prompt stringlengths 134 5.79k | completion stringlengths 5 2.16k | meta stringlengths 120 394 |
|---|---|---|
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"} |
Ongoing Proof History:
```
Proof.
revert is is'.
induction i as [ | i].
-
intros is is'.
rewrite natminuseqn.
rewrite natminuseqn.
apply is.
-
intros is is'.
induction (natlthorgeh n (S i)) as [H | H].
+
assert (e : n - S i = 0).
{
apply minuseq0.
exact (natlthtoleh _ _ H).
}
rewrite e.
apply (natlthandplusrinv _ _ (S i)).
rewrite natplusl0.
rewrite minusplusnmm.
apply is'.
apply natlthtoleh.
apply is'.
+
apply (natlthandplusrinv _ _ (S i)).
rewrite (minusplusnmm m (S i)).
```
Proof State:
```
2 subgoals
n, m, i : nat IHi : n < m β i < m β n - i < m - i
is : n < m is' : S i < m H : n β₯ S i
============================
n - S i + S i < m
subgoal 2 is:
m β₯ S i
```
Next Tactic:
```
| *
``` | {"package":"coq-unimath","file":"\/home\/tpr\/.opam\/PackageTable~8.13~2022.01.csv.formatted\/lib\/coq\/user-contrib\/UniMath\/Foundations\/NaturalNumbers.v"} |
Ongoing Proof History:
```
Proof.
split=> [[B eqrUV injB homB] | [f injf homf defV]].
have: \rank (U *m val_submod (in_submod U 1%:M *m B)) = \rank U.
do 2!rewrite mulmxA mxrankMfree ?row_base_free //.
by rewrite -(eqmxMr _ (val_submod1 U)) -in_submodE val_submodK mxrank1.
case/complete_unitmx => f injf defUf; exists f => //.
apply/hom_mxP=> x Gx; rewrite -defUf -2!mulmxA -(val_submodJ modV) //.
```
Proof State:
```
3 subgoals
F : fieldType gT : finGroupType G : {group gT}
n : nat rG : mx_representation F G n U, V : 'M_n
modU : mxmodule (G:=G) rG U
modV : mxmodule (G:=G) rG V
B : 'M_(\rank U, \rank V)
eqrUV : \rank U = \rank V
injB : row_free (F:=F) B
homB : forall x : gT,
x \in G ->
submod_repr (G:=G) (rG:=rG) (U:=U) modU x *m B =
B *m submod_repr (G:=G) (rG:=rG) (U:=V) modV x
f : 'M_n
injf : f \in unitmx
defUf : U *m val_submod (in_submod U 1%:M *m B) = U *m f
x : gT
Gx : x \in G
============================
U *m (rG x *m f) =
U *m val_submod
(in_submod U 1%:M *m B *m submod_mx (G:=G) (rG:=rG) (U:=V) modV x)
subgoal 2 is:
(U *m f :=: V)%MS
subgoal 3 is:
mx_rsim (submod_repr (G:=G) (rG:=rG) (U:=U) modU)
(submod_repr (G:=G) (rG:=rG) (U:=V) modV)
```
Next Tactic:
```
| rewrite -(<LOOKUP>mulmxA : forall [R : ringType] [m n p q : nat] (A : 'M_(m, n)) (B : 'M_(n, p)) (C : 'M_(p, q)), A *m (B *m C) = A *m B *m C</LOOKUP> _ B) -homB // <LOOKUP>val_submodE : forall [n : nat] [U : 'M_n] [m : nat] (W : 'M_(m, \rank U)), val_submod W = W *m val_submod 1%:M</LOOKUP> 3!(<LOOKUP>mulmxA : forall [R : ringType] [m n p q : nat] (A : 'M_(m, n)) (B : 'M_(n, p)) (C : 'M_(p, q)), A *m (B *m C) = A *m B *m C</LOOKUP> U) (<LOOKUP>mulmxA : forall [R : ringType] [m n p q : nat] (A : 'M_(m, n)) (B : 'M_(n, p)) (C : 'M_(p, q)), A *m (B *m C) = A *m B *m C</LOOKUP> _ _ B).
``` | {"package":"coq-mathcomp-character","file":"\/home\/tpr\/.opam\/PackageTable~8.13~2022.01.csv.formatted\/lib\/coq\/user-contrib\/mathcomp\/character\/mxrepresentation.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
```
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.
+ intros H H0; rewrite (Permutation_nil (Permutation_sym H0)); auto with datatypes.
simpl in |- *; intros; discriminate.
intros n0 H l1; case l1.
- intros l2 H0 H1; rewrite perm_nil_inv with (1 := permutation_sym _ _ H1) in H0; discriminate.
+ intros l2 H0 H1; rewrite (Permutation_nil H1) in H0; discriminate.
clear l1; intros a1 l1 l2 H1 H2.
case (split_one_in_ex a1 l2); auto.
- apply permutation_in with (1 := H2); auto with datatypes.
+ apply Permutation_in with (1 := H2); auto with datatypes.
intros x H0.
apply in_flat_map with (b := (a1, x)); auto.
apply in_map; simpl in |- *.
apply H; auto.
apply eq_add_S.
apply trans_equal with (1 := H1).
change (length l2 = length (a1 :: x)) in |- *.
? apply permutation_length; auto.
? apply permutation_sym; apply split_one_permutation; auto.
? apply permutation_inv with (a := a1).
```
Proof State:
```
2 goals
A : Set n, n0 : nat
H : forall l1 l2 : list A,
n0 = length l2 ->
Permutation l1 l2 -> In l1 (all_permutations_aux l2 n0)
a1 : A l1, l2 : list A
H1 : S n0 = length l2
H2 : Permutation (a1 :: l1) l2
x : list A
H0 : In (a1, x) (split_one l2)
============================
length l2 = length (a1 :: x)
goal 2 is:
Permutation l1 x
```
Next Tactic:
```
| apply <LOOKUP>Permutation_length : forall [A : Type] [l l' : list A], Permutation l l' -> length l = length l'</LOOKUP>; auto.
``` | {"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.
ginit.
? gcofix CIH.
? intros x y H.
? rewrite itree_eta, (itree_eta (translate h y)), !unfold_translate, <-!itree_eta.
```
Proof State:
```
1 subgoal
E : Type -> Type F : Type -> Type R : Type
h : forall T : Type, E T -> F T
============================
forall x y : itree E R,
x β
y ->
gpaco2 (eqit_ eq false false id) (eqitC eq false false) bot2 bot2
(translate h x) (translate h y)
```
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.
```
Next Tactic:
```
| pcofix CIH.
``` | {"final_commit":"7f00358aeb230f7970571fddaa8b256375910c62","project":"InteractionTrees","repair_file":"\/data\/coq-pearls\/repairs\/InteractionTrees\/repair-InteractionTrees-672193e2507cf3326b834f5fa47b7518af829b4b-7f00358aeb230f7970571fddaa8b256375910c62-8_10_2-8_10_2-93.git.json","initial_commit":"672193e2507cf3326b834f5fa47b7518af829b4b","file":"theories\/Interp\/TranslateFacts.v"} |
Ongoing Proof History:
```
Proof.
repeat split.
-
intros X Y f g Ξ± ; cbn.
rewrite !vassocr.
rewrite vcomp_lunitor.
rewrite !vassocl.
rewrite rinvunitor_natural.
rewrite rwhisker_hcomp.
apply idpath.
-
intros X ; cbn.
rewrite id2_left.
rewrite !vassocr.
rewrite vcomp_lunitor.
rewrite !vassocl.
rewrite lunitor_runitor_identity.
rewrite rinvunitor_natural.
rewrite lunitor_V_id_is_left_unit_V_id.
rewrite rwhisker_hcomp.
apply idpath.
-
intros X Y Z f g ; cbn.
rewrite id2_left.
rewrite <- lwhisker_vcomp, <- rwhisker_vcomp.
```
Proof State:
```
1 subgoal
Bβ, Bβ : bicat F : psfunctor Bβ Bβ X, Y, Z : Bβ
f : Bβ β¦ X, Y β§ g : Bβ β¦ Y, Z β§
============================
(idβ (F X) β (pr222 (pr1 F)) X Y Z f g)
β’ (lunitor (# F (f Β· g)) β’ rinvunitor (# F (f Β· g))) =
((((lassociator (idβ (F X)) (# F f) (# F g)
β’ ((lunitor (# F f) βΉ # F g) β’ (rinvunitor (# F f) βΉ # F g)))
β’ rassociator (# F f) (idβ (F Y)) (# F g))
β’ ((# F f β lunitor (# F g)) β’ (# F f β rinvunitor (# F g))))
β’ lassociator (# F f) (# F g) (idβ (F Z)))
β’ ((pr222 (pr1 F)) X Y Z f g βΉ idβ (F Z))
```
Next Tactic:
```
| rewrite !<LOOKUP>vassocr : β (C : prebicat) (a b : C) (f g h k : C β¦ a, b β§) (x : f ==> g) (y : g ==> h) (z : h ==> k), x β’ (y β’ z) = (x β’ y) β’ z</LOOKUP>.
``` | {"package":"coq-unimath","file":"\/home\/tpr\/.opam\/PackageTable~8.13~2022.01.csv.formatted\/lib\/coq\/user-contrib\/UniMath\/Bicategories\/Transformations\/Examples\/Unitality.v"} |
Ongoing Proof History:
```
intros l1 l2 H1 Hi; case ulist_incl_permutation with ( 2 := Hi ); auto.
intros l3 Hl3; rewrite Permutation_length with ( 1 := Hl3 ); auto.
? rewrite app_length; simpl; auto with arith.
? Qed.
```
Proof State:
```
1 goal
A : Set eqA_dec : forall a b : A, {a = b} + {a <> b} l1, l2 : list A
H1 : ulist l1
Hi : incl l1 l2
l3 : list A
Hl3 : Permutation l2 (l1 ++ l3)
============================
length l1 <= length (l1 ++ l3)
```
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..5083f1b 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.
```
Next Tactic:
```
| rewrite <LOOKUP>length_app : forall (A : Set) (l1 l2 : list A), length (l1 ++ l2) = length l1 + length l2</LOOKUP>; simpl; auto with arith.
``` | {"final_commit":"427448011e3341d35aa5b9e00f675f6582fcd477","project":"sudoku","repair_file":"\/data\/coq-pearls\/repairs\/sudoku\/repair-sudoku-dc8d4294a32dbb6e0a642dc553b64495f27e6d60-427448011e3341d35aa5b9e00f675f6582fcd477-8_15_2-8_15_2-2.git.json","initial_commit":"dc8d4294a32dbb6e0a642dc553b64495f27e6d60","file":"UList.v"} |
Ongoing Proof History:
```
case (cross1_correct (y, z)); intros tmp _; case (tmp HH1); clear tmp HH1.
intros y1 (z1, (V1, (V2, V3))); injection V3; intros; subst; clear V3.
case (gen_column_correct l y1 z1).
intros tmp _; case (tmp Hl); clear tmp.
intros x1 (Hl1, Hy); subst.
- repeat (split; auto); case (in_indexes y1); auto.
+ repeat (split; auto); case (in_indexes y1)...
intros Hn; case (fold_clause_insert1 _ (fun iz => gen_rect (fst iz) (snd iz))) with (1 := Hn); clear Hn.
intros ((i,z), (n1, (HH1, HH2))); simpl in HH2.
injection HH2; intros; subst; clear HH2.
case (cross1_correct (i, z)); intros tmp _; case (tmp HH1); clear tmp HH1.
intros i1 (z1, (V1, (V2, V3))); injection V3; intros; subst; clear V3.
case (gen_rect_correct l i1 z1).
- case (in_indexes i1); auto.
+ case (in_indexes i1)...
intros tmp _; case (tmp Hl); clear tmp.
intros x1 (y1, (Hl1, (Hl2, Hl3))); subst.
repeat (split; auto).
unfold size.
- repeat rewrite (mult_comm h); apply mult_lt_plus; auto.
+ repeat rewrite (mult_comm h); apply mult_lt_plus...
- apply div_lt; auto.
+ apply div_lt...
- case (in_indexes i1); auto.
+ case (in_indexes i1)...
unfold size.
- repeat rewrite (mult_comm w); apply mult_lt_plus; auto.
+ repeat rewrite (mult_comm w); apply mult_lt_plus...
- apply mod_lt; auto.
+ apply mod_lt...
? apply le_lt_trans with (2 := Hl2); auto with arith.
? unfold all_cell; intros Hn; case fold_clause_insert1 with (1 := Hn); clear Hn.
? intros ((x, y), (n1, (HH1, HH2))); simpl in HH2.
```
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...
