member(ordered_pair(X::'a,Y),cross_product(universal_class::'a,universal_class)) & equal(sum_class(range_of(X)),Y) --> member(ordered_pair(X::'a,Y),union_of_range_map)) &
1610 (\<forall>X Y. equal(apply(recursion(X::'a,successor_relation,union_of_range_map),Y),ordinal_add(X::'a,Y))) &
1611 (\<forall>X Y. equal(recursion(null_class::'a,apply(add_relation::'a,X),union_of_range_map),ordinal_multiply(X::'a,Y))) &
1612 (\<forall>X. member(X::'a,omega) --> equal(integer_of(X),X)) &
1613 (\<forall>X. member(X::'a,omega) | equal(integer_of(X),null_class))"
1615 abbreviation "NUM004_0_eq well_ordering transitive section irreflexive
1616 connected asymmetric symmetrization_of segment rest_of
1617 recursion_equation_functions recursion ordinal_multiply ordinal_add
1618 not_well_ordering least integer_of equal \<equiv>
1619 (\<forall>D E. equal(D::'a,E) --> equal(integer_of(D),integer_of(E))) &
1620 (\<forall>F' G H. equal(F'::'a,G) --> equal(least(F'::'a,H),least(G::'a,H))) &
1621 (\<forall>I' K' J. equal(I'::'a,J) --> equal(least(K'::'a,I'),least(K'::'a,J))) &
1622 (\<forall>L M N. equal(L::'a,M) --> equal(not_well_ordering(L::'a,N),not_well_ordering(M::'a,N))) &
1623 (\<forall>O' Q P. equal(O'::'a,P) --> equal(not_well_ordering(Q::'a,O'),not_well_ordering(Q::'a,P))) &
1624 (\<forall>R S' T'. equal(R::'a,S') --> equal(ordinal_add(R::'a,T'),ordinal_add(S'::'a,T'))) &
1625 (\<forall>U W V. equal(U::'a,V) --> equal(ordinal_add(W::'a,U),ordinal_add(W::'a,V))) &
1626 (\<forall>X Y Z. equal(X::'a,Y) --> equal(ordinal_multiply(X::'a,Z),ordinal_multiply(Y::'a,Z))) &
1627 (\<forall>A1 C1 B1. equal(A1::'a,B1) --> equal(ordinal_multiply(C1::'a,A1),ordinal_multiply(C1::'a,B1))) &
1628 (\<forall>F1 G1 H1 I1. equal(F1::'a,G1) --> equal(recursion(F1::'a,H1,I1),recursion(G1::'a,H1,I1))) &
1629 (\<forall>J1 L1 K1 M1. equal(J1::'a,K1) --> equal(recursion(L1::'a,J1,M1),recursion(L1::'a,K1,M1))) &
1630 (\<forall>N1 P1 Q1 O1. equal(N1::'a,O1) --> equal(recursion(P1::'a,Q1,N1),recursion(P1::'a,Q1,O1))) &
1631 (\<forall>R1 S1. equal(R1::'a,S1) --> equal(recursion_equation_functions(R1),recursion_equation_functions(S1))) &
1632 (\<forall>T1 U1. equal(T1::'a,U1) --> equal(rest_of(T1),rest_of(U1))) &
1633 (\<forall>V1 W1 X1 Y1. equal(V1::'a,W1) --> equal(segment(V1::'a,X1,Y1),segment(W1::'a,X1,Y1))) &
1634 (\<forall>Z1 B2 A2 C2. equal(Z1::'a,A2) --> equal(segment(B2::'a,Z1,C2),segment(B2::'a,A2,C2))) &
1635 (\<forall>D2 F2 G2 E2. equal(D2::'a,E2) --> equal(segment(F2::'a,G2,D2),segment(F2::'a,G2,E2))) &
1636 (\<forall>H2 I2. equal(H2::'a,I2) --> equal(symmetrization_of(H2),symmetrization_of(I2))) &
1637 (\<forall>J2 K2 L2. equal(J2::'a,K2) & asymmetric(J2::'a,L2) --> asymmetric(K2::'a,L2)) &
1638 (\<forall>M2 O2 N2. equal(M2::'a,N2) & asymmetric(O2::'a,M2) --> asymmetric(O2::'a,N2)) &
1639 (\<forall>P2 Q2 R2. equal(P2::'a,Q2) & connected(P2::'a,R2) --> connected(Q2::'a,R2)) &
1640 (\<forall>S2 U2 T2. equal(S2::'a,T2) & connected(U2::'a,S2) --> connected(U2::'a,T2)) &
1641 (\<forall>V2 W2 X2. equal(V2::'a,W2) & irreflexive(V2::'a,X2) --> irreflexive(W2::'a,X2)) &
1642 (\<forall>Y2 A3 Z2. equal(Y2::'a,Z2) & irreflexive(A3::'a,Y2) --> irreflexive(A3::'a,Z2)) &
1643 (\<forall>B3 C3 D3 E3. equal(B3::'a,C3) & section(B3::'a,D3,E3) --> section(C3::'a,D3,E3)) &
1644 (\<forall>F3 H3 G3 I3. equal(F3::'a,G3) & section(H3::'a,F3,I3) --> section(H3::'a,G3,I3)) &
1645 (\<forall>J3 L3 M3 K3. equal(J3::'a,K3) & section(L3::'a,M3,J3) --> section(L3::'a,M3,K3)) &
1646 (\<forall>N3 O3 P3. equal(N3::'a,O3) & transitive(N3::'a,P3) --> transitive(O3::'a,P3)) &
1647 (\<forall>Q3 S3 R3. equal(Q3::'a,R3) & transitive(S3::'a,Q3) --> transitive(S3::'a,R3)) &
1648 (\<forall>T3 U3 V3. equal(T3::'a,U3) & well_ordering(T3::'a,V3) --> well_ordering(U3::'a,V3)) &
1649 (\<forall>W3 Y3 X3. equal(W3::'a,X3) & well_ordering(Y3::'a,W3) --> well_ordering(Y3::'a,X3))"
1651 (*1345 inferences so far. Searching to depth 7. 23.3 secs. BIG*)
1653 "EQU001_0_ax equal &
1654 SET004_0_ax not_homomorphism2 not_homomorphism1
1655 homomorphism compatible operation cantor diagonalise subset_relation
1656 one_to_one choice apply regular function identity_relation
1657 single_valued_class compos powerClass sum_class omega inductive
1658 successor_relation successor image' rng domain range_of INVERSE flip
1659 rot domain_of null_class restrct difference union complement
1660 intersection element_relation second first cross_product ordered_pair
1661 singleton unordered_pair equal universal_class not_subclass_element
1663 SET004_0_eq subclass single_valued_class operation
1664 one_to_one member inductive homomorphism function compatible
1665 unordered_pair union sum_class successor singleton second rot restrct
1666 regular range_of rng powerClass ordered_pair not_subclass_element
1667 not_homomorphism2 not_homomorphism1 INVERSE intersection image' flip
1668 first domain_of domain difference diagonalise cross_product compos
1669 complement cantor apply equal &
1670 SET004_1_ax range_of function maps apply
1671 application_function singleton_relation element_relation complement
1672 intersection single_valued3 singleton image' domain single_valued2
1673 second single_valued1 identity_relation INVERSE not_subclass_element
1674 first domain_of domain_relation composition_function compos equal
1675 ordered_pair member universal_class cross_product compose_class
1677 SET004_1_eq maps single_valued3 single_valued2 single_valued1 compose_class equal &
1678 NUM004_0_ax integer_of omega ordinal_multiply
1679 add_relation ordinal_add recursion apply range_of union_of_range_map
1680 function recursion_equation_functions rest_relation rest_of
1681 limit_ordinals kind_1_ordinals successor_relation image'
1682 universal_class sum_class element_relation ordinal_numbers section
1683 not_well_ordering ordered_pair least member well_ordering singleton
1684 domain_of segment null_class intersection asymmetric compos transitive
1685 cross_product connected identity_relation complement restrct subclass
1686 irreflexive symmetrization_of INVERSE union equal &
1687 NUM004_0_eq well_ordering transitive section irreflexive
1688 connected asymmetric symmetrization_of segment rest_of
1689 recursion_equation_functions recursion ordinal_multiply ordinal_add
1690 not_well_ordering least integer_of equal &
1691 (~subclass(limit_ordinals::'a,ordinal_numbers)) --> False"
1695 (*0 inferences so far. Searching to depth 0. 16.8 secs. BIG*)
1697 "EQU001_0_ax equal &
1698 SET004_0_ax not_homomorphism2 not_homomorphism1
1699 homomorphism compatible operation cantor diagonalise subset_relation
1700 one_to_one choice apply regular function identity_relation
1701 single_valued_class compos powerClass sum_class omega inductive
1702 successor_relation successor image' rng domain range_of INVERSE flip
1703 rot domain_of null_class restrct difference union complement
1704 intersection element_relation second first cross_product ordered_pair
1705 singleton unordered_pair equal universal_class not_subclass_element
1707 SET004_0_eq subclass single_valued_class operation
1708 one_to_one member inductive homomorphism function compatible
1709 unordered_pair union sum_class successor singleton second rot restrct
1710 regular range_of rng powerClass ordered_pair not_subclass_element
1711 not_homomorphism2 not_homomorphism1 INVERSE intersection image' flip
1712 first domain_of domain difference diagonalise cross_product compos
1713 complement cantor apply equal &
1714 SET004_1_ax range_of function maps apply
1715 application_function singleton_relation element_relation complement
1716 intersection single_valued3 singleton image' domain single_valued2
1717 second single_valued1 identity_relation INVERSE not_subclass_element
1718 first domain_of domain_relation composition_function compos equal
1719 ordered_pair member universal_class cross_product compose_class
1721 SET004_1_eq maps single_valued3 single_valued2 single_valued1 compose_class equal &
1722 NUM004_0_ax integer_of omega ordinal_multiply
1723 add_relation ordinal_add recursion apply range_of union_of_range_map
1724 function recursion_equation_functions rest_relation rest_of
1725 limit_ordinals kind_1_ordinals successor_relation image'
1726 universal_class sum_class element_relation ordinal_numbers section
1727 not_well_ordering ordered_pair least member well_ordering singleton
1728 domain_of segment null_class intersection asymmetric compos transitive
1729 cross_product connected identity_relation complement restrct subclass
1730 irreflexive symmetrization_of INVERSE union equal &
1731 NUM004_0_eq well_ordering transitive section irreflexive
1732 connected asymmetric symmetrization_of segment rest_of
1733 recursion_equation_functions recursion ordinal_multiply ordinal_add
1734 not_well_ordering least integer_of equal &
1736 (~equal(recursion_equation_functions(z),null_class)) --> False"
1740 (*4868 inferences so far. Searching to depth 12. 4.3 secs*)
1742 "(\<forall>Situation1 Situation2. warm(Situation1) | cold(Situation2)) &
1743 (\<forall>Situation. at(a::'a,Situation) --> at(b::'a,walk(b::'a,Situation))) &
1744 (\<forall>Situation. at(a::'a,Situation) --> at(b::'a,drive(b::'a,Situation))) &
1745 (\<forall>Situation. at(b::'a,Situation) --> at(a::'a,walk(a::'a,Situation))) &
1746 (\<forall>Situation. at(b::'a,Situation) --> at(a::'a,drive(a::'a,Situation))) &
1747 (\<forall>Situation. cold(Situation) & at(b::'a,Situation) --> at(c::'a,skate(c::'a,Situation))) &
1748 (\<forall>Situation. cold(Situation) & at(c::'a,Situation) --> at(b::'a,skate(b::'a,Situation))) &
1749 (\<forall>Situation. warm(Situation) & at(b::'a,Situation) --> at(d::'a,climb(d::'a,Situation))) &
1750 (\<forall>Situation. warm(Situation) & at(d::'a,Situation) --> at(b::'a,climb(b::'a,Situation))) &
1751 (\<forall>Situation. at(c::'a,Situation) --> at(d::'a,go(d::'a,Situation))) &
1752 (\<forall>Situation. at(d::'a,Situation) --> at(c::'a,go(c::'a,Situation))) &
1753 (\<forall>Situation. at(c::'a,Situation) --> at(e::'a,go(e::'a,Situation))) &
1754 (\<forall>Situation. at(e::'a,Situation) --> at(c::'a,go(c::'a,Situation))) &
1755 (\<forall>Situation. at(d::'a,Situation) --> at(f::'a,go(f::'a,Situation))) &
1756 (\<forall>Situation. at(f::'a,Situation) --> at(d::'a,go(d::'a,Situation))) &
1758 (\<forall>S'. ~at(a::'a,S')) --> False"
1761 abbreviation "PLA001_0_ax putdown on pickup do holding table differ clear EMPTY and' holds \<equiv>
1762 (\<forall>X Y State. holds(X::'a,State) & holds(Y::'a,State) --> holds(and'(X::'a,Y),State)) &
1763 (\<forall>State X. holds(EMPTY::'a,State) & holds(clear(X),State) & differ(X::'a,table) --> holds(holding(X),do(pickup(X),State))) &
1764 (\<forall>Y X State. holds(on(X::'a,Y),State) & holds(clear(X),State) & holds(EMPTY::'a,State) --> holds(clear(Y),do(pickup(X),State))) &
1765 (\<forall>Y State X Z. holds(on(X::'a,Y),State) & differ(X::'a,Z) --> holds(on(X::'a,Y),do(pickup(Z),State))) &
1766 (\<forall>State X Z. holds(clear(X),State) & differ(X::'a,Z) --> holds(clear(X),do(pickup(Z),State))) &
1767 (\<forall>X Y State. holds(holding(X),State) & holds(clear(Y),State) --> holds(EMPTY::'a,do(putdown(X::'a,Y),State))) &
1768 (\<forall>X Y State. holds(holding(X),State) & holds(clear(Y),State) --> holds(on(X::'a,Y),do(putdown(X::'a,Y),State))) &
1769 (\<forall>X Y State. holds(holding(X),State) & holds(clear(Y),State) --> holds(clear(X),do(putdown(X::'a,Y),State))) &
1770 (\<forall>Z W X Y State. holds(on(X::'a,Y),State) --> holds(on(X::'a,Y),do(putdown(Z::'a,W),State))) &
1771 (\<forall>X State Z Y. holds(clear(Z),State) & differ(Z::'a,Y) --> holds(clear(Z),do(putdown(X::'a,Y),State)))"
1773 abbreviation "PLA001_1_ax EMPTY clear s0 on holds table d c b a differ \<equiv>
1774 (\<forall>Y X. differ(Y::'a,X) --> differ(X::'a,Y)) &
1778 (differ(a::'a,table)) &
1781 (differ(b::'a,table)) &
1783 (differ(c::'a,table)) &
1784 (differ(d::'a,table)) &
1785 (holds(on(a::'a,table),s0)) &
1786 (holds(on(b::'a,table),s0)) &
1787 (holds(on(c::'a,d),s0)) &
1788 (holds(on(d::'a,table),s0)) &
1789 (holds(clear(a),s0)) &
1790 (holds(clear(b),s0)) &
1791 (holds(clear(c),s0)) &
1792 (holds(EMPTY::'a,s0)) &
1793 (\<forall>State. holds(clear(table),State))"
1795 (*190 inferences so far. Searching to depth 10. 0.6 secs*)
