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Proof State:
```
2 goals
A : Set a : A l : list A
H : forall (a : A) (l1 : list A),
In (a, l1) (split_one l) -> Permutation (a :: l1) l
a0 : A l1 : list A
H0 : (a, l) = (a0, l1)
============================
Permutation (a0 :: l1) (a :: l)
goal 2 is:
Permutation (a0 :: l1) (a :: l)
```
Commit Diff:
```
diff --git a/Div.v b/Div.v
index 7e7d694..e83d601 100644
--- a/Div.v
+++ b/Div.v
@@ -57,2 +57,2 @@ n) p) (c2 := mod n p); try apply mod_lt; auto with arith.
-rewrite mult_plus_distr_r; rewrite <- plus_assoc; repeat
-rewrite <- div_mod_correct; auto with arith.
+rewrite mult_plus_distr_r, <- plus_assoc; repeat rewrite
+<- div_mod_correct; auto with arith.
@@ -67,4 +67,2 @@ intros n m p H; apply plus_reg_l with (div (p * m + n) p * p).
-rewrite <- div_mod_correct; auto.
-rewrite div_mult_comp; auto.
-rewrite mult_plus_distr_r; rewrite (mult_comm p); rewrite <- plus_assoc.
-eq_tac; apply div_mod_correct; auto.
+rewrite <- div_mod_correct, div_mult_comp, mult_plus_distr_r, (mult_comm p), <- plus_assoc by assumption.
+f_equal. now apply div_mod_correct.
diff --git a/ListAux.v b/ListAux.v
index 82a8d82..986e88b 100644
--- a/ListAux.v
+++ b/ListAux.v
@@ -76 +76 @@ exists (nil (A:=A)); exists l; simpl; auto.
-eq_tac; auto.
+f_equal; auto.
@@ -78 +78 @@ case H; auto; intros l1 [l2 Hl2]; exists (a1 :: l1); exists l2; simpl; auto.
-eq_tac; auto.
+f_equal; auto.
@@ -146 +146 @@ intros a l H l2 l3 l4 a0; case l3; simpl.
-intros H0; left; exists l; eq_tac; injection H0; auto.
+intros H0; left; exists l; f_equal; injection H0; auto.
@@ -150 +150 @@ intros [l5 H1].
-left; exists l5; eq_tac; injection H0; auto.
+left; exists l5; f_equal; injection H0; auto.
@@ -165,2 +165,2 @@ intros H0; right; right; injection H0; split; auto.
-eq_tac; auto.
-intros b0 l0 H0; left; exists l0; injection H0; intros; (repeat eq_tac); auto.
+f_equal; auto.
+intros b0 l0 H0; left; exists l0; injection H0; intros; (repeat f_equal); auto.
@@ -169 +169 @@ injection H0; auto.
-intros [l5 HH1]; left; exists l5; eq_tac; auto; injection H0; auto.
+intros [l5 HH1]; left; exists l5; f_equal; auto; injection H0; auto.
@@ -171 +171 @@ intros [H1|[H1 H2]]; auto.
-right; right; split; auto; eq_tac; auto; injection H0; auto.
+right; right; split; auto; f_equal; auto; injection H0; auto.
@@ -190 +190 @@ exists (b :: l4); exists l5; exists b1; (repeat (simpl; split; auto)).
-eq_tac; auto.
+f_equal; auto.
@@ -237 +237 @@ intros H4 H5; split; auto.
-eq_tac; auto.
+f_equal; auto.
@@ -278 +278 @@ intros A l1 l2; elim l1; simpl; auto.
-intros a l H; rewrite length_app; rewrite length_map; rewrite H; auto.
+intros a l H; rewrite length_app, length_map, H; auto.
diff --git a/ListOp.v b/ListOp.v
index b9797d9..e1936ca 100644
--- a/ListOp.v
+++ b/ListOp.v
@@ -138,2 +138 @@ intros a b c; elim c; simpl; auto.
-intros n H; rewrite jump_nil; rewrite
-take_nil; rewrite H; auto with arith.
+intros n H; rewrite jump_nil, take_nil, H; auto with arith.
@@ -146,2 +145,2 @@ intros k Rec i j s H;
-rewrite length_app; rewrite length_take1; auto with arith.
-eq_tac; auto.
+rewrite length_app, length_take1; auto with arith.
+f_equal; auto.
@@ -199 +198 @@ intros l; elim l; simpl; auto with datatypes.
-intros; eq_tac; auto.
+intros; f_equal; auto.
@@ -206 +205 @@ intros H; contradict H; auto with arith.
-intros n1 H; eq_tac; auto with arith.
+intros n1 H; f_equal; auto with arith.
@@ -219 +218 @@ intros a l1 Rec n; case n; auto; clear n.
-simpl length; intros n H; simpl; eq_tac; auto with arith.
+simpl length; intros n H; simpl; f_equal; auto with arith.
diff --git a/OrderedList.v b/OrderedList.v
index f2762b8..3136749 100644
--- a/OrderedList.v
+++ b/OrderedList.v
@@ -624 +624 @@ intros; discriminate.
-intros; eq_tac; auto.
+intros; f_equal; auto.
diff --git a/Permutation.v b/Permutation.v
index ecfa699..3f52951 100644
--- a/Permutation.v
+++ b/Permutation.v
@@ -33,2 +32,0 @@ Variable A : Set.
-Definition permutation (l1 l2 : list A) := @Permutation A l1 l2.
-
@@ -37,66 +34,0 @@ Hint Constructors Permutation : core.
-Definition permutation_trans := perm_trans.
-
-
-
-
-
-Theorem permutation_refl : forall l : list A, permutation l l.
-Proof.
-simple induction l.
-apply perm_nil.
-intros a l1 H.
-apply perm_skip with (1 := H).
-Qed.
-Hint Resolve permutation_refl : core.
-
-
-
```
Ongoing Proof History:
```
Proof.
intros a l1 l2; generalize a l1; elim l2; clear a l1 l2; simpl in |- *; auto.
intros a l1 H1; case H1.
intros a l H a0 l1 [H0| H0].
? injection H0; intros H1 H2; rewrite H2; rewrite H1; auto.
? generalize H H0; elim (split_one l); simpl in |- *; auto.
? intros H1 H2; case H2.
```
Next Tactic:
```
|
injection H0; intros H1 H2; rewrite H2, H1; auto.
```
|
{"final_commit":"82eb0678991955dfa8f12720d652ad5adb465c3e","project":"sudoku","repair_file":"\/data\/coq-pearls\/repairs\/sudoku\/repair-sudoku-b5543771d6bdeab13ec58ac6c8d281815d96ed62-82eb0678991955dfa8f12720d652ad5adb465c3e-8_15_2-8_15_2-35.git.json","initial_commit":"b5543771d6bdeab13ec58ac6c8d281815d96ed62","file":"Permutation.v"}
|
Ongoing Proof History:
```
Proof.
intros n m p Hp H.
apply gcd_unique; trivial.
-
apply H.
apply divide_refl.
```
Proof State:
```
This subproof is complete, but there are some unfocused goals.
Focus next goal with bullet -.
2 subgoals
subgoal 1 is:
(p | m)
subgoal 2 is:
forall q : t, (q | n) -> (q | m) -> (q | p)
```
Next Tactic:
```
|
-
```
|
{"package":"coq","file":"\/home\/tpr\/.opam\/PackageTable~8.13~2022.01.csv.formatted\/lib\/coq\/theories\/Numbers\/NatInt\/NZGcd.v"}
|
Proof State:
```
1 subgoal
E : Type -> Type R : Type S : Type
classicT : forall P : Type, P + (P -> False)
r : itrace E S -> itrace E R -> Prop
CIH : forall (b : itrace E R) (t : itree E S) (f : S -> itree E R),
b β ITree.bind t f -> r (peel b t) b
b : itrace E R t : itree E S f : S -> itree E R
A0 : Type e1 : E A0 ans : A0 k : unit -> itree (EvAns E) R
k0 : A0 -> itree E R t0 : itree E S k' : A0 -> itree E S
Hvis : eqit_ eq true true id (upaco2 (eqit_ eq true true id) bot2)
(Vis e1 (fun x : A0 => ITree.bind (k' x) f))
(Vis e1 k0)
Href : rutt_ (REvRef E) (RAnsRef E) eq
(upaco2 (rutt_ (REvRef E) (RAnsRef E) eq) bot2)
(Vis (evans A0 e1 ans) k)
(Vis e1 (fun x : A0 => ITree.bind (k' x) f))
Ht0 : eqitF eq true true id (upaco2 (eqit_ eq true true id) bot2)
(observe t0) (VisF e1 k')
H6 : forall (a : unit) (b : A0),
RAnsRef E unit A0 (evans A0 e1 ans) a e1 b ->
upaco2 (rutt_ (REvRef E) (RAnsRef E) eq) bot2
(k a) (ITree.bind (k' b) f)
k4 : A0 -> itree E R
REL : forall v : A0, paco2 (eqit_ eq true true id) bot2 (k4 v) (k0 v)
============================
trace_prefixF (upaco2 trace_prefix_ r)
(observe (peel_ (VisF (evans A0 e1 ans) k) (observe t0)))
(VisF (evans A0 e1 ans) k)
```
Ongoing Proof History:
```
cbn in *.
destruct (observe t0) eqn : Heqt0; try discriminate.
+
right.
exists r0.
pfold.
red.
cbn.
rewrite Heqt0.
- auto.
+ auto with itree.
+
injection x as Ht1.
symmetry in Heqt0.
apply simpobs in Heqt0.
setoid_rewrite Heqt0.
setoid_rewrite tau_eutt.
eapply IHHvis; eauto.
rewrite Ht1.
auto.
}
destruct H as [ [B [k' [e1 Ht0] ] ] | [s Ht0] ].
+
rewrite Heqt in Href.
rewrite tau_eutt in Href.
rewrite Ht0 in Href.
rewrite bind_vis in Href.
pinversion Href.
subst; ddestruction; subst.
rewrite Ht0 in Hvis.
rewrite bind_vis in Hvis.
pinversion Hvis.
subst; ddestruction; subst.
clear Heqt Heqb.
punfold Ht0.
red in Ht0.
cbn in *.
destruct e.
*
inv H1.
ddestruction; subst.
cbn.
constructor.
? eapply trace_prefix_vis_evans; eauto.
? *
? eapply trace_prefix_vis_evempty; eauto.
```
Commit Diff:
```
diff --git a/theories/Basics/Basics.v b/theories/Basics/Basics.v
index ff16faf..1f991e2 100644
--- a/theories/Basics/Basics.v
+++ b/theories/Basics/Basics.v
@@ -57,0 +58,2 @@ Definition state (s a : Type) := s -> prod s a.
+Definition run_stateT {s m a} (x : stateT s m a) : s -> m (s * a)%type := x.
+
diff --git a/theories/Basics/HeterogeneousRelations.v b/theories/Basics/HeterogeneousRelations.v
index 8894184..ad37cca 100644
--- a/theories/Basics/HeterogeneousRelations.v
+++ b/theories/Basics/HeterogeneousRelations.v
@@ -751 +751,21 @@ cbn in *. unfold diagonal_prop in *; tauto.
-Qed.
\ No newline at end of file
+Qed.
+
+
+
+
+
+Definition option_rel {X : Type} (R :
+relation X) : relation (option X) :=
+fun mx my => match mx,my with
+| Some x, Some y => R
+x y | None, None => True
+| _, _ => False end.
+Hint Unfold option_rel : core.
+
+Lemma option_rel_eq : forall {A : Type},
+eq_rel (@eq (option A)) (option_rel eq).
+Proof.
+intros ?; split; intros [] [] EQ; subst; try inv EQ; cbn; auto.
+Qed.
+
+#[global] Hint Unfold option_rel : core.
\ No newline at end of file
diff --git a/theories/Core/ITreeDefinition.v b/theories/Core/ITreeDefinition.v
index 3f88726..eff8f67 100644
--- a/theories/Core/ITreeDefinition.v
+++ b/theories/Core/ITreeDefinition.v
@@ -69,0 +70,2 @@ Arguments itreeF _ _ : clear implicits.
+Create HintDb itree.
+
diff --git a/theories/Core/KTreeFacts.v b/theories/Core/KTreeFacts.v
index a1c2431..9949acf 100644
--- a/theories/Core/KTreeFacts.v
+++ b/theories/Core/KTreeFacts.v
@@ -113 +113 @@ intros ? ? ?. red.
-induction 1; auto.
+induction 1; auto with itree.
diff --git a/theories/Dijkstra/ITreeDijkstra.v b/theories/Dijkstra/ITreeDijkstra.v
```
Next Tactic:
```
|
eapply <LOOKUP>trace_prefix_vis_evans : forall (E : Type -> Type) (R S : Type) (r : itrace E S -> itrace E R -> Prop) (A0 : Type) (ev : E A0) (ans : A0) (k : unit -> itree (EvAns E) R) (k' : A0 -> itree E S) (t0 : itree E S) (f : S -> itree E R), (forall (a : unit) (b : A0), RAnsRef E unit A0 (evans A0 ev ans) a ev b -> id (upaco2 (rutt_ (REvRef E) (RAnsRef E) eq) bot2) (k a) (ITree.bind (k' b) f)) -> t0 β Vis ev k' -> (forall (b : itrace E R) (t : itree E S) (f0 : S -> itree E R), b β ITree.bind t f0 -> r (peel b t) b) -> trace_prefixF (upaco2 trace_prefix_ r) (observe (peel_ (VisF (evans A0 ev ans) k) (observe t0))) (VisF (evans A0 ev ans) k)</LOOKUP>; eauto with itree.
```
|
{"final_commit":"8fcf6f5f81ec9289d56e777b3885e715b3328486","project":"InteractionTrees","repair_file":"\/data\/coq-pearls\/repairs\/InteractionTrees\/repair-InteractionTrees-8447a4e3e92c94e5ec1f23f35b6ecd63fae0c880-8fcf6f5f81ec9289d56e777b3885e715b3328486-8_10_2-8_12_2-83.git.json","initial_commit":"8447a4e3e92c94e5ec1f23f35b6ecd63fae0c880","file":"theories\/ITrace\/ITraceBind.v"}
|
Ongoing Proof History:
```
Proof.
induction p.
-
intro x.
simpl.
destruct (stop (f x)) eqn:des1.
exists xH.
split.
reflexivity.
split.
```
Proof State:
```
3 subgoals
X : Type f : X β X stop : X β bool p : positive
IHp : β x : X,
β q : positive,
iterate_stop p x β‘ iterate q x
β§ (β r : positive, (r < q)%positive β stop (iterate r x) β‘ false)
β§ (p β‘ q β¨ (q < p)%positive β§ stop (iterate q x) β‘ true)
x : X
des1 : stop (f x) β‘ true
============================
β r : positive, (r < 1)%positive β stop (iterate r x) β‘ false
subgoal 2 is:
(p~1)%positive β‘ 1%positive β¨ (1 < p~1)%positive β§ stop (iterate 1 x) β‘ true
subgoal 3 is:
β q : positive,
(if stop (iterate_stop p (f x))
then iterate_stop p (f x)
else iterate_stop p (iterate_stop p (f x))) β‘
iterate q x
β§ (β r : positive, (r < q)%positive β stop (iterate r x) β‘ false)
β§ ((p~1)%positive β‘ q β¨ (q < p~1)%positive β§ stop (iterate q x) β‘ true)
```
Next Tactic:
```
|
intros.
```
|
{"package":"coq-corn","file":"\/home\/tpr\/.opam\/PackageTable~8.13~2022.01.csv.formatted\/lib\/coq\/user-contrib\/CoRN\/reals\/fast\/CRstreams.v"}
|
Proof State:
```
1 subgoal
A : Type t1, t2 : btree A l1, l2, l3 : list (btree A)
H : ordered_cover l1 t1
H0 : forall l2 : list (btree A),
length l1 = length l2 -> exists t2 : btree A, subst_pred l1 l2 t1 t2
H1 : ordered_cover l2 t2
H2 : forall l3 : list (btree A),
length l2 = length l3 -> exists t3 : btree A, subst_pred l2 l3 t2 t3
l0 : list (btree A)
H3 : length (l1 ++ l2) = length l0
t4 : btree A
HH1 : subst_pred l1 (firstn (length l1) l0) t1 t4
============================
exists t0 : btree A, subst_pred (l1 ++ l2) l0 (node t1 t2) t0
```
Ongoing Proof History:
```
Proof using.
intros t1 l1 l2 H; generalize l2; elim H; clear t1 l1 l2 H.
intros t l l2; case l2.
simpl in |- *; intros; discriminate.
intros b l0; case l0; simpl in |- *; auto.
intros H; exists b; auto.
intros; discriminate.
intros t1 t2 l1 l2 l3 H H0 H1 H2 l0 H3.
- case (H0 (first_n l0 (length l1))); auto.
+ case (H0 (firstn (length l1) l0)); auto.
- rewrite first_n_length; auto; (rewrite <- H3; rewrite length_app; auto with arith).
+ rewrite firstn_le_length_eq; auto; (rewrite <- H3; rewrite app_length; auto with arith).
intros t4 HH1.
? case (H2 (skip_n l0 (length l1))); auto.
? rewrite skip_n_length; auto; (rewrite <- H3; rewrite length_app; rewrite minus_plus; auto with arith).
? intros t5 HH2.
```
Commit Diff:
```
diff --git a/theories/Aux.v b/theories/Aux.v
index c7e8186..355e02d 100644
--- a/theories/Aux.v
+++ b/theories/Aux.v
@@ -132 +132,2 @@ End LeBool.
-Section fold.
+Section Fold.
+
@@ -137,7 +137,0 @@ Variable h : A -> A.
-Variable eqA_dec : forall a b : A, {a = b} + {a <> b}.
-
-Theorem fold_left_app : forall a l1 l2, fold_left f (l1 ++
-l2) a = fold_left f l2 (fold_left f l1 a).
-Proof using.
-intros a l1; generalize a; elim l1; simpl in |- *; auto; clear a l1.
-Qed.
@@ -161,8 +154,0 @@ Qed.
-Theorem fold_right_app : forall a l1 l2,
-fold_right g a (l1 ++ l2) =
-fold_right g (fold_right g a l2) l1.
-Proof using.
-intros a l1; generalize a; elim l1; simpl in |- *; auto; clear a l1.
-intros a l H a0 l2; rewrite H; auto.
-Qed.
-
@@ -178 +164 @@ Qed.
-End fold.
+End Fold.
@@ -181,0 +168 @@ Section List.
+
@@ -227,31 +213,0 @@ Qed.
-
-Theorem length_app : forall l1 l2 : list A, length
-(l1 ++ l2) = length l1 + length l2.
-Proof using.
-intros l1; elim l1; simpl in |- *; auto.
-Qed.
-
-Theorem app_inv_head : forall l1 l2 l3 : list A, l1
-++ l2 = l1 ++ l3 -> l2 = l3.
-Proof using.
-intros l1; elim l1; simpl in |- *; auto.
-intros a l H l2 l3 H0; apply H; injection H0; auto.
-Qed.
-
-Theorem app_inv_tail : forall l1 l2 l3 : list A, l2
-++ l1 = l3 ++ l1 -> l2 = l3.
-Proof using.
-intros l1 l2; generalize l1; elim l2; clear l1 l2; simpl in |- *; auto.
-intros l1 l3; case l3; auto.
-intros b l H; absurd (length ((b :: l) ++ l1) <= length l1).
-simpl in |- *; rewrite length_app; auto with arith.
-rewrite <- H; auto with arith.
-intros a l H l1 l3; case l3.
-simpl in |- *; intros H1; absurd (length (a :: l ++ l1) <= length l1).
-simpl in |- *; rewrite length_app; auto with arith.
-rewrite H1; auto with arith.
-simpl in |- *; intros b l0 H0; injection H0.
-intros H1 H2; apply f_equal2 with (f := cons (A:=A)); auto.
-apply H with (1 := H1); auto.
-Qed.
-
@@ -341,4 +296,0 @@ Qed.
-Theorem length_map : forall l, length (map f l) = length l.
-Proof using.
-intros l; elim l; simpl in |- *; auto.
-Qed.
@@ -346,8 +298 @@ Qed.
-Theorem map_app : forall l1 l2, map f (l1 ++ l2) = map f l1 ++ map f l2.
-Proof using.
-intros l; elim l; simpl in |- *; auto.
-intros a l0 H l2; apply f_equal2 with (f := cons (A:=B)); auto.
-Qed.
-
-
-Theorem in_flat_map : forall (l : list B) (f :
+Theorem in_flat_map_in : forall (l : list B) (f :
@@ -357,4 +302 @@ Proof using.
-intros l g; elim l; simpl in |- *; auto.
-intros a l0 H a0 b H0 [H1| H1]; apply in_or_app; auto.
-left; rewrite H1; auto.
-right; apply H with (b := b); auto.
+intros; apply in_flat_map; exists b; split; auto.
@@ -367,6 +309 @@ Proof using.
-intros l g; elim l; simpl in |- *; auto.
-intros a H; case H.
-intros a l0 H a0 H0; case in_app_or with (1 := H0); simpl in |- *; auto.
-intros H1; exists a; auto.
-intros H1; case H with (1 := H1).
-intros b (H2, H3); exists b; simpl in |- *; auto.
+intros; apply in_flat_map; auto.
@@ -416,14 +353,3 @@ Variable A : Type.
-Fixpoint first_n (l : list
-A) (n : nat) {struct
-n} : list A :=
-match n with | O
-=> [] | S n1
-=> match l with |
-[] => [] | a
-:: l1 => a ::
```
Next Tactic:
```
|
case (H2 (<LOOKUP>skipn : forall A : Type, nat -> list A -> list A</LOOKUP> (<LOOKUP>length : forall A : Type, list A -> nat</LOOKUP> l1) l0)); auto.
```
|
{"final_commit":"b6befe10be8f4acefcd6424a3eec659dbd39c0c4","project":"huffman","repair_file":"\/data\/coq-pearls\/repairs\/huffman\/repair-huffman-6e3f37b2113e34f74c6b5a6ff2c7346a9cdf9e4e-b6befe10be8f4acefcd6424a3eec659dbd39c0c4-8_11_2-8_11_2-18.git.json","initial_commit":"6e3f37b2113e34f74c6b5a6ff2c7346a9cdf9e4e","file":"theories\/SubstPred.v"}
|
Proof State:
```
1 subgoal
X, Y : MetricSpace
XY := ProductMS X Y : MetricSpace
f : X --> Complete Y
plX : PrelengthSpace X
plFEX : PrelengthSpace (FinEnum X)
x : Complete X y : Complete Y s : Compact X
H0 : inCompact x s
H1 : msp_eq y (Cbind plX f x)
============================
inCompact (Couple (PairMS x (Cbind plX f x))) (CompactGraph_b plFEX s)
```
Ongoing Proof History:
```
Proof.
intros plX plFEX x y s.
split; intros H.
split; rewrite <- (CoupleCorrect2 x y).
apply (@CompactGraph_b_correct2 plFEX).
exact H.
symmetry.
transitivity (Csnd (Couple (x,y))).
refine (CompactGraph_b_correct3 _ _).
apply H.
apply CoupleCorrect3.
destruct H as [H0 H1].
change (x, y) with (PairMS x y).
rewrite -> H1.
```
Next Tactic:
```
|
apply <LOOKUP>CompactGraph_b_correct1 : forall [plX : PrelengthSpace X] (plFEX : PrelengthSpace (FinEnum X)) [x : Complete X] [s : Compact X], inCompact x s -> inCompact (Couple (x, Cbind plX f x)) (CompactGraph_b plFEX s)</LOOKUP>.
```
|
{"package":"coq-corn","file":"\/home\/tpr\/.opam\/PackageTable~8.13~2022.01.csv.formatted\/lib\/coq\/user-contrib\/CoRN\/metric2\/Graph.v"}
|
Commit Diff:
```
diff --git a/theories/Basics/CategoryFacts.v b/theories/Basics/CategoryFacts.v
index 732b8d2..a684e54 100644
--- a/theories/Basics/CategoryFacts.v
+++ b/theories/Basics/CategoryFacts.v
@@ -121,0 +122,13 @@ Qed.
+Context {t : obj}.
+Context {Terminal_t : Terminal C t}.
+Context {TerminalObject_t : TerminalObject C t}.
+
+
+Lemma terminal_unique : forall a (f g
+: C a t), f β©― g.
+Proof.
+intros.
+rewrite (terminal_object f), (terminal_object g).
+reflexivity.
+Qed.
+
@@ -402,0 +416,281 @@ End CoproductFacts.
+
+
+
+Section ProductFacts.
+
+Context {obj : Type} {C : Hom obj}.
+
+Context {Eq2_C : Eq2 C}.
+Context {E_Eq2_C : forall a b, @Equivalence (C a b) eq2}.
+
+Context {Id_C : Id_ C} {Cat_C : Cat C}.
+
+Context {Category_C : Category C}.
+
+Context {bif : binop obj}.
+Context {Prod_C : Pair C bif}
+{Fst_C : Fst C bif}
+{Snd_C : Snd C bif}.
+Context {Product_C : Product C bif}.
+
+Lemma pair_fst' {a b c d} (ca : C c a) (cb : C c b) (dc
+: C d c) : dc >>> pair_ ca cb >>> fst_ β©― dc >>> ca.
+Proof.
+rewrite cat_assoc, pair_fst. reflexivity.
+Qed.
+
+Lemma pair_snd' {a b c d} (ca : C c a) (cb : C c b) (dc
+: C d c) : dc >>> pair_ ca cb >>> snd_ β©― dc >>> cb.
+Proof.
+rewrite cat_assoc, pair_snd. reflexivity.
+Qed.
+
+
+Lemma pair_cat {a b c d} (ab : C a b) (bc : C b c) (bd :
+C b d) : (ab >>> pair_ bc bd β©― pair_ (ab >>> bc) (ab >>> bd)).
+Proof.
+apply pair_universal.
+- rewrite pair_fst'; reflexivity.
+- rewrite pair_snd'; reflexivity.
+Qed.
+
+
+Corollary pair_eta {a b} : id_ (bif a b) β©― pair_ fst_ snd_.
+Proof.
+apply pair_universal; rewrite cat_id_l; reflexivity.
+Qed.
+
+Lemma pair_eta' {a b c} (f : C a (bif b
+c)) : f β©― pair_ (f >>> fst_) (f >>> snd_).
+Proof.
+eapply pair_universal; reflexivity.
+Qed.
+
+
+
+Lemma pair_split {a b c} (f g : C
+a (bif b c)) : (f >>> fst_
+β©― g >>> fst_) -> (f >>> snd_
+β©― g >>> snd_) -> f β©― g.
+Proof.
+intros. rewrite (pair_eta' g).
+apply pair_universal; assumption.
+Qed.
+
+Existing Instance Bimap_Product.
+Existing Instance Swap_Product.
+Existing Instance AssocR_Product.
+Existing Instance AssocL_Product.
+Existing Instance UnitL_Product.
+Existing Instance UnitL'_Product.
+Existing Instance UnitR_Product.
+Existing Instance UnitR'_Product.
+
+Ltac unfold_product
+:= unfold
+bimap, Bimap_Product,
+assoc_r, AssocR_Product,
+assoc_l, AssocL_Product,
+unit_l, UnitL_Product,
+unit_l', UnitL'_Product,
+unit_r, UnitR_Product,
+unit_r', UnitR'_Product,
+swap, Swap_Product.
+
+
+Ltac cat_auto_simpl_prod := match goal with
+| [ |- eq2 ?lhs ?rhs
+] => repeat (rewrite cat_id_l ||
+rewrite cat_id_r || rewrite pair_fst ||
+rewrite <- (cat_assoc (pair_ _ _)
+fst_), pair_fst || rewrite pair_snd ||
+rewrite <- (cat_assoc (pair_ _ _)
+snd_), pair_snd || rewrite !cat_assoc); reflexivity
```
Ongoing Proof History:
```
Proof.
unfold map_default_eq, interp_map.
repeat intro.
revert x y H s1 s2 H0.
einit.
ecofix CH.
intros.
rewrite! unfold_interp_state.
punfold H0.
red in H0.
revert s1 s2 H1.
induction H0; intros; subst; simpl; pclearbot.
-
eret.
-
etau.
-
ebind.
apply pbc_intro_h with (RU := prod_rel (@eq_map _ _ _ _ d) eq).
{
? unfold over.
? destruct (case e).
? -
```
Proof State:
```
1 subgoal
K, V : Type map : Type M : Map K V map MOk : MapOk eq M
Kdec : RelDec eq
KdecOk : RelDec_Correct Kdec
R : Type E : Type -> Type d : V RR : R -> R -> Prop
gL' : itree E (map * R) -> itree E (map * R) -> Prop
INCL : forall x0 x1 : itree E (map * R), bot2 x0 x1 -> gL' x0 x1
gH' : itree E (map * R) -> itree E (map * R) -> Prop
INCH : forall x0 x1 : itree E (map * R), bot2 x0 x1 -> gH' x0 x1
CHL : forall x x0 : itree (mapE K d +' E) R,
eutt RR x x0 ->
forall x2 x3 : map,
eq_map x2 x3 ->
gL' (interp_state (case_ (handle_map (V:=V)) pure_state) x x2)
(interp_state (case_ (handle_map (V:=V)) pure_state) x0 x3)
CHH : forall x x0 : itree (mapE K d +' E) R,
eutt RR x x0 ->
forall x2 x3 : map,
eq_map x2 x3 ->
gH' (interp_state (case_ (handle_map (V:=V)) pure_state) x x2)
(interp_state (case_ (handle_map (V:=V)) pure_state) x0 x3)
x, y : itree (mapE K d +' E) R u : Type e : (mapE K d +' E) u
k1, k2 : u -> itree (mapE K d +' E) R
REL : forall v : u, paco2 (eqit_ RR true true id) bot2 (k1 v) (k2 v)
s1, s2 : map
H1 : eq_map s1 s2
============================
eqit (prod_rel eq_map eq) true true
(case_ (handle_map (V:=V)) pure_state u e s1)
(case_ (handle_map (V:=V)) pure_state u e s2)
```
Next Tactic:
```
|
unfold case_.
```
|
{"final_commit":"81a1ac45e6eac7285942c92f0079e632d7200904","project":"InteractionTrees","repair_file":"\/data\/coq-pearls\/repairs\/InteractionTrees\/repair-InteractionTrees-440b18da2c80560f3454b2cbc1736209a76d64ad-81a1ac45e6eac7285942c92f0079e632d7200904-8_10_2-8_10_2-3.git.json","initial_commit":"440b18da2c80560f3454b2cbc1736209a76d64ad","file":"theories\/Events\/MapDefaultFacts.v"}
|
Ongoing Proof History:
```
Proof.
intro.
```
Proof State:
```
1 subgoal
I : interval pI : proper I F, G : PartFunct IR
diffF : Diffble I pI F diffG : Diffble I pI G
X : bnd_away_zero_in_P G I
============================
Diffble I pI (F {/} G)
```
Next Tactic:
```
|
apply <LOOKUP>Diffble_wd : forall (I : interval) (pI : proper I) (F H : PartFunct IR), Feq I F H -> Diffble I pI F -> Diffble I pI H</LOOKUP> with (F{*}{1/}G).
```
|
{"package":"coq-corn","file":"\/home\/tpr\/.opam\/PackageTable~8.13~2022.01.csv.formatted\/lib\/coq\/user-contrib\/CoRN\/ftc\/MoreFunctions.v"}
|
Ongoing Proof History:
```
Proof.
intros C HnotEM.
set (R := fun A b => A /\ true = b \/ ~ A /\ false = b).
assert (H : exists f : Prop -> bool, (forall A:Prop, R A (f A))).
apply unique_choice.
intro A.
destruct (classic A) as [Ha| Hnota].
exists true; split.
left; split; [ assumption | reflexivity ].
intros y [[_ Hy]| [Hna _]].
assumption.
contradiction.
exists false; split.
right; split; [ assumption | reflexivity ].
intros y [[Ha _]| [_ Hy]].
contradiction.
assumption.
```
Proof State:
```
1 subgoal
C : Prop
HnotEM : (forall P : Prop, {P} + {~ P}) -> C
R := fun (A : Prop) (b : bool) => A /\ true = b \/ ~ A /\ false = b
: Prop -> bool -> Prop
H : exists f : Prop -> bool, forall A : Prop, R A (f A)
============================
C
```
Next Tactic:
```
|
destruct H as [f Hf].
