--- a/src/HOL/Predicate_Compile_Examples/Specialisation_Examples.thy Tue Aug 13 14:20:22 2013 +0200
+++ b/src/HOL/Predicate_Compile_Examples/Specialisation_Examples.thy Tue Aug 13 16:25:47 2013 +0200
@@ -93,15 +93,15 @@
"[]\<langle>i\<rangle> = \<bottom>"
| "(x # xs)\<langle>i\<rangle> = (case i of 0 \<Rightarrow> \<lfloor>x\<rfloor> | Suc j \<Rightarrow> xs \<langle>j\<rangle>)"
-primrec assoc :: "('a \<times> 'b) list \<Rightarrow> 'a \<Rightarrow> 'b option" ("_\<langle>_\<rangle>\<^isub>?" [90, 0] 91)
+primrec assoc :: "('a \<times> 'b) list \<Rightarrow> 'a \<Rightarrow> 'b option" ("_\<langle>_\<rangle>\<^sub>?" [90, 0] 91)
where
- "[]\<langle>a\<rangle>\<^isub>? = \<bottom>"
-| "(x # xs)\<langle>a\<rangle>\<^isub>? = (if fst x = a then \<lfloor>snd x\<rfloor> else xs\<langle>a\<rangle>\<^isub>?)"
+ "[]\<langle>a\<rangle>\<^sub>? = \<bottom>"
+| "(x # xs)\<langle>a\<rangle>\<^sub>? = (if fst x = a then \<lfloor>snd x\<rfloor> else xs\<langle>a\<rangle>\<^sub>?)"
primrec unique :: "('a \<times> 'b) list \<Rightarrow> bool"
where
"unique [] = True"
-| "unique (x # xs) = (xs\<langle>fst x\<rangle>\<^isub>? = \<bottom> \<and> unique xs)"
+| "unique (x # xs) = (xs\<langle>fst x\<rangle>\<^sub>? = \<bottom> \<and> unique xs)"
datatype type =
TVar nat
@@ -132,66 +132,66 @@
| Abs type trm ("(3\<lambda>:_./ _)" [0, 10] 10)
| TAbs type trm ("(3\<lambda><:_./ _)" [0, 10] 10)
| App trm trm (infixl "\<bullet>" 200)
- | TApp trm type (infixl "\<bullet>\<^isub>\<tau>" 200)
+ | TApp trm type (infixl "\<bullet>\<^sub>\<tau>" 200)
-primrec liftT :: "nat \<Rightarrow> nat \<Rightarrow> type \<Rightarrow> type" ("\<up>\<^isub>\<tau>")
+primrec liftT :: "nat \<Rightarrow> nat \<Rightarrow> type \<Rightarrow> type" ("\<up>\<^sub>\<tau>")
where
- "\<up>\<^isub>\<tau> n k (TVar i) = (if i < k then TVar i else TVar (i + n))"
-| "\<up>\<^isub>\<tau> n k Top = Top"
-| "\<up>\<^isub>\<tau> n k (T \<rightarrow> U) = \<up>\<^isub>\<tau> n k T \<rightarrow> \<up>\<^isub>\<tau> n k U"
-| "\<up>\<^isub>\<tau> n k (\<forall><:T. U) = (\<forall><:\<up>\<^isub>\<tau> n k T. \<up>\<^isub>\<tau> n (k + 1) U)"
+ "\<up>\<^sub>\<tau> n k (TVar i) = (if i < k then TVar i else TVar (i + n))"
+| "\<up>\<^sub>\<tau> n k Top = Top"
+| "\<up>\<^sub>\<tau> n k (T \<rightarrow> U) = \<up>\<^sub>\<tau> n k T \<rightarrow> \<up>\<^sub>\<tau> n k U"
