src/HOL/ex/Meson_Test.thy
 author wenzelm Wed Jun 22 10:09:20 2016 +0200 (2016-06-22) changeset 63343 fb5d8a50c641 parent 61933 cf58b5b794b2 child 64267 b9a1486e79be permissions -rw-r--r--
bundle lifting_syntax;
```     1
```
```     2 section \<open>Meson test cases\<close>
```
```     3
```
```     4 theory Meson_Test
```
```     5 imports Main
```
```     6 begin
```
```     7
```
```     8 text \<open>
```
```     9   WARNING: there are many potential conflicts between variables used
```
```    10   below and constants declared in HOL!
```
```    11 \<close>
```
```    12
```
```    13 hide_const (open) implies union inter subset quotient
```
```    14
```
```    15 text \<open>
```
```    16   Test data for the MESON proof procedure
```
```    17   (Excludes the equality problems 51, 52, 56, 58)
```
```    18 \<close>
```
```    19
```
```    20
```
```    21 subsection \<open>Interactive examples\<close>
```
```    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 \<open>
```
```    27     val ctxt = @{context};
```
```    28     val prem25 = Thm.assume @{cprop "\<not> ?thesis"};
```
```    29     val nnf25 = Meson.make_nnf ctxt prem25;
```
```    30     val xsko25 = Meson.skolemize ctxt nnf25;
```
```    31 \<close>
```
```    32   apply (tactic \<open>cut_tac xsko25 1 THEN REPEAT (eresolve_tac @{context} [exE] 1)\<close>)
```
```    33   ML_val \<open>
```
```    34     val ctxt = @{context};
```
```    35     val [_, sko25] = #prems (#1 (Subgoal.focus ctxt 1 NONE (#goal @{Isar.goal})));
```
```    36     val clauses25 = Meson.make_clauses ctxt [sko25];   (*7 clauses*)
```
```    37     val horns25 = Meson.make_horns clauses25;     (*16 Horn clauses*)
```
```    38     val go25 :: _ = Meson.gocls clauses25;
```
```    39
```
```    40     val ctxt' = fold Thm.declare_hyps (maps Thm.chyps_of (go25 :: horns25)) ctxt;
```
```    41     Goal.prove ctxt' [] [] @{prop False} (fn _ =>
```
```    42       resolve_tac ctxt' [go25] 1 THEN
```
```    43       Meson.depth_prolog_tac ctxt' horns25);
```
```    44 \<close>
```
```    45   oops
```
```    46
```
```    47 lemma problem_26:
```
```    48   "((\<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))"
```
```    49   apply (rule ccontr)
```
```    50   ML_prf \<open>
```
```    51     val ctxt = @{context};
```
```    52     val prem26 = Thm.assume @{cprop "\<not> ?thesis"}
```
```    53     val nnf26 = Meson.make_nnf ctxt prem26;
```
```    54     val xsko26 = Meson.skolemize ctxt nnf26;
```
```    55 \<close>
```
```    56   apply (tactic \<open>cut_tac xsko26 1 THEN REPEAT (eresolve_tac @{context} [exE] 1)\<close>)
```
```    57   ML_val \<open>
```
```    58     val ctxt = @{context};
```
```    59     val [_, sko26] = #prems (#1 (Subgoal.focus ctxt 1 NONE (#goal @{Isar.goal})));
```
```    60     val clauses26 = Meson.make_clauses ctxt [sko26];
```
```    61     val _ = @{assert} (length clauses26 = 9);
```
```    62     val horns26 = Meson.make_horns clauses26;
```
```    63     val _ = @{assert} (length horns26 = 24);
```
```    64     val go26 :: _ = Meson.gocls clauses26;
```
```    65
```
```    66     val ctxt' = fold Thm.declare_hyps (maps Thm.chyps_of (go26 :: horns26)) ctxt;
```
```    67     Goal.prove ctxt' [] [] @{prop False} (fn _ =>
```
```    68       resolve_tac ctxt' [go26] 1 THEN
```
```    69       Meson.depth_prolog_tac ctxt' horns26);  (*7 ms*)
```
```    70     (*Proof is of length 107!!*)
```
```    71 \<close>
```
```    72   oops
```
```    73
```
```    74 lemma problem_43:  \<comment> "NOW PROVED AUTOMATICALLY!!"  (*16 Horn clauses*)
```
```    75   "(\<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)))"
```
```    76   apply (rule ccontr)
```
```    77   ML_prf \<open>
```
```    78     val ctxt = @{context};
```
```    79     val prem43 = Thm.assume @{cprop "\<not> ?thesis"};
```
```    80     val nnf43 = Meson.make_nnf ctxt prem43;
```
```    81     val xsko43 = Meson.skolemize ctxt nnf43;
```
```    82 \<close>
```
```    83   apply (tactic \<open>cut_tac xsko43 1 THEN REPEAT (eresolve_tac @{context} [exE] 1)\<close>)
```
```    84   ML_val \<open>
```
```    85     val ctxt = @{context};
```
```    86     val [_, sko43] = #prems (#1 (Subgoal.focus ctxt 1 NONE (#goal @{Isar.goal})));
```
```    87     val clauses43 = Meson.make_clauses ctxt [sko43];
```
```    88     val _ = @{assert} (length clauses43 = 6);
```
```    89     val horns43 = Meson.make_horns clauses43;
```
```    90     val _ = @{assert} (length horns43 = 16);
```
```    91     val go43 :: _ = Meson.gocls clauses43;
```
```    92
```
```    93     val ctxt' = fold Thm.declare_hyps (maps Thm.chyps_of (go43 :: horns43)) ctxt;
```
```    94     Goal.prove ctxt' [] [] @{prop False} (fn _ =>
```
```    95       resolve_tac ctxt' [go43] 1 THEN
```
```    96       Meson.best_prolog_tac ctxt' Meson.size_of_subgoals horns43);   (*7ms*)
```
```    97 \<close>
```
```    98   oops
```
```    99
```
```   100 (*
```
```   101 #1  (q x xa ==> ~ q x xa) ==> q xa x
```
```   102 #2  (q xa x ==> ~ q xa x) ==> q x xa
```
```   103 #3  (~ q x xa ==> q x xa) ==> ~ q xa x
```
```   104 #4  (~ q xa x ==> q xa x) ==> ~ q x xa
```
```   105 #5  [| ~ q ?U ?V ==> q ?U ?V; ~ p ?W ?U ==> p ?W ?U |] ==> p ?W ?V
```
```   106 #6  [| ~ p ?W ?U ==> p ?W ?U; p ?W ?V ==> ~ p ?W ?V |] ==> ~ q ?U ?V
```
```   107 #7  [| p ?W ?V ==> ~ p ?W ?V; ~ q ?U ?V ==> q ?U ?V |] ==> ~ p ?W ?U
```
```   108 #8  [| ~ q ?U ?V ==> q ?U ?V; ~ p ?W ?V ==> p ?W ?V |] ==> p ?W ?U
```
```   109 #9  [| ~ p ?W ?V ==> p ?W ?V; p ?W ?U ==> ~ p ?W ?U |] ==> ~ q ?U ?V
```
```   110 #10 [| p ?W ?U ==> ~ p ?W ?U; ~ q ?U ?V ==> q ?U ?V |] ==> ~ p ?W ?V
```
```   111 #11 [| p (xb ?U ?V) ?U ==> ~ p (xb ?U ?V) ?U;
```
```   112        p (xb ?U ?V) ?V ==> ~ p (xb ?U ?V) ?V |] ==> q ?U ?V
```
```   113 #12 [| p (xb ?U ?V) ?V ==> ~ p (xb ?U ?V) ?V; q ?U ?V ==> ~ q ?U ?V |] ==>
```
```   114     p (xb ?U ?V) ?U
```
```   115 #13 [| q ?U ?V ==> ~ q ?U ?V; p (xb ?U ?V) ?U ==> ~ p (xb ?U ?V) ?U |] ==>
```
```   116     p (xb ?U ?V) ?V
```
```   117 #14 [| ~ p (xb ?U ?V) ?U ==> p (xb ?U ?V) ?U;
```
```   118        ~ p (xb ?U ?V) ?V ==> p (xb ?U ?V) ?V |] ==> q ?U ?V
```
```   119 #15 [| ~ p (xb ?U ?V) ?V ==> p (xb ?U ?V) ?V; q ?U ?V ==> ~ q ?U ?V |] ==>
```
```   120     ~ p (xb ?U ?V) ?U
```
```   121 #16 [| q ?U ?V ==> ~ q ?U ?V; ~ p (xb ?U ?V) ?U ==> p (xb ?U ?V) ?U |] ==>
```
```   122     ~ p (xb ?U ?V) ?V
```
```   123
```
```   124 And here is the proof! (Unkn is the start state after use of goal clause)
```
```   125 [Unkn, Res ([Thm "#14"], false, 1), Res ([Thm "#5"], false, 1),
```
```   126    Res ([Thm "#1"], false, 1), Asm 1, Res ([Thm "#13"], false, 1), Asm 2,
```
```   127    Asm 1, Res ([Thm "#13"], false, 1), Asm 1, Res ([Thm "#10"], false, 1),
```
```   128    Res ([Thm "#16"], false, 1), Asm 2, Asm 1, Res ([Thm "#1"], false, 1),
```
```   129    Asm 1, Res ([Thm "#14"], false, 1), Res ([Thm "#5"], false, 1),
```
```   130    Res ([Thm "#2"], false, 1), Asm 1, Res ([Thm "#13"], false, 1), Asm 2,
```
```   131    Asm 1, Res ([Thm "#8"], false, 1), Res ([Thm "#2"], false, 1), Asm 1,
```
```   132    Res ([Thm "#12"], false, 1), Asm 2, Asm 1] : lderiv list
```
```   133 *)
```
```   134
```
```   135
```
```   136 text \<open>
```
```   137   MORE and MUCH HARDER test data for the MESON proof procedure
```
```   138   (courtesy John Harrison).
```
```   139 \<close>
```
```   140
```
```   141 (* ========================================================================= *)
```
```   142 (* 100 problems selected from the TPTP library                               *)
```
```   143 (* ========================================================================= *)
```
```   144
```
```   145 (*
```
```   146  * Original timings for John Harrison's MESON_TAC.
```
```   147  * Timings below on a 600MHz Pentium III (perch)
```
```   148  * Some timings below refer to griffon, which is a dual 2.5GHz Power Mac G5.
```
```   149  *
```
```   150  * A few variable names have been changed to avoid clashing with constants.
```
```   151  *
```
```   152  * Changed numeric constants e.g. 0, 1, 2... to num0, num1, num2...
```
```   153  *
```
```   154  * Here's a list giving typical CPU times, as well as common names and
```
```   155  * literature references.
```
```   156  *
```
```   157  * BOO003-1     34.6    B2 part 1 [McCharen, et al., 1976]; Lemma proved [Overbeek, et al., 1976]; prob2_part1.ver1.in [ANL]
```
```   158  * BOO004-1     36.7    B2 part 2 [McCharen, et al., 1976]; Lemma proved [Overbeek, et al., 1976]; prob2_part2.ver1 [ANL]
```
```   159  * 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]
```
```   160  * 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]
```
```   161  * BOO011-1     19.0    B7 [McCharen, et al., 1976]; prob7.ver1 [ANL]
```
```   162  * CAT001-3     45.2    C1 [McCharen, et al., 1976]; p1.ver3.in [ANL]
```
```   163  * CAT003-3     10.5    C3 [McCharen, et al., 1976]; p3.ver3.in [ANL]
```
```   164  * CAT005-1    480.1    C5 [McCharen, et al., 1976]; p5.ver1.in [ANL]
```
```   165  * CAT007-1     11.9    C7 [McCharen, et al., 1976]; p7.ver1.in [ANL]
```
```   166  * CAT018-1     81.3    p18.ver1.in [ANL]
```
```   167  * COL001-2     16.0    C1 [Wos & McCune, 1988]
```
```   168  * COL023-1      5.1    [McCune & Wos, 1988]
```
```   169  * COL032-1     15.8    [McCune & Wos, 1988]
```
```   170  * COL052-2     13.2    bird4.ver2.in [ANL]
```
```   171  * COL075-2    116.9    [Jech, 1994]
```
```   172  * COM001-1      1.7    shortburst [Wilson & Minker, 1976]
```
```   173  * COM002-1      4.4    burstall [Wilson & Minker, 1976]
```
```   174  * COM002-2      7.4
```
```   175  * COM003-2     22.1    [Brushi, 1991]
```
```   176  * COM004-1     45.1
```
```   177  * GEO003-1     71.7    T3 [McCharen, et al., 1976]; t3.ver1.in [ANL]
```
```   178  * GEO017-2     78.8    D4.1 [Quaife, 1989]
```
```   179  * GEO027-3    181.5    D10.1 [Quaife, 1989]
```
```   180  * GEO058-2    104.0    R4 [Quaife, 1989]
```
```   181  * GEO079-1      2.4    GEOMETRY THEOREM [Slagle, 1967]
```
```   182  * 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]
```
```   183  * GRP008-1     50.4    Problem 4 [Wos, 1965]; wos4 [Wilson & Minker, 1976]
```
```   184  * GRP013-1     40.2    Problem 11 [Wos, 1965]; wos11 [Wilson & Minker, 1976]
```
```   185  * GRP037-3     43.8    Problem 17 [Wos, 1965]; wos17 [Wilson & Minker, 1976]
```
```   186  * GRP031-2      3.2    ls23 [Lawrence & Starkey, 1974]; ls23 [Wilson & Minker, 1976]
```
```   187  * GRP034-4      2.5    ls26 [Lawrence & Starkey, 1974]; ls26 [Wilson & Minker, 1976]
```
```   188  * GRP047-2     11.7    [Veroff, 1992]
```
```   189  * GRP130-1    170.6    Bennett QG8 [TPTP]; QG8 [Slaney, 1993]
```
```   190  * GRP156-1     48.7    ax_mono1c [Schulz, 1995]
```
```   191  * GRP168-1    159.1    p01a [Schulz, 1995]
```
```   192  * HEN003-3     39.9    HP3 [McCharen, et al., 1976]
```
```   193  * HEN007-2    125.7    H7 [McCharen, et al., 1976]
```
```   194  * HEN008-4     62.0    H8 [McCharen, et al., 1976]
```
```   195  * HEN009-5    136.3    H9 [McCharen, et al., 1976]; hp9.ver3.in [ANL]
```
```   196  * HEN012-3     48.5    new.ver2.in [ANL]
```
```   197  * LCL010-1    370.9    EC-73 [McCune & Wos, 1992]; ec_yq.in [OTTER]
```
```   198  * LCL077-2     51.6    morgan.two.ver1.in [ANL]
```
```   199  * LCL082-1     14.6    IC-1.1 [Wos, et al., 1990]; IC-65 [McCune & Wos, 1992]; ls2 [SETHEO]; S1 [Pfenning, 1988]
```
```   200  * 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]
```
```   201  * LCL143-1     10.9    Lattice structure theorem 2 [Bonacina, 1991]
```
```   202  * LCL182-1    271.6    Problem 2.16 [Whitehead & Russell, 1927]
```
```   203  * LCL200-1     12.0    Problem 2.46 [Whitehead & Russell, 1927]
```
```   204  * LCL215-1    214.4    Problem 2.62 [Whitehead & Russell, 1927]; Problem 2.63 [Whitehead & Russell, 1927]
```
```   205  * LCL230-2      0.2    Pelletier 5 [Pelletier, 1986]
```
```   206  * LDA003-1     68.5    Problem 3 [Jech, 1993]
```
```   207  * MSC002-1      9.2    DBABHP [Michie, et al., 1972]; DBABHP [Wilson & Minker, 1976]
```
```   208  * MSC003-1      3.2    HASPARTS-T1 [Wilson & Minker, 1976]
```
```   209  * MSC004-1      9.3    HASPARTS-T2 [Wilson & Minker, 1976]
```
```   210  * MSC005-1      1.8    Problem 5.1 [Plaisted, 1982]
```
```   211  * MSC006-1     39.0    nonob.lop [SETHEO]
```
```   212  * NUM001-1     14.0    Chang-Lee-10a [Chang, 1970]; ls28 [Lawrence & Starkey, 1974]; ls28 [Wilson & Minker, 1976]
```
```   213  * NUM021-1     52.3    ls65 [Lawrence & Starkey, 1974]; ls65 [Wilson & Minker, 1976]
```
```   214  * NUM024-1     64.6    ls75 [Lawrence & Starkey, 1974]; ls75 [Wilson & Minker, 1976]
```
```   215  * NUM180-1    621.2    LIM2.1 [Quaife]
```
```   216  * NUM228-1    575.9    TRECDEF4 cor. [Quaife]
```
```   217  * PLA002-1     37.4    Problem 5.7 [Plaisted, 1982]
```
```   218  * PLA006-1      7.2    [Segre & Elkan, 1994]
```
```   219  * PLA017-1    484.8    [Segre & Elkan, 1994]
```
```   220  * PLA022-1     19.1    [Segre & Elkan, 1994]
```
```   221  * PLA022-2     19.7    [Segre & Elkan, 1994]
```
```   222  * PRV001-1     10.3    PV1 [McCharen, et al., 1976]
```
```   223  * PRV003-1      3.9    E2 [McCharen, et al., 1976]; v2.lop [SETHEO]
```
```   224  * PRV005-1      4.3    E4 [McCharen, et al., 1976]; v4.lop [SETHEO]
```
```   225  * PRV006-1      6.0    E5 [McCharen, et al., 1976]; v5.lop [SETHEO]
```
```   226  * PRV009-1      2.2    Hoares FIND [Bledsoe, 1977]; Problem 5.5 [Plaisted, 1982]
```
```   227  * PUZ012-1      3.5    Boxes-of-fruit [Wos, 1988]; Boxes-of-fruit [Wos, et al., 1992]; boxes.ver1.in [ANL]
```
```   228  * PUZ020-1     56.6    knightknave.in [ANL]
```
```   229  * PUZ025-1     58.4    Problem 35 [Smullyan, 1978]; tandl35.ver1.in [ANL]
```
```   230  * PUZ029-1      5.1    pigs.ver1.in [ANL]
```
```   231  * RNG001-3     82.4    EX6-T? [Wilson & Minker, 1976]; ex6.lop [SETHEO]; Example 6a [Fleisig, et al., 1974]; FEX6T1 [SPRFN]; FEX6T2 [SPRFN]
```
```   232  * RNG001-5    399.8    Problem 21 [Wos, 1965]; wos21 [Wilson & Minker, 1976]
```
```   233  * RNG011-5      8.4    CADE-11 Competition Eq-10 [Overbeek, 1990]; PROBLEM 10 [Zhang, 1993]; THEOREM EQ-10 [Lusk & McCune, 1993]
```
```   234  * RNG023-6      9.1    [Stevens, 1987]
```
```   235  * RNG028-2      9.3    PROOF III [Anantharaman & Hsiang, 1990]
```
```   236  * RNG038-2     16.2    Problem 27 [Wos, 1965]; wos27 [Wilson & Minker, 1976]
```
```   237  * RNG040-2    180.5    Problem 29 [Wos, 1965]; wos29 [Wilson & Minker, 1976]
```
```   238  * RNG041-1     35.8    Problem 30 [Wos, 1965]; wos30 [Wilson & Minker, 1976]
```
```   239  * ROB010-1    205.0    Lemma 3.3 [Winker, 1990]; RA2 [Lusk & Wos, 1992]
```
```   240  * ROB013-1     23.6    Lemma 3.5 [Winker, 1990]
```
```   241  * ROB016-1     15.2    Corollary 3.7 [Winker, 1990]
```
```   242  * ROB021-1    230.4    [McCune, 1992]
```
```   243  * SET005-1    192.2    ls108 [Lawrence & Starkey, 1974]; ls108 [Wilson & Minker, 1976]
```
```   244  * SET009-1     10.5    ls116 [Lawrence & Starkey, 1974]; ls116 [Wilson & Minker, 1976]
```
```   245  * SET025-4    694.7    Lemma 10 [Boyer, et al, 1986]
```
```   246  * SET046-5      2.3    p42.in [ANL]; Pelletier 42 [Pelletier, 1986]
```
```   247  * SET047-5      3.7    p43.in [ANL]; Pelletier 43 [Pelletier, 1986]
```
```   248  * SYN034-1      2.8    QW [Michie, et al., 1972]; QW [Wilson & Minker, 1976]
```
```   249  * SYN071-1      1.9    Pelletier 48 [Pelletier, 1986]
```
```   250  * SYN349-1     61.7    Ch17N5 [Tammet, 1994]
```
```   251  * SYN352-1      5.5    Ch18N4 [Tammet, 1994]
```
```   252  * TOP001-2     61.1    Lemma 1a [Wick & McCune, 1989]
```
```   253  * TOP002-2      0.4    Lemma 1b [Wick & McCune, 1989]
```
```   254  * TOP004-1    181.6    Lemma 1d [Wick & McCune, 1989]
```
```   255  * TOP004-2      9.0    Lemma 1d [Wick & McCune, 1989]
```
```   256  * TOP005-2    139.8    Lemma 1e [Wick & McCune, 1989]
```
```   257  *)
```
```   258
```
```   259 abbreviation "EQU001_0_ax equal \<equiv> (\<forall>X. equal(X::'a,X)) &
```
```   260   (\<forall>Y X. equal(X::'a,Y) --> equal(Y::'a,X)) &
```
```   261   (\<forall>Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z))"
```
```   262
```
```   263 abbreviation "BOO002_0_ax equal INVERSE multiplicative_identity
```
```   264   additive_identity multiply product add sum \<equiv>
```
```   265   (\<forall>X Y. sum(X::'a,Y,add(X::'a,Y))) &
```
```   266   (\<forall>X Y. product(X::'a,Y,multiply(X::'a,Y))) &
```
```   267   (\<forall>Y X Z. sum(X::'a,Y,Z) --> sum(Y::'a,X,Z)) &
```
```   268   (\<forall>Y X Z. product(X::'a,Y,Z) --> product(Y::'a,X,Z)) &
```
```   269   (\<forall>X. sum(additive_identity::'a,X,X)) &
```
```   270   (\<forall>X. sum(X::'a,additive_identity,X)) &
```
```   271   (\<forall>X. product(multiplicative_identity::'a,X,X)) &
```
```   272   (\<forall>X. product(X::'a,multiplicative_identity,X)) &
```
```   273   (\<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)) &
```
```   274   (\<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)) &
```
```   275   (\<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)) &
```
```   276   (\<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)) &
```
```   277   (\<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)) &
```
```   278   (\<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)) &
```
```   279   (\<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)) &
```
```   280   (\<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)) &
```
```   281   (\<forall>X. sum(INVERSE(X),X,multiplicative_identity)) &
```
```   282   (\<forall>X. sum(X::'a,INVERSE(X),multiplicative_identity)) &
```
```   283   (\<forall>X. product(INVERSE(X),X,additive_identity)) &
```
```   284   (\<forall>X. product(X::'a,INVERSE(X),additive_identity)) &
```
```   285   (\<forall>X Y U V. sum(X::'a,Y,U) & sum(X::'a,Y,V) --> equal(U::'a,V)) &
```
```   286   (\<forall>X Y U V. product(X::'a,Y,U) & product(X::'a,Y,V) --> equal(U::'a,V))"
```
```   287
```
```   288 abbreviation "BOO002_0_eq INVERSE multiply add product sum equal \<equiv>
```
```   289   (\<forall>X Y W Z. equal(X::'a,Y) & sum(X::'a,W,Z) --> sum(Y::'a,W,Z)) &
```
```   290   (\<forall>X W Y Z. equal(X::'a,Y) & sum(W::'a,X,Z) --> sum(W::'a,Y,Z)) &
```
```   291   (\<forall>X W Z Y. equal(X::'a,Y) & sum(W::'a,Z,X) --> sum(W::'a,Z,Y)) &
```
```   292   (\<forall>X Y W Z. equal(X::'a,Y) & product(X::'a,W,Z) --> product(Y::'a,W,Z)) &
```
```   293   (\<forall>X W Y Z. equal(X::'a,Y) & product(W::'a,X,Z) --> product(W::'a,Y,Z)) &
```
```   294   (\<forall>X W Z Y. equal(X::'a,Y) & product(W::'a,Z,X) --> product(W::'a,Z,Y)) &
```
```   295   (\<forall>X Y W. equal(X::'a,Y) --> equal(add(X::'a,W),add(Y::'a,W))) &
```
```   296   (\<forall>X W Y. equal(X::'a,Y) --> equal(add(W::'a,X),add(W::'a,Y))) &
```
```   297   (\<forall>X Y W. equal(X::'a,Y) --> equal(multiply(X::'a,W),multiply(Y::'a,W))) &
```
```   298   (\<forall>X W Y. equal(X::'a,Y) --> equal(multiply(W::'a,X),multiply(W::'a,Y))) &
```
```   299   (\<forall>X Y. equal(X::'a,Y) --> equal(INVERSE(X),INVERSE(Y)))"
```
```   300
```
```   301 (*51194 inferences so far.  Searching to depth 13.  232.9 secs*)
```
```   302 lemma BOO003_1:
```
```   303   "EQU001_0_ax equal &
```
```   304   BOO002_0_ax equal INVERSE multiplicative_identity additive_identity multiply product add sum &
```
```   305   BOO002_0_eq INVERSE multiply add product sum equal &
```
```   306   (~product(x::'a,x,x)) --> False"
```
```   307   oops
```
```   308
```
```   309 (*51194 inferences so far.  Searching to depth 13. 204.6 secs
```
```   310   Strange! The previous problem also has 51194 inferences at depth 13.  They
```
```   311    must be very similar!*)
```
```   312 lemma BOO004_1:
```
```   313   "EQU001_0_ax equal &
```
```   314   BOO002_0_ax equal INVERSE multiplicative_identity additive_identity multiply product add sum &
```
```   315   BOO002_0_eq INVERSE multiply add product sum equal &
```
```   316   (~sum(x::'a,x,x)) --> False"
```
```   317   oops
```
```   318
```
```   319 (*74799 inferences so far.  Searching to depth 13.  290.0 secs*)
```
```   320 lemma BOO005_1:
```
```   321   "EQU001_0_ax equal &
```
```   322   BOO002_0_ax equal INVERSE multiplicative_identity additive_identity multiply product add sum &
```
```   323   BOO002_0_eq INVERSE multiply add product sum equal &
```
```   324   (~sum(x::'a,multiplicative_identity,multiplicative_identity)) --> False"
```
```   325   oops
```
```   326
```
```   327 (*74799 inferences so far.  Searching to depth 13.  314.6 secs*)
```
```   328 lemma BOO006_1:
```
```   329   "EQU001_0_ax equal &
```
```   330   BOO002_0_ax equal INVERSE multiplicative_identity additive_identity multiply product add sum &
```
```   331   BOO002_0_eq INVERSE multiply add product sum equal &
```
```   332   (~product(x::'a,additive_identity,additive_identity)) --> False"
```
```   333   oops
```
```   334
```
```   335 (*5 inferences so far.  Searching to depth 5.  1.3 secs*)
```
```   336 lemma BOO011_1:
```
```   337   "EQU001_0_ax equal &
```
```   338   BOO002_0_ax equal INVERSE multiplicative_identity additive_identity multiply product add sum &
```
```   339   BOO002_0_eq INVERSE multiply add product sum equal &
```
```   340   (~equal(INVERSE(additive_identity),multiplicative_identity)) --> False"
```
```   341   by meson
```
```   342
```
```   343 abbreviation "CAT003_0_ax f1 compos codomain domain equal there_exists equivalent \<equiv>
```
```   344   (\<forall>Y X. equivalent(X::'a,Y) --> there_exists(X)) &
```
```   345   (\<forall>X Y. equivalent(X::'a,Y) --> equal(X::'a,Y)) &
```
```   346   (\<forall>X Y. there_exists(X) & equal(X::'a,Y) --> equivalent(X::'a,Y)) &
```
```   347   (\<forall>X. there_exists(domain(X)) --> there_exists(X)) &
```
```   348   (\<forall>X. there_exists(codomain(X)) --> there_exists(X)) &
```
```   349   (\<forall>Y X. there_exists(compos(X::'a,Y)) --> there_exists(domain(X))) &
```
```   350   (\<forall>X Y. there_exists(compos(X::'a,Y)) --> equal(domain(X),codomain(Y))) &
```
```   351   (\<forall>X Y. there_exists(domain(X)) & equal(domain(X),codomain(Y)) --> there_exists(compos(X::'a,Y))) &
```
```   352   (\<forall>X Y Z. equal(compos(X::'a,compos(Y::'a,Z)),compos(compos(X::'a,Y),Z))) &
```
```   353   (\<forall>X. equal(compos(X::'a,domain(X)),X)) &
```
```   354   (\<forall>X. equal(compos(codomain(X),X),X)) &
```
```   355   (\<forall>X Y. equivalent(X::'a,Y) --> there_exists(Y)) &
```
```   356   (\<forall>X Y. there_exists(X) & there_exists(Y) & equal(X::'a,Y) --> equivalent(X::'a,Y)) &
```
```   357   (\<forall>Y X. there_exists(compos(X::'a,Y)) --> there_exists(codomain(X))) &
```
```   358   (\<forall>X Y. there_exists(f1(X::'a,Y)) | equal(X::'a,Y)) &
```
```   359   (\<forall>X Y. equal(X::'a,f1(X::'a,Y)) | equal(Y::'a,f1(X::'a,Y)) | equal(X::'a,Y)) &
```
```   360   (\<forall>X Y. equal(X::'a,f1(X::'a,Y)) & equal(Y::'a,f1(X::'a,Y)) --> equal(X::'a,Y))"
```
```   361
```
```   362 abbreviation "CAT003_0_eq f1 compos codomain domain equivalent there_exists equal \<equiv>
```
```   363   (\<forall>X Y. equal(X::'a,Y) & there_exists(X) --> there_exists(Y)) &
```
```   364   (\<forall>X Y Z. equal(X::'a,Y) & equivalent(X::'a,Z) --> equivalent(Y::'a,Z)) &
```
```   365   (\<forall>X Z Y. equal(X::'a,Y) & equivalent(Z::'a,X) --> equivalent(Z::'a,Y)) &
```
```   366   (\<forall>X Y. equal(X::'a,Y) --> equal(domain(X),domain(Y))) &
```
```   367   (\<forall>X Y. equal(X::'a,Y) --> equal(codomain(X),codomain(Y))) &
```
```   368   (\<forall>X Y Z. equal(X::'a,Y) --> equal(compos(X::'a,Z),compos(Y::'a,Z))) &
```
```   369   (\<forall>X Z Y. equal(X::'a,Y) --> equal(compos(Z::'a,X),compos(Z::'a,Y))) &
```
```   370   (\<forall>A B C. equal(A::'a,B) --> equal(f1(A::'a,C),f1(B::'a,C))) &
```
```   371   (\<forall>D F' E. equal(D::'a,E) --> equal(f1(F'::'a,D),f1(F'::'a,E)))"
```
```   372
```
```   373 (*4007 inferences so far.  Searching to depth 9.  13 secs*)
```
```   374 lemma CAT001_3:
```
```   375   "EQU001_0_ax equal &
```
```   376   CAT003_0_ax f1 compos codomain domain equal there_exists equivalent &
```
```   377   CAT003_0_eq f1 compos codomain domain equivalent there_exists equal &
```
```   378   (there_exists(compos(a::'a,b))) &
```
```   379   (\<forall>Y X Z. equal(compos(compos(a::'a,b),X),Y) & equal(compos(compos(a::'a,b),Z),Y) --> equal(X::'a,Z)) &
```
```   380   (there_exists(compos(b::'a,h))) &
```
```   381   (equal(compos(b::'a,h),compos(b::'a,g))) &
```
```   382   (~equal(h::'a,g)) --> False"
```
```   383    by meson
```
```   384
```
```   385 (*245 inferences so far.  Searching to depth 7.  1.0 secs*)
```
```   386 lemma CAT003_3:
```
```   387   "EQU001_0_ax equal &
```
```   388   CAT003_0_ax f1 compos codomain domain equal there_exists equivalent &
```
```   389   CAT003_0_eq f1 compos codomain domain equivalent there_exists equal &
```
```   390   (there_exists(compos(a::'a,b))) &
```
```   391   (\<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)) &
```
```   392   (there_exists(h)) &
```
```   393   (equal(compos(h::'a,a),compos(g::'a,a))) &
```
```   394   (~equal(g::'a,h)) --> False"
```
```   395   by meson
```
```   396
```
```   397 abbreviation "CAT001_0_ax equal codomain domain identity_map compos product defined \<equiv>
```
```   398   (\<forall>X Y. defined(X::'a,Y) --> product(X::'a,Y,compos(X::'a,Y))) &
```
```   399   (\<forall>Z X Y. product(X::'a,Y,Z) --> defined(X::'a,Y)) &
```
```   400   (\<forall>X Xy Y Z. product(X::'a,Y,Xy) & defined(Xy::'a,Z) --> defined(Y::'a,Z)) &
```
```   401   (\<forall>Y Xy Z X Yz. product(X::'a,Y,Xy) & product(Y::'a,Z,Yz) & defined(Xy::'a,Z) --> defined(X::'a,Yz)) &
```
```   402   (\<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)) &
```
```   403   (\<forall>Z Yz X Y. product(Y::'a,Z,Yz) & defined(X::'a,Yz) --> defined(X::'a,Y)) &
```
```   404   (\<forall>Y X Yz Xy Z. product(Y::'a,Z,Yz) & product(X::'a,Y,Xy) & defined(X::'a,Yz) --> defined(Xy::'a,Z)) &
```
```   405   (\<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)) &
```
```   406   (\<forall>Y X Z. defined(X::'a,Y) & defined(Y::'a,Z) & identity_map(Y) --> defined(X::'a,Z)) &
```
```   407   (\<forall>X. identity_map(domain(X))) &
```
```   408   (\<forall>X. identity_map(codomain(X))) &
```
```   409   (\<forall>X. defined(X::'a,domain(X))) &
```
```   410   (\<forall>X. defined(codomain(X),X)) &
```
```   411   (\<forall>X. product(X::'a,domain(X),X)) &
```
```   412   (\<forall>X. product(codomain(X),X,X)) &
```
```   413   (\<forall>X Y. defined(X::'a,Y) & identity_map(X) --> product(X::'a,Y,Y)) &
```
```   414   (\<forall>Y X. defined(X::'a,Y) & identity_map(Y) --> product(X::'a,Y,X)) &
```
```   415   (\<forall>X Y Z W. product(X::'a,Y,Z) & product(X::'a,Y,W) --> equal(Z::'a,W))"
```
```   416
```
```   417 abbreviation "CAT001_0_eq compos defined identity_map codomain domain product equal \<equiv>
```
