Commit ceb39ea6 authored by Lennard Gäher's avatar Lennard Gäher
Browse files

systemf mu

parent 5c40c02d
......@@ -49,6 +49,15 @@ theories/systemf/free_theorems.v
theories/systemf/binary_logrel.v
theories/systemf/existential_invariants.v
# SystemF-Mu
theories/systemf_mu/lang.v
theories/systemf_mu/notation.v
theories/systemf_mu/types.v
theories/systemf_mu/tactics.v
theories/systemf_mu/pure.v
theories/systemf_mu/untyped_encoding.v
# By removing the # below, you can add the exercise sheets to make
# theories/warmup/sheet0.v
......
This diff is collapsed.
From semantics.systemf_mu Require Export lang.
Set Default Proof Using "Type".
(** Coercions to make programs easier to type. *)
Coercion of_val : val >-> expr.
Coercion LitInt : Z >-> base_lit.
Coercion LitBool : bool >-> base_lit.
Coercion App : expr >-> Funclass.
Coercion Var : string >-> expr.
(** Define some derived forms. *)
Notation Let x e1 e2 := (App (Lam x e2) e1) (only parsing).
Notation Seq e1 e2 := (Let BAnon e1 e2) (only parsing).
Notation Match e0 x1 e1 x2 e2 := (Case e0 (Lam x1 e1) (Lam x2 e2)) (only parsing).
(* No scope for the values, does not conflict and scope is often not inferred
properly. *)
Notation "# l" := (LitV l%Z%V%stdpp) (at level 8, format "# l").
Notation "# l" := (Lit l%Z%E%stdpp) (at level 8, format "# l") : expr_scope.
(** Syntax inspired by Coq/Ocaml. Constructions with higher precedence come
first. *)
Notation "( e1 , e2 , .. , en )" := (Pair .. (Pair e1 e2) .. en) : expr_scope.
Notation "( e1 , e2 , .. , en )" := (PairV .. (PairV e1 e2) .. en) : val_scope.
Notation "'match:' e0 'with' 'InjL' x1 => e1 | 'InjR' x2 => e2 'end'" :=
(Match e0 x1%binder e1 x2%binder e2)
(e0, x1, e1, x2, e2 at level 200,
format "'[hv' 'match:' e0 'with' '/ ' '[' 'InjL' x1 => '/ ' e1 ']' '/' '[' | 'InjR' x2 => '/ ' e2 ']' '/' 'end' ']'") : expr_scope.
Notation "'match:' e0 'with' 'InjR' x1 => e1 | 'InjL' x2 => e2 'end'" :=
(Match e0 x2%binder e2 x1%binder e1)
(e0, x1, e1, x2, e2 at level 200, only parsing) : expr_scope.
Notation "()" := LitUnit : val_scope.
Notation "e1 + e2" := (BinOp PlusOp e1%E e2%E) : expr_scope.
Notation "e1 - e2" := (BinOp MinusOp e1%E e2%E) : expr_scope.
Notation "e1 * e2" := (BinOp MultOp e1%E e2%E) : expr_scope.
Notation "e1 ≤ e2" := (BinOp LeOp e1%E e2%E) : expr_scope.
Notation "e1 = e2" := (BinOp EqOp e1%E e2%E) : expr_scope.
Notation "e1 < e2" := (BinOp LtOp e1%E e2%E) : expr_scope.
(*Notation "~ e" := (UnOp NegOp e%E) (at level 75, right associativity) : expr_scope.*)
Notation "'if:' e1 'then' e2 'else' e3" := (If e1%E e2%E e3%E)
(at level 200, e1, e2, e3 at level 200) : expr_scope.
Notation "λ: x , e" := (Lam x%binder e%E)
(at level 200, x at level 1, e at level 200,
format "'[' 'λ:' x , '/ ' e ']'") : expr_scope.
Notation "λ: x y .. z , e" := (Lam x%binder (Lam y%binder .. (Lam z%binder e%E) ..))
