Well typed programs don't go wrong

prototyping/FFI/Data/Vector.agda

@@ -34,11 +34,12 @@ postulate
 {-# COMPILE GHC snoc = \_ -> Data.Vector.snoc #-}
 
 postulate length-empty : ∀ {A} → (length (empty {A}) ≡ 0)
+postulate lookup-empty : ∀ {A} n → (lookup (empty {A}) n ≡ nothing)
 postulate lookup-snoc : ∀ {A} (x : A) (v : Vector A) → (lookup (snoc v x) (length v) ≡ just x)
 postulate lookup-snoc-empty : ∀ {A} (x : A) → (lookup (snoc empty x) 0 ≡ just x)
 postulate lookup-snoc-not : ∀ {A n} (x : A) (v : Vector A) → (n ≢ length v) → (lookup v n ≡ lookup (snoc v x) n)
 
-{-# REWRITE length-empty lookup-snoc lookup-snoc-empty #-}
+{-# REWRITE length-empty lookup-snoc lookup-snoc-empty lookup-empty #-}
 
 head : ∀ {A} → (Vector A) → (Maybe A)
 head vec with null vec

prototyping/Properties/StrictMode.agda

@@ -5,7 +5,7 @@ module Properties.StrictMode where
 import Agda.Builtin.Equality.Rewrite
 open import Agda.Builtin.Equality using (_≡_; refl)
 open import FFI.Data.Maybe using (Maybe; just; nothing)
-open import Luau.Heap using (Heap; Object; function_is_end; defn; alloc; ok; next; lookup-not-allocated) renaming (_≡_⊕_↦_ to _≡ᴴ_⊕_↦_; _[_] to _[_]ᴴ)
+open import Luau.Heap using (Heap; Object; function_is_end; defn; alloc; ok; next; lookup-not-allocated) renaming (_≡_⊕_↦_ to _≡ᴴ_⊕_↦_; _[_] to _[_]ᴴ; ∅ to ∅ᴴ)
 open import Luau.StrictMode using (Warningᴱ; Warningᴮ; Warningᴼ; Warningᴴᴱ; Warningᴴᴮ; UnallocatedAddress; UnboundVariable; app₀; app₁; app₂; block; return; local₀; local₁; local₂; function₀; function₁; function₂; heap; expr; addr)
 open import Luau.Substitution using (_[_/_]ᴮ; _[_/_]ᴱ; _[_/_]ᴮunless_; var_[_/_]ᴱwhenever_)
 open import Luau.Syntax using (Expr; yes; var; var_∈_; _⟨_⟩∈_; _$_; addr; nil; function_is_end; block_is_end; done; return; local_←_; _∙_; fun; arg)
@@ -21,7 +21,7 @@ open import Properties.Equality using (_≢_; sym; cong; trans; subst₁)
 open import Properties.Dec using (Dec; yes; no)
 open import Properties.Contradiction using (CONTRADICTION)
 open import Properties.TypeCheck(strict) using (typeOfᴼ; typeOfᴹᴼ; typeOfⱽ; typeOfᴱ; typeOfᴮ; typeOfᴱⱽ; typeCheckᴱ; typeCheckᴮ; typeCheckᴼ; typeCheckᴴᴱ; typeCheckᴴᴮ)
-open import Luau.OpSem using (_⊢_⟶ᴮ_⊣_; _⊢_⟶ᴱ_⊣_; app; function; beta; return; block; done; local; subst)
+open import Luau.OpSem using (_⊢_⟶*_⊣_; _⊢_⟶ᴮ_⊣_; _⊢_⟶ᴱ_⊣_; app; function; beta; return; block; done; local; subst; refl; step)
 open import Luau.RuntimeError using (RuntimeErrorᴱ; RuntimeErrorᴮ; NilIsNotAFunction; UnboundVariable; SEGV; app; block; local; return)
 
 src = Luau.Type.src strict
@@ -249,6 +249,10 @@ reflectᴴᴮ (return s) (heap W′) with reflectᴴᴱ s (heap W′)
 reflectᴴᴮ (return s) (heap W′) | heap W = heap W
 reflectᴴᴮ (return s) (heap W′) | expr W = block (return W)
 
