theory Specialisation_Examples
imports Main Predicate_Compile_Alternative_Defs
begin
section {* Specialisation Examples *}
fun nth_el'
where
"nth_el' [] i = None"
| "nth_el' (x # xs) i = (case i of 0 => Some x | Suc j => nth_el' xs j)"
definition
"greater_than_index xs = (\<forall>i x. nth_el' xs i = Some x --> x > i)"
code_pred (expected_modes: i => bool) [inductify, skip_proof, specialise] greater_than_index .
ML {* Predicate_Compile_Core.intros_of @{context} @{const_name specialised_nth_el'P} *}
thm greater_than_index.equation
values [expected "{()}"] "{x. greater_than_index [1,2,4,6]}"
values [expected "{}"] "{x. greater_than_index [0,2,3,2]}"
subsection {* Common subterms *}
text {* If a predicate is called with common subterms as arguments,
this predicate should be specialised.
*}
definition max_nat :: "nat => nat => nat"
where "max_nat a b = (if a <= b then b else a)"
lemma [code_pred_inline]:
"max = max_nat"
by (simp add: expand_fun_eq max_def max_nat_def)
definition
"max_of_my_Suc x = max x (Suc x)"
text {* In this example, max is specialised, hence the mode o => i => bool is possible *}
code_pred (modes: o => i => bool) [inductify, specialise, skip_proof] max_of_my_Suc .
thm max_of_my_SucP.equation
ML {* Predicate_Compile_Core.intros_of @{context} @{const_name specialised_max_natP} *}
values "{x. max_of_my_SucP x 6}"
subsection {* Sorts *}
code_pred [inductify] sorted .
thm sorted.equation
section {* Specialisation in POPLmark theory *}
notation
Some ("\<lfloor>_\<rfloor>")
notation
None ("\<bottom>")
notation
length ("\<parallel>_\<parallel>")
notation
Cons ("_ \<Colon>/ _" [66, 65] 65)
primrec
nth_el :: "'a list \<Rightarrow> nat \<Rightarrow> 'a option" ("_\<langle>_\<rangle>" [90, 0] 91)
where
"[]\<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)
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>?)"
primrec unique :: "('a \<times> 'b) list \<Rightarrow> bool"
where
"unique [] = True"
| "unique (x # xs) = (xs\<langle>fst x\<rangle>\<^isub>? = \<bottom> \<and> unique xs)"
datatype type =
TVar nat
| Top
| Fun type type (infixr "\<rightarrow>" 200)
| TyAll type type ("(3\<forall><:_./ _)" [0, 10] 10)
datatype binding = VarB type | TVarB type
types env = "binding list"
primrec is_TVarB :: "binding \<Rightarrow> bool"
where
"is_TVarB (VarB T) = False"
| "is_TVarB (TVarB T) = True"
primrec type_ofB :: "binding \<Rightarrow> type"
where
"type_ofB (VarB T) = T"
| "type_ofB (TVarB T) = T"
primrec mapB :: "(type \<Rightarrow> type) \<Rightarrow> binding \<Rightarrow> binding"
where
"mapB f (VarB T) = VarB (f T)"
| "mapB f (TVarB T) = TVarB (f T)"
datatype trm =
Var nat
| 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)
primrec liftT :: "nat \<Rightarrow> nat \<Rightarrow> type \<Rightarrow> type" ("\<up>\<^isub>\<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)"
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 (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"
primrec substTT :: "type \<Rightarrow> nat \<Rightarrow> type \<Rightarrow> type" ("_[_ \<mapsto>\<^isub>\<tau> _]\<^isub>\<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>)"
primrec decT :: "nat \<Rightarrow> nat \<Rightarrow> type \<Rightarrow> type" ("\<down>\<^isub>\<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>)"
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])"
primrec substT :: "trm \<Rightarrow> nat \<Rightarrow> type \<Rightarrow> trm" ("_[_ \<mapsto>\<^isub>\<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])"
