Skip to content

GitLab

Commit a6d2564f authored by David Swasey's avatar David Swasey
Browse files
Options

Got rid of iris_unsafe.v (htUnsafe in iris_wp.v, robust_safety in iris_meta.v).

parent 730a9f4a
Hide whitespace changes
Inline Side-by-side
Showing with 196 additions and 249 deletions
Makefile
View file @ a6d2564f
......@@ -14,7 +14,7 @@
#
# This Makefile was generated by the command line :
# coq_makefile lib/ModuRes -R lib/ModuRes ModuRes core_lang.v iris_core.v iris_unsafe.v iris_vs.v iris_wp.v lang.v masks.v world_prop.v -o Makefile
# coq_makefile lib/ModuRes -R lib/ModuRes ModuRes core_lang.v iris_core.v iris_meta.v iris_vs.v iris_wp.v lang.v masks.v world_prop.v -o Makefile
#
.DEFAULT_GOAL := all
......@@ -82,10 +82,9 @@ endif
VFILES:=core_lang.v\
iris_core.v\
iris_unsafe.v\
iris_meta.v\
iris_vs.v\
iris_wp.v\
iris_meta.v\
lang.v\
masks.v\
world_prop.v
......
README.txt
View file @ a6d2564f
......@@ -63,7 +63,7 @@ CONTENTS
* iris_wp.v defines weakest preconditions and proves the rules for
Hoare triples
* iris_unsafe.v proves rules for unsafe Hoare triples
* iris_meta.v proves adequacy, robust safety, and the lifting lemmas
The development uses ModuRes, a Coq library by Sieczkowski et al. to
solve the recursive domain equation (see the paper for a reference)
......
iris_core.v
View file @ a6d2564f
......@@ -417,4 +417,28 @@ Module IrisCore (RL : RA_T) (C : CORE_LANG).
- intros w n r; apply Hp; exact I.
Qed.
(*
Simple monotonicity tactics for props and wsat.
The tactic propsM H proves P w' n' r' given H : P w n r when
w ⊑ w', n' <= n, r ⊑ r'
are immediate.
The tactic wsatM is similar.
*)
Lemma propsM {P w n r w' n' r'}
(HP : P w n r) (HSw : w ⊑ w') (HLe : n' <= n) (HSr : r ⊑ r') :
P w' n' r'.
Proof. by apply: (mu_mono _ _ P _ _ HSw); exact: (uni_pred _ _ _ _ _ HLe HSr). Qed.
Ltac propsM H := solve [ done | apply (propsM H); solve [ done | reflexivity | omega ] ].
Lemma wsatM {σ m} {r : res} {w n k}
(HW : wsat σ m r w @ n) (HLe : k <= n) :
wsat σ m r w @ k.
Proof. by exact: (uni_pred _ _ _ _ _ HLe). Qed.
Ltac wsatM H := solve [done | apply (wsatM H); solve [done | omega] ].
End IrisCore.
iris_meta.v
View file @ a6d2564f
......@@ -198,6 +198,153 @@ Module IrisMeta (RL : RA_T) (C : CORE_LANG).
End Adequacy.
Set Bullet Behavior "None". (* PDS: Ridiculous. *)
Section RobustSafety.
Implicit Types (P : Props) (i n k : nat) (safe : bool) (m : mask) (e : expr) (Q : vPred) (r : pres) (w : Wld) (σ : state).
Program Definition restrictV (Q : expr -n> Props) : vPred :=
n[(fun v => Q (` v))].
Next Obligation.
move=> v v' Hv w k r; move: Hv.
case: n=>[_ Hk | n]; first by exfalso; omega.
by move=> /= ->.
Qed.
(*
* Primitive reductions are either pure (do not change the state)
* or atomic (step to a value).
*)
Hypothesis atomic_dec : forall e, atomic e + ~atomic e.
Hypothesis pure_step : forall e σ e' σ',
~ atomic e ->
prim_step (e, σ) (e', σ') ->
σ = σ'.
Variable E : expr -n> Props.
(* Compatibility for those expressions wp cares about. *)
Hypothesis forkE : forall e, E (fork e) == E e.
Hypothesis fillE : forall K e, E (K [[e]]) == E e * E (K [[fork_ret]]).
(* One can prove forkE, fillE as valid internal equalities. *)
Remark valid_intEq {P P' : Props} (H : valid(P === P')) : P == P'.
