Theory Specialisation_Examples

theory Specialisation_Examples
imports Main "HOL-Library.Predicate_Compile_Alternative_Defs"
begin

declare [[values_timeout = 960.0]]

section ‹Specialisation Examples›

primrec 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 = (∀i x. nth_el' xs i = Some x --> x > i)"

code_pred (expected_modes: i => bool) [inductify, skip_proof, specialise] greater_than_index .
ML_val ‹Core_Data.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: fun_eq_iff 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_val ‹Core_Data.intros_of context const_name‹specialised_max_natP››

values "{x. max_of_my_SucP x 6}"

subsection ‹Sorts›

inductive sorted :: "'a::linorder list ⇒ bool" where
  Nil [simp]: "sorted []"
| Cons: "∀y∈set xs. x ≤ y ⟹ sorted xs ⟹ sorted (x # xs)"

lemma sorted_single [simp]: "sorted [x]"
by (rule sorted.Cons) auto

lemma sorted_many: "x ≤ y ⟹ sorted (y # zs) ⟹ sorted (x # y # zs)"
by (rule sorted.Cons) (cases "y # zs" rule: sorted.cases, auto)

lemma sorted_many_eq [simp]:
  "sorted (x # y # zs) ⟷ x ≤ y ∧ sorted (y # zs)"
by (auto intro: sorted_many elim: sorted.cases)

declare sorted.Nil [code_pred_intro]
  sorted_single [code_pred_intro]
  sorted_many [code_pred_intro]

code_pred sorted
proof -
  assume "sorted xa"
  assume 1: "xa = [] ⟹ thesis"
  assume 2: "⋀x. xa = [x] ⟹ thesis"
  assume 3: "⋀x y zs. xa = x # y # zs ⟹ x ≤ y ⟹ sorted (y # zs) ⟹ thesis"
  show thesis proof (cases xa)
    case Nil with 1 show ?thesis .
  next
    case (Cons x xs) show ?thesis proof (cases xs)
      case Nil with Cons 2 show ?thesis by simp
    next
      case (Cons y zs)
      show ?thesis
      proof (rule 3)
        from Cons ‹xa = x # xs› show "xa = x # y # zs" by simp
        with ‹sorted xa› show "x ≤ y" and "sorted (y # zs)" by simp_all
      qed
    qed
  qed
qed
thm sorted.equation


section ‹Specialisation in POPLmark theory›

notation
  Some ("⌊_⌋")

notation
  None ("⊥")

notation
  length ("∥_∥")

notation
  Cons ("_ ∷/ _" [66, 65] 65)

primrec
  nth_el :: "'a list ⇒ nat ⇒ 'a option" ("_⟨_⟩" [90, 0] 91)
where
  "[]⟨i⟩ = ⊥"
| "(x # xs)⟨i⟩ = (case i of 0 ⇒ ⌊x⌋ | Suc j ⇒ xs ⟨j⟩)"

primrec assoc :: "('a × 'b) list ⇒ 'a ⇒ 'b option" ("_⟨_⟩?" [90, 0] 91)
where
  "[]⟨a⟩? = ⊥"
| "(x # xs)⟨a⟩? = (if fst x = a then ⌊snd x⌋ else xs⟨a⟩?)"

primrec unique :: "('a × 'b) list ⇒ bool"
where
  "unique [] = True"
| "unique (x # xs) = (xs⟨fst x⟩? = ⊥ ∧ unique xs)"

datatype type =
    TVar nat
  | Top
  | Fun type type    (infixr "→" 200)
  | TyAll type type  ("(3∀<:_./ _)" [0, 10] 10)

datatype binding = VarB type | TVarB type
type_synonym env = "binding list"

primrec is_TVarB :: "binding ⇒ bool"
where
  "is_TVarB (VarB T) = False"
| "is_TVarB (TVarB T) = True"

primrec type_ofB :: "binding ⇒ type"
where
  "type_ofB (VarB T) = T"
| "type_ofB (TVarB T) = T"

primrec mapB :: "(type ⇒ type) ⇒ binding ⇒ binding"
where
  "mapB f (VarB T) = VarB (f T)"
| "mapB f (TVarB T) = TVarB (f T)"