```
Proof State:
```
2 goals
h, w : nat s : list nat
H : length s = size * size
H1 : empty s
z1, i1, x1, y1 : nat
Hl : In (L (Pos (h * div i1 h + x1) (w * mod i1 h + y1)) z1)
(gen_rect i1 z1)
n1 : nat
V1 : In i1 indexes
V2 : In z1 ref_list
Hl2 : x1 < h
Hl3 : y1 < w
============================
0 < h
goal 2 is:
In (n, c) all_cell -> valid_lit l s
```
Next Tactic:
```
| apply le_lt_trans with (2 := Hl2)...
``` | {"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.
```
Proof State:
```
1 subgoal
K : fieldType vT : vectType K
vs2mxP : forall U V, reflect (U = V) (vs2mx U == vs2mx V)%MS
memvK : forall v U, (v \in U) = (v2r v <= vs2mx U)%MS
mem_r2v : forall (rv : 'rV_(Vector.dim vT)) U,
(r2v rv \in U) = (rv <= vs2mx U)%MS
vs2mx0 : vs2mx 0 = 0
vs2mxD : forall U V, vs2mx (U + V) = (vs2mx U + vs2mx V)%MS
vs2mx_sum := big_morph vs2mx vs2mxD vs2mx0
: forall (I : Type) (r : seq I) (P : pred I) (F : I -> {vspace vT}),
vs2mx (\sum_(i <- r | P i) F i) = (\sum_(i <- r | P i) vs2mx (F i))%MS
vs2mxI : forall U V, vs2mx (U :&: V) = (vs2mx U :&: vs2mx V)%MS
vs2mxF : vs2mx fullv = 1%:M
row_b2mx : forall (n : nat) (X : n.-tuple vT) (i : 'I_n),
row i (b2mx X) = v2r X`_i
span_b2mx : forall (n : nat) (X : n.-tuple vT), <<X>>%VS = mx2vs (b2mx X)
mul_b2mx : forall (n : nat) (X : n.-tuple vT) (rk : 'rV_n),
\sum_i rk 0 i *: X`_i = r2v (rk *m b2mx X)
lin_b2mx : forall (n : nat) (X : n.-tuple vT) (k : 'I_n -> K),
\sum_(i < n) k i *: X`_i = r2v (\row_i k i *m b2mx X)
free_b2mx : forall (n : nat) (X : n.-tuple vT),
free X = row_free (F:=K) (b2mx X)
U, V : {vspace vT}
============================
(U :\: V + U :&: V)%VS = U
```
Next Tactic:
```
| apply/vs2mxP; rewrite vs2mxD -<LOOKUP>genmx_adds : forall [F : fieldType] [m1 m2 n : nat] (A : 'M_(m1, n)) (B : 'M_(m2, n)), <<(A + B)%MS>>%MS = (<<A>> + <<B>>)%MS</LOOKUP> !<LOOKUP>genmxE : forall [F : fieldType] [m n : nat] (A : 'M_(m, n)), (<<A>> :=: A)%MS</LOOKUP>.
``` | {"package":"coq-mathcomp-algebra","file":"\/home\/tpr\/.opam\/PackageTable~8.13~2022.01.csv.formatted\/lib\/coq\/user-contrib\/mathcomp\/algebra\/vector.v"} |
Proof State:
```
1 subgoal
cs : compspecs t' : type n : Z a : attr
sh : Share.t v : list (reptype t') p : val
H : n >= 0
H0 : field_compatible (Tarray t' n a) nil p
H1 : Zlength v = Z.max 0 n
H2 : Forall (value_fits t') v
============================
Zlength v = n
```
Ongoing Proof History:
```
Proof.
intros.
eapply derives_trans; [apply data_array_at_local_facts |].
apply prop_derives.
intros [? [? ?]]; split3; auto.
```
Next Tactic:
```
| rewrite <LOOKUP>Z.max_r : forall n m : Z, n <= m -> Z.max n m = m</LOOKUP> in H1 by lia.
``` | {"package":"coq-vst","file":"\/home\/tpr\/.opam\/PackageTable~8.13~2022.01.csv.formatted\/lib\/coq\/user-contrib\/VST\/floyd\/field_at.v"} |
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.
+ intros H H0; rewrite (Permutation_nil (Permutation_sym H0)); auto with datatypes.
simpl in |- *; intros; discriminate.
intros n0 H l1; case l1.
- intros l2 H0 H1; rewrite perm_nil_inv with (1 := permutation_sym _ _ H1) in H0; discriminate.
+ intros l2 H0 H1; rewrite (Permutation_nil H1) in H0; discriminate.
clear l1; intros a1 l1 l2 H1 H2.
case (split_one_in_ex a1 l2); auto.
? apply permutation_in with (1 := H2); auto with datatypes.
? intros x H0.
? apply in_flat_map with (b := (a1, x)); auto.
```
Proof State:
```
2 goals
A : Set n, n0 : nat
H : forall l1 l2 : list A,
n0 = length l2 ->
Permutation l1 l2 -> In l1 (all_permutations_aux l2 n0)
a1 : A l1, l2 : list A
H1 : S n0 = length l2
H2 : Permutation (a1 :: l1) l2
============================
In a1 l2
goal 2 is:
forall x : list A,
In (a1, x) (split_one l2) ->
In (a1 :: l1)
(flat_map
(fun p : A * list A =>
map (cons (fst p)) (all_permutations_aux (snd p) n0))
(split_one l2))
```
Commit Diff:
```
diff --git a/Permutation.v b/Permutation.v
index ecfa699..e499555 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 |- *.
```
Next Tactic:
```
| apply <LOOKUP>Permutation_in : forall [A : Type] [l l' : list A] (x : A), Permutation l l' -> In x l -> In x l'</LOOKUP> with (1 := H2); auto with datatypes.
``` | {"final_commit":"5a29a619b6acc23531ef31506b909056de730bf6","project":"sudoku","repair_file":"\/data\/coq-pearls\/repairs\/sudoku\/repair-sudoku-b5543771d6bdeab13ec58ac6c8d281815d96ed62-5a29a619b6acc23531ef31506b909056de730bf6-8_15_2-8_15_2-17.git.json","initial_commit":"b5543771d6bdeab13ec58ac6c8d281815d96ed62","file":"Permutation.v"} |
Commit Diff:
```
diff --git a/Eval.v b/Eval.v
index bc7147c..0f3077e 100644
--- a/Eval.v
+++ b/Eval.v
@@ -42,0 +43,13 @@ v]fields)] Ο | None => Ο end.
+
+Inductive Result : Type
+:= | Timeout |
+Error | Success
+: Value ->
+Store -> Result.
+Inductive Result_l : Type
+:= | Timeout_l |
+Error_l | Success_l :
+(list Value) ->
+Store -> Result_l.
+
+
@@ -74 +87 @@ el _β§(Ο1, Ο, v)(n)) with |
-Success_list args_val Ο2 => let Ο1
+Success_l args_val Ο2 => let Ο1
@@ -82 +95 @@ end) | new C args =>
-v)(n)) with | Success_list args_val Ο1
+v)(n)) with | Success_l args_val Ο1
@@ -89,37 +102,39 @@ Some Ο3 => (Success I Ο3)
-| None => Error end )
-| _ => Error end) |
-asgn e1 x e2 e' =>
-( match (β¦e1β§(Ο, Ο, v)(n)) with
-| Success v1 Ο1 => match
-(β¦e2β§(Ο1, Ο, v)(n)) with | Success
-v2 Ο2 => ( let Ο3
-:= (assign v1 x v2 Ο2)
-in β¦e'β§(Ο3, Ο, v)(n)) |
-_ => Error end |
-_ => Error end )
-end end where "'β¦' e
-'β§' '(' Ο ',' Ο
-',' v ')(' k ')'"
-:= (eval e Ο Ο
-v k) with eval_list (e_l:
-list Expr) (Ο: Store) (Ο:
-Env) (v: Value) (k: nat)
-: Result := match k
-with | 0 => Timeout
-| S n => fold_left
-(eval_list_aux Ο v n) e_l
-(Success_list [] Ο) end where
-"'β¦_' e '_β§' '(' Ο
-',' Ο ',' v ')('
-k ')'" := (eval_list e
-Ο Ο v k) with
-eval_list_aux (Ο: Env) (v: Value)
-(k: nat) (acc: Result) (e:
-Expr) := match k with
-| 0 => Timeout |
-S n => match acc
-with | Success_list vs Ο1
-=> match (β¦eβ§(Ο1, Ο, v)(n))
-with | Success v Ο2
-=> Success_list (v::vs) Ο2 |
-z => z end |
+| None => Error end
+) | _ => Error
+end) | asgn e1 x
+e2 e' => ( match
```
Ongoing Proof History:
```
{
induction el0; intros; simpl in H4.
+
invert_constructor_equalities; subst; split => //.
+
destruct n; [rewrite_anywhere foldLeft_constant => // |].
simpl in H4.
+ destruct_eval.
? destruct (β¦ a β§ (Ο1, Ο, Ο )( n)) eqn:A; try solve [rewrite_anywhere foldLeft_constant => //; try eval_not_success_list].
? unshelve epose proof (H n _ _ _ _ _ _ _ A _ _); try lia ; eauto with wf; destruct_and.
? assert (dom Ο <= dom s) by eauto using PeanoNat.Nat.le_trans with pM.
```
Proof State:
```
1 subgoal
n : nat
H : forall k : nat,
k < S (S (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 el : list Expr Ο : list Obj Ο : Env
Ο, l0 : Value Ο0 : Store Ο' : list Value Ο_n : Store
H0 : fold_left (eval_list_aux Ο Ο (S n)) el (Success_l [] Ο0) =
Success_l Ο' Ο_n
H1 : wf Ο
H2 : (codom Ο βͺ {Ο}) βͺ½ Ο
H3 : (β¦ e0 β§ (Ο, Ο, Ο )( S (S n))) = Success l0 Ο0
H_scope : (Ο, codom Ο βͺ {Ο}) β (Ο0, {l0})
H_preserv : Ο β Ο0 β codom Ο βͺ {Ο}
a : Expr el0 : list Expr
IHel0 : forall (Ο' : list Value) (Ο1 Ο2 : Store) (acc : list Value),
fold_left (eval_list_aux Ο Ο (S n)) el0 (Success_l acc Ο1) =
Success_l Ο' Ο2 ->
wf Ο1 ->
dom Ο <= dom Ο1 ->
Ο β Ο1 β codom Ο βͺ {Ο} ->
(codom acc βͺ {l0}) βͺ½ Ο1 ->
(Ο, codom Ο βͺ {Ο}) β (Ο1, codom acc βͺ {l0}) ->
((Ο, codom Ο βͺ {Ο}) β (Ο2, codom Ο' βͺ {l0})) /\
(Ο β Ο2 β codom Ο βͺ {Ο})
Ο'0 : list Value Ο1, Ο2 : Store acc : list Value
v : Value s : Store
H10 : (β¦ a β§ (Ο1, Ο, Ο )( n)) = Success v s
H4 : fold_left
(fun (acc : Result_l) (e : Expr) =>
match acc with
| Success_l vs Ο1 =>
match β¦ e β§ (Ο1, Ο, Ο )( n) with
| Timeout => Timeout_l
| Error => Error_l
| Success v Ο2 => Success_l (v :: vs) Ο2
end
| _ => acc
end) el0 (Success_l (v :: acc) s) = Success_l Ο'0 Ο2
H5 : wf Ο1
H6 : dom Ο <= dom Ο1
H7 : Ο β Ο1 β codom Ο βͺ {Ο}
H8 : (codom acc βͺ {l0}) βͺ½ Ο1
H9 : (Ο, codom Ο βͺ {Ο}) β (Ο1, codom acc βͺ {l0})
============================
((Ο, codom Ο βͺ {Ο}) β (Ο2, codom Ο'0 βͺ {l0})) /\ (Ο β Ο2 β codom Ο βͺ {Ο})
```
Next Tactic:
```
| unshelve epose proof (H n _ _ _ _ _ _ _ H10 _ _); try lia ; eauto with wf; destruct_and.
``` | {"final_commit":"81a8b9e957eb20b980dd0b63d6bd418225e4e1f8","project":"celsius","repair_file":"\/data\/coq-pearls\/repairs\/celsius\/repair-celsius-ac60f930c564d720bea462f088eb6b63d28c97cc-81a8b9e957eb20b980dd0b63d6bd418225e4e1f8-8_13_2-8_13_2-6.git.json","initial_commit":"ac60f930c564d720bea462f088eb6b63d28c97cc","file":"Scopability.v"} |
Proof State:
```
1 subgoal
P : Ghost a1, a2 : option G m1, m2 : list (option G)
a3 : option G m3 : list (option G)
H : join a1 a2 a3
H0 : list_join m1 m2 m3
IHlist_join : forall n : nat,
join (nth_error m1 n) (nth_error m2 n) (nth_error m3 n)
n : nat
============================
join (nth_error (a1 :: m1) n) (nth_error (a2 :: m2) n)
(nth_error (a3 :: m3) n)
```
Ongoing Proof History:
```
Proof.
intros; revert n.
induction H; intro.
