Require
Import
Arith.
Require
Import
Omega. Require
Import
Compare_dec.
Require
Import
Relations.
Require
Import
Wellfounded.
Require
Import
Tools.
Require
Import
More_nat.
Require
Import
AccP.
Require
Import
not_decreasing.
Require
Import
ArithRing.
Require
Import
Wf_nat.
Set Implicit
Arguments.
cons a n b represents omega^a *(S n) + b |
Inductive
T1 : Set :=
zero : T1
| cons : T1 -> nat -> T1 -> T1.
some abreviations |
omega^x * (S k) |
Definition
omega_term (a:T1)(k:nat) :=
cons a k zero.
Definition
phi0 a := cons a 0 zero.
Definition
finite (n:nat) : T1 :=
match n with 0 => zero
| S p => cons zero p zero
end.
Notation
"'F' n" := (finite n)(at level 29) : cantor_scope.
Definition
omega := cons (cons zero 0 zero) 0 zero.
Definition
log a := match a with
| zero => zero
| cons a _ _ => a
end.
Fixpoint
omega_tower (n:nat) : T1 :=
match n with 0 => finite 1
| S p => cons (omega_tower p) 0 zero
end.
Inductive
lt : T1 -> T1 -> Prop :=
| zero_lt : forall a n b, zero < cons a n b
| head_lt :
forall a a' n n' b b', a < a' ->
cons a n b < cons a' n' b'
| coeff_lt : forall a n n' b b', (n < n')%nat ->
cons a n b < cons a n' b'
| tail_lt : forall a n b b', b < b' ->
cons a n b < cons a n b'
where "o < o'" := (lt o o') : cantor_scope.
Hint
Resolve zero_lt head_lt coeff_lt tail_lt : T1.
Open Scope cantor_scope.
Delimit Scope cantor_scope with ca.
Definition
le (alpha beta :T1) := alpha = beta \/ alpha < beta.
Notation
"alpha <= beta" := (le alpha beta) : cantor_scope.
Hint
Unfold le : T1.
Inductive
ap : T1 -> Prop:=
ap_intro : forall a, ap (cons a 0 zero).
Lemma
ap_phi0 : forall a, ap (phi0 a).
Proof
.
unfold phi0;constructor.
Qed
.
Lemma
ap_phi0R : forall a, ap a ->{b : T1 | a = phi0 b}.
Proof
.
destruct a.
intro; elimtype False.
inversion H.
exists a1.
inversion H.
unfold phi0;auto.
Qed
.
Fixpoint
compare (c c':T1){struct c'}:comparison :=
match c,c' with
zero, zero => Eq
| zero, cons a' n' b' => Lt
| _ , zero => Gt
| (cons a n b),(cons a' n' b') =>
(match compare a a' with
| Lt => Lt
| Gt => Gt
| Eq => (match lt_eq_lt_dec n n' with
inleft (left _) => Lt
| inright _ => Gt
| _ => compare b b'
end)
end)
end.
Definition
max a b := match compare a b with Lt => b | _ => a end.
Inductive
nf : T1 -> Prop :=
| zero_nf : nf zero
| single_nf : forall a n, nf a -> nf (cons a n zero)
| cons_nf : forall a n a' n' b, a' < a ->
nf a ->
nf(cons a' n' b)->
nf(cons a n (cons a' n' b)).
Hint
Resolve zero_nf single_nf cons_nf : T1.
Inductive
nf2 : T1 -> T1 -> Prop :=
| nf2_z : forall a, nf2 zero a
| nf2_c : forall a a' n' b', a' < a ->
nf2 (cons a' n' b') a.
Hint
Resolve nf2_z nf2_c : T1.
Fixpoint
succ (c:T1) : T1 :=
match c with zero => finite 1
| cons zero n _ => cons zero (S n) zero
| cons a n b => cons a n (succ b)
end.
Fixpoint
pred (c:T1) : option T1 :=
match c with zero => None
| cons zero 0 _ => Some zero
| cons zero (S n) _ => Some (cons zero n zero)
| cons a n b =>
match (pred b) with None => None
| Some c => Some (cons a n c)
end
end.
Fixpoint
plus (c1 c2 : T1) {struct c1}:T1 :=
match c1,c2 with
| zero, y => y
| cons zero n _, zero => cons zero n zero
| x, zero => x
| cons zero n _, cons zero n' _ => cons zero (S (n+n')) zero
| cons a n b, cons a' n' b' =>
(match compare a a' with
| Lt => cons a' n' b'
| Gt => (cons a n (plus b (cons a' n' b')))
| Eq => (cons a (S(n+n')) b')
end)
end
where "a + b" := (plus a b) : cantor_scope.
Fixpoint
minus (c1 c2 : T1) {struct c1}:T1 :=
match c1,c2 with
| zero, y => zero
| cons zero n _, cons zero n' _ =>
if (le_lt_dec n n')
then zero
else cons zero (Peano.pred (n-n')) zero
| cons zero n _, zero => cons zero n zero
| cons zero n _, y => zero
| cons a n b, zero => cons a n b
| cons a n b, cons a' n' b' =>
(match compare a a' with
| Lt => zero
| Gt => cons a n b
| Eq => (match (lt_eq_lt_dec n n') with
| inleft (left _) => zero
| inright _ => (cons a (Peano.pred (n-n')) b')
| _ => b - b'
end)
end)
end
where "c1 - c2" := (minus c1 c2):cantor_scope.
Lemma
omega_minus_one : omega - F 1 = omega.
Proof
.
reflexivity.
Qed
.
Fixpoint
mult (c1 c2 : T1) {struct c2}:T1 :=
match c1,c2 with
| zero, y => zero
| x, zero => zero
| cons zero n _, cons zero n' _ =>
cons zero (Peano.pred((S n) * (S n'))) zero
| cons a n b, cons zero n' b' =>
cons a (Peano.pred((S n) * (S n'))) b
| cons a n b, cons a' n' b' =>
cons (a + a') n' ((cons a n b) * b')
end
where "c1 * c2" := (mult c1 c2) : cantor_scope.
Fixpoint
exp_F (a:T1)(n:nat){struct n}:T1 :=
match n with
| 0 => F 1
| S p => a * (exp_F a p)
end.
Fixpoint
exp (a b : T1) {struct b}:T1 :=
match a,b with
| x, zero => cons zero 0 zero
| cons zero 0 _ , _ => cons zero 0 zero
| zero, y => zero
| x , cons zero n' _ => exp_F x (S n')
| cons zero n _, cons (cons zero 0 zero) n' b' =>
((cons (cons zero n' zero) 0 zero) *
((cons zero n zero) ^ b'))
| cons zero n _, cons a' n' b' =>
(omega_term
(omega_term (a' - (F 1)) n')
0) *
((cons zero n zero) ^ b')
| cons a n b, cons a' n' b' =>
((omega_term (a * (cons a' n' zero))
0) *
((cons a n b) ^ b'))
end
where "a ^ b" := (exp a b) : cantor_scope.
