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pkg_rhl.v
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pkg_rhl.v
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(** Package Relational Hoar Logic
This file connects packages to the relational Hoare logic and provides
basic crypto-style reasoning notions.
*)
From Coq Require Import Utf8.
From SSProve.Relational Require Import OrderEnrichedCategory
OrderEnrichedRelativeMonadExamples.
Set Warnings "-ambiguous-paths,-notation-overridden,-notation-incompatible-format".
From mathcomp Require Import ssrnat ssreflect ssrfun ssrbool ssrnum eqtype
choice reals distr seq all_algebra fintype realsum.
Set Warnings "ambiguous-paths,notation-overridden,notation-incompatible-format".
From extructures Require Import ord fset fmap.
From SSProve.Mon Require Import SPropBase.
From SSProve.Crypt Require Import Prelude Axioms ChoiceAsOrd SubDistr Couplings
RulesStateProb UniformStateProb UniformDistrLemmas StateTransfThetaDens
StateTransformingLaxMorph choice_type pkg_core_definition pkg_notation
pkg_tactics pkg_composition pkg_heap pkg_semantics pkg_lookup pkg_advantage
pkg_invariants pkg_distr Casts.
Require Import Equations.Prop.DepElim.
From Equations Require Import Equations.
(* Must come after importing Equations.Equations, who knows why. *)
From SSProve.Crypt Require Import FreeProbProg.
Import Num.Theory.
Set Equations With UIP.
Set Equations Transparent.
Import SPropNotations.
Import PackageNotation.
Import RSemanticNotation.
Set Bullet Behavior "Strict Subproofs".
Set Default Goal Selector "!".
Set Primitive Projections.
#[local] Open Scope rsemantic_scope.
#[local] Open Scope fset.
#[local] Open Scope fset_scope.
#[local] Open Scope type_scope.
#[local] Open Scope package_scope.
#[local] Open Scope ring_scope.
#[local] Open Scope real_scope.
Notation " r⊨ ⦃ pre ⦄ c1 ≈ c2 ⦃ post ⦄ " :=
(semantic_judgement _ _ (repr c1) (repr c2) (fromPrePost pre post))
: package_scope.
Theorem rbind_rule :
∀ {A1 A2 B1 B2 : ord_choiceType} {f1 f2} m1 m2
(pre : precond) (mid : postcond A1 A2) (post : postcond B1 B2),
r⊨ ⦃ pre ⦄ m1 ≈ m2 ⦃ mid ⦄ →
(∀ a1 a2,
r⊨ ⦃ λ '(s1, s2), mid (a1, s1) (a2, s2) ⦄ f1 a1 ≈ f2 a2 ⦃ post ⦄
) →
r⊨ ⦃ pre ⦄ bind m1 f1 ≈ bind m2 f2 ⦃ post ⦄.
Proof.
intros A1 A2 B1 B2 f1 f2 m1 m2 pre mid post hm hf.
rewrite !repr_bind.
apply (bind_rule_pp (repr m1) (repr m2) pre mid post hm hf).
Qed.
Lemma get_case :
∀ LA (I : heap_choiceType * heap_choiceType → Prop) ℓ,
INV LA I →
ℓ \in LA →
r⊨ ⦃ λ '(s₀, s₃), I (s₀, s₃) ⦄
x ← get ℓ ;; ret x ≈ x ← get ℓ ;; ret x
⦃ λ '(b₁, s₁) '(b₂, s₂), b₁ = b₂ ∧ I (s₁, s₂) ⦄.
Proof.
intros LA I ℓ hinv hin. intros [s₁ s₂]. simpl.
rewrite /SpecificationMonads.MonoCont_bind /=.
rewrite /SpecificationMonads.MonoCont_order
/SPropMonadicStructures.SProp_op_order
/Morphisms.pointwise_relation /Basics.flip
/SPropMonadicStructures.SProp_order /=.
intuition auto.
assert (get_heap s₁ ℓ = get_heap s₂ ℓ) as Hv.
{ unfold INV in hinv.
specialize (hinv s₁ s₂). destruct hinv as [hinv _].
eapply hinv. all: auto.
}
pose v := (SDistr_unit _ (((get_heap s₁ ℓ), s₁),
((get_heap s₂ ℓ), s₂))).
exists v. split.
- apply SDistr_unit_F_choice_prod_coupling.
reflexivity.
- intros [b₁ s₃] [b₂ s₄]. intro hd.
apply H1. rewrite dunit1E in hd.
assert (
(get_heap s₁ ℓ, s₁, (get_heap s₂ ℓ, s₂)) = (b₁, s₃, (b₂, s₄))
) as e.
{ destruct ((get_heap s₁ ℓ, s₁, (get_heap s₂ ℓ, s₂)) == (b₁, s₃, (b₂, s₄))) eqn:e.
- move: e => /eqP e. assumption.
- rewrite e in hd. cbn in hd.
rewrite order.Order.POrderTheory.ltxx in hd. discriminate.
}
inversion e. subst. intuition auto.
Qed.
Lemma put_case :
∀ {LA} (I : heap_choiceType * heap_choiceType → Prop) l v,
INV LA I →
l \in LA →
r⊨ ⦃ λ '(s0, s3), I (s0, s3) ⦄
putr l v (ret tt) ≈ putr l v (ret tt)
⦃ λ '(b1, s1) '(b2, s2), b1 = b2 ∧ I (s1, s2) ⦄.
Proof.
intros LA I l v hinv hin.
intros [s1 s2]. simpl.
rewrite /SpecificationMonads.MonoCont_bind /=.
rewrite /SpecificationMonads.MonoCont_order /SPropMonadicStructures.SProp_op_order
/Morphisms.pointwise_relation /Basics.flip /SPropMonadicStructures.SProp_order /=.
intuition eauto.
eexists (SDistr_unit _ _).
split.
+ apply SDistr_unit_F_choice_prod_coupling.
reflexivity.
+ intros [b1 s3] [b2 s4].
intros Hd.
apply H1.
unfold SDistr_unit in Hd.
rewrite dunit1E in Hd.
assert ((tt, set_heap s1 l v, (tt, set_heap s2 l v)) = (b1, s3, (b2, s4))) as Heqs.
{ destruct ((tt, set_heap s1 l v, (tt, set_heap s2 l v)) == (b1, s3, (b2, s4))) eqn:Heqd.