-
rewrite nth_error_nil; constructor.
-
rewrite nth_error_nil; constructor.
-
```
Next Tactic:
```
| destruct n; simpl; auto.
``` | {"package":"coq-vst","file":"\/home\/tpr\/.opam\/PackageTable~8.13~2022.01.csv.formatted\/lib\/coq\/user-contrib\/VST\/veric\/invariants.v"} |
Ongoing Proof History:
```
Proof.
intros s s'; revert s.
induction s' as [ | x' s' IH]; intros [ | x s] Hs Hs'; simpl; auto.
split; try red; intros; auto.
split; intros H.
discriminate.
assert (In x nil) by (apply H; auto).
inv.
split; try red; intros; auto.
inv.
inv.
elim_compare x x' as C.
rewrite IH; auto.
split; intros S y; specialize (S y).
rewrite !InA_cons, C.
intuition.
rewrite !InA_cons, C in S.
intuition; try sort_inf_in; order.
split; intros S.
discriminate.
assert (In x (x'::s')) by (apply S; auto).
inv; try sort_inf_in; order.
rewrite IH; auto.
```
Proof State:
```
1 subgoal
x' : elt s' : list elt
IH : forall s : t, Ok s -> Ok s' -> subset s s' = true <-> Subset s s'
x : elt s : list elt
H : Sorted X.lt s'
H0 : Inf x' s'
H1 : Sorted X.lt s
H2 : Inf x s
C : X.lt x' x
============================
Subset (x :: s) s' <-> Subset (x :: s) (x' :: s')
```
Next Tactic:
```
| split; intros S y; specialize (<LOOKUP>S : nat -> nat</LOOKUP> y).
``` | {"package":"coq","file":"\/home\/tpr\/.opam\/PackageTable~8.13~2022.01.csv.formatted\/lib\/coq\/theories\/MSets\/MSetList.v"} |
Proof State:
```
1 subgoal
Bβ, Bβ, Bβ : bicat Fβ, Fβ : psfunctor Bβ Bβ
G : psfunctor Bβ Bβ Ξ· : pstrans Fβ Fβ Dβ : disp_bicat Bβ
Dβ : disp_bicat Bβ Dβ : disp_bicat Bβ FFβ : disp_psfunctor Dβ Dβ Fβ
FFβ : disp_psfunctor Dβ Dβ Fβ GG : disp_psfunctor Dβ Dβ G
Ξ·Ξ· : disp_pstrans FFβ FFβ Ξ·
PP := total_psfunctor Dβ Dβ G GG β
total_pstrans FFβ FFβ Ξ· Ξ·Ξ·
: pstrans
(comp_psfunctor (total_psfunctor Dβ Dβ G GG)
(total_psfunctor Dβ Dβ Fβ FFβ))
(comp_psfunctor (total_psfunctor Dβ Dβ G GG)
(total_psfunctor Dβ Dβ Fβ FFβ))
PP2 := pstrans_to_is_pstrans PP : is_pstrans (pstrans_to_pstrans_data PP)
X : pstrans_to_pstrans_data PP =
total_pstrans_data (disp_pseudo_comp Fβ G Dβ Dβ Dβ FFβ GG)
(disp_pseudo_comp Fβ G Dβ Dβ Dβ FFβ GG) (G β
Ξ·)
disp_left_whisker_data
============================
is_pstrans
(total_pstrans_data (disp_pseudo_comp Fβ G Dβ Dβ Dβ FFβ GG)
(disp_pseudo_comp Fβ G Dβ Dβ Dβ FFβ GG) (G β
Ξ·) disp_left_whisker_data)
```
Ongoing Proof History:
```
Proof.
apply is_disp_pstrans_from_total.
pose (PP := (total_psfunctor _ _ _ GG) β
total_pstrans _ _ _ Ξ·Ξ·).
pose (PP2 := pstrans_to_is_pstrans PP).
assert (pstrans_to_pstrans_data PP = total_pstrans_data _ _ _ disp_left_whisker_data).
-
use total2_paths_f.
+
apply idpath.
+
apply funextsec.
intro x.
apply funextsec.
intro y.
apply funextsec.
intro f.
use subtypePath.
{
intro.
apply isaprop_is_invertible_2cell.
}
apply idpath.
-
```
Next Tactic:
```
| exact (<LOOKUP>transportf : β (X : UU) (P : X β UU) (x x' : X), x = x' β P x β P x'</LOOKUP> <LOOKUP>is_pstrans : β (C D : bicat) (F G : psfunctor C D), pstrans_data F G β UU</LOOKUP> X PP2).
``` | {"package":"coq-unimath","file":"\/home\/tpr\/.opam\/PackageTable~8.13~2022.01.csv.formatted\/lib\/coq\/user-contrib\/UniMath\/Bicategories\/DisplayedBicats\/DispTransformation.v"} |
Proof State:
```
3 subgoals
l, r : tree
IHl : forall x y : elt, min_elt l = Some x -> mem y l = true -> ~ E.lt y x
e : elt
IHr : forall x y : elt, Some e = Some x -> mem y r = true -> ~ E.lt y x
z : positive
H' : mem z~0 (Node l false r) = true
Hp : Empty l
============================
~ E.lt z~0 e~1
subgoal 2 is:
~ E.lt 1 e~1
subgoal 3 is:
~ E.lt y x
```
Ongoing Proof History:
```
Proof.
unfold In.
induction s as [|l IHl o r IHr]; intros x y H H'.
discriminate.
simpl in H.
case_eq (min_elt l).
intros p Hp.
rewrite Hp in H.
injection H as [= <-].
destruct y as [z|z|]; simpl; intro; trivial.
apply (IHl p z); trivial.
intro Hp; rewrite Hp in H.
apply min_elt_3 in Hp.
destruct o.
injection H as [= <-].
intros Hl.
destruct y as [z|z|]; simpl; trivial.
elim (Hp _ H').
destruct (min_elt r).
injection H as [= <-].
destruct y as [z|z|].
apply (IHr e z); trivial.
```
Next Tactic:
```
| elim (Hp _ H').
``` | {"package":"coq","file":"\/home\/tpr\/.opam\/PackageTable~8.13~2022.01.csv.formatted\/lib\/coq\/theories\/FSets\/FSetPositive.v"} |
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.
+ remember (VisF e k3) as y.
? hinduction EQVr before r; intros; subst; try dependent destruction Heqy; try inv CHECK; eauto.
? econstructor.
? intros.
```
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..9024a7a 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.
```
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 y : itree' E R2
Heqy : y = VisF e k3
EQVr : eqitF RR2 b2 false id (upaco2 (eqit_ RR2 b2 false id) bot2)
(observe t2) y
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:
```
| hinduction EQVr before r; intros; try discriminate Heqy; eauto; inv_Vis.
``` | {"final_commit":"7f00358aeb230f7970571fddaa8b256375910c62","project":"InteractionTrees","repair_file":"\/data\/coq-pearls\/repairs\/InteractionTrees\/repair-InteractionTrees-672193e2507cf3326b834f5fa47b7518af829b4b-7f00358aeb230f7970571fddaa8b256375910c62-8_10_2-8_10_2-1.git.json","initial_commit":"672193e2507cf3326b834f5fa47b7518af829b4b","file":"theories\/Eq\/Eq.v"} |
Commit Diff:
```
diff --git a/theories/Basics/CategoryKleisliFacts.v b/theories/Basics/CategoryKleisliFacts.v
index e0b955f..d6eecd6 100644
--- a/theories/Basics/CategoryKleisliFacts.v
+++ b/theories/Basics/CategoryKleisliFacts.v
@@ -59,0 +60,8 @@ Qed.
+
+
+
+
+
+
+
+
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
```
Proof State:
```
1 subgoal
c : Type -> Type T : Type a, b : Type -> Type
f : forall T : Type, a T -> itree (a +' b) T
g : forall T : Type, b T -> itree c T a0 : a T
gL : itree c T -> itree c T -> Prop
CIH : forall t : itree (a +' b) T,
gL
(interp g
(Recursion.interp_mrec (fun (T : Type) (e : a T) => Tau (f T e))
t))
(Recursion.interp_mrec
(fun (R : Type) (e : a R) =>
interp
(fun (T : Type) (ab : (a +' b) T) =>
match ab with
| inl1 a0 =>
interp
(fun (T0 : Type) (e0 : a T0) =>
ITree.trigger (inl1 e0)) (ITree.trigger a0)
| inr1 b =>
interp
(fun (T0 : Type) (e0 : c T0) =>
ITree.trigger (inr1 e0)) (g T b)
end) (Tau (f R e)))
(interp
(fun (T : Type) (ab : (a +' b) T) =>
match ab with
| inl1 a0 =>
interp
(fun (T0 : Type) (e : a T0) => ITree.trigger (inl1 e))
(ITree.trigger a0)
| inr1 b =>
interp
(fun (T0 : Type) (e : c T0) => ITree.trigger (inr1 e))
(g T b)
end) t))
gH : itree c T -> itree c T -> Prop
CIH0 : forall t : itree (a +' b) T,
gH
(interp g
(Recursion.interp_mrec
(fun (T : Type) (e : a T) => Tau (f T e)) t))
(Recursion.interp_mrec
(fun (R : Type) (e : a R) =>
interp
(fun (T : Type) (ab : (a +' b) T) =>
match ab with
| inl1 a0 =>
interp
(fun (T0 : Type) (e0 : a T0) =>
ITree.trigger (inl1 e0))
(ITree.trigger a0)
| inr1 b =>
interp
(fun (T0 : Type) (e0 : c T0) =>
ITree.trigger (inr1 e0))
(g T b)
end) (Tau (f R e)))
(interp
(fun (T : Type) (ab : (a +' b) T) =>
match ab with
| inl1 a0 =>
interp
(fun (T0 : Type) (e : a T0) => ITree.trigger (inl1 e))
(ITree.trigger a0)
| inr1 b =>
interp
(fun (T0 : Type) (e : c T0) => ITree.trigger (inr1 e))
(g T b)
end) t))
t : itree (a +' b) T X : Type a1 : a X k : X -> itree (a +' b) T
============================
euttG eq bot2 gL gL gH
(interp g
(Recursion.interp_mrec (fun (T0 : Type) (e : a T0) => Tau (f T0 e))
(x <- f X a1;; k x)))
(Recursion.interp_mrec
(fun (R : Type) (e : a R) =>
interp
(fun (T0 : Type) (ab : (a +' b) T0) =>
match ab with
| inl1 a2 =>
interp
(fun (T1 : Type) (e0 : a T1) => ITree.trigger (inl1 e0))
(ITree.trigger a2)
| inr1 b0 =>
interp
(fun (T1 : Type) (e0 : c T1) => ITree.trigger (inr1 e0))
(g T0 b0)
end) (Tau (f R e)))
(x <-
interp
(fun (T0 : Type) (ab : (a +' b) T0) =>
match ab with
| inl1 a2 =>
interp (fun (T1 : Type) (e : a T1) => ITree.trigger (inl1 e))
(ITree.trigger a2)
| inr1 b0 =>
interp (fun (T1 : Type) (e : c T1) => ITree.trigger (inr1 e))
(g T0 b0)
end) (f X a1);;
Tau
(interp
(fun (T0 : Type) (ab : (a +' b) T0) =>
match ab with
| inl1 a2 =>
interp
(fun (T1 : Type) (e : a T1) => ITree.trigger (inl1 e))
(ITree.trigger a2)
| inr1 b0 =>
interp
(fun (T1 : Type) (e : c T1) => ITree.trigger (inr1 e))
(g T0 b0)
end) (k x))))
```
Ongoing Proof History:
```
unfold Recursion.mrec.
rewrite !interp_tau.
rewrite (unfold_interp_mrec _ _ (Tau _)); cbn.
rewrite !bind_tau.
etau.
- rewrite tau_eutt, <- interp_bind, <- 2 interp_mrec_bind.
+ rewrite tau_euttge, <- interp_bind, <- 2 interp_mrec_bind.
? setoid_rewrite (tau_eutt (interp _ _)).
? rewrite <- interp_bind.
? auto with paco.
```
Next Tactic:
```
| setoid_rewrite (<LOOKUP>tau_euttge : forall (E : Type -> Type) (R : Type) (t : itree E R), Tau t β³ t</LOOKUP> (<LOOKUP>interp : forall E M : Type -> Type, Functor.Functor M -> Monad.Monad M -> MonadIter M -> (forall T : Type, E T -> M T) -> forall T : Type, itree E T -> M T</LOOKUP> _ _)).