Proof. move=> w n r; exact: H. Qed.
(* View shifts or atomic triples for every primitive reduction. *)
Variable w₀ : Wld.
Definition valid₀ P := forall w n r (HSw₀ : w₀ ⊑ w), P w n r.
Hypothesis pureE : forall {e σ e'},
prim_step (e,σ) (e',σ) ->
valid₀ (vs mask_full mask_full (E e) (E e')).
Hypothesis atomicE : forall {e},
atomic e ->
valid₀ (ht false mask_full (E e) e (restrictV E)).
Lemma robust_safety {e} : valid₀(ht false mask_full (E e) e (restrictV E)).
Proof.
move=> wz nz rz HSw₀ w HSw n r HLe _ He.
have {HSw₀ HSw} HSw₀ : w₀ ⊑ w by transitivity wz.
(* For e = K[fork e'] we'll have to prove wp(e', ⊤), so the IH takes a post. *)
pose post Q := forall (v : value) w n r, (E (`v)) w n r -> (Q v) w n r.
set Q := restrictV E; have HQ: post Q by done.
move: {HLe} HSw₀ He HQ; move: n e w r Q; elim/wf_nat_ind;
move=> {wz nz rz} n IH e w r Q HSw₀ He HQ.
apply unfold_wp; move=> w' k rf mf σ HSw HLt HD HW.
split; [| split; [| split; [| done] ] ]; first 2 last.
(* e forks: fillE, IH (twice), forkE *)
- move=> e' K HDec.
have {He} He: (E e) w' k r by propsM He.
move: He; rewrite HDec fillE; move=> [re' [rK [Hr [He' HK] ] ] ].
exists w' re' rK; split; first by reflexivity.
have {IH} IH: forall Q, post Q ->
forall e r, (E e) w' k r -> wp false mask_full e Q w' k r.
+ by move=> Q0 HQ0 e0 r0 He0; apply: (IH _ HLt); first by transitivity w.
split; [exact: IH | split]; last first.
+ by move: HW; rewrite -Hr => HW; wsatM HW.
have Htop: post (umconst ⊤) by done.
by apply: (IH _ Htop e' re'); move: He'; rewrite forkE.
(* e value: done *)
- move=> {IH} HV; exists w' r; split; [by reflexivity | split; [| done] ].
by apply: HQ; propsM He.
(* e steps: fillE, atomic_dec *)
move=> σ' ei ei' K HDec HStep.
have {HSw₀} HSw₀ : w₀ ⊑ w' by transitivity w.
move: He; rewrite HDec fillE; move=> [rei [rK [Hr [Hei HK] ] ] ].
move: HW; rewrite -Hr => HW.
(* bookkeeping common to both cases. *)
have {Hei} Hei: (E ei) w' (S k) rei by propsM Hei.
have HSw': w' ⊑ w' by reflexivity.
have HLe: S k <= S k by omega.
have H1ei: ra_pos_unit ⊑ rei by apply: unit_min.
have HLt': k < S k by omega.
move: HW; rewrite {1}mask_full_union -{1}(mask_full_union mask_emp) -assoc => HW.
case: (atomic_dec ei) => HA; last first.
(* ei pure: pureE, IH, fillE *)
- move: (pure_step _ _ _ _ HA HStep) => {HA} Hσ.
rewrite Hσ in HStep HW => {Hσ}.
move: (pureE _ _ _ HStep) => {HStep} He.
move: {He} (He w' (S k) r HSw₀) => He.
move: {He HLe H1ei Hei} (He _ HSw' _ _ HLe H1ei Hei) => He.
move: {HD} (mask_emp_disjoint (mask_full ∪ mask_full)) => HD.
move: {He HSw' HW} (He _ _ _ _ _ HSw' HLt' HD HW) => [w'' [r' [HSw' [Hei' HW] ] ] ].
move: HW; rewrite assoc=>HW.
pose↓ α := (ra_proj r' · ra_proj rK).
+ by apply wsat_valid in HW; auto_valid.
have {HSw₀} HSw₀: w₀ ⊑ w'' by transitivity w'.
exists w'' α; split; [done| split]; last first.
