(* Title: Sequents/LK0.thy
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1993 University of Cambridge
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
class "term"
default_sort "term"
consts
Trueprop :: "two_seqi"
True :: o
False :: o
equal :: "['a,'a] => o" (infixl "=" 50)
Not :: "o => o" ("~ _" [40] 40)
conj :: "[o,o] => o" (infixr "&" 35)
disj :: "[o,o] => o" (infixr "|" 30)
imp :: "[o,o] => o" (infixr "-->" 25)
iff :: "[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)
parse_translation {* [(@{syntax_const "_Trueprop"}, K (two_seq_tr @{const_syntax Trueprop}))] *}
print_translation {* [(@{const_syntax Trueprop}, K (two_seq_tr' @{syntax_const "_Trueprop"}))] *}
abbreviation
not_equal (infixl "~=" 50) where
"x ~= y == ~ (x = y)"
notation (xsymbols)
Not ("\<not> _" [40] 40) and
conj (infixr "\<and>" 35) and
disj (infixr "\<or>" 30) and
imp (infixr "\<longrightarrow>" 25) and
iff (infixr "\<longleftrightarrow>" 25) and
All (binder "\<forall>" 10) and
Ex (binder "\<exists>" 10) and
not_equal (infixl "\<noteq>" 50)
notation (HTML output)
Not ("\<not> _" [40] 40) and
conj (infixr "\<and>" 35) and
disj (infixr "\<or>" 30) and
All (binder "\<forall>" 10) and
Ex (binder "\<exists>" 10) and
not_equal (infixl "\<noteq>" 50)
axiomatization where
(*Structural rules: contraction, thinning, exchange [Soren Heilmann] *)
contRS: "$H |- $E, $S, $S, $F ==> $H |- $E, $S, $F" and
contLS: "$H, $S, $S, $G |- $E ==> $H, $S, $G |- $E" and
thinRS: "$H |- $E, $F ==> $H |- $E, $S, $F" and
thinLS: "$H, $G |- $E ==> $H, $S, $G |- $E" and
exchRS: "$H |- $E, $R, $S, $F ==> $H |- $E, $S, $R, $F" and
exchLS: "$H, $R, $S, $G |- $E ==> $H, $S, $R, $G |- $E" and
cut: "[| $H |- $E, P; $H, P |- $E |] ==> $H |- $E" and
(*Propositional rules*)
basic: "$H, P, $G |- $E, P, $F" and
conjR: "[| $H|- $E, P, $F; $H|- $E, Q, $F |] ==> $H|- $E, P&Q, $F" and
conjL: "$H, P, Q, $G |- $E ==> $H, P & Q, $G |- $E" and
disjR: "$H |- $E, P, Q, $F ==> $H |- $E, P|Q, $F" and
disjL: "[| $H, P, $G |- $E; $H, Q, $G |- $E |] ==> $H, P|Q, $G |- $E" and
impR: "$H, P |- $E, Q, $F ==> $H |- $E, P-->Q, $F" and
impL: "[| $H,$G |- $E,P; $H, Q, $G |- $E |] ==> $H, P-->Q, $G |- $E" and
notR: "$H, P |- $E, $F ==> $H |- $E, ~P, $F" and
notL: "$H, $G |- $E, P ==> $H, ~P, $G |- $E" and
FalseL: "$H, False, $G |- $E" and
True_def: "True == False-->False" and
iff_def: "P<->Q == (P-->Q) & (Q-->P)"
axiomatization where
(*Quantifiers*)
allR: "(!!x.$H |- $E, P(x), $F) ==> $H |- $E, ALL x. P(x), $F" and
allL: "$H, P(x), $G, ALL x. P(x) |- $E ==> $H, ALL x. P(x), $G |- $E" and
exR: "$H |- $E, P(x), $F, EX x. P(x) ==> $H |- $E, EX x. P(x), $F" and
exL: "(!!x.$H, P(x), $G |- $E) ==> $H, EX x. P(x), $G |- $E" and
(*Equality*)
refl: "$H |- $E, a=a, $F" and
subst: "\<And>G H E. $H(a), $G(a) |- $E(a) ==> $H(b), a=b, $G(b) |- $E(b)"
(* Reflection *)