1797 "PLA001_0_ax putdown on pickup do holding table differ clear EMPTY and' holds &
1798 PLA001_1_ax EMPTY clear s0 on holds table d c b a differ &
1799 (\<forall>State. ~holds(on(c::'a,table),State)) --> False"
1802 (*190 inferences so far. Searching to depth 10. 0.5 secs*)
1804 "PLA001_0_ax putdown on pickup do holding table differ clear EMPTY and' holds &
1805 PLA001_1_ax EMPTY clear s0 on holds table d c b a differ &
1806 (\<forall>State. ~holds(on(a::'a,c),State)) --> False"
1809 (*13732 inferences so far. Searching to depth 16. 9.8 secs*)
1811 "PLA001_0_ax putdown on pickup do holding table differ clear EMPTY and' holds &
1812 PLA001_1_ax EMPTY clear s0 on holds table d c b a differ &
1813 (\<forall>State. ~holds(and'(on(c::'a,d),on(a::'a,c)),State)) --> False"
1816 (*217 inferences so far. Searching to depth 13. 0.7 secs*)
1818 "PLA001_0_ax putdown on pickup do holding table differ clear EMPTY and' holds &
1819 PLA001_1_ax EMPTY clear s0 on holds table d c b a differ &
1820 (\<forall>State. ~holds(and'(on(a::'a,c),on(c::'a,d)),State)) --> False"
1823 (*948 inferences so far. Searching to depth 18. 1.1 secs*)
1825 "(\<forall>X Y Z. q1(X::'a,Y,Z) & mless_or_equal(X::'a,Y) --> q2(X::'a,Y,Z)) &
1826 (\<forall>X Y Z. q1(X::'a,Y,Z) --> mless_or_equal(X::'a,Y) | q3(X::'a,Y,Z)) &
1827 (\<forall>Z X Y. q2(X::'a,Y,Z) --> q4(X::'a,Y,Y)) &
1828 (\<forall>Z Y X. q3(X::'a,Y,Z) --> q4(X::'a,Y,X)) &
1829 (\<forall>X. mless_or_equal(X::'a,X)) &
1830 (\<forall>X Y. mless_or_equal(X::'a,Y) & mless_or_equal(Y::'a,X) --> equal(X::'a,Y)) &
1831 (\<forall>Y X Z. mless_or_equal(X::'a,Y) & mless_or_equal(Y::'a,Z) --> mless_or_equal(X::'a,Z)) &
1832 (\<forall>Y X. mless_or_equal(X::'a,Y) | mless_or_equal(Y::'a,X)) &
1833 (\<forall>X Y. equal(X::'a,Y) --> mless_or_equal(X::'a,Y)) &
1834 (\<forall>X Y Z. equal(X::'a,Y) & mless_or_equal(X::'a,Z) --> mless_or_equal(Y::'a,Z)) &
1835 (\<forall>X Z Y. equal(X::'a,Y) & mless_or_equal(Z::'a,X) --> mless_or_equal(Z::'a,Y)) &
1837 (\<forall>W. ~(q4(a::'a,b,W) & mless_or_equal(a::'a,W) & mless_or_equal(b::'a,W) & mless_or_equal(W::'a,a))) &
1838 (\<forall>W. ~(q4(a::'a,b,W) & mless_or_equal(a::'a,W) & mless_or_equal(b::'a,W) & mless_or_equal(W::'a,b))) --> False"
1841 (*PRV is now called SWV (software verification) *)
1842 abbreviation "SWV001_1_ax mless_THAN successor predecessor equal \<equiv>
1843 (\<forall>X. equal(predecessor(successor(X)),X)) &
1844 (\<forall>X. equal(successor(predecessor(X)),X)) &
1845 (\<forall>X Y. equal(predecessor(X),predecessor(Y)) --> equal(X::'a,Y)) &
1846 (\<forall>X Y. equal(successor(X),successor(Y)) --> equal(X::'a,Y)) &
1847 (\<forall>X. mless_THAN(predecessor(X),X)) &
1848 (\<forall>X. mless_THAN(X::'a,successor(X))) &
1849 (\<forall>X Y Z. mless_THAN(X::'a,Y) & mless_THAN(Y::'a,Z) --> mless_THAN(X::'a,Z)) &
1850 (\<forall>X Y. mless_THAN(X::'a,Y) | mless_THAN(Y::'a,X) | equal(X::'a,Y)) &
1851 (\<forall>X. ~mless_THAN(X::'a,X)) &
1852 (\<forall>Y X. ~(mless_THAN(X::'a,Y) & mless_THAN(Y::'a,X))) &
1853 (\<forall>Y X Z. equal(X::'a,Y) & mless_THAN(X::'a,Z) --> mless_THAN(Y::'a,Z)) &
1854 (\<forall>Y Z X. equal(X::'a,Y) & mless_THAN(Z::'a,X) --> mless_THAN(Z::'a,Y))"
1856 abbreviation "SWV001_0_eq a successor predecessor equal \<equiv>
1857 (\<forall>X Y. equal(X::'a,Y) --> equal(predecessor(X),predecessor(Y))) &
1858 (\<forall>X Y. equal(X::'a,Y) --> equal(successor(X),successor(Y))) &
1859 (\<forall>X Y. equal(X::'a,Y) --> equal(a(X),a(Y)))"
1861 (*21 inferences so far. Searching to depth 5. 0.4 secs*)
1863 "EQU001_0_ax equal &
1864 SWV001_1_ax mless_THAN successor predecessor equal &
1865 SWV001_0_eq a successor predecessor equal &
1866 (~mless_THAN(n::'a,j)) &
1867 (mless_THAN(k::'a,j)) &
1868 (~mless_THAN(k::'a,i)) &
1869 (mless_THAN(i::'a,n)) &
1870 (mless_THAN(a(j),a(k))) &
1871 (\<forall>X. mless_THAN(X::'a,j) & mless_THAN(a(X),a(k)) --> mless_THAN(X::'a,i)) &
1872 (\<forall>X. mless_THAN(One::'a,i) & mless_THAN(a(X),a(predecessor(i))) --> mless_THAN(X::'a,i) | mless_THAN(n::'a,X)) &
1873 (\<forall>X. ~(mless_THAN(One::'a,X) & mless_THAN(X::'a,i) & mless_THAN(a(X),a(predecessor(X))))) &
1874 (mless_THAN(j::'a,i)) --> False"
1877 (*584 inferences so far. Searching to depth 7. 1.1 secs*)
1879 "EQU001_0_ax equal &
1880 SWV001_1_ax mless_THAN successor predecessor equal &
1881 SWV001_0_eq a successor predecessor equal &
1882 (~mless_THAN(n::'a,k)) &
1883 (~mless_THAN(k::'a,l)) &
1884 (~mless_THAN(k::'a,i)) &
1885 (mless_THAN(l::'a,n)) &
1886 (mless_THAN(One::'a,l)) &
1887 (mless_THAN(a(k),a(predecessor(l)))) &
1888 (\<forall>X. mless_THAN(X::'a,successor(n)) & mless_THAN(a(X),a(k)) --> mless_THAN(X::'a,l)) &
1889 (\<forall>X. mless_THAN(One::'a,l) & mless_THAN(a(X),a(predecessor(l))) --> mless_THAN(X::'a,l) | mless_THAN(n::'a,X)) &
1890 (\<forall>X. ~(mless_THAN(One::'a,X) & mless_THAN(X::'a,l) & mless_THAN(a(X),a(predecessor(X))))) --> False"
1893 (*2343 inferences so far. Searching to depth 8. 3.5 secs*)
1895 "EQU001_0_ax equal &
1896 SWV001_1_ax mless_THAN successor predecessor equal &
1897 SWV001_0_eq a successor predecessor equal &
1898 (~mless_THAN(n::'a,m)) &
1899 (mless_THAN(i::'a,m)) &
1900 (mless_THAN(i::'a,n)) &
1901 (~mless_THAN(i::'a,One)) &
1902 (mless_THAN(a(i),a(m))) &
1903 (\<forall>X. mless_THAN(X::'a,successor(n)) & mless_THAN(a(X),a(m)) --> mless_THAN(X::'a,i)) &
1904 (\<forall>X. mless_THAN(One::'a,i) & mless_THAN(a(X),a(predecessor(i))) --> mless_THAN(X::'a,i) | mless_THAN(n::'a,X)) &
1905 (\<forall>X. ~(mless_THAN(One::'a,X) & mless_THAN(X::'a,i) & mless_THAN(a(X),a(predecessor(X))))) --> False"
1908 (*86 inferences so far. Searching to depth 14. 0.1 secs*)
1910 "(\<forall>Y X. mless_or_equal(X::'a,Y) | mless(Y::'a,X)) &
1912 (mless_or_equal(m::'a,p)) &
1913 (mless_or_equal(p::'a,q)) &
1914 (mless_or_equal(q::'a,n)) &
1915 (\<forall>X Y. mless_or_equal(m::'a,X) & mless(X::'a,i) & mless(j::'a,Y) & mless_or_equal(Y::'a,n) --> mless_or_equal(a(X),a(Y))) &
1916 (\<forall>X Y. mless_or_equal(m::'a,X) & mless_or_equal(X::'a,Y) & mless_or_equal(Y::'a,j) --> mless_or_equal(a(X),a(Y))) &
1917 (\<forall>X Y. mless_or_equal(i::'a,X) & mless_or_equal(X::'a,Y) & mless_or_equal(Y::'a,n) --> mless_or_equal(a(X),a(Y))) &
1918 (~mless_or_equal(a(p),a(q))) --> False"
1921 (*222 inferences so far. Searching to depth 8. 0.4 secs*)
1923 "(\<forall>X. equal_fruits(X::'a,X)) &
1924 (\<forall>X. equal_boxes(X::'a,X)) &
1925 (\<forall>X Y. ~(label(X::'a,Y) & contains(X::'a,Y))) &
1926 (\<forall>X. contains(boxa::'a,X) | contains(boxb::'a,X) | contains(boxc::'a,X)) &
1927 (\<forall>X. contains(X::'a,apples) | contains(X::'a,bananas) | contains(X::'a,oranges)) &
1928 (\<forall>X Y Z. contains(X::'a,Y) & contains(X::'a,Z) --> equal_fruits(Y::'a,Z)) &
1929 (\<forall>Y X Z. contains(X::'a,Y) & contains(Z::'a,Y) --> equal_boxes(X::'a,Z)) &
1930 (~equal_boxes(boxa::'a,boxb)) &
1931 (~equal_boxes(boxb::'a,boxc)) &
1932 (~equal_boxes(boxa::'a,boxc)) &
1933 (~equal_fruits(apples::'a,bananas)) &
1934 (~equal_fruits(bananas::'a,oranges)) &
1935 (~equal_fruits(apples::'a,oranges)) &
1936 (label(boxa::'a,apples)) &
1937 (label(boxb::'a,oranges)) &
1938 (label(boxc::'a,bananas)) &
1939 (contains(boxb::'a,apples)) &
1940 (~(contains(boxa::'a,bananas) & contains(boxc::'a,oranges))) --> False"
1943 (*35 inferences so far. Searching to depth 5. 3.2 secs*)
1945 "EQU001_0_ax equal &
1946 (\<forall>A B. equal(A::'a,B) --> equal(statement_by(A),statement_by(B))) &
1947 (\<forall>X. person(X) --> knight(X) | knave(X)) &
1948 (\<forall>X. ~(person(X) & knight(X) & knave(X))) &
1949 (\<forall>X Y. says(X::'a,Y) & a_truth(Y) --> a_truth(Y)) &
1950 (\<forall>X Y. ~(says(X::'a,Y) & equal(X::'a,Y))) &
1951 (\<forall>Y X. says(X::'a,Y) --> equal(Y::'a,statement_by(X))) &
1952 (\<forall>X Y. ~(person(X) & equal(X::'a,statement_by(Y)))) &
1953 (\<forall>X. person(X) & a_truth(statement_by(X)) --> knight(X)) &
1954 (\<forall>X. person(X) --> a_truth(statement_by(X)) | knave(X)) &
1955 (\<forall>X Y. equal(X::'a,Y) & knight(X) --> knight(Y)) &
1956 (\<forall>X Y. equal(X::'a,Y) & knave(X) --> knave(Y)) &
1957 (\<forall>X Y. equal(X::'a,Y) & person(X) --> person(Y)) &
1958 (\<forall>X Y Z. equal(X::'a,Y) & says(X::'a,Z) --> says(Y::'a,Z)) &
1959 (\<forall>X Z Y. equal(X::'a,Y) & says(Z::'a,X) --> says(Z::'a,Y)) &
1960 (\<forall>X Y. equal(X::'a,Y) & a_truth(X) --> a_truth(Y)) &
1961 (\<forall>X Y. knight(X) & says(X::'a,Y) --> a_truth(Y)) &
1962 (\<forall>X Y. ~(knave(X) & says(X::'a,Y) & a_truth(Y))) &
1965 (~equal(husband::'a,wife)) &
1966 (says(husband::'a,statement_by(husband))) &
1967 (a_truth(statement_by(husband)) & knight(husband) --> knight(wife)) &
1968 (knight(husband) --> a_truth(statement_by(husband))) &
1969 (a_truth(statement_by(husband)) | knight(wife)) &
1970 (knight(wife) --> a_truth(statement_by(husband))) &
1971 (~knight(husband)) --> False"
1974 (*121806 inferences so far. Searching to depth 17. 63.0 secs*)
1976 "(\<forall>X. a_truth(truthteller(X)) | a_truth(liar(X))) &
1977 (\<forall>X. ~(a_truth(truthteller(X)) & a_truth(liar(X)))) &
1978 (\<forall>Truthteller Statement. a_truth(truthteller(Truthteller)) & a_truth(says(Truthteller::'a,Statement)) --> a_truth(Statement)) &
1979 (\<forall>Liar Statement. ~(a_truth(liar(Liar)) & a_truth(says(Liar::'a,Statement)) & a_truth(Statement))) &
1980 (\<forall>Statement Truthteller. a_truth(Statement) & a_truth(says(Truthteller::'a,Statement)) --> a_truth(truthteller(Truthteller))) &
1981 (\<forall>Statement Liar. a_truth(says(Liar::'a,Statement)) --> a_truth(Statement) | a_truth(liar(Liar))) &
1982 (\<forall>Z X Y. people(X::'a,Y,Z) & a_truth(liar(X)) & a_truth(liar(Y)) --> a_truth(equal_type(X::'a,Y))) &
1983 (\<forall>Z X Y. people(X::'a,Y,Z) & a_truth(truthteller(X)) & a_truth(truthteller(Y)) --> a_truth(equal_type(X::'a,Y))) &
1984 (\<forall>X Y. a_truth(equal_type(X::'a,Y)) & a_truth(truthteller(X)) --> a_truth(truthteller(Y))) &
1985 (\<forall>X Y. a_truth(equal_type(X::'a,Y)) & a_truth(liar(X)) --> a_truth(liar(Y))) &
1986 (\<forall>X Y. a_truth(truthteller(X)) --> a_truth(equal_type(X::'a,Y)) | a_truth(liar(Y))) &
1987 (\<forall>X Y. a_truth(liar(X)) --> a_truth(equal_type(X::'a,Y)) | a_truth(truthteller(Y))) &
1988 (\<forall>Y X. a_truth(equal_type(X::'a,Y)) --> a_truth(equal_type(Y::'a,X))) &
1989 (\<forall>X Y. ask_1_if_2(X::'a,Y) & a_truth(truthteller(X)) & a_truth(Y) --> answer(yes)) &
1990 (\<forall>X Y. ask_1_if_2(X::'a,Y) & a_truth(truthteller(X)) --> a_truth(Y) | answer(no)) &
1991 (\<forall>X Y. ask_1_if_2(X::'a,Y) & a_truth(liar(X)) & a_truth(Y) --> answer(no)) &
1992 (\<forall>X Y. ask_1_if_2(X::'a,Y) & a_truth(liar(X)) --> a_truth(Y) | answer(yes)) &
1993 (people(b::'a,c,a)) &
1994 (people(a::'a,b,a)) &
1995 (people(a::'a,c,b)) &
1996 (people(c::'a,b,a)) &
1997 (a_truth(says(a::'a,equal_type(b::'a,c)))) &
1998 (ask_1_if_2(c::'a,equal_type(a::'a,b))) &
1999 (\<forall>Answer. ~answer(Answer)) --> False"
2003 (*621 inferences so far. Searching to depth 18. 0.2 secs*)
2005 "(\<forall>X. dances_on_tightropes(X) | eats_pennybuns(X) | old(X)) &
2006 (\<forall>X. pig(X) & liable_to_giddiness(X) --> treated_with_respect(X)) &
2007 (\<forall>X. wise(X) & balloonist(X) --> has_umbrella(X)) &
2008 (\<forall>X. ~(looks_ridiculous(X) & eats_pennybuns(X) & eats_lunch_in_public(X))) &
2009 (\<forall>X. balloonist(X) & young(X) --> liable_to_giddiness(X)) &
2010 (\<forall>X. fat(X) & looks_ridiculous(X) --> dances_on_tightropes(X) | eats_lunch_in_public(X)) &
2011 (\<forall>X. ~(liable_to_giddiness(X) & wise(X) & dances_on_tightropes(X))) &
2012 (\<forall>X. pig(X) & has_umbrella(X) --> looks_ridiculous(X)) &
2013 (\<forall>X. treated_with_respect(X) --> dances_on_tightropes(X) | fat(X)) &
2014 (\<forall>X. young(X) | old(X)) &
2015 (\<forall>X. ~(young(X) & old(X))) &
2019 (balloonist(piggy)) --> False"
2022 abbreviation "RNG001_0_ax equal additive_inverse add multiply product additive_identity sum \<equiv>
2023 (\<forall>X. sum(additive_identity::'a,X,X)) &
2024 (\<forall>X. sum(X::'a,additive_identity,X)) &
2025 (\<forall>X Y. product(X::'a,Y,multiply(X::'a,Y))) &
2026 (\<forall>X Y. sum(X::'a,Y,add(X::'a,Y))) &
2027 (\<forall>X. sum(additive_inverse(X),X,additive_identity)) &
2028 (\<forall>X. sum(X::'a,additive_inverse(X),additive_identity)) &
