src/HOL/ex/mesontest2.ML

cleaned up;

(*  Title:      HOL/ex/mesontest2
    ID:         $Id$
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   2000  University of Cambridge

MORE and MUCH HARDER test data for the MESON proof procedure

Courtesy John Harrison 
*)

(*All but the fastest are ignored to reduce build time*)
val even_hard_ones = false;

(*Prod.thy instead of HOL.thy regards arguments as tuples.
  But Main.thy would allow clashes with many other constants*)
fun prove (s,tac) = prove_goal Prod.thy s (fn [] => [tac]);

fun prove_hard arg = if even_hard_ones then prove arg else TrueI;

val meson_tac = safe_meson_tac 1;

set timing;

(* ========================================================================= *)
(* 100 problems selected from the TPTP library                               *)
(* ========================================================================= *)

(*
 * Original timings for John Harrison's MESON_TAC.
 * Timings below on a 600MHz Pentium III (perch)
 * 
 * A few variable names have been primed to avoid clashing with constants.
 *
 * Changed numeric constants e.g. 0, 1, 2... to num0, num1, num2...
 *
 * Here's a list giving typical CPU times, as well as common names and
 * literature references.
 *
 * BOO003-1     34.6    B2 part 1 [McCharen, et al., 1976]; Lemma proved [Overbeek, et al., 1976]; prob2_part1.ver1.in [ANL]
 * BOO004-1     36.7    B2 part 2 [McCharen, et al., 1976]; Lemma proved [Overbeek, et al., 1976]; prob2_part2.ver1 [ANL]
 * 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]
 * 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]
 * BOO011-1     19.0    B7 [McCharen, et al., 1976]; prob7.ver1 [ANL]
 * CAT001-3     45.2    C1 [McCharen, et al., 1976]; p1.ver3.in [ANL]
 * CAT003-3     10.5    C3 [McCharen, et al., 1976]; p3.ver3.in [ANL]
 * CAT005-1    480.1    C5 [McCharen, et al., 1976]; p5.ver1.in [ANL]
 * CAT007-1     11.9    C7 [McCharen, et al., 1976]; p7.ver1.in [ANL]
 * CAT018-1     81.3    p18.ver1.in [ANL]
 * COL001-2     16.0    C1 [Wos & McCune, 1988]
 * COL023-1      5.1    [McCune & Wos, 1988]
 * COL032-1     15.8    [McCune & Wos, 1988]
 * COL052-2     13.2    bird4.ver2.in [ANL]
 * COL075-2    116.9    [Jech, 1994]
 * COM001-1      1.7    shortburst [Wilson & Minker, 1976]
 * COM002-1      4.4    burstall [Wilson & Minker, 1976]
 * COM002-2      7.4
 * COM003-2     22.1    [Brushi, 1991]
 * COM004-1     45.1
 * GEO003-1     71.7    T3 [McCharen, et al., 1976]; t3.ver1.in [ANL]
 * GEO017-2     78.8    D4.1 [Quaife, 1989]
 * GEO027-3    181.5    D10.1 [Quaife, 1989]
 * GEO058-2    104.0    R4 [Quaife, 1989]
 * GEO079-1      2.4    GEOMETRY THEOREM [Slagle, 1967]
 * 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]
 * GRP008-1     50.4    Problem 4 [Wos, 1965]; wos4 [Wilson & Minker, 1976]
 * GRP013-1     40.2    Problem 11 [Wos, 1965]; wos11 [Wilson & Minker, 1976]
 * GRP037-3     43.8    Problem 17 [Wos, 1965]; wos17 [Wilson & Minker, 1976]
 * GRP031-2      3.2    ls23 [Lawrence & Starkey, 1974]; ls23 [Wilson & Minker, 1976]
 * GRP034-4      2.5    ls26 [Lawrence & Starkey, 1974]; ls26 [Wilson & Minker, 1976]
 * GRP047-2     11.7    [Veroff, 1992]
 * GRP130-1    170.6    Bennett QG8 [TPTP]; QG8 [Slaney, 1993]
 * GRP156-1     48.7    ax_mono1c [Schulz, 1995]
 * GRP168-1    159.1    p01a [Schulz, 1995]
 * HEN003-3     39.9    HP3 [McCharen, et al., 1976]
 * HEN007-2    125.7    H7 [McCharen, et al., 1976]
 * HEN008-4     62.0    H8 [McCharen, et al., 1976]
 * HEN009-5    136.3    H9 [McCharen, et al., 1976]; hp9.ver3.in [ANL]
 * HEN012-3     48.5    new.ver2.in [ANL]
 * LCL010-1    370.9    EC-73 [McCune & Wos, 1992]; ec_yq.in [OTTER]
 * LCL077-2     51.6    morgan.two.ver1.in [ANL]
 * LCL082-1     14.6    IC-1.1 [Wos, et al., 1990]; IC-65 [McCune & Wos, 1992]; ls2 [SETHEO]; S1 [Pfenning, 1988]
 * 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]
 * LCL143-1     10.9    Lattice structure theorem 2 [Bonacina, 1991]
 * LCL182-1    271.6    Problem 2.16 [Whitehead & Russell, 1927]
 * LCL200-1     12.0    Problem 2.46 [Whitehead & Russell, 1927]
 * LCL215-1    214.4    Problem 2.62 [Whitehead & Russell, 1927]; Problem 2.63 [Whitehead & Russell, 1927]
 * LCL230-2      0.2    Pelletier 5 [Pelletier, 1986]
 * LDA003-1     68.5    Problem 3 [Jech, 1993]
 * MSC002-1      9.2    DBABHP [Michie, et al., 1972]; DBABHP [Wilson & Minker, 1976]
 * MSC003-1      3.2    HASPARTS-T1 [Wilson & Minker, 1976]
 * MSC004-1      9.3    HASPARTS-T2 [Wilson & Minker, 1976]
 * MSC005-1      1.8    Problem 5.1 [Plaisted, 1982]
 * MSC006-1     39.0    nonob.lop [SETHEO]
 * NUM001-1     14.0    Chang-Lee-10a [Chang, 1970]; ls28 [Lawrence & Starkey, 1974]; ls28 [Wilson & Minker, 1976]
 * NUM021-1     52.3    ls65 [Lawrence & Starkey, 1974]; ls65 [Wilson & Minker, 1976]
 * NUM024-1     64.6    ls75 [Lawrence & Starkey, 1974]; ls75 [Wilson & Minker, 1976]
 * NUM180-1    621.2    LIM2.1 [Quaife]
 * NUM228-1    575.9    TRECDEF4 cor. [Quaife]
 * PLA002-1     37.4    Problem 5.7 [Plaisted, 1982]
 * PLA006-1      7.2    [Segre & Elkan, 1994]
 * PLA017-1    484.8    [Segre & Elkan, 1994]
 * PLA022-1     19.1    [Segre & Elkan, 1994]
 * PLA022-2     19.7    [Segre & Elkan, 1994]
 * PRV001-1     10.3    PV1 [McCharen, et al., 1976]
 * PRV003-1      3.9    E2 [McCharen, et al., 1976]; v2.lop [SETHEO]
 * PRV005-1      4.3    E4 [McCharen, et al., 1976]; v4.lop [SETHEO]
 * PRV006-1      6.0    E5 [McCharen, et al., 1976]; v5.lop [SETHEO]
 * PRV009-1      2.2    Hoares FIND [Bledsoe, 1977]; Problem 5.5 [Plaisted, 1982]
 * PUZ012-1      3.5    Boxes-of-fruit [Wos, 1988]; Boxes-of-fruit [Wos, et al., 1992]; boxes.ver1.in [ANL]
 * PUZ020-1     56.6    knightknave.in [ANL]
 * PUZ025-1     58.4    Problem 35 [Smullyan, 1978]; tandl35.ver1.in [ANL]
 * PUZ029-1      5.1    pigs.ver1.in [ANL]
 * RNG001-3     82.4    EX6-T? [Wilson & Minker, 1976]; ex6.lop [SETHEO]; Example 6a [Fleisig, et al., 1974]; FEX6T1 [SPRFN]; FEX6T2 [SPRFN]
 * RNG001-5    399.8    Problem 21 [Wos, 1965]; wos21 [Wilson & Minker, 1976]
 * RNG011-5      8.4    CADE-11 Competition Eq-10 [Overbeek, 1990]; PROBLEM 10 [Zhang, 1993]; THEOREM EQ-10 [Lusk & McCune, 1993]
 * RNG023-6      9.1    [Stevens, 1987]
 * RNG028-2      9.3    PROOF III [Anantharaman & Hsiang, 1990]
 * RNG038-2     16.2    Problem 27 [Wos, 1965]; wos27 [Wilson & Minker, 1976]
 * RNG040-2    180.5    Problem 29 [Wos, 1965]; wos29 [Wilson & Minker, 1976]
 * RNG041-1     35.8    Problem 30 [Wos, 1965]; wos30 [Wilson & Minker, 1976]
 * ROB010-1    205.0    Lemma 3.3 [Winker, 1990]; RA2 [Lusk & Wos, 1992]
 * ROB013-1     23.6    Lemma 3.5 [Winker, 1990]
 * ROB016-1     15.2    Corollary 3.7 [Winker, 1990]
 * ROB021-1    230.4    [McCune, 1992]
 * SET005-1    192.2    ls108 [Lawrence & Starkey, 1974]; ls108 [Wilson & Minker, 1976]
 * SET009-1     10.5    ls116 [Lawrence & Starkey, 1974]; ls116 [Wilson & Minker, 1976]
 * SET025-4    694.7    Lemma 10 [Boyer, et al, 1986]
 * SET046-5      2.3    p42.in [ANL]; Pelletier 42 [Pelletier, 1986]
 * SET047-5      3.7    p43.in [ANL]; Pelletier 43 [Pelletier, 1986]
 * SYN034-1      2.8    QW [Michie, et al., 1972]; QW [Wilson & Minker, 1976]
 * SYN071-1      1.9    Pelletier 48 [Pelletier, 1986]
 * SYN349-1     61.7    Ch17N5 [Tammet, 1994]
 * SYN352-1      5.5    Ch18N4 [Tammet, 1994]
 * TOP001-2     61.1    Lemma 1a [Wick & McCune, 1989]
 * TOP002-2      0.4    Lemma 1b [Wick & McCune, 1989]
 * TOP004-1    181.6    Lemma 1d [Wick & McCune, 1989]
 * TOP004-2      9.0    Lemma 1d [Wick & McCune, 1989]
 * TOP005-2    139.8    Lemma 1e [Wick & McCune, 1989]
 *)

(*51194 inferences so far.  Searching to depth 13.  232.9 secs*)
val BOO003_1 = prove_hard
 ("(! X. equal(X::'a,X)) &  \
\  (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
\  (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) &  \
\  (! X Y. sum(X::'a,Y,add(X::'a,Y))) & \
\  (! X Y. product(X::'a,Y,multiply(X::'a,Y))) &        \
\  (! Y X Z. sum(X::'a,Y,Z) --> sum(Y::'a,X,Z)) &       \
\  (! Y X Z. product(X::'a,Y,Z) --> product(Y::'a,X,Z)) &       \
\  (! X. sum(additive_identity::'a,X,X)) &  \
\  (! X. sum(X::'a,additive_identity,X)) &  \
\  (! X. product(multiplicative_identity::'a,X,X)) &        \
\  (! X. product(X::'a,multiplicative_identity,X)) &        \
\  (! 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)) &        \
\  (! 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)) &        \
\  (! 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)) &        \
\  (! 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)) &        \
\  (! 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)) &    \
\  (! 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)) &    \
\  (! 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)) &    \
\  (! 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)) &    \
\  (! X. sum(inverse(X),X,multiplicative_identity)) &   \
\  (! X. sum(X::'a,inverse(X),multiplicative_identity)) &   \
\  (! X. product(inverse(X),X,additive_identity)) &     \
\  (! X. product(X::'a,inverse(X),additive_identity)) &     \
\  (! X Y U V. sum(X::'a,Y,U) & sum(X::'a,Y,V) --> equal(U::'a,V)) &        \
\  (! X Y U V. product(X::'a,Y,U) & product(X::'a,Y,V) --> equal(U::'a,V)) &        \
\  (! X Y W Z. equal(X::'a,Y) & sum(X::'a,W,Z) --> sum(Y::'a,W,Z)) &        \
\  (! X W Y Z. equal(X::'a,Y) & sum(W::'a,X,Z) --> sum(W::'a,Y,Z)) &        \
\  (! X W Z Y. equal(X::'a,Y) & sum(W::'a,Z,X) --> sum(W::'a,Z,Y)) &        \
\  (! X Y W Z. equal(X::'a,Y) & product(X::'a,W,Z) --> product(Y::'a,W,Z)) &        \
\  (! X W Y Z. equal(X::'a,Y) & product(W::'a,X,Z) --> product(W::'a,Y,Z)) &        \
\  (! X W Z Y. equal(X::'a,Y) & product(W::'a,Z,X) --> product(W::'a,Z,Y)) &        \
\  (! X Y W. equal(X::'a,Y) --> equal(add(X::'a,W),add(Y::'a,W))) & \
\  (! X W Y. equal(X::'a,Y) --> equal(add(W::'a,X),add(W::'a,Y))) & \
\  (! X Y W. equal(X::'a,Y) --> equal(multiply(X::'a,W),multiply(Y::'a,W))) &       \
\  (! X W Y. equal(X::'a,Y) --> equal(multiply(W::'a,X),multiply(W::'a,Y))) &       \
\  (! X Y. equal(X::'a,Y) --> equal(inverse(X),inverse(Y))) &       \
\  (~product(x::'a,x,x)) --> False",
  meson_tac);

(*51194 inferences so far.  Searching to depth 13. 204.6 secs
  Strange!  The previous problem also has 51194 inferences at depth 13.  They
   must be very similar!*)
val BOO004_1 = prove_hard
 ("(! X. equal(X::'a,X)) &  \
\  (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
\  (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) &  \
\  (! X Y. sum(X::'a,Y,add(X::'a,Y))) & \
\  (! X Y. product(X::'a,Y,multiply(X::'a,Y))) &        \
\  (! Y X Z. sum(X::'a,Y,Z) --> sum(Y::'a,X,Z)) &       \
\  (! Y X Z. product(X::'a,Y,Z) --> product(Y::'a,X,Z)) &       \
\  (! X. sum(additive_identity::'a,X,X)) &  \
\  (! X. sum(X::'a,additive_identity,X)) &  \
\  (! X. product(multiplicative_identity::'a,X,X)) &        \
\  (! X. product(X::'a,multiplicative_identity,X)) &        \
\  (! 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)) &        \
\  (! 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)) &        \
\  (! 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)) &        \
\  (! 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)) &        \
\  (! 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)) &    \
\  (! 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)) &    \
\  (! 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)) &    \
\  (! 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)) &    \
\  (! X. sum(inverse(X),X,multiplicative_identity)) &   \
\  (! X. sum(X::'a,inverse(X),multiplicative_identity)) &   \
\  (! X. product(inverse(X),X,additive_identity)) &     \
\  (! X. product(X::'a,inverse(X),additive_identity)) &     \
\  (! X Y U V. sum(X::'a,Y,U) & sum(X::'a,Y,V) --> equal(U::'a,V)) &        \
\  (! X Y U V. product(X::'a,Y,U) & product(X::'a,Y,V) --> equal(U::'a,V)) &        \
\  (! X Y W Z. equal(X::'a,Y) & sum(X::'a,W,Z) --> sum(Y::'a,W,Z)) &        \
\  (! X W Y Z. equal(X::'a,Y) & sum(W::'a,X,Z) --> sum(W::'a,Y,Z)) &        \
\  (! X W Z Y. equal(X::'a,Y) & sum(W::'a,Z,X) --> sum(W::'a,Z,Y)) &        \
\  (! X Y W Z. equal(X::'a,Y) & product(X::'a,W,Z) --> product(Y::'a,W,Z)) &        \
\  (! X W Y Z. equal(X::'a,Y) & product(W::'a,X,Z) --> product(W::'a,Y,Z)) &        \
\  (! X W Z Y. equal(X::'a,Y) & product(W::'a,Z,X) --> product(W::'a,Z,Y)) &        \
\  (! X Y W. equal(X::'a,Y) --> equal(add(X::'a,W),add(Y::'a,W))) & \
\  (! X W Y. equal(X::'a,Y) --> equal(add(W::'a,X),add(W::'a,Y))) & \
\  (! X Y W. equal(X::'a,Y) --> equal(multiply(X::'a,W),multiply(Y::'a,W))) &       \
\  (! X W Y. equal(X::'a,Y) --> equal(multiply(W::'a,X),multiply(W::'a,Y))) &       \
\  (! X Y. equal(X::'a,Y) --> equal(inverse(X),inverse(Y))) &       \
\  (~sum(x::'a,x,x)) --> False",
  meson_tac);

(*74799 inferences so far.  Searching to depth 13.  290.0 secs*)
val BOO005_1 = prove_hard
 ("(! X. equal(X::'a,X)) &  \
\  (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
\  (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) &  \
\  (! X Y. sum(X::'a,Y,add(X::'a,Y))) & \
\  (! X Y. product(X::'a,Y,multiply(X::'a,Y))) &        \
\  (! Y X Z. sum(X::'a,Y,Z) --> sum(Y::'a,X,Z)) &       \
\  (! Y X Z. product(X::'a,Y,Z) --> product(Y::'a,X,Z)) &       \
\  (! X. sum(additive_identity::'a,X,X)) &  \
\  (! X. sum(X::'a,additive_identity,X)) &  \
\  (! X. product(multiplicative_identity::'a,X,X)) &        \
\  (! X. product(X::'a,multiplicative_identity,X)) &        \
\  (! 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)) &        \
\  (! 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)) &        \
\  (! 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)) &        \
\  (! 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)) &        \
\  (! 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)) &    \
\  (! 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)) &    \
\  (! 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)) &    \
\  (! 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)) &    \
\  (! X. sum(inverse(X),X,multiplicative_identity)) &   \
\  (! X. sum(X::'a,inverse(X),multiplicative_identity)) &   \
\  (! X. product(inverse(X),X,additive_identity)) &     \
\  (! X. product(X::'a,inverse(X),additive_identity)) &     \
\  (! X Y U V. sum(X::'a,Y,U) & sum(X::'a,Y,V) --> equal(U::'a,V)) &        \
\  (! X Y U V. product(X::'a,Y,U) & product(X::'a,Y,V) --> equal(U::'a,V)) &        \
\  (! X Y W Z. equal(X::'a,Y) & sum(X::'a,W,Z) --> sum(Y::'a,W,Z)) &        \
\  (! X W Y Z. equal(X::'a,Y) & sum(W::'a,X,Z) --> sum(W::'a,Y,Z)) &        \
\  (! X W Z Y. equal(X::'a,Y) & sum(W::'a,Z,X) --> sum(W::'a,Z,Y)) &        \
\  (! X Y W Z. equal(X::'a,Y) & product(X::'a,W,Z) --> product(Y::'a,W,Z)) &        \
\  (! X W Y Z. equal(X::'a,Y) & product(W::'a,X,Z) --> product(W::'a,Y,Z)) &        \
\  (! X W Z Y. equal(X::'a,Y) & product(W::'a,Z,X) --> product(W::'a,Z,Y)) &        \
\  (! X Y W. equal(X::'a,Y) --> equal(add(X::'a,W),add(Y::'a,W))) & \
\  (! X W Y. equal(X::'a,Y) --> equal(add(W::'a,X),add(W::'a,Y))) & \
\  (! X Y W. equal(X::'a,Y) --> equal(multiply(X::'a,W),multiply(Y::'a,W))) &       \
\  (! X W Y. equal(X::'a,Y) --> equal(multiply(W::'a,X),multiply(W::'a,Y))) &       \
\  (! X Y. equal(X::'a,Y) --> equal(inverse(X),inverse(Y))) &       \
\  (~sum(x::'a,multiplicative_identity,multiplicative_identity)) --> False",
  meson_tac);