(at level 200, x, y, z at level 1, e at level 200,
format "'[' 'λ:' x y .. z , '/ ' e ']'") : expr_scope.
Notation "λ: x , e" := (LamV x%binder e%E)
(at level 200, x at level 1, e at level 200,
format "'[' 'λ:' x , '/ ' e ']'") : val_scope.
Notation "λ: x y .. z , e" := (LamV x%binder (Lam y%binder .. (Lam z%binder e%E) .. ))
(at level 200, x, y, z at level 1, e at level 200,
format "'[' 'λ:' x y .. z , '/ ' e ']'") : val_scope.
Notation "'let:' x := e1 'in' e2" := (Lam x%binder e2%E e1%E)
(at level 200, x at level 1, e1, e2 at level 200,
format "'[' 'let:' x := '[' e1 ']' 'in' '/' e2 ']'") : expr_scope.
Notation "e1 ;; e2" := (Lam BAnon e2%E e1%E)
(at level 100, e2 at level 200,
format "'[' '[hv' '[' e1 ']' ;; ']' '/' e2 ']'") : expr_scope.
Notation "'Λ' , e" := (TLam e%E)
(at level 200, e at level 200,
format "'[' 'Λ' , '/ ' e ']'") : expr_scope.
Notation "'Λ' , e" := (TLamV e%E)
(at level 200, e at level 200,
format "'[' 'Λ' , '/ ' e ']'") : val_scope.
(* the [e] always needs to be paranthesized, due to the level
(chosen to make this cooperate with the [App] coercion) *)
Notation "e '<>'" := (TApp e%E)
(at level 10) : expr_scope.
(*Check ((Λ, #4) <>)%E.*)
(*Check (((λ: "x", "x") #5) <>)%E.*)
Notation "'pack' e" := (Pack e%E)
(at level 200, e at level 200) : expr_scope.
Notation "'pack' v" := (PackV v%V)
(at level 200, v at level 200) : val_scope.
Notation "'unpack' e1 'as' x 'in' e2" := (Unpack x%binder e1%E e2%E)
(at level 200, e1, e2 at level 200, x at level 1) : expr_scope.
Notation "'roll' e" := (Roll e%E)
(at level 200, e at level 200) : expr_scope.
Notation "'roll' v" := (RollV v%E)
(at level 200, v at level 200) : val_scope.
Notation "'unroll' e" := (Unroll e%E)
(at level 200, e at level 200) : expr_scope.
From stdpp Require Import gmap base relations.
From iris Require Import prelude.
From semantics.lib Require Import maps.
From semantics.systemf_mu Require Import lang notation.
Lemma contextual_ectx_step_case K e e' :
contextual_step (fill K e) e'
( e'', e' = fill K e'' contextual_step e e'') is_val e.
Proof.
(* FIXME: exercise for you :) *)
(*Qed.*)
Admitted.
(** ** Deterministic reduction *)
Record det_step (e1 e2 : expr) := {
det_step_safe : reducible e1;
det_step_det e2' :
contextual_step e1 e2' e2' = e2
}.
Record det_base_step (e1 e2 : expr) := {
det_base_step_safe : base_reducible e1;
det_base_step_det e2' :
base_step e1 e2' e2' = e2
}.
Lemma det_base_step_det_step e1 e2 : det_base_step e1 e2 det_step e1 e2.
Proof.
intros [Hp1 Hp2]. split.
- destruct Hp1 as (e2' & ?).
eexists e2'. by apply base_contextual_step.
- intros e2' ?%base_reducible_contextual_step; [ | done]. by apply Hp2.
Qed.
(** *** Pure execution lemmas *)
Local Ltac inv_step :=
repeat match goal with
| H : to_val _ = Some _ |- _ => apply of_to_val in H
| H : base_step ?e ?e2 |- _ =>
try (is_var e; fail 1); (* inversion yields many goals if [e] is a variable
and should thus better be avoided. *)
inversion H; subst; clear H
end.
