doc-src/Logics/HOL-rules.txt
 author lcp Wed Nov 10 05:00:57 1993 +0100 (1993-11-10) changeset 104 d8205bb279a7 permissions -rw-r--r--
Initial revision
```     1 ruleshell.ML lemmas.ML set.ML fun.ML subset.ML equalities.ML prod.ML sum.ML wf.ML mono.ML fixedpt.ML nat.ML list.ML
```
```     2 ----------------------------------------------------------------
```
```     3 ruleshell.ML
```
```     4
```
```     5 \idx{refl}      t = t::'a
```
```     6 \idx{subst}     [| s = t; P(s) |] ==> P(t::'a)
```
```     7 \idx{abs},!!x::'a. f(x)::'b = g(x)) ==> (%x.f(x)) = (%x.g(x)))
```
```     8 \idx{disch}     (P ==> Q) ==> P-->Q
```
```     9 \idx{mp}        [| P-->Q;  P |] ==> Q
```
```    10
```
```    11 \idx{True_def}  True = ((%x.x)=(%x.x))
```
```    12 \idx{All_def}   All  = (%P. P = (%x.True))
```
```    13 \idx{Ex_def}    Ex   = (%P. P(Eps(P)))
```
```    14 \idx{False_def} False = (!P.P)
```
```    15 \idx{not_def}   not  = (%P. P-->False)
```
```    16 \idx{and_def}   op & = (%P Q. !R. (P-->Q-->R) --> R)
```
```    17 \idx{or_def}    op | = (%P Q. !R. (P-->R) --> (Q-->R) --> R)
```
```    18 \idx{Ex1_def}   Ex1 == (%P. ? x. P(x) & (! y. P(y) --> y=x))
```
```    19
```
```    20 \idx{iff}       (P-->Q) --> (Q-->P) --> (P=Q)
```
```    21 \idx{True_or_False}     (P=True) | (P=False)
```
```    22 \idx{select}    P(x::'a) --> P(Eps(P))
```
```    23
```
```    24 \idx{Inv_def}   Inv = (%(f::'a=>'b) y. @x. f(x)=y)
```
```    25 \idx{o_def}     op o = (%(f::'b=>'c) g (x::'a). f(g(x)))
```
```    26 \idx{Cond_def}  Cond = (%P x y.@z::'a. (P=True --> z=x) & (P=False --> z=y))
```
```    27
```
```    28 ----------------------------------------------------------------
```
```    29 lemmas.ML
```
```    30
```
```    31 \idx{sym}    s=t ==> t=s
```
```    32 \idx{trans}    [| r=s; s=t |] ==> r=t
```
```    33 \idx{box_equals}
```
```    34     [| a=b;  a=c;  b=d |] ==> c=d
```
```    35 \idx{ap_term}    s=t ==> f(s)=f(t)
```
```    36 \idx{ap_thm}    s::'a=>'b = t ==> s(x)=t(x)
```
```    37 \idx{cong}
```
```    38    [| f = g; x::'a = y |] ==> f(x) = g(y)
```
```    39 \idx{iffI}
```
```    40    [| P ==> Q;  Q ==> P |] ==> P=Q
```
```    41 \idx{iffD1}    [| P=Q; Q |] ==> P
```
```    42 \idx{iffE}
```
```    43     [| P=Q; [| P --> Q; Q --> P |] ==> R |] ==> R
```
```    44 \idx{eqTrueI}    P ==> P=True
```
```    45 \idx{eqTrueE}    P=True ==> P
```
```    46 \idx{allI}    (!!x::'a. P(x)) ==> !x. P(x)
```
```    47 \idx{spec}    !x::'a.P(x) ==> P(x)
```
```    48 \idx{allE}    [| !x.P(x);  P(x) ==> R |] ==> R
```
```    49 \idx{all_dupE}
```
```    50     [| ! x.P(x);  [| P(x); ! x.P(x) |] ==> R
