--- a/src/HOL/Nominal/Examples/Contexts.thy Tue Apr 29 19:55:02 2008 +0200
+++ b/src/HOL/Nominal/Examples/Contexts.thy Fri May 02 02:16:10 2008 +0200
@@ -7,14 +7,15 @@
text {*
We show here that the Plotkin-style of defining
- reduction relations (based on congruence rules) is
+ beta-reduction (based on congruence rules) is
equivalent to the Felleisen-Hieb-style representation
- (based on contexts), and show cbv-evaluation via a
- CK-machine described by Roshan James.
+ (based on contexts). We also define cbv-evaluation
+ via a CK-machine described by Roshan James.
- The interesting point is that contexts do not contain
- any binders. On the other hand the operation of filling
- a term into a context produces an alpha-equivalent term.
+ The interesting point in this theory is that contexts
+ do not contain any binders. On the other hand the operation
+ of filling a term into a context produces an alpha-equivalent
+ term.
*}
@@ -29,9 +30,10 @@
| Lam "\<guillemotleft>name\<guillemotright>lam" ("Lam [_]._" [100,100] 100)
text {*
- Contexts - the lambda case in contexts does not bind a name,
- even if we introduce the notation [_]._ for CLam.
+ Contexts - the lambda case in contexts does not bind
+ a name, even if we introduce the notation [_]._ for CLam.
*}
+
nominal_datatype ctx =
Hole ("\<box>" 1000)
| CAppL "ctx" "lam"
@@ -58,22 +60,23 @@
*}
consts
- filling :: "ctx \<Rightarrow> lam \<Rightarrow> lam" ("_<_>" [100,100] 100)
+ filling :: "ctx \<Rightarrow> lam \<Rightarrow> lam" ("_\<lbrakk>_\<rbrakk>" [100,100] 100)
nominal_primrec
- "\<box><t> = t"
- "(CAppL E t')<t> = App (E<t>) t'"
- "(CAppR t' E)<t> = App t' (E<t>)"
- "(CLam [x].E)<t> = Lam [x].(E<t>)"
+ "\<box>\<lbrakk>t\<rbrakk> = t"
+ "(CAppL E t')\<lbrakk>t\<rbrakk> = App (E\<lbrakk>t\<rbrakk>) t'"
+ "(CAppR t' E)\<lbrakk>t\<rbrakk> = App t' (E\<lbrakk>t\<rbrakk>)"
+ "(CLam [x].E)\<lbrakk>t\<rbrakk> = Lam [x].(E\<lbrakk>t\<rbrakk>)"
by (rule TrueI)+
text {*
While contexts themselves are not alpha-equivalence classes, the
filling operation produces an alpha-equivalent lambda-term.
*}
+
lemma alpha_test:
shows "x\<noteq>y \<Longrightarrow> (CLam [x].\<box>) \<noteq> (CLam [y].\<box>)"
- and "(CLam [x].\<box>)<Var x> = (CLam [y].\<box>)<Var y>"
+ and "(CLam [x].\<box>)\<lbrakk>Var x\<rbrakk> = (CLam [y].\<box>)\<lbrakk>Var y\<rbrakk>"
by (auto simp add: alpha ctx.inject lam.inject calc_atm fresh_atm)
text {* The composition of two contexts. *}
@@ -89,7 +92,7 @@
by (rule TrueI)+
lemma ctx_compose:
- shows "E1<E2<t>> = (E1 \<circ> E2)<t>"
+ shows "(E1 \<circ> E2)\<lbrakk>t\<rbrakk> = E1\<lbrakk>E2\<lbrakk>t\<rbrakk>\<rbrakk>"
by (induct E1 rule: ctx.weak_induct) (auto)
text {* Beta-reduction via contexts in the Felleisen-Hieb style. *}
@@ -97,7 +100,7 @@
inductive
ctx_red :: "lam\<Rightarrow>lam\<Rightarrow>bool" ("_ \<longrightarrow>x _" [80,80] 80)
where
- xbeta[intro]: "E<App (Lam [x].t) t'> \<longrightarrow>x E<t[x::=t']>"
+ xbeta[intro]: "E\<lbrakk>App (Lam [x].t) t'\<rbrakk> \<longrightarrow>x E\<lbrakk>t[x::=t']\<rbrakk>"
text {* Beta-reduction via congruence rules in the Plotkin style. *}
@@ -113,21 +116,20 @@
lemma cong_red_in_ctx:
assumes a: "t \<longrightarrow>o t'"
- shows "E<t> \<longrightarrow>o E<t'>"
+ shows "E\<lbrakk>t\<rbrakk> \<longrightarrow>o E\<lbrakk>t'\<rbrakk>"
using a
by (induct E rule: ctx.weak_induct) (auto)
lemma ctx_red_in_ctx:
assumes a: "t \<longrightarrow>x t'"
- shows "E<t> \<longrightarrow>x E<t'>"
-using a
-by (induct) (auto simp add: ctx_compose)
+ shows "E\<lbrakk>t\<rbrakk> \<longrightarrow>x E\<lbrakk>t'\<rbrakk>"
+using a
+by (induct) (auto simp add: ctx_compose[symmetric])
theorem ctx_red_implies_cong_red:
assumes a: "t \<longrightarrow>x t'"
shows "t \<longrightarrow>o t'"
-using a
-by (induct) (auto intro!: cong_red_in_ctx)
+using a by (induct) (auto intro: cong_red_in_ctx)
theorem cong_red_implies_ctx_red:
assumes a: "t \<longrightarrow>o t'"
@@ -135,13 +137,14 @@
using a
proof (induct)
case (obeta x t' t)
- have "\<box><App (Lam [x].t) t'> \<longrightarrow>x \<box><t[x::=t']>" by (rule xbeta)
+ have "\<box>\<lbrakk>App (Lam [x].t) t'\<rbrakk> \<longrightarrow>x \<box>\<lbrakk>t[x::=t']\<rbrakk>" by (rule xbeta)
then show "App (Lam [x].t) t' \<longrightarrow>x t[x::=t']" by simp
qed (metis ctx_red_in_ctx filling.simps)+ (* found by SledgeHammer *)
-section {* We now use context in a CBV list machine *}
-text {* First the function that composes a list of contexts *}
+section {* We now use context in a CBV list machine. *}
+
+text {* First, a function that composes a list of contexts. *}
primrec
ctx_composes :: "ctx list \<Rightarrow> ctx" ("_\<down>" [80] 80)
@@ -150,6 +153,7 @@
| "(E#Es)\<down> = (Es\<down>) \<circ> E"
text {* Values *}
+
inductive
val :: "lam\<Rightarrow>bool"
where
@@ -157,10 +161,11 @@
| v_Var[intro]: "val (Var x)"
text {* CBV-reduction using contexts *}
+
inductive
cbv :: "lam\<Rightarrow>lam\<Rightarrow>bool" ("_ \<longrightarrow>cbv _" [80,80] 80)
where
- cbv_beta[intro]: "val v \<Longrightarrow> E<App (Lam [x].e) v> \<longrightarrow>cbv E<e[x::=v]>"
+ cbv_beta[intro]: "val v \<Longrightarrow> E\<lbrakk>App (Lam [x].e) v\<rbrakk> \<longrightarrow>cbv E\<lbrakk>e[x::=v]\<rbrakk>"
text {* reflexive, transitive closure of CBV reduction *}
inductive
@@ -171,6 +176,7 @@
| cbvs3[intro,trans]: "\<lbrakk>e1\<longrightarrow>cbv* e2; e2 \<longrightarrow>cbv* e3\<rbrakk> \<Longrightarrow> e1 \<longrightarrow>cbv* e3"
text {* A little CK-machine, which builds up a list of evaluation contexts. *}
+
inductive
machine :: "lam\<Rightarrow>ctx list\<Rightarrow>lam\<Rightarrow>ctx list\<Rightarrow>bool" ("<_,_> \<mapsto> <_,_>" [60,60,60,60] 60)
where
@@ -180,15 +186,13 @@
lemma machine_red_implies_cbv_reds_aux:
assumes a: "<e,Es> \<mapsto> <e',Es'>"
- shows "(Es\<down>)<e> \<longrightarrow>cbv* (Es'\<down>)<e'>"
-using a
-by (induct) (auto simp add: ctx_compose[symmetric])
+ shows "(Es\<down>)\<lbrakk>e\<rbrakk> \<longrightarrow>cbv* (Es'\<down>)\<lbrakk>e'\<rbrakk>"
+using a by (induct) (auto simp add: ctx_compose)
lemma machine_execution_implies_cbv_reds:
assumes a: "<e,[]> \<mapsto> <e',[]>"
shows "e \<longrightarrow>cbv* e'"
-using a
-by (auto dest: machine_red_implies_cbv_reds_aux)
+using a by (auto dest: machine_red_implies_cbv_reds_aux)
text {*
One would now like to show something about the opposite