- move: Heqd. move /eqP => Heqd. assumption.
- rewrite Heqd in Hd. simpl in Hd.
rewrite order.Order.POrderTheory.ltxx in Hd. discriminate.
}
inversion Heqs.
intuition eauto.
eapply hinv. all: eauto.
Qed.
(* TODO MOVE? *)
Lemma destruct_pair_eq {R : ringType} {A B : eqType} {a b : A} {c d : B} :
((a, c) == (b, d))%:R = (a == b)%:R * (c == d)%:R :> R.
Proof.
destruct (a == b) eqn:ab, (c == d) eqn:cd.
all: cbn; rewrite ab cd /=; try rewrite GRing.mulr1; try rewrite GRing.mulr0; reflexivity.
Qed.
Lemma summable_eq {A : choiceType} {s : A} :
realsum.summable (T:=A) (R:=R) (λ x, (x == s)%:R).
Proof.
match goal with
| |- realsum.summable ?f => eassert (f = _) as Hf end.
{ extensionality x. rewrite eq_sym.
rewrite -dunit1E. reflexivity. }
rewrite Hf. clear Hf.
apply summable_mu.
Qed.
Lemma summable_pair_eq {A : choiceType} {B C : eqType} (f1 f3 : A -> B) (f2 f4 : A -> C)
(h1 : realsum.summable (T:=A) (R:=R) (λ x, (f1 x == f3 x)%:R))
(h2 : realsum.summable (T:=_) (R:=R) (λ x, (f2 x == f4 x)%:R))
:
realsum.summable (T:=_) (R:=R) (λ x, ((f1 x, f2 x) == (f3 x, f4 x))%:R).
Proof.
match goal with
| |- realsum.summable ?f => eassert (f = _) as Hf end.
{ extensionality x.
instantiate (1 := fun x1 => (f1 x1 == f3 x1)%:R * (f2 x1 == f4 x1)%:R).
simpl.
exact: (destruct_pair_eq (a:= f1 x) (b:=f3 x) (c:= f2 x) (d := f4 x)). }
rewrite Hf.
apply realsum.summableM. all: assumption.
Qed.
Lemma psum_exists {R : realType} {A : choiceType} {f : A -> R}
(H : 0 < realsum.psum (T:=A) (R:=R) f) (Hpos : forall x, 0 <= f x) :
exists x, 0 < f x.
Proof.
assert (realsum.psum f ≠ 0) as Hneq.
{ intros Hgt.
rewrite Hgt in H.
rewrite order.Order.POrderTheory.ltxx in H.
auto. }
destruct (realsum.neq0_psum (R:=R) Hneq) as [x Hx].
exists x. specialize (Hpos x).
rewrite order.Order.POrderTheory.le_eqVlt in Hpos.
move: Hpos. move /orP => [H1 | H2].
- move: H1. move /eqP => H1. rewrite -H1.
rewrite order.Order.POrderTheory.ltxx. auto.
- assumption.
Qed.
Lemma pmulr_me (x y : R) : 0 <= y -> (0 < y * x) -> (0 < x).
Proof.
rewrite le0r => /orP[/eqP->|].
- by rewrite GRing.mul0r order.Order.POrderTheory.ltxx.
- intros. by rewrite -(pmulr_rgt0 x b).
Qed.
Lemma ge0_eq {R : realType} {A : eqType} {x y : A} (H : 0 < ((x == y)%:R) :> R) :
x = y.
Proof.
destruct (x == y) eqn:Heq.
- move: Heq. by move /eqP.
- by rewrite order.Order.POrderTheory.ltxx in H.
Qed.
Lemma ne0_eq {R : ringType} {A : eqType} {x y : A} (H : ((x == y)%:R) ≠ (0 : R)) :
x = y.
Proof.
destruct (x == y) eqn:Heq.
- move: Heq. by move /eqP.
- cbn in H. intuition.
Qed.
(* TODO MOVE *)
Lemma dlet_f_equal :
∀ {R : realType} {T U : choiceType} (m : {distr T / R}) (f g : T → {distr U / R}),
f =1 g →
\dlet_(x <- m) f x =1 \dlet_(x <- m) g x.
Proof.
intros R T U m f g e x.
apply functional_extensionality in e. subst.
reflexivity.
Qed.
(* TODO This proof is really the same as cmd_sample_preserve_pre *)
Lemma sampler_case :
∀ {LA} (I : heap_choiceType * heap_choiceType -> Prop) op,
INV LA I →
r⊨ ⦃ λ '(s0, s3), I (s0, s3) ⦄
sampler op [eta ret (A:=Arit op)] ≈ sampler op [eta ret (A:=Arit op)]
⦃ λ '(b1, s1) '(b2, s2), b1 = b2 ∧ I (s1, s2) ⦄.
Proof.
intros LA I op HINV.
intros [s₀ s₁]. hnf. intro P. hnf.
intros [hpre hpost]. simpl.
destruct op as [opA opB].
pose (d :=
SDistr_bind (λ x, SDistr_unit _ ((x, s₀), (x, s₁)))
(Theta_dens.unary_ThetaDens0 _ (ropr (opA ; opB) (λ x : chElement opA, retrFree x)))
).
exists d. split.
- unfold coupling. split.
+ unfold lmg. unfold dfst.
apply distr_ext. intro. simpl.
rewrite dlet_dlet.
simpl.
unfold SDistr_bind, SDistr_unit.
rewrite dlet_dlet.
apply dlet_f_equal. intro.
apply distr_ext. intro.
rewrite dlet_unit. rewrite dlet_unit. simpl. reflexivity.
+ unfold rmg. unfold dsnd.
apply distr_ext. intro. simpl.
rewrite dlet_dlet.
simpl.
unfold SDistr_bind, SDistr_unit.
rewrite dlet_dlet.
apply dlet_f_equal. intro.
apply distr_ext. intro.
rewrite dlet_unit. rewrite dlet_unit. simpl. reflexivity.
- intros [] [] e. subst d. simpl in e.
rewrite SDistr_rightneutral in e. simpl in e.
unfold SDistr_bind, SDistr_unit in e.
rewrite dletE in e.
erewrite eq_psum in e.
2:{
intro. rewrite dunit1E. reflexivity.
}
apply psum_exists in e.
2:{
intro. apply mulr_ge0.
- auto.
- apply ler0n.