``` | {"final_commit":"6b6e1763da94788e6204f549039421936cad4437","project":"InteractionTrees","repair_file":"\/data\/coq-pearls\/repairs\/InteractionTrees\/repair-InteractionTrees-75a5a716bb6a18a6fdca492391f694fb344ea589-6b6e1763da94788e6204f549039421936cad4437-8_10_2-8_10_2-17.git.json","initial_commit":"75a5a716bb6a18a6fdca492391f694fb344ea589","file":"theories\/Interp\/HandlerFacts.v"} |
Proof State:
```
3 goals
h, w : nat s : list nat
H : length s = size * size
H1 : empty s
l : lit z1, i1 : nat
Hl : In l (gen_rect i1 z1)
n1 : nat
V1 : In i1 indexes
V2 : In z1 ref_list
============================
i1 < size
goal 2 is:
(In l (gen_rect i1 z1) ->
exists i2 j1 : nat,
l = L (Pos (h * div i1 h + i2) (w * mod i1 h + j1)) z1 /\
i2 < h /\ j1 < w) ->
((exists i2 j1 : nat,
l = L (Pos (h * div i1 h + i2) (w * mod i1 h + j1)) z1 /\
i2 < h /\ j1 < w) -> In l (gen_rect i1 z1)) ->
valid_lit l s
goal 3 is:
In (n, c) all_cell -> valid_lit l s
```
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...
```
Ongoing Proof History:
```
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.
+ repeat (split; auto); case (in_indexes x1)...
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.
injection HH2; intros; subst; clear HH2.
case (cross1_correct (y, z)); intros tmp _; case (tmp HH1); clear tmp HH1.
intros y1 (z1, (V1, (V2, V3))); injection V3; intros; subst; clear V3.
case (gen_column_correct l y1 z1).
intros tmp _; case (tmp Hl); clear tmp.
intros x1 (Hl1, Hy); subst.
- repeat (split; auto); case (in_indexes y1); auto.
+ repeat (split; auto); case (in_indexes y1)...
intros Hn; case (fold_clause_insert1 _ (fun iz => gen_rect (fst iz) (snd iz))) with (1 := Hn); clear Hn.
intros ((i,z), (n1, (HH1, HH2))); simpl in HH2.
injection HH2; intros; subst; clear HH2.
case (cross1_correct (i, z)); intros tmp _; case (tmp HH1); clear tmp HH1.
intros i1 (z1, (V1, (V2, V3))); injection V3; intros; subst; clear V3.
case (gen_rect_correct l i1 z1).
? case (in_indexes i1); auto.
? intros tmp _; case (tmp Hl); clear tmp.
? intros x1 (y1, (Hl1, (Hl2, Hl3))); subst.
```
Next Tactic:
```
| case (<LOOKUP>in_indexes : forall i : nat, In i indexes <-> i < size</LOOKUP> i1)...
``` | {"final_commit":"5337edcaf31b9f036a0a6651b2f733d3068b8f8f","project":"sudoku","repair_file":"\/data\/coq-pearls\/repairs\/sudoku\/repair-sudoku-427448011e3341d35aa5b9e00f675f6582fcd477-5337edcaf31b9f036a0a6651b2f733d3068b8f8f-8_15_2-8_15_2-8.git.json","initial_commit":"427448011e3341d35aa5b9e00f675f6582fcd477","file":"Sudoku.v"} |
Ongoing Proof History:
```
Proof.
intros.
apply pred_ext.
+
```
Proof State:
```
1 subgoal
A : Type ND : NatDed A P : Prop Q : A
H : P
============================
!! P --> Q |-- Q
```
Next Tactic:
```
| eapply <LOOKUP>derives_trans : forall {A : Type} {NatDed : NatDed A} (P Q R : A), P |-- Q -> Q |-- R -> P |-- R</LOOKUP>; [| apply <LOOKUP>modus_ponens : forall {A : Type} {ND : NatDed A} (P Q : A), P && (P --> Q) |-- Q</LOOKUP>].
``` | {"package":"coq-vst","file":"\/home\/tpr\/.opam\/PackageTable~8.13~2022.01.csv.formatted\/lib\/coq\/user-contrib\/VST\/msl\/log_normalize.v"} |
Ongoing Proof History:
```
+ apply ulist_perm with ref_list; auto.
- apply Permutation_sym...
+ apply Permutation_sym; auto.
- repeat rewrite <- get_column; try rewrite H2...
+ repeat rewrite <- get_column; try rewrite H2; auto.
case (gen_rect_correct l (div x h * h + div y w) z).
- apply rect_aux1...
+ apply rect_aux1; auto.
intros tmp _; case tmp; auto; clear tmp.
- apply (rm_incl _ lit_test) with (l1 := L (Pos x y) z::nil)...
+ apply (rm_incl _ lit_test) with (l1 := L (Pos x y) z::nil); auto.
intros x1 (y1, (H5, (H6, H7))); subst l.
simpl in H4.
match type of H4 with get (Pos ?X ?Y) _ = _ => generalize H3 H4; clear H3 H4; replace (Pos X Y) with (Pos (h * div x h + x1) (w * div y w + y1)); [intros H3 H4 | idtac] end.
match type of H4 with get (Pos ?X ?Y) _ = _ => assert (X = x /\ Y = y) end.
- apply rect_aux2...
+ apply rect_aux2; 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.
- rewrite get_rect in H4...
+ rewrite get_rect in H4; auto.
? rewrite get_rect in H2...
? generalize H4; clear H4.
? repeat rewrite (fun x => mult_comm x h).
```
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)...
```
Proof State:
```
7 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
H3 : In (L (Pos (h * div x h + x1) (w * div y w + y1)) z)
(rm_cont lit lit_test (fun l2 : list lit => l2)
(L (Pos x y) z) (gen_rect (div x h * h + div y w) z))
H4 : nth (mod (h * div x h + x1) h * w + mod (w * div y w + y1) w)
(rect (div (h * div x h + x1) h * h + div (w * div y w + y1) w) s)
out = z
============================
mod (h * div x h + x1) h * w + mod (w * div y w + y1) w =
mod x h * w + mod y w
goal 2 is:
h * div x h + x1 < size
goal 3 is:
w * div y w + y1 < size
goal 4 is:
div (h * div x h + x1) h * h + div (w * div y w + y1) w =
div x h * h + div y w
goal 5 is:
False
goal 6 is:
Pos (h * div x h + x1) (w * div y w + y1) =
Pos (h * div (div x h * h + div y w) h + x1)
(w * mod (div x h * h + div y w) h + y1)
goal 7 is:
False
```
Next Tactic:
```
| rewrite <LOOKUP>get_rect : forall (x y : nat) (s : list nat), x < size -> y < size -> get (Pos x y) s = nth (mod x h * w + mod y w) (rect (div x h * h + div y w) s) out</LOOKUP> in H2; 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"} |
Ongoing Proof History:
```
Proof.
induction m.
simpl; auto.
destruct o; simpl; intros.
apply (SortA_app (eqA:=eq_key_elt)).
1-2: auto with typeclass_instances.
constructor; auto.
```
Proof State:
```
3 subgoals
A : Type m, m', m'' : t A x, y, z : key e, e' : A
t0_1 : tree A a : A t0_2 : tree A
IHt0_1 : forall p : key, Sorted lt_key (xelements t0_1 p)
IHt0_2 : forall p : key, Sorted lt_key (xelements t0_2 p)
p : key
============================
HdRel lt_key (p, a) (xelements t0_2 (append p 3))
subgoal 2 is:
forall x0 y0 : key * A,
InA eq_key_elt x0 (xelements t0_1 (append p 2)) ->
InA eq_key_elt y0 ((p, a) :: xelements t0_2 (append p 3)) -> lt_key x0 y0
subgoal 3 is:
Sorted lt_key (xelements t0_1 (append p 2) ++ xelements t0_2 (append p 3))
```
Next Tactic:
```
| apply <LOOKUP>In_InfA : forall [A : Type] [ltA : A -> A -> Prop] [l : list A] [x : A], (forall y : A, In y l -> ltA x y) -> HdRel ltA x l</LOOKUP>; intros.
``` | {"package":"coq","file":"\/home\/tpr\/.opam\/PackageTable~8.13~2022.01.csv.formatted\/lib\/coq\/theories\/FSets\/FMapPositive.v"} |
Ongoing Proof History:
```
Proof.
destruct n; intros.
- omega.
+ lia.
simpl.
? replace (n - 0) with n; [ | omega ].
? reflexivity.
? Qed.
```
Proof State:
```
1 goal
T : Type n : nat a : T b : list T
H : 0 < S n
============================
a :: firstn n b = a :: firstn (n - 0) b
```
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..54bfc69 100644
--- a/theories/Data/ListFirstnSkipn.v
+++ b/theories/Data/ListFirstnSkipn.v
@@ -2,0 +3 @@ Require Import Coq.ZArith.ZArith.
+Require Import Coq.micromega.Lia.
@@ -9,2 +10,2 @@ induction n; destruct a; simpl in *; intros; auto.
-exfalso; omega.
-f_equal. eapply IHn; eauto. omega.
+exfalso; lia.
+f_equal. eapply IHn; eauto. lia.
@@ -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.
@@ -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 }.
+Global Instance Monad_list : Monad list := {
```
Next Tactic:
```
| replace (n - 0) with n; [ | 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_1-5.git.json","initial_commit":"7532d92d1d6a9d7c9c45e111766f65421f6e09ba","file":"theories\/Data\/ListFirstnSkipn.v"} |
Ongoing Proof History:
```
Proof.
intros D H r Dr q Hq.
generalize (le_eqorltNonnegativeRationals _ _ Hq) ; clear Hq.
apply sumofmaps ; intros H0.
-
rewrite H0.
exact Dr.
-
rewrite ltNonnegativeRationals_correct in H0.
generalize (pr2 (pr2 H) _ _ H0).
apply hinhuniv, sumofmaps ; intros Dq.
```
Proof State:
```
2 subgoals
D : hq β hProp H : isOneSided D r : NonnegativeRationals
Dr : D (pr1 r) q : NonnegativeRationals H0 : pr1 q < pr1 r
Dq : D (pr1 q)
============================
D (pr1 q)
subgoal 2 is:
D (pr1 q)
```
Next Tactic:
```
| +
``` | {"package":"coq-unimath","file":"\/home\/tpr\/.opam\/PackageTable~8.13~2022.01.csv.formatted\/lib\/coq\/user-contrib\/UniMath\/RealNumbers\/DedekindCuts.v"} |
Ongoing Proof History:
```
-
remember (RetF r2) as ot.
hinduction INR before CIH; intros; inv Heqot; eauto with paco.
-
assert (DEC: (exists m3, ot3 = TauF m3) \/ (forall m3, ot3 <> TauF m3)).
{
destruct ot3; eauto; right; red; intros; inv H.
}
destruct DEC as [EQ | EQ].
+
destruct EQ as [m3 ?]; subst.
econstructor.
right.
pclearbot.
eapply CIH; eauto with paco.
- eapply eqit_inv_tauLR.
+ eapply eqit_inv_Tau.
eauto.
+
inv INR; try (exfalso; eapply EQ; eauto; fail).
econstructor; eauto.
pclearbot.
punfold REL.
red in REL.
hinduction REL0 before CIH; intros; try (exfalso; eapply EQ; eauto; fail).
*
remember (RetF r1) as ot.
hinduction REL0 before CIH; intros; inv Heqot; eauto with paco.
*
remember (VisF e k1) as ot.
- hinduction REL0 before CIH; intros; dependent destruction Heqot; eauto with paco.
+ hinduction REL0 before CIH; intros; try discriminate; eauto; inv_Vis.
econstructor.
intros.
right.
destruct (REL v), (REL0 v); try contradiction.
eauto.
*
eapply IHREL0; eauto.
pstep_reverse.
destruct b1; inv CHECK0.
? apply eqit_inv_tauR.
? eauto.
? -
```
Commit Diff:
```
diff --git a/theories/Basics/Basics.v b/theories/Basics/Basics.v
index 6298c6d..d772a5d 100644
--- a/theories/Basics/Basics.v
+++ b/theories/Basics/Basics.v
@@ -20 +20,3 @@ Data.Monads.EitherMonad.
-Import MonadNotation.
+Import
+FunctorNotation
+MonadNotation.
@@ -55,0 +58,3 @@ Definition state (s a : Type) := s -> prod s a.
+Definition liftState {s a f} `{Functor f} (fa : f a) :
+Monads.stateT s f a := fun s => pair s <$> fa.
+
@@ -157 +162,32 @@ r -> iter_Prop step i r.
-Polymorphic Instance MonadIter_Prop : MonadIter Ensembles.Ensemble := @iter_Prop.
\ No newline at end of file
+Polymorphic Instance MonadIter_Prop : MonadIter Ensembles.Ensemble := @iter_Prop.
