src/HOL/ex/Meson_Test.thy
author haftmann
Sat Aug 28 16:14:32 2010 +0200 (2010-08-28)
changeset 38864 4abe644fcea5
parent 38786 e46e7a9cb622
child 38867 23af89f419bb
permissions -rw-r--r--
formerly unnamed infix equality now named HOL.eq
     1 
     2 header {* Meson test cases *}
     3 
     4 theory Meson_Test
     5 imports Main
     6 begin
     7 
     8 text {*
     9   WARNING: there are many potential conflicts between variables used
    10   below and constants declared in HOL!
    11 *}
    12 
    13 hide_const (open) eq implies union inter subset sum quotient 
    14 
    15 text {*
    16   Test data for the MESON proof procedure
    17   (Excludes the equality problems 51, 52, 56, 58)
    18 *}
    19 
    20 
    21 subsection {* Interactive examples *}
    22 
    23 lemma problem_25:
    24   "(\<exists>x. P x) & (\<forall>x. L x --> ~ (M x & R x)) & (\<forall>x. P x --> (M x & L x)) & ((\<forall>x. P x --> Q x) | (\<exists>x. P x & R x)) --> (\<exists>x. Q x & P x)"
    25   apply (rule ccontr)
    26   ML_prf {*
    27     val prem25 = Thm.assume @{cprop "\<not> ?thesis"};
    28     val nnf25 = Meson.make_nnf @{context} prem25;
    29     val xsko25 = Meson.skolemize @{context} nnf25;
    30   *}
    31   apply (tactic {* cut_facts_tac [xsko25] 1 THEN REPEAT (etac exE 1) *})
    32   ML_val {*
    33     val [_, sko25] = #prems (#1 (Subgoal.focus @{context} 1 (#goal @{Isar.goal})));
    34     val clauses25 = Meson.make_clauses [sko25];   (*7 clauses*)
    35     val horns25 = Meson.make_horns clauses25;     (*16 Horn clauses*)
    36     val go25 :: _ = Meson.gocls clauses25;
    37 
    38     Goal.prove @{context} [] [] @{prop False} (fn _ =>
    39       rtac go25 1 THEN
    40       Meson.depth_prolog_tac horns25);
    41   *}
    42   oops
    43 
    44 lemma problem_26:
    45   "((\<exists>x. p x) = (\<exists>x. q x)) & (\<forall>x. \<forall>y. p x & q y --> (r x = s y)) --> ((\<forall>x. p x --> r x) = (\<forall>x. q x --> s x))"
    46   apply (rule ccontr)
    47   ML_prf {*
    48     val prem26 = Thm.assume @{cprop "\<not> ?thesis"}
    49     val nnf26 = Meson.make_nnf @{context} prem26;
    50     val xsko26 = Meson.skolemize @{context} nnf26;
    51   *}
    52   apply (tactic {* cut_facts_tac [xsko26] 1 THEN REPEAT (etac exE 1) *})
    53   ML_val {*
    54     val [_, sko26] = #prems (#1 (Subgoal.focus @{context} 1 (#goal @{Isar.goal})));
    55     val clauses26 = Meson.make_clauses [sko26];                   (*9 clauses*)
    56     val horns26 = Meson.make_horns clauses26;                     (*24 Horn clauses*)
    57     val go26 :: _ = Meson.gocls clauses26;
    58 
    59     Goal.prove @{context} [] [] @{prop False} (fn _ =>
    60       rtac go26 1 THEN
    61       Meson.depth_prolog_tac horns26);  (*7 ms*)
    62     (*Proof is of length 107!!*)
    63   *}
    64   oops
    65 
    66 lemma problem_43:  -- "NOW PROVED AUTOMATICALLY!!"  (*16 Horn clauses*)
    67   "(\<forall>x. \<forall>y. q x y = (\<forall>z. p z x = (p z y::bool))) --> (\<forall>x. (\<forall>y. q x y = (q y x::bool)))"
    68   apply (rule ccontr)
    69   ML_prf {*
    70     val prem43 = Thm.assume @{cprop "\<not> ?thesis"};
    71     val nnf43 = Meson.make_nnf @{context} prem43;
    72     val xsko43 = Meson.skolemize @{context} nnf43;
    73   *}
    74   apply (tactic {* cut_facts_tac [xsko43] 1 THEN REPEAT (etac exE 1) *})
    75   ML_val {*
    76     val [_, sko43] = #prems (#1 (Subgoal.focus @{context} 1 (#goal @{Isar.goal})));
    77     val clauses43 = Meson.make_clauses [sko43];   (*6*)
    78     val horns43 = Meson.make_horns clauses43;     (*16*)
    79     val go43 :: _ = Meson.gocls clauses43;
    80 
    81     Goal.prove @{context} [] [] @{prop False} (fn _ =>
    82       rtac go43 1 THEN
    83       Meson.best_prolog_tac Meson.size_of_subgoals horns43);   (*7ms*)
    84     *}
    85   oops
    86 
    87 (*
    88 #1  (q x xa ==> ~ q x xa) ==> q xa x
    89 #2  (q xa x ==> ~ q xa x) ==> q x xa
    90 #3  (~ q x xa ==> q x xa) ==> ~ q xa x
    91 #4  (~ q xa x ==> q xa x) ==> ~ q x xa
    92 #5  [| ~ q ?U ?V ==> q ?U ?V; ~ p ?W ?U ==> p ?W ?U |] ==> p ?W ?V
    93 #6  [| ~ p ?W ?U ==> p ?W ?U; p ?W ?V ==> ~ p ?W ?V |] ==> ~ q ?U ?V
    94 #7  [| p ?W ?V ==> ~ p ?W ?V; ~ q ?U ?V ==> q ?U ?V |] ==> ~ p ?W ?U
    95 #8  [| ~ q ?U ?V ==> q ?U ?V; ~ p ?W ?V ==> p ?W ?V |] ==> p ?W ?U
    96 #9  [| ~ p ?W ?V ==> p ?W ?V; p ?W ?U ==> ~ p ?W ?U |] ==> ~ q ?U ?V
    97 #10 [| p ?W ?U ==> ~ p ?W ?U; ~ q ?U ?V ==> q ?U ?V |] ==> ~ p ?W ?V
    98 #11 [| p (xb ?U ?V) ?U ==> ~ p (xb ?U ?V) ?U;
    99        p (xb ?U ?V) ?V ==> ~ p (xb ?U ?V) ?V |] ==> q ?U ?V
   100 #12 [| p (xb ?U ?V) ?V ==> ~ p (xb ?U ?V) ?V; q ?U ?V ==> ~ q ?U ?V |] ==>
   101     p (xb ?U ?V) ?U
   102 #13 [| q ?U ?V ==> ~ q ?U ?V; p (xb ?U ?V) ?U ==> ~ p (xb ?U ?V) ?U |] ==>
   103     p (xb ?U ?V) ?V
   104 #14 [| ~ p (xb ?U ?V) ?U ==> p (xb ?U ?V) ?U;
   105        ~ p (xb ?U ?V) ?V ==> p (xb ?U ?V) ?V |] ==> q ?U ?V
   106 #15 [| ~ p (xb ?U ?V) ?V ==> p (xb ?U ?V) ?V; q ?U ?V ==> ~ q ?U ?V |] ==>
   107     ~ p (xb ?U ?V) ?U
   108 #16 [| q ?U ?V ==> ~ q ?U ?V; ~ p (xb ?U ?V) ?U ==> p (xb ?U ?V) ?U |] ==>
   109     ~ p (xb ?U ?V) ?V
   110 
   111 And here is the proof! (Unkn is the start state after use of goal clause)
   112 [Unkn, Res ([Thm "#14"], false, 1), Res ([Thm "#5"], false, 1),
   113    Res ([Thm "#1"], false, 1), Asm 1, Res ([Thm "#13"], false, 1), Asm 2,
   114    Asm 1, Res ([Thm "#13"], false, 1), Asm 1, Res ([Thm "#10"], false, 1),
   115    Res ([Thm "#16"], false, 1), Asm 2, Asm 1, Res ([Thm "#1"], false, 1),
   116    Asm 1, Res ([Thm "#14"], false, 1), Res ([Thm "#5"], false, 1),
   117    Res ([Thm "#2"], false, 1), Asm 1, Res ([Thm "#13"], false, 1), Asm 2,
   118    Asm 1, Res ([Thm "#8"], false, 1), Res ([Thm "#2"], false, 1), Asm 1,
   119    Res ([Thm "#12"], false, 1), Asm 2, Asm 1] : lderiv list
   120 *)
   121 
   122 
   123 text {*
   124   MORE and MUCH HARDER test data for the MESON proof procedure
   125   (courtesy John Harrison).
   126 *}
   127 
   128 (* ========================================================================= *)
   129 (* 100 problems selected from the TPTP library                               *)
   130 (* ========================================================================= *)
   131 
   132 (*
   133  * Original timings for John Harrison's MESON_TAC.
   134  * Timings below on a 600MHz Pentium III (perch)
   135  * Some timings below refer to griffon, which is a dual 2.5GHz Power Mac G5.
   136  *
   137  * A few variable names have been changed to avoid clashing with constants.
   138  *
   139  * Changed numeric constants e.g. 0, 1, 2... to num0, num1, num2...
   140  *
   141  * Here's a list giving typical CPU times, as well as common names and
   142  * literature references.
   143  *
   144  * BOO003-1     34.6    B2 part 1 [McCharen, et al., 1976]; Lemma proved [Overbeek, et al., 1976]; prob2_part1.ver1.in [ANL]
   145  * BOO004-1     36.7    B2 part 2 [McCharen, et al., 1976]; Lemma proved [Overbeek, et al., 1976]; prob2_part2.ver1 [ANL]
   146  * BOO005-1     47.4    B3 part 1 [McCharen, et al., 1976]; B5 [McCharen, et al., 1976]; Lemma proved [Overbeek, et al., 1976]; prob3_part1.ver1.in [ANL]
   147  * BOO006-1     48.4    B3 part 2 [McCharen, et al., 1976]; B6 [McCharen, et al., 1976]; Lemma proved [Overbeek, et al., 1976]; prob3_part2.ver1 [ANL]
   148  * BOO011-1     19.0    B7 [McCharen, et al., 1976]; prob7.ver1 [ANL]
   149  * CAT001-3     45.2    C1 [McCharen, et al., 1976]; p1.ver3.in [ANL]
   150  * CAT003-3     10.5    C3 [McCharen, et al., 1976]; p3.ver3.in [ANL]
   151  * CAT005-1    480.1    C5 [McCharen, et al., 1976]; p5.ver1.in [ANL]
   152  * CAT007-1     11.9    C7 [McCharen, et al., 1976]; p7.ver1.in [ANL]
   153  * CAT018-1     81.3    p18.ver1.in [ANL]
   154  * COL001-2     16.0    C1 [Wos & McCune, 1988]
   155  * COL023-1      5.1    [McCune & Wos, 1988]
   156  * COL032-1     15.8    [McCune & Wos, 1988]
   157  * COL052-2     13.2    bird4.ver2.in [ANL]
   158  * COL075-2    116.9    [Jech, 1994]
   159  * COM001-1      1.7    shortburst [Wilson & Minker, 1976]
   160  * COM002-1      4.4    burstall [Wilson & Minker, 1976]
   161  * COM002-2      7.4
   162  * COM003-2     22.1    [Brushi, 1991]
   163  * COM004-1     45.1
   164  * GEO003-1     71.7    T3 [McCharen, et al., 1976]; t3.ver1.in [ANL]
   165  * GEO017-2     78.8    D4.1 [Quaife, 1989]
   166  * GEO027-3    181.5    D10.1 [Quaife, 1989]
   167  * GEO058-2    104.0    R4 [Quaife, 1989]
   168  * GEO079-1      2.4    GEOMETRY THEOREM [Slagle, 1967]
   169  * GRP001-1     47.8    CADE-11 Competition 1 [Overbeek, 1990]; G1 [McCharen, et al., 1976]; THEOREM 1 [Lusk & McCune, 1993]; wos10 [Wilson & Minker, 1976]; xsquared.ver1.in [ANL]; [Robinson, 1963]
   170  * GRP008-1     50.4    Problem 4 [Wos, 1965]; wos4 [Wilson & Minker, 1976]
   171  * GRP013-1     40.2    Problem 11 [Wos, 1965]; wos11 [Wilson & Minker, 1976]
   172  * GRP037-3     43.8    Problem 17 [Wos, 1965]; wos17 [Wilson & Minker, 1976]
   173  * GRP031-2      3.2    ls23 [Lawrence & Starkey, 1974]; ls23 [Wilson & Minker, 1976]
   174  * GRP034-4      2.5    ls26 [Lawrence & Starkey, 1974]; ls26 [Wilson & Minker, 1976]
   175  * GRP047-2     11.7    [Veroff, 1992]
   176  * GRP130-1    170.6    Bennett QG8 [TPTP]; QG8 [Slaney, 1993]
   177  * GRP156-1     48.7    ax_mono1c [Schulz, 1995]
   178  * GRP168-1    159.1    p01a [Schulz, 1995]
   179  * HEN003-3     39.9    HP3 [McCharen, et al., 1976]
   180  * HEN007-2    125.7    H7 [McCharen, et al., 1976]
   181  * HEN008-4     62.0    H8 [McCharen, et al., 1976]
   182  * HEN009-5    136.3    H9 [McCharen, et al., 1976]; hp9.ver3.in [ANL]
   183  * HEN012-3     48.5    new.ver2.in [ANL]
   184  * LCL010-1    370.9    EC-73 [McCune & Wos, 1992]; ec_yq.in [OTTER]
   185  * LCL077-2     51.6    morgan.two.ver1.in [ANL]
   186  * LCL082-1     14.6    IC-1.1 [Wos, et al., 1990]; IC-65 [McCune & Wos, 1992]; ls2 [SETHEO]; S1 [Pfenning, 1988]
   187  * LCL111-1    585.6    CADE-11 Competition 6 [Overbeek, 1990]; mv25.in [OTTER]; MV-57 [McCune & Wos, 1992]; mv.in part 2 [OTTER]; ovb6 [SETHEO]; THEOREM 6 [Lusk & McCune, 1993]
   188  * LCL143-1     10.9    Lattice structure theorem 2 [Bonacina, 1991]
   189  * LCL182-1    271.6    Problem 2.16 [Whitehead & Russell, 1927]
   190  * LCL200-1     12.0    Problem 2.46 [Whitehead & Russell, 1927]
   191  * LCL215-1    214.4    Problem 2.62 [Whitehead & Russell, 1927]; Problem 2.63 [Whitehead & Russell, 1927]
   192  * LCL230-2      0.2    Pelletier 5 [Pelletier, 1986]
   193  * LDA003-1     68.5    Problem 3 [Jech, 1993]
   194  * MSC002-1      9.2    DBABHP [Michie, et al., 1972]; DBABHP [Wilson & Minker, 1976]
   195  * MSC003-1      3.2    HASPARTS-T1 [Wilson & Minker, 1976]
   196  * MSC004-1      9.3    HASPARTS-T2 [Wilson & Minker, 1976]
   197  * MSC005-1      1.8    Problem 5.1 [Plaisted, 1982]
   198  * MSC006-1     39.0    nonob.lop [SETHEO]
   199  * NUM001-1     14.0    Chang-Lee-10a [Chang, 1970]; ls28 [Lawrence & Starkey, 1974]; ls28 [Wilson & Minker, 1976]
   200  * NUM021-1     52.3    ls65 [Lawrence & Starkey, 1974]; ls65 [Wilson & Minker, 1976]
   201  * NUM024-1     64.6    ls75 [Lawrence & Starkey, 1974]; ls75 [Wilson & Minker, 1976]
   202  * NUM180-1    621.2    LIM2.1 [Quaife]
   203  * NUM228-1    575.9    TRECDEF4 cor. [Quaife]
   204  * PLA002-1     37.4    Problem 5.7 [Plaisted, 1982]
   205  * PLA006-1      7.2    [Segre & Elkan, 1994]
   206  * PLA017-1    484.8    [Segre & Elkan, 1994]
   207  * PLA022-1     19.1    [Segre & Elkan, 1994]
   208  * PLA022-2     19.7    [Segre & Elkan, 1994]
   209  * PRV001-1     10.3    PV1 [McCharen, et al., 1976]
   210  * PRV003-1      3.9    E2 [McCharen, et al., 1976]; v2.lop [SETHEO]
   211  * PRV005-1      4.3    E4 [McCharen, et al., 1976]; v4.lop [SETHEO]
   212  * PRV006-1      6.0    E5 [McCharen, et al., 1976]; v5.lop [SETHEO]
   213  * PRV009-1      2.2    Hoares FIND [Bledsoe, 1977]; Problem 5.5 [Plaisted, 1982]
   214  * PUZ012-1      3.5    Boxes-of-fruit [Wos, 1988]; Boxes-of-fruit [Wos, et al., 1992]; boxes.ver1.in [ANL]
   215  * PUZ020-1     56.6    knightknave.in [ANL]
   216  * PUZ025-1     58.4    Problem 35 [Smullyan, 1978]; tandl35.ver1.in [ANL]
   217  * PUZ029-1      5.1    pigs.ver1.in [ANL]
   218  * RNG001-3     82.4    EX6-T? [Wilson & Minker, 1976]; ex6.lop [SETHEO]; Example 6a [Fleisig, et al., 1974]; FEX6T1 [SPRFN]; FEX6T2 [SPRFN]
   219  * RNG001-5    399.8    Problem 21 [Wos, 1965]; wos21 [Wilson & Minker, 1976]
   220  * RNG011-5      8.4    CADE-11 Competition Eq-10 [Overbeek, 1990]; PROBLEM 10 [Zhang, 1993]; THEOREM EQ-10 [Lusk & McCune, 1993]
   221  * RNG023-6      9.1    [Stevens, 1987]
   222  * RNG028-2      9.3    PROOF III [Anantharaman & Hsiang, 1990]
   223  * RNG038-2     16.2    Problem 27 [Wos, 1965]; wos27 [Wilson & Minker, 1976]
   224  * RNG040-2    180.5    Problem 29 [Wos, 1965]; wos29 [Wilson & Minker, 1976]
   225  * RNG041-1     35.8    Problem 30 [Wos, 1965]; wos30 [Wilson & Minker, 1976]
   226  * ROB010-1    205.0    Lemma 3.3 [Winker, 1990]; RA2 [Lusk & Wos, 1992]
   227  * ROB013-1     23.6    Lemma 3.5 [Winker, 1990]
   228  * ROB016-1     15.2    Corollary 3.7 [Winker, 1990]
   229  * ROB021-1    230.4    [McCune, 1992]
   230  * SET005-1    192.2    ls108 [Lawrence & Starkey, 1974]; ls108 [Wilson & Minker, 1976]
   231  * SET009-1     10.5    ls116 [Lawrence & Starkey, 1974]; ls116 [Wilson & Minker, 1976]
   232  * SET025-4    694.7    Lemma 10 [Boyer, et al, 1986]
   233  * SET046-5      2.3    p42.in [ANL]; Pelletier 42 [Pelletier, 1986]
   234  * SET047-5      3.7    p43.in [ANL]; Pelletier 43 [Pelletier, 1986]
   235  * SYN034-1      2.8    QW [Michie, et al., 1972]; QW [Wilson & Minker, 1976]
   236  * SYN071-1      1.9    Pelletier 48 [Pelletier, 1986]
   237  * SYN349-1     61.7    Ch17N5 [Tammet, 1994]
   238  * SYN352-1      5.5    Ch18N4 [Tammet, 1994]
   239  * TOP001-2     61.1    Lemma 1a [Wick & McCune, 1989]
   240  * TOP002-2      0.4    Lemma 1b [Wick & McCune, 1989]
   241  * TOP004-1    181.6    Lemma 1d [Wick & McCune, 1989]
   242  * TOP004-2      9.0    Lemma 1d [Wick & McCune, 1989]
   243  * TOP005-2    139.8    Lemma 1e [Wick & McCune, 1989]
   244  *)
   245 
   246 abbreviation "EQU001_0_ax equal \<equiv> (\<forall>X. equal(X::'a,X)) &
   247   (\<forall>Y X. equal(X::'a,Y) --> equal(Y::'a,X)) &
   248   (\<forall>Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z))"
   249 
   250 abbreviation "BOO002_0_ax equal INVERSE multiplicative_identity
   251   additive_identity multiply product add sum \<equiv>
   252   (\<forall>X Y. sum(X::'a,Y,add(X::'a,Y))) &
   253   (\<forall>X Y. product(X::'a,Y,multiply(X::'a,Y))) &
   254   (\<forall>Y X Z. sum(X::'a,Y,Z) --> sum(Y::'a,X,Z)) &
   255   (\<forall>Y X Z. product(X::'a,Y,Z) --> product(Y::'a,X,Z)) &
   256   (\<forall>X. sum(additive_identity::'a,X,X)) &
   257   (\<forall>X. sum(X::'a,additive_identity,X)) &
   258   (\<forall>X. product(multiplicative_identity::'a,X,X)) &
   259   (\<forall>X. product(X::'a,multiplicative_identity,X)) &
   260   (\<forall>Y Z X V3 V1 V2 V4. product(X::'a,Y,V1) & product(X::'a,Z,V2) & sum(Y::'a,Z,V3) & product(X::'a,V3,V4) --> sum(V1::'a,V2,V4)) &
   261   (\<forall>Y Z V1 V2 X V3 V4. product(X::'a,Y,V1) & product(X::'a,Z,V2) & sum(Y::'a,Z,V3) & sum(V1::'a,V2,V4) --> product(X::'a,V3,V4)) &
   262   (\<forall>Y Z V3 X V1 V2 V4. product(Y::'a,X,V1) & product(Z::'a,X,V2) & sum(Y::'a,Z,V3) & product(V3::'a,X,V4) --> sum(V1::'a,V2,V4)) &
   263   (\<forall>Y Z V1 V2 V3 X V4. product(Y::'a,X,V1) & product(Z::'a,X,V2) & sum(Y::'a,Z,V3) & sum(V1::'a,V2,V4) --> product(V3::'a,X,V4)) &
   264   (\<forall>Y Z X V3 V1 V2 V4. sum(X::'a,Y,V1) & sum(X::'a,Z,V2) & product(Y::'a,Z,V3) & sum(X::'a,V3,V4) --> product(V1::'a,V2,V4)) &
   265   (\<forall>Y Z V1 V2 X V3 V4. sum(X::'a,Y,V1) & sum(X::'a,Z,V2) & product(Y::'a,Z,V3) & product(V1::'a,V2,V4) --> sum(X::'a,V3,V4)) &
   266   (\<forall>Y Z V3 X V1 V2 V4. sum(Y::'a,X,V1) & sum(Z::'a,X,V2) & product(Y::'a,Z,V3) & sum(V3::'a,X,V4) --> product(V1::'a,V2,V4)) &
   267   (\<forall>Y Z V1 V2 V3 X V4. sum(Y::'a,X,V1) & sum(Z::'a,X,V2) & product(Y::'a,Z,V3) & product(V1::'a,V2,V4) --> sum(V3::'a,X,V4)) &
   268   (\<forall>X. sum(INVERSE(X),X,multiplicative_identity)) &
   269   (\<forall>X. sum(X::'a,INVERSE(X),multiplicative_identity)) &
   270   (\<forall>X. product(INVERSE(X),X,additive_identity)) &
   271   (\<forall>X. product(X::'a,INVERSE(X),additive_identity)) &
   272   (\<forall>X Y U V. sum(X::'a,Y,U) & sum(X::'a,Y,V) --> equal(U::'a,V)) &
   273   (\<forall>X Y U V. product(X::'a,Y,U) & product(X::'a,Y,V) --> equal(U::'a,V))"
   274 
   275 abbreviation "BOO002_0_eq INVERSE multiply add product sum equal \<equiv>
   276   (\<forall>X Y W Z. equal(X::'a,Y) & sum(X::'a,W,Z) --> sum(Y::'a,W,Z)) &
   277   (\<forall>X W Y Z. equal(X::'a,Y) & sum(W::'a,X,Z) --> sum(W::'a,Y,Z)) &
   278   (\<forall>X W Z Y. equal(X::'a,Y) & sum(W::'a,Z,X) --> sum(W::'a,Z,Y)) &
   279   (\<forall>X Y W Z. equal(X::'a,Y) & product(X::'a,W,Z) --> product(Y::'a,W,Z)) &
   280   (\<forall>X W Y Z. equal(X::'a,Y) & product(W::'a,X,Z) --> product(W::'a,Y,Z)) &
   281   (\<forall>X W Z Y. equal(X::'a,Y) & product(W::'a,Z,X) --> product(W::'a,Z,Y)) &
   282   (\<forall>X Y W. equal(X::'a,Y) --> equal(add(X::'a,W),add(Y::'a,W))) &
   283   (\<forall>X W Y. equal(X::'a,Y) --> equal(add(W::'a,X),add(W::'a,Y))) &
   284   (\<forall>X Y W. equal(X::'a,Y) --> equal(multiply(X::'a,W),multiply(Y::'a,W))) &
   285   (\<forall>X W Y. equal(X::'a,Y) --> equal(multiply(W::'a,X),multiply(W::'a,Y))) &
   286   (\<forall>X Y. equal(X::'a,Y) --> equal(INVERSE(X),INVERSE(Y)))"
   287 
   288 (*51194 inferences so far.  Searching to depth 13.  232.9 secs*)
   289 lemma BOO003_1:
   290   "EQU001_0_ax equal &
   291   BOO002_0_ax equal INVERSE multiplicative_identity additive_identity multiply product add sum &
   292   BOO002_0_eq INVERSE multiply add product sum equal &
   293   (~product(x::'a,x,x)) --> False"
   294   oops
   295 
   296 (*51194 inferences so far.  Searching to depth 13. 204.6 secs
   297   Strange! The previous problem also has 51194 inferences at depth 13.  They
   298    must be very similar!*)
   299 lemma BOO004_1:
   300   "EQU001_0_ax equal &
   301   BOO002_0_ax equal INVERSE multiplicative_identity additive_identity multiply product add sum &
   302   BOO002_0_eq INVERSE multiply add product sum equal &
   303   (~sum(x::'a,x,x)) --> False"
   304   oops
   305 
   306 (*74799 inferences so far.  Searching to depth 13.  290.0 secs*)
   307 lemma BOO005_1:
   308   "EQU001_0_ax equal &
   309   BOO002_0_ax equal INVERSE multiplicative_identity additive_identity multiply product add sum &
   310   BOO002_0_eq INVERSE multiply add product sum equal &
   311   (~sum(x::'a,multiplicative_identity,multiplicative_identity)) --> False"
   312   oops
   313 
   314 (*74799 inferences so far.  Searching to depth 13.  314.6 secs*)
   315 lemma BOO006_1:
   316   "EQU001_0_ax equal &
   317   BOO002_0_ax equal INVERSE multiplicative_identity additive_identity multiply product add sum &
   318   BOO002_0_eq INVERSE multiply add product sum equal &
   319   (~product(x::'a,additive_identity,additive_identity)) --> False"
   320   oops
   321 
   322 (*5 inferences so far.  Searching to depth 5.  1.3 secs*)
   323 lemma BOO011_1:
   324   "EQU001_0_ax equal &
   325   BOO002_0_ax equal INVERSE multiplicative_identity additive_identity multiply product add sum &
   326   BOO002_0_eq INVERSE multiply add product sum equal &
   327   (~equal(INVERSE(additive_identity),multiplicative_identity)) --> False"
   328   by meson
   329 
   330 abbreviation "CAT003_0_ax f1 compos codomain domain equal there_exists equivalent \<equiv>
   331   (\<forall>Y X. equivalent(X::'a,Y) --> there_exists(X)) &
   332   (\<forall>X Y. equivalent(X::'a,Y) --> equal(X::'a,Y)) &
   333   (\<forall>X Y. there_exists(X) & equal(X::'a,Y) --> equivalent(X::'a,Y)) &
   334   (\<forall>X. there_exists(domain(X)) --> there_exists(X)) &
   335   (\<forall>X. there_exists(codomain(X)) --> there_exists(X)) &
   336   (\<forall>Y X. there_exists(compos(X::'a,Y)) --> there_exists(domain(X))) &
   337   (\<forall>X Y. there_exists(compos(X::'a,Y)) --> equal(domain(X),codomain(Y))) &
   338   (\<forall>X Y. there_exists(domain(X)) & equal(domain(X),codomain(Y)) --> there_exists(compos(X::'a,Y))) &
   339   (\<forall>X Y Z. equal(compos(X::'a,compos(Y::'a,Z)),compos(compos(X::'a,Y),Z))) &
   340   (\<forall>X. equal(compos(X::'a,domain(X)),X)) &
   341   (\<forall>X. equal(compos(codomain(X),X),X)) &
   342   (\<forall>X Y. equivalent(X::'a,Y) --> there_exists(Y)) &
   343   (\<forall>X Y. there_exists(X) & there_exists(Y) & equal(X::'a,Y) --> equivalent(X::'a,Y)) &
   344   (\<forall>Y X. there_exists(compos(X::'a,Y)) --> there_exists(codomain(X))) &
   345   (\<forall>X Y. there_exists(f1(X::'a,Y)) | equal(X::'a,Y)) &
   346   (\<forall>X Y. equal(X::'a,f1(X::'a,Y)) | equal(Y::'a,f1(X::'a,Y)) | equal(X::'a,Y)) &
   347   (\<forall>X Y. equal(X::'a,f1(X::'a,Y)) & equal(Y::'a,f1(X::'a,Y)) --> equal(X::'a,Y))"
   348 
   349 abbreviation "CAT003_0_eq f1 compos codomain domain equivalent there_exists equal \<equiv>
   350   (\<forall>X Y. equal(X::'a,Y) & there_exists(X) --> there_exists(Y)) &
   351   (\<forall>X Y Z. equal(X::'a,Y) & equivalent(X::'a,Z) --> equivalent(Y::'a,Z)) &
   352   (\<forall>X Z Y. equal(X::'a,Y) & equivalent(Z::'a,X) --> equivalent(Z::'a,Y)) &
   353   (\<forall>X Y. equal(X::'a,Y) --> equal(domain(X),domain(Y))) &
   354   (\<forall>X Y. equal(X::'a,Y) --> equal(codomain(X),codomain(Y))) &
   355   (\<forall>X Y Z. equal(X::'a,Y) --> equal(compos(X::'a,Z),compos(Y::'a,Z))) &
   356   (\<forall>X Z Y. equal(X::'a,Y) --> equal(compos(Z::'a,X),compos(Z::'a,Y))) &
   357   (\<forall>A B C. equal(A::'a,B) --> equal(f1(A::'a,C),f1(B::'a,C))) &
   358   (\<forall>D F' E. equal(D::'a,E) --> equal(f1(F'::'a,D),f1(F'::'a,E)))"
   359 
   360 (*4007 inferences so far.  Searching to depth 9.  13 secs*)
   361 lemma CAT001_3:
   362   "EQU001_0_ax equal &
   363   CAT003_0_ax f1 compos codomain domain equal there_exists equivalent &
   364   CAT003_0_eq f1 compos codomain domain equivalent there_exists equal &
   365   (there_exists(compos(a::'a,b))) &
   366   (\<forall>Y X Z. equal(compos(compos(a::'a,b),X),Y) & equal(compos(compos(a::'a,b),Z),Y) --> equal(X::'a,Z)) &
   367   (there_exists(compos(b::'a,h))) &
   368   (equal(compos(b::'a,h),compos(b::'a,g))) &
   369   (~equal(h::'a,g)) --> False"
   370    by meson
   371 
   372 (*245 inferences so far.  Searching to depth 7.  1.0 secs*)
   373 lemma CAT003_3:
   374   "EQU001_0_ax equal &
   375   CAT003_0_ax f1 compos codomain domain equal there_exists equivalent &
   376   CAT003_0_eq f1 compos codomain domain equivalent there_exists equal &
   377   (there_exists(compos(a::'a,b))) &
   378   (\<forall>Y X Z. equal(compos(X::'a,compos(a::'a,b)),Y) & equal(compos(Z::'a,compos(a::'a,b)),Y) --> equal(X::'a,Z)) &
   379   (there_exists(h)) &
   380   (equal(compos(h::'a,a),compos(g::'a,a))) &
   381   (~equal(g::'a,h)) --> False"
   382   by meson
   383 
   384 abbreviation "CAT001_0_ax equal codomain domain identity_map compos product defined \<equiv>
   385   (\<forall>X Y. defined(X::'a,Y) --> product(X::'a,Y,compos(X::'a,Y))) &
   386   (\<forall>Z X Y. product(X::'a,Y,Z) --> defined(X::'a,Y)) &
   387   (\<forall>X Xy Y Z. product(X::'a,Y,Xy) & defined(Xy::'a,Z) --> defined(Y::'a,Z)) &
   388   (\<forall>Y Xy Z X Yz. product(X::'a,Y,Xy) & product(Y::'a,Z,Yz) & defined(Xy::'a,Z) --> defined(X::'a,Yz)) &
   389   (\<forall>Xy Y Z X Yz Xyz. product(X::'a,Y,Xy) & product(Xy::'a,Z,Xyz) & product(Y::'a,Z,Yz) --> product(X::'a,Yz,Xyz)) &
   390   (\<forall>Z Yz X Y. product(Y::'a,Z,Yz) & defined(X::'a,Yz) --> defined(X::'a,Y)) &
   391   (\<forall>Y X Yz Xy Z. product(Y::'a,Z,Yz) & product(X::'a,Y,Xy) & defined(X::'a,Yz) --> defined(Xy::'a,Z)) &
   392   (\<forall>Yz X Y Xy Z Xyz. product(Y::'a,Z,Yz) & product(X::'a,Yz,Xyz) & product(X::'a,Y,Xy) --> product(Xy::'a,Z,Xyz)) &
   393   (\<forall>Y X Z. defined(X::'a,Y) & defined(Y::'a,Z) & identity_map(Y) --> defined(X::'a,Z)) &
   394   (\<forall>X. identity_map(domain(X))) &
   395   (\<forall>X. identity_map(codomain(X))) &
   396   (\<forall>X. defined(X::'a,domain(X))) &
   397   (\<forall>X. defined(codomain(X),X)) &
   398   (\<forall>X. product(X::'a,domain(X),X)) &
   399   (\<forall>X. product(codomain(X),X,X)) &
   400   (\<forall>X Y. defined(X::'a,Y) & identity_map(X) --> product(X::'a,Y,Y)) &
   401   (\<forall>Y X. defined(X::'a,Y) & identity_map(Y) --> product(X::'a,Y,X)) &
   402   (\<forall>X Y Z W. product(X::'a,Y,Z) & product(X::'a,Y,W) --> equal(Z::'a,W))"
   403 
   404 abbreviation "CAT001_0_eq compos defined identity_map codomain domain product equal \<equiv>
   405   (\<forall>X Y Z W. equal(X::'a,Y) & product(X::'a,Z,W) --> product(Y::'a,Z,W)) &
   406   (\<forall>X Z Y W. equal(X::'a,Y) & product(Z::'a,X,W) --> product(Z::'a,Y,W)) &
   407   (\<forall>X Z W Y. equal(X::'a,Y) & product(Z::'a,W,X) --> product(Z::'a,W,Y)) &
   408   (\<forall>X Y. equal(X::'a,Y) --> equal(domain(X),domain(Y))) &
   409   (\<forall>X Y. equal(X::'a,Y) --> equal(codomain(X),codomain(Y))) &
   410   (\<forall>X Y. equal(X::'a,Y) & identity_map(X) --> identity_map(Y)) &
   411   (\<forall>X Y Z. equal(X::'a,Y) & defined(X::'a,Z) --> defined(Y::'a,Z)) &
   412   (\<forall>X Z Y. equal(X::'a,Y) & defined(Z::'a,X) --> defined(Z::'a,Y)) &
   413   (\<forall>X Z Y. equal(X::'a,Y) --> equal(compos(Z::'a,X),compos(Z::'a,Y))) &
   414   (\<forall>X Y Z. equal(X::'a,Y) --> equal(compos(X::'a,Z),compos(Y::'a,Z)))"
   415 
   416 (*54288 inferences so far.  Searching to depth 14.  118.0 secs*)
   417 lemma CAT005_1:
   418   "EQU001_0_ax equal &
   419   CAT001_0_ax equal codomain domain identity_map compos product defined &
   420   CAT001_0_eq compos defined identity_map codomain domain product equal &
   421   (defined(a::'a,d)) &
   422   (identity_map(d)) &
   423   (~equal(domain(a),d)) --> False"
   424   oops
   425 
   426 (*1728 inferences so far.  Searching to depth 10.  5.8 secs*)
   427 lemma CAT007_1:
   428   "EQU001_0_ax equal &
   429   CAT001_0_ax equal codomain domain identity_map compos product defined &
   430   CAT001_0_eq compos defined identity_map codomain domain product equal &
   431   (equal(domain(a),codomain(b))) &
   432   (~defined(a::'a,b)) --> False"
   433   by meson
   434 
   435 (*82895 inferences so far.  Searching to depth 13.  355 secs*)
   436 lemma CAT018_1:
   437   "EQU001_0_ax equal &
   438   CAT001_0_ax equal codomain domain identity_map compos product defined &
   439   CAT001_0_eq compos defined identity_map codomain domain product equal &
   440   (defined(a::'a,b)) &
   441   (defined(b::'a,c)) &
   442   (~defined(a::'a,compos(b::'a,c))) --> False"
   443   oops
   444 
   445 (*1118 inferences so far.  Searching to depth 8.  2.3 secs*)
   446 lemma COL001_2:
   447   "EQU001_0_ax equal &
   448   (\<forall>X Y Z. equal(apply(apply(apply(s::'a,X),Y),Z),apply(apply(X::'a,Z),apply(Y::'a,Z)))) &
   449   (\<forall>Y X. equal(apply(apply(k::'a,X),Y),X)) &
   450   (\<forall>X Y Z. equal(apply(apply(apply(b::'a,X),Y),Z),apply(X::'a,apply(Y::'a,Z)))) &
   451   (\<forall>X. equal(apply(i::'a,X),X)) &
   452   (\<forall>A B C. equal(A::'a,B) --> equal(apply(A::'a,C),apply(B::'a,C))) &
   453   (\<forall>D F' E. equal(D::'a,E) --> equal(apply(F'::'a,D),apply(F'::'a,E))) &
   454   (\<forall>X. equal(apply(apply(apply(s::'a,apply(b::'a,X)),i),apply(apply(s::'a,apply(b::'a,X)),i)),apply(x::'a,apply(apply(apply(s::'a,apply(b::'a,X)),i),apply(apply(s::'a,apply(b::'a,X)),i))))) &
   455   (\<forall>Y. ~equal(Y::'a,apply(combinator::'a,Y))) --> False"