Local Ltac solve_exec_safe := intros; subst; eexists; econstructor; eauto.
Local Ltac solve_exec_detdet := simpl; intros; inv_step; try done.
Local Ltac solve_det_exec :=
subst; intros; apply det_base_step_det_step;
constructor; [solve_exec_safe | solve_exec_detdet].
Lemma det_step_beta x e e2 :
is_val e2
det_step (App (@Lam x e) e2) (subst' x e2 e).
Proof. solve_det_exec. Qed.
Lemma det_step_tbeta e :
det_step ((Λ, e) <>) e.
Proof. solve_det_exec. Qed.
Lemma det_step_unpack e1 e2 x :
is_val e1
det_step (unpack (pack e1) as x in e2) (subst' x e1 e2).
Proof. solve_det_exec. Qed.
Lemma det_step_unop op e v v' :
to_val e = Some v
un_op_eval op v = Some v'
det_step (UnOp op e) v'.
Proof. solve_det_exec. by simplify_eq. Qed.
Lemma det_step_binop op e1 v1 e2 v2 v' :
to_val e1 = Some v1
to_val e2 = Some v2
bin_op_eval op v1 v2 = Some v'
det_step (BinOp op e1 e2) v'.
Proof. solve_det_exec. by simplify_eq. Qed.
Lemma det_step_if_true e1 e2 :
det_step (if: #true then e1 else e2) e1.
Proof. solve_det_exec. Qed.
Lemma det_step_if_false e1 e2 :
det_step (if: #false then e1 else e2) e2.
Proof. solve_det_exec. Qed.
Lemma det_step_fst e1 e2 :
is_val e1
is_val e2
det_step (Fst (e1, e2)) e1.
Proof. solve_det_exec. Qed.
Lemma det_step_snd e1 e2 :
is_val e1
is_val e2
det_step (Snd (e1, e2)) e2.
Proof. solve_det_exec. Qed.
Lemma det_step_casel e e1 e2 :
is_val e
det_step (Case (InjL e) e1 e2) (e1 e).
Proof. solve_det_exec. Qed.
Lemma det_step_caser e e1 e2 :
is_val e
det_step (Case (InjR e) e1 e2) (e2 e).
Proof. solve_det_exec. Qed.
Lemma det_step_unroll e :
is_val e
det_step (unroll (roll e)) e.
Proof. solve_det_exec. Qed.
(** ** n-step reduction *)
(** Reduce in n steps to an irreducible expression.
(this is ⇝^n from the lecture notes)
*)
Definition red_nsteps (n : nat) (e e' : expr) := nsteps contextual_step n e e' irreducible e'.
Lemma det_step_red e e' e'' n :
det_step e e'
red_nsteps n e e''
1 n red_nsteps (n - 1) e' e''.
Proof.
intros [Hprog Hstep] Hred.
inversion Hprog; subst.
destruct Hred as [Hred Hirred].
destruct n as [ | n].
{ inversion Hred; subst.
exfalso; eapply not_reducible; done.
}
inversion Hred; subst. simpl.
apply Hstep in H as ->. apply Hstep in H1 as ->.
split; first lia.
replace (n - 0) with n by lia. done.
Qed.
Lemma contextual_step_red_nsteps n e e' e'' :
contextual_step e e'
red_nsteps n e' e''
red_nsteps (S n) e e''.
Proof.
intros Hstep [Hsteps Hirred].
split; last done.
by econstructor.
Qed.
Lemma nsteps_val_inv n v e' :
red_nsteps n (of_val v) e' n = 0 e' = of_val v.
Proof.
intros [Hred Hirred]; cbn in *.
destruct n as [ | n].
- inversion Hred; subst. done.
- inversion Hred; subst. exfalso. eapply val_irreducible; last done.
rewrite to_of_val. eauto.
Qed.