+reflect* : ∀ {H H′ B B′} → (H ⊢ B ⟶* B′ ⊣ H′) → Warningᴴᴮ H′ (typeCheckᴴᴮ H′ ∅ B′) → Warningᴴᴮ H (typeCheckᴴᴮ H ∅ B)
+reflect* refl W = W
+reflect* (step s t) W = reflectᴴᴮ s (reflect* t W)
+
 bot-not-obj : ∀ O → bot ≢ typeOfᴼ O
 bot-not-obj (function f ⟨ var x ∈ T ⟩∈ U is B end) ()
 
@@ -267,126 +271,7 @@ runtimeWarningᴱ (block b err) = block b (runtimeWarningᴮ err)
 runtimeWarningᴮ (local (var x ∈ T) err) = local₁ (runtimeWarningᴱ err)
 runtimeWarningᴮ (return err) = return (runtimeWarningᴱ err)
 
-
+wellTypedProgramsDontGoWrong : ∀ {H′ B B′} → (∅ᴴ ⊢ B ⟶* B′ ⊣ H′) → (RuntimeErrorᴮ H′ B′) → Warningᴮ ∅ᴴ (typeCheckᴮ ∅ᴴ ∅ B)
-
+wellTypedProgramsDontGoWrong t err with reflect* t (block (runtimeWarningᴮ err))
-
+wellTypedProgramsDontGoWrong t err | heap (addr a refl ())
-
+wellTypedProgramsDontGoWrong t err | block W = W
--- reflectᴱ (function {F = f ⟨ var x ∈ S ⟩∈ T} defn) (bot ())
--- reflectᴱ (function defn) (addr a T q) = CONTRADICTION (q refl)
--- reflectᴱ (app s) (bot x) = {!x!}
--- reflectᴱ (app s) (app₁ W) = app₁ {!reflectᴱ s W!}
--- reflectᴱ (app s) (app₂ W) = {!!}
--- reflectᴱ (beta x) W = {!!}
--- reflectᴱ (block x) W = {!!}
--- reflectᴱ (return x) W = {!!}
--- reflectᴱ done W = {!!}
-
--- heap-miss : ∀ {Σ HV T} → (Σ ▷ HV ∈ T) → (HV ≡ nothing) → (T ≡ bot)
--- heap-miss nothing refl = refl
-
--- data ProgressResultᴱ {Σ Γ S M T Δ} (H : Heap yes) (D : Γ ⊢ᴱ S ∋ M ∈ T ⊣ Δ) : Set
--- data ProgressResultᴮ {Σ Γ S B T Δ} (H : Heap yes) (D : Γ ⊢ᴮ S ∋ B ∈ T ⊣ Δ) : Set
-
--- data ProgressResultᴱ {Σ Γ S M T Δ} H D where
-
---   value : ∀ V → (M ≡ val V) → ProgressResultᴱ H D
---   warning : (Warningᴱ Σ D) → ProgressResultᴱ H D
---   step : ∀ {M′ H′} → (H ⊢ M ⟶ᴱ M′ ⊣ H′) → ProgressResultᴱ H D
-
--- data ProgressResultᴮ {Σ Γ S B T Δ} H D where
-
---   done : (B ≡ done) → ProgressResultᴮ H D
---   return : ∀ V {C} → (B ≡ (return (val V) ∙ C)) → ProgressResultᴮ H D
---   warning : (Warningᴮ Σ D) → ProgressResultᴮ H D
---   step : ∀ {B′ H′} → (H ⊢ B ⟶ᴮ B′ ⊣ H′) → ProgressResultᴮ H D
-
--- progressᴱ : ∀ {Σ Γ S M T Δ} H → (Σ ▷ H ✓) → (D : Σ ▷ Γ ⊢ᴱ S ∋ M ∈ T ⊣ Δ) → (Γ ≡ ∅) → ProgressResultᴱ H D
--- progressᴮ : ∀ {Σ Γ S B T Δ} H → (Σ ▷ H ✓) → (D : Σ ▷ Γ ⊢ᴮ S ∋ B ∈ T ⊣ Δ) → (Γ ≡ ∅) → ProgressResultᴮ H D
-
--- progressᴱ H h nil _ = value nil refl
--- progressᴱ H h (var x p) refl = warning (bot p)
--- progressᴱ H h (addr a refl) _ = value (addr a) refl
--- progressᴱ H h (app D₁ D₂) p with progressᴱ H h D₁ p
--- progressᴱ H h (app nil D₂) p | value nil refl = warning (bot refl)
--- progressᴱ H h (app (var _ _) D₂) p | value nil ()