```   418   (\<forall>X Y Z W. equal(X::'a,Y) & product(X::'a,Z,W) --> product(Y::'a,Z,W)) &
```
```   419   (\<forall>X Z Y W. equal(X::'a,Y) & product(Z::'a,X,W) --> product(Z::'a,Y,W)) &
```
```   420   (\<forall>X Z W Y. equal(X::'a,Y) & product(Z::'a,W,X) --> product(Z::'a,W,Y)) &
```
```   421   (\<forall>X Y. equal(X::'a,Y) --> equal(domain(X),domain(Y))) &
```
```   422   (\<forall>X Y. equal(X::'a,Y) --> equal(codomain(X),codomain(Y))) &
```
```   423   (\<forall>X Y. equal(X::'a,Y) & identity_map(X) --> identity_map(Y)) &
```
```   424   (\<forall>X Y Z. equal(X::'a,Y) & defined(X::'a,Z) --> defined(Y::'a,Z)) &
```
```   425   (\<forall>X Z Y. equal(X::'a,Y) & defined(Z::'a,X) --> defined(Z::'a,Y)) &
```
```   426   (\<forall>X Z Y. equal(X::'a,Y) --> equal(compos(Z::'a,X),compos(Z::'a,Y))) &
```
```   427   (\<forall>X Y Z. equal(X::'a,Y) --> equal(compos(X::'a,Z),compos(Y::'a,Z)))"
```
```   428
```
```   429 (*54288 inferences so far.  Searching to depth 14.  118.0 secs*)
```
```   430 lemma CAT005_1:
```
```   431   "EQU001_0_ax equal &
```
```   432   CAT001_0_ax equal codomain domain identity_map compos product defined &
```
```   433   CAT001_0_eq compos defined identity_map codomain domain product equal &
```
```   434   (defined(a::'a,d)) &
```
```   435   (identity_map(d)) &
```
```   436   (~equal(domain(a),d)) --> False"
```
```   437   oops
```
```   438
```
```   439 (*1728 inferences so far.  Searching to depth 10.  5.8 secs*)
```
```   440 lemma CAT007_1:
```
```   441   "EQU001_0_ax equal &
```
```   442   CAT001_0_ax equal codomain domain identity_map compos product defined &
```
```   443   CAT001_0_eq compos defined identity_map codomain domain product equal &
```
```   444   (equal(domain(a),codomain(b))) &
```
```   445   (~defined(a::'a,b)) --> False"
```
```   446   by meson
```
```   447
```
```   448 (*82895 inferences so far.  Searching to depth 13.  355 secs*)
```
```   449 lemma CAT018_1:
```
```   450   "EQU001_0_ax equal &
```
```   451   CAT001_0_ax equal codomain domain identity_map compos product defined &
```
```   452   CAT001_0_eq compos defined identity_map codomain domain product equal &
```
```   453   (defined(a::'a,b)) &
```
```   454   (defined(b::'a,c)) &
```
```   455   (~defined(a::'a,compos(b::'a,c))) --> False"
```
```   456   oops
```
```   457
```
```   458 (*1118 inferences so far.  Searching to depth 8.  2.3 secs*)
```
```   459 lemma COL001_2:
```
```   460   "EQU001_0_ax equal &
```
```   461   (\<forall>X Y Z. equal(apply(apply(apply(s::'a,X),Y),Z),apply(apply(X::'a,Z),apply(Y::'a,Z)))) &
```
```   462   (\<forall>Y X. equal(apply(apply(k::'a,X),Y),X)) &
```
```   463   (\<forall>X Y Z. equal(apply(apply(apply(b::'a,X),Y),Z),apply(X::'a,apply(Y::'a,Z)))) &
```
```   464   (\<forall>X. equal(apply(i::'a,X),X)) &
```
```   465   (\<forall>A B C. equal(A::'a,B) --> equal(apply(A::'a,C),apply(B::'a,C))) &
```
```   466   (\<forall>D F' E. equal(D::'a,E) --> equal(apply(F'::'a,D),apply(F'::'a,E))) &
```
```   467   (\<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))))) &
```
```   468   (\<forall>Y. ~equal(Y::'a,apply(combinator::'a,Y))) --> False"
```
```   469   by meson
```
```   470
```
```   471 (*500 inferences so far.  Searching to depth 8.  0.9 secs*)
```
```   472 lemma COL023_1:
```
```   473   "EQU001_0_ax equal &
```
```   474   (\<forall>X Y Z. equal(apply(apply(apply(b::'a,X),Y),Z),apply(X::'a,apply(Y::'a,Z)))) &
```
```   475   (\<forall>X Y Z. equal(apply(apply(apply(n::'a,X),Y),Z),apply(apply(apply(X::'a,Z),Y),Z))) &
```
```   476   (\<forall>A B C. equal(A::'a,B) --> equal(apply(A::'a,C),apply(B::'a,C))) &
```
```   477   (\<forall>D F' E. equal(D::'a,E) --> equal(apply(F'::'a,D),apply(F'::'a,E))) &
```
```   478   (\<forall>Y. ~equal(Y::'a,apply(combinator::'a,Y))) --> False"
```
```   479   by meson
```
```   480
```
```   481 (*3018 inferences so far.  Searching to depth 10.  4.3 secs*)
```
```   482 lemma COL032_1:
```
```   483   "EQU001_0_ax equal &
```
```   484   (\<forall>X. equal(apply(m::'a,X),apply(X::'a,X))) &
```
```   485   (\<forall>Y X Z. equal(apply(apply(apply(q::'a,X),Y),Z),apply(Y::'a,apply(X::'a,Z)))) &
```
```   486   (\<forall>A B C. equal(A::'a,B) --> equal(apply(A::'a,C),apply(B::'a,C))) &
```
```   487   (\<forall>D F' E. equal(D::'a,E) --> equal(apply(F'::'a,D),apply(F'::'a,E))) &
```
```   488   (\<forall>G H. equal(G::'a,H) --> equal(f(G),f(H))) &
```
```   489   (\<forall>Y. ~equal(apply(Y::'a,f(Y)),apply(f(Y),apply(Y::'a,f(Y))))) --> False"
```
```   490   by meson
```
```   491
```
```   492 (*381878 inferences so far.  Searching to depth 13.  670.4 secs*)
```
```   493 lemma COL052_2:
```
```   494   "EQU001_0_ax equal &
```
```   495   (\<forall>X Y W. equal(response(compos(X::'a,Y),W),response(X::'a,response(Y::'a,W)))) &
```
```   496   (\<forall>X Y. agreeable(X) --> equal(response(X::'a,common_bird(Y)),response(Y::'a,common_bird(Y)))) &
```
```   497   (\<forall>Z X. equal(response(X::'a,Z),response(compatible(X),Z)) --> agreeable(X)) &
```
```   498   (\<forall>A B. equal(A::'a,B) --> equal(common_bird(A),common_bird(B))) &
```
```   499   (\<forall>C D. equal(C::'a,D) --> equal(compatible(C),compatible(D))) &
```
```   500   (\<forall>Q R. equal(Q::'a,R) & agreeable(Q) --> agreeable(R)) &
```
```   501   (\<forall>A B C. equal(A::'a,B) --> equal(compos(A::'a,C),compos(B::'a,C))) &
```
```   502   (\<forall>D F' E. equal(D::'a,E) --> equal(compos(F'::'a,D),compos(F'::'a,E))) &
```
```   503   (\<forall>G H I'. equal(G::'a,H) --> equal(response(G::'a,I'),response(H::'a,I'))) &
```
```   504   (\<forall>J L K'. equal(J::'a,K') --> equal(response(L::'a,J),response(L::'a,K'))) &
```
```   505   (agreeable(c)) &
```
```   506   (~agreeable(a)) &
```
```   507   (equal(c::'a,compos(a::'a,b))) --> False"
```
```   508   oops
```
```   509
```
```   510 (*13201 inferences so far.  Searching to depth 11.  31.9 secs*)
```
```   511 lemma COL075_2:
```
```   512   "EQU001_0_ax equal &
```
```   513   (\<forall>Y X. equal(apply(apply(k::'a,X),Y),X)) &
```
```   514   (\<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)))) &
```
```   515   (\<forall>D E F'. equal(D::'a,E) --> equal(apply(D::'a,F'),apply(E::'a,F'))) &
```
```   516   (\<forall>G I' H. equal(G::'a,H) --> equal(apply(I'::'a,G),apply(I'::'a,H))) &
```
```   517   (\<forall>A B. equal(A::'a,B) --> equal(b(A),b(B))) &
```
```   518   (\<forall>C D. equal(C::'a,D) --> equal(c(C),c(D))) &
```
```   519   (\<forall>Y. ~equal(apply(apply(Y::'a,b(Y)),c(Y)),apply(b(Y),b(Y)))) --> False"
```
```   520   oops
```
```   521
```
```   522 (*33 inferences so far.  Searching to depth 7.  0.1 secs*)
```
```   523 lemma COM001_1:
```
```   524   "(\<forall>Goal_state Start_state. follows(Goal_state::'a,Start_state) --> succeeds(Goal_state::'a,Start_state)) &
```
```   525   (\<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)) &
```
```   526   (\<forall>Start_state Label Goal_state. has(Start_state::'a,goto(Label)) & labels(Label::'a,Goal_state) --> succeeds(Goal_state::'a,Start_state)) &
```
```   527   (\<forall>Start_state Condition Goal_state. has(Start_state::'a,ifthen(Condition::'a,Goal_state)) --> succeeds(Goal_state::'a,Start_state)) &
```
```   528   (labels(loop::'a,p3)) &
```
```   529   (has(p3::'a,ifthen(equal(register_j::'a,n),p4))) &
```
```   530   (has(p4::'a,goto(out))) &
```
```   531   (follows(p5::'a,p4)) &
```
```   532   (follows(p8::'a,p3)) &
```
```   533   (has(p8::'a,goto(loop))) &
```
```   534   (~succeeds(p3::'a,p3)) --> False"
```
```   535   by meson
```
```   536
```
```   537 (*533 inferences so far.  Searching to depth 13.  0.3 secs*)
```
```   538 lemma COM002_1:
```
```   539   "(\<forall>Goal_state Start_state. follows(Goal_state::'a,Start_state) --> succeeds(Goal_state::'a,Start_state)) &
```
```   540   (\<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)) &
```
```   541   (\<forall>Start_state Label Goal_state. has(Start_state::'a,goto(Label)) & labels(Label::'a,Goal_state) --> succeeds(Goal_state::'a,Start_state)) &
```
```   542   (\<forall>Start_state Condition Goal_state. has(Start_state::'a,ifthen(Condition::'a,Goal_state)) --> succeeds(Goal_state::'a,Start_state)) &
```
```   543   (has(p1::'a,assign(register_j::'a,num0))) &
```
```   544   (follows(p2::'a,p1)) &
```
```   545   (has(p2::'a,assign(register_k::'a,num1))) &
```
```   546   (labels(loop::'a,p3)) &
```
```   547   (follows(p3::'a,p2)) &
```
```   548   (has(p3::'a,ifthen(equal(register_j::'a,n),p4))) &
```
```   549   (has(p4::'a,goto(out))) &
```
```   550   (follows(p5::'a,p4)) &
```
```   551   (follows(p6::'a,p3)) &
```
```   552   (has(p6::'a,assign(register_k::'a,mtimes(num2::'a,register_k)))) &
```
```   553   (follows(p7::'a,p6)) &
```
```   554   (has(p7::'a,assign(register_j::'a,mplus(register_j::'a,num1)))) &
```
```   555   (follows(p8::'a,p7)) &
```
```   556   (has(p8::'a,goto(loop))) &
```
```   557   (~succeeds(p3::'a,p3)) --> False"
```
```   558   by meson
```
```   559
```
```   560 (*4821 inferences so far.  Searching to depth 14.  1.3 secs*)
```
```   561 lemma COM002_2:
```
```   562   "(\<forall>Goal_state Start_state. ~(fails(Goal_state::'a,Start_state) & follows(Goal_state::'a,Start_state))) &
```
```   563   (\<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)) &
```
```   564   (\<forall>Start_state Label Goal_state. ~(fails(Goal_state::'a,Start_state) & has(Start_state::'a,goto(Label)) & labels(Label::'a,Goal_state))) &
```
```   565   (\<forall>Start_state Condition Goal_state. ~(fails(Goal_state::'a,Start_state) & has(Start_state::'a,ifthen(Condition::'a,Goal_state)))) &
```
```   566   (has(p1::'a,assign(register_j::'a,num0))) &
```
```   567   (follows(p2::'a,p1)) &
```
```   568   (has(p2::'a,assign(register_k::'a,num1))) &
```
```   569   (labels(loop::'a,p3)) &
```
```   570   (follows(p3::'a,p2)) &
```
```   571   (has(p3::'a,ifthen(equal(register_j::'a,n),p4))) &
```
```   572   (has(p4::'a,goto(out))) &
```
```   573   (follows(p5::'a,p4)) &
```
```   574   (follows(p6::'a,p3)) &
```
```   575   (has(p6::'a,assign(register_k::'a,mtimes(num2::'a,register_k)))) &
```
```   576   (follows(p7::'a,p6)) &
```
```   577   (has(p7::'a,assign(register_j::'a,mplus(register_j::'a,num1)))) &
```
```   578   (follows(p8::'a,p7)) &
```
```   579   (has(p8::'a,goto(loop))) &
```
```   580   (fails(p3::'a,p3)) --> False"
```
```   581   by meson
```
```   582
```
```   583 (*98 inferences so far.  Searching to depth 10.  1.1 secs*)
```
```   584 lemma COM003_2:
```
```   585   "(\<forall>X Y Z. program_decides(X) & program(Y) --> decides(X::'a,Y,Z)) &
```
```   586   (\<forall>X. program_decides(X) | program(f2(X))) &
```
```   587   (\<forall>X. decides(X::'a,f2(X),f1(X)) --> program_decides(X)) &
```
```   588   (\<forall>X. program_program_decides(X) --> program(X)) &
```
```   589   (\<forall>X. program_program_decides(X) --> program_decides(X)) &
```
```   590   (\<forall>X. program(X) & program_decides(X) --> program_program_decides(X)) &
```
```   591   (\<forall>X. algorithm_program_decides(X) --> algorithm(X)) &
```
```   592   (\<forall>X. algorithm_program_decides(X) --> program_decides(X)) &
```
```   593   (\<forall>X. algorithm(X) & program_decides(X) --> algorithm_program_decides(X)) &
```
```   594   (\<forall>Y X. program_halts2(X::'a,Y) --> program(X)) &
```
```   595   (\<forall>X Y. program_halts2(X::'a,Y) --> halts2(X::'a,Y)) &
```
```   596   (\<forall>X Y. program(X) & halts2(X::'a,Y) --> program_halts2(X::'a,Y)) &
```
```   597   (\<forall>W X Y Z. halts3_outputs(X::'a,Y,Z,W) --> halts3(X::'a,Y,Z)) &
```
```   598   (\<forall>Y Z X W. halts3_outputs(X::'a,Y,Z,W) --> outputs(X::'a,W)) &
```
```   599   (\<forall>Y Z X W. halts3(X::'a,Y,Z) & outputs(X::'a,W) --> halts3_outputs(X::'a,Y,Z,W)) &
```
```   600   (\<forall>Y X. program_not_halts2(X::'a,Y) --> program(X)) &
```
```   601   (\<forall>X Y. ~(program_not_halts2(X::'a,Y) & halts2(X::'a,Y))) &
```
```   602   (\<forall>X Y. program(X) --> program_not_halts2(X::'a,Y) | halts2(X::'a,Y)) &
```
```   603   (\<forall>W X Y. halts2_outputs(X::'a,Y,W) --> halts2(X::'a,Y)) &
```
```   604   (\<forall>Y X W. halts2_outputs(X::'a,Y,W) --> outputs(X::'a,W)) &
```
```   605   (\<forall>Y X W. halts2(X::'a,Y) & outputs(X::'a,W) --> halts2_outputs(X::'a,Y,W)) &
```
```   606   (\<forall>X W Y Z. program_halts2_halts3_outputs(X::'a,Y,Z,W) --> program_halts2(Y::'a,Z)) &
```
```   607   (\<forall>X Y Z W. program_halts2_halts3_outputs(X::'a,Y,Z,W) --> halts3_outputs(X::'a,Y,Z,W)) &
```
```   608   (\<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)) &
```
```   609   (\<forall>X W Y Z. program_not_halts2_halts3_outputs(X::'a,Y,Z,W) --> program_not_halts2(Y::'a,Z)) &
```
```   610   (\<forall>X Y Z W. program_not_halts2_halts3_outputs(X::'a,Y,Z,W) --> halts3_outputs(X::'a,Y,Z,W)) &
```
```   611   (\<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)) &
```
```   612   (\<forall>X W Y. program_halts2_halts2_outputs(X::'a,Y,W) --> program_halts2(Y::'a,Y)) &
```
```   613   (\<forall>X Y W. program_halts2_halts2_outputs(X::'a,Y,W) --> halts2_outputs(X::'a,Y,W)) &
```
```   614   (\<forall>X Y W. program_halts2(Y::'a,Y) & halts2_outputs(X::'a,Y,W) --> program_halts2_halts2_outputs(X::'a,Y,W)) &
```
```   615   (\<forall>X W Y. program_not_halts2_halts2_outputs(X::'a,Y,W) --> program_not_halts2(Y::'a,Y)) &
```
```   616   (\<forall>X Y W. program_not_halts2_halts2_outputs(X::'a,Y,W) --> halts2_outputs(X::'a,Y,W)) &
```
```   617   (\<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)) &
```
```   618   (\<forall>X. algorithm_program_decides(X) --> program_program_decides(c1)) &
```
```   619   (\<forall>W Y Z. program_program_decides(W) --> program_halts2_halts3_outputs(W::'a,Y,Z,good)) &
```
```   620   (\<forall>W Y Z. program_program_decides(W) --> program_not_halts2_halts3_outputs(W::'a,Y,Z,bad)) &
```
```   621   (\<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)) &
```
```   622   (\<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)) &
```
```   623   (\<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)) &
```
```   624   (\<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)) &
```
```   625   (\<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)) &
```
```   626   (\<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)) &
```
```   627   (algorithm_program_decides(c4)) --> False"
```
```   628   by meson
```
```   629
```
```   630 (*2100398 inferences so far.  Searching to depth 12.
```
```   631   1256s (21 mins) on griffon*)
```
```   632 lemma COM004_1:
```
```   633   "EQU001_0_ax equal &
```
```   634   (\<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))) &
```
```   635   (\<forall>X. contradictory(negate(X),X)) &
```
```   636   (\<forall>X. contradictory(X::'a,negate(X))) &
```
```   637   (\<forall>X. siblings(left_child_of(X),right_child_of(X))) &
```
```   638   (\<forall>D E. equal(D::'a,E) --> equal(left_child_of(D),left_child_of(E))) &
```
```   639   (\<forall>F' G. equal(F'::'a,G) --> equal(negate(F'),negate(G))) &
```
```   640   (\<forall>H I' J. equal(H::'a,I') --> equal(or(H::'a,J),or(I'::'a,J))) &
```
```   641   (\<forall>K' M L. equal(K'::'a,L) --> equal(or(M::'a,K'),or(M::'a,L))) &
```
```   642   (\<forall>N O' P. equal(N::'a,O') --> equal(parent_of(N::'a,P),parent_of(O'::'a,P))) &
```
```   643   (\<forall>Q S' R. equal(Q::'a,R) --> equal(parent_of(S'::'a,Q),parent_of(S'::'a,R))) &
```
```   644   (\<forall>T' U. equal(T'::'a,U) --> equal(right_child_of(T'),right_child_of(U))) &
```
```   645   (\<forall>V W X. equal(V::'a,W) & contradictory(V::'a,X) --> contradictory(W::'a,X)) &
```
```   646   (\<forall>Y A1 Z. equal(Y::'a,Z) & contradictory(A1::'a,Y) --> contradictory(A1::'a,Z)) &
```
```   647   (\<forall>B1 C1 D1. equal(B1::'a,C1) & failure_node(B1::'a,D1) --> failure_node(C1::'a,D1)) &
```
```   648   (\<forall>E1 G1 F1. equal(E1::'a,F1) & failure_node(G1::'a,E1) --> failure_node(G1::'a,F1)) &
```
```   649   (\<forall>H1 I1 J1. equal(H1::'a,I1) & siblings(H1::'a,J1) --> siblings(I1::'a,J1)) &
```
```   650   (\<forall>K1 M1 L1. equal(K1::'a,L1) & siblings(M1::'a,K1) --> siblings(M1::'a,L1)) &
```
```   651   (failure_node(n_left::'a,or(EMPTY::'a,atom))) &
```
```   652   (failure_node(n_right::'a,or(EMPTY::'a,negate(atom)))) &
```
```   653   (equal(n_left::'a,left_child_of(n))) &
```
```   654   (equal(n_right::'a,right_child_of(n))) &
```
```   655   (\<forall>Z. ~failure_node(Z::'a,or(EMPTY::'a,EMPTY))) --> False"
```
```   656   oops
```
```   657
```
```   658 abbreviation "GEO001_0_ax continuous lower_dimension_point_3 lower_dimension_point_2
```
```   659   lower_dimension_point_1 extension euclid2 euclid1 outer_pasch equidistant equal between \<equiv>
```
```   660   (\<forall>X Y. between(X::'a,Y,X) --> equal(X::'a,Y)) &
```
```   661   (\<forall>V X Y Z. between(X::'a,Y,V) & between(Y::'a,Z,V) --> between(X::'a,Y,Z)) &
```
```   662   (\<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)) &
```
```   663   (\<forall>Y X. equidistant(X::'a,Y,Y,X)) &
```
```   664   (\<forall>Z X Y. equidistant(X::'a,Y,Z,Z) --> equal(X::'a,Y)) &
```
```   665   (\<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)) &
```
```   666   (\<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)) &
```
```   667   (\<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))) &
```
```   668   (\<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))) &
```
```   669   (\<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))) &
```
```   670   (\<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))) &
```
```   671   (\<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)) &
```
```   672   (\<forall>X Y W V. between(X::'a,Y,extension(X::'a,Y,W,V))) &
```
```   673   (\<forall>X Y W V. equidistant(Y::'a,extension(X::'a,Y,W,V),W,V)) &
```
```   674   (~between(lower_dimension_point_1::'a,lower_dimension_point_2,lower_dimension_point_3)) &
```
```   675   (~between(lower_dimension_point_2::'a,lower_dimension_point_3,lower_dimension_point_1)) &
```
```   676   (~between(lower_dimension_point_3::'a,lower_dimension_point_1,lower_dimension_point_2)) &
```
```   677   (\<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)) &
```
```   678   (\<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))) &
```
```   679   (\<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))"
```
```   680
```
```   681 abbreviation "GEO001_0_eq continuous extension euclid2 euclid1 outer_pasch equidistant
```
```   682   between equal \<equiv>
```
```   683   (\<forall>X Y W Z. equal(X::'a,Y) & between(X::'a,W,Z) --> between(Y::'a,W,Z)) &
```
```   684   (\<forall>X W Y Z. equal(X::'a,Y) & between(W::'a,X,Z) --> between(W::'a,Y,Z)) &
```
```   685   (\<forall>X W Z Y. equal(X::'a,Y) & between(W::'a,Z,X) --> between(W::'a,Z,Y)) &
```
```   686   (\<forall>X Y V W Z. equal(X::'a,Y) & equidistant(X::'a,V,W,Z) --> equidistant(Y::'a,V,W,Z)) &
```
```   687   (\<forall>X V Y W Z. equal(X::'a,Y) & equidistant(V::'a,X,W,Z) --> equidistant(V::'a,Y,W,Z)) &
```
```   688   (\<forall>X V W Y Z. equal(X::'a,Y) & equidistant(V::'a,W,X,Z) --> equidistant(V::'a,W,Y,Z)) &
```
```   689   (\<forall>X V W Z Y. equal(X::'a,Y) & equidistant(V::'a,W,Z,X) --> equidistant(V::'a,W,Z,Y)) &
```
```   690   (\<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))) &
```
```   691   (\<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))) &
```
```   692   (\<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))) &
```
```   693   (\<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))) &
```
```   694   (\<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))) &
```
```   695   (\<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'))) &
```
```   696   (\<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))) &
```
```   697   (\<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))) &
```
```   698   (\<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))) &
```
```   699   (\<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))) &
```
```   700   (\<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))) &
```
```   701   (\<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))) &
```
```   702   (\<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))) &
```
```   703   (\<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))) &
```
```   704   (\<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))) &
```
```   705   (\<forall>X Y V1 V2 V3. equal(X::'a,Y) --> equal(extension(X::'a,V1,V2,V3),extension(Y::'a,V1,V2,V3))) &
```
```   706   (\<forall>X V1 Y V2 V3. equal(X::'a,Y) --> equal(extension(V1::'a,X,V2,V3),extension(V1::'a,Y,V2,V3))) &
```
```   707   (\<forall>X V1 V2 Y V3. equal(X::'a,Y) --> equal(extension(V1::'a,V2,X,V3),extension(V1::'a,V2,Y,V3))) &
```
```   708   (\<forall>X V1 V2 V3 Y. equal(X::'a,Y) --> equal(extension(V1::'a,V2,V3,X),extension(V1::'a,V2,V3,Y))) &
```
```   709   (\<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))) &
```
```   710   (\<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))) &
```
```   711   (\<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))) &
```
```   712   (\<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))) &
```
```   713   (\<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))) &
```
```   714   (\<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)))"
```
```   715
```
```   716
```
```   717 (*179 inferences so far.  Searching to depth 7.  3.9 secs*)
```
```   718 lemma GEO003_1:
```
```   719   "EQU001_0_ax equal &
```
```   720   GEO001_0_ax continuous lower_dimension_point_3 lower_dimension_point_2
```
```   721     lower_dimension_point_1 extension euclid2 euclid1 outer_pasch equidistant equal between &
```
```   722   GEO001_0_eq continuous extension euclid2 euclid1 outer_pasch equidistant between equal &
```
```   723   (~between(a::'a,b,b)) --> False"
```
```   724   by meson
```
```   725
```
```   726 abbreviation "GEO002_ax_eq continuous euclid2 euclid1 lower_dimension_point_3
```
```   727   lower_dimension_point_2 lower_dimension_point_1 inner_pasch extension
```
```   728   between equal equidistant \<equiv>
```
```   729   (\<forall>Y X. equidistant(X::'a,Y,Y,X)) &
```
```   730   (\<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)) &
```
```   731   (\<forall>Z X Y. equidistant(X::'a,Y,Z,Z) --> equal(X::'a,Y)) &
```
```   732   (\<forall>X Y W V. between(X::'a,Y,extension(X::'a,Y,W,V))) &
```
```   733   (\<forall>X Y W V. equidistant(Y::'a,extension(X::'a,Y,W,V),W,V)) &
```
```   734   (\<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)) &
```
```   735   (\<forall>X Y. between(X::'a,Y,X) --> equal(X::'a,Y)) &
```
```   736   (\<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)) &
```
```   737   (\<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)) &
```
```   738   (~between(lower_dimension_point_1::'a,lower_dimension_point_2,lower_dimension_point_3)) &
```
```   739   (~between(lower_dimension_point_2::'a,lower_dimension_point_3,lower_dimension_point_1)) &
```
```   740   (~between(lower_dimension_point_3::'a,lower_dimension_point_1,lower_dimension_point_2)) &
```
```   741   (\<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)) &
```
```   742   (\<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))) &
```
```   743   (\<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))) &
```
```   744   (\<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))) &
```
```   745   (\<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)) &
```
```   746   (\<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))) &
```
```   747   (\<forall>X Y W Z. equal(X::'a,Y) & between(X::'a,W,Z) --> between(Y::'a,W,Z)) &
```
```   748   (\<forall>X W Y Z. equal(X::'a,Y) & between(W::'a,X,Z) --> between(W::'a,Y,Z)) &
```
```   749   (\<forall>X W Z Y. equal(X::'a,Y) & between(W::'a,Z,X) --> between(W::'a,Z,Y)) &
```
```   750   (\<forall>X Y V W Z. equal(X::'a,Y) & equidistant(X::'a,V,W,Z) --> equidistant(Y::'a,V,W,Z)) &
```
```   751   (\<forall>X V Y W Z. equal(X::'a,Y) & equidistant(V::'a,X,W,Z) --> equidistant(V::'a,Y,W,Z)) &
```
```   752   (\<forall>X V W Y Z. equal(X::'a,Y) & equidistant(V::'a,W,X,Z) --> equidistant(V::'a,W,Y,Z)) &
```
```   753   (\<forall>X V W Z Y. equal(X::'a,Y) & equidistant(V::'a,W,Z,X) --> equidistant(V::'a,W,Z,Y)) &
```
```   754   (\<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))) &
```
```   755   (\<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))) &
```
```   756   (\<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))) &
```
```   757   (\<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))) &
```
```   758   (\<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))) &
```
```   759   (\<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'))) &
```
```   760   (\<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))) &
```
```   761   (\<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))) &
```
```   762   (\<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))) &
```
```   763   (\<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))) &
```
```   764   (\<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))) &
```
```   765   (\<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))) &
```
```   766   (\<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))) &
```
```   767   (\<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))) &
```
```   768   (\<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))) &
```
```   769   (\<forall>X Y V1 V2 V3. equal(X::'a,Y) --> equal(extension(X::'a,V1,V2,V3),extension(Y::'a,V1,V2,V3))) &
```
```   770   (\<forall>X V1 Y V2 V3. equal(X::'a,Y) --> equal(extension(V1::'a,X,V2,V3),extension(V1::'a,Y,V2,V3))) &
```
```   771   (\<forall>X V1 V2 Y V3. equal(X::'a,Y) --> equal(extension(V1::'a,V2,X,V3),extension(V1::'a,V2,Y,V3))) &
```
```   772   (\<forall>X V1 V2 V3 Y. equal(X::'a,Y) --> equal(extension(V1::'a,V2,V3,X),extension(V1::'a,V2,V3,Y))) &
```
```   773   (\<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))) &
```
```   774   (\<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))) &
```
```   775   (\<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))) &
```
```   776   (\<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))) &
```
```   777   (\<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))) &
```
```   778   (\<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)))"
```
```   779
```
```   780 (*4272 inferences so far.  Searching to depth 10.  29.4 secs*)
```
```   781 lemma GEO017_2:
```
```   782   "EQU001_0_ax equal &
```
```   783   GEO002_ax_eq continuous euclid2 euclid1 lower_dimension_point_3
```
```   784     lower_dimension_point_2 lower_dimension_point_1 inner_pasch extension
```
```   785     between equal equidistant &
```
```   786   (equidistant(u::'a,v,w,x)) &
```
```   787   (~equidistant(u::'a,v,x,w)) --> False"
```
```   788   oops
```
```   789
```
```   790 (*382903 inferences so far.  Searching to depth 9. 1022s (17 mins) on griffon*)
