src/HOL/Library/Old_Recdef.thy
changeset 67091 1393c2340eec
parent 60523 be2d9f5ddc76
child 69605 a96320074298
     1.1 --- a/src/HOL/Library/Old_Recdef.thy	Sun Nov 26 13:19:52 2017 +0100
     1.2 +++ b/src/HOL/Library/Old_Recdef.thy	Sun Nov 26 21:08:32 2017 +0100
     1.3 @@ -13,8 +13,8 @@
     1.4  
     1.5  subsection \<open>Lemmas for TFL\<close>
     1.6  
     1.7 -lemma tfl_wf_induct: "ALL R. wf R -->
     1.8 -       (ALL P. (ALL x. (ALL y. (y,x):R --> P y) --> P x) --> (ALL x. P x))"
     1.9 +lemma tfl_wf_induct: "\<forall>R. wf R \<longrightarrow>
    1.10 +       (\<forall>P. (\<forall>x. (\<forall>y. (y,x)\<in>R \<longrightarrow> P y) \<longrightarrow> P x) \<longrightarrow> (\<forall>x. P x))"
    1.11  apply clarify
    1.12  apply (rule_tac r = R and P = P and a = x in wf_induct, assumption, blast)
    1.13  done
    1.14 @@ -22,39 +22,39 @@
    1.15  lemma tfl_cut_def: "cut f r x \<equiv> (\<lambda>y. if (y,x) \<in> r then f y else undefined)"
    1.16    unfolding cut_def .
    1.17  
    1.18 -lemma tfl_cut_apply: "ALL f R. (x,a):R --> (cut f R a)(x) = f(x)"
    1.19 +lemma tfl_cut_apply: "\<forall>f R. (x,a)\<in>R \<longrightarrow> (cut f R a)(x) = f(x)"
    1.20  apply clarify
    1.21  apply (rule cut_apply, assumption)
    1.22  done
    1.23  
    1.24  lemma tfl_wfrec:
    1.25 -     "ALL M R f. (f=wfrec R M) --> wf R --> (ALL x. f x = M (cut f R x) x)"
    1.26 +     "\<forall>M R f. (f=wfrec R M) \<longrightarrow> wf R \<longrightarrow> (\<forall>x. f x = M (cut f R x) x)"
    1.27  apply clarify
    1.28  apply (erule wfrec)
    1.29  done
    1.30  
    1.31 -lemma tfl_eq_True: "(x = True) --> x"
    1.32 +lemma tfl_eq_True: "(x = True) \<longrightarrow> x"
    1.33    by blast
    1.34  
    1.35 -lemma tfl_rev_eq_mp: "(x = y) --> y --> x"
    1.36 +lemma tfl_rev_eq_mp: "(x = y) \<longrightarrow> y \<longrightarrow> x"
    1.37    by blast
    1.38  
    1.39 -lemma tfl_simp_thm: "(x --> y) --> (x = x') --> (x' --> y)"
    1.40 +lemma tfl_simp_thm: "(x \<longrightarrow> y) \<longrightarrow> (x = x') \<longrightarrow> (x' \<longrightarrow> y)"
    1.41    by blast
    1.42  
    1.43 -lemma tfl_P_imp_P_iff_True: "P ==> P = True"
    1.44 +lemma tfl_P_imp_P_iff_True: "P \<Longrightarrow> P = True"
    1.45    by blast
    1.46  
    1.47 -lemma tfl_imp_trans: "(A --> B) ==> (B --> C) ==> (A --> C)"
    1.48 +lemma tfl_imp_trans: "(A \<longrightarrow> B) \<Longrightarrow> (B \<longrightarrow> C) \<Longrightarrow> (A \<longrightarrow> C)"
    1.49    by blast
    1.50  
    1.51 -lemma tfl_disj_assoc: "(a \<or> b) \<or> c == a \<or> (b \<or> c)"
    1.52 +lemma tfl_disj_assoc: "(a \<or> b) \<or> c \<equiv> a \<or> (b \<or> c)"
    1.53    by simp
    1.54  
    1.55 -lemma tfl_disjE: "P \<or> Q ==> P --> R ==> Q --> R ==> R"
    1.56 +lemma tfl_disjE: "P \<or> Q \<Longrightarrow> P \<longrightarrow> R \<Longrightarrow> Q \<longrightarrow> R \<Longrightarrow> R"
    1.57    by blast
    1.58  
    1.59 -lemma tfl_exE: "\<exists>x. P x ==> \<forall>x. P x --> Q ==> Q"
    1.60 +lemma tfl_exE: "\<exists>x. P x \<Longrightarrow> \<forall>x. P x \<longrightarrow> Q \<Longrightarrow> Q"
    1.61    by blast
    1.62  
    1.63  ML_file "old_recdef.ML"