```
|
{"package":"coq","file":"\/home\/tpr\/.opam\/PackageTable~8.13~2022.01.csv.formatted\/lib\/coq\/theories\/Logic\/ClassicalUniqueChoice.v"}
|
Ongoing Proof History:
```
Proof.
repeat red.
intros.
subst.
split.
-
revert_until y1.
pcofix CIH.
intros.
pstep.
- punfold H1.
+ punfold H0.
- red in H1.
+ red in H0.
red.
? hinduction H1 before CIH; intros; eauto.
? +
? apply EqRet.
```
Proof State:
```
1 subgoal
E : Type -> Type R1 : Type R2 : Type x, y : relationH R1 R2
H : eq_rel x y
y0, y1 : bool r : rel2 (itree E R1) (fun _ : itree E R1 => itree E R2)
CIH0 : forall (a0 : itree E R1) (a1 : itree E R2), bot2 a0 a1 -> r a0 a1
CIH : forall (x0 : itree E R1) (x1 : itree E R2),
eqit x y0 y1 x0 x1 -> r x0 x1
y2 : itree E R1 y3 : itree E R2
H0 : eqitF x y0 y1 id (upaco2 (eqit_ x y0 y1 id) bot2)
(observe y2) (observe y3)
============================
eqitF y y0 y1 id (upaco2 (eqit_ y y0 y1 id) r) (observe y2) (observe y3)
```
Commit Diff:
```
diff --git a/theories/Core/ITreeDefinition.v b/theories/Core/ITreeDefinition.v
index b8604b4..1d0cd49 100644
--- a/theories/Core/ITreeDefinition.v
+++ b/theories/Core/ITreeDefinition.v
@@ -253,2 +253,2 @@ t2)) (at level 61, t1 at next level, right associativity) : itree_scope.
-Notation "` x : t <- t1 ;; t2" := (ITree.bind t1 (fun x : t => t2))
-(at level 61, t at next level, t1 at next level, x ident, right associativity) : itree_scope.
+Notation "` x : t <- t1 ;; t2" := (ITree.bind t1 (fun x : t => t2)) (at
+level 61, t at next level, t1 at next level, x ident, right associativity, only parsing) : itree_scope.
diff --git a/theories/Core/KTreeFacts.v b/theories/Core/KTreeFacts.v
index b911a90..32f56a6 100644
--- a/theories/Core/KTreeFacts.v
+++ b/theories/Core/KTreeFacts.v
@@ -6,2 +6 @@ Require Import
-Program Classes.Morphisms
-Setoids.Setoid
+Classes.Morphisms Setoids.Setoid
@@ -24 +23,2 @@ Eq.Eq
-Eq.UpToTaus.
+Eq.UpToTaus
+Eq.Paco2.
@@ -68 +68 @@ Proof.
-ginit. gcofix CIH. intros.
+ginit. pcofix CIH. intros.
@@ -205,2 +205,2 @@ Proof.
-revert A B C f g a0.
-ginit. gcofix CIH. intros.
+revert f g a0.
+ginit. pcofix CIH. intros.
@@ -268 +268 @@ revert a0.
-ginit. gcofix CIH. intros.
+ginit. pcofix CIH. intros.
@@ -277 +277 @@ revert a.
-gcofix CIH'. intros.
+pcofix CIH'. intros.
@@ -337,2 +337 @@ etau.
-specialize (CIH xa).
-cbn in CIH.
+specialize (CIHL xa). cbn in CIHL.
diff --git a/theories/Eq/Eq.v b/theories/Eq/Eq.v
index 21ac7e0..9f80143 100644
--- a/theories/Eq/Eq.v
+++ b/theories/Eq/Eq.v
@@ -29 +29,2 @@ Basics.Basics Basics.HeterogeneousRelations
-Core.ITreeDefinition.
+Core.ITreeDefinition
+Eq.Paco2.
@@ -165,0 +167,54 @@ Hint Unfold id: core.
+Lemma eqitF_VisF_inv_r {E R1 R2} (RR : R1 -> R2 -> Prop) {b1 b2 vclo sim}
+t1 X2 (e2 : E X2) (k2 : X2 -> _) : eqitF RR b1
+b2 vclo sim t1 (VisF e2 k2) -> (exists k1, t1 = VisF e2 k1
+/\ forall v, vclo sim (k1 v) (k2 v)) \/ (b1 /\ exists t1', t1
+= TauF t1' /\ eqitF RR b1 b2 vclo sim (observe t1') (VisF e2 k2)).
+Proof.
+refine (fun H => match H
+in eqitF _ _ _ _
+_ _ t2 return match t2
+return Prop with | VisF e2
+k2 => _ | _
+=> True end with |
+EqVis _ _ _ _
+_ _ _ _ _
+=> _ | _ =>
+_ end); try exact I.
+- left; eauto.
+- destruct i0; eauto.
+Qed.
+
+Lemma eqitF_VisF_inv {E R1 R2} (RR : R1 -> R2 -> Prop) {b1 b2 vclo sim} X1
+(e1 : E X1) (k1 : X1 -> _) X2 (e2 : E X2) (k2 : X2
+-> _) : eqitF RR b1 b2 vclo sim (VisF e1 k1) (VisF e2 k2) -> exists
+p : X1 = X2, eqeq E p e1 e2 /\ pweqeq (vclo sim) p k1 k2.
+Proof.
+refine (fun H => match H in
+eqitF _ _ _ _ _ t1
+t2 return match t1, t2 return Prop
+with | VisF e1 k1, VisF e2
+k2 => _ | _, _ =>
+True end with | EqVis _
+_ _ _ _ _ _
+_ _ => _ | _
+=> _ end); try exact I.
+- exists eq_refl; cbn; eauto.
+- destruct i; exact I.
+Qed.
+
+Lemma eqitF_VisF_inv_strong {E R1 R2} (RR : R1 -> R2 -> Prop) {b1 b2
+vclo sim} X (e : E X) (k1 : X -> _) (k2
+: X -> _) : eqitF RR b1 b2 vclo sim (VisF e
+k1) (VisF e k2) -> forall x, vclo sim (k1 x) (k2 x).
+Proof.
+intros H. dependent destruction H. assumption.
+Qed.
+
+Lemma eqitF_VisF {E R1 R2} {RR : R1 -> R2 -> Prop} {b1 b2 vclo sim} {X1
+X2} (p : X1 = X2) (e1 : E X1) (k1 : X1 -> _) (e2 :
+E X2) (k2 : X2 -> _) : eqeq E p e1 e2 -> pweqeq (vclo sim)
```
Next Tactic:
```
|
hinduction H0 before CIH; intros; eauto.
```
|
{"final_commit":"f3853150994e962978fa4cf0ed18dcb8c42e3262","project":"InteractionTrees","repair_file":"\/data\/coq-pearls\/repairs\/InteractionTrees\/repair-InteractionTrees-fbba5fd82470aa824ec0eab20c275063ca38fb75-f3853150994e962978fa4cf0ed18dcb8c42e3262-8_10_2-8_11_2-0.git.json","initial_commit":"fbba5fd82470aa824ec0eab20c275063ca38fb75","file":"theories\/Eq\/Eq.v"}
|
Proof State:
```
1 subgoal
x, y : t
H : y <= x
============================
match x ?= y with
| Lt => y
| _ => x
end == x
```
Ongoing Proof History:
```
Proof.
intros.
unfold max, gmax.
```
Next Tactic:
```
|
case <LOOKUP>compare_spec : forall x y : t, CompareSpec (x == y) (x < y) (y < x) (x ?= y)</LOOKUP>; auto with relations.
```
|
{"package":"coq","file":"\/home\/tpr\/.opam\/PackageTable~8.13~2022.01.csv.formatted\/lib\/coq\/theories\/Structures\/GenericMinMax.v"}
|
Commit Diff:
```
diff --git a/theories/Basics/Basics.v b/theories/Basics/Basics.v
index ff16faf..1f991e2 100644
--- a/theories/Basics/Basics.v
+++ b/theories/Basics/Basics.v
@@ -57,0 +58,2 @@ Definition state (s a : Type) := s -> prod s a.
+Definition run_stateT {s m a} (x : stateT s m a) : s -> m (s * a)%type := x.
+
diff --git a/theories/Basics/HeterogeneousRelations.v b/theories/Basics/HeterogeneousRelations.v
index 8894184..ad37cca 100644
--- a/theories/Basics/HeterogeneousRelations.v
+++ b/theories/Basics/HeterogeneousRelations.v
@@ -751 +751,21 @@ cbn in *. unfold diagonal_prop in *; tauto.
-Qed.
\ No newline at end of file
+Qed.
+
+
+
+
+
+Definition option_rel {X : Type} (R :
+relation X) : relation (option X) :=
+fun mx my => match mx,my with
+| Some x, Some y => R
+x y | None, None => True
+| _, _ => False end.
+Hint Unfold option_rel : core.
+
+Lemma option_rel_eq : forall {A : Type},
+eq_rel (@eq (option A)) (option_rel eq).
+Proof.
+intros ?; split; intros [] [] EQ; subst; try inv EQ; cbn; auto.
+Qed.
+
+#[global] Hint Unfold option_rel : core.
\ No newline at end of file
diff --git a/theories/Core/ITreeDefinition.v b/theories/Core/ITreeDefinition.v
index 3f88726..eff8f67 100644
--- a/theories/Core/ITreeDefinition.v
+++ b/theories/Core/ITreeDefinition.v
@@ -69,0 +70,2 @@ Arguments itreeF _ _ : clear implicits.
+Create HintDb itree.
+
diff --git a/theories/Core/KTreeFacts.v b/theories/Core/KTreeFacts.v
index a1c2431..9949acf 100644
--- a/theories/Core/KTreeFacts.v
+++ b/theories/Core/KTreeFacts.v
@@ -113 +113 @@ intros ? ? ?. red.
-induction 1; auto.
+induction 1; auto with itree.
diff --git a/theories/Dijkstra/ITreeDijkstra.v b/theories/Dijkstra/ITreeDijkstra.v
index 9529757..b25a16e 100644
--- a/theories/Dijkstra/ITreeDijkstra.v
+++ b/theories/Dijkstra/ITreeDijkstra.v
@@ -189 +189 @@ is_inf_cons (h : A) (t : stream A) : F t -> is_infF F (ConsF h t).
-Hint Constructors is_infF : core.
+Hint Constructors is_infF : itree.
@@ -198 +198 @@ Proof.
-red. intros. red in IN. red. induction IN; auto.
+red. intros. red in IN. red. induction IN; auto with itree.
@@ -217 +217 @@ s2 : stream A) : F s1 s2 -> bisimF F (ConsF h s1) (ConsF h s2).
-Hint Constructors bisimF : core.
+Hint Constructors bisimF : itree.
@@ -226 +226 @@ Proof.
-red. intros. red in IN. red. induction IN; auto.
+red. intros. red in IN. red. induction IN; auto with itree.
@@ -234 +234 @@ constructor; red.
-- pcofix CIH. intros. pfold. red. destruct (observe_stream x); auto.
+- pcofix CIH. intros. pfold. red. destruct (observe_stream x); auto with itree.
@@ -237 +237 @@ pfold. red.
-pinversion H0; subst; auto.
+pinversion H0; subst; auto with itree.
@@ -239 +239 @@ pinversion H0; subst; auto.
-pinversion H0; pinversion H1; auto.
+pinversion H0; pinversion H1; auto with itree.
@@ -250 +250 @@ pfold. red. unfold app. pinversion H12.
-- simpl. destruct s3. destruct s4. pinversion H34; simpl in *; subst; auto.
+- simpl. destruct s3. destruct s4. pinversion H34; simpl in *; subst; auto with itree.
@@ -282 +282 @@ stream' A -> Prop := | forall_nil : forall_streamF P F NilF | forall_cons (h : A
-Hint Constructors forall_streamF : core.
+Hint Constructors forall_streamF : itree.
@@ -289 +289 @@ Proof.
-red. intros. red. red in IN. destruct IN; auto.
+red. intros. red. red in IN. destruct IN; auto with itree.
@@ -301 +301 @@ F (observe_stream t) -> inf_manyF P F (ConsF h t) | cons_found (h : A) (t
-Hint Constructors inf_manyF : core.
+Hint Constructors inf_manyF : itree.
@@ -308 +308 @@ Proof.
-red. intros. red in IN. red. induction IN; auto.
+red. intros. red in IN. red. induction IN; auto with itree.
@@ -320,2 +320,2 @@ punfold Him. red in Him. pfold. red.
-induction Him; auto. pclearbot.
-auto.
+induction Him; auto with itree. pclearbot.
+auto with itree.
@@ -355 +355 @@ sim ot1 (observe t2) -> eqitEF RR sim ot1 (TauF t2).
-Hint Constructors eqitEF : core.
+Hint Constructors eqitEF : itree.
@@ -367 +367 @@ repeat red. intros. rename x0 into t1. rename x1 into t2.
-induction IN; eauto.
+induction IN; eauto with itree.
@@ -378 +378 @@ r) | eventlessTau (t : itree E R) : F t -> eventlessF F (TauF t).
-Hint Constructors eventlessF : core.
+Hint Constructors eventlessF : itree.
@@ -383 +383 @@ Prop) : itree E R -> Prop := fun t => eventlessF F (observe t).
-Hint Unfold eventless_ : core.
+Hint Unfold eventless_ : itree.
@@ -390 +390 @@ Proof.
-red. intros. red in IN. red. inversion IN; auto.
+red. intros. red in IN. red. inversion IN; auto with itree.
```
Proof State:
```
1 subgoal
E : Type -> Type R : Type eq_ : itree E R -> itree E R -> Prop
============================
Equivalence eq_ -> Equivalence (going eq_)
```
Ongoing Proof History:
```
Proof.
? intros []; constructor; cbv; eauto.
? -
? intros ? ? []; auto.
```
Next Tactic:
```
|
intros []; constructor; cbv; eauto with itree.
```
|
{"final_commit":"8fcf6f5f81ec9289d56e777b3885e715b3328486","project":"InteractionTrees","repair_file":"\/data\/coq-pearls\/repairs\/InteractionTrees\/repair-InteractionTrees-8447a4e3e92c94e5ec1f23f35b6ecd63fae0c880-8fcf6f5f81ec9289d56e777b3885e715b3328486-8_10_2-8_12_2-5.git.json","initial_commit":"8447a4e3e92c94e5ec1f23f35b6ecd63fae0c880","file":"theories\/Eq\/Shallow.v"}
|
Ongoing Proof History:
```
apply Rle_pow.
unfold C; apply RmaxLess1.
replace (4 * N)%nat with (2 * (2 * N))%nat; [ idtac | ring ].
apply (fun m n p:nat => mult_le_compat_l p n m).
replace (2 * N)%nat with (S (N + pred N)).
apply le_n_S.
apply plus_le_compat_l; assumption.
rewrite pred_of_minus.
lia.
apply Rle_trans with (sum_f_R0 (fun k:nat => sum_f_R0 (fun l:nat => C ^ (4 * N) * Rsqr (/ INR (fact (S (N + k))))) (pred (N - k))) (pred N)).
apply sum_Rle; intros.
apply sum_Rle; intros.
rewrite <- (Rmult_comm (C ^ (4 * N))).
apply Rmult_le_compat_l.
apply pow_le.
left; apply Rlt_le_trans with 1.
apply Rlt_0_1.
unfold C; apply RmaxLess1.
replace (/ INR (fact (2 * S (n0 + n)) * fact (2 * (N - n0)))) with (Binomial.C (2 * S (N + n)) (2 * S (n0 + n)) / INR (fact (2 * S (N + n)))).
apply Rle_trans with (Binomial.C (2 * S (N + n)) (S (N + n)) / INR (fact (2 * S (N + n)))).
unfold Rdiv; do 2 rewrite <- (Rmult_comm (/ INR (fact (2 * S (N + n))))).
apply Rmult_le_compat_l.
left; apply Rinv_0_lt_compat; apply INR_fact_lt_0.
apply C_maj.
lia.
right.
unfold Rdiv; rewrite Rmult_comm.
unfold Binomial.C.
unfold Rdiv; repeat rewrite <- Rmult_assoc.
rewrite <- Rinv_l_sym.
rewrite Rmult_1_l.
replace (2 * S (N + n) - S (N + n))%nat with (S (N + n)).
rewrite Rinv_mult_distr.
unfold Rsqr; reflexivity.
apply INR_fact_neq_0.
apply INR_fact_neq_0.
lia.
apply INR_fact_neq_0.
unfold Rdiv; rewrite Rmult_comm.
unfold Binomial.C.
unfold Rdiv; repeat rewrite <- Rmult_assoc.
rewrite <- Rinv_l_sym.
rewrite Rmult_1_l.
replace (2 * S (N + n) - 2 * S (n0 + n))%nat with (2 * (N - n0))%nat.
rewrite mult_INR.
reflexivity.
lia.
apply INR_fact_neq_0.
apply Rle_trans with (sum_f_R0 (fun k:nat => INR N / INR (fact (S N)) * C ^ (4 * N)) (pred N)).
apply sum_Rle; intros.
rewrite <- (scal_sum (fun _:nat => C ^ (4 * N)) (pred (N - n)) (Rsqr (/ INR (fact (S (N + n)))))).
rewrite sum_cte.
rewrite <- Rmult_assoc.
do 2 rewrite <- (Rmult_comm (C ^ (4 * N))).
rewrite Rmult_assoc.
apply Rmult_le_compat_l.
apply pow_le.
left; apply Rlt_le_trans with 1.
apply Rlt_0_1.
unfold C; apply RmaxLess1.
apply Rle_trans with (Rsqr (/ INR (fact (S (N + n)))) * INR N).
apply Rmult_le_compat_l.
apply Rle_0_sqr.
apply le_INR.
lia.
rewrite Rmult_comm; unfold Rdiv; apply Rmult_le_compat_l.
apply pos_INR.
apply Rle_trans with (/ INR (fact (S (N + n)))).
pattern (/ INR (fact (S (N + n)))) at 2; rewrite <- Rmult_1_r.
unfold Rsqr.
apply Rmult_le_compat_l.
left; apply Rinv_0_lt_compat; apply INR_fact_lt_0.
apply Rmult_le_reg_l with (INR (fact (S (N + n)))).
apply INR_fact_lt_0.
rewrite <- Rinv_r_sym.
rewrite Rmult_1_r.
apply (le_INR 1).
apply lt_le_S.
apply INR_lt; apply INR_fact_lt_0.
apply INR_fact_neq_0.
apply Rmult_le_reg_l with (INR (fact (S (N + n)))).
apply INR_fact_lt_0.
rewrite <- Rinv_r_sym.
apply Rmult_le_reg_l with (INR (fact (S N))).
apply INR_fact_lt_0.
rewrite Rmult_1_r.
rewrite (Rmult_comm (INR (fact (S N)))).
rewrite Rmult_assoc.
rewrite <- Rinv_l_sym.
rewrite Rmult_1_r.
apply le_INR.
apply fact_le.
apply le_n_S.
apply le_plus_l.
apply INR_fact_neq_0.
apply INR_fact_neq_0.
rewrite sum_cte.
apply Rle_trans with (C ^ (4 * N) / INR (fact (pred N))).
rewrite <- (Rmult_comm (C ^ (4 * N))).
unfold Rdiv; rewrite Rmult_assoc; apply Rmult_le_compat_l.
apply pow_le.
left; apply Rlt_le_trans with 1.
apply Rlt_0_1.
unfold C; apply RmaxLess1.
cut (S (pred N) = N).
intro; rewrite H0.
pattern N at 2; rewrite <- H0.
do 2 rewrite fact_simpl.
rewrite H0.
repeat rewrite mult_INR.
repeat rewrite Rinv_mult_distr.
rewrite (Rmult_comm (/ INR (S N))).
repeat rewrite <- Rmult_assoc.
rewrite <- Rinv_r_sym.
rewrite Rmult_1_l.
pattern (/ INR (fact (pred N))) at 2; rewrite <- Rmult_1_r.
rewrite Rmult_assoc.
apply Rmult_le_compat_l.
left; apply Rinv_0_lt_compat; apply INR_fact_lt_0.
apply Rmult_le_reg_l with (INR (S N)).
apply lt_INR_0; apply lt_O_Sn.
rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym.
rewrite Rmult_1_r; rewrite Rmult_1_l.
apply le_INR; apply le_n_Sn.
apply not_O_INR; discriminate.
apply not_O_INR.
red; intro; rewrite H1 in H; elim (lt_irrefl _ H).
apply not_O_INR.
red; intro; rewrite H1 in H; elim (lt_irrefl _ H).
```
Proof State:
```
5 subgoals
x, y : R N : nat
H : (0 < N)%nat
C := Rmax 1 (Rmax (Rabs x) (Rabs y)) : R
H0 : S (Init.Nat.pred N) = N
============================
INR (fact (Init.Nat.pred N)) <> 0
subgoal 2 is:
INR (S N) <> 0
subgoal 3 is:
INR N * INR (fact (Init.Nat.pred N)) <> 0
subgoal 4 is:
S (Init.Nat.pred N) = N
subgoal 5 is:
C ^ (4 * N) / INR (fact (Init.Nat.pred N)) <= Majxy x y (Init.Nat.pred N)
```
Next Tactic:
```
|
apply <LOOKUP>INR_fact_neq_0 : forall n : nat, INR (fact n) <> 0</LOOKUP>.
```
|
{"package":"coq","file":"\/home\/tpr\/.opam\/PackageTable~8.13~2022.01.csv.formatted\/lib\/coq\/theories\/Reals\/Cos_plus.v"}
|
Commit Diff:
```
diff --git a/Reachability.v b/Reachability.v
index 5c47635..f125ad9 100644
--- a/Reachability.v
+++ b/Reachability.v
@@ -18,5 +18,20 @@ Import Eval.Evaluator.
-
-Inductive reachability : Store -> Loc -> Loc ->Prop := |rch_heap : forall l Ο, l < (dom Ο) -> (reachability Ο
-l l) |rch_trans : forall l0 l1 l2 C Ο Ο, (reachability Ο l0 l1) -> (getObj Ο l1 = Some
-(C, Ο)) -> (exists f, (getVal Ο f = Some l2)) -> (l2 < dom Ο) -> (reachability Ο l0 l2).
-Notation "Ο β¨ l1 β l2" := (reachability Ο l1 l2) (at level 80, l1 at level 80, l2 at level 80).
+Reserved Notation "Ο β¨ l1 β l2" (at level 80, l1 at level 80, l2 at level 80).
+Inductive reachability : Store -> Loc
+-> Loc ->Prop := | rch_heap:
+forall l Ο, l < (dom
+Ο) -> Ο β¨ l β
+l | rch_step: forall l0 l1
+C f Ο Ο, l1 <
+(dom Ο) -> getObj Ο l0
+= Some (C, Ο) -> getVal
+Ο f = Some l1 ->
+Ο β¨ l0 β l1
+| rch_trans: forall l0 l1
+l2 Ο, Ο β¨ l0
+β l1 -> Ο β¨
+l1 β l2 -> Ο
+β¨ l0 β l2 where
+"Ο β¨ l1 β l2"
+:= (reachability Ο l1 l2).
+Hint Resolve rch_heap rch_step rch_trans: rch.
+Hint Rewrite update_dom: rch.
@@ -41,3 +56 @@ Proof.
-intros Ο l1 l2 l3 H1 H2.
-induction H2 => //.
-apply (rch_trans l1 l2 l3 C Ο Ο (IHreachability H1) H H0 H3).
+eauto using rch_trans.
@@ -44,0 +58 @@ Qed.
+
@@ -60 +74 @@ intros.
-induction H; steps.
+induction H; repeat steps || eapply_anywhere getObj_dom.
@@ -64,11 +78,11 @@ Qed.
-Lemma reachability_rev: forall Ο l l', Ο β¨ l
-β l' -> (l = l' /\ l <
-dom Ο) \/ (exists C Ο f l0, getObj
-Ο l = Some(C, Ο) /\ getVal Ο f
-= Some l0 /\ Ο β¨ l0 β l').
-Proof.
-intros.
-induction H; steps.
-+ right. repeat eexists; eauto using rch_heap.
-+ right. repeat eexists; eauto using rch_heap, rch_trans.
-Qed.
+
+
+
+
+
+
+
+
+
+
+
@@ -85,3 +98,0 @@ induction H2; eauto.
-steps.
-pose proof (H1 l1 l2 C Ο f).
-eapply H0; eauto.
@@ -99,25 +110,4 @@ intros Ο x C Ο l Hx Hobj.
-apply reachability_rev_ind; intros.
-+ left; rewrite_anywhere update_dom; eauto using rch_heap.
-+ steps; eauto using reachability_trans.
-+ destruct (PeanoNat.Nat.eq_dec l0 x) as [Heq | Hneq]; steps.
-++ move: (H1) => H1cp.
-rewrite_anywhere getObj_update1 => //.
-invert_constructor_equalities; subst.
-unfold getVal in *.
-assert (f < length Ο \/ f = length Ο) as [Hf | Hf]. {
-apply Lt.le_lt_or_eq, Lt.lt_n_Sm_le.
-pose proof (nth_error_Some (Ο ++ [l]) f) as Hf.
-rewrite app_length PeanoNat.Nat.add_1_r in Hf.
-apply Hf. steps.
-}
-+++ rewrite_anywhere nth_error_app1; steps.
-rewrite_anywhere update_dom.
-left. eapply rch_trans; eauto using rch_heap.
-+++ rewrite nth_error_app2 in H2; steps.
-rewrite PeanoNat.Nat.sub_diag in H2. simpl in H2.
-steps. rewrite_anywhere update_dom.
-right; eauto
-using rch_heap.
-++ rewrite_anywhere getObj_update2; steps.
-rewrite_anywhere update_dom.
-left. eapply rch_trans; eauto using rch_heap, getObj_dom.
+apply reachability_rev_ind; steps; eauto with rch;
+rewrite_anywhere update_dom; eauto with rch.
+eapply_anywhere getObj_update3; steps; eauto with rch.
+eapply_anywhere getVal_add; steps; eauto with rch.
@@ -127,2 +116,0 @@ Qed.
-
-
@@ -138,10 +126,3 @@ Proof.
```
Proof State:
```
1 subgoal
Ο : list (ClN * list Value) C : ClN L : LocSet l1 : Loc l : nat
H : S l <= dom Ο
C0 : ClN Ο : Env
H1 : S l1 <= S (dom Ο)
H2 : getObj (Ο ++ [(C, [])]) l1 = Some (C0, Ο)
f : nat
H0 : getVal Ο f = Some l
H4 : S l <= S (dom Ο)
============================
Ο β¨ l1 β l
```
Ongoing Proof History:
```
Proof.
intros.
inversion H0 as [l1 [Hl1 Hrch]].
exists l1; split => //.
apply reachable_path_reachability in Hrch as [Hrch | [p Hrch]].
+
- steps; eauto using rch_heap.
+ steps; eauto with rch.
+
clear Hl1 H0.
generalize dependent l.
generalize dependent l1.
induction p; intros.
++
steps; unfold reachable_one_step in *; steps.
- unfold dom in H1.
+ repeat rewrite_anywhere dom_app.
? repeat rewrite_anywhere app_length ; steps.
? pose proof (getObj_last_empty _ _ _ _ _ _ _ H2 H0) as [Hobj Hl1].
? eapply rch_trans; eauto using rch_trans, rch_heap.
```
Next Tactic:
```
|
eapply <LOOKUP>getObj_last_empty : forall (Ο : list (ClN * list Value)) (C C' : ClN) (Ο : Env) (l f : nat) (v : Value), getObj (Ο ++ [(C, [])]) l = Some (C', Ο) -> getVal Ο f = Some v -> getObj Ο l = Some (C', Ο) /\ l < dom Ο</LOOKUP> in H2; eauto; steps.
```
|
{"final_commit":"2dfd0b6c87b32ebd6bd1dd83e76e654e73dd3696","project":"celsius","repair_file":"\/data\/coq-pearls\/repairs\/celsius\/repair-celsius-a726b4d083864e7cb26d15bceeb49fbc418b2811-2dfd0b6c87b32ebd6bd1dd83e76e654e73dd3696-8_13_2-8_13_2-16.git.json","initial_commit":"a726b4d083864e7cb26d15bceeb49fbc418b2811","file":"Reachability.v"}
|
Ongoing Proof History:
```
Proof.
intros.
unfold decode_val in H.
revert H; case_eq (getN 4 i b); intros.
inv H.
unfold getN in H.
destruct l; inv H.
destruct (proj_bytes (ZMap.get i b :: ZMap.get (i + 1) b :: ZMap.get (i + 1 + 1) b :: ZMap.get (i + 1 + 1 + 1) b :: nil)) eqn:PB.
*
simpl proj_bytes in PB.
destruct (ZMap.get i b); inv PB.
destruct (ZMap.get (i+1) b); inv H2.
destruct (ZMap.get (i+1+1) b); inv H3.
destruct (ZMap.get (i+1+1+1) b); inv H2.
unfold decode_int in H1.
assert (Int.repr (int_of_bytes (rev_if_be (i0 :: i1 :: i2 :: i3 :: nil))) = Int.repr 0) by (forget (Int.repr (int_of_bytes (rev_if_be (i0 :: i1 :: i2 :: i3 :: nil)))) as foo; inv H1; auto).
clear H1.
assert (forall b0 b1 b2 b3, Int.repr (int_of_bytes (b0::b1::b2::b3::nil)) = Int.repr 0 -> (Byte.unsigned b0=0/\Byte.unsigned b1=0/\Byte.unsigned b2=0/\Byte.unsigned b3=0)).
clear.
intros.
simpl in H.
pose proof (Byte.unsigned_range b0).
pose proof (Byte.unsigned_range b1).
pose proof (Byte.unsigned_range b2).
pose proof (Byte.unsigned_range b3).
replace (Byte.modulus) with 256 in * by reflexivity.
pose proof (Int.unsigned_repr (Byte.unsigned b0 + (Byte.unsigned b1 + (Byte.unsigned b2 + (Byte.unsigned b3 + 0) * 256) * 256) * 256)).
spec H4.
clear H.
rewrite max_unsigned_eq; lia.
```
Proof State:
```
2 subgoals
b0, b1, b2, b3 : byte
H : Int.repr
(Byte.unsigned b0 +
(Byte.unsigned b1 +
(Byte.unsigned b2 + (Byte.unsigned b3 + 0) * 256) * 256) * 256) =
Int.repr 0
H0 : 0 <= Byte.unsigned b0 < 256
H1 : 0 <= Byte.unsigned b1 < 256
H2 : 0 <= Byte.unsigned b2 < 256
H3 : 0 <= Byte.unsigned b3 < 256
H4 : Int.unsigned
(Int.repr
(Byte.unsigned b0 +
(Byte.unsigned b1 +
(Byte.unsigned b2 + (Byte.unsigned b3 + 0) * 256) * 256) * 256)) =
Byte.unsigned b0 +
(Byte.unsigned b1 +
(Byte.unsigned b2 + (Byte.unsigned b3 + 0) * 256) * 256) * 256
============================
Byte.unsigned b0 = 0 /\
Byte.unsigned b1 = 0 /\ Byte.unsigned b2 = 0 /\ Byte.unsigned b3 = 0
subgoal 2 is:
nth (Z.to_nat (j - i)) (Byte i0 :: Byte i1 :: Byte i2 :: Byte i3 :: nil)
Undef = Byte Byte.zero
```
Next Tactic:
```
|
rewrite H in H4.
```
|
{"package":"coq-vst","file":"\/home\/tpr\/.opam\/PackageTable~8.13~2022.01.csv.formatted\/lib\/coq\/user-contrib\/VST\/veric\/initialize.v"}
|
Commit Diff:
```
diff --git a/theories/Basics/Basics.v b/theories/Basics/Basics.v
index ff16faf..1f991e2 100644
--- a/theories/Basics/Basics.v
+++ b/theories/Basics/Basics.v
@@ -57,0 +58,2 @@ Definition state (s a : Type) := s -> prod s a.
+Definition run_stateT {s m a} (x : stateT s m a) : s -> m (s * a)%type := x.