}
destruct e as [? e].
apply pmulr_me in e. 2: auto.
apply ge0_eq in e. noconf e.
eapply hpost. intuition auto.
Qed.
(** Syntactic judgment *)
(* It's the same as the semantic one, but we're abstracting it away. *)
Inductive rel_jdg {A B : choiceType} (pre : precond) (post : postcond A B)
(p : raw_code A) (q : raw_code B) :=
_from_sem_jdg : locked (r⊨ ⦃ pre ⦄ p ≈ q ⦃ post ⦄) → rel_jdg pre post p q.
Notation "⊢ ⦃ pre ⦄ c1 ≈ c2 ⦃ post ⦄" :=
(rel_jdg pre post c1 c2)
(format "⊢ ⦃ pre ⦄ '/ ' '[' c1 ']' '/' ≈ '/ ' '[' c2 ']' '/' ⦃ post ⦄")
: package_scope.
Lemma from_sem_jdg :
∀ {A B : choiceType} pre (post : postcond A B) p q,
r⊨ ⦃ pre ⦄ p ≈ q ⦃ post ⦄ →
⊢ ⦃ pre ⦄ p ≈ q ⦃ post ⦄.
Proof.
intros A B pre post p q h.
constructor. rewrite -lock. auto.
Qed.
Lemma to_sem_jdg :
∀ {A B : choiceType} pre (post : postcond A B) p q,
⊢ ⦃ pre ⦄ p ≈ q ⦃ post ⦄ →
r⊨ ⦃ pre ⦄ p ≈ q ⦃ post ⦄.
Proof.
intros A B pre post p q [h].
rewrite -lock in h.
exact h.
Qed.
(** Equivalence of packages in the program logic *)
Definition eq_up_to_inv (E : Interface) (I : precond) (p₀ p₁ : raw_package) :=
∀ (id : ident) (S T : choice_type) (x : S),
(id, (S, T)) \in E →
⊢ ⦃ λ '(s₀, s₁), I (s₀, s₁) ⦄
get_op_default p₀ (id, (S, T)) x ≈ get_op_default p₁ (id, (S, T)) x
⦃ λ '(b₀, s₀) '(b₁, s₁), b₀ = b₁ ∧ I (s₀, s₁) ⦄.
Lemma Pr_eq_empty :
∀ {X Y : ord_choiceType}
{A : pred (X * heap_choiceType)} {B : pred (Y * heap_choiceType)}
Ψ ϕ
(c1 : @FrStP heap_choiceType X) (c2 : @FrStP heap_choiceType Y),
⊨ ⦃ Ψ ⦄ c1 ≈ c2 ⦃ ϕ ⦄ →
Ψ (empty_heap, empty_heap) →
(∀ x y, ϕ x y → (A x) ↔ (B y)) →
\P_[ θ_dens (θ0 c1 empty_heap) ] A =
\P_[ θ_dens (θ0 c2 empty_heap) ] B.
Proof.
intros X Y A B Ψ Φ c1 c2 ? ? ?.
apply (Pr_eq Ψ Φ). all: assumption.
Qed.
Lemma eq_up_to_inv_adversary_link :
∀ {L₀ L₁ LA E} (p₀ p₁ : raw_package) (I : precond) {B} (A : raw_code B)
`{ValidPackage L₀ Game_import E p₀}
`{ValidPackage L₁ Game_import E p₁}
`{@ValidCode LA E B A},
INV LA I →
eq_up_to_inv E I p₀ p₁ →
r⊨ ⦃ I ⦄ code_link A p₀ ≈ code_link A p₁
⦃ λ '(b₀, s₀) '(b₁, s₁), b₀ = b₁ ∧ I (s₀, s₁) ⦄.
Proof.
intros L₀ L₁ LA E p₀ p₁ I B A vp₀ vp₁ vA hLA hp.
induction A in vA |- *.
- cbn - [semantic_judgement].
eapply weaken_rule. 1: apply ret_rule.
intros [h₀ h₁] post.
cbn. unfold SPropMonadicStructures.SProp_op_order.
unfold Basics.flip, SPropMonadicStructures.SProp_order.
intros [HI Hp].
apply Hp. intuition auto.
- cbn - [semantic_judgement lookup_op].
apply inversion_valid_opr in vA as hA. destruct hA as [hi vk].
destruct o as [id [S T]].
eapply from_valid_package in vp₀.
specialize (vp₀ _ hi). simpl in vp₀.
destruct vp₀ as [f₀ [e₀ h₀]].
eapply from_valid_package in vp₁.
specialize (vp₁ _ hi). simpl in vp₁.
destruct vp₁ as [f₁ [e₁ h₁]].
erewrite lookup_op_spec_inv. 2: eauto.
erewrite lookup_op_spec_inv. 2: eauto.
specialize (hp id S T x hi).
erewrite get_op_default_spec in hp. 2: eauto.
erewrite get_op_default_spec in hp. 2: eauto.
rewrite !repr_bind.
eapply bind_rule_pp. 1:{ eapply to_sem_jdg in hp. exact hp. }
cbn - [semantic_judgement].
intros a₀ a₁.
apply pre_hypothesis_rule.
intros s₀ s₁ [? ?]. subst.
eapply pre_weaken_rule. 1: eapply H.
+ eapply vk.
+ cbn. intros s₀' s₁' [? ?]. subst. auto.
- cbn - [semantic_judgement bindrFree].
apply inversion_valid_getr in vA as hA. destruct hA as [hi vk].
match goal with
| |- ⊨ ⦃ ?pre_ ⦄ bindrFree ?m_ ?f1_ ≈ bindrFree _ ?f2_ ⦃ ?post_ ⦄ =>
eapply (bind_rule_pp (f1 := f1_) (f2 := f2_) m_ m_ pre_ _ post_)
end.
+ eapply (get_case LA). all: auto.
+ intros a₀ a₁. cbn - [semantic_judgement].
eapply pre_hypothesis_rule.
intros s₀ s₁ [? ?]. subst a₁.
eapply pre_weaken_rule. 1: eapply H.
* eapply vk.
* cbn. intros s₀' s₁' [? ?]. subst. auto.
