(* Title: HOL/Induct/PropLog.thy ID: $Id$ Author: Tobias Nipkow Copyright 1994 TU Muenchen & University of Cambridge *) header {* Meta-theory of propositional logic *} theory PropLog imports Main begin text {* Datatype definition of propositional logic formulae and inductive definition of the propositional tautologies. Inductive definition of propositional logic. Soundness and completeness w.r.t.\ truth-tables. Prove: If @{text "H |= p"} then @{text "G |= p"} where @{text "G \ Fin(H)"} *} subsection {* The datatype of propositions *} datatype 'a pl = false | var 'a ("#_" [1000]) | "->" "'a pl" "'a pl" (infixr 90) subsection {* The proof system *} consts thms :: "'a pl set => 'a pl set" abbreviation thm_rel :: "['a pl set, 'a pl] => bool" (infixl "|-" 50) where "H |- p == p \ thms H" inductive "thms(H)" intros H [intro]: "p\H ==> H |- p" K: "H |- p->q->p" S: "H |- (p->q->r) -> (p->q) -> p->r" DN: "H |- ((p->false) -> false) -> p" MP: "[| H |- p->q; H |- p |] ==> H |- q" subsection {* The semantics *} subsubsection {* Semantics of propositional logic. *} consts eval :: "['a set, 'a pl] => bool" ("_[[_]]" [100,0] 100) primrec "tt[[false]] = False" "tt[[#v]] = (v \ tt)" eval_imp: "tt[[p->q]] = (tt[[p]] --> tt[[q]])" text {* A finite set of hypotheses from @{text t} and the @{text Var}s in @{text p}. *} consts hyps :: "['a pl, 'a set] => 'a pl set" primrec "hyps false tt = {}" "hyps (#v) tt = {if v \ tt then #v else #v->false}" "hyps (p->q) tt = hyps p tt Un hyps q tt" subsubsection {* Logical consequence *} text {* For every valuation, if all elements of @{text H} are true then so is @{text p}. *} definition sat :: "['a pl set, 'a pl] => bool" (infixl "|=" 50) where "H |= p = (\tt. (\q\H. tt[[q]]) --> tt[[p]])" subsection {* Proof theory of propositional logic *} lemma thms_mono: "G<=H ==> thms(G) <= thms(H)" apply (unfold thms.defs ) apply (rule lfp_mono) apply (assumption | rule basic_monos)+ done lemma thms_I: "H |- p->p" -- {*Called @{text I} for Identity Combinator, not for Introduction. *} by (best intro: thms.K thms.S thms.MP) subsubsection {* Weakening, left and right *} lemma weaken_left: "[| G \ H; G|-p |] ==> H|-p" -- {* Order of premises is convenient with @{text THEN} *} by (erule thms_mono [THEN subsetD]) lemmas weaken_left_insert = subset_insertI [THEN weaken_left] lemmas weaken_left_Un1 = Un_upper1 [THEN weaken_left] lemmas weaken_left_Un2 = Un_upper2 [THEN weaken_left] lemma weaken_right: "H |- q ==> H |- p->q" by (fast intro: thms.K thms.MP) subsubsection {* The deduction theorem *} theorem deduction: "insert p H |- q ==> H |- p->q" apply (induct set: thms) apply (fast intro: thms_I thms.H thms.K thms.S thms.DN thms.S [THEN thms.MP, THEN thms.MP] weaken_right)+ done subsubsection {* The cut rule *} lemmas cut = deduction [THEN thms.MP] lemmas thms_falseE = weaken_right [THEN thms.DN [THEN thms.MP]] lemmas thms_notE = thms.MP [THEN thms_falseE, standard] subsubsection {* Soundness of the rules wrt truth-table semantics *} theorem soundness: "H |- p ==> H |= p" apply (unfold sat_def) apply (induct set: thms) apply auto done subsection {* Completeness *} subsubsection {* Towards the completeness proof *} lemma false_imp: "H |- p->false ==> H |- p->q" apply (rule deduction) apply (erule weaken_left_insert [THEN thms_notE]) apply blast done lemma imp_false: "[| H |- p; H |- q->false |] ==> H |- (p->q)->false" apply (rule deduction) apply (blast intro: weaken_left_insert thms.MP thms.H) done lemma hyps_thms_if: "hyps p tt |- (if tt[[p]] then p