datatype trm =
    Var nat
  | Abs type trm   ("(3λ:_./ _)" [0, 10] 10)
  | TAbs type trm  ("(3λ<:_./ _)" [0, 10] 10)
  | App trm trm    (infixl "∙" 200)
  | TApp trm type  (infixl "∙τ" 200)

primrec liftT :: "nat ⇒ nat ⇒ type ⇒ type" ("↑τ")
where
  "↑τ n k (TVar i) = (if i < k then TVar i else TVar (i + n))"
| "↑τ n k Top = Top"
| "↑τ n k (T → U) = ↑τ n k T → ↑τ n k U"
| "↑τ n k (∀<:T. U) = (∀<:↑τ n k T. ↑τ n (k + 1) U)"

primrec lift :: "nat ⇒ nat ⇒ trm ⇒ trm" ("↑")
where
  "↑ n k (Var i) = (if i < k then Var i else Var (i + n))"
| "↑ n k (λ:T. t) = (λ:↑τ n k T. ↑ n (k + 1) t)"
| "↑ n k (λ<:T. t) = (λ<:↑τ n k T. ↑ n (k + 1) t)"
| "↑ n k (s ∙ t) = ↑ n k s ∙ ↑ n k t"
| "↑ n k (t ∙τ T) = ↑ n k t ∙τ ↑τ n k T"

primrec substTT :: "type ⇒ nat ⇒ type ⇒ type"  ("_[_ ↦τ _]τ" [300, 0, 0] 300)
where
  "(TVar i)[k ↦τ S]τ =
     (if k < i then TVar (i - 1) else if i = k then ↑τ k 0 S else TVar i)"
| "Top[k ↦τ S]τ = Top"
| "(T → U)[k ↦τ S]τ = T[k ↦τ S]τ → U[k ↦τ S]τ"
| "(∀<:T. U)[k ↦τ S]τ = (∀<:T[k ↦τ S]τ. U[k+1 ↦τ S]τ)"

primrec decT :: "nat ⇒ nat ⇒ type ⇒ type"  ("↓τ")
where
  "↓τ 0 k T = T"
| "↓τ (Suc n) k T = ↓τ n k (T[k ↦τ Top]τ)"

primrec subst :: "trm ⇒ nat ⇒ trm ⇒ trm"  ("_[_ ↦ _]" [300, 0, 0] 300)
where
  "(Var i)[k ↦ s] = (if k < i then Var (i - 1) else if i = k then ↑ k 0 s else Var i)"
| "(t ∙ u)[k ↦ s] = t[k ↦ s] ∙ u[k ↦ s]"
| "(t ∙τ T)[k ↦ s] = t[k ↦ s] ∙τ ↓τ 1 k T"
| "(λ:T. t)[k ↦ s] = (λ:↓τ 1 k T. t[k+1 ↦ s])"
| "(λ<:T. t)[k ↦ s] = (λ<:↓τ 1 k T. t[k+1 ↦ s])"

primrec substT :: "trm ⇒ nat ⇒ type ⇒ trm"    ("_[_ ↦τ _]" [300, 0, 0] 300)
where
  "(Var i)[k ↦τ S] = (if k < i then Var (i - 1) else Var i)"
| "(t ∙ u)[k ↦τ S] = t[k ↦τ S] ∙ u[k ↦τ S]"
| "(t ∙τ T)[k ↦τ S] = t[k ↦τ S] ∙τ T[k ↦τ S]τ"
| "(λ:T. t)[k ↦τ S] = (λ:T[k ↦τ S]τ. t[k+1 ↦τ S])"
| "(λ<:T. t)[k ↦τ S] = (λ<:T[k ↦τ S]τ. t[k+1 ↦τ S])"

primrec liftE :: "nat ⇒ nat ⇒ env ⇒ env" ("↑e")
where
  "↑e n k [] = []"
| "↑e n k (B ∷ Γ) = mapB (↑τ n (k + ∥Γ∥)) B ∷ ↑e n k Γ"

primrec substE :: "env ⇒ nat ⇒ type ⇒ env"  ("_[_ ↦τ _]e" [300, 0, 0] 300)
where
  "[][k ↦τ T]e = []"
| "(B ∷ Γ)[k ↦τ T]e = mapB (λU. U[k + ∥Γ∥ ↦τ T]τ) B ∷ Γ[k ↦τ T]e"

