Booleans |
The type bool is defined in the prelude as
|
Interpretation of booleans as Proposition |
Definition
Is_true := [b:bool](Cases b of
true => True
| false => False
end).
Hints
Unfold Is_true : bool.
Lemma
Is_true_eq_left : (x:bool)x=true -> (Is_true x).
Proof
.
Intros; Rewrite H; Auto with bool.
Save
.
Lemma
Is_true_eq_right : (x:bool)true=x -> (Is_true x).
Proof
.
Intros; Rewrite <- H; Auto with bool.
Save
.
Hints
Immediate
Is_true_eq_right Is_true_eq_left : bool.
Discrimination |
Lemma
diff_true_false : ~true=false.
Goal.
Unfold not; Intro contr; Change (Is_true false).
Elim contr; Simpl; Trivial with bool.
Save
.
Hints
Resolve diff_true_false : bool v62.
Lemma
diff_false_true : ~false=true.
Goal.
Red; Intros H; Apply diff_true_false.
Symmetry.
Assumption.
Save
.
Hints
Resolve diff_false_true : bool v62.
Lemma
eq_true_false_abs : (b:bool)(b=true)->(b=false)->False.
Intros b H; Rewrite H; Auto with bool.
Save
.
Hints
Resolve eq_true_false_abs : bool.
Lemma
not_true_is_false : (b:bool)~b=true->b=false.
NewDestruct b.
Intros.
Red in H; Elim H.
Reflexivity.
Intros abs.
Reflexivity.
Save
.
Lemma
not_false_is_true : (b:bool)~b=false->b=true.
NewDestruct b.
Intros.
Reflexivity.
Intro H; Red in H; Elim H.
Reflexivity.
Save
.
Order on booleans |
Definition
leb := [b1,b2:bool]
Cases b1 of
| true => b2=true
| false => True
end.
Hints
Unfold leb : bool v62.
Equality |
Definition
eqb : bool->bool->bool :=
[b1,b2:bool]
Cases b1 b2 of
true true => true
| true false => false
| false true => false
| false false => true
end.
Lemma
eqb_refl : (x:bool)(Is_true (eqb x x)).
NewDestruct x; Simpl; Auto with bool.
Save
.
Lemma
eqb_eq : (x,y:bool)(Is_true (eqb x y))->x=y.
NewDestruct x; NewDestruct y; Simpl; Tauto.
Save
.
Lemma
Is_true_eq_true : (x:bool) (Is_true x) -> x=true.
NewDestruct x; Simpl; Tauto.
Save
.
Lemma
Is_true_eq_true2 : (x:bool) x=true -> (Is_true x).
NewDestruct x; Simpl; Auto with bool.
Save
.
Lemma
eqb_subst :
(P:bool->Prop)(b1,b2:bool)(eqb b1 b2)=true->(P b1)->(P b2).
Unfold eqb .
Intros P b1.
Intros b2.
Case b1.
Case b2.
Trivial with bool.
Intros H.
Inversion_clear H.
Case b2.
Intros H.
Inversion_clear H.
Trivial with bool.
Save
.
Lemma
eqb_reflx : (b:bool)(eqb b b)=true.
Intro b.
Case b.
Trivial with bool.
Trivial with bool.
Save
.
Lemma
eqb_prop : (a,b:bool)(eqb a b)=true -> a=b.
NewDestruct a; NewDestruct b; Simpl; Intro;
Discriminate H Orelse Reflexivity.
Save
.
Logical combinators |
Definition
ifb : bool -> bool -> bool -> bool
:= [b1,b2,b3:bool](Cases b1 of true => b2 | false => b3 end).
Definition
andb : bool -> bool -> bool
:= [b1,b2:bool](ifb b1 b2 false).
Definition
orb : bool -> bool -> bool
:= [b1,b2:bool](ifb b1 true b2).
Definition
implb : bool -> bool -> bool
:= [b1,b2:bool](ifb b1 b2 true).
Definition
xorb : bool -> bool -> bool
:= [b1,b2:bool]
Cases b1 b2 of
true true => false
| true false => true
| false true => true
| false false => false
end.
