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IPPGrammarTheorems.v
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IPPGrammarTheorems.v
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Require Import MyUtils.
Require Export IPPGrammar.
Section IPPGrammarTheorems.
Create HintDb IPPGrammar.
Hint Resolve CPrio_infix_infix_1 CPrio_infix_infix_2 CPrio_prefix_infix CPrio_infix_prefix CLeft_prefix_infix
CRight_infix_prefix CRight_postfix_infix CRight_postfix_infix CPrio_postfix_infix CPrio_postfix_prefix
CPrio_infix_postfix CLeft_infix_postfix CPrio_prefix_postfix CLeft_prefix_postfix CRight_postfix_prefix CLeft CRight
HMatch InfixMatch PrefixMatch PostfixMatch Atomic_wf Infix_wf Prefix_wf Postfix_wf Atomic_cf Infix_cf Prefix_cf
Postfix_cf Atomic_drmcf Infix_drmcf Prefix_drmcf Postfix_drmcf Match_rm InfixMatch_rm PrefixMatch_rm InfixMatch_drm
PrefixMatch_drm PostfixMatch_drm Atomic_dlmcf Infix_dlmcf Prefix_dlmcf Postfix_dlmcf Match_lm InfixMatch_lm
PostfixMatch_lm InfixMatch_dlm PrefixMatch_dlm PostfixMatch_dlm
: IPPGrammar.
(*
##############################################
##############################################
##############################################
*)
(* The following lemmas relate to the functions boolean functions that check for conflicts in trees,
such as [is_i_conflict_pattern]. *)
Lemma is_i_conflict_pattern_true {g} (pr : drules g) q :
i_conflict_pattern pr q <-> is_i_conflict_pattern pr q = true.
Proof.
split; intro.
- inv H; simpl; auto using decide_True, decide_False.
+ destruct (decide (prio pr (InfixProd o1) (InfixProd o2))); auto using decide_True, decide_False.
+ destruct (decide (prio pr (InfixProd o1) (InfixProd o2))); auto using decide_True, decide_False.
+ destruct (decide (prio pr (PrefixProd o1) (InfixProd o2))); auto using decide_True, decide_False.
+ destruct (decide (prio pr (PostfixProd o1) (InfixProd o2))); auto using decide_True, decide_False.
- destruct q; inv H.
+ destruct q1, q2; inv H1.
* destruct q2_1, q2_2; inv H0.
destruct (decide (prio pr (InfixProd o) (InfixProd o0))); eauto with IPPGrammar.
destruct (decide (left_a pr (InfixProd o) (InfixProd o0))); eauto with IPPGrammar.
inv H1.
* destruct q1_1, q1_2; inv H0.
destruct (decide (prio pr (InfixProd o) (InfixProd o0))); eauto with IPPGrammar.
destruct (decide (right_a pr (InfixProd o) (InfixProd o0))); eauto with IPPGrammar.
inv H1.
* destruct q1_1, q1_2; inv H0.
* destruct q1_1, q1_2; inv H0.
* destruct q1_1, q1_2; inv H0.
+ destruct q; inv H1.
destruct q1, q2; inv H0.
destruct (decide (prio pr (PrefixProd o) (InfixProd o0))); eauto with IPPGrammar.
destruct (decide (left_a pr (PrefixProd o) (InfixProd o0))); eauto with IPPGrammar.
inv H1.
+ destruct q; inv H1.
destruct q1, q2; inv H0.
destruct (decide (prio pr (PostfixProd o) (InfixProd o0))); eauto with IPPGrammar.
destruct (decide (right_a pr (PostfixProd o) (InfixProd o0))); eauto with IPPGrammar.
inv H1.
Qed.
Lemma is_i_conflict_pattern_false {g} (pr : drules g) q :
~ i_conflict_pattern pr q <-> is_i_conflict_pattern pr q = false.
Proof.
split; intro.
- destruct (is_i_conflict_pattern pr q) eqn:E; auto.
exfalso. destruct H. apply is_i_conflict_pattern_true. assumption.
- intro. apply is_i_conflict_pattern_true in H0. rewrite H in H0. inv H0.
Qed.
Lemma is_lm_conflict_pattern_true {g} (pr : drules g) q :
lm_conflict_pattern pr q <-> is_lm_conflict_pattern pr q = true.
Proof.
split; intro.
- inv H; simpl; auto using decide_True, decide_False.
+ destruct (decide (prio pr (InfixProd o1) (PostfixProd o2))); auto using decide_True, decide_False.
+ destruct (decide (prio pr (PrefixProd o1) (PostfixProd o2))); auto using decide_True, decide_False.
- destruct q; inv H.
+ destruct q1, q2; inv H1.
destruct q2; inv H0.
destruct (decide (prio pr (InfixProd o) (PostfixProd o0))); eauto with IPPGrammar.
destruct (decide (left_a pr (InfixProd o) (PostfixProd o0))); eauto with IPPGrammar.
inv H1.
+ destruct q; inv H1.
destruct q; inv H0.
destruct (decide (prio pr (PrefixProd o) (PostfixProd o0))); eauto with IPPGrammar.
destruct (decide (left_a pr (PrefixProd o) (PostfixProd o0))); eauto with IPPGrammar.
inv H1.
Qed.
Lemma is_lm_conflict_pattern_false {g} (pr : drules g) q :
~ lm_conflict_pattern pr q <-> is_lm_conflict_pattern pr q = false.
Proof.
split; intro.
- destruct (is_lm_conflict_pattern pr q) eqn:E; auto.
exfalso. destruct H. apply is_lm_conflict_pattern_true. assumption.
- intro. apply is_lm_conflict_pattern_true in H0. rewrite H in H0. inv H0.
Qed.
Lemma has_infix_lm_conflicts_true {g} (pr : drules g) o t2 :
has_infix_lm_conflicts pr o t2 = true <->
exists x, lm_conflict_pattern pr (CR_infix_postfix o x) /\ matches_lm t2 (PostfixPatt HPatt x).
Proof.
induction t2; split; intros.
- inv H.
- inv H. inv H0. inv H1. inv H0.
- apply IHt2_1 in H. inv H. inv H0. eauto with IPPGrammar.
- simpl. inv H. inv H0.
inv H1.
+ inv H0.
+ apply IHt2_1. eauto with IPPGrammar.
- inv H.
- inv H. inv H0. inv H1. inv H0.
- cbn [has_infix_lm_conflicts] in H.
destruct (is_lm_conflict_pattern pr (CR_infix_postfix o o0)) eqn:E.
+ apply is_lm_conflict_pattern_true in E. eauto with IPPGrammar.
+ apply IHt2 in H. inv H. inv H0. eauto with IPPGrammar.
- inv H. inv H0. cbn [has_infix_lm_conflicts].
destruct (is_lm_conflict_pattern pr (CR_infix_postfix o o0)) eqn:E; auto.
inv H1.
+ inv H0. apply is_lm_conflict_pattern_true in H. rewrite H in E. inv E.
+ rewrite IHt2. eauto with IPPGrammar.
Qed.
Lemma has_infix_lm_conflicts_false {g} (pr : drules g) o t2 :
has_infix_lm_conflicts pr o t2 = false <->
(forall x, matches_lm t2 (PostfixPatt HPatt x) -> ~ lm_conflict_pattern pr (CR_infix_postfix o x)).
Proof.
split; intros.
