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Require Import Coq.Strings.String.
From stdpp Require Import gmap.
From mininix Require Import
expr relations sem interpreter complete sound shared.
Theorem correct e v' : (∃ n, eval n false e = Some v') ↔ e -->* v'.
Proof.
split.
- intros [n Heval]. by apply (eval_sound_strong n false).
- intros Heval. by apply eval_complete.
Qed.
(* Top-level program reduction/evaluation:
we make the prelude available here. *)
Definition with_prelude (e : expr) : expr :=
subst (closed prelude) e.
Definition tl_reds (e e' : expr) :=
with_prelude e -->* e'.
Definition tl_eval (n : nat) (e : expr) : option value :=
eval n false (subst (closed prelude) e).
Definition tl_evals e e' := ∃ n, tl_eval n e = Some e'.
(* Macros *)
Definition μ_neq e1 e2 := X_Cond (X_Op O_Eq e1 e2) false true.
Definition μ_or e1 e2 := X_Cond e1 true e2.
Definition μ_and e1 e2 := X_Cond e1 e2 false.
Definition μ_impl e1 e2 := X_Cond e1 e2 true.
Definition μ_neg e := X_Cond e false true.
Definition μ_le n m := μ_or (X_Op O_Eq n m) (X_Op O_Lt n m).
Definition μ_gt n m := X_Op O_Lt m n.
Definition μ_ge n m := μ_or (X_Op O_Eq n m) (μ_gt n m).
(* Tests/examples *)
Definition ex1 := X_LetBinding {[ "a" := B_Rec 1%Z ]} "a".
(* [let a = 1; in a] gives 1 *)
Theorem ex1_step : tl_reds ex1 1%Z.
Proof.
unfold ex1.
eapply rtc_l.
- by eapply E_Ctx with (E := id).
- simplify_option_eq.
eapply rtc_once.
by eapply E_Ctx with (E := id).
Qed.
Example ex1_eval : tl_evals ex1 (V_Int 1).
Proof. by exists 3. Qed.
(* Definition ex2 := <{ let a = 1 in with { a = 2 }; a }>. *)
Definition ex2 := X_LetBinding {[ "a" := B_Rec 1%Z ]}
(X_Incl (V_Attrset {[ "a" := X_V 2%Z ]}) "a").
(* [let a = 1; in with { a = 2; }; a] gives 1 *)
Theorem ex2_step : tl_reds ex2 1%Z.
Proof.
unfold ex2.
eapply rtc_l.
- by eapply E_Ctx with (E := id).
- simplify_option_eq.
eapply rtc_l.
+ by eapply E_Ctx with (E := id).
+ simpl. eapply rtc_once.
by eapply E_Ctx with (E := id).
Qed.
Example ex2_eval : tl_evals ex2 (V_Int 1).
Proof. by exists 4. Qed.
(* [with { a = 1; }; with { a = 2; }; a] gives 2 *)
Definition ex3 :=
X_Incl (V_Attrset {[ "a" := X_V 1%Z ]})
(X_Incl (V_Attrset {[ "a" := X_V 2%Z ]}) "a").
Theorem ex3_step : tl_reds ex3 2%Z.
Proof.
unfold ex3.
eapply rtc_l.
- eapply E_Ctx with (E := id); [done | apply E_With].
- simpl. eapply rtc_l.
+ by eapply E_Ctx with (E := id).
+ simplify_option_eq.
eapply rtc_once.
by eapply E_Ctx with (E := id).
Qed.
Example ex3_eval : tl_evals ex3 (V_Int 2).
Proof. by exists 4. Qed.
(* [({ x, y ? x } : y) { x = 1; }] gives 1 *)
Definition ex4 :=
X_Apply
(V_AttrsetFn
(M_Matcher
{[ "x" := M_Mandatory;
"y" := M_Optional "x"
]}
true)
"y")
(V_Attrset {[ "x" := X_V 1%Z ]}).
Example ex4_eval : tl_evals ex4 (V_Int 1).
Proof. by exists 7. Qed.
(* [({ x ? y, y ? x } : y) { x = 1; }] gives 1 *)
Definition ex5 :=
X_Apply
(V_AttrsetFn
(M_Matcher
{[ "x" := M_Optional "y";
"y" := M_Optional "x"
]}
true)
"y")
(V_Attrset {[ "x" := X_V 1%Z ]}).
Example ex5_eval : tl_evals ex5 (V_Int 1).
Proof. by exists 7. Qed.