2029 (\<forall>Y U Z X V W. sum(X::'a,Y,U) & sum(Y::'a,Z,V) & sum(U::'a,Z,W) --> sum(X::'a,V,W)) &
2030 (\<forall>Y X V U Z W. sum(X::'a,Y,U) & sum(Y::'a,Z,V) & sum(X::'a,V,W) --> sum(U::'a,Z,W)) &
2031 (\<forall>Y X Z. sum(X::'a,Y,Z) --> sum(Y::'a,X,Z)) &
2032 (\<forall>Y U Z X V W. product(X::'a,Y,U) & product(Y::'a,Z,V) & product(U::'a,Z,W) --> product(X::'a,V,W)) &
2033 (\<forall>Y X V U Z W. product(X::'a,Y,U) & product(Y::'a,Z,V) & product(X::'a,V,W) --> product(U::'a,Z,W)) &
2034 (\<forall>Y Z X V3 V1 V2 V4. product(X::'a,Y,V1) & product(X::'a,Z,V2) & sum(Y::'a,Z,V3) & product(X::'a,V3,V4) --> sum(V1::'a,V2,V4)) &
2035 (\<forall>Y Z V1 V2 X V3 V4. product(X::'a,Y,V1) & product(X::'a,Z,V2) & sum(Y::'a,Z,V3) & sum(V1::'a,V2,V4) --> product(X::'a,V3,V4)) &
2036 (\<forall>Y Z V3 X V1 V2 V4. product(Y::'a,X,V1) & product(Z::'a,X,V2) & sum(Y::'a,Z,V3) & product(V3::'a,X,V4) --> sum(V1::'a,V2,V4)) &
2037 (\<forall>Y Z V1 V2 V3 X V4. product(Y::'a,X,V1) & product(Z::'a,X,V2) & sum(Y::'a,Z,V3) & sum(V1::'a,V2,V4) --> product(V3::'a,X,V4)) &
2038 (\<forall>X Y U V. sum(X::'a,Y,U) & sum(X::'a,Y,V) --> equal(U::'a,V)) &
2039 (\<forall>X Y U V. product(X::'a,Y,U) & product(X::'a,Y,V) --> equal(U::'a,V))"
2041 abbreviation "RNG001_0_eq product multiply sum add additive_inverse equal \<equiv>
2042 (\<forall>X Y. equal(X::'a,Y) --> equal(additive_inverse(X),additive_inverse(Y))) &
2043 (\<forall>X Y W. equal(X::'a,Y) --> equal(add(X::'a,W),add(Y::'a,W))) &
2044 (\<forall>X W Y. equal(X::'a,Y) --> equal(add(W::'a,X),add(W::'a,Y))) &
2045 (\<forall>X Y W Z. equal(X::'a,Y) & sum(X::'a,W,Z) --> sum(Y::'a,W,Z)) &
2046 (\<forall>X W Y Z. equal(X::'a,Y) & sum(W::'a,X,Z) --> sum(W::'a,Y,Z)) &
2047 (\<forall>X W Z Y. equal(X::'a,Y) & sum(W::'a,Z,X) --> sum(W::'a,Z,Y)) &
2048 (\<forall>X Y W. equal(X::'a,Y) --> equal(multiply(X::'a,W),multiply(Y::'a,W))) &
2049 (\<forall>X W Y. equal(X::'a,Y) --> equal(multiply(W::'a,X),multiply(W::'a,Y))) &
2050 (\<forall>X Y W Z. equal(X::'a,Y) & product(X::'a,W,Z) --> product(Y::'a,W,Z)) &
2051 (\<forall>X W Y Z. equal(X::'a,Y) & product(W::'a,X,Z) --> product(W::'a,Y,Z)) &
2052 (\<forall>X W Z Y. equal(X::'a,Y) & product(W::'a,Z,X) --> product(W::'a,Z,Y))"
2054 (*93620 inferences so far. Searching to depth 24. 65.9 secs*)
2056 "(\<forall>X. sum(additive_identity::'a,X,X)) &
2057 (\<forall>X. sum(additive_inverse(X),X,additive_identity)) &
2058 (\<forall>Y U Z X V W. sum(X::'a,Y,U) & sum(Y::'a,Z,V) & sum(U::'a,Z,W) --> sum(X::'a,V,W)) &
2059 (\<forall>Y X V U Z W. sum(X::'a,Y,U) & sum(Y::'a,Z,V) & sum(X::'a,V,W) --> sum(U::'a,Z,W)) &
2060 (\<forall>X Y. product(X::'a,Y,multiply(X::'a,Y))) &
2061 (\<forall>Y Z X V3 V1 V2 V4. product(X::'a,Y,V1) & product(X::'a,Z,V2) & sum(Y::'a,Z,V3) & product(X::'a,V3,V4) --> sum(V1::'a,V2,V4)) &
2062 (\<forall>Y Z V1 V2 X V3 V4. product(X::'a,Y,V1) & product(X::'a,Z,V2) & sum(Y::'a,Z,V3) & sum(V1::'a,V2,V4) --> product(X::'a,V3,V4)) &
2063 (~product(a::'a,additive_identity,additive_identity)) --> False"
2066 abbreviation "RNG_other_ax multiply add equal product additive_identity additive_inverse sum \<equiv>
2067 (\<forall>X. sum(X::'a,additive_inverse(X),additive_identity)) &
2068 (\<forall>Y U Z X V W. sum(X::'a,Y,U) & sum(Y::'a,Z,V) & sum(U::'a,Z,W) --> sum(X::'a,V,W)) &
2069 (\<forall>Y X V U Z W. sum(X::'a,Y,U) & sum(Y::'a,Z,V) & sum(X::'a,V,W) --> sum(U::'a,Z,W)) &
2070 (\<forall>Y X Z. sum(X::'a,Y,Z) --> sum(Y::'a,X,Z)) &
2071 (\<forall>Y U Z X V W. product(X::'a,Y,U) & product(Y::'a,Z,V) & product(U::'a,Z,W) --> product(X::'a,V,W)) &
2072 (\<forall>Y X V U Z W. product(X::'a,Y,U) & product(Y::'a,Z,V) & product(X::'a,V,W) --> product(U::'a,Z,W)) &
2073 (\<forall>Y Z X V3 V1 V2 V4. product(X::'a,Y,V1) & product(X::'a,Z,V2) & sum(Y::'a,Z,V3) & product(X::'a,V3,V4) --> sum(V1::'a,V2,V4)) &
2074 (\<forall>Y Z V1 V2 X V3 V4. product(X::'a,Y,V1) & product(X::'a,Z,V2) & sum(Y::'a,Z,V3) & sum(V1::'a,V2,V4) --> product(X::'a,V3,V4)) &
2075 (\<forall>Y Z V3 X V1 V2 V4. product(Y::'a,X,V1) & product(Z::'a,X,V2) & sum(Y::'a,Z,V3) & product(V3::'a,X,V4) --> sum(V1::'a,V2,V4)) &
2076 (\<forall>Y Z V1 V2 V3 X V4. product(Y::'a,X,V1) & product(Z::'a,X,V2) & sum(Y::'a,Z,V3) & sum(V1::'a,V2,V4) --> product(V3::'a,X,V4)) &
2077 (\<forall>X Y U V. sum(X::'a,Y,U) & sum(X::'a,Y,V) --> equal(U::'a,V)) &
2078 (\<forall>X Y U V. product(X::'a,Y,U) & product(X::'a,Y,V) --> equal(U::'a,V)) &
2079 (\<forall>X Y. equal(X::'a,Y) --> equal(additive_inverse(X),additive_inverse(Y))) &
2080 (\<forall>X Y W. equal(X::'a,Y) --> equal(add(X::'a,W),add(Y::'a,W))) &
2081 (\<forall>X Y W Z. equal(X::'a,Y) & sum(X::'a,W,Z) --> sum(Y::'a,W,Z)) &
2082 (\<forall>X W Y Z. equal(X::'a,Y) & sum(W::'a,X,Z) --> sum(W::'a,Y,Z)) &
2083 (\<forall>X W Z Y. equal(X::'a,Y) & sum(W::'a,Z,X) --> sum(W::'a,Z,Y)) &
2084 (\<forall>X Y W. equal(X::'a,Y) --> equal(multiply(X::'a,W),multiply(Y::'a,W))) &
2085 (\<forall>X Y W Z. equal(X::'a,Y) & product(X::'a,W,Z) --> product(Y::'a,W,Z)) &
2086 (\<forall>X W Y Z. equal(X::'a,Y) & product(W::'a,X,Z) --> product(W::'a,Y,Z)) &
2087 (\<forall>X W Z Y. equal(X::'a,Y) & product(W::'a,Z,X) --> product(W::'a,Z,Y))"
2090 (****************SLOW
2091 76385914 inferences so far. Searching to depth 18
2092 No proof after 5 1/2 hours! (griffon)
2095 "EQU001_0_ax equal &
2096 (\<forall>X. sum(additive_identity::'a,X,X)) &
2097 (\<forall>X. sum(X::'a,additive_identity,X)) &
2098 (\<forall>X Y. product(X::'a,Y,multiply(X::'a,Y))) &
2099 (\<forall>X Y. sum(X::'a,Y,add(X::'a,Y))) &
2100 (\<forall>X. sum(additive_inverse(X),X,additive_identity)) &
2101 RNG_other_ax multiply add equal product additive_identity additive_inverse sum &
2102 (~product(a::'a,additive_identity,additive_identity)) --> False"
2105 (*0 inferences so far. Searching to depth 0. 0.5 secs*)
2107 "EQU001_0_ax equal &
2108 (\<forall>A B C. equal(A::'a,B) --> equal(add(A::'a,C),add(B::'a,C))) &
2109 (\<forall>D F' E. equal(D::'a,E) --> equal(add(F'::'a,D),add(F'::'a,E))) &
2110 (\<forall>G H. equal(G::'a,H) --> equal(additive_inverse(G),additive_inverse(H))) &
2111 (\<forall>I' J K'. equal(I'::'a,J) --> equal(multiply(I'::'a,K'),multiply(J::'a,K'))) &
2112 (\<forall>L N M. equal(L::'a,M) --> equal(multiply(N::'a,L),multiply(N::'a,M))) &
2113 (\<forall>A B C D. equal(A::'a,B) --> equal(associator(A::'a,C,D),associator(B::'a,C,D))) &
2114 (\<forall>E G F' H. equal(E::'a,F') --> equal(associator(G::'a,E,H),associator(G::'a,F',H))) &
2115 (\<forall>I' K' L J. equal(I'::'a,J) --> equal(associator(K'::'a,L,I'),associator(K'::'a,L,J))) &
2116 (\<forall>M N O'. equal(M::'a,N) --> equal(commutator(M::'a,O'),commutator(N::'a,O'))) &
2117 (\<forall>P R Q. equal(P::'a,Q) --> equal(commutator(R::'a,P),commutator(R::'a,Q))) &
2118 (\<forall>Y X. equal(add(X::'a,Y),add(Y::'a,X))) &
2119 (\<forall>X Y Z. equal(add(add(X::'a,Y),Z),add(X::'a,add(Y::'a,Z)))) &
2120 (\<forall>X. equal(add(X::'a,additive_identity),X)) &
2121 (\<forall>X. equal(add(additive_identity::'a,X),X)) &
2122 (\<forall>X. equal(add(X::'a,additive_inverse(X)),additive_identity)) &
2123 (\<forall>X. equal(add(additive_inverse(X),X),additive_identity)) &
2124 (equal(additive_inverse(additive_identity),additive_identity)) &
2125 (\<forall>X Y. equal(add(X::'a,add(additive_inverse(X),Y)),Y)) &
2126 (\<forall>X Y. equal(additive_inverse(add(X::'a,Y)),add(additive_inverse(X),additive_inverse(Y)))) &
2127 (\<forall>X. equal(additive_inverse(additive_inverse(X)),X)) &
2128 (\<forall>X. equal(multiply(X::'a,additive_identity),additive_identity)) &
2129 (\<forall>X. equal(multiply(additive_identity::'a,X),additive_identity)) &
2130 (\<forall>X Y. equal(multiply(additive_inverse(X),additive_inverse(Y)),multiply(X::'a,Y))) &
2131 (\<forall>X Y. equal(multiply(X::'a,additive_inverse(Y)),additive_inverse(multiply(X::'a,Y)))) &
2132 (\<forall>X Y. equal(multiply(additive_inverse(X),Y),additive_inverse(multiply(X::'a,Y)))) &
2133 (\<forall>Y X Z. equal(multiply(X::'a,add(Y::'a,Z)),add(multiply(X::'a,Y),multiply(X::'a,Z)))) &
2134 (\<forall>X Y Z. equal(multiply(add(X::'a,Y),Z),add(multiply(X::'a,Z),multiply(Y::'a,Z)))) &
2135 (\<forall>X Y. equal(multiply(multiply(X::'a,Y),Y),multiply(X::'a,multiply(Y::'a,Y)))) &
2136 (\<forall>X Y Z. equal(associator(X::'a,Y,Z),add(multiply(multiply(X::'a,Y),Z),additive_inverse(multiply(X::'a,multiply(Y::'a,Z)))))) &
2137 (\<forall>X Y. equal(commutator(X::'a,Y),add(multiply(Y::'a,X),additive_inverse(multiply(X::'a,Y))))) &
2138 (\<forall>X Y. equal(multiply(multiply(associator(X::'a,X,Y),X),associator(X::'a,X,Y)),additive_identity)) &
2139 (~equal(multiply(multiply(associator(a::'a,a,b),a),associator(a::'a,a,b)),additive_identity)) --> False"
2142 (*202 inferences so far. Searching to depth 8. 0.6 secs*)
2144 "EQU001_0_ax equal &
2145 (\<forall>Y X. equal(add(X::'a,Y),add(Y::'a,X))) &
2146 (\<forall>X Y Z. equal(add(X::'a,add(Y::'a,Z)),add(add(X::'a,Y),Z))) &
2147 (\<forall>X. equal(add(additive_identity::'a,X),X)) &
2148 (\<forall>X. equal(add(X::'a,additive_identity),X)) &
2149 (\<forall>X. equal(multiply(additive_identity::'a,X),additive_identity)) &
2150 (\<forall>X. equal(multiply(X::'a,additive_identity),additive_identity)) &
2151 (\<forall>X. equal(add(additive_inverse(X),X),additive_identity)) &
2152 (\<forall>X. equal(add(X::'a,additive_inverse(X)),additive_identity)) &
2153 (\<forall>Y X Z. equal(multiply(X::'a,add(Y::'a,Z)),add(multiply(X::'a,Y),multiply(X::'a,Z)))) &
2154 (\<forall>X Y Z. equal(multiply(add(X::'a,Y),Z),add(multiply(X::'a,Z),multiply(Y::'a,Z)))) &
2155 (\<forall>X. equal(additive_inverse(additive_inverse(X)),X)) &
2156 (\<forall>X Y. equal(multiply(multiply(X::'a,Y),Y),multiply(X::'a,multiply(Y::'a,Y)))) &
2157 (\<forall>X Y. equal(multiply(multiply(X::'a,X),Y),multiply(X::'a,multiply(X::'a,Y)))) &
2158 (\<forall>X Y Z. equal(associator(X::'a,Y,Z),add(multiply(multiply(X::'a,Y),Z),additive_inverse(multiply(X::'a,multiply(Y::'a,Z)))))) &
2159 (\<forall>X Y. equal(commutator(X::'a,Y),add(multiply(Y::'a,X),additive_inverse(multiply(X::'a,Y))))) &
2160 (\<forall>D E F'. equal(D::'a,E) --> equal(add(D::'a,F'),add(E::'a,F'))) &
2161 (\<forall>G I' H. equal(G::'a,H) --> equal(add(I'::'a,G),add(I'::'a,H))) &
2162 (\<forall>J K'. equal(J::'a,K') --> equal(additive_inverse(J),additive_inverse(K'))) &
2163 (\<forall>L M N O'. equal(L::'a,M) --> equal(associator(L::'a,N,O'),associator(M::'a,N,O'))) &
2164 (\<forall>P R Q S'. equal(P::'a,Q) --> equal(associator(R::'a,P,S'),associator(R::'a,Q,S'))) &
2165 (\<forall>T' V W U. equal(T'::'a,U) --> equal(associator(V::'a,W,T'),associator(V::'a,W,U))) &
2166 (\<forall>X Y Z. equal(X::'a,Y) --> equal(commutator(X::'a,Z),commutator(Y::'a,Z))) &
2167 (\<forall>A1 C1 B1. equal(A1::'a,B1) --> equal(commutator(C1::'a,A1),commutator(C1::'a,B1))) &
2168 (\<forall>D1 E1 F1. equal(D1::'a,E1) --> equal(multiply(D1::'a,F1),multiply(E1::'a,F1))) &
2169 (\<forall>G1 I1 H1. equal(G1::'a,H1) --> equal(multiply(I1::'a,G1),multiply(I1::'a,H1))) &
2170 (~equal(associator(x::'a,x,y),additive_identity)) --> False"
2173 (*0 inferences so far. Searching to depth 0. 0.6 secs*)
2175 "EQU001_0_ax equal &
2176 (\<forall>X. equal(add(additive_identity::'a,X),X)) &
2177 (\<forall>X. equal(multiply(additive_identity::'a,X),additive_identity)) &
2178 (\<forall>X. equal(multiply(X::'a,additive_identity),additive_identity)) &
2179 (\<forall>X. equal(add(additive_inverse(X),X),additive_identity)) &
2180 (\<forall>X Y. equal(additive_inverse(add(X::'a,Y)),add(additive_inverse(X),additive_inverse(Y)))) &
2181 (\<forall>X. equal(additive_inverse(additive_inverse(X)),X)) &
2182 (\<forall>Y X Z. equal(multiply(X::'a,add(Y::'a,Z)),add(multiply(X::'a,Y),multiply(X::'a,Z)))) &
2183 (\<forall>X Y Z. equal(multiply(add(X::'a,Y),Z),add(multiply(X::'a,Z),multiply(Y::'a,Z)))) &
2184 (\<forall>X Y. equal(multiply(multiply(X::'a,Y),Y),multiply(X::'a,multiply(Y::'a,Y)))) &
2185 (\<forall>X Y. equal(multiply(multiply(X::'a,X),Y),multiply(X::'a,multiply(X::'a,Y)))) &
2186 (\<forall>X Y. equal(multiply(additive_inverse(X),Y),additive_inverse(multiply(X::'a,Y)))) &
2187 (\<forall>X Y. equal(multiply(X::'a,additive_inverse(Y)),additive_inverse(multiply(X::'a,Y)))) &
2188 (equal(additive_inverse(additive_identity),additive_identity)) &
2189 (\<forall>Y X. equal(add(X::'a,Y),add(Y::'a,X))) &