```
```    51     |] ==> R
```
```    52 \idx{FalseE}    False ==> P
```
```    53 \idx{False_neq_True}    False=True ==> P
```
```    54 \idx{notI}    (P ==> False) ==> ~P
```
```    55 \idx{notE}    [| ~P;  P |] ==> R
```
```    56 \idx{impE}    [| P-->Q;  P;  Q ==> R |] ==> R
```
```    57 \idx{rev_mp}    [| P;  P --> Q |] ==> Q
```
```    58 \idx{contrapos}    [| ~Q;  P==>Q |] ==> ~P
```
```    59 \idx{exI}    P(x) ==> ? x::'a.P(x)
```
```    60 \idx{exE}    [| ? x::'a.P(x); !!x. P(x) ==> Q |] ==> Q
```
```    61
```
```    62 \idx{conjI}    [| P; Q |] ==> P&Q
```
```    63 \idx{conjunct1}    [| P & Q |] ==> P
```
```    64 \idx{conjunct2}    [| P & Q |] ==> Q
```
```    65 \idx{conjE}    [| P&Q;  [| P; Q |] ==> R |] ==> R
```
```    66 \idx{disjI1}    P ==> P|Q
```
```    67 \idx{disjI2}    Q ==> P|Q
```
```    68 \idx{disjE}    [| P | Q; P ==> R; Q ==> R |] ==> R
```
```    69 \idx{ccontr}    (~P ==> False) ==> P
```
```    70 \idx{classical}    (~P ==> P) ==> P
```
```    71 \idx{notnotD}    ~~P ==> P
```
```    72 \idx{ex1I}
```
```    73     [| P(a);  !!x. P(x) ==> x=a |] ==> ?! x. P(x)
```
```    74 \idx{ex1E}
```
```    75     [| ?! x.P(x);  !!x. [| P(x);  ! y. P(y) --> y=x |] ==> R |] ==> R
```
```    76 \idx{select_equality}
```
```    77     [| P(a);  !!x. P(x) ==> x=a |] ==> (@x.P(x)) = a
```
```    78 \idx{disjCI}    (~Q ==> P) ==> P|Q
```
```    79 \idx{excluded_middle}    ~P | P
```
```    80 \idx{impCE}    [| P-->Q; ~P ==> R; Q ==> R |] ==> R
```
```    81 \idx{iffCE}
```
```    82     [| P=Q;  [| P; Q |] ==> R;  [| ~P; ~Q |] ==> R |] ==> R
```
```    83 \idx{exCI}    (! x. ~P(x) ==> P(a)) ==> ? x.P(x)
```
```    84 \idx{swap}    ~P ==> (~Q ==> P) ==> Q
```
```    85
```
```    86 ----------------------------------------------------------------
```
```    87 simpdata.ML
```
```    88
```
```    89 \idx{if_True}    Cond(True,x,y) = x
```
```    90 \idx{if_False}    Cond(False,x,y) = y
```
```    91 \idx{if_P}    P ==> Cond(P,x,y) = x
```
```    92 \idx{if_not_P}    ~P ==> Cond(P,x,y) = y
```
```    93 \idx{expand_if}
```
```    94     P(Cond(Q,x,y)) = ((Q --> P(x)) & (~Q --> P(y)))
```
```    95
```
```    96 ----------------------------------------------------------------
```
```    97 \idx{set.ML}
```
```    98
```
```    99 \idx{CollectI}          [| P(a) |] ==> a : \{x.P(x)\}
```
```   100 \idx{CollectD}          [| a : \{x.P(x)\} |] ==> P(a)
```
```   101 \idx{set_ext}           [| !!x. (x:A) = (x:B) |] ==> A = B
```
```   102
```
```   103 \idx{Ball_def}          Ball(A,P)  == ! x. x:A --> P(x)
```
```   104 \idx{Bex_def}           Bex(A,P)   == ? x. x:A & P(x)
```
