src/FOLP/IFOLP.thy
 changeset 17480 fd19f77dcf60 parent 14854 61bdf2ae4dc5 child 26322 eaf634e975fa
```     1.1 --- a/src/FOLP/IFOLP.thy	Sat Sep 17 20:49:14 2005 +0200
1.2 +++ b/src/FOLP/IFOLP.thy	Sun Sep 18 14:25:48 2005 +0200
1.3 @@ -2,30 +2,32 @@
1.4      ID:         \$Id\$
1.5      Author:     Martin D Coen, Cambridge University Computer Laboratory
1.6      Copyright   1992  University of Cambridge
1.7 -
1.8 -Intuitionistic First-Order Logic with Proofs
1.9  *)
1.10
1.11 -IFOLP = Pure +
1.12 +header {* Intuitionistic First-Order Logic with Proofs *}
1.13 +
1.14 +theory IFOLP
1.15 +imports Pure
1.16 +begin
1.17
1.18  global
1.19
1.20 -classes term
1.21 -default term
1.22 +classes "term"
1.23 +defaultsort "term"
1.24
1.25 -types
1.26 -  p
1.27 -  o
1.28 +typedecl p
1.29 +typedecl o
1.30
1.31 -consts
1.32 +consts
1.33        (*** Judgements ***)
1.34   "@Proof"       ::   "[p,o]=>prop"      ("(_ /: _)" [51,10] 5)
1.35   Proof          ::   "[o,p]=>prop"
1.36   EqProof        ::   "[p,p,o]=>prop"    ("(3_ /= _ :/ _)" [10,10,10] 5)
1.37 -
1.38 +
1.39        (*** Logical Connectives -- Type Formers ***)
1.40   "="            ::      "['a,'a] => o"  (infixl 50)
1.41 - True,False     ::      "o"
1.42 + True           ::      "o"
1.43 + False          ::      "o"
1.44   Not            ::      "o => o"        ("~ _" [40] 40)
1.45   "&"            ::      "[o,o] => o"    (infixr 35)
1.46   "|"            ::      "[o,o] => o"    (infixr 30)
1.47 @@ -42,10 +44,12 @@
1.48        (*** Proof Term Formers: precedence must exceed 50 ***)
1.49   tt             :: "p"
1.50   contr          :: "p=>p"
1.51 - fst,snd        :: "p=>p"
1.52 + fst            :: "p=>p"
1.53 + snd            :: "p=>p"
1.54   pair           :: "[p,p]=>p"           ("(1<_,/_>)")
1.55   split          :: "[p, [p,p]=>p] =>p"
1.56 - inl,inr        :: "p=>p"
1.57 + inl            :: "p=>p"
1.58 + inr            :: "p=>p"
1.59   when           :: "[p, p=>p, p=>p]=>p"
1.60   lambda         :: "(p => p) => p"      (binder "lam " 55)
1.61   "`"            :: "[p,p]=>p"           (infixl 60)
1.62 @@ -55,98 +59,103 @@
1.63   xsplit         :: "[p,['a,p]=>p]=>p"
1.64   ideq           :: "'a=>p"
1.65   idpeel         :: "[p,'a=>p]=>p"
1.66 - nrm, NRM       :: "p"
1.67 + nrm            :: p
1.68 + NRM            :: p
1.69
1.70  local
1.71
1.72 -rules
1.73 +ML {*
1.74 +
1.75 +(*show_proofs:=true displays the proof terms -- they are ENORMOUS*)
1.76 +val show_proofs = ref false;
1.77 +
1.78 +fun proof_tr [p,P] = Const("Proof",dummyT) \$ P \$ p;
1.79 +
1.80 +fun proof_tr' [P,p] =
1.81 +    if !show_proofs then Const("@Proof",dummyT) \$ p \$ P
