src/Sequents/LK0.thy
 author wenzelm Tue, 13 Jun 2006 23:41:39 +0200 changeset 19876 11d447d5d68c parent 17481 75166ebb619b child 21426 87ac12bed1ab permissions -rw-r--r--
tuned;
```
(*  Title:      LK/LK0.thy
ID:         \$Id\$
Author:     Lawrence C Paulson, Cambridge University Computer Laboratory

There may be printing problems if a seqent is in expanded normal form
(eta-expanded, beta-contracted)
*)

header {* Classical First-Order Sequent Calculus *}

theory LK0
imports Sequents
begin

global

classes "term"
defaultsort "term"

consts

Trueprop       :: "two_seqi"

True         :: o
False        :: o
"="          :: "['a,'a] => o"     (infixl 50)
Not          :: "o => o"           ("~ _" [40] 40)
"&"          :: "[o,o] => o"       (infixr 35)
"|"          :: "[o,o] => o"       (infixr 30)
"-->"        :: "[o,o] => o"       (infixr 25)
"<->"        :: "[o,o] => o"       (infixr 25)
The          :: "('a => o) => 'a"  (binder "THE " 10)
All          :: "('a => o) => o"   (binder "ALL " 10)
Ex           :: "('a => o) => o"   (binder "EX " 10)

syntax
"@Trueprop"    :: "two_seqe" ("((_)/ |- (_))" [6,6] 5)
"_not_equal" :: "['a, 'a] => o"              (infixl "~=" 50)

parse_translation {* [("@Trueprop", two_seq_tr "Trueprop")] *}
print_translation {* [("Trueprop", two_seq_tr' "@Trueprop")] *}

translations
"x ~= y"      == "~ (x = y)"

syntax (xsymbols)
Not           :: "o => o"               ("\<not> _" [40] 40)
"op &"        :: "[o, o] => o"          (infixr "\<and>" 35)
"op |"        :: "[o, o] => o"          (infixr "\<or>" 30)
"op -->"      :: "[o, o] => o"          (infixr "\<longrightarrow>" 25)
"op <->"      :: "[o, o] => o"          (infixr "\<longleftrightarrow>" 25)
"ALL "        :: "[idts, o] => o"       ("(3\<forall>_./ _)" [0, 10] 10)
"EX "         :: "[idts, o] => o"       ("(3\<exists>_./ _)" [0, 10] 10)
"EX! "        :: "[idts, o] => o"       ("(3\<exists>!_./ _)" [0, 10] 10)
"_not_equal"  :: "['a, 'a] => o"        (infixl "\<noteq>" 50)

syntax (HTML output)
Not           :: "o => o"               ("\<not> _" [40] 40)
"op &"        :: "[o, o] => o"          (infixr "\<and>" 35)
"op |"        :: "[o, o] => o"          (infixr "\<or>" 30)
"ALL "        :: "[idts, o] => o"       ("(3\<forall>_./ _)" [0, 10] 10)
"EX "         :: "[idts, o] => o"       ("(3\<exists>_./ _)" [0, 10] 10)
"EX! "        :: "[idts, o] => o"       ("(3\<exists>!_./ _)" [0, 10] 10)
"_not_equal"  :: "['a, 'a] => o"        (infixl "\<noteq>" 50)

local

axioms

(*Structural rules: contraction, thinning, exchange [Soren Heilmann] *)

contRS: "\$H |- \$E, \$S, \$S, \$F ==> \$H |- \$E, \$S, \$F"
contLS: "\$H, \$S, \$S, \$G |- \$E ==> \$H, \$S, \$G |- \$E"

thinRS: "\$H |- \$E, \$F ==> \$H |- \$E, \$S, \$F"
thinLS: "\$H, \$G |- \$E ==> \$H, \$S, \$G |- \$E"

exchRS: "\$H |- \$E, \$R, \$S, \$F ==> \$H |- \$E, \$S, \$R, \$F"
exchLS: "\$H, \$R, \$S, \$G |- \$E ==> \$H, \$S, \$R, \$G |- \$E"

cut:   "[| \$H |- \$E, P;  \$H, P |- \$E |] ==> \$H |- \$E"

(*Propositional rules*)

basic: "\$H, P, \$G |- \$E, P, \$F"

conjR: "[| \$H|- \$E, P, \$F;  \$H|- \$E, Q, \$F |] ==> \$H|- \$E, P&Q, \$F"
conjL: "\$H, P, Q, \$G |- \$E ==> \$H, P & Q, \$G |- \$E"

disjR: "\$H |- \$E, P, Q, \$F ==> \$H |- \$E, P|Q, \$F"
disjL: "[| \$H, P, \$G |- \$E;  \$H, Q, \$G |- \$E |] ==> \$H, P|Q, \$G |- \$E"

impR:  "\$H, P |- \$E, Q, \$F ==> \$H |- \$E, P-->Q, \$F"
impL:  "[| \$H,\$G |- \$E,P;  \$H, Q, \$G |- \$E |] ==> \$H, P-->Q, \$G |- \$E"

notR:  "\$H, P |- \$E, \$F ==> \$H |- \$E, ~P, \$F"
notL:  "\$H, \$G |- \$E, P ==> \$H, ~P, \$G |- \$E"

FalseL: "\$H, False, \$G |- \$E"

True_def: "True == False-->False"
iff_def:  "P<->Q == (P-->Q) & (Q-->P)"

(*Quantifiers*)

allR:  "(!!x.\$H |- \$E, P(x), \$F) ==> \$H |- \$E, ALL x. P(x), \$F"
allL:  "\$H, P(x), \$G, ALL x. P(x) |- \$E ==> \$H, ALL x. P(x), \$G |- \$E"

exR:   "\$H |- \$E, P(x), \$F, EX x. P(x) ==> \$H |- \$E, EX x. P(x), \$F"
exL:   "(!!x.\$H, P(x), \$G |- \$E) ==> \$H, EX x. P(x), \$G |- \$E"

(*Equality*)

refl:  "\$H |- \$E, a=a, \$F"
subst: "\$H(a), \$G(a) |- \$E(a) ==> \$H(b), a=b, \$G(b) |- \$E(b)"

(* Reflection *)

eq_reflection:  "|- x=y ==> (x==y)"
iff_reflection: "|- P<->Q ==> (P==Q)"

(*Descriptions*)

The: "[| \$H |- \$E, P(a), \$F;  !!x.\$H, P(x) |- \$E, x=a, \$F |] ==>
\$H |- \$E, P(THE x. P(x)), \$F"

constdefs
If :: "[o, 'a, 'a] => 'a"   ("(if (_)/ then (_)/ else (_))" 10)
"If(P,x,y) == THE z::'a. (P --> z=x) & (~P --> z=y)"

setup
prover_setup

ML {* use_legacy_bindings (the_context ()) *}

end

```