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LogicTest.v
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Set Warnings "-notation-overridden,-parsing".
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 :=
first [
match eval compute in A with
| ?T => idtac "Score:" T
end
| idtac "Score: Ungraded"].
End Check.
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 score_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 score_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 "------------------- beq_nat_false_iff --------------------".
idtac " ".
idtac "#> beq_nat_false_iff".
idtac "Possible points: 1".
check_type @beq_nat_false_iff ((forall x y : nat, beq_nat x y = false <-> x <> y)).
idtac "Assumptions:".
Abort.
Print Assumptions beq_nat_false_iff.
Goal True.
idtac " ".
idtac "------------------- beq_list --------------------".
idtac " ".
idtac "#> beq_list".
idtac "Possible points: 3".
check_type @beq_list ((forall A : Type, (A -> A -> bool) -> list A -> list A -> bool)).
idtac "Assumptions:".
Abort.
Print Assumptions beq_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".
Abort.