--- a/src/ZF/Induct/Binary_Trees.thy Tue Sep 27 13:34:54 2022 +0200
+++ b/src/ZF/Induct/Binary_Trees.thy Tue Sep 27 16:51:35 2022 +0100
@@ -17,7 +17,7 @@
declare bt.intros [simp]
-lemma Br_neq_left: "l \<in> bt(A) ==> Br(x, l, r) \<noteq> l"
+lemma Br_neq_left: "l \<in> bt(A) \<Longrightarrow> Br(x, l, r) \<noteq> l"
by (induct arbitrary: x r set: bt) auto
lemma Br_iff: "Br(a, l, r) = Br(a', l', r') \<longleftrightarrow> a = a' & l = l' & r = r'"
@@ -32,7 +32,7 @@
definitions.
\<close>
-lemma bt_mono: "A \<subseteq> B ==> bt(A) \<subseteq> bt(B)"
+lemma bt_mono: "A \<subseteq> B \<Longrightarrow> bt(A) \<subseteq> bt(B)"
apply (unfold bt.defs)
apply (rule lfp_mono)
apply (rule bt.bnd_mono)+
@@ -46,18 +46,18 @@
apply (fast intro!: zero_in_univ Inl_in_univ Inr_in_univ Pair_in_univ)
done
-lemma bt_subset_univ: "A \<subseteq> univ(B) ==> bt(A) \<subseteq> univ(B)"
+lemma bt_subset_univ: "A \<subseteq> univ(B) \<Longrightarrow> bt(A) \<subseteq> univ(B)"
apply (rule subset_trans)
apply (erule bt_mono)
apply (rule bt_univ)
done
lemma bt_rec_type:
- "[| t \<in> bt(A);
+ "\<lbrakk>t \<in> bt(A);
c \<in> C(Lf);
- !!x y z r s. [| x \<in> A; y \<in> bt(A); z \<in> bt(A); r \<in> C(y); s \<in> C(z) |] ==>
+ \<And>x y z r s. \<lbrakk>x \<in> A; y \<in> bt(A); z \<in> bt(A); r \<in> C(y); s \<in> C(z)\<rbrakk> \<Longrightarrow>
h(x, y, z, r, s) \<in> C(Br(x, y, z))
- |] ==> bt_rec(c, h, t) \<in> C(t)"
+\<rbrakk> \<Longrightarrow> bt_rec(c, h, t) \<in> C(t)"
\<comment> \<open>Type checking for recursor -- example only; not really needed.\<close>
apply (induct_tac t)
apply simp_all
@@ -71,7 +71,7 @@
"n_nodes(Lf) = 0"
"n_nodes(Br(a, l, r)) = succ(n_nodes(l) #+ n_nodes(r))"
-lemma n_nodes_type [simp]: "t \<in> bt(A) ==> n_nodes(t) \<in> nat"
+lemma n_nodes_type [simp]: "t \<in> bt(A) \<Longrightarrow> n_nodes(t) \<in> nat"
by (induct set: bt) auto
consts n_nodes_aux :: "i => i"
@@ -81,7 +81,7 @@
(\<lambda>k \<in> nat. n_nodes_aux(r) ` (n_nodes_aux(l) ` succ(k)))"
lemma n_nodes_aux_eq:
- "t \<in> bt(A) ==> k \<in> nat ==> n_nodes_aux(t)`k = n_nodes(t) #+ k"
+ "t \<in> bt(A) \<Longrightarrow> k \<in> nat \<Longrightarrow> n_nodes_aux(t)`k = n_nodes(t) #+ k"
apply (induct arbitrary: k set: bt)
apply simp
apply (atomize, simp)
@@ -89,9 +89,9 @@
definition
n_nodes_tail :: "i => i" where
- "n_nodes_tail(t) == n_nodes_aux(t) ` 0"
+ "n_nodes_tail(t) \<equiv> n_nodes_aux(t) ` 0"
-lemma "t \<in> bt(A) ==> n_nodes_tail(t) = n_nodes(t)"
+lemma "t \<in> bt(A) \<Longrightarrow> n_nodes_tail(t) = n_nodes(t)"
by (simp add: n_nodes_tail_def n_nodes_aux_eq)
@@ -103,7 +103,7 @@
"n_leaves(Lf) = 1"
"n_leaves(Br(a, l, r)) = n_leaves(l) #+ n_leaves(r)"
-lemma n_leaves_type [simp]: "t \<in> bt(A) ==> n_leaves(t) \<in> nat"
+lemma n_leaves_type [simp]: "t \<in> bt(A) \<Longrightarrow> n_leaves(t) \<in> nat"
by (induct set: bt) auto
@@ -115,24 +115,24 @@
"bt_reflect(Lf) = Lf"
"bt_reflect(Br(a, l, r)) = Br(a, bt_reflect(r), bt_reflect(l))"
-lemma bt_reflect_type [simp]: "t \<in> bt(A) ==> bt_reflect(t) \<in> bt(A)"
+lemma bt_reflect_type [simp]: "t \<in> bt(A) \<Longrightarrow> bt_reflect(t) \<in> bt(A)"
by (induct set: bt) auto
text \<open>
\medskip Theorems about \<^term>\<open>n_leaves\<close>.
\<close>
-lemma n_leaves_reflect: "t \<in> bt(A) ==> n_leaves(bt_reflect(t)) = n_leaves(t)"
+lemma n_leaves_reflect: "t \<in> bt(A) \<Longrightarrow> n_leaves(bt_reflect(t)) = n_leaves(t)"
by (induct set: bt) (simp_all add: add_commute)
-lemma n_leaves_nodes: "t \<in> bt(A) ==> n_leaves(t) = succ(n_nodes(t))"
+lemma n_leaves_nodes: "t \<in> bt(A) \<Longrightarrow> n_leaves(t) = succ(n_nodes(t))"
by (induct set: bt) simp_all
text \<open>
Theorems about \<^term>\<open>bt_reflect\<close>.
\<close>
-lemma bt_reflect_bt_reflect_ident: "t \<in> bt(A) ==> bt_reflect(bt_reflect(t)) = t"
+lemma bt_reflect_bt_reflect_ident: "t \<in> bt(A) \<Longrightarrow> bt_reflect(bt_reflect(t)) = t"
by (induct set: bt) simp_all
end