(*74799 inferences so far.  Searching to depth 13.  314.6 secs*)
val BOO006_1 = prove_hard
 ("(! X. equal(X::'a,X)) &  \
\  (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
\  (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) &  \
\  (! X Y. sum(X::'a,Y,add(X::'a,Y))) & \
\  (! X Y. product(X::'a,Y,multiply(X::'a,Y))) &        \
\  (! Y X Z. sum(X::'a,Y,Z) --> sum(Y::'a,X,Z)) &       \
\  (! Y X Z. product(X::'a,Y,Z) --> product(Y::'a,X,Z)) &       \
\  (! X. sum(additive_identity::'a,X,X)) &  \
\  (! X. sum(X::'a,additive_identity,X)) &  \
\  (! X. product(multiplicative_identity::'a,X,X)) &        \
\  (! X. product(X::'a,multiplicative_identity,X)) &        \
\  (! 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)) &        \
\  (! 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)) &        \
\  (! 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)) &        \
\  (! 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)) &        \
\  (! 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)) &    \
\  (! 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)) &    \
\  (! 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)) &    \
\  (! 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)) &    \
\  (! X. sum(inverse(X),X,multiplicative_identity)) &   \
\  (! X. sum(X::'a,inverse(X),multiplicative_identity)) &   \
\  (! X. product(inverse(X),X,additive_identity)) &     \
\  (! X. product(X::'a,inverse(X),additive_identity)) &     \
\  (! X Y U V. sum(X::'a,Y,U) & sum(X::'a,Y,V) --> equal(U::'a,V)) &        \
\  (! X Y U V. product(X::'a,Y,U) & product(X::'a,Y,V) --> equal(U::'a,V)) &        \
\  (! X Y W Z. equal(X::'a,Y) & sum(X::'a,W,Z) --> sum(Y::'a,W,Z)) &        \
\  (! X W Y Z. equal(X::'a,Y) & sum(W::'a,X,Z) --> sum(W::'a,Y,Z)) &        \
\  (! X W Z Y. equal(X::'a,Y) & sum(W::'a,Z,X) --> sum(W::'a,Z,Y)) &        \
\  (! X Y W Z. equal(X::'a,Y) & product(X::'a,W,Z) --> product(Y::'a,W,Z)) &        \
\  (! X W Y Z. equal(X::'a,Y) & product(W::'a,X,Z) --> product(W::'a,Y,Z)) &        \
\  (! X W Z Y. equal(X::'a,Y) & product(W::'a,Z,X) --> product(W::'a,Z,Y)) &        \
\  (! X Y W. equal(X::'a,Y) --> equal(add(X::'a,W),add(Y::'a,W))) & \
\  (! X W Y. equal(X::'a,Y) --> equal(add(W::'a,X),add(W::'a,Y))) & \
\  (! X Y W. equal(X::'a,Y) --> equal(multiply(X::'a,W),multiply(Y::'a,W))) &       \
\  (! X W Y. equal(X::'a,Y) --> equal(multiply(W::'a,X),multiply(W::'a,Y))) &       \
\  (! X Y. equal(X::'a,Y) --> equal(inverse(X),inverse(Y))) &       \
\  (~product(x::'a,additive_identity,additive_identity)) --> False",
  meson_tac);

(*5 inferences so far.  Searching to depth 5.  1.3 secs*)
val BOO011_1 = prove
 ("(! X. equal(X::'a,X)) &  \
\  (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
\  (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) &  \
\  (! X Y. sum(X::'a,Y,add(X::'a,Y))) & \
\  (! X Y. product(X::'a,Y,multiply(X::'a,Y))) &        \
\  (! Y X Z. sum(X::'a,Y,Z) --> sum(Y::'a,X,Z)) &       \
\  (! Y X Z. product(X::'a,Y,Z) --> product(Y::'a,X,Z)) &       \
\  (! X. sum(additive_identity::'a,X,X)) &  \
\  (! X. sum(X::'a,additive_identity,X)) &  \
\  (! X. product(multiplicative_identity::'a,X,X)) &        \
\  (! X. product(X::'a,multiplicative_identity,X)) &        \
\  (! 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)) &        \
\  (! 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)) &        \
\  (! 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)) &        \
\  (! 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)) &        \
\  (! 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)) &    \
\  (! 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)) &    \
\  (! 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)) &    \
\  (! 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)) &    \
\  (! X. sum(inverse(X),X,multiplicative_identity)) &   \
\  (! X. sum(X::'a,inverse(X),multiplicative_identity)) &   \
\  (! X. product(inverse(X),X,additive_identity)) &     \
\  (! X. product(X::'a,inverse(X),additive_identity)) &     \
\  (! X Y U V. sum(X::'a,Y,U) & sum(X::'a,Y,V) --> equal(U::'a,V)) &        \
\  (! X Y U V. product(X::'a,Y,U) & product(X::'a,Y,V) --> equal(U::'a,V)) &        \
\  (! X Y W Z. equal(X::'a,Y) & sum(X::'a,W,Z) --> sum(Y::'a,W,Z)) &        \
\  (! X W Y Z. equal(X::'a,Y) & sum(W::'a,X,Z) --> sum(W::'a,Y,Z)) &        \
\  (! X W Z Y. equal(X::'a,Y) & sum(W::'a,Z,X) --> sum(W::'a,Z,Y)) &        \
\  (! X Y W Z. equal(X::'a,Y) & product(X::'a,W,Z) --> product(Y::'a,W,Z)) &        \
\  (! X W Y Z. equal(X::'a,Y) & product(W::'a,X,Z) --> product(W::'a,Y,Z)) &        \
\  (! X W Z Y. equal(X::'a,Y) & product(W::'a,Z,X) --> product(W::'a,Z,Y)) &        \
\  (! X Y W. equal(X::'a,Y) --> equal(add(X::'a,W),add(Y::'a,W))) & \
\  (! X W Y. equal(X::'a,Y) --> equal(add(W::'a,X),add(W::'a,Y))) & \
\  (! X Y W. equal(X::'a,Y) --> equal(multiply(X::'a,W),multiply(Y::'a,W))) &       \
\  (! X W Y. equal(X::'a,Y) --> equal(multiply(W::'a,X),multiply(W::'a,Y))) &       \
\  (! X Y. equal(X::'a,Y) --> equal(inverse(X),inverse(Y))) &       \
\  (~equal(inverse(additive_identity),multiplicative_identity)) --> False",
  meson_tac);

(*4007 inferences so far.  Searching to depth 9.  13 secs*)
val CAT001_3 = prove_hard
 ("(! X. equal(X::'a,X)) &  \
\  (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
\  (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) &  \
\  (! Y X. equivalent(X::'a,Y) --> there_exists(X)) &       \
\  (! X Y. equivalent(X::'a,Y) --> equal(X::'a,Y)) &    \
\  (! X Y. there_exists(X) & equal(X::'a,Y) --> equivalent(X::'a,Y)) &  \
\  (! X. there_exists(domain(X)) --> there_exists(X)) & \
\  (! X. there_exists(codomain(X)) --> there_exists(X)) &       \
\  (! Y X. there_exists(compos(X::'a,Y)) --> there_exists(domain(X))) &    \
\  (! X Y. there_exists(compos(X::'a,Y)) --> equal(domain(X),codomain(Y))) &       \
\  (! X Y. there_exists(domain(X)) & equal(domain(X),codomain(Y)) --> there_exists(compos(X::'a,Y))) &     \
\  (! X Y Z. equal(compos(X::'a,compos(Y::'a,Z)),compos(compos(X::'a,Y),Z))) &  \
\  (! X. equal(compos(X::'a,domain(X)),X)) &       \
\  (! X. equal(compos(codomain(X),X),X)) &     \
\  (! X Y. equivalent(X::'a,Y) --> there_exists(Y)) &       \
\  (! X Y. there_exists(X) & there_exists(Y) & equal(X::'a,Y) --> equivalent(X::'a,Y)) &        \
\  (! Y X. there_exists(compos(X::'a,Y)) --> there_exists(codomain(X))) &  \
\  (! X Y. there_exists(f1(X::'a,Y)) | equal(X::'a,Y)) &       \
\  (! X Y. equal(X::'a,f1(X::'a,Y)) | equal(Y::'a,f1(X::'a,Y)) | equal(X::'a,Y)) &        \
\  (! X Y. equal(X::'a,f1(X::'a,Y)) & equal(Y::'a,f1(X::'a,Y)) --> equal(X::'a,Y)) &        \
\  (! X Y. equal(X::'a,Y) & there_exists(X) --> there_exists(Y)) &  \
\  (! X Y Z. equal(X::'a,Y) & equivalent(X::'a,Z) --> equivalent(Y::'a,Z)) &        \
\  (! X Z Y. equal(X::'a,Y) & equivalent(Z::'a,X) --> equivalent(Z::'a,Y)) &        \
\  (! X Y. equal(X::'a,Y) --> equal(domain(X),domain(Y))) & \
\  (! X Y. equal(X::'a,Y) --> equal(codomain(X),codomain(Y))) &     \
\  (! X Y Z. equal(X::'a,Y) --> equal(compos(X::'a,Z),compos(Y::'a,Z))) & \
\  (! X Z Y. equal(X::'a,Y) --> equal(compos(Z::'a,X),compos(Z::'a,Y))) & \
\  (! A B C. equal(A::'a,B) --> equal(f1(A::'a,C),f1(B::'a,C))) &   \
\  (! D F' E. equal(D::'a,E) --> equal(f1(F'::'a,D),f1(F'::'a,E))) &        \
\  (there_exists(compos(a::'a,b))) &       \
\  (! Y X Z. equal(compos(compos(a::'a,b),X),Y) & equal(compos(compos(a::'a,b),Z),Y) --> equal(X::'a,Z)) &      \
\  (there_exists(compos(b::'a,h))) &       \
\  (equal(compos(b::'a,h),compos(b::'a,g))) & \
\  (~equal(h::'a,g)) --> False",
  meson_tac);

(*245 inferences so far.  Searching to depth 7.  1.0 secs*)
val CAT003_3 = prove
 ("(! X. equal(X::'a,X)) &  \
\  (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
\  (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) &  \
\  (! Y X. equivalent(X::'a,Y) --> there_exists(X)) &       \
\  (! X Y. equivalent(X::'a,Y) --> equal(X::'a,Y)) &    \
\  (! X Y. there_exists(X) & equal(X::'a,Y) --> equivalent(X::'a,Y)) &  \
\  (! X. there_exists(domain(X)) --> there_exists(X)) & \
\  (! X. there_exists(codomain(X)) --> there_exists(X)) &       \
\  (! Y X. there_exists(compos(X::'a,Y)) --> there_exists(domain(X))) &    \
\  (! X Y. there_exists(compos(X::'a,Y)) --> equal(domain(X),codomain(Y))) &       \
\  (! X Y. there_exists(domain(X)) & equal(domain(X),codomain(Y)) --> there_exists(compos(X::'a,Y))) &     \
\  (! X Y Z. equal(compos(X::'a,compos(Y::'a,Z)),compos(compos(X::'a,Y),Z))) &  \
\  (! X. equal(compos(X::'a,domain(X)),X)) &       \
\  (! X. equal(compos(codomain(X),X),X)) &     \
\  (! X Y. equivalent(X::'a,Y) --> there_exists(Y)) &       \
\  (! X Y. there_exists(X) & there_exists(Y) & equal(X::'a,Y) --> equivalent(X::'a,Y)) &        \
\  (! Y X. there_exists(compos(X::'a,Y)) --> there_exists(codomain(X))) &  \
\  (! X Y. there_exists(f1(X::'a,Y)) | equal(X::'a,Y)) &       \
\  (! X Y. equal(X::'a,f1(X::'a,Y)) | equal(Y::'a,f1(X::'a,Y)) | equal(X::'a,Y)) &        \
\  (! X Y. equal(X::'a,f1(X::'a,Y)) & equal(Y::'a,f1(X::'a,Y)) --> equal(X::'a,Y)) &        \
\  (! X Y. equal(X::'a,Y) & there_exists(X) --> there_exists(Y)) &  \
\  (! X Y Z. equal(X::'a,Y) & equivalent(X::'a,Z) --> equivalent(Y::'a,Z)) &        \
\  (! X Z Y. equal(X::'a,Y) & equivalent(Z::'a,X) --> equivalent(Z::'a,Y)) &        \
\  (! X Y. equal(X::'a,Y) --> equal(domain(X),domain(Y))) & \
\  (! X Y. equal(X::'a,Y) --> equal(codomain(X),codomain(Y))) &     \
\  (! X Y Z. equal(X::'a,Y) --> equal(compos(X::'a,Z),compos(Y::'a,Z))) & \
\  (! X Z Y. equal(X::'a,Y) --> equal(compos(Z::'a,X),compos(Z::'a,Y))) & \
\  (! A B C. equal(A::'a,B) --> equal(f1(A::'a,C),f1(B::'a,C))) &   \
\  (! D F' E. equal(D::'a,E) --> equal(f1(F'::'a,D),f1(F'::'a,E))) &        \
\  (there_exists(compos(a::'a,b))) &       \
\  (! 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)) &      \
\  (there_exists(h)) &  \
\  (equal(compos(h::'a,a),compos(g::'a,a))) & \
\  (~equal(g::'a,h)) --> False",
  meson_tac);

(*54288 inferences so far.  Searching to depth 14.  118.0 secs*)
val CAT005_1 = prove_hard
 ("(! X. equal(X::'a,X)) &  \
\  (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
\  (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) &  \
\  (! X Y. defined(X::'a,Y) --> product(X::'a,Y,compos(X::'a,Y))) &        \
\  (! Z X Y. product(X::'a,Y,Z) --> defined(X::'a,Y)) & \
\  (! X Xy Y Z. product(X::'a,Y,Xy) & defined(Xy::'a,Z) --> defined(Y::'a,Z)) &     \
\  (! Y Xy Z X Yz. product(X::'a,Y,Xy) & product(Y::'a,Z,Yz) & defined(Xy::'a,Z) --> defined(X::'a,Yz)) &       \
\  (! 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)) &   \
\  (! Z Yz X Y. product(Y::'a,Z,Yz) & defined(X::'a,Yz) --> defined(X::'a,Y)) &     \
\  (! Y X Yz Xy Z. product(Y::'a,Z,Yz) & product(X::'a,Y,Xy) & defined(X::'a,Yz) --> defined(Xy::'a,Z)) &       \
\  (! 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)) &   \
\  (! Y X Z. defined(X::'a,Y) & defined(Y::'a,Z) & identity_map(Y) --> defined(X::'a,Z)) &  \
\  (! X. identity_map(domain(X))) &     \
\  (! X. identity_map(codomain(X))) &   \
\  (! X. defined(X::'a,domain(X))) &        \
\  (! X. defined(codomain(X),X)) &      \
\  (! X. product(X::'a,domain(X),X)) &      \
\  (! X. product(codomain(X),X,X)) &    \
\  (! X Y. defined(X::'a,Y) & identity_map(X) --> product(X::'a,Y,Y)) & \
\  (! Y X. defined(X::'a,Y) & identity_map(Y) --> product(X::'a,Y,X)) & \
\  (! X Y Z W. product(X::'a,Y,Z) & product(X::'a,Y,W) --> equal(Z::'a,W)) &        \
\  (! X Y Z W. equal(X::'a,Y) & product(X::'a,Z,W) --> product(Y::'a,Z,W)) &        \
\  (! X Z Y W. equal(X::'a,Y) & product(Z::'a,X,W) --> product(Z::'a,Y,W)) &        \
\  (! X Z W Y. equal(X::'a,Y) & product(Z::'a,W,X) --> product(Z::'a,W,Y)) &        \
\  (! X Y. equal(X::'a,Y) --> equal(domain(X),domain(Y))) & \
\  (! X Y. equal(X::'a,Y) --> equal(codomain(X),codomain(Y))) &     \
\  (! X Y. equal(X::'a,Y) & identity_map(X) --> identity_map(Y)) &  \
\  (! X Y Z. equal(X::'a,Y) & defined(X::'a,Z) --> defined(Y::'a,Z)) &      \
\  (! X Z Y. equal(X::'a,Y) & defined(Z::'a,X) --> defined(Z::'a,Y)) &      \
\  (! X Z Y. equal(X::'a,Y) --> equal(compos(Z::'a,X),compos(Z::'a,Y))) & \
\  (! X Y Z. equal(X::'a,Y) --> equal(compos(X::'a,Z),compos(Y::'a,Z))) & \
\  (defined(a::'a,d)) &     \
\  (identity_map(d)) &  \
\  (~equal(domain(a),d)) --> False",
  meson_tac);