+
diff --git a/theories/Basics/HeterogeneousRelations.v b/theories/Basics/HeterogeneousRelations.v
index 8894184..ad37cca 100644
--- a/theories/Basics/HeterogeneousRelations.v
+++ b/theories/Basics/HeterogeneousRelations.v
@@ -751 +751,21 @@ cbn in *. unfold diagonal_prop in *; tauto.
-Qed.
\ No newline at end of file
+Qed.
+
+
+
+
+
+Definition option_rel {X : Type} (R :
+relation X) : relation (option X) :=
+fun mx my => match mx,my with
+| Some x, Some y => R
+x y | None, None => True
+| _, _ => False end.
+Hint Unfold option_rel : core.
+
+Lemma option_rel_eq : forall {A : Type},
+eq_rel (@eq (option A)) (option_rel eq).
+Proof.
+intros ?; split; intros [] [] EQ; subst; try inv EQ; cbn; auto.
+Qed.
+
+#[global] Hint Unfold option_rel : core.
\ No newline at end of file
diff --git a/theories/Core/ITreeDefinition.v b/theories/Core/ITreeDefinition.v
index 3f88726..eff8f67 100644
--- a/theories/Core/ITreeDefinition.v
+++ b/theories/Core/ITreeDefinition.v
@@ -69,0 +70,2 @@ Arguments itreeF _ _ : clear implicits.
+Create HintDb itree.
+
diff --git a/theories/Core/KTreeFacts.v b/theories/Core/KTreeFacts.v
index a1c2431..9949acf 100644
--- a/theories/Core/KTreeFacts.v
+++ b/theories/Core/KTreeFacts.v
@@ -113 +113 @@ intros ? ? ?. red.
-induction 1; auto.
+induction 1; auto with itree.
diff --git a/theories/Dijkstra/ITreeDijkstra.v b/theories/Dijkstra/ITreeDijkstra.v
index 9529757..b25a16e 100644
--- a/theories/Dijkstra/ITreeDijkstra.v
+++ b/theories/Dijkstra/ITreeDijkstra.v
@@ -189 +189 @@ is_inf_cons (h : A) (t : stream A) : F t -> is_infF F (ConsF h t).
-Hint Constructors is_infF : core.
+Hint Constructors is_infF : itree.
@@ -198 +198 @@ Proof.
-red. intros. red in IN. red. induction IN; auto.
+red. intros. red in IN. red. induction IN; auto with itree.
@@ -217 +217 @@ s2 : stream A) : F s1 s2 -> bisimF F (ConsF h s1) (ConsF h s2).
-Hint Constructors bisimF : core.
+Hint Constructors bisimF : itree.
@@ -226 +226 @@ Proof.
-red. intros. red in IN. red. induction IN; auto.
+red. intros. red in IN. red. induction IN; auto with itree.
@@ -234 +234 @@ constructor; red.
-- pcofix CIH. intros. pfold. red. destruct (observe_stream x); auto.
+- pcofix CIH. intros. pfold. red. destruct (observe_stream x); auto with itree.
@@ -237 +237 @@ pfold. red.
-pinversion H0; subst; auto.
+pinversion H0; subst; auto with itree.
@@ -239 +239 @@ pinversion H0; subst; auto.
-pinversion H0; pinversion H1; auto.
+pinversion H0; pinversion H1; auto with itree.
@@ -250 +250 @@ pfold. red. unfold app. pinversion H12.
-- simpl. destruct s3. destruct s4. pinversion H34; simpl in *; subst; auto.
+- simpl. destruct s3. destruct s4. pinversion H34; simpl in *; subst; auto with itree.
@@ -282 +282 @@ stream' A -> Prop := | forall_nil : forall_streamF P F NilF | forall_cons (h : A
-Hint Constructors forall_streamF : core.
+Hint Constructors forall_streamF : itree.
@@ -289 +289 @@ Proof.
-red. intros. red. red in IN. destruct IN; auto.
+red. intros. red. red in IN. destruct IN; auto with itree.
@@ -301 +301 @@ F (observe_stream t) -> inf_manyF P F (ConsF h t) | cons_found (h : A) (t
-Hint Constructors inf_manyF : core.
+Hint Constructors inf_manyF : itree.
@@ -308 +308 @@ Proof.
-red. intros. red in IN. red. induction IN; auto.
```
Proof State:
```
1 subgoal
E : Type -> Type A : Type B : Type R : A -> B -> Prop
a : A u : itree E A
FIN : a β u
IHFIN : forall u0 : itree E B, eutt R u u0 -> exists b : B, b β u0 /\ R a b
u2 : itree E B i : itree' E A
Heqi : i = TauF u
ou2 : itree' E B
Heqou2 : ou2 = observe u2
EQ : eqitF R true true id (upaco2 (eqit_ R true true id) bot2) i ou2
============================
exists b : B, b β u2 /\ R a b
```
Ongoing Proof History:
```
Proof.
intros * EQ FIN; revert u EQ.
induction FIN; intros u2 EQ.
-
punfold EQ.
red in EQ; rewrite H in EQ; clear H t.
remember (RetF a); genobs u2 ou.
hinduction EQ before R; intros; try now discriminate.
+ +
- inv Heqi; eauto.
+ inv Heqi; eauto with itree.
+ +
- edestruct IHEQ as (b & IN & HR); eauto.
+ edestruct IHEQ as (b & IN & HR); eauto with itree.
-
punfold EQ; red in EQ; rewrite H in EQ; clear H t.
remember (TauF u); genobs u2 ou2.
? hinduction EQ before R; intros; try discriminate; pclearbot; inv Heqi; eauto.
? edestruct IHFIN as (? & ? & ?); eauto.
? edestruct IHEQ as (? & ? & ?); eauto.
```
Next Tactic:
```
|
hinduction EQ before <LOOKUP>R : A -> B -> Prop</LOOKUP>; intros; try discriminate; pclearbot; inv Heqi.
```
|
{"final_commit":"8fcf6f5f81ec9289d56e777b3885e715b3328486","project":"InteractionTrees","repair_file":"\/data\/coq-pearls\/repairs\/InteractionTrees\/repair-InteractionTrees-1158ff1cbb628aeaa69973bf098058744d458c36-8fcf6f5f81ec9289d56e777b3885e715b3328486-8_10_2-8_12_2-92.git.json","initial_commit":"1158ff1cbb628aeaa69973bf098058744d458c36","file":"theories\/Props\/Leaf.v"}
|
Ongoing Proof History:
```
Proof.
intros until m0.
intros NR H8 ? ?.
destruct (eq_dec loc (nextblock m0, 0)).
*
subst loc.
unfold initial_core.
unfold inflate_initial_mem.
rewrite !resource_at_make_rmap.
unfold inflate_initial_mem'.
rewrite !resource_at_make_rmap.
rewrite nextblock_access_empty by (apply Pos2Nat.inj_ge; lia).
split; intros _; [ |apply NO_identity].
unfold Genv.alloc_global in H.
destruct (alloc m0 0 1) eqn:?.
assert (H9: 0 <= 0 < 1) by (clear; lia).
assert (H6 := alloc_result _ _ _ _ _ Heqp); subst b.
assert (H1 := perm_drop_1 _ _ _ _ _ _ H 0 Cur H9).
destruct (perm_mem_access _ _ _ _ H1) as [p [H4 H5]].
assert (H2 := perm_drop_2 _ _ _ _ _ _ H 0 Cur p H9).
rewrite H5.
unfold perm in *.
unfold access_at in H5.
simpl in H5.
destruct ((mem_access m) !! (nextblock m0) 0 Cur); inv H5.
spec H2; [constructor | ].
destruct p; try solve [inv H2].
unfold initial_core'.
simpl.
rewrite Genv.find_invert_symbol with (id:=i) by auto.
destruct (list_in_map_inv _ _ _ H0) as [[i' fd] [H10 H11]]; simpl in H10, H11.
subst i'.
rewrite find_id_i with (fs:=fd); auto.
destruct fd.
apply PURE_identity.
apply in_app.
right; auto.
*
clear NR.
unfold initial_core.
unfold inflate_initial_mem.
rewrite !resource_at_make_rmap.
unfold inflate_initial_mem'.
rewrite !resource_at_make_rmap.
pose proof (Pos.ltb_spec (fst loc) (nextblock m0)).
destruct ((fst loc <? nextblock m0)%positive); inv H1.
destruct (alloc_global_old _ _ _ _ H loc H2) as [? ?].
rewrite H3.
rewrite H1; split; intro; auto.
destruct loc as [b ofs].
simpl fst in *; simpl snd in *.
rewrite (nextblock_access_empty m0) by (apply Pos.le_ge; auto).
split; intros _; [ |apply NO_identity].
replace (access_at m (b,ofs) Cur) with (@None permission).
apply NO_identity.
symmetry.
unfold Genv.alloc_global in H.
destruct (alloc m0 0 1) eqn:?.
assert (H6 := alloc_result _ _ _ _ _ Heqp); subst b0.
clear - n0 H2 Heqp H.
assert (b <> nextblock m0 \/ ofs <> 0).
{
```
Proof State:
```
1 subgoal
m : Memory.mem b : block ofs : Z m0, m1 : mem'
H : drop_perm m1 (nextblock m0) 0 1 Nonempty = Some m
Heqp : alloc m0 0 1 = (m1, nextblock m0)
n0 : (b, ofs) <> (nextblock m0, 0)
H2 : (nextblock m0 <= b)%positive
============================
b <> nextblock m0 \/ ofs <> 0
```
Next Tactic:
```
|
destruct (<LOOKUP>eq_block : forall x y : positive, {x = y} + {x <> y}</LOOKUP> b (<LOOKUP>nextblock : mem' -> block</LOOKUP> m0)).
```
|
{"package":"coq-vst","file":"\/home\/tpr\/.opam\/PackageTable~8.13~2022.01.csv.formatted\/lib\/coq\/user-contrib\/VST\/veric\/initialize.v"}
|
Proof State:
```
1 subgoal
p : positive x : nat
H : p = Pos.of_succ_nat x
============================
(- Pos.to_nat p)%Z = Z.neg p
```
Ongoing Proof History:
```
Proof.
intro p.
elim (p_is_some_anti_convert p).
intros x H.
```
Next Tactic:
```
|
rewrite H.
```
|
{"package":"coq-corn","file":"\/home\/tpr\/.opam\/PackageTable~8.13~2022.01.csv.formatted\/lib\/coq\/user-contrib\/CoRN\/logic\/CornBasics.v"}
|
Ongoing Proof History:
```
Proof.
intros.
punfold H.
unfold eqit_ in H.
cbn in *.
remember (ITree.bind ma kab) as tl.
assert (tl β
(ITree.bind ma kab)) by (subst; reflexivity).
clear Heqtl.
genobs tl tl'.
remember (VisF e kxc) as tr.
revert ma kab tl Heqtl' H0 kxc Heqtr.
induction H.
-
intros.
inv Heqtr.
-
intros.
inv Heqtr.
-
intros.
rewrite unfold_bind in H0.
destruct (observe ma) eqn: Hobma; cbn in *; rewrite itree_eta in H0; rewrite <- Heqtl' in H0.
+
right.
exists r.
split.
rewrite itree_eta.
rewrite Hobma.
reflexivity.
rewrite <- H0.
apply eqit_Vis.
intros.
destruct (REL u0); auto.
inv H.
+
symmetry in H0.
apply eqitree_inv_tau_vis in H0.
contradiction.
+
setoid_rewrite itree_eta at 1.
rewrite Hobma.
clear Hobma Heqtl'.
- red in H0.
+ inv Heqtr; auto_inj_pair2; subst.
- apply eqit_inv_vis_weak in H0.
+ apply eq_itree_inv_vis in H0.
? destruct H0 as [<- [<- H0]]; cbn in H0.
? inv_Vis.
? left.
```
Proof State:
```
1 subgoal
A : Type B : Type C : Type E : Type -> Type
X : Type RR : B -> C -> Prop b1, b2 : bool
e : E X k1 : X -> itree E B kxc : X -> itree E C
REL : forall v : X,
id
(upaco2
(fun (sim : itree E B -> itree E C -> Prop)
(t1 : itree E B) (t2 : itree E C) =>
eqitF RR b1 b2 id sim (observe t1) (observe t2)) bot2)
(k1 v) (kxc v)
ma : itree E A kab : A -> itree E B tl : itree E B
X0 : Type e1 : E X0 k : X0 -> itree E A
H0 : exists k' : X0 -> itree E B,
observe (Vis e k1) = VisF e1 k' /\
(forall u : X0, k' u β
x <- k u;; kab x)
============================
(exists kxa : X -> itree E A,
eqit eq b1 b2 (Vis e1 k) (Vis e kxa) /\
(forall x : X, eqit RR b1 b2 (x <- kxa x;; kab x) (kxc x))) \/
(exists a : A,
eqit eq b1 b2 ma (Ret a) /\ eqit RR b1 b2 (kab a) (Vis e kxc))
```
Commit Diff:
```
diff --git a/theories/Core/KTreeFacts.v b/theories/Core/KTreeFacts.v
index 32f56a6..b911a90 100644
--- a/theories/Core/KTreeFacts.v
+++ b/theories/Core/KTreeFacts.v
@@ -6 +6,2 @@ Require Import
-Classes.Morphisms Setoids.Setoid
+Program Classes.Morphisms
+Setoids.Setoid
@@ -23,2 +24 @@ Eq.Eq
-Eq.UpToTaus
-Eq.Paco2.
+Eq.UpToTaus.
@@ -68 +68 @@ Proof.
-ginit. pcofix CIH. intros.
+ginit. gcofix CIH. intros.
@@ -205,2 +205,2 @@ Proof.
-revert f g a0.
-ginit. pcofix CIH. intros.
+revert A B C f g a0.
+ginit. gcofix CIH. intros.
@@ -268 +268 @@ revert a0.
-ginit. pcofix CIH. intros.
+ginit. gcofix CIH. intros.
@@ -277 +277 @@ revert a.
-pcofix CIH'. intros.
+gcofix CIH'. intros.
@@ -337 +337,2 @@ etau.
-specialize (CIHL xa). cbn in CIHL.
+specialize (CIH xa).
+cbn in CIH.
diff --git a/theories/Eq/Eq.v b/theories/Eq/Eq.v
index 42a3038..1c90dcb 100644
--- a/theories/Eq/Eq.v
+++ b/theories/Eq/Eq.v
@@ -29,2 +29 @@ Basics.Basics Basics.HeterogeneousRelations
-Core.ITreeDefinition
-Eq.Paco2.
+Core.ITreeDefinition.
@@ -167,54 +165,0 @@ Hint Unfold id: core.
-Lemma eqitF_VisF_inv_r {E R1 R2} (RR : R1 -> R2 -> Prop) {b1 b2 vclo sim}
-t1 X2 (e2 : E X2) (k2 : X2 -> _) : eqitF RR b1
-b2 vclo sim t1 (VisF e2 k2) -> (exists k1, t1 = VisF e2 k1
-/\ forall v, vclo sim (k1 v) (k2 v)) \/ (b1 /\ exists t1', t1
-= TauF t1' /\ eqitF RR b1 b2 vclo sim (observe t1') (VisF e2 k2)).
-Proof.
-refine (fun H => match H
-in eqitF _ _ _ _
-_ _ t2 return match t2
-return Prop with | VisF e2
-k2 => _ | _
-=> True end with |
-EqVis _ _ _ _
-_ _ _ _ _
-=> _ | _ =>
-_ end); try exact I.
-- left; eauto.
-- destruct i0; eauto.
-Qed.
-
-Lemma eqitF_VisF_inv {E R1 R2} (RR : R1 -> R2 -> Prop) {b1 b2 vclo sim} X1
-(e1 : E X1) (k1 : X1 -> _) X2 (e2 : E X2) (k2 : X2
--> _) : eqitF RR b1 b2 vclo sim (VisF e1 k1) (VisF e2 k2) -> exists
-p : X1 = X2, eqeq E p e1 e2 /\ pweqeq (vclo sim) p k1 k2.
-Proof.
-refine (fun H => match H in
-eqitF _ _ _ _ _ t1
-t2 return match t1, t2 return Prop
-with | VisF e1 k1, VisF e2
-k2 => _ | _, _ =>
-True end with | EqVis _
-_ _ _ _ _ _
-_ _ => _ | _
-=> _ end); try exact I.
-- exists eq_refl; cbn; eauto.
-- destruct i; exact I.
-Qed.
-
-Lemma eqitF_VisF_inv_strong {E R1 R2} (RR : R1 -> R2 -> Prop) {b1 b2
-vclo sim} X (e : E X) (k1 : X -> _) (k2
```
Next Tactic:
```
|
destruct H0 as (? & ? & ?).
```
|
{"final_commit":"f4bacabbc5b1556a723e3cb4efb1cd614c156d8f","project":"InteractionTrees","repair_file":"\/data\/coq-pearls\/repairs\/InteractionTrees\/repair-InteractionTrees-f3853150994e962978fa4cf0ed18dcb8c42e3262-f4bacabbc5b1556a723e3cb4efb1cd614c156d8f-8_11_2-8_10_2-13.git.json","initial_commit":"f3853150994e962978fa4cf0ed18dcb8c42e3262","file":"theories\/Eq\/Eq.v"}
|
Ongoing Proof History:
```
Proof.
split; try (split; try apply _).
intros ?? E1 ?? E2.
now rewrite !le_correct, E1, E2.
```
Proof State:
```
5 subgoals
L : Type Ae : Equiv L Ameet : Meet L
H : MeetSemiLattice L
H0 : Le L
le_correct : β x y : L, x β€ y β x β y = x
============================
PreOrder le
subgoal 2 is:
AntiSymmetric le
subgoal 3 is:
β x y : L, x β y β€ x
subgoal 4 is:
β x y : L, x β y β€ y
subgoal 5 is:
β x y z : L, z β€ x β z β€ y β z β€ x β y
```
Next Tactic:
```
|
split.
```
|
{"package":"coq-math-classes","file":"\/home\/tpr\/.opam\/PackageTable~8.13~2022.01.csv.formatted\/lib\/coq\/user-contrib\/MathClasses\/orders\/lattices.v"}
|
Commit Diff:
```
diff --git a/theories/Core/KTreeFacts.v b/theories/Core/KTreeFacts.v
index b911a90..32f56a6 100644
--- a/theories/Core/KTreeFacts.v
+++ b/theories/Core/KTreeFacts.v
@@ -6,2 +6 @@ Require Import
-Program Classes.Morphisms
-Setoids.Setoid
+Classes.Morphisms Setoids.Setoid
@@ -24 +23,2 @@ Eq.Eq
-Eq.UpToTaus.
+Eq.UpToTaus
+Eq.Paco2.
@@ -68 +68 @@ Proof.
-ginit. gcofix CIH. intros.
+ginit. pcofix CIH. intros.
@@ -205,2 +205,2 @@ Proof.
-revert A B C f g a0.
-ginit. gcofix CIH. intros.
+revert f g a0.
+ginit. pcofix CIH. intros.
@@ -268 +268 @@ revert a0.
-ginit. gcofix CIH. intros.
+ginit. pcofix CIH. intros.
@@ -277 +277 @@ revert a.
-gcofix CIH'. intros.
+pcofix CIH'. intros.
@@ -337,2 +337 @@ etau.
-specialize (CIH xa).
-cbn in CIH.
+specialize (CIHL xa). cbn in CIHL.
diff --git a/theories/Eq/Eq.v b/theories/Eq/Eq.v
index 21ac7e0..4f26bd0 100644
--- a/theories/Eq/Eq.v
+++ b/theories/Eq/Eq.v
@@ -29 +29,2 @@ Basics.Basics Basics.HeterogeneousRelations
-Core.ITreeDefinition.
+Core.ITreeDefinition
+Eq.Paco2.
@@ -165,0 +167,54 @@ Hint Unfold id: core.
+Lemma eqitF_VisF_inv_r {E R1 R2} (RR : R1 -> R2 -> Prop) {b1 b2 vclo sim}
+t1 X2 (e2 : E X2) (k2 : X2 -> _) : eqitF RR b1
+b2 vclo sim t1 (VisF e2 k2) -> (exists k1, t1 = VisF e2 k1
+/\ forall v, vclo sim (k1 v) (k2 v)) \/ (b1 /\ exists t1', t1
+= TauF t1' /\ eqitF RR b1 b2 vclo sim (observe t1') (VisF e2 k2)).
+Proof.
+refine (fun H => match H
+in eqitF _ _ _ _
+_ _ t2 return match t2
+return Prop with | VisF e2
+k2 => _ | _
+=> True end with |
+EqVis _ _ _ _
+_ _ _ _ _
+=> _ | _ =>
+_ end); try exact I.
+- left; eauto.
+- destruct i0; eauto.
+Qed.
+
+Lemma eqitF_VisF_inv {E R1 R2} (RR : R1 -> R2 -> Prop) {b1 b2 vclo sim} X1
+(e1 : E X1) (k1 : X1 -> _) X2 (e2 : E X2) (k2 : X2
+-> _) : eqitF RR b1 b2 vclo sim (VisF e1 k1) (VisF e2 k2) -> exists
+p : X1 = X2, eqeq E p e1 e2 /\ pweqeq (vclo sim) p k1 k2.
+Proof.
+refine (fun H => match H in
```
Ongoing Proof History:
```
Proof.
econstructor.
pmonauto.
intros.
- dependent destruction PR.
+ destruct PR.
punfold EQVl.
punfold EQVr.
unfold_eqit.
hinduction REL before r; intros; clear t1' t2'.
-
remember (RetF r1) as x.
hinduction EQVl before r; intros; subst; try inv Heqx; eauto.
remember (RetF r3) as y.
hinduction EQVr before r; intros; subst; try inv Heqy; eauto.
-
remember (TauF m1) as x.
hinduction EQVl before r; intros; subst; try inv Heqx; try inv CHECK; eauto.
remember (TauF m3) as y.
hinduction EQVr before r; intros; subst; try inv Heqy; try inv CHECK; eauto.
pclearbot.
econstructor.
gclo.
- econstructor; cycle -1; eauto with paco.
+ econstructor; eauto with paco.
-
remember (VisF e k1) as x.
- hinduction EQVl before r; intros; subst; try dependent destruction Heqx; try inv CHECK; eauto.
+ hinduction EQVl before r; intros; try discriminate Heqx; eauto; inv_Vis.
? remember (VisF e0 k3) as y.
? hinduction EQVr before r; intros; subst; try dependent destruction Heqy; try inv CHECK; eauto.
? econstructor.
```
Proof State:
```
1 subgoal
E : Type -> Type R1 : Type R2 : Type RR : R1 -> R2 -> Prop
b1, b2 : bool
vclo : rel2 (itree E R1) (fun _ : itree E R1 => itree E R2) ->
rel2 (itree E R1) (fun _ : itree E R1 => itree E R2)
MON : monotone2 vclo
CMP : forall (x0 : rel2 (itree E R1) (fun _ : itree E R1 => itree E R2))
(x1 : itree E R1) (x2 : itree E R2),
(eqitC b1 b2 β vclo) x0 x1 x2 -> (vclo β eqitC b1 b2) x0 x1 x2
r : rel2 (itree E R1) (fun _ : itree E R1 => itree E R2)
t1 : itree E R1 RR1 : R1 -> R1 -> Prop u : Type
e : E u k1, k2 : u -> itree E R1
REL : forall v : u, id (upaco2 (eqit_ RR1 b1 false id) bot2) (k1 v) (k2 v)
k3 : u -> itree E R2
REL0 : forall v : u, vclo r (k2 v) (k3 v)
t2 : itree E R2 RR2 : R2 -> R2 -> Prop
EQVr : eqitF RR2 b2 false id (upaco2 (eqit_ RR2 b2 false id) bot2)
(observe t2) (VisF e k3)
LERR1 : forall (x x' : R1) (y : R2), RR1 x x' -> RR x' y -> RR x y
LERR2 : forall (x : R1) (y y' : R2), RR2 y y' -> RR x y' -> RR x y
============================
eqitF RR b1 b2 vclo (gupaco2 (eqit_ RR b1 b2 vclo) (eqitC b1 b2) r)
(VisF e k1) (observe t2)
```
Next Tactic:
```
|
remember (<LOOKUP>VisF : forall (E : Type -> Type) (R itree X : Type), E X -> (X -> itree) -> itreeF E R itree</LOOKUP> e k3) as y.
```
|
{"final_commit":"f3853150994e962978fa4cf0ed18dcb8c42e3262","project":"InteractionTrees","repair_file":"\/data\/coq-pearls\/repairs\/InteractionTrees\/repair-InteractionTrees-1fcd2d39e27da8068431211b7952365e163782cc-f3853150994e962978fa4cf0ed18dcb8c42e3262-8_10_2-8_11_2-57.git.json","initial_commit":"1fcd2d39e27da8068431211b7952365e163782cc","file":"theories\/Eq\/Eq.v"}
|
Commit Diff:
```
diff --git a/ListAux.v b/ListAux.v
index 986e88b..769776a 100644
--- a/ListAux.v
+++ b/ListAux.v
@@ -131,6 +130,0 @@ Qed.
-Theorem length_app: forall (l1 l2 : list A), length
-(l1 ++ l2) = length l1 + length l2.
-Proof.
-apply app_length.
-Qed.
-
@@ -278 +272 @@ intros A l1 l2; elim l1; simpl; auto.
-intros a l H; rewrite length_app, length_map, H; auto.
+intros a l H; rewrite app_length, length_map, H; auto.
diff --git a/ListOp.v b/ListOp.v
index e1936ca..7a13f04 100644
--- a/ListOp.v
+++ b/ListOp.v
@@ -145 +145 @@ intros k Rec i j s H;
-rewrite length_app, length_take1; auto with arith.
+rewrite app_length, length_take1; auto with arith.
diff --git a/Sudoku.v b/Sudoku.v
index 5083f1b..9b556eb 100644
--- a/Sudoku.v
+++ b/Sudoku.v
@@ -675 +675 @@ assert (F1: div y w < h).
-apply div_lt; rewrite mult_comm; auto.
+apply div_lt; rewrite mult_comm...
@@ -677,6 +677,2 @@ apply div_lt; rewrite mult_comm; auto.
-repeat (rewrite (fun x =>
-mult_comm x h)); rewrite mod_mult_comp...
-rewrite div_mult_comp; auto.
-rewrite (mod_small (div y w) h); auto.
-rewrite (div_is_0 (div y w) h); auto.
-rewrite plus_0_r.
+repeat (rewrite (fun x => mult_comm x h)); rewrite mod_mult_comp...
+rewrite div_mult_comp, (mod_small (div y w) h), (div_is_0 (div y w) h), plus_0_r...
@@ -703,3 +699,3 @@ cut (w <= size).
-intros U3 n; generalize h; elim n; simpl;
-auto with arith; clear n H1 H2 U1.
-+ intros h1; case h1; simpl; auto.
+intros U3 n; generalize h; elim n; simpl...
+clear n H1 H2 U1.
++ intros h1; case h1; simpl...
@@ -713,3 +709,2 @@ apply sym_equal; apply take_nth...
-** apply le_trans with (1 := H1)...
-apply le_trans with (2 := U3).
-apply lt_le_weak; apply mod_lt...
+** pose proof (mod_lt y w).
+lia.
@@ -726 +721 @@ apply mod_lt...
-case (le_or_lt (length l) w); intros H1.
+case (le_or_lt (length l) w); intros H3.
@@ -728,2 +723 @@ case (le_or_lt (length l) w); intros H1.
-** rewrite length_take_small...
-rewrite jump_too_far.
+** rewrite length_take_small, jump_too_far...
@@ -732,2 +726,2 @@ repeat rewrite nth_default...
-apply le_trans with (1 := H1)...
-*** apply le_trans with (1 := H1)...
+apply le_trans with (1 := H3)...
+*** apply le_trans with (1 := H3)...
@@ -738,2 +732 @@ repeat rewrite <- plus_assoc; rewrite minus_plus...
-rewrite jump_nth; rewrite jump_add.
-rewrite <- jump_nth...
+rewrite jump_nth, jump_add, <- jump_nth...
@@ -750 +743 @@ generalize (w_pos _ H); intros U2.
-rewrite length_take_and_jump; auto.
+rewrite length_take_and_jump...
@@ -767,2 +760 @@ rewrite (plus_assoc w).
-pattern w at 1; rewrite <- mult_1_r; rewrite <- mult_plus_distr_l.
-rewrite le_plus_minus_r...
+pattern w at 1; rewrite <- mult_1_r, <- mult_plus_distr_l, le_plus_minus_r...
@@ -854,2 +846,2 @@ l2 l3 -> lit_test l1 l3 = lit_test l1 l2.
-Proof.
-intros l1 l2 l3; case l1; case l2; case l3; simpl; auto.
+Proof with auto.
+intros l1 l2 l3; case l1; case l2; case l3; simpl...
@@ -859 +851 @@ case_eq (pos_test p2 p3); intros H1.
-rewrite (pos_test_trans p1 p2 p3); rewrite H; auto.
+rewrite (pos_test_trans p1 p2 p3); rewrite H...
@@ -861,2 +853,2 @@ assert (p2 = p3); subst.
-apply pos_test_exact with (1 := H1); auto.
-rewrite H; auto.
```
Proof State:
```
2 goals
h, w : nat s : list nat
H : length s = size * size
H1 : empty s
z1, x1, y1 : nat
Hl : In (L (Pos x1 y1) z1) (gen_row x1 z1)
n1 : nat
V1 : In x1 indexes
V2 : In z1 ref_list
Hy : y1 < size
============================
valid_lit (L (Pos x1 y1) z1) s
goal 2 is:
In (n, c)
(fold_right
(fun (iz : nat * nat) (l0 : clauses) =>
clause_insert (gen_column (fst iz) (snd iz)) l0)
(fold_right
(fun (iz : nat * nat) (l0 : clauses) =>
clause_insert (gen_rect (fst iz) (snd iz)) l0) all_cell cross1)
cross1) -> valid_lit l s
```
Ongoing Proof History:
```
- Proof.
+ Proof with auto with arith.
unfold init_c; intros s H H1 n c Hn l Hl.
case (fold_clause_insert1 _ (fun iz => gen_row (fst iz) (snd iz))) with (1 := Hn); clear Hn.
intros ((x,z), (n1, (HH1, HH2))); simpl in HH2.
injection HH2; intros; subst; clear HH2.
case (cross1_correct (x, z)); intros tmp _; case (tmp HH1); clear tmp HH1.
intros x1 (z1, (V1, (V2, V3))); injection V3; intros; subst; clear V3.
case (gen_row_correct l x1 z1).
intros tmp _; case (tmp Hl); clear tmp.
intros y1 (Hl1, Hy); subst.
? repeat (split; auto); case (in_indexes x1); auto.
? intros Hn; case (fold_clause_insert1 _ (fun iz => gen_column (fst iz) (snd iz))) with (1 := Hn); clear Hn.
? intros ((y,z), (n1, (HH1, HH2))); simpl in HH2.
```
Next Tactic:
```
|
repeat (split; auto); case (<LOOKUP>in_indexes : forall i : nat, In i indexes <-> i < size</LOOKUP> x1)...
```
|
{"final_commit":"5337edcaf31b9f036a0a6651b2f733d3068b8f8f","project":"sudoku","repair_file":"\/data\/coq-pearls\/repairs\/sudoku\/repair-sudoku-427448011e3341d35aa5b9e00f675f6582fcd477-5337edcaf31b9f036a0a6651b2f733d3068b8f8f-8_15_2-8_15_2-36.git.json","initial_commit":"427448011e3341d35aa5b9e00f675f6582fcd477","file":"Sudoku.v"}
|
Ongoing Proof History:
```
Proof.
cut (forall (x1 x2 : Q2), msp_eq x1 x2 -> forall x3 x4 : Q2, msp_eq x3 x4 -> forall y, InFinEnumC y (CentersOfPixels bm x1 x3) -> InFinEnumC y (CentersOfPixels bm x2 x4)).
{
intro L.
split.
discriminate.
split.
intros q H1 abs.
contradiction (abs q).
split.
exact (L x y H x0 y0 H0 q H1).
reflexivity.
intros q H1 abs.
contradiction (abs q).
split.
symmetry in H, H0.
exact (L y x H y0 x0 H0 q H1).
reflexivity.