--- progressᴱ H h (app (app _ _) D₂) p | value nil ()
--- progressᴱ H h (app (function _) D₂) p | value nil ()
--- progressᴱ H h (app (block _ _) D₂) p | value nil ()
--- progressᴱ H h (app (addr _ refl) D₂) p | value (addr a) refl with remember(H [ a ]ᴴ)
--- progressᴱ H h (app (addr _ refl) D₂) p | value (addr a) refl | (nothing , r) = warning (bot (cong tgt (heap-miss (h a) r)))
--- progressᴱ H h (app (addr _ refl) D₂) p | value (addr a) refl | (just(function f ⟨ var x ∈ S ⟩∈ T is B end) , r) = step (beta r)
--- progressᴱ H h (app D₁ D₂) p | warning W = warning (app₁ W)
--- progressᴱ H h (app D₁ D₂) p | step S = step (app S)
--- progressᴱ H h (function D) _ with alloc H _
--- progressᴱ H h (function D) _ | ok a H′ r = step (function r)
--- progressᴱ H h (block b D) q with progressᴮ H h D q
--- progressᴱ H h (block b D) q | done refl = step done
--- progressᴱ H h (block b D) q | return V refl = step (return refl)
--- progressᴱ H h (block b D) q | warning W = warning (block b W)
--- progressᴱ H h (block b D) q | step S = step (block S)
-
--- progressᴮ H h done q = done refl
--- progressᴮ H h (return D₁ D₂) q with progressᴱ H h D₁ q
--- progressᴮ H h (return D₁ D₂) q | value V refl = return V refl
--- progressᴮ H h (return D₁ D₂) q | warning W = warning (return W)
--- progressᴮ H h (return D₁ D₂) q | step S = step (return S)
--- progressᴮ H h (local D₁ D₂) q with progressᴱ H h D₁ q
--- progressᴮ H h (local D₁ D₂) q | value V refl = step subst
--- progressᴮ H h (local D₁ D₂) q | warning W = warning (local₁ W)
--- progressᴮ H h (local D₁ D₂) q | step S = step (local S)
--- progressᴮ H h (function D₁ D₂) q with alloc H _
--- progressᴮ H h (function D₁ D₂) q | ok a H′ r = step (function r)
-
--- data LookupResult {Σ V S} (D : Σ ▷ V ∈ S) : Set where
-
---   function : ∀ f {x B T U W} →
---     (S ≡ (T ⇒ U)) →
---     (V ≡ just(function f ⟨ var x ∈ T ⟩∈ U is B end)) →
---     (Σ ▷ (x ↦ T) ⊢ᴮ U ∋ B ∈ U ⊣ (x ↦ W)) →
---     LookupResult D
-
---   warningᴴ :
---     Warningᴴ(D) →
---     LookupResult D
-
--- lookup : ∀ {Σ V T} (D : Σ ▷ V ∈ T) → LookupResult D
--- lookup nothing = warningᴴ nothing
--- lookup (function f {U = U} {V = V} D) with U ≡ᵀ V
--- lookup (function f D) | yes refl = function f refl refl D
--- lookup (function f D) | no p = warningᴴ (function f (disagree p))
-
--- data PreservationResultᴮ {Σ S Δ T H B} (D : ∅ ⊢ᴮ S ∋ B ∈ T ⊣ Δ) M′ H′ : Set where
-
---   ok : ∀ {Δ′} → (∅ ⊢ᴮ S ∋ M′ ∈ T ⊣ Δ′) → PreservationResultᴮ D M′ H′
---   warning : Warningᴮ Σ D → PreservationResultᴮ D M′ H′
-