- intro. assert (has_infix_lm_conflicts pr o t2 = true). { apply has_infix_lm_conflicts_true. eauto. }
rewrite H in H2. inv H2.
- destruct (has_infix_lm_conflicts pr o t2) eqn:E; auto.
apply has_infix_lm_conflicts_true in E. inv E. inv H0. exfalso. eapply H; eauto.
Qed.
Lemma has_prefix_lm_conflicts_true {g} (pr : drules g) o t2 :
has_prefix_lm_conflicts pr o t2 = true <->
exists x, lm_conflict_pattern pr (CR_prefix_postfix o x) /\ matches_lm t2 (PostfixPatt HPatt x).
Proof.
induction t2; split; intros.
- inv H.
- inv H. inv H0. inv H1. inv H0.
- apply IHt2_1 in H. inv H. inv H0. eauto with IPPGrammar.
- simpl. inv H. inv H0.
inv H1.
+ inv H0.
+ apply IHt2_1. eauto with IPPGrammar.
- inv H.
- inv H. inv H0. inv H1. inv H0.
- cbn [has_prefix_lm_conflicts] in H.
destruct (is_lm_conflict_pattern pr (CR_prefix_postfix o o0)) eqn:E.
+ apply is_lm_conflict_pattern_true in E. eauto with IPPGrammar.
+ apply IHt2 in H. inv H. inv H0. eauto with IPPGrammar.
- inv H. inv H0. cbn [has_prefix_lm_conflicts].
destruct (is_lm_conflict_pattern pr (CR_prefix_postfix o o0)) eqn:E; auto.
inv H1.
+ inv H0. apply is_lm_conflict_pattern_true in H. rewrite H in E. inv E.
+ rewrite IHt2. eauto with IPPGrammar.
Qed.
Lemma has_prefix_lm_conflicts_false {g} (pr : drules g) o t2 :
has_prefix_lm_conflicts pr o t2 = false <->
(forall x, matches_lm t2 (PostfixPatt HPatt x) -> ~ lm_conflict_pattern pr (CR_prefix_postfix o x)).
Proof.
split; intros.
- intro. assert (has_prefix_lm_conflicts pr o t2 = true). { apply has_prefix_lm_conflicts_true. eauto. }
rewrite H in H2. inv H2.
- destruct (has_prefix_lm_conflicts pr o t2) eqn:E; auto.
apply has_prefix_lm_conflicts_true in E. inv E. inv H0. exfalso. eapply H; eauto.
Qed.
Lemma has_postfix_rm_conflicts_true {g} (pr : drules g) t1 o :
has_postfix_rm_conflicts pr t1 o = true <->
exists x, rm_conflict_pattern pr (CL_postfix_prefix o x) /\ matches_rm t1 (PrefixPatt x HPatt).
Proof.
induction t1; split; intros.
- inv H.
- inv H. inv H0. inv H1. inv H0.
- simpl in H. apply IHt1_2 in H. inv H. inv H0. eauto with IPPGrammar.
- simpl. inv H. inv H0.
inv H1.
+ inv H0.
+ apply IHt1_2. eauto with IPPGrammar.
- simpl in H. destruct (decide (prio pr (PostfixProd o) (PrefixProd o0))).
+ eexists. eauto with IPPGrammar.
+ destruct (decide (right_a pr (PostfixProd o) (PrefixProd o0))).
* eexists. eauto with IPPGrammar.
* apply IHt1 in H. inv H. inv H0. eexists. eauto with IPPGrammar.
- simpl. destruct (decide (prio pr (PostfixProd o) (PrefixProd o0))),
(decide (right_a pr (PostfixProd o) (PrefixProd o0))); auto.
inv H. inv H0.
inv H1.
+ inv H0. exfalso. inv H; auto.
+ apply IHt1. eexists. eauto.
- inv H.
- inv H. inv H0. inv H1. inv H0.
Qed.
Lemma has_postfix_rm_conflicts_false {g} (pr : drules g) t1 o :
has_postfix_rm_conflicts pr t1 o = false <->
(forall x, matches_rm t1 (PrefixPatt x HPatt) -> ~ rm_conflict_pattern pr (CL_postfix_prefix o x)).
Proof.
split; intros.
- intro. assert (has_postfix_rm_conflicts pr t1 o = true). { apply has_postfix_rm_conflicts_true. eauto. }
rewrite H in H2. inv H2.
- destruct (has_postfix_rm_conflicts pr t1 o) eqn:E; auto.
apply has_postfix_rm_conflicts_true in E. inv E. inv H0. edestruct H; eauto.
Qed.
(*
##############################################
##############################################
##############################################
*)
(* The following lemmas relate safety to relations between conflict patterns. *)
Lemma safe_infix_infix {g} (pr : drules g) o1 o2 :
safe_pr pr ->
i_conflict_pattern pr (CL_infix_infix o1 o2) ->
i_conflict_pattern pr (CR_infix_infix o2 o1) ->
False.
Proof.
intros H_safe H_CL H_CR. unfold safe_pr in H_safe. inv H_CL; inv H_CR; eauto.
Qed.
Lemma safe_infix_prefix {g} (pr : drules g) o1 o2 :
safe_pr pr ->
rm_conflict_pattern pr (CL_infix_prefix o1 o2) ->
i_conflict_pattern pr (CR_prefix_infix o2 o1) ->
False.
Proof.
intros. unfold safe_pr in H. inv H0; inv H1; eauto.
Qed.
Lemma safe_infix_postfix {g} (pr : drules g) o1 o2 :
safe_pr pr ->
lm_conflict_pattern pr (CR_infix_postfix o1 o2) ->
i_conflict_pattern pr (CL_postfix_infix o2 o1) ->
False.
Proof.
intros. unfold safe_pr in H. inv H0; inv H1; eauto.
Qed.
Lemma safe_prefix_postfix {g} (pr : drules g) o1 o2 :
safe_pr pr ->
lm_conflict_pattern pr (CR_prefix_postfix o1 o2) ->
rm_conflict_pattern pr (CL_postfix_prefix o2 o1) ->
False.
Proof.
intros. unfold safe_pr in H. inv H0; inv H1; eauto.
Qed.
(*
##############################################
##############################################
##############################################
*)
(* Helper tactics for simplifying specific repair terms. *)
(* Simplifies the term [insert_in pr t1 o (InfixNode t21 o2 t22)] *)
Ltac insert_in_inode_destruct pr o t21 o2 t22 :=
cbn [insert_in] in *;
destruct (is_i_conflict_pattern pr (CR_infix_infix o o2)) eqn:E;
[ apply is_i_conflict_pattern_true in E |
apply is_i_conflict_pattern_false in E;
destruct (has_infix_lm_conflicts pr o (InfixNode t21 o2 t22)) eqn:E2;
[ apply has_infix_lm_conflicts_true in E2 |
assert (forall x, matches_lm (InfixNode t21 o2 t22) (PostfixPatt HPatt x) ->
~ lm_conflict_pattern pr (CR_infix_postfix o x));
[apply has_infix_lm_conflicts_false; assumption|]
]
].
(* Simplifies the term [insert_in pr t1 o (PostfixNode t21 o2)] *)
Ltac insert_in_pnode_destruct pr o t21 o2 :=
cbn [insert_in] in *;
destruct (has_infix_lm_conflicts pr o (PostfixNode t21 o2)) eqn:E;
[apply has_infix_lm_conflicts_true in E |
assert (forall x, matches_lm (PostfixNode t21 o2) (PostfixPatt HPatt x) ->
~ lm_conflict_pattern pr (CR_infix_postfix o x));
[apply has_infix_lm_conflicts_false; assumption|]
].