1.82 +    else P  (*this case discards the proof term*);
1.83 +*}
1.84 +
1.85 +parse_translation {* [("@Proof", proof_tr)] *}
1.86 +print_translation {* [("Proof", proof_tr')] *}
1.87 +
1.88 +axioms
1.89
1.90  (**** Propositional logic ****)
1.91
1.92  (*Equality*)
1.93  (* Like Intensional Equality in MLTT - but proofs distinct from terms *)
1.94
1.95 -ieqI      "ideq(a) : a=a"
1.96 -ieqE      "[| p : a=b;  !!x. f(x) : P(x,x) |] ==> idpeel(p,f) : P(a,b)"
1.97 +ieqI:      "ideq(a) : a=a"
1.98 +ieqE:      "[| p : a=b;  !!x. f(x) : P(x,x) |] ==> idpeel(p,f) : P(a,b)"
1.99
1.100  (* Truth and Falsity *)
1.101
1.102 -TrueI     "tt : True"
1.103 -FalseE    "a:False ==> contr(a):P"
1.104 +TrueI:     "tt : True"
1.105 +FalseE:    "a:False ==> contr(a):P"
1.106
1.107  (* Conjunction *)
1.108
1.109 -conjI     "[| a:P;  b:Q |] ==> <a,b> : P&Q"
1.110 -conjunct1 "p:P&Q ==> fst(p):P"
1.111 -conjunct2 "p:P&Q ==> snd(p):Q"
1.112 +conjI:     "[| a:P;  b:Q |] ==> <a,b> : P&Q"
1.113 +conjunct1: "p:P&Q ==> fst(p):P"
1.114 +conjunct2: "p:P&Q ==> snd(p):Q"
1.115
1.116  (* Disjunction *)
1.117
1.118 -disjI1    "a:P ==> inl(a):P|Q"
1.119 -disjI2    "b:Q ==> inr(b):P|Q"
1.120 -disjE     "[| a:P|Q;  !!x. x:P ==> f(x):R;  !!x. x:Q ==> g(x):R
1.121 -          |] ==> when(a,f,g):R"
1.122 +disjI1:    "a:P ==> inl(a):P|Q"
1.123 +disjI2:    "b:Q ==> inr(b):P|Q"
1.124 +disjE:     "[| a:P|Q;  !!x. x:P ==> f(x):R;  !!x. x:Q ==> g(x):R
1.125 +           |] ==> when(a,f,g):R"
1.126
1.127  (* Implication *)
1.128
1.129 -impI      "(!!x. x:P ==> f(x):Q) ==> lam x. f(x):P-->Q"
1.130 -mp        "[| f:P-->Q;  a:P |] ==> f`a:Q"
1.131 +impI:      "(!!x. x:P ==> f(x):Q) ==> lam x. f(x):P-->Q"
1.132 +mp:        "[| f:P-->Q;  a:P |] ==> f`a:Q"
1.133
1.134  (*Quantifiers*)
1.135
1.136 -allI      "(!!x. f(x) : P(x)) ==> all x. f(x) : ALL x. P(x)"
1.137 -spec      "(f:ALL x. P(x)) ==> f^x : P(x)"
1.138 +allI:      "(!!x. f(x) : P(x)) ==> all x. f(x) : ALL x. P(x)"
1.139 +spec:      "(f:ALL x. P(x)) ==> f^x : P(x)"
1.140
1.141 -exI       "p : P(x) ==> [x,p] : EX x. P(x)"
1.142 -exE       "[| p: EX x. P(x);  !!x u. u:P(x) ==> f(x,u) : R |] ==> xsplit(p,f):R"
1.143 +exI:       "p : P(x) ==> [x,p] : EX x. P(x)"
1.144 +exE:       "[| p: EX x. P(x);  !!x u. u:P(x) ==> f(x,u) : R |] ==> xsplit(p,f):R"
1.145
1.146  (**** Equality between proofs ****)
1.147
1.148 -prefl     "a : P ==> a = a : P"
1.149 -psym      "a = b : P ==> b = a : P"
1.150 -ptrans    "[| a = b : P;  b = c : P |] ==> a = c : P"
1.151 +prefl:     "a : P ==> a = a : P"
1.152 +psym:      "a = b : P ==> b = a : P"
1.153 +ptrans:    "[| a = b : P;  b = c : P |] ==> a = c : P"
1.154