Definition
get_decomposition : forall c, lt zero c ->
{a:T1 & {n:nat & {b:T1 | c = cons a n b}}}.
intro c; case c.
intro H; elimtype False; inversion H.
intros c0 n c1; exists c0;exists n;exists c1;auto.
Defined
.
Ltac
decomp_exhib H a n b e:=
let Ha := fresh in
let Hn := fresh in
let Hb := fresh in
match type of H
with lt zero ?c =>
case (get_decomposition H);
intros a Ha;
case Ha;intros n Hn; case Hn; intros b e;
clear Ha Hn
end.
Theorem
not_lt_zero : forall a, ~ a < zero.
Proof
.
red; inversion_clear 1.
Qed
.
Hint
Resolve not_lt_zero : T1.
Theorem
lt_inv : forall a n b a' n' b',
cons a n b < cons a' n' b' ->
a < a' \/
a = a' /\ (n < n')%nat \/
a = a' /\ n = n' /\ b < b'.
Proof
.
inversion_clear 1; auto.
Qed
.
Theorem
lt_irr : forall a, ~ a < a.
Proof
.
induction a.
red;inversion_clear 1.
intro H; case (lt_inv H); intuition.
Qed
.
Hint
Resolve lt_irr : T1.
Lemma
lt_inv_nb : forall a n n' b b',
cons a n b < cons a n' b' ->
(n<n')%nat \/ n=n' /\ b < b'.
Proof
.
inversion_clear 1; auto with T1.
elim (lt_irr (a:=a)); auto.
Qed
.
Lemma
lt_inv_b : forall a n b b',
cons a n b < cons a n b' -> b < b'.
Proof
.
inversion_clear 1; auto with arith T1.
elim (lt_irr (a:=a));auto.
elim (lt_irrefl n);auto.
Qed
.
Theorem
lt_trans : forall a b, a < b ->
forall c, b < c -> a < c.
Proof
.
induction 1.
inversion 1; auto with T1.
inversion_clear 1; auto with T1.
inversion_clear 1; eauto with T1 arith.
inversion_clear 1; auto with T1.
Qed
.
Theorem
lt_not_gt : forall a b, a < b -> ~ b < a.
Proof
.
intros o1 o2 H H0.
generalize (lt_trans H H0).
intro H2; case (lt_irr H2).
Qed
.
Lemma
finite_lt : forall n p : nat, (n < p)%nat ->
F n < F p.
Proof
.
destruct n;simpl.
destruct p.
inversion 1.
simpl;auto with T1.
destruct p.
inversion 1.
simpl;auto with T1.
intro; assert (n<p)%nat ; auto with arith T1.
Qed
.
Lemma
finite_ltR : forall n p : nat,
F n < F p ->
(n < p)%nat.
Proof
.
destruct n;simpl.
destruct p.
inversion 1.
auto with arith.
destruct p.
inversion 1.
simpl.
intro H; case (lt_inv_nb H).
auto with arith.
intros (_,H0); case (lt_irr H0).
Qed
.
Lemma
lt_a_phi0_a : forall a, a < phi0 a.
Proof
.
induction a;simpl.
compute; constructor.
unfold phi0.
constructor 2.
assert (le (cons a1 0 zero) (cons a1 n a2)).
case n.
case a2.
left.
trivial.
right;constructor 4;constructor.
right;constructor 3.
auto with arith.
case H.
injection 1.
intros; subst a2;subst n;auto.
intro; eapply lt_trans; eauto.
Qed
.
Lemma
phi0_lt : forall a b, a < b -> phi0 a < phi0 b.
Proof
.
unfold phi0;intros c c' H.
constructor 2;trivial.
Qed
.
Lemma
phi0_ltR : forall a b, phi0 a < phi0 b -> a < b.
Proof
.
unfold phi0;intros c c' H.
case (lt_inv H).
trivial.
intros [(_,i)|(_,(_,i))]; inversion i.
Qed
.
Section
lt_not_well_founded.
Let
f := (fix f (i:nat): T1 :=
match i with 0 => (phi0 (F 2))
| S i => cons (F 1) 0 (f i)
end).
Lemma
f_not_in_normal_form :
forall i, ~ (nf (f (S i))).
Proof
.
induction i; red; simpl.
inversion 1.
inversion H3.
inversion H9.
inversion H9.
simpl in IHi.
inversion 1.
case (lt_irr H3).
Qed
.
Lemma
f_decreases : forall i, f (S i) < f i.
Proof
.
induction i; compute; auto with T1.
Qed
.
Theorem
lt_not_wf : ~ (well_founded lt).
Proof
.
red; intro wf.
case (not_decreasing _ lt).
auto.
exists f.
exact f_decreases.
Qed
.
End
lt_not_well_founded.
Theorem
zero_le : forall a, zero <= a.
Proof
.
unfold le.
destruct a; [left|right];repeat constructor.
Qed
.
Theorem
le_trans : forall a b c, a <= b -> b <= c -> a <= c.
Proof
.
destruct 1.
subst b;auto.
destruct 1.
subst b;right;auto.
right;eapply lt_trans;eauto.
Qed
.
Theorem
le_lt_trans : forall a b c, a <= b -> b < c -> a < c.
Proof
.
destruct 1.
subst b;auto.
intros;eapply lt_trans;eauto.
Qed
.
Theorem
lt_le_trans : forall a b c, a < b -> b <= c -> a < c.
Proof
.
destruct 2.
subst b;auto.
eapply lt_trans;eauto.
Qed
.
Theorem
le_inv : forall a n b a' n' b',
cons a n b <= cons a' n' b' ->
a < a' \/
a = a' /\ (n < n')%nat \/
a = a' /\ n = n' /\ b <= b'.
Proof
.
intros a n b a' n' b' H; case H.
injection 1; right.
right; subst a; subst n ; subst b; auto with T1.
intro H0; generalize (lt_inv H0).
intro H1; case H1; auto.
intros [(H2,H3) | (H2,(H3,H4))].
auto.
right;right;auto with T1.
Qed
.
Lemma
lt_not_le: forall a b, a < b -> ~ b <= a.
Proof
.
red; unfold le.
intros a b H H0; case H0;intro.
subst b; absurd (lt a a);auto with T1.
absurd (lt a a); eauto with T1.
eapply lt_trans;eauto.
Qed
.
Lemma
lt_inv_le : forall a n b a' n' b',
cons a n b < cons a' n' b' ->
a <= a'.