(*1728 inferences so far.  Searching to depth 10.  5.8 secs*)
val CAT007_1 = prove_hard
 ("(! X. equal(X::'a,X)) &  \
\  (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
\  (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) &  \
\  (! X Y. defined(X::'a,Y) --> product(X::'a,Y,compos(X::'a,Y))) &        \
\  (! Z X Y. product(X::'a,Y,Z) --> defined(X::'a,Y)) & \
\  (! X Xy Y Z. product(X::'a,Y,Xy) & defined(Xy::'a,Z) --> defined(Y::'a,Z)) &     \
\  (! Y Xy Z X Yz. product(X::'a,Y,Xy) & product(Y::'a,Z,Yz) & defined(Xy::'a,Z) --> defined(X::'a,Yz)) &       \
\  (! 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)) &   \
\  (! Z Yz X Y. product(Y::'a,Z,Yz) & defined(X::'a,Yz) --> defined(X::'a,Y)) &     \
\  (! Y X Yz Xy Z. product(Y::'a,Z,Yz) & product(X::'a,Y,Xy) & defined(X::'a,Yz) --> defined(Xy::'a,Z)) &       \
\  (! 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)) &   \
\  (! Y X Z. defined(X::'a,Y) & defined(Y::'a,Z) & identity_map(Y) --> defined(X::'a,Z)) &  \
\  (! X. identity_map(domain(X))) &     \
\  (! X. identity_map(codomain(X))) &   \
\  (! X. defined(X::'a,domain(X))) &        \
\  (! X. defined(codomain(X),X)) &      \
\  (! X. product(X::'a,domain(X),X)) &      \
\  (! X. product(codomain(X),X,X)) &    \
\  (! X Y. defined(X::'a,Y) & identity_map(X) --> product(X::'a,Y,Y)) & \
\  (! Y X. defined(X::'a,Y) & identity_map(Y) --> product(X::'a,Y,X)) & \
\  (! X Y Z W. product(X::'a,Y,Z) & product(X::'a,Y,W) --> equal(Z::'a,W)) &        \
\  (! X Y Z W. equal(X::'a,Y) & product(X::'a,Z,W) --> product(Y::'a,Z,W)) &        \
\  (! X Z Y W. equal(X::'a,Y) & product(Z::'a,X,W) --> product(Z::'a,Y,W)) &        \
\  (! X Z W Y. equal(X::'a,Y) & product(Z::'a,W,X) --> product(Z::'a,W,Y)) &        \
\  (! X Y. equal(X::'a,Y) --> equal(domain(X),domain(Y))) & \
\  (! X Y. equal(X::'a,Y) --> equal(codomain(X),codomain(Y))) &     \
\  (! X Y. equal(X::'a,Y) & identity_map(X) --> identity_map(Y)) &  \
\  (! X Y Z. equal(X::'a,Y) & defined(X::'a,Z) --> defined(Y::'a,Z)) &      \
\  (! X Z Y. equal(X::'a,Y) & defined(Z::'a,X) --> defined(Z::'a,Y)) &      \
\  (! X Z Y. equal(X::'a,Y) --> equal(compos(Z::'a,X),compos(Z::'a,Y))) & \
\  (! X Y Z. equal(X::'a,Y) --> equal(compos(X::'a,Z),compos(Y::'a,Z))) & \
\  (equal(domain(a),codomain(b))) &     \
\  (~defined(a::'a,b)) --> False",
  meson_tac);

(*82895 inferences so far.  Searching to depth 13.  355 secs*)
val CAT018_1 = prove_hard
 ("(! X. equal(X::'a,X)) &  \
\  (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
\  (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) &  \
\  (! X Y. defined(X::'a,Y) --> product(X::'a,Y,compos(X::'a,Y))) &        \
\  (! Z X Y. product(X::'a,Y,Z) --> defined(X::'a,Y)) & \
\  (! X Xy Y Z. product(X::'a,Y,Xy) & defined(Xy::'a,Z) --> defined(Y::'a,Z)) &     \
\  (! Y Xy Z X Yz. product(X::'a,Y,Xy) & product(Y::'a,Z,Yz) & defined(Xy::'a,Z) --> defined(X::'a,Yz)) &       \
\  (! 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)) &   \
\  (! Z Yz X Y. product(Y::'a,Z,Yz) & defined(X::'a,Yz) --> defined(X::'a,Y)) &     \
\  (! Y X Yz Xy Z. product(Y::'a,Z,Yz) & product(X::'a,Y,Xy) & defined(X::'a,Yz) --> defined(Xy::'a,Z)) &       \
\  (! 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)) &   \
\  (! Y X Z. defined(X::'a,Y) & defined(Y::'a,Z) & identity_map(Y) --> defined(X::'a,Z)) &  \
\  (! X. identity_map(domain(X))) &     \
\  (! X. identity_map(codomain(X))) &   \
\  (! X. defined(X::'a,domain(X))) &        \
\  (! X. defined(codomain(X),X)) &      \
\  (! X. product(X::'a,domain(X),X)) &      \
\  (! X. product(codomain(X),X,X)) &    \
\  (! X Y. defined(X::'a,Y) & identity_map(X) --> product(X::'a,Y,Y)) & \
\  (! Y X. defined(X::'a,Y) & identity_map(Y) --> product(X::'a,Y,X)) & \
\  (! X Y Z W. product(X::'a,Y,Z) & product(X::'a,Y,W) --> equal(Z::'a,W)) &        \
\  (! X Y Z W. equal(X::'a,Y) & product(X::'a,Z,W) --> product(Y::'a,Z,W)) &        \
\  (! X Z Y W. equal(X::'a,Y) & product(Z::'a,X,W) --> product(Z::'a,Y,W)) &        \
\  (! X Z W Y. equal(X::'a,Y) & product(Z::'a,W,X) --> product(Z::'a,W,Y)) &        \
\  (! X Y. equal(X::'a,Y) --> equal(domain(X),domain(Y))) & \
\  (! X Y. equal(X::'a,Y) --> equal(codomain(X),codomain(Y))) &     \
\  (! X Y. equal(X::'a,Y) & identity_map(X) --> identity_map(Y)) &  \
\  (! X Y Z. equal(X::'a,Y) & defined(X::'a,Z) --> defined(Y::'a,Z)) &      \
\  (! X Z Y. equal(X::'a,Y) & defined(Z::'a,X) --> defined(Z::'a,Y)) &      \
\  (! X Z Y. equal(X::'a,Y) --> equal(compos(Z::'a,X),compos(Z::'a,Y))) & \
\  (! X Y Z. equal(X::'a,Y) --> equal(compos(X::'a,Z),compos(Y::'a,Z))) & \
\  (defined(a::'a,b)) &     \
\  (defined(b::'a,c)) &     \
\  (~defined(a::'a,compos(b::'a,c))) --> False",
  meson_tac);

(*1118 inferences so far.  Searching to depth 8.  2.3 secs*)
val COL001_2 = prove
 ("(! X. equal(X::'a,X)) &  \
\  (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
\  (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) &  \
\  (! X Y Z. equal(apply(apply(apply(s::'a,X),Y),Z),apply(apply(X::'a,Z),apply(Y::'a,Z)))) &        \
\  (! Y X. equal(apply(apply(k::'a,X),Y),X)) &      \
\  (! X Y Z. equal(apply(apply(apply(b::'a,X),Y),Z),apply(X::'a,apply(Y::'a,Z)))) & \
\  (! X. equal(apply(i::'a,X),X)) & \
\  (! A B C. equal(A::'a,B) --> equal(apply(A::'a,C),apply(B::'a,C))) &     \
\  (! D F' E. equal(D::'a,E) --> equal(apply(F'::'a,D),apply(F'::'a,E))) &  \
\  (! 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))))) &   \
\  (! Y. ~equal(Y::'a,apply(combinator::'a,Y))) --> False",
  meson_tac);

(*500 inferences so far.  Searching to depth 8.  0.9 secs*)
val COL023_1 = prove
 ("(! X. equal(X::'a,X)) &  \
\  (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
\  (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) &  \
\  (! X Y Z. equal(apply(apply(apply(b::'a,X),Y),Z),apply(X::'a,apply(Y::'a,Z)))) & \
\  (! X Y Z. equal(apply(apply(apply(n::'a,X),Y),Z),apply(apply(apply(X::'a,Z),Y),Z))) &        \
\  (! A B C. equal(A::'a,B) --> equal(apply(A::'a,C),apply(B::'a,C))) &     \
\  (! D F' E. equal(D::'a,E) --> equal(apply(F'::'a,D),apply(F'::'a,E))) &  \
\  (! Y. ~equal(Y::'a,apply(combinator::'a,Y))) --> False",
  meson_tac);

(*3018 inferences so far.  Searching to depth 10.  4.3 secs*)
val COL032_1 = prove_hard
 ("(! X. equal(X::'a,X)) &  \
\  (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
\  (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) &  \
\  (! X. equal(apply(m::'a,X),apply(X::'a,X))) &        \
\  (! Y X Z. equal(apply(apply(apply(q::'a,X),Y),Z),apply(Y::'a,apply(X::'a,Z)))) & \
\  (! A B C. equal(A::'a,B) --> equal(apply(A::'a,C),apply(B::'a,C))) &     \
\  (! D F' E. equal(D::'a,E) --> equal(apply(F'::'a,D),apply(F'::'a,E))) &  \
\  (! G H. equal(G::'a,H) --> equal(f(G),f(H))) &   \
\  (! Y. ~equal(apply(Y::'a,f(Y)),apply(f(Y),apply(Y::'a,f(Y))))) --> False",
  meson_tac);

(*381878 inferences so far.  Searching to depth 13.  670.4 secs*)
val COL052_2 = prove_hard
 ("(! X. equal(X::'a,X)) &  \
\  (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
\  (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) &  \
\  (! X Y W. equal(response(compos(X::'a,Y),W),response(X::'a,response(Y::'a,W)))) &       \
\  (! X Y. agreeable(X) --> equal(response(X::'a,common_bird(Y)),response(Y::'a,common_bird(Y)))) &     \
\  (! Z X. equal(response(X::'a,Z),response(compatible(X),Z)) --> agreeable(X)) &   \
\  (! A B. equal(A::'a,B) --> equal(common_bird(A),common_bird(B))) &       \
\  (! C D. equal(C::'a,D) --> equal(compatible(C),compatible(D))) & \
\  (! Q R. equal(Q::'a,R) & agreeable(Q) --> agreeable(R)) &        \
\  (! A B C. equal(A::'a,B) --> equal(compos(A::'a,C),compos(B::'a,C))) & \
\  (! D F' E. equal(D::'a,E) --> equal(compos(F'::'a,D),compos(F'::'a,E))) &      \
\  (! G H I'. equal(G::'a,H) --> equal(response(G::'a,I'),response(H::'a,I'))) &    \
\  (! J L K'. equal(J::'a,K') --> equal(response(L::'a,J),response(L::'a,K'))) &    \
\  (agreeable(c)) &     \
\  (~agreeable(a)) &    \
\  (equal(c::'a,compos(a::'a,b))) --> False",
  meson_tac);

(*13201 inferences so far.  Searching to depth 11.  31.9 secs*)
val COL075_2 = prove_hard
 ("(! X. equal(X::'a,X)) &  \
\  (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
\  (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) &  \
\  (! Y X. equal(apply(apply(k::'a,X),Y),X)) &      \
\  (! X Y Z. equal(apply(apply(apply(abstraction::'a,X),Y),Z),apply(apply(X::'a,apply(k::'a,Z)),apply(Y::'a,Z)))) &     \
\  (! D E F'. equal(D::'a,E) --> equal(apply(D::'a,F'),apply(E::'a,F'))) &  \
\  (! G I' H. equal(G::'a,H) --> equal(apply(I'::'a,G),apply(I'::'a,H))) &  \
\  (! A B. equal(A::'a,B) --> equal(b(A),b(B))) &   \
\  (! C D. equal(C::'a,D) --> equal(c(C),c(D))) &   \
\  (! Y. ~equal(apply(apply(Y::'a,b(Y)),c(Y)),apply(b(Y),b(Y)))) --> False",
  meson_tac);

(*33 inferences so far.  Searching to depth 7.  0.1 secs*)
val COM001_1 = prove
 ("(! Goal_state Start_state. follows(Goal_state::'a,Start_state) --> succeeds(Goal_state::'a,Start_state)) &   \
\  (! Goal_state Intermediate_state Start_state. succeeds(Goal_state::'a,Intermediate_state) & succeeds(Intermediate_state::'a,Start_state) --> succeeds(Goal_state::'a,Start_state)) &     \
\  (! Start_state Label Goal_state. has(Start_state::'a,goto(Label)) & labels(Label::'a,Goal_state) --> succeeds(Goal_state::'a,Start_state)) &     \
\  (! Start_state Condition Goal_state. has(Start_state::'a,ifthen(Condition::'a,Goal_state)) --> succeeds(Goal_state::'a,Start_state)) &   \
\  (labels(loop::'a,p3)) &  \
\  (has(p3::'a,ifthen(equal(register_j::'a,n),p4))) &   \
\  (has(p4::'a,goto(out))) &        \
\  (follows(p5::'a,p4)) &   \
\  (follows(p8::'a,p3)) &   \
\  (has(p8::'a,goto(loop))) &       \
\  (~succeeds(p3::'a,p3)) --> False",
  meson_tac);

(*533 inferences so far.  Searching to depth 13.  0.3 secs*)
val COM002_1 = prove
 ("(! Goal_state Start_state. follows(Goal_state::'a,Start_state) --> succeeds(Goal_state::'a,Start_state)) &   \
\  (! Goal_state Intermediate_state Start_state. succeeds(Goal_state::'a,Intermediate_state) & succeeds(Intermediate_state::'a,Start_state) --> succeeds(Goal_state::'a,Start_state)) &     \
\  (! Start_state Label Goal_state. has(Start_state::'a,goto(Label)) & labels(Label::'a,Goal_state) --> succeeds(Goal_state::'a,Start_state)) &     \
\  (! Start_state Condition Goal_state. has(Start_state::'a,ifthen(Condition::'a,Goal_state)) --> succeeds(Goal_state::'a,Start_state)) &   \
\  (has(p1::'a,assign(register_j::'a,num0))) &     \
\  (follows(p2::'a,p1)) &   \
\  (has(p2::'a,assign(register_k::'a,num1))) &     \
\  (labels(loop::'a,p3)) &  \
\  (follows(p3::'a,p2)) &   \
\  (has(p3::'a,ifthen(equal(register_j::'a,n),p4))) &   \
\  (has(p4::'a,goto(out))) &        \
\  (follows(p5::'a,p4)) &   \
\  (follows(p6::'a,p3)) &   \
\  (has(p6::'a,assign(register_k::'a,times(num2::'a,register_k)))) &   \
\  (follows(p7::'a,p6)) &   \
\  (has(p7::'a,assign(register_j::'a,plus(register_j::'a,num1)))) &    \
\  (follows(p8::'a,p7)) &   \
\  (has(p8::'a,goto(loop))) &       \
\  (~succeeds(p3::'a,p3)) --> False",
  meson_tac);

(*4821 inferences so far.  Searching to depth 14.  1.3 secs*)
val COM002_2 = prove
 ("(! Goal_state Start_state. ~(fails(Goal_state::'a,Start_state) & follows(Goal_state::'a,Start_state))) &     \
\  (! Goal_state Intermediate_state Start_state. fails(Goal_state::'a,Start_state) --> fails(Goal_state::'a,Intermediate_state) | fails(Intermediate_state::'a,Start_state)) &     \
\  (! Start_state Label Goal_state. ~(fails(Goal_state::'a,Start_state) & has(Start_state::'a,goto(Label)) & labels(Label::'a,Goal_state))) &       \
\  (! Start_state Condition Goal_state. ~(fails(Goal_state::'a,Start_state) & has(Start_state::'a,ifthen(Condition::'a,Goal_state)))) &     \
\  (has(p1::'a,assign(register_j::'a,num0))) &     \
\  (follows(p2::'a,p1)) &   \
\  (has(p2::'a,assign(register_k::'a,num1))) &     \
\  (labels(loop::'a,p3)) &  \
\  (follows(p3::'a,p2)) &   \
\  (has(p3::'a,ifthen(equal(register_j::'a,n),p4))) &   \
\  (has(p4::'a,goto(out))) &        \
\  (follows(p5::'a,p4)) &   \
\  (follows(p6::'a,p3)) &   \
\  (has(p6::'a,assign(register_k::'a,times(num2::'a,register_k)))) &   \
\  (follows(p7::'a,p6)) &   \
\  (has(p7::'a,assign(register_j::'a,plus(register_j::'a,num1)))) &    \
\  (follows(p8::'a,p7)) &   \
\  (has(p8::'a,goto(loop))) &       \
\  (fails(p3::'a,p3)) --> False",
  meson_tac);

(*98 inferences so far.  Searching to depth 10.  1.1 secs*)
val COM003_2 = prove
 ("(! X Y Z. program_decides(X) & program(Y) --> decides(X::'a,Y,Z)) &      \
\  (! X. program_decides(X) | program(f2(X))) &        \
\  (! X. decides(X::'a,f2(X),f1(X)) --> program_decides(X)) &       \
\  (! X. program_program_decides(X) --> program(X)) &   \
\  (! X. program_program_decides(X) --> program_decides(X)) &   \
\  (! X. program(X) & program_decides(X) --> program_program_decides(X)) &      \
\  (! X. algorithm_program_decides(X) --> algorithm(X)) &       \
\  (! X. algorithm_program_decides(X) --> program_decides(X)) & \
\  (! X. algorithm(X) & program_decides(X) --> algorithm_program_decides(X)) &  \
\  (! Y X. program_halts2(X::'a,Y) --> program(X)) &        \
\  (! X Y. program_halts2(X::'a,Y) --> halts2(X::'a,Y)) &       \
\  (! X Y. program(X) & halts2(X::'a,Y) --> program_halts2(X::'a,Y)) &  \
\  (! W X Y Z. halts3_outputs(X::'a,Y,Z,W) --> halts3(X::'a,Y,Z)) &     \
\  (! Y Z X W. halts3_outputs(X::'a,Y,Z,W) --> outputs(X::'a,W)) &      \
\  (! Y Z X W. halts3(X::'a,Y,Z) & outputs(X::'a,W) --> halts3_outputs(X::'a,Y,Z,W)) &      \
\  (! Y X. program_not_halts2(X::'a,Y) --> program(X)) &    \
\  (! X Y. ~(program_not_halts2(X::'a,Y) & halts2(X::'a,Y))) &  \
\  (! X Y. program(X) --> program_not_halts2(X::'a,Y) | halts2(X::'a,Y)) &     \
\  (! W X Y. halts2_outputs(X::'a,Y,W) --> halts2(X::'a,Y)) &   \
\  (! Y X W. halts2_outputs(X::'a,Y,W) --> outputs(X::'a,W)) &  \
\  (! Y X W. halts2(X::'a,Y) & outputs(X::'a,W) --> halts2_outputs(X::'a,Y,W)) &    \
\  (! X W Y Z. program_halts2_halts3_outputs(X::'a,Y,Z,W) --> program_halts2(Y::'a,Z)) &        \
\  (! X Y Z W. program_halts2_halts3_outputs(X::'a,Y,Z,W) --> halts3_outputs(X::'a,Y,Z,W)) &    \
\  (! 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)) &      \
\  (! X W Y Z. program_not_halts2_halts3_outputs(X::'a,Y,Z,W) --> program_not_halts2(Y::'a,Z)) &        \
\  (! X Y Z W. program_not_halts2_halts3_outputs(X::'a,Y,Z,W) --> halts3_outputs(X::'a,Y,Z,W)) &        \
\  (! 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)) &      \
\  (! X W Y. program_halts2_halts2_outputs(X::'a,Y,W) --> program_halts2(Y::'a,Y)) &    \
\  (! X Y W. program_halts2_halts2_outputs(X::'a,Y,W) --> halts2_outputs(X::'a,Y,W)) &  \
\  (! X Y W. program_halts2(Y::'a,Y) & halts2_outputs(X::'a,Y,W) --> program_halts2_halts2_outputs(X::'a,Y,W)) &    \
\  (! X W Y. program_not_halts2_halts2_outputs(X::'a,Y,W) --> program_not_halts2(Y::'a,Y)) &    \
\  (! X Y W. program_not_halts2_halts2_outputs(X::'a,Y,W) --> halts2_outputs(X::'a,Y,W)) &      \
\  (! X Y W. program_not_halts2(Y::'a,Y) & halts2_outputs(X::'a,Y,W) --> program_not_halts2_halts2_outputs(X::'a,Y,W)) &    \
\  (! X. algorithm_program_decides(X) --> program_program_decides(c1)) &        \
\  (! W Y Z. program_program_decides(W) --> program_halts2_halts3_outputs(W::'a,Y,Z,good)) &        \
\  (! W Y Z. program_program_decides(W) --> program_not_halts2_halts3_outputs(W::'a,Y,Z,bad)) &     \
\  (! 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)) &       \
\  (! 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)) &        \
\  (! 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)) &     \
\  (! V. program(V) & program_halts2_halts2_outputs(V::'a,f4(V),good) & program_not_halts2_halts2_outputs(V::'a,f4(V),bad) --> program(c3)) &   \
\  (! 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)) &  \
\  (! 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)) & \
\  (algorithm_program_decides(c4)) --> False",
  meson_tac);