2190 (\<forall>X Y Z. equal(add(X::'a,add(Y::'a,Z)),add(add(X::'a,Y),Z))) &
2191 (\<forall>Z X Y. equal(add(X::'a,Z),add(Y::'a,Z)) --> equal(X::'a,Y)) &
2192 (\<forall>Z X Y. equal(add(Z::'a,X),add(Z::'a,Y)) --> equal(X::'a,Y)) &
2193 (\<forall>D E F'. equal(D::'a,E) --> equal(add(D::'a,F'),add(E::'a,F'))) &
2194 (\<forall>G I' H. equal(G::'a,H) --> equal(add(I'::'a,G),add(I'::'a,H))) &
2195 (\<forall>J K'. equal(J::'a,K') --> equal(additive_inverse(J),additive_inverse(K'))) &
2196 (\<forall>D1 E1 F1. equal(D1::'a,E1) --> equal(multiply(D1::'a,F1),multiply(E1::'a,F1))) &
2197 (\<forall>G1 I1 H1. equal(G1::'a,H1) --> equal(multiply(I1::'a,G1),multiply(I1::'a,H1))) &
2198 (\<forall>X Y Z. equal(associator(X::'a,Y,Z),add(multiply(multiply(X::'a,Y),Z),additive_inverse(multiply(X::'a,multiply(Y::'a,Z)))))) &
2199 (\<forall>L M N O'. equal(L::'a,M) --> equal(associator(L::'a,N,O'),associator(M::'a,N,O'))) &
2200 (\<forall>P R Q S'. equal(P::'a,Q) --> equal(associator(R::'a,P,S'),associator(R::'a,Q,S'))) &
2201 (\<forall>T' V W U. equal(T'::'a,U) --> equal(associator(V::'a,W,T'),associator(V::'a,W,U))) &
2202 (\<forall>X Y. ~equal(multiply(multiply(Y::'a,X),Y),multiply(Y::'a,multiply(X::'a,Y)))) &
2203 (\<forall>X Y Z. ~equal(associator(Y::'a,X,Z),additive_inverse(associator(X::'a,Y,Z)))) &
2204 (\<forall>X Y Z. ~equal(associator(Z::'a,Y,X),additive_inverse(associator(X::'a,Y,Z)))) &
2205 (~equal(multiply(multiply(cx::'a,multiply(cy::'a,cx)),cz),multiply(cx::'a,multiply(cy::'a,multiply(cx::'a,cz))))) --> False"
2208 (*209 inferences so far. Searching to depth 9. 1.2 secs*)
2210 "(\<forall>X. sum(X::'a,additive_identity,X)) &
2211 (\<forall>X Y. product(X::'a,Y,multiply(X::'a,Y))) &
2212 (\<forall>X Y. sum(X::'a,Y,add(X::'a,Y))) &
2213 RNG_other_ax multiply add equal product additive_identity additive_inverse sum &
2214 (\<forall>X. product(additive_identity::'a,X,additive_identity)) &
2215 (\<forall>X. product(X::'a,additive_identity,additive_identity)) &
2216 (\<forall>X Y. equal(X::'a,additive_identity) --> product(X::'a,h(X::'a,Y),Y)) &
2217 (product(a::'a,b,additive_identity)) &
2218 (~equal(a::'a,additive_identity)) &
2219 (~equal(b::'a,additive_identity)) --> False"
2222 (*2660 inferences so far. Searching to depth 10. 7.0 secs*)
2224 "EQU001_0_ax equal &
2225 RNG001_0_eq product multiply sum add additive_inverse equal &
2226 (\<forall>X. sum(additive_identity::'a,X,X)) &
2227 (\<forall>X. sum(X::'a,additive_identity,X)) &
2228 (\<forall>X Y. product(X::'a,Y,multiply(X::'a,Y))) &
2229 (\<forall>X Y. sum(X::'a,Y,add(X::'a,Y))) &
2230 (\<forall>X. sum(additive_inverse(X),X,additive_identity)) &
2231 (\<forall>X. sum(X::'a,additive_inverse(X),additive_identity)) &
2232 (\<forall>Y U Z X V W. sum(X::'a,Y,U) & sum(Y::'a,Z,V) & sum(U::'a,Z,W) --> sum(X::'a,V,W)) &
2233 (\<forall>Y X V U Z W. sum(X::'a,Y,U) & sum(Y::'a,Z,V) & sum(X::'a,V,W) --> sum(U::'a,Z,W)) &
2234 (\<forall>Y X Z. sum(X::'a,Y,Z) --> sum(Y::'a,X,Z)) &
2235 (\<forall>Y U Z X V W. product(X::'a,Y,U) & product(Y::'a,Z,V) & product(U::'a,Z,W) --> product(X::'a,V,W)) &
2236 (\<forall>Y X V U Z W. product(X::'a,Y,U) & product(Y::'a,Z,V) & product(X::'a,V,W) --> product(U::'a,Z,W)) &
2237 (\<forall>Y Z X V3 V1 V2 V4. product(X::'a,Y,V1) & product(X::'a,Z,V2) & sum(Y::'a,Z,V3) & product(X::'a,V3,V4) --> sum(V1::'a,V2,V4)) &
2238 (\<forall>Y Z V1 V2 X V3 V4. product(X::'a,Y,V1) & product(X::'a,Z,V2) & sum(Y::'a,Z,V3) & sum(V1::'a,V2,V4) --> product(X::'a,V3,V4)) &
2239 (\<forall>X Y U V. sum(X::'a,Y,U) & sum(X::'a,Y,V) --> equal(U::'a,V)) &
2240 (\<forall>X Y U V. product(X::'a,Y,U) & product(X::'a,Y,V) --> equal(U::'a,V)) &
2241 (\<forall>A. product(A::'a,multiplicative_identity,A)) &
2242 (\<forall>A. product(multiplicative_identity::'a,A,A)) &
2243 (\<forall>A. product(A::'a,h(A),multiplicative_identity) | equal(A::'a,additive_identity)) &
2244 (\<forall>A. product(h(A),A,multiplicative_identity) | equal(A::'a,additive_identity)) &
2245 (\<forall>B A C. product(A::'a,B,C) --> product(B::'a,A,C)) &
2246 (\<forall>A B. equal(A::'a,B) --> equal(h(A),h(B))) &
2248 (product(d::'a,a,additive_identity)) &
2249 (product(b::'a,a,l)) &
2250 (product(c::'a,a,n)) &
2251 (~sum(l::'a,n,additive_identity)) --> False"
2254 (*8991 inferences so far. Searching to depth 9. 22.2 secs*)
2256 "EQU001_0_ax equal &
2257 RNG001_0_ax equal additive_inverse add multiply product additive_identity sum &
2258 RNG001_0_eq product multiply sum add additive_inverse equal &
2259 (\<forall>A B. equal(A::'a,B) --> equal(h(A),h(B))) &
2260 (\<forall>A. product(additive_identity::'a,A,additive_identity)) &
2261 (\<forall>A. product(A::'a,additive_identity,additive_identity)) &
2262 (\<forall>A. product(A::'a,multiplicative_identity,A)) &
2263 (\<forall>A. product(multiplicative_identity::'a,A,A)) &
2264 (\<forall>A. product(A::'a,h(A),multiplicative_identity) | equal(A::'a,additive_identity)) &
2265 (\<forall>A. product(h(A),A,multiplicative_identity) | equal(A::'a,additive_identity)) &
2266 (product(a::'a,b,additive_identity)) &
2267 (~equal(a::'a,additive_identity)) &
2268 (~equal(b::'a,additive_identity)) --> False"
2271 (*101319 inferences so far. Searching to depth 14. 76.0 secs*)
2273 "EQU001_0_ax equal &
2274 (\<forall>Y X. equal(add(X::'a,Y),add(Y::'a,X))) &
2275 (\<forall>X Y Z. equal(add(add(X::'a,Y),Z),add(X::'a,add(Y::'a,Z)))) &
2276 (\<forall>Y X. equal(negate(add(negate(add(X::'a,Y)),negate(add(X::'a,negate(Y))))),X)) &
2277 (\<forall>A B C. equal(A::'a,B) --> equal(add(A::'a,C),add(B::'a,C))) &
2278 (\<forall>D F' E. equal(D::'a,E) --> equal(add(F'::'a,D),add(F'::'a,E))) &
2279 (\<forall>G H. equal(G::'a,H) --> equal(negate(G),negate(H))) &
2280 (equal(negate(add(a::'a,negate(b))),c)) &
2281 (~equal(negate(add(c::'a,negate(add(b::'a,a)))),a)) --> False"
2285 (*6933 inferences so far. Searching to depth 12. 5.1 secs*)
2287 "EQU001_0_ax equal &
2288 (\<forall>Y X. equal(add(X::'a,Y),add(Y::'a,X))) &
2289 (\<forall>X Y Z. equal(add(add(X::'a,Y),Z),add(X::'a,add(Y::'a,Z)))) &
2290 (\<forall>Y X. equal(negate(add(negate(add(X::'a,Y)),negate(add(X::'a,negate(Y))))),X)) &
2291 (\<forall>A B C. equal(A::'a,B) --> equal(add(A::'a,C),add(B::'a,C))) &
2292 (\<forall>D F' E. equal(D::'a,E) --> equal(add(F'::'a,D),add(F'::'a,E))) &
2293 (\<forall>G H. equal(G::'a,H) --> equal(negate(G),negate(H))) &
2294 (equal(negate(add(a::'a,b)),c)) &
2295 (~equal(negate(add(c::'a,negate(add(negate(b),a)))),a)) --> False"
2298 (*6614 inferences so far. Searching to depth 11. 20.4 secs*)
2300 "EQU001_0_ax equal &
2301 (\<forall>Y X. equal(add(X::'a,Y),add(Y::'a,X))) &
2302 (\<forall>X Y Z. equal(add(add(X::'a,Y),Z),add(X::'a,add(Y::'a,Z)))) &
2303 (\<forall>Y X. equal(negate(add(negate(add(X::'a,Y)),negate(add(X::'a,negate(Y))))),X)) &
2304 (\<forall>A B C. equal(A::'a,B) --> equal(add(A::'a,C),add(B::'a,C))) &
2305 (\<forall>D F' E. equal(D::'a,E) --> equal(add(F'::'a,D),add(F'::'a,E))) &
2306 (\<forall>G H. equal(G::'a,H) --> equal(negate(G),negate(H))) &
2307 (\<forall>J K' L. equal(J::'a,K') --> equal(multiply(J::'a,L),multiply(K'::'a,L))) &
2308 (\<forall>M O' N. equal(M::'a,N) --> equal(multiply(O'::'a,M),multiply(O'::'a,N))) &
2309 (\<forall>P Q. equal(P::'a,Q) --> equal(successor(P),successor(Q))) &
2310 (\<forall>R S'. equal(R::'a,S') & positive_integer(R) --> positive_integer(S')) &
2311 (\<forall>X. equal(multiply(One::'a,X),X)) &
2312 (\<forall>V X. positive_integer(X) --> equal(multiply(successor(V),X),add(X::'a,multiply(V::'a,X)))) &
2313 (positive_integer(One)) &
2314 (\<forall>X. positive_integer(X) --> positive_integer(successor(X))) &
2315 (equal(negate(add(d::'a,e)),negate(e))) &
2316 (positive_integer(k)) &
2317 (\<forall>Vk X Y. equal(negate(add(negate(Y),negate(add(X::'a,negate(Y))))),X) & positive_integer(Vk) --> equal(negate(add(Y::'a,multiply(Vk::'a,add(X::'a,negate(add(X::'a,negate(Y))))))),negate(Y))) &
2318 (~equal(negate(add(e::'a,multiply(k::'a,add(d::'a,negate(add(d::'a,negate(e))))))),negate(e))) --> False"
2321 (*14077 inferences so far. Searching to depth 11. 32.8 secs*)
2323 "EQU001_0_ax equal &
2324 (\<forall>Y X. equal(add(X::'a,Y),add(Y::'a,X))) &
2325 (\<forall>X Y Z. equal(add(add(X::'a,Y),Z),add(X::'a,add(Y::'a,Z)))) &
2326 (\<forall>Y X. equal(negate(add(negate(add(X::'a,Y)),negate(add(X::'a,negate(Y))))),X)) &
2327 (\<forall>A B C. equal(A::'a,B) --> equal(add(A::'a,C),add(B::'a,C))) &
2328 (\<forall>D F' E. equal(D::'a,E) --> equal(add(F'::'a,D),add(F'::'a,E))) &
2329 (\<forall>G H. equal(G::'a,H) --> equal(negate(G),negate(H))) &
2330 (\<forall>X Y. equal(negate(X),negate(Y)) --> equal(X::'a,Y)) &
2331 (~equal(add(negate(add(a::'a,negate(b))),negate(add(negate(a),negate(b)))),b)) --> False"
2334 (*35532 inferences so far. Searching to depth 19. 54.3 secs*)
2336 "(\<forall>Subset Element Superset. member(Element::'a,Subset) & subset(Subset::'a,Superset) --> member(Element::'a,Superset)) &
2337 (\<forall>Superset Subset. subset(Subset::'a,Superset) | member(member_of_1_not_of_2(Subset::'a,Superset),Subset)) &
2338 (\<forall>Subset Superset. member(member_of_1_not_of_2(Subset::'a,Superset),Superset) --> subset(Subset::'a,Superset)) &
2339 (\<forall>Subset Superset. equal_sets(Subset::'a,Superset) --> subset(Subset::'a,Superset)) &
2340 (\<forall>Subset Superset. equal_sets(Superset::'a,Subset) --> subset(Subset::'a,Superset)) &
2341 (\<forall>Set2 Set1. subset(Set1::'a,Set2) & subset(Set2::'a,Set1) --> equal_sets(Set2::'a,Set1)) &
2342 (\<forall>Set2 Intersection Element Set1. intersection(Set1::'a,Set2,Intersection) & member(Element::'a,Intersection) --> member(Element::'a,Set1)) &
2343 (\<forall>Set1 Intersection Element Set2. intersection(Set1::'a,Set2,Intersection) & member(Element::'a,Intersection) --> member(Element::'a,Set2)) &
2344 (\<forall>Set2 Set1 Element Intersection. intersection(Set1::'a,Set2,Intersection) & member(Element::'a,Set2) & member(Element::'a,Set1) --> member(Element::'a,Intersection)) &
2345 (\<forall>Set2 Intersection Set1. member(h(Set1::'a,Set2,Intersection),Intersection) | intersection(Set1::'a,Set2,Intersection) | member(h(Set1::'a,Set2,Intersection),Set1)) &
2346 (\<forall>Set1 Intersection Set2. member(h(Set1::'a,Set2,Intersection),Intersection) | intersection(Set1::'a,Set2,Intersection) | member(h(Set1::'a,Set2,Intersection),Set2)) &
2347 (\<forall>Set1 Set2 Intersection. member(h(Set1::'a,Set2,Intersection),Intersection) & member(h(Set1::'a,Set2,Intersection),Set2) & member(h(Set1::'a,Set2,Intersection),Set1) --> intersection(Set1::'a,Set2,Intersection)) &
2348 (intersection(a::'a,b,aIb)) &
2349 (intersection(b::'a,c,bIc)) &
2350 (intersection(a::'a,bIc,aIbIc)) &
2351 (~intersection(aIb::'a,c,aIbIc)) --> False"
2355 (*6450 inferences so far. Searching to depth 14. 4.2 secs*)
2357 "(\<forall>Subset Element Superset. member(Element::'a,Subset) & ssubset(Subset::'a,Superset) --> member(Element::'a,Superset)) &
2358 (\<forall>Superset Subset. ssubset(Subset::'a,Superset) | member(member_of_1_not_of_2(Subset::'a,Superset),Subset)) &
2359 (\<forall>Subset Superset. member(member_of_1_not_of_2(Subset::'a,Superset),Superset) --> ssubset(Subset::'a,Superset)) &
2360 (\<forall>Subset Superset. equal_sets(Subset::'a,Superset) --> ssubset(Subset::'a,Superset)) &
2361 (\<forall>Subset Superset. equal_sets(Superset::'a,Subset) --> ssubset(Subset::'a,Superset)) &
2362 (\<forall>Set2 Set1. ssubset(Set1::'a,Set2) & ssubset(Set2::'a,Set1) --> equal_sets(Set2::'a,Set1)) &
2363 (\<forall>Set2 Difference Element Set1. difference(Set1::'a,Set2,Difference) & member(Element::'a,Difference) --> member(Element::'a,Set1)) &
2364 (\<forall>Element A_set Set1 Set2. ~(member(Element::'a,Set1) & member(Element::'a,Set2) & difference(A_set::'a,Set1,Set2))) &
2365 (\<forall>Set1 Difference Element Set2. member(Element::'a,Set1) & difference(Set1::'a,Set2,Difference) --> member(Element::'a,Difference) | member(Element::'a,Set2)) &
2366 (\<forall>Set1 Set2 Difference. difference(Set1::'a,Set2,Difference) | member(k(Set1::'a,Set2,Difference),Set1) | member(k(Set1::'a,Set2,Difference),Difference)) &
2367 (\<forall>Set1 Set2 Difference. member(k(Set1::'a,Set2,Difference),Set2) --> member(k(Set1::'a,Set2,Difference),Difference) | difference(Set1::'a,Set2,Difference)) &