   456   by meson
   457 
   458 (*500 inferences so far.  Searching to depth 8.  0.9 secs*)
   459 lemma COL023_1:
   460   "EQU001_0_ax equal &
   461   (\<forall>X Y Z. equal(apply(apply(apply(b::'a,X),Y),Z),apply(X::'a,apply(Y::'a,Z)))) &
   462   (\<forall>X Y Z. equal(apply(apply(apply(n::'a,X),Y),Z),apply(apply(apply(X::'a,Z),Y),Z))) &
   463   (\<forall>A B C. equal(A::'a,B) --> equal(apply(A::'a,C),apply(B::'a,C))) &
   464   (\<forall>D F' E. equal(D::'a,E) --> equal(apply(F'::'a,D),apply(F'::'a,E))) &
   465   (\<forall>Y. ~equal(Y::'a,apply(combinator::'a,Y))) --> False"
   466   by meson
   467 
   468 (*3018 inferences so far.  Searching to depth 10.  4.3 secs*)
   469 lemma COL032_1:
   470   "EQU001_0_ax equal &
   471   (\<forall>X. equal(apply(m::'a,X),apply(X::'a,X))) &
   472   (\<forall>Y X Z. equal(apply(apply(apply(q::'a,X),Y),Z),apply(Y::'a,apply(X::'a,Z)))) &
   473   (\<forall>A B C. equal(A::'a,B) --> equal(apply(A::'a,C),apply(B::'a,C))) &
   474   (\<forall>D F' E. equal(D::'a,E) --> equal(apply(F'::'a,D),apply(F'::'a,E))) &
   475   (\<forall>G H. equal(G::'a,H) --> equal(f(G),f(H))) &
   476   (\<forall>Y. ~equal(apply(Y::'a,f(Y)),apply(f(Y),apply(Y::'a,f(Y))))) --> False"
   477   by meson
   478 
   479 (*381878 inferences so far.  Searching to depth 13.  670.4 secs*)
   480 lemma COL052_2:
   481   "EQU001_0_ax equal &
   482   (\<forall>X Y W. equal(response(compos(X::'a,Y),W),response(X::'a,response(Y::'a,W)))) &
   483   (\<forall>X Y. agreeable(X) --> equal(response(X::'a,common_bird(Y)),response(Y::'a,common_bird(Y)))) &
   484   (\<forall>Z X. equal(response(X::'a,Z),response(compatible(X),Z)) --> agreeable(X)) &
   485   (\<forall>A B. equal(A::'a,B) --> equal(common_bird(A),common_bird(B))) &
   486   (\<forall>C D. equal(C::'a,D) --> equal(compatible(C),compatible(D))) &
   487   (\<forall>Q R. equal(Q::'a,R) & agreeable(Q) --> agreeable(R)) &
   488   (\<forall>A B C. equal(A::'a,B) --> equal(compos(A::'a,C),compos(B::'a,C))) &
   489   (\<forall>D F' E. equal(D::'a,E) --> equal(compos(F'::'a,D),compos(F'::'a,E))) &
   490   (\<forall>G H I'. equal(G::'a,H) --> equal(response(G::'a,I'),response(H::'a,I'))) &
   491   (\<forall>J L K'. equal(J::'a,K') --> equal(response(L::'a,J),response(L::'a,K'))) &
   492   (agreeable(c)) &
   493   (~agreeable(a)) &
   494   (equal(c::'a,compos(a::'a,b))) --> False"
   495   oops
   496 
   497 (*13201 inferences so far.  Searching to depth 11.  31.9 secs*)
   498 lemma COL075_2:
   499   "EQU001_0_ax equal &
   500   (\<forall>Y X. equal(apply(apply(k::'a,X),Y),X)) &
   501   (\<forall>X Y Z. equal(apply(apply(apply(abstraction::'a,X),Y),Z),apply(apply(X::'a,apply(k::'a,Z)),apply(Y::'a,Z)))) &
   502   (\<forall>D E F'. equal(D::'a,E) --> equal(apply(D::'a,F'),apply(E::'a,F'))) &
   503   (\<forall>G I' H. equal(G::'a,H) --> equal(apply(I'::'a,G),apply(I'::'a,H))) &
   504   (\<forall>A B. equal(A::'a,B) --> equal(b(A),b(B))) &
   505   (\<forall>C D. equal(C::'a,D) --> equal(c(C),c(D))) &
   506   (\<forall>Y. ~equal(apply(apply(Y::'a,b(Y)),c(Y)),apply(b(Y),b(Y)))) --> False"
   507   oops
   508 
   509 (*33 inferences so far.  Searching to depth 7.  0.1 secs*)
   510 lemma COM001_1:
   511   "(\<forall>Goal_state Start_state. follows(Goal_state::'a,Start_state) --> succeeds(Goal_state::'a,Start_state)) &
   512   (\<forall>Goal_state Intermediate_state Start_state. succeeds(Goal_state::'a,Intermediate_state) & succeeds(Intermediate_state::'a,Start_state) --> succeeds(Goal_state::'a,Start_state)) &
   513   (\<forall>Start_state Label Goal_state. has(Start_state::'a,goto(Label)) & labels(Label::'a,Goal_state) --> succeeds(Goal_state::'a,Start_state)) &
   514   (\<forall>Start_state Condition Goal_state. has(Start_state::'a,ifthen(Condition::'a,Goal_state)) --> succeeds(Goal_state::'a,Start_state)) &
   515   (labels(loop::'a,p3)) &
   516   (has(p3::'a,ifthen(equal(register_j::'a,n),p4))) &
   517   (has(p4::'a,goto(out))) &
   518   (follows(p5::'a,p4)) &
   519   (follows(p8::'a,p3)) &
   520   (has(p8::'a,goto(loop))) &
   521   (~succeeds(p3::'a,p3)) --> False"
   522   by meson
   523 
   524 (*533 inferences so far.  Searching to depth 13.  0.3 secs*)
   525 lemma COM002_1:
   526   "(\<forall>Goal_state Start_state. follows(Goal_state::'a,Start_state) --> succeeds(Goal_state::'a,Start_state)) &
   527   (\<forall>Goal_state Intermediate_state Start_state. succeeds(Goal_state::'a,Intermediate_state) & succeeds(Intermediate_state::'a,Start_state) --> succeeds(Goal_state::'a,Start_state)) &
   528   (\<forall>Start_state Label Goal_state. has(Start_state::'a,goto(Label)) & labels(Label::'a,Goal_state) --> succeeds(Goal_state::'a,Start_state)) &
   529   (\<forall>Start_state Condition Goal_state. has(Start_state::'a,ifthen(Condition::'a,Goal_state)) --> succeeds(Goal_state::'a,Start_state)) &
   530   (has(p1::'a,assign(register_j::'a,num0))) &
   531   (follows(p2::'a,p1)) &
   532   (has(p2::'a,assign(register_k::'a,num1))) &
   533   (labels(loop::'a,p3)) &
   534   (follows(p3::'a,p2)) &
   535   (has(p3::'a,ifthen(equal(register_j::'a,n),p4))) &
   536   (has(p4::'a,goto(out))) &
   537   (follows(p5::'a,p4)) &
   538   (follows(p6::'a,p3)) &
   539   (has(p6::'a,assign(register_k::'a,mtimes(num2::'a,register_k)))) &
   540   (follows(p7::'a,p6)) &
   541   (has(p7::'a,assign(register_j::'a,mplus(register_j::'a,num1)))) &
   542   (follows(p8::'a,p7)) &
   543   (has(p8::'a,goto(loop))) &
   544   (~succeeds(p3::'a,p3)) --> False"
   545   by meson
   546 
   547 (*4821 inferences so far.  Searching to depth 14.  1.3 secs*)
   548 lemma COM002_2:
   549   "(\<forall>Goal_state Start_state. ~(fails(Goal_state::'a,Start_state) & follows(Goal_state::'a,Start_state))) &
   550   (\<forall>Goal_state Intermediate_state Start_state. fails(Goal_state::'a,Start_state) --> fails(Goal_state::'a,Intermediate_state) | fails(Intermediate_state::'a,Start_state)) &
   551   (\<forall>Start_state Label Goal_state. ~(fails(Goal_state::'a,Start_state) & has(Start_state::'a,goto(Label)) & labels(Label::'a,Goal_state))) &
   552   (\<forall>Start_state Condition Goal_state. ~(fails(Goal_state::'a,Start_state) & has(Start_state::'a,ifthen(Condition::'a,Goal_state)))) &
   553   (has(p1::'a,assign(register_j::'a,num0))) &
   554   (follows(p2::'a,p1)) &
   555   (has(p2::'a,assign(register_k::'a,num1))) &
   556   (labels(loop::'a,p3)) &
   557   (follows(p3::'a,p2)) &
   558   (has(p3::'a,ifthen(equal(register_j::'a,n),p4))) &
   559   (has(p4::'a,goto(out))) &
   560   (follows(p5::'a,p4)) &
   561   (follows(p6::'a,p3)) &
   562   (has(p6::'a,assign(register_k::'a,mtimes(num2::'a,register_k)))) &
   563   (follows(p7::'a,p6)) &
   564   (has(p7::'a,assign(register_j::'a,mplus(register_j::'a,num1)))) &
   565   (follows(p8::'a,p7)) &
   566   (has(p8::'a,goto(loop))) &
   567   (fails(p3::'a,p3)) --> False"
   568   by meson
   569 
   570 (*98 inferences so far.  Searching to depth 10.  1.1 secs*)
   571 lemma COM003_2:
   572   "(\<forall>X Y Z. program_decides(X) & program(Y) --> decides(X::'a,Y,Z)) &
   573   (\<forall>X. program_decides(X) | program(f2(X))) &
   574   (\<forall>X. decides(X::'a,f2(X),f1(X)) --> program_decides(X)) &
   575   (\<forall>X. program_program_decides(X) --> program(X)) &
   576   (\<forall>X. program_program_decides(X) --> program_decides(X)) &
   577   (\<forall>X. program(X) & program_decides(X) --> program_program_decides(X)) &
   578   (\<forall>X. algorithm_program_decides(X) --> algorithm(X)) &
   579   (\<forall>X. algorithm_program_decides(X) --> program_decides(X)) &
   580   (\<forall>X. algorithm(X) & program_decides(X) --> algorithm_program_decides(X)) &
   581   (\<forall>Y X. program_halts2(X::'a,Y) --> program(X)) &
   582   (\<forall>X Y. program_halts2(X::'a,Y) --> halts2(X::'a,Y)) &
   583   (\<forall>X Y. program(X) & halts2(X::'a,Y) --> program_halts2(X::'a,Y)) &
   584   (\<forall>W X Y Z. halts3_outputs(X::'a,Y,Z,W) --> halts3(X::'a,Y,Z)) &
   585   (\<forall>Y Z X W. halts3_outputs(X::'a,Y,Z,W) --> outputs(X::'a,W)) &
   586   (\<forall>Y Z X W. halts3(X::'a,Y,Z) & outputs(X::'a,W) --> halts3_outputs(X::'a,Y,Z,W)) &
   587   (\<forall>Y X. program_not_halts2(X::'a,Y) --> program(X)) &
   588   (\<forall>X Y. ~(program_not_halts2(X::'a,Y) & halts2(X::'a,Y))) &
   589   (\<forall>X Y. program(X) --> program_not_halts2(X::'a,Y) | halts2(X::'a,Y)) &
   590   (\<forall>W X Y. halts2_outputs(X::'a,Y,W) --> halts2(X::'a,Y)) &
   591   (\<forall>Y X W. halts2_outputs(X::'a,Y,W) --> outputs(X::'a,W)) &
   592   (\<forall>Y X W. halts2(X::'a,Y) & outputs(X::'a,W) --> halts2_outputs(X::'a,Y,W)) &
   593   (\<forall>X W Y Z. program_halts2_halts3_outputs(X::'a,Y,Z,W) --> program_halts2(Y::'a,Z)) &
   594   (\<forall>X Y Z W. program_halts2_halts3_outputs(X::'a,Y,Z,W) --> halts3_outputs(X::'a,Y,Z,W)) &
   595   (\<forall>X Y Z W. program_halts2(Y::'a,Z) & halts3_outputs(X::'a,Y,Z,W) --> program_halts2_halts3_outputs(X::'a,Y,Z,W)) &
   596   (\<forall>X W Y Z. program_not_halts2_halts3_outputs(X::'a,Y,Z,W) --> program_not_halts2(Y::'a,Z)) &
   597   (\<forall>X Y Z W. program_not_halts2_halts3_outputs(X::'a,Y,Z,W) --> halts3_outputs(X::'a,Y,Z,W)) &
   598   (\<forall>X Y Z W. program_not_halts2(Y::'a,Z) & halts3_outputs(X::'a,Y,Z,W) --> program_not_halts2_halts3_outputs(X::'a,Y,Z,W)) &
   599   (\<forall>X W Y. program_halts2_halts2_outputs(X::'a,Y,W) --> program_halts2(Y::'a,Y)) &
   600   (\<forall>X Y W. program_halts2_halts2_outputs(X::'a,Y,W) --> halts2_outputs(X::'a,Y,W)) &
   601   (\<forall>X Y W. program_halts2(Y::'a,Y) & halts2_outputs(X::'a,Y,W) --> program_halts2_halts2_outputs(X::'a,Y,W)) &
   602   (\<forall>X W Y. program_not_halts2_halts2_outputs(X::'a,Y,W) --> program_not_halts2(Y::'a,Y)) &
   603   (\<forall>X Y W. program_not_halts2_halts2_outputs(X::'a,Y,W) --> halts2_outputs(X::'a,Y,W)) &
   604   (\<forall>X Y W. program_not_halts2(Y::'a,Y) & halts2_outputs(X::'a,Y,W) --> program_not_halts2_halts2_outputs(X::'a,Y,W)) &
   605   (\<forall>X. algorithm_program_decides(X) --> program_program_decides(c1)) &
   606   (\<forall>W Y Z. program_program_decides(W) --> program_halts2_halts3_outputs(W::'a,Y,Z,good)) &
   607   (\<forall>W Y Z. program_program_decides(W) --> program_not_halts2_halts3_outputs(W::'a,Y,Z,bad)) &
   608   (\<forall>W. program(W) & program_halts2_halts3_outputs(W::'a,f3(W),f3(W),good) & program_not_halts2_halts3_outputs(W::'a,f3(W),f3(W),bad) --> program(c2)) &
   609   (\<forall>W Y. program(W) & program_halts2_halts3_outputs(W::'a,f3(W),f3(W),good) & program_not_halts2_halts3_outputs(W::'a,f3(W),f3(W),bad) --> program_halts2_halts2_outputs(c2::'a,Y,good)) &
   610   (\<forall>W Y. program(W) & program_halts2_halts3_outputs(W::'a,f3(W),f3(W),good) & program_not_halts2_halts3_outputs(W::'a,f3(W),f3(W),bad) --> program_not_halts2_halts2_outputs(c2::'a,Y,bad)) &
   611   (\<forall>V. program(V) & program_halts2_halts2_outputs(V::'a,f4(V),good) & program_not_halts2_halts2_outputs(V::'a,f4(V),bad) --> program(c3)) &
   612   (\<forall>V Y. program(V) & program_halts2_halts2_outputs(V::'a,f4(V),good) & program_not_halts2_halts2_outputs(V::'a,f4(V),bad) & program_halts2(Y::'a,Y) --> halts2(c3::'a,Y)) &
   613   (\<forall>V Y. program(V) & program_halts2_halts2_outputs(V::'a,f4(V),good) & program_not_halts2_halts2_outputs(V::'a,f4(V),bad) --> program_not_halts2_halts2_outputs(c3::'a,Y,bad)) &
   614   (algorithm_program_decides(c4)) --> False"
   615   by meson
   616 
   617 (*2100398 inferences so far.  Searching to depth 12.
   618   1256s (21 mins) on griffon*)
   619 lemma COM004_1:
   620   "EQU001_0_ax equal &
   621   (\<forall>C D P Q X Y. failure_node(X::'a,or(C::'a,P)) & failure_node(Y::'a,or(D::'a,Q)) & contradictory(P::'a,Q) & siblings(X::'a,Y) --> failure_node(parent_of(X::'a,Y),or(C::'a,D))) &
   622   (\<forall>X. contradictory(negate(X),X)) &
   623   (\<forall>X. contradictory(X::'a,negate(X))) &
   624   (\<forall>X. siblings(left_child_of(X),right_child_of(X))) &
   625   (\<forall>D E. equal(D::'a,E) --> equal(left_child_of(D),left_child_of(E))) &
   626   (\<forall>F' G. equal(F'::'a,G) --> equal(negate(F'),negate(G))) &
   627   (\<forall>H I' J. equal(H::'a,I') --> equal(or(H::'a,J),or(I'::'a,J))) &
   628   (\<forall>K' M L. equal(K'::'a,L) --> equal(or(M::'a,K'),or(M::'a,L))) &
   629   (\<forall>N O' P. equal(N::'a,O') --> equal(parent_of(N::'a,P),parent_of(O'::'a,P))) &
   630   (\<forall>Q S' R. equal(Q::'a,R) --> equal(parent_of(S'::'a,Q),parent_of(S'::'a,R))) &
   631   (\<forall>T' U. equal(T'::'a,U) --> equal(right_child_of(T'),right_child_of(U))) &
   632   (\<forall>V W X. equal(V::'a,W) & contradictory(V::'a,X) --> contradictory(W::'a,X)) &
   633   (\<forall>Y A1 Z. equal(Y::'a,Z) & contradictory(A1::'a,Y) --> contradictory(A1::'a,Z)) &
   634   (\<forall>B1 C1 D1. equal(B1::'a,C1) & failure_node(B1::'a,D1) --> failure_node(C1::'a,D1)) &
   635   (\<forall>E1 G1 F1. equal(E1::'a,F1) & failure_node(G1::'a,E1) --> failure_node(G1::'a,F1)) &
   636   (\<forall>H1 I1 J1. equal(H1::'a,I1) & siblings(H1::'a,J1) --> siblings(I1::'a,J1)) &
   637   (\<forall>K1 M1 L1. equal(K1::'a,L1) & siblings(M1::'a,K1) --> siblings(M1::'a,L1)) &
   638   (failure_node(n_left::'a,or(EMPTY::'a,atom))) &
   639   (failure_node(n_right::'a,or(EMPTY::'a,negate(atom)))) &
   640   (equal(n_left::'a,left_child_of(n))) &
   641   (equal(n_right::'a,right_child_of(n))) &
   642   (\<forall>Z. ~failure_node(Z::'a,or(EMPTY::'a,EMPTY))) --> False"
   643   oops
   644 
   645 abbreviation "GEO001_0_ax continuous lower_dimension_point_3 lower_dimension_point_2
   646   lower_dimension_point_1 extension euclid2 euclid1 outer_pasch equidistant equal between \<equiv>
   647   (\<forall>X Y. between(X::'a,Y,X) --> equal(X::'a,Y)) &
   648   (\<forall>V X Y Z. between(X::'a,Y,V) & between(Y::'a,Z,V) --> between(X::'a,Y,Z)) &
   649   (\<forall>Y X V Z. between(X::'a,Y,Z) & between(X::'a,Y,V) --> equal(X::'a,Y) | between(X::'a,Z,V) | between(X::'a,V,Z)) &
   650   (\<forall>Y X. equidistant(X::'a,Y,Y,X)) &
   651   (\<forall>Z X Y. equidistant(X::'a,Y,Z,Z) --> equal(X::'a,Y)) &
   652   (\<forall>X Y Z V V2 W. equidistant(X::'a,Y,Z,V) & equidistant(X::'a,Y,V2,W) --> equidistant(Z::'a,V,V2,W)) &
   653   (\<forall>W X Z V Y. between(X::'a,W,V) & between(Y::'a,V,Z) --> between(X::'a,outer_pasch(W::'a,X,Y,Z,V),Y)) &
   654   (\<forall>W X Y Z V. between(X::'a,W,V) & between(Y::'a,V,Z) --> between(Z::'a,W,outer_pasch(W::'a,X,Y,Z,V))) &
   655   (\<forall>W X Y Z V. between(X::'a,V,W) & between(Y::'a,V,Z) --> equal(X::'a,V) | between(X::'a,Z,euclid1(W::'a,X,Y,Z,V))) &
   656   (\<forall>W X Y Z V. between(X::'a,V,W) & between(Y::'a,V,Z) --> equal(X::'a,V) | between(X::'a,Y,euclid2(W::'a,X,Y,Z,V))) &
   657   (\<forall>W X Y Z V. between(X::'a,V,W) & between(Y::'a,V,Z) --> equal(X::'a,V) | between(euclid1(W::'a,X,Y,Z,V),W,euclid2(W::'a,X,Y,Z,V))) &
   658   (\<forall>X1 Y1 X Y Z V Z1 V1. equidistant(X::'a,Y,X1,Y1) & equidistant(Y::'a,Z,Y1,Z1) & equidistant(X::'a,V,X1,V1) & equidistant(Y::'a,V,Y1,V1) & between(X::'a,Y,Z) & between(X1::'a,Y1,Z1) --> equal(X::'a,Y) | equidistant(Z::'a,V,Z1,V1)) &
   659   (\<forall>X Y W V. between(X::'a,Y,extension(X::'a,Y,W,V))) &
   660   (\<forall>X Y W V. equidistant(Y::'a,extension(X::'a,Y,W,V),W,V)) &
   661   (~between(lower_dimension_point_1::'a,lower_dimension_point_2,lower_dimension_point_3)) &
   662   (~between(lower_dimension_point_2::'a,lower_dimension_point_3,lower_dimension_point_1)) &
   663   (~between(lower_dimension_point_3::'a,lower_dimension_point_1,lower_dimension_point_2)) &
   664   (\<forall>Z X Y W V. equidistant(X::'a,W,X,V) & equidistant(Y::'a,W,Y,V) & equidistant(Z::'a,W,Z,V) --> between(X::'a,Y,Z) | between(Y::'a,Z,X) | between(Z::'a,X,Y) | equal(W::'a,V)) &
   665   (\<forall>X Y Z X1 Z1 V. equidistant(V::'a,X,V,X1) & equidistant(V::'a,Z,V,Z1) & between(V::'a,X,Z) & between(X::'a,Y,Z) --> equidistant(V::'a,Y,Z,continuous(X::'a,Y,Z,X1,Z1,V))) &
   666   (\<forall>X Y Z X1 V Z1. equidistant(V::'a,X,V,X1) & equidistant(V::'a,Z,V,Z1) & between(V::'a,X,Z) & between(X::'a,Y,Z) --> between(X1::'a,continuous(X::'a,Y,Z,X1,Z1,V),Z1))"
   667 
   668 abbreviation "GEO001_0_eq continuous extension euclid2 euclid1 outer_pasch equidistant
   669   between equal \<equiv>
   670   (\<forall>X Y W Z. equal(X::'a,Y) & between(X::'a,W,Z) --> between(Y::'a,W,Z)) &
   671   (\<forall>X W Y Z. equal(X::'a,Y) & between(W::'a,X,Z) --> between(W::'a,Y,Z)) &
   672   (\<forall>X W Z Y. equal(X::'a,Y) & between(W::'a,Z,X) --> between(W::'a,Z,Y)) &
   673   (\<forall>X Y V W Z. equal(X::'a,Y) & equidistant(X::'a,V,W,Z) --> equidistant(Y::'a,V,W,Z)) &
   674   (\<forall>X V Y W Z. equal(X::'a,Y) & equidistant(V::'a,X,W,Z) --> equidistant(V::'a,Y,W,Z)) &
   675   (\<forall>X V W Y Z. equal(X::'a,Y) & equidistant(V::'a,W,X,Z) --> equidistant(V::'a,W,Y,Z)) &
   676   (\<forall>X V W Z Y. equal(X::'a,Y) & equidistant(V::'a,W,Z,X) --> equidistant(V::'a,W,Z,Y)) &
   677   (\<forall>X Y V1 V2 V3 V4. equal(X::'a,Y) --> equal(outer_pasch(X::'a,V1,V2,V3,V4),outer_pasch(Y::'a,V1,V2,V3,V4))) &
   678   (\<forall>X V1 Y V2 V3 V4. equal(X::'a,Y) --> equal(outer_pasch(V1::'a,X,V2,V3,V4),outer_pasch(V1::'a,Y,V2,V3,V4))) &
   679   (\<forall>X V1 V2 Y V3 V4. equal(X::'a,Y) --> equal(outer_pasch(V1::'a,V2,X,V3,V4),outer_pasch(V1::'a,V2,Y,V3,V4))) &
   680   (\<forall>X V1 V2 V3 Y V4. equal(X::'a,Y) --> equal(outer_pasch(V1::'a,V2,V3,X,V4),outer_pasch(V1::'a,V2,V3,Y,V4))) &
   681   (\<forall>X V1 V2 V3 V4 Y. equal(X::'a,Y) --> equal(outer_pasch(V1::'a,V2,V3,V4,X),outer_pasch(V1::'a,V2,V3,V4,Y))) &
   682   (\<forall>A B C D E F'. equal(A::'a,B) --> equal(euclid1(A::'a,C,D,E,F'),euclid1(B::'a,C,D,E,F'))) &
   683   (\<forall>G I' H J K' L. equal(G::'a,H) --> equal(euclid1(I'::'a,G,J,K',L),euclid1(I'::'a,H,J,K',L))) &
   684   (\<forall>M O' P N Q R. equal(M::'a,N) --> equal(euclid1(O'::'a,P,M,Q,R),euclid1(O'::'a,P,N,Q,R))) &
   685   (\<forall>S' U V W T' X. equal(S'::'a,T') --> equal(euclid1(U::'a,V,W,S',X),euclid1(U::'a,V,W,T',X))) &
   686   (\<forall>Y A1 B1 C1 D1 Z. equal(Y::'a,Z) --> equal(euclid1(A1::'a,B1,C1,D1,Y),euclid1(A1::'a,B1,C1,D1,Z))) &
   687   (\<forall>E1 F1 G1 H1 I1 J1. equal(E1::'a,F1) --> equal(euclid2(E1::'a,G1,H1,I1,J1),euclid2(F1::'a,G1,H1,I1,J1))) &
   688   (\<forall>K1 M1 L1 N1 O1 P1. equal(K1::'a,L1) --> equal(euclid2(M1::'a,K1,N1,O1,P1),euclid2(M1::'a,L1,N1,O1,P1))) &
   689   (\<forall>Q1 S1 T1 R1 U1 V1. equal(Q1::'a,R1) --> equal(euclid2(S1::'a,T1,Q1,U1,V1),euclid2(S1::'a,T1,R1,U1,V1))) &
   690   (\<forall>W1 Y1 Z1 A2 X1 B2. equal(W1::'a,X1) --> equal(euclid2(Y1::'a,Z1,A2,W1,B2),euclid2(Y1::'a,Z1,A2,X1,B2))) &
   691   (\<forall>C2 E2 F2 G2 H2 D2. equal(C2::'a,D2) --> equal(euclid2(E2::'a,F2,G2,H2,C2),euclid2(E2::'a,F2,G2,H2,D2))) &
   692   (\<forall>X Y V1 V2 V3. equal(X::'a,Y) --> equal(extension(X::'a,V1,V2,V3),extension(Y::'a,V1,V2,V3))) &
   693   (\<forall>X V1 Y V2 V3. equal(X::'a,Y) --> equal(extension(V1::'a,X,V2,V3),extension(V1::'a,Y,V2,V3))) &
   694   (\<forall>X V1 V2 Y V3. equal(X::'a,Y) --> equal(extension(V1::'a,V2,X,V3),extension(V1::'a,V2,Y,V3))) &
   695   (\<forall>X V1 V2 V3 Y. equal(X::'a,Y) --> equal(extension(V1::'a,V2,V3,X),extension(V1::'a,V2,V3,Y))) &
   696   (\<forall>X Y V1 V2 V3 V4 V5. equal(X::'a,Y) --> equal(continuous(X::'a,V1,V2,V3,V4,V5),continuous(Y::'a,V1,V2,V3,V4,V5))) &
   697   (\<forall>X V1 Y V2 V3 V4 V5. equal(X::'a,Y) --> equal(continuous(V1::'a,X,V2,V3,V4,V5),continuous(V1::'a,Y,V2,V3,V4,V5))) &
   698   (\<forall>X V1 V2 Y V3 V4 V5. equal(X::'a,Y) --> equal(continuous(V1::'a,V2,X,V3,V4,V5),continuous(V1::'a,V2,Y,V3,V4,V5))) &
   699   (\<forall>X V1 V2 V3 Y V4 V5. equal(X::'a,Y) --> equal(continuous(V1::'a,V2,V3,X,V4,V5),continuous(V1::'a,V2,V3,Y,V4,V5))) &
   700   (\<forall>X V1 V2 V3 V4 Y V5. equal(X::'a,Y) --> equal(continuous(V1::'a,V2,V3,V4,X,V5),continuous(V1::'a,V2,V3,V4,Y,V5))) &
   701   (\<forall>X V1 V2 V3 V4 V5 Y. equal(X::'a,Y) --> equal(continuous(V1::'a,V2,V3,V4,V5,X),continuous(V1::'a,V2,V3,V4,V5,Y)))"
   702 
   703 
   704 (*179 inferences so far.  Searching to depth 7.  3.9 secs*)
   705 lemma GEO003_1:
   706   "EQU001_0_ax equal &
   707   GEO001_0_ax continuous lower_dimension_point_3 lower_dimension_point_2
   708     lower_dimension_point_1 extension euclid2 euclid1 outer_pasch equidistant equal between &
   709   GEO001_0_eq continuous extension euclid2 euclid1 outer_pasch equidistant between equal &
   710   (~between(a::'a,b,b)) --> False"
   711   by meson
   712 
   713 abbreviation "GEO002_ax_eq continuous euclid2 euclid1 lower_dimension_point_3
   714   lower_dimension_point_2 lower_dimension_point_1 inner_pasch extension
   715   between equal equidistant \<equiv>
   716   (\<forall>Y X. equidistant(X::'a,Y,Y,X)) &
   717   (\<forall>X Y Z V V2 W. equidistant(X::'a,Y,Z,V) & equidistant(X::'a,Y,V2,W) --> equidistant(Z::'a,V,V2,W)) &
   718   (\<forall>Z X Y. equidistant(X::'a,Y,Z,Z) --> equal(X::'a,Y)) &
   719   (\<forall>X Y W V. between(X::'a,Y,extension(X::'a,Y,W,V))) &
   720   (\<forall>X Y W V. equidistant(Y::'a,extension(X::'a,Y,W,V),W,V)) &
   721   (\<forall>X1 Y1 X Y Z V Z1 V1. equidistant(X::'a,Y,X1,Y1) & equidistant(Y::'a,Z,Y1,Z1) & equidistant(X::'a,V,X1,V1) & equidistant(Y::'a,V,Y1,V1) & between(X::'a,Y,Z) & between(X1::'a,Y1,Z1) --> equal(X::'a,Y) | equidistant(Z::'a,V,Z1,V1)) &
   722   (\<forall>X Y. between(X::'a,Y,X) --> equal(X::'a,Y)) &
   723   (\<forall>U V W X Y. between(U::'a,V,W) & between(Y::'a,X,W) --> between(V::'a,inner_pasch(U::'a,V,W,X,Y),Y)) &
   724   (\<forall>V W X Y U. between(U::'a,V,W) & between(Y::'a,X,W) --> between(X::'a,inner_pasch(U::'a,V,W,X,Y),U)) &
   725   (~between(lower_dimension_point_1::'a,lower_dimension_point_2,lower_dimension_point_3)) &
   726   (~between(lower_dimension_point_2::'a,lower_dimension_point_3,lower_dimension_point_1)) &
   727   (~between(lower_dimension_point_3::'a,lower_dimension_point_1,lower_dimension_point_2)) &
   728   (\<forall>Z X Y W V. equidistant(X::'a,W,X,V) & equidistant(Y::'a,W,Y,V) & equidistant(Z::'a,W,Z,V) --> between(X::'a,Y,Z) | between(Y::'a,Z,X) | between(Z::'a,X,Y) | equal(W::'a,V)) &
   729   (\<forall>U V W X Y. between(U::'a,W,Y) & between(V::'a,W,X) --> equal(U::'a,W) | between(U::'a,V,euclid1(U::'a,V,W,X,Y))) &
   730   (\<forall>U V W X Y. between(U::'a,W,Y) & between(V::'a,W,X) --> equal(U::'a,W) | between(U::'a,X,euclid2(U::'a,V,W,X,Y))) &
   731   (\<forall>U V W X Y. between(U::'a,W,Y) & between(V::'a,W,X) --> equal(U::'a,W) | between(euclid1(U::'a,V,W,X,Y),Y,euclid2(U::'a,V,W,X,Y))) &
   732   (\<forall>U V V1 W X X1. equidistant(U::'a,V,U,V1) & equidistant(U::'a,X,U,X1) & between(U::'a,V,X) & between(V::'a,W,X) --> between(V1::'a,continuous(U::'a,V,V1,W,X,X1),X1)) &
   733   (\<forall>U V V1 W X X1. equidistant(U::'a,V,U,V1) & equidistant(U::'a,X,U,X1) & between(U::'a,V,X) & between(V::'a,W,X) --> equidistant(U::'a,W,U,continuous(U::'a,V,V1,W,X,X1))) &
   734   (\<forall>X Y W Z. equal(X::'a,Y) & between(X::'a,W,Z) --> between(Y::'a,W,Z)) &
   735   (\<forall>X W Y Z. equal(X::'a,Y) & between(W::'a,X,Z) --> between(W::'a,Y,Z)) &
   736   (\<forall>X W Z Y. equal(X::'a,Y) & between(W::'a,Z,X) --> between(W::'a,Z,Y)) &
   737   (\<forall>X Y V W Z. equal(X::'a,Y) & equidistant(X::'a,V,W,Z) --> equidistant(Y::'a,V,W,Z)) &
   738   (\<forall>X V Y W Z. equal(X::'a,Y) & equidistant(V::'a,X,W,Z) --> equidistant(V::'a,Y,W,Z)) &
   739   (\<forall>X V W Y Z. equal(X::'a,Y) & equidistant(V::'a,W,X,Z) --> equidistant(V::'a,W,Y,Z)) &
   740   (\<forall>X V W Z Y. equal(X::'a,Y) & equidistant(V::'a,W,Z,X) --> equidistant(V::'a,W,Z,Y)) &
   741   (\<forall>X Y V1 V2 V3 V4. equal(X::'a,Y) --> equal(inner_pasch(X::'a,V1,V2,V3,V4),inner_pasch(Y::'a,V1,V2,V3,V4))) &
   742   (\<forall>X V1 Y V2 V3 V4. equal(X::'a,Y) --> equal(inner_pasch(V1::'a,X,V2,V3,V4),inner_pasch(V1::'a,Y,V2,V3,V4))) &
   743   (\<forall>X V1 V2 Y V3 V4. equal(X::'a,Y) --> equal(inner_pasch(V1::'a,V2,X,V3,V4),inner_pasch(V1::'a,V2,Y,V3,V4))) &
   744   (\<forall>X V1 V2 V3 Y V4. equal(X::'a,Y) --> equal(inner_pasch(V1::'a,V2,V3,X,V4),inner_pasch(V1::'a,V2,V3,Y,V4))) &
   745   (\<forall>X V1 V2 V3 V4 Y. equal(X::'a,Y) --> equal(inner_pasch(V1::'a,V2,V3,V4,X),inner_pasch(V1::'a,V2,V3,V4,Y))) &
   746   (\<forall>A B C D E F'. equal(A::'a,B) --> equal(euclid1(A::'a,C,D,E,F'),euclid1(B::'a,C,D,E,F'))) &
   747   (\<forall>G I' H J K' L. equal(G::'a,H) --> equal(euclid1(I'::'a,G,J,K',L),euclid1(I'::'a,H,J,K',L))) &
   748   (\<forall>M O' P N Q R. equal(M::'a,N) --> equal(euclid1(O'::'a,P,M,Q,R),euclid1(O'::'a,P,N,Q,R))) &
   749   (\<forall>S' U V W T' X. equal(S'::'a,T') --> equal(euclid1(U::'a,V,W,S',X),euclid1(U::'a,V,W,T',X))) &
   750   (\<forall>Y A1 B1 C1 D1 Z. equal(Y::'a,Z) --> equal(euclid1(A1::'a,B1,C1,D1,Y),euclid1(A1::'a,B1,C1,D1,Z))) &
   751   (\<forall>E1 F1 G1 H1 I1 J1. equal(E1::'a,F1) --> equal(euclid2(E1::'a,G1,H1,I1,J1),euclid2(F1::'a,G1,H1,I1,J1))) &
   752   (\<forall>K1 M1 L1 N1 O1 P1. equal(K1::'a,L1) --> equal(euclid2(M1::'a,K1,N1,O1,P1),euclid2(M1::'a,L1,N1,O1,P1))) &
   753   (\<forall>Q1 S1 T1 R1 U1 V1. equal(Q1::'a,R1) --> equal(euclid2(S1::'a,T1,Q1,U1,V1),euclid2(S1::'a,T1,R1,U1,V1))) &
   754   (\<forall>W1 Y1 Z1 A2 X1 B2. equal(W1::'a,X1) --> equal(euclid2(Y1::'a,Z1,A2,W1,B2),euclid2(Y1::'a,Z1,A2,X1,B2))) &
   755   (\<forall>C2 E2 F2 G2 H2 D2. equal(C2::'a,D2) --> equal(euclid2(E2::'a,F2,G2,H2,C2),euclid2(E2::'a,F2,G2,H2,D2))) &
   756   (\<forall>X Y V1 V2 V3. equal(X::'a,Y) --> equal(extension(X::'a,V1,V2,V3),extension(Y::'a,V1,V2,V3))) &
   757   (\<forall>X V1 Y V2 V3. equal(X::'a,Y) --> equal(extension(V1::'a,X,V2,V3),extension(V1::'a,Y,V2,V3))) &
   758   (\<forall>X V1 V2 Y V3. equal(X::'a,Y) --> equal(extension(V1::'a,V2,X,V3),extension(V1::'a,V2,Y,V3))) &
   759   (\<forall>X V1 V2 V3 Y. equal(X::'a,Y) --> equal(extension(V1::'a,V2,V3,X),extension(V1::'a,V2,V3,Y))) &
   760   (\<forall>X Y V1 V2 V3 V4 V5. equal(X::'a,Y) --> equal(continuous(X::'a,V1,V2,V3,V4,V5),continuous(Y::'a,V1,V2,V3,V4,V5))) &
   761   (\<forall>X V1 Y V2 V3 V4 V5. equal(X::'a,Y) --> equal(continuous(V1::'a,X,V2,V3,V4,V5),continuous(V1::'a,Y,V2,V3,V4,V5))) &
   762   (\<forall>X V1 V2 Y V3 V4 V5. equal(X::'a,Y) --> equal(continuous(V1::'a,V2,X,V3,V4,V5),continuous(V1::'a,V2,Y,V3,V4,V5))) &
   763   (\<forall>X V1 V2 V3 Y V4 V5. equal(X::'a,Y) --> equal(continuous(V1::'a,V2,V3,X,V4,V5),continuous(V1::'a,V2,V3,Y,V4,V5))) &
   764   (\<forall>X V1 V2 V3 V4 Y V5. equal(X::'a,Y) --> equal(continuous(V1::'a,V2,V3,V4,X,V5),continuous(V1::'a,V2,V3,V4,Y,V5))) &
   765   (\<forall>X V1 V2 V3 V4 V5 Y. equal(X::'a,Y) --> equal(continuous(V1::'a,V2,V3,V4,V5,X),continuous(V1::'a,V2,V3,V4,V5,Y)))"
   766 
   767 (*4272 inferences so far.  Searching to depth 10.  29.4 secs*)
   768 lemma GEO017_2:
   769   "EQU001_0_ax equal &
   770   GEO002_ax_eq continuous euclid2 euclid1 lower_dimension_point_3
   771     lower_dimension_point_2 lower_dimension_point_1 inner_pasch extension
   772     between equal equidistant &
   773   (equidistant(u::'a,v,w,x)) &
   774   (~equidistant(u::'a,v,x,w)) --> False"
   775   oops
   776 
   777 (*382903 inferences so far.  Searching to depth 9. 1022s (17 mins) on griffon*)
   778 lemma GEO027_3:
   779   "EQU001_0_ax equal &
   780   GEO002_ax_eq continuous euclid2 euclid1 lower_dimension_point_3
   781     lower_dimension_point_2 lower_dimension_point_1 inner_pasch extension
   782     between equal equidistant &
   783   (\<forall>U V. equal(reflection(U::'a,V),extension(U::'a,V,U,V))) &
   784   (\<forall>X Y Z. equal(X::'a,Y) --> equal(reflection(X::'a,Z),reflection(Y::'a,Z))) &
   785   (\<forall>A1 C1 B1. equal(A1::'a,B1) --> equal(reflection(C1::'a,A1),reflection(C1::'a,B1))) &
   786   (\<forall>U V. equidistant(U::'a,V,U,V)) &
   787   (\<forall>W X U V. equidistant(U::'a,V,W,X) --> equidistant(W::'a,X,U,V)) &
