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proofs1.v
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proofs1.v
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Theorem plus_O_n: forall n : nat , (plus O n) = n.
Proof.
intros n. simpl. reflexivity. Qed.
Theorem plus_O_n': forall n : nat , (O + n) = n.
Proof.
intros n. simpl. reflexivity. Qed.
Theorem plus_O_n'': forall n : nat, (plus O n) = n.
Proof.
intros n. reflexivity. Qed.
Theorem plus_1_1 : forall n : nat, ((S O) + n) = (S n).
Proof.
intros n. simpl. reflexivity. Qed.
Theorem mult_0_1: forall n:nat, (O * n) = O.
Proof.
intros n. reflexivity. Qed.
Theorem mult_0_11: forall n:nat, (mult O n) = O.
Proof.
intros n. reflexivity. Qed.
Theorem plus_n_0: forall n:nat, n = (n + O).
Proof.
intros n.
Admitted.
Theorem plus_id_example: forall n m :nat,
n = m ->
n + n = m + m.
Proof.
Admitted.
Theorem plus_id_exercise: forall n m o : nat,
n = m -> m = o -> n + m = m + o.
Proof.
intros n m o.
intros H.
intros J.
rewrite H.
rewrite J.
reflexivity.
Qed.
Theorem mult_0_plus : forall n m: nat,
(O + n) * m = n * m.
Proof.
intros n m.
simpl.
reflexivity.
Qed.
Theorem mult_0_plus' : forall n m: nat,
(O + n) * m = n * m.
Proof.
intros n m.
rewrite mult_0_plus.
reflexivity.
Qed.
Theorem mult_0_plus'' : forall n m: nat,
(O + n) * m = n * m.
Proof.
intros n m.
rewrite plus_O_n'.
reflexivity.
Qed.
Theorem mult_S_1 : forall n m : nat,
m = S n ->
m * ((S O) + n) = m * m.
Proof.
intros n m.
intros H.
rewrite plus_1_1.
rewrite <- H.
reflexivity.
Qed.
Theorem plus_1_neq_0_firsttry : forall n : nat,
beq_nat (n + (S O)) O = false.
Proof.
intros n.
destruct n as [| n'].
reflexivity.
reflexivity.
Qed.
Theorem plus_1_neq_0_firsttry' : forall n : nat,
beq_nat (n + (S O)) O = false.
Proof.
intros n.
destruct n as [| n'].
simpl.
reflexivity.
simpl.
reflexivity.
Qed.
Theorem negb_involutive : forall b : bool,
(negb (negb b)) = b.
Proof.
intro b.
simpl.
destruct b.
- simpl.
reflexivity.
- simpl.
reflexivity.
Qed.
Theorem andb_commutative : forall b c: bool,
andb b c = andb c b.
Proof.
intros b c.
destruct b.
destruct c.
simpl.
reflexivity.
simpl.
reflexivity.
destruct c.
simpl.
reflexivity.
simpl.
reflexivity.
Qed.
Theorem andb_commutative_with_braces : forall b c:bool,
andb b c = andb c b.
Proof.
intros b c.
destruct b.
{ destruct c.
{ simpl. reflexivity. }
{ simpl. reflexivity. } }
{ destruct c.
{ simpl. reflexivity. }
{ simpl. reflexivity. } }
Qed.
Theorem andb3_exchange: forall b c d,
andb (andb b c) d = andb (andb b d) c.
intros b c d.
{ destruct b.
{ destruct c.
{ destruct d.
{ simpl.
reflexivity. }
{ simpl.
reflexivity. } }
{ destruct d.
{ simpl.
reflexivity. }
{ simpl.
reflexivity. } }
{ destruct c.
{ destruct d.
{ simpl.
reflexivity. }
{ simpl.
reflexivity. } }
{ destruct d.
{ simpl.
reflexivity. }
{ simpl.
reflexivity. } } } } }
Qed.
Theorem plus_1_neq_0''' : forall n : nat,
beq_nat (n + (S O)) O = false.
Proof.
intros [|n].
reflexivity.
reflexivity.
Qed.
Theorem andb_true_elim2: forall b c: bool,
andb b c = true -> c = true.
Proof.
intros b c.
{ destruct b.
{ destruct c.
{ simpl.
intros H1.
reflexivity. }
{ simpl.
intros H2.
rewrite H2.
reflexivity. } }
{ destruct c.
{ simpl.
intros H3.
reflexivity. }
{ simpl.
intros H4.
rewrite H4.
reflexivity. } } }
Qed.
Theorem zero_nbeq_plus_1 : forall n: nat,
beq_nat O (S n) = false.
Proof.
intros n.
{ simpl.
reflexivity. }
Qed.
Theorem identity_fn_applied_twice : forall (f : bool -> bool),
(forall (x : bool), f x = x) ->
forall (b : bool),
f (f b) = b.
Proof.
intros f.
intros H.
intros b0.
rewrite H.
rewrite H.
reflexivity.
Qed.
Theorem andb_eq_orb:
forall (b c : bool),
(andb b c) = (orb b c) ->
b=c.
Proof.
intros b0.
intros c.
destruct b0.
destruct c.
simpl.
intros H.
reflexivity.
simpl.
intros H1.
rewrite H1.
reflexivity.
destruct c.
simpl.
intros H2.
rewrite H2.
reflexivity.
simpl.
intros H3.
reflexivity.
Qed.