- cbn - [semantic_judgement bindrFree].
apply inversion_valid_putr in vA as hA. destruct hA as [hi vk].
match goal with
| |- ⊨ ⦃ ?pre_ ⦄ bindrFree ?m_ ?f1_ ≈ bindrFree _ ?f2_ ⦃ ?post_ ⦄ =>
eapply (bind_rule_pp (f1 := f1_) (f2 := f2_) m_ m_ pre_ _ post_)
end.
+ eapply (@put_case LA). all: auto.
+ intros a₀ a₁. cbn - [semantic_judgement].
eapply pre_hypothesis_rule.
intros s₀ s₁ [? ?]. subst a₁.
eapply pre_weaken_rule. 1: eapply IHA.
* eapply vk.
* cbn. intros s₀' s₁' [? ?]. subst. auto.
- cbn - [semantic_judgement bindrFree].
eapply bind_rule_pp.
+ eapply (@sampler_case LA). auto.
+ intros a₀ a₁. cbn - [semantic_judgement].
eapply pre_hypothesis_rule.
intros s₀ s₁ [? ?]. subst a₁.
eapply pre_weaken_rule. 1: eapply H.
* eapply inversion_valid_sampler. eauto.
* cbn. intros s₀' s₁' [? ?]. subst. auto.
Qed.
Lemma eq_upto_inv_perf_ind :
∀ {L₀ L₁ LA E} (p₀ p₁ : raw_package) (I : precond) (A : raw_package)
`{ValidPackage L₀ Game_import E p₀}
`{ValidPackage L₁ Game_import E p₁}
`{ValidPackage LA E A_export A},
INV' L₀ L₁ I →
I (empty_heap, empty_heap) →
fdisjoint LA L₀ →
fdisjoint LA L₁ →
eq_up_to_inv E I p₀ p₁ →
AdvantageE p₀ p₁ A = 0.
Proof.
intros L₀ L₁ LA E p₀ p₁ I A vp₀ vp₁ vA hI' hIe hd₀ hd₁ hp.
unfold AdvantageE, Pr.
pose r := get_op_default A RUN tt.
assert (hI : INV LA I).
{ unfold INV. intros s₀ s₁. split.
- intros hi l hin. apply hI'.
+ assumption.
+ move: hd₀ => /fdisjointP hd₀. apply hd₀. assumption.
+ move: hd₁ => /fdisjointP hd₁. apply hd₁. assumption.
- intros hi l v hin. apply hI'.
+ assumption.
+ move: hd₀ => /fdisjointP hd₀. apply hd₀. assumption.
+ move: hd₁ => /fdisjointP hd₁. apply hd₁. assumption.
}
unshelve epose proof (eq_up_to_inv_adversary_link p₀ p₁ I r hI hp) as h.
1:{
eapply valid_get_op_default.
- eauto.
- auto_in_fset.
}
assert (
∀ x y : tgt RUN * heap_choiceType,
(let '(b₀, s₀) := x in λ '(b₁, s₁), b₀ = b₁ ∧ I (s₀, s₁)) y →
(fst x == true) ↔ (fst y == true)
) as Ha.
{ intros [b₀ s₀] [b₁ s₁]. simpl.
intros [e ?]. rewrite e. intuition auto.
}
unfold Pr_op.
unshelve epose (rhs := thetaFstd _ (repr (code_link r p₀)) empty_heap).
simpl in rhs.
epose (lhs := Pr_op (A ∘ p₀) RUN tt empty_heap).
assert (lhs = rhs) as he.
{ subst lhs rhs.
unfold Pr_op. unfold Pr_code.
unfold thetaFstd. simpl. apply f_equal2. 2: reflexivity.
apply f_equal. apply f_equal.
rewrite get_op_default_link. reflexivity.
}
unfold lhs in he. unfold Pr_op in he.
rewrite he.
unshelve epose (rhs' := thetaFstd _ (repr (code_link r p₁)) empty_heap).
simpl in rhs'.
epose (lhs' := Pr_op (A ∘ p₁) RUN tt empty_heap).
assert (lhs' = rhs') as e'.
{ subst lhs' rhs'.
unfold Pr_op. unfold Pr_code.
unfold thetaFstd. simpl. apply f_equal2. 2: reflexivity.
apply f_equal. apply f_equal.
rewrite get_op_default_link. reflexivity.
}
unfold lhs' in e'. unfold Pr_op in e'.
rewrite e'.
unfold rhs', rhs.
unfold SDistr_bind. unfold SDistr_unit.
rewrite !dletE.
assert (
∀ x : bool_choiceType * heap_choiceType,
((let '(b, _) := x in dunit (R:=R) (T:=bool_choiceType) b) true) ==
(x.1 == true)%:R
) as h1.
{ intros [b s].
simpl. rewrite dunit1E. apply/eqP. reflexivity.
}
assert (
∀ y,
(λ x : prod_choiceType (tgt RUN) heap_choiceType, (y x) * (let '(b, _) := x in dunit (R:=R) (T:=tgt RUN) b) true) =
(λ x : prod_choiceType (tgt RUN) heap_choiceType, (x.1 == true)%:R * (y x))
) as Hrew.
{ intros y. extensionality x.
destruct x as [x1 x2].
rewrite dunit1E.
simpl. rewrite GRing.mulrC. reflexivity.
}
rewrite !Hrew.
unfold TransformingLaxMorph.rlmm_from_lmla_obligation_1. simpl.
unfold SubDistr.SDistr_obligation_2. simpl.
unfold OrderEnrichedRelativeAdjunctionsExamples.ToTheS_obligation_1.
rewrite !SDistr_rightneutral. simpl.
pose proof (Pr_eq_empty _ _ _ _ h hIe Ha) as Heq.
simpl in Heq.
unfold θ_dens in Heq.
simpl in Heq. unfold pr in Heq.
simpl in Heq.
rewrite Heq.
rewrite /StateTransfThetaDens.unaryStateBeta'_obligation_1.
assert (∀ (x : R), `|x - x| = 0) as Hzero.
{ intros x.
assert (x - x = 0) as H3.
{ apply /eqP. rewrite GRing.subr_eq0. intuition. }
rewrite H3. apply normr0.
}
apply Hzero.
Qed.
Lemma eq_rel_perf_ind :
∀ {L₀ L₁ E} (p₀ p₁ : raw_package) (inv : precond)
`{ValidPackage L₀ Game_import E p₀}
`{ValidPackage L₁ Game_import E p₁},
Invariant L₀ L₁ inv →
eq_up_to_inv E inv p₀ p₁ →
p₀ ≈₀ p₁.