(****************SLOW
2100398 inferences so far.  Searching to depth 12.  No proof after 30 mins.
val COM004_1 = prove
 ("(! X. equal(X::'a,X)) &  \
\  (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
\  (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) &  \
\  (! 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))) &   \
\  (! X. contradictory(negate(X),X)) &  \
\  (! X. contradictory(X::'a,negate(X))) &  \
\  (! X. siblings(left_child_of(X),right_child_of(X))) &        \
\  (! D E. equal(D::'a,E) --> equal(left_child_of(D),left_child_of(E))) &   \
\  (! F' G. equal(F'::'a,G) --> equal(negate(F'),negate(G))) &      \
\  (! H I' J. equal(H::'a,I') --> equal(or(H::'a,J),or(I'::'a,J))) &        \
\  (! K' M L. equal(K'::'a,L) --> equal(or(M::'a,K'),or(M::'a,L))) &        \
\  (! N O_ P. equal(N::'a,O_) --> equal(parent_of(N::'a,P),parent_of(O_::'a,P))) &     \
\  (! Q S' R. equal(Q::'a,R) --> equal(parent_of(S'::'a,Q),parent_of(S'::'a,R))) &  \
\  (! T' U. equal(T'::'a,U) --> equal(right_child_of(T'),right_child_of(U))) &      \
\  (! V W X. equal(V::'a,W) & contradictory(V::'a,X) --> contradictory(W::'a,X)) &  \
\  (! Y A1 Z. equal(Y::'a,Z) & contradictory(A1::'a,Y) --> contradictory(A1::'a,Z)) &       \
\  (! B1 C1 D1. equal(B1::'a,C1) & failure_node(B1::'a,D1) --> failure_node(C1::'a,D1)) &   \
\  (! E1 G1 F1. equal(E1::'a,F1) & failure_node(G1::'a,E1) --> failure_node(G1::'a,F1)) &   \
\  (! H1 I1 J1. equal(H1::'a,I1) & siblings(H1::'a,J1) --> siblings(I1::'a,J1)) &   \
\  (! K1 M1 L1. equal(K1::'a,L1) & siblings(M1::'a,K1) --> siblings(M1::'a,L1)) &   \
\  (failure_node(n_left::'a,or(empty::'a,atom))) &      \
\  (failure_node(n_right::'a,or(empty::'a,negate(atom)))) &     \
\  (equal(n_left::'a,left_child_of(n))) &   \
\  (equal(n_right::'a,right_child_of(n))) & \
\  (! Z. ~failure_node(Z::'a,or(empty::'a,empty))) --> False",
  meson_tac);
****************)

(*179 inferences so far.  Searching to depth 7.  3.9 secs*)
val GEO003_1 = prove_hard
 ("(! X. equal(X::'a,X)) &  \
\  (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
\  (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) &  \
\  (! X Y. between(X::'a,Y,X) --> equal(X::'a,Y)) &     \
\  (! V X Y Z. between(X::'a,Y,V) & between(Y::'a,Z,V) --> between(X::'a,Y,Z)) &    \
\  (! 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)) &    \
\  (! Y X. equidistant(X::'a,Y,Y,X)) &      \
\  (! Z X Y. equidistant(X::'a,Y,Z,Z) --> equal(X::'a,Y)) &     \
\  (! X Y Z V V2 W. equidistant(X::'a,Y,Z,V) & equidistant(X::'a,Y,V2,W) --> equidistant(Z::'a,V,V2,W)) &   \
\  (! 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)) &     \
\  (! 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))) &     \
\  (! 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))) &   \
\  (! 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))) &   \
\  (! 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))) &  \
\  (! 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)) &   \
\  (! X Y W V. between(X::'a,Y,extension(X::'a,Y,W,V))) &       \
\  (! X Y W V. equidistant(Y::'a,extension(X::'a,Y,W,V),W,V)) & \
\  (~between(lower_dimension_point_1::'a,lower_dimension_point_2,lower_dimension_point_3)) &        \
\  (~between(lower_dimension_point_2::'a,lower_dimension_point_3,lower_dimension_point_1)) &        \
\  (~between(lower_dimension_point_3::'a,lower_dimension_point_1,lower_dimension_point_2)) &        \
\  (! 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)) &     \
\  (! 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))) &        \
\  (! 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)) &    \
\  (! X Y W Z. equal(X::'a,Y) & between(X::'a,W,Z) --> between(Y::'a,W,Z)) &        \
\  (! X W Y Z. equal(X::'a,Y) & between(W::'a,X,Z) --> between(W::'a,Y,Z)) &        \
\  (! X W Z Y. equal(X::'a,Y) & between(W::'a,Z,X) --> between(W::'a,Z,Y)) &        \
\  (! X Y V W Z. equal(X::'a,Y) & equidistant(X::'a,V,W,Z) --> equidistant(Y::'a,V,W,Z)) &  \
\  (! X V Y W Z. equal(X::'a,Y) & equidistant(V::'a,X,W,Z) --> equidistant(V::'a,Y,W,Z)) &  \
\  (! X V W Y Z. equal(X::'a,Y) & equidistant(V::'a,W,X,Z) --> equidistant(V::'a,W,Y,Z)) &  \
\  (! X V W Z Y. equal(X::'a,Y) & equidistant(V::'a,W,Z,X) --> equidistant(V::'a,W,Z,Y)) &  \
\  (! 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))) &   \
\  (! 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))) &   \
\  (! 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))) &   \
\  (! 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))) &   \
\  (! 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))) &   \
\  (! 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'))) &    \
\  (! 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))) & \
\  (! 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))) &       \
\  (! 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))) & \
\  (! 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))) &   \
\  (! 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))) &     \
\  (! 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))) &     \
\  (! 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))) &     \
\  (! 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))) &     \
\  (! 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))) &     \
\  (! X Y V1 V2 V3. equal(X::'a,Y) --> equal(extension(X::'a,V1,V2,V3),extension(Y::'a,V1,V2,V3))) &        \
\  (! X V1 Y V2 V3. equal(X::'a,Y) --> equal(extension(V1::'a,X,V2,V3),extension(V1::'a,Y,V2,V3))) &        \
\  (! X V1 V2 Y V3. equal(X::'a,Y) --> equal(extension(V1::'a,V2,X,V3),extension(V1::'a,V2,Y,V3))) &        \
\  (! X V1 V2 V3 Y. equal(X::'a,Y) --> equal(extension(V1::'a,V2,V3,X),extension(V1::'a,V2,V3,Y))) &        \
\  (! 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))) &    \
\  (! 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))) &    \
\  (! 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))) &    \
\  (! 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))) &    \
\  (! 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))) &    \
\  (! 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))) &    \
\  (~between(a::'a,b,b)) --> False",
  meson_tac);

(*4272 inferences so far.  Searching to depth 10.  29.4 secs*)
val GEO017_2 = prove_hard
 ("(! X. equal(X::'a,X)) &  \
\  (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
\  (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) &  \
\  (! Y X. equidistant(X::'a,Y,Y,X)) &      \
\  (! X Y Z V V2 W. equidistant(X::'a,Y,Z,V) & equidistant(X::'a,Y,V2,W) --> equidistant(Z::'a,V,V2,W)) &   \
\  (! Z X Y. equidistant(X::'a,Y,Z,Z) --> equal(X::'a,Y)) &     \
\  (! X Y W V. between(X::'a,Y,extension(X::'a,Y,W,V))) &       \
\  (! X Y W V. equidistant(Y::'a,extension(X::'a,Y,W,V),W,V)) & \
\  (! 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)) &   \
\  (! X Y. between(X::'a,Y,X) --> equal(X::'a,Y)) &     \
\  (! 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)) &     \
\  (! 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)) &     \
\  (~between(lower_dimension_point_1::'a,lower_dimension_point_2,lower_dimension_point_3)) &        \
\  (~between(lower_dimension_point_2::'a,lower_dimension_point_3,lower_dimension_point_1)) &        \
\  (~between(lower_dimension_point_3::'a,lower_dimension_point_1,lower_dimension_point_2)) &        \
\  (! 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)) &     \
\  (! 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))) &   \
\  (! 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))) &   \
\  (! 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))) &  \
\  (! 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)) &    \
\  (! 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))) &        \
\  (! X Y W Z. equal(X::'a,Y) & between(X::'a,W,Z) --> between(Y::'a,W,Z)) &        \
\  (! X W Y Z. equal(X::'a,Y) & between(W::'a,X,Z) --> between(W::'a,Y,Z)) &        \
\  (! X W Z Y. equal(X::'a,Y) & between(W::'a,Z,X) --> between(W::'a,Z,Y)) &        \
\  (! X Y V W Z. equal(X::'a,Y) & equidistant(X::'a,V,W,Z) --> equidistant(Y::'a,V,W,Z)) &  \
\  (! X V Y W Z. equal(X::'a,Y) & equidistant(V::'a,X,W,Z) --> equidistant(V::'a,Y,W,Z)) &  \
\  (! X V W Y Z. equal(X::'a,Y) & equidistant(V::'a,W,X,Z) --> equidistant(V::'a,W,Y,Z)) &  \
\  (! X V W Z Y. equal(X::'a,Y) & equidistant(V::'a,W,Z,X) --> equidistant(V::'a,W,Z,Y)) &  \
\  (! 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))) &   \
\  (! 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))) &   \
\  (! 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))) &   \
\  (! 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))) &   \
\  (! 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))) &   \
\  (! 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'))) &    \
\  (! 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))) & \
\  (! 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))) &       \
\  (! 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))) & \
\  (! 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))) &   \
\  (! 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))) &     \
\  (! 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))) &     \
\  (! 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))) &     \
\  (! 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))) &     \
\  (! 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))) &     \
\  (! X Y V1 V2 V3. equal(X::'a,Y) --> equal(extension(X::'a,V1,V2,V3),extension(Y::'a,V1,V2,V3))) &        \
\  (! X V1 Y V2 V3. equal(X::'a,Y) --> equal(extension(V1::'a,X,V2,V3),extension(V1::'a,Y,V2,V3))) &        \
\  (! X V1 V2 Y V3. equal(X::'a,Y) --> equal(extension(V1::'a,V2,X,V3),extension(V1::'a,V2,Y,V3))) &        \
\  (! X V1 V2 V3 Y. equal(X::'a,Y) --> equal(extension(V1::'a,V2,V3,X),extension(V1::'a,V2,V3,Y))) &        \
\  (! 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))) &    \
\  (! 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))) &    \
\  (! 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))) &    \
\  (! 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))) &    \
\  (! 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))) &    \
\  (! 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))) &    \
\  (equidistant(u::'a,v,w,x)) &     \
\  (~equidistant(u::'a,v,x,w)) --> False",
  meson_tac);

(****************SLOW
382903 inferences so far.  Searching to depth 9.  No proof after 35 minutes.
val GEO027_3 = prove_hard
 ("(! X. equal(X::'a,X)) &  \
\  (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
\  (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) &  \
\  (! Y X. equidistant(X::'a,Y,Y,X)) &      \
\  (! X Y Z V V2 W. equidistant(X::'a,Y,Z,V) & equidistant(X::'a,Y,V2,W) --> equidistant(Z::'a,V,V2,W)) &   \
\  (! Z X Y. equidistant(X::'a,Y,Z,Z) --> equal(X::'a,Y)) &     \
\  (! X Y W V. between(X::'a,Y,extension(X::'a,Y,W,V))) &       \
\  (! X Y W V. equidistant(Y::'a,extension(X::'a,Y,W,V),W,V)) & \
\  (! 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)) &   \
\  (! X Y. between(X::'a,Y,X) --> equal(X::'a,Y)) &     \
\  (! 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)) &     \
\  (! 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)) &     \
\  (~between(lower_dimension_point_1::'a,lower_dimension_point_2,lower_dimension_point_3)) &        \
\  (~between(lower_dimension_point_2::'a,lower_dimension_point_3,lower_dimension_point_1)) &        \
\  (~between(lower_dimension_point_3::'a,lower_dimension_point_1,lower_dimension_point_2)) &        \
\  (! 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)) &     \
\  (! 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))) &   \
\  (! 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))) &   \
\  (! 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))) &  \
\  (! 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)) &    \
\  (! 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))) &        \
\  (! X Y W Z. equal(X::'a,Y) & between(X::'a,W,Z) --> between(Y::'a,W,Z)) &        \
\  (! X W Y Z. equal(X::'a,Y) & between(W::'a,X,Z) --> between(W::'a,Y,Z)) &        \
\  (! X W Z Y. equal(X::'a,Y) & between(W::'a,Z,X) --> between(W::'a,Z,Y)) &        \
\  (! X Y V W Z. equal(X::'a,Y) & equidistant(X::'a,V,W,Z) --> equidistant(Y::'a,V,W,Z)) &  \
\  (! X V Y W Z. equal(X::'a,Y) & equidistant(V::'a,X,W,Z) --> equidistant(V::'a,Y,W,Z)) &  \
\  (! X V W Y Z. equal(X::'a,Y) & equidistant(V::'a,W,X,Z) --> equidistant(V::'a,W,Y,Z)) &  \
\  (! X V W Z Y. equal(X::'a,Y) & equidistant(V::'a,W,Z,X) --> equidistant(V::'a,W,Z,Y)) &  \
\  (! 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))) &   \
\  (! 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))) &   \
\  (! 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))) &   \
\  (! 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))) &   \
\  (! 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))) &   \
\  (! 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'))) &    \
\  (! 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))) & \
\  (! 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))) &       \
\  (! 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))) & \
\  (! 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))) &   \
\  (! 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))) &     \
\  (! 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))) &     \
\  (! 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))) &     \
\  (! 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))) &     \
\  (! 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))) &     \
\  (! X Y V1 V2 V3. equal(X::'a,Y) --> equal(extension(X::'a,V1,V2,V3),extension(Y::'a,V1,V2,V3))) &        \
\  (! X V1 Y V2 V3. equal(X::'a,Y) --> equal(extension(V1::'a,X,V2,V3),extension(V1::'a,Y,V2,V3))) &        \
\  (! X V1 V2 Y V3. equal(X::'a,Y) --> equal(extension(V1::'a,V2,X,V3),extension(V1::'a,V2,Y,V3))) &        \
\  (! X V1 V2 V3 Y. equal(X::'a,Y) --> equal(extension(V1::'a,V2,V3,X),extension(V1::'a,V2,V3,Y))) &        \
\  (! 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))) &    \
\  (! 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))) &    \
\  (! 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))) &    \
\  (! 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))) &    \
\  (! 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))) &    \
\  (! 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))) &    \
\  (! U V. equal(reflection(U::'a,V),extension(U::'a,V,U,V))) & \
\  (! X Y Z. equal(X::'a,Y) --> equal(reflection(X::'a,Z),reflection(Y::'a,Z))) &   \
\  (! A1 C1 B1. equal(A1::'a,B1) --> equal(reflection(C1::'a,A1),reflection(C1::'a,B1))) &  \
\  (! U V. equidistant(U::'a,V,U,V)) &      \
\  (! W X U V. equidistant(U::'a,V,W,X) --> equidistant(W::'a,X,U,V)) & \
\  (! V U W X. equidistant(U::'a,V,W,X) --> equidistant(V::'a,U,W,X)) & \
\  (! U V X W. equidistant(U::'a,V,W,X) --> equidistant(U::'a,V,X,W)) & \
\  (! V U X W. equidistant(U::'a,V,W,X) --> equidistant(V::'a,U,X,W)) & \
\  (! W X V U. equidistant(U::'a,V,W,X) --> equidistant(W::'a,X,V,U)) & \
\  (! X W U V. equidistant(U::'a,V,W,X) --> equidistant(X::'a,W,U,V)) & \
\  (! X W V U. equidistant(U::'a,V,W,X) --> equidistant(X::'a,W,V,U)) & \
\  (! W X U V Y Z. equidistant(U::'a,V,W,X) & equidistant(W::'a,X,Y,Z) --> equidistant(U::'a,V,Y,Z)) &      \
\  (! U V W. equal(V::'a,extension(U::'a,V,W,W))) &     \
\  (! W X U V Y. equal(Y::'a,extension(U::'a,V,W,X)) --> between(U::'a,V,Y)) &      \
\  (! U V. between(U::'a,V,reflection(U::'a,V))) &      \
\  (! U V. equidistant(V::'a,reflection(U::'a,V),U,V)) &        \
\  (! U V. equal(U::'a,V) --> equal(V::'a,reflection(U::'a,V))) &   \
\  (! U. equal(U::'a,reflection(U::'a,U))) &    \
\  (! U V. equal(V::'a,reflection(U::'a,V)) --> equal(U::'a,V)) &   \
\  (! U V. equidistant(U::'a,U,V,V)) &      \
\  (! 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)) &        \
\  (! 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)) &   \
\  (between(u::'a,v,w)) &   \
\  (~equal(u::'a,v)) &      \
\  (~equal(w::'a,extension(u::'a,v,v,w))) --> False",
  meson_tac);
****************)