+| "\<up>\<^sub>\<tau> n k (\<forall><:T. U) = (\<forall><:\<up>\<^sub>\<tau> n k T. \<up>\<^sub>\<tau> n (k + 1) U)"
primrec lift :: "nat \<Rightarrow> nat \<Rightarrow> trm \<Rightarrow> trm" ("\<up>")
where
"\<up> n k (Var i) = (if i < k then Var i else Var (i + n))"
-| "\<up> n k (\<lambda>:T. t) = (\<lambda>:\<up>\<^isub>\<tau> n k T. \<up> n (k + 1) t)"
-| "\<up> n k (\<lambda><:T. t) = (\<lambda><:\<up>\<^isub>\<tau> n k T. \<up> n (k + 1) t)"
+| "\<up> n k (\<lambda>:T. t) = (\<lambda>:\<up>\<^sub>\<tau> n k T. \<up> n (k + 1) t)"
+| "\<up> n k (\<lambda><:T. t) = (\<lambda><:\<up>\<^sub>\<tau> n k T. \<up> n (k + 1) t)"
| "\<up> n k (s \<bullet> t) = \<up> n k s \<bullet> \<up> n k t"
-| "\<up> n k (t \<bullet>\<^isub>\<tau> T) = \<up> n k t \<bullet>\<^isub>\<tau> \<up>\<^isub>\<tau> n k T"
+| "\<up> n k (t \<bullet>\<^sub>\<tau> T) = \<up> n k t \<bullet>\<^sub>\<tau> \<up>\<^sub>\<tau> n k T"
-primrec substTT :: "type \<Rightarrow> nat \<Rightarrow> type \<Rightarrow> type" ("_[_ \<mapsto>\<^isub>\<tau> _]\<^isub>\<tau>" [300, 0, 0] 300)
+primrec substTT :: "type \<Rightarrow> nat \<Rightarrow> type \<Rightarrow> type" ("_[_ \<mapsto>\<^sub>\<tau> _]\<^sub>\<tau>" [300, 0, 0] 300)
where
- "(TVar i)[k \<mapsto>\<^isub>\<tau> S]\<^isub>\<tau> =
- (if k < i then TVar (i - 1) else if i = k then \<up>\<^isub>\<tau> k 0 S else TVar i)"
-| "Top[k \<mapsto>\<^isub>\<tau> S]\<^isub>\<tau> = Top"
-| "(T \<rightarrow> U)[k \<mapsto>\<^isub>\<tau> S]\<^isub>\<tau> = T[k \<mapsto>\<^isub>\<tau> S]\<^isub>\<tau> \<rightarrow> U[k \<mapsto>\<^isub>\<tau> S]\<^isub>\<tau>"
-| "(\<forall><:T. U)[k \<mapsto>\<^isub>\<tau> S]\<^isub>\<tau> = (\<forall><:T[k \<mapsto>\<^isub>\<tau> S]\<^isub>\<tau>. U[k+1 \<mapsto>\<^isub>\<tau> S]\<^isub>\<tau>)"
+ "(TVar i)[k \<mapsto>\<^sub>\<tau> S]\<^sub>\<tau> =
+ (if k < i then TVar (i - 1) else if i = k then \<up>\<^sub>\<tau> k 0 S else TVar i)"
+| "Top[k \<mapsto>\<^sub>\<tau> S]\<^sub>\<tau> = Top"
+| "(T \<rightarrow> U)[k \<mapsto>\<^sub>\<tau> S]\<^sub>\<tau> = T[k \<mapsto>\<^sub>\<tau> S]\<^sub>\<tau> \<rightarrow> U[k \<mapsto>\<^sub>\<tau> S]\<^sub>\<tau>"
+| "(\<forall><:T. U)[k \<mapsto>\<^sub>\<tau> S]\<^sub>\<tau> = (\<forall><:T[k \<mapsto>\<^sub>\<tau> S]\<^sub>\<tau>. U[k+1 \<mapsto>\<^sub>\<tau> S]\<^sub>\<tau>)"