   788   (\<forall>V U W X. equidistant(U::'a,V,W,X) --> equidistant(V::'a,U,W,X)) &
   789   (\<forall>U V X W. equidistant(U::'a,V,W,X) --> equidistant(U::'a,V,X,W)) &
   790   (\<forall>V U X W. equidistant(U::'a,V,W,X) --> equidistant(V::'a,U,X,W)) &
   791   (\<forall>W X V U. equidistant(U::'a,V,W,X) --> equidistant(W::'a,X,V,U)) &
   792   (\<forall>X W U V. equidistant(U::'a,V,W,X) --> equidistant(X::'a,W,U,V)) &
   793   (\<forall>X W V U. equidistant(U::'a,V,W,X) --> equidistant(X::'a,W,V,U)) &
   794   (\<forall>W X U V Y Z. equidistant(U::'a,V,W,X) & equidistant(W::'a,X,Y,Z) --> equidistant(U::'a,V,Y,Z)) &
   795   (\<forall>U V W. equal(V::'a,extension(U::'a,V,W,W))) &
   796   (\<forall>W X U V Y. equal(Y::'a,extension(U::'a,V,W,X)) --> between(U::'a,V,Y)) &
   797   (\<forall>U V. between(U::'a,V,reflection(U::'a,V))) &
   798   (\<forall>U V. equidistant(V::'a,reflection(U::'a,V),U,V)) &
   799   (\<forall>U V. equal(U::'a,V) --> equal(V::'a,reflection(U::'a,V))) &
   800   (\<forall>U. equal(U::'a,reflection(U::'a,U))) &
   801   (\<forall>U V. equal(V::'a,reflection(U::'a,V)) --> equal(U::'a,V)) &
   802   (\<forall>U V. equidistant(U::'a,U,V,V)) &
   803   (\<forall>V V1 U W U1 W1. equidistant(U::'a,V,U1,V1) & equidistant(V::'a,W,V1,W1) & between(U::'a,V,W) & between(U1::'a,V1,W1) --> equidistant(U::'a,W,U1,W1)) &
   804   (\<forall>U V W X. between(U::'a,V,W) & between(U::'a,V,X) & equidistant(V::'a,W,V,X) --> equal(U::'a,V) | equal(W::'a,X)) &
   805   (between(u::'a,v,w)) &
   806   (~equal(u::'a,v)) &
   807   (~equal(w::'a,extension(u::'a,v,v,w))) --> False"
   808   oops
   809 
   810 (*313884 inferences so far.  Searching to depth 10.  887 secs: 15 mins.*)
   811 lemma GEO058_2:
   812   "EQU001_0_ax equal &
   813   GEO002_ax_eq continuous euclid2 euclid1 lower_dimension_point_3
   814     lower_dimension_point_2 lower_dimension_point_1 inner_pasch extension
   815     between equal equidistant &
   816   (\<forall>U V. equal(reflection(U::'a,V),extension(U::'a,V,U,V))) &
   817   (\<forall>X Y Z. equal(X::'a,Y) --> equal(reflection(X::'a,Z),reflection(Y::'a,Z))) &
   818   (\<forall>A1 C1 B1. equal(A1::'a,B1) --> equal(reflection(C1::'a,A1),reflection(C1::'a,B1))) &
   819   (equal(v::'a,reflection(u::'a,v))) &
   820   (~equal(u::'a,v)) --> False"
   821   oops
   822 
   823 (*0 inferences so far.  Searching to depth 0.  0.2 secs*)
   824 lemma GEO079_1:
   825   "(\<forall>U V W X Y Z. right_angle(U::'a,V,W) & right_angle(X::'a,Y,Z) --> eq(U::'a,V,W,X,Y,Z)) &
   826   (\<forall>U V W X Y Z. CONGRUENT(U::'a,V,W,X,Y,Z) --> eq(U::'a,V,W,X,Y,Z)) &
   827   (\<forall>V W U X. trapezoid(U::'a,V,W,X) --> parallel(V::'a,W,U,X)) &
   828   (\<forall>U V X Y. parallel(U::'a,V,X,Y) --> eq(X::'a,V,U,V,X,Y)) &
   829   (trapezoid(a::'a,b,c,d)) &
   830   (~eq(a::'a,c,b,c,a,d)) --> False"
   831    by meson
   832 
   833 abbreviation "GRP003_0_ax equal multiply INVERSE identity product \<equiv>
   834   (\<forall>X. product(identity::'a,X,X)) &
   835   (\<forall>X. product(X::'a,identity,X)) &
   836   (\<forall>X. product(INVERSE(X),X,identity)) &
   837   (\<forall>X. product(X::'a,INVERSE(X),identity)) &
   838   (\<forall>X Y. product(X::'a,Y,multiply(X::'a,Y))) &
   839   (\<forall>X Y Z W. product(X::'a,Y,Z) & product(X::'a,Y,W) --> equal(Z::'a,W)) &
   840   (\<forall>Y U Z X V W. product(X::'a,Y,U) & product(Y::'a,Z,V) & product(U::'a,Z,W) --> product(X::'a,V,W)) &
   841   (\<forall>Y X V U Z W. product(X::'a,Y,U) & product(Y::'a,Z,V) & product(X::'a,V,W) --> product(U::'a,Z,W))"
   842 
   843 abbreviation "GRP003_0_eq product multiply INVERSE equal \<equiv>
   844   (\<forall>X Y. equal(X::'a,Y) --> equal(INVERSE(X),INVERSE(Y))) &
   845   (\<forall>X Y W. equal(X::'a,Y) --> equal(multiply(X::'a,W),multiply(Y::'a,W))) &
   846   (\<forall>X W Y. equal(X::'a,Y) --> equal(multiply(W::'a,X),multiply(W::'a,Y))) &
   847   (\<forall>X Y W Z. equal(X::'a,Y) & product(X::'a,W,Z) --> product(Y::'a,W,Z)) &
   848   (\<forall>X W Y Z. equal(X::'a,Y) & product(W::'a,X,Z) --> product(W::'a,Y,Z)) &
   849   (\<forall>X W Z Y. equal(X::'a,Y) & product(W::'a,Z,X) --> product(W::'a,Z,Y))"
   850 
   851 (*2032008 inferences so far. Searching to depth 16. 658s (11 mins) on griffon*)
   852 lemma GRP001_1:
   853   "EQU001_0_ax equal &
   854   GRP003_0_ax equal multiply INVERSE identity product &
   855   GRP003_0_eq product multiply INVERSE equal &
   856   (\<forall>X. product(X::'a,X,identity)) &
   857   (product(a::'a,b,c)) &
   858   (~product(b::'a,a,c)) --> False"
   859   oops
   860 
   861 (*2386 inferences so far.  Searching to depth 11.  8.7 secs*)
   862 lemma GRP008_1:
   863   "EQU001_0_ax equal &
   864   GRP003_0_ax equal multiply INVERSE identity product &
   865   GRP003_0_eq product multiply INVERSE equal &
   866   (\<forall>A B. equal(A::'a,B) --> equal(h(A),h(B))) &
   867   (\<forall>C D. equal(C::'a,D) --> equal(j(C),j(D))) &
   868   (\<forall>A B. equal(A::'a,B) & q(A) --> q(B)) &
   869   (\<forall>B A C. q(A) & product(A::'a,B,C) --> product(B::'a,A,C)) &
   870   (\<forall>A. product(j(A),A,h(A)) | product(A::'a,j(A),h(A)) | q(A)) &
   871   (\<forall>A. product(j(A),A,h(A)) & product(A::'a,j(A),h(A)) --> q(A)) &
   872   (~q(identity)) --> False"
   873   by meson
   874 
   875 (*8625 inferences so far.  Searching to depth 11.  20 secs*)
   876 lemma GRP013_1:
   877   "EQU001_0_ax equal &
   878   GRP003_0_ax equal multiply INVERSE identity product &
   879   GRP003_0_eq product multiply INVERSE equal &
   880   (\<forall>A. product(A::'a,A,identity)) &
   881   (product(a::'a,b,c)) &
   882   (product(INVERSE(a),INVERSE(b),d)) &
   883   (\<forall>A C B. product(INVERSE(A),INVERSE(B),C) --> product(A::'a,C,B)) &
   884   (~product(c::'a,d,identity)) --> False"
   885   oops
   886 
   887 (*2448 inferences so far.  Searching to depth 10.  7.2 secs*)
   888 lemma GRP037_3:
   889   "EQU001_0_ax equal &
   890   GRP003_0_ax equal multiply INVERSE identity product &
   891   GRP003_0_eq product multiply INVERSE equal &
   892   (\<forall>A B C. subgroup_member(A) & subgroup_member(B) & product(A::'a,INVERSE(B),C) --> subgroup_member(C)) &
   893   (\<forall>A B. equal(A::'a,B) & subgroup_member(A) --> subgroup_member(B)) &
   894   (\<forall>A. subgroup_member(A) --> product(Gidentity::'a,A,A)) &
   895   (\<forall>A. subgroup_member(A) --> product(A::'a,Gidentity,A)) &
   896   (\<forall>A. subgroup_member(A) --> product(A::'a,Ginverse(A),Gidentity)) &
   897   (\<forall>A. subgroup_member(A) --> product(Ginverse(A),A,Gidentity)) &
   898   (\<forall>A. subgroup_member(A) --> subgroup_member(Ginverse(A))) &
   899   (\<forall>A B. equal(A::'a,B) --> equal(Ginverse(A),Ginverse(B))) &
   900   (\<forall>A C D B. product(A::'a,B,C) & product(A::'a,D,C) --> equal(D::'a,B)) &
   901   (\<forall>B C D A. product(A::'a,B,C) & product(D::'a,B,C) --> equal(D::'a,A)) &
   902   (subgroup_member(a)) &
   903   (subgroup_member(Gidentity)) &
   904   (~equal(INVERSE(a),Ginverse(a))) --> False"
   905   by meson
   906 
   907 (*163 inferences so far.  Searching to depth 11.  0.3 secs*)
   908 lemma GRP031_2:
   909   "(\<forall>X Y. product(X::'a,Y,multiply(X::'a,Y))) &
   910   (\<forall>X Y Z W. product(X::'a,Y,Z) & product(X::'a,Y,W) --> equal(Z::'a,W)) &
   911   (\<forall>Y U Z X V W. product(X::'a,Y,U) & product(Y::'a,Z,V) & product(U::'a,Z,W) --> product(X::'a,V,W)) &
   912   (\<forall>Y X V U Z W. product(X::'a,Y,U) & product(Y::'a,Z,V) & product(X::'a,V,W) --> product(U::'a,Z,W)) &
   913   (\<forall>A. product(A::'a,INVERSE(A),identity)) &
   914   (\<forall>A. product(A::'a,identity,A)) &
   915   (\<forall>A. ~product(A::'a,a,identity)) --> False"
   916    by meson
   917 
   918 (*47 inferences so far.  Searching to depth 11.   0.2 secs*)
   919 lemma GRP034_4:
   920   "(\<forall>X Y. product(X::'a,Y,multiply(X::'a,Y))) &
   921   (\<forall>X. product(identity::'a,X,X)) &
   922   (\<forall>X. product(X::'a,identity,X)) &
   923   (\<forall>X. product(X::'a,INVERSE(X),identity)) &
   924   (\<forall>Y U Z X V W. product(X::'a,Y,U) & product(Y::'a,Z,V) & product(U::'a,Z,W) --> product(X::'a,V,W)) &
   925   (\<forall>Y X V U Z W. product(X::'a,Y,U) & product(Y::'a,Z,V) & product(X::'a,V,W) --> product(U::'a,Z,W)) &
   926   (\<forall>B A C. subgroup_member(A) & subgroup_member(B) & product(B::'a,INVERSE(A),C) --> subgroup_member(C)) &
   927   (subgroup_member(a)) &
   928   (~subgroup_member(INVERSE(a))) --> False"
   929   by meson
   930 
   931 (*3287 inferences so far.  Searching to depth 14.  3.5 secs*)
   932 lemma GRP047_2:
   933   "(\<forall>X. product(identity::'a,X,X)) &
   934   (\<forall>X. product(INVERSE(X),X,identity)) &
   935   (\<forall>X Y. product(X::'a,Y,multiply(X::'a,Y))) &
   936   (\<forall>X Y Z W. product(X::'a,Y,Z) & product(X::'a,Y,W) --> equal(Z::'a,W)) &
   937   (\<forall>Y U Z X V W. product(X::'a,Y,U) & product(Y::'a,Z,V) & product(U::'a,Z,W) --> product(X::'a,V,W)) &
   938   (\<forall>Y X V U Z W. product(X::'a,Y,U) & product(Y::'a,Z,V) & product(X::'a,V,W) --> product(U::'a,Z,W)) &
   939   (\<forall>X W Z Y. equal(X::'a,Y) & product(W::'a,Z,X) --> product(W::'a,Z,Y)) &
   940   (equal(a::'a,b)) &
   941   (~equal(multiply(c::'a,a),multiply(c::'a,b))) --> False"
   942   by meson
   943 
   944 (*25559 inferences so far.  Searching to depth 19.  16.9 secs*)
   945 lemma GRP130_1_002:
   946   "(group_element(e_1)) &
   947   (group_element(e_2)) &
   948   (~equal(e_1::'a,e_2)) &
   949   (~equal(e_2::'a,e_1)) &
   950   (\<forall>X Y. group_element(X) & group_element(Y) --> product(X::'a,Y,e_1) | product(X::'a,Y,e_2)) &
   951   (\<forall>X Y W Z. product(X::'a,Y,W) & product(X::'a,Y,Z) --> equal(W::'a,Z)) &
   952   (\<forall>X Y W Z. product(X::'a,W,Y) & product(X::'a,Z,Y) --> equal(W::'a,Z)) &
   953   (\<forall>Y X W Z. product(W::'a,Y,X) & product(Z::'a,Y,X) --> equal(W::'a,Z)) &
   954   (\<forall>Z1 Z2 Y X. product(X::'a,Y,Z1) & product(X::'a,Z1,Z2) --> product(Z2::'a,Y,X)) --> False"
   955   oops
   956 
   957 abbreviation "GRP004_0_ax INVERSE identity multiply equal \<equiv>
   958   (\<forall>X. equal(multiply(identity::'a,X),X)) &
   959   (\<forall>X. equal(multiply(INVERSE(X),X),identity)) &
   960   (\<forall>X Y Z. equal(multiply(multiply(X::'a,Y),Z),multiply(X::'a,multiply(Y::'a,Z)))) &
   961   (\<forall>A B. equal(A::'a,B) --> equal(INVERSE(A),INVERSE(B))) &
   962   (\<forall>C D E. equal(C::'a,D) --> equal(multiply(C::'a,E),multiply(D::'a,E))) &
   963   (\<forall>F' H G. equal(F'::'a,G) --> equal(multiply(H::'a,F'),multiply(H::'a,G)))"
   964 
   965 abbreviation "GRP004_2_ax multiply least_upper_bound greatest_lower_bound equal \<equiv>
   966   (\<forall>Y X. equal(greatest_lower_bound(X::'a,Y),greatest_lower_bound(Y::'a,X))) &
   967   (\<forall>Y X. equal(least_upper_bound(X::'a,Y),least_upper_bound(Y::'a,X))) &
   968   (\<forall>X Y Z. equal(greatest_lower_bound(X::'a,greatest_lower_bound(Y::'a,Z)),greatest_lower_bound(greatest_lower_bound(X::'a,Y),Z))) &
   969   (\<forall>X Y Z. equal(least_upper_bound(X::'a,least_upper_bound(Y::'a,Z)),least_upper_bound(least_upper_bound(X::'a,Y),Z))) &
   970   (\<forall>X. equal(least_upper_bound(X::'a,X),X)) &
   971   (\<forall>X. equal(greatest_lower_bound(X::'a,X),X)) &
   972   (\<forall>Y X. equal(least_upper_bound(X::'a,greatest_lower_bound(X::'a,Y)),X)) &
   973   (\<forall>Y X. equal(greatest_lower_bound(X::'a,least_upper_bound(X::'a,Y)),X)) &
   974   (\<forall>Y X Z. equal(multiply(X::'a,least_upper_bound(Y::'a,Z)),least_upper_bound(multiply(X::'a,Y),multiply(X::'a,Z)))) &
   975   (\<forall>Y X Z. equal(multiply(X::'a,greatest_lower_bound(Y::'a,Z)),greatest_lower_bound(multiply(X::'a,Y),multiply(X::'a,Z)))) &
   976   (\<forall>Y Z X. equal(multiply(least_upper_bound(Y::'a,Z),X),least_upper_bound(multiply(Y::'a,X),multiply(Z::'a,X)))) &
   977   (\<forall>Y Z X. equal(multiply(greatest_lower_bound(Y::'a,Z),X),greatest_lower_bound(multiply(Y::'a,X),multiply(Z::'a,X)))) &
   978   (\<forall>A B C. equal(A::'a,B) --> equal(greatest_lower_bound(A::'a,C),greatest_lower_bound(B::'a,C))) &
   979   (\<forall>A C B. equal(A::'a,B) --> equal(greatest_lower_bound(C::'a,A),greatest_lower_bound(C::'a,B))) &
   980   (\<forall>A B C. equal(A::'a,B) --> equal(least_upper_bound(A::'a,C),least_upper_bound(B::'a,C))) &
   981   (\<forall>A C B. equal(A::'a,B) --> equal(least_upper_bound(C::'a,A),least_upper_bound(C::'a,B))) &
   982   (\<forall>A B C. equal(A::'a,B) --> equal(multiply(A::'a,C),multiply(B::'a,C))) &
   983   (\<forall>A C B. equal(A::'a,B) --> equal(multiply(C::'a,A),multiply(C::'a,B)))"
   984 
   985 (*3468 inferences so far.  Searching to depth 10.  9.1 secs*)
   986 lemma GRP156_1:
   987   "EQU001_0_ax equal &
   988   GRP004_0_ax INVERSE identity multiply equal &
   989   GRP004_2_ax multiply least_upper_bound greatest_lower_bound equal &
   990   (equal(least_upper_bound(a::'a,b),b)) &
   991   (~equal(greatest_lower_bound(multiply(a::'a,c),multiply(b::'a,c)),multiply(a::'a,c))) --> False"
   992     by meson
   993 
   994 (*4394 inferences so far.  Searching to depth 10.  8.2 secs*)
   995 lemma GRP168_1:
   996   "EQU001_0_ax equal &
   997   GRP004_0_ax INVERSE identity multiply equal &
   998   GRP004_2_ax multiply least_upper_bound greatest_lower_bound equal &
   999   (equal(least_upper_bound(a::'a,b),b)) &
  1000   (~equal(least_upper_bound(multiply(INVERSE(c),multiply(a::'a,c)),multiply(INVERSE(c),multiply(b::'a,c))),multiply(INVERSE(c),multiply(b::'a,c)))) --> False"
  1001   by meson
  1002 
  1003 abbreviation "HEN002_0_ax identity Zero Divide equal mless_equal \<equiv>
  1004   (\<forall>X Y. mless_equal(X::'a,Y) --> equal(Divide(X::'a,Y),Zero)) &
  1005   (\<forall>X Y. equal(Divide(X::'a,Y),Zero) --> mless_equal(X::'a,Y)) &
  1006   (\<forall>Y X. mless_equal(Divide(X::'a,Y),X)) &
  1007   (\<forall>X Y Z. mless_equal(Divide(Divide(X::'a,Z),Divide(Y::'a,Z)),Divide(Divide(X::'a,Y),Z))) &
  1008   (\<forall>X. mless_equal(Zero::'a,X)) &
  1009   (\<forall>X Y. mless_equal(X::'a,Y) & mless_equal(Y::'a,X) --> equal(X::'a,Y)) &
  1010   (\<forall>X. mless_equal(X::'a,identity))"
  1011 
  1012 abbreviation "HEN002_0_eq mless_equal Divide equal \<equiv>
  1013   (\<forall>A B C. equal(A::'a,B) --> equal(Divide(A::'a,C),Divide(B::'a,C))) &
  1014   (\<forall>D F' E. equal(D::'a,E) --> equal(Divide(F'::'a,D),Divide(F'::'a,E))) &
  1015   (\<forall>G H I'. equal(G::'a,H) & mless_equal(G::'a,I') --> mless_equal(H::'a,I')) &
  1016   (\<forall>J L K'. equal(J::'a,K') & mless_equal(L::'a,J) --> mless_equal(L::'a,K'))"
  1017 
  1018 (*250258 inferences so far.  Searching to depth 16.  406.2 secs*)
  1019 lemma HEN003_3:
  1020   "EQU001_0_ax equal &
  1021   HEN002_0_ax identity Zero Divide equal mless_equal &
  1022   HEN002_0_eq mless_equal Divide equal &
  1023   (~equal(Divide(a::'a,a),Zero)) --> False"
  1024   oops
  1025 
  1026 (*202177 inferences so far.  Searching to depth 14.  451 secs*)
  1027 lemma HEN007_2:
  1028   "EQU001_0_ax equal &
  1029   (\<forall>X Y. mless_equal(X::'a,Y) --> quotient(X::'a,Y,Zero)) &
  1030   (\<forall>X Y. quotient(X::'a,Y,Zero) --> mless_equal(X::'a,Y)) &
  1031   (\<forall>Y Z X. quotient(X::'a,Y,Z) --> mless_equal(Z::'a,X)) &
  1032   (\<forall>Y X V3 V2 V1 Z V4 V5. quotient(X::'a,Y,V1) & quotient(Y::'a,Z,V2) & quotient(X::'a,Z,V3) & quotient(V3::'a,V2,V4) & quotient(V1::'a,Z,V5) --> mless_equal(V4::'a,V5)) &
  1033   (\<forall>X. mless_equal(Zero::'a,X)) &
  1034   (\<forall>X Y. mless_equal(X::'a,Y) & mless_equal(Y::'a,X) --> equal(X::'a,Y)) &
  1035   (\<forall>X. mless_equal(X::'a,identity)) &
  1036   (\<forall>X Y. quotient(X::'a,Y,Divide(X::'a,Y))) &
  1037   (\<forall>X Y Z W. quotient(X::'a,Y,Z) & quotient(X::'a,Y,W) --> equal(Z::'a,W)) &
  1038   (\<forall>X Y W Z. equal(X::'a,Y) & quotient(X::'a,W,Z) --> quotient(Y::'a,W,Z)) &
  1039   (\<forall>X W Y Z. equal(X::'a,Y) & quotient(W::'a,X,Z) --> quotient(W::'a,Y,Z)) &
  1040   (\<forall>X W Z Y. equal(X::'a,Y) & quotient(W::'a,Z,X) --> quotient(W::'a,Z,Y)) &
  1041   (\<forall>X Z Y. equal(X::'a,Y) & mless_equal(Z::'a,X) --> mless_equal(Z::'a,Y)) &
  1042   (\<forall>X Y Z. equal(X::'a,Y) & mless_equal(X::'a,Z) --> mless_equal(Y::'a,Z)) &
  1043   (\<forall>X Y W. equal(X::'a,Y) --> equal(Divide(X::'a,W),Divide(Y::'a,W))) &
  1044   (\<forall>X W Y. equal(X::'a,Y) --> equal(Divide(W::'a,X),Divide(W::'a,Y))) &
  1045   (\<forall>X. quotient(X::'a,identity,Zero)) &
  1046   (\<forall>X. quotient(Zero::'a,X,Zero)) &
  1047   (\<forall>X. quotient(X::'a,X,Zero)) &
  1048   (\<forall>X. quotient(X::'a,Zero,X)) &
  1049   (\<forall>Y X Z. mless_equal(X::'a,Y) & mless_equal(Y::'a,Z) --> mless_equal(X::'a,Z)) &
  1050   (\<forall>W1 X Z W2 Y. quotient(X::'a,Y,W1) & mless_equal(W1::'a,Z) & quotient(X::'a,Z,W2) --> mless_equal(W2::'a,Y)) &
  1051   (mless_equal(x::'a,y)) &
  1052   (quotient(z::'a,y,zQy)) &
  1053   (quotient(z::'a,x,zQx)) &
  1054   (~mless_equal(zQy::'a,zQx)) --> False"
  1055   oops
  1056 
  1057 (*60026 inferences so far.  Searching to depth 12.  42.2 secs*)
  1058 lemma HEN008_4:
  1059   "EQU001_0_ax equal &
  1060   HEN002_0_ax identity Zero Divide equal mless_equal &
  1061   HEN002_0_eq mless_equal Divide equal &
  1062   (\<forall>X. equal(Divide(X::'a,identity),Zero)) &
  1063   (\<forall>X. equal(Divide(Zero::'a,X),Zero)) &
  1064   (\<forall>X. equal(Divide(X::'a,X),Zero)) &
  1065   (equal(Divide(a::'a,Zero),a)) &
  1066   (\<forall>Y X Z. mless_equal(X::'a,Y) & mless_equal(Y::'a,Z) --> mless_equal(X::'a,Z)) &
  1067   (\<forall>X Z Y. mless_equal(Divide(X::'a,Y),Z) --> mless_equal(Divide(X::'a,Z),Y)) &
  1068   (\<forall>Y Z X. mless_equal(X::'a,Y) --> mless_equal(Divide(Z::'a,Y),Divide(Z::'a,X))) &
  1069   (mless_equal(a::'a,b)) &
  1070   (~mless_equal(Divide(a::'a,c),Divide(b::'a,c))) --> False"
  1071   oops
  1072 
  1073 (*3160 inferences so far.  Searching to depth 11.  3.5 secs*)
  1074 lemma HEN009_5:
  1075   "EQU001_0_ax equal &
  1076   (\<forall>Y X. equal(Divide(Divide(X::'a,Y),X),Zero)) &
  1077   (\<forall>X Y Z. equal(Divide(Divide(Divide(X::'a,Z),Divide(Y::'a,Z)),Divide(Divide(X::'a,Y),Z)),Zero)) &
  1078   (\<forall>X. equal(Divide(Zero::'a,X),Zero)) &
  1079   (\<forall>X Y. equal(Divide(X::'a,Y),Zero) & equal(Divide(Y::'a,X),Zero) --> equal(X::'a,Y)) &
  1080   (\<forall>X. equal(Divide(X::'a,identity),Zero)) &
  1081   (\<forall>A B C. equal(A::'a,B) --> equal(Divide(A::'a,C),Divide(B::'a,C))) &
  1082   (\<forall>D F' E. equal(D::'a,E) --> equal(Divide(F'::'a,D),Divide(F'::'a,E))) &
  1083   (\<forall>Y X Z. equal(Divide(X::'a,Y),Zero) & equal(Divide(Y::'a,Z),Zero) --> equal(Divide(X::'a,Z),Zero)) &
  1084   (\<forall>X Z Y. equal(Divide(Divide(X::'a,Y),Z),Zero) --> equal(Divide(Divide(X::'a,Z),Y),Zero)) &
  1085   (\<forall>Y Z X. equal(Divide(X::'a,Y),Zero) --> equal(Divide(Divide(Z::'a,Y),Divide(Z::'a,X)),Zero)) &
  1086   (~equal(Divide(identity::'a,a),Divide(identity::'a,Divide(identity::'a,Divide(identity::'a,a))))) &
  1087   (equal(Divide(identity::'a,a),b)) &
  1088   (equal(Divide(identity::'a,b),c)) &
  1089   (equal(Divide(identity::'a,c),d)) &
  1090   (~equal(b::'a,d)) --> False"
  1091   by meson
  1092 
  1093 (*970373 inferences so far.  Searching to depth 17.  890.0 secs*)
  1094 lemma HEN012_3:
  1095   "EQU001_0_ax equal &
  1096   HEN002_0_ax identity Zero Divide equal mless_equal &
  1097   HEN002_0_eq mless_equal Divide equal &
  1098   (~mless_equal(a::'a,a)) --> False"
  1099   oops
  1100 
  1101 
  1102 (*1063 inferences so far.  Searching to depth 20.  1.0 secs*)
  1103 lemma LCL010_1:
  1104  "(\<forall>X Y. is_a_theorem(equivalent(X::'a,Y)) & is_a_theorem(X) --> is_a_theorem(Y)) &
  1105   (\<forall>X Z Y. is_a_theorem(equivalent(equivalent(X::'a,Y),equivalent(equivalent(X::'a,Z),equivalent(Z::'a,Y))))) &
  1106   (~is_a_theorem(equivalent(equivalent(a::'a,b),equivalent(equivalent(c::'a,b),equivalent(a::'a,c))))) --> False"
  1107   by meson
  1108 
  1109 (*2549 inferences so far.  Searching to depth 12.  1.4 secs*)
  1110 lemma LCL077_2:
  1111  "(\<forall>X Y. is_a_theorem(implies(X,Y)) & is_a_theorem(X) --> is_a_theorem(Y)) &
  1112   (\<forall>Y X. is_a_theorem(implies(X,implies(Y,X)))) &
  1113   (\<forall>Y X Z. is_a_theorem(implies(implies(X,implies(Y,Z)),implies(implies(X,Y),implies(X,Z))))) &
  1114   (\<forall>Y X. is_a_theorem(implies(implies(not(X),not(Y)),implies(Y,X)))) &
  1115   (\<forall>X2 X1 X3. is_a_theorem(implies(X1,X2)) & is_a_theorem(implies(X2,X3)) --> is_a_theorem(implies(X1,X3))) &
  1116   (~is_a_theorem(implies(not(not(a)),a))) --> False"
  1117   by meson
  1118 
  1119 (*2036 inferences so far.  Searching to depth 20.  1.5 secs*)
  1120 lemma LCL082_1:
  1121  "(\<forall>X Y. is_a_theorem(implies(X::'a,Y)) & is_a_theorem(X) --> is_a_theorem(Y)) &
  1122   (\<forall>Y Z U X. is_a_theorem(implies(implies(implies(X::'a,Y),Z),implies(implies(Z::'a,X),implies(U::'a,X))))) &
  1123   (~is_a_theorem(implies(a::'a,implies(b::'a,a)))) --> False"
  1124   by meson
  1125 
  1126 (*1100 inferences so far.  Searching to depth 13.  1.0 secs*)
  1127 lemma LCL111_1:
  1128  "(\<forall>X Y. is_a_theorem(implies(X,Y)) & is_a_theorem(X) --> is_a_theorem(Y)) &
  1129   (\<forall>Y X. is_a_theorem(implies(X,implies(Y,X)))) &
  1130   (\<forall>Y X Z. is_a_theorem(implies(implies(X,Y),implies(implies(Y,Z),implies(X,Z))))) &
  1131   (\<forall>Y X. is_a_theorem(implies(implies(implies(X,Y),Y),implies(implies(Y,X),X)))) &
  1132   (\<forall>Y X. is_a_theorem(implies(implies(not(X),not(Y)),implies(Y,X)))) &
  1133   (~is_a_theorem(implies(implies(a,b),implies(implies(c,a),implies(c,b))))) --> False"
  1134   by meson
  1135 
  1136 (*667 inferences so far.  Searching to depth 9.  1.4 secs*)
  1137 lemma LCL143_1:
  1138  "(\<forall>X. equal(X,X)) &
  1139   (\<forall>Y X. equal(X,Y) --> equal(Y,X)) &
  1140   (\<forall>Y X Z. equal(X,Y) & equal(Y,Z) --> equal(X,Z)) &
  1141   (\<forall>X. equal(implies(true,X),X)) &
  1142   (\<forall>Y X Z. equal(implies(implies(X,Y),implies(implies(Y,Z),implies(X,Z))),true)) &
  1143   (\<forall>Y X. equal(implies(implies(X,Y),Y),implies(implies(Y,X),X))) &
  1144   (\<forall>Y X. equal(implies(implies(not(X),not(Y)),implies(Y,X)),true)) &
  1145   (\<forall>A B C. equal(A,B) --> equal(implies(A,C),implies(B,C))) &
  1146   (\<forall>D F' E. equal(D,E) --> equal(implies(F',D),implies(F',E))) &
  1147   (\<forall>G H. equal(G,H) --> equal(not(G),not(H))) &
  1148   (\<forall>X Y. equal(big_V(X,Y),implies(implies(X,Y),Y))) &
  1149   (\<forall>X Y. equal(big_hat(X,Y),not(big_V(not(X),not(Y))))) &
  1150   (\<forall>X Y. ordered(X,Y) --> equal(implies(X,Y),true)) &
  1151   (\<forall>X Y. equal(implies(X,Y),true) --> ordered(X,Y)) &
  1152   (\<forall>A B C. equal(A,B) --> equal(big_V(A,C),big_V(B,C))) &
  1153   (\<forall>D F' E. equal(D,E) --> equal(big_V(F',D),big_V(F',E))) &
  1154   (\<forall>G H I'. equal(G,H) --> equal(big_hat(G,I'),big_hat(H,I'))) &
  1155   (\<forall>J L K'. equal(J,K') --> equal(big_hat(L,J),big_hat(L,K'))) &
  1156   (\<forall>M N O'. equal(M,N) & ordered(M,O') --> ordered(N,O')) &
  1157   (\<forall>P R Q. equal(P,Q) & ordered(R,P) --> ordered(R,Q)) &
  1158   (ordered(x,y)) &
  1159   (~ordered(implies(z,x),implies(z,y))) --> False"
  1160   by meson
  1161 
  1162 (*5245 inferences so far.  Searching to depth 12.  4.6 secs*)
  1163 lemma LCL182_1:
  1164  "(\<forall>A. axiom(or(not(or(A,A)),A))) &
  1165   (\<forall>B A. axiom(or(not(A),or(B,A)))) &
  1166   (\<forall>B A. axiom(or(not(or(A,B)),or(B,A)))) &
  1167   (\<forall>B A C. axiom(or(not(or(A,or(B,C))),or(B,or(A,C))))) &
  1168   (\<forall>A C B. axiom(or(not(or(not(A),B)),or(not(or(C,A)),or(C,B))))) &
  1169   (\<forall>X. axiom(X) --> theorem(X)) &
  1170   (\<forall>X Y. axiom(or(not(Y),X)) & theorem(Y) --> theorem(X)) &
  1171   (\<forall>X Y Z. axiom(or(not(X),Y)) & theorem(or(not(Y),Z)) --> theorem(or(not(X),Z))) &
  1172   (~theorem(or(not(or(not(p),q)),or(not(not(q)),not(p))))) --> False"
  1173   by meson
  1174 
  1175 (*410 inferences so far.  Searching to depth 10.  0.3 secs*)
  1176 lemma LCL200_1:
  1177  "(\<forall>A. axiom(or(not(or(A,A)),A))) &
  1178   (\<forall>B A. axiom(or(not(A),or(B,A)))) &
  1179   (\<forall>B A. axiom(or(not(or(A,B)),or(B,A)))) &
  1180   (\<forall>B A C. axiom(or(not(or(A,or(B,C))),or(B,or(A,C))))) &
  1181   (\<forall>A C B. axiom(or(not(or(not(A),B)),or(not(or(C,A)),or(C,B))))) &
  1182   (\<forall>X. axiom(X) --> theorem(X)) &
  1183   (\<forall>X Y. axiom(or(not(Y),X)) & theorem(Y) --> theorem(X)) &
  1184   (\<forall>X Y Z. axiom(or(not(X),Y)) & theorem(or(not(Y),Z)) --> theorem(or(not(X),Z))) &
  1185   (~theorem(or(not(not(or(p,q))),not(q)))) --> False"
  1186   by meson
  1187 
  1188 (*5849 inferences so far.  Searching to depth 12.  5.6 secs*)
  1189 lemma LCL215_1:
  1190  "(\<forall>A. axiom(or(not(or(A,A)),A))) &
  1191   (\<forall>B A. axiom(or(not(A),or(B,A)))) &
  1192   (\<forall>B A. axiom(or(not(or(A,B)),or(B,A)))) &
  1193   (\<forall>B A C. axiom(or(not(or(A,or(B,C))),or(B,or(A,C))))) &
  1194   (\<forall>A C B. axiom(or(not(or(not(A),B)),or(not(or(C,A)),or(C,B))))) &
  1195   (\<forall>X. axiom(X) --> theorem(X)) &
  1196   (\<forall>X Y. axiom(or(not(Y),X)) & theorem(Y) --> theorem(X)) &
  1197   (\<forall>X Y Z. axiom(or(not(X),Y)) & theorem(or(not(Y),Z)) --> theorem(or(not(X),Z))) &
  1198   (~theorem(or(not(or(not(p),q)),or(not(or(p,q)),q)))) --> False"
  1199   by meson
  1200 
  1201 (*0 secs.  Not sure that a search even starts!*)
  1202 lemma LCL230_2:
  1203   "(q --> p | r) &
  1204   (~p) &
  1205   (q) &
  1206   (~r) --> False"
  1207   by meson
  1208 
  1209 (*119585 inferences so far.  Searching to depth 14.  262.4 secs*)
  1210 lemma LDA003_1:
  1211   "EQU001_0_ax equal &
  1212   (\<forall>Y X Z. equal(f(X::'a,f(Y::'a,Z)),f(f(X::'a,Y),f(X::'a,Z)))) &
  1213   (\<forall>X Y. left(X::'a,f(X::'a,Y))) &
  1214   (\<forall>Y X Z. left(X::'a,Y) & left(Y::'a,Z) --> left(X::'a,Z)) &
  1215   (equal(num2::'a,f(num1::'a,num1))) &
  1216   (equal(num3::'a,f(num2::'a,num1))) &
  1217   (equal(u::'a,f(num2::'a,num2))) &
  1218   (\<forall>A B C. equal(A::'a,B) --> equal(f(A::'a,C),f(B::'a,C))) &
  1219   (\<forall>D F' E. equal(D::'a,E) --> equal(f(F'::'a,D),f(F'::'a,E))) &
  1220   (\<forall>G H I'. equal(G::'a,H) & left(G::'a,I') --> left(H::'a,I')) &
  1221   (\<forall>J L K'. equal(J::'a,K') & left(L::'a,J) --> left(L::'a,K')) &
  1222   (~left(num3::'a,u)) --> False"
  1223   oops
  1224 
  1225 
  1226 (*2392 inferences so far.  Searching to depth 12.  2.2 secs*)
  1227 lemma MSC002_1:
  1228  "(at(something::'a,here,now)) &
  1229   (\<forall>Place Situation. hand_at(Place::'a,Situation) --> hand_at(Place::'a,let_go(Situation))) &
  1230   (\<forall>Place Another_place Situation. hand_at(Place::'a,Situation) --> hand_at(Another_place::'a,go(Another_place::'a,Situation))) &
  1231   (\<forall>Thing Situation. ~held(Thing::'a,let_go(Situation))) &
  1232   (\<forall>Situation Thing. at(Thing::'a,here,Situation) --> red(Thing)) &
  1233   (\<forall>Thing Place Situation. at(Thing::'a,Place,Situation) --> at(Thing::'a,Place,let_go(Situation))) &
  1234   (\<forall>Thing Place Situation. at(Thing::'a,Place,Situation) --> at(Thing::'a,Place,pick_up(Situation))) &
  1235   (\<forall>Thing Place Situation. at(Thing::'a,Place,Situation) --> grabbed(Thing::'a,pick_up(go(Place::'a,let_go(Situation))))) &
  1236   (\<forall>Thing Situation. red(Thing) & put(Thing::'a,there,Situation) --> answer(Situation)) &
  1237   (\<forall>Place Thing Another_place Situation. at(Thing::'a,Place,Situation) & grabbed(Thing::'a,Situation) --> put(Thing::'a,Another_place,go(Another_place::'a,Situation))) &
  1238   (\<forall>Thing Place Another_place Situation. at(Thing::'a,Place,Situation) --> held(Thing::'a,Situation) | at(Thing::'a,Place,go(Another_place::'a,Situation))) &