(*313884 inferences so far.  Searching to depth 10.  887 secs: 15 mins.*)
val GEO058_2 = prove_hard
 ("(! X. equal(X::'a,X)) &  \
\  (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
\  (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) &  \
\  (! Y X. equidistant(X::'a,Y,Y,X)) &      \
\  (! X Y Z V V2 W. equidistant(X::'a,Y,Z,V) & equidistant(X::'a,Y,V2,W) --> equidistant(Z::'a,V,V2,W)) &   \
\  (! Z X Y. equidistant(X::'a,Y,Z,Z) --> equal(X::'a,Y)) &     \
\  (! X Y W V. between(X::'a,Y,extension(X::'a,Y,W,V))) &       \
\  (! X Y W V. equidistant(Y::'a,extension(X::'a,Y,W,V),W,V)) & \
\  (! 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)) &   \
\  (! X Y. between(X::'a,Y,X) --> equal(X::'a,Y)) &     \
\  (! 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)) &     \
\  (! 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)) &     \
\  (~between(lower_dimension_point_1::'a,lower_dimension_point_2,lower_dimension_point_3)) &        \
\  (~between(lower_dimension_point_2::'a,lower_dimension_point_3,lower_dimension_point_1)) &        \
\  (~between(lower_dimension_point_3::'a,lower_dimension_point_1,lower_dimension_point_2)) &        \
\  (! 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)) &     \
\  (! 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))) &   \
\  (! 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))) &   \
\  (! 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))) &  \
\  (! 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)) &    \
\  (! 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))) &        \
\  (! X Y W Z. equal(X::'a,Y) & between(X::'a,W,Z) --> between(Y::'a,W,Z)) &        \
\  (! X W Y Z. equal(X::'a,Y) & between(W::'a,X,Z) --> between(W::'a,Y,Z)) &        \
\  (! X W Z Y. equal(X::'a,Y) & between(W::'a,Z,X) --> between(W::'a,Z,Y)) &        \
\  (! X Y V W Z. equal(X::'a,Y) & equidistant(X::'a,V,W,Z) --> equidistant(Y::'a,V,W,Z)) &  \
\  (! X V Y W Z. equal(X::'a,Y) & equidistant(V::'a,X,W,Z) --> equidistant(V::'a,Y,W,Z)) &  \
\  (! X V W Y Z. equal(X::'a,Y) & equidistant(V::'a,W,X,Z) --> equidistant(V::'a,W,Y,Z)) &  \
\  (! X V W Z Y. equal(X::'a,Y) & equidistant(V::'a,W,Z,X) --> equidistant(V::'a,W,Z,Y)) &  \
\  (! 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))) &   \
\  (! 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))) &   \
\  (! 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))) &   \
\  (! 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))) &   \
\  (! 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))) &   \
\  (! 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'))) &    \
\  (! 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))) & \
\  (! 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))) &       \
\  (! 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))) & \
\  (! 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))) &   \
\  (! 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))) &     \
\  (! 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))) &     \
\  (! 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))) &     \
\  (! 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))) &     \
\  (! 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))) &     \
\  (! X Y V1 V2 V3. equal(X::'a,Y) --> equal(extension(X::'a,V1,V2,V3),extension(Y::'a,V1,V2,V3))) &        \
\  (! X V1 Y V2 V3. equal(X::'a,Y) --> equal(extension(V1::'a,X,V2,V3),extension(V1::'a,Y,V2,V3))) &        \
\  (! X V1 V2 Y V3. equal(X::'a,Y) --> equal(extension(V1::'a,V2,X,V3),extension(V1::'a,V2,Y,V3))) &        \
\  (! X V1 V2 V3 Y. equal(X::'a,Y) --> equal(extension(V1::'a,V2,V3,X),extension(V1::'a,V2,V3,Y))) &        \
\  (! 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))) &    \
\  (! 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))) &    \
\  (! 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))) &    \
\  (! 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))) &    \
\  (! 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))) &    \
\  (! 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))) &    \
\  (! U V. equal(reflection(U::'a,V),extension(U::'a,V,U,V))) & \
\  (! X Y Z. equal(X::'a,Y) --> equal(reflection(X::'a,Z),reflection(Y::'a,Z))) &   \
\  (! A1 C1 B1. equal(A1::'a,B1) --> equal(reflection(C1::'a,A1),reflection(C1::'a,B1))) &  \
\  (equal(v::'a,reflection(u::'a,v))) & \
\  (~equal(u::'a,v)) --> False",
  meson_tac);

(*0 inferences so far.  Searching to depth 0.  0.2 secs*)
val GEO079_1 = prove
 ("(! 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)) &       \
\  (! U V W X Y Z. congruent(U::'a,V,W,X,Y,Z) --> eq(U::'a,V,W,X,Y,Z)) &        \
\  (! V W U X. trapezoid(U::'a,V,W,X) --> parallel(V::'a,W,U,X)) &      \
\  (! U V X Y. parallel(U::'a,V,X,Y) --> eq(X::'a,V,U,V,X,Y)) & \
\  (trapezoid(a::'a,b,c,d)) &       \
\  (~eq(a::'a,c,b,c,a,d)) --> False",
  meson_tac);

(****************SLOW
2032008 inferences so far.  Searching to depth 16.  No proof after 30 minutes.
val GRP001_1 = prove_hard
 ("(! X. equal(X::'a,X)) &  \
\  (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
\  (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) &  \
\  (! X. product(identity::'a,X,X)) &       \
\  (! X. product(X::'a,identity,X)) &       \
\  (! X. product(inverse(X),X,identity)) &      \
\  (! X. product(X::'a,inverse(X),identity)) &      \
\  (! X Y. product(X::'a,Y,multiply(X::'a,Y))) &        \
\  (! X Y Z W. product(X::'a,Y,Z) & product(X::'a,Y,W) --> equal(Z::'a,W)) &        \
\  (! 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)) &       \
\  (! 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)) &       \
\  (! X Y. equal(X::'a,Y) --> equal(inverse(X),inverse(Y))) &       \
\  (! X Y W. equal(X::'a,Y) --> equal(multiply(X::'a,W),multiply(Y::'a,W))) &       \
\  (! X W Y. equal(X::'a,Y) --> equal(multiply(W::'a,X),multiply(W::'a,Y))) &       \
\  (! X Y W Z. equal(X::'a,Y) & product(X::'a,W,Z) --> product(Y::'a,W,Z)) &        \
\  (! X W Y Z. equal(X::'a,Y) & product(W::'a,X,Z) --> product(W::'a,Y,Z)) &        \
\  (! X W Z Y. equal(X::'a,Y) & product(W::'a,Z,X) --> product(W::'a,Z,Y)) &        \
\  (! X. product(X::'a,X,identity)) &       \
\  (product(a::'a,b,c)) &   \
\  (~product(b::'a,a,c)) --> False",
  meson_tac);
****************)

(*2386 inferences so far.  Searching to depth 11.  8.7 secs*)
val GRP008_1 = prove_hard
 ("(! X. equal(X::'a,X)) &  \
\  (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
\  (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) &  \
\  (! X. product(identity::'a,X,X)) &       \
\  (! X. product(X::'a,identity,X)) &       \
\  (! X. product(inverse(X),X,identity)) &      \
\  (! X. product(X::'a,inverse(X),identity)) &      \
\  (! X Y. product(X::'a,Y,multiply(X::'a,Y))) &        \
\  (! X Y Z W. product(X::'a,Y,Z) & product(X::'a,Y,W) --> equal(Z::'a,W)) &        \
\  (! 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)) &       \
\  (! 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)) &       \
\  (! X Y. equal(X::'a,Y) --> equal(inverse(X),inverse(Y))) &       \
\  (! X Y W. equal(X::'a,Y) --> equal(multiply(X::'a,W),multiply(Y::'a,W))) &       \
\  (! X W Y. equal(X::'a,Y) --> equal(multiply(W::'a,X),multiply(W::'a,Y))) &       \
\  (! X Y W Z. equal(X::'a,Y) & product(X::'a,W,Z) --> product(Y::'a,W,Z)) &        \
\  (! X W Y Z. equal(X::'a,Y) & product(W::'a,X,Z) --> product(W::'a,Y,Z)) &        \
\  (! X W Z Y. equal(X::'a,Y) & product(W::'a,Z,X) --> product(W::'a,Z,Y)) &        \
\  (! A B. equal(A::'a,B) --> equal(h(A),h(B))) &   \
\  (! C D. equal(C::'a,D) --> equal(j(C),j(D))) &   \
\  (! A B. equal(A::'a,B) & q(A) --> q(B)) &        \
\  (! B A C. q(A) & product(A::'a,B,C) --> product(B::'a,A,C)) &        \
\  (! A. product(j(A),A,h(A)) | product(A::'a,j(A),h(A)) | q(A)) &        \
\  (! A. product(j(A),A,h(A)) & product(A::'a,j(A),h(A)) --> q(A)) &        \
\  (~q(identity)) --> False",
  meson_tac);

(*8625 inferences so far.  Searching to depth 11.  20 secs*)
val GRP013_1 = prove_hard
 ("(! X. equal(X::'a,X)) &  \
\  (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
\  (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) &  \
\  (! X. product(identity::'a,X,X)) &       \
\  (! X. product(X::'a,identity,X)) &       \
\  (! X. product(inverse(X),X,identity)) &      \
\  (! X. product(X::'a,inverse(X),identity)) &      \
\  (! X Y. product(X::'a,Y,multiply(X::'a,Y))) &        \
\  (! X Y Z W. product(X::'a,Y,Z) & product(X::'a,Y,W) --> equal(Z::'a,W)) &        \
\  (! 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)) &       \
\  (! 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)) &       \
\  (! X Y. equal(X::'a,Y) --> equal(inverse(X),inverse(Y))) &       \
\  (! X Y W. equal(X::'a,Y) --> equal(multiply(X::'a,W),multiply(Y::'a,W))) &       \
\  (! X W Y. equal(X::'a,Y) --> equal(multiply(W::'a,X),multiply(W::'a,Y))) &       \
\  (! X Y W Z. equal(X::'a,Y) & product(X::'a,W,Z) --> product(Y::'a,W,Z)) &        \
\  (! X W Y Z. equal(X::'a,Y) & product(W::'a,X,Z) --> product(W::'a,Y,Z)) &        \
\  (! X W Z Y. equal(X::'a,Y) & product(W::'a,Z,X) --> product(W::'a,Z,Y)) &        \
\  (! A. product(A::'a,A,identity)) &       \
\  (product(a::'a,b,c)) &   \
\  (product(inverse(a),inverse(b),d)) & \
\  (! A C B. product(inverse(A),inverse(B),C) --> product(A::'a,C,B)) &     \
\  (~product(c::'a,d,identity)) --> False",
  meson_tac);

(*2448 inferences so far.  Searching to depth 10.  7.2 secs*)
val GRP037_3 = prove_hard
 ("(! X. equal(X::'a,X)) &  \
\  (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
\  (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) &  \
\  (! X. product(identity::'a,X,X)) &       \
\  (! X. product(X::'a,identity,X)) &       \
\  (! X. product(inverse(X),X,identity)) &      \
\  (! X. product(X::'a,inverse(X),identity)) &      \
\  (! X Y. product(X::'a,Y,multiply(X::'a,Y))) &        \
\  (! X Y Z W. product(X::'a,Y,Z) & product(X::'a,Y,W) --> equal(Z::'a,W)) &        \
\  (! 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)) &       \
\  (! 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)) &       \
\  (! X Y. equal(X::'a,Y) --> equal(inverse(X),inverse(Y))) &       \
\  (! X Y W. equal(X::'a,Y) --> equal(multiply(X::'a,W),multiply(Y::'a,W))) &       \
\  (! X W Y. equal(X::'a,Y) --> equal(multiply(W::'a,X),multiply(W::'a,Y))) &       \
\  (! X Y W Z. equal(X::'a,Y) & product(X::'a,W,Z) --> product(Y::'a,W,Z)) &        \
\  (! X W Y Z. equal(X::'a,Y) & product(W::'a,X,Z) --> product(W::'a,Y,Z)) &        \
\  (! X W Z Y. equal(X::'a,Y) & product(W::'a,Z,X) --> product(W::'a,Z,Y)) &        \
\  (! A B C. subgroup_member(A) & subgroup_member(B) & product(A::'a,inverse(B),C) --> subgroup_member(C)) &        \
\  (! A B. equal(A::'a,B) & subgroup_member(A) --> subgroup_member(B)) &    \
\  (! A. subgroup_member(A) --> product(another_identity::'a,A,A)) &        \
\  (! A. subgroup_member(A) --> product(A::'a,another_identity,A)) &        \
\  (! A. subgroup_member(A) --> product(A::'a,another_inverse(A),another_identity)) &       \
\  (! A. subgroup_member(A) --> product(another_inverse(A),A,another_identity)) &       \
\  (! A. subgroup_member(A) --> subgroup_member(another_inverse(A))) &  \
\  (! A B. equal(A::'a,B) --> equal(another_inverse(A),another_inverse(B))) &       \
\  (! A C D B. product(A::'a,B,C) & product(A::'a,D,C) --> equal(D::'a,B)) &        \
\  (! B C D A. product(A::'a,B,C) & product(D::'a,B,C) --> equal(D::'a,A)) &        \
\  (subgroup_member(a)) &       \
\  (subgroup_member(another_identity)) &        \
\  (~equal(inverse(a),another_inverse(a))) --> False",
  meson_tac);

(*163 inferences so far.  Searching to depth 11.  0.3 secs*)
val GRP031_2 = prove
 ("(! X Y. product(X::'a,Y,multiply(X::'a,Y))) &        \
\  (! X Y Z W. product(X::'a,Y,Z) & product(X::'a,Y,W) --> equal(Z::'a,W)) &        \
\  (! 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)) &       \
\  (! 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)) &       \
\  (! A. product(A::'a,inverse(A),identity)) &      \
\  (! A. product(A::'a,identity,A)) &       \
\  (! A. ~product(A::'a,a,identity)) --> False",
  meson_tac);

(*47 inferences so far.  Searching to depth 11.   0.2 secs*)
val GRP034_4 = prove
 ("(! X Y. product(X::'a,Y,multiply(X::'a,Y))) &        \
\  (! X. product(identity::'a,X,X)) &       \
\  (! X. product(X::'a,identity,X)) &       \
\  (! X. product(X::'a,inverse(X),identity)) &      \
\  (! 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)) &       \
\  (! 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)) &       \
\  (! B A C. subgroup_member(A) & subgroup_member(B) & product(B::'a,inverse(A),C) --> subgroup_member(C)) &        \
\  (subgroup_member(a)) &       \
\  (~subgroup_member(inverse(a))) --> False",
  meson_tac);

(*3287 inferences so far.  Searching to depth 14.  3.5 secs*)
val GRP047_2 = prove_hard
 ("(! X. product(identity::'a,X,X)) &       \
\  (! X. product(inverse(X),X,identity)) &      \
\  (! X Y. product(X::'a,Y,multiply(X::'a,Y))) &        \
\  (! X Y Z W. product(X::'a,Y,Z) & product(X::'a,Y,W) --> equal(Z::'a,W)) &        \
\  (! 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)) &       \
\  (! 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)) &       \
\  (! X W Z Y. equal(X::'a,Y) & product(W::'a,Z,X) --> product(W::'a,Z,Y)) &        \
\  (equal(a::'a,b)) &       \
\  (~equal(multiply(c::'a,a),multiply(c::'a,b))) --> False",
  meson_tac);

(*25559 inferences so far.  Searching to depth 19.  16.9 secs*)
val GRP130_1_002 = prove_hard
 ("(group_element(e_1)) &       \
\  (group_element(e_2)) &       \
\  (~equal(e_1::'a,e_2)) &  \
\  (~equal(e_2::'a,e_1)) &  \
\  (! X Y. group_element(X) & group_element(Y) --> product(X::'a,Y,e_1) | product(X::'a,Y,e_2)) &      \
\  (! X Y W Z. product(X::'a,Y,W) & product(X::'a,Y,Z) --> equal(W::'a,Z)) &        \
\  (! X Y W Z. product(X::'a,W,Y) & product(X::'a,Z,Y) --> equal(W::'a,Z)) &        \
\  (! Y X W Z. product(W::'a,Y,X) & product(Z::'a,Y,X) --> equal(W::'a,Z)) &        \
\  (! Z1 Z2 Y X. product(X::'a,Y,Z1) & product(X::'a,Z1,Z2) --> product(Z2::'a,Y,X)) --> False",
  meson_tac);

(*3468 inferences so far.  Searching to depth 10.  9.1 secs*)
val GRP156_1 = prove_hard
 ("(! X. equal(X::'a,X)) &  \
\  (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
\  (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) &  \
\  (! X. equal(multiply(identity::'a,X),X)) &       \
\  (! X. equal(multiply(inverse(X),X),identity)) &      \
\  (! X Y Z. equal(multiply(multiply(X::'a,Y),Z),multiply(X::'a,multiply(Y::'a,Z)))) &      \
\  (! A B. equal(A::'a,B) --> equal(inverse(A),inverse(B))) &       \
\  (! C D E. equal(C::'a,D) --> equal(multiply(C::'a,E),multiply(D::'a,E))) &       \
\  (! F' H G. equal(F'::'a,G) --> equal(multiply(H::'a,F'),multiply(H::'a,G))) &    \
\  (! Y X. equal(greatest_lower_bound(X::'a,Y),greatest_lower_bound(Y::'a,X))) &        \
\  (! Y X. equal(least_upper_bound(X::'a,Y),least_upper_bound(Y::'a,X))) &      \
\  (! 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))) &      \
\  (! 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))) &  \
\  (! X. equal(least_upper_bound(X::'a,X),X)) &     \
\  (! X. equal(greatest_lower_bound(X::'a,X),X)) &  \
\  (! Y X. equal(least_upper_bound(X::'a,greatest_lower_bound(X::'a,Y)),X)) &   \
\  (! Y X. equal(greatest_lower_bound(X::'a,least_upper_bound(X::'a,Y)),X)) &   \
\  (! Y X Z. equal(multiply(X::'a,least_upper_bound(Y::'a,Z)),least_upper_bound(multiply(X::'a,Y),multiply(X::'a,Z)))) &        \
\  (! Y X Z. equal(multiply(X::'a,greatest_lower_bound(Y::'a,Z)),greatest_lower_bound(multiply(X::'a,Y),multiply(X::'a,Z)))) &  \
\  (! Y Z X. equal(multiply(least_upper_bound(Y::'a,Z),X),least_upper_bound(multiply(Y::'a,X),multiply(Z::'a,X)))) &        \
\  (! Y Z X. equal(multiply(greatest_lower_bound(Y::'a,Z),X),greatest_lower_bound(multiply(Y::'a,X),multiply(Z::'a,X)))) &  \
\  (! A B C. equal(A::'a,B) --> equal(greatest_lower_bound(A::'a,C),greatest_lower_bound(B::'a,C))) &       \
\  (! A C B. equal(A::'a,B) --> equal(greatest_lower_bound(C::'a,A),greatest_lower_bound(C::'a,B))) &       \
\  (! A B C. equal(A::'a,B) --> equal(least_upper_bound(A::'a,C),least_upper_bound(B::'a,C))) &     \
\  (! A C B. equal(A::'a,B) --> equal(least_upper_bound(C::'a,A),least_upper_bound(C::'a,B))) &     \
\  (! A B C. equal(A::'a,B) --> equal(multiply(A::'a,C),multiply(B::'a,C))) &       \
\  (! A C B. equal(A::'a,B) --> equal(multiply(C::'a,A),multiply(C::'a,B))) &       \
\  (equal(least_upper_bound(a::'a,b),b)) &  \
\  (~equal(greatest_lower_bound(multiply(a::'a,c),multiply(b::'a,c)),multiply(a::'a,c))) --> False",
  meson_tac);

