rename HOL-Multivariate_Analysis to HOL-Analysis.
(* 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).
*)
section \<open>Classical First-Order Sequent Calculus\<close>
theory LK0
imports Sequents
begin
class "term"
default_sort "term"
consts
Trueprop :: "two_seqi"
True :: o
False :: o
equal :: "['a,'a] \<Rightarrow> o" (infixl "=" 50)
Not :: "o \<Rightarrow> o" ("\<not> _" [40] 40)
conj :: "[o,o] \<Rightarrow> o" (infixr "\<and>" 35)
disj :: "[o,o] \<Rightarrow> o" (infixr "\<or>" 30)
imp :: "[o,o] \<Rightarrow> o" (infixr "\<longrightarrow>" 25)
iff :: "[o,o] \<Rightarrow> o" (infixr "\<longleftrightarrow>" 25)
The :: "('a \<Rightarrow> o) \<Rightarrow> 'a" (binder "THE " 10)
All :: "('a \<Rightarrow> o) \<Rightarrow> o" (binder "\<forall>" 10)
Ex :: "('a \<Rightarrow> o) \<Rightarrow> o" (binder "\<exists>" 10)
syntax
"_Trueprop" :: "two_seqe" ("((_)/ \<turnstile> (_))" [6,6] 5)
parse_translation \<open>[(@{syntax_const "_Trueprop"}, K (two_seq_tr @{const_syntax Trueprop}))]\<close>
print_translation \<open>[(@{const_syntax Trueprop}, K (two_seq_tr' @{syntax_const "_Trueprop"}))]\<close>
abbreviation
not_equal (infixl "\<noteq>" 50) where
"x \<noteq> y \<equiv> \<not> (x = y)"
axiomatization where
(*Structural rules: contraction, thinning, exchange [Soren Heilmann] *)
contRS: "$H \<turnstile> $E, $S, $S, $F \<Longrightarrow> $H \<turnstile> $E, $S, $F" and
contLS: "$H, $S, $S, $G \<turnstile> $E \<Longrightarrow> $H, $S, $G \<turnstile> $E" and
thinRS: "$H \<turnstile> $E, $F \<Longrightarrow> $H \<turnstile> $E, $S, $F" and
thinLS: "$H, $G \<turnstile> $E \<Longrightarrow> $H, $S, $G \<turnstile> $E" and
exchRS: "$H \<turnstile> $E, $R, $S, $F \<Longrightarrow> $H \<turnstile> $E, $S, $R, $F" and
exchLS: "$H, $R, $S, $G \<turnstile> $E \<Longrightarrow> $H, $S, $R, $G \<turnstile> $E" and
cut: "\<lbrakk>$H \<turnstile> $E, P; $H, P \<turnstile> $E\<rbrakk> \<Longrightarrow> $H \<turnstile> $E" and
(*Propositional rules*)
basic: "$H, P, $G \<turnstile> $E, P, $F" and
conjR: "\<lbrakk>$H\<turnstile> $E, P, $F; $H\<turnstile> $E, Q, $F\<rbrakk> \<Longrightarrow> $H\<turnstile> $E, P \<and> Q, $F" and
conjL: "$H, P, Q, $G \<turnstile> $E \<Longrightarrow> $H, P \<and> Q, $G \<turnstile> $E" and
disjR: "$H \<turnstile> $E, P, Q, $F \<Longrightarrow> $H \<turnstile> $E, P \<or> Q, $F" and
disjL: "\<lbrakk>$H, P, $G \<turnstile> $E; $H, Q, $G \<turnstile> $E\<rbrakk> \<Longrightarrow> $H, P \<or> Q, $G \<turnstile> $E" and
impR: "$H, P \<turnstile> $E, Q, $F \<Longrightarrow> $H \<turnstile> $E, P \<longrightarrow> Q, $F" and
impL: "\<lbrakk>$H,$G \<turnstile> $E,P; $H, Q, $G \<turnstile> $E\<rbrakk> \<Longrightarrow> $H, P \<longrightarrow> Q, $G \<turnstile> $E" and
notR: "$H, P \<turnstile> $E, $F \<Longrightarrow> $H \<turnstile> $E, \<not> P, $F" and
notL: "$H, $G \<turnstile> $E, P \<Longrightarrow> $H, \<not> P, $G \<turnstile> $E" and
FalseL: "$H, False, $G \<turnstile> $E" and
True_def: "True \<equiv> False \<longrightarrow> False" and
iff_def: "P \<longleftrightarrow> Q \<equiv> (P \<longrightarrow> Q) \<and> (Q \<longrightarrow> P)"
axiomatization where
(*Quantifiers*)
allR: "(\<And>x. $H \<turnstile> $E, P(x), $F) \<Longrightarrow> $H \<turnstile> $E, \<forall>x. P(x), $F" and
allL: "$H, P(x), $G, \<forall>x. P(x) \<turnstile> $E \<Longrightarrow> $H, \<forall>x. P(x), $G \<turnstile> $E" and
exR: "$H \<turnstile> $E, P(x), $F, \<exists>x. P(x) \<Longrightarrow> $H \<turnstile> $E, \<exists>x. P(x), $F" and