axiomatization where
eq_reflection: "|- x=y ==> (x==y)" and
iff_reflection: "|- P<->Q ==> (P==Q)"
(*Descriptions*)
axiomatization where
The: "[| $H |- $E, P(a), $F; !!x.$H, P(x) |- $E, x=a, $F |] ==>
$H |- $E, P(THE x. P(x)), $F"
definition If :: "[o, 'a, 'a] => 'a" ("(if (_)/ then (_)/ else (_))" 10)
where "If(P,x,y) == THE z::'a. (P --> z=x) & (~P --> z=y)"
(** Structural Rules on formulas **)
(*contraction*)
lemma contR: "$H |- $E, P, P, $F ==> $H |- $E, P, $F"
by (rule contRS)
lemma contL: "$H, P, P, $G |- $E ==> $H, P, $G |- $E"
by (rule contLS)
(*thinning*)
lemma thinR: "$H |- $E, $F ==> $H |- $E, P, $F"
by (rule thinRS)
lemma thinL: "$H, $G |- $E ==> $H, P, $G |- $E"
by (rule thinLS)
(*exchange*)
lemma exchR: "$H |- $E, Q, P, $F ==> $H |- $E, P, Q, $F"
by (rule exchRS)
lemma exchL: "$H, Q, P, $G |- $E ==> $H, P, Q, $G |- $E"
by (rule exchLS)
ML {*
(*Cut and thin, replacing the right-side formula*)
fun cutR_tac ctxt s i =
res_inst_tac ctxt [(("P", 0), s) ] @{thm cut} i THEN rtac @{thm thinR} i
(*Cut and thin, replacing the left-side formula*)
fun cutL_tac ctxt s i =
res_inst_tac ctxt [(("P", 0), s)] @{thm cut} i THEN rtac @{thm thinL} (i+1)
*}
(** If-and-only-if rules **)
lemma iffR:
"[| $H,P |- $E,Q,$F; $H,Q |- $E,P,$F |] ==> $H |- $E, P <-> Q, $F"
apply (unfold iff_def)
apply (assumption | rule conjR impR)+
done
lemma iffL:
"[| $H,$G |- $E,P,Q; $H,Q,P,$G |- $E |] ==> $H, P <-> Q, $G |- $E"
apply (unfold iff_def)
apply (assumption | rule conjL impL basic)+
done
lemma iff_refl: "$H |- $E, (P <-> P), $F"
apply (rule iffR basic)+
done
lemma TrueR: "$H |- $E, True, $F"
apply (unfold True_def)
apply (rule impR)
apply (rule basic)
done
(*Descriptions*)
lemma the_equality:
assumes p1: "$H |- $E, P(a), $F"
and p2: "!!x. $H, P(x) |- $E, x=a, $F"
shows "$H |- $E, (THE x. P(x)) = a, $F"
apply (rule cut)
apply (rule_tac [2] p2)
apply (rule The, rule thinR, rule exchRS, rule p1)
apply (rule thinR, rule exchRS, rule p2)
done
(** Weakened quantifier rules. Incomplete, they let the search terminate.**)
lemma allL_thin: "$H, P(x), $G |- $E ==> $H, ALL x. P(x), $G |- $E"
apply (rule allL)
apply (erule thinL)
done
lemma exR_thin: "$H |- $E, P(x), $F ==> $H |- $E, EX x. P(x), $F"
apply (rule exR)
apply (erule thinR)
done
(*The rules of LK*)
lemmas [safe] =
iffR iffL
notR notL
impR impL
disjR disjL
conjR conjL
FalseL TrueR
refl basic
ML {* val prop_pack = Cla.get_pack @{context} *}
lemmas [safe] = exL allR
lemmas [unsafe] = the_equality exR_thin allL_thin
ML {* val LK_pack = Cla.get_pack @{context} *}
ML {*
val LK_dup_pack =
Cla.put_pack prop_pack @{context}
|> fold_rev Cla.add_safe @{thms allR exL}
|> fold_rev Cla.add_unsafe @{thms allL exR the_equality}
|> Cla.get_pack;
*}
method_setup fast_prop =
{* Scan.succeed (fn ctxt => SIMPLE_METHOD' (Cla.fast_tac (Cla.put_pack prop_pack ctxt))) *}
method_setup fast_dup =