-primrec decT :: "nat \<Rightarrow> nat \<Rightarrow> type \<Rightarrow> type" ("\<down>\<^isub>\<tau>")
+primrec decT :: "nat \<Rightarrow> nat \<Rightarrow> type \<Rightarrow> type" ("\<down>\<^sub>\<tau>")
where
- "\<down>\<^isub>\<tau> 0 k T = T"
-| "\<down>\<^isub>\<tau> (Suc n) k T = \<down>\<^isub>\<tau> n k (T[k \<mapsto>\<^isub>\<tau> Top]\<^isub>\<tau>)"
+ "\<down>\<^sub>\<tau> 0 k T = T"
+| "\<down>\<^sub>\<tau> (Suc n) k T = \<down>\<^sub>\<tau> n k (T[k \<mapsto>\<^sub>\<tau> Top]\<^sub>\<tau>)"
primrec subst :: "trm \<Rightarrow> nat \<Rightarrow> trm \<Rightarrow> trm" ("_[_ \<mapsto> _]" [300, 0, 0] 300)
where
"(Var i)[k \<mapsto> s] = (if k < i then Var (i - 1) else if i = k then \<up> k 0 s else Var i)"
| "(t \<bullet> u)[k \<mapsto> s] = t[k \<mapsto> s] \<bullet> u[k \<mapsto> s]"
-| "(t \<bullet>\<^isub>\<tau> T)[k \<mapsto> s] = t[k \<mapsto> s] \<bullet>\<^isub>\<tau> \<down>\<^isub>\<tau> 1 k T"
-| "(\<lambda>:T. t)[k \<mapsto> s] = (\<lambda>:\<down>\<^isub>\<tau> 1 k T. t[k+1 \<mapsto> s])"
-| "(\<lambda><:T. t)[k \<mapsto> s] = (\<lambda><:\<down>\<^isub>\<tau> 1 k T. t[k+1 \<mapsto> s])"
+| "(t \<bullet>\<^sub>\<tau> T)[k \<mapsto> s] = t[k \<mapsto> s] \<bullet>\<^sub>\<tau> \<down>\<^sub>\<tau> 1 k T"
+| "(\<lambda>:T. t)[k \<mapsto> s] = (\<lambda>:\<down>\<^sub>\<tau> 1 k T. t[k+1 \<mapsto> s])"
+| "(\<lambda><:T. t)[k \<mapsto> s] = (\<lambda><:\<down>\<^sub>\<tau> 1 k T. t[k+1 \<mapsto> s])"
-primrec substT :: "trm \<Rightarrow> nat \<Rightarrow> type \<Rightarrow> trm" ("_[_ \<mapsto>\<^isub>\<tau> _]" [300, 0, 0] 300)
+primrec substT :: "trm \<Rightarrow> nat \<Rightarrow> type \<Rightarrow> trm" ("_[_ \<mapsto>\<^sub>\<tau> _]" [300, 0, 0] 300)
where
- "(Var i)[k \<mapsto>\<^isub>\<tau> S] = (if k < i then Var (i - 1) else Var i)"
-| "(t \<bullet> u)[k \<mapsto>\<^isub>\<tau> S] = t[k \<mapsto>\<^isub>\<tau> S] \<bullet> u[k \<mapsto>\<^isub>\<tau> S]"
-| "(t \<bullet>\<^isub>\<tau> T)[k \<mapsto>\<^isub>\<tau> S] = t[k \<mapsto>\<^isub>\<tau> S] \<bullet>\<^isub>\<tau> T[k \<mapsto>\<^isub>\<tau> S]\<^isub>\<tau>"
-| "(\<lambda>:T. t)[k \<mapsto>\<^isub>\<tau> S] = (\<lambda>:T[k \<mapsto>\<^isub>\<tau> S]\<^isub>\<tau>. t[k+1 \<mapsto>\<^isub>\<tau> S])"
-| "(\<lambda><:T. t)[k \<mapsto>\<^isub>\<tau> S] = (\<lambda><:T[k \<mapsto>\<^isub>\<tau> S]\<^isub>\<tau>. t[k+1 \<mapsto>\<^isub>\<tau> S])"
