1.1 --- a/src/HOL/IsaMakefile Wed Mar 22 13:23:57 2000 +0100
1.2 +++ b/src/HOL/IsaMakefile Thu Mar 23 10:22:08 2000 +0100
1.3 @@ -418,7 +418,8 @@
1.4 ex/Primrec.ML ex/Primrec.thy \
1.5 ex/Puzzle.ML ex/Puzzle.thy ex/Qsort.ML ex/Qsort.thy \
1.6 ex/ROOT.ML ex/Recdefs.ML ex/Recdefs.thy ex/cla.ML ex/meson.ML \
1.7 - ex/mesontest.ML ex/set.ML ex/Group.ML ex/Group.thy ex/IntRing.ML \
1.8 + ex/mesontest.ML ex/mesontest2.ML ex/set.ML \
1.9 + ex/Group.ML ex/Group.thy ex/IntRing.ML \
1.10 ex/IntRing.thy ex/IntRingDefs.ML ex/IntRingDefs.thy ex/Lagrange.ML \
1.11 ex/Lagrange.thy ex/Ring.ML ex/Ring.thy ex/StringEx.ML \
1.12 ex/StringEx.thy ex/Tarski.ML ex/Tarski.thy \
2.1 --- a/src/HOL/ex/ROOT.ML Wed Mar 22 13:23:57 2000 +0100
2.2 +++ b/src/HOL/ex/ROOT.ML Thu Mar 23 10:22:08 2000 +0100
2.3 @@ -21,6 +21,7 @@
2.4 time_use "cla.ML";
2.5 time_use "meson.ML";
2.6 time_use "mesontest.ML";
2.7 +time_use "mesontest2.ML";
2.8 time_use_thy "BT";
2.9 time_use_thy "InSort";
2.10 time_use_thy "Qsort";
3.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
3.2 +++ b/src/HOL/ex/mesontest2.ML Thu Mar 23 10:22:08 2000 +0100
3.3 @@ -0,0 +1,3467 @@
3.4 +(* Title: HOL/ex/mesontest2
3.5 + ID: $Id$
3.6 + Author: Lawrence C Paulson, Cambridge University Computer Laboratory
3.7 + Copyright 2000 University of Cambridge
3.8 +
3.9 +MORE and MUCH HARDER test data for the MESON proof procedure
3.10 +
3.11 +Courtesy John Harrison
3.12 +*)
3.13 +
3.14 +(*All but the fastest are ignored to reduce build time*)
3.15 +val even_hard_ones = false;
3.16 +
3.17 +(*Prod.thy instead of HOL.thy regards arguments as tuples.
3.18 + But Main.thy would allow clashes with many other constants*)
3.19 +fun prove (s,tac) = prove_goal Prod.thy s (fn [] => [tac]);
3.20 +
3.21 +fun prove_hard arg = if even_hard_ones then prove arg else TrueI;
3.22 +
3.23 +val meson_tac = safe_meson_tac 1;
3.24 +
3.25 +set proof_timing;
3.26 +
3.27 +(* ========================================================================= *)
3.28 +(* 100 problems selected from the TPTP library *)
3.29 +(* ========================================================================= *)
3.30 +
3.31 +(*
3.32 + * Original timings for John Harrison's MESON_TAC.
3.33 + * Timings below on a 600MHz Pentium III (perch)
3.34 + *
3.35 + * A few variable names have been primed to avoid clashing with constants.
3.36 + *
3.37 + * Changed numeric constants e.g. 0, 1, 2... to num0, num1, num2...
3.38 + *
3.39 + * Here's a list giving typical CPU times, as well as common names and
3.40 + * literature references.
3.41 + *
3.42 + * BOO003-1 34.6 B2 part 1 [McCharen, et al., 1976]; Lemma proved [Overbeek, et al., 1976]; prob2_part1.ver1.in [ANL]
3.43 + * BOO004-1 36.7 B2 part 2 [McCharen, et al., 1976]; Lemma proved [Overbeek, et al., 1976]; prob2_part2.ver1 [ANL]
3.44 + * 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]
3.45 + * 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]
3.46 + * BOO011-1 19.0 B7 [McCharen, et al., 1976]; prob7.ver1 [ANL]
3.47 + * CAT001-3 45.2 C1 [McCharen, et al., 1976]; p1.ver3.in [ANL]
3.48 + * CAT003-3 10.5 C3 [McCharen, et al., 1976]; p3.ver3.in [ANL]
3.49 + * CAT005-1 480.1 C5 [McCharen, et al., 1976]; p5.ver1.in [ANL]
3.50 + * CAT007-1 11.9 C7 [McCharen, et al., 1976]; p7.ver1.in [ANL]
3.51 + * CAT018-1 81.3 p18.ver1.in [ANL]
3.52 + * COL001-2 16.0 C1 [Wos & McCune, 1988]
3.53 + * COL023-1 5.1 [McCune & Wos, 1988]
3.54 + * COL032-1 15.8 [McCune & Wos, 1988]
3.55 + * COL052-2 13.2 bird4.ver2.in [ANL]
3.56 + * COL075-2 116.9 [Jech, 1994]
3.57 + * COM001-1 1.7 shortburst [Wilson & Minker, 1976]
3.58 + * COM002-1 4.4 burstall [Wilson & Minker, 1976]
3.59 + * COM002-2 7.4
3.60 + * COM003-2 22.1 [Brushi, 1991]
3.61 + * COM004-1 45.1
3.62 + * GEO003-1 71.7 T3 [McCharen, et al., 1976]; t3.ver1.in [ANL]
3.63 + * GEO017-2 78.8 D4.1 [Quaife, 1989]
3.64 + * GEO027-3 181.5 D10.1 [Quaife, 1989]
3.65 + * GEO058-2 104.0 R4 [Quaife, 1989]
3.66 + * GEO079-1 2.4 GEOMETRY THEOREM [Slagle, 1967]
3.67 + * 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]
3.68 + * GRP008-1 50.4 Problem 4 [Wos, 1965]; wos4 [Wilson & Minker, 1976]
3.69 + * GRP013-1 40.2 Problem 11 [Wos, 1965]; wos11 [Wilson & Minker, 1976]
3.70 + * GRP037-3 43.8 Problem 17 [Wos, 1965]; wos17 [Wilson & Minker, 1976]
3.71 + * GRP031-2 3.2 ls23 [Lawrence & Starkey, 1974]; ls23 [Wilson & Minker, 1976]
3.72 + * GRP034-4 2.5 ls26 [Lawrence & Starkey, 1974]; ls26 [Wilson & Minker, 1976]
3.73 + * GRP047-2 11.7 [Veroff, 1992]
3.74 + * GRP130-1 170.6 Bennett QG8 [TPTP]; QG8 [Slaney, 1993]
3.75 + * GRP156-1 48.7 ax_mono1c [Schulz, 1995]
3.76 + * GRP168-1 159.1 p01a [Schulz, 1995]
3.77 + * HEN003-3 39.9 HP3 [McCharen, et al., 1976]
3.78 + * HEN007-2 125.7 H7 [McCharen, et al., 1976]
3.79 + * HEN008-4 62.0 H8 [McCharen, et al., 1976]
3.80 + * HEN009-5 136.3 H9 [McCharen, et al., 1976]; hp9.ver3.in [ANL]
3.81 + * HEN012-3 48.5 new.ver2.in [ANL]
3.82 + * LCL010-1 370.9 EC-73 [McCune & Wos, 1992]; ec_yq.in [OTTER]
3.83 + * LCL077-2 51.6 morgan.two.ver1.in [ANL]
3.84 + * LCL082-1 14.6 IC-1.1 [Wos, et al., 1990]; IC-65 [McCune & Wos, 1992]; ls2 [SETHEO]; S1 [Pfenning, 1988]
3.85 + * 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]
3.86 + * LCL143-1 10.9 Lattice structure theorem 2 [Bonacina, 1991]
3.87 + * LCL182-1 271.6 Problem 2.16 [Whitehead & Russell, 1927]
3.88 + * LCL200-1 12.0 Problem 2.46 [Whitehead & Russell, 1927]
3.89 + * LCL215-1 214.4 Problem 2.62 [Whitehead & Russell, 1927]; Problem 2.63 [Whitehead & Russell, 1927]
3.90 + * LCL230-2 0.2 Pelletier 5 [Pelletier, 1986]
3.91 + * LDA003-1 68.5 Problem 3 [Jech, 1993]
3.92 + * MSC002-1 9.2 DBABHP [Michie, et al., 1972]; DBABHP [Wilson & Minker, 1976]
3.93 + * MSC003-1 3.2 HASPARTS-T1 [Wilson & Minker, 1976]
3.94 + * MSC004-1 9.3 HASPARTS-T2 [Wilson & Minker, 1976]
3.95 + * MSC005-1 1.8 Problem 5.1 [Plaisted, 1982]
3.96 + * MSC006-1 39.0 nonob.lop [SETHEO]
3.97 + * NUM001-1 14.0 Chang-Lee-10a [Chang, 1970]; ls28 [Lawrence & Starkey, 1974]; ls28 [Wilson & Minker, 1976]
3.98 + * NUM021-1 52.3 ls65 [Lawrence & Starkey, 1974]; ls65 [Wilson & Minker, 1976]
3.99 + * NUM024-1 64.6 ls75 [Lawrence & Starkey, 1974]; ls75 [Wilson & Minker, 1976]
3.100 + * NUM180-1 621.2 LIM2.1 [Quaife]
3.101 + * NUM228-1 575.9 TRECDEF4 cor. [Quaife]
3.102 + * PLA002-1 37.4 Problem 5.7 [Plaisted, 1982]
3.103 + * PLA006-1 7.2 [Segre & Elkan, 1994]
3.104 + * PLA017-1 484.8 [Segre & Elkan, 1994]
3.105 + * PLA022-1 19.1 [Segre & Elkan, 1994]
3.106 + * PLA022-2 19.7 [Segre & Elkan, 1994]
3.107 + * PRV001-1 10.3 PV1 [McCharen, et al., 1976]
3.108 + * PRV003-1 3.9 E2 [McCharen, et al., 1976]; v2.lop [SETHEO]
3.109 + * PRV005-1 4.3 E4 [McCharen, et al., 1976]; v4.lop [SETHEO]
3.110 + * PRV006-1 6.0 E5 [McCharen, et al., 1976]; v5.lop [SETHEO]
3.111 + * PRV009-1 2.2 Hoares FIND [Bledsoe, 1977]; Problem 5.5 [Plaisted, 1982]
3.112 + * PUZ012-1 3.5 Boxes-of-fruit [Wos, 1988]; Boxes-of-fruit [Wos, et al., 1992]; boxes.ver1.in [ANL]
3.113 + * PUZ020-1 56.6 knightknave.in [ANL]
3.114 + * PUZ025-1 58.4 Problem 35 [Smullyan, 1978]; tandl35.ver1.in [ANL]
3.115 + * PUZ029-1 5.1 pigs.ver1.in [ANL]
3.116 + * RNG001-3 82.4 EX6-T? [Wilson & Minker, 1976]; ex6.lop [SETHEO]; Example 6a [Fleisig, et al., 1974]; FEX6T1 [SPRFN]; FEX6T2 [SPRFN]
3.117 + * RNG001-5 399.8 Problem 21 [Wos, 1965]; wos21 [Wilson & Minker, 1976]
3.118 + * RNG011-5 8.4 CADE-11 Competition Eq-10 [Overbeek, 1990]; PROBLEM 10 [Zhang, 1993]; THEOREM EQ-10 [Lusk & McCune, 1993]
3.119 + * RNG023-6 9.1 [Stevens, 1987]
3.120 + * RNG028-2 9.3 PROOF III [Anantharaman & Hsiang, 1990]
3.121 + * RNG038-2 16.2 Problem 27 [Wos, 1965]; wos27 [Wilson & Minker, 1976]
3.122 + * RNG040-2 180.5 Problem 29 [Wos, 1965]; wos29 [Wilson & Minker, 1976]
3.123 + * RNG041-1 35.8 Problem 30 [Wos, 1965]; wos30 [Wilson & Minker, 1976]
3.124 + * ROB010-1 205.0 Lemma 3.3 [Winker, 1990]; RA2 [Lusk & Wos, 1992]
3.125 + * ROB013-1 23.6 Lemma 3.5 [Winker, 1990]
3.126 + * ROB016-1 15.2 Corollary 3.7 [Winker, 1990]
3.127 + * ROB021-1 230.4 [McCune, 1992]
3.128 + * SET005-1 192.2 ls108 [Lawrence & Starkey, 1974]; ls108 [Wilson & Minker, 1976]
3.129 + * SET009-1 10.5 ls116 [Lawrence & Starkey, 1974]; ls116 [Wilson & Minker, 1976]
3.130 + * SET025-4 694.7 Lemma 10 [Boyer, et al, 1986]
3.131 + * SET046-5 2.3 p42.in [ANL]; Pelletier 42 [Pelletier, 1986]
3.132 + * SET047-5 3.7 p43.in [ANL]; Pelletier 43 [Pelletier, 1986]
3.133 + * SYN034-1 2.8 QW [Michie, et al., 1972]; QW [Wilson & Minker, 1976]
3.134 + * SYN071-1 1.9 Pelletier 48 [Pelletier, 1986]
3.135 + * SYN349-1 61.7 Ch17N5 [Tammet, 1994]
3.136 + * SYN352-1 5.5 Ch18N4 [Tammet, 1994]
3.137 + * TOP001-2 61.1 Lemma 1a [Wick & McCune, 1989]
3.138 + * TOP002-2 0.4 Lemma 1b [Wick & McCune, 1989]
3.139 + * TOP004-1 181.6 Lemma 1d [Wick & McCune, 1989]
3.140 + * TOP004-2 9.0 Lemma 1d [Wick & McCune, 1989]
3.141 + * TOP005-2 139.8 Lemma 1e [Wick & McCune, 1989]
3.142 + *)
3.143 +
3.144 +(*51194 inferences so far. Searching to depth 13. 232.9 secs*)
3.145 +val BOO003_1 = prove_hard
3.146 + ("(! X. equal(X::'a,X)) & \
3.147 +\ (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
3.148 +\ (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) & \
3.149 +\ (! X Y. sum(X::'a,Y,add(X::'a,Y))) & \
3.150 +\ (! X Y. product(X::'a,Y,multiply(X::'a,Y))) & \
3.151 +\ (! Y X Z. sum(X::'a,Y,Z) --> sum(Y::'a,X,Z)) & \
3.152 +\ (! Y X Z. product(X::'a,Y,Z) --> product(Y::'a,X,Z)) & \
3.153 +\ (! X. sum(additive_identity::'a,X,X)) & \
3.154 +\ (! X. sum(X::'a,additive_identity,X)) & \
3.155 +\ (! X. product(multiplicative_identity::'a,X,X)) & \
3.156 +\ (! X. product(X::'a,multiplicative_identity,X)) & \
3.157 +\ (! 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)) & \
3.158 +\ (! 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)) & \
3.159 +\ (! 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)) & \
3.160 +\ (! 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)) & \
3.161 +\ (! 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)) & \
3.162 +\ (! 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)) & \
3.163 +\ (! 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)) & \
3.164 +\ (! 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)) & \
3.165 +\ (! X. sum(inverse(X),X,multiplicative_identity)) & \
3.166 +\ (! X. sum(X::'a,inverse(X),multiplicative_identity)) & \
3.167 +\ (! X. product(inverse(X),X,additive_identity)) & \
3.168 +\ (! X. product(X::'a,inverse(X),additive_identity)) & \
3.169 +\ (! X Y U V. sum(X::'a,Y,U) & sum(X::'a,Y,V) --> equal(U::'a,V)) & \
3.170 +\ (! X Y U V. product(X::'a,Y,U) & product(X::'a,Y,V) --> equal(U::'a,V)) & \
3.171 +\ (! X Y W Z. equal(X::'a,Y) & sum(X::'a,W,Z) --> sum(Y::'a,W,Z)) & \
3.172 +\ (! X W Y Z. equal(X::'a,Y) & sum(W::'a,X,Z) --> sum(W::'a,Y,Z)) & \
3.173 +\ (! X W Z Y. equal(X::'a,Y) & sum(W::'a,Z,X) --> sum(W::'a,Z,Y)) & \
3.174 +\ (! X Y W Z. equal(X::'a,Y) & product(X::'a,W,Z) --> product(Y::'a,W,Z)) & \
3.175 +\ (! X W Y Z. equal(X::'a,Y) & product(W::'a,X,Z) --> product(W::'a,Y,Z)) & \
3.176 +\ (! X W Z Y. equal(X::'a,Y) & product(W::'a,Z,X) --> product(W::'a,Z,Y)) & \
3.177 +\ (! X Y W. equal(X::'a,Y) --> equal(add(X::'a,W),add(Y::'a,W))) & \
3.178 +\ (! X W Y. equal(X::'a,Y) --> equal(add(W::'a,X),add(W::'a,Y))) & \
3.179 +\ (! X Y W. equal(X::'a,Y) --> equal(multiply(X::'a,W),multiply(Y::'a,W))) & \
3.180 +\ (! X W Y. equal(X::'a,Y) --> equal(multiply(W::'a,X),multiply(W::'a,Y))) & \
3.181 +\ (! X Y. equal(X::'a,Y) --> equal(inverse(X),inverse(Y))) & \
3.182 +\ (~product(x::'a,x,x)) --> False",
3.183 + meson_tac);
3.184 +
3.185 +(*51194 inferences so far. Searching to depth 13. 204.6 secs
3.186 + Strange! The previous problem also has 51194 inferences at depth 13. They
3.187 + must be very similar!*)
3.188 +val BOO004_1 = prove_hard
3.189 + ("(! X. equal(X::'a,X)) & \
3.190 +\ (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
3.191 +\ (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) & \
3.192 +\ (! X Y. sum(X::'a,Y,add(X::'a,Y))) & \
3.193 +\ (! X Y. product(X::'a,Y,multiply(X::'a,Y))) & \
3.194 +\ (! Y X Z. sum(X::'a,Y,Z) --> sum(Y::'a,X,Z)) & \
3.195 +\ (! Y X Z. product(X::'a,Y,Z) --> product(Y::'a,X,Z)) & \
3.196 +\ (! X. sum(additive_identity::'a,X,X)) & \
3.197 +\ (! X. sum(X::'a,additive_identity,X)) & \
3.198 +\ (! X. product(multiplicative_identity::'a,X,X)) & \
3.199 +\ (! X. product(X::'a,multiplicative_identity,X)) & \
3.200 +\ (! 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)) & \
3.201 +\ (! 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)) & \
3.202 +\ (! 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)) & \
3.203 +\ (! 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)) & \
3.204 +\ (! 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)) & \
3.205 +\ (! 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)) & \
3.206 +\ (! 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)) & \
3.207 +\ (! 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)) & \
3.208 +\ (! X. sum(inverse(X),X,multiplicative_identity)) & \
3.209 +\ (! X. sum(X::'a,inverse(X),multiplicative_identity)) & \
3.210 +\ (! X. product(inverse(X),X,additive_identity)) & \
3.211 +\ (! X. product(X::'a,inverse(X),additive_identity)) & \
3.212 +\ (! X Y U V. sum(X::'a,Y,U) & sum(X::'a,Y,V) --> equal(U::'a,V)) & \
3.213 +\ (! X Y U V. product(X::'a,Y,U) & product(X::'a,Y,V) --> equal(U::'a,V)) & \
3.214 +\ (! X Y W Z. equal(X::'a,Y) & sum(X::'a,W,Z) --> sum(Y::'a,W,Z)) & \
3.215 +\ (! X W Y Z. equal(X::'a,Y) & sum(W::'a,X,Z) --> sum(W::'a,Y,Z)) & \
3.216 +\ (! X W Z Y. equal(X::'a,Y) & sum(W::'a,Z,X) --> sum(W::'a,Z,Y)) & \
3.217 +\ (! X Y W Z. equal(X::'a,Y) & product(X::'a,W,Z) --> product(Y::'a,W,Z)) & \
3.218 +\ (! X W Y Z. equal(X::'a,Y) & product(W::'a,X,Z) --> product(W::'a,Y,Z)) & \
3.219 +\ (! X W Z Y. equal(X::'a,Y) & product(W::'a,Z,X) --> product(W::'a,Z,Y)) & \
3.220 +\ (! X Y W. equal(X::'a,Y) --> equal(add(X::'a,W),add(Y::'a,W))) & \
3.221 +\ (! X W Y. equal(X::'a,Y) --> equal(add(W::'a,X),add(W::'a,Y))) & \
3.222 +\ (! X Y W. equal(X::'a,Y) --> equal(multiply(X::'a,W),multiply(Y::'a,W))) & \
3.223 +\ (! X W Y. equal(X::'a,Y) --> equal(multiply(W::'a,X),multiply(W::'a,Y))) & \
3.224 +\ (! X Y. equal(X::'a,Y) --> equal(inverse(X),inverse(Y))) & \
3.225 +\ (~sum(x::'a,x,x)) --> False",
3.226 + meson_tac);
3.227 +
3.228 +(*74799 inferences so far. Searching to depth 13. 290.0 secs*)
3.229 +val BOO005_1 = prove_hard
3.230 + ("(! X. equal(X::'a,X)) & \
3.231 +\ (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
3.232 +\ (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) & \
3.233 +\ (! X Y. sum(X::'a,Y,add(X::'a,Y))) & \
3.234 +\ (! X Y. product(X::'a,Y,multiply(X::'a,Y))) & \
3.235 +\ (! Y X Z. sum(X::'a,Y,Z) --> sum(Y::'a,X,Z)) & \
3.236 +\ (! Y X Z. product(X::'a,Y,Z) --> product(Y::'a,X,Z)) & \
3.237 +\ (! X. sum(additive_identity::'a,X,X)) & \
3.238 +\ (! X. sum(X::'a,additive_identity,X)) & \
3.239 +\ (! X. product(multiplicative_identity::'a,X,X)) & \
3.240 +\ (! X. product(X::'a,multiplicative_identity,X)) & \
3.241 +\ (! 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)) & \
3.242 +\ (! 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)) & \
3.243 +\ (! 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)) & \
3.244 +\ (! 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)) & \
3.245 +\ (! 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)) & \
3.246 +\ (! 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)) & \
3.247 +\ (! 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)) & \
3.248 +\ (! 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)) & \
3.249 +\ (! X. sum(inverse(X),X,multiplicative_identity)) & \
3.250 +\ (! X. sum(X::'a,inverse(X),multiplicative_identity)) & \
3.251 +\ (! X. product(inverse(X),X,additive_identity)) & \
3.252 +\ (! X. product(X::'a,inverse(X),additive_identity)) & \
3.253 +\ (! X Y U V. sum(X::'a,Y,U) & sum(X::'a,Y,V) --> equal(U::'a,V)) & \
3.254 +\ (! X Y U V. product(X::'a,Y,U) & product(X::'a,Y,V) --> equal(U::'a,V)) & \
3.255 +\ (! X Y W Z. equal(X::'a,Y) & sum(X::'a,W,Z) --> sum(Y::'a,W,Z)) & \
3.256 +\ (! X W Y Z. equal(X::'a,Y) & sum(W::'a,X,Z) --> sum(W::'a,Y,Z)) & \
3.257 +\ (! X W Z Y. equal(X::'a,Y) & sum(W::'a,Z,X) --> sum(W::'a,Z,Y)) & \
3.258 +\ (! X Y W Z. equal(X::'a,Y) & product(X::'a,W,Z) --> product(Y::'a,W,Z)) & \
3.259 +\ (! X W Y Z. equal(X::'a,Y) & product(W::'a,X,Z) --> product(W::'a,Y,Z)) & \
3.260 +\ (! X W Z Y. equal(X::'a,Y) & product(W::'a,Z,X) --> product(W::'a,Z,Y)) & \
3.261 +\ (! X Y W. equal(X::'a,Y) --> equal(add(X::'a,W),add(Y::'a,W))) & \
3.262 +\ (! X W Y. equal(X::'a,Y) --> equal(add(W::'a,X),add(W::'a,Y))) & \
3.263 +\ (! X Y W. equal(X::'a,Y) --> equal(multiply(X::'a,W),multiply(Y::'a,W))) & \
3.264 +\ (! X W Y. equal(X::'a,Y) --> equal(multiply(W::'a,X),multiply(W::'a,Y))) & \
3.265 +\ (! X Y. equal(X::'a,Y) --> equal(inverse(X),inverse(Y))) & \
3.266 +\ (~sum(x::'a,multiplicative_identity,multiplicative_identity)) --> False",
3.267 + meson_tac);
3.268 +
3.269 +(*74799 inferences so far. Searching to depth 13. 314.6 secs*)
3.270 +val BOO006_1 = prove_hard
3.271 + ("(! X. equal(X::'a,X)) & \
3.272 +\ (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
3.273 +\ (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) & \
3.274 +\ (! X Y. sum(X::'a,Y,add(X::'a,Y))) & \
3.275 +\ (! X Y. product(X::'a,Y,multiply(X::'a,Y))) & \
3.276 +\ (! Y X Z. sum(X::'a,Y,Z) --> sum(Y::'a,X,Z)) & \
3.277 +\ (! Y X Z. product(X::'a,Y,Z) --> product(Y::'a,X,Z)) & \
3.278 +\ (! X. sum(additive_identity::'a,X,X)) & \
3.279 +\ (! X. sum(X::'a,additive_identity,X)) & \
3.280 +\ (! X. product(multiplicative_identity::'a,X,X)) & \
3.281 +\ (! X. product(X::'a,multiplicative_identity,X)) & \
3.282 +\ (! 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)) & \
3.283 +\ (! 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)) & \
3.284 +\ (! 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)) & \
3.285 +\ (! 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)) & \
3.286 +\ (! 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)) & \
3.287 +\ (! 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)) & \
3.288 +\ (! 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)) & \
3.289 +\ (! 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)) & \
3.290 +\ (! X. sum(inverse(X),X,multiplicative_identity)) & \
3.291 +\ (! X. sum(X::'a,inverse(X),multiplicative_identity)) & \
3.292 +\ (! X. product(inverse(X),X,additive_identity)) & \
3.293 +\ (! X. product(X::'a,inverse(X),additive_identity)) & \
3.294 +\ (! X Y U V. sum(X::'a,Y,U) & sum(X::'a,Y,V) --> equal(U::'a,V)) & \
3.295 +\ (! X Y U V. product(X::'a,Y,U) & product(X::'a,Y,V) --> equal(U::'a,V)) & \
3.296 +\ (! X Y W Z. equal(X::'a,Y) & sum(X::'a,W,Z) --> sum(Y::'a,W,Z)) & \
3.297 +\ (! X W Y Z. equal(X::'a,Y) & sum(W::'a,X,Z) --> sum(W::'a,Y,Z)) & \
3.298 +\ (! X W Z Y. equal(X::'a,Y) & sum(W::'a,Z,X) --> sum(W::'a,Z,Y)) & \
3.299 +\ (! X Y W Z. equal(X::'a,Y) & product(X::'a,W,Z) --> product(Y::'a,W,Z)) & \
3.300 +\ (! X W Y Z. equal(X::'a,Y) & product(W::'a,X,Z) --> product(W::'a,Y,Z)) & \
3.301 +\ (! X W Z Y. equal(X::'a,Y) & product(W::'a,Z,X) --> product(W::'a,Z,Y)) & \
3.302 +\ (! X Y W. equal(X::'a,Y) --> equal(add(X::'a,W),add(Y::'a,W))) & \
3.303 +\ (! X W Y. equal(X::'a,Y) --> equal(add(W::'a,X),add(W::'a,Y))) & \
3.304 +\ (! X Y W. equal(X::'a,Y) --> equal(multiply(X::'a,W),multiply(Y::'a,W))) & \
3.305 +\ (! X W Y. equal(X::'a,Y) --> equal(multiply(W::'a,X),multiply(W::'a,Y))) & \
3.306 +\ (! X Y. equal(X::'a,Y) --> equal(inverse(X),inverse(Y))) & \
3.307 +\ (~product(x::'a,additive_identity,additive_identity)) --> False",
3.308 + meson_tac);
3.309 +
3.310 +(*5 inferences so far. Searching to depth 5. 1.3 secs*)
3.311 +val BOO011_1 = prove
3.312 + ("(! X. equal(X::'a,X)) & \
3.313 +\ (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
3.314 +\ (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) & \
3.315 +\ (! X Y. sum(X::'a,Y,add(X::'a,Y))) & \
3.316 +\ (! X Y. product(X::'a,Y,multiply(X::'a,Y))) & \
3.317 +\ (! Y X Z. sum(X::'a,Y,Z) --> sum(Y::'a,X,Z)) & \
3.318 +\ (! Y X Z. product(X::'a,Y,Z) --> product(Y::'a,X,Z)) & \
3.319 +\ (! X. sum(additive_identity::'a,X,X)) & \
3.320 +\ (! X. sum(X::'a,additive_identity,X)) & \
3.321 +\ (! X. product(multiplicative_identity::'a,X,X)) & \
3.322 +\ (! X. product(X::'a,multiplicative_identity,X)) & \
3.323 +\ (! 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)) & \
3.324 +\ (! 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)) & \
3.325 +\ (! 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)) & \
3.326 +\ (! 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)) & \
3.327 +\ (! 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)) & \
3.328 +\ (! 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)) & \
3.329 +\ (! 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)) & \
3.330 +\ (! 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)) & \
3.331 +\ (! X. sum(inverse(X),X,multiplicative_identity)) & \
3.332 +\ (! X. sum(X::'a,inverse(X),multiplicative_identity)) & \
3.333 +\ (! X. product(inverse(X),X,additive_identity)) & \
3.334 +\ (! X. product(X::'a,inverse(X),additive_identity)) & \
3.335 +\ (! X Y U V. sum(X::'a,Y,U) & sum(X::'a,Y,V) --> equal(U::'a,V)) & \
3.336 +\ (! X Y U V. product(X::'a,Y,U) & product(X::'a,Y,V) --> equal(U::'a,V)) & \
3.337 +\ (! X Y W Z. equal(X::'a,Y) & sum(X::'a,W,Z) --> sum(Y::'a,W,Z)) & \
3.338 +\ (! X W Y Z. equal(X::'a,Y) & sum(W::'a,X,Z) --> sum(W::'a,Y,Z)) & \
3.339 +\ (! X W Z Y. equal(X::'a,Y) & sum(W::'a,Z,X) --> sum(W::'a,Z,Y)) & \
3.340 +\ (! X Y W Z. equal(X::'a,Y) & product(X::'a,W,Z) --> product(Y::'a,W,Z)) & \
3.341 +\ (! X W Y Z. equal(X::'a,Y) & product(W::'a,X,Z) --> product(W::'a,Y,Z)) & \
3.342 +\ (! X W Z Y. equal(X::'a,Y) & product(W::'a,Z,X) --> product(W::'a,Z,Y)) & \
3.343 +\ (! X Y W. equal(X::'a,Y) --> equal(add(X::'a,W),add(Y::'a,W))) & \
3.344 +\ (! X W Y. equal(X::'a,Y) --> equal(add(W::'a,X),add(W::'a,Y))) & \
3.345 +\ (! X Y W. equal(X::'a,Y) --> equal(multiply(X::'a,W),multiply(Y::'a,W))) & \
3.346 +\ (! X W Y. equal(X::'a,Y) --> equal(multiply(W::'a,X),multiply(W::'a,Y))) & \
3.347 +\ (! X Y. equal(X::'a,Y) --> equal(inverse(X),inverse(Y))) & \
3.348 +\ (~equal(inverse(additive_identity),multiplicative_identity)) --> False",
3.349 + meson_tac);
3.350 +
3.351 +(*4007 inferences so far. Searching to depth 9. 13 secs*)
3.352 +val CAT001_3 = prove_hard
3.353 + ("(! X. equal(X::'a,X)) & \
3.354 +\ (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
3.355 +\ (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) & \
3.356 +\ (! Y X. equivalent(X::'a,Y) --> there_exists(X)) & \
3.357 +\ (! X Y. equivalent(X::'a,Y) --> equal(X::'a,Y)) & \
3.358 +\ (! X Y. there_exists(X) & equal(X::'a,Y) --> equivalent(X::'a,Y)) & \
3.359 +\ (! X. there_exists(domain(X)) --> there_exists(X)) & \
3.360 +\ (! X. there_exists(codomain(X)) --> there_exists(X)) & \
3.361 +\ (! Y X. there_exists(compos(X::'a,Y)) --> there_exists(domain(X))) & \
3.362 +\ (! X Y. there_exists(compos(X::'a,Y)) --> equal(domain(X),codomain(Y))) & \
3.363 +\ (! X Y. there_exists(domain(X)) & equal(domain(X),codomain(Y)) --> there_exists(compos(X::'a,Y))) & \
3.364 +\ (! X Y Z. equal(compos(X::'a,compos(Y::'a,Z)),compos(compos(X::'a,Y),Z))) & \
3.365 +\ (! X. equal(compos(X::'a,domain(X)),X)) & \
3.366 +\ (! X. equal(compos(codomain(X),X),X)) & \
3.367 +\ (! X Y. equivalent(X::'a,Y) --> there_exists(Y)) & \
3.368 +\ (! X Y. there_exists(X) & there_exists(Y) & equal(X::'a,Y) --> equivalent(X::'a,Y)) & \
3.369 +\ (! Y X. there_exists(compos(X::'a,Y)) --> there_exists(codomain(X))) & \
3.370 +\ (! X Y. there_exists(f1(X::'a,Y)) | equal(X::'a,Y)) & \
3.371 +\ (! X Y. equal(X::'a,f1(X::'a,Y)) | equal(Y::'a,f1(X::'a,Y)) | equal(X::'a,Y)) & \
3.372 +\ (! X Y. equal(X::'a,f1(X::'a,Y)) & equal(Y::'a,f1(X::'a,Y)) --> equal(X::'a,Y)) & \
3.373 +\ (! X Y. equal(X::'a,Y) & there_exists(X) --> there_exists(Y)) & \
3.374 +\ (! X Y Z. equal(X::'a,Y) & equivalent(X::'a,Z) --> equivalent(Y::'a,Z)) & \
3.375 +\ (! X Z Y. equal(X::'a,Y) & equivalent(Z::'a,X) --> equivalent(Z::'a,Y)) & \
3.376 +\ (! X Y. equal(X::'a,Y) --> equal(domain(X),domain(Y))) & \
3.377 +\ (! X Y. equal(X::'a,Y) --> equal(codomain(X),codomain(Y))) & \
3.378 +\ (! X Y Z. equal(X::'a,Y) --> equal(compos(X::'a,Z),compos(Y::'a,Z))) & \
3.379 +\ (! X Z Y. equal(X::'a,Y) --> equal(compos(Z::'a,X),compos(Z::'a,Y))) & \
3.380 +\ (! A B C. equal(A::'a,B) --> equal(f1(A::'a,C),f1(B::'a,C))) & \
3.381 +\ (! D F' E. equal(D::'a,E) --> equal(f1(F'::'a,D),f1(F'::'a,E))) & \
3.382 +\ (there_exists(compos(a::'a,b))) & \
3.383 +\ (! Y X Z. equal(compos(compos(a::'a,b),X),Y) & equal(compos(compos(a::'a,b),Z),Y) --> equal(X::'a,Z)) & \
3.384 +\ (there_exists(compos(b::'a,h))) & \
3.385 +\ (equal(compos(b::'a,h),compos(b::'a,g))) & \
3.386 +\ (~equal(h::'a,g)) --> False",
3.387 + meson_tac);
3.388 +
3.389 +(*245 inferences so far. Searching to depth 7. 1.0 secs*)
3.390 +val CAT003_3 = prove
3.391 + ("(! X. equal(X::'a,X)) & \
3.392 +\ (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
3.393 +\ (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) & \
3.394 +\ (! Y X. equivalent(X::'a,Y) --> there_exists(X)) & \
3.395 +\ (! X Y. equivalent(X::'a,Y) --> equal(X::'a,Y)) & \
3.396 +\ (! X Y. there_exists(X) & equal(X::'a,Y) --> equivalent(X::'a,Y)) & \
3.397 +\ (! X. there_exists(domain(X)) --> there_exists(X)) & \
3.398 +\ (! X. there_exists(codomain(X)) --> there_exists(X)) & \
3.399 +\ (! Y X. there_exists(compos(X::'a,Y)) --> there_exists(domain(X))) & \
3.400 +\ (! X Y. there_exists(compos(X::'a,Y)) --> equal(domain(X),codomain(Y))) & \
3.401 +\ (! X Y. there_exists(domain(X)) & equal(domain(X),codomain(Y)) --> there_exists(compos(X::'a,Y))) & \
3.402 +\ (! X Y Z. equal(compos(X::'a,compos(Y::'a,Z)),compos(compos(X::'a,Y),Z))) & \
3.403 +\ (! X. equal(compos(X::'a,domain(X)),X)) & \
3.404 +\ (! X. equal(compos(codomain(X),X),X)) & \
3.405 +\ (! X Y. equivalent(X::'a,Y) --> there_exists(Y)) & \
3.406 +\ (! X Y. there_exists(X) & there_exists(Y) & equal(X::'a,Y) --> equivalent(X::'a,Y)) & \
3.407 +\ (! Y X. there_exists(compos(X::'a,Y)) --> there_exists(codomain(X))) & \
3.408 +\ (! X Y. there_exists(f1(X::'a,Y)) | equal(X::'a,Y)) & \
3.409 +\ (! X Y. equal(X::'a,f1(X::'a,Y)) | equal(Y::'a,f1(X::'a,Y)) | equal(X::'a,Y)) & \
3.410 +\ (! X Y. equal(X::'a,f1(X::'a,Y)) & equal(Y::'a,f1(X::'a,Y)) --> equal(X::'a,Y)) & \
3.411 +\ (! X Y. equal(X::'a,Y) & there_exists(X) --> there_exists(Y)) & \
3.412 +\ (! X Y Z. equal(X::'a,Y) & equivalent(X::'a,Z) --> equivalent(Y::'a,Z)) & \
3.413 +\ (! X Z Y. equal(X::'a,Y) & equivalent(Z::'a,X) --> equivalent(Z::'a,Y)) & \
3.414 +\ (! X Y. equal(X::'a,Y) --> equal(domain(X),domain(Y))) & \
3.415 +\ (! X Y. equal(X::'a,Y) --> equal(codomain(X),codomain(Y))) & \
3.416 +\ (! X Y Z. equal(X::'a,Y) --> equal(compos(X::'a,Z),compos(Y::'a,Z))) & \
3.417 +\ (! X Z Y. equal(X::'a,Y) --> equal(compos(Z::'a,X),compos(Z::'a,Y))) & \
3.418 +\ (! A B C. equal(A::'a,B) --> equal(f1(A::'a,C),f1(B::'a,C))) & \
3.419 +\ (! D F' E. equal(D::'a,E) --> equal(f1(F'::'a,D),f1(F'::'a,E))) & \
3.420 +\ (there_exists(compos(a::'a,b))) & \
3.421 +\ (! 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)) & \
3.422 +\ (there_exists(h)) & \
3.423 +\ (equal(compos(h::'a,a),compos(g::'a,a))) & \
3.424 +\ (~equal(g::'a,h)) --> False",
3.425 + meson_tac);
3.426 +
3.427 +(*54288 inferences so far. Searching to depth 14. 118.0 secs*)
3.428 +val CAT005_1 = prove_hard
3.429 + ("(! X. equal(X::'a,X)) & \
3.430 +\ (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
3.431 +\ (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) & \
3.432 +\ (! X Y. defined(X::'a,Y) --> product(X::'a,Y,compos(X::'a,Y))) & \
3.433 +\ (! Z X Y. product(X::'a,Y,Z) --> defined(X::'a,Y)) & \
3.434 +\ (! X Xy Y Z. product(X::'a,Y,Xy) & defined(Xy::'a,Z) --> defined(Y::'a,Z)) & \
3.435 +\ (! Y Xy Z X Yz. product(X::'a,Y,Xy) & product(Y::'a,Z,Yz) & defined(Xy::'a,Z) --> defined(X::'a,Yz)) & \
3.436 +\ (! 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)) & \
3.437 +\ (! Z Yz X Y. product(Y::'a,Z,Yz) & defined(X::'a,Yz) --> defined(X::'a,Y)) & \
3.438 +\ (! Y X Yz Xy Z. product(Y::'a,Z,Yz) & product(X::'a,Y,Xy) & defined(X::'a,Yz) --> defined(Xy::'a,Z)) & \
3.439 +\ (! 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)) & \
3.440 +\ (! Y X Z. defined(X::'a,Y) & defined(Y::'a,Z) & identity_map(Y) --> defined(X::'a,Z)) & \
3.441 +\ (! X. identity_map(domain(X))) & \
3.442 +\ (! X. identity_map(codomain(X))) & \
3.443 +\ (! X. defined(X::'a,domain(X))) & \
3.444 +\ (! X. defined(codomain(X),X)) & \
3.445 +\ (! X. product(X::'a,domain(X),X)) & \
3.446 +\ (! X. product(codomain(X),X,X)) & \
3.447 +\ (! X Y. defined(X::'a,Y) & identity_map(X) --> product(X::'a,Y,Y)) & \
3.448 +\ (! Y X. defined(X::'a,Y) & identity_map(Y) --> product(X::'a,Y,X)) & \
3.449 +\ (! X Y Z W. product(X::'a,Y,Z) & product(X::'a,Y,W) --> equal(Z::'a,W)) & \
3.450 +\ (! X Y Z W. equal(X::'a,Y) & product(X::'a,Z,W) --> product(Y::'a,Z,W)) & \
3.451 +\ (! X Z Y W. equal(X::'a,Y) & product(Z::'a,X,W) --> product(Z::'a,Y,W)) & \
3.452 +\ (! X Z W Y. equal(X::'a,Y) & product(Z::'a,W,X) --> product(Z::'a,W,Y)) & \
3.453 +\ (! X Y. equal(X::'a,Y) --> equal(domain(X),domain(Y))) & \
3.454 +\ (! X Y. equal(X::'a,Y) --> equal(codomain(X),codomain(Y))) & \
3.455 +\ (! X Y. equal(X::'a,Y) & identity_map(X) --> identity_map(Y)) & \
3.456 +\ (! X Y Z. equal(X::'a,Y) & defined(X::'a,Z) --> defined(Y::'a,Z)) & \
3.457 +\ (! X Z Y. equal(X::'a,Y) & defined(Z::'a,X) --> defined(Z::'a,Y)) & \
3.458 +\ (! X Z Y. equal(X::'a,Y) --> equal(compos(Z::'a,X),compos(Z::'a,Y))) & \
3.459 +\ (! X Y Z. equal(X::'a,Y) --> equal(compos(X::'a,Z),compos(Y::'a,Z))) & \
3.460 +\ (defined(a::'a,d)) & \
3.461 +\ (identity_map(d)) & \
3.462 +\ (~equal(domain(a),d)) --> False",
3.463 + meson_tac);
3.464 +
3.465 +
3.466 +(*1728 inferences so far. Searching to depth 10. 5.8 secs*)
3.467 +val CAT007_1 = prove_hard
3.468 + ("(! X. equal(X::'a,X)) & \
3.469 +\ (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
3.470 +\ (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) & \
3.471 +\ (! X Y. defined(X::'a,Y) --> product(X::'a,Y,compos(X::'a,Y))) & \
3.472 +\ (! Z X Y. product(X::'a,Y,Z) --> defined(X::'a,Y)) & \
3.473 +\ (! X Xy Y Z. product(X::'a,Y,Xy) & defined(Xy::'a,Z) --> defined(Y::'a,Z)) & \
3.474 +\ (! Y Xy Z X Yz. product(X::'a,Y,Xy) & product(Y::'a,Z,Yz) & defined(Xy::'a,Z) --> defined(X::'a,Yz)) & \
3.475 +\ (! 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)) & \
3.476 +\ (! Z Yz X Y. product(Y::'a,Z,Yz) & defined(X::'a,Yz) --> defined(X::'a,Y)) & \
3.477 +\ (! Y X Yz Xy Z. product(Y::'a,Z,Yz) & product(X::'a,Y,Xy) & defined(X::'a,Yz) --> defined(Xy::'a,Z)) & \
3.478 +\ (! 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)) & \
3.479 +\ (! Y X Z. defined(X::'a,Y) & defined(Y::'a,Z) & identity_map(Y) --> defined(X::'a,Z)) & \
3.480 +\ (! X. identity_map(domain(X))) & \
3.481 +\ (! X. identity_map(codomain(X))) & \
3.482 +\ (! X. defined(X::'a,domain(X))) & \
3.483 +\ (! X. defined(codomain(X),X)) & \
3.484 +\ (! X. product(X::'a,domain(X),X)) & \
3.485 +\ (! X. product(codomain(X),X,X)) & \
3.486 +\ (! X Y. defined(X::'a,Y) & identity_map(X) --> product(X::'a,Y,Y)) & \
3.487 +\ (! Y X. defined(X::'a,Y) & identity_map(Y) --> product(X::'a,Y,X)) & \
3.488 +\ (! X Y Z W. product(X::'a,Y,Z) & product(X::'a,Y,W) --> equal(Z::'a,W)) & \
3.489 +\ (! X Y Z W. equal(X::'a,Y) & product(X::'a,Z,W) --> product(Y::'a,Z,W)) & \
3.490 +\ (! X Z Y W. equal(X::'a,Y) & product(Z::'a,X,W) --> product(Z::'a,Y,W)) & \
3.491 +\ (! X Z W Y. equal(X::'a,Y) & product(Z::'a,W,X) --> product(Z::'a,W,Y)) & \
3.492 +\ (! X Y. equal(X::'a,Y) --> equal(domain(X),domain(Y))) & \
3.493 +\ (! X Y. equal(X::'a,Y) --> equal(codomain(X),codomain(Y))) & \
3.494 +\ (! X Y. equal(X::'a,Y) & identity_map(X) --> identity_map(Y)) & \
3.495 +\ (! X Y Z. equal(X::'a,Y) & defined(X::'a,Z) --> defined(Y::'a,Z)) & \
3.496 +\ (! X Z Y. equal(X::'a,Y) & defined(Z::'a,X) --> defined(Z::'a,Y)) & \
3.497 +\ (! X Z Y. equal(X::'a,Y) --> equal(compos(Z::'a,X),compos(Z::'a,Y))) & \
3.498 +\ (! X Y Z. equal(X::'a,Y) --> equal(compos(X::'a,Z),compos(Y::'a,Z))) & \
3.499 +\ (equal(domain(a),codomain(b))) & \
3.500 +\ (~defined(a::'a,b)) --> False",
3.501 + meson_tac);
3.502 +
3.503 +(*82895 inferences so far. Searching to depth 13. 355 secs*)
3.504 +val CAT018_1 = prove_hard
3.505 + ("(! X. equal(X::'a,X)) & \
3.506 +\ (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
3.507 +\ (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) & \
3.508 +\ (! X Y. defined(X::'a,Y) --> product(X::'a,Y,compos(X::'a,Y))) & \
3.509 +\ (! Z X Y. product(X::'a,Y,Z) --> defined(X::'a,Y)) & \
3.510 +\ (! X Xy Y Z. product(X::'a,Y,Xy) & defined(Xy::'a,Z) --> defined(Y::'a,Z)) & \
3.511 +\ (! Y Xy Z X Yz. product(X::'a,Y,Xy) & product(Y::'a,Z,Yz) & defined(Xy::'a,Z) --> defined(X::'a,Yz)) & \
3.512 +\ (! 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)) & \
3.513 +\ (! Z Yz X Y. product(Y::'a,Z,Yz) & defined(X::'a,Yz) --> defined(X::'a,Y)) & \
3.514 +\ (! Y X Yz Xy Z. product(Y::'a,Z,Yz) & product(X::'a,Y,Xy) & defined(X::'a,Yz) --> defined(Xy::'a,Z)) & \
3.515 +\ (! 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)) & \
3.516 +\ (! Y X Z. defined(X::'a,Y) & defined(Y::'a,Z) & identity_map(Y) --> defined(X::'a,Z)) & \
3.517 +\ (! X. identity_map(domain(X))) & \
3.518 +\ (! X. identity_map(codomain(X))) & \
3.519 +\ (! X. defined(X::'a,domain(X))) & \
3.520 +\ (! X. defined(codomain(X),X)) & \
3.521 +\ (! X. product(X::'a,domain(X),X)) & \
3.522 +\ (! X. product(codomain(X),X,X)) & \
3.523 +\ (! X Y. defined(X::'a,Y) & identity_map(X) --> product(X::'a,Y,Y)) & \
3.524 +\ (! Y X. defined(X::'a,Y) & identity_map(Y) --> product(X::'a,Y,X)) & \
3.525 +\ (! X Y Z W. product(X::'a,Y,Z) & product(X::'a,Y,W) --> equal(Z::'a,W)) & \
3.526 +\ (! X Y Z W. equal(X::'a,Y) & product(X::'a,Z,W) --> product(Y::'a,Z,W)) & \
3.527 +\ (! X Z Y W. equal(X::'a,Y) & product(Z::'a,X,W) --> product(Z::'a,Y,W)) & \
3.528 +\ (! X Z W Y. equal(X::'a,Y) & product(Z::'a,W,X) --> product(Z::'a,W,Y)) & \
3.529 +\ (! X Y. equal(X::'a,Y) --> equal(domain(X),domain(Y))) & \
3.530 +\ (! X Y. equal(X::'a,Y) --> equal(codomain(X),codomain(Y))) & \
3.531 +\ (! X Y. equal(X::'a,Y) & identity_map(X) --> identity_map(Y)) & \
3.532 +\ (! X Y Z. equal(X::'a,Y) & defined(X::'a,Z) --> defined(Y::'a,Z)) & \
3.533 +\ (! X Z Y. equal(X::'a,Y) & defined(Z::'a,X) --> defined(Z::'a,Y)) & \