}
intros [x1l x1r] x2 Hx [y1l y1r] y2 Hy z Hz.
destruct (@InStrengthen _ _ Hz) as [[ax ay] [Ha0 Ha1]].
destruct (InterpRaster_correct2 _ _ _ _ _ _ _ bmWf Ha0) as [[bx by'] [Hb0 [Hb1 Hb2]]].
rewrite Hb1 in Ha1.
rewrite Hb2 in Ha1.
unfold snd, fst in Ha1.
destruct x2 as [x2l x2r].
destruct y2 as [y2l y2r].
assert (L0:msp_eq z ((x2l + (y2l - x2l) * (2 * Z.of_nat by' + 1 # 1) / (2 * Zpos n # 1)), (x2r + (y2r - x2r) * (2 * Z.of_nat bx + 1 # 1) / (2 * Zpos m # 1)))).
transitivity ((x1l + (y1l - x1l) * (2 * Z.of_nat by' + 1 # 1) / (2 * Zpos n # 1)), (x1r + (y1r - x1r) * (2 * Z.of_nat bx + 1 # 1) / (2 * Zpos m # 1))).
auto.
clear - Hx Hy.
destruct Hx as [Hx1 Hx2].
destruct Hy as [Hy1 Hy2].
split; unfold fst,snd in *.
apply Qball_0 in Hx1.
apply Qball_0 in Hy1.
rewrite -> Hx1, Hy1.
reflexivity.
apply Qball_0 in Hx2.
apply Qball_0 in Hy2.
rewrite -> Hx2, Hy2.
reflexivity.
```
Proof State:
```
1 subgoal
n, m : positive bm : raster n m
bmWf : raster_well_formed bm
x1l, x1r, x2l, x2r : Q_as_MetricSpace
Hx : msp_eq (x1l, x1r) (x2l, x2r)
y1l, y1r, y2l, y2r : Q_as_MetricSpace
Hy : msp_eq (y1l, y1r) (y2l, y2r)
z : Q2
Hz : InFinEnumC z (CentersOfPixels bm (x1l, x1r) (y1l, y1r))
ax, ay : Q_as_MetricSpace
Ha0 : In (ax, ay) (CentersOfPixels bm (x1l, x1r) (y1l, y1r))
bx, by' : nat
Ha1 : msp_eq z
(x1l + (y1l - x1l) * (2 * Z.of_nat by' + 1 # 1) / (2 * Z.pos n # 1),
x1r + (y1r - x1r) * (2 * Z.of_nat bx + 1 # 1) / (2 * Z.pos m # 1))
Hb0 : Is_true (RasterIndex bm (fst (bx, by')) (snd (bx, by')))
Hb1 : ax =
x1l +
(y1l - x1l) * (2 * Z.of_nat (snd (bx, by')) + 1 # 1) /
(2 * Z.pos n # 1)
Hb2 : ay =
x1r +
(y1r - x1r) * (2 * Z.of_nat (fst (bx, by')) + 1 # 1) /
(2 * Z.pos m # 1)
L0 : msp_eq z
(x2l + (y2l - x2l) * (2 * Z.of_nat by' + 1 # 1) / (2 * Z.pos n # 1),
x2r + (y2r - x2r) * (2 * Z.of_nat bx + 1 # 1) / (2 * Z.pos m # 1))
============================
InFinEnumC z (CentersOfPixels bm (x2l, x2r) (y2l, y2r))
```
Next Tactic:
```
|
unfold InFinEnumC.
```
|
{"package":"coq-corn","file":"\/home\/tpr\/.opam\/PackageTable~8.13~2022.01.csv.formatted\/lib\/coq\/user-contrib\/CoRN\/reals\/fast\/RasterQ.v"}
|
Ongoing Proof History:
```
Proof.
repeat red.
intros until T.
ginit.
gcofix CIH.
intros.
rewrite !unfold_interp.
punfold H1.
red in H1.
induction H1; intros; subst; pclearbot; simpl.
-
gstep.
constructor.
eauto.
-
gstep.
constructor.
eauto with paco.
-
guclo eqit_clo_bind; econstructor; [apply H|].
intros; subst.
gstep; constructor; eauto with paco.
-
- rewrite tau_eutt, unfold_interp.
+ rewrite tau_euttge, unfold_interp.
auto.
-
? rewrite tau_eutt, unfold_interp.
? auto.
? Qed.
```
Proof State:
```
1 subgoal
F : Type -> Type T : Type E : Type -> Type x, y : Handler E F
H : eq2 x y
r : itree F T -> itree F T -> Prop
CIH : forall a1 a2 : itree E T, a1 β a2 -> r (interp x a1) (interp y a2)
a1, a2 : itree E T ot1 : itree' E T t2 : itree E T
CHECK : true
H1 : eqitF eq true true id (upaco2 (eqit_ eq true true id) bot2) ot1
(observe t2)
IHeqitF : gpaco2 (eqit_ eq true true id) (eqitC eq true true) bot2 r
(_interp x ot1) (_interp y (observe t2))
============================
gpaco2 (eqit_ eq true true id) (eqitC eq true true) bot2 r
(_interp x ot1) (Tau (interp y t2))
```
Commit Diff:
```
diff --git a/theories/Core/ITreeDefinition.v b/theories/Core/ITreeDefinition.v
index 57c2013..9b33167 100644
--- a/theories/Core/ITreeDefinition.v
+++ b/theories/Core/ITreeDefinition.v
@@ -242 +242 @@ Notation "t1 >>= k2" := (ITree.bind t1 k2)
-(at level 50, left associativity) : itree_scope.
+(at level 58, left associativity) : itree_scope.
diff --git a/theories/Core/KTree.v b/theories/Core/KTree.v
index 7740c5f..49acfb6 100644
--- a/theories/Core/KTree.v
+++ b/theories/Core/KTree.v
@@ -69,4 +69 @@ Local Notation ktree := (ktree E).
-End Operations.
-
-Notation iter := (@iter _ (ktree _) sum _ _ _).
-Notation loop := (@loop _ (ktree _) sum _ _ _ _ _ _ _ _ _).
\ No newline at end of file
+End Operations.
\ No newline at end of file
diff --git a/theories/Core/KTreeFacts.v b/theories/Core/KTreeFacts.v
index 37d0ea8..1c27491 100644
--- a/theories/Core/KTreeFacts.v
+++ b/theories/Core/KTreeFacts.v
@@ -47 +47 @@ intros [a | b] _ [].
-- rewrite bind_ret_l, tau_eutt.
+- rewrite bind_ret_l, tau_euttge.
@@ -129,4 +129,4 @@ Qed.
-Definition eutt_iter_gen {F A B R S} : @Proper
-((A -> itree F (A + B)) -> A
--> itree F B) ((R ==> eutt (sum_rel
-R S)) ==> R ==> (eutt S)) KTree.iter.
+Definition eutt_iter_gen {F A B R S} : @Proper ((A
+-> itree F (A + B)) -> A -> itree
+F B) ((R ==> eutt (sum_rel R S)) ==>
+R ==> (eutt S)) (iter (C := ktree F)).
@@ -185,18 +185,18 @@ Qed.
-Lemma iter_dinatural_ktree {E A B C}
-(f : ktree E A (C
-+ B)) (g : ktree E
-C (A + B)) (a0 :
-A) : iter (fun a =>
-cb <- f a ;; match
-cb with | inl c =>
-Tau (g c) | inr b
-=> Ret (inr b) end) a0
-β
cb <- f a0 ;;
-match cb with | inl c0
-=> Tau (iter (fun c =>
-ab <- g c ;; match
-ab with | inl a
-=> Tau (f a) |
-inr b => Ret (inr
-b) end) c0) | inr
-b => Ret b end.
+Lemma iter_dinatural_ktree {E A B C} (f
+: ktree E A (C + B))
+(g : ktree E C (A +
+B)) (a0 : A) : iter
+(C := ktree E) (fun a
+=> cb <- f a ;;
+match cb with | inl c
+=> Tau (g c) | inr
+b => Ret (inr b) end)
+a0 β
cb <- f a0
+;; match cb with | inl
+c0 => Tau (iter (C :=
+ktree E) (fun c => ab
+<- g c ;; match ab
+with | inl a => Tau
+(f a) | inr b =>
+Ret (inr b) end) c0) |
+inr b => Ret b end.
@@ -230,6 +230,6 @@ cbn.
-transitivity (iter (fun t =>
-x <- f t;; match
-x with | inl a1
-=> Tau (g a1) |
-inr b0 => Ret
-(inr b0) end) a0).
+transitivity (iter (C := ktree E)
+(fun t => x <- f
+t;; match x with |
+inl a1 => Tau (g
+a1) | inr b0 =>
+Ret (inr b0) end) a0).
@@ -256,9 +256,9 @@ Qed.
-Lemma iter_codiagonal_ktree {E A B} (f :
-ktree E A (A + (A +
-B))) (a0 : A) : iter (iter
-f) a0 β
iter (fun a =>
-r <- f a ;; match r
-with | inl a' => Ret (inl
-a') | inr (inl a') => Ret
-(inl a') | inr (inr b)
-=> Ret (inr b) end) a0.
+Lemma iter_codiagonal_ktree {E A B} (f : ktree
+E A (A + (A + B))) (a0
+: A) : iter (iter f) a0
+β
iter (C := ktree _) (fun
+a => r <- f a ;;
+match r with | inl a' =>
+Ret (inl a') | inr (inl a')
```
Next Tactic:
```
|
rewrite <LOOKUP>tau_euttge : forall (E : Type -> Type) (R : Type) (t : itree E R), Tau t β³ t</LOOKUP>, <LOOKUP>unfold_interp : forall (E F : Type -> Type) (R : Type) (f : forall T : Type, E T -> itree F T) (t : itree E R), interp f t β
_interp f (observe t)</LOOKUP>.
```
|
{"final_commit":"4e1301e834381725b933593fcf1f3b973986b630","project":"InteractionTrees","repair_file":"\/data\/coq-pearls\/repairs\/InteractionTrees\/repair-InteractionTrees-66f56a8a4d02a262dbe75a676a4064f18ea7c3a0-4e1301e834381725b933593fcf1f3b973986b630-8_10_2-8_10_2-6.git.json","initial_commit":"66f56a8a4d02a262dbe75a676a4064f18ea7c3a0","file":"theories\/Interp\/InterpFacts.v"}
|
Proof State:
```
This subproof is complete, but there are some unfocused goals.
Focus next goal with bullet -.
3 subgoals
subgoal 1 is:
x ^Z Z.pos_sub n m = x ^ Pos.to_nat n * / x ^ Pos.to_nat m
subgoal 2 is:
x ^Z Z.pos_sub m n = / x ^ Pos.to_nat n * x ^ Pos.to_nat m
subgoal 3 is:
/ x ^ Pos.to_nat (n + m) = / x ^ Pos.to_nat n * / x ^ Pos.to_nat m
```
Ongoing Proof History:
```
Proof.
intros x [|n|n] [|m|m]; simpl; intros; auto with real.
-
rewrite Pos2Nat.inj_add; auto with real.
```
Next Tactic:
```
|
-
```
|
{"package":"coq","file":"\/home\/tpr\/.opam\/PackageTable~8.13~2022.01.csv.formatted\/lib\/coq\/theories\/Reals\/Rfunctions.v"}
|
Commit Diff:
```
diff --git a/theories/Core/KTreeFacts.v b/theories/Core/KTreeFacts.v
index 32f56a6..b911a90 100644
--- a/theories/Core/KTreeFacts.v
+++ b/theories/Core/KTreeFacts.v
@@ -6 +6,2 @@ Require Import
-Classes.Morphisms Setoids.Setoid
+Program Classes.Morphisms
+Setoids.Setoid
@@ -23,2 +24 @@ Eq.Eq
-Eq.UpToTaus
-Eq.Paco2.
+Eq.UpToTaus.
@@ -68 +68 @@ Proof.
-ginit. pcofix CIH. intros.
+ginit. gcofix CIH. intros.
@@ -205,2 +205,2 @@ Proof.
-revert f g a0.
-ginit. pcofix CIH. intros.
+revert A B C f g a0.
+ginit. gcofix CIH. intros.
@@ -268 +268 @@ revert a0.
-ginit. pcofix CIH. intros.
+ginit. gcofix CIH. intros.
@@ -277 +277 @@ revert a.
-pcofix CIH'. intros.
+gcofix CIH'. intros.
@@ -337 +337,2 @@ etau.
-specialize (CIHL xa). cbn in CIHL.
+specialize (CIH xa).
+cbn in CIH.
diff --git a/theories/Eq/Eq.v b/theories/Eq/Eq.v
index 42a3038..21ac7e0 100644
--- a/theories/Eq/Eq.v
+++ b/theories/Eq/Eq.v
@@ -29,2 +29 @@ Basics.Basics Basics.HeterogeneousRelations
-Core.ITreeDefinition
-Eq.Paco2.
+Core.ITreeDefinition.
@@ -167,54 +165,0 @@ Hint Unfold id: core.
-Lemma eqitF_VisF_inv_r {E R1 R2} (RR : R1 -> R2 -> Prop) {b1 b2 vclo sim}
-t1 X2 (e2 : E X2) (k2 : X2 -> _) : eqitF RR b1
-b2 vclo sim t1 (VisF e2 k2) -> (exists k1, t1 = VisF e2 k1
-/\ forall v, vclo sim (k1 v) (k2 v)) \/ (b1 /\ exists t1', t1
-= TauF t1' /\ eqitF RR b1 b2 vclo sim (observe t1') (VisF e2 k2)).
-Proof.
-refine (fun H => match H
-in eqitF _ _ _ _
-_ _ t2 return match t2
-return Prop with | VisF e2
-k2 => _ | _
-=> True end with |
-EqVis _ _ _ _
-_ _ _ _ _
-=> _ | _ =>
-_ end); try exact I.
-- left; eauto.
-- destruct i0; eauto.
-Qed.
-
-Lemma eqitF_VisF_inv {E R1 R2} (RR : R1 -> R2 -> Prop) {b1 b2 vclo sim} X1
-(e1 : E X1) (k1 : X1 -> _) X2 (e2 : E X2) (k2 : X2
--> _) : eqitF RR b1 b2 vclo sim (VisF e1 k1) (VisF e2 k2) -> exists
-p : X1 = X2, eqeq E p e1 e2 /\ pweqeq (vclo sim) p k1 k2.
-Proof.
-refine (fun H => match H in
-eqitF _ _ _ _ _ t1
-t2 return match t1, t2 return Prop
-with | VisF e1 k1, VisF e2
-k2 => _ | _, _ =>
-True end with | EqVis _
-_ _ _ _ _ _
-_ _ => _ | _
-=> _ end); try exact I.
-- exists eq_refl; cbn; eauto.
-- destruct i; exact I.
-Qed.
-
-Lemma eqitF_VisF_inv_strong {E R1 R2} (RR : R1 -> R2 -> Prop) {b1 b2
-vclo sim} X (e : E X) (k1 : X -> _) (k2
-: X -> _) : eqitF RR b1 b2 vclo sim (VisF e
-k1) (VisF e k2) -> forall x, vclo sim (k1 x) (k2 x).
-Proof.
-intros H. dependent destruction H. assumption.
-Qed.
-
-Lemma eqitF_VisF {E R1 R2} {RR : R1 -> R2 -> Prop} {b1 b2 vclo sim} {X1
-X2} (p : X1 = X2) (e1 : E X1) (k1 : X1 -> _) (e2 :
-E X2) (k2 : X2 -> _) : eqeq E p e1 e2 -> pweqeq (vclo sim)
-p k1 k2 -> eqitF RR b1 b2 vclo sim (VisF e1 k1) (VisF e2 k2).
-Proof.
-destruct p; intros <-; cbn; constructor; auto.
-Qed.
-
@@ -270,2 +215,2 @@ split.
-pstep. punfold H0. red in H0. red.
-hinduction H0 before CIH; intros; eauto.
+pstep. punfold H1. red in H1. red.
+hinduction H1 before CIH; intros; eauto.
@@ -278,2 +223,2 @@ red in REL. pclearbot. pinversion REL.
-pstep. punfold H0. red in H0. red.
-hinduction H0 before CIH; intros; eauto.
+pstep. punfold H1. red in H1. red.
+hinduction H1 before CIH; intros; eauto.
@@ -381 +326 @@ econstructor. pmonauto.
-intros. destruct PR.
```
Proof State:
```
1 subgoal
E : Type -> Type R2 : Type R1 : Type RR : R1 -> R2 -> Prop
rH, rL, gL, gH, x : itree E R1 -> itree E R2 -> Prop
H : forall (x0 : itree E R1) (x1 : itree E R2),
x x0 x1 -> euttG rH rL (gL \2/ x) (gH \2/ x) x0 x1
r : itree E R1 -> itree E R2 -> Prop
CIH0 : forall (x0 : itree E R1) (x1 : itree E R2),
gL x0 x1 \/ transU RR rH x0 x1 \/ rL x0 x1 -> r x0 x1
CIH : forall (x2 : itree E R1) (x3 : itree E R2), x x2 x3 -> r x2 x3
x2 : itree E R1 x3 : itree E R2
PR : x x2 x3
============================
gpaco2 (eqit_ RR true true (euttVC RR gH)) (transD RR)
(transU RR rH \2/ rL) r x2 x3
```
Ongoing Proof History:
```
Proof.
intros.
apply euttG_gen.
econstructor.
revert x0 x1 PR.
- pcofix CIH.
+ gcofix CIH.
+ intros.
? intros t1 t2 PR.
? apply H in PR.
? destruct PR as [IN].
```
Next Tactic:
```
|
apply H in PR.
```
|
{"final_commit":"f4bacabbc5b1556a723e3cb4efb1cd614c156d8f","project":"InteractionTrees","repair_file":"\/data\/coq-pearls\/repairs\/InteractionTrees\/repair-InteractionTrees-f3853150994e962978fa4cf0ed18dcb8c42e3262-f4bacabbc5b1556a723e3cb4efb1cd614c156d8f-8_11_2-8_10_2-18.git.json","initial_commit":"f3853150994e962978fa4cf0ed18dcb8c42e3262","file":"theories\/Eq\/UpToTaus.v"}
|
Commit Diff:
```
diff --git a/theories/Core/KTreeFacts.v b/theories/Core/KTreeFacts.v
index 32f56a6..b911a90 100644
--- a/theories/Core/KTreeFacts.v
+++ b/theories/Core/KTreeFacts.v
@@ -6 +6,2 @@ Require Import
-Classes.Morphisms Setoids.Setoid
+Program Classes.Morphisms
+Setoids.Setoid
@@ -23,2 +24 @@ Eq.Eq
-Eq.UpToTaus
-Eq.Paco2.
+Eq.UpToTaus.
@@ -68 +68 @@ Proof.
-ginit. pcofix CIH. intros.
+ginit. gcofix CIH. intros.
@@ -205,2 +205,2 @@ Proof.
-revert f g a0.
-ginit. pcofix CIH. intros.
+revert A B C f g a0.
+ginit. gcofix CIH. intros.
@@ -268 +268 @@ revert a0.
-ginit. pcofix CIH. intros.
+ginit. gcofix CIH. intros.
@@ -277 +277 @@ revert a.
-pcofix CIH'. intros.
+gcofix CIH'. intros.
@@ -337 +337,2 @@ etau.
-specialize (CIHL xa). cbn in CIHL.
+specialize (CIH xa).
+cbn in CIH.
diff --git a/theories/Eq/Eq.v b/theories/Eq/Eq.v
index 42a3038..21ac7e0 100644
--- a/theories/Eq/Eq.v
+++ b/theories/Eq/Eq.v
@@ -29,2 +29 @@ Basics.Basics Basics.HeterogeneousRelations
-Core.ITreeDefinition
-Eq.Paco2.
+Core.ITreeDefinition.
@@ -167,54 +165,0 @@ Hint Unfold id: core.
-Lemma eqitF_VisF_inv_r {E R1 R2} (RR : R1 -> R2 -> Prop) {b1 b2 vclo sim}
-t1 X2 (e2 : E X2) (k2 : X2 -> _) : eqitF RR b1
-b2 vclo sim t1 (VisF e2 k2) -> (exists k1, t1 = VisF e2 k1
-/\ forall v, vclo sim (k1 v) (k2 v)) \/ (b1 /\ exists t1', t1
-= TauF t1' /\ eqitF RR b1 b2 vclo sim (observe t1') (VisF e2 k2)).
-Proof.
-refine (fun H => match H
-in eqitF _ _ _ _
-_ _ t2 return match t2
-return Prop with | VisF e2
-k2 => _ | _
-=> True end with |
-EqVis _ _ _ _
-_ _ _ _ _
-=> _ | _ =>
-_ end); try exact I.
-- left; eauto.
-- destruct i0; eauto.
-Qed.
-
-Lemma eqitF_VisF_inv {E R1 R2} (RR : R1 -> R2 -> Prop) {b1 b2 vclo sim} X1
-(e1 : E X1) (k1 : X1 -> _) X2 (e2 : E X2) (k2 : X2
--> _) : eqitF RR b1 b2 vclo sim (VisF e1 k1) (VisF e2 k2) -> exists
-p : X1 = X2, eqeq E p e1 e2 /\ pweqeq (vclo sim) p k1 k2.
-Proof.
-refine (fun H => match H in
-eqitF _ _ _ _ _ t1
-t2 return match t1, t2 return Prop
-with | VisF e1 k1, VisF e2
-k2 => _ | _, _ =>
-True end with | EqVis _
-_ _ _ _ _ _
-_ _ => _ | _
-=> _ end); try exact I.
-- exists eq_refl; cbn; eauto.
-- destruct i; exact I.
-Qed.
-
-Lemma eqitF_VisF_inv_strong {E R1 R2} (RR : R1 -> R2 -> Prop) {b1 b2
-vclo sim} X (e : E X) (k1 : X -> _) (k2
-: X -> _) : eqitF RR b1 b2 vclo sim (VisF e
-k1) (VisF e k2) -> forall x, vclo sim (k1 x) (k2 x).
-Proof.
-intros H. dependent destruction H. assumption.
-Qed.
-
-Lemma eqitF_VisF {E R1 R2} {RR : R1 -> R2 -> Prop} {b1 b2 vclo sim} {X1
-X2} (p : X1 = X2) (e1 : E X1) (k1 : X1 -> _) (e2 :
-E X2) (k2 : X2 -> _) : eqeq E p e1 e2 -> pweqeq (vclo sim)
-p k1 k2 -> eqitF RR b1 b2 vclo sim (VisF e1 k1) (VisF e2 k2).
-Proof.
-destruct p; intros <-; cbn; constructor; auto.
-Qed.
-
@@ -270,2 +215,2 @@ split.
-pstep. punfold H0. red in H0. red.
-hinduction H0 before CIH; intros; eauto.
+pstep. punfold H1. red in H1. red.
+hinduction H1 before CIH; intros; eauto.
@@ -278,2 +223,2 @@ red in REL. pclearbot. pinversion REL.
-pstep. punfold H0. red in H0. red.
-hinduction H0 before CIH; intros; eauto.
+pstep. punfold H1. red in H1. red.
```
Ongoing Proof History:
```
Proof.
intros.
apply euttG_gen.
econstructor.
revert x0 x1 PR.
- pcofix CIH.
+ gcofix CIH.
- intros t1 t2 PR.
+ intros.
apply H in PR.
? destruct PR as [IN].
? revert t1 t2 IN.
? pcofix CIH.
```
Proof State:
```
1 subgoal
E : Type -> Type R2 : Type R1 : Type RR : R1 -> R2 -> Prop
rH, rL, gL, gH, x : itree E R1 -> itree E R2 -> Prop
H : forall (x0 : itree E R1) (x1 : itree E R2),
x x0 x1 -> euttG rH rL (gL \2/ x) (gH \2/ x) x0 x1
r : itree E R1 -> itree E R2 -> Prop
CIH0 : forall (x0 : itree E R1) (x1 : itree E R2),
gL x0 x1 \/ transU RR rH x0 x1 \/ rL x0 x1 -> r x0 x1
CIH : forall (x2 : itree E R1) (x3 : itree E R2), x x2 x3 -> r x2 x3
x2 : itree E R1 x3 : itree E R2
PR : euttG rH rL (gL \2/ x) (gH \2/ x) x2 x3
============================
gpaco2 (eqit_ RR true true (euttVC RR gH)) (transD RR)
(transU RR rH \2/ rL) r x2 x3
```
Next Tactic:
```
|
destruct PR.
```
|
{"final_commit":"f4bacabbc5b1556a723e3cb4efb1cd614c156d8f","project":"InteractionTrees","repair_file":"\/data\/coq-pearls\/repairs\/InteractionTrees\/repair-InteractionTrees-f3853150994e962978fa4cf0ed18dcb8c42e3262-f4bacabbc5b1556a723e3cb4efb1cd614c156d8f-8_11_2-8_10_2-18.git.json","initial_commit":"f3853150994e962978fa4cf0ed18dcb8c42e3262","file":"theories\/Eq\/UpToTaus.v"}
|
Ongoing Proof History:
```
intros l1; elim l1; simpl; auto.
intros l2 H H0; exists l2; simpl; auto.
intros a l H l2 H0 H1; auto.
case (in_permutation_ex _ a l2); auto with datatypes.
intros l3 Hl3.
case (H l3); auto.
apply ulist_inv with ( 1 := H0 ); auto.
intros b Hb.
assert (H2: In b (a :: l3)).
- apply permutation_in with ( 1 := permutation_sym _ _ _ Hl3 ); auto with datatypes.
+ apply Permutation_in with ( 1 := Permutation_sym Hl3 ); auto with datatypes.
simpl in H2 |-; case H2; intros H3; simpl; auto.
inversion_clear H0 as [|c lc Hk1]; auto.
case Hk1; subst a; auto.
intros l4 H4; exists l4.
? apply permutation_trans with (a :: l3); auto.
? apply permutation_sym; auto.
? Qed.
```
Proof State:
```
1 goal
A : Set eqA_dec : forall a b : A, {a = b} + {a <> b}
l1 : list A a : A l : list A
H : forall l2 : list A,
ulist l -> incl l l2 -> exists l3 : list A, Permutation l2 (l ++ l3)
l2 : list A
H0 : ulist (a :: l)
H1 : incl (a :: l) l2
l3 : list A
Hl3 : Permutation (a :: l3) l2
l4 : list A
H4 : Permutation l3 (l ++ l4)
============================
Permutation l2 (a :: l ++ l4)
```
Commit Diff:
```
diff --git a/Permutation.v b/Permutation.v
index ecfa699..af80c37 100644
--- a/Permutation.v
+++ b/Permutation.v
@@ -33,2 +32,0 @@ Variable A : Set.
-Definition permutation (l1 l2 : list A) := @Permutation A l1 l2.
-
@@ -37,66 +34,0 @@ Hint Constructors Permutation : core.
-Definition permutation_trans := perm_trans.
-
-
-
-
-
-Theorem permutation_refl : forall l : list A, permutation l l.
-Proof.
-simple induction l.
-apply perm_nil.
-intros a l1 H.
-apply perm_skip with (1 := H).
-Qed.
-Hint Resolve permutation_refl : core.
-
-
-
-
-
-Theorem permutation_sym : forall l m : list
-A, permutation l m -> permutation m l.
-Proof.
-apply Permutation_sym.
-Qed.
-
-
-
-
-
-Theorem permutation_length : forall l m : list A,
-permutation l m -> length l = length m.
-Proof.
-apply Permutation_length.
-Qed.
-
-
-
-
-
-Theorem perm_nil_inv : forall l : list A, permutation l nil -> l = nil.
-Proof.
-intros l H.
-apply Permutation_nil.
-apply Permutation_sym; assumption.
-Qed.
-
-
-
-
-Theorem permutation_one_inv : forall (a : A) (l : list A),
-permutation (a :: nil) l -> l = a :: nil.
-Proof.
-apply Permutation_length_1_inv.
-Qed.
-
-
-
-
-
-Theorem permutation_in : forall (a : A) (l m : list A),
-permutation l m -> In a l -> In a m.
-Proof.
-intros a l m H H0.
-eapply Permutation_in; eassumption.
-Qed.
-
@@ -107,26 +39,3 @@ Qed.
-Theorem permutation_app_comp : forall l1 l2 l3 l4,
-permutation l1 l2 -> permutation l3 l4 ->
-permutation (l1 ++ l3) (l2 ++ l4).
-Proof.
-intros l1 l2 l3 l4 H H0.
-apply Permutation_app; auto.
-Qed.
-Hint Resolve permutation_app_comp : core.
-
-
-
-
-
-Theorem permutation_app_swap : forall l1 l2, permutation
-(l1 ++ l2) (l2 ++ l1).
-Proof.
-apply Permutation_app_comm.
-Qed.
-
-
-
-
-
-Theorem perm_transposition : forall a b l1 l2 l3, permutation
-(l1 ++ a :: l2 ++ b :: l3)
-(l1 ++ b :: l2 ++ a :: l3).
+Theorem perm_transposition : forall a b (l1 l2 l3 : list
+A), Permutation (l1 ++ a :: l2 ++ b ::
+l3) (l1 ++ b :: l2 ++ a :: l3).
@@ -135,2 +44,2 @@ intros a b l1 l2 l3.
-apply permutation_app_comp; auto.
-change (permutation ((a :: nil) ++ l2 ++ (b
+apply Permutation_app; auto.
+change (Permutation ((a :: nil) ++ l2 ++ (b
@@ -140 +49 @@ repeat rewrite <- app_ass.
-apply permutation_app_comp; auto.
+apply Permutation_app; auto.
@@ -142 +51 @@ apply perm_trans with ((b :: nil) ++ (a :: nil) ++ l2); auto.
-apply permutation_app_swap; auto.
+apply Permutation_app_comm; auto.
@@ -144,2 +53,2 @@ repeat rewrite app_ass.
-apply permutation_app_comp; auto.
-apply permutation_app_swap; auto.
+apply Permutation_app; auto.
+apply Permutation_app_comm; auto.
@@ -153 +62 @@ Theorem in_permutation_ex : forall a l, In a l ->
-exists l1 : list A, permutation (a :: l1) l.
+exists l1 : list A, Permutation (a :: l1) l.
@@ -167,13 +75,0 @@ Qed.
-Theorem permutation_inv : forall (a : A) (l1
-l2 : list A), permutation (a :: l1)
-(a :: l2) -> permutation l1 l2.
-Proof.
-intros a l1 l2 H.
-eapply Permutation_cons_inv.
-eassumption.
-Qed.
-
-
-
-
-
@@ -195 +91 @@ l2 : list A), In (a, l1) (split_one
-l2) -> permutation (a :: l1) l2.
+l2) -> Permutation (a :: l1) l2.
@@ -256 +152 @@ Lemma all_permutations_aux_permutation : forall (n : nat) (l1 l2
-l1 (all_permutations_aux l2 n) -> permutation l1 l2.
+l1 (all_permutations_aux l2 n) -> Permutation l1 l2.
@@ -276,2 +172,2 @@ change (length l2 = length (a1 :: l3)) in |- *.
-apply permutation_length; auto.
```
Next Tactic:
```
|
apply <LOOKUP>perm_trans : forall [A : Type] [l l' l'' : list A], Permutation l l' -> Permutation l' l'' -> Permutation l l''</LOOKUP> with (a :: l3); auto.
```
|
{"final_commit":"5a29a619b6acc23531ef31506b909056de730bf6","project":"sudoku","repair_file":"\/data\/coq-pearls\/repairs\/sudoku\/repair-sudoku-b5543771d6bdeab13ec58ac6c8d281815d96ed62-5a29a619b6acc23531ef31506b909056de730bf6-8_15_2-8_15_2-7.git.json","initial_commit":"b5543771d6bdeab13ec58ac6c8d281815d96ed62","file":"UList.v"}
|
Commit Diff:
```
diff --git a/theories/Aux.v b/theories/Aux.v
index d240627..c7e8186 100644
--- a/theories/Aux.v
+++ b/theories/Aux.v
@@ -27 +26,0 @@
-
@@ -30,0 +30 @@ Require Export List.
+Export ListNotations.
@@ -85,11 +84,0 @@ Hint Resolve le_minus: arith.
-Section EqBool.
