--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Nominal/Examples/Contexts.thy Wed Nov 28 18:39:53 2007 +0100
@@ -0,0 +1,191 @@
+(* $Id: *)
+
+theory Contexts
+imports "../Nominal"
+begin
+
+text {*
+
+ We show here that the Plotkin-style of defining
+ reductions relation based on congruence rules is
+ equivalent to the Felleisen-Hieb-style representation
+ based on contexts.
+
+ The interesting point is that contexts do not bind
+ anything. On the other hand the operation of replacing
+ a term into a context produces an alpha-equivalent term.
+
+*}
+
+atom_decl name
+
+text {* terms *}
+
+nominal_datatype lam =
+ Var "name"
+ | App "lam" "lam"
+ | Lam "\<guillemotleft>name\<guillemotright>lam" ("Lam [_]._" [100,100] 100)
+
+text {* contexts - the context-lambda does not bind anything *}
+
+nominal_datatype ctx =
+ Hole
+ | CAppL "ctx" "lam"
+ | CAppR "lam" "ctx"
+ | CLam "name" "ctx" ("CLam [_]._" [100,100] 100)
+
+text {* Capture-avoiding substitution and three lemmas *}
+
+consts subst :: "lam \<Rightarrow> name \<Rightarrow> lam \<Rightarrow> lam" ("_[_::=_]" [100,100,100] 100)
+
+nominal_primrec
+ "(Var x)[y::=s] = (if x=y then s else (Var x))"
+ "(App t\<^isub>1 t\<^isub>2)[y::=s] = App (t\<^isub>1[y::=s]) (t\<^isub>2[y::=s])"
+ "x\<sharp>(y,s) \<Longrightarrow> (Lam [x].t)[y::=s] = Lam [x].(t[y::=s])"
+apply(finite_guess)+
+apply(rule TrueI)+
+apply(simp add: abs_fresh)
+apply(fresh_guess)+
+done
+
+lemma subst_eqvt[eqvt]:
+ fixes pi::"name prm"
+ shows "pi\<bullet>t1[x::=t2] = (pi\<bullet>t1)[(pi\<bullet>x)::=(pi\<bullet>t2)]"
+by (nominal_induct t1 avoiding: x t2 rule: lam.induct)
+ (auto simp add: perm_bij fresh_atm fresh_bij)
+
+lemma subst_fresh:
+ fixes x y::"name"
+ and t t'::"lam"
+ shows "y\<sharp>([x].t,t') \<Longrightarrow> y\<sharp>t[x::=t']"
+by (nominal_induct t avoiding: x y t' rule: lam.inducts)
+ (auto simp add: abs_fresh fresh_prod fresh_atm)
+
+lemma subst_swap:
+ fixes x y::"name"
+ and t t'::"lam"
+ shows "y\<sharp>t \<Longrightarrow> ([(y,x)]\<bullet>t)[y::=t'] = t[x::=t']"
+by (nominal_induct t avoiding: x y t' rule: lam.inducts)
+ (auto simp add: lam.inject calc_atm fresh_atm abs_fresh)
+
+text {*
+ The operation that fills one term into a hole. While
+ contexts are not alpha-equivalence classes, the filling
+ operation produces an alpha-equivalent lambda-term.