(* Simplifies the term [insert_pre pr o (InfixNode t21 o2 t22)] *)
Ltac insert_pre_inode_destruct pr o t21 o2 t22 :=
cbn [insert_pre] in *;
destruct (is_i_conflict_pattern pr (CR_prefix_infix o o2)) eqn:E;
[ apply is_i_conflict_pattern_true in E |
apply is_i_conflict_pattern_false in E;
destruct (has_prefix_lm_conflicts pr o (InfixNode t21 o2 t22)) eqn:E2;
[ apply has_prefix_lm_conflicts_true in E2 |
assert (forall x, matches_lm (InfixNode t21 o2 t22) (PostfixPatt HPatt x) ->
~ lm_conflict_pattern pr (CR_prefix_postfix o x));
[apply has_prefix_lm_conflicts_false; assumption|]
]
].
(* Simplifies the term [insert_pre pr o (PostfixNode t21 o2)] *)
Ltac insert_pre_pnode_destruct pr o t21 o2 :=
cbn [insert_pre] in *;
destruct (has_prefix_lm_conflicts pr o (PostfixNode t21 o2)) eqn:E;
[apply has_prefix_lm_conflicts_true in E |
assert (forall x, matches_lm (PostfixNode t21 o2) (PostfixPatt HPatt x) ->
~ lm_conflict_pattern pr (CR_prefix_postfix o x));
[apply has_prefix_lm_conflicts_false; assumption|]
].
(* Simplifies the term [insert_post pr (InfixNode t11 o1 t12) o] *)
Ltac insert_post_inode_destruct pr t11 o1 t12 o :=
cbn [insert_post] in *;
destruct (is_i_conflict_pattern pr (CL_postfix_infix o o1)) eqn:E;
[ apply is_i_conflict_pattern_true in E |
apply is_i_conflict_pattern_false in E;
destruct (has_postfix_rm_conflicts pr (InfixNode t11 o1 t12) o) eqn:E2;
[ apply has_postfix_rm_conflicts_true in E2 |
assert (forall x, matches_rm (InfixNode t11 o1 t12) (PrefixPatt x HPatt) ->
~ rm_conflict_pattern pr (CL_postfix_prefix o x));
[apply has_postfix_rm_conflicts_false; assumption|]
]
].
(* Simplifies the term [insert_post pr (PrefixNode o1 t12) o] *)
Ltac insert_post_pnode_destruct pr o1 t12 o :=
cbn [insert_post] in *;
destruct (has_postfix_rm_conflicts pr (PrefixNode o1 t12) o) eqn:E;
[ apply has_postfix_rm_conflicts_true in E |
assert (forall x, matches_rm (PrefixNode o1 t12) (PrefixPatt x HPatt) ->
~ rm_conflict_pattern pr (CL_postfix_prefix o x));
[apply has_postfix_rm_conflicts_false; assumption|]
].
(*
##############################################
##############################################
##############################################
*)
(* Lemmas that show well-formedness of [repair] *)
Lemma insert_in_wf {g} (pr : drules g) t1 o t2 :
wf_parse_tree g t1 -> wf_parse_tree g t2 -> g.(prods) (InfixProd (inl o)) ->
wf_parse_tree g (insert_in pr t1 (inl o) t2).
Proof.
intros. induction t2.
- simpl. auto with IPPGrammar.
- inv H0.
insert_in_inode_destruct pr (@inl (OPinpre g) (OPpost g) o) t2_1 (@inl (OPinpre g) (OPpost g) o1) t2_2;
auto with IPPGrammar.
- simpl. inv H0. auto with IPPGrammar.
- inv H0.
insert_in_pnode_destruct pr (@inl (OPinpre g) (OPpost g) o) t2 (@inr (OPinpre g) (OPpost g) o1);
auto with IPPGrammar.
Qed.
Lemma insert_pre_wf {g} (pr : drules g) o t2 :
wf_parse_tree g t2 -> g.(prods) (PrefixProd (inl o)) ->
wf_parse_tree g (insert_pre pr (inl o) t2).
Proof.
intros. induction H; eauto with IPPGrammar.
- insert_pre_inode_destruct pr (@inl (OPinpre g) (OPpost g) o) t1 (@inl (OPinpre g) (OPpost g) o0) t2;
auto with IPPGrammar.
- insert_pre_pnode_destruct pr (@inl (OPinpre g) (OPpost g) o) t (@inr (OPinpre g) (OPpost g) o0);
auto with IPPGrammar.
Qed.
Lemma repair_in_wf {g} (pr : drules g) t1 o t2 :
wf_parse_tree g t1 -> wf_parse_tree g t2 -> g.(prods) (InfixProd (inl o)) ->
wf_parse_tree g (repair_in pr t1 (inl o) t2).
Proof.
intro. revert o t2. induction H; intros; simpl; auto using insert_in_wf, insert_pre_wf with IPPGrammar.
Qed.
Lemma insert_post_wf {g} (pr : drules g) t1 o :
wf_parse_tree g t1 -> g.(prods) (PostfixProd (inr o)) ->
wf_parse_tree g (insert_post pr t1 (inr o)).
Proof.
intros. induction H; eauto with IPPGrammar.
- insert_post_inode_destruct pr t1 (@inl (OPinpre g) (OPpost g) o0) t2 (@inr (OPinpre g) (OPpost g) o);
auto with IPPGrammar.
- insert_post_pnode_destruct pr (@inl (OPinpre g) (OPpost g) o0) t (@inr (OPinpre g) (OPpost g) o);
auto with IPPGrammar.
Qed.
Lemma repair_wf {g} (pr : drules g) t :
wf_parse_tree g t ->
wf_parse_tree g (repair pr t).
Proof.
intro. induction H; simpl; auto using repair_in_wf, insert_pre_wf, insert_post_wf with IPPGrammar.
Qed.
(*
##############################################
##############################################
##############################################
*)
(* Lemmas that show yield-preservation of [repair] *)
Lemma insert_pre_yield_preserve {g} (pr : drules g) o t2 :
yield (insert_pre pr o t2) = inr o :: yield t2.
Proof.
induction t2; try reflexivity.
- insert_pre_inode_destruct pr o t2_1 o0 t2_2; auto; simpl; rewrite IHt2_1; auto.
- insert_pre_pnode_destruct pr o t2 o0; simpl; auto. rewrite IHt2. reflexivity.
Qed.
Lemma insert_in_yield_preserve {g} (pr : drules g) t1 o t2 :
yield (insert_in pr t1 o t2) = yield t1 ++ inr o :: yield t2.
Proof.
induction t2; try reflexivity.
- insert_in_inode_destruct pr o t2_1 o0 t2_2; simpl; auto.
+ rewrite IHt2_1. simplify_list_eq. reflexivity.
+ rewrite IHt2_1. simplify_list_eq. reflexivity.
- insert_in_pnode_destruct pr o t2 o0; simpl; auto. rewrite IHt2. simplify_list_eq. reflexivity.
Qed.
Lemma repair_in_yield_preserve {g} (pr : drules g) t1 o t2 :
yield (repair_in pr t1 o t2) = yield t1 ++ inr o :: yield t2.
Proof.
revert o t2. induction t1; intros.