  1239   (\<forall>One_place Thing Place Situation. hand_at(One_place::'a,Situation) & held(Thing::'a,Situation) --> at(Thing::'a,Place,go(Place::'a,Situation))) &
  1240   (\<forall>Place Thing Situation. hand_at(Place::'a,Situation) & at(Thing::'a,Place,Situation) --> held(Thing::'a,pick_up(Situation))) &
  1241   (\<forall>Situation. ~answer(Situation)) --> False"
  1242   by meson
  1243 
  1244 (*73 inferences so far.  Searching to depth 10.  0.2 secs*)
  1245 lemma MSC003_1:
  1246   "(\<forall>Number_of_small_parts Small_part Big_part Number_of_mid_parts Mid_part. has_parts(Big_part::'a,Number_of_mid_parts,Mid_part) --> in'(object_in(Big_part::'a,Mid_part,Small_part,Number_of_mid_parts,Number_of_small_parts),Mid_part) | has_parts(Big_part::'a,mtimes(Number_of_mid_parts::'a,Number_of_small_parts),Small_part)) &
  1247   (\<forall>Big_part Mid_part Number_of_mid_parts Number_of_small_parts Small_part. has_parts(Big_part::'a,Number_of_mid_parts,Mid_part) & has_parts(object_in(Big_part::'a,Mid_part,Small_part,Number_of_mid_parts,Number_of_small_parts),Number_of_small_parts,Small_part) --> has_parts(Big_part::'a,mtimes(Number_of_mid_parts::'a,Number_of_small_parts),Small_part)) &
  1248   (in'(john::'a,boy)) &
  1249   (\<forall>X. in'(X::'a,boy) --> in'(X::'a,human)) &
  1250   (\<forall>X. in'(X::'a,hand) --> has_parts(X::'a,num5,fingers)) &
  1251   (\<forall>X. in'(X::'a,human) --> has_parts(X::'a,num2,arm)) &
  1252   (\<forall>X. in'(X::'a,arm) --> has_parts(X::'a,num1,hand)) &
  1253   (~has_parts(john::'a,mtimes(num2::'a,num1),hand)) --> False"
  1254   by meson
  1255 
  1256 (*1486 inferences so far.  Searching to depth 20.  1.2 secs*)
  1257 lemma MSC004_1:
  1258  "(\<forall>Number_of_small_parts Small_part Big_part Number_of_mid_parts Mid_part. has_parts(Big_part::'a,Number_of_mid_parts,Mid_part) --> in'(object_in(Big_part::'a,Mid_part,Small_part,Number_of_mid_parts,Number_of_small_parts),Mid_part) | has_parts(Big_part::'a,mtimes(Number_of_mid_parts::'a,Number_of_small_parts),Small_part)) &
  1259   (\<forall>Big_part Mid_part Number_of_mid_parts Number_of_small_parts Small_part. has_parts(Big_part::'a,Number_of_mid_parts,Mid_part) & has_parts(object_in(Big_part::'a,Mid_part,Small_part,Number_of_mid_parts,Number_of_small_parts),Number_of_small_parts,Small_part) --> has_parts(Big_part::'a,mtimes(Number_of_mid_parts::'a,Number_of_small_parts),Small_part)) &
  1260   (in'(john::'a,boy)) &
  1261   (\<forall>X. in'(X::'a,boy) --> in'(X::'a,human)) &
  1262   (\<forall>X. in'(X::'a,hand) --> has_parts(X::'a,num5,fingers)) &
  1263   (\<forall>X. in'(X::'a,human) --> has_parts(X::'a,num2,arm)) &
  1264   (\<forall>X. in'(X::'a,arm) --> has_parts(X::'a,num1,hand)) &
  1265   (~has_parts(john::'a,mtimes(mtimes(num2::'a,num1),num5),fingers)) --> False"
  1266   by meson
  1267 
  1268 (*100 inferences so far.  Searching to depth 12.  0.1 secs*)
  1269 lemma MSC005_1:
  1270   "(value(truth::'a,truth)) &
  1271   (value(falsity::'a,falsity)) &
  1272   (\<forall>X Y. value(X::'a,truth) & value(Y::'a,truth) --> value(xor(X::'a,Y),falsity)) &
  1273   (\<forall>X Y. value(X::'a,truth) & value(Y::'a,falsity) --> value(xor(X::'a,Y),truth)) &
  1274   (\<forall>X Y. value(X::'a,falsity) & value(Y::'a,truth) --> value(xor(X::'a,Y),truth)) &
  1275   (\<forall>X Y. value(X::'a,falsity) & value(Y::'a,falsity) --> value(xor(X::'a,Y),falsity)) &
  1276   (\<forall>Value. ~value(xor(xor(xor(xor(truth::'a,falsity),falsity),truth),falsity),Value)) --> False"
  1277   by meson
  1278 
  1279 (*19116 inferences so far.  Searching to depth 16.  15.9 secs*)
  1280 lemma MSC006_1:
  1281  "(\<forall>Y X Z. p(X::'a,Y) & p(Y::'a,Z) --> p(X::'a,Z)) &
  1282   (\<forall>Y X Z. q(X::'a,Y) & q(Y::'a,Z) --> q(X::'a,Z)) &
  1283   (\<forall>Y X. q(X::'a,Y) --> q(Y::'a,X)) &
  1284   (\<forall>X Y. p(X::'a,Y) | q(X::'a,Y)) &
  1285   (~p(a::'a,b)) &
  1286   (~q(c::'a,d)) --> False"
  1287    by meson
  1288 
  1289 (*1713 inferences so far.  Searching to depth 10.  2.8 secs*)
  1290 lemma NUM001_1:
  1291   "(\<forall>A. equal(A::'a,A)) &
  1292   (\<forall>B A C. equal(A::'a,B) & equal(B::'a,C) --> equal(A::'a,C)) &
  1293   (\<forall>B A. equal(add(A::'a,B),add(B::'a,A))) &
  1294   (\<forall>A B C. equal(add(A::'a,add(B::'a,C)),add(add(A::'a,B),C))) &
  1295   (\<forall>B A. equal(subtract(add(A::'a,B),B),A)) &
  1296   (\<forall>A B. equal(A::'a,subtract(add(A::'a,B),B))) &
  1297   (\<forall>A C B. equal(add(subtract(A::'a,B),C),subtract(add(A::'a,C),B))) &
  1298   (\<forall>A C B. equal(subtract(add(A::'a,B),C),add(subtract(A::'a,C),B))) &
  1299   (\<forall>A C B D. equal(A::'a,B) & equal(C::'a,add(A::'a,D)) --> equal(C::'a,add(B::'a,D))) &
  1300   (\<forall>A C D B. equal(A::'a,B) & equal(C::'a,add(D::'a,A)) --> equal(C::'a,add(D::'a,B))) &
  1301   (\<forall>A C B D. equal(A::'a,B) & equal(C::'a,subtract(A::'a,D)) --> equal(C::'a,subtract(B::'a,D))) &
  1302   (\<forall>A C D B. equal(A::'a,B) & equal(C::'a,subtract(D::'a,A)) --> equal(C::'a,subtract(D::'a,B))) &
  1303   (~equal(add(add(a::'a,b),c),add(a::'a,add(b::'a,c)))) --> False"
  1304   by meson
  1305 
  1306 abbreviation "NUM001_0_ax multiply successor num0 add equal \<equiv>
  1307   (\<forall>A. equal(add(A::'a,num0),A)) &
  1308   (\<forall>A B. equal(add(A::'a,successor(B)),successor(add(A::'a,B)))) &
  1309   (\<forall>A. equal(multiply(A::'a,num0),num0)) &
  1310   (\<forall>B A. equal(multiply(A::'a,successor(B)),add(multiply(A::'a,B),A))) &
  1311   (\<forall>A B. equal(successor(A),successor(B)) --> equal(A::'a,B)) &
  1312   (\<forall>A B. equal(A::'a,B) --> equal(successor(A),successor(B)))"
  1313 
  1314 abbreviation "NUM001_1_ax predecessor_of_1st_minus_2nd successor add equal mless \<equiv>
  1315   (\<forall>A C B. mless(A::'a,B) & mless(C::'a,A) --> mless(C::'a,B)) &
  1316   (\<forall>A B C. equal(add(successor(A),B),C) --> mless(B::'a,C)) &
  1317   (\<forall>A B. mless(A::'a,B) --> equal(add(successor(predecessor_of_1st_minus_2nd(B::'a,A)),A),B))"
  1318 
  1319 abbreviation "NUM001_2_ax equal mless divides \<equiv>
  1320   (\<forall>A B. divides(A::'a,B) --> mless(A::'a,B) | equal(A::'a,B)) &
  1321   (\<forall>A B. mless(A::'a,B) --> divides(A::'a,B)) &
  1322   (\<forall>A B. equal(A::'a,B) --> divides(A::'a,B))"
  1323 
  1324 (*20717 inferences so far.  Searching to depth 11.  13.7 secs*)
  1325 lemma NUM021_1:
  1326   "EQU001_0_ax equal &
  1327   NUM001_0_ax multiply successor num0 add equal &
  1328   NUM001_1_ax predecessor_of_1st_minus_2nd successor add equal mless &
  1329   NUM001_2_ax equal mless divides &
  1330   (mless(b::'a,c)) &
  1331    (~mless(b::'a,a)) &
  1332    (divides(c::'a,a)) &
  1333    (\<forall>A. ~equal(successor(A),num0)) --> False"
  1334   by meson
  1335 
  1336 (*26320 inferences so far.  Searching to depth 10.  26.4 secs*)
  1337 lemma NUM024_1:
  1338   "EQU001_0_ax equal &
  1339   NUM001_0_ax multiply successor num0 add equal &
  1340   NUM001_1_ax predecessor_of_1st_minus_2nd successor add equal mless &
  1341   (\<forall>B A. equal(add(A::'a,B),add(B::'a,A))) &
  1342   (\<forall>B A C. equal(add(A::'a,B),add(C::'a,B)) --> equal(A::'a,C)) &
  1343   (mless(a::'a,a)) &
  1344   (\<forall>A. ~equal(successor(A),num0)) --> False"
  1345   oops
  1346 
  1347 abbreviation "SET004_0_ax not_homomorphism2 not_homomorphism1
  1348     homomorphism compatible operation cantor diagonalise subset_relation
  1349     one_to_one choice apply regular function identity_relation
  1350     single_valued_class compos powerClass sum_class omega inductive
  1351     successor_relation successor image' rng domain range_of INVERSE flip
  1352     rot domain_of null_class restrct difference union complement
  1353     intersection element_relation second first cross_product ordered_pair
  1354     singleton unordered_pair equal universal_class not_subclass_element
  1355     member subclass \<equiv>
  1356   (\<forall>X U Y. subclass(X::'a,Y) & member(U::'a,X) --> member(U::'a,Y)) &
  1357   (\<forall>X Y. member(not_subclass_element(X::'a,Y),X) | subclass(X::'a,Y)) &
  1358   (\<forall>X Y. member(not_subclass_element(X::'a,Y),Y) --> subclass(X::'a,Y)) &
  1359   (\<forall>X. subclass(X::'a,universal_class)) &
  1360   (\<forall>X Y. equal(X::'a,Y) --> subclass(X::'a,Y)) &
  1361   (\<forall>Y X. equal(X::'a,Y) --> subclass(Y::'a,X)) &
  1362   (\<forall>X Y. subclass(X::'a,Y) & subclass(Y::'a,X) --> equal(X::'a,Y)) &
  1363   (\<forall>X U Y. member(U::'a,unordered_pair(X::'a,Y)) --> equal(U::'a,X) | equal(U::'a,Y)) &
  1364   (\<forall>X Y. member(X::'a,universal_class) --> member(X::'a,unordered_pair(X::'a,Y))) &
  1365   (\<forall>X Y. member(Y::'a,universal_class) --> member(Y::'a,unordered_pair(X::'a,Y))) &
  1366   (\<forall>X Y. member(unordered_pair(X::'a,Y),universal_class)) &
  1367   (\<forall>X. equal(unordered_pair(X::'a,X),singleton(X))) &
  1368   (\<forall>X Y. equal(unordered_pair(singleton(X),unordered_pair(X::'a,singleton(Y))),ordered_pair(X::'a,Y))) &
  1369   (\<forall>V Y U X. member(ordered_pair(U::'a,V),cross_product(X::'a,Y)) --> member(U::'a,X)) &
  1370   (\<forall>U X V Y. member(ordered_pair(U::'a,V),cross_product(X::'a,Y)) --> member(V::'a,Y)) &
  1371   (\<forall>U V X Y. member(U::'a,X) & member(V::'a,Y) --> member(ordered_pair(U::'a,V),cross_product(X::'a,Y))) &
  1372   (\<forall>X Y Z. member(Z::'a,cross_product(X::'a,Y)) --> equal(ordered_pair(first(Z),second(Z)),Z)) &
  1373   (subclass(element_relation::'a,cross_product(universal_class::'a,universal_class))) &
  1374   (\<forall>X Y. member(ordered_pair(X::'a,Y),element_relation) --> member(X::'a,Y)) &
  1375   (\<forall>X Y. member(ordered_pair(X::'a,Y),cross_product(universal_class::'a,universal_class)) & member(X::'a,Y) --> member(ordered_pair(X::'a,Y),element_relation)) &
  1376   (\<forall>Y Z X. member(Z::'a,intersection(X::'a,Y)) --> member(Z::'a,X)) &
  1377   (\<forall>X Z Y. member(Z::'a,intersection(X::'a,Y)) --> member(Z::'a,Y)) &
  1378   (\<forall>Z X Y. member(Z::'a,X) & member(Z::'a,Y) --> member(Z::'a,intersection(X::'a,Y))) &
  1379   (\<forall>Z X. ~(member(Z::'a,complement(X)) & member(Z::'a,X))) &
  1380   (\<forall>Z X. member(Z::'a,universal_class) --> member(Z::'a,complement(X)) | member(Z::'a,X)) &
  1381   (\<forall>X Y. equal(complement(intersection(complement(X),complement(Y))),union(X::'a,Y))) &
  1382   (\<forall>X Y. equal(intersection(complement(intersection(X::'a,Y)),complement(intersection(complement(X),complement(Y)))),difference(X::'a,Y))) &
  1383   (\<forall>Xr X Y. equal(intersection(Xr::'a,cross_product(X::'a,Y)),restrct(Xr::'a,X,Y))) &
  1384   (\<forall>Xr X Y. equal(intersection(cross_product(X::'a,Y),Xr),restrct(Xr::'a,X,Y))) &
  1385   (\<forall>Z X. ~(equal(restrct(X::'a,singleton(Z),universal_class),null_class) & member(Z::'a,domain_of(X)))) &
  1386   (\<forall>Z X. member(Z::'a,universal_class) --> equal(restrct(X::'a,singleton(Z),universal_class),null_class) | member(Z::'a,domain_of(X))) &
  1387   (\<forall>X. subclass(rot(X),cross_product(cross_product(universal_class::'a,universal_class),universal_class))) &
  1388   (\<forall>V W U X. member(ordered_pair(ordered_pair(U::'a,V),W),rot(X)) --> member(ordered_pair(ordered_pair(V::'a,W),U),X)) &
  1389   (\<forall>U V W X. member(ordered_pair(ordered_pair(V::'a,W),U),X) & member(ordered_pair(ordered_pair(U::'a,V),W),cross_product(cross_product(universal_class::'a,universal_class),universal_class)) --> member(ordered_pair(ordered_pair(U::'a,V),W),rot(X))) &
  1390   (\<forall>X. subclass(flip(X),cross_product(cross_product(universal_class::'a,universal_class),universal_class))) &
  1391   (\<forall>V U W X. member(ordered_pair(ordered_pair(U::'a,V),W),flip(X)) --> member(ordered_pair(ordered_pair(V::'a,U),W),X)) &
  1392   (\<forall>U V W X. member(ordered_pair(ordered_pair(V::'a,U),W),X) & member(ordered_pair(ordered_pair(U::'a,V),W),cross_product(cross_product(universal_class::'a,universal_class),universal_class)) --> member(ordered_pair(ordered_pair(U::'a,V),W),flip(X))) &
  1393   (\<forall>Y. equal(domain_of(flip(cross_product(Y::'a,universal_class))),INVERSE(Y))) &
  1394   (\<forall>Z. equal(domain_of(INVERSE(Z)),range_of(Z))) &
  1395   (\<forall>Z X Y. equal(first(not_subclass_element(restrct(Z::'a,X,singleton(Y)),null_class)),domain(Z::'a,X,Y))) &
  1396   (\<forall>Z X Y. equal(second(not_subclass_element(restrct(Z::'a,singleton(X),Y),null_class)),rng(Z::'a,X,Y))) &
  1397   (\<forall>Xr X. equal(range_of(restrct(Xr::'a,X,universal_class)),image'(Xr::'a,X))) &
  1398   (\<forall>X. equal(union(X::'a,singleton(X)),successor(X))) &
  1399   (subclass(successor_relation::'a,cross_product(universal_class::'a,universal_class))) &
  1400   (\<forall>X Y. member(ordered_pair(X::'a,Y),successor_relation) --> equal(successor(X),Y)) &
  1401   (\<forall>X Y. equal(successor(X),Y) & member(ordered_pair(X::'a,Y),cross_product(universal_class::'a,universal_class)) --> member(ordered_pair(X::'a,Y),successor_relation)) &
  1402   (\<forall>X. inductive(X) --> member(null_class::'a,X)) &
  1403   (\<forall>X. inductive(X) --> subclass(image'(successor_relation::'a,X),X)) &
  1404   (\<forall>X. member(null_class::'a,X) & subclass(image'(successor_relation::'a,X),X) --> inductive(X)) &
  1405   (inductive(omega)) &
  1406   (\<forall>Y. inductive(Y) --> subclass(omega::'a,Y)) &
  1407   (member(omega::'a,universal_class)) &
  1408   (\<forall>X. equal(domain_of(restrct(element_relation::'a,universal_class,X)),sum_class(X))) &
  1409   (\<forall>X. member(X::'a,universal_class) --> member(sum_class(X),universal_class)) &
  1410   (\<forall>X. equal(complement(image'(element_relation::'a,complement(X))),powerClass(X))) &
  1411   (\<forall>U. member(U::'a,universal_class) --> member(powerClass(U),universal_class)) &
  1412   (\<forall>Yr Xr. subclass(compos(Yr::'a,Xr),cross_product(universal_class::'a,universal_class))) &
  1413   (\<forall>Z Yr Xr Y. member(ordered_pair(Y::'a,Z),compos(Yr::'a,Xr)) --> member(Z::'a,image'(Yr::'a,image'(Xr::'a,singleton(Y))))) &
  1414   (\<forall>Y Z Yr Xr. member(Z::'a,image'(Yr::'a,image'(Xr::'a,singleton(Y)))) & member(ordered_pair(Y::'a,Z),cross_product(universal_class::'a,universal_class)) --> member(ordered_pair(Y::'a,Z),compos(Yr::'a,Xr))) &
  1415   (\<forall>X. single_valued_class(X) --> subclass(compos(X::'a,INVERSE(X)),identity_relation)) &
  1416   (\<forall>X. subclass(compos(X::'a,INVERSE(X)),identity_relation) --> single_valued_class(X)) &
  1417   (\<forall>Xf. function(Xf) --> subclass(Xf::'a,cross_product(universal_class::'a,universal_class))) &
  1418   (\<forall>Xf. function(Xf) --> subclass(compos(Xf::'a,INVERSE(Xf)),identity_relation)) &
  1419   (\<forall>Xf. subclass(Xf::'a,cross_product(universal_class::'a,universal_class)) & subclass(compos(Xf::'a,INVERSE(Xf)),identity_relation) --> function(Xf)) &
  1420   (\<forall>Xf X. function(Xf) & member(X::'a,universal_class) --> member(image'(Xf::'a,X),universal_class)) &
  1421   (\<forall>X. equal(X::'a,null_class) | member(regular(X),X)) &
  1422   (\<forall>X. equal(X::'a,null_class) | equal(intersection(X::'a,regular(X)),null_class)) &
  1423   (\<forall>Xf Y. equal(sum_class(image'(Xf::'a,singleton(Y))),apply(Xf::'a,Y))) &
  1424   (function(choice)) &
  1425   (\<forall>Y. member(Y::'a,universal_class) --> equal(Y::'a,null_class) | member(apply(choice::'a,Y),Y)) &
  1426   (\<forall>Xf. one_to_one(Xf) --> function(Xf)) &
  1427   (\<forall>Xf. one_to_one(Xf) --> function(INVERSE(Xf))) &
  1428   (\<forall>Xf. function(INVERSE(Xf)) & function(Xf) --> one_to_one(Xf)) &
  1429   (equal(intersection(cross_product(universal_class::'a,universal_class),intersection(cross_product(universal_class::'a,universal_class),complement(compos(complement(element_relation),INVERSE(element_relation))))),subset_relation)) &
  1430   (equal(intersection(INVERSE(subset_relation),subset_relation),identity_relation)) &
  1431   (\<forall>Xr. equal(complement(domain_of(intersection(Xr::'a,identity_relation))),diagonalise(Xr))) &
  1432   (\<forall>X. equal(intersection(domain_of(X),diagonalise(compos(INVERSE(element_relation),X))),cantor(X))) &
  1433   (\<forall>Xf. operation(Xf) --> function(Xf)) &
  1434   (\<forall>Xf. operation(Xf) --> equal(cross_product(domain_of(domain_of(Xf)),domain_of(domain_of(Xf))),domain_of(Xf))) &
  1435   (\<forall>Xf. operation(Xf) --> subclass(range_of(Xf),domain_of(domain_of(Xf)))) &
  1436   (\<forall>Xf. function(Xf) & equal(cross_product(domain_of(domain_of(Xf)),domain_of(domain_of(Xf))),domain_of(Xf)) & subclass(range_of(Xf),domain_of(domain_of(Xf))) --> operation(Xf)) &
  1437   (\<forall>Xf1 Xf2 Xh. compatible(Xh::'a,Xf1,Xf2) --> function(Xh)) &
  1438   (\<forall>Xf2 Xf1 Xh. compatible(Xh::'a,Xf1,Xf2) --> equal(domain_of(domain_of(Xf1)),domain_of(Xh))) &
  1439   (\<forall>Xf1 Xh Xf2. compatible(Xh::'a,Xf1,Xf2) --> subclass(range_of(Xh),domain_of(domain_of(Xf2)))) &
  1440   (\<forall>Xh Xh1 Xf1 Xf2. function(Xh) & equal(domain_of(domain_of(Xf1)),domain_of(Xh)) & subclass(range_of(Xh),domain_of(domain_of(Xf2))) --> compatible(Xh1::'a,Xf1,Xf2)) &
  1441   (\<forall>Xh Xf2 Xf1. homomorphism(Xh::'a,Xf1,Xf2) --> operation(Xf1)) &
  1442   (\<forall>Xh Xf1 Xf2. homomorphism(Xh::'a,Xf1,Xf2) --> operation(Xf2)) &
  1443   (\<forall>Xh Xf1 Xf2. homomorphism(Xh::'a,Xf1,Xf2) --> compatible(Xh::'a,Xf1,Xf2)) &
  1444   (\<forall>Xf2 Xh Xf1 X Y. homomorphism(Xh::'a,Xf1,Xf2) & member(ordered_pair(X::'a,Y),domain_of(Xf1)) --> equal(apply(Xf2::'a,ordered_pair(apply(Xh::'a,X),apply(Xh::'a,Y))),apply(Xh::'a,apply(Xf1::'a,ordered_pair(X::'a,Y))))) &
  1445   (\<forall>Xh Xf1 Xf2. operation(Xf1) & operation(Xf2) & compatible(Xh::'a,Xf1,Xf2) --> member(ordered_pair(not_homomorphism1(Xh::'a,Xf1,Xf2),not_homomorphism2(Xh::'a,Xf1,Xf2)),domain_of(Xf1)) | homomorphism(Xh::'a,Xf1,Xf2)) &
  1446   (\<forall>Xh Xf1 Xf2. operation(Xf1) & operation(Xf2) & compatible(Xh::'a,Xf1,Xf2) & equal(apply(Xf2::'a,ordered_pair(apply(Xh::'a,not_homomorphism1(Xh::'a,Xf1,Xf2)),apply(Xh::'a,not_homomorphism2(Xh::'a,Xf1,Xf2)))),apply(Xh::'a,apply(Xf1::'a,ordered_pair(not_homomorphism1(Xh::'a,Xf1,Xf2),not_homomorphism2(Xh::'a,Xf1,Xf2))))) --> homomorphism(Xh::'a,Xf1,Xf2))"
  1447 
  1448 abbreviation "SET004_0_eq subclass single_valued_class operation
  1449     one_to_one member inductive homomorphism function compatible
  1450     unordered_pair union sum_class successor singleton second rot restrct
  1451     regular range_of rng powerClass ordered_pair not_subclass_element
  1452     not_homomorphism2 not_homomorphism1 INVERSE intersection image' flip
  1453     first domain_of domain difference diagonalise cross_product compos
  1454     complement cantor apply equal \<equiv>
  1455   (\<forall>D E F'. equal(D::'a,E) --> equal(apply(D::'a,F'),apply(E::'a,F'))) &
  1456   (\<forall>G I' H. equal(G::'a,H) --> equal(apply(I'::'a,G),apply(I'::'a,H))) &
  1457   (\<forall>J K'. equal(J::'a,K') --> equal(cantor(J),cantor(K'))) &
  1458   (\<forall>L M. equal(L::'a,M) --> equal(complement(L),complement(M))) &
  1459   (\<forall>N O' P. equal(N::'a,O') --> equal(compos(N::'a,P),compos(O'::'a,P))) &
  1460   (\<forall>Q S' R. equal(Q::'a,R) --> equal(compos(S'::'a,Q),compos(S'::'a,R))) &
  1461   (\<forall>T' U V. equal(T'::'a,U) --> equal(cross_product(T'::'a,V),cross_product(U::'a,V))) &
  1462   (\<forall>W Y X. equal(W::'a,X) --> equal(cross_product(Y::'a,W),cross_product(Y::'a,X))) &
  1463   (\<forall>Z A1. equal(Z::'a,A1) --> equal(diagonalise(Z),diagonalise(A1))) &
  1464   (\<forall>B1 C1 D1. equal(B1::'a,C1) --> equal(difference(B1::'a,D1),difference(C1::'a,D1))) &
  1465   (\<forall>E1 G1 F1. equal(E1::'a,F1) --> equal(difference(G1::'a,E1),difference(G1::'a,F1))) &
  1466   (\<forall>H1 I1 J1 K1. equal(H1::'a,I1) --> equal(domain(H1::'a,J1,K1),domain(I1::'a,J1,K1))) &
  1467   (\<forall>L1 N1 M1 O1. equal(L1::'a,M1) --> equal(domain(N1::'a,L1,O1),domain(N1::'a,M1,O1))) &
  1468   (\<forall>P1 R1 S1 Q1. equal(P1::'a,Q1) --> equal(domain(R1::'a,S1,P1),domain(R1::'a,S1,Q1))) &
  1469   (\<forall>T1 U1. equal(T1::'a,U1) --> equal(domain_of(T1),domain_of(U1))) &
  1470   (\<forall>V1 W1. equal(V1::'a,W1) --> equal(first(V1),first(W1))) &
  1471   (\<forall>X1 Y1. equal(X1::'a,Y1) --> equal(flip(X1),flip(Y1))) &
  1472   (\<forall>Z1 A2 B2. equal(Z1::'a,A2) --> equal(image'(Z1::'a,B2),image'(A2::'a,B2))) &
  1473   (\<forall>C2 E2 D2. equal(C2::'a,D2) --> equal(image'(E2::'a,C2),image'(E2::'a,D2))) &
  1474   (\<forall>F2 G2 H2. equal(F2::'a,G2) --> equal(intersection(F2::'a,H2),intersection(G2::'a,H2))) &
  1475   (\<forall>I2 K2 J2. equal(I2::'a,J2) --> equal(intersection(K2::'a,I2),intersection(K2::'a,J2))) &
  1476   (\<forall>L2 M2. equal(L2::'a,M2) --> equal(INVERSE(L2),INVERSE(M2))) &
  1477   (\<forall>N2 O2 P2 Q2. equal(N2::'a,O2) --> equal(not_homomorphism1(N2::'a,P2,Q2),not_homomorphism1(O2::'a,P2,Q2))) &
  1478   (\<forall>R2 T2 S2 U2. equal(R2::'a,S2) --> equal(not_homomorphism1(T2::'a,R2,U2),not_homomorphism1(T2::'a,S2,U2))) &
  1479   (\<forall>V2 X2 Y2 W2. equal(V2::'a,W2) --> equal(not_homomorphism1(X2::'a,Y2,V2),not_homomorphism1(X2::'a,Y2,W2))) &
  1480   (\<forall>Z2 A3 B3 C3. equal(Z2::'a,A3) --> equal(not_homomorphism2(Z2::'a,B3,C3),not_homomorphism2(A3::'a,B3,C3))) &
  1481   (\<forall>D3 F3 E3 G3. equal(D3::'a,E3) --> equal(not_homomorphism2(F3::'a,D3,G3),not_homomorphism2(F3::'a,E3,G3))) &
  1482   (\<forall>H3 J3 K3 I3. equal(H3::'a,I3) --> equal(not_homomorphism2(J3::'a,K3,H3),not_homomorphism2(J3::'a,K3,I3))) &
  1483   (\<forall>L3 M3 N3. equal(L3::'a,M3) --> equal(not_subclass_element(L3::'a,N3),not_subclass_element(M3::'a,N3))) &
  1484   (\<forall>O3 Q3 P3. equal(O3::'a,P3) --> equal(not_subclass_element(Q3::'a,O3),not_subclass_element(Q3::'a,P3))) &
  1485   (\<forall>R3 S3 T3. equal(R3::'a,S3) --> equal(ordered_pair(R3::'a,T3),ordered_pair(S3::'a,T3))) &
  1486   (\<forall>U3 W3 V3. equal(U3::'a,V3) --> equal(ordered_pair(W3::'a,U3),ordered_pair(W3::'a,V3))) &
  1487   (\<forall>X3 Y3. equal(X3::'a,Y3) --> equal(powerClass(X3),powerClass(Y3))) &
  1488   (\<forall>Z3 A4 B4 C4. equal(Z3::'a,A4) --> equal(rng(Z3::'a,B4,C4),rng(A4::'a,B4,C4))) &
  1489   (\<forall>D4 F4 E4 G4. equal(D4::'a,E4) --> equal(rng(F4::'a,D4,G4),rng(F4::'a,E4,G4))) &
  1490   (\<forall>H4 J4 K4 I4. equal(H4::'a,I4) --> equal(rng(J4::'a,K4,H4),rng(J4::'a,K4,I4))) &
  1491   (\<forall>L4 M4. equal(L4::'a,M4) --> equal(range_of(L4),range_of(M4))) &
  1492   (\<forall>N4 O4. equal(N4::'a,O4) --> equal(regular(N4),regular(O4))) &
  1493   (\<forall>P4 Q4 R4 S4. equal(P4::'a,Q4) --> equal(restrct(P4::'a,R4,S4),restrct(Q4::'a,R4,S4))) &
  1494   (\<forall>T4 V4 U4 W4. equal(T4::'a,U4) --> equal(restrct(V4::'a,T4,W4),restrct(V4::'a,U4,W4))) &
  1495   (\<forall>X4 Z4 A5 Y4. equal(X4::'a,Y4) --> equal(restrct(Z4::'a,A5,X4),restrct(Z4::'a,A5,Y4))) &
  1496   (\<forall>B5 C5. equal(B5::'a,C5) --> equal(rot(B5),rot(C5))) &
  1497   (\<forall>D5 E5. equal(D5::'a,E5) --> equal(second(D5),second(E5))) &
  1498   (\<forall>F5 G5. equal(F5::'a,G5) --> equal(singleton(F5),singleton(G5))) &
  1499   (\<forall>H5 I5. equal(H5::'a,I5) --> equal(successor(H5),successor(I5))) &
  1500   (\<forall>J5 K5. equal(J5::'a,K5) --> equal(sum_class(J5),sum_class(K5))) &
  1501   (\<forall>L5 M5 N5. equal(L5::'a,M5) --> equal(union(L5::'a,N5),union(M5::'a,N5))) &
  1502   (\<forall>O5 Q5 P5. equal(O5::'a,P5) --> equal(union(Q5::'a,O5),union(Q5::'a,P5))) &
  1503   (\<forall>R5 S5 T5. equal(R5::'a,S5) --> equal(unordered_pair(R5::'a,T5),unordered_pair(S5::'a,T5))) &
  1504   (\<forall>U5 W5 V5. equal(U5::'a,V5) --> equal(unordered_pair(W5::'a,U5),unordered_pair(W5::'a,V5))) &
  1505   (\<forall>X5 Y5 Z5 A6. equal(X5::'a,Y5) & compatible(X5::'a,Z5,A6) --> compatible(Y5::'a,Z5,A6)) &
  1506   (\<forall>B6 D6 C6 E6. equal(B6::'a,C6) & compatible(D6::'a,B6,E6) --> compatible(D6::'a,C6,E6)) &
  1507   (\<forall>F6 H6 I6 G6. equal(F6::'a,G6) & compatible(H6::'a,I6,F6) --> compatible(H6::'a,I6,G6)) &
  1508   (\<forall>J6 K6. equal(J6::'a,K6) & function(J6) --> function(K6)) &
  1509   (\<forall>L6 M6 N6 O6. equal(L6::'a,M6) & homomorphism(L6::'a,N6,O6) --> homomorphism(M6::'a,N6,O6)) &
  1510   (\<forall>P6 R6 Q6 S6. equal(P6::'a,Q6) & homomorphism(R6::'a,P6,S6) --> homomorphism(R6::'a,Q6,S6)) &
  1511   (\<forall>T6 V6 W6 U6. equal(T6::'a,U6) & homomorphism(V6::'a,W6,T6) --> homomorphism(V6::'a,W6,U6)) &
  1512   (\<forall>X6 Y6. equal(X6::'a,Y6) & inductive(X6) --> inductive(Y6)) &
  1513   (\<forall>Z6 A7 B7. equal(Z6::'a,A7) & member(Z6::'a,B7) --> member(A7::'a,B7)) &
  1514   (\<forall>C7 E7 D7. equal(C7::'a,D7) & member(E7::'a,C7) --> member(E7::'a,D7)) &
  1515   (\<forall>F7 G7. equal(F7::'a,G7) & one_to_one(F7) --> one_to_one(G7)) &
  1516   (\<forall>H7 I7. equal(H7::'a,I7) & operation(H7) --> operation(I7)) &
  1517   (\<forall>J7 K7. equal(J7::'a,K7) & single_valued_class(J7) --> single_valued_class(K7)) &
  1518   (\<forall>L7 M7 N7. equal(L7::'a,M7) & subclass(L7::'a,N7) --> subclass(M7::'a,N7)) &
  1519   (\<forall>O7 Q7 P7. equal(O7::'a,P7) & subclass(Q7::'a,O7) --> subclass(Q7::'a,P7))"
  1520 
  1521 abbreviation "SET004_1_ax range_of function maps apply
  1522     application_function singleton_relation element_relation complement
  1523     intersection single_valued3 singleton image' domain single_valued2
  1524     second single_valued1 identity_relation INVERSE not_subclass_element
  1525     first domain_of domain_relation composition_function compos equal
  1526     ordered_pair member universal_class cross_product compose_class
  1527     subclass \<equiv>
  1528   (\<forall>X. subclass(compose_class(X),cross_product(universal_class::'a,universal_class))) &
  1529   (\<forall>X Y Z. member(ordered_pair(Y::'a,Z),compose_class(X)) --> equal(compos(X::'a,Y),Z)) &
  1530   (\<forall>Y Z X. member(ordered_pair(Y::'a,Z),cross_product(universal_class::'a,universal_class)) & equal(compos(X::'a,Y),Z) --> member(ordered_pair(Y::'a,Z),compose_class(X))) &
  1531   (subclass(composition_function::'a,cross_product(universal_class::'a,cross_product(universal_class::'a,universal_class)))) &
  1532   (\<forall>X Y Z. member(ordered_pair(X::'a,ordered_pair(Y::'a,Z)),composition_function) --> equal(compos(X::'a,Y),Z)) &
  1533   (\<forall>X Y. member(ordered_pair(X::'a,Y),cross_product(universal_class::'a,universal_class)) --> member(ordered_pair(X::'a,ordered_pair(Y::'a,compos(X::'a,Y))),composition_function)) &
  1534   (subclass(domain_relation::'a,cross_product(universal_class::'a,universal_class))) &
  1535   (\<forall>X Y. member(ordered_pair(X::'a,Y),domain_relation) --> equal(domain_of(X),Y)) &
  1536   (\<forall>X. member(X::'a,universal_class) --> member(ordered_pair(X::'a,domain_of(X)),domain_relation)) &
  1537   (\<forall>X. equal(first(not_subclass_element(compos(X::'a,INVERSE(X)),identity_relation)),single_valued1(X))) &
  1538   (\<forall>X. equal(second(not_subclass_element(compos(X::'a,INVERSE(X)),identity_relation)),single_valued2(X))) &
  1539   (\<forall>X. equal(domain(X::'a,image'(INVERSE(X),singleton(single_valued1(X))),single_valued2(X)),single_valued3(X))) &
  1540   (equal(intersection(complement(compos(element_relation::'a,complement(identity_relation))),element_relation),singleton_relation)) &
  1541   (subclass(application_function::'a,cross_product(universal_class::'a,cross_product(universal_class::'a,universal_class)))) &
  1542   (\<forall>Z Y X. member(ordered_pair(X::'a,ordered_pair(Y::'a,Z)),application_function) --> member(Y::'a,domain_of(X))) &
  1543   (\<forall>X Y Z. member(ordered_pair(X::'a,ordered_pair(Y::'a,Z)),application_function) --> equal(apply(X::'a,Y),Z)) &
  1544   (\<forall>Z X Y. member(ordered_pair(X::'a,ordered_pair(Y::'a,Z)),cross_product(universal_class::'a,cross_product(universal_class::'a,universal_class))) & member(Y::'a,domain_of(X)) --> member(ordered_pair(X::'a,ordered_pair(Y::'a,apply(X::'a,Y))),application_function)) &
