src/CTT/CTT.thy
author paulson
Tue Nov 14 13:26:48 2000 +0100 (2000-11-14)
changeset 10467 e6e7205e9e91
parent 3837 d7f033c74b38
child 12110 f8b4b11cd79d
permissions -rw-r--r--
x-symbol support for Pi, Sigma, -->, : (membership)
note that "lam" is displayed as TWO lambda-symbols
     1 (*  Title:      CTT/ctt.thy
     2     ID:         $Id$
     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     4     Copyright   1993  University of Cambridge
     5 
     6 Constructive Type Theory
     7 *)
     8 
     9 CTT = Pure +
    10 
    11 types
    12   i
    13   t
    14   o
    15 
    16 arities
    17    i,t,o :: logic
    18 
    19 consts
    20   (*Types*)
    21   F,T       :: "t"          (*F is empty, T contains one element*)
    22   contr     :: "i=>i"
    23   tt        :: "i"
    24   (*Natural numbers*)
    25   N         :: "t"
    26   succ      :: "i=>i"
    27   rec       :: "[i, i, [i,i]=>i] => i"
    28   (*Unions*)
    29   inl,inr   :: "i=>i"
    30   when      :: "[i, i=>i, i=>i]=>i"
    31   (*General Sum and Binary Product*)
    32   Sum       :: "[t, i=>t]=>t"
    33   fst,snd   :: "i=>i"
    34   split     :: "[i, [i,i]=>i] =>i"
    35   (*General Product and Function Space*)
    36   Prod      :: "[t, i=>t]=>t"
    37   (*Equality type*)
    38   Eq        :: "[t,i,i]=>t"
    39   eq        :: "i"
    40   (*Judgements*)
    41   Type      :: "t => prop"          ("(_ type)" [10] 5)
    42   Eqtype    :: "[t,t]=>prop"        ("(_ =/ _)" [10,10] 5)
    43   Elem      :: "[i, t]=>prop"       ("(_ /: _)" [10,10] 5)
    44   Eqelem    :: "[i,i,t]=>prop"      ("(_ =/ _ :/ _)" [10,10,10] 5)
    45   Reduce    :: "[i,i]=>prop"        ("Reduce[_,_]")
    46   (*Types*)
    47   "@PROD"   :: "[idt,t,t]=>t"       ("(3PROD _:_./ _)" 10)
    48   "@SUM"    :: "[idt,t,t]=>t"       ("(3SUM _:_./ _)" 10)
    49   "+"       :: "[t,t]=>t"           (infixr 40)
    50   (*Invisible infixes!*)
    51   "@-->"    :: "[t,t]=>t"           ("(_ -->/ _)" [31,30] 30)
    52   "@*"      :: "[t,t]=>t"           ("(_ */ _)" [51,50] 50)
    53   (*Functions*)
    54   lambda    :: "(i => i) => i"      (binder "lam " 10)
    55   "`"       :: "[i,i]=>i"           (infixl 60)
    56   (*Natural numbers*)
    57   "0"       :: "i"                  ("0")
    58   (*Pairing*)
    59   pair      :: "[i,i]=>i"           ("(1<_,/_>)")
    60 
    61 translations
    62   "PROD x:A. B" => "Prod(A, %x. B)"
    63   "A --> B"     => "Prod(A, _K(B))"
    64   "SUM x:A. B"  => "Sum(A, %x. B)"
    65   "A * B"       => "Sum(A, _K(B))"
    66 
    67 syntax (xsymbols)
    68   "@-->"    :: "[t,t]=>t"           ("(_ \\<longrightarrow>/ _)" [31,30] 30)
    69   "@*"      :: "[t,t]=>t"           ("(_ \\<times>/ _)"          [51,50] 50)
    70 
    71 syntax (symbols)
    72   Elem      :: "[i, t]=>prop"     ("(_ /\\<in> _)" [10,10] 5)
    73   Eqelem    :: "[i,i,t]=>prop"    ("(2_ =/ _ \\<in>/ _)" [10,10,10] 5)
    74   "@SUM"    :: "[idt,t,t] => t"   ("(3\\<Sigma> _\\<in>_./ _)" 10)
    75   "@PROD"   :: "[idt,t,t] => t"   ("(3\\<Pi> _\\<in>_./ _)"    10)
    76   "lam "    :: "[idts, i] => i"   ("(3\\<lambda>\\<lambda>_./ _)" 10)
    77 
    78 rules
    79 
    80   (*Reduction: a weaker notion than equality;  a hack for simplification.