```
```   791 lemma GEO027_3:
```
```   792   "EQU001_0_ax equal &
```
```   793   GEO002_ax_eq continuous euclid2 euclid1 lower_dimension_point_3
```
```   794     lower_dimension_point_2 lower_dimension_point_1 inner_pasch extension
```
```   795     between equal equidistant &
```
```   796   (\<forall>U V. equal(reflection(U::'a,V),extension(U::'a,V,U,V))) &
```
```   797   (\<forall>X Y Z. equal(X::'a,Y) --> equal(reflection(X::'a,Z),reflection(Y::'a,Z))) &
```
```   798   (\<forall>A1 C1 B1. equal(A1::'a,B1) --> equal(reflection(C1::'a,A1),reflection(C1::'a,B1))) &
```
```   799   (\<forall>U V. equidistant(U::'a,V,U,V)) &
```
```   800   (\<forall>W X U V. equidistant(U::'a,V,W,X) --> equidistant(W::'a,X,U,V)) &
```
```   801   (\<forall>V U W X. equidistant(U::'a,V,W,X) --> equidistant(V::'a,U,W,X)) &
```
```   802   (\<forall>U V X W. equidistant(U::'a,V,W,X) --> equidistant(U::'a,V,X,W)) &
```
```   803   (\<forall>V U X W. equidistant(U::'a,V,W,X) --> equidistant(V::'a,U,X,W)) &
```
```   804   (\<forall>W X V U. equidistant(U::'a,V,W,X) --> equidistant(W::'a,X,V,U)) &
```
```   805   (\<forall>X W U V. equidistant(U::'a,V,W,X) --> equidistant(X::'a,W,U,V)) &
```
```   806   (\<forall>X W V U. equidistant(U::'a,V,W,X) --> equidistant(X::'a,W,V,U)) &
```
```   807   (\<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)) &
```
```   808   (\<forall>U V W. equal(V::'a,extension(U::'a,V,W,W))) &
```
```   809   (\<forall>W X U V Y. equal(Y::'a,extension(U::'a,V,W,X)) --> between(U::'a,V,Y)) &
```
```   810   (\<forall>U V. between(U::'a,V,reflection(U::'a,V))) &
```
```   811   (\<forall>U V. equidistant(V::'a,reflection(U::'a,V),U,V)) &
```
```   812   (\<forall>U V. equal(U::'a,V) --> equal(V::'a,reflection(U::'a,V))) &
```
```   813   (\<forall>U. equal(U::'a,reflection(U::'a,U))) &
```
```   814   (\<forall>U V. equal(V::'a,reflection(U::'a,V)) --> equal(U::'a,V)) &
```
```   815   (\<forall>U V. equidistant(U::'a,U,V,V)) &
```
```   816   (\<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)) &
```
```   817   (\<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)) &
```
```   818   (between(u::'a,v,w)) &
```
```   819   (~equal(u::'a,v)) &
```
```   820   (~equal(w::'a,extension(u::'a,v,v,w))) --> False"
```
```   821   oops
```
```   822
```
```   823 (*313884 inferences so far.  Searching to depth 10.  887 secs: 15 mins.*)
```
```   824 lemma GEO058_2:
```
```   825   "EQU001_0_ax equal &
```
```   826   GEO002_ax_eq continuous euclid2 euclid1 lower_dimension_point_3
```
```   827     lower_dimension_point_2 lower_dimension_point_1 inner_pasch extension
```
```   828     between equal equidistant &
```
```   829   (\<forall>U V. equal(reflection(U::'a,V),extension(U::'a,V,U,V))) &
```
```   830   (\<forall>X Y Z. equal(X::'a,Y) --> equal(reflection(X::'a,Z),reflection(Y::'a,Z))) &
```
```   831   (\<forall>A1 C1 B1. equal(A1::'a,B1) --> equal(reflection(C1::'a,A1),reflection(C1::'a,B1))) &
```
```   832   (equal(v::'a,reflection(u::'a,v))) &
```
```   833   (~equal(u::'a,v)) --> False"
```
```   834   oops
```
```   835
```
```   836 (*0 inferences so far.  Searching to depth 0.  0.2 secs*)
```
```   837 lemma GEO079_1:
```
```   838   "(\<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)) &
```
```   839   (\<forall>U V W X Y Z. CONGRUENT(U::'a,V,W,X,Y,Z) --> eq(U::'a,V,W,X,Y,Z)) &
```
```   840   (\<forall>V W U X. trapezoid(U::'a,V,W,X) --> parallel(V::'a,W,U,X)) &
```
```   841   (\<forall>U V X Y. parallel(U::'a,V,X,Y) --> eq(X::'a,V,U,V,X,Y)) &
```
```   842   (trapezoid(a::'a,b,c,d)) &
```
```   843   (~eq(a::'a,c,b,c,a,d)) --> False"
```
```   844    by meson
```
```   845
```
```   846 abbreviation "GRP003_0_ax equal multiply INVERSE identity product \<equiv>
```
```   847   (\<forall>X. product(identity::'a,X,X)) &
```
```   848   (\<forall>X. product(X::'a,identity,X)) &
```
```   849   (\<forall>X. product(INVERSE(X),X,identity)) &
```
```   850   (\<forall>X. product(X::'a,INVERSE(X),identity)) &
```
```   851   (\<forall>X Y. product(X::'a,Y,multiply(X::'a,Y))) &
```
```   852   (\<forall>X Y Z W. product(X::'a,Y,Z) & product(X::'a,Y,W) --> equal(Z::'a,W)) &
```
```   853   (\<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)) &
```
```   854   (\<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))"
```
```   855
```
```   856 abbreviation "GRP003_0_eq product multiply INVERSE equal \<equiv>
```
```   857   (\<forall>X Y. equal(X::'a,Y) --> equal(INVERSE(X),INVERSE(Y))) &
```
```   858   (\<forall>X Y W. equal(X::'a,Y) --> equal(multiply(X::'a,W),multiply(Y::'a,W))) &
```
```   859   (\<forall>X W Y. equal(X::'a,Y) --> equal(multiply(W::'a,X),multiply(W::'a,Y))) &
```
```   860   (\<forall>X Y W Z. equal(X::'a,Y) & product(X::'a,W,Z) --> product(Y::'a,W,Z)) &
```
```   861   (\<forall>X W Y Z. equal(X::'a,Y) & product(W::'a,X,Z) --> product(W::'a,Y,Z)) &
```
```   862   (\<forall>X W Z Y. equal(X::'a,Y) & product(W::'a,Z,X) --> product(W::'a,Z,Y))"
```
```   863
```
```   864 (*2032008 inferences so far. Searching to depth 16. 658s (11 mins) on griffon*)
```
```   865 lemma GRP001_1:
```
```   866   "EQU001_0_ax equal &
```
```   867   GRP003_0_ax equal multiply INVERSE identity product &
```
```   868   GRP003_0_eq product multiply INVERSE equal &
```
```   869   (\<forall>X. product(X::'a,X,identity)) &
```
```   870   (product(a::'a,b,c)) &
```
```   871   (~product(b::'a,a,c)) --> False"
```
```   872   oops
```
```   873
```
```   874 (*2386 inferences so far.  Searching to depth 11.  8.7 secs*)
```
```   875 lemma GRP008_1:
```
```   876   "EQU001_0_ax equal &
```
```   877   GRP003_0_ax equal multiply INVERSE identity product &
```
```   878   GRP003_0_eq product multiply INVERSE equal &
```
```   879   (\<forall>A B. equal(A::'a,B) --> equal(h(A),h(B))) &
```
```   880   (\<forall>C D. equal(C::'a,D) --> equal(j(C),j(D))) &
```
```   881   (\<forall>A B. equal(A::'a,B) & q(A) --> q(B)) &
```
```   882   (\<forall>B A C. q(A) & product(A::'a,B,C) --> product(B::'a,A,C)) &
```
```   883   (\<forall>A. product(j(A),A,h(A)) | product(A::'a,j(A),h(A)) | q(A)) &
```
```   884   (\<forall>A. product(j(A),A,h(A)) & product(A::'a,j(A),h(A)) --> q(A)) &
```
```   885   (~q(identity)) --> False"
```
```   886   by meson
```
```   887
```
```   888 (*8625 inferences so far.  Searching to depth 11.  20 secs*)
```
```   889 lemma GRP013_1:
```
```   890   "EQU001_0_ax equal &
```
```   891   GRP003_0_ax equal multiply INVERSE identity product &
```
```   892   GRP003_0_eq product multiply INVERSE equal &
```
```   893   (\<forall>A. product(A::'a,A,identity)) &
```
```   894   (product(a::'a,b,c)) &
```
```   895   (product(INVERSE(a),INVERSE(b),d)) &
```
```   896   (\<forall>A C B. product(INVERSE(A),INVERSE(B),C) --> product(A::'a,C,B)) &
```
```   897   (~product(c::'a,d,identity)) --> False"
```
```   898   oops
```
```   899
```
```   900 (*2448 inferences so far.  Searching to depth 10.  7.2 secs*)
```
```   901 lemma GRP037_3:
```
```   902   "EQU001_0_ax equal &
```
```   903   GRP003_0_ax equal multiply INVERSE identity product &
```
```   904   GRP003_0_eq product multiply INVERSE equal &
```
```   905   (\<forall>A B C. subgroup_member(A) & subgroup_member(B) & product(A::'a,INVERSE(B),C) --> subgroup_member(C)) &
```
```   906   (\<forall>A B. equal(A::'a,B) & subgroup_member(A) --> subgroup_member(B)) &
```
```   907   (\<forall>A. subgroup_member(A) --> product(Gidentity::'a,A,A)) &
```
```   908   (\<forall>A. subgroup_member(A) --> product(A::'a,Gidentity,A)) &
```
```   909   (\<forall>A. subgroup_member(A) --> product(A::'a,Ginverse(A),Gidentity)) &
```
```   910   (\<forall>A. subgroup_member(A) --> product(Ginverse(A),A,Gidentity)) &
```
```   911   (\<forall>A. subgroup_member(A) --> subgroup_member(Ginverse(A))) &
```
```   912   (\<forall>A B. equal(A::'a,B) --> equal(Ginverse(A),Ginverse(B))) &
```
```   913   (\<forall>A C D B. product(A::'a,B,C) & product(A::'a,D,C) --> equal(D::'a,B)) &
```
```   914   (\<forall>B C D A. product(A::'a,B,C) & product(D::'a,B,C) --> equal(D::'a,A)) &
```
```   915   (subgroup_member(a)) &
```
```   916   (subgroup_member(Gidentity)) &
```
```   917   (~equal(INVERSE(a),Ginverse(a))) --> False"
```
```   918   by meson
```
```   919
```
```   920 (*163 inferences so far.  Searching to depth 11.  0.3 secs*)
```
```   921 lemma GRP031_2:
```
```   922   "(\<forall>X Y. product(X::'a,Y,multiply(X::'a,Y))) &
```
```   923   (\<forall>X Y Z W. product(X::'a,Y,Z) & product(X::'a,Y,W) --> equal(Z::'a,W)) &
```
```   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>A. product(A::'a,INVERSE(A),identity)) &
```
```   927   (\<forall>A. product(A::'a,identity,A)) &
```
```   928   (\<forall>A. ~product(A::'a,a,identity)) --> False"
```
```   929    by meson
```
```   930
```
```   931 (*47 inferences so far.  Searching to depth 11.   0.2 secs*)
```
```   932 lemma GRP034_4:
```
```   933   "(\<forall>X Y. product(X::'a,Y,multiply(X::'a,Y))) &
```
```   934   (\<forall>X. product(identity::'a,X,X)) &
```
```   935   (\<forall>X. product(X::'a,identity,X)) &
```
```   936   (\<forall>X. product(X::'a,INVERSE(X),identity)) &
```
```   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>B A C. subgroup_member(A) & subgroup_member(B) & product(B::'a,INVERSE(A),C) --> subgroup_member(C)) &
```
```   940   (subgroup_member(a)) &
```
```   941   (~subgroup_member(INVERSE(a))) --> False"
```
```   942   by meson
```
```   943
```
```   944 (*3287 inferences so far.  Searching to depth 14.  3.5 secs*)
```
```   945 lemma GRP047_2:
```
```   946   "(\<forall>X. product(identity::'a,X,X)) &
```
```   947   (\<forall>X. product(INVERSE(X),X,identity)) &
```
```   948   (\<forall>X Y. product(X::'a,Y,multiply(X::'a,Y))) &
```
```   949   (\<forall>X Y Z W. product(X::'a,Y,Z) & product(X::'a,Y,W) --> equal(Z::'a,W)) &
```
```   950   (\<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)) &
```
```   951   (\<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)) &
```
```   952   (\<forall>X W Z Y. equal(X::'a,Y) & product(W::'a,Z,X) --> product(W::'a,Z,Y)) &
```
```   953   (equal(a::'a,b)) &
```
```   954   (~equal(multiply(c::'a,a),multiply(c::'a,b))) --> False"
```
```   955   by meson
```
```   956
```
```   957 (*25559 inferences so far.  Searching to depth 19.  16.9 secs*)
```
```   958 lemma GRP130_1_002:
```
```   959   "(group_element(e_1)) &
```
```   960   (group_element(e_2)) &
```
```   961   (~equal(e_1::'a,e_2)) &
```
```   962   (~equal(e_2::'a,e_1)) &
```
```   963   (\<forall>X Y. group_element(X) & group_element(Y) --> product(X::'a,Y,e_1) | product(X::'a,Y,e_2)) &
```
```   964   (\<forall>X Y W Z. product(X::'a,Y,W) & product(X::'a,Y,Z) --> equal(W::'a,Z)) &
```
```   965   (\<forall>X Y W Z. product(X::'a,W,Y) & product(X::'a,Z,Y) --> equal(W::'a,Z)) &
```
```   966   (\<forall>Y X W Z. product(W::'a,Y,X) & product(Z::'a,Y,X) --> equal(W::'a,Z)) &
```
```   967   (\<forall>Z1 Z2 Y X. product(X::'a,Y,Z1) & product(X::'a,Z1,Z2) --> product(Z2::'a,Y,X)) --> False"
```
```   968   oops
```
```   969
```
```   970 abbreviation "GRP004_0_ax INVERSE identity multiply equal \<equiv>
```
```   971   (\<forall>X. equal(multiply(identity::'a,X),X)) &
```
```   972   (\<forall>X. equal(multiply(INVERSE(X),X),identity)) &
```
```   973   (\<forall>X Y Z. equal(multiply(multiply(X::'a,Y),Z),multiply(X::'a,multiply(Y::'a,Z)))) &
```
```   974   (\<forall>A B. equal(A::'a,B) --> equal(INVERSE(A),INVERSE(B))) &
```
```   975   (\<forall>C D E. equal(C::'a,D) --> equal(multiply(C::'a,E),multiply(D::'a,E))) &
```
```   976   (\<forall>F' H G. equal(F'::'a,G) --> equal(multiply(H::'a,F'),multiply(H::'a,G)))"
```
```   977
```
```   978 abbreviation "GRP004_2_ax multiply least_upper_bound greatest_lower_bound equal \<equiv>
```
```   979   (\<forall>Y X. equal(greatest_lower_bound(X::'a,Y),greatest_lower_bound(Y::'a,X))) &
```
```   980   (\<forall>Y X. equal(least_upper_bound(X::'a,Y),least_upper_bound(Y::'a,X))) &
```
```   981   (\<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))) &
```
```   982   (\<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))) &
```
```   983   (\<forall>X. equal(least_upper_bound(X::'a,X),X)) &
```
```   984   (\<forall>X. equal(greatest_lower_bound(X::'a,X),X)) &
```
```   985   (\<forall>Y X. equal(least_upper_bound(X::'a,greatest_lower_bound(X::'a,Y)),X)) &
```
```   986   (\<forall>Y X. equal(greatest_lower_bound(X::'a,least_upper_bound(X::'a,Y)),X)) &
```
```   987   (\<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)))) &
```
```   988   (\<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)))) &
```
```   989   (\<forall>Y Z X. equal(multiply(least_upper_bound(Y::'a,Z),X),least_upper_bound(multiply(Y::'a,X),multiply(Z::'a,X)))) &
```
```   990   (\<forall>Y Z X. equal(multiply(greatest_lower_bound(Y::'a,Z),X),greatest_lower_bound(multiply(Y::'a,X),multiply(Z::'a,X)))) &
```
```   991   (\<forall>A B C. equal(A::'a,B) --> equal(greatest_lower_bound(A::'a,C),greatest_lower_bound(B::'a,C))) &
```
```   992   (\<forall>A C B. equal(A::'a,B) --> equal(greatest_lower_bound(C::'a,A),greatest_lower_bound(C::'a,B))) &
```
```   993   (\<forall>A B C. equal(A::'a,B) --> equal(least_upper_bound(A::'a,C),least_upper_bound(B::'a,C))) &
```
```   994   (\<forall>A C B. equal(A::'a,B) --> equal(least_upper_bound(C::'a,A),least_upper_bound(C::'a,B))) &
```
```   995   (\<forall>A B C. equal(A::'a,B) --> equal(multiply(A::'a,C),multiply(B::'a,C))) &
```
```   996   (\<forall>A C B. equal(A::'a,B) --> equal(multiply(C::'a,A),multiply(C::'a,B)))"
```
```   997
```
```   998 (*3468 inferences so far.  Searching to depth 10.  9.1 secs*)
```
```   999 lemma GRP156_1:
```
```  1000   "EQU001_0_ax equal &
```
```  1001   GRP004_0_ax INVERSE identity multiply equal &
```
```  1002   GRP004_2_ax multiply least_upper_bound greatest_lower_bound equal &
```
```  1003   (equal(least_upper_bound(a::'a,b),b)) &
```
```  1004   (~equal(greatest_lower_bound(multiply(a::'a,c),multiply(b::'a,c)),multiply(a::'a,c))) --> False"
```
```  1005     by meson
```
```  1006
```
```  1007 (*4394 inferences so far.  Searching to depth 10.  8.2 secs*)
```
```  1008 lemma GRP168_1:
```
```  1009   "EQU001_0_ax equal &
```
```  1010   GRP004_0_ax INVERSE identity multiply equal &
```
```  1011   GRP004_2_ax multiply least_upper_bound greatest_lower_bound equal &
```
```  1012   (equal(least_upper_bound(a::'a,b),b)) &
```
```  1013   (~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"
```
```  1014   by meson
```
```  1015
```
```  1016 abbreviation "HEN002_0_ax identity Zero Divide equal mless_equal \<equiv>
```
```  1017   (\<forall>X Y. mless_equal(X::'a,Y) --> equal(Divide(X::'a,Y),Zero)) &
```
```  1018   (\<forall>X Y. equal(Divide(X::'a,Y),Zero) --> mless_equal(X::'a,Y)) &
```
```  1019   (\<forall>Y X. mless_equal(Divide(X::'a,Y),X)) &
```
```  1020   (\<forall>X Y Z. mless_equal(Divide(Divide(X::'a,Z),Divide(Y::'a,Z)),Divide(Divide(X::'a,Y),Z))) &
```
```  1021   (\<forall>X. mless_equal(Zero::'a,X)) &
```
```  1022   (\<forall>X Y. mless_equal(X::'a,Y) & mless_equal(Y::'a,X) --> equal(X::'a,Y)) &
```
```  1023   (\<forall>X. mless_equal(X::'a,identity))"
```
```  1024
```
```  1025 abbreviation "HEN002_0_eq mless_equal Divide equal \<equiv>
```
```  1026   (\<forall>A B C. equal(A::'a,B) --> equal(Divide(A::'a,C),Divide(B::'a,C))) &
```
```  1027   (\<forall>D F' E. equal(D::'a,E) --> equal(Divide(F'::'a,D),Divide(F'::'a,E))) &
```
```  1028   (\<forall>G H I'. equal(G::'a,H) & mless_equal(G::'a,I') --> mless_equal(H::'a,I')) &
```
```  1029   (\<forall>J L K'. equal(J::'a,K') & mless_equal(L::'a,J) --> mless_equal(L::'a,K'))"
```
```  1030
```
```  1031 (*250258 inferences so far.  Searching to depth 16.  406.2 secs*)
```
```  1032 lemma HEN003_3:
```
```  1033   "EQU001_0_ax equal &
```
```  1034   HEN002_0_ax identity Zero Divide equal mless_equal &
```
```  1035   HEN002_0_eq mless_equal Divide equal &
```
```  1036   (~equal(Divide(a::'a,a),Zero)) --> False"
```
```  1037   oops
```
```  1038
```
```  1039 (*202177 inferences so far.  Searching to depth 14.  451 secs*)
```
```  1040 lemma HEN007_2:
```
```  1041   "EQU001_0_ax equal &
```
```  1042   (\<forall>X Y. mless_equal(X::'a,Y) --> quotient(X::'a,Y,Zero)) &
```
```  1043   (\<forall>X Y. quotient(X::'a,Y,Zero) --> mless_equal(X::'a,Y)) &
```
```  1044   (\<forall>Y Z X. quotient(X::'a,Y,Z) --> mless_equal(Z::'a,X)) &
```
```  1045   (\<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)) &
```
```  1046   (\<forall>X. mless_equal(Zero::'a,X)) &
```
```  1047   (\<forall>X Y. mless_equal(X::'a,Y) & mless_equal(Y::'a,X) --> equal(X::'a,Y)) &
```
```  1048   (\<forall>X. mless_equal(X::'a,identity)) &
```
```  1049   (\<forall>X Y. quotient(X::'a,Y,Divide(X::'a,Y))) &
```
```  1050   (\<forall>X Y Z W. quotient(X::'a,Y,Z) & quotient(X::'a,Y,W) --> equal(Z::'a,W)) &
```
```  1051   (\<forall>X Y W Z. equal(X::'a,Y) & quotient(X::'a,W,Z) --> quotient(Y::'a,W,Z)) &
```
```  1052   (\<forall>X W Y Z. equal(X::'a,Y) & quotient(W::'a,X,Z) --> quotient(W::'a,Y,Z)) &
```
```  1053   (\<forall>X W Z Y. equal(X::'a,Y) & quotient(W::'a,Z,X) --> quotient(W::'a,Z,Y)) &
```
```  1054   (\<forall>X Z Y. equal(X::'a,Y) & mless_equal(Z::'a,X) --> mless_equal(Z::'a,Y)) &
```
```  1055   (\<forall>X Y Z. equal(X::'a,Y) & mless_equal(X::'a,Z) --> mless_equal(Y::'a,Z)) &
```
```  1056   (\<forall>X Y W. equal(X::'a,Y) --> equal(Divide(X::'a,W),Divide(Y::'a,W))) &
```
```  1057   (\<forall>X W Y. equal(X::'a,Y) --> equal(Divide(W::'a,X),Divide(W::'a,Y))) &
```
```  1058   (\<forall>X. quotient(X::'a,identity,Zero)) &
```
```  1059   (\<forall>X. quotient(Zero::'a,X,Zero)) &
```
```  1060   (\<forall>X. quotient(X::'a,X,Zero)) &
```
```  1061   (\<forall>X. quotient(X::'a,Zero,X)) &
```
```  1062   (\<forall>Y X Z. mless_equal(X::'a,Y) & mless_equal(Y::'a,Z) --> mless_equal(X::'a,Z)) &
```
```  1063   (\<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)) &
```
```  1064   (mless_equal(x::'a,y)) &
```
```  1065   (quotient(z::'a,y,zQy)) &
```
```  1066   (quotient(z::'a,x,zQx)) &
```
```  1067   (~mless_equal(zQy::'a,zQx)) --> False"
```
```  1068   oops
```
```  1069
```
```  1070 (*60026 inferences so far.  Searching to depth 12.  42.2 secs*)
```
```  1071 lemma HEN008_4:
```
```  1072   "EQU001_0_ax equal &
```
```  1073   HEN002_0_ax identity Zero Divide equal mless_equal &
```
```  1074   HEN002_0_eq mless_equal Divide equal &
```
```  1075   (\<forall>X. equal(Divide(X::'a,identity),Zero)) &
```
```  1076   (\<forall>X. equal(Divide(Zero::'a,X),Zero)) &
```
```  1077   (\<forall>X. equal(Divide(X::'a,X),Zero)) &
```
```  1078   (equal(Divide(a::'a,Zero),a)) &
```
```  1079   (\<forall>Y X Z. mless_equal(X::'a,Y) & mless_equal(Y::'a,Z) --> mless_equal(X::'a,Z)) &
```
```  1080   (\<forall>X Z Y. mless_equal(Divide(X::'a,Y),Z) --> mless_equal(Divide(X::'a,Z),Y)) &
```
```  1081   (\<forall>Y Z X. mless_equal(X::'a,Y) --> mless_equal(Divide(Z::'a,Y),Divide(Z::'a,X))) &
```
```  1082   (mless_equal(a::'a,b)) &
```
```  1083   (~mless_equal(Divide(a::'a,c),Divide(b::'a,c))) --> False"
```
```  1084   oops
```
```  1085
```
```  1086 (*3160 inferences so far.  Searching to depth 11.  3.5 secs*)
```
```  1087 lemma HEN009_5:
```
```  1088   "EQU001_0_ax equal &
```
```  1089   (\<forall>Y X. equal(Divide(Divide(X::'a,Y),X),Zero)) &
```
```  1090   (\<forall>X Y Z. equal(Divide(Divide(Divide(X::'a,Z),Divide(Y::'a,Z)),Divide(Divide(X::'a,Y),Z)),Zero)) &
```
```  1091   (\<forall>X. equal(Divide(Zero::'a,X),Zero)) &
```
```  1092   (\<forall>X Y. equal(Divide(X::'a,Y),Zero) & equal(Divide(Y::'a,X),Zero) --> equal(X::'a,Y)) &
```
```  1093   (\<forall>X. equal(Divide(X::'a,identity),Zero)) &
```
```  1094   (\<forall>A B C. equal(A::'a,B) --> equal(Divide(A::'a,C),Divide(B::'a,C))) &
```
```  1095   (\<forall>D F' E. equal(D::'a,E) --> equal(Divide(F'::'a,D),Divide(F'::'a,E))) &
```
```  1096   (\<forall>Y X Z. equal(Divide(X::'a,Y),Zero) & equal(Divide(Y::'a,Z),Zero) --> equal(Divide(X::'a,Z),Zero)) &
```
```  1097   (\<forall>X Z Y. equal(Divide(Divide(X::'a,Y),Z),Zero) --> equal(Divide(Divide(X::'a,Z),Y),Zero)) &
```
```  1098   (\<forall>Y Z X. equal(Divide(X::'a,Y),Zero) --> equal(Divide(Divide(Z::'a,Y),Divide(Z::'a,X)),Zero)) &
```
```  1099   (~equal(Divide(identity::'a,a),Divide(identity::'a,Divide(identity::'a,Divide(identity::'a,a))))) &
```
```  1100   (equal(Divide(identity::'a,a),b)) &
```
```  1101   (equal(Divide(identity::'a,b),c)) &
```
```  1102   (equal(Divide(identity::'a,c),d)) &
```
```  1103   (~equal(b::'a,d)) --> False"
```
```  1104   by meson
```
```  1105
```
```  1106 (*970373 inferences so far.  Searching to depth 17.  890.0 secs*)
```
```  1107 lemma HEN012_3:
```
```  1108   "EQU001_0_ax equal &
```
```  1109   HEN002_0_ax identity Zero Divide equal mless_equal &
```
```  1110   HEN002_0_eq mless_equal Divide equal &
```
```  1111   (~mless_equal(a::'a,a)) --> False"
```
```  1112   oops
```
```  1113
```
```  1114
```
```  1115 (*1063 inferences so far.  Searching to depth 20.  1.0 secs*)
```
```  1116 lemma LCL010_1:
```
```  1117  "(\<forall>X Y. is_a_theorem(equivalent(X::'a,Y)) & is_a_theorem(X) --> is_a_theorem(Y)) &
```
```  1118   (\<forall>X Z Y. is_a_theorem(equivalent(equivalent(X::'a,Y),equivalent(equivalent(X::'a,Z),equivalent(Z::'a,Y))))) &
```
```  1119   (~is_a_theorem(equivalent(equivalent(a::'a,b),equivalent(equivalent(c::'a,b),equivalent(a::'a,c))))) --> False"
```
```  1120   by meson
```
```  1121
```
```  1122 (*2549 inferences so far.  Searching to depth 12.  1.4 secs*)
```
```  1123 lemma LCL077_2:
```
```  1124  "(\<forall>X Y. is_a_theorem(implies(X,Y)) & is_a_theorem(X) --> is_a_theorem(Y)) &
```
```  1125   (\<forall>Y X. is_a_theorem(implies(X,implies(Y,X)))) &
```
```  1126   (\<forall>Y X Z. is_a_theorem(implies(implies(X,implies(Y,Z)),implies(implies(X,Y),implies(X,Z))))) &
```
```  1127   (\<forall>Y X. is_a_theorem(implies(implies(not(X),not(Y)),implies(Y,X)))) &
```
```  1128   (\<forall>X2 X1 X3. is_a_theorem(implies(X1,X2)) & is_a_theorem(implies(X2,X3)) --> is_a_theorem(implies(X1,X3))) &
```
```  1129   (~is_a_theorem(implies(not(not(a)),a))) --> False"
```
```  1130   by meson
```
```  1131
```
```  1132 (*2036 inferences so far.  Searching to depth 20.  1.5 secs*)
```
```  1133 lemma LCL082_1:
```
```  1134  "(\<forall>X Y. is_a_theorem(implies(X::'a,Y)) & is_a_theorem(X) --> is_a_theorem(Y)) &
```
```  1135   (\<forall>Y Z U X. is_a_theorem(implies(implies(implies(X::'a,Y),Z),implies(implies(Z::'a,X),implies(U::'a,X))))) &
```
```  1136   (~is_a_theorem(implies(a::'a,implies(b::'a,a)))) --> False"
```
```  1137   by meson
```
```  1138
```
```  1139 (*1100 inferences so far.  Searching to depth 13.  1.0 secs*)
```
```  1140 lemma LCL111_1:
```
```  1141  "(\<forall>X Y. is_a_theorem(implies(X,Y)) & is_a_theorem(X) --> is_a_theorem(Y)) &
```
```  1142   (\<forall>Y X. is_a_theorem(implies(X,implies(Y,X)))) &
```
```  1143   (\<forall>Y X Z. is_a_theorem(implies(implies(X,Y),implies(implies(Y,Z),implies(X,Z))))) &
```
```  1144   (\<forall>Y X. is_a_theorem(implies(implies(implies(X,Y),Y),implies(implies(Y,X),X)))) &
```
```  1145   (\<forall>Y X. is_a_theorem(implies(implies(not(X),not(Y)),implies(Y,X)))) &
```
```  1146   (~is_a_theorem(implies(implies(a,b),implies(implies(c,a),implies(c,b))))) --> False"
```
```  1147   by meson
```
```  1148
```
```  1149 (*667 inferences so far.  Searching to depth 9.  1.4 secs*)
```
```  1150 lemma LCL143_1:
```
```  1151  "(\<forall>X. equal(X,X)) &
```
```  1152   (\<forall>Y X. equal(X,Y) --> equal(Y,X)) &
```
```  1153   (\<forall>Y X Z. equal(X,Y) & equal(Y,Z) --> equal(X,Z)) &
```
```  1154   (\<forall>X. equal(implies(true,X),X)) &
```
```  1155   (\<forall>Y X Z. equal(implies(implies(X,Y),implies(implies(Y,Z),implies(X,Z))),true)) &
```
```  1156   (\<forall>Y X. equal(implies(implies(X,Y),Y),implies(implies(Y,X),X))) &
```
```  1157   (\<forall>Y X. equal(implies(implies(not(X),not(Y)),implies(Y,X)),true)) &
```
```  1158   (\<forall>A B C. equal(A,B) --> equal(implies(A,C),implies(B,C))) &
```
```  1159   (\<forall>D F' E. equal(D,E) --> equal(implies(F',D),implies(F',E))) &
```
```  1160   (\<forall>G H. equal(G,H) --> equal(not(G),not(H))) &
```
```  1161   (\<forall>X Y. equal(big_V(X,Y),implies(implies(X,Y),Y))) &
```
```  1162   (\<forall>X Y. equal(big_hat(X,Y),not(big_V(not(X),not(Y))))) &
```
```  1163   (\<forall>X Y. ordered(X,Y) --> equal(implies(X,Y),true)) &
```
```  1164   (\<forall>X Y. equal(implies(X,Y),true) --> ordered(X,Y)) &
```
```  1165   (\<forall>A B C. equal(A,B) --> equal(big_V(A,C),big_V(B,C))) &
```
```  1166   (\<forall>D F' E. equal(D,E) --> equal(big_V(F',D),big_V(F',E))) &
```
```  1167   (\<forall>G H I'. equal(G,H) --> equal(big_hat(G,I'),big_hat(H,I'))) &
```
```  1168   (\<forall>J L K'. equal(J,K') --> equal(big_hat(L,J),big_hat(L,K'))) &
```
```  1169   (\<forall>M N O'. equal(M,N) & ordered(M,O') --> ordered(N,O')) &
```
```  1170   (\<forall>P R Q. equal(P,Q) & ordered(R,P) --> ordered(R,Q)) &
```
```  1171   (ordered(x,y)) &
```
```  1172   (~ordered(implies(z,x),implies(z,y))) --> False"
```
```  1173   by meson
```
```  1174
```
```  1175 (*5245 inferences so far.  Searching to depth 12.  4.6 secs*)
```
```  1176 lemma LCL182_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(or(not(p),q)),or(not(not(q)),not(p))))) --> False"
```
```  1186   by meson
```
```  1187
```
```  1188 (*410 inferences so far.  Searching to depth 10.  0.3 secs*)
```
```  1189 lemma LCL200_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(not(or(p,q))),not(q)))) --> False"
```
```  1199   by meson
```
```  1200
```
```  1201 (*5849 inferences so far.  Searching to depth 12.  5.6 secs*)
```
```  1202 lemma LCL215_1:
```
```  1203  "(\<forall>A. axiom(or(not(or(A,A)),A))) &
```
```  1204   (\<forall>B A. axiom(or(not(A),or(B,A)))) &
```
```  1205   (\<forall>B A. axiom(or(not(or(A,B)),or(B,A)))) &
```
```  1206   (\<forall>B A C. axiom(or(not(or(A,or(B,C))),or(B,or(A,C))))) &
```
```  1207   (\<forall>A C B. axiom(or(not(or(not(A),B)),or(not(or(C,A)),or(C,B))))) &
```
```  1208   (\<forall>X. axiom(X) --> theorem(X)) &
```
```  1209   (\<forall>X Y. axiom(or(not(Y),X)) & theorem(Y) --> theorem(X)) &
```
```  1210   (\<forall>X Y Z. axiom(or(not(X),Y)) & theorem(or(not(Y),Z)) --> theorem(or(not(X),Z))) &
```
```  1211   (~theorem(or(not(or(not(p),q)),or(not(or(p,q)),q)))) --> False"
```
```  1212   by meson
```
```  1213
```
```  1214 (*0 secs.  Not sure that a search even starts!*)
```
```  1215 lemma LCL230_2:
```
```  1216   "(q --> p | r) &
```
```  1217   (~p) &
```
```  1218   (q) &
```
```  1219   (~r) --> False"
```
```  1220   by meson
```
```  1221
```
```  1222 (*119585 inferences so far.  Searching to depth 14.  262.4 secs*)
```
```  1223 lemma LDA003_1:
```
```  1224   "EQU001_0_ax equal &
```
```  1225   (\<forall>Y X Z. equal(f(X::'a,f(Y::'a,Z)),f(f(X::'a,Y),f(X::'a,Z)))) &
```
```  1226   (\<forall>X Y. left(X::'a,f(X::'a,Y))) &