  1545   (\<forall>X Y Xf. maps(Xf::'a,X,Y) --> function(Xf)) &
  1546   (\<forall>Y Xf X. maps(Xf::'a,X,Y) --> equal(domain_of(Xf),X)) &
  1547   (\<forall>X Xf Y. maps(Xf::'a,X,Y) --> subclass(range_of(Xf),Y)) &
  1548   (\<forall>Xf Y. function(Xf) & subclass(range_of(Xf),Y) --> maps(Xf::'a,domain_of(Xf),Y))"
  1549 
  1550 abbreviation "SET004_1_eq maps single_valued3 single_valued2 single_valued1 compose_class equal \<equiv>
  1551   (\<forall>L M. equal(L::'a,M) --> equal(compose_class(L),compose_class(M))) &
  1552   (\<forall>N2 O2. equal(N2::'a,O2) --> equal(single_valued1(N2),single_valued1(O2))) &
  1553   (\<forall>P2 Q2. equal(P2::'a,Q2) --> equal(single_valued2(P2),single_valued2(Q2))) &
  1554   (\<forall>R2 S2. equal(R2::'a,S2) --> equal(single_valued3(R2),single_valued3(S2))) &
  1555   (\<forall>X2 Y2 Z2 A3. equal(X2::'a,Y2) & maps(X2::'a,Z2,A3) --> maps(Y2::'a,Z2,A3)) &
  1556   (\<forall>B3 D3 C3 E3. equal(B3::'a,C3) & maps(D3::'a,B3,E3) --> maps(D3::'a,C3,E3)) &
  1557   (\<forall>F3 H3 I3 G3. equal(F3::'a,G3) & maps(H3::'a,I3,F3) --> maps(H3::'a,I3,G3))"
  1558 
  1559 abbreviation "NUM004_0_ax integer_of omega ordinal_multiply
  1560     add_relation ordinal_add recursion apply range_of union_of_range_map
  1561     function recursion_equation_functions rest_relation rest_of
  1562     limit_ordinals kind_1_ordinals successor_relation image'
  1563     universal_class sum_class element_relation ordinal_numbers section
  1564     not_well_ordering ordered_pair least member well_ordering singleton
  1565     domain_of segment null_class intersection asymmetric compos transitive
  1566     cross_product connected identity_relation complement restrct subclass
  1567     irreflexive symmetrization_of INVERSE union equal \<equiv>
  1568   (\<forall>X. equal(union(X::'a,INVERSE(X)),symmetrization_of(X))) &
  1569   (\<forall>X Y. irreflexive(X::'a,Y) --> subclass(restrct(X::'a,Y,Y),complement(identity_relation))) &
  1570   (\<forall>X Y. subclass(restrct(X::'a,Y,Y),complement(identity_relation)) --> irreflexive(X::'a,Y)) &
  1571   (\<forall>Y X. connected(X::'a,Y) --> subclass(cross_product(Y::'a,Y),union(identity_relation::'a,symmetrization_of(X)))) &
  1572   (\<forall>X Y. subclass(cross_product(Y::'a,Y),union(identity_relation::'a,symmetrization_of(X))) --> connected(X::'a,Y)) &
  1573   (\<forall>Xr Y. transitive(Xr::'a,Y) --> subclass(compos(restrct(Xr::'a,Y,Y),restrct(Xr::'a,Y,Y)),restrct(Xr::'a,Y,Y))) &
  1574   (\<forall>Xr Y. subclass(compos(restrct(Xr::'a,Y,Y),restrct(Xr::'a,Y,Y)),restrct(Xr::'a,Y,Y)) --> transitive(Xr::'a,Y)) &
  1575   (\<forall>Xr Y. asymmetric(Xr::'a,Y) --> equal(restrct(intersection(Xr::'a,INVERSE(Xr)),Y,Y),null_class)) &
  1576   (\<forall>Xr Y. equal(restrct(intersection(Xr::'a,INVERSE(Xr)),Y,Y),null_class) --> asymmetric(Xr::'a,Y)) &
  1577   (\<forall>Xr Y Z. equal(segment(Xr::'a,Y,Z),domain_of(restrct(Xr::'a,Y,singleton(Z))))) &
  1578   (\<forall>X Y. well_ordering(X::'a,Y) --> connected(X::'a,Y)) &
  1579   (\<forall>Y Xr U. well_ordering(Xr::'a,Y) & subclass(U::'a,Y) --> equal(U::'a,null_class) | member(least(Xr::'a,U),U)) &
  1580   (\<forall>Y V Xr U. well_ordering(Xr::'a,Y) & subclass(U::'a,Y) & member(V::'a,U) --> member(least(Xr::'a,U),U)) &
  1581   (\<forall>Y Xr U. well_ordering(Xr::'a,Y) & subclass(U::'a,Y) --> equal(segment(Xr::'a,U,least(Xr::'a,U)),null_class)) &
  1582   (\<forall>Y V U Xr. ~(well_ordering(Xr::'a,Y) & subclass(U::'a,Y) & member(V::'a,U) & member(ordered_pair(V::'a,least(Xr::'a,U)),Xr))) &
  1583   (\<forall>Xr Y. connected(Xr::'a,Y) & equal(not_well_ordering(Xr::'a,Y),null_class) --> well_ordering(Xr::'a,Y)) &
  1584   (\<forall>Xr Y. connected(Xr::'a,Y) --> subclass(not_well_ordering(Xr::'a,Y),Y) | well_ordering(Xr::'a,Y)) &
  1585   (\<forall>V Xr Y. member(V::'a,not_well_ordering(Xr::'a,Y)) & equal(segment(Xr::'a,not_well_ordering(Xr::'a,Y),V),null_class) & connected(Xr::'a,Y) --> well_ordering(Xr::'a,Y)) &
  1586   (\<forall>Xr Y Z. section(Xr::'a,Y,Z) --> subclass(Y::'a,Z)) &
  1587   (\<forall>Xr Z Y. section(Xr::'a,Y,Z) --> subclass(domain_of(restrct(Xr::'a,Z,Y)),Y)) &
  1588   (\<forall>Xr Y Z. subclass(Y::'a,Z) & subclass(domain_of(restrct(Xr::'a,Z,Y)),Y) --> section(Xr::'a,Y,Z)) &
  1589   (\<forall>X. member(X::'a,ordinal_numbers) --> well_ordering(element_relation::'a,X)) &
  1590   (\<forall>X. member(X::'a,ordinal_numbers) --> subclass(sum_class(X),X)) &
  1591   (\<forall>X. well_ordering(element_relation::'a,X) & subclass(sum_class(X),X) & member(X::'a,universal_class) --> member(X::'a,ordinal_numbers)) &
  1592   (\<forall>X. well_ordering(element_relation::'a,X) & subclass(sum_class(X),X) --> member(X::'a,ordinal_numbers) | equal(X::'a,ordinal_numbers)) &
  1593   (equal(union(singleton(null_class),image'(successor_relation::'a,ordinal_numbers)),kind_1_ordinals)) &
  1594   (equal(intersection(complement(kind_1_ordinals),ordinal_numbers),limit_ordinals)) &
  1595   (\<forall>X. subclass(rest_of(X),cross_product(universal_class::'a,universal_class))) &
  1596   (\<forall>V U X. member(ordered_pair(U::'a,V),rest_of(X)) --> member(U::'a,domain_of(X))) &
  1597   (\<forall>X U V. member(ordered_pair(U::'a,V),rest_of(X)) --> equal(restrct(X::'a,U,universal_class),V)) &
  1598   (\<forall>U V X. member(U::'a,domain_of(X)) & equal(restrct(X::'a,U,universal_class),V) --> member(ordered_pair(U::'a,V),rest_of(X))) &
  1599   (subclass(rest_relation::'a,cross_product(universal_class::'a,universal_class))) &
  1600   (\<forall>X Y. member(ordered_pair(X::'a,Y),rest_relation) --> equal(rest_of(X),Y)) &
  1601   (\<forall>X. member(X::'a,universal_class) --> member(ordered_pair(X::'a,rest_of(X)),rest_relation)) &
  1602   (\<forall>X Z. member(X::'a,recursion_equation_functions(Z)) --> function(Z)) &
  1603   (\<forall>Z X. member(X::'a,recursion_equation_functions(Z)) --> function(X)) &
  1604   (\<forall>Z X. member(X::'a,recursion_equation_functions(Z)) --> member(domain_of(X),ordinal_numbers)) &
  1605   (\<forall>Z X. member(X::'a,recursion_equation_functions(Z)) --> equal(compos(Z::'a,rest_of(X)),X)) &
  1606   (\<forall>X Z. function(Z) & function(X) & member(domain_of(X),ordinal_numbers) & equal(compos(Z::'a,rest_of(X)),X) --> member(X::'a,recursion_equation_functions(Z))) &
  1607   (subclass(union_of_range_map::'a,cross_product(universal_class::'a,universal_class))) &
  1608   (\<forall>X Y. member(ordered_pair(X::'a,Y),union_of_range_map) --> equal(sum_class(range_of(X)),Y)) &
  1609   (\<forall>X Y. member(ordered_pair(X::'a,Y),cross_product(universal_class::'a,universal_class)) & equal(sum_class(range_of(X)),Y) --> member(ordered_pair(X::'a,Y),union_of_range_map)) &
  1610   (\<forall>X Y. equal(apply(recursion(X::'a,successor_relation,union_of_range_map),Y),ordinal_add(X::'a,Y))) &
  1611   (\<forall>X Y. equal(recursion(null_class::'a,apply(add_relation::'a,X),union_of_range_map),ordinal_multiply(X::'a,Y))) &
  1612   (\<forall>X. member(X::'a,omega) --> equal(integer_of(X),X)) &
  1613   (\<forall>X. member(X::'a,omega) | equal(integer_of(X),null_class))"
  1614 
  1615 abbreviation "NUM004_0_eq well_ordering transitive section irreflexive
  1616     connected asymmetric symmetrization_of segment rest_of
  1617     recursion_equation_functions recursion ordinal_multiply ordinal_add
  1618     not_well_ordering least integer_of equal \<equiv>
  1619   (\<forall>D E. equal(D::'a,E) --> equal(integer_of(D),integer_of(E))) &
  1620   (\<forall>F' G H. equal(F'::'a,G) --> equal(least(F'::'a,H),least(G::'a,H))) &
  1621   (\<forall>I' K' J. equal(I'::'a,J) --> equal(least(K'::'a,I'),least(K'::'a,J))) &
  1622   (\<forall>L M N. equal(L::'a,M) --> equal(not_well_ordering(L::'a,N),not_well_ordering(M::'a,N))) &
  1623   (\<forall>O' Q P. equal(O'::'a,P) --> equal(not_well_ordering(Q::'a,O'),not_well_ordering(Q::'a,P))) &
  1624   (\<forall>R S' T'. equal(R::'a,S') --> equal(ordinal_add(R::'a,T'),ordinal_add(S'::'a,T'))) &
  1625   (\<forall>U W V. equal(U::'a,V) --> equal(ordinal_add(W::'a,U),ordinal_add(W::'a,V))) &
  1626   (\<forall>X Y Z. equal(X::'a,Y) --> equal(ordinal_multiply(X::'a,Z),ordinal_multiply(Y::'a,Z))) &
  1627   (\<forall>A1 C1 B1. equal(A1::'a,B1) --> equal(ordinal_multiply(C1::'a,A1),ordinal_multiply(C1::'a,B1))) &
  1628   (\<forall>F1 G1 H1 I1. equal(F1::'a,G1) --> equal(recursion(F1::'a,H1,I1),recursion(G1::'a,H1,I1))) &
  1629   (\<forall>J1 L1 K1 M1. equal(J1::'a,K1) --> equal(recursion(L1::'a,J1,M1),recursion(L1::'a,K1,M1))) &
  1630   (\<forall>N1 P1 Q1 O1. equal(N1::'a,O1) --> equal(recursion(P1::'a,Q1,N1),recursion(P1::'a,Q1,O1))) &
  1631   (\<forall>R1 S1. equal(R1::'a,S1) --> equal(recursion_equation_functions(R1),recursion_equation_functions(S1))) &
  1632   (\<forall>T1 U1. equal(T1::'a,U1) --> equal(rest_of(T1),rest_of(U1))) &
  1633   (\<forall>V1 W1 X1 Y1. equal(V1::'a,W1) --> equal(segment(V1::'a,X1,Y1),segment(W1::'a,X1,Y1))) &
  1634   (\<forall>Z1 B2 A2 C2. equal(Z1::'a,A2) --> equal(segment(B2::'a,Z1,C2),segment(B2::'a,A2,C2))) &
  1635   (\<forall>D2 F2 G2 E2. equal(D2::'a,E2) --> equal(segment(F2::'a,G2,D2),segment(F2::'a,G2,E2))) &
  1636   (\<forall>H2 I2. equal(H2::'a,I2) --> equal(symmetrization_of(H2),symmetrization_of(I2))) &
  1637   (\<forall>J2 K2 L2. equal(J2::'a,K2) & asymmetric(J2::'a,L2) --> asymmetric(K2::'a,L2)) &
  1638   (\<forall>M2 O2 N2. equal(M2::'a,N2) & asymmetric(O2::'a,M2) --> asymmetric(O2::'a,N2)) &
  1639   (\<forall>P2 Q2 R2. equal(P2::'a,Q2) & connected(P2::'a,R2) --> connected(Q2::'a,R2)) &
  1640   (\<forall>S2 U2 T2. equal(S2::'a,T2) & connected(U2::'a,S2) --> connected(U2::'a,T2)) &
  1641   (\<forall>V2 W2 X2. equal(V2::'a,W2) & irreflexive(V2::'a,X2) --> irreflexive(W2::'a,X2)) &
  1642   (\<forall>Y2 A3 Z2. equal(Y2::'a,Z2) & irreflexive(A3::'a,Y2) --> irreflexive(A3::'a,Z2)) &
  1643   (\<forall>B3 C3 D3 E3. equal(B3::'a,C3) & section(B3::'a,D3,E3) --> section(C3::'a,D3,E3)) &
  1644   (\<forall>F3 H3 G3 I3. equal(F3::'a,G3) & section(H3::'a,F3,I3) --> section(H3::'a,G3,I3)) &
  1645   (\<forall>J3 L3 M3 K3. equal(J3::'a,K3) & section(L3::'a,M3,J3) --> section(L3::'a,M3,K3)) &
  1646   (\<forall>N3 O3 P3. equal(N3::'a,O3) & transitive(N3::'a,P3) --> transitive(O3::'a,P3)) &
  1647   (\<forall>Q3 S3 R3. equal(Q3::'a,R3) & transitive(S3::'a,Q3) --> transitive(S3::'a,R3)) &
  1648   (\<forall>T3 U3 V3. equal(T3::'a,U3) & well_ordering(T3::'a,V3) --> well_ordering(U3::'a,V3)) &
  1649   (\<forall>W3 Y3 X3. equal(W3::'a,X3) & well_ordering(Y3::'a,W3) --> well_ordering(Y3::'a,X3))"
  1650 
  1651 (*1345 inferences so far.  Searching to depth 7.  23.3 secs.  BIG*)
  1652 lemma NUM180_1:
  1653   "EQU001_0_ax equal &
  1654   SET004_0_ax not_homomorphism2 not_homomorphism1
  1655     homomorphism compatible operation cantor diagonalise subset_relation
  1656     one_to_one choice apply regular function identity_relation
  1657     single_valued_class compos powerClass sum_class omega inductive
  1658     successor_relation successor image' rng domain range_of INVERSE flip
  1659     rot domain_of null_class restrct difference union complement
  1660     intersection element_relation second first cross_product ordered_pair
  1661     singleton unordered_pair equal universal_class not_subclass_element
  1662     member subclass &
  1663   SET004_0_eq subclass single_valued_class operation
  1664     one_to_one member inductive homomorphism function compatible
  1665     unordered_pair union sum_class successor singleton second rot restrct
  1666     regular range_of rng powerClass ordered_pair not_subclass_element
  1667     not_homomorphism2 not_homomorphism1 INVERSE intersection image' flip
  1668     first domain_of domain difference diagonalise cross_product compos
  1669     complement cantor apply equal &
  1670   SET004_1_ax range_of function maps apply
  1671     application_function singleton_relation element_relation complement
  1672     intersection single_valued3 singleton image' domain single_valued2
  1673     second single_valued1 identity_relation INVERSE not_subclass_element
  1674     first domain_of domain_relation composition_function compos equal
  1675     ordered_pair member universal_class cross_product compose_class
  1676     subclass &
  1677   SET004_1_eq maps single_valued3 single_valued2 single_valued1 compose_class equal &
  1678   NUM004_0_ax integer_of omega ordinal_multiply
  1679     add_relation ordinal_add recursion apply range_of union_of_range_map
  1680     function recursion_equation_functions rest_relation rest_of
  1681     limit_ordinals kind_1_ordinals successor_relation image'
  1682     universal_class sum_class element_relation ordinal_numbers section
  1683     not_well_ordering ordered_pair least member well_ordering singleton
  1684     domain_of segment null_class intersection asymmetric compos transitive
  1685     cross_product connected identity_relation complement restrct subclass
  1686     irreflexive symmetrization_of INVERSE union equal &
  1687   NUM004_0_eq well_ordering transitive section irreflexive
  1688     connected asymmetric symmetrization_of segment rest_of
  1689     recursion_equation_functions recursion ordinal_multiply ordinal_add
  1690     not_well_ordering least integer_of equal &
  1691   (~subclass(limit_ordinals::'a,ordinal_numbers)) --> False"
  1692   by meson
  1693 
  1694 
  1695 (*0 inferences so far.  Searching to depth 0.  16.8 secs.  BIG*)
  1696 lemma NUM228_1:
  1697   "EQU001_0_ax equal &
  1698   SET004_0_ax not_homomorphism2 not_homomorphism1
  1699     homomorphism compatible operation cantor diagonalise subset_relation
  1700     one_to_one choice apply regular function identity_relation
  1701     single_valued_class compos powerClass sum_class omega inductive
  1702     successor_relation successor image' rng domain range_of INVERSE flip
  1703     rot domain_of null_class restrct difference union complement
  1704     intersection element_relation second first cross_product ordered_pair
  1705     singleton unordered_pair equal universal_class not_subclass_element
  1706     member subclass &
  1707   SET004_0_eq subclass single_valued_class operation
  1708     one_to_one member inductive homomorphism function compatible
  1709     unordered_pair union sum_class successor singleton second rot restrct
  1710     regular range_of rng powerClass ordered_pair not_subclass_element
  1711     not_homomorphism2 not_homomorphism1 INVERSE intersection image' flip
  1712     first domain_of domain difference diagonalise cross_product compos
  1713     complement cantor apply equal &
  1714   SET004_1_ax range_of function maps apply
  1715     application_function singleton_relation element_relation complement
  1716     intersection single_valued3 singleton image' domain single_valued2
  1717     second single_valued1 identity_relation INVERSE not_subclass_element
  1718     first domain_of domain_relation composition_function compos equal
  1719     ordered_pair member universal_class cross_product compose_class
  1720     subclass &
  1721   SET004_1_eq maps single_valued3 single_valued2 single_valued1 compose_class equal &
  1722   NUM004_0_ax integer_of omega ordinal_multiply
  1723     add_relation ordinal_add recursion apply range_of union_of_range_map
  1724     function recursion_equation_functions rest_relation rest_of
  1725     limit_ordinals kind_1_ordinals successor_relation image'
  1726     universal_class sum_class element_relation ordinal_numbers section
  1727     not_well_ordering ordered_pair least member well_ordering singleton
  1728     domain_of segment null_class intersection asymmetric compos transitive
  1729     cross_product connected identity_relation complement restrct subclass
  1730     irreflexive symmetrization_of INVERSE union equal &
  1731   NUM004_0_eq well_ordering transitive section irreflexive
  1732     connected asymmetric symmetrization_of segment rest_of
  1733     recursion_equation_functions recursion ordinal_multiply ordinal_add
  1734     not_well_ordering least integer_of equal &
  1735   (~function(z)) &
  1736     (~equal(recursion_equation_functions(z),null_class)) --> False"
  1737   by meson
  1738 
  1739 
  1740 (*4868 inferences so far.  Searching to depth 12.  4.3 secs*)
  1741 lemma PLA002_1:
  1742   "(\<forall>Situation1 Situation2. warm(Situation1) | cold(Situation2)) &
  1743   (\<forall>Situation. at(a::'a,Situation) --> at(b::'a,walk(b::'a,Situation))) &
  1744   (\<forall>Situation. at(a::'a,Situation) --> at(b::'a,drive(b::'a,Situation))) &
  1745   (\<forall>Situation. at(b::'a,Situation) --> at(a::'a,walk(a::'a,Situation))) &
  1746   (\<forall>Situation. at(b::'a,Situation) --> at(a::'a,drive(a::'a,Situation))) &
  1747   (\<forall>Situation. cold(Situation) & at(b::'a,Situation) --> at(c::'a,skate(c::'a,Situation))) &
  1748   (\<forall>Situation. cold(Situation) & at(c::'a,Situation) --> at(b::'a,skate(b::'a,Situation))) &
  1749   (\<forall>Situation. warm(Situation) & at(b::'a,Situation) --> at(d::'a,climb(d::'a,Situation))) &
  1750   (\<forall>Situation. warm(Situation) & at(d::'a,Situation) --> at(b::'a,climb(b::'a,Situation))) &
  1751   (\<forall>Situation. at(c::'a,Situation) --> at(d::'a,go(d::'a,Situation))) &
  1752   (\<forall>Situation. at(d::'a,Situation) --> at(c::'a,go(c::'a,Situation))) &
  1753   (\<forall>Situation. at(c::'a,Situation) --> at(e::'a,go(e::'a,Situation))) &
  1754   (\<forall>Situation. at(e::'a,Situation) --> at(c::'a,go(c::'a,Situation))) &
  1755   (\<forall>Situation. at(d::'a,Situation) --> at(f::'a,go(f::'a,Situation))) &
  1756   (\<forall>Situation. at(f::'a,Situation) --> at(d::'a,go(d::'a,Situation))) &
  1757   (at(f::'a,s0)) &
  1758   (\<forall>S'. ~at(a::'a,S')) --> False"
  1759   by meson
  1760 
  1761 abbreviation "PLA001_0_ax putdown on pickup do holding table differ clear EMPTY and' holds \<equiv>
  1762   (\<forall>X Y State. holds(X::'a,State) & holds(Y::'a,State) --> holds(and'(X::'a,Y),State)) &
  1763   (\<forall>State X. holds(EMPTY::'a,State) & holds(clear(X),State) & differ(X::'a,table) --> holds(holding(X),do(pickup(X),State))) &
  1764   (\<forall>Y X State. holds(on(X::'a,Y),State) & holds(clear(X),State) & holds(EMPTY::'a,State) --> holds(clear(Y),do(pickup(X),State))) &
  1765   (\<forall>Y State X Z. holds(on(X::'a,Y),State) & differ(X::'a,Z) --> holds(on(X::'a,Y),do(pickup(Z),State))) &
  1766   (\<forall>State X Z. holds(clear(X),State) & differ(X::'a,Z) --> holds(clear(X),do(pickup(Z),State))) &
  1767   (\<forall>X Y State. holds(holding(X),State) & holds(clear(Y),State) --> holds(EMPTY::'a,do(putdown(X::'a,Y),State))) &
  1768   (\<forall>X Y State. holds(holding(X),State) & holds(clear(Y),State) --> holds(on(X::'a,Y),do(putdown(X::'a,Y),State))) &
  1769   (\<forall>X Y State. holds(holding(X),State) & holds(clear(Y),State) --> holds(clear(X),do(putdown(X::'a,Y),State))) &
  1770   (\<forall>Z W X Y State. holds(on(X::'a,Y),State) --> holds(on(X::'a,Y),do(putdown(Z::'a,W),State))) &
  1771   (\<forall>X State Z Y. holds(clear(Z),State) & differ(Z::'a,Y) --> holds(clear(Z),do(putdown(X::'a,Y),State)))"
  1772 
  1773 abbreviation "PLA001_1_ax EMPTY clear s0 on holds table d c b a differ \<equiv>
  1774   (\<forall>Y X. differ(Y::'a,X) --> differ(X::'a,Y)) &
  1775   (differ(a::'a,b)) &
  1776   (differ(a::'a,c)) &
  1777   (differ(a::'a,d)) &
  1778   (differ(a::'a,table)) &
  1779   (differ(b::'a,c)) &
  1780   (differ(b::'a,d)) &
  1781   (differ(b::'a,table)) &
  1782   (differ(c::'a,d)) &
  1783   (differ(c::'a,table)) &
  1784   (differ(d::'a,table)) &
  1785   (holds(on(a::'a,table),s0)) &
  1786   (holds(on(b::'a,table),s0)) &
  1787   (holds(on(c::'a,d),s0)) &
  1788   (holds(on(d::'a,table),s0)) &
  1789   (holds(clear(a),s0)) &
  1790   (holds(clear(b),s0)) &
  1791   (holds(clear(c),s0)) &
  1792   (holds(EMPTY::'a,s0)) &
  1793   (\<forall>State. holds(clear(table),State))"
  1794 
  1795 (*190 inferences so far.  Searching to depth 10.  0.6 secs*)
  1796 lemma PLA006_1:
  1797   "PLA001_0_ax putdown on pickup do holding table differ clear EMPTY and' holds &
  1798   PLA001_1_ax EMPTY clear s0 on holds table d c b a differ &
  1799   (\<forall>State. ~holds(on(c::'a,table),State)) --> False"
  1800   by meson
  1801 
  1802 (*190 inferences so far.  Searching to depth 10.  0.5 secs*)
  1803 lemma PLA017_1:
  1804   "PLA001_0_ax putdown on pickup do holding table differ clear EMPTY and' holds &
  1805   PLA001_1_ax EMPTY clear s0 on holds table d c b a differ &
  1806   (\<forall>State. ~holds(on(a::'a,c),State)) --> False"
  1807   by meson
  1808 
  1809 (*13732 inferences so far.  Searching to depth 16.  9.8 secs*)
  1810 lemma PLA022_1:
  1811   "PLA001_0_ax putdown on pickup do holding table differ clear EMPTY and' holds &
  1812   PLA001_1_ax EMPTY clear s0 on holds table d c b a differ &
  1813   (\<forall>State. ~holds(and'(on(c::'a,d),on(a::'a,c)),State)) --> False"
  1814   by meson
  1815 
  1816 (*217 inferences so far.  Searching to depth 13.  0.7 secs*)
  1817 lemma PLA022_2:
  1818   "PLA001_0_ax putdown on pickup do holding table differ clear EMPTY and' holds &
  1819   PLA001_1_ax EMPTY clear s0 on holds table d c b a differ &
  1820   (\<forall>State. ~holds(and'(on(a::'a,c),on(c::'a,d)),State)) --> False"
  1821   by meson
  1822 
  1823 (*948 inferences so far.  Searching to depth 18.  1.1 secs*)
  1824 lemma PRV001_1:
  1825  "(\<forall>X Y Z. q1(X::'a,Y,Z) & mless_or_equal(X::'a,Y) --> q2(X::'a,Y,Z)) &
  1826   (\<forall>X Y Z. q1(X::'a,Y,Z) --> mless_or_equal(X::'a,Y) | q3(X::'a,Y,Z)) &
  1827   (\<forall>Z X Y. q2(X::'a,Y,Z) --> q4(X::'a,Y,Y)) &
  1828   (\<forall>Z Y X. q3(X::'a,Y,Z) --> q4(X::'a,Y,X)) &
  1829   (\<forall>X. mless_or_equal(X::'a,X)) &
  1830   (\<forall>X Y. mless_or_equal(X::'a,Y) & mless_or_equal(Y::'a,X) --> equal(X::'a,Y)) &
  1831   (\<forall>Y X Z. mless_or_equal(X::'a,Y) & mless_or_equal(Y::'a,Z) --> mless_or_equal(X::'a,Z)) &
  1832   (\<forall>Y X. mless_or_equal(X::'a,Y) | mless_or_equal(Y::'a,X)) &
  1833   (\<forall>X Y. equal(X::'a,Y) --> mless_or_equal(X::'a,Y)) &
  1834   (\<forall>X Y Z. equal(X::'a,Y) & mless_or_equal(X::'a,Z) --> mless_or_equal(Y::'a,Z)) &
  1835   (\<forall>X Z Y. equal(X::'a,Y) & mless_or_equal(Z::'a,X) --> mless_or_equal(Z::'a,Y)) &
  1836   (q1(a::'a,b,c)) &
  1837   (\<forall>W. ~(q4(a::'a,b,W) & mless_or_equal(a::'a,W) & mless_or_equal(b::'a,W) & mless_or_equal(W::'a,a))) &
  1838   (\<forall>W. ~(q4(a::'a,b,W) & mless_or_equal(a::'a,W) & mless_or_equal(b::'a,W) & mless_or_equal(W::'a,b))) --> False"
  1839   by meson
  1840 
  1841 (*PRV is now called SWV (software verification) *)
  1842 abbreviation "SWV001_1_ax mless_THAN successor predecessor equal \<equiv>
  1843   (\<forall>X. equal(predecessor(successor(X)),X)) &
  1844   (\<forall>X. equal(successor(predecessor(X)),X)) &
  1845   (\<forall>X Y. equal(predecessor(X),predecessor(Y)) --> equal(X::'a,Y)) &
  1846   (\<forall>X Y. equal(successor(X),successor(Y)) --> equal(X::'a,Y)) &
  1847   (\<forall>X. mless_THAN(predecessor(X),X)) &
  1848   (\<forall>X. mless_THAN(X::'a,successor(X))) &
  1849   (\<forall>X Y Z. mless_THAN(X::'a,Y) & mless_THAN(Y::'a,Z) --> mless_THAN(X::'a,Z)) &
  1850   (\<forall>X Y. mless_THAN(X::'a,Y) | mless_THAN(Y::'a,X) | equal(X::'a,Y)) &
  1851   (\<forall>X. ~mless_THAN(X::'a,X)) &
  1852   (\<forall>Y X. ~(mless_THAN(X::'a,Y) & mless_THAN(Y::'a,X))) &
  1853   (\<forall>Y X Z. equal(X::'a,Y) & mless_THAN(X::'a,Z) --> mless_THAN(Y::'a,Z)) &
  1854   (\<forall>Y Z X. equal(X::'a,Y) & mless_THAN(Z::'a,X) --> mless_THAN(Z::'a,Y))"
  1855 
  1856 abbreviation "SWV001_0_eq a successor predecessor equal \<equiv>
  1857   (\<forall>X Y. equal(X::'a,Y) --> equal(predecessor(X),predecessor(Y))) &
  1858   (\<forall>X Y. equal(X::'a,Y) --> equal(successor(X),successor(Y))) &
  1859   (\<forall>X Y. equal(X::'a,Y) --> equal(a(X),a(Y)))"
  1860 
  1861 (*21 inferences so far.  Searching to depth 5.  0.4 secs*)
  1862 lemma PRV003_1:
  1863   "EQU001_0_ax equal &
  1864   SWV001_1_ax mless_THAN successor predecessor equal &
  1865   SWV001_0_eq a successor predecessor equal &
  1866   (~mless_THAN(n::'a,j)) &
  1867   (mless_THAN(k::'a,j)) &
  1868   (~mless_THAN(k::'a,i)) &
  1869   (mless_THAN(i::'a,n)) &
  1870   (mless_THAN(a(j),a(k))) &
  1871   (\<forall>X. mless_THAN(X::'a,j) & mless_THAN(a(X),a(k)) --> mless_THAN(X::'a,i)) &
  1872   (\<forall>X. mless_THAN(One::'a,i) & mless_THAN(a(X),a(predecessor(i))) --> mless_THAN(X::'a,i) | mless_THAN(n::'a,X)) &
  1873   (\<forall>X. ~(mless_THAN(One::'a,X) & mless_THAN(X::'a,i) & mless_THAN(a(X),a(predecessor(X))))) &
  1874   (mless_THAN(j::'a,i)) --> False"
  1875   by meson
  1876 
  1877 (*584 inferences so far.  Searching to depth 7.  1.1 secs*)
  1878 lemma PRV005_1:
  1879   "EQU001_0_ax equal &
  1880   SWV001_1_ax mless_THAN successor predecessor equal &
  1881   SWV001_0_eq a successor predecessor equal &
  1882   (~mless_THAN(n::'a,k)) &
  1883   (~mless_THAN(k::'a,l)) &
  1884   (~mless_THAN(k::'a,i)) &
  1885   (mless_THAN(l::'a,n)) &
  1886   (mless_THAN(One::'a,l)) &
  1887   (mless_THAN(a(k),a(predecessor(l)))) &
  1888   (\<forall>X. mless_THAN(X::'a,successor(n)) & mless_THAN(a(X),a(k)) --> mless_THAN(X::'a,l)) &
  1889   (\<forall>X. mless_THAN(One::'a,l) & mless_THAN(a(X),a(predecessor(l))) --> mless_THAN(X::'a,l) | mless_THAN(n::'a,X)) &
  1890   (\<forall>X. ~(mless_THAN(One::'a,X) & mless_THAN(X::'a,l) & mless_THAN(a(X),a(predecessor(X))))) --> False"
  1891   by meson
  1892 
  1893 (*2343 inferences so far.  Searching to depth 8.  3.5 secs*)
  1894 lemma PRV006_1:
  1895   "EQU001_0_ax equal &
  1896   SWV001_1_ax mless_THAN successor predecessor equal &
  1897   SWV001_0_eq a successor predecessor equal &
  1898   (~mless_THAN(n::'a,m)) &
  1899   (mless_THAN(i::'a,m)) &
  1900   (mless_THAN(i::'a,n)) &
  1901   (~mless_THAN(i::'a,One)) &
  1902   (mless_THAN(a(i),a(m))) &
  1903   (\<forall>X. mless_THAN(X::'a,successor(n)) & mless_THAN(a(X),a(m)) --> mless_THAN(X::'a,i)) &
  1904   (\<forall>X. mless_THAN(One::'a,i) & mless_THAN(a(X),a(predecessor(i))) --> mless_THAN(X::'a,i) | mless_THAN(n::'a,X)) &
  1905   (\<forall>X. ~(mless_THAN(One::'a,X) & mless_THAN(X::'a,i) & mless_THAN(a(X),a(predecessor(X))))) --> False"
  1906   by meson
  1907 
  1908 (*86 inferences so far.  Searching to depth 14.  0.1 secs*)
  1909 lemma PRV009_1:
  1910   "(\<forall>Y X. mless_or_equal(X::'a,Y) | mless(Y::'a,X)) &
  1911   (mless(j::'a,i)) &
  1912   (mless_or_equal(m::'a,p)) &
  1913   (mless_or_equal(p::'a,q)) &
  1914   (mless_or_equal(q::'a,n)) &
  1915   (\<forall>X Y. mless_or_equal(m::'a,X) & mless(X::'a,i) & mless(j::'a,Y) & mless_or_equal(Y::'a,n) --> mless_or_equal(a(X),a(Y))) &
  1916   (\<forall>X Y. mless_or_equal(m::'a,X) & mless_or_equal(X::'a,Y) & mless_or_equal(Y::'a,j) --> mless_or_equal(a(X),a(Y))) &
  1917   (\<forall>X Y. mless_or_equal(i::'a,X) & mless_or_equal(X::'a,Y) & mless_or_equal(Y::'a,n) --> mless_or_equal(a(X),a(Y))) &
  1918   (~mless_or_equal(a(p),a(q))) --> False"
  1919   by meson
  1920 
  1921 (*222 inferences so far.  Searching to depth 8.  0.4 secs*)
  1922 lemma PUZ012_1:
  1923   "(\<forall>X. equal_fruits(X::'a,X)) &
  1924   (\<forall>X. equal_boxes(X::'a,X)) &
  1925   (\<forall>X Y. ~(label(X::'a,Y) & contains(X::'a,Y))) &
  1926   (\<forall>X. contains(boxa::'a,X) | contains(boxb::'a,X) | contains(boxc::'a,X)) &
  1927   (\<forall>X. contains(X::'a,apples) | contains(X::'a,bananas) | contains(X::'a,oranges)) &
  1928   (\<forall>X Y Z. contains(X::'a,Y) & contains(X::'a,Z) --> equal_fruits(Y::'a,Z)) &
  1929   (\<forall>Y X Z. contains(X::'a,Y) & contains(Z::'a,Y) --> equal_boxes(X::'a,Z)) &
  1930   (~equal_boxes(boxa::'a,boxb)) &
  1931   (~equal_boxes(boxb::'a,boxc)) &
  1932   (~equal_boxes(boxa::'a,boxc)) &
  1933   (~equal_fruits(apples::'a,bananas)) &
  1934   (~equal_fruits(bananas::'a,oranges)) &