3.534 +\ (! X Z Y. equal(X::'a,Y) --> equal(compos(Z::'a,X),compos(Z::'a,Y))) & \
3.535 +\ (! X Y Z. equal(X::'a,Y) --> equal(compos(X::'a,Z),compos(Y::'a,Z))) & \
3.536 +\ (defined(a::'a,b)) & \
3.537 +\ (defined(b::'a,c)) & \
3.538 +\ (~defined(a::'a,compos(b::'a,c))) --> False",
3.539 + meson_tac);
3.540 +
3.541 +(*1118 inferences so far. Searching to depth 8. 2.3 secs*)
3.542 +val COL001_2 = prove
3.543 + ("(! X. equal(X::'a,X)) & \
3.544 +\ (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
3.545 +\ (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) & \
3.546 +\ (! X Y Z. equal(apply(apply(apply(s::'a,X),Y),Z),apply(apply(X::'a,Z),apply(Y::'a,Z)))) & \
3.547 +\ (! Y X. equal(apply(apply(k::'a,X),Y),X)) & \
3.548 +\ (! X Y Z. equal(apply(apply(apply(b::'a,X),Y),Z),apply(X::'a,apply(Y::'a,Z)))) & \
3.549 +\ (! X. equal(apply(i::'a,X),X)) & \
3.550 +\ (! A B C. equal(A::'a,B) --> equal(apply(A::'a,C),apply(B::'a,C))) & \
3.551 +\ (! D F' E. equal(D::'a,E) --> equal(apply(F'::'a,D),apply(F'::'a,E))) & \
3.552 +\ (! 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))))) & \
3.553 +\ (! Y. ~equal(Y::'a,apply(combinator::'a,Y))) --> False",
3.554 + meson_tac);
3.555 +
3.556 +(*500 inferences so far. Searching to depth 8. 0.9 secs*)
3.557 +val COL023_1 = prove
3.558 + ("(! X. equal(X::'a,X)) & \
3.559 +\ (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
3.560 +\ (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) & \
3.561 +\ (! X Y Z. equal(apply(apply(apply(b::'a,X),Y),Z),apply(X::'a,apply(Y::'a,Z)))) & \
3.562 +\ (! X Y Z. equal(apply(apply(apply(n::'a,X),Y),Z),apply(apply(apply(X::'a,Z),Y),Z))) & \
3.563 +\ (! A B C. equal(A::'a,B) --> equal(apply(A::'a,C),apply(B::'a,C))) & \
3.564 +\ (! D F' E. equal(D::'a,E) --> equal(apply(F'::'a,D),apply(F'::'a,E))) & \
3.565 +\ (! Y. ~equal(Y::'a,apply(combinator::'a,Y))) --> False",
3.566 + meson_tac);
3.567 +
3.568 +(*3018 inferences so far. Searching to depth 10. 4.3 secs*)
3.569 +val COL032_1 = prove_hard
3.570 + ("(! X. equal(X::'a,X)) & \
3.571 +\ (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
3.572 +\ (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) & \
3.573 +\ (! X. equal(apply(m::'a,X),apply(X::'a,X))) & \
3.574 +\ (! Y X Z. equal(apply(apply(apply(q::'a,X),Y),Z),apply(Y::'a,apply(X::'a,Z)))) & \
3.575 +\ (! A B C. equal(A::'a,B) --> equal(apply(A::'a,C),apply(B::'a,C))) & \
3.576 +\ (! D F' E. equal(D::'a,E) --> equal(apply(F'::'a,D),apply(F'::'a,E))) & \
3.577 +\ (! G H. equal(G::'a,H) --> equal(f(G),f(H))) & \
3.578 +\ (! Y. ~equal(apply(Y::'a,f(Y)),apply(f(Y),apply(Y::'a,f(Y))))) --> False",
3.579 + meson_tac);
3.580 +
3.581 +(*381878 inferences so far. Searching to depth 13. 670.4 secs*)
3.582 +val COL052_2 = prove_hard
3.583 + ("(! X. equal(X::'a,X)) & \
3.584 +\ (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
3.585 +\ (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) & \
3.586 +\ (! X Y W. equal(response(compos(X::'a,Y),W),response(X::'a,response(Y::'a,W)))) & \
3.587 +\ (! X Y. agreeable(X) --> equal(response(X::'a,common_bird(Y)),response(Y::'a,common_bird(Y)))) & \
3.588 +\ (! Z X. equal(response(X::'a,Z),response(compatible(X),Z)) --> agreeable(X)) & \
3.589 +\ (! A B. equal(A::'a,B) --> equal(common_bird(A),common_bird(B))) & \
3.590 +\ (! C D. equal(C::'a,D) --> equal(compatible(C),compatible(D))) & \
3.591 +\ (! Q R. equal(Q::'a,R) & agreeable(Q) --> agreeable(R)) & \
3.592 +\ (! A B C. equal(A::'a,B) --> equal(compos(A::'a,C),compos(B::'a,C))) & \
3.593 +\ (! D F' E. equal(D::'a,E) --> equal(compos(F'::'a,D),compos(F'::'a,E))) & \
3.594 +\ (! G H I'. equal(G::'a,H) --> equal(response(G::'a,I'),response(H::'a,I'))) & \
3.595 +\ (! J L K'. equal(J::'a,K') --> equal(response(L::'a,J),response(L::'a,K'))) & \
3.596 +\ (agreeable(c)) & \
3.597 +\ (~agreeable(a)) & \
3.598 +\ (equal(c::'a,compos(a::'a,b))) --> False",
3.599 + meson_tac);
3.600 +
3.601 +(*13201 inferences so far. Searching to depth 11. 31.9 secs*)
3.602 +val COL075_2 = prove_hard
3.603 + ("(! X. equal(X::'a,X)) & \
3.604 +\ (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
3.605 +\ (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) & \
3.606 +\ (! Y X. equal(apply(apply(k::'a,X),Y),X)) & \
3.607 +\ (! X Y Z. equal(apply(apply(apply(abstraction::'a,X),Y),Z),apply(apply(X::'a,apply(k::'a,Z)),apply(Y::'a,Z)))) & \
3.608 +\ (! D E F'. equal(D::'a,E) --> equal(apply(D::'a,F'),apply(E::'a,F'))) & \
3.609 +\ (! G I' H. equal(G::'a,H) --> equal(apply(I'::'a,G),apply(I'::'a,H))) & \
3.610 +\ (! A B. equal(A::'a,B) --> equal(b(A),b(B))) & \
3.611 +\ (! C D. equal(C::'a,D) --> equal(c(C),c(D))) & \
3.612 +\ (! Y. ~equal(apply(apply(Y::'a,b(Y)),c(Y)),apply(b(Y),b(Y)))) --> False",
3.613 + meson_tac);
3.614 +
3.615 +(*33 inferences so far. Searching to depth 7. 0.1 secs*)
3.616 +val COM001_1 = prove
3.617 + ("(! Goal_state Start_state. follows(Goal_state::'a,Start_state) --> succeeds(Goal_state::'a,Start_state)) & \
3.618 +\ (! Goal_state Intermediate_state Start_state. succeeds(Goal_state::'a,Intermediate_state) & succeeds(Intermediate_state::'a,Start_state) --> succeeds(Goal_state::'a,Start_state)) & \
3.619 +\ (! Start_state Label Goal_state. has(Start_state::'a,goto(Label)) & labels(Label::'a,Goal_state) --> succeeds(Goal_state::'a,Start_state)) & \
3.620 +\ (! Start_state Condition Goal_state. has(Start_state::'a,ifthen(Condition::'a,Goal_state)) --> succeeds(Goal_state::'a,Start_state)) & \
3.621 +\ (labels(loop::'a,p3)) & \
3.622 +\ (has(p3::'a,ifthen(equal(register_j::'a,n),p4))) & \
3.623 +\ (has(p4::'a,goto(out))) & \
3.624 +\ (follows(p5::'a,p4)) & \
3.625 +\ (follows(p8::'a,p3)) & \
3.626 +\ (has(p8::'a,goto(loop))) & \
3.627 +\ (~succeeds(p3::'a,p3)) --> False",
3.628 + meson_tac);
3.629 +
3.630 +(*533 inferences so far. Searching to depth 13. 0.3 secs*)
3.631 +val COM002_1 = prove
3.632 + ("(! Goal_state Start_state. follows(Goal_state::'a,Start_state) --> succeeds(Goal_state::'a,Start_state)) & \
3.633 +\ (! Goal_state Intermediate_state Start_state. succeeds(Goal_state::'a,Intermediate_state) & succeeds(Intermediate_state::'a,Start_state) --> succeeds(Goal_state::'a,Start_state)) & \
3.634 +\ (! Start_state Label Goal_state. has(Start_state::'a,goto(Label)) & labels(Label::'a,Goal_state) --> succeeds(Goal_state::'a,Start_state)) & \
3.635 +\ (! Start_state Condition Goal_state. has(Start_state::'a,ifthen(Condition::'a,Goal_state)) --> succeeds(Goal_state::'a,Start_state)) & \
3.636 +\ (has(p1::'a,assign(register_j::'a,num0))) & \
3.637 +\ (follows(p2::'a,p1)) & \
3.638 +\ (has(p2::'a,assign(register_k::'a,num1))) & \
3.639 +\ (labels(loop::'a,p3)) & \
3.640 +\ (follows(p3::'a,p2)) & \
3.641 +\ (has(p3::'a,ifthen(equal(register_j::'a,n),p4))) & \
3.642 +\ (has(p4::'a,goto(out))) & \
3.643 +\ (follows(p5::'a,p4)) & \
3.644 +\ (follows(p6::'a,p3)) & \
3.645 +\ (has(p6::'a,assign(register_k::'a,times(num2::'a,register_k)))) & \
3.646 +\ (follows(p7::'a,p6)) & \
3.647 +\ (has(p7::'a,assign(register_j::'a,plus(register_j::'a,num1)))) & \
3.648 +\ (follows(p8::'a,p7)) & \
3.649 +\ (has(p8::'a,goto(loop))) & \
3.650 +\ (~succeeds(p3::'a,p3)) --> False",
3.651 + meson_tac);
3.652 +
3.653 +(*4821 inferences so far. Searching to depth 14. 1.3 secs*)
3.654 +val COM002_2 = prove
3.655 + ("(! Goal_state Start_state. ~(fails(Goal_state::'a,Start_state) & follows(Goal_state::'a,Start_state))) & \
3.656 +\ (! Goal_state Intermediate_state Start_state. fails(Goal_state::'a,Start_state) --> fails(Goal_state::'a,Intermediate_state) | fails(Intermediate_state::'a,Start_state)) & \
3.657 +\ (! Start_state Label Goal_state. ~(fails(Goal_state::'a,Start_state) & has(Start_state::'a,goto(Label)) & labels(Label::'a,Goal_state))) & \
3.658 +\ (! Start_state Condition Goal_state. ~(fails(Goal_state::'a,Start_state) & has(Start_state::'a,ifthen(Condition::'a,Goal_state)))) & \
3.659 +\ (has(p1::'a,assign(register_j::'a,num0))) & \
3.660 +\ (follows(p2::'a,p1)) & \
3.661 +\ (has(p2::'a,assign(register_k::'a,num1))) & \
3.662 +\ (labels(loop::'a,p3)) & \
3.663 +\ (follows(p3::'a,p2)) & \
3.664 +\ (has(p3::'a,ifthen(equal(register_j::'a,n),p4))) & \
3.665 +\ (has(p4::'a,goto(out))) & \
3.666 +\ (follows(p5::'a,p4)) & \
3.667 +\ (follows(p6::'a,p3)) & \
3.668 +\ (has(p6::'a,assign(register_k::'a,times(num2::'a,register_k)))) & \
3.669 +\ (follows(p7::'a,p6)) & \
3.670 +\ (has(p7::'a,assign(register_j::'a,plus(register_j::'a,num1)))) & \
3.671 +\ (follows(p8::'a,p7)) & \
3.672 +\ (has(p8::'a,goto(loop))) & \
3.673 +\ (fails(p3::'a,p3)) --> False",
3.674 + meson_tac);
3.675 +
3.676 +(*98 inferences so far. Searching to depth 10. 1.1 secs*)
3.677 +val COM003_2 = prove
3.678 + ("(! X Y Z. program_decides(X) & program(Y) --> decides(X::'a,Y,Z)) & \
3.679 +\ (! X. program_decides(X) | program(f2(X))) & \
3.680 +\ (! X. decides(X::'a,f2(X),f1(X)) --> program_decides(X)) & \
3.681 +\ (! X. program_program_decides(X) --> program(X)) & \
3.682 +\ (! X. program_program_decides(X) --> program_decides(X)) & \
3.683 +\ (! X. program(X) & program_decides(X) --> program_program_decides(X)) & \
3.684 +\ (! X. algorithm_program_decides(X) --> algorithm(X)) & \
3.685 +\ (! X. algorithm_program_decides(X) --> program_decides(X)) & \
3.686 +\ (! X. algorithm(X) & program_decides(X) --> algorithm_program_decides(X)) & \
3.687 +\ (! Y X. program_halts2(X::'a,Y) --> program(X)) & \
3.688 +\ (! X Y. program_halts2(X::'a,Y) --> halts2(X::'a,Y)) & \
3.689 +\ (! X Y. program(X) & halts2(X::'a,Y) --> program_halts2(X::'a,Y)) & \
3.690 +\ (! W X Y Z. halts3_outputs(X::'a,Y,Z,W) --> halts3(X::'a,Y,Z)) & \
3.691 +\ (! Y Z X W. halts3_outputs(X::'a,Y,Z,W) --> outputs(X::'a,W)) & \
3.692 +\ (! Y Z X W. halts3(X::'a,Y,Z) & outputs(X::'a,W) --> halts3_outputs(X::'a,Y,Z,W)) & \
3.693 +\ (! Y X. program_not_halts2(X::'a,Y) --> program(X)) & \
3.694 +\ (! X Y. ~(program_not_halts2(X::'a,Y) & halts2(X::'a,Y))) & \
3.695 +\ (! X Y. program(X) --> program_not_halts2(X::'a,Y) | halts2(X::'a,Y)) & \
3.696 +\ (! W X Y. halts2_outputs(X::'a,Y,W) --> halts2(X::'a,Y)) & \
3.697 +\ (! Y X W. halts2_outputs(X::'a,Y,W) --> outputs(X::'a,W)) & \
3.698 +\ (! Y X W. halts2(X::'a,Y) & outputs(X::'a,W) --> halts2_outputs(X::'a,Y,W)) & \
3.699 +\ (! X W Y Z. program_halts2_halts3_outputs(X::'a,Y,Z,W) --> program_halts2(Y::'a,Z)) & \
3.700 +\ (! X Y Z W. program_halts2_halts3_outputs(X::'a,Y,Z,W) --> halts3_outputs(X::'a,Y,Z,W)) & \
3.701 +\ (! 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)) & \
3.702 +\ (! X W Y Z. program_not_halts2_halts3_outputs(X::'a,Y,Z,W) --> program_not_halts2(Y::'a,Z)) & \
3.703 +\ (! X Y Z W. program_not_halts2_halts3_outputs(X::'a,Y,Z,W) --> halts3_outputs(X::'a,Y,Z,W)) & \
3.704 +\ (! 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)) & \
3.705 +\ (! X W Y. program_halts2_halts2_outputs(X::'a,Y,W) --> program_halts2(Y::'a,Y)) & \
3.706 +\ (! X Y W. program_halts2_halts2_outputs(X::'a,Y,W) --> halts2_outputs(X::'a,Y,W)) & \
3.707 +\ (! X Y W. program_halts2(Y::'a,Y) & halts2_outputs(X::'a,Y,W) --> program_halts2_halts2_outputs(X::'a,Y,W)) & \
3.708 +\ (! X W Y. program_not_halts2_halts2_outputs(X::'a,Y,W) --> program_not_halts2(Y::'a,Y)) & \
3.709 +\ (! X Y W. program_not_halts2_halts2_outputs(X::'a,Y,W) --> halts2_outputs(X::'a,Y,W)) & \
3.710 +\ (! X Y W. program_not_halts2(Y::'a,Y) & halts2_outputs(X::'a,Y,W) --> program_not_halts2_halts2_outputs(X::'a,Y,W)) & \
3.711 +\ (! X. algorithm_program_decides(X) --> program_program_decides(c1)) & \
3.712 +\ (! W Y Z. program_program_decides(W) --> program_halts2_halts3_outputs(W::'a,Y,Z,good)) & \
3.713 +\ (! W Y Z. program_program_decides(W) --> program_not_halts2_halts3_outputs(W::'a,Y,Z,bad)) & \
3.714 +\ (! 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)) & \
3.715 +\ (! 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)) & \
3.716 +\ (! 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)) & \
3.717 +\ (! V. program(V) & program_halts2_halts2_outputs(V::'a,f4(V),good) & program_not_halts2_halts2_outputs(V::'a,f4(V),bad) --> program(c3)) & \
3.718 +\ (! 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)) & \
3.719 +\ (! 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)) & \
3.720 +\ (algorithm_program_decides(c4)) --> False",
3.721 + meson_tac);
3.722 +
3.723 +(****************SLOW
3.724 +2100398 inferences so far. Searching to depth 12. No proof after 30 mins.
3.725 +val COM004_1 = prove
3.726 + ("(! X. equal(X::'a,X)) & \
3.727 +\ (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
3.728 +\ (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) & \
3.729 +\ (! 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))) & \
3.730 +\ (! X. contradictory(negate(X),X)) & \
3.731 +\ (! X. contradictory(X::'a,negate(X))) & \
3.732 +\ (! X. siblings(left_child_of(X),right_child_of(X))) & \
3.733 +\ (! D E. equal(D::'a,E) --> equal(left_child_of(D),left_child_of(E))) & \
3.734 +\ (! F' G. equal(F'::'a,G) --> equal(negate(F'),negate(G))) & \
3.735 +\ (! H I' J. equal(H::'a,I') --> equal(or(H::'a,J),or(I'::'a,J))) & \
3.736 +\ (! K' M L. equal(K'::'a,L) --> equal(or(M::'a,K'),or(M::'a,L))) & \
3.737 +\ (! N O_ P. equal(N::'a,O_) --> equal(parent_of(N::'a,P),parent_of(O_::'a,P))) & \
3.738 +\ (! Q S' R. equal(Q::'a,R) --> equal(parent_of(S'::'a,Q),parent_of(S'::'a,R))) & \
3.739 +\ (! T' U. equal(T'::'a,U) --> equal(right_child_of(T'),right_child_of(U))) & \
3.740 +\ (! V W X. equal(V::'a,W) & contradictory(V::'a,X) --> contradictory(W::'a,X)) & \
3.741 +\ (! Y A1 Z. equal(Y::'a,Z) & contradictory(A1::'a,Y) --> contradictory(A1::'a,Z)) & \
3.742 +\ (! B1 C1 D1. equal(B1::'a,C1) & failure_node(B1::'a,D1) --> failure_node(C1::'a,D1)) & \
3.743 +\ (! E1 G1 F1. equal(E1::'a,F1) & failure_node(G1::'a,E1) --> failure_node(G1::'a,F1)) & \
3.744 +\ (! H1 I1 J1. equal(H1::'a,I1) & siblings(H1::'a,J1) --> siblings(I1::'a,J1)) & \
3.745 +\ (! K1 M1 L1. equal(K1::'a,L1) & siblings(M1::'a,K1) --> siblings(M1::'a,L1)) & \
3.746 +\ (failure_node(n_left::'a,or(empty::'a,atom))) & \
3.747 +\ (failure_node(n_right::'a,or(empty::'a,negate(atom)))) & \
3.748 +\ (equal(n_left::'a,left_child_of(n))) & \
3.749 +\ (equal(n_right::'a,right_child_of(n))) & \
3.750 +\ (! Z. ~failure_node(Z::'a,or(empty::'a,empty))) --> False",
3.751 + meson_tac);
3.752 +****************)
3.753 +
3.754 +(*179 inferences so far. Searching to depth 7. 3.9 secs*)
3.755 +val GEO003_1 = prove_hard
3.756 + ("(! X. equal(X::'a,X)) & \
3.757 +\ (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
3.758 +\ (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) & \
3.759 +\ (! X Y. between(X::'a,Y,X) --> equal(X::'a,Y)) & \
3.760 +\ (! V X Y Z. between(X::'a,Y,V) & between(Y::'a,Z,V) --> between(X::'a,Y,Z)) & \
3.761 +\ (! 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)) & \
3.762 +\ (! Y X. equidistant(X::'a,Y,Y,X)) & \
3.763 +\ (! Z X Y. equidistant(X::'a,Y,Z,Z) --> equal(X::'a,Y)) & \
3.764 +\ (! X Y Z V V2 W. equidistant(X::'a,Y,Z,V) & equidistant(X::'a,Y,V2,W) --> equidistant(Z::'a,V,V2,W)) & \
3.765 +\ (! 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)) & \
3.766 +\ (! 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))) & \
3.767 +\ (! 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))) & \
3.768 +\ (! 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))) & \
3.769 +\ (! 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))) & \
3.770 +\ (! 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)) & \
3.771 +\ (! X Y W V. between(X::'a,Y,extension(X::'a,Y,W,V))) & \
3.772 +\ (! X Y W V. equidistant(Y::'a,extension(X::'a,Y,W,V),W,V)) & \
3.773 +\ (~between(lower_dimension_point_1::'a,lower_dimension_point_2,lower_dimension_point_3)) & \
3.774 +\ (~between(lower_dimension_point_2::'a,lower_dimension_point_3,lower_dimension_point_1)) & \
3.775 +\ (~between(lower_dimension_point_3::'a,lower_dimension_point_1,lower_dimension_point_2)) & \
3.776 +\ (! 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)) & \
3.777 +\ (! 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))) & \
3.778 +\ (! 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)) & \
3.779 +\ (! X Y W Z. equal(X::'a,Y) & between(X::'a,W,Z) --> between(Y::'a,W,Z)) & \
3.780 +\ (! X W Y Z. equal(X::'a,Y) & between(W::'a,X,Z) --> between(W::'a,Y,Z)) & \
3.781 +\ (! X W Z Y. equal(X::'a,Y) & between(W::'a,Z,X) --> between(W::'a,Z,Y)) & \
3.782 +\ (! X Y V W Z. equal(X::'a,Y) & equidistant(X::'a,V,W,Z) --> equidistant(Y::'a,V,W,Z)) & \
3.783 +\ (! X V Y W Z. equal(X::'a,Y) & equidistant(V::'a,X,W,Z) --> equidistant(V::'a,Y,W,Z)) & \
3.784 +\ (! X V W Y Z. equal(X::'a,Y) & equidistant(V::'a,W,X,Z) --> equidistant(V::'a,W,Y,Z)) & \
3.785 +\ (! X V W Z Y. equal(X::'a,Y) & equidistant(V::'a,W,Z,X) --> equidistant(V::'a,W,Z,Y)) & \
3.786 +\ (! 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))) & \
3.787 +\ (! 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))) & \
3.788 +\ (! 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))) & \
3.789 +\ (! 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))) & \
3.790 +\ (! 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))) & \
3.791 +\ (! 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'))) & \
3.792 +\ (! 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))) & \
3.793 +\ (! 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))) & \
3.794 +\ (! 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))) & \
3.795 +\ (! 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))) & \
3.796 +\ (! 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))) & \
3.797 +\ (! 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))) & \
3.798 +\ (! 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))) & \
3.799 +\ (! 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))) & \
3.800 +\ (! 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))) & \
3.801 +\ (! X Y V1 V2 V3. equal(X::'a,Y) --> equal(extension(X::'a,V1,V2,V3),extension(Y::'a,V1,V2,V3))) & \
3.802 +\ (! X V1 Y V2 V3. equal(X::'a,Y) --> equal(extension(V1::'a,X,V2,V3),extension(V1::'a,Y,V2,V3))) & \
3.803 +\ (! X V1 V2 Y V3. equal(X::'a,Y) --> equal(extension(V1::'a,V2,X,V3),extension(V1::'a,V2,Y,V3))) & \
3.804 +\ (! X V1 V2 V3 Y. equal(X::'a,Y) --> equal(extension(V1::'a,V2,V3,X),extension(V1::'a,V2,V3,Y))) & \
3.805 +\ (! 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))) & \
3.806 +\ (! 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))) & \
3.807 +\ (! 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))) & \
3.808 +\ (! 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))) & \
3.809 +\ (! 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))) & \
3.810 +\ (! 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))) & \
3.811 +\ (~between(a::'a,b,b)) --> False",
3.812 + meson_tac);
3.813 +
3.814 +(*4272 inferences so far. Searching to depth 10. 29.4 secs*)
3.815 +val GEO017_2 = prove_hard
3.816 + ("(! X. equal(X::'a,X)) & \
3.817 +\ (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
3.818 +\ (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) & \
3.819 +\ (! Y X. equidistant(X::'a,Y,Y,X)) & \
3.820 +\ (! X Y Z V V2 W. equidistant(X::'a,Y,Z,V) & equidistant(X::'a,Y,V2,W) --> equidistant(Z::'a,V,V2,W)) & \
3.821 +\ (! Z X Y. equidistant(X::'a,Y,Z,Z) --> equal(X::'a,Y)) & \
3.822 +\ (! X Y W V. between(X::'a,Y,extension(X::'a,Y,W,V))) & \
3.823 +\ (! X Y W V. equidistant(Y::'a,extension(X::'a,Y,W,V),W,V)) & \
3.824 +\ (! 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)) & \
3.825 +\ (! X Y. between(X::'a,Y,X) --> equal(X::'a,Y)) & \
3.826 +\ (! 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)) & \
3.827 +\ (! 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)) & \
3.828 +\ (~between(lower_dimension_point_1::'a,lower_dimension_point_2,lower_dimension_point_3)) & \
3.829 +\ (~between(lower_dimension_point_2::'a,lower_dimension_point_3,lower_dimension_point_1)) & \
3.830 +\ (~between(lower_dimension_point_3::'a,lower_dimension_point_1,lower_dimension_point_2)) & \
3.831 +\ (! 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)) & \
3.832 +\ (! 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))) & \
3.833 +\ (! 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))) & \
3.834 +\ (! 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))) & \
3.835 +\ (! 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)) & \
3.836 +\ (! 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))) & \
3.837 +\ (! X Y W Z. equal(X::'a,Y) & between(X::'a,W,Z) --> between(Y::'a,W,Z)) & \
3.838 +\ (! X W Y Z. equal(X::'a,Y) & between(W::'a,X,Z) --> between(W::'a,Y,Z)) & \
3.839 +\ (! X W Z Y. equal(X::'a,Y) & between(W::'a,Z,X) --> between(W::'a,Z,Y)) & \
3.840 +\ (! X Y V W Z. equal(X::'a,Y) & equidistant(X::'a,V,W,Z) --> equidistant(Y::'a,V,W,Z)) & \
3.841 +\ (! X V Y W Z. equal(X::'a,Y) & equidistant(V::'a,X,W,Z) --> equidistant(V::'a,Y,W,Z)) & \
3.842 +\ (! X V W Y Z. equal(X::'a,Y) & equidistant(V::'a,W,X,Z) --> equidistant(V::'a,W,Y,Z)) & \
3.843 +\ (! X V W Z Y. equal(X::'a,Y) & equidistant(V::'a,W,Z,X) --> equidistant(V::'a,W,Z,Y)) & \
3.844 +\ (! 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))) & \
3.845 +\ (! 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))) & \
3.846 +\ (! 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))) & \
3.847 +\ (! 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))) & \
3.848 +\ (! 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))) & \
3.849 +\ (! 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'))) & \
3.850 +\ (! 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))) & \
3.851 +\ (! 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))) & \
3.852 +\ (! 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))) & \
3.853 +\ (! 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))) & \
3.854 +\ (! 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))) & \
3.855 +\ (! 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))) & \
3.856 +\ (! 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))) & \
3.857 +\ (! 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))) & \
3.858 +\ (! 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))) & \
3.859 +\ (! X Y V1 V2 V3. equal(X::'a,Y) --> equal(extension(X::'a,V1,V2,V3),extension(Y::'a,V1,V2,V3))) & \
3.860 +\ (! X V1 Y V2 V3. equal(X::'a,Y) --> equal(extension(V1::'a,X,V2,V3),extension(V1::'a,Y,V2,V3))) & \
3.861 +\ (! X V1 V2 Y V3. equal(X::'a,Y) --> equal(extension(V1::'a,V2,X,V3),extension(V1::'a,V2,Y,V3))) & \
3.862 +\ (! X V1 V2 V3 Y. equal(X::'a,Y) --> equal(extension(V1::'a,V2,V3,X),extension(V1::'a,V2,V3,Y))) & \
3.863 +\ (! 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))) & \
3.864 +\ (! 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))) & \
3.865 +\ (! 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))) & \
3.866 +\ (! 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))) & \
3.867 +\ (! 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))) & \
3.868 +\ (! 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))) & \
3.869 +\ (equidistant(u::'a,v,w,x)) & \
3.870 +\ (~equidistant(u::'a,v,x,w)) --> False",
3.871 + meson_tac);
3.872 +
3.873 +(****************SLOW
3.874 +382903 inferences so far. Searching to depth 9. No proof after 35 minutes.
3.875 +val GEO027_3 = prove_hard
3.876 + ("(! X. equal(X::'a,X)) & \
3.877 +\ (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
3.878 +\ (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) & \
3.879 +\ (! Y X. equidistant(X::'a,Y,Y,X)) & \
3.880 +\ (! X Y Z V V2 W. equidistant(X::'a,Y,Z,V) & equidistant(X::'a,Y,V2,W) --> equidistant(Z::'a,V,V2,W)) & \
3.881 +\ (! Z X Y. equidistant(X::'a,Y,Z,Z) --> equal(X::'a,Y)) & \
3.882 +\ (! X Y W V. between(X::'a,Y,extension(X::'a,Y,W,V))) & \
3.883 +\ (! X Y W V. equidistant(Y::'a,extension(X::'a,Y,W,V),W,V)) & \
3.884 +\ (! 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)) & \
3.885 +\ (! X Y. between(X::'a,Y,X) --> equal(X::'a,Y)) & \
3.886 +\ (! 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)) & \
3.887 +\ (! 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)) & \
3.888 +\ (~between(lower_dimension_point_1::'a,lower_dimension_point_2,lower_dimension_point_3)) & \
3.889 +\ (~between(lower_dimension_point_2::'a,lower_dimension_point_3,lower_dimension_point_1)) & \
3.890 +\ (~between(lower_dimension_point_3::'a,lower_dimension_point_1,lower_dimension_point_2)) & \
3.891 +\ (! 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)) & \
3.892 +\ (! 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))) & \
3.893 +\ (! 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))) & \
3.894 +\ (! 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))) & \
3.895 +\ (! 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)) & \
3.896 +\ (! 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))) & \
3.897 +\ (! X Y W Z. equal(X::'a,Y) & between(X::'a,W,Z) --> between(Y::'a,W,Z)) & \
3.898 +\ (! X W Y Z. equal(X::'a,Y) & between(W::'a,X,Z) --> between(W::'a,Y,Z)) & \
3.899 +\ (! X W Z Y. equal(X::'a,Y) & between(W::'a,Z,X) --> between(W::'a,Z,Y)) & \
3.900 +\ (! X Y V W Z. equal(X::'a,Y) & equidistant(X::'a,V,W,Z) --> equidistant(Y::'a,V,W,Z)) & \
3.901 +\ (! X V Y W Z. equal(X::'a,Y) & equidistant(V::'a,X,W,Z) --> equidistant(V::'a,Y,W,Z)) & \
3.902 +\ (! X V W Y Z. equal(X::'a,Y) & equidistant(V::'a,W,X,Z) --> equidistant(V::'a,W,Y,Z)) & \
3.903 +\ (! X V W Z Y. equal(X::'a,Y) & equidistant(V::'a,W,Z,X) --> equidistant(V::'a,W,Z,Y)) & \
3.904 +\ (! 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))) & \
3.905 +\ (! 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))) & \
3.906 +\ (! 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))) & \
3.907 +\ (! 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))) & \
3.908 +\ (! 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))) & \
3.909 +\ (! 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'))) & \
3.910 +\ (! 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))) & \
3.911 +\ (! 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))) & \
3.912 +\ (! 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))) & \
3.913 +\ (! 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))) & \
3.914 +\ (! 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))) & \
3.915 +\ (! 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))) & \
3.916 +\ (! 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))) & \
3.917 +\ (! 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))) & \
3.918 +\ (! 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))) & \
3.919 +\ (! X Y V1 V2 V3. equal(X::'a,Y) --> equal(extension(X::'a,V1,V2,V3),extension(Y::'a,V1,V2,V3))) & \
3.920 +\ (! X V1 Y V2 V3. equal(X::'a,Y) --> equal(extension(V1::'a,X,V2,V3),extension(V1::'a,Y,V2,V3))) & \
3.921 +\ (! X V1 V2 Y V3. equal(X::'a,Y) --> equal(extension(V1::'a,V2,X,V3),extension(V1::'a,V2,Y,V3))) & \
3.922 +\ (! X V1 V2 V3 Y. equal(X::'a,Y) --> equal(extension(V1::'a,V2,V3,X),extension(V1::'a,V2,V3,Y))) & \
3.923 +\ (! 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))) & \
3.924 +\ (! 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))) & \
3.925 +\ (! 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))) & \
3.926 +\ (! 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))) & \
3.927 +\ (! 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))) & \
3.928 +\ (! 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))) & \
3.929 +\ (! U V. equal(reflection(U::'a,V),extension(U::'a,V,U,V))) & \
3.930 +\ (! X Y Z. equal(X::'a,Y) --> equal(reflection(X::'a,Z),reflection(Y::'a,Z))) & \
3.931 +\ (! A1 C1 B1. equal(A1::'a,B1) --> equal(reflection(C1::'a,A1),reflection(C1::'a,B1))) & \
3.932 +\ (! U V. equidistant(U::'a,V,U,V)) & \
3.933 +\ (! W X U V. equidistant(U::'a,V,W,X) --> equidistant(W::'a,X,U,V)) & \
3.934 +\ (! V U W X. equidistant(U::'a,V,W,X) --> equidistant(V::'a,U,W,X)) & \
3.935 +\ (! U V X W. equidistant(U::'a,V,W,X) --> equidistant(U::'a,V,X,W)) & \
3.936 +\ (! V U X W. equidistant(U::'a,V,W,X) --> equidistant(V::'a,U,X,W)) & \
3.937 +\ (! W X V U. equidistant(U::'a,V,W,X) --> equidistant(W::'a,X,V,U)) & \
3.938 +\ (! X W U V. equidistant(U::'a,V,W,X) --> equidistant(X::'a,W,U,V)) & \
3.939 +\ (! X W V U. equidistant(U::'a,V,W,X) --> equidistant(X::'a,W,V,U)) & \
3.940 +\ (! W X U V Y Z. equidistant(U::'a,V,W,X) & equidistant(W::'a,X,Y,Z) --> equidistant(U::'a,V,Y,Z)) & \
3.941 +\ (! U V W. equal(V::'a,extension(U::'a,V,W,W))) & \
3.942 +\ (! W X U V Y. equal(Y::'a,extension(U::'a,V,W,X)) --> between(U::'a,V,Y)) & \
3.943 +\ (! U V. between(U::'a,V,reflection(U::'a,V))) & \
3.944 +\ (! U V. equidistant(V::'a,reflection(U::'a,V),U,V)) & \
3.945 +\ (! U V. equal(U::'a,V) --> equal(V::'a,reflection(U::'a,V))) & \
3.946 +\ (! U. equal(U::'a,reflection(U::'a,U))) & \
3.947 +\ (! U V. equal(V::'a,reflection(U::'a,V)) --> equal(U::'a,V)) & \
3.948 +\ (! U V. equidistant(U::'a,U,V,V)) & \
3.949 +\ (! 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)) & \
3.950 +\ (! 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)) & \
3.951 +\ (between(u::'a,v,w)) & \
3.952 +\ (~equal(u::'a,v)) & \
3.953 +\ (~equal(w::'a,extension(u::'a,v,v,w))) --> False",
3.954 + meson_tac);
3.955 +****************)
3.956 +
3.957 +(*313884 inferences so far. Searching to depth 10. 887 secs: 15 mins.*)
3.958 +val GEO058_2 = prove_hard
3.959 + ("(! X. equal(X::'a,X)) & \
3.960 +\ (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
3.961 +\ (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) & \
3.962 +\ (! Y X. equidistant(X::'a,Y,Y,X)) & \
3.963 +\ (! X Y Z V V2 W. equidistant(X::'a,Y,Z,V) & equidistant(X::'a,Y,V2,W) --> equidistant(Z::'a,V,V2,W)) & \
3.964 +\ (! Z X Y. equidistant(X::'a,Y,Z,Z) --> equal(X::'a,Y)) & \
3.965 +\ (! X Y W V. between(X::'a,Y,extension(X::'a,Y,W,V))) & \
3.966 +\ (! X Y W V. equidistant(Y::'a,extension(X::'a,Y,W,V),W,V)) & \
3.967 +\ (! 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)) & \
3.968 +\ (! X Y. between(X::'a,Y,X) --> equal(X::'a,Y)) & \
3.969 +\ (! 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)) & \
3.970 +\ (! 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)) & \
3.971 +\ (~between(lower_dimension_point_1::'a,lower_dimension_point_2,lower_dimension_point_3)) & \
3.972 +\ (~between(lower_dimension_point_2::'a,lower_dimension_point_3,lower_dimension_point_1)) & \
3.973 +\ (~between(lower_dimension_point_3::'a,lower_dimension_point_1,lower_dimension_point_2)) & \
3.974 +\ (! 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)) & \
3.975 +\ (! 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))) & \
3.976 +\ (! 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))) & \
3.977 +\ (! 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))) & \
3.978 +\ (! 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)) & \
3.979 +\ (! 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))) & \
3.980 +\ (! X Y W Z. equal(X::'a,Y) & between(X::'a,W,Z) --> between(Y::'a,W,Z)) & \
3.981 +\ (! X W Y Z. equal(X::'a,Y) & between(W::'a,X,Z) --> between(W::'a,Y,Z)) & \
3.982 +\ (! X W Z Y. equal(X::'a,Y) & between(W::'a,Z,X) --> between(W::'a,Z,Y)) & \
3.983 +\ (! X Y V W Z. equal(X::'a,Y) & equidistant(X::'a,V,W,Z) --> equidistant(Y::'a,V,W,Z)) & \
3.984 +\ (! X V Y W Z. equal(X::'a,Y) & equidistant(V::'a,X,W,Z) --> equidistant(V::'a,Y,W,Z)) & \
3.985 +\ (! X V W Y Z. equal(X::'a,Y) & equidistant(V::'a,W,X,Z) --> equidistant(V::'a,W,Y,Z)) & \
3.986 +\ (! X V W Z Y. equal(X::'a,Y) & equidistant(V::'a,W,Z,X) --> equidistant(V::'a,W,Z,Y)) & \
3.987 +\ (! 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))) & \
3.988 +\ (! 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))) & \
3.989 +\ (! 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))) & \
3.990 +\ (! 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))) & \
3.991 +\ (! 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))) & \
3.992 +\ (! 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'))) & \
3.993 +\ (! 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))) & \
3.994 +\ (! 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))) & \
3.995 +\ (! 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))) & \
3.996 +\ (! 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))) & \
3.997 +\ (! 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))) & \
3.998 +\ (! 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))) & \
3.999 +\ (! 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))) & \
3.1000 +\ (! 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))) & \
3.1001 +\ (! 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))) & \
3.1002 +\ (! X Y V1 V2 V3. equal(X::'a,Y) --> equal(extension(X::'a,V1,V2,V3),extension(Y::'a,V1,V2,V3))) & \
3.1003 +\ (! X V1 Y V2 V3. equal(X::'a,Y) --> equal(extension(V1::'a,X,V2,V3),extension(V1::'a,Y,V2,V3))) & \
3.1004 +\ (! X V1 V2 Y V3. equal(X::'a,Y) --> equal(extension(V1::'a,V2,X,V3),extension(V1::'a,V2,Y,V3))) & \
3.1005 +\ (! X V1 V2 V3 Y. equal(X::'a,Y) --> equal(extension(V1::'a,V2,V3,X),extension(V1::'a,V2,V3,Y))) & \
3.1006 +\ (! 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))) & \
3.1007 +\ (! 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))) & \
3.1008 +\ (! 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))) & \
3.1009 +\ (! 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))) & \
3.1010 +\ (! 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))) & \
3.1011 +\ (! 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))) & \
3.1012 +\ (! U V. equal(reflection(U::'a,V),extension(U::'a,V,U,V))) & \
3.1013 +\ (! X Y Z. equal(X::'a,Y) --> equal(reflection(X::'a,Z),reflection(Y::'a,Z))) & \
3.1014 +\ (! A1 C1 B1. equal(A1::'a,B1) --> equal(reflection(C1::'a,A1),reflection(C1::'a,B1))) & \
3.1015 +\ (equal(v::'a,reflection(u::'a,v))) & \
3.1016 +\ (~equal(u::'a,v)) --> False",
3.1017 + meson_tac);
3.1018 +
3.1019 +(*0 inferences so far. Searching to depth 0. 0.2 secs*)
3.1020 +val GEO079_1 = prove
3.1021 + ("(! 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)) & \
3.1022 +\ (! U V W X Y Z. congruent(U::'a,V,W,X,Y,Z) --> eq(U::'a,V,W,X,Y,Z)) & \
3.1023 +\ (! V W U X. trapezoid(U::'a,V,W,X) --> parallel(V::'a,W,U,X)) & \
3.1024 +\ (! U V X Y. parallel(U::'a,V,X,Y) --> eq(X::'a,V,U,V,X,Y)) & \
3.1025 +\ (trapezoid(a::'a,b,c,d)) & \
3.1026 +\ (~eq(a::'a,c,b,c,a,d)) --> False",
3.1027 + meson_tac);
3.1028 +
3.1029 +(****************SLOW
3.1030 +2032008 inferences so far. Searching to depth 16. No proof after 30 minutes.
