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Complete Basics.v

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Oystein Kristoffer Tveit
2026-03-11 16:34:29 +09:00
parent 7d744693f6
commit 8c0ec374e5
1 changed files with 243 additions and 105 deletions
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Basics.v
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@@ -345,18 +345,35 @@ Compute (invert bw_white).
over [simpl] and go directly to [reflexivity]. We'll explain
what's happening later in the chapter. *)
Definition nandb (b1:bool) (b2:bool) : bool
(* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted.
Definition nandb (b1:bool) (b2:bool) : bool :=
match b1 with
| true => negb b2
| false => true
end.
Example test_nandb1: (nandb true false) = true.
(* FILL IN HERE *) Admitted.
Proof.
simpl.
reflexivity.
Qed.
Example test_nandb2: (nandb false false) = true.
(* FILL IN HERE *) Admitted.
Proof.
simpl.
reflexivity.
Qed.
Example test_nandb3: (nandb false true) = true.
(* FILL IN HERE *) Admitted.
Proof.
simpl.
reflexivity.
Qed.
Example test_nandb4: (nandb true true) = false.
(* FILL IN HERE *) Admitted.
(** [] *)
Proof.
simpl.
reflexivity.
Qed.
(** **** Exercise: 1 star, standard (andb3)
@@ -364,18 +381,36 @@ Example test_nandb4: (nandb true true) = false.
return [true] when all of its inputs are [true], and [false]
otherwise. *)
Definition andb3 (b1:bool) (b2:bool) (b3:bool) : bool
(* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted.
Definition andb3 (b1:bool) (b2:bool) (b3:bool) : bool :=
match b1, b2, b3 with
| true, true, true => true
| _, _, _ => false
end.
Example test_andb31: (andb3 true true true) = true.
(* FILL IN HERE *) Admitted.
Proof.
simpl.
reflexivity.
Qed.
Example test_andb32: (andb3 false true true) = false.
(* FILL IN HERE *) Admitted.
Proof.
simpl.
reflexivity.
Qed.
Example test_andb33: (andb3 true false true) = false.
(* FILL IN HERE *) Admitted.
Proof.
simpl.
reflexivity.
Qed.
Example test_andb34: (andb3 true true false) = false.
(* FILL IN HERE *) Admitted.
(** [] *)
Proof.
simpl.
reflexivity.
Qed.
(* ================================================================= *)
(** ** Types *)
@@ -816,14 +851,23 @@ Fixpoint exp (base power : nat) : nat :=
factorial was not found in the current environment," it means
you've forgotten the [:=]. *)
Fixpoint factorial (n:nat) : nat
(* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted.
Fixpoint factorial (n:nat) : nat :=
match n with
| O => S O
| S p => mult (S p) (factorial p)
end.
Example test_factorial1: (factorial 3) = 6.
(* FILL IN HERE *) Admitted.
Proof.
simpl.
reflexivity.
Qed.
Example test_factorial2: (factorial 5) = (mult 10 12).
(* FILL IN HERE *) Admitted.
(** [] *)
Proof.
simpl.
reflexivity.
Qed.
(** Again, we can make numerical expressions easier to read and write
by introducing notations for addition, subtraction, and
@@ -917,18 +961,36 @@ Proof. simpl. reflexivity. Qed.
function. It can be done with just one previously defined
function, but you can use two if you want. *)
Definition ltb (n m : nat) : bool
(* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted.
Definition ltb (n m : nat) : bool :=
if m <=? n then false else true.
(* match n, m with *)
(* | S n', S m' => ltb n' m' *)
(* | _, O => false *)
(* | O, _ => true *)
(* end. *)
Notation "x <? y" := (ltb x y) (at level 70) : nat_scope.
Example test_ltb1: (ltb 2 2) = false.
(* FILL IN HERE *) Admitted.
Proof.
simpl.
reflexivity.
Qed.
Example test_ltb2: (ltb 2 4) = true.
(* FILL IN HERE *) Admitted.
Proof.
simpl.
reflexivity.
Qed.
Example test_ltb3: (ltb 4 2) = false.
(* FILL IN HERE *) Admitted.
(** [] *)
Proof.
simpl.
reflexivity.
Qed.
(* ################################################################# *)
(** * Proof by Simplification *)
@@ -1091,8 +1153,10 @@ Proof.