```
```  1227   (\<forall>Y X Z. left(X::'a,Y) & left(Y::'a,Z) --> left(X::'a,Z)) &
```
```  1228   (equal(num2::'a,f(num1::'a,num1))) &
```
```  1229   (equal(num3::'a,f(num2::'a,num1))) &
```
```  1230   (equal(u::'a,f(num2::'a,num2))) &
```
```  1231   (\<forall>A B C. equal(A::'a,B) --> equal(f(A::'a,C),f(B::'a,C))) &
```
```  1232   (\<forall>D F' E. equal(D::'a,E) --> equal(f(F'::'a,D),f(F'::'a,E))) &
```
```  1233   (\<forall>G H I'. equal(G::'a,H) & left(G::'a,I') --> left(H::'a,I')) &
```
```  1234   (\<forall>J L K'. equal(J::'a,K') & left(L::'a,J) --> left(L::'a,K')) &
```
```  1235   (~left(num3::'a,u)) --> False"
```
```  1236   oops
```
```  1237
```
```  1238
```
```  1239 (*2392 inferences so far.  Searching to depth 12.  2.2 secs*)
```
```  1240 lemma MSC002_1:
```
```  1241  "(at(something::'a,here,now)) &
```
```  1242   (\<forall>Place Situation. hand_at(Place::'a,Situation) --> hand_at(Place::'a,let_go(Situation))) &
```
```  1243   (\<forall>Place Another_place Situation. hand_at(Place::'a,Situation) --> hand_at(Another_place::'a,go(Another_place::'a,Situation))) &
```
```  1244   (\<forall>Thing Situation. ~held(Thing::'a,let_go(Situation))) &
```
```  1245   (\<forall>Situation Thing. at(Thing::'a,here,Situation) --> red(Thing)) &
```
```  1246   (\<forall>Thing Place Situation. at(Thing::'a,Place,Situation) --> at(Thing::'a,Place,let_go(Situation))) &
```
```  1247   (\<forall>Thing Place Situation. at(Thing::'a,Place,Situation) --> at(Thing::'a,Place,pick_up(Situation))) &
```
```  1248   (\<forall>Thing Place Situation. at(Thing::'a,Place,Situation) --> grabbed(Thing::'a,pick_up(go(Place::'a,let_go(Situation))))) &
```
```  1249   (\<forall>Thing Situation. red(Thing) & put(Thing::'a,there,Situation) --> answer(Situation)) &
```
```  1250   (\<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))) &
```
```  1251   (\<forall>Thing Place Another_place Situation. at(Thing::'a,Place,Situation) --> held(Thing::'a,Situation) | at(Thing::'a,Place,go(Another_place::'a,Situation))) &
```
```  1252   (\<forall>One_place Thing Place Situation. hand_at(One_place::'a,Situation) & held(Thing::'a,Situation) --> at(Thing::'a,Place,go(Place::'a,Situation))) &
```
```  1253   (\<forall>Place Thing Situation. hand_at(Place::'a,Situation) & at(Thing::'a,Place,Situation) --> held(Thing::'a,pick_up(Situation))) &
```
```  1254   (\<forall>Situation. ~answer(Situation)) --> False"
```
```  1255   by meson
```
```  1256
```
```  1257 (*73 inferences so far.  Searching to depth 10.  0.2 secs*)
```
```  1258 lemma MSC003_1:
```
```  1259   "(\<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)) &
```
```  1260   (\<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)) &
```
```  1261   (in'(john::'a,boy)) &
```
```  1262   (\<forall>X. in'(X::'a,boy) --> in'(X::'a,human)) &
```
```  1263   (\<forall>X. in'(X::'a,hand) --> has_parts(X::'a,num5,fingers)) &
```
```  1264   (\<forall>X. in'(X::'a,human) --> has_parts(X::'a,num2,arm)) &
```
```  1265   (\<forall>X. in'(X::'a,arm) --> has_parts(X::'a,num1,hand)) &
```
```  1266   (~has_parts(john::'a,mtimes(num2::'a,num1),hand)) --> False"
```
```  1267   by meson
```
```  1268
```
```  1269 (*1486 inferences so far.  Searching to depth 20.  1.2 secs*)
```
```  1270 lemma MSC004_1:
```
```  1271  "(\<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)) &
```
```  1272   (\<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)) &
```
```  1273   (in'(john::'a,boy)) &
```
```  1274   (\<forall>X. in'(X::'a,boy) --> in'(X::'a,human)) &
```
```  1275   (\<forall>X. in'(X::'a,hand) --> has_parts(X::'a,num5,fingers)) &
```
```  1276   (\<forall>X. in'(X::'a,human) --> has_parts(X::'a,num2,arm)) &
```
```  1277   (\<forall>X. in'(X::'a,arm) --> has_parts(X::'a,num1,hand)) &
```
```  1278   (~has_parts(john::'a,mtimes(mtimes(num2::'a,num1),num5),fingers)) --> False"
```
```  1279   by meson
```
```  1280
```
```  1281 (*100 inferences so far.  Searching to depth 12.  0.1 secs*)
```
```  1282 lemma MSC005_1:
```
```  1283   "(value(truth::'a,truth)) &
```
```  1284   (value(falsity::'a,falsity)) &
```
```  1285   (\<forall>X Y. value(X::'a,truth) & value(Y::'a,truth) --> value(xor(X::'a,Y),falsity)) &
```
```  1286   (\<forall>X Y. value(X::'a,truth) & value(Y::'a,falsity) --> value(xor(X::'a,Y),truth)) &
```
```  1287   (\<forall>X Y. value(X::'a,falsity) & value(Y::'a,truth) --> value(xor(X::'a,Y),truth)) &
```
```  1288   (\<forall>X Y. value(X::'a,falsity) & value(Y::'a,falsity) --> value(xor(X::'a,Y),falsity)) &
```
```  1289   (\<forall>Value. ~value(xor(xor(xor(xor(truth::'a,falsity),falsity),truth),falsity),Value)) --> False"
```
```  1290   by meson
```
```  1291
```
```  1292 (*19116 inferences so far.  Searching to depth 16.  15.9 secs*)
```
```  1293 lemma MSC006_1:
```
```  1294  "(\<forall>Y X Z. p(X::'a,Y) & p(Y::'a,Z) --> p(X::'a,Z)) &
```
```  1295   (\<forall>Y X Z. q(X::'a,Y) & q(Y::'a,Z) --> q(X::'a,Z)) &
```
```  1296   (\<forall>Y X. q(X::'a,Y) --> q(Y::'a,X)) &
```
```  1297   (\<forall>X Y. p(X::'a,Y) | q(X::'a,Y)) &
```
```  1298   (~p(a::'a,b)) &
```
```  1299   (~q(c::'a,d)) --> False"
```
```  1300    by meson
```
```  1301
```
```  1302 (*1713 inferences so far.  Searching to depth 10.  2.8 secs*)
```
```  1303 lemma NUM001_1:
```
```  1304   "(\<forall>A. equal(A::'a,A)) &
```
```  1305   (\<forall>B A C. equal(A::'a,B) & equal(B::'a,C) --> equal(A::'a,C)) &
```
```  1306   (\<forall>B A. equal(add(A::'a,B),add(B::'a,A))) &
```
```  1307   (\<forall>A B C. equal(add(A::'a,add(B::'a,C)),add(add(A::'a,B),C))) &
```
```  1308   (\<forall>B A. equal(subtract(add(A::'a,B),B),A)) &
```
```  1309   (\<forall>A B. equal(A::'a,subtract(add(A::'a,B),B))) &
```
```  1310   (\<forall>A C B. equal(add(subtract(A::'a,B),C),subtract(add(A::'a,C),B))) &
```
```  1311   (\<forall>A C B. equal(subtract(add(A::'a,B),C),add(subtract(A::'a,C),B))) &
```
```  1312   (\<forall>A C B D. equal(A::'a,B) & equal(C::'a,add(A::'a,D)) --> equal(C::'a,add(B::'a,D))) &
```
```  1313   (\<forall>A C D B. equal(A::'a,B) & equal(C::'a,add(D::'a,A)) --> equal(C::'a,add(D::'a,B))) &
```
```  1314   (\<forall>A C B D. equal(A::'a,B) & equal(C::'a,subtract(A::'a,D)) --> equal(C::'a,subtract(B::'a,D))) &
```
```  1315   (\<forall>A C D B. equal(A::'a,B) & equal(C::'a,subtract(D::'a,A)) --> equal(C::'a,subtract(D::'a,B))) &
```
```  1316   (~equal(add(add(a::'a,b),c),add(a::'a,add(b::'a,c)))) --> False"
```
```  1317   by meson
```
```  1318
```
```  1319 abbreviation "NUM001_0_ax multiply successor num0 add equal \<equiv>
```
```  1320   (\<forall>A. equal(add(A::'a,num0),A)) &
```
```  1321   (\<forall>A B. equal(add(A::'a,successor(B)),successor(add(A::'a,B)))) &
```
```  1322   (\<forall>A. equal(multiply(A::'a,num0),num0)) &
```
```  1323   (\<forall>B A. equal(multiply(A::'a,successor(B)),add(multiply(A::'a,B),A))) &
```
```  1324   (\<forall>A B. equal(successor(A),successor(B)) --> equal(A::'a,B)) &
```
```  1325   (\<forall>A B. equal(A::'a,B) --> equal(successor(A),successor(B)))"
```
```  1326
```
```  1327 abbreviation "NUM001_1_ax predecessor_of_1st_minus_2nd successor add equal mless \<equiv>
```
```  1328   (\<forall>A C B. mless(A::'a,B) & mless(C::'a,A) --> mless(C::'a,B)) &
```
```  1329   (\<forall>A B C. equal(add(successor(A),B),C) --> mless(B::'a,C)) &
```
```  1330   (\<forall>A B. mless(A::'a,B) --> equal(add(successor(predecessor_of_1st_minus_2nd(B::'a,A)),A),B))"
```
```  1331
```
```  1332 abbreviation "NUM001_2_ax equal mless divides \<equiv>
```
```  1333   (\<forall>A B. divides(A::'a,B) --> mless(A::'a,B) | equal(A::'a,B)) &
```
```  1334   (\<forall>A B. mless(A::'a,B) --> divides(A::'a,B)) &
```
```  1335   (\<forall>A B. equal(A::'a,B) --> divides(A::'a,B))"
```
```  1336
```
```  1337 (*20717 inferences so far.  Searching to depth 11.  13.7 secs*)
```
```  1338 lemma NUM021_1:
```
```  1339   "EQU001_0_ax equal &
```
```  1340   NUM001_0_ax multiply successor num0 add equal &
```
```  1341   NUM001_1_ax predecessor_of_1st_minus_2nd successor add equal mless &
```
```  1342   NUM001_2_ax equal mless divides &
```
```  1343   (mless(b::'a,c)) &
```
```  1344    (~mless(b::'a,a)) &
```
```  1345    (divides(c::'a,a)) &
```
```  1346    (\<forall>A. ~equal(successor(A),num0)) --> False"
```
```  1347   by meson
```
```  1348
```
```  1349 (*26320 inferences so far.  Searching to depth 10.  26.4 secs*)
```
```  1350 lemma NUM024_1:
```
```  1351   "EQU001_0_ax equal &
```
```  1352   NUM001_0_ax multiply successor num0 add equal &
```
```  1353   NUM001_1_ax predecessor_of_1st_minus_2nd successor add equal mless &
```
```  1354   (\<forall>B A. equal(add(A::'a,B),add(B::'a,A))) &
```
```  1355   (\<forall>B A C. equal(add(A::'a,B),add(C::'a,B)) --> equal(A::'a,C)) &
```
```  1356   (mless(a::'a,a)) &
```
```  1357   (\<forall>A. ~equal(successor(A),num0)) --> False"
```
```  1358   oops
```
```  1359
```
```  1360 abbreviation "SET004_0_ax not_homomorphism2 not_homomorphism1
```
```  1361     homomorphism compatible operation cantor diagonalise subset_relation
```
```  1362     one_to_one choice apply regular function identity_relation
```
```  1363     single_valued_class compos powerClass sum_class omega inductive
```
```  1364     successor_relation successor image' rng domain range_of INVERSE flip
```
```  1365     rot domain_of null_class restrct difference union complement
```
```  1366     intersection element_relation second first cross_product ordered_pair
```
```  1367     singleton unordered_pair equal universal_class not_subclass_element
```
```  1368     member subclass \<equiv>
```
```  1369   (\<forall>X U Y. subclass(X::'a,Y) & member(U::'a,X) --> member(U::'a,Y)) &
```
```  1370   (\<forall>X Y. member(not_subclass_element(X::'a,Y),X) | subclass(X::'a,Y)) &
```
```  1371   (\<forall>X Y. member(not_subclass_element(X::'a,Y),Y) --> subclass(X::'a,Y)) &
```
```  1372   (\<forall>X. subclass(X::'a,universal_class)) &
```
```  1373   (\<forall>X Y. equal(X::'a,Y) --> subclass(X::'a,Y)) &
```
```  1374   (\<forall>Y X. equal(X::'a,Y) --> subclass(Y::'a,X)) &
```
```  1375   (\<forall>X Y. subclass(X::'a,Y) & subclass(Y::'a,X) --> equal(X::'a,Y)) &
```
```  1376   (\<forall>X U Y. member(U::'a,unordered_pair(X::'a,Y)) --> equal(U::'a,X) | equal(U::'a,Y)) &
```
```  1377   (\<forall>X Y. member(X::'a,universal_class) --> member(X::'a,unordered_pair(X::'a,Y))) &
```
```  1378   (\<forall>X Y. member(Y::'a,universal_class) --> member(Y::'a,unordered_pair(X::'a,Y))) &
```
```  1379   (\<forall>X Y. member(unordered_pair(X::'a,Y),universal_class)) &
```
```  1380   (\<forall>X. equal(unordered_pair(X::'a,X),singleton(X))) &
```
```  1381   (\<forall>X Y. equal(unordered_pair(singleton(X),unordered_pair(X::'a,singleton(Y))),ordered_pair(X::'a,Y))) &
```
```  1382   (\<forall>V Y U X. member(ordered_pair(U::'a,V),cross_product(X::'a,Y)) --> member(U::'a,X)) &
```
```  1383   (\<forall>U X V Y. member(ordered_pair(U::'a,V),cross_product(X::'a,Y)) --> member(V::'a,Y)) &
```
```  1384   (\<forall>U V X Y. member(U::'a,X) & member(V::'a,Y) --> member(ordered_pair(U::'a,V),cross_product(X::'a,Y))) &
```
```  1385   (\<forall>X Y Z. member(Z::'a,cross_product(X::'a,Y)) --> equal(ordered_pair(first(Z),second(Z)),Z)) &
```
```  1386   (subclass(element_relation::'a,cross_product(universal_class::'a,universal_class))) &
```
```  1387   (\<forall>X Y. member(ordered_pair(X::'a,Y),element_relation) --> member(X::'a,Y)) &
```
```  1388   (\<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)) &
```
```  1389   (\<forall>Y Z X. member(Z::'a,intersection(X::'a,Y)) --> member(Z::'a,X)) &
```
```  1390   (\<forall>X Z Y. member(Z::'a,intersection(X::'a,Y)) --> member(Z::'a,Y)) &
```
```  1391   (\<forall>Z X Y. member(Z::'a,X) & member(Z::'a,Y) --> member(Z::'a,intersection(X::'a,Y))) &
```
```  1392   (\<forall>Z X. ~(member(Z::'a,complement(X)) & member(Z::'a,X))) &
```
```  1393   (\<forall>Z X. member(Z::'a,universal_class) --> member(Z::'a,complement(X)) | member(Z::'a,X)) &
```
```  1394   (\<forall>X Y. equal(complement(intersection(complement(X),complement(Y))),union(X::'a,Y))) &
```
```  1395   (\<forall>X Y. equal(intersection(complement(intersection(X::'a,Y)),complement(intersection(complement(X),complement(Y)))),difference(X::'a,Y))) &
```
```  1396   (\<forall>Xr X Y. equal(intersection(Xr::'a,cross_product(X::'a,Y)),restrct(Xr::'a,X,Y))) &
```
```  1397   (\<forall>Xr X Y. equal(intersection(cross_product(X::'a,Y),Xr),restrct(Xr::'a,X,Y))) &
```
```  1398   (\<forall>Z X. ~(equal(restrct(X::'a,singleton(Z),universal_class),null_class) & member(Z::'a,domain_of(X)))) &
```
```  1399   (\<forall>Z X. member(Z::'a,universal_class) --> equal(restrct(X::'a,singleton(Z),universal_class),null_class) | member(Z::'a,domain_of(X))) &
```
```  1400   (\<forall>X. subclass(rot(X),cross_product(cross_product(universal_class::'a,universal_class),universal_class))) &
```
```  1401   (\<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)) &
```
```  1402   (\<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))) &
```
```  1403   (\<forall>X. subclass(flip(X),cross_product(cross_product(universal_class::'a,universal_class),universal_class))) &
```
```  1404   (\<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)) &
```
```  1405   (\<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))) &
```
```  1406   (\<forall>Y. equal(domain_of(flip(cross_product(Y::'a,universal_class))),INVERSE(Y))) &
```
```  1407   (\<forall>Z. equal(domain_of(INVERSE(Z)),range_of(Z))) &
```
```  1408   (\<forall>Z X Y. equal(first(not_subclass_element(restrct(Z::'a,X,singleton(Y)),null_class)),domain(Z::'a,X,Y))) &
```
```  1409   (\<forall>Z X Y. equal(second(not_subclass_element(restrct(Z::'a,singleton(X),Y),null_class)),rng(Z::'a,X,Y))) &
```
```  1410   (\<forall>Xr X. equal(range_of(restrct(Xr::'a,X,universal_class)),image'(Xr::'a,X))) &
```
```  1411   (\<forall>X. equal(union(X::'a,singleton(X)),successor(X))) &
```
```  1412   (subclass(successor_relation::'a,cross_product(universal_class::'a,universal_class))) &
```
```  1413   (\<forall>X Y. member(ordered_pair(X::'a,Y),successor_relation) --> equal(successor(X),Y)) &
```
```  1414   (\<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)) &
```
```  1415   (\<forall>X. inductive(X) --> member(null_class::'a,X)) &
```
```  1416   (\<forall>X. inductive(X) --> subclass(image'(successor_relation::'a,X),X)) &
```
```  1417   (\<forall>X. member(null_class::'a,X) & subclass(image'(successor_relation::'a,X),X) --> inductive(X)) &
```
```  1418   (inductive(omega)) &
```
```  1419   (\<forall>Y. inductive(Y) --> subclass(omega::'a,Y)) &
```
```  1420   (member(omega::'a,universal_class)) &
```
```  1421   (\<forall>X. equal(domain_of(restrct(element_relation::'a,universal_class,X)),sum_class(X))) &
```
```  1422   (\<forall>X. member(X::'a,universal_class) --> member(sum_class(X),universal_class)) &
```
```  1423   (\<forall>X. equal(complement(image'(element_relation::'a,complement(X))),powerClass(X))) &
```
```  1424   (\<forall>U. member(U::'a,universal_class) --> member(powerClass(U),universal_class)) &
```
```  1425   (\<forall>Yr Xr. subclass(compos(Yr::'a,Xr),cross_product(universal_class::'a,universal_class))) &
```
```  1426   (\<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))))) &
```
```  1427   (\<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))) &
```
```  1428   (\<forall>X. single_valued_class(X) --> subclass(compos(X::'a,INVERSE(X)),identity_relation)) &
```
```  1429   (\<forall>X. subclass(compos(X::'a,INVERSE(X)),identity_relation) --> single_valued_class(X)) &
```
```  1430   (\<forall>Xf. function(Xf) --> subclass(Xf::'a,cross_product(universal_class::'a,universal_class))) &
```
```  1431   (\<forall>Xf. function(Xf) --> subclass(compos(Xf::'a,INVERSE(Xf)),identity_relation)) &
```
```  1432   (\<forall>Xf. subclass(Xf::'a,cross_product(universal_class::'a,universal_class)) & subclass(compos(Xf::'a,INVERSE(Xf)),identity_relation) --> function(Xf)) &
```
```  1433   (\<forall>Xf X. function(Xf) & member(X::'a,universal_class) --> member(image'(Xf::'a,X),universal_class)) &
```
```  1434   (\<forall>X. equal(X::'a,null_class) | member(regular(X),X)) &
```
```  1435   (\<forall>X. equal(X::'a,null_class) | equal(intersection(X::'a,regular(X)),null_class)) &
```
```  1436   (\<forall>Xf Y. equal(sum_class(image'(Xf::'a,singleton(Y))),apply(Xf::'a,Y))) &
```
```  1437   (function(choice)) &
```
```  1438   (\<forall>Y. member(Y::'a,universal_class) --> equal(Y::'a,null_class) | member(apply(choice::'a,Y),Y)) &
```
```  1439   (\<forall>Xf. one_to_one(Xf) --> function(Xf)) &
```
```  1440   (\<forall>Xf. one_to_one(Xf) --> function(INVERSE(Xf))) &
```
```  1441   (\<forall>Xf. function(INVERSE(Xf)) & function(Xf) --> one_to_one(Xf)) &
```
```  1442   (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)) &
```
```  1443   (equal(intersection(INVERSE(subset_relation),subset_relation),identity_relation)) &
```
```  1444   (\<forall>Xr. equal(complement(domain_of(intersection(Xr::'a,identity_relation))),diagonalise(Xr))) &
```
```  1445   (\<forall>X. equal(intersection(domain_of(X),diagonalise(compos(INVERSE(element_relation),X))),cantor(X))) &
```
```  1446   (\<forall>Xf. operation(Xf) --> function(Xf)) &
```
```  1447   (\<forall>Xf. operation(Xf) --> equal(cross_product(domain_of(domain_of(Xf)),domain_of(domain_of(Xf))),domain_of(Xf))) &
```
```  1448   (\<forall>Xf. operation(Xf) --> subclass(range_of(Xf),domain_of(domain_of(Xf)))) &
```
```  1449   (\<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)) &
```
```  1450   (\<forall>Xf1 Xf2 Xh. compatible(Xh::'a,Xf1,Xf2) --> function(Xh)) &
```
```  1451   (\<forall>Xf2 Xf1 Xh. compatible(Xh::'a,Xf1,Xf2) --> equal(domain_of(domain_of(Xf1)),domain_of(Xh))) &
```
```  1452   (\<forall>Xf1 Xh Xf2. compatible(Xh::'a,Xf1,Xf2) --> subclass(range_of(Xh),domain_of(domain_of(Xf2)))) &
```
```  1453   (\<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)) &
```
```  1454   (\<forall>Xh Xf2 Xf1. homomorphism(Xh::'a,Xf1,Xf2) --> operation(Xf1)) &
```
```  1455   (\<forall>Xh Xf1 Xf2. homomorphism(Xh::'a,Xf1,Xf2) --> operation(Xf2)) &
```
```  1456   (\<forall>Xh Xf1 Xf2. homomorphism(Xh::'a,Xf1,Xf2) --> compatible(Xh::'a,Xf1,Xf2)) &
```
```  1457   (\<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))))) &
```
```  1458   (\<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)) &
```
```  1459   (\<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))"
```
```  1460
```
```  1461 abbreviation "SET004_0_eq subclass single_valued_class operation
```
```  1462     one_to_one member inductive homomorphism function compatible
```
```  1463     unordered_pair union sum_class successor singleton second rot restrct
```
```  1464     regular range_of rng powerClass ordered_pair not_subclass_element
```
```  1465     not_homomorphism2 not_homomorphism1 INVERSE intersection image' flip
```
```  1466     first domain_of domain difference diagonalise cross_product compos
```
```  1467     complement cantor apply equal \<equiv>
```
```  1468   (\<forall>D E F'. equal(D::'a,E) --> equal(apply(D::'a,F'),apply(E::'a,F'))) &
```
```  1469   (\<forall>G I' H. equal(G::'a,H) --> equal(apply(I'::'a,G),apply(I'::'a,H))) &
```
```  1470   (\<forall>J K'. equal(J::'a,K') --> equal(cantor(J),cantor(K'))) &
```
```  1471   (\<forall>L M. equal(L::'a,M) --> equal(complement(L),complement(M))) &
```
```  1472   (\<forall>N O' P. equal(N::'a,O') --> equal(compos(N::'a,P),compos(O'::'a,P))) &
```
```  1473   (\<forall>Q S' R. equal(Q::'a,R) --> equal(compos(S'::'a,Q),compos(S'::'a,R))) &
```
```  1474   (\<forall>T' U V. equal(T'::'a,U) --> equal(cross_product(T'::'a,V),cross_product(U::'a,V))) &
```
```  1475   (\<forall>W Y X. equal(W::'a,X) --> equal(cross_product(Y::'a,W),cross_product(Y::'a,X))) &
```
```  1476   (\<forall>Z A1. equal(Z::'a,A1) --> equal(diagonalise(Z),diagonalise(A1))) &
```
```  1477   (\<forall>B1 C1 D1. equal(B1::'a,C1) --> equal(difference(B1::'a,D1),difference(C1::'a,D1))) &
```
```  1478   (\<forall>E1 G1 F1. equal(E1::'a,F1) --> equal(difference(G1::'a,E1),difference(G1::'a,F1))) &
```
```  1479   (\<forall>H1 I1 J1 K1. equal(H1::'a,I1) --> equal(domain(H1::'a,J1,K1),domain(I1::'a,J1,K1))) &
```
```  1480   (\<forall>L1 N1 M1 O1. equal(L1::'a,M1) --> equal(domain(N1::'a,L1,O1),domain(N1::'a,M1,O1))) &
```
```  1481   (\<forall>P1 R1 S1 Q1. equal(P1::'a,Q1) --> equal(domain(R1::'a,S1,P1),domain(R1::'a,S1,Q1))) &
```
```  1482   (\<forall>T1 U1. equal(T1::'a,U1) --> equal(domain_of(T1),domain_of(U1))) &
```
```  1483   (\<forall>V1 W1. equal(V1::'a,W1) --> equal(first(V1),first(W1))) &
```
```  1484   (\<forall>X1 Y1. equal(X1::'a,Y1) --> equal(flip(X1),flip(Y1))) &
```
```  1485   (\<forall>Z1 A2 B2. equal(Z1::'a,A2) --> equal(image'(Z1::'a,B2),image'(A2::'a,B2))) &
```
```  1486   (\<forall>C2 E2 D2. equal(C2::'a,D2) --> equal(image'(E2::'a,C2),image'(E2::'a,D2))) &
```
```  1487   (\<forall>F2 G2 H2. equal(F2::'a,G2) --> equal(intersection(F2::'a,H2),intersection(G2::'a,H2))) &
```
```  1488   (\<forall>I2 K2 J2. equal(I2::'a,J2) --> equal(intersection(K2::'a,I2),intersection(K2::'a,J2))) &
```
```  1489   (\<forall>L2 M2. equal(L2::'a,M2) --> equal(INVERSE(L2),INVERSE(M2))) &
```
```  1490   (\<forall>N2 O2 P2 Q2. equal(N2::'a,O2) --> equal(not_homomorphism1(N2::'a,P2,Q2),not_homomorphism1(O2::'a,P2,Q2))) &
```
```  1491   (\<forall>R2 T2 S2 U2. equal(R2::'a,S2) --> equal(not_homomorphism1(T2::'a,R2,U2),not_homomorphism1(T2::'a,S2,U2))) &
```
```  1492   (\<forall>V2 X2 Y2 W2. equal(V2::'a,W2) --> equal(not_homomorphism1(X2::'a,Y2,V2),not_homomorphism1(X2::'a,Y2,W2))) &
```
```  1493   (\<forall>Z2 A3 B3 C3. equal(Z2::'a,A3) --> equal(not_homomorphism2(Z2::'a,B3,C3),not_homomorphism2(A3::'a,B3,C3))) &
```
```  1494   (\<forall>D3 F3 E3 G3. equal(D3::'a,E3) --> equal(not_homomorphism2(F3::'a,D3,G3),not_homomorphism2(F3::'a,E3,G3))) &
```
```  1495   (\<forall>H3 J3 K3 I3. equal(H3::'a,I3) --> equal(not_homomorphism2(J3::'a,K3,H3),not_homomorphism2(J3::'a,K3,I3))) &
```
```  1496   (\<forall>L3 M3 N3. equal(L3::'a,M3) --> equal(not_subclass_element(L3::'a,N3),not_subclass_element(M3::'a,N3))) &
```
```  1497   (\<forall>O3 Q3 P3. equal(O3::'a,P3) --> equal(not_subclass_element(Q3::'a,O3),not_subclass_element(Q3::'a,P3))) &
```
```  1498   (\<forall>R3 S3 T3. equal(R3::'a,S3) --> equal(ordered_pair(R3::'a,T3),ordered_pair(S3::'a,T3))) &
```
```  1499   (\<forall>U3 W3 V3. equal(U3::'a,V3) --> equal(ordered_pair(W3::'a,U3),ordered_pair(W3::'a,V3))) &
```
```  1500   (\<forall>X3 Y3. equal(X3::'a,Y3) --> equal(powerClass(X3),powerClass(Y3))) &
```
```  1501   (\<forall>Z3 A4 B4 C4. equal(Z3::'a,A4) --> equal(rng(Z3::'a,B4,C4),rng(A4::'a,B4,C4))) &
```
```  1502   (\<forall>D4 F4 E4 G4. equal(D4::'a,E4) --> equal(rng(F4::'a,D4,G4),rng(F4::'a,E4,G4))) &
```
```  1503   (\<forall>H4 J4 K4 I4. equal(H4::'a,I4) --> equal(rng(J4::'a,K4,H4),rng(J4::'a,K4,I4))) &
```
```  1504   (\<forall>L4 M4. equal(L4::'a,M4) --> equal(range_of(L4),range_of(M4))) &
```
```  1505   (\<forall>N4 O4. equal(N4::'a,O4) --> equal(regular(N4),regular(O4))) &
```
```  1506   (\<forall>P4 Q4 R4 S4. equal(P4::'a,Q4) --> equal(restrct(P4::'a,R4,S4),restrct(Q4::'a,R4,S4))) &
```
```  1507   (\<forall>T4 V4 U4 W4. equal(T4::'a,U4) --> equal(restrct(V4::'a,T4,W4),restrct(V4::'a,U4,W4))) &
```
```  1508   (\<forall>X4 Z4 A5 Y4. equal(X4::'a,Y4) --> equal(restrct(Z4::'a,A5,X4),restrct(Z4::'a,A5,Y4))) &
```
```  1509   (\<forall>B5 C5. equal(B5::'a,C5) --> equal(rot(B5),rot(C5))) &
```
```  1510   (\<forall>D5 E5. equal(D5::'a,E5) --> equal(second(D5),second(E5))) &
```
```  1511   (\<forall>F5 G5. equal(F5::'a,G5) --> equal(singleton(F5),singleton(G5))) &
```
```  1512   (\<forall>H5 I5. equal(H5::'a,I5) --> equal(successor(H5),successor(I5))) &
```
```  1513   (\<forall>J5 K5. equal(J5::'a,K5) --> equal(sum_class(J5),sum_class(K5))) &
```
```  1514   (\<forall>L5 M5 N5. equal(L5::'a,M5) --> equal(union(L5::'a,N5),union(M5::'a,N5))) &
```
```  1515   (\<forall>O5 Q5 P5. equal(O5::'a,P5) --> equal(union(Q5::'a,O5),union(Q5::'a,P5))) &
```
```  1516   (\<forall>R5 S5 T5. equal(R5::'a,S5) --> equal(unordered_pair(R5::'a,T5),unordered_pair(S5::'a,T5))) &
```
```  1517   (\<forall>U5 W5 V5. equal(U5::'a,V5) --> equal(unordered_pair(W5::'a,U5),unordered_pair(W5::'a,V5))) &
```
```  1518   (\<forall>X5 Y5 Z5 A6. equal(X5::'a,Y5) & compatible(X5::'a,Z5,A6) --> compatible(Y5::'a,Z5,A6)) &
```
```  1519   (\<forall>B6 D6 C6 E6. equal(B6::'a,C6) & compatible(D6::'a,B6,E6) --> compatible(D6::'a,C6,E6)) &
```
```  1520   (\<forall>F6 H6 I6 G6. equal(F6::'a,G6) & compatible(H6::'a,I6,F6) --> compatible(H6::'a,I6,G6)) &
```
```  1521   (\<forall>J6 K6. equal(J6::'a,K6) & function(J6) --> function(K6)) &
```
```  1522   (\<forall>L6 M6 N6 O6. equal(L6::'a,M6) & homomorphism(L6::'a,N6,O6) --> homomorphism(M6::'a,N6,O6)) &
```
```  1523   (\<forall>P6 R6 Q6 S6. equal(P6::'a,Q6) & homomorphism(R6::'a,P6,S6) --> homomorphism(R6::'a,Q6,S6)) &
```
```  1524   (\<forall>T6 V6 W6 U6. equal(T6::'a,U6) & homomorphism(V6::'a,W6,T6) --> homomorphism(V6::'a,W6,U6)) &
```
```  1525   (\<forall>X6 Y6. equal(X6::'a,Y6) & inductive(X6) --> inductive(Y6)) &
```
```  1526   (\<forall>Z6 A7 B7. equal(Z6::'a,A7) & member(Z6::'a,B7) --> member(A7::'a,B7)) &
```
```  1527   (\<forall>C7 E7 D7. equal(C7::'a,D7) & member(E7::'a,C7) --> member(E7::'a,D7)) &
```
```  1528   (\<forall>F7 G7. equal(F7::'a,G7) & one_to_one(F7) --> one_to_one(G7)) &
```
```  1529   (\<forall>H7 I7. equal(H7::'a,I7) & operation(H7) --> operation(I7)) &
```
```  1530   (\<forall>J7 K7. equal(J7::'a,K7) & single_valued_class(J7) --> single_valued_class(K7)) &
```
```  1531   (\<forall>L7 M7 N7. equal(L7::'a,M7) & subclass(L7::'a,N7) --> subclass(M7::'a,N7)) &
```
```  1532   (\<forall>O7 Q7 P7. equal(O7::'a,P7) & subclass(Q7::'a,O7) --> subclass(Q7::'a,P7))"
```
```  1533
```
```  1534 abbreviation "SET004_1_ax range_of function maps apply
```
```  1535     application_function singleton_relation element_relation complement
```
```  1536     intersection single_valued3 singleton image' domain single_valued2
```
```  1537     second single_valued1 identity_relation INVERSE not_subclass_element
```
```  1538     first domain_of domain_relation composition_function compos equal
```
```  1539     ordered_pair member universal_class cross_product compose_class
```
```  1540     subclass \<equiv>
```
```  1541   (\<forall>X. subclass(compose_class(X),cross_product(universal_class::'a,universal_class))) &
```
```  1542   (\<forall>X Y Z. member(ordered_pair(Y::'a,Z),compose_class(X)) --> equal(compos(X::'a,Y),Z)) &
```
```  1543   (\<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))) &
```
```  1544   (subclass(composition_function::'a,cross_product(universal_class::'a,cross_product(universal_class::'a,universal_class)))) &
```
```  1545   (\<forall>X Y Z. member(ordered_pair(X::'a,ordered_pair(Y::'a,Z)),composition_function) --> equal(compos(X::'a,Y),Z)) &