-
-Definition eq_bool_dec : forall a b : bool, {a = b} + {a <> b}.
-Proof.
-intros a b; case a; case b; simpl in |- *; auto.
-right; red in |- *; intros; discriminate.
-Defined.
-
-End EqBool.
-
-
@@ -231 +220 @@ intros a1 l H [H1| H1]; auto.
-exists (nil (A:=A)); exists l; simpl in |- *; auto.
+exists []; exists l; simpl in |- *; auto.
@@ -270,4 +259,4 @@ Theorem app_inv_app : forall l1 l2 l3 l4 a,
-l1 ++ l2 = l3 ++ a :: l4
--> (exists l5 : list A, l1 = l3
-++ a :: l5) \/ (exists l5 :
-_, l2 = l5 ++ a :: l4).
+l1 ++ l2 = l3 ++ a ::
+l4 -> (exists l5 : list A, l1
+= l3 ++ a :: l5) \/ (exists
+l5, l2 = l5 ++ a :: l4).
@@ -288,2 +277,2 @@ Theorem app_inv_app2 : forall l1 l2 l3 l4 a b, l1
-(exists l5 : list A, l1 = l3 ++ a ::
-b :: l5) \/ (exists l5 : _, l2 = l5
+(exists l5 : list A, l1 = l3 ++ a
+:: b :: l5) \/ (exists l5, l2 = l5
@@ -291 +280 @@ b :: l5) \/ (exists l5 : _, l2 = l5
-++ a :: nil /\ l2 = b :: l4.
+++ a :: [] /\ l2 = b :: l4.
@@ -310,7 +299,7 @@ Qed.
-Theorem same_length_ex : forall (a : A) l1
-l2 l3, length (l1 ++ a :: l2)
-= length l3 -> exists l4 :
-_, (exists l5 : _, (exists b
-: B, length l1 = length l4
-/\ length l2 = length l5 /\
-l3 = l4 ++ b :: l5)).
+Theorem same_length_ex : forall (a : A)
+l1 l2 l3, length (l1 ++ a
+:: l2) = length l3 -> exists
+l4, (exists l5, (exists b : B,
+length l1 = length l4 /\ length
+l2 = length l5 /\ l3
+= l4 ++ b :: l5)).
@@ -320 +309 @@ intros l2 l3; case l3; simpl in |- *; try (intros; discriminate).
-intros b l H; exists (nil (A:=B)); exists l; exists b; repeat (split; auto).
+intros b l H; exists []; exists l; exists b; repeat (split; auto).
@@ -341,3 +330,3 @@ Qed.
-Theorem in_map_fst_inv : forall a (l : list (B
-* C)), In a (map (fst (B:=_)) l) ->
-exists c : _, In (a, c) l.
+Theorem in_map_fst_inv : forall a (l : list
+(B * C)), In a (map (fst (B:=_))
+l) -> exists c, In (a, c) l.
```
Ongoing Proof History:
```
apply trans_equal with (1 := E1); rewrite HH; auto.
intros (HH1, HH2).
- cut (ln0 = nil /\ ln4 = nil /\ a = S n); [ intros (HH3, (HH4, HH5)) | generalize HH1; rewrite H7; case ln0; simpl in |- *; [ case ln4; try (intros; discriminate); (intros HH6; injection HH6; auto) | intros n0 l; case l; simpl in |- *; intros; discriminate ] ].
+ cut (ln0 = [] /\ ln4 = [] /\ a = S n); [ intros (HH3, (HH4, HH5)) | generalize HH1; rewrite H7; case ln0; simpl in |- *; [ case ln4; try (intros; discriminate); (intros HH6; injection HH6; auto) | intros n0 l; case l; simpl in |- *; intros; discriminate ] ].
generalize E1 H1; case ln2; simpl in |- *; auto; clear E1 H1.
intros E1 H1; case height_pred_not_nil2 with (1 := H1); auto.
generalize (height_pred_length _ _ _ _ H1); case l2; simpl in |- *; auto; intros; discriminate.
intros n0 ln5 E1 H1; case height_pred_larger_strict with (n1 := n0) (1 := H1); simpl in |- *; auto with datatypes.
intros HH6; Contradict HH6; apply le_not_lt; rewrite <- HH5; apply H5; rewrite E1; auto with datatypes.
? intros (H8, H9); left; exists (nil (A:=nat)); injection H8.
? intros HH7 HH8; rewrite HH5; rewrite <- HH8; rewrite <- HH7; rewrite E1; rewrite HH4; auto.
? intros (ln4, H7); auto.
```
Proof State:
```
2 subgoals
A : Type f : A -> nat eqA_dec : forall a b : A, {a = b} + {a <> b}
n : nat ln1, ln2 : list nat t1, t2 : btree A
l1, l2 : list (btree A)
H : height_pred (S n) ln1 l1 t1
ln0, ln3 : list nat a : nat
H3 : ln1 ++ ln2 = ln0 ++ a :: ln3
H4 : forall n1 : nat, In n1 ln0 -> n1 < a
H5 : forall n1 : nat, In n1 ln3 -> n1 <= a
ln4 : list nat
H7 : ln1 = ln0 ++ a :: ln4
HH1 : ln1 = [S n]
HH2 : l1 = [t1]
HH3 : ln0 = []
HH4 : ln4 = []
HH5 : a = S n
n0 : nat ln5 : list nat
E1 : ln3 = ln4 ++ n0 :: ln5
H1 : height_pred (S n) (n0 :: ln5) l2 t2
============================
n0 :: ln5 = [S n] /\ l2 = [t2] ->
(exists ln6 : list nat, ln3 = a :: ln6) \/
ln1 ++ n0 :: ln5 = [n] /\ l1 ++ l2 = [node t1 t2]
subgoal 2 is:
(exists l5 : list nat, ln2 = l5 ++ a :: ln3) ->
(exists ln4 : list nat, ln3 = a :: ln4) \/
ln1 ++ ln2 = [n] /\ l1 ++ l2 = [node t1 t2]
```
Next Tactic:
```
|
intros (H8, H9); left; exists []; injection H8.
```
|
{"final_commit":"6e3f37b2113e34f74c6b5a6ff2c7346a9cdf9e4e","project":"huffman","repair_file":"\/data\/coq-pearls\/repairs\/huffman\/repair-huffman-773c1feea03edc5e845ecafc7c9752fa403e7faf-6e3f37b2113e34f74c6b5a6ff2c7346a9cdf9e4e-8_11_2-8_11_2-27.git.json","initial_commit":"773c1feea03edc5e845ecafc7c9752fa403e7faf","file":"theories\/HeightPred.v"}
|
Commit Diff:
```
diff --git a/theories/Core/KTreeFacts.v b/theories/Core/KTreeFacts.v
index b911a90..32f56a6 100644
--- a/theories/Core/KTreeFacts.v
+++ b/theories/Core/KTreeFacts.v
@@ -6,2 +6 @@ Require Import
-Program Classes.Morphisms
-Setoids.Setoid
+Classes.Morphisms Setoids.Setoid
@@ -24 +23,2 @@ Eq.Eq
-Eq.UpToTaus.
+Eq.UpToTaus
+Eq.Paco2.
@@ -68 +68 @@ Proof.
-ginit. gcofix CIH. intros.
+ginit. pcofix CIH. intros.
@@ -205,2 +205,2 @@ Proof.
-revert A B C f g a0.
-ginit. gcofix CIH. intros.
+revert f g a0.
+ginit. pcofix CIH. intros.
@@ -268 +268 @@ revert a0.
-ginit. gcofix CIH. intros.
+ginit. pcofix CIH. intros.
@@ -277 +277 @@ revert a.
-gcofix CIH'. intros.
+pcofix CIH'. intros.
@@ -337,2 +337 @@ etau.
-specialize (CIH xa).
-cbn in CIH.
+specialize (CIHL xa). cbn in CIHL.
diff --git a/theories/Eq/Eq.v b/theories/Eq/Eq.v
index 21ac7e0..7f12983 100644
--- a/theories/Eq/Eq.v
+++ b/theories/Eq/Eq.v
@@ -29 +29,2 @@ Basics.Basics Basics.HeterogeneousRelations
-Core.ITreeDefinition.
+Core.ITreeDefinition
+Eq.Paco2.
@@ -165,0 +167,54 @@ Hint Unfold id: core.
+Lemma eqitF_inv_VisF_r {E R1 R2} (RR : R1 -> R2 -> Prop) {b1 b2 vclo sim}
+t1 X2 (e2 : E X2) (k2 : X2 -> _) : eqitF RR b1
+b2 vclo sim t1 (VisF e2 k2) -> (exists k1, t1 = VisF e2 k1
+/\ forall v, vclo sim (k1 v) (k2 v)) \/ (b1 /\ exists t1', t1
+= TauF t1' /\ eqitF RR b1 b2 vclo sim (observe t1') (VisF e2 k2)).
+Proof.
+refine (fun H => match H
+in eqitF _ _ _ _
+_ _ t2 return match t2
+return Prop with | VisF e2
+k2 => _ | _
+=> True end with |
+EqVis _ _ _ _
+_ _ _ _ _
+=> _ | _ =>
+_ end); try exact I.
+- left; eauto.
+- destruct i0; eauto.
+Qed.
+
+Lemma eqitF_inv_VisF_weak {E R1 R2} (RR : R1 -> R2 -> Prop) {b1 b2 vclo sim} X1
+(e1 : E X1) (k1 : X1 -> _) X2 (e2 : E X2) (k2 : X2
+-> _) : eqitF RR b1 b2 vclo sim (VisF e1 k1) (VisF e2 k2) -> exists
+p : X1 = X2, eqeq E p e1 e2 /\ pweqeq (vclo sim) p k1 k2.
+Proof.
+refine (fun H => match H in
+eqitF _ _ _ _ _ t1
+t2 return match t1, t2 return Prop
+with | VisF e1 k1, VisF e2
```
Proof State:
```
1 subgoal
E : Type -> Type X : Type R1 : Type R2 : Type
RR : R1 -> R2 -> Prop b1, b2 : bool t1 : itree E R1
CHECK : b1
b : R2 ma : itree E X
IHeqitF : t1 β
ma;; t1 ->
forall b0 : R2,
RetF b = RetF b0 ->
exists a : X,
eqit eq b1 b2 ma (Ret a) /\
eqit RR b1 b2 ((fun _ : X => t1) a) (Ret b)
H : eqitF RR b1 b2 id
(upaco2
(fun (sim : itree E R1 -> itree E R2 -> Prop)
(t1 : itree E R1) (t2 : itree E R2) =>
eqitF RR b1 b2 id sim (observe t1) (observe t2)) bot2)
(observe t1) (RetF b)
kb : X -> itree E R1 r : X
Hobma : observe ma = RetF r
H0 : Tau t1 β
kb r
a : X
H1 : eqit eq b1 b2 (Ret r) (Ret a)
H2 : eqit RR b1 b2 t1 (Ret b)
============================
eqit RR b1 b2 (kb a) (Ret b)
```
Ongoing Proof History:
```
revert ma kb tl Heqtl' H0 b Heqtr.
induction H.
-
intros; subst.
inv Heqtr.
destruct (observe tl) eqn: Hobtl; inv Heqtl'.
rewrite unfold_bind in H0.
destruct (observe ma) eqn: Hobma.
*
exists r0.
split.
rewrite <- Hobma.
tau_steps.
reflexivity.
cbn in *.
rewrite <- H0.
rewrite itree_eta, Hobtl.
apply eqit_Ret; auto.
*
cbn in H0.
rewrite itree_eta in H0.
rewrite Hobtl in H0.
- apply eqitree_inv_ret_tau in H0.
+ apply eqit_inv in H0.
contradiction.
*
cbn in H0.
rewrite itree_eta, Hobtl in H0.
- apply eqitree_inv_ret_vis in H0.
+ apply eqit_inv in H0.
contradiction.
-
intros.
inversion Heqtr.
-
intros.
inversion Heqtr.
-
intros.
subst.
apply simpobs in Heqtl'.
rewrite Heqtl' in H0; clear tl Heqtl'.
rewrite unfold_bind in H0.
destruct (observe ma) eqn: Hobma.
+
cbn in *.
specialize (IHeqitF ma (fun _ => t1) t1 eq_refl).
edestruct IHeqitF as (a & ? & ?);[| reflexivity |].
*
setoid_rewrite itree_eta at 4.
rewrite Hobma, bind_ret_l.
reflexivity.
*
exists a; split; auto.
rewrite itree_eta, Hobma in H1.
? apply eqit_inv_ret in H1; subst.
? rewrite <- H0.
? destruct b1; [| inv CHECK].
```
Next Tactic:
```
|
apply <LOOKUP>eqit_inv_Ret : forall (E : Type -> Type) (R1 R2 : Type) (RR : R1 -> R2 -> Prop) (b1 b2 : bool) (r1 : R1) (r2 : R2), eqit RR b1 b2 (Ret r1) (Ret r2) -> RR r1 r2</LOOKUP> in H1; subst.
```
|
{"final_commit":"7f00358aeb230f7970571fddaa8b256375910c62","project":"InteractionTrees","repair_file":"\/data\/coq-pearls\/repairs\/InteractionTrees\/repair-InteractionTrees-672193e2507cf3326b834f5fa47b7518af829b4b-7f00358aeb230f7970571fddaa8b256375910c62-8_10_2-8_10_2-15.git.json","initial_commit":"672193e2507cf3326b834f5fa47b7518af829b4b","file":"theories\/Eq\/Eq.v"}
|
Proof State:
```
2 goals
h, w : nat
Eq1 : ulist ref_list
Eq2 : length ref_list = size
s : list nat
H : length s = size * size
H1 : sat init_c s
HH0 : forall i : nat, i < size -> incl ref_list (row i s)
HH1 : forall i : nat, i < size -> incl ref_list (column i s)
HH2 : forall i : nat, i < size -> incl ref_list (rect i s)
HH3 : forall p : pos, valid_pos p -> In (get p s) ref_list
============================
length s = size * size /\
(forall i : nat, i < size -> Permutation (row i s) ref_list) /\
(forall i : nat, i < size -> Permutation (column i s) ref_list) /\
(forall i : nat, i < size -> Permutation (rect i s) ref_list)
goal 2 is:
sat init_c s
```
Ongoing Proof History:
```
- Proof with auto.
+ Proof.
generalize (ref_list_ulist); intros Eq1.
generalize (ref_list_length); intros Eq2.
intros s H; case (init_c_sat s); auto; intros HH0 HH1; split; intros H1.
case HH0; clear HH0 HH1; auto; intros HH0 (HH1, (HH2, HH3)); unfold sudoku.
? split...
? repeat split; intros i U; apply Permutation_sym; apply ulist_eq_permutation...
? rewrite length_row...
```
Commit Diff:
```
diff --git a/ListAux.v b/ListAux.v
index 769776a..986e88b 100644
--- a/ListAux.v
+++ b/ListAux.v
@@ -130,0 +131,6 @@ Qed.
+Theorem length_app: forall (l1 l2 : list A), length
+(l1 ++ l2) = length l1 + length l2.
+Proof.
+apply app_length.
+Qed.
+
@@ -272 +278 @@ intros A l1 l2; elim l1; simpl; auto.
-intros a l H; rewrite app_length, length_map, H; auto.
+intros a l H; rewrite length_app, length_map, H; auto.
diff --git a/ListOp.v b/ListOp.v
index 7a13f04..e1936ca 100644
--- a/ListOp.v
+++ b/ListOp.v
@@ -145 +145 @@ intros k Rec i j s H;
-rewrite app_length, length_take1; auto with arith.
+rewrite length_app, length_take1; auto with arith.
diff --git a/Sudoku.v b/Sudoku.v
index f9e372e..3a01485 100644
--- a/Sudoku.v
+++ b/Sudoku.v
@@ -675 +675 @@ assert (F1: div y w < h).
-apply div_lt; rewrite mult_comm...
+apply div_lt; rewrite mult_comm; auto.
@@ -677,2 +677,6 @@ apply div_lt; rewrite mult_comm...
-repeat (rewrite (fun x => mult_comm x h)); rewrite mod_mult_comp...
-rewrite div_mult_comp, (mod_small (div y w) h), (div_is_0 (div y w) h), plus_0_r...
+repeat (rewrite (fun x =>
+mult_comm x h)); rewrite mod_mult_comp...
+rewrite div_mult_comp; auto.
+rewrite (mod_small (div y w) h); auto.
+rewrite (div_is_0 (div y w) h); auto.
+rewrite plus_0_r.
@@ -699,3 +703,3 @@ cut (w <= size).
-intros U3 n; generalize h; elim n; simpl...
-clear n H1 H2 U1.
-+ intros h1; case h1; simpl...
+intros U3 n; generalize h; elim n; simpl;
+auto with arith; clear n H1 H2 U1.
++ intros h1; case h1; simpl; auto.
@@ -709,2 +713,3 @@ apply sym_equal; apply take_nth...
-** pose proof (mod_lt y w).
-lia.
+** apply le_trans with (1 := H1)...
+apply le_trans with (2 := U3).
+apply lt_le_weak; apply mod_lt...
@@ -721 +726 @@ apply mod_lt...
-case (le_or_lt (length l) w); intros H3.
+case (le_or_lt (length l) w); intros H1.
@@ -723 +728,2 @@ case (le_or_lt (length l) w); intros H3.
-** rewrite length_take_small, jump_too_far...
+** rewrite length_take_small...
+rewrite jump_too_far.
@@ -726,2 +732,2 @@ repeat rewrite nth_default...
-apply le_trans with (1 := H3)...
-*** apply le_trans with (1 := H3)...
+apply le_trans with (1 := H1)...
+*** apply le_trans with (1 := H1)...
@@ -732 +738,2 @@ repeat rewrite <- plus_assoc; rewrite minus_plus...
-rewrite jump_nth, jump_add, <- jump_nth...
+rewrite jump_nth; rewrite jump_add.
+rewrite <- jump_nth...
@@ -743 +750 @@ generalize (w_pos _ H); intros U2.
-rewrite length_take_and_jump...
+rewrite length_take_and_jump; auto.
@@ -760 +767,2 @@ rewrite (plus_assoc w).
-pattern w at 1; rewrite <- mult_1_r, <- mult_plus_distr_l, le_plus_minus_r...
+pattern w at 1; rewrite <- mult_1_r; rewrite <- mult_plus_distr_l.
+rewrite le_plus_minus_r...
@@ -846,2 +854,2 @@ l2 l3 -> lit_test l1 l3 = lit_test l1 l2.
-Proof with auto.
-intros l1 l2 l3; case l1; case l2; case l3; simpl...
+Proof.
+intros l1 l2 l3; case l1; case l2; case l3; simpl; auto.
@@ -851 +859 @@ case_eq (pos_test p2 p3); intros H1.
-rewrite (pos_test_trans p1 p2 p3); rewrite H...
+rewrite (pos_test_trans p1 p2 p3); rewrite H; auto.
@@ -853,2 +861,2 @@ assert (p2 = p3); subst.
-apply pos_test_exact with (1 := H1)...
-rewrite H...
+apply pos_test_exact with (1 := H1); auto.
+rewrite H; auto.
@@ -856,4 +864,4 @@ intros; discriminate.
-replace p1 with p2...
-case_eq (pos_test p2 p3); intros H1...
-intros; apply test_trans...
-apply sym_equal; apply pos_test_exact with (1 := H)...
+replace p1 with p2; auto.
+case_eq (pos_test p2 p3); intros H1; auto.
+intros; apply test_trans; auto.
+apply sym_equal; apply pos_test_exact with (1 := H); auto.
@@ -863,3 +871,3 @@ assert (p2 = p3); subst.
```
Next Tactic:
```
|
split; auto.
```
|
{"final_commit":"427448011e3341d35aa5b9e00f675f6582fcd477","project":"sudoku","repair_file":"\/data\/coq-pearls\/repairs\/sudoku\/repair-sudoku-dc8d4294a32dbb6e0a642dc553b64495f27e6d60-427448011e3341d35aa5b9e00f675f6582fcd477-8_15_2-8_15_2-72.git.json","initial_commit":"dc8d4294a32dbb6e0a642dc553b64495f27e6d60","file":"Sudoku.v"}
|
Commit Diff:
```
diff --git a/src/DblibTactics.v b/src/DblibTactics.v
index 2a08b3e..2aa54e9 100644
--- a/src/DblibTactics.v
+++ b/src/DblibTactics.v
@@ -2,3 +2 @@ Set Implicit Arguments.
-Require Import Compare_dec.
-Require Import Peano_dec.
-Require Import Lia.
+Require Export Omega.
@@ -12,7 +10,8 @@ Hint Extern 1 => f_equal : f_equal.
-Hint Extern 1 (_ = _ :> nat) => reflexivity : lia.
-Hint Extern 3 (_ = _ :> nat) => lia : lia.
-Hint Extern 3 (_ <> _ :> nat) => lia : lia.
-Hint Extern 3 (_ < _) => lia : lia.
-Hint Extern 3 (_ > _) => lia : lia.
-Hint Extern 3 (_ <= _) => lia : lia.
-Hint Extern 3 (_ >= _) => lia : lia.
+Hint Extern 1 (_ = _ :> nat) => reflexivity : omega.
+
+Hint Extern 1 (_ = _ :> nat) => omega : omega.
+Hint Extern 1 (_ <> _ :> nat) => omega : omega.
+Hint Extern 1 (_ < _) => omega : omega.
+Hint Extern 1 (_ > _) => omega : omega.
+Hint Extern 1 (_ <= _) => omega : omega.
+Hint Extern 1 (_ >= _) => omega : omega.
@@ -43 +42 @@ Ltac dblib_by_cases := repeat dblib_inspect_cases; try solve
-[ intros; exfalso; lia ]; intros.
\ No newline at end of file
+[ intros; exfalso; omega ]; intros.
\ No newline at end of file
diff --git a/src/DeBruijn.v b/src/DeBruijn.v
index 45bfa51..7a58a05 100644
--- a/src/DeBruijn.v
+++ b/src/DeBruijn.v
@@ -3,2 +2,0 @@ Generalizable All Variables.
-Require Import Arith.
-Require Import Lia.
@@ -286,3 +284,3 @@ intros.
-replace k with (ws + (k - ws)) by lia.
-erewrite <- lift_lift by lia.
-replace (ws + (k - ws) - ws) with (k - ws) by lia.
+replace k with (ws + (k - ws)) by omega.
+erewrite <- lift_lift by omega.
+replace (ws + (k - ws) - ws) with (k - ws) by omega.
@@ -456,2 +454,2 @@ eapply traverse_relative with (p := 0).
-intros m ?. replace (m + 0) with m by lia. eauto.
-lia.
+intros m ?. replace (m + 0) with m by omega. eauto.
+omega.
@@ -483,4 +481,3 @@ Qed.
-Ltac just_do_it := unfold
-subst, Subst_idx, subst_idx, lift,
-Lift_idx, var, Var_idx; intros;
-dblib_by_cases; eauto with lia.
+Ltac just_do_it := unfold subst, Subst_idx, subst_idx,
+lift, Lift_idx, var, Var_idx; intros; solve [
+dblib_by_cases; eauto with f_equal omega ].
@@ -512,2 +509,2 @@ Ltac lift_idx := first [ rewrite @lift_idx_recent by
-solve [ lia | eauto with lift_idx_hints
-] | rewrite @lift_idx_old by lia ].
+solve [ omega | eauto with lift_idx_hints
+] | rewrite @lift_idx_old by omega ].
@@ -518,2 +515,2 @@ Ltac lift_idx_in h := first [ rewrite @lift_idx_recent in
-h by solve [ lia | eauto with lift_idx_hints
-] | rewrite @lift_idx_old in h by lia ].
+h by solve [ omega | eauto with lift_idx_hints
+] | rewrite @lift_idx_old in h by omega ].
@@ -522,2 +519,2 @@ Ltac lift_idx_all := first [ rewrite @lift_idx_recent in *
-by solve [ lia | eauto with lift_idx_hints ]
-| rewrite @lift_idx_old in * by lia ].
+by solve [ omega | eauto with lift_idx_hints ]
+| rewrite @lift_idx_old in * by omega ].
@@ -587,2 +584,2 @@ Ltac subst_idx := first
-lia | rewrite
-@subst_idx_miss_1 by lia
+omega | rewrite
+@subst_idx_miss_1 by omega
@@ -590 +587 @@ lia | rewrite
-by lia ].
+by omega ].
@@ -594,2 +591,2 @@ Ltac subst_idx_in h := first
-by lia | rewrite @subst_idx_miss_1
-in h by lia
+by omega | rewrite @subst_idx_miss_1
+in h by omega
@@ -597 +594 @@ in h by lia
-h by lia ].
+h by omega ].
@@ -601,2 +598,2 @@ rewrite @subst_idx_identity in * by
-lia | rewrite @subst_idx_miss_1
-in * by lia
+omega | rewrite @subst_idx_miss_1
+in * by omega
@@ -604 +601 @@ in * by lia
-* by lia ].
+* by omega ].
@@ -667 +664 @@ intros. subst. simpl.
-eapply traverse_relative; [ | instantiate (1 := k1); lia ].
+eapply traverse_relative; [ | instantiate (1 := k1); omega ].
@@ -820,2 +817,2 @@ intros. f_equal.
-rewrite lift_lift by lia.
-f_equal. lia.
+rewrite lift_lift by omega.
+f_equal. omega.
```
Proof State:
```
1 subgoal
A : Type s : nat
IHs : forall (k : nat) (a b : option A) (e : env A),
k <= s ->
raw_insert k a (raw_insert s b e) =
raw_insert (1 + s) b (raw_insert k a e)
k : nat a, b : option A e : env A
H : S k <= S s
============================
raw_insert (S k) a (raw_insert (S s) b e) =
raw_insert (1 + S s) b (raw_insert (S k) a e)
```
Ongoing Proof History:
```
Proof.
intros ? k s.
generalize s k; clear s k.
induction s; intros.
-
- destruct k; [ | elimtype False; lia ].
+ destruct k; [ | elimtype False; omega ].
reflexivity.
-
{
destruct k.
-
reflexivity.
-
? destruct e; replace (1 + S s) with (S (1 + s)) by lia; simpl; f_equal; intuition.
? }
? Qed.
```
Next Tactic:
```
|
destruct e; replace (1 + <LOOKUP>S : nat -> nat</LOOKUP> s) with (<LOOKUP>S : nat -> nat</LOOKUP> (1 + s)) by omega; simpl; f_equal; eauto with omega.
```
|
{"final_commit":"3de04cadb63c28bfcffb194cf9e64e75a0c1f697","project":"dblib","repair_file":"\/data\/coq-pearls\/repairs\/dblib\/repair-dblib-d4ece98bb63f75c2a53e9d125338454e6e1393b2-3de04cadb63c28bfcffb194cf9e64e75a0c1f697-8_12_2-8_12_2-24.git.json","initial_commit":"d4ece98bb63f75c2a53e9d125338454e6e1393b2","file":"src\/Environments.v"}
|
Ongoing Proof History:
```
Proof.
intros.
repeat rewrite two_power_nat_two_p in *.
unfold Z.divide.
exists (two_p (Z.of_nat m - Z.of_nat n)).
assert ((Z.of_nat m) = (Z.of_nat m - Z.of_nat n) + Z.of_nat n) by lia.
rewrite H0 at 1.
assert (Z.of_nat m >= 0) by lia.
```
Proof State:
```
1 subgoal
n, m : nat
H : two_p (Z.of_nat n) <= two_p (Z.of_nat m)
H0 : Z.of_nat m = Z.of_nat m - Z.of_nat n + Z.of_nat n
H1 : Z.of_nat m >= 0
============================
two_p (Z.of_nat m - Z.of_nat n + Z.of_nat n) =
two_p (Z.of_nat m - Z.of_nat n) * two_p (Z.of_nat n)
```
Next Tactic:
```
|
assert (<LOOKUP>Z.of_nat : nat -> Z</LOOKUP> n >= 0) by lia.
```
|
{"package":"coq-vst","file":"\/home\/tpr\/.opam\/PackageTable~8.13~2022.01.csv.formatted\/lib\/coq\/user-contrib\/VST\/veric\/coqlib4.v"}
|
Ongoing Proof History:
```
Proof.
intros.
transitivity (PROPx P (LOCALx Q (EX x:A, SEPx (replace_nth n R (S x))))).
*
f_equal.
f_equal.
unfold SEPx.
simpl.
extensionality rho.
revert R H; induction n; destruct R; intros.
unfold replace_nth, fold_right.
simpl.
unfold nth in H.
rewrite H; clear H.
apply pred_ext.
apply exp_left; intro x.
apply exp_right with x.
apply exp_right with x.
auto.
apply exp_left; intro x.
auto.
unfold replace_nth, nth in *.
subst m.
unfold fold_right_sepcon.
fold (fold_right_sepcon R).
normalize.
unfold nth in H.
unfold replace_nth.
```
Proof State:
```
2 subgoals
n : nat P : list Prop Q : list localdef A : Type
S : A -> mpred rho : environ
IHn : forall R : list mpred,
nth n R emp = EX y, S y ->
fold_right_sepcon R =
EX x : A, fold_right_sepcon (replace_nth n R (S x))
H : emp = EX y, S y
============================
fold_right_sepcon nil = EX _ : A, fold_right_sepcon nil
subgoal 2 is:
fold_right_sepcon (m :: R) =
EX x : A, fold_right_sepcon (replace_nth (Datatypes.S n) (m :: R) (S x))
```
Next Tactic:
```
|
simpl.
```
|
{"package":"coq-vst","file":"\/home\/tpr\/.opam\/PackageTable~8.13~2022.01.csv.formatted\/lib\/coq\/user-contrib\/VST\/floyd\/client_lemmas.v"}
|
Proof State:
```
1 subgoal
s, s', s'' : t x, y, z : elt
============================
choose empty = None
```
Ongoing Proof History:
```
Proof.
```
Next Tactic:
```
|
generalize (@choose_1 <LOOKUP>empty : t</LOOKUP>).
```
|
{"package":"coq","file":"\/home\/tpr\/.opam\/PackageTable~8.13~2022.01.csv.formatted\/lib\/coq\/theories\/FSets\/FSetEqProperties.v"}
|
Ongoing Proof History:
```
+ unfold size; rewrite (mult_comm h); apply mult_lt_plus; auto.
- apply div_lt; rewrite mult_comm...
+ apply div_lt; rewrite mult_comm; auto.
apply ref_list_ulist.
repeat rewrite (fun x => mult_comm x h).
- unfold out in H4; rewrite H4...
+ unfold out in H4; rewrite H4; auto.
- repeat rewrite (mult_comm h)...
+ repeat rewrite (mult_comm h); auto.
- unfold size; repeat rewrite (mult_comm h); apply mult_lt_plus...
+ unfold size; repeat rewrite (mult_comm h); apply mult_lt_plus; auto.
- unfold size; rewrite (mult_comm w); apply mult_lt_plus...
+ unfold size; rewrite (mult_comm w); apply mult_lt_plus; auto.
- apply div_lt; rewrite mult_comm...
+ apply div_lt; rewrite mult_comm; auto.
repeat rewrite div_mult_comp; auto with arith.
rewrite (div_is_0 x1); auto with arith.
rewrite (div_is_0 y1); auto with arith.
case H; clear H; intros V1 V2; rewrite V1 in H3; rewrite V2 in H3.
match goal with H:(In ?X _) |- _ => apply (rm_not_in _ lit_test) with (a := X) (l1 := X::nil) (l2 := gen_rect (div x h * h + div y w) z); auto with datatypes end.
exact lit_test_trans.
intros; apply lit_test_anti_sym.
exact lit_test_exact.
apply olist_one.
apply gen_rect_ordered.
repeat rewrite (fun x => mult_comm x h).
repeat ((rewrite mod_mult_comp || rewrite div_mult_comp); auto).
- rewrite (div_is_0 (div y w) h)...
+ rewrite (div_is_0 (div y w) h); auto.
- rewrite (mod_small (div y w) h)...
+ rewrite (mod_small (div y w) h); auto.
? apply div_lt; rewrite (mult_comm w)...
? apply div_lt; rewrite (mult_comm w)...
? case (gen_cell_correct l x y); auto; intros tmp; case tmp; auto; clear tmp.