+ by move: HW; rewrite 2! mask_full_union => HW; wsatM HW.
apply: (IH _ HLt _ _ _ _ HSw₀); last done.
rewrite fillE; exists r' rK; split; [exact: equivR | split; [by propsM Hei' |] ].
have {HSw} HSw: w ⊑ w'' by transitivity w'.
by propsM HK.
(* ei atomic: atomicE, IH, fillE *)
move: (atomic_step _ _ _ _ HA HStep) => HV.
move: (atomicE _ HA) => He {HA}.
move: {He} (He w' (S k) rei HSw₀) => He.
move: {He HLe H1ei Hei} (He _ HSw' _ _ HLe H1ei Hei) => He.
(* unroll wp(ei,E)—step case—to get wp(ei',E) *)
move: He; rewrite {1}unfold_wp => He.
move: {HD} (mask_emp_disjoint mask_full) => HD.
move: {He HSw' HLt' HW} (He _ _ _ _ _ HSw' HLt' HD HW) => [_ [HS _] ].
have Hεei: ei = ε[[ei]] by rewrite fill_empty.
move: {HS Hεei HStep} (HS _ _ _ _ Hεei HStep) => [w'' [r' [HSw' [He' HW] ] ] ].
(* unroll wp(ei',E)—value case—to get E ei' *)
move: He'; rewrite {1}unfold_wp => He'.
move: HW; case Hk': k => [| k'] HW.
- by exists w'' r'; split; [done | split; [exact: wpO | done] ].
have HSw'': w'' ⊑ w'' by reflexivity.
have HLt': k' < k by omega.
move: {He' HSw'' HLt' HD HW} (He' _ _ _ _ _ HSw'' HLt' HD HW) => [Hv _].
move: HV; rewrite -(fill_empty ei') => HV.
move: {Hv} (Hv HV) => [w''' [rei' [HSw'' [Hei' HW] ] ] ].
(* now IH *)
move: HW; rewrite assoc => HW.
pose↓ α := (ra_proj rei' · ra_proj rK).
+ by apply wsat_valid in HW; auto_valid.
exists w''' α. split; first by transitivity w''.
split; last by rewrite mask_full_union -(mask_full_union mask_emp).
rewrite/= in Hei'; rewrite fill_empty -Hk' in Hei' * => {Hk'}.
have {HSw₀} HSw₀ : w₀ ⊑ w''' by transitivity w''; first by transitivity w'.
apply: (IH _ HLt _ _ _ _ HSw₀); last done.
rewrite fillE; exists rei' rK; split; [exact: equivR | split; [done |] ].
have {HSw HSw' HSw''} HSw: w ⊑ w''' by transitivity w''; first by transitivity w'.
by propsM HK.
Qed.
End RobustSafety.
Set Bullet Behavior "Strict Subproofs".
Section Lifting.
Implicit Types (P : Props) (i : nat) (safe : bool) (m : mask) (e : expr) (Q R : vPred) (r : pres).
......
iris_unsafe.v deleted 100644 → 0
View file @ 730a9f4a
Set Automatic Coercions Import.
Require Import ssreflect ssrfun ssrbool eqtype.
Require Import core_lang masks iris_wp.
Require Import ModuRes.RA ModuRes.UPred ModuRes.BI ModuRes.PreoMet ModuRes.Finmap.
Module Unsafety (RL : RA_T) (C : CORE_LANG).
Module Export Iris := IrisWP RL C.
Local Open Scope lang_scope.
Local Open Scope ra_scope.
Local Open Scope bi_scope.
Local Open Scope iris_scope.
(* PDS: Hoist, somewhere. *)
Program Definition restrictV (Q : expr -n> Props) : vPred :=
n[(fun v => Q (` v))].
Solve Obligations using resp_set.
Next Obligation.
move=> v v' Hv w k r; move: Hv.
case: n=>[_ Hk | n]; first by exfalso; omega.
by move=> /= ->.
Qed.
Implicit Types (P : Props) (i n k : nat) (safe : bool) (m : mask) (e : expr) (Q : vPred) (r : pres) (w : Wld) (σ : state).
(* PDS: Move to iris_wp.v *)
Lemma htUnsafe {m P e Q} : ht true m P e Q ⊑ ht false m P e Q.