exL: "(\<And>x. $H, P(x), $G \<turnstile> $E) \<Longrightarrow> $H, \<exists>x. P(x), $G \<turnstile> $E" and
(*Equality*)
refl: "$H \<turnstile> $E, a = a, $F" and
subst: "\<And>G H E. $H(a), $G(a) \<turnstile> $E(a) \<Longrightarrow> $H(b), a=b, $G(b) \<turnstile> $E(b)"
(* Reflection *)
axiomatization where
eq_reflection: "\<turnstile> x = y \<Longrightarrow> (x \<equiv> y)" and
iff_reflection: "\<turnstile> P \<longleftrightarrow> Q \<Longrightarrow> (P \<equiv> Q)"
(*Descriptions*)
axiomatization where
The: "\<lbrakk>$H \<turnstile> $E, P(a), $F; \<And>x.$H, P(x) \<turnstile> $E, x=a, $F\<rbrakk> \<Longrightarrow>
$H \<turnstile> $E, P(THE x. P(x)), $F"
definition If :: "[o, 'a, 'a] \<Rightarrow> 'a" ("(if (_)/ then (_)/ else (_))" 10)
where "If(P,x,y) \<equiv> THE z::'a. (P \<longrightarrow> z = x) \<and> (\<not> P \<longrightarrow> z = y)"
(** Structural Rules on formulas **)
(*contraction*)
lemma contR: "$H \<turnstile> $E, P, P, $F \<Longrightarrow> $H \<turnstile> $E, P, $F"
by (rule contRS)
lemma contL: "$H, P, P, $G \<turnstile> $E \<Longrightarrow> $H, P, $G \<turnstile> $E"
by (rule contLS)
(*thinning*)
lemma thinR: "$H \<turnstile> $E, $F \<Longrightarrow> $H \<turnstile> $E, P, $F"
by (rule thinRS)
lemma thinL: "$H, $G \<turnstile> $E \<Longrightarrow> $H, P, $G \<turnstile> $E"
by (rule thinLS)
(*exchange*)
lemma exchR: "$H \<turnstile> $E, Q, P, $F \<Longrightarrow> $H \<turnstile> $E, P, Q, $F"
by (rule exchRS)
lemma exchL: "$H, Q, P, $G \<turnstile> $E \<Longrightarrow> $H, P, Q, $G \<turnstile> $E"
by (rule exchLS)
ML \<open>
(*Cut and thin, replacing the right-side formula*)
fun cutR_tac ctxt s i =
Rule_Insts.res_inst_tac ctxt [((("P", 0), Position.none), s)] [] @{thm cut} i THEN
resolve_tac ctxt @{thms thinR} i
(*Cut and thin, replacing the left-side formula*)
fun cutL_tac ctxt s i =
Rule_Insts.res_inst_tac ctxt [((("P", 0), Position.none), s)] [] @{thm cut} i THEN
resolve_tac ctxt @{thms thinL} (i + 1)
\<close>
(** If-and-only-if rules **)
lemma iffR: "\<lbrakk>$H,P \<turnstile> $E,Q,$F; $H,Q \<turnstile> $E,P,$F\<rbrakk> \<Longrightarrow> $H \<turnstile> $E, P \<longleftrightarrow> Q, $F"
apply (unfold iff_def)
apply (assumption | rule conjR impR)+
done
lemma iffL: "\<lbrakk>$H,$G \<turnstile> $E,P,Q; $H,Q,P,$G \<turnstile> $E\<rbrakk> \<Longrightarrow> $H, P \<longleftrightarrow> Q, $G \<turnstile> $E"
apply (unfold iff_def)
apply (assumption | rule conjL impL basic)+
done
lemma iff_refl: "$H \<turnstile> $E, (P \<longleftrightarrow> P), $F"
apply (rule iffR basic)+
done
lemma TrueR: "$H \<turnstile> $E, True, $F"
apply (unfold True_def)
apply (rule impR)
apply (rule basic)
done
(*Descriptions*)
lemma the_equality:
assumes p1: "$H \<turnstile> $E, P(a), $F"
and p2: "\<And>x. $H, P(x) \<turnstile> $E, x=a, $F"
shows "$H \<turnstile> $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 \<turnstile> $E \<Longrightarrow> $H, \<forall>x. P(x), $G \<turnstile> $E"
apply (rule allL)
apply (erule thinL)
done
lemma exR_thin: "$H \<turnstile> $E, P(x), $F \<Longrightarrow> $H \<turnstile> $E, \<exists>x. P(x), $F"
apply (rule exR)
apply (erule thinR)
done
(*The rules of LK*)
lemmas [safe] =
iffR iffL
notR notL
impR impL
disjR disjL
conjR conjL
FalseL TrueR
refl basic
ML \<open>val prop_pack = Cla.get_pack @{context}\<close>
lemmas [safe] = exL allR
lemmas [unsafe] = the_equality exR_thin allL_thin
ML \<open>val LK_pack = Cla.get_pack @{context}\<close>
ML \<open>
val LK_dup_pack =
Cla.put_pack prop_pack @{context}
|> fold_rev Cla.add_safe @{thms allR exL}
