src/HOL/Induct/PropLog.thy
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(*  Title:      HOL/Induct/PropLog.thy
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    Author:     Tobias Nipkow
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    Copyright   1994  TU Muenchen & University of Cambridge
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*)
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header {* Meta-theory of propositional logic *}
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theory PropLog imports Main begin
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text {*
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  Datatype definition of propositional logic formulae and inductive
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  definition of the propositional tautologies.
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  Inductive definition of propositional logic.  Soundness and
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  completeness w.r.t.\ truth-tables.
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  Prove: If @{text "H |= p"} then @{text "G |= p"} where @{text "G \<in>
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  Fin(H)"}
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*}
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subsection {* The datatype of propositions *}
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datatype 'a pl =
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  false |
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  var 'a ("#_" [1000]) |
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  imp "'a pl" "'a pl" (infixr "->" 90)
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subsection {* The proof system *}
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inductive thms :: "['a pl set, 'a pl] => bool"  (infixl "|-" 50)
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  for H :: "'a pl set"
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where
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  H: "p\<in>H ==> H |- p"
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| K: "H |- p->q->p"
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| S: "H |- (p->q->r) -> (p->q) -> p->r"
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| DN: "H |- ((p->false) -> false) -> p"
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| MP: "[| H |- p->q; H |- p |] ==> H |- q"
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subsection {* The semantics *}
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subsubsection {* Semantics of propositional logic. *}
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primrec eval :: "['a set, 'a pl] => bool"  ("_[[_]]" [100,0] 100)
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where
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  "tt[[false]] = False"
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| "tt[[#v]] = (v \<in> tt)"
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| eval_imp: "tt[[p->q]] = (tt[[p]] --> tt[[q]])"
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text {*
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  A finite set of hypotheses from @{text t} and the @{text Var}s in
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  @{text p}.
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*}
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primrec hyps :: "['a pl, 'a set] => 'a pl set"
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where
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  "hyps false  tt = {}"
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| "hyps (#v)   tt = {if v \<in> tt then #v else #v->false}"
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| "hyps (p->q) tt = hyps p tt Un hyps q tt"
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subsubsection {* Logical consequence *}
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text {*
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  For every valuation, if all elements of @{text H} are true then so
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  is @{text p}.
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*}
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definition sat :: "['a pl set, 'a pl] => bool"  (infixl "|=" 50)
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  where "H |= p = (\<forall>tt. (\<forall>q\<in>H. tt[[q]]) --> tt[[p]])"
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subsection {* Proof theory of propositional logic *}
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lemma thms_mono: "G<=H ==> thms(G) <= thms(H)"
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apply (rule predicate1I)
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apply (erule thms.induct)
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apply (auto intro: thms.intros)
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done
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lemma thms_I: "H |- p->p"
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  -- {*Called @{text I} for Identity Combinator, not for Introduction. *}
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by (best intro: thms.K thms.S thms.MP)
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subsubsection {* Weakening, left and right *}
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lemma weaken_left: "[| G \<subseteq> H;  G|-p |] ==> H|-p"
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  -- {* Order of premises is convenient with @{text THEN} *}
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  by (erule thms_mono [THEN predicate1D])
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lemma weaken_left_insert: "G |- p \<Longrightarrow> insert a G |- p"
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by (rule weaken_left) (rule subset_insertI)
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lemma weaken_left_Un1: "G |- p \<Longrightarrow> G \<union> B |- p"
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by (rule weaken_left) (rule Un_upper1)
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lemma weaken_left_Un2: "G |- p \<Longrightarrow> A \<union> G |- p"
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by (rule weaken_left) (rule Un_upper2)
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lemma weaken_right: "H |- q ==> H |- p->q"
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by (fast intro: thms.K thms.MP)
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subsubsection {* The deduction theorem *}
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theorem deduction: "insert p H |- q  ==>  H |- p->q"
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apply (induct set: thms)
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apply (fast intro: thms_I thms.H thms.K thms.S thms.DN
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                   thms.S [THEN thms.MP, THEN thms.MP] weaken_right)+
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done
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subsubsection {* The cut rule *}
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lemma cut: "insert p H |- q \<Longrightarrow> H |- p \<Longrightarrow> H |- q"
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by (rule thms.MP) (rule deduction)
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lemma thms_falseE: "H |- false \<Longrightarrow> H |- q"
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by (rule thms.MP, rule thms.DN) (rule weaken_right)
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lemma thms_notE: "H |- p -> false \<Longrightarrow> H |- p \<Longrightarrow> H |- q"
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by (rule thms_falseE) (rule thms.MP)
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subsubsection {* Soundness of the rules wrt truth-table semantics *}
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theorem soundness: "H |- p ==> H |= p"
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by (induct set: thms) (auto simp: sat_def)
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subsection {* Completeness *}
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subsubsection {* Towards the completeness proof *}
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lemma false_imp: "H |- p->false ==> H |- p->q"
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apply (rule deduction)
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apply (metis H insert_iff weaken_left_insert thms_notE)
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done
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lemma imp_false:
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    "[| H |- p;  H |- q->false |] ==> H |- (p->q)->false"
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apply (rule deduction)
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apply (metis H MP insert_iff weaken_left_insert)
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done
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lemma hyps_thms_if: "hyps p tt |- (if tt[[p]] then p else p->false)"
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  -- {* Typical example of strengthening the induction statement. *}
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apply simp
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apply (induct p)
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apply (simp_all add: thms_I thms.H)
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apply (blast intro: weaken_left_Un1 weaken_left_Un2 weaken_right
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                    imp_false false_imp)
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done
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lemma sat_thms_p: "{} |= p ==> hyps p tt |- p"
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  -- {* Key lemma for completeness; yields a set of assumptions
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        satisfying @{text p} *}
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apply (unfold sat_def)
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apply (drule spec, erule mp [THEN if_P, THEN subst],
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       rule_tac [2] hyps_thms_if, simp)
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done
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text {*
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  For proving certain theorems in our new propositional logic.