Proof
.
intros a n b a' n' b' H.
case (lt_inv H).
auto with T1.
intros [(e,i)|(e,(e',i))].
subst a; auto with T1.
subst a; auto with T1.
Qed
.
Theorem
le_zero_inv : forall a, a <= zero -> a = zero.
Proof
.
destruct 1.
auto.
absurd (a < zero);auto with T1.
Qed
.
Theorem
le_tail : forall a n b b', b <= b' ->
cons a n b <= cons a n b'.
Proof
.
destruct 1.
subst b; left; auto.
right; auto with T1.
Qed
.
Hint
Resolve zero_le le_tail : T1.
Lemma
head_lt_cons : forall a n b, a < cons a n b.
Proof
.
induction a.
constructor.
constructor 2; auto.
Qed
.
Definition
T1_eq_dec : forall (a b : T1), {a = b}+{a <> b}.
Proof
.
decide equality.
apply eq_nat_dec.
Defined
.
Definition
trichotomy_inf : forall a b, {a < b}+{a = b}+{b < a}.
Proof
.
induction a; destruct b; auto with T1.
case (IHa1 b1);intros s.
case s;intros.
auto with T1.
subst b1; case (lt_eq_lt_dec n n0).
destruct 1.
auto with T1.
subst n;
case (IHa2 b2); auto with T1.
destruct 1;[idtac| subst b2];auto with T1.
right;auto with T1.
auto with T1.
Defined
.
Definition
max' a b :=
if trichotomy_inf a b
then b else a.
Goal forall a b, a < b -> max' a b = b.
intros a b H; unfold max';
case (trichotomy_inf a b);auto.
intro;case (lt_not_gt H);auto.
Qed
.
Goal max' omega (omega + omega) = omega + omega.
trivial.
Qed
.
Definition
lt_le_dec : forall a b, {a < b}+{b <= a}.
Proof
.
intros a b; case (trichotomy_inf a b).
destruct 1.
left;auto.
right;left; auto.
right;right; auto.
Defined
.
Lemma
nf_inv1 : forall a n b, nf (cons a n b) -> nf a.
Proof
.
inversion_clear 1; auto.
Qed
.
Lemma
nf_inv2 : forall a n b, nf (cons a n b) -> nf b.
Proof
.
inversion_clear 1; auto with T1.
Qed
.
Hint
Resolve nf_inv1 nf_inv2 : T1.
Ltac
nf_inv := (eapply nf_inv1; progress eauto)||
(eapply nf_inv2; progress eauto).
Lemma
nf_tail_lt_nf : forall b b', b' < b -> nf b' ->
forall a n, nf (cons a n b) ->
nf (cons a n b').
Proof
.
induction 1.
constructor; eauto with T1.
constructor.
apply lt_trans with a'; auto.
inversion H1; eauto with T1.
eauto with T1.
assumption.
constructor.
inversion H1;eauto with T1.
eauto with T1.
assumption.
constructor.
inversion H1;eauto with T1.
eauto with T1.
assumption.
Qed
.
Lemma
tail_lt_cons : forall b n a,
nf (cons a n b)->
b < cons a n b.
Proof
.
induction b.
constructor.
constructor 2.
inversion H;auto.
Qed
.
Lemma
nf_intro : forall a n b, nf a ->
nf b ->
nf2 b a ->
nf (cons a n b).
Proof
.
destruct 3; constructor; auto.
Qed
.
Lemma
nf2_intro : forall a n b, nf (cons a n b) ->
nf2 b a.
Proof
.
inversion 1 ; constructor; auto.
Qed
.
Lemma
nf2_phi0 : forall a b, nf2 b a ->
b < phi0 a.
Proof
.
induction 1; compute; auto with T1.
Qed
.
Lemma
nf2_phi0R : forall a b, b < phi0 a -> nf2 b a.
Proof
.
induction b.
constructor.
inversion_clear 1.
constructor;auto.
inversion H0.
inversion H0.
Qed
.
Lemma
nf_coeff_irrelevance : forall a b n p, nf (cons a n b) ->
nf (cons a p b).
Proof
.
intros; apply nf_intro; try nf_inv.
eapply nf2_intro;eauto.
Qed
.
Lemma
log_nf : forall a, nf a -> nf (log a).
Proof
.
destruct a;unfold log;simpl.
constructor; eauto with T1.
eauto with T1.
Qed
.
Lemma
nf_of_finite : forall n b, nf (cons zero n b) ->
b = zero.
Proof
.
intros n b H; inversion_clear H.
trivial.
case (not_lt_zero (a:=a'));auto.
Qed
.
Lemma
ordinal_finite : forall n, nf (F n).
Proof
.
unfold finite; intro n;case n; auto with T1 arith.
Qed
.
Lemma
nf_omega : nf omega.
Proof
.
unfold omega; auto with T1.
Qed
.
Theorem
nf_phi0 : forall a, nf a -> nf (phi0 a).
compute;auto with T1.
Qed
.
Lemma
nf_tower : forall n, nf (omega_tower n).
induction n; simpl; auto with T1.
Qed
.
Definition
nf_rect : forall P : T1 -> Type,
P zero ->
(forall n: nat, P (cons zero n zero)) ->
(forall a n b n' b', nf (cons a n b) ->
P (cons a n b)->
nf2 b' (cons a n b) ->
nf b' ->
P b' ->
P (cons (cons a n b) n' b')) ->
forall a, nf a -> P a.
Proof
.
intros P H0 Hfinite Hcons.
induction a.
trivial.
generalize IHa1;case a1.
intros IHc0 H.
rewrite (nf_of_finite H).
apply Hfinite.
intros c n0 c0 IHc0 H2; apply Hcons.
eauto with T1.
apply IHc0; eauto with T1.
eapply nf2_intro. eauto.
nf_inv; eauto.
apply IHa2.
nf_inv.
Defined
.
Lemma
compare_ok_1:
forall a a', (compare a a' = Lt <-> a < a') /\
(compare a a' = Eq <-> a = a') .