+ "(Var i)[k \<mapsto>\<^sub>\<tau> S] = (if k < i then Var (i - 1) else Var i)"
+| "(t \<bullet> u)[k \<mapsto>\<^sub>\<tau> S] = t[k \<mapsto>\<^sub>\<tau> S] \<bullet> u[k \<mapsto>\<^sub>\<tau> S]"
+| "(t \<bullet>\<^sub>\<tau> T)[k \<mapsto>\<^sub>\<tau> S] = t[k \<mapsto>\<^sub>\<tau> S] \<bullet>\<^sub>\<tau> T[k \<mapsto>\<^sub>\<tau> S]\<^sub>\<tau>"
+| "(\<lambda>:T. t)[k \<mapsto>\<^sub>\<tau> S] = (\<lambda>:T[k \<mapsto>\<^sub>\<tau> S]\<^sub>\<tau>. t[k+1 \<mapsto>\<^sub>\<tau> S])"
+| "(\<lambda><:T. t)[k \<mapsto>\<^sub>\<tau> S] = (\<lambda><:T[k \<mapsto>\<^sub>\<tau> S]\<^sub>\<tau>. t[k+1 \<mapsto>\<^sub>\<tau> S])"
-primrec liftE :: "nat \<Rightarrow> nat \<Rightarrow> env \<Rightarrow> env" ("\<up>\<^isub>e")
+primrec liftE :: "nat \<Rightarrow> nat \<Rightarrow> env \<Rightarrow> env" ("\<up>\<^sub>e")
where
- "\<up>\<^isub>e n k [] = []"
-| "\<up>\<^isub>e n k (B \<Colon> \<Gamma>) = mapB (\<up>\<^isub>\<tau> n (k + \<parallel>\<Gamma>\<parallel>)) B \<Colon> \<up>\<^isub>e n k \<Gamma>"
+ "\<up>\<^sub>e n k [] = []"
+| "\<up>\<^sub>e n k (B \<Colon> \<Gamma>) = mapB (\<up>\<^sub>\<tau> n (k + \<parallel>\<Gamma>\<parallel>)) B \<Colon> \<up>\<^sub>e n k \<Gamma>"
-primrec substE :: "env \<Rightarrow> nat \<Rightarrow> type \<Rightarrow> env" ("_[_ \<mapsto>\<^isub>\<tau> _]\<^isub>e" [300, 0, 0] 300)
+primrec substE :: "env \<Rightarrow> nat \<Rightarrow> type \<Rightarrow> env" ("_[_ \<mapsto>\<^sub>\<tau> _]\<^sub>e" [300, 0, 0] 300)
where
- "[][k \<mapsto>\<^isub>\<tau> T]\<^isub>e = []"
-| "(B \<Colon> \<Gamma>)[k \<mapsto>\<^isub>\<tau> T]\<^isub>e = mapB (\<lambda>U. U[k + \<parallel>\<Gamma>\<parallel> \<mapsto>\<^isub>\<tau> T]\<^isub>\<tau>) B \<Colon> \<Gamma>[k \<mapsto>\<^isub>\<tau> T]\<^isub>e"
+ "[][k \<mapsto>\<^sub>\<tau> T]\<^sub>e = []"
+| "(B \<Colon> \<Gamma>)[k \<mapsto>\<^sub>\<tau> T]\<^sub>e = mapB (\<lambda>U. U[k + \<parallel>\<Gamma>\<parallel> \<mapsto>\<^sub>\<tau> T]\<^sub>\<tau>) B \<Colon> \<Gamma>[k \<mapsto>\<^sub>\<tau> T]\<^sub>e"
-primrec decE :: "nat \<Rightarrow> nat \<Rightarrow> env \<Rightarrow> env" ("\<down>\<^isub>e")
+primrec decE :: "nat \<Rightarrow> nat \<Rightarrow> env \<Rightarrow> env" ("\<down>\<^sub>e")
where
- "\<down>\<^isub>e 0 k \<Gamma> = \<Gamma>"