Proof.
move=> wz nz rz He w HSw n r HLe Hr HP.
move: {He P wz nz rz HSw HLe Hr HP} (He _ HSw _ _ HLe Hr HP).
move: n e Q w r; elim/wf_nat_ind; move=> n IH e Q w r He.
rewrite unfold_wp; move=> w' k rf mf σ HSw HLt HD Hw.
move: {IH} (IH _ HLt) => IH.
move: He => /unfold_wp He; move: {He HSw HLt HD Hw} (He _ _ _ _ _ HSw HLt HD Hw) => [HV [HS [HF _] ] ].
split; [done | clear HV; split; [clear HF | split; [clear HS | done] ] ].
- move=> σ' ei ei' K HK Hs.
move: {HS HK Hs} (HS _ _ _ _ HK Hs) => [w'' [r' [HSw' [He' Hw'] ] ] ].
exists w'' r'; split; [done | split; [exact: IH | done] ].
move=> e' K HK.
move: {HF HK} (HF _ _ HK) => [w'' [rfk [rret [HSw' [Hk [He' Hw'] ] ] ] ] ].
exists w'' rfk rret; split; [done | split; [exact: IH | split; [exact: IH | done] ] ].
Qed.
(*
Adjustments.
PDS: Should be moved or discarded.
*)
Lemma wpO {safe m e Q w r} : wp safe m e Q w O r.
Proof.
rewrite unfold_wp.
move=> w' k rf mf σ HSw HLt HD HW.
by exfalso; omega.
Qed.
(*
Simple monotonicity tactics for props and wsat.
The tactic propsM H proves P w' n' r' given H : P w n r when
w ⊑ w', n' <= n, r ⊑ r'
are immediate.
The tactic wsatM is similar.
PDS: Should be moved.
*)
Lemma propsM {P w n r w' n' r'}
(HP : P w n r) (HSw : w ⊑ w') (HLe : n' <= n) (HSr : r ⊑ r') :
P w' n' r'.
Proof. by apply: (mu_mono _ _ P _ _ HSw); exact: (uni_pred _ _ _ _ _ HLe HSr). Qed.
Ltac propsM H := solve [ done | apply (propsM H); solve [ done | reflexivity | omega ] ].
Lemma wsatM {σ m} {r : res} {w n k}
(HW : wsat σ m r w @ n) (HLe : k <= n) :
wsat σ m r w @ k.
Proof. by exact: (uni_pred _ _ _ _ _ HLe). Qed.
Ltac wsatM H := solve [done | apply (wsatM H); solve [done | omega] ].
(*
* Robust safety:
*
* Assume E : Exp → Prop satisfies
*
* E(fork e) = E e
* E(K[e]) = E e * E(K[()])
*
* and there exist Γ₀, Θ₀ s.t.
*
* (i) for every pure reduction ς; e → ς; e',
* Γ₀ | □Θ₀ ⊢ E e ==>>_⊤ E e'
*
* (ii) for every atomic reduction ς; e → ς'; e',
* Γ₀ | □Θ₀ ⊢ [E e] e [v. E v]_⊤
*
* Then, for every e, Γ₀ | □Θ₀ ⊢ [E e] e [v. E v]_⊤.
*
* The theorem has applications to security (hence the name).
*)
Section RobustSafety.
(*
* Assume primitive reductions are either pure (do not change the
* state) or atomic (step to a value).
*
* PDS: I suspect we need these to prove the lifting lemmas. If
* so, they should be in core_lang.v.
*)
Axiom atomic_dec : forall e, atomic e + ~atomic e.
Axiom pure_step : forall e σ e' σ',
~ atomic e ->
prim_step (e, σ) (e', σ') ->
σ = σ'.
Parameter E : expr -n> Props.
(* Compatibility for those expressions wp cares about. *)
Axiom forkE : forall e, E (fork e) == E e.
Axiom fillE : forall K e, E (K [[e]]) == E e * E (K [[fork_ret]]).
(* One can prove forkE, fillE as valid internal equalities. *)
Remark valid_intEq {P P' : Props} (H : valid(P === P')) : P == P'.
Proof. move=> w n r; exact: H. Qed.
(* View shifts or atomic triples for every primitive reduction. *)
Parameter w₀ : Wld.
Definition valid₀ P := forall {w n r} (HSw₀ : w₀ ⊑ w), P w n r.
Axiom pureE : forall {e σ e'},
prim_step (e,σ) (e',σ) ->
valid₀ (vs mask_full mask_full (E e) (E e')).