    81     Reduce[a,b] means either that  a=b:A  for some A or else that "a" and "b"
    82     are textually identical.*)
    83 
    84   (*does not verify a:A!  Sound because only trans_red uses a Reduce premise
    85     No new theorems can be proved about the standard judgements.*)
    86   refl_red "Reduce[a,a]"
    87   red_if_equal "a = b : A ==> Reduce[a,b]"
    88   trans_red "[| a = b : A;  Reduce[b,c] |] ==> a = c : A"
    89 
    90   (*Reflexivity*)
    91 
    92   refl_type "A type ==> A = A"
    93   refl_elem "a : A ==> a = a : A"
    94 
    95   (*Symmetry*)
    96 
    97   sym_type  "A = B ==> B = A"
    98   sym_elem  "a = b : A ==> b = a : A"
    99 
   100   (*Transitivity*)
   101 
   102   trans_type   "[| A = B;  B = C |] ==> A = C"
   103   trans_elem   "[| a = b : A;  b = c : A |] ==> a = c : A"
   104 
   105   equal_types  "[| a : A;  A = B |] ==> a : B"
   106   equal_typesL "[| a = b : A;  A = B |] ==> a = b : B"
   107 
   108   (*Substitution*)
   109 
   110   subst_type   "[| a : A;  !!z. z:A ==> B(z) type |] ==> B(a) type"
   111   subst_typeL  "[| a = c : A;  !!z. z:A ==> B(z) = D(z) |] ==> B(a) = D(c)"
   112 
   113   subst_elem   "[| a : A;  !!z. z:A ==> b(z):B(z) |] ==> b(a):B(a)"
   114   subst_elemL
   115     "[| a=c : A;  !!z. z:A ==> b(z)=d(z) : B(z) |] ==> b(a)=d(c) : B(a)"
   116 
   117 
   118   (*The type N -- natural numbers*)
   119 
   120   NF "N type"
   121   NI0 "0 : N"
   122   NI_succ "a : N ==> succ(a) : N"
   123   NI_succL  "a = b : N ==> succ(a) = succ(b) : N"
   124 
   125   NE
   126    "[| p: N;  a: C(0);  !!u v. [| u: N; v: C(u) |] ==> b(u,v): C(succ(u)) |] 
   127    ==> rec(p, a, %u v. b(u,v)) : C(p)"
   128 
   129   NEL
   130    "[| p = q : N;  a = c : C(0);  
   131       !!u v. [| u: N; v: C(u) |] ==> b(u,v) = d(u,v): C(succ(u)) |] 
   132    ==> rec(p, a, %u v. b(u,v)) = rec(q,c,d) : C(p)"
   133 
   134   NC0
   135    "[| a: C(0);  !!u v. [| u: N; v: C(u) |] ==> b(u,v): C(succ(u)) |] 
   136    ==> rec(0, a, %u v. b(u,v)) = a : C(0)"
   137 
   138   NC_succ
   139    "[| p: N;  a: C(0);  
   140        !!u v. [| u: N; v: C(u) |] ==> b(u,v): C(succ(u)) |] ==>  
   141    rec(succ(p), a, %u v. b(u,v)) = b(p, rec(p, a, %u v. b(u,v))) : C(succ(p))"
   142 
   143   (*The fourth Peano axiom.  See page 91 of Martin-Lof's book*)
   144   zero_ne_succ
   145     "[| a: N;  0 = succ(a) : N |] ==> 0: F"
   146 
   147 
   148   (*The Product of a family of types*)
   149 
   150   ProdF  "[| A type; !!x. x:A ==> B(x) type |] ==> PROD x:A. B(x) type"
   151 
   152   ProdFL
   153    "[| A = C;  !!x. x:A ==> B(x) = D(x) |] ==> 
   154    PROD x:A. B(x) = PROD x:C. D(x)"
   155 
   156   ProdI
   157    "[| A type;  !!x. x:A ==> b(x):B(x)|] ==> lam x. b(x) : PROD x:A. B(x)"
   158 
   159   ProdIL
   160    "[| A type;  !!x. x:A ==> b(x) = c(x) : B(x)|] ==> 
   161    lam x. b(x) = lam x. c(x) : PROD x:A. B(x)"
   162 
   163   ProdE  "[| p : PROD x:A. B(x);  a : A |] ==> p`a : B(a)"
   164   ProdEL "[| p=q: PROD x:A. B(x);  a=b : A |] ==> p`a = q`b : B(a)"
   165 
   166   ProdC
   167    "[| a : A;  !!x. x:A ==> b(x) : B(x)|] ==> 
   168    (lam x. b(x)) ` a = b(a) : B(a)"
   169 
   170   ProdC2
   171    "p : PROD x:A. B(x) ==> (lam x. p`x) = p : PROD x:A. B(x)"
   172 
   173 
   174   (*The Sum of a family of types*)
   175 
   176   SumF  "[| A type;  !!x. x:A ==> B(x) type |] ==> SUM x:A. B(x) type"