2368 (\<forall>Set1 Set2 Difference. member(k(Set1::'a,Set2,Difference),Difference) & member(k(Set1::'a,Set2,Difference),Set1) --> member(k(Set1::'a,Set2,Difference),Set2) | difference(Set1::'a,Set2,Difference)) &
2369 (ssubset(d::'a,a)) &
2370 (difference(b::'a,a,bDa)) &
2371 (difference(b::'a,d,bDd)) &
2372 (~ssubset(bDa::'a,bDd)) --> False"
2375 (*34726 inferences so far. Searching to depth 6. 2420 secs: 40 mins! BIG*)
2377 "EQU001_0_ax equal &
2378 (\<forall>Y X. member(X::'a,Y) --> little_set(X)) &
2379 (\<forall>X Y. little_set(f1(X::'a,Y)) | equal(X::'a,Y)) &
2380 (\<forall>X Y. member(f1(X::'a,Y),X) | member(f1(X::'a,Y),Y) | equal(X::'a,Y)) &
2381 (\<forall>X Y. member(f1(X::'a,Y),X) & member(f1(X::'a,Y),Y) --> equal(X::'a,Y)) &
2382 (\<forall>X U Y. member(U::'a,non_ordered_pair(X::'a,Y)) --> equal(U::'a,X) | equal(U::'a,Y)) &
2383 (\<forall>Y U X. little_set(U) & equal(U::'a,X) --> member(U::'a,non_ordered_pair(X::'a,Y))) &
2384 (\<forall>X U Y. little_set(U) & equal(U::'a,Y) --> member(U::'a,non_ordered_pair(X::'a,Y))) &
2385 (\<forall>X Y. little_set(non_ordered_pair(X::'a,Y))) &
2386 (\<forall>X. equal(singleton_set(X),non_ordered_pair(X::'a,X))) &
2387 (\<forall>X Y. equal(ordered_pair(X::'a,Y),non_ordered_pair(singleton_set(X),non_ordered_pair(X::'a,Y)))) &
2388 (\<forall>X. ordered_pair_predicate(X) --> little_set(f2(X))) &
2389 (\<forall>X. ordered_pair_predicate(X) --> little_set(f3(X))) &
2390 (\<forall>X. ordered_pair_predicate(X) --> equal(X::'a,ordered_pair(f2(X),f3(X)))) &
2391 (\<forall>X Y Z. little_set(Y) & little_set(Z) & equal(X::'a,ordered_pair(Y::'a,Z)) --> ordered_pair_predicate(X)) &
2392 (\<forall>Z X. member(Z::'a,first(X)) --> little_set(f4(Z::'a,X))) &
2393 (\<forall>Z X. member(Z::'a,first(X)) --> little_set(f5(Z::'a,X))) &
2394 (\<forall>Z X. member(Z::'a,first(X)) --> equal(X::'a,ordered_pair(f4(Z::'a,X),f5(Z::'a,X)))) &
2395 (\<forall>Z X. member(Z::'a,first(X)) --> member(Z::'a,f4(Z::'a,X))) &
2396 (\<forall>X V Z U. little_set(U) & little_set(V) & equal(X::'a,ordered_pair(U::'a,V)) & member(Z::'a,U) --> member(Z::'a,first(X))) &
2397 (\<forall>Z X. member(Z::'a,second(X)) --> little_set(f6(Z::'a,X))) &
2398 (\<forall>Z X. member(Z::'a,second(X)) --> little_set(f7(Z::'a,X))) &
2399 (\<forall>Z X. member(Z::'a,second(X)) --> equal(X::'a,ordered_pair(f6(Z::'a,X),f7(Z::'a,X)))) &
2400 (\<forall>Z X. member(Z::'a,second(X)) --> member(Z::'a,f7(Z::'a,X))) &
2401 (\<forall>X U Z V. little_set(U) & little_set(V) & equal(X::'a,ordered_pair(U::'a,V)) & member(Z::'a,V) --> member(Z::'a,second(X))) &
2402 (\<forall>Z. member(Z::'a,estin) --> ordered_pair_predicate(Z)) &
2403 (\<forall>Z. member(Z::'a,estin) --> member(first(Z),second(Z))) &
2404 (\<forall>Z. little_set(Z) & ordered_pair_predicate(Z) & member(first(Z),second(Z)) --> member(Z::'a,estin)) &
2405 (\<forall>Y Z X. member(Z::'a,intersection(X::'a,Y)) --> member(Z::'a,X)) &
2406 (\<forall>X Z Y. member(Z::'a,intersection(X::'a,Y)) --> member(Z::'a,Y)) &
2407 (\<forall>X Z Y. member(Z::'a,X) & member(Z::'a,Y) --> member(Z::'a,intersection(X::'a,Y))) &
2408 (\<forall>Z X. ~(member(Z::'a,complement(X)) & member(Z::'a,X))) &
2409 (\<forall>Z X. little_set(Z) --> member(Z::'a,complement(X)) | member(Z::'a,X)) &
2410 (\<forall>X Y. equal(union(X::'a,Y),complement(intersection(complement(X),complement(Y))))) &
2411 (\<forall>Z X. member(Z::'a,domain_of(X)) --> ordered_pair_predicate(f8(Z::'a,X))) &
2412 (\<forall>Z X. member(Z::'a,domain_of(X)) --> member(f8(Z::'a,X),X)) &
2413 (\<forall>Z X. member(Z::'a,domain_of(X)) --> equal(Z::'a,first(f8(Z::'a,X)))) &
2414 (\<forall>X Z Xp. little_set(Z) & ordered_pair_predicate(Xp) & member(Xp::'a,X) & equal(Z::'a,first(Xp)) --> member(Z::'a,domain_of(X))) &
2415 (\<forall>X Y Z. member(Z::'a,cross_product(X::'a,Y)) --> ordered_pair_predicate(Z)) &
2416 (\<forall>Y Z X. member(Z::'a,cross_product(X::'a,Y)) --> member(first(Z),X)) &
2417 (\<forall>X Z Y. member(Z::'a,cross_product(X::'a,Y)) --> member(second(Z),Y)) &
2418 (\<forall>X Z Y. little_set(Z) & ordered_pair_predicate(Z) & member(first(Z),X) & member(second(Z),Y) --> member(Z::'a,cross_product(X::'a,Y))) &
2419 (\<forall>X Z. member(Z::'a,inv1 X) --> ordered_pair_predicate(Z)) &
2420 (\<forall>Z X. member(Z::'a,inv1 X) --> member(ordered_pair(second(Z),first(Z)),X)) &
2421 (\<forall>Z X. little_set(Z) & ordered_pair_predicate(Z) & member(ordered_pair(second(Z),first(Z)),X) --> member(Z::'a,inv1 X)) &
2422 (\<forall>Z X. member(Z::'a,rot_right(X)) --> little_set(f9(Z::'a,X))) &
2423 (\<forall>Z X. member(Z::'a,rot_right(X)) --> little_set(f10(Z::'a,X))) &
2424 (\<forall>Z X. member(Z::'a,rot_right(X)) --> little_set(f11(Z::'a,X))) &
2425 (\<forall>Z X. member(Z::'a,rot_right(X)) --> equal(Z::'a,ordered_pair(f9(Z::'a,X),ordered_pair(f10(Z::'a,X),f11(Z::'a,X))))) &
2426 (\<forall>Z X. member(Z::'a,rot_right(X)) --> member(ordered_pair(f10(Z::'a,X),ordered_pair(f11(Z::'a,X),f9(Z::'a,X))),X)) &
2427 (\<forall>Z V W U X. little_set(Z) & little_set(U) & little_set(V) & little_set(W) & equal(Z::'a,ordered_pair(U::'a,ordered_pair(V::'a,W))) & member(ordered_pair(V::'a,ordered_pair(W::'a,U)),X) --> member(Z::'a,rot_right(X))) &
2428 (\<forall>Z X. member(Z::'a,flip_range_of(X)) --> little_set(f12(Z::'a,X))) &
2429 (\<forall>Z X. member(Z::'a,flip_range_of(X)) --> little_set(f13(Z::'a,X))) &
2430 (\<forall>Z X. member(Z::'a,flip_range_of(X)) --> little_set(f14(Z::'a,X))) &
2431 (\<forall>Z X. member(Z::'a,flip_range_of(X)) --> equal(Z::'a,ordered_pair(f12(Z::'a,X),ordered_pair(f13(Z::'a,X),f14(Z::'a,X))))) &
2432 (\<forall>Z X. member(Z::'a,flip_range_of(X)) --> member(ordered_pair(f12(Z::'a,X),ordered_pair(f14(Z::'a,X),f13(Z::'a,X))),X)) &
2433 (\<forall>Z U W V X. little_set(Z) & little_set(U) & little_set(V) & little_set(W) & equal(Z::'a,ordered_pair(U::'a,ordered_pair(V::'a,W))) & member(ordered_pair(U::'a,ordered_pair(W::'a,V)),X) --> member(Z::'a,flip_range_of(X))) &
2434 (\<forall>X. equal(successor(X),union(X::'a,singleton_set(X)))) &
2435 (\<forall>Z. ~member(Z::'a,empty_set)) &
2436 (\<forall>Z. little_set(Z) --> member(Z::'a,universal_set)) &
2437 (little_set(infinity)) &
2438 (member(empty_set::'a,infinity)) &
2439 (\<forall>X. member(X::'a,infinity) --> member(successor(X),infinity)) &
2440 (\<forall>Z X. member(Z::'a,sigma(X)) --> member(f16(Z::'a,X),X)) &
2441 (\<forall>Z X. member(Z::'a,sigma(X)) --> member(Z::'a,f16(Z::'a,X))) &
2442 (\<forall>X Z Y. member(Y::'a,X) & member(Z::'a,Y) --> member(Z::'a,sigma(X))) &
2443 (\<forall>U. little_set(U) --> little_set(sigma(U))) &
2444 (\<forall>X U Y. ssubset(X::'a,Y) & member(U::'a,X) --> member(U::'a,Y)) &
2445 (\<forall>Y X. ssubset(X::'a,Y) | member(f17(X::'a,Y),X)) &
2446 (\<forall>X Y. member(f17(X::'a,Y),Y) --> ssubset(X::'a,Y)) &
2447 (\<forall>X Y. proper_subset(X::'a,Y) --> ssubset(X::'a,Y)) &
2448 (\<forall>X Y. ~(proper_subset(X::'a,Y) & equal(X::'a,Y))) &
2449 (\<forall>X Y. ssubset(X::'a,Y) --> proper_subset(X::'a,Y) | equal(X::'a,Y)) &
2450 (\<forall>Z X. member(Z::'a,powerset(X)) --> ssubset(Z::'a,X)) &
2451 (\<forall>Z X. little_set(Z) & ssubset(Z::'a,X) --> member(Z::'a,powerset(X))) &
2452 (\<forall>U. little_set(U) --> little_set(powerset(U))) &
2453 (\<forall>Z X. relation(Z) & member(X::'a,Z) --> ordered_pair_predicate(X)) &
2454 (\<forall>Z. relation(Z) | member(f18(Z),Z)) &
2455 (\<forall>Z. ordered_pair_predicate(f18(Z)) --> relation(Z)) &
2456 (\<forall>U X V W. single_valued_set(X) & little_set(U) & little_set(V) & little_set(W) & member(ordered_pair(U::'a,V),X) & member(ordered_pair(U::'a,W),X) --> equal(V::'a,W)) &
2457 (\<forall>X. single_valued_set(X) | little_set(f19(X))) &
2458 (\<forall>X. single_valued_set(X) | little_set(f20(X))) &
2459 (\<forall>X. single_valued_set(X) | little_set(f21(X))) &
2460 (\<forall>X. single_valued_set(X) | member(ordered_pair(f19(X),f20(X)),X)) &
2461 (\<forall>X. single_valued_set(X) | member(ordered_pair(f19(X),f21(X)),X)) &
2462 (\<forall>X. equal(f20(X),f21(X)) --> single_valued_set(X)) &
2463 (\<forall>Xf. function(Xf) --> relation(Xf)) &
2464 (\<forall>Xf. function(Xf) --> single_valued_set(Xf)) &
2465 (\<forall>Xf. relation(Xf) & single_valued_set(Xf) --> function(Xf)) &
2466 (\<forall>Z X Xf. member(Z::'a,image'(X::'a,Xf)) --> ordered_pair_predicate(f22(Z::'a,X,Xf))) &
2467 (\<forall>Z X Xf. member(Z::'a,image'(X::'a,Xf)) --> member(f22(Z::'a,X,Xf),Xf)) &
2468 (\<forall>Z Xf X. member(Z::'a,image'(X::'a,Xf)) --> member(first(f22(Z::'a,X,Xf)),X)) &
2469 (\<forall>X Xf Z. member(Z::'a,image'(X::'a,Xf)) --> equal(second(f22(Z::'a,X,Xf)),Z)) &
2470 (\<forall>Xf X Y Z. little_set(Z) & ordered_pair_predicate(Y) & member(Y::'a,Xf) & member(first(Y),X) & equal(second(Y),Z) --> member(Z::'a,image'(X::'a,Xf))) &
2471 (\<forall>X Xf. little_set(X) & function(Xf) --> little_set(image'(X::'a,Xf))) &
2472 (\<forall>X U Y. ~(disjoint(X::'a,Y) & member(U::'a,X) & member(U::'a,Y))) &
2473 (\<forall>Y X. disjoint(X::'a,Y) | member(f23(X::'a,Y),X)) &
2474 (\<forall>X Y. disjoint(X::'a,Y) | member(f23(X::'a,Y),Y)) &
2475 (\<forall>X. equal(X::'a,empty_set) | member(f24(X),X)) &
2476 (\<forall>X. equal(X::'a,empty_set) | disjoint(f24(X),X)) &
2478 (\<forall>X. little_set(X) --> equal(X::'a,empty_set) | member(f26(X),X)) &
2479 (\<forall>X. little_set(X) --> equal(X::'a,empty_set) | member(ordered_pair(X::'a,f26(X)),f25)) &
2480 (\<forall>Z X. member(Z::'a,range_of(X)) --> ordered_pair_predicate(f27(Z::'a,X))) &
2481 (\<forall>Z X. member(Z::'a,range_of(X)) --> member(f27(Z::'a,X),X)) &
2482 (\<forall>Z X. member(Z::'a,range_of(X)) --> equal(Z::'a,second(f27(Z::'a,X)))) &
2483 (\<forall>X Z Xp. little_set(Z) & ordered_pair_predicate(Xp) & member(Xp::'a,X) & equal(Z::'a,second(Xp)) --> member(Z::'a,range_of(X))) &
2484 (\<forall>Z. member(Z::'a,identity_relation) --> ordered_pair_predicate(Z)) &
2485 (\<forall>Z. member(Z::'a,identity_relation) --> equal(first(Z),second(Z))) &
2486 (\<forall>Z. little_set(Z) & ordered_pair_predicate(Z) & equal(first(Z),second(Z)) --> member(Z::'a,identity_relation)) &
2487 (\<forall>X Y. equal(restrct(X::'a,Y),intersection(X::'a,cross_product(Y::'a,universal_set)))) &
2488 (\<forall>Xf. one_to_one_function(Xf) --> function(Xf)) &
2489 (\<forall>Xf. one_to_one_function(Xf) --> function(inv1 Xf)) &
2490 (\<forall>Xf. function(Xf) & function(inv1 Xf) --> one_to_one_function(Xf)) &
2491 (\<forall>Z Xf Y. member(Z::'a,apply(Xf::'a,Y)) --> ordered_pair_predicate(f28(Z::'a,Xf,Y))) &
2492 (\<forall>Z Y Xf. member(Z::'a,apply(Xf::'a,Y)) --> member(f28(Z::'a,Xf,Y),Xf)) &
2493 (\<forall>Z Xf Y. member(Z::'a,apply(Xf::'a,Y)) --> equal(first(f28(Z::'a,Xf,Y)),Y)) &
2494 (\<forall>Z Xf Y. member(Z::'a,apply(Xf::'a,Y)) --> member(Z::'a,second(f28(Z::'a,Xf,Y)))) &
2495 (\<forall>Xf Y Z W. ordered_pair_predicate(W) & member(W::'a,Xf) & equal(first(W),Y) & member(Z::'a,second(W)) --> member(Z::'a,apply(Xf::'a,Y))) &
2496 (\<forall>Xf X Y. equal(apply_to_two_arguments(Xf::'a,X,Y),apply(Xf::'a,ordered_pair(X::'a,Y)))) &
2497 (\<forall>X Y Xf. maps(Xf::'a,X,Y) --> function(Xf)) &
2498 (\<forall>Y Xf X. maps(Xf::'a,X,Y) --> equal(domain_of(Xf),X)) &
2499 (\<forall>X Xf Y. maps(Xf::'a,X,Y) --> ssubset(range_of(Xf),Y)) &
2500 (\<forall>X Xf Y. function(Xf) & equal(domain_of(Xf),X) & ssubset(range_of(Xf),Y) --> maps(Xf::'a,X,Y)) &
2501 (\<forall>Xf Xs. closed(Xs::'a,Xf) --> little_set(Xs)) &
2502 (\<forall>Xs Xf. closed(Xs::'a,Xf) --> little_set(Xf)) &
2503 (\<forall>Xf Xs. closed(Xs::'a,Xf) --> maps(Xf::'a,cross_product(Xs::'a,Xs),Xs)) &
2504 (\<forall>Xf Xs. little_set(Xs) & little_set(Xf) & maps(Xf::'a,cross_product(Xs::'a,Xs),Xs) --> closed(Xs::'a,Xf)) &
2505 (\<forall>Z Xf Xg. member(Z::'a,composition(Xf::'a,Xg)) --> little_set(f29(Z::'a,Xf,Xg))) &
2506 (\<forall>Z Xf Xg. member(Z::'a,composition(Xf::'a,Xg)) --> little_set(f30(Z::'a,Xf,Xg))) &
2507 (\<forall>Z Xf Xg. member(Z::'a,composition(Xf::'a,Xg)) --> little_set(f31(Z::'a,Xf,Xg))) &
2508 (\<forall>Z Xf Xg. member(Z::'a,composition(Xf::'a,Xg)) --> equal(Z::'a,ordered_pair(f29(Z::'a,Xf,Xg),f30(Z::'a,Xf,Xg)))) &
2509 (\<forall>Z Xg Xf. member(Z::'a,composition(Xf::'a,Xg)) --> member(ordered_pair(f29(Z::'a,Xf,Xg),f31(Z::'a,Xf,Xg)),Xf)) &
2510 (\<forall>Z Xf Xg. member(Z::'a,composition(Xf::'a,Xg)) --> member(ordered_pair(f31(Z::'a,Xf,Xg),f30(Z::'a,Xf,Xg)),Xg)) &