--- data PreservationResultᴱ {Σ S Δ T M} (D : ∅ ⊢ᴱ S ∋ M ∈ T ⊣ Δ) M′ : Set where
-
---   ok : ∀ {Δ′} → (∅ ⊢ᴱ S ∋ M′ ∈ T ⊣ Δ′) → PreservationResultᴱ D M′ H′
---   warning : Warningᴱ Σ D → PreservationResultᴱ D M′ H′
-
--- preservationᴱ : ∀ {Σ S Δ T H H′ M M′} → (D : ∅ ⊢ᴱ S ∋ M ∈ T ⊣ Δ) → (s : H ⊢ M ⟶ᴱ M′ ⊣ H′) → PreservationResultᴱ D M′ H′
--- preservationᴱ {S = S} {T = T} h D s with S ≡ᵀ T
--- preservationᴱ h D s | no p = warning (disagree p)
--- preservationᴱ h (app D₁ D₂) (app s) | yes refl with preservationᴱ h D₁ s
--- preservationᴱ h (app D₁ D₂) (app s) | yes refl | ok h′ D₁′ = ok h′ (app D₁′ {!D₂!})
--- preservationᴱ h (app D₁ D₂) (app s) | yes refl | warning W = warning (app₁ W)
--- -- preservationᴱ h (app (addr a p) D₂) (beta q) | yes refl with lookup (h a)
--- -- preservationᴱ h (app (addr a p) D₂) (beta q) | yes refl | function f r₁ r₂ D₁ with trans p r₁ | trans (sym q) r₂
--- -- preservationᴱ h (app (addr a p) D₂) (beta q) | yes refl | function f r₁ r₂ D₁ | refl | refl = ok h (block f (local D₂ D₁))
--- -- preservationᴱ h (app (addr a p) D₂) (beta q) | yes refl | warningᴴ W = warningᴴ a W
-
--- -- preservationᴱ h (app {T = T} {U = U} (addr a p) D₂) (beta q) with subst₂ (λ X Y → _ ▷ X ∈ Y) q (sym p) (h a)
--- -- preservationᴱ {S = S} h (app {T = T} {U = U} (addr a p) D₂) (beta q) | function f {T = T′} {U = U′} {V = V′} D with S ≡ᵀ U′ | U′ ≡ᵀ V′
--- -- preservationᴱ h (app {T = T} {U = U} (addr a p) D₂) (beta q) | function f {T = T′} {U = U′} {V = V′} D | yes refl | yes refl = ok h (block _ (local D₂ D))
--- -- preservationᴱ h (app {T = T} {U = U} (addr a p) D₂) (beta q) | function f {T = T′} {U = U′} D | no r | _ = warningᴱ (disagree r)
--- -- preservationᴱ h (app {T = T} {U = U} (addr a p) D₂) (beta q) | function f {T = T′} {U = U′} D | yes refl | no r = warningᴴ a {!function f (disagree r)!} -- (subst₁ Warningᴴ {!!} {!(function f (disagree r))!}) -- (function f (disagree r))
-
--- -- with src T ≡ᵀ U --  = ok h (block f (local {!D₂!} {!!}))
--- -- preservationᴱ h (app {T = T} {U = U} D₁ D₂) (beta {F = f ⟨ var x ∈ _ ⟩∈ R} p) | yes refl = ok h (block f (local D₂ {!!})) -- with src T ≡ᵀ S --  = ok h (block f (local {!D₂!} {!!}))
--- -- preservationᴱ h (app {T = T} {U = U} D₁ D₂) (beta {F = f ⟨ var x ∈ S ⟩∈ R} p) | no q = warning (app₀ {!q!}) -- with src T ≡ᵀ S --  = ok h (block f (local {!D₂!} {!!}))
--- preservationᴱ h D s | yes refl = {!!}
--- preservationᴱ h (function D) (function p) = {!x!}
--- preservationᴱ h (block b D) (block s) = {!!}
--- preservationᴱ h (block b D) (return p) = {!x!}
--- preservationᴱ h (block b D) done = {!!}