+
+
+Definition equiv_pred {A : Type} (R S: A -> Prop):
+Prop := forall a, R a <-> S a.
+
+Definition sum_pred {A B : Type} (PA : A -> Prop) (PB : B -> Prop) : A + B ->
+Prop := fun x => match x with | inl a => PA a | inr b => PB b end.
+
+Definition prod_pred {A B : Type} (PA : A -> Prop) (PB : B -> Prop)
+: A * B -> Prop := fun '(a,b) => PA a /\ PB b.
+
+Definition TT {A : Type} : A -> Prop := fun _ => True.
+Global Hint Unfold TT sum_pred prod_pred: core.
+
+Global Instance equiv_pred_refl {A} : Reflexive (@equiv_pred A).
+Proof.
+split; auto.
+Qed.
+Global Instance equiv_pred_symm {A} : Symmetric (@equiv_pred A).
+
+Proof.
+red; intros * EQ; split; intros; eapply EQ; auto.
+Qed.
+Global Instance equiv_pred_trans {A} : Transitive (@equiv_pred A).
+Proof.
+red; intros * EQ1 EQ2; split; intros; (apply EQ1,EQ2 || apply EQ2,EQ1); auto.
+Qed.
+Global Instance equiv_pred_equiv {A} : Equivalence (@equiv_pred A).
+Proof.
+split; typeclasses eauto.
+Qed.
\ No newline at end of file
diff --git a/theories/Basics/CategoryFacts.v b/theories/Basics/CategoryFacts.v
index 63cfb1d..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.
+
@@ -124 +137 @@ End CategoryFacts.
-Hint Resolve @initial_unique : cat.
```
Proof State:
```
1 subgoal
E : Type -> Type R1 : Type R2 : Type R3 : Type
RR1 : R1 -> R2 -> Prop RR2 : R2 -> R3 -> Prop
b2 : bool r : rel2 (itree E R1) (fun _ : itree E R1 => itree E R3)
CIH0 : forall (a0 : itree E R1) (a1 : itree E R3), bot2 a0 a1 -> r a0 a1
CIH : forall (x : itree E R1) (x0 : itree E R2) (x1 : itree E R3),
eqit RR1 true b2 x x0 -> eqit RR2 true b2 x0 x1 -> r x x1
m2, t1 : itree E R2 ot2 : itree' E R3
CHECK : true
REL0 : eqitF RR2 true b2 id (upaco2 (eqit_ RR2 true b2 id) bot2)
(observe t1) ot2
IHREL0 : forall m1 : itree E R1,
eqitF RR1 true b2 id (upaco2 (eqit_ RR1 true b2 id) bot2)
(observe m1) (observe t1) ->
(forall m3 : itree E R3, ot2 <> TauF m3) ->
true ->
eqitF (rcompose RR1 RR2) true b2 id
(upaco2 (eqit_ (rcompose RR1 RR2) true b2 id) r)
(observe m1) ot2
m1 : itree E R1
REL : eqitF RR1 true b2 id (upaco2 (eqit_ RR1 true b2 id) bot2)
(observe m1) (TauF t1)
EQ : forall m3 : itree E R3, ot2 <> TauF m3
============================
paco2 (eqit_ RR1 true b2 id) bot2 m1 t1
```
Next Tactic:
```
| apply <LOOKUP>eqit_inv_Tau_r : forall (E : Type -> Type) (R1 R2 : Type) (RR : R1 -> R2 -> Prop) (b2 : bool) (t1 : itree E R1) (t2 : itree E R2), eqit RR true b2 t1 (Tau t2) -> eqit RR true b2 t1 t2</LOOKUP>.
``` | {"final_commit":"f0a99a2e4751c033f4db1d29683e26aa0bab9d79","project":"InteractionTrees","repair_file":"\/data\/coq-pearls\/repairs\/InteractionTrees\/repair-InteractionTrees-6f69653dfbd363117616c670169aaf80e9ff19a6-f0a99a2e4751c033f4db1d29683e26aa0bab9d79-8_13_2-8_10_2-19.git.json","initial_commit":"6f69653dfbd363117616c670169aaf80e9ff19a6","file":"theories\/Eq\/Eq.v"} |
Proof State:
```
2 subgoals
Bn : nat -> R l : R
H : Un_cv Bn l
An := fun _ : nat => 1 : nat -> R
H0 : forall n : nat, 0 < An n
H1 : forall n : nat, 0 < sum_f_R0 An n
M : R
H2 : 0 < M
m := up M : Z
H3 : IZR (up M) > M
H4 : IZR (up M) - M <= 1
H5 : (0 <= m)%Z
============================
exists N : nat, forall n : nat, (N <= n)%nat -> M < sum_f_R0 An n
subgoal 2 is:
Un_cv (fun n : nat => sum_f_R0 Bn (Init.Nat.pred n) / INR n) l
```
Ongoing Proof History:
```
Proof with trivial.
intros Bn l H; set (An := fun _:nat => 1)...
assert (H0 : forall n:nat, 0 < An n)...
intro; unfold An; apply Rlt_0_1...
assert (H1 : forall n:nat, 0 < sum_f_R0 An n)...
intro; apply tech1...
assert (H2 : cv_infty (fun n:nat => sum_f_R0 An n))...
unfold cv_infty; intro; destruct (Rle_dec M 0) as [Hle|Hnle]...
exists 0%nat; intros; apply Rle_lt_trans with 0...
assert (H2 : 0 < M)...
auto with real...
clear Hnle; set (m := up M); elim (archimed M); intros; assert (H5 : (0 <= m)%Z)...
apply le_IZR; unfold m; simpl; left; apply Rlt_trans with M...
```
Next Tactic:
```
| elim (<LOOKUP>IZN : forall n : Z, (0 <= n)%Z -> exists m : nat, n = Z.of_nat m</LOOKUP> _ H5); intros; exists x; intros; unfold An; rewrite <LOOKUP>sum_cte : forall (x : R) (N : nat), sum_f_R0 (fun _ : nat => x) N = x * INR (S N)</LOOKUP>; rewrite <LOOKUP>Rmult_1_l : forall r : R, 1 * r = r</LOOKUP>; apply <LOOKUP>Rlt_trans : forall r1 r2 r3 : R, r1 < r2 -> r2 < r3 -> r1 < r3</LOOKUP> with (<LOOKUP>IZR : Z -> R</LOOKUP> (<LOOKUP>up : R -> Z</LOOKUP> M))...
``` | {"package":"coq","file":"\/home\/tpr\/.opam\/PackageTable~8.13~2022.01.csv.formatted\/lib\/coq\/theories\/Reals\/SeqSeries.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..dd17260 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.
@@ -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.
@@ -401,2 +401,2 @@ unfold_eqit. assert (Hev' : eventless t1); auto. punfold Hev.
-dependent induction Heutt; subst; auto.
-- rewrite <- x. auto.
+dependent induction Heutt; subst; auto with itree.
```
Proof State:
```
1 subgoal
A : Type E : Type -> Type r : stream A -> stream A -> Prop
CIH : forall x y : stream A, bisim x y -> r y x
x, y : stream A
H0 : bisim x y
============================
bisimF (upaco2 bisim_ r) (observe_stream y) (observe_stream x)
```
Ongoing Proof History:
```
Proof.
constructor; red.
-
pcofix CIH.
intros.
pfold.
red.
- destruct (observe_stream x); auto.
+ destruct (observe_stream x); auto with itree.
-
pcofix CIH.
intros.
pfold.
red.
? pinversion H0; subst; auto.
? -
? pcofix CIH.
```
Next Tactic:
```
| pinversion H0; subst; auto with itree.
``` | {"final_commit":"8fcf6f5f81ec9289d56e777b3885e715b3328486","project":"InteractionTrees","repair_file":"\/data\/coq-pearls\/repairs\/InteractionTrees\/repair-InteractionTrees-1158ff1cbb628aeaa69973bf098058744d458c36-8fcf6f5f81ec9289d56e777b3885e715b3328486-8_10_2-8_12_2-97.git.json","initial_commit":"1158ff1cbb628aeaa69973bf098058744d458c36","file":"theories\/Dijkstra\/ITreeDijkstra.v"} |
Proof State:
```
2 subgoals
A : Type B : Type JB : Join B PB : Perm_alg B SB : Sep_alg B
CB : Disj_alg B
Bdec : forall x : B, {identity x} + {~ identity x}
X : Cross_alg B X0 : Cross_alg (A -> option (lifted JB))
a : A -> option (lifted JB)
Ha : finMap a
b : A -> option (lifted JB)
Hb : finMap b
c : A -> option (lifted JB)
Hc : finMap c
d : A -> option (lifted JB)
Hd : finMap d
z : A -> option (lifted JB)
Hz : finMap z
H : join (exist (finMap (B:=lifted JB)) a Ha)
(exist (finMap (B:=lifted JB)) b Hb)
(exist (finMap (B:=lifted JB)) z Hz)
H0 : join (exist (finMap (B:=lifted JB)) c Hc)
(exist (finMap (B:=lifted JB)) d Hd)
(exist (finMap (B:=lifted JB)) z Hz)
s, p, q, r : A -> option (lifted JB)
H1 : join s p a
H3 : join q r b
H2 : join s q c
H5 : join p r d
============================
finMap s
subgoal 2 is:
{x
: fpm A (lifted JB) * fpm A (lifted JB) * fpm A (lifted JB) *
fpm A (lifted JB) &
let (p0, bd) := x in
let (p1, bc) := p0 in
let (ac, ad) := p1 in
join ac ad (exist (finMap (B:=lifted JB)) a Ha) /\
join bc bd (exist (finMap (B:=lifted JB)) b Hb) /\
join ac bc (exist (finMap (B:=lifted JB)) c Hc) /\
join ad bd (exist (finMap (B:=lifted JB)) d Hd)}
```
Ongoing Proof History:
```
Proof.
intros.
assert (Cross_alg (A -> option (lifted JB))).
apply cross_split_fun.
apply Cross_smash; auto.
hnf.
intros [a Ha] [b Hb] [c Hc] [d Hd] [z Hz].
simpl; intros.
destruct (X0 a b c d z); auto.
destruct x as [[[s p] q] r].
decompose [and] y; clear y.
assert (Hs : finMap s).
```
Next Tactic:
```
| destruct Ha.
``` | {"package":"coq-vst","file":"\/home\/tpr\/.opam\/PackageTable~8.13~2022.01.csv.formatted\/lib\/coq\/user-contrib\/VST\/msl\/cross_split.v"} |
Ongoing Proof History:
```
Proof.
constructor.
repeat intro.
reflexivity.
intros ? ? E ? ?.
symmetry.
```
Proof State:
```
2 subgoals
sign : Signature x, y : Object sign x0, y0 : x βΆ y
E : x0 = y0
b : sorts sign a : x b
============================
(` x0) b a = (` y0) b a
subgoal 2 is:
Transitive equiv
```
Next Tactic:
```
| apply E.
``` | {"package":"coq-math-classes","file":"\/home\/tpr\/.opam\/PackageTable~8.13~2022.01.csv.formatted\/lib\/coq\/user-contrib\/MathClasses\/categories\/algebras.v"} |
Proof State:
```
1 subgoal
R : commring p : prime_ideal R x : ringmultabmonoid R
============================
hProp
```
Ongoing Proof History:
```
Proof.
use make_submonoid.
-
intro x.
```
Next Tactic:
```
| exact (Β¬ p x).
``` | {"package":"coq-unimath","file":"\/home\/tpr\/.opam\/PackageTable~8.13~2022.01.csv.formatted\/lib\/coq\/user-contrib\/UniMath\/Algebra\/RigsAndRings\/Ideals.v"} |
Ongoing Proof History:
```
lia.
}
rewrite jump_add.
match goal with |- context [jump ?X s] => generalize (jump X s) end.
intros l; rewrite <- jump_nth; generalize l; clear l.
generalize (mod_lt x h U1).
generalize (mod x h).
cut (w <= size).
-
generalize size; intros m.
- intros U3 n; generalize h; elim n; simpl...
+ intros U3 n; generalize h; elim n; simpl; auto with arith; clear n H1 H2 U1.
- clear n H1 H2 U1.
+
- intros h1; case h1; simpl...
+ intros h1; case h1; simpl; auto.
*
intros HH; contradict HH...
*
intros n1 _ l.
case (le_or_lt (length l) (mod y w)); intros H1.
++
rewrite jump_too_far.
**
rewrite take_and_jump_nil.
rewrite <- app_nil_end.
apply sym_equal; apply take_nth...
**
- pose proof (mod_lt y w).
+ apply le_trans with (1 := H1)...
- lia.
+ apply le_trans with (2 := U3).
+ apply lt_le_weak; apply mod_lt...
++
rewrite nth_app_l.
**
apply sym_equal; apply take_nth...
left; apply mod_lt...
