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Basics.v
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Basics.v
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Inductive day : Type :=
| monday : day
| tuesday : day
| wednesday : day
| thursday : day
| friday : day
| saturday : day
| sunday : day.
Definition next_weekday (d : day) : day :=
match d with
| monday => tuesday
| tuesday => wednesday
| wednesday => thursday
| thursday => friday
| friday => saturday
| saturday => sunday
| sunday => monday
end.
Inductive bool : Type :=
| true : bool
| false : bool.
Definition negb (b : bool) : bool :=
match b with
| true => false
| false => true
end.
Definition andb (b1 : bool) (b2 : bool) :=
match b1 with
| true => b2
| false => false
end.
Definition orb (b1 : bool) (b2 : bool) :=
match b1 with
| true => true
| false => b2
end.
Fixpoint evenb (n:nat) : bool :=
match n with
| O => true
| S O => false
| S (S n') => evenb n'
end.
Eval simpl in (next_weekday (next_weekday saturday)).
Example test_orb1: (orb true false) = true.
Proof. simpl. reflexivity. Qed.
Example test_orb2: (orb false false) = false.
Proof. simpl. reflexivity. Qed.
Example test_orb3: (orb false true ) = true.
Proof. simpl. reflexivity. Qed.
Example test_orb4: (orb true true ) = true.
Proof. simpl. reflexivity. Qed.
Definition nandb (b1 : bool) (b2 : bool) : bool :=
match b1, b2 with
| true, true => false
| false, true => true
| true, false => true
| false, false => true
end.
Example test_nandb1: (nandb true false) = true.
Proof. simpl. reflexivity. Qed.
Example test_nandb2: (nandb false false) = true.
Proof. simpl. reflexivity. Qed.
Example test_nandb3: (nandb false true) = true.
Proof. simpl. reflexivity. Qed.
Example test_nandb4: (nandb true true) = false.
Proof. simpl. reflexivity. Qed.
Check (negb true).
Check negb.
Theorem plus_0_n : forall n : nat, 0 + n = n.
Proof.
simpl. reflexivity. Qed.
Eval simpl in (forall n : nat, n + 0 = n).
Eval simpl in (forall n : nat, 0 + n = n).
Fixpoint beq_nat (n m : nat) : bool :=
match n with
| O => match m with
| O => true
| S m' => false
end
| S n' => match m with
| O => false
| S m' => beq_nat n' m'
end
end.
Theorem plus_0_n'' : forall n : nat, 0 + n = n.
Proof.
intros n. reflexivity. Qed.
Theorem plus_id_example : forall n m : nat,
n = m ->
n + n = m + m.
Proof.
intros n m. intros H. rewrite -> H.
reflexivity. Qed.
Theorem plus_id_exercise : forall n m o : nat,
n = m -> m = o -> n + m = m + o.
Proof.
intros m n o. intros H. intros I.
rewrite -> H. rewrite -> I.
reflexivity.
Qed.
Theorem mult_0_plus : forall n m : nat,
(0 + n) * m = n * m.
Proof.
intros n m.
rewrite -> plus_O_n.
reflexivity.
Qed.
Theorem plus_1_n : forall n : nat, 1 + n = S n.
Proof.
intros n. reflexivity.
Qed.
Theorem mult_1_plus : forall n m : nat,
(1 + n) * m = m + (n * m).
Proof.
intros n m.
rewrite -> plus_1_n.
reflexivity.
Qed.
Theorem plus_1_neq_0_firsttry : forall n : nat,
beq_nat (n + 1) 0 = false.
Proof.
intros n. destruct n as [| n'].
reflexivity.
reflexivity.
Qed.
Theorem negb_involutive : forall b : bool,
negb (negb b) = b.
Proof.
intros b. destruct b.
reflexivity.
reflexivity.
Qed.
Theorem zero_nbeq_plus_1 : forall n : nat,
beq_nat 0 (n + 1) = false.
