394 lines
9.9 KiB
Coq
394 lines
9.9 KiB
Coq
Set Warnings "-notation-overridden,-parsing".
|
|
From Coq Require Export String.
|
|
From LF Require Import Logic.
|
|
|
|
Parameter MISSING: Type.
|
|
|
|
Module Check.
|
|
|
|
Ltac check_type A B :=
|
|
match type of A with
|
|
| context[MISSING] => idtac "Missing:" A
|
|
| ?T => first [unify T B; idtac "Type: ok" | idtac "Type: wrong - should be (" B ")"]
|
|
end.
|
|
|
|
Ltac print_manual_grade A :=
|
|
match eval compute in A with
|
|
| Some (_ ?S ?C) =>
|
|
idtac "Score:" S;
|
|
match eval compute in C with
|
|
| ""%string => idtac "Comment: None"
|
|
| _ => idtac "Comment:" C
|
|
end
|
|
| None =>
|
|
idtac "Score: Ungraded";
|
|
idtac "Comment: None"
|
|
end.
|
|
|
|
End Check.
|
|
|
|
From LF Require Import Logic.
|
|
Import Check.
|
|
|
|
Goal True.
|
|
|
|
idtac "------------------- and_exercise --------------------".
|
|
idtac " ".
|
|
|
|
idtac "#> and_exercise".
|
|
idtac "Possible points: 2".
|
|
check_type @and_exercise ((forall n m : nat, n + m = 0 -> n = 0 /\ m = 0)).
|
|
idtac "Assumptions:".
|
|
Abort.
|
|
Print Assumptions and_exercise.
|
|
Goal True.
|
|
idtac " ".
|
|
|
|
idtac "------------------- and_assoc --------------------".
|
|
idtac " ".
|
|
|
|
idtac "#> and_assoc".
|
|
idtac "Possible points: 2".
|
|
check_type @and_assoc ((forall P Q R : Prop, P /\ Q /\ R -> (P /\ Q) /\ R)).
|
|
idtac "Assumptions:".
|
|
Abort.
|
|
Print Assumptions and_assoc.
|
|
Goal True.
|
|
idtac " ".
|
|
|
|
idtac "------------------- mult_eq_0 --------------------".
|
|
idtac " ".
|
|
|
|
idtac "#> mult_eq_0".
|
|
idtac "Possible points: 1".
|
|
check_type @mult_eq_0 ((forall n m : nat, n * m = 0 -> n = 0 \/ m = 0)).
|
|
idtac "Assumptions:".
|
|
Abort.
|
|
Print Assumptions mult_eq_0.
|
|
Goal True.
|
|
idtac " ".
|
|
|
|
idtac "------------------- or_commut --------------------".
|
|
idtac " ".
|
|
|
|
idtac "#> or_commut".
|
|
idtac "Possible points: 1".
|
|
check_type @or_commut ((forall P Q : Prop, P \/ Q -> Q \/ P)).
|
|
idtac "Assumptions:".
|
|
Abort.
|
|
Print Assumptions or_commut.
|
|
Goal True.
|
|
idtac " ".
|
|
|
|
idtac "------------------- double_neg_inf --------------------".
|
|
idtac " ".
|
|
|
|
idtac "#> Manually graded: double_neg_inf".
|
|
idtac "Advanced".
|
|
idtac "Possible points: 2".
|
|
print_manual_grade manual_grade_for_double_neg_inf.
|
|
idtac " ".
|
|
|
|
idtac "------------------- contrapositive --------------------".
|
|
idtac " ".
|
|
|
|
idtac "#> contrapositive".
|
|
idtac "Possible points: 2".
|
|
check_type @contrapositive ((forall P Q : Prop, (P -> Q) -> ~ Q -> ~ P)).
|
|
idtac "Assumptions:".
|
|
Abort.
|
|
Print Assumptions contrapositive.
|
|
Goal True.
|
|
idtac " ".
|
|
|
|
idtac "------------------- not_both_true_and_false --------------------".
|
|
idtac " ".