Axiom atomicE : forall {e},
atomic e ->
valid₀ (ht false mask_full (E e) e (restrictV E)).
Lemma robust_safety {e} : valid₀(ht false mask_full (E e) e (restrictV E)).
Proof.
move=> wz nz rz HSw₀ w HSw n r HLe _ He.
have {HSw₀ HSw} HSw₀ : w₀ ⊑ w by transitivity wz.
(* For e = K[fork e'] we'll have to prove wp(e', ⊤), so the IH takes a post. *)
pose post Q := forall (v : value) w n r, (E (`v)) w n r -> (Q v) w n r.
set Q := restrictV E; have HQ: post Q by done.
move: {HLe} HSw₀ He HQ; move: n e w r Q; elim/wf_nat_ind;
move=> {wz nz rz} n IH e w r Q HSw₀ He HQ.
apply unfold_wp; move=> w' k rf mf σ HSw HLt HD HW.
split; [| split; [| split; [| done] ] ]; first 2 last.
(* e forks: fillE, IH (twice), forkE *)
- move=> e' K HDec.
have {He} He: (E e) w' k r by propsM He.
move: He; rewrite HDec fillE; move=> [re' [rK [Hr [He' HK] ] ] ].
exists w' re' rK; split; first by reflexivity.
have {IH} IH: forall Q, post Q ->
forall e r, (E e) w' k r -> wp false mask_full e Q w' k r.
+ by move=> Q0 HQ0 e0 r0 He0; apply: (IH _ HLt); first by transitivity w.
split; [exact: IH | split]; last first.
+ by move: HW; rewrite -Hr => HW; wsatM HW.
have Htop: post (umconst ⊤) by done.
by apply: (IH _ Htop e' re'); move: He'; rewrite forkE.
(* e value: done *)
- move=> {IH} HV; exists w' r; split; [by reflexivity | split; [| done] ].
by apply: HQ; propsM He.
(* e steps: fillE, atomic_dec *)
move=> σ' ei ei' K HDec HStep.
have {HSw₀} HSw₀ : w₀ ⊑ w' by transitivity w.
move: He; rewrite HDec fillE; move=> [rei [rK [Hr [Hei HK] ] ] ].
move: HW; rewrite -Hr => HW.
(* bookkeeping common to both cases. *)
have {Hei} Hei: (E ei) w' (S k) rei by propsM Hei.
have HSw': w' ⊑ w' by reflexivity.
have HLe: S k <= S k by omega.
have H1ei: ra_pos_unit ⊑ rei by apply: unit_min.
have HLt': k < S k by omega.
move: HW; rewrite
{1}mask_full_union -{1}(mask_full_union mask_emp)
-assoc
=> HW.
case: (atomic_dec ei) => HA; last first.
(* ei pure: pureE, IH, fillE *)
- move: (pure_step _ _ _ _ HA HStep) => {HA} Hσ.
rewrite Hσ in HStep HW => {Hσ}.
move: (pureE HStep) => {HStep} He.
move: {He} (He w' (S k) r HSw₀) => He.
move: {He HLe H1ei Hei} (He _ HSw' _ _ HLe H1ei Hei) => He.
move: {HD} (mask_emp_disjoint (mask_full ∪ mask_full)) => HD.
move: {He HSw' HW} (He _ _ _ _ _ HSw' HLt' HD HW) => [w'' [r' [HSw' [Hei' HW] ] ] ].
move: HW; rewrite assoc=>HW.
pose↓ α := (ra_proj r' · ra_proj rK).
+ by apply wsat_valid in HW; auto_valid.
have {HSw₀} HSw₀: w₀ ⊑ w'' by transitivity w'.
exists w'' α; split; [done| split]; last first.
+ by move: HW; rewrite 2! mask_full_union => HW; wsatM HW.
apply: (IH _ HLt _ _ _ _ HSw₀); last done.
rewrite fillE; exists r' rK; split; [exact: equivR | split; [by propsM Hei' |] ].
have {HSw} HSw: w ⊑ w'' by transitivity w'.
by propsM HK.
(* ei atomic: atomicE, IH, fillE *)
move: (atomic_step _ _ _ _ HA HStep) => HV.
move: (atomicE HA) => He {HA}.
move: {He} (He w' (S k) rei HSw₀) => He.
move: {He HLe H1ei Hei} (He _ HSw' _ _ HLe H1ei Hei) => He.