else p->false)" -- {* Typical example of strengthening the induction statement. *} apply simp apply (induct p) apply (simp_all add: thms_I thms.H) apply (blast intro: weaken_left_Un1 weaken_left_Un2 weaken_right imp_false false_imp) done lemma sat_thms_p: "{} |= p ==> hyps p tt |- p" -- {* Key lemma for completeness; yields a set of assumptions satisfying @{text p} *} apply (unfold sat_def) apply (drule spec, erule mp [THEN if_P, THEN subst], rule_tac [2] hyps_thms_if, simp) done text {* For proving certain theorems in our new propositional logic. *} declare deduction [intro!] declare thms.H [THEN thms.MP, intro] text {* The excluded middle in the form of an elimination rule. *} lemma thms_excluded_middle: "H |- (p->q) -> ((p->false)->q) -> q" apply (rule deduction [THEN deduction]) apply (rule thms.DN [THEN thms.MP], best) done lemma thms_excluded_middle_rule: "[| insert p H |- q; insert (p->false) H |- q |] ==> H |- q" -- {* Hard to prove directly because it requires cuts *} by (rule thms_excluded_middle [THEN thms.MP, THEN thms.MP], auto) subsection{* Completeness -- lemmas for reducing the set of assumptions*} text {* For the case @{prop "hyps p t - insert #v Y |- p"} we also have @{prop "hyps p t - {#v} \ hyps p (t-{v})"}. *} lemma hyps_Diff: "hyps p (t-{v}) <= insert (#v->false) ((hyps p t)-{#v})" by (induct p) auto text {* For the case @{prop "hyps p t - insert (#v -> Fls) Y |- p"} we also have @{prop "hyps p t-{#v->Fls} \ hyps p (insert v t)"}. *} lemma hyps_insert: "hyps p (insert v t) <= insert (#v) (hyps p t-{#v->false})" by (induct p) auto text {* Two lemmas for use with @{text weaken_left} *} lemma insert_Diff_same: "B-C <= insert a (B-insert a C)" by fast lemma insert_Diff_subset2: "insert a (B-{c}) - D <= insert a (B-insert c D)" by fast text {* The set @{term "hyps p t"} is finite, and elements have the form @{term "#v"} or @{term "#v->Fls"}. *} lemma hyps_finite: "finite(hyps p t)" by (induct p) auto lemma hyps_subset: "hyps p t <= (UN v. {#v, #v->false})" by (induct p) auto lemmas Diff_weaken_left = Diff_mono [OF _ subset_refl, THEN weaken_left] subsubsection {* Completeness theorem *} text {* Induction on the finite set of assumptions @{term "hyps p t0"}. We may repeatedly subtract assumptions until none are left! *} lemma completeness_0_lemma: "{} |= p ==> \t. hyps p t - hyps p t0 |- p" apply (rule hyps_subset [THEN hyps_finite [THEN finite_subset_induct]]) apply (simp add: sat_thms_p, safe) txt{*Case @{text"hyps p t-insert(#v,Y) |- p"} *} apply (iprover intro: thms_excluded_middle_rule insert_Diff_same [THEN weaken_left] insert_Diff_subset2 [THEN weaken_left] hyps_Diff [THEN Diff_weaken_left]) txt{*Case @{text"hyps p t-insert(#v -> false,Y) |- p"} *} apply (iprover intro: thms_excluded_middle_rule insert_Diff_same [THEN weaken_left] insert_Diff_subset2 [THEN weaken_left] hyps_insert [THEN Diff_weaken_left]) done text{*The base case for completeness*} lemma completeness_0: "{} |= p ==> {} |- p" apply (rule Diff_cancel [THEN subst]) apply (erule completeness_0_lemma [THEN spec]) done text{*A semantic analogue of the Deduction Theorem*} lemma sat_imp: "insert p H |= q ==> H |= p->q" by (unfold sat_def, auto) theorem completeness: "finite H ==> H |= p ==> H |- p" apply (induct arbitrary: p rule: finite_induct) apply (blast intro: completeness_0) apply (iprover intro: sat_imp thms.H insertI1 weaken_left_insert [THEN thms.MP]) done theorem syntax_iff_semantics: "finite H ==> (H |- p) = (H |= p)" by (blast intro: soundness completeness) end