|
|
|
|
idtac "#> not_both_true_and_false".
|
|
idtac "Possible points: 1".
|
|
check_type @not_both_true_and_false ((forall P : Prop, ~ (P /\ ~ P))).
|
|
idtac "Assumptions:".
|
|
Abort.
|
|
Print Assumptions not_both_true_and_false.
|
|
Goal True.
|
|
idtac " ".
|
|
|
|
idtac "------------------- informal_not_PNP --------------------".
|
|
idtac " ".
|
|
|
|
idtac "#> Manually graded: informal_not_PNP".
|
|
idtac "Advanced".
|
|
idtac "Possible points: 1".
|
|
print_manual_grade manual_grade_for_informal_not_PNP.
|
|
idtac " ".
|
|
|
|
idtac "------------------- or_distributes_over_and --------------------".
|
|
idtac " ".
|
|
|
|
idtac "#> or_distributes_over_and".
|
|
idtac "Possible points: 3".
|
|
check_type @or_distributes_over_and (
|
|
(forall P Q R : Prop, P \/ Q /\ R <-> (P \/ Q) /\ (P \/ R))).
|
|
idtac "Assumptions:".
|
|
Abort.
|
|
Print Assumptions or_distributes_over_and.
|
|
Goal True.
|
|
idtac " ".
|
|
|
|
idtac "------------------- dist_not_exists --------------------".
|
|
idtac " ".
|
|
|
|
idtac "#> dist_not_exists".
|
|
idtac "Possible points: 1".
|
|
check_type @dist_not_exists (
|
|
(forall (X : Type) (P : X -> Prop),
|
|
(forall x : X, P x) -> ~ (exists x : X, ~ P x))).
|
|
idtac "Assumptions:".
|
|
Abort.
|
|
Print Assumptions dist_not_exists.
|
|
Goal True.
|
|
idtac " ".
|
|
|
|
idtac "------------------- dist_exists_or --------------------".
|
|
idtac " ".
|
|
|
|
idtac "#> dist_exists_or".
|
|
idtac "Possible points: 2".
|
|
check_type @dist_exists_or (
|
|
(forall (X : Type) (P Q : X -> Prop),
|
|
(exists x : X, P x \/ Q x) <-> (exists x : X, P x) \/ (exists x : X, Q x))).
|
|
idtac "Assumptions:".
|
|
Abort.
|
|
Print Assumptions dist_exists_or.
|
|
Goal True.
|
|
idtac " ".
|
|
|
|
idtac "------------------- In_map_iff --------------------".
|
|
idtac " ".
|
|
|
|
idtac "#> In_map_iff".
|
|
idtac "Possible points: 2".
|
|
check_type @In_map_iff (
|
|
(forall (A B : Type) (f : A -> B) (l : list A) (y : B),
|
|
@In B y (@map A B f l) <-> (exists x : A, f x = y /\ @In A x l))).
|
|
idtac "Assumptions:".
|
|
Abort.
|
|
Print Assumptions In_map_iff.
|
|
Goal True.
|
|
idtac " ".
|
|
|
|
idtac "------------------- In_app_iff --------------------".
|
|
idtac " ".
|
|
|
|
idtac "#> In_app_iff".
|
|
idtac "Possible points: 2".
|
|
check_type @In_app_iff (
|
|
(forall (A : Type) (l l' : list A) (a : A),
|
|
@In A a (l ++ l') <-> @In A a l \/ @In A a l')).
|
|
idtac "Assumptions:".
|
|
Abort.
|
|
Print Assumptions In_app_iff.
|
|
Goal True.
|
|
idtac " ".
|
|
|
|
idtac "------------------- All --------------------".
|
|
idtac " ".
|
|
|
|
idtac "#> All".
|
|
idtac "Possible points: 3".
|
|
check_type @All ((forall T : Type, (T -> Prop) -> list T -> Prop)).
|
|
idtac "Assumptions:".
|
|
Abort.
|
|
Print Assumptions All.
|
|
Goal True.
|
|
idtac " ".
|
|
|
|
idtac "------------------- combine_odd_even --------------------".