Lemma nsteps_val_inv' n v e e' :
to_val e = Some v
red_nsteps n e e' n = 0 e' = of_val v.
Proof. intros Ht. rewrite -(of_to_val _ _ Ht). apply nsteps_val_inv. Qed.
Lemma red_nsteps_fill K k e e' :
red_nsteps k (fill K e) e'
j e'', j k
red_nsteps j e e''
red_nsteps (k - j) (fill K e'') e'.
Proof.
(* FIXME: this is an exercise :) *)
Admitted.
(** Additionally useful stepping lemmas *)
Lemma app_step_r (e1 e2 e2': expr) :
contextual_step e2 e2' contextual_step (e1 e2) (e1 e2').
Proof. by apply (fill_contextual_step [AppRCtx _]). Qed.
Lemma app_step_l (e1 e1' e2: expr) :
contextual_step e1 e1' is_val e2 contextual_step (e1 e2) (e1' e2).
Proof.
intros ? (v & Hv)%is_val_spec.
rewrite <-(of_to_val _ _ Hv).
by apply (fill_contextual_step [AppLCtx _]).
Qed.
Lemma app_step_beta (x: string) (e e': expr) :
is_val e' is_closed [x] e contextual_step ((λ: x, e) e') (lang.subst x e' e).
Proof.
intros Hval Hclosed. eapply base_contextual_step, BetaS; eauto.
Qed.
Lemma unroll_roll_step (e: expr) :
is_val e contextual_step (unroll (roll e)) e.
Proof.
intros ?; by eapply base_contextual_step, UnrollS.
Qed.
Lemma fill_reducible K e :
reducible e reducible (fill K e).
Proof.
intros (e' & Hstep).
exists (fill K e'). eapply fill_contextual_step. done.
Qed.
Lemma reducible_contextual_step_case K e e' :
contextual_step (fill K e) (e')
reducible e
e'', e' = fill K e'' contextual_step e e''.
Proof.
intros [ | Hval]%contextual_ectx_step_case Hred; first done.
exfalso. apply is_val_spec in Hval as (v & Hval).
apply reducible_not_val in Hred. congruence.
Qed.
(** Contextual lifting lemmas for deterministic reduction *)
Tactic Notation "lift_det" uconstr(ctx) :=
intros;
let Hs := fresh in
match goal with
| H : det_step _ _ |- _ => destruct H as [? Hs]
end;
simplify_val; econstructor;
[intros; by eapply (fill_reducible ctx) |
intros ? (? & -> & ->%Hs)%(reducible_contextual_step_case ctx); done ].
Lemma det_step_pair_r e1 e2 e2' :
det_step e2 e2'
det_step (e1, e2)%E (e1, e2')%E.
Proof. lift_det [PairRCtx _]. Qed.
Lemma det_step_pair_l e1 e1' e2 :
is_val e2
det_step e1 e1'
det_step (e1, e2)%E (e1', e2)%E.
Proof. lift_det [PairLCtx _]. Qed.
Lemma det_step_binop_r e1 e2 e2' op :
det_step e2 e2'
det_step (BinOp op e1 e2)%E (BinOp op e1 e2')%E.
Proof. lift_det [BinOpRCtx _ _]. Qed.
Lemma det_step_binop_l e1 e1' e2 op :
is_val e2
det_step e1 e1'
det_step (BinOp op e1 e2)%E (BinOp op e1' e2)%E.
Proof. lift_det [BinOpLCtx _ _]. Qed.
Lemma det_step_if e e' e1 e2 :
det_step e e'
det_step (If e e1 e2)%E (If e' e1 e2)%E.
Proof. lift_det [IfCtx _ _]. Qed.
Lemma det_step_app_r e1 e2 e2' :
det_step e2 e2'
det_step (App e1 e2)%E (App e1 e2')%E.
Proof. lift_det [AppRCtx _]. Qed.
Lemma det_step_app_l e1 e1' e2 :
is_val e2
det_step e1 e1'
det_step (App e1 e2)%E (App e1' e2)%E.