```
```  1546   (\<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)) &
```
```  1547   (subclass(domain_relation::'a,cross_product(universal_class::'a,universal_class))) &
```
```  1548   (\<forall>X Y. member(ordered_pair(X::'a,Y),domain_relation) --> equal(domain_of(X),Y)) &
```
```  1549   (\<forall>X. member(X::'a,universal_class) --> member(ordered_pair(X::'a,domain_of(X)),domain_relation)) &
```
```  1550   (\<forall>X. equal(first(not_subclass_element(compos(X::'a,INVERSE(X)),identity_relation)),single_valued1(X))) &
```
```  1551   (\<forall>X. equal(second(not_subclass_element(compos(X::'a,INVERSE(X)),identity_relation)),single_valued2(X))) &
```
```  1552   (\<forall>X. equal(domain(X::'a,image'(INVERSE(X),singleton(single_valued1(X))),single_valued2(X)),single_valued3(X))) &
```
```  1553   (equal(intersection(complement(compos(element_relation::'a,complement(identity_relation))),element_relation),singleton_relation)) &
```
```  1554   (subclass(application_function::'a,cross_product(universal_class::'a,cross_product(universal_class::'a,universal_class)))) &
```
```  1555   (\<forall>Z Y X. member(ordered_pair(X::'a,ordered_pair(Y::'a,Z)),application_function) --> member(Y::'a,domain_of(X))) &
```
```  1556   (\<forall>X Y Z. member(ordered_pair(X::'a,ordered_pair(Y::'a,Z)),application_function) --> equal(apply(X::'a,Y),Z)) &
```
```  1557   (\<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)) &
```
```  1558   (\<forall>X Y Xf. maps(Xf::'a,X,Y) --> function(Xf)) &
```
```  1559   (\<forall>Y Xf X. maps(Xf::'a,X,Y) --> equal(domain_of(Xf),X)) &
```
```  1560   (\<forall>X Xf Y. maps(Xf::'a,X,Y) --> subclass(range_of(Xf),Y)) &
```
```  1561   (\<forall>Xf Y. function(Xf) & subclass(range_of(Xf),Y) --> maps(Xf::'a,domain_of(Xf),Y))"
```
```  1562
```
```  1563 abbreviation "SET004_1_eq maps single_valued3 single_valued2 single_valued1 compose_class equal \<equiv>
```
```  1564   (\<forall>L M. equal(L::'a,M) --> equal(compose_class(L),compose_class(M))) &
```
```  1565   (\<forall>N2 O2. equal(N2::'a,O2) --> equal(single_valued1(N2),single_valued1(O2))) &
```
```  1566   (\<forall>P2 Q2. equal(P2::'a,Q2) --> equal(single_valued2(P2),single_valued2(Q2))) &
```
```  1567   (\<forall>R2 S2. equal(R2::'a,S2) --> equal(single_valued3(R2),single_valued3(S2))) &
```
```  1568   (\<forall>X2 Y2 Z2 A3. equal(X2::'a,Y2) & maps(X2::'a,Z2,A3) --> maps(Y2::'a,Z2,A3)) &
```
```  1569   (\<forall>B3 D3 C3 E3. equal(B3::'a,C3) & maps(D3::'a,B3,E3) --> maps(D3::'a,C3,E3)) &
```
```  1570   (\<forall>F3 H3 I3 G3. equal(F3::'a,G3) & maps(H3::'a,I3,F3) --> maps(H3::'a,I3,G3))"
```
```  1571
```
```  1572 abbreviation "NUM004_0_ax integer_of omega ordinal_multiply
```
```  1573     add_relation ordinal_add recursion apply range_of union_of_range_map
```
```  1574     function recursion_equation_functions rest_relation rest_of
```
```  1575     limit_ordinals kind_1_ordinals successor_relation image'
```
```  1576     universal_class sum_class element_relation ordinal_numbers section
```
```  1577     not_well_ordering ordered_pair least member well_ordering singleton
```
```  1578     domain_of segment null_class intersection asymmetric compos transitive
```
```  1579     cross_product connected identity_relation complement restrct subclass
```
```  1580     irreflexive symmetrization_of INVERSE union equal \<equiv>
```
```  1581   (\<forall>X. equal(union(X::'a,INVERSE(X)),symmetrization_of(X))) &
```
```  1582   (\<forall>X Y. irreflexive(X::'a,Y) --> subclass(restrct(X::'a,Y,Y),complement(identity_relation))) &
```
```  1583   (\<forall>X Y. subclass(restrct(X::'a,Y,Y),complement(identity_relation)) --> irreflexive(X::'a,Y)) &
```
```  1584   (\<forall>Y X. connected(X::'a,Y) --> subclass(cross_product(Y::'a,Y),union(identity_relation::'a,symmetrization_of(X)))) &
```
```  1585   (\<forall>X Y. subclass(cross_product(Y::'a,Y),union(identity_relation::'a,symmetrization_of(X))) --> connected(X::'a,Y)) &
```
```  1586   (\<forall>Xr Y. transitive(Xr::'a,Y) --> subclass(compos(restrct(Xr::'a,Y,Y),restrct(Xr::'a,Y,Y)),restrct(Xr::'a,Y,Y))) &
```
```  1587   (\<forall>Xr Y. subclass(compos(restrct(Xr::'a,Y,Y),restrct(Xr::'a,Y,Y)),restrct(Xr::'a,Y,Y)) --> transitive(Xr::'a,Y)) &
```
```  1588   (\<forall>Xr Y. asymmetric(Xr::'a,Y) --> equal(restrct(intersection(Xr::'a,INVERSE(Xr)),Y,Y),null_class)) &
```
```  1589   (\<forall>Xr Y. equal(restrct(intersection(Xr::'a,INVERSE(Xr)),Y,Y),null_class) --> asymmetric(Xr::'a,Y)) &
```
```  1590   (\<forall>Xr Y Z. equal(segment(Xr::'a,Y,Z),domain_of(restrct(Xr::'a,Y,singleton(Z))))) &
```
```  1591   (\<forall>X Y. well_ordering(X::'a,Y) --> connected(X::'a,Y)) &
```
```  1592   (\<forall>Y Xr U. well_ordering(Xr::'a,Y) & subclass(U::'a,Y) --> equal(U::'a,null_class) | member(least(Xr::'a,U),U)) &
```
```  1593   (\<forall>Y V Xr U. well_ordering(Xr::'a,Y) & subclass(U::'a,Y) & member(V::'a,U) --> member(least(Xr::'a,U),U)) &
```
```  1594   (\<forall>Y Xr U. well_ordering(Xr::'a,Y) & subclass(U::'a,Y) --> equal(segment(Xr::'a,U,least(Xr::'a,U)),null_class)) &
```
```  1595   (\<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))) &
```
```  1596   (\<forall>Xr Y. connected(Xr::'a,Y) & equal(not_well_ordering(Xr::'a,Y),null_class) --> well_ordering(Xr::'a,Y)) &
```
```  1597   (\<forall>Xr Y. connected(Xr::'a,Y) --> subclass(not_well_ordering(Xr::'a,Y),Y) | well_ordering(Xr::'a,Y)) &
```
```  1598   (\<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)) &
```
```  1599   (\<forall>Xr Y Z. section(Xr::'a,Y,Z) --> subclass(Y::'a,Z)) &
```
```  1600   (\<forall>Xr Z Y. section(Xr::'a,Y,Z) --> subclass(domain_of(restrct(Xr::'a,Z,Y)),Y)) &
```
```  1601   (\<forall>Xr Y Z. subclass(Y::'a,Z) & subclass(domain_of(restrct(Xr::'a,Z,Y)),Y) --> section(Xr::'a,Y,Z)) &
```
```  1602   (\<forall>X. member(X::'a,ordinal_numbers) --> well_ordering(element_relation::'a,X)) &
```
```  1603   (\<forall>X. member(X::'a,ordinal_numbers) --> subclass(sum_class(X),X)) &
```
```  1604   (\<forall>X. well_ordering(element_relation::'a,X) & subclass(sum_class(X),X) & member(X::'a,universal_class) --> member(X::'a,ordinal_numbers)) &
```
```  1605   (\<forall>X. well_ordering(element_relation::'a,X) & subclass(sum_class(X),X) --> member(X::'a,ordinal_numbers) | equal(X::'a,ordinal_numbers)) &
```
```  1606   (equal(union(singleton(null_class),image'(successor_relation::'a,ordinal_numbers)),kind_1_ordinals)) &
```
```  1607   (equal(intersection(complement(kind_1_ordinals),ordinal_numbers),limit_ordinals)) &
```
```  1608   (\<forall>X. subclass(rest_of(X),cross_product(universal_class::'a,universal_class))) &
```
```  1609   (\<forall>V U X. member(ordered_pair(U::'a,V),rest_of(X)) --> member(U::'a,domain_of(X))) &
```
```  1610   (\<forall>X U V. member(ordered_pair(U::'a,V),rest_of(X)) --> equal(restrct(X::'a,U,universal_class),V)) &
```
```  1611   (\<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))) &
```
```  1612   (subclass(rest_relation::'a,cross_product(universal_class::'a,universal_class))) &
```
```  1613   (\<forall>X Y. member(ordered_pair(X::'a,Y),rest_relation) --> equal(rest_of(X),Y)) &
```
```  1614   (\<forall>X. member(X::'a,universal_class) --> member(ordered_pair(X::'a,rest_of(X)),rest_relation)) &
```
```  1615   (\<forall>X Z. member(X::'a,recursion_equation_functions(Z)) --> function(Z)) &
```
```  1616   (\<forall>Z X. member(X::'a,recursion_equation_functions(Z)) --> function(X)) &
```
```  1617   (\<forall>Z X. member(X::'a,recursion_equation_functions(Z)) --> member(domain_of(X),ordinal_numbers)) &
```
```  1618   (\<forall>Z X. member(X::'a,recursion_equation_functions(Z)) --> equal(compos(Z::'a,rest_of(X)),X)) &
```
```  1619   (\<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))) &
```
```  1620   (subclass(union_of_range_map::'a,cross_product(universal_class::'a,universal_class))) &
```
```  1621   (\<forall>X Y. member(ordered_pair(X::'a,Y),union_of_range_map) --> equal(sum_class(range_of(X)),Y)) &
```
```  1622   (\<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)) &
```
```  1623   (\<forall>X Y. equal(apply(recursion(X::'a,successor_relation,union_of_range_map),Y),ordinal_add(X::'a,Y))) &
```
```  1624   (\<forall>X Y. equal(recursion(null_class::'a,apply(add_relation::'a,X),union_of_range_map),ordinal_multiply(X::'a,Y))) &
```
```  1625   (\<forall>X. member(X::'a,omega) --> equal(integer_of(X),X)) &
```
```  1626   (\<forall>X. member(X::'a,omega) | equal(integer_of(X),null_class))"
```
```  1627
```
```  1628 abbreviation "NUM004_0_eq well_ordering transitive section irreflexive
```
```  1629     connected asymmetric symmetrization_of segment rest_of
```
```  1630     recursion_equation_functions recursion ordinal_multiply ordinal_add
```
```  1631     not_well_ordering least integer_of equal \<equiv>
```
```  1632   (\<forall>D E. equal(D::'a,E) --> equal(integer_of(D),integer_of(E))) &
```
```  1633   (\<forall>F' G H. equal(F'::'a,G) --> equal(least(F'::'a,H),least(G::'a,H))) &
```
```  1634   (\<forall>I' K' J. equal(I'::'a,J) --> equal(least(K'::'a,I'),least(K'::'a,J))) &
```
```  1635   (\<forall>L M N. equal(L::'a,M) --> equal(not_well_ordering(L::'a,N),not_well_ordering(M::'a,N))) &
```
```  1636   (\<forall>O' Q P. equal(O'::'a,P) --> equal(not_well_ordering(Q::'a,O'),not_well_ordering(Q::'a,P))) &
```
```  1637   (\<forall>R S' T'. equal(R::'a,S') --> equal(ordinal_add(R::'a,T'),ordinal_add(S'::'a,T'))) &
```
```  1638   (\<forall>U W V. equal(U::'a,V) --> equal(ordinal_add(W::'a,U),ordinal_add(W::'a,V))) &
```
```  1639   (\<forall>X Y Z. equal(X::'a,Y) --> equal(ordinal_multiply(X::'a,Z),ordinal_multiply(Y::'a,Z))) &
```
```  1640   (\<forall>A1 C1 B1. equal(A1::'a,B1) --> equal(ordinal_multiply(C1::'a,A1),ordinal_multiply(C1::'a,B1))) &
```
```  1641   (\<forall>F1 G1 H1 I1. equal(F1::'a,G1) --> equal(recursion(F1::'a,H1,I1),recursion(G1::'a,H1,I1))) &
```
```  1642   (\<forall>J1 L1 K1 M1. equal(J1::'a,K1) --> equal(recursion(L1::'a,J1,M1),recursion(L1::'a,K1,M1))) &
```
```  1643   (\<forall>N1 P1 Q1 O1. equal(N1::'a,O1) --> equal(recursion(P1::'a,Q1,N1),recursion(P1::'a,Q1,O1))) &
```
```  1644   (\<forall>R1 S1. equal(R1::'a,S1) --> equal(recursion_equation_functions(R1),recursion_equation_functions(S1))) &
```
```  1645   (\<forall>T1 U1. equal(T1::'a,U1) --> equal(rest_of(T1),rest_of(U1))) &
```
```  1646   (\<forall>V1 W1 X1 Y1. equal(V1::'a,W1) --> equal(segment(V1::'a,X1,Y1),segment(W1::'a,X1,Y1))) &
```
```  1647   (\<forall>Z1 B2 A2 C2. equal(Z1::'a,A2) --> equal(segment(B2::'a,Z1,C2),segment(B2::'a,A2,C2))) &
```
```  1648   (\<forall>D2 F2 G2 E2. equal(D2::'a,E2) --> equal(segment(F2::'a,G2,D2),segment(F2::'a,G2,E2))) &
```
```  1649   (\<forall>H2 I2. equal(H2::'a,I2) --> equal(symmetrization_of(H2),symmetrization_of(I2))) &
```
```  1650   (\<forall>J2 K2 L2. equal(J2::'a,K2) & asymmetric(J2::'a,L2) --> asymmetric(K2::'a,L2)) &
```
```  1651   (\<forall>M2 O2 N2. equal(M2::'a,N2) & asymmetric(O2::'a,M2) --> asymmetric(O2::'a,N2)) &
```
```  1652   (\<forall>P2 Q2 R2. equal(P2::'a,Q2) & connected(P2::'a,R2) --> connected(Q2::'a,R2)) &
```
```  1653   (\<forall>S2 U2 T2. equal(S2::'a,T2) & connected(U2::'a,S2) --> connected(U2::'a,T2)) &
```
```  1654   (\<forall>V2 W2 X2. equal(V2::'a,W2) & irreflexive(V2::'a,X2) --> irreflexive(W2::'a,X2)) &
```
```  1655   (\<forall>Y2 A3 Z2. equal(Y2::'a,Z2) & irreflexive(A3::'a,Y2) --> irreflexive(A3::'a,Z2)) &
```
```  1656   (\<forall>B3 C3 D3 E3. equal(B3::'a,C3) & section(B3::'a,D3,E3) --> section(C3::'a,D3,E3)) &
```
```  1657   (\<forall>F3 H3 G3 I3. equal(F3::'a,G3) & section(H3::'a,F3,I3) --> section(H3::'a,G3,I3)) &
```
```  1658   (\<forall>J3 L3 M3 K3. equal(J3::'a,K3) & section(L3::'a,M3,J3) --> section(L3::'a,M3,K3)) &
```
```  1659   (\<forall>N3 O3 P3. equal(N3::'a,O3) & transitive(N3::'a,P3) --> transitive(O3::'a,P3)) &
```
```  1660   (\<forall>Q3 S3 R3. equal(Q3::'a,R3) & transitive(S3::'a,Q3) --> transitive(S3::'a,R3)) &
```
```  1661   (\<forall>T3 U3 V3. equal(T3::'a,U3) & well_ordering(T3::'a,V3) --> well_ordering(U3::'a,V3)) &
```
```  1662   (\<forall>W3 Y3 X3. equal(W3::'a,X3) & well_ordering(Y3::'a,W3) --> well_ordering(Y3::'a,X3))"
```
```  1663
```
```  1664 (*1345 inferences so far.  Searching to depth 7.  23.3 secs.  BIG*)
```
```  1665 lemma NUM180_1:
```
```  1666   "EQU001_0_ax equal &
```
```  1667   SET004_0_ax not_homomorphism2 not_homomorphism1
```
```  1668     homomorphism compatible operation cantor diagonalise subset_relation
```
```  1669     one_to_one choice apply regular function identity_relation
```
```  1670     single_valued_class compos powerClass sum_class omega inductive
```
```  1671     successor_relation successor image' rng domain range_of INVERSE flip
```
```  1672     rot domain_of null_class restrct difference union complement
```
```  1673     intersection element_relation second first cross_product ordered_pair
```
```  1674     singleton unordered_pair equal universal_class not_subclass_element
```
```  1675     member subclass &
```
```  1676   SET004_0_eq subclass single_valued_class operation
```
```  1677     one_to_one member inductive homomorphism function compatible
```
```  1678     unordered_pair union sum_class successor singleton second rot restrct
```
```  1679     regular range_of rng powerClass ordered_pair not_subclass_element
```
```  1680     not_homomorphism2 not_homomorphism1 INVERSE intersection image' flip
```
```  1681     first domain_of domain difference diagonalise cross_product compos
```
```  1682     complement cantor apply equal &
```
```  1683   SET004_1_ax range_of function maps apply
```
```  1684     application_function singleton_relation element_relation complement
```
```  1685     intersection single_valued3 singleton image' domain single_valued2
```
```  1686     second single_valued1 identity_relation INVERSE not_subclass_element
```
```  1687     first domain_of domain_relation composition_function compos equal
```
```  1688     ordered_pair member universal_class cross_product compose_class
```
```  1689     subclass &
```
```  1690   SET004_1_eq maps single_valued3 single_valued2 single_valued1 compose_class equal &
```
```  1691   NUM004_0_ax integer_of omega ordinal_multiply
```
```  1692     add_relation ordinal_add recursion apply range_of union_of_range_map
```
```  1693     function recursion_equation_functions rest_relation rest_of
```
```  1694     limit_ordinals kind_1_ordinals successor_relation image'
```
```  1695     universal_class sum_class element_relation ordinal_numbers section
```
```  1696     not_well_ordering ordered_pair least member well_ordering singleton
```
```  1697     domain_of segment null_class intersection asymmetric compos transitive
```
```  1698     cross_product connected identity_relation complement restrct subclass
```
```  1699     irreflexive symmetrization_of INVERSE union equal &
```
```  1700   NUM004_0_eq well_ordering transitive section irreflexive
```
```  1701     connected asymmetric symmetrization_of segment rest_of
```
```  1702     recursion_equation_functions recursion ordinal_multiply ordinal_add
```
```  1703     not_well_ordering least integer_of equal &
```
```  1704   (~subclass(limit_ordinals::'a,ordinal_numbers)) --> False"
```
```  1705   by meson
```
```  1706
```
```  1707
```
```  1708 (*0 inferences so far.  Searching to depth 0.  16.8 secs.  BIG*)
```
```  1709 lemma NUM228_1:
```
```  1710   "EQU001_0_ax equal &
```
```  1711   SET004_0_ax not_homomorphism2 not_homomorphism1
```
```  1712     homomorphism compatible operation cantor diagonalise subset_relation
```
```  1713     one_to_one choice apply regular function identity_relation
```
```  1714     single_valued_class compos powerClass sum_class omega inductive
```
```  1715     successor_relation successor image' rng domain range_of INVERSE flip
```
```  1716     rot domain_of null_class restrct difference union complement
```
```  1717     intersection element_relation second first cross_product ordered_pair
```
```  1718     singleton unordered_pair equal universal_class not_subclass_element
```
```  1719     member subclass &
```
```  1720   SET004_0_eq subclass single_valued_class operation
```
```  1721     one_to_one member inductive homomorphism function compatible
```
```  1722     unordered_pair union sum_class successor singleton second rot restrct
```
```  1723     regular range_of rng powerClass ordered_pair not_subclass_element
```
```  1724     not_homomorphism2 not_homomorphism1 INVERSE intersection image' flip
```
```  1725     first domain_of domain difference diagonalise cross_product compos
```
```  1726     complement cantor apply equal &
```
```  1727   SET004_1_ax range_of function maps apply
```
```  1728     application_function singleton_relation element_relation complement
```
```  1729     intersection single_valued3 singleton image' domain single_valued2
```
```  1730     second single_valued1 identity_relation INVERSE not_subclass_element
```
```  1731     first domain_of domain_relation composition_function compos equal
```
```  1732     ordered_pair member universal_class cross_product compose_class
```
```  1733     subclass &
```
```  1734   SET004_1_eq maps single_valued3 single_valued2 single_valued1 compose_class equal &
```
```  1735   NUM004_0_ax integer_of omega ordinal_multiply
```
```  1736     add_relation ordinal_add recursion apply range_of union_of_range_map
```
```  1737     function recursion_equation_functions rest_relation rest_of
```
```  1738     limit_ordinals kind_1_ordinals successor_relation image'
```
```  1739     universal_class sum_class element_relation ordinal_numbers section
```
```  1740     not_well_ordering ordered_pair least member well_ordering singleton
```
```  1741     domain_of segment null_class intersection asymmetric compos transitive
```
```  1742     cross_product connected identity_relation complement restrct subclass
```
```  1743     irreflexive symmetrization_of INVERSE union equal &
```
```  1744   NUM004_0_eq well_ordering transitive section irreflexive
```
```  1745     connected asymmetric symmetrization_of segment rest_of
```
```  1746     recursion_equation_functions recursion ordinal_multiply ordinal_add
```
```  1747     not_well_ordering least integer_of equal &
```
```  1748   (~function(z)) &
```
```  1749     (~equal(recursion_equation_functions(z),null_class)) --> False"
```
```  1750   by meson
```
```  1751
```
```  1752
```
```  1753 (*4868 inferences so far.  Searching to depth 12.  4.3 secs*)
```
```  1754 lemma PLA002_1:
```
```  1755   "(\<forall>Situation1 Situation2. warm(Situation1) | cold(Situation2)) &
```
```  1756   (\<forall>Situation. at(a::'a,Situation) --> at(b::'a,walk(b::'a,Situation))) &
```
```  1757   (\<forall>Situation. at(a::'a,Situation) --> at(b::'a,drive(b::'a,Situation))) &
```
```  1758   (\<forall>Situation. at(b::'a,Situation) --> at(a::'a,walk(a::'a,Situation))) &
```
```  1759   (\<forall>Situation. at(b::'a,Situation) --> at(a::'a,drive(a::'a,Situation))) &
```
```  1760   (\<forall>Situation. cold(Situation) & at(b::'a,Situation) --> at(c::'a,skate(c::'a,Situation))) &
```
```  1761   (\<forall>Situation. cold(Situation) & at(c::'a,Situation) --> at(b::'a,skate(b::'a,Situation))) &
```
```  1762   (\<forall>Situation. warm(Situation) & at(b::'a,Situation) --> at(d::'a,climb(d::'a,Situation))) &
```
```  1763   (\<forall>Situation. warm(Situation) & at(d::'a,Situation) --> at(b::'a,climb(b::'a,Situation))) &
```
```  1764   (\<forall>Situation. at(c::'a,Situation) --> at(d::'a,go(d::'a,Situation))) &
```
```  1765   (\<forall>Situation. at(d::'a,Situation) --> at(c::'a,go(c::'a,Situation))) &
```
```  1766   (\<forall>Situation. at(c::'a,Situation) --> at(e::'a,go(e::'a,Situation))) &
```
```  1767   (\<forall>Situation. at(e::'a,Situation) --> at(c::'a,go(c::'a,Situation))) &
```
```  1768   (\<forall>Situation. at(d::'a,Situation) --> at(f::'a,go(f::'a,Situation))) &
```
```  1769   (\<forall>Situation. at(f::'a,Situation) --> at(d::'a,go(d::'a,Situation))) &
```
```  1770   (at(f::'a,s0)) &
```
```  1771   (\<forall>S'. ~at(a::'a,S')) --> False"
```
```  1772   by meson
```
```  1773
```
```  1774 abbreviation "PLA001_0_ax putdown on pickup do holding table differ clear EMPTY and' holds \<equiv>
```
```  1775   (\<forall>X Y State. holds(X::'a,State) & holds(Y::'a,State) --> holds(and'(X::'a,Y),State)) &
```
```  1776   (\<forall>State X. holds(EMPTY::'a,State) & holds(clear(X),State) & differ(X::'a,table) --> holds(holding(X),do(pickup(X),State))) &
```
```  1777   (\<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))) &
```
```  1778   (\<forall>Y State X Z. holds(on(X::'a,Y),State) & differ(X::'a,Z) --> holds(on(X::'a,Y),do(pickup(Z),State))) &
```
```  1779   (\<forall>State X Z. holds(clear(X),State) & differ(X::'a,Z) --> holds(clear(X),do(pickup(Z),State))) &
```
```  1780   (\<forall>X Y State. holds(holding(X),State) & holds(clear(Y),State) --> holds(EMPTY::'a,do(putdown(X::'a,Y),State))) &
```
```  1781   (\<forall>X Y State. holds(holding(X),State) & holds(clear(Y),State) --> holds(on(X::'a,Y),do(putdown(X::'a,Y),State))) &
```
```  1782   (\<forall>X Y State. holds(holding(X),State) & holds(clear(Y),State) --> holds(clear(X),do(putdown(X::'a,Y),State))) &
```
```  1783   (\<forall>Z W X Y State. holds(on(X::'a,Y),State) --> holds(on(X::'a,Y),do(putdown(Z::'a,W),State))) &
```
```  1784   (\<forall>X State Z Y. holds(clear(Z),State) & differ(Z::'a,Y) --> holds(clear(Z),do(putdown(X::'a,Y),State)))"
```
```  1785
```
```  1786 abbreviation "PLA001_1_ax EMPTY clear s0 on holds table d c b a differ \<equiv>
```
```  1787   (\<forall>Y X. differ(Y::'a,X) --> differ(X::'a,Y)) &
```
```  1788   (differ(a::'a,b)) &
```
```  1789   (differ(a::'a,c)) &
```
```  1790   (differ(a::'a,d)) &
```
```  1791   (differ(a::'a,table)) &
```
```  1792   (differ(b::'a,c)) &
```
```  1793   (differ(b::'a,d)) &
```
```  1794   (differ(b::'a,table)) &
```
```  1795   (differ(c::'a,d)) &
```
```  1796   (differ(c::'a,table)) &
```
```  1797   (differ(d::'a,table)) &
```
```  1798   (holds(on(a::'a,table),s0)) &
```
```  1799   (holds(on(b::'a,table),s0)) &
```
```  1800   (holds(on(c::'a,d),s0)) &
```
```  1801   (holds(on(d::'a,table),s0)) &
```
```  1802   (holds(clear(a),s0)) &
```
```  1803   (holds(clear(b),s0)) &
```
```  1804   (holds(clear(c),s0)) &
```
```  1805   (holds(EMPTY::'a,s0)) &
```
```  1806   (\<forall>State. holds(clear(table),State))"
```
```  1807
```
```  1808 (*190 inferences so far.  Searching to depth 10.  0.6 secs*)
```
```  1809 lemma PLA006_1:
```
```  1810   "PLA001_0_ax putdown on pickup do holding table differ clear EMPTY and' holds &
```
```  1811   PLA001_1_ax EMPTY clear s0 on holds table d c b a differ &
```
```  1812   (\<forall>State. ~holds(on(c::'a,table),State)) --> False"
```
```  1813   by meson
```
```  1814
```
```  1815 (*190 inferences so far.  Searching to depth 10.  0.5 secs*)
```
```  1816 lemma PLA017_1:
```
```  1817   "PLA001_0_ax putdown on pickup do holding table differ clear EMPTY and' holds &
```
```  1818   PLA001_1_ax EMPTY clear s0 on holds table d c b a differ &
```
```  1819   (\<forall>State. ~holds(on(a::'a,c),State)) --> False"
```
```  1820   by meson
```
```  1821
```
```  1822 (*13732 inferences so far.  Searching to depth 16.  9.8 secs*)
```
```  1823 lemma PLA022_1:
```
```  1824   "PLA001_0_ax putdown on pickup do holding table differ clear EMPTY and' holds &
```
```  1825   PLA001_1_ax EMPTY clear s0 on holds table d c b a differ &
```
```  1826   (\<forall>State. ~holds(and'(on(c::'a,d),on(a::'a,c)),State)) --> False"
```
```  1827   by meson
```
```  1828
```
```  1829 (*217 inferences so far.  Searching to depth 13.  0.7 secs*)
```
```  1830 lemma PLA022_2:
```
```  1831   "PLA001_0_ax putdown on pickup do holding table differ clear EMPTY and' holds &
```
```  1832   PLA001_1_ax EMPTY clear s0 on holds table d c b a differ &
```
```  1833   (\<forall>State. ~holds(and'(on(a::'a,c),on(c::'a,d)),State)) --> False"
```
```  1834   by meson
```
```  1835
```
```  1836 (*948 inferences so far.  Searching to depth 18.  1.1 secs*)
```
```  1837 lemma PRV001_1:
```
```  1838  "(\<forall>X Y Z. q1(X::'a,Y,Z) & mless_or_equal(X::'a,Y) --> q2(X::'a,Y,Z)) &
```
```  1839   (\<forall>X Y Z. q1(X::'a,Y,Z) --> mless_or_equal(X::'a,Y) | q3(X::'a,Y,Z)) &
```
```  1840   (\<forall>Z X Y. q2(X::'a,Y,Z) --> q4(X::'a,Y,Y)) &
```
```  1841   (\<forall>Z Y X. q3(X::'a,Y,Z) --> q4(X::'a,Y,X)) &
```
```  1842   (\<forall>X. mless_or_equal(X::'a,X)) &
```
```  1843   (\<forall>X Y. mless_or_equal(X::'a,Y) & mless_or_equal(Y::'a,X) --> equal(X::'a,Y)) &
```
```  1844   (\<forall>Y X Z. mless_or_equal(X::'a,Y) & mless_or_equal(Y::'a,Z) --> mless_or_equal(X::'a,Z)) &
```
```  1845   (\<forall>Y X. mless_or_equal(X::'a,Y) | mless_or_equal(Y::'a,X)) &
```
```  1846   (\<forall>X Y. equal(X::'a,Y) --> mless_or_equal(X::'a,Y)) &
```
```  1847   (\<forall>X Y Z. equal(X::'a,Y) & mless_or_equal(X::'a,Z) --> mless_or_equal(Y::'a,Z)) &
```
```  1848   (\<forall>X Z Y. equal(X::'a,Y) & mless_or_equal(Z::'a,X) --> mless_or_equal(Z::'a,Y)) &
```
```  1849   (q1(a::'a,b,c)) &
```
```  1850   (\<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))) &
```
```  1851   (\<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"
```
```  1852   by meson
```
```  1853
```
```  1854 (*PRV is now called SWV (software verification) *)
```
```  1855 abbreviation "SWV001_1_ax mless_THAN successor predecessor equal \<equiv>
```
```  1856   (\<forall>X. equal(predecessor(successor(X)),X)) &
```
```  1857   (\<forall>X. equal(successor(predecessor(X)),X)) &
```
```  1858   (\<forall>X Y. equal(predecessor(X),predecessor(Y)) --> equal(X::'a,Y)) &
```
```  1859   (\<forall>X Y. equal(successor(X),successor(Y)) --> equal(X::'a,Y)) &
```
```  1860   (\<forall>X. mless_THAN(predecessor(X),X)) &
```
```  1861   (\<forall>X. mless_THAN(X::'a,successor(X))) &
```
```  1862   (\<forall>X Y Z. mless_THAN(X::'a,Y) & mless_THAN(Y::'a,Z) --> mless_THAN(X::'a,Z)) &
```
```  1863   (\<forall>X Y. mless_THAN(X::'a,Y) | mless_THAN(Y::'a,X) | equal(X::'a,Y)) &
```
```  1864   (\<forall>X. ~mless_THAN(X::'a,X)) &
```
```  1865   (\<forall>Y X. ~(mless_THAN(X::'a,Y) & mless_THAN(Y::'a,X))) &
```
```  1866   (\<forall>Y X Z. equal(X::'a,Y) & mless_THAN(X::'a,Z) --> mless_THAN(Y::'a,Z)) &
```
```  1867   (\<forall>Y Z X. equal(X::'a,Y) & mless_THAN(Z::'a,X) --> mless_THAN(Z::'a,Y))"
```
```  1868
```
```  1869 abbreviation "SWV001_0_eq a successor predecessor equal \<equiv>
```
```  1870   (\<forall>X Y. equal(X::'a,Y) --> equal(predecessor(X),predecessor(Y))) &
```
```  1871   (\<forall>X Y. equal(X::'a,Y) --> equal(successor(X),successor(Y))) &
```
```  1872   (\<forall>X Y. equal(X::'a,Y) --> equal(a(X),a(Y)))"
```
```  1873
```
```  1874 (*21 inferences so far.  Searching to depth 5.  0.4 secs*)
```
```  1875 lemma PRV003_1:
```
```  1876   "EQU001_0_ax equal &
```
```  1877   SWV001_1_ax mless_THAN successor predecessor equal &