primrec liftE :: "nat \<Rightarrow> nat \<Rightarrow> env \<Rightarrow> env" ("\<up>\<^isub>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>"
primrec substE :: "env \<Rightarrow> nat \<Rightarrow> type \<Rightarrow> env" ("_[_ \<mapsto>\<^isub>\<tau> _]\<^isub>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"
primrec decE :: "nat \<Rightarrow> nat \<Rightarrow> env \<Rightarrow> env" ("\<down>\<^isub>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)"
inductive
well_formed :: "env \<Rightarrow> type \<Rightarrow> bool" ("_ \<turnstile>\<^bsub>wf\<^esub> _" [50, 50] 50)
where
wf_TVar: "\<Gamma>\<langle>i\<rangle> = \<lfloor>TVarB T\<rfloor> \<Longrightarrow> \<Gamma> \<turnstile>\<^bsub>wf\<^esub> TVar i"
| wf_Top: "\<Gamma> \<turnstile>\<^bsub>wf\<^esub> Top"
| wf_arrow: "\<Gamma> \<turnstile>\<^bsub>wf\<^esub> T \<Longrightarrow> \<Gamma> \<turnstile>\<^bsub>wf\<^esub> U \<Longrightarrow> \<Gamma> \<turnstile>\<^bsub>wf\<^esub> T \<rightarrow> U"
| wf_all: "\<Gamma> \<turnstile>\<^bsub>wf\<^esub> T \<Longrightarrow> TVarB T \<Colon> \<Gamma> \<turnstile>\<^bsub>wf\<^esub> U \<Longrightarrow> \<Gamma> \<turnstile>\<^bsub>wf\<^esub> (\<forall><:T. U)"
inductive
well_formedE :: "env \<Rightarrow> bool" ("_ \<turnstile>\<^bsub>wf\<^esub>" [50] 50)
and well_formedB :: "env \<Rightarrow> binding \<Rightarrow> bool" ("_ \<turnstile>\<^bsub>wfB\<^esub> _" [50, 50] 50)
where
"\<Gamma> \<turnstile>\<^bsub>wfB\<^esub> B \<equiv> \<Gamma> \<turnstile>\<^bsub>wf\<^esub> type_ofB B"
| wf_Nil: "[] \<turnstile>\<^bsub>wf\<^esub>"
| wf_Cons: "\<Gamma> \<turnstile>\<^bsub>wfB\<^esub> B \<Longrightarrow> \<Gamma> \<turnstile>\<^bsub>wf\<^esub> \<Longrightarrow> B \<Colon> \<Gamma> \<turnstile>\<^bsub>wf\<^esub>"
inductive_cases well_formed_cases:
"\<Gamma> \<turnstile>\<^bsub>wf\<^esub> TVar i"
"\<Gamma> \<turnstile>\<^bsub>wf\<^esub> Top"
"\<Gamma> \<turnstile>\<^bsub>wf\<^esub> T \<rightarrow> U"
"\<Gamma> \<turnstile>\<^bsub>wf\<^esub> (\<forall><:T. U)"
inductive_cases well_formedE_cases:
"B \<Colon> \<Gamma> \<turnstile>\<^bsub>wf\<^esub>"
inductive
subtyping :: "env \<Rightarrow> type \<Rightarrow> type \<Rightarrow> bool" ("_ \<turnstile> _ <: _" [50, 50, 50] 50)
where
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)"
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>1\<^isub>1 \<rightarrow> T\<^isub>1\<^isub>2 \<Longrightarrow> \<Gamma> \<turnstile> t\<^isub>2 : T\<^isub>1\<^isub>1 \<Longrightarrow> \<Gamma> \<turnstile> t\<^isub>1 \<bullet> t\<^isub>2 : T\<^isub>1\<^isub>2"
| 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>1\<^isub>1. T\<^isub>1\<^isub>2) \<Longrightarrow> \<Gamma> \<turnstile> T\<^isub>2 <: T\<^isub>1\<^isub>1 \<Longrightarrow>
\<Gamma> \<turnstile> t\<^isub>1 \<bullet>\<^isub>\<tau> T\<^isub>2 : T\<^isub>1\<^isub>2[0 \<mapsto>\<^isub>\<tau> T\<^isub>2]\<^isub>\<tau>"
| T_Sub: "\<Gamma> \<turnstile> t : S \<Longrightarrow> \<Gamma> \<turnstile> S <: T \<Longrightarrow> \<Gamma> \<turnstile> t : T"
code_pred [inductify, skip_proof, specialise] typing .
thm typing.equation
values 6 "{(E, t, T). typing E t T}"
subsection {* Higher-order predicate *}
code_pred [inductify] mapB .
subsection {* Multiple instances *}
inductive subtype_refl' where
"\<Gamma> \<turnstile> t : T ==> \<not> (\<Gamma> \<turnstile> T <: T) ==> subtype_refl' t T"
code_pred (modes: i => i => bool, o => i => bool, i => o => bool, o => o => bool) [inductify] subtype_refl' .
thm subtype_refl'.equation
end