src/FOLP/ifolp.thy
author paulson
Mon Apr 26 13:25:49 1999 +0200 (1999-04-26)
changeset 6509 9f7f4fd05b1f
parent 283 76caebd18756
permissions -rw-r--r--
fixed a bug many years old in rule plusEC
     1 IFOLP = Pure +
     2 
     3 classes term < logic
     4 
     5 default term
     6 
     7 types
     8   p
     9   o
    10 
    11 arities
    12   p,o :: logic
    13 
    14 consts	
    15       (*** Judgements ***)
    16  "@Proof"   	::   "[p,o]=>prop"	("(_ /: _)" [10,10] 5)
    17  Proof  	::   "[o,p]=>prop"
    18  EqProof        ::   "[p,p,o]=>prop"    ("(3_ /= _ :/ _)" [10,10,10] 5)
    19         
    20       (*** Logical Connectives -- Type Formers ***)
    21  "="		::	"['a,'a] => o"	(infixl 50)
    22  True,False	::	"o"
    23  "Not"		::	"o => o"	("~ _" [40] 40)
    24  "&"		::	"[o,o] => o"	(infixr 35)
    25  "|"		::	"[o,o] => o"	(infixr 30)
    26  "-->"		::	"[o,o] => o"	(infixr 25)
    27  "<->"		::	"[o,o] => o"	(infixr 25)
    28       (*Quantifiers*)
    29  All		::	"('a => o) => o"	(binder "ALL " 10)
    30  Ex		::	"('a => o) => o"	(binder "EX " 10)
    31  Ex1		::	"('a => o) => o"	(binder "EX! " 10)
    32       (*Rewriting gadgets*)
    33  NORM		::	"o => o"
    34  norm		::	"'a => 'a"
    35 
    36       (*** Proof Term Formers ***)
    37  tt		:: "p"
    38  contr		:: "p=>p"
    39  fst,snd	:: "p=>p"
    40  pair		:: "[p,p]=>p"		("(1<_,/_>)")
    41  split		:: "[p, [p,p]=>p] =>p"
    42  inl,inr	:: "p=>p"
    43  when		:: "[p, p=>p, p=>p]=>p"
    44  lambda		:: "(p => p) => p"	(binder "lam " 20)
    45  "`"		:: "[p,p]=>p"		(infixl 60)
    46  alll           :: "['a=>p]=>p"         (binder "all " 15)
    47  "^"            :: "[p,'a]=>p"          (infixl 50)
    48  exists		:: "['a,p]=>p"		("(1[_,/_])")
    49  xsplit         :: "[p,['a,p]=>p]=>p"
    50  ideq           :: "'a=>p"
    51  idpeel         :: "[p,'a=>p]=>p"
    52  nrm, NRM       :: "p"
    53 
    54 rules
    55 
    56 (**** Propositional logic ****)
    57 
    58 (*Equality*)
    59 (* Like Intensional Equality in MLTT - but proofs distinct from terms *)
    60 
    61 ieqI	  "ideq(a) : a=a"
    62 ieqE      "[| p : a=b;  !!x.f(x) : P(x,x) |] ==> idpeel(p,f) : P(a,b)"
    63 
    64 (* Truth and Falsity *)
    65 
    66 TrueI     "tt : True"
    67 FalseE    "a:False ==> contr(a):P"
    68 
    69 (* Conjunction *)
    70 
    71 conjI     "[| a:P;  b:Q |] ==> <a,b> : P&Q"
    72 conjunct1 "p:P&Q ==> fst(p):P"
    73 conjunct2 "p:P&Q ==> snd(p):Q"
    74 
    75 (* Disjunction *)
    76 
    77 disjI1    "a:P ==> inl(a):P|Q"
    78 disjI2    "b:Q ==> inr(b):P|Q"
    79 disjE     "[| a:P|Q;  !!x.x:P ==> f(x):R;  !!x.x:Q ==> g(x):R \
    80 \          |] ==> when(a,f,g):R"
    81 
    82 (* Implication *)
    83 
    84 impI      "(!!x.x:P ==> f(x):Q) ==> lam x.f(x):P-->Q"
    85 mp        "[| f:P-->Q;  a:P |] ==> f`a:Q"
    86 
    87 (*Quantifiers*)
    88 
    89 allI	  "(!!x. f(x) : P(x)) ==> all x.f(x) : ALL x.P(x)"
    90 spec	  "(f:ALL x.P(x)) ==> f^x : P(x)"
    91 
    92 exI	  "p : P(x) ==> [x,p] : EX x.P(x)"
    93 exE	  "[| p: EX x.P(x);  !!x u. u:P(x) ==> f(x,u) : R |] ==> xsplit(p,f):R"
    94 
    95 (**** Equality between proofs ****)
    96 
    97 prefl     "a : P ==> a = a : P"
    98 psym      "a = b : P ==> b = a : P"
    99 ptrans    "[| a = b : P;  b = c : P |] ==> a = c : P"
   100 
   101 idpeelB   "[| !!x.f(x) : P(x,x) |] ==> idpeel(ideq(a),f) = f(a) : P(a,a)"
   102 
   103 fstB      "a:P ==> fst(<a,b>) = a : P"
   104 sndB      "b:Q ==> snd(<a,b>) = b : Q"
   105 pairEC    "p:P&Q ==> p = <fst(p),snd(p)> : P&Q"
   106 
   107 whenBinl  "[| a:P;  !!x.x:P ==> f(x) : Q |] ==> when(inl(a),f,g) = f(a) : Q"
   108 whenBinr  "[| b:P;  !!x.x:P ==> g(x) : Q |] ==> when(inr(b),f,g) = g(b) : Q"
   109 plusEC    "a:P|Q ==> when(a,%x.inl(x),%y.inr(y)) = p : P|Q"
   110 
   111 applyB     "[| a:P;  !!x.x:P ==> b(x) : Q |] ==> (lam x.b(x)) ` a = b(a) : Q"
   112 funEC      "f:P ==> f = lam x.f`x : P"
   113 
   114 specB      "[| !!x.f(x) : P(x) |] ==> (all x.f(x)) ^ a = f(a) : P(a)"
   115 
   116 
   117 (**** Definitions ****)
   118 
   119 not_def 	     "~P == P-->False"
   120 iff_def         "P<->Q == (P-->Q) & (Q-->P)"
   121 
   122 (*Unique existence*)
   123 ex1_def   "EX! x. P(x) == EX x. P(x) & (ALL y. P(y) --> y=x)"
   124 
   125 (*Rewriting -- special constants to flag normalized terms and formulae*)
   126 norm_eq	"nrm : norm(x) = x"
   127 NORM_iff	"NRM : NORM(P) <-> P"
   128 
   129 end
   130 
   131 ML
   132 
   133 (*show_proofs:=true displays the proof terms -- they are ENORMOUS*)
   134 val show_proofs = ref false;
   135 
   136 fun proof_tr [p,P] = Const("Proof",dummyT) $ P $ p;
   137 
   138 fun proof_tr' [P,p] = 
   139     if !show_proofs then Const("@Proof",dummyT) $ p $ P 
   140     else P  (*this case discards the proof term*);
   141 
   142 val  parse_translation = [("@Proof", proof_tr)];
   143 val print_translation  = [("Proof", proof_tr')];
   144