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"
```