3.1031 +val GRP001_1 = prove_hard
3.1032 + ("(! X. equal(X::'a,X)) & \
3.1033 +\ (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
3.1034 +\ (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) & \
3.1035 +\ (! X. product(identity::'a,X,X)) & \
3.1036 +\ (! X. product(X::'a,identity,X)) & \
3.1037 +\ (! X. product(inverse(X),X,identity)) & \
3.1038 +\ (! X. product(X::'a,inverse(X),identity)) & \
3.1039 +\ (! X Y. product(X::'a,Y,multiply(X::'a,Y))) & \
3.1040 +\ (! X Y Z W. product(X::'a,Y,Z) & product(X::'a,Y,W) --> equal(Z::'a,W)) & \
3.1041 +\ (! 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)) & \
3.1042 +\ (! 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)) & \
3.1043 +\ (! X Y. equal(X::'a,Y) --> equal(inverse(X),inverse(Y))) & \
3.1044 +\ (! X Y W. equal(X::'a,Y) --> equal(multiply(X::'a,W),multiply(Y::'a,W))) & \
3.1045 +\ (! X W Y. equal(X::'a,Y) --> equal(multiply(W::'a,X),multiply(W::'a,Y))) & \
3.1046 +\ (! X Y W Z. equal(X::'a,Y) & product(X::'a,W,Z) --> product(Y::'a,W,Z)) & \
3.1047 +\ (! X W Y Z. equal(X::'a,Y) & product(W::'a,X,Z) --> product(W::'a,Y,Z)) & \
3.1048 +\ (! X W Z Y. equal(X::'a,Y) & product(W::'a,Z,X) --> product(W::'a,Z,Y)) & \
3.1049 +\ (! X. product(X::'a,X,identity)) & \
3.1050 +\ (product(a::'a,b,c)) & \
3.1051 +\ (~product(b::'a,a,c)) --> False",
3.1052 + meson_tac);
3.1053 +****************)
3.1054 +
3.1055 +(*2386 inferences so far. Searching to depth 11. 8.7 secs*)
3.1056 +val GRP008_1 = prove_hard
3.1057 + ("(! X. equal(X::'a,X)) & \
3.1058 +\ (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
3.1059 +\ (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) & \
3.1060 +\ (! X. product(identity::'a,X,X)) & \
3.1061 +\ (! X. product(X::'a,identity,X)) & \
3.1062 +\ (! X. product(inverse(X),X,identity)) & \
3.1063 +\ (! X. product(X::'a,inverse(X),identity)) & \
3.1064 +\ (! X Y. product(X::'a,Y,multiply(X::'a,Y))) & \
3.1065 +\ (! X Y Z W. product(X::'a,Y,Z) & product(X::'a,Y,W) --> equal(Z::'a,W)) & \
3.1066 +\ (! 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)) & \
3.1067 +\ (! 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)) & \
3.1068 +\ (! X Y. equal(X::'a,Y) --> equal(inverse(X),inverse(Y))) & \
3.1069 +\ (! X Y W. equal(X::'a,Y) --> equal(multiply(X::'a,W),multiply(Y::'a,W))) & \
3.1070 +\ (! X W Y. equal(X::'a,Y) --> equal(multiply(W::'a,X),multiply(W::'a,Y))) & \
3.1071 +\ (! X Y W Z. equal(X::'a,Y) & product(X::'a,W,Z) --> product(Y::'a,W,Z)) & \
3.1072 +\ (! X W Y Z. equal(X::'a,Y) & product(W::'a,X,Z) --> product(W::'a,Y,Z)) & \
3.1073 +\ (! X W Z Y. equal(X::'a,Y) & product(W::'a,Z,X) --> product(W::'a,Z,Y)) & \
3.1074 +\ (! A B. equal(A::'a,B) --> equal(h(A),h(B))) & \
3.1075 +\ (! C D. equal(C::'a,D) --> equal(j(C),j(D))) & \
3.1076 +\ (! A B. equal(A::'a,B) & q(A) --> q(B)) & \
3.1077 +\ (! B A C. q(A) & product(A::'a,B,C) --> product(B::'a,A,C)) & \
3.1078 +\ (! A. product(j(A),A,h(A)) | product(A::'a,j(A),h(A)) | q(A)) & \
3.1079 +\ (! A. product(j(A),A,h(A)) & product(A::'a,j(A),h(A)) --> q(A)) & \
3.1080 +\ (~q(identity)) --> False",
3.1081 + meson_tac);
3.1082 +
3.1083 +(*8625 inferences so far. Searching to depth 11. 20 secs*)
3.1084 +val GRP013_1 = prove_hard
3.1085 + ("(! X. equal(X::'a,X)) & \
3.1086 +\ (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
3.1087 +\ (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) & \
3.1088 +\ (! X. product(identity::'a,X,X)) & \
3.1089 +\ (! X. product(X::'a,identity,X)) & \
3.1090 +\ (! X. product(inverse(X),X,identity)) & \
3.1091 +\ (! X. product(X::'a,inverse(X),identity)) & \
3.1092 +\ (! X Y. product(X::'a,Y,multiply(X::'a,Y))) & \
3.1093 +\ (! X Y Z W. product(X::'a,Y,Z) & product(X::'a,Y,W) --> equal(Z::'a,W)) & \
3.1094 +\ (! 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)) & \
3.1095 +\ (! 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)) & \
3.1096 +\ (! X Y. equal(X::'a,Y) --> equal(inverse(X),inverse(Y))) & \
3.1097 +\ (! X Y W. equal(X::'a,Y) --> equal(multiply(X::'a,W),multiply(Y::'a,W))) & \
3.1098 +\ (! X W Y. equal(X::'a,Y) --> equal(multiply(W::'a,X),multiply(W::'a,Y))) & \
3.1099 +\ (! X Y W Z. equal(X::'a,Y) & product(X::'a,W,Z) --> product(Y::'a,W,Z)) & \
3.1100 +\ (! X W Y Z. equal(X::'a,Y) & product(W::'a,X,Z) --> product(W::'a,Y,Z)) & \
3.1101 +\ (! X W Z Y. equal(X::'a,Y) & product(W::'a,Z,X) --> product(W::'a,Z,Y)) & \
3.1102 +\ (! A. product(A::'a,A,identity)) & \
3.1103 +\ (product(a::'a,b,c)) & \
3.1104 +\ (product(inverse(a),inverse(b),d)) & \
3.1105 +\ (! A C B. product(inverse(A),inverse(B),C) --> product(A::'a,C,B)) & \
3.1106 +\ (~product(c::'a,d,identity)) --> False",
3.1107 + meson_tac);
3.1108 +
3.1109 +(*2448 inferences so far. Searching to depth 10. 7.2 secs*)
3.1110 +val GRP037_3 = prove_hard
3.1111 + ("(! X. equal(X::'a,X)) & \
3.1112 +\ (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
3.1113 +\ (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) & \
3.1114 +\ (! X. product(identity::'a,X,X)) & \
3.1115 +\ (! X. product(X::'a,identity,X)) & \
3.1116 +\ (! X. product(inverse(X),X,identity)) & \
3.1117 +\ (! X. product(X::'a,inverse(X),identity)) & \
3.1118 +\ (! X Y. product(X::'a,Y,multiply(X::'a,Y))) & \
3.1119 +\ (! X Y Z W. product(X::'a,Y,Z) & product(X::'a,Y,W) --> equal(Z::'a,W)) & \
3.1120 +\ (! 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)) & \
3.1121 +\ (! 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)) & \
3.1122 +\ (! X Y. equal(X::'a,Y) --> equal(inverse(X),inverse(Y))) & \
3.1123 +\ (! X Y W. equal(X::'a,Y) --> equal(multiply(X::'a,W),multiply(Y::'a,W))) & \
3.1124 +\ (! X W Y. equal(X::'a,Y) --> equal(multiply(W::'a,X),multiply(W::'a,Y))) & \
3.1125 +\ (! X Y W Z. equal(X::'a,Y) & product(X::'a,W,Z) --> product(Y::'a,W,Z)) & \
3.1126 +\ (! X W Y Z. equal(X::'a,Y) & product(W::'a,X,Z) --> product(W::'a,Y,Z)) & \
3.1127 +\ (! X W Z Y. equal(X::'a,Y) & product(W::'a,Z,X) --> product(W::'a,Z,Y)) & \
3.1128 +\ (! A B C. subgroup_member(A) & subgroup_member(B) & product(A::'a,inverse(B),C) --> subgroup_member(C)) & \
3.1129 +\ (! A B. equal(A::'a,B) & subgroup_member(A) --> subgroup_member(B)) & \
3.1130 +\ (! A. subgroup_member(A) --> product(another_identity::'a,A,A)) & \
3.1131 +\ (! A. subgroup_member(A) --> product(A::'a,another_identity,A)) & \
3.1132 +\ (! A. subgroup_member(A) --> product(A::'a,another_inverse(A),another_identity)) & \
3.1133 +\ (! A. subgroup_member(A) --> product(another_inverse(A),A,another_identity)) & \
3.1134 +\ (! A. subgroup_member(A) --> subgroup_member(another_inverse(A))) & \
3.1135 +\ (! A B. equal(A::'a,B) --> equal(another_inverse(A),another_inverse(B))) & \
3.1136 +\ (! A C D B. product(A::'a,B,C) & product(A::'a,D,C) --> equal(D::'a,B)) & \
3.1137 +\ (! B C D A. product(A::'a,B,C) & product(D::'a,B,C) --> equal(D::'a,A)) & \
3.1138 +\ (subgroup_member(a)) & \
3.1139 +\ (subgroup_member(another_identity)) & \
3.1140 +\ (~equal(inverse(a),another_inverse(a))) --> False",
3.1141 + meson_tac);
3.1142 +
3.1143 +(*163 inferences so far. Searching to depth 11. 0.3 secs*)
3.1144 +val GRP031_2 = prove
3.1145 + ("(! X Y. product(X::'a,Y,multiply(X::'a,Y))) & \
3.1146 +\ (! X Y Z W. product(X::'a,Y,Z) & product(X::'a,Y,W) --> equal(Z::'a,W)) & \
3.1147 +\ (! 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)) & \
3.1148 +\ (! 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)) & \
3.1149 +\ (! A. product(A::'a,inverse(A),identity)) & \
3.1150 +\ (! A. product(A::'a,identity,A)) & \
3.1151 +\ (! A. ~product(A::'a,a,identity)) --> False",
3.1152 + meson_tac);
3.1153 +
3.1154 +(*47 inferences so far. Searching to depth 11. 0.2 secs*)
3.1155 +val GRP034_4 = prove
3.1156 + ("(! X Y. product(X::'a,Y,multiply(X::'a,Y))) & \
3.1157 +\ (! X. product(identity::'a,X,X)) & \
3.1158 +\ (! X. product(X::'a,identity,X)) & \
3.1159 +\ (! X. product(X::'a,inverse(X),identity)) & \
3.1160 +\ (! 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)) & \
3.1161 +\ (! 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)) & \
3.1162 +\ (! B A C. subgroup_member(A) & subgroup_member(B) & product(B::'a,inverse(A),C) --> subgroup_member(C)) & \
3.1163 +\ (subgroup_member(a)) & \
3.1164 +\ (~subgroup_member(inverse(a))) --> False",
3.1165 + meson_tac);
3.1166 +
3.1167 +(*3287 inferences so far. Searching to depth 14. 3.5 secs*)
3.1168 +val GRP047_2 = prove_hard
3.1169 + ("(! X. product(identity::'a,X,X)) & \
3.1170 +\ (! X. product(inverse(X),X,identity)) & \
3.1171 +\ (! X Y. product(X::'a,Y,multiply(X::'a,Y))) & \
3.1172 +\ (! X Y Z W. product(X::'a,Y,Z) & product(X::'a,Y,W) --> equal(Z::'a,W)) & \
3.1173 +\ (! 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)) & \
3.1174 +\ (! 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)) & \
3.1175 +\ (! X W Z Y. equal(X::'a,Y) & product(W::'a,Z,X) --> product(W::'a,Z,Y)) & \
3.1176 +\ (equal(a::'a,b)) & \
3.1177 +\ (~equal(multiply(c::'a,a),multiply(c::'a,b))) --> False",
3.1178 + meson_tac);
3.1179 +
3.1180 +(*25559 inferences so far. Searching to depth 19. 16.9 secs*)
3.1181 +val GRP130_1_002 = prove_hard
3.1182 + ("(group_element(e_1)) & \
3.1183 +\ (group_element(e_2)) & \
3.1184 +\ (~equal(e_1::'a,e_2)) & \
3.1185 +\ (~equal(e_2::'a,e_1)) & \
3.1186 +\ (! X Y. group_element(X) & group_element(Y) --> product(X::'a,Y,e_1) | product(X::'a,Y,e_2)) & \
3.1187 +\ (! X Y W Z. product(X::'a,Y,W) & product(X::'a,Y,Z) --> equal(W::'a,Z)) & \
3.1188 +\ (! X Y W Z. product(X::'a,W,Y) & product(X::'a,Z,Y) --> equal(W::'a,Z)) & \
3.1189 +\ (! Y X W Z. product(W::'a,Y,X) & product(Z::'a,Y,X) --> equal(W::'a,Z)) & \
3.1190 +\ (! Z1 Z2 Y X. product(X::'a,Y,Z1) & product(X::'a,Z1,Z2) --> product(Z2::'a,Y,X)) --> False",
3.1191 + meson_tac);
3.1192 +
3.1193 +(*3468 inferences so far. Searching to depth 10. 9.1 secs*)
3.1194 +val GRP156_1 = prove_hard
3.1195 + ("(! X. equal(X::'a,X)) & \
3.1196 +\ (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
3.1197 +\ (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) & \
3.1198 +\ (! X. equal(multiply(identity::'a,X),X)) & \
3.1199 +\ (! X. equal(multiply(inverse(X),X),identity)) & \
3.1200 +\ (! X Y Z. equal(multiply(multiply(X::'a,Y),Z),multiply(X::'a,multiply(Y::'a,Z)))) & \
3.1201 +\ (! A B. equal(A::'a,B) --> equal(inverse(A),inverse(B))) & \
3.1202 +\ (! C D E. equal(C::'a,D) --> equal(multiply(C::'a,E),multiply(D::'a,E))) & \
3.1203 +\ (! F' H G. equal(F'::'a,G) --> equal(multiply(H::'a,F'),multiply(H::'a,G))) & \
3.1204 +\ (! Y X. equal(greatest_lower_bound(X::'a,Y),greatest_lower_bound(Y::'a,X))) & \
3.1205 +\ (! Y X. equal(least_upper_bound(X::'a,Y),least_upper_bound(Y::'a,X))) & \
3.1206 +\ (! 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))) & \
3.1207 +\ (! 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))) & \
3.1208 +\ (! X. equal(least_upper_bound(X::'a,X),X)) & \
3.1209 +\ (! X. equal(greatest_lower_bound(X::'a,X),X)) & \
3.1210 +\ (! Y X. equal(least_upper_bound(X::'a,greatest_lower_bound(X::'a,Y)),X)) & \
3.1211 +\ (! Y X. equal(greatest_lower_bound(X::'a,least_upper_bound(X::'a,Y)),X)) & \
3.1212 +\ (! Y X Z. equal(multiply(X::'a,least_upper_bound(Y::'a,Z)),least_upper_bound(multiply(X::'a,Y),multiply(X::'a,Z)))) & \
3.1213 +\ (! Y X Z. equal(multiply(X::'a,greatest_lower_bound(Y::'a,Z)),greatest_lower_bound(multiply(X::'a,Y),multiply(X::'a,Z)))) & \
3.1214 +\ (! Y Z X. equal(multiply(least_upper_bound(Y::'a,Z),X),least_upper_bound(multiply(Y::'a,X),multiply(Z::'a,X)))) & \
3.1215 +\ (! Y Z X. equal(multiply(greatest_lower_bound(Y::'a,Z),X),greatest_lower_bound(multiply(Y::'a,X),multiply(Z::'a,X)))) & \
3.1216 +\ (! A B C. equal(A::'a,B) --> equal(greatest_lower_bound(A::'a,C),greatest_lower_bound(B::'a,C))) & \
3.1217 +\ (! A C B. equal(A::'a,B) --> equal(greatest_lower_bound(C::'a,A),greatest_lower_bound(C::'a,B))) & \
3.1218 +\ (! A B C. equal(A::'a,B) --> equal(least_upper_bound(A::'a,C),least_upper_bound(B::'a,C))) & \
3.1219 +\ (! A C B. equal(A::'a,B) --> equal(least_upper_bound(C::'a,A),least_upper_bound(C::'a,B))) & \
3.1220 +\ (! A B C. equal(A::'a,B) --> equal(multiply(A::'a,C),multiply(B::'a,C))) & \
3.1221 +\ (! A C B. equal(A::'a,B) --> equal(multiply(C::'a,A),multiply(C::'a,B))) & \
3.1222 +\ (equal(least_upper_bound(a::'a,b),b)) & \
3.1223 +\ (~equal(greatest_lower_bound(multiply(a::'a,c),multiply(b::'a,c)),multiply(a::'a,c))) --> False",
3.1224 + meson_tac);
3.1225 +
3.1226 +(*4394 inferences so far. Searching to depth 10. 8.2 secs*)
3.1227 +val GRP168_1 = prove_hard
3.1228 + ("(! X. equal(X::'a,X)) & \
3.1229 +\ (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
3.1230 +\ (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) & \
3.1231 +\ (! X. equal(multiply(identity::'a,X),X)) & \
3.1232 +\ (! X. equal(multiply(inverse(X),X),identity)) & \
3.1233 +\ (! X Y Z. equal(multiply(multiply(X::'a,Y),Z),multiply(X::'a,multiply(Y::'a,Z)))) & \
3.1234 +\ (! A B. equal(A::'a,B) --> equal(inverse(A),inverse(B))) & \
3.1235 +\ (! C D E. equal(C::'a,D) --> equal(multiply(C::'a,E),multiply(D::'a,E))) & \
3.1236 +\ (! F' H G. equal(F'::'a,G) --> equal(multiply(H::'a,F'),multiply(H::'a,G))) & \
3.1237 +\ (! Y X. equal(greatest_lower_bound(X::'a,Y),greatest_lower_bound(Y::'a,X))) & \
3.1238 +\ (! Y X. equal(least_upper_bound(X::'a,Y),least_upper_bound(Y::'a,X))) & \
3.1239 +\ (! 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))) & \
3.1240 +\ (! 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))) & \
3.1241 +\ (! X. equal(least_upper_bound(X::'a,X),X)) & \
3.1242 +\ (! X. equal(greatest_lower_bound(X::'a,X),X)) & \
3.1243 +\ (! Y X. equal(least_upper_bound(X::'a,greatest_lower_bound(X::'a,Y)),X)) & \
3.1244 +\ (! Y X. equal(greatest_lower_bound(X::'a,least_upper_bound(X::'a,Y)),X)) & \
3.1245 +\ (! Y X Z. equal(multiply(X::'a,least_upper_bound(Y::'a,Z)),least_upper_bound(multiply(X::'a,Y),multiply(X::'a,Z)))) & \
3.1246 +\ (! Y X Z. equal(multiply(X::'a,greatest_lower_bound(Y::'a,Z)),greatest_lower_bound(multiply(X::'a,Y),multiply(X::'a,Z)))) & \
3.1247 +\ (! Y Z X. equal(multiply(least_upper_bound(Y::'a,Z),X),least_upper_bound(multiply(Y::'a,X),multiply(Z::'a,X)))) & \
3.1248 +\ (! Y Z X. equal(multiply(greatest_lower_bound(Y::'a,Z),X),greatest_lower_bound(multiply(Y::'a,X),multiply(Z::'a,X)))) & \
3.1249 +\ (! A B C. equal(A::'a,B) --> equal(greatest_lower_bound(A::'a,C),greatest_lower_bound(B::'a,C))) & \
3.1250 +\ (! A C B. equal(A::'a,B) --> equal(greatest_lower_bound(C::'a,A),greatest_lower_bound(C::'a,B))) & \
3.1251 +\ (! A B C. equal(A::'a,B) --> equal(least_upper_bound(A::'a,C),least_upper_bound(B::'a,C))) & \
3.1252 +\ (! A C B. equal(A::'a,B) --> equal(least_upper_bound(C::'a,A),least_upper_bound(C::'a,B))) & \
3.1253 +\ (! A B C. equal(A::'a,B) --> equal(multiply(A::'a,C),multiply(B::'a,C))) & \
3.1254 +\ (! A C B. equal(A::'a,B) --> equal(multiply(C::'a,A),multiply(C::'a,B))) & \
3.1255 +\ (equal(least_upper_bound(a::'a,b),b)) & \
3.1256 +\ (~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",
3.1257 + meson_tac);
3.1258 +
3.1259 +(*250258 inferences so far. Searching to depth 16. 406.2 secs*)
3.1260 +val HEN003_3 = prove_hard
3.1261 + ("(! X. equal(X::'a,X)) & \
3.1262 +\ (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
3.1263 +\ (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) & \
3.1264 +\ (! X Y. less_equal(X::'a,Y) --> equal(divide(X::'a,Y),zero)) & \
3.1265 +\ (! X Y. equal(divide(X::'a,Y),zero) --> less_equal(X::'a,Y)) & \
3.1266 +\ (! Y X. less_equal(divide(X::'a,Y),X)) & \
3.1267 +\ (! X Y Z. less_equal(divide(divide(X::'a,Z),divide(Y::'a,Z)),divide(divide(X::'a,Y),Z))) & \
3.1268 +\ (! X. less_equal(zero::'a,X)) & \
3.1269 +\ (! X Y. less_equal(X::'a,Y) & less_equal(Y::'a,X) --> equal(X::'a,Y)) & \
3.1270 +\ (! X. less_equal(X::'a,identity)) & \
3.1271 +\ (! A B C. equal(A::'a,B) --> equal(divide(A::'a,C),divide(B::'a,C))) & \
3.1272 +\ (! D F' E. equal(D::'a,E) --> equal(divide(F'::'a,D),divide(F'::'a,E))) & \
3.1273 +\ (! G H I'. equal(G::'a,H) & less_equal(G::'a,I') --> less_equal(H::'a,I')) & \
3.1274 +\ (! J L K'. equal(J::'a,K') & less_equal(L::'a,J) --> less_equal(L::'a,K')) & \
3.1275 +\ (~equal(divide(a::'a,a),zero)) --> False",
3.1276 + meson_tac);
3.1277 +
3.1278 +
3.1279 +(*202177 inferences so far. Searching to depth 14. 451 secs*)
3.1280 +val HEN007_2 = prove_hard
3.1281 + ("(! X. equal(X::'a,X)) & \
3.1282 +\ (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
3.1283 +\ (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) & \
3.1284 +\ (! X Y. less_equal(X::'a,Y) --> quotient(X::'a,Y,zero)) & \
3.1285 +\ (! X Y. quotient(X::'a,Y,zero) --> less_equal(X::'a,Y)) & \
3.1286 +\ (! Y Z X. quotient(X::'a,Y,Z) --> less_equal(Z::'a,X)) & \
3.1287 +\ (! 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) --> less_equal(V4::'a,V5)) & \
3.1288 +\ (! X. less_equal(zero::'a,X)) & \
3.1289 +\ (! X Y. less_equal(X::'a,Y) & less_equal(Y::'a,X) --> equal(X::'a,Y)) & \
3.1290 +\ (! X. less_equal(X::'a,identity)) & \
3.1291 +\ (! X Y. quotient(X::'a,Y,divide(X::'a,Y))) & \
3.1292 +\ (! X Y Z W. quotient(X::'a,Y,Z) & quotient(X::'a,Y,W) --> equal(Z::'a,W)) & \
3.1293 +\ (! X Y W Z. equal(X::'a,Y) & quotient(X::'a,W,Z) --> quotient(Y::'a,W,Z)) & \
3.1294 +\ (! X W Y Z. equal(X::'a,Y) & quotient(W::'a,X,Z) --> quotient(W::'a,Y,Z)) & \
3.1295 +\ (! X W Z Y. equal(X::'a,Y) & quotient(W::'a,Z,X) --> quotient(W::'a,Z,Y)) & \
3.1296 +\ (! X Z Y. equal(X::'a,Y) & less_equal(Z::'a,X) --> less_equal(Z::'a,Y)) & \
3.1297 +\ (! X Y Z. equal(X::'a,Y) & less_equal(X::'a,Z) --> less_equal(Y::'a,Z)) & \
3.1298 +\ (! X Y W. equal(X::'a,Y) --> equal(divide(X::'a,W),divide(Y::'a,W))) & \
3.1299 +\ (! X W Y. equal(X::'a,Y) --> equal(divide(W::'a,X),divide(W::'a,Y))) & \
3.1300 +\ (! X. quotient(X::'a,identity,zero)) & \
3.1301 +\ (! X. quotient(zero::'a,X,zero)) & \
3.1302 +\ (! X. quotient(X::'a,X,zero)) & \
3.1303 +\ (! X. quotient(X::'a,zero,X)) & \
3.1304 +\ (! Y X Z. less_equal(X::'a,Y) & less_equal(Y::'a,Z) --> less_equal(X::'a,Z)) & \
3.1305 +\ (! W1 X Z W2 Y. quotient(X::'a,Y,W1) & less_equal(W1::'a,Z) & quotient(X::'a,Z,W2) --> less_equal(W2::'a,Y)) & \
3.1306 +\ (less_equal(x::'a,y)) & \
3.1307 +\ (quotient(z::'a,y,zQy)) & \
3.1308 +\ (quotient(z::'a,x,zQx)) & \
3.1309 +\ (~less_equal(zQy::'a,zQx)) --> False",
3.1310 + meson_tac);
3.1311 +
3.1312 +(*60026 inferences so far. Searching to depth 12. 42.2 secs*)
3.1313 +val HEN008_4 = prove_hard
3.1314 + ("(! X. equal(X::'a,X)) & \
3.1315 +\ (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
3.1316 +\ (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) & \
3.1317 +\ (! X Y. less_equal(X::'a,Y) --> equal(divide(X::'a,Y),zero)) & \
3.1318 +\ (! X Y. equal(divide(X::'a,Y),zero) --> less_equal(X::'a,Y)) & \
3.1319 +\ (! Y X. less_equal(divide(X::'a,Y),X)) & \
3.1320 +\ (! X Y Z. less_equal(divide(divide(X::'a,Z),divide(Y::'a,Z)),divide(divide(X::'a,Y),Z))) & \
3.1321 +\ (! X. less_equal(zero::'a,X)) & \
3.1322 +\ (! X Y. less_equal(X::'a,Y) & less_equal(Y::'a,X) --> equal(X::'a,Y)) & \
3.1323 +\ (! X. less_equal(X::'a,identity)) & \
3.1324 +\ (! A B C. equal(A::'a,B) --> equal(divide(A::'a,C),divide(B::'a,C))) & \
3.1325 +\ (! D F' E. equal(D::'a,E) --> equal(divide(F'::'a,D),divide(F'::'a,E))) & \
3.1326 +\ (! G H I'. equal(G::'a,H) & less_equal(G::'a,I') --> less_equal(H::'a,I')) & \
3.1327 +\ (! J L K'. equal(J::'a,K') & less_equal(L::'a,J) --> less_equal(L::'a,K')) & \
3.1328 +\ (! X. equal(divide(X::'a,identity),zero)) & \
3.1329 +\ (! X. equal(divide(zero::'a,X),zero)) & \
3.1330 +\ (! X. equal(divide(X::'a,X),zero)) & \
3.1331 +\ (equal(divide(a::'a,zero),a)) & \
3.1332 +\ (! Y X Z. less_equal(X::'a,Y) & less_equal(Y::'a,Z) --> less_equal(X::'a,Z)) & \
3.1333 +\ (! X Z Y. less_equal(divide(X::'a,Y),Z) --> less_equal(divide(X::'a,Z),Y)) & \
3.1334 +\ (! Y Z X. less_equal(X::'a,Y) --> less_equal(divide(Z::'a,Y),divide(Z::'a,X))) & \
3.1335 +\ (less_equal(a::'a,b)) & \
3.1336 +\ (~less_equal(divide(a::'a,c),divide(b::'a,c))) --> False",
3.1337 + meson_tac);
3.1338 +
3.1339 +
3.1340 +(*3160 inferences so far. Searching to depth 11. 3.5 secs*)
3.1341 +val HEN009_5 = prove_hard
3.1342 + ("(! X. equal(X::'a,X)) & \
3.1343 +\ (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
3.1344 +\ (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) & \
3.1345 +\ (! Y X. equal(divide(divide(X::'a,Y),X),zero)) & \
3.1346 +\ (! X Y Z. equal(divide(divide(divide(X::'a,Z),divide(Y::'a,Z)),divide(divide(X::'a,Y),Z)),zero)) & \
3.1347 +\ (! X. equal(divide(zero::'a,X),zero)) & \
3.1348 +\ (! X Y. equal(divide(X::'a,Y),zero) & equal(divide(Y::'a,X),zero) --> equal(X::'a,Y)) & \
3.1349 +\ (! X. equal(divide(X::'a,identity),zero)) & \
3.1350 +\ (! A B C. equal(A::'a,B) --> equal(divide(A::'a,C),divide(B::'a,C))) & \
3.1351 +\ (! D F' E. equal(D::'a,E) --> equal(divide(F'::'a,D),divide(F'::'a,E))) & \
3.1352 +\ (! Y X Z. equal(divide(X::'a,Y),zero) & equal(divide(Y::'a,Z),zero) --> equal(divide(X::'a,Z),zero)) & \
3.1353 +\ (! X Z Y. equal(divide(divide(X::'a,Y),Z),zero) --> equal(divide(divide(X::'a,Z),Y),zero)) & \
3.1354 +\ (! Y Z X. equal(divide(X::'a,Y),zero) --> equal(divide(divide(Z::'a,Y),divide(Z::'a,X)),zero)) & \
3.1355 +\ (~equal(divide(identity::'a,a),divide(identity::'a,divide(identity::'a,divide(identity::'a,a))))) & \
3.1356 +\ (equal(divide(identity::'a,a),b)) & \
3.1357 +\ (equal(divide(identity::'a,b),c)) & \
3.1358 +\ (equal(divide(identity::'a,c),d)) & \
3.1359 +\ (~equal(b::'a,d)) --> False",
3.1360 + meson_tac);
3.1361 +
3.1362 +(*970373 inferences so far. Searching to depth 17. 890.0 secs*)
3.1363 +val HEN012_3 = prove_hard
3.1364 + ("(! X. equal(X::'a,X)) & \
3.1365 +\ (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
3.1366 +\ (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) & \
3.1367 +\ (! X Y. less_equal(X::'a,Y) --> equal(divide(X::'a,Y),zero)) & \
3.1368 +\ (! X Y. equal(divide(X::'a,Y),zero) --> less_equal(X::'a,Y)) & \
3.1369 +\ (! Y X. less_equal(divide(X::'a,Y),X)) & \
3.1370 +\ (! X Y Z. less_equal(divide(divide(X::'a,Z),divide(Y::'a,Z)),divide(divide(X::'a,Y),Z))) & \
3.1371 +\ (! X. less_equal(zero::'a,X)) & \
3.1372 +\ (! X Y. less_equal(X::'a,Y) & less_equal(Y::'a,X) --> equal(X::'a,Y)) & \
3.1373 +\ (! X. less_equal(X::'a,identity)) & \
3.1374 +\ (! A B C. equal(A::'a,B) --> equal(divide(A::'a,C),divide(B::'a,C))) & \
3.1375 +\ (! D F' E. equal(D::'a,E) --> equal(divide(F'::'a,D),divide(F'::'a,E))) & \
3.1376 +\ (! G H I'. equal(G::'a,H) & less_equal(G::'a,I') --> less_equal(H::'a,I')) & \
3.1377 +\ (! J L K'. equal(J::'a,K') & less_equal(L::'a,J) --> less_equal(L::'a,K')) & \
3.1378 +\ (~less_equal(a::'a,a)) --> False",
3.1379 + meson_tac);
3.1380 +
3.1381 +
3.1382 +(*1063 inferences so far. Searching to depth 20. 1.0 secs*)
3.1383 +val LCL010_1 = prove
3.1384 + ("(! X Y. is_a_theorem(equivalent(X::'a,Y)) & is_a_theorem(X) --> is_a_theorem(Y)) & \
3.1385 +\ (! X Z Y. is_a_theorem(equivalent(equivalent(X::'a,Y),equivalent(equivalent(X::'a,Z),equivalent(Z::'a,Y))))) & \
3.1386 +\ (~is_a_theorem(equivalent(equivalent(a::'a,b),equivalent(equivalent(c::'a,b),equivalent(a::'a,c))))) --> False",
3.1387 + meson_tac);
3.1388 +
3.1389 +(*2549 inferences so far. Searching to depth 12. 1.4 secs*)
3.1390 +val LCL077_2 = prove
3.1391 + ("(! X Y. is_a_theorem(implies(X,Y)) & is_a_theorem(X) --> is_a_theorem(Y)) & \
3.1392 +\ (! Y X. is_a_theorem(implies(X,implies(Y,X)))) & \
3.1393 +\ (! Y X Z. is_a_theorem(implies(implies(X,implies(Y,Z)),implies(implies(X,Y),implies(X,Z))))) & \
3.1394 +\ (! Y X. is_a_theorem(implies(implies(not(X),not(Y)),implies(Y,X)))) & \
3.1395 +\ (! X2 X1 X3. is_a_theorem(implies(X1,X2)) & is_a_theorem(implies(X2,X3)) --> is_a_theorem(implies(X1,X3))) & \
3.1396 +\ (~is_a_theorem(implies(not(not(a)),a))) --> False",
3.1397 + meson_tac);
3.1398 +
3.1399 +(*2036 inferences so far. Searching to depth 20. 1.5 secs*)
3.1400 +val LCL082_1 = prove
3.1401 + ("(! X Y. is_a_theorem(implies(X::'a,Y)) & is_a_theorem(X) --> is_a_theorem(Y)) & \
3.1402 +\ (! Y Z U X. is_a_theorem(implies(implies(implies(X::'a,Y),Z),implies(implies(Z::'a,X),implies(U::'a,X))))) & \
3.1403 +\ (~is_a_theorem(implies(a::'a,implies(b::'a,a)))) --> False",
3.1404 + meson_tac);
3.1405 +
3.1406 +(*1100 inferences so far. Searching to depth 13. 1.0 secs*)
3.1407 +val LCL111_1 = prove
3.1408 + ("(! X Y. is_a_theorem(implies(X,Y)) & is_a_theorem(X) --> is_a_theorem(Y)) & \
3.1409 +\ (! Y X. is_a_theorem(implies(X,implies(Y,X)))) & \
3.1410 +\ (! Y X Z. is_a_theorem(implies(implies(X,Y),implies(implies(Y,Z),implies(X,Z))))) & \
3.1411 +\ (! Y X. is_a_theorem(implies(implies(implies(X,Y),Y),implies(implies(Y,X),X)))) & \
3.1412 +\ (! Y X. is_a_theorem(implies(implies(not(X),not(Y)),implies(Y,X)))) & \
3.1413 +\ (~is_a_theorem(implies(implies(a,b),implies(implies(c,a),implies(c,b))))) --> False",
3.1414 + meson_tac);
3.1415 +
3.1416 +(*667 inferences so far. Searching to depth 9. 1.4 secs*)
3.1417 +val LCL143_1 = prove
3.1418 + ("(! X. equal(X,X)) & \
3.1419 +\ (! Y X. equal(X,Y) --> equal(Y,X)) & \
3.1420 +\ (! Y X Z. equal(X,Y) & equal(Y,Z) --> equal(X,Z)) & \
3.1421 +\ (! X. equal(implies(true,X),X)) & \
3.1422 +\ (! Y X Z. equal(implies(implies(X,Y),implies(implies(Y,Z),implies(X,Z))),true)) & \
3.1423 +\ (! Y X. equal(implies(implies(X,Y),Y),implies(implies(Y,X),X))) & \
3.1424 +\ (! Y X. equal(implies(implies(not(X),not(Y)),implies(Y,X)),true)) & \
3.1425 +\ (! A B C. equal(A,B) --> equal(implies(A,C),implies(B,C))) & \
3.1426 +\ (! D F' E. equal(D,E) --> equal(implies(F',D),implies(F',E))) & \
3.1427 +\ (! G H. equal(G,H) --> equal(not(G),not(H))) & \
3.1428 +\ (! X Y. equal(big_V(X,Y),implies(implies(X,Y),Y))) & \
3.1429 +\ (! X Y. equal(big_hat(X,Y),not(big_V(not(X),not(Y))))) & \
3.1430 +\ (! X Y. ordered(X,Y) --> equal(implies(X,Y),true)) & \
3.1431 +\ (! X Y. equal(implies(X,Y),true) --> ordered(X,Y)) & \
3.1432 +\ (! A B C. equal(A,B) --> equal(big_V(A,C),big_V(B,C))) & \
3.1433 +\ (! D F' E. equal(D,E) --> equal(big_V(F',D),big_V(F',E))) & \
3.1434 +\ (! G H I'. equal(G,H) --> equal(big_hat(G,I'),big_hat(H,I'))) & \
3.1435 +\ (! J L K'. equal(J,K') --> equal(big_hat(L,J),big_hat(L,K'))) & \
3.1436 +\ (! M N O_. equal(M,N) & ordered(M,O_) --> ordered(N,O_)) & \
3.1437 +\ (! P R Q. equal(P,Q) & ordered(R,P) --> ordered(R,Q)) & \
3.1438 +\ (ordered(x,y)) & \
3.1439 +\ (~ordered(implies(z,x),implies(z,y))) --> False",
3.1440 + meson_tac);
3.1441 +
3.1442 +(*5245 inferences so far. Searching to depth 12. 4.6 secs*)
3.1443 +val LCL182_1 = prove_hard
3.1444 + ("(! A. axiom(or(not(or(A,A)),A))) & \
3.1445 +\ (! B A. axiom(or(not(A),or(B,A)))) & \
3.1446 +\ (! B A. axiom(or(not(or(A,B)),or(B,A)))) & \
3.1447 +\ (! B A C. axiom(or(not(or(A,or(B,C))),or(B,or(A,C))))) & \
3.1448 +\ (! A C B. axiom(or(not(or(not(A),B)),or(not(or(C,A)),or(C,B))))) & \
3.1449 +\ (! X. axiom(X) --> theorem(X)) & \
3.1450 +\ (! X Y. axiom(or(not(Y),X)) & theorem(Y) --> theorem(X)) & \
3.1451 +\ (! X Y Z. axiom(or(not(X),Y)) & theorem(or(not(Y),Z)) --> theorem(or(not(X),Z))) & \
3.1452 +\ (~theorem(or(not(or(not(p),q)),or(not(not(q)),not(p))))) --> False",
3.1453 + meson_tac);
3.1454 +
3.1455 +(*410 inferences so far. Searching to depth 10. 0.3 secs*)
3.1456 +val LCL200_1 = prove
3.1457 + ("(! A. axiom(or(not(or(A,A)),A))) & \
3.1458 +\ (! B A. axiom(or(not(A),or(B,A)))) & \
3.1459 +\ (! B A. axiom(or(not(or(A,B)),or(B,A)))) & \
3.1460 +\ (! B A C. axiom(or(not(or(A,or(B,C))),or(B,or(A,C))))) & \
3.1461 +\ (! A C B. axiom(or(not(or(not(A),B)),or(not(or(C,A)),or(C,B))))) & \
3.1462 +\ (! X. axiom(X) --> theorem(X)) & \
3.1463 +\ (! X Y. axiom(or(not(Y),X)) & theorem(Y) --> theorem(X)) & \
3.1464 +\ (! X Y Z. axiom(or(not(X),Y)) & theorem(or(not(Y),Z)) --> theorem(or(not(X),Z))) & \
3.1465 +\ (~theorem(or(not(not(or(p,q))),not(q)))) --> False",
3.1466 + meson_tac);
3.1467 +
3.1468 +(*5849 inferences so far. Searching to depth 12. 5.6 secs*)
3.1469 +val LCL215_1 = prove_hard
3.1470 + ("(! A. axiom(or(not(or(A,A)),A))) & \
3.1471 +\ (! B A. axiom(or(not(A),or(B,A)))) & \
3.1472 +\ (! B A. axiom(or(not(or(A,B)),or(B,A)))) & \
3.1473 +\ (! B A C. axiom(or(not(or(A,or(B,C))),or(B,or(A,C))))) & \
3.1474 +\ (! A C B. axiom(or(not(or(not(A),B)),or(not(or(C,A)),or(C,B))))) & \
3.1475 +\ (! X. axiom(X) --> theorem(X)) & \