  1935   (~equal_fruits(apples::'a,oranges)) &
  1936   (label(boxa::'a,apples)) &
  1937   (label(boxb::'a,oranges)) &
  1938   (label(boxc::'a,bananas)) &
  1939   (contains(boxb::'a,apples)) &
  1940   (~(contains(boxa::'a,bananas) & contains(boxc::'a,oranges))) --> False"
  1941   by meson
  1942 
  1943 (*35 inferences so far.  Searching to depth 5.  3.2 secs*)
  1944 lemma PUZ020_1:
  1945   "EQU001_0_ax equal &
  1946   (\<forall>A B. equal(A::'a,B) --> equal(statement_by(A),statement_by(B))) &
  1947   (\<forall>X. person(X) --> knight(X) | knave(X)) &
  1948   (\<forall>X. ~(person(X) & knight(X) & knave(X))) &
  1949   (\<forall>X Y. says(X::'a,Y) & a_truth(Y) --> a_truth(Y)) &
  1950   (\<forall>X Y. ~(says(X::'a,Y) & equal(X::'a,Y))) &
  1951   (\<forall>Y X. says(X::'a,Y) --> equal(Y::'a,statement_by(X))) &
  1952   (\<forall>X Y. ~(person(X) & equal(X::'a,statement_by(Y)))) &
  1953   (\<forall>X. person(X) & a_truth(statement_by(X)) --> knight(X)) &
  1954   (\<forall>X. person(X) --> a_truth(statement_by(X)) | knave(X)) &
  1955   (\<forall>X Y. equal(X::'a,Y) & knight(X) --> knight(Y)) &
  1956   (\<forall>X Y. equal(X::'a,Y) & knave(X) --> knave(Y)) &
  1957   (\<forall>X Y. equal(X::'a,Y) & person(X) --> person(Y)) &
  1958   (\<forall>X Y Z. equal(X::'a,Y) & says(X::'a,Z) --> says(Y::'a,Z)) &
  1959   (\<forall>X Z Y. equal(X::'a,Y) & says(Z::'a,X) --> says(Z::'a,Y)) &
  1960   (\<forall>X Y. equal(X::'a,Y) & a_truth(X) --> a_truth(Y)) &
  1961   (\<forall>X Y. knight(X) & says(X::'a,Y) --> a_truth(Y)) &
  1962   (\<forall>X Y. ~(knave(X) & says(X::'a,Y) & a_truth(Y))) &
  1963   (person(husband)) &
  1964   (person(wife)) &
  1965   (~equal(husband::'a,wife)) &
  1966   (says(husband::'a,statement_by(husband))) &
  1967   (a_truth(statement_by(husband)) & knight(husband) --> knight(wife)) &
  1968   (knight(husband) --> a_truth(statement_by(husband))) &
  1969   (a_truth(statement_by(husband)) | knight(wife)) &
  1970   (knight(wife) --> a_truth(statement_by(husband))) &
  1971   (~knight(husband)) --> False"
  1972   by meson
  1973 
  1974 (*121806 inferences so far.  Searching to depth 17.  63.0 secs*)
  1975 lemma PUZ025_1:
  1976   "(\<forall>X. a_truth(truthteller(X)) | a_truth(liar(X))) &
  1977   (\<forall>X. ~(a_truth(truthteller(X)) & a_truth(liar(X)))) &
  1978   (\<forall>Truthteller Statement. a_truth(truthteller(Truthteller)) & a_truth(says(Truthteller::'a,Statement)) --> a_truth(Statement)) &
  1979   (\<forall>Liar Statement. ~(a_truth(liar(Liar)) & a_truth(says(Liar::'a,Statement)) & a_truth(Statement))) &
  1980   (\<forall>Statement Truthteller. a_truth(Statement) & a_truth(says(Truthteller::'a,Statement)) --> a_truth(truthteller(Truthteller))) &
  1981   (\<forall>Statement Liar. a_truth(says(Liar::'a,Statement)) --> a_truth(Statement) | a_truth(liar(Liar))) &
  1982   (\<forall>Z X Y. people(X::'a,Y,Z) & a_truth(liar(X)) & a_truth(liar(Y)) --> a_truth(equal_type(X::'a,Y))) &
  1983   (\<forall>Z X Y. people(X::'a,Y,Z) & a_truth(truthteller(X)) & a_truth(truthteller(Y)) --> a_truth(equal_type(X::'a,Y))) &
  1984   (\<forall>X Y. a_truth(equal_type(X::'a,Y)) & a_truth(truthteller(X)) --> a_truth(truthteller(Y))) &
  1985   (\<forall>X Y. a_truth(equal_type(X::'a,Y)) & a_truth(liar(X)) --> a_truth(liar(Y))) &
  1986   (\<forall>X Y. a_truth(truthteller(X)) --> a_truth(equal_type(X::'a,Y)) | a_truth(liar(Y))) &
  1987   (\<forall>X Y. a_truth(liar(X)) --> a_truth(equal_type(X::'a,Y)) | a_truth(truthteller(Y))) &
  1988   (\<forall>Y X. a_truth(equal_type(X::'a,Y)) --> a_truth(equal_type(Y::'a,X))) &
  1989   (\<forall>X Y. ask_1_if_2(X::'a,Y) & a_truth(truthteller(X)) & a_truth(Y) --> answer(yes)) &
  1990   (\<forall>X Y. ask_1_if_2(X::'a,Y) & a_truth(truthteller(X)) --> a_truth(Y) | answer(no)) &
  1991   (\<forall>X Y. ask_1_if_2(X::'a,Y) & a_truth(liar(X)) & a_truth(Y) --> answer(no)) &
  1992   (\<forall>X Y. ask_1_if_2(X::'a,Y) & a_truth(liar(X)) --> a_truth(Y) | answer(yes)) &
  1993   (people(b::'a,c,a)) &
  1994   (people(a::'a,b,a)) &
  1995   (people(a::'a,c,b)) &
  1996   (people(c::'a,b,a)) &
  1997   (a_truth(says(a::'a,equal_type(b::'a,c)))) &
  1998   (ask_1_if_2(c::'a,equal_type(a::'a,b))) &
  1999   (\<forall>Answer. ~answer(Answer)) --> False"
  2000   oops
  2001 
  2002 
  2003 (*621 inferences so far.  Searching to depth 18.  0.2 secs*)
  2004 lemma PUZ029_1:
  2005  "(\<forall>X. dances_on_tightropes(X) | eats_pennybuns(X) | old(X)) &
  2006   (\<forall>X. pig(X) & liable_to_giddiness(X) --> treated_with_respect(X)) &
  2007   (\<forall>X. wise(X) & balloonist(X) --> has_umbrella(X)) &
  2008   (\<forall>X. ~(looks_ridiculous(X) & eats_pennybuns(X) & eats_lunch_in_public(X))) &
  2009   (\<forall>X. balloonist(X) & young(X) --> liable_to_giddiness(X)) &
  2010   (\<forall>X. fat(X) & looks_ridiculous(X) --> dances_on_tightropes(X) | eats_lunch_in_public(X)) &
  2011   (\<forall>X. ~(liable_to_giddiness(X) & wise(X) & dances_on_tightropes(X))) &
  2012   (\<forall>X. pig(X) & has_umbrella(X) --> looks_ridiculous(X)) &
  2013   (\<forall>X. treated_with_respect(X) --> dances_on_tightropes(X) | fat(X)) &
  2014   (\<forall>X. young(X) | old(X)) &
  2015   (\<forall>X. ~(young(X) & old(X))) &
  2016   (wise(piggy)) &
  2017   (young(piggy)) &
  2018   (pig(piggy)) &
  2019   (balloonist(piggy)) --> False"
  2020   by meson
  2021 
  2022 abbreviation "RNG001_0_ax equal additive_inverse add multiply product additive_identity sum \<equiv>
  2023   (\<forall>X. sum(additive_identity::'a,X,X)) &
  2024   (\<forall>X. sum(X::'a,additive_identity,X)) &
  2025   (\<forall>X Y. product(X::'a,Y,multiply(X::'a,Y))) &
  2026   (\<forall>X Y. sum(X::'a,Y,add(X::'a,Y))) &
  2027   (\<forall>X. sum(additive_inverse(X),X,additive_identity)) &
  2028   (\<forall>X. sum(X::'a,additive_inverse(X),additive_identity)) &
  2029   (\<forall>Y U Z X V W. sum(X::'a,Y,U) & sum(Y::'a,Z,V) & sum(U::'a,Z,W) --> sum(X::'a,V,W)) &
  2030   (\<forall>Y X V U Z W. sum(X::'a,Y,U) & sum(Y::'a,Z,V) & sum(X::'a,V,W) --> sum(U::'a,Z,W)) &
  2031   (\<forall>Y X Z. sum(X::'a,Y,Z) --> sum(Y::'a,X,Z)) &
  2032   (\<forall>Y U Z X V W. product(X::'a,Y,U) & product(Y::'a,Z,V) & product(U::'a,Z,W) --> product(X::'a,V,W)) &
  2033   (\<forall>Y X V U Z W. product(X::'a,Y,U) & product(Y::'a,Z,V) & product(X::'a,V,W) --> product(U::'a,Z,W)) &
  2034   (\<forall>Y Z X V3 V1 V2 V4. product(X::'a,Y,V1) & product(X::'a,Z,V2) & sum(Y::'a,Z,V3) & product(X::'a,V3,V4) --> sum(V1::'a,V2,V4)) &
  2035   (\<forall>Y Z V1 V2 X V3 V4. product(X::'a,Y,V1) & product(X::'a,Z,V2) & sum(Y::'a,Z,V3) & sum(V1::'a,V2,V4) --> product(X::'a,V3,V4)) &
  2036   (\<forall>Y Z V3 X V1 V2 V4. product(Y::'a,X,V1) & product(Z::'a,X,V2) & sum(Y::'a,Z,V3) & product(V3::'a,X,V4) --> sum(V1::'a,V2,V4)) &
  2037   (\<forall>Y Z V1 V2 V3 X V4. product(Y::'a,X,V1) & product(Z::'a,X,V2) & sum(Y::'a,Z,V3) & sum(V1::'a,V2,V4) --> product(V3::'a,X,V4)) &
  2038   (\<forall>X Y U V. sum(X::'a,Y,U) & sum(X::'a,Y,V) --> equal(U::'a,V)) &
  2039   (\<forall>X Y U V. product(X::'a,Y,U) & product(X::'a,Y,V) --> equal(U::'a,V))"
  2040 
  2041 abbreviation "RNG001_0_eq product multiply sum add additive_inverse equal \<equiv>
  2042   (\<forall>X Y. equal(X::'a,Y) --> equal(additive_inverse(X),additive_inverse(Y))) &
  2043   (\<forall>X Y W. equal(X::'a,Y) --> equal(add(X::'a,W),add(Y::'a,W))) &
  2044   (\<forall>X W Y. equal(X::'a,Y) --> equal(add(W::'a,X),add(W::'a,Y))) &
  2045   (\<forall>X Y W Z. equal(X::'a,Y) & sum(X::'a,W,Z) --> sum(Y::'a,W,Z)) &
  2046   (\<forall>X W Y Z. equal(X::'a,Y) & sum(W::'a,X,Z) --> sum(W::'a,Y,Z)) &
  2047   (\<forall>X W Z Y. equal(X::'a,Y) & sum(W::'a,Z,X) --> sum(W::'a,Z,Y)) &
  2048   (\<forall>X Y W. equal(X::'a,Y) --> equal(multiply(X::'a,W),multiply(Y::'a,W))) &
  2049   (\<forall>X W Y. equal(X::'a,Y) --> equal(multiply(W::'a,X),multiply(W::'a,Y))) &
  2050   (\<forall>X Y W Z. equal(X::'a,Y) & product(X::'a,W,Z) --> product(Y::'a,W,Z)) &
  2051   (\<forall>X W Y Z. equal(X::'a,Y) & product(W::'a,X,Z) --> product(W::'a,Y,Z)) &
  2052   (\<forall>X W Z Y. equal(X::'a,Y) & product(W::'a,Z,X) --> product(W::'a,Z,Y))"
  2053 
  2054 (*93620 inferences so far.  Searching to depth 24.  65.9 secs*)
  2055 lemma RNG001_3:
  2056  "(\<forall>X. sum(additive_identity::'a,X,X)) &
  2057   (\<forall>X. sum(additive_inverse(X),X,additive_identity)) &
  2058   (\<forall>Y U Z X V W. sum(X::'a,Y,U) & sum(Y::'a,Z,V) & sum(U::'a,Z,W) --> sum(X::'a,V,W)) &
  2059   (\<forall>Y X V U Z W. sum(X::'a,Y,U) & sum(Y::'a,Z,V) & sum(X::'a,V,W) --> sum(U::'a,Z,W)) &
  2060   (\<forall>X Y. product(X::'a,Y,multiply(X::'a,Y))) &
  2061   (\<forall>Y Z X V3 V1 V2 V4. product(X::'a,Y,V1) & product(X::'a,Z,V2) & sum(Y::'a,Z,V3) & product(X::'a,V3,V4) --> sum(V1::'a,V2,V4)) &
  2062   (\<forall>Y Z V1 V2 X V3 V4. product(X::'a,Y,V1) & product(X::'a,Z,V2) & sum(Y::'a,Z,V3) & sum(V1::'a,V2,V4) --> product(X::'a,V3,V4)) &
  2063   (~product(a::'a,additive_identity,additive_identity)) --> False"
  2064   oops
  2065 
  2066 abbreviation "RNG_other_ax multiply add equal product additive_identity additive_inverse sum \<equiv>
  2067   (\<forall>X. sum(X::'a,additive_inverse(X),additive_identity)) &
  2068   (\<forall>Y U Z X V W. sum(X::'a,Y,U) & sum(Y::'a,Z,V) & sum(U::'a,Z,W) --> sum(X::'a,V,W)) &
  2069   (\<forall>Y X V U Z W. sum(X::'a,Y,U) & sum(Y::'a,Z,V) & sum(X::'a,V,W) --> sum(U::'a,Z,W)) &
  2070   (\<forall>Y X Z. sum(X::'a,Y,Z) --> sum(Y::'a,X,Z)) &
  2071   (\<forall>Y U Z X V W. product(X::'a,Y,U) & product(Y::'a,Z,V) & product(U::'a,Z,W) --> product(X::'a,V,W)) &
  2072   (\<forall>Y X V U Z W. product(X::'a,Y,U) & product(Y::'a,Z,V) & product(X::'a,V,W) --> product(U::'a,Z,W)) &
  2073   (\<forall>Y Z X V3 V1 V2 V4. product(X::'a,Y,V1) & product(X::'a,Z,V2) & sum(Y::'a,Z,V3) & product(X::'a,V3,V4) --> sum(V1::'a,V2,V4)) &
  2074   (\<forall>Y Z V1 V2 X V3 V4. product(X::'a,Y,V1) & product(X::'a,Z,V2) & sum(Y::'a,Z,V3) & sum(V1::'a,V2,V4) --> product(X::'a,V3,V4)) &
  2075   (\<forall>Y Z V3 X V1 V2 V4. product(Y::'a,X,V1) & product(Z::'a,X,V2) & sum(Y::'a,Z,V3) & product(V3::'a,X,V4) --> sum(V1::'a,V2,V4)) &
  2076   (\<forall>Y Z V1 V2 V3 X V4. product(Y::'a,X,V1) & product(Z::'a,X,V2) & sum(Y::'a,Z,V3) & sum(V1::'a,V2,V4) --> product(V3::'a,X,V4)) &
  2077   (\<forall>X Y U V. sum(X::'a,Y,U) & sum(X::'a,Y,V) --> equal(U::'a,V)) &
  2078   (\<forall>X Y U V. product(X::'a,Y,U) & product(X::'a,Y,V) --> equal(U::'a,V)) &
  2079   (\<forall>X Y. equal(X::'a,Y) --> equal(additive_inverse(X),additive_inverse(Y))) &
  2080   (\<forall>X Y W. equal(X::'a,Y) --> equal(add(X::'a,W),add(Y::'a,W))) &
  2081   (\<forall>X Y W Z. equal(X::'a,Y) & sum(X::'a,W,Z) --> sum(Y::'a,W,Z)) &
  2082   (\<forall>X W Y Z. equal(X::'a,Y) & sum(W::'a,X,Z) --> sum(W::'a,Y,Z)) &
  2083   (\<forall>X W Z Y. equal(X::'a,Y) & sum(W::'a,Z,X) --> sum(W::'a,Z,Y)) &
  2084   (\<forall>X Y W. equal(X::'a,Y) --> equal(multiply(X::'a,W),multiply(Y::'a,W))) &
  2085   (\<forall>X Y W Z. equal(X::'a,Y) & product(X::'a,W,Z) --> product(Y::'a,W,Z)) &
  2086   (\<forall>X W Y Z. equal(X::'a,Y) & product(W::'a,X,Z) --> product(W::'a,Y,Z)) &
  2087   (\<forall>X W Z Y. equal(X::'a,Y) & product(W::'a,Z,X) --> product(W::'a,Z,Y))"
  2088 
  2089 
  2090 (****************SLOW
  2091 76385914 inferences so far.  Searching to depth 18
  2092 No proof after 5 1/2 hours! (griffon)
  2093 ****************)
  2094 lemma RNG001_5:
  2095   "EQU001_0_ax equal &
  2096   (\<forall>X. sum(additive_identity::'a,X,X)) &
  2097   (\<forall>X. sum(X::'a,additive_identity,X)) &
  2098   (\<forall>X Y. product(X::'a,Y,multiply(X::'a,Y))) &
  2099   (\<forall>X Y. sum(X::'a,Y,add(X::'a,Y))) &
  2100   (\<forall>X. sum(additive_inverse(X),X,additive_identity)) &
  2101   RNG_other_ax multiply add equal product additive_identity additive_inverse sum &
  2102   (~product(a::'a,additive_identity,additive_identity)) --> False"
  2103   oops
  2104 
  2105 (*0 inferences so far.  Searching to depth 0.  0.5 secs*)
  2106 lemma RNG011_5:
  2107   "EQU001_0_ax equal &
  2108   (\<forall>A B C. equal(A::'a,B) --> equal(add(A::'a,C),add(B::'a,C))) &
  2109   (\<forall>D F' E. equal(D::'a,E) --> equal(add(F'::'a,D),add(F'::'a,E))) &
  2110   (\<forall>G H. equal(G::'a,H) --> equal(additive_inverse(G),additive_inverse(H))) &
  2111   (\<forall>I' J K'. equal(I'::'a,J) --> equal(multiply(I'::'a,K'),multiply(J::'a,K'))) &
  2112   (\<forall>L N M. equal(L::'a,M) --> equal(multiply(N::'a,L),multiply(N::'a,M))) &
  2113   (\<forall>A B C D. equal(A::'a,B) --> equal(associator(A::'a,C,D),associator(B::'a,C,D))) &
  2114   (\<forall>E G F' H. equal(E::'a,F') --> equal(associator(G::'a,E,H),associator(G::'a,F',H))) &
  2115   (\<forall>I' K' L J. equal(I'::'a,J) --> equal(associator(K'::'a,L,I'),associator(K'::'a,L,J))) &
  2116   (\<forall>M N O'. equal(M::'a,N) --> equal(commutator(M::'a,O'),commutator(N::'a,O'))) &
  2117   (\<forall>P R Q. equal(P::'a,Q) --> equal(commutator(R::'a,P),commutator(R::'a,Q))) &
  2118   (\<forall>Y X. equal(add(X::'a,Y),add(Y::'a,X))) &
  2119   (\<forall>X Y Z. equal(add(add(X::'a,Y),Z),add(X::'a,add(Y::'a,Z)))) &
  2120   (\<forall>X. equal(add(X::'a,additive_identity),X)) &
  2121   (\<forall>X. equal(add(additive_identity::'a,X),X)) &
  2122   (\<forall>X. equal(add(X::'a,additive_inverse(X)),additive_identity)) &
  2123   (\<forall>X. equal(add(additive_inverse(X),X),additive_identity)) &
  2124   (equal(additive_inverse(additive_identity),additive_identity)) &
  2125   (\<forall>X Y. equal(add(X::'a,add(additive_inverse(X),Y)),Y)) &
  2126   (\<forall>X Y. equal(additive_inverse(add(X::'a,Y)),add(additive_inverse(X),additive_inverse(Y)))) &
  2127   (\<forall>X. equal(additive_inverse(additive_inverse(X)),X)) &
  2128   (\<forall>X. equal(multiply(X::'a,additive_identity),additive_identity)) &
  2129   (\<forall>X. equal(multiply(additive_identity::'a,X),additive_identity)) &
  2130   (\<forall>X Y. equal(multiply(additive_inverse(X),additive_inverse(Y)),multiply(X::'a,Y))) &
  2131   (\<forall>X Y. equal(multiply(X::'a,additive_inverse(Y)),additive_inverse(multiply(X::'a,Y)))) &
  2132   (\<forall>X Y. equal(multiply(additive_inverse(X),Y),additive_inverse(multiply(X::'a,Y)))) &
  2133   (\<forall>Y X Z. equal(multiply(X::'a,add(Y::'a,Z)),add(multiply(X::'a,Y),multiply(X::'a,Z)))) &
  2134   (\<forall>X Y Z. equal(multiply(add(X::'a,Y),Z),add(multiply(X::'a,Z),multiply(Y::'a,Z)))) &
  2135   (\<forall>X Y. equal(multiply(multiply(X::'a,Y),Y),multiply(X::'a,multiply(Y::'a,Y)))) &
  2136   (\<forall>X Y Z. equal(associator(X::'a,Y,Z),add(multiply(multiply(X::'a,Y),Z),additive_inverse(multiply(X::'a,multiply(Y::'a,Z)))))) &
  2137   (\<forall>X Y. equal(commutator(X::'a,Y),add(multiply(Y::'a,X),additive_inverse(multiply(X::'a,Y))))) &
  2138   (\<forall>X Y. equal(multiply(multiply(associator(X::'a,X,Y),X),associator(X::'a,X,Y)),additive_identity)) &
  2139   (~equal(multiply(multiply(associator(a::'a,a,b),a),associator(a::'a,a,b)),additive_identity)) --> False"
  2140   by meson
  2141 
  2142 (*202 inferences so far.  Searching to depth 8.  0.6 secs*)
  2143 lemma RNG023_6:
  2144   "EQU001_0_ax equal &
  2145   (\<forall>Y X. equal(add(X::'a,Y),add(Y::'a,X))) &
  2146   (\<forall>X Y Z. equal(add(X::'a,add(Y::'a,Z)),add(add(X::'a,Y),Z))) &
  2147   (\<forall>X. equal(add(additive_identity::'a,X),X)) &
  2148   (\<forall>X. equal(add(X::'a,additive_identity),X)) &
  2149   (\<forall>X. equal(multiply(additive_identity::'a,X),additive_identity)) &
  2150   (\<forall>X. equal(multiply(X::'a,additive_identity),additive_identity)) &
  2151   (\<forall>X. equal(add(additive_inverse(X),X),additive_identity)) &
  2152   (\<forall>X. equal(add(X::'a,additive_inverse(X)),additive_identity)) &
  2153   (\<forall>Y X Z. equal(multiply(X::'a,add(Y::'a,Z)),add(multiply(X::'a,Y),multiply(X::'a,Z)))) &
  2154   (\<forall>X Y Z. equal(multiply(add(X::'a,Y),Z),add(multiply(X::'a,Z),multiply(Y::'a,Z)))) &
  2155   (\<forall>X. equal(additive_inverse(additive_inverse(X)),X)) &
  2156   (\<forall>X Y. equal(multiply(multiply(X::'a,Y),Y),multiply(X::'a,multiply(Y::'a,Y)))) &
  2157   (\<forall>X Y. equal(multiply(multiply(X::'a,X),Y),multiply(X::'a,multiply(X::'a,Y)))) &
  2158   (\<forall>X Y Z. equal(associator(X::'a,Y,Z),add(multiply(multiply(X::'a,Y),Z),additive_inverse(multiply(X::'a,multiply(Y::'a,Z)))))) &
  2159   (\<forall>X Y. equal(commutator(X::'a,Y),add(multiply(Y::'a,X),additive_inverse(multiply(X::'a,Y))))) &
  2160   (\<forall>D E F'. equal(D::'a,E) --> equal(add(D::'a,F'),add(E::'a,F'))) &
  2161   (\<forall>G I' H. equal(G::'a,H) --> equal(add(I'::'a,G),add(I'::'a,H))) &
  2162   (\<forall>J K'. equal(J::'a,K') --> equal(additive_inverse(J),additive_inverse(K'))) &
  2163   (\<forall>L M N O'. equal(L::'a,M) --> equal(associator(L::'a,N,O'),associator(M::'a,N,O'))) &
  2164   (\<forall>P R Q S'. equal(P::'a,Q) --> equal(associator(R::'a,P,S'),associator(R::'a,Q,S'))) &
  2165   (\<forall>T' V W U. equal(T'::'a,U) --> equal(associator(V::'a,W,T'),associator(V::'a,W,U))) &
  2166   (\<forall>X Y Z. equal(X::'a,Y) --> equal(commutator(X::'a,Z),commutator(Y::'a,Z))) &
  2167   (\<forall>A1 C1 B1. equal(A1::'a,B1) --> equal(commutator(C1::'a,A1),commutator(C1::'a,B1))) &
  2168   (\<forall>D1 E1 F1. equal(D1::'a,E1) --> equal(multiply(D1::'a,F1),multiply(E1::'a,F1))) &
  2169   (\<forall>G1 I1 H1. equal(G1::'a,H1) --> equal(multiply(I1::'a,G1),multiply(I1::'a,H1))) &
  2170   (~equal(associator(x::'a,x,y),additive_identity)) --> False"
  2171   by meson
  2172 
  2173 (*0 inferences so far.  Searching to depth 0.  0.6 secs*)
  2174 lemma RNG028_2:
  2175   "EQU001_0_ax equal &
  2176   (\<forall>X. equal(add(additive_identity::'a,X),X)) &
  2177   (\<forall>X. equal(multiply(additive_identity::'a,X),additive_identity)) &
  2178   (\<forall>X. equal(multiply(X::'a,additive_identity),additive_identity)) &
  2179   (\<forall>X. equal(add(additive_inverse(X),X),additive_identity)) &
  2180   (\<forall>X Y. equal(additive_inverse(add(X::'a,Y)),add(additive_inverse(X),additive_inverse(Y)))) &
  2181   (\<forall>X. equal(additive_inverse(additive_inverse(X)),X)) &
  2182   (\<forall>Y X Z. equal(multiply(X::'a,add(Y::'a,Z)),add(multiply(X::'a,Y),multiply(X::'a,Z)))) &
  2183   (\<forall>X Y Z. equal(multiply(add(X::'a,Y),Z),add(multiply(X::'a,Z),multiply(Y::'a,Z)))) &
  2184   (\<forall>X Y. equal(multiply(multiply(X::'a,Y),Y),multiply(X::'a,multiply(Y::'a,Y)))) &
  2185   (\<forall>X Y. equal(multiply(multiply(X::'a,X),Y),multiply(X::'a,multiply(X::'a,Y)))) &
  2186   (\<forall>X Y. equal(multiply(additive_inverse(X),Y),additive_inverse(multiply(X::'a,Y)))) &
  2187   (\<forall>X Y. equal(multiply(X::'a,additive_inverse(Y)),additive_inverse(multiply(X::'a,Y)))) &
  2188   (equal(additive_inverse(additive_identity),additive_identity)) &
  2189   (\<forall>Y X. equal(add(X::'a,Y),add(Y::'a,X))) &
  2190   (\<forall>X Y Z. equal(add(X::'a,add(Y::'a,Z)),add(add(X::'a,Y),Z))) &
  2191   (\<forall>Z X Y. equal(add(X::'a,Z),add(Y::'a,Z)) --> equal(X::'a,Y)) &
  2192   (\<forall>Z X Y. equal(add(Z::'a,X),add(Z::'a,Y)) --> equal(X::'a,Y)) &
  2193   (\<forall>D E F'. equal(D::'a,E) --> equal(add(D::'a,F'),add(E::'a,F'))) &
  2194   (\<forall>G I' H. equal(G::'a,H) --> equal(add(I'::'a,G),add(I'::'a,H))) &
  2195   (\<forall>J K'. equal(J::'a,K') --> equal(additive_inverse(J),additive_inverse(K'))) &
  2196   (\<forall>D1 E1 F1. equal(D1::'a,E1) --> equal(multiply(D1::'a,F1),multiply(E1::'a,F1))) &
  2197   (\<forall>G1 I1 H1. equal(G1::'a,H1) --> equal(multiply(I1::'a,G1),multiply(I1::'a,H1))) &
  2198   (\<forall>X Y Z. equal(associator(X::'a,Y,Z),add(multiply(multiply(X::'a,Y),Z),additive_inverse(multiply(X::'a,multiply(Y::'a,Z)))))) &
  2199   (\<forall>L M N O'. equal(L::'a,M) --> equal(associator(L::'a,N,O'),associator(M::'a,N,O'))) &
  2200   (\<forall>P R Q S'. equal(P::'a,Q) --> equal(associator(R::'a,P,S'),associator(R::'a,Q,S'))) &
  2201   (\<forall>T' V W U. equal(T'::'a,U) --> equal(associator(V::'a,W,T'),associator(V::'a,W,U))) &
  2202   (\<forall>X Y. ~equal(multiply(multiply(Y::'a,X),Y),multiply(Y::'a,multiply(X::'a,Y)))) &
  2203   (\<forall>X Y Z. ~equal(associator(Y::'a,X,Z),additive_inverse(associator(X::'a,Y,Z)))) &
  2204   (\<forall>X Y Z. ~equal(associator(Z::'a,Y,X),additive_inverse(associator(X::'a,Y,Z)))) &
  2205   (~equal(multiply(multiply(cx::'a,multiply(cy::'a,cx)),cz),multiply(cx::'a,multiply(cy::'a,multiply(cx::'a,cz))))) --> False"
  2206   by meson
  2207 
  2208 (*209 inferences so far.  Searching to depth 9.  1.2 secs*)
  2209 lemma RNG038_2:
  2210   "(\<forall>X. sum(X::'a,additive_identity,X)) &
  2211   (\<forall>X Y. product(X::'a,Y,multiply(X::'a,Y))) &
  2212   (\<forall>X Y. sum(X::'a,Y,add(X::'a,Y))) &
  2213   RNG_other_ax multiply add equal product additive_identity additive_inverse sum &
  2214   (\<forall>X. product(additive_identity::'a,X,additive_identity)) &
  2215   (\<forall>X. product(X::'a,additive_identity,additive_identity)) &
  2216   (\<forall>X Y. equal(X::'a,additive_identity) --> product(X::'a,h(X::'a,Y),Y)) &
  2217   (product(a::'a,b,additive_identity)) &
  2218   (~equal(a::'a,additive_identity)) &
  2219   (~equal(b::'a,additive_identity)) --> False"
  2220   by meson
  2221 
  2222 (*2660 inferences so far.  Searching to depth 10.  7.0 secs*)
  2223 lemma RNG040_2:
  2224   "EQU001_0_ax equal &
  2225   RNG001_0_eq product multiply sum add additive_inverse equal &
  2226   (\<forall>X. sum(additive_identity::'a,X,X)) &
  2227   (\<forall>X. sum(X::'a,additive_identity,X)) &
  2228   (\<forall>X Y. product(X::'a,Y,multiply(X::'a,Y))) &
  2229   (\<forall>X Y. sum(X::'a,Y,add(X::'a,Y))) &
  2230   (\<forall>X. sum(additive_inverse(X),X,additive_identity)) &
  2231   (\<forall>X. sum(X::'a,additive_inverse(X),additive_identity)) &
  2232   (\<forall>Y U Z X V W. sum(X::'a,Y,U) & sum(Y::'a,Z,V) & sum(U::'a,Z,W) --> sum(X::'a,V,W)) &
  2233   (\<forall>Y X V U Z W. sum(X::'a,Y,U) & sum(Y::'a,Z,V) & sum(X::'a,V,W) --> sum(U::'a,Z,W)) &
  2234   (\<forall>Y X Z. sum(X::'a,Y,Z) --> sum(Y::'a,X,Z)) &
  2235   (\<forall>Y U Z X V W. product(X::'a,Y,U) & product(Y::'a,Z,V) & product(U::'a,Z,W) --> product(X::'a,V,W)) &
  2236   (\<forall>Y X V U Z W. product(X::'a,Y,U) & product(Y::'a,Z,V) & product(X::'a,V,W) --> product(U::'a,Z,W)) &
  2237   (\<forall>Y Z X V3 V1 V2 V4. product(X::'a,Y,V1) & product(X::'a,Z,V2) & sum(Y::'a,Z,V3) & product(X::'a,V3,V4) --> sum(V1::'a,V2,V4)) &
  2238   (\<forall>Y Z V1 V2 X V3 V4. product(X::'a,Y,V1) & product(X::'a,Z,V2) & sum(Y::'a,Z,V3) & sum(V1::'a,V2,V4) --> product(X::'a,V3,V4)) &
  2239   (\<forall>X Y U V. sum(X::'a,Y,U) & sum(X::'a,Y,V) --> equal(U::'a,V)) &
  2240   (\<forall>X Y U V. product(X::'a,Y,U) & product(X::'a,Y,V) --> equal(U::'a,V)) &
  2241   (\<forall>A. product(A::'a,multiplicative_identity,A)) &
  2242   (\<forall>A. product(multiplicative_identity::'a,A,A)) &
  2243   (\<forall>A. product(A::'a,h(A),multiplicative_identity) | equal(A::'a,additive_identity)) &
  2244   (\<forall>A. product(h(A),A,multiplicative_identity) | equal(A::'a,additive_identity)) &
  2245   (\<forall>B A C. product(A::'a,B,C) --> product(B::'a,A,C)) &
  2246   (\<forall>A B. equal(A::'a,B) --> equal(h(A),h(B))) &
  2247   (sum(b::'a,c,d)) &
  2248   (product(d::'a,a,additive_identity)) &
  2249   (product(b::'a,a,l)) &
  2250   (product(c::'a,a,n)) &
  2251   (~sum(l::'a,n,additive_identity)) --> False"
  2252   by meson
  2253 
  2254 (*8991 inferences so far.  Searching to depth 9.  22.2 secs*)
  2255 lemma RNG041_1:
  2256   "EQU001_0_ax equal &
  2257   RNG001_0_ax equal additive_inverse add multiply product additive_identity sum &
  2258   RNG001_0_eq product multiply sum add additive_inverse equal &
  2259   (\<forall>A B. equal(A::'a,B) --> equal(h(A),h(B))) &
  2260   (\<forall>A. product(additive_identity::'a,A,additive_identity)) &
  2261   (\<forall>A. product(A::'a,additive_identity,additive_identity)) &
  2262   (\<forall>A. product(A::'a,multiplicative_identity,A)) &
  2263   (\<forall>A. product(multiplicative_identity::'a,A,A)) &
  2264   (\<forall>A. product(A::'a,h(A),multiplicative_identity) | equal(A::'a,additive_identity)) &
  2265   (\<forall>A. product(h(A),A,multiplicative_identity) | equal(A::'a,additive_identity)) &
  2266   (product(a::'a,b,additive_identity)) &
  2267   (~equal(a::'a,additive_identity)) &
  2268   (~equal(b::'a,additive_identity)) --> False"
  2269   oops
  2270 
  2271 (*101319 inferences so far.  Searching to depth 14.  76.0 secs*)
  2272 lemma ROB010_1:
  2273   "EQU001_0_ax equal &
  2274   (\<forall>Y X. equal(add(X::'a,Y),add(Y::'a,X))) &
  2275   (\<forall>X Y Z. equal(add(add(X::'a,Y),Z),add(X::'a,add(Y::'a,Z)))) &
  2276   (\<forall>Y X. equal(negate(add(negate(add(X::'a,Y)),negate(add(X::'a,negate(Y))))),X)) &
  2277   (\<forall>A B C. equal(A::'a,B) --> equal(add(A::'a,C),add(B::'a,C))) &
  2278   (\<forall>D F' E. equal(D::'a,E) --> equal(add(F'::'a,D),add(F'::'a,E))) &
  2279   (\<forall>G H. equal(G::'a,H) --> equal(negate(G),negate(H))) &
  2280   (equal(negate(add(a::'a,negate(b))),c)) &
  2281   (~equal(negate(add(c::'a,negate(add(b::'a,a)))),a)) --> False"
  2282   oops
  2283 
  2284 
  2285 (*6933 inferences so far.  Searching to depth 12.  5.1 secs*)
  2286 lemma ROB013_1:
  2287   "EQU001_0_ax equal &
  2288   (\<forall>Y X. equal(add(X::'a,Y),add(Y::'a,X))) &
  2289   (\<forall>X Y Z. equal(add(add(X::'a,Y),Z),add(X::'a,add(Y::'a,Z)))) &
  2290   (\<forall>Y X. equal(negate(add(negate(add(X::'a,Y)),negate(add(X::'a,negate(Y))))),X)) &
  2291   (\<forall>A B C. equal(A::'a,B) --> equal(add(A::'a,C),add(B::'a,C))) &
  2292   (\<forall>D F' E. equal(D::'a,E) --> equal(add(F'::'a,D),add(F'::'a,E))) &
  2293   (\<forall>G H. equal(G::'a,H) --> equal(negate(G),negate(H))) &
  2294   (equal(negate(add(a::'a,b)),c)) &
  2295   (~equal(negate(add(c::'a,negate(add(negate(b),a)))),a)) --> False"
  2296   by meson
  2297 
  2298 (*6614 inferences so far.  Searching to depth 11.  20.4 secs*)
  2299 lemma ROB016_1:
  2300   "EQU001_0_ax equal &
  2301   (\<forall>Y X. equal(add(X::'a,Y),add(Y::'a,X))) &
  2302   (\<forall>X Y Z. equal(add(add(X::'a,Y),Z),add(X::'a,add(Y::'a,Z)))) &
  2303   (\<forall>Y X. equal(negate(add(negate(add(X::'a,Y)),negate(add(X::'a,negate(Y))))),X)) &
  2304   (\<forall>A B C. equal(A::'a,B) --> equal(add(A::'a,C),add(B::'a,C))) &
  2305   (\<forall>D F' E. equal(D::'a,E) --> equal(add(F'::'a,D),add(F'::'a,E))) &
  2306   (\<forall>G H. equal(G::'a,H) --> equal(negate(G),negate(H))) &
  2307   (\<forall>J K' L. equal(J::'a,K') --> equal(multiply(J::'a,L),multiply(K'::'a,L))) &
  2308   (\<forall>M O' N. equal(M::'a,N) --> equal(multiply(O'::'a,M),multiply(O'::'a,N))) &
  2309   (\<forall>P Q. equal(P::'a,Q) --> equal(successor(P),successor(Q))) &
  2310   (\<forall>R S'. equal(R::'a,S') & positive_integer(R) --> positive_integer(S')) &
  2311   (\<forall>X. equal(multiply(One::'a,X),X)) &
  2312   (\<forall>V X. positive_integer(X) --> equal(multiply(successor(V),X),add(X::'a,multiply(V::'a,X)))) &
  2313   (positive_integer(One)) &
  2314   (\<forall>X. positive_integer(X) --> positive_integer(successor(X))) &
  2315   (equal(negate(add(d::'a,e)),negate(e))) &
  2316   (positive_integer(k)) &
  2317   (\<forall>Vk X Y. equal(negate(add(negate(Y),negate(add(X::'a,negate(Y))))),X) & positive_integer(Vk) --> equal(negate(add(Y::'a,multiply(Vk::'a,add(X::'a,negate(add(X::'a,negate(Y))))))),negate(Y))) &
  2318   (~equal(negate(add(e::'a,multiply(k::'a,add(d::'a,negate(add(d::'a,negate(e))))))),negate(e))) --> False"
  2319   oops
  2320 
  2321 (*14077 inferences so far.  Searching to depth 11.  32.8 secs*)
  2322 lemma ROB021_1:
  2323   "EQU001_0_ax equal &
  2324   (\<forall>Y X. equal(add(X::'a,Y),add(Y::'a,X))) &
  2325   (\<forall>X Y Z. equal(add(add(X::'a,Y),Z),add(X::'a,add(Y::'a,Z)))) &