-| "\<down>\<^isub>e (Suc n) k \<Gamma> = \<down>\<^isub>e n k (\<Gamma>[k \<mapsto>\<^isub>\<tau> Top]\<^isub>e)"
+ "\<down>\<^sub>e 0 k \<Gamma> = \<Gamma>"
+| "\<down>\<^sub>e (Suc n) k \<Gamma> = \<down>\<^sub>e n k (\<Gamma>[k \<mapsto>\<^sub>\<tau> Top]\<^sub>e)"
inductive
well_formed :: "env \<Rightarrow> type \<Rightarrow> bool" ("_ \<turnstile>\<^bsub>wf\<^esub> _" [50, 50] 50)
@@ -224,20 +224,20 @@
SA_Top: "\<Gamma> \<turnstile>\<^bsub>wf\<^esub> \<Longrightarrow> \<Gamma> \<turnstile>\<^bsub>wf\<^esub> S \<Longrightarrow> \<Gamma> \<turnstile> S <: Top"
| SA_refl_TVar: "\<Gamma> \<turnstile>\<^bsub>wf\<^esub> \<Longrightarrow> \<Gamma> \<turnstile>\<^bsub>wf\<^esub> TVar i \<Longrightarrow> \<Gamma> \<turnstile> TVar i <: TVar i"
| SA_trans_TVar: "\<Gamma>\<langle>i\<rangle> = \<lfloor>TVarB U\<rfloor> \<Longrightarrow>
- \<Gamma> \<turnstile> \<up>\<^isub>\<tau> (Suc i) 0 U <: T \<Longrightarrow> \<Gamma> \<turnstile> TVar i <: T"
-| SA_arrow: "\<Gamma> \<turnstile> T\<^isub>1 <: S\<^isub>1 \<Longrightarrow> \<Gamma> \<turnstile> S\<^isub>2 <: T\<^isub>2 \<Longrightarrow> \<Gamma> \<turnstile> S\<^isub>1 \<rightarrow> S\<^isub>2 <: T\<^isub>1 \<rightarrow> T\<^isub>2"
-| SA_all: "\<Gamma> \<turnstile> T\<^isub>1 <: S\<^isub>1 \<Longrightarrow> TVarB T\<^isub>1 \<Colon> \<Gamma> \<turnstile> S\<^isub>2 <: T\<^isub>2 \<Longrightarrow>
- \<Gamma> \<turnstile> (\<forall><:S\<^isub>1. S\<^isub>2) <: (\<forall><:T\<^isub>1. T\<^isub>2)"
+ \<Gamma> \<turnstile> \<up>\<^sub>\<tau> (Suc i) 0 U <: T \<Longrightarrow> \<Gamma> \<turnstile> TVar i <: T"
+| SA_arrow: "\<Gamma> \<turnstile> T\<^sub>1 <: S\<^sub>1 \<Longrightarrow> \<Gamma> \<turnstile> S\<^sub>2 <: T\<^sub>2 \<Longrightarrow> \<Gamma> \<turnstile> S\<^sub>1 \<rightarrow> S\<^sub>2 <: T\<^sub>1 \<rightarrow> T\<^sub>2"
+| SA_all: "\<Gamma> \<turnstile> T\<^sub>1 <: S\<^sub>1 \<Longrightarrow> TVarB T\<^sub>1 \<Colon> \<Gamma> \<turnstile> S\<^sub>2 <: T\<^sub>2 \<Longrightarrow>
+ \<Gamma> \<turnstile> (\<forall><:S\<^sub>1. S\<^sub>2) <: (\<forall><:T\<^sub>1. T\<^sub>2)"
inductive
typing :: "env \<Rightarrow> trm \<Rightarrow> type \<Rightarrow> bool" ("_ \<turnstile> _ : _" [50, 50, 50] 50)
where
- T_Var: "\<Gamma> \<turnstile>\<^bsub>wf\<^esub> \<Longrightarrow> \<Gamma>\<langle>i\<rangle> = \<lfloor>VarB U\<rfloor> \<Longrightarrow> T = \<up>\<^isub>\<tau> (Suc i) 0 U \<Longrightarrow> \<Gamma> \<turnstile> Var i : T"