+*}
+
+consts
+ replace :: "ctx \<Rightarrow> lam \<Rightarrow> lam" ("_<_>" [100,100] 100)
+
+nominal_primrec
+ "Hole<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>)"
+by (rule TrueI)+
+
+lemma alpha_test:
+ shows "(CLam [x].Hole)<Var x> = (CLam [y].Hole)<Var y>"
+by (auto simp add: alpha lam.inject calc_atm fresh_atm)
+
+lemma replace_eqvt[eqvt]:
+ fixes pi:: "name prm"
+ shows "pi\<bullet>(E<e>) = (pi\<bullet>E)<(pi\<bullet>e)>"
+by (nominal_induct E rule: ctx.inducts) (auto)
+
+lemma replace_fresh:
+ fixes x::"name"
+ and E::"ctx"
+ and t::"lam"
+ shows "x\<sharp>(E,t) \<Longrightarrow> x\<sharp>E<t>"
+by (induct E rule: ctx.weak_induct)
+ (auto simp add: fresh_prod abs_fresh)
+
+text {* composition of two contexts *}
+
+consts
+ ctx_replace :: "ctx \<Rightarrow> ctx \<Rightarrow> ctx" ("_ \<circ> _" [100,100] 100)
+
+nominal_primrec
+ "Hole \<circ> E' = E'"
+ "(CAppL E t') \<circ> E' = CAppL (E \<circ> E') t'"
+ "(CAppR t' E) \<circ> E' = CAppR t' (E \<circ> E')"
+ "(CLam [x].E) \<circ> E' = CLam [x].(E \<circ> E')"
+by (rule TrueI)+
+
+lemma ctx_compose:
+ shows "E1<E2<t>> = (E1 \<circ> E2)<t>"
+by (induct E1 rule: ctx.weak_induct) (auto)
+
+lemma ctx_compose_fresh:
+ fixes x::"name"
+ and E1 E2::"ctx"
+ shows "x\<sharp>(E1,E2) \<Longrightarrow> x\<sharp>(E1\<circ>E2)"
+by (induct E1 rule: ctx.weak_induct)
+ (auto simp add: fresh_prod)
+
+text {* beta-reduction via contexts *}
+
+inductive
+ ctx_red :: "lam\<Rightarrow>lam\<Rightarrow>bool" ("_ \<longrightarrow>x _" [80,80] 80)
+where
+ xbeta[intro]: "x\<sharp>(E,t') \<Longrightarrow> E<App (Lam [x].t) t'> \<longrightarrow>x E<t[x::=t']>"
+
+equivariance ctx_red
+
+nominal_inductive ctx_red
+ by (simp_all add: replace_fresh subst_fresh abs_fresh)
+
+text {* beta-reduction via congruence rules *}
+
+inductive
+ cong_red :: "lam\<Rightarrow>lam\<Rightarrow>bool" ("_ \<longrightarrow>o _" [80,80] 80)
+where
+ obeta[intro]: "x\<sharp>t' \<Longrightarrow> App (Lam [x].t) t' \<longrightarrow>o t[x::=t']"
+| oapp1[intro]: "t \<longrightarrow>o t' \<Longrightarrow> App t t2 \<longrightarrow>o App t' t2"
+| oapp2[intro]: "t \<longrightarrow>o t' \<Longrightarrow> App t2 t \<longrightarrow>o App t2 t'"
+| olam[intro]: "t \<longrightarrow>o t' \<Longrightarrow> Lam [x].t \<longrightarrow>o Lam [x].t'"
+
+equivariance cong_red
+
+nominal_inductive cong_red
+ by (simp_all add: subst_fresh abs_fresh)
+
+text {* the proof that shows both relations are equal *}
+
+lemma cong_red_ctx:
+ assumes a: "t \<longrightarrow>o t'"
+ shows "E<t> \<longrightarrow>o E<t'>"
+using a
+by (induct E rule: ctx.weak_induct) (auto)
+
+lemma ctx_red_ctx:
+ assumes a: "t \<longrightarrow>x t'"
+ shows "E<t> \<longrightarrow>x E<t'>"
+using a
+by (nominal_induct t t' avoiding: E rule: ctx_red.strong_induct)
+ (auto simp add: ctx_compose ctx_compose_fresh)
+
+lemma ctx_red_hole:
+ assumes a: "Hole<t> \<longrightarrow>x Hole<t'>"
+ shows "t \<longrightarrow>x t'"
+using a by simp
+
+theorem ctx_red_cong_red:
+ assumes a: "t \<longrightarrow>x t'"
+ shows "t \<longrightarrow>o t'"
+using a
+by (induct) (auto intro!: cong_red_ctx)
+
+theorem cong_red_ctx_red:
+ assumes a: "t \<longrightarrow>o t'"
+ shows "t \<longrightarrow>x t'"
+using a
+apply(induct)
+apply(rule ctx_red_hole)
+apply(rule xbeta)
+apply(simp)
+apply(metis ctx_red_ctx replace.simps)+ (* found by SledgeHammer *)
+done
+
+end