Theorem plus_id_exercise : forall n m o : nat,
n = m -> m = o -> n + m = m + o.
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
intros n m o H1 H2.
rewrite -> H1, H2.
reflexivity.
Qed.
(** The [Admitted] command tells Rocq that we want to skip trying
to prove this theorem and just accept it as a given. This is
@@ -1142,9 +1206,11 @@ Proof.
Theorem mult_n_1 : forall p : nat,
p * 1 = p.
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
intros n.
rewrite <- mult_n_Sm.
rewrite <- mult_n_O.
reflexivity.
Qed.
(* ################################################################# *)
(** * Proof by Case Analysis *)
@@ -1348,8 +1414,16 @@ Qed.
Theorem andb_true_elim2 : forall b c : bool,
andb b c = true -> c = true.
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
intros b c.
destruct b eqn:Eb.
- simpl.
intros H.
rewrite <- H.
reflexivity.
- simpl.
intros H.
discriminate H.
Qed.
(** Before closing the chapter, we should mention one final
convenience. As you may have noticed, many proofs perform case
@@ -1389,8 +1463,10 @@ Qed.
Theorem zero_nbeq_plus_1 : forall n : nat,
0 =? (n + 1) = false.
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
intros [].
- reflexivity.
- reflexivity.
Qed.
(* ================================================================= *)
(** ** More on Notation (Optional) *)
@@ -1474,9 +1550,14 @@ Fixpoint plus' (n : nat) (m : nat) : nat :=
homework assignment, make sure you comment out your solution so
that it doesn't cause Rocq to reject the whole file!) *)
(* FILL IN HERE
[] *)
(*
Fixpoint rejectme (a b: nat) : nat :=
match a, b with
| O, _ => b
| _, O => a
| S a', S b' => if even a then rejectme a' b else rejectme a b'
end.
*)
(* ################################################################# *)
(** * More Exercises *)
@@ -1494,9 +1575,10 @@ Theorem identity_fn_applied_twice :
(forall (x : bool), f x = x) ->
forall (b : bool), f (f b) = b.
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
intros f H b.
rewrite -> H, H.
reflexivity.
Qed.
(** **** Exercise: 1 star, standard (negation_fn_applied_twice)
@@ -1504,7 +1586,16 @@ Proof.
to the previous one but where the hypothesis says that the
function [f] has the property that [f x = negb x]. *)
(* FILL IN HERE *)
Theorem negation_fn_applied_twice :
forall (f : bool -> bool),
(forall (x : bool), f x = negb x) ->
forall (b : bool), f (f b) = b.
Proof.
intros f H b.
rewrite -> H, H.
rewrite -> negb_involutive.
reflexivity.
Qed.
(* Do not modify the following line: *)
Definition manual_grade_for_negation_fn_applied_twice : option (nat*string) := None.
@@ -1519,12 +1610,48 @@ Definition manual_grade_for_negation_fn_applied_twice : option (nat*string) := N
[destruct] and [rewrite], but destructing everything in sight is
not the best way.) *)
Theorem orb_true : forall (b : bool), true || b = true.
Proof.
destruct b.
- reflexivity.
- reflexivity.
Qed.
Theorem andb_true : forall (b : bool), true && b = b.
Proof.
destruct b.
- reflexivity.
- reflexivity.
Qed.
Theorem orb_false : forall (b : bool), false || b = b.
Proof.
destruct b.
- reflexivity.
- reflexivity.
Qed.
Theorem andb_false : forall (b : bool), false && b = false.
Proof.
destruct b.
- reflexivity.
- reflexivity.
Qed.
Theorem andb_eq_orb :
forall (b c : bool),
(andb b c = orb b c) ->
b = c.
Proof.
(* FILL IN HERE *) Admitted.
intros b c H.
destruct b.
- rewrite andb_true, orb_true in H.
rewrite <- H.
reflexivity.
- rewrite andb_false, orb_false in H.
rewrite <- H.
reflexivity.
Qed.
(** [] *)
@@ -1627,8 +1754,13 @@ Compute letter_comparison B F.
Theorem letter_comparison_Eq :
forall l, letter_comparison l l = Eq.
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
destruct l.
reflexivity.
reflexivity.
reflexivity.
reflexivity.
reflexivity.