3.1476 +\ (! X Y. axiom(or(not(Y),X)) & theorem(Y) --> theorem(X)) & \
3.1477 +\ (! X Y Z. axiom(or(not(X),Y)) & theorem(or(not(Y),Z)) --> theorem(or(not(X),Z))) & \
3.1478 +\ (~theorem(or(not(or(not(p),q)),or(not(or(p,q)),q)))) --> False",
3.1479 + meson_tac);
3.1480 +
3.1481 +(*0 secs. Not sure that a search even starts!*)
3.1482 +val LCL230_2 = prove
3.1483 + ("(q --> p | r) & \
3.1484 +\ (~p) & \
3.1485 +\ (q) & \
3.1486 +\ (~r) --> False",
3.1487 + meson_tac);
3.1488 +
3.1489 +(*119585 inferences so far. Searching to depth 14. 262.4 secs*)
3.1490 +val LDA003_1 = prove_hard
3.1491 + ("(! X. equal(X::'a,X)) & \
3.1492 +\ (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
3.1493 +\ (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) & \
3.1494 +\ (! Y X Z. equal(f(X::'a,f(Y::'a,Z)),f(f(X::'a,Y),f(X::'a,Z)))) & \
3.1495 +\ (! X Y. left(X::'a,f(X::'a,Y))) & \
3.1496 +\ (! Y X Z. left(X::'a,Y) & left(Y::'a,Z) --> left(X::'a,Z)) & \
3.1497 +\ (equal(num2::'a,f(num1::'a,num1))) & \
3.1498 +\ (equal(num3::'a,f(num2::'a,num1))) & \
3.1499 +\ (equal(u::'a,f(num2::'a,num2))) & \
3.1500 +\ (! A B C. equal(A::'a,B) --> equal(f(A::'a,C),f(B::'a,C))) & \
3.1501 +\ (! D F' E. equal(D::'a,E) --> equal(f(F'::'a,D),f(F'::'a,E))) & \
3.1502 +\ (! G H I'. equal(G::'a,H) & left(G::'a,I') --> left(H::'a,I')) & \
3.1503 +\ (! J L K'. equal(J::'a,K') & left(L::'a,J) --> left(L::'a,K')) & \
3.1504 +\ (~left(num3::'a,u)) --> False",
3.1505 + meson_tac);
3.1506 +
3.1507 +
3.1508 +(*2392 inferences so far. Searching to depth 12. 2.2 secs*)
3.1509 +val MSC002_1 = prove
3.1510 + ("(at(something::'a,here,now)) & \
3.1511 +\ (! Place Situation. hand_at(Place::'a,Situation) --> hand_at(Place::'a,let_go(Situation))) & \
3.1512 +\ (! Place Another_place Situation. hand_at(Place::'a,Situation) --> hand_at(Another_place::'a,go(Another_place::'a,Situation))) & \
3.1513 +\ (! Thing Situation. ~held(Thing::'a,let_go(Situation))) & \
3.1514 +\ (! Situation Thing. at(Thing::'a,here,Situation) --> red(Thing)) & \
3.1515 +\ (! Thing Place Situation. at(Thing::'a,Place,Situation) --> at(Thing::'a,Place,let_go(Situation))) & \
3.1516 +\ (! Thing Place Situation. at(Thing::'a,Place,Situation) --> at(Thing::'a,Place,pick_up(Situation))) & \
3.1517 +\ (! Thing Place Situation. at(Thing::'a,Place,Situation) --> grabbed(Thing::'a,pick_up(go(Place::'a,let_go(Situation))))) & \
3.1518 +\ (! Thing Situation. red(Thing) & put(Thing::'a,there,Situation) --> answer(Situation)) & \
3.1519 +\ (! Place Thing Another_place Situation. at(Thing::'a,Place,Situation) & grabbed(Thing::'a,Situation) --> put(Thing::'a,Another_place,go(Another_place::'a,Situation))) & \
3.1520 +\ (! Thing Place Another_place Situation. at(Thing::'a,Place,Situation) --> held(Thing::'a,Situation) | at(Thing::'a,Place,go(Another_place::'a,Situation))) & \
3.1521 +\ (! One_place Thing Place Situation. hand_at(One_place::'a,Situation) & held(Thing::'a,Situation) --> at(Thing::'a,Place,go(Place::'a,Situation))) & \
3.1522 +\ (! Place Thing Situation. hand_at(Place::'a,Situation) & at(Thing::'a,Place,Situation) --> held(Thing::'a,pick_up(Situation))) & \
3.1523 +\ (! Situation. ~answer(Situation)) --> False",
3.1524 + meson_tac);
3.1525 +
3.1526 +(*73 inferences so far. Searching to depth 10. 0.2 secs*)
3.1527 +val MSC003_1 = prove
3.1528 + ("(! 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,times(Number_of_mid_parts::'a,Number_of_small_parts),Small_part)) & \
3.1529 +\ (! 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,times(Number_of_mid_parts::'a,Number_of_small_parts),Small_part)) & \
3.1530 +\ (in'(john::'a,boy)) & \
3.1531 +\ (! X. in'(X::'a,boy) --> in'(X::'a,human)) & \
3.1532 +\ (! X. in'(X::'a,hand) --> has_parts(X::'a,num5,fingers)) & \
3.1533 +\ (! X. in'(X::'a,human) --> has_parts(X::'a,num2,arm)) & \
3.1534 +\ (! X. in'(X::'a,arm) --> has_parts(X::'a,num1,hand)) & \
3.1535 +\ (~has_parts(john::'a,times(num2::'a,num1),hand)) --> False",
3.1536 + meson_tac);
3.1537 +
3.1538 +(*1486 inferences so far. Searching to depth 20. 1.2 secs*)
3.1539 +val MSC004_1 = prove
3.1540 + ("(! 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,times(Number_of_mid_parts::'a,Number_of_small_parts),Small_part)) & \
3.1541 +\ (! 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,times(Number_of_mid_parts::'a,Number_of_small_parts),Small_part)) & \
3.1542 +\ (in'(john::'a,boy)) & \
3.1543 +\ (! X. in'(X::'a,boy) --> in'(X::'a,human)) & \
3.1544 +\ (! X. in'(X::'a,hand) --> has_parts(X::'a,num5,fingers)) & \
3.1545 +\ (! X. in'(X::'a,human) --> has_parts(X::'a,num2,arm)) & \
3.1546 +\ (! X. in'(X::'a,arm) --> has_parts(X::'a,num1,hand)) & \
3.1547 +\ (~has_parts(john::'a,times(times(num2::'a,num1),num5),fingers)) --> False",
3.1548 + meson_tac);
3.1549 +
3.1550 +(*100 inferences so far. Searching to depth 12. 0.1 secs*)
3.1551 +val MSC005_1 = prove
3.1552 + ("(value(truth::'a,truth)) & \
3.1553 +\ (value(falsity::'a,falsity)) & \
3.1554 +\ (! X Y. value(X::'a,truth) & value(Y::'a,truth) --> value(xor(X::'a,Y),falsity)) & \
3.1555 +\ (! X Y. value(X::'a,truth) & value(Y::'a,falsity) --> value(xor(X::'a,Y),truth)) & \
3.1556 +\ (! X Y. value(X::'a,falsity) & value(Y::'a,truth) --> value(xor(X::'a,Y),truth)) & \
3.1557 +\ (! X Y. value(X::'a,falsity) & value(Y::'a,falsity) --> value(xor(X::'a,Y),falsity)) & \
3.1558 +\ (! Value. ~value(xor(xor(xor(xor(truth::'a,falsity),falsity),truth),falsity),Value)) --> False",
3.1559 + meson_tac);
3.1560 +
3.1561 +(*19116 inferences so far. Searching to depth 16. 15.9 secs*)
3.1562 +val MSC006_1 = prove_hard
3.1563 + ("(! Y X Z. p(X::'a,Y) & p(Y::'a,Z) --> p(X::'a,Z)) & \
3.1564 +\ (! Y X Z. q(X::'a,Y) & q(Y::'a,Z) --> q(X::'a,Z)) & \
3.1565 +\ (! Y X. q(X::'a,Y) --> q(Y::'a,X)) & \
3.1566 +\ (! X Y. p(X::'a,Y) | q(X::'a,Y)) & \
3.1567 +\ (~p(a::'a,b)) & \
3.1568 +\ (~q(c::'a,d)) --> False",
3.1569 + meson_tac);
3.1570 +
3.1571 +(*1713 inferences so far. Searching to depth 10. 2.8 secs*)
3.1572 +val NUM001_1 = prove
3.1573 + ("(! A. equal(A::'a,A)) & \
3.1574 +\ (! B A C. equal(A::'a,B) & equal(B::'a,C) --> equal(A::'a,C)) & \
3.1575 +\ (! B A. equal(add(A::'a,B),add(B::'a,A))) & \
3.1576 +\ (! A B C. equal(add(A::'a,add(B::'a,C)),add(add(A::'a,B),C))) & \
3.1577 +\ (! B A. equal(subtract(add(A::'a,B),B),A)) & \
3.1578 +\ (! A B. equal(A::'a,subtract(add(A::'a,B),B))) & \
3.1579 +\ (! A C B. equal(add(subtract(A::'a,B),C),subtract(add(A::'a,C),B))) & \
3.1580 +\ (! A C B. equal(subtract(add(A::'a,B),C),add(subtract(A::'a,C),B))) & \
3.1581 +\ (! A C B D. equal(A::'a,B) & equal(C::'a,add(A::'a,D)) --> equal(C::'a,add(B::'a,D))) & \
3.1582 +\ (! A C D B. equal(A::'a,B) & equal(C::'a,add(D::'a,A)) --> equal(C::'a,add(D::'a,B))) & \
3.1583 +\ (! A C B D. equal(A::'a,B) & equal(C::'a,subtract(A::'a,D)) --> equal(C::'a,subtract(B::'a,D))) & \
3.1584 +\ (! A C D B. equal(A::'a,B) & equal(C::'a,subtract(D::'a,A)) --> equal(C::'a,subtract(D::'a,B))) & \
3.1585 +\ (~equal(add(add(a::'a,b),c),add(a::'a,add(b::'a,c)))) --> False",
3.1586 + meson_tac);
3.1587 +
3.1588 +(*20717 inferences so far. Searching to depth 11. 13.7 secs*)
3.1589 +val NUM021_1 = prove_hard
3.1590 + ("(! X. equal(X::'a,X)) & \
3.1591 +\ (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
3.1592 +\ (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) & \
3.1593 +\ (! A. equal(add(A::'a,num0),A)) & \
3.1594 +\ (! A B. equal(add(A::'a,successor(B)),successor(add(A::'a,B)))) & \
3.1595 +\ (! A. equal(multiply(A::'a,num0),num0)) & \
3.1596 +\ (! B A. equal(multiply(A::'a,successor(B)),add(multiply(A::'a,B),A))) & \
3.1597 +\ (! A B. equal(successor(A),successor(B)) --> equal(A::'a,B)) & \
3.1598 +\ (! A B. equal(A::'a,B) --> equal(successor(A),successor(B))) & \
3.1599 +\ (! A C B. less(A::'a,B) & less(C::'a,A) --> less(C::'a,B)) & \
3.1600 +\ (! A B C. equal(add(successor(A),B),C) --> less(B::'a,C)) & \
3.1601 +\ (! A B. less(A::'a,B) --> equal(add(successor(predecessor_of_1st_minus_2nd(B::'a,A)),A),B)) & \
3.1602 +\ (! A B. divides(A::'a,B) --> less(A::'a,B) | equal(A::'a,B)) & \
3.1603 +\ (! A B. less(A::'a,B) --> divides(A::'a,B)) & \
3.1604 +\ (! A B. equal(A::'a,B) --> divides(A::'a,B)) & \
3.1605 +\ (less(b::'a,c)) & \
3.1606 +\ (~less(b::'a,a)) & \
3.1607 +\ (divides(c::'a,a)) & \
3.1608 +\ (! A. ~equal(successor(A),num0)) --> False",
3.1609 + meson_tac);
3.1610 +
3.1611 +(*26320 inferences so far. Searching to depth 10. 26.4 secs*)
3.1612 +val NUM024_1 = prove_hard
3.1613 + ("(! X. equal(X::'a,X)) & \
3.1614 +\ (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
3.1615 +\ (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) & \
3.1616 +\ (! A. equal(add(A::'a,num0),A)) & \
3.1617 +\ (! A B. equal(add(A::'a,successor(B)),successor(add(A::'a,B)))) & \
3.1618 +\ (! A. equal(multiply(A::'a,num0),num0)) & \
3.1619 +\ (! B A. equal(multiply(A::'a,successor(B)),add(multiply(A::'a,B),A))) & \
3.1620 +\ (! A B. equal(successor(A),successor(B)) --> equal(A::'a,B)) & \
3.1621 +\ (! A B. equal(A::'a,B) --> equal(successor(A),successor(B))) & \
3.1622 +\ (! A C B. less(A::'a,B) & less(C::'a,A) --> less(C::'a,B)) & \
3.1623 +\ (! A B C. equal(add(successor(A),B),C) --> less(B::'a,C)) & \
3.1624 +\ (! A B. less(A::'a,B) --> equal(add(successor(predecessor_of_1st_minus_2nd(B::'a,A)),A),B)) & \
3.1625 +\ (! B A. equal(add(A::'a,B),add(B::'a,A))) & \
3.1626 +\ (! B A C. equal(add(A::'a,B),add(C::'a,B)) --> equal(A::'a,C)) & \
3.1627 +\ (less(a::'a,a)) & \
3.1628 +\ (! A. ~equal(successor(A),num0)) --> False",
3.1629 + meson_tac);
3.1630 +
3.1631 +
3.1632 +(*1345 inferences so far. Searching to depth 7. 23.3 secs. BIG*)
3.1633 +val NUM180_1 = prove_hard
3.1634 + ("(! X. equal(X::'a,X)) & \
3.1635 +\ (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
3.1636 +\ (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) & \
3.1637 +\ (! X U Y. subclass(X::'a,Y) & member(U::'a,X) --> member(U::'a,Y)) & \
3.1638 +\ (! X Y. member(not_subclass_element(X::'a,Y),X) | subclass(X::'a,Y)) & \
3.1639 +\ (! X Y. member(not_subclass_element(X::'a,Y),Y) --> subclass(X::'a,Y)) & \
3.1640 +\ (! X. subclass(X::'a,universal_class)) & \
3.1641 +\ (! X Y. equal(X::'a,Y) --> subclass(X::'a,Y)) & \
3.1642 +\ (! Y X. equal(X::'a,Y) --> subclass(Y::'a,X)) & \
3.1643 +\ (! X Y. subclass(X::'a,Y) & subclass(Y::'a,X) --> equal(X::'a,Y)) & \
3.1644 +\ (! X U Y. member(U::'a,unordered_pair(X::'a,Y)) --> equal(U::'a,X) | equal(U::'a,Y)) & \
3.1645 +\ (! X Y. member(X::'a,universal_class) --> member(X::'a,unordered_pair(X::'a,Y))) & \
3.1646 +\ (! X Y. member(Y::'a,universal_class) --> member(Y::'a,unordered_pair(X::'a,Y))) & \
3.1647 +\ (! X Y. member(unordered_pair(X::'a,Y),universal_class)) & \
3.1648 +\ (! X. equal(unordered_pair(X::'a,X),singleton(X))) & \
3.1649 +\ (! X Y. equal(unordered_pair(singleton(X),unordered_pair(X::'a,singleton(Y))),ordered_pair(X::'a,Y))) & \
3.1650 +\ (! V Y U X. member(ordered_pair(U::'a,V),cross_product(X::'a,Y)) --> member(U::'a,X)) & \
3.1651 +\ (! U X V Y. member(ordered_pair(U::'a,V),cross_product(X::'a,Y)) --> member(V::'a,Y)) & \
3.1652 +\ (! U V X Y. member(U::'a,X) & member(V::'a,Y) --> member(ordered_pair(U::'a,V),cross_product(X::'a,Y))) & \
3.1653 +\ (! X Y Z. member(Z::'a,cross_product(X::'a,Y)) --> equal(ordered_pair(first(Z),second(Z)),Z)) & \
3.1654 +\ (subclass(element_relation::'a,cross_product(universal_class::'a,universal_class))) & \
3.1655 +\ (! X Y. member(ordered_pair(X::'a,Y),element_relation) --> member(X::'a,Y)) & \
3.1656 +\ (! 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)) & \
3.1657 +\ (! Y Z X. member(Z::'a,intersection(X::'a,Y)) --> member(Z::'a,X)) & \
3.1658 +\ (! X Z Y. member(Z::'a,intersection(X::'a,Y)) --> member(Z::'a,Y)) & \
3.1659 +\ (! Z X Y. member(Z::'a,X) & member(Z::'a,Y) --> member(Z::'a,intersection(X::'a,Y))) & \
3.1660 +\ (! Z X. ~(member(Z::'a,complement(X)) & member(Z::'a,X))) & \
3.1661 +\ (! Z X. member(Z::'a,universal_class) --> member(Z::'a,complement(X)) | member(Z::'a,X)) & \
3.1662 +\ (! X Y. equal(complement(intersection(complement(X),complement(Y))),union(X::'a,Y))) & \
3.1663 +\ (! X Y. equal(intersection(complement(intersection(X::'a,Y)),complement(intersection(complement(X),complement(Y)))),difference(X::'a,Y))) & \
3.1664 +\ (! Xr X Y. equal(intersection(Xr::'a,cross_product(X::'a,Y)),restrct(Xr::'a,X,Y))) & \
3.1665 +\ (! Xr X Y. equal(intersection(cross_product(X::'a,Y),Xr),restrct(Xr::'a,X,Y))) & \
3.1666 +\ (! Z X. ~(equal(restrct(X::'a,singleton(Z),universal_class),null_class) & member(Z::'a,domain_of(X)))) & \
3.1667 +\ (! Z X. member(Z::'a,universal_class) --> equal(restrct(X::'a,singleton(Z),universal_class),null_class) | member(Z::'a,domain_of(X))) & \
3.1668 +\ (! X. subclass(rotate(X),cross_product(cross_product(universal_class::'a,universal_class),universal_class))) & \
3.1669 +\ (! V W U X. member(ordered_pair(ordered_pair(U::'a,V),W),rotate(X)) --> member(ordered_pair(ordered_pair(V::'a,W),U),X)) & \
3.1670 +\ (! 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),rotate(X))) & \
3.1671 +\ (! X. subclass(flip(X),cross_product(cross_product(universal_class::'a,universal_class),universal_class))) & \
3.1672 +\ (! 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)) & \
3.1673 +\ (! 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))) & \
3.1674 +\ (! Y. equal(domain_of(flip(cross_product(Y::'a,universal_class))),inverse(Y))) & \
3.1675 +\ (! Z. equal(domain_of(inverse(Z)),range_of(Z))) & \
3.1676 +\ (! Z X Y. equal(first(not_subclass_element(restrct(Z::'a,X,singleton(Y)),null_class)),domain(Z::'a,X,Y))) & \
3.1677 +\ (! Z X Y. equal(second(not_subclass_element(restrct(Z::'a,singleton(X),Y),null_class)),rng(Z::'a,X,Y))) & \
3.1678 +\ (! Xr X. equal(range_of(restrct(Xr::'a,X,universal_class)),image_(Xr::'a,X))) & \
3.1679 +\ (! X. equal(union(X::'a,singleton(X)),successor(X))) & \
3.1680 +\ (subclass(successor_relation::'a,cross_product(universal_class::'a,universal_class))) & \
3.1681 +\ (! X Y. member(ordered_pair(X::'a,Y),successor_relation) --> equal(successor(X),Y)) & \
3.1682 +\ (! 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)) & \
3.1683 +\ (! X. inductive(X) --> member(null_class::'a,X)) & \
3.1684 +\ (! X. inductive(X) --> subclass(image_(successor_relation::'a,X),X)) & \
3.1685 +\ (! X. member(null_class::'a,X) & subclass(image_(successor_relation::'a,X),X) --> inductive(X)) & \
3.1686 +\ (inductive(omega)) & \
3.1687 +\ (! Y. inductive(Y) --> subclass(omega::'a,Y)) & \
3.1688 +\ (member(omega::'a,universal_class)) & \
3.1689 +\ (! X. equal(domain_of(restrct(element_relation::'a,universal_class,X)),sum_class(X))) & \
3.1690 +\ (! X. member(X::'a,universal_class) --> member(sum_class(X),universal_class)) & \
3.1691 +\ (! X. equal(complement(image_(element_relation::'a,complement(X))),powerClass(X))) & \
3.1692 +\ (! U. member(U::'a,universal_class) --> member(powerClass(U),universal_class)) & \
3.1693 +\ (! Yr Xr. subclass(compos(Yr::'a,Xr),cross_product(universal_class::'a,universal_class))) & \
3.1694 +\ (! 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))))) & \
3.1695 +\ (! 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))) & \
3.1696 +\ (! X. single_valued_class(X) --> subclass(compos(X::'a,inverse(X)),identity_relation)) & \
3.1697 +\ (! X. subclass(compos(X::'a,inverse(X)),identity_relation) --> single_valued_class(X)) & \
3.1698 +\ (! Xf. function(Xf) --> subclass(Xf::'a,cross_product(universal_class::'a,universal_class))) & \
3.1699 +\ (! Xf. function(Xf) --> subclass(compos(Xf::'a,inverse(Xf)),identity_relation)) & \
3.1700 +\ (! Xf. subclass(Xf::'a,cross_product(universal_class::'a,universal_class)) & subclass(compos(Xf::'a,inverse(Xf)),identity_relation) --> function(Xf)) & \
3.1701 +\ (! Xf X. function(Xf) & member(X::'a,universal_class) --> member(image_(Xf::'a,X),universal_class)) & \
3.1702 +\ (! X. equal(X::'a,null_class) | member(regular(X),X)) & \
3.1703 +\ (! X. equal(X::'a,null_class) | equal(intersection(X::'a,regular(X)),null_class)) & \
3.1704 +\ (! Xf Y. equal(sum_class(image_(Xf::'a,singleton(Y))),apply(Xf::'a,Y))) & \
3.1705 +\ (function(choice)) & \
3.1706 +\ (! Y. member(Y::'a,universal_class) --> equal(Y::'a,null_class) | member(apply(choice::'a,Y),Y)) & \
3.1707 +\ (! Xf. one_to_one(Xf) --> function(Xf)) & \
3.1708 +\ (! Xf. one_to_one(Xf) --> function(inverse(Xf))) & \
3.1709 +\ (! Xf. function(inverse(Xf)) & function(Xf) --> one_to_one(Xf)) & \
3.1710 +\ (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)) & \
3.1711 +\ (equal(intersection(inverse(subset_relation),subset_relation),identity_relation)) & \
3.1712 +\ (! Xr. equal(complement(domain_of(intersection(Xr::'a,identity_relation))),diagonalise(Xr))) & \
3.1713 +\ (! X. equal(intersection(domain_of(X),diagonalise(compos(inverse(element_relation),X))),cantor(X))) & \
3.1714 +\ (! Xf. operation(Xf) --> function(Xf)) & \
3.1715 +\ (! Xf. operation(Xf) --> equal(cross_product(domain_of(domain_of(Xf)),domain_of(domain_of(Xf))),domain_of(Xf))) & \
3.1716 +\ (! Xf. operation(Xf) --> subclass(range_of(Xf),domain_of(domain_of(Xf)))) & \
3.1717 +\ (! 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)) & \
3.1718 +\ (! Xf1 Xf2 Xh. compatible(Xh::'a,Xf1,Xf2) --> function(Xh)) & \
3.1719 +\ (! Xf2 Xf1 Xh. compatible(Xh::'a,Xf1,Xf2) --> equal(domain_of(domain_of(Xf1)),domain_of(Xh))) & \
3.1720 +\ (! Xf1 Xh Xf2. compatible(Xh::'a,Xf1,Xf2) --> subclass(range_of(Xh),domain_of(domain_of(Xf2)))) & \
3.1721 +\ (! 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)) & \
3.1722 +\ (! Xh Xf2 Xf1. homomorphism(Xh::'a,Xf1,Xf2) --> operation(Xf1)) & \
3.1723 +\ (! Xh Xf1 Xf2. homomorphism(Xh::'a,Xf1,Xf2) --> operation(Xf2)) & \
3.1724 +\ (! Xh Xf1 Xf2. homomorphism(Xh::'a,Xf1,Xf2) --> compatible(Xh::'a,Xf1,Xf2)) & \
3.1725 +\ (! 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))))) & \
3.1726 +\ (! 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)) & \
3.1727 +\ (! 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)) & \
3.1728 +\ (! D E F'. equal(D::'a,E) --> equal(apply(D::'a,F'),apply(E::'a,F'))) & \
3.1729 +\ (! G I' H. equal(G::'a,H) --> equal(apply(I'::'a,G),apply(I'::'a,H))) & \
3.1730 +\ (! J K'. equal(J::'a,K') --> equal(cantor(J),cantor(K'))) & \
3.1731 +\ (! L M. equal(L::'a,M) --> equal(complement(L),complement(M))) & \
3.1732 +\ (! N O_ P. equal(N::'a,O_) --> equal(compos(N::'a,P),compos(O_::'a,P))) & \
3.1733 +\ (! Q S' R. equal(Q::'a,R) --> equal(compos(S'::'a,Q),compos(S'::'a,R))) & \
3.1734 +\ (! T' U V. equal(T'::'a,U) --> equal(cross_product(T'::'a,V),cross_product(U::'a,V))) & \
3.1735 +\ (! W Y X. equal(W::'a,X) --> equal(cross_product(Y::'a,W),cross_product(Y::'a,X))) & \
3.1736 +\ (! Z A1. equal(Z::'a,A1) --> equal(diagonalise(Z),diagonalise(A1))) & \
3.1737 +\ (! B1 C1 D1. equal(B1::'a,C1) --> equal(difference(B1::'a,D1),difference(C1::'a,D1))) & \
3.1738 +\ (! E1 G1 F1. equal(E1::'a,F1) --> equal(difference(G1::'a,E1),difference(G1::'a,F1))) & \
3.1739 +\ (! H1 I1 J1 K1. equal(H1::'a,I1) --> equal(domain(H1::'a,J1,K1),domain(I1::'a,J1,K1))) & \
3.1740 +\ (! L1 N1 M1 O1. equal(L1::'a,M1) --> equal(domain(N1::'a,L1,O1),domain(N1::'a,M1,O1))) & \
3.1741 +\ (! P1 R1 S1 Q1. equal(P1::'a,Q1) --> equal(domain(R1::'a,S1,P1),domain(R1::'a,S1,Q1))) & \
3.1742 +\ (! T1 U1. equal(T1::'a,U1) --> equal(domain_of(T1),domain_of(U1))) & \
3.1743 +\ (! V1 W1. equal(V1::'a,W1) --> equal(first(V1),first(W1))) & \
3.1744 +\ (! X1 Y1. equal(X1::'a,Y1) --> equal(flip(X1),flip(Y1))) & \
3.1745 +\ (! Z1 A2 B2. equal(Z1::'a,A2) --> equal(image_(Z1::'a,B2),image_(A2::'a,B2))) & \
3.1746 +\ (! C2 E2 D2. equal(C2::'a,D2) --> equal(image_(E2::'a,C2),image_(E2::'a,D2))) & \
3.1747 +\ (! F2 G2 H2. equal(F2::'a,G2) --> equal(intersection(F2::'a,H2),intersection(G2::'a,H2))) & \
3.1748 +\ (! I2 K2 J2. equal(I2::'a,J2) --> equal(intersection(K2::'a,I2),intersection(K2::'a,J2))) & \
3.1749 +\ (! L2 M2. equal(L2::'a,M2) --> equal(inverse(L2),inverse(M2))) & \
3.1750 +\ (! N2 O2 P2 Q2. equal(N2::'a,O2) --> equal(not_homomorphism1(N2::'a,P2,Q2),not_homomorphism1(O2::'a,P2,Q2))) & \
3.1751 +\ (! R2 T2 S2 U2. equal(R2::'a,S2) --> equal(not_homomorphism1(T2::'a,R2,U2),not_homomorphism1(T2::'a,S2,U2))) & \
3.1752 +\ (! V2 X2 Y2 W2. equal(V2::'a,W2) --> equal(not_homomorphism1(X2::'a,Y2,V2),not_homomorphism1(X2::'a,Y2,W2))) & \
3.1753 +\ (! Z2 A3 B3 C3. equal(Z2::'a,A3) --> equal(not_homomorphism2(Z2::'a,B3,C3),not_homomorphism2(A3::'a,B3,C3))) & \
3.1754 +\ (! D3 F3 E3 G3. equal(D3::'a,E3) --> equal(not_homomorphism2(F3::'a,D3,G3),not_homomorphism2(F3::'a,E3,G3))) & \
3.1755 +\ (! H3 J3 K3 I3. equal(H3::'a,I3) --> equal(not_homomorphism2(J3::'a,K3,H3),not_homomorphism2(J3::'a,K3,I3))) & \
3.1756 +\ (! L3 M3 N3. equal(L3::'a,M3) --> equal(not_subclass_element(L3::'a,N3),not_subclass_element(M3::'a,N3))) & \
3.1757 +\ (! O3 Q3 P3. equal(O3::'a,P3) --> equal(not_subclass_element(Q3::'a,O3),not_subclass_element(Q3::'a,P3))) & \
3.1758 +\ (! R3 S3 T3. equal(R3::'a,S3) --> equal(ordered_pair(R3::'a,T3),ordered_pair(S3::'a,T3))) & \
3.1759 +\ (! U3 W3 V3. equal(U3::'a,V3) --> equal(ordered_pair(W3::'a,U3),ordered_pair(W3::'a,V3))) & \
3.1760 +\ (! X3 Y3. equal(X3::'a,Y3) --> equal(powerClass(X3),powerClass(Y3))) & \
3.1761 +\ (! Z3 A4 B4 C4. equal(Z3::'a,A4) --> equal(rng(Z3::'a,B4,C4),rng(A4::'a,B4,C4))) & \
3.1762 +\ (! D4 F4 E4 G4. equal(D4::'a,E4) --> equal(rng(F4::'a,D4,G4),rng(F4::'a,E4,G4))) & \
3.1763 +\ (! H4 J4 K4 I4. equal(H4::'a,I4) --> equal(rng(J4::'a,K4,H4),rng(J4::'a,K4,I4))) & \
3.1764 +\ (! L4 M4. equal(L4::'a,M4) --> equal(range_of(L4),range_of(M4))) & \
3.1765 +\ (! N4 O4. equal(N4::'a,O4) --> equal(regular(N4),regular(O4))) & \
3.1766 +\ (! P4 Q4 R4 S4. equal(P4::'a,Q4) --> equal(restrct(P4::'a,R4,S4),restrct(Q4::'a,R4,S4))) & \
3.1767 +\ (! T4 V4 U4 W4. equal(T4::'a,U4) --> equal(restrct(V4::'a,T4,W4),restrct(V4::'a,U4,W4))) & \
3.1768 +\ (! X4 Z4 A5 Y4. equal(X4::'a,Y4) --> equal(restrct(Z4::'a,A5,X4),restrct(Z4::'a,A5,Y4))) & \
3.1769 +\ (! B5 C5. equal(B5::'a,C5) --> equal(rotate(B5),rotate(C5))) & \
3.1770 +\ (! D5 E5. equal(D5::'a,E5) --> equal(second(D5),second(E5))) & \
3.1771 +\ (! F5 G5. equal(F5::'a,G5) --> equal(singleton(F5),singleton(G5))) & \
3.1772 +\ (! H5 I5. equal(H5::'a,I5) --> equal(successor(H5),successor(I5))) & \
3.1773 +\ (! J5 K5. equal(J5::'a,K5) --> equal(sum_class(J5),sum_class(K5))) & \
3.1774 +\ (! L5 M5 N5. equal(L5::'a,M5) --> equal(union(L5::'a,N5),union(M5::'a,N5))) & \
3.1775 +\ (! O5 Q5 P5. equal(O5::'a,P5) --> equal(union(Q5::'a,O5),union(Q5::'a,P5))) & \
3.1776 +\ (! R5 S5 T5. equal(R5::'a,S5) --> equal(unordered_pair(R5::'a,T5),unordered_pair(S5::'a,T5))) & \
3.1777 +\ (! U5 W5 V5. equal(U5::'a,V5) --> equal(unordered_pair(W5::'a,U5),unordered_pair(W5::'a,V5))) & \
3.1778 +\ (! X5 Y5 Z5 A6. equal(X5::'a,Y5) & compatible(X5::'a,Z5,A6) --> compatible(Y5::'a,Z5,A6)) & \
3.1779 +\ (! B6 D6 C6 E6. equal(B6::'a,C6) & compatible(D6::'a,B6,E6) --> compatible(D6::'a,C6,E6)) & \
3.1780 +\ (! F6 H6 I6 G6. equal(F6::'a,G6) & compatible(H6::'a,I6,F6) --> compatible(H6::'a,I6,G6)) & \
3.1781 +\ (! J6 K6. equal(J6::'a,K6) & function(J6) --> function(K6)) & \
3.1782 +\ (! L6 M6 N6 O6. equal(L6::'a,M6) & homomorphism(L6::'a,N6,O6) --> homomorphism(M6::'a,N6,O6)) & \
3.1783 +\ (! P6 R6 Q6 S6. equal(P6::'a,Q6) & homomorphism(R6::'a,P6,S6) --> homomorphism(R6::'a,Q6,S6)) & \
3.1784 +\ (! T6 V6 W6 U6. equal(T6::'a,U6) & homomorphism(V6::'a,W6,T6) --> homomorphism(V6::'a,W6,U6)) & \
3.1785 +\ (! X6 Y6. equal(X6::'a,Y6) & inductive(X6) --> inductive(Y6)) & \
3.1786 +\ (! Z6 A7 B7. equal(Z6::'a,A7) & member(Z6::'a,B7) --> member(A7::'a,B7)) & \
3.1787 +\ (! C7 E7 D7. equal(C7::'a,D7) & member(E7::'a,C7) --> member(E7::'a,D7)) & \
3.1788 +\ (! F7 G7. equal(F7::'a,G7) & one_to_one(F7) --> one_to_one(G7)) & \
3.1789 +\ (! H7 I7. equal(H7::'a,I7) & operation(H7) --> operation(I7)) & \
3.1790 +\ (! J7 K7. equal(J7::'a,K7) & single_valued_class(J7) --> single_valued_class(K7)) & \
3.1791 +\ (! L7 M7 N7. equal(L7::'a,M7) & subclass(L7::'a,N7) --> subclass(M7::'a,N7)) & \
3.1792 +\ (! O7 Q7 P7. equal(O7::'a,P7) & subclass(Q7::'a,O7) --> subclass(Q7::'a,P7)) & \
3.1793 +\ (! X. subclass(compose_class(X),cross_product(universal_class::'a,universal_class))) & \
3.1794 +\ (! X Y Z. member(ordered_pair(Y::'a,Z),compose_class(X)) --> equal(compos(X::'a,Y),Z)) & \
3.1795 +\ (! 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))) & \
3.1796 +\ (subclass(composition_function::'a,cross_product(universal_class::'a,cross_product(universal_class::'a,universal_class)))) & \
3.1797 +\ (! X Y Z. member(ordered_pair(X::'a,ordered_pair(Y::'a,Z)),composition_function) --> equal(compos(X::'a,Y),Z)) & \
3.1798 +\ (! 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)) & \
3.1799 +\ (subclass(domain_relation::'a,cross_product(universal_class::'a,universal_class))) & \
3.1800 +\ (! X Y. member(ordered_pair(X::'a,Y),domain_relation) --> equal(domain_of(X),Y)) & \
3.1801 +\ (! X. member(X::'a,universal_class) --> member(ordered_pair(X::'a,domain_of(X)),domain_relation)) & \
3.1802 +\ (! X. equal(first(not_subclass_element(compos(X::'a,inverse(X)),identity_relation)),single_valued1(X))) & \
3.1803 +\ (! X. equal(second(not_subclass_element(compos(X::'a,inverse(X)),identity_relation)),single_valued2(X))) & \
3.1804 +\ (! X. equal(domain(X::'a,image_(inverse(X),singleton(single_valued1(X))),single_valued2(X)),single_valued3(X))) & \
3.1805 +\ (equal(intersection(complement(compos(element_relation::'a,complement(identity_relation))),element_relation),singleton_relation)) & \
3.1806 +\ (subclass(application_function::'a,cross_product(universal_class::'a,cross_product(universal_class::'a,universal_class)))) & \
3.1807 +\ (! Z Y X. member(ordered_pair(X::'a,ordered_pair(Y::'a,Z)),application_function) --> member(Y::'a,domain_of(X))) & \
3.1808 +\ (! X Y Z. member(ordered_pair(X::'a,ordered_pair(Y::'a,Z)),application_function) --> equal(apply(X::'a,Y),Z)) & \
3.1809 +\ (! 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)) & \
3.1810 +\ (! X Y Xf. maps(Xf::'a,X,Y) --> function(Xf)) & \
3.1811 +\ (! Y Xf X. maps(Xf::'a,X,Y) --> equal(domain_of(Xf),X)) & \
3.1812 +\ (! X Xf Y. maps(Xf::'a,X,Y) --> subclass(range_of(Xf),Y)) & \
3.1813 +\ (! Xf Y. function(Xf) & subclass(range_of(Xf),Y) --> maps(Xf::'a,domain_of(Xf),Y)) & \
3.1814 +\ (! L M. equal(L::'a,M) --> equal(compose_class(L),compose_class(M))) & \
3.1815 +\ (! N2 O2. equal(N2::'a,O2) --> equal(single_valued1(N2),single_valued1(O2))) & \
3.1816 +\ (! P2 Q2. equal(P2::'a,Q2) --> equal(single_valued2(P2),single_valued2(Q2))) & \
3.1817 +\ (! R2 S2. equal(R2::'a,S2) --> equal(single_valued3(R2),single_valued3(S2))) & \
3.1818 +\ (! X2 Y2 Z2 A3. equal(X2::'a,Y2) & maps(X2::'a,Z2,A3) --> maps(Y2::'a,Z2,A3)) & \
3.1819 +\ (! B3 D3 C3 E3. equal(B3::'a,C3) & maps(D3::'a,B3,E3) --> maps(D3::'a,C3,E3)) & \
3.1820 +\ (! F3 H3 I3 G3. equal(F3::'a,G3) & maps(H3::'a,I3,F3) --> maps(H3::'a,I3,G3)) & \
3.1821 +\ (! X. equal(union(X::'a,inverse(X)),symmetrization_of(X))) & \
3.1822 +\ (! X Y. irreflexive(X::'a,Y) --> subclass(restrct(X::'a,Y,Y),complement(identity_relation))) & \
3.1823 +\ (! X Y. subclass(restrct(X::'a,Y,Y),complement(identity_relation)) --> irreflexive(X::'a,Y)) & \
3.1824 +\ (! Y X. connected(X::'a,Y) --> subclass(cross_product(Y::'a,Y),union(identity_relation::'a,symmetrization_of(X)))) & \
3.1825 +\ (! X Y. subclass(cross_product(Y::'a,Y),union(identity_relation::'a,symmetrization_of(X))) --> connected(X::'a,Y)) & \
3.1826 +\ (! Xr Y. transitive(Xr::'a,Y) --> subclass(compos(restrct(Xr::'a,Y,Y),restrct(Xr::'a,Y,Y)),restrct(Xr::'a,Y,Y))) & \
3.1827 +\ (! Xr Y. subclass(compos(restrct(Xr::'a,Y,Y),restrct(Xr::'a,Y,Y)),restrct(Xr::'a,Y,Y)) --> transitive(Xr::'a,Y)) & \
3.1828 +\ (! Xr Y. asymmetric(Xr::'a,Y) --> equal(restrct(intersection(Xr::'a,inverse(Xr)),Y,Y),null_class)) & \
3.1829 +\ (! Xr Y. equal(restrct(intersection(Xr::'a,inverse(Xr)),Y,Y),null_class) --> asymmetric(Xr::'a,Y)) & \
3.1830 +\ (! Xr Y Z. equal(segment(Xr::'a,Y,Z),domain_of(restrct(Xr::'a,Y,singleton(Z))))) & \
3.1831 +\ (! X Y. well_ordering(X::'a,Y) --> connected(X::'a,Y)) & \
3.1832 +\ (! Y Xr U. well_ordering(Xr::'a,Y) & subclass(U::'a,Y) --> equal(U::'a,null_class) | member(least(Xr::'a,U),U)) & \