primrec decE :: "nat ⇒ nat ⇒ env ⇒ env"  ("↓e")
where
  "↓e 0 k Γ = Γ"
| "↓e (Suc n) k Γ = ↓e n k (Γ[k ↦τ Top]e)"

inductive
  well_formed :: "env ⇒ type ⇒ bool"  ("_ ⊢wf _" [50, 50] 50)
where
  wf_TVar: "Γ⟨i⟩ = ⌊TVarB T⌋ ⟹ Γ ⊢wf TVar i"
| wf_Top: "Γ ⊢wf Top"
| wf_arrow: "Γ ⊢wf T ⟹ Γ ⊢wf U ⟹ Γ ⊢wf T → U"
| wf_all: "Γ ⊢wf T ⟹ TVarB T ∷ Γ ⊢wf U ⟹ Γ ⊢wf (∀<:T. U)"

inductive
  well_formedE :: "env ⇒ bool"  ("_ ⊢wf" [50] 50)
  and well_formedB :: "env ⇒ binding ⇒ bool"  ("_ ⊢wfB _" [50, 50] 50)
where
  "Γ ⊢wfB B ≡ Γ ⊢wf type_ofB B"
| wf_Nil: "[] ⊢wf"
| wf_Cons: "Γ ⊢wfB B ⟹ Γ ⊢wf ⟹ B ∷ Γ ⊢wf"

inductive_cases well_formed_cases:
  "Γ ⊢wf TVar i"
  "Γ ⊢wf Top"
  "Γ ⊢wf T → U"
  "Γ ⊢wf (∀<:T. U)"

inductive_cases well_formedE_cases:
  "B ∷ Γ ⊢wf"

inductive
  subtyping :: "env ⇒ type ⇒ type ⇒ bool"  ("_ ⊢ _ <: _" [50, 50, 50] 50)
where
  SA_Top: "Γ ⊢wf ⟹ Γ ⊢wf S ⟹ Γ ⊢ S <: Top"
| SA_refl_TVar: "Γ ⊢wf ⟹ Γ ⊢wf TVar i ⟹ Γ ⊢ TVar i <: TVar i"
| SA_trans_TVar: "Γ⟨i⟩ = ⌊TVarB U⌋ ⟹
    Γ ⊢ ↑τ (Suc i) 0 U <: T ⟹ Γ ⊢ TVar i <: T"
| SA_arrow: "Γ ⊢ T1 <: S1 ⟹ Γ ⊢ S2 <: T2 ⟹ Γ ⊢ S1 → S2 <: T1 → T2"
| SA_all: "Γ ⊢ T1 <: S1 ⟹ TVarB T1 ∷ Γ ⊢ S2 <: T2 ⟹
    Γ ⊢ (∀<:S1. S2) <: (∀<:T1. T2)"

inductive
  typing :: "env ⇒ trm ⇒ type ⇒ bool"    ("_ ⊢ _ : _" [50, 50, 50] 50)
where
  T_Var: "Γ ⊢wf ⟹ Γ⟨i⟩ = ⌊VarB U⌋ ⟹ T = ↑τ (Suc i) 0 U ⟹ Γ ⊢ Var i : T"
| T_Abs: "VarB T1 ∷ Γ ⊢ t2 : T2 ⟹ Γ ⊢ (λ:T1. t2) : T1 → ↓τ 1 0 T2"
| T_App: "Γ ⊢ t1 : T11 → T12 ⟹ Γ ⊢ t2 : T11 ⟹ Γ ⊢ t1 ∙ t2 : T12"
| T_TAbs: "TVarB T1 ∷ Γ ⊢ t2 : T2 ⟹ Γ ⊢ (λ<:T1. t2) : (∀<:T1. T2)"
| T_TApp: "Γ ⊢ t1 : (∀<:T11. T12) ⟹ Γ ⊢ T2 <: T11 ⟹
    Γ ⊢ t1 ∙τ T2 : T12[0 ↦τ T2]τ"
| T_Sub: "Γ ⊢ t : S ⟹ Γ ⊢ S <: T ⟹ Γ ⊢ 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
  "Γ ⊢ t : T ==> ¬ (Γ ⊢ 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