Proof
.
induction a;simpl.
destruct a';simpl.
split;split.
discriminate 1.
inversion 1.
auto.
auto.
split;split.
auto with T1.
auto.
discriminate 1.
discriminate 1.
destruct a'; simpl.
split; split.
discriminate 1.
inversion 1.
inversion 1.
discriminate 1.
caseEq (compare a1 a'1).
intros.
case (lt_eq_lt_dec n n0).
destruct s.
repeat split.
replace a1 with a'1.
constructor 3;auto.
case (IHa1 a'1).
intros.
case H1;auto.
intros;symmetry.
auto.
discriminate 1.
injection 1.
intros; subst n0.
absurd (n<n)%nat;auto with arith.
split.
repeat split.
intro.
subst n0.
replace a1 with a'1.
constructor 4.
case (IHa2 a'2);intros.
auto.
case H1;auto.
elim (IHa1 a'1);intros.
case H2;auto.
symmetry;auto.
subst n0.
replace a1 with a'1.
intro.
assert (lt a2 a'2).
eapply lt_inv_b;eauto.
case (IHa2 a'2);intros.
case H2;auto.
case (IHa1 a'1);intros.
case H1;auto.
intros;symmetry;auto.
replace a1 with a'1.
subst n0.
repeat split.
intros.
case (IHa2 a'2); intros.
replace a2 with a'2.
auto.
case H2;intros.
symmetry;auto.
injection 1.
intro;subst a'2.
case (IHa2 a2).
intros.
case H2;auto.
case (IHa1 a'1);intros.
case H1;intros;symmetry;auto.
intro;repeat split.
discriminate 1.
intro H0.
absurd (cons a1 n a2 < cons a'1 n0 a'2);auto.
apply lt_not_gt.
replace a1 with a'1.
constructor 3;auto.
elim (IHa1 a'1);intros.
case H2;auto.
symmetry;auto.
discriminate 1.
injection 1.
intros; subst n0; absurd (n<n)%nat;auto with arith.
intros; repeat split.
constructor 2.
case (IHa1 a'1);intros.
case H1;auto.
discriminate 1.
injection 1;intros.
subst a'1;absurd (lt a1 a1).
apply lt_irr.
case (IHa1 a1);intros.
case H3;auto.
intro;repeat split.
discriminate 1.
inversion 1.
case (IHa1 a'1);intros.
case H8;intros.
rewrite <- (H11 H2).
symmetry;auto.
absurd (cons a'1 n0 a'2 < cons a1 n a2).
apply lt_not_gt;auto.
case (IHa1 a'1);intros.
case H9;intros.
rewrite H in H11.
generalize (H11 H5).
discriminate 1.
case (IHa1 a'1);intros.
case H9;intros.
rewrite H in H11.
generalize (H11 H5).
discriminate 1.
discriminate 1.
injection 1;intros.
subst a'1; case (IHa1 a1);intros.
case H4;intros.
rewrite H in H6.
auto.
Qed
.
Lemma
compare_reflect : forall a a', match compare a a' with
| Lt => a < a'
| Eq => a = a'
| Gt => a' < a
end.
Proof
.
intros c c'; case (compare_ok_1 c c');intros H0 H1; case H0; case H1;
intros;
caseEq (compare c c'); auto.
intro comp; case (trichotomy_inf c c').
intro x; case x; intro H'.
rewrite (H4 H') in comp;discriminate comp.
rewrite (H2 H') in comp;discriminate comp.
trivial.
Qed
.
Lemma
compare_rw1 : forall a b, a < b -> compare a b = Lt.
Proof
.
intros c1 c2; generalize (compare_reflect c1 c2).
case (compare c1 c2);auto.
intros e H;subst c2;case (lt_irr H).
intros H1 H2;case (lt_not_gt H2);auto.
Qed
.
Lemma
compare_rw2 : forall a, compare a a = Eq.
Proof
.
intro c; generalize (compare_reflect c c).
case (compare c c);auto;
intro H;case (lt_irr H);auto.
Qed
.
Lemma
compare_rw3 : forall a b, b < a -> compare a b = Gt.
Proof
.
intros c1 c2; generalize (compare_reflect c1 c2).
case (compare c1 c2);auto.
intros e H;subst c2;case (lt_irr H).
intros H1 H2;case (lt_not_gt H2);auto.
Qed
.
Theorem
compare_reflectR : forall a b : T1,
(match trichotomy_inf a b with
inleft (left _) => Lt
| inleft (right _) => Eq
| inright _ => Gt
end)
= compare a b.
Proof
.
intros c1 c2;case (trichotomy_inf c1 c2).
destruct s; auto; try discriminate.
rewrite compare_rw1;auto.
subst c1;rewrite compare_rw2;auto.
intro; rewrite compare_rw3;auto.
Qed
.
Lemma
max_le_1 : forall a b, a <= max a b.
Proof
.
unfold max.
intros.
rewrite <- compare_reflectR.
case (trichotomy_inf a b);auto with T1.
destruct s;auto with T1.
Qed
.
Lemma
max_comm : forall a b, max a b = max b a.
Proof
.
unfold max.
intros a b; repeat rewrite <- compare_reflectR.
case ( trichotomy_inf a b); case (trichotomy_inf b a);auto with T1.
destruct s; destruct s; auto with T1.
case (lt_not_gt l);auto.
destruct s;auto with T1.
destruct s;auto with T1.
intros H H0; case (lt_not_gt H);auto.
Qed
.
Lemma
lt_intro : forall a b, compare a b = Lt -> a < b.
Proof
.
intros a b; rewrite <- compare_reflectR.
case (trichotomy_inf a b);auto with T1.
destruct s; auto.
discriminate 1.
discriminate 2.
Qed
.
Lemma
max_dec : forall a b, {max a b = a}+{max a b = b}.
Proof
.
unfold max; intros a b; case (trichotomy_inf a b);auto.
destruct 1.
repeat rewrite compare_rw1;auto.
subst b;repeat rewrite compare_rw2;auto.
intro; repeat rewrite compare_rw3;auto.
Qed
.
Lemma
max_nf : forall a b, nf a -> nf b -> nf (max a b).
Proof
.
intros c c'; case (max_dec c c');
intro H;rewrite H;auto.
Qed
.
Lemma
max_assoc : forall a b c, max (max a b) c =
max a (max b c).
Proof
.
intros c1 c2 c3.
unfold max.
case (trichotomy_inf c1 c2).
destruct 1.
repeat (rewrite (compare_rw1 l)).
case (trichotomy_inf c2 c3).
destruct 1.
repeat (rewrite (compare_rw1 l0)).
assert (lt c1 c3).
eapply lt_trans;eauto.
rewrite (compare_rw1 H);auto.
subst c3.
repeat rewrite compare_rw2.
rewrite compare_rw1;auto.
intro c'.
repeat (rewrite (compare_rw3 c')).
rewrite compare_rw1;auto.
subst c2;
repeat (rewrite (compare_rw2));auto.
case (trichotomy_inf c1 c3).
destruct 1.
repeat (rewrite (compare_rw1 l));auto.
subst c3;repeat rewrite compare_rw2;auto.
intro c;repeat (rewrite (compare_rw3 c));auto.
repeat rewrite compare_rw2;auto.
intro c; repeat rewrite (compare_rw3 c);auto.
case (trichotomy_inf c1 c3).
destruct 1.
rewrite (compare_rw1 l);auto.
assert (lt c2 c3).
eapply lt_trans;eauto.
repeat rewrite (compare_rw1 H);auto.
rewrite (compare_rw1 l);auto.
subst c3;rewrite (compare_rw2);auto.
repeat rewrite (compare_rw1 c);auto.
rewrite compare_rw2;auto.
intro.
rewrite (compare_rw3 l).
case (trichotomy_inf c2 c3).
destruct 1.
repeat rewrite (compare_rw1 l0);auto.
repeat rewrite (compare_rw3 l);auto.
subst c3;repeat rewrite (compare_rw2);auto.
repeat rewrite (compare_rw3 l);auto.
intro c';repeat rewrite (compare_rw3 c');auto.
repeat rewrite (compare_rw3 c);auto.