Qed.
(** We can follow the same strategy to define the comparison operation
for two grade modifiers. We consider them to be ordered as
@@ -1658,29 +1790,34 @@ Definition modifier_comparison (m1 m2 : modifier) : comparison :=
of a suitable call to [letter_comparison] to end up with just [3]
possibilities. *)
Definition grade_comparison (g1 g2 : grade) : comparison
(* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted.
Definition grade_comparison (g1 g2 : grade) : comparison :=
match g1, g2 with
| Grade l1 m1, Grade l2 m2 =>
match letter_comparison l1 l2 with
| Gt => Gt
| Lt => Lt
| Eq => modifier_comparison m1 m2
end
end.
(** The following "unit tests" of your [grade_comparison] function
should pass once you have defined it correctly. *)
Example test_grade_comparison1 :
(grade_comparison (Grade A Minus) (Grade B Plus)) = Gt.
(* FILL IN HERE *) Admitted.
Proof. reflexivity. Qed.
Example test_grade_comparison2 :
(grade_comparison (Grade A Minus) (Grade A Plus)) = Lt.
(* FILL IN HERE *) Admitted.
Proof. reflexivity. Qed.
Example test_grade_comparison3 :
(grade_comparison (Grade F Plus) (Grade F Plus)) = Eq.
(* FILL IN HERE *) Admitted.
Proof. reflexivity. Qed.
Example test_grade_comparison4 :
(grade_comparison (Grade B Minus) (Grade C Plus)) = Gt.
(* FILL IN HERE *) Admitted.
(** [] *)
Proof. reflexivity. Qed.
(** Now that we have a definition of grades and how they compare to
one another, let us implement a late-penalty fuction. *)
@@ -1736,9 +1873,14 @@ Theorem lower_letter_lowers:
letter_comparison F l = Lt ->
letter_comparison (lower_letter l) l = Lt.
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
intros l H.
destruct l.
- simpl. reflexivity.
- simpl. reflexivity.
- simpl. reflexivity.
- simpl. reflexivity.
- rewrite <- H. simpl. reflexivity.
Qed.
(** **** Exercise: 2 stars, standard (lower_grade)
@@ -1757,61 +1899,47 @@ Proof.
cases.
Our solution is under 10 lines of code total. *)
Definition lower_grade (g : grade) : grade
(* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted.
Definition lower_grade (g : grade) : grade :=
match g with
| Grade F Minus => Grade F Minus
| Grade l Plus => Grade l Natural
| Grade l Natural => Grade l Minus
| Grade l Minus => Grade (lower_letter l) Plus
end.
Example lower_grade_A_Plus :
lower_grade (Grade A Plus) = (Grade A Natural).
Proof.
(* FILL IN HERE *) Admitted.
Proof. reflexivity. Qed.
Example lower_grade_A_Natural :
lower_grade (Grade A Natural) = (Grade A Minus).
Proof.
(* FILL IN HERE *) Admitted.
Proof. reflexivity. Qed.
Example lower_grade_A_Minus :
lower_grade (Grade A Minus) = (Grade B Plus).
Proof.
(* FILL IN HERE *) Admitted.
Proof. reflexivity. Qed.
Example lower_grade_B_Plus :
lower_grade (Grade B Plus) = (Grade B Natural).
Proof.
(* FILL IN HERE *) Admitted.
Proof. reflexivity. Qed.
Example lower_grade_F_Natural :
lower_grade (Grade F Natural) = (Grade F Minus).
Proof.
(* FILL IN HERE *) Admitted.
Proof. reflexivity. Qed.
Example lower_grade_twice :
lower_grade (lower_grade (Grade B Minus)) = (Grade C Natural).
Proof.
(* FILL IN HERE *) Admitted.
Proof. reflexivity. Qed.
Example lower_grade_thrice :
lower_grade (lower_grade (lower_grade (Grade B Minus))) = (Grade C Minus).
Proof.
(* FILL IN HERE *) Admitted.
Proof. reflexivity. Qed.
(** Rocq makes no distinction between an [Example] and a [Theorem]. We
state the following as a [Theorem] only as a hint that we will use
it in proofs below. *)
Theorem lower_grade_F_Minus : lower_grade (Grade F Minus) = (Grade F Minus).
Proof.