(* unroll wp(ei,E)—step case—to get wp(ei',E) *)
move: He; rewrite {1}unfold_wp => He.
move: {HD} (mask_emp_disjoint mask_full) => HD.
move: {He HSw' HLt' HW} (He _ _ _ _ _ HSw' HLt' HD HW) => [_ [HS _] ].
have Hεei: ei = ε[[ei]] by rewrite fill_empty.
move: {HS Hεei HStep} (HS _ _ _ _ Hεei HStep) => [w'' [r' [HSw' [He' HW] ] ] ].
(* unroll wp(ei',E)—value case—to get E ei' *)
move: He'; rewrite {1}unfold_wp => He'.
move: HW; case Hk': k => [| k'] HW.
- by exists w'' r'; split; [done | split; [exact: wpO | done] ].
have HSw'': w'' ⊑ w'' by reflexivity.
have HLt': k' < k by omega.
move: {He' HSw'' HLt' HD HW} (He' _ _ _ _ _ HSw'' HLt' HD HW) => [Hv _].
move: HV; rewrite -(fill_empty ei') => HV.
move: {Hv} (Hv HV) => [w''' [rei' [HSw'' [Hei' HW] ] ] ].
(* now IH *)
move: HW; rewrite assoc => HW.
pose↓ α := (ra_proj rei' · ra_proj rK).
+ by apply wsat_valid in HW; auto_valid.
exists w''' α. split; first by transitivity w''.
split; last by rewrite mask_full_union -(mask_full_union mask_emp).
rewrite/= in Hei'; rewrite fill_empty -Hk' in Hei' * => {Hk'}.
have {HSw₀} HSw₀ : w₀ ⊑ w''' by transitivity w''; first by transitivity w'.
apply: (IH _ HLt _ _ _ _ HSw₀); last done.
rewrite fillE; exists rei' rK; split; [exact: equivR | split; [done |] ].
have {HSw HSw' HSw''} HSw: w ⊑ w''' by transitivity w''; first by transitivity w'.
by propsM HK.
Qed.
End RobustSafety.
End Unsafety.
iris_wp.v
View file @ a6d2564f
......@@ -230,7 +230,7 @@ Module IrisWP (RL : RA_T) (C : CORE_LANG).
- unfold safeExpr. auto.
Qed.
Lemma wpO safe m e φ w r : wp safe m e φ w O r.
Lemma wpO {safe m e Q w r} : wp safe m e Q w O r.
Proof.
rewrite unfold_wp; intros w'; intros; now inversion HLt.
Qed.
......@@ -544,6 +544,27 @@ Module IrisWP (RL : RA_T) (C : CORE_LANG).
- right; right; exists e empty_ctx; rewrite ->fill_empty; reflexivity.
Qed.
Set Bullet Behavior "None". (* PDS: Ridiculous. *)
Lemma htUnsafe {m P e Q} : ht true m P e Q ⊑ ht false m P e Q.
Proof.
move=> wz nz rz He w HSw n r HLe Hr HP.
move: {He P wz nz rz HSw HLe Hr HP} (He _ HSw _ _ HLe Hr HP).
move: n e Q w r; elim/wf_nat_ind; move=> n IH e Q w r He.
rewrite unfold_wp; move=> w' k rf mf σ HSw HLt HD Hw.
move: {IH} (IH _ HLt) => IH.
move: He => /unfold_wp He; move: {He HSw HLt HD Hw} (He _ _ _ _ _ HSw HLt HD Hw) => [HV [HS [HF _] ] ].
split; [done | clear HV; split; [clear HF | split; [clear HS | done] ] ].
- move=> σ' ei ei' K HK Hs.
move: {HS HK Hs} (HS _ _ _ _ HK Hs) => [w'' [r' [HSw' [He' Hw'] ] ] ].
exists w'' r'; split; [done | split; [exact: IH | done] ].
move=> e' K HK.
move: {HF HK} (HF _ _ HK) => [w'' [rfk [rret [HSw' [Hk [He' Hw'] ] ] ] ] ].
exists w'' rfk rret; split; [done | split; [exact: IH | split; [exact: IH | done] ] ].
Qed.
Set Bullet Behavior "Strict Subproofs".
End HoareTripleProperties.
Section DerivedRules.
......
Markdown is supported
0% or .
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!
Please register or to comment