   177   SumFL
   178     "[| A = C;  !!x. x:A ==> B(x) = D(x) |] ==> SUM x:A. B(x) = SUM x:C. D(x)"
   179 
   180   SumI  "[| a : A;  b : B(a) |] ==> <a,b> : SUM x:A. B(x)"
   181   SumIL "[| a=c:A;  b=d:B(a) |] ==> <a,b> = <c,d> : SUM x:A. B(x)"
   182 
   183   SumE
   184     "[| p: SUM x:A. B(x);  !!x y. [| x:A; y:B(x) |] ==> c(x,y): C(<x,y>) |] 
   185     ==> split(p, %x y. c(x,y)) : C(p)"
   186 
   187   SumEL
   188     "[| p=q : SUM x:A. B(x); 
   189        !!x y. [| x:A; y:B(x) |] ==> c(x,y)=d(x,y): C(<x,y>)|] 
   190     ==> split(p, %x y. c(x,y)) = split(q, % x y. d(x,y)) : C(p)"
   191 
   192   SumC
   193     "[| a: A;  b: B(a);  !!x y. [| x:A; y:B(x) |] ==> c(x,y): C(<x,y>) |] 
   194     ==> split(<a,b>, %x y. c(x,y)) = c(a,b) : C(<a,b>)"
   195 
   196   fst_def   "fst(a) == split(a, %x y. x)"
   197   snd_def   "snd(a) == split(a, %x y. y)"
   198 
   199 
   200   (*The sum of two types*)
   201 
   202   PlusF   "[| A type;  B type |] ==> A+B type"
   203   PlusFL  "[| A = C;  B = D |] ==> A+B = C+D"
   204 
   205   PlusI_inl   "[| a : A;  B type |] ==> inl(a) : A+B"
   206   PlusI_inlL "[| a = c : A;  B type |] ==> inl(a) = inl(c) : A+B"
   207 
   208   PlusI_inr   "[| A type;  b : B |] ==> inr(b) : A+B"
   209   PlusI_inrL "[| A type;  b = d : B |] ==> inr(b) = inr(d) : A+B"
   210 
   211   PlusE
   212     "[| p: A+B;  !!x. x:A ==> c(x): C(inl(x));  
   213                 !!y. y:B ==> d(y): C(inr(y)) |] 
   214     ==> when(p, %x. c(x), %y. d(y)) : C(p)"
   215 
   216   PlusEL
   217     "[| p = q : A+B;  !!x. x: A ==> c(x) = e(x) : C(inl(x));   
   218                      !!y. y: B ==> d(y) = f(y) : C(inr(y)) |] 
   219     ==> when(p, %x. c(x), %y. d(y)) = when(q, %x. e(x), %y. f(y)) : C(p)"
   220 
   221   PlusC_inl
   222     "[| a: A;  !!x. x:A ==> c(x): C(inl(x));  
   223               !!y. y:B ==> d(y): C(inr(y)) |] 
   224     ==> when(inl(a), %x. c(x), %y. d(y)) = c(a) : C(inl(a))"
   225 
   226   PlusC_inr
   227     "[| b: B;  !!x. x:A ==> c(x): C(inl(x));  
   228               !!y. y:B ==> d(y): C(inr(y)) |] 
   229     ==> when(inr(b), %x. c(x), %y. d(y)) = d(b) : C(inr(b))"
   230 
   231 
   232   (*The type Eq*)
   233 
   234   EqF    "[| A type;  a : A;  b : A |] ==> Eq(A,a,b) type"
   235   EqFL "[| A=B;  a=c: A;  b=d : A |] ==> Eq(A,a,b) = Eq(B,c,d)"
   236   EqI "a = b : A ==> eq : Eq(A,a,b)"
   237   EqE "p : Eq(A,a,b) ==> a = b : A"
   238 
   239   (*By equality of types, can prove C(p) from C(eq), an elimination rule*)
   240   EqC "p : Eq(A,a,b) ==> p = eq : Eq(A,a,b)"
   241 
   242   (*The type F*)
   243 
   244   FF "F type"
   245   FE "[| p: F;  C type |] ==> contr(p) : C"
   246   FEL  "[| p = q : F;  C type |] ==> contr(p) = contr(q) : C"
   247 
   248   (*The type T
   249      Martin-Lof's book (page 68) discusses elimination and computation.
   250      Elimination can be derived by computation and equality of types,
   251      but with an extra premise C(x) type x:T.
   252      Also computation can be derived from elimination. *)
   253 
   254   TF "T type"
   255   TI "tt : T"
   256   TE "[| p : T;  c : C(tt) |] ==> c : C(p)"
   257   TEL "[| p = q : T;  c = d : C(tt) |] ==> c = d : C(p)"
   258   TC "p : T ==> p = tt : T"
   259 end
   260 
   261 
   262 ML
   263 
   264 val print_translation =
   265   [("Prod", dependent_tr' ("@PROD", "@-->")),
   266    ("Sum", dependent_tr' ("@SUM", "@*"))];
   267