src/Sequents/LK0.thy
 changeset 17481 75166ebb619b parent 16019 0e1405402d53 child 21426 87ac12bed1ab
```     1.1 --- a/src/Sequents/LK0.thy	Sun Sep 18 14:25:48 2005 +0200
1.2 +++ b/src/Sequents/LK0.thy	Sun Sep 18 15:20:08 2005 +0200
1.3 @@ -1,136 +1,139 @@
1.4 -(*  Title:      LK/LK0
1.5 +(*  Title:      LK/LK0.thy
1.6      ID:         \$Id\$
1.7      Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
1.8      Copyright   1993  University of Cambridge
1.9
1.10 -Classical First-Order Sequent Calculus
1.11 -
1.12  There may be printing problems if a seqent is in expanded normal form
1.13 -	(eta-expanded, beta-contracted)
1.14 +        (eta-expanded, beta-contracted)
1.15  *)
1.16
1.17 -LK0 = Sequents +
1.18 +header {* Classical First-Order Sequent Calculus *}
1.19 +
1.20 +theory LK0
1.21 +imports Sequents
1.22 +begin
1.23
1.24  global
1.25
1.26 -classes term
1.27 -default term
1.28 +classes "term"
1.29 +defaultsort "term"
1.30
1.31  consts
1.32
1.33 - Trueprop	:: "two_seqi"
1.34 + Trueprop       :: "two_seqi"
1.35
1.36 -  True,False   :: o
1.37 -  "="          :: ['a,'a] => o       (infixl 50)
1.38 -  Not          :: o => o             ("~ _" [40] 40)
1.39 -  "&"          :: [o,o] => o         (infixr 35)
1.40 -  "|"          :: [o,o] => o         (infixr 30)
1.41 -  "-->","<->"  :: [o,o] => o         (infixr 25)
1.42 -  The          :: ('a => o) => 'a    (binder "THE " 10)
1.43 -  All          :: ('a => o) => o     (binder "ALL " 10)
1.44 -  Ex           :: ('a => o) => o     (binder "EX " 10)
1.45 +  True         :: o
1.46 +  False        :: o
1.47 +  "="          :: "['a,'a] => o"     (infixl 50)
1.48 +  Not          :: "o => o"           ("~ _" [40] 40)
1.49 +  "&"          :: "[o,o] => o"       (infixr 35)
1.50 +  "|"          :: "[o,o] => o"       (infixr 30)
1.51 +  "-->"        :: "[o,o] => o"       (infixr 25)
1.52 +  "<->"        :: "[o,o] => o"       (infixr 25)
1.53 +  The          :: "('a => o) => 'a"  (binder "THE " 10)
1.54 +  All          :: "('a => o) => o"   (binder "ALL " 10)
1.55 +  Ex           :: "('a => o) => o"   (binder "EX " 10)
1.56
1.57  syntax
1.58 - "@Trueprop"	:: "two_seqe" ("((_)/ |- (_))" [6,6] 5)
1.59 -  "_not_equal" :: ['a, 'a] => o                (infixl "~=" 50)
1.60 + "@Trueprop"    :: "two_seqe" ("((_)/ |- (_))" [6,6] 5)
1.61 +  "_not_equal" :: "['a, 'a] => o"              (infixl "~=" 50)
1.62 +
1.63 +parse_translation {* [("@Trueprop", two_seq_tr "Trueprop")] *}
1.64 +print_translation {* [("Trueprop", two_seq_tr' "@Trueprop")] *}
1.65
1.66  translations
1.67    "x ~= y"      == "~ (x = y)"
1.68
1.69  syntax (xsymbols)
1.70 -  Not           :: o => o               ("\\<not> _" [40] 40)
1.71 -  "op &"        :: [o, o] => o          (infixr "\\<and>" 35)
1.72 -  "op |"        :: [o, o] => o          (infixr "\\<or>" 30)
1.73 -  "op -->"      :: [o, o] => o          (infixr "\\<longrightarrow>" 25)
1.74 -  "op <->"      :: [o, o] => o          (infixr "\\<longleftrightarrow>" 25)
1.75 -  "ALL "        :: [idts, o] => o       ("(3\\<forall>_./ _)" [0, 10] 10)
1.76 -  "EX "         :: [idts, o] => o       ("(3\\<exists>_./ _)" [0, 10] 10)
1.77 -  "EX! "        :: [idts, o] => o       ("(3\\<exists>!_./ _)" [0, 10] 10)