Qed
.
Section
well_founded.
Let
R := restrict T1 nf lt.
Hint
Unfold restrict R.
Lemma
Acc_zero : Acc R zero.
Proof
.
split.
unfold R, restrict.
intros y (H1, (H2,H3)).
inversion_clear H2.
Qed
.
Theorem
b_Acc :
forall a n b, Acc R a ->
(forall y : T1, R y a ->
forall (n : nat) (b : T1), nf (cons y n b) ->
Acc R (cons y n b)) ->
nf (cons a n b) -> Acc R b.
intros a n b Aa Ha Hnf.
assert (nf b).
nf_inv.
assert (R b (phi0 a)).
repeat split;auto.
apply nf2_phi0; apply nf2_intro with n; auto.
apply nf_phi0; auto.
nf_inv.
generalize H0;pattern b; case b.
intro;apply Acc_zero.
intros.
case H1.
destruct 2.
unfold phi0 in H3.
case (lt_inv H3).
intro.
generalize H2;case n.
inversion 1.
intros; apply Ha.
split.
eauto with T1.
split; eauto with T1.
constructor.
auto.
apply Ha.
split;auto.
constructor;eauto with T1.
subst t0; assumption.
intros;apply Ha.
split;eauto with T1.
assumption.
destruct 1.
case H5; inversion 2.
case H5;intros _ (_,H6);inversion H6.
Qed
.
Lemma
Accons : forall a, Acc R a ->
forall n b, nf (cons a n b) -> Acc R (cons a n b).
Proof
.
intros a Aa; pattern a.
eapply Acc_ind with (R:= R).
2:trivial.
clear a Aa.
intros a Aa Ha.
intros n b Hnf.
assert (Acc R a).
apply Acc_intro;auto.
assert (Acc R b).
apply b_Acc with a n; auto.
assert (Acc Peano.lt n).
apply lt_wf.
generalize H H0 Ha.
clear Hnf Aa Ha.
apply AccElim3 with (2:= H) (3:=H1) (4:= H0).
clear H1 H0 H b n a.
intros a n b Hreca Hrecn Hrecb Aa Ab Ha.
apply Acc_intro; intros y Hy.
unfold R, restrict in Hy; decompose [and] Hy.
inversion H1.
apply Acc_zero.
assert (Ha': forall y : T1,
R y a0 ->
forall (n1 : nat) (b1 : T1), nf (cons y n1 b1) -> Acc R (cons y n1 b1)).
intros y1 Hr n1 b1 Hn; apply Ha; auto.
split;eauto with T1.
split.
apply lt_trans with a0;auto.
unfold R, restrict in Hr; decompose [and] Hr; auto.
nf_inv.
assert (R a0 a).
subst y; split.
nf_inv.
split;auto.
nf_inv.
assert (Acc R a0).
subst y.
apply Acc_inv with a.
assumption.
assumption.
assert (Acc R b0).
subst y; apply b_Acc with a0 n0; auto.
subst y; apply Hreca;auto.
apply lt_wf.
subst y a0.
assert (Acc R b0).
apply b_Acc with a n0; auto.
apply Hrecn; auto.
subst y a0 n0.
assert (R b0 b).
split.
nf_inv.
split;try nf_inv;auto.
apply Hrecb; auto.
apply Acc_inv with b; auto.
Qed
.
Theorem
nf_Acc : forall a, nf a -> Acc R a.
Proof
.
intro o;elim o.
intro; apply Acc_zero.
intros; eapply Accons.
2:eauto.
apply H; try nf_inv.
Qed
.
Lemma
nf_Wf : well_founded_P _ nf lt.
Proof
.
unfold well_founded_P.
intros.
apply nf_Acc.
auto.
Qed
.
Definition
transfinite_induction :
forall (P:T1 -> Type),
(forall x:T1, nf x ->
(forall y:T1, nf y -> y < x -> P y) -> P x) ->
forall a, nf a -> P a.
Proof
.
intros; eapply P_well_founded_induction_type; eauto.
eexact nf_Wf;auto.
Defined
.
Definition
transfinite_induction_Q :
forall (P : T1 -> Type) (Q : T1 -> Prop),
(forall x:T1, Q x -> nf x ->
(forall y:T1, Q y -> nf y -> y < x -> P y) -> P x) ->
forall a, nf a -> Q a -> P a.
Proof
.
intros.
eapply P_well_founded_induction_type with (R:=lt)(P:=fun a => nf a /\ Q a).
3:split;auto.
2:destruct 1; intros; eapply X; eauto.
unfold well_founded_P.
intros.
apply Acc_incl with (restrict _ nf lt).
unfold inclusion; intros.
unfold restrict.
unfold restrict in H2.
tauto.
apply nf_Acc.
case H1;auto.
Defined
.
End
well_founded.
Lemma
succ_nf2 : forall c a n b, nf2 c (cons a n b) ->
nf2 (succ c) (cons a n b).
Proof
.
induction c.
simpl.
repeat constructor.
simpl.
case c1.
repeat constructor.
intros t n0 t0 a n1 b H.
inversion_clear H.
constructor; auto.
Qed
.
Lemma
succ_nf : forall a, nf a -> nf (succ a).
Proof
.
induction a.
simpl.
repeat constructor; auto with arith.
simpl.
generalize IHa1 ; case a1.
simpl;repeat constructor; auto.
intros c n0 c0 H H0.
apply nf_intro.
nf_inv.
apply IHa2; nf_inv.
apply succ_nf2.
eapply nf2_intro; eauto.
Qed
.
Lemma
lt_succ : forall a, a < succ a.
Proof
.
intro c; elim c; simpl; auto with T1.
intro c0; case c0; simpl; auto with T1.
Qed
.
Lemma
phi0_log : forall a, a < phi0 (succ (log a)).
Proof
.
destruct a.
simpl.
compute.
constructor.
simpl.
unfold phi0.
constructor 2.
apply lt_succ.
Qed
.
Lemma
plus_zero : forall a, zero + a = a.
Proof
.
simpl; intro a; case a; auto.
Qed
.