(*4394 inferences so far.  Searching to depth 10.  8.2 secs*)
val GRP168_1 = prove_hard
 ("(! X. equal(X::'a,X)) &  \
\  (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
\  (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) &  \
\  (! X. equal(multiply(identity::'a,X),X)) &       \
\  (! X. equal(multiply(inverse(X),X),identity)) &      \
\  (! X Y Z. equal(multiply(multiply(X::'a,Y),Z),multiply(X::'a,multiply(Y::'a,Z)))) &      \
\  (! A B. equal(A::'a,B) --> equal(inverse(A),inverse(B))) &       \
\  (! C D E. equal(C::'a,D) --> equal(multiply(C::'a,E),multiply(D::'a,E))) &       \
\  (! F' H G. equal(F'::'a,G) --> equal(multiply(H::'a,F'),multiply(H::'a,G))) &    \
\  (! Y X. equal(greatest_lower_bound(X::'a,Y),greatest_lower_bound(Y::'a,X))) &        \
\  (! Y X. equal(least_upper_bound(X::'a,Y),least_upper_bound(Y::'a,X))) &      \
\  (! 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))) &      \
\  (! 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))) &  \
\  (! X. equal(least_upper_bound(X::'a,X),X)) &     \
\  (! X. equal(greatest_lower_bound(X::'a,X),X)) &  \
\  (! Y X. equal(least_upper_bound(X::'a,greatest_lower_bound(X::'a,Y)),X)) &   \
\  (! Y X. equal(greatest_lower_bound(X::'a,least_upper_bound(X::'a,Y)),X)) &   \
\  (! Y X Z. equal(multiply(X::'a,least_upper_bound(Y::'a,Z)),least_upper_bound(multiply(X::'a,Y),multiply(X::'a,Z)))) &        \
\  (! Y X Z. equal(multiply(X::'a,greatest_lower_bound(Y::'a,Z)),greatest_lower_bound(multiply(X::'a,Y),multiply(X::'a,Z)))) &  \
\  (! Y Z X. equal(multiply(least_upper_bound(Y::'a,Z),X),least_upper_bound(multiply(Y::'a,X),multiply(Z::'a,X)))) &        \
\  (! Y Z X. equal(multiply(greatest_lower_bound(Y::'a,Z),X),greatest_lower_bound(multiply(Y::'a,X),multiply(Z::'a,X)))) &  \
\  (! A B C. equal(A::'a,B) --> equal(greatest_lower_bound(A::'a,C),greatest_lower_bound(B::'a,C))) &       \
\  (! A C B. equal(A::'a,B) --> equal(greatest_lower_bound(C::'a,A),greatest_lower_bound(C::'a,B))) &       \
\  (! A B C. equal(A::'a,B) --> equal(least_upper_bound(A::'a,C),least_upper_bound(B::'a,C))) &     \
\  (! A C B. equal(A::'a,B) --> equal(least_upper_bound(C::'a,A),least_upper_bound(C::'a,B))) &     \
\  (! A B C. equal(A::'a,B) --> equal(multiply(A::'a,C),multiply(B::'a,C))) &       \
\  (! A C B. equal(A::'a,B) --> equal(multiply(C::'a,A),multiply(C::'a,B))) &       \
\  (equal(least_upper_bound(a::'a,b),b)) &  \
\  (~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",
  meson_tac);

(*250258 inferences so far.  Searching to depth 16.  406.2 secs*)
val HEN003_3 = prove_hard
 ("(! X. equal(X::'a,X)) &  \
\  (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
\  (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) &  \
\  (! X Y. less_equal(X::'a,Y) --> equal(divide(X::'a,Y),zero)) &       \
\  (! X Y. equal(divide(X::'a,Y),zero) --> less_equal(X::'a,Y)) &       \
\  (! Y X. less_equal(divide(X::'a,Y),X)) & \
\  (! X Y Z. less_equal(divide(divide(X::'a,Z),divide(Y::'a,Z)),divide(divide(X::'a,Y),Z))) &       \
\  (! X. less_equal(zero::'a,X)) &  \
\  (! X Y. less_equal(X::'a,Y) & less_equal(Y::'a,X) --> equal(X::'a,Y)) &  \
\  (! X. less_equal(X::'a,identity)) &      \
\  (! A B C. equal(A::'a,B) --> equal(divide(A::'a,C),divide(B::'a,C))) &   \
\  (! D F' E. equal(D::'a,E) --> equal(divide(F'::'a,D),divide(F'::'a,E))) &        \
\  (! G H I'. equal(G::'a,H) & less_equal(G::'a,I') --> less_equal(H::'a,I')) &     \
\  (! J L K'. equal(J::'a,K') & less_equal(L::'a,J) --> less_equal(L::'a,K')) &     \
\  (~equal(divide(a::'a,a),zero)) --> False",
  meson_tac);


(*202177 inferences so far.  Searching to depth 14.  451 secs*)
val HEN007_2 = prove_hard
 ("(! X. equal(X::'a,X)) &  \
\  (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
\  (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) &  \
\  (! X Y. less_equal(X::'a,Y) --> quotient(X::'a,Y,zero)) &    \
\  (! X Y. quotient(X::'a,Y,zero) --> less_equal(X::'a,Y)) &    \
\  (! Y Z X. quotient(X::'a,Y,Z) --> less_equal(Z::'a,X)) &     \
\  (! 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)) &    \
\  (! X. less_equal(zero::'a,X)) &  \
\  (! X Y. less_equal(X::'a,Y) & less_equal(Y::'a,X) --> equal(X::'a,Y)) &  \
\  (! X. less_equal(X::'a,identity)) &      \
\  (! X Y. quotient(X::'a,Y,divide(X::'a,Y))) & \
\  (! X Y Z W. quotient(X::'a,Y,Z) & quotient(X::'a,Y,W) --> equal(Z::'a,W)) &      \
\  (! X Y W Z. equal(X::'a,Y) & quotient(X::'a,W,Z) --> quotient(Y::'a,W,Z)) &      \
\  (! X W Y Z. equal(X::'a,Y) & quotient(W::'a,X,Z) --> quotient(W::'a,Y,Z)) &      \
\  (! X W Z Y. equal(X::'a,Y) & quotient(W::'a,Z,X) --> quotient(W::'a,Z,Y)) &      \
\  (! X Z Y. equal(X::'a,Y) & less_equal(Z::'a,X) --> less_equal(Z::'a,Y)) &        \
\  (! X Y Z. equal(X::'a,Y) & less_equal(X::'a,Z) --> less_equal(Y::'a,Z)) &        \
\  (! X Y W. equal(X::'a,Y) --> equal(divide(X::'a,W),divide(Y::'a,W))) &   \
\  (! X W Y. equal(X::'a,Y) --> equal(divide(W::'a,X),divide(W::'a,Y))) &   \
\  (! X. quotient(X::'a,identity,zero)) &   \
\  (! X. quotient(zero::'a,X,zero)) &       \
\  (! X. quotient(X::'a,X,zero)) &  \
\  (! X. quotient(X::'a,zero,X)) &  \
\  (! Y X Z. less_equal(X::'a,Y) & less_equal(Y::'a,Z) --> less_equal(X::'a,Z)) &   \
\  (! 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)) &       \
\  (less_equal(x::'a,y)) &  \
\  (quotient(z::'a,y,zQy)) &        \
\  (quotient(z::'a,x,zQx)) &        \
\  (~less_equal(zQy::'a,zQx)) --> False",
  meson_tac);

(*60026 inferences so far.  Searching to depth 12.  42.2 secs*)
val HEN008_4 = prove_hard
 ("(! X. equal(X::'a,X)) &  \
\  (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
\  (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) &  \
\  (! X Y. less_equal(X::'a,Y) --> equal(divide(X::'a,Y),zero)) &       \
\  (! X Y. equal(divide(X::'a,Y),zero) --> less_equal(X::'a,Y)) &       \
\  (! Y X. less_equal(divide(X::'a,Y),X)) & \
\  (! X Y Z. less_equal(divide(divide(X::'a,Z),divide(Y::'a,Z)),divide(divide(X::'a,Y),Z))) &       \
\  (! X. less_equal(zero::'a,X)) &  \
\  (! X Y. less_equal(X::'a,Y) & less_equal(Y::'a,X) --> equal(X::'a,Y)) &  \
\  (! X. less_equal(X::'a,identity)) &      \
\  (! A B C. equal(A::'a,B) --> equal(divide(A::'a,C),divide(B::'a,C))) &   \
\  (! D F' E. equal(D::'a,E) --> equal(divide(F'::'a,D),divide(F'::'a,E))) &        \
\  (! G H I'. equal(G::'a,H) & less_equal(G::'a,I') --> less_equal(H::'a,I')) &     \
\  (! J L K'. equal(J::'a,K') & less_equal(L::'a,J) --> less_equal(L::'a,K')) &     \
\  (! X. equal(divide(X::'a,identity),zero)) &      \
\  (! X. equal(divide(zero::'a,X),zero)) &  \
\  (! X. equal(divide(X::'a,X),zero)) &     \
\  (equal(divide(a::'a,zero),a)) &  \
\  (! Y X Z. less_equal(X::'a,Y) & less_equal(Y::'a,Z) --> less_equal(X::'a,Z)) &   \
\  (! X Z Y. less_equal(divide(X::'a,Y),Z) --> less_equal(divide(X::'a,Z),Y)) & \
\  (! Y Z X. less_equal(X::'a,Y) --> less_equal(divide(Z::'a,Y),divide(Z::'a,X))) & \
\  (less_equal(a::'a,b)) &  \
\  (~less_equal(divide(a::'a,c),divide(b::'a,c))) --> False",
  meson_tac);


(*3160 inferences so far.  Searching to depth 11.  3.5 secs*)
val HEN009_5 = prove_hard
 ("(! X. equal(X::'a,X)) &  \
\  (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
\  (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) &  \
\  (! Y X. equal(divide(divide(X::'a,Y),X),zero)) & \
\  (! X Y Z. equal(divide(divide(divide(X::'a,Z),divide(Y::'a,Z)),divide(divide(X::'a,Y),Z)),zero)) &       \
\  (! X. equal(divide(zero::'a,X),zero)) &  \
\  (! X Y. equal(divide(X::'a,Y),zero) & equal(divide(Y::'a,X),zero) --> equal(X::'a,Y)) &  \
\  (! X. equal(divide(X::'a,identity),zero)) &      \
\  (! A B C. equal(A::'a,B) --> equal(divide(A::'a,C),divide(B::'a,C))) &   \
\  (! D F' E. equal(D::'a,E) --> equal(divide(F'::'a,D),divide(F'::'a,E))) &        \
\  (! Y X Z. equal(divide(X::'a,Y),zero) & equal(divide(Y::'a,Z),zero) --> equal(divide(X::'a,Z),zero)) &   \
\  (! X Z Y. equal(divide(divide(X::'a,Y),Z),zero) --> equal(divide(divide(X::'a,Z),Y),zero)) & \
\  (! Y Z X. equal(divide(X::'a,Y),zero) --> equal(divide(divide(Z::'a,Y),divide(Z::'a,X)),zero)) & \
\  (~equal(divide(identity::'a,a),divide(identity::'a,divide(identity::'a,divide(identity::'a,a))))) &  \
\  (equal(divide(identity::'a,a),b)) &      \
\  (equal(divide(identity::'a,b),c)) &      \
\  (equal(divide(identity::'a,c),d)) &      \
\  (~equal(b::'a,d)) --> False",
  meson_tac);

(*970373 inferences so far.  Searching to depth 17.  890.0 secs*)
val HEN012_3 = prove_hard
 ("(! X. equal(X::'a,X)) &  \
\  (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
\  (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) &  \
\  (! X Y. less_equal(X::'a,Y) --> equal(divide(X::'a,Y),zero)) &       \
\  (! X Y. equal(divide(X::'a,Y),zero) --> less_equal(X::'a,Y)) &       \
\  (! Y X. less_equal(divide(X::'a,Y),X)) & \
\  (! X Y Z. less_equal(divide(divide(X::'a,Z),divide(Y::'a,Z)),divide(divide(X::'a,Y),Z))) &       \
\  (! X. less_equal(zero::'a,X)) &  \
\  (! X Y. less_equal(X::'a,Y) & less_equal(Y::'a,X) --> equal(X::'a,Y)) &  \
\  (! X. less_equal(X::'a,identity)) &      \
\  (! A B C. equal(A::'a,B) --> equal(divide(A::'a,C),divide(B::'a,C))) &   \
\  (! D F' E. equal(D::'a,E) --> equal(divide(F'::'a,D),divide(F'::'a,E))) &        \
\  (! G H I'. equal(G::'a,H) & less_equal(G::'a,I') --> less_equal(H::'a,I')) &     \
\  (! J L K'. equal(J::'a,K') & less_equal(L::'a,J) --> less_equal(L::'a,K')) &     \
\  (~less_equal(a::'a,a)) --> False",
  meson_tac);


(*1063 inferences so far.  Searching to depth 20.  1.0 secs*)
val LCL010_1 = prove
 ("(! X Y. is_a_theorem(equivalent(X::'a,Y)) & is_a_theorem(X) --> is_a_theorem(Y)) &       \
\  (! X Z Y. is_a_theorem(equivalent(equivalent(X::'a,Y),equivalent(equivalent(X::'a,Z),equivalent(Z::'a,Y))))) &   \
\  (~is_a_theorem(equivalent(equivalent(a::'a,b),equivalent(equivalent(c::'a,b),equivalent(a::'a,c))))) --> False",
  meson_tac);

(*2549 inferences so far.  Searching to depth 12.  1.4 secs*)
val LCL077_2 = prove
 ("(! X Y. is_a_theorem(implies(X,Y)) & is_a_theorem(X) --> is_a_theorem(Y)) &  \
\  (! Y X. is_a_theorem(implies(X,implies(Y,X)))) &     \
\  (! Y X Z. is_a_theorem(implies(implies(X,implies(Y,Z)),implies(implies(X,Y),implies(X,Z))))) &       \
\  (! Y X. is_a_theorem(implies(implies(not(X),not(Y)),implies(Y,X)))) &        \
\  (! X2 X1 X3. is_a_theorem(implies(X1,X2)) & is_a_theorem(implies(X2,X3)) --> is_a_theorem(implies(X1,X3))) & \
\  (~is_a_theorem(implies(not(not(a)),a))) --> False",
  meson_tac);

(*2036 inferences so far.  Searching to depth 20.  1.5 secs*)
val LCL082_1 = prove
 ("(! X Y. is_a_theorem(implies(X::'a,Y)) & is_a_theorem(X) --> is_a_theorem(Y)) &  \
\  (! Y Z U X. is_a_theorem(implies(implies(implies(X::'a,Y),Z),implies(implies(Z::'a,X),implies(U::'a,X))))) &     \
\  (~is_a_theorem(implies(a::'a,implies(b::'a,a)))) --> False",
  meson_tac);

(*1100 inferences so far.  Searching to depth 13.  1.0 secs*)
val LCL111_1 = prove
 ("(! X Y. is_a_theorem(implies(X,Y)) & is_a_theorem(X) --> is_a_theorem(Y)) &  \
\  (! Y X. is_a_theorem(implies(X,implies(Y,X)))) &     \
\  (! Y X Z. is_a_theorem(implies(implies(X,Y),implies(implies(Y,Z),implies(X,Z))))) &  \
\  (! Y X. is_a_theorem(implies(implies(implies(X,Y),Y),implies(implies(Y,X),X)))) &    \
\  (! Y X. is_a_theorem(implies(implies(not(X),not(Y)),implies(Y,X)))) &        \
\  (~is_a_theorem(implies(implies(a,b),implies(implies(c,a),implies(c,b))))) --> False",
  meson_tac);

(*667 inferences so far.  Searching to depth 9.  1.4 secs*)
val LCL143_1 = prove
 ("(! X. equal(X,X)) &  \
\  (! Y X. equal(X,Y) --> equal(Y,X)) & \
\  (! Y X Z. equal(X,Y) & equal(Y,Z) --> equal(X,Z)) &  \
\  (! X. equal(implies(true,X),X)) &    \
\  (! Y X Z. equal(implies(implies(X,Y),implies(implies(Y,Z),implies(X,Z))),true)) &    \
\  (! Y X. equal(implies(implies(X,Y),Y),implies(implies(Y,X),X))) &    \
\  (! Y X. equal(implies(implies(not(X),not(Y)),implies(Y,X)),true)) &  \
\  (! A B C. equal(A,B) --> equal(implies(A,C),implies(B,C))) & \
\  (! D F' E. equal(D,E) --> equal(implies(F',D),implies(F',E))) &      \
\  (! G H. equal(G,H) --> equal(not(G),not(H))) &       \
\  (! X Y. equal(big_V(X,Y),implies(implies(X,Y),Y))) & \
\  (! X Y. equal(big_hat(X,Y),not(big_V(not(X),not(Y))))) &     \
\  (! X Y. ordered(X,Y) --> equal(implies(X,Y),true)) & \
\  (! X Y. equal(implies(X,Y),true) --> ordered(X,Y)) & \
\  (! A B C. equal(A,B) --> equal(big_V(A,C),big_V(B,C))) &     \
\  (! D F' E. equal(D,E) --> equal(big_V(F',D),big_V(F',E))) &  \
\  (! G H I'. equal(G,H) --> equal(big_hat(G,I'),big_hat(H,I'))) &      \
\  (! J L K'. equal(J,K') --> equal(big_hat(L,J),big_hat(L,K'))) &      \
\  (! M N O_. equal(M,N) & ordered(M,O_) --> ordered(N,O_)) &      \
\  (! P R Q. equal(P,Q) & ordered(R,P) --> ordered(R,Q)) &      \
\  (ordered(x,y)) &     \
\  (~ordered(implies(z,x),implies(z,y))) --> False",
  meson_tac);

(*5245 inferences so far.  Searching to depth 12.  4.6 secs*)
val LCL182_1 = prove_hard
 ("(! A. axiom(or(not(or(A,A)),A))) &   \
\  (! B A. axiom(or(not(A),or(B,A)))) & \
\  (! B A. axiom(or(not(or(A,B)),or(B,A)))) &   \
\  (! B A C. axiom(or(not(or(A,or(B,C))),or(B,or(A,C))))) &     \
\  (! A C B. axiom(or(not(or(not(A),B)),or(not(or(C,A)),or(C,B))))) &   \
\  (! X. axiom(X) --> theorem(X)) &     \
\  (! X Y. axiom(or(not(Y),X)) & theorem(Y) --> theorem(X)) &   \
\  (! X Y Z. axiom(or(not(X),Y)) & theorem(or(not(Y),Z)) --> theorem(or(not(X),Z))) &   \
\  (~theorem(or(not(or(not(p),q)),or(not(not(q)),not(p))))) --> False",
  meson_tac);