3.1833 +\ (! Y V Xr U. well_ordering(Xr::'a,Y) & subclass(U::'a,Y) & member(V::'a,U) --> member(least(Xr::'a,U),U)) & \
3.1834 +\ (! Y Xr U. well_ordering(Xr::'a,Y) & subclass(U::'a,Y) --> equal(segment(Xr::'a,U,least(Xr::'a,U)),null_class)) & \
3.1835 +\ (! 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))) & \
3.1836 +\ (! Xr Y. connected(Xr::'a,Y) & equal(not_well_ordering(Xr::'a,Y),null_class) --> well_ordering(Xr::'a,Y)) & \
3.1837 +\ (! Xr Y. connected(Xr::'a,Y) --> subclass(not_well_ordering(Xr::'a,Y),Y) | well_ordering(Xr::'a,Y)) & \
3.1838 +\ (! 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)) & \
3.1839 +\ (! Xr Y Z. section(Xr::'a,Y,Z) --> subclass(Y::'a,Z)) & \
3.1840 +\ (! Xr Z Y. section(Xr::'a,Y,Z) --> subclass(domain_of(restrct(Xr::'a,Z,Y)),Y)) & \
3.1841 +\ (! Xr Y Z. subclass(Y::'a,Z) & subclass(domain_of(restrct(Xr::'a,Z,Y)),Y) --> section(Xr::'a,Y,Z)) & \
3.1842 +\ (! X. member(X::'a,ordinal_numbers) --> well_ordering(element_relation::'a,X)) & \
3.1843 +\ (! X. member(X::'a,ordinal_numbers) --> subclass(sum_class(X),X)) & \
3.1844 +\ (! X. well_ordering(element_relation::'a,X) & subclass(sum_class(X),X) & member(X::'a,universal_class) --> member(X::'a,ordinal_numbers)) & \
3.1845 +\ (! X. well_ordering(element_relation::'a,X) & subclass(sum_class(X),X) --> member(X::'a,ordinal_numbers) | equal(X::'a,ordinal_numbers)) & \
3.1846 +\ (equal(union(singleton(null_class),image_(successor_relation::'a,ordinal_numbers)),kind_1_ordinals)) & \
3.1847 +\ (equal(intersection(complement(kind_1_ordinals),ordinal_numbers),limit_ordinals)) & \
3.1848 +\ (! X. subclass(rest_of(X),cross_product(universal_class::'a,universal_class))) & \
3.1849 +\ (! V U X. member(ordered_pair(U::'a,V),rest_of(X)) --> member(U::'a,domain_of(X))) & \
3.1850 +\ (! X U V. member(ordered_pair(U::'a,V),rest_of(X)) --> equal(restrct(X::'a,U,universal_class),V)) & \
3.1851 +\ (! 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))) & \
3.1852 +\ (subclass(rest_relation::'a,cross_product(universal_class::'a,universal_class))) & \
3.1853 +\ (! X Y. member(ordered_pair(X::'a,Y),rest_relation) --> equal(rest_of(X),Y)) & \
3.1854 +\ (! X. member(X::'a,universal_class) --> member(ordered_pair(X::'a,rest_of(X)),rest_relation)) & \
3.1855 +\ (! X Z. member(X::'a,recursion_equation_functions(Z)) --> function(Z)) & \
3.1856 +\ (! Z X. member(X::'a,recursion_equation_functions(Z)) --> function(X)) & \
3.1857 +\ (! Z X. member(X::'a,recursion_equation_functions(Z)) --> member(domain_of(X),ordinal_numbers)) & \
3.1858 +\ (! Z X. member(X::'a,recursion_equation_functions(Z)) --> equal(compos(Z::'a,rest_of(X)),X)) & \
3.1859 +\ (! 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))) & \
3.1860 +\ (subclass(union_of_range_map::'a,cross_product(universal_class::'a,universal_class))) & \
3.1861 +\ (! X Y. member(ordered_pair(X::'a,Y),union_of_range_map) --> equal(sum_class(range_of(X)),Y)) & \
3.1862 +\ (! 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)) & \
3.1863 +\ (! X Y. equal(apply(recursion(X::'a,successor_relation,union_of_range_map),Y),ordinal_add(X::'a,Y))) & \
3.1864 +\ (! X Y. equal(recursion(null_class::'a,apply(add_relation::'a,X),union_of_range_map),ordinal_multiply(X::'a,Y))) & \
3.1865 +\ (! X. member(X::'a,omega) --> equal(integer_of(X),X)) & \
3.1866 +\ (! X. member(X::'a,omega) | equal(integer_of(X),null_class)) & \
3.1867 +\ (! D E. equal(D::'a,E) --> equal(integer_of(D),integer_of(E))) & \
3.1868 +\ (! F' G H. equal(F'::'a,G) --> equal(least(F'::'a,H),least(G::'a,H))) & \
3.1869 +\ (! I' K' J. equal(I'::'a,J) --> equal(least(K'::'a,I'),least(K'::'a,J))) & \
3.1870 +\ (! L M N. equal(L::'a,M) --> equal(not_well_ordering(L::'a,N),not_well_ordering(M::'a,N))) & \
3.1871 +\ (! O_ Q P. equal(O_::'a,P) --> equal(not_well_ordering(Q::'a,O_),not_well_ordering(Q::'a,P))) & \
3.1872 +\ (! R S' T'. equal(R::'a,S') --> equal(ordinal_add(R::'a,T'),ordinal_add(S'::'a,T'))) & \
3.1873 +\ (! U W V. equal(U::'a,V) --> equal(ordinal_add(W::'a,U),ordinal_add(W::'a,V))) & \
3.1874 +\ (! X Y Z. equal(X::'a,Y) --> equal(ordinal_multiply(X::'a,Z),ordinal_multiply(Y::'a,Z))) & \
3.1875 +\ (! A1 C1 B1. equal(A1::'a,B1) --> equal(ordinal_multiply(C1::'a,A1),ordinal_multiply(C1::'a,B1))) & \
3.1876 +\ (! F1 G1 H1 I1. equal(F1::'a,G1) --> equal(recursion(F1::'a,H1,I1),recursion(G1::'a,H1,I1))) & \
3.1877 +\ (! J1 L1 K1 M1. equal(J1::'a,K1) --> equal(recursion(L1::'a,J1,M1),recursion(L1::'a,K1,M1))) & \
3.1878 +\ (! N1 P1 Q1 O1. equal(N1::'a,O1) --> equal(recursion(P1::'a,Q1,N1),recursion(P1::'a,Q1,O1))) & \
3.1879 +\ (! R1 S1. equal(R1::'a,S1) --> equal(recursion_equation_functions(R1),recursion_equation_functions(S1))) & \
3.1880 +\ (! T1 U1. equal(T1::'a,U1) --> equal(rest_of(T1),rest_of(U1))) & \
3.1881 +\ (! V1 W1 X1 Y1. equal(V1::'a,W1) --> equal(segment(V1::'a,X1,Y1),segment(W1::'a,X1,Y1))) & \
3.1882 +\ (! Z1 B2 A2 C2. equal(Z1::'a,A2) --> equal(segment(B2::'a,Z1,C2),segment(B2::'a,A2,C2))) & \
3.1883 +\ (! D2 F2 G2 E2. equal(D2::'a,E2) --> equal(segment(F2::'a,G2,D2),segment(F2::'a,G2,E2))) & \
3.1884 +\ (! H2 I2. equal(H2::'a,I2) --> equal(symmetrization_of(H2),symmetrization_of(I2))) & \
3.1885 +\ (! J2 K2 L2. equal(J2::'a,K2) & asymmetric(J2::'a,L2) --> asymmetric(K2::'a,L2)) & \
3.1886 +\ (! M2 O2 N2. equal(M2::'a,N2) & asymmetric(O2::'a,M2) --> asymmetric(O2::'a,N2)) & \
3.1887 +\ (! P2 Q2 R2. equal(P2::'a,Q2) & connected(P2::'a,R2) --> connected(Q2::'a,R2)) & \
3.1888 +\ (! S2 U2 T2. equal(S2::'a,T2) & connected(U2::'a,S2) --> connected(U2::'a,T2)) & \
3.1889 +\ (! V2 W2 X2. equal(V2::'a,W2) & irreflexive(V2::'a,X2) --> irreflexive(W2::'a,X2)) & \
3.1890 +\ (! Y2 A3 Z2. equal(Y2::'a,Z2) & irreflexive(A3::'a,Y2) --> irreflexive(A3::'a,Z2)) & \
3.1891 +\ (! B3 C3 D3 E3. equal(B3::'a,C3) & section(B3::'a,D3,E3) --> section(C3::'a,D3,E3)) & \
3.1892 +\ (! F3 H3 G3 I3. equal(F3::'a,G3) & section(H3::'a,F3,I3) --> section(H3::'a,G3,I3)) & \
3.1893 +\ (! J3 L3 M3 K3. equal(J3::'a,K3) & section(L3::'a,M3,J3) --> section(L3::'a,M3,K3)) & \
3.1894 +\ (! N3 O3 P3. equal(N3::'a,O3) & transitive(N3::'a,P3) --> transitive(O3::'a,P3)) & \
3.1895 +\ (! Q3 S3 R3. equal(Q3::'a,R3) & transitive(S3::'a,Q3) --> transitive(S3::'a,R3)) & \
3.1896 +\ (! T3 U3 V3. equal(T3::'a,U3) & well_ordering(T3::'a,V3) --> well_ordering(U3::'a,V3)) & \
3.1897 +\ (! W3 Y3 X3. equal(W3::'a,X3) & well_ordering(Y3::'a,W3) --> well_ordering(Y3::'a,X3)) & \
3.1898 +\ (~subclass(limit_ordinals::'a,ordinal_numbers)) --> False",
3.1899 + meson_tac);
3.1900 +
3.1901 +
3.1902 +(*0 inferences so far. Searching to depth 0. 16.8 secs. BIG*)
3.1903 +val NUM228_1 = prove_hard
3.1904 + ("(! X. equal(X::'a,X)) & \
3.1905 +\ (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
3.1906 +\ (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) & \
3.1907 +\ (! X U Y. subclass(X::'a,Y) & member(U::'a,X) --> member(U::'a,Y)) & \
3.1908 +\ (! X Y. member(not_subclass_element(X::'a,Y),X) | subclass(X::'a,Y)) & \
3.1909 +\ (! X Y. member(not_subclass_element(X::'a,Y),Y) --> subclass(X::'a,Y)) & \
3.1910 +\ (! X. subclass(X::'a,universal_class)) & \
3.1911 +\ (! X Y. equal(X::'a,Y) --> subclass(X::'a,Y)) & \
3.1912 +\ (! Y X. equal(X::'a,Y) --> subclass(Y::'a,X)) & \
3.1913 +\ (! X Y. subclass(X::'a,Y) & subclass(Y::'a,X) --> equal(X::'a,Y)) & \
3.1914 +\ (! X U Y. member(U::'a,unordered_pair(X::'a,Y)) --> equal(U::'a,X) | equal(U::'a,Y)) & \
3.1915 +\ (! X Y. member(X::'a,universal_class) --> member(X::'a,unordered_pair(X::'a,Y))) & \
3.1916 +\ (! X Y. member(Y::'a,universal_class) --> member(Y::'a,unordered_pair(X::'a,Y))) & \
3.1917 +\ (! X Y. member(unordered_pair(X::'a,Y),universal_class)) & \
3.1918 +\ (! X. equal(unordered_pair(X::'a,X),singleton(X))) & \
3.1919 +\ (! X Y. equal(unordered_pair(singleton(X),unordered_pair(X::'a,singleton(Y))),ordered_pair(X::'a,Y))) & \
3.1920 +\ (! V Y U X. member(ordered_pair(U::'a,V),cross_product(X::'a,Y)) --> member(U::'a,X)) & \
3.1921 +\ (! U X V Y. member(ordered_pair(U::'a,V),cross_product(X::'a,Y)) --> member(V::'a,Y)) & \
3.1922 +\ (! U V X Y. member(U::'a,X) & member(V::'a,Y) --> member(ordered_pair(U::'a,V),cross_product(X::'a,Y))) & \
3.1923 +\ (! X Y Z. member(Z::'a,cross_product(X::'a,Y)) --> equal(ordered_pair(first(Z),second(Z)),Z)) & \
3.1924 +\ (subclass(element_relation::'a,cross_product(universal_class::'a,universal_class))) & \
3.1925 +\ (! X Y. member(ordered_pair(X::'a,Y),element_relation) --> member(X::'a,Y)) & \
3.1926 +\ (! 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)) & \
3.1927 +\ (! Y Z X. member(Z::'a,intersection(X::'a,Y)) --> member(Z::'a,X)) & \
3.1928 +\ (! X Z Y. member(Z::'a,intersection(X::'a,Y)) --> member(Z::'a,Y)) & \
3.1929 +\ (! Z X Y. member(Z::'a,X) & member(Z::'a,Y) --> member(Z::'a,intersection(X::'a,Y))) & \
3.1930 +\ (! Z X. ~(member(Z::'a,complement(X)) & member(Z::'a,X))) & \
3.1931 +\ (! Z X. member(Z::'a,universal_class) --> member(Z::'a,complement(X)) | member(Z::'a,X)) & \
3.1932 +\ (! X Y. equal(complement(intersection(complement(X),complement(Y))),union(X::'a,Y))) & \
3.1933 +\ (! X Y. equal(intersection(complement(intersection(X::'a,Y)),complement(intersection(complement(X),complement(Y)))),difference(X::'a,Y))) & \
3.1934 +\ (! Xr X Y. equal(intersection(Xr::'a,cross_product(X::'a,Y)),restrct(Xr::'a,X,Y))) & \
3.1935 +\ (! Xr X Y. equal(intersection(cross_product(X::'a,Y),Xr),restrct(Xr::'a,X,Y))) & \
3.1936 +\ (! Z X. ~(equal(restrct(X::'a,singleton(Z),universal_class),null_class) & member(Z::'a,domain_of(X)))) & \
3.1937 +\ (! Z X. member(Z::'a,universal_class) --> equal(restrct(X::'a,singleton(Z),universal_class),null_class) | member(Z::'a,domain_of(X))) & \
3.1938 +\ (! X. subclass(rotate(X),cross_product(cross_product(universal_class::'a,universal_class),universal_class))) & \
3.1939 +\ (! V W U X. member(ordered_pair(ordered_pair(U::'a,V),W),rotate(X)) --> member(ordered_pair(ordered_pair(V::'a,W),U),X)) & \
3.1940 +\ (! 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),rotate(X))) & \
3.1941 +\ (! X. subclass(flip(X),cross_product(cross_product(universal_class::'a,universal_class),universal_class))) & \
3.1942 +\ (! 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)) & \
3.1943 +\ (! 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))) & \
3.1944 +\ (! Y. equal(domain_of(flip(cross_product(Y::'a,universal_class))),inverse(Y))) & \
3.1945 +\ (! Z. equal(domain_of(inverse(Z)),range_of(Z))) & \
3.1946 +\ (! Z X Y. equal(first(not_subclass_element(restrct(Z::'a,X,singleton(Y)),null_class)),domain(Z::'a,X,Y))) & \
3.1947 +\ (! Z X Y. equal(second(not_subclass_element(restrct(Z::'a,singleton(X),Y),null_class)),rng(Z::'a,X,Y))) & \
3.1948 +\ (! Xr X. equal(range_of(restrct(Xr::'a,X,universal_class)),image_(Xr::'a,X))) & \
3.1949 +\ (! X. equal(union(X::'a,singleton(X)),successor(X))) & \
3.1950 +\ (subclass(successor_relation::'a,cross_product(universal_class::'a,universal_class))) & \
3.1951 +\ (! X Y. member(ordered_pair(X::'a,Y),successor_relation) --> equal(successor(X),Y)) & \
3.1952 +\ (! 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)) & \
3.1953 +\ (! X. inductive(X) --> member(null_class::'a,X)) & \
3.1954 +\ (! X. inductive(X) --> subclass(image_(successor_relation::'a,X),X)) & \
3.1955 +\ (! X. member(null_class::'a,X) & subclass(image_(successor_relation::'a,X),X) --> inductive(X)) & \
3.1956 +\ (inductive(omega)) & \
3.1957 +\ (! Y. inductive(Y) --> subclass(omega::'a,Y)) & \
3.1958 +\ (member(omega::'a,universal_class)) & \
3.1959 +\ (! X. equal(domain_of(restrct(element_relation::'a,universal_class,X)),sum_class(X))) & \
3.1960 +\ (! X. member(X::'a,universal_class) --> member(sum_class(X),universal_class)) & \
3.1961 +\ (! X. equal(complement(image_(element_relation::'a,complement(X))),powerClass(X))) & \
3.1962 +\ (! U. member(U::'a,universal_class) --> member(powerClass(U),universal_class)) & \
3.1963 +\ (! Yr Xr. subclass(compos(Yr::'a,Xr),cross_product(universal_class::'a,universal_class))) & \
3.1964 +\ (! 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))))) & \
3.1965 +\ (! 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))) & \
3.1966 +\ (! X. single_valued_class(X) --> subclass(compos(X::'a,inverse(X)),identity_relation)) & \
3.1967 +\ (! X. subclass(compos(X::'a,inverse(X)),identity_relation) --> single_valued_class(X)) & \
3.1968 +\ (! Xf. function(Xf) --> subclass(Xf::'a,cross_product(universal_class::'a,universal_class))) & \
3.1969 +\ (! Xf. function(Xf) --> subclass(compos(Xf::'a,inverse(Xf)),identity_relation)) & \
3.1970 +\ (! Xf. subclass(Xf::'a,cross_product(universal_class::'a,universal_class)) & subclass(compos(Xf::'a,inverse(Xf)),identity_relation) --> function(Xf)) & \
3.1971 +\ (! Xf X. function(Xf) & member(X::'a,universal_class) --> member(image_(Xf::'a,X),universal_class)) & \
3.1972 +\ (! X. equal(X::'a,null_class) | member(regular(X),X)) & \
3.1973 +\ (! X. equal(X::'a,null_class) | equal(intersection(X::'a,regular(X)),null_class)) & \
3.1974 +\ (! Xf Y. equal(sum_class(image_(Xf::'a,singleton(Y))),apply(Xf::'a,Y))) & \
3.1975 +\ (function(choice)) & \
3.1976 +\ (! Y. member(Y::'a,universal_class) --> equal(Y::'a,null_class) | member(apply(choice::'a,Y),Y)) & \
3.1977 +\ (! Xf. one_to_one(Xf) --> function(Xf)) & \
3.1978 +\ (! Xf. one_to_one(Xf) --> function(inverse(Xf))) & \
3.1979 +\ (! Xf. function(inverse(Xf)) & function(Xf) --> one_to_one(Xf)) & \
3.1980 +\ (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)) & \
3.1981 +\ (equal(intersection(inverse(subset_relation),subset_relation),identity_relation)) & \
3.1982 +\ (! Xr. equal(complement(domain_of(intersection(Xr::'a,identity_relation))),diagonalise(Xr))) & \
3.1983 +\ (! X. equal(intersection(domain_of(X),diagonalise(compos(inverse(element_relation),X))),cantor(X))) & \
3.1984 +\ (! Xf. operation(Xf) --> function(Xf)) & \
3.1985 +\ (! Xf. operation(Xf) --> equal(cross_product(domain_of(domain_of(Xf)),domain_of(domain_of(Xf))),domain_of(Xf))) & \
3.1986 +\ (! Xf. operation(Xf) --> subclass(range_of(Xf),domain_of(domain_of(Xf)))) & \
3.1987 +\ (! 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)) & \
3.1988 +\ (! Xf1 Xf2 Xh. compatible(Xh::'a,Xf1,Xf2) --> function(Xh)) & \
3.1989 +\ (! Xf2 Xf1 Xh. compatible(Xh::'a,Xf1,Xf2) --> equal(domain_of(domain_of(Xf1)),domain_of(Xh))) & \
3.1990 +\ (! Xf1 Xh Xf2. compatible(Xh::'a,Xf1,Xf2) --> subclass(range_of(Xh),domain_of(domain_of(Xf2)))) & \
3.1991 +\ (! 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)) & \
3.1992 +\ (! Xh Xf2 Xf1. homomorphism(Xh::'a,Xf1,Xf2) --> operation(Xf1)) & \
3.1993 +\ (! Xh Xf1 Xf2. homomorphism(Xh::'a,Xf1,Xf2) --> operation(Xf2)) & \
3.1994 +\ (! Xh Xf1 Xf2. homomorphism(Xh::'a,Xf1,Xf2) --> compatible(Xh::'a,Xf1,Xf2)) & \
3.1995 +\ (! 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))))) & \
3.1996 +\ (! 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)) & \
3.1997 +\ (! 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)) & \
3.1998 +\ (! D E F'. equal(D::'a,E) --> equal(apply(D::'a,F'),apply(E::'a,F'))) & \
3.1999 +\ (! G I' H. equal(G::'a,H) --> equal(apply(I'::'a,G),apply(I'::'a,H))) & \
3.2000 +\ (! J K'. equal(J::'a,K') --> equal(cantor(J),cantor(K'))) & \
3.2001 +\ (! L M. equal(L::'a,M) --> equal(complement(L),complement(M))) & \
3.2002 +\ (! N O_ P. equal(N::'a,O_) --> equal(compos(N::'a,P),compos(O_::'a,P))) & \
3.2003 +\ (! Q S' R. equal(Q::'a,R) --> equal(compos(S'::'a,Q),compos(S'::'a,R))) & \
3.2004 +\ (! T' U V. equal(T'::'a,U) --> equal(cross_product(T'::'a,V),cross_product(U::'a,V))) & \
3.2005 +\ (! W Y X. equal(W::'a,X) --> equal(cross_product(Y::'a,W),cross_product(Y::'a,X))) & \
3.2006 +\ (! Z A1. equal(Z::'a,A1) --> equal(diagonalise(Z),diagonalise(A1))) & \
3.2007 +\ (! B1 C1 D1. equal(B1::'a,C1) --> equal(difference(B1::'a,D1),difference(C1::'a,D1))) & \
3.2008 +\ (! E1 G1 F1. equal(E1::'a,F1) --> equal(difference(G1::'a,E1),difference(G1::'a,F1))) & \
3.2009 +\ (! H1 I1 J1 K1. equal(H1::'a,I1) --> equal(domain(H1::'a,J1,K1),domain(I1::'a,J1,K1))) & \
3.2010 +\ (! L1 N1 M1 O1. equal(L1::'a,M1) --> equal(domain(N1::'a,L1,O1),domain(N1::'a,M1,O1))) & \
3.2011 +\ (! P1 R1 S1 Q1. equal(P1::'a,Q1) --> equal(domain(R1::'a,S1,P1),domain(R1::'a,S1,Q1))) & \
3.2012 +\ (! T1 U1. equal(T1::'a,U1) --> equal(domain_of(T1),domain_of(U1))) & \
3.2013 +\ (! V1 W1. equal(V1::'a,W1) --> equal(first(V1),first(W1))) & \
3.2014 +\ (! X1 Y1. equal(X1::'a,Y1) --> equal(flip(X1),flip(Y1))) & \
3.2015 +\ (! Z1 A2 B2. equal(Z1::'a,A2) --> equal(image_(Z1::'a,B2),image_(A2::'a,B2))) & \
3.2016 +\ (! C2 E2 D2. equal(C2::'a,D2) --> equal(image_(E2::'a,C2),image_(E2::'a,D2))) & \
3.2017 +\ (! F2 G2 H2. equal(F2::'a,G2) --> equal(intersection(F2::'a,H2),intersection(G2::'a,H2))) & \
3.2018 +\ (! I2 K2 J2. equal(I2::'a,J2) --> equal(intersection(K2::'a,I2),intersection(K2::'a,J2))) & \
3.2019 +\ (! L2 M2. equal(L2::'a,M2) --> equal(inverse(L2),inverse(M2))) & \
3.2020 +\ (! N2 O2 P2 Q2. equal(N2::'a,O2) --> equal(not_homomorphism1(N2::'a,P2,Q2),not_homomorphism1(O2::'a,P2,Q2))) & \
3.2021 +\ (! R2 T2 S2 U2. equal(R2::'a,S2) --> equal(not_homomorphism1(T2::'a,R2,U2),not_homomorphism1(T2::'a,S2,U2))) & \
3.2022 +\ (! V2 X2 Y2 W2. equal(V2::'a,W2) --> equal(not_homomorphism1(X2::'a,Y2,V2),not_homomorphism1(X2::'a,Y2,W2))) & \
3.2023 +\ (! Z2 A3 B3 C3. equal(Z2::'a,A3) --> equal(not_homomorphism2(Z2::'a,B3,C3),not_homomorphism2(A3::'a,B3,C3))) & \
3.2024 +\ (! D3 F3 E3 G3. equal(D3::'a,E3) --> equal(not_homomorphism2(F3::'a,D3,G3),not_homomorphism2(F3::'a,E3,G3))) & \
3.2025 +\ (! H3 J3 K3 I3. equal(H3::'a,I3) --> equal(not_homomorphism2(J3::'a,K3,H3),not_homomorphism2(J3::'a,K3,I3))) & \
3.2026 +\ (! L3 M3 N3. equal(L3::'a,M3) --> equal(not_subclass_element(L3::'a,N3),not_subclass_element(M3::'a,N3))) & \
3.2027 +\ (! O3 Q3 P3. equal(O3::'a,P3) --> equal(not_subclass_element(Q3::'a,O3),not_subclass_element(Q3::'a,P3))) & \
3.2028 +\ (! R3 S3 T3. equal(R3::'a,S3) --> equal(ordered_pair(R3::'a,T3),ordered_pair(S3::'a,T3))) & \
3.2029 +\ (! U3 W3 V3. equal(U3::'a,V3) --> equal(ordered_pair(W3::'a,U3),ordered_pair(W3::'a,V3))) & \
3.2030 +\ (! X3 Y3. equal(X3::'a,Y3) --> equal(powerClass(X3),powerClass(Y3))) & \
3.2031 +\ (! Z3 A4 B4 C4. equal(Z3::'a,A4) --> equal(rng(Z3::'a,B4,C4),rng(A4::'a,B4,C4))) & \
3.2032 +\ (! D4 F4 E4 G4. equal(D4::'a,E4) --> equal(rng(F4::'a,D4,G4),rng(F4::'a,E4,G4))) & \
3.2033 +\ (! H4 J4 K4 I4. equal(H4::'a,I4) --> equal(rng(J4::'a,K4,H4),rng(J4::'a,K4,I4))) & \
3.2034 +\ (! L4 M4. equal(L4::'a,M4) --> equal(range_of(L4),range_of(M4))) & \
3.2035 +\ (! N4 O4. equal(N4::'a,O4) --> equal(regular(N4),regular(O4))) & \
3.2036 +\ (! P4 Q4 R4 S4. equal(P4::'a,Q4) --> equal(restrct(P4::'a,R4,S4),restrct(Q4::'a,R4,S4))) & \
3.2037 +\ (! T4 V4 U4 W4. equal(T4::'a,U4) --> equal(restrct(V4::'a,T4,W4),restrct(V4::'a,U4,W4))) & \
3.2038 +\ (! X4 Z4 A5 Y4. equal(X4::'a,Y4) --> equal(restrct(Z4::'a,A5,X4),restrct(Z4::'a,A5,Y4))) & \
3.2039 +\ (! B5 C5. equal(B5::'a,C5) --> equal(rotate(B5),rotate(C5))) & \
3.2040 +\ (! D5 E5. equal(D5::'a,E5) --> equal(second(D5),second(E5))) & \
3.2041 +\ (! F5 G5. equal(F5::'a,G5) --> equal(singleton(F5),singleton(G5))) & \
3.2042 +\ (! H5 I5. equal(H5::'a,I5) --> equal(successor(H5),successor(I5))) & \
3.2043 +\ (! J5 K5. equal(J5::'a,K5) --> equal(sum_class(J5),sum_class(K5))) & \
3.2044 +\ (! L5 M5 N5. equal(L5::'a,M5) --> equal(union(L5::'a,N5),union(M5::'a,N5))) & \
3.2045 +\ (! O5 Q5 P5. equal(O5::'a,P5) --> equal(union(Q5::'a,O5),union(Q5::'a,P5))) & \
3.2046 +\ (! R5 S5 T5. equal(R5::'a,S5) --> equal(unordered_pair(R5::'a,T5),unordered_pair(S5::'a,T5))) & \
3.2047 +\ (! U5 W5 V5. equal(U5::'a,V5) --> equal(unordered_pair(W5::'a,U5),unordered_pair(W5::'a,V5))) & \
3.2048 +\ (! X5 Y5 Z5 A6. equal(X5::'a,Y5) & compatible(X5::'a,Z5,A6) --> compatible(Y5::'a,Z5,A6)) & \
3.2049 +\ (! B6 D6 C6 E6. equal(B6::'a,C6) & compatible(D6::'a,B6,E6) --> compatible(D6::'a,C6,E6)) & \
3.2050 +\ (! F6 H6 I6 G6. equal(F6::'a,G6) & compatible(H6::'a,I6,F6) --> compatible(H6::'a,I6,G6)) & \
3.2051 +\ (! J6 K6. equal(J6::'a,K6) & function(J6) --> function(K6)) & \
3.2052 +\ (! L6 M6 N6 O6. equal(L6::'a,M6) & homomorphism(L6::'a,N6,O6) --> homomorphism(M6::'a,N6,O6)) & \
3.2053 +\ (! P6 R6 Q6 S6. equal(P6::'a,Q6) & homomorphism(R6::'a,P6,S6) --> homomorphism(R6::'a,Q6,S6)) & \
3.2054 +\ (! T6 V6 W6 U6. equal(T6::'a,U6) & homomorphism(V6::'a,W6,T6) --> homomorphism(V6::'a,W6,U6)) & \
3.2055 +\ (! X6 Y6. equal(X6::'a,Y6) & inductive(X6) --> inductive(Y6)) & \
3.2056 +\ (! Z6 A7 B7. equal(Z6::'a,A7) & member(Z6::'a,B7) --> member(A7::'a,B7)) & \
3.2057 +\ (! C7 E7 D7. equal(C7::'a,D7) & member(E7::'a,C7) --> member(E7::'a,D7)) & \
3.2058 +\ (! F7 G7. equal(F7::'a,G7) & one_to_one(F7) --> one_to_one(G7)) & \
3.2059 +\ (! H7 I7. equal(H7::'a,I7) & operation(H7) --> operation(I7)) & \
3.2060 +\ (! J7 K7. equal(J7::'a,K7) & single_valued_class(J7) --> single_valued_class(K7)) & \
3.2061 +\ (! L7 M7 N7. equal(L7::'a,M7) & subclass(L7::'a,N7) --> subclass(M7::'a,N7)) & \
3.2062 +\ (! O7 Q7 P7. equal(O7::'a,P7) & subclass(Q7::'a,O7) --> subclass(Q7::'a,P7)) & \
3.2063 +\ (! X. subclass(compose_class(X),cross_product(universal_class::'a,universal_class))) & \
3.2064 +\ (! X Y Z. member(ordered_pair(Y::'a,Z),compose_class(X)) --> equal(compos(X::'a,Y),Z)) & \
3.2065 +\ (! 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))) & \
3.2066 +\ (subclass(composition_function::'a,cross_product(universal_class::'a,cross_product(universal_class::'a,universal_class)))) & \
3.2067 +\ (! X Y Z. member(ordered_pair(X::'a,ordered_pair(Y::'a,Z)),composition_function) --> equal(compos(X::'a,Y),Z)) & \
3.2068 +\ (! 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)) & \
3.2069 +\ (subclass(domain_relation::'a,cross_product(universal_class::'a,universal_class))) & \
3.2070 +\ (! X Y. member(ordered_pair(X::'a,Y),domain_relation) --> equal(domain_of(X),Y)) & \
3.2071 +\ (! X. member(X::'a,universal_class) --> member(ordered_pair(X::'a,domain_of(X)),domain_relation)) & \
3.2072 +\ (! X. equal(first(not_subclass_element(compos(X::'a,inverse(X)),identity_relation)),single_valued1(X))) & \
3.2073 +\ (! X. equal(second(not_subclass_element(compos(X::'a,inverse(X)),identity_relation)),single_valued2(X))) & \
3.2074 +\ (! X. equal(domain(X::'a,image_(inverse(X),singleton(single_valued1(X))),single_valued2(X)),single_valued3(X))) & \
3.2075 +\ (equal(intersection(complement(compos(element_relation::'a,complement(identity_relation))),element_relation),singleton_relation)) & \
3.2076 +\ (subclass(application_function::'a,cross_product(universal_class::'a,cross_product(universal_class::'a,universal_class)))) & \
3.2077 +\ (! Z Y X. member(ordered_pair(X::'a,ordered_pair(Y::'a,Z)),application_function) --> member(Y::'a,domain_of(X))) & \
3.2078 +\ (! X Y Z. member(ordered_pair(X::'a,ordered_pair(Y::'a,Z)),application_function) --> equal(apply(X::'a,Y),Z)) & \
3.2079 +\ (! 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)) & \
3.2080 +\ (! X Y Xf. maps(Xf::'a,X,Y) --> function(Xf)) & \
3.2081 +\ (! Y Xf X. maps(Xf::'a,X,Y) --> equal(domain_of(Xf),X)) & \
3.2082 +\ (! X Xf Y. maps(Xf::'a,X,Y) --> subclass(range_of(Xf),Y)) & \
3.2083 +\ (! Xf Y. function(Xf) & subclass(range_of(Xf),Y) --> maps(Xf::'a,domain_of(Xf),Y)) & \
3.2084 +\ (! L M. equal(L::'a,M) --> equal(compose_class(L),compose_class(M))) & \
3.2085 +\ (! N2 O2. equal(N2::'a,O2) --> equal(single_valued1(N2),single_valued1(O2))) & \
3.2086 +\ (! P2 Q2. equal(P2::'a,Q2) --> equal(single_valued2(P2),single_valued2(Q2))) & \
3.2087 +\ (! R2 S2. equal(R2::'a,S2) --> equal(single_valued3(R2),single_valued3(S2))) & \
3.2088 +\ (! X2 Y2 Z2 A3. equal(X2::'a,Y2) & maps(X2::'a,Z2,A3) --> maps(Y2::'a,Z2,A3)) & \
3.2089 +\ (! B3 D3 C3 E3. equal(B3::'a,C3) & maps(D3::'a,B3,E3) --> maps(D3::'a,C3,E3)) & \
3.2090 +\ (! F3 H3 I3 G3. equal(F3::'a,G3) & maps(H3::'a,I3,F3) --> maps(H3::'a,I3,G3)) & \
3.2091 +\ (! X. equal(union(X::'a,inverse(X)),symmetrization_of(X))) & \
3.2092 +\ (! X Y. irreflexive(X::'a,Y) --> subclass(restrct(X::'a,Y,Y),complement(identity_relation))) & \
3.2093 +\ (! X Y. subclass(restrct(X::'a,Y,Y),complement(identity_relation)) --> irreflexive(X::'a,Y)) & \
3.2094 +\ (! Y X. connected(X::'a,Y) --> subclass(cross_product(Y::'a,Y),union(identity_relation::'a,symmetrization_of(X)))) & \
3.2095 +\ (! X Y. subclass(cross_product(Y::'a,Y),union(identity_relation::'a,symmetrization_of(X))) --> connected(X::'a,Y)) & \
3.2096 +\ (! Xr Y. transitive(Xr::'a,Y) --> subclass(compos(restrct(Xr::'a,Y,Y),restrct(Xr::'a,Y,Y)),restrct(Xr::'a,Y,Y))) & \
3.2097 +\ (! Xr Y. subclass(compos(restrct(Xr::'a,Y,Y),restrct(Xr::'a,Y,Y)),restrct(Xr::'a,Y,Y)) --> transitive(Xr::'a,Y)) & \
3.2098 +\ (! Xr Y. asymmetric(Xr::'a,Y) --> equal(restrct(intersection(Xr::'a,inverse(Xr)),Y,Y),null_class)) & \
3.2099 +\ (! Xr Y. equal(restrct(intersection(Xr::'a,inverse(Xr)),Y,Y),null_class) --> asymmetric(Xr::'a,Y)) & \
3.2100 +\ (! Xr Y Z. equal(segment(Xr::'a,Y,Z),domain_of(restrct(Xr::'a,Y,singleton(Z))))) & \
3.2101 +\ (! X Y. well_ordering(X::'a,Y) --> connected(X::'a,Y)) & \
3.2102 +\ (! Y Xr U. well_ordering(Xr::'a,Y) & subclass(U::'a,Y) --> equal(U::'a,null_class) | member(least(Xr::'a,U),U)) & \
3.2103 +\ (! Y V Xr U. well_ordering(Xr::'a,Y) & subclass(U::'a,Y) & member(V::'a,U) --> member(least(Xr::'a,U),U)) & \
3.2104 +\ (! Y Xr U. well_ordering(Xr::'a,Y) & subclass(U::'a,Y) --> equal(segment(Xr::'a,U,least(Xr::'a,U)),null_class)) & \
3.2105 +\ (! 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))) & \
3.2106 +\ (! Xr Y. connected(Xr::'a,Y) & equal(not_well_ordering(Xr::'a,Y),null_class) --> well_ordering(Xr::'a,Y)) & \
3.2107 +\ (! Xr Y. connected(Xr::'a,Y) --> subclass(not_well_ordering(Xr::'a,Y),Y) | well_ordering(Xr::'a,Y)) & \
3.2108 +\ (! 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)) & \
3.2109 +\ (! Xr Y Z. section(Xr::'a,Y,Z) --> subclass(Y::'a,Z)) & \
3.2110 +\ (! Xr Z Y. section(Xr::'a,Y,Z) --> subclass(domain_of(restrct(Xr::'a,Z,Y)),Y)) & \
3.2111 +\ (! Xr Y Z. subclass(Y::'a,Z) & subclass(domain_of(restrct(Xr::'a,Z,Y)),Y) --> section(Xr::'a,Y,Z)) & \
3.2112 +\ (! X. member(X::'a,ordinal_numbers) --> well_ordering(element_relation::'a,X)) & \
3.2113 +\ (! X. member(X::'a,ordinal_numbers) --> subclass(sum_class(X),X)) & \
3.2114 +\ (! X. well_ordering(element_relation::'a,X) & subclass(sum_class(X),X) & member(X::'a,universal_class) --> member(X::'a,ordinal_numbers)) & \
3.2115 +\ (! X. well_ordering(element_relation::'a,X) & subclass(sum_class(X),X) --> member(X::'a,ordinal_numbers) | equal(X::'a,ordinal_numbers)) & \
3.2116 +\ (equal(union(singleton(null_class),image_(successor_relation::'a,ordinal_numbers)),kind_1_ordinals)) & \
3.2117 +\ (equal(intersection(complement(kind_1_ordinals),ordinal_numbers),limit_ordinals)) & \
3.2118 +\ (! X. subclass(rest_of(X),cross_product(universal_class::'a,universal_class))) & \
3.2119 +\ (! V U X. member(ordered_pair(U::'a,V),rest_of(X)) --> member(U::'a,domain_of(X))) & \
3.2120 +\ (! X U V. member(ordered_pair(U::'a,V),rest_of(X)) --> equal(restrct(X::'a,U,universal_class),V)) & \
3.2121 +\ (! 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))) & \
3.2122 +\ (subclass(rest_relation::'a,cross_product(universal_class::'a,universal_class))) & \
3.2123 +\ (! X Y. member(ordered_pair(X::'a,Y),rest_relation) --> equal(rest_of(X),Y)) & \
3.2124 +\ (! X. member(X::'a,universal_class) --> member(ordered_pair(X::'a,rest_of(X)),rest_relation)) & \
3.2125 +\ (! X Z. member(X::'a,recursion_equation_functions(Z)) --> function(Z)) & \
3.2126 +\ (! Z X. member(X::'a,recursion_equation_functions(Z)) --> function(X)) & \
3.2127 +\ (! Z X. member(X::'a,recursion_equation_functions(Z)) --> member(domain_of(X),ordinal_numbers)) & \
3.2128 +\ (! Z X. member(X::'a,recursion_equation_functions(Z)) --> equal(compos(Z::'a,rest_of(X)),X)) & \
3.2129 +\ (! 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))) & \
3.2130 +\ (subclass(union_of_range_map::'a,cross_product(universal_class::'a,universal_class))) & \
3.2131 +\ (! X Y. member(ordered_pair(X::'a,Y),union_of_range_map) --> equal(sum_class(range_of(X)),Y)) & \
3.2132 +\ (! 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)) & \
3.2133 +\ (! X Y. equal(apply(recursion(X::'a,successor_relation,union_of_range_map),Y),ordinal_add(X::'a,Y))) & \