```
```  1878   SWV001_0_eq a successor predecessor equal &
```
```  1879   (~mless_THAN(n::'a,j)) &
```
```  1880   (mless_THAN(k::'a,j)) &
```
```  1881   (~mless_THAN(k::'a,i)) &
```
```  1882   (mless_THAN(i::'a,n)) &
```
```  1883   (mless_THAN(a(j),a(k))) &
```
```  1884   (\<forall>X. mless_THAN(X::'a,j) & mless_THAN(a(X),a(k)) --> mless_THAN(X::'a,i)) &
```
```  1885   (\<forall>X. mless_THAN(One::'a,i) & mless_THAN(a(X),a(predecessor(i))) --> mless_THAN(X::'a,i) | mless_THAN(n::'a,X)) &
```
```  1886   (\<forall>X. ~(mless_THAN(One::'a,X) & mless_THAN(X::'a,i) & mless_THAN(a(X),a(predecessor(X))))) &
```
```  1887   (mless_THAN(j::'a,i)) --> False"
```
```  1888   by meson
```
```  1889
```
```  1890 (*584 inferences so far.  Searching to depth 7.  1.1 secs*)
```
```  1891 lemma PRV005_1:
```
```  1892   "EQU001_0_ax equal &
```
```  1893   SWV001_1_ax mless_THAN successor predecessor equal &
```
```  1894   SWV001_0_eq a successor predecessor equal &
```
```  1895   (~mless_THAN(n::'a,k)) &
```
```  1896   (~mless_THAN(k::'a,l)) &
```
```  1897   (~mless_THAN(k::'a,i)) &
```
```  1898   (mless_THAN(l::'a,n)) &
```
```  1899   (mless_THAN(One::'a,l)) &
```
```  1900   (mless_THAN(a(k),a(predecessor(l)))) &
```
```  1901   (\<forall>X. mless_THAN(X::'a,successor(n)) & mless_THAN(a(X),a(k)) --> mless_THAN(X::'a,l)) &
```
```  1902   (\<forall>X. mless_THAN(One::'a,l) & mless_THAN(a(X),a(predecessor(l))) --> mless_THAN(X::'a,l) | mless_THAN(n::'a,X)) &
```
```  1903   (\<forall>X. ~(mless_THAN(One::'a,X) & mless_THAN(X::'a,l) & mless_THAN(a(X),a(predecessor(X))))) --> False"
```
```  1904   by meson
```
```  1905
```
```  1906 (*2343 inferences so far.  Searching to depth 8.  3.5 secs*)
```
```  1907 lemma PRV006_1:
```
```  1908   "EQU001_0_ax equal &
```
```  1909   SWV001_1_ax mless_THAN successor predecessor equal &
```
```  1910   SWV001_0_eq a successor predecessor equal &
```
```  1911   (~mless_THAN(n::'a,m)) &
```
```  1912   (mless_THAN(i::'a,m)) &
```
```  1913   (mless_THAN(i::'a,n)) &
```
```  1914   (~mless_THAN(i::'a,One)) &
```
```  1915   (mless_THAN(a(i),a(m))) &
```
```  1916   (\<forall>X. mless_THAN(X::'a,successor(n)) & mless_THAN(a(X),a(m)) --> mless_THAN(X::'a,i)) &
```
```  1917   (\<forall>X. mless_THAN(One::'a,i) & mless_THAN(a(X),a(predecessor(i))) --> mless_THAN(X::'a,i) | mless_THAN(n::'a,X)) &
```
```  1918   (\<forall>X. ~(mless_THAN(One::'a,X) & mless_THAN(X::'a,i) & mless_THAN(a(X),a(predecessor(X))))) --> False"
```
```  1919   by meson
```
```  1920
```
```  1921 (*86 inferences so far.  Searching to depth 14.  0.1 secs*)
```
```  1922 lemma PRV009_1:
```
```  1923   "(\<forall>Y X. mless_or_equal(X::'a,Y) | mless(Y::'a,X)) &
```
```  1924   (mless(j::'a,i)) &
```
```  1925   (mless_or_equal(m::'a,p)) &
```
```  1926   (mless_or_equal(p::'a,q)) &
```
```  1927   (mless_or_equal(q::'a,n)) &
```
```  1928   (\<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))) &
```
```  1929   (\<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))) &
```
```  1930   (\<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))) &
```
```  1931   (~mless_or_equal(a(p),a(q))) --> False"
```
```  1932   by meson
```
```  1933
```
```  1934 (*222 inferences so far.  Searching to depth 8.  0.4 secs*)
```
```  1935 lemma PUZ012_1:
```
```  1936   "(\<forall>X. equal_fruits(X::'a,X)) &
```
```  1937   (\<forall>X. equal_boxes(X::'a,X)) &
```
```  1938   (\<forall>X Y. ~(label(X::'a,Y) & contains(X::'a,Y))) &
```
```  1939   (\<forall>X. contains(boxa::'a,X) | contains(boxb::'a,X) | contains(boxc::'a,X)) &
```
```  1940   (\<forall>X. contains(X::'a,apples) | contains(X::'a,bananas) | contains(X::'a,oranges)) &
```
```  1941   (\<forall>X Y Z. contains(X::'a,Y) & contains(X::'a,Z) --> equal_fruits(Y::'a,Z)) &
```
```  1942   (\<forall>Y X Z. contains(X::'a,Y) & contains(Z::'a,Y) --> equal_boxes(X::'a,Z)) &
```
```  1943   (~equal_boxes(boxa::'a,boxb)) &
```
```  1944   (~equal_boxes(boxb::'a,boxc)) &
```
```  1945   (~equal_boxes(boxa::'a,boxc)) &
```
```  1946   (~equal_fruits(apples::'a,bananas)) &
```
```  1947   (~equal_fruits(bananas::'a,oranges)) &
```
```  1948   (~equal_fruits(apples::'a,oranges)) &
```
```  1949   (label(boxa::'a,apples)) &
```
```  1950   (label(boxb::'a,oranges)) &
```
```  1951   (label(boxc::'a,bananas)) &
```
```  1952   (contains(boxb::'a,apples)) &
```
```  1953   (~(contains(boxa::'a,bananas) & contains(boxc::'a,oranges))) --> False"
```
```  1954   by meson
```
```  1955
```
```  1956 (*35 inferences so far.  Searching to depth 5.  3.2 secs*)
```
```  1957 lemma PUZ020_1:
```
```  1958   "EQU001_0_ax equal &
```
```  1959   (\<forall>A B. equal(A::'a,B) --> equal(statement_by(A),statement_by(B))) &
```
```  1960   (\<forall>X. person(X) --> knight(X) | knave(X)) &
```
```  1961   (\<forall>X. ~(person(X) & knight(X) & knave(X))) &
```
```  1962   (\<forall>X Y. says(X::'a,Y) & a_truth(Y) --> a_truth(Y)) &
```
```  1963   (\<forall>X Y. ~(says(X::'a,Y) & equal(X::'a,Y))) &
```
```  1964   (\<forall>Y X. says(X::'a,Y) --> equal(Y::'a,statement_by(X))) &
```
```  1965   (\<forall>X Y. ~(person(X) & equal(X::'a,statement_by(Y)))) &
```
```  1966   (\<forall>X. person(X) & a_truth(statement_by(X)) --> knight(X)) &
```
```  1967   (\<forall>X. person(X) --> a_truth(statement_by(X)) | knave(X)) &
```
```  1968   (\<forall>X Y. equal(X::'a,Y) & knight(X) --> knight(Y)) &
```
```  1969   (\<forall>X Y. equal(X::'a,Y) & knave(X) --> knave(Y)) &
```
```  1970   (\<forall>X Y. equal(X::'a,Y) & person(X) --> person(Y)) &
```
```  1971   (\<forall>X Y Z. equal(X::'a,Y) & says(X::'a,Z) --> says(Y::'a,Z)) &
```
```  1972   (\<forall>X Z Y. equal(X::'a,Y) & says(Z::'a,X) --> says(Z::'a,Y)) &
```
```  1973   (\<forall>X Y. equal(X::'a,Y) & a_truth(X) --> a_truth(Y)) &
```
```  1974   (\<forall>X Y. knight(X) & says(X::'a,Y) --> a_truth(Y)) &
```
```  1975   (\<forall>X Y. ~(knave(X) & says(X::'a,Y) & a_truth(Y))) &
```
```  1976   (person(husband)) &
```
```  1977   (person(wife)) &
```
```  1978   (~equal(husband::'a,wife)) &
```
```  1979   (says(husband::'a,statement_by(husband))) &
```
```  1980   (a_truth(statement_by(husband)) & knight(husband) --> knight(wife)) &
```
```  1981   (knight(husband) --> a_truth(statement_by(husband))) &
```
```  1982   (a_truth(statement_by(husband)) | knight(wife)) &
```
```  1983   (knight(wife) --> a_truth(statement_by(husband))) &
```
```  1984   (~knight(husband)) --> False"
```
```  1985   by meson
```
```  1986
```
```  1987 (*121806 inferences so far.  Searching to depth 17.  63.0 secs*)
```
```  1988 lemma PUZ025_1:
```
```  1989   "(\<forall>X. a_truth(truthteller(X)) | a_truth(liar(X))) &
```
```  1990   (\<forall>X. ~(a_truth(truthteller(X)) & a_truth(liar(X)))) &
```
```  1991   (\<forall>Truthteller Statement. a_truth(truthteller(Truthteller)) & a_truth(says(Truthteller::'a,Statement)) --> a_truth(Statement)) &
```
```  1992   (\<forall>Liar Statement. ~(a_truth(liar(Liar)) & a_truth(says(Liar::'a,Statement)) & a_truth(Statement))) &
```
```  1993   (\<forall>Statement Truthteller. a_truth(Statement) & a_truth(says(Truthteller::'a,Statement)) --> a_truth(truthteller(Truthteller))) &
```
```  1994   (\<forall>Statement Liar. a_truth(says(Liar::'a,Statement)) --> a_truth(Statement) | a_truth(liar(Liar))) &
```
```  1995   (\<forall>Z X Y. people(X::'a,Y,Z) & a_truth(liar(X)) & a_truth(liar(Y)) --> a_truth(equal_type(X::'a,Y))) &
```
```  1996   (\<forall>Z X Y. people(X::'a,Y,Z) & a_truth(truthteller(X)) & a_truth(truthteller(Y)) --> a_truth(equal_type(X::'a,Y))) &
```
```  1997   (\<forall>X Y. a_truth(equal_type(X::'a,Y)) & a_truth(truthteller(X)) --> a_truth(truthteller(Y))) &
```
```  1998   (\<forall>X Y. a_truth(equal_type(X::'a,Y)) & a_truth(liar(X)) --> a_truth(liar(Y))) &
```
```  1999   (\<forall>X Y. a_truth(truthteller(X)) --> a_truth(equal_type(X::'a,Y)) | a_truth(liar(Y))) &
```
```  2000   (\<forall>X Y. a_truth(liar(X)) --> a_truth(equal_type(X::'a,Y)) | a_truth(truthteller(Y))) &
```
```  2001   (\<forall>Y X. a_truth(equal_type(X::'a,Y)) --> a_truth(equal_type(Y::'a,X))) &
```
```  2002   (\<forall>X Y. ask_1_if_2(X::'a,Y) & a_truth(truthteller(X)) & a_truth(Y) --> answer(yes)) &
```
```  2003   (\<forall>X Y. ask_1_if_2(X::'a,Y) & a_truth(truthteller(X)) --> a_truth(Y) | answer(no)) &
```
```  2004   (\<forall>X Y. ask_1_if_2(X::'a,Y) & a_truth(liar(X)) & a_truth(Y) --> answer(no)) &
```
```  2005   (\<forall>X Y. ask_1_if_2(X::'a,Y) & a_truth(liar(X)) --> a_truth(Y) | answer(yes)) &
```
```  2006   (people(b::'a,c,a)) &
```
```  2007   (people(a::'a,b,a)) &
```
```  2008   (people(a::'a,c,b)) &
```
```  2009   (people(c::'a,b,a)) &
```
```  2010   (a_truth(says(a::'a,equal_type(b::'a,c)))) &
```
```  2011   (ask_1_if_2(c::'a,equal_type(a::'a,b))) &
```
```  2012   (\<forall>Answer. ~answer(Answer)) --> False"
```
```  2013   oops
```
```  2014
```
```  2015
```
```  2016 (*621 inferences so far.  Searching to depth 18.  0.2 secs*)
```
```  2017 lemma PUZ029_1:
```
```  2018  "(\<forall>X. dances_on_tightropes(X) | eats_pennybuns(X) | old(X)) &
```
```  2019   (\<forall>X. pig(X) & liable_to_giddiness(X) --> treated_with_respect(X)) &
```
```  2020   (\<forall>X. wise(X) & balloonist(X) --> has_umbrella(X)) &
```
```  2021   (\<forall>X. ~(looks_ridiculous(X) & eats_pennybuns(X) & eats_lunch_in_public(X))) &
```
```  2022   (\<forall>X. balloonist(X) & young(X) --> liable_to_giddiness(X)) &
```
```  2023   (\<forall>X. fat(X) & looks_ridiculous(X) --> dances_on_tightropes(X) | eats_lunch_in_public(X)) &
```
```  2024   (\<forall>X. ~(liable_to_giddiness(X) & wise(X) & dances_on_tightropes(X))) &
```
```  2025   (\<forall>X. pig(X) & has_umbrella(X) --> looks_ridiculous(X)) &
```
```  2026   (\<forall>X. treated_with_respect(X) --> dances_on_tightropes(X) | fat(X)) &
```
```  2027   (\<forall>X. young(X) | old(X)) &
```
```  2028   (\<forall>X. ~(young(X) & old(X))) &
```
```  2029   (wise(piggy)) &
```
```  2030   (young(piggy)) &
```
```  2031   (pig(piggy)) &
```
```  2032   (balloonist(piggy)) --> False"
```
```  2033   by meson
```
```  2034
```
```  2035 abbreviation "RNG001_0_ax equal additive_inverse add multiply product additive_identity sum \<equiv>
```
```  2036   (\<forall>X. sum(additive_identity::'a,X,X)) &
```
```  2037   (\<forall>X. sum(X::'a,additive_identity,X)) &
```
```  2038   (\<forall>X Y. product(X::'a,Y,multiply(X::'a,Y))) &
```
```  2039   (\<forall>X Y. sum(X::'a,Y,add(X::'a,Y))) &
```
```  2040   (\<forall>X. sum(additive_inverse(X),X,additive_identity)) &
```
```  2041   (\<forall>X. sum(X::'a,additive_inverse(X),additive_identity)) &
```
```  2042   (\<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)) &
```
```  2043   (\<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)) &
```
```  2044   (\<forall>Y X Z. sum(X::'a,Y,Z) --> sum(Y::'a,X,Z)) &
```
```  2045   (\<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)) &
```
```  2046   (\<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)) &
```
```  2047   (\<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)) &
```
```  2048   (\<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)) &
```
```  2049   (\<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)) &
```
```  2050   (\<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)) &
```
```  2051   (\<forall>X Y U V. sum(X::'a,Y,U) & sum(X::'a,Y,V) --> equal(U::'a,V)) &
```
```  2052   (\<forall>X Y U V. product(X::'a,Y,U) & product(X::'a,Y,V) --> equal(U::'a,V))"
```
```  2053
```
```  2054 abbreviation "RNG001_0_eq product multiply sum add additive_inverse equal \<equiv>
```
```  2055   (\<forall>X Y. equal(X::'a,Y) --> equal(additive_inverse(X),additive_inverse(Y))) &
```
```  2056   (\<forall>X Y W. equal(X::'a,Y) --> equal(add(X::'a,W),add(Y::'a,W))) &
```
```  2057   (\<forall>X W Y. equal(X::'a,Y) --> equal(add(W::'a,X),add(W::'a,Y))) &
```
```  2058   (\<forall>X Y W Z. equal(X::'a,Y) & sum(X::'a,W,Z) --> sum(Y::'a,W,Z)) &
```
```  2059   (\<forall>X W Y Z. equal(X::'a,Y) & sum(W::'a,X,Z) --> sum(W::'a,Y,Z)) &
```
```  2060   (\<forall>X W Z Y. equal(X::'a,Y) & sum(W::'a,Z,X) --> sum(W::'a,Z,Y)) &
```
```  2061   (\<forall>X Y W. equal(X::'a,Y) --> equal(multiply(X::'a,W),multiply(Y::'a,W))) &
```
```  2062   (\<forall>X W Y. equal(X::'a,Y) --> equal(multiply(W::'a,X),multiply(W::'a,Y))) &
```
```  2063   (\<forall>X Y W Z. equal(X::'a,Y) & product(X::'a,W,Z) --> product(Y::'a,W,Z)) &
```
```  2064   (\<forall>X W Y Z. equal(X::'a,Y) & product(W::'a,X,Z) --> product(W::'a,Y,Z)) &
```
```  2065   (\<forall>X W Z Y. equal(X::'a,Y) & product(W::'a,Z,X) --> product(W::'a,Z,Y))"
```
```  2066
```
```  2067 (*93620 inferences so far.  Searching to depth 24.  65.9 secs*)
```
```  2068 lemma RNG001_3:
```
```  2069  "(\<forall>X. sum(additive_identity::'a,X,X)) &
```
```  2070   (\<forall>X. sum(additive_inverse(X),X,additive_identity)) &
```
```  2071   (\<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)) &
```
```  2072   (\<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)) &
```
```  2073   (\<forall>X Y. product(X::'a,Y,multiply(X::'a,Y))) &
```
```  2074   (\<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)) &
```
```  2075   (\<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)) &
```
```  2076   (~product(a::'a,additive_identity,additive_identity)) --> False"
```
```  2077   oops
```
```  2078
```
```  2079 abbreviation "RNG_other_ax multiply add equal product additive_identity additive_inverse sum \<equiv>
```
```  2080   (\<forall>X. sum(X::'a,additive_inverse(X),additive_identity)) &
```
```  2081   (\<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)) &
```
```  2082   (\<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)) &
```
```  2083   (\<forall>Y X Z. sum(X::'a,Y,Z) --> sum(Y::'a,X,Z)) &
```
```  2084   (\<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)) &
```
```  2085   (\<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)) &
```
```  2086   (\<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)) &
```
```  2087   (\<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)) &
```
```  2088   (\<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)) &
```
```  2089   (\<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)) &
```
```  2090   (\<forall>X Y U V. sum(X::'a,Y,U) & sum(X::'a,Y,V) --> equal(U::'a,V)) &
```
```  2091   (\<forall>X Y U V. product(X::'a,Y,U) & product(X::'a,Y,V) --> equal(U::'a,V)) &
```
```  2092   (\<forall>X Y. equal(X::'a,Y) --> equal(additive_inverse(X),additive_inverse(Y))) &
```
```  2093   (\<forall>X Y W. equal(X::'a,Y) --> equal(add(X::'a,W),add(Y::'a,W))) &
```
```  2094   (\<forall>X Y W Z. equal(X::'a,Y) & sum(X::'a,W,Z) --> sum(Y::'a,W,Z)) &
```
```  2095   (\<forall>X W Y Z. equal(X::'a,Y) & sum(W::'a,X,Z) --> sum(W::'a,Y,Z)) &
```
```  2096   (\<forall>X W Z Y. equal(X::'a,Y) & sum(W::'a,Z,X) --> sum(W::'a,Z,Y)) &
```
```  2097   (\<forall>X Y W. equal(X::'a,Y) --> equal(multiply(X::'a,W),multiply(Y::'a,W))) &
```
```  2098   (\<forall>X Y W Z. equal(X::'a,Y) & product(X::'a,W,Z) --> product(Y::'a,W,Z)) &
```
```  2099   (\<forall>X W Y Z. equal(X::'a,Y) & product(W::'a,X,Z) --> product(W::'a,Y,Z)) &
```
```  2100   (\<forall>X W Z Y. equal(X::'a,Y) & product(W::'a,Z,X) --> product(W::'a,Z,Y))"
```
```  2101
```
```  2102
```
```  2103 (****************SLOW
```
```  2104 76385914 inferences so far.  Searching to depth 18
```
```  2105 No proof after 5 1/2 hours! (griffon)
```
```  2106 ****************)
```
```  2107 lemma RNG001_5:
```
```  2108   "EQU001_0_ax equal &
```
```  2109   (\<forall>X. sum(additive_identity::'a,X,X)) &
```
```  2110   (\<forall>X. sum(X::'a,additive_identity,X)) &
```
```  2111   (\<forall>X Y. product(X::'a,Y,multiply(X::'a,Y))) &
```
```  2112   (\<forall>X Y. sum(X::'a,Y,add(X::'a,Y))) &
```
```  2113   (\<forall>X. sum(additive_inverse(X),X,additive_identity)) &
```
```  2114   RNG_other_ax multiply add equal product additive_identity additive_inverse sum &
```
```  2115   (~product(a::'a,additive_identity,additive_identity)) --> False"
```
```  2116   oops
```
```  2117
```
```  2118 (*0 inferences so far.  Searching to depth 0.  0.5 secs*)
```
```  2119 lemma RNG011_5:
```
```  2120   "EQU001_0_ax equal &
```
```  2121   (\<forall>A B C. equal(A::'a,B) --> equal(add(A::'a,C),add(B::'a,C))) &
```
```  2122   (\<forall>D F' E. equal(D::'a,E) --> equal(add(F'::'a,D),add(F'::'a,E))) &
```
```  2123   (\<forall>G H. equal(G::'a,H) --> equal(additive_inverse(G),additive_inverse(H))) &
```
```  2124   (\<forall>I' J K'. equal(I'::'a,J) --> equal(multiply(I'::'a,K'),multiply(J::'a,K'))) &
```
```  2125   (\<forall>L N M. equal(L::'a,M) --> equal(multiply(N::'a,L),multiply(N::'a,M))) &
```
```  2126   (\<forall>A B C D. equal(A::'a,B) --> equal(associator(A::'a,C,D),associator(B::'a,C,D))) &
```
```  2127   (\<forall>E G F' H. equal(E::'a,F') --> equal(associator(G::'a,E,H),associator(G::'a,F',H))) &
```
```  2128   (\<forall>I' K' L J. equal(I'::'a,J) --> equal(associator(K'::'a,L,I'),associator(K'::'a,L,J))) &
```
```  2129   (\<forall>M N O'. equal(M::'a,N) --> equal(commutator(M::'a,O'),commutator(N::'a,O'))) &
```
```  2130   (\<forall>P R Q. equal(P::'a,Q) --> equal(commutator(R::'a,P),commutator(R::'a,Q))) &
```
```  2131   (\<forall>Y X. equal(add(X::'a,Y),add(Y::'a,X))) &
```
```  2132   (\<forall>X Y Z. equal(add(add(X::'a,Y),Z),add(X::'a,add(Y::'a,Z)))) &
```
```  2133   (\<forall>X. equal(add(X::'a,additive_identity),X)) &
```
```  2134   (\<forall>X. equal(add(additive_identity::'a,X),X)) &
```
```  2135   (\<forall>X. equal(add(X::'a,additive_inverse(X)),additive_identity)) &
```
```  2136   (\<forall>X. equal(add(additive_inverse(X),X),additive_identity)) &
```
```  2137   (equal(additive_inverse(additive_identity),additive_identity)) &
```
```  2138   (\<forall>X Y. equal(add(X::'a,add(additive_inverse(X),Y)),Y)) &
```
```  2139   (\<forall>X Y. equal(additive_inverse(add(X::'a,Y)),add(additive_inverse(X),additive_inverse(Y)))) &
```
```  2140   (\<forall>X. equal(additive_inverse(additive_inverse(X)),X)) &
```
```  2141   (\<forall>X. equal(multiply(X::'a,additive_identity),additive_identity)) &
```
```  2142   (\<forall>X. equal(multiply(additive_identity::'a,X),additive_identity)) &
```
```  2143   (\<forall>X Y. equal(multiply(additive_inverse(X),additive_inverse(Y)),multiply(X::'a,Y))) &
```
```  2144   (\<forall>X Y. equal(multiply(X::'a,additive_inverse(Y)),additive_inverse(multiply(X::'a,Y)))) &
```
```  2145   (\<forall>X Y. equal(multiply(additive_inverse(X),Y),additive_inverse(multiply(X::'a,Y)))) &
```
```  2146   (\<forall>Y X Z. equal(multiply(X::'a,add(Y::'a,Z)),add(multiply(X::'a,Y),multiply(X::'a,Z)))) &
```
```  2147   (\<forall>X Y Z. equal(multiply(add(X::'a,Y),Z),add(multiply(X::'a,Z),multiply(Y::'a,Z)))) &
```
```  2148   (\<forall>X Y. equal(multiply(multiply(X::'a,Y),Y),multiply(X::'a,multiply(Y::'a,Y)))) &
```
```  2149   (\<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)))))) &
```
```  2150   (\<forall>X Y. equal(commutator(X::'a,Y),add(multiply(Y::'a,X),additive_inverse(multiply(X::'a,Y))))) &
```
```  2151   (\<forall>X Y. equal(multiply(multiply(associator(X::'a,X,Y),X),associator(X::'a,X,Y)),additive_identity)) &
```
```  2152   (~equal(multiply(multiply(associator(a::'a,a,b),a),associator(a::'a,a,b)),additive_identity)) --> False"
```
```  2153   by meson
```
```  2154
```
```  2155 (*202 inferences so far.  Searching to depth 8.  0.6 secs*)
```
```  2156 lemma RNG023_6:
```
```  2157   "EQU001_0_ax equal &
```
```  2158   (\<forall>Y X. equal(add(X::'a,Y),add(Y::'a,X))) &
```
```  2159   (\<forall>X Y Z. equal(add(X::'a,add(Y::'a,Z)),add(add(X::'a,Y),Z))) &
```
```  2160   (\<forall>X. equal(add(additive_identity::'a,X),X)) &
```
```  2161   (\<forall>X. equal(add(X::'a,additive_identity),X)) &
```
```  2162   (\<forall>X. equal(multiply(additive_identity::'a,X),additive_identity)) &
```
```  2163   (\<forall>X. equal(multiply(X::'a,additive_identity),additive_identity)) &
```
```  2164   (\<forall>X. equal(add(additive_inverse(X),X),additive_identity)) &
```
```  2165   (\<forall>X. equal(add(X::'a,additive_inverse(X)),additive_identity)) &
```
```  2166   (\<forall>Y X Z. equal(multiply(X::'a,add(Y::'a,Z)),add(multiply(X::'a,Y),multiply(X::'a,Z)))) &
```
```  2167   (\<forall>X Y Z. equal(multiply(add(X::'a,Y),Z),add(multiply(X::'a,Z),multiply(Y::'a,Z)))) &
```
```  2168   (\<forall>X. equal(additive_inverse(additive_inverse(X)),X)) &
```
```  2169   (\<forall>X Y. equal(multiply(multiply(X::'a,Y),Y),multiply(X::'a,multiply(Y::'a,Y)))) &
```
```  2170   (\<forall>X Y. equal(multiply(multiply(X::'a,X),Y),multiply(X::'a,multiply(X::'a,Y)))) &
```
```  2171   (\<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)))))) &
```
```  2172   (\<forall>X Y. equal(commutator(X::'a,Y),add(multiply(Y::'a,X),additive_inverse(multiply(X::'a,Y))))) &
```
```  2173   (\<forall>D E F'. equal(D::'a,E) --> equal(add(D::'a,F'),add(E::'a,F'))) &
```
```  2174   (\<forall>G I' H. equal(G::'a,H) --> equal(add(I'::'a,G),add(I'::'a,H))) &
```
```  2175   (\<forall>J K'. equal(J::'a,K') --> equal(additive_inverse(J),additive_inverse(K'))) &
```
```  2176   (\<forall>L M N O'. equal(L::'a,M) --> equal(associator(L::'a,N,O'),associator(M::'a,N,O'))) &
```
```  2177   (\<forall>P R Q S'. equal(P::'a,Q) --> equal(associator(R::'a,P,S'),associator(R::'a,Q,S'))) &
```
```  2178   (\<forall>T' V W U. equal(T'::'a,U) --> equal(associator(V::'a,W,T'),associator(V::'a,W,U))) &
```
```  2179   (\<forall>X Y Z. equal(X::'a,Y) --> equal(commutator(X::'a,Z),commutator(Y::'a,Z))) &
```
```  2180   (\<forall>A1 C1 B1. equal(A1::'a,B1) --> equal(commutator(C1::'a,A1),commutator(C1::'a,B1))) &
```
```  2181   (\<forall>D1 E1 F1. equal(D1::'a,E1) --> equal(multiply(D1::'a,F1),multiply(E1::'a,F1))) &
```
```  2182   (\<forall>G1 I1 H1. equal(G1::'a,H1) --> equal(multiply(I1::'a,G1),multiply(I1::'a,H1))) &
```
```  2183   (~equal(associator(x::'a,x,y),additive_identity)) --> False"
```
```  2184   by meson
```
```  2185
```
```  2186 (*0 inferences so far.  Searching to depth 0.  0.6 secs*)
```
```  2187 lemma RNG028_2:
```
```  2188   "EQU001_0_ax equal &
```
```  2189   (\<forall>X. equal(add(additive_identity::'a,X),X)) &
```
```  2190   (\<forall>X. equal(multiply(additive_identity::'a,X),additive_identity)) &
```
```  2191   (\<forall>X. equal(multiply(X::'a,additive_identity),additive_identity)) &
```
```  2192   (\<forall>X. equal(add(additive_inverse(X),X),additive_identity)) &
```
```  2193   (\<forall>X Y. equal(additive_inverse(add(X::'a,Y)),add(additive_inverse(X),additive_inverse(Y)))) &
```
```  2194   (\<forall>X. equal(additive_inverse(additive_inverse(X)),X)) &
```
```  2195   (\<forall>Y X Z. equal(multiply(X::'a,add(Y::'a,Z)),add(multiply(X::'a,Y),multiply(X::'a,Z)))) &
```
```  2196   (\<forall>X Y Z. equal(multiply(add(X::'a,Y),Z),add(multiply(X::'a,Z),multiply(Y::'a,Z)))) &
```
```  2197   (\<forall>X Y. equal(multiply(multiply(X::'a,Y),Y),multiply(X::'a,multiply(Y::'a,Y)))) &
```
```  2198   (\<forall>X Y. equal(multiply(multiply(X::'a,X),Y),multiply(X::'a,multiply(X::'a,Y)))) &
```
```  2199   (\<forall>X Y. equal(multiply(additive_inverse(X),Y),additive_inverse(multiply(X::'a,Y)))) &
```
```  2200   (\<forall>X Y. equal(multiply(X::'a,additive_inverse(Y)),additive_inverse(multiply(X::'a,Y)))) &
```
```  2201   (equal(additive_inverse(additive_identity),additive_identity)) &
```
```  2202   (\<forall>Y X. equal(add(X::'a,Y),add(Y::'a,X))) &
```
```  2203   (\<forall>X Y Z. equal(add(X::'a,add(Y::'a,Z)),add(add(X::'a,Y),Z))) &
```
```  2204   (\<forall>Z X Y. equal(add(X::'a,Z),add(Y::'a,Z)) --> equal(X::'a,Y)) &
```
```  2205   (\<forall>Z X Y. equal(add(Z::'a,X),add(Z::'a,Y)) --> equal(X::'a,Y)) &
```
```  2206   (\<forall>D E F'. equal(D::'a,E) --> equal(add(D::'a,F'),add(E::'a,F'))) &
```
```  2207   (\<forall>G I' H. equal(G::'a,H) --> equal(add(I'::'a,G),add(I'::'a,H))) &
```
```  2208   (\<forall>J K'. equal(J::'a,K') --> equal(additive_inverse(J),additive_inverse(K'))) &
```
```  2209   (\<forall>D1 E1 F1. equal(D1::'a,E1) --> equal(multiply(D1::'a,F1),multiply(E1::'a,F1))) &
```
```  2210   (\<forall>G1 I1 H1. equal(G1::'a,H1) --> equal(multiply(I1::'a,G1),multiply(I1::'a,H1))) &
```
```  2211   (\<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)))))) &
```
```  2212   (\<forall>L M N O'. equal(L::'a,M) --> equal(associator(L::'a,N,O'),associator(M::'a,N,O'))) &
```
```  2213   (\<forall>P R Q S'. equal(P::'a,Q) --> equal(associator(R::'a,P,S'),associator(R::'a,Q,S'))) &
```
```  2214   (\<forall>T' V W U. equal(T'::'a,U) --> equal(associator(V::'a,W,T'),associator(V::'a,W,U))) &
```
```  2215   (\<forall>X Y. ~equal(multiply(multiply(Y::'a,X),Y),multiply(Y::'a,multiply(X::'a,Y)))) &
```
```  2216   (\<forall>X Y Z. ~equal(associator(Y::'a,X,Z),additive_inverse(associator(X::'a,Y,Z)))) &
```
```  2217   (\<forall>X Y Z. ~equal(associator(Z::'a,Y,X),additive_inverse(associator(X::'a,Y,Z)))) &
```
```  2218   (~equal(multiply(multiply(cx::'a,multiply(cy::'a,cx)),cz),multiply(cx::'a,multiply(cy::'a,multiply(cx::'a,cz))))) --> False"
```
```  2219   by meson
```
```  2220
```
```  2221 (*209 inferences so far.  Searching to depth 9.  1.2 secs*)
```
```  2222 lemma RNG038_2:
```
```  2223   "(\<forall>X. sum(X::'a,additive_identity,X)) &
```
```  2224   (\<forall>X Y. product(X::'a,Y,multiply(X::'a,Y))) &
```
```  2225   (\<forall>X Y. sum(X::'a,Y,add(X::'a,Y))) &
```
```  2226   RNG_other_ax multiply add equal product additive_identity additive_inverse sum &
```
```  2227   (\<forall>X. product(additive_identity::'a,X,additive_identity)) &
```
```  2228   (\<forall>X. product(X::'a,additive_identity,additive_identity)) &
```
```  2229   (\<forall>X Y. equal(X::'a,additive_identity) --> product(X::'a,h(X::'a,Y),Y)) &
```
```  2230   (product(a::'a,b,additive_identity)) &
```
```  2231   (~equal(a::'a,additive_identity)) &
```
```  2232   (~equal(b::'a,additive_identity)) --> False"
```
```  2233   by meson
```
```  2234
```
```  2235 (*2660 inferences so far.  Searching to depth 10.  7.0 secs*)
```
```  2236 lemma RNG040_2:
```
```  2237   "EQU001_0_ax equal &
```
```  2238   RNG001_0_eq product multiply sum add additive_inverse equal &
```
```  2239   (\<forall>X. sum(additive_identity::'a,X,X)) &
```
```  2240   (\<forall>X. sum(X::'a,additive_identity,X)) &
```
```  2241   (\<forall>X Y. product(X::'a,Y,multiply(X::'a,Y))) &
```
```  2242   (\<forall>X Y. sum(X::'a,Y,add(X::'a,Y))) &
```
```  2243   (\<forall>X. sum(additive_inverse(X),X,additive_identity)) &
```
```  2244   (\<forall>X. sum(X::'a,additive_inverse(X),additive_identity)) &
```
```  2245   (\<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)) &
```
```  2246   (\<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)) &
```
```  2247   (\<forall>Y X Z. sum(X::'a,Y,Z) --> sum(Y::'a,X,Z)) &
```
```  2248   (\<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)) &
```
```  2249   (\<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)) &