```   105 \idx{subset_def}        A <= B     == ! x:A. x:B
```
```   106 \idx{Un_def}            A Un B     == \{x.x:A | x:B\}
```
```   107 \idx{Int_def}           A Int B    == \{x.x:A & x:B\}
```
```   108 \idx{Compl_def}         Compl(A)   == \{x. ~x:A\}
```
```   109 \idx{Inter_def}         Inter(S)   == \{x. ! A:S. x:A\}
```
```   110 \idx{Union_def}         Union(S)   == \{x. ? A:S. x:A\}
```
```   111 \idx{INTER_def}         INTER(A,B) == \{y. ! x:A. y: B(x)\}
```
```   112 \idx{UNION_def}         UNION(A,B) == \{y. ? x:A. y: B(x)\}
```
```   113 \idx{mono_def}          mono(f)    == (!A B. A <= B --> f(A) <= f(B))
```
```   114 \idx{image_def}         f``A       == \{y. ? x:A. y=f(x)\}
```
```   115 \idx{singleton_def}     \{a\}      == \{x.x=a\}
```
```   116 \idx{range_def}         range(f)   == \{y. ? x. y=f(x)\}
```
```   117 \idx{One_One_def}       One_One(f) == ! x y. f(x)=f(y) --> x=y
```
```   118 \idx{One_One_on_def}    One_One_on(f,A) == !x y. x:A --> y:A --> f(x)=f(y) --> x=y
```
```   119 \idx{Onto_def}          Onto(f) == ! y. ? x. y=f(x)
```
```   120
```
```   121
```
```   122 \idx{Collect_cong}    [| !!x. P(x)=Q(x) |] ==> \{x. P(x)\} = \{x. Q(x)\}
```
```   123
```
```   124 \idx{ballI}    [| !!x. x:A ==> P(x) |] ==> ! x:A. P(x)
```
```   125 \idx{bspec}    [| ! x:A. P(x);  x:A |] ==> P(x)
```
```   126 \idx{ballE}    [| ! x:A. P(x);  P(x) ==> Q;  ~ x:A ==> Q |] ==> Q
```
```   127
```
```   128 \idx{bexI}     [| P(x);  x:A |] ==> ? x:A. P(x)
```
```   129 \idx{bexCI}    [| ! x:A. ~P(x) ==> P(a);  a:A |] ==> ? x:A.P(x)
```
```   130 \idx{bexE}     [| ? x:A. P(x);  !!x. [| x:A; P(x) |] ==> Q  |] ==> Q
```
```   131
```
```   132 \idx{ball_cong}
```
```   133     [| A=A';  !!x. x:A' ==> P(x) = P'(x) |] ==>
```
```   134     (! x:A. P(x)) = (! x:A'. P'(x))
```
```   135
```
```   136 \idx{bex_cong}
```
```   137     [| A=A';  !!x. x:A' ==> P(x) = P'(x) |] ==>
```
```   138     (? x:A. P(x)) = (? x:A'. P'(x))
```
```   139
```
```   140 \idx{subsetI}         (!!x.x:A ==> x:B) ==> A <= B
```
```   141 \idx{subsetD}         [| A <= B;  c:A |] ==> c:B
```
```   142 \idx{subsetCE}        [| A <= B;  ~(c:A) ==> P;  c:B ==> P |] ==> P
```
```   143
```
```   144 \idx{subset_refl}     A <= A
```
```   145 \idx{subset_antisym}  [| A <= B;  B <= A |] ==> A = B
```
```   146 \idx{subset_trans}    [| A<=B;  B<=C |] ==> A<=C
```
```   147
```
```   148 \idx{equalityD1}      A = B ==> A<=B
```
```   149 \idx{equalityD2}      A = B ==> B<=A
```
```   150 \idx{equalityE}       [| A = B;  [| A<=B; B<=A |] ==> P |]  ==>  P
```
```   151
```
```   152 \idx{singletonI}      a : \{a\}
```