(* [let binToString = n:
if n == 0
then "0"
else if n == 1
then "1"
else binToString (n / 2) + (if isEven n then "0" else "1");
isEven = n: n != 1 && (n = 0 || isEven (n - 2));
test = { x, y ? attrs.x, ... } @ attrs:
"x: " + x + ", y: " + y + ", z: " + attrs.z or "(no z)"
in test { x = binToString 6; }] gives "x: 110, y: 110, z: (no z)" *)
Definition ex6 :=
X_LetBinding
{[ "binToString" := B_Rec $ V_Fn "n" $
X_Cond
(X_Op O_Eq "n" 0%Z)
(V_Str "0")
(X_Cond
(X_Op O_Eq "n" 1%Z)
(V_Str "1")
(X_Op O_Plus
(X_Apply
"binToString"
(X_Op O_Div "n" 2%Z))
(X_Cond
(X_Apply "isEven" "n")
(V_Str "0")
(V_Str "1"))));
"isEven" := B_Rec $ V_Fn "n" $
μ_and
(μ_neq "n" 1%Z)
(μ_or
(X_Op O_Eq "n" 0%Z)
(X_Apply "isEven" (X_Op O_Min "n" 2%Z)));
"test" := B_Rec $ V_Fn "attrs" $
X_Apply
(V_AttrsetFn
(M_Matcher
{[ "x" := M_Mandatory;
"y" := M_Optional
(X_Select "attrs"
(nonempty_singleton "x"))
]} false)
(X_Op O_Plus
(X_Op O_Plus
(X_Op O_Plus
(X_Op O_Plus
(X_Op O_Plus
(V_Str "x: ")
"x")
(V_Str ", y: "))
"y")
(V_Str ", z: "))
(X_SelectOr
"attrs"
(nonempty_singleton "z")
(V_Str "(no z)"))))
"attrs"
]}
(X_Apply "test" $ V_Attrset
{[ "x" := X_Apply "binToString" 6%Z ]}).
Example ex6_eval : tl_evals ex6 (V_Str "x: 110, y: 110, z: (no z)").
Proof. by exists 37. Qed.
(* Important check of if placeholders work correctly:
[with { x = 1; }; let foo = y: let x = 2; in y; foo x]
gives 1 *)
Definition ex7 := X_Incl
(V_Attrset {[ "x" := X_V 1%Z ]})
(X_LetBinding
{[ "foo" := B_Rec $ V_Fn "y" $
X_LetBinding {[ "x" := B_Rec 2%Z ]} "y"
]}
(X_Apply "foo" "x")).
Example ex7_eval : tl_evals ex7 (V_Int 1).
Proof. by exists 7. Qed.
Definition ex8 :=
X_LetBinding
{[ "divide" := B_Rec $ V_Fn "a" $ V_Fn "b" $
X_Assert
(μ_and (μ_ge "a" 0%Z) (μ_gt "b" 0%Z))
(X_Cond
(X_Op O_Lt "a" "b")
0
(X_Op
O_Plus
(X_Apply
(X_Apply
"divide"
(X_Op O_Min "a" "b"))
"b")
1));
"divider" := B_Rec $ X_Attrset
{[ "__functor" := B_Nonrec $ V_Fn "self" $ V_Fn "x" $
X_Op
O_Upd
"self"
(X_Attrset
{[ "value" := B_Nonrec $
X_Apply
(X_Apply
"divide"
(X_Select "self" $ nonempty_singleton "value"))
"x"
]})
]};
"mkDivider" := B_Rec $ V_Fn "value" $
X_Op
O_Upd
"divider"
(X_Attrset {[ "value" := B_Nonrec "value" ]})
]}%Z
(X_Select
(X_Apply (X_Apply (X_Apply "mkDivider" 100) 5) 4)
(nonempty_singleton "value"))%Z.
Example ex8_eval : tl_evals ex8 (V_Int 5).
Proof. by exists 170. Qed.
Example ex9 :=
X_Apply
(X_Apply
(V_Attrset
{[ "__functor" := X_V $ V_Fn "self" $ V_Fn "f" $
X_Apply "f" (X_Apply "self" "f")
]})
(V_Fn "go" $ V_Fn "n" $
X_Cond
(μ_le "n" 1)
"n"
(X_Op
O_Plus
(X_Apply "go" (X_Op O_Min "n" 1))
(X_Apply "go" (X_Op O_Min "n" 2)))))%Z
15%Z.
Example ex9_eval : tl_evals ex9 (V_Int 610).
Proof. by exists 78. Qed.
Example ex10 :=
X_LetBinding
{[ "true" := B_Rec 42 ]}%Z
(X_Op O_Eq "true" 42)%Z.
Example ex10_eval : tl_evals ex10 (V_Bool true).
Proof. by exists 4. Qed.
Definition ex11 :=
X_LetBinding
{[ "x" := B_Rec "y" ]}%Z
(X_LetBinding {[ "y" := B_Rec 10 ]} "x")%Z.
Example ex11_eval : tl_eval 1000 ex11 = None.
Proof. done. Qed.
Definition ex12 :=
X_LetBinding
{[ "pkgs" := B_Rec $ V_Attrset
{[ "x" := X_Incl (V_Attrset {[ "y" := X_V 1%Z ]}) "y" ]}
]}
(X_Select
(X_Attrset
{[ "x" := B_Rec $ X_Select "pkgs" (nonempty_singleton "x");
"y" := B_Rec 3%Z
]})
(nonempty_singleton "x")).
Example ex12_eval : tl_eval 1000 ex12 = Some (V_Int 1).
Proof. done. Qed.
(* Aio, quantitas magna frumentorum est. *)
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