--- a/src/HOL/Nominal/Examples/Height.thy Wed Oct 18 23:06:51 2006 +0200
+++ b/src/HOL/Nominal/Examples/Height.thy Wed Oct 18 23:15:16 2006 +0200
@@ -3,7 +3,7 @@
(* Simple, but artificial, problem suggested by D. Wang *)
theory Height
-imports Nominal
+imports "Nominal"
begin
atom_decl name
@@ -12,12 +12,6 @@
| App "lam" "lam"
| Lam "\<guillemotleft>name\<guillemotright>lam" ("Lam [_]._" [100,100] 100)
-thm lam.recs
-
-types 'a f1_ty = "name\<Rightarrow>('a::pt_name)"
- 'a f2_ty = "lam\<Rightarrow>lam\<Rightarrow>'a\<Rightarrow>'a\<Rightarrow>('a::pt_name)"
- 'a f3_ty = "name\<Rightarrow>lam\<Rightarrow>'a\<Rightarrow>('a::pt_name)"
-
text {* definition of the height-function by "structural recursion" ;o) *}
constdefs
@@ -47,22 +41,16 @@
done
text {* derive the characteristic equations for height from the iteration combinator *}
-lemma height_Var:
+lemma height[simp]:
shows "height (Var c) = 1"
+ and "height (App t1 t2) = (max (height t1) (height t2))+1"
+ and "height (Lam [a].t) = (height t)+1"
apply(simp add: height_def)
apply(simp add: lam.recs[where P="\<lambda>_. True", simplified, OF fin_supp_height, OF fcb_height_Lam])
apply(simp add: height_Var_def)
-done
-
-lemma height_App:
- shows "height (App t1 t2) = (max (height t1) (height t2))+1"
apply(simp add: height_def)
apply(simp add: lam.recs[where P="\<lambda>_. True", simplified, OF fin_supp_height, OF fcb_height_Lam])
apply(simp add: height_App_def)
-done
-
-lemma height_Lam:
- shows "height (Lam [a].t) = (height t)+1"
apply(simp add: height_def)
apply(rule trans)
apply(rule lam.recs[where P="\<lambda>_. True", simplified, OF fin_supp_height, OF fcb_height_Lam])
@@ -73,10 +61,8 @@
apply(simp add: height_Lam_def)
done
-text {* add the characteristic equations of height to the simplifier *}
-declare height_Var[simp] height_App[simp] height_Lam[simp]
+text {* define capture-avoiding substitution *}
-text {* define capture-avoiding substitution *}
constdefs
subst_Var :: "name \<Rightarrow> lam \<Rightarrow> name \<Rightarrow> lam"
"subst_Var x t' \<equiv> \<lambda>y. (if y=x then t' else (Var y))"
@@ -87,8 +73,8 @@
subst_Lam :: "name \<Rightarrow> lam \<Rightarrow> name \<Rightarrow> lam \<Rightarrow> lam \<Rightarrow> lam"
"subst_Lam x t' \<equiv> \<lambda>a _ r. Lam [a].r"
- subst_lam :: "name \<Rightarrow> lam \<Rightarrow> lam \<Rightarrow> lam"
- "subst_lam x t' \<equiv> lam_rec (subst_Var x t') (subst_App x t') (subst_Lam x t')"
+ subst_lam :: "lam \<Rightarrow> name \<Rightarrow> lam \<Rightarrow> lam" ("_[_::=_]" [100,100,100] 100)
+ "t[x::=t'] \<equiv> (lam_rec (subst_Var x t') (subst_App x t') (subst_Lam x t')) t"
lemma supports_subst_Var:
shows "((supp (x,t))::name set) supports (subst_Var x t)"
@@ -115,12 +101,6 @@
shows "a\<sharp>(subst_Lam y t') a t r"
by (simp add: subst_Lam_def abs_fresh)
-syntax
- subst_lam_syn :: "lam \<Rightarrow> name \<Rightarrow> lam \<Rightarrow> lam" ("_[_::=_]" [100,100,100] 100)
-
-translations
- "t1[y::=t2]" \<rightleftharpoons> "subst_lam y t2 t1"
-
lemma subst_lam[simp]:
shows "(Var x)[y::=t'] = (if x=y then t' else (Var x))"
and "(App t1 t2)[y::=t'] = App (t1[y::=t']) (t2[y::=t'])"
@@ -143,10 +123,11 @@
lemma height_ge_one:
shows "1 \<le> (height e)"
- by (nominal_induct e rule: lam.induct) (simp | arith)+
+ by (nominal_induct e rule: lam.induct)
+ (simp | arith)+
-text {* unlike the proplem suggested by Wang, the theorem is formulated
- here entirely by using functions *}
+text {* unlike the proplem suggested by Wang, however, the
+ theorem is formulated here entirely by using functions *}
theorem height_subst:
shows "height (e[x::=e']) \<le> (((height e) - 1) + (height e'))"
@@ -158,7 +139,7 @@
case (Lam y e1)
hence ih: "height (e1[x::=e']) \<le> (((height e1) - 1) + (height e'))" by simp
moreover
- have fresh: "y\<sharp>x" "y\<sharp>e'" by fact
+ have vc: "y\<sharp>x" "y\<sharp>e'" by fact (* usual variable convention *)
ultimately show "height ((Lam [y].e1)[x::=e']) \<le> height (Lam [y].e1) - 1 + height e'" by simp
next
case (App e1 e2)