- simpl. rewrite insert_in_yield_preserve. reflexivity.
- simplify_list_eq. rewrite <- IHt1_2. rewrite <- IHt1_1. reflexivity.
- simpl. rewrite <- IHt1. rewrite insert_pre_yield_preserve. reflexivity.
- simpl. rewrite insert_in_yield_preserve. reflexivity.
Qed.
Lemma insert_post_yield_preserve {g} (pr : drules g) t1 o :
yield (insert_post pr t1 o) = yield t1 ++ [inr o].
Proof.
induction t1; try reflexivity.
- insert_post_inode_destruct pr t1_1 o0 t1_2 o; auto; simplify_list_eq; rewrite IHt1_2; auto.
- insert_post_pnode_destruct pr o0 t1 o; auto; simplify_list_eq; rewrite IHt1; auto.
Qed.
Lemma repair_yield_preserve {g} (pr : drules g) t :
yield (repair pr t) = yield t.
Proof.
induction t; auto; simpl.
- rewrite repair_in_yield_preserve. rewrite IHt1. rewrite IHt2. reflexivity.
- rewrite insert_pre_yield_preserve. rewrite IHt. reflexivity.
- rewrite insert_post_yield_preserve. rewrite IHt. reflexivity.
Qed.
(*
##############################################
##############################################
##############################################
*)
(* Auxiliary lemmas and tactics that help proving safety. *)
Lemma i_conflict_pattern_cases {g} (pr : drules g) q :
i_conflict_pattern pr q -> exists o1 o2,
q = CR_infix_infix o1 o2 \/ q = CL_infix_infix o1 o2 \/ q = CR_prefix_infix o1 o2 \/ q = CL_postfix_infix o1 o2.
Proof.
intros. inv H; eauto 7.
Qed.
Ltac icp_cases H :=
apply i_conflict_pattern_cases in H as T; destruct T as [? T1]; destruct T1 as [? T2]; destruct T2 as [T5|T3];
[|destruct T3 as [T5|T4]; [|destruct T4 as [T5|T5]]]; rewrite T5 in *; clear T5.
Lemma rm_conflict_pattern_cases {g} (pr : drules g) q :
rm_conflict_pattern pr q -> exists o1 o2,
q = CL_infix_prefix o1 o2 \/ q = CL_postfix_prefix o1 o2.
Proof.
intros. inv H; eauto.
Qed.
Ltac rcp_cases H :=
apply rm_conflict_pattern_cases in H as T; destruct T as [? T1]; destruct T1 as [? T2]; destruct T2; subst.
Lemma lm_conflict_pattern_cases {g} (pr : drules g) q :
lm_conflict_pattern pr q -> exists o1 o2,
q = CR_infix_postfix o1 o2 \/ q = CR_prefix_postfix o1 o2.
Proof.
intros. inv H; eauto.
Qed.
Ltac lcp_cases H :=
apply lm_conflict_pattern_cases in H as T; destruct T as [? T1]; destruct T1 as [? T2]; destruct T2; subst.
(*
##############################################
##############################################
##############################################
*)
(* Proving safety of shallow conflict patterns (i_conflict_patterns). *)
Lemma insert_in_top {g} (pr : drules g) t1 o t2 :
matches (insert_in pr t1 o t2) (InfixPatt HPatt o HPatt) \/
(exists o2, matches t2 (InfixPatt HPatt o2 HPatt) /\ matches (insert_in pr t1 o t2) (InfixPatt HPatt o2 HPatt)) \/
(exists o2, matches t2 (PostfixPatt HPatt o2) /\ matches (insert_in pr t1 o t2) (PostfixPatt HPatt o2)).
Proof.
destruct t2; eauto 6 with IPPGrammar.
- insert_in_inode_destruct pr o t2_1 o0 t2_2; eauto 7 with IPPGrammar.
- insert_in_pnode_destruct pr o t2 o0; eauto 7 with IPPGrammar.
Qed.
Lemma insert_in_top_unchanged {g} (pr : drules g) t1 o t2 x :
matches_lm t2 (PostfixPatt HPatt x) ->
lm_conflict_pattern pr (CR_infix_postfix o x) ->
(exists o2, matches t2 (InfixPatt HPatt o2 HPatt) /\ matches (insert_in pr t1 o t2) (InfixPatt HPatt o2 HPatt)) \/
(exists o2, matches t2 (PostfixPatt HPatt o2) /\ matches (insert_in pr t1 o t2) (PostfixPatt HPatt o2)).
Proof.
intros. destruct t2.
- inv H. inv H1.
- inv H. inv H1. left.
eexists. split; auto with IPPGrammar.
insert_in_inode_destruct pr o t2_1 o0 t2_2; auto with IPPGrammar. exfalso. apply H with x; auto with IPPGrammar.
- inv H. inv H1.
- right. eexists. split; auto with IPPGrammar.
insert_in_pnode_destruct pr o t2 o0; auto with IPPGrammar. exfalso. apply H1 with x; auto.
Qed.
Lemma insert_in_icfree {g} (pr : drules g) t1 o t2 :
safe_pr pr ->
i_conflict_free (i_conflict_pattern pr) t1 ->
i_conflict_free (i_conflict_pattern pr) t2 ->
(forall o1, i_conflict_pattern pr (CL_infix_infix o o1) -> ~ matches t1 (InfixPatt HPatt o1 HPatt)) ->
i_conflict_free (i_conflict_pattern pr) (insert_in pr t1 o t2).
Proof.
intros. induction t2 as [l2|t21 ? o2 t22|o2 t22|t21 ? o2].
- simpl. apply Infix_cf; auto. intro. inv H3. inv H4. icp_cases H3; inv H5. inv H12. eapply H2; eauto.
- insert_in_inode_destruct pr o t21 o2 t22.
+ inv H1. apply Infix_cf; auto. intro. inv H1. inv H3. icp_cases H1; inv H4.
* destruct H6. eexists. eauto with IPPGrammar.
* inv H9. decompose [or] (insert_in_top pr t1 o t21); rewrite <- H3 in *.
**inv H4. eauto using safe_infix_infix.
**inv H5. inv H4. inv H9. destruct H6. eexists. eauto with IPPGrammar.
**inv H5. inv H4. inv H9.
+ inv H1. apply Infix_cf; auto with IPPGrammar. intro. inv H1. inv H3. icp_cases H1; inv H4.
* destruct H6. eexists. eauto with IPPGrammar.
* inv E2. inv H3. inv H5. inv H3.
inv H9. apply insert_in_top_unchanged with pr t1 o t21 x2 in H13; auto. destruct H13; rewrite <- H3 in *.
**inv H5. inv H9. inv H10. destruct H6. eexists. eauto with IPPGrammar.
**inv H5. inv H9. inv H10.
+ apply Infix_cf; auto. intro. inv H4. inv H5. icp_cases H4; inv H6.
* inv H13. contradiction.
* inv H8. apply H2 with x1; auto with IPPGrammar.
- simpl. apply Infix_cf; auto with IPPGrammar. intro. inv H3. inv H4. icp_cases H3; inv H5. inv H12. inv H1.
eapply H2; eauto with IPPGrammar.
- insert_in_pnode_destruct pr o t21 o2.
+ inv H1. apply Postfix_cf; auto with IPPGrammar. intro. inv H1. inv H3. icp_cases H1; inv H4. inv H7. inv E.
inv H4. inv H8.