Definition
negb := [b:bool]Cases b of
true => false
| false => true
end.
Lemmas about negb
|
Lemma
negb_intro : (b:bool)b=(negb (negb b)).
Goal.
Induction b; Reflexivity.
Save
.
Lemma
negb_elim : (b:bool)(negb (negb b))=b.
Goal.
Induction b; Reflexivity.
Save
.
Lemma
negb_orb : (b1,b2:bool)
(negb (orb b1 b2)) = (andb (negb b1) (negb b2)).
Proof
.
NewDestruct b1; NewDestruct b2; Simpl; Reflexivity.
Qed
.
Lemma
negb_andb : (b1,b2:bool)
(negb (andb b1 b2)) = (orb (negb b1) (negb b2)).
Proof
.
NewDestruct b1; NewDestruct b2; Simpl; Reflexivity.
Qed
.
Lemma
negb_sym : (b,b':bool)(b'=(negb b))->(b=(negb b')).
Goal.
Induction b; Induction b'; Intros; Simpl; Trivial with bool.
Save
.
Lemma
no_fixpoint_negb : (b:bool)~(negb b)=b.
Goal.
Induction b; Simpl; Unfold not; Intro; Apply diff_true_false; Auto with bool.
Save
.
Lemma
eqb_negb1 : (b:bool)(eqb (negb b) b)=false.
NewDestruct b.
Trivial with bool.
Trivial with bool.
Save
.
Lemma
eqb_negb2 : (b:bool)(eqb b (negb b))=false.
NewDestruct b.
Trivial with bool.
Trivial with bool.
Save
.
Lemma
if_negb : (A:Set) (b:bool) (x,y:A) (if (negb b) then x else y)=(if b then y else x).
Proof
.
Induction b;Trivial.
Qed
.
A few lemmas about or
|
Lemma
orb_prop :
(a,b:bool)(orb a b)=true -> (a = true)\/(b = true).
Induction a; Induction b; Simpl; Try (Intro H;Discriminate H); Auto with bool.
Save
.
Lemma
orb_prop2 :
(a,b:bool)(Is_true (orb a b)) -> (Is_true a)\/(Is_true b).
Induction a; Induction b; Simpl; Try (Intro H;Discriminate H); Auto with bool.
Save
.
Lemma
orb_true_intro
: (b1,b2:bool)(b1=true)\/(b2=true)->(orb b1 b2)=true.
NewDestruct b1; Auto with bool.
NewDestruct 1; Intros.
Elim diff_true_false; Auto with bool.
Rewrite H; Trivial with bool.
Save
.
Hints
Resolve orb_true_intro : bool v62.
Lemma
orb_b_true : (b:bool)(orb b true)=true.
Auto with bool.
Save
.
Hints
Resolve orb_b_true : bool v62.
Lemma
orb_true_b : (b:bool)(orb true b)=true.
Trivial with bool.
Save
.
Lemma
orb_true_elim : (b1,b2:bool)(orb b1 b2)=true -> {b1=true}+{b2=true}.
NewDestruct b1; Simpl; Auto with bool.
Save
.
Lemma
orb_false_intro
: (b1,b2:bool)(b1=false)->(b2=false)->(orb b1 b2)=false.
Intros b1 b2 H1 H2; Rewrite H1; Rewrite H2; Trivial with bool.
Save
.
Hints
Resolve orb_false_intro : bool v62.
Lemma
orb_b_false : (b:bool)(orb b false)=b.
Proof
.
NewDestruct b; Trivial with bool.
Save
.
Hints
Resolve orb_b_false : bool v62.
Lemma
orb_false_b : (b:bool)(orb false b)=b.
Proof
.
NewDestruct b; Trivial with bool.
Save
.
Hints
Resolve orb_false_b : bool v62.
Lemma
orb_false_elim :
(b1,b2:bool)(orb b1 b2)=false -> (b1=false)/\(b2=false).
Proof
.
NewDestruct b1.