**
case (le_or_lt (length l) w); intros H2.
***
rewrite length_take_small...
***
rewrite length_take...
apply mod_lt...
+
intros n1 Rec h1; case h1; simpl.
*
intros HH; contradict HH...
*
intros h2 HH l.
- case (le_or_lt (length l) w); intros H3.
+ case (le_or_lt (length l) w); intros H1.
++
rewrite nth_app_r.
**
? rewrite length_take_small, jump_too_far...
? ***
? rewrite take_and_jump_nil.
```
Proof State:
```
1 goal
h, w, x, y : nat s : list nat
U2 : 0 < w
F1 : div y w < h
m : nat
U3 : w <= m
n1 : nat
Rec : forall h : nat,
n1 < h ->
forall l : list nat,
nth (n1 * m + mod y w) l out =
nth (n1 * w + mod y w) (take_and_jump w m h l) out
h1, h2 : nat
HH : S n1 < S h2
l : list nat
H1 : length l <= w
============================
nth (m + n1 * m + mod y w) l out =
nth (w + n1 * w + mod y w - length (take w l))
(take_and_jump w m h2 (jump m l)) out
```
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..8e8aed4 100644
--- a/Sudoku.v
+++ b/Sudoku.v
@@ -709 +709,2 @@ apply sym_equal; apply take_nth...
-** pose proof (mod_lt y w).
+**
+pose proof (mod_lt y w).
@@ -743 +744 @@ generalize (w_pos _ H); intros U2.
-rewrite length_take_and_jump...
+rewrite length_take_and_jump; auto.
@@ -760 +761,2 @@ 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 +848,2 @@ 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 +853 @@ 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 +855,2 @@ 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 +858,4 @@ 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 +865,3 @@ 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 +1447,2 @@ 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.
```
Next Tactic:
```
| rewrite <LOOKUP>length_take_small : forall (A : Set) (i : nat) (l : list A), length l <= i -> length (take i l) = length l</LOOKUP>...
``` | {"final_commit":"427448011e3341d35aa5b9e00f675f6582fcd477","project":"sudoku","repair_file":"\/data\/coq-pearls\/repairs\/sudoku\/repair-sudoku-dc8d4294a32dbb6e0a642dc553b64495f27e6d60-427448011e3341d35aa5b9e00f675f6582fcd477-8_15_2-8_15_2-29.git.json","initial_commit":"dc8d4294a32dbb6e0a642dc553b64495f27e6d60","file":"Sudoku.v"} |
Ongoing Proof History:
```
Proof.
intros x'; destruct (decide (x' = x)) as [->|].
-
by rewrite discrete_fun_lookup_op !discrete_fun_lookup_singleton.
-
```
Proof State:
```
1 subgoal
A : Type EqDecision0 : EqDecision A B : A β ucmra
x : A y1, y2 : B x x' : A
n : x' β x
============================
(discrete_fun_singleton x y1 β
discrete_fun_singleton x y2) x'
β‘ discrete_fun_singleton x (y1 β
y2) x'
```
Next Tactic:
```
| by rewrite <LOOKUP>discrete_fun_lookup_op : β (A : Type) (B : A β ucmra) (f g : discrete_fun (Ξ» x : A, B x)) (x : A), (f β
g) x = f x β
g x</LOOKUP> !<LOOKUP>discrete_fun_lookup_singleton_ne : β x x' (y : B x), x β x' β discrete_fun_singleton x y x' = Ξ΅</LOOKUP> // <LOOKUP>left_id : β (A : Type) (R : relation A) (i : A) (f : A β A β A), LeftId R i f β β x : A, R (f i x) x</LOOKUP>.
``` | {"package":"coq-iris","file":"\/home\/tpr\/.opam\/PackageTable~8.13~2022.01.csv.formatted\/lib\/coq\/user-contrib\/iris\/algebra\/functions.v"} |
Ongoing Proof History:
```
Proof.
intros.
induction Q.
+
simpl; auto.
+
simpl.
split.
-
apply H; simpl; auto.
```
Proof State:
```
This subproof is complete, but there are some unfocused goals.
Focus next goal with bullet -.
1 subgoal
subgoal 1 is:
fold_right (` and) (` True) Q rho
```
Next Tactic:
```
| -
``` | {"package":"coq-vst","file":"\/home\/tpr\/.opam\/PackageTable~8.13~2022.01.csv.formatted\/lib\/coq\/user-contrib\/VST\/floyd\/local2ptree_denote.v"} |
Ongoing Proof History:
```
Proof using.
intros.
```
Proof State:
```
1 subgoal
A : Type m, m', m'' : t A x, y, z : key e, e' : A
p : positive p0 : key q : positive m0 : t A
v : A
H : List.In (p0, v) (xelements m0 (append p q~1))
============================
E.bits_lt p p0
```
Next Tactic:
```
| generalize (<LOOKUP>xelements_complete : forall (i j : key) (m : t A) (v : A), List.In (i, v) (xelements m j) -> xfind i j m = Some v</LOOKUP> _ _ _ _ H); clear H; intros.
``` | {"package":"coq","file":"\/home\/tpr\/.opam\/PackageTable~8.13~2022.01.csv.formatted\/lib\/coq\/theories\/FSets\/FMapPositive.v"} |
Ongoing Proof History:
```
Proof.
apply Build_Component with OKp; trivial.
+
apply Condition1.
+
apply LNR_VGI.
+
apply LNR_E.
+
apply LNR_Exports.
+
apply Condition2.
+
apply G_dom.
+
apply G_E.
+
apply G_justified.
+
apply G_Exports.
```
Proof State:
```
This subproof is complete, but there are some unfocused goals.
Focus next goal with bullet +.
1 subgoal
subgoal 1 is:
forall gv : globals,
globals_ok gv -> InitGPred (Vardefs p) gv |-- GP1 gv * GP2 gv
```
Next Tactic:
```
| +
``` | {"package":"coq-vst","file":"\/home\/tpr\/.opam\/PackageTable~8.13~2022.01.csv.formatted\/lib\/coq\/user-contrib\/VST\/floyd\/Component.v"} |
Proof State:
```
2 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:
False
```
Ongoing Proof History:
```
repeat rewrite (fun x => mult_comm x h).
- unfold out in H4; rewrite H4; auto.
+ unfold out in H4; rewrite H4...
- repeat rewrite (mult_comm h); auto.
+ repeat rewrite (mult_comm h)...
- unfold size; repeat rewrite (mult_comm h); apply mult_lt_plus; auto.
+ unfold size; repeat rewrite (mult_comm h); apply mult_lt_plus...
- unfold size; rewrite (mult_comm w); apply mult_lt_plus; auto.
+ unfold size; rewrite (mult_comm w); apply mult_lt_plus...
- apply div_lt; rewrite mult_comm; auto.
+ apply div_lt; rewrite mult_comm...
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); auto.
+ rewrite (div_is_0 (div y w) h)...
- rewrite (mod_small (div y w) h); auto.
+ rewrite (mod_small (div y w) h)...
- apply div_lt; rewrite (mult_comm w); auto.
+ apply div_lt; rewrite (mult_comm w)...
? apply div_lt; rewrite (mult_comm w); auto.
? case (gen_cell_correct l x y); auto; intros tmp; case tmp; auto; clear tmp.
? apply (rm_incl _ lit_test) with (l1 := L (Pos x y) z::nil) (l2 := gen_cell (Pos x y)); auto.
```
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..abf9251 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...
```
Next Tactic:
```
| apply <LOOKUP>div_lt : forall a b c : nat, a < b * c -> div a b < c</LOOKUP>; rewrite (mult_comm w)...
``` | {"final_commit":"5337edcaf31b9f036a0a6651b2f733d3068b8f8f","project":"sudoku","repair_file":"\/data\/coq-pearls\/repairs\/sudoku\/repair-sudoku-427448011e3341d35aa5b9e00f675f6582fcd477-5337edcaf31b9f036a0a6651b2f733d3068b8f8f-8_15_2-8_15_2-41.git.json","initial_commit":"427448011e3341d35aa5b9e00f675f6582fcd477","file":"Sudoku.v"} |
Ongoing Proof History:
```
+
gstep.
red.
simpobs.
econstructor.
gbase.
destruct REL.
*
eapply CIH.
econstructor; [|eauto using paco2_mon with paco|].
--
eapply eqit_trans; [apply REL0|].
rewrite tau_eutt.
reflexivity.
--
auto_ctrans.
*
eapply CIH0.
apply CLOL.
econstructor; [|eauto|].
--
eapply eqit_trans; [apply REL0|].
rewrite tau_eutt.
reflexivity.
--
auto_ctrans.
+
punfold REL0.
red in REL0.
simpl in *.
remember (VisF e k1) as ot.
genobs m1 ot2.
hinduction REL0 before CIH; intros; subst; try dependent destruction Heqot; cycle 1.
*
gclo; econstructor; auto_ctrans_eq; try reflexivity.
? rewrite (simpobs Heqot1), tau_eutt.
? reflexivity.
? *
```
Commit Diff:
```
diff --git a/theories/Basics/CategoryKleisliFacts.v b/theories/Basics/CategoryKleisliFacts.v
index e0b955f..d6eecd6 100644
--- a/theories/Basics/CategoryKleisliFacts.v
+++ b/theories/Basics/CategoryKleisliFacts.v
@@ -59,0 +60,8 @@ Qed.
+
+
+
+
+
+
+
+
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.
```
Proof State:
```
1 subgoal
E : Type -> Type R2 : Type R1 : Type RR : R1 -> R2 -> Prop
vclo : rel2 (itree E R1) (fun _ : itree E R1 => itree E R2) ->
rel2 (itree E R1) (fun _ : itree E R1 => itree E R2)
r : itree E R1 -> itree E R2 -> Prop
MON : monotone2 vclo
COMP : wcompatible2 (eqit_ RR true true vclo) (transD RR)
CLOV : forall r : itree E R1 -> itree E R2 -> Prop,
(forall (x0 : itree E R1) (x1 : itree E R2),
transL r x0 x1 -> r x0 x1) ->
forall (x0 : itree E R1) (x1 : itree E R2),
transL (vclo r) x0 x1 -> vclo r x0 x1
CLOL : forall (x0 : itree E R1) (x1 : itree E R2),
transL r x0 x1 -> r x0 x1
CLOD : forall (x0 : itree E R1) (x1 : itree E R2),
transD RR r x0 x1 -> r x0 x1
r0 : itree E R1 -> itree E R2 -> Prop
CIH0 : forall (x0 : itree E R1) (x1 : itree E R2), r x0 x1 -> r0 x0 x1
CIH : forall (x2 : itree E R1) (x3 : itree E R2),
transL (gupaco2 (eqit_ RR true true vclo) (transD RR) r) x2 x3 ->
r0 x2 x3
RR1 : R1 -> R1 -> Prop t1 : itree E R1
CHECK : true
u : Type e : E u k1 : u -> itree E R1
IHREL0 : forall (u0 : Type) (e0 : E u0) (k2 : u0 -> itree E R1)
(k3 : u0 -> itree E R2),
(forall v : u0,
vclo (upaco2 (eqit_ RR true true vclo) r) (k2 v) (k3 v)) ->
forall m1 t' : itree E R1,
VisF e k1 = VisF e0 k2 ->
VisF e k1 = observe t' ->
observe t1 = observe m1 ->
forall (t1 : itree E R1) (t2 : itree E R2),
TauF m1 = observe t1 ->
(forall (x x' : R1) (y : R2), RR1 x x' -> RR x' y -> RR x y) ->
VisF e0 k3 = observe t2 ->
gpaco2 (eqit_ RR true true vclo) (transD RR) r r0 t1 t2
REL0 : eqitF RR1 true true id (upaco2 (eqit_ RR1 true true id) bot2)
(observe t1) (VisF e k1)
k2 : u -> itree E R2
REL : forall v : u, vclo (upaco2 (eqit_ RR true true vclo) r) (k1 v) (k2 v)
m1, t' : itree E R1
Heqot' : VisF e k1 = observe t'
Heqot0 : TauF t1 = observe m1
t0 : itree E R1 t2 : itree E R2
Heqot1 : TauF m1 = observe t0
LERR : forall (x x' : R1) (y : R2), RR1 x x' -> RR x' y -> RR x y
Heqot2 : VisF e k2 = observe t2
============================
eqit eq true false t0 m1
```
Next Tactic:
```
| rewrite (<LOOKUP>simpobs : forall (E : Type -> Type) (R : Type) (ot : itree' E R) (t : itree E R), ot = observe t -> t β
{| _observe := ot |}</LOOKUP> Heqot1), <LOOKUP>tau_euttge : forall (E : Type -> Type) (R : Type) (t : itree E R), Tau t β³ t</LOOKUP>.