|
|
idtac " ".
|
|
|
|
idtac "#> combine_odd_even".
|
|
idtac "Possible points: 3".
|
|
check_type @combine_odd_even (((nat -> Prop) -> (nat -> Prop) -> nat -> Prop)).
|
|
idtac "Assumptions:".
|
|
Abort.
|
|
Print Assumptions combine_odd_even.
|
|
Goal True.
|
|
idtac " ".
|
|
|
|
idtac "------------------- tr_rev_correct --------------------".
|
|
idtac " ".
|
|
|
|
idtac "#> tr_rev_correct".
|
|
idtac "Possible points: 4".
|
|
check_type @tr_rev_correct ((forall X : Type, @tr_rev X = @rev X)).
|
|
idtac "Assumptions:".
|
|
Abort.
|
|
Print Assumptions tr_rev_correct.
|
|
Goal True.
|
|
idtac " ".
|
|
|
|
idtac "------------------- evenb_double_conv --------------------".
|
|
idtac " ".
|
|
|
|
idtac "#> evenb_double_conv".
|
|
idtac "Possible points: 3".
|
|
check_type @evenb_double_conv (
|
|
(forall n : nat,
|
|
exists k : nat, n = (if evenb n then double k else S (double k)))).
|
|
idtac "Assumptions:".
|
|
Abort.
|
|
Print Assumptions evenb_double_conv.
|
|
Goal True.
|
|
idtac " ".
|
|
|
|
idtac "------------------- logical_connectives --------------------".
|
|
idtac " ".
|
|
|
|
idtac "#> andb_true_iff".
|
|
idtac "Possible points: 1".
|
|
check_type @andb_true_iff (
|
|
(forall b1 b2 : bool, b1 && b2 = true <-> b1 = true /\ b2 = true)).
|
|
idtac "Assumptions:".
|
|
Abort.
|
|
Print Assumptions andb_true_iff.
|
|
Goal True.
|
|
idtac " ".
|
|
|
|
idtac "#> orb_true_iff".
|
|
idtac "Possible points: 1".
|
|
check_type @orb_true_iff (
|
|
(forall b1 b2 : bool, b1 || b2 = true <-> b1 = true \/ b2 = true)).
|
|
idtac "Assumptions:".
|
|
Abort.
|
|
Print Assumptions orb_true_iff.
|
|
Goal True.
|
|
idtac " ".
|
|
|
|
idtac "------------------- eqb_neq --------------------".
|
|
idtac " ".
|
|
|
|
idtac "#> eqb_neq".
|
|
idtac "Possible points: 1".
|
|
check_type @eqb_neq ((forall x y : nat, (x =? y) = false <-> x <> y)).
|
|
idtac "Assumptions:".
|
|
Abort.
|
|
Print Assumptions eqb_neq.
|
|
Goal True.
|
|
idtac " ".
|
|
|
|
idtac "------------------- eqb_list --------------------".
|
|
idtac " ".
|
|
|
|
idtac "#> eqb_list".
|
|
idtac "Possible points: 3".
|
|
check_type @eqb_list ((forall A : Type, (A -> A -> bool) -> list A -> list A -> bool)).
|
|
idtac "Assumptions:".
|
|
Abort.
|
|
Print Assumptions eqb_list.
|
|
Goal True.
|
|
idtac " ".
|
|
|
|
idtac "------------------- All_forallb --------------------".
|
|
idtac " ".
|
|
|
|
idtac "#> forallb_true_iff".
|
|
idtac "Possible points: 2".
|
|
check_type @forallb_true_iff (
|
|
(forall (X : Type) (test : X -> bool) (l : list X),
|
|
@forallb X test l = true <-> @All X (fun x : X => test x = true) l)).
|
|
idtac "Assumptions:".
|
|
Abort.
|
|
Print Assumptions forallb_true_iff.
|
|
Goal True.
|
|
idtac " ".
|
|
|
|
idtac "------------------- excluded_middle_irrefutable --------------------".
|
|
idtac " ".