Proof. lift_det [AppLCtx _]. Qed.
Lemma det_step_snd_lift e e' :
det_step e e'
det_step (Snd e)%E (Snd e')%E.
Proof. lift_det [SndCtx]. Qed.
Lemma det_step_fst_lift e e' :
det_step e e'
det_step (Fst e)%E (Fst e')%E.
Proof. lift_det [FstCtx]. Qed.
#[global]
Hint Resolve app_step_r app_step_l app_step_beta unroll_roll_step : core.
#[global]
Hint Extern 1 (is_val _) => (simpl; fast_done) : core.
#[global]
Hint Immediate is_val_of_val : core.
#[global]
Hint Resolve det_step_beta det_step_tbeta det_step_unpack det_step_unop det_step_binop det_step_if_true det_step_if_false det_step_fst det_step_snd det_step_casel det_step_caser det_step_unroll : core.
#[global]
Hint Resolve det_step_pair_r det_step_pair_l det_step_binop_r det_step_binop_l det_step_if det_step_app_r det_step_app_l det_step_snd_lift det_step_fst_lift : core.
#[global]
Hint Constructors nsteps : core.
#[global]
Hint Extern 1 (is_val _) => simpl : core.
(** Prove a single deterministic step using the lemmas we just proved *)
Ltac do_det_step :=
match goal with
| |- nsteps det_step _ _ _ => econstructor 2; first do_det_step
| |- det_step _ _ => simpl; solve[eauto 10]
end.
From stdpp Require Import gmap base relations.
From iris Require Import prelude.
From semantics.lib Require Export facts maps sets.
From semantics.systemf_mu Require Export lang notation types.
Ltac map_solver :=
repeat match goal with
| |- ( _) !! _ = Some _ =>
rewrite fmap_insert
| |- <[ ?p := _ ]> _ !! ?p = Some _ =>
apply lookup_insert
| |- <[ _ := _ ]> _ !! _ = Some _ =>
rewrite lookup_insert_ne; [ | congruence]
end.
Ltac solve_typing :=
intros;
repeat match goal with
| |- TY _ ; _ ?e : ?A => first [eassumption | econstructor]
(* heuristic to solve tapp goals where we need to pick the right type for the substitution *)
| |- TY _ ; _ ?e <> : ?A => eapply typed_tapp'; [solve_typing | | asimpl; reflexivity]
| |- TY _ ; _ Unroll ?e : ?A => eapply typed_unroll'; [solve_typing | asimpl; reflexivity]
| |- bin_op_typed _ _ _ _ => econstructor
| |- un_op_typed _ _ _ _ => econstructor
| |- type_wf _ ?e => assert_fails (is_evar e); eassumption
| |- type_wf _ ?e => assert_fails (is_evar e); econstructor
| |- type_wf _ (subst _ ?A) => eapply type_wf_subst; [ | intros; simpl]
| |- type_wf _ ?e => assert_fails (is_evar e); eapply type_wf_mono; [eassumption | lia]
(* conditions spawned by the tyvar case of [type_wf] *)
| |- _ < _ => lia
(* conditions spawned by the variable case *)
| |- _ !! _ = Some _ => map_solver
end.
Tactic Notation "unify_type" uconstr(A) :=
match goal with
| |- TY ?n; ?Γ ?e : ?B => unify A B
end.
Tactic Notation "replace_type" uconstr(A) :=
match goal with
| |- TY ?n; ?Γ ?e : ?B => replace B%ty with A%ty; cycle -1; first try by asimpl
end.
Ltac simplify_list_elem :=
simpl;
repeat match goal with
| |- ?x ?y :: ?l => apply elem_of_cons; first [left; reflexivity | right]
| |- _ [] => apply not_elem_of_nil
| |- _ _ :: _ => apply not_elem_of_cons; split
end; try fast_done.