``` | {"final_commit":"6b6e1763da94788e6204f549039421936cad4437","project":"InteractionTrees","repair_file":"\/data\/coq-pearls\/repairs\/InteractionTrees\/repair-InteractionTrees-75a5a716bb6a18a6fdca492391f694fb344ea589-6b6e1763da94788e6204f549039421936cad4437-8_10_2-8_10_2-1.git.json","initial_commit":"75a5a716bb6a18a6fdca492391f694fb344ea589","file":"theories\/Eq\/UpToTaus.v"} |
Ongoing Proof History:
```
+ intros U3 n; generalize h; elim n; simpl...
+ clear n H1 H2 U1.
+
- intros h1; case h1; simpl; auto.
+ intros h1; case h1; simpl...
*
intros HH; contradict HH...
*
intros n1 _ l.
case (le_or_lt (length l) (mod y w)); intros H1.
++
rewrite jump_too_far.
**
rewrite take_and_jump_nil.
rewrite <- app_nil_end.
apply sym_equal; apply take_nth...
**
- apply le_trans with (1 := H1)...
+ pose proof (mod_lt y w).
- apply le_trans with (2 := U3).
+ lia.
- apply lt_le_weak; apply mod_lt...
++
rewrite nth_app_l.
**
apply sym_equal; apply take_nth...
left; apply mod_lt...
**
case (le_or_lt (length l) w); intros H2.
***
rewrite length_take_small...
***
rewrite length_take...
apply mod_lt...
+
intros n1 Rec h1; case h1; simpl.
*
intros HH; contradict HH...
*
intros h2 HH l.
- case (le_or_lt (length l) w); intros H1.
+ case (le_or_lt (length l) w); intros H3.
++
rewrite nth_app_r.
**
- rewrite length_take_small...
+ rewrite length_take_small, jump_too_far...
- rewrite jump_too_far.
***
rewrite take_and_jump_nil.
repeat rewrite nth_default...
- apply le_trans with (1 := H1)...
+ apply le_trans with (1 := H3)...
***
- apply le_trans with (1 := H1)...
+ apply le_trans with (1 := H3)...
**
rewrite length_take_small...
++
rewrite nth_app_r.
**
rewrite length_take...
repeat rewrite <- plus_assoc; rewrite minus_plus...
- rewrite jump_nth; rewrite jump_add.
+ rewrite jump_nth, jump_add, <- jump_nth...
? rewrite <- jump_nth...
? **
? rewrite length_take...
```
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..e294c2a 100644
--- a/Sudoku.v
+++ b/Sudoku.v
@@ -713,2 +713 @@ apply sym_equal; apply take_nth...
-** apply le_trans with (1 := H1)...
-apply le_trans with (2 := U3).
+** apply le_trans with (1 := H1)... apply le_trans with (2 := U3).
@@ -750 +749 @@ generalize (w_pos _ H); intros U2.
-rewrite length_take_and_jump; auto.
+rewrite length_take_and_jump...
@@ -767,2 +766 @@ 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 +852,2 @@ 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 +857 @@ 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 +859,2 @@ assert (p2 = p3); subst.
-apply pos_test_exact with (1 := H1); auto.
-rewrite H; auto.
+apply pos_test_exact with (1 := H1)...
+rewrite H...
@@ -864,4 +862,4 @@ intros; discriminate.
-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.
+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)...
@@ -871,3 +869,3 @@ assert (p2 = p3); subst.
-apply pos_test_exact with (1 := H1); auto.
-rewrite H; auto.
-rewrite (pos_test_trans p1 p2 p3); rewrite H; auto.
+apply pos_test_exact with (1 := H1)...
+rewrite H...
+rewrite (pos_test_trans p1 p2 p3); rewrite H...
@@ -1453,2 +1451,2 @@ ordered cs -> ordered (clauses_update l c cs).
-Proof.
-intros l c cs; elim cs; simpl; auto.
+Proof with auto with datatypes.
+intros l c cs; elim cs; simpl...
@@ -1456,7 +1454,7 @@ intros (n1, c1) cs1 Rec H H1; case (lit_is_in l c1).
-apply Rec; auto.
-intros n2 c2 H2; apply (H1 n2 c2); auto with datatypes.
-apply clause_insert_ordered; auto.
-apply lit_rm_ordered; auto.
-apply (H1 n1 c1); auto with datatypes.
-apply Rec; auto.
-intros n2 c2 H2; apply (H1 n2 c2); auto with datatypes.
+apply Rec...
```
Proof State:
```
This subproof is complete, but there are some unfocused goals.
Focus next goal with bullet **.
2 goals
goal 1 is:
length (take w l) <= w + n1 * w + mod y w
goal 2 is:
w <= size
```
Next Tactic:
```
| **
``` | {"final_commit":"5337edcaf31b9f036a0a6651b2f733d3068b8f8f","project":"sudoku","repair_file":"\/data\/coq-pearls\/repairs\/sudoku\/repair-sudoku-427448011e3341d35aa5b9e00f675f6582fcd477-5337edcaf31b9f036a0a6651b2f733d3068b8f8f-8_15_2-8_15_2-31.git.json","initial_commit":"427448011e3341d35aa5b9e00f675f6582fcd477","file":"Sudoku.v"} |
Proof State:
```
1 subgoal
m : hz is : hzneq 0 m a, b : fpscommring hz
n : nat
IHn : hzquotientmod m is (precarry (a + b) n) =
hzquotientmod m is (precarry a n) + hzquotientmod m is (precarry b n) +
hzquotientmod m is (precarry (carry a + carry b) n)
pa := precarry a (S n) : hz
pb := precarry b (S n) : hz
ab := precarry (carry a + carry b) : nat β hz
============================
(hzremaindermod m is pa +
hzremaindermod m is (pb + hzquotientmod m is (ab n)))%hz =
(hzremaindermod m is pa +
hzremaindermod m is (carry b (S n) + hzquotientmod m is (ab n))%ring)%hz
```
Ongoing Proof History:
```
Proof.
intros.
induction n.
-
simpl.
change ( hzquotientmod m is ( a 0%nat + b 0%nat ) = hzquotientmod m is (a 0%nat) + hzquotientmod m is (b 0%nat) + hzquotientmod m is ( hzremaindermod m is ( a 0%nat ) + hzremaindermod m is ( b 0%nat ) ) ).
rewrite hzquotientmodandplus.
apply idpath.
-
change ( hzquotientmod m is ( a ( S n ) + b ( S n ) + hzquotientmod m is ( precarry (a + b) n ) ) = hzquotientmod m is (precarry a (S n)) + hzquotientmod m is (precarry b (S n)) + hzquotientmod m is (carry a ( S n ) + carry b ( S n ) + hzquotientmod m is ( precarry (carry a + carry b) n)) ).
rewrite IHn.
rewrite ( ringassoc1 hz ( a ( S n ) ) ( b ( S n ) ) _ ).
rewrite <- ( ringassoc1 hz ( b ( S n ) ) ).
rewrite ( ringcomm1 hz ( b ( S n ) ) _ ).
rewrite <- 3! ( ringassoc1 hz ( a ( S n ) ) _ _ ).
change ( a ( S n ) + hzquotientmod m is ( precarry a n ) ) with ( precarry a ( S n ) ).
set ( pa := precarry a ( S n ) ).
rewrite ( ringassoc1 hz pa _ ( b ( S n ) ) ).
rewrite ( ringcomm1 hz _ ( b ( S n ) ) ).
change ( b ( S n ) + hzquotientmod m is ( precarry b n ) ) with ( precarry b ( S n ) ).
set ( pb := precarry b ( S n ) ).
set ( ab := precarry ( carry a + carry b ) ).
rewrite ( ringassoc1 hz ( carry a ( S n ) ) ( carry b ( S n ) ) ( hzquotientmod m is ( ab n ) ) ).
rewrite ( hzquotientmodandplus m is ( carry a ( S n ) ) _ ).
unfold carry at 1.
rewrite <- hzqrandremainderq.
rewrite hzplusl0.
rewrite ( hzquotientmodandplus m is ( carry b ( S n ) ) _ ).
unfold carry at 1.
rewrite <- hzqrandremainderq.
rewrite hzplusl0.
rewrite ( ringassoc1 hz pa pb _ ).
rewrite ( hzquotientmodandplus m is pa _ ).
change (pb + hzquotientmod m is (ab n)) with (pb + hzquotientmod m is (ab n))%hz.
rewrite ( hzquotientmodandplus m is pb ( hzquotientmod m is ( ab n ) ) ).
rewrite <- 2! ( ringassoc1 hz ( hzquotientmod m is pa ) _ _ ).
rewrite <- 2! ( ringassoc1 hz ( hzquotientmod m is pa + hzquotientmod m is pb ) _ ).
rewrite 2! ( ringassoc1 hz ( hzquotientmod m is pa + hzquotientmod m is pb + hzquotientmod m is (hzquotientmod m is (ab n)) ) _ _ ).
apply ( maponpaths ( fun x : hz => hzquotientmod m is pa + hzquotientmod m is pb + hzquotientmod m is (hzquotientmod m is (ab n)) + x ) ).
unfold carry at 1 2.
rewrite 2! hzremaindermoditerated.
change ( precarry b ( S n ) ) with pb.
change ( precarry a ( S n ) ) with pa.
apply ( maponpaths ( fun x : hz => ( hzquotientmod m is (hzremaindermod m is pb + hzremaindermod m is (hzquotientmod m is (ab n)))%hz ) + x ) ).
apply maponpaths.
```
Next Tactic:
```
| apply ( <LOOKUP>maponpaths : β (T1 T2 : UU) (f : T1 β T2) (t1 t2 : T1), t1 = t2 β f t1 = f t2</LOOKUP> ( fun x : <LOOKUP>hz : commring</LOOKUP> => <LOOKUP>hzremaindermod : β p : hz, hzneq 0 p β hz β hz</LOOKUP> m is pa + x ) ).
``` | {"package":"coq-unimath","file":"\/home\/tpr\/.opam\/PackageTable~8.13~2022.01.csv.formatted\/lib\/coq\/user-contrib\/UniMath\/PAdics\/padics.v"} |
Ongoing Proof History:
```
Proof.
repeat red.
pcofix CIH.
intros t1 t2 Heutt Hev.
pfold.
punfold Heutt.
red.
unfold_eqit.
assert (Hev' : eventless t1); auto.
punfold Hev.
- dependent induction Heutt; subst; auto.
+ dependent induction Heutt; subst; auto with itree.
-
rewrite <- x.
- auto.
+ auto with itree.
-
rewrite <- x.
constructor.
right.
eapply CIH; eauto.
specialize (itree_eta t1) as Ht1.
rewrite <- x0 in Ht1.
rewrite Ht1.
rewrite tau_eutt.
pclearbot.
auto.
-
red in Hev.
inversion Hev; subst.
+
rewrite <- H0 in x0.
discriminate.
+
rewrite <- H in x0.
discriminate.
-
red in Hev.
rewrite <- x in Hev.
inversion Hev; subst.
pclearbot.
eapply IHHeutt; try apply H0; eauto.
red.
punfold H0.
-
rewrite <- x.
constructor.
right.
? eapply CIH; eauto.
? Qed.
```
Proof State:
```
1 subgoal
E1 : Type -> Type R : Type E : Type -> Type
r : itree E1 R -> Prop t1, t0 : itree E1 R
CHECK : is_true true
Heutt : eqitF eq true true id (upaco2 (eqit_ eq true true id) bot2)
(observe t1) (observe t0)
IHHeutt : (forall x y : itree E1 R, x β y -> eventless x -> r y) ->
forall t2 t3 : itree E1 R,
R = R ->
eq ~= eq ->
true = true ->
true = true ->
id ~= id ->
upaco2 (eqit_ eq true true id) bot2 ~=
upaco2 (eqit_ eq true true id) bot2 ->
observe t1 = observe t2 ->
observe t0 ~= observe t3 ->
eventless_ (upaco1 eventless_ bot1) t2 ->
eventless t2 -> eventlessF (upaco1 eventless_ r) (observe t3)
CIH : forall x y : itree E1 R, x β y -> eventless x -> r y
t2 : itree E1 R
x : TauF t0 = observe t2
Hev : eventless_ (upaco1 eventless_ bot1) t1
Hev' : eventless t1
============================
r t0
```
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..875d3b0 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.
```
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_2-104.git.json","initial_commit":"1158ff1cbb628aeaa69973bf098058744d458c36","file":"theories\/Dijkstra\/ITreeDijkstra.v"} |
Ongoing Proof History:
```
Proof.
```
Proof State:
```
1 subgoal
============================
forall a b : Share.t,
Share.glb a b = Share.bot -> Share.glb (Share.comp a) b = b
```
Next Tactic:
```
| intros.