{* Scan.succeed (fn ctxt => SIMPLE_METHOD' (Cla.fast_tac (Cla.put_pack LK_dup_pack ctxt))) *}
method_setup best_dup =
{* Scan.succeed (fn ctxt => SIMPLE_METHOD' (Cla.best_tac (Cla.put_pack LK_dup_pack ctxt))) *}
method_setup lem = {*
Attrib.thm >> (fn th => fn _ =>
SIMPLE_METHOD' (fn i =>
rtac (@{thm thinR} RS @{thm cut}) i THEN
REPEAT (rtac @{thm thinL} i) THEN
rtac th i))
*}
lemma mp_R:
assumes major: "$H |- $E, $F, P --> Q"
and minor: "$H |- $E, $F, P"
shows "$H |- $E, Q, $F"
apply (rule thinRS [THEN cut], rule major)
apply step
apply (rule thinR, rule minor)
done
lemma mp_L:
assumes major: "$H, $G |- $E, P --> Q"
and minor: "$H, $G, Q |- $E"
shows "$H, P, $G |- $E"
apply (rule thinL [THEN cut], rule major)
apply step
apply (rule thinL, rule minor)
done
(** Two rules to generate left- and right- rules from implications **)
lemma R_of_imp:
assumes major: "|- P --> Q"
and minor: "$H |- $E, $F, P"
shows "$H |- $E, Q, $F"
apply (rule mp_R)
apply (rule_tac [2] minor)
apply (rule thinRS, rule major [THEN thinLS])
done
lemma L_of_imp:
assumes major: "|- P --> Q"
and minor: "$H, $G, Q |- $E"
shows "$H, P, $G |- $E"
apply (rule mp_L)
apply (rule_tac [2] minor)
apply (rule thinRS, rule major [THEN thinLS])
done
(*Can be used to create implications in a subgoal*)
lemma backwards_impR:
assumes prem: "$H, $G |- $E, $F, P --> Q"
shows "$H, P, $G |- $E, Q, $F"
apply (rule mp_L)
apply (rule_tac [2] basic)
apply (rule thinR, rule prem)
done
lemma conjunct1: "|-P&Q ==> |-P"
apply (erule thinR [THEN cut])
apply fast
done
lemma conjunct2: "|-P&Q ==> |-Q"
apply (erule thinR [THEN cut])
apply fast
done
lemma spec: "|- (ALL x. P(x)) ==> |- P(x)"
apply (erule thinR [THEN cut])
apply fast
done
(** Equality **)
lemma sym: "|- a=b --> b=a"
by (safe add!: subst)
lemma trans: "|- a=b --> b=c --> a=c"
by (safe add!: subst)
(* Symmetry of equality in hypotheses *)
lemmas symL = sym [THEN L_of_imp]
(* Symmetry of equality in hypotheses *)
lemmas symR = sym [THEN R_of_imp]
lemma transR: "[| $H|- $E, $F, a=b; $H|- $E, $F, b=c |] ==> $H|- $E, a=c, $F"
by (rule trans [THEN R_of_imp, THEN mp_R])
(* Two theorms for rewriting only one instance of a definition:
the first for definitions of formulae and the second for terms *)
lemma def_imp_iff: "(A == B) ==> |- A <-> B"
apply unfold
apply (rule iff_refl)
done
lemma meta_eq_to_obj_eq: "(A == B) ==> |- A = B"
apply unfold
apply (rule refl)
done
(** if-then-else rules **)
lemma if_True: "|- (if True then x else y) = x"
unfolding If_def by fast
lemma if_False: "|- (if False then x else y) = y"
unfolding If_def by fast
lemma if_P: "|- P ==> |- (if P then x else y) = x"
apply (unfold If_def)
apply (erule thinR [THEN cut])
apply fast
done
lemma if_not_P: "|- ~P ==> |- (if P then x else y) = y";
apply (unfold If_def)
apply (erule thinR [THEN cut])
apply fast
done
end