Ltac simplify_list_subseteq :=
simpl;
repeat match goal with
| |- ?x :: _ ?x :: _ => apply list_subseteq_cons_l
| |- ?x :: _ _ => apply list_subseteq_cons_elem; first solve [simplify_list_elem]
| |- elements _ elements _ => apply elements_subseteq; set_solver
| |- [] _ => apply list_subseteq_nil
| |- ?x :b: _ ?x :b: _ => apply list_subseteq_cons_binder
(* NOTE: this might make the goal unprovable *)
(*| |- elements _ ⊆ _ :: _ => apply list_subseteq_cons_r*)
end;
try fast_done.
(* Try to solve [is_closed] goals using a number of heuristics (that shouldn't make the goal unprovable) *)
Ltac simplify_closed :=
simpl; intros;
repeat match goal with
| |- closed _ _ => unfold closed; simpl
| |- Is_true (is_closed [] _) => first [assumption | done]
| |- Is_true (is_closed _ (lang.subst _ _ _)) => rewrite subst_is_closed_nil; last solve[simplify_closed]
| |- Is_true (is_closed ?X ?v) => assert_fails (is_evar X); eapply is_closed_weaken
| |- Is_true (is_closed _ _) => eassumption
| |- Is_true (_ && true) => rewrite andb_true_r
| |- Is_true (true && _) => rewrite andb_true_l
| |- Is_true (?a && ?a) => rewrite andb_diag
| |- Is_true (_ && _) => simpl; rewrite !andb_True; split_and!
| |- _ ?A => match type of A with | list _ => simplify_list_subseteq end
| |- _ ?A => match type of A with | list _ => simplify_list_elem end
| |- _ _ => congruence
| H : closed _ _ |- _ => unfold closed in H; simpl in H
| H: Is_true (_ && _) |- _ => simpl in H; apply andb_True in H
| H : _ _ |- _ => destruct H
| |- Is_true (bool_decide (_ _)) => apply bool_decide_pack; set_solver
end; try fast_done.
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From stdpp Require Import gmap base relations tactics.
From iris Require Import prelude.
From semantics.systemf_mu Require Import lang notation types pure tactics.
From Autosubst Require Import Autosubst.
(** * Encoding of the untyped lambda calculus *)
Definition D := (μ: #0 #0)%ty.
Definition lame (x : string) (e : expr) : val := RollV (λ: x, e).
Definition appe (e1 e2 : expr) : expr := (unroll e1) e2.
Lemma lame_typed n Γ x e :
TY n; (<[x := D]> Γ) e : D
TY n; Γ lame x e : D.
Proof. solve_typing. Qed.
Lemma app_typed n Γ e1 e2 :
TY n; Γ e1 : D
TY n; Γ e2 : D
TY n; Γ appe e1 e2 : D.
Proof.
solve_typing.
Qed.
Lemma appe_step_l e1 e1' (v : val) :
contextual_step e1 e1'
contextual_step (appe e1 v) (appe e1' v).
Proof.
intros Hstep. unfold appe.
by eapply (fill_contextual_step [UnrollCtx; AppLCtx _]).
Qed.
Lemma appe_step_r e1 e2 e2' :
contextual_step e2 e2'
contextual_step (appe e1 e2) (appe e1 e2').
Proof.
intros Hstep. unfold appe.
by eapply (fill_contextual_step [AppRCtx _]).
Qed.
Lemma lame_step_beta x e (v : val) :
rtc contextual_step (appe (lame x e) v) (lang.subst x v e).
Proof.
unfold appe, lame.
econstructor 2.
{ eapply (fill_contextual_step [AppLCtx _]).
eapply base_contextual_step. by econstructor.
}
econstructor 2.
{ eapply base_contextual_step. econstructor; eauto. }
reflexivity.
Qed.
(* Divergence *)
Definition ω : expr :=
roll (λ: "x", (unroll "x") "x").
Definition Ω := ((unroll ω) ω)%E.