|> fold_rev Cla.add_unsafe @{thms allL exR the_equality}
|> Cla.get_pack;
\<close>
method_setup fast_prop =
\<open>Scan.succeed (fn ctxt => SIMPLE_METHOD' (Cla.fast_tac (Cla.put_pack prop_pack ctxt)))\<close>
method_setup fast_dup =
\<open>Scan.succeed (fn ctxt => SIMPLE_METHOD' (Cla.fast_tac (Cla.put_pack LK_dup_pack ctxt)))\<close>
method_setup best_dup =
\<open>Scan.succeed (fn ctxt => SIMPLE_METHOD' (Cla.best_tac (Cla.put_pack LK_dup_pack ctxt)))\<close>
method_setup lem = \<open>
Attrib.thm >> (fn th => fn ctxt =>
SIMPLE_METHOD' (fn i =>
resolve_tac ctxt [@{thm thinR} RS @{thm cut}] i THEN
REPEAT (resolve_tac ctxt @{thms thinL} i) THEN
resolve_tac ctxt [th] i))
\<close>
lemma mp_R:
assumes major: "$H \<turnstile> $E, $F, P \<longrightarrow> Q"
and minor: "$H \<turnstile> $E, $F, P"
shows "$H \<turnstile> $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 \<turnstile> $E, P \<longrightarrow> Q"
and minor: "$H, $G, Q \<turnstile> $E"
shows "$H, P, $G \<turnstile> $E"
apply (rule thinL [THEN cut], rule major)
apply step
apply (rule thinL, rule minor)
done
(** Two rules to generate left- and right- rules from implications **)
lemma R_of_imp:
assumes major: "\<turnstile> P \<longrightarrow> Q"
and minor: "$H \<turnstile> $E, $F, P"
shows "$H \<turnstile> $E, Q, $F"
apply (rule mp_R)
apply (rule_tac [2] minor)
apply (rule thinRS, rule major [THEN thinLS])
done
lemma L_of_imp:
assumes major: "\<turnstile> P \<longrightarrow> Q"
and minor: "$H, $G, Q \<turnstile> $E"
shows "$H, P, $G \<turnstile> $E"
apply (rule mp_L)
apply (rule_tac [2] minor)
apply (rule thinRS, rule major [THEN thinLS])
done
(*Can be used to create implications in a subgoal*)
lemma backwards_impR:
assumes prem: "$H, $G \<turnstile> $E, $F, P \<longrightarrow> Q"
shows "$H, P, $G \<turnstile> $E, Q, $F"
apply (rule mp_L)
apply (rule_tac [2] basic)
apply (rule thinR, rule prem)
done
lemma conjunct1: "\<turnstile>P \<and> Q \<Longrightarrow> \<turnstile>P"
apply (erule thinR [THEN cut])
apply fast
done
lemma conjunct2: "\<turnstile>P \<and> Q \<Longrightarrow> \<turnstile>Q"
apply (erule thinR [THEN cut])
apply fast
done
lemma spec: "\<turnstile> (\<forall>x. P(x)) \<Longrightarrow> \<turnstile> P(x)"
apply (erule thinR [THEN cut])
apply fast
done
(** Equality **)
lemma sym: "\<turnstile> a = b \<longrightarrow> b = a"
by (safe add!: subst)
lemma trans: "\<turnstile> a = b \<longrightarrow> b = c \<longrightarrow> a = c"
by (safe add!: subst)
(* Symmetry of equality in hypotheses *)
lemmas symL = sym [THEN L_of_imp]
(* Symmetry of equality in hypotheses *)
lemmas symR = sym [THEN R_of_imp]
lemma transR: "\<lbrakk>$H\<turnstile> $E, $F, a = b; $H\<turnstile> $E, $F, b=c\<rbrakk> \<Longrightarrow> $H\<turnstile> $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 \<equiv> B) \<Longrightarrow> \<turnstile> A \<longleftrightarrow> B"
apply unfold
apply (rule iff_refl)
done
lemma meta_eq_to_obj_eq: "(A \<equiv> B) \<Longrightarrow> \<turnstile> A = B"
apply unfold
apply (rule refl)
done
(** if-then-else rules **)
lemma if_True: "\<turnstile> (if True then x else y) = x"
unfolding If_def by fast
lemma if_False: "\<turnstile> (if False then x else y) = y"
unfolding If_def by fast
lemma if_P: "\<turnstile> P \<Longrightarrow> \<turnstile> (if P then x else y) = x"
apply (unfold If_def)
apply (erule thinR [THEN cut])
apply fast
done
lemma if_not_P: "\<turnstile> \<not> P \<Longrightarrow> \<turnstile> (if P then x else y) = y"
apply (unfold If_def)
apply (erule thinR [THEN cut])
apply fast
done
end