```   153 \idx{singletonD}      b : \{a\} ==> b=a
```
```   154
```
```   155 \idx{imageI}    [| x:A |] ==> f(x) : f``A
```
```   156 \idx{imageE}    [| b : f``A;  !!x.[| b=f(x);  x:A |] ==> P |] ==> P
```
```   157
```
```   158 \idx{rangeI}    f(x) : range(f)
```
```   159 \idx{rangeE}    [| b : range(f);  !!x.[| b=f(x) |] ==> P |] ==> P
```
```   160
```
```   161 \idx{UnionI}    [| X:C;  A:X |] ==> A : Union(C)
```
```   162 \idx{UnionE}    [| A : Union(C);  !!X.[| A:X;  X:C |] ==> R |] ==> R
```
```   163
```
```   164 \idx{InterI}    [| !!X. X:C ==> A:X |] ==> A : Inter(C)
```
```   165 \idx{InterD}    [| A : Inter(C);  X:C |] ==> A:X
```
```   166 \idx{InterE}    [| A : Inter(C);  A:X ==> R;  ~ X:C ==> R |] ==> R
```
```   167
```
```   168 \idx{UN_I}    [| a:A;  b: B(a) |] ==> b: (UN x:A. B(x))
```
```   169 \idx{UN_E}    [| b : (UN x:A. B(x));  !!x.[| x:A;  b: B(x) |] ==> R |] ==> R
```
```   170
```
```   171 \idx{INT_I}    (!!x. x:A ==> b: B(x)) ==> b : (INT x:A. B(x))
```
```   172 \idx{INT_D}    [| b : (INT x:A. B(x));  a:A |] ==> b: B(a)
```
```   173 \idx{INT_E}    [| b : (INT x:A. B(x));  b: B(a) ==> R;  ~ a:A ==> R |] ==> R
```
```   174
```
```   175 \idx{UnI1}    c:A ==> c : A Un B
```
```   176 \idx{UnI2}    c:B ==> c : A Un B
```
```   177 \idx{UnCI}    (~c:B ==> c:A) ==> c : A Un B
```
```   178 \idx{UnE}    [| c : A Un B;  c:A ==> P;  c:B ==> P |] ==> P
```
```   179
```
```   180 \idx{IntI}    [| c:A;  c:B |] ==> c : A Int B
```
```   181 \idx{IntD1}    c : A Int B ==> c:A
```
```   182 \idx{IntD2}    c : A Int B ==> c:B
```
```   183 \idx{IntE}    [| c : A Int B;  [| c:A; c:B |] ==> P |] ==> P
```
```   184
```
```   185 \idx{ComplI}    [| c:A ==> False |] ==> c : Compl(A)
```
```   186 \idx{ComplD}    [| c : Compl(A) |] ==> ~c:A
```
```   187
```
```   188 \idx{monoI}    [| !!A B. A <= B ==> f(A) <= f(B) |] ==> mono(f)
```
```   189 \idx{monoD}    [| mono(f);  A <= B |] ==> f(A) <= f(B)
```
```   190
```
```   191
```
```   192 ----------------------------------------------------------------
```
```   193 \idx{fun.ML}
```
```   194
```
```   195 \idx{One_OneI}            [| !! x y. f(x) = f(y) ==> x=y |] ==> One_One(f)
```
```   196 \idx{One_One_inverseI}    (!!x. g(f(x)) = x) ==> One_One(f)
```
```   197 \idx{One_OneD}            [| One_One(f); f(x) = f(y) |] ==> x=y
```
```   198
```
```   199 \idx{Inv_f_f}             One_One(f)   ==> Inv(f,f(x)) = x
```
```   200 \idx{f_Inv_f}             y : range(f) ==> f(Inv(f,y)) = y
```
```   201
```
```   202 \idx{Inv_injective}
```
```   203     [| Inv(f,x)=Inv(f,y); x: range(f);  y: range(f) |] ==> x=y
```
```   204
```