```
```  2250   (\<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)) &
```
```  2251   (\<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)) &
```
```  2252   (\<forall>X Y U V. sum(X::'a,Y,U) & sum(X::'a,Y,V) --> equal(U::'a,V)) &
```
```  2253   (\<forall>X Y U V. product(X::'a,Y,U) & product(X::'a,Y,V) --> equal(U::'a,V)) &
```
```  2254   (\<forall>A. product(A::'a,multiplicative_identity,A)) &
```
```  2255   (\<forall>A. product(multiplicative_identity::'a,A,A)) &
```
```  2256   (\<forall>A. product(A::'a,h(A),multiplicative_identity) | equal(A::'a,additive_identity)) &
```
```  2257   (\<forall>A. product(h(A),A,multiplicative_identity) | equal(A::'a,additive_identity)) &
```
```  2258   (\<forall>B A C. product(A::'a,B,C) --> product(B::'a,A,C)) &
```
```  2259   (\<forall>A B. equal(A::'a,B) --> equal(h(A),h(B))) &
```
```  2260   (sum(b::'a,c,d)) &
```
```  2261   (product(d::'a,a,additive_identity)) &
```
```  2262   (product(b::'a,a,l)) &
```
```  2263   (product(c::'a,a,n)) &
```
```  2264   (~sum(l::'a,n,additive_identity)) --> False"
```
```  2265   by meson
```
```  2266
```
```  2267 (*8991 inferences so far.  Searching to depth 9.  22.2 secs*)
```
```  2268 lemma RNG041_1:
```
```  2269   "EQU001_0_ax equal &
```
```  2270   RNG001_0_ax equal additive_inverse add multiply product additive_identity sum &
```
```  2271   RNG001_0_eq product multiply sum add additive_inverse equal &
```
```  2272   (\<forall>A B. equal(A::'a,B) --> equal(h(A),h(B))) &
```
```  2273   (\<forall>A. product(additive_identity::'a,A,additive_identity)) &
```
```  2274   (\<forall>A. product(A::'a,additive_identity,additive_identity)) &
```
```  2275   (\<forall>A. product(A::'a,multiplicative_identity,A)) &
```
```  2276   (\<forall>A. product(multiplicative_identity::'a,A,A)) &
```
```  2277   (\<forall>A. product(A::'a,h(A),multiplicative_identity) | equal(A::'a,additive_identity)) &
```
```  2278   (\<forall>A. product(h(A),A,multiplicative_identity) | equal(A::'a,additive_identity)) &
```
```  2279   (product(a::'a,b,additive_identity)) &
```
```  2280   (~equal(a::'a,additive_identity)) &
```
```  2281   (~equal(b::'a,additive_identity)) --> False"
```
```  2282   oops
```
```  2283
```
```  2284 (*101319 inferences so far.  Searching to depth 14.  76.0 secs*)
```
```  2285 lemma ROB010_1:
```
```  2286   "EQU001_0_ax equal &
```
```  2287   (\<forall>Y X. equal(add(X::'a,Y),add(Y::'a,X))) &
```
```  2288   (\<forall>X Y Z. equal(add(add(X::'a,Y),Z),add(X::'a,add(Y::'a,Z)))) &
```
```  2289   (\<forall>Y X. equal(negate(add(negate(add(X::'a,Y)),negate(add(X::'a,negate(Y))))),X)) &
```
```  2290   (\<forall>A B C. equal(A::'a,B) --> equal(add(A::'a,C),add(B::'a,C))) &
```
```  2291   (\<forall>D F' E. equal(D::'a,E) --> equal(add(F'::'a,D),add(F'::'a,E))) &
```
```  2292   (\<forall>G H. equal(G::'a,H) --> equal(negate(G),negate(H))) &
```
```  2293   (equal(negate(add(a::'a,negate(b))),c)) &
```
```  2294   (~equal(negate(add(c::'a,negate(add(b::'a,a)))),a)) --> False"
```
```  2295   oops
```
```  2296
```
```  2297
```
```  2298 (*6933 inferences so far.  Searching to depth 12.  5.1 secs*)
```
```  2299 lemma ROB013_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   (equal(negate(add(a::'a,b)),c)) &
```
```  2308   (~equal(negate(add(c::'a,negate(add(negate(b),a)))),a)) --> False"
```
```  2309   by meson
```
```  2310
```
```  2311 (*6614 inferences so far.  Searching to depth 11.  20.4 secs*)
```
```  2312 lemma ROB016_1:
```
```  2313   "EQU001_0_ax equal &
```
```  2314   (\<forall>Y X. equal(add(X::'a,Y),add(Y::'a,X))) &
```
```  2315   (\<forall>X Y Z. equal(add(add(X::'a,Y),Z),add(X::'a,add(Y::'a,Z)))) &
```
```  2316   (\<forall>Y X. equal(negate(add(negate(add(X::'a,Y)),negate(add(X::'a,negate(Y))))),X)) &
```
```  2317   (\<forall>A B C. equal(A::'a,B) --> equal(add(A::'a,C),add(B::'a,C))) &
```
```  2318   (\<forall>D F' E. equal(D::'a,E) --> equal(add(F'::'a,D),add(F'::'a,E))) &
```
```  2319   (\<forall>G H. equal(G::'a,H) --> equal(negate(G),negate(H))) &
```
```  2320   (\<forall>J K' L. equal(J::'a,K') --> equal(multiply(J::'a,L),multiply(K'::'a,L))) &
```
```  2321   (\<forall>M O' N. equal(M::'a,N) --> equal(multiply(O'::'a,M),multiply(O'::'a,N))) &
```
```  2322   (\<forall>P Q. equal(P::'a,Q) --> equal(successor(P),successor(Q))) &
```
```  2323   (\<forall>R S'. equal(R::'a,S') & positive_integer(R) --> positive_integer(S')) &
```
```  2324   (\<forall>X. equal(multiply(One::'a,X),X)) &
```
```  2325   (\<forall>V X. positive_integer(X) --> equal(multiply(successor(V),X),add(X::'a,multiply(V::'a,X)))) &
```
```  2326   (positive_integer(One)) &
```
```  2327   (\<forall>X. positive_integer(X) --> positive_integer(successor(X))) &
```
```  2328   (equal(negate(add(d::'a,e)),negate(e))) &
```
```  2329   (positive_integer(k)) &
```
```  2330   (\<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))) &
```
```  2331   (~equal(negate(add(e::'a,multiply(k::'a,add(d::'a,negate(add(d::'a,negate(e))))))),negate(e))) --> False"
```
```  2332   oops
```
```  2333
```
```  2334 (*14077 inferences so far.  Searching to depth 11.  32.8 secs*)
```
```  2335 lemma ROB021_1:
```
```  2336   "EQU001_0_ax equal &
```
```  2337   (\<forall>Y X. equal(add(X::'a,Y),add(Y::'a,X))) &
```
```  2338   (\<forall>X Y Z. equal(add(add(X::'a,Y),Z),add(X::'a,add(Y::'a,Z)))) &
```
```  2339   (\<forall>Y X. equal(negate(add(negate(add(X::'a,Y)),negate(add(X::'a,negate(Y))))),X)) &
```
```  2340   (\<forall>A B C. equal(A::'a,B) --> equal(add(A::'a,C),add(B::'a,C))) &
```
```  2341   (\<forall>D F' E. equal(D::'a,E) --> equal(add(F'::'a,D),add(F'::'a,E))) &
```
```  2342   (\<forall>G H. equal(G::'a,H) --> equal(negate(G),negate(H))) &
```
```  2343   (\<forall>X Y. equal(negate(X),negate(Y)) --> equal(X::'a,Y)) &
```
```  2344   (~equal(add(negate(add(a::'a,negate(b))),negate(add(negate(a),negate(b)))),b)) --> False"
```
```  2345   oops
```
```  2346
```
```  2347 (*35532 inferences so far.  Searching to depth 19.  54.3 secs*)
```
```  2348 lemma SET005_1:
```
```  2349  "(\<forall>Subset Element Superset. member(Element::'a,Subset) & subset(Subset::'a,Superset) --> member(Element::'a,Superset)) &
```
```  2350   (\<forall>Superset Subset. subset(Subset::'a,Superset) | member(member_of_1_not_of_2(Subset::'a,Superset),Subset)) &
```
```  2351   (\<forall>Subset Superset. member(member_of_1_not_of_2(Subset::'a,Superset),Superset) --> subset(Subset::'a,Superset)) &
```
```  2352   (\<forall>Subset Superset. equal_sets(Subset::'a,Superset) --> subset(Subset::'a,Superset)) &
```
```  2353   (\<forall>Subset Superset. equal_sets(Superset::'a,Subset) --> subset(Subset::'a,Superset)) &
```
```  2354   (\<forall>Set2 Set1. subset(Set1::'a,Set2) & subset(Set2::'a,Set1) --> equal_sets(Set2::'a,Set1)) &
```
```  2355   (\<forall>Set2 Intersection Element Set1. intersection(Set1::'a,Set2,Intersection) & member(Element::'a,Intersection) --> member(Element::'a,Set1)) &
```
```  2356   (\<forall>Set1 Intersection Element Set2. intersection(Set1::'a,Set2,Intersection) & member(Element::'a,Intersection) --> member(Element::'a,Set2)) &
```
```  2357   (\<forall>Set2 Set1 Element Intersection. intersection(Set1::'a,Set2,Intersection) & member(Element::'a,Set2) & member(Element::'a,Set1) --> member(Element::'a,Intersection)) &
```
```  2358   (\<forall>Set2 Intersection Set1. member(h(Set1::'a,Set2,Intersection),Intersection) | intersection(Set1::'a,Set2,Intersection) | member(h(Set1::'a,Set2,Intersection),Set1)) &
```
```  2359   (\<forall>Set1 Intersection Set2. member(h(Set1::'a,Set2,Intersection),Intersection) | intersection(Set1::'a,Set2,Intersection) | member(h(Set1::'a,Set2,Intersection),Set2)) &
```
```  2360   (\<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)) &
```
```  2361   (intersection(a::'a,b,aIb)) &
```
```  2362   (intersection(b::'a,c,bIc)) &
```
```  2363   (intersection(a::'a,bIc,aIbIc)) &
```
```  2364   (~intersection(aIb::'a,c,aIbIc)) --> False"
```
```  2365   oops
```
```  2366
```
```  2367
```
```  2368 (*6450 inferences so far.  Searching to depth 14.  4.2 secs*)
```
```  2369 lemma SET009_1:
```
```  2370   "(\<forall>Subset Element Superset. member(Element::'a,Subset) & ssubset(Subset::'a,Superset) --> member(Element::'a,Superset)) &
```
```  2371   (\<forall>Superset Subset. ssubset(Subset::'a,Superset) | member(member_of_1_not_of_2(Subset::'a,Superset),Subset)) &
```
```  2372   (\<forall>Subset Superset. member(member_of_1_not_of_2(Subset::'a,Superset),Superset) --> ssubset(Subset::'a,Superset)) &
```
```  2373   (\<forall>Subset Superset. equal_sets(Subset::'a,Superset) --> ssubset(Subset::'a,Superset)) &
```
```  2374   (\<forall>Subset Superset. equal_sets(Superset::'a,Subset) --> ssubset(Subset::'a,Superset)) &
```
```  2375   (\<forall>Set2 Set1. ssubset(Set1::'a,Set2) & ssubset(Set2::'a,Set1) --> equal_sets(Set2::'a,Set1)) &
```
```  2376   (\<forall>Set2 Difference Element Set1. difference(Set1::'a,Set2,Difference) & member(Element::'a,Difference) --> member(Element::'a,Set1)) &
```
```  2377   (\<forall>Element A_set Set1 Set2. ~(member(Element::'a,Set1) & member(Element::'a,Set2) & difference(A_set::'a,Set1,Set2))) &
```
```  2378   (\<forall>Set1 Difference Element Set2. member(Element::'a,Set1) & difference(Set1::'a,Set2,Difference) --> member(Element::'a,Difference) | member(Element::'a,Set2)) &
```
```  2379   (\<forall>Set1 Set2 Difference. difference(Set1::'a,Set2,Difference) | member(k(Set1::'a,Set2,Difference),Set1) | member(k(Set1::'a,Set2,Difference),Difference)) &
```
```  2380   (\<forall>Set1 Set2 Difference. member(k(Set1::'a,Set2,Difference),Set2) --> member(k(Set1::'a,Set2,Difference),Difference) | difference(Set1::'a,Set2,Difference)) &
```
```  2381   (\<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)) &
```
```  2382   (ssubset(d::'a,a)) &
```
```  2383   (difference(b::'a,a,bDa)) &
```
```  2384   (difference(b::'a,d,bDd)) &
```
```  2385   (~ssubset(bDa::'a,bDd)) --> False"
```
```  2386   by meson
```
```  2387
```
```  2388 (*34726 inferences so far.  Searching to depth 6.  2420 secs: 40 mins! BIG*)
```
```  2389 lemma SET025_4:
```
```  2390   "EQU001_0_ax equal &
```
```  2391   (\<forall>Y X. member(X::'a,Y) --> little_set(X)) &
```
```  2392   (\<forall>X Y. little_set(f1(X::'a,Y)) | equal(X::'a,Y)) &
```
```  2393   (\<forall>X Y. member(f1(X::'a,Y),X) | member(f1(X::'a,Y),Y) | equal(X::'a,Y)) &
```
```  2394   (\<forall>X Y. member(f1(X::'a,Y),X) & member(f1(X::'a,Y),Y) --> equal(X::'a,Y)) &
```
```  2395   (\<forall>X U Y. member(U::'a,non_ordered_pair(X::'a,Y)) --> equal(U::'a,X) | equal(U::'a,Y)) &
```
```  2396   (\<forall>Y U X. little_set(U) & equal(U::'a,X) --> member(U::'a,non_ordered_pair(X::'a,Y))) &
```
```  2397   (\<forall>X U Y. little_set(U) & equal(U::'a,Y) --> member(U::'a,non_ordered_pair(X::'a,Y))) &
```
```  2398   (\<forall>X Y. little_set(non_ordered_pair(X::'a,Y))) &
```
```  2399   (\<forall>X. equal(singleton_set(X),non_ordered_pair(X::'a,X))) &
```
```  2400   (\<forall>X Y. equal(ordered_pair(X::'a,Y),non_ordered_pair(singleton_set(X),non_ordered_pair(X::'a,Y)))) &
```
```  2401   (\<forall>X. ordered_pair_predicate(X) --> little_set(f2(X))) &
```
```  2402   (\<forall>X. ordered_pair_predicate(X) --> little_set(f3(X))) &
```
```  2403   (\<forall>X. ordered_pair_predicate(X) --> equal(X::'a,ordered_pair(f2(X),f3(X)))) &
```
```  2404   (\<forall>X Y Z. little_set(Y) & little_set(Z) & equal(X::'a,ordered_pair(Y::'a,Z)) --> ordered_pair_predicate(X)) &
```
```  2405   (\<forall>Z X. member(Z::'a,first(X)) --> little_set(f4(Z::'a,X))) &
```
```  2406   (\<forall>Z X. member(Z::'a,first(X)) --> little_set(f5(Z::'a,X))) &
```
```  2407   (\<forall>Z X. member(Z::'a,first(X)) --> equal(X::'a,ordered_pair(f4(Z::'a,X),f5(Z::'a,X)))) &
```
```  2408   (\<forall>Z X. member(Z::'a,first(X)) --> member(Z::'a,f4(Z::'a,X))) &
```
```  2409   (\<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))) &
```
```  2410   (\<forall>Z X. member(Z::'a,second(X)) --> little_set(f6(Z::'a,X))) &
```
```  2411   (\<forall>Z X. member(Z::'a,second(X)) --> little_set(f7(Z::'a,X))) &
```
```  2412   (\<forall>Z X. member(Z::'a,second(X)) --> equal(X::'a,ordered_pair(f6(Z::'a,X),f7(Z::'a,X)))) &
```
```  2413   (\<forall>Z X. member(Z::'a,second(X)) --> member(Z::'a,f7(Z::'a,X))) &
```
```  2414   (\<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))) &
```
```  2415   (\<forall>Z. member(Z::'a,estin) --> ordered_pair_predicate(Z)) &
```
```  2416   (\<forall>Z. member(Z::'a,estin) --> member(first(Z),second(Z))) &
```
```  2417   (\<forall>Z. little_set(Z) & ordered_pair_predicate(Z) & member(first(Z),second(Z)) --> member(Z::'a,estin)) &
```
```  2418   (\<forall>Y Z X. member(Z::'a,intersection(X::'a,Y)) --> member(Z::'a,X)) &
```
```  2419   (\<forall>X Z Y. member(Z::'a,intersection(X::'a,Y)) --> member(Z::'a,Y)) &
```
```  2420   (\<forall>X Z Y. member(Z::'a,X) & member(Z::'a,Y) --> member(Z::'a,intersection(X::'a,Y))) &
```
```  2421   (\<forall>Z X. ~(member(Z::'a,complement(X)) & member(Z::'a,X))) &
```
```  2422   (\<forall>Z X. little_set(Z) --> member(Z::'a,complement(X)) | member(Z::'a,X)) &
```
```  2423   (\<forall>X Y. equal(union(X::'a,Y),complement(intersection(complement(X),complement(Y))))) &
```
```  2424   (\<forall>Z X. member(Z::'a,domain_of(X)) --> ordered_pair_predicate(f8(Z::'a,X))) &
```
```  2425   (\<forall>Z X. member(Z::'a,domain_of(X)) --> member(f8(Z::'a,X),X)) &
```
```  2426   (\<forall>Z X. member(Z::'a,domain_of(X)) --> equal(Z::'a,first(f8(Z::'a,X)))) &
```
```  2427   (\<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))) &
```
```  2428   (\<forall>X Y Z. member(Z::'a,cross_product(X::'a,Y)) --> ordered_pair_predicate(Z)) &
```
```  2429   (\<forall>Y Z X. member(Z::'a,cross_product(X::'a,Y)) --> member(first(Z),X)) &
```
```  2430   (\<forall>X Z Y. member(Z::'a,cross_product(X::'a,Y)) --> member(second(Z),Y)) &
```
```  2431   (\<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))) &
```
```  2432   (\<forall>X Z. member(Z::'a,inv1 X) --> ordered_pair_predicate(Z)) &
```
```  2433   (\<forall>Z X. member(Z::'a,inv1 X) --> member(ordered_pair(second(Z),first(Z)),X)) &
```
```  2434   (\<forall>Z X. little_set(Z) & ordered_pair_predicate(Z) & member(ordered_pair(second(Z),first(Z)),X) --> member(Z::'a,inv1 X)) &
```
```  2435   (\<forall>Z X. member(Z::'a,rot_right(X)) --> little_set(f9(Z::'a,X))) &
```
```  2436   (\<forall>Z X. member(Z::'a,rot_right(X)) --> little_set(f10(Z::'a,X))) &
```
```  2437   (\<forall>Z X. member(Z::'a,rot_right(X)) --> little_set(f11(Z::'a,X))) &
```
```  2438   (\<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))))) &
```
```  2439   (\<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)) &
```
```  2440   (\<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))) &
```
```  2441   (\<forall>Z X. member(Z::'a,flip_range_of(X)) --> little_set(f12(Z::'a,X))) &
```
```  2442   (\<forall>Z X. member(Z::'a,flip_range_of(X)) --> little_set(f13(Z::'a,X))) &
```
```  2443   (\<forall>Z X. member(Z::'a,flip_range_of(X)) --> little_set(f14(Z::'a,X))) &
```
```  2444   (\<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))))) &
```
```  2445   (\<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)) &
```
```  2446   (\<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))) &
```
```  2447   (\<forall>X. equal(successor(X),union(X::'a,singleton_set(X)))) &
```
```  2448   (\<forall>Z. ~member(Z::'a,empty_set)) &
```
```  2449   (\<forall>Z. little_set(Z) --> member(Z::'a,universal_set)) &
```
```  2450   (little_set(infinity)) &
```
```  2451   (member(empty_set::'a,infinity)) &
```
```  2452   (\<forall>X. member(X::'a,infinity) --> member(successor(X),infinity)) &
```
```  2453   (\<forall>Z X. member(Z::'a,sigma(X)) --> member(f16(Z::'a,X),X)) &
```
```  2454   (\<forall>Z X. member(Z::'a,sigma(X)) --> member(Z::'a,f16(Z::'a,X))) &
```
```  2455   (\<forall>X Z Y. member(Y::'a,X) & member(Z::'a,Y) --> member(Z::'a,sigma(X))) &
```
```  2456   (\<forall>U. little_set(U) --> little_set(sigma(U))) &
```
```  2457   (\<forall>X U Y. ssubset(X::'a,Y) & member(U::'a,X) --> member(U::'a,Y)) &
```
```  2458   (\<forall>Y X. ssubset(X::'a,Y) | member(f17(X::'a,Y),X)) &
```
```  2459   (\<forall>X Y. member(f17(X::'a,Y),Y) --> ssubset(X::'a,Y)) &
```
```  2460   (\<forall>X Y. proper_subset(X::'a,Y) --> ssubset(X::'a,Y)) &
```
```  2461   (\<forall>X Y. ~(proper_subset(X::'a,Y) & equal(X::'a,Y))) &
```
```  2462   (\<forall>X Y. ssubset(X::'a,Y) --> proper_subset(X::'a,Y) | equal(X::'a,Y)) &
```
```  2463   (\<forall>Z X. member(Z::'a,powerset(X)) --> ssubset(Z::'a,X)) &
```
```  2464   (\<forall>Z X. little_set(Z) & ssubset(Z::'a,X) --> member(Z::'a,powerset(X))) &
```
```  2465   (\<forall>U. little_set(U) --> little_set(powerset(U))) &
```
```  2466   (\<forall>Z X. relation(Z) & member(X::'a,Z) --> ordered_pair_predicate(X)) &
```
```  2467   (\<forall>Z. relation(Z) | member(f18(Z),Z)) &
```
```  2468   (\<forall>Z. ordered_pair_predicate(f18(Z)) --> relation(Z)) &
```
```  2469   (\<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)) &
```
```  2470   (\<forall>X. single_valued_set(X) | little_set(f19(X))) &
```
```  2471   (\<forall>X. single_valued_set(X) | little_set(f20(X))) &
```
```  2472   (\<forall>X. single_valued_set(X) | little_set(f21(X))) &
```
```  2473   (\<forall>X. single_valued_set(X) | member(ordered_pair(f19(X),f20(X)),X)) &
```
```  2474   (\<forall>X. single_valued_set(X) | member(ordered_pair(f19(X),f21(X)),X)) &
```
```  2475   (\<forall>X. equal(f20(X),f21(X)) --> single_valued_set(X)) &
```
```  2476   (\<forall>Xf. function(Xf) --> relation(Xf)) &
```
```  2477   (\<forall>Xf. function(Xf) --> single_valued_set(Xf)) &
```
```  2478   (\<forall>Xf. relation(Xf) & single_valued_set(Xf) --> function(Xf)) &
```
```  2479   (\<forall>Z X Xf. member(Z::'a,image'(X::'a,Xf)) --> ordered_pair_predicate(f22(Z::'a,X,Xf))) &
```
```  2480   (\<forall>Z X Xf. member(Z::'a,image'(X::'a,Xf)) --> member(f22(Z::'a,X,Xf),Xf)) &
```
```  2481   (\<forall>Z Xf X. member(Z::'a,image'(X::'a,Xf)) --> member(first(f22(Z::'a,X,Xf)),X)) &
```
```  2482   (\<forall>X Xf Z. member(Z::'a,image'(X::'a,Xf)) --> equal(second(f22(Z::'a,X,Xf)),Z)) &
```
```  2483   (\<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))) &
```
```  2484   (\<forall>X Xf. little_set(X) & function(Xf) --> little_set(image'(X::'a,Xf))) &
```
```  2485   (\<forall>X U Y. ~(disjoint(X::'a,Y) & member(U::'a,X) & member(U::'a,Y))) &
```
```  2486   (\<forall>Y X. disjoint(X::'a,Y) | member(f23(X::'a,Y),X)) &
```
```  2487   (\<forall>X Y. disjoint(X::'a,Y) | member(f23(X::'a,Y),Y)) &
```
```  2488   (\<forall>X. equal(X::'a,empty_set) | member(f24(X),X)) &
```
```  2489   (\<forall>X. equal(X::'a,empty_set) | disjoint(f24(X),X)) &
```
```  2490   (function(f25)) &
```
```  2491   (\<forall>X. little_set(X) --> equal(X::'a,empty_set) | member(f26(X),X)) &
```
```  2492   (\<forall>X. little_set(X) --> equal(X::'a,empty_set) | member(ordered_pair(X::'a,f26(X)),f25)) &
```
```  2493   (\<forall>Z X. member(Z::'a,range_of(X)) --> ordered_pair_predicate(f27(Z::'a,X))) &
```
```  2494   (\<forall>Z X. member(Z::'a,range_of(X)) --> member(f27(Z::'a,X),X)) &
```
```  2495   (\<forall>Z X. member(Z::'a,range_of(X)) --> equal(Z::'a,second(f27(Z::'a,X)))) &
```
```  2496   (\<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))) &
```
```  2497   (\<forall>Z. member(Z::'a,identity_relation) --> ordered_pair_predicate(Z)) &
```
```  2498   (\<forall>Z. member(Z::'a,identity_relation) --> equal(first(Z),second(Z))) &
```
```  2499   (\<forall>Z. little_set(Z) & ordered_pair_predicate(Z) & equal(first(Z),second(Z)) --> member(Z::'a,identity_relation)) &
```
```  2500   (\<forall>X Y. equal(restrct(X::'a,Y),intersection(X::'a,cross_product(Y::'a,universal_set)))) &
```
```  2501   (\<forall>Xf. one_to_one_function(Xf) --> function(Xf)) &
```
```  2502   (\<forall>Xf. one_to_one_function(Xf) --> function(inv1 Xf)) &
```
```  2503   (\<forall>Xf. function(Xf) & function(inv1 Xf) --> one_to_one_function(Xf)) &
```
```  2504   (\<forall>Z Xf Y. member(Z::'a,apply(Xf::'a,Y)) --> ordered_pair_predicate(f28(Z::'a,Xf,Y))) &
```
```  2505   (\<forall>Z Y Xf. member(Z::'a,apply(Xf::'a,Y)) --> member(f28(Z::'a,Xf,Y),Xf)) &
```
```  2506   (\<forall>Z Xf Y. member(Z::'a,apply(Xf::'a,Y)) --> equal(first(f28(Z::'a,Xf,Y)),Y)) &
```
```  2507   (\<forall>Z Xf Y. member(Z::'a,apply(Xf::'a,Y)) --> member(Z::'a,second(f28(Z::'a,Xf,Y)))) &
```
```  2508   (\<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))) &
```
```  2509   (\<forall>Xf X Y. equal(apply_to_two_arguments(Xf::'a,X,Y),apply(Xf::'a,ordered_pair(X::'a,Y)))) &
```
```  2510   (\<forall>X Y Xf. maps(Xf::'a,X,Y) --> function(Xf)) &
```
```  2511   (\<forall>Y Xf X. maps(Xf::'a,X,Y) --> equal(domain_of(Xf),X)) &
```
```  2512   (\<forall>X Xf Y. maps(Xf::'a,X,Y) --> ssubset(range_of(Xf),Y)) &
```
```  2513   (\<forall>X Xf Y. function(Xf) & equal(domain_of(Xf),X) & ssubset(range_of(Xf),Y) --> maps(Xf::'a,X,Y)) &
```
```  2514   (\<forall>Xf Xs. closed(Xs::'a,Xf) --> little_set(Xs)) &
```
```  2515   (\<forall>Xs Xf. closed(Xs::'a,Xf) --> little_set(Xf)) &
```
```  2516   (\<forall>Xf Xs. closed(Xs::'a,Xf) --> maps(Xf::'a,cross_product(Xs::'a,Xs),Xs)) &
```
```  2517   (\<forall>Xf Xs. little_set(Xs) & little_set(Xf) & maps(Xf::'a,cross_product(Xs::'a,Xs),Xs) --> closed(Xs::'a,Xf)) &
```
```  2518   (\<forall>Z Xf Xg. member(Z::'a,composition(Xf::'a,Xg)) --> little_set(f29(Z::'a,Xf,Xg))) &
```
```  2519   (\<forall>Z Xf Xg. member(Z::'a,composition(Xf::'a,Xg)) --> little_set(f30(Z::'a,Xf,Xg))) &
```
```  2520   (\<forall>Z Xf Xg. member(Z::'a,composition(Xf::'a,Xg)) --> little_set(f31(Z::'a,Xf,Xg))) &
```
```  2521   (\<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)))) &
```
```  2522   (\<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)) &
```
```  2523   (\<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)) &
```
```  2524   (\<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))) &
```
```  2525   (\<forall>Xh Xs2 Xf2 Xs1 Xf1. homomorphism(Xh::'a,Xs1,Xf1,Xs2,Xf2) --> closed(Xs1::'a,Xf1)) &
```
```  2526   (\<forall>Xh Xs1 Xf1 Xs2 Xf2. homomorphism(Xh::'a,Xs1,Xf1,Xs2,Xf2) --> closed(Xs2::'a,Xf2)) &
```
```  2527   (\<forall>Xf1 Xf2 Xh Xs1 Xs2. homomorphism(Xh::'a,Xs1,Xf1,Xs2,Xf2) --> maps(Xh::'a,Xs1,Xs2)) &
```
```  2528   (\<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)))) &
```
```  2529   (\<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)) &
```
```  2530   (\<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)) &
```
```  2531   (\<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)) &
```
```  2532   (\<forall>A B C. equal(A::'a,B) --> equal(f1(A::'a,C),f1(B::'a,C))) &
```
```  2533   (\<forall>D F' E. equal(D::'a,E) --> equal(f1(F'::'a,D),f1(F'::'a,E))) &
```
```  2534   (\<forall>A2 B2. equal(A2::'a,B2) --> equal(f2(A2),f2(B2))) &
```
```  2535   (\<forall>G4 H4. equal(G4::'a,H4) --> equal(f3(G4),f3(H4))) &
```
```  2536   (\<forall>O7 P7 Q7. equal(O7::'a,P7) --> equal(f4(O7::'a,Q7),f4(P7::'a,Q7))) &
```
```  2537   (\<forall>R7 T7 S7. equal(R7::'a,S7) --> equal(f4(T7::'a,R7),f4(T7::'a,S7))) &
```
```  2538   (\<forall>U7 V7 W7. equal(U7::'a,V7) --> equal(f5(U7::'a,W7),f5(V7::'a,W7))) &
```
```  2539   (\<forall>X7 Z7 Y7. equal(X7::'a,Y7) --> equal(f5(Z7::'a,X7),f5(Z7::'a,Y7))) &
```
```  2540   (\<forall>A8 B8 C8. equal(A8::'a,B8) --> equal(f6(A8::'a,C8),f6(B8::'a,C8))) &
```
```  2541   (\<forall>D8 F8 E8. equal(D8::'a,E8) --> equal(f6(F8::'a,D8),f6(F8::'a,E8))) &
```
```  2542   (\<forall>G8 H8 I8. equal(G8::'a,H8) --> equal(f7(G8::'a,I8),f7(H8::'a,I8))) &
```
```  2543   (\<forall>J8 L8 K8. equal(J8::'a,K8) --> equal(f7(L8::'a,J8),f7(L8::'a,K8))) &
```
```  2544   (\<forall>M8 N8 O8. equal(M8::'a,N8) --> equal(f8(M8::'a,O8),f8(N8::'a,O8))) &
```
```  2545   (\<forall>P8 R8 Q8. equal(P8::'a,Q8) --> equal(f8(R8::'a,P8),f8(R8::'a,Q8))) &
```
```  2546   (\<forall>S8 T8 U8. equal(S8::'a,T8) --> equal(f9(S8::'a,U8),f9(T8::'a,U8))) &
```
```  2547   (\<forall>V8 X8 W8. equal(V8::'a,W8) --> equal(f9(X8::'a,V8),f9(X8::'a,W8))) &
```
```  2548   (\<forall>G H I'. equal(G::'a,H) --> equal(f10(G::'a,I'),f10(H::'a,I'))) &
```
```  2549   (\<forall>J L K'. equal(J::'a,K') --> equal(f10(L::'a,J),f10(L::'a,K'))) &
```
```  2550   (\<forall>M N O'. equal(M::'a,N) --> equal(f11(M::'a,O'),f11(N::'a,O'))) &
```
```  2551   (\<forall>P R Q. equal(P::'a,Q) --> equal(f11(R::'a,P),f11(R::'a,Q))) &
```
```  2552   (\<forall>S' T' U. equal(S'::'a,T') --> equal(f12(S'::'a,U),f12(T'::'a,U))) &
```
```  2553   (\<forall>V X W. equal(V::'a,W) --> equal(f12(X::'a,V),f12(X::'a,W))) &
```
```  2554   (\<forall>Y Z A1. equal(Y::'a,Z) --> equal(f13(Y::'a,A1),f13(Z::'a,A1))) &
```
```  2555   (\<forall>B1 D1 C1. equal(B1::'a,C1) --> equal(f13(D1::'a,B1),f13(D1::'a,C1))) &
```
```  2556   (\<forall>E1 F1 G1. equal(E1::'a,F1) --> equal(f14(E1::'a,G1),f14(F1::'a,G1))) &
```
```  2557   (\<forall>H1 J1 I1. equal(H1::'a,I1) --> equal(f14(J1::'a,H1),f14(J1::'a,I1))) &
```
```  2558   (\<forall>K1 L1 M1. equal(K1::'a,L1) --> equal(f16(K1::'a,M1),f16(L1::'a,M1))) &
```
```  2559   (\<forall>N1 P1 O1. equal(N1::'a,O1) --> equal(f16(P1::'a,N1),f16(P1::'a,O1))) &
```
```  2560   (\<forall>Q1 R1 S1. equal(Q1::'a,R1) --> equal(f17(Q1::'a,S1),f17(R1::'a,S1))) &
```
```  2561   (\<forall>T1 V1 U1. equal(T1::'a,U1) --> equal(f17(V1::'a,T1),f17(V1::'a,U1))) &
```
```  2562   (\<forall>W1 X1. equal(W1::'a,X1) --> equal(f18(W1),f18(X1))) &
```
```  2563   (\<forall>Y1 Z1. equal(Y1::'a,Z1) --> equal(f19(Y1),f19(Z1))) &
```
```  2564   (\<forall>C2 D2. equal(C2::'a,D2) --> equal(f20(C2),f20(D2))) &
```
```  2565   (\<forall>E2 F2. equal(E2::'a,F2) --> equal(f21(E2),f21(F2))) &
```
```  2566   (\<forall>G2 H2 I2 J2. equal(G2::'a,H2) --> equal(f22(G2::'a,I2,J2),f22(H2::'a,I2,J2))) &
```
```  2567   (\<forall>K2 M2 L2 N2. equal(K2::'a,L2) --> equal(f22(M2::'a,K2,N2),f22(M2::'a,L2,N2))) &
```
```  2568   (\<forall>O2 Q2 R2 P2. equal(O2::'a,P2) --> equal(f22(Q2::'a,R2,O2),f22(Q2::'a,R2,P2))) &
```
```  2569   (\<forall>S2 T2 U2. equal(S2::'a,T2) --> equal(f23(S2::'a,U2),f23(T2::'a,U2))) &
```
```  2570   (\<forall>V2 X2 W2. equal(V2::'a,W2) --> equal(f23(X2::'a,V2),f23(X2::'a,W2))) &
```
```  2571   (\<forall>Y2 Z2. equal(Y2::'a,Z2) --> equal(f24(Y2),f24(Z2))) &
```
```  2572   (\<forall>A3 B3. equal(A3::'a,B3) --> equal(f26(A3),f26(B3))) &
```
```  2573   (\<forall>C3 D3 E3. equal(C3::'a,D3) --> equal(f27(C3::'a,E3),f27(D3::'a,E3))) &
```
```  2574   (\<forall>F3 H3 G3. equal(F3::'a,G3) --> equal(f27(H3::'a,F3),f27(H3::'a,G3))) &
```
```  2575   (\<forall>I3 J3 K3 L3. equal(I3::'a,J3) --> equal(f28(I3::'a,K3,L3),f28(J3::'a,K3,L3))) &
```
```  2576   (\<forall>M3 O3 N3 P3. equal(M3::'a,N3) --> equal(f28(O3::'a,M3,P3),f28(O3::'a,N3,P3))) &
```
```  2577   (\<forall>Q3 S3 T3 R3. equal(Q3::'a,R3) --> equal(f28(S3::'a,T3,Q3),f28(S3::'a,T3,R3))) &