```
Commit Diff:
```
diff --git a/ListAux.v b/ListAux.v
index 769776a..986e88b 100644
--- a/ListAux.v
+++ b/ListAux.v
@@ -130,0 +131,6 @@ Qed.
+Theorem length_app: forall (l1 l2 : list A), length
+(l1 ++ l2) = length l1 + length l2.
+Proof.
+apply app_length.
+Qed.
+
@@ -272 +278 @@ intros A l1 l2; elim l1; simpl; auto.
-intros a l H; rewrite app_length, length_map, H; auto.
+intros a l H; rewrite length_app, length_map, H; auto.
diff --git a/ListOp.v b/ListOp.v
index 7a13f04..e1936ca 100644
--- a/ListOp.v
+++ b/ListOp.v
@@ -145 +145 @@ intros k Rec i j s H;
-rewrite app_length, length_take1; auto with arith.
+rewrite length_app, length_take1; auto with arith.
diff --git a/Sudoku.v b/Sudoku.v
index f9e372e..838c038 100644
--- a/Sudoku.v
+++ b/Sudoku.v
@@ -675 +675 @@ assert (F1: div y w < h).
-apply div_lt; rewrite mult_comm...
+apply div_lt; rewrite mult_comm; auto.
@@ -677,2 +677,6 @@ apply div_lt; rewrite mult_comm...
-repeat (rewrite (fun x => mult_comm x h)); rewrite mod_mult_comp...
-rewrite div_mult_comp, (mod_small (div y w) h), (div_is_0 (div y w) h), plus_0_r...
+repeat (rewrite (fun x =>
+mult_comm x h)); rewrite mod_mult_comp...
+rewrite div_mult_comp; auto.
+rewrite (mod_small (div y w) h); auto.
+rewrite (div_is_0 (div y w) h); auto.
+rewrite plus_0_r.
@@ -699,3 +703,3 @@ cut (w <= size).
-intros U3 n; generalize h; elim n; simpl...
-clear n H1 H2 U1.
-+ intros h1; case h1; simpl...
+intros U3 n; generalize h; elim n; simpl;
+auto with arith; clear n H1 H2 U1.
++ intros h1; case h1; simpl; auto.
@@ -709,2 +713,3 @@ apply sym_equal; apply take_nth...
-** pose proof (mod_lt y w).
-lia.
+** apply le_trans with (1 := H1)...
+apply le_trans with (2 := U3).
+apply lt_le_weak; apply mod_lt...
@@ -721 +726 @@ apply mod_lt...
-case (le_or_lt (length l) w); intros H3.
+case (le_or_lt (length l) w); intros H1.
@@ -723 +728,2 @@ case (le_or_lt (length l) w); intros H3.
-** rewrite length_take_small, jump_too_far...
+** rewrite length_take_small...
+rewrite jump_too_far.
@@ -726,2 +732,2 @@ repeat rewrite nth_default...
-apply le_trans with (1 := H3)...
-*** apply le_trans with (1 := H3)...
+apply le_trans with (1 := H1)...
+*** apply le_trans with (1 := H1)...
@@ -732 +738,2 @@ repeat rewrite <- plus_assoc; rewrite minus_plus...
-rewrite jump_nth, jump_add, <- jump_nth...
+rewrite jump_nth; rewrite jump_add.
+rewrite <- jump_nth...
@@ -743 +750 @@ generalize (w_pos _ H); intros U2.
-rewrite length_take_and_jump...
+rewrite length_take_and_jump; auto.
@@ -760 +767,2 @@ rewrite (plus_assoc w).
-pattern w at 1; rewrite <- mult_1_r, <- mult_plus_distr_l, le_plus_minus_r...
+pattern w at 1; rewrite <- mult_1_r; rewrite <- mult_plus_distr_l.
+rewrite le_plus_minus_r...
```
Proof State:
```
3 goals
h, w : nat
Eq1 : ulist ref_list
x, y, z : nat s : list nat
H1 : sudoku s
U1 : x < size
U2 : y < size
H2 : get (Pos x y) s = z
x1, y1 : nat
U3 : 0 < h
U4 : 0 < w
U5 : forall a b c : nat, a < b * c -> div a b < c
U6 : forall n m : nat, 0 < m -> mod n m < m
H6 : x1 < h
H7 : y1 < w
============================
div y w < h
goal 2 is:
div y w < h
goal 3 is:
False
```
Next Tactic:
```
|
apply <LOOKUP>div_lt : forall a b c : nat, a < b * c -> div a b < c</LOOKUP>; rewrite (mult_comm w); auto.
```
|
{"final_commit":"427448011e3341d35aa5b9e00f675f6582fcd477","project":"sudoku","repair_file":"\/data\/coq-pearls\/repairs\/sudoku\/repair-sudoku-dc8d4294a32dbb6e0a642dc553b64495f27e6d60-427448011e3341d35aa5b9e00f675f6582fcd477-8_15_2-8_15_2-16.git.json","initial_commit":"dc8d4294a32dbb6e0a642dc553b64495f27e6d60","file":"Sudoku.v"}
|
Proof State:
```
1 subgoal
E : Type -> Type R : Type r1, r2 : R
============================
Ret r1 β Ret r2 -> r1 = r2
```
Commit Diff:
```
diff --git a/theories/Core/KTreeFacts.v b/theories/Core/KTreeFacts.v
index b911a90..32f56a6 100644
--- a/theories/Core/KTreeFacts.v
+++ b/theories/Core/KTreeFacts.v
@@ -6,2 +6 @@ Require Import
-Program Classes.Morphisms
-Setoids.Setoid
+Classes.Morphisms Setoids.Setoid
@@ -24 +23,2 @@ Eq.Eq
-Eq.UpToTaus.
+Eq.UpToTaus
+Eq.Paco2.
@@ -68 +68 @@ Proof.
-ginit. gcofix CIH. intros.
+ginit. pcofix CIH. intros.
@@ -205,2 +205,2 @@ Proof.
-revert A B C f g a0.
-ginit. gcofix CIH. intros.
+revert f g a0.
+ginit. pcofix CIH. intros.
@@ -268 +268 @@ revert a0.
-ginit. gcofix CIH. intros.
+ginit. pcofix CIH. intros.
@@ -277 +277 @@ revert a.
-gcofix CIH'. intros.
+pcofix CIH'. intros.
@@ -337,2 +337 @@ etau.
-specialize (CIH xa).
-cbn in CIH.
+specialize (CIHL xa). cbn in CIHL.
diff --git a/theories/Eq/Eq.v b/theories/Eq/Eq.v
index 21ac7e0..b6665d0 100644
--- a/theories/Eq/Eq.v
+++ b/theories/Eq/Eq.v
@@ -29 +29,2 @@ Basics.Basics Basics.HeterogeneousRelations
-Core.ITreeDefinition.
+Core.ITreeDefinition
+Eq.Paco2.
@@ -165,0 +167,54 @@ Hint Unfold id: core.
+Lemma eqitF_inv_VisF_r {E R1 R2} (RR : R1 -> R2 -> Prop) {b1 b2 vclo sim}
+t1 X2 (e2 : E X2) (k2 : X2 -> _) : eqitF RR b1
+b2 vclo sim t1 (VisF e2 k2) -> (exists k1, t1 = VisF e2 k1
+/\ forall v, vclo sim (k1 v) (k2 v)) \/ (b1 /\ exists t1', t1
+= TauF t1' /\ eqitF RR b1 b2 vclo sim (observe t1') (VisF e2 k2)).
+Proof.
+refine (fun H => match H
+in eqitF _ _ _ _
+_ _ t2 return match t2
+return Prop with | VisF e2
+k2 => _ | _
+=> True end with |
+EqVis _ _ _ _
+_ _ _ _ _
+=> _ | _ =>
+_ end); try exact I.
+- left; eauto.
+- destruct i0; eauto.
+Qed.
+
+Lemma eqitF_inv_VisF_weak {E R1 R2} (RR : R1 -> R2 -> Prop) {b1 b2 vclo sim} X1
+(e1 : E X1) (k1 : X1 -> _) X2 (e2 : E X2) (k2 : X2
+-> _) : eqitF RR b1 b2 vclo sim (VisF e1 k1) (VisF e2 k2) -> exists
+p : X1 = X2, eqeq E p e1 e2 /\ pweqeq (vclo sim) p k1 k2.
+Proof.
+refine (fun H => match H in
+eqitF _ _ _ _ _ t1
+t2 return match t1, t2 return Prop
+with | VisF e1 k1, VisF e2
+k2 => _ | _, _ =>
+True end with | EqVis _
+_ _ _ _ _ _
+_ _ => _ | _
+=> _ end); try exact I.
+- exists eq_refl; cbn; eauto.
+- destruct i; exact I.
+Qed.
+
+Lemma eqitF_inv_VisF {E R1 R2} (RR : R1 -> R2 -> Prop) {b1 b2
+vclo sim} X (e : E X) (k1 : X -> _) (k2
+: X -> _) : eqitF RR b1 b2 vclo sim (VisF e
+k1) (VisF e k2) -> forall x, vclo sim (k1 x) (k2 x).
+Proof.
+intros H. dependent destruction H. assumption.
+Qed.
+
+Lemma eqitF_VisF_gen {E R1 R2} {RR : R1 -> R2 -> Prop} {b1 b2 vclo sim} {X1
+X2} (p : X1 = X2) (e1 : E X1) (k1 : X1 -> _) (e2 :
+E X2) (k2 : X2 -> _) : eqeq E p e1 e2 -> pweqeq (vclo sim)
+p k1 k2 -> eqitF RR b1 b2 vclo sim (VisF e1 k1) (VisF e2 k2).
+Proof.
+destruct p; intros <-; cbn; constructor; auto.
+Qed.
+
@@ -215,2 +270,2 @@ split.
-pstep. punfold H1. red in H1. red.
-hinduction H1 before CIH; intros; eauto.
+pstep. punfold H0. red in H0. red.
+hinduction H0 before CIH; intros; eauto.
@@ -223,2 +278,2 @@ red in REL. pclearbot. pinversion REL.
-pstep. punfold H1. red in H1. red.
-hinduction H1 before CIH; intros; eauto.
+pstep. punfold H0. red in H0. red.
+hinduction H0 before CIH; intros; eauto.
@@ -326 +381 @@ econstructor. pmonauto.
-intros. dependent destruction PR.
+intros. destruct PR.
@@ -338 +393 @@ pclearbot. econstructor. gclo.
-econstructor; cycle -1; eauto with paco.
+econstructor; eauto with paco.
@@ -340,3 +395,3 @@ econstructor; cycle -1; eauto with paco.
-hinduction EQVl before r; intros; subst; try dependent destruction Heqx; try inv CHECK; eauto.
-remember (VisF e0 k3) as y.
-hinduction EQVr before r; intros; subst; try dependent destruction Heqy; try inv CHECK; eauto.
+hinduction EQVl before r; intros; try discriminate Heqx; eauto; inv_Vis.
+remember (VisF e k3) as y.
+hinduction EQVr before r; intros; try discriminate Heqy; eauto; inv_Vis.
@@ -425 +480 @@ Proof.
-gcofix CIH. gstep; intros.
+pcofix CIH. gstep; intros.
@@ -443 +498 @@ Proof.
-ginit. gcofix CIH. intros.
```
Ongoing Proof History:
```
Proof.
? apply ITree.Eq.Eq.eqit_inv_ret.
? Qed.
```
Next Tactic:
```
|
apply ITree.Eq.Eq.eqit_inv_Ret.
```
|
{"final_commit":"7f00358aeb230f7970571fddaa8b256375910c62","project":"InteractionTrees","repair_file":"\/data\/coq-pearls\/repairs\/InteractionTrees\/repair-InteractionTrees-672193e2507cf3326b834f5fa47b7518af829b4b-7f00358aeb230f7970571fddaa8b256375910c62-8_10_2-8_10_2-58.git.json","initial_commit":"672193e2507cf3326b834f5fa47b7518af829b4b","file":"theories\/Simple.v"}
|
Ongoing Proof History:
```
Proof.
```
Proof State:
```
1 subgoal
R : ringType a : R
============================
'X - a%:P = [:: - a; 1]
```
Next Tactic:
```
|
by rewrite -['X]<LOOKUP>mul1r : forall [R : ringType], left_id 1 *%R</LOOKUP> -<LOOKUP>polyCN : {morph polyC : c / - c >-> - c}</LOOKUP> -<LOOKUP>cons_poly_def : forall p a, cons_poly a p = p * 'X + a%:P</LOOKUP> <LOOKUP>polyseq_cons : forall c p, cons_poly c p = (if ~~ nilp p then c :: p else c%:P)</LOOKUP> <LOOKUP>polyseq1 : (1 : {poly R}) = [:: 1]</LOOKUP>.
```
|
{"package":"coq-mathcomp-algebra","file":"\/home\/tpr\/.opam\/PackageTable~8.13~2022.01.csv.formatted\/lib\/coq\/user-contrib\/mathcomp\/algebra\/poly.v"}
|
Proof State:
```
3 focused subgoals
(shelved: 4)
f : R -> R r, r1 : R r2 : Rlist
H0 : forall (l2 lf1 lf2 : Rlist) (a b : R),
a <= b ->
adapted_couple f a b (r1 :: r2) lf1 ->
adapted_couple_opt f a b l2 lf2 ->
Int_SF lf1 (r1 :: r2) = Int_SF lf2 l2
l2 : Rlist r3 : R lf1 : Rlist r4 : R lf2 : Rlist a, b : R
H1 : a <= b
H2 : adapted_couple f a b (r :: r1 :: r2) (r3 :: lf1)
s1, s2 : R s3 : Rlist
H : adapted_couple f a b (s1 :: s2 :: s3) (r4 :: lf2)
H3 : (forall i : nat,
(i < Init.Nat.pred (Rlength (r4 :: lf2)))%nat ->
pos_Rl (r4 :: lf2) i <> pos_Rl (r4 :: lf2) (S i) \/
f (pos_Rl l2 (S i)) <> pos_Rl (r4 :: lf2) i) /\
(forall i : nat,
(i < Init.Nat.pred (Rlength l2))%nat ->
pos_Rl l2 i <> pos_Rl l2 (S i))
H4 : a <> b
Hyp_min : Rmin a b = a
Hyp_max : Rmax a b = b
H5 : l2 = s1 :: s2 :: s3
H6 : r = s1
H7 : r <> r1
============================
adapted_couple f a b (?s1 :: s2 :: ?s3) (?r4 :: ?lf2)
subgoal 2 is:
(forall i : nat,
(i < Init.Nat.pred (Rlength (?r4 :: ?lf2)))%nat ->
pos_Rl (?r4 :: ?lf2) i <> pos_Rl (?r4 :: ?lf2) (S i) \/
f (pos_Rl (?s1 :: s2 :: ?s3) (S i)) <> pos_Rl (?r4 :: ?lf2) i) /\
(forall i : nat,
(i < Init.Nat.pred (Rlength (?s1 :: s2 :: ?s3)))%nat ->
pos_Rl (?s1 :: s2 :: ?s3) i <> pos_Rl (?s1 :: s2 :: ?s3) (S i))
subgoal 3 is:
Int_SF (r3 :: lf1) (r :: r1 :: r2) = Int_SF (r4 :: lf2) (s1 :: s2 :: s3)
```
Ongoing Proof History:
```
Proof.
induction l1 as [ | r r0 H0].
intros l2 lf1 lf2 a b Hyp H H0; unfold adapted_couple in H; decompose [and] H; clear H H0 H2 H3 H1 H6; simpl in H4; discriminate.
induction r0 as [|r1 r2 H].
intros; case (Req_dec a b); intro.
unfold adapted_couple_opt in H2; elim H2; intros; rewrite (StepFun_P8 H4 H3); rewrite (StepFun_P8 H1 H3); reflexivity.
assert (H4 := StepFun_P9 H1 H3); simpl in H4; elim (le_Sn_O _ (le_S_n _ _ H4)).
intros; clear H; unfold adapted_couple_opt in H3; elim H3; clear H3; intros; case (Req_dec a b); intro.
rewrite (StepFun_P8 H2 H4); rewrite (StepFun_P8 H H4); reflexivity.
assert (Hyp_min : Rmin a b = a).
unfold Rmin; decide (Rle_dec a b) with H1; reflexivity.
assert (Hyp_max : Rmax a b = b).
unfold Rmax; decide (Rle_dec a b) with H1; reflexivity.
elim (RList_P20 _ (StepFun_P9 H H4)); intros s1 [s2 [s3 H5]]; rewrite H5 in H; rewrite H5; induction lf1 as [| r3 lf1 Hreclf1].
unfold adapted_couple in H2; decompose [and] H2; clear H H2 H4 H5 H3 H6 H8 H7 H11; simpl in H9; discriminate.
clear Hreclf1; induction lf2 as [| r4 lf2 Hreclf2].
unfold adapted_couple in H; decompose [and] H; clear H H2 H4 H5 H3 H6 H8 H7 H11; simpl in H9; discriminate.
clear Hreclf2; assert (H6 : r = s1).
unfold adapted_couple in H, H2; decompose [and] H; decompose [and] H2; clear H H2; simpl in H13; simpl in H8; rewrite H13; rewrite H8; reflexivity.
assert (H7 : r3 = r4 \/ r = r1).
case (Req_dec r r1); intro.
right; assumption.
left; cut (r1 <= s2).
intro; unfold adapted_couple in H2, H; decompose [and] H; decompose [and] H2; clear H H2; set (x := (r + r1) / 2); assert (H18 := H14 0%nat); assert (H20 := H19 0%nat); unfold constant_D_eq, open_interval in H18, H20; simpl in H18; simpl in H20; rewrite <- (H18 (lt_O_Sn _) x).
rewrite <- (H20 (lt_O_Sn _) x).
reflexivity.
assert (H21 := H13 0%nat (lt_O_Sn _)); simpl in H21; elim H21; intro; [ idtac | elim H7; assumption ]; unfold x; split.
apply Rmult_lt_reg_l with 2; [ prove_sup0 | unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym; [ rewrite Rmult_1_l; rewrite double; apply Rplus_lt_compat_l; apply H | discrR ] ].
apply Rmult_lt_reg_l with 2; [ prove_sup0 | unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym; [ rewrite Rmult_1_l; rewrite <- (Rplus_comm r1); rewrite double; apply Rplus_lt_compat_l; apply H | discrR ] ].
rewrite <- H6; assert (H21 := H13 0%nat (lt_O_Sn _)); simpl in H21; elim H21; intro; [ idtac | elim H7; assumption ]; unfold x; split.
apply Rmult_lt_reg_l with 2; [ prove_sup0 | unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym; [ rewrite Rmult_1_l; rewrite double; apply Rplus_lt_compat_l; apply H | discrR ] ].
apply Rlt_le_trans with r1; [ apply Rmult_lt_reg_l with 2; [ prove_sup0 | unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym; [ rewrite Rmult_1_l; rewrite <- (Rplus_comm r1); rewrite double; apply Rplus_lt_compat_l; apply H | discrR ] ] | assumption ].
eapply StepFun_P13.
apply H4.
apply H2.
unfold adapted_couple_opt; split.
```
Next Tactic:
```
|
apply H.
```
|
{"package":"coq","file":"\/home\/tpr\/.opam\/PackageTable~8.13~2022.01.csv.formatted\/lib\/coq\/theories\/Reals\/RiemannInt_SF.v"}
|
Proof State:
```
1 subgoal
E : Type -> Type S : Type R : Type A : Type
e0 : E A ans : A ks : unit -> itrace E S
k2 : unit -> itree (EvAns E) S
REL : forall v : unit, paco2 (eqit_ eq true true id) bot2 (ks v) (k2 v)
m2 : itree (EvAns E) S r : itrace E R -> itrace E S -> Prop
k : unit -> itree (EvAns E) R
H : paco2 trace_prefix_ bot2 (k tt) (ks tt)
m1 : itree (EvAns E) S
x : VisF (evans A e0 ans) k2 = observe m2
x0 : VisF (evans A e0 ans) ks = observe m1
CIH : forall (b : itrace E R) (b1 b2 : itrace E S),
b1 β b2 -> trace_prefix b b1 -> r b b2
============================
r (k tt) (k2 tt)
```
Ongoing Proof History:
```
Proof.
intros E S R m1 m2 REL r X e k H1 CIH.
punfold REL.
red in REL.
dependent induction H1.
-
eapply IHtrace_prefixF; auto.
rewrite <- x in REL.
assert (Tau bs β m2).
{
pfold.
auto.
}
rewrite tau_eutt in H.
punfold H.
-
rewrite <- x in REL.
dependent induction REL.
+
rewrite <- x.
constructor.
+
rewrite <- x.
constructor.
eapply IHREL; eauto.
-
pclearbot.
rewrite <- x in REL.
dependent induction REL.
+
rewrite <- x.
constructor.
right.
pclearbot.
? eapply CIH; eauto.
? +
? rewrite <- x.
```
Commit Diff:
```
diff --git a/theories/Basics/Basics.v b/theories/Basics/Basics.v
index ff16faf..1f991e2 100644
--- a/theories/Basics/Basics.v
+++ b/theories/Basics/Basics.v
@@ -57,0 +58,2 @@ Definition state (s a : Type) := s -> prod s a.
+Definition run_stateT {s m a} (x : stateT s m a) : s -> m (s * a)%type := x.
+
diff --git a/theories/Basics/HeterogeneousRelations.v b/theories/Basics/HeterogeneousRelations.v
index 8894184..ad37cca 100644
--- a/theories/Basics/HeterogeneousRelations.v
+++ b/theories/Basics/HeterogeneousRelations.v
@@ -751 +751,21 @@ cbn in *. unfold diagonal_prop in *; tauto.
-Qed.
\ No newline at end of file
+Qed.
+
+
+
+
+
+Definition option_rel {X : Type} (R :
+relation X) : relation (option X) :=
+fun mx my => match mx,my with
+| Some x, Some y => R
+x y | None, None => True
+| _, _ => False end.
+Hint Unfold option_rel : core.
+
+Lemma option_rel_eq : forall {A : Type},
+eq_rel (@eq (option A)) (option_rel eq).
+Proof.
+intros ?; split; intros [] [] EQ; subst; try inv EQ; cbn; auto.
+Qed.
+
+#[global] Hint Unfold option_rel : core.
\ No newline at end of file
diff --git a/theories/Core/ITreeDefinition.v b/theories/Core/ITreeDefinition.v
index 3f88726..eff8f67 100644
--- a/theories/Core/ITreeDefinition.v
+++ b/theories/Core/ITreeDefinition.v
@@ -69,0 +70,2 @@ Arguments itreeF _ _ : clear implicits.
+Create HintDb itree.
+
diff --git a/theories/Core/KTreeFacts.v b/theories/Core/KTreeFacts.v
index a1c2431..9949acf 100644
--- a/theories/Core/KTreeFacts.v
+++ b/theories/Core/KTreeFacts.v
@@ -113 +113 @@ intros ? ? ?. red.
-induction 1; auto.
+induction 1; auto with itree.
diff --git a/theories/Dijkstra/ITreeDijkstra.v b/theories/Dijkstra/ITreeDijkstra.v
index 9529757..b25a16e 100644
--- a/theories/Dijkstra/ITreeDijkstra.v
+++ b/theories/Dijkstra/ITreeDijkstra.v
@@ -189 +189 @@ is_inf_cons (h : A) (t : stream A) : F t -> is_infF F (ConsF h t).
-Hint Constructors is_infF : core.
+Hint Constructors is_infF : itree.
@@ -198 +198 @@ Proof.
-red. intros. red in IN. red. induction IN; auto.
+red. intros. red in IN. red. induction IN; auto with itree.
@@ -217 +217 @@ s2 : stream A) : F s1 s2 -> bisimF F (ConsF h s1) (ConsF h s2).
-Hint Constructors bisimF : core.
+Hint Constructors bisimF : itree.
@@ -226 +226 @@ Proof.
-red. intros. red in IN. red. induction IN; auto.
+red. intros. red in IN. red. induction IN; auto with itree.
@@ -234 +234 @@ constructor; red.
-- pcofix CIH. intros. pfold. red. destruct (observe_stream x); auto.
+- pcofix CIH. intros. pfold. red. destruct (observe_stream x); auto with itree.
@@ -237 +237 @@ pfold. red.
-pinversion H0; subst; auto.
+pinversion H0; subst; auto with itree.
@@ -239 +239 @@ pinversion H0; subst; auto.
-pinversion H0; pinversion H1; auto.
+pinversion H0; pinversion H1; auto with itree.
@@ -250 +250 @@ pfold. red. unfold app. pinversion H12.
-- simpl. destruct s3. destruct s4. pinversion H34; simpl in *; subst; auto.
+- simpl. destruct s3. destruct s4. pinversion H34; simpl in *; subst; auto with itree.
@@ -282 +282 @@ stream' A -> Prop := | forall_nil : forall_streamF P F NilF | forall_cons (h : A
-Hint Constructors forall_streamF : core.
+Hint Constructors forall_streamF : itree.
@@ -289 +289 @@ Proof.
-red. intros. red. red in IN. destruct IN; auto.
+red. intros. red. red in IN. destruct IN; auto with itree.
@@ -301 +301 @@ F (observe_stream t) -> inf_manyF P F (ConsF h t) | cons_found (h : A) (t
-Hint Constructors inf_manyF : core.
+Hint Constructors inf_manyF : itree.
```
Next Tactic:
```
|
eapply CIH; eauto with itree.
```
|
{"final_commit":"8fcf6f5f81ec9289d56e777b3885e715b3328486","project":"InteractionTrees","repair_file":"\/data\/coq-pearls\/repairs\/InteractionTrees\/repair-InteractionTrees-1158ff1cbb628aeaa69973bf098058744d458c36-8fcf6f5f81ec9289d56e777b3885e715b3328486-8_10_2-8_12_2-76.git.json","initial_commit":"1158ff1cbb628aeaa69973bf098058744d458c36","file":"theories\/ITrace\/ITracePrefix.v"}
|
Commit Diff:
```
diff --git a/theories/Basics/Basics.v b/theories/Basics/Basics.v
index ff16faf..1f991e2 100644
--- a/theories/Basics/Basics.v
+++ b/theories/Basics/Basics.v
@@ -57,0 +58,2 @@ Definition state (s a : Type) := s -> prod s a.
+Definition run_stateT {s m a} (x : stateT s m a) : s -> m (s * a)%type := x.
+
diff --git a/theories/Basics/HeterogeneousRelations.v b/theories/Basics/HeterogeneousRelations.v
index 8894184..ad37cca 100644
--- a/theories/Basics/HeterogeneousRelations.v
+++ b/theories/Basics/HeterogeneousRelations.v
@@ -751 +751,21 @@ cbn in *. unfold diagonal_prop in *; tauto.
-Qed.
\ No newline at end of file
+Qed.
+
+
+
+
+
+Definition option_rel {X : Type} (R :
+relation X) : relation (option X) :=
+fun mx my => match mx,my with
+| Some x, Some y => R
+x y | None, None => True
+| _, _ => False end.
+Hint Unfold option_rel : core.
+
+Lemma option_rel_eq : forall {A : Type},
+eq_rel (@eq (option A)) (option_rel eq).
+Proof.
+intros ?; split; intros [] [] EQ; subst; try inv EQ; cbn; auto.
+Qed.
+
+#[global] Hint Unfold option_rel : core.
\ No newline at end of file
diff --git a/theories/Core/ITreeDefinition.v b/theories/Core/ITreeDefinition.v
index 3f88726..eff8f67 100644
--- a/theories/Core/ITreeDefinition.v
+++ b/theories/Core/ITreeDefinition.v
@@ -69,0 +70,2 @@ Arguments itreeF _ _ : clear implicits.
+Create HintDb itree.
+
diff --git a/theories/Core/KTreeFacts.v b/theories/Core/KTreeFacts.v
index a1c2431..9949acf 100644
--- a/theories/Core/KTreeFacts.v
+++ b/theories/Core/KTreeFacts.v
@@ -113 +113 @@ intros ? ? ?. red.
-induction 1; auto.
+induction 1; auto with itree.
diff --git a/theories/Dijkstra/ITreeDijkstra.v b/theories/Dijkstra/ITreeDijkstra.v
index 9529757..b25a16e 100644
--- a/theories/Dijkstra/ITreeDijkstra.v
+++ b/theories/Dijkstra/ITreeDijkstra.v
@@ -189 +189 @@ is_inf_cons (h : A) (t : stream A) : F t -> is_infF F (ConsF h t).
-Hint Constructors is_infF : core.
+Hint Constructors is_infF : itree.
@@ -198 +198 @@ Proof.
-red. intros. red in IN. red. induction IN; auto.
+red. intros. red in IN. red. induction IN; auto with itree.
@@ -217 +217 @@ s2 : stream A) : F s1 s2 -> bisimF F (ConsF h s1) (ConsF h s2).
-Hint Constructors bisimF : core.
+Hint Constructors bisimF : itree.
@@ -226 +226 @@ Proof.
-red. intros. red in IN. red. induction IN; auto.
+red. intros. red in IN. red. induction IN; auto with itree.
@@ -234 +234 @@ constructor; red.
-- pcofix CIH. intros. pfold. red. destruct (observe_stream x); auto.
+- pcofix CIH. intros. pfold. red. destruct (observe_stream x); auto with itree.
```
Proof State:
```
1 subgoal
E : Type -> Type S : Type R : Type A : Type
e0 : E A ans : A kr : unit -> itrace E R
k2 : unit -> itree (EvAns E) R
REL0 : forall v : unit, paco2 (eqit_ eq true true id) bot2 (kr v) (k2 v)
m2 : itree (EvAns E) R r : itrace E R -> itrace E S -> Prop
b1, b2 : itrace E R b : itrace E S k : unit -> itree (EvAns E) S
H : paco2 trace_prefix_ bot2 (kr tt) (k tt)
m1 : itree (EvAns E) R
REL : paco2 (eqit_ eq true true id) bot2 m1 m2
CIH : forall (b1 b2 : itrace E R) (b : itrace E S),
b1 β b2 -> trace_prefix b1 b -> r b2 b
x0 : m1 β
Vis (evans A e0 ans) kr
x : VisF (evans A e0 ans) k2 = observe m2
============================
r (k2 tt) (k tt)
```
Ongoing Proof History:
```
apply simpobs in x.
assert (m1 β m2); auto.
rewrite x in H.
rewrite tau_eutt in H.
auto.
+
constructor.
right.
eapply CIH; eauto.
eapply trace_prefix_tau_inv; eauto.
+
constructor.
clear Heqb.
inv Hbp.
dependent induction H0.
*
apply simpobs in x0.
assert (m1 β m2); auto.
rewrite x0 in H.
punfold H.
red in H.
cbn in *.
dependent induction H.
++
rewrite <- x.
apply trace_prefix_ret.
++
rewrite <- x.
constructor.
eapply IHeqitF; eauto.
assert (m1 β m2); auto.
apply simpobs in x.
rewrite x in H0.
rewrite tau_eutt in H0.
auto.
*
eapply IHtrace_prefixF; auto.
assert (m1 β m2); auto.
apply simpobs in x.
rewrite x in H.
rewrite tau_eutt in H.
auto.
*
assert (m1 β m2); auto.
apply simpobs in x.
rewrite x in H0.
punfold H0.
red in H0.
cbn in *.
dependent induction H0.
++
rewrite <- x.
constructor.
++
rewrite <- x.
constructor.
eapply IHeqitF; eauto.
assert (m1 β m2); auto.
apply simpobs in x.
rewrite x in H1.
rewrite tau_eutt in H1.
auto.
*
pclearbot.
apply simpobs in x.
assert (m1 β m2); auto.
rewrite x in H0.
punfold H0.
red in H0.
cbn in *.
dependent induction H0.
++
rewrite <- x.
constructor.
right.
pclearbot.
? eapply CIH; eauto.
? ++
? rewrite <- x.
```
Next Tactic:
```
|
eapply CIH; eauto with itree.
```
|
{"final_commit":"8fcf6f5f81ec9289d56e777b3885e715b3328486","project":"InteractionTrees","repair_file":"\/data\/coq-pearls\/repairs\/InteractionTrees\/repair-InteractionTrees-1158ff1cbb628aeaa69973bf098058744d458c36-8fcf6f5f81ec9289d56e777b3885e715b3328486-8_10_2-8_12_2-75.git.json","initial_commit":"1158ff1cbb628aeaa69973bf098058744d458c36","file":"theories\/ITrace\/ITracePrefix.v"}
|
Proof State:
```
1 subgoal
m, n : nat p : t m IHn : t (n + m)
============================
t (S n + m)
```
Ongoing Proof History:
```
Proof.
induction n as [|n IHn].