  2326   (\<forall>Y X. equal(negate(add(negate(add(X::'a,Y)),negate(add(X::'a,negate(Y))))),X)) &
  2327   (\<forall>A B C. equal(A::'a,B) --> equal(add(A::'a,C),add(B::'a,C))) &
  2328   (\<forall>D F' E. equal(D::'a,E) --> equal(add(F'::'a,D),add(F'::'a,E))) &
  2329   (\<forall>G H. equal(G::'a,H) --> equal(negate(G),negate(H))) &
  2330   (\<forall>X Y. equal(negate(X),negate(Y)) --> equal(X::'a,Y)) &
  2331   (~equal(add(negate(add(a::'a,negate(b))),negate(add(negate(a),negate(b)))),b)) --> False"
  2332   oops
  2333 
  2334 (*35532 inferences so far.  Searching to depth 19.  54.3 secs*)
  2335 lemma SET005_1:
  2336  "(\<forall>Subset Element Superset. member(Element::'a,Subset) & subset(Subset::'a,Superset) --> member(Element::'a,Superset)) &
  2337   (\<forall>Superset Subset. subset(Subset::'a,Superset) | member(member_of_1_not_of_2(Subset::'a,Superset),Subset)) &
  2338   (\<forall>Subset Superset. member(member_of_1_not_of_2(Subset::'a,Superset),Superset) --> subset(Subset::'a,Superset)) &
  2339   (\<forall>Subset Superset. equal_sets(Subset::'a,Superset) --> subset(Subset::'a,Superset)) &
  2340   (\<forall>Subset Superset. equal_sets(Superset::'a,Subset) --> subset(Subset::'a,Superset)) &
  2341   (\<forall>Set2 Set1. subset(Set1::'a,Set2) & subset(Set2::'a,Set1) --> equal_sets(Set2::'a,Set1)) &
  2342   (\<forall>Set2 Intersection Element Set1. intersection(Set1::'a,Set2,Intersection) & member(Element::'a,Intersection) --> member(Element::'a,Set1)) &
  2343   (\<forall>Set1 Intersection Element Set2. intersection(Set1::'a,Set2,Intersection) & member(Element::'a,Intersection) --> member(Element::'a,Set2)) &
  2344   (\<forall>Set2 Set1 Element Intersection. intersection(Set1::'a,Set2,Intersection) & member(Element::'a,Set2) & member(Element::'a,Set1) --> member(Element::'a,Intersection)) &
  2345   (\<forall>Set2 Intersection Set1. member(h(Set1::'a,Set2,Intersection),Intersection) | intersection(Set1::'a,Set2,Intersection) | member(h(Set1::'a,Set2,Intersection),Set1)) &
  2346   (\<forall>Set1 Intersection Set2. member(h(Set1::'a,Set2,Intersection),Intersection) | intersection(Set1::'a,Set2,Intersection) | member(h(Set1::'a,Set2,Intersection),Set2)) &
  2347   (\<forall>Set1 Set2 Intersection. member(h(Set1::'a,Set2,Intersection),Intersection) & member(h(Set1::'a,Set2,Intersection),Set2) & member(h(Set1::'a,Set2,Intersection),Set1) --> intersection(Set1::'a,Set2,Intersection)) &
  2348   (intersection(a::'a,b,aIb)) &
  2349   (intersection(b::'a,c,bIc)) &
  2350   (intersection(a::'a,bIc,aIbIc)) &
  2351   (~intersection(aIb::'a,c,aIbIc)) --> False"
  2352   oops
  2353 
  2354 
  2355 (*6450 inferences so far.  Searching to depth 14.  4.2 secs*)
  2356 lemma SET009_1:
  2357   "(\<forall>Subset Element Superset. member(Element::'a,Subset) & ssubset(Subset::'a,Superset) --> member(Element::'a,Superset)) &
  2358   (\<forall>Superset Subset. ssubset(Subset::'a,Superset) | member(member_of_1_not_of_2(Subset::'a,Superset),Subset)) &
  2359   (\<forall>Subset Superset. member(member_of_1_not_of_2(Subset::'a,Superset),Superset) --> ssubset(Subset::'a,Superset)) &
  2360   (\<forall>Subset Superset. equal_sets(Subset::'a,Superset) --> ssubset(Subset::'a,Superset)) &
  2361   (\<forall>Subset Superset. equal_sets(Superset::'a,Subset) --> ssubset(Subset::'a,Superset)) &
  2362   (\<forall>Set2 Set1. ssubset(Set1::'a,Set2) & ssubset(Set2::'a,Set1) --> equal_sets(Set2::'a,Set1)) &
  2363   (\<forall>Set2 Difference Element Set1. difference(Set1::'a,Set2,Difference) & member(Element::'a,Difference) --> member(Element::'a,Set1)) &
  2364   (\<forall>Element A_set Set1 Set2. ~(member(Element::'a,Set1) & member(Element::'a,Set2) & difference(A_set::'a,Set1,Set2))) &
  2365   (\<forall>Set1 Difference Element Set2. member(Element::'a,Set1) & difference(Set1::'a,Set2,Difference) --> member(Element::'a,Difference) | member(Element::'a,Set2)) &
  2366   (\<forall>Set1 Set2 Difference. difference(Set1::'a,Set2,Difference) | member(k(Set1::'a,Set2,Difference),Set1) | member(k(Set1::'a,Set2,Difference),Difference)) &
  2367   (\<forall>Set1 Set2 Difference. member(k(Set1::'a,Set2,Difference),Set2) --> member(k(Set1::'a,Set2,Difference),Difference) | difference(Set1::'a,Set2,Difference)) &
  2368   (\<forall>Set1 Set2 Difference. member(k(Set1::'a,Set2,Difference),Difference) & member(k(Set1::'a,Set2,Difference),Set1) --> member(k(Set1::'a,Set2,Difference),Set2) | difference(Set1::'a,Set2,Difference)) &
  2369   (ssubset(d::'a,a)) &
  2370   (difference(b::'a,a,bDa)) &
  2371   (difference(b::'a,d,bDd)) &
  2372   (~ssubset(bDa::'a,bDd)) --> False"
  2373   by meson
  2374 
  2375 (*34726 inferences so far.  Searching to depth 6.  2420 secs: 40 mins! BIG*)
  2376 lemma SET025_4:
  2377   "EQU001_0_ax equal &
  2378   (\<forall>Y X. member(X::'a,Y) --> little_set(X)) &
  2379   (\<forall>X Y. little_set(f1(X::'a,Y)) | equal(X::'a,Y)) &
  2380   (\<forall>X Y. member(f1(X::'a,Y),X) | member(f1(X::'a,Y),Y) | equal(X::'a,Y)) &
  2381   (\<forall>X Y. member(f1(X::'a,Y),X) & member(f1(X::'a,Y),Y) --> equal(X::'a,Y)) &
  2382   (\<forall>X U Y. member(U::'a,non_ordered_pair(X::'a,Y)) --> equal(U::'a,X) | equal(U::'a,Y)) &
  2383   (\<forall>Y U X. little_set(U) & equal(U::'a,X) --> member(U::'a,non_ordered_pair(X::'a,Y))) &
  2384   (\<forall>X U Y. little_set(U) & equal(U::'a,Y) --> member(U::'a,non_ordered_pair(X::'a,Y))) &
  2385   (\<forall>X Y. little_set(non_ordered_pair(X::'a,Y))) &
  2386   (\<forall>X. equal(singleton_set(X),non_ordered_pair(X::'a,X))) &
  2387   (\<forall>X Y. equal(ordered_pair(X::'a,Y),non_ordered_pair(singleton_set(X),non_ordered_pair(X::'a,Y)))) &
  2388   (\<forall>X. ordered_pair_predicate(X) --> little_set(f2(X))) &
  2389   (\<forall>X. ordered_pair_predicate(X) --> little_set(f3(X))) &
  2390   (\<forall>X. ordered_pair_predicate(X) --> equal(X::'a,ordered_pair(f2(X),f3(X)))) &
  2391   (\<forall>X Y Z. little_set(Y) & little_set(Z) & equal(X::'a,ordered_pair(Y::'a,Z)) --> ordered_pair_predicate(X)) &
  2392   (\<forall>Z X. member(Z::'a,first(X)) --> little_set(f4(Z::'a,X))) &
  2393   (\<forall>Z X. member(Z::'a,first(X)) --> little_set(f5(Z::'a,X))) &
  2394   (\<forall>Z X. member(Z::'a,first(X)) --> equal(X::'a,ordered_pair(f4(Z::'a,X),f5(Z::'a,X)))) &
  2395   (\<forall>Z X. member(Z::'a,first(X)) --> member(Z::'a,f4(Z::'a,X))) &
  2396   (\<forall>X V Z U. little_set(U) & little_set(V) & equal(X::'a,ordered_pair(U::'a,V)) & member(Z::'a,U) --> member(Z::'a,first(X))) &
  2397   (\<forall>Z X. member(Z::'a,second(X)) --> little_set(f6(Z::'a,X))) &
  2398   (\<forall>Z X. member(Z::'a,second(X)) --> little_set(f7(Z::'a,X))) &
  2399   (\<forall>Z X. member(Z::'a,second(X)) --> equal(X::'a,ordered_pair(f6(Z::'a,X),f7(Z::'a,X)))) &
  2400   (\<forall>Z X. member(Z::'a,second(X)) --> member(Z::'a,f7(Z::'a,X))) &
  2401   (\<forall>X U Z V. little_set(U) & little_set(V) & equal(X::'a,ordered_pair(U::'a,V)) & member(Z::'a,V) --> member(Z::'a,second(X))) &
  2402   (\<forall>Z. member(Z::'a,estin) --> ordered_pair_predicate(Z)) &
  2403   (\<forall>Z. member(Z::'a,estin) --> member(first(Z),second(Z))) &
  2404   (\<forall>Z. little_set(Z) & ordered_pair_predicate(Z) & member(first(Z),second(Z)) --> member(Z::'a,estin)) &
  2405   (\<forall>Y Z X. member(Z::'a,intersection(X::'a,Y)) --> member(Z::'a,X)) &
  2406   (\<forall>X Z Y. member(Z::'a,intersection(X::'a,Y)) --> member(Z::'a,Y)) &
  2407   (\<forall>X Z Y. member(Z::'a,X) & member(Z::'a,Y) --> member(Z::'a,intersection(X::'a,Y))) &
  2408   (\<forall>Z X. ~(member(Z::'a,complement(X)) & member(Z::'a,X))) &
  2409   (\<forall>Z X. little_set(Z) --> member(Z::'a,complement(X)) | member(Z::'a,X)) &
  2410   (\<forall>X Y. equal(union(X::'a,Y),complement(intersection(complement(X),complement(Y))))) &
  2411   (\<forall>Z X. member(Z::'a,domain_of(X)) --> ordered_pair_predicate(f8(Z::'a,X))) &
  2412   (\<forall>Z X. member(Z::'a,domain_of(X)) --> member(f8(Z::'a,X),X)) &
  2413   (\<forall>Z X. member(Z::'a,domain_of(X)) --> equal(Z::'a,first(f8(Z::'a,X)))) &
  2414   (\<forall>X Z Xp. little_set(Z) & ordered_pair_predicate(Xp) & member(Xp::'a,X) & equal(Z::'a,first(Xp)) --> member(Z::'a,domain_of(X))) &
  2415   (\<forall>X Y Z. member(Z::'a,cross_product(X::'a,Y)) --> ordered_pair_predicate(Z)) &
  2416   (\<forall>Y Z X. member(Z::'a,cross_product(X::'a,Y)) --> member(first(Z),X)) &
  2417   (\<forall>X Z Y. member(Z::'a,cross_product(X::'a,Y)) --> member(second(Z),Y)) &
  2418   (\<forall>X Z Y. little_set(Z) & ordered_pair_predicate(Z) & member(first(Z),X) & member(second(Z),Y) --> member(Z::'a,cross_product(X::'a,Y))) &
  2419   (\<forall>X Z. member(Z::'a,inv1 X) --> ordered_pair_predicate(Z)) &
  2420   (\<forall>Z X. member(Z::'a,inv1 X) --> member(ordered_pair(second(Z),first(Z)),X)) &
  2421   (\<forall>Z X. little_set(Z) & ordered_pair_predicate(Z) & member(ordered_pair(second(Z),first(Z)),X) --> member(Z::'a,inv1 X)) &
  2422   (\<forall>Z X. member(Z::'a,rot_right(X)) --> little_set(f9(Z::'a,X))) &
  2423   (\<forall>Z X. member(Z::'a,rot_right(X)) --> little_set(f10(Z::'a,X))) &
  2424   (\<forall>Z X. member(Z::'a,rot_right(X)) --> little_set(f11(Z::'a,X))) &
  2425   (\<forall>Z X. member(Z::'a,rot_right(X)) --> equal(Z::'a,ordered_pair(f9(Z::'a,X),ordered_pair(f10(Z::'a,X),f11(Z::'a,X))))) &
  2426   (\<forall>Z X. member(Z::'a,rot_right(X)) --> member(ordered_pair(f10(Z::'a,X),ordered_pair(f11(Z::'a,X),f9(Z::'a,X))),X)) &
  2427   (\<forall>Z V W U X. little_set(Z) & little_set(U) & little_set(V) & little_set(W) & equal(Z::'a,ordered_pair(U::'a,ordered_pair(V::'a,W))) & member(ordered_pair(V::'a,ordered_pair(W::'a,U)),X) --> member(Z::'a,rot_right(X))) &
  2428   (\<forall>Z X. member(Z::'a,flip_range_of(X)) --> little_set(f12(Z::'a,X))) &
  2429   (\<forall>Z X. member(Z::'a,flip_range_of(X)) --> little_set(f13(Z::'a,X))) &
  2430   (\<forall>Z X. member(Z::'a,flip_range_of(X)) --> little_set(f14(Z::'a,X))) &
  2431   (\<forall>Z X. member(Z::'a,flip_range_of(X)) --> equal(Z::'a,ordered_pair(f12(Z::'a,X),ordered_pair(f13(Z::'a,X),f14(Z::'a,X))))) &
  2432   (\<forall>Z X. member(Z::'a,flip_range_of(X)) --> member(ordered_pair(f12(Z::'a,X),ordered_pair(f14(Z::'a,X),f13(Z::'a,X))),X)) &
  2433   (\<forall>Z U W V X. little_set(Z) & little_set(U) & little_set(V) & little_set(W) & equal(Z::'a,ordered_pair(U::'a,ordered_pair(V::'a,W))) & member(ordered_pair(U::'a,ordered_pair(W::'a,V)),X) --> member(Z::'a,flip_range_of(X))) &
  2434   (\<forall>X. equal(successor(X),union(X::'a,singleton_set(X)))) &
  2435   (\<forall>Z. ~member(Z::'a,empty_set)) &
  2436   (\<forall>Z. little_set(Z) --> member(Z::'a,universal_set)) &
  2437   (little_set(infinity)) &
  2438   (member(empty_set::'a,infinity)) &
  2439   (\<forall>X. member(X::'a,infinity) --> member(successor(X),infinity)) &
  2440   (\<forall>Z X. member(Z::'a,sigma(X)) --> member(f16(Z::'a,X),X)) &
  2441   (\<forall>Z X. member(Z::'a,sigma(X)) --> member(Z::'a,f16(Z::'a,X))) &
  2442   (\<forall>X Z Y. member(Y::'a,X) & member(Z::'a,Y) --> member(Z::'a,sigma(X))) &
  2443   (\<forall>U. little_set(U) --> little_set(sigma(U))) &
  2444   (\<forall>X U Y. ssubset(X::'a,Y) & member(U::'a,X) --> member(U::'a,Y)) &
  2445   (\<forall>Y X. ssubset(X::'a,Y) | member(f17(X::'a,Y),X)) &
  2446   (\<forall>X Y. member(f17(X::'a,Y),Y) --> ssubset(X::'a,Y)) &
  2447   (\<forall>X Y. proper_subset(X::'a,Y) --> ssubset(X::'a,Y)) &
  2448   (\<forall>X Y. ~(proper_subset(X::'a,Y) & equal(X::'a,Y))) &
  2449   (\<forall>X Y. ssubset(X::'a,Y) --> proper_subset(X::'a,Y) | equal(X::'a,Y)) &
  2450   (\<forall>Z X. member(Z::'a,powerset(X)) --> ssubset(Z::'a,X)) &
  2451   (\<forall>Z X. little_set(Z) & ssubset(Z::'a,X) --> member(Z::'a,powerset(X))) &
  2452   (\<forall>U. little_set(U) --> little_set(powerset(U))) &
  2453   (\<forall>Z X. relation(Z) & member(X::'a,Z) --> ordered_pair_predicate(X)) &
  2454   (\<forall>Z. relation(Z) | member(f18(Z),Z)) &
  2455   (\<forall>Z. ordered_pair_predicate(f18(Z)) --> relation(Z)) &
  2456   (\<forall>U X V W. single_valued_set(X) & little_set(U) & little_set(V) & little_set(W) & member(ordered_pair(U::'a,V),X) & member(ordered_pair(U::'a,W),X) --> equal(V::'a,W)) &
  2457   (\<forall>X. single_valued_set(X) | little_set(f19(X))) &
  2458   (\<forall>X. single_valued_set(X) | little_set(f20(X))) &
  2459   (\<forall>X. single_valued_set(X) | little_set(f21(X))) &
  2460   (\<forall>X. single_valued_set(X) | member(ordered_pair(f19(X),f20(X)),X)) &
  2461   (\<forall>X. single_valued_set(X) | member(ordered_pair(f19(X),f21(X)),X)) &
  2462   (\<forall>X. equal(f20(X),f21(X)) --> single_valued_set(X)) &
  2463   (\<forall>Xf. function(Xf) --> relation(Xf)) &
  2464   (\<forall>Xf. function(Xf) --> single_valued_set(Xf)) &
  2465   (\<forall>Xf. relation(Xf) & single_valued_set(Xf) --> function(Xf)) &
  2466   (\<forall>Z X Xf. member(Z::'a,image'(X::'a,Xf)) --> ordered_pair_predicate(f22(Z::'a,X,Xf))) &
  2467   (\<forall>Z X Xf. member(Z::'a,image'(X::'a,Xf)) --> member(f22(Z::'a,X,Xf),Xf)) &
  2468   (\<forall>Z Xf X. member(Z::'a,image'(X::'a,Xf)) --> member(first(f22(Z::'a,X,Xf)),X)) &
  2469   (\<forall>X Xf Z. member(Z::'a,image'(X::'a,Xf)) --> equal(second(f22(Z::'a,X,Xf)),Z)) &
  2470   (\<forall>Xf X Y Z. little_set(Z) & ordered_pair_predicate(Y) & member(Y::'a,Xf) & member(first(Y),X) & equal(second(Y),Z) --> member(Z::'a,image'(X::'a,Xf))) &
  2471   (\<forall>X Xf. little_set(X) & function(Xf) --> little_set(image'(X::'a,Xf))) &
  2472   (\<forall>X U Y. ~(disjoint(X::'a,Y) & member(U::'a,X) & member(U::'a,Y))) &
  2473   (\<forall>Y X. disjoint(X::'a,Y) | member(f23(X::'a,Y),X)) &
  2474   (\<forall>X Y. disjoint(X::'a,Y) | member(f23(X::'a,Y),Y)) &
  2475   (\<forall>X. equal(X::'a,empty_set) | member(f24(X),X)) &
  2476   (\<forall>X. equal(X::'a,empty_set) | disjoint(f24(X),X)) &
  2477   (function(f25)) &
  2478   (\<forall>X. little_set(X) --> equal(X::'a,empty_set) | member(f26(X),X)) &
  2479   (\<forall>X. little_set(X) --> equal(X::'a,empty_set) | member(ordered_pair(X::'a,f26(X)),f25)) &
  2480   (\<forall>Z X. member(Z::'a,range_of(X)) --> ordered_pair_predicate(f27(Z::'a,X))) &
  2481   (\<forall>Z X. member(Z::'a,range_of(X)) --> member(f27(Z::'a,X),X)) &
  2482   (\<forall>Z X. member(Z::'a,range_of(X)) --> equal(Z::'a,second(f27(Z::'a,X)))) &
  2483   (\<forall>X Z Xp. little_set(Z) & ordered_pair_predicate(Xp) & member(Xp::'a,X) & equal(Z::'a,second(Xp)) --> member(Z::'a,range_of(X))) &
  2484   (\<forall>Z. member(Z::'a,identity_relation) --> ordered_pair_predicate(Z)) &
  2485   (\<forall>Z. member(Z::'a,identity_relation) --> equal(first(Z),second(Z))) &
  2486   (\<forall>Z. little_set(Z) & ordered_pair_predicate(Z) & equal(first(Z),second(Z)) --> member(Z::'a,identity_relation)) &
  2487   (\<forall>X Y. equal(restrct(X::'a,Y),intersection(X::'a,cross_product(Y::'a,universal_set)))) &
  2488   (\<forall>Xf. one_to_one_function(Xf) --> function(Xf)) &
  2489   (\<forall>Xf. one_to_one_function(Xf) --> function(inv1 Xf)) &
  2490   (\<forall>Xf. function(Xf) & function(inv1 Xf) --> one_to_one_function(Xf)) &
  2491   (\<forall>Z Xf Y. member(Z::'a,apply(Xf::'a,Y)) --> ordered_pair_predicate(f28(Z::'a,Xf,Y))) &
  2492   (\<forall>Z Y Xf. member(Z::'a,apply(Xf::'a,Y)) --> member(f28(Z::'a,Xf,Y),Xf)) &
  2493   (\<forall>Z Xf Y. member(Z::'a,apply(Xf::'a,Y)) --> equal(first(f28(Z::'a,Xf,Y)),Y)) &
  2494   (\<forall>Z Xf Y. member(Z::'a,apply(Xf::'a,Y)) --> member(Z::'a,second(f28(Z::'a,Xf,Y)))) &
  2495   (\<forall>Xf Y Z W. ordered_pair_predicate(W) & member(W::'a,Xf) & equal(first(W),Y) & member(Z::'a,second(W)) --> member(Z::'a,apply(Xf::'a,Y))) &
  2496   (\<forall>Xf X Y. equal(apply_to_two_arguments(Xf::'a,X,Y),apply(Xf::'a,ordered_pair(X::'a,Y)))) &
  2497   (\<forall>X Y Xf. maps(Xf::'a,X,Y) --> function(Xf)) &
  2498   (\<forall>Y Xf X. maps(Xf::'a,X,Y) --> equal(domain_of(Xf),X)) &
  2499   (\<forall>X Xf Y. maps(Xf::'a,X,Y) --> ssubset(range_of(Xf),Y)) &
  2500   (\<forall>X Xf Y. function(Xf) & equal(domain_of(Xf),X) & ssubset(range_of(Xf),Y) --> maps(Xf::'a,X,Y)) &
  2501   (\<forall>Xf Xs. closed(Xs::'a,Xf) --> little_set(Xs)) &
  2502   (\<forall>Xs Xf. closed(Xs::'a,Xf) --> little_set(Xf)) &
  2503   (\<forall>Xf Xs. closed(Xs::'a,Xf) --> maps(Xf::'a,cross_product(Xs::'a,Xs),Xs)) &
  2504   (\<forall>Xf Xs. little_set(Xs) & little_set(Xf) & maps(Xf::'a,cross_product(Xs::'a,Xs),Xs) --> closed(Xs::'a,Xf)) &
  2505   (\<forall>Z Xf Xg. member(Z::'a,composition(Xf::'a,Xg)) --> little_set(f29(Z::'a,Xf,Xg))) &
  2506   (\<forall>Z Xf Xg. member(Z::'a,composition(Xf::'a,Xg)) --> little_set(f30(Z::'a,Xf,Xg))) &
  2507   (\<forall>Z Xf Xg. member(Z::'a,composition(Xf::'a,Xg)) --> little_set(f31(Z::'a,Xf,Xg))) &
  2508   (\<forall>Z Xf Xg. member(Z::'a,composition(Xf::'a,Xg)) --> equal(Z::'a,ordered_pair(f29(Z::'a,Xf,Xg),f30(Z::'a,Xf,Xg)))) &
  2509   (\<forall>Z Xg Xf. member(Z::'a,composition(Xf::'a,Xg)) --> member(ordered_pair(f29(Z::'a,Xf,Xg),f31(Z::'a,Xf,Xg)),Xf)) &
  2510   (\<forall>Z Xf Xg. member(Z::'a,composition(Xf::'a,Xg)) --> member(ordered_pair(f31(Z::'a,Xf,Xg),f30(Z::'a,Xf,Xg)),Xg)) &
  2511   (\<forall>Z X Xf W Y Xg. little_set(Z) & little_set(X) & little_set(Y) & little_set(W) & equal(Z::'a,ordered_pair(X::'a,Y)) & member(ordered_pair(X::'a,W),Xf) & member(ordered_pair(W::'a,Y),Xg) --> member(Z::'a,composition(Xf::'a,Xg))) &
  2512   (\<forall>Xh Xs2 Xf2 Xs1 Xf1. homomorphism(Xh::'a,Xs1,Xf1,Xs2,Xf2) --> closed(Xs1::'a,Xf1)) &
  2513   (\<forall>Xh Xs1 Xf1 Xs2 Xf2. homomorphism(Xh::'a,Xs1,Xf1,Xs2,Xf2) --> closed(Xs2::'a,Xf2)) &
  2514   (\<forall>Xf1 Xf2 Xh Xs1 Xs2. homomorphism(Xh::'a,Xs1,Xf1,Xs2,Xf2) --> maps(Xh::'a,Xs1,Xs2)) &
  2515   (\<forall>Xs2 Xs1 Xf1 Xf2 X Xh Y. homomorphism(Xh::'a,Xs1,Xf1,Xs2,Xf2) & member(X::'a,Xs1) & member(Y::'a,Xs1) --> equal(apply(Xh::'a,apply_to_two_arguments(Xf1::'a,X,Y)),apply_to_two_arguments(Xf2::'a,apply(Xh::'a,X),apply(Xh::'a,Y)))) &
  2516   (\<forall>Xh Xf1 Xs2 Xf2 Xs1. closed(Xs1::'a,Xf1) & closed(Xs2::'a,Xf2) & maps(Xh::'a,Xs1,Xs2) --> homomorphism(Xh::'a,Xs1,Xf1,Xs2,Xf2) | member(f32(Xh::'a,Xs1,Xf1,Xs2,Xf2),Xs1)) &
  2517   (\<forall>Xh Xf1 Xs2 Xf2 Xs1. closed(Xs1::'a,Xf1) & closed(Xs2::'a,Xf2) & maps(Xh::'a,Xs1,Xs2) --> homomorphism(Xh::'a,Xs1,Xf1,Xs2,Xf2) | member(f33(Xh::'a,Xs1,Xf1,Xs2,Xf2),Xs1)) &
  2518   (\<forall>Xh Xs1 Xf1 Xs2 Xf2. closed(Xs1::'a,Xf1) & closed(Xs2::'a,Xf2) & maps(Xh::'a,Xs1,Xs2) & equal(apply(Xh::'a,apply_to_two_arguments(Xf1::'a,f32(Xh::'a,Xs1,Xf1,Xs2,Xf2),f33(Xh::'a,Xs1,Xf1,Xs2,Xf2))),apply_to_two_arguments(Xf2::'a,apply(Xh::'a,f32(Xh::'a,Xs1,Xf1,Xs2,Xf2)),apply(Xh::'a,f33(Xh::'a,Xs1,Xf1,Xs2,Xf2)))) --> homomorphism(Xh::'a,Xs1,Xf1,Xs2,Xf2)) &
  2519   (\<forall>A B C. equal(A::'a,B) --> equal(f1(A::'a,C),f1(B::'a,C))) &
  2520   (\<forall>D F' E. equal(D::'a,E) --> equal(f1(F'::'a,D),f1(F'::'a,E))) &
  2521   (\<forall>A2 B2. equal(A2::'a,B2) --> equal(f2(A2),f2(B2))) &
  2522   (\<forall>G4 H4. equal(G4::'a,H4) --> equal(f3(G4),f3(H4))) &
  2523   (\<forall>O7 P7 Q7. equal(O7::'a,P7) --> equal(f4(O7::'a,Q7),f4(P7::'a,Q7))) &
  2524   (\<forall>R7 T7 S7. equal(R7::'a,S7) --> equal(f4(T7::'a,R7),f4(T7::'a,S7))) &
  2525   (\<forall>U7 V7 W7. equal(U7::'a,V7) --> equal(f5(U7::'a,W7),f5(V7::'a,W7))) &
  2526   (\<forall>X7 Z7 Y7. equal(X7::'a,Y7) --> equal(f5(Z7::'a,X7),f5(Z7::'a,Y7))) &
  2527   (\<forall>A8 B8 C8. equal(A8::'a,B8) --> equal(f6(A8::'a,C8),f6(B8::'a,C8))) &
  2528   (\<forall>D8 F8 E8. equal(D8::'a,E8) --> equal(f6(F8::'a,D8),f6(F8::'a,E8))) &
  2529   (\<forall>G8 H8 I8. equal(G8::'a,H8) --> equal(f7(G8::'a,I8),f7(H8::'a,I8))) &
  2530   (\<forall>J8 L8 K8. equal(J8::'a,K8) --> equal(f7(L8::'a,J8),f7(L8::'a,K8))) &
  2531   (\<forall>M8 N8 O8. equal(M8::'a,N8) --> equal(f8(M8::'a,O8),f8(N8::'a,O8))) &
  2532   (\<forall>P8 R8 Q8. equal(P8::'a,Q8) --> equal(f8(R8::'a,P8),f8(R8::'a,Q8))) &
  2533   (\<forall>S8 T8 U8. equal(S8::'a,T8) --> equal(f9(S8::'a,U8),f9(T8::'a,U8))) &
  2534   (\<forall>V8 X8 W8. equal(V8::'a,W8) --> equal(f9(X8::'a,V8),f9(X8::'a,W8))) &
  2535   (\<forall>G H I'. equal(G::'a,H) --> equal(f10(G::'a,I'),f10(H::'a,I'))) &
  2536   (\<forall>J L K'. equal(J::'a,K') --> equal(f10(L::'a,J),f10(L::'a,K'))) &
  2537   (\<forall>M N O'. equal(M::'a,N) --> equal(f11(M::'a,O'),f11(N::'a,O'))) &
  2538   (\<forall>P R Q. equal(P::'a,Q) --> equal(f11(R::'a,P),f11(R::'a,Q))) &
  2539   (\<forall>S' T' U. equal(S'::'a,T') --> equal(f12(S'::'a,U),f12(T'::'a,U))) &
  2540   (\<forall>V X W. equal(V::'a,W) --> equal(f12(X::'a,V),f12(X::'a,W))) &
  2541   (\<forall>Y Z A1. equal(Y::'a,Z) --> equal(f13(Y::'a,A1),f13(Z::'a,A1))) &
  2542   (\<forall>B1 D1 C1. equal(B1::'a,C1) --> equal(f13(D1::'a,B1),f13(D1::'a,C1))) &
  2543   (\<forall>E1 F1 G1. equal(E1::'a,F1) --> equal(f14(E1::'a,G1),f14(F1::'a,G1))) &
  2544   (\<forall>H1 J1 I1. equal(H1::'a,I1) --> equal(f14(J1::'a,H1),f14(J1::'a,I1))) &
  2545   (\<forall>K1 L1 M1. equal(K1::'a,L1) --> equal(f16(K1::'a,M1),f16(L1::'a,M1))) &
  2546   (\<forall>N1 P1 O1. equal(N1::'a,O1) --> equal(f16(P1::'a,N1),f16(P1::'a,O1))) &
  2547   (\<forall>Q1 R1 S1. equal(Q1::'a,R1) --> equal(f17(Q1::'a,S1),f17(R1::'a,S1))) &
  2548   (\<forall>T1 V1 U1. equal(T1::'a,U1) --> equal(f17(V1::'a,T1),f17(V1::'a,U1))) &
  2549   (\<forall>W1 X1. equal(W1::'a,X1) --> equal(f18(W1),f18(X1))) &
  2550   (\<forall>Y1 Z1. equal(Y1::'a,Z1) --> equal(f19(Y1),f19(Z1))) &
  2551   (\<forall>C2 D2. equal(C2::'a,D2) --> equal(f20(C2),f20(D2))) &
  2552   (\<forall>E2 F2. equal(E2::'a,F2) --> equal(f21(E2),f21(F2))) &
  2553   (\<forall>G2 H2 I2 J2. equal(G2::'a,H2) --> equal(f22(G2::'a,I2,J2),f22(H2::'a,I2,J2))) &
  2554   (\<forall>K2 M2 L2 N2. equal(K2::'a,L2) --> equal(f22(M2::'a,K2,N2),f22(M2::'a,L2,N2))) &
  2555   (\<forall>O2 Q2 R2 P2. equal(O2::'a,P2) --> equal(f22(Q2::'a,R2,O2),f22(Q2::'a,R2,P2))) &
  2556   (\<forall>S2 T2 U2. equal(S2::'a,T2) --> equal(f23(S2::'a,U2),f23(T2::'a,U2))) &
  2557   (\<forall>V2 X2 W2. equal(V2::'a,W2) --> equal(f23(X2::'a,V2),f23(X2::'a,W2))) &
  2558   (\<forall>Y2 Z2. equal(Y2::'a,Z2) --> equal(f24(Y2),f24(Z2))) &
  2559   (\<forall>A3 B3. equal(A3::'a,B3) --> equal(f26(A3),f26(B3))) &
  2560   (\<forall>C3 D3 E3. equal(C3::'a,D3) --> equal(f27(C3::'a,E3),f27(D3::'a,E3))) &
  2561   (\<forall>F3 H3 G3. equal(F3::'a,G3) --> equal(f27(H3::'a,F3),f27(H3::'a,G3))) &
  2562   (\<forall>I3 J3 K3 L3. equal(I3::'a,J3) --> equal(f28(I3::'a,K3,L3),f28(J3::'a,K3,L3))) &
  2563   (\<forall>M3 O3 N3 P3. equal(M3::'a,N3) --> equal(f28(O3::'a,M3,P3),f28(O3::'a,N3,P3))) &
  2564   (\<forall>Q3 S3 T3 R3. equal(Q3::'a,R3) --> equal(f28(S3::'a,T3,Q3),f28(S3::'a,T3,R3))) &
  2565   (\<forall>U3 V3 W3 X3. equal(U3::'a,V3) --> equal(f29(U3::'a,W3,X3),f29(V3::'a,W3,X3))) &
  2566   (\<forall>Y3 A4 Z3 B4. equal(Y3::'a,Z3) --> equal(f29(A4::'a,Y3,B4),f29(A4::'a,Z3,B4))) &
  2567   (\<forall>C4 E4 F4 D4. equal(C4::'a,D4) --> equal(f29(E4::'a,F4,C4),f29(E4::'a,F4,D4))) &
  2568   (\<forall>I4 J4 K4 L4. equal(I4::'a,J4) --> equal(f30(I4::'a,K4,L4),f30(J4::'a,K4,L4))) &
  2569   (\<forall>M4 O4 N4 P4. equal(M4::'a,N4) --> equal(f30(O4::'a,M4,P4),f30(O4::'a,N4,P4))) &
  2570   (\<forall>Q4 S4 T4 R4. equal(Q4::'a,R4) --> equal(f30(S4::'a,T4,Q4),f30(S4::'a,T4,R4))) &
  2571   (\<forall>U4 V4 W4 X4. equal(U4::'a,V4) --> equal(f31(U4::'a,W4,X4),f31(V4::'a,W4,X4))) &
  2572   (\<forall>Y4 A5 Z4 B5. equal(Y4::'a,Z4) --> equal(f31(A5::'a,Y4,B5),f31(A5::'a,Z4,B5))) &
  2573   (\<forall>C5 E5 F5 D5. equal(C5::'a,D5) --> equal(f31(E5::'a,F5,C5),f31(E5::'a,F5,D5))) &
  2574   (\<forall>G5 H5 I5 J5 K5 L5. equal(G5::'a,H5) --> equal(f32(G5::'a,I5,J5,K5,L5),f32(H5::'a,I5,J5,K5,L5))) &
  2575   (\<forall>M5 O5 N5 P5 Q5 R5. equal(M5::'a,N5) --> equal(f32(O5::'a,M5,P5,Q5,R5),f32(O5::'a,N5,P5,Q5,R5))) &
  2576   (\<forall>S5 U5 V5 T5 W5 X5. equal(S5::'a,T5) --> equal(f32(U5::'a,V5,S5,W5,X5),f32(U5::'a,V5,T5,W5,X5))) &
  2577   (\<forall>Y5 A6 B6 C6 Z5 D6. equal(Y5::'a,Z5) --> equal(f32(A6::'a,B6,C6,Y5,D6),f32(A6::'a,B6,C6,Z5,D6))) &
  2578   (\<forall>E6 G6 H6 I6 J6 F6. equal(E6::'a,F6) --> equal(f32(G6::'a,H6,I6,J6,E6),f32(G6::'a,H6,I6,J6,F6))) &
  2579   (\<forall>K6 L6 M6 N6 O6 P6. equal(K6::'a,L6) --> equal(f33(K6::'a,M6,N6,O6,P6),f33(L6::'a,M6,N6,O6,P6))) &
  2580   (\<forall>Q6 S6 R6 T6 U6 V6. equal(Q6::'a,R6) --> equal(f33(S6::'a,Q6,T6,U6,V6),f33(S6::'a,R6,T6,U6,V6))) &
  2581   (\<forall>W6 Y6 Z6 X6 A7 B7. equal(W6::'a,X6) --> equal(f33(Y6::'a,Z6,W6,A7,B7),f33(Y6::'a,Z6,X6,A7,B7))) &
  2582   (\<forall>C7 E7 F7 G7 D7 H7. equal(C7::'a,D7) --> equal(f33(E7::'a,F7,G7,C7,H7),f33(E7::'a,F7,G7,D7,H7))) &
  2583   (\<forall>I7 K7 L7 M7 N7 J7. equal(I7::'a,J7) --> equal(f33(K7::'a,L7,M7,N7,I7),f33(K7::'a,L7,M7,N7,J7))) &
  2584   (\<forall>A B C. equal(A::'a,B) --> equal(apply(A::'a,C),apply(B::'a,C))) &
  2585   (\<forall>D F' E. equal(D::'a,E) --> equal(apply(F'::'a,D),apply(F'::'a,E))) &
  2586   (\<forall>G H I' J. equal(G::'a,H) --> equal(apply_to_two_arguments(G::'a,I',J),apply_to_two_arguments(H::'a,I',J))) &
  2587   (\<forall>K' M L N. equal(K'::'a,L) --> equal(apply_to_two_arguments(M::'a,K',N),apply_to_two_arguments(M::'a,L,N))) &
  2588   (\<forall>O' Q R P. equal(O'::'a,P) --> equal(apply_to_two_arguments(Q::'a,R,O'),apply_to_two_arguments(Q::'a,R,P))) &
  2589   (\<forall>S' T'. equal(S'::'a,T') --> equal(complement(S'),complement(T'))) &
  2590   (\<forall>U V W. equal(U::'a,V) --> equal(composition(U::'a,W),composition(V::'a,W))) &
  2591   (\<forall>X Z Y. equal(X::'a,Y) --> equal(composition(Z::'a,X),composition(Z::'a,Y))) &
  2592   (\<forall>A1 B1. equal(A1::'a,B1) --> equal(inv1 A1,inv1 B1)) &
  2593   (\<forall>C1 D1 E1. equal(C1::'a,D1) --> equal(cross_product(C1::'a,E1),cross_product(D1::'a,E1))) &
  2594   (\<forall>F1 H1 G1. equal(F1::'a,G1) --> equal(cross_product(H1::'a,F1),cross_product(H1::'a,G1))) &
  2595   (\<forall>I1 J1. equal(I1::'a,J1) --> equal(domain_of(I1),domain_of(J1))) &
  2596   (\<forall>I10 J10. equal(I10::'a,J10) --> equal(first(I10),first(J10))) &
  2597   (\<forall>Q10 R10. equal(Q10::'a,R10) --> equal(flip_range_of(Q10),flip_range_of(R10))) &
  2598   (\<forall>S10 T10 U10. equal(S10::'a,T10) --> equal(image'(S10::'a,U10),image'(T10::'a,U10))) &
  2599   (\<forall>V10 X10 W10. equal(V10::'a,W10) --> equal(image'(X10::'a,V10),image'(X10::'a,W10))) &
  2600   (\<forall>Y10 Z10 A11. equal(Y10::'a,Z10) --> equal(intersection(Y10::'a,A11),intersection(Z10::'a,A11))) &
  2601   (\<forall>B11 D11 C11. equal(B11::'a,C11) --> equal(intersection(D11::'a,B11),intersection(D11::'a,C11))) &
  2602   (\<forall>E11 F11 G11. equal(E11::'a,F11) --> equal(non_ordered_pair(E11::'a,G11),non_ordered_pair(F11::'a,G11))) &
  2603   (\<forall>H11 J11 I11. equal(H11::'a,I11) --> equal(non_ordered_pair(J11::'a,H11),non_ordered_pair(J11::'a,I11))) &
  2604   (\<forall>K11 L11 M11. equal(K11::'a,L11) --> equal(ordered_pair(K11::'a,M11),ordered_pair(L11::'a,M11))) &
  2605   (\<forall>N11 P11 O11. equal(N11::'a,O11) --> equal(ordered_pair(P11::'a,N11),ordered_pair(P11::'a,O11))) &
  2606   (\<forall>Q11 R11. equal(Q11::'a,R11) --> equal(powerset(Q11),powerset(R11))) &
  2607   (\<forall>S11 T11. equal(S11::'a,T11) --> equal(range_of(S11),range_of(T11))) &
  2608   (\<forall>U11 V11 W11. equal(U11::'a,V11) --> equal(restrct(U11::'a,W11),restrct(V11::'a,W11))) &
  2609   (\<forall>X11 Z11 Y11. equal(X11::'a,Y11) --> equal(restrct(Z11::'a,X11),restrct(Z11::'a,Y11))) &
  2610   (\<forall>A12 B12. equal(A12::'a,B12) --> equal(rot_right(A12),rot_right(B12))) &
  2611   (\<forall>C12 D12. equal(C12::'a,D12) --> equal(second(C12),second(D12))) &
  2612   (\<forall>K12 L12. equal(K12::'a,L12) --> equal(sigma(K12),sigma(L12))) &
  2613   (\<forall>M12 N12. equal(M12::'a,N12) --> equal(singleton_set(M12),singleton_set(N12))) &
  2614   (\<forall>O12 P12. equal(O12::'a,P12) --> equal(successor(O12),successor(P12))) &
  2615   (\<forall>Q12 R12 S12. equal(Q12::'a,R12) --> equal(union(Q12::'a,S12),union(R12::'a,S12))) &