Proof.
intros L₀ L₁ E p₀ p₁ inv v₀ v₁ [? ?] he.
intros LA A vA hd₀ hd₁.
eapply eq_upto_inv_perf_ind. all: eauto.
Qed.
(* Special case where the invariant is equality of state. *)
Corollary eq_rel_perf_ind_eq :
∀ {L₀ L₁ E} (p₀ p₁ : raw_package)
`{ValidPackage L₀ Game_import E p₀}
`{ValidPackage L₁ Game_import E p₁},
eq_up_to_inv E (λ '(h₀, h₁), h₀ = h₁) p₀ p₁ →
p₀ ≈₀ p₁.
Proof.
intros L₀ L₁ E p₀ p₁ v₀ v₁ h.
eapply eq_rel_perf_ind with (λ '(h₀, h₁), h₀ = h₁).
- exact _.
- assumption.
Qed.
(* Special case where the invariant is heap_ignore. *)
Corollary eq_rel_perf_ind_ignore :
∀ L {L₀ L₁ E} (p₀ p₁ : raw_package)
`{ValidPackage L₀ Game_import E p₀}
`{ValidPackage L₁ Game_import E p₁},
fsubset L (L₀ :|: L₁) →
eq_up_to_inv E (heap_ignore L) p₀ p₁ →
p₀ ≈₀ p₁.
Proof.
intros L L₀ L₁ E p₀ p₁ v₀ v₁ hs h.
eapply eq_rel_perf_ind with (heap_ignore L).
- ssprove_invariant.
- assumption.
Qed.
(** Rules for packages *)
(* same as in RulesStateprob.v with `r` at the beginning *)
(* Alternative to rbind_rule *)
Lemma r_bind :
∀ {A₀ A₁ B₀ B₁ : ord_choiceType} m₀ m₁ f₀ f₁
(pre : precond) (mid : postcond A₀ A₁) (post : postcond B₀ B₁),
⊢ ⦃ pre ⦄ m₀ ≈ m₁ ⦃ mid ⦄ →
(∀ a₀ a₁, ⊢ ⦃ λ '(s₀, s₁), mid (a₀, s₀) (a₁, s₁) ⦄ f₀ a₀ ≈ f₁ a₁ ⦃ post ⦄) →
⊢ ⦃ pre ⦄ bind m₀ f₀ ≈ bind m₁ f₁ ⦃ post ⦄.
Proof.
intros A₀ A₁ B₀ B₁ m₀ m₁ f₀ f₁ pre mid post hm hf.
eapply from_sem_jdg. eapply rbind_rule.
- eapply to_sem_jdg. exact hm.
- intros a₀ a₁. eapply to_sem_jdg. eapply hf.
Qed.
(* Pre-condition manipulating rules *)
Theorem rpre_weaken_rule :
∀ {A₀ A₁ : ord_choiceType} {p₀ : raw_code A₀} {p₁ : raw_code A₁}
(pre pre' : precond) post,
⊢ ⦃ pre ⦄ p₀ ≈ p₁ ⦃ post ⦄ →
(∀ s₀ s₁, pre' (s₀, s₁) → pre (s₀, s₁)) →
⊢ ⦃ pre' ⦄ p₀ ≈ p₁ ⦃ post ⦄.
Proof.
intros A₀ A₁ p₀ p₁ pre pre' post he hi.
eapply from_sem_jdg.
eapply pre_weaken_rule.
- eapply to_sem_jdg. eauto.
- eauto.
Qed.
Theorem rpre_hypothesis_rule :
∀ {A₀ A₁ : ord_choiceType} {p₀ : raw_code A₀} {p₁ : raw_code A₁}
(pre : precond) post,
(∀ s₀ s₁,
pre (s₀, s₁) → ⊢ ⦃ λ s, s.1 = s₀ ∧ s.2 = s₁ ⦄ p₀ ≈ p₁ ⦃ post ⦄
) →
⊢ ⦃ pre ⦄ p₀ ≈ p₁ ⦃ post ⦄.
Proof.
intros A₀ A₁ p₀ p₁ pre post h.
eapply from_sem_jdg.
eapply pre_hypothesis_rule.
intros. eapply to_sem_jdg.
apply h. auto.
Qed.
Theorem rpre_strong_hypothesis_rule :
∀ {A₀ A₁ : ord_choiceType} {p₀ : raw_code A₀} {p₁ : raw_code A₁}
(pre : precond) post,
(∀ s₀ s₁, pre (s₀, s₁)) →
⊢ ⦃ λ _, True ⦄ p₀ ≈ p₁ ⦃ post ⦄ →
⊢ ⦃ pre ⦄ p₀ ≈ p₁ ⦃ post ⦄.
Proof.
intros A₀ A₁ p₀ p₁ pre post hs h.
eapply from_sem_jdg.
eapply pre_strong_hypothesis_rule.
- eauto.
- eapply to_sem_jdg. auto.
Qed.
Theorem rpost_weaken_rule :
∀ {A₀ A₁ : ord_choiceType} {p₀ : raw_code A₀} {p₁ : raw_code A₁}
(pre : precond) (post1 post2 : postcond A₀ A₁),
⊢ ⦃ pre ⦄ p₀ ≈ p₁ ⦃ post1 ⦄ →
(∀ a₀ a₁, post1 a₀ a₁ → post2 a₀ a₁) →
⊢ ⦃ pre ⦄ p₀ ≈ p₁ ⦃ post2 ⦄.
Proof.
intros A₀ A₁ p₀ p₁ pre post1 post2 h hi.
eapply from_sem_jdg.
eapply post_weaken_rule.
- eapply to_sem_jdg. eauto.
- eauto.
Qed.
Lemma rreflexivity_rule :
∀ {A : ord_choiceType} (c : raw_code A),
⊢ ⦃ λ '(s₀, s₁), s₀ = s₁ ⦄ c ≈ c ⦃ eq ⦄.
Proof.
intros A c.
eapply from_sem_jdg.
apply (reflexivity_rule (repr c)).
Qed.
(* TODO MOVE? *)
(* bindrFree_and_ret is too constrained *)
Lemma bindrFree_ret :
∀ S P A (m : rFreeF S P A),
bindrFree m (λ x, retrFree x) = m.
Proof.
intros S P A m.
induction m.
- reflexivity.