-
exact p.
-
```
Next Tactic:
```
|
exact ((fix LS k (<LOOKUP>p : t m</LOOKUP>: <LOOKUP>t : nat -> Set</LOOKUP> k) := match <LOOKUP>p : t m</LOOKUP> with |@F1 k' => @F1 (<LOOKUP>S : nat -> nat</LOOKUP> k') |FS p' => <LOOKUP>FS : forall {n : nat}, t n -> t (S n)</LOOKUP> (LS _ p') end) _ IHn).
```
|
{"package":"coq","file":"\/home\/tpr\/.opam\/PackageTable~8.13~2022.01.csv.formatted\/lib\/coq\/theories\/Vectors\/Fin.v"}
|
Proof State:
```
1 subgoal
S : ident -> Prop i : ident
H : ~ S i
============================
closed_wrt_vars S
(fun rho : environ => env_set (globals_only rho) ret_temp (eval_id i rho))
```
Ongoing Proof History:
```
Proof.
intros.
unfold get_result1.
simpl.
```
Next Tactic:
```
|
hnf; intros.
```
|
{"package":"coq-vst","file":"\/home\/tpr\/.opam\/PackageTable~8.13~2022.01.csv.formatted\/lib\/coq\/user-contrib\/VST\/floyd\/closed_lemmas.v"}
|
Ongoing Proof History:
```
Proof.
induction n; destruct a; simpl in *; intros; auto.
- exfalso; omega.
+ exfalso; lia.
f_equal.
eapply IHn; eauto.
? omega.
? Qed.
```
Commit Diff:
```
diff --git a/examples/ConsiderDemo.v b/examples/ConsiderDemo.v
index 2151e52..9490a21 100644
--- a/examples/ConsiderDemo.v
+++ b/examples/ConsiderDemo.v
@@ -7,0 +8 @@ Require Import Coq.ZArith.ZArith.
+Require Import Coq.micromega.Lia.
@@ -18 +19 @@ consider (ltb x y && ltb y z).
-consider (ltb x z); auto. intros. exfalso. omega.
+consider (ltb x z); auto. intros. exfalso. lia.
@@ -27 +28 @@ intros. consider (ltb x y); consider (ltb y z); consider (ltb x z); intros; auto
-exfalso; omega.
+exfalso; lia.
diff --git a/theories/Data/ListFirstnSkipn.v b/theories/Data/ListFirstnSkipn.v
index cf126d1..dda9dff 100644
--- a/theories/Data/ListFirstnSkipn.v
+++ b/theories/Data/ListFirstnSkipn.v
@@ -2,0 +3 @@ Require Import Coq.ZArith.ZArith.
+Require Import Coq.micromega.Lia.
@@ -18,2 +19,2 @@ induction n; destruct a; simpl in *; intros; auto.
-exfalso; omega.
-f_equal. eapply IHn; eauto. omega.
+exfalso; lia.
+f_equal. eapply IHn; eauto. lia.
@@ -27,2 +28,2 @@ induction n; destruct a; simpl; intros; auto.
-exfalso; omega.
-simpl. f_equal. eapply IHn; omega.
+exfalso; lia.
+simpl. f_equal. eapply IHn; lia.
@@ -43,2 +44,2 @@ destruct n; intros.
-omega.
-simpl. replace (n - 0) with n; [ | omega ]. reflexivity.
+lia.
+simpl. replace (n - 0) with n; [ | lia ]. reflexivity.
@@ -47 +48 @@ Qed.
-Hint Rewrite firstn_app_L firstn_app_R firstn_all firstn_0 firstn_cons using omega : list_rw.
+Hint Rewrite firstn_app_L firstn_app_R firstn_all firstn_0 firstn_cons using lia : list_rw.
@@ -54,2 +55,2 @@ induction n; destruct a; simpl in *; intros; auto.
-exfalso; omega.
-eapply IHn. omega.
+exfalso; lia.
+eapply IHn. lia.
@@ -63,2 +64,2 @@ induction n; destruct a; simpl in *; intros; auto.
-exfalso; omega.
-eapply IHn. omega.
+exfalso; lia.
+eapply IHn. lia.
@@ -79,2 +80,2 @@ induction n; destruct a; simpl in *; intros; auto.
-exfalso; omega.
-apply IHn; omega.
+exfalso; lia.
+apply IHn; lia.
@@ -88,2 +89,2 @@ destruct n; intros.
-omega.
-simpl. replace (n - 0) with n; [ | omega ]. reflexivity.
+lia.
+simpl. replace (n - 0) with n; [ | lia ]. reflexivity.
@@ -92 +93 @@ Qed.
-Hint Rewrite skipn_app_L skipn_app_R skipn_0 skipn_all skipn_cons using omega : list_rw.
\ No newline at end of file
+Hint Rewrite skipn_app_L skipn_app_R skipn_0 skipn_all skipn_cons using lia : list_rw.
\ No newline at end of file
diff --git a/theories/Data/ListNth.v b/theories/Data/ListNth.v
index d6a09d0..c2394fd 100644
--- a/theories/Data/ListNth.v
+++ b/theories/Data/ListNth.v
@@ -1,0 +2 @@ Require Import Coq.Lists.List.
+Require Import Coq.Arith.Lt Coq.Arith.Plus.
diff --git a/theories/Data/Map/FMapAList.v b/theories/Data/Map/FMapAList.v
index 9d25baf..6d62e97 100644
--- a/theories/Data/Map/FMapAList.v
+++ b/theories/Data/Map/FMapAList.v
@@ -8,0 +9,9 @@ Require Import ExtLib.Structures.Functor.
+From Coq Require
+Import Basics.
+From ExtLib
+Require Import
+Extras OptionMonad.
+Import
+FunNotation
+FunctorNotation.
+Open Scope program_scope.
@@ -37,0 +47,14 @@ alist_find k ms end.
+Definition alist_find' (k: K) : alist -> option V
+:= fmap snd β find (rel_dec k β fst).
+
+Lemma alist_find_alt (m: alist) : forall k: K,
+alist_find k m = alist_find' k m.
+Proof.
+induction m; intuition.
+unfold alist_find', compose.
+simpl.
+destruct (k ?[ R ] a0) eqn:Heq; intuition.
+rewrite IHm.
+reflexivity.
+Qed.
+
@@ -51,0 +75,13 @@ in fold_alist acc m end.
+
+Definition fold_alist' : T -> alist -> T
+:= flip $ fold_left (flip $ uncurry f).
+
+Lemma fold_alist_alt (map: alist) : forall acc: T,
+fold_alist acc map = fold_alist' acc map.
+Proof.
+induction map; intuition.
+simpl.
+rewrite IHmap.
+reflexivity.
+Qed.
+
diff --git a/theories/Data/Monads/ListMonad.v b/theories/Data/Monads/ListMonad.v
index fd6906f..7023710 100644
--- a/theories/Data/Monads/ListMonad.v
+++ b/theories/Data/Monads/ListMonad.v
@@ -7,8 +7,4 @@ Set Strict Implicit.
-Global Instance Monad_list : Monad list :=
-{ ret := fun _ v
-=> v :: nil ; bind
-:= fun _ _ => fix
-recur c1 c2 := match c1
-with | nil => nil |
-a :: b => c2 a
-++ recur b c2 end }.
```
Proof State:
```
1 goal
T : Type n : nat
IHn : forall a b : list T, n <= length a -> firstn n (a ++ b) = firstn n a
t : T a, b : list T
H : S n <= S (length a)
============================
n <= length a
```
Next Tactic:
```
|
lia.
```
|
{"final_commit":"8f0f0228332007a1b73abb01fb9bf828023007fa","project":"coq-ext-lib","repair_file":"\/data\/coq-pearls\/repairs\/coq-ext-lib\/repair-coq-ext-lib-7532d92d1d6a9d7c9c45e111766f65421f6e09ba-8f0f0228332007a1b73abb01fb9bf828023007fa-8_12_2-8_14_1-2.git.json","initial_commit":"7532d92d1d6a9d7c9c45e111766f65421f6e09ba","file":"theories\/Data\/ListFirstnSkipn.v"}
|
Ongoing Proof History:
```
Proof.
induction e; simpl; intros.
-
- replace n with k by lia.
+ replace n with k by omega.
assumption.
-
? eauto using agree_insert with lia.
? Qed.
```
Commit Diff:
```
diff --git a/src/DblibTactics.v b/src/DblibTactics.v
index 2a08b3e..2aa54e9 100644
--- a/src/DblibTactics.v
+++ b/src/DblibTactics.v
@@ -2,3 +2 @@ Set Implicit Arguments.
-Require Import Compare_dec.
-Require Import Peano_dec.
-Require Import Lia.
+Require Export Omega.
@@ -12,7 +10,8 @@ Hint Extern 1 => f_equal : f_equal.
-Hint Extern 1 (_ = _ :> nat) => reflexivity : lia.
-Hint Extern 3 (_ = _ :> nat) => lia : lia.
-Hint Extern 3 (_ <> _ :> nat) => lia : lia.
-Hint Extern 3 (_ < _) => lia : lia.
-Hint Extern 3 (_ > _) => lia : lia.
-Hint Extern 3 (_ <= _) => lia : lia.
-Hint Extern 3 (_ >= _) => lia : lia.
+Hint Extern 1 (_ = _ :> nat) => reflexivity : omega.
+
+Hint Extern 1 (_ = _ :> nat) => omega : omega.
+Hint Extern 1 (_ <> _ :> nat) => omega : omega.
+Hint Extern 1 (_ < _) => omega : omega.
+Hint Extern 1 (_ > _) => omega : omega.
+Hint Extern 1 (_ <= _) => omega : omega.
+Hint Extern 1 (_ >= _) => omega : omega.
@@ -43 +42 @@ Ltac dblib_by_cases := repeat dblib_inspect_cases; try solve
-[ intros; exfalso; lia ]; intros.
\ No newline at end of file
+[ intros; exfalso; omega ]; intros.
\ No newline at end of file
diff --git a/src/DeBruijn.v b/src/DeBruijn.v
index 45bfa51..7a58a05 100644
--- a/src/DeBruijn.v
+++ b/src/DeBruijn.v
@@ -3,2 +2,0 @@ Generalizable All Variables.
-Require Import Arith.
-Require Import Lia.
@@ -286,3 +284,3 @@ intros.
-replace k with (ws + (k - ws)) by lia.
-erewrite <- lift_lift by lia.
-replace (ws + (k - ws) - ws) with (k - ws) by lia.
+replace k with (ws + (k - ws)) by omega.
+erewrite <- lift_lift by omega.
+replace (ws + (k - ws) - ws) with (k - ws) by omega.
@@ -456,2 +454,2 @@ eapply traverse_relative with (p := 0).
-intros m ?. replace (m + 0) with m by lia. eauto.
-lia.
+intros m ?. replace (m + 0) with m by omega. eauto.
+omega.
@@ -483,4 +481,3 @@ Qed.
-Ltac just_do_it := unfold
-subst, Subst_idx, subst_idx, lift,
-Lift_idx, var, Var_idx; intros;
-dblib_by_cases; eauto with lia.
+Ltac just_do_it := unfold subst, Subst_idx, subst_idx,
+lift, Lift_idx, var, Var_idx; intros; solve [
+dblib_by_cases; eauto with f_equal omega ].
@@ -512,2 +509,2 @@ Ltac lift_idx := first [ rewrite @lift_idx_recent by
-solve [ lia | eauto with lift_idx_hints
-] | rewrite @lift_idx_old by lia ].
+solve [ omega | eauto with lift_idx_hints
+] | rewrite @lift_idx_old by omega ].
@@ -518,2 +515,2 @@ Ltac lift_idx_in h := first [ rewrite @lift_idx_recent in
-h by solve [ lia | eauto with lift_idx_hints
-] | rewrite @lift_idx_old in h by lia ].
+h by solve [ omega | eauto with lift_idx_hints
+] | rewrite @lift_idx_old in h by omega ].
@@ -522,2 +519,2 @@ Ltac lift_idx_all := first [ rewrite @lift_idx_recent in *
-by solve [ lia | eauto with lift_idx_hints ]
-| rewrite @lift_idx_old in * by lia ].
+by solve [ omega | eauto with lift_idx_hints ]
+| rewrite @lift_idx_old in * by omega ].
@@ -587,2 +584,2 @@ Ltac subst_idx := first
-lia | rewrite
-@subst_idx_miss_1 by lia
+omega | rewrite
+@subst_idx_miss_1 by omega
@@ -590 +587 @@ lia | rewrite
-by lia ].
+by omega ].
@@ -594,2 +591,2 @@ Ltac subst_idx_in h := first
-by lia | rewrite @subst_idx_miss_1
-in h by lia
+by omega | rewrite @subst_idx_miss_1
+in h by omega
@@ -597 +594 @@ in h by lia
-h by lia ].
+h by omega ].
@@ -601,2 +598,2 @@ rewrite @subst_idx_identity in * by
-lia | rewrite @subst_idx_miss_1
-in * by lia
+omega | rewrite @subst_idx_miss_1
+in * by omega
@@ -604 +601 @@ in * by lia
-* by lia ].
+* by omega ].
@@ -667 +664 @@ intros. subst. simpl.
-eapply traverse_relative; [ | instantiate (1 := k1); lia ].
+eapply traverse_relative; [ | instantiate (1 := k1); omega ].
@@ -820,2 +817,2 @@ intros. f_equal.
-rewrite lift_lift by lia.
-f_equal. lia.
+rewrite lift_lift by omega.
+f_equal. omega.
@@ -835 +832 @@ intros. f_equal.
-rewrite lift_lift_fuse by lia. reflexivity.
+rewrite lift_lift_fuse by omega. reflexivity.
```
Proof State:
```
1 subgoal
A : Type a : A e : list A
IHe : forall (e1 e2 : env A) (k n : nat),
agree e1 e2 k ->
k + length e = n -> agree (concat e1 e) (concat e2 e) n
e1, e2 : env A k, n : nat
H : agree e1 e2 k
H0 : k + S (length e) = n
============================
agree (concat (insert 0 a e1) e) (concat (insert 0 a e2) e) n
```
Next Tactic:
```
|
eauto using <LOOKUP>agree_insert : forall [A : Type] [e1 e2 : env A] [k : nat], agree e1 e2 k -> forall (x : nat) (o : option A), x <= k -> agree (raw_insert x o e1) (raw_insert x o e2) (1 + k)</LOOKUP> with omega.
```
|
{"final_commit":"3de04cadb63c28bfcffb194cf9e64e75a0c1f697","project":"dblib","repair_file":"\/data\/coq-pearls\/repairs\/dblib\/repair-dblib-d4ece98bb63f75c2a53e9d125338454e6e1393b2-3de04cadb63c28bfcffb194cf9e64e75a0c1f697-8_12_2-8_12_2-36.git.json","initial_commit":"d4ece98bb63f75c2a53e9d125338454e6e1393b2","file":"src\/Environments.v"}
|
Proof State:
```
4 subgoals
x, h, eps, l1, alp_f2, alp_f2t2 : R eps_f2 : posreal f2 : R -> R
H : 0 < eps
H0 : f2 x <> 0
H1 : f2 (x + h) <> 0
H2 : forall a : R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) < Rabs (eps * (f2 x)Β² / (8 * l1))
H3 : forall a : R,
Rabs a < Rmin eps_f2 alp_f2 -> / Rabs (f2 (x + a)) < 2 / Rabs (f2 x)
H4 : h <> 0
H5 : Rabs h < alp_f2t2
H6 : Rabs h < Rmin eps_f2 alp_f2
H7 : l1 <> 0
H8 : / Rabs (f2 (x + h)) < 2 / Rabs (f2 x)
H9 : Rabs (f2 (x + h) - f2 x) < Rabs (eps * (f2 x)Β² / (8 * l1))
============================
2 * (Rabs l1 * (/ Rabs (f2 x) * / Rabs (f2 x))) *
(eps * (Rabs (f2 x) * Rabs (f2 x)) * (/ 8 * / Rabs l1)) =
eps * / 4
subgoal 2 is:
2 = Rabs 2
subgoal 3 is:
8 = Rabs 8
subgoal 4 is:
eps = Rabs eps
```
Ongoing Proof History:
```
Proof.
intros.
assert (H8 := H3 h H6).
assert (H9 := H2 h H5).
apply Rle_lt_trans with (Rabs (l1 / (f2 x * f2 (x + h))) * Rabs (eps * Rsqr (f2 x) / (8 * l1))).
rewrite Rabs_mult; apply Rmult_le_compat_l.
apply Rabs_pos.
rewrite <- (Rabs_Ropp (f2 x - f2 (x + h))); rewrite Ropp_minus_distr.
left; apply H9.
apply Rlt_le_trans with (Rabs (2 * (l1 / (f2 x * f2 x))) * Rabs (eps * Rsqr (f2 x) / (8 * l1))).
apply Rmult_lt_compat_r.
apply Rabs_pos_lt.
unfold Rdiv; unfold Rsqr; repeat apply prod_neq_R0; try assumption || discrR.
red; intro H10; rewrite H10 in H; elim (Rlt_irrefl _ H).
apply Rinv_neq_0_compat; apply prod_neq_R0; try assumption || discrR.
unfold Rdiv.
repeat rewrite Rinv_mult_distr; try assumption.
repeat rewrite Rabs_mult.
replace (Rabs 2) with 2.
rewrite (Rmult_comm 2).
replace (Rabs l1 * (Rabs (/ f2 x) * Rabs (/ f2 x)) * 2) with (Rabs l1 * (Rabs (/ f2 x) * (Rabs (/ f2 x) * 2))); [ idtac | ring ].
repeat apply Rmult_lt_compat_l.
apply Rabs_pos_lt; assumption.
apply Rabs_pos_lt; apply Rinv_neq_0_compat; assumption.
repeat rewrite Rabs_Rinv; try assumption.
rewrite <- (Rmult_comm 2).
unfold Rdiv in H8; exact H8.
symmetry ; apply Rabs_right; left; prove_sup0.
right.
unfold Rsqr, Rdiv.
do 1 rewrite Rinv_mult_distr; try assumption || discrR.
do 1 rewrite Rinv_mult_distr; try assumption || discrR.
repeat rewrite Rabs_mult.
repeat rewrite Rabs_Rinv; try assumption || discrR.
replace (Rabs eps) with eps.
replace (Rabs 8) with 8.
replace (Rabs 2) with 2.
```
Next Tactic:
```
|
replace 8 with (4 * 2); [ idtac | ring ].
```
|
{"package":"coq","file":"\/home\/tpr\/.opam\/PackageTable~8.13~2022.01.csv.formatted\/lib\/coq\/theories\/Reals\/Ranalysis2.v"}
|
Proof State:
```
1 subgoal
n : nat r : AV.address -> resource m1, m2 : ghost
s2 : {g : ghost.Ghost & {a : ghost.G | ghost.valid a}}
p2 : preds
g0 : list
(option ({g : ghost.Ghost & {a : ghost.G | ghost.valid a}} * preds))
IHghost_join : forall g1 : ghost,
ghost_fmap (approx n) (approx n) g0 =
ghost_fmap (approx n) (approx n) g1 ->
ghost_fmap (approx n) (approx n) m1 = m1
H0 : ghost_join m1 m2 (ghost_fmap (approx n) (approx n) g0)
s : {g : ghost.Ghost & {a : ghost.G | ghost.valid a}}
p : preds s0 : {g : ghost.Ghost & {a : ghost.G | ghost.valid a}}
p0 : preds
H : join s s0 s2
H2 : join p p0 (preds_fmap (approx n) (approx n) p2)
============================
Some (s, preds_fmap (approx n) (approx n) p) = Some (s, p)
```
Ongoing Proof History:
```
Proof.
intros.
rewrite rmap_level_eq.
unfold ghost_of in *.
revert H; case_eq (unsquash phi); intros n ? ?.
generalize H; rewrite <- (squash_unsquash phi).
rewrite H.
rewrite unsquash_squash.
simpl; intros.
injection H0.
clear H0.
intro.
clear phi H.
rewrite <- H0 in H1.
clear H0.
unfold rmap_fmap in *.
destruct r.
simpl in H1; destruct H1.
remember (ghost_fmap (approx n) (approx n) g0) as g'.
revert dependent g0; induction H; auto; intros; subst.
-
rewrite ghost_fmap_fmap, approx_oo_approx; auto.
-
destruct g0; inv Heqg'.
simpl; f_equal; eauto.
inv H; auto; simpl.
+
destruct o as [[]|]; auto; simpl.
rewrite preds_fmap_fmap, approx_oo_approx; auto.
+
destruct a0, a3, a4; inv H4; simpl in *.
destruct o as [[]|]; inv H1.
```
Next Tactic:
```
|
inv H2.
```
|
{"package":"coq-vst","file":"\/home\/tpr\/.opam\/PackageTable~8.13~2022.01.csv.formatted\/lib\/coq\/user-contrib\/VST\/veric\/rmaps_lemmas.v"}
|
Proof State:
```
2 focused subgoals
(shelved: 1)
n : nat
H_strong2 : forall k : nat,
k < S n ->
forall (e : Expr) (Ο Ο' : Store) (Ο : Env) (Ο l : Value),
(β¦ e β§ (Ο, Ο, Ο )( k)) = Success l Ο' ->
wf Ο ->
(codom Ο βͺ {Ο}) βͺ½ Ο ->
((Ο, codom Ο βͺ {Ο}) β (Ο', {l})) /\ (Ο β Ο' β codom Ο βͺ {Ο})
e0 : Expr f : Var Ο, Ο' : Store Ο : Env Ο, l, v : Value
matched : (β¦ e0 β§ (Ο, Ο, Ο )( n)) = Success v Ο'
c : ClN e : Env
matched0 : getObj Ο' v = Some (c, e)
matched2 : getVal e f = Some l
H_wf : wf Ο
H_codom : (codom Ο βͺ {Ο}) βͺ½ Ο
H_strong : forall (e : Expr) (Ο Ο' : Store) (Ο : Env) (Ο l : Value),
(β¦ e β§ (Ο, Ο, Ο )( n)) = Success l Ο' ->
wf Ο ->
(codom Ο βͺ {Ο}) βͺ½ Ο ->
((Ο, codom Ο βͺ {Ο}) β (Ο', {l})) /\ (Ο β Ο' β codom Ο βͺ {Ο})
H : dom Ο <= dom Ο'
A2 : (Ο, codom Ο βͺ {Ο}) β (Ο', {v})
A3 : Ο β Ο' β codom Ο βͺ {Ο}
============================
Ο' β¨ v β ?l1
subgoal 2 is:
Ο' β¨ ?l1 β l
```
Commit Diff:
```
diff --git a/Reachability.v b/Reachability.v
index 5c47635..1b96ea3 100644
--- a/Reachability.v
+++ b/Reachability.v
@@ -18,5 +18,20 @@ Import Eval.Evaluator.
-
-Inductive reachability : Store -> Loc -> Loc ->Prop := |rch_heap : forall l Ο, l < (dom Ο) -> (reachability Ο
-l l) |rch_trans : forall l0 l1 l2 C Ο Ο, (reachability Ο l0 l1) -> (getObj Ο l1 = Some
-(C, Ο)) -> (exists f, (getVal Ο f = Some l2)) -> (l2 < dom Ο) -> (reachability Ο l0 l2).
-Notation "Ο β¨ l1 β l2" := (reachability Ο l1 l2) (at level 80, l1 at level 80, l2 at level 80).
+Reserved Notation "Ο β¨ l1 β l2" (at level 80, l1 at level 80, l2 at level 80).
+Inductive reachability : Store -> Loc
+-> Loc ->Prop := | rch_heap:
+forall l Ο, l < (dom
+Ο) -> Ο β¨ l β
+l | rch_step: forall l0 l1
+C f Ο Ο, l1 <
+(dom Ο) -> getObj Ο l0
+= Some (C, Ο) -> getVal
+Ο f = Some l1 ->
+Ο β¨ l0 β l1
+| rch_trans: forall l0 l1
+l2 Ο, Ο β¨ l0
+β l1 -> Ο β¨
+l1 β l2 -> Ο
+β¨ l0 β l2 where
+"Ο β¨ l1 β l2"
+:= (reachability Ο l1 l2).
+Hint Resolve rch_heap rch_step rch_trans: rch.
+Hint Rewrite update_dom: rch.
@@ -41,3 +56 @@ Proof.
-intros Ο l1 l2 l3 H1 H2.
-induction H2 => //.
-apply (rch_trans l1 l2 l3 C Ο Ο (IHreachability H1) H H0 H3).
+eauto using rch_trans.
@@ -44,0 +58 @@ Qed.
+
@@ -60 +74 @@ intros.
-induction H; steps.
+induction H; repeat steps || eapply_anywhere getObj_dom.
@@ -64,11 +78,11 @@ Qed.
-Lemma reachability_rev: forall Ο l l', Ο β¨ l
-β l' -> (l = l' /\ l <
-dom Ο) \/ (exists C Ο f l0, getObj
-Ο l = Some(C, Ο) /\ getVal Ο f
-= Some l0 /\ Ο β¨ l0 β l').
-Proof.
-intros.
-induction H; steps.
-+ right. repeat eexists; eauto using rch_heap.
-+ right. repeat eexists; eauto using rch_heap, rch_trans.
-Qed.
+
+
+
+
+
+
+
+
+
+
+
@@ -85,3 +98,0 @@ induction H2; eauto.
-steps.
-pose proof (H1 l1 l2 C Ο f).
-eapply H0; eauto.
```
Ongoing Proof History:
```
Proof.
apply strong_induction.
unfold ScopabilityProp.
intros n H_strong2 e Ο Ο' Ο Ο l H_success H_wf H_codom.
destruct n => //.
move: (H_strong2 n (le_n (S n)) ) => H_strong.
destruct e as [x | this | e0 f | e0 m el | C el | e0 f e1 e2]; simpl in H_success; repeat destruct_match; try discriminate; try invert_constructor_equalities; subst.
+
split.
unfold scoping; steps.
exists l; steps.
apply Union_introl.
unfold codom, In.
eauto using nth_error_In.
apply_anywhere reachability_singleton => //.
unfold scoping_preservation; steps.
+
split.
unfold scoping; steps.
unfold reachability_set in *; steps.
inSingleton.
exists l.
steps; eauto using Union_intror.
unfold scoping_preservation; steps.
+
unfold scoping; intros; simpl.
pose proof (PartialMonotonicity.partialMonotonicity_theorem_dom _ _ _ _ _ _ _ matched).
move : (H_strong _ _ _ _ _ _ matched H_wf H_codom) => [A2 A3].
assert ((Ο', {v}) β (Ο', {l})) as B1.
{
apply scoping_reachability.
eapply rch_trans; eauto.
? apply rch_heap; eauto using getObj_dom.
? eapply wellformedness_conserved in matched0; eauto.
? }
```
Next Tactic:
```
|
eauto using <LOOKUP>getObj_dom : forall (Ο : Store) (o : Obj) (l : nat), getObj Ο l = Some o -> l < dom Ο</LOOKUP> with rch.
```
|
{"final_commit":"2dfd0b6c87b32ebd6bd1dd83e76e654e73dd3696","project":"celsius","repair_file":"\/data\/coq-pearls\/repairs\/celsius\/repair-celsius-a726b4d083864e7cb26d15bceeb49fbc418b2811-2dfd0b6c87b32ebd6bd1dd83e76e654e73dd3696-8_13_2-8_13_2-17.git.json","initial_commit":"a726b4d083864e7cb26d15bceeb49fbc418b2811","file":"Scopability.v"}
|
Commit Diff:
```
diff --git a/Div.v b/Div.v
index 7e7d694..e83d601 100644
--- a/Div.v
+++ b/Div.v
@@ -57,2 +57,2 @@ n) p) (c2 := mod n p); try apply mod_lt; auto with arith.
-rewrite mult_plus_distr_r; rewrite <- plus_assoc; repeat
-rewrite <- div_mod_correct; auto with arith.
+rewrite mult_plus_distr_r, <- plus_assoc; repeat rewrite
+<- div_mod_correct; auto with arith.
@@ -67,4 +67,2 @@ intros n m p H; apply plus_reg_l with (div (p * m + n) p * p).
-rewrite <- div_mod_correct; auto.
-rewrite div_mult_comp; auto.
-rewrite mult_plus_distr_r; rewrite (mult_comm p); rewrite <- plus_assoc.
-eq_tac; apply div_mod_correct; auto.
+rewrite <- div_mod_correct, div_mult_comp, mult_plus_distr_r, (mult_comm p), <- plus_assoc by assumption.
+f_equal. now apply div_mod_correct.
diff --git a/ListAux.v b/ListAux.v
index 82a8d82..986e88b 100644
--- a/ListAux.v
+++ b/ListAux.v
@@ -76 +76 @@ exists (nil (A:=A)); exists l; simpl; auto.
-eq_tac; auto.
+f_equal; auto.
@@ -78 +78 @@ case H; auto; intros l1 [l2 Hl2]; exists (a1 :: l1); exists l2; simpl; auto.
-eq_tac; auto.
+f_equal; auto.
@@ -146 +146 @@ intros a l H l2 l3 l4 a0; case l3; simpl.
-intros H0; left; exists l; eq_tac; injection H0; auto.
+intros H0; left; exists l; f_equal; injection H0; auto.
@@ -150 +150 @@ intros [l5 H1].
-left; exists l5; eq_tac; injection H0; auto.
+left; exists l5; f_equal; injection H0; auto.
@@ -165,2 +165,2 @@ intros H0; right; right; injection H0; split; auto.
-eq_tac; auto.
-intros b0 l0 H0; left; exists l0; injection H0; intros; (repeat eq_tac); auto.
+f_equal; auto.
+intros b0 l0 H0; left; exists l0; injection H0; intros; (repeat f_equal); auto.
@@ -169 +169 @@ injection H0; auto.
-intros [l5 HH1]; left; exists l5; eq_tac; auto; injection H0; auto.
+intros [l5 HH1]; left; exists l5; f_equal; auto; injection H0; auto.
@@ -171 +171 @@ intros [H1|[H1 H2]]; auto.
-right; right; split; auto; eq_tac; auto; injection H0; auto.
+right; right; split; auto; f_equal; auto; injection H0; auto.
@@ -190 +190 @@ exists (b :: l4); exists l5; exists b1; (repeat (simpl; split; auto)).
-eq_tac; auto.
+f_equal; auto.
@@ -237 +237 @@ intros H4 H5; split; auto.
-eq_tac; auto.
+f_equal; auto.
@@ -278 +278 @@ intros A l1 l2; elim l1; simpl; auto.
-intros a l H; rewrite length_app; rewrite length_map; rewrite H; auto.
+intros a l H; rewrite length_app, length_map, H; auto.
diff --git a/ListOp.v b/ListOp.v
index b9797d9..e1936ca 100644
--- a/ListOp.v
+++ b/ListOp.v
@@ -138,2 +138 @@ intros a b c; elim c; simpl; auto.
-intros n H; rewrite jump_nil; rewrite
-take_nil; rewrite H; auto with arith.
+intros n H; rewrite jump_nil, take_nil, H; auto with arith.
@@ -146,2 +145,2 @@ intros k Rec i j s H;
-rewrite length_app; rewrite length_take1; auto with arith.
-eq_tac; auto.
+rewrite length_app, length_take1; auto with arith.
+f_equal; auto.
@@ -199 +198 @@ intros l; elim l; simpl; auto with datatypes.
-intros; eq_tac; auto.
+intros; f_equal; auto.
@@ -206 +205 @@ intros H; contradict H; auto with arith.
-intros n1 H; eq_tac; auto with arith.
+intros n1 H; f_equal; auto with arith.
@@ -219 +218 @@ intros a l1 Rec n; case n; auto; clear n.
-simpl length; intros n H; simpl; eq_tac; auto with arith.
+simpl length; intros n H; simpl; f_equal; auto with arith.
diff --git a/OrderedList.v b/OrderedList.v
index f2762b8..3136749 100644
--- a/OrderedList.v
+++ b/OrderedList.v
@@ -624 +624 @@ intros; discriminate.