2511 (\<forall>Z X Xf W Y Xg. little_set(Z) & little_set(X) & little_set(Y) & little_set(W) & equal(Z::'a,ordered_pair(X::'a,Y)) & member(ordered_pair(X::'a,W),Xf) & member(ordered_pair(W::'a,Y),Xg) --> member(Z::'a,composition(Xf::'a,Xg))) &
2512 (\<forall>Xh Xs2 Xf2 Xs1 Xf1. homomorphism(Xh::'a,Xs1,Xf1,Xs2,Xf2) --> closed(Xs1::'a,Xf1)) &
2513 (\<forall>Xh Xs1 Xf1 Xs2 Xf2. homomorphism(Xh::'a,Xs1,Xf1,Xs2,Xf2) --> closed(Xs2::'a,Xf2)) &
2514 (\<forall>Xf1 Xf2 Xh Xs1 Xs2. homomorphism(Xh::'a,Xs1,Xf1,Xs2,Xf2) --> maps(Xh::'a,Xs1,Xs2)) &
2515 (\<forall>Xs2 Xs1 Xf1 Xf2 X Xh Y. homomorphism(Xh::'a,Xs1,Xf1,Xs2,Xf2) & member(X::'a,Xs1) & member(Y::'a,Xs1) --> equal(apply(Xh::'a,apply_to_two_arguments(Xf1::'a,X,Y)),apply_to_two_arguments(Xf2::'a,apply(Xh::'a,X),apply(Xh::'a,Y)))) &
2516 (\<forall>Xh Xf1 Xs2 Xf2 Xs1. closed(Xs1::'a,Xf1) & closed(Xs2::'a,Xf2) & maps(Xh::'a,Xs1,Xs2) --> homomorphism(Xh::'a,Xs1,Xf1,Xs2,Xf2) | member(f32(Xh::'a,Xs1,Xf1,Xs2,Xf2),Xs1)) &
2517 (\<forall>Xh Xf1 Xs2 Xf2 Xs1. closed(Xs1::'a,Xf1) & closed(Xs2::'a,Xf2) & maps(Xh::'a,Xs1,Xs2) --> homomorphism(Xh::'a,Xs1,Xf1,Xs2,Xf2) | member(f33(Xh::'a,Xs1,Xf1,Xs2,Xf2),Xs1)) &
2518 (\<forall>Xh Xs1 Xf1 Xs2 Xf2. closed(Xs1::'a,Xf1) & closed(Xs2::'a,Xf2) & maps(Xh::'a,Xs1,Xs2) & equal(apply(Xh::'a,apply_to_two_arguments(Xf1::'a,f32(Xh::'a,Xs1,Xf1,Xs2,Xf2),f33(Xh::'a,Xs1,Xf1,Xs2,Xf2))),apply_to_two_arguments(Xf2::'a,apply(Xh::'a,f32(Xh::'a,Xs1,Xf1,Xs2,Xf2)),apply(Xh::'a,f33(Xh::'a,Xs1,Xf1,Xs2,Xf2)))) --> homomorphism(Xh::'a,Xs1,Xf1,Xs2,Xf2)) &
2519 (\<forall>A B C. equal(A::'a,B) --> equal(f1(A::'a,C),f1(B::'a,C))) &
2520 (\<forall>D F' E. equal(D::'a,E) --> equal(f1(F'::'a,D),f1(F'::'a,E))) &
2521 (\<forall>A2 B2. equal(A2::'a,B2) --> equal(f2(A2),f2(B2))) &
2522 (\<forall>G4 H4. equal(G4::'a,H4) --> equal(f3(G4),f3(H4))) &
2523 (\<forall>O7 P7 Q7. equal(O7::'a,P7) --> equal(f4(O7::'a,Q7),f4(P7::'a,Q7))) &
2524 (\<forall>R7 T7 S7. equal(R7::'a,S7) --> equal(f4(T7::'a,R7),f4(T7::'a,S7))) &
2525 (\<forall>U7 V7 W7. equal(U7::'a,V7) --> equal(f5(U7::'a,W7),f5(V7::'a,W7))) &
2526 (\<forall>X7 Z7 Y7. equal(X7::'a,Y7) --> equal(f5(Z7::'a,X7),f5(Z7::'a,Y7))) &
2527 (\<forall>A8 B8 C8. equal(A8::'a,B8) --> equal(f6(A8::'a,C8),f6(B8::'a,C8))) &
2528 (\<forall>D8 F8 E8. equal(D8::'a,E8) --> equal(f6(F8::'a,D8),f6(F8::'a,E8))) &
2529 (\<forall>G8 H8 I8. equal(G8::'a,H8) --> equal(f7(G8::'a,I8),f7(H8::'a,I8))) &
2530 (\<forall>J8 L8 K8. equal(J8::'a,K8) --> equal(f7(L8::'a,J8),f7(L8::'a,K8))) &
2531 (\<forall>M8 N8 O8. equal(M8::'a,N8) --> equal(f8(M8::'a,O8),f8(N8::'a,O8))) &
2532 (\<forall>P8 R8 Q8. equal(P8::'a,Q8) --> equal(f8(R8::'a,P8),f8(R8::'a,Q8))) &
2533 (\<forall>S8 T8 U8. equal(S8::'a,T8) --> equal(f9(S8::'a,U8),f9(T8::'a,U8))) &
2534 (\<forall>V8 X8 W8. equal(V8::'a,W8) --> equal(f9(X8::'a,V8),f9(X8::'a,W8))) &
2535 (\<forall>G H I'. equal(G::'a,H) --> equal(f10(G::'a,I'),f10(H::'a,I'))) &
2536 (\<forall>J L K'. equal(J::'a,K') --> equal(f10(L::'a,J),f10(L::'a,K'))) &
2537 (\<forall>M N O'. equal(M::'a,N) --> equal(f11(M::'a,O'),f11(N::'a,O'))) &
2538 (\<forall>P R Q. equal(P::'a,Q) --> equal(f11(R::'a,P),f11(R::'a,Q))) &
2539 (\<forall>S' T' U. equal(S'::'a,T') --> equal(f12(S'::'a,U),f12(T'::'a,U))) &
2540 (\<forall>V X W. equal(V::'a,W) --> equal(f12(X::'a,V),f12(X::'a,W))) &
2541 (\<forall>Y Z A1. equal(Y::'a,Z) --> equal(f13(Y::'a,A1),f13(Z::'a,A1))) &
2542 (\<forall>B1 D1 C1. equal(B1::'a,C1) --> equal(f13(D1::'a,B1),f13(D1::'a,C1))) &
2543 (\<forall>E1 F1 G1. equal(E1::'a,F1) --> equal(f14(E1::'a,G1),f14(F1::'a,G1))) &
2544 (\<forall>H1 J1 I1. equal(H1::'a,I1) --> equal(f14(J1::'a,H1),f14(J1::'a,I1))) &
2545 (\<forall>K1 L1 M1. equal(K1::'a,L1) --> equal(f16(K1::'a,M1),f16(L1::'a,M1))) &
2546 (\<forall>N1 P1 O1. equal(N1::'a,O1) --> equal(f16(P1::'a,N1),f16(P1::'a,O1))) &
2547 (\<forall>Q1 R1 S1. equal(Q1::'a,R1) --> equal(f17(Q1::'a,S1),f17(R1::'a,S1))) &
2548 (\<forall>T1 V1 U1. equal(T1::'a,U1) --> equal(f17(V1::'a,T1),f17(V1::'a,U1))) &
2549 (\<forall>W1 X1. equal(W1::'a,X1) --> equal(f18(W1),f18(X1))) &
2550 (\<forall>Y1 Z1. equal(Y1::'a,Z1) --> equal(f19(Y1),f19(Z1))) &
2551 (\<forall>C2 D2. equal(C2::'a,D2) --> equal(f20(C2),f20(D2))) &
2552 (\<forall>E2 F2. equal(E2::'a,F2) --> equal(f21(E2),f21(F2))) &
2553 (\<forall>G2 H2 I2 J2. equal(G2::'a,H2) --> equal(f22(G2::'a,I2,J2),f22(H2::'a,I2,J2))) &
2554 (\<forall>K2 M2 L2 N2. equal(K2::'a,L2) --> equal(f22(M2::'a,K2,N2),f22(M2::'a,L2,N2))) &
2555 (\<forall>O2 Q2 R2 P2. equal(O2::'a,P2) --> equal(f22(Q2::'a,R2,O2),f22(Q2::'a,R2,P2))) &
2556 (\<forall>S2 T2 U2. equal(S2::'a,T2) --> equal(f23(S2::'a,U2),f23(T2::'a,U2))) &
2557 (\<forall>V2 X2 W2. equal(V2::'a,W2) --> equal(f23(X2::'a,V2),f23(X2::'a,W2))) &
2558 (\<forall>Y2 Z2. equal(Y2::'a,Z2) --> equal(f24(Y2),f24(Z2))) &
2559 (\<forall>A3 B3. equal(A3::'a,B3) --> equal(f26(A3),f26(B3))) &
2560 (\<forall>C3 D3 E3. equal(C3::'a,D3) --> equal(f27(C3::'a,E3),f27(D3::'a,E3))) &
2561 (\<forall>F3 H3 G3. equal(F3::'a,G3) --> equal(f27(H3::'a,F3),f27(H3::'a,G3))) &
2562 (\<forall>I3 J3 K3 L3. equal(I3::'a,J3) --> equal(f28(I3::'a,K3,L3),f28(J3::'a,K3,L3))) &
2563 (\<forall>M3 O3 N3 P3. equal(M3::'a,N3) --> equal(f28(O3::'a,M3,P3),f28(O3::'a,N3,P3))) &
2564 (\<forall>Q3 S3 T3 R3. equal(Q3::'a,R3) --> equal(f28(S3::'a,T3,Q3),f28(S3::'a,T3,R3))) &
2565 (\<forall>U3 V3 W3 X3. equal(U3::'a,V3) --> equal(f29(U3::'a,W3,X3),f29(V3::'a,W3,X3))) &
2566 (\<forall>Y3 A4 Z3 B4. equal(Y3::'a,Z3) --> equal(f29(A4::'a,Y3,B4),f29(A4::'a,Z3,B4))) &
2567 (\<forall>C4 E4 F4 D4. equal(C4::'a,D4) --> equal(f29(E4::'a,F4,C4),f29(E4::'a,F4,D4))) &
2568 (\<forall>I4 J4 K4 L4. equal(I4::'a,J4) --> equal(f30(I4::'a,K4,L4),f30(J4::'a,K4,L4))) &
2569 (\<forall>M4 O4 N4 P4. equal(M4::'a,N4) --> equal(f30(O4::'a,M4,P4),f30(O4::'a,N4,P4))) &
2570 (\<forall>Q4 S4 T4 R4. equal(Q4::'a,R4) --> equal(f30(S4::'a,T4,Q4),f30(S4::'a,T4,R4))) &
2571 (\<forall>U4 V4 W4 X4. equal(U4::'a,V4) --> equal(f31(U4::'a,W4,X4),f31(V4::'a,W4,X4))) &
2572 (\<forall>Y4 A5 Z4 B5. equal(Y4::'a,Z4) --> equal(f31(A5::'a,Y4,B5),f31(A5::'a,Z4,B5))) &
2573 (\<forall>C5 E5 F5 D5. equal(C5::'a,D5) --> equal(f31(E5::'a,F5,C5),f31(E5::'a,F5,D5))) &
2574 (\<forall>G5 H5 I5 J5 K5 L5. equal(G5::'a,H5) --> equal(f32(G5::'a,I5,J5,K5,L5),f32(H5::'a,I5,J5,K5,L5))) &
2575 (\<forall>M5 O5 N5 P5 Q5 R5. equal(M5::'a,N5) --> equal(f32(O5::'a,M5,P5,Q5,R5),f32(O5::'a,N5,P5,Q5,R5))) &
2576 (\<forall>S5 U5 V5 T5 W5 X5. equal(S5::'a,T5) --> equal(f32(U5::'a,V5,S5,W5,X5),f32(U5::'a,V5,T5,W5,X5))) &
2577 (\<forall>Y5 A6 B6 C6 Z5 D6. equal(Y5::'a,Z5) --> equal(f32(A6::'a,B6,C6,Y5,D6),f32(A6::'a,B6,C6,Z5,D6))) &
2578 (\<forall>E6 G6 H6 I6 J6 F6. equal(E6::'a,F6) --> equal(f32(G6::'a,H6,I6,J6,E6),f32(G6::'a,H6,I6,J6,F6))) &
2579 (\<forall>K6 L6 M6 N6 O6 P6. equal(K6::'a,L6) --> equal(f33(K6::'a,M6,N6,O6,P6),f33(L6::'a,M6,N6,O6,P6))) &
2580 (\<forall>Q6 S6 R6 T6 U6 V6. equal(Q6::'a,R6) --> equal(f33(S6::'a,Q6,T6,U6,V6),f33(S6::'a,R6,T6,U6,V6))) &
2581 (\<forall>W6 Y6 Z6 X6 A7 B7. equal(W6::'a,X6) --> equal(f33(Y6::'a,Z6,W6,A7,B7),f33(Y6::'a,Z6,X6,A7,B7))) &
2582 (\<forall>C7 E7 F7 G7 D7 H7. equal(C7::'a,D7) --> equal(f33(E7::'a,F7,G7,C7,H7),f33(E7::'a,F7,G7,D7,H7))) &
2583 (\<forall>I7 K7 L7 M7 N7 J7. equal(I7::'a,J7) --> equal(f33(K7::'a,L7,M7,N7,I7),f33(K7::'a,L7,M7,N7,J7))) &
2584 (\<forall>A B C. equal(A::'a,B) --> equal(apply(A::'a,C),apply(B::'a,C))) &
2585 (\<forall>D F' E. equal(D::'a,E) --> equal(apply(F'::'a,D),apply(F'::'a,E))) &
2586 (\<forall>G H I' J. equal(G::'a,H) --> equal(apply_to_two_arguments(G::'a,I',J),apply_to_two_arguments(H::'a,I',J))) &
2587 (\<forall>K' M L N. equal(K'::'a,L) --> equal(apply_to_two_arguments(M::'a,K',N),apply_to_two_arguments(M::'a,L,N))) &
2588 (\<forall>O' Q R P. equal(O'::'a,P) --> equal(apply_to_two_arguments(Q::'a,R,O'),apply_to_two_arguments(Q::'a,R,P))) &
2589 (\<forall>S' T'. equal(S'::'a,T') --> equal(complement(S'),complement(T'))) &
2590 (\<forall>U V W. equal(U::'a,V) --> equal(composition(U::'a,W),composition(V::'a,W))) &
2591 (\<forall>X Z Y. equal(X::'a,Y) --> equal(composition(Z::'a,X),composition(Z::'a,Y))) &
2592 (\<forall>A1 B1. equal(A1::'a,B1) --> equal(inv1 A1,inv1 B1)) &
2593 (\<forall>C1 D1 E1. equal(C1::'a,D1) --> equal(cross_product(C1::'a,E1),cross_product(D1::'a,E1))) &
2594 (\<forall>F1 H1 G1. equal(F1::'a,G1) --> equal(cross_product(H1::'a,F1),cross_product(H1::'a,G1))) &
2595 (\<forall>I1 J1. equal(I1::'a,J1) --> equal(domain_of(I1),domain_of(J1))) &
2596 (\<forall>I10 J10. equal(I10::'a,J10) --> equal(first(I10),first(J10))) &
2597 (\<forall>Q10 R10. equal(Q10::'a,R10) --> equal(flip_range_of(Q10),flip_range_of(R10))) &
2598 (\<forall>S10 T10 U10. equal(S10::'a,T10) --> equal(image'(S10::'a,U10),image'(T10::'a,U10))) &
2599 (\<forall>V10 X10 W10. equal(V10::'a,W10) --> equal(image'(X10::'a,V10),image'(X10::'a,W10))) &
2600 (\<forall>Y10 Z10 A11. equal(Y10::'a,Z10) --> equal(intersection(Y10::'a,A11),intersection(Z10::'a,A11))) &
2601 (\<forall>B11 D11 C11. equal(B11::'a,C11) --> equal(intersection(D11::'a,B11),intersection(D11::'a,C11))) &
2602 (\<forall>E11 F11 G11. equal(E11::'a,F11) --> equal(non_ordered_pair(E11::'a,G11),non_ordered_pair(F11::'a,G11))) &
2603 (\<forall>H11 J11 I11. equal(H11::'a,I11) --> equal(non_ordered_pair(J11::'a,H11),non_ordered_pair(J11::'a,I11))) &
2604 (\<forall>K11 L11 M11. equal(K11::'a,L11) --> equal(ordered_pair(K11::'a,M11),ordered_pair(L11::'a,M11))) &
2605 (\<forall>N11 P11 O11. equal(N11::'a,O11) --> equal(ordered_pair(P11::'a,N11),ordered_pair(P11::'a,O11))) &
2606 (\<forall>Q11 R11. equal(Q11::'a,R11) --> equal(powerset(Q11),powerset(R11))) &
2607 (\<forall>S11 T11. equal(S11::'a,T11) --> equal(range_of(S11),range_of(T11))) &
2608 (\<forall>U11 V11 W11. equal(U11::'a,V11) --> equal(restrct(U11::'a,W11),restrct(V11::'a,W11))) &
2609 (\<forall>X11 Z11 Y11. equal(X11::'a,Y11) --> equal(restrct(Z11::'a,X11),restrct(Z11::'a,Y11))) &
2610 (\<forall>A12 B12. equal(A12::'a,B12) --> equal(rot_right(A12),rot_right(B12))) &
2611 (\<forall>C12 D12. equal(C12::'a,D12) --> equal(second(C12),second(D12))) &
2612 (\<forall>K12 L12. equal(K12::'a,L12) --> equal(sigma(K12),sigma(L12))) &
2613 (\<forall>M12 N12. equal(M12::'a,N12) --> equal(singleton_set(M12),singleton_set(N12))) &
2614 (\<forall>O12 P12. equal(O12::'a,P12) --> equal(successor(O12),successor(P12))) &
2615 (\<forall>Q12 R12 S12. equal(Q12::'a,R12) --> equal(union(Q12::'a,S12),union(R12::'a,S12))) &
2616 (\<forall>T12 V12 U12. equal(T12::'a,U12) --> equal(union(V12::'a,T12),union(V12::'a,U12))) &
2617 (\<forall>W12 X12 Y12. equal(W12::'a,X12) & closed(W12::'a,Y12) --> closed(X12::'a,Y12)) &
2618 (\<forall>Z12 B13 A13. equal(Z12::'a,A13) & closed(B13::'a,Z12) --> closed(B13::'a,A13)) &
2619 (\<forall>C13 D13 E13. equal(C13::'a,D13) & disjoint(C13::'a,E13) --> disjoint(D13::'a,E13)) &
2620 (\<forall>F13 H13 G13. equal(F13::'a,G13) & disjoint(H13::'a,F13) --> disjoint(H13::'a,G13)) &
2621 (\<forall>I13 J13. equal(I13::'a,J13) & function(I13) --> function(J13)) &
2622 (\<forall>K13 L13 M13 N13 O13 P13. equal(K13::'a,L13) & homomorphism(K13::'a,M13,N13,O13,P13) --> homomorphism(L13::'a,M13,N13,O13,P13)) &
2623 (\<forall>Q13 S13 R13 T13 U13 V13. equal(Q13::'a,R13) & homomorphism(S13::'a,Q13,T13,U13,V13) --> homomorphism(S13::'a,R13,T13,U13,V13)) &
2624 (\<forall>W13 Y13 Z13 X13 A14 B14. equal(W13::'a,X13) & homomorphism(Y13::'a,Z13,W13,A14,B14) --> homomorphism(Y13::'a,Z13,X13,A14,B14)) &
2625 (\<forall>C14 E14 F14 G14 D14 H14. equal(C14::'a,D14) & homomorphism(E14::'a,F14,G14,C14,H14) --> homomorphism(E14::'a,F14,G14,D14,H14)) &
2626 (\<forall>I14 K14 L14 M14 N14 J14. equal(I14::'a,J14) & homomorphism(K14::'a,L14,M14,N14,I14) --> homomorphism(K14::'a,L14,M14,N14,J14)) &
2627 (\<forall>O14 P14. equal(O14::'a,P14) & little_set(O14) --> little_set(P14)) &
2628 (\<forall>Q14 R14 S14 T14. equal(Q14::'a,R14) & maps(Q14::'a,S14,T14) --> maps(R14::'a,S14,T14)) &
2629 (\<forall>U14 W14 V14 X14. equal(U14::'a,V14) & maps(W14::'a,U14,X14) --> maps(W14::'a,V14,X14)) &
2630 (\<forall>Y14 A15 B15 Z14. equal(Y14::'a,Z14) & maps(A15::'a,B15,Y14) --> maps(A15::'a,B15,Z14)) &
2631 (\<forall>C15 D15 E15. equal(C15::'a,D15) & member(C15::'a,E15) --> member(D15::'a,E15)) &