1.155 -idpeelB   "[| !!x. f(x) : P(x,x) |] ==> idpeel(ideq(a),f) = f(a) : P(a,a)"
1.156 +idpeelB:   "[| !!x. f(x) : P(x,x) |] ==> idpeel(ideq(a),f) = f(a) : P(a,a)"
1.157
1.158 -fstB      "a:P ==> fst(<a,b>) = a : P"
1.159 -sndB      "b:Q ==> snd(<a,b>) = b : Q"
1.160 -pairEC    "p:P&Q ==> p = <fst(p),snd(p)> : P&Q"
1.161 +fstB:      "a:P ==> fst(<a,b>) = a : P"
1.162 +sndB:      "b:Q ==> snd(<a,b>) = b : Q"
1.163 +pairEC:    "p:P&Q ==> p = <fst(p),snd(p)> : P&Q"
1.164
1.165 -whenBinl  "[| a:P;  !!x. x:P ==> f(x) : Q |] ==> when(inl(a),f,g) = f(a) : Q"
1.166 -whenBinr  "[| b:P;  !!x. x:P ==> g(x) : Q |] ==> when(inr(b),f,g) = g(b) : Q"
1.167 -plusEC    "a:P|Q ==> when(a,%x. inl(x),%y. inr(y)) = a : P|Q"
1.168 +whenBinl:  "[| a:P;  !!x. x:P ==> f(x) : Q |] ==> when(inl(a),f,g) = f(a) : Q"
1.169 +whenBinr:  "[| b:P;  !!x. x:P ==> g(x) : Q |] ==> when(inr(b),f,g) = g(b) : Q"
1.170 +plusEC:    "a:P|Q ==> when(a,%x. inl(x),%y. inr(y)) = a : P|Q"
1.171
1.172 -applyB     "[| a:P;  !!x. x:P ==> b(x) : Q |] ==> (lam x. b(x)) ` a = b(a) : Q"
1.173 -funEC      "f:P ==> f = lam x. f`x : P"
1.174 +applyB:     "[| a:P;  !!x. x:P ==> b(x) : Q |] ==> (lam x. b(x)) ` a = b(a) : Q"
1.175 +funEC:      "f:P ==> f = lam x. f`x : P"
1.176
1.177 -specB      "[| !!x. f(x) : P(x) |] ==> (all x. f(x)) ^ a = f(a) : P(a)"
1.178 +specB:      "[| !!x. f(x) : P(x) |] ==> (all x. f(x)) ^ a = f(a) : P(a)"
1.179
1.180
1.181  (**** Definitions ****)
1.182
1.183 -not_def              "~P == P-->False"
1.184 -iff_def         "P<->Q == (P-->Q) & (Q-->P)"
1.185 +not_def:              "~P == P-->False"
1.186 +iff_def:         "P<->Q == (P-->Q) & (Q-->P)"
1.187
1.188  (*Unique existence*)
1.189 -ex1_def   "EX! x. P(x) == EX x. P(x) & (ALL y. P(y) --> y=x)"
1.190 +ex1_def:   "EX! x. P(x) == EX x. P(x) & (ALL y. P(y) --> y=x)"
1.191
1.192  (*Rewriting -- special constants to flag normalized terms and formulae*)
1.193 -norm_eq "nrm : norm(x) = x"
1.194 -NORM_iff        "NRM : NORM(P) <-> P"
1.195 +norm_eq: "nrm : norm(x) = x"
1.196 +NORM_iff:        "NRM : NORM(P) <-> P"
1.197 +
1.198 +ML {* use_legacy_bindings (the_context ()) *}
1.199
1.200  end
1.201
1.202 -ML
1.203
1.204 -(*show_proofs:=true displays the proof terms -- they are ENORMOUS*)
1.205 -val show_proofs = ref false;
1.206 -
1.207 -fun proof_tr [p,P] = Const("Proof",dummyT) \$ P \$ p;
1.208 -
1.209 -fun proof_tr' [P,p] =
1.210 -    if !show_proofs then Const("@Proof",dummyT) \$ p \$ P
1.211 -    else P  (*this case discards the proof term*);
1.212 -
1.213 -val  parse_translation = [("@Proof", proof_tr)];
1.214 -val print_translation  = [("Proof", proof_tr')];
1.215 -
```