```
```  2578   (\<forall>U3 V3 W3 X3. equal(U3::'a,V3) --> equal(f29(U3::'a,W3,X3),f29(V3::'a,W3,X3))) &
```
```  2579   (\<forall>Y3 A4 Z3 B4. equal(Y3::'a,Z3) --> equal(f29(A4::'a,Y3,B4),f29(A4::'a,Z3,B4))) &
```
```  2580   (\<forall>C4 E4 F4 D4. equal(C4::'a,D4) --> equal(f29(E4::'a,F4,C4),f29(E4::'a,F4,D4))) &
```
```  2581   (\<forall>I4 J4 K4 L4. equal(I4::'a,J4) --> equal(f30(I4::'a,K4,L4),f30(J4::'a,K4,L4))) &
```
```  2582   (\<forall>M4 O4 N4 P4. equal(M4::'a,N4) --> equal(f30(O4::'a,M4,P4),f30(O4::'a,N4,P4))) &
```
```  2583   (\<forall>Q4 S4 T4 R4. equal(Q4::'a,R4) --> equal(f30(S4::'a,T4,Q4),f30(S4::'a,T4,R4))) &
```
```  2584   (\<forall>U4 V4 W4 X4. equal(U4::'a,V4) --> equal(f31(U4::'a,W4,X4),f31(V4::'a,W4,X4))) &
```
```  2585   (\<forall>Y4 A5 Z4 B5. equal(Y4::'a,Z4) --> equal(f31(A5::'a,Y4,B5),f31(A5::'a,Z4,B5))) &
```
```  2586   (\<forall>C5 E5 F5 D5. equal(C5::'a,D5) --> equal(f31(E5::'a,F5,C5),f31(E5::'a,F5,D5))) &
```
```  2587   (\<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))) &
```
```  2588   (\<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))) &
```
```  2589   (\<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))) &
```
```  2590   (\<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))) &
```
```  2591   (\<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))) &
```
```  2592   (\<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))) &
```
```  2593   (\<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))) &
```
```  2594   (\<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))) &
```
```  2595   (\<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))) &
```
```  2596   (\<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))) &
```
```  2597   (\<forall>A B C. equal(A::'a,B) --> equal(apply(A::'a,C),apply(B::'a,C))) &
```
```  2598   (\<forall>D F' E. equal(D::'a,E) --> equal(apply(F'::'a,D),apply(F'::'a,E))) &
```
```  2599   (\<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))) &
```
```  2600   (\<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))) &
```
```  2601   (\<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))) &
```
```  2602   (\<forall>S' T'. equal(S'::'a,T') --> equal(complement(S'),complement(T'))) &
```
```  2603   (\<forall>U V W. equal(U::'a,V) --> equal(composition(U::'a,W),composition(V::'a,W))) &
```
```  2604   (\<forall>X Z Y. equal(X::'a,Y) --> equal(composition(Z::'a,X),composition(Z::'a,Y))) &
```
```  2605   (\<forall>A1 B1. equal(A1::'a,B1) --> equal(inv1 A1,inv1 B1)) &
```
```  2606   (\<forall>C1 D1 E1. equal(C1::'a,D1) --> equal(cross_product(C1::'a,E1),cross_product(D1::'a,E1))) &
```
```  2607   (\<forall>F1 H1 G1. equal(F1::'a,G1) --> equal(cross_product(H1::'a,F1),cross_product(H1::'a,G1))) &
```
```  2608   (\<forall>I1 J1. equal(I1::'a,J1) --> equal(domain_of(I1),domain_of(J1))) &
```
```  2609   (\<forall>I10 J10. equal(I10::'a,J10) --> equal(first(I10),first(J10))) &
```
```  2610   (\<forall>Q10 R10. equal(Q10::'a,R10) --> equal(flip_range_of(Q10),flip_range_of(R10))) &
```
```  2611   (\<forall>S10 T10 U10. equal(S10::'a,T10) --> equal(image'(S10::'a,U10),image'(T10::'a,U10))) &
```
```  2612   (\<forall>V10 X10 W10. equal(V10::'a,W10) --> equal(image'(X10::'a,V10),image'(X10::'a,W10))) &
```
```  2613   (\<forall>Y10 Z10 A11. equal(Y10::'a,Z10) --> equal(intersection(Y10::'a,A11),intersection(Z10::'a,A11))) &
```
```  2614   (\<forall>B11 D11 C11. equal(B11::'a,C11) --> equal(intersection(D11::'a,B11),intersection(D11::'a,C11))) &
```
```  2615   (\<forall>E11 F11 G11. equal(E11::'a,F11) --> equal(non_ordered_pair(E11::'a,G11),non_ordered_pair(F11::'a,G11))) &
```
```  2616   (\<forall>H11 J11 I11. equal(H11::'a,I11) --> equal(non_ordered_pair(J11::'a,H11),non_ordered_pair(J11::'a,I11))) &
```
```  2617   (\<forall>K11 L11 M11. equal(K11::'a,L11) --> equal(ordered_pair(K11::'a,M11),ordered_pair(L11::'a,M11))) &
```
```  2618   (\<forall>N11 P11 O11. equal(N11::'a,O11) --> equal(ordered_pair(P11::'a,N11),ordered_pair(P11::'a,O11))) &
```
```  2619   (\<forall>Q11 R11. equal(Q11::'a,R11) --> equal(powerset(Q11),powerset(R11))) &
```
```  2620   (\<forall>S11 T11. equal(S11::'a,T11) --> equal(range_of(S11),range_of(T11))) &
```
```  2621   (\<forall>U11 V11 W11. equal(U11::'a,V11) --> equal(restrct(U11::'a,W11),restrct(V11::'a,W11))) &
```
```  2622   (\<forall>X11 Z11 Y11. equal(X11::'a,Y11) --> equal(restrct(Z11::'a,X11),restrct(Z11::'a,Y11))) &
```
```  2623   (\<forall>A12 B12. equal(A12::'a,B12) --> equal(rot_right(A12),rot_right(B12))) &
```
```  2624   (\<forall>C12 D12. equal(C12::'a,D12) --> equal(second(C12),second(D12))) &
```
```  2625   (\<forall>K12 L12. equal(K12::'a,L12) --> equal(sigma(K12),sigma(L12))) &
```
```  2626   (\<forall>M12 N12. equal(M12::'a,N12) --> equal(singleton_set(M12),singleton_set(N12))) &
```
```  2627   (\<forall>O12 P12. equal(O12::'a,P12) --> equal(successor(O12),successor(P12))) &
```
```  2628   (\<forall>Q12 R12 S12. equal(Q12::'a,R12) --> equal(union(Q12::'a,S12),union(R12::'a,S12))) &
```
```  2629   (\<forall>T12 V12 U12. equal(T12::'a,U12) --> equal(union(V12::'a,T12),union(V12::'a,U12))) &
```
```  2630   (\<forall>W12 X12 Y12. equal(W12::'a,X12) & closed(W12::'a,Y12) --> closed(X12::'a,Y12)) &
```
```  2631   (\<forall>Z12 B13 A13. equal(Z12::'a,A13) & closed(B13::'a,Z12) --> closed(B13::'a,A13)) &
```
```  2632   (\<forall>C13 D13 E13. equal(C13::'a,D13) & disjoint(C13::'a,E13) --> disjoint(D13::'a,E13)) &
```
```  2633   (\<forall>F13 H13 G13. equal(F13::'a,G13) & disjoint(H13::'a,F13) --> disjoint(H13::'a,G13)) &
```
```  2634   (\<forall>I13 J13. equal(I13::'a,J13) & function(I13) --> function(J13)) &
```
```  2635   (\<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)) &
```
```  2636   (\<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)) &
```
```  2637   (\<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)) &
```
```  2638   (\<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)) &
```
```  2639   (\<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)) &
```
```  2640   (\<forall>O14 P14. equal(O14::'a,P14) & little_set(O14) --> little_set(P14)) &
```
```  2641   (\<forall>Q14 R14 S14 T14. equal(Q14::'a,R14) & maps(Q14::'a,S14,T14) --> maps(R14::'a,S14,T14)) &
```
```  2642   (\<forall>U14 W14 V14 X14. equal(U14::'a,V14) & maps(W14::'a,U14,X14) --> maps(W14::'a,V14,X14)) &
```
```  2643   (\<forall>Y14 A15 B15 Z14. equal(Y14::'a,Z14) & maps(A15::'a,B15,Y14) --> maps(A15::'a,B15,Z14)) &
```
```  2644   (\<forall>C15 D15 E15. equal(C15::'a,D15) & member(C15::'a,E15) --> member(D15::'a,E15)) &
```
```  2645   (\<forall>F15 H15 G15. equal(F15::'a,G15) & member(H15::'a,F15) --> member(H15::'a,G15)) &
```
```  2646   (\<forall>I15 J15. equal(I15::'a,J15) & one_to_one_function(I15) --> one_to_one_function(J15)) &
```
```  2647   (\<forall>K15 L15. equal(K15::'a,L15) & ordered_pair_predicate(K15) --> ordered_pair_predicate(L15)) &
```
```  2648   (\<forall>M15 N15 O15. equal(M15::'a,N15) & proper_subset(M15::'a,O15) --> proper_subset(N15::'a,O15)) &
```
```  2649   (\<forall>P15 R15 Q15. equal(P15::'a,Q15) & proper_subset(R15::'a,P15) --> proper_subset(R15::'a,Q15)) &
```
```  2650   (\<forall>S15 T15. equal(S15::'a,T15) & relation(S15) --> relation(T15)) &
```
```  2651   (\<forall>U15 V15. equal(U15::'a,V15) & single_valued_set(U15) --> single_valued_set(V15)) &
```
```  2652   (\<forall>W15 X15 Y15. equal(W15::'a,X15) & ssubset(W15::'a,Y15) --> ssubset(X15::'a,Y15)) &
```
```  2653   (\<forall>Z15 B16 A16. equal(Z15::'a,A16) & ssubset(B16::'a,Z15) --> ssubset(B16::'a,A16)) &
```
```  2654   (~little_set(ordered_pair(a::'a,b))) --> False"
```
```  2655   oops
```
```  2656
```
```  2657
```
```  2658 (*13 inferences so far.  Searching to depth 8.  0 secs*)
```
```  2659 lemma SET046_5:
```
```  2660  "(\<forall>Y X. ~(element(X::'a,a) & element(X::'a,Y) & element(Y::'a,X))) &
```
```  2661   (\<forall>X. element(X::'a,f(X)) | element(X::'a,a)) &
```
```  2662   (\<forall>X. element(f(X),X) | element(X::'a,a)) --> False"
```
```  2663    by meson
```
```  2664
```
```  2665 (*33 inferences so far.  Searching to depth 9.  0.2 secs*)
```
```  2666 lemma SET047_5:
```
```  2667  "(\<forall>X Z Y. set_equal(X::'a,Y) & element(Z::'a,X) --> element(Z::'a,Y)) &
```
```  2668   (\<forall>Y Z X. set_equal(X::'a,Y) & element(Z::'a,Y) --> element(Z::'a,X)) &
```
```  2669   (\<forall>X Y. element(f(X::'a,Y),X) | element(f(X::'a,Y),Y) | set_equal(X::'a,Y)) &
```
```  2670   (\<forall>X Y. element(f(X::'a,Y),Y) & element(f(X::'a,Y),X) --> set_equal(X::'a,Y)) &
```
```  2671   (set_equal(a::'a,b) | set_equal(b::'a,a)) &
```
```  2672   (~(set_equal(b::'a,a) & set_equal(a::'a,b))) --> False"
```
```  2673   by meson
```
```  2674
```
```  2675 (*311 inferences so far.  Searching to depth 12.  0.1 secs*)
```
```  2676 lemma SYN034_1:
```
```  2677  "(\<forall>A. p(A::'a,a) | p(A::'a,f(A))) &
```
```  2678   (\<forall>A. p(A::'a,a) | p(f(A),A)) &
```
```  2679   (\<forall>A B. ~(p(A::'a,B) & p(B::'a,A) & p(B::'a,a))) --> False"
```
```  2680   by meson
```
```  2681
```
```  2682 (*30 inferences so far.  Searching to depth 6.  0.2 secs*)
```
```  2683 lemma SYN071_1:
```
```  2684   "EQU001_0_ax equal &
```
```  2685   (equal(a::'a,b) | equal(c::'a,d)) &
```
```  2686   (equal(a::'a,c) | equal(b::'a,d)) &
```
```  2687   (~equal(a::'a,d)) &
```
```  2688   (~equal(b::'a,c)) --> False"
```
```  2689   by meson
```
```  2690
```
```  2691 (*1897410 inferences so far.  Searching to depth 48
```
```  2692   206s, nearly 4 mins on griffon.*)
```
```  2693 lemma SYN349_1:
```
```  2694  "(\<forall>X Y. f(w(X),g(X::'a,Y)) --> f(X::'a,g(X::'a,Y))) &
```
```  2695   (\<forall>X Y. f(X::'a,g(X::'a,Y)) --> f(w(X),g(X::'a,Y))) &
```
```  2696   (\<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))) &
```
```  2697   (\<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))) &
```
```  2698   (\<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))) &
```
```  2699   (\<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))) &
```
```  2700   (\<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))) &
```
```  2701   (\<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))) &
```
```  2702   (\<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)) &
```
```  2703   (\<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"
```
```  2704    oops
```
```  2705
```
```  2706 (*398 inferences so far.  Searching to depth 12.  0.4 secs*)
```
```  2707 lemma SYN352_1:
```
```  2708  "(f(a::'a,b)) &
```
```  2709   (\<forall>X Y. f(X::'a,Y) --> f(b::'a,z(X::'a,Y)) | f(Y::'a,z(X::'a,Y))) &
```
```  2710   (\<forall>X Y. f(X::'a,Y) | f(z(X::'a,Y),z(X::'a,Y))) &
```
```  2711   (\<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))) &
```
```  2712   (\<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))) &
```
```  2713   (\<forall>X Y. ~(f(X::'a,Y) & f(X::'a,z(X::'a,Y)) & f(Y::'a,z(X::'a,Y)))) &
```
```  2714   (\<forall>X Y. f(X::'a,Y) --> f(X::'a,z(X::'a,Y)) | f(Y::'a,z(X::'a,Y))) --> False"
```
```  2715   by meson
```
```  2716
```
```  2717 (*5336 inferences so far.  Searching to depth 15.  5.3 secs*)
```
```  2718 lemma TOP001_2:
```
```  2719  "(\<forall>Vf U. element_of_set(U::'a,union_of_members(Vf)) --> element_of_set(U::'a,f1(Vf::'a,U))) &
```
```  2720   (\<forall>U Vf. element_of_set(U::'a,union_of_members(Vf)) --> element_of_collection(f1(Vf::'a,U),Vf)) &
```
```  2721   (\<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))) &
```
```  2722   (\<forall>Vf X. basis(X::'a,Vf) --> equal_sets(union_of_members(Vf),X)) &
```
```  2723   (\<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))) &
```
```  2724   (\<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)) &
```
```  2725   (\<forall>X. subset_sets(X::'a,X)) &
```
```  2726   (\<forall>X U Y. subset_sets(X::'a,Y) & element_of_set(U::'a,X) --> element_of_set(U::'a,Y)) &
```
```  2727   (\<forall>X Y. equal_sets(X::'a,Y) --> subset_sets(X::'a,Y)) &
```
```  2728   (\<forall>Y X. subset_sets(X::'a,Y) | element_of_set(in_1st_set(X::'a,Y),X)) &
```
```  2729   (\<forall>X Y. element_of_set(in_1st_set(X::'a,Y),Y) --> subset_sets(X::'a,Y)) &
```
```  2730   (basis(cx::'a,f)) &
```
```  2731   (~subset_sets(union_of_members(top_of_basis(f)),cx)) --> False"
```
```  2732   by meson
```
```  2733
```
```  2734 (*0 inferences so far.  Searching to depth 0.  0 secs*)
```
```  2735 lemma TOP002_2:
```
```  2736  "(\<forall>Vf U. element_of_collection(U::'a,top_of_basis(Vf)) | element_of_set(f11(Vf::'a,U),U)) &
```
```  2737   (\<forall>X. ~element_of_set(X::'a,empty_set)) &
```
```  2738   (~element_of_collection(empty_set::'a,top_of_basis(f))) --> False"
```
```  2739   by meson
```
```  2740
```
```  2741 (*0 inferences so far.  Searching to depth 0.  6.5 secs.  BIG*)
```
```  2742 lemma TOP004_1:
```
```  2743  "(\<forall>Vf U. element_of_set(U::'a,union_of_members(Vf)) --> element_of_set(U::'a,f1(Vf::'a,U))) &
```
```  2744   (\<forall>U Vf. element_of_set(U::'a,union_of_members(Vf)) --> element_of_collection(f1(Vf::'a,U),Vf)) &
```
```  2745   (\<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))) &
```
```  2746   (\<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)) &
```
```  2747   (\<forall>U Vf. element_of_set(U::'a,intersection_of_members(Vf)) | element_of_collection(f2(Vf::'a,U),Vf)) &
```
```  2748   (\<forall>Vf U. element_of_set(U::'a,f2(Vf::'a,U)) --> element_of_set(U::'a,intersection_of_members(Vf))) &
```
```  2749   (\<forall>Vt X. topological_space(X::'a,Vt) --> equal_sets(union_of_members(Vt),X)) &
```
```  2750   (\<forall>X Vt. topological_space(X::'a,Vt) --> element_of_collection(empty_set::'a,Vt)) &
```
```  2751   (\<forall>X Vt. topological_space(X::'a,Vt) --> element_of_collection(X::'a,Vt)) &
```
```  2752   (\<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)) &
```
```  2753   (\<forall>X Vf Vt. topological_space(X::'a,Vt) & subset_collections(Vf::'a,Vt) --> element_of_collection(union_of_members(Vf),Vt)) &
```
```  2754   (\<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)) &
```
```  2755   (\<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)) &
```
```  2756   (\<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)) &
```
```  2757   (\<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)) &
```
```  2758   (\<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)) &
```
```  2759   (\<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)) &
```
```  2760   (\<forall>U X Vt. open(U::'a,X,Vt) --> topological_space(X::'a,Vt)) &
```
```  2761   (\<forall>X U Vt. open(U::'a,X,Vt) --> element_of_collection(U::'a,Vt)) &
```
```  2762   (\<forall>X U Vt. topological_space(X::'a,Vt) & element_of_collection(U::'a,Vt) --> open(U::'a,X,Vt)) &
```
```  2763   (\<forall>U X Vt. closed(U::'a,X,Vt) --> topological_space(X::'a,Vt)) &
```
```  2764   (\<forall>U X Vt. closed(U::'a,X,Vt) --> open(relative_complement_sets(U::'a,X),X,Vt)) &
```
```  2765   (\<forall>U X Vt. topological_space(X::'a,Vt) & open(relative_complement_sets(U::'a,X),X,Vt) --> closed(U::'a,X,Vt)) &
```
```  2766   (\<forall>Vs X Vt. finer(Vt::'a,Vs,X) --> topological_space(X::'a,Vt)) &
```
```  2767   (\<forall>Vt X Vs. finer(Vt::'a,Vs,X) --> topological_space(X::'a,Vs)) &
```
```  2768   (\<forall>X Vs Vt. finer(Vt::'a,Vs,X) --> subset_collections(Vs::'a,Vt)) &
```
```  2769   (\<forall>X Vs Vt. topological_space(X::'a,Vt) & topological_space(X::'a,Vs) & subset_collections(Vs::'a,Vt) --> finer(Vt::'a,Vs,X)) &
```
```  2770   (\<forall>Vf X. basis(X::'a,Vf) --> equal_sets(union_of_members(Vf),X)) &
```
```  2771   (\<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))) &
```
```  2772   (\<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)) &
```
```  2773   (\<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))) &
```
```  2774   (\<forall>Vf X. equal_sets(union_of_members(Vf),X) --> basis(X::'a,Vf) | element_of_set(f7(X::'a,Vf),X)) &
```
```  2775   (\<forall>X Vf. equal_sets(union_of_members(Vf),X) --> basis(X::'a,Vf) | element_of_collection(f8(X::'a,Vf),Vf)) &
```
```  2776   (\<forall>X Vf. equal_sets(union_of_members(Vf),X) --> basis(X::'a,Vf) | element_of_collection(f9(X::'a,Vf),Vf)) &
```
```  2777   (\<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)))) &
```
```  2778   (\<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)) &
```
```  2779   (\<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))) &
```
```  2780   (\<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)) &
```
```  2781   (\<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)) &
```
```  2782   (\<forall>Vf U. element_of_collection(U::'a,top_of_basis(Vf)) | element_of_set(f11(Vf::'a,U),U)) &
```
```  2783   (\<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))) &
```
```  2784   (\<forall>U Y X Vt. element_of_collection(U::'a,subspace_topology(X::'a,Vt,Y)) --> topological_space(X::'a,Vt)) &
```
```  2785   (\<forall>U Vt Y X. element_of_collection(U::'a,subspace_topology(X::'a,Vt,Y)) --> subset_sets(Y::'a,X)) &
```
```  2786   (\<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)) &
```
```  2787   (\<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)))) &
```
```  2788   (\<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))) &
```
```  2789   (\<forall>U Y X Vt. element_of_set(U::'a,interior(Y::'a,X,Vt)) --> topological_space(X::'a,Vt)) &
```
```  2790   (\<forall>U Vt Y X. element_of_set(U::'a,interior(Y::'a,X,Vt)) --> subset_sets(Y::'a,X)) &
```
```  2791   (\<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))) &
```
```  2792   (\<forall>X Vt U Y. element_of_set(U::'a,interior(Y::'a,X,Vt)) --> subset_sets(f13(Y::'a,X,Vt,U),Y)) &
```
```  2793   (\<forall>Y U X Vt. element_of_set(U::'a,interior(Y::'a,X,Vt)) --> open(f13(Y::'a,X,Vt,U),X,Vt)) &
```
```  2794   (\<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))) &
```
```  2795   (\<forall>U Y X Vt. element_of_set(U::'a,closure(Y::'a,X,Vt)) --> topological_space(X::'a,Vt)) &
```
```  2796   (\<forall>U Vt Y X. element_of_set(U::'a,closure(Y::'a,X,Vt)) --> subset_sets(Y::'a,X)) &
```
```  2797   (\<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)) &
```
```  2798   (\<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))) &
```
```  2799   (\<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)) &
```
```  2800   (\<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))) &
```
```  2801   (\<forall>U Y X Vt. neighborhood(U::'a,Y,X,Vt) --> topological_space(X::'a,Vt)) &
```
```  2802   (\<forall>Y U X Vt. neighborhood(U::'a,Y,X,Vt) --> open(U::'a,X,Vt)) &
```
```  2803   (\<forall>X Vt Y U. neighborhood(U::'a,Y,X,Vt) --> element_of_set(Y::'a,U)) &
```
```  2804   (\<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)) &
```
```  2805   (\<forall>Z Y X Vt. limit_point(Z::'a,Y,X,Vt) --> topological_space(X::'a,Vt)) &
```
```  2806   (\<forall>Z Vt Y X. limit_point(Z::'a,Y,X,Vt) --> subset_sets(Y::'a,X)) &
```
```  2807   (\<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))) &
```
```  2808   (\<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))) &
```
```  2809   (\<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)) &
```
```  2810   (\<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)) &
```
```  2811   (\<forall>U Y X Vt. element_of_set(U::'a,boundary(Y::'a,X,Vt)) --> topological_space(X::'a,Vt)) &
```
```  2812   (\<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))) &
```
```  2813   (\<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))) &
```
```  2814   (\<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))) &
```
```  2815   (\<forall>X Vt. hausdorff(X::'a,Vt) --> topological_space(X::'a,Vt)) &
```
```  2816   (\<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)) &
```
```  2817   (\<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)) &
```
```  2818   (\<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))) &
```
```  2819   (\<forall>Vt X. topological_space(X::'a,Vt) --> hausdorff(X::'a,Vt) | element_of_set(f19(X::'a,Vt),X)) &
```
```  2820   (\<forall>Vt X. topological_space(X::'a,Vt) --> hausdorff(X::'a,Vt) | element_of_set(f20(X::'a,Vt),X)) &
```
```  2821   (\<forall>X Vt. topological_space(X::'a,Vt) & eq_p(f19(X::'a,Vt),f20(X::'a,Vt)) --> hausdorff(X::'a,Vt)) &
```
```  2822   (\<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)) &
```
```  2823   (\<forall>Va1 Va2 X Vt. separation(Va1::'a,Va2,X,Vt) --> topological_space(X::'a,Vt)) &
```
```  2824   (\<forall>Va2 X Vt Va1. ~(separation(Va1::'a,Va2,X,Vt) & equal_sets(Va1::'a,empty_set))) &
```
```  2825   (\<forall>Va1 X Vt Va2. ~(separation(Va1::'a,Va2,X,Vt) & equal_sets(Va2::'a,empty_set))) &
```
```  2826   (\<forall>Va2 X Va1 Vt. separation(Va1::'a,Va2,X,Vt) --> element_of_collection(Va1::'a,Vt)) &
```
```  2827   (\<forall>Va1 X Va2 Vt. separation(Va1::'a,Va2,X,Vt) --> element_of_collection(Va2::'a,Vt)) &
```
```  2828   (\<forall>Vt Va1 Va2 X. separation(Va1::'a,Va2,X,Vt) --> equal_sets(union_of_sets(Va1::'a,Va2),X)) &
```
```  2829   (\<forall>X Vt Va1 Va2. separation(Va1::'a,Va2,X,Vt) --> disjoint_s(Va1::'a,Va2)) &
```
```  2830   (\<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)) &
```
```  2831   (\<forall>X Vt. connected_space(X::'a,Vt) --> topological_space(X::'a,Vt)) &
```
```  2832   (\<forall>Va1 Va2 X Vt. ~(connected_space(X::'a,Vt) & separation(Va1::'a,Va2,X,Vt))) &
```
```  2833   (\<forall>X Vt. topological_space(X::'a,Vt) --> connected_space(X::'a,Vt) | separation(f21(X::'a,Vt),f22(X::'a,Vt),X,Vt)) &
```
```  2834   (\<forall>Va X Vt. connected_set(Va::'a,X,Vt) --> topological_space(X::'a,Vt)) &
```
```  2835   (\<forall>Vt Va X. connected_set(Va::'a,X,Vt) --> subset_sets(Va::'a,X)) &
```
```  2836   (\<forall>X Vt Va. connected_set(Va::'a,X,Vt) --> connected_space(Va::'a,subspace_topology(X::'a,Vt,Va))) &
```
```  2837   (\<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)) &
```
```  2838   (\<forall>Vf X Vt. open_covering(Vf::'a,X,Vt) --> topological_space(X::'a,Vt)) &
```
```  2839   (\<forall>X Vf Vt. open_covering(Vf::'a,X,Vt) --> subset_collections(Vf::'a,Vt)) &
```
```  2840   (\<forall>Vt Vf X. open_covering(Vf::'a,X,Vt) --> equal_sets(union_of_members(Vf),X)) &
```
```  2841   (\<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)) &
```
```  2842   (\<forall>X Vt. compact_space(X::'a,Vt) --> topological_space(X::'a,Vt)) &
```
```  2843   (\<forall>X Vt Vf1. compact_space(X::'a,Vt) & open_covering(Vf1::'a,X,Vt) --> finite'(f23(X::'a,Vt,Vf1))) &
```
```  2844   (\<forall>X Vt Vf1. compact_space(X::'a,Vt) & open_covering(Vf1::'a,X,Vt) --> subset_collections(f23(X::'a,Vt,Vf1),Vf1)) &
```
```  2845   (\<forall>Vf1 X Vt. compact_space(X::'a,Vt) & open_covering(Vf1::'a,X,Vt) --> open_covering(f23(X::'a,Vt,Vf1),X,Vt)) &
```
```  2846   (\<forall>X Vt. topological_space(X::'a,Vt) --> compact_space(X::'a,Vt) | open_covering(f24(X::'a,Vt),X,Vt)) &
```
```  2847   (\<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)) &
```
```  2848   (\<forall>Va X Vt. compact_set(Va::'a,X,Vt) --> topological_space(X::'a,Vt)) &
```
```  2849   (\<forall>Vt Va X. compact_set(Va::'a,X,Vt) --> subset_sets(Va::'a,X)) &
```
```  2850   (\<forall>X Vt Va. compact_set(Va::'a,X,Vt) --> compact_space(Va::'a,subspace_topology(X::'a,Vt,Va))) &
```
```  2851   (\<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)) &
```
```  2852   (basis(cx::'a,f)) &
```
```  2853   (\<forall>U. element_of_collection(U::'a,top_of_basis(f))) &
```
```  2854   (\<forall>V. element_of_collection(V::'a,top_of_basis(f))) &
```
```  2855   (\<forall>U V. ~element_of_collection(intersection_of_sets(U::'a,V),top_of_basis(f))) --> False"
```
```  2856   by meson
```
```  2857
```
```  2858
```
```  2859 (*0 inferences so far.  Searching to depth 0.  0.8 secs*)
```
```  2860 lemma TOP004_2:
```
```  2861  "(\<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))) &
```
```  2862   (\<forall>Vf X. basis(X::'a,Vf) --> equal_sets(union_of_members(Vf),X)) &
```
```  2863   (\<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))) &
```
```  2864   (\<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)) &
```
```  2865   (\<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))) &
```
```  2866   (\<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))) &
```
```  2867   (\<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)) &
```
```  2868   (\<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)) &
```
```  2869   (\<forall>Vf U. element_of_collection(U::'a,top_of_basis(Vf)) | element_of_set(f11(Vf::'a,U),U)) &
```
```  2870   (\<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))) &
```
```  2871   (\<forall>Y X Z. subset_sets(X::'a,Y) & subset_sets(Y::'a,Z) --> subset_sets(X::'a,Z)) &
```
```  2872   (\<forall>Y Z X. element_of_set(Z::'a,intersection_of_sets(X::'a,Y)) --> element_of_set(Z::'a,X)) &
```
```  2873   (\<forall>X Z Y. element_of_set(Z::'a,intersection_of_sets(X::'a,Y)) --> element_of_set(Z::'a,Y)) &
```
```  2874   (\<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))) &
```
```  2875   (\<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))) &
```
```  2876   (\<forall>X Z Y. equal_sets(X::'a,Y) & element_of_set(Z::'a,X) --> element_of_set(Z::'a,Y)) &
```
```  2877   (\<forall>Y X. equal_sets(intersection_of_sets(X::'a,Y),intersection_of_sets(Y::'a,X))) &
```
```  2878   (basis(cx::'a,f)) &
```
```  2879   (\<forall>U. element_of_collection(U::'a,top_of_basis(f))) &
```
```  2880   (\<forall>V. element_of_collection(V::'a,top_of_basis(f))) &
```
```  2881   (\<forall>U V. ~element_of_collection(intersection_of_sets(U::'a,V),top_of_basis(f))) --> False"
```
```  2882   by meson
```
```  2883
```
```  2884 (*53777 inferences so far.  Searching to depth 20.  68.7 secs*)
```
```  2885 lemma TOP005_2:
```
```  2886  "(\<forall>Vf U. element_of_set(U::'a,union_of_members(Vf)) --> element_of_set(U::'a,f1(Vf::'a,U))) &
```
```  2887   (\<forall>U Vf. element_of_set(U::'a,union_of_members(Vf)) --> element_of_collection(f1(Vf::'a,U),Vf)) &
```
```  2888   (\<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))) &
```
```  2889   (\<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)) &
```
```  2890   (\<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)) &
```
```  2891   (\<forall>Vf U. element_of_collection(U::'a,top_of_basis(Vf)) | element_of_set(f11(Vf::'a,U),U)) &
```
```  2892   (\<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))) &
```
```  2893   (\<forall>X U Y. element_of_set(U::'a,X) --> subset_sets(X::'a,Y) | element_of_set(U::'a,Y)) &
```
```  2894   (\<forall>Y X Z. subset_sets(X::'a,Y) & element_of_collection(Y::'a,Z) --> subset_sets(X::'a,union_of_members(Z))) &
```
```  2895   (\<forall>X U Y. subset_collections(X::'a,Y) & element_of_collection(U::'a,X) --> element_of_collection(U::'a,Y)) &
```
```  2896   (subset_collections(g::'a,top_of_basis(f))) &
```
```  2897   (~element_of_collection(union_of_members(g),top_of_basis(f))) --> False"
```
```  2898   oops
```
```  2899
```
```  2900 end
```