- cbn. f_equal. extensionality x. auto.
Qed.
Theorem rpost_conclusion_rule :
∀ {A₀ A₁ B : ord_choiceType} {pre : precond}
{c₀ : raw_code A₀} {c₁ : raw_code A₁}
(f₀ : A₀ → B) (f₁ : A₁ → B),
⊢ ⦃ pre ⦄
x₀ ← c₀ ;; ret x₀ ≈ x₁ ← c₁ ;; ret x₁
⦃ λ '(a₀, s₀) '(a₁, s₁), s₀ = s₁ ∧ f₀ a₀ = f₁ a₁ ⦄ →
⊢ ⦃ pre ⦄ x₀ ← c₀ ;; ret (f₀ x₀) ≈ x₁ ← c₁ ;; ret (f₁ x₁) ⦃ eq ⦄.
Proof.
intros A₀ A₁ B pre c₀ c₁ f₀ f₁ h.
eapply from_sem_jdg. eapply to_sem_jdg in h.
rewrite !repr_bind in h. rewrite !repr_bind.
eapply bind_rule_pp.
- simpl (repr (ret _)) in h.
rewrite !bindrFree_ret in h. exact h.
- intros a₀ a₁. eapply to_sem_jdg.
eapply rpre_hypothesis_rule. intros s s' [? e]. subst s'.
rewrite e. eapply rpre_weaken_rule. 1: eapply rreflexivity_rule.
cbn. intros ? ? [? ?]. subst. reflexivity.
Qed.
Theorem rpost_conclusion_rule_cmd :
∀ {A₀ A₁ B : ord_choiceType} {pre : precond}
{c₀ : command A₀} {c₁ : command A₁}
(f₀ : A₀ → B) (f₁ : A₁ → B),
⊢ ⦃ pre ⦄
x₀ ← cmd c₀ ;; ret x₀ ≈
x₁ ← cmd c₁ ;; ret x₁
⦃ λ '(a₀, s₀) '(a₁, s₁), s₀ = s₁ ∧ f₀ a₀ = f₁ a₁ ⦄ →
⊢ ⦃ pre ⦄ x₀ ← cmd c₀ ;; ret (f₀ x₀) ≈ x₁ ← cmd c₁ ;; ret (f₁ x₁) ⦃ eq ⦄.
Proof.
intros A₀ A₁ B pre c₀ c₁ f₀ f₁ h.
eapply from_sem_jdg. eapply to_sem_jdg in h.
rewrite !repr_cmd_bind in h. rewrite !repr_cmd_bind.
eapply bind_rule_pp.
- simpl (repr (ret _)) in h.
rewrite !bindrFree_ret in h. exact h.
- intros a₀ a₁. eapply to_sem_jdg.
eapply rpre_hypothesis_rule. intros s s' [? e]. subst s'.
rewrite e. eapply rpre_weaken_rule. 1: eapply rreflexivity_rule.
cbn. intros ? ? [? ?]. subst. reflexivity.
Qed.
Lemma r_ret :
∀ {A₀ A₁ : ord_choiceType} u₀ u₁ (pre : precond) (post : postcond A₀ A₁),
(∀ s₀ s₁, pre (s₀, s₁) → post (u₀, s₀) (u₁, s₁)) →
⊢ ⦃ pre ⦄ ret u₀ ≈ ret u₁ ⦃ post ⦄.
Proof.
intros A₀ A₁ u₀ u₁ pre post h.
eapply from_sem_jdg. simpl.
eapply weaken_rule. 1: eapply ret_rule.
intros [s₀ s₁] P [hpre hpost]. simpl.
eapply hpost. eapply h. apply hpre.
Qed.
Theorem rif_rule :
∀ {A₀ A₁ : ord_choiceType}
(c₀ c₀' : raw_code A₀) (c₁ c₁' : raw_code A₁)
{b₀ b₁}
{pre : precond} {post : postcond A₀ A₁},
(∀ s, pre s → b₀ = b₁) →
⊢ ⦃ λ s, pre s ∧ b₀ = true ⦄ c₀ ≈ c₁ ⦃ post ⦄ →
⊢ ⦃ λ s, pre s ∧ b₀ = false ⦄ c₀' ≈ c₁' ⦃ post ⦄ →
⊢ ⦃ pre ⦄ if b₀ then c₀ else c₀' ≈ if b₁ then c₁ else c₁' ⦃ post ⦄.
Proof.
intros A₀ A₁ c₀ c₀' c₁ c₁' b₀ b₁ pre post hb ht hf.
eapply from_sem_jdg. eapply to_sem_jdg in ht, hf.
rewrite !repr_if.
eapply if_rule. all: eauto.
Qed.
(* TODO: asymmetric variants of if_rule: if_ruleL and if_ruleR *)
(* skipped for now:
Theorem bounded_do_while_rule *)
(*TODO: asymmetric variants of bounded_do_while --
Rem.: low priority as not useful for our examples *)
Section For_loop_rule.
(* for i = 0 to N : do c *)
Fixpoint for_loop (c : nat → raw_code 'unit) (N : nat) : raw_code 'unit :=
match N with
| 0 => c 0%nat
| S m => for_loop c m ;; c (S m)
end.
Context (I : nat → precond) (N : nat).
Context (c₀ c₁ : nat → raw_code 'unit).
(* hypothesis : *)
(* body maintains the loop invariant I *)
(* to ease the proof we forget about this condition (0 <= n <= N)%nat -> *)
Lemma for_loop_rule :
(∀ i, ⊢ ⦃ I i ⦄ c₀ i ≈ c₁ i ⦃ λ '(_, s₀) '(_, s₁), I i.+1 (s₀,s₁) ⦄) →
⊢ ⦃ I 0%nat ⦄
for_loop c₀ N ≈ for_loop c₁ N
⦃ λ '(_,s₀) '(_,s₁), I N.+1 (s₀,s₁) ⦄.
Proof.
intros h.
induction N as [| n ih].
- simpl. apply h.
- simpl. eapply r_bind.
+ eapply ih.
+ simpl. intros _ _.
eapply rpre_weaken_rule. 1: eapply h.
auto.
Qed.
End For_loop_rule.
Lemma valid_for_loop :
∀ L I c N,
(∀ i, ValidCode L I (c i)) →
ValidCode L I (for_loop c N).
Proof.
intros L I c N h.
induction N. all: simpl.