Lemma
plus_a_zero : forall a, nf a -> a + zero = a.
Proof
.
intro c; case c;simpl.
trivial.
intro c0;case c0;simpl;auto.
intros n c1 H1; rewrite (nf_of_finite H1); auto with T1.
Qed
.
Lemma
plus_fin_omega : forall n ,F n + omega = omega.
Proof
.
destruct n;simpl;auto.
Qed
.
Lemma
plus_not_comm : {a:T1 & {b :T1 |
nf a /\ nf a /\ a + b <> b + a}}.
Proof
.
exists (finite 1); exists omega.
split.
simpl;repeat constructor.
split.
compute;repeat constructor.
compute.
discriminate 1.
Defined
.
Lemma
succ_is_plus_one: forall a, nf a -> succ a = a + F 1.
Proof
.
unfold finite.
intro c; elim c.
compute; trivial.
intro c0;case c0.
simpl.
intros H n c1 H0 H1.
rewrite <- plus_n_O; trivial.
intros c1 n c2 H n0 c3 H0 H1.
simpl.
rewrite H0; [trivial | nf_inv].
Qed
.
Lemma
plus_cons_cons_rw1 : forall a n b a' n' b',
a < a' ->
cons a n b + cons a' n' b' = cons a' n' b'.
Proof
.
simpl.
destruct a.
destruct a'.
inversion 1.
intros; rewrite compare_rw1; auto with T1.
destruct a'.
inversion 1.
intros n' b' H; rewrite (compare_rw1);auto.
Qed
.
Lemma
plus_cons_cons_rw2 : forall a n b n' b',
nf (cons a n b) ->
nf (cons a n' b') ->
plus (cons a n b) (cons a n' b')=
cons a (S (n + n') ) b'.
Proof
.
simpl.
destruct a.
intros.
rewrite (nf_of_finite H0).
auto.
rewrite (compare_rw2).
auto.
Qed
.
Lemma
plus_cons_cons_rw3 : forall a n b a' n' b',
a' < a ->
nf (cons a n b) ->
nf (cons a' n' b') ->
cons a n b + cons a' n' b'=
cons a n (b + (cons a' n' b')).
Proof
.
simpl.
destruct a.
inversion 1.
destruct a'.
rewrite compare_rw3.
auto.
constructor.
intros;
rewrite compare_rw3.
auto.
auto.
Qed
.
Lemma
ap_plus : forall a, ap a ->
forall b c, nf b -> nf c -> b < a -> c < a -> b + c < a.
Proof
.
destruct 1.
intro b; elim b.
intro c; elim c;intros.
simpl; auto with T1.
simpl.
auto.
intros c H0.
intros.
generalize c0 H2 H4.
destruct c1.
rewrite (plus_a_zero).
auto.
auto.
intros.
case (trichotomy_inf c c1_1).
destruct 1.
rewrite (plus_cons_cons_rw1 n t n0 c1_2 l).
auto.
subst c1_1.
rewrite (plus_cons_cons_rw2 H1 H5).
constructor 2.
inversion H3;auto.
inversion H8. inversion H8.
intro H7.
rewrite (plus_cons_cons_rw3).
constructor 2.
inversion_clear H3;auto.
inversion H8.
inversion H8. auto.
auto.
auto.
Qed
.
Lemma
ap_plusR : forall c, nf c -> c <> zero ->
(forall a b, nf a -> nf b -> a < c ->
b < c -> a + b < c) ->
ap c.
destruct c.
intros; absurd (zero = zero); auto.
case c2.
case n.
constructor.
intros.
generalize (H1 (cons c1 0 zero) (cons c1 n0 zero)).
clear H1;intros.
assert (nf (cons c1 0 zero)).
eapply nf_coeff_irrelevance;eauto.
assert (nf (cons c1 n0 zero)).
eapply nf_coeff_irrelevance;eauto.
assert (lt (cons c1 0 zero) (cons c1 (S n0) zero)).
constructor 3;auto with arith.
assert (lt (cons c1 n0 zero) (cons c1 (S n0) zero) ).
constructor 3;auto with arith.
generalize (H1 H2 H3 H4 H5).
intro.
rewrite plus_cons_cons_rw2 in H6.
simpl in H6.
case (lt_irr H6).
auto.
auto.
intros.
assert (nf (cons c1 n zero)).
constructor.
nf_inv.
assert (nf (cons t n0 t0)).
nf_inv.
assert (cons c1 n zero < cons c1 n (cons t n0 t0)).
constructor 4;auto with T1.
assert (lt (cons t n0 t0) (cons c1 n (cons t n0 t0))).
constructor 2.
inversion H;auto.
generalize (H1 _ _ H2 H3 H4 H5).
clear H1 H4 H5;intro.
rewrite plus_cons_cons_rw3 in H1.
simpl in H1.
case (lt_irr H1).
inversion H;auto.
auto.
auto.
Qed
.
Lemma
plus_nf0 : forall a, nf a -> forall b c, b < phi0 a ->
c < phi0 a ->
nf b -> nf c ->
nf (b + c).
Proof
.
intros a Ha ; pattern a.
apply transfinite_induction.
2:assumption.
intros x Cx Hx.
destruct b; destruct c.
simpl;constructor.
simpl;auto.
intros;rewrite plus_a_zero; auto.
intros.
case (trichotomy_inf b1 c1).
destruct 1.
rewrite plus_cons_cons_rw1.
auto.
auto.
subst c1.
rewrite plus_cons_cons_rw2.
eapply nf_coeff_irrelevance;eauto.
auto.
auto.
intro; rewrite plus_cons_cons_rw3;auto.
apply nf_intro.
nf_inv.
eapply Hx with b1.
nf_inv.
inversion_clear H; auto.
inversion H3.
inversion H3.
inversion H1.
compute;auto with T1.
unfold phi0.
constructor 2;auto.
unfold phi0.
constructor 2;auto.
nf_inv.
auto.
apply nf2_phi0R.
apply ap_plus.
unfold phi0;constructor.
nf_inv.
auto.
apply nf2_phi0.
eapply nf2_intro.
eauto.
unfold phi0; constructor 2; auto.
Qed
.
Lemma
plus_nf : forall a, nf a -> forall b, nf b -> nf (a + b).
Proof
.
intros.
case (trichotomy_inf a b).
destruct 1.
apply plus_nf0 with b; auto.
apply lt_trans with b; auto.
apply lt_a_phi0_a.
apply lt_a_phi0_a.
subst b.
apply plus_nf0 with a; auto.
apply lt_a_phi0_a.
apply lt_a_phi0_a.
intro; apply plus_nf0 with a; auto.
apply lt_a_phi0_a.
apply lt_trans with a.
auto.
apply lt_a_phi0_a.