```   205 \idx{One_One_onI}
```
```   206     (!! x y. [| f(x) = f(y); x:A; y:A |] ==> x=y) ==> One_One_on(f,A)
```
```   207
```
```   208 \idx{One_One_on_inverseI}
```
```   209     (!!x. x:A ==> g(f(x)) = x) ==> One_One_on(f,A)
```
```   210
```
```   211 \idx{One_One_onD}
```
```   212     [| One_One_on(f,A);  f(x)=f(y);  x:A;  y:A |] ==> x=y
```
```   213
```
```   214 \idx{One_One_on_contraD}
```
```   215     [| One_One_on(f,A);  ~x=y;  x:A;  y:A |] ==> ~ f(x)=f(y)
```
```   216
```
```   217
```
```   218 ----------------------------------------------------------------
```
```   219 \idx{subset.ML}
```
```   220
```
```   221 \idx{Union_upper}     B:A ==> B <= Union(A)
```
```   222 \idx{Union_least}     [| !!X. X:A ==> X<=C |] ==> Union(A) <= C
```
```   223
```
```   224 \idx{Inter_lower}     B:A ==> Inter(A) <= B
```
```   225 \idx{Inter_greatest}  [| !!X. X:A ==> C<=X |] ==> C <= Inter(A)
```
```   226
```
```   227 \idx{Un_upper1}       A <= A Un B
```
```   228 \idx{Un_upper2}       B <= A Un B
```
```   229 \idx{Un_least}        [| A<=C;  B<=C |] ==> A Un B <= C
```
```   230
```
```   231 \idx{Int_lower1}      A Int B <= A
```
```   232 \idx{Int_lower2}      A Int B <= B
```
```   233 \idx{Int_greatest}    [| C<=A;  C<=B |] ==> C <= A Int B
```
```   234
```
```   235
```
```   236 ----------------------------------------------------------------
```
```   237 \idx{equalities.ML}
```
```   238
```
```   239 \idx{Int_absorb}        A Int A = A
```
```   240 \idx{Int_commute}       A Int B  =  B Int A
```
```   241 \idx{Int_assoc}         (A Int B) Int C  =  A Int (B Int C)
```
```   242 \idx{Int_Un_distrib}    (A Un B) Int C  =  (A Int C) Un (B Int C)
```
```   243
```
```   244 \idx{Un_absorb}         A Un A = A
```
```   245 \idx{Un_commute}        A Un B  =  B Un A
```
```   246 \idx{Un_assoc}          (A Un B) Un C  =  A Un (B Un C)
```
```   247 \idx{Un_Int_distrib}    (A Int B) Un C  =  (A Un C) Int (B Un C)
```
```   248
```
```   249 \idx{Compl_disjoint}    A Int Compl(A) = \{x.False\}
```
```   250 \idx{Compl_partition    A Un Compl(A) = \{x.True\}
```
```   251 \idx{double_complement} Compl(Compl(A)) = A
```
```   252
```
```   253
```
```   254 \idx{Compl_Un}          Compl(A Un B) = Compl(A) Int Compl(B)
```
```   255 \idx{Compl_Int}         Compl(A Int B) = Compl(A) Un Compl(B)
```
```   256
```
```   257 \idx{Union_Un_distrib}  Union(A Un B) = Union(A) Un Union(B)
```
```   258 \idx{Int_Union_image}   A Int Union(B) = (UN C:B. A Int C)
```
```   259 \idx{Un_Union_image}    (UN x:C. A(x) Un B(x)) = Union(A``C)  Un  Union(B``C)
```
```   260
```