  2616   (\<forall>T12 V12 U12. equal(T12::'a,U12) --> equal(union(V12::'a,T12),union(V12::'a,U12))) &
  2617   (\<forall>W12 X12 Y12. equal(W12::'a,X12) & closed(W12::'a,Y12) --> closed(X12::'a,Y12)) &
  2618   (\<forall>Z12 B13 A13. equal(Z12::'a,A13) & closed(B13::'a,Z12) --> closed(B13::'a,A13)) &
  2619   (\<forall>C13 D13 E13. equal(C13::'a,D13) & disjoint(C13::'a,E13) --> disjoint(D13::'a,E13)) &
  2620   (\<forall>F13 H13 G13. equal(F13::'a,G13) & disjoint(H13::'a,F13) --> disjoint(H13::'a,G13)) &
  2621   (\<forall>I13 J13. equal(I13::'a,J13) & function(I13) --> function(J13)) &
  2622   (\<forall>K13 L13 M13 N13 O13 P13. equal(K13::'a,L13) & homomorphism(K13::'a,M13,N13,O13,P13) --> homomorphism(L13::'a,M13,N13,O13,P13)) &
  2623   (\<forall>Q13 S13 R13 T13 U13 V13. equal(Q13::'a,R13) & homomorphism(S13::'a,Q13,T13,U13,V13) --> homomorphism(S13::'a,R13,T13,U13,V13)) &
  2624   (\<forall>W13 Y13 Z13 X13 A14 B14. equal(W13::'a,X13) & homomorphism(Y13::'a,Z13,W13,A14,B14) --> homomorphism(Y13::'a,Z13,X13,A14,B14)) &
  2625   (\<forall>C14 E14 F14 G14 D14 H14. equal(C14::'a,D14) & homomorphism(E14::'a,F14,G14,C14,H14) --> homomorphism(E14::'a,F14,G14,D14,H14)) &
  2626   (\<forall>I14 K14 L14 M14 N14 J14. equal(I14::'a,J14) & homomorphism(K14::'a,L14,M14,N14,I14) --> homomorphism(K14::'a,L14,M14,N14,J14)) &
  2627   (\<forall>O14 P14. equal(O14::'a,P14) & little_set(O14) --> little_set(P14)) &
  2628   (\<forall>Q14 R14 S14 T14. equal(Q14::'a,R14) & maps(Q14::'a,S14,T14) --> maps(R14::'a,S14,T14)) &
  2629   (\<forall>U14 W14 V14 X14. equal(U14::'a,V14) & maps(W14::'a,U14,X14) --> maps(W14::'a,V14,X14)) &
  2630   (\<forall>Y14 A15 B15 Z14. equal(Y14::'a,Z14) & maps(A15::'a,B15,Y14) --> maps(A15::'a,B15,Z14)) &
  2631   (\<forall>C15 D15 E15. equal(C15::'a,D15) & member(C15::'a,E15) --> member(D15::'a,E15)) &
  2632   (\<forall>F15 H15 G15. equal(F15::'a,G15) & member(H15::'a,F15) --> member(H15::'a,G15)) &
  2633   (\<forall>I15 J15. equal(I15::'a,J15) & one_to_one_function(I15) --> one_to_one_function(J15)) &
  2634   (\<forall>K15 L15. equal(K15::'a,L15) & ordered_pair_predicate(K15) --> ordered_pair_predicate(L15)) &
  2635   (\<forall>M15 N15 O15. equal(M15::'a,N15) & proper_subset(M15::'a,O15) --> proper_subset(N15::'a,O15)) &
  2636   (\<forall>P15 R15 Q15. equal(P15::'a,Q15) & proper_subset(R15::'a,P15) --> proper_subset(R15::'a,Q15)) &
  2637   (\<forall>S15 T15. equal(S15::'a,T15) & relation(S15) --> relation(T15)) &
  2638   (\<forall>U15 V15. equal(U15::'a,V15) & single_valued_set(U15) --> single_valued_set(V15)) &
  2639   (\<forall>W15 X15 Y15. equal(W15::'a,X15) & ssubset(W15::'a,Y15) --> ssubset(X15::'a,Y15)) &
  2640   (\<forall>Z15 B16 A16. equal(Z15::'a,A16) & ssubset(B16::'a,Z15) --> ssubset(B16::'a,A16)) &
  2641   (~little_set(ordered_pair(a::'a,b))) --> False"
  2642   oops
  2643 
  2644 
  2645 (*13 inferences so far.  Searching to depth 8.  0 secs*)
  2646 lemma SET046_5:
  2647  "(\<forall>Y X. ~(element(X::'a,a) & element(X::'a,Y) & element(Y::'a,X))) &
  2648   (\<forall>X. element(X::'a,f(X)) | element(X::'a,a)) &
  2649   (\<forall>X. element(f(X),X) | element(X::'a,a)) --> False"
  2650    by meson
  2651 
  2652 (*33 inferences so far.  Searching to depth 9.  0.2 secs*)
  2653 lemma SET047_5:
  2654  "(\<forall>X Z Y. set_equal(X::'a,Y) & element(Z::'a,X) --> element(Z::'a,Y)) &
  2655   (\<forall>Y Z X. set_equal(X::'a,Y) & element(Z::'a,Y) --> element(Z::'a,X)) &
  2656   (\<forall>X Y. element(f(X::'a,Y),X) | element(f(X::'a,Y),Y) | set_equal(X::'a,Y)) &
  2657   (\<forall>X Y. element(f(X::'a,Y),Y) & element(f(X::'a,Y),X) --> set_equal(X::'a,Y)) &
  2658   (set_equal(a::'a,b) | set_equal(b::'a,a)) &
  2659   (~(set_equal(b::'a,a) & set_equal(a::'a,b))) --> False"
  2660   by meson
  2661 
  2662 (*311 inferences so far.  Searching to depth 12.  0.1 secs*)
  2663 lemma SYN034_1:
  2664  "(\<forall>A. p(A::'a,a) | p(A::'a,f(A))) &
  2665   (\<forall>A. p(A::'a,a) | p(f(A),A)) &
  2666   (\<forall>A B. ~(p(A::'a,B) & p(B::'a,A) & p(B::'a,a))) --> False"
  2667   by meson
  2668 
  2669 (*30 inferences so far.  Searching to depth 6.  0.2 secs*)
  2670 lemma SYN071_1:
  2671   "EQU001_0_ax equal &
  2672   (equal(a::'a,b) | equal(c::'a,d)) &
  2673   (equal(a::'a,c) | equal(b::'a,d)) &
  2674   (~equal(a::'a,d)) &
  2675   (~equal(b::'a,c)) --> False"
  2676   by meson
  2677 
  2678 (*1897410 inferences so far.  Searching to depth 48
  2679   206s, nearly 4 mins on griffon.*)
  2680 lemma SYN349_1:
  2681  "(\<forall>X Y. f(w(X),g(X::'a,Y)) --> f(X::'a,g(X::'a,Y))) &
  2682   (\<forall>X Y. f(X::'a,g(X::'a,Y)) --> f(w(X),g(X::'a,Y))) &
  2683   (\<forall>Y X. f(X::'a,g(X::'a,Y)) & f(Y::'a,g(X::'a,Y)) --> f(g(X::'a,Y),Y) | f(g(X::'a,Y),w(X))) &
  2684   (\<forall>Y X. f(g(X::'a,Y),Y) & f(Y::'a,g(X::'a,Y)) --> f(X::'a,g(X::'a,Y)) | f(g(X::'a,Y),w(X))) &
  2685   (\<forall>Y X. f(X::'a,g(X::'a,Y)) | f(g(X::'a,Y),Y) | f(Y::'a,g(X::'a,Y)) | f(g(X::'a,Y),w(X))) &
  2686   (\<forall>Y X. f(X::'a,g(X::'a,Y)) & f(g(X::'a,Y),Y) --> f(Y::'a,g(X::'a,Y)) | f(g(X::'a,Y),w(X))) &
  2687   (\<forall>Y X. f(X::'a,g(X::'a,Y)) & f(g(X::'a,Y),w(X)) --> f(g(X::'a,Y),Y) | f(Y::'a,g(X::'a,Y))) &
  2688   (\<forall>Y X. f(g(X::'a,Y),Y) & f(g(X::'a,Y),w(X)) --> f(X::'a,g(X::'a,Y)) | f(Y::'a,g(X::'a,Y))) &
  2689   (\<forall>Y X. f(Y::'a,g(X::'a,Y)) & f(g(X::'a,Y),w(X)) --> f(X::'a,g(X::'a,Y)) | f(g(X::'a,Y),Y)) &
  2690   (\<forall>Y X. ~(f(X::'a,g(X::'a,Y)) & f(g(X::'a,Y),Y) & f(Y::'a,g(X::'a,Y)) & f(g(X::'a,Y),w(X)))) --> False"
  2691    oops
  2692 
  2693 (*398 inferences so far.  Searching to depth 12.  0.4 secs*)
  2694 lemma SYN352_1:
  2695  "(f(a::'a,b)) &
  2696   (\<forall>X Y. f(X::'a,Y) --> f(b::'a,z(X::'a,Y)) | f(Y::'a,z(X::'a,Y))) &
  2697   (\<forall>X Y. f(X::'a,Y) | f(z(X::'a,Y),z(X::'a,Y))) &
  2698   (\<forall>X Y. f(b::'a,z(X::'a,Y)) | f(X::'a,z(X::'a,Y)) | f(z(X::'a,Y),z(X::'a,Y))) &
  2699   (\<forall>X Y. f(b::'a,z(X::'a,Y)) & f(X::'a,z(X::'a,Y)) --> f(z(X::'a,Y),z(X::'a,Y))) &
  2700   (\<forall>X Y. ~(f(X::'a,Y) & f(X::'a,z(X::'a,Y)) & f(Y::'a,z(X::'a,Y)))) &
  2701   (\<forall>X Y. f(X::'a,Y) --> f(X::'a,z(X::'a,Y)) | f(Y::'a,z(X::'a,Y))) --> False"
  2702   by meson
  2703 
  2704 (*5336 inferences so far.  Searching to depth 15.  5.3 secs*)
  2705 lemma TOP001_2:
  2706  "(\<forall>Vf U. element_of_set(U::'a,union_of_members(Vf)) --> element_of_set(U::'a,f1(Vf::'a,U))) &
  2707   (\<forall>U Vf. element_of_set(U::'a,union_of_members(Vf)) --> element_of_collection(f1(Vf::'a,U),Vf)) &
  2708   (\<forall>U Uu1 Vf. element_of_set(U::'a,Uu1) & element_of_collection(Uu1::'a,Vf) --> element_of_set(U::'a,union_of_members(Vf))) &
  2709   (\<forall>Vf X. basis(X::'a,Vf) --> equal_sets(union_of_members(Vf),X)) &
  2710   (\<forall>Vf U X. element_of_collection(U::'a,top_of_basis(Vf)) & element_of_set(X::'a,U) --> element_of_set(X::'a,f10(Vf::'a,U,X))) &
  2711   (\<forall>U X Vf. element_of_collection(U::'a,top_of_basis(Vf)) & element_of_set(X::'a,U) --> element_of_collection(f10(Vf::'a,U,X),Vf)) &
  2712   (\<forall>X. subset_sets(X::'a,X)) &
  2713   (\<forall>X U Y. subset_sets(X::'a,Y) & element_of_set(U::'a,X) --> element_of_set(U::'a,Y)) &
  2714   (\<forall>X Y. equal_sets(X::'a,Y) --> subset_sets(X::'a,Y)) &
  2715   (\<forall>Y X. subset_sets(X::'a,Y) | element_of_set(in_1st_set(X::'a,Y),X)) &
  2716   (\<forall>X Y. element_of_set(in_1st_set(X::'a,Y),Y) --> subset_sets(X::'a,Y)) &
  2717   (basis(cx::'a,f)) &
  2718   (~subset_sets(union_of_members(top_of_basis(f)),cx)) --> False"
  2719   by meson
  2720 
  2721 (*0 inferences so far.  Searching to depth 0.  0 secs*)
  2722 lemma TOP002_2:
  2723  "(\<forall>Vf U. element_of_collection(U::'a,top_of_basis(Vf)) | element_of_set(f11(Vf::'a,U),U)) &
  2724   (\<forall>X. ~element_of_set(X::'a,empty_set)) &
  2725   (~element_of_collection(empty_set::'a,top_of_basis(f))) --> False"
  2726   by meson
  2727 
  2728 (*0 inferences so far.  Searching to depth 0.  6.5 secs.  BIG*)
  2729 lemma TOP004_1:
  2730  "(\<forall>Vf U. element_of_set(U::'a,union_of_members(Vf)) --> element_of_set(U::'a,f1(Vf::'a,U))) &
  2731   (\<forall>U Vf. element_of_set(U::'a,union_of_members(Vf)) --> element_of_collection(f1(Vf::'a,U),Vf)) &
  2732   (\<forall>U Uu1 Vf. element_of_set(U::'a,Uu1) & element_of_collection(Uu1::'a,Vf) --> element_of_set(U::'a,union_of_members(Vf))) &
  2733   (\<forall>Vf U Va. element_of_set(U::'a,intersection_of_members(Vf)) & element_of_collection(Va::'a,Vf) --> element_of_set(U::'a,Va)) &
  2734   (\<forall>U Vf. element_of_set(U::'a,intersection_of_members(Vf)) | element_of_collection(f2(Vf::'a,U),Vf)) &
  2735   (\<forall>Vf U. element_of_set(U::'a,f2(Vf::'a,U)) --> element_of_set(U::'a,intersection_of_members(Vf))) &
  2736   (\<forall>Vt X. topological_space(X::'a,Vt) --> equal_sets(union_of_members(Vt),X)) &
  2737   (\<forall>X Vt. topological_space(X::'a,Vt) --> element_of_collection(empty_set::'a,Vt)) &
  2738   (\<forall>X Vt. topological_space(X::'a,Vt) --> element_of_collection(X::'a,Vt)) &
  2739   (\<forall>X Y Z Vt. topological_space(X::'a,Vt) & element_of_collection(Y::'a,Vt) & element_of_collection(Z::'a,Vt) --> element_of_collection(intersection_of_sets(Y::'a,Z),Vt)) &
  2740   (\<forall>X Vf Vt. topological_space(X::'a,Vt) & subset_collections(Vf::'a,Vt) --> element_of_collection(union_of_members(Vf),Vt)) &
  2741   (\<forall>X Vt. equal_sets(union_of_members(Vt),X) & element_of_collection(empty_set::'a,Vt) & element_of_collection(X::'a,Vt) --> topological_space(X::'a,Vt) | element_of_collection(f3(X::'a,Vt),Vt) | subset_collections(f5(X::'a,Vt),Vt)) &
  2742   (\<forall>X Vt. equal_sets(union_of_members(Vt),X) & element_of_collection(empty_set::'a,Vt) & element_of_collection(X::'a,Vt) & element_of_collection(union_of_members(f5(X::'a,Vt)),Vt) --> topological_space(X::'a,Vt) | element_of_collection(f3(X::'a,Vt),Vt)) &
  2743   (\<forall>X Vt. equal_sets(union_of_members(Vt),X) & element_of_collection(empty_set::'a,Vt) & element_of_collection(X::'a,Vt) --> topological_space(X::'a,Vt) | element_of_collection(f4(X::'a,Vt),Vt) | subset_collections(f5(X::'a,Vt),Vt)) &
  2744   (\<forall>X Vt. equal_sets(union_of_members(Vt),X) & element_of_collection(empty_set::'a,Vt) & element_of_collection(X::'a,Vt) & element_of_collection(union_of_members(f5(X::'a,Vt)),Vt) --> topological_space(X::'a,Vt) | element_of_collection(f4(X::'a,Vt),Vt)) &
  2745   (\<forall>X Vt. equal_sets(union_of_members(Vt),X) & element_of_collection(empty_set::'a,Vt) & element_of_collection(X::'a,Vt) & element_of_collection(intersection_of_sets(f3(X::'a,Vt),f4(X::'a,Vt)),Vt) --> topological_space(X::'a,Vt) | subset_collections(f5(X::'a,Vt),Vt)) &
  2746   (\<forall>X Vt. equal_sets(union_of_members(Vt),X) & element_of_collection(empty_set::'a,Vt) & element_of_collection(X::'a,Vt) & element_of_collection(intersection_of_sets(f3(X::'a,Vt),f4(X::'a,Vt)),Vt) & element_of_collection(union_of_members(f5(X::'a,Vt)),Vt) --> topological_space(X::'a,Vt)) &
  2747   (\<forall>U X Vt. open(U::'a,X,Vt) --> topological_space(X::'a,Vt)) &
  2748   (\<forall>X U Vt. open(U::'a,X,Vt) --> element_of_collection(U::'a,Vt)) &
  2749   (\<forall>X U Vt. topological_space(X::'a,Vt) & element_of_collection(U::'a,Vt) --> open(U::'a,X,Vt)) &
  2750   (\<forall>U X Vt. closed(U::'a,X,Vt) --> topological_space(X::'a,Vt)) &
  2751   (\<forall>U X Vt. closed(U::'a,X,Vt) --> open(relative_complement_sets(U::'a,X),X,Vt)) &
  2752   (\<forall>U X Vt. topological_space(X::'a,Vt) & open(relative_complement_sets(U::'a,X),X,Vt) --> closed(U::'a,X,Vt)) &
  2753   (\<forall>Vs X Vt. finer(Vt::'a,Vs,X) --> topological_space(X::'a,Vt)) &
  2754   (\<forall>Vt X Vs. finer(Vt::'a,Vs,X) --> topological_space(X::'a,Vs)) &
  2755   (\<forall>X Vs Vt. finer(Vt::'a,Vs,X) --> subset_collections(Vs::'a,Vt)) &
  2756   (\<forall>X Vs Vt. topological_space(X::'a,Vt) & topological_space(X::'a,Vs) & subset_collections(Vs::'a,Vt) --> finer(Vt::'a,Vs,X)) &
  2757   (\<forall>Vf X. basis(X::'a,Vf) --> equal_sets(union_of_members(Vf),X)) &
  2758   (\<forall>X Vf Y Vb1 Vb2. basis(X::'a,Vf) & element_of_set(Y::'a,X) & element_of_collection(Vb1::'a,Vf) & element_of_collection(Vb2::'a,Vf) & element_of_set(Y::'a,intersection_of_sets(Vb1::'a,Vb2)) --> element_of_set(Y::'a,f6(X::'a,Vf,Y,Vb1,Vb2))) &
  2759   (\<forall>X Y Vb1 Vb2 Vf. basis(X::'a,Vf) & element_of_set(Y::'a,X) & element_of_collection(Vb1::'a,Vf) & element_of_collection(Vb2::'a,Vf) & element_of_set(Y::'a,intersection_of_sets(Vb1::'a,Vb2)) --> element_of_collection(f6(X::'a,Vf,Y,Vb1,Vb2),Vf)) &
  2760   (\<forall>X Vf Y Vb1 Vb2. basis(X::'a,Vf) & element_of_set(Y::'a,X) & element_of_collection(Vb1::'a,Vf) & element_of_collection(Vb2::'a,Vf) & element_of_set(Y::'a,intersection_of_sets(Vb1::'a,Vb2)) --> subset_sets(f6(X::'a,Vf,Y,Vb1,Vb2),intersection_of_sets(Vb1::'a,Vb2))) &
  2761   (\<forall>Vf X. equal_sets(union_of_members(Vf),X) --> basis(X::'a,Vf) | element_of_set(f7(X::'a,Vf),X)) &
  2762   (\<forall>X Vf. equal_sets(union_of_members(Vf),X) --> basis(X::'a,Vf) | element_of_collection(f8(X::'a,Vf),Vf)) &
  2763   (\<forall>X Vf. equal_sets(union_of_members(Vf),X) --> basis(X::'a,Vf) | element_of_collection(f9(X::'a,Vf),Vf)) &
  2764   (\<forall>X Vf. equal_sets(union_of_members(Vf),X) --> basis(X::'a,Vf) | element_of_set(f7(X::'a,Vf),intersection_of_sets(f8(X::'a,Vf),f9(X::'a,Vf)))) &
  2765   (\<forall>Uu9 X Vf. equal_sets(union_of_members(Vf),X) & element_of_set(f7(X::'a,Vf),Uu9) & element_of_collection(Uu9::'a,Vf) & subset_sets(Uu9::'a,intersection_of_sets(f8(X::'a,Vf),f9(X::'a,Vf))) --> basis(X::'a,Vf)) &
  2766   (\<forall>Vf U X. element_of_collection(U::'a,top_of_basis(Vf)) & element_of_set(X::'a,U) --> element_of_set(X::'a,f10(Vf::'a,U,X))) &
  2767   (\<forall>U X Vf. element_of_collection(U::'a,top_of_basis(Vf)) & element_of_set(X::'a,U) --> element_of_collection(f10(Vf::'a,U,X),Vf)) &
  2768   (\<forall>Vf X U. element_of_collection(U::'a,top_of_basis(Vf)) & element_of_set(X::'a,U) --> subset_sets(f10(Vf::'a,U,X),U)) &
  2769   (\<forall>Vf U. element_of_collection(U::'a,top_of_basis(Vf)) | element_of_set(f11(Vf::'a,U),U)) &
  2770   (\<forall>Vf Uu11 U. element_of_set(f11(Vf::'a,U),Uu11) & element_of_collection(Uu11::'a,Vf) & subset_sets(Uu11::'a,U) --> element_of_collection(U::'a,top_of_basis(Vf))) &
  2771   (\<forall>U Y X Vt. element_of_collection(U::'a,subspace_topology(X::'a,Vt,Y)) --> topological_space(X::'a,Vt)) &
  2772   (\<forall>U Vt Y X. element_of_collection(U::'a,subspace_topology(X::'a,Vt,Y)) --> subset_sets(Y::'a,X)) &
  2773   (\<forall>X Y U Vt. element_of_collection(U::'a,subspace_topology(X::'a,Vt,Y)) --> element_of_collection(f12(X::'a,Vt,Y,U),Vt)) &
  2774   (\<forall>X Vt Y U. element_of_collection(U::'a,subspace_topology(X::'a,Vt,Y)) --> equal_sets(U::'a,intersection_of_sets(Y::'a,f12(X::'a,Vt,Y,U)))) &
  2775   (\<forall>X Vt U Y Uu12. topological_space(X::'a,Vt) & subset_sets(Y::'a,X) & element_of_collection(Uu12::'a,Vt) & equal_sets(U::'a,intersection_of_sets(Y::'a,Uu12)) --> element_of_collection(U::'a,subspace_topology(X::'a,Vt,Y))) &
  2776   (\<forall>U Y X Vt. element_of_set(U::'a,interior(Y::'a,X,Vt)) --> topological_space(X::'a,Vt)) &
  2777   (\<forall>U Vt Y X. element_of_set(U::'a,interior(Y::'a,X,Vt)) --> subset_sets(Y::'a,X)) &
  2778   (\<forall>Y X Vt U. element_of_set(U::'a,interior(Y::'a,X,Vt)) --> element_of_set(U::'a,f13(Y::'a,X,Vt,U))) &
  2779   (\<forall>X Vt U Y. element_of_set(U::'a,interior(Y::'a,X,Vt)) --> subset_sets(f13(Y::'a,X,Vt,U),Y)) &
  2780   (\<forall>Y U X Vt. element_of_set(U::'a,interior(Y::'a,X,Vt)) --> open(f13(Y::'a,X,Vt,U),X,Vt)) &
  2781   (\<forall>U Y Uu13 X Vt. topological_space(X::'a,Vt) & subset_sets(Y::'a,X) & element_of_set(U::'a,Uu13) & subset_sets(Uu13::'a,Y) & open(Uu13::'a,X,Vt) --> element_of_set(U::'a,interior(Y::'a,X,Vt))) &
  2782   (\<forall>U Y X Vt. element_of_set(U::'a,closure(Y::'a,X,Vt)) --> topological_space(X::'a,Vt)) &
  2783   (\<forall>U Vt Y X. element_of_set(U::'a,closure(Y::'a,X,Vt)) --> subset_sets(Y::'a,X)) &
  2784   (\<forall>Y X Vt U V. element_of_set(U::'a,closure(Y::'a,X,Vt)) & subset_sets(Y::'a,V) & closed(V::'a,X,Vt) --> element_of_set(U::'a,V)) &
  2785   (\<forall>Y X Vt U. topological_space(X::'a,Vt) & subset_sets(Y::'a,X) --> element_of_set(U::'a,closure(Y::'a,X,Vt)) | subset_sets(Y::'a,f14(Y::'a,X,Vt,U))) &
  2786   (\<forall>Y U X Vt. topological_space(X::'a,Vt) & subset_sets(Y::'a,X) --> element_of_set(U::'a,closure(Y::'a,X,Vt)) | closed(f14(Y::'a,X,Vt,U),X,Vt)) &
  2787   (\<forall>Y X Vt U. topological_space(X::'a,Vt) & subset_sets(Y::'a,X) & element_of_set(U::'a,f14(Y::'a,X,Vt,U)) --> element_of_set(U::'a,closure(Y::'a,X,Vt))) &
  2788   (\<forall>U Y X Vt. neighborhood(U::'a,Y,X,Vt) --> topological_space(X::'a,Vt)) &
  2789   (\<forall>Y U X Vt. neighborhood(U::'a,Y,X,Vt) --> open(U::'a,X,Vt)) &
  2790   (\<forall>X Vt Y U. neighborhood(U::'a,Y,X,Vt) --> element_of_set(Y::'a,U)) &
  2791   (\<forall>X Vt Y U. topological_space(X::'a,Vt) & open(U::'a,X,Vt) & element_of_set(Y::'a,U) --> neighborhood(U::'a,Y,X,Vt)) &
  2792   (\<forall>Z Y X Vt. limit_point(Z::'a,Y,X,Vt) --> topological_space(X::'a,Vt)) &
  2793   (\<forall>Z Vt Y X. limit_point(Z::'a,Y,X,Vt) --> subset_sets(Y::'a,X)) &
  2794   (\<forall>Z X Vt U Y. limit_point(Z::'a,Y,X,Vt) & neighborhood(U::'a,Z,X,Vt) --> element_of_set(f15(Z::'a,Y,X,Vt,U),intersection_of_sets(U::'a,Y))) &
  2795   (\<forall>Y X Vt U Z. ~(limit_point(Z::'a,Y,X,Vt) & neighborhood(U::'a,Z,X,Vt) & eq_p(f15(Z::'a,Y,X,Vt,U),Z))) &
  2796   (\<forall>Y Z X Vt. topological_space(X::'a,Vt) & subset_sets(Y::'a,X) --> limit_point(Z::'a,Y,X,Vt) | neighborhood(f16(Z::'a,Y,X,Vt),Z,X,Vt)) &
  2797   (\<forall>X Vt Y Uu16 Z. topological_space(X::'a,Vt) & subset_sets(Y::'a,X) & element_of_set(Uu16::'a,intersection_of_sets(f16(Z::'a,Y,X,Vt),Y)) --> limit_point(Z::'a,Y,X,Vt) | eq_p(Uu16::'a,Z)) &
  2798   (\<forall>U Y X Vt. element_of_set(U::'a,boundary(Y::'a,X,Vt)) --> topological_space(X::'a,Vt)) &
  2799   (\<forall>U Y X Vt. element_of_set(U::'a,boundary(Y::'a,X,Vt)) --> element_of_set(U::'a,closure(Y::'a,X,Vt))) &
  2800   (\<forall>U Y X Vt. element_of_set(U::'a,boundary(Y::'a,X,Vt)) --> element_of_set(U::'a,closure(relative_complement_sets(Y::'a,X),X,Vt))) &
  2801   (\<forall>U Y X Vt. topological_space(X::'a,Vt) & element_of_set(U::'a,closure(Y::'a,X,Vt)) & element_of_set(U::'a,closure(relative_complement_sets(Y::'a,X),X,Vt)) --> element_of_set(U::'a,boundary(Y::'a,X,Vt))) &
  2802   (\<forall>X Vt. hausdorff(X::'a,Vt) --> topological_space(X::'a,Vt)) &
  2803   (\<forall>X_2 X_1 X Vt. hausdorff(X::'a,Vt) & element_of_set(X_1::'a,X) & element_of_set(X_2::'a,X) --> eq_p(X_1::'a,X_2) | neighborhood(f17(X::'a,Vt,X_1,X_2),X_1,X,Vt)) &
  2804   (\<forall>X_1 X_2 X Vt. hausdorff(X::'a,Vt) & element_of_set(X_1::'a,X) & element_of_set(X_2::'a,X) --> eq_p(X_1::'a,X_2) | neighborhood(f18(X::'a,Vt,X_1,X_2),X_2,X,Vt)) &
  2805   (\<forall>X Vt X_1 X_2. hausdorff(X::'a,Vt) & element_of_set(X_1::'a,X) & element_of_set(X_2::'a,X) --> eq_p(X_1::'a,X_2) | disjoint_s(f17(X::'a,Vt,X_1,X_2),f18(X::'a,Vt,X_1,X_2))) &
  2806   (\<forall>Vt X. topological_space(X::'a,Vt) --> hausdorff(X::'a,Vt) | element_of_set(f19(X::'a,Vt),X)) &
  2807   (\<forall>Vt X. topological_space(X::'a,Vt) --> hausdorff(X::'a,Vt) | element_of_set(f20(X::'a,Vt),X)) &
  2808   (\<forall>X Vt. topological_space(X::'a,Vt) & eq_p(f19(X::'a,Vt),f20(X::'a,Vt)) --> hausdorff(X::'a,Vt)) &
  2809   (\<forall>X Vt Uu19 Uu20. topological_space(X::'a,Vt) & neighborhood(Uu19::'a,f19(X::'a,Vt),X,Vt) & neighborhood(Uu20::'a,f20(X::'a,Vt),X,Vt) & disjoint_s(Uu19::'a,Uu20) --> hausdorff(X::'a,Vt)) &
  2810   (\<forall>Va1 Va2 X Vt. separation(Va1::'a,Va2,X,Vt) --> topological_space(X::'a,Vt)) &
  2811   (\<forall>Va2 X Vt Va1. ~(separation(Va1::'a,Va2,X,Vt) & equal_sets(Va1::'a,empty_set))) &
  2812   (\<forall>Va1 X Vt Va2. ~(separation(Va1::'a,Va2,X,Vt) & equal_sets(Va2::'a,empty_set))) &
  2813   (\<forall>Va2 X Va1 Vt. separation(Va1::'a,Va2,X,Vt) --> element_of_collection(Va1::'a,Vt)) &
  2814   (\<forall>Va1 X Va2 Vt. separation(Va1::'a,Va2,X,Vt) --> element_of_collection(Va2::'a,Vt)) &
  2815   (\<forall>Vt Va1 Va2 X. separation(Va1::'a,Va2,X,Vt) --> equal_sets(union_of_sets(Va1::'a,Va2),X)) &
  2816   (\<forall>X Vt Va1 Va2. separation(Va1::'a,Va2,X,Vt) --> disjoint_s(Va1::'a,Va2)) &
  2817   (\<forall>Vt X Va1 Va2. topological_space(X::'a,Vt) & element_of_collection(Va1::'a,Vt) & element_of_collection(Va2::'a,Vt) & equal_sets(union_of_sets(Va1::'a,Va2),X) & disjoint_s(Va1::'a,Va2) --> separation(Va1::'a,Va2,X,Vt) | equal_sets(Va1::'a,empty_set) | equal_sets(Va2::'a,empty_set)) &
  2818   (\<forall>X Vt. connected_space(X::'a,Vt) --> topological_space(X::'a,Vt)) &
  2819   (\<forall>Va1 Va2 X Vt. ~(connected_space(X::'a,Vt) & separation(Va1::'a,Va2,X,Vt))) &
  2820   (\<forall>X Vt. topological_space(X::'a,Vt) --> connected_space(X::'a,Vt) | separation(f21(X::'a,Vt),f22(X::'a,Vt),X,Vt)) &
  2821   (\<forall>Va X Vt. connected_set(Va::'a,X,Vt) --> topological_space(X::'a,Vt)) &
  2822   (\<forall>Vt Va X. connected_set(Va::'a,X,Vt) --> subset_sets(Va::'a,X)) &
  2823   (\<forall>X Vt Va. connected_set(Va::'a,X,Vt) --> connected_space(Va::'a,subspace_topology(X::'a,Vt,Va))) &
  2824   (\<forall>X Vt Va. topological_space(X::'a,Vt) & subset_sets(Va::'a,X) & connected_space(Va::'a,subspace_topology(X::'a,Vt,Va)) --> connected_set(Va::'a,X,Vt)) &
  2825   (\<forall>Vf X Vt. open_covering(Vf::'a,X,Vt) --> topological_space(X::'a,Vt)) &
  2826   (\<forall>X Vf Vt. open_covering(Vf::'a,X,Vt) --> subset_collections(Vf::'a,Vt)) &
  2827   (\<forall>Vt Vf X. open_covering(Vf::'a,X,Vt) --> equal_sets(union_of_members(Vf),X)) &
  2828   (\<forall>Vt Vf X. topological_space(X::'a,Vt) & subset_collections(Vf::'a,Vt) & equal_sets(union_of_members(Vf),X) --> open_covering(Vf::'a,X,Vt)) &
  2829   (\<forall>X Vt. compact_space(X::'a,Vt) --> topological_space(X::'a,Vt)) &
  2830   (\<forall>X Vt Vf1. compact_space(X::'a,Vt) & open_covering(Vf1::'a,X,Vt) --> finite'(f23(X::'a,Vt,Vf1))) &
  2831   (\<forall>X Vt Vf1. compact_space(X::'a,Vt) & open_covering(Vf1::'a,X,Vt) --> subset_collections(f23(X::'a,Vt,Vf1),Vf1)) &
  2832   (\<forall>Vf1 X Vt. compact_space(X::'a,Vt) & open_covering(Vf1::'a,X,Vt) --> open_covering(f23(X::'a,Vt,Vf1),X,Vt)) &
  2833   (\<forall>X Vt. topological_space(X::'a,Vt) --> compact_space(X::'a,Vt) | open_covering(f24(X::'a,Vt),X,Vt)) &
  2834   (\<forall>Uu24 X Vt. topological_space(X::'a,Vt) & finite'(Uu24) & subset_collections(Uu24::'a,f24(X::'a,Vt)) & open_covering(Uu24::'a,X,Vt) --> compact_space(X::'a,Vt)) &
  2835   (\<forall>Va X Vt. compact_set(Va::'a,X,Vt) --> topological_space(X::'a,Vt)) &
  2836   (\<forall>Vt Va X. compact_set(Va::'a,X,Vt) --> subset_sets(Va::'a,X)) &
  2837   (\<forall>X Vt Va. compact_set(Va::'a,X,Vt) --> compact_space(Va::'a,subspace_topology(X::'a,Vt,Va))) &
  2838   (\<forall>X Vt Va. topological_space(X::'a,Vt) & subset_sets(Va::'a,X) & compact_space(Va::'a,subspace_topology(X::'a,Vt,Va)) --> compact_set(Va::'a,X,Vt)) &
  2839   (basis(cx::'a,f)) &
  2840   (\<forall>U. element_of_collection(U::'a,top_of_basis(f))) &
  2841   (\<forall>V. element_of_collection(V::'a,top_of_basis(f))) &
  2842   (\<forall>U V. ~element_of_collection(intersection_of_sets(U::'a,V),top_of_basis(f))) --> False"
  2843   by meson
  2844 
  2845 
  2846 (*0 inferences so far.  Searching to depth 0.  0.8 secs*)
  2847 lemma TOP004_2:
  2848  "(\<forall>U Uu1 Vf. element_of_set(U::'a,Uu1) & element_of_collection(Uu1::'a,Vf) --> element_of_set(U::'a,union_of_members(Vf))) &
  2849   (\<forall>Vf X. basis(X::'a,Vf) --> equal_sets(union_of_members(Vf),X)) &
  2850   (\<forall>X Vf Y Vb1 Vb2. basis(X::'a,Vf) & element_of_set(Y::'a,X) & element_of_collection(Vb1::'a,Vf) & element_of_collection(Vb2::'a,Vf) & element_of_set(Y::'a,intersection_of_sets(Vb1::'a,Vb2)) --> element_of_set(Y::'a,f6(X::'a,Vf,Y,Vb1,Vb2))) &
  2851   (\<forall>X Y Vb1 Vb2 Vf. basis(X::'a,Vf) & element_of_set(Y::'a,X) & element_of_collection(Vb1::'a,Vf) & element_of_collection(Vb2::'a,Vf) & element_of_set(Y::'a,intersection_of_sets(Vb1::'a,Vb2)) --> element_of_collection(f6(X::'a,Vf,Y,Vb1,Vb2),Vf)) &
  2852   (\<forall>X Vf Y Vb1 Vb2. basis(X::'a,Vf) & element_of_set(Y::'a,X) & element_of_collection(Vb1::'a,Vf) & element_of_collection(Vb2::'a,Vf) & element_of_set(Y::'a,intersection_of_sets(Vb1::'a,Vb2)) --> subset_sets(f6(X::'a,Vf,Y,Vb1,Vb2),intersection_of_sets(Vb1::'a,Vb2))) &
  2853   (\<forall>Vf U X. element_of_collection(U::'a,top_of_basis(Vf)) & element_of_set(X::'a,U) --> element_of_set(X::'a,f10(Vf::'a,U,X))) &
  2854   (\<forall>U X Vf. element_of_collection(U::'a,top_of_basis(Vf)) & element_of_set(X::'a,U) --> element_of_collection(f10(Vf::'a,U,X),Vf)) &
  2855   (\<forall>Vf X U. element_of_collection(U::'a,top_of_basis(Vf)) & element_of_set(X::'a,U) --> subset_sets(f10(Vf::'a,U,X),U)) &
  2856   (\<forall>Vf U. element_of_collection(U::'a,top_of_basis(Vf)) | element_of_set(f11(Vf::'a,U),U)) &
  2857   (\<forall>Vf Uu11 U. element_of_set(f11(Vf::'a,U),Uu11) & element_of_collection(Uu11::'a,Vf) & subset_sets(Uu11::'a,U) --> element_of_collection(U::'a,top_of_basis(Vf))) &
  2858   (\<forall>Y X Z. subset_sets(X::'a,Y) & subset_sets(Y::'a,Z) --> subset_sets(X::'a,Z)) &
  2859   (\<forall>Y Z X. element_of_set(Z::'a,intersection_of_sets(X::'a,Y)) --> element_of_set(Z::'a,X)) &
  2860   (\<forall>X Z Y. element_of_set(Z::'a,intersection_of_sets(X::'a,Y)) --> element_of_set(Z::'a,Y)) &
  2861   (\<forall>X Z Y. element_of_set(Z::'a,X) & element_of_set(Z::'a,Y) --> element_of_set(Z::'a,intersection_of_sets(X::'a,Y))) &
  2862   (\<forall>X U Y V. subset_sets(X::'a,Y) & subset_sets(U::'a,V) --> subset_sets(intersection_of_sets(X::'a,U),intersection_of_sets(Y::'a,V))) &
  2863   (\<forall>X Z Y. equal_sets(X::'a,Y) & element_of_set(Z::'a,X) --> element_of_set(Z::'a,Y)) &
  2864   (\<forall>Y X. equal_sets(intersection_of_sets(X::'a,Y),intersection_of_sets(Y::'a,X))) &
  2865   (basis(cx::'a,f)) &
  2866   (\<forall>U. element_of_collection(U::'a,top_of_basis(f))) &
  2867   (\<forall>V. element_of_collection(V::'a,top_of_basis(f))) &
  2868   (\<forall>U V. ~element_of_collection(intersection_of_sets(U::'a,V),top_of_basis(f))) --> False"
  2869   by meson
  2870 
  2871 (*53777 inferences so far.  Searching to depth 20.  68.7 secs*)
  2872 lemma TOP005_2:
  2873  "(\<forall>Vf U. element_of_set(U::'a,union_of_members(Vf)) --> element_of_set(U::'a,f1(Vf::'a,U))) &
  2874   (\<forall>U Vf. element_of_set(U::'a,union_of_members(Vf)) --> element_of_collection(f1(Vf::'a,U),Vf)) &
  2875   (\<forall>Vf U X. element_of_collection(U::'a,top_of_basis(Vf)) & element_of_set(X::'a,U) --> element_of_set(X::'a,f10(Vf::'a,U,X))) &
  2876   (\<forall>U X Vf. element_of_collection(U::'a,top_of_basis(Vf)) & element_of_set(X::'a,U) --> element_of_collection(f10(Vf::'a,U,X),Vf)) &
  2877   (\<forall>Vf X U. element_of_collection(U::'a,top_of_basis(Vf)) & element_of_set(X::'a,U) --> subset_sets(f10(Vf::'a,U,X),U)) &
  2878   (\<forall>Vf U. element_of_collection(U::'a,top_of_basis(Vf)) | element_of_set(f11(Vf::'a,U),U)) &
  2879   (\<forall>Vf Uu11 U. element_of_set(f11(Vf::'a,U),Uu11) & element_of_collection(Uu11::'a,Vf) & subset_sets(Uu11::'a,U) --> element_of_collection(U::'a,top_of_basis(Vf))) &
  2880   (\<forall>X U Y. element_of_set(U::'a,X) --> subset_sets(X::'a,Y) | element_of_set(U::'a,Y)) &
  2881   (\<forall>Y X Z. subset_sets(X::'a,Y) & element_of_collection(Y::'a,Z) --> subset_sets(X::'a,union_of_members(Z))) &
  2882   (\<forall>X U Y. subset_collections(X::'a,Y) & element_of_collection(U::'a,X) --> element_of_collection(U::'a,Y)) &
  2883   (subset_collections(g::'a,top_of_basis(f))) &
  2884   (~element_of_collection(union_of_members(g),top_of_basis(f))) --> False"
  2885   oops
  2886 
  2887 end