* inv H4. decompose [or] (insert_in_top pr t1 o t21); rewrite <- H3 in *.
**inv H4. eauto using safe_infix_postfix.
**inv H8. inv H4. inv H12. destruct H5. eexists. eauto with IPPGrammar.
**inv H8. inv H4. inv H12.
* apply insert_in_top_unchanged with pr t1 o t21 x2 in H13; auto. destruct H13; auto; rewrite <- H3 in *.
**inv H4. inv H8. inv H10. destruct H5. eexists. eauto with IPPGrammar.
**inv H4. inv H8. inv H10.
+ apply Infix_cf; auto with IPPGrammar. intro. inv H4. inv H5. icp_cases H4; inv H6. inv H13. inv H8.
eapply H2; eauto with IPPGrammar.
Qed.
Lemma insert_pre_top {g} (pr : drules g) o t2 :
matches (insert_pre pr o t2) (PrefixPatt o HPatt) \/
(exists o2, matches t2 (InfixPatt HPatt o2 HPatt) /\ matches (insert_pre pr o t2) (InfixPatt HPatt o2 HPatt)) \/
(exists o2, matches t2 (PostfixPatt HPatt o2) /\ matches (insert_pre pr o t2) (PostfixPatt HPatt o2)).
Proof.
destruct t2; eauto with IPPGrammar.
- insert_pre_inode_destruct pr o t2_1 o0 t2_2; eauto 7 with IPPGrammar.
- insert_pre_pnode_destruct pr o t2 o0; eauto 7 with IPPGrammar.
Qed.
Lemma insert_pre_top_unchanged {g} (pr : drules g) o t2 x :
matches_lm t2 (PostfixPatt HPatt x) ->
lm_conflict_pattern pr (CR_prefix_postfix o x) ->
(exists o2, matches t2 (InfixPatt HPatt o2 HPatt) /\ matches (insert_pre pr o t2) (InfixPatt HPatt o2 HPatt)) \/
(exists o2, matches t2 (PostfixPatt HPatt o2) /\ matches (insert_pre pr o t2) (PostfixPatt HPatt o2)).
Proof.
intros. destruct t2.
- inv H. inv H1.
- inv H. inv H1. left.
eexists. split; auto with IPPGrammar.
insert_pre_inode_destruct pr o t2_1 o0 t2_2; auto with IPPGrammar. exfalso. apply H with x; auto with IPPGrammar.
- inv H. inv H1.
- right. eexists. split; auto with IPPGrammar.
insert_pre_pnode_destruct pr o t2 o0; auto with IPPGrammar. exfalso. apply H1 with x; auto.
Qed.
Lemma insert_pre_icfree {g} (pr : drules g) o t2 :
safe_pr pr ->
i_conflict_free (i_conflict_pattern pr) t2 ->
i_conflict_free (i_conflict_pattern pr) (insert_pre pr o t2).
Proof.
intros. induction t2 as [l2|t21 ? o2 t22|o2 t22|t21 ? o2].
- simpl. apply Prefix_cf; auto. intro. inv H1. inv H2. icp_cases H1; inv H3. inv H4.
- insert_pre_inode_destruct pr o t21 o2 t22.
+ inv H0. apply Infix_cf; auto. intro. inv H0. inv H1. icp_cases H0; inv H2.
* inv H12. destruct H4. eexists. eauto with IPPGrammar.
* decompose [or] (insert_pre_top pr o t21).
**inv H7. rewrite <- H2 in *. inv H1.
**inv H2. inv H1. inv H7. rewrite <- H1 in *. inv H3. destruct H4. eexists. eauto with IPPGrammar.
**inv H2. inv H1. inv H7. rewrite <- H1 in *. inv H3.
+ inv H0. apply Infix_cf; auto. intro. inv H0. inv H1. inv E2. inv H1. inv H7. inv H1. icp_cases H0; inv H2.
* destruct H4. eexists. eauto with IPPGrammar.
* apply insert_pre_top_unchanged with pr o t21 x0 in H11; auto. destruct H11.
**inv H1. inv H2. inv H8. rewrite <- H2 in *. inv H7. destruct H4. eexists. eauto with IPPGrammar.
**inv H1. inv H2. inv H8. rewrite <- H2 in *. inv H7.
+ apply Prefix_cf; auto. intro. inv H2. inv H3. icp_cases H2; inv H4. inv H5. contradiction.
- simpl. apply Prefix_cf; auto. intro. inv H1. inv H2. icp_cases H1; inv H3. inv H4.
- insert_pre_pnode_destruct pr o t21 o2.
+ inv H0. apply Postfix_cf; auto. inv E. inv H0. intro. inv H0. inv H5. icp_cases H0; inv H6. inv H7. inv H2.
* inv H6. decompose [or] (insert_pre_top pr o t21); rewrite <- H5 in *.
** inv H2.
** inv H6. inv H2. inv H8. destruct H3. eexists. eauto with IPPGrammar.
** inv H6. inv H2. inv H8.
* apply insert_pre_top_unchanged with pr o t21 x in H10; auto.
destruct H10; rewrite <- H5 in *.
** inv H2. inv H6. inv H7. destruct H3. eexists. eauto with IPPGrammar.
** inv H2. inv H6. inv H7.
+ apply Prefix_cf; auto.
intro. inv H2. inv H3. icp_cases H2; inv H4. inv H5.
Qed.
Lemma repair_in_icfree {g} (pr : drules g) t1 o t2 :
safe_pr pr ->
i_conflict_free (i_conflict_pattern pr) t1 ->
i_conflict_free (i_conflict_pattern pr) t2 ->
i_conflict_free (i_conflict_pattern pr) (repair_in pr t1 o t2).
Proof.
intro. revert o t2. induction t1 as [l1|t11 ? o1 t12|o1 t12|t11 ? o1]; intros; simpl.
- apply insert_in_icfree; auto with IPPGrammar. intros. intro. inv H3.
- inv H0. auto.
- inv H0. apply insert_pre_icfree; auto.
- apply insert_in_icfree; auto with IPPGrammar. intros. intro. inv H3.
Qed.
Lemma insert_post_top {g} (pr : drules g) t1 o :
matches (insert_post pr t1 o) (PostfixPatt HPatt o) \/
(exists o1, matches t1 (InfixPatt HPatt o1 HPatt) /\ matches (insert_post pr t1 o) (InfixPatt HPatt o1 HPatt)) \/
(exists o1, matches t1 (PrefixPatt o1 HPatt) /\ matches (insert_post pr t1 o) (PrefixPatt o1 HPatt)).
Proof.
destruct t1; eauto with IPPGrammar.
- insert_post_inode_destruct pr t1_1 o0 t1_2 o; eauto 7 with IPPGrammar.
- insert_post_pnode_destruct pr o0 t1 o; eauto 7 with IPPGrammar.
Qed.
Lemma insert_post_top_unchanged {g} (pr : drules g) t1 o x :
matches_rm t1 (PrefixPatt x HPatt) ->
rm_conflict_pattern pr (CL_postfix_prefix o x) ->
(exists o1, matches t1 (InfixPatt HPatt o1 HPatt) /\ matches (insert_post pr t1 o) (InfixPatt HPatt o1 HPatt)) \/
(exists o1, matches t1 (PrefixPatt o1 HPatt) /\ matches (insert_post pr t1 o) (PrefixPatt o1 HPatt)).