(* FILL IN HERE *) Admitted.
(* GRADE_THEOREM 0.25: lower_grade_A_Plus *)
(* GRADE_THEOREM 0.25: lower_grade_A_Natural *)
(* GRADE_THEOREM 0.25: lower_grade_A_Minus *)
(* GRADE_THEOREM 0.25: lower_grade_B_Plus *)
(* GRADE_THEOREM 0.25: lower_grade_F_Natural *)
(* GRADE_THEOREM 0.25: lower_grade_twice *)
(* GRADE_THEOREM 0.25: lower_grade_thrice *)
(* GRADE_THEOREM 0.25: lower_grade_F_Minus
[] *)
Proof. reflexivity. Qed.
(** **** Exercise: 3 stars, standard (lower_grade_lowers)
@@ -1830,9 +1958,10 @@ Theorem lower_grade_lowers :
grade_comparison (Grade F Minus) g = Lt ->
grade_comparison (lower_grade g) g = Lt.
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
intros [].
simpl.
destruct l, m; simpl; intros H; apply H.
Qed.
(** Now that we have implemented and tested a function that lowers a
grade by one step, we can implement a specific late-days policy.
@@ -1886,9 +2015,10 @@ Theorem no_penalty_for_mostly_on_time :
(late_days <? 9 = true) ->
apply_late_policy late_days g = g.
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
intros.
rewrite -> apply_late_policy_unfold, H.
reflexivity.
Qed.
(** The following theorem states that, if a student has between 9 and
16 late days, their final grade is lowered by one step. *)
@@ -1900,9 +2030,11 @@ Theorem grade_lowered_once :
(late_days <? 17 = true) ->
(apply_late_policy late_days g) = (lower_grade g).
Proof.
(* FILL IN HERE *) Admitted.
intros.
rewrite -> apply_late_policy_unfold, H, H0.
reflexivity.
Qed.
(** [] *)
End LateDays.
(* ================================================================= *)
@@ -1945,11 +2077,19 @@ Inductive bin : Type :=
for binary numbers, and a function [bin_to_nat] to convert
binary numbers to unary numbers. *)
Fixpoint incr (m:bin) : bin
(* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted.
Fixpoint incr (m:bin) : bin :=
match m with
| Z => B1 Z
| B0 b => B1 b
| B1 b => B0 (incr b)
end.
Fixpoint bin_to_nat (m:bin) : nat
(* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted.
Fixpoint bin_to_nat (m:bin) : nat :=
match m with
| Z => O
| B0 m' => 2 * (bin_to_nat m')
| B1 m' => S (2 * (bin_to_nat m'))
end.
(** The following "unit tests" of your increment and binary-to-unary
functions should pass after you have defined those functions correctly.
@@ -1958,29 +2098,27 @@ Fixpoint bin_to_nat (m:bin) : nat
next chapter. *)
Example test_bin_incr1 : (incr (B1 Z)) = B0 (B1 Z).
(* FILL IN HERE *) Admitted.
Proof. reflexivity. Qed.
Example test_bin_incr2 : (incr (B0 (B1 Z))) = B1 (B1 Z).
(* FILL IN HERE *) Admitted.
Proof. reflexivity. Qed.
Example test_bin_incr3 : (incr (B1 (B1 Z))) = B0 (B0 (B1 Z)).
(* FILL IN HERE *) Admitted.
Proof. reflexivity. Qed.
Example test_bin_incr4 : bin_to_nat (B0 (B1 Z)) = 2.
(* FILL IN HERE *) Admitted.
Proof. reflexivity. Qed.
Example test_bin_incr5 :
bin_to_nat (incr (B1 Z)) = 1 + bin_to_nat (B1 Z).
(* FILL IN HERE *) Admitted.
Proof. reflexivity. Qed.
Example test_bin_incr6 :
bin_to_nat (incr (incr (B1 Z))) = 2 + bin_to_nat (B1 Z).
(* FILL IN HERE *) Admitted.
Proof. reflexivity. Qed.
Example test_bin_incr7 : bin_to_nat (B0 (B0 (B0 (B1 Z)))) = 8.
(* FILL IN HERE *) Admitted.
(** [] *)
Proof. reflexivity. Qed.
(* ################################################################# *)
(** * Optional: Testing Your Solutions *)