Proof.
intros n. destruct n as [O | S n'].
reflexivity.
reflexivity.
Qed.
Require String. Open Scope string_scope.
Ltac move_to_top x :=
match reverse goal with
| H : _ |- _ => try move x after H
end.
Tactic Notation "assert_eq" ident(x) constr (v) :=
let H := fresh in
assert (x = v) as H by reflexivity;
clear H.
Tactic Notation "Case_aux" ident(x) constr(name) :=
first [
set (x := name); move_to_top x
| assert_eq x name; move_to_top x
| fail 1 "because we are working on a different case" ].
Tactic Notation "Case" constr(name) := Case_aux Case name.
Tactic Notation "SCase" constr(name) := Case_aux SCase name.
Tactic Notation "SSCase" constr(name) := Case_aux SSCase name.
Tactic Notation "SSSCase" constr(name) := Case_aux SSSCase name.
Tactic Notation "SSSSCase" constr(name) := Case_aux SSSSCase name.
Tactic Notation "SSSSSCase" constr(name) := Case_aux SSSSSCase name.
Tactic Notation "SSSSSSCase" constr(name) := Case_aux SSSSSSCase name.
Tactic Notation "SSSSSSSCase" constr(name) := Case_aux SSSSSSSCase name.
Theorem andb_true_elim1 : forall b c : bool,
andb b c = true -> b = true.
Proof.
intros b c H.
destruct b.
Case "b = true". reflexivity.
Case "b = false". rewrite <- H. reflexivity.
Qed.
Theorem andb_true_elim2 : forall b c : bool,
andb b c = true -> c = true.
Proof.
intros [] [].
- reflexivity.
- simpl. intros H. rewrite -> H. reflexivity.
- reflexivity.
- simpl. intros H. rewrite -> H. reflexivity.
Qed.
Theorem plus_0_r : forall n : nat, n + 0 = n.
Proof.
intros n. induction n as [| n'].
Case "n = 0". reflexivity.
Case "n = S n". simpl. rewrite -> IHn'. reflexivity.
Qed.
Theorem minus_diag : forall n,
minus n n = 0.
Proof.
intros n.
induction n as [| n'].
Case "n = 0".
simpl. reflexivity.
Case "n = S n'".
simpl. rewrite -> IHn'. reflexivity.
Qed.
Theorem mult_0_r : forall n : nat,
n * 0 = 0.
Proof.
intros n.
induction n as [| n'].
Case "n = 0".
simpl.
reflexivity.
Case "n = S n'".
simpl.
rewrite -> IHn'.
reflexivity.
Qed.
Theorem plus_n_Sm : forall n m : nat,
S (n + m) = n + (S m).
Proof.
intros n m.
induction n as [| n'].
Case "n = 0".
simpl.
reflexivity.
Case "n = S n'".
simpl.
rewrite -> IHn'.
reflexivity.
Qed.
Theorem plus_0 : forall n : nat,
n + 0 = n.
Proof.
intros n.
induction n as [| n'].
Case "n = 0".
simpl.
reflexivity.
Case "n = S n'".
simpl.
rewrite -> IHn'.
reflexivity.
Qed.
Theorem plus_comm : forall n m : nat,
n + m = m + n.
Proof.
intros n m.
induction n as [ | n'].
Case "n = 0".
simpl.
rewrite -> plus_0.
reflexivity.
Case "n = S n'".
simpl.
rewrite -> IHn'.
rewrite <- plus_n_Sm.
reflexivity.
Qed.
Fixpoint double (n : nat) :=
match n with
| O => O
| S n' => S (S (double n'))
end.
Lemma double_plus : forall n, double n = n + n.
Proof.
intros n.
induction n as [ | n'].
Case "n = 0".
simpl.
reflexivity.
Case "n = S n'".
simpl.
rewrite -> IHn'.
rewrite -> plus_n_Sm.
reflexivity.
Qed.