Qed
.
Lemma
plus_to_cons: forall a n b,
nf (cons a n b) -> omega_term a n + b =
cons a n b.
Proof
.
simpl.
destruct a.
intros n b H.
rewrite (nf_of_finite H); auto.
destruct b.
auto.
inversion_clear 1.
case (trichotomy_inf (cons a1 n a2) b1).
destruct 1.
absurd (lt b1 b1);eauto with T1.
eapply lt_trans;eauto.
subst b1.
case (lt_irr H0).
intro; rewrite compare_rw3; auto.
Qed
.
Lemma
plus_is_zero : forall a b, nf a -> nf b ->
a + b = zero -> a = zero /\
b = zero.
Proof
.
destruct a;destruct b.
compute.
auto.
simpl.
discriminate 3.
intro;rewrite plus_a_zero.
discriminate 2.
auto.
simpl.
case a1;case b1.
discriminate 3.
intros c n1 c0 H H0; rewrite compare_rw1.
discriminate 1.
constructor.
intros c n1 c0 H H0; rewrite compare_rw3.
discriminate 1.
constructor.
intros c n1 c0 c3 n2 c4 H H0;
case (compare (cons c3 n2 c4) (cons c n1 c0));discriminate 1.
Qed
.
Lemma
lt_succ_succ : forall a b,
a < b -> nf a -> nf b ->
succ a < succ b.
Proof
.
induction 1.
simpl.
case a.
constructor 3;auto with arith.
constructor 2.
constructor.
generalize H; simpl.
case a.
case a'.
inversion 1.
constructor 2.
constructor 1.
case a'.
inversion 1.
constructor 2;auto.
simpl.
case a.
constructor 3;auto with arith.
constructor 3;auto.
simpl.
case a.
intros.
assert (b'=zero).
inversion H1.
auto.
inversion H5.
subst b';inversion H.
constructor 4.
apply IHlt; nf_inv.
Qed
.
Lemma
lt_phi0_phi0 : forall a b, a < b -> phi0 a < phi0 b.
Proof
.
unfold phi0.
constructor 2.
auto.
Qed
.
Lemma
le_phi0_phi0 : forall a b, a <= b -> phi0 a <= phi0 b.
Proof
.
destruct 1.
subst b;left;auto.
right;unfold phi0;constructor 2.
auto.
Qed
.
Lemma
le_succ_succ : forall a b, nf a -> nf b ->
a <= b -> succ a <= succ b.
Proof
.
destruct 3.
subst a;left;auto.
right.
apply lt_succ_succ;auto.
Qed
.
Lemma
lt_succ_le_R : forall a, nf a -> forall b, nf b ->
succ a <= b -> a < b .
intros c Hc; elim Hc using nf_rect.
compute.
intros.
case H0;intros.
subst b;auto with T1.
eapply lt_trans.
2:eexact H1.
auto with T1.
intros.
inversion_clear H0.
subst b; simpl; auto with T1.
simpl in H1.
eapply lt_trans.
2:eauto.
auto with T1.
intros.
simpl in H5.
case H5.
intro; subst b0.
constructor 4.
apply lt_succ;auto.
intro.
eapply lt_trans.
2:eauto.
constructor 4.
apply lt_succ;auto.
Qed
.
Lemma
lt_succ_le_2 : forall a, nf a -> forall b, nf b ->
a < succ b -> a <= b.
intros c Hc; elim Hc using nf_rect.
intros;apply zero_le.
intros.
generalize H0; case b;simpl.
intros.
generalize (lt_inv_nb H1).
destruct 1.
inversion H2.
case H2;intros.
inversion H4.
destruct t.
inversion 1.
inversion H3.
assert (n = n0 \/ (n < n0)%nat).
omega.
destruct H8.
rewrite H8.
case t.
left.
auto.
right;constructor 4.
auto with T1.
right;constructor 3;auto.
inversion H3.
right;constructor 2;auto with T1.
destruct b0.
simpl.
inversion 2.
inversion H7.
simpl.
case b0_1;simpl.
inversion 2.
inversion H7.
intros.
inversion_clear H5.
right;constructor 2;auto.
right;constructor 3;auto.
case (H3 b0_2 (nf_inv2 H4) H6).
intro; subst b';left;auto.
right;constructor 4;auto.
Qed
.
Lemma
lt_succ_le : forall a, nf a -> forall b, nf b ->
a < b -> succ a <= b.
induction a.
intros H0 c'; case c'.
inversion 2.
destruct t.
destruct n.
intros.
inversion_clear H.
simpl.
left;auto.
inversion H2.
simpl;case n.
right;constructor 3;auto with arith.
right; constructor 3;auto with arith.
simpl;right;constructor 2; auto with T1.
inversion 3.
simpl;constructor 2;auto with T1.
generalize H6;case a1.
constructor 2;auto with T1.
constructor 2.
auto.
simpl.
case a1.
assert (S n = n' \/ (S n < n')%nat).
omega.
case H7.
intro; subst n'.
case b'.
left;auto.
right;constructor 4;auto with T1.
right;constructor 3;auto.
intros.
right;constructor 3;auto.
subst b;generalize H0;case a1.
intro.
assert (b'=zero).
inversion_clear H3.
auto.
inversion H7.
subst b'; inversion H6.
simpl.
intros.
case (IHa2 (nf_inv2 H) b').
eapply nf_inv2;eauto.
auto.
destruct 1;left;auto.
intro;apply le_tail.
right;auto.
Qed
.
Lemma
minus_lt : forall a b, a < b -> a - b = zero.
induction 1;simpl.
auto.
generalize H;case a.
case a';simpl;auto.
intro H0;case (lt_irr H0).
intros.
rewrite (compare_rw1 H0).
auto.
case a.
case (le_lt_dec n n').
auto.
intro H0; absurd (n < n)%nat;auto with arith.
eauto with arith.
intros;rewrite (compare_rw2).
case (lt_eq_lt_dec n n').
destruct s.
auto.
subst n';absurd (n < n)%nat;auto with arith.
intro; absurd (n<n)%nat;eauto with arith.
case a.
case (le_lt_dec n n).
auto.
intro; absurd (n>n);auto with arith.
intros; rewrite (compare_rw2).
case (lt_eq_lt_dec n n).
destruct s.
auto.
auto.
intro; absurd (n < n)%nat;auto with arith.
Qed
.
Lemma
minus_a_a : forall a, a - a = zero.
Proof
.
induction a;simpl;auto.
case a1.
case (le_lt_dec n n).
auto.
intros; absurd (n < n)%nat; auto with arith.
case (lt_eq_lt_dec n n ).
destruct s.
absurd (n < n)%nat;auto with arith.
intros;rewrite compare_rw2.
rewrite IHa2;auto.
intro; absurd (n<n)%nat;auto with arith.