1.78 -  "_not_equal"  :: ['a, 'a] => o        (infixl "\\<noteq>" 50)
1.79 +  Not           :: "o => o"               ("\<not> _" [40] 40)
1.80 +  "op &"        :: "[o, o] => o"          (infixr "\<and>" 35)
1.81 +  "op |"        :: "[o, o] => o"          (infixr "\<or>" 30)
1.82 +  "op -->"      :: "[o, o] => o"          (infixr "\<longrightarrow>" 25)
1.83 +  "op <->"      :: "[o, o] => o"          (infixr "\<longleftrightarrow>" 25)
1.84 +  "ALL "        :: "[idts, o] => o"       ("(3\<forall>_./ _)" [0, 10] 10)
1.85 +  "EX "         :: "[idts, o] => o"       ("(3\<exists>_./ _)" [0, 10] 10)
1.86 +  "EX! "        :: "[idts, o] => o"       ("(3\<exists>!_./ _)" [0, 10] 10)
1.87 +  "_not_equal"  :: "['a, 'a] => o"        (infixl "\<noteq>" 50)
1.88
1.89  syntax (HTML output)
1.90 -  Not           :: o => o               ("\\<not> _" [40] 40)
1.91 -  "op &"        :: [o, o] => o          (infixr "\\<and>" 35)
1.92 -  "op |"        :: [o, o] => o          (infixr "\\<or>" 30)
1.93 -  "ALL "        :: [idts, o] => o       ("(3\\<forall>_./ _)" [0, 10] 10)
1.94 -  "EX "         :: [idts, o] => o       ("(3\\<exists>_./ _)" [0, 10] 10)
1.95 -  "EX! "        :: [idts, o] => o       ("(3\\<exists>!_./ _)" [0, 10] 10)
1.96 -  "_not_equal"  :: ['a, 'a] => o        (infixl "\\<noteq>" 50)
1.97 -
1.98 +  Not           :: "o => o"               ("\<not> _" [40] 40)
1.99 +  "op &"        :: "[o, o] => o"          (infixr "\<and>" 35)
1.100 +  "op |"        :: "[o, o] => o"          (infixr "\<or>" 30)
1.101 +  "ALL "        :: "[idts, o] => o"       ("(3\<forall>_./ _)" [0, 10] 10)
1.102 +  "EX "         :: "[idts, o] => o"       ("(3\<exists>_./ _)" [0, 10] 10)
1.103 +  "EX! "        :: "[idts, o] => o"       ("(3\<exists>!_./ _)" [0, 10] 10)
1.104 +  "_not_equal"  :: "['a, 'a] => o"        (infixl "\<noteq>" 50)
1.105
1.106  local
1.107 -
1.108 -rules
1.109 +
1.110 +axioms
1.111
1.112    (*Structural rules: contraction, thinning, exchange [Soren Heilmann] *)
1.113
1.114 -  contRS "\$H |- \$E, \$S, \$S, \$F ==> \$H |- \$E, \$S, \$F"
1.115 -  contLS "\$H, \$S, \$S, \$G |- \$E ==> \$H, \$S, \$G |- \$E"
1.116 +  contRS: "\$H |- \$E, \$S, \$S, \$F ==> \$H |- \$E, \$S, \$F"
1.117 +  contLS: "\$H, \$S, \$S, \$G |- \$E ==> \$H, \$S, \$G |- \$E"
1.118
1.119 -  thinRS "\$H |- \$E, \$F ==> \$H |- \$E, \$S, \$F"
1.120 -  thinLS "\$H, \$G |- \$E ==> \$H, \$S, \$G |- \$E"
1.121 +  thinRS: "\$H |- \$E, \$F ==> \$H |- \$E, \$S, \$F"
1.122 +  thinLS: "\$H, \$G |- \$E ==> \$H, \$S, \$G |- \$E"
1.123
1.124 -  exchRS "\$H |- \$E, \$R, \$S, \$F ==> \$H |- \$E, \$S, \$R, \$F"
1.125 -  exchLS "\$H, \$R, \$S, \$G |- \$E ==> \$H, \$S, \$R, \$G |- \$E"
1.126 +  exchRS: "\$H |- \$E, \$R, \$S, \$F ==> \$H |- \$E, \$S, \$R, \$F"
1.127 +  exchLS: "\$H, \$R, \$S, \$G |- \$E ==> \$H, \$S, \$R, \$G |- \$E"
1.128
1.129 -  cut   "[| \$H |- \$E, P;  \$H, P |- \$E |] ==> \$H |- \$E"
1.130 +  cut:   "[| \$H |- \$E, P;  \$H, P |- \$E |] ==> \$H |- \$E"
1.131
1.132    (*Propositional rules*)
1.133
1.134 -  basic "\$H, P, \$G |- \$E, P, \$F"
1.135 +  basic: "\$H, P, \$G |- \$E, P, \$F"
1.136
1.137 -  conjR "[| \$H|- \$E, P, \$F;  \$H|- \$E, Q, \$F |] ==> \$H|- \$E, P&Q, \$F"
1.138 -  conjL "\$H, P, Q, \$G |- \$E ==> \$H, P & Q, \$G |- \$E"