Proof.
intros. destruct t1.
- inv H. inv H1.
- inv H. inv H1. left.
eexists. split; auto with IPPGrammar.
insert_post_inode_destruct pr t1_1 o0 t1_2 o; auto with IPPGrammar. exfalso. apply H with x; auto with IPPGrammar.
- right. eexists. split; auto with IPPGrammar.
insert_post_pnode_destruct pr o0 t1 o; auto with IPPGrammar. exfalso. apply H1 with x; auto.
- inv H. inv H1.
Qed.
Lemma insert_post_icfree {g} (pr : drules g) t1 o :
safe_pr pr ->
i_conflict_free (i_conflict_pattern pr) t1 ->
i_conflict_free (i_conflict_pattern pr) (insert_post pr t1 o).
Proof.
intros. induction t1 as [l1|t11 ? o1 t12|o1 t12|t11 ? o1].
- simpl. apply Postfix_cf; auto. intro. inv H1. inv H2. icp_cases H1; inv H3. inv H4.
- insert_post_inode_destruct pr t11 o1 t12 o.
+ inv H0. apply Infix_cf; auto. intro. inv H0. inv H1. icp_cases H0; inv H2.
* decompose [or] (insert_post_top pr t12 o).
**inv H12. rewrite <- H2 in *. inv H1.
**inv H2. inv H1. inv H12. rewrite <- H1 in *. inv H3. destruct H4. eexists. eauto with IPPGrammar.
**inv H2. inv H1. inv H12. rewrite <- H1 in *. inv H3.
* inv H12. destruct H4. eexists. eauto with IPPGrammar.
+ inv H0. apply Infix_cf; auto. intro. inv H0. inv H1. inv E2. inv H1. inv H7. inv H1. icp_cases H0; inv H2.
* apply insert_post_top_unchanged with pr t12 o x0 in H11; auto. destruct H11.
**inv H1. inv H2. inv H7. rewrite <- H2 in *. inv H14. destruct H4. eexists. eauto with IPPGrammar.
**inv H1. inv H2. inv H7. rewrite <- H2 in *. inv H14.
* destruct H4. eexists. eauto with IPPGrammar.
+ apply Postfix_cf; auto. intro. inv H2. inv H3. icp_cases H2; inv H4. inv H5. contradiction.
- insert_post_pnode_destruct pr o1 t12 o.
+ inv H0. apply Prefix_cf; auto. inv E. inv H0. intro. inv H0. inv H5. icp_cases H0; inv H6. inv H7. inv H2.
* inv H6. decompose [or] (insert_post_top pr t12 o); rewrite <- H5 in *.
**inv H2.
**inv H6. inv H2. inv H8. destruct H3. eexists. eauto with IPPGrammar.
**inv H6. inv H2. inv H8.
* apply insert_post_top_unchanged with pr t12 o x in H10; auto.
destruct H10; rewrite <- H5 in *.
**inv H2. inv H6. inv H7. destruct H3. eexists. eauto with IPPGrammar.
**inv H2. inv H6. inv H7.
+ apply Postfix_cf; auto. intro. inv H2. inv H3. icp_cases H2; inv H4. inv H5.
- simpl. apply Postfix_cf; auto. intro. inv H1. inv H2. icp_cases H1; inv H3. inv H4.
Qed.
Lemma repair_icfree {g} (pr : drules g) t :
safe_pr pr ->
i_conflict_free (i_conflict_pattern pr) (repair pr t).
Proof.
intro. induction t; simpl; auto using repair_in_icfree, insert_pre_icfree, insert_post_icfree with IPPGrammar.
Qed.
(*
##############################################
##############################################
##############################################
*)
(* Proving safety of deep rm conflict patterns. *)
Lemma insert_in_matches_rm {g} (pr : drules g) t1 o1 t2 o2 :
matches_rm (insert_in pr t1 o1 t2) (PrefixPatt o2 HPatt) ->
matches_rm t2 (PrefixPatt o2 HPatt).
Proof.
intros. destruct t2.
- simpl in H. inv H. inv H0. inv H4. inv H.
- insert_in_inode_destruct pr o1 t2_1 o t2_2.
+ inv H. inv H0. auto with IPPGrammar.
+ inv H. inv H0. auto with IPPGrammar.
+ inv H. inv H1. assumption.
- simpl in H. inv H. inv H0. assumption.
- insert_in_pnode_destruct pr o1 t2 o.
+ inv H. inv H0.
+ inv H. inv H1. inv H5. inv H.
Qed.
Lemma insert_in_drmcfree {g} (pr : drules g) t1 o t2 :
safe_pr pr ->
drm_conflict_free (rm_conflict_pattern pr) t1 ->
drm_conflict_free (rm_conflict_pattern pr) t2 ->
(forall x, rm_conflict_pattern pr (CL_infix_prefix o x) -> ~ matches_rm t1 (PrefixPatt x HPatt)) ->
drm_conflict_free (rm_conflict_pattern pr) (insert_in pr t1 o t2).
Proof.
intros. induction t2 as [l2|t21 ? o2 t22|o2 t22|t21 ? o2].
- simpl. apply Infix_drmcf; auto. intro. inv H3. inv H4. rcp_cases H3; inv H5. eapply H2; eauto.
- insert_in_inode_destruct pr o t21 o2 t22.
+ inv H1. apply Infix_drmcf; auto. intro. inv H1. inv H3. rcp_cases H1; inv H4.
destruct H6. eexists. eauto using insert_in_matches_rm with IPPGrammar.
+ inv H1. apply Infix_drmcf; auto. intro. inv H1. inv H3. rcp_cases H1; inv H4.
destruct H6. eexists. eauto using insert_in_matches_rm with IPPGrammar.
+ apply Infix_drmcf; auto with IPPGrammar. intro. inv H4. inv H5. rcp_cases H4; inv H6. eapply H2; eauto.
- simpl. apply Infix_drmcf; auto with IPPGrammar. intro. inv H3. inv H4. rcp_cases H3; inv H5. eapply H2; eauto.
- insert_in_pnode_destruct pr o t21 o2.
+ inv H1. apply Postfix_drmcf; auto. intro. inv H1. inv H3. rcp_cases H1; inv H4.
destruct H5. eexists. eauto using insert_in_matches_rm with IPPGrammar.
+ apply Infix_drmcf; auto with IPPGrammar. intro. inv H4. inv H5. rcp_cases H4; inv H6. eapply H2; eauto.
Qed.
Lemma prefixnode_single_drmcfree {g} (pr : drules g) o l2 :
drm_conflict_free (rm_conflict_pattern pr) (PrefixNode o (AtomicNode l2)).
Proof.
apply Prefix_drmcf; auto using Atomic_drmcf. intro. destruct H as [q]. inv H. rcp_cases H0; inv H1.
Qed.
Lemma insert_pre_matches_rm {g} (pr : drules g) o t2 o2 :
matches_rm (insert_pre pr o t2) (PrefixPatt o2 HPatt) ->
matches_rm t2 (PrefixPatt o2 HPatt) \/ o2 = o.
Proof.
intros. destruct t2.
- simpl in H. inv H.
+ inv H0. auto.
+ inv H3. inv H.
- insert_pre_inode_destruct pr o t2_1 o0 t2_2.
+ inv H. inv H0. left. auto with IPPGrammar.
+ inv H. inv H0. left. auto with IPPGrammar.
+ inv H; auto. inv H1. auto.
