src/ZF/Induct/PropLog.thy

   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)"}
+  Prove: If \<open>H |= p\<close> then \<open>G |= p\<close> where \<open>G \<in>
+  Fin(H)\<close>
 \<close>
 
 
 subsection \<open>The datatype of propositions\<close>
 

   "is_true_fun(p=>q, t) = (if is_true_fun(p,t) = 1 then is_true_fun(q,t) else 1)"
 
 definition
   is_true :: "[i,i] => o"  where
   "is_true(p,t) == is_true_fun(p,t) = 1"
-  -- \<open>this definition is required since predicates can't be recursive\<close>
+  \<comment> \<open>this definition is required since predicates can't be recursive\<close>
 
 lemma is_true_Fls [simp]: "is_true(Fls,t) \<longleftrightarrow> False"
   by (simp add: is_true_def)
 
 lemma is_true_Var [simp]: "is_true(#v,t) \<longleftrightarrow> v \<in> t"

 
 
 subsubsection \<open>Logical consequence\<close>
 
 text \<open>
-  For every valuation, if all elements of @{text H} are true then so
-  is @{text p}.
+  For every valuation, if all elements of \<open>H\<close> are true then so
+  is \<open>p\<close>.
 \<close>
 
 definition
   logcon :: "[i,i] => o"    (infixl "|=" 50)  where
   "H |= p == \<forall>t. (\<forall>q \<in> H. is_true(q,t)) \<longrightarrow> is_true(p,t)"
 
 
 text \<open>
-  A finite set of hypotheses from @{text t} and the @{text Var}s in
-  @{text p}.
+  A finite set of hypotheses from \<open>t\<close> and the \<open>Var\<close>s in
+  \<open>p\<close>.
 \<close>
 
 consts
   hyps :: "[i,i] => i"
 primrec

 lemmas thms_in_pl = thms.dom_subset [THEN subsetD]
 
 inductive_cases ImpE: "p=>q \<in> propn"
 
 lemma thms_MP: "[| H |- p=>q;  H |- p |] ==> H |- q"
-  -- \<open>Stronger Modus Ponens rule: no typechecking!\<close>
+  \<comment> \<open>Stronger Modus Ponens rule: no typechecking!\<close>
   apply (rule thms.MP)
      apply (erule asm_rl thms_in_pl thms_in_pl [THEN ImpE])+
   done
 
 lemma thms_I: "p \<in> propn ==> H |- p=>p"
-  -- \<open>Rule is called @{text I} for Identity Combinator, not for Introduction.\<close>
+  \<comment> \<open>Rule is called \<open>I\<close> for Identity Combinator, not for Introduction.\<close>
   apply (rule thms.S [THEN thms_MP, THEN thms_MP])
       apply (rule_tac [5] thms.K)
        apply (rule_tac [4] thms.K)
          apply simp_all
   done
 
 
 subsubsection \<open>Weakening, left and right\<close>
 
 lemma weaken_left: "[| G \<subseteq> H;  G|-p |] ==> H|-p"
-  -- \<open>Order of premises is convenient with @{text THEN}\<close>
+  \<comment> \<open>Order of premises is convenient with \<open>THEN\<close>\<close>
   by (erule thms_mono [THEN subsetD])
 
 lemma weaken_left_cons: "H |- p ==> cons(a,H) |- p"
   by (erule subset_consI [THEN weaken_left])
 

     apply (blast intro: weaken_left_cons elim: ImpE)+
   done
 
 lemma hyps_thms_if:
     "p \<in> propn ==> hyps(p,t) |- (if is_true(p,t) then p else p=>Fls)"
-  -- \<open>Typical example of strengthening the induction statement.\<close>
+  \<comment> \<open>Typical example of strengthening the induction statement.\<close>
   apply simp
   apply (induct_tac p)
     apply (simp_all add: thms_I thms.H)
   apply (safe elim!: Fls_Imp [THEN weaken_left_Un1] Fls_Imp [THEN weaken_left_Un2])
   apply (blast intro: weaken_left_Un1 weaken_left_Un2 weaken_right Imp_Fls)+
   done
 
 lemma logcon_thms_p: "[| p \<in> propn;  0 |= p |] ==> hyps(p,t) |- p"
-  -- \<open>Key lemma for completeness; yields a set of assumptions satisfying @{text p}\<close>
+  \<comment> \<open>Key lemma for completeness; yields a set of assumptions satisfying \<open>p\<close>\<close>
   apply (drule hyps_thms_if)
   apply (simp add: logcon_def)
   done
 
 text \<open>

      apply (best intro!: propn_SIs intro: propn_Is)+
   done
 
 lemma thms_excluded_middle_rule:
   "[| cons(p,H) |- q;  cons(p=>Fls,H) |- q;  p \<in> propn |] ==> H |- q"
-  -- \<open>Hard to prove directly because it requires cuts\<close>
+  \<comment> \<open>Hard to prove directly because it requires cuts\<close>
   apply (rule thms_excluded_middle [THEN thms_MP, THEN thms_MP])
      apply (blast intro!: propn_SIs intro: propn_Is)+
   done
 
 

 
 lemma hyps_cons:
     "p \<in> propn ==> hyps(p, cons(v,t)) \<subseteq> cons(#v, hyps(p,t)-{#v=>Fls})"
   by (induct set: propn) auto
 
-text \<open>Two lemmas for use with @{text weaken_left}\<close>
+text \<open>Two lemmas for use with \<open>weaken_left\<close>\<close>
 
 lemma cons_Diff_same: "B-C \<subseteq> cons(a, B-cons(a,C))"
   by blast
 
 lemma cons_Diff_subset2: "cons(a, B-{c}) - D \<subseteq> cons(a, B-cons(c,D))"

 
 
 subsubsection \<open>Completeness theorem\<close>
 
 lemma completeness_0: "[| p \<in> propn;  0 |= p |] ==> 0 |- p"
-  -- \<open>The base case for completeness\<close>
+  \<comment> \<open>The base case for completeness\<close>
   apply (rule Diff_cancel [THEN subst])
   apply (blast intro: completeness_0_lemma)
   done
 
 lemma logcon_Imp: "[| cons(p,H) |= q |] ==> H |= p=>q"
-  -- \<open>A semantic analogue of the Deduction Theorem\<close>
+  \<comment> \<open>A semantic analogue of the Deduction Theorem\<close>
   by (simp add: logcon_def)
 
 lemma completeness:
      "H \<in> Fin(propn) ==> p \<in> propn \<Longrightarrow> H |= p \<Longrightarrow> H |- p"
   apply (induct arbitrary: p set: Fin)