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*}
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declare deduction [intro!]
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declare thms.H [THEN thms.MP, intro]
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text {*
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  The excluded middle in the form of an elimination rule.
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*}
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lemma thms_excluded_middle: "H |- (p->q) -> ((p->false)->q) -> q"
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apply (rule deduction [THEN deduction])
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apply (rule thms.DN [THEN thms.MP], best intro: H)
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done
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lemma thms_excluded_middle_rule:
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    "[| insert p H |- q;  insert (p->false) H |- q |] ==> H |- q"
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  -- {* Hard to prove directly because it requires cuts *}
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by (rule thms_excluded_middle [THEN thms.MP, THEN thms.MP], auto)
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subsection{* Completeness -- lemmas for reducing the set of assumptions*}
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text {*
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  For the case @{prop "hyps p t - insert #v Y |- p"} we also have @{prop
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  "hyps p t - {#v} \<subseteq> hyps p (t-{v})"}.
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*}
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lemma hyps_Diff: "hyps p (t-{v}) <= insert (#v->false) ((hyps p t)-{#v})"
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by (induct p) auto
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text {*
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  For the case @{prop "hyps p t - insert (#v -> Fls) Y |- p"} we also have
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  @{prop "hyps p t-{#v->Fls} \<subseteq> hyps p (insert v t)"}.
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*}
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lemma hyps_insert: "hyps p (insert v t) <= insert (#v) (hyps p t-{#v->false})"
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by (induct p) auto
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text {* Two lemmas for use with @{text weaken_left} *}
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lemma insert_Diff_same: "B-C <= insert a (B-insert a C)"
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by fast
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lemma insert_Diff_subset2: "insert a (B-{c}) - D <= insert a (B-insert c D)"
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by fast
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text {*
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  The set @{term "hyps p t"} is finite, and elements have the form
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  @{term "#v"} or @{term "#v->Fls"}.
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*}
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lemma hyps_finite: "finite(hyps p t)"
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by (induct p) auto
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lemma hyps_subset: "hyps p t <= (UN v. {#v, #v->false})"
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by (induct p) auto
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lemma Diff_weaken_left: "A \<subseteq> C \<Longrightarrow> A - B |- p \<Longrightarrow> C - B |- p"
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  by (rule Diff_mono [OF _ subset_refl, THEN weaken_left])
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subsubsection {* Completeness theorem *}
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text {*
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  Induction on the finite set of assumptions @{term "hyps p t0"}.  We
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  may repeatedly subtract assumptions until none are left!
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*}
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lemma completeness_0_lemma:
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    "{} |= p ==> \<forall>t. hyps p t - hyps p t0 |- p"
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apply (rule hyps_subset [THEN hyps_finite [THEN finite_subset_induct]])
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 apply (simp add: sat_thms_p, safe)
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 txt{*Case @{text"hyps p t-insert(#v,Y) |- p"} *}
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 apply (iprover intro: thms_excluded_middle_rule
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                     insert_Diff_same [THEN weaken_left]
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                     insert_Diff_subset2 [THEN weaken_left]
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                     hyps_Diff [THEN Diff_weaken_left])
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txt{*Case @{text"hyps p t-insert(#v -> false,Y) |- p"} *}
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 apply (iprover intro: thms_excluded_middle_rule
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                     insert_Diff_same [THEN weaken_left]
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                     insert_Diff_subset2 [THEN weaken_left]
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                     hyps_insert [THEN Diff_weaken_left])
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done
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text{*The base case for completeness*}
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lemma completeness_0:  "{} |= p ==> {} |- p"
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apply (rule Diff_cancel [THEN subst])
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apply (erule completeness_0_lemma [THEN spec])
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done
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text{*A semantic analogue of the Deduction Theorem*}
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lemma sat_imp: "insert p H |= q ==> H |= p->q"
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by (auto simp: sat_def)
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theorem completeness: "finite H ==> H |= p ==> H |- p"
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apply (induct arbitrary: p rule: finite_induct)
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 apply (blast intro: completeness_0)
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apply (iprover intro: sat_imp thms.H insertI1 weaken_left_insert [THEN thms.MP])
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done
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theorem syntax_iff_semantics: "finite H ==> (H |- p) = (H |= p)"
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by (blast intro: soundness completeness)
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end