- eapply h.
- eapply valid_bind. all: eauto.
Qed.
#[export] Hint Extern 1 (ValidCode ?L ?I (for_loop ?c ?N)) =>
eapply valid_for_loop ;
intro
: typeclass_instances ssprove_valid_db.
Lemma rcoupling_eq :
∀ {A : ord_choiceType} (K₀ K₁ : raw_code A) (ψ : precond),
⊢ ⦃ ψ ⦄ K₀ ≈ K₁ ⦃ eq ⦄ →
∀ s₀ s₁,
ψ (s₀, s₁) →
θ_dens (θ0 (repr K₀) s₀) = θ_dens (θ0 (repr K₁) s₁).
Proof.
intros A K₀ K₁ ψ h s₀ s₁ hψ.
eapply to_sem_jdg in h.
eapply coupling_eq. all: eauto.
Qed.
Lemma rrewrite_eqDistrL :
∀ {A₀ A₁ : ord_choiceType} {P Q}
(c₀ c₀' : raw_code A₀) (c₁ : raw_code A₁),
⊢ ⦃ P ⦄ c₀ ≈ c₁ ⦃ Q ⦄ →
(∀ s, θ_dens (θ0 (repr c₀) s) = θ_dens (θ0 (repr c₀') s)) →
⊢ ⦃ P ⦄ c₀' ≈ c₁ ⦃ Q ⦄.
Proof.
intros A₀ A₁ P Q c₀ c₀' c₁ h hθ.
eapply to_sem_jdg in h.
eapply from_sem_jdg.
eapply rewrite_eqDistrL. all: eauto.
Qed.
Lemma rrewrite_eqDistrR :
∀ {A₀ A₁ : ord_choiceType} {P Q}
(c₀ : raw_code A₀) (c₁ c₁' : raw_code A₁),
⊢ ⦃ P ⦄ c₀ ≈ c₁ ⦃ Q ⦄ →
(∀ s, θ_dens (θ0 (repr c₁) s) = θ_dens (θ0 (repr c₁') s)) →
⊢ ⦃ P ⦄ c₀ ≈ c₁' ⦃ Q ⦄.
Proof.
intros A₀ A₁ P Q c₀ c₁ c₁' h hθ.
eapply to_sem_jdg in h.
eapply from_sem_jdg.
eapply rewrite_eqDistrR. all: eauto.
Qed.
Theorem rswap_rule :
∀ {A₀ A₁ : ord_choiceType} {I : precond} {post : postcond A₀ A₁}
(c₀ : raw_code A₀) (c₁ : raw_code A₁),
⊢ ⦃ I ⦄ c₀ ≈ c₁
⦃ λ '(a₀, s₀) '(a₁, s₁), I (s₀, s₁) ∧ post (a₀, s₀) (a₁, s₁) ⦄ →
⊢ ⦃ I ⦄ c₁ ≈ c₀
⦃ λ '(a₁, s₁) '(a₀, s₀), I (s₀, s₁) ∧ post (a₀, s₀) (a₁, s₁) ⦄ →
⊢ ⦃ I ⦄ c₀ ;; c₁ ≈ c₁ ;; c₀
⦃ λ '(a₁, s₁) '(a₀, s₀), I (s₀, s₁) ∧ post (a₀, s₀) (a₁, s₁) ⦄.
Proof.
intros A₀ A₁ I post c₀ c₁ h1 h2.
eapply to_sem_jdg in h1. eapply to_sem_jdg in h2.
eapply from_sem_jdg.
rewrite !repr_bind.
eapply (swap_rule (repr c₀) (repr c₁)). all: auto.
Qed.
(** TW: I guess this to allow going under binders.
We might be better off defining some morphisms on semantic judgments
to use setoid_rewrite.
*)
Theorem rswap_ruleL :
∀ {A₀ A₁ B : ord_choiceType} {pre I : precond} {post : postcond A₁ A₀}
(l : raw_code B) (c₀ : raw_code A₀) (c₁ : raw_code A₁),
⊢ ⦃ pre ⦄ l ≈ l ⦃ λ '(b₀, s₀) '(b₁, s₁), I (s₀, s₁) ⦄ →
⊢ ⦃ I ⦄ c₀ ≈ c₁ ⦃ λ '(a₀, s₀) '(a₁, s₁), I (s₀, s₁) ∧ post (a₁, s₁) (a₀, s₀) ⦄ →
⊢ ⦃ I ⦄ c₁ ≈ c₀ ⦃ λ '(a₁, s₁) '(a₀, s₀), I (s₀, s₁) ∧ post (a₁, s₁) (a₀, s₀) ⦄ →
⊢ ⦃ pre ⦄ l ;; c₀ ;; c₁ ≈ l ;; c₁ ;; c₀ ⦃ post ⦄.
Proof.
intros A₀ A₁ B pre I post l c₀ c₁ hl h0 h1.
eapply to_sem_jdg in h0, h1, hl.
eapply from_sem_jdg.
rewrite !repr_bind.
eapply swap_ruleL. all: eauto.
Qed.
Theorem rswap_ruleR :
∀ {A₀ A₁ B : ord_choiceType} {post : postcond B B}
(c₀ : raw_code A₀) (c₁ : raw_code A₁) (r : A₀ → A₁ → raw_code B),
(∀ b b', b = b' → post b b') →
(∀ a₀ a₁, ⊢ ⦃ λ '(s₁, s₀), s₀ = s₁ ⦄ r a₀ a₁ ≈ r a₀ a₁ ⦃ post ⦄) →
⊢ ⦃ λ '(h₀, h₁), h₀ = h₁ ⦄
a₀ ← c₀ ;; a₁ ← c₁ ;; ret (a₀, a₁) ≈
a₁ ← c₁ ;; a₀ ← c₀ ;; ret (a₀, a₁)
⦃ eq ⦄ →
⊢ ⦃ λ '(h₀, h₁), h₀ = h₁ ⦄
a₀ ← c₀ ;; a₁ ← c₁ ;; r a₀ a₁ ≈
a₁ ← c₁ ;; a₀ ← c₀ ;; r a₀ a₁
⦃ post ⦄.
Proof.
intros A₀ A₁ B post c₀ c₁ r postr hr h.
eapply from_sem_jdg.
repeat setoid_rewrite repr_bind. simpl.
eapply (swap_ruleR (λ a₀ a₁, repr (r a₀ a₁)) (repr c₀) (repr c₁)).