Intros; Elim diff_true_false; Auto with bool.
NewDestruct b2.
Intros; Elim diff_true_false; Auto with bool.
Auto with bool.
Save
.
Lemma
orb_neg_b :
(b:bool)(orb b (negb b))=true.
Proof
.
NewDestruct b; Reflexivity.
Save
.
Hints
Resolve orb_neg_b : bool v62.
Lemma
orb_sym : (b1,b2:bool)(orb b1 b2)=(orb b2 b1).
NewDestruct b1; NewDestruct b2; Reflexivity.
Save
.
Lemma
orb_assoc : (b1,b2,b3:bool)(orb b1 (orb b2 b3))=(orb (orb b1 b2) b3).
Proof
.
NewDestruct b1; NewDestruct b2; NewDestruct b3; Reflexivity.
Save
.
Hints
Resolve orb_sym orb_assoc orb_b_false orb_false_b : bool v62.
A few lemmas about and
|
Lemma
andb_prop :
(a,b:bool)(andb a b) = true -> (a = true)/\(b = true).
Proof
.
Induction a; Induction b; Simpl; Try (Intro H;Discriminate H);
Auto with bool.
Save
.
Hints
Resolve andb_prop : bool v62.
Lemma
andb_prop2 :
(a,b:bool)(Is_true (andb a b)) -> (Is_true a)/\(Is_true b).
Proof
.
Induction a; Induction b; Simpl; Try (Intro H;Discriminate H);
Auto with bool.
Save
.
Hints
Resolve andb_prop2 : bool v62.
Lemma
andb_true_intro : (b1,b2:bool)(b1=true)/\(b2=true)->(andb b1 b2)=true.
Proof
.
NewDestruct b1; NewDestruct b2; Simpl; Tauto Orelse Auto with bool.
Save
.
Hints
Resolve andb_true_intro : bool v62.
Lemma
andb_true_intro2 :
(b1,b2:bool)(Is_true b1)->(Is_true b2)->(Is_true (andb b1 b2)).
Proof
.
NewDestruct b1; NewDestruct b2; Simpl; Tauto.
Save
.
Hints
Resolve andb_true_intro2 : bool v62.
Lemma
andb_false_intro1
: (b1,b2:bool)(b1=false)->(andb b1 b2)=false.
NewDestruct b1; NewDestruct b2; Simpl; Tauto Orelse Auto with bool.
Save
.
Lemma
andb_false_intro2
: (b1,b2:bool)(b2=false)->(andb b1 b2)=false.
NewDestruct b1; NewDestruct b2; Simpl; Tauto Orelse Auto with bool.
Save
.
Lemma
andb_b_false : (b:bool)(andb b false)=false.
NewDestruct b; Auto with bool.
Save
.
Lemma
andb_false_b : (b:bool)(andb false b)=false.
Trivial with bool.
Save
.
Lemma
andb_b_true : (b:bool)(andb b true)=b.
NewDestruct b; Auto with bool.
Save
.
Lemma
andb_true_b : (b:bool)(andb true b)=b.
Trivial with bool.
Save
.
Lemma
andb_false_elim :
(b1,b2:bool)(andb b1 b2)=false -> {b1=false}+{b2=false}.
NewDestruct b1; Simpl; Auto with bool.
Save
.
Hints
Resolve andb_false_elim : bool v62.
Lemma
andb_neg_b :
(b:bool)(andb b (negb b))=false.
NewDestruct b; Reflexivity.
Save
.
Hints
Resolve andb_neg_b : bool v62.
Lemma
andb_sym : (b1,b2:bool)(andb b1 b2)=(andb b2 b1).
NewDestruct b1; NewDestruct b2; Reflexivity.
Save
.
Lemma
andb_assoc : (b1,b2,b3:bool)(andb b1 (andb b2 b3))=(andb (andb b1 b2) b3).
NewDestruct b1; NewDestruct b2; NewDestruct b3; Reflexivity.
Save
.
Hints
Resolve andb_sym andb_assoc : bool v62.