|
|
|
|
idtac "#> excluded_middle_irrefutable".
|
|
idtac "Possible points: 3".
|
|
check_type @excluded_middle_irrefutable ((forall P : Prop, ~ ~ (P \/ ~ P))).
|
|
idtac "Assumptions:".
|
|
Abort.
|
|
Print Assumptions excluded_middle_irrefutable.
|
|
Goal True.
|
|
idtac " ".
|
|
|
|
idtac "------------------- not_exists_dist --------------------".
|
|
idtac " ".
|
|
|
|
idtac "#> not_exists_dist".
|
|
idtac "Advanced".
|
|
idtac "Possible points: 3".
|
|
check_type @not_exists_dist (
|
|
(excluded_middle ->
|
|
forall (X : Type) (P : X -> Prop),
|
|
~ (exists x : X, ~ P x) -> forall x : X, P x)).
|
|
idtac "Assumptions:".
|
|
Abort.
|
|
Print Assumptions not_exists_dist.
|
|
Goal True.
|
|
idtac " ".
|
|
|
|
idtac " ".
|
|
|
|
idtac "Max points - standard: 43".
|
|
idtac "Max points - advanced: 49".
|
|
idtac "".
|
|
idtac "********** Summary **********".
|
|
idtac "".
|
|
idtac "********** Standard **********".
|
|
idtac "---------- and_exercise ---------".
|
|
Print Assumptions and_exercise.
|
|
idtac "---------- and_assoc ---------".
|
|
Print Assumptions and_assoc.
|
|
idtac "---------- mult_eq_0 ---------".
|
|
Print Assumptions mult_eq_0.
|
|
idtac "---------- or_commut ---------".
|
|
Print Assumptions or_commut.
|
|
idtac "---------- contrapositive ---------".
|
|
Print Assumptions contrapositive.
|
|
idtac "---------- not_both_true_and_false ---------".
|
|
Print Assumptions not_both_true_and_false.
|
|
idtac "---------- or_distributes_over_and ---------".
|
|
Print Assumptions or_distributes_over_and.
|
|
idtac "---------- dist_not_exists ---------".
|
|
Print Assumptions dist_not_exists.
|
|
idtac "---------- dist_exists_or ---------".
|
|
Print Assumptions dist_exists_or.
|
|
idtac "---------- In_map_iff ---------".
|
|
Print Assumptions In_map_iff.
|
|
idtac "---------- In_app_iff ---------".
|
|
Print Assumptions In_app_iff.
|
|
idtac "---------- All ---------".
|
|
Print Assumptions All.
|
|
idtac "---------- combine_odd_even ---------".
|
|
Print Assumptions combine_odd_even.
|
|
idtac "---------- tr_rev_correct ---------".
|
|
Print Assumptions tr_rev_correct.
|
|
idtac "---------- evenb_double_conv ---------".
|
|
Print Assumptions evenb_double_conv.
|
|
idtac "---------- andb_true_iff ---------".
|
|
Print Assumptions andb_true_iff.
|
|
idtac "---------- orb_true_iff ---------".
|
|
Print Assumptions orb_true_iff.
|
|
idtac "---------- eqb_neq ---------".
|
|
Print Assumptions eqb_neq.
|
|
idtac "---------- eqb_list ---------".
|
|
Print Assumptions eqb_list.
|
|
idtac "---------- forallb_true_iff ---------".
|
|
Print Assumptions forallb_true_iff.
|
|
idtac "---------- excluded_middle_irrefutable ---------".
|
|
Print Assumptions excluded_middle_irrefutable.
|
|
idtac "".
|
|
idtac "********** Advanced **********".
|
|
idtac "---------- double_neg_inf ---------".
|
|
idtac "MANUAL".
|
|
idtac "---------- informal_not_PNP ---------".
|
|
idtac "MANUAL".
|
|
idtac "---------- not_exists_dist ---------".
|
|
Print Assumptions not_exists_dist.
|
|
Abort.
|
|
|
|
(* Wed Jan 9 12:02:13 EST 2019 *)
|