```   261 \idx{Inter_Un_distrib}  Inter(A Un B) = Inter(A) Int Inter(B)
```
```   262 \idx{Un_Inter_image}    A Un Inter(B) = (INT C:B. A Un C)
```
```   263 \idx{Int_Inter_image}   (INT x:C. A(x) Int B(x)) = Inter(A``C) Int Inter(B``C)
```
```   264
```
```   265
```
```   266 ----------------------------------------------------------------
```
```   267 prod.ML
```
```   268
```
```   269       mixfix = [ Delimfix((1<_,/_>), ['a,'b] => ('a,'b)prod, Pair),
```
```   270                  TInfixl(*, prod, 20) ],
```
```   271 thy = extend_theory Set.thy Prod
```
```   272   [([prod],([[term],[term]],term))],
```
```   273    ([fst],              'a * 'b => 'a),
```
```   274    ([snd],              'a * 'b => 'b),
```
```   275    ([split],            ['a * 'b, ['a,'b]=>'c] => 'c)],
```
```   276 \idx{fst_def}             fst(p) == @a. ? b. p = <a,b>),
```
```   277 \idx{snd_def}             snd(p) == @b. ? a. p = <a,b>),
```
```   278 \idx{split_def}           split(p,c) == c(fst(p),snd(p)))
```
```   279
```
```   280 \idx{Pair_inject}  [| <a, b> = <a',b'>;  [| a=a';  b=b' |] ==> R |] ==> R
```
```   281
```
```   282 \idx{fst_conv}     fst(<a,b>) = a
```
```   283 \idx{snd_conv}     snd(<a,b>) = b
```
```   284 \idx{split_conv}   split(<a,b>, c) = c(a,b)
```
```   285
```
```   286 \idx{surjective_pairing}    p = <fst(p),snd(p)>
```
```   287
```
```   288 ----------------------------------------------------------------
```
```   289 sum.ML
```
```   290
```
```   291       mixfix = [TInfixl(+, sum, 10)],
```
```   292 thy = extend_theory Prod.thy sum
```
```   293   [([sum], ([[term],[term]],term))],
```
```   294  [Inl],              'a => 'a+'b),
```
```   295  [Inr],              'b => 'a+'b),
```
```   296  [when],             ['a+'b, 'a=>'c, 'b=>'c] =>'c)],
```
```   297 \idx{when_def}    when == (%p f g. @z.  (!x. p=Inl(x) --> z=f(x))
```
```   298                                     & (!y. p=Inr(y) --> z=g(y))))
```
```   299
```
```   300 \idx{Inl_not_Inr}    ~ (Inl(a) = Inr(b))
```
```   301
```
```   302 \idx{One_One_Inl}    One_One(Inl)
```
```   303
```
```   304 \idx{One_One_Inr}    One_One(Inr)
```
```   305
```
```   306 \idx{when_Inl_conv}    when(Inl(x), f, g) = f(x)
```
```   307
```
```   308 \idx{when_Inr_conv}    when(Inr(x), f, g) = g(x)
```
```   309
```
```   310 \idx{sumE}
```
```   311     [| !!x::'a. P(Inl(x));  !!y::'b. P(Inr(y))
```
```   312     |] ==> P(s)
```
```   313
```
```   314 \idx{surjective_sum}    when(s, %x::'a. f(Inl(x)), %y::'b. f(Inr(y))) = f(s)
```
```   315
```
```   316
```
```   317 ????????????????????????????????????????????????????????????????
```
```   318 trancl?