``` | {"package":"coq-vst","file":"\/home\/tpr\/.opam\/PackageTable~8.13~2022.01.csv.formatted\/lib\/coq\/user-contrib\/VST\/veric\/shares.v"} |
Proof State:
```
1 subgoal
Ξ£ : gFunctors heapGS0 : heapGS Ξ£ l : loc dq : dfrac
v : val vs : list val
============================
l β¦β{dq} (v :: vs) β£β’ l β¦{dq} v β (l +β 1) β¦β{dq} vs
```
Ongoing Proof History:
```
Proof.
```
Next Tactic:
```
| rewrite /<LOOKUP>array : β Ξ£ : gFunctors, heapGS Ξ£ β loc β dfrac β list val β iProp Ξ£</LOOKUP> <LOOKUP>big_sepL_cons : β (PROP : bi) (A : Type) (Ξ¦ : nat β A β PROP) (x : A) (l : list A), ([β list] kβ¦y β (x :: l), Ξ¦ k y) β£β’ Ξ¦ 0 x β ([β list] kβ¦y β l, Ξ¦ (S k) y)</LOOKUP> <LOOKUP>loc_add_0 : β l, l +β 0 = l</LOOKUP>.
``` | {"package":"coq-iris-heap-lang","file":"\/home\/tpr\/.opam\/PackageTable~8.13~2022.01.csv.formatted\/lib\/coq\/user-contrib\/iris\/heap_lang\/derived_laws.v"} |
Proof State:
```
4 goals
h, w : nat
Eq1 : ulist ref_list
x, y, z : nat s : list nat
U1 : x < size
U2 : y < size
H2 : get (Pos x y) s = z
l : lit
H4 : lit_sat l s
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
H3 : In l
(rm_cont lit lit_test (fun l2 : list lit => l2)
(L (Pos x y) z) (gen_column y z))
V1 : length s = size * size
V2 : forall i : nat, i < size -> Permutation (column i s) ref_list
============================
In l (gen_column y z)
goal 2 is:
forall x0 : nat, l = L (Pos x0 y) z /\ x0 < size -> False
goal 3 is:
False
goal 4 is:
False
```
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..abf9251 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.
```
Ongoing Proof History:
```
case (gen_row_correct l x z); intros tmp _; case tmp; auto; clear tmp.
- apply (rm_incl _ lit_test) with (l1 := L (Pos x y) z::nil) (l2 := gen_row x z); auto.
+ apply (rm_incl _ lit_test) with (l1 := L (Pos x y) z::nil) (l2 := gen_row x z)...
intros y1 (H5, H6); subst l.
simpl in H4; absurd (y = y1).
intros H7; apply (rm_not_in _ lit_test) with (a := L (Pos x y) z) (l1 := L (Pos x y) z::nil) (l2 := gen_row x z); subst; auto with datatypes.
exact lit_test_trans.
intros; apply lit_test_anti_sym.
exact lit_test_exact.
apply olist_one.
apply gen_row_ordered.
- apply nth_ulist with (a := 0) (l := row x s); auto.
+ apply nth_ulist with (a := 0) (l := row x s)...
- rewrite length_row; auto.
+ rewrite length_row...
- rewrite length_row; auto.
+ rewrite length_row...
- apply ulist_perm with ref_list; auto.
+ apply ulist_perm with ref_list...
- apply Permutation_sym; auto.
+ apply Permutation_sym...
- repeat rewrite <- get_row; try rewrite H2; auto.
+ repeat rewrite <- get_row; try rewrite H2...
case H1; clear H1; intros V1 (_, (V2, _)).
case (gen_column_correct l y z); intros tmp _; case tmp; auto; clear tmp.
? apply (rm_incl _ lit_test) with (l1 := L (Pos x y) z::nil) (l2 := gen_column y z); auto.
? intros x1 (H5, H6); subst l.
? simpl in H4; absurd (x = x1).
```
Next Tactic:
```
| apply (<LOOKUP>rm_incl : forall (A : Set) (weight : A -> A -> cmp) (l1 l2 : list A), incl (rm A weight l1 l2) l2</LOOKUP> _ <LOOKUP>lit_test : lit -> lit -> cmp</LOOKUP>) with (l1 := <LOOKUP>L : pos -> nat -> lit</LOOKUP> (<LOOKUP>Pos : nat -> nat -> pos</LOOKUP> x y) z::<LOOKUP>nil : forall {A : Type}, list A</LOOKUP>) (l2 := <LOOKUP>gen_column : nat -> nat -> clause</LOOKUP> y z)...
``` | {"final_commit":"5337edcaf31b9f036a0a6651b2f733d3068b8f8f","project":"sudoku","repair_file":"\/data\/coq-pearls\/repairs\/sudoku\/repair-sudoku-427448011e3341d35aa5b9e00f675f6582fcd477-5337edcaf31b9f036a0a6651b2f733d3068b8f8f-8_15_2-8_15_2-41.git.json","initial_commit":"427448011e3341d35aa5b9e00f675f6582fcd477","file":"Sudoku.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'.
- inv Heqtr; auto_inj_pair2; subst.
+ red in H0.
- apply eq_itree_inv_vis in H0.
+ apply eqit_inv_vis_weak in H0.
- destruct H0 as (? & ? & ?).
+ destruct H0 as [<- [<- H0]]; cbn in H0.
+ inv_Vis.
+ left.
+ exists k.
? inv H; auto_inj_pair2; subst.
? left.
? exists k.
```
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..46d9145 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.
+
```
Proof State:
```
1 subgoal
A : Type B : Type C : Type E : Type -> Type
RR : B -> C -> Prop b1, b2 : bool u : Type
e0 : E u k1 : u -> itree E B k2 : u -> itree E C
REL : forall v : u,
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) (k2 v)
ma : itree E A kab : A -> itree E B tl : itree E B k : u -> itree E A
H0 : forall x : u, eqit eq false false (k1 x) (x <- k x;; kab x)
============================
eqit eq b1 b2 (Vis e0 k) (Vis e0 k) /\
(forall x : u, eqit RR b1 b2 (x <- k x;; kab x) (k2 x))
```
Next Tactic:
```
| split; [reflexivity |].
``` | {"final_commit":"f3853150994e962978fa4cf0ed18dcb8c42e3262","project":"InteractionTrees","repair_file":"\/data\/coq-pearls\/repairs\/InteractionTrees\/repair-InteractionTrees-1fcd2d39e27da8068431211b7952365e163782cc-f3853150994e962978fa4cf0ed18dcb8c42e3262-8_10_2-8_11_2-13.git.json","initial_commit":"1fcd2d39e27da8068431211b7952365e163782cc","file":"theories\/Eq\/Eq.v"} |
Proof State:
```
This subproof is complete, but there are some unfocused goals.
Focus next goal with bullet --.
5 subgoals
subgoal 1 is:
address_mapsto_zeros sh n (b, Z.succ ofs)
|-- (ALL y,
jam (adr_range_dec (b, Z.succ ofs) (Z.of_nat n))
(fun loc : address =>
yesat NoneP
(VAL
match Z.to_nat (snd loc - ofs) with
| 0%nat => Byte Byte.zero
| S m => nth m (repeat (Byte Byte.zero) n) Undef
end) sh loc) noat y) && noghost
subgoal 2 is:
exists resp : resource -> address -> nat -> Prop,
is_resource_pred
(fun loc : address =>
yesat NoneP
(VAL
(nth (Z.to_nat (snd loc - ofs)) (repeat (Byte Byte.zero) (S n))
Undef)) sh loc) resp
subgoal 3 is:
exists resp : resource -> address -> nat -> Prop,
is_resource_pred
(fun loc : address =>
yesat NoneP
(VAL
(nth (Z.to_nat (snd loc - ofs)) (repeat (Byte Byte.zero) (S n))
Undef)) sh loc) resp
subgoal 4 is:
exists resp : resource -> address -> nat -> Prop,
is_resource_pred
(fun loc : address =>
yesat NoneP
(VAL
(nth (Z.to_nat (snd loc - ofs)) (repeat (Byte Byte.zero) (S n))
Undef)) sh loc) resp
subgoal 5 is:
forall (l : address) (m : rmap) (sh0 : rshare) (k : AV.kind),
adr_range (b, ofs) (Z.succ (Z.of_nat n)) l ->
yesat NoneP
(VAL (nth (Z.to_nat (snd l - ofs)) (repeat (Byte Byte.zero) (S n)) Undef))
sh l m -> res_option (m @ l) = Some (sh0, k) -> isVAL k \/ isFUN k
```
Ongoing Proof History:
```
Proof.
intros until t.
intros H2 H.
unfold mapsto_zeros, mapsto.
simpl.
rewrite andb_false_r by auto.
rewrite (prop_true_andp (is_pointer_or_null _)) by auto.
apply prop_andp_left; intros [? ?].
rewrite prop_true_andp by auto.
rewrite if_true by auto.
apply orp_right1.
unfold address_mapsto.
apply exp_right with (encode_val (if Archi.ptr64 then Mint64 else Mint32) nullval).
rewrite prop_true_andp by (split3; simpl; [rewrite encode_nullval; reflexivity | exact decode_encode_nullval | auto]).
forget (Ptrofs.unsigned z) as ofs; clear z.
replace (encode_val (if Archi.ptr64 then Mint64 else Mint32) nullval) with (repeat (Byte Byte.zero) (size_chunk_nat Mptr)) by (unfold size_chunk_nat, size_chunk, Mptr, encode_val, nullval; simpl; destruct Archi.ptr64; simpl; change (Int64.unsigned Int64.zero) with 0; change (Int.unsigned Int.zero) with 0; unfold encode_int, inj_bytes; simpl; compute; destruct Archi.big_endian; simpl; reflexivity).
rewrite size_chunk_conv, Nat2Z.id.
clear - H2.
simpl snd.
revert ofs; induction (size_chunk_nat Mptr); intros.
*
unfold address_mapsto_zeros.
apply andp_right.
apply allp_right; intro y.
rewrite jam_false.
intros ? ?.
do 3 red.
do 3 red in H.
apply resource_at_identity; auto.
simpl; destruct y; intros [? ?]; lia.
intros ? ?.
do 3 red in H|-*.
apply ghost_of_identity; auto.
*
rewrite inj_S.
simpl snd in *.
rewrite allp_jam_split2 with (q := (fun loc : address => yesat NoneP (VAL (nth (Z.to_nat (snd loc - ofs)) (repeat (Byte Byte.zero) (S n)) Undef)) sh loc)) (r := (fun loc : address => yesat NoneP (VAL (nth (Z.to_nat (snd loc - ofs)) (repeat (Byte Byte.zero) (S n)) Undef)) sh loc)) (Q_DEC := adr_range_dec (b,ofs) 1) (R_DEC := adr_range_dec ( b, Z.succ ofs) (Z.of_nat n)); auto.
5:{ split; intros.
destruct a; split; intros.
destruct H; subst b0.
destruct (zeq z ofs); [left|right]; split; auto; lia.
destruct H; destruct H; subst b0; split; auto; lia.
destruct a; destruct H,H0; subst; lia.
}
simpl.
apply sepcon_derives.
--
clear IHn.
unfold address_mapsto.
apply exp_left; intro bl.
rewrite andp_assoc.
apply prop_andp_left.
intros [? ?].
apply andp_derives; auto.
apply allp_derives; intro y.
simpl.
destruct H0.
destruct bl; inv H.
destruct bl; inv H4.
destruct (adr_range_dec (b, ofs) 1 y).
++
rewrite !jam_true by auto.
destruct y; destruct a; subst b0.
assert (z=ofs) by lia.
subst z.
simpl snd.
rewrite Z.sub_diag.
simpl.
replace m with (Byte Byte.zero); auto.
clear - H0.
destruct m; try discriminate.
rewrite decode_byte_val in H0.
apply Vint_inj in H0.
f_equal.
rewrite zero_ext_inrange in H0.
unfold Int.zero in H0.
apply repr_inj_unsigned in H0.
apply (f_equal Byte.repr) in H0.
rewrite Byte.repr_unsigned in H0.
auto.
pose proof (Byte.unsigned_range i).
change Byte.modulus with 256 in H.
split; try lia.
apply Z.le_trans with 256; try lia.
compute; congruence.
split.
lia.
compute; congruence.
rewrite Int.unsigned_repr.
pose proof (Byte.unsigned_range i).
change Byte.modulus with 256 in H.
simpl.
lia.
pose proof (Byte.unsigned_range i).
assert (Byte.modulus < Int.max_unsigned) by reflexivity.
lia.
++
rewrite !jam_false by auto.
auto.
```
Next Tactic:
```
| --
``` | {"package":"coq-vst","file":"\/home\/tpr\/.opam\/PackageTable~8.13~2022.01.csv.formatted\/lib\/coq\/user-contrib\/VST\/veric\/mapsto_memory_block.v"} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.