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src/Sequents/LK0.thy

author | haftmann |

Thu, 08 Jul 2010 16:19:24 +0200 | |

changeset 37744 | 3daaf23b9ab4 |

parent 36452 | d37c6eed8117 |

child 38499 | 8f0cd11238a7 |

permissions | -rw-r--r-- |

tuned titles

(* Title: LK/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 global classes "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"}, two_seq_tr @{const_syntax Trueprop})] *} print_translation {* [(@{const_syntax Trueprop}, 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) 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)), $F" definition If :: "[o, 'a, 'a] => 'a" ("(if (_)/ then (_)/ else (_))" 10) where "If(P,x,y) == THE z::'a. (P --> z=x) & (~P --> z=y)" (** Structural Rules on formulas **) (*contraction*) lemma contR: "$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), $F" apply (rule exR) apply (erule thinR) done (*The rules of LK*) ML {* val prop_pack = empty_pack add_safes [thm "basic", thm "refl", thm "TrueR", thm "FalseL", thm "conjL", thm "conjR", thm "disjL", thm "disjR", thm "impL", thm "impR", thm "notL", thm "notR", thm "iffL", thm "iffR"]; val LK_pack = prop_pack add_safes [thm "allR", thm "exL"] add_unsafes [thm "allL_thin", thm "exR_thin", thm "the_equality"]; val LK_dup_pack = prop_pack add_safes [thm "allR", thm "exL"] add_unsafes [thm "allL", thm "exR", thm "the_equality"]; local val thinR = thm "thinR" val thinL = thm "thinL" val cut = thm "cut" in fun lemma_tac th i = rtac (thinR RS cut) i THEN REPEAT (rtac thinL i) THEN rtac th i; end; *} method_setup fast_prop = {* Scan.succeed (K (SIMPLE_METHOD' (fast_tac prop_pack))) *} "propositional reasoning" method_setup fast = {* Scan.succeed (K (SIMPLE_METHOD' (fast_tac LK_pack))) *} "classical reasoning" method_setup fast_dup = {* Scan.succeed (K (SIMPLE_METHOD' (fast_tac LK_dup_pack))) *} "classical reasoning" method_setup best = {* Scan.succeed (K (SIMPLE_METHOD' (best_tac LK_pack))) *} "classical reasoning" method_setup best_dup = {* Scan.succeed (K (SIMPLE_METHOD' (best_tac LK_dup_pack))) *} "classical reasoning" lemma mp_R: assumes major: "$H |- $E, $F, P --> Q" and minor: "$H |- $E, $F, P" shows "$H |- $E, Q, $F" apply (rule thinRS [THEN cut], rule major) apply (tactic "step_tac LK_pack 1") apply (rule thinR, rule minor) done lemma mp_L: assumes major: "$H, $G |- $E, P --> Q" and minor: "$H, $G, Q |- $E" shows "$H, P, $G |- $E" apply (rule thinL [THEN cut], rule major) apply (tactic "step_tac LK_pack 1") apply (rule thinL, rule minor) done (** Two rules to generate left- and right- rules from implications **) lemma R_of_imp: assumes major: "|- P --> Q" and minor: "$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 (tactic {* safe_tac (LK_pack add_safes [thm "subst"]) 1 *}) lemma trans: "|- a=b --> b=c --> a=c" by (tactic {* safe_tac (LK_pack add_safes [thm "subst"]) 1 *}) (* Symmetry of equality in hypotheses *) lemmas symL = sym [THEN L_of_imp, standard] (* Symmetry of equality in hypotheses *) lemmas symR = sym [THEN R_of_imp, standard] 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