-intros; eq_tac; auto.
+intros; f_equal; auto.
diff --git a/Permutation.v b/Permutation.v
index ecfa699..ccd8195 100644
--- a/Permutation.v
+++ b/Permutation.v
@@ -33,2 +32,0 @@ Variable A : Set.
-Definition permutation (l1 l2 : list A) := @Permutation A l1 l2.
-
```
Ongoing Proof History:
```
Proof.
intros n; elim n; simpl in |- *; auto.
intros l1 l2; case l2.
? intros H H0; rewrite perm_nil_inv with (1 := H0); auto with datatypes.
? simpl in |- *; intros; discriminate.
? intros n0 H l1; case l1.
```
Proof State:
```
3 goals
A : Set n : nat l1, l2 : list A
============================
0 = length nil -> Permutation l1 nil -> nil = l1 \/ False
goal 2 is:
forall (a : A) (l : list A),
0 = length (a :: l) -> Permutation l1 (a :: l) -> nil = l1 \/ False
goal 3 is:
forall n0 : nat,
(forall l1 l2 : list A,
n0 = length l2 -> Permutation l1 l2 -> In l1 (all_permutations_aux l2 n0)) ->
forall l1 l2 : list A,
S n0 = length l2 ->
Permutation l1 l2 ->
In l1
(flat_map
(fun p : A * list A =>
map (cons (fst p)) (all_permutations_aux (snd p) n0))
(split_one l2))
```
Next Tactic:
```
|
intros H H0; rewrite (<LOOKUP>Permutation_nil : forall [A : Type] [l : list A], Permutation nil l -> l = nil</LOOKUP> (<LOOKUP>Permutation_sym : forall [A : Type] [l l' : list A], Permutation l l' -> Permutation l' l</LOOKUP> H0)); auto with datatypes.
```
|
{"final_commit":"82eb0678991955dfa8f12720d652ad5adb465c3e","project":"sudoku","repair_file":"\/data\/coq-pearls\/repairs\/sudoku\/repair-sudoku-b5543771d6bdeab13ec58ac6c8d281815d96ed62-82eb0678991955dfa8f12720d652ad5adb465c3e-8_15_2-8_15_2-37.git.json","initial_commit":"b5543771d6bdeab13ec58ac6c8d281815d96ed62","file":"Permutation.v"}
|
Ongoing Proof History:
```
+ Proof.
? Proof with auto.
? generalize (ref_list_ulist); intros Eq1.
? generalize (ref_list_length); intros Eq2.
```
Proof State:
```
1 goal
h, w : nat
============================
forall s : list nat, length s = size * size -> sat init_c s <-> sudoku s
```
Commit Diff:
```
diff --git a/ListAux.v b/ListAux.v
index 769776a..986e88b 100644
--- a/ListAux.v
+++ b/ListAux.v
@@ -130,0 +131,6 @@ Qed.
+Theorem length_app: forall (l1 l2 : list A), length
+(l1 ++ l2) = length l1 + length l2.
+Proof.
+apply app_length.
+Qed.
+
@@ -272 +278 @@ intros A l1 l2; elim l1; simpl; auto.
-intros a l H; rewrite app_length, length_map, H; auto.
+intros a l H; rewrite length_app, length_map, H; auto.
diff --git a/ListOp.v b/ListOp.v
index 7a13f04..e1936ca 100644
--- a/ListOp.v
+++ b/ListOp.v
@@ -145 +145 @@ intros k Rec i j s H;
-rewrite app_length, length_take1; auto with arith.
+rewrite length_app, length_take1; auto with arith.
diff --git a/Sudoku.v b/Sudoku.v
index f9e372e..3a01485 100644
--- a/Sudoku.v
+++ b/Sudoku.v
@@ -675 +675 @@ assert (F1: div y w < h).
-apply div_lt; rewrite mult_comm...
+apply div_lt; rewrite mult_comm; auto.
@@ -677,2 +677,6 @@ apply div_lt; rewrite mult_comm...
-repeat (rewrite (fun x => mult_comm x h)); rewrite mod_mult_comp...
-rewrite div_mult_comp, (mod_small (div y w) h), (div_is_0 (div y w) h), plus_0_r...
+repeat (rewrite (fun x =>
+mult_comm x h)); rewrite mod_mult_comp...
+rewrite div_mult_comp; auto.
+rewrite (mod_small (div y w) h); auto.
+rewrite (div_is_0 (div y w) h); auto.
+rewrite plus_0_r.
@@ -699,3 +703,3 @@ cut (w <= size).
-intros U3 n; generalize h; elim n; simpl...
-clear n H1 H2 U1.
-+ intros h1; case h1; simpl...
+intros U3 n; generalize h; elim n; simpl;
+auto with arith; clear n H1 H2 U1.
++ intros h1; case h1; simpl; auto.
@@ -709,2 +713,3 @@ apply sym_equal; apply take_nth...
-** pose proof (mod_lt y w).
-lia.
+** apply le_trans with (1 := H1)...
+apply le_trans with (2 := U3).
+apply lt_le_weak; apply mod_lt...
@@ -721 +726 @@ apply mod_lt...
-case (le_or_lt (length l) w); intros H3.
+case (le_or_lt (length l) w); intros H1.
@@ -723 +728,2 @@ case (le_or_lt (length l) w); intros H3.
-** rewrite length_take_small, jump_too_far...
+** rewrite length_take_small...
+rewrite jump_too_far.
@@ -726,2 +732,2 @@ repeat rewrite nth_default...
-apply le_trans with (1 := H3)...
-*** apply le_trans with (1 := H3)...
+apply le_trans with (1 := H1)...
+*** apply le_trans with (1 := H1)...
@@ -732 +738,2 @@ repeat rewrite <- plus_assoc; rewrite minus_plus...
-rewrite jump_nth, jump_add, <- jump_nth...
+rewrite jump_nth; rewrite jump_add.
+rewrite <- jump_nth...
@@ -743 +750 @@ generalize (w_pos _ H); intros U2.
-rewrite length_take_and_jump...
+rewrite length_take_and_jump; auto.
@@ -760 +767,2 @@ rewrite (plus_assoc w).
-pattern w at 1; rewrite <- mult_1_r, <- mult_plus_distr_l, le_plus_minus_r...
+pattern w at 1; rewrite <- mult_1_r; rewrite <- mult_plus_distr_l.
+rewrite le_plus_minus_r...
@@ -846,2 +854,2 @@ l2 l3 -> lit_test l1 l3 = lit_test l1 l2.
-Proof with auto.
-intros l1 l2 l3; case l1; case l2; case l3; simpl...
+Proof.
+intros l1 l2 l3; case l1; case l2; case l3; simpl; auto.
@@ -851 +859 @@ case_eq (pos_test p2 p3); intros H1.
-rewrite (pos_test_trans p1 p2 p3); rewrite H...
+rewrite (pos_test_trans p1 p2 p3); rewrite H; auto.
@@ -853,2 +861,2 @@ assert (p2 = p3); subst.
-apply pos_test_exact with (1 := H1)...
-rewrite H...
+apply pos_test_exact with (1 := H1); auto.
+rewrite H; auto.
@@ -856,4 +864,4 @@ intros; discriminate.
-replace p1 with p2...
-case_eq (pos_test p2 p3); intros H1...
-intros; apply test_trans...
-apply sym_equal; apply pos_test_exact with (1 := H)...
+replace p1 with p2; auto.
+case_eq (pos_test p2 p3); intros H1; auto.
+intros; apply test_trans; auto.
+apply sym_equal; apply pos_test_exact with (1 := H); auto.
@@ -863,3 +871,3 @@ assert (p2 = p3); subst.
-apply pos_test_exact with (1 := H1)...
-rewrite H...
-rewrite (pos_test_trans p1 p2 p3); rewrite H...
+apply pos_test_exact with (1 := H1); auto.
+rewrite H; auto.
+rewrite (pos_test_trans p1 p2 p3); rewrite H; auto.
@@ -1445,2 +1453,2 @@ ordered cs -> ordered (clauses_update l c cs).
-Proof with auto with datatypes.
-intros l c cs; elim cs; simpl...
+Proof.
+intros l c cs; elim cs; simpl; auto.
@@ -1448,7 +1456,7 @@ intros (n1, c1) cs1 Rec H H1; case (lit_is_in l c1).
-apply Rec...
-intros n2 c2 H2; apply (H1 n2 c2)...
-apply clause_insert_ordered...
-apply lit_rm_ordered...
-apply (H1 n1 c1)...
-apply Rec...
-intros n2 c2 H2; apply (H1 n2 c2)...
+apply Rec; auto.
```
Next Tactic:
```
|
generalize (<LOOKUP>ref_list_ulist : ulist ref_list</LOOKUP>); intros Eq1.
```
|
{"final_commit":"427448011e3341d35aa5b9e00f675f6582fcd477","project":"sudoku","repair_file":"\/data\/coq-pearls\/repairs\/sudoku\/repair-sudoku-dc8d4294a32dbb6e0a642dc553b64495f27e6d60-427448011e3341d35aa5b9e00f675f6582fcd477-8_15_2-8_15_2-23.git.json","initial_commit":"dc8d4294a32dbb6e0a642dc553b64495f27e6d60","file":"Sudoku.v"}
|
Ongoing Proof History:
```
Proof using A_coll B_coll A_conn B_conn L_AB.
exists iso.
split.
-
exact: Bijective can_iso_inv can_iso.
-
move => x a.
apply/B_coll => w.
rewrite -[_ (iso x) a]/(delta (iso x) [::a]).
by rewrite -delta_cat -!delta_iso_inv !can_iso_inv.
-
move => x.
by rewrite -[iso x]/(delta _ [::]) delta_iso.
-
```
Proof State:
```
1 subgoal
char : finType A, B : dfa
L_AB : dfa_lang A =i dfa_lang B
A_coll : collapsed A
B_coll : collapsed B
A_conn : connected A
B_conn : connected B
============================
iso (dfa_s A) = dfa_s B
```
Next Tactic:
```
|
apply/B_coll => w.
```
|
{"package":"coq-reglang","file":"\/home\/tpr\/.opam\/PackageTable~8.13~2022.01.csv.formatted\/lib\/coq\/user-contrib\/RegLang\/minimization.v"}
|
Proof State:
```
1 subgoal
cs : compspecs Qtemp : PTree.t val Qvar : PTree.t (type * val)
GV : option globals a : expr Delta : tycontext
P : list Prop Q : list localdef R, R' : list mpred
fs : funspec argsig : list type retty' : type
cc : calling_convention ts : list Type A : rmaps.TypeTree
Pre : forall ts : list Type,
functors.MixVariantFunctor._functor
(rmaps.dependent_type_functor_rec ts (ArgsTT A)) mpred
Post : forall ts : list Type,
functors.MixVariantFunctor._functor
(rmaps.dependent_type_functor_rec ts (AssertTT A)) mpred
NEPre : args_super_non_expansive Pre
NEPost : super_non_expansive Post
witness : functors.MixVariantFunctor._functor
(rmaps.dependent_type_functor_rec ts A) mpred
Frame : list mpred bl : list expr Ppre : list Prop
Rpre : list mpred GV' : option globals gv : list globals
args, vl : list val
SETUP : call_setup2 cs Qtemp Qvar GV a Delta P Q R R' fs argsig retty' cc
ts A Pre Post NEPre NEPost bl vl witness Frame Ppre Rpre GV' gv
args
retty : type Espec : OracleKind ret : ident
ret' : positive Post2 : environ -> mpred Qnew : list localdef
B : Type Ppost : B -> list Prop F : B -> val
Rpost : B -> list mpred
RETinit : (temp_types Delta) ! ret' = Some retty'
OKretty : check_retty retty
OKretty' : check_retty retty'
NEUTRAL : is_neutral_cast retty' retty = true
NEret : ret <> ret'
POST1 : Post ts witness =
(EX vret : B,
PROPx (Ppost vret) (RETURN (F vret) (SEPx (Rpost vret)))%assert3)%argsassert
DELETE : remove_localdef_temp ret Q = Qnew
DELETE' : remove_localdef_temp ret' Q = Q
H0 : Post2 =
(EX vret : B,
PROPx (P ++ Ppost vret)
(LOCALx (temp ret (F vret) :: Qnew) (SEPx (Rpost vret ++ Frame))))%argsassert
PPRE : fold_right_and True Ppre
vret : B rho : environ
H : tc_environ Delta rho
H1 : fold_right and True (P ++ Ppost vret)
H2 : (` and) ((` (eq (F vret))) (eval_id ret')) (` (F vret <> Vundef)) rho
H3 : fold_right
(fun (x x0 : environ -> Prop) (x1 : environ) => x x1 /\ x0 x1)
(fun _ : environ => True) (map locald_denote Q) rho
============================
fold_right_sepcon (Rpost vret ++ Frame)
|-- denote_tc_assert
(tc_andp
(tc_bool (is_neutral_cast (implicit_deref retty) retty)
(invalid_cast retty retty))
(isCastResultType (implicit_deref retty) retty
(Ecast (Etempvar ret' retty') retty))) rho
```
Ongoing Proof History:
```
Proof.
intros.
eapply semax_seq'.
eapply semax_call_id1_wow; try eassumption; auto.
unfold typeof_temp; rewrite RETinit; reflexivity.
apply extract_exists_pre; intro vret.
eapply semax_pre_post'; [ | | apply semax_set_forward].
+
eapply derives_trans; [ | apply now_later ].
instantiate (1:= (PROPx (P ++ Ppost vret) (LOCALx (temp ret' (F vret) :: Qnew) (SEPx (Rpost vret ++ Frame))))).
apply andp_right; [apply andp_right |].
-
unfold tc_expr.
simpl typecheck_expr.
rewrite RETinit.
simpl @fst.
replace ((is_neutral_cast retty' retty' || same_base_type retty' retty')%bool) with true by (clear- OKretty'; destruct retty' as [ | [ | | |] [| ]| [|] | [ | ] | | | | | ]; try contradiction; unfold is_neutral_cast; rewrite ?eqb_type_refl; reflexivity).
rewrite denote_tc_assert_andp.
apply andp_right; [| intros rho; apply neutral_isCastResultType; auto].
apply PQR_denote_tc_initialized; auto.
-
unfold tc_temp_id, typecheck_temp_id.
unfold typeof_temp in TYret.
destruct ((temp_types Delta) ! ret); inversion TYret; clear TYret; try subst t.
go_lowerx.
```
Next Tactic:
```
|
repeat rewrite <LOOKUP>denote_tc_assert_andp : forall {CS : compspecs} (a b : tc_assert), denote_tc_assert (tc_andp a b) = denote_tc_assert a && denote_tc_assert b</LOOKUP>; simpl.
```
|
{"package":"coq-vst","file":"\/home\/tpr\/.opam\/PackageTable~8.13~2022.01.csv.formatted\/lib\/coq\/user-contrib\/VST\/floyd\/call_lemmas.v"}
|
Ongoing Proof History:
```
Proof.
unfold map_default_eq, interp_map.
repeat intro.
revert x y H s1 s2 H0.
einit.
ecofix CH.
intros.
rewrite! unfold_interp_state.
punfold H0.
red in H0.
revert s1 s2 H1.
induction H0; intros; subst; simpl; pclearbot.
-
eret.
-
etau.
-
ebind.
apply pbc_intro_h with (RU := prod_rel (@eq_map _ _ _ _ d) eq).
{
- unfold case_.
+ unfold over.
- unfold Case_sum1, case_sum1.
+ destruct (case e).
- destruct e.
+ -
apply handle_map_eq.
assumption.
+ -
- unfold pure_state.
+ unfold trigger, Trigger_State, trigger, Trigger_ITree, ITree.trigger.
pstep.
+ cbn.
+ red.
+ cbn.
econstructor.
intros.
constructor.
pfold.
? econstructor.
? constructor; auto.
? }
```
Commit Diff:
```
diff --git a/theories/Core/Subevent.v b/theories/Core/Subevent.v
index 42a9534..e0c634e 100644
--- a/theories/Core/Subevent.v
+++ b/theories/Core/Subevent.v
@@ -4,0 +5,3 @@
+From Coq Require
+Import Setoid Morphisms.
+
@@ -8 +11,6 @@ Require Import
-Basics.Basics Basics.CategoryOps
+Basics.Basics
+Basics.CategoryOps
+Basics.CategoryTheory
+Basics.CategoryFacts
+Basics.Function
+Basics.FunctionFacts
@@ -11 +19,72 @@ Indexed.Sum
-Indexed.Function.
+Indexed.Function
+Indexed.FunctionFacts.
+
+
+From ExtLib Require
+Import Monad.
+
+Import MonadNotation.
+Import CatNotations.
+Open Scope cat_scope.
+
+
+
+Set Implicit Arguments.
+
+
+Inductive option1 (A : Type
+-> Type) X :Type
+:= | None1 |
+Some1 (_: A X).
+Arguments None1 {_ _}.
+Arguments Some1 {_} [_].
+
+Section Subevent.
+
+
+Class Subevent {A B C :
+Type -> Type} : Type :=
+{ split_E : B ~> A
++' C ; merge_E : (A
++' C) ~> B }.
+
+Class Subevent_wf {A B C}
+(sub: @Subevent A B C):
+Prop := { sub_iso :>
+Iso _ split_E merge_E }.
+
+Arguments Subevent : clear implicits.
+Arguments split_E {_ _ _ _} [_].
+Arguments merge_E {_ _ _ _} [_].
+Definition inj1 {A B C} `{Subevent A B C} : A ~> B := inl_ >>> merge_E.
+Definition inj2 {A B C} `{Subevent A B C} : C ~> B := inr_ >>> merge_E.
+Definition case {A B C} `{Subevent A B C} : B ~> (A +' C) := split_E.
+
+End Subevent.
+
+Arguments Subevent : clear implicits.
+Arguments case {_ _ _ _} [_].
+Arguments inj1 {_ _ _ _} [_].
+Arguments inj2 {_ _ _ _} [_].
+
+Notation "A +? C -< B" := (Subevent A
+B C) (at level 89, left associativity) : type_scope.
+
+Lemma case_inj1: forall {A B C: Type -> Type} `{Sub: A +? C -< B}
+{SubWF: Subevent_wf Sub} {T} (e: A T), case (inj1 e) = inl_ _ e.
+Proof.
+intros.
+pose proof (iso_epi IFun T (inl_ _ e)).
+auto.
+Qed.
+
+Lemma case_inj2: forall {A B C: Type -> Type} `{Sub: A +? C -< B}
+{SubWF: Subevent_wf Sub} {T} (e: C T), case (inj2 e) = inr_ _ e.
+Proof.
+intros.
+pose proof (iso_epi IFun T (inr_ _ e)).
+auto.
+Qed.
+
+Section Trigger.
+
@@ -16,0 +96 @@ Indexed.Function.
+Class Trigger (E: Type -> Type) (M: Type -> Type) := trigger: E ~> M.
@@ -17,0 +98,2 @@ Indexed.Function.
+End Trigger.
+Arguments trigger {E M Trigger} [T].
@@ -19,0 +102 @@ Indexed.Function.
+Notation vis e k := (Vis (inj1 e) k).
@@ -25,0 +109,35 @@ Indexed.Function.
+Definition over {A B C M : Type -> Type}
+{S:A +? C -< B} {T:Trigger C M} (f :
+A ~> M) : B ~> M := fun t
+b => match case b with | inl1 a =>
+f _ a | inr1 c => trigger c end.
+
```
Proof State:
```
1 subgoal
K, V : Type map : Type M : Map K V map MOk : MapOk eq M
Kdec : RelDec eq
KdecOk : RelDec_Correct Kdec
R : Type E : Type -> Type F : Type -> Type
d : V SE : mapE K d +? F -< E RR : R -> R -> Prop
gL' : itree F (map * R) -> itree F (map * R) -> Prop
INCL : forall x0 x1 : itree F (map * R), bot2 x0 x1 -> gL' x0 x1
gH' : itree F (map * R) -> itree F (map * R) -> Prop
INCH : forall x0 x1 : itree F (map * R), bot2 x0 x1 -> gH' x0 x1
CHL : forall x x0 : itree E R,
eutt RR x x0 ->
forall x2 x3 : map,
eq_map x2 x3 ->
gL' (interp_state (over (handle_map (V:=V))) x x2)
(interp_state (over (handle_map (V:=V))) x0 x3)
CHH : forall x x0 : itree E R,
eutt RR x x0 ->
forall x2 x3 : map,
eq_map x2 x3 ->
gH' (interp_state (over (handle_map (V:=V))) x x2)
(interp_state (over (handle_map (V:=V))) x0 x3)
x, y : itree E R u : Type e : E u k1, k2 : u -> itree E R
REL : forall v : u, paco2 (eqit_ RR true true id) bot2 (k1 v) (k2 v)
s1, s2 : map
H1 : eq_map s1 s2
f : F u v : u
============================
eqit_ (prod_rel eq_map eq) true true id
(upaco2 (eqit_ (prod_rel eq_map eq) true true id) bot2)
(ITree.subst (fun t : u => Ret (s1, t)) (Ret v))
(ITree.subst (fun t : u => Ret (s2, t)) (Ret v))
```
Next Tactic:
```
|
red; cbn.
```
|
{"final_commit":"440b18da2c80560f3454b2cbc1736209a76d64ad","project":"InteractionTrees","repair_file":"\/data\/coq-pearls\/repairs\/InteractionTrees\/repair-InteractionTrees-fe0513e4d0c57354d47c6ed811e0d6a0ce65670d-440b18da2c80560f3454b2cbc1736209a76d64ad-8_10_2-8_10_2-1.git.json","initial_commit":"fe0513e4d0c57354d47c6ed811e0d6a0ce65670d","file":"theories\/Events\/MapDefaultFacts.v"}
|
Proof State:
```
6 subgoals
h : R
H : Rabs h <= 1
H0 : 0 <= 1 + h
Hlt : h < 0
============================
sqrt (1 + h)Β² - 1 < 0
subgoal 2 is:
sqrt (1 + h) - 1 < 0
subgoal 3 is:
Rabs (sqrt (1 + h) - 1) <= Rabs (sqrt (1 + h)Β² - 1)
subgoal 4 is:
Rabs (sqrt (1 + h) - 1) <= Rabs (sqrt (1 + h)Β² - 1)
subgoal 5 is:
Rabs (sqrt (1 + h)Β² - 1) <= Rabs h
subgoal 6 is:
0 <= 1 + h
```
Ongoing Proof History:
```
Proof.
intros; cut (0 <= 1 + h).
intro; apply Rle_trans with (Rabs (sqrt (Rsqr (1 + h)) - 1)).
destruct (total_order_T h 0) as [[Hlt|Heq]|Hgt].
repeat rewrite Rabs_left.
unfold Rminus; do 2 rewrite <- (Rplus_comm (-1)).
change (-1) with (-(1)).
do 2 rewrite Ropp_plus_distr; rewrite Ropp_involutive; apply Rplus_le_compat_l.
apply Ropp_le_contravar; apply sqrt_le_1.
apply Rle_0_sqr.
apply H0.
pattern (1 + h) at 2; rewrite <- Rmult_1_r; unfold Rsqr; apply Rmult_le_compat_l.
apply H0.
pattern 1 at 2; rewrite <- Rplus_0_r; apply Rplus_le_compat_l; left; assumption.
```
Next Tactic:
```
|
apply <LOOKUP>Rplus_lt_reg_l : forall r r1 r2 : R, r + r1 < r + r2 -> r1 < r2</LOOKUP> with 1; rewrite <LOOKUP>Rplus_0_r : forall r : R, r + 0 = r</LOOKUP>; rewrite <LOOKUP>Rplus_comm : forall r1 r2 : R, r1 + r2 = r2 + r1</LOOKUP>; unfold Rminus; rewrite <LOOKUP>Rplus_assoc : forall r1 r2 r3 : R, r1 + r2 + r3 = r1 + (r2 + r3)</LOOKUP>; rewrite <LOOKUP>Rplus_opp_l : forall r : R, - r + r = 0</LOOKUP>; rewrite <LOOKUP>Rplus_0_r : forall r : R, r + 0 = r</LOOKUP>.
```
|
{"package":"coq","file":"\/home\/tpr\/.opam\/PackageTable~8.13~2022.01.csv.formatted\/lib\/coq\/theories\/Reals\/Sqrt_reg.v"}
|
Ongoing Proof History:
```
Proof.
intros.
apply semax_seq_inv in H.
destruct H as [Q [? ?]].
eapply (AuxDefs.semax_loop _ P Q).
```
Proof State:
```
2 subgoals
CS : compspecs Espec : OracleKind Delta : tycontext
P : environ -> mpred body, incr : statement
R : ret_assert Q : environ -> mpred
H : semax Delta P body (overridePost Q (loop_nocontinue_ret_assert P R))
H0 : semax Delta Q incr (loop_nocontinue_ret_assert P R)
============================
AuxDefs.semax Delta P body (loop1_ret_assert Q R)
subgoal 2 is:
AuxDefs.semax Delta Q incr (loop2_ret_assert P R)
```
Next Tactic:
```
|
+
```
|
{"package":"coq-vst","file":"\/home\/tpr\/.opam\/PackageTable~8.13~2022.01.csv.formatted\/lib\/coq\/user-contrib\/VST\/floyd\/SeparationLogicAsLogic.v"}
|
Commit Diff:
```
diff --git a/theories/Aux.v b/theories/Aux.v
index d240627..c7e8186 100644
--- a/theories/Aux.v
+++ b/theories/Aux.v
@@ -27 +26,0 @@
-
@@ -30,0 +30 @@ Require Export List.
+Export ListNotations.
@@ -85,11 +84,0 @@ Hint Resolve le_minus: arith.
-Section EqBool.
-
-Definition eq_bool_dec : forall a b : bool, {a = b} + {a <> b}.
-Proof.
-intros a b; case a; case b; simpl in |- *; auto.
-right; red in |- *; intros; discriminate.
-Defined.
-
-End EqBool.
-
-
@@ -231 +220 @@ intros a1 l H [H1| H1]; auto.
-exists (nil (A:=A)); exists l; simpl in |- *; auto.
+exists []; exists l; simpl in |- *; auto.
@@ -270,4 +259,4 @@ Theorem app_inv_app : forall l1 l2 l3 l4 a,
-l1 ++ l2 = l3 ++ a :: l4
--> (exists l5 : list A, l1 = l3
-++ a :: l5) \/ (exists l5 :
-_, l2 = l5 ++ a :: l4).
+l1 ++ l2 = l3 ++ a ::
+l4 -> (exists l5 : list A, l1
+= l3 ++ a :: l5) \/ (exists
+l5, l2 = l5 ++ a :: l4).
@@ -288,2 +277,2 @@ Theorem app_inv_app2 : forall l1 l2 l3 l4 a b, l1
-(exists l5 : list A, l1 = l3 ++ a ::
-b :: l5) \/ (exists l5 : _, l2 = l5
+(exists l5 : list A, l1 = l3 ++ a
+:: b :: l5) \/ (exists l5, l2 = l5
@@ -291 +280 @@ b :: l5) \/ (exists l5 : _, l2 = l5
-++ a :: nil /\ l2 = b :: l4.
+++ a :: [] /\ l2 = b :: l4.
@@ -310,7 +299,7 @@ Qed.
-Theorem same_length_ex : forall (a : A) l1
-l2 l3, length (l1 ++ a :: l2)
-= length l3 -> exists l4 :
-_, (exists l5 : _, (exists b
-: B, length l1 = length l4
-/\ length l2 = length l5 /\
-l3 = l4 ++ b :: l5)).
+Theorem same_length_ex : forall (a : A)
+l1 l2 l3, length (l1 ++ a
+:: l2) = length l3 -> exists
+l4, (exists l5, (exists b : B,
+length l1 = length l4 /\ length
+l2 = length l5 /\ l3
+= l4 ++ b :: l5)).
@@ -320 +309 @@ intros l2 l3; case l3; simpl in |- *; try (intros; discriminate).
-intros b l H; exists (nil (A:=B)); exists l; exists b; repeat (split; auto).
+intros b l H; exists []; exists l; exists b; repeat (split; auto).
@@ -341,3 +330,3 @@ Qed.
-Theorem in_map_fst_inv : forall a (l : list (B
-* C)), In a (map (fst (B:=_)) l) ->
-exists c : _, In (a, c) l.
+Theorem in_map_fst_inv : forall a (l : list
+(B * C)), In a (map (fst (B:=_))
+l) -> exists c, In (a, c) l.
@@ -375,2 +364,2 @@ Theorem in_flat_map_ex : forall (l : list B) (f : B
--> list C) a, In a (flat_map f l) -> exists
-b : _, In b l /\ In a (f b).
+-> list C) a, In a (flat_map f l) ->
+exists b, In b l /\ In a (f b).
@@ -397 +386 @@ list C := fun l2
-nil => nil | a
+[] => [] | a
@@ -399 +388 @@ nil => nil | a
-with | nil => nil
+with | [] => []
@@ -431 +420 @@ match n with | O
-=> nil | S n1
+=> [] | S n1
@@ -433 +422 @@ match n with | O
-nil => nil | a
+[] => [] | a
@@ -482 +471 @@ match n with | O
-nil => nil | a
+[] => [] | a
@@ -512 +501 @@ Qed.
-Theorem skip_n_id : forall l : list A, skip_n l (length l) = nil.
+Theorem skip_n_id : forall l : list A, skip_n l (length l) = [].
@@ -533 +522 @@ Theorem exist_first_max : forall l : list
-nat, l <> nil -> exists a
+nat, l <> [] -> exists a
@@ -544,2 +533,2 @@ intros a l0; case l0.
-intros H H0; exists a; exists (nil (A:=nat)); exists (nil
-(A:=nat)); repeat (split; simpl in |- *; auto with datatypes).
+intros H H0; exists a; exists []; exists [];
+repeat (split; simpl in |- *; auto with datatypes).
@@ -552,2 +541,2 @@ case (le_or_lt a1 a); intros HH4; auto.
-exists a; exists (nil (A:=nat)); exists (n
-:: l1); repeat (split; auto with datatypes).
+exists a; exists []; exists (n ::
+l1); repeat (split; auto with datatypes).
```
Proof State:
```
2 subgoals
A : Type l1 : list A a : A l : list A
H : forall l2 : list A, permutation (l ++ l2) (l2 ++ l)
l2 : list A
============================
permutation (l ++ [a] ++ l2) (l ++ l2 ++ [a])
subgoal 2 is:
(l2 ++ [a]) ++ l = l2 ++ a :: l
```
Ongoing Proof History:
```
Proof using.
intros l1; elim l1; auto.
intros; rewrite <- app_nil_end; auto.
intros a l H l2.
- replace (l2 ++ a :: l) with ((l2 ++ a :: nil) ++ l).
+ replace (l2 ++ a :: l) with ((l2 ++ a :: []) ++ l).
- apply permutation_trans with (l ++ l2 ++ a :: nil); auto.
+ apply permutation_trans with (l ++ l2 ++ a :: []); auto.
- apply permutation_trans with (((a :: nil) ++ l2) ++ l); auto.
+ apply permutation_trans with (((a :: []) ++ l2) ++ l); auto.
simpl in |- *; auto.
- apply permutation_trans with (l ++ (a :: nil) ++ l2); auto.
+ apply permutation_trans with (l ++ (a :: []) ++ l2); auto.
apply permutation_sym; auto.
? replace (l2 ++ a :: l) with ((l2 ++ a :: nil) ++ l).
? apply permutation_app_comp; auto.
? elim l2; simpl in |- *; auto.
```
Next Tactic:
```
|
replace (l2 ++ a :: l) with ((l2 ++ a :: []) ++ l).
```
|
{"final_commit":"6e3f37b2113e34f74c6b5a6ff2c7346a9cdf9e4e","project":"huffman","repair_file":"\/data\/coq-pearls\/repairs\/huffman\/repair-huffman-773c1feea03edc5e845ecafc7c9752fa403e7faf-6e3f37b2113e34f74c6b5a6ff2c7346a9cdf9e4e-8_11_2-8_11_2-4.git.json","initial_commit":"773c1feea03edc5e845ecafc7c9752fa403e7faf","file":"theories\/Permutation.v"}
|
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