3.2134 +\ (! X Y. equal(recursion(null_class::'a,apply(add_relation::'a,X),union_of_range_map),ordinal_multiply(X::'a,Y))) & \
3.2135 +\ (! X. member(X::'a,omega) --> equal(integer_of(X),X)) & \
3.2136 +\ (! X. member(X::'a,omega) | equal(integer_of(X),null_class)) & \
3.2137 +\ (! D E. equal(D::'a,E) --> equal(integer_of(D),integer_of(E))) & \
3.2138 +\ (! F' G H. equal(F'::'a,G) --> equal(least(F'::'a,H),least(G::'a,H))) & \
3.2139 +\ (! I' K' J. equal(I'::'a,J) --> equal(least(K'::'a,I'),least(K'::'a,J))) & \
3.2140 +\ (! L M N. equal(L::'a,M) --> equal(not_well_ordering(L::'a,N),not_well_ordering(M::'a,N))) & \
3.2141 +\ (! O_ Q P. equal(O_::'a,P) --> equal(not_well_ordering(Q::'a,O_),not_well_ordering(Q::'a,P))) & \
3.2142 +\ (! R S' T'. equal(R::'a,S') --> equal(ordinal_add(R::'a,T'),ordinal_add(S'::'a,T'))) & \
3.2143 +\ (! U W V. equal(U::'a,V) --> equal(ordinal_add(W::'a,U),ordinal_add(W::'a,V))) & \
3.2144 +\ (! X Y Z. equal(X::'a,Y) --> equal(ordinal_multiply(X::'a,Z),ordinal_multiply(Y::'a,Z))) & \
3.2145 +\ (! A1 C1 B1. equal(A1::'a,B1) --> equal(ordinal_multiply(C1::'a,A1),ordinal_multiply(C1::'a,B1))) & \
3.2146 +\ (! F1 G1 H1 I1. equal(F1::'a,G1) --> equal(recursion(F1::'a,H1,I1),recursion(G1::'a,H1,I1))) & \
3.2147 +\ (! J1 L1 K1 M1. equal(J1::'a,K1) --> equal(recursion(L1::'a,J1,M1),recursion(L1::'a,K1,M1))) & \
3.2148 +\ (! N1 P1 Q1 O1. equal(N1::'a,O1) --> equal(recursion(P1::'a,Q1,N1),recursion(P1::'a,Q1,O1))) & \
3.2149 +\ (! R1 S1. equal(R1::'a,S1) --> equal(recursion_equation_functions(R1),recursion_equation_functions(S1))) & \
3.2150 +\ (! T1 U1. equal(T1::'a,U1) --> equal(rest_of(T1),rest_of(U1))) & \
3.2151 +\ (! V1 W1 X1 Y1. equal(V1::'a,W1) --> equal(segment(V1::'a,X1,Y1),segment(W1::'a,X1,Y1))) & \
3.2152 +\ (! Z1 B2 A2 C2. equal(Z1::'a,A2) --> equal(segment(B2::'a,Z1,C2),segment(B2::'a,A2,C2))) & \
3.2153 +\ (! D2 F2 G2 E2. equal(D2::'a,E2) --> equal(segment(F2::'a,G2,D2),segment(F2::'a,G2,E2))) & \
3.2154 +\ (! H2 I2. equal(H2::'a,I2) --> equal(symmetrization_of(H2),symmetrization_of(I2))) & \
3.2155 +\ (! J2 K2 L2. equal(J2::'a,K2) & asymmetric(J2::'a,L2) --> asymmetric(K2::'a,L2)) & \
3.2156 +\ (! M2 O2 N2. equal(M2::'a,N2) & asymmetric(O2::'a,M2) --> asymmetric(O2::'a,N2)) & \
3.2157 +\ (! P2 Q2 R2. equal(P2::'a,Q2) & connected(P2::'a,R2) --> connected(Q2::'a,R2)) & \
3.2158 +\ (! S2 U2 T2. equal(S2::'a,T2) & connected(U2::'a,S2) --> connected(U2::'a,T2)) & \
3.2159 +\ (! V2 W2 X2. equal(V2::'a,W2) & irreflexive(V2::'a,X2) --> irreflexive(W2::'a,X2)) & \
3.2160 +\ (! Y2 A3 Z2. equal(Y2::'a,Z2) & irreflexive(A3::'a,Y2) --> irreflexive(A3::'a,Z2)) & \
3.2161 +\ (! B3 C3 D3 E3. equal(B3::'a,C3) & section(B3::'a,D3,E3) --> section(C3::'a,D3,E3)) & \
3.2162 +\ (! F3 H3 G3 I3. equal(F3::'a,G3) & section(H3::'a,F3,I3) --> section(H3::'a,G3,I3)) & \
3.2163 +\ (! J3 L3 M3 K3. equal(J3::'a,K3) & section(L3::'a,M3,J3) --> section(L3::'a,M3,K3)) & \
3.2164 +\ (! N3 O3 P3. equal(N3::'a,O3) & transitive(N3::'a,P3) --> transitive(O3::'a,P3)) & \
3.2165 +\ (! Q3 S3 R3. equal(Q3::'a,R3) & transitive(S3::'a,Q3) --> transitive(S3::'a,R3)) & \
3.2166 +\ (! T3 U3 V3. equal(T3::'a,U3) & well_ordering(T3::'a,V3) --> well_ordering(U3::'a,V3)) & \
3.2167 +\ (! W3 Y3 X3. equal(W3::'a,X3) & well_ordering(Y3::'a,W3) --> well_ordering(Y3::'a,X3)) & \
3.2168 +\ (~function(z)) & \
3.2169 +\ (~equal(recursion_equation_functions(z),null_class)) --> False",
3.2170 + meson_tac);
3.2171 +
3.2172 +
3.2173 +(*4868 inferences so far. Searching to depth 12. 4.3 secs*)
3.2174 +val PLA002_1 = prove_hard
3.2175 + ("(! Situation1 Situation2. warm(Situation1) | cold(Situation2)) & \
3.2176 +\ (! Situation. at(a::'a,Situation) --> at(b::'a,walk(b::'a,Situation))) & \
3.2177 +\ (! Situation. at(a::'a,Situation) --> at(b::'a,drive(b::'a,Situation))) & \
3.2178 +\ (! Situation. at(b::'a,Situation) --> at(a::'a,walk(a::'a,Situation))) & \
3.2179 +\ (! Situation. at(b::'a,Situation) --> at(a::'a,drive(a::'a,Situation))) & \
3.2180 +\ (! Situation. cold(Situation) & at(b::'a,Situation) --> at(c::'a,skate(c::'a,Situation))) & \
3.2181 +\ (! Situation. cold(Situation) & at(c::'a,Situation) --> at(b::'a,skate(b::'a,Situation))) & \
3.2182 +\ (! Situation. warm(Situation) & at(b::'a,Situation) --> at(d::'a,climb(d::'a,Situation))) & \
3.2183 +\ (! Situation. warm(Situation) & at(d::'a,Situation) --> at(b::'a,climb(b::'a,Situation))) & \
3.2184 +\ (! Situation. at(c::'a,Situation) --> at(d::'a,go(d::'a,Situation))) & \
3.2185 +\ (! Situation. at(d::'a,Situation) --> at(c::'a,go(c::'a,Situation))) & \
3.2186 +\ (! Situation. at(c::'a,Situation) --> at(e::'a,go(e::'a,Situation))) & \
3.2187 +\ (! Situation. at(e::'a,Situation) --> at(c::'a,go(c::'a,Situation))) & \
3.2188 +\ (! Situation. at(d::'a,Situation) --> at(f::'a,go(f::'a,Situation))) & \
3.2189 +\ (! Situation. at(f::'a,Situation) --> at(d::'a,go(d::'a,Situation))) & \
3.2190 +\ (at(f::'a,s0)) & \
3.2191 +\ (! S'. ~at(a::'a,S')) --> False",
3.2192 + meson_tac);
3.2193 +
3.2194 +(*190 inferences so far. Searching to depth 10. 0.6 secs*)
3.2195 +val PLA006_1 = prove
3.2196 + ("(! X Y State. holds(X::'a,State) & holds(Y::'a,State) --> holds(and'(X::'a,Y),State)) & \
3.2197 +\ (! State X. holds(empty::'a,State) & holds(clear(X),State) & differ(X::'a,table) --> holds(holding(X),do(pickup(X),State))) & \
3.2198 +\ (! Y X State. holds(on(X::'a,Y),State) & holds(clear(X),State) & holds(empty::'a,State) --> holds(clear(Y),do(pickup(X),State))) & \
3.2199 +\ (! Y State X Z. holds(on(X::'a,Y),State) & differ(X::'a,Z) --> holds(on(X::'a,Y),do(pickup(Z),State))) & \
3.2200 +\ (! State X Z. holds(clear(X),State) & differ(X::'a,Z) --> holds(clear(X),do(pickup(Z),State))) & \
3.2201 +\ (! X Y State. holds(holding(X),State) & holds(clear(Y),State) --> holds(empty::'a,do(putdown(X::'a,Y),State))) & \
3.2202 +\ (! X Y State. holds(holding(X),State) & holds(clear(Y),State) --> holds(on(X::'a,Y),do(putdown(X::'a,Y),State))) & \
3.2203 +\ (! X Y State. holds(holding(X),State) & holds(clear(Y),State) --> holds(clear(X),do(putdown(X::'a,Y),State))) & \
3.2204 +\ (! Z W X Y State. holds(on(X::'a,Y),State) --> holds(on(X::'a,Y),do(putdown(Z::'a,W),State))) & \
3.2205 +\ (! X State Z Y. holds(clear(Z),State) & differ(Z::'a,Y) --> holds(clear(Z),do(putdown(X::'a,Y),State))) & \
3.2206 +\ (! Y X. differ(Y::'a,X) --> differ(X::'a,Y)) & \
3.2207 +\ (differ(a::'a,b)) & \
3.2208 +\ (differ(a::'a,c)) & \
3.2209 +\ (differ(a::'a,d)) & \
3.2210 +\ (differ(a::'a,table)) & \
3.2211 +\ (differ(b::'a,c)) & \
3.2212 +\ (differ(b::'a,d)) & \
3.2213 +\ (differ(b::'a,table)) & \
3.2214 +\ (differ(c::'a,d)) & \
3.2215 +\ (differ(c::'a,table)) & \
3.2216 +\ (differ(d::'a,table)) & \
3.2217 +\ (holds(on(a::'a,table),s0)) & \
3.2218 +\ (holds(on(b::'a,table),s0)) & \
3.2219 +\ (holds(on(c::'a,d),s0)) & \
3.2220 +\ (holds(on(d::'a,table),s0)) & \
3.2221 +\ (holds(clear(a),s0)) & \
3.2222 +\ (holds(clear(b),s0)) & \
3.2223 +\ (holds(clear(c),s0)) & \
3.2224 +\ (holds(empty::'a,s0)) & \
3.2225 +\ (! State. holds(clear(table),State)) & \
3.2226 +\ (! State. ~holds(on(c::'a,table),State)) --> False",
3.2227 + meson_tac);
3.2228 +
3.2229 +(*190 inferences so far. Searching to depth 10. 0.5 secs*)
3.2230 +val PLA017_1 = prove
3.2231 + ("(! X Y State. holds(X::'a,State) & holds(Y::'a,State) --> holds(and'(X::'a,Y),State)) & \
3.2232 +\ (! State X. holds(empty::'a,State) & holds(clear(X),State) & differ(X::'a,table) --> holds(holding(X),do(pickup(X),State))) & \
3.2233 +\ (! Y X State. holds(on(X::'a,Y),State) & holds(clear(X),State) & holds(empty::'a,State) --> holds(clear(Y),do(pickup(X),State))) & \
3.2234 +\ (! Y State X Z. holds(on(X::'a,Y),State) & differ(X::'a,Z) --> holds(on(X::'a,Y),do(pickup(Z),State))) & \
3.2235 +\ (! State X Z. holds(clear(X),State) & differ(X::'a,Z) --> holds(clear(X),do(pickup(Z),State))) & \
3.2236 +\ (! X Y State. holds(holding(X),State) & holds(clear(Y),State) --> holds(empty::'a,do(putdown(X::'a,Y),State))) & \
3.2237 +\ (! X Y State. holds(holding(X),State) & holds(clear(Y),State) --> holds(on(X::'a,Y),do(putdown(X::'a,Y),State))) & \
3.2238 +\ (! X Y State. holds(holding(X),State) & holds(clear(Y),State) --> holds(clear(X),do(putdown(X::'a,Y),State))) & \
3.2239 +\ (! Z W X Y State. holds(on(X::'a,Y),State) --> holds(on(X::'a,Y),do(putdown(Z::'a,W),State))) & \
3.2240 +\ (! X State Z Y. holds(clear(Z),State) & differ(Z::'a,Y) --> holds(clear(Z),do(putdown(X::'a,Y),State))) & \
3.2241 +\ (! Y X. differ(Y::'a,X) --> differ(X::'a,Y)) & \
3.2242 +\ (differ(a::'a,b)) & \
3.2243 +\ (differ(a::'a,c)) & \
3.2244 +\ (differ(a::'a,d)) & \
3.2245 +\ (differ(a::'a,table)) & \
3.2246 +\ (differ(b::'a,c)) & \
3.2247 +\ (differ(b::'a,d)) & \
3.2248 +\ (differ(b::'a,table)) & \
3.2249 +\ (differ(c::'a,d)) & \
3.2250 +\ (differ(c::'a,table)) & \
3.2251 +\ (differ(d::'a,table)) & \
3.2252 +\ (holds(on(a::'a,table),s0)) & \
3.2253 +\ (holds(on(b::'a,table),s0)) & \
3.2254 +\ (holds(on(c::'a,d),s0)) & \
3.2255 +\ (holds(on(d::'a,table),s0)) & \
3.2256 +\ (holds(clear(a),s0)) & \
3.2257 +\ (holds(clear(b),s0)) & \
3.2258 +\ (holds(clear(c),s0)) & \
3.2259 +\ (holds(empty::'a,s0)) & \
3.2260 +\ (! State. holds(clear(table),State)) & \
3.2261 +\ (! State. ~holds(on(a::'a,c),State)) --> False",
3.2262 + meson_tac);
3.2263 +
3.2264 +(*13732 inferences so far. Searching to depth 16. 9.8 secs*)
3.2265 +val PLA022_1 = prove_hard
3.2266 + ("(! X Y State. holds(X::'a,State) & holds(Y::'a,State) --> holds(and'(X::'a,Y),State)) & \
3.2267 +\ (! State X. holds(empty::'a,State) & holds(clear(X),State) & differ(X::'a,table) --> holds(holding(X),do(pickup(X),State))) & \
3.2268 +\ (! Y X State. holds(on(X::'a,Y),State) & holds(clear(X),State) & holds(empty::'a,State) --> holds(clear(Y),do(pickup(X),State))) & \
3.2269 +\ (! Y State X Z. holds(on(X::'a,Y),State) & differ(X::'a,Z) --> holds(on(X::'a,Y),do(pickup(Z),State))) & \
3.2270 +\ (! State X Z. holds(clear(X),State) & differ(X::'a,Z) --> holds(clear(X),do(pickup(Z),State))) & \
3.2271 +\ (! X Y State. holds(holding(X),State) & holds(clear(Y),State) --> holds(empty::'a,do(putdown(X::'a,Y),State))) & \
3.2272 +\ (! X Y State. holds(holding(X),State) & holds(clear(Y),State) --> holds(on(X::'a,Y),do(putdown(X::'a,Y),State))) & \
3.2273 +\ (! X Y State. holds(holding(X),State) & holds(clear(Y),State) --> holds(clear(X),do(putdown(X::'a,Y),State))) & \
3.2274 +\ (! Z W X Y State. holds(on(X::'a,Y),State) --> holds(on(X::'a,Y),do(putdown(Z::'a,W),State))) & \
3.2275 +\ (! X State Z Y. holds(clear(Z),State) & differ(Z::'a,Y) --> holds(clear(Z),do(putdown(X::'a,Y),State))) & \
3.2276 +\ (! Y X. differ(Y::'a,X) --> differ(X::'a,Y)) & \
3.2277 +\ (differ(a::'a,b)) & \
3.2278 +\ (differ(a::'a,c)) & \
3.2279 +\ (differ(a::'a,d)) & \
3.2280 +\ (differ(a::'a,table)) & \
3.2281 +\ (differ(b::'a,c)) & \
3.2282 +\ (differ(b::'a,d)) & \
3.2283 +\ (differ(b::'a,table)) & \
3.2284 +\ (differ(c::'a,d)) & \
3.2285 +\ (differ(c::'a,table)) & \
3.2286 +\ (differ(d::'a,table)) & \
3.2287 +\ (holds(on(a::'a,table),s0)) & \
3.2288 +\ (holds(on(b::'a,table),s0)) & \
3.2289 +\ (holds(on(c::'a,d),s0)) & \
3.2290 +\ (holds(on(d::'a,table),s0)) & \
3.2291 +\ (holds(clear(a),s0)) & \
3.2292 +\ (holds(clear(b),s0)) & \
3.2293 +\ (holds(clear(c),s0)) & \
3.2294 +\ (holds(empty::'a,s0)) & \
3.2295 +\ (! State. holds(clear(table),State)) & \
3.2296 +\ (! State. ~holds(and'(on(c::'a,d),on(a::'a,c)),State)) --> False",
3.2297 + meson_tac);
3.2298 +
3.2299 +(*217 inferences so far. Searching to depth 13. 0.7 secs*)
3.2300 +val PLA022_2 = prove
3.2301 + ("(! X Y State. holds(X::'a,State) & holds(Y::'a,State) --> holds(and'(X::'a,Y),State)) & \
3.2302 +\ (! State X. holds(empty::'a,State) & holds(clear(X),State) & differ(X::'a,table) --> holds(holding(X),do(pickup(X),State))) & \
3.2303 +\ (! Y X State. holds(on(X::'a,Y),State) & holds(clear(X),State) & holds(empty::'a,State) --> holds(clear(Y),do(pickup(X),State))) & \
3.2304 +\ (! Y State X Z. holds(on(X::'a,Y),State) & differ(X::'a,Z) --> holds(on(X::'a,Y),do(pickup(Z),State))) & \
3.2305 +\ (! State X Z. holds(clear(X),State) & differ(X::'a,Z) --> holds(clear(X),do(pickup(Z),State))) & \
3.2306 +\ (! X Y State. holds(holding(X),State) & holds(clear(Y),State) --> holds(empty::'a,do(putdown(X::'a,Y),State))) & \
3.2307 +\ (! X Y State. holds(holding(X),State) & holds(clear(Y),State) --> holds(on(X::'a,Y),do(putdown(X::'a,Y),State))) & \
3.2308 +\ (! X Y State. holds(holding(X),State) & holds(clear(Y),State) --> holds(clear(X),do(putdown(X::'a,Y),State))) & \
3.2309 +\ (! Z W X Y State. holds(on(X::'a,Y),State) --> holds(on(X::'a,Y),do(putdown(Z::'a,W),State))) & \
3.2310 +\ (! X State Z Y. holds(clear(Z),State) & differ(Z::'a,Y) --> holds(clear(Z),do(putdown(X::'a,Y),State))) & \
3.2311 +\ (! Y X. differ(Y::'a,X) --> differ(X::'a,Y)) & \
3.2312 +\ (differ(a::'a,b)) & \
3.2313 +\ (differ(a::'a,c)) & \
3.2314 +\ (differ(a::'a,d)) & \
3.2315 +\ (differ(a::'a,table)) & \
3.2316 +\ (differ(b::'a,c)) & \
3.2317 +\ (differ(b::'a,d)) & \
3.2318 +\ (differ(b::'a,table)) & \
3.2319 +\ (differ(c::'a,d)) & \
3.2320 +\ (differ(c::'a,table)) & \
3.2321 +\ (differ(d::'a,table)) & \
3.2322 +\ (holds(on(a::'a,table),s0)) & \
3.2323 +\ (holds(on(b::'a,table),s0)) & \
3.2324 +\ (holds(on(c::'a,d),s0)) & \
3.2325 +\ (holds(on(d::'a,table),s0)) & \
3.2326 +\ (holds(clear(a),s0)) & \
3.2327 +\ (holds(clear(b),s0)) & \
3.2328 +\ (holds(clear(c),s0)) & \
3.2329 +\ (holds(empty::'a,s0)) & \
3.2330 +\ (! State. holds(clear(table),State)) & \
3.2331 +\ (! State. ~holds(and'(on(a::'a,c),on(c::'a,d)),State)) --> False",
3.2332 + meson_tac);
3.2333 +
3.2334 +(*948 inferences so far. Searching to depth 18. 1.1 secs*)
3.2335 +val PRV001_1 = prove
3.2336 + ("(! X Y Z. q1(X::'a,Y,Z) & less_or_equal(X::'a,Y) --> q2(X::'a,Y,Z)) & \
3.2337 +\ (! X Y Z. q1(X::'a,Y,Z) --> less_or_equal(X::'a,Y) | q3(X::'a,Y,Z)) & \
3.2338 +\ (! Z X Y. q2(X::'a,Y,Z) --> q4(X::'a,Y,Y)) & \
3.2339 +\ (! Z Y X. q3(X::'a,Y,Z) --> q4(X::'a,Y,X)) & \
3.2340 +\ (! X. less_or_equal(X::'a,X)) & \
3.2341 +\ (! X Y. less_or_equal(X::'a,Y) & less_or_equal(Y::'a,X) --> equal(X::'a,Y)) & \
3.2342 +\ (! Y X Z. less_or_equal(X::'a,Y) & less_or_equal(Y::'a,Z) --> less_or_equal(X::'a,Z)) & \
3.2343 +\ (! Y X. less_or_equal(X::'a,Y) | less_or_equal(Y::'a,X)) & \
3.2344 +\ (! X Y. equal(X::'a,Y) --> less_or_equal(X::'a,Y)) & \
3.2345 +\ (! X Y Z. equal(X::'a,Y) & less_or_equal(X::'a,Z) --> less_or_equal(Y::'a,Z)) & \
3.2346 +\ (! X Z Y. equal(X::'a,Y) & less_or_equal(Z::'a,X) --> less_or_equal(Z::'a,Y)) & \
3.2347 +\ (q1(a::'a,b,c)) & \
3.2348 +\ (! W. ~(q4(a::'a,b,W) & less_or_equal(a::'a,W) & less_or_equal(b::'a,W) & less_or_equal(W::'a,a))) & \
3.2349 +\ (! W. ~(q4(a::'a,b,W) & less_or_equal(a::'a,W) & less_or_equal(b::'a,W) & less_or_equal(W::'a,b))) --> False",
3.2350 + meson_tac);
3.2351 +
3.2352 +(*21 inferences so far. Searching to depth 5. 0.4 secs*)
3.2353 +val PRV003_1 = prove
3.2354 + ("(! X. equal(X::'a,X)) & \
3.2355 +\ (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
3.2356 +\ (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) & \
3.2357 +\ (! X. equal(predecessor(successor(X)),X)) & \
3.2358 +\ (! X. equal(successor(predecessor(X)),X)) & \
3.2359 +\ (! X Y. equal(predecessor(X),predecessor(Y)) --> equal(X::'a,Y)) & \
3.2360 +\ (! X Y. equal(successor(X),successor(Y)) --> equal(X::'a,Y)) & \
3.2361 +\ (! X. less_than(predecessor(X),X)) & \
3.2362 +\ (! X. less_than(X::'a,successor(X))) & \
3.2363 +\ (! X Y Z. less_than(X::'a,Y) & less_than(Y::'a,Z) --> less_than(X::'a,Z)) & \
3.2364 +\ (! X Y. less_than(X::'a,Y) | less_than(Y::'a,X) | equal(X::'a,Y)) & \
3.2365 +\ (! X. ~less_than(X::'a,X)) & \
3.2366 +\ (! Y X. ~(less_than(X::'a,Y) & less_than(Y::'a,X))) & \
3.2367 +\ (! Y X Z. equal(X::'a,Y) & less_than(X::'a,Z) --> less_than(Y::'a,Z)) & \
3.2368 +\ (! Y Z X. equal(X::'a,Y) & less_than(Z::'a,X) --> less_than(Z::'a,Y)) & \
3.2369 +\ (! X Y. equal(X::'a,Y) --> equal(predecessor(X),predecessor(Y))) & \
3.2370 +\ (! X Y. equal(X::'a,Y) --> equal(successor(X),successor(Y))) & \
3.2371 +\ (! X Y. equal(X::'a,Y) --> equal(a(X),a(Y))) & \
3.2372 +\ (~less_than(n::'a,j)) & \
3.2373 +\ (less_than(k::'a,j)) & \
3.2374 +\ (~less_than(k::'a,i)) & \
3.2375 +\ (less_than(i::'a,n)) & \
3.2376 +\ (less_than(a(j),a(k))) & \
3.2377 +\ (! X. less_than(X::'a,j) & less_than(a(X),a(k)) --> less_than(X::'a,i)) & \
3.2378 +\ (! X. less_than(one::'a,i) & less_than(a(X),a(predecessor(i))) --> less_than(X::'a,i) | less_than(n::'a,X)) & \
3.2379 +\ (! X. ~(less_than(one::'a,X) & less_than(X::'a,i) & less_than(a(X),a(predecessor(X))))) & \
3.2380 +\ (less_than(j::'a,i)) --> False",
3.2381 + meson_tac);
3.2382 +
3.2383 +(*584 inferences so far. Searching to depth 7. 1.1 secs*)
3.2384 +val PRV005_1 = prove
3.2385 + ("(! X. equal(X::'a,X)) & \
3.2386 +\ (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
3.2387 +\ (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) & \
3.2388 +\ (! X. equal(predecessor(successor(X)),X)) & \
3.2389 +\ (! X. equal(successor(predecessor(X)),X)) & \
3.2390 +\ (! X Y. equal(predecessor(X),predecessor(Y)) --> equal(X::'a,Y)) & \
3.2391 +\ (! X Y. equal(successor(X),successor(Y)) --> equal(X::'a,Y)) & \
3.2392 +\ (! X. less_than(predecessor(X),X)) & \
3.2393 +\ (! X. less_than(X::'a,successor(X))) & \
3.2394 +\ (! X Y Z. less_than(X::'a,Y) & less_than(Y::'a,Z) --> less_than(X::'a,Z)) & \
3.2395 +\ (! X Y. less_than(X::'a,Y) | less_than(Y::'a,X) | equal(X::'a,Y)) & \
3.2396 +\ (! X. ~less_than(X::'a,X)) & \
3.2397 +\ (! Y X. ~(less_than(X::'a,Y) & less_than(Y::'a,X))) & \
3.2398 +\ (! Y X Z. equal(X::'a,Y) & less_than(X::'a,Z) --> less_than(Y::'a,Z)) & \
3.2399 +\ (! Y Z X. equal(X::'a,Y) & less_than(Z::'a,X) --> less_than(Z::'a,Y)) & \
3.2400 +\ (! X Y. equal(X::'a,Y) --> equal(predecessor(X),predecessor(Y))) & \
3.2401 +\ (! X Y. equal(X::'a,Y) --> equal(successor(X),successor(Y))) & \
3.2402 +\ (! X Y. equal(X::'a,Y) --> equal(a(X),a(Y))) & \
3.2403 +\ (~less_than(n::'a,k)) & \
3.2404 +\ (~less_than(k::'a,l)) & \
3.2405 +\ (~less_than(k::'a,i)) & \
3.2406 +\ (less_than(l::'a,n)) & \
3.2407 +\ (less_than(one::'a,l)) & \
3.2408 +\ (less_than(a(k),a(predecessor(l)))) & \
3.2409 +\ (! X. less_than(X::'a,successor(n)) & less_than(a(X),a(k)) --> less_than(X::'a,l)) & \
3.2410 +\ (! X. less_than(one::'a,l) & less_than(a(X),a(predecessor(l))) --> less_than(X::'a,l) | less_than(n::'a,X)) & \
3.2411 +\ (! X. ~(less_than(one::'a,X) & less_than(X::'a,l) & less_than(a(X),a(predecessor(X))))) --> False",
3.2412 + meson_tac);
3.2413 +
3.2414 +(*2343 inferences so far. Searching to depth 8. 3.5 secs*)
3.2415 +val PRV006_1 = prove_hard
3.2416 + ("(! X. equal(X::'a,X)) & \
3.2417 +\ (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
3.2418 +\ (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) & \
3.2419 +\ (! X. equal(predecessor(successor(X)),X)) & \
3.2420 +\ (! X. equal(successor(predecessor(X)),X)) & \
3.2421 +\ (! X Y. equal(predecessor(X),predecessor(Y)) --> equal(X::'a,Y)) & \
3.2422 +\ (! X Y. equal(successor(X),successor(Y)) --> equal(X::'a,Y)) & \
3.2423 +\ (! X. less_than(predecessor(X),X)) & \
3.2424 +\ (! X. less_than(X::'a,successor(X))) & \
3.2425 +\ (! X Y Z. less_than(X::'a,Y) & less_than(Y::'a,Z) --> less_than(X::'a,Z)) & \
3.2426 +\ (! X Y. less_than(X::'a,Y) | less_than(Y::'a,X) | equal(X::'a,Y)) & \
3.2427 +\ (! X. ~less_than(X::'a,X)) & \
3.2428 +\ (! Y X. ~(less_than(X::'a,Y) & less_than(Y::'a,X))) & \
3.2429 +\ (! Y X Z. equal(X::'a,Y) & less_than(X::'a,Z) --> less_than(Y::'a,Z)) & \
3.2430 +\ (! Y Z X. equal(X::'a,Y) & less_than(Z::'a,X) --> less_than(Z::'a,Y)) & \
3.2431 +\ (! X Y. equal(X::'a,Y) --> equal(predecessor(X),predecessor(Y))) & \
3.2432 +\ (! X Y. equal(X::'a,Y) --> equal(successor(X),successor(Y))) & \
3.2433 +\ (! X Y. equal(X::'a,Y) --> equal(a(X),a(Y))) & \
3.2434 +\ (~less_than(n::'a,m)) & \
3.2435 +\ (less_than(i::'a,m)) & \
3.2436 +\ (less_than(i::'a,n)) & \
3.2437 +\ (~less_than(i::'a,one)) & \
3.2438 +\ (less_than(a(i),a(m))) & \
3.2439 +\ (! X. less_than(X::'a,successor(n)) & less_than(a(X),a(m)) --> less_than(X::'a,i)) & \
3.2440 +\ (! X. less_than(one::'a,i) & less_than(a(X),a(predecessor(i))) --> less_than(X::'a,i) | less_than(n::'a,X)) & \
3.2441 +\ (! X. ~(less_than(one::'a,X) & less_than(X::'a,i) & less_than(a(X),a(predecessor(X))))) --> False",
3.2442 + meson_tac);
3.2443 +
3.2444 +(*86 inferences so far. Searching to depth 14. 0.1 secs*)
3.2445 +val PRV009_1 = prove
3.2446 + ("(! Y X. less_or_equal(X::'a,Y) | less(Y::'a,X)) & \
3.2447 +\ (less(j::'a,i)) & \
3.2448 +\ (less_or_equal(m::'a,p)) & \
3.2449 +\ (less_or_equal(p::'a,q)) & \
3.2450 +\ (less_or_equal(q::'a,n)) & \
3.2451 +\ (! X Y. less_or_equal(m::'a,X) & less(X::'a,i) & less(j::'a,Y) & less_or_equal(Y::'a,n) --> less_or_equal(a(X),a(Y))) & \
3.2452 +\ (! X Y. less_or_equal(m::'a,X) & less_or_equal(X::'a,Y) & less_or_equal(Y::'a,j) --> less_or_equal(a(X),a(Y))) & \
3.2453 +\ (! X Y. less_or_equal(i::'a,X) & less_or_equal(X::'a,Y) & less_or_equal(Y::'a,n) --> less_or_equal(a(X),a(Y))) & \
3.2454 +\ (~less_or_equal(a(p),a(q))) --> False",
3.2455 + meson_tac);
3.2456 +
3.2457 +(*222 inferences so far. Searching to depth 8. 0.4 secs*)
3.2458 +val PUZ012_1 = prove
3.2459 + ("(! X. equal_fruits(X::'a,X)) & \
3.2460 +\ (! X. equal_boxes(X::'a,X)) & \
3.2461 +\ (! X Y. ~(label(X::'a,Y) & contains(X::'a,Y))) & \
3.2462 +\ (! X. contains(boxa::'a,X) | contains(boxb::'a,X) | contains(boxc::'a,X)) & \
3.2463 +\ (! X. contains(X::'a,apples) | contains(X::'a,bananas) | contains(X::'a,oranges)) & \
3.2464 +\ (! X Y Z. contains(X::'a,Y) & contains(X::'a,Z) --> equal_fruits(Y::'a,Z)) & \
3.2465 +\ (! Y X Z. contains(X::'a,Y) & contains(Z::'a,Y) --> equal_boxes(X::'a,Z)) & \
3.2466 +\ (~equal_boxes(boxa::'a,boxb)) & \
3.2467 +\ (~equal_boxes(boxb::'a,boxc)) & \
3.2468 +\ (~equal_boxes(boxa::'a,boxc)) & \
3.2469 +\ (~equal_fruits(apples::'a,bananas)) & \
3.2470 +\ (~equal_fruits(bananas::'a,oranges)) & \
3.2471 +\ (~equal_fruits(apples::'a,oranges)) & \
3.2472 +\ (label(boxa::'a,apples)) & \
3.2473 +\ (label(boxb::'a,oranges)) & \
3.2474 +\ (label(boxc::'a,bananas)) & \
3.2475 +\ (contains(boxb::'a,apples)) & \
3.2476 +\ (~(contains(boxa::'a,bananas) & contains(boxc::'a,oranges))) --> False",
3.2477 + meson_tac);
3.2478 +
3.2479 +(*35 inferences so far. Searching to depth 5. 3.2 secs*)
3.2480 +val PUZ020_1 = prove
3.2481 + ("(! X. equal(X::'a,X)) & \
3.2482 +\ (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
3.2483 +\ (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) & \
3.2484 +\ (! A B. equal(A::'a,B) --> equal(statement_by(A),statement_by(B))) & \
3.2485 +\ (! X. person(X) --> knight(X) | knave(X)) & \
3.2486 +\ (! X. ~(person(X) & knight(X) & knave(X))) & \
3.2487 +\ (! X Y. says(X::'a,Y) & a_truth(Y) --> a_truth(Y)) & \
3.2488 +\ (! X Y. ~(says(X::'a,Y) & equal(X::'a,Y))) & \
3.2489 +\ (! Y X. says(X::'a,Y) --> equal(Y::'a,statement_by(X))) & \
3.2490 +\ (! X Y. ~(person(X) & equal(X::'a,statement_by(Y)))) & \
3.2491 +\ (! X. person(X) & a_truth(statement_by(X)) --> knight(X)) & \
3.2492 +\ (! X. person(X) --> a_truth(statement_by(X)) | knave(X)) & \
3.2493 +\ (! X Y. equal(X::'a,Y) & knight(X) --> knight(Y)) & \
3.2494 +\ (! X Y. equal(X::'a,Y) & knave(X) --> knave(Y)) & \
3.2495 +\ (! X Y. equal(X::'a,Y) & person(X) --> person(Y)) & \
3.2496 +\ (! X Y Z. equal(X::'a,Y) & says(X::'a,Z) --> says(Y::'a,Z)) & \
3.2497 +\ (! X Z Y. equal(X::'a,Y) & says(Z::'a,X) --> says(Z::'a,Y)) & \
3.2498 +\ (! X Y. equal(X::'a,Y) & a_truth(X) --> a_truth(Y)) & \
3.2499 +\ (! X Y. knight(X) & says(X::'a,Y) --> a_truth(Y)) & \
3.2500 +\ (! X Y. ~(knave(X) & says(X::'a,Y) & a_truth(Y))) & \
3.2501 +\ (person(husband)) & \
3.2502 +\ (person(wife)) & \
3.2503 +\ (~equal(husband::'a,wife)) & \
3.2504 +\ (says(husband::'a,statement_by(husband))) & \
3.2505 +\ (a_truth(statement_by(husband)) & knight(husband) --> knight(wife)) & \
3.2506 +\ (knight(husband) --> a_truth(statement_by(husband))) & \
3.2507 +\ (a_truth(statement_by(husband)) | knight(wife)) & \
3.2508 +\ (knight(wife) --> a_truth(statement_by(husband))) & \
3.2509 +\ (~knight(husband)) --> False",
3.2510 + meson_tac);
3.2511 +
3.2512 +(*121806 inferences so far. Searching to depth 17. 63.0 secs*)
3.2513 +val PUZ025_1 = prove_hard
3.2514 + ("(! X. a_truth(truthteller(X)) | a_truth(liar(X))) & \
3.2515 +\ (! X. ~(a_truth(truthteller(X)) & a_truth(liar(X)))) & \
3.2516 +\ (! Truthteller Statement. a_truth(truthteller(Truthteller)) & a_truth(says(Truthteller::'a,Statement)) --> a_truth(Statement)) & \
3.2517 +\ (! Liar Statement. ~(a_truth(liar(Liar)) & a_truth(says(Liar::'a,Statement)) & a_truth(Statement))) & \
3.2518 +\ (! Statement Truthteller. a_truth(Statement) & a_truth(says(Truthteller::'a,Statement)) --> a_truth(truthteller(Truthteller))) & \
3.2519 +\ (! Statement Liar. a_truth(says(Liar::'a,Statement)) --> a_truth(Statement) | a_truth(liar(Liar))) & \
3.2520 +\ (! Z X Y. people(X::'a,Y,Z) & a_truth(liar(X)) & a_truth(liar(Y)) --> a_truth(equal_type(X::'a,Y))) & \
3.2521 +\ (! Z X Y. people(X::'a,Y,Z) & a_truth(truthteller(X)) & a_truth(truthteller(Y)) --> a_truth(equal_type(X::'a,Y))) & \
3.2522 +\ (! X Y. a_truth(equal_type(X::'a,Y)) & a_truth(truthteller(X)) --> a_truth(truthteller(Y))) & \
3.2523 +\ (! X Y. a_truth(equal_type(X::'a,Y)) & a_truth(liar(X)) --> a_truth(liar(Y))) & \
3.2524 +\ (! X Y. a_truth(truthteller(X)) --> a_truth(equal_type(X::'a,Y)) | a_truth(liar(Y))) & \
3.2525 +\ (! X Y. a_truth(liar(X)) --> a_truth(equal_type(X::'a,Y)) | a_truth(truthteller(Y))) & \
3.2526 +\ (! Y X. a_truth(equal_type(X::'a,Y)) --> a_truth(equal_type(Y::'a,X))) & \
3.2527 +\ (! X Y. ask_1_if_2(X::'a,Y) & a_truth(truthteller(X)) & a_truth(Y) --> answer(yes)) & \
3.2528 +\ (! X Y. ask_1_if_2(X::'a,Y) & a_truth(truthteller(X)) --> a_truth(Y) | answer(no)) & \
3.2529 +\ (! X Y. ask_1_if_2(X::'a,Y) & a_truth(liar(X)) & a_truth(Y) --> answer(no)) & \
3.2530 +\ (! X Y. ask_1_if_2(X::'a,Y) & a_truth(liar(X)) --> a_truth(Y) | answer(yes)) & \
3.2531 +\ (people(b::'a,c,a)) & \
3.2532 +\ (people(a::'a,b,a)) & \
3.2533 +\ (people(a::'a,c,b)) & \
3.2534 +\ (people(c::'a,b,a)) & \
3.2535 +\ (a_truth(says(a::'a,equal_type(b::'a,c)))) & \
3.2536 +\ (ask_1_if_2(c::'a,equal_type(a::'a,b))) & \
3.2537 +\ (! Answer. ~answer(Answer)) --> False",
3.2538 + meson_tac);
3.2539 +
3.2540 +
3.2541 +(*621 inferences so far. Searching to depth 18. 0.2 secs*)
3.2542 +val PUZ029_1 = prove
3.2543 + ("(! X. dances_on_tightropes(X) | eats_pennybuns(X) | old(X)) & \
3.2544 +\ (! X. pig(X) & liable_to_giddiness(X) --> treated_with_respect(X)) & \
3.2545 +\ (! X. wise(X) & balloonist(X) --> has_umbrella(X)) & \
3.2546 +\ (! X. ~(looks_ridiculous(X) & eats_pennybuns(X) & eats_lunch_in_public(X))) & \
3.2547 +\ (! X. balloonist(X) & young(X) --> liable_to_giddiness(X)) & \
3.2548 +\ (! X. fat(X) & looks_ridiculous(X) --> dances_on_tightropes(X) | eats_lunch_in_public(X)) & \
3.2549 +\ (! X. ~(liable_to_giddiness(X) & wise(X) & dances_on_tightropes(X))) & \
3.2550 +\ (! X. pig(X) & has_umbrella(X) --> looks_ridiculous(X)) & \
3.2551 +\ (! X. treated_with_respect(X) --> dances_on_tightropes(X) | fat(X)) & \
3.2552 +\ (! X. young(X) | old(X)) & \
3.2553 +\ (! X. ~(young(X) & old(X))) & \
3.2554 +\ (wise(piggy)) & \
3.2555 +\ (young(piggy)) & \
3.2556 +\ (pig(piggy)) & \
3.2557 +\ (balloonist(piggy)) --> False",
3.2558 + meson_tac);
3.2559 +
3.2560 +(*93620 inferences so far. Searching to depth 24. 65.9 secs*)
3.2561 +val RNG001_3 = prove_hard
3.2562 + ("(! X. sum(additive_identity::'a,X,X)) & \
3.2563 +\ (! X. sum(additive_inverse(X),X,additive_identity)) & \
3.2564 +\ (! 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)) & \
3.2565 +\ (! 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)) & \
3.2566 +\ (! X Y. product(X::'a,Y,multiply(X::'a,Y))) & \
3.2567 +\ (! 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)) & \
3.2568 +\ (! 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)) & \
3.2569 +\ (~product(a::'a,additive_identity,additive_identity)) --> False",
3.2570 + meson_tac);
3.2571 +
3.2572 +
3.2573 +(****************SLOW
3.2574 +3057170 inferences so far. Searching to depth 16. No proof after 45 minutes.