-| T_Abs: "VarB T\<^isub>1 \<Colon> \<Gamma> \<turnstile> t\<^isub>2 : T\<^isub>2 \<Longrightarrow> \<Gamma> \<turnstile> (\<lambda>:T\<^isub>1. t\<^isub>2) : T\<^isub>1 \<rightarrow> \<down>\<^isub>\<tau> 1 0 T\<^isub>2"
-| T_App: "\<Gamma> \<turnstile> t\<^isub>1 : T\<^isub>11 \<rightarrow> T\<^isub>12 \<Longrightarrow> \<Gamma> \<turnstile> t\<^isub>2 : T\<^isub>11 \<Longrightarrow> \<Gamma> \<turnstile> t\<^isub>1 \<bullet> t\<^isub>2 : T\<^isub>12"
-| T_TAbs: "TVarB T\<^isub>1 \<Colon> \<Gamma> \<turnstile> t\<^isub>2 : T\<^isub>2 \<Longrightarrow> \<Gamma> \<turnstile> (\<lambda><:T\<^isub>1. t\<^isub>2) : (\<forall><:T\<^isub>1. T\<^isub>2)"
-| T_TApp: "\<Gamma> \<turnstile> t\<^isub>1 : (\<forall><:T\<^isub>11. T\<^isub>12) \<Longrightarrow> \<Gamma> \<turnstile> T\<^isub>2 <: T\<^isub>11 \<Longrightarrow>
- \<Gamma> \<turnstile> t\<^isub>1 \<bullet>\<^isub>\<tau> T\<^isub>2 : T\<^isub>12[0 \<mapsto>\<^isub>\<tau> T\<^isub>2]\<^isub>\<tau>"
+ T_Var: "\<Gamma> \<turnstile>\<^bsub>wf\<^esub> \<Longrightarrow> \<Gamma>\<langle>i\<rangle> = \<lfloor>VarB U\<rfloor> \<Longrightarrow> T = \<up>\<^sub>\<tau> (Suc i) 0 U \<Longrightarrow> \<Gamma> \<turnstile> Var i : T"
+| T_Abs: "VarB T\<^sub>1 \<Colon> \<Gamma> \<turnstile> t\<^sub>2 : T\<^sub>2 \<Longrightarrow> \<Gamma> \<turnstile> (\<lambda>:T\<^sub>1. t\<^sub>2) : T\<^sub>1 \<rightarrow> \<down>\<^sub>\<tau> 1 0 T\<^sub>2"
+| T_App: "\<Gamma> \<turnstile> t\<^sub>1 : T\<^sub>11 \<rightarrow> T\<^sub>12 \<Longrightarrow> \<Gamma> \<turnstile> t\<^sub>2 : T\<^sub>11 \<Longrightarrow> \<Gamma> \<turnstile> t\<^sub>1 \<bullet> t\<^sub>2 : T\<^sub>12"
+| T_TAbs: "TVarB T\<^sub>1 \<Colon> \<Gamma> \<turnstile> t\<^sub>2 : T\<^sub>2 \<Longrightarrow> \<Gamma> \<turnstile> (\<lambda><:T\<^sub>1. t\<^sub>2) : (\<forall><:T\<^sub>1. T\<^sub>2)"
+| T_TApp: "\<Gamma> \<turnstile> t\<^sub>1 : (\<forall><:T\<^sub>11. T\<^sub>12) \<Longrightarrow> \<Gamma> \<turnstile> T\<^sub>2 <: T\<^sub>11 \<Longrightarrow>
+ \<Gamma> \<turnstile> t\<^sub>1 \<bullet>\<^sub>\<tau> T\<^sub>2 : T\<^sub>12[0 \<mapsto>\<^sub>\<tau> T\<^sub>2]\<^sub>\<tau>"
| T_Sub: "\<Gamma> \<turnstile> t : S \<Longrightarrow> \<Gamma> \<turnstile> S <: T \<Longrightarrow> \<Gamma> \<turnstile> t : T"
code_pred [inductify, skip_proof, specialise] typing .