1.139 +  conjR: "[| \$H|- \$E, P, \$F;  \$H|- \$E, Q, \$F |] ==> \$H|- \$E, P&Q, \$F"
1.140 +  conjL: "\$H, P, Q, \$G |- \$E ==> \$H, P & Q, \$G |- \$E"
1.141
1.142 -  disjR "\$H |- \$E, P, Q, \$F ==> \$H |- \$E, P|Q, \$F"
1.143 -  disjL "[| \$H, P, \$G |- \$E;  \$H, Q, \$G |- \$E |] ==> \$H, P|Q, \$G |- \$E"
1.144 +  disjR: "\$H |- \$E, P, Q, \$F ==> \$H |- \$E, P|Q, \$F"
1.145 +  disjL: "[| \$H, P, \$G |- \$E;  \$H, Q, \$G |- \$E |] ==> \$H, P|Q, \$G |- \$E"
1.146
1.147 -  impR  "\$H, P |- \$E, Q, \$F ==> \$H |- \$E, P-->Q, \$F"
1.148 -  impL  "[| \$H,\$G |- \$E,P;  \$H, Q, \$G |- \$E |] ==> \$H, P-->Q, \$G |- \$E"
1.149 +  impR:  "\$H, P |- \$E, Q, \$F ==> \$H |- \$E, P-->Q, \$F"
1.150 +  impL:  "[| \$H,\$G |- \$E,P;  \$H, Q, \$G |- \$E |] ==> \$H, P-->Q, \$G |- \$E"
1.151
1.152 -  notR  "\$H, P |- \$E, \$F ==> \$H |- \$E, ~P, \$F"
1.153 -  notL  "\$H, \$G |- \$E, P ==> \$H, ~P, \$G |- \$E"
1.154 +  notR:  "\$H, P |- \$E, \$F ==> \$H |- \$E, ~P, \$F"
1.155 +  notL:  "\$H, \$G |- \$E, P ==> \$H, ~P, \$G |- \$E"
1.156
1.157 -  FalseL "\$H, False, \$G |- \$E"
1.158 +  FalseL: "\$H, False, \$G |- \$E"
1.159
1.160 -  True_def "True == False-->False"
1.161 -  iff_def  "P<->Q == (P-->Q) & (Q-->P)"
1.162 +  True_def: "True == False-->False"
1.163 +  iff_def:  "P<->Q == (P-->Q) & (Q-->P)"
1.164
1.165    (*Quantifiers*)
1.166
1.167 -  allR  "(!!x.\$H |- \$E, P(x), \$F) ==> \$H |- \$E, ALL x. P(x), \$F"
1.168 -  allL  "\$H, P(x), \$G, ALL x. P(x) |- \$E ==> \$H, ALL x. P(x), \$G |- \$E"
1.169 +  allR:  "(!!x.\$H |- \$E, P(x), \$F) ==> \$H |- \$E, ALL x. P(x), \$F"
1.170 +  allL:  "\$H, P(x), \$G, ALL x. P(x) |- \$E ==> \$H, ALL x. P(x), \$G |- \$E"
1.171
1.172 -  exR   "\$H |- \$E, P(x), \$F, EX x. P(x) ==> \$H |- \$E, EX x. P(x), \$F"
1.173 -  exL   "(!!x.\$H, P(x), \$G |- \$E) ==> \$H, EX x. P(x), \$G |- \$E"
1.174 +  exR:   "\$H |- \$E, P(x), \$F, EX x. P(x) ==> \$H |- \$E, EX x. P(x), \$F"
1.175 +  exL:   "(!!x.\$H, P(x), \$G |- \$E) ==> \$H, EX x. P(x), \$G |- \$E"
1.176
1.177    (*Equality*)
1.178
1.179 -  refl  "\$H |- \$E, a=a, \$F"
1.180 -  subst "\$H(a), \$G(a) |- \$E(a) ==> \$H(b), a=b, \$G(b) |- \$E(b)"
1.181 +  refl:  "\$H |- \$E, a=a, \$F"
1.182 +  subst: "\$H(a), \$G(a) |- \$E(a) ==> \$H(b), a=b, \$G(b) |- \$E(b)"
1.183
1.184    (* Reflection *)
1.185
1.186 -  eq_reflection  "|- x=y ==> (x==y)"
1.187 -  iff_reflection "|- P<->Q ==> (P==Q)"
1.188 +  eq_reflection:  "|- x=y ==> (x==y)"
1.189 +  iff_reflection: "|- P<->Q ==> (P==Q)"
1.190
1.191    (*Descriptions*)
1.192
1.193 -  The "[| \$H |- \$E, P(a), \$F;  !!x.\$H, P(x) |- \$E, x=a, \$F |] ==>
1.194 +  The: "[| \$H |- \$E, P(a), \$F;  !!x.\$H, P(x) |- \$E, x=a, \$F |] ==>
1.195            \$H |- \$E, P(THE x. P(x)), \$F"
1.196
1.197  constdefs
1.198 -  If :: [o, 'a, 'a] => 'a   ("(if (_)/ then (_)/ else (_))" 10)
1.199 +  If :: "[o, 'a, 'a] => 'a"   ("(if (_)/ then (_)/ else (_))" 10)
1.200     "If(P,x,y) == THE z::'a. (P --> z=x) & (~P --> z=y)"
1.201
1.202  setup
1.203    prover_setup
1.204
1.205 +ML {* use_legacy_bindings (the_context ()) *}
1.206 +
1.207  end
1.208
1.209 -ML
1.210
1.211 -
1.212 -val parse_translation = [("@Trueprop",Sequents.two_seq_tr "Trueprop")];
1.213 -val print_translation = [("Trueprop",Sequents.two_seq_tr' "@Trueprop")];
1.214 -
```