- inv H; auto. inv H0. auto.
- insert_pre_pnode_destruct pr o t2 o0.
+ inv H. inv H0.
+ inv H; auto. inv H1. auto.
Qed.
Lemma insert_pre_drmcfree {g} (pr : drules g) o t2 :
safe_pr pr ->
drm_conflict_free (rm_conflict_pattern pr) t2 ->
drm_conflict_free (rm_conflict_pattern pr) (insert_pre pr o t2).
Proof.
intros. induction t2 as [l2|t21 ? o2 t22|o2 t22|t21 ? o2].
- simpl. apply prefixnode_single_drmcfree.
- insert_pre_inode_destruct pr o t21 o2 t22.
+ inv H0. apply Infix_drmcf; auto. intro. inv H0. inv H1. rcp_cases H0; inv H2.
apply insert_pre_matches_rm in H7. inv H7.
* destruct H4. eexists. eauto with IPPGrammar.
* eapply safe_infix_prefix; eauto using CPrio_infix_prefix.
+ inv H0. apply Infix_drmcf; auto. intro. inv H0. inv H1. rcp_cases H0; inv H2. inv E2. inv H1. inv H3. inv H1.
destruct t21.
**inv H11. inv H1.
**inv H11. inv H1. rename E into E'. insert_pre_inode_destruct pr o t21_1 o0 t21_2.
***inv H7. inv H1. destruct H4. eexists. eauto with IPPGrammar.
***inv H7. inv H1. destruct H4. eexists. eauto with IPPGrammar.
***eapply H1; eauto with IPPGrammar.
**inv H11. inv H1.
**rename E into E'. insert_pre_pnode_destruct pr o t21 o0.
***inv H7. inv H1.
***eapply H1; eauto.
+ apply Prefix_drmcf; auto. intro. inv H2. inv H3. rcp_cases H2; inv H4.
- simpl. apply Prefix_drmcf; auto. intro. inv H1. inv H2. rcp_cases H1; inv H3.
- insert_pre_pnode_destruct pr o t21 o2.
+ inv H0. apply Postfix_drmcf; auto. intro. inv H0. inv H1. rcp_cases H0; inv H2. inv E. inv H1. inv H6.
* inv H1. destruct t21.
**simpl in H5. inv H5.
***inv H1. eauto using safe_prefix_postfix.
***inv H9. inv H1.
**insert_pre_inode_destruct pr o t21_1 o0 t21_2.
***inv H5. inv H1. destruct H3. eexists. eauto with IPPGrammar.
***inv H5. inv H1. destruct H3. eexists. eauto with IPPGrammar.
***inv H5.
****inv H6. eauto using safe_prefix_postfix.
****inv H10. inv H5. destruct H3. eexists. eauto with IPPGrammar.
**simpl in H5. inv H5.
***inv H1. eauto using safe_prefix_postfix.
***destruct H3. eexists. eauto with IPPGrammar.
**insert_pre_pnode_destruct pr o t21 o0.
***inv H5. inv H1.
***inv H5.
****inv H6. eauto using safe_prefix_postfix.
****inv H10. inv H5.
* destruct t21.
**inv H9. inv H1.
**inv H9. inv H1. insert_pre_inode_destruct pr o t21_1 o0 t21_2.
***inv H5. inv H1. destruct H3. eexists. eauto with IPPGrammar.
***inv H5. inv H1. destruct H3. eexists. eauto with IPPGrammar.
***inv H5.
****inv H6. eapply H1; eauto with IPPGrammar.
****inv H9. inv H5. destruct H3. eexists. eauto with IPPGrammar.
**inv H9. inv H1.
**insert_pre_pnode_destruct pr o t21 o0.
***inv H5. inv H1.
***eapply H1; eauto.
+ apply Prefix_drmcf; auto. intro. inv H2. inv H3. rcp_cases H2; inv H4.
Qed.
Lemma postfixnode_single_drmcfree {g} (pr : drules g) o l1 :
drm_conflict_free (rm_conflict_pattern pr) (PostfixNode (AtomicNode l1) o).
Proof.
apply Postfix_drmcf; auto using Atomic_drmcf. intro. destruct H as [q]. inv H. rcp_cases H0; inv H1. inv H2. inv H.
Qed.
Lemma insert_post_drmcfree {g} (pr : drules g) t1 o :
safe_pr pr ->
drm_conflict_free (rm_conflict_pattern pr) t1 ->
drm_conflict_free (rm_conflict_pattern pr) (insert_post pr t1 o).
Proof.
intros. induction t1 as [l1|t11 ? o1 t12|o1 t12|t11 ? o1].
- simpl. apply Postfix_drmcf; auto. intro. inv H1. inv H2. rcp_cases H1; inv H3. inv H4. inv H2.
- insert_post_inode_destruct pr t11 o1 t12 o.
+ inv H0. apply Infix_drmcf; auto. intro. inv H0. inv H1. rcp_cases H0; inv H2.
destruct H4. eexists. eauto with IPPGrammar.
+ inv H0. apply Infix_drmcf; auto. intro. inv H0. inv H1. rcp_cases H0; inv H2.
destruct H4. eexists. eauto with IPPGrammar.
+ apply Postfix_drmcf; auto. intro. inv H2. inv H3. rcp_cases H2; inv H4. inv H5. inv H3.
eapply H1; eauto with IPPGrammar.
- insert_post_pnode_destruct pr o1 t12 o.
+ inv H0. apply Prefix_drmcf; auto. intro. inv H0. inv H1. rcp_cases H0; inv H2.
+ apply Postfix_drmcf; auto. intro. inv H2. inv H3. rcp_cases H2; inv H4. eapply H1; eauto.
- simpl. apply Postfix_drmcf; auto. intro. inv H1. inv H2. rcp_cases H1; inv H3. inv H4. inv H2.
Qed.
Lemma repair_in_drmcfree {g} (pr : drules g) t1 o t2 :
safe_pr pr ->
drm_conflict_free (rm_conflict_pattern pr) t1 ->
drm_conflict_free (rm_conflict_pattern pr) t2 ->
drm_conflict_free (rm_conflict_pattern pr) (repair_in pr t1 o t2).
Proof.
intro. revert o t2. induction t1 as [l1|t11 ? o1 t12|o1 t12|t11 ? o1]; intros; simpl.
- apply insert_in_drmcfree; auto with IPPGrammar. intros. intro. inv H3. inv H4.
- inv H0. auto.
- inv H0. apply insert_pre_drmcfree; auto.
- apply insert_in_drmcfree; auto with IPPGrammar. intros. intro. inv H3. inv H4.
Qed.
Lemma repair_drmcfree {g} (pr : drules g) t :
safe_pr pr ->
drm_conflict_free (rm_conflict_pattern pr) (repair pr t).
Proof.
intro. induction t; simpl; auto using repair_in_drmcfree, insert_pre_drmcfree, insert_post_drmcfree with IPPGrammar.
Qed.
(*
##############################################
##############################################
##############################################
*)
(* Proving safety of deep lm conflict patterns. *)
Lemma insert_in_dlmcfree {g} (pr : drules g) t1 o t2 :
safe_pr pr ->
dlm_conflict_free (lm_conflict_pattern pr) t1 ->
dlm_conflict_free (lm_conflict_pattern pr) t2 ->
dlm_conflict_free (lm_conflict_pattern pr) (insert_in pr t1 o t2).