3.2575 +val RNG001_5 = prove_hard
3.2576 + ("(! X. equal(X::'a,X)) & \
3.2577 +\ (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
3.2578 +\ (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) & \
3.2579 +\ (! X. sum(additive_identity::'a,X,X)) & \
3.2580 +\ (! X. sum(X::'a,additive_identity,X)) & \
3.2581 +\ (! X Y. product(X::'a,Y,multiply(X::'a,Y))) & \
3.2582 +\ (! X Y. sum(X::'a,Y,add(X::'a,Y))) & \
3.2583 +\ (! X. sum(additive_inverse(X),X,additive_identity)) & \
3.2584 +\ (! X. sum(X::'a,additive_inverse(X),additive_identity)) & \
3.2585 +\ (! 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)) & \
3.2586 +\ (! 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)) & \
3.2587 +\ (! Y X Z. sum(X::'a,Y,Z) --> sum(Y::'a,X,Z)) & \
3.2588 +\ (! 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)) & \
3.2589 +\ (! 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)) & \
3.2590 +\ (! 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)) & \
3.2591 +\ (! 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)) & \
3.2592 +\ (! 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)) & \
3.2593 +\ (! 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)) & \
3.2594 +\ (! X Y U V. sum(X::'a,Y,U) & sum(X::'a,Y,V) --> equal(U::'a,V)) & \
3.2595 +\ (! X Y U V. product(X::'a,Y,U) & product(X::'a,Y,V) --> equal(U::'a,V)) & \
3.2596 +\ (! X Y. equal(X::'a,Y) --> equal(additive_inverse(X),additive_inverse(Y))) & \
3.2597 +\ (! X Y W. equal(X::'a,Y) --> equal(add(X::'a,W),add(Y::'a,W))) & \
3.2598 +\ (! X Y W Z. equal(X::'a,Y) & sum(X::'a,W,Z) --> sum(Y::'a,W,Z)) & \
3.2599 +\ (! X W Y Z. equal(X::'a,Y) & sum(W::'a,X,Z) --> sum(W::'a,Y,Z)) & \
3.2600 +\ (! X W Z Y. equal(X::'a,Y) & sum(W::'a,Z,X) --> sum(W::'a,Z,Y)) & \
3.2601 +\ (! X Y W. equal(X::'a,Y) --> equal(multiply(X::'a,W),multiply(Y::'a,W))) & \
3.2602 +\ (! X Y W Z. equal(X::'a,Y) & product(X::'a,W,Z) --> product(Y::'a,W,Z)) & \
3.2603 +\ (! X W Y Z. equal(X::'a,Y) & product(W::'a,X,Z) --> product(W::'a,Y,Z)) & \
3.2604 +\ (! X W Z Y. equal(X::'a,Y) & product(W::'a,Z,X) --> product(W::'a,Z,Y)) & \
3.2605 +\ (~product(a::'a,additive_identity,additive_identity)) --> False",
3.2606 + meson_tac);
3.2607 +****************)
3.2608 +
3.2609 +(*0 inferences so far. Searching to depth 0. 0.5 secs*)
3.2610 +val RNG011_5 = prove
3.2611 + ("(! X. equal(X::'a,X)) & \
3.2612 +\ (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
3.2613 +\ (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) & \
3.2614 +\ (! A B C. equal(A::'a,B) --> equal(add(A::'a,C),add(B::'a,C))) & \
3.2615 +\ (! D F' E. equal(D::'a,E) --> equal(add(F'::'a,D),add(F'::'a,E))) & \
3.2616 +\ (! G H. equal(G::'a,H) --> equal(additive_inverse(G),additive_inverse(H))) & \
3.2617 +\ (! I' J K'. equal(I'::'a,J) --> equal(multiply(I'::'a,K'),multiply(J::'a,K'))) & \
3.2618 +\ (! L N M. equal(L::'a,M) --> equal(multiply(N::'a,L),multiply(N::'a,M))) & \
3.2619 +\ (! A B C D. equal(A::'a,B) --> equal(associator(A::'a,C,D),associator(B::'a,C,D))) & \
3.2620 +\ (! E G F' H. equal(E::'a,F') --> equal(associator(G::'a,E,H),associator(G::'a,F',H))) & \
3.2621 +\ (! I' K' L J. equal(I'::'a,J) --> equal(associator(K'::'a,L,I'),associator(K'::'a,L,J))) & \
3.2622 +\ (! M N O_. equal(M::'a,N) --> equal(commutator(M::'a,O_),commutator(N::'a,O_))) & \
3.2623 +\ (! P R Q. equal(P::'a,Q) --> equal(commutator(R::'a,P),commutator(R::'a,Q))) & \
3.2624 +\ (! Y X. equal(add(X::'a,Y),add(Y::'a,X))) & \
3.2625 +\ (! X Y Z. equal(add(add(X::'a,Y),Z),add(X::'a,add(Y::'a,Z)))) & \
3.2626 +\ (! X. equal(add(X::'a,additive_identity),X)) & \
3.2627 +\ (! X. equal(add(additive_identity::'a,X),X)) & \
3.2628 +\ (! X. equal(add(X::'a,additive_inverse(X)),additive_identity)) & \
3.2629 +\ (! X. equal(add(additive_inverse(X),X),additive_identity)) & \
3.2630 +\ (equal(additive_inverse(additive_identity),additive_identity)) & \
3.2631 +\ (! X Y. equal(add(X::'a,add(additive_inverse(X),Y)),Y)) & \
3.2632 +\ (! X Y. equal(additive_inverse(add(X::'a,Y)),add(additive_inverse(X),additive_inverse(Y)))) & \
3.2633 +\ (! X. equal(additive_inverse(additive_inverse(X)),X)) & \
3.2634 +\ (! X. equal(multiply(X::'a,additive_identity),additive_identity)) & \
3.2635 +\ (! X. equal(multiply(additive_identity::'a,X),additive_identity)) & \
3.2636 +\ (! X Y. equal(multiply(additive_inverse(X),additive_inverse(Y)),multiply(X::'a,Y))) & \
3.2637 +\ (! X Y. equal(multiply(X::'a,additive_inverse(Y)),additive_inverse(multiply(X::'a,Y)))) & \
3.2638 +\ (! X Y. equal(multiply(additive_inverse(X),Y),additive_inverse(multiply(X::'a,Y)))) & \
3.2639 +\ (! Y X Z. equal(multiply(X::'a,add(Y::'a,Z)),add(multiply(X::'a,Y),multiply(X::'a,Z)))) & \
3.2640 +\ (! X Y Z. equal(multiply(add(X::'a,Y),Z),add(multiply(X::'a,Z),multiply(Y::'a,Z)))) & \
3.2641 +\ (! X Y. equal(multiply(multiply(X::'a,Y),Y),multiply(X::'a,multiply(Y::'a,Y)))) & \
3.2642 +\ (! 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)))))) & \
3.2643 +\ (! X Y. equal(commutator(X::'a,Y),add(multiply(Y::'a,X),additive_inverse(multiply(X::'a,Y))))) & \
3.2644 +\ (! X Y. equal(multiply(multiply(associator(X::'a,X,Y),X),associator(X::'a,X,Y)),additive_identity)) & \
3.2645 +\ (~equal(multiply(multiply(associator(a::'a,a,b),a),associator(a::'a,a,b)),additive_identity)) --> False",
3.2646 + meson_tac);
3.2647 +
3.2648 +(*202 inferences so far. Searching to depth 8. 0.6 secs*)
3.2649 +val RNG023_6 = prove
3.2650 + ("(! X. equal(X::'a,X)) & \
3.2651 +\ (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
3.2652 +\ (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) & \
3.2653 +\ (! Y X. equal(add(X::'a,Y),add(Y::'a,X))) & \
3.2654 +\ (! X Y Z. equal(add(X::'a,add(Y::'a,Z)),add(add(X::'a,Y),Z))) & \
3.2655 +\ (! X. equal(add(additive_identity::'a,X),X)) & \
3.2656 +\ (! X. equal(add(X::'a,additive_identity),X)) & \
3.2657 +\ (! X. equal(multiply(additive_identity::'a,X),additive_identity)) & \
3.2658 +\ (! X. equal(multiply(X::'a,additive_identity),additive_identity)) & \
3.2659 +\ (! X. equal(add(additive_inverse(X),X),additive_identity)) & \
3.2660 +\ (! X. equal(add(X::'a,additive_inverse(X)),additive_identity)) & \
3.2661 +\ (! Y X Z. equal(multiply(X::'a,add(Y::'a,Z)),add(multiply(X::'a,Y),multiply(X::'a,Z)))) & \
3.2662 +\ (! X Y Z. equal(multiply(add(X::'a,Y),Z),add(multiply(X::'a,Z),multiply(Y::'a,Z)))) & \
3.2663 +\ (! X. equal(additive_inverse(additive_inverse(X)),X)) & \
3.2664 +\ (! X Y. equal(multiply(multiply(X::'a,Y),Y),multiply(X::'a,multiply(Y::'a,Y)))) & \
3.2665 +\ (! X Y. equal(multiply(multiply(X::'a,X),Y),multiply(X::'a,multiply(X::'a,Y)))) & \
3.2666 +\ (! 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)))))) & \
3.2667 +\ (! X Y. equal(commutator(X::'a,Y),add(multiply(Y::'a,X),additive_inverse(multiply(X::'a,Y))))) & \
3.2668 +\ (! D E F'. equal(D::'a,E) --> equal(add(D::'a,F'),add(E::'a,F'))) & \
3.2669 +\ (! G I' H. equal(G::'a,H) --> equal(add(I'::'a,G),add(I'::'a,H))) & \
3.2670 +\ (! J K'. equal(J::'a,K') --> equal(additive_inverse(J),additive_inverse(K'))) & \
3.2671 +\ (! L M N O_. equal(L::'a,M) --> equal(associator(L::'a,N,O_),associator(M::'a,N,O_))) & \
3.2672 +\ (! P R Q S'. equal(P::'a,Q) --> equal(associator(R::'a,P,S'),associator(R::'a,Q,S'))) & \
3.2673 +\ (! T' V W U. equal(T'::'a,U) --> equal(associator(V::'a,W,T'),associator(V::'a,W,U))) & \
3.2674 +\ (! X Y Z. equal(X::'a,Y) --> equal(commutator(X::'a,Z),commutator(Y::'a,Z))) & \
3.2675 +\ (! A1 C1 B1. equal(A1::'a,B1) --> equal(commutator(C1::'a,A1),commutator(C1::'a,B1))) & \
3.2676 +\ (! D1 E1 F1. equal(D1::'a,E1) --> equal(multiply(D1::'a,F1),multiply(E1::'a,F1))) & \
3.2677 +\ (! G1 I1 H1. equal(G1::'a,H1) --> equal(multiply(I1::'a,G1),multiply(I1::'a,H1))) & \
3.2678 +\ (~equal(associator(x::'a,x,y),additive_identity)) --> False",
3.2679 + meson_tac);
3.2680 +
3.2681 +(*0 inferences so far. Searching to depth 0. 0.6 secs*)
3.2682 +val RNG028_2 = prove
3.2683 + ("(! X. equal(X::'a,X)) & \
3.2684 +\ (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
3.2685 +\ (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) & \
3.2686 +\ (! X. equal(add(additive_identity::'a,X),X)) & \
3.2687 +\ (! X. equal(multiply(additive_identity::'a,X),additive_identity)) & \
3.2688 +\ (! X. equal(multiply(X::'a,additive_identity),additive_identity)) & \
3.2689 +\ (! X. equal(add(additive_inverse(X),X),additive_identity)) & \
3.2690 +\ (! X Y. equal(additive_inverse(add(X::'a,Y)),add(additive_inverse(X),additive_inverse(Y)))) & \
3.2691 +\ (! X. equal(additive_inverse(additive_inverse(X)),X)) & \
3.2692 +\ (! Y X Z. equal(multiply(X::'a,add(Y::'a,Z)),add(multiply(X::'a,Y),multiply(X::'a,Z)))) & \
3.2693 +\ (! X Y Z. equal(multiply(add(X::'a,Y),Z),add(multiply(X::'a,Z),multiply(Y::'a,Z)))) & \
3.2694 +\ (! X Y. equal(multiply(multiply(X::'a,Y),Y),multiply(X::'a,multiply(Y::'a,Y)))) & \
3.2695 +\ (! X Y. equal(multiply(multiply(X::'a,X),Y),multiply(X::'a,multiply(X::'a,Y)))) & \
3.2696 +\ (! X Y. equal(multiply(additive_inverse(X),Y),additive_inverse(multiply(X::'a,Y)))) & \
3.2697 +\ (! X Y. equal(multiply(X::'a,additive_inverse(Y)),additive_inverse(multiply(X::'a,Y)))) & \
3.2698 +\ (equal(additive_inverse(additive_identity),additive_identity)) & \
3.2699 +\ (! Y X. equal(add(X::'a,Y),add(Y::'a,X))) & \
3.2700 +\ (! X Y Z. equal(add(X::'a,add(Y::'a,Z)),add(add(X::'a,Y),Z))) & \
3.2701 +\ (! Z X Y. equal(add(X::'a,Z),add(Y::'a,Z)) --> equal(X::'a,Y)) & \
3.2702 +\ (! Z X Y. equal(add(Z::'a,X),add(Z::'a,Y)) --> equal(X::'a,Y)) & \
3.2703 +\ (! D E F'. equal(D::'a,E) --> equal(add(D::'a,F'),add(E::'a,F'))) & \
3.2704 +\ (! G I' H. equal(G::'a,H) --> equal(add(I'::'a,G),add(I'::'a,H))) & \
3.2705 +\ (! J K'. equal(J::'a,K') --> equal(additive_inverse(J),additive_inverse(K'))) & \
3.2706 +\ (! D1 E1 F1. equal(D1::'a,E1) --> equal(multiply(D1::'a,F1),multiply(E1::'a,F1))) & \
3.2707 +\ (! G1 I1 H1. equal(G1::'a,H1) --> equal(multiply(I1::'a,G1),multiply(I1::'a,H1))) & \
3.2708 +\ (! 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)))))) & \
3.2709 +\ (! L M N O_. equal(L::'a,M) --> equal(associator(L::'a,N,O_),associator(M::'a,N,O_))) & \
3.2710 +\ (! P R Q S'. equal(P::'a,Q) --> equal(associator(R::'a,P,S'),associator(R::'a,Q,S'))) & \
3.2711 +\ (! T' V W U. equal(T'::'a,U) --> equal(associator(V::'a,W,T'),associator(V::'a,W,U))) & \
3.2712 +\ (! X Y. ~equal(multiply(multiply(Y::'a,X),Y),multiply(Y::'a,multiply(X::'a,Y)))) & \
3.2713 +\ (! X Y Z. ~equal(associator(Y::'a,X,Z),additive_inverse(associator(X::'a,Y,Z)))) & \
3.2714 +\ (! X Y Z. ~equal(associator(Z::'a,Y,X),additive_inverse(associator(X::'a,Y,Z)))) & \
3.2715 +\ (~equal(multiply(multiply(cx::'a,multiply(cy::'a,cx)),cz),multiply(cx::'a,multiply(cy::'a,multiply(cx::'a,cz))))) --> False",
3.2716 + meson_tac);
3.2717 +
3.2718 +(*209 inferences so far. Searching to depth 9. 1.2 secs*)
3.2719 +val RNG038_2 = prove
3.2720 + ("(! X. sum(X::'a,additive_identity,X)) & \
3.2721 +\ (! X Y. product(X::'a,Y,multiply(X::'a,Y))) & \
3.2722 +\ (! X Y. sum(X::'a,Y,add(X::'a,Y))) & \
3.2723 +\ (! X. sum(X::'a,additive_inverse(X),additive_identity)) & \
3.2724 +\ (! 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)) & \
3.2725 +\ (! 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)) & \
3.2726 +\ (! Y X Z. sum(X::'a,Y,Z) --> sum(Y::'a,X,Z)) & \
3.2727 +\ (! 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)) & \
3.2728 +\ (! 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)) & \
3.2729 +\ (! 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)) & \
3.2730 +\ (! 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)) & \
3.2731 +\ (! 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)) & \
3.2732 +\ (! 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)) & \
3.2733 +\ (! X Y U V. sum(X::'a,Y,U) & sum(X::'a,Y,V) --> equal(U::'a,V)) & \
3.2734 +\ (! X Y U V. product(X::'a,Y,U) & product(X::'a,Y,V) --> equal(U::'a,V)) & \
3.2735 +\ (! X Y. equal(X::'a,Y) --> equal(additive_inverse(X),additive_inverse(Y))) & \
3.2736 +\ (! X Y W. equal(X::'a,Y) --> equal(add(X::'a,W),add(Y::'a,W))) & \
3.2737 +\ (! X Y W Z. equal(X::'a,Y) & sum(X::'a,W,Z) --> sum(Y::'a,W,Z)) & \
3.2738 +\ (! X W Y Z. equal(X::'a,Y) & sum(W::'a,X,Z) --> sum(W::'a,Y,Z)) & \
3.2739 +\ (! X W Z Y. equal(X::'a,Y) & sum(W::'a,Z,X) --> sum(W::'a,Z,Y)) & \
3.2740 +\ (! X Y W. equal(X::'a,Y) --> equal(multiply(X::'a,W),multiply(Y::'a,W))) & \
3.2741 +\ (! X Y W Z. equal(X::'a,Y) & product(X::'a,W,Z) --> product(Y::'a,W,Z)) & \
3.2742 +\ (! X W Y Z. equal(X::'a,Y) & product(W::'a,X,Z) --> product(W::'a,Y,Z)) & \
3.2743 +\ (! X W Z Y. equal(X::'a,Y) & product(W::'a,Z,X) --> product(W::'a,Z,Y)) & \
3.2744 +\ (! X. product(additive_identity::'a,X,additive_identity)) & \
3.2745 +\ (! X. product(X::'a,additive_identity,additive_identity)) & \
3.2746 +\ (! X Y. equal(X::'a,additive_identity) --> product(X::'a,h(X::'a,Y),Y)) & \
3.2747 +\ (product(a::'a,b,additive_identity)) & \
3.2748 +\ (~equal(a::'a,additive_identity)) & \
3.2749 +\ (~equal(b::'a,additive_identity)) --> False",
3.2750 + meson_tac);
3.2751 +
3.2752 +(*2660 inferences so far. Searching to depth 10. 7.0 secs*)
3.2753 +val RNG040_2 = prove_hard
3.2754 + ("(! X. equal(X::'a,X)) & \
3.2755 +\ (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
3.2756 +\ (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) & \
3.2757 +\ (! X Y. equal(X::'a,Y) --> equal(additive_inverse(X),additive_inverse(Y))) & \
3.2758 +\ (! X Y W. equal(X::'a,Y) --> equal(add(X::'a,W),add(Y::'a,W))) & \
3.2759 +\ (! X W Y. equal(X::'a,Y) --> equal(add(W::'a,X),add(W::'a,Y))) & \
3.2760 +\ (! X Y W Z. equal(X::'a,Y) & sum(X::'a,W,Z) --> sum(Y::'a,W,Z)) & \
3.2761 +\ (! X W Y Z. equal(X::'a,Y) & sum(W::'a,X,Z) --> sum(W::'a,Y,Z)) & \
3.2762 +\ (! X W Z Y. equal(X::'a,Y) & sum(W::'a,Z,X) --> sum(W::'a,Z,Y)) & \
3.2763 +\ (! X Y W. equal(X::'a,Y) --> equal(multiply(X::'a,W),multiply(Y::'a,W))) & \
3.2764 +\ (! X W Y. equal(X::'a,Y) --> equal(multiply(W::'a,X),multiply(W::'a,Y))) & \
3.2765 +\ (! X Y W Z. equal(X::'a,Y) & product(X::'a,W,Z) --> product(Y::'a,W,Z)) & \
3.2766 +\ (! X W Y Z. equal(X::'a,Y) & product(W::'a,X,Z) --> product(W::'a,Y,Z)) & \
3.2767 +\ (! X W Z Y. equal(X::'a,Y) & product(W::'a,Z,X) --> product(W::'a,Z,Y)) & \
3.2768 +\ (! X. sum(additive_identity::'a,X,X)) & \
3.2769 +\ (! X. sum(X::'a,additive_identity,X)) & \
3.2770 +\ (! X Y. product(X::'a,Y,multiply(X::'a,Y))) & \
3.2771 +\ (! X Y. sum(X::'a,Y,add(X::'a,Y))) & \
3.2772 +\ (! X. sum(additive_inverse(X),X,additive_identity)) & \
3.2773 +\ (! X. sum(X::'a,additive_inverse(X),additive_identity)) & \
3.2774 +\ (! 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)) & \
3.2775 +\ (! 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)) & \
3.2776 +\ (! Y X Z. sum(X::'a,Y,Z) --> sum(Y::'a,X,Z)) & \
3.2777 +\ (! 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)) & \
3.2778 +\ (! 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)) & \
3.2779 +\ (! 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)) & \
3.2780 +\ (! 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)) & \
3.2781 +\ (! X Y U V. sum(X::'a,Y,U) & sum(X::'a,Y,V) --> equal(U::'a,V)) & \
3.2782 +\ (! X Y U V. product(X::'a,Y,U) & product(X::'a,Y,V) --> equal(U::'a,V)) & \
3.2783 +\ (! A. product(A::'a,multiplicative_identity,A)) & \
3.2784 +\ (! A. product(multiplicative_identity::'a,A,A)) & \
3.2785 +\ (! A. product(A::'a,h(A),multiplicative_identity) | equal(A::'a,additive_identity)) & \
3.2786 +\ (! A. product(h(A),A,multiplicative_identity) | equal(A::'a,additive_identity)) & \
3.2787 +\ (! B A C. product(A::'a,B,C) --> product(B::'a,A,C)) & \
3.2788 +\ (! A B. equal(A::'a,B) --> equal(h(A),h(B))) & \
3.2789 +\ (sum(b::'a,c,d)) & \
3.2790 +\ (product(d::'a,a,additive_identity)) & \
3.2791 +\ (product(b::'a,a,l)) & \
3.2792 +\ (product(c::'a,a,n)) & \
3.2793 +\ (~sum(l::'a,n,additive_identity)) --> False",
3.2794 + meson_tac);
3.2795 +
3.2796 +(*8991 inferences so far. Searching to depth 9. 22.2 secs*)
3.2797 +val RNG041_1 = prove_hard
3.2798 + ("(! X. equal(X::'a,X)) & \
3.2799 +\ (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
3.2800 +\ (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) & \
3.2801 +\ (! X. sum(additive_identity::'a,X,X)) & \
3.2802 +\ (! X. sum(X::'a,additive_identity,X)) & \
3.2803 +\ (! X Y. product(X::'a,Y,multiply(X::'a,Y))) & \
3.2804 +\ (! X Y. sum(X::'a,Y,add(X::'a,Y))) & \
3.2805 +\ (! X. sum(additive_inverse(X),X,additive_identity)) & \
3.2806 +\ (! X. sum(X::'a,additive_inverse(X),additive_identity)) & \
3.2807 +\ (! 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)) & \
3.2808 +\ (! 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)) & \
3.2809 +\ (! Y X Z. sum(X::'a,Y,Z) --> sum(Y::'a,X,Z)) & \
3.2810 +\ (! 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)) & \
3.2811 +\ (! 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)) & \
3.2812 +\ (! 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)) & \
3.2813 +\ (! 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)) & \
3.2814 +\ (! 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)) & \
3.2815 +\ (! 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)) & \
3.2816 +\ (! X Y U V. sum(X::'a,Y,U) & sum(X::'a,Y,V) --> equal(U::'a,V)) & \
3.2817 +\ (! X Y U V. product(X::'a,Y,U) & product(X::'a,Y,V) --> equal(U::'a,V)) & \
3.2818 +\ (! X Y. equal(X::'a,Y) --> equal(additive_inverse(X),additive_inverse(Y))) & \
3.2819 +\ (! X Y W. equal(X::'a,Y) --> equal(add(X::'a,W),add(Y::'a,W))) & \
3.2820 +\ (! X W Y. equal(X::'a,Y) --> equal(add(W::'a,X),add(W::'a,Y))) & \
3.2821 +\ (! X Y W Z. equal(X::'a,Y) & sum(X::'a,W,Z) --> sum(Y::'a,W,Z)) & \
3.2822 +\ (! X W Y Z. equal(X::'a,Y) & sum(W::'a,X,Z) --> sum(W::'a,Y,Z)) & \
3.2823 +\ (! X W Z Y. equal(X::'a,Y) & sum(W::'a,Z,X) --> sum(W::'a,Z,Y)) & \
3.2824 +\ (! X Y W. equal(X::'a,Y) --> equal(multiply(X::'a,W),multiply(Y::'a,W))) & \
3.2825 +\ (! X W Y. equal(X::'a,Y) --> equal(multiply(W::'a,X),multiply(W::'a,Y))) & \
3.2826 +\ (! X Y W Z. equal(X::'a,Y) & product(X::'a,W,Z) --> product(Y::'a,W,Z)) & \
3.2827 +\ (! X W Y Z. equal(X::'a,Y) & product(W::'a,X,Z) --> product(W::'a,Y,Z)) & \
3.2828 +\ (! X W Z Y. equal(X::'a,Y) & product(W::'a,Z,X) --> product(W::'a,Z,Y)) & \
3.2829 +\ (! A B. equal(A::'a,B) --> equal(h(A),h(B))) & \
3.2830 +\ (! A. product(additive_identity::'a,A,additive_identity)) & \
3.2831 +\ (! A. product(A::'a,additive_identity,additive_identity)) & \
3.2832 +\ (! A. product(A::'a,multiplicative_identity,A)) & \
3.2833 +\ (! A. product(multiplicative_identity::'a,A,A)) & \
3.2834 +\ (! A. product(A::'a,h(A),multiplicative_identity) | equal(A::'a,additive_identity)) & \
3.2835 +\ (! A. product(h(A),A,multiplicative_identity) | equal(A::'a,additive_identity)) & \
3.2836 +\ (product(a::'a,b,additive_identity)) & \
3.2837 +\ (~equal(a::'a,additive_identity)) & \
3.2838 +\ (~equal(b::'a,additive_identity)) --> False",
3.2839 + meson_tac);
3.2840 +
3.2841 +(*101319 inferences so far. Searching to depth 14. 76.0 secs*)
3.2842 +val ROB010_1 = prove_hard
3.2843 + ("(! X. equal(X::'a,X)) & \
3.2844 +\ (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
3.2845 +\ (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) & \
3.2846 +\ (! Y X. equal(add(X::'a,Y),add(Y::'a,X))) & \
3.2847 +\ (! X Y Z. equal(add(add(X::'a,Y),Z),add(X::'a,add(Y::'a,Z)))) & \
3.2848 +\ (! Y X. equal(negate(add(negate(add(X::'a,Y)),negate(add(X::'a,negate(Y))))),X)) & \
3.2849 +\ (! A B C. equal(A::'a,B) --> equal(add(A::'a,C),add(B::'a,C))) & \
3.2850 +\ (! D F' E. equal(D::'a,E) --> equal(add(F'::'a,D),add(F'::'a,E))) & \
3.2851 +\ (! G H. equal(G::'a,H) --> equal(negate(G),negate(H))) & \
3.2852 +\ (equal(negate(add(a::'a,negate(b))),c)) & \
3.2853 +\ (~equal(negate(add(c::'a,negate(add(b::'a,a)))),a)) --> False",
3.2854 + meson_tac);
3.2855 +
3.2856 +
3.2857 +(*6933 inferences so far. Searching to depth 12. 5.1 secs*)
3.2858 +val ROB013_1 = prove_hard
3.2859 + ("(! X. equal(X::'a,X)) & \
3.2860 +\ (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
3.2861 +\ (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) & \
3.2862 +\ (! Y X. equal(add(X::'a,Y),add(Y::'a,X))) & \
3.2863 +\ (! X Y Z. equal(add(add(X::'a,Y),Z),add(X::'a,add(Y::'a,Z)))) & \
3.2864 +\ (! Y X. equal(negate(add(negate(add(X::'a,Y)),negate(add(X::'a,negate(Y))))),X)) & \
3.2865 +\ (! A B C. equal(A::'a,B) --> equal(add(A::'a,C),add(B::'a,C))) & \
3.2866 +\ (! D F' E. equal(D::'a,E) --> equal(add(F'::'a,D),add(F'::'a,E))) & \
3.2867 +\ (! G H. equal(G::'a,H) --> equal(negate(G),negate(H))) & \
3.2868 +\ (equal(negate(add(a::'a,b)),c)) & \
3.2869 +\ (~equal(negate(add(c::'a,negate(add(negate(b),a)))),a)) --> False",
3.2870 + meson_tac);
3.2871 +
3.2872 +(*6614 inferences so far. Searching to depth 11. 20.4 secs*)
3.2873 +val ROB016_1 = prove_hard
3.2874 + ("(! X. equal(X::'a,X)) & \
3.2875 +\ (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
3.2876 +\ (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) & \
3.2877 +\ (! Y X. equal(add(X::'a,Y),add(Y::'a,X))) & \
3.2878 +\ (! X Y Z. equal(add(add(X::'a,Y),Z),add(X::'a,add(Y::'a,Z)))) & \
3.2879 +\ (! Y X. equal(negate(add(negate(add(X::'a,Y)),negate(add(X::'a,negate(Y))))),X)) & \
3.2880 +\ (! A B C. equal(A::'a,B) --> equal(add(A::'a,C),add(B::'a,C))) & \
3.2881 +\ (! D F' E. equal(D::'a,E) --> equal(add(F'::'a,D),add(F'::'a,E))) & \
3.2882 +\ (! G H. equal(G::'a,H) --> equal(negate(G),negate(H))) & \
3.2883 +\ (! J K' L. equal(J::'a,K') --> equal(multiply(J::'a,L),multiply(K'::'a,L))) & \
3.2884 +\ (! M O_ N. equal(M::'a,N) --> equal(multiply(O_::'a,M),multiply(O_::'a,N))) & \
3.2885 +\ (! P Q. equal(P::'a,Q) --> equal(successor(P),successor(Q))) & \
3.2886 +\ (! R S'. equal(R::'a,S') & positive_integer(R) --> positive_integer(S')) & \
3.2887 +\ (! X. equal(multiply(one::'a,X),X)) & \
3.2888 +\ (! V X. positive_integer(X) --> equal(multiply(successor(V),X),add(X::'a,multiply(V::'a,X)))) & \
3.2889 +\ (positive_integer(one)) & \
3.2890 +\ (! X. positive_integer(X) --> positive_integer(successor(X))) & \
3.2891 +\ (equal(negate(add(d::'a,e)),negate(e))) & \
3.2892 +\ (positive_integer(k)) & \
3.2893 +\ (! 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))) & \
3.2894 +\ (~equal(negate(add(e::'a,multiply(k::'a,add(d::'a,negate(add(d::'a,negate(e))))))),negate(e))) --> False",
3.2895 + meson_tac);
3.2896 +
3.2897 +(*14077 inferences so far. Searching to depth 11. 32.8 secs*)
3.2898 +val ROB021_1 = prove_hard
3.2899 + ("(! X. equal(X::'a,X)) & \
3.2900 +\ (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
3.2901 +\ (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) & \
3.2902 +\ (! Y X. equal(add(X::'a,Y),add(Y::'a,X))) & \
3.2903 +\ (! X Y Z. equal(add(add(X::'a,Y),Z),add(X::'a,add(Y::'a,Z)))) & \
3.2904 +\ (! Y X. equal(negate(add(negate(add(X::'a,Y)),negate(add(X::'a,negate(Y))))),X)) & \
3.2905 +\ (! A B C. equal(A::'a,B) --> equal(add(A::'a,C),add(B::'a,C))) & \
3.2906 +\ (! D F' E. equal(D::'a,E) --> equal(add(F'::'a,D),add(F'::'a,E))) & \
3.2907 +\ (! G H. equal(G::'a,H) --> equal(negate(G),negate(H))) & \
3.2908 +\ (! X Y. equal(negate(X),negate(Y)) --> equal(X::'a,Y)) & \
3.2909 +\ (~equal(add(negate(add(a::'a,negate(b))),negate(add(negate(a),negate(b)))),b)) --> False",
3.2910 + meson_tac);
3.2911 +
3.2912 +(*35532 inferences so far. Searching to depth 19. 54.3 secs*)
3.2913 +val SET005_1 = prove_hard
3.2914 + ("(! Subset Element Superset. member(Element::'a,Subset) & subset(Subset::'a,Superset) --> member(Element::'a,Superset)) & \
3.2915 +\ (! Superset Subset. subset(Subset::'a,Superset) | member(member_of_1_not_of_2(Subset::'a,Superset),Subset)) & \
3.2916 +\ (! Subset Superset. member(member_of_1_not_of_2(Subset::'a,Superset),Superset) --> subset(Subset::'a,Superset)) & \
3.2917 +\ (! Subset Superset. equal_sets(Subset::'a,Superset) --> subset(Subset::'a,Superset)) & \
3.2918 +\ (! Subset Superset. equal_sets(Superset::'a,Subset) --> subset(Subset::'a,Superset)) & \
3.2919 +\ (! Set2 Set1. subset(Set1::'a,Set2) & subset(Set2::'a,Set1) --> equal_sets(Set2::'a,Set1)) & \
3.2920 +\ (! Set2 Intersection Element Set1. intersection(Set1::'a,Set2,Intersection) & member(Element::'a,Intersection) --> member(Element::'a,Set1)) & \
3.2921 +\ (! Set1 Intersection Element Set2. intersection(Set1::'a,Set2,Intersection) & member(Element::'a,Intersection) --> member(Element::'a,Set2)) & \
3.2922 +\ (! Set2 Set1 Element Intersection. intersection(Set1::'a,Set2,Intersection) & member(Element::'a,Set2) & member(Element::'a,Set1) --> member(Element::'a,Intersection)) & \
3.2923 +\ (! Set2 Intersection Set1. member(h(Set1::'a,Set2,Intersection),Intersection) | intersection(Set1::'a,Set2,Intersection) | member(h(Set1::'a,Set2,Intersection),Set1)) & \
3.2924 +\ (! Set1 Intersection Set2. member(h(Set1::'a,Set2,Intersection),Intersection) | intersection(Set1::'a,Set2,Intersection) | member(h(Set1::'a,Set2,Intersection),Set2)) & \
3.2925 +\ (! 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)) & \
3.2926 +\ (intersection(a::'a,b,aIb)) & \
3.2927 +\ (intersection(b::'a,c,bIc)) & \
3.2928 +\ (intersection(a::'a,bIc,aIbIc)) & \
3.2929 +\ (~intersection(aIb::'a,c,aIbIc)) --> False",
3.2930 + meson_tac);
3.2931 +
3.2932 +
3.2933 +(*6450 inferences so far. Searching to depth 14. 4.2 secs*)
3.2934 +val SET009_1 = prove_hard
3.2935 + ("(! Subset Element Superset. member(Element::'a,Subset) & subset(Subset::'a,Superset) --> member(Element::'a,Superset)) & \
3.2936 +\ (! Superset Subset. subset(Subset::'a,Superset) | member(member_of_1_not_of_2(Subset::'a,Superset),Subset)) & \
3.2937 +\ (! Subset Superset. member(member_of_1_not_of_2(Subset::'a,Superset),Superset) --> subset(Subset::'a,Superset)) & \
3.2938 +\ (! Subset Superset. equal_sets(Subset::'a,Superset) --> subset(Subset::'a,Superset)) & \
3.2939 +\ (! Subset Superset. equal_sets(Superset::'a,Subset) --> subset(Subset::'a,Superset)) & \
3.2940 +\ (! Set2 Set1. subset(Set1::'a,Set2) & subset(Set2::'a,Set1) --> equal_sets(Set2::'a,Set1)) & \
3.2941 +\ (! Set2 Difference Element Set1. difference(Set1::'a,Set2,Difference) & member(Element::'a,Difference) --> member(Element::'a,Set1)) & \
3.2942 +\ (! Element A_set Set1 Set2. ~(member(Element::'a,Set1) & member(Element::'a,Set2) & difference(A_set::'a,Set1,Set2))) & \
3.2943 +\ (! Set1 Difference Element Set2. member(Element::'a,Set1) & difference(Set1::'a,Set2,Difference) --> member(Element::'a,Difference) | member(Element::'a,Set2)) & \
3.2944 +\ (! Set1 Set2 Difference. difference(Set1::'a,Set2,Difference) | member(k(Set1::'a,Set2,Difference),Set1) | member(k(Set1::'a,Set2,Difference),Difference)) & \
3.2945 +\ (! Set1 Set2 Difference. member(k(Set1::'a,Set2,Difference),Set2) --> member(k(Set1::'a,Set2,Difference),Difference) | difference(Set1::'a,Set2,Difference)) & \
3.2946 +\ (! 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)) & \
3.2947 +\ (subset(d::'a,a)) & \
3.2948 +\ (difference(b::'a,a,bDa)) & \
3.2949 +\ (difference(b::'a,d,bDd)) & \
3.2950 +\ (~subset(bDa::'a,bDd)) --> False",
3.2951 + meson_tac);
3.2952 +
3.2953 +(*34726 inferences so far. Searching to depth 6. 2420 secs: 40 mins! BIG*)
3.2954 +val SET025_4 = prove_hard
3.2955 + ("(! X. equal(X::'a,X)) & \
3.2956 +\ (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
3.2957 +\ (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) & \
3.2958 +\ (! Y X. member(X::'a,Y) --> little_set(X)) & \
3.2959 +\ (! X Y. little_set(f1(X::'a,Y)) | equal(X::'a,Y)) & \
3.2960 +\ (! X Y. member(f1(X::'a,Y),X) | member(f1(X::'a,Y),Y) | equal(X::'a,Y)) & \
3.2961 +\ (! X Y. member(f1(X::'a,Y),X) & member(f1(X::'a,Y),Y) --> equal(X::'a,Y)) & \
3.2962 +\ (! X U Y. member(U::'a,non_ordered_pair(X::'a,Y)) --> equal(U::'a,X) | equal(U::'a,Y)) & \
3.2963 +\ (! Y U X. little_set(U) & equal(U::'a,X) --> member(U::'a,non_ordered_pair(X::'a,Y))) & \
3.2964 +\ (! X U Y. little_set(U) & equal(U::'a,Y) --> member(U::'a,non_ordered_pair(X::'a,Y))) & \
3.2965 +\ (! X Y. little_set(non_ordered_pair(X::'a,Y))) & \
3.2966 +\ (! X. equal(singleton_set(X),non_ordered_pair(X::'a,X))) & \
3.2967 +\ (! X Y. equal(ordered_pair(X::'a,Y),non_ordered_pair(singleton_set(X),non_ordered_pair(X::'a,Y)))) & \
3.2968 +\ (! X. ordered_pair_predicate(X) --> little_set(f2(X))) & \
3.2969 +\ (! X. ordered_pair_predicate(X) --> little_set(f3(X))) & \
3.2970 +\ (! X. ordered_pair_predicate(X) --> equal(X::'a,ordered_pair(f2(X),f3(X)))) & \
3.2971 +\ (! X Y Z. little_set(Y) & little_set(Z) & equal(X::'a,ordered_pair(Y::'a,Z)) --> ordered_pair_predicate(X)) & \
3.2972 +\ (! Z X. member(Z::'a,first(X)) --> little_set(f4(Z::'a,X))) & \
3.2973 +\ (! Z X. member(Z::'a,first(X)) --> little_set(f5(Z::'a,X))) & \
3.2974 +\ (! Z X. member(Z::'a,first(X)) --> equal(X::'a,ordered_pair(f4(Z::'a,X),f5(Z::'a,X)))) & \
3.2975 +\ (! Z X. member(Z::'a,first(X)) --> member(Z::'a,f4(Z::'a,X))) & \
3.2976 +\ (! 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))) & \
3.2977 +\ (! Z X. member(Z::'a,second(X)) --> little_set(f6(Z::'a,X))) & \
3.2978 +\ (! Z X. member(Z::'a,second(X)) --> little_set(f7(Z::'a,X))) & \
3.2979 +\ (! Z X. member(Z::'a,second(X)) --> equal(X::'a,ordered_pair(f6(Z::'a,X),f7(Z::'a,X)))) & \
3.2980 +\ (! Z X. member(Z::'a,second(X)) --> member(Z::'a,f7(Z::'a,X))) & \
3.2981 +\ (! 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))) & \
3.2982 +\ (! Z. member(Z::'a,estin) --> ordered_pair_predicate(Z)) & \
3.2983 +\ (! Z. member(Z::'a,estin) --> member(first(Z),second(Z))) & \
3.2984 +\ (! Z. little_set(Z) & ordered_pair_predicate(Z) & member(first(Z),second(Z)) --> member(Z::'a,estin)) & \
3.2985 +\ (! Y Z X. member(Z::'a,intersection(X::'a,Y)) --> member(Z::'a,X)) & \
3.2986 +\ (! X Z Y. member(Z::'a,intersection(X::'a,Y)) --> member(Z::'a,Y)) & \
3.2987 +\ (! X Z Y. member(Z::'a,X) & member(Z::'a,Y) --> member(Z::'a,intersection(X::'a,Y))) & \
3.2988 +\ (! Z X. ~(member(Z::'a,complement(X)) & member(Z::'a,X))) & \
3.2989 +\ (! Z X. little_set(Z) --> member(Z::'a,complement(X)) | member(Z::'a,X)) & \
3.2990 +\ (! X Y. equal(union(X::'a,Y),complement(intersection(complement(X),complement(Y))))) & \
3.2991 +\ (! Z X. member(Z::'a,domain_of(X)) --> ordered_pair_predicate(f8(Z::'a,X))) & \
3.2992 +\ (! Z X. member(Z::'a,domain_of(X)) --> member(f8(Z::'a,X),X)) & \
3.2993 +\ (! Z X. member(Z::'a,domain_of(X)) --> equal(Z::'a,first(f8(Z::'a,X)))) & \
3.2994 +\ (! 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))) & \
3.2995 +\ (! X Y Z. member(Z::'a,cross_product(X::'a,Y)) --> ordered_pair_predicate(Z)) & \
3.2996 +\ (! Y Z X. member(Z::'a,cross_product(X::'a,Y)) --> member(first(Z),X)) & \
3.2997 +\ (! X Z Y. member(Z::'a,cross_product(X::'a,Y)) --> member(second(Z),Y)) & \
3.2998 +\ (! 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))) & \
3.2999 +\ (! X Z. member(Z::'a,inv1 X) --> ordered_pair_predicate(Z)) & \
3.3000 +\ (! Z X. member(Z::'a,inv1 X) --> member(ordered_pair(second(Z),first(Z)),X)) & \
3.3001 +\ (! Z X. little_set(Z) & ordered_pair_predicate(Z) & member(ordered_pair(second(Z),first(Z)),X) --> member(Z::'a,inv1 X)) & \
3.3002 +\ (! Z X. member(Z::'a,rotate_right(X)) --> little_set(f9(Z::'a,X))) & \
3.3003 +\ (! Z X. member(Z::'a,rotate_right(X)) --> little_set(f10(Z::'a,X))) & \
3.3004 +\ (! Z X. member(Z::'a,rotate_right(X)) --> little_set(f11(Z::'a,X))) & \
3.3005 +\ (! Z X. member(Z::'a,rotate_right(X)) --> equal(Z::'a,ordered_pair(f9(Z::'a,X),ordered_pair(f10(Z::'a,X),f11(Z::'a,X))))) & \
3.3006 +\ (! Z X. member(Z::'a,rotate_right(X)) --> member(ordered_pair(f10(Z::'a,X),ordered_pair(f11(Z::'a,X),f9(Z::'a,X))),X)) & \
3.3007 +\ (! 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,rotate_right(X))) & \
3.3008 +\ (! Z X. member(Z::'a,flip_range_of(X)) --> little_set(f12(Z::'a,X))) & \
3.3009 +\ (! Z X. member(Z::'a,flip_range_of(X)) --> little_set(f13(Z::'a,X))) & \
3.3010 +\ (! Z X. member(Z::'a,flip_range_of(X)) --> little_set(f14(Z::'a,X))) & \
3.3011 +\ (! 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))))) & \
3.3012 +\ (! 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)) & \
3.3013 +\ (! 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))) & \
3.3014 +\ (! X. equal(successor(X),union(X::'a,singleton_set(X)))) & \
3.3015 +\ (! Z. ~member(Z::'a,empty_set)) & \
3.3016 +\ (! Z. little_set(Z) --> member(Z::'a,universal_set)) & \
3.3017 +\ (little_set(infinity)) & \
3.3018 +\ (member(empty_set::'a,infinity)) & \
3.3019 +\ (! X. member(X::'a,infinity) --> member(successor(X),infinity)) & \
3.3020 +\ (! Z X. member(Z::'a,sigma(X)) --> member(f16(Z::'a,X),X)) & \
3.3021 +\ (! Z X. member(Z::'a,sigma(X)) --> member(Z::'a,f16(Z::'a,X))) & \
3.3022 +\ (! X Z Y. member(Y::'a,X) & member(Z::'a,Y) --> member(Z::'a,sigma(X))) & \
3.3023 +\ (! U. little_set(U) --> little_set(sigma(U))) & \
3.3024 +\ (! X U Y. subset(X::'a,Y) & member(U::'a,X) --> member(U::'a,Y)) & \
3.3025 +\ (! Y X. subset(X::'a,Y) | member(f17(X::'a,Y),X)) & \
3.3026 +\ (! X Y. member(f17(X::'a,Y),Y) --> subset(X::'a,Y)) & \
3.3027 +\ (! X Y. proper_subset(X::'a,Y) --> subset(X::'a,Y)) & \
3.3028 +\ (! X Y. ~(proper_subset(X::'a,Y) & equal(X::'a,Y))) & \
3.3029 +\ (! X Y. subset(X::'a,Y) --> proper_subset(X::'a,Y) | equal(X::'a,Y)) & \
3.3030 +\ (! Z X. member(Z::'a,powerset(X)) --> subset(Z::'a,X)) & \
3.3031 +\ (! Z X. little_set(Z) & subset(Z::'a,X) --> member(Z::'a,powerset(X))) & \
3.3032 +\ (! U. little_set(U) --> little_set(powerset(U))) & \
3.3033 +\ (! Z X. relation(Z) & member(X::'a,Z) --> ordered_pair_predicate(X)) & \
3.3034 +\ (! Z. relation(Z) | member(f18(Z),Z)) & \
3.3035 +\ (! Z. ordered_pair_predicate(f18(Z)) --> relation(Z)) & \
3.3036 +\ (! 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)) & \
3.3037 +\ (! X. single_valued_set(X) | little_set(f19(X))) & \
3.3038 +\ (! X. single_valued_set(X) | little_set(f20(X))) & \
3.3039 +\ (! X. single_valued_set(X) | little_set(f21(X))) & \
3.3040 +\ (! X. single_valued_set(X) | member(ordered_pair(f19(X),f20(X)),X)) & \
3.3041 +\ (! X. single_valued_set(X) | member(ordered_pair(f19(X),f21(X)),X)) & \