(*410 inferences so far.  Searching to depth 10.  0.3 secs*)
val LCL200_1 = prove
 ("(! A. axiom(or(not(or(A,A)),A))) &   \
\  (! B A. axiom(or(not(A),or(B,A)))) & \
\  (! B A. axiom(or(not(or(A,B)),or(B,A)))) &   \
\  (! B A C. axiom(or(not(or(A,or(B,C))),or(B,or(A,C))))) &     \
\  (! A C B. axiom(or(not(or(not(A),B)),or(not(or(C,A)),or(C,B))))) &   \
\  (! X. axiom(X) --> theorem(X)) &     \
\  (! X Y. axiom(or(not(Y),X)) & theorem(Y) --> theorem(X)) &   \
\  (! X Y Z. axiom(or(not(X),Y)) & theorem(or(not(Y),Z)) --> theorem(or(not(X),Z))) &   \
\  (~theorem(or(not(not(or(p,q))),not(q)))) --> False",
  meson_tac);

(*5849 inferences so far.  Searching to depth 12.  5.6 secs*)
val LCL215_1 = prove_hard
 ("(! A. axiom(or(not(or(A,A)),A))) &   \
\  (! B A. axiom(or(not(A),or(B,A)))) & \
\  (! B A. axiom(or(not(or(A,B)),or(B,A)))) &   \
\  (! B A C. axiom(or(not(or(A,or(B,C))),or(B,or(A,C))))) &     \
\  (! A C B. axiom(or(not(or(not(A),B)),or(not(or(C,A)),or(C,B))))) &   \
\  (! X. axiom(X) --> theorem(X)) &     \
\  (! X Y. axiom(or(not(Y),X)) & theorem(Y) --> theorem(X)) &   \
\  (! X Y Z. axiom(or(not(X),Y)) & theorem(or(not(Y),Z)) --> theorem(or(not(X),Z))) &   \
\  (~theorem(or(not(or(not(p),q)),or(not(or(p,q)),q)))) --> False",
  meson_tac);

(*0 secs.  Not sure that a search even starts!*)
val LCL230_2 = prove
 ("(q --> p | r) &     \
\  (~p) &       \
\  (q) &        \
\  (~r) --> False",
  meson_tac);

(*119585 inferences so far.  Searching to depth 14.  262.4 secs*)
val LDA003_1 = prove_hard
 ("(! X. equal(X::'a,X)) &  \
\  (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
\  (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) &  \
\  (! Y X Z. equal(f(X::'a,f(Y::'a,Z)),f(f(X::'a,Y),f(X::'a,Z)))) &     \
\  (! X Y. left(X::'a,f(X::'a,Y))) &    \
\  (! Y X Z. left(X::'a,Y) & left(Y::'a,Z) --> left(X::'a,Z)) &     \
\  (equal(num2::'a,f(num1::'a,num1))) &  \
\  (equal(num3::'a,f(num2::'a,num1))) &  \
\  (equal(u::'a,f(num2::'a,num2))) &  \
\  (! A B C. equal(A::'a,B) --> equal(f(A::'a,C),f(B::'a,C))) &     \
\  (! D F' E. equal(D::'a,E) --> equal(f(F'::'a,D),f(F'::'a,E))) &  \
\  (! G H I'. equal(G::'a,H) & left(G::'a,I') --> left(H::'a,I')) & \
\  (! J L K'. equal(J::'a,K') & left(L::'a,J) --> left(L::'a,K')) & \
\  (~left(num3::'a,u)) --> False",
  meson_tac);


(*2392 inferences so far.  Searching to depth 12.  2.2 secs*)
val MSC002_1 = prove
 ("(at(something::'a,here,now)) &   \
\  (! Place Situation. hand_at(Place::'a,Situation) --> hand_at(Place::'a,let_go(Situation))) & \
\  (! Place Another_place Situation. hand_at(Place::'a,Situation) --> hand_at(Another_place::'a,go(Another_place::'a,Situation))) & \
\  (! Thing Situation. ~held(Thing::'a,let_go(Situation))) &        \
\  (! Situation Thing. at(Thing::'a,here,Situation) --> red(Thing)) &       \
\  (! Thing Place Situation. at(Thing::'a,Place,Situation) --> at(Thing::'a,Place,let_go(Situation))) & \
\  (! Thing Place Situation. at(Thing::'a,Place,Situation) --> at(Thing::'a,Place,pick_up(Situation))) &        \
\  (! Thing Place Situation. at(Thing::'a,Place,Situation) --> grabbed(Thing::'a,pick_up(go(Place::'a,let_go(Situation))))) &       \
\  (! Thing Situation. red(Thing) & put(Thing::'a,there,Situation) --> answer(Situation)) & \
\  (! Place Thing Another_place Situation. at(Thing::'a,Place,Situation) & grabbed(Thing::'a,Situation) --> put(Thing::'a,Another_place,go(Another_place::'a,Situation))) &     \
\  (! Thing Place Another_place Situation. at(Thing::'a,Place,Situation) --> held(Thing::'a,Situation) | at(Thing::'a,Place,go(Another_place::'a,Situation))) &        \
\  (! One_place Thing Place Situation. hand_at(One_place::'a,Situation) & held(Thing::'a,Situation) --> at(Thing::'a,Place,go(Place::'a,Situation))) &  \
\  (! Place Thing Situation. hand_at(Place::'a,Situation) & at(Thing::'a,Place,Situation) --> held(Thing::'a,pick_up(Situation))) & \
\  (! Situation. ~answer(Situation)) --> False",
  meson_tac);

(*73 inferences so far.  Searching to depth 10.  0.2 secs*)
val MSC003_1 = prove
 ("(! 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)) &        \
\  (! 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)) &   \
\  (in'(john::'a,boy)) &    \
\  (! X. in'(X::'a,boy) --> in'(X::'a,human)) & \
\  (! X. in'(X::'a,hand) --> has_parts(X::'a,num5,fingers)) &      \
\  (! X. in'(X::'a,human) --> has_parts(X::'a,num2,arm)) & \
\  (! X. in'(X::'a,arm) --> has_parts(X::'a,num1,hand)) &  \
\  (~has_parts(john::'a,times(num2::'a,num1),hand)) --> False",
  meson_tac);

(*1486 inferences so far.  Searching to depth 20.  1.2 secs*)
val MSC004_1 = prove
 ("(! 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)) &        \
\  (! 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)) &   \
\  (in'(john::'a,boy)) &    \
\  (! X. in'(X::'a,boy) --> in'(X::'a,human)) & \
\  (! X. in'(X::'a,hand) --> has_parts(X::'a,num5,fingers)) &      \
\  (! X. in'(X::'a,human) --> has_parts(X::'a,num2,arm)) & \
\  (! X. in'(X::'a,arm) --> has_parts(X::'a,num1,hand)) &  \
\  (~has_parts(john::'a,times(times(num2::'a,num1),num5),fingers)) --> False",
  meson_tac);

(*100 inferences so far.  Searching to depth 12.  0.1 secs*)
val MSC005_1 = prove
 ("(value(truth::'a,truth)) &       \
\  (value(falsity::'a,falsity)) &   \
\  (! X Y. value(X::'a,truth) & value(Y::'a,truth) --> value(xor(X::'a,Y),falsity)) &       \
\  (! X Y. value(X::'a,truth) & value(Y::'a,falsity) --> value(xor(X::'a,Y),truth)) &       \
\  (! X Y. value(X::'a,falsity) & value(Y::'a,truth) --> value(xor(X::'a,Y),truth)) &       \
\  (! X Y. value(X::'a,falsity) & value(Y::'a,falsity) --> value(xor(X::'a,Y),falsity)) &   \
\  (! Value. ~value(xor(xor(xor(xor(truth::'a,falsity),falsity),truth),falsity),Value)) --> False",
  meson_tac);

(*19116 inferences so far.  Searching to depth 16.  15.9 secs*)
val MSC006_1 = prove_hard
 ("(! Y X Z. p(X::'a,Y) & p(Y::'a,Z) --> p(X::'a,Z)) &      \
\  (! Y X Z. q(X::'a,Y) & q(Y::'a,Z) --> q(X::'a,Z)) &      \
\  (! Y X. q(X::'a,Y) --> q(Y::'a,X)) & \
\  (! X Y. p(X::'a,Y) | q(X::'a,Y)) &  \
\  (~p(a::'a,b)) &  \
\  (~q(c::'a,d)) --> False",
  meson_tac);

(*1713 inferences so far.  Searching to depth 10.  2.8 secs*)
val NUM001_1 = prove
 ("(! A. equal(A::'a,A)) &                                                \
\  (! B A C. equal(A::'a,B) & equal(B::'a,C) --> equal(A::'a,C)) &        \
\  (! B A. equal(add(A::'a,B),add(B::'a,A))) &                            \
\  (! A B C. equal(add(A::'a,add(B::'a,C)),add(add(A::'a,B),C))) &        \
\  (! B A. equal(subtract(add(A::'a,B),B),A)) &                           \
\  (! A B. equal(A::'a,subtract(add(A::'a,B),B))) &                       \
\  (! A C B. equal(add(subtract(A::'a,B),C),subtract(add(A::'a,C),B))) &  \
\  (! A C B. equal(subtract(add(A::'a,B),C),add(subtract(A::'a,C),B))) &  \
\  (! A C B D. equal(A::'a,B) & equal(C::'a,add(A::'a,D)) --> equal(C::'a,add(B::'a,D))) &  \
\  (! A C D B. equal(A::'a,B) & equal(C::'a,add(D::'a,A)) --> equal(C::'a,add(D::'a,B))) &  \
\  (! A C B D. equal(A::'a,B) & equal(C::'a,subtract(A::'a,D)) --> equal(C::'a,subtract(B::'a,D))) &        \
\  (! A C D B. equal(A::'a,B) & equal(C::'a,subtract(D::'a,A)) --> equal(C::'a,subtract(D::'a,B))) &        \
\  (~equal(add(add(a::'a,b),c),add(a::'a,add(b::'a,c)))) --> False",
  meson_tac);

(*20717 inferences so far.  Searching to depth 11.  13.7 secs*)
val NUM021_1 = prove_hard
 ("(! X. equal(X::'a,X)) &  \
\  (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
\  (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) &  \
\  (! A. equal(add(A::'a,num0),A)) &   \
\  (! A B. equal(add(A::'a,successor(B)),successor(add(A::'a,B)))) &    \
\  (! A. equal(multiply(A::'a,num0),num0)) &      \
\  (! B A. equal(multiply(A::'a,successor(B)),add(multiply(A::'a,B),A))) &      \
\  (! A B. equal(successor(A),successor(B)) --> equal(A::'a,B)) &   \
\  (! A B. equal(A::'a,B) --> equal(successor(A),successor(B))) &   \
\  (! A C B. less(A::'a,B) & less(C::'a,A) --> less(C::'a,B)) &     \
\  (! A B C. equal(add(successor(A),B),C) --> less(B::'a,C)) &      \
\  (! A B. less(A::'a,B) --> equal(add(successor(predecessor_of_1st_minus_2nd(B::'a,A)),A),B)) &        \
\  (! A B. divides(A::'a,B) --> less(A::'a,B) | equal(A::'a,B)) &  \
\  (! A B. less(A::'a,B) --> divides(A::'a,B)) &        \
\  (! A B. equal(A::'a,B) --> divides(A::'a,B)) &       \
\  (less(b::'a,c)) &        \
\  (~less(b::'a,a)) &       \
\  (divides(c::'a,a)) &     \
\  (! A. ~equal(successor(A),num0)) --> False",
  meson_tac);

(*26320 inferences so far.  Searching to depth 10.  26.4 secs*)
val NUM024_1 = prove_hard
 ("(! X. equal(X::'a,X)) &  \
\  (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
\  (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) &  \
\  (! A. equal(add(A::'a,num0),A)) &   \
\  (! A B. equal(add(A::'a,successor(B)),successor(add(A::'a,B)))) &    \
\  (! A. equal(multiply(A::'a,num0),num0)) &      \
\  (! B A. equal(multiply(A::'a,successor(B)),add(multiply(A::'a,B),A))) &      \
\  (! A B. equal(successor(A),successor(B)) --> equal(A::'a,B)) &   \
\  (! A B. equal(A::'a,B) --> equal(successor(A),successor(B))) &   \
\  (! A C B. less(A::'a,B) & less(C::'a,A) --> less(C::'a,B)) &     \
\  (! A B C. equal(add(successor(A),B),C) --> less(B::'a,C)) &      \
\  (! A B. less(A::'a,B) --> equal(add(successor(predecessor_of_1st_minus_2nd(B::'a,A)),A),B)) &        \
\  (! B A. equal(add(A::'a,B),add(B::'a,A))) &  \
\  (! B A C. equal(add(A::'a,B),add(C::'a,B)) --> equal(A::'a,C)) & \
\  (less(a::'a,a)) &        \
\  (! A. ~equal(successor(A),num0)) --> False",
  meson_tac);


