--- a/src/HOL/Induct/PropLog.thy Tue Apr 02 13:47:01 2002 +0200
+++ b/src/HOL/Induct/PropLog.thy Tue Apr 02 14:28:28 2002 +0200
@@ -1,45 +1,276 @@
-(* Title: HOL/ex/PropLog.thy
+(* Title: HOL/Induct/PropLog.thy
ID: $Id$
Author: Tobias Nipkow
- Copyright 1994 TU Muenchen
-
-Inductive definition of propositional logic.
+ Copyright 1994 TU Muenchen & University of Cambridge
*)
-PropLog = Main +
+header {* Meta-theory of propositional logic *}
+
+theory PropLog = Main:
+
+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 \<in>
+ Fin(H)"}
+*}
+
+subsection {* The datatype of propositions *}
datatype
- 'a pl = false | var 'a ("#_" [1000]) | "->" ('a pl) ('a pl) (infixr 90)
+ 'a pl = false | var 'a ("#_" [1000]) | "->" "'a pl" "'a pl" (infixr 90)
+
+subsection {* The proof system *}
+
consts
- thms :: 'a pl set => 'a pl set
- "|-" :: ['a pl set, 'a pl] => bool (infixl 50)
- "|=" :: ['a pl set, 'a pl] => bool (infixl 50)
- eval :: ['a set, 'a pl] => bool ("_[[_]]" [100,0] 100)
- hyps :: ['a pl, 'a set] => 'a pl set
+ thms :: "'a pl set => 'a pl set"
+ "|-" :: "['a pl set, 'a pl] => bool" (infixl 50)
translations
- "H |- p" == "p : thms(H)"
+ "H |- p" == "p \<in> thms(H)"
inductive "thms(H)"
- intrs
- H "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"
+ intros
+ H [intro]: "p\<in>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. *}
-defs
- sat_def "H |= p == (!tt. (!q:H. tt[[q]]) --> tt[[p]])"
+consts
+ eval :: "['a set, 'a pl] => bool" ("_[[_]]" [100,0] 100)
+
+primrec "tt[[false]] = False"
+ "tt[[#v]] = (v \<in> tt)"
+ eval_imp: "tt[[p->q]] = (tt[[p]] --> tt[[q]])"
-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 (#v) tt = {if v \<in> 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}.
+*}
+
+constdefs
+ sat :: "['a pl set, 'a pl] => bool" (infixl "|=" 50)
+ "H |= p == (\<forall>tt. (\<forall>q\<in>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 \<subseteq> 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 (erule thms.induct)
+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 (erule thms.induct, 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_tac "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} \<subseteq> hyps p (t-{v})"}.
+*}
+
+lemma hyps_Diff: "hyps p (t-{v}) <= insert (#v->false) ((hyps p t)-{#v})"
+by (induct_tac "p", auto)
+
+text {*
+ For the case @{prop "hyps p t - insert (#v -> Fls) Y |- p"} we also have
+ @{prop "hyps p t-{#v->Fls} \<subseteq> hyps p (insert v t)"}.
+*}
+
+lemma hyps_insert: "hyps p (insert v t) <= insert (#v) (hyps p t-{#v->false})"
+by (induct_tac "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_tac "p", auto)
+
+lemma hyps_subset: "hyps p t <= (UN v. {#v, #v->false})"
+by (induct_tac "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 ==> \<forall>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)
+(*Case hyps p t-insert(#v,Y) |- p *)
+ apply (rule thms_excluded_middle_rule)
+ apply (rule insert_Diff_same [THEN weaken_left])
+ apply (erule spec)
+ apply (rule insert_Diff_subset2 [THEN weaken_left])
+ apply (rule hyps_Diff [THEN Diff_weaken_left])
+ apply (erule spec)
+(*Case hyps p t-insert(#v -> false,Y) |- p *)
+apply (rule thms_excluded_middle_rule)
+ apply (rule_tac [2] insert_Diff_same [THEN weaken_left])
+ apply (erule_tac [2] spec)
+apply (rule insert_Diff_subset2 [THEN weaken_left])
+apply (rule hyps_insert [THEN Diff_weaken_left])
+apply (erule spec)
+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 [rule_format]: "finite H ==> \<forall>p. H |= p --> H |- p"
+apply (erule finite_induct)
+apply (safe intro!: completeness_0)
+apply (drule sat_imp)
+apply (drule spec)
+apply (blast intro: weaken_left_insert [THEN thms.MP])
+done
+
+theorem syntax_iff_semantics: "finite H ==> (H |- p) = (H |= p)"
+by (fast elim!: soundness completeness)
+
end