Qed
.
Lemma
minus_le : forall a b, a <= b -> a - b = zero.
Proof
.
destruct 1.
subst b; apply minus_a_a.
apply minus_lt;auto.
Qed
.
Lemma
mult_fin_omega : forall n,
(F (S n)) * omega = omega.
Proof
.
simpl.
unfold omega;auto.
Qed
.
Lemma
phi0_plus_mult : forall a b, nf a -> nf b ->
phi0 (a + b) = phi0 a * phi0 b.
Proof
.
simpl.
intro a; case a.
intro b; case b;simpl.
compute;trivial.
compute; trivial.
intros until b;case b;simpl.
case t;simpl;auto.
intro H; rewrite (nf_of_finite H).
compute;trivial.
case t;simpl.
compute;auto.
unfold phi0;auto.
Qed
.
operations on T1 extend operations on nat |
Lemma
succ_compat : forall n:nat, succ (F n) = F (S n).
Proof
.
destruct n; compute ; trivial.
Qed
.
Lemma
plus_compat: forall n p, F n + F p = F (n + p)%nat.
Proof
.
induction n; destruct p; simpl ; auto.
rewrite <- (plus_n_O n);auto.
rewrite plus_n_Sm;auto.
Qed
.
Lemma
mult_compat : forall n p, (F n) * (F p) =
F (n * p)%nat.
Proof
.
induction n; destruct p; simpl; auto.
rewrite (mult_0_r n).
compute; trivial.
Qed
.
Lemma
exp_F_compat :
forall p n, exp_F (F n) p =
F (n ^ p)%nat.
Proof
.
induction p;simpl.
trivial.
intro n.
rewrite (IHp n).
rewrite mult_compat; trivial.
Qed
.
Lemma
exp_compat : forall p n, (F n) ^ (F p) =
F (n ^ p)%nat.
Proof
.
induction p.
destruct n;simpl.
auto.
destruct n;auto.
destruct n;simpl.
auto.
case n.
simpl.
rewrite power_of_1.
rewrite <- plus_n_O;auto.
simpl.
intros.
replace (cons zero (S n0) zero) with (finite (S (S n0))).
2:simpl;auto.
rewrite exp_F_compat.
rewrite mult_compat.
assert ( (S (S n0) * S (S n0) ^ p) =
(S (S n0) ^ p + (S (S n0) ^ p + n0 * S (S n0) ^ p)))%nat.
ring_nat.
rewrite H;trivial.
Qed
.
Lemma
mult_0_a : forall a, zero * a = zero.
Proof
.
induction a;simpl;auto.
Qed
.
Lemma
mult_a_0 : forall a, a * zero = zero.
Proof
.
simple induction a; simpl.
auto.
destruct t;auto.
Qed
.
Lemma
mult_1_a : forall a, nf a -> (F 1) * a = a.
induction a.
simpl.
trivial.
simpl.
simpl in IHa2.
intro.
caseEq a1.
intro.
subst a1.
rewrite (nf_of_finite H).
rewrite <- (plus_n_O n).
auto.
intros.
subst a1.
unfold finite in IHa2.
rewrite IHa2.
auto.
nf_inv.
Qed
.
Lemma
mult_a_1 : forall a, nf a -> a * (F 1) = a.
induction a.
simpl.
trivial.
simpl.
simpl in IHa2.
intro.
caseEq a1.
intro.
subst a1.
rewrite mult_1_r.
rewrite (nf_of_finite H).
auto.
intros.
subst a1.
rewrite mult_1_r;auto.
Qed
.
Lemma
exp_fin_omega : forall n, (F (S (S n)))^ omega = omega.
Proof
.
reflexivity.
Qed
.
Lemma
omega_exp_rw : forall a, nf a -> omega ^ a = phi0 a.
Proof
.
unfold omega, phi0;simpl.
intro a; elim a; simpl.
trivial.
destruct t.
simpl.
intros.
generalize (nf_of_finite H1).
intro; subst t.
case n;simpl.
auto.
simpl.
induction n0;simpl.
auto.
rewrite IHn0.
simpl.
auto.
intros.
unfold omega_term.
rewrite H0.
fold (phi0 t).
fold (phi0 (cons (cons t1 n t2) n0 t)).
fold (phi0 (cons (cons t1 n t2) n0 zero)).
rewrite <- (plus_to_cons H1).
rewrite phi0_plus_mult.
unfold omega_term;auto.
unfold omega_term;constructor.
nf_inv.
nf_inv.
nf_inv.
Qed
.
Lemma
omega_term_ambiguity : forall a n, nf a -> omega_term a n =
(omega ^ a) * (F (S n)).
Proof
.
intros a n H; rewrite omega_exp_rw.
simpl.
case a; simpl; unfold omega_term; auto.
rewrite <- (plus_n_O n).
auto.
rewrite <- (plus_n_O n).
auto.
auto.
Qed
.
Lemma
cons_ambiguity : forall a n b, nf(cons a n b) ->
cons a n b = (omega^a)*(F (S n))+b.
Proof
.
intros.
rewrite <- plus_to_cons.
rewrite omega_term_ambiguity; auto.
nf_inv.
auto.
Qed
.
Lemma
cons_unicity : forall a n b a' n' b',
nf (cons a n b) -> nf (cons a' n' b') ->
(omega^a)*(F (S n))+b = (omega^a')*(F (S n'))+b' ->
a=a' /\ n = n' /\ b = b'.
Proof
.
intros a n b a' n' b' N N'.
rewrite <- (cons_ambiguity N).
rewrite <- (cons_ambiguity N').
injection 1;auto.
Qed
.
Theorem
Cantor_normal_form :
forall o, zero < o -> nf o ->
{a:T1 & {n: nat &{b : T1 | o = omega ^ a * (F (S n)) + b /\
nf (cons a n b) /\
(forall a' n' b', nf (cons a' n' b') ->
o = omega ^ a' * (F (S n')) + b' ->
a = a' /\ n=n' /\ b = b' )}}}.
Proof
.
intro ; case o.
intro i; case (lt_irr i).
intros a n b H H0.
exists a;exists n;exists b; split.
apply cons_ambiguity;auto.
split;[auto|intros a' n' b' H' e'].
apply cons_unicity;auto.
rewrite <- e'.
symmetry;apply cons_ambiguity;auto.
Defined
.
Lemma
trichotomy : forall a b, a < b \/ a = b \/ b < a.
Proof
.
intros a b; case (trichotomy_inf a b); auto.
destruct 1;auto.
Qed
.
Ltac
tricho t t' Hname := case (trichotomy t t');
[auto with T1 |
auto with T1 |
intro Hname |
intros [Hname|Hname]].