Proof.
intros. induction t2 as [l2|t21 ? o2 t22|o2 t22|t21 ? o2].
- simpl. apply Infix_dlmcf; auto. intro. inv H2. inv H3. lcp_cases H2; inv H4. inv H11. inv H3.
- insert_in_inode_destruct pr o t21 o2 t22.
+ inv H1. apply Infix_dlmcf; auto. intro. inv H1. inv H2. lcp_cases H1; inv H3.
destruct H5. eexists. eauto with IPPGrammar.
+ inv H1. apply Infix_dlmcf; auto. intro. inv H1. inv H2. lcp_cases H1; inv H3.
destruct H5. eexists. eauto with IPPGrammar.
+ apply Infix_dlmcf; auto. intro. inv H3. inv H4. lcp_cases H3; inv H5. inv H12. inv H4.
eapply H2; eauto with IPPGrammar.
- simpl. apply Infix_dlmcf; auto with IPPGrammar. intro. inv H2. inv H3. lcp_cases H2; inv H4. inv H11. inv H3.
- insert_in_pnode_destruct pr o t21 o2.
+ inv H1. apply Postfix_dlmcf; auto. intro. inv H1. inv H2. lcp_cases H1; inv H3.
+ apply Infix_dlmcf; auto with IPPGrammar. intro. inv H3. inv H4. lcp_cases H3; inv H5. eapply H2; eauto.
Qed.
Lemma insert_pre_dlmcfree {g} (pr : drules g) o t2 :
safe_pr pr ->
dlm_conflict_free (lm_conflict_pattern pr) t2 ->
dlm_conflict_free (lm_conflict_pattern pr) (insert_pre pr o t2).
Proof.
intros. induction t2 as [l2|t21 ? o2 t22|o2 t22|t21 ? o2].
- simpl. apply Prefix_dlmcf; auto. intro. inv H1. inv H2. lcp_cases H1; inv H3. inv H4. inv H2.
- insert_pre_inode_destruct pr o t21 o2 t22.
+ inv H0. apply Infix_dlmcf; auto. intro. inv H0. inv H1. lcp_cases H0; inv H2.
destruct H4. eexists. eauto with IPPGrammar.
+ inv H0. apply Infix_dlmcf; auto. intro. inv H0. inv H1. lcp_cases H0; inv H2.
destruct H4. eexists. eauto with IPPGrammar.
+ apply Prefix_dlmcf; auto. intro. inv H2. inv H3. lcp_cases H2; inv H4. inv H5. inv H3.
eapply H1; eauto with IPPGrammar.
- simpl. apply Prefix_dlmcf; auto. intro. inv H1. inv H2. lcp_cases H1; inv H3. inv H4. inv H2.
- insert_pre_pnode_destruct pr o t21 o2.
+ inv H0. apply Postfix_dlmcf; auto. intro. inv H0. inv H1. lcp_cases H0; inv H2.
+ apply Prefix_dlmcf; auto with IPPGrammar. intro. inv H2. inv H3. lcp_cases H2; inv H4. eapply H1; eauto.
Qed.
Lemma postfixnode_single_dlmcfree {g} (pr : drules g) o l1 :
dlm_conflict_free (lm_conflict_pattern pr) (PostfixNode (AtomicNode l1) o).
Proof.
apply Postfix_dlmcf; auto with IPPGrammar. intro. destruct H as [q]. inv H. lcp_cases H0; inv H1.
Qed.
Lemma insert_post_matches_lm {g} (pr : drules g) o t1 o1 :
matches_lm (insert_post pr t1 o) (PostfixPatt HPatt o1) ->
matches_lm t1 (PostfixPatt HPatt o1) \/ o1 = o.
Proof.
intros. destruct t1.
- simpl in H. inv H.
+ inv H0. auto.
+ inv H3. inv H.
- insert_post_inode_destruct pr t1_1 o0 t1_2 o.
+ inv H. inv H0. left. auto with IPPGrammar.
+ inv H. inv H0. left. auto with IPPGrammar.
+ inv H; auto. inv H1. auto.
- insert_post_pnode_destruct pr o0 t1 o.
+ inv H. inv H0.
+ inv H; auto. inv H1; auto.
- inv H; auto. inv H0. auto.
Qed.
Lemma insert_post_dlmcfree {g} (pr : drules g) t1 o :
safe_pr pr ->
dlm_conflict_free (lm_conflict_pattern pr) t1 ->
dlm_conflict_free (lm_conflict_pattern pr) (insert_post pr t1 o).
Proof.
intros. induction t1 as [l1|t11 ? o1 t12|o1 t12|t11 ? o1].
- simpl. apply postfixnode_single_dlmcfree.
- insert_post_inode_destruct pr t11 o1 t12 o.
+ inv H0. apply Infix_dlmcf; auto. intro. inv H0. inv H1. lcp_cases H0; inv H2.
apply insert_post_matches_lm in H12. inv H12.
* destruct H4. eexists. eauto with IPPGrammar.
* eauto using safe_infix_postfix.
+ inv H0. apply Infix_dlmcf; auto. intro. inv H0. inv H1. lcp_cases H0; inv H2. inv E2. inv H1. inv H3. inv H1.
destruct t12.
**inv H11. inv H1.
**inv H11. inv H1. rename E into E'. insert_post_inode_destruct pr t12_1 o0 t12_2 o.
***inv H12. inv H1. destruct H4. eexists. eauto with IPPGrammar.
***inv H12. inv H1. destruct H4. eexists. eauto with IPPGrammar.
***eapply H1; eauto with IPPGrammar.
**rename E into E'. insert_post_pnode_destruct pr o0 t12 o.
***inv H12. inv H1.
***eapply H1; eauto.
**inv H11. inv H1.
+ apply Postfix_dlmcf; auto. intro. inv H2. inv H3. lcp_cases H2; inv H4.
- insert_post_pnode_destruct pr o1 t12 o.
+ inv H0. apply Prefix_dlmcf; auto. intro. inv H0. inv H1. lcp_cases H0; inv H2. inv E. inv H1. inv H6.
* inv H1. destruct t12.
**simpl in H5. inv H5.
***inv H1. eauto using safe_prefix_postfix.
***inv H9. inv H1.
**insert_post_inode_destruct pr t12_1 o0 t12_2 o.
***inv H5. inv H1. destruct H3. eexists. eauto with IPPGrammar.
***inv H5. inv H1. destruct H3. eexists. eauto with IPPGrammar.
***inv H5.
****inv H6. eauto using safe_prefix_postfix.
****inv H10. inv H5. destruct H3. eexists. eauto with IPPGrammar.
**insert_post_pnode_destruct pr o0 t12 o.
***inv H5. inv H1.
***inv H5.
****inv H6. eauto using safe_prefix_postfix.
****inv H10. inv H5.
**simpl in H5. inv H5.
***inv H1. eauto using safe_prefix_postfix.
***destruct H3. eexists. eauto with IPPGrammar.
* destruct t12.
**inv H9. inv H1.
**inv H9. inv H1. insert_post_inode_destruct pr t12_1 o0 t12_2 o.
***inv H5. inv H1. destruct H3. eexists. eauto with IPPGrammar.
***inv H5. inv H1. destruct H3. eexists. eauto with IPPGrammar.
***inv H5.
****inv H6. eapply H1; eauto with IPPGrammar.
****inv H9. inv H5. destruct H3. eexists. eauto with IPPGrammar.