(*1345 inferences so far.  Searching to depth 7.  23.3 secs.  BIG*)
val NUM180_1 = prove_hard
 ("(! X. equal(X::'a,X)) &  \
\  (! Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & \
\  (! Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z)) &  \
\  (! X U Y. subclass(X::'a,Y) & member(U::'a,X) --> member(U::'a,Y)) &     \
\  (! X Y. member(not_subclass_element(X::'a,Y),X) | subclass(X::'a,Y)) &      \
\  (! X Y. member(not_subclass_element(X::'a,Y),Y) --> subclass(X::'a,Y)) &     \
\  (! X. subclass(X::'a,universal_class)) & \
\  (! X Y. equal(X::'a,Y) --> subclass(X::'a,Y)) &      \
\  (! Y X. equal(X::'a,Y) --> subclass(Y::'a,X)) &      \
\  (! X Y. subclass(X::'a,Y) & subclass(Y::'a,X) --> equal(X::'a,Y)) &      \
\  (! X U Y. member(U::'a,unordered_pair(X::'a,Y)) --> equal(U::'a,X) | equal(U::'a,Y)) &      \
\  (! X Y. member(X::'a,universal_class) --> member(X::'a,unordered_pair(X::'a,Y))) &       \
\  (! X Y. member(Y::'a,universal_class) --> member(Y::'a,unordered_pair(X::'a,Y))) &       \
\  (! X Y. member(unordered_pair(X::'a,Y),universal_class)) &       \
\  (! X. equal(unordered_pair(X::'a,X),singleton(X))) &     \
\  (! X Y. equal(unordered_pair(singleton(X),unordered_pair(X::'a,singleton(Y))),ordered_pair(X::'a,Y))) &      \
\  (! V Y U X. member(ordered_pair(U::'a,V),cross_product(X::'a,Y)) --> member(U::'a,X)) &  \
\  (! U X V Y. member(ordered_pair(U::'a,V),cross_product(X::'a,Y)) --> member(V::'a,Y)) &  \
\  (! U V X Y. member(U::'a,X) & member(V::'a,Y) --> member(ordered_pair(U::'a,V),cross_product(X::'a,Y))) &    \
\  (! X Y Z. member(Z::'a,cross_product(X::'a,Y)) --> equal(ordered_pair(first(Z),second(Z)),Z)) &      \
\  (subclass(element_relation::'a,cross_product(universal_class::'a,universal_class))) &        \
\  (! X Y. member(ordered_pair(X::'a,Y),element_relation) --> member(X::'a,Y)) &        \
\  (! 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)) &     \
\  (! Y Z X. member(Z::'a,intersection(X::'a,Y)) --> member(Z::'a,X)) &     \
\  (! X Z Y. member(Z::'a,intersection(X::'a,Y)) --> member(Z::'a,Y)) &     \
\  (! Z X Y. member(Z::'a,X) & member(Z::'a,Y) --> member(Z::'a,intersection(X::'a,Y))) &       \
\  (! Z X. ~(member(Z::'a,complement(X)) & member(Z::'a,X))) &  \
\  (! Z X. member(Z::'a,universal_class) --> member(Z::'a,complement(X)) | member(Z::'a,X)) &      \
\  (! X Y. equal(complement(intersection(complement(X),complement(Y))),union(X::'a,Y))) &   \
\  (! X Y. equal(intersection(complement(intersection(X::'a,Y)),complement(intersection(complement(X),complement(Y)))),difference(X::'a,Y))) &  \
\  (! Xr X Y. equal(intersection(Xr::'a,cross_product(X::'a,Y)),restrct(Xr::'a,X,Y))) &    \
\  (! Xr X Y. equal(intersection(cross_product(X::'a,Y),Xr),restrct(Xr::'a,X,Y))) &    \
\  (! Z X. ~(equal(restrct(X::'a,singleton(Z),universal_class),null_class) & member(Z::'a,domain_of(X)))) &    \
\  (! Z X. member(Z::'a,universal_class) --> equal(restrct(X::'a,singleton(Z),universal_class),null_class) | member(Z::'a,domain_of(X))) &        \
\  (! X. subclass(rotate(X),cross_product(cross_product(universal_class::'a,universal_class),universal_class))) &   \
\  (! 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)) &   \
\  (! 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))) & \
\  (! X. subclass(flip(X),cross_product(cross_product(universal_class::'a,universal_class),universal_class))) &     \
\  (! 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)) &     \
\  (! 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))) &   \
\  (! Y. equal(domain_of(flip(cross_product(Y::'a,universal_class))),inverse(Y))) & \
\  (! Z. equal(domain_of(inverse(Z)),range_of(Z))) &    \
\  (! Z X Y. equal(first(not_subclass_element(restrct(Z::'a,X,singleton(Y)),null_class)),domain(Z::'a,X,Y))) & \
\  (! Z X Y. equal(second(not_subclass_element(restrct(Z::'a,singleton(X),Y),null_class)),rng(Z::'a,X,Y))) & \
\  (! Xr X. equal(range_of(restrct(Xr::'a,X,universal_class)),image_(Xr::'a,X))) &      \
\  (! X. equal(union(X::'a,singleton(X)),successor(X))) &   \
\  (subclass(successor_relation::'a,cross_product(universal_class::'a,universal_class))) &      \
\  (! X Y. member(ordered_pair(X::'a,Y),successor_relation) --> equal(successor(X),Y)) &    \
\  (! 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)) & \
\  (! X. inductive(X) --> member(null_class::'a,X)) &       \
\  (! X. inductive(X) --> subclass(image_(successor_relation::'a,X),X)) &    \
\  (! X. member(null_class::'a,X) & subclass(image_(successor_relation::'a,X),X) --> inductive(X)) &     \
\  (inductive(omega)) & \
\  (! Y. inductive(Y) --> subclass(omega::'a,Y)) &  \
\  (member(omega::'a,universal_class)) &    \
\  (! X. equal(domain_of(restrct(element_relation::'a,universal_class,X)),sum_class(X))) & \
\  (! X. member(X::'a,universal_class) --> member(sum_class(X),universal_class)) &  \
\  (! X. equal(complement(image_(element_relation::'a,complement(X))),powerClass(X))) &     \
\  (! U. member(U::'a,universal_class) --> member(powerClass(U),universal_class)) &        \
\  (! Yr Xr. subclass(compos(Yr::'a,Xr),cross_product(universal_class::'a,universal_class))) & \
\  (! 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))))) &     \
\  (! 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))) &  \
\  (! X. single_valued_class(X) --> subclass(compos(X::'a,inverse(X)),identity_relation)) &        \
\  (! X. subclass(compos(X::'a,inverse(X)),identity_relation) --> single_valued_class(X)) &        \
\  (! Xf. function(Xf) --> subclass(Xf::'a,cross_product(universal_class::'a,universal_class))) &       \
\  (! Xf. function(Xf) --> subclass(compos(Xf::'a,inverse(Xf)),identity_relation)) &       \
\  (! Xf. subclass(Xf::'a,cross_product(universal_class::'a,universal_class)) & subclass(compos(Xf::'a,inverse(Xf)),identity_relation) --> function(Xf)) & \
\  (! Xf X. function(Xf) & member(X::'a,universal_class) --> member(image_(Xf::'a,X),universal_class)) & \
\  (! X. equal(X::'a,null_class) | member(regular(X),X)) & \
\  (! X. equal(X::'a,null_class) | equal(intersection(X::'a,regular(X)),null_class)) & \
\  (! Xf Y. equal(sum_class(image_(Xf::'a,singleton(Y))),apply(Xf::'a,Y))) &     \
\  (function(choice)) & \
\  (! Y. member(Y::'a,universal_class) --> equal(Y::'a,null_class) | member(apply(choice::'a,Y),Y)) &      \
\  (! Xf. one_to_one(Xf) --> function(Xf)) &    \
\  (! Xf. one_to_one(Xf) --> function(inverse(Xf))) &   \
\  (! Xf. function(inverse(Xf)) & function(Xf) --> one_to_one(Xf)) &    \
\  (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)) &     \
\  (equal(intersection(inverse(subset_relation),subset_relation),identity_relation)) &  \
\  (! Xr. equal(complement(domain_of(intersection(Xr::'a,identity_relation))),diagonalise(Xr))) &   \
\  (! X. equal(intersection(domain_of(X),diagonalise(compos(inverse(element_relation),X))),cantor(X))) &       \
\  (! Xf. operation(Xf) --> function(Xf)) &     \
\  (! Xf. operation(Xf) --> equal(cross_product(domain_of(domain_of(Xf)),domain_of(domain_of(Xf))),domain_of(Xf))) &    \
\  (! Xf. operation(Xf) --> subclass(range_of(Xf),domain_of(domain_of(Xf)))) &  \
\  (! 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)) &   \
\  (! Xf1 Xf2 Xh. compatible(Xh::'a,Xf1,Xf2) --> function(Xh)) &    \
\  (! Xf2 Xf1 Xh. compatible(Xh::'a,Xf1,Xf2) --> equal(domain_of(domain_of(Xf1)),domain_of(Xh))) &  \
\  (! Xf1 Xh Xf2. compatible(Xh::'a,Xf1,Xf2) --> subclass(range_of(Xh),domain_of(domain_of(Xf2)))) &        \
\  (! 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)) &   \
\  (! Xh Xf2 Xf1. homomorphism(Xh::'a,Xf1,Xf2) --> operation(Xf1)) &        \
\  (! Xh Xf1 Xf2. homomorphism(Xh::'a,Xf1,Xf2) --> operation(Xf2)) &        \
\  (! Xh Xf1 Xf2. homomorphism(Xh::'a,Xf1,Xf2) --> compatible(Xh::'a,Xf1,Xf2)) &        \
\  (! 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))))) & \
\  (! 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)) &  \
\  (! 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)) &     \
\  (! D E F'. equal(D::'a,E) --> equal(apply(D::'a,F'),apply(E::'a,F'))) &  \
\  (! G I' H. equal(G::'a,H) --> equal(apply(I'::'a,G),apply(I'::'a,H))) &  \
\  (! J K'. equal(J::'a,K') --> equal(cantor(J),cantor(K'))) &      \
\  (! L M. equal(L::'a,M) --> equal(complement(L),complement(M))) & \
\  (! N O_ P. equal(N::'a,O_) --> equal(compos(N::'a,P),compos(O_::'a,P))) & \
\  (! Q S' R. equal(Q::'a,R) --> equal(compos(S'::'a,Q),compos(S'::'a,R))) &      \
\  (! T' U V. equal(T'::'a,U) --> equal(cross_product(T'::'a,V),cross_product(U::'a,V))) &  \
\  (! W Y X. equal(W::'a,X) --> equal(cross_product(Y::'a,W),cross_product(Y::'a,X))) &     \
\  (! Z A1. equal(Z::'a,A1) --> equal(diagonalise(Z),diagonalise(A1))) &    \
\  (! B1 C1 D1. equal(B1::'a,C1) --> equal(difference(B1::'a,D1),difference(C1::'a,D1))) &  \
\  (! E1 G1 F1. equal(E1::'a,F1) --> equal(difference(G1::'a,E1),difference(G1::'a,F1))) &  \
\  (! H1 I1 J1 K1. equal(H1::'a,I1) --> equal(domain(H1::'a,J1,K1),domain(I1::'a,J1,K1))) & \
\  (! L1 N1 M1 O1. equal(L1::'a,M1) --> equal(domain(N1::'a,L1,O1),domain(N1::'a,M1,O1))) & \
\  (! P1 R1 S1 Q1. equal(P1::'a,Q1) --> equal(domain(R1::'a,S1,P1),domain(R1::'a,S1,Q1))) & \
\  (! T1 U1. equal(T1::'a,U1) --> equal(domain_of(T1),domain_of(U1))) &     \
\  (! V1 W1. equal(V1::'a,W1) --> equal(first(V1),first(W1))) &     \
\  (! X1 Y1. equal(X1::'a,Y1) --> equal(flip(X1),flip(Y1))) &       \
\  (! Z1 A2 B2. equal(Z1::'a,A2) --> equal(image_(Z1::'a,B2),image_(A2::'a,B2))) &    \
\  (! C2 E2 D2. equal(C2::'a,D2) --> equal(image_(E2::'a,C2),image_(E2::'a,D2))) &    \
\  (! F2 G2 H2. equal(F2::'a,G2) --> equal(intersection(F2::'a,H2),intersection(G2::'a,H2))) &      \
\  (! I2 K2 J2. equal(I2::'a,J2) --> equal(intersection(K2::'a,I2),intersection(K2::'a,J2))) &      \
\  (! L2 M2. equal(L2::'a,M2) --> equal(inverse(L2),inverse(M2))) & \
\  (! N2 O2 P2 Q2. equal(N2::'a,O2) --> equal(not_homomorphism1(N2::'a,P2,Q2),not_homomorphism1(O2::'a,P2,Q2))) &   \
\  (! R2 T2 S2 U2. equal(R2::'a,S2) --> equal(not_homomorphism1(T2::'a,R2,U2),not_homomorphism1(T2::'a,S2,U2))) &   \
\  (! V2 X2 Y2 W2. equal(V2::'a,W2) --> equal(not_homomorphism1(X2::'a,Y2,V2),not_homomorphism1(X2::'a,Y2,W2))) &   \
\  (! Z2 A3 B3 C3. equal(Z2::'a,A3) --> equal(not_homomorphism2(Z2::'a,B3,C3),not_homomorphism2(A3::'a,B3,C3))) &   \
\  (! D3 F3 E3 G3. equal(D3::'a,E3) --> equal(not_homomorphism2(F3::'a,D3,G3),not_homomorphism2(F3::'a,E3,G3))) &   \
\  (! H3 J3 K3 I3. equal(H3::'a,I3) --> equal(not_homomorphism2(J3::'a,K3,H3),not_homomorphism2(J3::'a,K3,I3))) &   \
\  (! L3 M3 N3. equal(L3::'a,M3) --> equal(not_subclass_element(L3::'a,N3),not_subclass_element(M3::'a,N3))) &      \
\  (! O3 Q3 P3. equal(O3::'a,P3) --> equal(not_subclass_element(Q3::'a,O3),not_subclass_element(Q3::'a,P3))) &      \
\  (! R3 S3 T3. equal(R3::'a,S3) --> equal(ordered_pair(R3::'a,T3),ordered_pair(S3::'a,T3))) &      \
\  (! U3 W3 V3. equal(U3::'a,V3) --> equal(ordered_pair(W3::'a,U3),ordered_pair(W3::'a,V3))) &      \
\  (! X3 Y3. equal(X3::'a,Y3) --> equal(powerClass(X3),powerClass(Y3))) & \
\  (! Z3 A4 B4 C4. equal(Z3::'a,A4) --> equal(rng(Z3::'a,B4,C4),rng(A4::'a,B4,C4))) &   \
\  (! D4 F4 E4 G4. equal(D4::'a,E4) --> equal(rng(F4::'a,D4,G4),rng(F4::'a,E4,G4))) &   \
\  (! H4 J4 K4 I4. equal(H4::'a,I4) --> equal(rng(J4::'a,K4,H4),rng(J4::'a,K4,I4))) &   \
\  (! L4 M4. equal(L4::'a,M4) --> equal(range_of(L4),range_of(M4))) &       \
\  (! N4 O4. equal(N4::'a,O4) --> equal(regular(N4),regular(O4))) & \
\  (! P4 Q4 R4 S4. equal(P4::'a,Q4) --> equal(restrct(P4::'a,R4,S4),restrct(Q4::'a,R4,S4))) &     \
\  (! T4 V4 U4 W4. equal(T4::'a,U4) --> equal(restrct(V4::'a,T4,W4),restrct(V4::'a,U4,W4))) &     \
\  (! X4 Z4 A5 Y4. equal(X4::'a,Y4) --> equal(restrct(Z4::'a,A5,X4),restrct(Z4::'a,A5,Y4))) &     \
\  (! B5 C5. equal(B5::'a,C5) --> equal(rotate(B5),rotate(C5))) &   \
\  (! D5 E5. equal(D5::'a,E5) --> equal(second(D5),second(E5))) &   \
\  (! F5 G5. equal(F5::'a,G5) --> equal(singleton(F5),singleton(G5))) &     \
\  (! H5 I5. equal(H5::'a,I5) --> equal(successor(H5),successor(I5))) &     \
\  (! J5 K5. equal(J5::'a,K5) --> equal(sum_class(J5),sum_class(K5))) &     \
\  (! L5 M5 N5. equal(L5::'a,M5) --> equal(union(L5::'a,N5),union(M5::'a,N5))) &    \
\  (! O5 Q5 P5. equal(O5::'a,P5) --> equal(union(Q5::'a,O5),union(Q5::'a,P5))) &    \
\  (! R5 S5 T5. equal(R5::'a,S5) --> equal(unordered_pair(R5::'a,T5),unordered_pair(S5::'a,T5))) &  \
\  (! U5 W5 V5. equal(U5::'a,V5) --> equal(unordered_pair(W5::'a,U5),unordered_pair(W5::'a,V5))) &  \
\  (! X5 Y5 Z5 A6. equal(X5::'a,Y5) & compatible(X5::'a,Z5,A6) --> compatible(Y5::'a,Z5,A6)) &      \
\  (! B6 D6 C6 E6. equal(B6::'a,C6) & compatible(D6::'a,B6,E6) --> compatible(D6::'a,C6,E6)) &      \
\  (! F6 H6 I6 G6. equal(F6::'a,G6) & compatible(H6::'a,I6,F6) --> compatible(H6::'a,I6,G6)) &      \
\  (! J6 K6. equal(J6::'a,K6) & function(J6) --> function(K6)) &    \
\  (! L6 M6 N6 O6. equal(L6::'a,M6) & homomorphism(L6::'a,N6,O6) --> homomorphism(M6::'a,N6,O6)) &  \
\  (! P6 R6 Q6 S6. equal(P6::'a,Q6) & homomorphism(R6::'a,P6,S6) --> homomorphism(R6::'a,Q6,S6)) &  \
\  (! T6 V6 W6 U6. equal(T6::'a,U6) & homomorphism(V6::'a,W6,T6) --> homomorphism(V6::'a,W6,U6)) &  \
\  (! X6 Y6. equal(X6::'a,Y6) & inductive(X6) --> inductive(Y6)) &  \
\  (! Z6 A7 B7. equal(Z6::'a,A7) & member(Z6::'a,B7) --> member(A7::'a,B7)) &       \
\  (! C7 E7 D7. equal(C7::'a,D7) & member(E7::'a,C7) --> member(E7::'a,D7)) &       \
\  (! F7 G7. equal(F7::'a,G7) & one_to_one(F7) --> one_to_one(G7)) &        \
\  (! H7 I7. equal(H7::'a,I7) & operation(H7) --> operation(I7)) &  \
\  (! J7 K7. equal(J7::'a,K7) & single_valued_class(J7) --> single_valued_class(K7)) &      \
\  (! L7 M7 N7. equal(L7::'a,M7) & subclass(L7::'a,N7) --> subclass(M7::'a,N7)) &   \
\  (! O7 Q7 P7. equal(O7::'a,P7) & subclass(Q7::'a,O7) --> subclass(Q7::'a,P7)) &   \
\  (! X. subclass(compose_class(X),cross_product(universal_class::'a,universal_class))) &   \
\  (! X Y Z. member(ordered_pair(Y::'a,Z),compose_class(X)) --> equal(compos(X::'a,Y),Z)) &    \
\  (! 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))) & \
\  (subclass(composition_function::'a,cross_product(universal_class::'a,cross_product(universal_class::'a,universal_class)))) &     \
\  (! X Y Z. member(ordered_pair(X::'a,ordered_pair(Y::'a,Z)),composition_function) --> equal(compos(X::'a,Y),Z)) &        \
\  (! 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)) &    \
\  (subclass(domain_relation::'a,cross_product(universal_class::'a,universal_class))) & \
\  (! X Y. member(ordered_pair(X::'a,Y),domain_relation) --> equal(domain_of(X),Y)) &       \
\  (! X. member(X::'a,universal_class) --> member(ordered_pair(X::'a,domain_of(X)),domain_relation)) &  \
\  (! X. equal(first(not_subclass_element(compos(X::'a,inverse(X)),identity_relation)),single_valued1(X))) &       \
\  (! X. equal(second(not_subclass_element(compos(X::'a,inverse(X)),identity_relation)),single_valued2(X))) &      \
\  (! X. equal(domain(X::'a,image_(inverse(X),singleton(single_valued1(X))),single_valued2(X)),single_valued3(X))) & \
\  (equal(intersection(complement(compos(element_relation::'a,complement(identity_relation))),element_relation),singleton_relation)) &     \
\  (subclass(application_function::'a,cross_product(universal_class::'a,cross_product(universal_class::'a,universal_class)))) &     \
\  (! Z Y X. member(ordered_pair(X::'a,ordered_pair(Y::'a,Z)),application_function) --> member(Y::'a,domain_of(X))) &       \
\  (! X Y Z. member(ordered_pair(X::'a,ordered_pair(Y::'a,Z)),application_function) --> equal(apply(X::'a,Y),Z)) &  \
\  (! 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)) &    \
\  (! X Y Xf. maps(Xf::'a,X,Y) --> function(Xf)) &  \
\  (! Y Xf X. maps(Xf::'a,X,Y) --> equal(domain_of(Xf),X)) &        \
\  (! X Xf Y. maps(Xf::'a,X,Y) --> subclass(range_of(Xf),Y)) &      \
\  (! Xf Y. function(Xf) & subclass(range_of(Xf),Y) --> maps(Xf::'a,domain_of(Xf),Y)) &     \
\  (! L M. equal(L::'a,M) --> equal(compose_class(L),compose_class(M))) &   \
\  (! N2 O2. equal(N2::'a,O2) --> equal(single_valued1(N2),single_valued1(O2))) &   \
\  (! P2 Q2. equal(P2::'a,Q2) --> equal(single_valued2(P2),single_valued2(Q2))) &   \
\  (! R2 S2. equal(R2::'a,S2) --> equal(single_valued3(R2),single_valued3(S2))) &   \
\  (! X2 Y2 Z2 A3. equal(X2::'a,Y2) & maps(X2::'a,Z2,A3) --> maps(Y2::'a,Z2,A3)) &  \
\  (! B3 D3 C3 E3. equal(B3::'a,C3) & maps(D3::'a,B3,E3) --> maps(D3::'a,C3,E3)) &  \
\  (! F3 H3 I3 G3. equal(F3::'a,G3) & maps(H3::'a,I3,F3) --> maps(H3::'a,I3,G3)) &  \
\  (! X. equal(union(X::'a,inverse(X)),symmetrization_of(X))) &     \
\  (! X Y. irreflexive(X::'a,Y) --> subclass(restrct(X::'a,Y,Y),complement(identity_relation))) &      \
\  (! X Y. subclass(restrct(X::'a,Y,Y),complement(identity_relation)) --> irreflexive(X::'a,Y)) &      \
\  (! Y X. connected(X::'a,Y) --> subclass(cross_product(Y::'a,Y),union(identity_relation::'a,symmetrization_of(X)))) &     \
\  (! X Y. subclass(cross_product(Y::'a,Y),union(identity_relation::'a,symmetrization_of(X))) --> connected(X::'a,Y)) &     \
\  (! Xr Y. transitive(Xr::'a,Y) --> subclass(compos(restrct(Xr::'a,Y,Y),restrct(Xr::'a,Y,Y)),restrct(Xr::'a,Y,Y))) &       \
\  (! Xr Y. subclass(compos(restrct(Xr::'a,Y,Y),restrct(Xr::'a,Y,Y)),restrct(Xr::'a,Y,Y)) --> transitive(Xr::'a,Y)) &       \
\  (! Xr Y. asymmetric(Xr::'a,Y) --> equal(restrct(intersection(Xr::'a,inverse(Xr)),Y,Y),null_class)) &        \
\  (! Xr Y. equal(restrct(intersection(Xr::'a,inverse(Xr)),Y,Y),null_class) --> asymmetric(Xr::'a,Y)) &        \
\  (! Xr Y Z. equal(segment(Xr::'a,Y,Z),domain_of(restrct(Xr::'a,Y,singleton(Z))))) &  \
\  (! X Y. well_ordering(X::'a,Y) --> connected(X::'a,Y)) &     \
\  (! Y Xr U. well_ordering(Xr::'a,Y) & subclass(U::'a,Y) --> equal(U::'a,null_class) | member(least(Xr::'a,U),U)) &   \
\  (! Y V Xr U. well_ordering(Xr::'a,Y) & subclass(U::'a,Y) & member(V::'a,U) --> member(least(Xr::'a,U),U)) &  \