Properties of xorb
|
Lemma
xorb_false : (b:bool) (xorb b false)=b.
Proof
.
Induction b; Trivial.
Qed
.
Lemma
false_xorb : (b:bool) (xorb false b)=b.
Proof
.
Induction b; Trivial.
Qed
.
Lemma
xorb_true : (b:bool) (xorb b true)=(negb b).
Proof
.
Trivial.
Qed
.
Lemma
true_xorb : (b:bool) (xorb true b)=(negb b).
Proof
.
Induction b; Trivial.
Qed
.
Lemma
xorb_nilpotent : (b:bool) (xorb b b)=false.
Proof
.
Induction b; Trivial.
Qed
.
Lemma
xorb_comm : (b,b':bool) (xorb b b')=(xorb b' b).
Proof
.
Induction b; Induction b'; Trivial.
Qed
.
Lemma
xorb_assoc : (b,b',b'':bool) (xorb (xorb b b') b'')=(xorb b (xorb b' b'')).
Proof
.
Induction b; Induction b'; Induction b''; Trivial.
Qed
.
Lemma
xorb_eq : (b,b':bool) (xorb b b')=false -> b=b'.
Proof
.
Induction b; Induction b'; Trivial.
Unfold xorb. Intros. Rewrite H. Reflexivity.
Qed
.
Lemma
xorb_move_l_r_1 : (b,b',b'':bool) (xorb b b')=b'' -> b'=(xorb b b'').
Proof
.
Intros. Rewrite <- (false_xorb b'). Rewrite <- (xorb_nilpotent b). Rewrite xorb_assoc.
Rewrite H. Reflexivity.
Qed
.
Lemma
xorb_move_l_r_2 : (b,b',b'':bool) (xorb b b')=b'' -> b=(xorb b'' b').
Proof
.
Intros. Rewrite (xorb_comm b b') in H. Rewrite (xorb_move_l_r_1 b' b b'' H). Apply xorb_comm.
Qed
.
Lemma
xorb_move_r_l_1 : (b,b',b'':bool) b=(xorb b' b'') -> (xorb b' b)=b''.
Proof
.
Intros. Rewrite H. Rewrite <- xorb_assoc. Rewrite xorb_nilpotent. Apply false_xorb.
Qed
.
Lemma
xorb_move_r_l_2 : (b,b',b'':bool) b=(xorb b' b'') -> (xorb b b'')=b'.
Proof
.
Intros. Rewrite H. Rewrite xorb_assoc. Rewrite xorb_nilpotent. Apply xorb_false.
Qed
.
De Morgan's law |
Lemma
demorgan1 : (b1,b2,b3:bool)
(andb b1 (orb b2 b3)) = (orb (andb b1 b2) (andb b1 b3)).
NewDestruct b1; NewDestruct b2; NewDestruct b3; Reflexivity.
Save
.
Lemma
demorgan2 : (b1,b2,b3:bool)
(andb (orb b1 b2) b3) = (orb (andb b1 b3) (andb b2 b3)).
NewDestruct b1; NewDestruct b2; NewDestruct b3; Reflexivity.
Save
.
Lemma
demorgan3 : (b1,b2,b3:bool)
(orb b1 (andb b2 b3)) = (andb (orb b1 b2) (orb b1 b3)).
NewDestruct b1; NewDestruct b2; NewDestruct b3; Reflexivity.
Save
.
Lemma
demorgan4 : (b1,b2,b3:bool)
(orb (andb b1 b2) b3) = (andb (orb b1 b3) (orb b2 b3)).
NewDestruct b1; NewDestruct b2; NewDestruct b3; Reflexivity.
Save
.
Lemma
absoption_andb : (b1,b2:bool)
(andb b1 (orb b1 b2)) = b1.
Proof
.
NewDestruct b1; NewDestruct b2; Simpl; Reflexivity.
Qed
.
Lemma
absoption_orb : (b1,b2:bool)
(orb b1 (andb b1 b2)) = b1.
Proof
.
NewDestruct b1; NewDestruct b2; Simpl; Reflexivity.
Qed
.