3.3042 +\ (! X. equal(f20(X),f21(X)) --> single_valued_set(X)) & \
3.3043 +\ (! Xf. function(Xf) --> relation(Xf)) & \
3.3044 +\ (! Xf. function(Xf) --> single_valued_set(Xf)) & \
3.3045 +\ (! Xf. relation(Xf) & single_valued_set(Xf) --> function(Xf)) & \
3.3046 +\ (! Z X Xf. member(Z::'a,image_(X::'a,Xf)) --> ordered_pair_predicate(f22(Z::'a,X,Xf))) & \
3.3047 +\ (! Z X Xf. member(Z::'a,image_(X::'a,Xf)) --> member(f22(Z::'a,X,Xf),Xf)) & \
3.3048 +\ (! Z Xf X. member(Z::'a,image_(X::'a,Xf)) --> member(first(f22(Z::'a,X,Xf)),X)) & \
3.3049 +\ (! X Xf Z. member(Z::'a,image_(X::'a,Xf)) --> equal(second(f22(Z::'a,X,Xf)),Z)) & \
3.3050 +\ (! 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))) & \
3.3051 +\ (! X Xf. little_set(X) & function(Xf) --> little_set(image_(X::'a,Xf))) & \
3.3052 +\ (! X U Y. ~(disjoint(X::'a,Y) & member(U::'a,X) & member(U::'a,Y))) & \
3.3053 +\ (! Y X. disjoint(X::'a,Y) | member(f23(X::'a,Y),X)) & \
3.3054 +\ (! X Y. disjoint(X::'a,Y) | member(f23(X::'a,Y),Y)) & \
3.3055 +\ (! X. equal(X::'a,empty_set) | member(f24(X),X)) & \
3.3056 +\ (! X. equal(X::'a,empty_set) | disjoint(f24(X),X)) & \
3.3057 +\ (function(f25)) & \
3.3058 +\ (! X. little_set(X) --> equal(X::'a,empty_set) | member(f26(X),X)) & \
3.3059 +\ (! X. little_set(X) --> equal(X::'a,empty_set) | member(ordered_pair(X::'a,f26(X)),f25)) & \
3.3060 +\ (! Z X. member(Z::'a,range_of(X)) --> ordered_pair_predicate(f27(Z::'a,X))) & \
3.3061 +\ (! Z X. member(Z::'a,range_of(X)) --> member(f27(Z::'a,X),X)) & \
3.3062 +\ (! Z X. member(Z::'a,range_of(X)) --> equal(Z::'a,second(f27(Z::'a,X)))) & \
3.3063 +\ (! 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))) & \
3.3064 +\ (! Z. member(Z::'a,identity_relation) --> ordered_pair_predicate(Z)) & \
3.3065 +\ (! Z. member(Z::'a,identity_relation) --> equal(first(Z),second(Z))) & \
3.3066 +\ (! Z. little_set(Z) & ordered_pair_predicate(Z) & equal(first(Z),second(Z)) --> member(Z::'a,identity_relation)) & \
3.3067 +\ (! X Y. equal(restrct(X::'a,Y),intersection(X::'a,cross_product(Y::'a,universal_set)))) & \
3.3068 +\ (! Xf. one_to_one_function(Xf) --> function(Xf)) & \
3.3069 +\ (! Xf. one_to_one_function(Xf) --> function(inv1 Xf)) & \
3.3070 +\ (! Xf. function(Xf) & function(inv1 Xf) --> one_to_one_function(Xf)) & \
3.3071 +\ (! Z Xf Y. member(Z::'a,apply(Xf::'a,Y)) --> ordered_pair_predicate(f28(Z::'a,Xf,Y))) & \
3.3072 +\ (! Z Y Xf. member(Z::'a,apply(Xf::'a,Y)) --> member(f28(Z::'a,Xf,Y),Xf)) & \
3.3073 +\ (! Z Xf Y. member(Z::'a,apply(Xf::'a,Y)) --> equal(first(f28(Z::'a,Xf,Y)),Y)) & \
3.3074 +\ (! Z Xf Y. member(Z::'a,apply(Xf::'a,Y)) --> member(Z::'a,second(f28(Z::'a,Xf,Y)))) & \
3.3075 +\ (! 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))) & \
3.3076 +\ (! Xf X Y. equal(apply_to_two_arguments(Xf::'a,X,Y),apply(Xf::'a,ordered_pair(X::'a,Y)))) & \
3.3077 +\ (! X Y Xf. maps(Xf::'a,X,Y) --> function(Xf)) & \
3.3078 +\ (! Y Xf X. maps(Xf::'a,X,Y) --> equal(domain_of(Xf),X)) & \
3.3079 +\ (! X Xf Y. maps(Xf::'a,X,Y) --> subset(range_of(Xf),Y)) & \
3.3080 +\ (! X Xf Y. function(Xf) & equal(domain_of(Xf),X) & subset(range_of(Xf),Y) --> maps(Xf::'a,X,Y)) & \
3.3081 +\ (! Xf Xs. closed(Xs::'a,Xf) --> little_set(Xs)) & \
3.3082 +\ (! Xs Xf. closed(Xs::'a,Xf) --> little_set(Xf)) & \
3.3083 +\ (! Xf Xs. closed(Xs::'a,Xf) --> maps(Xf::'a,cross_product(Xs::'a,Xs),Xs)) & \
3.3084 +\ (! Xf Xs. little_set(Xs) & little_set(Xf) & maps(Xf::'a,cross_product(Xs::'a,Xs),Xs) --> closed(Xs::'a,Xf)) & \
3.3085 +\ (! Z Xf Xg. member(Z::'a,composition(Xf::'a,Xg)) --> little_set(f29(Z::'a,Xf,Xg))) & \
3.3086 +\ (! Z Xf Xg. member(Z::'a,composition(Xf::'a,Xg)) --> little_set(f30(Z::'a,Xf,Xg))) & \
3.3087 +\ (! Z Xf Xg. member(Z::'a,composition(Xf::'a,Xg)) --> little_set(f31(Z::'a,Xf,Xg))) & \
3.3088 +\ (! 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)))) & \
3.3089 +\ (! Z Xg Xf. member(Z::'a,composition(Xf::'a,Xg)) --> member(ordered_pair(f29(Z::'a,Xf,Xg),f31(Z::'a,Xf,Xg)),Xf)) & \
3.3090 +\ (! Z Xf Xg. member(Z::'a,composition(Xf::'a,Xg)) --> member(ordered_pair(f31(Z::'a,Xf,Xg),f30(Z::'a,Xf,Xg)),Xg)) & \
3.3091 +\ (! 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))) & \
3.3092 +\ (! Xh Xs2 Xf2 Xs1 Xf1. homomorphism(Xh::'a,Xs1,Xf1,Xs2,Xf2) --> closed(Xs1::'a,Xf1)) & \
3.3093 +\ (! Xh Xs1 Xf1 Xs2 Xf2. homomorphism(Xh::'a,Xs1,Xf1,Xs2,Xf2) --> closed(Xs2::'a,Xf2)) & \
3.3094 +\ (! Xf1 Xf2 Xh Xs1 Xs2. homomorphism(Xh::'a,Xs1,Xf1,Xs2,Xf2) --> maps(Xh::'a,Xs1,Xs2)) & \
3.3095 +\ (! 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)))) & \
3.3096 +\ (! 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)) & \
3.3097 +\ (! 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)) & \
3.3098 +\ (! 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)) & \
3.3099 +\ (! A B C. equal(A::'a,B) --> equal(f1(A::'a,C),f1(B::'a,C))) & \
3.3100 +\ (! D F' E. equal(D::'a,E) --> equal(f1(F'::'a,D),f1(F'::'a,E))) & \
3.3101 +\ (! A2 B2. equal(A2::'a,B2) --> equal(f2(A2),f2(B2))) & \
3.3102 +\ (! G4 H4. equal(G4::'a,H4) --> equal(f3(G4),f3(H4))) & \
3.3103 +\ (! O7 P7 Q7. equal(O7::'a,P7) --> equal(f4(O7::'a,Q7),f4(P7::'a,Q7))) & \
3.3104 +\ (! R7 T7 S7. equal(R7::'a,S7) --> equal(f4(T7::'a,R7),f4(T7::'a,S7))) & \
3.3105 +\ (! U7 V7 W7. equal(U7::'a,V7) --> equal(f5(U7::'a,W7),f5(V7::'a,W7))) & \
3.3106 +\ (! X7 Z7 Y7. equal(X7::'a,Y7) --> equal(f5(Z7::'a,X7),f5(Z7::'a,Y7))) & \
3.3107 +\ (! A8 B8 C8. equal(A8::'a,B8) --> equal(f6(A8::'a,C8),f6(B8::'a,C8))) & \
3.3108 +\ (! D8 F8 E8. equal(D8::'a,E8) --> equal(f6(F8::'a,D8),f6(F8::'a,E8))) & \
3.3109 +\ (! G8 H8 I8. equal(G8::'a,H8) --> equal(f7(G8::'a,I8),f7(H8::'a,I8))) & \
3.3110 +\ (! J8 L8 K8. equal(J8::'a,K8) --> equal(f7(L8::'a,J8),f7(L8::'a,K8))) & \
3.3111 +\ (! M8 N8 O8. equal(M8::'a,N8) --> equal(f8(M8::'a,O8),f8(N8::'a,O8))) & \
3.3112 +\ (! P8 R8 Q8. equal(P8::'a,Q8) --> equal(f8(R8::'a,P8),f8(R8::'a,Q8))) & \
3.3113 +\ (! S8 T8 U8. equal(S8::'a,T8) --> equal(f9(S8::'a,U8),f9(T8::'a,U8))) & \
3.3114 +\ (! V8 X8 W8. equal(V8::'a,W8) --> equal(f9(X8::'a,V8),f9(X8::'a,W8))) & \
3.3115 +\ (! G H I'. equal(G::'a,H) --> equal(f10(G::'a,I'),f10(H::'a,I'))) & \
3.3116 +\ (! J L K'. equal(J::'a,K') --> equal(f10(L::'a,J),f10(L::'a,K'))) & \
3.3117 +\ (! M N O_. equal(M::'a,N) --> equal(f11(M::'a,O_),f11(N::'a,O_))) & \
3.3118 +\ (! P R Q. equal(P::'a,Q) --> equal(f11(R::'a,P),f11(R::'a,Q))) & \
3.3119 +\ (! S' T' U. equal(S'::'a,T') --> equal(f12(S'::'a,U),f12(T'::'a,U))) & \
3.3120 +\ (! V X W. equal(V::'a,W) --> equal(f12(X::'a,V),f12(X::'a,W))) & \
3.3121 +\ (! Y Z A1. equal(Y::'a,Z) --> equal(f13(Y::'a,A1),f13(Z::'a,A1))) & \
3.3122 +\ (! B1 D1 C1. equal(B1::'a,C1) --> equal(f13(D1::'a,B1),f13(D1::'a,C1))) & \
3.3123 +\ (! E1 F1 G1. equal(E1::'a,F1) --> equal(f14(E1::'a,G1),f14(F1::'a,G1))) & \
3.3124 +\ (! H1 J1 I1. equal(H1::'a,I1) --> equal(f14(J1::'a,H1),f14(J1::'a,I1))) & \
3.3125 +\ (! K1 L1 M1. equal(K1::'a,L1) --> equal(f16(K1::'a,M1),f16(L1::'a,M1))) & \
3.3126 +\ (! N1 P1 O1. equal(N1::'a,O1) --> equal(f16(P1::'a,N1),f16(P1::'a,O1))) & \
3.3127 +\ (! Q1 R1 S1. equal(Q1::'a,R1) --> equal(f17(Q1::'a,S1),f17(R1::'a,S1))) & \
3.3128 +\ (! T1 V1 U1. equal(T1::'a,U1) --> equal(f17(V1::'a,T1),f17(V1::'a,U1))) & \
3.3129 +\ (! W1 X1. equal(W1::'a,X1) --> equal(f18(W1),f18(X1))) & \
3.3130 +\ (! Y1 Z1. equal(Y1::'a,Z1) --> equal(f19(Y1),f19(Z1))) & \
3.3131 +\ (! C2 D2. equal(C2::'a,D2) --> equal(f20(C2),f20(D2))) & \
3.3132 +\ (! E2 F2. equal(E2::'a,F2) --> equal(f21(E2),f21(F2))) & \
3.3133 +\ (! G2 H2 I2 J2. equal(G2::'a,H2) --> equal(f22(G2::'a,I2,J2),f22(H2::'a,I2,J2))) & \
3.3134 +\ (! K2 M2 L2 N2. equal(K2::'a,L2) --> equal(f22(M2::'a,K2,N2),f22(M2::'a,L2,N2))) & \
3.3135 +\ (! O2 Q2 R2 P2. equal(O2::'a,P2) --> equal(f22(Q2::'a,R2,O2),f22(Q2::'a,R2,P2))) & \
3.3136 +\ (! S2 T2 U2. equal(S2::'a,T2) --> equal(f23(S2::'a,U2),f23(T2::'a,U2))) & \
3.3137 +\ (! V2 X2 W2. equal(V2::'a,W2) --> equal(f23(X2::'a,V2),f23(X2::'a,W2))) & \
3.3138 +\ (! Y2 Z2. equal(Y2::'a,Z2) --> equal(f24(Y2),f24(Z2))) & \
3.3139 +\ (! A3 B3. equal(A3::'a,B3) --> equal(f26(A3),f26(B3))) & \
3.3140 +\ (! C3 D3 E3. equal(C3::'a,D3) --> equal(f27(C3::'a,E3),f27(D3::'a,E3))) & \
3.3141 +\ (! F3 H3 G3. equal(F3::'a,G3) --> equal(f27(H3::'a,F3),f27(H3::'a,G3))) & \
3.3142 +\ (! I3 J3 K3 L3. equal(I3::'a,J3) --> equal(f28(I3::'a,K3,L3),f28(J3::'a,K3,L3))) & \
3.3143 +\ (! M3 O3 N3 P3. equal(M3::'a,N3) --> equal(f28(O3::'a,M3,P3),f28(O3::'a,N3,P3))) & \
3.3144 +\ (! Q3 S3 T3 R3. equal(Q3::'a,R3) --> equal(f28(S3::'a,T3,Q3),f28(S3::'a,T3,R3))) & \
3.3145 +\ (! U3 V3 W3 X3. equal(U3::'a,V3) --> equal(f29(U3::'a,W3,X3),f29(V3::'a,W3,X3))) & \
3.3146 +\ (! Y3 A4 Z3 B4. equal(Y3::'a,Z3) --> equal(f29(A4::'a,Y3,B4),f29(A4::'a,Z3,B4))) & \
3.3147 +\ (! C4 E4 F4 D4. equal(C4::'a,D4) --> equal(f29(E4::'a,F4,C4),f29(E4::'a,F4,D4))) & \
3.3148 +\ (! I4 J4 K4 L4. equal(I4::'a,J4) --> equal(f30(I4::'a,K4,L4),f30(J4::'a,K4,L4))) & \
3.3149 +\ (! M4 O4 N4 P4. equal(M4::'a,N4) --> equal(f30(O4::'a,M4,P4),f30(O4::'a,N4,P4))) & \
3.3150 +\ (! Q4 S4 T4 R4. equal(Q4::'a,R4) --> equal(f30(S4::'a,T4,Q4),f30(S4::'a,T4,R4))) & \
3.3151 +\ (! U4 V4 W4 X4. equal(U4::'a,V4) --> equal(f31(U4::'a,W4,X4),f31(V4::'a,W4,X4))) & \
3.3152 +\ (! Y4 A5 Z4 B5. equal(Y4::'a,Z4) --> equal(f31(A5::'a,Y4,B5),f31(A5::'a,Z4,B5))) & \
3.3153 +\ (! C5 E5 F5 D5. equal(C5::'a,D5) --> equal(f31(E5::'a,F5,C5),f31(E5::'a,F5,D5))) & \
3.3154 +\ (! 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))) & \
3.3155 +\ (! 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))) & \
3.3156 +\ (! 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))) & \
3.3157 +\ (! 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))) & \
3.3158 +\ (! 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))) & \
3.3159 +\ (! 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))) & \
3.3160 +\ (! 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))) & \
3.3161 +\ (! 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))) & \
3.3162 +\ (! 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))) & \
3.3163 +\ (! 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))) & \
3.3164 +\ (! A B C. equal(A::'a,B) --> equal(apply(A::'a,C),apply(B::'a,C))) & \
3.3165 +\ (! D F' E. equal(D::'a,E) --> equal(apply(F'::'a,D),apply(F'::'a,E))) & \
3.3166 +\ (! 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))) & \
3.3167 +\ (! 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))) & \
3.3168 +\ (! 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))) & \
3.3169 +\ (! S' T'. equal(S'::'a,T') --> equal(complement(S'),complement(T'))) & \
3.3170 +\ (! U V W. equal(U::'a,V) --> equal(composition(U::'a,W),composition(V::'a,W))) & \
3.3171 +\ (! X Z Y. equal(X::'a,Y) --> equal(composition(Z::'a,X),composition(Z::'a,Y))) & \
3.3172 +\ (! A1 B1. equal(A1::'a,B1) --> equal(inv1 A1,inv1 B1)) & \
3.3173 +\ (! C1 D1 E1. equal(C1::'a,D1) --> equal(cross_product(C1::'a,E1),cross_product(D1::'a,E1))) & \
3.3174 +\ (! F1 H1 G1. equal(F1::'a,G1) --> equal(cross_product(H1::'a,F1),cross_product(H1::'a,G1))) & \
3.3175 +\ (! I1 J1. equal(I1::'a,J1) --> equal(domain_of(I1),domain_of(J1))) & \
3.3176 +\ (! I10 J10. equal(I10::'a,J10) --> equal(first(I10),first(J10))) & \
3.3177 +\ (! Q10 R10. equal(Q10::'a,R10) --> equal(flip_range_of(Q10),flip_range_of(R10))) & \
3.3178 +\ (! S10 T10 U10. equal(S10::'a,T10) --> equal(image_(S10::'a,U10),image_(T10::'a,U10))) & \
3.3179 +\ (! V10 X10 W10. equal(V10::'a,W10) --> equal(image_(X10::'a,V10),image_(X10::'a,W10))) & \
3.3180 +\ (! Y10 Z10 A11. equal(Y10::'a,Z10) --> equal(intersection(Y10::'a,A11),intersection(Z10::'a,A11))) & \
3.3181 +\ (! B11 D11 C11. equal(B11::'a,C11) --> equal(intersection(D11::'a,B11),intersection(D11::'a,C11))) & \
3.3182 +\ (! E11 F11 G11. equal(E11::'a,F11) --> equal(non_ordered_pair(E11::'a,G11),non_ordered_pair(F11::'a,G11))) & \
3.3183 +\ (! H11 J11 I11. equal(H11::'a,I11) --> equal(non_ordered_pair(J11::'a,H11),non_ordered_pair(J11::'a,I11))) & \
3.3184 +\ (! K11 L11 M11. equal(K11::'a,L11) --> equal(ordered_pair(K11::'a,M11),ordered_pair(L11::'a,M11))) & \
3.3185 +\ (! N11 P11 O11. equal(N11::'a,O11) --> equal(ordered_pair(P11::'a,N11),ordered_pair(P11::'a,O11))) & \
3.3186 +\ (! Q11 R11. equal(Q11::'a,R11) --> equal(powerset(Q11),powerset(R11))) & \
3.3187 +\ (! S11 T11. equal(S11::'a,T11) --> equal(range_of(S11),range_of(T11))) & \
3.3188 +\ (! U11 V11 W11. equal(U11::'a,V11) --> equal(restrct(U11::'a,W11),restrct(V11::'a,W11))) & \
3.3189 +\ (! X11 Z11 Y11. equal(X11::'a,Y11) --> equal(restrct(Z11::'a,X11),restrct(Z11::'a,Y11))) & \
3.3190 +\ (! A12 B12. equal(A12::'a,B12) --> equal(rotate_right(A12),rotate_right(B12))) & \
3.3191 +\ (! C12 D12. equal(C12::'a,D12) --> equal(second(C12),second(D12))) & \
3.3192 +\ (! K12 L12. equal(K12::'a,L12) --> equal(sigma(K12),sigma(L12))) & \
3.3193 +\ (! M12 N12. equal(M12::'a,N12) --> equal(singleton_set(M12),singleton_set(N12))) & \
3.3194 +\ (! O12 P12. equal(O12::'a,P12) --> equal(successor(O12),successor(P12))) & \
3.3195 +\ (! Q12 R12 S12. equal(Q12::'a,R12) --> equal(union(Q12::'a,S12),union(R12::'a,S12))) & \
3.3196 +\ (! T12 V12 U12. equal(T12::'a,U12) --> equal(union(V12::'a,T12),union(V12::'a,U12))) & \
3.3197 +\ (! W12 X12 Y12. equal(W12::'a,X12) & closed(W12::'a,Y12) --> closed(X12::'a,Y12)) & \
3.3198 +\ (! Z12 B13 A13. equal(Z12::'a,A13) & closed(B13::'a,Z12) --> closed(B13::'a,A13)) & \
3.3199 +\ (! C13 D13 E13. equal(C13::'a,D13) & disjoint(C13::'a,E13) --> disjoint(D13::'a,E13)) & \
3.3200 +\ (! F13 H13 G13. equal(F13::'a,G13) & disjoint(H13::'a,F13) --> disjoint(H13::'a,G13)) & \
3.3201 +\ (! I13 J13. equal(I13::'a,J13) & function(I13) --> function(J13)) & \
3.3202 +\ (! K13 L13 M13 N13 O13 P13. equal(K13::'a,L13) & homomorphism(K13::'a,M13,N13,O13,P13) --> homomorphism(L13::'a,M13,N13,O13,P13)) & \
3.3203 +\ (! Q13 S13 R13 T13 U13 V13. equal(Q13::'a,R13) & homomorphism(S13::'a,Q13,T13,U13,V13) --> homomorphism(S13::'a,R13,T13,U13,V13)) & \
3.3204 +\ (! W13 Y13 Z13 X13 A14 B14. equal(W13::'a,X13) & homomorphism(Y13::'a,Z13,W13,A14,B14) --> homomorphism(Y13::'a,Z13,X13,A14,B14)) & \
3.3205 +\ (! C14 E14 F14 G14 D14 H14. equal(C14::'a,D14) & homomorphism(E14::'a,F14,G14,C14,H14) --> homomorphism(E14::'a,F14,G14,D14,H14)) & \
3.3206 +\ (! I14 K14 L14 M14 N14 J14. equal(I14::'a,J14) & homomorphism(K14::'a,L14,M14,N14,I14) --> homomorphism(K14::'a,L14,M14,N14,J14)) & \
3.3207 +\ (! O14 P14. equal(O14::'a,P14) & little_set(O14) --> little_set(P14)) & \
3.3208 +\ (! Q14 R14 S14 T14. equal(Q14::'a,R14) & maps(Q14::'a,S14,T14) --> maps(R14::'a,S14,T14)) & \
3.3209 +\ (! U14 W14 V14 X14. equal(U14::'a,V14) & maps(W14::'a,U14,X14) --> maps(W14::'a,V14,X14)) & \
3.3210 +\ (! Y14 A15 B15 Z14. equal(Y14::'a,Z14) & maps(A15::'a,B15,Y14) --> maps(A15::'a,B15,Z14)) & \
3.3211 +\ (! C15 D15 E15. equal(C15::'a,D15) & member(C15::'a,E15) --> member(D15::'a,E15)) & \
3.3212 +\ (! F15 H15 G15. equal(F15::'a,G15) & member(H15::'a,F15) --> member(H15::'a,G15)) & \
3.3213 +\ (! I15 J15. equal(I15::'a,J15) & one_to_one_function(I15) --> one_to_one_function(J15)) & \
3.3214 +\ (! K15 L15. equal(K15::'a,L15) & ordered_pair_predicate(K15) --> ordered_pair_predicate(L15)) & \
3.3215 +\ (! M15 N15 O15. equal(M15::'a,N15) & proper_subset(M15::'a,O15) --> proper_subset(N15::'a,O15)) & \
3.3216 +\ (! P15 R15 Q15. equal(P15::'a,Q15) & proper_subset(R15::'a,P15) --> proper_subset(R15::'a,Q15)) & \
3.3217 +\ (! S15 T15. equal(S15::'a,T15) & relation(S15) --> relation(T15)) & \
3.3218 +\ (! U15 V15. equal(U15::'a,V15) & single_valued_set(U15) --> single_valued_set(V15)) & \
3.3219 +\ (! W15 X15 Y15. equal(W15::'a,X15) & subset(W15::'a,Y15) --> subset(X15::'a,Y15)) & \
3.3220 +\ (! Z15 B16 A16. equal(Z15::'a,A16) & subset(B16::'a,Z15) --> subset(B16::'a,A16)) & \
3.3221 +\ (~little_set(ordered_pair(a::'a,b))) --> False",
3.3222 + meson_tac);
3.3223 +
3.3224 +
3.3225 +(*13 inferences so far. Searching to depth 8. 0 secs*)
3.3226 +val SET046_5 = prove
3.3227 + ("(! Y X. ~(element(X::'a,a) & element(X::'a,Y) & element(Y::'a,X))) & \
3.3228 +\ (! X. element(X::'a,f(X)) | element(X::'a,a)) & \
3.3229 +\ (! X. element(f(X),X) | element(X::'a,a)) --> False",
3.3230 + meson_tac);
3.3231 +
3.3232 +(*33 inferences so far. Searching to depth 9. 0.2 secs*)
3.3233 +val SET047_5 = prove
3.3234 + ("(! X Z Y. set_equal(X::'a,Y) & element(Z::'a,X) --> element(Z::'a,Y)) & \
3.3235 +\ (! Y Z X. set_equal(X::'a,Y) & element(Z::'a,Y) --> element(Z::'a,X)) & \
3.3236 +\ (! X Y. element(f(X::'a,Y),X) | element(f(X::'a,Y),Y) | set_equal(X::'a,Y)) & \
3.3237 +\ (! X Y. element(f(X::'a,Y),Y) & element(f(X::'a,Y),X) --> set_equal(X::'a,Y)) & \
3.3238 +\ (set_equal(a::'a,b) | set_equal(b::'a,a)) & \
3.3239 +\ (~(set_equal(b::'a,a) & set_equal(a::'a,b))) --> False",
3.3240 + meson_tac);
3.3241 +
3.3242 +(*311 inferences so far. Searching to depth 12. 0.1 secs*)
3.3243 +val SYN034_1 = prove
3.3244 + ("(! A. p(A::'a,a) | p(A::'a,f(A))) & \
3.3245 +\ (! A. p(A::'a,a) | p(f(A),A)) & \
3.3246 +\ (! A B. ~(p(A::'a,B) & p(B::'a,A) & p(B::'a,a))) --> False",
3.3247 + meson_tac);
3.3248 +
3.3249 +(*30 inferences so far. Searching to depth 6. 0.2 secs*)
3.3250 +val SYN071_1 = prove
3.3251 + ("(! X. equal(X::'a,X)) & \
3.3252 +\ (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
3.3253 +\ (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) & \
3.3254 +\ (equal(a::'a,b) | equal(c::'a,d)) & \
3.3255 +\ (equal(a::'a,c) | equal(b::'a,d)) & \
3.3256 +\ (~equal(a::'a,d)) & \
3.3257 +\ (~equal(b::'a,c)) --> False",
3.3258 + meson_tac);
3.3259 +
3.3260 +(****************SLOW
3.3261 +655670 inferences so far. Searching to depth 44. No proof after 10 minutes.
3.3262 +val SYN349_1 = prove_hard
3.3263 + ("(! X Y. f(w(X),g(X::'a,Y)) --> f(X::'a,g(X::'a,Y))) & \
3.3264 +\ (! X Y. f(X::'a,g(X::'a,Y)) --> f(w(X),g(X::'a,Y))) & \
3.3265 +\ (! 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))) & \
3.3266 +\ (! 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))) & \
3.3267 +\ (! 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))) & \
3.3268 +\ (! 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))) & \
3.3269 +\ (! 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))) & \
3.3270 +\ (! 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))) & \
3.3271 +\ (! 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)) & \
3.3272 +\ (! 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",
3.3273 + meson_tac);
3.3274 +****************)
3.3275 +
3.3276 +(*398 inferences so far. Searching to depth 12. 0.4 secs*)
3.3277 +val SYN352_1 = prove
3.3278 + ("(f(a::'a,b)) & \
3.3279 +\ (! X Y. f(X::'a,Y) --> f(b::'a,z(X::'a,Y)) | f(Y::'a,z(X::'a,Y))) & \
3.3280 +\ (! X Y. f(X::'a,Y) | f(z(X::'a,Y),z(X::'a,Y))) & \
3.3281 +\ (! 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))) & \
3.3282 +\ (! 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))) & \
3.3283 +\ (! X Y. ~(f(X::'a,Y) & f(X::'a,z(X::'a,Y)) & f(Y::'a,z(X::'a,Y)))) & \
3.3284 +\ (! X Y. f(X::'a,Y) --> f(X::'a,z(X::'a,Y)) | f(Y::'a,z(X::'a,Y))) --> False",
3.3285 + meson_tac);
3.3286 +
3.3287 +(*5336 inferences so far. Searching to depth 15. 5.3 secs*)
3.3288 +val TOP001_2 = prove_hard
3.3289 + ("(! Vf U. element_of_set(U::'a,union_of_members(Vf)) --> element_of_set(U::'a,f1(Vf::'a,U))) & \
3.3290 +\ (! U Vf. element_of_set(U::'a,union_of_members(Vf)) --> element_of_collection(f1(Vf::'a,U),Vf)) & \
3.3291 +\ (! U Uu1 Vf. element_of_set(U::'a,Uu1) & element_of_collection(Uu1::'a,Vf) --> element_of_set(U::'a,union_of_members(Vf))) & \
3.3292 +\ (! Vf X. basis(X::'a,Vf) --> equal_sets(union_of_members(Vf),X)) & \
3.3293 +\ (! 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))) & \
3.3294 +\ (! 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)) & \
3.3295 +\ (! X. subset_sets(X::'a,X)) & \
3.3296 +\ (! X U Y. subset_sets(X::'a,Y) & element_of_set(U::'a,X) --> element_of_set(U::'a,Y)) & \
3.3297 +\ (! X Y. equal_sets(X::'a,Y) --> subset_sets(X::'a,Y)) & \
3.3298 +\ (! Y X. subset_sets(X::'a,Y) | element_of_set(in_1st_set(X::'a,Y),X)) & \
3.3299 +\ (! X Y. element_of_set(in_1st_set(X::'a,Y),Y) --> subset_sets(X::'a,Y)) & \
3.3300 +\ (basis(cx::'a,f)) & \
3.3301 +\ (~subset_sets(union_of_members(top_of_basis(f)),cx)) --> False",
3.3302 + meson_tac);
3.3303 +
3.3304 +(*0 inferences so far. Searching to depth 0. 0 secs*)
3.3305 +val TOP002_2 = prove
3.3306 + ("(! Vf U. element_of_collection(U::'a,top_of_basis(Vf)) | element_of_set(f11(Vf::'a,U),U)) & \
3.3307 +\ (! X. ~element_of_set(X::'a,empty_set)) & \
3.3308 +\ (~element_of_collection(empty_set::'a,top_of_basis(f))) --> False",
3.3309 + meson_tac);
3.3310 +
3.3311 +(*0 inferences so far. Searching to depth 0. 6.5 secs. BIG*)
3.3312 +val TOP004_1 = prove_hard
3.3313 + ("(! Vf U. element_of_set(U::'a,union_of_members(Vf)) --> element_of_set(U::'a,f1(Vf::'a,U))) & \
3.3314 +\ (! U Vf. element_of_set(U::'a,union_of_members(Vf)) --> element_of_collection(f1(Vf::'a,U),Vf)) & \
3.3315 +\ (! U Uu1 Vf. element_of_set(U::'a,Uu1) & element_of_collection(Uu1::'a,Vf) --> element_of_set(U::'a,union_of_members(Vf))) & \
3.3316 +\ (! Vf U Va. element_of_set(U::'a,intersection_of_members(Vf)) & element_of_collection(Va::'a,Vf) --> element_of_set(U::'a,Va)) & \
3.3317 +\ (! U Vf. element_of_set(U::'a,intersection_of_members(Vf)) | element_of_collection(f2(Vf::'a,U),Vf)) & \
3.3318 +\ (! Vf U. element_of_set(U::'a,f2(Vf::'a,U)) --> element_of_set(U::'a,intersection_of_members(Vf))) & \
3.3319 +\ (! Vt X. topological_space(X::'a,Vt) --> equal_sets(union_of_members(Vt),X)) & \
3.3320 +\ (! X Vt. topological_space(X::'a,Vt) --> element_of_collection(empty_set::'a,Vt)) & \
3.3321 +\ (! X Vt. topological_space(X::'a,Vt) --> element_of_collection(X::'a,Vt)) & \
3.3322 +\ (! 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)) & \
3.3323 +\ (! X Vf Vt. topological_space(X::'a,Vt) & subset_collections(Vf::'a,Vt) --> element_of_collection(union_of_members(Vf),Vt)) & \
3.3324 +\ (! 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)) & \
3.3325 +\ (! 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)) & \
3.3326 +\ (! 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)) & \
3.3327 +\ (! 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)) & \
3.3328 +\ (! 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)) & \
3.3329 +\ (! 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)) & \
3.3330 +\ (! U X Vt. open(U::'a,X,Vt) --> topological_space(X::'a,Vt)) & \
3.3331 +\ (! X U Vt. open(U::'a,X,Vt) --> element_of_collection(U::'a,Vt)) & \
3.3332 +\ (! X U Vt. topological_space(X::'a,Vt) & element_of_collection(U::'a,Vt) --> open(U::'a,X,Vt)) & \
3.3333 +\ (! U X Vt. closed(U::'a,X,Vt) --> topological_space(X::'a,Vt)) & \
3.3334 +\ (! U X Vt. closed(U::'a,X,Vt) --> open(relative_complement_sets(U::'a,X),X,Vt)) & \
3.3335 +\ (! U X Vt. topological_space(X::'a,Vt) & open(relative_complement_sets(U::'a,X),X,Vt) --> closed(U::'a,X,Vt)) & \
3.3336 +\ (! Vs X Vt. finer(Vt::'a,Vs,X) --> topological_space(X::'a,Vt)) & \
3.3337 +\ (! Vt X Vs. finer(Vt::'a,Vs,X) --> topological_space(X::'a,Vs)) & \
3.3338 +\ (! X Vs Vt. finer(Vt::'a,Vs,X) --> subset_collections(Vs::'a,Vt)) & \
3.3339 +\ (! X Vs Vt. topological_space(X::'a,Vt) & topological_space(X::'a,Vs) & subset_collections(Vs::'a,Vt) --> finer(Vt::'a,Vs,X)) & \
3.3340 +\ (! Vf X. basis(X::'a,Vf) --> equal_sets(union_of_members(Vf),X)) & \
3.3341 +\ (! 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))) & \
3.3342 +\ (! 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)) & \
3.3343 +\ (! 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))) & \
3.3344 +\ (! Vf X. equal_sets(union_of_members(Vf),X) --> basis(X::'a,Vf) | element_of_set(f7(X::'a,Vf),X)) & \
3.3345 +\ (! X Vf. equal_sets(union_of_members(Vf),X) --> basis(X::'a,Vf) | element_of_collection(f8(X::'a,Vf),Vf)) & \
3.3346 +\ (! X Vf. equal_sets(union_of_members(Vf),X) --> basis(X::'a,Vf) | element_of_collection(f9(X::'a,Vf),Vf)) & \
3.3347 +\ (! 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)))) & \
3.3348 +\ (! 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)) & \
3.3349 +\ (! 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))) & \
3.3350 +\ (! 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)) & \
3.3351 +\ (! 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)) & \
3.3352 +\ (! Vf U. element_of_collection(U::'a,top_of_basis(Vf)) | element_of_set(f11(Vf::'a,U),U)) & \
3.3353 +\ (! 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))) & \
3.3354 +\ (! U Y X Vt. element_of_collection(U::'a,subspace_topology(X::'a,Vt,Y)) --> topological_space(X::'a,Vt)) & \
3.3355 +\ (! U Vt Y X. element_of_collection(U::'a,subspace_topology(X::'a,Vt,Y)) --> subset_sets(Y::'a,X)) & \
3.3356 +\ (! 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)) & \
3.3357 +\ (! 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)))) & \
3.3358 +\ (! 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))) & \
3.3359 +\ (! U Y X Vt. element_of_set(U::'a,interior(Y::'a,X,Vt)) --> topological_space(X::'a,Vt)) & \
3.3360 +\ (! U Vt Y X. element_of_set(U::'a,interior(Y::'a,X,Vt)) --> subset_sets(Y::'a,X)) & \
3.3361 +\ (! 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))) & \
3.3362 +\ (! X Vt U Y. element_of_set(U::'a,interior(Y::'a,X,Vt)) --> subset_sets(f13(Y::'a,X,Vt,U),Y)) & \
3.3363 +\ (! Y U X Vt. element_of_set(U::'a,interior(Y::'a,X,Vt)) --> open(f13(Y::'a,X,Vt,U),X,Vt)) & \
3.3364 +\ (! 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))) & \
3.3365 +\ (! U Y X Vt. element_of_set(U::'a,closure(Y::'a,X,Vt)) --> topological_space(X::'a,Vt)) & \
3.3366 +\ (! U Vt Y X. element_of_set(U::'a,closure(Y::'a,X,Vt)) --> subset_sets(Y::'a,X)) & \
3.3367 +\ (! 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)) & \
3.3368 +\ (! 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))) & \
3.3369 +\ (! 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)) & \
3.3370 +\ (! 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))) & \
3.3371 +\ (! U Y X Vt. neighborhood(U::'a,Y,X,Vt) --> topological_space(X::'a,Vt)) & \
3.3372 +\ (! Y U X Vt. neighborhood(U::'a,Y,X,Vt) --> open(U::'a,X,Vt)) & \
3.3373 +\ (! X Vt Y U. neighborhood(U::'a,Y,X,Vt) --> element_of_set(Y::'a,U)) & \
3.3374 +\ (! 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)) & \
3.3375 +\ (! Z Y X Vt. limit_point(Z::'a,Y,X,Vt) --> topological_space(X::'a,Vt)) & \
3.3376 +\ (! Z Vt Y X. limit_point(Z::'a,Y,X,Vt) --> subset_sets(Y::'a,X)) & \
3.3377 +\ (! 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))) & \
3.3378 +\ (! 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))) & \
3.3379 +\ (! 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)) & \
3.3380 +\ (! 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)) & \
3.3381 +\ (! U Y X Vt. element_of_set(U::'a,boundary(Y::'a,X,Vt)) --> topological_space(X::'a,Vt)) & \
3.3382 +\ (! U Y X Vt. element_of_set(U::'a,boundary(Y::'a,X,Vt)) --> element_of_set(U::'a,closure(Y::'a,X,Vt))) & \
3.3383 +\ (! 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))) & \
3.3384 +\ (! 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))) & \
3.3385 +\ (! X Vt. hausdorff(X::'a,Vt) --> topological_space(X::'a,Vt)) & \
3.3386 +\ (! 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)) & \
3.3387 +\ (! 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)) & \
3.3388 +\ (! 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))) & \
3.3389 +\ (! Vt X. topological_space(X::'a,Vt) --> hausdorff(X::'a,Vt) | element_of_set(f19(X::'a,Vt),X)) & \
3.3390 +\ (! Vt X. topological_space(X::'a,Vt) --> hausdorff(X::'a,Vt) | element_of_set(f20(X::'a,Vt),X)) & \
3.3391 +\ (! X Vt. topological_space(X::'a,Vt) & eq_p(f19(X::'a,Vt),f20(X::'a,Vt)) --> hausdorff(X::'a,Vt)) & \
3.3392 +\ (! 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)) & \
3.3393 +\ (! Va1 Va2 X Vt. separation(Va1::'a,Va2,X,Vt) --> topological_space(X::'a,Vt)) & \
3.3394 +\ (! Va2 X Vt Va1. ~(separation(Va1::'a,Va2,X,Vt) & equal_sets(Va1::'a,empty_set))) & \
3.3395 +\ (! Va1 X Vt Va2. ~(separation(Va1::'a,Va2,X,Vt) & equal_sets(Va2::'a,empty_set))) & \
3.3396 +\ (! Va2 X Va1 Vt. separation(Va1::'a,Va2,X,Vt) --> element_of_collection(Va1::'a,Vt)) & \
3.3397 +\ (! Va1 X Va2 Vt. separation(Va1::'a,Va2,X,Vt) --> element_of_collection(Va2::'a,Vt)) & \
3.3398 +\ (! Vt Va1 Va2 X. separation(Va1::'a,Va2,X,Vt) --> equal_sets(union_of_sets(Va1::'a,Va2),X)) & \
3.3399 +\ (! X Vt Va1 Va2. separation(Va1::'a,Va2,X,Vt) --> disjoint_s(Va1::'a,Va2)) & \
3.3400 +\ (! 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)) & \
3.3401 +\ (! X Vt. connected_space(X::'a,Vt) --> topological_space(X::'a,Vt)) & \
3.3402 +\ (! Va1 Va2 X Vt. ~(connected_space(X::'a,Vt) & separation(Va1::'a,Va2,X,Vt))) & \
3.3403 +\ (! X Vt. topological_space(X::'a,Vt) --> connected_space(X::'a,Vt) | separation(f21(X::'a,Vt),f22(X::'a,Vt),X,Vt)) & \
3.3404 +\ (! Va X Vt. connected_set(Va::'a,X,Vt) --> topological_space(X::'a,Vt)) & \
3.3405 +\ (! Vt Va X. connected_set(Va::'a,X,Vt) --> subset_sets(Va::'a,X)) & \
3.3406 +\ (! X Vt Va. connected_set(Va::'a,X,Vt) --> connected_space(Va::'a,subspace_topology(X::'a,Vt,Va))) & \
3.3407 +\ (! 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)) & \
3.3408 +\ (! Vf X Vt. open_covering(Vf::'a,X,Vt) --> topological_space(X::'a,Vt)) & \
3.3409 +\ (! X Vf Vt. open_covering(Vf::'a,X,Vt) --> subset_collections(Vf::'a,Vt)) & \
3.3410 +\ (! Vt Vf X. open_covering(Vf::'a,X,Vt) --> equal_sets(union_of_members(Vf),X)) & \
3.3411 +\ (! 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)) & \
3.3412 +\ (! X Vt. compact_space(X::'a,Vt) --> topological_space(X::'a,Vt)) & \
3.3413 +\ (! X Vt Vf1. compact_space(X::'a,Vt) & open_covering(Vf1::'a,X,Vt) --> finite(f23(X::'a,Vt,Vf1))) & \
3.3414 +\ (! X Vt Vf1. compact_space(X::'a,Vt) & open_covering(Vf1::'a,X,Vt) --> subset_collections(f23(X::'a,Vt,Vf1),Vf1)) & \
3.3415 +\ (! Vf1 X Vt. compact_space(X::'a,Vt) & open_covering(Vf1::'a,X,Vt) --> open_covering(f23(X::'a,Vt,Vf1),X,Vt)) & \
3.3416 +\ (! X Vt. topological_space(X::'a,Vt) --> compact_space(X::'a,Vt) | open_covering(f24(X::'a,Vt),X,Vt)) & \
3.3417 +\ (! 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)) & \
3.3418 +\ (! Va X Vt. compact_set(Va::'a,X,Vt) --> topological_space(X::'a,Vt)) & \
3.3419 +\ (! Vt Va X. compact_set(Va::'a,X,Vt) --> subset_sets(Va::'a,X)) & \
3.3420 +\ (! X Vt Va. compact_set(Va::'a,X,Vt) --> compact_space(Va::'a,subspace_topology(X::'a,Vt,Va))) & \
3.3421 +\ (! 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)) & \
3.3422 +\ (basis(cx::'a,f)) & \
3.3423 +\ (! U. element_of_collection(U::'a,top_of_basis(f))) & \
3.3424 +\ (! V. element_of_collection(V::'a,top_of_basis(f))) & \
3.3425 +\ (! U V. ~element_of_collection(intersection_of_sets(U::'a,V),top_of_basis(f))) --> False",
3.3426 + meson_tac);
3.3427 +
3.3428 +
3.3429 +(*0 inferences so far. Searching to depth 0. 0.8 secs*)
3.3430 +val TOP004_2 = prove
3.3431 + ("(! U Uu1 Vf. element_of_set(U::'a,Uu1) & element_of_collection(Uu1::'a,Vf) --> element_of_set(U::'a,union_of_members(Vf))) & \
3.3432 +\ (! Vf X. basis(X::'a,Vf) --> equal_sets(union_of_members(Vf),X)) & \
3.3433 +\ (! 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))) & \
3.3434 +\ (! 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)) & \
3.3435 +\ (! 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))) & \
3.3436 +\ (! 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))) & \
3.3437 +\ (! 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)) & \
3.3438 +\ (! 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)) & \
3.3439 +\ (! Vf U. element_of_collection(U::'a,top_of_basis(Vf)) | element_of_set(f11(Vf::'a,U),U)) & \
3.3440 +\ (! 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))) & \
3.3441 +\ (! Y X Z. subset_sets(X::'a,Y) & subset_sets(Y::'a,Z) --> subset_sets(X::'a,Z)) & \
3.3442 +\ (! Y Z X. element_of_set(Z::'a,intersection_of_sets(X::'a,Y)) --> element_of_set(Z::'a,X)) & \
3.3443 +\ (! X Z Y. element_of_set(Z::'a,intersection_of_sets(X::'a,Y)) --> element_of_set(Z::'a,Y)) & \
3.3444 +\ (! 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))) & \
3.3445 +\ (! 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))) & \
3.3446 +\ (! X Z Y. equal_sets(X::'a,Y) & element_of_set(Z::'a,X) --> element_of_set(Z::'a,Y)) & \
3.3447 +\ (! Y X. equal_sets(intersection_of_sets(X::'a,Y),intersection_of_sets(Y::'a,X))) & \
3.3448 +\ (basis(cx::'a,f)) & \
3.3449 +\ (! U. element_of_collection(U::'a,top_of_basis(f))) & \
3.3450 +\ (! V. element_of_collection(V::'a,top_of_basis(f))) & \
3.3451 +\ (! U V. ~element_of_collection(intersection_of_sets(U::'a,V),top_of_basis(f))) --> False",
3.3452 + meson_tac);
3.3453 +
3.3454 +(*53777 inferences so far. Searching to depth 20. 68.7 secs*)
3.3455 +val TOP005_2 = prove_hard
3.3456 + ("(! Vf U. element_of_set(U::'a,union_of_members(Vf)) --> element_of_set(U::'a,f1(Vf::'a,U))) & \
3.3457 +\ (! U Vf. element_of_set(U::'a,union_of_members(Vf)) --> element_of_collection(f1(Vf::'a,U),Vf)) & \
3.3458 +\ (! 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))) & \
3.3459 +\ (! 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)) & \
3.3460 +\ (! 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)) & \
3.3461 +\ (! Vf U. element_of_collection(U::'a,top_of_basis(Vf)) | element_of_set(f11(Vf::'a,U),U)) & \
3.3462 +\ (! 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))) & \
3.3463 +\ (! X U Y. element_of_set(U::'a,X) --> subset_sets(X::'a,Y) | element_of_set(U::'a,Y)) & \
3.3464 +\ (! Y X Z. subset_sets(X::'a,Y) & element_of_collection(Y::'a,Z) --> subset_sets(X::'a,union_of_members(Z))) & \
3.3465 +\ (! X U Y. subset_collections(X::'a,Y) & element_of_collection(U::'a,X) --> element_of_collection(U::'a,Y)) & \
3.3466 +\ (subset_collections(g::'a,top_of_basis(f))) & \
3.3467 +\ (~element_of_collection(union_of_members(g),top_of_basis(f))) --> False",
3.3468 + meson_tac);
3.3469 +
3.3470 +