2632 (\<forall>F15 H15 G15. equal(F15::'a,G15) & member(H15::'a,F15) --> member(H15::'a,G15)) &
2633 (\<forall>I15 J15. equal(I15::'a,J15) & one_to_one_function(I15) --> one_to_one_function(J15)) &
2634 (\<forall>K15 L15. equal(K15::'a,L15) & ordered_pair_predicate(K15) --> ordered_pair_predicate(L15)) &
2635 (\<forall>M15 N15 O15. equal(M15::'a,N15) & proper_subset(M15::'a,O15) --> proper_subset(N15::'a,O15)) &
2636 (\<forall>P15 R15 Q15. equal(P15::'a,Q15) & proper_subset(R15::'a,P15) --> proper_subset(R15::'a,Q15)) &
2637 (\<forall>S15 T15. equal(S15::'a,T15) & relation(S15) --> relation(T15)) &
2638 (\<forall>U15 V15. equal(U15::'a,V15) & single_valued_set(U15) --> single_valued_set(V15)) &
2639 (\<forall>W15 X15 Y15. equal(W15::'a,X15) & ssubset(W15::'a,Y15) --> ssubset(X15::'a,Y15)) &
2640 (\<forall>Z15 B16 A16. equal(Z15::'a,A16) & ssubset(B16::'a,Z15) --> ssubset(B16::'a,A16)) &
2641 (~little_set(ordered_pair(a::'a,b))) --> False"
2645 (*13 inferences so far. Searching to depth 8. 0 secs*)
2647 "(\<forall>Y X. ~(element(X::'a,a) & element(X::'a,Y) & element(Y::'a,X))) &
2648 (\<forall>X. element(X::'a,f(X)) | element(X::'a,a)) &
2649 (\<forall>X. element(f(X),X) | element(X::'a,a)) --> False"
2652 (*33 inferences so far. Searching to depth 9. 0.2 secs*)
2654 "(\<forall>X Z Y. set_equal(X::'a,Y) & element(Z::'a,X) --> element(Z::'a,Y)) &
2655 (\<forall>Y Z X. set_equal(X::'a,Y) & element(Z::'a,Y) --> element(Z::'a,X)) &
2656 (\<forall>X Y. element(f(X::'a,Y),X) | element(f(X::'a,Y),Y) | set_equal(X::'a,Y)) &
2657 (\<forall>X Y. element(f(X::'a,Y),Y) & element(f(X::'a,Y),X) --> set_equal(X::'a,Y)) &
2658 (set_equal(a::'a,b) | set_equal(b::'a,a)) &
2659 (~(set_equal(b::'a,a) & set_equal(a::'a,b))) --> False"
2662 (*311 inferences so far. Searching to depth 12. 0.1 secs*)
2664 "(\<forall>A. p(A::'a,a) | p(A::'a,f(A))) &
2665 (\<forall>A. p(A::'a,a) | p(f(A),A)) &
2666 (\<forall>A B. ~(p(A::'a,B) & p(B::'a,A) & p(B::'a,a))) --> False"
2669 (*30 inferences so far. Searching to depth 6. 0.2 secs*)
2671 "EQU001_0_ax equal &
2672 (equal(a::'a,b) | equal(c::'a,d)) &
2673 (equal(a::'a,c) | equal(b::'a,d)) &
2675 (~equal(b::'a,c)) --> False"
2678 (*1897410 inferences so far. Searching to depth 48
2679 206s, nearly 4 mins on griffon.*)
2681 "(\<forall>X Y. f(w(X),g(X::'a,Y)) --> f(X::'a,g(X::'a,Y))) &
2682 (\<forall>X Y. f(X::'a,g(X::'a,Y)) --> f(w(X),g(X::'a,Y))) &
2683 (\<forall>Y X. f(X::'a,g(X::'a,Y)) & f(Y::'a,g(X::'a,Y)) --> f(g(X::'a,Y),Y) | f(g(X::'a,Y),w(X))) &
2684 (\<forall>Y X. f(g(X::'a,Y),Y) & f(Y::'a,g(X::'a,Y)) --> f(X::'a,g(X::'a,Y)) | f(g(X::'a,Y),w(X))) &
2685 (\<forall>Y X. f(X::'a,g(X::'a,Y)) | f(g(X::'a,Y),Y) | f(Y::'a,g(X::'a,Y)) | f(g(X::'a,Y),w(X))) &
2686 (\<forall>Y X. f(X::'a,g(X::'a,Y)) & f(g(X::'a,Y),Y) --> f(Y::'a,g(X::'a,Y)) | f(g(X::'a,Y),w(X))) &
2687 (\<forall>Y X. f(X::'a,g(X::'a,Y)) & f(g(X::'a,Y),w(X)) --> f(g(X::'a,Y),Y) | f(Y::'a,g(X::'a,Y))) &
2688 (\<forall>Y X. f(g(X::'a,Y),Y) & f(g(X::'a,Y),w(X)) --> f(X::'a,g(X::'a,Y)) | f(Y::'a,g(X::'a,Y))) &
2689 (\<forall>Y X. f(Y::'a,g(X::'a,Y)) & f(g(X::'a,Y),w(X)) --> f(X::'a,g(X::'a,Y)) | f(g(X::'a,Y),Y)) &
2690 (\<forall>Y X. ~(f(X::'a,g(X::'a,Y)) & f(g(X::'a,Y),Y) & f(Y::'a,g(X::'a,Y)) & f(g(X::'a,Y),w(X)))) --> False"
2693 (*398 inferences so far. Searching to depth 12. 0.4 secs*)
2696 (\<forall>X Y. f(X::'a,Y) --> f(b::'a,z(X::'a,Y)) | f(Y::'a,z(X::'a,Y))) &
2697 (\<forall>X Y. f(X::'a,Y) | f(z(X::'a,Y),z(X::'a,Y))) &
2698 (\<forall>X Y. f(b::'a,z(X::'a,Y)) | f(X::'a,z(X::'a,Y)) | f(z(X::'a,Y),z(X::'a,Y))) &
2699 (\<forall>X Y. f(b::'a,z(X::'a,Y)) & f(X::'a,z(X::'a,Y)) --> f(z(X::'a,Y),z(X::'a,Y))) &
2700 (\<forall>X Y. ~(f(X::'a,Y) & f(X::'a,z(X::'a,Y)) & f(Y::'a,z(X::'a,Y)))) &
2701 (\<forall>X Y. f(X::'a,Y) --> f(X::'a,z(X::'a,Y)) | f(Y::'a,z(X::'a,Y))) --> False"
2704 (*5336 inferences so far. Searching to depth 15. 5.3 secs*)
2706 "(\<forall>Vf U. element_of_set(U::'a,union_of_members(Vf)) --> element_of_set(U::'a,f1(Vf::'a,U))) &
2707 (\<forall>U Vf. element_of_set(U::'a,union_of_members(Vf)) --> element_of_collection(f1(Vf::'a,U),Vf)) &
2708 (\<forall>U Uu1 Vf. element_of_set(U::'a,Uu1) & element_of_collection(Uu1::'a,Vf) --> element_of_set(U::'a,union_of_members(Vf))) &
2709 (\<forall>Vf X. basis(X::'a,Vf) --> equal_sets(union_of_members(Vf),X)) &
2710 (\<forall>Vf U X. element_of_collection(U::'a,top_of_basis(Vf)) & element_of_set(X::'a,U) --> element_of_set(X::'a,f10(Vf::'a,U,X))) &
2711 (\<forall>U X Vf. element_of_collection(U::'a,top_of_basis(Vf)) & element_of_set(X::'a,U) --> element_of_collection(f10(Vf::'a,U,X),Vf)) &
2712 (\<forall>X. subset_sets(X::'a,X)) &
2713 (\<forall>X U Y. subset_sets(X::'a,Y) & element_of_set(U::'a,X) --> element_of_set(U::'a,Y)) &
2714 (\<forall>X Y. equal_sets(X::'a,Y) --> subset_sets(X::'a,Y)) &
2715 (\<forall>Y X. subset_sets(X::'a,Y) | element_of_set(in_1st_set(X::'a,Y),X)) &
2716 (\<forall>X Y. element_of_set(in_1st_set(X::'a,Y),Y) --> subset_sets(X::'a,Y)) &
2718 (~subset_sets(union_of_members(top_of_basis(f)),cx)) --> False"
2721 (*0 inferences so far. Searching to depth 0. 0 secs*)
2723 "(\<forall>Vf U. element_of_collection(U::'a,top_of_basis(Vf)) | element_of_set(f11(Vf::'a,U),U)) &
2724 (\<forall>X. ~element_of_set(X::'a,empty_set)) &
2725 (~element_of_collection(empty_set::'a,top_of_basis(f))) --> False"
2728 (*0 inferences so far. Searching to depth 0. 6.5 secs. BIG*)
2730 "(\<forall>Vf U. element_of_set(U::'a,union_of_members(Vf)) --> element_of_set(U::'a,f1(Vf::'a,U))) &
2731 (\<forall>U Vf. element_of_set(U::'a,union_of_members(Vf)) --> element_of_collection(f1(Vf::'a,U),Vf)) &
2732 (\<forall>U Uu1 Vf. element_of_set(U::'a,Uu1) & element_of_collection(Uu1::'a,Vf) --> element_of_set(U::'a,union_of_members(Vf))) &
2733 (\<forall>Vf U Va. element_of_set(U::'a,intersection_of_members(Vf)) & element_of_collection(Va::'a,Vf) --> element_of_set(U::'a,Va)) &
2734 (\<forall>U Vf. element_of_set(U::'a,intersection_of_members(Vf)) | element_of_collection(f2(Vf::'a,U),Vf)) &
2735 (\<forall>Vf U. element_of_set(U::'a,f2(Vf::'a,U)) --> element_of_set(U::'a,intersection_of_members(Vf))) &
2736 (\<forall>Vt X. topological_space(X::'a,Vt) --> equal_sets(union_of_members(Vt),X)) &
2737 (\<forall>X Vt. topological_space(X::'a,Vt) --> element_of_collection(empty_set::'a,Vt)) &
2738 (\<forall>X Vt. topological_space(X::'a,Vt) --> element_of_collection(X::'a,Vt)) &
2739 (\<forall>X Y Z Vt. topological_space(X::'a,Vt) & element_of_collection(Y::'a,Vt) & element_of_collection(Z::'a,Vt) --> element_of_collection(intersection_of_sets(Y::'a,Z),Vt)) &
2740 (\<forall>X Vf Vt. topological_space(X::'a,Vt) & subset_collections(Vf::'a,Vt) --> element_of_collection(union_of_members(Vf),Vt)) &
2741 (\<forall>X Vt. equal_sets(union_of_members(Vt),X) & element_of_collection(empty_set::'a,Vt) & element_of_collection(X::'a,Vt) --> topological_space(X::'a,Vt) | element_of_collection(f3(X::'a,Vt),Vt) | subset_collections(f5(X::'a,Vt),Vt)) &
2742 (\<forall>X Vt. equal_sets(union_of_members(Vt),X) & element_of_collection(empty_set::'a,Vt) & element_of_collection(X::'a,Vt) & element_of_collection(union_of_members(f5(X::'a,Vt)),Vt) --> topological_space(X::'a,Vt) | element_of_collection(f3(X::'a,Vt),Vt)) &
2743 (\<forall>X Vt. equal_sets(union_of_members(Vt),X) & element_of_collection(empty_set::'a,Vt) & element_of_collection(X::'a,Vt) --> topological_space(X::'a,Vt) | element_of_collection(f4(X::'a,Vt),Vt) | subset_collections(f5(X::'a,Vt),Vt)) &
2744 (\<forall>X Vt. equal_sets(union_of_members(Vt),X) & element_of_collection(empty_set::'a,Vt) & element_of_collection(X::'a,Vt) & element_of_collection(union_of_members(f5(X::'a,Vt)),Vt) --> topological_space(X::'a,Vt) | element_of_collection(f4(X::'a,Vt),Vt)) &
2745 (\<forall>X Vt. equal_sets(union_of_members(Vt),X) & element_of_collection(empty_set::'a,Vt) & element_of_collection(X::'a,Vt) & element_of_collection(intersection_of_sets(f3(X::'a,Vt),f4(X::'a,Vt)),Vt) --> topological_space(X::'a,Vt) | subset_collections(f5(X::'a,Vt),Vt)) &
2746 (\<forall>X Vt. equal_sets(union_of_members(Vt),X) & element_of_collection(empty_set::'a,Vt) & element_of_collection(X::'a,Vt) & element_of_collection(intersection_of_sets(f3(X::'a,Vt),f4(X::'a,Vt)),Vt) & element_of_collection(union_of_members(f5(X::'a,Vt)),Vt) --> topological_space(X::'a,Vt)) &
2747 (\<forall>U X Vt. open(U::'a,X,Vt) --> topological_space(X::'a,Vt)) &
2748 (\<forall>X U Vt. open(U::'a,X,Vt) --> element_of_collection(U::'a,Vt)) &
2749 (\<forall>X U Vt. topological_space(X::'a,Vt) & element_of_collection(U::'a,Vt) --> open(U::'a,X,Vt)) &
2750 (\<forall>U X Vt. closed(U::'a,X,Vt) --> topological_space(X::'a,Vt)) &
2751 (\<forall>U X Vt. closed(U::'a,X,Vt) --> open(relative_complement_sets(U::'a,X),X,Vt)) &
2752 (\<forall>U X Vt. topological_space(X::'a,Vt) & open(relative_complement_sets(U::'a,X),X,Vt) --> closed(U::'a,X,Vt)) &
2753 (\<forall>Vs X Vt. finer(Vt::'a,Vs,X) --> topological_space(X::'a,Vt)) &
2754 (\<forall>Vt X Vs. finer(Vt::'a,Vs,X) --> topological_space(X::'a,Vs)) &
2755 (\<forall>X Vs Vt. finer(Vt::'a,Vs,X) --> subset_collections(Vs::'a,Vt)) &
2756 (\<forall>X Vs Vt. topological_space(X::'a,Vt) & topological_space(X::'a,Vs) & subset_collections(Vs::'a,Vt) --> finer(Vt::'a,Vs,X)) &
2757 (\<forall>Vf X. basis(X::'a,Vf) --> equal_sets(union_of_members(Vf),X)) &
2758 (\<forall>X Vf Y Vb1 Vb2. basis(X::'a,Vf) & element_of_set(Y::'a,X) & element_of_collection(Vb1::'a,Vf) & element_of_collection(Vb2::'a,Vf) & element_of_set(Y::'a,intersection_of_sets(Vb1::'a,Vb2)) --> element_of_set(Y::'a,f6(X::'a,Vf,Y,Vb1,Vb2))) &
2759 (\<forall>X Y Vb1 Vb2 Vf. basis(X::'a,Vf) & element_of_set(Y::'a,X) & element_of_collection(Vb1::'a,Vf) & element_of_collection(Vb2::'a,Vf) & element_of_set(Y::'a,intersection_of_sets(Vb1::'a,Vb2)) --> element_of_collection(f6(X::'a,Vf,Y,Vb1,Vb2),Vf)) &
2760 (\<forall>X Vf Y Vb1 Vb2. basis(X::'a,Vf) & element_of_set(Y::'a,X) & element_of_collection(Vb1::'a,Vf) & element_of_collection(Vb2::'a,Vf) & element_of_set(Y::'a,intersection_of_sets(Vb1::'a,Vb2)) --> subset_sets(f6(X::'a,Vf,Y,Vb1,Vb2),intersection_of_sets(Vb1::'a,Vb2))) &
2761 (\<forall>Vf X. equal_sets(union_of_members(Vf),X) --> basis(X::'a,Vf) | element_of_set(f7(X::'a,Vf),X)) &
2762 (\<forall>X Vf. equal_sets(union_of_members(Vf),X) --> basis(X::'a,Vf) | element_of_collection(f8(X::'a,Vf),Vf)) &
2763 (\<forall>X Vf. equal_sets(union_of_members(Vf),X) --> basis(X::'a,Vf) | element_of_collection(f9(X::'a,Vf),Vf)) &
2764 (\<forall>X Vf. equal_sets(union_of_members(Vf),X) --> basis(X::'a,Vf) | element_of_set(f7(X::'a,Vf),intersection_of_sets(f8(X::'a,Vf),f9(X::'a,Vf)))) &
2765 (\<forall>Uu9 X Vf. equal_sets(union_of_members(Vf),X) & element_of_set(f7(X::'a,Vf),Uu9) & element_of_collection(Uu9::'a,Vf) & subset_sets(Uu9::'a,intersection_of_sets(f8(X::'a,Vf),f9(X::'a,Vf))) --> basis(X::'a,Vf)) &
2766 (\<forall>Vf U X. element_of_collection(U::'a,top_of_basis(Vf)) & element_of_set(X::'a,U) --> element_of_set(X::'a,f10(Vf::'a,U,X))) &
2767 (\<forall>U X Vf. element_of_collection(U::'a,top_of_basis(Vf)) & element_of_set(X::'a,U) --> element_of_collection(f10(Vf::'a,U,X),Vf)) &
2768 (\<forall>Vf X U. element_of_collection(U::'a,top_of_basis(Vf)) & element_of_set(X::'a,U) --> subset_sets(f10(Vf::'a,U,X),U)) &
2769 (\<forall>Vf U. element_of_collection(U::'a,top_of_basis(Vf)) | element_of_set(f11(Vf::'a,U),U)) &
2770 (\<forall>Vf Uu11 U. element_of_set(f11(Vf::'a,U),Uu11) & element_of_collection(Uu11::'a,Vf) & subset_sets(Uu11::'a,U) --> element_of_collection(U::'a,top_of_basis(Vf))) &
2771 (\<forall>U Y X Vt. element_of_collection(U::'a,subspace_topology(X::'a,Vt,Y)) --> topological_space(X::'a,Vt)) &
2772 (\<forall>U Vt Y X. element_of_collection(U::'a,subspace_topology(X::'a,Vt,Y)) --> subset_sets(Y::'a,X)) &
2773 (\<forall>X Y U Vt. element_of_collection(U::'a,subspace_topology(X::'a,Vt,Y)) --> element_of_collection(f12(X::'a,Vt,Y,U),Vt)) &
2774 (\<forall>X Vt Y U. element_of_collection(U::'a,subspace_topology(X::'a,Vt,Y)) --> equal_sets(U::'a,intersection_of_sets(Y::'a,f12(X::'a,Vt,Y,U)))) &
2775 (\<forall>X Vt U Y Uu12. topological_space(X::'a,Vt) & subset_sets(Y::'a,X) & element_of_collection(Uu12::'a,Vt) & equal_sets(U::'a,intersection_of_sets(Y::'a,Uu12)) --> element_of_collection(U::'a,subspace_topology(X::'a,Vt,Y))) &