```
```   319
```
```   320 ----------------------------------------------------------------
```
```   321 nat.ML
```
```   322
```
```   323   Sext\{mixfix=[Delimfix(0, nat, 0),
```
```   324                Infixl(<,[nat,nat] => bool,50)],
```
```   325 thy = extend_theory Trancl.thy Nat
```
```   326 [nat], ([],term))
```
```   327 [nat_case],          [nat, 'a, nat=>'a] =>'a),
```
```   328 [pred_nat],nat*nat) set),
```
```   329 [nat_rec],           [nat, 'a, [nat, 'a]=>'a] => 'a)
```
```   330
```
```   331 \idx{nat_case_def}        nat_case == (%n a f. @z.  (n=0 --> z=a)
```
```   332                                           & (!x. n=Suc(x) --> z=f(x)))),
```
```   333 \idx{pred_nat_def}        pred_nat == \{p. ? n. p = <n, Suc(n)>\} ),
```
```   334 \idx{less_def} m<n == <m,n>:trancl(pred_nat)),
```
```   335 \idx{nat_rec_def}
```
```   336    nat_rec(n,c,d) == wfrec(trancl(pred_nat),
```
```   337                         %rec l. nat_case(l, c, %m. d(m,rec(m))),
```
```   338                         n) )
```
```   339
```
```   340 \idx{nat_induct}    [| P(0); !!k. [| P(k) |] ==> P(Suc(k)) |]  ==> P(n)
```
```   341
```
```   342
```
```   343 \idx{Suc_not_Zero}    ~ (Suc(m) = 0)
```
```   344 \idx{One_One_Suc}    One_One(Suc)
```
```   345 \idx{n_not_Suc_n}    ~(n=Suc(n))
```
```   346
```
```   347 \idx{nat_case_0_conv}    nat_case(0, a, f) = a
```
```   348
```
```   349 \idx{nat_case_Suc_conv}    nat_case(Suc(k), a, f) = f(k)
```
```   350
```
```   351 \idx{pred_natI}    <n, Suc(n)> : pred_nat
```
```   352 \idx{pred_natE}
```
```   353     [| p : pred_nat;  !!x n. [| p = <n, Suc(n)> |] ==> R
```
```   354     |] ==> R
```
```   355
```
```   356 \idx{wf_pred_nat}    wf(pred_nat)
```
```   357
```
```   358 \idx{nat_rec_0_conv}    nat_rec(0,c,h) = c
```
```   359
```
```   360 \idx{nat_rec_Suc_conv}    nat_rec(Suc(n), c, h) = h(n, nat_rec(n,c,h))
```
```   361
```
```   362
```
```   363 (*** Basic properties of less than ***)
```
```   364 \idx{less_trans}     [| i<j;  j<k |] ==> i<k
```
```   365 \idx{lessI}          n < Suc(n)
```
```   366 \idx{zero_less_Suc}  0 < Suc(n)
```
```   367
```
```   368 \idx{less_not_sym}   n<m --> ~m<n
```
```   369 \idx{less_not_refl}  ~ (n<n)
```
```   370 \idx{not_less0}      ~ (n<0)
```
```   371
```
```   372 \idx{Suc_less_eq}    (Suc(m) < Suc(n)) = (m<n)
```
```   373 \idx{less_induct}    [| !!n. [| ! m. m<n --> P(m) |] ==> P(n) |]  ==>  P(n)
```
```   374
```
```   375 \idx{less_linear}    m<n | m=n | n<m
```
```   376
```
```   377
```
```   378 ----------------------------------------------------------------
```
```   379 list.ML
```
```   380
```
```   381  [([list], ([[term]],term))],
```
```   382   ([Nil],       'a list),
```
```   383   ([Cons],      ['a, 'a list] => 'a list),
```
```   384   ([list_rec],        ['a list, 'b, ['a ,'a list, 'b]=>'b] => 'b),
```
```   385   ([list_all],        ('a => bool) => ('a list => bool)),
```
```   386   ([map],               ('a=>'b) => ('a list => 'b list))
```
```   387
```
```   388 \idx{map_def}     map(f,xs) == list_rec(xs, Nil, %x l r. Cons(f(x), r)) )
```
```   389
```
```   390 \idx{list_induct}
```
```   391     [| P(Nil);
```
```   392        !!x xs. [| P(xs) |] ==> P(Cons(x,xs)) |]  ==> P(l)
```
```   393
```
```   394 \idx{Cons_not_Nil}   ~ Cons(x,xs) = Nil
```
```   395 \idx{Cons_Cons_eq}   (Cons(x,xs)=Cons(y,ys)) = (x=y & xs=ys)
```
```   396
```
```   397 \idx{list_rec_Nil_conv}    list_rec(Nil,c,h) = c
```
```   398 \idx{list_rec_Cons_conv}   list_rec(Cons(a,l), c, h) =
```
```   399                                h(a, l, list_rec(l,c,h))
```
```   400
```
```   401 \idx{map_Nil_conv}   map(f,Nil) = Nil
```
```   402 \idx{map_Cons_conv}  map(f, Cons(x,xs)) = Cons(f(x), map(f,xs))
```
```   403
```