- intros. eapply to_sem_jdg. apply hr.
- apply postr.
- intro s.
unshelve eapply coupling_eq.
+ exact (λ '(h₀, h₁), h₀ = h₁).
+ eapply to_sem_jdg in h. repeat setoid_rewrite repr_bind in h.
apply h.
+ reflexivity.
Qed.
Lemma rsym_pre :
∀ {A₀ A₁ : ord_choiceType} {pre : precond} {post}
{c₀ : raw_code A₀} {c₁ : raw_code A₁},
(∀ h₀ h₁, pre (h₀, h₁) → pre (h₁, h₀)) →
⊢ ⦃ λ '(h₀, h₁), pre (h₁, h₀) ⦄ c₀ ≈ c₁ ⦃ post ⦄ →
⊢ ⦃ pre ⦄ c₀ ≈ c₁ ⦃ post ⦄.
Proof.
intros A₀ A₁ pre post c₀ c₁ pre_sym h.
unshelve eapply rpre_weaken_rule. 2: eassumption.
assumption.
Qed.
Lemma rsymmetry :
∀ {A₀ A₁ : ord_choiceType} {pre : precond} {post}
{c₀ : raw_code A₀} {c₁ : raw_code A₁},
⊢ ⦃ λ '(h₁, h₀), pre (h₀, h₁) ⦄ c₁ ≈ c₀
⦃ λ '(a₁, h₁) '(a₀, h₀), post (a₀, h₀) (a₁, h₁) ⦄ →
⊢ ⦃ pre ⦄ c₀ ≈ c₁ ⦃ post ⦄.
Proof.
intros A₀ A₁ pre post c₀ c₁ h.
eapply from_sem_jdg.
eapply symmetry_rule. eapply to_sem_jdg. auto.
Qed.
Definition spl (o : Op) :=
@callrFree (@ops_StP heap_choiceType) (@ar_StP heap_choiceType) (op_iota o).
Lemma rsamplerC :
∀ {A : ord_choiceType} (o : Op) (c : raw_code A),
⊢ ⦃ λ '(h₀, h₁), h₀ = h₁ ⦄
a ← c ;; r ← (r ← sample o ;; ret r) ;; ret (a, r) ≈
r ← (r ← sample o ;; ret r) ;; a ← c ;; ret (a, r)
⦃ eq ⦄.
Proof.
intros A o c.
eapply rrewrite_eqDistrL.
- eapply rreflexivity_rule.
- intro s.
assert (
repr_sample_c :
repr (r ← (r ← sample o ;; ret r) ;; a ← c ;; ret (a, r)) =
bindrFree (spl o) (λ r, bindrFree (repr c) (λ a, retrFree (a,r)))
).
{ rewrite !repr_bind. f_equal. extensionality r.
rewrite !repr_bind. reflexivity.
}
assert (
repr_c_sample :
repr (a ← c ;; r ← (r ← sample o ;; ret r) ;; ret (a, r)) =
bindrFree (repr c) (λ a, bindrFree (spl o) (λ r, retrFree (a,r)))
).
{ rewrite repr_bind. reflexivity. }
rewrite repr_c_sample repr_sample_c.
pose proof (sample_c_is_c_sample o (repr c) s) as hlp.
unfold sample_c in hlp. unfold c_sample in hlp.
apply hlp.
Qed.
Lemma rsamplerC_sym' :
∀ {A : ord_choiceType} (o : Op) (c : raw_code A),
⊢ ⦃ λ '(h₀, h₁), h₀ = h₁ ⦄
a ← c ;; r ← (r ← sample o ;; ret r) ;; (ret (r, a)) ≈
r ← (r ← sample o ;; ret r) ;; a ← c ;; (ret (r, a))
⦃ eq ⦄.
Proof.
intros A o c.
unshelve eapply rswap_ruleR.
- auto.
- intros a r. apply rsym_pre. 1: auto.
apply rreflexivity_rule.
- apply rsamplerC.
Qed.
Lemma rsamplerC' :
∀ {A : ord_choiceType} (o : Op) (c : raw_code A),
⊢ ⦃ λ '(h₀, h₁), h₀ = h₁ ⦄
r ← (r ← sample o ;; ret r) ;; a ← c ;; ret (r, a) ≈
a ← c ;; r ← (r ← sample o ;; ret r) ;; ret (r, a)
⦃ eq ⦄.
Proof.
intros A o c.
eapply rsymmetry. eapply rsym_pre. 1: auto.
eapply rpost_weaken_rule.
- apply rsamplerC_sym'.
- intros [? ?] [? ?] e. inversion e. intuition auto.
Qed.
(* TODO: generalize the corresponding rule in RulesStateProb.v *)
Theorem rswap_rule_ctx :
∀ {A : ord_choiceType} {I pre} {post Q : postcond A A}
(l r c₀ c₁ : raw_code A),
⊢ ⦃ pre ⦄ l ≈ l ⦃ λ '(a₀, s₀) '(a₁, s₁), I (s₀, s₁) ⦄ →
(∀ a₀ a₁, ⊢ ⦃ λ '(s₁, s₀), Q (a₀,s₀) (a₁,s₁) ⦄ r ≈ r ⦃ post ⦄) →
⊢ ⦃ I ⦄ c₀ ≈ c₁ ⦃ λ '(a₀, s₀) '(a₁, s₁), I (s₀, s₁) ∧ Q (a₀, s₀) (a₁, s₁) ⦄ →
⊢ ⦃ I ⦄ c₁ ≈ c₀ ⦃ λ '(a₁, s₁) '(a₀, s₀), I (s₀, s₁) ∧ Q (a₀, s₀) (a₁, s₁) ⦄ →
⊢ ⦃ pre ⦄ l ;; c₀ ;; c₁ ;; r ≈ l ;; c₁ ;; c₀ ;; r ⦃ post ⦄.
Proof.
intros A I pre post Q l r c₀ c₁ hl hr h₀ h₁.
eapply from_sem_jdg.
rewrite !repr_bind.
eapply swap_rule_ctx.
1:{ eapply to_sem_jdg. exact hl. }
2:{ eapply to_sem_jdg. exact h₀. }
2:{ eapply to_sem_jdg. exact h₁. }
intros a₀ a₁. eapply to_sem_jdg. eapply hr.