2776 (\<forall>U Y X Vt. element_of_set(U::'a,interior(Y::'a,X,Vt)) --> topological_space(X::'a,Vt)) &
2777 (\<forall>U Vt Y X. element_of_set(U::'a,interior(Y::'a,X,Vt)) --> subset_sets(Y::'a,X)) &
2778 (\<forall>Y X Vt U. element_of_set(U::'a,interior(Y::'a,X,Vt)) --> element_of_set(U::'a,f13(Y::'a,X,Vt,U))) &
2779 (\<forall>X Vt U Y. element_of_set(U::'a,interior(Y::'a,X,Vt)) --> subset_sets(f13(Y::'a,X,Vt,U),Y)) &
2780 (\<forall>Y U X Vt. element_of_set(U::'a,interior(Y::'a,X,Vt)) --> open(f13(Y::'a,X,Vt,U),X,Vt)) &
2781 (\<forall>U Y Uu13 X Vt. topological_space(X::'a,Vt) & subset_sets(Y::'a,X) & element_of_set(U::'a,Uu13) & subset_sets(Uu13::'a,Y) & open(Uu13::'a,X,Vt) --> element_of_set(U::'a,interior(Y::'a,X,Vt))) &
2782 (\<forall>U Y X Vt. element_of_set(U::'a,closure(Y::'a,X,Vt)) --> topological_space(X::'a,Vt)) &
2783 (\<forall>U Vt Y X. element_of_set(U::'a,closure(Y::'a,X,Vt)) --> subset_sets(Y::'a,X)) &
2784 (\<forall>Y X Vt U V. element_of_set(U::'a,closure(Y::'a,X,Vt)) & subset_sets(Y::'a,V) & closed(V::'a,X,Vt) --> element_of_set(U::'a,V)) &
2785 (\<forall>Y X Vt U. topological_space(X::'a,Vt) & subset_sets(Y::'a,X) --> element_of_set(U::'a,closure(Y::'a,X,Vt)) | subset_sets(Y::'a,f14(Y::'a,X,Vt,U))) &
2786 (\<forall>Y U X Vt. topological_space(X::'a,Vt) & subset_sets(Y::'a,X) --> element_of_set(U::'a,closure(Y::'a,X,Vt)) | closed(f14(Y::'a,X,Vt,U),X,Vt)) &
2787 (\<forall>Y X Vt U. topological_space(X::'a,Vt) & subset_sets(Y::'a,X) & element_of_set(U::'a,f14(Y::'a,X,Vt,U)) --> element_of_set(U::'a,closure(Y::'a,X,Vt))) &
2788 (\<forall>U Y X Vt. neighborhood(U::'a,Y,X,Vt) --> topological_space(X::'a,Vt)) &
2789 (\<forall>Y U X Vt. neighborhood(U::'a,Y,X,Vt) --> open(U::'a,X,Vt)) &
2790 (\<forall>X Vt Y U. neighborhood(U::'a,Y,X,Vt) --> element_of_set(Y::'a,U)) &
2791 (\<forall>X Vt Y U. topological_space(X::'a,Vt) & open(U::'a,X,Vt) & element_of_set(Y::'a,U) --> neighborhood(U::'a,Y,X,Vt)) &
2792 (\<forall>Z Y X Vt. limit_point(Z::'a,Y,X,Vt) --> topological_space(X::'a,Vt)) &
2793 (\<forall>Z Vt Y X. limit_point(Z::'a,Y,X,Vt) --> subset_sets(Y::'a,X)) &
2794 (\<forall>Z X Vt U Y. limit_point(Z::'a,Y,X,Vt) & neighborhood(U::'a,Z,X,Vt) --> element_of_set(f15(Z::'a,Y,X,Vt,U),intersection_of_sets(U::'a,Y))) &
2795 (\<forall>Y X Vt U Z. ~(limit_point(Z::'a,Y,X,Vt) & neighborhood(U::'a,Z,X,Vt) & eq_p(f15(Z::'a,Y,X,Vt,U),Z))) &
2796 (\<forall>Y Z X Vt. topological_space(X::'a,Vt) & subset_sets(Y::'a,X) --> limit_point(Z::'a,Y,X,Vt) | neighborhood(f16(Z::'a,Y,X,Vt),Z,X,Vt)) &
2797 (\<forall>X Vt Y Uu16 Z. topological_space(X::'a,Vt) & subset_sets(Y::'a,X) & element_of_set(Uu16::'a,intersection_of_sets(f16(Z::'a,Y,X,Vt),Y)) --> limit_point(Z::'a,Y,X,Vt) | eq_p(Uu16::'a,Z)) &
2798 (\<forall>U Y X Vt. element_of_set(U::'a,boundary(Y::'a,X,Vt)) --> topological_space(X::'a,Vt)) &
2799 (\<forall>U Y X Vt. element_of_set(U::'a,boundary(Y::'a,X,Vt)) --> element_of_set(U::'a,closure(Y::'a,X,Vt))) &
2800 (\<forall>U Y X Vt. element_of_set(U::'a,boundary(Y::'a,X,Vt)) --> element_of_set(U::'a,closure(relative_complement_sets(Y::'a,X),X,Vt))) &
2801 (\<forall>U Y X Vt. topological_space(X::'a,Vt) & element_of_set(U::'a,closure(Y::'a,X,Vt)) & element_of_set(U::'a,closure(relative_complement_sets(Y::'a,X),X,Vt)) --> element_of_set(U::'a,boundary(Y::'a,X,Vt))) &
2802 (\<forall>X Vt. hausdorff(X::'a,Vt) --> topological_space(X::'a,Vt)) &
2803 (\<forall>X_2 X_1 X Vt. hausdorff(X::'a,Vt) & element_of_set(X_1::'a,X) & element_of_set(X_2::'a,X) --> eq_p(X_1::'a,X_2) | neighborhood(f17(X::'a,Vt,X_1,X_2),X_1,X,Vt)) &
2804 (\<forall>X_1 X_2 X Vt. hausdorff(X::'a,Vt) & element_of_set(X_1::'a,X) & element_of_set(X_2::'a,X) --> eq_p(X_1::'a,X_2) | neighborhood(f18(X::'a,Vt,X_1,X_2),X_2,X,Vt)) &
2805 (\<forall>X Vt X_1 X_2. hausdorff(X::'a,Vt) & element_of_set(X_1::'a,X) & element_of_set(X_2::'a,X) --> eq_p(X_1::'a,X_2) | disjoint_s(f17(X::'a,Vt,X_1,X_2),f18(X::'a,Vt,X_1,X_2))) &
2806 (\<forall>Vt X. topological_space(X::'a,Vt) --> hausdorff(X::'a,Vt) | element_of_set(f19(X::'a,Vt),X)) &
2807 (\<forall>Vt X. topological_space(X::'a,Vt) --> hausdorff(X::'a,Vt) | element_of_set(f20(X::'a,Vt),X)) &
2808 (\<forall>X Vt. topological_space(X::'a,Vt) & eq_p(f19(X::'a,Vt),f20(X::'a,Vt)) --> hausdorff(X::'a,Vt)) &
2809 (\<forall>X Vt Uu19 Uu20. topological_space(X::'a,Vt) & neighborhood(Uu19::'a,f19(X::'a,Vt),X,Vt) & neighborhood(Uu20::'a,f20(X::'a,Vt),X,Vt) & disjoint_s(Uu19::'a,Uu20) --> hausdorff(X::'a,Vt)) &
2810 (\<forall>Va1 Va2 X Vt. separation(Va1::'a,Va2,X,Vt) --> topological_space(X::'a,Vt)) &
2811 (\<forall>Va2 X Vt Va1. ~(separation(Va1::'a,Va2,X,Vt) & equal_sets(Va1::'a,empty_set))) &
2812 (\<forall>Va1 X Vt Va2. ~(separation(Va1::'a,Va2,X,Vt) & equal_sets(Va2::'a,empty_set))) &
2813 (\<forall>Va2 X Va1 Vt. separation(Va1::'a,Va2,X,Vt) --> element_of_collection(Va1::'a,Vt)) &
2814 (\<forall>Va1 X Va2 Vt. separation(Va1::'a,Va2,X,Vt) --> element_of_collection(Va2::'a,Vt)) &
2815 (\<forall>Vt Va1 Va2 X. separation(Va1::'a,Va2,X,Vt) --> equal_sets(union_of_sets(Va1::'a,Va2),X)) &
2816 (\<forall>X Vt Va1 Va2. separation(Va1::'a,Va2,X,Vt) --> disjoint_s(Va1::'a,Va2)) &
2817 (\<forall>Vt X Va1 Va2. topological_space(X::'a,Vt) & element_of_collection(Va1::'a,Vt) & element_of_collection(Va2::'a,Vt) & equal_sets(union_of_sets(Va1::'a,Va2),X) & disjoint_s(Va1::'a,Va2) --> separation(Va1::'a,Va2,X,Vt) | equal_sets(Va1::'a,empty_set) | equal_sets(Va2::'a,empty_set)) &
2818 (\<forall>X Vt. connected_space(X::'a,Vt) --> topological_space(X::'a,Vt)) &
2819 (\<forall>Va1 Va2 X Vt. ~(connected_space(X::'a,Vt) & separation(Va1::'a,Va2,X,Vt))) &
2820 (\<forall>X Vt. topological_space(X::'a,Vt) --> connected_space(X::'a,Vt) | separation(f21(X::'a,Vt),f22(X::'a,Vt),X,Vt)) &
2821 (\<forall>Va X Vt. connected_set(Va::'a,X,Vt) --> topological_space(X::'a,Vt)) &
2822 (\<forall>Vt Va X. connected_set(Va::'a,X,Vt) --> subset_sets(Va::'a,X)) &
2823 (\<forall>X Vt Va. connected_set(Va::'a,X,Vt) --> connected_space(Va::'a,subspace_topology(X::'a,Vt,Va))) &
2824 (\<forall>X Vt Va. topological_space(X::'a,Vt) & subset_sets(Va::'a,X) & connected_space(Va::'a,subspace_topology(X::'a,Vt,Va)) --> connected_set(Va::'a,X,Vt)) &
2825 (\<forall>Vf X Vt. open_covering(Vf::'a,X,Vt) --> topological_space(X::'a,Vt)) &
2826 (\<forall>X Vf Vt. open_covering(Vf::'a,X,Vt) --> subset_collections(Vf::'a,Vt)) &
2827 (\<forall>Vt Vf X. open_covering(Vf::'a,X,Vt) --> equal_sets(union_of_members(Vf),X)) &
2828 (\<forall>Vt Vf X. topological_space(X::'a,Vt) & subset_collections(Vf::'a,Vt) & equal_sets(union_of_members(Vf),X) --> open_covering(Vf::'a,X,Vt)) &
2829 (\<forall>X Vt. compact_space(X::'a,Vt) --> topological_space(X::'a,Vt)) &
2830 (\<forall>X Vt Vf1. compact_space(X::'a,Vt) & open_covering(Vf1::'a,X,Vt) --> finite'(f23(X::'a,Vt,Vf1))) &
2831 (\<forall>X Vt Vf1. compact_space(X::'a,Vt) & open_covering(Vf1::'a,X,Vt) --> subset_collections(f23(X::'a,Vt,Vf1),Vf1)) &
2832 (\<forall>Vf1 X Vt. compact_space(X::'a,Vt) & open_covering(Vf1::'a,X,Vt) --> open_covering(f23(X::'a,Vt,Vf1),X,Vt)) &
2833 (\<forall>X Vt. topological_space(X::'a,Vt) --> compact_space(X::'a,Vt) | open_covering(f24(X::'a,Vt),X,Vt)) &
2834 (\<forall>Uu24 X Vt. topological_space(X::'a,Vt) & finite'(Uu24) & subset_collections(Uu24::'a,f24(X::'a,Vt)) & open_covering(Uu24::'a,X,Vt) --> compact_space(X::'a,Vt)) &
2835 (\<forall>Va X Vt. compact_set(Va::'a,X,Vt) --> topological_space(X::'a,Vt)) &
2836 (\<forall>Vt Va X. compact_set(Va::'a,X,Vt) --> subset_sets(Va::'a,X)) &
2837 (\<forall>X Vt Va. compact_set(Va::'a,X,Vt) --> compact_space(Va::'a,subspace_topology(X::'a,Vt,Va))) &
2838 (\<forall>X Vt Va. topological_space(X::'a,Vt) & subset_sets(Va::'a,X) & compact_space(Va::'a,subspace_topology(X::'a,Vt,Va)) --> compact_set(Va::'a,X,Vt)) &
2840 (\<forall>U. element_of_collection(U::'a,top_of_basis(f))) &
2841 (\<forall>V. element_of_collection(V::'a,top_of_basis(f))) &
2842 (\<forall>U V. ~element_of_collection(intersection_of_sets(U::'a,V),top_of_basis(f))) --> False"
2846 (*0 inferences so far. Searching to depth 0. 0.8 secs*)
2848 "(\<forall>U Uu1 Vf. element_of_set(U::'a,Uu1) & element_of_collection(Uu1::'a,Vf) --> element_of_set(U::'a,union_of_members(Vf))) &
2849 (\<forall>Vf X. basis(X::'a,Vf) --> equal_sets(union_of_members(Vf),X)) &
2850 (\<forall>X Vf Y Vb1 Vb2. basis(X::'a,Vf) & element_of_set(Y::'a,X) & element_of_collection(Vb1::'a,Vf) & element_of_collection(Vb2::'a,Vf) & element_of_set(Y::'a,intersection_of_sets(Vb1::'a,Vb2)) --> element_of_set(Y::'a,f6(X::'a,Vf,Y,Vb1,Vb2))) &
2851 (\<forall>X Y Vb1 Vb2 Vf. basis(X::'a,Vf) & element_of_set(Y::'a,X) & element_of_collection(Vb1::'a,Vf) & element_of_collection(Vb2::'a,Vf) & element_of_set(Y::'a,intersection_of_sets(Vb1::'a,Vb2)) --> element_of_collection(f6(X::'a,Vf,Y,Vb1,Vb2),Vf)) &
2852 (\<forall>X Vf Y Vb1 Vb2. basis(X::'a,Vf) & element_of_set(Y::'a,X) & element_of_collection(Vb1::'a,Vf) & element_of_collection(Vb2::'a,Vf) & element_of_set(Y::'a,intersection_of_sets(Vb1::'a,Vb2)) --> subset_sets(f6(X::'a,Vf,Y,Vb1,Vb2),intersection_of_sets(Vb1::'a,Vb2))) &
2853 (\<forall>Vf U X. element_of_collection(U::'a,top_of_basis(Vf)) & element_of_set(X::'a,U) --> element_of_set(X::'a,f10(Vf::'a,U,X))) &
2854 (\<forall>U X Vf. element_of_collection(U::'a,top_of_basis(Vf)) & element_of_set(X::'a,U) --> element_of_collection(f10(Vf::'a,U,X),Vf)) &
2855 (\<forall>Vf X U. element_of_collection(U::'a,top_of_basis(Vf)) & element_of_set(X::'a,U) --> subset_sets(f10(Vf::'a,U,X),U)) &
2856 (\<forall>Vf U. element_of_collection(U::'a,top_of_basis(Vf)) | element_of_set(f11(Vf::'a,U),U)) &
2857 (\<forall>Vf Uu11 U. element_of_set(f11(Vf::'a,U),Uu11) & element_of_collection(Uu11::'a,Vf) & subset_sets(Uu11::'a,U) --> element_of_collection(U::'a,top_of_basis(Vf))) &
2858 (\<forall>Y X Z. subset_sets(X::'a,Y) & subset_sets(Y::'a,Z) --> subset_sets(X::'a,Z)) &
2859 (\<forall>Y Z X. element_of_set(Z::'a,intersection_of_sets(X::'a,Y)) --> element_of_set(Z::'a,X)) &
2860 (\<forall>X Z Y. element_of_set(Z::'a,intersection_of_sets(X::'a,Y)) --> element_of_set(Z::'a,Y)) &
2861 (\<forall>X Z Y. element_of_set(Z::'a,X) & element_of_set(Z::'a,Y) --> element_of_set(Z::'a,intersection_of_sets(X::'a,Y))) &
2862 (\<forall>X U Y V. subset_sets(X::'a,Y) & subset_sets(U::'a,V) --> subset_sets(intersection_of_sets(X::'a,U),intersection_of_sets(Y::'a,V))) &
2863 (\<forall>X Z Y. equal_sets(X::'a,Y) & element_of_set(Z::'a,X) --> element_of_set(Z::'a,Y)) &
2864 (\<forall>Y X. equal_sets(intersection_of_sets(X::'a,Y),intersection_of_sets(Y::'a,X))) &
2866 (\<forall>U. element_of_collection(U::'a,top_of_basis(f))) &
2867 (\<forall>V. element_of_collection(V::'a,top_of_basis(f))) &
2868 (\<forall>U V. ~element_of_collection(intersection_of_sets(U::'a,V),top_of_basis(f))) --> False"
2871 (*53777 inferences so far. Searching to depth 20. 68.7 secs*)
2873 "(\<forall>Vf U. element_of_set(U::'a,union_of_members(Vf)) --> element_of_set(U::'a,f1(Vf::'a,U))) &
2874 (\<forall>U Vf. element_of_set(U::'a,union_of_members(Vf)) --> element_of_collection(f1(Vf::'a,U),Vf)) &
2875 (\<forall>Vf U X. element_of_collection(U::'a,top_of_basis(Vf)) & element_of_set(X::'a,U) --> element_of_set(X::'a,f10(Vf::'a,U,X))) &
2876 (\<forall>U X Vf. element_of_collection(U::'a,top_of_basis(Vf)) & element_of_set(X::'a,U) --> element_of_collection(f10(Vf::'a,U,X),Vf)) &
2877 (\<forall>Vf X U. element_of_collection(U::'a,top_of_basis(Vf)) & element_of_set(X::'a,U) --> subset_sets(f10(Vf::'a,U,X),U)) &
2878 (\<forall>Vf U. element_of_collection(U::'a,top_of_basis(Vf)) | element_of_set(f11(Vf::'a,U),U)) &
2879 (\<forall>Vf Uu11 U. element_of_set(f11(Vf::'a,U),Uu11) & element_of_collection(Uu11::'a,Vf) & subset_sets(Uu11::'a,U) --> element_of_collection(U::'a,top_of_basis(Vf))) &
2880 (\<forall>X U Y. element_of_set(U::'a,X) --> subset_sets(X::'a,Y) | element_of_set(U::'a,Y)) &
2881 (\<forall>Y X Z. subset_sets(X::'a,Y) & element_of_collection(Y::'a,Z) --> subset_sets(X::'a,union_of_members(Z))) &
2882 (\<forall>X U Y. subset_collections(X::'a,Y) & element_of_collection(U::'a,X) --> element_of_collection(U::'a,Y)) &
2883 (subset_collections(g::'a,top_of_basis(f))) &
2884 (~element_of_collection(union_of_members(g),top_of_basis(f))) --> False"